YAI E UNIVERSITY LIBRARY 3 9002 06494 9564 i vVsi jV've : . • ; •, . ■ a -v •>• ,'V \ ft>;■ V'i\’v; I-:; A}:j ■ ■m i. . ■■>. ■ ; :kK'un vvivVi\Mv;VV^ ^I ww ' ■. - •v.'l' <■''■.■ • i«.: '■.■■ ... V .VViV i:VU‘ :i' v V^'foWlW: 1 : •‘■\v .vrA/, >, . , ■ \ \ •■ •■';'. ' . ' ' ' • ■ :, i: . ’■ *1;; VV'H ;,.; Lr; V' ■ . ', V. v V ■ V. f, \ Vi' ;■ !i; :. . iAfO^U^, THE FIELD ENGINEER: Mantis 33ooft oi Practice IN THE SURVEY, LOCATION, AND TRACK-WORK OF RAILROADS; CONTAINING A LARGE COLLECTION OP RULES AND TABLES, ORIGINAL AND SELECTED, APPLICABLE TO BOTH THE STANDARD AND THE NARROW GAUGE, AND PREPARED WITH BPECIATj REFERENCE TO THE WANTS OF THE YOUNG ENGINEER. HY WILLIAM FINDLAY SIIUNK, C.E. SECOND EDITION, REVISED AND CORRECTED. NEW YORK: D. VAN NOSTRAND, PUBLISHER, 23 Mubbay and 27 Wabken Stbkkts. 1883.Copyright, 1879, By D. VAN NOSTRAND.THE AUTHOR Slffecttonatelg ©etucatess tfjfe Book TO ALBERT J. SCHERZER, C.E., ©10 ©omraOe anO ©car JFricnO, LX TOKEN OF ESTEEM FOli HIS PROFESSIONAL ATTAINMENTS AND RESPECT FOR IIIS MANLY CHARACTER.PREFACE, The author’s principal aim in preparing this volume has been, as its title indicates, to serve that large class of young engineers who, like himself, have not had the advantage of a technical education before going out for their livelihood. The initial chapters are, therefore, given to a compendious exposition of those mathematical truths and methods which they must needs become familiar with from the beginning. Plane Trigonometry, Logarithms, and propositions relating to the circle, are tools of the craft in constant use; ready han- dling of them is an indispensable condition of excellence. Be not discouraged by obscurities and difficulties at the outset; light will gradually break on the scrutinizing eye, and a way always open to manful effort. These chapters are followed by instructions as to the adjust- ment and use of instruments, and hints concerning field rou- tine, which it is thought will be found acceptable to the inexperienced learner. The same may be said of the articles on staking chit work, and those on track problems, with which the text of the book closes. They have been written with the, author’s own early ignorance in mind, ancl with a wish to set the subjects forth as plainly as possible, disembarrassed of hard words in the description, and of unpractical niceties in the operation. The chapter on field location is believed to include all the problems likely to occur. The author, in compiling it, has taken those only which have arisen in his own practice, and which, therefore, may arise in the practice of others. His Vvi PREFACE. own practice having, been unusually large and diversified, probably the examples given will prove adequate, directly or indirectly, to all contingencies. No attempt has been made to swell the bulk of the volume with imaginary cases; the object being, not to provide barren mathematical exercises, but to teach useful knowledge. Problems, also, affecting location in its economical aspects, — the balancing of physical and financial conditions, equating of alternative lines, and the like, — do not come within the scope of the-work, and are therefore not treated. Considerable pains have been spent on the tables. However far the young engineer may eventually outgo his teacher as re- gards the text of the book, these are implements of his art which never become antiquated, and can never fall into dis use. Those herein contained which are original will, it is hoped, be esteemed worthy of place with their well-approved associates. The author invites friendly criticism: he would be pleased to receive suggestions, both for the improvement of the book, and for the correction of possible errors in it, should another edition be called for. In dismissing the work from his hands, the precarious snatches of time occupied in its preparation, by day and by night, during the past two years, which might have been more agreeably spent in reading, talking, or musing, recur to the writer’s mind; and the thought arises, To what end or from what motive do people undertake these technical labors? Why should Forney and Bourne toil to simplify steam for our ap- prehension; Nystrom to compile mechanical, Molesworth and Trautwine to epitomize civil engineering; Henck to prepare his elegant manual of field mathematics; Box to illustrate hydraulics; and Shreve, with lucid pen, to make clear for us the strains in truss or arch? The ordinary motives to en- deavor here have no place. There is neither fame nor profit in these drudging'enlerprises. At best the author gives namePREFACE. vii to liis book; he remains impersonal, — known but indirectly, and but to a class. How, then, shall we account for his labors? I take it, the Father of mankind has not only made our minds to hunger for knowledge as our bodies for food, but has also imposed upon us a kindly law of communion, by virtue whereof we cannot do otherwise, without violence to generous nature, than share with our fellows whatsoever we have learned that seems new and useful. Under this law these beneficial works would appear to have had their being, and thus pure are they from the stain of selfishness. Though the present writer would not arrogate equal fellow- ship in the eminent brotherhood named, yet he may justly claim like pureness from unworthy motive, and certainly feels like comfort at heart to that which they must know, for having discharged, in what measure it has been laid upon him, the divine obligation. WM. F. SHUNK. Rahway, N.J.ABBREVIATIONS, -f- Increased by. — Diminished by. X Multiplied by. -f- Divided by. = Equal to. \ • Since, or seeing that. . •. IJence, or therefore. ; Indicates the quotient of one divided by the other of the quantities it connects, called sometimes the ratio of the quan- tities. :: Indicates an equality of ratios, and connects equal ratios in a.proportion. Thus, a ’. b \ \ c '. d indicates that a-^-b = c — d; or it may be read, a is to b as c is to d. ( ) Brackets indicate that the operations embraced by them shall first be performed, and the result treated as a single factor in the remaining processes required by a formula. Thus, (a X b) -r- (a + b) requires that the product of a and b shall be divided by their sum. A2. A small secondary figure annexed thus to an expression is called its exponent. It requires the principal to which it is attached to be used as many times in continued multiplication as there are units in the exponent. Thus, A2 = A X A; A3 = A X A X A, which is called the cube, or third power, of A. V This is called the square root sign: it signifies that the square root of the quantity covered by it is to be taken. ■\/ If preceded by a small secondary figure, called the index, as in the marginal figure, it indicates that the cube root of the quantity covered by it shall be taken; and so on. If the index be fractional, as in the marginal figure, it requires that the square root of the third power of the quantity covered shall be taken. B. M. Bench-mark : any fixed reference point for the level, ixX ABBREVIA TIONS. as outcropping ledge, water-table of building, or other perma- nent object. Usually a blunt conical seat for the rod, hewn on a buttressed tree-base, having a small nail sometimes driven flush in the top of it, and a blaze opposite, on which the eleva- tion is marked with kiel. T. P. Turning-point: usually marked O in the field-book. P. I. Point of intersection: as of tangents, which are to be connected by a curve. A. D. Apex distance: i.e., the distance from the P. I. to the point where a curve merges iu the tangent. P. C. Point of curve: the stake-mark at the beginning of a curve. P. T. Point of tangent: the stake-mark at the end of a curve. P. C. C. Point of compound curvature: the stake-mark where a curve merges in another of different curvature, turn- ing in the same direction. P. R. C. Point of reverse curvature: the stake-mark where a curve merges in another turning in the opposite direction. B. S. Backsight, in transit work; or the reading of the rod to ascertain the instrument height in levelling. F. S. Foresight, in transit work; or the reading of the rod to ascertain elevations in levelling. H. I. Height of instrument: elevation of the level above the datum or zero plane. H. W. High water. L. W. Low water.TABLE OF CONTENTS. PAGE Logabithms. I. Definitions and principles ...... 3 II. Manner of using the tables.....................4 To find the logarithm of any number . . 4 To find the number corresponding to a given logarithm . ....................5 Multiplication by means of logarithms . . 5 Division by means of logarithms ... 6 To raise a number to any power by means of logarithms....................................7 To extract roots by means of logarithms . . 7 Plane Trigonometry. III. Definitions..................................11 IY. Natural sines..................................12 V. Logarithmic sines, &c..........................13 YI. General propositions...........................15 VII. Solution of plane triangles...................16 VIII. Right-angled plane triangles . . . .18 Adjustment and Use of Instruments. IX. General remarks on adjustment . . . .23 X. The level.....................................24 To bring the intersection of the cross-hairs into the optical axis of the telescope ... 24 To bring the level bubble parallel with the tele- scope axis.....................................25 To adjust the wyes; or, in other words, to bring the telescope into a position at right angles to the vertical axis of the instrument ... 25 XI. Levelling ......... 26 Correction for the earth’s curvature and refrac- tion ..........................................28 To find differences in elevation by means of the barometer . . . . . . .29 Heights by the thermometer.....................29 xixii TABLE OF CONTENTS. PAGE XII. Setting slope stakes.......................... XIII. Vertical curves............................... XIV. The transit................................... To adjust the level tubes...................... To adjust the vertical hair so that it shall re- volve in a plane, and mark backsight and fore- sight points in the same straight line To adjust the needle........................... XV. Miscellaneous . . ....................... The vernier...............................• To read an angle............................... To re-inagnetize a needle...................... To replace cross-hairs......................... To fix a true meridian......................... Propositions and Problems relating to the Circle. XVI. Propositions relating to the circle . .49 XVII. Circular curves on railroads .... 50 XVIII. To find the radius, the apex distance, the length, the degree, &c., of a curve ... 52 Givtfn the intersection angle I and radius R, to find the tangent T...........................52 Given the intersection angle I and tangent T, to find the radius R............................53 Given the intersection angle I and chord AB=C, connecting the tangent points, to find the radius R............................‘ . .54 Given the intersection angle I and the degree of curvature or deflection angle D, with 100-feet chords, to determine the length of the long chord C, the versed sine V, the external secant S, or the tangent T.............................54 Given C, V, S, or T, of any curve, and D, the degree of curvature, to find the intersection angle I.........................................55 Given the intersection angle I and deflection angle D, to find the length of the curve . . 55 Given any radius R and chord C, to find the de- flection angle D..................................57 57 Given any radius R and chord C, to find the tangential angle T..............................57 Given any radius R and chord C, to find the tangential distance t..........................58 Given any radius R and chord C, to find the de- flection distance d.................................. tU kU ^ kCk k£k k^ ^k*i (*i*)TABLE OF CONTENTS. xiii PAGE XIX. Ordinates................................... .58 Given any radius R and chord C, to find the . middle ordinate M............................58 Given the radius R, chord C, and middle ordi- nate M, to find any other ordinate . . . 59 Ordinates of a 1° curve, chord 100-feet . . 60 Tracing Curves and turning Obstacles in the Field. XX. To trace a curve on the ground with the chain only..........................................63 XXI. To trace a curve on the ground with transit and 100-feet chain................................66 XXn. Turning obstacles to vision in tangent . . 71 XXIII. Turning obstacles to measurement in tangent . 73 Suggestions as to Field-Work and Location-Projects. XXIV. Suggestions concerning field-work ... 79 XXV. The curve-protractor and the projecting of loca- tions ................................................84 Table showing the distance, D, in feet, at which a straight line must pass from the nearest point of any curve struck with radius r, in order that a terminal branch having a radius R=2 r, and consuming a given angle, x, may merge in said straight line .... 88 Table showing the distance, d, in feet, at which curves of contrary flexure must be placed asunder, in order that the connecting tangent, T, may be 300 feet long.......................89 Problems in Field Location. XXVI. How to proceed when the P. C. is inaccessible . 93 XXVII. How to proceed when the P. C. C. is inaccessible, 95 XXVIII. To shift a P. C. so that the curve shall termi- nate in a given tangent..................................96 XXIX. To substitute for a curve already located one of different radius, beginning at the same point, containing the same angle, and ending in a fixed terminal tangent........................97 XXX. Having located a curve A B C, to find the point B at which to compound into another curve of given radius, which shall end in tangent E F, parallel to the terminal tangent of the original curve, and a given distance from it . 98 XXXI. To shift a P. C. C. so that the terminal branch of a curve shall end in a given tangent . . 99xiv TABLE OF CONTENTS. PAGE XXXII. Having located a tangent, A B, intersecting a curve, C D, from the concave side, to find the point E on said curve, at which to begin a curve of given radius which shall merge in the located tangent..........................102 XXXIII. Having located a tangent, A B, intersecting a curve, C D, from the convex side, to find the point E on said curve at which to begin a curve of given radius which shall merge in the located tangent..............................103 XXXIY. To locate a Y.....................................103 XXXV. To' locate a tangent to a curve from an outside fixed point..................................107 XXXVI. To substitute a curve of given radius for a tan- gent connecting two curves .... 108 XXXVH. To run a tangent to two curves already located . 109 Track Problems. XXXVIII. Reversed curves...................................115 XXXIX. To connect two parallel tangents by a reversed curve having equal radii.....................115 XL. To connect two parallel tangents by a reversed curve having unequal radii .... 117 XLI. A reversed curve having unequal augles . . 119 XLII. A reversed curve between fixed points . . 120 XLIII. To connect two divergent tangents by a reversed curve........................................123 XLIV. To shift a P. R. C. so that the terminal tangent shall merge in a given tangent .... 125 XLV. To pass a curve through a fixed point, the angle . of intersection being given .... 127 XLVI. Frogs and switches..............................129 To find the radius of a turnout curve, the frog angles, and the distances from the toe of switch to the frog points...........................129 To find the angle of switch-rail with main track. 130 To find the distance from toe of switch to point of main frog............................... 130 To find the radius of outer rail of turnout curve, 131 To find the main frog angle, the radius of outer rail being known.............................131 To find the angle of the middle frog in the case of a double turnout..........................131 To find the distance from toe of switch to point of middle frog . . . . . . 131 Turnout tables...........................135, 136TABLE OF CONTENTS. XV PAGE XLYII. To locate a turnout..............................137 139 139 141 142 143 144 XLVIII. Crossings on straight lines.................... XLIX. Crossings on curves............................. L. Elevation of the outer rail on curves . Table for same................................ LI. Trackmen’s table of curves and spring of rails . Explanation of same........................... List of Tables. I. Time of meridian passage of North Star above the pole for the year 1870, and on..................149 II. Time of extreme elongation of North Star for the year 1870, lat. 40°, and on.....................150 III. Azimuths of the North Star, and their natural tan- gents ................................................150 IY. Roods and perches in decimal parts of an acre . . 151 V. Decimals of an acre in one chain length of 100 feet, and of various widths...........................151 YI. Acres, roods, and perches in square feet . . . 152 VII. Circular arcs to radius of 1.....................152 VIII. Feet in decimals of a mile........................153 IX. Inches reduced to decimal parts of a foot . . . 153 X. Radii and their logarithms, middle ordinates, and deflection distances............................155 XI. Squares, cubes, &c., of numbers from 1 to 1042 . . 161 XII. Logarithms of numbers, 1 to 1000................. 179 XIII. Logarithmic sines, cosines, tangents, and cotan- gents ............................................197 XIV. Natural sines and tangents........................243 XVI. Chords, versed sines, external secants, and tangents of a one-degree curve...........................269 XV. Slopes for topography*...........................315 XVII. Rise per mile of various grades.....................317LOGARITHMS. I.-II.LOGARITHMS. i. DEFINITIONS AND PRINCIPLES. 1. The logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number to produce the given number; that is to say, it represents the number of times a fixed number must be multiplied by itself in order to produce any given number. The fixed number is called the base of the system. In the common system, -this base is 10. It follows from the above, that the logarithm of any power of 10 is equal to the exponent of that power. If, therefore, a number is an exact power of 10, its logarithm is a whole number. If a number is not an exact power of 10, its logarithm will not be a whole number, but will be made up of an entire part plus a fractional part, which is generally expressed decimally. The entire part of the logarithm is called the characteristic ; the decimal part is called the mantissa. 2. The characteristic of the logarithm of a whole number is positive, and numerically 1 less than the number of places of figures in the given number. Thus, if a number lies between 1 and 10, its logarithm lies between 0 and 1; that is, it is equal to 0 plus a decimal. If a number lies between 10 and 100, its logarithm is equal to 1 plus a decimal; and so on. 3. The characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the number of 0’s that immediately follow the decimal point. The characteristic alone, in this case, is negative, the man- 34 MANNER OF USING THE TABLES. tissa being always positive. This is indicated by writing the negative sign over the characteristic: thus, 2.880211 is equiva- lent to — 2 -f .380211. 4. The characteristic of the logarithm of a mixed number is the same as that of its entire part. Thus the mixed number 74.103 lies between 10 and 100; hence its logarithm lies be- tween 1 and 2, as does the logarithm of 74. 5. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. The. logarithm of a quotient is equal to the. logarithm of the dividend diminished by that of the divisor. The logarithm of any power of a number is equal to the loga- rithm of the number multiplied by the exponent of the power. The logarithm of any root of a number is equal to the loga- rithm of the number divided by the index of the root. 6. The preceding principles enable us to abridge labor in arithmetical calculations, by using simple addition and sub- traction instead of multiplication and division. II. MANNER OF USING THE TABLES. TO FIND THE LOGARITHM OF ANT NUMBER. 1. First find the characteristic by rule 2, 3, or 4, given above. 2. Then, if the number be less than 100, look in column N of the table for 10 times or 100 times the amount of it; oppo- site this multiple, in column O, will be found the mantissa. Thus the logarithm of 6 is 0.778151; that of S4 is 1.924279. 3. If the number lie between 100 and 10000, find the first three figures of it in column N; then pass along a horizontal line until you come to the column headed with the fourth figure of the number. At this place will be found the mantissa. Thus the logarithm of 7200 is 3.857332; that of 8536 is 3.931254.MANNER OF USING THE TABLES. 5 4. If the number be greater than 10000, place a decimal point after the fourth figure, thus converting the number into a mixed number. Find the mantissa of the entire part by the method last given. Then take from column D the correspond- ing tabular difference, multiply this by the decimal part, and add the product to the mantissa just found. The principle employed is that the differences of numbers are proportional to the differences of their logarithms, when these differences are small. Thus the logarithm of 672887 is 5.827943; that of 43467 is 4.638160. 5. If the number be a decimal, drop the decimal point, thus reducing it to a whole number. Find the mantissa of the log- arithm of this number, and it will be the mantissa required. Thus the logarithm of .0327 is 2.514548; that of 378.024 is 2.577520. TO FIND THE NUMBER CORRESPONDING TO A GIVEN LOGARITHM. 6. The rule is the reverse of those just given. Look in the table for the mantissa of the given logarithm. If it cannot be found, take out the next less mantissa, and also the corre- sponding number, which set aside. Find the difference be- tween the mantissa taken out and that of the given logarithm; annex as many 0’s as may be necessary, and divide this result by the corresponding number in column D. Annex the quo- tient to the number set aside, and then point off from the left hand a number of places of figures equal to the characteristic plus 1; the result will be the number required. If the char- acteristic is negative, the result will be a pure decimal, and the number of 0’s which immediately follow the decimal point will be one less than the number of units in the charac- teristic. Thus the number corresponding to the logarithm 5.233568 is 171225.296; that corresponding to the logarithm 2.23356S is .0171225. MULTIPLICATION BY MEANS OF LOGARITHMS. 7. Find the logarithms of the factors, and take their sum; then find the number corresponding to the resulting logariihin, and it will be the product required.6 MANNER OF USING THE TABLES. Example. Find the continued product of 3.902, 597.16, and 0.0314728. Operation. Log. 3.902. . . 0.591287 Log. 597.16 . . . 2.776091 Log. 0.0314728 . 2.497936 1.S65314 = log. 73.3354, the product. Here the 2 cancels the -j- 2, autl the 1 carried from the deci- mal part is set down. DIVISION BY MEANS OF LOGARITHMS. 8. Find the logarithms of the dividend and the divisor, and subtract the latter from the former; then find the number corresponding to the resulting logarithm, and it will be the quotient required. Example 1. Divide 24163 by 4567. Operation. , Log. 24163 . . . 4.383151 Log. 4567 . . . 3.659631 0.723520 = log. 5.29078, the quotient. Example 2. Divide 0.7438 by 12.9476. Operation. Log. 0.7438. . . 1.871456 Log. 12.9476. . . 1.112189 2.759267 = log. 0.057447, the quotient. Here 1 taken from _ gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic way. 9. The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of the arithmetical complement. The arithmetical complement of a logarithm (written a. c.)MANNER OF USING THE TABLES. 7 is the result obtained by subtracting it from 10: it may be written out by commencing at the left hand, and subtracting each figure from 9 until the last significant figure is reached, which must be taken from 10. Thus 8.130456 is the arithmet- ical complement of 1.869544. To divide one number by another by means of the arith- metical complement, find the logarithm of the dividend and the arithmetical complement of the logarithm of the divisor; add them together, and diminish the sum by 10; the number corresponding to the resulting logarithm will be the quotient required. Example. Multiply 358884 by 5672, and divide the product by 89721. Operation. Log. 358884 . . . 5.554954 Log. 5672 ... 3.753736 (a.c.) Log. 89721 . . . 5.047106 4.355796 = log. 22688, the result. The operation of subtracting 10 is performed mentally. TO RAISE A NUMBER TO ANT POWER BY MEANS OF LOGA- RITHMS. 10. Find the logarithm of the number, and multiply it by the exponent of the power; then find the number correspond- ing to the resulting logarithm, and it will be the power required. Example. Find the 5th power of 9. Operation. Log. 9 ........... 0.954243 5 4.771215=log. 59049, the power. TO EXTRACT ROOTS BY MEANS OF LOGARITHMS. 11. Find the logarithm of the number, and divide it by the index of the root; then find the number corresponding to the resulting logarithm, and it will be the root required.8 MANNER OF USING THE TABLES. Example. Find the cube root of 4,096. Operation. Log. 4,096, 3.612360; one-third of this is 1.204120, to which the corresponding number is 16, which is the root sought. 12. When the characteristic is negative, and not divisible by the index, add to it the smallest negative number that will make it divisible, and then prefix the same number, with a plus sign, to the mantissa. Example. Find the 4tli root of .00000081. The logarithm of this num- ber is 7.908485, which is equal to 8 -+- 1.908485, and one-fourth of this is 2.477121; the number corresponding to this logarithm is .03: hence .03 is the root required.PLANE TEIGONOMETBY. III.-VIII.PLANE TRIGONOMETRY, in. DEFINITIONS. 1. Plane Trigonometry treats of the solution of plane tri- angles. In every plane triangle there are six parts, — three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts is called the solu- tion of the triangle. 2. A plane angle is measured by the arc of a circle included between its sides; the centre of the circle being at the vertex, and its radius being 1. The circle, for convenience, is divided into 360 equal parts called degrees; 90 of these parts are included in a quadrant, which includes one-quarter of the circle, and is the measure of a right angle. Each degree is further divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds. Degrees, minutes, and seconds are de- noted by the symbols tl___________ b ________________Y °, thus the ex- pression 7° 22' 33" is read, 7 degrees, 22 minutes, and 33 sec- onds. 3. The complement of an angle is the dif- ference between that angle and a right angle. 4. The supplement of an angle is the difference between that angle and two right angles. 1112 NATURAL SINES, ETC. 5. Instead of employing the arcs themselves, certain func- tions of the arcs are usually employed, as explained below. A function of a quantity is something which depends upon that quantity for its value. The sine of an angle is the distance from one extremity of the arc enclosing it, to the diameter, through the other extrem- ity. Thus P M is the sine of the angle M O A. The cosine of an angle is the sine of the complement of the angle. Thus NM = OP is the cosine of the angle MOA. The tangent of an angle is a right line which touches the enclosing arc at one extremity, and is limited by a right line drawn from the centre of the circle through the other extrem- ity: the sloping line which thus limits the tangent is called the secant of the angle. A T is the tangent and O T the secant of the angle MOA. The versed sine of an angle is that part of the diameter AP which is intercepted between the foot of the sine and the ex- tremity of the enclosing arc. The cotangent of an angle is the tangent of the complement of that angle; the co-versed sine and cosecant are similarly defined. Thus BT', BN, and OT' are respectively the co- tangent, co-versed sine, and cosecant of the angle MOA. These terms are in practice indicated by obvious contractions; as, sin. A for the sine of A, cos. A for the cosine of A, &c. 6. The above definitions have been made with reference to a radius of 1. Anyfunction of an arc whose radius is R is equal to the corresponding function of an arc whose radius is 1, multiplied by the radius R. So also any function of an arc whose radius is 1 is equal to the corresponding function of an arc whose radius is R, divided by that radius. IV. NATURAL SINES, ETC. 1. Natural sines, cosines, tangents, or cotangents are those which are referred to a radius of 1. They may be used for all the purposes of trigonometrical computation; but it is found more convenient, in many cases, to employ a table of logarith- mic sines.LOGARITHMIC SINES, ETC. 13 V. LOGARITHMIC SINES, ETC. 1. Logarithmic sines, cosines, tangents, or cotangents are re- ferred to a radius of 10,000,000,000, of which the logarithm is 10. TO FIND THE T.OGARITHMTC FUNCTIONS OF AN ARC WHICH IS EXPRESSED IN DEGREES AND MINUTES. 2. If the arc is less than 45°, look for the degrees at the top of the page, and for the minutes in the left-hand column; then follow the corresponding horizontal line till you come to the column designated at the top by sine, cosine, tang., or cotang., as the case may be; the number there found is the logarithm sought. Thus, log. sin. 19° 55'.... 9.532312 log. tang. 19° oo'. . . . 9.559097 3. If the angle is greater than 45°, look for the degrees at the bottom of the page, and for the minutes in the right-hand column; then follow the corresponding line towards the left, till you come to the column designated at the bottom by sine, cosine, tang, or cotang, as the case may he; the number there found is the logarithm sought. Thus, log. cos. 52° 18' ... . 9.786416 log. tan. 52° 18' ... . 10.111884 4. If the arc is expressed in degrees, minutes, and seconds, proceed as before with the degrees and minutes; then multiply the corresponding number taken from column D by the num- ber of seconds, and add the product to the preceding result, for the sine or tangent, and subtract it therefrom for the cosine or cotangent. Examx>le. Find the logarithmic sine of 40° 26/ 28".14 LOGARITIIMIC SINES, ETC. Operation. Log. sine 40° 26'...............9.811952 Tabular diff. 2.47 No. of seconds, 28 Product . 69.16 to be added . 69 Log. sine 40° 26' 28"........... 9.812021 5. If the arc is greater than 90°, find the required function of its supplement. Thus the logarithmic tangent of 118° 18* 25", is equivalent to that of its supplement, or 61° 41' 35", and is 10.268732. Also the logarithmic cosine of 95° 18' 24" is 8.966080, and the log. cot. of 126° 23' 50" is 9.867579. TO FIND THE ABC CORRESPONDING TO ANY LOGARITHMIC FUNCTION. 6. This is done by a reverse process. Look in the proper column of the table for the given logarithm; if it is found there, the degrees are to be taken from the top or bottom, and the minutes from the left or right hand column, as the case may be. If the given logarithm is not found in the table, find the next less logarithm, take from the table the corresponding degrees and minutes, and set them aside. Subtract the loga- rithm found in the table from the given logarithm, and divide the remainder by the corresponding tabular difference. The quotient will be seconds, which must be added to the degrees and minutes set aside, in the case of a sine or tangeut, and subtracted in the case of a cosine or cotangent. Example. Find the arc corresponding to log. sin. 9.422248. Operation. Given logarithm . . . 9.422248 Next less in table . . . 9.421857 . . . .15° 19' Tabular diff. . . 7.68) 391(51" to be added. Hence the required arc is 15° 19' 51". 7. By analogous process, the arc corresponding to log. cos. 9.427485 will be found to be 74° 28' 43".GENERAL PROPOSITIONS. 15 VI. GENERAL PROPOSITIONS. 1. In any right-angled triangle the hypothenuse Is to one of the legs as the radius to the sine of the angle opposite to that leg. And one of the legs is to the other as the radius to the tan- gent of the angle opposite to the latter. 2. In any plane triangle, as one of the sides is to another, so is the sine of the angle opposite to the former to the sine of the angle opposite to the latter. 3. In any plane triangle, as the sum of the sides about the vertical angle is to their difference, so is the tangent of half the sum of the angles at the base to the tangent of half their difference. 4. In any plane triangle, as the cosine of half the difference of the angles at the base is to the cosine of half their sum, so is the sum of the sides about the vertical angle to the third side, or base. Also, as the sine of half the difference of the angles at the base is to the sine of half their sum, so is the difference of the sides about the vertical angle to the third side, or base. 5. In any plane triangle, as the base is to the sum of the other two sides, so is the difference of those sides to the difference of the segments of the base made by a perpendicular let fall from the vertical angle. 6. In any plane triangle, as twice the rectangle under any two sides is to the difference of the sum of the squares of those two sides and the square of the base, so is the radius to the cosine of the angle contained by the two sides.16 SOLUTION OF PLANE TRIANGLES. VII. SOLUTION OF PLANE TRIANGLES. 1. It is usually, though not always, best to work the propor- tions in trigonometry by means of logarithms, taking the logarithm of the first term from the sum of the logarithms of the second and third terras, to obtain the logarithm of the fourth term; or adding the arithmetical complement of the logarithm of the first term to the logarithms of the other two, to obtain that of the fourth. 2. There are three distinct cases in which separate rules are required. CASE I. 3. When a side and an angle are two of the given parts, the solution may be effected by proposition 2 of the preceding section. If a side be required, say, — As the sine of the given angle is to its opposite side, So is the sine of either of the other angles to its opposite side. 4. If an angle be required, say, — As one of the given sides is to the sine of its opposite angle, So is the other given side to the sine of its opposite angle. The third angle becomes known by taking the sum of the two former from 1S0°. c Example 1. Given angle A = 24° 26'; angle B = 36° 43'; side 6 = 137.6: to find B side a. As sin. B . . log. 9.776598 Is to sin. A . log. 9.616616 So is b . . . log. 2.13S618 11.755234, sum of 2d and 3d terms, To a, 95.2 log. 1.978636 less 1st term.SOLUTION OF PLANE TRIANGLES. 17 Example 2. Given, sides a and 6, as above, and angle A; to find angle B. As side a................(a. c.) log. 8.021364 Is to sin. A..................log. 9.616616 So is side b..................log. 2.138618 To sin. B = 36° 43'...........log. 9.776598 sum. CASE it. 5. When two sides and the included angle are given, the solution may be effected by means of propositions 3 and 4. Thus, take the given angle from 180°; the remainder will be the sum of the other two angles. Then, by proposition 3, — As the sum of the given sides is to their difference, So is the tangent of half the sum of the remaining angles to the tangent of half their difference. Half the sum of the remaining angles added to half their difference will give the larger of them, and half their sum diminished by half their difference will give the lesser of them. The solution may be completed either by proposition 4, or by proposition 2, as in Case I. Example. Given side a=95.2, side 6 = 137.6, and the included angle c = 118° 51'; to find the remaining angles. Here 180.00 — 118° 51'=61° 09', the sum of the remaining angles. As sum of given sides, 232.8 .................log. 2.366983 Is to their difference, 42.4...............log. 1.627366 So is tang. \ sflin of rem. angles, 30° 34 . log. 9.771447 To tang, i their difference = 0° 08$' . . . log. 9.031830 Adding half the difference to half the sum, 30° 34|'-j-6° 08$' = 36° 43', = the larger angle, B. Deducting half the difference from half the sum =24° 26'= the smaller angle, A. This case is susceptible of solution. also by means of propo- sition 6.18 RIGHT-ANGLED PLANE TRIANGLES. CASE III. 6. When the three sides of a plane triangle are given, to find the angles. First Method. Assume the longest of the three sides as base; then say, conformably with proposition 5, — As the base is to the sum of the two other sides, So is the difference of those sides to the difference of the segments of the base. Half the base added to half the said difference gives the greater segment, and diminished by it gives the less; thus, by means of the perpendicular from the vertical angle, the original triangle is divided into two, each of which falls under the first case. Or they may be solved by the simpler methods applica- ble to right-angled triangles. Second Method. 7. Find any one of the angles by means of proposition 6, and the remaining angles either by a repetition of the same rule, or by the relation of sides to the sines of their opposite angles. VIII. RIGHT-ANGLED PLANE TRIANGLES. 1. Right angles may be solved by the rules applicable to all plane triangles; and it will be found, since a right angle is always one of the data, that the rule usually becomes simplified in its application. 2. When two of the sides are given, the third may be found by means of the rule that the square of the hypolhenuse is equal to the sum of the squares of the remaining sides. 3. Another method for solving right-angled triangles is as follows: — To find a side. Call any one of the sides radius, and write upon it the word “ radius.” Observe whether the other sidesRIGHT-ANGLED PLANE TRIANGLES. 19 become sines, tangents, cosines, or the like, and write upon them the proper designations accordingly. Then say, As the name of the given side is to the given side, So is the name of the required side to the required side. 4. To find an angle. Assume one side to be radius, and mark the remaining sides as before. Then say, As the side made radius is to radius, So is the other given side to the name of that side; Which determines the opposite angle. 5. Applying this method to the right- angled triangle ABC, and calling the hypothenuse a radius, we shall have, c = a sin. C -i- R; hence sin. C = Rc -f- a. b = a cos. C -T- R; hence cos. C = Rb -j- a. Then, assuming the side b to be radius, we shall have, c = b tang. C -i- R; hence tang. C = Rc -f- b. If radius be called 1, the natural sines and cosines will be used in the application of these formulas; they are often more convenient than logarithms in railroad practice, especially when the numbers which measure the sides of the triangle are either less thau 12, or are resolvable into factors less than 12.ADJUSTMENT AND USE OF INSTRUMENTS. IX.-XV.ADJUSTMENT AND USE OF INSTRUMENTS. IX. GENERAL REMARKS ON ADJUSTMENTS. 1. Care should be taken in all instrumental adjustments, where screws work in pairs, to loosen one before tightening its opposite. 2. Remember that the eye-piece inverts the image of the cross-hairs, and that consequently any movement of it, by means of the small capstan head screws on the outside of the telescope-barrel, should be in the direction which would seem to increase the error requiring correction. 3. Before beginning the adjustments, screw the object-glass close home, and make a pin-scratcli across its rim and the end of the tube, by which to mark its proper place; draw out the eye-piece until the cross-hairs are exactly in focus; that is to say, until no movement of the eye shall appear to displace them, and bring the object to be observed clearly into view. 4. Never permit the glasses to be rubbed with a gritty fabric. To remove the dust from them, use a soft, cleau handkerchief, and cliauge often the part applied. 2324 THE LEVEL. X. THE LEVEL. TO BRING THE INTERSECTION OF THE CROSS-HAIRS INTO THE OPTICAL AXIS OF THE TELESCOPE. 1. Set the instrument firmly, cast loose the Mryes, and, by levelling and tangent screws, bring either of the cross-hairs to coincide with a well-defined object, distant from 400 to 000 feet, or as much farther as distinct vision can be had free from heat ripple. Gently rotate the telescope half-way around in the wyes. If the cross-hair selected for treatment then fails to coincide with the object, reduce the error one-lnilf by means of the small capstan head screws at right angles to it on the telescope-barrel. Bring the spider-line again to coincide with Ihe object by means of the levelling and tangent screws, and, if necessary, repeat the operation. Proceed in the same man- ner with the other cross-hair. If the error is large, bring both nearly right before undertaking their final adjustment. 2. Having thus adjusted the line of collimation upon a dis- tant point, requiring the object-tube to be drawn well in, select a point close by, which shall require it to be thrust out almost to its limit. If any error appears, correct half of it with the small screws provided for the purpose, a little forward of the diaphragm, and usually protected by a movable sleeve on the outside; correct the other half with the levelling-screws. After completing this adjustment, test the former one on a distant object, and. if necessary, repeat the operations. 3. In the transit, the small guide-ring screws used for this adjustment are covered by the bulb of the cross-bar in which the telescope is fixed, and are therefore inaccessible. The adjustment, however, is one not liable to become deranged in either instrument, and, in the transit, is of comparatively small importance. 4. The young practitioner should bear in mind that the intersection of the cross-hairs may coincide with the 'optical axis of the telescope, and yet be out of centre as regards the field of view. Such eccentricity does not affect the working accuracy of the instrument, which depends upon the positionTHE LEVEL. 25 of the object-piece solely. It may be removed by manipulation of the small screws securing the inner end of the eye-piece. TO BRING THE LEVEL BUBBLE PARALLEL WITH THE TELE- SCOPE AXIS. 5. Clamp the instrument over either pair of levelling screws, and bring the bubble to the middle of its tube. Turn the tele- scope slightly on its bearings, so that the bubble-case shall project a little ou one side or the other. If the bubble slips, correct half its movement by means of the small lateral capstan bead screws at one end of the case. Return the telescope to its first position, level up again, and repeat the operation until the erroneous movement ceases. This adjustment brings the telescope and level into the same vertical plane. G. Next, the bubble being at the middle of its tube, carefully lift the telescope out of the wyes, turn it end for end, and replace it. If the bubble settles away from the middle, bring it half-way back by means of the capstan-heads, working up and down at one end of the case. Again middle it with the levelling screws, and repeat the operation until the error is corrected. XD ADJUST TIIE WYES ; OR, IN OTHER WORDS, TO BRING THE TELESCOPE INTO A POSITION AT RIGHT ANGLES TO THE VERTICAL AXIS OF THE INSTRUMENT. 7. Close the wyes. Unclamp. Set the telescope directly over two of the levelling screws, and with them bring the bubble to the middle of the tube. Then rotate the telescope horizontally, until it stands over the same pair of screws, changed end for end. If the bubble errs, correct one-half of the deviation with the capstan head nuts at the end of the bar, and one-half with the levelling screws. Place the tele- scope over the other pair of levelling screws. Repeat the operation there; and continue the corrections, over one and the other pair of levelling screws alternately, until the bubble stands without varying during an entire revolution of the instrument upon its vertical axis. 8. The capstan head nuts on the cross-bar should be moved by gradual stress, not by pounding. They are a rude device. With so short a leverage as the length of the common adjust- ing-pin supplies, it is almost impossible to give them a smooth,26 LEVELLING. manageable motion. They reproach the instrument-maker’s art as unchecked hydrophobia and cancer do that of medicine, or mercenary villany that of law, and should be supplanted by belter practice. 9. Having thus completed the principal adjustments in their proper order, bring the telescope and its bubble-case as nearly vertical in the wye bearings as hand and eye can make them, and by reference to a plumb-line, or the corner of a well-built house, see if the vertical hair is out of true. If so, slightly loosen two opposite screws of the diaphragm, and correct (be error by turning it. Try again the adjustment of the line of collimation before piuning up the wyes. XI. LEVELLING. 1. Suppose O the starting-point; 1, 2, 3, &c., the stakes of survey; and A the initial bench-mark. Wherever convenient the elevation of A above mean tide should be ascertained. It is to be regretted that this was not done from the outset, under statute provisions requiring maps and profiles also to be filed at the several State capitals. In that case, not only would much after labor and expense by way of duplicate sur- veys have been spared, but the older Commonwealths would now have in hand materials for the preparation of physio- graphical maps, the value of which to science, to the engineer, and to the economical geologist, it were hard to measure.LEVELLING. 27 2. For the purposes of a railroad-survey, however, such determination is not needful. Any elevation may be assumed for A, taking care only that it be large enough to avoid the possibility of having minus levels, which would be inconven- ient. Zero of the datum should be below the lowest probable ground on the contemplated line. 3. Let the elevation of the initial bench-mark, A, in the figure, be taken at +200. Set the level at 13, and suppose the rod on the BM to read 2.22. The “ instrument height ” then is 202.22. If the rod at sta. O reads 8.4, the elevation at that point is 202.22—-8.4 = 193.8. The rod being L9 at sta. 1, the elevation there is 202.2 — 1.9 = 200.3. If desirable to turn at sta. 2, drive a pin nearly to the ground at that stake; sup- pose the rod on it to read 0.81. The elevation then is 202.22 — 0.81 = 201.41. Now move the instrument to C, and, sighting back to sta. 2, let the rod standing on the pin read 2.64. This makes the new “ instrument height” at C = 201.41, the height of sta. 2, + 2.64 = 204.05, and the elevations at 3, 4, 5, or other points observed from C are found by deducting the “ rods” at those points from the ascertained instrument height at the new point of observation. 4. It thus appears how simple is the rule of levelling, namely: Find the “instrument height” by adding the “back- sight” to the elevation of the point upon which the rod stands for that purpose: from the “instrument height” thus found deduct the “foresights,” severally, in order to find the eleva- tions of the points at which such foresights are taken. 5. The foregoing example would appear in the field-book as follows: — Sta. B. S. Inst. F. S. Eleva. Remarks. B M 2!22 202!22 200.00 B M on \V. Oak. 40 It. N. of Sta. O. 0 , , 8.4 193.8 1 1.9 200.3 2 2.k 204!05 0.81 201.41 3 3.7 200.3 4 3.2 2!Ml.8 5 10.36 193.69 6. In levelling where great exactness is necessary, the rod at turning-points should be read to thousandths, and the reading checked by the leveller. Before taking it down, after clamp-28 LEVELLING. ing the target fast, it should be swayed slowly to and fro in the direction of the instrument to make sure of getting the full height. In foul weather the rodman should lake care that the foot of the rod does not ball up with mud or snow. The leveller should have his cross-hairs free from parallax, the tar- get in focus, and see his bubble true at the moment of obser- vation. He should also set the instrument about half-way between turning-points when practicable, balancing largely unequal sights by subsequent ones similarly unequal in the opposite direction; and his turning-points, even on favorable ground, ought not to be more than 600 or 800 feet asunder. 7. On ordinary railroad field work such nicety as is implied in most of these rules is not required. To read to the nearest tenth is sufficient, especially where the progress ot the party depends in a good degree on the level; as, for example, in run- ning grade lines on preliminary survey. The location levels are usually carried along more carefully; but even then the writer’s practice has been to turn to hundredths only. S. The Philadelphia Rod is the best for our service. The sliding halves are unconnected except by brass sleeves or clips, which guide them, and are therefore, not liable to bind in wet weather. They are made by William J. Young’s Sons, who some years ago, at the writer’s suggestion, supplied what seemed to be their only defect by adopting rivets for fastening the clips instead of wood screws: the screws had a tendency to work loose in the field, and cause the parts to chafe or jam. These rods are clearly figured, so as to be legible at a distance of several hundred feet; the leveller is thus enabled to take intermediate elevations rapidly, and, when necccsary, to do his work with the aid of an unlettered rodman. 9. CORRECTION FOR THE EARTH’S CURVATURE ANI) REFRAC- TION. The correction for a 100-feet “station” is .00021"): for one mile, 0.6. It is to be added to the calculated elevation of the point observed, or to be deducted from the “rod” before calculating the elevation, in the case of a long unbalanced sight. It varies as the square of the distance. Calling the required correction A, for any given distance D, then A = .000215 X D2 if L) is in “stations,” and A = 0.6 X D '2 if D is in miles. Thus the correction for 10 stations would beLEVELLING. 29 .0215; for 50 stations, 0.5375; for 10 miles, 60 feet, ami a spire or treetop apparently level with the instrument at that dis- tance would really be 60 feet above it. Transposing the equa- tion we have D = y/ A4-0.6. In this form it is applicable to the determination of distances at sea. The Peak of Teneriffe, for example, 16,000 feet high, should be just visible from the sea-level at a distance = y/16000 4- 0.0— say 163 miles. 10. TO FIND DIFFERENCES IN ELEVATION BY MEANS OF TIIE BAROMETER. Call the required difference D; the barometrical reading at the lower stand, L; that at the upper stand, U. Then, D=[ (L — U)4-(L + U) | X 55000. Example. L = 26.64; U = 20.82. Then, L —U= 5.82 .... log. 0.764923 L + U = 47.40 .... log. 1.676328 0.1226 .............Diff. —1.088595 And 0.1226 X 55000 = 6743, the required difference of elevation in feet. 11. A closer approximation is thought to be attainable by using a thermometer in connection with t he mercurial barome- ter. In that case, having found the difference as above, add of the result for each degree by which the mean tempera- ture of the air at the two stands exceeds 55°; subtract the like proportion if the mean temperature be below 55°. When the upper thermometer reads highest, for “subtract” say “add,” and vice versa in the foregoing rule. 12. The naked formula, however, will usually be sufficient for the engineer. He can prescribe gradients by it for surveys, which shall develop the ground to he occupied, and can decide between summits well differenced in height. If not so differ- enced, questions of detour, of approaches, and the like, will contribute to determine the expediency of making an instru- mental examination. 13. HEIGHTS BY THE THERMOMETER. T = the difference, in degrees Fahrenheit, between 212°, the temperature of boiling water at the sea level, and that at the place of observation.30 SETTING SLOPE STAKES. H = the height of place of observation above or below the sea in feet. H = 513 T + T2. Example. T -- 212° — 20S° = 4°. H = (513 x 4) + 42 = 2068 feet. XII. SETTING SLOPE STAKES. 1. Like swallowing, this is more easily done than described. To no detail of field service does the proverb more fitly apply, that “ work makes the workman.” 2. The problem is, to find where a formation slope of given inclination, beginning at the side of the road-bed, must needs intersect the ground surface. Formation slopes are usually stated in parts horizontal to one part vertical. Thus a slope of 45° is “ 1 to 1.” A slope of “ 2 to 1 ” has a horizontal reach of two feet to each foot vertical. The carriages of a stairway with twelve-inch treads and eight-inch risers would have a slope of “ 1$ to 1.” 3. To fix the point where any proposed formation slope must meet the surface on level ground, is quite simple; the distance from the centre line being obviously made up of half the width of road-bed added to the horizontal distance due from the slope, to the depth of cut or height of fill. Thus, with 20 feet road-bed, 9 feet cut, and slope of 1$ to 1, the distance out would be 10 -f- 9 + 4$ = 23$ feet, as shown in the annexed diagram. <. 23.6 X___10___X >3.0______>___ lO K 10 4. On slant or broken ground, the solution is more difficult: recourse must then be had to the level, with a rodman, a tape- man, and, for good speed, an axeman to assist.SETTING SLOPE STAKES. 31 Example No. 1. 5. Let the accompanying figure represent the cross-section at any given point of a proposed excavation; road-bed 20 feet wide, cutting at centre stake 12 feet, and formation slopes 1 to 1. —4._. 6. The first step is to set the level, as at A, commanding, let us suppose, the lower slope, and to ascertain its height above grade at the proposed section. This is usually done by refer- ence to the nearest bench, and pegging from stake to stake as the work progresses. Unless the ground is very steep, and the slope-stakes largely different in elevation, labor will be saved and likelihood of error reduced by levelling over the centre line beforehand, as a separate job, and marking on centre stakes the cuts, fills, and grade points, that is to say, the points where excavation passes into embankment. The rods should be taken carefully at the stakes, and the latter marked on their backs to the nearest tenth, as “ grade,” “ C 12,” signify- ing cut 12 feet, or “ F 6.2,” signifying fill 6.2 feet, for ex- ample. This being done, each centre stake serves as a bench- mark for slope staking at that section, and each cross section can be staked out independently. 7. Instrument height, in the example treated, being by either method fixed at 15.5 above grade, the next step is a guess how far out from the centre stake the formation slope would proba- bly meet the ground surface. The closeness of the guess will correspond to the experience and natural skill of the leveller: the young engineer should not be discouraged if he misses U ! mark rather widely in his early trials.32 GETTING SLOPE STAKES. 8. It is true, that, on a uniform declivity, he might aid con- jecture by taking a rod distant half the width of road-bed, or 10 feet, from the centre stake, ascertain thus the slope of the ground surface as well as the cutting at that point; and with these data, knowing also the formation slope, approximate the point sought by solving the terminal triangle of the pro- posed section, indicated by dotted lines in the figure. But, in practice, he will find it the quicker and better way to approxi- mate the point by vividly imagining the underground forma- tion lines; or by vividly imagining a level section, the upper surface of which shall coincide with his instrument height, 15.5 feet above grade. This gives him a point in the air, 10 + 15.5 = 25.5 feet out from the centre stake, level with the instrument, as the limit of the imaginary section; and from that point he can pretty well judge where a line corresponding to the formation slope must meet the ground. t). Suppose him, by either method, or even by random guess, to think that 10 feet for half the road-bed, and 10 more for the slope, looks about right. The formation slope being 1 to 1, this implies a cutting of 10 feet at the side stake, and a rod, therefore, of 15.5 — 10.0 = 5.5 feet. Taking a rod accordingly, 20 feet out, measured horizontally from the centre stake, he finds it to be 11.0 instead of 5.5, indicating that he has gone too far down hill. Let him now reason that the rod of 11.0 corresponds to a cutting of 15.5 — 11.0 = 4.5 feet, and that a cutting of 4.5 feet corresponds to a distance out of 10 -f- 4.5 = 14.5 feet. Try, then, a rod 14.5 feet out. It proves to be ‘J.0, corresponding to a cutting of 15.5 — 9.0 = 6.5, instead of 4.5 feet, and a distance out of 16.5 instead of 14.5 feet. Try, next, 16.5 feet out; the rod there, of 10.0 instead of 9.0, show's him again to be in error on the down-hill side of his object; but the lessening error shows also that he is approaching it, and that a few more like trials will reach it. 10. Recurring to his first error with the 11.0 feet rod, he cannot fail to observe after a little practice, since the ground ascends thence toward the centre line, that the side stake must fall farther out than the point where his second trial was made; that its true position, in fact, divides the distance be- tween those points of observation into two parts which are to one another directly as the inclinations of the formation slope and the ground surface. By degrees he will grow skilful in divining this true position, and, becoming ineauwhile quick inSETTING SLOPE STAKES. 33 observation, will place a slope stake on the second or third trial, without conscious effort of mind. 11. Next, suppose the level at B, 25.5 feet above grade, com- manding the upper slope. Note the change of ground 11 feet out, and take a rod there, recording the observation. The cutting at that point is 25.5 — 9.5 = 16 feet, corresponding to a distance out for the side stake of 10 + 16 = 26 feet, if the ground were level. A trial rod 26 feet out reads 7.8, corresponding to a cutting of 25.5 — 7.8 = 17.7 feet, and a distance out for the side stake of 10 -+-17.7 = 27.7 feet, showing that the point sought is still beyond. A repetition of such trials will finally fix it; but, as in the case of the lower slope, practice will speedily lessen the number and abridge the labor of them. 12. The foregoing section would be noted in the field book as follows: — St a. Dia. Left. Centre Right. | Area. C. Yds 258 50 + 5.8 15.8 + 12.0 + 16.0 11.0 + 18.0 28.0 Example No. 2. 13. In the annexed figure, representing an embankment 14 feet wide on top, with side slopes of 1$ to 1, the first thing to attract attention is that the instrument is 1 foot below c two straight lines intersecting at E. Lay oft equal distances, EA, EB; erect perpendiculars at A and and G A or GB, the radius, R. Call the deflection angle to a chord of 100 feet D, as before. 2. By XVI. 3 and 4, angle EAB = EB A = AGE = EGB = il. 3. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, TO FIND THE TANGENT T. T = R X tan. * I. Example. R = 1,910.1, I = 35° 24'. Then T = R tan. i I = 1,910.1 X 0.3191 = 609.5. 4. Measure from the P. I. equal distances, E M, E F, along the tangents. Measure, also, M F and E K, the distance from E to the middle point of M F. Then, by reason of similarity in the triangles M E K, E A G, B, meeting at G, and con- nect A B, E G. From the centre G, with radius G A, draw the curve A H B. G The point E will be the P. I.; A and B, tangent points; E A, EB, the tan- gents, or apex distances, which denote by T; E.H, the external secant, or S; H N, the versed sine, or V. Let the long chord AB, connecting the tan- gent points, be called C, M K : E K:: A G : A E ;: R: T .•.T = RXEKvMK.TO FIND THE RADIOS OF A CUR VE. 53 Example. Let M K = 190.5, E K = 60.8, R = 1910.1. Then R = 1910.1 E K = 60.8 M K= 190.5 (a.c.) 3.281056 1.783904 7.720105 T = 609.6 2.785065 5. If 100-feet chords be used, find the tangent in Table XYI. corresponding to the given angle I. Divide that tabular tan- gent by the degree of curvature corresponding to the given radius: the quotient will be the required tangent.' Thus, Tab. Tan. corresponding to 35° 24' = 1,828.7, which, divided by 3, the degree of curvature, gives 009.6, the tangent sought. 6. GIVEN TIIE INTERSECTION ANGLE I AND TANGENT T, TO FIND RADIUS R. Transposing the equation in (3), R = T -r tan. i I = T X cot. 11. Example. 7=009.6,7=35°24' R = Tcot. * 1=609.6 X 3.1334 = 1910.1. By a like transposition of the equation in (4), R = TXM K + E K. 7. If 100-feet chords be used, find in Table XYI. the tangent corresponding to the given angle I. Divide that tabular tan- gent by tlie given tangent; the quotient will be the degree of curvature in degrees and decimals. The radius corresponding to this degree of curvature may be found by (12), by Table X., or, with sufficient accuracy for ordinary practice, by dividing 5,7:10, the radius of a 1° curve, by it. Thus, in the foregoing example, the tabular tangent cor- responding to 35° 24' is 1,828.7. Dividing by 609.6, we have 3 for the degree of curvature; and 5,730 divided by 3 gives R = 1,910 feet.54 TO FIND THE RADIUS OF A CURVE. S. GIVEN TIIE INTERSECTION ANGEE I AND CHORD A B = C, CONNECTING THE TANGENT POINTS, TO FIND RADIUS E. AN = \AB=^C; A G N —i I. A G = A N -r- sin. AG N ; 2.6., It = ^ C -f- sin. I. Example. I = 35° 24', C = 11G1.4. Then It = -J- C -f- sin. ^ I, = 580.7 -j- 0.304 = 1910.2. 9. If 100-feet chords be used, find in Table XVI. the chord corresponding to the given angle I. Divide that chord by the given chord, for the degree of curvature in degrees and deci- mals. Determine the corresponding radius by (17), by Table X., or, for ordinary practice, by dividing 5,730 by it. Thus, in the foregoing example, the tabular chord corre- sponding to angle 35° 24' would be 3,4S4.2, which, divided by the given chord, 1;161.4, gives 3 for the degree of curvature, and 5,730 divided by 3 makes the radius E = 1,910 feet. 10. GIVEN THE INTERSECTION ANGLE I AND THE DEGREE OF CURVATURE OR DEFLECTION ANGLE D, WITH 100-FEET CHORDS, TO DETERMINE THE LENGTH OF THE LONG CHORD C, THE VERSED SINE V, THE EXTERNAL SECANT S, OR THE TANGENT T. Take from the proper column in Table XVI., the number corresponding to the intersection angle, and divide it by the degree of curvature: the quotient will be the length required. Example. A 4° curve, 1 = 50° 16'; to find the several functions above named. Table XVI. gives the designated functions of a 1° curve as follows: C 4,867.3, V 542.4, S 599.3, T 2,688.2. Dividing by 4 the degree of curvature, we have for the corresponding func- tions of a 4° curve as follows: C 1,216.8, V 135.6, S 149.8, T 672.0.RADII, DEFLECTION ANGLES, ETC. 55 11. GIVEN C, V, S, on T, OF ANY CURVE, AND D, THE DE- GREE OF CURVATURE, TO FIND THE INTERSECTION ANGLE, I. Multiply the given function C, V, S, or T, by the degree of curvature, D: the product will be found in the proper col- umn of Table XVI., corresponding to the required angle. Example 1. Given T = 515, D = 5°; to find I. Then T X D = 2,575, which corresponds in Table XVI. to 48° 24' -- I. . Example 2. Given C = 1,656, D = 3°; to find I. Then C X D = 4,968, which corresponds in Table XVI. to 51° 23' = I. 12. GIVEN C, V, S, OR T, OF ANY CURVE, AND THE INTER- SECTION ANGLE I, TO FIND THE DEGREE OF CURVATURE D. Take from the proper column of Table XVI. the number corresponding to the given angle I, and divide that tabular number by the length of the given part; the quotient will be D, in degrees and decimals. Example 1. Given T = 587, I = 22° 26'; to find D. The Tan. corresponding to I in Table XVI. is 1,136.3. Then 1,136.3 -4- 587 = 1.935 = 1° 56' = D. Example 2. Given S = 64, I = 30° 25', to find D. The Ex. Sec. corresponding to I in Table XVI is 208. Then 208 64 = 3.25 = 3° 15' = D. 13. GIVEN THE INTERSECTION ANGLE I, AND DEFLECTION ANGLE D, TO FIND THE LENGTH OF THE CURVE. Divide I by D: the quotient will be the number of chord lengths in the curve. If the degree of curvature is a whole number, the more con- venient method of effecting the division is, first, to reduce the56 RADII, DEFLECTION ANGLES, ETC. minutes, if any, in I to decimals of a degree; then divide by the degree of curvature. Example 1. I = 20° 40', D = 3°. 20° 40' is equivalent to 20.67 degrees. Dividing by 3, we have 6.89 chord lengths for the length of the curve. If the chords, as is usual, are each 100 feet long, the length of the curve in this case will be 6S9 feet. If the chord lengths were 50 feet each, the length of the curve would be half this number of feet. 14. If the degree of curvature is fractional, the more con- venient method of effecting the division is, first, to reduce both I and D to minutes; then divide the former by the latter. Example 2. I =.30° 22', D = 2° 45'. These are equivalent, respectively, to 1,822 and 165 minutes. Dividing the former by the latter, we have 1,104 feet for the length of the curve. 15. The ingenious assistant who will attentively consider the preceding figures cannot fail to detect other obvious analo- gies which it has not been thought necessary to include in this compendium. 16. In railroad field practice it is usually sufficient to deter- mine angles to the nearest minute, and distances to the nearest foot. The nicety of seconds and tenths appears generally to be quite superfluous; the time consumed on them were better employed in pushing ahead. 17. GIVEN ANY DEFLECTION AN- GLE D, AND CHORD C, TO FIND RADIUS R. FB -r- sin. i A L B = B L ; i.e., i C-~ sin. % D = R. Example. Let (7=100 feet, D = 4°. Then R = i C sin. £ D = 50 -f- .0349 = 1432.7. If the chords are 100 feet long, as is usual in railroad prac- tice, radius may be found with sufficient accuracy by dividingRADII, DEFLECTION ANGLES, ETC. 57 5,730, the radius of a 1° curve, by the deflection angle, or de- gree of curvature. Thus, in the foregoing example, 5,730 -f- 4 = 1,432.5. 18. GIVEN ANV RADIUS R, AND CHORD C, TO FIND THE DE- FLECTION ANGLE D. From the preceding equation and example: — Siti i D = | C -f- 11 = 50 -r-1,432.7 = .0349 = sin 2° = i D ... D = 4°. 19. GIVEN ANY RADIUS It, AND CHORD C, TO FIND THE DE- FLECTION DISTANCE d. First find the deflection angle by above method (18). Then, angle IIAB in the figure being made equal to D, and HA = BA, BII will be llie deflection distance. Draw AK to the middle point of n B. Then cZ = IIB = 2KB = 2AB X sin K A B = 2 C X sin *D. Example. Let II = 1,140 feet, C = 100 feet. By (18) D will be found = 5°. Then d = 2 C X sin ^ D = 200 X .0436 = S.72 feet. 20. If the chords are 100 feet long, as is usual in field meas- urement, divide the constant number 10,000 by the radius in feet: the quotient will be the deflection distance. The deflec- tion distance with radius of 10,000 feet, and chord of 100 feet is one foot: this rule is based upon the principle that deflection distances, the chord length being fixed, will vary inversely as the radii. Thus, in the foregoing example, 10,000 1,140 = 8.72. 21. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN- GENTIAL ANGLE T. The angle T is equal to j D by construction; for mode of determining it, see preceding section (18).58 ORDINATES. 22. GIVEN ANY HADIUS R, AND CIIOJJD C, TO FIND THE TAN- GENTIAL DISTANCE t. First find the tangential angle, as above directed. Then, angle B AE in the figure being made equal to T, and AE = AB, BE will be the tangential distance. Draw AN to the middle point of BE. Then t = EB=2XB = 2AB X sin NAB = 2 C X .sin £ T. Example. Let It = 1,14G feet, C = 100 feet. By sect. 1, T will be found = 2° CO'. Then t = 2 C X sin £ T = 200 X .021S = 4.36 feet. 23. In ordinary railroad practice the tangential distance may be considered equal to half the dellecLion distance. XIX. ORDINATES. 1. GIVEN ANY IIADIUS R, AND CIIOKD C, TO FIND TIIE MID- DLE OltDINATE M. In the annexed figure, IIN = M, HG = R, A B = C. ♦ NG = Va G2 — AN* = Ve2 — iC2; II N = II G — NG, i.e., M = R — VRj —i C2.OliVmATJCS. m Example. R = S19, C=100; to find the middle ordinate, M. M = SI 9 — V670761 —2500 = 1.53. 2. Angle IIAN = iHGB; HGB = JAGB, HAN = JAGB. JIN = AN X tan. HAN; i.e., M = | C X tan. { D; D being the central angle subtended by the chord. Example. D = 7°, C = 100; to find M, the middle ordinate. M = 1CX tan. £ D = 50 X 0.03055 = 1.528. 3. GIVEN THE RADIUS R, CHORD C, AND MIDDLE ORDINATE M, TO FIND ANY OTHER ORDINATE E K = M', DISTANT (1 FROM N, THE MIDDLE POINT OF THE CHORD. KL = NG; NK = GL; EK = EL — NG. E L = Vg E2 —N K2 = Vk- — d3; NG (1) = VR2 —± C2. Then EK = M' = VR2 — d2 — VR2 — * C2. 4. It is a property of the parabola, that ordinates vary as the products of their abscissas. This property may be assigned to the circle in cases where the arc encloses a small angle. Applying it here we have — HN:EK::AN X N B : A K X KB. Call any segments AK, KB, of the chord, a and b. Then M : M':: £C*: ab, .'. M' = M X 4 ab C2. Example. M = 1.528, C = 100, a = 60, b = 40; to find M'. M' = 1.528 X 9600 -f-10000 = 1.528 X 0.96 = 1.467. 5. Multiply the corresponding ordinate of a 1° curve from the annexed table by the degree of curvature: the product will be the ordinate sought.GO ORDINATES. ORDINATES OF A 1° CURVE, CHORD 100 FEET. Distances of the Ordinates from the End of the 100-feet Chord. Middle Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 50 ,45 40 35 30 25 20 15 10 5 Lengths of the Ordinates in Feet. .218 .218 .209 .193 .183 .164 .140 .111 .078 .041 Example. What is the ordinate of a G° curve, 30 feet from the end of the 100-feet chord? The corresponding tabular ordinate of a.l° curve is .1S3; which, multiplied by 6, gives 1.09S, tbe required ordinate. 6. A quick way of laying off ordinates on the ground, and one sufficiently exact for the field, is, after fixing the point II by means of the middle ordinate IIX, to stretch a line from H to A, and make the middle ordinate F O = | IIX; then from F to A and F to H, making the middle ordinates = £ FO; and so on. 7. A good track-layer will seldom require points at shorter intervals than 25 feet.TRACING CURVES AND TURNING OBSTACLES IN THE FIELD. XX.-XXIIITRACING CURVES AND TURNING OBSTACLES IN THE FIELD. XX. TO TRACE A CURVE OX TITE GROUND WITH THE CHAIN ONLY. 1. This is best taught by an example. Example. From a point B, IS feet in advance of A, on tangent A B, to trace a curve of 367 feet radius to the right, with chords 66 feet long, and consuming an angle of 34° 27'. 6364 TO TRACE A CURVE ON THE GROUND. 2. First, dividing half the unit chord, or 33 feet, by the radius, 367 feet (XVIII., 18), we have 0.09+ for the sine of the tangential angle, corresponding to an angle of 5° 10': the de- flection angle, therefore, is 10° 20'. The tangential distance corresponding to the angle 5° 10', and chord 66 feet, is equal (XVIII., 22) to twice the chord multiplied by the sine of half the tangential angle, = 132 X 0.04507 = 5.95 feet. The deflec- tion distance (XVIII., 19) is equal to twice the chord multi- plied by the sine of half the deflection angle, = 132 X 0.09+ = 11.88, say 11.9 feet. 3. To find the length of the curve (XVIII., 13): Divide the total central angle by the degree of curvature. The central angle, 34° 27', is equivalent to 2067 minutes; dividing by 620, the number of minutes in the deflection angle, we have 3.33, the number of chord lengths in the curve, = 3^ chains = 220 feet. If A be a stake numbered 2, then the point of curvature, B, will be 2.18, and the "point of tangent, F, will fall at 2.18 + 3.22 = stake 5.40. 4. To determine the tangential distance C P, to the first stake on the curve, either of two methods may be used: — First, The sine of any tangential angle is equal to half the chord which limits the angle on one side divided by radius. The limiting chord B C in this instance is equal to 66 — 18 = 48 feet; half of 48, therefore, or 24 feet, divided by radius, 367 feet, gives 0.0654, the sine of 3° 45' = tangential angle P B C. The sirre of half this angle multiplied by twice the given chord = 0.0327 X 96 = 3.14 feet, the tangential distance CP. 5. Secondly, CP may be found as follows, assuming that arc C P. Prolong B C to D. E F may be taken as the tangen- tial distance due to the whole chord BF, and PC the tangen- tial distance due to the sub-chord B C. the functions of small angles vary directly as the angles themselves, and vice versa. Let BF be a portion of the curve. Make the tangent B E equal to the A chord B C, and strike theTO TRACE A CURVE ON THE GROUND. 65 Then PC : ED :: BC : BD or BF; and, by the foregoing supposition, E D : E F :: B C : B F. Combining these propor- tions, and cancelling E D, we have PC : EF :: BC2 : BF2 .*. PC*=EF X (BC-^BF)2. In words, the tangential distance for a sub-chord is to that for a whole chord as the square of the sub-chord is to the square of the whole chord. The same is true of deflection dis- tances. 6. In the example we are treating, the tangential distance for the whole chord of 66 feet has been found to be 5.95 feet; that for 48 feet, therefore, is 5.95 X 482 662 = 5.95 X 0.528 = 3.14, as before. Stretch the 48 feet of chain from B to P, in prolongation of tangent A B, and mark the point P; then step aside, and stretch from B to C, making the distance PC = 3.14 feet: C will be a stake on the curve. 7. Next, run out the whole chain length from C to O in the range B C. To find O D, suppose the line N C T to be drawn tangent to the curve at C. Then N D may be considered the tangential distance due to the whole chord, = 5.95, as above determined. The angle O C N = T C B = P B C (XVI., 4); and (5) ON:ND;:BC:CD.l.ON = NDXBC-rCD; i.e., O D = ND + ON = ND+[NDX (B C-=-C D)] =5.95 X [1 + (48 -=-66)] =5.95 X 1.727 = 10.27. 8. The point N may be fixed otherwise by laying off B T = C P, and running out the chain length C N in the range C T. The point D on the curve may then be fixed by making N D equal to 5.95 feet, the tangential distance. Next run out the chain to M, in the range CD; make ME equal to the deflection distance, 11.9 feet, and fix the point E. The points C, D, and E will be stakes 3, 4, and 5 on the curve. 9. To set the point of tangent, F, at stake 5.40, prolong the chord line D E for 40 feet to L, and suppose V E to be drawn tangent to the curve at E. Then the angle L E V is equal to the tangential angle of the curve; and the sub-tangential dis- tance L V is to the whole tangential distance due to the 66- feet chord, as the sub-chord is to the whole chord (5); i.e., L V = 5.95 X 40 -i- 66 = 3.6 feet. By the method illustrated in (6), the distance FY will be66 TO TRACE A CURVE ON THE GROUND. equal to 5.95 X 401 2 662 = 5.95 X 0.367 = 2.18 feet. With the distance LF = 3.6 + 2.18 = 5.78 feet, thus obtained, and the sub-chord E F = 40 feet, the point of tangent F may be established. 10. Next, set off UE = F V = 2.18 feet, and lay out FK in prolongation of the range UF; FK will be in the line of the terminal tangent. 11. This analysis has been somewhat minute and detailed, in Order that the subject may be thoroughly understood. An instrument for measuring angles should always be used in rail- road service: it greatly simplifies and abridges the labor of tracing field-curves, and gives more exact results. But occa- sions sometimes rise, in miscellaneous practice, when strict accuracy is not required, and the chain only can be had: the young engineer should qualify against such occasions. XXI. TO TRACE A CURVE ON THE GROUND WITH TRANSIT AND 100-FEET CHAIN. 1. This, also, is best taught by an example. Let it be a general rule, in locating, to fix the intersection of tangents, and to set the tangent points, or the P. C. at least, from the P. I. There are exceptional conditions, as a steep hillside, timber or broken ground, a very long arc, unimpor- tance of exact conformity to the project, and the like, which warrant its omission; but where these conditions do not obtain or are not prohibitory, and a snug fit is desirable, time will usually be saved by fixing the P. I. It often proves serviceable as a reference point during construction: on the location, i( gives confidence in the work and an assurance of safe progress', which are well worth a little painstaking beforehand. 2. Having established the P. I., and found the intersectign angle to measure, say, 66° 45r, the first step is to find the apex distances so called, or tangent lengths IB, IF. These are each equal to R X tan. $ I. If a 7° 30' curve be prescribed to close the angle, R X tan. i I = 764 X 0.659 = 503 feet.TO TRACE A CURVE ON THE GROUND. 67 Or, referring to Ta- ble XVI., the tangent corresponding to 66° 45' is found by inter- polation to be 3774.6; dividing by 7.5, the rate of curvature in degrees and decimals, we have lor the apex distance 503 feet, as above. 3. Before disturbing the instrument, which is presumed to stand in the Tange of the terminal tangent, measure I F, = 503 feet, and set the P. T. at F. Then direct the telescope to the last point fixed on the ini- tial range AB, meas- ure I B, = 503 feet, and set the P. C. at B. Move to B. 4. Suppose the P. C. to have fallen at a stake 2.50. In order to find the length of the curve, divide the intersection angle by the degree of curva- ture, having first re- duced the minutes in each to hundreths of a degree by multiplying by 10 and dividing the product by 6. Thua the intersection angle becomes 66.75°, and the degree of curva- ture 7.5°: dividing the68 TO TRACE A CURVE ON THE GROUND. former by the latter, we have 890 feet for the length of the curve. Or, the intersection angle 66° 45' is equivalent to 4005', and the degree of curvature 7° 30' is equivalent to 450': dividing the former by the latter, we have 890 feet for the length of the curve, as before. 5. Adding 8.90 to 2.50, the number of the P. C., the P. T. is found to fall at stake 11.40. Let the rear chainman make a note of this, that there may be no mistake in the terminal plus. 6. Next, to determine the proper deflections from the line of tangent at B, bear in mind that the deflection for a whole chain is half the degree of curvature ; and that, in field-curves of more than 300 feet radius, the deflections for sub-chords, or plusses, may, without material error, beheld to vary directly as the sub-chords themselves; that is to say, the sub-deflec- tions due to 30, 60, and 80 feet, for instance, will be, to the deflection due to 100 feet, as 30, 60, and 80 are to 100. 7. Thus, in the example, 7° 30' being the degree of curva- ture, one-half of this, or 3° 45', will be the deflection due to a chord of 100 feet; and of this, or a deflection of 1° 52^ from the line of tangent at B, will fix stake 3, 50 feet distant on the curve. 8. The following is a simple rule for finding sub-deflec- tions:— Multiply the sub-chord in feet by the rate. of curvature in degrees and decimals: three-tenths of the product will be the sub-deflection in minutes. Thus, in the example, 50 X 7.5 = 375, and 375 X 0.3 = 112.5' = 1° 52*', as before. 9. Having set stake 3, stakes 4 and 5 will be fixed by succes- sive deflections of 3° 45'. In establishing stake 5, the index will read, 1° 52^ + 3° 45' + 3° 45' = 9°-22f = angle C B 5. 10. Suppose the instrument moved to 5. See that the ver- nier has not been disturbed, backsight to B, and deflect 9° 22£/ right; i.e., double the index angle. The index will now read, 18° 45' = the angle 1C D; and the telescope will be directed along the line C D, tangent to the curve at 5, for the reason that the angle B 5 C has been made equal to the angle C B 5 (XVI. 4). Proceed with successive deflections of 3° 45' from this tan- gent, and set stakes 6, 7, 8, and 9, at intervals of 100 feet. 11. Suppose the instrument moved to 9. In fixing thisTO TRACE A CURVE ON THE GROUND. 69 stake, the index will read, 18° 45' + 4 times the constant angle 3° 45', = 18° 45' + 15° = angle IC D + angle D 59, = 33° 45'. In order to place the telescope in the line D E, tangent to the curve at 9, it is now necessary to turn an angle to the right, from backsight to 5, equal toD95 = D59 = 15°; i.e., the vernier should be moved from 33° 45' to 33° 45' + 15° = 48° 45'. The telescope will then be in tangent at 9. 12. A .simple rule for finding the index angle which shall place the telescope in tangent at any point on the curve is as follows: — From double the index angle which fixed the given point, sub- tract the index reading in tangent at the last turning-point: the remainder will be the required index angle. Thus the index angle which established stake 9 was 33° 45'. Double this angle will be 67° 30'; subtracting 18° 45', the reading in tangent at the last turning-point, we have 48° 45', the required index angle, as before. The reasons for the rule will be obvious from an examina- tion of the figure. 13. Being in tangent, then, at 9, and the index reading 48° 45', a deflection of 3° 45' will fix 10: a further deflection of 3° 45' will fix 11, and the index will stand at 48° 45' -f- 7° 30' = 56° 15'. 14. To find the deflection corresponding to the sub-chord 11 F, =40 feet: by. the foregoing rule (8), the degree of curva- ture, 7.5, multiplied by 40, the length of the sub-chord in feet, gives a product of 300, three-tenths of which amount to 90 minutes = 1° 30'. Adding 1° 30' to 56° 15', makes the index angle 57° 45' to fix the P. T. at 11.40. 15. Move to the P. T. at 11.40, see that the vernier has not been disturbed, and backsight to 9. By the foregoing rule (12), double the index angle, 57° 45', less the angle in tangent at 9, the last turning-point, 48° 45', = 115° 30' — 48° 45', = 66° 45', = the index angle in tangent at the P. T., = the total angle consumed by the curve. The work thus proves itself. 16. The preceding example would appear in the field-book as follows: —70 TO TRACE A CURVE ON THE GROUND. QkftOlOlQiOkAl* • COrtO^CQr^^l* • e e ole o o o o o o iH O 05 I tO © CO C* © t"» Oi (O oil lO Ifl |Q iO O U5 U5 if) kO kO O * • 'O ^ ^ Tjk ^k ^k CO • • MNn^aeNa9>dHHc|TURNING OBSTACLES TO VISION IN TANGENT. 71 17. This mode of running curves secures a record of each step in the proceeding; so that, if any error occurs, it can readily be detected. At each turning-point, the number in the “ tangent” column must correspond with the central angle due to the length of curve to that point; and at the P. T. that number must correspond with the total central angle. The work can thus be checked with facility during its progress, and checks itself at the end. 18. The young transitman is recommended to rule blanks after the pattern given, and exercise himself thoroughly in computing the parts, and recording the field-notes of various curves assumed at will: drawings are not necessary. XXII. TURNING OBSTACLES TO VISION IN TANGENT. 1. Draw CF parallel to AB. these parallels at equal inclinations. Call this angle I. Then B C = CE = FG. BE = BD-f DE = 2BD. But BD = BC cos. I, BG = EG + BE Let lines BC, CE, FG, cut B E = 2 B C cos. I. C F + 2 B C cos. I. EG = CF. Example. Suppose B to be a stake 24.50 on the tangent AB, and that a deflection left of 10° be made there for 200 feet to. a point C. Set transit at C, vernier reading 10° left. B S to B, and deflect 20° right. Vernier will now read 10° right, and telescope will be in line C E. Make C E = 200 feet. Move to E. See that vernier still reads 10° right. BS to C, and turn 10° left. Ver- nier will now read zero, and telescope will ‘be in line E G, or tangent A B prolonged. Distance BE = 2 B C cos. 1 = 2 (200 cos. 10°) =400 X .985 = 394 feet. Then E = 24.50 -j- 394, = stake 28.44 on tangent A B prolonged.72 TURNING OBSTACLES TO VISION IN TANGENT. If a parallel line C F were run, a deflection of 10° right would be made at each of the points C and F. If C F were 250 feet, then B G would be = 250 -f- 394 = 644 feet, and the point G would fall at stake 30.94 on tangent A B prolonged. 2. If angle I = 60°, the other conditions of above method being observed, triangle BHE will be equilateral, and BE = BH = HE. If the parallel D C or DF be run, BE = BD -f DC, and BG = BD + DF. For field work see last example. 3. In turning obstacles by either of these methods, the angles should be measured with extreme niceness. Handle the instrument lightly, to avoid jarring the vernier; and, if possible, observe well-defined distant objects in the several short Tanges, that the lines of foresight and backsight may accurately coincide. In locating, the following method is preferable to those given above, and should always be used on long tangents. 4. Having established points A and B on the centre line, the farther apart the better within limits of distinct vision, set off the equal £ - p rectangular d i s - "j T tances A E, B F, | j ranging clear of a b the obstacle. Place the transit at E or F, fix points G and H on the forward range, and, rectangularly to these points, establish others on the forward range of the centre line at C and D. The offset distances should be measured very carefully with the rod, or with a steel tape if they exceed in length the pocket rule which every engineer should have about him.TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 73 XXIII. TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 1. Fix a point on tangent A B prolonged at E. Lay off at B a perpendicular of any convenient length. Move the instru- ment to D, make the angle B D A = B D E, and mark the point of intersection A. By reason of symmetry in the triangles ADB, B D E, A B == B E, and may be measured on the ground. 2. Or, fix the point E, and lay off the perpendicular BD as before. Move to D, direct the telescope to E, turn a right angle E D C, mark the point of intersection C, and measure CB. Then, by reason of simi- larity in the triangles C BD, DBE, CB : BD :: BD : BE, BE = BDs-rBC. Example. Suppose B D to be 60 feet, and B C 40 feet. Then B D 2 -r- BC = 3600 40 = 90 feet = BE. 3. Or, with the instrument at D, measure the angle BDE. Then BE = BD tan. BDE. Example. BD = 120 feet, angle BDE = 54° 40'. BD tan. BDE = 120 X 1.41 = 169.2 feet = B E. 4. Or, without an instrument, lay off any convenient lines B F or C H. Mark the middle point D. Line out H G, parallel to AB. Mark on it the point G in range with D and E. Then GF = BE, or GH = C E. 5. Should the use of a right angle be inconvenient, turn any angle EBD = *, measure BD about equal by estimation to B E, if the ground permits, and set a point D. Move to D, and measure angle B D E =» y.74 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. Then the angle B E D, or z, = 180 — (x -(- y), and, by trigonom- etry, sin. z : sin. y :: BD : BE, . \ BE = BX) sin. y -j- sin. z. Example. Let x = 44° 02', y = 71° 48', B D = 300 feet. Then z = 180° — (x + y) = 180° — (44° 02' + 71° 48') = 180° — 115° 50' = 64° 10'. BE = BD sin. y 4- sin. z = 300 sin. 71° 48' -i- sin. 64° 10' = 300 X .95 -i- .90 = 316.6 feet. The calculation by logarithms would be as follows: — Log. 300 ............................... 2.477121 / Log. sin. 71° 48'....................... 9.977711 Sum .............................. 12.454832 Log. sin. 64° 10'........................ 9.954274 Log. 316.6. Diff....................... 2.500558 If E is invisible from B, extend the line D B towards C, until a line C E clears the obstacle. The point E must then he established by intersection of the sides CE, DE, in triangle C D E. Supposing the extension B C to have been 120 feet, the side CD will be 420 feet, the angle y 71° 48'; and, by a calculation similar to the above, the side DE, opposite angle x in the lesser triangle, identical with DE in the larger one, will be found to be 231.7 feet. The sum of the angles at the base C E of the triangle C D E = 180° — y = 180° — 71° 48' = 108° 12'. By trigonometry, two sides and the included angle being known in any plane triangle, the sum of the known sides is to their difference as the tangent of the half sum of angles at base is to the tangent of half their difference. In triangle C D E, therefore, CD -f DE or 651.7 : CD — DE or 188.3 :: tan. 108.12 2 or tan. 54° 06' : tan. 54° 06' X 188.3 651.7 = .399 = tan. 21° 45', = half the difference of the angles at the base. Log. 188.3............................. 2.274850 Tan. 54° 06'........................... 0.140334 Sum................................2.415184 Log. 651.7 ............................ 2.814048 _Tam_21° 45'. Diff.. . 9.6Q1136TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 75 The angle at C, being evidently the lesser of the two angles at the base, is equal to the half sum of these angles decreased by their half difference, = 54° 06' — 21° 45' = 32° 21'. Set the transit, then, at C, foresight to D, deflect 32° 21' left, and fix in that range two points F and G, between which a cord may be stretched, and as nearly as can be judged on opposite sides of E. Move to D, foresight to C, deflect 71° 4# right, and establish a point E at intersection with F G. Cross to E, B S to D, and deflect the angle z = 64° 10' into the line of the tangent A B prolonged.SUGGESTIONS AS TO FIELD-WORK AND LOCATION-PROJECTS. XXIV.-XXV.SUGGESTIONS AS TO FIELD-WORK AND LOCATION-PROJECTS. XXIV. SUGGESTIONS CONCERNING FIELD-WORK. 1. The Chief Engineer, after conference with his em- ployers in regard to the character of the work contemplated* and its general route, should, before organizing field-corps, go over the ground in both directions, and, aided by the best attainable maps, qualify himself by actual observation-to in- struct his assistants as to the conduct of the survey. Equipped with hand-level, pocket-compass, and in rough regions with the aneroid, he can often not only prescribe lines for examina- tion, but indicate the gradients to be tried, thus saving a vast amount of random labor and needless expense. Such thorough preliminary exploration is due both to himself and his princi- pals: it is too often omitted, or done with a perfunctory rush. In broken topography, no maps, notes, or information derived from others can supply the want of personal acquaintance with the ground itself. He must indispensably make that acquaint- ance, in order to project an intelligent location, — a work which should rarely be delegated; being capital service, it comes within the special function of the chief engineer, and only the necessary distribution of labor attending a great charge should relieve him from its direct performance. 2. A Field-Corps in settled regions generally consists of one senior assistant or chief of corps, one transitman, one leveller, one rodman, two chainmen, one slopeman, and two or more axemen. The following notes in regard to the' allotment of duties and the conduct of work may be acceptable. They are copied from the writer’s memoranda for the guidance of his field- parties, with the addition of some detail, and practical hints here and there, to aid the inexperienced. 7980 SUGGESTIONS CONCERNING FIELD-WORK. 3. Titr Senior Assistant will receive instructions from the principal assistant in charge, or the chief engineer, and will act exclusively under his direction. He will be held responsible for the good conduct of the corps, and for the rapid, exact, and economical performance of the work. Indecent or blasphemous outcries in the field should be prohibited. The writer’s various travel by land and sea has brought him acquainted with many cultivated, estimable, energetic, profane fellows, but not one in whom swearing was a grace; nor has he ever seen an instance where it forwarded work. Those considerate of others’ pride and self-respect will generally find that a good leader makes good followers. The senior assistant is empowered to appoint and dismiss employes below the rank of rodman, and will report any inefficiency or neglect of duty in the ranks above to his chief. He will pay the authorized expenses of the corps for sup- plies, repairs, transportation, and subsistence, taking duplicate vouchers. Accommodations should be sought near the work. When not thus obtainable, transportation to and from the field is to be regarded as a measure of economy for the com- pany, compensating the expense incurred by saving time and labor. He will superintend field operations in person, keeping in advance of the transit to direct and expedite the work, and establish the turning-points. On preliminary surveys, the axe should be little used; and on alternative locations, or such as may be subject to revision, trees over four inches in diameter need rarely be felled. He should be patient with sensitive landholders. He will find exercise for that amiable virtue, also, with the field vis- itors who so often spare time from useful toil to tell him he is on the wrong line, and to show him where the right one is. Note for record the kind and quality of material to be moved, observing quarries, wells, or other indications for the purpose; the timber and rock in the country traversed, with a view to their use in construction, and the widths of passage to be pro- vided for streams, together with the character of their banks and beds. Note the names of residents in the immediate vicinity of the work on survey; and, on location, cause the property-lines to be observed and recorded also when convenient.SUGGESTIONS CONCERNING FIELD-WORK. 81 Always begin grade-lines at the summit, and work down. For such service, carry habitually a slip of profile paper, say six inches wide and two feet long. Rule the proposed grade- line on it, assume a summit cut, mark the stations, and start down. When at fault, the elevation can be spotted on the profile, which will show at a glance, without any calculation, how you stand in relation to grade. The work of each day should be compiled and recorded in the evening, that no delay may result from the loss or deface- ment of a field-book. FORM FOR SURVEY RECORD. Sta. DlS. Deflec. Course M.C. Eleva. Slope. Remarks. FORM FOR LOCATION RECORD. Station. Distance. Course. Mag. Course. Elevation. Gradient. 0 « C*5 Variation. Slope. Remarks. On location, check the transitman’s calculation of the length of each curve and the fractional deflections. The senior assistant must be qualified to locate a line accu- rately on the ground from the project furnished him. Lateral deviations exceeding five feet on ten-degree slopes, three feet on fifteen-degree slopes, and two feet on twenty-degree slopes, will be considered errors requiring correction. Measurements to the experimental line should be made and noted frequently, in order not only to check the field-work, but that the line may by means of them be laid down on the map. The senior assistant will supply himself with drawing in- struments, cold's, brushes, and the like personal furniture of an engineer. He will take care also that the stationery, field- books, instruments, and other articles of outfit supplied by the company, are not misused. His field equipment should always include a hand-level and a pocket-compass: to these may be82 SUGGESTIONS CONCERNING FIELD-WORK. added a straight, round staff, five or six feet long, steel pointed; it will be found exceedingly useful. If without a topographer, he should make sketches of irregu- lar ground, of streams, buildings, roads, and the like, to help in compiling the map. In hilly or wooded districts, the front chainman carries the flag on survey, and is at the head of the line. In open, plain country, work is greatly forwarded by detaching an axeman with flag, to accompany the senior in advance, and set turning- points for the transit. The transitman follows as rapidly as possible, and the chainmen come after, lining in their stakes by the eye from point to point. The whole force is thus kept pretty steadily in motion. On wide plains, a set of chain-pins may he used, and survey- stakes placed five hundred or a thousand feet asunder. Very often stakes at intervals of two hundred feet are sufficient, the levels being taken every hundred. Location stakes are put in every hundred feet. 4. The Transitman will be expected to keep his instru- ment in adjustment, and to be quick and accurate in its manip- ulation. It is not needful to plant it as if for eternity. On the contrary, it should be set gently, the legs thrust but slightly into the ground, and the screws worked without straining. On long tangents it is a good plan to reverse the instrument at each new point, putting the north and south ends forward alternately. Small errors in adjustment are thus balanced in some measure. Select also, in such a case, some distant object in range, when practicable, to run by. The telescope, in wind or sun, will sometimes warp a little out of line. Never omit to note both the calculated and magnetic hear- ings of the lines on survey, and of the tangents on location. Guard against the error of reading deflections or bearings from the wrong ten mark; as, for instance, 34 instead of 26. At the beginning of a curve, let the rear chainman know the plus of the P. T. Tell the front chainman the degree of curve, and instruct him how, by multiplying 1.75 by the degree, he can find the distance of each full station from the range of the last two. A quick fellow will soon pick this up, and become wonderfully skilful in practice. Thus accomplished, he is a check on wrong deflections. In running curves, a tangential angle of fifteen degrees from one point should seldom be exceeded: twenty degrees is to be regarded as a maximum.SUGGESTIONS CONCERNING FIELD-WORK. 83 Carry a pocket-compass, and observe with it the magnetic bearings of streams and roads crossed. Record daily each day’s run; fill out the distance column, transcribe the chain-book, and, on location, record the apex distances also in the column of remarks. On survey, do not erase from the field-book the notes of abandoned lines. Simply cancel, and mark them “ aban- doned,” in such manner that they may still be legible. When required by the senior assistant, the transitman will aid in the making of maps. 5. The Leveller must be familiar with the adjustments of his instrument, keep it in order, and handle it rapidly. On survey, establish and mark benches at half-mile intervals; on location, four to the mile when practicable. Note the surface elevations, the depths, and the flood heights, of all considerable streams crossed. Take a rod in the beds of small streams. Six hundred feet each way should be regarded as the maxi- mum sweep of the level. Carry a hand-level, and thus save the time required to peg across narrow hollows, or over heights which can be turned with the instrument. The leveller should record his work, and make up the profile daily. 6. The Rodman will give his intermediates close by the sta- tions, observing the number of each one as a check on the chainmen, and calling it out to the leveller. He should have an eye to abrupt irregularities in the ground, and give plus elevations when necessary. He will keep note of bench-marks and turning-pegs, describ- ing the latter occasionally with reference to the nearest stake, that the levels may be taken up speedily in case of a revision of the line. When unaccompanied by an axeman, the rodman is equipped with belt and hatchet. Sometimes he is furnished also with a steel pin for turning on. The pin has a ring through the head, by which it may be hung to a spring hook in the belt. The rodman will assist the leveller, at record and profiles, and transcribe the slope-book daily. If stakes of survey are set at intervals of two hundred feet, give rods every hundred feet, as nearly as the midway points can be guessed.84 TUB CURVE-PROTRACTOR. 7. The Shoreman will give backsights, and take the cross slopes for one hundred feet on each side of the line at every station. 8. The Rear Chainman will carry a book in which to note the turning-points, the crossings of roads, streams, swamps, woodland, and, when convenient, property lines also. He will hand it in daily to the transitman for record. As each succes- sive chain is stretched, the rear chainman calls out the number of the stake it is stretched from: this assures the selection of the right number for the stake ahead. 9. One Axeman will be detailed to make stakes, another to mark and drive them. Additional axemen may be employed at the discretion of the senior assistant, as the work requires them. Wanton destruction of timber, fences, growing crops, or other property, should not be allowed. Axemen must be careful, in passing through the country, to do as little damage as possible. XXV. THE CURVE-PROTRACTOR, AND THE PROJECTING OF LOCATIONS. 1. The curve-prbtractor is simply an eight-inch, semi-circu- lar horn protractor, upon which a series of twenty-three curves, from half a degree up to eight degrees, is finely engraved, with radii of 400 feet to an inch. After some years’ use in his own practice, the contrivance was transmitted by the writer to the well-known firm of James W. Queen & Co., mathematical- instrument makers, New York and Philadelphia, by whom,, it is now manufactured. It greatly facilitates the projecting of lines and solution of field-problems on location. It enables the engineer, for example, by a short, graphical process and rapid inspection, to find the curve which shall close an angle between tangents, or terminate a compound curve, and pass at the same time through some fixed intermediate point, without liability to the errors, and free from the loss of time, involved in a tedious calculation. Other applications, such as the nice adjustment of line among buildings, on precipitous steeps, and the like, will suggest themselves to the experienced reader.THE CURVE-PROTRACTOR. 85 2. For office use, the writer prefers a home-made curve- protractor of isinglass, prepared as follows: Take a thin, clear sheet, say six by ten inches, free from bubbles and cracks. Block it securely on the drawing-table with thumb-tacks, set- ting the shanks close against the edge of the sheet, but not piercing it, and the heads lapping its edge. From a centre, midway of one of the long sides and near its margin, strike the curves from 12° or less, varying outwards by half-degrees to 6°; thence by quarter-degrees to 4°; and thence by ten-minute differences to 2^°. This covers one side of the sheet, the scale being 400 feet to an inch. Now release the sheet, turn it over, and on its other face strike the remaining curves, down to ten minutes, from centres on the table, in the reverse direc- tion, so that they shall cross the first series at a large angle. Space them about three-eighths of an inch asunder at the mid- dle. Use a needle-point centre for the first series, to avoid boring a large hole in the sheet. Add also, on that face, two radial lines drawn towards the corners. Score the fractional curves very lightly, the full figure curves a little deeper, but all of them with steadiness and delicate stress. Practise beforehand on a separate slip, for the right intensity of stroke. Engrave the numbers with a stiff steel point on the opposite side of the sheet to that upon which the corresponding curve is traced. Bring the work out by rubbing it with India ink. If preferred, the flat curves on the reverse side may be colored with carmine. Duplicate protractors will be found useful in projecting compound curves. Clip off the four corners, re- enforce the edges with a narrow ribbon of tracing-linen, folded over them and glued fast, and the article is complete. It is perfect for its use; durable, flexible, spotlessly transparent, not liable to warp or change dimensions with changes in the temperature or moisture of the air, and, withal, takes and pre- serves a visible line, thin as the gossamer. 3. To experienced locating engineers, the curve-protractor needs no wordy commendation. Contrasted with the incon- venient appliances of the old method, —cardboard, veneer, glass, or dividers, — its advantages will be manifest. A few hints as to the manner of using it may-be in place. 4. First of all, let the experimental line approximate to the probable line of location; and, upon that base, construct a contour map, with reference to which special observations should be made in the field, and the chaining done with care.66 THE CURVE-PROTRACTOR. Extreme accuracy in the contours need not be attempted. Note the courses of streams, ravines, and ridges, the average slopes at frequent intervals, and, on irregular ground, make illustrative sketches to aid in utilizing the other notes. Prac- tice gradually teaches how to observe critical points intelli- gently, and to record them briefly. In valleys or plains, where the location indicated is made up of long tangents and easy curves, little detail is required; but on bluffy, tortuous ground, with unavoidable divides to overcome, and long reaches of maximum gradient to be fitted, the method by contours is uot only the simplest and clearest way of compiling necessary information, but is an aid to the engineer in projecting the right line, which no substitute can fully replace. 5. The writer is forced by the strong constraint of experi- ence to differ on this subject with Mr. Trautwine. The dif- ference, however, is a permissible one, and implies no lack of grateful respect for that veteran engineer, whose hooks are our handy-books, and to whose genius we are all debtors. 6. Having made the map, with ten-foot contours, suppose, for example, that a continuous gradient five miles long is to be located. Spread the dividers to 500 feet by the scale, start at the foot of the ascent, and step up, complying with the general trend of the ground, to the summit. This needful preliminary gives about the distance you have to work on, which cannot in many cases be derived from the experimental line directly. The profile furnishes the height to be overcome; and you are thus prepared to assume a summit cut, and determine the gradient. Having adopted one, say, of 66 feet per mile, observe that this rises five feet in 400 feet. Spread the dividers, then, to 400 feet by scale, and stand one leg on or near the summit, at a point corresponding to a five or ten unit in the elevation of the gradient. That is to say, if the grade elevation at the summit be 362, for instance, stand the leg of the dividers a little beyond or a little short of the summit, at a point where the grade elevation is 365 or 360. Thence, exer- cising good judgment to conform in a general way to what the location ought to be, and to make no angular indirections which cannot be closed with the maximum curvature, step forward down the incline. Name each step mentally as it is made, 355, 350, 345, 340, &c., and spot at the same time with a pencil- point the contour or half space, directly opposite, correspond-THE CURVE-PROTRACTOR. 87 ing to it in elevation. Connect the pencil-marks with a faint dotted line. 7. Were the ground a straight, regular hillside, the steps would be made directly from contour to half-space, thence to the next contour below, and the dotted line would mark out a tangent conforming exactly to the ground surface. On devious slopes, rounding within the limit of the sharpest permissible curve, the same exact conformity could be obtained, if desired, and a grade-line laid down which should require the least possible expense in building. On irregular, winding ground, an approximation only to the dotted line can be made: it is nevertheless a guide to go by; and, the more nearly the loca- tion project approaches it, the lighter will the work of con- struction be. The dotted line, in short, is analogous to a profile; and the engineer can prescribe his cuts and fills with reference to it, by means of curve or tangent, just as on the profile he does the same by means of grade-lines. A fairly correct map will enable him to construct a profile from the project, and to amend its errors without the trouble and ex- pense of tentative field-work. The writer’s habitual practice has been to base his preliminary estimates on a profile thus deduced from the map; and he recommends the practice to others. They will be surprised to observe the likeness between such a profile, tolerably well done, and that of the subsequent location. 8. It is a good custom, and one which cannot prudently be neglected where long reaches of maximum gradient are en- countered, to “slacken grade ” on the curves. In making this adjustment, the contour map is exceedingly useful. An ap- proximate project is first required, in order to determine the curvature, and, from that, the varying gradient. The location can then be laid down on the map with satisfactory precision. Opinions differ as to the right allowance per degree of curva- ture, and no experiments seem to have been made from which to deduce an authoritative rule. Some say 0.025 per degree per 100 feet; others, 0.05; others, variously between the two. Probably 0.05 is the safer rate. This amounts to 2.64 feet on a mile of continuous one-degree curve, and makes a nine-degree curve, about the curve of double resistance at ordinary passen- ger speeds. 9. In projecting locations, the better way generally is to strike the curves first.88 THE CURVE-PROTRACTOR. . 10. The following tables may be of assistance. It was need- ful, calculating them at all, to calculate them right; but of course such exactness as the figures would seem to indicate is unattainable in practice. 11. TABLE SHOWING THE DISTANCE, D, IN FEET, AT WHICH A STRAIGHT LINE MUST PASS FROM THE NEAREST POINT OF ANY CURVE, STRUCK WITH RADIUS r, IN ORDER THAT A TERMINAL BRANCH HAVING RADIUS R = 2 r, AND CONSUMING A GIVEN ANGLE, X, MAY MERGE IN SAID STRAIGHT LINE. D = (R — r) X (1 — cos. x). Degree of the Main Curve. Angle 2° 3° 4° 5° 6° 7° 8° 9° 10° X. D. 2° 1.72 1.15 0.86 0.69 0.57 0.49 0.43 0.38 0.34 3° 4.01 2.67 2.00 1.60 1.34 1.15 1.00 0.89 0.80 4° 5° 6.88 4.58 3.44 2.75 2.29 1.96 1.72 1.53 1.37 10.89 7.29 5.44 4.35 3.63 3.11 2.72 2.42 2.18 6° . 15.76 10.50 7.88 6.30 5.25 4.50 3.94 3 50 3.15 7° 21.49 14,32 10.74 8.59 7.10 6.14 5.37 4.77 4.30 8° 23.36 18.91 14.18 11.35 9.45 11.73 8.10 7.09 6 30 5.67 9° 35.24 23.49 17.62 14.09 10.07 8.S1 7.83 7.05 10° 43.55 29.13 21.77 17.42 14.52 12.44 10.89 9.68 8.71THE CURVE-PROTRACTOR. 89 If R = 1| r, use half the tabular distance; if R = 3 r, use twice the tabular distance; if R = 4 r, use three times the tabular distance, and so on. 12. TABLE SHOWING THE DISTANCE, d, IN FEET, AT WHICH CURVES OF CONTRARY FLEXURE MUST BE PLACED ASUNDER IN ORDER THAT THE CONNECTING TANGENT, T, MAY BE 300 FEET LONG. Degree of Curve. H tf u o 0 % 1“ 2° 3° 4° 5° 6* 7° 8° 9° 10° 0 w I d. « © w 1° 3.9 5.24 5.92 6.29 6.35 6.68 6.86 7.00 7.08 7.18 1° 2° 7.84 9.43 10.38 11.20 11.70 12.20 12.55 12.80 13.06 2° 3° 11.77 13.43 14.64 15.68 16.45 17.09 17.61 18.05 3° 4° . . 15.65 17.39 18.76 19.90 20.82 21.64 22.31 4° 5° .. 19.54 21.22 22.76 24.01 25 07 25.97 5° 6" • • 23.32 25.20 26.70 28.00 29.13 6° •• 27.25 29.01 30.58 31.93 7° 8° 31.05 32.82 34.41 8° 9° 34.82 36.31 9° 10° - 38.56 10. Examples. A 7° and 4° should be 19.9 feet asunder; a 5° and 9° should be 25.07 feet asunder. As approximations, for a connecting tangent 400 feet long, take twice the tabular distance: fora connecting tangent 200 feet long, take half the tabular distance.PROBLEMS IN FIELD LOCATION. XXVI. - XXXVII.PROBLEMS IN FIELD LOCATION. XXVI. HOW TO PROCEED WHEN THE P. C. IS INACCES- SIBLE. 1. Suppose, for example, a pro- jected 5° curve, beginning at stake 24.20, or B in the diagram. First Method. — At any point A, which we will assume to be stake 23.40, set up the transit. Let it be judged that stake 27, marked D in the diagram, must fall on ac- cessible ground. Then the distance B D, around the curve, is 280 feet, corresponding to an angle E B D of 7° at the circumference, or an angle of 14° at the centre. The chord of a 1° curve consuming this angle, by Table XVI., is 1,396.6 feet; that of a 5° curve, B D in the figure, is one-fifth of this, or 279.3 feet. In the triangle A B D are thus known the sides A B, B D, and the sum of the angles at A and D, which sum is equal to the angle E B D. Hence, by trigonometry, — As the sum of the sides given=359.3 A C .... 7.444543 Is to their difference =199.3..............v 2.299507 So is tan. £ sum of angles at base =3° 30'. . . . 8.786486 To tan. £ their difference = 1° 56J' . . . 8.530536 Adding half the difference to half the sum, the larger angle, A, is found to be 5° 26£'; subtracting half the difference from half the sum, the smaller angle, D, is found to be 1° 33^. The 9394 BOW TO PROCEED WEEN TEE P. C. IS INACCESSIBLE. length of the side A D may he found in like manner by trigo- nometrical proportion; or, perhaps more simply, thus: — BD X nat. cos. D = DF = 279.2. BAX nat. cos. A = A F = 79.6. AF + FD = AD = 358.8. We are now prepared, from our point A, to deflect the angle 5° 26J' R, and lay out the line A D to the point D on the curve. Moving the instrument to that point, and backsighting to A, a deflection of 1° 33^ R places the telescope on line DB; a fur- ther deflection of 7° places it in tangent at D, and the curve may thence be traced in both directions. 2. Second Method. — Having, as in the first method, judged that stake 27, marked D, must fall on accessible ground, and thus determined the central angle subtended by the arc B D, refer to Table XVI. for the tangent of a 1° curve, corre- sponding to 14°, the given angle. It proves to be 703.5 feet. One-fifth of this, 140.7 feet, is the tangent or apex distance, BC, of a 5° curve, which may be measured on the ground. Moving the instrument to C, turning 14° R, and laying off the line C D = B C, the point D on the curve is ascertained. 3. The preceding methods are manifestly applicable to the ends also of curves, as well as the beginnings. A case not unfrequent in practice may be added in conclusion of the subject. Suppose a 2° curve terminating at C, in marsh or stream not measurable directly. Let C fall at stake 32.20. At any con- venient point A, say stake 29, place the transit with telescope in tangent. The arc A C, = 320 feet, includes an angle of 6° 24'. The tangent of a 1° curve corre- sponding to this angle in Table XYI. is 320.34 feet; that of a 2° curve is therefore 160.2 = A B. Move to B, deflect 6° 24' R into the range of the terminal tangent, and fix E on the opposite shore. Fix also D, and note the angle EBD. Move to E. Measure the angle DEB, and the distance DE. The tri- angle BED may then be solved. If BE is found to be 670 feet, C E = 670 — 160.2 = 509.8, and stake E = 32.20 + 509.8, = say 37.30.HOW TO PROCEED WHEN THE P. C. C. IS INACCESSIBLE. 93 XXVII. HOW TO PROCEED WHEN THE P. C. C. IS INAC- CESSIBLE. 1. Suppose a 4°. curve, A B, compounding at B into a 6° curve B C. Fikst Method. — Place the transit at any point A, say stake 34. Let the pro- posed P. C. C. fall at stake 36.25. Assume that we wish to reach C, on the second curve, by means of the straight line A D C. The arc A B, covering 225 feet of a 4° curve, subtends an angle of 9°. A D is half the chord of twice this angle. By Table XVI., the chord of 18° on a 1° curve is 1,792.7 feet. That of a 4° curve is therefore 448.2 feet, half of which = 224.1, =AD. The versin. of 18° on a 1° curve, by the same table, is 70.54 feet; one-fourth of which, or 17.635, is the versin. B D, corresponding to the same angle on a 4° curve. In order to find what angle on the 6° curve this versin. BD, = 17.635 feet, corresponds to, multiply it by 6, and seek the product, 105.S1, in Table XVI., where it is found, nearly enough for field-practice, opposite the angle 22° 04'. The chord of that angle, on a 1° curve, is seen at the same time in the adjoining column to be 2,193.2 feet; on a 6° curve it is therefore 365.5 feet, one-half of which, = 182.75 feet, = DC, and one-half of 22° 04' =11° 02', = the angle covered by the arc B C. Thus are found the angle at A = 9°, the angle at C = 11° 02', and the distance AC = 224.1 -f- 182.75, = 406.85 feet. The angle 11^02'corresponds to a length of 1.84 feet on the 6° curve; C, therefore, falls at stake 36.25 -f- 1.84 = 38.09. With these data the field-work is obvious. 2. Second Method. — Having reached the point A, and determined the arc A B = 9°, as above, find in Table XVI. the tangent 450.95 feet, corresponding to the given arc, one-fourth90 TO SEIFT A P. C. of which = 112.7 feet, = tan. AE for the 4° curve. Move to E, deflect 9° R; range out the line E F, made up of E B = A E = 112.7 feet, and BF any convenient distance, say 90 feet. This 90 feet is the assumed tangent of some unknown angle on the 6° curve. To find the angle, multiply 90 by 6, and seek the product, 540, in the tan. column of Table XVI., where it is found opposite 10° 46'. By moving then to F, deflecting 10° 46' R, and measuring F C = 90 feet, the point C is fixed on the second curve. 3. Should unexpected obstacles be met in carrying out either of these plans, the triangles A G C or E G F may be solved, and the point C fixed by means of the lines AG, G C. 4. The application of the foregoing methods to turning obstacles on simple curves needs no special instance. XXVIII. TO SHIFT A F. C. SO THAT THE CURVE SHALL TERMINATE IN A GIVEN TANGENT. 1. Suppose a 3° curve AB to have been located, containing an angle of 44° 26', and ending in tangent B E: required, that it shall end in tangent D F, parallel to B E. It is plain, from the diagram, that if the curve and its initial tangent be moved forward, like the blade of a skate, until the terminal -tangent merges in D F, the P. T. will have traversed the line BD, equal and parallel to AC. If, there- fore, on the ground at B, the angle E B D, equal to the whole angle consumed by the curve, in this case 44° 26', be laid off to the right, and the distance B D to the range of the proposed terminal tangent be measured, the equal distance AC, from the original to the required P. C., is thus directly ascertained. Should such direct measurement be impracticable, range out the tangent BE, and, at any convenient point, measure the distance from it square across to the proposed terminal tan- gent DF, say 56 feet. Then in the right triangle BED, mak- ing BD radius, we have given the angle at B = 44° 26', andTO SUBSTITUTE A CURVE OF DIFFERENT RADIUS. 97 the sine E D = 56 feet. Hence, by trigonometry, ED-r not. sin. 44° 26', or 56 -7- 0.7, = B D = 80 feet, = distance A C along the initial tangent, from the erroneous to the correct P. C. 2. This problem occurs more fi’equently than any other in the field; and the young engineer should have it by heart, that the distance square across between terminal tangents, divided by the natural sine of the total angle turned, will give him the distance he is to advance or recede with his P. C. to make a fit. 3. Excepting on precarious rocky steeps, city streets, or like exact confines, to strike within two feet of any point desig- nated in the project, may be considered striking the mark. Astronomical nicety, whether with transit or level, in an ordi- nary railroad location, is mere waste of time. 4. The observant reader will not fail to perceive that the foregoing rule applies to systems of curves, or to compound lines also, the angle EBD being the angle included between the initial and terminal tangents, let what flexures or indirec- tions soever have been interposed; and that, if the angle re- ferred to be either 180° or 360°, adjustment by shift of P. C. is impracticable. In those cases, a change of radius becomes necessary. XXIX. TO SUBSTITUTE FOR A CURVE ALREADY LOCATED, ONE OF DIFFERENT RADIUS, BEGINNING AT THE SAME POINT, CONTAINING THE SAME ANGLE, AND ENDING IN A FIXED TERMINAL TANGENT. 1. Suppose the 4° curve AB, containing an angle of 32° 20', to have been located, and that it is required to substitute for it an- other curve A C, which shall end in a parallel tangent C F, 60 feet to the right. Fikst Method. — Find the length of the long chord AC, = AB-f BC. Referring to Table XVI., the chord of a 1° curve for 32° 20' is seen to be 3,190.8 feet; that of a 4° curve, there-98 TO FIND THE POINT AT WHICH TO COMPOUND. fore, = 797.7 feet, say 798 feet, = AB. To find BC, solve the triangle BDC, observing that the angle DBC = BAI = one-half of the central angle 32° 20', = 16° 10', and that D C = 60 feet. Then DC nat. sin. 16° 10' = 60 -f- .278 = say 216 feet, = BC. Hence AC = AB + B C = 798 + 216 = 1,014 feet. Having thus found the length of chord A C, the radius and rate of curvature may be deduced as in X. Or, dividing the tabular chord of 32° 20' by chord A C = 1,014, the degree of the required curve is ascertained directly to be 3.15, equivalent to 3° 09/. 2. Second Method.—Find the apex distance AH, = AI -f IH. The tabular tangent of 32° 20' divided by 4 gives AI = 415 feet. In the triangle KDC, the side DC -r nat. sin. K = 60 -T- nat. sin. 32° 20', = 112 feet = K C = I H. Then AH = AI -f IH = 415 -(- 112= 527 feet; and the tabular tangent 1,661 -f- 527 gives 3.15, equivalent to 3° 09', the degree of the required curve A C, as before. XXX. HAVING LOCATED A CURVE ABC, TO FIND THE POINT B AT WHICH TO COMPOUND INTO ANOTHER CURVE OF GIVEN RADIUS, WHICH SHALL END IN TANGENT E F, PARALLEL TO THE TERMINAL TANGENT OF THE ORIGINAL CURVE, AND A GIVEN DISTANCE FROM IT. —|- F C, — t -(- D; i.e a = R — r — D, 1. To find B, the angle BIC must be found. Call the given distance between tangents D; the larger radius, R; the smaller one, r; the required angle, a. Then, referring to the figure, observe that in the triangle IH K, IH being ra- dius, IK is the cosine a; i.e., IK -f- IH = nat. cosine a. But IH = R — r; IK = IC — KC, and KC = KF or HE IK = R — r — D. Hence nat. cosine R —r = 1 — [D-h (R —r)].TO SHIFT A P. C. C. 99 The same reasoning would apply if A BE were the curve first located, and a terminal curve of larger radius required to be put in. 2. We have, then, the following general rule for such cases: Divide the perpendicular distance between terminal tangents by the difference of the radii, and subtract the quotient from unity; the remainder is the natural cosine of the angle of re- treat along the located curve to the required P. C. C. Example. 3. A 3° curve on the ground, to find the P. C. C. of a 5° curve striking 27 feet to the right. Here D = 27; R — r = 1,910 — 1,146, = 764; D-f-R — r = 27 -r- 764, = .03534; and 1 — .03534 = .96466 = nat. cosine 15° 17'. We must go back, therefore, 509 feet on the 3° curve, to compound into the 5° curve. Had the 5° curve been located first, we must have gone back 306 feet to begin the 3° curve which should strike 27 feet to the left. In either case, time might be saved by moving directly from E to C, or the reverse, and spotting in the curve backwards. To do this, we have in the right triangle F E C, the angle E = half of 15° 19', = 7° 38^, and the side F C = 27 feet. Then E C = 27 -r- nat. sin. 7° 38|', = 203 feet; and if E were stake 54.20 on the 5° curve, B would fall at stake 54.20 — 3.06, = 51.14; and C, the P. T. of the 3° curve, at 51.14 -|- 5.09, = stake 56.23. XXXI. TO SHIFT A P. C. C., SO THAT THE TERMINAL BRANCH OF THE CURVE SHALL END IN A GIVEN TANGENT. First Case: the terminal branch having the shorter radius. 1. Suppose the compound curve A C N located, and that it is required to fix a new P. C. C. at B, from which the terminal branch BM shall merge in tangent M L, a given distance from N O. To fix B, the central angle B H M of the new terminal branch must be found, and substituted for CIN. Call the longer radius R; the shorter one, r; the dis- k------100 TO SHIFT A P. C. C. tance asunder of the terminal tangents, D; the central angle, CIN, = IE K, of the located terminal branch, b; and the central angle, B H M, = H E F, to be substituted for it, a. In the right triangle, EIK,EK = EI cos. IEK= (R — r) cos. b. In the right triangle HFE, EF = EH cos. HEF=(R — r) COS* CL, Also, FK = LO = D, since each is equal to r — KL. Then EF = EK — FK; i.e., (R — r) cos. a= (R — r) cos. b — D. Hence nat. cosine a = nat. cosine b— [D-f- (R — r)]. Were the curve B M located, and the curve C N to be substi- tuted for it, — that is to say, were a given and b required,— we should have, by transposition, nat. cos. b = nat. cos. a -}- [D-f- (R — r)] Example. A 3°, compounding into a 5° curve at C, which consumes an angle CIN, = 30° 22', and ends in a tangent, N O, which is found, by measurement of L O, to be 34 feet too far to the left. Here, D = 34, R = 1,910, r = 1,146, b = 30° 22'; and, by the solution, nat. cos. a = nat. cos. 30° 22' — (34 -j- (1,910 — 1,1461) = 0.8628 — (34764). 34.................log. 1.531479 764 ................log. 2.S83093 .0445 ................log. 2.6483S6 Then 0.8628 — 0.0445 = 0.8183 = cos. 35° 05', = angle a ; a — i» = BHM — C IN = BEC = the angle of retreat from the erroneous P. C. C. = 35° 05' — 30° 22' = 4° 43', equivalent to 157 feet, on the 3° curve, from C to B. 2. Second Case: the terminal branch having the longer radius. Let B N represent the terminal branch located with central angle IKO = 6, and suppose it required to determine the new arc CM, with central angle IEF = a. Call the longer radius R, the shorter one r; the distance L N between tangents, D. In theTO SHIFT A P. C. C. 101 right triangle IKO, KO =KI, cos. IKO = (R— r) cos. b. In the right triangle FIE, EF = El, cos. IEF = (R— r) cos. a. Also, E H = L N = D, since each is equal to R — KL. Then EF = EH + HF = EII-f KO; i.e., (R —r) cos. a=(R—r) cos. b -f- D. Hence nat. cos. a = nat. cos. 6-(- ID-HR-r)]. Were the curve CM located, and the curve BN to be sub- stituted for it, that is to say, were a given and b required, we should have, by transposition, nat. cos. b = nat. cos. a — p-HR-r)]. Example. A 5° compounding into a 3° curve at B, which consumes an angle of 44° 20', and terminates at N, 28 feet too far to the left. Here D = 28, R = 1,910, r = 1,146, b = 44° 20; and, by the solution, nat. cos. a = nat. cos. 44° 20' -f- (28 -f- 764). The nat. cos. 44° 20'_= 0.69883 ; 28 764 = log. 1.447158 — log. 2.883093 = log. 2.564065, corresponding to the decimal 0.03665, which, being added to nat. cos. 44° 20', gives 0.73548, the nat. cos. 42° 29'. Then B K N — C E M = 44° 20' — 42° 29* = 1° 51' = angle BIC, equivalent on a 5° curve to 37 feet, which therefore is the distance around the arc from B, the erroneous P.C.C., to C, the correct one. 3. From these formulas the following general rule may be drawn: Divide the distance between terminal tangents by the difference of the radii, and call the quotient Q. Find the nat- ural cosine of the terminal arc already located, and call it C. The sum or the difference of Q and C will be the natural cosine of the terminal arc to be substituted for that already located. With radii in the order R, r, should the terminal tangent located strike j | the proposed tangent; or, with radii in the order r, R, should the terminal tangent located strike j j the proposed tangent, — take the 1 | of Q and C for the required cosine. ( difference ) n102 TO FIND TUB POINT AT WHICH TO BEGIN A CURVE. XXXIT. HAVING LOCATED A TANGENT, A B, INTERSECTING A CURVE, C B, FROM THE CONCAVE SIDE, TO FIND THE POINT E ON SAID CURVE AT WHICH TO BEGIN A CURVE OF GIVEN RADIUS WHICH SHALL MERGE IN THE LOCATED TANGENT. 1. Place llie transit at the intersection point B. Set points at equal distances therefrom in both directions on the curve already located, by means of which the direc- tion of a tangent to that curve at B may he fixed, and the angle F B A measured. Call that angle a; and, as shown in the figure, suppose the lo- cated curve to be prolonged in- to a terminal tangent, parallel with the newly located tan- gent A B. Complete the dia- gram. Call the larger radius R; the proposed radius, r; the central angle of the proposed curve, x. Then, obviously, the line A G = R cos. a. It is also equal to (R — r) cos. x -f- r. That is to say, R cos. a = (R — r) cos. x + r. Hence cos. x = (R cos. a — r) -5- (R — r); and x — a = angle B G E, sub- tended by the arc BE, from which the length of the arc may be deduced, and the point E ascertained. Example. DC, a 1° curve; angle a = 64° 32': to connect with a 4° curve. Here cos. x = (5,730 X 0.43) — 1,433 -4- (5,730 — 1,433) = 0.24 = cos. 70° 06'; and x — a = 11° 34', equivalent to a distance from B around the 1° curve of 1,157 feet to E, the point at which to begin the 4° curve.TO LOCATE A Y. 103 XXXIII. HAVING LOCATED A TANGENT, A B, INTERSECTING A CURVE, C D, FROM THE CONVEX SIDE, TO FIND THE POINT E ON SAID CURVE AT WHICH TO BEGIN A CURVE OF GIVEN RADIUS WHICH SHALL MERGE IN THE LOCATED TANGENT. 1. This problem is analo- gous to the preceding one. The preparatory steps are the same in both. Having found the angle a, however, it will he manifest to the attentive reader, that, in this case, R cos. a = (R -(- r) cos. x -f- r. Hence cos. x = (R cos. a — r) -MR + *'). Example. 2. D C, a 1° curve; angle a = 64° 32': to connect with a 4° curve. Here cos. x = (5,730 X 0.43) — 1,433 -r- (5,730 + 1,433) = 0.1439 = cos. 81° 43'; and x — a = 17° 11', equivalent to a distance from 13 around the 1° curve of 1,718 feet to E, the point at which to start the 4° curve. XXXIV. TO LOCATE A Y. tersecting the tangent BA. 1. The processes of the two former problems may be adopted. In this case the angle a vanishes, and the cos. x clearly is equal to (R —r) (R + r). 2. Another solution of the Y problem is as follows: — Draw the tangent E D in- Then is B D = D A, for the rea-104 TO LOCATE A Y. son that each is equal to DE. Make GF = R-(-r, the diame- ter of a semicircle. Said semicircle touches tangent B A at D, its middle point; and D E being perpendicular to G F, we have by geometry GE : DE :: DE : EF; i.e., GE X EF, or R X r, = D E‘J. Hence DE = BD = DA = y/R X r = R tan. | x, and we are thus enabled to fix the points E and A. 3. In the two foregoing problems, the angle consumed by curve E A is = 1S0° — x. Example. BE, a 2|° curve located; BA, a tangent: to complete the Y with a 6° curve, E A. By the first method, cos. x = (R — r) -r- (R + r) = (2,292 — 955) -f- (2,292 + 955) = 1,337 -i- 3,247 = log. 3.126131 — log. 3.5114S2 = 1.014649, which corresponds to log. cos. 9.614649, or to the decimal number 0.4118, indicating in either case the angle 65° 41' — x. DE = BD = DA = R tan. ix = 2,292 X 0.6455 = 1,479.4. DE may be found also by reference to Table XVI., where the tangent of a 1° curve for 65° 41' is seen to be 3,698.6. Dividing this number by 2J, we have 1,479.4, as above. Or, by the second method, — DE = VB X r = V21SS860 = 1,479.4. Having thus the means of fixing points E, D, and A, the curve E A can be laid down. 4. If B A is curved con- vex to the Y, construct the figure as in margin, and reason thus: — I11 the trjangle EGF, formed by lines connect- ing the curve-centres, the sides are respectively equal to the sums of the contiguous radii: the angles may therefore be found as in Case 111., Trigonometry. Lines drawn bisecting the central angles of the severalTO LOCATE A Y. 105 curves will pass through the points of intersection of the tan- gents to those curves severally. But lines so drawn in this case bisect also the angles of a triangle, and, demonstrably by geometry, meet in one point equidistant from the three sides of the triangle. That point, therefore, must be a com- mon P. I. for all the curves, and that equidistance the “ tan- gent ” length common to them all. Example. Given B A, a 3°, and B C, a4° curve: to complete the Y with a 5° curve, C A. EF = 1,910 + 1,140 = 3,056. G F = 1,433 + 1,146 = 2,579. E G = 1,910 + 1,433 = 3,343. Then, by Case III., Trigonometry, — As EG, 3,343. . . . log. (a. c.) 6.475864 Is to E F -f G F, 5,635 .... log. . . 3.750894 SoisEF—GF, 477 . . . . log. . . 2.678518 To diff. of segments of E G, 804 ........ 2.905276 Adding half the difference to half the sum of the segments of the base EG, we shall have the greater of them; i.e., (3,343 -|- 804) -j- 2 = 2,073.5, which is the co.s. E, EF being radius. Hence 2,073.5 -f- 3,056 = log. 3.316704 — log. 3.485153 = 9.831551 = cos. 47° 16' = E. By Table XVI., the tangent of a 1° curve corresponding to this angle is 2,507.3: that of a 3° curve, therefore, is 835.8 = the common tangent BD or DA. Multiplying the common tangent by 4, we shall find opposite the product in Table XVI. the central angle of the 4° curve to be 60° 32'; multiplying it by 5, we find, in like manner, the central angle of the 5° curve to be 72° 12'. Arc B A, = 47° 16', is equivalent to 1,575 feet on the 3° curve; arc B C, = 60° 32', is equivalent to 1,513 feet on the 4° curve. Points being thus fixed at A and C, curve C A can be laid on the ground. 5. If curve BA is dbncave to the Y, the radii being given, construct the figure as follows: — First draw the triangle GFE, the sides of which are obvi- ously derived from the given radii. Prolong the sides E G and E F indefinitely. Bisect the exterior angles at G and F with106 TO LOCATE A Y. lines meeting at D, and from D let fall perpendiculars on EB, EA, and GF. Then, comparing triangles GBD, GCD, the angles at G are equal by construction; the angles at B and C are right angles, the side G D common. Hence the triangles are equal in all their parts: BG = GC, and BD = DC. By like reasoning, it appears that CF = FA, and DA = DC. The point D being equidistant from the right lines EB, EA, which limit angle E, a line bisecting that angle will strike point D. 6. It may be remarked, therefore, that lines bisecting the vertical angle and the exterior angles contained between the base and the prolongation of the sides of any triangle, will meet in a point equidistant from the base and the said prolon- gations. We thus have in the figure all the conditions for fit- ness of the curves. It remains only to solve the triangle G F E, seeing that from its angles the required central angles can be obtained. Example. B A, a 1°, B C, a 6° curve, located: to complete the T with an 8° curve, C A.TO LOCATE A TANGENT TO A CURVE. 107 In triangle G F E, — EF = 5,730 — 717 = 5,013. EG-5,730 — 955 — 4,775. G F = 955 + 717 = 1,672. Then, by Case III., Trigonometry, — AsEF . . . 5,013. . . . log. (a. c.) 6.299902 Is to E G + G F, 6,447 .... log. . . 3.809358 SoisEG — GF, 3,103 .... log. . . 3.491782 To diff. seg. of base, 3,991 . . . log. . . 3.601042 The longer segment, therefore, is 4,502; the shorter, 511. Cos. E = the longer segment divided by E G = 4,502 -j- 4,775 = log. 3.653405 — 3.678973 = 9.974432 = cos. 19° 28' = angle E. Cos. GFE = the shorter segment divided by GF = 511 1,672 = log. 2.708421 — log. 3.223236 = 9.485185 = cos. 72° 12' = angle GFE. The central angle, B G C, of the 6° curve, is equal to 180 — F G E = the sum of the angles at E and F = 72° 12' -|- 19° 28' = 91° 40', making the arc B C = 1,528 feet. The arc B A, equivalent to 19° 28' of a 1° curve, = 1,947 feet. Points C and A being thus ascertained, curve AC may be located. It will consume an angle = 1S0° — 72° 12' = 107° 48', equivalent, on an 8° curve, to 1,347.5 feet. XXXV. TO LOCATE A TANGENT TO A CURVE FROM AN OUTSIDE FIXED POINT. 1. If the ground is open, and the curve can be seen from the fixed point, it may be marked by stakes or poles at short inter- vals, and the tangent laid off without more ado. 2. Suppose, however, that on cumbered ground a trial tan- gent, A B, has been run out, intersecting the curve at B: it is required then to find the angle BAE, in order that the true tangent A E may be laid down.106 TO SUBSTITUTE A CURVE. Example. A B = 1,500 feet; D H B, a 4® curve; angle F B D = 20° 13'. First, the angle FBD, between a tangent and a chord, is equal to half the central angle subtended by the same chord. Angle D C B, therefore, = 40° 26'. By Table XVI., the chord of 40° 26', for a 1° curve, = 3,960.2 feet; for a 4° curve, it is, say, 990 feet = D B; and DI = IB = 495 feet. The versin. H I is, in like manner, found to be 88.25 feet. Deducting this from the radius of the 4° curve, we have IC = 1,344.4 feet. Then IC-MA = tan. 1AC; i.e., 1,344.4-^ (495 + 1,500) = 0.674 = tan. 33® 507 = angle IA C. dZ- Next, by geometry, the proposed tangent AE = VAD X AB = *>/2,490 X 1,500 = 1,932.6; and E C -1-A E = e the natural tangent of the angle of retreat or advance on the first curve required to make the tangent fit. 5. A still closer adjustment would be, after determining the angle approximately as above, to find the “tangents” corre- sponding to it for the two curves in Table XYI. Subtract the sum of these tangents from the length of the trial line, if it cuts the objective curve; add the sum, if it passes outside. With the number thus found, divide the measured amount of error for the tangent of the angle of retreat or advance, as the case may be. 6. Suppose, for illustration, that a trial tangent, bearing by needle N. 54° 30' E., is run out from stake 24.80 of a 4° curve, intending to touch a 6°, but is found to cut it. Suppose fur- ther that the objective G° curve was laid down and numbered in the direction of approach towards the 4° curve; that its P. C. is stake 25.10, and the magnetic bearing of its initial tan- gent S. 30° 30' W. The angle, then, between the bearing of the trial tangent and that of the initial tangent of the 6° curve, is 24°, corresponding to a distance of 400 feet on the latter curve. At stake 25.10 -j- 4.0 = 29.10, therefore, a tangent to the 6° curve would be parallel to the trial tangent. Go forward on the trial tangent, accordingly, to a point opposite 29.10, and measure the distance square across to that plus on the 6° curve. Assuming the trial tangent to be 2,500 feet long, and the amount of the miss to be 87 feet, the nat. tan. of the angle of error is 0.0348 = tan. 2°. By the method in (4), this calls for a shift of the P. T. 50 feet ahead on the 4° curve, making the new P. T. 24.80 -f- 0.50 = stake 25.30, and ad- vances the P. T. of the 6° curve to stake 29.43 of that numera- tion. The method in (5), applied to this case, brings the angle of error 2° 02', instead of 2°, equivalent to a deviation of 1J feet scant in half a mile from the line corrected by the method in (4), and agreeing exactly with the correction determined by the method in (2).TRACK PROBLEMS. XXXVIII.-LI.TRACK PROBLEMS. XXXVIII. REVERSED CURVES. The following problems will be useful in laying off turnouts, the adjustment of tracks near stations or shops, and the like ; but reversed curves should never be used on the main line between stations, where they are both objectionable and unne- cessary. Ground which allows any permissible location at all will allow straight reaches of at least two hundred or three hundred feet between curves of contrary flexure; and in every case it is worth the small additional outlay to make such a location. XXXIX. TO CONNECT TWO PARALLEL TANGENTS BY A REVERSED CURVE HAVING EQUAL RADII. 1. The radius R, and the perpendicular distance D, between the tangents given. 115116 TO CONNECT TWO PARALLEL TANGENTS. Draw the tangents, radii, and curves, fixing the P. R. C. midway of D. Draw the chords G I, IE, the line B F perpendicular to G I, and the line E H in prolongation of radius C E to an intersec- tion with B H passed through centre B parallel to tangents. That 1 falls midway of D, follows from the necessary sym- metry of the figure; and GIE must he a straight line, because the radii B I, IC, perpendicular to a common tangent at the same point, form a straight line, to which the chords GI, IE, are equally inclined. C H-v- C B = cos. A; but C H.= 2 R —D, and C B = 2 R. . \ cos. A = (2 R — D) -i- 2 R. BH = BC sin. A = 2 R sin. A; GF = R sin. i A; GE = 4 GF. . •. G E = 4 R sin. i A, and GIorIE = 2R sin. 1 A. Observe, that, in the right triangles GKE and B G F, the angles at G and B are each equal to i A: hence the triangles are similar. Example. R — 800 feet, D = 24 feet. To find angle A. Cos. A = (2 R — D) 2 R = 1,576 -7- 1,600 = 0.985 == nat. cos. 9° 56'. BH may then he found = 2R sin. A = 1,600 X 0.1725 270 feet, and laid off from the P. C. at G to K, the point E being fixed by a right angle from K. Or G E may be found = 4 R sin. ■} A = 3,200 X 0.866 — 277.1 feet, and laid off from G to E, the point I being fixed 138.5 feet from G, and angle K G E made equal to half of A = 4° 58'. 2. The distances G K and D given, to find R. In triangle GKE, KE = D. D -7- GK = tan. \ A; D -j- sin. i A = GE; and GE sin. i A = 4 R. Or, having found GE, we have from the congruity of trian- gles GKE, BFG, D : GE :: i GE or GF : R. R = GE2-^4D.TO CONNECT TWO 'PARALLEL TANGENTS. 117 Example. G K = 300 feet, D = 28 feet. D-^GK.................Log. 28 ... . 1.447158 Log. 300 ... . 2.477121 = Tan. | A.. 5° 20' . . . . 8.970037 D -f- sin. i A . . . . Log. 28 ... . 1.447158 Sin. 5° 20' . . . . 8.968249 = G E . . 301.24 .... 2.478909 GE-f- sin. i A . . . Sin. 5° 20' . . . . 8.968249 = 4K. . 3,241 .... 3.510660 .-. R = 810.2. XL. TO CONNECT TWO PARALLEL TANGENTS BY A REVERSED CURVE HAVING UNEQUAL RADII. / >' 1. Given the perpendicular distance, D, between two paral- lel tangents, and the unequal radii, R and r, of a reversed curve, to find the central angles, A, the chords, and the straight reach, G K, of the curve.118 TO CONNECT TWO PARALLEL TANGENTS. Cos. A = C H -f- B C; but C H = (R + r) — D, and BC = R + r. .*. Cos. A = (It + r — D) -f- (R + r). The straight reach GK = BH = (R-|-r) sin. A. The sum of the chords 6E = 6K-|- cos. \ A. GI = 2 R sin. $ A. IE = 2 r sin. i A = GE — GI. Example. D = 28, R = 955, r = 574. Cos. A = (R + r — D) (R + r) = 1,501 1,529. 1,501 . . . log. 3.176381 1,529 . . . log. 3.184407 . Cos. A, 10° 59/.............. 9.991974 GK=(R-(-r) sin. A. . R + r, 1,529 . . . log. 3.184407 Sin. A, 10° 59' . . . log. 9.279948 GK = 291.3 ................ 2.464355 G E = G K -r cos. { A. GK, 291.3 . . . log. 2.464355 Cos. i A, 5° 291' • • • log. 9.998014 GE = 292.6 ................ 2.466341 GI = 2 R sin. 1 A. 2 R, 1,910 . . . log. 3.281033 Sin. i A, 5° 291' . . . log. 8.980916 GI = 182.8 ................ 2.261949 IE = GE — GI = 292.6 — 182.8 = 109.8. 2. The distances GK and D, and one of the unequal radii, R, given, to find the other radius, r, and the central angles, A.REVERSED CURVE WITH UNEQUAL ANGLES. 119 Example. GK = 422, D = 30, R == 2,292. Tan. | A = D -r G K. D = 30 . . . log. 1.477121 G K = 422 . . . log. 2.625312 Tan. * A, 4° 04' . .... 8.851809 .*. A = 8° 08'. G E = D -7- sin. -J A. D = 30 . . . log. 1.477i21 Sin. | A, 4° 04' . . . log. 8.S50751 GE —423 . GI = 2 R sin. i A. 2R = 4,584 . . . log. 3.661245 Sin. i A, 4° 04' . . . log. 8.850751 GI —325.1 . 2.511996 G E — GI = 423 - - 325 = 98 = IE. r = i IE -r- sin. £ A. *IE = 49 '. .* . log. 1.690196 Sin. i A, 4° 04' . . . log. 8.850751 r — 691 . .... 2.839445 XLI. A REVERSED CURVE HAVING UNEQUAL ANGLES. Given the angles A and B, and the length A B of a straight line connecting two diverging tangents, to find theradius of a reversed curve to close the angles. AI = R X tan. i A; BI = R X tan. i B. .*. AB — R X (tan. i A + tan. £ B). R = AB-f- (tan. J A -J- tan. £ B).120 REVERSED CURVE BETWEEN FIXED POINTS. A Example. 16°, B — 10°, A B = 840. AB, 840 ...............................log. 2.924279 A = 8°, nat. tan. 0.14054 i B = 5°, nat. tan. 0.08749 Tan. i A + tan. £ B = 0.22803 . . log. —1.357992 R = 3,600 .................................. 3.566287 Xtll. A REVERSED CURVE BETWEEN FIXED POINTS. Given the angles N and K, and the length of the straight line E F connecting two divergent tangents, to find the radius of a reversed curve from E to F, connecting the tangents. 1. Denote the angle EIC or DIF byl; the angle CEI, complement of N, by n; and the angle DFI, complement of K, by k. Then, in triangle E C I, — EC : CI :: sin. I : sin. n. .•. EC X sin. n = CI X sin. I.REVERSED CURVE BETWEEN FIXED POINTS. 121 Also, in triangle D F I, — D F : DI :: sin. I : sin. k. . \ D F X sin. k = DI X sin. I. Adding these equations, we have — E C X sin. n + DFX sin. k = (CI + D I) X sin. I. N / But E C and D F are each equal to It; sin. n = cos. N; sin. k = cos. K; and CI-|-DI = 2R. Hence the equation becomes, — R X (cos. N -f- cos. K) = 2 E X sin. I. sin. I = (cos. N + cos. K) -4- 2. The foregoing elegant solution is abridged from Henck. 2. Angle A = 180 — (n -|- I); angle B = 180 — ( fc -f- I). To find radius, draw F H parallel, and E H perpendicular, to CD. Then EH = EF X sin. I. But EH = EG -f- GH; EG = EX sin. A; and G H = E X sin. B. EF X sin. I = RX (sin. A + sin. B). R = E F X sin. I -f- (sin. A -J- sin. B).122 REVERSED CURVE BETWEEN FIXED POINTS. Example. E F = 1,400, N = 30°, K = 20°. Sin. I = (cos. N + cos. K) -f- 2. N = 30°, nat. cos............................ 0.86603 K = 20°, nat. cos............................ 0.93969 1.80572 1.80572 + 2 = 0.90286 = nat. sin. 64° 32'. .-.1 = 64° 32'. A — 180° — (n + I) = 180° — (60° + 64° 32') -- - 55° 28'. B = 180° — (k + I) = 180° — (70° + 64° 32') = 45° 28'. R = E F X sin. I -j- (sin. A -J- sin. B). EF = 1,400 ............................log. 3.146128 Nat. sin. I, 0.90286................log. —1.955621 EG = 1,264 .................................. 3.101749 A = 55° 28' nat. sin........... 0.82380 B = 45° 28' nat. sin...........0.712S4 Sin. A + sin. B................ 1.53664 log. 0.186579 R = 822.6 ................................... 2.915170 3. The young student should bear in mind that the addition or subtraction of the logarithms of two natural numbers gives a logarithm representing, not the sum or difference, hut the product or quotient, of such numbers. When, therefore, as in the two foregoing cases, the sum or difference of two or more trigonometric functions — sines, tangents, and the like — is sought, the logarithm of the sum of the natural functions, and not the sum of their logarithms, is to be used. If, for example, sin. A X sin. B is required, the log. sin. A + log. sin. B = the logarithm of the product of the sines designated; but, if sin. A + sin. B is sought, the natural sines of those angles must be added together, and the logarithm of the sum of these natural functions must be used in making logarithmic calculations.'JVO CONNECT TWO DIVERGENT TANGENTS. 123 XLIII. TO CONNECT TWO DIVERGENT TANGENTS BY A REVERSED CURVE. 1. ADVANCING TOWARDS THE INTERSECTION OF TANGENTS. Given the angle of divergence, N, the initial P. C. at G, the distance GH, and the radii R, r, to find the central angles A and B. G K = C G X cos. N = R cos. N. G L = G H X sin. N. GK — GL = LKorEF, CF being drawn parallel to LE. Cos. B = DF-fDC = (r + EF) -f- (R+r). Angle GCK = 90° — N; angle DCF = 90° — B. Angle A = GCK — DCF = (90° — N) — (90° — B) = B — N. Example. N = 24° 30', G H = 854, R = 1,440, r = 1,146. G K = R cos. N. R = 1,440 . ; . log. 3.158362 Cos. N, 24° 30' . . . log. 9.959023 GK = 1,310 3.117385124 TO CONNECT TWO DIVERGENT TANGENTS. G L = G H .X sin. N. GH = 854 . . . log. 2.931458 Sin. 'N, 24° 30' . . . log. 9.617727 GL = 354 .............. 2.549185 LKorEF = GK —GL = 1,310 — 354 = 956. Cos. B = (r + EF)^-(R + r). r + EF = 2,102 . . . log. 3.322633 R + r = 2,586 . . . log. 3.412629 Cos. B, 35° 38'............ 9.910004 B = 35° 38'. A = B — N = 35° 38' — 24° 30' = 11° 08'. ______If 2. RECEDING FROM THE INTERSECTION OF TANGENTS. Given the angle of divergence, N, the initial P. C. at G, the distance G H, and the radii R, r, to find the central angles A and B. G K = G H X tan. N. KC = GC — GK-R — GK. LC or EF = KC X cos. N, the line CF being drawn paral- lel to LE. Cos. B = DF-rCD=(r-(-EF) -f- (R + r). Angle A manifestly = B + N.TO SHIFT A P. It. C. 125 Example. N = 18° 30', GH = 920, R = 955, r = 819. GK = GH X tan. N. G H = 920 . . . log. 2.963788 Tan. 18° 30' . . . log. 9.524520 GK —307.8 . . . . . 2.488308 KC = R — GK = 955 — 307.8 = 647.2. LCorEF = KC X cos. N. KC = 647.2 . . . log. 2.811039 Cos. N, 18° 30' . . . log. 9.976957 EF —613.8 . . . . . 2.787996 Cos. B = (r + EF)-HR + r). r + EF = 1,432.8 . . . log. 3.156185 R + r = 1,774 . . . log. 3.248954 Cos. B, 36° 08' . . . . . 9.907231 B = 36° 08 A = B + N = 36° 08' + 18° 30' = 54° 38'. xliv; TO SHIFT A R R. C. SO THAT THE TERMINAL TANGENT SHALL MERGE IN A GIVEN TANGENT PARALLEL THERETO. Given the reversed curve EFG, ending in tangent GV: to find the angle of retreat, A, on the first branch, and the angle C of the second branch, ending in tangent LIT, parallel to G V. Measure the error TG = D, perpendicular to the terminal tangent.126 TO SHIFT A P. R. O. In the figure, draw L K parallel to G V, and passing through centre of first branch. ______1. Then MK = (R + r)X cos. B. NL = (R + r) X cos. C. WL = GK. NL = r + D-f GK. MK = r + GK. N L — M K = D. (R + r) X cos. C — (R + r) X cos. B = D. (R + r) X cos. C = (R + r) X cos. B + D. Cos. C = [(R + r) X cos. B + D] -j- (R + r). A = (90° — C) — (90° — B) = B — C. Example. R = 1,433, r = 819, B = 34° 20', D = 94. Cos. C = [(R + r) cos. B + D] -j- (R + r). R + r = 2,252 . . . log. 3.352568 B = 34° 20', cos. . . . log. 9.916859 (R + r) cos. B = 1,860 ........... 3.269427 Add D 94 1,954 . . . log. 3.290925 (R + r) . . . log. 3.352568 Cos. C, 29° 49'...........9.93S357 A = B — C = 34° 20' — 29° 49' = 4° 31'.CUR YE THROUGH A FIXED POINT. 127 XLV. TO PASS A CURVE THROUGH A FIXED POINT, THE ANGLE OF INTERSECTION BEING GIVEN. Given the intersection angle, A, of two tangents, to'find the radius, R, of a curve which shall pass through a point, C; the position of said point, with reference to the tangents or the point of intersection, being known. 1. By what data soever point C is located, they may be com- muted by simple processes to the form shown in the figure; namely, the ordinate B C and the distance IC to apex. Call the angle BIC a, and complete the triangle ICO. In this triangle, x = — a~ (£ A + a). Also, C O : I O :: sin. x : sin. z. R R But C O = R; 10 =-----------r-r-. .\ R:----j-t- :: sin. x : sin. z. cos. I A cos. £ A Hence sin. z = a. The triangle ICO may then be C06‘* y A solved, and the radius ascertained.128 CURVE THROUGH A FIXED POINT. Example. A = 40°, B C = 32 feet, I B = 80 feet. Then BC -j- IB = 32 80 = 0.4 nat. tan. 21° 49'; and IC = B C -f- nat. sin. 21° 49' = 32 -r- .372 = S6 feet. Also, x = 90° — (} A -f a) = 90° — (41° 49') = 48° 11'. Next, sin. sc, 48° 11' . . . log. 9.872321 Divided by cos. | A, 20° . . . log. 9.972986 = sin. z, 127° 31' . . . log. 9.899335 Or, since the sine of any angle is equal to the sine of its sup- plement, the supplement in this case, 52° 29', may be taken directly from the logarithmic table, from which supplement deducting x, or 48° 11', the remainder is the angle y = 4° 18'. Finally, IC = 86 Multiplied by sin. x, 48° 11' log. 1.934498 log. 9.872321 = CD And C D divided by sin. y, 4° 18' log. 1.806819 log. 8.874938 = C O = It = say, 855 feet log. 2.931881 2. In the case of a rectangular intersection, the solution is more simple. It is quite plain, from the figure, that — R2 — (R _ o)2 + (R — 6)2, from which equation, R = a -(- b -}- 2 ab.FROGS AND SWITCHES. 129 Example, a = 40, b = 80. Then R = 40 + SO + Vt>,400 = 200. 3. Cases of this kind are disposed of with great ease in the field by means of the curve-protractor. XLVI. FROGS AND SWITCHES. TO FIND THE RADIUS OF A TURNOUT CURVE, THE FROG ANGLES, AND THE DISTANCES FROM THE TOE OF SWITCH TO THE FROG POINTS. 1. Draw the figure as in margin, C being the centre of the turnout curve, C K parallel to main track, and O K, IE, L M, perpendicular to it. Call the angle of the frogs at O, F; that of the intermediate frog at I, 2 F'; the throw of the switch-rail for single turnout, D; its angle with main track, S; the gauge of the track, G; and radius of outer rail, R. 2. Usually the length and throw of switch-rail and the angles of the frogs at O are given. In that case, to find R, F', and the distances LO, LI, reason thus: —130 FROGS AND SWITCHES. 3. The angle HN W, between the line of the switch-rail pro- longed and a tangent to turnout curve at frog point O, = N O P — NIIYV = F — S. The angle NOL or NLO, be- tween chord and tangent, = half the intersection angle IINW = HF — S). The angle NOB = NOL + LOB. ButNOL is seen to be = ^ (F — S), and NOB = F; then LOB = NOB —NOL = F —J(F —S)=*(F + S). The distance LO, from toe of switch to point of main frog, = LB -r- sin. LOB = (G — D) -r- sin. i (F + S). 4. Again: the angle LCY = NLO=J(F — S);LY = i LO = i (G — D) -f- sin. i (F + S). LY -f- sin. LCY = LC; i.e., [$ (G — D) sin. i (F + S)| -f- sin. * (F — S) = R. 5. R may be found otherwise, as follows: — OK = OC cos. KOC = R cos. F; LM = LC cos. CLM = R cos. S; LM — OK = LB; i.e., R (cos. S — cos. F) = (G — D). Hence R = (G — D) -f- (nat. cos. S — nat. cos. F). 6. If R be known, to find F. This equation gives nat. cos. F = nat. cos. S — |(G — D) -f- R]. 7. To find the angle, 2 F', of the middle frog at I. IE = IP -f PE or OK; i.e., R cos. F' = i G -f R cos. F. Hence nat. cos. F' = nat. cos. F-f (|G-f R). 8. The angle LIY, by similar reasoning to that used in rela- tion to LOB, is found to be = | (F' -f- S). The distance LI, from toe of switch to point of middle frog, = LV-r sin. LI Y = a G — D) -r- sin. J (F' -f S). The preceding formulas translate into the following — KUL.ES FOIt FROGS AND SWITCHES. 9. To Jind the Angle of Switch-Rail with Main Track. Divide its throw, in decimals, by its length: the quotient will be the natural sine of the angle sought. 10. To find the Distance from Toe of Switch to Point of Main F)-og. Subtract the throw of switch-rail from the gauge of track, both in decimals; call the remainder a. Add together the angle of switch-rail with main track and the angle of the main frog; find the natural sine of half this sum, and call it b. Divide a by b: the quotient will be the distance sought-FROGS AND SWITCHES. 131 11. To find the Radius of Outer Rail of Turnout Curve. Subtract the throw of switch-rail from the gauge of track, both in decimals; call the remainder a. Subtract the natural cosine of the main frog angle from the natural cosine of the switch-rail angle; call the remainder b. Divide a by b: the quotient will be radius. 12. To find the Main Frog Angle, the Radius of the Outer Rail being known. Call the natural cosine of the switch-rail angle a. Subtract the throw of switch-rail from the gauge of track, both in deci- mals. Divide the remainder by radius; call the quotient b. Subtract b from a: the remainder will be the natural cosine of the main frog angle. 13. To find the Angle of the Middle Frog, in the Case of a Double Turnout. Call the natural cosine of the. main frog angle a. Divide half the gauge of track by the radius of outer rail of turnout curve; call the quotient b. Add a and b together. Their sum is the natural cosine of half the middle frog angle. 14. To find the Distance from Toe of Switch'to Point of Middle Frog. Subtract the throw of switch-rail from half the gauge of track, both in decimals; call the remainder a. Add together the switch-rail angle and half the middle frog angle. Find the natural sine of half this sum; call said natural sine b. Divide a by b: the quotient will be the distance sought. 15. The use of logarithms will be found convenient in work- ing these rules. Examples. 16. Switch-rail, 18 feet; throw, 5 inches = 0.42 feet; frog angle, 5° 44'; gauge, 4.71 feet. Sin. S = 0.42 18 = .02334 = sin. 1° 20'. LO = (G — D) -f- sin. i (F -j- S) = (4.71 — 0.42) sin. 3° 32' = 4.29 -v- 0.0616 = 69.64 feet. R = (G — D) -T- (nat. cos. S — nat. cos. F) = 4.29-1- 0.C0473 = 907 feet. Nat. cos. F7 = nat. cos. F -f- (£ G -f- li) = 0.995 -f- (2.354 -f-132 FROGS AND SWITCHES. 907) = 0.99739 = cos. 3° 5S}'. Hence the angle of the middle frog = 2 F = 7° 57'. LI= (i G-D)-r sin. i (F' + S) = (2.354 — 0,42) sin. i (3° 58*' + 1° 20') = 1.934 0.0403 = 41.S feet. 17. In ordinary practice, frogs maybe located with sufficient exactness by the following rules, deduced from the congruily of triangles. Great nicety in their location is not necessary. The important thing in practice is to lay the turnout cuiTe so that the approach to the frog shall be fair and regular. How trackmen may do this without the use of instruments, in a very simple way, will be shown hereafter. Not that frogs may be set hap-hazard, and the approaches forced to fit: they ought to be nearly where they mathematically belong, and they can be thus placed by means of the rules subjoined. 18. Let N stand for the number of the frog; L the length of switch-rail in feet; F the distance from toe of outer switch-rail to point of frog in feet. Then, for standard gauge, 4 feet Si inches, straight switch- rail, and 5 inches throw of switch. 8.6 L N ■ * — L + 0.42 N’ The above may be written roundly as a rule thus: — Multiply the length of switch-rail in feet by the number of the frog, and set down the product. Multiply that product by 8*, and call the result A. Next add together the length of switch-rail in feet and two-fifths of the frog number; call the sum B. Then divide A by B, and the quotient will be the dis- tance in feet from toe of outer switch-rail to point of frog. Example. Switch-rail, 20 feet long; frog, No. 9. Length of switch-rail................... 20 Multiplied by frog number .... 9 Product.............................180 Multiplied by........................ S* 1,530 = A. Length of switch-rail................... 20 Added to § frog No. 9................ 3.6 23.6 = B.FROGS AND SWITCHES. 133 A divided by B = 1,530 divided by 23.6 = G4,J3 feet, the frog distance; say, 65 feet. 19. If the switch-rail be curved, the formula would stand thus: — 8.6 L N * L -{- 0.84 N* Which may be made a written rule as follows: — Multiply the length of switch-rail in feet by the number of the frog, and their product by 8£; call the result A. Add together the length of switch-rail in feet and four-fifths of the frog number; call the sum B. Then divide A by B, and the quotient will be the distance from toe of outer switch-rail to point of frog in feet. 20. The foregoing rules are applicable to turnouts from curves, as well as from straight lines. 21. To find the radius of outer rail of a turnout curve from straight track. Data same as in previous rules for frogs; R the required radius iu feet. If the switch-rail be straight, R If the switch-rail be curved, R 8.6 L2 N2 — L2 —0.17 N2’ 8.6 L2 N2 — L2 — 0.68 N2' 22. To find the radius of the outer rail of a turnout curve from curved track, proceed thus: — First find the radius as for a turnout from straight track by the preceding rule; call it, as before, R. Call the radius of the main track R2, and the required radius of turnout curve r. Then, if the turnout be towards the coucave side of main track, — R2 X R r — r^Tk' If the turnout be towards the convex side of main track, — r Rj X R R-2^—R* More explicitly, in the first case, r.is equal to the product of the other radii divided by their sum; and, in the second case, r is equal to the product of the other radii divided by tlieir difference.134 FROGS AND SWITCHES. 23. The angle of a frog is equal to 3,440' divided by the frog number. 24. To find the frog distances and radii for a three-foot gauge, find them by the preceding rules for standard gauge, and take five-eighths of the result, using a switch-rail reduced in like measure. For a metre gauge, take seven-tenths of the result, using a switch-rail reduced in like measure. Or these radii and distances may be found from the appended tables for standard gauge by pro-rating as above. 25. Three frog patterns are enough for general service. They should be so proportioned, that, taken in couples, the less may fit as middle frogs on double turnouts. Numbers 5£, 7$, and 10| make an excellent suit; numbers 5, 7, and 9J also answer very well. 26. At the terminal stations, and about the shops of busy roads, patterns necessarily multiply. The better way in such cases is to plot the situation to a large scale, and to take the required distances and angles from the drawing.TURNOUT TABLE. 135 of II 51 ^3“ 8witch-rail,24ft Do.angle,1° 00' Switch-rail,20 ft Do.angle,1° 12^ Switch-rail,18 ft Do.angle,1° 30' 11 s£ Main frog diet. Rad. outer rail. Mid. frog diet. Mid. frog angle. Main frog dlat. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. 2-88 ^cccob 32.2 138.5 20.9 19° 17' ® tO GO CO c© cb oo oo 31.2 139.4 19.9 19*22' 30.2 140.0 19.0 19°28' £ © MAIN FROG NUMBERS AND ANGLES. 36.4 175.2 23.7 jl7°08' “o tO-ICO oWCHtn §iobb 35.4 176.0 22.7 17° 10' 34.6 170.9 22.0 17° 14' 33.5 178.9 20.9 17° 20' S.* 40.1 215.8 20.1 15° 25' ^ to CO to ?' 9* P 00 ot A ot 38.9 217.4 24.8 15° 30' 38.0 218.8 24.0 15° 34' 30.6 221.8 22.6 15°41' S- 00 °oioa>4 cb®®? ® © tO tO 43.5 267.1 27.8 14°00' 42.7 268.2 27.1 14° 02' © CO <3> 39.9 274.9 24.4 14° W S.o. tow- 47.7 312.9 30.6 12°54' 40.8 314.2 29.8 12“ 56' 45.9 315.7 28.9 12°58' 44.6 318.6 27.8 13° 02' 42.7 325.0 25.8 13° 12' CD r 50.9 362.0 32.8 11*58' 49.9 363.7 31.7 12°00' 48.9 365.8 30.8 12°02' *0*0 §»|^. ^ -1 cn M 00 CB <3^ CO CO 00 - t^iocoo ^""c co to o* o««« © ® © —t '“o CO CO CB M to to to © bob* © hoCOMCn © © 48.4 450.2 •28.9 11° 24' 00 58.1 485.3 37.0 10° 25' 56.8 488.4 35.0 10° 28' 55.5 492.1 34.5 10°30' ® w©cb CotO®M CD tO -1 50.9 615.0 30.1 10° 46' 61.8 '556.2 39.3 9° 42' 60.3 560.2 37.9 9“ 44' CO Ot I CD CB ® W © CB ° CO —1 CB tfc*©CBOO 1. CB CB © sboa | ^cbooo 53.7 595.8 31.0 10° 04' -t O °° 63.4 630.0 39.7 9° W 65.0 624.8 41.3 9° 10' ”“co S o» 1 ^Co£cn oo o> t-* 1 O* OO <0 ^bb 1 ^bob iiiz 66 8 713.5 41.0 8° 41' 68.7 707.0 43.3 8° 39' ?8pa I s?s38 asHs NtOOO 1 O Q) 1 NOOOO <* to © 10 CO _.i 00 ^ ° i-> GO ° C© g JB 00 JO M Co © O siob'b scobb 61.3 870.0 35.1 8° 40' 05.3 826.0 38.7 8° 27' 08.0 807.0 41.4 8° 21' © ¥ sjijpa g6pe31 ggpa ©spa 2gps ncoom \bbi-j 1 ^i-»oo ’si.bb to —i © CB 65.9 1,089 37.2 7° 55' 70.0 1,020 41.6 7° 40' 73.8 992.0 44.4 7° av 70.1 976.2 46.8 7° 30' 78.5 964.2 49.0 7° 27' CB O»-» to © oow*- 68.3 1,224 38.3 7° 38' 73.4 1,137 43.0 7° 22' 76.9 1,102 45.9 7° 15' 79-4 1,083 48.3 7° 12' 82.0 1,068 50.9 7° 09' CB ¥ lii 5° 00' 70.3 1,342 39.2 7° 21' 75.7, 1,263 43.8 7° 09' 79.4 1,196 47.6J 6° 57' 82:9 1,1(0 49.9 6*54' 84.8 1,157 52.9 6° 50' 2s’S.s CD CO NOtKC^ -NCO—tot n.*-* ©-* 's TURNOUT TABLE. — SWITCH-RAILS, STRAIGHT; GAUGE OF TRACK, 4 FEET 8£ INCHES; THROW OF RAIL, 5 INCHES.136 TURNOUT TABLE. h O * O « w H cc W w o 5 W w tf o ◄ o5 g W h w § g B >o O .. g B _ K O W > « P o «T B < * a u m W 3 fc g 03 PS w « 8 8 o PS pH 8 3 a? -t* COI'-OV 63.3 2,032 32.2 7*48' 68.5 1,648 36.6 7*20' 77.1 1,388 44.1 6*52' mKO ph O pH o O 54.7 3,135 26.2 ,8° 58', o d «© V. r5gr4 8 CO -CO© rH go *— oa cd 'v CO ^(0© 70.3 1,343 39,1 7*21' 74.6 1,256 43.0 7*08' a? UO TftOCJV gSgS « o> 59.9 1,549 31.1 8*18' S«-S" 72.2 1,144 41.9 7*24' _fc*0O o Cl rt o UO 52.0 2,032, 25.5 9° 26' 58.1 1,351 30.4 8*36' 62.6 1,169 34.3 8*10' OO O COS. SS-SS£ 60.6 1,032 40.7 7*43' s? uo 50.6 1,672 25.1 9° 42' 56.4 1,182 29.8 8*54' 60.5 1,040 33.5 8*29' 63.6 976.8 36.3 8*15' Hcouas. b^ddg ®f o O O <£> ^1 ®5^° iO -Cl o T-4 03 59.1 914.7 32.5 8*53' 61.2 865.2 35.1 8*39' n:™^Sb 00 00 Cl oa COHr-rt lOHCOo 00 oa 58.8 772.1 34.1 9*03' 00 00 00 S. PH cl CO 2 ©g eo ci ei n k-p oo 40.4 553.9 21.7 12*30' 44.0 487.1 24.9 11*52' 46.5 461.3 27.2 11*34' co oa cb 00 00 00 CO CO 00 ^ dodo'-' IO « CO^, ®3 OO 38.3 451.1 20.9 13*18' 41.5 405.7 23.8 12*42' b- co oo CO l- o N eo co tb d 00 ^ '"Sc,g 47.1 371.2 29.0 12*08' tD§S aa 36.4 375.8 20.1 14*08' CONl^^J oa c6 ci * «S«*« 41.3 330.7 24.6 13° 18' 42.7 324.0 25.9 13*10' ^S-e, -*»e5 kO o o 34.4 310.2 19.3 16*06' 36.9 288.1 21.6 14*36' 38.7 278.9 23.2 14*21' 39.9 274.0 24.4 14° 12' Cl CO b-^> ph © d ° ab -g 31.9 244.9 18.2 16*30' 34.1 230.8 20.2 16*02' 35.6 224.9 21.6 15*48' 36.6 221.8 22.6 15*41' 37.7 219.3 23.7 16*35' *6 lOSHtb sssg 31.4 184.8 18.8 17*40' 32.6 181.0 20.0 17*28' 33.5 178.9 20.8 17*20' 34.4 177.3 21.7 17*15' 5S 0> co oo ab iO fN ® lO o ^ &SSS5 d co oa ab O oa oo ^ ”3rHoa c!®lT£* ph oa oa M “SH» Main frog diet. Rad. outer rail. Mid. frog diet. Mid. frog angle. Main frog dlst. Rad. outer rail. Mid. frog diet. Mid. frog angle. Main frog dlst. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog diBt. Mid. frog angle. Main frog dlst. Rad. outer rail. Mid. frog dist. Mid. frog angle. Switch-rail, L’gth. 12 ft. Ang. 4*00'. Rad. 171.9. Switch-rail, L’gth. 16 ft. Ang. 3*00'. Rad. 312.7. Switch-rail, L’gth. 20 ft. Ang. 2“ 24'. Rad. 477.5. Switch-rail, L’gth. 24 ft. Ang. 2° 00'. Rad. 687.6. Switch-rail, L’gth. 30 ft. Ang. 1*36'. Rad. 1,074.TO LOCATE A TURNOUT. 137 XL VII. TO LOCATE A TURNOUT. 1. Let the heavy parallels in the figure represent the rails of the main track. 2. Stick a pin or drive a spike at A, the toe of switch, at a distance from the gauge side of the main-track rail equal to the throw of the switch-rail. Lay off the distances A C and AB (if a double turnout), taken from the foregoing tables, and place the frogs C and B, or mark those points. Stretch -the cord from A to B, and from B to C. Mark*the middle points of those stretches at H and P. Catch the cord at H with your forefinger, and pull it outwards until your finger, at E, lines with the switch-rail, and also with the right gauge side of frog B. Stick a pin at L, half-way between H and E. Let the cord spring in against L, so that it shall stretch straight from A to L, and from L to B. Opposite the middle points, V, of those stretches, stick pins on the outside at a distance from the cord equal to one-quarter of H L. In like manner, catch the cord at P, the point midway between B and C; stretch it to F, in line with the gauge sides of the frogs; and stick a pin at I, half-way between P and F.138 TO LOCATE A TURNOUT. 3. Next lay off the proposed line of the near rail of the side track, X D. Mark the point G on that line where the range of the proper gauge side of frog C strikes it. Measure C G. Set off G D, equal to C G, along the side-track line, and drive a pin at D. Stretch the cord from C to D. Mark the middle point of it at K, and drive a pin at N, half-way between K and G. Stretch the cord from C to N, and from N to D. Stick pins outside the middle points, M and O, of those stretches at a distance from those points, M and O, equal to one-quarter of KN. 4. These three sets of pins will fix the line of one rail of the turnout. The corresponding rail of a double turnout can be laid off from them, if required, by symmetrical measurements. 5. In the case of a single turnout, stretch the cord from the toe of switch, as above, to the point of frog, located by the foregoing tables; catch it at the middle, and pull it outwards to a point in range with the line of the switch-rail in one direction, and the gauge side of frog in the other direction. Half-way between that point and the middle of the cord, when straight, stick a pin. Measure that half-way distance, and divide it by 4; call the quotient the “quarter-distance.” Stretch the cord from the pin just set to the toe of switch in one direction, and to the point of frog in the other. Outside the middle points of these short stretches, lay off the “ quarter- distance,” as above found, and stick two other pins. These three pins will sufficiently mark the line of the outer rail of the turnout. 6. The same methods will apply in practice to turnouts from curves. In the latter case, the distance C G is to be calculated as follows: — Multiply the distance Y D, between the nearest rails of the parallel tracks, by the number of the frog, taken from the fore- going table. Thus, on the full gauge, with a space between tracks of 7 feet and a No. 6 frog, the distance C G would be 7 X 6 = 42 feet. Lay off C G, in range of the gauge side of the frog, and stick a pin at G. Measure out G D, equal to C G, and set another pin at D, making D Y the proper distance be- tween tracks. Then stretch the cord from D to C, and pro- ceed to stake off the curve C N D, as above directed.CROSSINGS ON STRAIGHT LINES. 139 XL VIII. CROSSINGS ON STRAIGHT LINES. 1. Having located frogs B and C by the preceding methods, stretch the cord any convenient distance, C D, in the range of the outer gauge side of the frog C. Set off E F parallel to CD, and distant the gauge-width from it. The intersection of said parallel at F with the near rail of the side track marks the spot for point of side-track frog; the curve F G, thence to toe of switch, corresponds to AC on the main track, and may he staked out in like manner. XLIX. CROSSINGS ON CURVES. 1. Having located frogs B and C by the preceding methods, set off the width of gauge, C D, from point of frog C, and square to its outer gauge side. Stick a pin at D. 2. Next calculate the distance D E to the point of side-track frog as follows: Subtract the gauge of track from the dis-140 CROSSINGS ON CURVES. tance, H I, between the gauge sides of the nearest rails of the main and side tracks; multiply the remainder hy the number of the frog, taken from foregoing tables. The product will be the distance from D to the point of side-track frog at E.ELEVATION OF THE OUTER RAIL ON CURVES. 141 3. Suppose, for example, the gauge sides of the nearest rails of the main and side tracks are 6 feet 6 inches asunder; gauge of track, 4 feet S£ inches; frog, a No. 9. Reducing inches to decimals, we have then the distance bet\v;een tracks 6.5 feet, less the gauge, 4.7 feet, = 1.8 feet; and 1.8 multiplied by 9, the number of the frog, gives 16.2 feet for the distance D E. The proper spring will be given to rail D E on the ground; and curve E G, from frog to toe of side-track switch, will be staked off as directed in the section on turnouts. L. ELEVATION OF THE OUTER RAIL ON CURVES. 1. Great precision in this adjustment is unattainable, owing to differences in the speed of trains and to the cost of track- maintenance, if it were attempted. The annexed table will be found convenient in practice. It has been calculated by the following simple rule: — 2. Divide the speed in miles per hour by 10; multiply the square of the quotient by the degree of curve. The product is the elevation in sixteenths of an inch for full gauge of 4 feet S£ inches. Take two-tliirds of it for 3-feet narrow gauge. 3. Moleswortli gives the following formula for determining the elevation of the outer rail with any gauge: — V = greatest velocity of trains in feet per second. G = gauge of railway in feet. C = length of chord whose middle ordinate will give the required elevation. Then C = i V Vg. A modification of this formula gives the following approximate rules: — To fix the elevation of the outer rail on the standard gauge of 4 feet 8£ inches, multiply the speed of trains in miles per hour by 5, and divide142 ELEVATION OF THE OUTER RAIL ON CURVES. the product by 3. This will give the length of tape, C, to stretch on the gauge side of the.outer rail; and the distance, e, from the middle of the tape to the gauge side of the rail, will be the proper elevation. For gauge of one metre, = 3.28 feet, make C equal to one and one-third times the speed of trains in miles per hour. For 3-feet gauge, make C equal to one and one-fourth times the speed of trains in miles per hour. TABLE OF ELEVATIONS FOR OUTER RAIL ON CURVES. SPEED IN MILES PER HOUR. 10 20 30 40 £ I ELEVATION OP OUTER RAIL IN INCHES AND FRACTIONS. I s E N r 6i !i <3 s' £ 3 . ® s ia 3 — c: -*« ! ® 3 |.= £ i ® i i C5 W =d S* g-s = —• M >-» >-• M M 4*COH«^a^tOOO‘4 0'COI^OOO*4CltCOH 2? o»i^»^^^09C>9eocotoiorokoi-»»-*^i-4 IOCO«-l^l>9CDa)iti-*cE)(3JCOi-*ODO»&90 HCJi-iH f—* 09 h-» t-J <-» t— 09 i—» l-*)-* K> 4*. t3 4»iU fed 4* tO 4“ 4* fcO 4- tO MMMl—IM 09t3i->i-»O«C0aD~4-‘9a90>O>*t»^-09tdt0i »—* CJ1 b-• CJ» ao to c© ® to <» 4 ^ 4 io ® 4 ® 4 ao 4 6o *09 09^-*^IC*9CJ» «■! CT» M H-» >—* M 03 Oi M CO CJ< U M TRACKMEN’S TABLE OF CURVES AND SPRING OF144 TRACKMEN'S TABLE OF CURVES. EXPLANATION OF THE FOREGOING TABLE. Columns 1 and 10 give the degree of curve. The use of column 2, containing the deflection distances, may be illustrated thus: Suppose stakes 4, 5, and 6 to be miss- ing from a 3-degree curve, and that stakes 2 and 3 are still standing 100 feet apart. To replace the missing stakes, pro- ceed as follows: Measure 100 feet from 3 to A, and make a mark at A exactly in range with 2 and 3. Find, in column 2 of the table, the deflection distance for a 3-degree curve, which is seen to be 5 feet 3 inches. Hold one end of the tape at A, and, stretching 5 feet 3 inches towards 4, nearly square to the range A-3, make a scratch on the ground three or four feet long, swinging the tape around A as a centre. Next lay ofE 100 feet from stake 3 to the scratch; where the end of that measurement strikes it, is the place for stake 4. By measuring 100 feet out to B on the range 3-4, and proceeding in like manner, stake 5 may be set; and so on. 3. If the centre line is already staked for track at points 100 feet asunder, and the degree of curve is wanted, range out the straight line between stakes, as above, to A or B, and measure across from those marks to the neighboring location-stake. Suppose the distance B-5, for example, to be 8 feet 9 inches. Referring, then, to column 2 of the table, we find that deflec- tiou distance to indicate a 5-degree curve. If the distanceTRACKMEN'S TABLE OF CURVES. 145 proved to be 4 feet 4 inches, we should soon discover that that distance was about half-way between 3 feet 6 inches and 5 feet 3 inches, the nearest numbers in the table corresponding respectively to a 2-degree and a 3-degree curve, and showing the located line to be a 24-degree curve. 4. Let A G B in the figure, which is drawn very much out of proportion in order to make the subject clear, represent the centre line of a curve. Suppose G H to be a chord 100 feet long, and G C or C H to be a chord 50 feet long. Then column 3 in the table gives the distance, C D, from the middle of the 100-feet chord to the rail, and column 4 gives the distance, E F, from the middle of the 50-feet chord to the rail, for the different degrees of curve. By the aid of these columns, pins can be set 25 feet apart on a curve where the location-stakes are 100 feet apart. Thus, for a 3-degree curve, C D is 8 inches, and E F 2 inches. If pins were wanted at the half-way marks, N, their distance from the dotted short chords would be one- quarter of E F. It must be an uncommon case, however, that calls for stakes closer together than 25 feet. 5. Columns 5, 6, and 7 give the spring of rails of different lengths for the various degrees of curve. 6. Columns 8 and 9 give figures for finding the degree of curve, by simple measurement of a straight line on the track, as follows: Suppose A C B and KIL to represent the rails of a curving track. From any point A, on the outer rail, sight tcross to a point B, on the same rail, along a line just touching the inner rail at I. Measure from A to B, and seek the dis- tance in column 8 or 9, according to the gauge of track. If the distance, for example, measured 232 feet on the full gauge, then the curve would be a 4-degree curve; if 249 feet, then it would indicate a 34-degree curve, for the reason that the146 TRACKMEN'S TABLE OF CURVES. measured distance falls half-way between the distances corre- sponding to a 3-degree and a 4-degree curve respectively. 7. The rate of curve can be found also very nearly by means of column 3. To do so, stretch a straight line, 100 feet long, between points on either rail; for, though they seem very dif- ferent in the figure, the two rails of a track have practically the same curvature. Measure from the middle of the line across to the gauge side of the rail, and seek the measured distance in column 3: opposite to it*, in column 1, will be found the degree of curve. 8. If, in any case, the exact figures sought are not found in the table, take out the next figure less and the next greater. Subtract one from the other, and divide the remainder by 4. Add the fourth part of the difference between them, thus determined, to the smaller number, and compare the sum with the number sought. If still too small, add another fourth part; and so on until the distance or the degree is ascertained to within a quarter part. 9. Suppose, for instance, a deflection distance measures 5 feet 7 inches. The nearest tabular numbers are 5 feet 3 inches and 7 feet. Their difference is 21 inches, one-fourth of w hich is 5} inches. Adding inches to the smaller number, 5 feet 3 inches, gives 5 feet inches, which indicates nearly enough a 3j-degree curve. Again: if a measurement of 175 feet is sought in column 9, the track is seen at once, without calcula- tion, to be a 4$-degree curve.TABLES.TABLES. TABLE I. TIME OP MERIDIAN PASSAGE OF NORTH STAR ABOVE TnE POLE, FOR THE YEAR 1870. Month. Day of Month. ist. nth. 2ISt. January .... H. M. 6 26 P.M. H. M. 5 46 P.M. H. M. 5 07 P.M. 3 05 “ I ebruary.... 4 =3 “ 3 44 “ March • • • 2 33 “ 1 54 “ 1 14 " April .... 0 31 “ II 52 A.M. II 13 A.M. May .... IO 33 A.M. 9 54 “ 9 15 “ June • 8 32 “ 7 S3 “ 7 13 “ July .... 6 34 “ 5 55 5 16 “ August .... 4 33 “ 3 54 " 3 14 " September 2 31 “ 1 52 “ I 12 “ October .... 0 33 “ II 50 P.M. II II P.M. 9 °9 " November IO 27 P.M. 9 48 “ December • • 8 29 “ 7 5° “ 7 n “ For the year 1880, add three minutes to the time; for 1890, add 7 minutes; for 1900, add 11 minutes. 149150 TABLES. TABLE n. TIME OF EXTREME ELONGATIONS OF NORTH STAB FOR THE TEAR 1870, —LAT. 40°. Day of Month. First. Eleventh. Twenty-first. Month. Time of Elongations. Eastern. Western. Eastern. Western. Eastern. Western. H. M. H. M. H. M. H. M. H. M. H. M. Jan. O.32 P.M. O.24 A.M. II. 52 A.M. 11.40 P.M. II.13 A.M. 11 .OI P.M. Feb. IO.29 A.M. IO.18 P.M. 9.5° “ 9.38 “ 9. II “ 8.59 '■ Mar. 8.39 “ 8.27 “ 8.00 “ 7.48 “ 7.20 “ 7.09 “ April. 6.37 “ 6.25 “ 5-58 “ 5.46 “ 5.19 “ 5.07 " May. 4-39 ‘ 4.27 “ 4.00 ‘ 3.48 “ 3.21 “ 3.09 “ June. 2.38 “ 2.26 “ 1.58 “ 1.47 “ 1.19 “ I.07 “ July. 0.39 “ O.28 “ 11.57 P*M* II.49 A-M- II.18 P.M. II .09 A.M. Aug. IO.35 P.M. IO.27 A.M. 9.56 “ 9.48 “ 9.16 “ 9.08 “ Sept. 8-33 “ 8.25 “ 7-54 “ 7.46 “ 7.14 “ 7.06 “ Oct. 6.35 “ 6.27 “ 5.56 “ 5.48 “ 5.17 “ 5.09 “ Nov. 4-33 “ 4.26 “ 3-54 “ 3.46 “ 3.15. “ 3.07 “ Dec. 2.35 “ 2.27 “ 1.56 “ 1.48 “ 1.17 “ 1.09 “ For the year 1880, add 3 minutes to the time ; for 1890, add 7 minutes ; for 1900, add 11 minutes. Change of latitude affects the tabular time very slightly. TABLE III. AZIMUTHS OF THE NORTH STAR, AND THEIR NATURAL TANGENTS. W Year. D h 1870. 1880. 1890. 1900. < Azi- Nat. Azi- Nat. Azi- Nat. Azi- Nat. muth. Tan. muth. Tan. muth. Tan. muth. Tan. 3° 1.36 .02793 O 1 1.32 .02677 1.28 .02560 O 1 1.25 •02473 32 1.38 .02851 1.34 .02735 1.30 .02619 1.27 .02531 34 I.40 •02910 1.36 .02793 1.32 .02677 1 .29 .02589 36 1.43 •02QQ7 1.39 .02881 I*3S .02764 1.31 .02648 38 1.45 •03055 1.41 .02939 i-37 .02822 1-33 .02706 40 1.48 •°3I43 1.44 .03026 1.40 •02910 1.36 .02793 42 1.52 .03259 1.47 .O3II4 1.43 .02997 1.39 .02881 44 i-55 .03346 I-51 •03230 1.46 .03084 1.42 .02968 46 1.59 .03463 1-55 .03346 I *50 .03201 1.46 .03084 48 2.04 .01600 1.59 .03463 1.54 •°33I7 1.50 .03201 5° 2.09 •03754 2.04 .03609 1*59 .03463 - 1.54 .03317TABLES. 151 TABLE IV. ROODS AND PERCHES IN DECIMAL PARTS OF AN ACRE. One Acre =4 Roods = 100 Perches = 4,840 Square Yards =43,560 Square Feet. f Ill X Roods. n « X Roods. cc (xl PU O 1 3 3 £ Ah 0 1 2 3 o • OOO .250 • 5°° •75° 21 .131 .381 .631 .881 I .006 .256 .506 .756 22 •'37 .387 .637 .887 2 • OI2 .262 .512 .762 23 .144 -394 .644 •894 3 .OI9 .269 .519 .769 24 .150 .400 .650 .900 4 • O25 •275 • 525 •775 25 .156 .406 .656 .906 5 .031 .281 •531 .781 26 .162 .412 .662 .912 6 .037 .287 •537 .787 27 • I69 ■4'9 .669 .919 7 .044 .294 •544 •794 28 •175 •425 •675 •925 8 .oqo . IOO .550 .800 29 .l8l .431 .681 .931 9 .056 .306 •556 .806 3° .187 •437 .687 •937 IO .062 .312 .562 .812 31 •'94 •444 .694 •944 1 [ • 069 .319 .569 .819 32 .200 .450 .700 .950 12 .075 •325 •575 .82s 33 • 206 .456 .706 .956 13 .081 •33' .581 .831 34 • 212 .462 •7'2 .962 h .087 -337 .587 .837 35 .219 .469 •7'9 .969 •5 .094 •344 •594 .844 36 .225 •475 •725 ■975 l6 • IOO •35° .600 .850 37 •231 .481 •73' .981 17 • 106 .356 .606 .856 38 .237 .487 •737 •987 18 .112 .362 .612 .862 39 ■A\ •494 •744 •994 '9 • 119 .369 .619 .869 40 • 25O • 500 .750 I .OOO 20 .125 •375 .625 .875 TABLE V. DECIMALS OF AN ACRE IN ONE CHAIN LENGTH OF 100 FEET, AND OF VARIOUS WIDTHS. Width in Rods. Decimals of an Acre per 100 Feet. Acres per Mile. Width in Rods. Decimals of an Acre per 100 Feet. Acres per Mile. % .018939 I 5% •208333 II 1 .037879 2 6 .227273 12 1% .036818 3 6'A .246212 '3 2 •075757 4 7 .265151 '4 2 A .094697 5 7% .284091 '5 3 •113636 6 8 .3O3O3O l6 3 Vi .132576 7 8% .321970 '7 4 .151515 8 9 .340909 l8 4 lA .170454 9 9% .359848 '9 5 .189394 IO IO .378788 20152 TABLES. TABLE VI. ACRES, ROODS, AND PERCHES IN SQUARE FEET. Acres. Square Feet. Roods. Square Feet. Perches. Square Feet. I 43560 I 10890 17 4628.25 2 87120 2 21780 18 49OO.5O 3 130680 3 32670 19 5172.75 4 174240 4 43560 20 5445.00 217800 261360 304920 21 57I7-75 5989.50 6261.75 6 7 Perches. Square Feet. 22 23 348480 .6534.00 6806.25 24 9 392040 I 272.25 25 io 435600 2 544.50 26 7078.50 II 479160 3 816.75 27 7350-75 7623.00 12 522720 566280 4 1089.00 28 13 s 1361.25 29 7895.25 8167.50 14 609840 6 1633.50 3° is 653400 7 1905.75 31 8439-75 l6 696969 8 2178.00 32 8712.00 *7 740520 783080 9 2450.25 33 8984.25 l8 IO 2722.50 34 9256.50 *9 827640 II 2994.75 35 9528.75 20 871200 12 3267.00 36 9801.00 21 916760 13 3539-25 37 10073*25 22 960320 14 3811.50 38 10345.50 is 4083.75 39 10617.75 l6 4356.00 40 I089O.OO TABLE Vn. CIRCULAR ARCS TO RADIUS OF 1. Degrees. Minutes. Seconds. J // I .01745329 I .00029089 1 .00000485 2 .03490658 2 .00058178 2 .OOOOO97O 3 .05235988 3 .00087266 3 .OOOOT454 4 .06981317 4 .00116355 4 .OOOOI939 5 .08726646 5 .00145444 5 .00002424 6 .10471975 6 .00174533 6 .00002909 7 .12217305 .13962634 7 .00203622 7 .OOOO3394 .OOOO3878 8 8 .00232711 .00261799 8 9 .15707963 9 9 .00004363TABLES. 153 TABLE Vin. FEET IN DECIMALS OF A MILE. Feet. Decimals of a Mile. I o.o 00189394 2 0.0 00378798 3 0.0 00568182 4 0.0 00757576 5 0.0 00946970 6 0.0 01136364 7 0.0 01325758 8 0.0 0 i 5 I 5 I 5 2 9 0.001704546 TABLE IX. INCHES REDUCED TO DECIMAL PARTS OF A FOOT. In. 0 I 2 3 4 5 6 7 8 9 10 XI In. 0 .0000 .0833 .1667 .2500 •3333 .4167 .5000 •5833 .6667 .7500 •8333 .9167 0 L8 .0052 .0855 .1719 .2352 .3385 •4219 •5052 •5885 .6719 •7552 .8385 •9219 i .0104 .0938 •177" .2604 •3438 .4271 .5104 •5938 .6771 .7604 .8438 .9271 i .0156 .O99O .1823 .2656 .3490 •4323 •5156 •599° .6823 .7656 .8490 •9323 iff i .0200 .1042 .1875 .2708 ■3542 ■4375 .5208 .6042 .6875 .7708 •8542 •9375 \ A .0260 .1094 .1927 .2760 •3594 •4427 .5260 .6094 .6927 .7760 •8594 •9427 T5ff 8 .0313 .1146 .1979 .2813 .3646 •4479 ■53"3 .6146 .6979 .7813 .8646 •9479 | Iff •°36S .1198 .2031 .2865 .3698 •453" •5365 .6198 .7031 •7865 .8698 •953" lV .0417 .1250 .2083 .2917 •375° •4583 •5417 .6250 .7083 •79"7 .8750 •9583 Tt 1^ .0469 .1302 •2"35 .2969 .3802 •4635 .5469 .6302 •7"35 .7969 .8802 •9635 T9ff * .0521 •1354 .2188 .3021 •3854 .4688 .5521 •6354 .7188 .8021 •8854 .9688 5. H •0573 .1406 .2240 •3073 .3906 .4740 ■5573 .6406 .7240 •8073 .8906 .9740 H 1 .0625 .1458 .2292 .3125 .3958 •4792 .5625 .6458 .7292 .8125 •8958 •9792 3 ia 1 6 .0677 .1510 •2344 •3"77 .4010 •4844 .5677 .6510 ■7344 ■8177 .9010 •9844 13 1 6 t .0729 .1563 .2396 .3229 .4063 .4896 •5729 •■6563 •7396 .8229 .9063 .9896 u .0781 .1615 .2448 .3281 ■4”5 •4948 .5781 .6615 •7448 .8281 .9115 •9948 nTABLE X. RADII AND THEIR LOGARITHMS, MIDDLE ORDI- NATES, AND DEFLECTION DISTANCES.156 RADII AND THEIR LOGARITHMS. Decree Logarithm Arithmetical Middle Deflec- Tangen* of Radius. of Comple- Ordinate, lion Dis- tial Dis- Curve. Radius. ment. Chord 100 Feet. tance. tance. o / 0 5 68754.9 4.837304 5.162696 .018 .145 .073 IO 34377*5 22918.3 17188.8 4.536274 5.463726 5.639818 .036 • 291 •145 1 i5 4.360182 -055 .436 .2X8 20 4.235246 5.764754 5.861665 .073 .582 • 291 25 13751-0 4.138335 .091 •727 .364 3° II459-2 4-°59I54 5.940846 .109 .873 .436 35 9822.2 3.992209 6.OO779I .127 1.02 •509 40 8594.4 3-9342I5 3.883066 6.065785 .145 1. l6 .582 45 7639-5 6.H6934 6.I62696 .164 1-31 .654 5° 6875.5 3.837304 .l82 i-4S •727 55 6250.5 3-795914 6.204086 .200 1.60 .800 1 O 57*9-6 3.758128 6.241872 .2l8 i-75 .873 5 S288.0 3-723365 6.276635 .236 1.89 •945 IO 4911.1 3.691179 3.661216 6.308821 •255 2.04 1 .02 15 4583-7 6.338784 .273 2.18 I.09 20 4297-3 3-633I95 6.366805 .291 2-33 I.l6 =s 4044.5 3.606864 6.393136 .309 2.47 2.62 I .24 3° 3819.8 3.582041 3-558565 6.417959 .327 1-31 35 3618.8 6.441435 •345 2.76 1.38 40 3437-9 3-536293 6.463707 6.484894 .364 2.91 1 -45 45 3274.2 3.515106 .382 3.05 1 *53 1.60 5° 3125.4 3.494906 6.505094 • 4OO 3-20 55 2989.5 3-475599 6.524401 .418 3-34 1.67 2 0 2864.9 3-457II4 6.542886 •436 3-49 1.74 5 2750.3 ' 3.439380 6.56062O •455 3-64 1.82 IO 2644.6 3-422359 6.577641 •473 3.78 1.89 15 2546.6 3.405961 6.594039 6.609825 •491 3-93 1.96 20 2455-7 3-39OI75 •509 4.07 2.04 25 2371.0 3*374932 3.360215 6.625068 .527 4.22 2.11 3° 2292.0 6.639785 6.654018 •545 .564 .582 • 600 4.36 2.18 35 2218.1 3-345982 4.51 2.25 40 2148.8 3.332196 6.667804 . 4-65 2-33 45 2083.7 3.318835 6.681165 4.80 2.40 5° 2022.4 3.305867 6.694133 .618 4.94 2.47 55 1964.6 3.293274 6.706726 .636 5-°9 2.54 3 0 1910.1 3.281056 6.718944 •655 5-23 2.62 d 1858.5 1809.6 3.269163 3.257584 6.730837 6.742416 .673 .691 5-38 5-53 2.69 2.76 15 1763.2 3.246301 6.753699 ■709 5-67 2.84 20 1719.1 1677.2 1637.3 3-235301 6.764699 •727 5.82 2.91 25 3.224585 6-775415 6.785871 •745 5.96 2.98 30 3.2I4I29 .764 6.11 3-°5 35 1599-2 3 * 20*3002 6.796098 6.806069 .782 6.25 3-r3 40 1562.9 3-193931 .800 6.40 3.20 45 1528.2 3.184180 6.815819 .818 6-54 3.27 5° 1495.0 3.174641 6.825359 .836 6.69 3-34 55 1463.2 3.165303 6.834607 .855 6.83 3-42 4 0 1432.7 3-156155 6.843845 .873 6.98 3-49 5 1403.5 3.147212 6.852788 .891 7.12 3-56 10 1375-4 3.138429 6.861571 6.870181 .909 7.27 3-63 15 1348.4 3.120810 .927 7.42 3-7i 20 1322.5 3.121395 6.878605 6.886859 ■945 7-56 3.78 25 1297.6 3.113141 .964 7.71 3-85 3° 1273.6 3-105033 6.894967 .982 7.85 3-93 35 1250.4 3.097048 6.902952 I. OO 8.00 4.00 40 1228.1 3.089233 6.910767 1.02 8.14 4.07RADII AMD THEIR LOGARITHMS. 157 Degree of Curve. Radius. Logarithm of Radius. Arithmetical Comple- ment. Middle Ordinate, Chord 100 Feet. Deflec- tion Dis- tance. Tangen- tial Dis- tance. 0 / 4 45 1206.6 3.081563 6.918437 1 .04 8.29 4.14 50 1183.8 3.O74OII 6.925989 1*05 8.43 4.22 55 1165.7 3.066587 6.9334I3 1 .07 8.58 4.29 5 0 1146.3 3.059299 6.940701 1.09 8.72 4.36 5 1127-5 3.O.52I 17 6.947883 l.II 8.87 4-43 10 ii°9-3 3.045050 6.954950 1.13 9*01 4-51 IS IO9I.7 3.038103 6.961897 1.15 9.16 4.58 20 1074.7 3.031287 6.968713 I. l6 9-3° 4.65 25 1058.2 3.024568 6.975432 I . l8 9-45 4.72 30 1042.I 3.0I7QI0 6.982090 1.20 9.60 4.80 35 1026.6 5.OII4OI 6.988599 1.22 9-74 4.87 40 IOII.5 3.004967 6.995033 I.24 9.89 4.94 45 996.9 2.008652 7.001348 1.25 10.0 5.02 5° 982.6 2.992377 7.007623 1.27 10.2 5-09 55 968.8 2.986234 7.013766 1.29 IO.3 5.16 '6 0 955-4 2.980185 7.019815 1.31 IO.5 5-23 5 942.3 2.974189 7.025811 . 1 *33 10.6 5-31 10 929.6 2.968296 7.031704 i-35 10.8 5.38 15 917.2 2.962464 7.037536 I-36 IO.9 5-45 20 905.1 2.956697 7.043303 1.38 11.O 5.52 25 893.4 2.951046 7.048954 1.40 II .2 5.60 30 882.0 2.945469 7.054531 I .42 11.3 5-67 35 870.8 2.939918 7.060082 1.44 n-5 5-74 40 859.9 2.934448 7.065552 1.45 11.6 5.81 45 849.3 2.929061 7.070939 1.47 11.8 5.89 5° 839.0 2.923762 7.076238 1.49 II .9 5.96 55 828.9 2.918502 7.O82498 r.51 12.1 6.03 7 0 819.0 2.913284 7.086716 1-53 12.2 6.10 5 809.4 2.908163 7.091837 i-55 I2.3 6.18 10 800.0 2.903090 7.O969IO 1.56 12.5 6.25 15 790.8 2.898067 7-IOI933 1.58 12.6 6.32 20 781.8 2.893096 7.IO69O4 1.60 12.8 6.39 25 773-1 2.888236 7.111764 1.62 12.9 6.47 3° 764.5 2.883377 7.116623 1.64 13.1 6-54 35 756.1 2.878579 7.I2I421 1.65 I3.2 6.61 40 747-9 2.873844 7.126156 . 1.67 13.4 6.68 45 739-9 2.869173 7.130827 1.69 13-5 6.76 5° 732.0 2.864511 7-135489 1.71 13.7 6.83 55 724-3 2.859918 7.140082 i-73 13.8 6.90 8 0 716.8 2-855398 7.144602 1-75 I4.O 6.98 5 709.4 2.850891 7.149109 1.76 14. I 7.05 IO 702.2 2.846461 7-I53539 1.78 I4.2 7.12 15 695.1 2.842047 7.157953 1.80 14.4 7-!9 20 688.2 2.837715 7.162285 1.82 14.5 7.27 25 681.3 2.833338 7.166662 1.84 14.7 7-34 3° 674.7 2.829m 7.170889 1.85 14.8 7.41 35 668.1 2.824841 7.I75I59 1.87 15*0 7.48 40 661.7 2.820661 7-179339 1.89 15.1 7.56 45 655.4 2.816506 7.183494 I .9I 15-3 7.63 5° 649.3 2.812445 7.187555 1.93 15.4 7.70 55 643.2 2.808346 7.181654 • i-95 I5-5 7-77 9 0 637.3 2.804344 7.195656 1.96 15-7 7.85 5 631.4 2.800305 7.199695 1.98 15.8 7.92 IO 625.7 . 2.796366 7.203634 2.00 16.0 7-99 15 620.1 2.792462 7.207538 2.02 16.1 8.06158 RADII AND THEIR LOGARITHMS. Degree of Curve. Radius. Logarithm of Radius. Arithmetical Comple- ment. Middle Ordinate, Chord 100 Feet. Deflec- tion Dis- tance. Tangen- tial Dis- tance. o / 9 20 614.6 2.788593 7.211407 2.04 16.3 8.14 25 609.1 2.784689 7-2*53ii 2.06 16.4 8.21 3° 603.8 2.780803 7.219107 2.07 16.6 8.28 35 598.6 2-777*37 7.222863 2.09 16.7 8.35 4° 593-4 2.773348 7.226652 2.II 16.8 8.43 45 588.4 2.769673 7.230327 2.13 17.0 8.50 5° 583.4 2.765966 7.234134 2.15 17.1 8.57 55 578-5 2.762303 7.237697 2.16 *7-3 8.64 10 o 573-7 2.7S8685 7-24*3*5 2.18 17.4 8.72 IO 564.3 2-75*5*0 7.248490 2.22 *7-7 8.86 20 555-2 2-744449 7-25555* 2.26 18.0 9.00 3° 546.4 2-7375*1 7.262489 2.29 18.3 9.15 40 537-9 2.730702 7.269298 2-33 18.6 9.30 1 5° 529-7 2.72403O 7.275970 2.36 18.9 9.44 11 0 521.7 2.7I742I 7.282579 2.40 19.2 9.58 IO 5*3-9 2.710879 7.289121 2.44 19.5 9-73 20 506.4 3.704494 7.295506 2.47 *0-7 9.87 3° 499.1 2.698188 7.301812 2.51 20.0 10.0 4° 492.0 2.69x965 7.308035 2-55 20.3 10.2 5° 485.1 2.685831 7.314169 2.58 20.6 IO.3 12 0 478.3 2.679700 7.320300 2.62 2O.9 IO.4 IO 471.8 2.673758 7.326242 2.66 21.2 10.6 20 465.5 2.667920 7.^2080 2.69 21.5 10.7 3° 459-3 2.662096 7.337904 2.73 21.8 10.9 40 453-3 2.656386 7.343614 2.77 22.1 11.0 5° 447-4 2.650696 7.349304 2.80 22.4 11.2 13 0 44T.7 2.645127 7-354873 2.84 22.6 **•3 IO 436.1 2.639586 7.360414 2.88 22.9 i*.s 20 430.7 2.634175 7.365825 2.91 23.2 11.6 3° 425.4 2.628797 7-37*203 2-95 23-5 11.7 4° 420.2 2.623456 7-376544 2.98 23.8 11.9 50 4*5-2 2.618257 7.381743 3.02 24.I 12.0 14 0 410.3 2.613102 7.386898 3.06 24.4 12.2 IO 405.5 2.607991 . 7.392009 3-°9 24.7 12.3 20 400.8 2.602928 7.397072 3-*3 25.O *2.5 3° 396.2 2.597914 7.402086 3-*7 25.2 12.6 40 39*-7 2.592954 7.407046 3.20 25-5 12.8 5° 387-3 2.588047 7-4**953 3-24 25.8 I2.9 15 0 383.1 2.583312 7.416688 3.28 26.1 I3.O IO 378.9 2-578525 7.421475 3-3* 26.4 13.2 20 374.8 2.573800 7.426200 3-35 26.7 *3-3 3° 370.8 2.569140 7.430860 3-39 27.O *3-5 40 366.9 2.564548 7-435452 3-42 27-3 13.6 50 363.0 2.559907 7.440093 3.46 27-5 13.8 16 0 359-3 2-555457 7-444543 3-5° 27.8 13.9 IO 355-6 2.550962 7.449038 3-53 28.1 14.I 20 352-0 2.546543 7-453457 3-57 28.4 14.2 30 348.4 2.542078 7-457922 3.61 28.7 14.3 40 345-o 2.537819 7.462181 3-64 29.O 14.5 5° 341-6 2-5335*8 7.466482 3-68 29.3 14.6 17 0 338.3 2.52Q302 7.470698 3-72 29.6 14.8 IO 335-° 2.525045 7-474955 3-75 29.9 14.9It ADIT AND TTTEIR LOGARITHMS. 159 Degree of Curve. Radius. Logarithm of Radius. Arithmetical Comple- ment. Middle Ordinate, Chord 100 Feet. Deflec- tion Dis- tance. Tangen- tial Dis- tance. o / 17 20 331.8 2.520876 7.479124 3-79 30.1 15.1 3° 328.7 2.516800 7.483200 3.82 30.4 15.2 4° 325.6 2.512684 7.487316 3-86 30.7 15-4 5° 322.6 2.508664 7.491336 3.90 31 .O 15-5 18 o 319.6 2.504607 7-495393 3-93 3i.3 15.6 io 316.7 2.500648 7.499352 3-97 31.6 15.8 20 3'3-9 2.496791 7-503209 4.01 31.9 15.9 30 311*1 2.492900 7.5O7IOO 4.04 32.1 16.1 40 308.3 2.488974 7.511026 4.08 32-4 16.2 5° 305.6 2.485153 7.514847 4.12 32.7 16.4 19 0 302.9 2.481299 7.518701 4.15 33-0 16.5 IO 300.3 2-47755S 7.522445 4.19 33-3 l6.6 20 297.8 2-473925 7.526075 4.23 33.6 16.8 30 295.2 2.4701l6 7.529884 4.26 33-9 16.9 40 292.8 2.46657I 7.53'3429 4.30 34.2 17.1 50 290.3 2.462847 7-537I53 4-34 34-4 17.2 20 0 '' 287.9 2.459242 7.540758 4-37 34-7 17.4TABLE XI. SQUARES, CUBES, ETC., OF NUMBERS FROM 1 TO 1042.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1G 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 TABLE OP SQUARE AND CUBE ROOTS OF NUMBERS Cubes* I Square Roots. I Cube Roots* I Reciprocals* 1 1 0000000 10000000 •100000000 8 1-4142136 1-2599210 -500000000 27 1-7320508 1-4422496 •333333333 64 20000000 1-5874011 -250000000 125 2-2360680 1-7099759 •200000000 216 2-44948.17 1-8171206 •166666667 343 2-6457513 1-9129312 •142857143 512 2-8284271 2-0000000 -125000000 729 3-0000000 2-0800837 •111111111 1000 3-J 622777 2-1544347 •100000000 1331 3-3166248 2-2239801 •090909091 1728 3-4641016 2-2894286 •083333333 2197 3-6055513 2-3513347 •076923077 2744 3-7416574 2-4101422 •071428571 3375 3-8729833 2 4662121 •066666667 4096 4-0000000 2-5198421 •062500000 4913 4-1231056 2-5712816 •058823529 5832 4-2426407 2-6207414 •055555556 6859 4-3588989 2-6684016 •052631579 8000 4-4721360 2-7144177 •050000000 9261 4-5825757 2-7589243 •047619048 10648 4-6904158 2-8020393 •045454545 12167 4-7958315 2-8438670 •043478261 13824 4-8989795 2-8844991 •041666667 15625 5 0000000 2-9240177 •040000000 17576 5-0990195 2-9624960 -038461538 19683 5 1961524 3-0000000 •037037037 21952 5-2915026 3 0365889 •035714286 24389 5-3851648 3 0723168 •034482759 27000 5-4772256 31072325 •033333333 29791 5-5677644 3 1413806 •032258065 32768 5-6568542 31748021 •031250000 35937 5-7445626 3-2075343 •030303030 39304 5-8309519 3-2396118 •029411765 42875 5-9160798 3-2710663 •028571429 46656 6-0000000 3-3019272 •027777778 50653 6 0827625 3-3322218 •027027027 54872 6 1644140 3-3619754 •026315789 59319 6-2449980 3-3912114 •025641026 64000 6-3245553 3-4199519 •025000000 68921 6-4031242 3-4482172 •024390244 74088 6-4807407 3-4760266 •023809524 70507 6-5574385 3-5033981 •023255814 85184 6-6332496 3-5303483 •022727273 91125 6-7082039 3-5568933 •022222222 97336 6-7823300 3-5830479 •021739130 103823 6-8556546 3-6088261 •021276600 110592 6-9282032 3-6342411 •020633333 117649 7 0000000 3-6593057 •020408163 125000 7 0710678 3-6840314 •020000000No. 51 52 5.1 54 55 56 57 58 59 60 61 62 61 64 65 66 67 68 69 70 71 72 71 74 75 76 77 78 ' 79 80 81 82 81 84 85 86 87 88 89 90 91 92 91 94 95 96 97 98 99 100 iOl 102 101 104 105 106 107 108 109 110 ill 112 CUBES, ETC., OF NUMBERS. 163 Cubes. 112651 140608 148877 157464 166175 175616 185191 195112 205179 216000 226981 238328 250047 262144 274625 287496 300763 314432 328509 341000 357911 373248 389017 405224 421875 438976 456513 474552 493039 512000 531441 551168 571787 592704 614125 636056 658503 681472 704969 729000 753571 778688 804357 830584 857375 884736 912673 941192 970299 1000000 1030101 1061208 1092727 1124864 1157625 1191016 1225043 1259712 1295029 1331000 1367631 1404928 Square Roots. 71414284 7-2111026 7-2801099 7-1484692 7-4161985 7 4833148 7-5498144 7-6157711 7-6811457 7-7459667 7-8102497 7-8740079 7- 9172519 8*0000000 8 0622577 8- 1240384 8-1851528 8-2462113 8-1066239 8-3666003 8-4261498 8-4852814 8-5440037 8-6023253 8 6602540 8-7177979 8-7749644 8-8117609 8-8881944 8- 9442719 9- 0000000 9 0553851 9 1104316 91651514 92195445 9-2716185 9 9271791 9-3808315 9-4139811 9 4868310 95393920 9-5916630 96416508 96953597 97467943 9-7979590 9-8188578 98994949 99498744 10 0000000 10 0498756 10 0995049 10 1488916 10 1980390 192469508 192956301 193440804 193923048 10-4403065 10-4880885 195356538 195830052 Cube Roots, 3-7084298 3-71251II 3-7502858 3-7797611 3-8929525 3-8258624 3-8485011 3-8708766 3-8929965 3-9148676 3-9334972 3-9578915 3- 9790571 4- 0000000 4 0207256 4 0412401 4-0615480 4-0816551 4 1015661 4 1212853 4 1408178 4 1601676 4 1793390 41983364 4-2171633 4 2358236 4-2543210 4-2726586 4-2908404 4-1088695 4-3-267487 4 1444815 4-9021)707 4 3795191 4-1908296 4-4140049 4-4310476 4-4479602 4-4647451 44814047 4-4979114 45143574 4-5306549 4-5468359 4-5629026 4-5788570 4-5947009 4-6104163 4-6-260650 4-6415888 4-6570095 4-6723287 4-6875182 4-7026694 4-7176940 4-7320235 4-7474594 4-7622032 4-7768562 4-7914199 4-8058995 4-8202845 Reciprocals. •019607843 •019210769 •018867925 •018518519 •018181818 •017857143 •017541860 •017241379 •016949153 ■016666667 016391443 016129032 •015871016 •01562.5000 015384615 015151515 •014925173 014705882 •014492754 •01428.5714 •014084517 •013888889 -013698630 •013513514 •013131333 •013157895 •012987013 •012820513 ■012658228 •012500000 •012:145679 012195122 012048191 •011904762 •011764706 •011627907 *011494253 -011161616 •0112:15955 ■0111 Hill •010989011 ■010869565 010752688 •0106:18298 -010526316 •010416667 •010309278 •010204082 -010101010 0100000110 •009900990 009803922 •0097087:18 009615:185 -009523810 009433962 -009345794 •009259259 •009174312 •009090909 •009009009 •008928571164 SQUARES, CUBES, ETC., OF NUMBERS. No. Square#. Cubes. Square Root9. Cube Roots. Reciprocal#. J13 12769 1442897 10 6301458 4-8345881 •008849558 114 12996 1481.544 10-6770783 4-8488076 •008771930 115 13225 1520875 10 7238053 4-8629442 •008695652 JIG 13456 1560896 10-7703296 4-8769990 ■008020690 117 13689 1601613 10‘R1fifi538 4-8909732 •008547009 118 13924 1643032 10-8627805 4-9048681 •008474576 J19 14161 1685159 10-9087121 4-9186847 •008403361 120 14400 1728000 10 9544512 4-9324242 •008333333 121 14641 1771561 H-0000000 4-9460874 •008264463 122 14a34 15129 1815848 11-0453610 4-9596757 ■008196721 123 1860867 11 0905365 4-9731898 •008130081 124 15376 1906624 ]| 1355287 4-9866310 •008064516 125 15625 1953125 11-1803399 5-0000000 •008000000 126 15876 2000376 11-2249722 11-2694277 5-0132979 •007936508 127 16129 2048383 5-0265257 •007874016 128 16384 2097152 11 3137085 5-0396842 •007812500 129 16641 2146689 11-3578167 5 0527743 •007751938 130 16900 2197000 11-4017543 5-0657970 •007692303 131 17161 2248091 ] 1-4455231 5-0787531 •00763.1588 132 17424 2299968 11 4891253 5-0916134 •007575758 133 17689 2352637 ] 1-5325626 5-1044687 •007518797 134 179.56 2406104 11-5758369 5-1172299 •007462687 1X5 18225 2460375 11-6189500 5-1299278 •007407407 136 18496 251.54.56 11-6619038 5-1425632 •0073.52941 137 138 18769 19044 2571353 2628072 11-7046999 11-7473401 5-1551367 ’ 5-1676493 •007299270 •007246377 •007194245 139 19321 2685619 11-7898261 5-1801015 140 19600 2744000 11 8321596 5-1924941 •007142857 141 19881 2803221 11-8743421 5-2048279 •007092199 142 20164 2863288 11-9163753 5-2171034 •00704-2254 143 144 20449 20736 2924207 2985984 11'9.582607 12-0000000 5-2293215 5-2414828 ' -006993007 ■006944444 145 21025 3048625 12-0415946 5-2535879 •006896552 146 21316 3112136 12 0830460 5-2656374 •006849315 147 21609 3176523 12 1243557 5-2776321 •006802721 148 149 21904 22201 3241792 3307949 12 1655251 12 2065556 5-2895725 5-3014592 •006756757 •006711409 150 22500 3375000 12-2474487 5-3132928 •006666667 151 22801 3442951 12-2882057 5-3250740 •006622517 152 23104 3511808 12-3288280 5-3368033 •006578947 153 23409 3581577 12-3693169 5-3484812 •006535948 154 23716 3652264 12-4096736 5-3601084 •006493506 155 24025 3723875 12-4498996 5 3716854 •006451613 156 24336 3796416 12-4899960 5-3832126 •006410256 , 157 24649 3869893 12-5299641 5-3946907 •006369427 ! 158 24964 3944312 12 5698051 5-4061202 •006329114 159 25281 4019679 12-6095202 5-4175015 •006289308 160 25600 4096000 12 6491106 5-4288352 •006250000 161 25921 4173281 12-6885775 5-4401218 •006211180 ■006172840 162 26244 4251528 12-7279221 5-4513618 163 26569 4330747 12-7671453 5-4625556 •006134969 164 26896 4410944 12-8062485 5-4737037 •006097561 165 27225 4492125 12-8452326 5-4848066 •006060606 '166 27556 4574296 12-8840987 5-4958647 •006024096 167 27889 4657463 12-9228480 5-5068784 •005988024 168 28224 4741632 129614814 5-5178484 •005952381 169 28561 4826809 13 0000000 5-5287748 •005917160 170 28900 4913000 13 0384048 5-5396583 •005882353 171 29241 5000211 13 0766968 5-5504991 •005847953 172 29584 5088448 13 1148770 5-5612978 •005813953 173 29929 5177717 13-1529464 5-5720546 •005780347 174 30276 5268024 13-1909060 5-5827702 •005747126SQUARES, CUBES, ETC., OF NUMBERS.. 165 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 175 30625 5359375 13-2287566 5-5934447 •005714286 176 30976 5451776 13-2664992 5-6040787 •005681818 177 31329 5545233 13-3041347 5-6146724 •005649718 178 31684 5639752 13 3416641 5 6252263 •005617978 179 32041 5735339 13-3790882 56357408 •005586592 180 32400 5832000 13-4164079 5-6462162 •005555556 381 | 32761 5929741 13-4536240 5-6566528 •005524862 182 33124 6028568 13-4907376 5 6670511 •005494505 183 33489 6128487 13 5277493 5-6774114 •005464481 184 33856 6229.504 13-5646600 5-6877340 •005434783 185 34225 633I625 13-6014705 56980192 •005405405 18C 34596 6434856 13-6381817 5-7082675 •005376344 187 34969 6539203 136747943 5-7184791 ■005347594 188 35344 6644672 137113092 5-7286543 •005319149 189 35721 6751269 13-7477271 5-7387936 •005291005 190 36100 6859000 13-7840488 5-7488971 •005263158 191 36481 6967871 138202750 5-7589652 •005*235602 192 36864 7077888 13-8564065 5-7 >89982 5-7789966 5-7889604 •005208333 193 37249 7189017 13-8924440 •005181347 194 37636 7301384 13-928:1883 •005154639 195 38025 7414875 139642400 5-7988900 •005128205 19G 38416 75295:16 14 0000000 5-8087857 •005102041 197 38809 7645373 140356688 5-8186479 5-8284767 •005076142 198 39204 7762392 14 071-2473 •005050505 199 39601 7880599 14-1067360 5 8382725 •005025126 200 40000 8000000 14 1421356 5 8480355 •005000000 201 40401 8120601 14 1774469 5-8577600 *04975124 202 40804 8242408 14-2126704 5 8674643 *04950495 203 41209 8365427 14-2478068 5-8771307 •004926108 204 41616 8489C64 14 2828569 58867653 •004901961 205 206 42025 42436 8015125 8741816 14-3178211 14-3527001 5-8963685 59059406 •004878049 •004854369 207 42849 8869743 14 3874946 5-9154817 •004830918 208 43264 8998912 144222051 5 9249921 •004807692 209 43681 9129329 14-4568323 5-9344721 •004784689 210 44100 9261000 14-4913767 5-9439220 •004761905 211 44521 9393931 145258390 5-9533418 5-9627320 •004739336 212 44944 9528128 145602198 •004716981 213 45369 9663597 145945195 5-9720926 •004694836 214 45796 9800344 14-6287388 5-9814240 •004672897 215 4G225 9936375 14-6628783 5-9907264 •004651163 216 46656 10077696 14G9G9385 c-ooooooo •004629630 217 47089 10218313 14 7309199 6 0092450 •004608295 218 219 47524 47961 10360232 10503459 14-7G48231 14-7986486 60184617 6 0276502 •1X14.587156 •004566210 220 48400 10648000 14-8323970 6 0368107 •004545455 221 48841 10793861 14-8660687 60459435 •004524887 OOP 49284 10941048 14-8996644 60550489 •004504505 223 49729 11089567 14-9331845 6-0641270 •004484305 224 50176 11239424 14-9666295 60731779 •004464286 225 50625 11390625 15-OuOOOOO 6 0822020 •004444444 226 51076 11543176 15 0332964 6 0911994 •004424779 227 51529 11097083 150665192 61001702 •004405286 228 51984 11852352 15 0996689 6-1091147 •004385965 229 52441 12008989 15 1327460 6 1180332 •004366812 230 52900 12167000 15-1657509 6-1269257 •004347826 231 53361 12326391 15 1986642 6 1357924 •004329004 232 511824 12487168 15 2315462 G-1446337 •004310345 233 54289 12649337 15-2643375 6-1534495 •004291845 234 54750 12812904 15-2970585 6-1622401 •004273504 235 55225 12977875 15-3297097 - 61710058 •004255319 236 55696 13144256 15-3622915 61797466 •004237288166 No. 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 CUBES, ETC., OF NUMBERS. Cubes. 13312053 13481272 13651919 13824000 13997521 14172488 14348907 14526784 14706125 14886936 15069223 15252992 15438249 15625000 15813251 16003008 16194277 16387064 16581375 16777216 16974593 17173512 17373979 17576000 17779581 17984728 18191447 18399744 18609625 18821096 19034163 19248832 19465109 1968:1000 19902511 20123648 20346117 20570824 20796875 21024576 21253933 21484952 21717639 21952000 22188041 22425768 22665187 22906304 23149125 23393656 23639903 23887872 24137569 24389000 24642171 24897088 25153757 25412184 25672375 25934336 26198073 26463592 Square Roots. 1 Cube Roots. I Reciprocals. 15 3948043 15-4272486 15-4596248 15-4919334 15-5241747 15-5563492 15-5884573 15-6204994 15-6524758 15-6843871 15 7162336 15-7480157 15-7797338 15-8113883 15-8429795 15-8745079 15-9059737 15-9373775 15- 9687194 16- 0000000 16-0312195 16 0623784 16 0934769 16 1245155 16 1554944 161864141 162172747 16-2480768 16-2788206 16-3095064 16-3401346 16-3707055 16 4012195 16-4316767 16-4620776 16-4924225 16-5227116 16-5529454 165831240 16-6132477 16-6433170 16-673332) 16-7032931 16-7332005 16-7630546 16-7928556 16-8226038 16-8522995 16 8819430 16-9115345 16-9410743 16- 9705627 17- 0000000 17-0293864 17-0587221 17-0880075 17-1172428 17-1464282 17-1755640 17-2046505 17-2336879 17-2626765 6-1884628 6-1971544 6-2058218 6-2144050 6-2230843 6-2316797 6-2402515 6-2487998 6-2573248 6-2658206 6-2743054 6-2827613 6-2911946 6-2996053 6-3079935 6-3163596 6-3247035 6-3330256 6-3413257 6-3496042 6-3578611 6-3660968 6-3743111 6-3825043 6-3906765 6-3988279 6-4069.585 6-4150687 6-4231583 6-4312276 6-4392767 6-4473057 6-4553148 6-4633041 6-4712736 6-4792236 6-4871541 6-4950653 6-5029572 6-5108300 6-5186839 6-5265189 6-5343351 6-5421326 6-5499116 6-5576722 6-5654144 6-5731385 6-5808443 6-5885323 6-5962023 66038545 66114890 66191060 6-0267054 6 6342874 6-6418522 6-6493998 6 6569302 6 0644437 6 6719403 6-6794200 •004219409 -004201681 -004184100 •004106667 •004149378 •004132231 •004115226 •004098361 •004081633 •004065041 •004048583 •0040.32258 ■004016064 •004000000 •003984064 •003968254 •003952569 •003937008 •003921569 •003906250 •00:1891051 •003875969 •003801004 •003846154 •003831418 •003816794 ■003802281 •003787879 •003773585 •003759398 •003745318 •003731343 •003717472 •003703704 •003690037 •00:1676471 •003663004 •003649635 •003636364 •003623188 •003610108 •003597122 •003.584229 ■003571429 •003558719 •003546099 •00:1533569 ■003522127 ■003508772 •003496503 •003484321 •003472222 •003460208 •003448276 •003436426 •003424658 ■003412969 •003401361 •003389831 •003378378 •003367003 •003355705SQUARES, CUBES, ETC., OF NUMBERS. 167 No. | Squares. | Cubes. Square Roots. | Cube Roots. Reciprocals. 209 89401 26730899 17-2916165 6-6868831 •003344482 " 300 90000 27000000 17-3205081 6-6943295 •003333333 301 90001 27270901 17-3493516 6-7017593 •003322259 302 91204 27543608 17-3781472 6-7091729 •003311258 303 91809 27318127 17-4068952 6-7165700 ■003301330 304 92416 28094464 17-4355958 6-7239508 •003289474 305 93025 28372625 17-4642492 6-7313155 •003278689 396 93636 28652616 17-4928557 6-7386641 •003267974 307 94249 28934443 17-5214155 6-7459967 *•003257329 308 94804 29218112 17-5499288 6-7533134 •003-240753 309 95481 29503629 17-5783958 6-7606143 •003230246 310 96100 29791000 17-6068169 6-7678995 •003225806 311 9G721 30080231 17-6351921 6-7751690 •003215434 312 97344 30371328 17-6635217 6-7824229 •003205128 313 97969 30664297 17-6918060 6-7896613 •003194888 314 98596 30959144 17-7200451 6 7968844 •003184713 315 99225 31255875 17-7482393 6-8040921 ■003174603 310 99856 31554496 17-7763883 6-8112847 •003164557 317 100489 31855013 17-8044938 6-8184G20 •003154574 318 101124 32157432 17-8325545 6-8256242 •003144654 319 101761 32461759 17-8605711 6-8327714 •003134796 320 102400 32768000 17-8885438 6-8399037 •003125000 321 103041 33076161 17-9164729 6-8470213 ■003115265 322 103684 33386248 17-9443584 6-8511240 •003105590 323 104329 33698267 17-9722008 6-8612120 •003095975 324 104976 34012224 18-0000000 6-8682855 ■003086420 325 105625 34328125 18-0277564 6-8753413 •00307G923 326 106276 34645976 18-0554701 6-8823888 003067485 327 106929 34965783 18-0831413 6-8894188 •003058104 328 107584 35287552 18-1107703 6-8964345 •003048780 329 108241 35611289 18-1383571 6-9034359 •003039514 330 108900 35937000 18-1659021 6-9104232 •003030303 331 109561 36264691 18-1934054 6-9173964 •003021148 332 110224 36594368 18-2208672 6-0243556 •003012048 333 110889 36926037 18-2482876 6-9313008 •003003003 334 111556 37259704 18-2756G69 6-9382321 •002994012 335 112225 37595375 18-3030052 G-9451496 •002985075 336 112896 37933056 18-3303028 6-9520533 •002976190 337 113569 38272753 18-3575598 6-9589434 ■002967359 338 114244 38614472 18-3847763 6-9658198 •002958580 339 114921 38958219 18-4119526 6-9726826 •002949853 340 115600 39304000 18-4390889 6-9795321 •002941176 341 116281 39651821 18 4661853 6-9863681 •002932551 342 116964 40001688 18-4932420 6-9931906 •002923977 343 117649 40353607 18-5202592 7-0000000 •002915452 344 118336 40707584 18-5472370 7-0067962 •002906977 345 119025 41063625 18-5741756 70135791 •002898551 346 119716 41421736 18-6010752 7-0203490 •002890173 347 120409 41781923 18-6279360 70271058 •002881844 348 121104 42144192 18-6547581 7(1338497 •002873563 349 121801 42508549 18-6815417 7-0405806 ■002865330 350 122.500 42875000 18-7082869 7-0472987 •002857143 351 123201 43243551 18-7349940 7-0540041 •002849003 352 123904 43614208 18-7616630 7-0606967 •002840909 353 124609 43986977 18-7882942 7-0673767 •002832861 354 125316 44361864 18-8148877 7-0740440 •002824859 355 126025 44738875 18-8414437 7-0806988 •002816901 356 126736 45118016 18-8679623 7 0873411 •002808989 357 127449 45499293 18-8944436 7 0939709 •002801120 358 128164 45882712 18-9208879 7 1005885 •00279:1296 359 128881 46268279 18-9472953 71071937 j •002785515 360 129600 46656000 18-9736660 71137866 1 ■008777778168 SQUARES, CUBES, ETC., OF NUMBERS. No. Square* I Cubes. Square Roots. Cube Roots. Reciprocal* 361 130321 47045881 19 0000000 71203674 •002770083- 362 131044 47437928 19 0262976 71269360 •002762431 363 131769 47832147 19 0525589 7 1334925 •002754821 364 132496 48228544 19 0787840 7-1400370 •002747253 365 133225 48627125 191049732 71465695 •0027:19726 366 133956 49027896 191311265 7-1530901 •002732240 367 134689 494.10863 19 1572441 7-1595988 •002724796 368 135124 49836032 19' 1833261 7-1660957 •002717391 363 1317161 5024:1409 192093727 71725809 •002710027 370 136900 50653000 192353841 71790544 •002702703 371 137641 51064811 192613603 7-1855162 •002695418 372 138384 51478848 192873015 7-1919663 •002688172 373 139129 51895117 193132079 7-1984050 •002680965 374 139876 52313024 19 3390796 7 2048322 ■002673797 375 140625 52734375 19 3649167 7-2112479 •002066667 376 141376 53157370 19 3907194 7 2176522 •002659574 377 142129 53582633 194164878 72240450 •002652520 378 142884 54010152 194422221 7-2304268 •002645503 379 143641 54439939 19 4079223 7 2367972 •002638321 380 144400 54872000 19 4935887 7 2431565 •002631579 381 145161 55306341 195192213 72495045 •002G24G72 382 145924 55742968 19 5448203 7 2558415 002017801 383 146689 56181887 19 5703858 7 2021675 •002C10960 384 147456 56623104 195959179 7-2684824 •002G041G7 385 148225 57066625 19 6214169 7-2747864 •002597403 386 148996 57512456 19 6468827 7-2810794 •002590674 387 149769 57960603 196723156 7-2873617 •002583979 •002577320 388 150544 58411072 19 0977156 72936330 389 151321 58863809 19 7230829 7-2998936 •002570694 390 152100 59319000 19 7484177 197737199 7-3061436 •002564103 391 152881 59776471 7-3123828 •002557545 392 153664 60236288 197989899 7-3186114 ■002551020 393 154449 60698457 19 8242276 7-3248295 •002544529 394 155236 61162984 19 8494332 733103G9 •0:)253807l 395 156025 61629875 19 8746069 7-3372339 •002531646 396 156816 62099136 198997487 7-3434205 •002525253 397 157609 62570773 19 9248.588 7-3495966 •002518892 398 158404 63044792 199499373 7-3557624 ■00251-2563 5,99 159201 6:1521199 19 9749844 7-3619178 •002506266 400 160900 64000000 20 0000000 7-3680630 •002500000 401 160801 64481201 20 0249844 7-3741979 •002493766 402 161604 64964808 20 0499377 7-3803227 •002487562 403 162409 65450827 20 0748599 7-3864373 •002481390 404 163216 65939264 20 0997512 7-3925418 •002475248 405 164025 66430125 20 1246118 7-3986363 002469130 406 164836 66923416 20 1494417 7-4047206 •002463054 407 165649 67419143 20 1742410 74107950 002457002 408 166464 67917312 20 1990099 7-4168595 •002450980 409 167281 68417929 20 2237484 7 4229142 •002444988 410 168100 68921000 20 2484507 7-4289589 •002439024 •002433090 411 168921 69426531 202731349 7 4349938 412 169744 69934528 20-2977831 7-4410189 •002427184 413 170569 70444997 20 3224014 74470342 •002121308 414 171396 70957944 20 3469899 7 4530399 •002115459 415 172225 71473375 20 3715488 7 4590359 ■002409639 416 173056 71991296 20-3960781 7 4650223 002403846 417 173889 72511713 20-4205779 7-4709991 •0023980 130055 3858 7037 140194 3327 140j140128 146438 l| 92191 9527 2]152288 152594 3 5330 4 8302 114277 7003 114011 7934 120903 121231 4178 4504 5 161308 6 4353 7 7317 8 170202 170555 9 3186 3478 5040 8604 161667 4050 7613 150:176091 1 8977 181844 4691 7521 176381 9264 182129 4975 7803 5 190332 190012 6j 3125 3403 7| 59001 6176 8' 80571 8932 9 201397i201670 7753 130977 4177 7354 140508 3039 146748 147058 9835 150142 5600 9870 014100 8284 022428 0533 030000 4028 8020 042576 0495 050380 4230 8040 061829 5580 9298 072985 0040 080200 3801 7420 090903 4471 7951 101403 4828 8227 111599 114944 8205 121560 4830 8070 1312981 4490 7071 140822 3951 001734 002106 0025981003029 0038 0460 6894 01030O 010724 011147 452] 4940 5300 8700 9110' 9532 022841 023252 0230144 6942 7350 7757 031004 031408 031812 5029 5430j 5830 9414 9811 7321 011570 5779 9947 024075 8104 032210 6230 040207 044148 8053 051924 57(H) 9503 9017 j 042969 043362'#43755 68851 7275: 7004 050766 051153 0515:18 46131 4996! 5379 8426! 8805' 9195 062206 062582 062959 063333 5953! 0320! 6099 7071 9008 070038 070407 073352; 37181 4085 7004 7308 7731 080020 080987 081347 4219; 4576 4934 77811 8130 8490 070770 4451 8094 081707 5291 8845 09J315 091007,092018 092370 092721 4820, 5169 5518 5806 0215 8298 8044| 8990 101747,102091,102434 5109! 5510 5851 8505' 8903| 9241 111934|112270|112005 115278 115011:115943 8595 8920 121889 5150 8399 9256 1222101122544 5481 5800 8722 9045 152900 5943 8905 101907 4947 7908 170848 3709 176070 9552 182415 5259 8084 190892 3681 6453 9206 201943 3205 0240 9200 102200 5244 8203 171141 40(H) 176959 9839 182700 5542 8306 191171 3959 6729 9481 202210 131019 1319391132200 48141 51331 5451 7987 '83031 8018 141130 141450,141703 4203 147307 150449 3510 6549 9507 102564 5541 8497 171434 4351 177248 180120 2985 5825 8047 191451 4237 7005 9755 202488 4574 4885 003401 7748 011993 6197 020301 4480 8571 032019 0029 040002 044540 8442 052309 42 U940 063709 7443 071145 4810 8457 082067 5647 9198 102777 103119 6191 9579 112940 110270 9580 122871 6131 9308 132580 5709 8934 142070 5196 1476701147985 148291 150750 151003 151370 3815 6852 9808 162803 5838 8792 171720 4041 177536 180413 3270 0108 8928 191730 4514 7281 2000*29 2701 4120 7154 100108 3101 6134 9080 4424 7457 100409 3400 6430 9380 172019 172311 4932 5222 177825 180099 3555 6391 9209 192010 4792 7556 Wo, I O I 1 I a I 3 I \ I 5 I 6 178113 180980 3839 6674 9490 192289 5009 7832 200303 200577j200850 ,3033| 3305| 3577 _ 0531 9910 113275 110608 9915 123198 0450 9090 132900 6086 9249 142:189 5507 148003 151070 4728 7759 1G0709 3758 0720 9074 172003 5512 178401 181272 4123 0950 9771 192507 5346 8107 0038911432 8174j428 0124151424 66161420 020775|410 4890:412 89781408 033021 404 70281400 0409981397 044932 393 8830;390 052094 38G 0524 383 000320 379 40831370 7815373 071514:370 51821366 88191363 082420! 360 0004 357 9552 355 093071 352 0502:349 1000201346 3402,343 6871 341 110253 338 3009 335 110940:333 1202451330 3525 328 67811325 130012;323 3219:321 6403 318 9564 316 142702 314 5818:311 148911 309 151982 307 5032 305 8001 303 101068 301 4055 299 7022 297 9968 295 172895 293 5802 291 178689 289 181558 287 4407 285 7239 283 190051 281 2846 279 5023 278 8382 276 201124 274 3848 272 8182 LOGARITHMS OF NUMBERS. M>l0|l|g[3|4|5|6|7|8|9|PiE 160 204120 2043911204603 204934 205204 205475 205746 206016 206286 206556|271 7096 7365 7634 7904 8173 8441 8710 8979 9247 269 1 2 3 4 5 6 7 8 9 170 1 2 3 4 5 6 7 8 9 180 1 2 3 4 5 6 8 9 6826 9515 212188 212454 190 1 o 3 4 4814 7484 220108 2716 5309 7887 239449 2996 5528 8046 240549 3038 5513 7973 250420 2853 255273 7079 260071 2451 4818 7172 9513 271842 4158 6462 278754 281033 3301 5557 7802 5 290035 2256 4466 6665 8853 200 1 2 3 4 5 6 7 8 9 210 1 2 3 4 5 6 7 8 9 5109 7747 220370 2976 5568 8144 230704 3250 5781 8297 240799 3286 5759 8219 250664 3096 255514 7918 260310 2688 5054 7406 9746 272074 4389 6692 278982 281261 3527 5782 8026 290257 2478 4687 6884 9071 301247 3412 5560 7710 9843 311966 4078 6180 8272 320354 301030 3196 5351 7496 9630 311754 3867 5970 8063 320146 322219 322426 4282 6336 8380 330414 2438 4454 6460 8456 340444 4488 6541 8583 2720 5373 8010 220631 3236 5826 8400 230960 3504 6033 8548 241048 3534 6006 8464 250908 3338 255755 8158 260548 2925 5290 7641 9980 272306 4620 6921 279211 281488 3753 6007 8249 290480 2699 4907 7104 9289 301464 3628 5781 7924 310056 2177 4289 6390 8481 320562 322633 4694 6745 8787 210319 2986 5638 8273 220892 3496 6084 8657' 231215 3757 6285 8799 241297 3782 6252 8709 251151 3580 255996 8398 260787 261025 330617 330819 2640 4655 6660 8656 340642 2842 4856 6860 8855 340841 210586 3252 5902 8536 221153 3755 6342 8913 231470 4011 0537 9049 241546 4030 6499 8954 251395 3822 256237 8637 3162 5525 7875 270213 2538 4850 7151 279439 281715 3979 6232 8473 290702 2920 5127 7323 9507 301681 3844 5996 8137 310268 2389 4499 6599 8689 320769 322839 4899 6950 8991 331022 3044 5057 7060 9054 341039 3399 5761 8110 270446 2770 5081 7380 279667 281942 4205 6456 8696 290925 3141 5347 7542 9725 301898 4059 6211 8351 210853 3518 6166 8798 221414 4015 6600 9170 231724 4264 6789 9299 241795 4277 6745 9198 251638 4064 256477 8877 261263 3636 5996 8344 270679 3001 5311 7609 279895 282169 4431 6681 8920 291147 3363 5567 7761 211121 3783 6430 9060 221675 4274 6858 •9426 231979 4517 7041 9550 242044 4525 6991 9443 251881 4306 256718 9116 261501 3873 6232 8578 270912 3233 5542 7838 280123 2396 4656 6905 9143 291369 3584 5787 7979 9943 300161 302114 4275 6425 8564 310481 310693 2600 4710 6809 8898 320977 323046 5105 7155 9194 331225 3246 5257 7260 9253 341237 2812 4920 7018 9106 321184 323252 5310 7359 9398 331427 3447 5458 7459 9451 341435 211388 4049 6694 9323 221936 4533 7115 9682 232234 4770 7292 9800 242293 4772 7237 9687 252125 4548 256958 9355 261739 4109 6467 8812 271144 3464 5772 8067 280351 2622 4882 7130 9366 291591 3804 6007 8198 300378 211654 4314 6957 9585 222196 4792 7372 9938 232488 5023 7544 211921 4579 7221 9846 222456 5051 7630 230193 232742 5276 7795 240050 240300 302331 4491 6639 8778 310906 3023 5130 7227 9314 321391 323458 5516 7563 9601 331630 3649 5658 7659 9650 341632 302547 4706 6854 8991 311118 3234 5340 7436 9522 321598 323665 5721 7767 9805 331832 3850 5859 7858 9849 341830 2541 5019 7482 9932 252368 4790 257198 9594 261976 4346 6702 9046 271377 3696 6002 8296 280578 2849 5107 7354 9539 291813 4025 6226 8416 300595 302764 4921 7068 9204 2790 5266 7128 250176 2610 5031 257439 9833 262214 4582 6937 9279 271609 3927 6232 8525 280806 228 267 266 264 262 261 259 258 256 255 253 252 250 249 248 246 245 243 242 241 239 238 237 235 234 233 232 230 229 3075 5332 7578 9812 292034 4246 6446 8635 300813 302980 5136 7282 9417 311330 311542 3445 5551 7646 9730 321805 323871 5926 7972 330008 2034 4051 6059 8058 340047 2028 3656 5760 7854 9938 322012 324077 6131 8176 330211 2236 4253 227 226 225, 223 222 221 220 219 218 217 216 215 213 212 211 210 209 208 207 206 205 204 203 202 202 62601201 8257 200 3402461199 2225|l98 No. | 0 I 1 l » | 3 | 4 | 5 | G | 7 | 8 9 ' Ditt.LOGARITHMS OF NUMBERS. 183 No. o 1 f a 3 4 5 6 7 8 9 DUE 220 342423 4392 342620 4589 342817 4785 343014 4981 343212 5178 343409 5374 343606 5570 343802 5766 343999 5962 344196 6157 197 196 2 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 195 3 8305 8500 8694 8889 9083 9278 9472 9666 9860 350054 194 4 350248 350442 350636 350829 351023 351216 351410 351603 351796 1989 193 5 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 6 4108 4301 4493 4685 4876 5068 5260 5452 5643 5831 192 7 6026 6217 6408 6599 6790 6981 7172 7363 7554 7744 191 8 7935 8125 8316 8506 8696 8886 9076 9206 9456 9646 190 9 9835 360025 360215 360404 360593 360783 360972 361161 361350 361539 189 230 361728 3012 361917 362105 362294 362482 362671 362859 363048 363236 363424 188 1 3800 3988 4176 4363 4551 4739 4926 5113 5301 188 2 5488 5675 5862 6049 6236 6423 6610 6796 6983 7169 187 3 7356 7542 7729 7915 8101 8287 8473 8659 8845 9030 186 4 9216 9401 9587 9772 9958 370143 370328 370513 370698 370883 185 5 371068 371253 371437 371622 371806 1991 2175 2360 2544 2728 184 C 2912 3096 3280 3464 3647 3831 4015 4198 4:182 4565 184 7 4748 4932 5115 5298 5481 5664 5840 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 380030 181 240 380211 380392 380573 380754 380934 381115 381296 381476 381656 381837 181 1 2017 2197 2377 2557 2737 2917 3097 3277 3456 3630 180 2 3815 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 3 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 4 7390 7568 7746 71-23 8101 8279 8456 8634 8811 8989 178 5 9166 9343 9520 9698 9875 390051 390228 390405 390582 390759 177 6 390935 391112 391288 391464 391641 1817 1993 2169 2345 2521 176 7 2697 2873 3048 3224 3400 3575 3751 3926 4101 4277 176 8 4452 4627 4802 4977 5152 5326 5501 5676 5850 6025 175 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 397940 398114 398287 398461 398634 398808 398981 399154 399328 399501 173 1 9674 9847 400020 400192 400365 400538 400711 400883 401056 401228 173 2 401401 401573 1745 1917 2089 2261 2433 2605 2777 2949 172 3 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 171 4 4834 5005 5170 5346 5517 5688 5858 6029 6199 6370 171 5 6540 6710 6881 7051 7221 7391 7561 7731 7901 8070 170 6 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 7 9933 410102 410271 410440 410609 410777 410946 411114 411283 411451 169 8 411620 1788 1956 2124 2293 2461 2629 2796 2964 3132 168 9 3300 3467 3635 3803 3970 4137 4305 4472 4639 4806 167 260 414973 415140 415307 415474 415641 415808 415974 416141 416308 416474 167 i 6641 6807 6973 7139 7306 7472 7638 7804 7970 8135 166 2 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 165 3 9956 420121 420286 420451 420616 420781 420945 421110 421275 421439 165 4 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 164 5 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 6 4882 5045 5208 5371 5534 5697 5860 6023 6186 6349 163 7 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 162 8 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 162 9 9752 9914 430075 430236 430398 430559 430720 430881 431042 431203 161 270 431364 431525 431685 431846 432007 432167 432328 432488 432649 432809 161 1 2969 3130 3290 3450 3610 3770 3930 •4090 4249 4409 160 2 4569 4729 4888 5048 5207 5307 5526 5685 5844 6004 7592 159 3 6163 6322 6481 6640 6799 6957 7110 7275 7433 159 4 7751 7909 8067 8226 8381 8542 8701 8859 9017 9175 158 5 9333 9491 9648 9806 9964 440122 440279 440137 440594 440752 158 6 440909 441066 441224 441381 441538 1695 1852 2009 2166 2323 157 7 24f“> 2637 2793 2950 *3106 3263 3419 3576 3732 3883 157 £ 4045! 4201 4357 4513 4669 4825 4981 5137 5293 5443 156 9 5604 1 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 No. I 0 1 1 Ji 3 4 5 1 6 7 1 8 1 9 1 DUE184 LOGARITHMS OF NUMBERS. No. o 1 a 3 4 5 G 7 8 9 Diff. 280 447158 447313 447408 447G23 447778 447933 448083 448242 448397 448552 155 1 870G 8801 9015 9170 9324 9478 9033 9787 1941 450095 154 o 450249 450403 450557 459711 450805 451018 451172 45132G 451479 1633 154 3 178G 1940 2093 2247 240.3 2553 2700 2859 3012 3165 153 4 3318 3471 3524 3777 3930 4082 4235 4387 4540 4G92 153 5 4845 4997 5150 5302 5454 5500 5758 5910 6062 6214 152 0 G3GG 6518 6070 6821 6973 7125 7270 7428 7579 7731 152 7 7882 8033 8184 8330 8187 8038 8789 8940 9091 9242 151 8 9392 9543 9534 9845 9395 4G0146 400200 460447 460597 4G0748 151 9 4G3898 461048 4G1198 461348 461499 1649 1799 1948 2098 2248 150 293 462338 462548 402537 402847 4G2997 403140 403290 463445 463594 463744 150 1 3893 4042 4191 4340 4490 4039 4788 4936 5085 5234 149 a 5383 5532 5580 5829 5977 0120 0274 6423 6571 6719 149 3 G8G8 7010 7104 7312 7400 7008 7750 7904 8052 8200 148 4 R14? 8495 8543 8793 8938 9085 9233 9380 9527 9675 148 5 9822 9909 470110 470263 470410 470557 470704 470851 470998 471145 147 7193 7272 7352 7431 7511 7590 7070 7749 7829 7908 79 7 7987 8007 8146 8225 8305 8384 8403 8543 8022 8701 79 8 8781 8800 8939 9018 9097 9177 9250 9335 9414 9493 79 9 9572 9651 9731 9810 9889 . 9968 740047 740120 740205 740284 79 550 740303 740442 740521 740600 740678 740757 740836 740915 740994 741073 79 1 1152 1230 1309 1388 1407 1546 1024 1703 1782 1800 79 o 1939 2018 2090 2175 2254 2332 2411 2489 2508 2047 79 3 2725 2804 2882 2901 3039 3118 3190 3275 3353 3431 78 4 3510 3588 3067 3745 3823 3902 3980 4058 4130 4215 78 5 4293 4371 4449 4528 4600 4084 4702 4840 4919 •4997 78 6 5075 5153 5231 5309 5387 5405 5543 5021 5099 5777 78 7 5855 5933 6011 6089 6107 6245 6323 6401 6479 0550 78 8 003-f 0712 6790 6868 6945 7023 7101 7179 7256 7334 78 9 7412 7489 7507 7645 7722 7800 7878 7955 8033 8110 78 500 748188 748260 748343 748421 748498 748576 748053 748731 748808 748885 77 1 8903 9940 9118 9195 9272 9350 9427 9504 9582 9059 77 2 9730 9814 9891 9908 750045 750123 750200 750277 750354 750431 77 3 750508 750580 750663 750740 0817 0894 0971 1048 1125 1202 77 4 1279 1350 1433 1510 1587 1004 1741 1818 1895 1972 77 5 2048 2125 2202 2279 2350 2433 2509 2586 2663 2740 77 6 2810 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 7 3583 3000 3736 3813 3889 3906 4042 4119 4195 4272 77 8 4348 4425 4501 4578 4054 4730 '4807 4883 4960 5030 70 9 5112 5189 5265 5341 5417 5494 5570 5046 5722 5799 76 no 755875 755951 750027 750103 750180 750256 750332 756408 750484 750560 76 i 0030 0712 6788 6804 0940 7010 7092 7168 7244 7320 76 2 7396 7472 7548 7024 7700 7775 7851 7927 8003 8079 76 3 8155 8230 8300 8382 8458 8533 8009 8685 8701 8836 76 4 8912 8988 9063 9139 9214 9290 9306 9441 9517 . 9592 76 5 9008 9743 9819 9894 9970 760045 700121 760190 760272 760347 75 6 760422 700498 700573 700049 760724 0799 0875 0950 1025 1101 75 7 1170 1251 1320 1402 1477 1552 1027 1702 1778 1853 75 8 1928 2003 2078 2153 2228 2303 2378 2453 2529 2661 75 9 2679 2754 2829 2904 2978 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4003 7 4139 4208 4276 4344 4412 4480 4548 4616 4685 8 4821 4889 4957 5025 5093 51G1 5229 5297 5365 9 5501 5569 5637 5705 5773 5841 5908 5976 6044 No. 0 1 a 1 3 4 5 6 7 I 8 | 9 | Diff. 75 75 75 7-1 74 74 74 74 74 74 4848 5594 6338 708i2 7823 8564 9303 770042 0778 771514 2248 2981 3713 4444 5173 5902 6629 7354 8079 778802 9524 780245 0965 1084 2401 3117 3832 4546 52^9 785970 6680 7390 8098 8804 9510 790215 0918 1620 2322 793022 3721 4418 5115 5811 6505 7198 7890 8582 9272 799961 800648 1335 2021 2705 3389 4071 4753 5433 6112 9 | D'tt ssssssss190 LOGARITHMS OF NUMBERS. Na 1 o 1 a 3 4 5 6 7 8 | 9 | 640 1 806180 6858 806218 6926 806316 6994 306384 7061 806451 7129 806519 7197 806587 7204 806655 7332 806723 7400 806790: 7467! 8143 o 7535 7603 7670 7738 7806 7873 7941 8008 8076 3 8211 8279 8346 8414 . 8481 8.549 8616 8684 8751 8818 4 8886 8953 9021 9088 9150 9223 9290 9358 9425 9492 5 9560 9627 9694 9762 9829 9896 9904 810031 810098 810165 6 8J0233 810300 810367 810434 810501 810569 810636 0703 0770 0837 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1503 8 1575 1642 1709 1776 1843 19'0 1977 2044 2111 2178 9 2243 2312 2379 2445 2512 2579 2640 2713 2780 2847 650 812913 3581 812980 3648 813047 813114 813181 813247 813314 813381 813448 813514 1 3714 3781 3848 3914 3081 4048 4114 4181 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 3 4913 4980 5046 5113 5179 5240 5312 5378 5445 5511 4 5578 ~ 5644 5711 5777 5843 5910 5970 6042 6109 6175 5 6241 6308 6374 0440 G506 0573 6039 0705 6771 0838 6 6904 6970 7030 7102 7169 7235 7301 7367 7433 7499 7 7565 8226 7031 7698 7704 7830 7896 7962 8028 8094 31G0 8 8292 8358 8424 8490 &556 8622 8088 8754 8820 9 8885 8951 9017 9083 9149 9215 9281 9340 9412 9478 6G0 819544 819610 8I9G70 819741 819807 819873 819939 820004 820070 820130 J 820201 820267 0924 820333 820399 820464 820530 820595 0601 0727 0792 2 0858 0989 1055 1120 1186 1251 1317 1382 1448 3 1514 1579 1645 1710 2304 1775 1841 1900 1972 2037 2103 4 2168 2822 2233 2299 2430 2495 2560 2020 2091 2750 5 2887 2952 3018 3083 3148 3213 3279 3344 3409 6 3474 3539 3605 3070 3735 3800 3865 3930 3990 4061 7 4126 4191 4256 4321 4386 4451 4510 4581 4640 4711 8 4776 4841 4906 4971 5030 5101 5100 5231 5290 5301 9 5426 5491 5556 5021 5680 5751 5815 5880 5945 6010 670 826075 826140 820204 820209 826334 820399 820464 820528 82G593 826058 1 0723 0787 6852 6917 6981 7040 7111 7175 7240 7305 2 7369 7434 7499 7503 7028 8273 7092 7757 '7821 7880 7951 3 8015 8080 8144 8209 8338 8402 8407 8531 8595 4 8060 8724 8789 8853 8918 8982 904G 9111 9175 9239 5 9304 9368 9-132 9497 9561 9625 9090 9754 9818 9882 6 9947 830011 0653 830075 830139 0781 830204 830208 830332 830390 830460 830525 7 830589 0717 0815 0909 0973 1037 1102 1100 8 1230 1294 1358 1422 I486 1550 1014 1678 1742 1800 9 1870 1934 1998 2002 2126 2189 2253 2317 2381 2445 68C 832509 832573 832037 832700 832704 832828 832892 832950 833020 833083 ' 1 3147 3211 3275 3338 3402 3466 3530 3593 3057 3721 2 3784 3848 3912 3975 4039 4675 4103 4166 4230 4294 4357 3 4421 4484 4548 4011 4739 4802 4806 4929 4993 4 51)50 5120 5183 5247 5310 5373 .5437 5500 5504 5027 5 5691 5754 5817 5881 5944 0007 G071 0134 0197 0201 6 6324 6397 0451 0514 0577 0641 6701 0707 0830 6894 7 6957 7020 7083 7140 7210 7273 7330 7399 7462 7525 8 7588 7052 7715 7778 7841 7904 7967 8030 8093 8I5G 9 8219 8282 8345 8408 8471 8534 8597 8G60 8723 8780 690 838849 838912 838975 839038 839101 839104 839227 839289 839352 839415 1 9478 9541 9604 9667 9729 9792 9355 9918 9981 840043 2 840116 840169 340232 84(8294 840357 840420 840482 840545 840608 0671 3 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 4 1359 1422 1485 2110 1547 1610 1672 1735 1797 I860 1922 5 1985 2047 2172 2235 2297 2360 2422 2484 2547 6 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 7 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 8 3855 3918 3980 4042 4104 4166 4229 4291 4353 4415 9 4477 4539 4601 4664 4726 4788 4850 4912 4974 5036 No, 0 1 a 3 * 5 ! 6 * 8 9 1 | Dijt 63 63 63 62 62 62 62 62 | Di& S3 SSSSSSSSSS ££££££££&£; SiSSSSSSSgSi SSIS8SS83LOGARITHMS OF NUMBERS. 191 No. | O | 1 | 3 | 3 1 * 1 5 1 6 1 7 1 8 l 9 | Diff. 700 845098 845160 845222 845281 845346 845408 845470 845532 845594 845656 m ] 57 Jf 5781 5842 5!MW 5966 602f 6001 6151 6212 6275 62 2 0337 6395 6461 652c 6585 6640 670f 677C 6832 6894 62 : 0055 7017 7079 7141 7202 7204 7326 738f 744S 7511 62 4 757' 7634 7090 775t 78U 7881 794- 8004 8066 812? 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No. 0 ' 1 a 3 I * | 5 I © | 7 I 8 I 9’ | Diffi, 192 LOGARITHMS OF NUMBERS. No | O 1 2 3 * 5 9 7 8 0 DiS 760 880814 880871 880: .28 880985 881042 681099 H81I5G 881213 881271 881328 57 1 1385 1442 1499 1556 1613 1G70 1727 1784 1841 1898 57 2 1953 2012 2069 2126 2183 2210 2297 2354 2411 2468 57 3 2525 2581 2638 2095 2752 2809 2866 2923 2980 3937 57 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 57 5 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 57 6 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 57 7 4795 4852 4909 4965 5022 5078 5135 5192 5248 5305 57 e 5361 5418 5474 5531 5587 5644 5700 5757 5813 5870 57 9 5926 5983 6039 6096 0152 6209 6265 6321 6378 6434 56 770 886491 886547 886904 886660 886716 886773 886829 886885 88G942 886998 56 1 7054 7111 7167 7223 7280 7336 7392 7449 7505 7561 56 2 7617 7674 7730 7786 7842 7898 7955 8011 8067 8123 56 3 8179 8236 82.12 8348 8404 8160 8516 8573 8629 -8685 56 4 8741 8797 8853 8009 8965 9021 9077 9134 9190 9246 56 5 9302 9358 9414 9470 9526 9582 9638 9094 9750 9806 56 6 9862 9918 9974 890030 890086 890141 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9602 9656 9711 9766 55 4 9821 9875 9930 9985 900039 900094 900149 900203 900258 900312 55 5 900367 900422, 900470 900531 0586 0640 0695 0749 0801 0859 55 6 0913 0968 1022 1077 1131 1180 1240 1295 1349 1404 55 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 8 2003 2057 2112 21GG 2221 2275 2329 2384 2438 2492 54 9 2547 2601 2055 2710 2764 2818 2873 2927 2981 3036 54 800 903090 903144 903io9 903253 903307 903361 903416 903470 903524 903578 54 1 3633i 3687 3741 3795 3819 3904 3958 4012 4066 4120 54 2 4171 4229 4283 4337 4391 4445 4499 4553 4607 4661 54 3 4710 4770 4824 4878 4932 4986 5040 5094 5148 5202 54 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 54 5 5796 5850 5904 5958 0012 6060 6119 6173 6227 6211 54 6 6335 6389 0443 6497 6551 6604 0658 6712 6766 6820 54 7 6874 6927 6981 7035 7089 7143 7796 7250 7304 7358 54 e 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 54 9 7949 8002 8056 8110 8163 8217 8270 8324 8378 8431 54 810 908485 908539 908592 908640 908699 908753 908807 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7190 7243 7295 7348 7400 7453 53 7 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 52 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 52 9 8555 8607 8659 8712 8704 8816 8809 8921 8973 9026 52 830 919078 919130 919183 919235 919287 919340 919392 919444 919496 919549 52 1 <1601 9653 9706| 9758 9810 9802 9914 99(57 920019 920071 52 2 920123 920176 920228 920280 920332 920384 920436 920489 0.541 0593 52 3 0645 0697 0749 0801 0853 0906 0958 1010 1002 1114 52 4 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 52 5 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 52 6 2206 2258 2310 2.302 2414 2406 251b 2570 2622 2674 52 7 2725 2777 2829 2881 2933 ■ 2985 3037 3089 3140 3192 52 8 3244 3296 3348 3399 3451 3503 3555 3007 3658 3710 52 9 3762 3814 3365 3917 3969 4021 4072 4124 4176 4228 52 840 924279 924331 924383 924434 924486 924538 924589 924641 924693 924744 52 1 4796 4848 4899 4951 5003 5054 5106 5157 5209 5201 52 2 5312 5364 5415 5467 5518 5570 5021 5673 5725 5770 52 3 5828 5879 5931 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934751 934801 934852 934902 934953 50 1 5003 5054 5104 5154 5205 5255 5306 5356 5400 5457 50 2 5507 5558 5008 5658 5709 5759 5809 5860 5910 5900 50 3 6011 6061 6111 6162 6212 6262 6313 6363 6413 6403 50 4 6514 6564 0014 6665 6715 6765 6815 6865 6916 6966 50 5 7016 7060 7117 7167 7217 7267 7317 7367 7418 7408 50 6 7518 7508 7018 7668 7718 7769 7819 7869 7919 7909 50 7 8019 8069 8119 8169 8219 8269 8320 8370 8420 8470 50 8 8520 8570 3020 8670 8720 8770 8820 8870 8920 8970 50 9 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 50 870 939519 939509 939019 939009 939719 939769 839819 939869 9399)8 939968 50 1 940018 940008 940118 940103 940218 940267 940317 940367 940417 940467 50 2 0510 0566 0016 0666 0710 0765 0815 0865 0915 0964 50 3 1014 1064 1114 1163 1213 1203 1313 1362 1412 1462 50 4 1511 1501 1611 1060 1710 1700 1809 1S59 1909 1958 50 5 2008 2058 2107 2157 2207 2250 2300 2355 2405 2455 50 6 2504 2554 2003 2653 2702 2752 2801 2851 2901 2950 50 7 3000 3049 3099 3148 3(98 3247 3297 3346 3396 3445 49 8 3495 3544 3593 3643 3092 3742 3791 3841 3890 3939 49 9 3989 4038 4088 4137 4180 4230 4285 4335 4384 4433 49 No. | 0 1 1 a 3 4 1 5 1 o 7 8 9 Di£194 LOGARITHMS OF NUMBERS. Na | 0 1 a 3 4 5 6 7 8 9 Diff. 88) 944483 944532,944581 944631 044(80 044720 044779 044828 944877 944927 49 1 4076 5025 5074 5124 5173 5222 5272 5321 5370 5419 49 2 54 GO 5518 5567 5616 5665 5715 5764 5813 5862 5912 49 3 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 49 4 6452 6501 6551 6600 6649 6608 6747 6796 6845 6894 4£ 5 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 49 6 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 7 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 49 8 8413 8462 8511 8560 8609 8657 8706 8755 8804 8853 49 9 8902 8951 8999 9048 9007 9146 9195 9244 9292 9341 49 890 1H9390 949439 949488 94953G 049585 949634 949683 949731 949780 949829 49 1 9878 9926 0975 950024 050073 950121 950170 950219 050267 950316 49 2 050365 950414 950462 0511 0560 0608 0657 0706 0754 0803 49 3 0851 0900 0049 0997 1046 1095 1143 1192 1240 1280 49 4 1338 1386 1435 1483 1532 1580 1629 1677 1726 1775 49 5 1823 1872 1020 1969 2017 2066 2114 2163 2211 2260 48 6 2308 2356 2405 2453 2502 2550 2509 2647 2606 2744 48 7 2702 2841 2889 2938 2086 3034 3083 3131 3180 3228 48 8 327G 3325 3373 3421 3470 3518 3566 3615 3663 3711 48 9 3760 3808 3856 3005 3953 4001 4040 4008 4146 4194 48 900 954243 954291 954330 954387 954435 954484 954532 954580 954628 954677 48 1 4725 4773 4821 4860 4918 4966 5014 5062 5)10 5158 48 2 5207 5255 5303 5351 5300 5447 5405 5543 5502 5640 48 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 6120 48 4 6168 6216 6265 6313 6361 6400 6457 6505 6553 6601 48 5 6649 6697 6745 6703 6840 6888 6936 6984 7032 7080 48 6 7128 7176 7224 7272 7320 7368 7416 7464 7512 7559 48 7 7607 7G55 7703 7751 7700 7847 7804 7942 7990 8038 48 8 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 48 9 8564 8612 8650 8707 8755 8803 8850 8808 8046 8904 48 910 059041 959089 959137 959185 959232 959280 959328 059375 959423 959471 48 1 9518 9566 9614 9661 0760 9757 9804 9852 0900 0947 48 2 0905 960042 960000 960138 960185 960233 960281 960328 960376 00042:) 48 3 960471 0518 0566 0613 0661 0709 0756 0804 0851 0890 48 4 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 48 5 1421 1460 1516 1563 1611 1658 1706 1753 1801 1848 47 6 1805 1943 1900 2038 2085 2132 2180 2227 2275 2322 47 7 2369 2417 2464 2511 2559 2606 2653 2701 2748 2705 47 8 2843 2890 2937 2085 3032 3079 3126 3174 3221 3268 47 9 3316 3363 3410 3457 3504 3552 3590 3646 3693 3741 47 920 963788 963835 963882 963929 063977 964024 964071 964118 954105 064212 47 J 4260 4307 41154 4401 4448 4495 4542 4500 4637 4684 47 n 4731 4778 4825 4872 4010 4066 5013 5061 5108 5155 47 3 5202 5249 5296 5343 5300 5437 5484 5531 5578 5625 47 4 5672 5719 5766 5813 5860 5037 5954 6001 6048 6005 47 5 G142 6189 6236 6283 6320 6376 6423 6470 6517 6504 47 fi G611 6658 6705 6752 6790 6845 6802 6930 6986 703:) 47 1 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 47 8 7548 7505 7642 7688 7735 7782 7820 7875 7922 7969 47 9 8016 8062 8109 8156 8203 8249 8296 8343 8390 8436 47 930 968483 068530 968576 968623 9G8670 968716 968763 968810 968856 968903 47 1 89.10 8906 9043 9090 9136 9183 9229 9276 9323 9369 47 o 9416 0463 9509 9556 9602 9640 9695 0742 9780 9835 47 3 0882 9928 9975 970021 970068 970114 970161 970207 970254 970300 47 4 970347 070393,970440 0486 0533 0579 0626 0672 0719 0765 46 5 0812 0858 0904 0951 0907 1044 1090 1137 1183 1220 46 6 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 46 7 1740 1786 1832 1879 1925 1971 2018 2064 2110 2157 46 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 46 9 2666 2712 2758 2801 2851 2807 2943 2989 3035 3082 46 No.) 0 1 1 3 4 5 6 7 8 9 Id* LOGARITHMS OF NUMBERS. 195 No. 0 1 a 3 * * 6 7 8 9 940 973128 973174 973220 f 1 973313 973359 973405 973451 973497 973543 J 3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 2 4051 4097 4143 4189 4235 4281 4327 4374 4420 4400 3 4512 4558 4604 4650 4696 4742 4788 4834 4880 4920 4 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 5 5432 5478 5524 5570 5616 ■ 5662 5707 5753 5799 5845 6 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 7 6350 6306 6442 6488 6533 6579 6625 6671 0717 6703 8 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 950 977724 977769 977815 977861 977906 977952 977998 978043 978089 978135 1 8181 8226 8272 8317 8363 8409 8454 8500 8540 8591 o 8637 8683 8728 8774 8819 8805 8911 8956 9002 9047 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 4 9548 9594 9039 9685 9730 9776 9821 9867 9912 9958 5 980003 980049 980094 980140 980185 980231 980276 980322 980367 980412 6 0458 0503 0549 0594 0640 0085 0730 0770 0821 0807 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 8 1366 1411 1450 1501 1547 1592 1637 1683 1728 1773 9 1819 JS04 1909 1954 2000 2045 2090 2135 2181 2220 960 982271 982:116 982362 982407 982452 982497 982543 982588 982633 982G78 1 2723 2709 2814 2859 2904 2949 2994 3040 3085 3130 2 3175 3220 3205 3310 335G 3401 3440 3491 3536 3581 3 3626 3671 3716 3702 3807 3852 3897 3942 3987 4032 4 4077 4122 4107 4212 4257 4302 4347 4392 4437 4482 5 4527 4572 4017 4002 4707 4752 4797 4842 4887 4932 6 4977 5022 5007 5112 5157 5202 5247 5292 5337 5382 7 5126 5471 5510 5501 5000 5051 5090 5741 5780 5830 8 5875 5920 5965 6010 6055 0100 6144 6189 6234 6279 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 980772 986817 986801 986906 986951 986996 987040 987085 987130 987175 i 7219 7204 7309 7353 7398 7443 7488 7532 7577 7022 2 7666 7711 7756 7800 7845 7890 7934 7979 8024 8008 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 4 8559 8004 8648 8093 8737 8782 8820 8871 8910 8960 5 9005 9049 9094 9138 9183 9227 9272 9316 9301 9405 6 9450 9494 9539 9583 9628 9072 9717 9761 9806 9850 7 9895 9939 9983 990028 990072 990117 990161 990206 990250 990294 8 990339 990383 990428 0472 0510 0501 0605 0050 0094 0738 9 0783 0827 0871 0910 0960 1004 1049 1093 1137 1182 980 991226 991270 991315 991359 991403 991448 991492 991536 991580 991625 1 1669 1713 1758 1802 1846 1890 1935 1979 2023 2007 2 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 3 2554 2598 2042 2086 2730 2774 2819 2863 2907 2951 4 2995 3039 3083 3127 3172 3210 3200 3304 3348 3392 5 3436 3480 3524 3508 3613 3657 3701 3745 3789 3833 6 3877 3921 3905 4009 4053 4097 4141 4185 4229 4273 7 4317 4301 4405 4449 4493 4537 4581 4625 4609 4713 8 4757 4801 4845 4889 4933 4977 5021 5005 5108 5152 9 5196 5240 5284 5328 5372 5416 5400 5504 5547 5591 990 995635 995079 995723 995767 995811 995854 995898 995942 995980 996030 J 6074 0117 0161 6205 6249 6293 6337 0380 6424 6408 2 6512 0555 6599 6043 G687 G731 6774 6818 6802 6900 3 6949 0993 7037 7080 7124 7168 7212 7255 7299 7343 4 7386 7430 7474 7517 7561 7605 7048 7092 7736 7779 5 7823 7807 7910 7954 7998 8041 8085 8129 8172 8216 6 8259 8303 8347 8390 8434 8477 8521 8504 8608 8052 7 8695 8739 8782 8820 8869 8913 8950 9000 9043 9087 e 9131 9174 9218 9201 9305 9348 9392 9435 9479 9522 9 9565 9609 9652 9096 9739 9783 - 9826 9870 9913 9957 No. 1 O 1 * 3 4 5 e 7 8 9 4 37 38 39 40 41 42 43 44 45 40 47 48 49 50 51 52 53 54 55 50 57 58 59 60 ! 60 | 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 6 5 4 3 o 1 _0 M. L 0 GA HITIIMIC SINES, COSINES, ETC. D. I Cosine 1 D. 1 Tenp. D. | Cotnnff.- HHI00000 0-000000 Intinile. 501717 000000 00 0-463726 501717 13-536274 293485 000000 00 764756 293483 235244 208231 000000 00 940847 208231 059153 101517 000000 00 7-065786 161517 12934214 131968 000000 00 162696 131969 837304 111575 9-999999 01 241878 111578 758122 96(53 999999 01 308825 99653 691175 85254 999999 01 366817 85254 633183 76263 999999 01 417970 76263 582030 68988 999998 01 463727 68988 536273 6298i 9-999998 01 7-505120 6298; 12-494880 57936 999997 91 542909 57933 457091 53641 999997 01 577672 53642 422328 49938 999996 01 609857 49939 390143 40' 14 999996 01 639820 46715 360180 43881 999995 - 01 667849 43882 332151 41372 999995 01 694179 41373 305821 39135 999994 01 719003 39136 280997 37127 999993 01 742484 37128 257516 35315 999993 01 764761 35136 235239 33072 9-999992 01 7-785951 33673 12-214049 32175 999991 01 806155 32176 193845 30805 999990 01 825460 30806 174.540 29547 999989 02 843944 29549 156056 28388 91*9988 02 861674 28390 138326 27317 999988 02 878708 27318 121292 20323 999987 (12 895099 26325 104901 25399 999986 02 910894 25401 089106 24538 999985 02 920134 24540 073866 23733 999983 02 940858 23735 059142 22980 9-999982 02 7-955100 22981 12-044900 22273 999981 02 968889 22275 031111 21008 999980 02 982253 21610 017747 20981 999979 02 995219 20983 004781 20390 999977 02 8007809 20392 11-992191 19831 999976 02 020045 19833 979955 19302 999975 02 031945 19305 968055 18801 999973 02 043527 18803 956473 18325 999972 02 054809 18327 945191 17872 999971 02 065806 17874 934194 17441 9-999969 02 8076531 17444 11-923469 17031 999968 02 086997 17034 913003 10039 999966 02 097217 16642 902783 10205 999964 03 107202 16268 892797 15908 999963 03 116963 15910 883037 15506 999961 03 126510 15568 873490 15238 999959 0.i 135851 15241 864149 14924 999958 63 144996 14927 855004 '14022 999956 03 153952 14627 846048 14333 999954 03 162727 14336 837273 14054 9-999952 03 8171328 14057 11-828672 13780 999950 03 179763 13790 820237 13529 999948 03 188036 13532 811964 13280 999946 03 I96156 13284 803844 13041 999944 03 204126 13044 795874 12810 999942 04 211953 12814 788047 12.587 999940 04 219641 12590 780359 12372 999938 04 227195 12376 •772805 12104 999936 04 234621 12168 765379 11903 999934 04 241921 11967 758079 Sme Cotang-. Tang. | 89 DegreesH. ~0 1 2 3 4 5 6 7 8 9 JO 11 12 13 14 15 10 17 18 19 20 21 22 23 24 25 20 27 28 29 30 31 32 33 34 35 30 37 38 39 40 41 42 43 44 45 40 47 48 49 50 51 52 53 54 55 50 57 58 59 GO 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 20 25 24 23 22 21 20 19 18 17 10 15 14 13 12 11 10 9 8 7 6 5 4 3 2 I (I M. 1ITHMIG SINES, COSINES, ETC. (1 Degree.) D. Cosine D. | Tang. D. Cotang. | 11903 9-099934 04 8-241921 11907 11-758079 11708 999932 04 249102 11772 750898 11580 999929 04 256105 11584 74:1835 11398 999927 04 • 203115 11402 736885 11221 999925 04 209956 11225 730044 11050 999922 04 270691 11054 723309 10883 999920 04 283323 10887 716677 10721 999918 04 i 289856 10726 710J44 10505 999915 04 290292 10570 703708 10413 999913 04 302034 10418 697306 10266 999910 04 308884 10270 691116 10122 9-999907 04 8-315046 10126 11-084954 9982 999905 04 321122 9987 678878 9847 999902 04 327114 9851 672886 9714 999899 05 333025 9719 666975 9580 999897 05 338856 9590 661144 9400 999894 05 344010 9405 655390 9338 99989L 05 350289 9343 649711 9219 999888 05 355895 9224 6441115 9103 999885 05 301430 9108 038570 8990 999882 05 300895 8995 633105 8880 9-999879 05 8-372292 8885 11-627708 8772 999870 05 377022 8777 622:178 8007 999873 05 382889 8072 617111 8504 999870 05 388092 8570 611908 8404 999807 05 393234 8470 606766 8300 991)804 05 398315 8371 601085 8271 999801 05 403338 8270 596062 8177 999858 05 408304 8182 591096 8080 999854 05 413213 8091 580787 7990 999851 06 418068 8002 581932 7909 9-999848 06 8-422809 7914 11-577131 7823 999844 06 427018 7830 572382 7740 999841 00 432315 7745 567085 7057 999838 00 430902 7003 563038 *0/ / 999834 06 441500 7583 558440 7499 999831 06 446110 7505 553890 7422 999827 00 450613 7428 549387 7346 999823 06 45.5070 7352 544930 7273 999820 06 459481 7279 540519 7200 999816 06 403849 7206 536151 7129 9-999812 06 8-408172 7135 11-531828 7000 999809 06 472454 7066 527546 6991 999805 06 476093 6998 523307 0924 999801 06 480892 6931 519108 0859 999797 07 485050 6865 514950 67D4 999793 07 489170 6801 510830 6731 999790 07 493250 6738 500750 6669 999786 07 497293 6076 5U2707 6608 999782 07 501298 6615 498702 6548 999778 07 505207 6555 494733 6489 9999774 07 8-509200 6496 11 -490800 6431 999709 07 513098 6439 486902 6375 999705 07 510901 6382 483039 6319 999701 07 520790 6320 479210 6204 999757 07 524586 0272 475414 6211 989753 07 528349 0218 471651 6158 999748 07 532080 0105 407920 6100 999744 07 535779 0113 464221 6055 999740 07 539447 6062 400553 6004 999735 07 543084 6012 450916 Sme Co tang. Tang. 88 Degrees.M- 0 1 2 3 4 5 6 7 R 797 26 08601)0 1751 914000 1721 996782 26 087050 1747 912950 1717 996766 26 0881198 1743 911902 1713 996751 26 089144 1738 910856 Siue Cotang. 1 Tang. 83 Degrees.M. 'O' 1 2 3 4 5 6 7 8 9 JO II 12 13 14 J5 J6 J7 J8 19 20 21 22 23 24 25 20 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 5G 57 58 59 60 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1L ’IITIIMIC SINES, COSINES, ETCt (7 Degrees.) D. | . Cosine 1 D. | Tang. D. | Coding 1713 9-996751 26 9 089144 1738 10-910856 1709 990735 26 090187 1734 909813 1704 996720 20 091228 1730 908772 1700 996704 26 092206 J727 907734 1696 990688 26 093302 1722 900098 1092 990073 26 094336 1719 905004 1688 990657 26 095367 1715 904033 1084 990641 26 090395 1711 903005 1080 990025 20 097422 1707 902578 1676 990010 26 09844G 1703 901554 1G73 990594 26 099408 1099 900532 1008 9-990578 27 9 100487 1095 10-899513 1005 990502 27 101.504 1091 898496 1001 990540 27 102519 1087 897481 1657 990530 27 103532 1084 89(i408 1653 990514 27 104542 1080 895458 1049 990498 27 105.550 1670 894450 1045 990482 27 1005.56 1072 893444 1041 990405 27 107559 1009 892441 1638 990449 27 108500 1005 891440 1034 996433 27 109.559 1001 890441 1030 9-990417 27 9-110556 1058 10-889444 1027 99(i400 27 111551 1054 888449 1023 990384 27 112543 1050 887457 1619 990300 1424 995316 31 169284 1455 830716 36 25 165454 1422 995297 31 170157 1453 829843 35 26 166307 1419 995278 31 171029 1450 828971 34 27 167159 1416 995260 31 171899 1447 828101 33 28 168008 1413 995241 32 172767 1444- 827233 32 29 168356 1410 995222 32 173634 1442 826366 31 30 169702 1407 995203 32 174499 1439 825501 30 31 9-170547 1405 9-995184 32 9-175362 1436 10-824638 29 32 171389 1402 995165 32 176224 1433 823776 28 33 172230 1399 995146 32 177084 1431 822910 27 34 173070 1396 995127 32 177942 1428 822058 26 35 173908 1394 995108 32 178799 1425 821201 25 36 174744 1391 995089 32 179655 1423 820345 24 37 175578 1388 995070 32 180508 1420 819492 23 38 176411 1386 995051 32 181360 1417 818640 22 39 177242 1383 995032 32 182211 1415 817789 21 40 178072 1380 995013 32 183059 1412 816941 20 41 9-178900 1377 9-994993 32 9 183907 1409 10-816093 19 42 179726 1374 994974 32 184752 1407 815248 18 43 180551 1372 994955 32 185597 1404 814403 17 44 181374 1369 994935 32 186439 1402 813561 16 45 182196 1366 994916 33 187280 1399 812720 15 46 183016 1364 994896 33 188120 1396 811880 14 47 183834 1361 994877 33 1889.58 1393 811042 13 48 184651 1359 994857 33 189794 1391 810206 12 49 185466 1356 994838 33 190629 1289 809371 11 50 186280 1353 994818 33. 191462 1386 808538 10 51 9 187092 1351 9-994798 33 9 192294 1384 10-807706 9 52 187903 1348 994779 33 193124 1281 806876 e 53 188712 1346 994759 33 193953 1379 806047 7 54 189519 1343 994739 ■ 33 194780 1376 805220 6 55 190325 1341 994719 33 195606 1374 804394 5 56 191130 1338 994700 33 196430 1371 803570 4 57 191933 '1336 994680 33 197253 1369 802747 3 58 192734 1333 994660 33 198074 1366 801926 2 59 193534 1330 994640 33 198894 1364 801106 1 60 194332 1328 994620 33 199713 1361 800287 0 Cosine Sum ' Cotan^. Tang. M. 61 Degrees.M. 0 1 2 3 4 5 6 7 8 9 10 il 12 J3 14 15 16 17 18 19 20 21 22 23 24 25 20 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 It 13 12 11 10 9 8 7 6 5 4 3 2 1 0 SINES, COSINES, ETC. (9 Degrees.) D. | Cosine D. Tan?. D. Cotang’. 1328 9-994620 33 9 199713 1361 10-800287 1326 994600 33 200529 1359 799471 1323 994580 33 201345 1356 798655 1321 994560 34 202159 1354 797841 1318 994540 34 202971 1352 797029 1316 994519 34 203782 1349 796218 1313 994499 34 204592 1347 795408 1311 994479 34 205400 1345 794600 1308 994459 34 206207 1342 793793 1306 994438 34 207012 1340 792987 1304 994418 34 207817 1338 792183 1301 9-994397 34 9-208619 1335 10i791381 1299 994377 34 209420 1333 7!10580 1296 994357 34 210220 1331 789780 1294 994336 34 211018 1328 768982 1292 994316 34 211815 1326 788185 1289 994295 34 212611 1324 787389 1287 994274 35 213405 1321 786595 1285 994254 35 214198 1319 785802 1282 994233 35 214989 1317 785011 1280 994212 35 215780 1315 784220 1278 9-994191 35 9-216568 1312 10-783432 1275 994171 35 217356 1310 782044 1273 994150 35 218142 1308 781858 1271 994129 35 218926 1305 781074 1268 994108 35 219710 1303 780290 1266 994087 35 220492 1301 779508 1264 994066 35 221272 1299 778728 ■ 1261 99*045 35 222052 1297 777948 1259 994024 35 222830 1294 777170 1257 994003 35 223606 1292 776394 1255 9-993981 35 9-224382 1290 10-775618 1253 993960 35 225156 1288 774844 1250 993939 35 225929 1286 774071 1248 993918 35 226700 1284 773300 1246 993896 36 227471 1281 772529 1244 993875 36 228239 1279 771761 1242 993854 36 229007 1277 770993 1239 993832 36 229773 1275 776227 1237 993811 36 230539 1273 769401 1235 993789 36 231302 1271 708698 1233 9-993768 36 9-232065 1269 10-707935 1231 993746 36 232826 1267 767174 1228 993725 30 233586 1265 766414 1226 993703 36 234:145 1262 765655 1224 993681 36 235103 1260 704897 1222 993660 36 235859 1258 764141 1220 993638 36 236614 1256 763380 1218 993616 36 237368 1254 762632 1216 993594 37 238120 1252 761880 1214 993572 37 238872 1250 761128 1212 9-993550 37 0-239622 1248 10-760378 1209 993528 37 240371 1246 759629 1207 993506 37 241118 1244 758682 1205 993484 37 241865 1242 758135 1203 993462 37 242610 1240 757390 1201 993440 37 243354 1238 756640 1199 993418 37 244097 1230 755903 119T 993396 37 244839 1234 755161 1195 993374 37 245579 1232 754421 1 1193 993351 37 246319 1230 753681 1 1 Sme | Coiang. Tang. 1 80 Degrees. 208 (10 Degrees.) LOGARITHMIC SIKES, COSINES, ETC. u. 0 1 2 3 4 5 6 7 8 .9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 58 59 60 | Sine : D. [ Cosine D. [ Tang. 1 D. 1 Cotang. 1 9-239670 1193 9-993351 37 9 240319 1230 10-750681 60 240386 1191 993329 37 247057 1228 752943 59 241101 1189 993307 37 247794 1226 752206 58 241814 1187 9!I3285 37 248530 1224 751470 57 242526 1185 993262 37 249204 1222 750736 56 243237 1183 993240 37 249998 1220 750002 55 243947 1181 993217 38 250730 1218 749270 54 244056 1179 993195 38 251461 1217 748539 53 245363 1177 993172 38 252191 1215 747809 52 240069 1175 993149 38 252920 1213 747080 51 240775 1173 993127 38 253048 1211 740352 50 9-247478 1171 9-993104 38 9-254374 1209 10-745020 49 248181 1109 993081 38 255100 1207 744900 48 248883 1107 933059 38 255824 1205 744170 47 249583 1105 993036 38 256547 1203 74M53 46 250282 1103 993013 38 257209 1201 742731 45 250980 1101 992990 38 257990 1200 742010 44 251077 1159 992907 38 258710 1198 741290 43 252373 1158 992944 38 259429 1196 740571 42 253007 1156 992921 38 200146 1194 739854 41 253701 1151 992898 38 200863 1192 739137 40 9254453 1152 9-992875 38 9-201578 1190 10-738422 39 255144 1150 992852 38 202292 1189 737708 38 255834 1148 992829 39 203005 1187 730995 37 250523 1146 992800 39 203717 1185 730283 36 257211 1144 992783 39 204428 1183 735572 35 257898 1142 992759 39 265138 1181 734802 34 258583 1141 992730 39 205847 1179 734153 33 259268 1139 992713 39 260555 1178 733445 32 259951 • 1137 992090 39 207201 1170 732739 31 200033 1135 992060 39 207907 1174 732033 30 9-201314 1133 9-992043 39 9-208671 1172 10-731329 29 20TJ04 1131 992019 39 209375 1170 730025 28 202073 - 1130 992596 39 270077 1109 729923 27 203351 1128 992572 39 270779 1107 729221 26 204027 1120 992549 39 271479 1105 728521 25 204703 1124 99-2525 39 272178 1104 727822 24 2G5377 1122 992591 39 272876 1102 727124 23 200051 1120 992478 40 . 273573 1100 720427 22 200723 1119 992454 40 274209 1158 725731 21 267395 1117 992430 40 274904 1157 725030 20 9-268065 1115 9-992406 40 9-275058 1155 10-724342 19 208734 1113 992382 40 270351 1153 723049 18 209402 1111 992359 40 277043 1151 722957 17 270009 1110 992335 40 277734 1150 722206 16 270735 1108 992311 40 278424 1148 721576 15 271400 1106 992287 40 279113 1147 720887 14 272004 1105 992203 40 279801 1145 720199 13 272720 1103 992239 40 280488 1143 719512 12 273388 1101 992214 40 281174 1141 718820 11 274049 1099 992190 40 281858 1140 718142 10 9-274708 1098 9-992106 40 9-282542 1138 10-717458 9 275307 1096 992142 40 283225 1130 710775 8 270024 1094 992117 41 283907 1135 710093 7 270081 1092 992093 41 284588 1133 715412 6 277337 1091 992009 41 285208 1131 714732 5 277991 1089 992044 41 285947 1130 714053 4 278644 1087 992020 41 280024 1128 713376 3 279297 1086 991996 41 287301 1120 712099 2 279948 1084 991971 41 287977 1125 712023 1 280599 1082 991947 41 288052 1123 711348 0 | Cname | | Buie | | CoUmg. | | Tang. | M. 79 Degrees.LOGARITHMIC SIRES, COSINES, ETC. (11 Degrees.) 209 M. | Sins D. Cosine D. Tang. D. Cotang. 0 9-280599 1082 9-991947 41 9-288652 1123 10-711348 60 1 281248 1081 991922 41 289326 1122 710674 59 2 281897 1079 991897 41 289999 1120 710001 58 3 282544 1077 991873 41 290671 1118 709329 57 4 283190 1076 991848 41 291342 1117 708058 56 S 283836 1074 991823 41 292013 1115 707987 55 6 284480 1072 991799 41 292682 1114 707318 54 7 285124 1071 991774 42 293350 1112 706650 53 8 285706 1069 991749 42 294017 1111 705983 52 9 286408 1067 991724 42 294684 1109 705316 51 10 287048 1066 991G99 42 295349 1107 704051 50 11 9-287087 1064 9-991674 42 9-296013 1106 10-703987 49 12 288326 1063 991649 42 296077 1104 703323 48 13 288964 1061 991624 42 297339 1103 702001 47 14 289600 1059 991599 42 298001 1101 701999 46 15 290236 1058 991574 42 298602 1100 701338 45 10 290870 1056 991549 42 299322 1098 700678 44 17 291504 1054 991524 42 299980 1096 700020 43 18 292137 1053 991498 42 300038 1095 699362 42 19 292708 1051 991473 42 301295 1093 698705 41 20 293399 1050 991448 42 301951 1092 698049 40 21 9-294029 1048 9-991422 42 9-302607 1090 10-697393 39 22 294058 1046 991397 42 303261 1089 696739 38 23 295286 1045 991372 43 303914 1087 690080 37 24 295913 1043 991346 43 304567 1086 695433 36 25 296539 1042 991321 43 305218 1084 694782 35 20 297104 1040 991295 43 305809 1083 694131 34 27 297788 1039 991270 43 306519 1081 693481 33 28 298412 1037 991244 43 307108 1080 692832 32 29 299034 1036 991218 43 307815 1078 692185 31 30 299055 1034 991193 43 308403 1077 691537 30 31 9-300276 1032 9-991167 43 9-309109 1075 10690891 29 32 300895 1031 991141 43 309754 1074 690246 28 33 301514 1029 991115 43 310398 1073 689602 27 34 302132 1028 991090 43 311042 1071 688958 26 35 302748 1026 991064 43 311685 1070 688315 25 30 303364 1025 991038 43 312327 1068 687673 24 37 303979 1023 991012 43 312967 1067 687033 23 38 304593 1022 990986 43 313G08 1065 680392 22 39 305207 1020 990900 43 314247 1004 685753 21 40 305819 1019 990934 44 314885 1062 085115 20 41 9-306430 1017 9-990908 44 9 315523 • 1061 10-684477 19 42 307041 1010 990882 44 316159 1000 683841 18 43 307050 1014 990855 44 316795 1058 683205 17 44 308259 1013 990829 44 317430 1057 682570 16 45 308867 1011 990803 44 318064 1055 681936 15 40 309474 1010 990777 44 318097 1054 681303 14 47 310080 1008 990750 44 319329 1053 680671 13 48 310085 1007 990724 44 319901 1051 680039 12 49 311289 1005 990097 44 320592 1050 679408 11 50 311893 1004 990671 44 321222 1048 678778 10 51 9-312495 1003 9-990044 44 9-321851 1047 10678149 9 52 313097 1001 990618 44 322479 1045 077521 8 53 313098 1000 990591 44 323100 1014 676894 7 54 314297 998 990565 44 323733 1043 670207 6 55 314897 997 990538 44 324358 1041 675042 5 50 315495 996 990511 45 324983 1040 675017 4 57 316092 994 990485 45 325607 1039 674393 3 58 316089 993 990458 45 320231 1037 073769 2 59 317284 991 990431 45 326853 1036 073147 1 60 317879 990 990404 45 327475 1035 072525 0 Conus | ! Sme# ColAUg’ Tang. M., 78 Degrees.210 (12 Degrees.) LOGARITHMIC SINES, COSINES, ETU. M. Sine D. Cosine D. Tang. 1 D. | Colnng. 0 9-317879 990 9-990404 45 9-327474 1035 10-672526 60 1 318473 988 990378 45 328095 1033 671905 59 2 319066 987 990351 45 328715 1032 671285 58 3 319658 986 990324 45 329334 1030' 670666 57 4 320249 984 990297 45 329953 1029 670047 56 5 320840 983 990270 45 330570 1028 669430 55 6 321430 982 990243 45 331187 1026 668813 54 7 322019 980 990215 45 331803 1025 668197 53 8 322607 979 990188 45 332418 1024 667582 52 9 323194 977 990161 45 333033 1023 666967 51 10 323780 976 990134 45 333646 1021 666354 50 11 9-324366 975 9-990107 46 9-334259 1020 10-665741 49 12 324950 973 990079 46 334871 1019 665129 48 13 325334 972 990052 46 335482 1017 664518 47 14 326117 970 990025 46 336093 1016 663907 46 15 326700 969 989997 46 336702 1015 663298 45 1G 327281 968 989970 46 337311 1013 662689 44 17 327862 966 989942 46 337919 1012 662081 43 18 328442 965 989915 46 338527 1011 661473 42 19 329021 964 989887 46 339133 1010 660867 41 20 329599 962 989860 46 339739 1008 660261 40 21 9-330176 961 9-989832 46 9-340344 1007 10-659656 39 22 330753 960 989804 46 340948 1006 659052 38 23 331329 958 989777 46 341552 1004 658448 37 24 331903 957 989749 47 342155 1003 657845 36 25 332478 956 989721 47 342757 1002 657243 35 26 333051 954 989693 47 313358 1000 656642 34 27 333624 953 989665 47 343958 999 656042 33 28 334195 952 989637 47 344558 998 655442 32 29 334766 950 989G09 47 345157 997 654843 31 30 335337 949 989582 47 345755 996 654245 30 31 9-335906 948 9-989553 47 9-346353 991 10-653647 29 32 336475 946 989525 47 346949 993 653051 28 33 337043 945 989497 47 347545 992 652455 27 34 337610 944 989469 47 348141 991 651859 26 35 338176 943 989441 47 348735 990 651265 25 36 338742 941 989413 47 349329 988 650671 24 37 339306 940 989384 47 349922 987 650078 23 38 339871 939 989356 47 350514 986 649486 22 39 340434 937 989328 47 351106 985 G48894 21 40 340996 936 989300 47 351697 983 648303 20 41 9-341558 935 9-989271 47 9-352287 982 10 647713 19 42 342119 934 989243 47 352876 981 647124 18 43 342679 932 989214 47 353465 980 646535 17 44 343239 931 989186 47 354053 979 645947 16 45 343797 930 989157 47 354640 977 645360 15 46 344355 929 989128 48 355227 976 644773 14 47 344912 927 989100 48 355813 975 644187 13 48 345469 926 989071 48 356398 974 643602 12 49 346024 925 989042 48 356982 973 643018 U 50 346579 924 989014 48 357566 971 642434 10 51 9347134 922 9-988985 48 9-358149 970 10-641851 9 52 347687 921 988956 48 358731 969 641269 8 53 348240 920 988927 48 359313 968 640687 7 54 348792 919 988898 48 359893 967 640107 6 55 349343 917 988869 48 360474 966 639526 5 56 349893 916 988840 48 361053 965 638947 4 57 350443 915 988811 49 361632 963 638368 3 58 350992 914 988782 49 362210 962 637790 2 59 351540 913 988753 49 362787 961 637213 1 60 352088 911 988724 49 363364 960 636636 0 J[ Caine Sine Gowng. l Tong. L M. 77 Degrees.LOGARITHMIC SINES, COSINES, ETC. (13 Degrees.) 211 u. | Sine | D. Cosine 1 D. 1 Tan 5. D. Cotang. 0 9-352088 911 9-988724 49 9-363364 960 10 036036 60 1 352635 910 988695 49 363940 959 636060 59 2 353181 909 988666 49 364515 958 6354H5 58 3 353726 908 988636 49 365090 957 634910 57 4 354271 907 988607 49 365664 955 634336 56 5 354815 905 988578 49 366237 954 633763 55 6 355358 904 988548 49 366810 953 633190 54 7 355901 903 988519 49 367382 952 632618 53 e 356443 902 988489 49 367953 951 632047 52 9 356984 901 988460 49 368524 950 631476 51 10 357524 899 988430 49 369094 949 630906 50 11 9-358064 898 9-988401 49 9-369663 948 10-630337 49 12 358603 897 988371 49 370232 946 629768 48 13 359141 896 9881142 49 370799 945 629201 47 14 359678 895 988:112 50 371307 944 628633 46 15 360215 893 988282 50 371933 943 628067 45 16 360752 892 988252 50 372499 942 627501 44 17 301287 891 988223 50 373064 941 026936 43 18 361822 890 988193 50 373029 940 626371 42 19 362356 889 988163 50 374193 939 625807 41 20 362889 888 988133 50 374750 938 625244 40 21 9-363422 887 9-988103 50 9-375319 937 10-624681 39 22 363954 885 .988073 50 375881 935 024119 38 23 364485 884 988043 50 370442 934 623558 37 24 365016 883 988013 50 377003 933 622997 36 25 365540 882 987983 50 377563 932 622437 35 26 306075 881 987953 50 378122 931 021878 34 27 366604 880 987922 50 378681 930 621319 33 28 367131 879 987892 50 379239 929 020761 32 29 367059 877 987862 50 379797 928 620203 31 30 368185 876 987832 51 380354 927 619646 30 31 9-368711 875 9-987801 51 9-380910' 926 10-619090 29 32 369236 874 987771 51 381466 925 618534 28 33 369761 873 987740 51 382020 924 617980 27 34 370285 872 987710 51 382575 923 617425 26 35 370808 871 987679 51 383129 922 616871 25 36 371330 870 987649 51 383682 921 616318 24 37 371852 869 987618 51 384234 920 615766 23 38 372373 867 987588 51 384786 919 615214 22 39 372894 866 987557 51 385337 918 614663 21 40 373414 865 987526 51 385888 917 614112 20 41 9-373933 864 9-987496 51 9-386438 915 10-613562 19 42 374452 863 987465 51 386987 914 613013 18 43 374970 862 987434 51 387536 •913 612464 17 44 375487 861 987403 52 388084 912 611916 16 45 376003 860 987372 52 388631 911 611369 15 46 376519 859 987341 52 389178 910 610822 14 47 377035 858 987310 52 389724 909 610276 13 48 377549 857 987279 52 390270 908 609730 12 49 378063 856 987248 52 390815 907 609185 11 50 378577 854 987217 52 391360 906 608640 10 51 9-379089 853 9-987186 52 9-391903 905 10-608097 9 52 379001 852 987155 52 392447 904 607553 8 53 380113 851 987124 52 392989 903 607011 7 54 380624 850 987092 52 393531 902 606469 6 55 381134 849 987061 52 394073 901 605927 5 56 381643 848 987030 52 394614 900 605386 4 57 382152 847 986:>98 52 395154 899 604846 3 58 382661 846 986967 52 395694 898 604306 2 59 383108 845 986936 52 396233 | 897 603767 1 60 383675 844 986904 52 396771 I 896 603229 0 l Come Cduuig. 4 | Tap*. 76 Dogreea212 (14 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. _ Sins D. Cosine D. 1 Tang. D. CoUng. - . 0 9-383675 844 9-986904 52 9-396771 8% 10-603229 60 1 384182 843 986873 53 397309 896 G02G91 59 2 384687 842 986841 53 397846 895 602154 58 3 385192 841 986809 53 398383 894 601617 57 4 385697 840 986778 53 398919 893 601081 56 5 386201 839 986746 53 399455 892 600545 55 6 38G704 838 986714 53 399990 891 600010 54 7 387207 837 986683 53 400524 890 599476 53 8 387709 ' 836 986651 53 401058 889 598942 52 9 388210 835 986G19 53 401591 888 598409 51 ]0 388711 834 986587 53 402124 887 597876 50 11 9-389211 833 9-986555 53 9-402656 886 10'597344 49 12 389711 832 986523 53 403187 885 59C813 48 13 390210 831 986491 53 403718 884 596282 47 14 390708 830 986459 53 404249 883 595751 46 15 391206 828 986427 53 404778 882 595222 45 16 391703 827 986395 53 405308 881 594692 44 17 392199 826 986363 54 405836 880 594164 43 18 392095 825 986331 54 406364 879 593636 42 19 393191 824 986299 54 406892 878 593108 41 20 393685 823 986266 54 407419 877 592581 40 21 9-394179 822 9-986234 54 9-407945 876 10-592055 39 22 394673 821 980202 54 408471 875 591529 38 23 395166 820 986169 54 408997 874 591003 37 24 305658 819 986137 54 409521 874 590479 36 25 396150 818 986104 54 410045 873 589955 35 26 396641 817 986072 54 410569 872 589431 34 27 397132 817 986039 54 411092 871 588908 33 28 397621 816 986007 54 411615 870 588385 32 29 398111 815 985974 54 412137 869 587863 31 30 398600 814 985942 54 412658 868 587342 30 31 9-399088 813 9-985909 55 9-413179 867 10-586821 29 32 399575 812 985876 55 413699 866 586301 28 33 400062 811 985843 55 414219 865 585781 27 34 400549 810 985811 55 414738 864 585262 26 35 401035 809 985778 55 415257 864 584743 25 3G 401520 808 985745 55 415775 863 584225 24 37 402005 807 985712 55 416293 862 583707 23 38 402489 806 985679 55 416810 861 583190 22 39 402972 805 985646 55 417326 860 582674 21 40 403455 804 985613 55 417842 859 582158 20 41 9-403938 803 9-985580 55 9-418358 858 10-581G42 19 42 404420 802 985547 55 418873 857 581127 18 43 404901 801 985514 55 419387 856 580613 17 44 405382 800 985480 55 419901 855 580099 16 45 405862 799 985447 55 420415 855 579585 15 46 406341 798 985414 56 420927 854 579073 14 47 406820 797 985380 56 421440 853 578560 13 48 407299 7% 985347 56 421952 852 578048 ■ 12 49 407777 795 985314 56 422163 851 577537 11 50 408254 794 985280 56 422974 850 577026 10 51 9-408731 794 9-985247 56 9-423484 849 10-576516 9 52 409207 793 985213 56 423993 848 576007 8 53 409682 792 985180 56 424503 848 575497 7 54 410157 791 985146 56 425011 847 574989 6 55 410632 790 985113 56 425519 846 574481 5 56 411106 789 985079 56 426027 845 573973 4 57 411579 788 985045 56 426534 844 573466 3 58 412052 787 985011 56 427041 843 572959 2 59 412524 786 984978 56 427547 843 572453 1 60 4129% 785 984944 56 428052 812 571948 0 | Cons* 1 Siue Coling. l l Tang. H. 76 Degrees.LOGARITHMIC SINES, COSINES, ETC. (15 Degrees.) 213 I*1' | Sine D. | Cosine 1 D. 1 Tan K. 1 D- | CoUng. 1 0 9-412996 785 I 9-984944 57 9-428052 842 10-571948 60 1 413467 784 984910 57 428557 841 571443 59 2 413938 783 984876 57 429062 840 570938 58 3 414408 783 984842 57 429566 839 570434 57 4 414878 782 984808 57 430070 838 569930 56 5 415347 781 984774 57 430573 838 5G9427 55 6 415815 780 984740 57 431075 837 568925 54 7 416283 779 984706 57 431577 836 568423 53 8 41C751 778 984072 57 432079 835 567921 52 9 417217 777 984637 57 432580 834 567420 51 10 417084 776 984603 57 433080 833 566920 50 11 9-418150 775 9-984569 57 9-433580 832 10‘50G420 49 12 418G15 774 984535 57 434080 832 5G5920 48 13 419079 773 984500 57 434579 831 565421 47 14 419544 773 984466 57 435078 830 564922 46 15 420007 772 984432 58 435576 829 564424 45 16 420470 771 984397 58 436073 828 503927 44 17 420933 770 984363 58 430570 828 563430 43 18 421395 769 984328 58 437067 827 562933 42 19 421857 768 984294 58 437563 826 562437 41 20 422318 7C7 984259 58 438059 825 561941 40 21 9-422778 767 9-984224 58 9-438554 824 10-561446 39 22 423238 766 984190 58 439048 823 500952 38 23 423697 765 984155 58 439543 823 560457 37 24 424156 764 984120 58 440036 822 559964 36 25 424G15 763 984085 58 440529 821 559471 35 26 425073 762 984050 58 441022 820 558978 34 27 425530 761 984015 58 441514 819 558486 33 28 425987 760 983981 58 442006 819 557994 32 29 42G443 760 983946 58 442497 818 557503 31 30 42G899 759 983911 58 442988 817 557012 30 31 9-427354 758 9-983875 58 9-443479 816 10-556521 29 32 427809 757 983840 59 443968 816 556032 28 33 4282G3 756 983805 59 444458 815 555542 27 34 428717 755 983770 59 444947 814 555053 26 35 429170 754 983735 59 445435 813 554565 25 36 429023 753 983700 59 445923 812 554077 24 37 430075 752 983664 59 446411 812 553589 23 38 430527 752 983629 59 446898 811 553102 22 39 430978 751 983594 59 447384 810 552616 21 40 431429 750 983558 59 447870 809 552130 20 41 9-431879 749 9-983523 59 9-448356 809 10-551644 19 42 432329 749 983487 59 448841 808 551159 18 43 432778 748 983452 59 449326 807 550074 17 44 433226 747 983416 59 449810 806 550190 16 45 433G75 746 983381 59 450294 806 549706 15 46 434122 745 983845 59 450777 805 549223 14 47 434569 744 983309 59 451260 804 548740 13 48 435016 744 983273 60 451743 803 548257 12 49 435462 743 983238 60 452225 802 547775 11 50 435908 742 983202 GO 452706 802 547294 10 51 9-43G353 741 9-983166 60 9-453187 801 10-546813 9 52 43G798 740 983130 60 453668 800 546332 8 53 437242 740 983094 60 454148 799 545852 7 54 437686 739 983058 GO 454628 799 545372 6 55 438129 738 983022 60 455107 798 544893 5 56 438572 737 9821)86 60 455586 797 544414 4 57 439014 736 982950 60 456064 796 543936 3 58 439456 736 982914 60 456542 796 543458 2 59 439897 735 982878 60 457019 795 542981 1 60 440338 734 982842 60 457496 794 542504 0' l Cams 1 Bins | 1 Colaug. | Tang. M. 74 Degrees.-0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1G 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 [60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 12 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 G 5 4 3 o 1 0 U. LOGARITHMIC SINES, COSINES, ETC D. | Cosine 1 D. 1 Tang. 1 D. | CoUng. 734 9-982842 60 9-457496 794 10*542504 733 982805 60 457973 793 542027 732 982769 61 458449 793 541551 731 982733 61 458925 792 541075 731 982696 61 459400 791 540600 730 982660 61 459875 790 540125 729 982624 61 460349 790 539651 728 982587 61 460823 789 539177 727 982551 61 461297 788 538703 727 982514 61 461770 788 538230 72G 982477 61 462242 787 537758 725 9-982441 61 9-462714 786 10-537286 724 982104 61 463186 785 536814 723 982367 61 463658 785 536342 723 982331 61 464129 784 535871 722 982294 61 464599 783 535401 721 982257 61 465069 783 534931 720 982220 62 465539 782 534461 720 982183 62 466008 781 533992 719 982146 62 466476 780 533524 718 982109 62 466945 780 533055 717 9-982072 62 9-467413 779 /0-532587 716 982035 62 467880 778 532120 716 981998 62 468347 778 531653 715 9819G1 62 468814 77.7 531186 714 981924 62 469280 776 530720 713 981886 62 469746 775 530254 713 981819 62 470211 775 529789 712 981812 62 470676 774 529324 711 981774 62 471141 773 528859 710 981737 62 471605 773 528395 710 9-981699 63 9-472068 772 10-527932 709 981GG2 63 472532 771 527468 708 981625 63 472995 771 527005 707 981587 63 473457 770 526543 707 981549 63 473919 769 526081 706 981512 63 474381 769 525619 705 981474 63 474842 768 525158 704 981436 63 475303 767 524697 704 981399 63 475763 767 524237 703 981361 63 476223 766 523777 702 9-981323 63 9-476683 765 10-523317 701 981285 63 477142 765 522858 701 981247 63 477601 764 522399 700 981209 63 478059 763 521941 699 981171 63 478517 763 521483 698 981133 64 478975 762 521025 698 981095 64 479432 761 520568 . 697 98J057 64 479889 761 520111 696 981019 64 480345 760 519655 695 980981 64 480801 759 519199 695 9-980942 64 9-481257 759 10-518743 694 980904 64 481712 758 518288 693 980866 64 482167 757 517833 693 980327 64 482G21 757 517379 692 980789 64 483075 75G 516925 691 980750 64 483529 755 516471 690 980712 64 483982 755 516018 690 980673 64 484435 754 515565 689 980635 64 484887 753 515113 688 980596 64 485339 753 514661 Sine | Coiaug. | 4 Twig. ■* 73 DegreesLOGARITHMIC SIRES, COSINES, ETC. (17 Degrees.) M. ( Sine D. Cosine D. Tans. D. | Cotftnjr. 0 0-465935 688 9-980596 64 9-485339 755 10-514001 1 460348 688 980558 G4 485791 752 514209 2 460761 687 980519 65 486242 751 5137.58 3 467173 686 980480 65 486693 751 513307 4 407585 685 980442 65 , 487143 750 512857 5 467996 685 980403 65 487593 749 512407 6 468407 684 980364 65 488043 749 511957 7 468817 683 980325 65 488492 748 511508 8 469227 683 980286 65 488941 747 511059 9 409637 682 980247 65 469390 747 510610 10 470046 681 960208 65 489838 746 510162 11 9-470455 680 9-980169 65 9-490286 746 10-5C9714 12 470863 680 960130 65 490733 745 569267 13 471271 679 980091 65 491180 744 508820 14 471079 678 980052 65 491627 744 508373 15 472086 678 980012 65 492073 743 507927 16 472492 677 979973 65 492519 743 507481 17 472898 676 979934 66 492965 742 507035 18 473304 676 979895 66 493410 741 506590 19 473710 675 979855 66 493854 740 506146 20 474115 674 97981G 66 494299 740 505701 21 9-474519 674 9-979776 66 9-494743 740 10-505257 22 474923 673 979737 66 495186 739 504814 23 475327 672 979697 66 495630 738 504370 24 475730 672 979658 66 496073 737 503927 25 476133 671 979618 G6 496515 737 503485 26 476536 G70 979579 66 496957 736 503043 27 476938 G69 979539 66 497399 736 502601 28 477340 609 979499 66 497841 735 502159 29 477741 668 979459 66 498282 734 501718 30 478142 667 979420 66 498722 734 501278 31 9-478542 G67 9-979380 66 9-499163 733 10-500837 32 478942 666 979340 66 499G03 733 500397 33 479342 665 979300 67 500042 732 499958 34 479741 665 979260 67 500481 731 499519 35 480140 664 979220 G7 500920 731 499080 36 480539 663 979180 67 601359 730 498641 37 480937 663 979140 G7 501797 730 496203 38 481334 662 979100 67 502235 729 497705 39 481731 G61 979059 67 502672 728 497328 40 482128 661 979019 67 503109 728 49G891 41 9-482525 660 9-978979 67 9-503546 727 10-496454 42 482921 659 978939 67 503982 727 496018 43 483316 659 978898 67 504418 726 495582 44 483712 058 978858 67 504854 725 495146 45 484107 657 9788J7 67 505289 725 494711 46 484501 657 978777 67 505724 • 724 494276 47 484895 656 978736 67 506159 724 493841 43 485289 655 978696 68 506593 723 493407 49 485082 655 978655 68 507027 722 492973 50 480075 654 978615 68 507460 722 492540 51 9-4804 G7 653 9-978574 68 9-507893 721 10-492107 52 480860 653 978533 68 508326 721 491674 53 487251 G52 978493 68 508759 720 491241 54 487643 651 978452 68 509191 719 vl90809 55 488034 G51 978411 68 509622 719 490378 56 488424 650 978370 68 510054 718 489946 57 488814 650 978329 68 510485 718 489515 58 489204 649 978288 68 510916 717 489084 59 489593 648 978247 68 511346 716 488654 60 489982 648 978206 68 511776 716 488224 | .Count 1 Sine 1 | CoULIIg. 1 1 Tang. 72 Degrees. 215 J___ 60 SO 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 33 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 ! | M.216 (18 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine 1 D. 1 Cosine | D. | Tang. | D. | Cotnn*. 0 9-489982 648 9-978206 68 9-511776 716 10-488224 1 490371 648 978165 68 512206 716 487794 2 490759 647 978124 68 512635 715 487365 3 491147 646 978083 69 513064 714 486936 4 491535 646 978042 69 513493 714 486507 5 491922 G45 978001 G9 513921 713 486079 6 492308 644 977959 69 514349 713 485651 7 492095 644 977918 69 514777 712 485223 8 493081 643 977877 69 515204 712 484796 9 4934G6 642 977835 69 515631 711 484309 10 493851 642 977794 • 69 516057 710 483943 11 9-494236 641 9-977752 69 9-516484 710 10*48351G 12 494G21 G41 977711 69 51G910 709 483090 13 495005 G40 977609 69 517335 709 482065 14 495388 G39 977028 69 5177G1 708 482239 15 495772 639 977586 69 518185 708 481815 1G 49G154 638 977544 70 518610 707 481390 17 49G537 637 977503 70 519034 706 480966 18 49G919 037 977401 70 519458 706 480542 19 497301 636 977419 70 519882 705 480118 20 497G82 G3G 977377 70 520305 705 479C95 21 9-498064 635 9-977335 70 9-520728 704 10*479272 22 498444 G34 977293 70 521151 703 478849 23 498825 034 977251 70 521573 703 478427 24 499204 633 977209 70 521995 703 478005 25 499584 632 977107 70 522417 702 477583 2G 499963 G32 977125 70 522838 702 477162 27 500342 631 977083 70 523259 701 476741 28 500721 631 977041 70 523G80 701 476320 29 501099 630 976999 70 524100 700 475900 30 501470 G29 970957 70 524520 699 475480 31 9-501854 629 9-970914 70 9-524939 699 10*475061 32 502231 628 970872 71 525359 698 474641 33 502007 628 970830 71 525778 698 474222 34 502984 627 970787 71 526197 697 473803 35 503300 G26 976745 71 526615 697 473385 3G 503735 626 976702 71 527033 696 472967 37 504110 G25 970000 71 527451 696 472549 38 504485 625 970617 71 527868 695 472132 39 504800 024 970574 71 528285 695 471715 40 505234 G23 970532 71 528702 694 471298 41 9-505008 623 9-976489 71 9-529119 693 10-470881 42 505981 G22 97(i446 71 529535 693 470465 43 500354 - 622 976404 71 529950 693 470050 44 500727 G21 976301 71 530366 692 4G9634 45 507099 020 070318 71 530781 , 691 469219 46 507471 020 976275 71 531196 691 468804 47 507843 619 970232 ■ 72 531G11 690 468389 48 508214 619 970189 72 532025 690 467975 49 508585 618 976146 72 532439 689 467561 50 508956 618 970103 72 532853 689 467147 51 9509326 617 9-976060 72 9-533266 688 10*466734 52 509096 616 970017 72 533679 688 4G6321 53 510005 616 975974 72 534092 687 465908 54 510434 G15 975930 72 534504 687 465496 55 510803 615 975887 72 534916 686 465084 56 511172 614 975844 72 535328 G86 464672 57. 511540 613 975800 72 535739 685 464261 58 511907 613 975757 72 536150 685 463850 59 512275 . 612 975714 72 536561 684 463439 GO 512G42 612 975670 72 536972 684 463028 GO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8. 7 6 5 4 3 2 1 0 I1 Sine I | Coiang. | |‘ Tang. I M. 71 Decrees. | Cosine | iSSgSSSESSSS 5£gg!38iJi!£l3i3£2 LOGARITHMIC SINES, COSINES, ETC. (19 Degrees.) 217 M. | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sine | D. | Cosine | D. 1 Tang. | D. Cotan*» | 9*512642 612 9*975670 73 9*530972 684 10*463028 513009 611 975627 73 537:182 683 462618 513375 611 975583 73 537792 083 402208 513741 610 "975539 73 538202 082 461798 514107 609 975496 73 538611 682 461389 514472 609 975452 73 539020 681 460980 514837 608 975408 73 539429 681 460571 515202 608 975‘165 73 539837 680 460163 515566 607 9/5321 73 540245 680 459755 515930 607 975277 73 540653 679 459347 516294 606 975233 73 541061 679 458939 9*516657 605 9*975189 73 9*541468 678 10*458532 517020 605 975145 73 541875 678 458125 517382 604 975101 73 542281 677 457719 517745 604 975057 73 542088 C77 457312 518107 603 975013 73 543094 676 456906 518468 603 974969 74 543499 676 456501 518829 002 974925 74 543905 675 450095 519190 601 974880 74 544310 675 455690 519551 601 974836 74 544715 674 455285 519911 600 974792 74 545119 674 454881 9-52,1271 600 9*974748 74 9*545524 673 10*454476 520631 599 974703 74 545928 673 451072 520990 599 974059 74 546331 672 453669 521349 598 974014 74 540735 672 453265 521707 593 974570 74 547138 671 452862 522066 597 974525 74 547540 671 452460 522424 596 974481 74 547943 670 452057 522781 596 974430 74 548345 070 451655 5231118 595 974391 74 548747 669 451*253 523495 595 974347 75 549149 669 450851 9 523852 594 9-974302 75 9*549550 668 10*450450 524208 594 974257 75 549951 668 450049 524504 593 974212 75 550352 667 449648 524920 593 974167 75 550752 667 449248 525275 592 974122 75 551152 666 448848 525630 591 974077 75 551552 666 448448 525984 591 974032 75 551952 665 448048 520339 590 973987 75 552351 665 447649 526693 590 973942 75 552750 665 447250 527046 589 973897 75 553149 664 446851 9*527400 589 9*973852 75 9*553548 664 10*446452 527753 588 973807 75 553946 663 446054 528105 588 973761 75 554344 663 445656 528458 587 973716 76 554741 662 445259 528810 587 973671 76 555139 662 444861 529161 586 9736*25 76 555.536 661 444464 529513 586 973580 76 555933 661 444007 529864 585 973535 76 556329 660 443671 530215 585 973489 76 556725 660 443275 530565 584 973444 76 557121 659 442879 9-530915 584 9*973398 76 9*557517 659 10*442483 531265 583 973352 76 557913 659 442987 531614 582 973307 76 558308 658 441692 531963 582 97326i 76 558702 658 441298 532312 581 973215 76 559097 657 440903 532661 581 973109 76 559491 657 440509 533009 580 973124 76 559885 656 440115 533357 580 973078 76 560279 656 439721 533704 579 973032 77 560673 655 439327 534052 578 972986 77 561066 655 438934 1 Coaine 1 line 1 | Cotang. Tang. 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 3 8 7 6 5 4 3 2 1 ___0^ I M. 70 Degrees. Cn CiCn&i W^biQ218 (20 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M» 1 Sine D. Cosine D. Tang. D. Cotan*. | 0 9-534052 578 9-972986 77 9 561066 655 10-438034 60 i 534399 577 972940 77 561459 054 438541 59 2 534745 577 972894 77 561851 654 438149 58 3 535092 577 972848 77 562244 653 437756 57 4 535438 576 972802 77 562636 053 437364 56 5 535783 576 972755 77 563028 653 436972 55 6 536129 575 972709 77 563419 652 436.581 54 7 536474 574 972663 77 563811 652 436189 53 8 53G818 574 972617 77 564202 651 435798 52 9 537163 573 972570 77 564592 651 435408 51 10 537507 573 972524 77 564983 650 415017 50 11 9-537851 572 9-972478 77 9-565373 650 10-434627 49 12 538194 572 972431 78 565763 649 434237 48 13 538538 571 972385 78 566153 649 433847 47 14 538880 571 972338 78 566542 649 433458 46 15 539223 570 972291 78 566932 648 433068 45 16 539565 570 972245 78 567320 648 432680 44 17 539907 569 972198 78 567709 647 432291 43 18 540249 569 972151 78 568098 647 431902 42 19 540590 568 972105 78 568486 646 431514 41 20 540931 5G8 972058 78 568873 646 431127 40 21 9-541272 567 9-972011 78 9-569261 645 10-430739 39 22 541613 567 971964 78 569648 645 430352 38 23 541953 566 971917 78 570035 645 429965 37 24 542293 566 971870 78 570422 644 429578 38 25 542632 5G5 971823 78 570809 644 429191 35 26 542971 565 971776 78 571195 643 428805 34 27 543310 564 971729 79 571581 643 428419 33 28 543649 564 971682 79 571967 642 428033 32 29 543987 563 971635 79 572352 642 427648 31 30 544325 563 971588 79 572738 642 427262 30 31 0-544663 562 9-971540 79 9-573123 641 10-420877 29 32 545000 562 971493 79 573507 641 426493 28 33 545338 561 971446 79 573892 640 426108 27 34 545674 561 971398 79 574276 640 425724 26 35 546011 560 971351 79 574660 639 425340 25 36 546347 560 971303 79 575044 639 424956 24 37 546683 559 971256 79 575427 639 424573 23 38 547019 559 971208 79 575810 638 424190 22 39 547354 558 971161 79 576193 638 423807 21 40 547689 558 971113 79 576576 637 423424 20 41 9-548024 557 9-971066 80 9-576958 637 10-423041 19 42 548359 557 971018 80 577341 636 422659 18 43 548693 556 970970 8C 577723 636 422277 17 44 549027 556 970922 80 578104 636 421896 16 45 549360 555 970874 80 578486 635 421514 15 46 549693 555 970827 80 578867 635 421133 14 47 550026 554 970779 80 579248 634 420752 13 48 550359 554 970731 80 579629 634 420371 12 49 550692 553 970683 80 580009 634 419991 11 50 551024 553 970635 80 580389 633 419611 10 51 9-551356 552 9-970586 80 9-580769 633 10-419231 9 52 551687 552 970538 ' 80 581149 632 418851 8 53 552018 552 970490 80 581528 632 418472 7 54 552349 551 970442 80 581907 632 418093 6 55 552680 551 970394 80 582286 631 417714 5 56 553010 550 970345 81 582665 631 417335 4 57 553341 550 970297 81 583043 630 410957 3 58 553670 549 970249 81 583422 630 416578 2 59 554000 549 970200 81 58:1800 629 410200 1 60 554329 548 970152 81 584177 629 415823 0 ' Cosine l Sine 1 | Cotang. 1 | Tong. 1 69 Degrees.LOGARITHMIC SINES, COSINES, ETC. (21 Degrees.) 219 ■. Sine D. Cosine D. | Tang1. D. Coding. ' 0 9-554329 548 9-970152 81 9-584177 029 10-415823 60 1 554658 548 970103 81 584555 629 415445 59 2 554987 547 970055 81 584932 628 4150G8 58 3 555315 547 97000G 81 585309 628 414691 57 4 555643 54G 969957 81 585686 627 414314 56 5 555971 546 969909 81 586062 627 413938 55 6 556299 545 969860 81 586439 027 413501 54 7 556626 545 909811 81 58G815 626 413185 53 8 556953 544 969702 81 587190 626 412810 52 9 557280 544 969714 81 587566 025 412434 51 10 557606 543 969665 81 587941 G25 412059 50 11 9557932 543 9-969616 82 9-588316 625 10-411684 49 12 558258 543 969507 82 5886D1 024 411309 48 13 558583 542 969518 82 589066 624 410934 47 14 558909 542 969469 82 589440 623 410560 46 15 559234 541 969420 82 589814 623 410186 45 16 559558 541 909370 82 590188 G23 409812 44 17 559883 540 909321 82 590562 622 409438 43 18 560207 540 969272 82 590935 622 409065 42 19 560531 539 969223 82 591308 622 408692 41 20 560855 539 969173 82 591681 621 408319 40 21 9-561178 538 9-969124 82 9-592054 621 10-407946 39 22 561501 538 969075 82 592426 020 407574 38 23 561824 537 969025 82 592798 620 407202 37 24 562146 537 968976 82 593170 619 406829 36 25 ■ 562468 536 96892G 83 593542 619 4C6458 35 26 562790 536 968877 83 593914 618 400086 34 27 563112 536 968827 83 594285 618 405715 33 28 563433 535 968777 83 594650 G18 405344 32 29 563755 535 968728 83 595027 617 404973 31 30 564075 534 968678 83 595398 617 404602 30 31 9-564396 534 9-968628 83 9-595768 617 10-404232 29 32 504716 533 968578 83 596138 616 403862 28 •33 565036 533 968528 83 596508 616 403492 27 34 565356 532 968479 83 596878 616 403122 26 35 565676 532 968429 83 597247 615 402753 25 36 565995 531 968379 83 597616 615 402384 24 37 566314 531 908329 83 597985 615 402015 23 38 566632 531 968278 83 598354 614 401646 22 39 566951 530 968228 84 . 598722 614 401278 21 40 567269 530 968178 84 599091 613 400909 20 41 9-567587 • 529 9-968128 84 9-599459 613 10-400541 19 42 567904 529 968078 84 599827 613 400173 18 43 568222 528 968027 84 600194 612 399806 17 44 568539 528 967977 84 600562 612 399438 16 45 568856 528 967927 84 600929 611 399071 15 46 569172 527 96787G 84 601296 611 398704 14 47 569488 527 967826 84 601662 611 398338 13 48 569804 520 967775 64 602029 610 397971 12 49 570120 526 967725 84 602395 610 397605 11 50 570435 525 967G74 84 602701 610 397239 10 51 9-570751 525 9-967624 84 9-603127 609 10-396873 9 52 571060 524 967573 84 603493 609 396507 8 53 571380 524 967522 85 603858 009 396142 7 54 571G95 523 967471 85 004223 608 395777 6 55 572009 523 967421 85 604588 608 395412 5 56 572323 523 967370 85 604953 607 395047 4 57 572030 522 967319 85 605317 607 394683 3 58 572950 522 967268 85 605682 607 394318 2 59 573263 521 967217 85 606046 606 393954 1 69 573575 521 967166 85 606410 606 393590 0 [ Come Sine | Coiang. 1 | Tang. G8 Degrees.220 (22 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine D. Cosine D. Tung. D. | Cotan». 0 9-573575 521 9-967166 85 9-606410 606 10 393590 G4 1 573888 520 907115 85 600773 606 393227 59 2 574200 520 967064 85 607137 605 392863 58 3 574512 519 9G7013 85 607500 605 392500 57 4 574824 519 966961 85 007863 604 392137 56 5 575136 519 9G6910 85 608225 604 391775 55 6 575447 518 966859 85 608588 604 391412 54 7 575758 518 966808 85 608950 603 391050 53 8 57G0G9 517 966756 86 609312 603 390688 52 9 57G379 517 966705 86 609674 603 390326 51 ]0 57GG89 516 966653 86 010036 602 389964 50 11 9-570999 516 9-966602 86 9610397 602 10-389603 49 12 577309 516 966550 86 610759 602 389241 48 13 577G18 515 9C6499 86 611120 601 388880 47 14 577927 515 966447 86 611480 601 388520 46 15 578236 • 514 960395 86 611841 601 388159 45 1G 578545 514 966344 86 612201 600 387799 44 17 578853 513 966292 86 612561 600 387439 43 18 5791G2 513 966240 86 612921 600 387079 42 19 579470 513 9GC188 86 613281 599 386719 41 20 579777 512 9G6I3G 86 613641 599 386359 40 21 9-580085 512 9 966085 87 9-614000 598 10-386000 39 22 580392 511 966033 87 614:159 598 385641 38 23 5S0G99 511 965981 87 614718 598 385282 37 24 581005 511 965928 87 615077 597 384923 36 25 581312 510 905876 87 615435 597 384565 35 2G 581G18 510 965824 87 615793 597 384207 34 27 581924 509 965772 87 610151 596 383849 33 28 532229 509 905720 87 616509 596 383491 32 29 582535 509 965608 87 616867 596 383133 31 30 582840 508 965015 87 617224 595 382776 30 31 9-583145 508 9-965563 87 9-617582 595 10-382418 29 32 583449 507 96551r 87 617939 595 382061 28 33 583754 507 905458 87 618295 594 381705 27 34 584058 506 965406 87 618652 594 381348 26 35 5843G1 506 965353 88 619008 594 380992 25 3G 5H4GG5 506 965301 88 619364 593 380636 24 37 5849G8 505 9G5248 88 619721' 593 380279 23 as 585272 505 965195 88 620076 593 379924 22 39 585574 504 965143 68 620432 592 379568 21 40 585877 504 965090 88 620787 592 379213 20 41 9586179 503 9-965037 88 9-621142 592 10-3788.58 19 42 580482 503 964984 88 621497 591 378503 18 43 53G783 503 964931 88 621852 591 378148 17 44 587085 502 964879 88 622207 590 377793 16 45 58738G 502 964826 88 622561 590 377439 15 40 587688 501 964773 88 622915 590 377085 14 47 587989 501 904719 88 623269 589 376731 13 48 588289 501 964006 89 623623 589 376377 12 49 588590 500 964013 89 623976 589 376024 11 50 588890 500 964560 89 624330 588 375670 10 51 9-589190 499 9-964507 89 9-624683 588 10-375317 9 52 539489 499 964454 89 62.5036 588 374964 8 53 589789 499 964400 89 625388 587 374612 7 54 590088 498 964347 89 625741 587 374259 6 55 590387 498 964294 89 C26093 587 373907 5 5G 590686 497 964240 89 626445 586 373555 4 57 590984 497 964187 89 626797 586 373203 3 58 591282 497 964133 89 627149 58G 372851 3 59 591580 496 964080 89 627501 585 372499 1 60 591878 496 964026 89 627852 585 372148 0 - l Coune | Sine | Cotang-. |- Tang. ) - 67 Degrees.LOGARITHMIC SINES, COSINES, ETC. (23 Degrees.) 221- M. Sine D. Cosine D. Tan?. D. Cotang. | 0 9-591878 496 9-964026 89 9-627852 585 10-372148 60 1 592176 495 963972 89 628203 585 371797 59 2 592473 495 963919 89 628554 585 371446 58 3 592770 495 963865 90 628905 584 371095 57 4 593067 494 963811 90 629255 584 370745 56 5 593363 494 963757 90 629606 583 370394 55 6 593659 493 963704 90 629950' 583 370044 54 7 593955 493 963650 90 630306 583 369694 53 8 594251 493 963596 90 630656 583 369344 52 9 594547 492 963542 90 631005 582 368995 51 10 594842 492 963488 90 631355 582 368645 50 ]1 9-595137 491 9-963434 90 9-631704 582 10-368296 49 12 595432 491 963379 90 632053 581 367947 48 13 595727 491 963:125 90 632401 581 367599 47 14 596021 490 963271 90 632750 581 367250 46 15 596315 490 963217 90 633098 580 366902 45 16 596609 489 963163 90 633447 580 366553 44 17 596903 489 963108 91 633795 580 366205 43 18 597196 489 963054 91 634143 579 365857 42 19 597490 488 962999 91 634490 579 365510 41 20 597783 488 962945 91 634838 579 365162 40 21 9-598075 487 9-962890 91 9-635185 578 10-364815 39 22 598368 487 962836 91 635532 578 364468 38 23 598660 487 962781 91 635879 578 364121 37 24 598952 486 •62727 91 636226 577 363774 36 25 599244 486 962672 91 636572 577 363428 35 26 599536 485 962617 91 636919 577 363081 34 27 599827 485 962562 91 637265 577 362735 33 28 600118 485 962508 91 637611 576 362389 32 29 600409 484 962453 91 637956 576 362044 31 30 600700 484 962398 92 638302 576 361698 30 31 9-600990 484 9-962343 ’ 92 9-638647 575 10-361353 29 32 601280 483 962288 92 638992 575 361008 28 33 601570 483 962233 92 639337 575 360663 27 34 601860 482 962178 92 639682 574 360318 26 35 602150 482 962123 92 640027 574 359973 25 36 602439 482 962067 92 640371 574 359629 24 37 602728 481 962012 92 640716 573 359284 23 38 603017 481 961957 92 641060 573 358940 22 39 603305 481 961902 92 641404 573 358596 21 40 603594 480 961846 92 641747 572 358253 20 41 9-603882 480 9-961791 92 9-642091 572 10 357909 19 42 604170 479 961735 92 642434 572 357566 18 43 604457 479 961680 92 642777 572 357223 17 44 604745 479 961624 93 643120 571 356880 16 45 605032 478 961569 93 643463 571 350537 15 46 605319 478 961513 93 643806 571 356194 14 47 605606 478 961458 93 644148 570 355852 13 48 605892 477 961402 93 644490 570 355510 12 49 606179 477 9ul346 93 644832 570 355168 11 50 606465 476 961290 93 645174 569 354826 10 51 9-606751 476 9-961235 93 9-645516 569 10-354484 9 52 607036 476 961179 93 645857 569 354143 8 53 607322 475 961123 93 646199 569 353801 7 54 607607 475 961067 93 646540 568 353460 6 55 607892 474 961011 93 646881 568 353119 5 56 608177 474 960955 93 647222 568 352778 4 57 608461 474 960899 93 647562 567 352438 3 58 608745 473 960843 94 647903 567 352097 2 59 609029 473 960786 94 648243 567 351757 1 00 609313 473 960730 94 648583 566 351417 0 [ Coane 1 | Sine | Cotang. _ | Tong. M, 66 Degrees.gfefcSSsfttfcSi gggtSgiSEBSS g£ 222 (24 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. | 0 ] 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Sine D. Coaioe D. Tang. | D. Coumg. 9-609313 473 9-960730 94 9-648583 566 10-351417 609597 472 960674 94 648923 566 351077 609880 472 960618 94 649263 566 350737 610164 472 960561 94 649602 566 350398 610447 471 960505 94 649942 565 350058 610729 471 960448 94 650281 565 349719 611012 470 960392 94 650620 565 349380 611294 470 960335 94 650959 564 349041 611576 470 960279 94 651297 564 348703 611858 469 960222 94 651636 564 348364 612140 469 960165 94 651974 563 348026 9-612421 469 9-960109 95 9-652312 563 10-347688 612702 •468 960052 95 652650 563 347350 6J2983 468 959995 95 652988 563 347012 613264 467 9599118 95 653326 562 346674 613545 467 959882 95 653663 562 346337 613825 467 959825 95 654000 562 346000 614105 466 959768 95 654337 561 345663 614385 466 959711 95 654674 561 345326 614665 466 959654 95 655011 561 344989 614944 465 959596 95 655348 561 344652 9-615223 465 9-959539 95 9-655684 560 10-344316 615502 465 959482 95 656020 560 343980 615781 464 959425 95 656356 560 343644 616060 464 959368 95 656092 559 343308 616338 464 959310 96 657028 £59 342972 616616 463 959253 96 657364 559 342636 616894 463 959195 96 657699 559 342301 617172 462 959138 96 658034 558 341966 617450 462 959081 96 6.58369 558 341631 617727 462 959023 96 658704 558 341296 9-618004 461 9-958965 96 9-659039 558 10-340961 618281 461 958908 96 659373 557 340627 618558 461 958850 96 659708 557 340292 618834 460 958792 96 660042 557 339958 619110 460 958734 96 660376 557 339624 619386 460 958677 96 660710 556 339290 619662 459 958619 96 661043 556 338957 619938 459 958561 96 661377 556 338623 620213 459 958503 97 661710 555 338290 620488 458 958445 97 662043 555 337957 9-620763 458 9-958387 97 9-662376 555 10-337624 621038 457 958329 97 662709 554 337291 621313 457 9.58271 97 663042 554 336958 621.587 457 958213 97 663375 554 336625 621861 456 958154 97 663707 554 336293 622135 456 958096 97 664039 553 335961 622409 456 958038 97 664371 553 335629 622682 455 957979 97 664703 553 335297 622956 455 957921 97 66.5035 553 334965 623229 455 957863 97 665366 552 334634 9-623502 454 9-957804 97 9-665607 652 10-334303 623774 454 957746 98 666029 552 333971 624047 4.54 957687 98 666360 551 333640 624319 453 957628 98 666691 551 3:13309 624591 453 957570 98 667021 551 332979 624863 453 957511 98 667352 551 332648 625135 452 957452 98 667682 550 332318 625406 452 957393 98 668013 550 331987 625677 452 957335 98 668343 550 331657 625948 451 957276 98 668672 550 331328 | Coaiiie | Sue Cotaog. l Tang. 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0^ I M. 65 Degrees. C>» trt V* to uM- 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2*5 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 41 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 M. SINES, COSINES, ETC. (25 Degrees.) D. Cosine .D- Tang. D. Cotang. | 451 9-957276 98 9-668673 550 10-331327 451 957217 98 669002 549 330998 451 957158 98 669332 549 3:10668 450 957099 98 669661 549 330339 450 957040 98 669991 548 330009 450 956981 98 670320 548 329680 449 956921 99 670649 548 329351 449 956862 99 670977 548 329023 449 956803 99 671306 547 328694 448 956744 99 671634 547 328366 448 956684 99 671963 547 328037 447 9-956625 99 9-672291 547 ' 10-327709 447 956566 99 672619 546 327:181 447 956506 99 672947 546 327053 446 956447 99 673274 546 326726 446 956:187 99 673602 546 326:198 446 956327 99 673929 545 326071 446 956268 99 674257 545 325743 445 956208 100 674584 545 325416 445 956148 100 674910 544 325090 445 956089 100 675237 544 324763 444 9-956029 100 9-675564 544 10-324436 444 955969 100 675890 544 324110 444 955909 100 676216 543 323784 443 955849 100 676543 543 323457 443 . 955789 100 676869 543 323131 443 955729 100 677194 543 322806 442 955669 100 677520 542 322480 442 955609 100 677846 542 322154 442 955548 100 678171 542 321829 441 955488 100 678496 542 321504 441 9-955428 101 9-678821 541 10-321179 440 955368 101 679146 541 320854 440 955307 101 679471 541 320529 440 955247 101 679795 541 320205 439 955186 101 680120 540 319880 439 955126 101 680444 540 319556 439 955065 101 680768 540 319232 438 955005 101 681092 540 318908 4:18 954944 101 681416 539 318584 438 954883 101 681740 539 318260 437 9-954823 101 9-682063 539 10-317937 437 954762 101 682387 539 317613 437 954701 101 682710 538 317290 437 954640 101 683033 538 316967 436 954579 101 683.356 538 316644 436 954518 102 683679 538 316321 436 954457 102 684001 537 315999 435 954396 102 684324 537 315676 435 954335 102 684646 537 315354 435 954274 102 684968 537 315032 434 9-954213 102 9-685290 536 10-314710 434 954152 102 685612 536 314388 434 954090 102 685934 536 314066 433 954029 102 686255 536 313745 433 953968 102 686577 535 313423 433 953906 102 686898 535 313102 432 953845 102 687219 535 312781 432 953783 102 687540 535 312460 432 953722 103 687861 534 312139 * 431 953660 103 688182 534 ' 311818 | Sine Coiang. 1 1 Tang. 64 Degrees.g£gt3gUi!i5i3 224 (26 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. I 0 1 2 3 4 5 6 7 e 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ! Sine D. Cosine D. Tang. D. Cotang. . 9-641842 431 9-953660 103 9-688182 534 10-311818 612101 431 953599 103 688502 534 311498 642300 421 953537 103 688823 534 311177 642018 430 953475 103 689143 533 310857 642877 430 953413 103 689463 533 310537 643135 430 953352 103 •689783 533 310217 643393 430 953290 103 690103 533 309897 643050 429 953228 103 690423 533 309577 643908 429 953106 103 690742 532 309258 644165 429 953104 103 6910G2 532 308938 644423 428 953042 103 691381 532 308619 9-044080 . 428 9-952980 104 9-691700 531 10-308300 614930 428 952918 104 692019 531 307981 645193 427 952855 104 692338 531 307602 645450 427 952793 104 692056 531 307344 645700 427 952731 104 692975 531 307025 645962 426 9520C9 104 693293 530 300707 640218 426 952CC6 104 693012 530 306388 646474 426 952544 104 693930 530 306070 646729 425 952481 104 694248 530 305752 640984 425 952419 104 694566 529 305434 9-647240 425 9-952356 104 9-694883 529 10-305117 647494 424 952294 104 695201 529 304799 647749 424 952231 104 695518 529 304482 648004 424 952108 105 69583G 529 304164 648258 424 95210G 105 G96I53 528 303847 648512 423 952043 105 690470 528 303530 648700 423 951980 105 690787 528 303213 649020 423 951917 105 697103 528 302897 649274 422 951854 105 697420 527 302580 649527 422 951791 105 697736 527 302264 9-649781 422 9-951728 105 9-698053 527 10-301947 650034 422 951005 105 698369 527 301631 650287 421 951002 105 698085 520 301315 650539 421 951539 105 699001 526 300999 650792 421 951476 105 699316 526 300684 651041 420 951412 105 699632 526 300368 651297 420 951349 106 699947 526 300053 651549 420 951286 106 700263 525 299737 651800 419 951222 106 700578 525 299422 652052 419 951159 106 700893 525 299107 9 652304 419 9-951096 10G 9-701208 524 10-298792 652555 418 951032 106 701523 524 298477 652806 416 950968 100 701837 524 298103 653057 418 950905 106 702152 524 297848 653308 418 950841 10G 702406 524 297534 653558 417 950778 106 702780 523 297220 653808 417 950714 100 703095 523 296905 654059 417 950650 106 703409 523 296591 . 654309 416 950586 106 703723 523 296277 654558 416 950522 107 704036 522 295964 654808 416 9-950458 107 9-704350 522 10-295650 655058 416 950394 107 704663 522 295337 655307 415 950330 107 704977 522 295023 655556 415 950266 107 705290 522 294710 655805 415 950202 107 705603 521 294397 656054 414 950138 107 705910 521 294084 656302 414 950074 107 706228 521 293772 656551 414 950010 107 706541 521 293459 650799 413 949945 107 706854 521 293146 1* 657047 413 949881 107 707166 520 292834 J Coame l | Sme 1 Coian£. Tang. j 63 Degrees. LOGARITHMIC SINES, COSINES, ETC. (27 Degrees.) 225 M. Sine D. | Cosine D. Tan g. D. Co tang*. 0 9-657047 413 9-949881 107 9-707166 520 10-292834 60 1 657295 413 949816 107 707478 520 292522 59 2 657542 412 949752 107 707790 520 292210 58 3 657790 412 949688 108 708102 520 291898 57 4 658037 412 949623 108 708414 519 291586 56 5 658284 412 949558 108 708726 519 291274 55 G 658531 411 949494 108 709037 519 290963 54 7 658778 411 949429 108 709349 519 290651 53 8 659025 411 949364 108 709660 519 290340 52 9 659271 410 949300 108 709971 518 290029 51 10 659517 410 949235 108 710282 518 289718 50 11 9-659763 410 9-949170 108 9-710593 518 10-289407 49 12 660009 409 949105 108 710904 518 289096 48 13 660255 409 949040 108 711215 518 288785 47 14 660501 409 948975 108 711525 517 288475 46 15 660746 409 948910 108 711836 517 288164 45 lli 660901 408 948845 108 712146 517 287854 44 17 661236 408 948780 109 712456 517 287544 43 H 661481 408 948715 109 712766 516 287234 42 19 661726 407 948650 109 713076 516 286924 41 20 661970 407 948584 109 713386 516 286614 40 21 9-662214 407 9-948519 109 9-713696 516 10-286304 39 22 662459 407 948454 109 714005 516 285995 38 23 662703 406 948388 109 714314 515 285686 37 24 662946 406 948323 109 714624 515 285376 36 25 663190 406 948257 109 714933 515 285067 35 26 663433 405 948192 109 715242 515 284758 34 27 663677 405 948126 109 715551 511 284449 33 28 663920 405 948060 109 715860 514 284140 32 29 664J63 405 947995 110 716168 514 283832 31 30 664406 404 947929 110 716477 514 283523 30 31 9-664648 404 9-947863 110 9-716785 514 JO-283215 29 32 664891 404 947797 no 717093 513 282907 28 33 665133 403 947731 no 717401 513 282599 27 34 665375 403 947665 no 717709 513 282291 26 35 665617 403 947600 no 718017 513 281983 25 3G 665859 402 947533 no 718325 513 281675 24 37 666100 402 947467 no 718633 512 281367 23 38 666342 402 947401 no 718940 512 281060 22 39 666583 402 947335 no 719248 512 280752 21 40 666824 401 947269 no 719555 512 280445 20 41 9-667065 401 9-947203 no 9-719862 512 10-280138 19 42 667305 401 94713G in 720169 511 279831 18 43 667546 401 947070 in 720476 511 279524 17 44 667786 400 947004 in 720783 511 279217 16 45 668027 400 946937 in 721089 511 278911 15 46 668267 400 946871 in 721396 511 278604 14 47 668506 399 946804 in 721702 510 278298 13 48 668746 399 946738 in 722009 510 277991 12 49 668986 399 946671 in 722315 510 277685 11 50 669225 399 946604 in 722621 510 277379 10 51 9-669464 398 9-946538 in 9-722927 510 10-277073 9 52 669703 398 946471 in 723232 509 276768 8 53 669942 398 946404 in 723538 509 276462 7 54 670181 397 946337 in 723844 509 276156 6 55 670419 397 946270 112 724149 509 275851 5 56 670658 397 946203 112 724454 509 275546 4 57 6711896 397 946136 112 724759 508 275241 3 58 671134 396 946069 112 725065 508 274935 2 59 671372 396 946002 112 725369 508 274631 1 60 671609 396 945935 112 725674 508 274326 0 Coaine Sioe | Coiaog. Tang. | M. 62 Degrees.38SESSS2 ££&££&*&&£ fegBJSgjSSggBSS ggigiSggSBSiS 220 (28 Degrees.) LOGARITHMIC SINES, COSINES, ETC. | Sine D. Cosine D. Tang. D. Colony. | 9-671609 396 9-945935 112 9-725674 508 10-274326 60 671847 395 945868 112 725979 508 274021 59 672084 395 945800 112 726284 507 273716 58 672321 395 945733 112 726588 507 273412 57 672558 395 945666 112 726892 507 273108 56 672705 394 945598 112 727197 507 272803 55 673032 394 945531 112 727501 507 272499 54 673268 394 945464 113 727805 506 272195 53 673505 394 945:196 113 728109 506 271891 52 673741 393 945.128 113 728412 506 271588 51 673977 393 945261 113 728716 506 271284 50 9-674213 393 9-945193 113 9-729020 506 10-270980 49 674448 392 945125 113 729323 505 270677 48 674684 392 945058 113 729626 505 270374 47 674019 392 944990 113 729929 505 270071 46 675155 392 944922 113 730233 505 269767 45 675390 391 944854 113 730535 ■505 269465 44 675624 391 944786 113 730838 504 269162 43 675859 391 944718 113 731141 504 268859 42 676094 391 944650 113 731444 504 268556 41 676328 390 944582 114 731746 504 268254 40 9-676562 390 9-944514 114 9-732048 504 10-267952 39 676796 390 944446 114 732351 503 267649 38 677030 390 944377 114 732653 503 267347 37 677264 389 944309 114 732955 503 267045 36 677498 389 944241 114 733257 503 266743 35 677731 389 944172 114 733558 503 266442 34 677964 388 944104 114 733860 502 266140 33 678197 388 944036 114 734162 502 265838 32 678430 388 943967 114 734463 502 265537 31 678663 388 943899 114 734764 502 265236 30 9-678805 387 9943830 114 9-735066 502 10-264934 29 670128 387 943761 114 735367 502 264633 28 679:160 387 943693 115 735668 501 264332 27 670502 387 943624 115 735969 501 264031 26 670824 386 943555 115 736269 501 263731 25 680056 386 943486 115 736570 501 263430 24 680288 386 943417 115 736871 501 263129 23 680519 385 943348 115 737171 500 262829 22 680750 385 94:1279 115 737471 500 262529 21 680982 385 943210 115 737771 500 262229 20 9681213 385 9943141 115 9-738071 500 10-261929 19 681443 384 943072 115 438371 500 261629 18 681674 384 943003 115 738671 499 261.'129 17 681905 384 942934 115 738971 499 261029 16 682135 384 942864 115 739271 499 260729 15 682365 383 942795 116 739570 499 260430 14 682595 383 942726 116 739870 499 260130 13 682825 383 942656 116 740169 499 259831 12 683055 383 942587 116 740468 498 259532 11 683284 382 942517 116 740767 498 259233 10 9-683514 382 9942448 116 9-741066 498 10-258934 9 683743 382 942:178 116 741365 498 258635 8 683972 382 942308 116 741664 498 258336 7 684201 381 942239 116 741962 497 258038 6 684430 381 942169 116 742261 497 257739 5 684658 381 942099 116 742559 497 257441 4 684887 380 942029 116 742858 497 257142 3 685115 380 941959 116 743156 497 256844 o 685343 380 941889 117 743454 497 250546 I 685571 380 941819 117 743752 496 256248 0 1 Cosine l Siue | Cuiang. 1 Tang. M. 61 Degrees.M. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 '18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 '23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 J) K LOGARITHMIC SINES, COSINES, ETC. (29 Degrees.) | D. | Cosine | D. | Tang. | D. | 380 9941819 117 9-743752 496 10-256248 379 941749 117 744050 406 255050 379 941679 117 • 744348 406 255652 379 941609 117 744645 496 255355 379 941539 117 744943 406 255057 378 941469 117 745240 496 251760 378 941308 117 745538 495 254462 378 941328 117 745835 405 254165 378 941258 117 746132 405 253868 377 941187 117 746429 405 253571 377 941117 117 746726 495 253274 377 9-941046 118 9-747023 494 10-252977 377 940975 118 747319 494 252681 376 940005 118 747616 494 252384 376 940834 118 747013 494 252087 376 940763 118 748209 494 251791 376 940603 118 748505 493 251495 375 940622 118 748801 493 251199 375 940551 118 749097 493 250003 375 940480 118 749393 493 250607 375 940409 118 749689 493 250311 374 9-940338 118 9-749985 493 10-250015 374 940267 118 750281 492 240719 374 040106 118 750576 492 240424 374 940125 119 750872 492 240128 373 940054 119 ■751167 492 248833 373 939982 119 751462 492 248538 373 939911 119 751757 492 248243 373 939840 119 752052 491 247048 372 930768 119 752347 491 247653 372 930697 119 752642 491 247358 372 9-939625 119 9-752937 491 10 247063 371 939554 119 753231 401 246769 371 939482 119 753526 491 246474 371 930410 119 753820 490 246180 371 930339 119 754115 490 245885 370 939267 120 754409 490 245591 370 939195 120 754703 490 245207 370 939123 120 754997 490 245003 370 939052 120 755291 490 244709 369 938980 120 755585 489 244415 369 9-938908 120 9-755878 489 10-244122 369 938836 120 756172 489 243828 369 938763 120 756465 489 243535 368 938691 120 756759 489 243241 368 938619 120 757052 489 242948 368 938547 120 757345 488 242655 368 938475 120 757638 488 242362 367 938402 121 757931 488 242069 367 938330 121 758224 488 241776 367 938258 121 758517 488 241483 367 9-938185 121 9-758810 488 10-241190 366 938113 121 759102 487 240898 366 938040 121 759395 487 240605 366 937967 121 759687 487 240313 366 937895 121 750979 487 240021 365 937822 121 760272 497 239728 365 937749 121 760564 487 239436 365 937676 121 760856 486 239144 365 937604 121 761148 486 238852 364 937531 121 761439 486 238561 Sine Cotang. Tin*. | 60 Degrees.228 (30 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine D. Cosine D. Tang. D. Cotang. 0 9 698970 364 9 937531 121 9-761439 486 10-238561 60 1 699189 364 937458 122 761731 486 238269 59 2 699407 364 937385 • 122 762023 486 237977 58 3 C99G26 364 937312 122 762314 486 237686 57 4 699844 363 937238 122 762606 485 237394 56 5 700062 363 937165 122 762897 485 237103 55 6 700280 363 937092 122 763188 485 236812 54 7 700498 363 937019 122 763479 485 236521 53 8 700716 363 936946 122 763770 485 236230 32 9 700933 362 936872 122 764061 485 235939 51 JO 701151 362 936799 122 764352 484 235648 50 11 9-701368 362 9-936725 122 9-764643 484 10-235357 49 J2 701585 362 93G652 123 764933 484 235067 48 13 701802 361 936578 123 765224 484 234776 47 J4 702019 361 936505 123 765514 484 234486 46 15 702236 361 936431 123 765805 484 234195 45 10 702452 361 936357 123 766095 484 233905 44 17 702669 360 936284 123 766:185 483 233615 43 18 702885 360 936210 1*3 766675 483 233325 42 19 703101 360 936136 123 766965 483 233035 41 20 703317 360 936062 123 767255 483 232745 40 21 9-703533 359 9-935988 123 9-7G7545 483 10-232455 39 22 703749 359 935914 123 767834 483 232166 38 23 703964 359 935840 123 768124 482 231876 37 24 7041.'9 359 915766 124 768413 482 231587 36 25 704395 359 935692 124 768703 482 231297 35 26 704610 358 935618 124 768992 482 231008 34 27 704825 358 935543 124 769281 482 230719 - 33 28 705040 358 9354G9 124 7G9570 482 230430 32 29 705254 358 935395 124 769860 481 230140 31 30 705469 357 935320 124 770148 481 229852 30 31 9-705683 357 9-935246 124 9-770437 481 10-229563 29 32 705898 357 935171 124 770726 481 229274 28 33 706112 357 935097 124 771015 481 228985 27 34 706326 356 935022 124 ' 771303 481 228697 26 35 706539 356 934948 124 771592 481 228408 25 36 706753 356 934873 124 771880 480 228120 24 37 706967 356 934798 125 772168 480 227832 23 38 707180 355 934723 125 772457 480 227543 22 39 707393 355 934649 125 772745 480 2272.55 21 40 707606 355 934574 125 773033 480 226967 20 41 0-707819 355 9-934499 125 9-773321 480 10-226679 19 42 708032 354 934424 125 773608 479 226392 18 43 708245 354 934349 125 773896 479 226104 17 44 708458 354 934274 125 774184 479 225816 16 45 708670 354 934199 125 774471 479 225.529 15 46 708882 353 934123 125 774759 479 225241 14 47 709094 353 934048 125 775046 479 224954 13 48 709306 353 933973 125 775:133 479 224667 12 49 709518 353 933898 126 775621 478 224379 11 50 709730 353 933822 126 775908 478 224092 10 51 9-709941 352 9-933747 126 9-776195 478 10-223805 9 52 710153 352 933671 126 776482 478 223518 8 53 710364 352 933596 12G 776769 478 223231 7 54 710575 352 933520 126 777055 478 222945 6 55 710786 351 933445 126 777342 478 222658 5 5G 710997 a51 933369 126 777628 477 222372 4 57 711208 351 93:1293 1£G 777915 477 222085 3 58 711419 351 933217 126 778201 477 221799 2 59 711629 350 933141 126 778487 477 221512 1 GO 711839 350 933066 126 778774 477 221226 0 | Coaiue | | Siue | | Cotang. I J Tang- J U* 69 DegreesLOGARITHMIC SINES, COSINES, ETC. (31 Degrees.) 229 H. Sine D. Cosine D. Trng. | D. Cotnng. 0 9-711839 350 9-933066 126 9-778774 477 10-221226 60 1 712050 350 932990 127 779060 477 2209)0 59 o 712260 350 932914 127 77934G 476 220654 58 3 71246!) 349 932838 127 779632 476 220368 57 4 712679 349 932762 127 779918 476 220082 56 5 712889 349 932685 127 780203 476 219797 55 6 713098 349 932609 127 780489 476 219511 54 7 713308 349 932533 127 780775 476 219225 53 8 713517 348 932457 127 781060 476 218940 52 9 713726 348 932380 127 781346 475 218654 51 10 713935 348 932304 127 781631 475 218369 50 11 9-714144 348 9-932228 127 9-781916 475 10-218084 49 12 714352 347 932151 127 782201 475 217799 48 13 714561 347 932075 128 78248G 475 217514 47 14 714769 347 931998 128 782771 475 217229 46 15 714978 347 931921 128 783056 475 216944 45 16 715186 347 931845 128 783341 475 216659 44 17 715394 346 931768 128 783620 474 216374 43 18 715602 '346 931691 128 783910 474 216090 42 19 715809 346 931G14 128 784195 474 215805 41 20 716017 346 931537 128 784479 474 215521 40 21 9-710224 345 9-931460 128 9-784764 474 10-215236 39 22 716432 345 931383 128 785048 474 214952 38 23 716639 345 931306 128 785332 473 214668 37 24 716846 345 931223 129 785616 473 214384 36 25 717053 345 931152 129 785900 473 214100 35 26 717259 344 931075 129 786184 473 213816 34 27 717466 344 930998 129 786468 473 213532 33 28 717673 344 930921 129 786752 473 213248 32 29 717879 344 930843 129 787036 473 212964 31 30 718085 343 93076G 129 787319 472 212681 30 31 9-7182DI 343 9-930688 129 9-787603 472 10*212397 29 32 718497 343 930611 129 787886 472 212114 28 33 718703 343 939533 129 788170 472 211830 27 34 718909 343 930456 129 788453 472 211547 26 35 719114 342 930378 129 788736 472 211264 25 36 719320 342 930300 130 789019 472 210981 24 37 719525 342 930223 130 789302 471 210G98 23 38 719730 342 930145 130 789585 471 210415 22 39 719935 341 930067 130 789868 471 210132 21 40 720140 341 929989 130 790151 471 209849 20 41 9-720345 341 9-929911 130 9-790433 471 10-209567 19 42 720549 341 929833 130 790716 471 209284 18 43 720754 340 929755 130 790999 471 209001 17 44 720958 340 929677 130 791281 471 208719 16 45 721162 340 929599 130 791563 470 208437 15 46 721366 340 929521 130 791846 470 208154 14 47 721570 340 929442 130 792128 470 207872 13 48 721774 339 929364 131 792410 470 207590 12 49 721978 339 929286 131 792692 470 207308 11 50 722181 339 929207 131 792974 470 207026 10 51 9-722385 339 9-929129 131 9-793256 470 10-206744 9 52 722588 339 929050 131 793538 469 206462 8 53 722791 338 928972 131 793819 469 206181 7 54 722994 338 928893 131 794101 469 205899 6 55 723197 338 928815 131 794383 469 205617 5 56 723400 338 928736 131 794664 469 205336 4 57 723603 337 928657 131 794945 469 205055 3 58 723805 337 928578 131 795227 469 204773 2 59 724007 337 928499 131 795508 468 204492 1 60 724210 337 928420 131 795789 468 204211 0 | Cosine Sine Cotang. 1 Tang. | M. 58 Degrees, 230 (32 Degrees.) LOGARITHMIC SINES, COSINES, ETC• M. | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 no Sine | D. Cosine | D. Tanff. D. Cotang, f 9-724210 337 9-928420 132 9-795789 468 10-204211 724412 337 928:142 132. 796070 468 203930 724614 336 928263 132 796351 468 203649 724816 336 928183 132 796632 468 203368 725017 336 928104 132 796913 468 203087 7252J9 336 928025 132 797194 468 202806 725420 335 927946 132 797475 468 202525 725622 335 927867 132 797755 468 202245 725823 335 927787 132 798036 467 201964 726024 335 927708 132 798316 467 201684 726225 335 927629 132 798596 467 201404 9-726426 334 9-927549 132 9*798877 467 10-201123 720626 334 927470 133 799157 467 200843 726827 334 927390 133 799437 467 200563 727027 334 927310 133 799717 467 200283 727228 334 927231 133 799997 466 200003 727423 333 927151 133 800277 466 199723 727628 333 927071 133 800557 466 199443 727828 333 926991 133 800836 466 199164 728027 333 926911 133 801116 466 198884 728227 333 926831 133 801396 466 198604 9-728427 332 9-926751 133 9-801675 466 10-198325 728626 332 926671 133 801955 466 198045 728825 332 926591 133 802234 465 197766 729024 332 926511 134 e02513 465 197487 729223 331 926431 134 802792 465 197208 729422 331 926351 134 803072 465 196928 729621 331 926270 134 803351 465 196649 729820 331 926190 134 803630 465 196370 730018 330 926110 134 803908 465 196092 730210 330 926029 134 804187 465 195813 9-730415 330 9-925949 134 9-804466 464 10195534 730613 330 925868 134 804745 464 195255 730811 330 925788 134 805023 464 194977 731009 329 925707 134 805302 464 194698 731206 329 925626 134 805580, 464 194420 731404 329 925545 135 805859 464 194141 731602 329 925465 135 806137 464 193863 731799 329 925384 135 806415 463 193585 731996 328 925303 135 806693 463 193307 732193 328 925222 135 806971 463 193029 9-732390 328 9-925141 135 9-807249 463 10192751 732587 328 925060 135 807527 403 192473 732784 328 924979 135 807805 463 192195 732980 327 924897 135 808083 463 191917 733177 327 924816 135 808361 463 191039 733373 327 924735 136 808638 462 1913G2 733569 327 924C54 136 808916 462 191084 733705 327 924572 136 809193 462 190807 733961 326 924491 136 809471 462 190529 734157 326 924409 136 809748 462 190252 9-734353 326 9-924328 136 9-810025 462 10-189975 734549 326 924246 136 810302 462 189698 734744 325 924164 136 810580 462 189420 734939 325 924083 136 810857 462 189143 735135 325 924001 136 811134 461 188866 735330 325 923919 136 811410 461 188590 735525 325 923837 136 811687 461 . 188313 735719 324 923755 137 811964 461 188036 735914 324 923673 137 812241 461 137759 736109 324 923591 137 812517 461 187483 | Cosine 1 [ Sine 1 Cotang; fiuigv 1 57 Degrees. 28BS8838S S2SSS2SLOGARITHMIC SIXES, COSINES, ETC. (33 Degrees.) 231 M. ' Sine D. Cosine 1 D. Tang. D. Cotang-. 1 0 9-736109 324 9-923591 137 9-812517 461 10-187482 00 1 736303 324 923509 137 8127114 461 187206 59 2 736498 324 923427 137 813070 461 186930 58 3 736692 323 923345 137 813:147 460 186653 57 4 736886 323 923263 137 813623 460 160377 ' 56 5 737080 323 923181 137 813899 460 186101 55 6 737274 323 923098 137 814175 460 185825 54 7 737467 323 923016 137 814452 460 185548 53 8 737661 322 922933 137 814728 460 185272 52 9 737855 322 922851 137 815004 460 184996 51 10 738048 322 922768 138 815279 460 184721 50 11 9-738241 320 9-922686 138 9*815555 459 10 184445 49 12 738434 322 922603 138 815831 459 184169 48 13 738627 321 922520 138 816107 ' 459 183893 47 14 738820 321 922438 138 816382 459 183618 46 15 739013 321 922355 138 816658 459 183342 45 16 739206 321 922272 138 816933 459 183067 44 17 739398 321 922189 138 817209 459 182791 43 18 739590 320 922106 138 817484 459 182516 42 19 739783 320 922023 138 817759 459 182241 41 20 739975 320 921940 138 818035 458 181965 40 21 9-740167 320 9-921857 139 9-818310 458 10-181690 39 22 740359 320 921774 139 818585 458 181415 38 23 740550 319 921691 139 818860 458 181140 37 24 740742 319 921607 139 819135 458 180865 36 25 740934 319 921524 139 819410 458 180590 35 26 741125 319 921441 139 819684 458 180316 34 27 741316 319 921357 139 819959 458 180041 33 28 741508 318 921274 139 820234 458 179766 32 29 741699 318 921190 139 820508 457 179492 31 30 741889 318 921107 139 820783 457 179217 30 31 9-742080 318 9-921023 139 9-821057 457 10-178943 29 32 742271 318 920939 140 821332 457 178668 28 33 742462 317 920856 140 821606 457 178394 27 34 742652 317 920772 140 821880 457 178120 26 35 742842 317 92C688 140 822154 457 177846 25 36 743033 317 920604 140 822429 457 177571 24 37 743223 317 920520 140 822703 457 177297 23' 38 743413 316 920436 140 822977 456 1770271 22 39 743602 316 920352 140 623250 456 170750 21 40 743792 316 920268 140 823524 456 176476 20 41 9-743982 316 9-920184 140 9-823798 456 10-176202 19 42 744171 316 920099 140 824072 456 175928 18 43 744361 315 920015 140 824345 456 175655 17 44 744550 315 919931 141 824619 456 175381 16 45 744739 315 919846 141 824893 456 175)07 15 46 744928 315 919762 141 825166 456 174834 14 47 745117 315 919677 141 825439 455 174561 13 48 745306 314 919593 141 825713 455 174287 12 49 745494 314 919508 141 825986 455 174014 11 50 745683 314 919424 141 826259 455 173741 10 51 9-745871 314 9-919339 141 9-826532 455 10-173468 9 52 746059 314 919254 141 826805 455 173195 8 53 746248 313 919169 141 827078 455 172322 7 54 746436 313 919085 141 827351 455 172G49 6 55 746624 313 919000 141 827624 455 172376 5 56 746812 313 918915 142 827897 454 172103 4 57 746999 313 918830 142 828170 454 171830 3 58 747187 312 918745 142 828442 454 171558 2 59 747374 312 918659 142 828715 454 171285 1 60 747562 312 918574 142 828987 454 171013 ' 0 | Coaine l 1 Sine | | Co ling. 1 | T«ng. 1 »*■ 66 Degrees-232 (34 Degrees.) LOGARITHMIC SINES, COSINES, ETC. H. Sine D. Cosine D. Targ. D. Cotang. 0 9-747562 312 ' 9-918574 142 9-828987 454 10171013 60 1 747749 312 918489 142 829260 454 170740 59 2 747936 312 918404 142 829532 454 170468 58 3 748123 311 918318 142 829805 454 170195 57 4 748310 311 918233 142 830077 454 169923 56 5 748497 311 918147 142 830349 453 169651 55 6 748G83 311 918062 142 830621 453 169379 54 7 748870 311 917976 143 830893 453 169107 53 8 749056 310 917891 143 831405 453 168835 52 9 749243 310 917805 143 831437 453 168563 51 10 749429 310 917719 143 831709 453 168291 50 11 9-749615 310 9-917034 143 9-831981 453 10-168019 49 12 749391 310 917548 143 832253 453 167747 48 13 749987 309 917462 143 832525 453 167475 47 14 759172 309 917376 143 832796 453 167204 46 15 750358 309 917290 143 833068 452 166932 45 1C 750543 309 917204 143 833339 452 166661 44 17 750729 309 917118 144 833011 452 166389 43 18 750914 308 917032 144 833882 452 166118 42 19 751099 308 916946 144 834154 452 165846 41 20 751284 308 916859 144 834425 452 165575 40 21 9-751469 308 9-910773 144 9-834696 452 10165304 39 22 751054 308 910687 144 834907 452 165033 38 23 751839 308 916600 144 835238 452 164762 37 24 752923 307 916514 144 835509 452 164491 36 25 752298 307 916427 144 835780 451 164220 35 26 752392 307 916341 144 836051 451 163949 34 27 752576 307 916254 144 836322 451 163678 33 28 752760 307 916167 145 836593 451 163407 32 29 752944 306 916081 145 836864 451 163136 31 30 753128 306 915994 145 837134 451 162866 30 31 9-753312 306 9-915907 145 9-837405 451 10-162595 29 32 753495 306 915820 145 837675 451 162325 28 33 753679 306 915733 145 837946 451 16-2054 27 34 753862 305 915646 145 838216 451 161784 26 35 754046 305 915559 145 838487 450 161513 25 36 754229 305 915472 145 838,757 450 161243 24 37 754412 305 915385 145 839027 450 160973 23 38 754595 305 915297 145 839297 450 160703 22 39 754778 304 915210 145 839568 450 160432 21 40 754960 394 915123 146 839838 450 160162 20 41 9-755143 304 9-915035 146 9-840108 450 10-159892 19 42 755326 304 914948 146 840378 450 159022 18 43 755508 304 914860 146 840647 450 159353 17 44 755690 304 914773 146 840917 449 159083 16 45 755872 303 914685 146 841187 449 158813 15 46 756054 303 914598 146 841457 449 158543 14 47 750236 303 914510 146 841726 449 158274 13 48 756418 303 914422 146 841996 449 158004 12 49 756600 303 914334 146 842266 449 157734 11 50 750782 302 914246 147 842535 449 157465 10 51 9-756963 302 9-914158 147 9-842805 449 10157195 9 52 757144 302 914070 147 843074 449 156926 8 53 757326 302 913982 147 843343 449 156657 7 54 757507 302 913894 147 843612 449 156388 6 55 757688 301 913806 147 843882 448 156118 5 56 757869 301 913718 147 844151 448 15.5849 4 57 758050 301 913630 147 844420 448 155580 3 .5e 758230 301 913541 147 844689 448 155311 2 59 758411 301 913453 147 844958 448 155042 1 60 758591 301 913365 147 845227 448 154773 0 | Cosine | Sine 1 | CoULDg* 1 | Tang. | U. 55 Degrees.H. 0 1 o 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 60 59 58 57 50 55 54 53 52 51 50 49 48 47 4G 45 44 43 42 41 40 39 38 37 30 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 C 5 4 13 12 11 10 9 8 7 6 5 4 3 2 1 0 ITHMIC SINES, COSINES, ETC. (35 Degrees.) D. Cosine D. Tung. D. Cotang. 301 9*913305- 147 9.845227 448 10*154773 300 913270 147 845496 448 154504 300 913187 148 845764 448 15423G 300 913099 148 846033 448 153967 300 913010 148 846302 448 153098 300 912922 148 846570 447 153430 299 912833 148 84G839 447 153101 299 912744 148 847107 447 152893 299 912055 148 847376 447 152G24 299 9125GG 148 847G44 447 15235G 299 912477 148 847913 447 152087 298 9-912388 148 9.848181 447 10-151819 298 912299 149 848449 447 151551 298 912210 149 848717 447 151283 298 912121 149 848986 447 151014 298 912031 149 849254 447 15074G 298 911942 149 849522 447 150478 297 911853 149 819790 44G 150210 297 9117G3 149 850058 446 149942 297 911G74 149 850325 446 149G75 297 911584 149 850593 446 149407 297 9-911495 149 9 8508G1 446 10149139 29G 911405 149 851129 446 148871 29G 911315 150 851396 446 148G04 29G 911226 150 851GG4 44G 14833G 29G 911130 150 851931 44G 146009 296 911046 150 852199 446 147801 290 910956 150 8524G6 446 147534 295 910806 150 852733 415 147267 295 910776 150 853001 445 140999 295 910686 150 853268 445 146732 295 9-910596 150 9-853535 445 10 1464G5 295 910506 150 853802 445 146198 294 910415 150 854069 445 145931 294 910325 151 8543J6 445 145664 294 910235 151 854003 445 145397 294 910144 151 854870 445 145130 294 910054 151 855137 445 144863 294 9099G3 151 855404 445 144596 293 909873 151 855071 444 144329 293 909782 151 855938 444 144002 293 9-909091 151 9-856204 444 10143796 293 909601 151 850471 444 143529 293 909510 151 85G737 444 143263 293 909419 151 e57004 444 142996 292 909328 152 857270 444 142730 292 909237 152 857537 444 142463 292 909146 152 857803 444 142197 2! 12 909055 152 858009 444 141931 292 908964 152 858336 444 141664 291 908873 152 858602 443 141398 291 9-908781 152 9-858863 443 10-141132 291 908690 152 859134 443 140806 291 908599 152 859400 443 140600 291 903507 152 859066 443 140334 290 908416 153 859932 443 140068 290 908324 153 860198 443 139802 290 908233 153 860464 443 139536 290 908141 153 860730 443 139270 290 908049 153 860995 443 139005 290 907958 153 861201 443 138739 | Sine | | Cotang. | | Ting. | 54 Degrees*234 (36 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M- | Sine I D. | Cosine | D. | Tang. | D. | Cotang. | 0 9769219 290 9-907958 153 9-861261 443 10 138739 60 1 769393 289 907866 153 861527 443 138473 59 2 769566 289 907774 153 861792 442 138208 58 3 769740 289 907682 353 662058 442 137942 57 4 769913 289 907590 153 862323 44j> 442 137677 56 5 770087 289 907498 153 862589 137411 55 6 770260 288 907406 153 862854 442 137146 54 7 770433 288 907314 154 863119 442 136881 53 8 770606 288 907222 154 863385 442 136615 52 9 770779 288 907129 154 863650 442 136350 51 10 770952 • 288 907037 154 863915 442 136085 50 11 9-771125 288 9-906945 154 9-864180 442 10135820 49 12 771298 287 906852 154 864445 442 135555 48 13 771470 287 906760 154 864710 442 135290 47 14 771643 287 906667 154 864975 441 135025 46 15 771815 » 287 906575 154 865240 441 134760 45 16 771987 287 906482 154 865505 441 134495 44 17 772159 287 906389 155 865770 441 134230 43 18 772331 286 906296 155 866035- 441 133965 42 19 772503 286 906204 155 866300 441 133700 41 20 772675 286 906111 155 866564 441 133436 40 21 9-772847 286 9-906018 155 9-866829 441 10-133171 39 22 773018 286 905925 155 867094 441 132906 38 23 773190 286 905832 155 867358 441 132642 37 24 773361 285 905739 155 867623 441 132377 36 25 773533 285 905645 155 8o7S87 441 132113 35 2G 773704 285 905552 155 868152 440 131848 34 27 773875 285 905459 155 868416 440 131584 33 28 774046 285 905366 156 86eoo0 440 131320 32 29 774217 285 905272 156 868945 440 131055 31 30 774388 284 905179 156 869209 440 130791 30 31 9-774558 284 9-905085 156 9-869473 440 10 130527 29 32 774729 284 904992 156 869737 440 130263 28 33 774899 284 904898 156 870001 440 129999 129735 27 34 775070 284 904804 156 870265 440 26 35 775240 284 904711 156 870529 440 129471 25 36 775410 283 904617 156 870793 440 129207 24 37 775580 283 904523 156 871057 440 128943 23 38 775750 283 904429 157 871321 440 128679 22 39 775920 283 904335 157 871585 440 128415 21 .40 776090 283 904241 157 871849 439- 128151 20 41 9-776250 283 9-904147 157 9-872112 439 10 127888 19 42 776429 282 904053 157 872376 439 127624 18 43 776598 282 903959 157 872640 439 127360 17 44 776768 282 903864 157 872903 439 127097 16 45 776937 282 903770 157 873167 439 126833 15 46 777106 282 903676 157 873430 439 126570 14 47 777275 281 903581 157 873694 439 126306 13 48 777444 281 903487 157 873957 439 126043 12 49 777613 281 903392 158 874220 439 125780 11 50 777781 281 903298 158 874484 439 125516 10 51 9-777950 281 9-903203 158 9-874747 439 10 125253 9 52 778119 281 903108 158 875010 439 124990 8 53 778287 280 903014 158 875273 438 124727 7 54 778455 280 902919 158 875536 438 124464 6 55 778624 280 902824 158 875800 438 124200 5 56 778792 280 902729 158 876063 438 123937 4 57 778960 280 902634 158 876326 438 123674 3 58 779128 280 902.539 159 876589 438 123411 2 50 779295 279 902444 159 876851 438 123149 1 60 779463 279 902349 159 e77114 438 122886 0 Cosine | Sine Cotang. [ Tang. M. 63 Degrees.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 10 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 __0_ M. SINES, COSINES, ETC. (37 Degrees.) D. | Coaine | D. I Tang. | D. 279 9-902349 159 9 877114 438 279 902253 159 877377 438 279 902158 159 877640 438 279 902063 159 877903 438 279 901967 159 878165 438 278 901872 159 878428 438 278 901776 159 878691 438 278 901681 159 878953 437 278 901585 159 879216 437 278 901490 159 879478 437 278 901394 160 879741 437 277 9-901298 160 9-880003 437 277 901202 160 880265 437 277 901106 160 880528 437 277 901010 160 880790 437 277 900914 160 881052 437 277 900818 160 88J314 437 276 900722 160 881575 437 276 900626 160 881839 437 276 900529 160 882101 437 276 900433 161 882363 436 276 9-900337 161 9-882625 436 276 900240 161 882887 436 275 900144 161 883148 436 275 900047 161 883410 436 275 899951 161 883672 436 275 899854 161 883934 436 275 899757 161 884196 436 275 899660 161 884457 436 274 899564 161 884719 436 274 899467 162 884980 436 274 9-899370 162 9-885242 436 274 899273 162 885503 436 274 899176 162 885765 436 274 899078 162 886026 436 273 898981 162 886288 436 273 898884 162 886549 435 273 898787 162 886810 435 273 898689 162 387072 435 273 898592 162 887333 435 273 898494 163 887594 435 272 9-898397 163 9-887855 435 272 898299 163 888116 435 272 898202 163 888377 435 272 898104 163 888639 435 272 898006 163 888900 435 272 897908 163 889160 435 271 897810 163 889421 435 271 897712 163 889082 435 271 897614 163 889943 435 271 897516 163 890204 434 271 9-897418 164 9-890465 434 271 897:120 164 890725 434 271 897222 164 8909H6 434 270 897123 164 891247 434 270 897025 164 891507 434 270 896926 164 891768 434 270 8968*28 164 892028 434 270 896729 164 892289 434 270 896631 164 892549 434 269 8965:12 164 892810 434 Sine Cotang. . 62 Degrees. | Cotang. | 10122886 122623 122360 122097 121835 121572 121309 121047 120784 120522 120259 10-119997 1197:15 119472 119210 118940 118686 118424 118161 117899 117637 10117375 117113 116852 116590 116328 116066 115804 115543 115281 115020 10114758 114497 114235 113974 113712 113451 113190 112928 112667 112406 10112145 111884 111623 111361 111100 110840 110579 110318 110057 109796 10-109535 109275 109014 108753 108493 108232 107972 107711 107451 107190 I Tang. |ggSSgK gg&SSftEisfefc ggg*3g8£i3lS£ 236 (38 Degrees;.) LOGARITHMIC SINES, COSINES, ETC. | Sine D. | Cosine D. Tang. | D. Cotang. 9-789342 269 9-896532 164 9-892810 434 10107190 60 789504 269 896433 165 89:1070 434 106930 59 789665 269 896335 165 893331 434 106669 58 789827 269 896236 165 893591 434 106409 57 789988 269 896137 165 893851 434 106149 56 790149 269 896038 165 894111 434 105889 55 790310 268 895939 165 894371 434 105629 54 790471 268 895840 165 894632 433 105368 53 790632 268 895741 165 894892 433 105108 52 790793 268 895641 165 895152 433 104848 51 790954 268 895542 165 895412 433 104588 50 9-791115 268 9-895443 166 9-895672 433 10-104328 49 791275 2n7 895343 ]6S 895932 433 104068 48 791436 267 895244 166 896192 433 103808 47 791596 267 895145 166 896452 433 103548 46 791757 267 895045 166 896712 433 103288 45 791917 267 894945 166 896971 433 103029 44 792077 267 894846 166 897231 433 102769 43 792237 266 894746 166 897491 433 102509 42 792397 266 894646 166 897751 433 102249 41 792557 266 894546 166 898010 433 101990 40 9-792716 266 9-894446 167 9-898270 433 10-101730 39 792876 266 894346 167 898530 433 101470 38 793035 266 894246 167 8987H9 433 101211 37 793195 265 894146 167 899049 432 100951 36 793354 265 894046 167 899308 432 100692 35 793514 265 893946 167 899568 432 100432 34 793673 265 893846 167 899827 432 100173 33 793832 265 893745 167 900086 432 099914 32 793991 265 893645 167 900346 432 099654 31 794150 264 893544 167 900605 432 099395 30 9-794308 264 9-893444 1G8 9-900864 432 10099136 29 794467 264 893343 168 901124 432 098876 28 794626 264 893243 168 901383 432 098617 27 794784 264 893142 168 901642 432 098358 26 794942 264 893041 168 901901 432 098099 25 795101 264 892940 168 902160 432 097840 24 795259 264 892839 168 902419 432 097581 23 795417 263 892739 168 902679 432 097321 22 795575 263 892638 168 902938 432 097062 21 795733 263 892536 168 903197 431 096803 20 9-795891 263 9-892435 169 9-903455 431 10 096545 19 796049 263 892334 169 903714 431 096286 18 796206 263 892233 169 903973 431 096027 17 796364 262 892132 169 904232 431 095768 16 796521 262 892030 169 904491 431 095509 15 796679 262 891929 169 904750 431 095250 14 796836 262 891827 169 905008 431 094992 13 796993 262 891726 169 905267 431 094733 12 797150 261 891624 169 905526 431 094474 11 797307 261 891523 170 905784 431 094216 10 9-797464 261 9-891421 170 9-906043 431 10093957 9 797621 261 891319 170 906302 431 093698 8 797777 261 891217 170 906560 431 093440 7 797934 261 891115 170 906819 431 093181 6 798091 261 891013 170 907077 431 092923 5 798247 261 890911 170 907336 431 092664 4 798403 260 890809 170 907594 431 092406 3 798560 260 890707 170 907852 431 092148 2 798716 260 890605 170 908111 430 091889 1 798872 260 890503 170 908369 430 091631 0 | Cosine 1 | Sine 1 Cotang. Tang. : k 51 Degrees.gggiSgSEtSSS §s LOGARITHMIC SINES, COSINES, ETC. (39 Degrees.) 237 | Sine D. Cosine D. Tnng-. D. CoUlDff. | 9-798872 260 9-890503 170 9-908369 430 10091631 799028 260 890400 171 908628 430 091372 799184 260 890298 171 908886 430 091114 799339 259 890195 171 909144 430 090856 799495 259 890093 171 909402 430 090598 799G51 259 889990 171 909GGO 430 090340 79980G 259 889888 171 909918 430 090082 799962 259 889785 171 910177 430 089823 800117 259 889682 171 910435 430 0895G5 800272 258 889579 171 910693 430 089307 800427 258 889477 171 910951 . 430 089049 9-800582 258 9-889374 172 9-911209 430 10-088791 800737 258 889271 172 911467 430 088533 800892 258 889168 172 911724 430 088276 801047 258 889064 172 911982 430 088018 801201 258 888961 172 912240 430 087760 80135G 257 888858 172 912498 430 087502 801511 257 888755 172 912756 430 087244 80IGG5 257 888G51 172 913014 429 080986 801819 257 888548 172 913271 429 086729 801973 257 888444 173 913529 429 086471 9-802128 257 9-888341 173 9-913787 429 10 08G213 802282 256 888237 173 914044 429 085956 802431? 256 888134 173 914302 420 085698 802589 256 888030 173 914560 429 085-140 802743 256 887926 173 914817 429 085183 802897 256 887822 173 915075 429 084925 803050 256 887718 173 915332 429 084668 803204 256 887614 173 915590 429 084410 803357 255 887510 173 915847 429 084153 803511 255 887406 174 916104 429 083896 9-803664 255 9-887302 174 9-9163G2 429 10-083638 803817 255 887198 174 916619 429 083381 803970 255 887093 174 91G877 429 08312J 804123 255 886989 174 917134 429 082866 80427G 254 886885 174 917391 429 082609 804428 254 886780 174 917648 429 082352 804581 254 886676 174 917905 429 082095 804734 254 886571 174 918163 428 081837 804886 254 886466 174 918420 428 081580 805039 254 886362 175 918677 428 081323 9-805191 254 9-886257 175 9-918934 428 10-081066 805343 253 886152 175 919191 428 0808:»9 805495 253 886047 175 919448 428 080552 805G47 253 885942 175 919705 428 080295 805799 253 885837 175 919962 428 080038 805951 253 885732 175 920219 428 079781 60G103 253 885627 175 920476 428 079524 80G254 253 885522 175 920733 428 079267 60640G 252 8854IG 175 920990 428 070010 806557 252 885311 176 921247 428 078753 9-806709 252 9-885205 176 9-921503 428 10‘078497 806860 252 885100 176 921760 428 078240 807011 252 884994 176 922017 428 077983 807163 252 884889 176 922274 428 077726 807314 252 884783 176 922530 428 077470 807465 251 884G77 176 922787 428 077213 807615 251 884572 176 923044 428 076956 807766 251 881466 176 923300 428 076700 807917 251 884360 176 923557 427 076443 808067 251 884254 177 923813 427 076187 \ Coawe | Sine Coiang. 1 | Taiig. GO 59 58 57 5G 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 GO Degrees. 238 (40 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine 1 D. Cosine 1 D. 1 Tang. | D. 1 Cot&n*. | t 0 9-808067 251 9-884254 177 9-923813 427 10-076187 60 1 808218 251 884148 177 924070 427 075930 59 2 808368 251 884042 177 924327 427 075073 58 3 808519 250 883936 177 924:583 427 075417 57 4 808009 250 883829 177 924840 427 075160 56 5 808819 250 883723 177 92.5096 427 074904 55 6 808909 250 883617 177 925352 427 074048 54 7 809119 250 883510 177 92.5609 427 074391 53 8 809209 250 883404 177 925805 427 074135 52 9 809419 249 883297 178 920122 427 073878 51 ]0 809509 249 883191 178 920378 427 073622 50 J1 9-809718 249 9-883084 178 9-920034 427 10-073366 49 12 809808 249 882977 178 920890 427 073110 48 j;t 810017 249 882871 178 927147 427 072853 47 J4 810167 249 882764 178 927403 427 072597 46 J5 810316 248 882057 178 927059 427 072341 45 16 810465 248 882550 178 927915 427 072085 44 J7 810014 248 882443 178 9-28171 427 071829 43 J8 810703 248 882336 179 928427 427 071573 42 19 810912 248 882229 179 928083 427 071317 41 20 811001 248 882121 179 928940 427 071060 40 21 9-811210 248 9-882014 179 9-929196 427 10-070804 39 22 8113.58 247 881907 179 929452 427 070548 38 23 811507 247 881799 179 929708 427 •70292 37 24 811055 247 8H1092 179 929964 426 070036 36 2.5 811804 247 881584 179 9:10220 426 069780 35 20 811952 247 881477 179 930475 426 069525 34 27 812100 , 247 881309 179 930731 426 009269 33 28 812248 247 881201 180 930987 426 009013 32 29 812390 240 881153 180 931243 426 068757 31 30 812544 240 881046 180 931499 426 068501 30 31 9-812092 246 9-880938 180 9-931755 426 10-068245 29 :i2 812840 240 880830 180 932010 42G 067993 28 33 812988 246 880722 180 9322G0 426 007734 27 34 813135 246 880613 180 922522 426 CC7478 26 35 813283 246 880.505 180 922778 426 CG7222 25 36 813430 245 880397 180 933033 426 06G9G7 24 37 813578 245 880289 181 923289 426 0GG7U 23 38 813725 245 880180 181 933545 426 06G455 22 39 813872 245 880072 181 933800 426 066200 21 40 814019 245 879963 181 934056 426 065944 20 41 9-814166 245 9-879855 181 9-934311 426 10-065689 19 42 814313 245 879746 181 934507 426 065433 18 43 814400 244 879637 181 934823 426 065177 17 44 814007 244 879529 181 935078 426 064922 16 45 814753 244 879420 181 935333 426 064667 15 46 814900 244 879311 181 935589 426 064411 14 47 815046 244 879202 182 935844 426 064156 13 48 815193 244 879093 182 9361 IK) 426 063900 12 49 815339 244 878984 182 930.155 426 063645 11 50 815485 243 878875 182 930010 426 063390 10 51 9-815031 243 9-878766 182 9-930866 425 10 063134 9 52 815778 243 878056 182 937121 425 062879 8 53 815924 243 878547 182 937376 425 062624 7 54 810009 243 878438 182 937632 425 062368 6 55 810215 243 878328 182 937887 425 062113 5 56 810301 243 87H219 183 938142 425 061858 4 57 810507 242 878109 183 9:18398 425 061602 3 58 810652 242 877999 183 938653 425 061347 2 59 810798 242 877890 183 938908 425 061092 1 60 810943 242 877780 183 939163 425 060837 0 | Cosine | Sine 1 | Coiaog. 1 | Tang. 1 49 Degrees.C£ W W C*5 CO QOiCutd LOGARITHMIC SINES, COSINES, ETC. (41 Degrees.) 239 M. 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | Sine D. Cosine D. Tang. D. Coiang. | 9-816943 242 9 877780 183 9-930163 425 10060837 817088 242 877670 183 030418 425 060582 817233 212 877560 183 930673 425 060327 817379 242 877450 183 030028 425 060072 817524 241 877340 183 940JW3 425 059817 817008 241 877230 184 940438 425 050562 817813 241 877120 184 940604 425 050306 817058 241 877010 184 940049 425 059051 818103 241 876809 184 941204 425 058796 818247 241 876789 184 941458 425 058542 818302 241 876678 184 041714 425 058286 9-818536 240 9876568 184 9-941968 425 10058032 818681 240 876457 184 042223 425 057777 818825 240 876347 184 942478 425 057522 818069 240 876236 185 942733 425 057267 810113 240 876125 185 942988 425 057012 810257 240 876014 185 943243 425 056757 810401 240 875904 185 943498 425 056502 810.545 239 875703 185 943752 425 056248 810089 230 875682 185 944007 425 055903 810832 239 875571 185 944262 425 055738 9-810976 239 9-875459 185 9-944517 425 10-055483 820120 230 B75348 185 944771 424 055220 820263 230 875237 185 945026 424 054074 820406 239 875126 186 945281 424 054719 820550 238 875014 186 945535 424 054465 820603 238 871903 186 945790 424 054210 820836 238 874791 186 946045 424 053955 820079 238 874680 186 946299 424 053701 821122 238 874568 186 946554 424 053446 821265 238 874456 186 946808 424 053102 9-821407 238 9-874344 186 9-947063 424 10-052937 821550 238 874232 187 047318 424 052682 821G03 237 874121 187 947572 424 052428 821835 237 874009 187 947826 424 052174 821077 237 873896 187 948081 424 051010 822120 237 873784 187 048336 424 051664 822202 237 873672 187 948500 424 051410 822404 237 873500 187 948844 424 051150 822546 237 873448 187 940009 424 050001 822688 236 873335 187 940353 424 050647 9-822830 236 9-873223 187 9-949607 424 10.050393 822072 236 873110 188 949862 424 050138 823114 236 872908 188 950116 424 040884 823255 236 872885 188 950370 424 049630 823307 236 872772 188 950625 424 040375 823539 236 872659 188 950879 424 040121 823680 235 872547 188 951133 424 048867 823821 235 872434 188 951388 424 048612 823063 235 872321 188 951642 424 048358 824104 235 872208 188 951896 424 048104 9-824245 235 9-872095 189 9-952150 424 10047850 824:186 235 871081 189 952405 424 047505 824527 235 871868 189 952659 424 047341 824668 234 871755 189 952013 424 047087 824808 2:14 871641 189 953167 423 046833 824049 234 871528 189 953421 423 046579 825000 234 871414 180 053675 423 046325 825230 2:14 871301 189 053020 423 046071 825.171 234 871187 189 954183 423 045817 82551l 234 871073 100 054.437 423 045563 | Cut mo | J Sioe | Coiang. | 20 19 18 17 16 15 14 13 12 11 10 9 a 7 6 5 4 3 2 1 0^ | Tang. ( M. 48 Degrees. sgsssgs? zzzszttt&t gggsBtsfiisiss 240 (« Degrees.) LOGARITHMIC SINES, COSINES, ETC. m. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 58 59 60 Sine D. Cotine D. Tang. D. | Cotang, ' 9-825511 234 9-871073 190 9-954437 423 10 045563 825651 233 870900 190 954691 423 045309 825791 233 870846 190 954945 423 045055 825931 233 870732 190 955200 423 044800 821)071 233 870618 190 955454 423 044546 826211 233 870504 190 955707 423 044293 826351 233 870390 190 955961 423 044039 826491 233 870276 190 956215 423 043785 826631 233 870161 190 956469 423 043531 826770 232 870047 191 956723 423 043277 826910 232 869933 191 956977 423 043023 9-827049 232 9-869818 191 9-957231 423 10042769 827189 232 869704 191 957485 423 042515 827328 232 869589 191 957739 423 042261 827467 232 869474 191 957993 423 042007 827606 232 869360 191 958246 423 041754 827745 232 869245 191 958500 423 041500 827884 231 869130 191 958754 423 041246 828023 231 869015 192 959008 423 040992 828162 231 868900 192 959262 423 040738 828301 231 868785 192 959516 423 040484 9828439 231 9-868670 192 9-959769 423 10 040231 828578 231 868555 192 960023 423 039977 828716 231 868-140 192 960277 423 039723 828855 230 868324 192 960531 423 039469 828993 230 868209 192 960784 423 039216 829131 230 868093 192 961038 423 038962 829269 230 867978 193 961291 423 0387C9 829407 230 867862 193 961545 423 038455 829545 230 867747 193 961799 423 038201 829683 230 867631 193 962052 423 037948 9-829821 229 9-867515 193 9-962306 423 10 037694 829959 229 867399 193 962560 423 037440 830097 229 867283 193 962813 423 037187 830234 229 867167 193 903067 423 036933 830372 229 867051 193 963320 423 036680 830509 229 866935 194 963574 423 036426 830646 229 866819 194 963827 423 036173 830784 229 866703 194 964031 423 035919 830921 228 866586 194 964335 423 035665 831058 228 866470 194 964588 422 035412 9-831195 228 9-866353 194 9-964842 422 10035158 831332 228 866237 194 965095 422 034905. 831469 228 866120 194 965349 422 034651 831006 228 866004 195 965602 422 034398 831742 228 865887 195 965855 422 034145 831879 228 865770 195 966109 422 033891 832015 227 865653 195 966362 422 033638 832152 227 865536 195 966616 422 033384 832288 227 865419 195 966869 422 033131 832425 227 865302 195 967123 422 032877 9-832561 227 9‘865185 195 9-967376 422 10032624 832697 227 865068 195 967629 422 032371 832833 227 864950 195 967883 422 032117 832969 226 864833 196 968136 422 031864 833105 226 864716 196 968389 422 031611 833241 226 864598 196 968643 422 031357 833377 226 864481 196 968896 422 031104 833512 226 864363 196 969149 422 030851 833648 226 864245 196 969403 422 030597 833783 226 864127 196 969656 422 030344 l Cosine > I Sine | | Coiang. | | Tang. I lit 47 Degrees. £i2!3S28§J388 S£SS£S;S£i£S gSiSSESSSSSsIw. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2G 27 28 29 30 31 32 33 34 35 30 37 38 39 40 41 42 43 44 45 4G 47 48 49 50 51 52 53 54 55 50 57 58 59 00 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 M. iiigigg ggggssgKgH jrilriir-iiiiiaiiti mm'ivm znmmM SIKES, COSINES, ETC. (43 Degrees.) D. 219 219 219 219 219 219 219 218 218 218 218 218 218 | Cosine D. Tang. 1 D. Cotang. 9-804127 196 9-969656 422 10-030344 804010 196 969909 422 030091 803892 197 970162 422 029838 863774 197 970416 422 029584 863656 197 970669 422 029331 863538 197 970922 422 029078 803419 197 971175 422 028825 863301 197 971429 422 028571 863183 197 971082 422 028318 863004 197 971935 422 028065 862946 198 972188 422 027812 9-802827 198 9-972441 422 10-027559 862709 198 972694 422 027306 802590 198 972948 422 027052 862471 198 973201 422 026799 802353 198 973454 422 026546 802234 198 973707 422 026293 862115 198 973960 422 026040 861996 198 974213 422 025787 801877 198 974466 422 025534 801758 199 974719 422 025281 9-861038 199 9-974973 422 10-025027 861519 199 975226 422 024774 801400 199 975479 422 024521 801280 199 975732 422 024268 861161 199 975985 422 024015 861041 199 976238 422 023762 800922 199 976491 422 023509 800802 199 976744 422 023256 860682 200 976997 422 023003 800562 200 977250 422 022750 9-860442 200 9-977503 422 10-022497 860322 200 977756 422 022244 800202 200 978009 422 021991 860082 200 978262 422 021738 859962 200 978515 422 021485 859842 200 978768 422 021232 859721 201 979021 422 020979 859601 201 979274 422 020726 859480 201 979527 422 020473 859360 201 979780 422 020220 9-859239 201 9-980033 422 10-019967 859119 201 980286 422 019714 858998 201 980538 422 019462 858877 201 980791 421 019209 858756 202 981044 421 018956 858635 202 981297 421 018703 858514 202 981550 421 018450 858393 202 981803 421 018197 858272 202 982056 421 (117944 858151 202 982309 421 017691 9-858029 202 9-982562 421 10-017438 857908 202 982814 421 017186 857786 202 983067 421 016933 857665 203 983320 421 016680 857543 203 983573 421 016127 857422 203 983826 421 016174 857300 203 984079 421 015921 857178 203 984331 421 015669 857056 203 984584 421 015416 856934 •' 203 984837 421 015163 | Sine | | Counig. I | Tang. | 40 Degrees.242 (44 Degrees.) LOGARITHMIC SINES, COSINES, ETC. . M. | Sine *> Co6ine D. Tan?. a | Cotang. 0 9-841771 •218 9-85G934 203 9-984837 421 10015163 60 1 841902 218 856812 203 985090 421 014910 59 2 842033 218 856690 204 985343 421 014057 58 3 842103 217 856568 204 985596 421 014404 57 4 842294 217 856446 204 985848 421 014152 56 5 842424 217 856323 204 986101 421 013899 55 C 842555 217 856201 204 980354 421 013046 54 7 842085 217 856078 204 986007 421 013393 53 8 842815 217 855956 204 986800 421 013140 52 9 842940 217 855833 204 987112 421 012888 51 10 84307G 217 855711 205 987365 421 012035 50 11 9-843206 216 9-855588 205 9-987618 421 10-012382 49 12 843336 216 8554G5 205 987871 421 012129 48 13 843406 216 855342 205 988123 421 011877 47 14 843595 216 855219 205 988376 421 911024 46 15 843725 216 855096 205 988029 421 011371 45 10 843855 216 854973 205 988882 421 011118 44 17 843984 216 854850 205 989134 421 010806 43 18 844114 215 854727 206 989387 421 010613 42 19 844243 215 854003 206 989640 421 010360 41 20 844372 215 854480 206 989893 421 010107 40 21 9-844502 215 9-854356 206 9-990145 421 10-009855 39 22 644031 215 854233 206 990398 421 009002 38 23 844760 215 854109 206 990051 421 009349 37 24 844889 215 853986 206 990903 421 009097 36 25 845018 215 853802 206 991150 421 008844 35 2G 845147 215 853738 206 991409 421 008591 34 27 845276 214 853014 207 991002 421 008338 33 23 845405 214 853490 207 991914 421 008080 32 29 845533 214 853300 207 992107 421 007833 31 30 845002 214 853242 207 992420 421 007580 30 31 9-845790 214 9-853118 207 9-992G72 421 10-007328 29 32 845919 214 852994 207 992925 421 007075 28 33 840047 214 852809 207 993178 421 000822 27 34 640175 214 852745 207 993430 421 000570 26 35 840304 214 852020 207 993083 421 000317 25 3G 840432 213 852490 208 993930 421 006004 24 37 8465G0 213 852371 208 994189 421 005811 23 38 840088 213 852247 208 994441 421 005559 22 39 846816 213 852122 208 994694 421 005306 21 40 840944 213 851997 208 994947 421 005053 20 41 9-847071 213 9-851872 208 9-995199 421 10-004801 19 42 847199 213 851747 208 995452 421 004548 18 43 847327 213 851022 208 995705 421 004295 17 44 847454 212 851497 209 995957 421 004043 16 45 847582 212 851372 209 990210 421 003790 15 40 847709 212 851246 209 990463 421 003537 14 47 847836 212 851121 209 990715 421 003285 13 48 847964 212 850996 209 996968 421 003032 12 49 848091 212 850870 209 997221 421 002779 11 50 848218 212 850745 209 997473 421 002527 10 51 9-848345 212 9-850619 209 9-997720 421 10 002274 9 52 848472 211 850493 210 997979 421 002021 8 53 848599 211 850368 210 998231 421 001709 7 54 848720 211 850242 210 998484 421 001516 6 55 848852 211 850110 210 998737 421 001203 5 50 848979 211 849990 210 998989 421 001011 4 57 849100 211 849804 210 999242 421 000758 3 58 849232 211 849738 210 999495 421 00U595 2 59 849359 211 849011 210 999748 421 000253 1 (HI 849485 211 849485 210 10 000000 421 000000 0 Cosine | Sine Ci-uuig. Tung. 1 1L 40 Decrees.TABLE XIV. NATURAL SINES AND TANGENTS./ 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 n 16 17 18 19 20 21 22 23 21 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 / / 60 59 58 67 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 1 0 / NATURAL SINES. 0° | 1° | 2° I ) 0000 -017 45211-034 8995' 2909, 7432I-03519021 5818-018 0341 4809! 8727| 3249 7716 0011636! 6158-0360623 4544 9066; 3530 7453-0191974 6437 0020362! 4883 9344 3271 61801-020 0699 90891 360S 0031908 6516 4907 9424 7815-021 2332 004 07241 5211 3633; 8149 6542'*0221057 77911-037 2251 9451 0052360 5263 3965 6873 97 SI 8177 -0232690 00610S6I 5158 8065 0380071 3S78 6785 9092 03S 2598 5505 8411 0401318 4224 3995 6904 9813 007 2721 5630 8539 008 1448i 4357 7265 0090174 3083 5992 8900 0101S09 7131 5598|-0410037 8506! 2944 •0241414 5850 4322 8757 7230 -042 1663 ■025 0138 4569 3046 7475 5954 -043 0382 8862 3288 0261769 61W 4677 9100 7585>044 2006 027 0493i 4912 3401! 7818 6309;-045 0724 4718 9216: 3630 7627 -028 2124i 6536 0110535 5032: 9442 3444 7940 :-046 2347 6353-02908471 5253 92611 3755: 8159 012 2170 6662-047 1065 5079| 9570: 3970 79871-030 2478! 6876 01308961 5385! 9781 3305 8293-048 2687 6713-0311200; 5592 9622 4108' 8498 014 2530 7015,049 1403 5439! 9922 4308 83481-0322830 7214 015 1256 5737-0500119 4165 8644 3024 7073-0331552 5929 9982: 4459 8835 016 28901 7366 -0511740 -5709,-034 0274 4645 8707! 3181 7550 017 1616 6088 -052 0455 4524 8995 3360 89° 88° | 87 3° 4° 5° 6° -052 3360-069 7565 •087 1557 •104 5285 • 6264 ■070 0467 4455 8178- 9169 3368 7353 •1051070 •053 2074 6270 •0880251 3963 4979 9171 3148 6856- 7883 •071 2073 6046 9748 •054 0788 4974 8943 •106 2641 3693 7876 •0891S40 5533 6597 9502 •072 0777 3678 4738 7635 8425- •1071318 ■055 2406 6580 •090 0532 4210 5311 94S1 3429 7102 • 8215 •073 23S2 6326 9994 •0561119 52S3 9223 •108 2885 4024 8184 •0912119 5777 6928 •07410S5 5016 8069 ■ 9832 39S6 7913 •If,91560 •057 2736 6887 •0920S09 4452 5640 9787 3706 7343- 8544 •0752688 6602 •1100234 •058 1448 5589 9499 3126 4352 8489 ■093 2395 6017 7250 ■076 1390 5291 8908 •0590100 4290 8187 •111 1799 3064 7190 •0941083 4689 5967 •077 0091 3979 75SO 8871 2991 6875 •112 0471 •060 1775 5891 9771 3361 4678 8791 •095 2666 6252 7582 •078 1691 5562 9142- •061 0485 4591 8458 •1132032 3389 7491 •0961353 4922 6292 •079 0391 4248 7812 9196 3290 7144 ■1140702 •062 2099 6190 097 0039 3592 5002 9090 2934 6482 7905 •080 1989 5829 9372 ■0630808 4889 8724 •115 2261 3711 7788 •098 1619 5151 6614 •081 0687 4514 8040 9517 3587 7408 •1160929 •0642420 6486 •099 0303 3818 5323 9385 3197 6707 8226 ■082 2284 6092 9596 •0651129 5183 8986 •117 2485 4031 8082 •1001881 5374 6934 •083 0981 4775 8263 9836 3880 7669 •1181151 •0662739 6778 •1010563 4040 5641 9677 3457 6928 8544 •0842576 6351 9816 •067 1446 5474 9245 •119 2704 4349 8373 •102 2138 5593 7251 •0851271 5032 8481 •068 0153 4169 7925 ■120 1368 3055 7067 •103 0819 4256 5957 9966 3712 7144 8859 •086 2864 6605 •1210031 •0691761 5762 9499 2919 4663 8660 •104 2392 5S06 7565 •087 1557 5285 8693 86° 85° 84° 83° NAT. COSINE./ 0 1 2 3 4 6 6 l 9 10 11 12 13 14 13 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 88 89 ■(0 <1 42 43 44 45 46 47 48 49 50 61 52 53 54 65 66 67 68 69 60 / / 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL SINES. 9° 5G4345 7218 57 0091 290." 5831 8708 581581 590197 80° 10° 11° 12° 13° 14° •173 6482 190 8090 207 9117 224 9511 241 9219 • 9346 191 0945 208 1962 225 2345 242 2041 • 174 2211 3801 4807 5179 4863 5075 6656 . 7652 8013 7685 7939 9510 2090497 226 0846 2430507 175 0803 192 2365 3341 3680 3329- 3667 5220 6186 6513 6150 6531 8074 9030 9346 8971 9395 1930928 2101874 227 2179 2441792- •176 2258 3782 4718 6012 4613 5121 6636 7561 7844 7433 7984 9490 •211 0405 228 0677 245 0254 •177 0847 •194 2344 3248 3509 3074 • 3710 5197 6091 6341 5894 6573 8050 8934 9172 8713 9435 •195 0903 •2121777 229 2004 2461533 • •178 2298 3756 4619 4835 4352 5160 6609 7462 7666 7171 8022 9461 •213 0304 230 0497 9990 •179 0884 •196 2314 3146 3328 247 2809 • 3746 5166 5988 6159 5627 6607 801S 8829 8989 8445 9469 •197 0870 •2141671 •2311819 •248 1263 •180 2330 3722 4512 4649 4081 • 5191 6573 7353 7479 6899 8052 9425 •215 0194 •2320309 9716 •181 0913 •198 2276 3035 3138 •2492533- 3774 5127 5876 5967 5350 6635 7978 8716 8796 8167 9495 ■199 0829 •2161556 •233 1625 •250 0984 ■1822355 3679 4396 4454 3800- 5215 6530 7236 7282 6616 8075 9380 •217 0076 •234 0110 9432 •183 0935 •200 2230 2915 2938 ■251 2248 3795 5080 5754 5766 5063 6654 7930 8593 8594 7879 9514 •2010779 •2181432 •2351421 •252 0694 ■184 2373 3629 4271 4248 3508 5232 6478 7110 7075 6323 8091 9327 9948 9902 9137 •185 0949 •2022176 •219 2786 •236 2729 •2531952 5 3808 5024 5624 5555 4766 4 6666 7873 8462 8381 7579 9524 •203 0721 •2201300 •237 1207 •2540393 S -186 2382 3569 4137 4033 3206 5 5240 6418 6974 6859 6019 2 8098 9265 9811 9684 8832 8-187 0956 ■204 2113 •221 2648 •238 2510 •255 1645 5 3813 4961 5485 5335 4458 1 6670 7808 8321 8159 7270 8 9528 •205 0655 •2221158 •2390984 •256 0082 4 -188 2385 3502 3994 3808 2894 0 5241 6349 6830 6633 5705 5 8098 9195 9666 9457 8517 1 -1890954 •2062042 •223 2501 ■240 2280 •257 1328 6 3811 4888 5337 6104 4139 2 6667 7734| 8172 7927 6950 7 9523 •207 0580 -2241007 ■241 0751 9760 2 -190 237S 3426 3842 3574 1 1 7 5234 6272 6676 639C 5381 2 809C 9117 9511 921£ 8190 79° 78° 77° 76° 75° 2 2 2 2 ■2 5 MAT. COSINE246 / o l 2 S 4 6 6 7 8 0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 61 62 63 64 66 66 67 63 69 60 / / 60 59 58 57 56 56 64 53 62 61 60 49 48 47 46 46 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 2 1 0 / NATURAL SINES. 18° 19° 20° •309 0170 325 5682 342 0201 •; 2936 8432 2935 5702 3261182 5CC8 8406 3932 8400 •: •3101234 6681 343 1133 3999 9430 3865 6764 327 2179 6597 9529 4928 9329 -I •311 2294 7676 3442060 5058 328 0424 4791 „7S22 3172 7521 •; •312 0586 6919 345 0252 3349 8666 2982 6112 •3291413 5712 8S75 4160 8441- •3131638 6906 3461171 4400 9653 3900 7163 •330 2398 6628 9925 5144 9357 • •314 268C 78S9 347 2085 6448 •331 0634 4812 8209 3379 7540- >•3150969 6123 •3480267 3730 8867 2994 6490 •3321611 5720 9250 4355 8447 • •316 2010 7098 •3491173 4770 9841 3898 7529 •333 2584 6624 •317 0288 6326 9349- 3047 8069 •350 2074 6805 •3340810 4798 8563 3552 7523- •3181321 6293 •3510246 4079 9034 2970 6836 •3351775 5693 9593 4516 8416- ■319 2350 7256 •3521139 5106 9996 3862 7863 •3362735 6584 •320 0619 5475 9306- 9 3374 8214 •3532027 6130 •337 0953 4748 2 8885 3691 7409- 2 -3211640 6429 •3540190 3 4395 9167 2910 3 7149 •3381905 5630 3 9903 4642 8350 3-3222657 7379 •3551070 3 6411 •339 0116 3789 2 8164 2852 6508 1-3230917 5589 9226 0 3670 8325 •3561944 8 6422 •3401060 4662 6 9174 3796 7380 4-3241926 6531 •357 0097 2 4678 9265 2814 9 7429 •341 2000 6531 6 -325 018C 4734 8248 3 2931 7468 •358 0964 0 5682 •342 0201 3679 71° 70° 69° 6395 9110 • 1825 4540 7254 9968 >2682 5395 8108 L0S21 3534 6246 8958 21669 4380 7091 9802 52512 5222 7932 10641 3351 6059 8768 51476 4184 6891 9599 5 2306 5012 7719 7 0425 3130 5836 8541 368 1246 3950 6654 9358 369 2061 4765 7468 370 0170 2S72 6574 8276 371 0977 3678 6379 9079 •372 1780 4479 7179 9878 ■373 2577 5275 7973 22° | 374 6066 -3 8763 3761459 3 4156. 6852 376 2243 4938 7032 377 0327 3021 6714 8408 3781101 •; 3794 6486 9178 3791870 4662 7253 9944 380 2C34 5324 8014 3810704 3393 6082 8770 •382 1459 4147 6834 9522 383 2209 4895 7582 384 0268 2953 5(39 8324 •3851008 3C93 6377 9060 •3861744 4427 7110 9792 •387 2474 6156 7837 •3880518 3199 6680 8560 •3891240 3919 6598 9277 •3 3 3 3 '3 .£ 4 ■( 374 0671 *390 1955 3369 4633 60661 7311 68° J 67° NAT. COSINE./ 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 31 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 51 52 53 64 55 66 57 68 69 60 / / CO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 33 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 13 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL SINES. •4120445 3096 5745 8395 •4131044 3693 6342 8990 •414 1638 4285 6932 9579 •415 2221 4872 7517 •416 0163 2808 5453 8097 •417 0741 3385 6028 8671 •4181313 3956 6597 9239 ■41918S0 4521 7161 9801 •420 2441 5080 7719 •4210358 2996 5634 8272 •4220909 3546 6183 65° 25° •422 6183 8819 ■423 1455 4099 6725 9360 ■4241994 4628 7262 9895 ■425 2528 5161 7793 ■4260425 3056 5687 8318 •427 0 9 49 3579 6208 8838 ■428 1467 4095 6723 9351 •4291979 46061 7233 9859 ■430 2485 5111 7736 6 -4310361 2986 5610 8234 •432 0857 3481 6103 8726 •433 1348 3970 6591 9212 •4341832 4453 7072 9692 •435 2311 4930 7548 •436 0166 2784 5401 8018 ■437 0634 3251 5866 8482 •4381097 3711 64° 26° 27° 28° 29° 30° •438 3711 •453 9905 •469 4716 •484 8096 -500 0000- 6326 •454 2497 7284 ■485 0640 2519 8940 5088 9852 3184 5037 •439 1553 7679 •470 2419 5727 7556 4166 •455 0269 4986 8270 ■501 0073 . 6779 2859 7553 •486 0812 2591 9392 5449 •471 0119 3354 5107 •440 2004 8038 2685 5895 7624 4615 •456 0627 5250 8436 •5020140 7227 3216 7815 •487 0977 2655 9838 5804 •472 0380 3517 5170 •441 2448 8392 2944 6057 7685 5059 •457 0979 5508 8597 •503 0199 7668 3566 8071 •4881136 2713 •442 0278 6153 •473 0634 3674 5227 2887 8739 3197 6212 7740 5496 ■458 1325 5759 8750 •5040-252 8104 3910 8321 •489 1288 2765 •443 0712 6496 •474 0882 3825 5276 3319 9080 3443 6361 7788 5927 •459 1665 6004 8897 •505 0298 8534 4248 8564 •490 1433 2809 •4441140 6832 •475 1124 3968 5319 3746 9415 3683 6503 7828 6352 •460 1998 6242 9038 •506 0338 8957 4580 8801 •4911572 2846 •445 1562 7162 •4761359 4105 5355 4167 9744 3917 6638 7863 6771 •461 2325 6474 9171 •507 0370 9375 4906 9031 •492 1704 2877 •446 1978 7486 •477 1588 4236 5384 4581 •4620066 4144 6767 7890 7184 2646 6700 9298 •508 0396 9786 5225 9255 •4931829 2901 •447 2388 7804 •478 1810 4359 5406 4990 •463 0382 4364 6889 7910 7591 2960 6919 9419 •5090414 •448 0192 5538 9472 •4941948 ■ 2918 2792 8115 •479 2026 4476 5421 5392 •464 0692 4579 7005 7924 7992 3269 7131 95321-510 0426 •449 0591 5845 9683 •495 2060 2928 3190 8420 •480 2235 4587 5429 6789 •465 0996 4786 7113 7930 8387 3571 7337 9639 ■5110431 ■450 0984 6145 9888 •496 2165 2931 3582 8719 ■481 2438 4690 5431 6179 •4661293 4987 7215 7930 8775 3866 7537 9740 •5120429 •4511372 6439 ■482 0086 •497 2264 2927 3967 9012 2634 4787 5425 6563 •467 1584 5182 7310 7923 9158 4156 7730 9833 •513 0420 ■4521753 6727 •483 0277 •498 2355 2916 4347 9298 2824 4877 5413 6941 •468 1869 5370 7399 7908 9535 4439 7916 9920 •514 0404 •453 2128 7009 ■484 0462 ■499 2441 2899 4721 9578 3007 4961 5393 7313 •469 2147 5552 7481 7887 9905 4716 8096 •500 0000 •515 0381 63° 62° 61° ' 60° 59° 515 0381 2874 ' 5367 7859 5160351 2842 5333 7824 517 0314 2804 5293 7782 518 0270 2758 5246 7733 519 0219 2705 5191 7676 520 0161 2646 5130 7613 521 0096 2579 5061 7543 •522 0024 2505 4986 7466 9945 ■523 2424 4903 7381 9859 •524 2336 4813 7290 976f •525 2241 4717 7191 9661 ■526 2139 4613 7085 9558 •527 2030 NAT. COS1NB.248 / o l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 66 67 68 69 60 / / 60 09 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL SIKES. 33° >44 6390 2956 5373 7790 •557 0206 2621 66° 34° 35° 36° 37° 38° •5591929 573 5764 587 7853 601 8150 615 6615 4340 8147 588 0206 602.0473 8907 6751 574 0529 2558 2795 6161198 9162 2911 4910 5117 3489 •5601572 5292 7262 7439 57 SO 3981 7672 9613 9760 8069 6390 575 0053 589 1964 603 2080 617 0359 8798 2432 4314 4400 2648 •5611206 4811 6663 6719 4936 3614 7190 9012 9038 7224 6021 9568 •5901361 604 1356 9511 8428 576.1946 3709 3674 6181798 •5620834 4323 6057 5991 4084 3239 6700 8404 8308 6370 5645 9076 •591 0750 605 0624 8655 8049 •577 1452 3096 2940 L619 0939 •563 0453 3827 5442 5255 3224 2857 6202 7787 7570 5507 5260 8576 •5920132 9884 7790 7663 •578 0950 2476 •606 2198 •6200073 •5640066 3323 4819 4511 2355 2467 5696 7163 6824 4636 4869 8069 9505 9136 6917 7270 •579 0440 •5931847 •607 1447 9198 9670 2812 4189 3758 •6211478 •565 2070 6183 6530 6069 3757 4469 7553 8871 8379 6036 6868 9923 •594 1211 •608 0689 8314 9267 •5S0 2292 3550 2998 ■6220592 •566.1665 4661 5889 5306 2870 4062 7030 8228 7614 5146 6459 9397 •595 0566 9922 7423 8856 •5811765 2904 •609 2229 9698 ■567 1252 4132 5241 4535 •623 1974 3648 6498 757J 6841 4248 6043 8864 9913 9147 6522 8437 •5821230 •596 2249 •6101452 8796 •568 0832 3595 4584 3756 •6241069 3225 6959 6918 6060 3342 6619 8323 9252 8363 5614 8011 •5830687 •597 1686 •6110666 7885 ■5690403 3050 3919 2969 •6250156 2795 5412 6251 5270 2427 5187 7774 8583 7572 4696 7577 •5840136 •598 0915 9873 6966 9968 2497 3246 •612 2173 9235 •570 2357 4857 5577 4473 •6261503 4747 7217 7906 6772 3771 7136 9577 •599 0236 9071 6038 9524 •5851986 2565 •6131369 8305 •5711912 4294 4893 366f •627 0571 4299 6652 7221 5964 2837 6686 9010 9549 8260 5102 9073 •5861367 •600 1 876 ■614 0556 7366 •572 1459 3724 4202 2852 9631 3844 6080 6528 5147 •6281894 6229 8435 8854 7442 4157 8614 •587 0790 •6011179 9736 6420 >•5730998 3145 3503 •615 2029 8682 3381 5499 5827 4322 0943 5764 7853 8150 6615 ■629 3204 65° 54° 53° 52° 51° •£ •£ ( •( MAT. COSINE,/ 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 46 46 47 48 49 60 61 62 63 64 65 66 67 68 69 60 f r 60 59 58 57 56 55 54 53 52 51 50 49 43 47 46 46 44 43 42 41 40 39 38 37 36 36 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL SINES. 8609 ■664 0810 3010 5209 7408 9607 ■665 1804 4002 6198 8395 •666 0590 49° 41° 42° 43° 44° i 45° 46° 6660590 669 1306 681 9984 694 6584 707 1068 719 3398 -1 2785 3468 682 2111 8676 3124 6418 4980 6628 4237 695 0767 6180 7438 7174 7789 6363 2858 7236 9457 9367 9948 8489 4949 9291 720 1476 •- 657 15C0 670 2108 683 0613 7039 7081345 3494 3752 4266 2738 9128 3398 5511 6944 6424 4861 69e 1217 5451 7528 8136 8582 6984 2105 7504 9544 658 0326 671 0739 9107 6392 9556 7211559 2516 2895 684 1229 7479 7091607 3574 4706 5051 3350 9565 3657 5589 6895 7206 5471 697 1651 5707 7602 9083 9361 7591 3736 7757 9615 659 1271 •672 1515 9711 5821 9806 722 1628 . 3458 3668 6851830 7905 7101854 3640 6645 6821 3948 9988 3901 5651 7831 7973 6066 698 2071 5948 7661 660 0017 •6730125 8184 4153 7995 9671 2202 2276 ■686 0300 6234 •7110041 •7231681- 4386 4427 2416 8315 2086 3690 6570 6577 4532 •699 0396 4130 5698 8754 8727 6647 2476 6174 7705 661 0936 •674 0876 8761 4555 8218 9712 3119 3024 •687 0875 6633 •7120260 ■7241719- 6300 5i72 2988 ■ 8711 2303 3724 7482 7319 6101 •7000789 4344 5729 9662 9466 7213 2866 6385 7734 6621842 •675 1 612 9325 4942 8426 9738 4022 3757 •688 1435 7018 •7130465 •7251741 6200 5902 3546 9093 2504 3744 8379 8046 5655 •7011167 4543 5746 •6630C57 •676 0190 7765 3241 6581 7747 2734 2333 9873 6314 8618 9748 4910 4476 •6891981 7387 •7140655 •7261748- 7087 6618 4089 9459 2691 3748 9262 8760 6196 •7021531 4727 5747 •6641437 ■677 0901 8302 3601 6762 7745 3613 3041 •690 0407 5672 8796 9743 5785 5181 2512 7741 •715 0830 ■727 1740 7959 7320 4617 9811 2863 3736 •665 0131 9459 6721 •7031879 4895 6732 2304 ■6781597 8824 3947 6927 7728 4475 3734 •6910927 6014 8959 9722 6646 6871 3029 8081 •7160989 •7281716 8817 8007 5131 •704 0147 3019 3710 •6660987 •679 0143 7232 2213 6049 6703 3156 2278 9332 4278 7078 7695 6325 4413 ■6921432 6342 9106 9686 7493 6547 3531 8406 ■717-1134 •729 1677 9661 8681 5630 •705 0469 3161 3668 •667 1828 •680 0813 7728 2532 6187 6657 3994 2946 9825 4594 7213 7646 616C 5078 •6931922 6655 9238 9635 8326 7208 4018 8716 •7181263 •7301623 •668 049C 9339 6114 •706 0776 3287 3610 265E •6811468 8209 2835 631C 5597 4818 3599 ■694 0304 4894 7335 7583 6981 5728 2398 6955 935f 9568 9144 7856 4491 9011 •7191377 •7311553 •669130( 9984 6584 •707 1068 3398 3537 48° 47° 46° 45° 44° 43® NAT. COSINB/ 0 1 2 3 4 5 6 7 8 e 10 n 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 61 52 53 54 55 56 57 68 69 60 / r 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 U 10 9 8 7 6 5 4 3 2 1 0 r NATURAL SINES. 48° •743 1448 3394 5340 7285 9229 •7441173 3115 5058 6999 8941 •745 0881 2821 4760 6699 8636 •7460574 2510 4446 6382 8317 9557 •749 1484 3411 6337 7262 9187 •7501111 3034 4957 6879 7980 9894 •7531808 3721 5634 7546 9457 •7541368 3278 6187 7096 41° 49° ■754 7096 9004 •755 0911 2818 4724 6630 8535 •756 0439 2342 4246 6148 8050 9951 ■757 1851 3751 5650 7548 9446 •7581343 3240 5136 7031 8926 •759 0820 2713 4606 6498 8389 •7600280 2170 4060 6949 7837 9724 •7611611 3497 53S3 7268 9152 •7621036 2919 4802 6683 8564 •7630445 2325 4204 6082 7960 9838 •7641714 3590 5465 7340 9214 •765 1087 2960 4832 6704 8574 •766 0444 40° 50° i 61° 52° 53° •7660444' •7771460 •788 0108 •798 6355 2314: 3290 1898 8105 4183 1 6120 3688 9855 6051 ! 6949 5477 ■799 1604 7918 , 8777 7266 3352 9785 i •778 0604 9054 5100 •767 1652 2431 •7890841 6847 3517 4258 2627 8593 5382 6084 4413 •800 0338 7246 7909 6198 2083 9110 9733 7983 3827 •768 0973 ; -7791557 9767 5571 2835 3380 •7001550 7314 4697 5202 3333 9056 6558 7024 6115 •8010797 8418 8845 6896 2538 •769 0278 ■780 0665 8676 4278 2137 2485 ■7910456 6018 3996 4304 2235 7756 5853 6123 4014 9495 7710 7940 5792 ■802 1232 9567 9757 7569 2969 •7701423 •781 1574 9345 4705 3278 3390 •7921121 6440 6132 5205 2896 8175 6986 7019 4671 9909 8840 8833 6445 ■803 1642 •771 0692 •782 0646 8218 3375 2544 2459 9990 6107 4395 4270 •7931762 6838 6246 6082 3533 8569 8096 7892 5304 •8040299 9945 9702 7074 2028 ■7721794 •7831511 8843 3756 3642 3320 •7940611 6484 5489 5127 2379 7211 7336 6035 4146 8938 9182 8741 5913 •805 0664 •7731027 •7840547 7678 2389 2872 2352 9444 4113 4716 4157 •7951208 6837 6559 5961 2972 7560 8402 7764 4735 9283 •7740244 9566 6497 •8061005 2086 •7851368 8259 2726 3926 3169 ■796 0020 4446 5767 4970 1780 6166 7606 6770 3540 7885 9445 8569 5299 9603 •7751283 •7860367 7058 •8071321 3121 2165 8815 3038 4957 3963 ■797 0572 4754 6794 5759 2329 6470 8629 7555 4084 8185 •7760464 9350 6839 9899 2298 •787 1145 7594 •8081612 4132 2939 9347 3325 5965 4732 •7981100 6037 7797 6524 2853 6749 9629 8316 4604 8460 •777 1460 •788 OIOS 6355 •809 0170 39° 38° 1 37° 36° •809 0170 1879 3588 5296 7004 8710 ■810 0416 2122 3826 6530 7234 8936 ■8110638 2339 4040 5740 7439 9137 •812 0S& 2532 4229 5925 7620 9314 •8131008 2701 4393 6084 7775 9466 •8141155 2844 4532 6220 7900 9593 •815127S 2963 4647 6330 8013 9695 •8161376 3056 4736 6416 8094 9772 •817 1449 3125 4801 6476 8151 9824 •8191497 3169 4841 6512 8182 9852 •8191520 35° HAT. COSINE./ 0 1 2 3 4 6 6 7 8 0 10 11 12 13 14 15 10 17 18 10 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 41 45 46 47 48 49 60 61 62 63 64 65 66 67 68 69 00 f 251 / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 f NATURAL SINES. 56° 57° 68° 59° 60° 61° •829 0376 •838 6706 •8480481 •857 1673 ■8660254 •874 6197 2002 8290 2022 3171 1708 7607 3628 9873 3562 4668 3161 9010 6252 •839 1455 5102 6164 4614 ■875 0425 6877 3037 6641 7060 6066 1832 8500 4618 8179 9155 7517 3239 •830 0123 6199 9717 ■858 0649 8967 4645 1745 7778 •849 1254 2143 •867 0417 6051 3366 9357 2790 3635 1866 7455 4987 ■840 0936 4325 6127 3314 8859 6607 2513 5860 6619 4762 •8760263 8226 4090 7394 8109 6209 1665 0845 5666 8927 9599 7055 3067 •8311463 7241 ■850 0459 •8591088 9100 44C8 3080 8816 1991 2570 ■868 0544 6868 4696 •8410390 3522 4064 1988 7268 6312 1963 5053 6551 3431 8666 7927 3536 6582 7037 4874 •877 0064 9541 5108 8111 8523 6315 1462 •8321155 6679 9639 •860 0007 7756 2858 2768 8249 •8511167 1491 9196 4254 4380 9S19 2693 2975 ■869 0636 5649 5991 ■842 1388 4219 4457 2074 7043 7602 2956 5745 5939 3512 8437 9212 4524 7269 7420 4949 9830 •833 0822 6091 8793 8901 6386 •8781222 2430 7657 •852 0316 •8610380 7821 2613 4038 9222 1839 1859 9256 4004 5646 •843 0787 3360 3337 •8700691 5394 7252 2351 4881 4815 2124 6783 8858 3914 6402 6292 3557 8171 •8340463 5477 7921 7768 4989 9559 2068 7039 9440 9243 6420 •879 094(5 3672 8600 •853 0958 •8620717 7851 2332 5275 •844 0161 2476 2191 9281 3717 6877 1720 3992 3664 •871 0710 6102 8479 3279 5508 5137 2138 6486 •835 0080 4838 7023 6608 3566 7869 1680 6395 8538 8079 4993 9251 3279 7952 •8540051 9549 6419 •880 0633 4878 9508 1564 •8631019 7844 2014 6476 •8451064 3077 2488 9269 3394 8074 2618 4588 3956 ■8720693 4774 9670 4172 6099 6423 2116 6152 •8361266 5726 7609 6889 3538 7530 2862 7278 9119 8355 4960 8907 4456 8830 •855 0627 9820 6381 •8810284 6050 •8460381 2135 ■8641284 7801 1660 7643 1932 3643 2748 9221 3035 9236 3481 5149 4211 ■873 0640 4409 •837 0827 5030 6655 5673 2058 5782 2418 6579 8160 7134 3475 7155 4009 8126 9664 8595 4891 8527 5598 9673 •8561168 ■865 0065 6307 9898 7187 •847 1219 2671 1514 7722 •882 1269 8775 2765 4173 2973 9137 2638 ■838 0363 4309 5674 4430 •874 0550 4007 1950 5853 7175 5887 1963 5376 3536 7397 8675 7344 3375 6743 5121 8939 •857 0174 8799 4786 8110 6706 ■848 0481 1673 ■866 0254 6197 9476 33° 32° 31° 30° 29° 28° NAT. COBINS.f 0 1 2 3 4 5 6 7 8 e 10 n 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 61 62 63 54 65 56 67 68 69 60 / / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL SINES. 63° .891 0065 1385 2705 4024 5342 CC59 7975 9291 •892 0606 1920 3234 4546 6858 7169 8480 9789 ■8931098 2406 3714 6021 6326 7632 8936 ■894 0240 1542 2844 4146 6446 6746 8045 0344 •895 0641 1938 3234 4529 6824 7118 8411 9703 .896 0994 2285 3575 4864 6153 7440 8727 •897 0014 1299 2584 3868 6151 6433 7715 8996 •898 0276 1555 2834 4112 6389 6665 7940 26° G4° 898 7940 9215 899 0489 1763 3035 4307 5578 6848 8117 9386 9000654 1921 3188 4453 6718 6982 8246 9508 •9010770 2031 3292 4551 5810 7068 8325 9582 •9020838 2092 3347 4600 6853 7105 8356 9606 ■903 0856 2105 3353 4600 5847 7093 8338 9582 ■9040825 2068 3310 4551 6792 7032 8271 9509 •905 0746 1983 3219 4454 6688 6922 8154 9386 •9060618 1848 3078 25° 65° 66° 07° ■906 3078 •913 5455 •920 6049 4307 6637 6185 6535 7S19 7320 6762 9001 6455 7989 •9140181 9589 9215 1361 .921 0722 •907 0440 2540 1854 1665 3718 2986 2888 4895 4116 4111 6072 6240 6333 7247 6375 6554 8422 7504 7775 9597 8632 8995 •915 0770 9758 •9080214 1943 •9220884 1432 3115 2010 2649 4266 3134 3866 6456 4258 5082 6626 6381 6297 7795 6503 7511 8963 7624 8725 ■9160130 8745 9938 1297 9865 •9091150 24C2 •923 0984 2361 3627 2102 3572 4791 3220 4781 6955 4336 6990 7118 5452 7199 8279 6567 8406 9440 7682 9613 •917 0601 8795 •910 0819 1760 9908 2024 2919 ■9241020 3228 4077 2131 4432 6234 3242 5635 6391 4351 6837 7546 5460 8038 8701 6568 9238 9855 7676 •9110438 •9181009 8762 1637 2161 9888 2835 3313 •925 0993 4033 4464 2097 6229 6614 3201 6425 6763 4303 7620 7912 5405 8815 9060 6506 •912 0006 •919 0207 7606 1201 1353 8706 2393 2499 9805 3584 3644 •9260902 4775 4788 2000 6965 6931 3096 7154 7073 4192 8342 8215 5286 9529 9356 6380 •913 0716 •920 0496 7474 1902 1635 8566 3087 2774 9658 4271 3912 •927 0748 5455 5049 1839 24° 23° 22° 68° •927 1839 2928 4016 5104 6191 7277 8363 9447 •9280531 1614 2696 377 S 4858 5938 7017 8096 9173 929 0250 1326 2401 3475 4549 5622 669+ 7765 8S35 9905 •9300974 2042 3109 4176 5241 6306 7370 8434 9496 •9310558 1619 2679 3739 5340 6390 7439 84S8 9535 ■9330562 1628 2073 3718 4761 5604 21° NAT. COSINE.0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 61 62 63 64 65 66 67 68 69 60 / 253 / 60 59 58 57 56 55 54 53 52 51 50 49 49 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4. 3 2 1 0 / NATURAL SINES. 70° 71° ■939 6926 •945 5186 7921 6132 8914 7078 9907 8023 940 0899 8968 1891 9911 2881 •9460854 3871 1795 4860 2736 6848 3677 6835 4616 7822 5555 8808 6493 9793 7430 941 0777 8366 1760 9301 2743 •947 0236 3724 1170 4705 2103 6686 3035 6665 3966 7644 4897 8621 6827 9598 6756 •942 0575 7684 1550 8612 2525 9538 3498 •9480464 4471 1389 6444 2313 - 6415 7386 8355 9324 •943 0293 1260 2227 3192 4157 6122 3237 4159 6081 6002 6922 7842 8700 9678 ■949 0595 1511 6869 7880 8889 9898 •9380906 ! 6085 7048 8010 8971 9931 1913 ' -944 0890 2920 : 3925 I 4930 5934 1 1849 2807 3764 4720 2426 3341 4255 6168 6080 6991 7902 8812 9721 ■9500629 6675 1536 6630 2443 7584 3348 8537 4253 9489 5157 9450441 6061 1391 6963 2341 7865 3290 8766 4238 9666 5186 19° •951 0565 18° 72° 73° 74° •9510565 •9563048 •961 2617 1464 3898 3418 2361 4747 4219 3258 6595 6019 4154 6443 5818 6050 7290 6616 5944 8136 7413 6838 8981 8210 7731 9825 9005 8623 •957 0669 9800 9514 1512 •9620594 •9520404 2354 1387 1294 3195 2180 2183 4035 2972 3071 4875 3762 3958 6714 4552 4844 6552 6342 6730 7389 6130 6615 8225 6917 7499 9060 7704 8382 9895 8490 9264 •958 0729 9275 •9530146 1562 •9630060 1027 2394 0843 1907 3226 1626 2786 4056 2408 3664 4886 3189 4542 5715 3969 6418 6543 4748 6294 7371 6527 7170 8197 6305 8044 9023 7081 8917 9848 7858 9790 •959 0672 8633 •954 0662 1496 9407 1533 2318 •954 0181 2403 3140 0954 3273 3961 1726 4141 4781 2497 5009 6600 3268 6876 6418 4037 6743 7230 4806 7608 8053 6574 8473 8869 6341 9336 9684 7108 •955 0199 0499 7873 1062 •9601312 8638 1923 2125 9402 2784 2937 •9650165 3643 3748 0927 4502 4558 1689 6361 6368 2449 6218 6177 3209 7074 6984 3968 7930 7792 4726 8785 8598 6484 9639 9403 6240 •9560492 •961 0208 6996 1345 1012 7751 2197 1815 8505 3048 2017 9258 17° | 16° 15° 75° •965 9258 •966 0011 0762 1513 2263 3012 3761 4508 6255 6001 6746 7490 8234 8977 9718 •967 0459 1200 1939 2678 3415 4152 4888 6624 6358 7092 7825 8557 9288 •968 0018 0748 8719 9438 •9690157 0875 1593 2309 3025 3740 4453 6167 NAT. COSINE.254 / o l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1G 17 18 19 20 21 22 23 24 25 20 27 28 29 30 31 32 33 34 35 36 37 38' 39 40 41 42 43 44 45 46 47 48 49 60 51 52 53 54 65 66 57 58 69 60 / / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 26 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 2 1 0 / NATURAL SINES. 76° •970 2957 36 ;o 4363 5065 5766 6466 7165 7863 8561 9268 77° ■974 3701 4355 6008 5660 6311 6962 7612 8261 8909 9556 ■975 0203 0849 1494 2138 2781 3423 4065 4706 5345 5985 6623 7260 7897 8533 9168 9802 •976 0435 1067 1699 2330 2960 3589 4217 4845 5472 6098 6723 7347 7970 8593 9215 9836 977 0456 1075 1693 2311 2928 3544! 4159 78° ■978 1476 2080 2684 3287 3889 4490 5090 5689 6288 6886 7483 8079 8674 9268 9S62 ■979 0455 1047 1638 2228 2818 3406 3994 4581 5167 5752 6337 6921 7504 8086 8667 9247 9827 •980 0405 0983 1560 2136 2712 3286 3860 4433 6005 5576 6147 6716 7285 7853 8420 8986 9552 4773 | -9810116 6386 6909 6611 7222 7832 8441 9050 9658 •978 0265 0871 1476 12° 0680 1243 1805 2366 2927 3486 4045 4603 6160 5716 6272 11° 79° ■981 6272 6826 7380 7933 8485 9037 9587 ■982 0137 0686 1234 1781 2327 2873 3417 3961 4504 6046 6587 6128 6668 7206 7744 8282 8818 9353 9888 •9830422 0955 1487 2019 2549 3079 3608 4136 4663 5189 5715 6239 6763 7286 7808 8330 8850 9370 9889 ■984 0407 0924 1441 1956 2471 2985 3498 4010 4521 5032 6542 6050 6558 7060 7572 8078 10° 80° ■0848 078 582 •9849086 589 9850 091 £qq 9851 093 . 593 1 ■9852092 590 9853087 583 9854079 574 9855 068 561 ■9856053 544 ■9857 035 524 •986ff013 501 988 •9859475 960 •9860 445 090 •9861 412 894 •9862375 856 •9863336 815 ■9864 293 770 •9865 246 722 •9866196 670 •9867 143 615 •9868 087 657 •9869027 496 964 •9870431 897 •9871 363 827 •9872 291 754 •9873 216 678 •9874138 598 •9875 057 514 972 •9876428 883 9° 81° 1 •9876 883 •9877 338 792 •9878 245 697 •9879 148 CQQ •9880048 497 945 •9881392 838 •9882 284 728 •9883172 615 •9884 057 498 939 ■9885 378 817 •9886 255 692 •9887 128 564 908 •9888 432 865 •9889 297 728 •9890159 588 •9891017 445 872 •9892 298 723 •9893 148 572 994 ■9894416 838 •9895258 677 •9896 096 514 931 •9697 347 762 •9898177 690 •9899 003 415 820 •9900 237 646 •9901 055 462 869 •9902 275 681 8° •c •i •j •s 687 •9907 083 478 873 ■9908 266 659 •9909 051 442 832 •9910 221 ■9918 204 574 944 •9919314 682 •9920049 416 7R9 •9921147 611 874 •9922 237 599 959 •9923 319 679 •9924 037 394 751 •9925 107 462 7° MAT. COSINE./ 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 32 33 34 36 36 37 38 39 40 41 42 43 44 46 46 47 48 49 60 61 62 63 64 66 66 67 68 69 60 / / 60 69 68 57 56 56 54 63 62 61 50 49 48 47 46 46 44 43 42 41 40 39 38 37 36 36 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 2 1 0 ' NATURAL SINES. 719 •9936047 375 703 •9937 029 355 679 ■9938 003 326 648 84° 85° 86° 87° 88° 89° ■9945 219 •9961 947 •9975 641 •9986 296 ■9993 908 •9998 477 523 •9962 200 843 447 •9994009 627 825 452 •9976045 698 110 677 •9946127 704 245 748 209 626 428 954 445 898 308 673 729 •9963 204 645 •9987 046 406 720 •9947 028 453 843 194 602 766 327 701 •9977 040 340 698 812 625 948 237 486 693 856 921 •9964195 433 631 788 900 •9048 217 440 627 775 881 942 613 685 821 919 974 984 807 929 •9978 015 •9988061 •9995060 ■9999 026 •9949 101 •9965172 207 203 157 065 393 414 399 344 247 106 685 666 680 -484 330 143 976 895 779 623 424 161 •9950 26G •9966135 968 761 612 218 656 374 •9979 156 899 699 264 844 612 343 •9989035 684 289 •9951132 *849 630 171 770 323 419 •9967 086 716 306 854 357 705 321 900 440 937 389 990 565 •9960 084 573 •9996020 421 •9952 274 789 267 706 101 452 557 •9968022 450 637 182 482 840 264 631 968 262 611 •9953122 485 811 •9990 098 341 £39 403 715 991 227 419 667 683 945 •9981170 355 497 693 962 •9969173 348 462 573 619 •9954 240 401 625 609 649 644 617 628 701 734 724 668 794 854 877 869 798 692 •9955 070 •9970080 •9982052 983 871 714 730 345 304 225 •9991106 043 620 628 398 228 •9997 015 756 893 750 570 350 086 776 •9956165 972 742 470 150 795 437 •9971193 912 690 224 813 708 413 •9983 082 709 292 831 978 633 260 827 360 847 •9957 247 851 418 944 426 863 515 •9972069 186 •9992060 492 878 783 286 761 176 666 892 •9958049 602 917 290 620 905 315 717 •9984061 404 663 917 580 931 245 617 745 928 844 •9973145 408 629 807 939 •9959107 357 670 740 867 949 370 669 731 861 927 958 631 780 891 960 •9993069 986 966 892 990 •9985 060 •9998 044 973 •9960152 •9974 199 209 177 101 979 411 408 367 284 157 985 669 615 624 390 213 989 926 822 680 495 267 993 •9961183 •9975 028 835 €00 321 990 438 233 989 704 374 998 693 437 •9986143 806 426 1-0000 000 947 641 295 908 477 000 l 6° 4° 3° 2° 1° 0° NAT. 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COTAN./ 0 1 2 3 4' 6 6 7 8' e 10 n 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 63 54 55 66 67 68 69 60 * NATURAL TANGENTS. 261 42° >00 4040 9309 >014580 9S54 >02 5131 >03 0411 6C93 >04 0979 6267 >051557 6S51 >062147 744C >07 2748 8053, >08 33C0 8671 >09 3984 9300 >10 4619 9940 >11 5265 >12 0592 6922 >131255 6591 >141929 7270 915 2615 7962 >163312 8665 917 4020 9379 918 4740 919 0104 6471 920 0641 6214 9211590 6969 922 2350 7734 923 3122 8512 924 3905 9301 925 4700 926 0102 6506 927 0914 6324 9281738 7154 929 2573 7996 930 3421 8849 931 4280 9714 932 5151 47° 43° 932 5161 933 0591 6034 9341479 6928 935 2380 7834 936 3292 8753 937 4216 9683 938 6153 939 0625 6101 940 1579 7061 9412545 8033 9423523 9017 9434513 944 0013 6516 945 1021 6530 946 2042 7556 947 3074 8595 948 4119 9646 949 5176 •950 0709 6245 9511784 7326 952 2871 8420 9533971 9526 •9546083 955 0644 6208 9561774 7344 957 2917 8494 ■958 4073 9655 •959 6241 •9600829 (UOl 961 2016 7614 •9623215 8819 9634427 9640037 5C51 9651268 6888 46° 44° •9C5 6888 ■966 2611 8137 •967 3767 9399 •968 6035 •969 0674 6316 •9701962 7610 •971 3262 8917 •972 4575 •973 0236 6901 •97415C9 7240 •975 2914 8591 •976 4272 9956 •977 6643 •9781333 7027 •9792724 8424 •980 4127 9833 •981 5543 •9821256 6973 ■983 2692 8415 •9844141 9871 45°' 1-00 00000 1' 06819 11642 17469 23298 29131 34968 40807 46651 62497 68348 64201 70058 75918 81782 87649 93520 99394 1-0105272 11153 17038 22925 28817 34712 40610 46512 62418 68326 64239 70155 •986 1339 7079 •987 2821 8567 •9884316 •9890069 6825 •9901584 7346 •9913112 8881 •9924654 •993 0429 6208 •9941991 76074 81997 87923 93853 99786 985 56031*02 05723 7777 1-03 01196 •995 3566 9358 •996 5154 •997 0953 6756 •9982562 8371 .999 4184 1-0000000 45° 11664 17608 23555 29506 35461 41419 47381 63346 69315 65287 71263 77243 83226 89212 95203 07194 13195 19199 25208 31220 37235 43254 49277 65303 44° 4C° •03 65303 61333 67367 73404 79446 85489 91538 97589 1-0403645 09704 15767 21833 27904 33977 40055 46136 62221 6S310 64402 70498 76598 82702 88809 94920 10501034 07153 13275 19401 25531 31664 37801 43942 60087 66235 62358 68544 74704 80367 87035 91206 99381 1-0605560 11742 17929 24119 30313 36511 42713 48918 65128 61341 67558 73779 80004 86233 92466> 08702 1-07 04943 11187 17435 23687 43° 47° I' 1-07 23687 00 29943 !59 36203[58 42467167 48734 56 65006 55 61282!54 67561 53 73845 52 80132 61 86423)60 92718149 99018 48 1*08 05321147 11628 46 17939 46 44 43 42 41 24254 4 30573‘ 36896 4 43223 4 49554 40 65889139 62228 38 68571137 74918136 81269 55 87624 34 95984133 109 00347 [32 06714131 13085130 19460 29 25840 32223 38610 46002 61397 67797 64201 70609 77020 83436 89857 96281 1-1002709 09141 15578 22019 28463 34912 41365 47823 54284 60750 67219 73693 80171 86653 93140 99630 1-1106126! 0 42° ' NAT. COTAN.262 / o l 2 3 4 6 e 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 21 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 46 46 47 48 49 60 61 62 63 64 66 66 67 68 69 60 J / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 46 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 IS 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 2 1 0 / NATURAL TANGENTS. 49° 50° 61° 52° 53° 115 03684 11917536 1-23 48972 1-27 99416 1-327044S 10145 24579 56319 1-28 07094 78483 17210 31626 63672 14776 86524 23979 38679 71030 22465 94571 30751 45736 78393 30160 1-3302624 37532 52799 85762 37860 10684 44316 69866 93136 45566 18750 51104 66938 1-2400515 63277 26822 67896 74015 07900 60995 34900 64693 81097 15290 68718 42984 71495 88184 22685 76447 61075 78301 95276 30086 84182 59172 85112 1-20 02373 37492 91922 67270 91927 09475 44903 99669 75386 98747 16581 62320 1-2907421 83502 116 05571 23693 69742 15179 91624 12400 30810 67169 22943 99753 19281 37932 74602 30713 1-3407888 26073 45058 82040 38488 16029 32916 62190 89484 46270 24177 39763 69327 96933 64057 32331 46615 66468 1-25 04388 61850 40492 63472 73615 11848 69649 48658 60334 80767 19313 77454 66832 67200 87924 26784 85265 65011 74071 95085 34260 93081 73198 809471-2102252 41742 1-30 00904 81390 87827 09424 49229 08733 89589 91712 16601 66721 16567 97794 1-17 01601 23783 64219 24407 1-35 06006 08196 30970 71723 32254 14224 15395 38162 79232 40106 22449 22298 45359 86747 47964 30680 29207 62562 94267 65828 38918 36120 69769 1-2601792 63699 47162 43038 66982 09323 71575 65413 49960 74199 16860 79457 63670 66888 81422 24402 87345 71934 63820 88650 31950 95239 80204 70756 95883 39503 1-3103140 88481 776981-2203121 47062 11046 96764 84644 10364 64626 18958 1-3605054 91595 17613 62196 26876 13350 98551 24866 69772 34801 21653 118 05512 32125 77353 42731 29963 12477 39389 84940 50668 38279 19447 46658 92532 68610 46602 26422 63932 1-27 00130 66559 64931 33402 61211 07733 74513 63267 40387 68496 15342 82474 71610 47376 76786 22957 90441 79959 64370 83081 30578 98414 88315 61369 90381 38204 1-32 06393 96678 68373 97687 45835 14379 1-37 05047 753821-2304997 63473 22370 13423 82395 12313 61116 30368 21800 ' 89414 19634 68765 38371 30195 96437 26961 76419 46381 38591 1-19 03465 34292 84079 54397 46994 10498 41629 91745 62420 65403 17536 48972 99416 70448 63819 40° 89° 38° 37° 36° 33571 42131 60698 69272 67852 76440 85034 93636 1-40 02245 10860 MAT. COTAN./ 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 67 58 69 60 / / GO 59 58 57 56 55 54 63 62 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL TANGENTS. 56° •48 25610 34916 44231 53554 62884 72223 81570 90925 •49 00288 09659 19039 28426 37822 47225 66637 66058 75486 84923 91367 •5003821 13282 22751 32229 4171C 61210 60713 70224 79743 89271 98807 i-5108352 17905 27466 37036 46614 56201 65796 75400 85012 94632 L-52 04261 13899 23545 33200 42863 62535 62215 71904 81602 91308 1-5301023 10746 20479 30219 39969 49727 59494 69270 79054 88848 98650 33° 67° 1-53 98650 1-5408460 18280 28108 37946 47792 67647 67510 77383 87264 97155 1-5507054 16963 26880 36806 46741 66685 66639 76601 86572 96552 1-5606542 16540 26548 36564 46590 56625 66669 76722 86784 96856 1-57 06936 17026 27120 37234 47352 67479 67615 77760 87915 98079 1-58 08253 18436 28628 38830 49041 59261 69491 79731 89979 1-59 00238 10505 20783 31070 41366 61672 61987 1-60 033451-66 42795 76094 86525 96966 1-6107417 17878 28349 38829 49320 70330 80850 91380 1-6201920 12469 23029 33599 44178 58° 13709 24082 34465 44858 65260 656721-67 08782 698201-68 08489 547681-69 09077 65368 75977 86597 97227 1-6307867 18517 29177 39847 50528 61218 71919 82630 93351 1-64 04082 14824 25576 47111 57893 68687 79490 90304 1-65 01128 11963 22808 33663 44529 55405 66292 77189 88097 99016 723121-6609945 82647 20884 92991 31834 1-60 03345) 42795 32° 31° 59° 53766 64748 75741 86744 97758 19818 30864 41921 62988 64067 ■7515G 8625G 97367 19621 307C5 41919 63085 64261 75449 86647 97850 20308 31550 42804 54069 65344 76631 87929 99238 1-7010559 21890 33233 44587 55953 67329 78717 90116 3633S1-7101527 12949 24382 35827 47283 58751 70230 81720 93222 1-7204736 16261 27797 39346 50905 62477 74060 85654 97260 1-7308878 20508 ; 30° 60° 1-73 20508 32149 43803 55468 67144 78833 90533 1-7402245 13969 25705 37453 49213 60984 72768 84564 96371 1-7508191 20023 31866 43722 55590 67470 79362 91267 1-7603183 15112 27053 39007 50972 62950 74940 86943 98958 1-7710985 23024 35070 47141 59218 71307 83409 95524 1-78 07651 19790 31943 44107 56285 68475 80678 92893 1-79 05121 17362 29616 41883 54162 66454 787 59 91077 1-80 03408 15751 28108 40478 29° 1 90628 1-8303275 15930 28610 41297 63999 66713 79442 92184 1-84 04940 1 HAT. COTAN264 / o l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 21 25 26 27 28 29 30 31 32 S3 34 35 36 37 38 39 40 41 42 43 44 45 46 47 / CO 69 58 57 56 55 64 63 52 61 60 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 26 24 23 22 21 20 19 16 17 16 16 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL TANGENTS. 63° 96 26106 40227 64364 68518 82688 96874 9711077 25296 39531 63782 68050 82334 96635 9810952 25286 39636 54003 68387 82787 26087 26° 64° 65° 66° 67* 2-05 03038 21445069 2-24 60368 235 58524 2 18186 61366 77962 77590 33349 77683 95580 96683 48531 94021 2-25 13221 23615801 2 63732 215 10378 30885 34946 78950 26757 48572 64118 94187 43156 66283 73316 2-06 09442 59575 84016 92540 24716 76015 2-26 01773 237 11791 2 40008 92476 19554 310C8 65318 21608958 37357 60372 70646 25460 65184 69703 85994 41983 73035 89060 2 2-07 01359 68527 90909 238 08444 16743 75091 227 08SO7 27855 32146 91677 26729 47293 47567 217 08283 44674 66758 63007 24911 62643 86250 2 78465 41559 80636 239 05769 93942 68229 98653 25316 20809438 74920 228 16693 44869 24953 91631 34756 64490 40487 218 08364 52846 84118 2 56039 25119 70959 240 03774 71610 41894 89096 23467 87200 58C91 229 07257 43168 2-09 02809 75510 25442 62906 18437 92349 43651 82672 34085 21909210 61885 24102465 49751 26093 80143 22286 65436 42997 98425 42136 81140 69923 23016732 62013 06864 76871 35064 81918 21012G07 93840 53420 2-4201851 28369 22010831 71801 21812 44150 27843 90206 41801 69951 44878 2-3108637 61819 75771 61934 27092 81864 91611 79012 45571 243 01938 211 07470 96112 64076 22041 23348 22113234 82606 42172 39246 30379 23201160 62331 65164 47545 19740 82519 71101 64733 38345 244 02736 87057 81944 66975 22982 121203031 99177 75630 43256 19030 2-22 16432 94311 63559 350461 33709 23313017 83891 610821 61009 31748 245 04252 67137 68331 50505 24642 5 832131 85676 69287 45061 2 99308 223 03043 68095 65510 7 213 15423, 20433 234 06928 85987 9 31559 37845 25787 2.46 06494 0 47714 65280 44672 27030 8 63890 72738 63582 47596 4 80085' 90218 82519 68191 8 96301 224 07721 235 01481 68816 0 214 125371 25247 20469 247 09470 0 28793 42796 39483 30155 8 45069 60368 58624 50869 26° I 24° 23° 22° 2' 2 2 2 2 2 5 5 NAT. COTAN./ 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 61 62 63 64 55 66 67 63 69 60 / 265 / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL TANGENTS. 69° 70° 71° 72° 73° 74° 75° 2-60 50891 2*74 74774 2-9042109 3-07 76835 3-27 08526 3-48 74144 3-73 20508 73558 99661 69576 308 07325 42588 3-49 12470 63980 96259 2-75 24588 97089 37869 76715 50874 3-7407546 2-6118995 49554 2-9124649 68468 3-28 10907 89356 51207 41766 74561 52256 99122 45164 3-50 27910 94963 64571 99608 79909 3-0929831 79487 66555 3-75 38815 87411 2-76 24695 2-9207610 60596 3-2913876 3-5105273 82763 2-6210286 49822 35358 91416 48330 44070 3-76 26807 33196 74990 63152 3-10 22291 82851 82946 70947 56141 2-77 00199 90995 63223 3-3017438 3-5221902 3-77 15185 79121 25148 2-9318885 84210 52091 60938 59519 2-63 02136 50738 46822 3-1115254 86811 3-5300054 3-78 03951 25186 76069 74807 46353 3-31 21598 39251 48481 48271 2-78 01440 2-94 02840 77509 56452 78528 93109 71392 26853 30921 3-1208722 91373 3-5417886 3-79 37836 94549 62307 59050 39991 3-3226362 67325 82661 2-6417741 77802 87227 71317 61419 96846 3-80 27585 40969 2-79 03339 2-9515453 3-13 02701 96543 3-55 36449 72609 64232 28917 43727 34141 3-3331736 76133 3-8117733 87531 54537 72050 65639 66997 3-5615900 62957 2-6610867 80198 2-9600422 97194 3-3402326 65749 3-82 08281 34238 2-8005901 28842 3-14 28807 37724 95681 53707 57645 31646 67312 60478 73191 3-57 35696 99233 81089 57433 85831 92207 3-35 08728 75794 3-83 44861 2-6604569 83263 2-97 14399 3-15 23994 44333 3-5815975 90591 28085 2*8109134 43016 55840 80008 56241 3-84 36424 51638 35048 71683 87744 3-3615753 96590 82358 75227 61004 2-98 00400 31619706 61568 3-5937024 3-85 28396 98853 87003 29167 51728 87453 77543 74537 2-67 22516 2-8213045 57983 83808 3-37 23408 3-6018146 3-86 20782 46215 39129 86850 3-1715948 59434 58835 67131 69951 65256 2-99 15766 48147 95531 99609 3-8713584 93723 91426 44734 80406 3-38 31699 3-6140469 60142 2-6817535 2-83 17639 73751 3-18 12724 67938 81415 3-88 06806 413S3 43896 3-00 02820 45102 3-3904249 3-62 22447 53574 65267 70196 31939 77540 40631 63566 3-8900448 89190 96539 61109 3-1910039 77085 3-63 04771 47429 2-6913149 2-84 22926 90330 42598 3-4013612 46064 94516 37147 49356 3-0119603 75217 60210 87444 3-90 41710 61181 75831 48926 3-20 07897 86882 3-64 28911 89011 85254 2-85 02349 78301 40638 3-41 23626 70467 3-91 36420 2-70 09364 28911 3-02 07728 73440 60443 3-6512111 83937 33513 55517 37207 3-2106304 97333 63844 3-92 31563 57699 82168 66737 39228 3-42 34297 95665 79297 81923 2-8608863 96320 72215 71334 3-6637575 3-93 27141 2-71 06186 35602 3-03 25954 3-2205263 3-43 08446 79675 75094 30487 62386 55641 38373 45631 3-67 21665 3-94 23157 51826 89215 85381 71546 82891 63845 71331 79204 2-87 16088 3-0415173 3-23 04780 3-44 20226 3-68 06115 3-9619615 2-72 03620 43007 45018 38078 67635 48475 68011 28076 69970 74915 71438 95120 90927 3-9616518 62569 96979 3-05 04866 3-2404860 3-45 32679 3-69 33469 65137 77102 2-88 24033 34870 38346 70315 76104 3-97 13868 2-73 01674 61132 64928 71895,3-4608026 3-7018830 62712 26284 78277 95038 3-2505508 45813 61648 3-9811669 50934 2-8905467 3-06 25203 39184 83676 3-7104558 60739 75623 32704 55421 72924.3-47 21616 47561 3-99 09924 2-74 00352 69986 85694 3-26 06728 59632 90658 59223 25120 87314 3-07 16020 40596 97726 3-72 33847 4-00 08636 49927 2-9014688 46400 74529 3-48 35896 77131 58165 74774 42109 76835 3-27 08526 74144 3-73 2Q508 4-01 07809 20° 19° 18° 17° 16° 15° 14° NAT. COTAN.266 / o l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 31 35 36 37 33 39 40 41 42 43 44 45 46 47 43 49 50 61 52 53 54 55 56 57 58 59 60 / / 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 31 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 / NATURAL TANGENTS. 77° •33 14759 72310 •34 30018 87866 •35 45861 :*36 04003 62293 ;-37 20731 79317 '•38 38054 95940 [■39 55977 L-4015164 74504 1-4133996 93641 1-42 53439 1-43 13392 73500 1-4433762 91181 1-45 51V56 1-4615189 76379 1-47 37428 9S636 1-48 60004 1-49 21532 83221 1-50 45072 1-5107085 69261 1-5231601 91105 4-53 56773 1-54 19608 82608 1-55 45776 1-5609111 72615 1-57 36287 4-58 00129 64141 4-59 28325 92680 4-60 57207 4-6121908 86783 4-62 51832 1-6317056 82457 1-6448031 4-65 13788 79721 4-6645832 4-67 12124 78595 4-68 45248 4-69 12083 79100 4-70 46301 12° 78° 4-7046301 1-7113686 81256 4-72 49012 4-7316354 85083 4-74 63401 4-75 21907 90603 4-7659190 4-77 28568 97837 4-78 67300 4-79 36957 4-80 06808 76854 4-8147096 4-8217536 88174 4-83 59010 4-8430045 4-85 01282 72719 4-86 44359 4-87 16201 88248 4-88 60199 4-89 32956 4-90 05620 78491 4-9151570 1-92 21859 98358 4-9372068 4-9145990 4-9520125 94174 4-9669037 4-97 43817 4-9818813 94027 4- 99 69459 5- 0045111 5-0120984 97078 5-0273395 5 0349935 5-04 26700 5-05 03690 80907 5-0658352 5-07 36025 5-0813928 92061 5-09 70426 510 49024 51127855 5-1206921 86224 5 13 65763 5-14 45540 11° 80° | 81° | 5-1445540 5-6712818 6 3137515 7 625557 605813 686311 767051 848035 929261 5-2 010738 092459 174428 256647 339116 421836 504809 588035 671517 755255 839251 923505 5-3 008018 092793 177830 263131 348696 431527 620626 606993 693630 780538 867718 955172 5-4 042901 130906 219188 307750 396592 485715 575121 809446 906394 5-7 003663 101256 199173 297416 395988 256601 376126 496092 616502 737359 858665 980422 494889 6-4 102633 694122 693688 793588 893825 994100 5-8095315 196572 225301 348428 472017 720591 845581 971043 298172 6-5096981 400117 502410 605051 708042 811386 915084 5-9019138 123550 228322 333455 438952 514815 651045 757614 951121 5-6045247 223396 350293 477672 605538 733892 862739 992080 6-6121919 2522581 383100 514449 646307 778677 911502 864614 6-7 014966 971957 178891 6-0 079676 313341 187772 448318 296247 583826 405103 719867 514343 856146 623967 993565 733979 6-8131227 844381 269437 955174 408196 6-1066360 647508 177943 687378 289923 827807 402303 968799 615085 6-9110359 628272 252489 741865 395192 855867 638473 970279 682335 6-2085106 826781 200347 971806 316007 7-0117441 432086 263662 548588 410482 665515 557905 782868 705934 900651 854573 6-3018866 7-1003826 1375151 153697 9° 8° NAT. COTAN./■ 0 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 61 62 63 54 55 66 f>7 68 69 60 / / 60 69 58 57 56 55 54 53 52 51 50 49 48 47 46 46 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 26 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 2 1 0 / NATURAL TANGENTS. 84° 410613 679068 949022 493475 768000 331713 600927 881732 448166 733823 310088 600724 893050 10018708 048283 078031 10*954 138054 168332 198789 229428 260249 291255 322447 353827 385397 417158 449112 481261 613607 646151 678895 611841 644992 678348 711913 745687 779673 813872 848288 882921 917775 952850 988150 11023676 059431 095416 131635 168089 204780 241712 278885 316304 353970 391885 430062 5° 85° 86° 87° 88° 89° 11.430052 L4-300666 19-081137 28-636263 57-289962 468474 360696 187930 877089 68-261174 507154 421230 295922 29-122006 69-265872 546093 482273 405133 371106 60-305820 685294 643833 515584 624499 61-382905 624761 606916 627296 882299 62-499154 664495 668529 740291 30-144619 63-656741 704500 731679 854591 411580 64-868008 744779 795372 970219 683307 66-105473 785333 859616 20-087199 959928 67-401864 826167 924417 206553 31-241577 68-750087 867282 989784 325308 528392 70-153346 908682 15-055723 446486 820516 71-615070 950370 122242 569115 32-118099 73-138991 992349 189349 693220 421295 74-729165 12.034622 257052 818828 730265 76-390009 077192 325358 945966 33-045173 78-126342 120062 394276 21-074664 366194 79-943430 163236 463814 204949 693509 81-847041 206716 633981 336861 34-027303 83-843607 250505 604784 470401 367771 85-939791 294609 676233 605630 715115 88-143572 339028 748337 742569 35 069546 90-463336 383768 821105 881251 431282 92-908487 428831 894645 22-021710 800553 95-489475 474221 968667 163980 36-177596 98-217943 519942 16-043482 308097 562659 10110690 665997 118998 454096 956001 104-17094 612390 195225 602015 37-357892 107-42648 659125 272174 751892 768613 110-89205 706205 349855 003766 38-188459 114-58866 753634 428279 23-057677 617738 118-54018 801417 507456 213666 39-056771 122-77396 849557 587390 371777 505895 127-32134 898058 668112 632052 966460 132-21851 946924 749614 694537 40-435837 137-50745 996160 831915 859277 917412 143-23712 13-045769 915025 24-026320 41-410588 149-46502 095757 998957 195714 915790 156-25908 146127 17-083724 367509 42-433464 163-70019 196883 169337 541758 964077 171-88540 248031 255809 718512 43 508122 180-93220 299574 343155 897826 44-066113 190-98419 351518 431385 25-079757 638596 202-21875 403867 520616 264361 45-226141 214-85762 456626 610559 451700 829351 229-18166 509799 701529 641832 46-448862 245-65198 663391 793442 834823 47-085343 264-44080 617409 886310 26030736 739501 286-47773 671856 980150 229638 48-412084 312-52137 726738 18-074977 431600 49*105881 343-77371 782060 170807 636690 815726 381-97099 837827 267654 844984 60-548506 429-71767 894045 365537 27-056557 51-303157 491-10600 950719 464471 271486 52-080673 572-95721 14-007856 564473 489863 882109 687-54887 065459 665562 711740 53-708587 859-43630 123536 767754 937233 54-561300 1145-9153 182092 871068 28 166422 55-441517 1718-8732 241134 975523 399397 66-350590 3437-7467 300666 19-081137 636263 57*289962 Infinite. 4° 3° 2° 1° 0° NAT. COTAN.TABLE XYI. CHORDS, YERSED SINES, EXTERNAL SECANTS, AND TANGENTS OF A ONE-DEGREE CURVE. The angles of the table are the intersection angles, I, equal to the total central angle included between the tangent points. To find the corresponding func- tion for any other curve, divide the tabular number by the de- gree of curvature. The unit chord is assumed to be one hundred feet long. By using radius of 5,730 feet, the chord column of the table can be made serviceable for plotting. 269270 CHORDS, VERSED SINES, EXTERNAL SECANTS. •h Min. 0 a 4© CO 0 Cl 4© 00 0 Cl 4© 00 0 « 4© CO 0 Cl 4© 00 0 N 4© 00 0 Tan. 0 s* m Q 0 r^mO smo smO nmo «>. m q s- m o r**. m o s n o 0 © rtO'O m 0 © m O © fOO'O m 0 NO MO'O ftO'O m 0 © r»0'O mo 0 ** m m© oo o h m m© co 0 ** m m© oo o m m m© oo 0 ** m m© oo 0 minminm m© © © © © © r>NMst> t-*oo co oo oo oo oo o* on o* on o* on 0 Ex. Sec. ao moo 4 0 n ^ n 0 oo oo c*. h**oo O' w m© O' ci © 0 iamvo m o oo m 4 m *+ m 4© oo O' m m m© oo o ci 4© on m m moo 0 m tnoo o m© oo ■* 4 s* c* ci ci « ci ci mmmmm44444mmm m© © © © tstsi*. r^oo oo oo ddddddddddddddddddddddddodddddd Ver. Sin. oo moo 4 0 r**. 4 ci 0 oo oo s* tvoo O' m m\o o» ci © 0 m w © m o oo m 4 m m m 4© oo on m m m© oo o ci 4© O' w m moo 0 m moo 0 m© oo n w ci c* ci ci ci mmmmmTf^Tj-^-^-mmm m© no no © n n n c^oo oo oo ddddddddddddddddddddddddoddddod Chord. o m© o m© o m© o mvo o m© 0 m© o mvo Q mvo 0 mvo o mvo ao 0 m© o mvo 0 mvo o mvo o mvo o mvo 0 mvo 0 mo 0 mo 0 mo on O mo O mo O mo 0 mo 0 mo O mo 0 mo 0 mo O mo 0 mo On 0OO^MMao t*. h*. KaO 0_ 400 o88o8oqoooooo>ooooo'ooo'HHMMHwHMeici dddoddddddddddddddddddddddddddo Chord. 0 mso mso mso mso mso m s. 0 m s% Q mso mso m t*** Q 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© 0 m© Q m© 0 h h m ci ci ci mmm444mm m© © © c**. r*. e>»co 00 co On on O' 0 Min. 0 Cl 4© 00 0 Cl 4© 00 0 Cl 4© ao 0 Cl 4© co 0 ci 4© ao 0 « 4© 00 0CHORDS, VERSED SINES, EXTERNAL SECANTS. 271 eo I 0 ft ^©OO 0 ft ^©00 o ft ^© 00 0 ft 'tf'VOOO 0 ft «*•© 00 0 ft -*©00 o h h h I-. m n ft ft n w m© ’3 n h oo m n O'© m o© mo o ft O'© mo t"- ^ 0 i-- ^ m oo m m os 0 r>. ^ 0 t-* 0 tN.^0 fs.^H r*» m 0 t-» ■*■ 0 r-. ^ o b- ^ o 0 0 m 'm m© oo o h m m© oo o m m m© oo o m m in© oo 0 h m m© oo o ir> m m m m m© vovovovovo i--. t>«. t--. t-> r-» r>»oo oo oo oo oo oo O' O' o o> o o o MHMHMMHMMHMMMHHHHHMMMMHHHHHMHMft 3 O' ^ O' ^ owo mo Ntn^mmmmmNO»M -^oo m t>. ft i*** ^ w oo m n vo 0 m o» ^oo moo ann n ft t-» w t-- ft r^. ft oo moo ^ O' m o © w t-* m O' O'OOO^Mftftmm^'^io to vo vO t-** t-soo oo o» O' o 0 « ft « m m ^ ^ HftftftftftftftftftftftftftWftftNWftftftmmmmmmmmm Ver. Sin. I ^•oo moo moo tn m ov© ^ m m ft ft ft '*•© oo 0 m r- w © m © mo vo o in O' ^oo moo « f-» ft r>» ft r>» ft t-*, ft r*. ft ao moo O' m 0 vo « r- m O' O' 0 o 0 ■-* h n ft mm^^m m© vo t>. p>-oo oo O' O' O 0 m ct n . m m ^ HftftftNftwNNwwNftftwftftftwwftftmmmmmmmmm Chord. vo O' ft m O' ft in co w inoo « moo m -a- i-* o ^ t-* 0 ^ t*- O 1*- 0 ■»$-1-- 0 O' ft VO O' ft vO O' ft VO O' ft VO O' ft VO O' ft © O' ft 'O O'NvO O' ft vO O' ft © O' O' m© O' mvo o> m© o» mvo o* m© O' m© O' mv© o» mvo O' m© o» m© o O' 0 0 0 m m h ft ft ft mmm-^-^-Tt-mm tnvo 'O'D nn iv.oo oo oo o* o* O' w m m m m m m mmmmmmmmmmmmmmmmmmmmmmmm 1 0 ft *♦© 00 0 ft ^©00 O ft ^©00 O ft 't'OOO 0 « •*■© oo o ft -^-VO 00 0 h w h h m ft ft ft w ft mmmmmtt’tttmm«nifl mvo b 1 0 ft ■*•© 00 0 ft 'd’VO 00 0 ft ^© 00 o ft '•$'© 00 o ft ^© oo O ft "tf-vo 00 0 i 0 rs» h oo m n owo n 0"0 mo t-» m o N'J-hoo m n oo m n 0"0 m 0 r-* 0 © mo© movo mO'O mo t^mo r-*mo smo t-* m o r-. m o r** o 0 m m m© op 0 « m mv© oo o m m mv© oo o h m mvo co o *-* m mv© oo 0 0 0 0 000 m h h m m m ft ft w ft ft ft mmmmmm^-^^-^-^^-m HHMHMHHHHMHMHIIHHHMMHHHHHMMMHHMHH Ex. Sec. m ft ft ft m yf-\o oo m ^-oo ft VO «oo ^ O' r->» m m n ft n w n ft ^© oo m t n 0 mvo o> ft moo ft moo ft m O' ft vo O' m r*» m loornNH mo'^swo oo O' O' O' O' 0 O o h m m ft ft ft m m m t»- ^ m m mvo 'O nn t-»oo oo O' O' Ver. Sin. m « w ft m <*-© oo m ^oo ft © ft oo ♦O'smrtfi ft ft « « ft •*■© oo h -t ts o mvo O'« moo ft moo ft m O' ft v© omNw m O' m r** m momNcivo oo O' v> O' O' 0 O o h h m ft ft ft mmm^'^'mm mvo vo ns r--oo oo O' O' Chord. oo W tNM ts. 0 mso mvo O'NvO O' ft v© 0»WV© O' ft VO O' ft © O' mv© O' mvo O' mv© O' m© o» mv© O' ft © O' ft © O' n v© ov ft © O' ft © o\ O' m© o» m© o* m© O' m© O' m© O' m© O' m© o> m© O' m© o» m© O' o\ 0 0 0 ** h m n ft w mmm^'TC^-mm m© © © r-. r-* r-.oo oo oo ov O' O' Hftftftftftftftftftftftftftftftftftftftftftftftftftftftftftft 1 o ft ^© 00 O ft ^© 00 0 ft ■’*■© OO O ft ^© oo o ft ^© oo o ft ^© 00 o m h m m m ft ft ft « ft mmmmm-«-'4-Tr^-«fmmmm m©272 CHORDS, VERSED SINES, EXTERNAL SECANTS. 1 0 Cl tvOCO o Cl tvO 00 o Cl tvO CO 0 Cl tvO 00 0 N tvO CO 0 Cl tv© 00 0 h h m m m ci ci n w ci m© * n«^ hoo m ci 0 vo to Ns t w eo t hoo mno Ns t h owo m 0 stna 0 h co in hoo in n oo inooo innoo m ci oo mM O' m ci os u-> ci owo ci O"# m H O m m mvo oo o m m m© oo o h m m© eo o h ni m© oo o h n m© oo o m in m m m m© vovonovovo n s n s n t--oo oo oo oo oo oo oioioioiooio WNWNWttWNWNNWNNNNOlWNNNNNWClWNCINNrt | Ex. Sec. oirtNH in oun h oo m N O ovoo N.© n.oo o h m© 0 too w NmumH infnooo m ci 0 oo m m h O'© t ci o oo © t m m o>oo © tnn o oo n.© t m© © N> 00 (JlOlO H N « (n t m© © N» 00 OiQ O H N n t lO VO VO No 00 in^inmininmm©©©©©©©©©©©© NNNNNNtsNNNN »0 Ver. 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Sec. ci m 0 o o 0 h mmN.O'mN.Hvo w n tnoo m t m w h o m m m n. o O' m h n. rn O' m m N*m O'© ci ov m ci oo mcioo mci 0"0 mo n t hoo m t mvo vo n. n*oo O' O' 0 o m ci ci m 11 mvo vo n»co ooo'OHMCimmt mmmmmmmmmtttttttttttttttmmmmmmm Ver. Sin. M 0 O'CO N. N» 00 o Cl 't'VO o ■too moo to N. t H o O'00 N»VO Ns00 0 « t O' m o vo ci oo -t- m Ns m 0"0 cioo m h oo m m oo mcioo mci ovvo m h oo m t mvo vo N. Nsoo O' O' 0 0 n ci ci mtt mvo vo n.oo coovOOMCimmt mmmmmmmmmtttttttttttttttmmmmmmrn 1 t N. o mvo O' ci vo O' ci moo h too h t Ns o mvo O' mvo o* ci moo h too O' ci vo O' ci m O' ci m o> ci m O' ci m o. ci m o' ci moo ci moo ci moo ci moo 5 O' mvo O' mvo o* mvo O' mvo ov mvo o> mvo O' mvo O' mvo O' mvo O' mvo O' O' o 0 0 m m h ci ci ci mmmtttmm mvo vo vo n* Ns n>oo oo oo O' O' O' mtttttttttttttttttttttttttttttt 1 0 Cl t'O 00 0 Cl t'O 00 0 Cl t'O 00 o ci tvo 00 0 Cl t'O oo 0 ci t'O oo o h h m h m ci ci ci ci w mmmmmtttttmmmm mvoCHORDSVERSED SIXES, EXTERNAL SECANTS. 273 1 0 cr ♦©« o cr ♦©<» o tr ♦©co o cr ♦©oo o tr ♦©oo o a ♦©*» Q m -h m mt m n n q 9 n ♦ m m*n *o«>© 3 ♦ h o© m hob mno tvtnfi o>© ♦ tr on ts ♦ tr o on intT)H o ♦ m NtHCt h oo in m oo in h o in in in in m m© © © © © © M^NtsN csaooooo oo good O' o O' O' o od w rantQpmonnn^nnrenn to r> w n m w m rj rti^ d tn n) ri fo h m^ mvo tsao on titiHRHRHRHHHHHHHHHHHHHHHHMHRHRM'H 1 E > On os O O h « m r tr m to ♦ in© ts rsiao OO h r in'© tsoo o> © m © vo r** on 0 h n to ♦ mvo tsao ov 0 m tr to ♦© tsoo O' O m cr to ♦ m tsoo on i HHRHRRHHHHHHHR MMMMMMMM 1 o to© co m ♦© oih to© o*m ♦ tsg^tr torso tototjo m mvo on m ♦ ts© 1 no O' ci in o r moo cr moo m moo m ♦« m ♦« m o>ononOnon©nO'On6 d d d o d d © M M M M M M MM 1 s >■ OOOOOOQOOOO’C'O* moo cr mi® Q to ♦ ♦ m in ^ m w m o a*oo ts,© m'tmmN « m m © © O'OnOnO'OnonO'OnO'Onqi (BO'OHNmmt m© tsoo OiO h r m^ m© © rsao o*0 h ci mt m© Is CsQQ a® OB 00 QO 00 00 00 00 00 00 O'OnO'OnO'O'AOnO'^^O O 9 O 0 0 (Q M M ml M M M M \ Chord. 0 mvo oo m ■*■© on cr -c rs o cr moo o cm'© on cr ctsO w moo 0 tn©oo © oo m r-nh r-no ^ n o ♦no cotso rn© 0 cn© 0 rn© o* m© on cr © o m© on rn© O' m© on m© <> rn© dv m© O mvd o* m© on n © O' cr © O o 0 0 © mm m cr cr cr mmmRttmm m© © © ts ts tsoo cd od O' o t> m© © © © © © © © © © © © © © © © © © © © © © © © © © © © © © s£ a 0 CR ♦©CD 0 CR ♦© CD 0 CR ♦© 00 0 CR ♦© 00 O CR ♦© 00 0 N ♦©00 O274 CHORDS, VERSED SfNES, EXTERNAL SECANTS. 85 s 0 d *-s© 00 0 N -*S© 00 0 d ^s© 00 0 04 **S© 00 0 d ^-v© 00 0 R sfrv© 00 O m h m h m d C4 d d d mrtfncnfo^^^^^ioioioui in'© z (S m m h ao s© o Mflnwoo'O mooovo ^ m w ©> ts.\© ^ d q oo s© m d ©>'© ro 0"0 ro O n m O ts o ts m* m n^hoo itihqo m d O"© o 0"0 ro 0 d m ts ©i *-• d ^S© NONI^ 0) ’«$■'© SO'H N ^vO ts O' w d ^s© s O' N m m m m m in\© 'O'O'O'O'O r>. rs ts is s soo ooooooooao O' O' O' O' O' O' 0 8 m ^oo m moo d m O' « v© o* m ts w m o* m is. m m o> -^ao mSMio o m o tsoo O' M n m m\o is, on 0 h m ^s© soo o h m ^ m tsoo 0 m m ^\© is.co •» w Ver. Sin. 1© O' d s© O* d s© O' d sO O'RVO 0 ^00 d v© 0 '♦OO Civ© o ^ O' rn Is. M m 0 \© s O'0 h m m is.00 O'h « ms© oo onh n n m*© oo O'0 « m ms© oo Chord. 0 d ^ is. O' m •♦s© oo m m moo 0 d m is. o' d ♦•s© O' w ms© oo 0 mmso m ^ ts. © m is. o mso o ms© o* ms© O' ci m O' d moo d mao m moo h ^oo O' d wo'N m O' n m O' d moo d moo d moo d moo d moo d moo d moo 00 0 6 *-* h* *-• d d c* mfom^-s^Tftnm mv© '©'© n is r>.oo oo oo o o o 00 O' O' O' O' O' O' O' O' © O' O' O' O' O' O' O' O' O' O' O' O' O' © O' © O' O' O' O' O' 1 0 Cl '♦s© 00 O N ♦s© OO O d >♦*© 00 O d ^v© 00 0 d ’♦'© 00 fc d ^s© 00 O m w h h mm d d d ci d + ms© 1 O Ct '♦so oo O d st-s© non st-v© oo o n "♦*© oo o d ^s© ooon •*•*© oo o 3 0 is. m o O"© ^moo m m 0 smn osinn 0 oo so «♦ h O' is m rt o oo m is, m 0 smO is. ^ o rs ■*■ m ts. ^ h rs st- w oo m m oo m d ao mw O"© ci O 0 ci •^•mis.O'O n m is. ©. 0 d sj- m is. 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Sec. so m m ossiofOH cjiNtfimN o O' Is* m d h O' inno m ^ m h o o»oo In m (n. O' 0 d ^-nO oo p'H mmtN.O'O d ^no oo 0 m mmt-.O'w m mvo oo 0 h h m d d d d d d mmmmm^'vi*n-Tt-'^mmmmm mvo 'O'O'O'O s mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm H Ver. Sin. O'fv’^d O' tv m d o in m m h O' cnvo d 0 oo tNvo ^ m h o O' inno m m m in O' o d -tf-vo oo O' h m mvo oo o d ■^'O so'h minsoo d ■«f'0 oo m h h m d d d d d d mmmmm'^'4'^^^^mmmm m\o 'O no no no m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m Chord. O' d moo m moo h ^ s o mso mvo O' d moo d moo h ^ no ^no m Sh ^ S H ^Sh N. m S h ^ S 0 ^ N 0 ^ In O ^ S 6 ^ In 0 ^ In o> o 0 0 h h ih d d d mmm-^^-^rmm mNO ovo nn inoo oo oo O' O' o* Mdddddddddddddddddddddddddddddd WWMNHMWMHMMHWMMHMMHHM’HMMMMMMWWM 2 S 0 d ^-NOOO 0 d ^NO 00 o d ^-VO 00 0 d ^NO 00 0 d ■^■'O 00 0 d ’*■'© oo o e» h h h m d d d d d mmmmm^^^^^-mmmm m^CHORDS, VERSED SINES, EXTERNAL SECANTS. 277 ! 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O' ci moo h t no mvo O' ci moo h ♦no mvo o> ei mao h ♦no mvo o» m O' ci m oi n movci moo n mao ci moo ci mao m moo ** moo m moo m + O' O' 0 o 0 *-• *+ •- ci ci ci mmm^-^-^-mm mvo vo vo tv tv tvoo oo oo O' ov ♦-■^mmmmmmmmmmmmmmmmmmmmmmmmmmmmm i 1 0 Cl ♦“VO 00 0 N ^*vO 00 0 Cl '♦vo 00 0 Cl •'♦'O 00 0 Cl '♦'O 00 0 Cl ♦■'O 00 Q i H I 0 Cl ^vO 00 0 Cl '♦’VO 00 0 Cl '♦■VO 00 0 Cl '♦■vO 00 0 W '♦■vO 00 0 Cl '♦'0 00 0 mmciciMMMOOO'O'OOv ovoo two m m ♦- m ci w h o o'oo tv two m m ci O"0 mo tv-^M tv ci oo m ci 0"0 mo tv m ao mciao m ci 0"0 m m mvo oo o ci m m tvoo o ci m m tvao 6 w ♦msoo ci ♦msO'O ci ^ : OOOO'-'MMHMMCicicicicicimmmmmm^-’vr'^^'^-Trmmm NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN Ex. Sec. m m mvo tvoo O' 0 « ci m ^*vo tvoo o m n mvo oo q ci '♦vo oo o (i ♦ 0 ci ♦■'O oo 0 ci ♦nO'm mmtvovH ^vo oo o n ♦■vo O' m mmtvo ci ♦■ mmmmm^-'^^-^-^-mmmm mvo vOvOvo tv tv tv tv tvoo oo oo oo ov O' O' Ver. Sin. m n ci m m'♦■’♦'m mvo tvoo O' 0 w ci m mvo tvoo o w m ♦■vo noiO n n O' h m m tv O' m m m tv O' m oo 0 ci ■♦'O oo 0 m m tv ov m m moo 0 ci ci mmmmm^^^Tt-^-mmm inv© vo vo vo vo tv tv tv tv tvoo oo oo oo O' Chord. vo O' ci moo m moo h ♦ s o mvo O' ci vo O' ci moo h ♦no m n o mvo O' vo O' mv© O' mvo O' mvo O' mvo O'Ci'O ovcivo O'Civo o> ci v© O' ci vo ov ci m O' o* 0 0 o « « ci ci ci mmm-^-^-Tt-mm mvo vo>o nn tvoo oo oo O' ov 1 0 Cl ’♦■vO 00 0 N '♦vo 00 0 Cl ’♦'© 00 0 Cl *-v© 00 0 Cl '♦‘O 00 0 Cl ’♦’VO 00 0278 CHORDS, VERSED SIXES, EXTERNAL SECANTS. 1 0 N ^vO 00 0 N ^-VO 00 0 N sJ-vO 00 O M sf-vO 00 0 N -*-vO 00 0 « -+VO 00 0 m m m h m « w w n w mvo i m mvo 'Ovo k n ts,oo oooo oo9<0 0 » « o nt mvo vo r>»oo o>o 0 ^ n mo n h oo m w O"0 m o in w owo mo n^hoo m n 0"0 ^ hoo in no oo O' m m sf vo oo O' « m mvo oo o « m m tx©o o n m m rsoo o n tviN m m mvo vo^ovovovo nsnn isoo ao oo oo oo co O'o O'o O'O'0 0 0 o o ooooaoaoooooooaoooooaoaoooooooooooooooaoaoaoooooooao O' O' o> O' O' Ex. Sec. ^•O'«n0'O hvo w in. moo m o* m w ts. oi ao ^ o vo m 0"0 n 0"0 n o> m n vo oo m O' m ^vo O' m -^vo O' n in. o' w m is. o w moo o mvo oo m mm^-^^^mmm mvo vp vo vo in. is is. isoo ao oo O' O' o O' 0 0 O 0 « m vOvOvOvO'OvOvOvO'OvOvOvOvOvOvOvOvOvOvOvOvOvO'OvOvO ts N Is N N ts H Ver. Sin. ■*-co moo moo « nn nn n moo ^•O'^O m m vo m NnO'U'O'O woo ov M ^*nO O' w ^-VO O' M ^-vO O' M T*-vO O' Cl vJ-NOiN Is. O' Cl IAN0 W IO n mmmm^^^^mmm mvo vo vo vo nn is isoo oo oo oo O' O' o> 0 0 0 vOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvOvO Is ts S Q O O' n moo m ts. o'« moo h >*• is o mvo o»« moo m mvo o» « moo * is msO m n o mvo 0 mvo 0 mvo 0 mvo O' mvo o> mvo O'W'O O' w vo O' ci O' O' 0 0 0 m m m ct n ci mmm^^^^mm mvo iovo nn rsoo oo oo O' VO VO MsIsNMsMstsNNNSNNNNNNNNSNMsNNNN 2 s 0 w ’^■'O 00 0 N ^-vo OOON sfvO WON -*-vO 00 0 N s*-vO WON s*-v£> 00 O 55 s 0 « ^vOOO 0 M -^VO 00 0 N sJ-vO 00 0 N <+vO 00 0 M ^vO 00 0 N s^vO 00 O h h m h m ot w n c* w mmmmm^-’^-^'^-^-mmmm mvo 2* £ O'O'tj-O'O'O'O'O'O'O'O'O'O'O'O'O'O'O 0 m h w n m m m ^ m m n owo mo n ^ n oo m w owo m o s ^ w 0"O mo in w oo mo* 0"0 m mvo oo o n m m ts.ao 0 « mmsO'O w t >o n o h n -**vo t-s ov h ci ^>vo 0 0 0 M M m m m h ci « m ci n w mmmmmms#*^-^'^-^^mmmm ooaoooooooaoaoooooooooooooooooaoooooooooooooooooooaooooooooooo H Ex. Sec. H •^•00 N vo 0 "*00 Cl vO 0 ^ O' moo N vo H mo ^O'^O'^O'^O'^O'S^ m m is. o w m is O' n is. o' m O' h ^vo O' m mvo oo « mvo oo m mvo VO VO VO Is N IS S IN. oo WWWO'O'O'O'OOOOHHHHNNNNmmrt lommmmmmmmmmmmmm mvo vovovovovovovovovovovovovovo Ver. Sin. vo O' mvo o m t-s o ^-nh ^-oo w vo 0 ^oo nvo o 'tov moo «vo « m o Is. O' N ** IN. O' H TCvO 00 M m moo 0 miONO N in S O' (1 t N(>N NOi »n m\6 vO VO VO Is N Is Is 00 WWWO'O'O'O'OOOOOhhhhNNNN mmmmmmmmmmmmmmmmm mvo vovovovovovovovovOvovovo Chord. | O' n moo h t so mvo O' w moo m ^no mvo O' w moo h tso mvo O' ■^-00 H ^00 M ’^■oo M sf In. M ’t N H 'I N n ^ N O ^ N O ^ N O *t N 0 m O' O' O o 0 M M M w n w mmm^-^*^mm mvo vo vo in. is tsoo ao oo O' O' m mvO 'OvOvOvO'OvOvOvOvOvOvO'OvOvO'OvOvOvOvOvOvOvOvOvO'O'OvOvO HHHMHHHMHHMHHIHHHMI-tHHHHHHHHHHHHH 2 2 0 « ^-vO WON st-vO oo o N s*-vO WON s|-\0 WON **vO 00 0 W ^vO 00 O h h h m ii w n ci ci w mmmmmsr-^*s|-ri*s|-mmmm tr ivoCHORDS, VERSED SINES, EXTERNAL SECANTS. 279 °o 1 0 « ^*» 0 0 w m mvo co O' M n mvO oo O w ^vo oo 0 w oo m m o o. mj- h oo m in o n ^ h oo m m o r>. m o"0 mow mw 0"0 -** 00 0 w N M}-v£> M>H N ^*vO 00 O' M m 4-vO 00 Q w m invO oo 6 mvo 'O'O'OvO'O t-* t-» t-» t-» r»» r**oo oo oo oo oo oo OO'O'O'O'O g g g g g h O' Qt O' Ov O' O' ^ Ot Os Os O' O' O' O' O' O' O' O' O' O' O' O' O' 0 0 O 0 0 0 0 Ex. Sec. m >*■ n m os oo t>. m -*#■ n h o g>oo ts. t>.vo m m m m n w h h m oo O'O' vO O'« in is. o rovO O'« moo 0 mvo O' w moo m r>. o mvo O' w moo 0 m O' O' O 0 0 M m m m n n oi mmrim^^^mm mvo sosovo nn t->oo oo N 0.00 ooooooooooooooooooooooooooooooooooooooooooooooooooooaooo Ver. Sin. ffiS'&s-asS'S^aa^g'SM as srjsw •s.?. R R 8 £j <2 & *■ tv o mvo ovh ^no mvo O' w m li il mi 4i4 ii 4Mt ttilii ^ mm €i t 1 ° ■ 2 a jss g s srggs 9>n>x9>%%5.T$%9>9>xs>%&280 CHORDS, VERSED SIKES, EXTEREAL SECAKTS.CITORDS, VERSED SINES, EXTERNAL SECANTS. 281 1 O M ^v© 00 O N ^VO 00 0 N *♦»© 00 0 M '♦'O 00 O N '♦'O OO O K ■♦vO OO 0 h h h h h cs d d d d m'O Tan. oo m d o tv ♦ d o> tv ♦ h owo ^hoo\o m h co m m o oo m d 0 m/in o i/) N (Ji H N ♦ 'O S O' H M ♦ vO 00 o M m in tvOO Q d ♦ m tv O' h d ♦vO 00 vO vO VO Is fs N N S tv.00 00 00 00 OO Oi O' O' O' O' O' O O 0 0 O 0 h h m m m MHMHHMMMHHHHHMHHMHHHddddddddddd 8 (/) y* ^ tv. h ♦oo m moo d iooinvo O' mvo 0 m tv. h ■♦oo m moo nio 0 m\o 0 t*v t>. 00 oo co 0 0 O' 6 0 0 h h m d m mvo vo O tv. tv. 00 » w w Ver. Sin. 0 « no ^ s 0 tv. h iw h ♦oo m •♦00 h moo h moo d moo d m ov d m m mvo ovo nn tvoo 00 00 oomoo 0 d « « m m d d mnmt Chord. 00 ►* m\o 00 k •♦vo 0 m ^so'W ♦* tv 0 n m tv 0 m mao 0 mv© 00 h mvo ♦00 m tNH ♦no ♦no m tv 0 m tv 0 mvo d mvd O' mvo O'N'O on 0000 O' 0 0 0 0 0 h m h d d d mmm^^^mmm mv© vO 'O n tv tvoo d e* d d d rtrtrofnMnPK’innnmrtmfnnifififnf'fOCimfnrtPi dWddddddddddddddddddddddddddddd 1 O e* '♦'O 00 0 d ♦■'O 00 0 N '♦'© 00 0 N '♦v© 00 0 N '♦v© 00 O N ♦'O 00 0 1 0 d ’♦'© OO O N ♦v© 00 O N '♦•VO 00 0 N ’♦v© 00 0 N ♦v© 00 O Cl ♦v© 00 O Tan. oo m m o tv. n ov vo ♦ h oo vo m o isioo on^m owo ♦ w oo v© m o oo m m tv on o d ♦mNO'M d ♦v© oo O' h m ♦v© oo o « m m tv.00 0 m ■♦ m m h h h d d d d w d rommmmm^^-v*-^-^mmmmm mvo v© v© >© - m d vo O' n vo O'ttvo O'N'O O' m vo O' mv© O' mvo O' mvo 0 m tv o mso 1 tv tv tv oo oo oo O' O' O' o O O h h m ti d d mtnfn^ + inm mvo vo vo tv tv 1 »• y W e* Ver. Sin. m v0 O' n moo m ♦no ♦no ♦no ♦no ♦no m tv o m tv o m tv o m m mvg vg vg t-» N* N»cg og> co g'^'^'0 6 0 m m m d w d'mmm^^^-m Chord. non moo 0 mvo O'*-* ♦nO'« m n o mvo oo m •♦vo O' d ♦no m moo vd 0 mvo O' mvo O'W'O O' d m o d m O' d moo d moo m moo m moo m ♦ oo O' O' O' O' 0 0 o h m mci d d mmm^^^mm mvo vo vo tv tv tvoo oo MMMMMdddddddddddddddddddddddddd ddddddddddddddddddddddddddddddd 1 0 W ♦'O 00 O N ♦■vO 00 0 N '♦vo 00 O Cl ♦'O 00 0 N ♦'O 00 0 N ’♦vO CO 0 h h h !-• h d d d d d mmmmm^^^^^mmmm mvo282 CHORDS, VERSED SINES, EXTERNAL SECANTS. z S O d ^■'O OO o N ♦'O 00 o N ♦vO 00 0 N ’**'0 00 0 N ♦vO 00 0 d ♦»© 00 O *-i h m m h n ch d w oi mvo * H m 0 00 tfihooo m m w oo vo m m oo vo m m 00 v© m w oo v© ♦ w OvvO ♦ m O' 0 cs MirtsoO d ♦>© r>. O' m m do o h n m hs.co 0 d ♦ m r-. O' hi d C*. f>. hs, fs» t>. r^»00 COOQCOOOOOO'O'O'O'O'OOOOOOhmnhhhNN d d d d d d d d ci d d d d w ci ci ci mfnmmmrnmnimrnmmmm 8 in y m m 0 mvo 0 ^00 d wo^nm in 0 mvo 0 ♦oo nno 0 -^oo d m O' m c-. 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I 00 ci inomo 0 rnts.o *00 w moo pi vo 0 m t*> w moo ci inonso -*-oo mvo 'O'O n hs.oo 00 oo O' O' O' 0 0 0 m w d d d m m m ♦ ♦ ♦ in mv© 'O vo Chord. ♦v© O' m ♦vO O' m ♦<© O' H ♦'O O' w ♦<© O m ♦»© O m ♦%© O w ♦v© O 0 mvo 0 mvo O' mvo O' d vo 0 d m 0 d moo ci moo m moo m -^*00 w ♦ is. 00 oo 00 O' O' O' O' 0 0 0 m m m d d d mmm^^-^mm mvo vo vo nnn ♦ ♦♦♦♦♦♦minmmminminmmminmmmminmmmmmmin dddddddddddddddddddddddddddddCId 5 s 0 Cl <+10 CO 0 ci ♦'OOO 0 ci ♦<© 00 0 d -♦'O 00 0 d ♦vo 00 0 d ♦lOOO 0 m h hi m * d d ci d d pmrtrtnt^^^tmminm mvo s s 0 d ♦vo CO 0 d ♦vo CO 0 d ♦O 00 0 d ♦O 00 0 d ♦O 00 0 d ♦vo 00 0 3 0 t>». ♦ d o» c-* ♦ d 0"0 ♦ m 0"0 ♦hcovo m m co 10 dMoo m m 0 co m m 00 on n ♦vo 00 6 m m m t^.00 6 d ♦mt^.O'w d ♦o 00 O' m m mvo 00 0 m h d d d d d mmmmmm^^^^-'j'^mmmmm mvo vo vo vo vo t>. ddddddddddddddddddddddddddddddd i v 0 ♦ ts. m moo d vo omno ♦oo h m O' d vo 0 ♦nm m O' d vo 0 + nm 000000 0 O' O' 0 O 0 m h d d d m m m ♦ ♦ m m mvo vo-o n tsoo ooco 0 d d d d d d mmmmmmmmmmmmmmmmmmmmmmmmm i w i Ver. Sin. d m O' d vo O' mvo 0 m t-* 0 ♦ m ♦oo m moo d vo O' mv© 0 ♦ t-* *-< ♦oo 1 m m mvo vo vo t-- t*-eo co 00 O' O' O' 0 0 0 n m m ci d d ^(^♦♦♦mmm ] d d •» d d d d d d d d d d d mmmmmmmmmmmmmmmmm l Chord. vo O' h ♦vo O' d ♦ t-» O' d inNO d moo 0 m moo m mvo 00 h ♦vo o» w ♦ j d m O' d moo d moo m moo m moo h ♦oo H^NH^NO^NOrtNO i 000000 o> 0 o> 0 0 0 *-i m r* ci d d mmm^^^mm mvo vo >o n s r-*co 1 cidciddddddddddddddddddddddddddcid £ a 0 d ♦vo CO 0 d ♦'O 00 0 d ♦VO 00 0 d ♦VO CO 0 d ♦vo 00 0 d ♦'O 00 0 m h hi hi m d d d d d rommmm^^^^^mmmm mvoCHORDS, VERSED SINES, EXTERNAL SECANTS. 283 0 CO 00 0 M -"t-vO 00 0 M -^-vO 00 0 N "*NO 00 O M ^f-NO 00 0 h m h h h co w c« co w mcnritofo^^^^^iniounn iono oo co c- m icioimnh m o ^oo n \o h id O' rooo co no o ID O' f’1 c» h »o o ^ co nn^^ + iD mvo 'O c** r^. t^oo oo O'O'O'O o m m co ci co 'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'OnO tsSts|sfstsNh,tsNNrsN CO NO 0 -**00 « SO 0 ^00 CO NO 0 ^-00 CO NO 0 *4*00 CO no O ^00 (4 NO O ’♦’OO CO oooo O'O'O'O o m m 4(i co nmn^^iDiD tnvo no nn i-'.ao oo c> on O' 6 0 CO ^NO 00 0 CO ^-nO 00 0 CO ^-nO 00 o CO ^*NO 00 0 co co co co co fnrinrKDtt^t^ifiiniDin mvo 0 CO <**0 00 0 CO '♦'O OO o CO -^NO 00 0 CO '♦'NO 00 0 co co co co co fncnmncD^^^^^iDiDiDiD mvo 00 H ID O' n N H >♦00 CO VO O ^SH ID O' ro N O ^00 CO VO 0 flNH V> O 0*1 no rs> rs. t>.oo oo O'O'O'O O m m h co co co com t + 'Cw udno vovo ns c-ao O CO ^NO 00 O CO -+NO 00 0 CO ♦’'O 00 0 CO '♦'O 00 o CO CO CO CO co rtrOmnnt^t^tiDiniDiD NO NO284 CHORDS, VERSED SINES, EXTERNAL SECANTS. Min. 0 W '^•'O 00 0 d ■+'0 00 O d **sO 00 O d -*sO 00 0 d ^VO 00 O d ^*vO 00 O m ih ci m m d d d d w tnfofotnn^^^^^ioioioio mso z H O' f-* d ooovo m *-» O'tN.md o oo so ^ w O'MnwOoovo ^d o Niori m nuifsO'O d ^vo h*. O' m m inso co 0 d nmNO»H d ^so oo o m mm aoaoooaocoO'O'O'O'O'O'OOOOOM'-iHMM^dddddfnmfnm OMHHHHHINMHMHHHHHHHHHMIHMMHINMHHMH £ c/3 x i in 0 ^ O' moo d ts. w in 0 m o* ^oo m t-» d t-* m so 0 m o> ^*oo rroo « nh | oo O' O' O' o 6 M w d d m m m ^ ^ in mso 'O t-» t-»oo aooo O'O'O 0 M *-* d 00 00 00 00 O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' 0 0 0 O o 1 w j et Ver. Sin. 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Sin. d so 0 ■'♦■oo d so 0 '^■oo mi>.M inoPiNH m O' ^oo d so 0 ^oo d so 0 m OOMMMddmmm^^inm inso 'O nn t-»oo ooo'0'000«-*»-'dd tsNNNh>NtsNNNNNNNNNt\NNNtststs r-.OO 00 00 00 00 00 00 | Chord. | *^so O' h mvo oo o nmNO d -*• t-% O' w ^so oo w m moo o d m t>* O' d d moo d mao m moo m ^oo m ^ s 0 ^ f-» o pino mso 0 mso O' d so O' ts fs. ts.oo aooo O' O' O' O 0 0 « h m d d d mmm-^-Tt-Tj-mmm mso 'O so NNtstsisNNf' t'-ao ooaoooooooooooooaoaoooooooaoaooooooooooooo ddddddddddddddddddddddddddddddd 2 S O d ^-sO 00 0 d ^SO 00 O d ^so 00 o d ^*sO 00 O d 't’O OO o d -*-sO oo OCHORDS, VERSED SINES, EXTERNAL SECANTS. 285 1 0 W 00 0 n 00 0 N '♦NO 00 O N 00 0 d ♦<© 00 0 N ♦vO 00 o ooovo ♦ « o oo >o ♦ Cl 0 00 VO ♦ Cl 0 00 NO ♦ Cl 0 00 v© m m M OMnmn on 0 ci ♦vo oo O' h m m tvoo 0 ci ♦<© tv o\ h m mvo oo o ci ♦ m tv O' m m oo9«o\0'OOtOQOOdOHHHHHHMoitHnNnmmnnnt^ IO V) (A in tO iovO vonOnOnOnOnOnOnOvOvOvOvOnO'OvOnOnOnOnO'OvOvOvO CO Ex. 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Sin. ♦oo m tv ci v© h m o ♦ O' moo ci tv h no o m o< ^oo ns«to h mo m o oo oo pNOvO 0 m h n ci ci nw tt m mvo vo so tv tvoo ooo'O'OOmwci OOOOHHHHUMMt-UMMMMlvUI-IMINMi-IUMCICICICICI NNNCIddddNNdNCIdddddNNNNdNdeiNdCINd Chord. no oo o « ♦'O oo o m m tv o' m mmtvONH m m tv o» w ♦»© oo o n ♦vo oo n in O' n moo h moo m tSH ^ s o m tv o mvo o* m\o O' ^ no O' n moo vo no no tv. tv tv oo co oo p'p'p'0 0 0 m m *-« c» ci ei ci mmm^^^mmm OOOOOOOOOOOOhmmhmmmhmHh«hhmwmmm m m m m m m m m m m m m m m m m m m m m m m mmmmmmmm m S s 0 Cl ♦vO 00 0 Cl ♦<© 00 0 Cl ♦<© oo 0 Cl ♦v© 00 0 Cl ♦NO 00 o Cl ♦'©00 o 1 0 Cl ♦'O 00 0 Cl ♦'O 00 0 N ♦v© 00 0 Cl ♦'O 00 0 M ♦'O 00 0 W ♦'O 00 0 i H m m 0 tv m m 0 oo vo ♦ ci ooono ♦ ci 0 tv m m M O' tv m m h O' tv m ci o m tvoo o ci ♦'O tv O' m m mvo oo 0 ci ^msotH n ♦so oo o ►« m m tv o' {^(^•^♦♦♦♦♦♦mmmm mvo vonO'O'O'O tv tv tv tv tvoo oo oo oo oo oo mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm s. Ex. Sec. hno h m o mot^ot moo m tv ci tv h vo w m o mo ♦ on ♦ O' ♦oo moo m w ci m m ♦ ♦ ♦ m mo vp tv tvco oo O'O'O 0 m h ci ci ci m m ♦ ♦ m mvo 00000000000000000*^hmhmhhmmhmmmm ClMCtCtCINCICICICICICICieiCICICICICICICICICICI Cl*" Cl Cl Cl Cl Cl Cl Ver. Sin. nvo h mo ♦oo ci v© m m O' ♦oo m tv h vo 0 m O' moo ci tv h m o ♦ on ♦ m mvo v© tv tv tvoo oo O'O»O'0 0 m m ci ci m m m ♦ ♦ m mvo vp ns tvoo O'O'O'O'O'ONO'OO'ONO'O'OOOOv'JOOpOOOpOOOOOOO Chord. m mm tv o ci ♦nooo 0 mmspin mv©oo o ci ♦no O' m m m tv o» ci ♦no v© O' c* m O' ci moo m moo m t nh ♦ tv o SSSrsNNNSNtst> °n Ex. Sec. ho w ts cm ts n n moo moo mo'-^O'^O m o m m no m no cm is cm oo moo ^>0 s tsoo ooo'O'OO^HNNmm^in mvo no n tsoo oo o» O' 0 0 m w mvo so © no MNMMMMMMMMMMMMMMMMMNMMMMNfIMMMNM « Ver. Sin. | 0 ^ O' ^ O' ^ O' moo moo cm ts cm nw\o ho hvo o mo mo o> ■+ o no no 'O ts tsoo oo O' O' 0 0 h m m cm m m ^ ^ m mvo no n tsoo oo oo O' O' 0 MMMMNMNMMMMMMMMMMMMMMMMNCinNMMMM Chord. I O' h m m tsoo o w ^no oo 0 cm ^n© ts O' m mmso'M m mv© mom -^no ^oo m ts o ^ ts o mvo 0 mvo O' w •no'N moo m moo m ts m ts o m m\o vo *o N n tsoo oo oo O' O' O O' O 0 0 « m m cm cm cs nrontt'Mn cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm mmmmmmmmmmmmmmmm m m m m m m m m m m> m m m m m m m m m m m m m m m m m m m m m 1 0 W '■*-'© MOM ^NO MON ''f'O MON ^n© MON -^•'O 00 0 CM ■'f-N© 00 0 h m h w m c» oi n cm cm mmmmm^'-xr^-^'^mmmm mNO 2 S 0 CM ^-NO MOM ^*NO MOM ThNO MOM ^n© MON <**n© MON - h cm 4*n© oo o m m m is ^^■^■Tt-mmmmm m\o no © no no ts ts ts ts ts tsoo oo oo oo oo ©* O' o> O' o> vO'OnO'O'OnOnOnOnOnO'OvOnOnOnOnO'OnOnO'OnOnOnO'O'OnOnO'O'OnOvO Si Ex. Sec. O'^O'^'O'^O'^O'^O'^O'^O'^O mo mo mo mo m w no w no h 0 m h cm cm m m m mvo no is tsoo o* O' 0 o m m cm cm mm^^m mN© mmmmmmmmmmmmmmmmmm^'«»-Tf-Tj-^Tj--^--^-^^^-^Tr NNNNNNNNNNMNNNNNNMNMNNNNNMNNNNN I Ver. Sin. O m 0* ■^■oo moo cm nhno h mo **■ O' ^oo ms« isctNO h mo mo mo C4 n w m m ^ m mNO no is tsoo oo oo O' O' 0 0 h m n N© MOM ^N© 00 0 N ^N© MOM ^NO 00 QCHORDS, VERSED SINES, EXTERNAL SECANTS. 287 si s 0 Cl 00 0 N "^VO 00 0 N -*->0 00 0 N ^vO 00 0 N ^vO 00 O O ^>0 00 0 MflrtN o 00 Ntnn« o oo MOWN o oo m O' Nvo OOO >000 o N in ts O' M m m<© 00 O N ^vO ts O' h m m Is 00 o CT ^>0 00 O' « 0 0 m m m m m m ct ct n ct ct mmmmmm^'5i^--*i'^minmmm mvo Ooaooooooocoooooooooooooaooooocoooooooooaoooooooooaoooooooaooo Ex. Sec. m vo w is moo ^O'tnO'O ct is m O' ^ 0 vim is ct ao O' in 0 vO N n ro o oo oo a> o> 0 0 m m n wn t m\© is isoo oo o> o> 0 0 ►- ct w m m ^ is is is isoo aoooaoooooooaoooooooooooaooocococo a>o>o>o>o>a>o>o>o> NNMNNOINNNNOIdMNNMNNNNNNNNCINNNNNN CO Ver. Sin. ct is ct n n is ct n n is ct is ct is ct is ct is ct is ct is ct n n is ct tattoo in mvo vo is isoo oo a> O' O o <» m n n mm mvo >o is koo ao o> o> 0 vOvOvOvOvOvOvDvOvOvO Isrstststststslstststststststststslsts IsaO «MNN(4NN«M(40o nO'h ro mvo oo 0 ct m m ts o' 0 ct •*■ vo o> n moo ct moo <- is « is o mvo 0 mvo o> ct >o o> ct moo *-« moo *- ^ m m mvo vo>0 nn isoo coooO'O'O'OOOO'xMuCTCTCTmmm^ ■^•^■■^■^^■■^'♦'♦■^^''♦^'•^■'^-'^•'^'^•mmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 1 0 W ^>0 00 0 N ^vOOO O N sf\© 00 o N •*>© 00 0 CT ^VO 00 o CT ^>©00 0 I 0 « ^vo 00 0 W ^>0 00 0 CT -^>0 00 0 CT ^vo 00 0 CT ■^•'O OO o N ^VO 00 O m m m m m ct ct ct ct w mmmmm^^^sf-^mmmm m>o i oo rs m m m o oo vo ^ m w o> isvo ^ w o o> is m ct 0 o> is m m ct o oo is. h m m is, o» m ct ^vo oo O m m m ts o ►< n ■♦vo oo Oct m m is o' m ^vo m m m m mvo vovO>OvO is is. is. is. is, isoo oo oo oo oo o> o> o> O' o> o> 0 0 0 0 IstsSMsIsNNSMsMsNNrsMsIsNNIsrMstsN Is00 00 00 00 * Ex. Sec. oo m o> ^ o mOvo w is. ct is. moo m o> ^ 0 m o >o m is ct oo no>^0 m m J n n m si sf m mvo vo is, tsoo oo o> 0> 0 m « ct n m m t»- in mvo is isoo vO'O'O'OvOvOvOvO'OvOvOvOvO'O'O'O IsMsIsIsNSIsMstsMsNIs CTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCTO w Ver. Sin. ^fOO CT Is CT Is CT Is N Is CT Is CT Is CT Is CT Is N Is CT Is CT Is CT Is CT Is CT Is CT OOMMNCTmm^-^m mv© vo is isoo oo o> o> 0 6 ►« * ct ct m m ti m mmmmmmmmmmmmmmmmmmm mvo vovovovo>ovovO>OvOvo ctctctctctctctctctctctctctctctctctctctctctctctctctctctctctctct Chord. >ooo o m miflMJ'H n ^vo oo o ct m m is o> m m -^vo ao 0 « ^ m is o> f O m is o m>o O' n 'O O' n moo w moo *-i t nh ^ n o m is o m>o o> ct vo m m m>o vo vo vo is is isoo ooooo>o>o>o O C n *+ m n ct ct mmmm^^ mmmmmmmmmmmmmmmm^^^'^'^'^^^si-^'^-^^^^ mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 1 0 CT ^vo 00 0 CT ^voeo O CT ^vOOO O CT sf-vO 00 O CT ->f*VO 00 0 CT ^*>0 00 Q288 CHORDS, VERSED SINES, EXTERNAL SECANTS. £ s 0 C4 ^-VO 00 0 N -**nOCO 0 04 ^SOOO 0 04 ^*vO 00 0 04 ^VO 00 0 04 •'♦’NO CO 0 s m H o 00 H On CO VO 0 O'00 vO H 0 CO S V) ^ f<1 H 0 tN. O' n 04 '♦vp oo o w moNOH m mvo go 0 n ■♦no oo o m ninsosH m m *- 04 w ti w n iono no no no no no tsts O' O' O' O' O' ^ ^ O' O' O' O' O' ^ O' O' ^ O' O' O' O' O' O' O' O' O' O' O' O' O' O' V. Ex. Sec. mCO o NO 04 CO ^ 0 V> M NMOnWh NWO'WNCO 0 NO woo 0 NO w « w m ^ ^ m iono ts ts.00 coo'O'OMHWNr'^-'^ w"0 no n t^oo O' O' o ^MMMMMHHMMHMHMWWWWWWWWWWWWWWWWm rtffirtfOMrtMrtPirtrtrtfOtnrtHrtPjnfOMfOOrtnwrtrtMMfn CO Ver. Sin. m no h i>. w tv moo m O' ^ O' m o m m no m nn t>. mco mo'tou'O'O w novo n tv co oo O' O' 0 o h m w nnttu' mvo nQ ts. r*-oo oo O' O' 0 m m w O'O'O'O'O'O'O'O'OOOOOOOOOOOOOOOOoOOhmhin w n ct w w w w w mmmmmmmmmromromcommmmmmmmm Chord. m in\o co O' m w ^-vo t-» o> 0 04 m mvo co 0 m m ^-no t>» O' h n in t-^oo 0 no O' w in co w in oo m o>. m -^no mvo o mvo O' « mco w moo m r*- t". m KinNj-^-NMnin mvo no vo n fs t^oo oocoo'p'0'0'OOOnw**wwNm nOnO'OnOnO'O'OnO'O'OnOnOnOnO'O'OnOnOnOnOnO NNSSNSCsNSN mmmmcnmmmmmmcnmmmmromrornmmmmmmmmmmm 1 O 04 ^NO 00 O 04 n^nO 00 0 04 ■^■'O 00 0 04 ''HO 00 0 01 -rJ-VO 00 0 04 tJ-nO CO 0 h m h h m C4 C4 04 C4 04 pxnrtnrtt^tttmu'nnifl m'O £ 3 0 04 ’♦'O CD 0 04 ^*NO 00 0 04 tJ-nO CO 0 04 ^*nO 00 0 04 -**NO 00 0 04 ^vO 00 O 3 oo no m m 04 0 m onoo no wnw 0 o> t^vo o oo t>. in m h n in s O' h w ^no co 0 oi mmi'-O'H m mvo co o m ^vo oo O' h m m t-» nOnononOnO t-s.t--r-.r-v t-.oo oo oo ao co oo O' O' O' O' O' 0 0 O 0 0 O m *- h •- ooooooaoooooooaoooooooooooooooooooaoooooco O'O'O'O'O'O'O'O'O'G' WUMWHMWHHUMMMUHrtwuMtHUMWMMMMHMMM Ex. Sec. O'^-O'O 04 t-% m O' m o no 0400 m O' m h no woo r** O m h snown t-s. m ^ mvo no n or.oo oo ao O m h n « n^^nn mvo n t-.oo oo o> on 0 h m n O'O'O'O'O'O'O'O'O'OOOOOOOOOOOOOOOOOOhhmm 04 C4 04 04 04 04 04 04 04 mmmmmmmmrommmmmmmmmfommm » Ver. Sin. ^ O' ^ O' m o m o no m no m t>. n t-. n oo moo mO'^O'^O mo m m no h 0 d m m w m m Rt- m mvo no t-» t-.oo 00 0 0 0 0 m h w m n 1110 uino oocDaoooaoooooooooaoooooooaooooooocoao O'O'O'O'O'O'O'O'O'O'O'O' C4 04 04 C4C4C404C4 04 C4C4C404C4C4C4 04 04 C404C4 04 C4C4C4C4C4C404C404 Chord. ^■*0 o^onO 04 ^ianoo 04 m os.oo o 04 m m Os.00 o m m mN© oo 0 h m *-> rj- t-. o o>» o mNQ on mvo O' w moo w moo m 't nh r* t>. o mNO 0 rvo ■t^^mm idno nonJno nn t>»oo ooooo'O'O'OOOMMMwwwmmm mmmmmmmmmmmmmmmmmm mNO vondnononOnondnonOnOnO mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm I 0 04 ^sO 00 0 04 ^NOOO 0 04 ^nO 00 0 0* ^*n£> 00 0 04 **nO OO 0 04 -*-nO 00 0CHORDS, VERSED SINES, EXTERNAL SECANTS. 2S9 i 1 0 Cl ♦'O ao 0 ei 'C-'O 00 o Cl 00 f* tl ^vO« 0 Cl d-sOW 0 tl ^voco o & h 0 O' tv\© O ooo Nin^nti m 0CO cw© m ci «-• o Oste *vm <> h « <4*0 00 d Cl 4-M3 tv O* M rtWN(>« r*l ^vO CO O Cl 4s© O0 O' *+ m m ( ci mv© *0 sO *© MO ncvNCsN tvoo oo od 1 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOl csciciciCTeiciMcicicineioicicicicieicicictwciwciwwwaw j Ex. Sec. is. m ©"O ci eo ^ h tv m o^vo ci o* m «h co NtnO'O motincioo %o n tv 00 0 0*0 H H ci 1/1 VO o woo CO OO 6 +* ci B i/ls© Is rs. ' tttlOi/lutinminiAlAlAlAUUAi/IU) mM3 'OvO'O'O'OvO'OvO'OnO'O'© i*i rt m n c*i c*i m ci m w ci ci ci m d n ci ci n ci 1 s > tv Cl 00 (1A*)0<0 Cl tsd O ^-O'O H tv rnco 0 tflH tv ci O0 "tOtCHHVO oo o o 0 o ** « ti m m ■+ <^ mvo vo tv tvoo oo o 0 o «-• *• wci nn tmn i ci ci ci i 1*1 11 M (1 IH 1*1 CO CO CO (O CO TO (1*1 CO to TO CO CO CO CO CO CO CO CO CO CO (O CO CO CO ! 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O m m ^vo tv O' o ci m mvo co o»w n tin tvoo o w m <^\o nosO ti m i/» h tno mvo o* mvo Os ci mao ~ moo m -t- tv o mv© O' ci vo O' ti m mvo vo s© tv tv tvoo oooo o> o O' 0 0 o 0 m <-• m ti ci tvrvCvtvrvtvtvrvtvCvtvtvtvtvtvTvCVTvtvtvTv tv 00 C' 0000 0000 00 00 00 (i(ifif>ntlrtrtciiiti(iiirt(icicicir»fifHi(ifi 00 0 W ♦vO 00 0 N ♦'O 00 O n ♦vO 00 O h m h m m w co co w w confonnt^i,ti,io»n«nin mv© % m co w o o O'co n.vo m ♦ m w m o ovoo n*\o m ♦ ♦ on w m o ovoo n.vo m Cl ♦'© CO O' w rO in ts O' W fOmNOiO Cl ♦'O CO 0 N 4-vO CO O m m m Ns O' ♦♦♦♦♦mmmm m\o O'O'O'o n s n is n»oo aocoooco O' O' O' O' O' O' WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW •h Ex. Sec. | ^hqo ♦ m n. ♦ m n* ♦ h ts ♦ *-» oo iohco m n o"o w 0"0 m O"© no n ts.oo ooo'OOnpifin't'r mo so n»oo co oiO O n ci n mo N. Ns OOOOOOQO O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O 0 O 0 O 0 0 O 00 o o nnnfnnmnnnrmnnmnmnfnfot^t'ft^'tft^tt Ver. Sin. O' ♦ O lOHVO W 00 ♦ 0 O WOO ♦ O'O W 00 ♦ 0 vO WOO ♦ 0 vO WOO ♦ 0 © n mvo vo n»oo oo O' O' 0 m w w rn 'f tmo imo n» n»oo oo O' o 0 VOVO'O'O'O'O'OVO'O'OVO'O'O NsN*N.N.N.Cs.rs»N.N.N.NsN»NsN.N. N»C0 00 co m m m m m co m m m m m to m m m m m m m m m m m m m m m m m m Chord. mvo ts. o» o m n co ♦vo ts.oo O'O h nt mvo n*oo own wtw n*co O' covO O' w m O' w moo m ♦ n. o fO N O mv© O' w moo w moo m ♦ ts. Q mvo m h m ci w w fOfom^^-'Cmm mv© 'O'e'd nn NsGO oooo ©v O' O' O O 0 OOOOOOOOOOOOOOOOOOOOOOOOOOOOmhh ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ £ 2 0 w ♦'O 00 O W ♦'O 00 0 w ♦vO 00 0 W ♦'O 00 0 w ♦»© 00 O W ♦<© 00 0 m ft ft ft c« w w w w fOfocorocos*-TfTt--*-T*-mmmm in'© 5 2 O W ♦© 00 O W ♦<© 00 O W ♦>© 00 O w ♦© 00 0 W ♦»© 00 0 W ♦'O 00 0 ft ft f* ft m ci w w w w nnincont^'t^^inioioin mv© 5 m ♦ m w m o ovoo n.vo m m w m o 000 n*vo iA^r>« h 0 O' t-svo m ♦ m msoiH m mvo 00 O w ♦v© op 0 w nnsoin m m n» o' *- w ♦*© 00 0 w 00 cb co O'O'O'O'O'© 0 OOO « m m m ft h w w w w w cortfontn^^ OOOOOOOOhhhhhhhhhhhhmhhhhhhhmhh WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW i Ex. Sec. N. ♦ 0 Nmoo cn 0"0 co 0"0 w O' m w 00 m m 00 h co ^noo ^hoo ♦ tsoo O'O'O h h n n n-t mvo \o N»ao 00 O' 0 0 *-< w w m ♦ ♦ \n\o vo N. 'OvO'O'O NsNsNsNsNsNsN.NsNsNsNsN.Ns.Ns ts.00 ooooooooeoooooaooooooo cocorocofococococococococococorococororocofocorocofofocororoco Ver. Sin. vo h n n co ♦ 0* m w N. moo ♦ O'© w m O' m m vo woo ♦ 0 m m N. m 0 mvo vo Ps Ns 00 00 0 d 6 >- m w mm4*-^-m m\o n. n.oo ooo'OOmhww ^■^■^Ti-sN^-Ti-Ti-mmmmmmmmmmmmmmmm mvo 'O^'O^'O^vO^'O^ Chord. mvo 00 O' 0 m m m Nsco O' *-« w m sfvo N.00 0 h n ^ mvo n. O' 0 h m ^ I O' w moo w moo >■* n 0 m n. 0 mvo O' w m O' w moo m n 0 n 0 m : O'S'O'O'O'O'O'O'O'O'O'O' 0'>S'N0''o''S' O' O' O'<0'<0't0' O'O'O'O 8800 ; m m m m m m m m m m m m m m m m m m m m m m m m m m ^ ^ ^ ^ ^ | — 2 2 1 0 W s*-\0 00 O W ^vOCO O w '♦vo OO 0 W '♦'OOO O W ♦vo 00 0 w •♦'OOO 0 ! m ft hi hi m w w w w w mmmmm^-Tr^-Tt''«j-mmmm mvo I 1CHORDS, VERSED SINES, EXTERNAL SECANTS. 291 I O « ^OCO 0 « J'gOO g « *OCO o « 0 « ?>ooo o - jvooo^g 3 h 0 0 O'® s rs.vo w h o 0 O'co oo is. rsvo vnn ^ m n n h m (s O' h cs ♦v© oo 0 ci ♦>© oo o n ♦no oo O' m mm soh rttfiNO'H mm m mo vO'O'O'O ts.ts.tsrs tsoo oo co oo oo oo o O' o O' o 0 0 0 0 0 *■> m m wcietcipieicicicicieipiciciwncicicicinpipimmmmmmmm dnOn«NMNMN«MNNNNCiCIN(iCIC4t4N(INflMNNN b Ex. Sec. no mo ts^Hoo m n 0"0 m o ® m ci 0"0 ^woo m n 0 is ♦ m ® no mo coo'OOMNNmt^’ m\o ts tsoo O' O' O m w ci m ♦ m in so ts tsoo o 0 ci w mmmmmmmmmmmmmmm^^^^^^^^^^^^^m Ver. Sin. ts m O'mh is ♦ o no woo ♦ m ts m O'm hoo ♦ o no ci ® m m ts m o m « oo O' O' 0 m h ci mm^-4- m no no ts tsoo O' O' 0 ** w « ci m ♦ ♦ m m\o ts O'O'O'OOOOOOOOOOOOOOOdMHMMMMHMHt-MN* g o U- m ci m ♦ mNO tsoo O' 0 m n m ♦ m mvo tsoo O' 0 * ci m ♦ mvo tsoo o* 0 0 mvo O' pi moo m ♦« h ^no mvo O' n moo ci moo h ^ s o mNO o» m 0 0 0 0 « w « ci ci mnntf + 'tinin mvo no no ts ts tsoo oo oo oo O' NNNNtlTlNdNdMNMdNNNMM NNNNNNNNNNNN z 2 0 Cl ♦'O® 0 PI ♦»©« 0 M ♦'O® 0 Cl ♦'O 00 0 Cl <4*NO 00 0 Cl ♦'O ® Q z s 0 Cl ♦>© 00 0 Cl ♦vO 00 0 N ♦vO 00 0 Cl ♦»© 00 0 Cl ♦'O 00 0 N ♦<© 00 O z‘ tS in^mmw h o O' O'® tsso m ♦ ♦ m « h o 0 O'® isn© no wtmmu m O' m mmtsO'M w ♦<© ® o pi ■♦no ® o ci ♦•© soh m m ts o « m m ts O' 0 0 0 0 0 ** *+ — « ■* ci « ci ci pt mmmmmm^-^^^^mmmm HCtPtCIPtPiCIPiCIPICtCtCICICICinPICINCtnCIPICIPiCICICICiei ctcicitieicicicictcictciciciciciciMCicicieictciciciciciciPici Ex. Sec. ts m o ts ♦ m ® m ti O'no m o no mo NtMoo m n o"0 mo ts ♦ m ® no tsoo O' O' 0 m m ci m m ♦ mvo vo is® ®O'00HCicim^*m mio ts tsoo OO0O*HMMMMMMMMMMHHCINPiPICIPICiPICI01CtCt(l ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ 3 Ver. Sin. no PI 00 ♦ O'© PI® ♦ O'© Pi® ♦ 0 © Cl® ♦ 0 © Cl® ♦ 0 © Cl « ♦ 0 ts QHMNmcoi,'t mNO no ts ts® O' O' 0 0 m ci ci m m ♦ m mNO no tsoo ® 00®®®®®®®®®®®®®®® O'O'O'O'O'O'O'O'O'O'O'O'O'O'O' mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm Chord. O' o m pi m ♦ mNO ® onO h n m^ in no ts® O' 0 m n ♦ in'© is® o' 0 h no 0 m«o O' ci moo w ♦« h ♦no mNO O' ci m O' ci m® m ♦ n o m ts o 0 m m m h n pi ci mmm-Nt^-^mmm m© no no ts ts tsoo ® ® O' O' O' 0 ♦ ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ 1 0 Cl ♦NO ® 0 Pi ♦©» 0 Ci ♦no® 0 pi ♦no ® 0 ci ♦no® 0 Ci ♦no® 0202 CHORDS, VERSED SINES, EXTERNAL SECANTS. z* 0 W '+- m -^vo ao o w 4*vO ao o w oo o w ^'O oo o n is is tN rs-co ooooooooo'O'O'O'O'OOOOO-HMNHficifiwcirti'i N(IOINM(I(IOIOINNOINOINNMOI««MM(«(IOI(IM(IMOIN Ex. Sec. h 0"0 *- oivO ^ m o«vO m O'VO ci O' ts m w O tsinmooo m m m oo N n n t in in\o ts.oo co o»0 H M N rnTj-Tf invo tsao aoO'0'-**wm^^- (sStsMsNNrsNN tsOO 00 00 00 00 00 00 00 00 00 00 00 00 O'O'O'O'OVO'O^ Ver. Sin. | w ao in m ao ^ o snoo w O' m w oo snoimoco m m is m ao m vo vO isoo ooo>oOHNNnn4-iA mvo is. isoo oo o*0 0 « « « + mnnfnnn^tt^ttt + ^^^^’t^t^ininininininininin I • invo is. is. oo mo 0 h o + invo vo tsao ovoowwmm^ mv© vo tsao moo m ■*■ ts o ^ is. o mvo O' w moo m ^ is. o no mvo O' w moo m. ^ is. oo ao O' O' O' o O 0 ►' h *-< w w w mmm^^^mmm mvo '0*0 isnn 2 S 0 Cl ^*vO 00 0 W ’♦'O 00 o N **vO 00 O W ^vO 00 0 W ^*vO 00 0 W ^*vO 00 0 h h m w w w w w w w rtrtnnmtttt ttfnomm mvo 1 0 Ct ^-vO 00 0 M ^VO 00 0 W *-v© 00 0 M ^vO 00 0 W 00 0 M ^-v© 00 Q l h o 0 O'oo ao ts |s.\o 'O inttmn« m m o 0 O' ovoo oo is. tsvo vo m m ^ m is O' © w ^*vq oo o w ^vo oo o ci ^vo oo 0 w m m is O' *-< nmNoiH m h w m w w w w w mmmmm-^-^^-st-^-mmmmm mvo vo 'O vo v© is. is. ?ff?yffyff??c?ererj?c?cfw>??w>w)cr5)w>c?{?«?e?c?cfc?w,w> Ex. Sec. 0 ts m w O' i> ^ h o"0 m m ao m m o ts m w O' is ^ w O' is ^ m 0"0 ^ h • OOnciNm^m mvo isoo ooO'OMt-wmm^ mvo vo isoo O' O' 0 “ w inmmmmmmmmmmmm mvo vOvO'O'O'O'OvOvO'OvO'O'O'O is n is Ver. Sin. woo ts m 0 vO w ov m m ao -*■ o is m owo woo m h ts^ovo m 0"0 w is tsOO O' O' 0 m m w W m st st mvO vo ts tsOO O' O' 0 m m w Wfl't 'f mvo MiHH*>-iwwwwwwwwwwwwwwwwmmmmmmmmmm Q O U 0 m w w m st- mvo is is oo O' 0 « w w m ^ mvo is tsao O'Onwwmsfm mvo O' w moo h ^ n 0 mvo 0 m<© O' w moo m ^ n o m ts o m»o O' w m O' O' O' 0 0 0 •- m m w w w mmmm^-s|*sfmm mv© v© v© ts ts is isoo oo w w w mmmmmmmmmmmmmmmmrommmmmmmmmmm SS s O W -*SO ao 0 W ^vO 00 0 W stv© OO 0 w stv© 00 O w ^v© 00 0 w ^*v© 00 0CHORDS, VERSED SINES, EXTERNAL SECANTS. 293 X S 0 N **© 00 0 Cl -*■© 00 o Cl >*■© 00 0 Cl ^© 00 O N ■*© 00 0 Cl ■♦© 00 0 Z H ci ci ci ci ci ci t* et ci ei m m m h h m n io n O' n to irt n » m n in ts O' h ntin ts O' h nin s O' n m in n O' h O' O' O' o> O' 0 o O 0 0 >-• i- h i- ** w ci « ci m ^■’f^’f^ioinininininininininininininuiininininininininininin nNMNN(inNMMMC4M(4MMM(4MNNNne«CIN(INNna % Ex. Sec. | m 0 oo © ci o oo so ei o oo © "t* ei o oo vo ci o oo © ^ « o oo no m oo O' O' 0 *h ei n fn ♦ in© ts 0.00 Oi0 h m n n^-m m© tsoo o o 0 *- w m m m ci w ei ci « ci ci w ci ci ci ci nfntnrtmnrtpmnnMrtttt inmininminioinininininininininininininininininininininininin * Ver.'Sin. « O' m w oo mnoo m w scno is. o> O — h ci m m ^ in mo o. noo q> O' 0 •-* •- ci m ^ ^ m tv ts. r>. (*> f- fs. ts t>.00 00 00 00 OO 00 OO OO OO OO OO OO OO OO OO O'O' O'O'O'O'O'O' Chord. ts. ts.00 oo O'O'O 0 m m ci m m m ^ ^ in mvo © Is*. tsoo ooo'O'OOhmci O' n moo •- ^*oo r>. o m© O' ci moo >■*>(■ t>. 0 cos© o> w m o« n moo m O n n (>.oo oo oo O' O' O'.© 0 0 O m m m c» ci ci mmmm^-^j-^-mm m«c mmmmmmmmm m© ©©©©©©©©©©©©©©©©©©©© S s 0 Cl '^'O 00 0 Cl "*■© 00 O N ^© CO 0 Cl "*© OO 0 N **•© 00 0 N ■*•© 00 0 t-i m h m m ch ci « ci ci (nfortncn^'t’i'^^inininm m© s s 0 « **■© 00 0 Cl **■© 00 0 Cl **•© 00 0 Cl •*"© 00 0 o **■© 00 0 Cl ■*■© 00 o 5' H mmmhmhooOO'O'©' onoo oo oo oo (>»(*>(>. ts.v© '©'©'©'© m m m m m ci ^© oo 0 ci ■'*■© oo ©> ^ ro m t>. O' f r*i m r>. O' ^ m m tx o* m m r>. o* >-« m© © © © © o. o. t>. o. r>.oo oo oo oo oo O' MMCICICICICICICICICICICICICICICICICICICICICICICICICICICICICI i Ex. Sec. oo © m ci otNinmooo'© ^ m O'Mnmooo'O m O' Cs m m m O' t>* m co ■*> ms© ts r>.oo o>0 h h n ms© r>op ooo'Oi*MCtm-^m mvo 0.00 O'O'O'O'O'O'O'OOOOOOOOOOOOOf-mmmhhmmmmm ^^'t^^^^unoinmininininioininininininininininininininin Ver. Sin. uimco ■*• m t*> 0 csfoo ts m o © m 0 © m 0"0 cn O"© ci O"© ci 0"n ci in'© s© 0.00 ooo'OOwNNfn^'i’ mNO »© (*> t*>oo O' O' 0 •- •- ci n m ^ m m m m m m m mv© v©'©'©'©'©'©'©'©'©'©'©'©'©'©'© h.NNNNtststs Chord. oo oio. tsO nts© m© O' ci mop m r>. o m© O' n © o» ci moo h m© O' tsoo oo oo O' O' O' O' o 0 0 « w m ci ci ci ci mrom-^-^-^-mm m© © © © ^.^.^.^.^.^^.^.mmmmmmmmmmmmmmmmmmmmmmm 1 0 Cl ^-© 00 0 Cl <+© 00 0 Cl ^© 00 0 Cl *4*© 00 0 N ^© 00 0 Cl *© oo 0 K.M m h h ci ci ci ci ci m©294 CHORDS, VERSED SINES, EXTERNAL SECANTS. s S O Cl 00 0 M 't-'O 00 0 M 00 0 M **vO 00 0 W ^VO 00 0 Cl '*■'£> 00 0 S invo v© vo vo nnn tsoo oo oo oo on o» > tn M lO is O' h M i/l N O' h PI i/l S O' H M (/) N O' h rtUlNO'M i/| fs O’ N h m.m m m n n n n n vnvo vO'O'O'O c* j V«'«V« « « « « « « Cl « Cl'S W Cl Cl'S W C* Cl'S n'n'n'n'n'n^'nn'n j !• i Ex. Sec. 1 ooo ts m m N o oo ts m m n o os tsvo ♦wh o oo ts m ci *h o\ oo «© m * , N ts00 O>0 H « N »n\0 ts tsOO O' 0 h N fl fl ■*•01'© ts00 OO O' O H N 'O'O'O'O tststststststststststs tsoo ooooaoooaooocooooooooo O' O' O' mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm Ver. Sin. o in « 0"0 mo n ^ h oo m m O"© mo n^-hoo inn O"© mo n^-moo j mvo ts c*»oo O'OOMdcim^-^’ mv© is tsoo O' O' o m m n m^*^* mv© »o HMMMMHnnnnnnnnnnnnnnnmmmmmmmmmm1 ininmmmmmmmmmmmmmmmmmmmmmmmmmmmmin | Chord. f*m + t^ + m«n mvo vovo sss tsoo aoooo'O'O'OOOOMMwnn n moo m i>. o mi© O' n moo m ts o rnvo as n m O' n moo m in. o m m m mvo 'Ovo nnn rsoo oo oo O' O' O' 0 0 0 0 M M *-■ n n w m m m ^ ^ tstststststststsrstsrsrstsrsrs tsoo cocoooooooooaoooaoooaoooooao £ 2 0 w ■^■'O oo 0 n -*-vo ao 0 n ^-v© oo o n s**vo oo o n -4-vo oo o n -*v© oo o h m m h m n n n n n mmmmm^-^^-^-^mmmm m>o 1 0 n Rf'O oo o n ^oo 0 n ^\ooo o n -4-vooO 0 n ^-'O oo o n ^-vooo 0 , i m m m c«« O' M m m ts ©v m m m ts on m nmSO'N m m t-*. o m m m is ov m m m m m in'© v© v© v© v© n n n c. ts.oo ao oo oo oo O' O' o> o» O' O o 0 0 5 m mmmmmmmmmmmmmmmmmmmmmmmm mvo 'O'O'OvO'O nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn °00 Ex. Sec. m h oi s m ^ n o oo *o m m m o» nc ^ n ooo ts m m n ooo ts in m n 0 n m m 4- mv© tsoo ooo'0-nnm^ mvd ts tsao O' 0 4 n n m 4* mvd ts ■^-Tl-^^‘^*^t-Tt-Tt-Tt-Tt*mmmmmmmmmmm mv© >©i©v©v©v©vOv©i© mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm Ver. Sin. ^ 0 n ^ m oo ^ h oo m n ao m n 0"0 n O"© m 0 v© mo ts ^ m ao m n o* invo vd tsoo ood'OOHCi«m4'4’ mvo vo tsoo o> O' 6 4 m n m m 4> m m O' O' O' O' O' O' O' OOOOOOOOOOOOOOOmwmwmmmmm ^■■^■^■■^■■^■^■^■mmmmmmmmmmmmmmmmmmmmmmmm Chord. n n m m m ^ m m mvo v© v© ts is tsoo ooO'O'O'OOOwwnnnmm 1 w ts o mvo O' n moo h t n o mvo O' n moo m moo m vno mv© O' n v© vO 'O ts ts ts ts oo 00 oO O' O' O' 0 0 0 0 *-< m m ci M ci mmm->*-^-Tj-,'*-m • vO'©'©'©'©'©'©'©'©'©'©'©'© lsMsNMsls|sNNNNSNtststs|s , ! I 0 w *■'© oo 0 n ^vo oo p n -t-v© oo o n **v© oo o n «*-v© oo o n ^*vo oo oCHORDS, VERSED SINES, EXTERNAL SECANTS. 295 1 o €1 ^-N© CO 0 (1 00 o « ***vOOO 0 CM ^vO 00 0 CM ^vO 00 O CM ^vO OO Q i OHwcMCMmm^-^i-m mvo ts. tsoo oo o o 0 o m cm n in'© \o ts n in n O' h fnintsoiw nmN o cm ^no oo o cm ^no oo 0 n t ntncnfi^ftM-^inminin iovO 'O'O'O n n is ts tsoo oo oo oo oo o> o> O' S Is ts N ts ts h» ts Is Is Is Is |s N Is, Is Is. |s Is. Is Is. |s ts |s fs |s |s Is ts ts CMMCMC1CTCMWCMCMWCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM | Ex. Sec. m m m h o O' ts\© in + nN on o onoo tsNO m met m o onoo t>v© m m cm oo 0*0 h n n nt ion© tsoo o>0 O h « fnOm 0 R CO O (-* HV M ►+ M N R R Cl W ‘O'O i 0> OV O h r ro^in m>o r^oo OQ h h » rot tr»o SCO ov o m r ci*funn vooo m m»<%ts»ovM p>mso*w ^-- rNoo oo coao OvOoocnQ 0 0 o 0 * h r- h h 09 00 09 00 CO 00 OO ao CO 00 00 oo ao to OO OO CO oo oo ao 00 ov o'- O' O' O* O* O' O' O' O' RRRRRRRRRRRRRRttRRRMRRNftRRRCtROClCi n Ex. Sec. tv psjko in^'tTON r ►-* o o on onco s r^\o no m m -4- 4- ro m r m o o ov R h* 4* mvo sot? Ov O ►* R mso sat? os 6 h « tn i- mvo i^-eo omd Q r>» r«, (»* s-fN. r-*. r-Noooo coco ao coco coco cooo ooo*OsO>ot>ovQsosO O NOsOvOsO'O'OvO'OvOOvOOOOOvOOsOOvOsOOsOO'OC'O'O v S S W 1 s > Oat? v)tn9tf torooco m r»0 oo m cn o oo m m O ao n hhco m rt 0 cono O N fOI*iO uvo SnOO 09 OsO H m R ro 4- 4 mvo s*. SO? 0*0 9H(f QQQQ9QOOQQO*,**+*,+ *,4MM’’+siMiNNMMRC?RfiRe*cf >0 no nO nOnOno no no no no no nO O NO nO nO no nO nO NO nO no C'OnO'OnO^'O^O vO Chord. mm RfRfsf-^tntnforow r r ►* •-» m o o o o oon oosaoaooo ssk mso onr moo m ^so mvo o>n moo •-» 4-fN.o m moo m 4f,vp mvo o»r m -* m r r r mroro^-^-sf^uNrsmvc>Ono s s s mdtsoo ovo»0>O*O mtnmmmmmmmmmmmmmmmmmmmmmmmmmmmmm 1 O R 4VOCO O R ^NOCO 0 R 4NOCO O R ^NOCO O R 4nO CO O R OO O « l-» H4 H* M R R R R r m m IOsO i O R ^NOCO O R 'fNOOO O R 4nOOO O R 4-nO 09 O R 4nO 09 O R 4-nOOO O i cn.oo aoaNOOwRRro44 uino no sc9 oo.ono m r r m 4 m mvo sco o 4vo co o m m Sn o» m ninsosH ro m s» o* m 4 no oo o r 4no eoON 4no i On o* Ov.0 O O O 0 h» w m m m n n n n (i mmmm44444mmmm • r>. Sn SnOO 09 CO 09 09 09 09 OO 00 OO 09 00 OO OO 09 00 00 00 09 09 09 CO 00 CD 00 00 OO 09 1 RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR | j •* Ex. Sec. R M O O ONOO fs.NO tfttfON N H O 909 fN.NO R M O On OXO Sn j mvo s fN.oo o»o h n nt mvo koo oo o' P w r co 4 mvo soo oo»o h m . 444444mmmmu->mmmmm mvo nonPnonononqnononqvq s s s ; NO nO nO nO nO nO nO NO nO vO nO NO nO nO n© nO ^ nO nO nO nO nO nOnOnOnOnOnOnOnO nO 1 •0 Ver. Sin. Onno »*)hoo m m O smci on s. 4 w onno m h oo m m O swn o»SNf» O O'OMRwri'Nt-m mvo s* (n»oo OX) O h n rort^ mvo no tNOO co 90 h « i fNOO 09 00 00 09 CO 00 OO 00 00 09 09 OO OVOnOnO'O'OnO'O'OnO'OnO'O'OnQ Q Q * mmmiommmmmmmmmmmmmmmmmmmmmmm mso no >o - Chord. fNtN.fN,fN.tN»tN.tN.tN.t>. ^n.nO nonOvOnOvOnOnONOnOnOnOnOnOnOnOnOnOnOnC) m msO os n mao m in* o fovo O' r moo « s*. o mso o» r moo h tso n ! « « r mrocON^^-^mmm msb voso ss fNtO oo oo oo P* O' O' O 0 0 h m OOOOOOOOOOOOOOOOOOOOOOOOOOmmwmm mmmmmmmmmmmmmmrnmmmmmmmmmmmmmmmm S s 0 Cl y*0 00 0 Cl -^NO 00 0 Cl ^NO 00 O R ^nO OO 0 N 00 O R -4-NO 00 0 H h m h m r r r ci r mcommro^-^-^-^-^-mmmm mvoCHORDS, VERSED SINES, EXTERNAL SECANTS. 297 l 0 01 ■**’'© 03 0 N -^-vO 00 0 04 OO O 04 -*"NO 00 0 W ^nO 00 0 <4 00 Q M m M H M N C4 04 04 04 U-)\0 £ co o»h n t-.oo o 0 n to in t--oo O' h n r>mNO i--oo o h n t mso 04 ^ t-. O' H niDSO'H -^vo OO o 04 '^■VO OO w fOW SOvn mO 00 0 04 ^VO CO OO OO OO O' 0> O O Q O O Q H m m M m N 04 04 04 04 m n n fO ^ ^ O'O'O'O'O'O»O'O'O*00 00000000000000000000 04 04 04 04 04 04 04 04 04 rtcnfO03fOrtco«nr>p>rtrtrtncn«nncnncnrt(n °«0 Ex. Sec. O' 04 O' O'OO 00 00 00 00 SSSNSNSNNSNtsS t-.v© NO NO NO N0 NO NO NO 0*0 N N 40 vO 0>00 0*0 H N nt lOvO t-»00 0*0 H N M t m*o 0^00 O' tsStsNNSt>SSSNtsSSSNSOsNNNNt\NKNNNMsN Ver. Sin. 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Sec. O' o*oo oooo nn t-.NO NOmmm^'*^mrOfnMC4C4MMHHOOOOOO' 0 h « m 4- m*o d-oo g*o « n iono t—oo o* o h « m 4- m*o d»co O' O' 0 0 0 0 OO OOOOhmhhhhHhmh04MP404C404C4C4 04 04 C4 tstStNtNt>|Nt-SCNtN4st>rsrstsf>SNIsNtsrsrstst«.|sSt-Nt>t-. c Ver. 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Sin. no m^^mci m h o 0 onoo ao tsvo no m^«4-mci ci h o 0 ooo oo ts ts ts oo h n mt mvo tsao ao do h « mi- ion© tsoo O' 0 h ci ci m *4 mvo ts tsoo coaoaoaoaoaoaoaoooao onOnO'O'O'OnO'OnOnOnQ Q Q 0 Q Q Q O 0 O' ^ A O' ^ ^ On O' ^ On On On O' ^ On On O' O' O' OOOOOOOOO M H M.M-M M M IH M Chord. m m oo no m « o»vo ^ h on ts ^ ct omon o ts m m o oo m m h oo vo m m ao h mvo On ci nC n o m moo w •nJ-vo O' ci mao 0 mvo O' h ^nO ci mao m 0 m m m m ci ci ci mmmmHj-sj--^-T}-ir)m mvo vo vo v© ts is tsoo ao oo oo O' 'O'j'O'O'O'^'^'?n0nO'O'O'O N0'^'?'^'0'?'0'^'O'OvOv? 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O' N ♦ In O' Cl ♦ In O' Cl ♦ In O' Cl 4- In O' ci 4 In m h h m ci ci ci ci mmmm-^-Tf-^-^-mmm mvo vo vo vo in in in inoo oo oo OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ b Ex. Sec. 0 ♦ O' moo ci vo h m o ♦oo m in ci vo o m ov ^00 mtNcivo hvo o m o ♦ m'O in O' 0 w m mvo oo O' 0 ci m mvo oo o» 0 « m mvo oo o«h n ^ mvo oo vO VO VO VO In In In In In In In CO 00 00 00 CO OO OO O'O'O'O'O'O'O'O O O O OO dddddNdddddNNddNdddddddNMfnncnnmn Ver. 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Sin. 4- mvo ts oo (jo h n m-t mvo tsoo o, as o m ci m 4- mvo tsoo O' o m m m 0'0'9'9'^9iOOOOOOOOOOOhmmhhmmmmi-icimmci Chord. h oo m m oo m ci 0"0 ci owo mo ts m o ts ♦ m co ^hoo m m O' m m owe ' no oo m 4-vo O' ci ♦ ts o m moo m mvo ovh ♦ ts ov ci m ts n m moo w mvo mm-^^^Tt-mm mNO vonO'O ts ts ts tsoo oo oo oo O' o o> o O 0 0 * m m tststststststststststststststststststststststs tsao 00 00 00 00 00 00 nOnOnO'O'OnOnOnOvOnOnO^'O'O'O'OnO'O'O'O'O'O'O'O'O'OnO'OvO'OnQ I O Cl ♦'O OO O Cl ♦'O 00 0 M ♦'O 00 O M ♦'O 00 O M ♦'O 00 0 M ♦vO 00 O h h h m m ci m ci ci ci mmmmm^-^-'^-^-^'mmmm mvoCHORDS, VERSED SINES, EXTERNAL SECANTS. 301 £ S O d ^n© 00 0 d 00 0 d •+'£> 00 O N <*«nO 00 O d 00 0 N *-NO 00 0 i. 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O' m d -4 m tvoo o d m mvo oo on 4 m 4v© oo O' 4 d 4n© n d 0 ci ^^■•^^■mmmmm mvo no so no so no no tv tv tv tv tv tvoo oo oo oo oo oo on on w Ver. Sin. 00 00 00 00 00 00 00 00 00 00 O'OnO'OnO'O'O'O'O'O'O 0 O 0 0 0 O 0 0 0 H m ^ mvo tv co O' 0 h d m ^ mvo tvoo do » d ^ mNO tvoo O' 0 h d m to m m m m m mvo vO'OsOvovovo'O'Ovo nssnisnnn tvoo oo oo oo oo Chord. oo h oo ^»h tv o smONO m 0"0 d O"0 d on m d oo m h oo tv*^ so O' d 4 tv o d moo o mvo oo h mvo on h 4 tv O' d mtvo m mod h mvd O' &* 000 m m m h d d d d mmmm^^^^-mm mvo no so so tv tv tv 00 00 O' On^n^^^n^1 O' O' 1 O d ^*NO 00 0 d ^NO 00 0 d Nj-vO 00 0 d *AO 00 O d '+'£> 00 0 d ^*nO 00 O308 CHORDS, VERSED SIXES, EXTERNAL SECANTS. s' 0 « CO 0 n "4-vO 00 0 N 00 O N **sO 00 O N TfsO 00 O N ^*SO 00 O 1 00 m m 0 MON O' t*s ^ w 0"0 (0000 W> CO 0 00 10(0000 m m o oo utmo ts 0 msO 00 H TfsO O'N ION0 msO 00 H 4- t*s Ox N mOO 0 COsO O'H ^SO msO SO sO vO ts ts ts tsOO 00 00 O' O' Os O' O 0 O 0 m m h. N p» f* N fOfOfO^ inmiOiOiOiOiOiOtOioiOiOiOiOiO lOvO sOsOsOsOsOsOsOvOvOsOsOsOsOsO % 1 Ex. Sec. n ^ w oo io« osso m 0 n^hoo m m o t*. h osso mo mon os so m w m m mso oo O m m uo tsoo 0 n mmsoo N ^ m t- O' m n ^so t- os m m OsO'OsOsOsQOQQQO'-'NiHHMiiC'INNNWNmPOcnmfOm^-'^' tO iO iO >0 i/i so so vO sO sO sOsOsOsOsOvOsOsOsOsOsOsOsOsOsOsOsOsOsOsOsO HHHMHHHMHHHHHHMHMHHHHHMHHHHMHHM r- Ver. Sin. J soso n ts® oooo Os os 0 o 0 h m n n n mso so ts tNoo oo o mso tsoo os 0 h o* m mso t-ao o 0 m n m Tf- mso p-.oo os 0 h n m ^ mso ^’t^ttuiioioinioioioui mso SOSOSOSOSOSOSOSOV N N N N N tN N NNNNNNtiNNNttNCINPiC'INNNNNCtNCINOtNOIMMt) Chord. 0 SO 0*00 "4-ONO 0* 00 0 SO 0* 00 ^ 0 SO 0* 00 Nf 0 SO N00 ^ 0 SO N 00 ”*■ 0 -fsO Os h -t NO»N t SO N U1N0 fO mOO 0 msO 00 m msO Os H ^-sO Os N tnnn^ tuiw mso so so so tN.ts.rs. tsoo aooooo os os os O' O o 0 0 h MHMHMHMMMMHMMMMHMtHMMMHMHHHNNNdC* tNtNfstsfstN.fstststststN.tNtN.tN.tstN.tstN.tstN.tN.fstsfstsfNtNtN.tsts 5 s 0 N ^so 00 0 N 'f so 00 0 N 'f'O 00 O N sO 00 O N ^sp 00 0 N sf sO 00 O m h h m h c* n n n n mmrnrom'f'f'f’^-’^-mmmm mso s IS 0 N NfsO 00 0 N Nfso 00 0 N NfsO 00 0 N sfsO 00 O N NfsO 00 0 N ^sO 00 0 H m h m m t* n n n n mfOfotnmtt'tt’tmmmm mso 3 H ts sf h oo m n osso ro o ts ro o ts m oo mmo ts sf m oo m n o nsThoo so os n -«f ts o n moo m nso o> m «*■ ts os n moo 0 mso oo w ^ sosn mts ts tsoo oooo O' O' O' os 0 o 0 0 >-» h h h w n t* mmmm^-'f^-'fmmm 'fNf'f'fTfTt'fTfsfmmmmmmmmmmmmmmmmmmmmmm CD h Ex. Sec. | m n os m n Os m n osmN os m n os m n osuinoo sf m oo m n osso won m ro ^so 00 O' H rO sf so 00 Os m m TfsO 00 Os m m -»f so OO Os H m sfsO 00 0 H ^-'t'fTf'f'fmmmmm mso so so so so so ts ts ts ts ts tsoo oo oo oo oo os os mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm HMMMMHMMHHMMMMHMMMMHHHMMMHHMWMM Ver. 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Sec. 0 ao m mo 00 h mo ao m 41««. O' w ianosn mao 0 mo O' n ^ w 0 mo m t«* 0 « 4*0 O' h m mao 0 w 4no»h mo ao 0 m m t»* o> w 4*© O' M m mm44444mmm mo 0000 w rs ts r>.ao ao 00 00 ao O'O'O'O'O 0 nwwwwwNNC'iwNwwwwMWCiwwcsiwwwwwwwrimm tItinCinntMinNNNNNNtINMNNMCIMNttdNNflNn 06 00 X CO K « 4 m t^ao 0 « m mo ao 0 m m 4vo eo O' m w 4-0 ts. O' 0 a 4* m t^ao 0 ao O' 0 h w 4* mo t>.co o» m e» m 4* mo ts. O' 0 m w m 4*0 t>.ao O' 0 m m OflMMMM»-iMMHMWwwwwwwwmmmmmmmmm4-4'4' OOOOOOOOOOOO OOOOOOOOOOOOOOOOOO 0 > g ao w0 0 mts»H m O'mi*sm <400 woo 400 m wO'^nh m O' w0 0 4 u 0000 nnnn fs.ao ooooaoo'OO'O'OOOOMMMMNwwwmm O'^O'O'O'O'O'O'O'O'O'O'O'O'O'O'O'O 0 0 O 0 0 0 0 0 0 000 0 NtstsN|sNNNt>WNNts.NNts NCO 00000000000000000000000000 1 ° « *'oa’ 2 2 ITS? 8 3 S-'S^g 8,3.9,£SLOPES, FOR TOPOGRAPHY. 315 TABLE XV. SLOPES, FOR TOPOGRAPHY. 1 j Degrees. VerticdUise in xoo Feet Horizontal. Horizontal Distance to a Rise of 10 Feet. Degrees. Vertical Rise in 300 Feet Horizontal. 1 Horizontal Distance ton Rise of so Feet. z *•75 572-9 *9 34-43 29*0 2 1 3-49 286.4 20 36.40 27.5 3 5.24 190.8 21 38.40 26.0 4 6.99 143.0 22 40.40 24.7 5 8.75 ”4-3 23 42-45 23-5 6 IO.5I 95.1 24 • 44 52 22.4 7 12.28 81.4 25 46.63 21.4 8 14.05 71.2 26 48.77 20.5 9 15.83 63.1 27 50.95 19.6 IO 17.63 56.7 28 53-17 18.8 XI 19.44 51.4 29 55-43 18.0 12 21.25 47-o 30 57-73 17-3 13 23.09 43-3 35 70.02 14*2 M 24.93 40.1 4° 83.91 IT.9 15 26.79 37-3 45 100.00 IO.O 16 28.67 34-9 5° 119.17 8.4 >7 3°-57 32.7 55 142.81 7.0 18 32.49 30.7 60 173.20 5-7TABLE XVII. RISE PER MILE OE VARIOUS GRADES.318 RISE PER MILE OF VARIOUS GRADES. Grade per Station. Rise per Mile. Grade per Station. Rise per Mile. Grade Station. Rise per Mile. Grade per Station. Rise per Mile. • OI .528 .6l 32.208 - 1.21 63.888 1.81 95-568 •02 1.056 • 62 32.736 1.22 64.416 1.82 96.O96 96.624 .03 1.584 .63 33.264 1.23 64.944 1.83 .04 2.112 .64 33.792 1.24 65.472 1.84 97-I52 .05 2.640 .65 34-320 1.25 66.000 1.85 97.680 .06 3.168 .66 34.848 1.26 66.528 1.86 98.208 .07 t.6q6 .67 35.376 1.27 67.056 1.87 98.736 .08 4.224 .68 35-904 1.28 67.584 1.88 00.264 .09 4-752 .69 36.432 1.29 68.112 1.89 99.792 • IO 5.280 . .70 36.960 1.30 68.640 1*90 100.320 • 11 5.808 .71 37.488 1.31 69.168 1.91 100.848 • 12 6.336 •72 38.016 1.32 6q.6q6 1.92 101.376 •«3 6.864 ■73 38.544 1.33 70.224 1.93 101.904 •M 7.392 •74 39.072 1.34 70.752 1 -94 102.432 •15 7.920 •75 39.600 i-35 71.280 1.95 102.960 103.488 104.016 • l6 8.448 .76 40.128 1.36 71.808 1.96 8.976 •77 40.656 i-37 72.336 1.97 • l8 9.504 .78 41.184 1.38 72.864 1.98 104.544 .19 10.032 •79 41.712 1.39 73.392 1.99 105.072 • 20 10.560 .80 42.240 I .4O 73.920 2.00 105.600 • 21 11.088 .81 42.768 1.41 74.448 2.10 no. 880 .22 11.616 .82 43.296 1.42 74.976 2.20 116.160 •23 I2.I44 .83 43.824 1.43 75-504 2.3O 121.440 .24 12.672 .84 44-352 44.880 1.44 76.032 2*40 126.720 •25 13*200 .85 1.45 76.560 2.50 132.000 .26 13.728 .86 45.408 1.46 77.088 2.60 137.280 •27 14.256 .87 45.936 46.464 1.47 77.616 78.144 78.672 2.7O 142.560 .28 14.784 .88 1.48 2.80 > 147.840 •29 15-312 .89 46.QQ2 1.49 2.9O 153.120 .30 15.840 .90 47.520 1.50 79.200 3«oo 158.400 •31 16.368 .91 48.048 1.51 79.728 3*10 163.680 •32 16.896 .92 48.576 1.52 80.256 3.20 168.960 •33 17.424 •93 49.104 1-53 80.784 3.30 174.240 •34 17.952 •94 49.632 1.54 81.312 3.40 179.520 •35 18.480 •95 50.160 1-55 81.840 3.50 184.800 .36 19.008 .96 50.688 1.56 82.368 3.60 190.080 ■ 37 19.536 ■97 51.216 1-57 82.896 3.70 195.360 .38 20.064 .98 51-744 1.58 83.424 3.80 200.640 •39 20.592 •99 52.272 1.59 83.952 3.90 20^.Q20 • 40 21.120 1.00 52.800 1.60 84.480 4.00 211.200 .41 21.648 1.01 53.328 1.6l 85.008 4.10 216.480 •42 22.176 1.02 53.856 1.62 85.536 4.20 221.760 •43 22.704 1.03 54.384 1.63 86.064 4.30 227.040 •44 23.232 23.760 24.288 I.04 54.912 1.64 86.592 4.40 232.32O •45 1.05 55.440 1.65 87.120 4.50 237.600 .46 1.06 55.968 1.66 87.648 4.60 242.880 •47 24.816 1.07 46.406 1.67 88.176 4.70 248.160 .48 25-344 1.08 57.024 1.68 88.704 4.80 253.440 •49 25.872 I.09 57-552 1.69 89.232 4.90 258.720 •5° 26.400 1.10 58.080 1.70 89.760 5*oo 264.000 •5i 2(2.928 1.11 58.608 1.71 90.288 5*io 269.280 •52 27-456 1.12 59-136 1.72 90.816 5.20 274.560 •53 27.984 1.13 59.664 i-73 91.344 5.30 279.840 •54 28.512 1.14 60.192 1.74 91.872 5.40 285.120 •55 29.O4O 1.15 60.720 i-75 92*400 5.50 29O.4OO .56 29.568 1 . l6 61.248 1.76 92.928 5.60 295.680 •57 30.096 1.17 61.776 *•77 93.456 5.70 5.80 300.960 .58 30.624 1.18 62.304 1.78 93.984 306.240 •59 31.152 1.19 62.832 1.79 94.512 5.90 311.520 .60 31.680 1.20 63.360 1.80 95.040 6.00 316.800IXDEX, PAGE Abbreviate jus explained.......................................... ix Acres, roods, and perches in square feet, Table VI................152 Adjustment and use of instruments..................................... 23 Angles of frogs, to find..............................................129 index, to find.................................................. 69 intersection, to find........................................... 55 plane........................................................... 12 to read on verniers............................................. 43 tangential and deflection....................................... 50 of switch-rails.................................................130 Apex distance of curves, to find........................' . 52 Arc, functions of, to find............................................ 13 Arithmetical complement............................................... 6 Axemen, duties of..................................................... 84 Azimuths of North Star, Table 111.....................................150 Barometer, levelling by... ......................................... 29 Bench-marks, proper intervals for................................... 83 Bubble, to adjust on level............................................ 25 to adjust on transit............................................ 40 Chain, to lay out curves with......................................... 63 Cliainman, duties of................................................. 82 Chief engineer, dutieB of............................................. 79 Chords, to calculate..............................................54, 58 Table XVI.......................................................269 Circle, propositions concerning....................................... 49 Circular arcs to radius of 1, Table VTI.............................. 152 Complement of an angle................................................ 12 arithmetical.................................................... 6 Compound curves. See Curves. Contour maps, utility of.......................................... 85 Correction for curvature and refraction in levelling.................. 28 Cosines defined....................................................... 12 Crossings, plain rules for laying off.................................139 Cross-hairs, to adjust....................................... 24, 26, 40 eccentricity of . . ............................................ 24 to put in new................................................. 44 319320 INDEX. PAQK Cross-sectioning. See Blope stakes. Cubes and cube roots of numbers, Table XI...........................161 Curves, circular, on railroad defined......................• . . . 51 to find radius, length, degree, apex distance, chord, versed sine, and external secant.........................................52, 56 form for field notes.............................................. 70 Curves, how to lay out on the ground, — with the chain only.............................................. 63 with transit and chain............................................ 66 hints as to field-work........................................ 82 protractor for................................................... 84 slackening grade on........................................... 87 terminal.......................................................... 88 Curves, simple, location of, — how to proceed when the P. C. is inaccessible................. 93 to shift the P. C. in order to strike a fixed tangent......... 96 to change radius from same P. C. in order to strike a fixed tan- gent ........................................................... 97 to triangulate on . . ........................................... 94 to pass through a fixed point.................................127,128 Curves, compound, — how to proceed when the P. C. C. is inaccessible.............. 95 to compound a curve in order to strike a fixed tangent .... 98 to shift a P. C. C. in order to strike a fixed tangent........ 99 summary of rules for............................................ 101 to compound into a tangent intersecting main curve on concave side........................................................102 to compound into a tangent intersecting main curve on convex side........................................................103 Curves, reversed, — parallel tangents, radii equal................................115 parallel tangents, radii unequal..............................117 angles unequal, radii equal...................................... 119 angles unequal, tangent points fixed, radii equal . . . . . . 120 divergent tangents, radii equal, advancing towards intersection . 123 receding from intersection..............................124 to shift a P. R. C. in order to strike a fixed tangent........125 Curves, miscellaneous,— elevation of outer rail.......................................141,142 degree of, to find by calculation.............................52, 55 to find on ground.......................................145,146 to connect curves of contrary flexure by short tangents ... 89 to locate a Y from a tangent..................................103 from a convex curve.....................................104 from a concave curve....................................106 to locate a tangent to a curve from a fixed point.............108 to two curves already located...........................109 to substitute a curve for a tangent connecting two curves . . . 109 termiual curves............................................. 88INDEX. 321 PAGE Curves, miscellaneous — continued. . trackmen’s table of curves and spring of rails .■..............143 vertical curves^ to. calculate ............................. 36 to project ............ ................................. 39 Datum in levelling .................................................... 27 Decimals of an acre per 100 feet for various widths, Table V...........151 Deflection angles and distances explained.............................. 50 to find '. . . . . . . '. . . '. . . . . . .... 57, 64, 68 short rule for sub-deflections .... ......................... 68 limit In field-practice . ............................................ 82 Degree of curve, to calculate .........................................52, 55 to find on ground...............................................145,146 Deviations from project admissible on location . •..................... 81 Distances, tangential and deflection, defined ......................... 50 table of .... ...............................................155 of frogs from toe of switch ....................................130,132 tables of . ............................................ 135,136 Elevation of outer rail on curves ...................................141 table of .........................................................142 Excavation and embankment, to stake out 30 External secants, to find ...................................... 54 of a 1° curve, Table XVT .....................................269 Extreme elongations of North Star, Table II. ......... 150 Feet in decimals of a-mile, Table VTH................................153 Field-work, suggestions concerning ; . . ............................79,85 Field-book, form of, for level . .................................... 27 for transit.................... :............................. 70 for slope stakes ...................................... 33, 34, 35 Frogs and switches..................................................... 129 rules for angles and distances 130 table of, switch-rails straight................................. 135 switch-rails curved ...................................... 136 plain rules for locating, switch-rails straight ..................132 switch-rails curved 133 on narrow gauges ...........................................134 patterns for 134 Functions, trigonometrical, defined...................................... 12 logarithmic, of arcs, to find................................... 14 General propositions in trigonometry ................................... 15 as to circles.................................................... 49 Grade, to slacken on curves............................................... 87 rise per mile, Table XVTI........................................317 Grade lines, how to project on map ..................................... 86 how to trace in field ........................................... . 81 Heights, to find by barometer and thermometer 29322 INDEX, PAGE Inches in decimals of a foot. Table IX............................ . 153 Index angles, to determine................. 69 Instruments, adjustment and use of ... ..................... , , 23 Intersection angles of tangents, to find.......................... 55 desirable to fix on ground.................................. 66 Level, to adjust................................................... 24 Leveller, duties of............................................... 83 Levelling, art of................................................... 26 by barometer and thermometer................................... 29 correction for curvature and refraction..................... 28 form for field-book............................................ 27 rules for exact work *..................... ................ 27 rules for survey and location.................................. 28 suggestions concerning......................................... 83 Location, problems in field......................................... 94 admissible errors on ground.................................... 81 form of record for ......................................... 81 projects, hints concerning..................................... 84 of terminal curves............................................ 88 of a Y............................................... 103,104,106 Logarithms explained.................................................. 3 multiplication by............................................... 5 division by..................................................... 6 of numbers, to find............................................ 4 Table XII. ................................ 179 roots and powers by............................................ 7 Logarithmic sines, tangents, &c., to find........................... 13 table of, XIII.................................................197 Maps, contour, utility of............................................ 85 notes for............................... 82, 83 not sufficient for intelligent projects....................... 79 Meridian, to establish............................................... 44 by equal shadows . ......................................... 45 by North Star................................................... ^ times of passage of North Star, Table I...................... . 149 Multiplication by logarithms......................................... 5 Natural sines, tangents, &c., defined............................... 12 Table XIV................................................. 243,256 Needle, magnetic, to adjust .....'................................... 41 to re-magnetize................................................ 44 hints as to management........................................ 44 bearinge should always be noted................................ 82 North Star, to establish meridian by................................. 45 times of meridian passage, Table 1.............................149 extreme elongations of, Table II...............................150 azimuths and natural tangents, Table 111.......................150INDEX. 323 PAGE Obstacles In the field to vision ............... 71 to measurement ................... 73 Ordinates of circular curves, to find . ............ 68, 69 of parabola, to find ......................... '. . 36 of a lc curve, Table X................ . 155 Parabola, ordinates of . . .... . . . . . . . . . . . . 36,59 Plane trigonometry ................................... 12 Powers and roots of numbers by logarithms . ......... 7 Propositions, general, in trigonometry ............. 15 Protractor for curves described ............... 84 how to make .. . . . . . . . .................. 85 Ralls, table of spring for trackmen ................143 Radius of a curve, how to find .................. 52, 54, 56 of a turnout curve ...............................129 plain rule for, on curves.........................133 for narrow gauges.......................... 134 Radii and their logarithms, Table X............ 155 Records, forms for..................................... 81 Refraction and curvature, correction for............... 28 Reversed curves. See Curves. Rise per mile of various grades, Table Xvil.... 317 Rod, levelling . . ;................................... 28 how to read...................................... 42 Rodman, duties of...................................... 83 Roods and perches in decimals of an acre, Table IV..151 Roots and powers of numbers by logarithms ......... 7 Senior assistant, duties of........................... 80 equipment for.......................-............. 81 Sines defined......................................... 12 Shadows, to fix true north by.................... 44 Slopes for topography, Table XV..................... 315 Slopeman, duties of................................... 84 Slope stakes, to set.................................. 30 for earth excavation.............................. 31 for embankment.................................... 33 for hillsides and rock............................ 35 field record of work.............................. 34 Spring of rails, table for trackmen ................. . 143 Squares, cubes, and roots of numbers, Table XI....... 161 Supplement of an angle............................. 12 Survey, form for record............................. 81 to facilitate.................................... 82 Switch-rails, angles of.............................. 130 tables of.....................................135,136 Tangent, or apex distance of curve, to find . . . . . . . . . . 52, 64324: index: PAGE Tangent of a 1“ curve, Table XVI.' . . . . . . . 269 to curve from a fixed point, bow to locate . >. .................108 to two. curves on the ground, how to locate .....................109 Tangential’angles and distances explained. ...................... 50 how to find........................................... 57,-58, 64 Thermometer, levelling by........................................ 29 Track problems ........... ... . 115 Trackmen’s plain rules for finding frog distances .. ..............132,133 tables of turnouts .. . .. . . . . . . ...................135,136 plain rules for laying off turnouts with tape-measure and pins . 137 crossings on straight lines and on curves............139- elevation of outer rail.. ... . .... . ... . 142 instructions how to put in missing stakes on curves with tape- . measure .......... ... •. '. . .' ." . '. . " . . ' . 144 table of curves and spring of rails................. . 143 explanation of the trackmen’s tables ......................- 144 how to find the degree of a curve......... . . . . . . . . 145,146 Transit, adjustment of.................. . ................... 40 cross-hairs................................................... 24 Transltman, duties of . . . ..................................... 82 Triangles, solution of, — . . two angles and a side given................................ 16 two sides and an angle given . . ....................... . . 17 three sides given . . . . ... ........ . . . 18 Triangles, right-angled, solution of ...... . ...... . . . . 19 Trigonometry,.plane . .......................................... 12 general propositions . ....................................' 15 Turnouts. See Trackmen. . Vernier explained............... ......... . . . .. . . . 42 on transit . ........................ ..................... 43 Versed sines defined ............. ............ . . . 12 to calculate . ...... . . ............................... . . 54,58 of a 1! curve, Table XVI . ... ............................ 269. Vertical curves, to calculate .... . . .......................... 36 to project ... . .......................................... 39 Tables:. . Ordinates of a Is curve .........- ........................... 60 For locating terminal curves.................................... 88 Tangents between ourves of contrary flexure................... 89 Turnouts, switch-rails straight............................... 135 switch-rails curved . ..................... ............136 Elevation of outer rail on curves ............................142 Curves and spring of rails . . ....................... . . . . 143 I. Time of meridian passage of North Star......................149 . II. Time of extreme elongations of North Star . . ....... 150INDEX. 325 TAGE ITT. Azimuths of North Star, and their natural tangents .... 150 IV. Roods and perches in decimal parts of an acre...............151 V. Decimals of an acre in one chain length of 100 feet, and of various widths.............................................151 VT. Acres, roods, and perches in square feet....................152 VII. Circular arcs to radius of 1.................................152 VIII. Feet in decimals of a mile...................................153 IX. Inches reduced to decimal parts of a foot...................153 X. Radii and their logarithms, middle ordinates, aud deflection distances................................................. 155 XI. Squares, cubes, roots, and reciprocals of numbers, from 1 to 1,042 .................................................... 161 XII. Logarithms of numbers from 1 to 10,000 179 XIII. Logarithmic sines, cosines, tangents, and cotangents .... 197 XIV. Natural sines and cosines....................................243 Natural tangents and cotangents .............................256 XVT. Chords, versed sines, external secants, and tangents of a Is curve......................................................269 XV. Slopes for topography......................................315 XVII. Rise per mile of various grades.............................317 SCIENTIFIC BOOKS, PUBLISHED BT D. VAN NOSTRAND, 23 MURRAY STREET, AND 27 WARREN STREET, 1TE-W TOBZ. WIESBACH (JULIUS). A Manual Of Theoretical Mechanics. Translated from the fourth augmented and improved German edition, with an Introduction 1o the Calculus, by Eckley B. Coxe, A.M., Mining Engineer. 1,100 pages and 002 woodcut illustrations. 8vo, cloth .... $10 00 FRANCIS (J. B.). Lowell Hydraulic Experiments. Being a selection from Experi- ments on Hydraulic Motors, on the Flow of Water over Weirs, in Open Canals of Uniform Rectangular Section, and through submerged Orifices and diverging Tubes. 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