GRISWOLD'S RAILROAD EN EER'POCK'ET COMPANION COMPRISING RULES FOR CALCULATING DEFLEXION DISTANCES AND ANGLES, TANGENTIAL DISTANCES AND ANGLES, AND ALL NECESSARY TABLES FOR ENGINEERS; ALSO, THE ART OF LEVELING FROM PRELIMINARY SURVEY TO THE CONSTRUCTION OF RAILROADS, INTENDED EXPRESSLY FOR THE YOUNG ENGINEER TOGETHER WITH NUMEROUS VALUABLE RULES AND EXAMPLES, BY W. GRISWOLD. PHILADELPHIA: HENRY CAREY BAIRD, INDUSTRIAL PUBLISHER, No. 406 WALNUT ST. 1866. Entered according to Act of Congress, in the year 1854, BY W. GRISWOLD, In the Clerk's Office of the District Court of the United States for the Northern District of New York. PREFACE. In offering this book to the patronage of the Assistant Engineer, I would wish to remark, that the book is composed of notes that I have long been collecting. Every Engineer has his private book of Rules, that should he want for memory in the field, he has only to refer to his book. This book I have intended for the same object. For the benefit of the young Engineer, I have inserted the art of leveling, running levels (as it is termed) in plain language, from preliminary surveys to the construction of a railroad; the manner of taking cross sections of the road bed, and setting slope stakes, with its rules. Every feature of theo book has a tendency to attract the attention of both the Assistant and the young Engineer. iv PREFACE. Tables that would be more used by the Assistant Engineer, are inserted, which can be relied upon as correct, as I have taken them from reliable authors. In the art of leveling, I have made no allowance for the Earth's curvature, as in practice upon railroads no allowance is ever made. CONTENTS. PAGE. Explanation of Letters and Terms,................... 5 CURVATURE. To find the Radius of Curves from the Deflexion Angles, 5 "...' " Distance, 5...." a Curve from the Deflexion Angle, 6.c cc a Segment of a Circle,....- 6 When the Angle at Vertex is given, and Radius, to find the T. P.,............................. 6 When the Distance from Vertex to T. P. and Angle given, to find Radius,......:......., 6 Having given the Internal Angle to find the T., P. of a given Radius,........-.........,. 7 Having given the Radius or Deflexion, and Internal Angle, to find the number of Chords that constitute the Curve,..-.-... — 7 Having the Angle at Vertex and Deflexion Angle given, to find the number of Chords that constitute the Curve, 7 With the distance from Vertex to P. C. or T. P. and Radius given to find the distance from Vertex to Curve, 7 With the distance from Vertex to P. C. or T. P. and distance from Vertex to Curve given, to find Radius,'...'8 To find the Radius corresponding to any given Angle of Deflexion and to equal Chords of any given length,,. -9 To find the. Circumference of a Circle' when the Diameter is given, or the Diameter when the Circumference is given,...-- -------- 9 To find the length of an Arc of a Circle containing any number of Degrees,............................... 9 ~ ~v Vi CONTENTS. PAGE. To find the Length of an Arc of a Circle,.............. 10 To find the Length of a Chord of half the Arc,..-..... 10 When the Chord, the Arc, and Chord of half the Arc are given, to find the length of an Arc,.-............... 10 To find the Circumference of an Ellipsis,. 13 To find the Radius corresponding to any given Angle of Deflexion, and to equal Chords of any given length,.. 12 With the Chord and Versed Sine given, to find the Radius,................ 12 With two Tangents of different Courses, an example to unite them with a Curve of a given Radius,. —...... 12 With the Angle at Vertex, and Degree of Curvature given, to find how many feet constitute the Curve,.-.... 14 If there is any number of feet less than 100 feet in your Curve, to find how many Degrees and Minutes to turn off,.....-.............. -.-.-.-........ 14 With a less number of Degrees and Minutes than you turn off in 100 feet, to find the number of feet necessary to measure,.............. 14 Given the Internal Angle, and distance from Vertex to P. C. or T. P. to find the Radius,................... 15 Example for Compounding a Curve,.................. 16 To unite a Reversed Curve with the greatest possible Radius,........................................... 17 ORDINATES. To find Ordinates or Chords of 100 feet,........ 19 Rule for getting the Ordinate for 50 feet and 25 feet approximately..................................... 19 To find the middle Ordinate to any given Radius and to any given Chord,.-_.._.-.. 19 To find the middle Ordinate to any given Radius, by Tabular Cosines...................................22 Given the middle Ordinate, to find any other,.-..- - - 22 To find the middle Ordinate approximately, Chords 100 feet,........... 23 To find the Ordinate for Railroad Bars,. -.....-23 An approximate Rule for calculating the middle Ordinate of a Sub-chord, when the middle Ordinate is given,- - - 24 CONTENTS. VIi PAGE. Mode of putting Ordinates for Track,..._ 24 When the Chord and Radius given, to find the middle Ordinate,..........................-..... 95 DEFLEXION DISTANCE. To find the Defiexion Distance 100 feet with any given Radius,..............-,,,,,,.,,,,,,, S) To find the Deflexion Distance for any given Radius for Chords,.. 25 To find the Deflexion Distance for any given Radius, Chords 100 feet,.... 25 Mode of laying off a Curve by Deflexion Distances,.... 26 To put in Intermediates at the end of a Curve,-..... 27 To find the Deflexion Distance for any number of feet less than 100 feet,..................27 To find the Deflexion Point for any number of feet, at commencement and ending of Curves, when the distance is less than 100 feet,-............................. 28 To form a Tangent to the Curve,..... 28 Examples to the Rules,.............................28 If two lines vary any number of Degrees, to find the distance approximately at their extremities, 30 To find the course of a Line that will connect two points on a given Straight Line, with the difference of Variation given,............................................ 31 DEFLEXION ANGLES. To find the Deflexion Angle corresponding to any given Radius.. —---------------------------- 32 To find the Deflexion Angle for any plus distance or less than 100 feet,................... —............. 32 TANGENTIAL DISTANCE. To find the Tangential Distance for any Radius and Chords 100 feet,......... 33 To find the Tangential Distance for any number of feet less than 100 feet, - -- 33 To find the Tangential Distance for any number of feet,. 33 Example of the Tangential diasmce on long Chords,.... 33 VIii CONTENTS. TANGENTIAL ANGLES. PAGE. To find the Tangential Angle for a Chord of 100 feet, with any given Radius,. -...-.35 To find the Tangential Angle for ally number of feet less than 100 feet,- -.. 35 To find the Tangential Angle for Chords of more than 100 feet...,... —.-............... 35 To find the length of long Chords,.................... 36 TRIGONOMETRY. The smallest Angle and Hypothenuse given, to find the Length of the shortest Leg, (by natural sines,).... — 37 To find the Length of the Base, (by the square root,).. 37 To find the Length of the Base, (by natural sines,) -.- 38 The Angles and length of the Base given, to find the Hypothenuse and shortest Leg, (by natural sines,)-. 38 To find the shortest Leg, (by natural sines,) 38 The Hypothenuse and one Leg given, to find the Angles and the other Leg, (by natural sines,)... - - 39 The Hypothenuse and smallest Angle given, to find the Length of the Base, (by cosines,) -.. 39 The Base and smallest Angle given, to find the shortest Leg, (by natural tangents,)................... —-- 40 The Angles and shortest Leg given, to find the Length of the Base, (by natural tangents,).. -.. -. 40 Solution of a Right Angled Triangle,.... -. 40 SURVEYING. To Measure the Distances to any inaccessible object by Triangulation,.. -...... 41 To Measure the Height of a Tower,..... 44 To get the Elevation of the top of a Hill, by Triangulation, _..-._..................._..... _._.._... 45 THE ART OF LEVELING. To adjust a Level,...........-..........46 Manner of keeping Field Book,-................. 55 Manner of keeping Grade Book.....................58 CONTENTS. ix PAGE. Manner of preserving Notes in the Field,......... 63 Manner of plutting in Slope Stakes with the Level-,.... 64 Rules for calculating the Length of the Base of any given Slope,...-...... 66 CROSS SECTIONS. Example how to take Cross Sections in leaving a Cut and entering an Embankment, or vice versa, -.. 68 BORROWING PITS. The Manner of Measuring Borrowing Pits to ascertain the Quantities taken out, -.. 69 Example of the Cross Section of Borrowing Pit,... 71 Manner of keeping Notes of Original and Final Cross Sections-,, —,,,.,., — —,,, 74 MENSURATION OF SURFACES. To find the Area of a Right Angled Triangle,....-.... 75 CC {' Triangle,....... 75 I.. " by its sides, -.....-.. 75... Rectangle or Square. 75... Rhombus or Rhomboid, 76 ".. Trapezoid,.-............ -... 76 Trapezium,. 76... t Regular Polygon, -..- 76.".. Irregular Polygon,....-...... 76.. -.. -....' Figure, bounded on one side by a Straight Line..........................77 To find the Area of a Circle when the Diameter and Circumforeiice are both known,.-..... -.... 77 To find the Area of a Sector of a Circle,.-. 77.... Semlnent of a Circle,-.-.-..... 77... an Ellipsis, 77.... a Circular Ring or Space included between two Concentric Circles,.. 78 To findd the Area of a Parabola or its Segment.-. 78 MENSURATION OF SOLIDS. To find the solid contents of a Cylinder,. 78 xr CONTEN'rS. PAGE. To find the solid contents of a Cone or Pyramid,.... 78..... " Frustum of a Cone,.. 79......... - Pyramid,... 79 To find the solidity of a Wedge, 79 " solid contents of a Prism,.................. 79 c CC " " Sphere or Globe,......... 80 it " " I. the Segment of a Sphere,.. 80 t" solidity of a Spheroid,....................80 it it Segment of a Spheroid,. 80 " solid contents of a Cylindric Ring,. —----- 80 superficial contents of a Board or Plank, 81 (" solidity of Timber, -......................81 Application to the Table of Flat or Board Measure,..-... 81'c.... the solidity of Timber,...... 82 MISCELLANIES. The different dimensions of Boxes, to correspond with the United States Standard Bushels, Gallons, etc.,... 84 To determine the amount of Imperial Gallons in a Vessel the shape of an Inverted Cone,..... 85 To determine the contents of Imperial Gallons in a Kettle forming the Segment of a Circle,. -.... 85 Manner of calculating the Natural Sines and Cosines in the Table,..-.........................-. 87 TABLES. Table of Radii,..-.. -. -1 " Ordinates,.............. 20, 21 " Deflexion Distances,.... 30 " Tangential "...................... 34 " Long Chords..............................36 " Timber, Flat, or Board Measure,....- 82 " the Solidity of Timber,..................... 83 " English Dry Measure,,.... 86 el Imperial Wine Measure ---- -----—, 86 the Dimensions of Drawing Paper........... 86 " Natural Sines, Cosines etc.,................. 90 RAILROAD ENGINEER'S POCKET COMPANION FOR THE FIELD. EXPLANATION OF LETTERS AND TERMS. P. C. Point of curve. E. C. End of curve. T. P. Tangent point. 1 Station is equal to 100 feet. A Plus Station is any number of feet less than 100. B. S. Back sight. Pi. S. Fore sight..%nt. S. Intermediate sight. H. Ins. Height of instrament. CURVATURE. To find the radius of curves, from the deflexion angles, from chord to chord. (Chord 100 ft.) RULE 1. As angle of deflexion Is to the length of the chord, So is 57.3 to radius. To find radius of curves, firom the deflexion distance, from chord to chord. (Chord 100 ft.) 6 ENGINEER S POCKET COMPANION. RULE 2. The square of the chord; divided by the deflexion distance. EXAMPLE.- Deflexion distance = 3.y- ft.; Square of chord 10,000; 3.49)10,000(2865, radius. To find the radius of a curve, from the deflexion angle, on chord of 100 ft. RULE 8. Divide the radius of a one degree curve (5,730) by the degrees of deflexion of 100 ft. To find radius of a segment of a circle. RULE 4. Square of half the chord, added to the square of versed sine, = square of chord of half the arc; and square of chord of half the arc, divided by versed sine, = diameter, - radius. When the angle at vertex is given, and radius, to find the tangent point. RULE 5. Multiply nat. tangent of half the whole angle in the curve, by radius of curve; will equal distance from vertex to tangent point. When the distance from vertex to tangent point, and angle given, to find radius. RULE 6. Subtract the angle a b c, (Fig. 1) which is half the angle a b d, from 90~; the remainder will be the angle b c a. Then say: As nat. sine of b c a is to nat. sine of a b c, so is a b or b d to the radius. CURVATURE. 7 Having given the angle a b d, (Fig. 1) it is required to find the point a or d, at which to commence a curve, of a given radius. RULE T. Subtract half the angle a b d from 90~, the remainder will be the angle b c a or b c d; then take the natural tangent of b c a or b c d, and multiply it by the given radius; the product will be b a or b d. Having the given radius (Fig. 1) a c, or deflexion angle for 100 ft., of a curve, and the angle a b d, it is required to find the number of chiords of 100 ft. that will constitute the curve. RULE 8. Subtract the angle a b d from 180~, and divide the remainder by the angle of deflexion in 100 ft. Having the angle at vertex (Fig. 1) e b d, (which is the number of degrees in the curve,) and deflexion angle for 100 ft., to find the number of chords in that curve. RULE 9. Divide the number of degrees at vertex by deflexion angle. With the distance a b or b d, and radius a c given, to find the distance fromf to b, (Fig. 1.) RULE 10. The square of the distance from vertex to P. C. divided by twice the radius. S ENGINEER'S POCKET COMPANION, Fig. 1 e EXAMPLE.-Suppose the distance from a to b511 ft, and radius = 1,910 ft., Then 5112= 261,121. 2 X 1,910 = 68.2, Answer. NOTE. —The square of any distance, divided by twice radius, will equal the distance from tangent to curve, very nearly. When the distance a b or b d is given, and distance f b, (Fig. 1) to find radius. RIFLE 11. Divide the square of the distance a b or b d by the distance f b, equals twice the length of radius, -= radius. EXAMPLE.-Suppose the distance f b equals 68.2, and a b or b d equal 511. CURVATURE. 9 STATEMENT.- 511' = 261,121 -- 68.2 = 3,820 -i2 = 1,910, radius. To find the radius corresponding to any given angle of deflexion, and to equal chords of any given length. tRULE 12. Subtract the.angle of deflexion from 1800; then say: as nat. sine of angle of defiexion, is to nat. sine of half the remainder, so is the given chord to the radius required. EXAMPLE.- Let the angle of deflexion be 20, and the chord 100 ft., required the radius. Then 2~- 180~=- 1780 -- 2-89~. N. S. 20. N. Side 890. Chord. STATEMENT. — 0.0349:.999848:: 100: 2865, radius. To find the circumference of a circle, when the diameter is given, or the diameter, when the circumferefice is given. RULE 18. Multiply the diameter by 3.1416, equals circumference; or, divide the circumference by 3.1416, equals diameter. 2d. As 7 is to 22, So is the diameter to the circumference. Or, as 22 is to 7, So is the circumference to the diameter. To find the length of an arc or circle, containing any number of degrees. RULE 14. Multiply the number of degrees in the given are, 10 ENGINEER'S POCKET COMPANION. by 0.0087266, and the product by the diameter of the circle. NOTE.- The circumference of a circle,. whose diameter is 1, is 3.1416; it follows, that if 3.1416 be divided by 360%, the quotient will be the length of an arc of i degree, 0.0087266. REMARK. —When the are contains degrees and minutes, reduce the minutes to a decimal of a degree. To find the length of any arc of a circle. RULE 15. Subtract the chord of the whole arc from 8 times the chord of half the arc, and 2 of the remainder is the length of the arc, nearly. When the chord of the are, and the ehord of half the arc, are given. RULE 16. From the square of the chord of half the arc, subtract the square of half the chord of the en.tire arc; will equal the square of the versed sine; extract the square root; will equal versed sine- the versed sine and the chord given-tlie square of ~ the length of the chord, added to the square of the versed sine, and square root of the remainder, will equal chord of ~ the arc; multiply the remainder by 8, subtract the chord of the whole arc, and divide by 3, equals length of arc. To find the circumference of an ellipses. RULE 1T. Half the sum of the two diameters, multiplied ib 3.1416; the product will equal circumference. CURVATURE. 11 TABLE OF RADII- CHORDS 100 FT. ~ 4 a' a i a -:, a t tt 015 22920 815 695'1 16 15 353 8 24 15 238'0 30 11460 30 674 G 30 3484 30 235 6 45 7640 45 655'5 45 343'3 45 233'3 1 5730 9 637]3 17 338'3 25 1231'0 15 45.84 15 62()2 15 333 7 15 228-7 30 3820 30 603'8 30 328-7 30 226'5 45 3274 45 588-4 45 324'8 45 224'3 2 2865 10 573'7 18 319'6 26 222 3 15 2547 15 559'7 15 315-2 15 220}'6 3f) 2292 30 546 4 30 311 0 30 21830 45 2084-0 45 533'8 45 306'9 45 216-0 3 1910 11 521-7 19 302 9 27 214'2 15 1763 15 510-1 15 2994 15 212 2 30 1637 30 499'1 30 295'3 30 210-3 45 152,8 45 488'5 45 291'5 45 208 5 4 1433 12 478.3 20 287.9 28 206-7 15 1348 15 468'7 15 284'4 29 199-7 30 1274 30 4593 30 280.9 30 19332 45 1207 45 450'3 45 277 6 31 1871 5 1146 13 441'7 21 274'4 32 181-4 1 5 1092 1 5 433'4 15 271'1 33 176'0 30 1042 30 425-5 30 268'0 34 171-0 45 996'8 45 417-7 45 26i50 35 166-3 6 95541 14 410'3 22 262'0 36 161 8 15 9170 15 403 1 15 26i00 37 157-6 30 882 0 30 396 2 30 257 4 38 1536 45 849 3 45 389-6 4-5 254-6 39 149-8 7 819-0 15 383-1 23 250(8 40 146-2 15 7908 15 3769 15 248-1 45 136-5 30 764 5 3() 370- 30 245-5 45 739 9 45 365-0 45 242-9 8 716-8 16 359-3 24 240 5 12 ENGINEER S POCKET COMPANION. To find the radius corresponding to any given angle of deflexion, and to equal chords of any given length. RULE 18. Subtract the angle of deflexion from 1800; then say: as nat. sine of angle of defiexion is to nat. sine of half the remainder, so is the given chord to the radius required. Fig. 2. With the chord and versed sine given, to find the radius. RULE 19. The square of half the chord divided by versed sine; to which add the versed sine, and divide by 2. EXAMPLE.- Suppose we have an are (Fig. 2) with a chord a d of five feet, and versed sine e b two feet, what is the radius a c? STATEMENT.- 5 2 = 25a2 = 6'25 -. 2 = 3'125 + 2 = 5'125.- 2 = 2-5625, radius a c. We have two tangents with their courses given. We wish to unite those tangents, with a curve of a given radius. CURVATURE. 13 Suppose we have a tangent whose course is N. 300 E., (Fig. 3) which we wish to unite with a 30 curve to a tangent whose course is S. 50~ E. We here find we have the difference of courses to be 100~. According to Rule 8, page 7, and Rule 20, page 14, we have 3,333-1-3; ft. to run. We start on the first gi,-en tangent, and run 3,333-3-%% ft. If our curve does not form a tangent of the line g e, but touches the point c, we measure in a line the same course of the first tangent, N. 30~ E., to its intersection, which distance you measure backward or forward for P. C. Fig. 3. EXAMPLE.-We have the tangent a b, NV: 300 E., and tangent e g, S. 50~ E.; we wish to join those two tangents with a 30 curve. STATEMENT.- T. NV; 300~ E. and T. S. 50 E. 100~, angle of deflexion, which makes 100~ in the curve; consequently, the number of feet in the curve 14 ENGINEER'S POCKET COMPANION. (chords 100 ft.) = 100~ -- 3 = 33 stations and 3313-3 ft.=3,333-1T-30- ft. We start from the tangent a b, at the point 1, and run 3,333-~-? ft., turning off for a 3~ curve, and find, when arriving at our tangent, we are 250 ft. from the line, as d c; we then return and measure the same distance on the tangent a b from 1. to P. C. With the angle at vertex given, and degree of curvature, to find how many feet constitutes the curve. RUILE 20. Divide the number of degrees at vertex, by degree of curvature. EXAMPLE.-We have the angle at vertex - 100~, and degree of curvature- 3~. STATEMENT.- 1000~ 3~ = 33 stations and 33 33l ft. = 3,33-1303 ft. If there is any number of feet less than 100 ft., in your curve, to find how many degrees and minutes to turn off. RULIE 21. Say: as 100 ft. is to the number of feet you wish to run, so is the number of degrees and minutes to the number of degrees and minutes you wish to find. EXAMPLE.-Suppose you turn off in every 100 ft., 10 30', how much will it be necessary to turn off in 33 ft.? STATEMENT.- 100:33::1~ 30':30', Answer. Suppose you have a less number of degrees and minutes, than you turn off in 100 ft., to find the number of feet necessary to measure. CURVATURE. 15 RULE 22. As the whole number of degrees and minutes is to the number of degrees and minutes you wish to turn off, so is the chord, 100 ft., to the number of feet required. EXAMPLE-We turn off in 100 ft. 1~ 30', we wish to find the number of feet to measure for 30'. STATEMENT.- 10 301: 30.::100: 33, Answer. Fig. 4. e With the angle a b d, (Fig. 4) and distance a b given, to find radius a c. RULE 28. Half the angle a b d taken from 900 = angle a c b. Then say: as nat. sine of a c b is to nat. sine of a b c, so is the given side a b or b d to the radius a c. 16 ENGINEER'S POCKET COMPANION. NOTE.-This rule is often used in compounding curves, where the curve you have run does not fit the tangent, by measuring from a given point, on a tangent, to curve already run, to the tangent you wish to connect, as b I (Fig. 5) and measuring the angle c d 1, and proceed according to rule 23. Thus in Fig. 5, we run our curve to n, and find that it does not come in tangentially to the tangent b 1, therefore to save loss of time, in long curves, we retrace our steps to the point c, and measure tangentially to curve a c, as c d to the tangent b l, and measure the angle c d 1, and form a new radius, as c g or e g. Fig. 5. We oftentimes have two different angles in a line, and in such close proximity that it is required to put in reversed curves (Fig. 6) that will connect with the greatest possible radius; we then wish to find CURVATURE. 17 the greatest radius a h and e g that will connect the tangents with a reversed curve. Fig. 6. RULE 24. Half the angle a b e taken from 90~ leaves the angle b h e or a h 6, and half the angle b n d taken from 900 leaves the angle e g n. From the table of nat. tangent take the nat. tangent of b h e or a h b and nat. tangent of the angle e g n and add them together. Then say: as the sum of these two nat. tangents is to the nat. tangent of b h e, so is the distance 6 n to b e. Again, in the triangle b h e, as the nat. sine of the angle b h e, opposite the given side b e just found, is to the nat. sine of the angle h b e, opposite the required side h e, so is b e to h e, the radius required. EXAMPLE.-Let the angle a b n be 71~ 40', the angle b n d 129~ 15', and the distance b n be 950 ft., what is the length of the radius h e or e g. 18 ENGINEER'S POCKET COMPANION. STATEMENT lst.-710 40' 2 = 35~ 50' 90 540 10'. 129~ 15'-.2 2=64~ 372' 90 =25 22k-' Nat. tangent of 540 10' = 13848. Nat. tangent of 25~ 22 —' = 0'4743 + 1'3848 1'8591. Sum. N. tang. 540 10/ b n be Then as 1'8591: 1'3848:: 950ft.: 707'63 ft. Again, 950- 707'63 = 242'37 ft. = e n. Sine b h e. N. sine h b e. STATEMENT 2d.- 0'8107:'5854:: 707'63:510'97, radius required. Hence, we have the distance h e and e g = 510'97 ft., and distance b e 707'63 ft., and distance e n 342'37 ft., and degree of curvature equal to 11~ 16' for chords of 100 f+ The radius of a curve is always at right angles withl its tangent. When running curves with a compass, or transit instrument, always turn off on the vernia, one-half the number of degrees as contains degrees of curvature; because when running with an instrument, you are running tangential angles instead of deflexion angles, and the tangential angle equals one-half the deflexion angle. ORDINATES. 19 ORDINATES. To find ordinates on chords of 100 ft. RULE 1. The product of the segment, divided by twice the radius. EXAMPLE. —Suppose the radius = 2'865 ft.; what would be the ordinate for 25 ft.? (Chords 100 ft.) 25 X 75. 5,730 - 0'327. Rule for getting the ordinate for 50 ft. and 25 ft. approximately. RULE 2. For an ordinate of 50 ft., divide the deflexion distance for 100 ft. by 8. For 25 ft. three-fourths of the ordinate for 50 ft. EXAMPLE.- Suppose the deflexion= 3, which deflexion- distance = 5'235 -- 8 = 0'654, ordinate for 50 ft., and three-fourths of 0'654 — 0491, ordinate for 25 ft. To find the middle ordinate to any given radius, and to any given chord. RULE 8. From the square of the radius, subtract the square of half the chord, and take the square root of the remainder from the radius = middle ordinate. EXAMPLE.-What is the length of the middle ordinate d e, (Fig. 1) the radius a c being 2'5625, and chord a b 5 ft. STATEMENT.-2'5625 - 25'2 = v 316406' 5625 - 2'5( 25 = 2 ft., middle ordinate. 20 ENGINEER S POCKET COMPANION. TABLE OF ORDINATES -CHORDS 100 FT. s0 oLength of Ordinates in Feet. I 50 45 40 35 30 25 20 15 10 5 0 5'018'018'017.016.015.014'012.009 -006 003 10 -036 *036'035.033 -031'027'023.019.013 0)7 15 054 054 *052'049 0046 041 035 028'019 010 20 073 *072'070 *066.061 055 047'037 *026 014 25 *091 J60 087 *082 *076 *068 058 *046.032'017 30 109'108 *105 099'002'082 070'055.039 )020O 35 *127,126 *123 *116 *108 096 *082 *005 045 024 40 *145 *144 *140'133 *123'110'093'074'052 02)7 45'163 161 *157'149'137 *123'105'0V3 *058 0.1 50'182 *180'175'166'153'138'117 092 065 034 55'200'198'192'182'168'151'128 10)2 071'018 1'218'216'209'198'183'164'140 1111'078'041 5'236'234'226'215'198'178'152 1120'085'044 10'254'252'244'231'214'191'163 1'0'091'048 15 *273'270'261'248'229'205'175 139'098'051 20'291'288'279'264'244'218'187'148'104'055 25'309'306'296'281 2159'232'198'157 1111'058 30'327'324'314'297'275'246'210'167'117'062 35'345'342'331'314'290'259'221'170'124'065 40'364'360'349'330'305'273'233 *185'130'069 45'382',378'366'347'321'287'245 195'137'072 50'400'396'384'364'336'300'256'204'144'076 55'418'414'401'380 *351'314'268'213'150'079 2'436'432'419'397'366'327'280'222'157'083 5'454'450'436'413'382'341'291'232'163'086 10'473'468'454'430'397'355'303'241'170'089 15'491'486'471'446'412'368'315'250'176'093 20'509'504'489'463'428 382'326'26()'183 09(i 25'527'522'506'480'443'396'338 269'190'100 30 7545.540'524'496'458'409'350'278'196'103 35'564'558'541'513'474'423 -361'288'203'107 40'582'576'559'529'489'436'373'297'209'110 45'600'594'576'546 504'450'384'306'216'114 50'618'612'594'562'519'464'396'.115'222'117 55'636'630'611'579'535'477'4()8'325'229 121 3'654'648'629'595'550'491'419'334 23.5'124 5'673'666'646'612'5(5'504'431'343'242 128 10'691'684'6G4'629'581'518'443'353'249'131 15'709'702'681 "645'596'532'454'362'255'134 20'727'720'699'662'611'545 466'371 262'138 25 *745'738'716'678'627'559'478'.30 *268'141 30'764'756'734'695'642'573 489'390'275'145 35'782'774'751'711'657'586'501'309'281' 148 40'800'792'709'728'673'600 -512'408'288'152 45'818'810'786'744'688'613'524'418'294'155 50'836 8s28 8014'761'703'627'536 427 301'159 55'854 8:46'821'778'718'641'547'436'308 *162 4'873 -864 *839'794'734'654'559'445'314 166 ORDINATES. 21 TABLE OF ORDINATES -CHORDS 100 FT. Length of Ordinates in Feet. s< 50 45 40 35 30 25 20 15 10 5 4 15''927'918 -891.844.780 *695'594 473 334 *176 30 *981.972'944 *893.825'736.629 -501 -354 i186 45 1.036 1-026 -996 -943 -871 -777 -664'-529.373 196 5 1-091 1-080 1-048 -993 -917 -818 *699 *557 1393 207 15 1-146 1-134 1.100 1-042 -963 *859'734.585 -413 *217 30 1.200 1-188 1-153 1-092 1-009.900'769 -613 -432 228 45 1*255 1'242 1'205 1'141 1-055'941'804 0640'452 -238 6 1-309 1-296 1-258 1-191 1'100'982'839'668'472'249 30 1-419 1-404 1-362 1,290 1-192 1-064 909 724 *511 2ti9 45 1-473 1-458 1-415 1-339 1-238 1-105'944'752'531'280 7 1-528 1-512 1-467 1-389 1-284 1'146'979'779 -551'290 30 1-637 1-620 1'572 1'488 1-375 1'228 1-048 -&35'590 -311 8 1-746 1-728 1'677 1'587 1-46'7 1 310 1-118'891'629'-32 30 1-855 1-836 1'782 1-687 1-559 1-392 1-188'946 -669 -353 9 1-965 1-944 1'886 1'787 1-651 1-474 1-258 1-002'708'373 30 2-074 2-052 1-991 1'887 1-742 1 556 1-328 1-057'748'394 10 2-183 2-161 2-096 1-9237 1-834 1-637 1-398 1-114'787 4'15 30 2-292 2-269 2-201 2'087 1-926 1-719 1-468 1'170'827 -436 ]1 2-401 2 377 2'306 24186 2-018 1 802 1-538 1-226'86&6'457 30 2-511 2-486 2-411 2-236 2-110 1-884 1G609 1-282 906'478 12 2-620 2-594 2-516 2-386 2-203 1-967 1680 1 3.39 -946 -499 13 2-839 2-811 2-726 2-1585 2-387 2-132 1-820 1-451 1-025'541 14 3-058 3-028 2-937 2-785 2-57i 2-297 1-961 1-564 1-105 -583 15 3-277 3-245 3-147 2984- 2-756 2-462 2-102 1-676 1-184'625 16 3-496 3-462 3-358 3 18. 2-941 2-627 2-243 1-789 1-264 66i7 17 3-716 3-680 3-569 3-384 3-125 2 791 2-384 1.902 1-344'709 18 3.935 3-897 3-779 3'584 3-310 2-958 2-525 2.014 1.424'751 19 4-155 4-115 3-990 3-784 3-495 3-123 2-666 2-127 1-504'793 20 4-375 4-3322 4-201 3.984 3680 3-2883 2-808 2-240 1-583'836 21 4-595 4-549 4-412 4'184 3-864 3-454 2-950 2-353 1' rs3'879 22 4-815 4-768 4-624 4 386 4050 3-620 3 093 2-467 1744'922 |3 5-035 4-986 4836 4-587 4-237 3-786 3 236 2'581 1-824 -965 24 5-255 5-204 5-048 4-789 4-423 3-952 3-379 2-695 1-905 1-008 25 5476 5-422 5-260 4-989 4-609 4 119 3-522 2-809 1 985 1 051 26 56897 5-642 5-473 5-192 4-798 4-286- 3-665 2-924 2-068 1-094 27 5918 5'860 5-685 5-393 4-984 4-454 3-808 3-039 2-150 1-137 28 6139 6-079 5-898 5.5'95 5171 4-622 3-952 3-154 2-232 1-181 29 6-361 6298 6-110 5-796 5-357 4-790 4-095 3-269 2-314 1-224 30 v6-582 6-517 6-323 5-999 5-544 4-958 4'-239 3-385 2-396 1-2;i8 31 6-t;)4 6-737 6537 6-202 5-733 5 127 4-384 3-502 2-481 1-312 32 7-027 6-957 6751 6-406 5-922 5-297 4530 3'619 2-565 1 356 33 7-249 7-178 6-965)5 6-609 6-111 5-467 4-676 3-737 2-649 1-401 34 7-472 7-398 7-179 6-813 6 300 5-637 4-822 3-854 2-733 1-445 35 7-694 7-619 7-39'3 7-017 6-489 5 807 4-968 3-972 2-817 1-400 36 7918 7-841 7'609 7.222 6 679 5 -978 5115 40901 2 901 1-535 37 8-143 8 063 7-825 7-427 6-870 6-149 5-262 4-209 2-985 1'581 38 8-307 8-286 8-041 7633 7-060 6-320 5-410 4- 327, 3-i9 1-626 22 ENGINEER'S POCKET COMPANION. RULE 4. Subtract the tabular cosine of the tangential angles from 1, and multiply the remainder by the radius. (Chords 100 ft.) EXAMPLE.- Radius 819 ft., angle of deflexion would be 7~ to chord of 100 ft; what will be the letngth of the middle ordinate? STATEMENT.- Here tabular cosine of the tangential angle 3~= -998135, which, subtracted from -1=.001865, which, multiplied by radius, 819 ft.,= — ordinate, 1528. Fig. 1. Having the middle ordinate d e, (Fig. 1) it is required to find any other ordinate, as i n. RULE 5. Subtract the middle ordinate d e, from the radius a c, the remainder will be e c; from- the square of the radius a c, subtract the square of the distance o i ORDINATES. 23 or e n, and extract the square root of the remainder; this square root will be o c; subtract e c friom o c; the remainder will be o e, which is equal to i n, the required ordinate. EXAMPLE. — The middle ordinate d e (Fig. 1) of a 100 ft. chord a b, to a radius of 819 ft. = 1 52; it is required to find the length of the ordinate i n 20 ft. from the middle one d e. STATEMENT.- 819 - 1'52S = 817'472. Again, 819'= 670761 202= Vr670361i —818.756 o c — 817472 = 1'284, will equal o e or i n, the required ordinate. To find middle ordinate approximately. Chord 100 ft. RULE 6. Multiply the ordinate of a 10 curve by the deflexion angle of 100 ft. Thlis rule is sufficiently close for curves of not less than 500 ft. radius. 2d. Multiply the chords together, and divide by twice the radius. Fig. 2. a d To find the ordinate for a railroad bar a c b (Fig. 2) 24 ft. long. RULE 7. Multiply one-half the length of the rail by onefourth its length, and divide by the radius. 24 ENGINEER' S POCKET COMPANION. EXAMPLE. —Rail 24 ft. long, radius 5730. STATEMENT.- 24 -- -12 = g the length of rail. 24 -. -= 6- = the length of rail. Then 6 X 12 = 72. 5730 = 0'01, ordinate required. 2d. Take one-fourth of the square of the length of the rail, and divide it by twice the radius. An approximate rule for calculating the middle ordinate of a sub-chord, when the middle ordinate is given. (Chord 100 ft.) RULE 8. As the square of the length of the whole chord is to the square of the length of the sub-chord, so is the middle ordinate of the chord to the middle ordinate of the sub-chord. EXAMPLE.-Chord 100 ft., middle ordinate'218, what will be the middle ordinate of the sub-chord s0 ft.? STATEMENT.- 100: 502:' 218:'0545. Fig. 3. In running a curve for track, after the grading is done, it is necessary to put in intermediate ordinates, if the curve exceeds 10; (Fig. 3) these intermediates are from 10 to 20 ft. apart, and instead of running these intermediates with an instrument, the best DEFLEXION DISTANCE. 25 method is, after your points are put in with an instrument 100 ft. apart, to draw a small cord or twine, as a b, and measure off your ordinates with a graduated rod, or with the leveling rod. NOTE.- Refer to Rule 1, page 19 for intermediates or to table. When the chord and radius are given, to find middle ordinate. (Chord 100 ft.) RULE 9. Divide the square of the chord by eight times the radius. DEFLEXION DISTANCE. To find the deflexion distance for 100 ft., with any given radius. It'LE 1. The square of the chord divided by the radius. RULE 2. Divide the constant number 10,000 (chords 100 ft.) by the radius in feet, equals deflexion distance. To find the deflexion distance for any given radius for chords of 100 ft. RULE 8. Divide the given chord by radius, will give the nat. sine of the deflexion angle, which, multiplied by the chord, will equal the required distance. NOTE. —The tangential distance for 100 ft. is equal to onehalf the deflexion distance, and the tangential angle is always equal to one-half the leflexion angle. 26 ENGINEER' S POCKET COMPANION. Fig. 1. n putting in curves by deflexion distances it is In putting in curves by deflexion distances it is quite necessary, for accuracy, to measure both on the line of deflexion and chord of the arc, as a 6 and a c, (Fig. 1) by first Ending your point b, and swinging your chain to c, and measuring your deflexion distance on the line b c. In commencing your curve on the tangent e d you DEFLEXION DISTANCE. 27 measure your distance f d, and lay off a d equal to one-half b c, or one-half the deflexion distance, as a d is the tangential distance, and the tangential distance is equal to one-half the deflexion distance. If you wish to put in intermediates, as it frequently occurs, at the end of a curve. RULE 4. Find your defiexion distance for 100 ft., string a line, and put in the required ordinate. To find the deflexion distance for any number of feet less than 100. IRULE 5. Take the deflexion distance for 100 ft. and multiply it by the required chord, and divide the product by the length of the whole chord, 100 ft., and subtract the ordinate corresponding in feet and degree. EXAMPLE.-We wish to get the deflexion distance for 25 ft., for a 15~ curve. STATEMENT.-Deflexion distance equals 26'11 ft., therefore: 26'11 X 25 = 652'75 -- 100 = 6'5275. Now the ordinate of a 150 curve for 25 ft.2'462, and 6'5275 —2462 =4'0655, deflexion distance for 25 ft. NOTE.-Tlhe above Rule is sufficiently close for all practical work. Where it is required to find the deflexion or tangential distance for more than 50 ft., subtract the distance to be found from 100, and find the ordinate corresponding to the remainder in feet. 3 28 ENGINEER'S POCKET COMPANION. To find the deflexion point for any number of feet, at commencement of curves and ending, where the distance is less than 100 ft. RULE 6. In commencing a curve, multiply the tangential distance in feet by the number of feet in the chord you wish to find, and divide the product by the length of the whole chord, 100 ft., and measure the distance on the end of a 100 ft. chord; then a line drawn from this point to the P. C., the length of the chord measured you desire to find, on this line, is the point of deflexion. RULE 7. To end your curve: Take half the deflexion distance for 100 ft., then multiply the remaining distance (which is the tangential distance,) by the number of feet you wish to find, and divide the product by the length of the whole chord, 100 ft., and measure on the tangential distance, the distance just found; then string a line from this point to the last station given, and measure, on this last given line, the distance required, will give the T. P. or E. C. To form a tangent to the curve: Measure as many feet more on the tangential distance, and a line drawn from the last given point to T. P. or E. C. will be the course of the tangent. EXAMPLE TO RULE 6. —Suppose in commencing a curve we wish to find the deflexion distance of 25 ft. as at r, (Fig. 2) for a 15~ curve. We take the tangential distance c f and multiply it by 25 ft., and divide by 100 ft., (the length of the whole chord,) will equal the distance c s; now if we measure in the DEFLEXION DISTANCE. 29 line 6 s 25 ft. from b to r, at r \ Fig. 2. will be the required point. We then measure 100 ft. from r to d, in line with 6 r, and measure the deflexion distance d e,,~ ecual to twice cf, for our next A full station. We then measure 100 ft. on the line r e from e to the point h, and measure the distance h i for our next station, so on to the last station. EXAMPLE TO RULE 7. —Sup- pose in ending our curve we wish to find the deflexioa dis- Hi Ad tance for 25 ft., to conclude the curve; (Fig. 2) suppose i to be the last station in the curve, and the point t 25 ft., to be the E. C.; we produce the line e i to k 100 ft. from i, and measure one-half the deflexion distance, k o = 1, then multiply the distance k 1 by 25 ft., and divide by the length of the whole chord i k, 100 ft., will equal the distance 1 m, and on the line drawn from m to i, 25 ft. measured from i to t, on this line i m, will be the required point. To form a tangent to the point t, measure mn y equal to 1 m, will form the tangent t n to the curve. 30 ENGINEER'S POCKET COMPANION. TABLE OF DEFLEX[ON DISTANCES. Length o' Chords in Feet. f- 100'75 50 25 0 30' 872 0'572 0-247 0-136 45 1 308 0'899 0'491 0 204 1 1'745 1'144 0'654 0-272 30 2'618 1'71.7 0'982 0408 45 3'054 2'003 1 145 0-476 2 3'490 2'290 1'309 0-545 30 4'363 2'863 1'636 0-681 45 4'799 3'419 1'799 0-749 3 5'235 3'435 1.963 0-818 30 6'108 4'008 2'290 0-954 45 6'544 4'295 2'454 1 023 4 6'980 4'581 2'617 1 091 30 7'853 5'153 i945 1227 45 8'289 5'440 31108 1.295 5 8.722 5'723 3'270 1-362 6 10'470 6'870 3'926 1 635 7 12'210 8'011 4'577 1906 8 13'950 9'152 5-229 21 77 9 15.680 10-286 5'875 2 446 10 17'430 11'435 6'532 2 720 11 19'170 12'575 7'184 2 990 12 20'940 13'738 7'850 3-268 13 22'640 14'848 8-481 3-528 14 24'370 15'980 9-127 3 795 15 26'110 17'120 9'778 4 065 If two lines vary any number of degrees, to find the distance approximately at their extremities. RULE 8. Say; If they vary 1'745 in 1~ for 100 ft, it will vary thirty times as much in an angle of 300 for DEFLEXI1O DISTANCE. 31 100 ft.; if it is more than 100 ft., make a second statement. EXAMPLE. —Suppose we have an angle of 30~ and 400 ft. long, what is the distance apart at their extremities? STATEMENT.-AS 10:30~::1-745:52'35; 52'35 is the difference for 100 ft. Then as 100 ft.:400 ft.::52'35:209'4, the difference for 400 ft. Suppose we run a line, on a given course, with the intention of striking a certain point, and find that we deviate from that point, to find the course of the second line that will unite these two points on a straig'ht line. RULE 9. Multiply the difference of variation in feet by 57 3,* and divide the product by the length of the line, the quotient either added or subtracted, as necessity requires, will be the course of the line that will unite the two points together. EXAMPLE. —Suppose the difference of variation= 209'4 ft., and length of line 400 ft., and course N. 29~ 59j-' E., what would be the course of the second line, if the point desired is NV. W: of the line run? STATEMENT.- 209'4 X 57'3 _. 400 = 29'9965 290 59;'. Then course N. 29~ 59-'E- 290 59'- =course N. 0~ E. * 57-3 is the radius of a circle (nearly) in such parts as the circumference contains 360. 32 ENGINEER'S POCKET COMPANION. DEFLEXION ANGLE. To find the deflexion angle corresponding to any given radius. (Chords 100 ft,) RULE 1. Divide the chord by the radius; the quotient will be the natural sine of the deflexion angle; therefore, the number of degrees corresponding to this sine, in the table of nat. sines, equals the deflexion angle. RULE 2. The deflexion angle may be found by dividing the radius of a 10 curve, 5730, by the radius in feet, (approximately.) To find the deflexion angle for any plus distance, or less than 100 ft. RULE 8. Multiply one-half the deflexion angle by the plus distance, and divide the product by 100 ft., (length of whole chord,) and add it to one-half the deflexion angle. EXAMPLE. —Suppose we are running a 15~ curve by deflexion distances; we wish to find the deflexion angle for 25 ft. STATEMENT. —15~ -. 2 = 7 30', one-half the deflexion angle. Then, 70 30' X 25=1870 30' - 100 = 1~ 52'. Then, 1~ 52k' + 7~ 30'= 9~ 22~', deflexion angle for 25 ft. For deflexion angles corresponding to any given radius, refer to table of radii, page 11. TANGENTIAL DISTANCE. 33 TANGENTIAL DISTANCE. To find the tangential distance for any radius, on chords of ] 00 ft. IRULE 1. Divide the square of half the chord (50 ft.) by the radius, and multiply the quotient by two. RULE 2. Divide the square of the whole chord by twice the radius. To find the tangential distance for any number of feet less than 100. RULE 8. Multiply the tangential distance for 100 ft. by the number of feet required, less than 100 ft., and divide the product by 100 ft., and from the quotient take the ordinate corresponding to the degree of curvature and feet; will equal the tangential distance for the required number of feet, less than 100 ft. To find the tangential distance for any number of feet. RULE 4. Divide the square of the chord given by twice the radius. In running curves, with equal chords on more than 100 ft., the tangential distances increase as the squares of the number of chords: thus, for 2, 3, 4, 5, 6 chords, 4, 9, 16, 25, 36, multiplied into the tangential distance of 1 chord, will equal each tangential distance respectively. 34 ENGINEER'S POCKET COMPANION. Or: the square of the length of the chord divided by twice the radius, will equal the tangential distance for any number of feet. TABLE OF TANGENTIAL DISTANCES. Length of Chords in Feet. 100 75 50 25 0 0 30 3. 0-436 0-245 0'109 0-027 45 0'654 0'367 0'164 0-040 1 0'873 0'490 0 218 0 054 30 1-309 0'736 0-327 0-081 45 1-527 0-858 0'381 0'095 2 1-745 0'981 0-436 0'109 30 2'82 1'227'O 546 0'136 45 2'399 1'349 0'599 0'150 3 2'618 1-472 0'655 0'163 30 3'054 1-717 0'763 0'190 45 3'272 1841 0'818 0'205 4 3-490 1'963 0'872 0'218 30 3'927 2'209 0-982 0'246 45 4-145 2'331 1'036 0'259 5 4'361 1 452 1'089 0'272 30 4'798 2'698 1'199 0'299 45 5015 28-20 1'252 0312 6 5-235 2'944 1-308 0-326 7 6'105 3-432 1-524 0-380 8 6'975 3'924 1-741 0)433 9 7-840 4'406 1'955 0'486 10 8'715 4'899 2'174 0'541 11 9'585 5'387 2391 0'594 12 10-470 5'885 2-615 0'650 13 11-340 6'373 2`831 0-703 14 12 210 6'860 3'047 0'755 15 13-080 7-348 3'263 0'808 TANGENTIAL ANGLES. 35 TANGENTIAL ANGLES. To find the tangential angle for a chord of 100 ft., with any given radius. RULE 1. Divide half the chord by the radius; the quotient will be the natural sine of the tangential angle; and the angle corresponding to this sine, in the table of nat. sines, is the angle required. To find the tangential angle for any number of feet less than 100 ft. RULE 2. Multiply the tangential angle by the number of feet given, and divide the product by the length of the whole chord, (100 ft.) EXAMPLE.- Suppose we have the tangential angle - 7~ 30', and wish to find the angle for 25 ft. STATEMENT.-7~ 30' X 25 ft. -'. 100 ft. = 1~ 521', tangential angle for 25 ft. Sometimes in running curves it is not necessary to set points in every chord, or 100 ft., and is more expedient, as running curves on a preliminary survey. They can be put in every 2, 3, or 400 ft., as you choose. We wish to find the tangential angle for any number of chords. RULE 8. Multiply the tangential angle for 100 ft. by the number of chords you wish to subtend, will equal the tangential angle required. 36 ENGINEER'S POCKET COMPANION. REMARK.-In running curves, the correct way of measuring with a chain for each station, is to measure around the curve. Instead of this the chain is stretchle(l across, forming a chord; the difference of distance is so comparatively small to a radii of 500 ft., that it is not necessary we should measure around, or make an allowance on the chain; but in running curves with long chords of 3, 4, or 500 ft., it is necessary, for accuracy, to make sufficient allowance, for which I will put in a table of long chords the lengths necessary to subtend fiom i to 4 stations. TABLE OF LONG CHORDS. Length of Chords in Feet required to Subtend. Station. 2 Stations. 3 Stations. 4 Stattions. 1 100 200 300 400 2 100 200 299'9 399'7 3 100 200 299-7 399'3 4 100 199'9 299-6 398'9 5 100 199'9 299'2 398'0 6 100 199'7 298'8 397-3 7 100 199'6 298'4 396-2 8 100 199'6 298'0 395'1 9 100 199-4 297-5 394-1 10 100 199-2 297'0 392-4 To find the length of long chords. RULE 4. Multiply the natural sine of the tangential angle of the given chord by twice the radius. EXAMPLFE. — The tangential angle for one station 5~, and radius= 573'7 ft; what would be the length of the chord of four stations. TRIGONOMETRY. 37 STATEMENT. —The tangential angle for four stations would equal 4 X 50 - 20~, and nat. sine of 20~ ='3420201; twice the radius = 1147'4 X'3420201 - 392'4, length of chord necessary to subtend an are of four stations. TRIGONOMETRY. The angle a given, (Fig. 1) and hypothenuse given, to find the leg c 6. RULE 1. By natural sines. —As unity, or one, is to the length of the hypothenuse, so is the natural sine of the smallest angle to the length of the shortest leg. Fig 1 e EXAMPLE. —Gven the angle b a c 350 30', and hypothenuse 25 rods; to find c 6b. STATEMENT.- 1:25::0580703: 14'5175. To find the length of the leg a b. 38 ENGINEER'S POCKET COMPANION. RULE 2. The difference of the sums of the squares of the legs a c and c b, and extract the square root; will equal the leg a b. EXAMPLE.- Given the leg a c 25 rods, and leg c b 14'5175; to find the leg a b. STATEMENT.- /252 - 1451752a 20'35. To find the leg a b, (Fig. 1.) RULE 8. By nat. sines. —As unity, or one, is to the nat. sine of the angle a c b, so is the hypothenuse to the leg a b. EXAMPLE.- Given the hypothenuse 25 rods, and angle a c b 540 30'; to find the leg a b. STATEMENT.-1:0'8141155::25:20'35. The angles and leg a b given, (Fig. 1) to find the hypothenuse a c, and leg b c. RULE 4. By nat. sines.-As the nat. sine of the angle opposite the given leg a b is to the length of given leg, so is unity, or one, to the length of the hypothenuse. EXAMPLE.-Given the angles a c b 540 30', and b a c 350 30', and leg a b 20-9-IL rods; to find the leg a c. STATEMENT.- 08141155:20'35::1:25. Refer to Rule 2, page 38. To find leg c b by nat. sines. TRIGONOMETRY. 39 RULE 5. As the nat. sine of the angle a c b, opposite the given leg, is to the given leg, so is the nat. sine of the angle b a c, opposite the required leg, to the leg c b. EXAMPLE.- Given the angle a c b (Fig. 1) 54~ 30', and angle b a c 350 30', and leg a b 20T 35 rods; to find the leg b c. STATEMENT.-0'8141155:20'35::0'580703: 14'52, leg b c. The hypothenuse and one leg given; to find the angles and the other leg. RULE 6. By nat. sines.-The angle opposite the given leg may be found by the following proportion: As the hypothenuse is to unity, or one, so is the given leg to the nat. sine of its opposite angle. EXAMPLE. —Given the hypothenuse a c 25 rods, and leg a b 20'35 rods; to find the angles. STATEMENT.-25: 1::20'35:8141155, nat. sine of the angle a c b; the nearest corresponding number of degrees and minutes in the table of nat. sines gives the angle a c b 540 30', and the angle a b c being 90~, the angle b a c would be 350 30', because in a right-angle triangle there is always 180~. The leg a c given, (Fig. 1) and angle b a c, to find the other leg, a b, by cosine. RULE 7. Multiply the cosine of the angle b a c by the hypothenuse a c. 40 ENGINEER'S POCKET COMPANION. EXAMPLE. — Given the hypothenuse a e 25 rods, and angle b a c 350 30'; to find the leg a b. STATEMENT.-0'8141155 X 25 _ 20'35, length of the leg a b. The leg a b found, (Fig. 1) to find the leg b c by nat. tangent. RULE 8. Multiply the base by the nat. tangent of the angle opposite the required leg. EXAMPLE -- Given the leg a h 20'35, and angle b a c 350 30'; to find the leg b c. STrATEN'ENT. —0713293 X 20'35 - 14'5, the required leg 6 c. The angle a c b, and leg b c, given, (Fig. 1) to find the leg a b, by nat. tangents. RULE 9. Multiply the nat. tangent of the angle a c b by the leg b c. EXAMPLE.- Given the leg b c 14'5, and angle a c b 540 30'; to find the leo a b. STATEMENT.- -1401948 X 14'5 = 20'35, length of leg required, a b. Solutionz of a Right-angled Triangle. The sine of the angle c equals the cosine of the angle a, and the sine of the angle a equals the cosine of the angle c. The tangent of the angle a equals the cotangent SURVEYING. 41 of the angle c, and the tangent of the angle c equals the cotangent of the angle a. Thie leg a b divided by the leg a c equals the nat. sine of the angle a c b, or the nat. cosine of the angle b a c; the leg b c divided by the leg a 6 equals the nat. tangent of the angle b a c, or the nat. cotangent of the angle a c b; the leg b c divided by the leg a c equals the nat. sine of the angle b a c, or the nat. cosine of the angle a c b. SURVEYING. In running lines, obstructions, viz: rivers, ponds, &c., occur, by which other means have to be resorted to, besides measuring with a chain. Fig 2. r The pnoccsorn6 we- wish' The point c occurs on our line b c, and we wish' to know the distance b c. 42 ENGINEER'S POCKET COMPANION. RULE 10. From b at right angles to the line b c measure any convenient distance, as a, and secure the point a; measure the angle b a c; then multiply the nat tangent of the angle b a c by the distance a b; will equal the distance b c. Fig. 3. c /n case you should not have a book of tables of In case you should not have a book of tables of nat. tangents, the above method could be resorted to, with nearly as much accuracy as the method given in Fig. 2. The point c occurring in the line, (Fig. 3) we wish to know the distance b c. ILULE 11. At right angles from b c measure on the line b d to a, and secure the point a, any convenient point, and measure any convenient distance, as d, and at SURVEYING. 48 right angles describe the line d e, with your instrument at a, on the line a c, produce it to e, intersecting the line d e. Thien with the distance b a, a d, and d e, given, the distance b c can be found proportionately to the triangle a d e. Say: As the distance a d is to a b, so is d e to b c. EXAMPLE.- Given the distance. b a 20 ft., distance a d 15 ft., and distance d e 11~ ft.; to find the distance b c. STATEMENT. —AS 1 5: 20::11:15 3 ft. Fig. 4. a d The above (Fig. 4) could be resorted to in preference to Fig. 3. Given b, the inaccessible object, and d b part of the line of survey; we wish to find the distance from a to b. RULE 12. Measure on the line a c (at right angles with a b,) any convenient distance, as at c; then at right angles - 4 44 ENGINEER'S POCKET COMPANION. to b c run the line c d to its intersection with the line d b at d, measure the distance a d, and the distance a c. Then say: The square of a c divided by a d equals a b, the distance required. EXAMPLE.-Given a c 26 ft., and a d 13i ft.; required the distance a b. STATEMENT. —26'. 13 = 50 ft., distance a b. -:_r-'-~ \ Fig. 5. a We wish to find the height of a tower, or building, as d c. Set your instrument any convenient distance, e, neither too great nor too small, in comparison to the altitude d c, and measure the angle b a c, and measuse the distance a b or e d; you then have, in the right-angle triangle, one side given, and the angle b a c. IRULE 18. Multiply the nat. tangent of the angle b a c by the distance a b; will equal b c. SURVEYING. 45 NOTE.-The point at b can be observed, and afterwards th distance b d can be measured, which, added to b c, will determine the distance c d. In finding the height of an object, let the observed angle be as near 450 as possible; for then a small error committed in taking it, makes the least error in the computed height of the object; because, if the observed angle, as at a, equals 450, the distance a b will equal b c. It very seldom occurs, in the construction of a railroad, to measure verticle heights with an instrument. In running levels, if the top of a hill is found inaccessible to find the elevation with a leveling instrument, we have to resort to the examples given by triangulation. Fig. 6 When it is necessary to determine the elevation of a hill as in Fig. 6, the elevation at d, is found with the leveling instrument, as will be explained in the Art of Leveling. EXAMPLE.- Set your instrument over d, and measure the angle b a c, and distance a b; you then have for the triangle b a c, the angle and one leg given, 46 ENGINEER'S POCKET COMPANION. to find the other leg, b c, as in Fig. 5; which, added to the height of your instrument, a from d, will equal e c; and added to your elevation d, will equal the elevation c. THE ART OF LEVELING. The first thing necessary in leveling, is to have the requisite instruments in adjustment. To Adjust a Level. In the common Y level there are three adjustments. 1st ADJUSTMENT.-Place the instrument in a firm position, and unclamp the Y's; place the horizontal hairs on some distant object, and revolve the telescope half around; if the -hair intersects the point first observed, the instrument is in adjustment, thus far; if not, move the hairs half way distant, between the two points of intersection, by means of the screws on the telescope, generally marked "Hairs," and by revolving the telescope, the hairs will intersect our given point. The vertical hairs can be adjusted the same way. 2d ADJtUSTMENT. —With the instrument firm, as before.-Fasten the telescope over the leveling screws, and level it exact; then take the telescope out of the Y's and reverse it; if the bubble is level, this adjustnment is correct; -if not, divide the difference of the THE ART OF LEVELING. 47 bubble (one-half,) by means of the screws under the bubble, and level the remainder by means of the leveling screws. This process for the second adjustment hardly ever proves correct the first time; therefore, repeat the above, until the telescope, when revolved in the Y's, on every screw, the bubble will be level. 3d ADJUSTMENT. —After the above adjustments, fasten the telescope on the Y's, by means of pins generally used; place your telescope over the leveling screws, and bring the bubble to a level, and repeat it on all the screws, so as to get the telescope as level as possible before commencing the adjustment; then placing the telescope over any two of the leveling screws, and level the bubble; reverse the telescope half way on the pivot, or, as near as possible over the same screws; if the bubble is level, the adjustment is correct, if not, move the bubble half way, by means of the screws under the leg of the Y, and level the remainder by means of the leveling screws. By continuing this process on all the screws, the adjustment can be perfected. NOTE. —The last adjustment is immaterial, only in saving time and trouble when using. The difference (if there is any,) is so comparatively small, that it is not observable. The third adjustment never will remain in adjustment on most of levels, so that no trouble need be borrowed when it is found that your level will not reverse correctly. The adjustment of the level now being complete, we will proceed to its use. In preliminary surveys, or location of railroads, 48 ENGINEER'S POCKET COMPANION. levels have to be run (as it is termed) to ascertain the exact surface of the ground, in order to establish grades. In commencing levels, an elevation is established upon a given point. This point is generally made by cutting on the root of a tree and is termed a "Bench." These benches are established on the entire length of the line, perhaps one-half to threequarters of a mile apart, for reference points. This elevation is generally estimated above, so as to reach the lowest point of the surface of the ground that should occur in your levels. For instance: The lowest point of ground we guess to be 50 ft. below the first established bench, and ior safety would call the bench elevation 60. We set up our level firm in the ground, not to exceed 400 ft. from the bench, and near the line we wish to run the levels over, and take what is termed a back sight (marked B. S.) on the bench, by holding the staff, or leveling rod, on the bench, and moving the target of the rod to its intersection with the horizontal hairs in the telescope; what the rod would read at this intersection, would show that our instrument would be that number of feet and parts above the bench. For instance: Suppose the rod read 3y-AA- ft., therefore, the elevation of the instrument would be 63'416. When we have the height of the instrument given, it shows very plainly that if you take a sight at any given point, the elevation would be as much less as the rod would read. For instance: Suppose Lhe rod at any point, or station, (as stations of 100 ft. are used in the location of railroad lines,) should read 10~-79o8%, which are termed fore, or intermediate sight, THE ART OF LEVELING. 49 (marked F. S.) the elevation of that point taken would be 53'218; or, 63'416 — 10.198= 53'218; consequently, these fore sights should be subtracted from the height of the instrument, or elevation of the instrument, to give the elevation of stations.* In order to keep up the same corresponding elevations, on the entire length of the line, we change our instrument on some substantial point, as a peg or stone, by holding the rod upon the peg; being a fore sight, we subtract it from the height of the instrument, which gives the elevation of the peg, and is the same as a bench. Suppose the rod reads on the peo 8.747, and instrument is 63'416; 63'416 - 8'747 = 54'669, elevation of peg. t We have the elevation of the peg, and can move our instrument further on, and set up our instrument firm in the ground, as before, and take a "back sight" on the peg, by holding the rod upon the peg, and notice the reading as before. Suppose the rod to read 1.201; it shows that our instrument is I foot and -2,0' above the peg; consequently, if we add it to the elevation of the instrument, thus: peg= 54'669, rod reads 1.201 + 54'669 = 55'870, height of instrument, and proceed as before, taking intermediate sights, subtracting every intermediate from the * These fore sights are more properly termed intermediate sights (marked I. S.) which we will hereafter term them, and fore sights will be termed as at changes of the instrument, after running the level of intermediates, not to exceed 400 it. on either side of your instrument. t In practice, whether you change your instrument upon a peg, stone, stump, or anything suitable, it is termed a peg. 50 ENGINEER S POCKET COMPANION. height of the instrument. It is not necessary to carry out the reading of the rod of the decimals, (when taking intermediates,) farther than hundredths, as 2'27, 4'15, 8'11, &c., and in most cases tenths is far enough, as 2'2, 4'1, 8'1, &c., &c. Intermediates, or plus stations, should be taken when the ground varies to any amount; discretion on your own part must govern that. It is evident that if you were running levels over an uneven ground, as Fig. 1, and should take the elevation at station 1 and station 2, that you lose, or there would be a loss in the estimate of the quantities, or if for the purpose of establishing grades, a correct line could not be drawn, as a man's discretion is governed by the correctness of the profile, or levels taken, therefore, a level should be taken at a, also at b c d, and the plus station noted in the book of levels, then you have the correct shape of the ground. Fig. 1. To explain the foregoing more intelligibly, we will refer to Figs. 2, 3, and 4. Our established bench elevation 60, is at A, on the root of a stump; our line to run is in the direction of d, consequently, we would set our level at c, not to exceed 400 ft. from the bench A; in directing our level at the target f, we find it reads 3'416; then TIHE ART OF LEVELING. 51 our level would be that number of feet and parts above the point a; bench elevation is 60, elevation of instrument would be 60 + 3'416 = 63'416; we will take the elevation of the ground at the stations 1, 2, 3, 4, 5, 6, &c., reading Figo 2. the nearest tenth of' a foot, on the rod. Station 1 reads 3'5, therefore, 63-416- -R 3'5 = 590916, or 59'9 _ elevation of station 1. Station 2 reads 3'3,..... __.. then 63'4 - 33 - 60'1. Station 3 reads c.._..___ 3 6, then 6384 - 3'6 = 59'8. Station 4._.. _...... reads 3'7, then 634 - 3.7 = 59-7. Sta-. —---- tion 5 reads 3'6, then 6334 - 3.6 = 59.8. Station 6 reads 3'9,. then 63'4 - 3.9 = 59'5. Station 7 reads O,. _ 4'8, then 63'4 - 48 58'6. Station 8 reads 5'8, then 634 - -58 = 57'6. Sta- h f tion 9 reads 8.4, then 63'4 - 84=55.0. We will change our Instrument at station 9, for convenience. We drive in a peg, firm into the ground, at or near the station; the rod reads 8'747, 52 ENGINEER'S POCKET COMPANION. then 63'416- 8'747-= 54'669, elevation of the peg at b. This being a secured point, we move our instrument, as in Fig. 3, tof. After firmly setting our instrument, we take a back sight on the peg b. SupFig. 3. pose it to read 1'201, the instru{ ment would be that number of feet and parts above the peg. *__ Elevation of the peg we found to be 54.669; the elevation of o_ the instrument would be 54'669 -- + 1-201 = 55'870. We now have the elevation of the instrument again. For intermediates,, Station 10 reads 2-2, then 55-8 - 2-2 = 53-6. Station 11 reads _.. - 3'1, then 55'8 - 3-1 = 52'7. Here we should take an elevat —--- tion at a. Suppose it to be 50 ft. from station 11, as found by — e_ measurement; then the plus station would be noted 11 + 50.' Station 11 + 50 reads 3'4, then t 558- 3'4 = 52'4. Station 12 -":'- reads 3.1, then 558 - 3'152-7. Station 13 reads 2'6, then 55-8 - 2-6 = 53'2. Station 14 reads 2'1, then 55-8- -21- / 53-7. Station 15 reads 1'8, then 55'8 - 1'8 = 54'0. Station 16 reads 2.8, then 55'8 - 28 = 53'0. Station 17 reads 3-2, then 55-8 - 3'2 = 52'6. We will change our instrument by driving a peg THIE ART OF LEVELING. 53 at or near station 17, and finding its elevation. Suppose the F. S. on the peg reads 3'775, then height instrument 55'870 - 3'775 = 52'095 -- elevation of peg; we then move our instrument tof, (Fig. 4) and take a B. S. on the peg. Suppose it to read 2'000; elevation peg 52-095 + 2'000 = 54.095 height of instrument. Then proceed as before. Fig. 4.......I e18 2 17 18 19 20 21 22 23 24 25 Station 18 reads 3'0, then 54'1 - 30 = 51'1. Station 19 reads 2'9, then 54'1 - 2'9 51.2. Station 20 reads 2'7, then 54'1 - 27- 51'4. Station 21 reads 2'5, then 54'1 - 25 =51'6. Station 22 reads 3'6, then 541 - 36- 50'5. Station 23 reads 4'4, then 54'1 - 4-4 49-7. Here at station 23 we have come to a stream, and it is quite necessary to get its shape, width, and character. We have got the elevation of the top of the slope of the river at station 23; we take an elevation at c, at the bottom of the slope of the creek, and notice its plus distance from station 23. Suppose it to be 25 ft.; then 23 + 25 would be the bottom * When changing, the elevations should be exact. 54 ENGINEER'S POCKET COMPANION. slope of the stream. We also get the elevation of the bottom of the slope on the opposite side d, and notice the plus station. Suppose it to be 85 ft. from station 23, then it would be 23 + 85, bottom slope of stream; and station 24 would also be taken, as we very seldom miss stations, wherever they may occur We have the top slope 23, and two bottom slopes 23 + 25, and 23 + 85, and to complete the levels of the creek we want the top slope of opposite side. Say it is 75 ft. from station 24; then elevation at 24 + 75 gives the shape of the river's banks and bottom. Also to govern the masonry, or bridging, over this stream, the elevation of high water mark is taken. This will be readily found by inquiry, if no signs on the banks are visible. Other very important things have to be observed and placed under remarks, which your own discretion must lead you. We will establish a bench before proceeding farther. Say, for instance, we cut a notch on the root of the stump g. Suppose our rod, or F. S., reads 4'005; height of instrument 54'095 - 4'005 = 50'090, elevation of the bench. The elevation of the benches are generally marked on the stump with either red chalk or paint. We secure this elevation at g, and along the line at intervals, to avoid the trouble of going back to the first established bench, A, Fig. 2, on the line. For instance: Suppose it is required to find the elevation of station 25, or wished to stake out a bridge, or stone culvert, in the stream at some future day. We, instead of running from the first established bench, take the established -bench g, by THE ART OF LEVELING. 55 setting our instrument say at f, the most convenient point, and do the work required. We proceed with our levels as before, and establish benches, or elevations, not to exceed three-quarters of a mile apart. It might be necessary to state, for accuracy in running levels, that the rod should be plumb, or point to the centre of the earth; also the bubble of the level should always be level. Accuracy depends both upon the leveler and rodman. I will here give the manner of keeping a fieldbook for running levels. NOTE.-In running levels, always add the back sights, which will give the elevation of the instrument, and stbtract the fore or intermediate sighllts, will give the elevation of the peg or bench. MANNER OF KEEPING FIELD-BOOK. NOTES FROM FORMER EXAMPLES. Sta'n. B. S. F. S. 1.S. H. Inst. Elev' n. Remarks. B'ch 3416 63'416 60')000 On root of' stump near stat'n 1. 1 3'5 599 2 3'3 60'1 3 3 6 598 4 3.7 59.7 5 3'6 59'8 6 3'9 59.5 7 4'8 58-6 8 5'8 57'6 9 8'4 55.0 Peg. 8'747 54'669 Elevat'n of peg at B. (Fig. 2.) 10 1201 22 55'870 53'6 11 3'1 52'7 +50 3'4 52.4 56 ENGINEER'S POCKET COMPAIION. Stat'n. B.S. F. S. I.S. H. Inst. Elevat'n. Remarks. 12 31 63'416 52.7 13 2-6 53'2 14 2'1 53'7 15. 18 54,0 16 2'8 53'0 17 3 2 52'6 Peg. 3'775 52'095 18 2000 3 0 54'095 51'1 19 2'9 51-2 20 2.7 51-4 21 25 51'6 22 36 505 23 4-4 49'7 Top of bank of stream. 4-25 5 2 18'9 Bottom of stream. +85 5'3 48'8 Bot. of stream, opposite side. 24 5'0 49'1 +- 7,5 44 4497 Top slope of opposite side. 4'6 49'5 High water mark. B'ch 4-005 50'090 On stump by station 26. In leaving the work at night, benches should be made, so that when you choose at any time to go to work, you have a convenient place to commence, and accurate. After the levels are run the work is plotted and grades are established. Grades vary in their ascent according to circumstances, and are governed by the discretion of the engineer in charge. They intend however to equalize the excavation and embankment as near as possible. Grades sometimes can be improved, and are governed by the contracts taken to grade the road. Grades have elevations as well as the surface of the ground. We will assume the grade at Station THE ART OF LEVELING. 57 1 to be 58'000 and descent is 0O-a-q per 100 feet, that it equalizes the excavation and embankment. We then have elevation at Station 1 = 58'000 Station 13 = 55'360 " 2 = 57'780 " 14 - 55'140 " 3 = 57'560 " 15 = 54'920 4 -57'340 " 16 = 54'700 " 5 = 57'120 " 17 =- 54480 " 6 - 56'900 " 18 = 54-260 " 7 - 56'680 " 19 = 54'040 " 8 = 56460 " 20 = 53820 " 9 = 56'240 " 21 - 53'600 10 = 56'020 " 22 53380 " 11 = 55'800 " 23 - 53'160 " 12= 55-580 " 24- 52'940 With the elevation of the surface of ground, and the elevation of grade, their difference would be the cut or fill. For instance - Elevation of the surface Station 1 = 59'9 grade ". 1 = 580 Difference = 1'9 The difference shows the cutting 1'9 as the surface elevation is the greatest, and at Station 11 Surface elevation = 52'7, Grade elevation 55'8, difference 3'1, fill, as the surface elevation is the least. When the estimates in cubic yards have to be made, (as they are generally made approximately from the profile) the cutting and filling is easily ascertained by calculating the grade for every 100 feet, and taking the difference of elevation as just shown. When a line is located, the grades are then es 68 ENGINEER'S POCKET COMPANION. tablished, and construction commences. We then have different field books, termed Grade Book, Cross Section Book and Monthly Estimate Book. MANNER OF KEEPING GRADE BOOK. Sta'n. Grade. Cut Fill. Elvn Remaks. 1 58T000 1 99.9 2 57-780 2-3 60-1 3 57-560 2'2 59 8 4 57.340 2'4 59 7 5 57-120 2-7 59 8 6 56.900 2'6 59*5 7 56'680 1.9 5816 8 56'460 1'2 57-6 9 56-240 12 55.0 10 56-020 24 536 11 55-800 3 1 52-7 +50 55'690 3-3 52 4 12 55580 28 5287 13 55'360 2 1 53-2 14 55-140 1-4 53*7 15 54 920 0 9 540( 16 54 700 1O 7 53 0 17 54-1480 1 8 526 18 5 1260( 31 511 19 54040 28 51-2 2)0 53.82U 24 51-4 21 53'600 20( 51 6 2' 53'380 2-9 50 5 23 53'160 34 497 Tcpof bankof stream. +25 53'10. 45 2 4859 Bottom of stream. +i5 52'9,3 41 4838 Bottom of stream opposite side. 24 52940 3-8 49-1 +75 52 775 3 0 49 7 Top of slope opposite side. The benches are all entered in the back part of the grade book for reference when necessary. In THE ART OF LEVELING. 59 the grade book we have the cuts and fills worked out, but in the staking out of the work, these cuts and fills are merely used for tests. Engineers generally have in all their work test points for reference, which all should have, to avoid the many mistakes that will occur. In staking out work the grade point or where the excavation and embankment commences, is always found with its distance from the station joining, and a stake put in to guide the contractors in commencing their work. The grade points are generally marked in the cross section book.* Cross sections are cuts and fills taken at right angles to the line of the road, any distance from the center line that should seem necessary by the engineer. Cross sections Care taken with the level, but more plus stations are taken, than in running levels, as the discretion of the engineer may direct him. Cross sections are taken to get as near as possible the amount of cubic vards excavation and embankment, as contracts are taken of the work, to complete at a given price per cubic yard. On very level ground, cross sections are taken at every station. On very uneven ground cross sections are taken as often as your judgment dictates. Before proceeding further, we will explain the manner of getting the cuts and fills with a level in the field. *This cross section book is more properly termed Original Cross Section Book, as there also is the Final Cross Section Book. 5 60 ENGINEER'S POCKET COMPANION. With the grade book we have the elevation of grade. We go into the field, say at station 21, and set up our level, and take a B. S. on the bench g, (Fig. 4,) and find the elevation of our instrument. Suppose the elevation of the bench to be 50.090, and rod reads 5'112, then 50'090 + 5'112 = 55'202, elevation of instrument. Suppose we wish to commence at station 19, with our cross sections. We would refer to the grade book at station 19 and find the elevation of grade at that point = 54'040. We now have the elevation of our instrument and the elevation of the grade at station 19. Now if we subtract the elevation of the grade from the elevation of the instrument, the difference shows that our instrument is that number of feet and parts above the grade line at station 19. For instance: Elevation of instrument - 55'202 grade = 54'040 Difference of elevation - 1'162 Then our instrument at station 19 is 1.162 feet above the grade line. We will get the cut or fill at station 19. Suppose our rod to read, at station 19, 4'0, (the nearest tenth of a foot,) it would show our instrument to be four feet above the surface of the ground, and if our instrument is 1'162, (or nearest tenth,) 1'2 above the grade line, and 4'0 above the surface, we see that if we take their differences, it will give the fill or cut at station 19. Then —, Elevation of instrument above surface of ground = 4'0 grade = 1'2 Difference = 2-8 THE ART OF LEVELING. 61 2'8 is then the fill at station 19. We then taKe our cross section at station 19, subtracting the height of instrument above the grade line, 1'2, from the height of the instrument above the surface, will equal the cut or fill at station 19, any distance at right angles from the station or center line. WVe see that our instrument at station 20 (by reference to a profile) would not be 1'162 above the grade line,' but would be more as the grade descends. We have found that the grade descends O. —%L4 per 100 feet. Then if we add 0-22 to the instrument above the grade line at station 19, will equal its height above the grade line at station 20. Then elevation of instrument at station 19 = 1'162; descends 0'22 per 100 feet, or station, then 0'22 + 1'162 = 1-382; height above station 20 = 1'382. We will suppose the rod reads at station 20, 4'0, the same as station 19, then 4'0 - 1'4 (nearest tenth of 1'38) = 2'6 fill, or 2 feet and 6 tenth is the difference between the grade line and the surface of the ground at station 20. We can continue this process to all the stations, until it becomes necessary to change the instrument. When we change, the elevation of the instrument we have preserved = 55'202, and change on a peg, as in running levels, subtracting the sight on the peg for its elevation, and after the instrument is set up again, add the sight to the elevation of the peg, and you have the height of your instrument again, and proceed as you commenced. Sometimes we find our instrument below the grade line. We will suppose that the rod reads on the 62 ENGINEER'S POCKET COMPANION. bench g, (Fig. 4,) 2'950; this added to the bench would give the elevation of the instrument; bench = 50'090 + 2'950 — 53'040, height of instrument. Grade at station 19 = 54'040. Now we have the elevation of the instrument, less than the elevation of the gradeThus, elevation of instrument = 53'040 " grade at stat. 19 = 54'040 Difference of elevation = 1000 Shows that our instrument is 1 foot below the grade line. Now if our instrument is below grade, the sights on the surface must be added to the elevation of the instrument, to give the difference or distances of the grade line to the surface. Suppose the rod, or sight, at station 19, reads 1'8. We would see that the surface of ground was 1 foot and 8 tenths below the instrument, and the instrument is 1 foot below the grade line, therefore the distance from the grade line to the surface of the ground, would be their sum. Thus, Elevation of instrument below grade = 100 g 49" above surface - 1'8 Difference of surface and grade line = 2-8 2'8 would be the fill at station 19. When you come to grade, the elevation of the instrument above the grade line, and its elevation above the surface of the ground, will be equal. For example, suppose the instrument be 1'2 above the grade line, and surface of the ground 1'2, the difference is 00, and is the point of grade. If the elevation of the surface and elevation of grade are equal at any point, that part of the surface ITHE ART OF LEVELING. 63 of the ground is grade, or the point where the excavation and embankment commences. Thus, Elevation of grade =- 54'040 Elevation of surface of ground = 54'040 Difference = 0'00 In taking cross sections, the elevation of the instrument should be preserved for changing- and to be correct, you should keep a table of both your elevation of peg and instrument and height of instrument above the grade line at each and every station. To show more plainly, we will make a sketch of the work already done: Bench = 50'090 B. S. = 5'112 H. Inst. - 55'202 Change Inst., F. S.- 2'211 Peg = 52 991 B. S. = 0.049 H. Inst. = 53'040 Also,- Grade at stat. 19 = 54'040 H. Inst. - 55'202 H. Inst. at stat. 19, above grade line = 1'162 + Grade descends 0'22 per 100 feet = 0.22 H. Inst. at stat. 20, above grade line = 1'382 + 0'22 H. Inst. at stat. 21, above grade line = 1'602 + 0'22 H. Inst. at sta. 22, above grade line = 1'822 + &c., &c. Change instrument. 64 ENGINEER'S POCKET COMPANION. We have the height of instrument again, 53'040, and will commence at station 19 again to show the manner of keeping notes with the instrument below grade: Grade at stat. 19 = 54'040 H. Inst. = 53'040 H. Inst. at sta. 19 = 1'000cc " 920 = 0'780" " 21 - 0'60" 22 - 0'340The sign of plus and minus can be annexed to the above, to show that the instrument is above or below grade line. Thus, + = above, - below. In running from station 19 down the grade, we notice that we add the descent, per 100 feet, to the height of the instrument when plus, and when minus we subtract, because when the instrument is above, the further we run on the descent the greater the height of the instrument from the grade line, as much greater as the descent per 100 feet; and when the instrument is minus, we subtract the descent, because we near the grade line every 100 feet, as the descent of the grade per 100 feet. Sometimes, in staking out work, it is not necessary Fig. 5. Jr... T...... THE ART OF LEVELING. 65 to cross section further than where the slope stakes would occur, as in Fig. 5. The above shows a cross section, a b c, with the slopes a e f c, determined, and slope stakes a c put in. The slope of a railroad is governed by the material of which it is composed. The excavations are governed the same as the slopes of embankments; the width at the top is governed by the width of track or material. We will assume it to be 14 feet. In making a cross section of the above, we will assume slopes of 1~- feet horizontal to one foot vertical, as the required slopes. Suppose we take a level at b, (surface of the ground and center of the road,) and find the fill to be four feet. We mark the stake b with red chalk, "4 feet fill," (this is to guide the contractor,) and measure out from b, on either side towards a, one-half the width of the road bed, 7 feet, and take the cutting or filling. From this filling, we can judge of the point for a slope stake a, if the ground does not vary too much; if it should, we keep trying until the exact point for the slope stake is found. For example, we take the level at a, and find that there is 3'8-3- feet filling, and, by calculation, slopes 1 to 1, we measure from the center 7 feet, and base of slope 5a-70, making in all to measure from the center line, (as one end of the tape or chain should always be kept at the center stake to avoid confusion in changing from one side to the other,) would equal 12'-. We take another sight if it varies much from the point in which we took our sight, and make the same calculations again. It may vary 66 ENGINEER'S POCKET COMPANION. 2 or 3 tenths in the distance from the point last found, but if the ground is level, or nearly so, the required distance can be measured for the slope stake. You go through the same process on the opposite side for the slope stake c. We now see that if the contractor fills in 4 feet at b to d, and 7 feet from d to e, and 7 feet from d to f, and carries the dirt out to the slope stakes a and c, that the slope of the cross sections would have 1- feet horizontal to 1 foot vertical. Slopes more general in use are 1~ feet horizontal to 1 vertical; 2 feet horizontal to I vertical; 1 horizontal to 1 vertical. Calculations for slopes will be found in the following rules: Slopes 1i to 1. -With the filling or cutting given, to find the length of the base of the slope. RULE 1. One half the cut or fill added to the cut or fill where it is taken. EXAMPLE.-Given the filling 3'8, to find the base of the slope. STATEMENT. — of 3'8 = 1'9 + 3'8 - 5'7. Slope 2 to 1.-With the filling or cutting given as before, to find the base of the slopes. RULTE 2. Multiply the cutting or filling by 2. EXAMPLE.- Given the filling 3'8 to find the base of the slope = 3'8 X 2 = 7.6. Slope 1 to 1 CROSS SECTIONS. 67 RULE S. The base is equal to the filling or cutting. EXAMPLE. —Given the filling or cutting 3'8; then the base would equal 3-8. CROSS SECTIONS. The number of cross sections to be taken will be as the engineer's judgment governs him, but we will give a few examples necessary to correctness in side hill ground entering from a cut to a fill. The following figures 6, 7 and 8 will represent the cross sections. In leaving the cut (Fig. 6) a cross section should be taken where the grade point occurs at the bottom slope as at a, also a cross section should be taken (Fig. 7) where the grade point occurs at the center B; also where the grade point occurs on the opposite side c. (Fig. 8.) Cross sections taken in this manner, give the shape of the ground as near as can be got at. We also have them in plane figures. The area of Fig. 6 to correspond with the area B c d (Fig. 7) and Pyd a B e, also the Pyd B c d, the area a B e to correspond with the area a b c (Fig. 8). The quantities in this manner are got as correct as is possible. Cross sections around curves should not exceed 50 ft. apart when the average form is used for calculating quantities. Henck gives a formula for estimating the quantities in curves, but it has been adopted only by a very few engineers, if any. My mode is most in practice 68 ENGINEER'S POCKET COMPANION. by eminent engineers at present, but there is no doubt that the manner in which Henck and also Trautwine do their work will come into general practice, as the immense quantities of earth that is estimated by the average form would be very materially diminished. C'F! i,,,~ Fig. 6. ie Fi 7. / Fig. 8. BORROWING PITS. 69 BORROWING PITS Consist of borrowed earth from a hill or side of a mountain, when the earth in the cut is not sufficient for the embankments. It very often occurs where earth has to be borrowed to finish up embankments; sometimes it is taken from ditches; sometimes knolls are cut off, and sometimes it is taken out of the side of hills and mountains. This amount taken ~out has to be ascertained in cubic yards. The Manner of Measuring Borrowing Pits, to ascertain the Quantities taken out. The object in the first place, is to make original surveys or cross sections; and secondly to make cross sections, (after the earth is taken out) to correspond with the original cross sections. In order to do this, points must be established in the original survey, that can be referred to when you wish to make the second survey. Sometimes it has to be measured monthly, to make monthly estimates. In order to get correct measurements we will establish'a line, (called a Base Line) along the base of the hill* between two points, that will cover the length of the area. For security, the points we will establish upon the roots of some stumps, and in a direct line between these two points, put in stations and plus stations as often as is considered necessary. From these stations, at right angles to the established * This is governed altogether by the engineer's judgment. 70 ENGINEER'S POCKET COMPANION. line, measure with the tape or chain, and where it is necessary take sights with the level, and ascertain the elevation, or cut or fill above a certain given point. Continue on this measurement and elevations back as far as will cover the area to be excavated in that direction; continue the same process at every station and plus station. The notes will be entered in the orioinal cross section book. Tf it is necessary to make a monthly estimate, the base line is found, and stations and plus stations are put in as before, and measurements are commenced as before. This is entered into the monthly cross section book. When the work is completed, a final measurement has to be made, by finding the base line, and proceed as in making the original survey. These elevations are taken with the level. A base line of levels is established for the pit, and when sights are taken, the notes show the elevation of the ground, either above or below the established base line of levels. When the last survey is made, where the earth has been taken out, the difference of elevation at their corresponding points would show the depth of earth taken at these points. When those measurements are made, they are plotted upon paper, which will show the area of the cross section. For example, suppose a b (Fig. 9) to be the base line of levels. The original levels commence at the base line of stations, 1 foot 5 tenths above the base line a b, and at the top of the hill, or the extent of excavation, the BORROWING PITS.'1t elevation is 13 feet 4 tenths above the base line. When the second measurement is made, we find at the station to be the same as the original, 1 fboo 5 Fig. 9.' 1-8+ 3'0-tc.20+ ---— 2 - + --- --------------- 672+ 2 1- i ------ \8 —3+ 3C3+ -- \-2+3 -7+ ------- 98+ o 4-1+ —-105 0 42- --- 11-4+ }_ ___ _ —----------— 13 4+ tenths, and find the last point to be 4 feet above the base line. The excavation we find at the top to be z72 ENGINEER'S POCKET COMPANION. 110 feet from the point at c, and at the bottom, e, or second survey, 100; this will make the slope d e, and complete the area c d e. The depth of excavation can be ascertained by the difference of the elevations. For instance, the depth of cuttino 80 feet from the station c equals the difference of the elevations 4'1 and 10'5 = 6'4; the cutting then at 80 feet distance is 6 feet 4 tenths. The remainder of the area is ascertained in the same manner. In the notes the sign of + is annexed to the sights when above the base line of levels, and - when below. If in two corresponding sights or elevations, one should be below the base line of levels and the other above, the depth of excavations would equal their sum. If in two corresponding sights or elevations, both should be below the base line of levels, their difference would equal the depth of excavation; if both should be above, their difference would equal the depth of excavations. NOTE.- All horizontal measurements, with a tape or chain, either in measuring for cross sections or land, should be measured perpendicular to the center of the earth. The area of the cross sections (Fig. 9) is easily determined, as the original elevations and final elevations correspond in their distance fiom the station c. It is not always that cross sections can be taken with equal distances (10, 20, 30 feet, &c.) from the station, as the ground may be very irregular both in the original and final measurement. The area of the section is ascertained by calculating the area of the BORROWING PITS. 73 original measurement to the base line of levels, and the area of the final measurement to the base line of levels; the difference of' these areas would be the area of the section. Fig. 10. 2' f +A....... e......... ~. —- 2'24!'+3+. —--- 737+ 2.-.... ~\ 1 1'8q77+ 2.6+ - a —R X ____m- ___ ________ ____.12'3+ For example, to ascertain the area of section a b c, (Fig. 10.) '74 ENGINEER'S POCKET COMPANION. Instead of the distances 10, 20, 30, &c., from the station at c, we have been obliged (on account of the unevenness of ground) to measure the distances 10, 25, 42, 65, 81, 100 and 120 from the station c, and in the final the distances 15, 50, 60, 77, 100 and 110. In this case the depth of excavation cannot be ascertained, but the area is easily found. We wish to get the area of the section a b c. By computing the area b c d e, and area b e d c a, the difference of their areas would equal the area a b c. The manner of keeping original cross section book and final cross section book, is similar. For example, we will arrange Fig. 9 and Fig. 10, supposing that Fig. 9 - stations 1, and Fig. 10 = stations 1 + 20. ORIGINAL CROSS SECTIONS. Stations 1..5+ 3-0+ 5-0+ 6.0+ 7.2+ 83 e- 9.2+ 9.8+ 10o5+ 114+ 1224- 13-4+ 10 20 30 40 50 60 70 80 90 100 110 Station 1 + 20. - 2 2-6+ 3.4+ 2-9 - 6-8+ 7-7 - 11-8+ 12 3+ 81 10 25 42 65 81 100 120 FINAL CROSS SECTIONS. 13 18+oJ 18+ 21~ Station 1. 1-'5+ 1'8- 1'8+ 21+ 2-2+ 2-5+3 33 3-7+ 4-1+ 42X- 4'0+ 10 20 30 40 50 60 70 80 90 100 Station 1 + 20. 2+ 13+ 2-0+ 10- 13 26+ 26 2-6f 15 50 60 77 100 110 When the above is plotted, it forms the figures 9 and 10. The number of cubic yards contained between MENSURATION OF SURFACES. 75 these two figures (according to the present mode of calculation) is equal to one-half the sum of their areas, multiplied by the distance they are apart, (20 feet,) and the quotient divided by 27. MENSURATION OF SURFACES. To find the area of a right-angled triangle. RULE 1. Multiply one-half the base by the perpendicular height, equal the area. To find the area of a triangle. RULE 2. Multiply the base by the perpendicular let fall on Lo it from the opposite angle, and one-half the product equals the area. To find the area of a triangle by its sides. RULE 8. From half the sum of the three sides subtract each side separately; then multiply the half sum and the three remainders continually together, and the square root of the product will equal the area. To find the area of a rectangle or a square. 6 76 ENGINEER'S POCKET COMPANION. RULE 4. Multiply the perpendicular height by the length, equal the area To find the area of a rhombus or a rhomboid. RULE 5. Multiply the length by the perpendicular distance let fall from its sides, equals the area. To find the area of a trapezoid. RULE 6. Multiply one-half the sum of the parallel sides by the perpendicular, equals area. To find the area of a trapezium. RULE 7. Multiply the diagonal by the sum of the two perpendiculars falling upon it from the opposite angles, and half the product equals area. To find the area of a regular polygon. RULE 8. Multiply one of its sides into half its perpendicular distance from the center, and this product into the number of sides, equals its area. To find the area of an irregular polygon. RULE 9. Draw diagonals to divide the figure into trapeziums and triangles; find the area of each separately, and the sum of the whole equals area. MENSURArION OF SURFACES. 77 To find the area of a long irregular figure, bounded on one side by a straight line, (Figs. 9 and 10 on borrowing pits.) RULE 10. Multiply one-half the sum of each succeeding height by their distance apart, the product will be the area between the two heights, the sum of all the areas will equal the area of the figure. To find the area of a circle when the diameter and circumference are both known. IULE 11. Multiply the square of the diameter by'7854, or the square of the circumference by'07958; or multiply the circumference by the diameter, and divide the product by 4, will equal area; or one-half the circumference by one-half the diameter. To find the area of a sector of a circle. RULE 12. Multiply the length of the arch by the radius of the circle, and half the product will equal the area, (nearly.) To find the area of a segment of a circle. RULE 18. Multiply the versed sine by the decimal'626; to the square of the product add the square of half the chord; multiply twice the square root of the sum by two-thirds of the versed sine, will equal area. To find the area of an ellipsis. 78 ENGINEER'S POCKET COMPANION. RULE 14. Multiply the transverse or longer diameter by the conjugate or shorter diameter, and by'7854,* will equal area. To find the area of a circular ring or space included between two concentric circles. RULE 15. Add the inside and outside diameters together, multiply the sum by their differences, and by'7854, will equal area. To find the area of a parabola or its segment. RULE 16. Multiply the base by the perpendicular height, and two-thirds of the product equals area. MENSURATION OF SOLIDS. To find the solid contents of a cylinder. RULE 1. Multiply the area of the base by the height of the cylinder, and the product is the solid contents. To find the solid contents of a cone or pyramid. * The area of a circle whose diameter is 1 =- 0'7854. MENSURATION OF SOLIDS. 79 RULE 2. Multiply the area of the base by the perpendicular height, and one-third of the product will equal the solid contents. To find the solid contents of a frustum of a cone. RULE 8. To the product of the diameters of the two ends add the sum of their squares; multiply this sum by the perpendicular height, and by -2618;* the product equals contents. To find the solid contents of a frustum of a pyramid. RULE 4. To the sum of the areas of the two ends add the square root of their product; multiply this sum by the perpendicular height, and one-third of the product equals the contents. To find the solidity of a wedge. RULE 5. To the length of the wedge add twice the length of the base; multiply that sum by the height, and by the breadth of the base, and one-sixth of the product equals contents. To find the solid contents of a prism. RULE 6. Multiply the area of the base by the length, equals contents. * The solidity of a cone 1 foot diameter and 1 foot high equals'2618. '80 ENGINEER S POCKET COMPANION. To find the solid contents of a sphere or globe. RULE 7. Multiply the cube of the diameter by'5236; the product equals contents. To find the solid contents of the segment of a sphere. RULE 8. Add the square root of the height to three times the square of the radius of the base; multiply that sum by the height, and by'5236; the product is the contents. To find the solidity of a spheroid. RULE 9. Multiply the square of the least diameter by the length of the greatest diameter, or a line drawn perpendicular to the least diameter, and by'5236; the product will be the solidity. To find the solidity of a segment of a spheroid, when the base is circular or parallel to the revolving axis or least diameter. RULE 10. From triple the fixed axis take double the height of the segment; multiply the difference by the square of the height, and by'5236. Then say, as the square of the fixed axis is to the square of the revolving axis, so is the former product to the solidity. To find the solid contents of a cylindric ring. MENSURATION OF SOLIDS. 81 RULE 11. To the thickness of the ring add the inner diameter; multiply that sum by the square of the thickness, and by 2'4674; the product will be the solid contents. To find the superficial contents of a board or plank. RULE 12. Multiply the length by the width. If the plank or board are of an unequal breadth at the ends. RULE 13. Multiply the average width of the ends by the length. To find the solidity of timber. RULE 14. Multiply the length in feet by the square of onefourth the girth in inches, gives the solidity in cubic feet. NOTE.-The above rules No. 12 and 13 only apply when all the dimensions are in feet. When either the length or breadth are given in inches, divide by 12, when all the dimensions are given in inches, divide by 144. Application to the table of flat or board measure. Multiply the length by the number in the table corresponding to any given width. EXAMPLE. -Given a board 161 feet in length and 9- inches in breadth. The number in the table opposite 91 inches= ~8125 X 16I - 13'4 square feet. 82 ENGINEER'S POCKET COMPANION. TABLE TO FACILITATE THE MENSURATION OF TIMBER, FLAT OR BROAD MEASURE. Breadth Area of a Breadth Area of a Breadth Area of a in Lineal i Lineal in Lineal inches. foot. inches. foot. inches. foot. i 0208'4 *3750 84 7292 i *0417 41.3958 9 *7500 0625 5 4167 9j 7708 1'0634 5j 4375 9' *7917 it'1042 5 [4'4583 94.8125 1~ *1250 4[ I4792 10 *8334 1j'1459 6'5000 10 *8542 2'1667 6j 5208 10'8750 2: /1875 6j.5416 10 *-8959 2]'2084 6 65625 11 I 9167 2]'2292 7' 5833 1li 9375 3.2500 7j.6042 I11.9583 3j'2708 7b *6250 11 *'9792 3j *2916 7 *|6458 3i] 3125 8' 6667 4 *3334 8j *6875 41 *3542 8| -7084 _ Application of the table of the solidity of timber. Multiply the area corresponding to the quarter girth in inches by the length in feet. EXAMPLE.-Given a piece of timber 20 feet long and 12 inches square. The numberopposite 12 inches = 1'000 X 20 = 20 cubic feet. MENSURATION OF SOLIDS. 83 TABLE TO FACILITATE THE MENSURATION OF TIHE SOLIDITY OF TIMBER. 1 qr gi rea in 1 qr. girth Area in I qr. girth Area in in incles. feat. ill inches. fee;. in inches. feet. 6. 250 12. 1.042 19 2.506 6-' 272 12 1 5 085 19 2'640 61' 294 12! 1'129 20 2'777 63' 317 13 1 174 204 2917 7 340 134 1'219 21 3-062 74' 364 132 1265 211 3209 71' 390 13! 1313 22 3362 73'417 14 1'361 22 3-516 8 ~ 444 144 1410 23 3673 8~ 472 142 1'460 234 3 835 8 501 144 1 511 24 4000 84 531 15 1-562 24k 4 168 9 562 154 1 615 25 4-340 9I 5~14 154 1'668 251 4516 99 *626 15! 1'722 26 4 694 9_ *659 16 1'777 26k 4~876 10' 694 16 1'833 27 5-062 104' 730 16i 1'890 271 5 252 10 *'766 16 1'948 28 5 444 104 803 17 2'006 284 5-640 11' 840 17i 2'066 29 5-840 114' 878 171 2'126 29k 6,044 11i'918 17! 2'187 30 6-250 11-'959 18 2-250 12 1 000 18k 2'376 Scantling is measured the same as timber, by multiplying the end area by the length. 84 ENGINEER'S POCKET COMPANION. MISCELLANIES. The dimensions of the United States standard bushel are 18~ inches inside diameter, and 8 in. deep. A box 24 inches by 16 inches square, and 28 inches deep, will contain a barrel, 5 bushels. A box 14 inches by 17 inches square, and 14 inches deep, will contain a half barrel. A box 26 inches by 15'2 inches square, and 8 inches deep, will contain 1 bushel. A box 12 inches by 11'2 inches square, and 8 inches deep, will contain one-half bushel. A box 8 inches by 8'4 inches square, and 8 inches deep, will contain 1 peck. A box 8 inches by 8 inches square, and 4'2 inches deep, will contain 1 gallon. A box 7 inches by 8 inches square, and 4'8 inches deep, will contain one-half gallon. A box 4 inches by 4 inches square, and 4'1 inches deep, will contain I quart. To get the number of bushels in any square crib. RULE 1. Find the number of cubic feet in the same, and multiply it by 8 and divide it by 10. Any area in feet multiplied by 6'232, the product is the number of imperial gallons at one foot in depth; or any area in inches multiplied by 0'4328 = gallons. MISCELLANIES. 85 Any area multiplied by'03704 = the number of cubic yards at one foot in depth. To determine the amount of imperial gallons in a vessel, the shape of an inverted cone. RULE 2. The square of the sum of the diameter at the top and bottom, of which subtract the quotient of the top and bottom; multiply the remainder by'7854, and by one-third the depth= cubic feet; and by 6'232 = imperial gallons. NOTE.- This rule applies where the dimensions are all given in feet. 2d. To the product of the inner diameters add the squares of the inner diameters; multiply the remainder by the depth, and by'2618; divide that by 277'274 = gallons, (nearly.) NOTE.- This rule applies where the dimensions are given in inches. To determine the contents of imperial gallons in a kettle forming the segment of a circle. RULE 8. Three times the square of half the diameter in inches at the mouth, added to the square of the depth, and multiplied by the depth, and by'5236; divide their product by 277'274, equals imperial gallons. The area of a circle in inches, multiplied by the length or thickness in inches, and by'263, equals the weight of cast iron in pounds. 86 ENGINEER'S POCKET COMPANION. The old English ale gallon contains 282 cubic inches, and the United States gallon contains 231. English Dry Measure. 8'665 cubic inches = 1 gill. 34'659' " = 1 pint. 69'318 " " -- 1 quart. 277'274 " " = 1 gallon. 554'548 " " = 1 peck. 1X2837 " feet = 1 bushel. English Imperial Wine'Ifeasure. 1'604 cubic feet = 1 anker. 2-888 " " = I runlet. 6'739 " " I tierce. 10'109 " " = I hogshead. 3478 " " -= I puncheon. 20'218 " "= 1 pipe. 40'435 " " - 1 tun. Dimensions of Drawing Paper. Wove Antiquarian, 4 feet 4 inches by 2 feet 7 inches. Double Elephant,. 3 " 4 " 2 " 2 2" Atlas, -- 2 " 9 " 2 " 2 Columbier, 2 " 91 " 1 "11'" Elephant, 2 " 3- " 1 "10 " Imperial, - 2 " 5 " 1 " 9 " Super Royal, - 2 " 3 " 1 " 7 " Royal, -- 2 " 0 " 1 " 7 " Medium, - - I " 10 " 1 " 6 " Demy, 1 " 7- " 1 " 3j- I" MISCELLANIES. 87 2fanner of Calculating the Natural Sines and Cosines in the Table. The radius of a circle being 1, it is known the semi-circumference will equal 3'141592653589 8, &c. Therefore if we divide it by the number of minutes in a semi-circumference (10800) it will equal the sine of 1 minute ='0002909 the first seven places in the table. The natural cosine equals V(1 —sine') = 9999999577. The natural sine and cosine given, the statement would be as follows:. 2 cosine 1' X sine 1' - 0' = sine 2' minute. 2 " 1 X " 2'1 -' " 3/'; 2 " 1' X < 3/ - 2' " 4/ " 2 " 1' X " 4/ - 3/- " 51 " This process can be run to any extent. With the sine of one minute, and cosine of one minute given, statement second would be as follows: 1': sine 2'- sine 1-':: sine 2' + sine 1': sine 3' 2/ " 3/'-,4 1': 31 + " 1/: " 4/ 3/ " 4' " 1' 41 +, 1': " 5' 4I' " 5' — " 1':.. 51 + " 1' " 61 The same statement can be applied for degrees: Thus: sine 1~: sine 20 - sine 1~:: sine 2~ + sine 1: sine 30, &c., &c. 88 ENGINEER S POCKET COMPANION. The natural sine of any number of degrees of de flexion with chords of 100 ft. may be found by dividing the chord by the radius corresponding to the angle of deflexion in the table of radii. EXAMPLE.-Given the deflexion angle 30, radius corresponding in the table of radii would equal 1910 feet and chord 100 feet. STATEMENT. —-100. 1910 - 0'052356, natural sine of 3~, NATURAL SINES AND TANGENTS TO A RADIUS 1. NATURAL SINES AND TANGENTS TO A RADIUS i.' 0 DGa. 0 DEG. O i SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0 -0000000 o.000 Infinite 1 000 60 31 0090174 *009017 110-8920 9999593 29 1.0002909.000291 3437'74(; 1 59 32.0093083 009308 107-4264.9999567 28 2 -0005818 -000582 1718-873 9999.8 58 33 0095992 009599 104-1709 99999539 27 3 0008727 000872 1145.915 9999996 57 34.009800 009890 101'1069'9999511 26 4'0011636 )01163 859'4363'9999993 56 35 0101809 010181 98-21794'9999,182 25 5 ~0014544 001454 687-5488'9999989 55 36' 0104718.010472 95.48947 9999452 24 6 0017453 *001745 572'9572'99985 54 37 0107(27 010763 92'90848 9999421 23 7 0320362 -002036 491'1060 9999979 53 38 0110535.011054'30-46333' 9999389 22 1! 8 0023271 (002327 429-7175'9999973 52 39 *0113444'011345 88-14357'9999357 21 9 (00261S0 -002618 381-9709 9999'66 51 10.0116353.011636 85'93979'9999323 20 10 00029089 002908 343'7737 -9999958 50 41 -0119261'011927 838,i350 -9999289 19 11'0031998'003199 312-5213'99.9949 49 42 -0122170'012217 81-84704 -9999254 18 12.0034907'003490 286'-4777'9999939 48 43.0125079'012508 79'94343.9939218 17 9d 13'0037815'008781 264.4408'99'9928 47 44'01279837'012799 78-12439'9999181 16 14 0040724'0040)72 245-5519 -9999917 46 45'0130896'013090 76'39000 -9999143 15 15 0 33 043633 00433 22931816 -9999005 45 46.0133835'013381 74'72916 -999105 14 16'0046542 004654 214'8576'9999892 44 47'0136713'013672 73-138939'993905 13 M 17.0049451 9004945 202-2187.9999878 43 48'0139622.013963 71-61507'9999025 12!Z 18'0052360'f0523 190'9841'999803 42 49'01425:30'014254 70-15334'9998984 11 Q 19'0055268 005526 1809322 9999847 41 50'0145439 014545 68'75008'9938942 10 20'0058177'006817 171'8854'9999831 40 51'0148348'014836 67-40185'9998900 9 21'0061086'006(108 10G37001'9 9813 39 52'0151256'015127 66'10547'9998&r6 8 W 22'0063995.00' 99 156.2590'9999795 38 53'0154165'015418 64-8580)0 9998812 7 o 23'0066904'006690 1494.650'9999776 37 54'0157073'015709 63-65674'9998766 6 r 24'0069813'006981 143 2371.9999756 36 55'0159982.016000 62-49915.9998720 5 25'0072721 2007272 137'5075'9999736 35 56'0162890.016291 61'38290'99f 8673 4 26'0075630'007563 132-2185'9999714 34 57'0165799'016582 60)30582'9998625 3 27'0078539'007854 127'3213'9999692 33 58'0168707'016873 59'2;6587'9998577 2 28'0081448'008145 122'7739'9999668 32 59'0171616'017164 58'26117'9998527 1 29'00&81357'008436 llS540)1'9999644 31 60'0174524'017455 57-28996'9998477 0 30.0087265'008726 114-5886 -9999619 30 I COSINE. COTANG. TANG. SINE. / c COSNI. COTANG. ITA'NG. SI INE. I DsG. 89. DoG. 89. NATURAL SINES AND TANGENTS TO A RADIUS 1.'1 DFG. 1 DEG. SINE. T.ANG. COTANG. COSINE. I / SINE. TANG. COTAN(T. COSINE. I 0 *0174524'017455 57 28996 *9998477 60 31'02fi4677 026477 37-76861'99964197 29 1 *0177432 *017746 56-33059 9.998426 59 32 *0267585 02876i8 37.35789'99S)419 28 2 10180:21.018037 55-44151.9998374 58 33.02704933 027059:395C 00 99'3341 27 3.0183249.018328 54-56130.9998321 57 34 -0273401 *027350 33-56265'996,22 26 4 1)18618.018610 53-7)858 *9998267 56 35 *027639 027641 36-17759 999182 25 5 *0189066.01,910 52.-,211.9998213 55 36 *0279216.027932 35fi00Z 999101 24 6 *0191974 -019201 52.08067 *9998157 54 37.0282124 0282~23 33-4312~3 *9998)020 23 7 -0194883.019492 51-30315 9998101 53 38.0285032'028514 35()i6954. 99,'5937 22 8'0197791'019783 50'54850'9998044 52 39'0287940 *.028805 34'71511 999554 21 9 *0200699 *020074 49-81.572'9997986 51 40'0290847 -029097 34-36777'9995770 20 Q 10'0203(608'02036(5 49-10388'9997927 50 41 *0293755 *029388 34'02730'9995684 19 11 *0206516'0206-56 4841208 899978677 49 42'0296662 -029679 3-.'93650'9995599 18 M 12 [0209424'020947 47'73950'99977807 48 43.0299570 *029970 33-.36619'9995.512 17 13.0212332'021238 47-08534'9997745 47 44'0302478 *0.30261 33-04517'9.)95424 16 14 *0215241'021529 46-44886'9997683 46 45 0305385 030552 32-73026.9995336 15 15'0218149'021820 45'82935 -9997620 45 46.0308293.030843 32-42129 9995247 14 16'0221057'022111 45-22614.9997556 44 47'0311200 *031135 32'11809'.995157 13 17'022396.5'022402 446.385g9 9997492 43 48.0314108 -031426 31'82051'9995()6 12 18'0226873'022693 44-0(8611.9997426 42 49'0317015 *031717 31-52839'9994974 11 19'0229781'022984 43350812'9997360 41 50'0319922 *0320.08 31-24157'9994881 10 H 20'0232690'023275 42-96407 *9997292 40 51 *0322830 *032299 3095992 9994788 9 O 21 *0235598'023566 42-43346'9997224 39 52'0325737'032591 30'68330'9994693 8 22'0238506'023857 41-91579'9997156 38 53'0328644 -032882 30'41158'9994598 7 23'0241414'024148 41-41058'9997086 37 54'0331552'033173 30'14461 9994502 6 24'0244322'024439 40-91741 9997015 36 55'0334459 *033464 29'88229'9994405 5 25 *0247230'024730 40-43583'9996943 35 56 0337366 *033755 29'62449'9994308 4 26 *0250138 -025021 39-96546.9996871 34 57'0340)274.034047 29-37110 -9994209 3 27 *0253046'025312 39-50589 *9996798 33 58'0343181'034338 29-12200'9994110 2 28'0255954 -025603 39-05677'9996724 32 59'0346088 034629 28-87708 -9994009 1 29'0258862 *025894.38-61773 9996649 31 6()'0348995'034020 28-63625'9993908 0 30 )0261769'026185 38-18845.9996573 30 I COSINE. COTAN I. TANG. SINE. I I COSINE. COTANG. TANG. SINE. I DEa RR DxGo. 88. - NATURAL SINES AND TANGENTS TO A RADIUS 1. co 2 DEG. 2 DEG. SINE. TANG. COTANG. COSINE. / I SINE. TANG. COTANG. COSINE. 0 *0348995 *034920 28-63625 *9993908 60 31 0439100 043952 22-75189 *9990355 29 1 *0351902 035212 28-39939 *9993806 59 32 *0442006 *044243 22-60201 *9990227 28 2 [0354809 *035503 28-16642 9993704 58 33'0444912'0445,35 22-45409 ]9990098 27 3 *0357716 *035794 27'93723'9993600 57 34'0447818 *044826 22-30809 19989968 26 4 *0360623 *036085 27-71174'9993495 56 35'0450724 *045118 22-16398 *9989837 25 5 *0363530 {036377 27 48985'999,3390 55 36'0453630'045409 22-02171 *9989706 24 3 6 *0366437 -036668 27-27148'9993284 54 37.0456536 *045701 21-88125 *9989573 23 m 7'0369344'036959 27-05655'9993177 53 38'0459442 [045992 2174256 ]9989440 22 8 *0372251 -037250 26-84498 *9993069 52 39'0462347'046284 21-60563 *9989306 21 9 *0375158 *037542 26-63669.9992960 51 40'0465253'046575 21-47040'9989171 20 10 *0378065 *037833 26-43160'9992851 50 41'0468159'046867 21-33685 *9989035 19 H 11 *0380971 *038124 26-22963'9992740 49 42'0471065'047158 21-20494 *9988899 18 r 12 *0383878 ]038416 26-03073'9992629 48 43'0473970'047450 21'07466'9988761 17: 13 *0386785 *038707 25'83482'9992517 47 44'0476876'047741 20-94596 998862 16 O 14'0389692 *038998 25-64183'9992404 46 45'0479781'048033 20-81882 9988484 15 15 *0392598 *039290 25-45170'9992290 45 46'0482687'048325 20-69322 *9988344 14 16'0395505 *039581 25-26436'9992176 44 47 *0485592'048616 20-56911 *9988203 13 M 17 *0398411 *039872 25-07975'9992060 43 48'0488498'048908 20-44648'9988061 12 Z 18 *0401318 *040164 24-89782'9991944 42 49'0491403'049199 20-32530 [9987919 11 Q 19'0404224 *040455 24-71851'9991827 41 50'0494308'049491 20-20555'9987775 10 o 20 -0407131 04()746 24-54175 -9991709 40 51'0497214'049782 20'08719 -9987631 9 M 21 *0410037'041038 24-36750'9991590 39 52'0500119'050074 19'97021 *9987486 8 t9 22'0412944 *041329 24-19571'9991470 38 53'0503024'050366 19'85459.9987340 7 7 23'0415850 *041621 24.02632'9991350 37 54'0505929'050657 19'74029 *9987194 6 _ 24'0418757 -041912 23'85927 9991228 36 55'0508835'050949 19'62729'9987046 5 25'0421663 -042203 2.3-69453'9991106 35 56 -0511740'051241 19-51558 *9986898 4 26'0424569'042495 23-53205'9990983 34 57'0514645'051532 19-40513'9986748 3 27 *0427475'042786 23-37177'9990859 33 58 -0517550'051824 19-29592 -9986598 2 28 *0430382 -043078 23-21366'9990734 32 59 -0520455 -052116 19-18793 *9986447 1 29 *0433288 -043369 23-05767'9990609 31 60'0523360 052407 19-08113 -9986295 0 30'0436194 -043660 22-90376'9990482 30 I COSINE. OOTAN. I TANG. SINE. I I COSINE. COTANG. TANG. SINE. I DIe. 87. DEG. 87. NATURAL SINES AND TANGENTS TO A4 IADIUS 1. 3 DEG. 3 DEG. SINVE. TANG. COTANG. COSINE. I SINE. TANG. COTAN';G. COSINE. S 0 *0523360 *052407 19-08113 9986295 60 31 0613389 *O64154 16-27217 *99S1170 29 1 10526264 052809 18-97552 9986143 59 32 0616292 061]746 16i19522 98S0991 28 2 0529169 *052991 18 87106( -99859S9 58 33 0618190 *062038 1]6(11889 *99S0811 27 3 80532074 *053282 1876775 *9985835 57 34 0622099 -062330 16 04348'9980631 026 4 05341) 9 053574 18 66556 *9985680 30i 35 -0625002 -062622 15-96866 -9980450 25 5 0537883 0.5 8G6 18 56447 -99895524 55 36 0627905 062914 15898854 9980207 24 6 0548805 -054158 18 40447 -988.5367 54 37 *0030808 *063206 15 82110 99800S4 123 7 05436493 054449 18-36553 *998524)9 53 38 0633711 063496 15374833 9979900 *22 > 8 *0546597 *054741 18 26705 998<5050 52 39 00636614 0603790 15367623 9979716 21 O 9 0549502 055033 18- 17080 *9984891 51 40 90639517 -064082 15360478 9979530 20 ( 10 018552406 9055325 18907497 9984731 9(1 41 0642420 064375 15-53398 9979343 19 W 11 0o555311 055616 17 98013 9984570 49 42 -0645323 *064667 15-40381 9979156 18 M 12 ()0v5,8215 *055908 17-888.31 *9984406 46 43 0648226 0064959 15-39427 *9978966 17 3 13 0561119 056200 17 79344 *9984245 47 44 0651129 *065251 15-32535 9978779 16 8 14 05N 1642 05i642 17 70152 *99S4081 46 45 *0654031 5065543 15-25705 *9978589 15 3 15 *0569028 056784 17-01055 *9983917 45 46 *0656934 *065835 15-198934 -9978399 14 16 059832 057075 17352031 998.3751 44 47 0659386 066127 15-12224 9975207 13 17 0572736 ((57367 17-43138 -9983585 43 48 -0662739 *066419 15-05572 9978015 12 P 18 0573640 *057659 17-34315 9983418 42 49 *0665641 066712 3498976 -9977821 11 19 *0(578,514 -057951 17-25580 9983250 41 5() 066&5-14 067004 14 9-141 *9977627 10 w 20 0.831440 -058243 17-16933 998.3062 40 51 *0671446 5067296 14 85961 9977433 9 O 21 0-3849352 0585335 17-08372 *9982912 39 52 90674349 06758S 14-79537 *9977237. 8 22 0587216 -058827 16-99895 *9982742 38 53 -0677251 067860 14-73167 -9977040 7 23 0359160 -059119 16-91502 99825870 37 54 -0680153 *068173 14-66852 9976843 6 24 -0593064 -059110 16-83191 *9982398 36 55 *0683055 -068465 14-60591'9976645 5 25 -0595967 *059702 16-74961 9982225 31 56 -068.5957 -068757 14-54383 9976445 4 26 *0598971 059994 16-66811 *9982052 34 57 *0688859 *069049 14-48227 *9976245 3 27 60601775 9060286 16-58739 *9981877 33 58 *0691761 6069342 14-42123 -9976045 2 28 -0604678 060578 16-5(1745 9981701 32 59 0694663 -069634 14-36069 -9975843 1 29 -0607582 -060870 16-42827 -9981525 31 60 -0697565 -069926 14-30066 -9975641 0 30 -0610485 -661162 16-34985 -9981348 30 _ _ OIE ___O.T _ _ _, / COSINE! COTANG. - TANG. SINE. / / COSINE. COTANG. TANG. SINE.|/ DEo. 86. DEG. 86. CO NATURAL SINES AND TANGENTS TO A RADIUS. 1. 4 DEG. 4 DEG. I SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I Q 0697565 069926 14-30066'9975641 60 31.0787491.078994 12-65912 *9968945 29 1'0700467'070219 14-24113'9975437 59 32 *0790391 *079287 12-61239'9968715 28 2'0702368 -070511 14-18209'9975233 58 33 *0793290 *079579 ] 256599'9968485 27 3.0706270 *070803 14-12353'9975028 57 34 *0796190.079872 12-51994'9968254 26 4'0709171 -071096 14-06545'9974822 56 35 -0799090'080165 12-47422'9968022 25 5 -0712073 *071388 14-00785'9974615 55 36 -0801989'080458 12-42883'9967789 24 s6'0714974'071680 13'95071'9974408 54 37 -0804889'080750 12-38376'9967555 23 z 7'0717876'071973 13-89404'9974199 53 38 -0807788.081043 12233902'9967321 22 M 8'0720777 -072265 13-83782'9973990 52 39'0810687'081336 12-29460'9967085 21 9'0723678'072558 13-78206'9973780 51 40 -0813587'081629 12-25050 9966849 20 10'0726580 -072850 13-72673 -9973569 50 41 *0816486 *081922 12-20671'9966612 19 11.0729451'073143 13-67185'9973357 49 42'0819.385 082215 12-16323'9966374 18 12'0732382 -073425 13-61740'9973145 48 43'0822284 -082507 12-12006'9966135 17 d 13 -0735283'073727 13-56339 -9972931 47 44 0825183'082800 12-07719'9965895 16 O 14 0738184 -074020 13-50979'9972717 46 45'0828082 -083093 12-03462'996(5655 15 15'0741085'074312 13-45662'9972502 45 46'0830981 -083386 11-99234'9965414 14 16 *0743986 *074605 13'40386'9972286 44 47'0833880 -083679 11-95037'396i5172 13 M 17 0746887'074897 13-35151 *9972069 43 48'0836778'083972 11-90868'9964929 12 Z 18.0749787 *075190 13'29957'9971851 42 49'039677'084265 11-86728'9965665 11 Q 19.0752688'075482 13'24803'9971633 41 50'0842576.084558 11-82616.9964440 10 20'0755,589'075775 13-19688'9971413 40 51 -0845474'084851 11-78533 99'64195 9 21 -0758489'076068 13314612'9971193 39 52'0845373 -085144 11-74477'9963948 8 M 22 -0761390'076360 13.09575'9970972 38 53'0851271'085437 11-70450'9963701 7 23'0764290'076653 13304576'9970750 37 54 -0854169'085730 11-66449'9963453 6' 24'0767190'076945 1.2'99616'9970528 36 55'0857067'086023 11-62476'9963204 5 25'0770091'0772.38 12'94692'9970304 35 56'0859966'086316 11-58529'9962954 4 26'0772991'077531 12-89805'9970080 34 57'0862864'086609 11-54609 -9962704 3 27'0775891'077823 12'84955' 9969854 33 58 *0865762'086902 11'50715'9962452 2 28'0778791 -078116 12'80141 -9969628 32 59'0868660'087195 11'46847'9962200 1 29'0781691'078409 12'75363'9969401 31 60'0871557'087488 11-43005'9961947 0 30'0784591 _078701 12-70620'9969173 30 I COSINEI. COTANG. TA. SINE. I I COSINE. COTANG. TANG. SINE. I DEG. 85. DEG. 85. NATURAL SINES AND TANGENTS TO A RADIUS 1. 5 DEG. 5 Dec. I COSINE. TANNG. COTANG. COSINE. I SINE. TANG. OT. COSINE. I 0'-0871557'08748S8 11-13005'9919417 60 31 09613'53'096i58S2 10-35382' 99.53683 29 1'0874455'087781 11'39188'9961693 59 32'0964248'096876 10-32244'9953403 28 2'0877353'088074 11'35397'99614138 58 33 0967144'097169 10-2912.'9953122 27 3'0880251'08368 11-31630'996i183 57 34 0970039 0974(63 10-26024'9952840 26 4'0883148'088661 11-27888'9960926 50 35 09729'34 097757 10-22942'9952557 25 5'0886046'088954 11-24171'9960669 55 36'0975'29' 098050 10-19878'9952274 24 6'0888943'089247 11'20478 9960411 54.37'078724'098344 10-16833'9951990 2.3 7'0891840'089540 11-16808'9960152 53:38 098161 i 9'098638 10-13805'9951705 22 8'0894738'089&34 11'13163'9959892 52 39'0963514'098932 10-10795'9951419 21 O 9'0807635'090127 11-09541'9959631 51 40'0.987408'095225 10-07803'9951132 20 6 10'0900532'090420 11'05943'9959:70 50 41'03900303'099519 10-04828'9950844 19 11'0903429'090713 11-02367'9959107 49 42'0993197'099813 10-01871'99501556 18 M 12 -0906326'091007 1098815 9958844 48 43 *0996;092 100107 9-989105'99502 17 13'0909223'091300 10-95285'9955880 47 4'4 0998986'100400 9'960072'9949976 16 14'0912119'091593 10.91777'9958315 46 45'1001881 -100694 9-931008'9949685 15 15'0915016'091887 10.88292 9958049 45 46'1004775.100988 9-902112!)949393.S 14 16'0917913'092180 10-84828'9957783 44 47'1007669'101282 9'873382'91.49101 13 17'0920809'092473 10-81387'9957515 43 48'1010563'101576 9-844816'1948907 12 18'0923706'092767 10-77967'9957247 42 49'1013457'101870 9-816414'19948513 11 Z 19 *0926602'093060 1.0-74568'9956978 41 50'1016:1'102164 9-788173'9948217 10 20'0929499'093354 10-71191'9956708 40 51'1019245'102458 9'760092'9947921 9 O 21'09132395'093647 10-67834'9956437 39 52'1022138'102752 9-732171'9947625 8 22'0935291'093940 10-64499'9956165 38 53'1025032'103046 9'704407 9947327' 7 23'0938187'094234 10-61184'9955892 37 54'1027925'10.4339 9'676800'9947028 6 24'0941083'094527 10'57889'9955620 36 55'10(30819'103634 9-649347'9946729 5 25'0943979'094821 10-54615'9955345 35 56'1053712'103928 9'622048'9946428 4 26'(0946875'095114 10-51360'9955070 34 57'1036605'104222 9-594902'9946127 3 27'0949771'095408 10-48126'9954794 33 58'1039499'104516 9.567906'9945825 2 28'0952666'095701 10-44911'9954517 32 59'1042392'104810 9'541061'9945523 1 29'0955562'095995 10-41715'9954210 31 60'104.5285'105104 9-514364'9945219 0 30'0958458'096289 10-38539'9953962 30 COSINE. COTAN. TANG. SINE. I COSNE. COTANG. TANG. I SINE. I DEG. 84. DEG. 84. C. NATURAL SINES AND TANGENTS TO A RADIUS 1. 6 DEG. 6 DEo. I - I SINE. TANG. COTANG. COSINE. I I SINE. T.NG. COTANG. COSINE. I 0 -1045285'105104 9-514364 -9945219 60 31'1134922 -114230 8-754246 -9935389 29 1 *1048178'105398 9'487814'9944914 59 32'1137812 *114525 8'731719'9935058 28 2'10,51070'10.5692 9'461411'9944609 58 33'1140702 -114819 8'709307 9934727 27 3'1053963'105986 9'435153'9944303 57 34'1143592 *115114 8-687008 -9934395 26 4'10568,56'106280 9-409038 -9943996 56 35'1146482'115409 8'664822'9934062 25 5 -1059748'106675 9-383066'9943688 55 36 -1149372'115703 8'642747'9933728 24 3 6'1062641'106869 9'357235 -994,3379 54 37 -1152261 *115998 8620783 -99.33393 23 W 7'1065533'107163 9-331545'9943070 53 38 -1155151'116293 8-598929'9933057 22 8'106S425 *107457 9'305993'9942760 52 39'1158040 -116588 8'577183'9932721 21 9'1071318 -107751 9'280580 89942448 51 40'1160929'116883 8-555546 -9932384 20 10 I074210'108046 92553(03'9942136 50 41 -1163818 -117178 8-534017'9932045 19 11'1077102 10S340 9-230162'9941823 49 42'1166707 -117473 8-512594'9931706 18 12'1079094 -108634 9'205156'9941510 48 43'1169596 -117767 8-491277 -9931367 17 13 -1082885 -108929 9'180283'9941195 47 44'1172485 -118062 8-470065' 9931026 16 O 14 -1085777 -109223 9-1,55543'9940880 46 45 -1175374 -118357 8-448957 -9930685 15 tl 15 -1088669 -109517 9-130934 -9940563 45 46 -1178263 -118652 8-427953 -9930,342 14 16 -1091560 -109812 9-106456 -9940246 44 47 -1181151 -118947 8-407051 -9929999 13 M 17 1094452 -110106 9-082107 -9939928 43 48 -1184040 -119242 8-386251 -99296i5 12 18 -1097343 -110401 90057886 9939610 42 49 -11S3928 -119537 8-365553 *9929310 11 Q 19 -1190234 -110695 9-033793 -9939290 41 50 -1189816 -119832 8-344955 -9928965 10 20 1103126 -110989 9-009826 -9938969 40 51 -1192704 -120127 8-324457 -9928618 9 M 21 -11()6017 -111284 8-9&5984 9938ti48 39 52 -1195593 -120423 8-304058 -9928271 8 1 22 -1108908 -111578 8-962266 -993S326 38 53 -1198481 -120718 8-283757 -9927922 7 ~ 23 -1111779 -111873 8-9.3672 -9938803 37 54 -1201368 -121013 8-263554 -9927573 6 24 -1114680 -112168 8-915200 -9'3937679 36 55 -1204256 -121308 8-243448 -9927224 5 25 -1117-580 -112462 8-891850 -9337.355 35 56 -1207144 -121603 8-223438 -9926873 4 26 -1120471 -112757 8-868620 -9937029 34 57 -1210031 -121898 8-203523 -9926521 3 27 -1123361 -113051 8-845510 -9936703 33 58 -1212919 -122194 8-183704 -9926169 2 28 -1126252 -113346 8-822518 -9936375 32 59 -1215806 -122489 8-I63978 -9925816 1 29 -1129142 -11.3641 8-799(44 -9)36047 31 60 -1218693 -122784 8-144346 -9925462 0 30 -1132032 -113935 8-77i887 -9935719 30 I COSINE. COTA.NG. TANG. SINE. / COSINE. COTANG. TANG. SINE. I DEG. 83. DEG. 83 NATURAL SINES AND TANGENTS TO A RADIUS 1. 7 DEG. 7 DEG. SINE. TANG. COTANG. COSINE. I SINE. TANG. COTANG. COSINE. I 0 1218693 122784 8-144346'9925462 60 31'1308146'131948 7-578717 -9914069 29 1 1221581'123079 8-124807.9925107 59 32 -1311030 *132244 7-5,1756 *9911,88 28 2 1224468 12'3775 8-105359 09924751 58 33'1313913 *132540 7-5448.9 I991,3306 27 3 1227355 3123670 8-086004.9924394 57 34 -1316797 -132836 7-528057 *9912923 26 4 1230241 -12.3965 8-066739.9924037 56 35 -1319681 *133132 7-511317 9912540 25 5 1233128 *124261 8-047564 *9923679 55 36 [ 1322564'133428 7-494651 *9912155 24 6 1236015 *124556 8-028479.9923319 54 37'1325447 -133724 7'478057 9911770 23 7 -1238901 124852 8-009483 *9922959 53 38 -1328330 *134020 7-461535 *9911384 22 8 1241788'125147 7-990575'9922599 52 39'1331213 -134.316 7-445085 9910997 21 O 9 1244674'125442 7-971755 *9922237 51 40 *1334096 134612 7-428706'9910610 20 ] 10 1247560 3125738 7-953(022 *9921874 50 41 31336979 134909 7'412397 -9910221 19 [ 11 1250446 *126033 7-934375 *9921511 49 42'1339862 135205 7-396159 -9909832 18 ] 12 1253332 *126329 7-915815 *9921147 48 43 *1342744 *135501 7-379990 9909442 17 [ 13 1256218 -126624 7-897339 *9920782 47 44'1:45627 *135797 7-363891 *9909051 16 14 1259104'126920 7-878948 *9920416 46 45.-13480(9 *136094 7-347861 /9908659 15 15 1261990 127216 7-860642 *9920049 45 46 1351392'136390 7-331898 *9908266 14 [ 16 -1264875'127511 7'842419 [9919682 44 47 -1354274 -136686 7-316004'9907873 13 17 1267761 *127807 7-824279 *9919314 43 48 -1357156 *136983 7'300(178'9907478 12 18 1270646'128103 7-806221 *9918944 42 49'1360038 *137279 7-284418'9907083 11 [ 19 1273531'128.398 7-788245 -9918574 41 50 1362919 *137575 7-268725 009906687 10 20 *1276416 3128694 7-770,350'9918204 40 51 31365801 3137872 7-253098 09906290 9 0 21 1279302'128990 7-752536 *9917832 39 52 31368683 *138168 7'237537'9905893 8 | 22 1282186'129285 7-734802 *9917459 38 53 -1371564'138465 7'222042 *9905494 7 23 1285071'129581 7'717148 *9917086 37 54 -1374445 *138761 7'206611 9005095 6 24 1287956 129877 7-699573 *9916712 36 55 *1377327 139058 7-191245 9904694 5 25 1290841'130173 7-682076 9916337 35 56 *1380208 *139354 7'175943 9904293 4 26 1293725 31.30469 7'664658 *9915961 34 57 1383089 *139651 7-160705 *9903891 3 27 1296609'130764 7-647317 *9915584 33 58 1385970 *139947 7145530 *9903489 2 28 1299494'131060 7-630053 -9915206 32 59 1388850 -140244 7-130419 9903083 1 29 *1302378 *131 356 7-612865 9914828 31 60 *1391731 *140540 7-115369 *9902681 0 30 1305262 131652 7-595754 *9914449 30 I COSINE. COT.ANG. ITANG. SINE. I/ /I COSINE. COTANG. TANG. SINE. DEG. 82. DEG. 82. NATURAL SINES AND TANGENTS TO A RADIUS 1. CS 8 DEC. 8 DEG. CO I SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0.1391731 140540 7115369 *9902681 60 31 *1480971.149748 6.677867 9889728 29 1 *1394,612.140837 7-100382 *9902275 59 32.1483,848 *150045 6-664630 *9889297 28 2 *1397492 *141134 7-08547.9901869 58 33 *1486724 *150343 6-651444.9888895 27 3 *1460372 *1414,30 7-070593 *990162 57 34 *1489601 *150640 6-638310 9888432 26 4 *1403252 *141727 7-055790.9901055 56 35 *1492477 *150938 6-625225.9887998 25 5'1406132 *142024 7-041048'9900646 55.36 1405353 *151235 6-612191 *9887564 24 6'1409012'142321 7-026366 )900237 54 37'1498230 *151533 6-599208'9887128 23 7 *1411892'142617 7-011744'9899826 53 38'1501106 *151830 6-586273'9886692 22 8'1414772'142914 6-997180'9899415 52 39'1503981 *152128 6-573389 *9886255 21 9'1417651'143211 6-982678'9899003 51 40 31506857 *152426 6-560553'9885817 20 10 *I420531'143508 6-968233'9898590 50 41'1509733 5152723 6-547767'9885378 19 9 11'1423410'143805 6-953847'9898177 49 42 -1512608 *153021 6'535029'9884939 18 r 12'1426289'144102 6-939519'9897762 48 43'1515484 *153319 6-522339'9884498 17 8 13'1429168'144.399 6-925248'9897347 47 44 31518.359 *153617 6-509698'9884057 16 14 -1432047'144696 6-911035'9896931 46 45'1521234 *153914 6-497104'988,3615 15 15'1434926'144993 6-896879'9896514 45 46'1524109 1]54212 6-484.558'9883172 14 16 -14,37805'145290 6-882780 -9896096.44 47'1526984 [154510 6-472059 9882728 13 M 17'1440)84'145587 6-868737'9895677 43 48'1529858 154808 6-459607'9882284 12 t 18'144:1562 -145884 6-854750 -9895258 42 49'1532733 155106 6-447201'988183 11 Q 19'1446440'146181 6 40819'9894838 41 50 *15.35607 *155404 6-434842 *98813'2 10 20 1449319'146478 6.8$6943'9894416 40 51'1538482 *155701 6-422530 988F0945 9 M 21'1452197 *146775 6 813122'9893)94 39 52'1541356 155999 6-410263'9880497 8 Ed 22 1455075 *147072 b'799356' 9893572 38 53 *1544230'156297 65398042'9880048 7 d 23 1457953'147369 6-785644'9893148 37 54'1547104 *156595 6-385866'9879599 6 24 1460830 147667 6-791986 -9892723 36 55'1549978 *156893 6-373735 -9879148 5 25 1463708 *147964 6-758382 *9892298 35 56.1552851'157191 6-361 650 *9878697 4 26.1466585.148261 6 744831'9891872 34 57'1555725 *157490 6'349609 -9878245 3 27 *1469463.148,559 6.731334.9891445 33 58 *1558598 *157788 6-337612'9877792 2 28 *1472340'148856 6'717889.9891017 32 59.1561472'158086 6-325660'9877338 1 29 *1475217 *149153 6-704496.9890588 31 60 *1564345 *158384 6-313751 998768i 3 0 30'1478094 *149451 6[691156 9890159 30 I COSINE. COTANG. I TANG. SINE. I COSINE. I COTANG. TANG. T SINE. I DEG. 81. DEG. 81. NATURAL SINES AND TANGENTS TO A RADIUS 1. 9 DEG. 9 DEG. I SINE. TANG. COTANGI. COSINE. I SI INE. TANG. COTANG. COSINE. I 0 -1564345 -158384 6-31,3751 -9876883 60 31 -1653345'167641 5-965104'9862375 29 1'1567218'158682 6-301886 -9876428 59 32'1656214'167940 5-954481'9861894 28 2'1570091'158980 6-290065.9875972 58 33'1659082'168239 5-94.3895 *9861412 27 3'1572963'159279 6-278286.9875514 57 34 -1661951'168539 5-933345. 9860929 26 4'15758.36 159577 6-266551.9875057 56 35'1664819 *168838 5-922832'9860445 25 5'1578708'159875 6-254858'9874598 55 36'1667687 -169137 5-912.355'9859960 24 6 -1581581 -160174 6-243208 -9874138 54 37 -1670556'169436 5-901913 *9859475 23 7'15,84453 -160472 6-231600'9873678 53 38'1673423'169735 5-891508'98589,Q8 22 8'1587325 -160770 6'220034'9873216 52 39 -1676291'170035 5-881138 -985&501 21 O 9'1590197'161069 6-208510 *9872754 51 40'1679159 -170334 5-870804'9858013 20 Q 10'1593069'161367 6-197027 *9872291 50 41 -1682026'1706i.33 5-860505 -9857524 19 W 11 -1595940'161666 6-18 5586'9871827 49 42'1684894 -170933 5-850241 9,8570,35 18 m 12 -1598812'161964 6-174186 -9871363 48 43'1687761'171232 5-840011'9856544 17'3 13'1601683 -162263 6-162827'9870897 47 44'1690628'171532 5-829817'9856053 16 14 -1604555 -162561 6-151508'9870431 46 45'1693495 -171831 5-819657'9855561 15 Q 15'1607426 162860 6-140230 98f69964 45 46'1696362'172130 5-809531 -98,55068 14 % 16 -1610297 -163159 6-128992 99869496 44 47'1699228 -172430 5-799440.9854574 13 17'1613167'163457 6-117794'9869027 43 48'1702095'172730 5-789382 -9854079 12 18 1616fi038 -163756 6-106636 *9868557 42 49'1704961'173029 5-779.358'985.583 11 Z 19'1618909'164055 6-095517'986i8087 41 50'1707828'173329 5-769368 -9853087 10 - 20 -1621779'164353 6-084438'9867615 40 51'1710694 -173628 5-759412 -9852590 9 O 21'1624650'164652 6-073397'97867143 39 52'1713560'173928 5-749488 -9852092 8 22 -1627520'164951 6-062396'9866670 38 53'1716425'174228 5-739598'9,851593 7 23 -1630390 *165250 6-051434'9866196 37 54'1719291 -174527 5-729741 -9851093 6 24'1633260'165548 6-040510 *9865722 36 55 -1722156 -174827 5-719917'9850593 5 25'1636129'165847 6-029624 -9865246 35 56 -1725022 -175127 5-710125'9850091 4 26 l1638999'166146 6-018777'98i64770 34 57 -1727887'175427 5-700366'9849.589 3 27 f1641868 -166445 6-007967 -9864293 33 58 -1730752'175727 5-690639 -98490086 2 28 -1644738 -166744 5-997195 -9863815 32 59 -1733617 -176027 5-680944 -984858r)2 1 29'1647607 -167043'5-986461 -9863336 31 60 -1736482 -176327 5-671281 -9848078 0 30 -1650476'167342 5-975764'9862856 30 i COSINE. COTANG. TANG. SINE. I I COSINE. COTANG. TANG. I SINE. I DEG. 80. DEG. 80. C3 NATURAL SINES AND TANGENTS TO A RADIUS 1. 10 DEG. 10 DEG. O SINE. TANG. COTANG. COSINE. / I SINE. TANG. COTANG. COSINE. 0 *1736482 *176327 5-671281 *9848078 60 31 *1825215 *185639 65-386771 9832019 29 1 *1739346 /176626 15661650 *9847572 59 32 182S075 *185940 5-o378053 o9831487 28 2 ]1742211 *176926 5-652051'9847066 58 33'1830935 *186241 5-369363 98&30955 27 3'1745075 *177226 5-642483' 9846558 57 34 *1833795 *186542 5-360699 9&830422 26 4'1747939 *177527 56632947'9846050 56 35'1836654 *186843 5-352062 98S29888 25 5'1750803 *177827 5'623442'9845542 55 36'1839514 *187144 5-343452 *9829353 24 [ 6 *1753667 *178127 5'613968'9845032 54 37 *1842373 *187446 5-334869'9828818 23 ] 7 *175i531 *178427 5-604524 *9844521 53 38 *1845232 *187747 5-326313 *9828282 22 M 8 *1759395 /178727 53595112 99844010 52 39 18&48091'188048 5-317783 98S27744 21 9 -1762258.179027 5-585730.9843498 51 40 *1850949 -188349 5-309279'9827206 20 10 I765121'179327 5.576378'9842985 50 41'1853808 188650 5-300801 *9826668 19 ] 11 -1767984.179628 5-567057.9842471 49 42'1856666.188952 5-292350'9826128 18 | 12'1770847 *179928 5-557766'9841956 48 43 *1859524 *189253 5-283925'9825587 17 ] 13 *1773710 *180228 5-548505'9841441 47 44'1862382 *189554 5-275525 -9825046 16 O 14 *17765735 180529 5-539274'9840924 46 45 *1865240 *189855 5-267151'9824504 15 15 *1779435 *180829 5-530072'9840407 45 46 *1868098 *190157 5-258803'9823961 14 16 *1782298 *181129 5-520900'9839889 44 47'1870956 -190458 5-250480'9823417 13 8 17 -1785160.181430 5-511757'9839370 43 48 *1873813 -190760 5-242183'9822873 12 t 18 *1788022 /181730 5-502644'9838850 42 49'1876670 *191061 5-233911 -9822327 11 1 19 *1790884' 1820.31 5-49560 98330 41 50 1879528 *191363 5-225664'9821781 10 ] 20 -179.3746 182331 5-484505'9837808 40 51'1882385'191664 5-217442 *9821234 9 M 21 *1796607 *182632 5-475478 *9837286 39 52 *1885241'191966 5-209245 -9820686 8 M 22'1799469 [182933 5-466481'9836763 38 53'1888098 *192268 5-201073 *9820137 7 d 23.18023530 *183233 5'457512 *9836239 37 54'1890954 *192569 5-192926'9819587 6 m 24 *1805191 *1835234 5-448571'9835715 36 55 |1893811 192871 5'184803 *9819037 5 25 *1808052 *183835 5'439659'9835189 35 56'1896667'193173 5-176705 *9818485 4 26 *1810913 188135 5'430775 *9834663 34 57 [1899523 *193474 5-168631'9817933 3 27 *1813774 *184436 5'421918'9834136 33 58 1902379'193776 5'160581'9817380 2 28 *1816635.184737 5-413)90'9833608 32 59'196X5234'194078 5-152555 *9816826 1 29.1819495 *185038 5'404290 983.3079 31 60 *1908090 7194.380 5-144554 *9816272 0 30 *1822355 *185339 52395517 98&32549 30 -I COSINE. COT ANG. TANG. SINE. COSINE. COTANG.I TANG. f SINE. / DEG. 79. DEG. 79. NATURPAL SINES AND TANGENTS TO A RADIUS 1. 11 DEG. 11 DEG. I SINE. TANG. COTANG. COSINE. I, SINE. TANG. COTANG. COSINE. / 0'1908090'194380 5-144554'9816272 60 31'19965.30 *203755 4 907849 *9798667 29 1 *1910945 *194682 5-13(6576 *9815716 59 32'1999., 204058 4-900)562 *9798086 28 2 *1913801 *194984 5-128622 *9815160 58 33 *2002230 *204361 4-893295 *9797504 27 3 *1916656 *195286 5-120692 *9814603 57 34 *2006080 *204004 4-886049 *9796921 26 4 *1919510'195588 5-112785 *9814045 56 31' 2007930 *204967 4-878824'9796337 25 5'1922365'195890 5-104902 *9813486 55 36'2010779 *205270 4'-71620 *9795752 24 6'1925220'196192 5-097042 -9812927 54 37'2013629 *205573 4-864435'9795167 23 7 1928074 196494 5-089206 *9812366 53 38 -2016478 -20)5876 4'857271'9794581 22 8'1930928 -196796 5-081392.9811805 52 39 2019327.206180 4-850128'9793994 21 O 9 -1933782'197098 5-073602 *9811243 51 40'2022176'206483 4-843004'9793406 20 0 10'1930636'197400 5-065835 *9810680 50 41 -2025024'206786 4-835901'9792818 19 8 11'1939490 -197703 5-058090 *9810116i 49 42'2027873 207090 4-828817'9792228 18 M 12'1942344'198005 5-050369 *9809552 48 43.2030721'207393 4-821753'9791638 17 e 13'1945197'198307 5-042670'9808986 47 44'2033569 -207696 4-814709'9791047 16 14'1948050'198610 5-034993.9808420 46 45 2036418 20800 4-07685'9790455 15 O 15 *1950903'198912 5-027339 *9807853 45 46.2039265.208303 4-800680'9789862 14 16 -1953756'199214 5-019707 -9807285 44 47 -2042113'208607 4-793695'9789268 13 17'1956609'199517 5.012098'9806716 43 48'2044961'208910 4-786730'9788674 12 18'1959461'199819 5-004511'9806147 42 49 -2047808 *209214 4-779783'9788079 11 19'1962314'200122 4-996945'9805576 41 50'2050655'209518 4'772856'9787483 10 20'1965166'200424 4-989402'9805005 40 51'2053502 209821 4-765949'9786886 9 21'1968018'200727 4-981881'9804433 39 52'2056349'210125 4-759060'9786288 8 22'1970870'201030 4-974381'9803860 38 53'2059195'210429 4-752190'9785689 7 23'1973722'201,332 4-966903'980.3286 37 54'2062042'2107,33 4-745340'9785090 6 24'1976573'201635 4-959447'9802712 36 55'2064888'211036 4-738508'9784490 5 25'1979425'201938 4-952012 -9802136 35 56'2067734'211,340 4-731695'9783889 4 26'198227G6 202240 4-944599'9801560 34 57 -2070580'211644 4-724901'9783287 3 27'1985127'202543 4-937206'9800983 | 58'2073426'211948 4-718125'9782684 2 28'19,87978'202846 4-929835'9800405 32 -59'2076272 *212252 4-711368'9782080 1 29'1990829'203149 4-922485'971827 31 60'2079117'212556 4-70630'9781476 0 30'1993679'203452 4-915157'9799247 30 I COSINE. COTANG. TANG. SINE. I I COSINE. COTANG. TANG. SINE. I IDEG. 78. DEG. 78.. NATURAL SINES AND TANGENTS TO A RADIUS 1. 12 DEG. 12 DEG. i SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0'2079117'212566 4'7046:30 9781476 60 31 -2167236'221999 4'504507'9762330 29 1.2081962.212S60 4'697910 3780871 59 32'2170076'222305 4'498322'9761699 28 2 -2084807.213164 4-691208 9780265 58 33'2172915.222610 4-492153.9761067 27 3.2087652.213468 41684524.9779658 57 34'2175754 -222915 4'486000'9760435 26 4 20)90497.213773 416778,59.9779050 56 35'2178593'223221 4'479863'9759802 25 5'2093341.214077 4'671212 9778441 55 36'2181432 223526 4-473742'9759168 24 3 6'2096186.214381 4'664583'9777832 54 37'2184271 -223831 4-467637'97585.33 23 4 7'2099030.214685 4'657972'9777222 53:38'2187110'224137 4'461548'9757897 22 9 8'2101874 -214990 46ti51378 -9776611 52 39'2189948 -224442 4-455475'9757260 21 9'2104718.215294 4-644803'9775999 51 40 -2192786'224748 4-449418'9756623 20 10'2107561.215598 4'638245 -9775386 50 41'21956(24'225054 4-443376'9755985 19 11.21104(05.215903 4-631705'9774773 49 42'2198462'225359 4-437350'975.5345 18 12'2113248.216207 4'6251853'9774159 48 43 -2201300'225665 4-431339'9754706 17 # 13 -2116091.216512 4-618678.9773544 47 44'2204137 225971 4-425343 9754065 16 O 14'2118934.216816 4'612190'9772928 46 45 -2206974'226276 4-419364'9753423 15 t3 15 -2121777.217121 4-605720 -9772311 45 46 -2209811'226582 4-413399'9752781 14 16'2124619.217425 4'599268'9771693 44 47'2212648 2266888 4'407450'9752138 13 9 17'2127462'217730 4'592832 -9771075 43 48'2215485 -227194 4-401516 9751494 12 M 18'2130304.218035 4'586414'9770456 42 49'2218321'227500 4'395597'9750849 11 Q 19 -2133146'218340 458(0012'9769836 41 50'2221158'227806 4-389694'9750203 10 H 20'2135988.218644 4-573628'9769215 40 51'2223994 -228112 4 383805 -9749556 9 m 21'2138829'218949 48567261'9768593 39 52'2226'30'228418 4-377931 -9748909 8 [t 22'2141671 -219254 4'560911'9767970 38 53'2229666 -228724 4-372073'9748261 7 9 23.2144512.21 9559 4-554577'9767347 37 54'2232501.229030 4'366229 9-747612 6 24.2147353.219864 4'548260' 9766723 306 55'2235,37.229336 4-360400.9746962 5 25.2150194 2290169) 4-541960'9766098 35 56'2238172.229642 4-.354586.9746311 4 26 -2153035'220474 4'535677'9765472 34 57'2241(07 -229949 45-.48786'9745660 3 27'2155876'220779 4'529410'9764845 33 58'2243842'230255 4-143001 -9745008 2 28'2158716'221084 4-523160'9764217 32 59'224;676'23)5i61 4-337231 -97443,5 1 29 -2161556 -221.89 4-516926'9763589 31 60'2249511'236868 4.331475 -9743701 0 30 *2164396 -221694 4'510708 -9762960 30) - COSINE. COTANG. TA.:G. SINE. / I C,_I-NE. COTANG. TA. SINE. D)E. 77 DEG. 77. NATURAL SINES AND TANGENTS TO A RADIUS 1. 13 DEG. 13 DEG. SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0'2249511'230868 4-331475'9743701 60) 31'2337282'240386 4-159968'9723020 29 1'2252345'231174 4-325734'9743046 59 32'2340110'240694 43154650 9722339 28 2 -2255179 -231481 4.3200()7.9742390 58 33 2.342938.241001 4.149344.9721658 27 3.2258013'231787 4-314295.9741734 57 34.2345766.241309 4-144051'9720976 26 4'2260846'232094 4-308597.974107'7 56 35.2348594.241617 4-138171'9720294 25 5 -2263680. 232400 4-302913.9740419 55 36.2,351421.241925 4-133504 -9719610 24 6 *22=6f513'232707 4-297244.9739760 54 37.2354248.242233 4-128249'9718926 23 7 2269.346.23.3014 4-291588. 9739100 53.38 2357075.242541 4-123007.9718240 22 8'2272179.233320 4-28,5947.9738439 52 39.2359902.242849 4-117778'9717554 21 o 9'2275012 -233627 4'280319.9737778 51 40.23,2729 -243157 4-112561'9716867 20 Q 10'2277844.233934 4-274706.9737116 50 41 -2365555 -243465 4-107356 -9716180 19 5 11'2280677.234241 4-269107.9736453 49 42.2368381'243773 4-102164 *9715491 18 5 12 2283509'234547 4-263521.97.35789 48 43.2371207.244081 4-096985.9714802 17 3 13'2286341'234854 4-257950.9735124 47 44.2374033.244390 4-091817'9714112 16 14'2289172'235161 4-252392 39734458 46 45'2376859.244698 4-086662'9713421 15 0 15'22920)4 2.35468 4-246848. 9733792 45 46 *2379684 245006 4081519 9712729 14 16'2294835'235775 4-241317 *9733125 44 47.2382510.245315 4-076389.9712036 13 17'2297666'236082 42.35800'9732457 43 48'2385335 *245623 4-071270 *9711343 12 18'2300897'236390 4-230297'9731789 42 49 -2388159 -245932 4'066164'9710649 11 19'2303328 -236697 4-224808.9731119 41 50'2390984'246240 4-061070'9709953 10 s 20'2306159'237004 4-2193.31'9730449 40 51'2393808 -246549 4-055987.9709258 9 0 21'2.308989'237311 4-213869'9729777 39 52'2396633 -246857 4-050917.9708561 8 4 22'2311819'237618 4-208419'9729105 38 53 -2399457'247166 4-045859.9707863 7 23'2314649'237926 4-202983'9728432 37 54'2402280'247475 4-040812.9707165 6 24 *2317479 23S8233 4-197560'9727759 36 55'2405104'247783 4'035777'9706466 5 25'2320309'238541 4'192151 *9727084 35 56'2407927'248092 40.30755'9705766 4 26'2323138'238848 4-186754 *9726409 34 57 -2410751'248401 4-02.5744 9705065 3 27'2325967'239156 4-181371 *9725733 33 58'2413574'248710 4'020744.9704363 2 28'2328796'239463 4'176001'9725(5i 32 59 -2416396'249019 4-015757 *9703660 1 29'2331125'239771 4-1706i-44.9724.378 31 60'2419219'249.328 4'010780.9702957 0 30'2334454.240078 4-165299 972.3699 30 I COSINE. COTANG. TANG. SINE. I I COSINE. COTANG. TANG. SINE. I I DEG. 76. DEG. 76. C10 NATURAL SINES AND TANGENTS TO A RADIUS 1. - 14 DEG. 14 DEG. l0 SINE. TANG. COTANG. COSINE. C N I I SINE. TANG. COTAKO. COSIN. 0 *2419219 *249328 4-010780 *9702957 60 31.2506616 *258928 3-862078 9680748 29 1 *2422041.249637 4-005816 -9702253 59 32 *2509432 *259238 3-857453 -9680018 28 2 *2424863 *249946 4-000863'9701548 58 33 *2512248 *259548 3-852839 *9679288 27 3 *2427685 *250255 3-995922'9700842 57 34'2515063'259859 3-848235 *9678557 26 4'2430507 *250564 3-990992'9700135 56 35 *2517879 *260169 3-843642 *9677825 25 5 -2433329'250873 3-986073'9699428 55 36'2520694 *260480 3-839059'9677092 24 S 6 *2436150'251182 3-981.166'9698720 54 37'.523508 *260791 3-834486 -9676358 23 t~? 7 *2438971 3251491 3-976271.9698011 53 38 -2526323 *261101 3-829923.9675624 22 bd 8 *2441792 3251801 3-971386.9697301 52 39'2529137 *261412 3-825370.9674885 21 9 *2444613 -252110 3'966513'9696591 51 40 32531952 *261723 3-820828'9674152 20 10 *2447433 *252420 3-961651 *9695879 50 41'2534766 *262034 3-816295 -9673415 19 n 11 *2450254 3252729 3-956801'9695167 49 42'2537579 -262345 3-811773'9672678 18 f 12 *2453074 -253038 3'951961'9694453 48 43'2540393 2ti2656 3-807260'9671939 17 P 13 *2455894 *253348 3'947133'9693740 47 44 32543206 262967 3-802758 9671200 16 O 14 *2458713 -253658 3-942315.9693025 46 45 32546019 *263278 3-798266'9670459 15 15 *2461533 -253967 3'937509 *9692309 45 46'2548832'263589 3-793783'9669718 14 16 *2464352'254277 3-932714'9691593 44 47 32551645 263909 3-789310'9668?977 13 M 17 52467171 -254587 3'927929 *9690875 43 48 32554458 *264211 3784848 *9668234 12 18 *2469990'254896 3-923156'9690157 42 49.2557270 *264522 3-780395 9667490 11 19 *2472809 *255206 3-918393'9689438 41 50 *2560082 *264833 3-775r1. 9666746 10 20 *2475627'255516 3-913642.9688719 40 51 -2562894 *265145 3-7715 18 9666001 9 M 21 *2478445'255826 3-908901.9687998 39 52 32565705 *265456 3-767('4 9665255 8 j 22 *2481263 3256136 3.904171.9687277 38 53'2568517 7265768 3-762(i; 0 *9664508 7. 23 *2484081 3256446 3-899451'9686555 37 54 32571328 *266079 3-7582- 6'9663761 6 s 24 *2486899 3256756 3-894742 *9685832 36 55'2574139 |266390 3-753t 81 9663012 5 25 *2489716 -257066 3'890044'9685108 35 56 32576950 *266702 3-7494 6'9662263 4 26 *24925.33 3257376 3-885357 *9684383 34 57 32579760 -267014 3-745] 20'9661513 3 27 *2495350'257686 3-880680'9683658 33 58 32582570 *267325 3-740.54 *9660762 2 28 *2498167'257997 3'876014 *9682931 32 59'2585381 *267637 3736-7 98'9660011 1 29 *2500984 3258307 3-871:58 9682204 31 60 *2588190 *267949 3-732050 *9659258 0 30 32503800 3258617 3-866713'9681476 30 I COSINE. COTANG. "TANG. I SINE. COSINE. COTANG. TANG#. SINE. _ DEG. 71. DEG. 75. NATURAL SINES AND TANGENTS TO A RADIUS 1. 15 DEG. 15 DEG I SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0'2588190'267949 3-732050'9659258 60 31'2675187'277637 3'601814'9625527 29 1'2591000'268261 3-727713.9658505 59 32'2677989'277951 3-597754 -9634748 28 2 -2593810'26M8572 3-723384.9657751 58 33'2680792'278264 3-59,3702 -9633969 27 3'2596619'268884 3-719065.9656996 57 34 *2683.594'278578 3-589659'9633189 26 4'2599428 -269196 3-714756.9656240 56 35'2686396'278891 3-585624'9632408 25 5'2602237'269508 3-710455.9655484 55 36'2689198.279205 3-581597'9631626 24 6'2605045'269820 3-706164 -9654726 54 37'2692000'279518 3-577579'9630843 23 7'2607853'270132 3-701883 -9653968 53 38'2694801'279832 3-573569 -9630060 22 8'2610662'270444 3-697610 -9653209 52 39'2697602 28(0145 3-569568 -9629275 21 O 9'2613469'270757 3-693346 *9652449 51 40 -2700403'280459 3-565574 -9628490 20 Q 10'2616277 -271069 3-689092 -9651689 50 41'2703204'280773 3-561590'9627704 19 W 11'2619085'271381 3-684847'9650927 49 42 -2706004'281087 3-557613'9626917 18 M 12'2621892'271694 3-680611'9650165 48 43'2708805'281401 3-553644'9626130 17 3 13'2624699'272006 3-676384'9649402 47 44 -2711605'281715 3-549684'9625342 16 14'2627506'272318 3-672166 -9648638 46 45.2714404.282029 3-545732'9624552 15 O 15'2630312'272631 3-667957 -9647873 45 46'2717204'282343 3-541788'9623762 14 16'2633118'272943 3-663757'9647108 44 47'2720003 -282657 3-537852 -9622972 13 17'2635925'273256 3-659566'9646341 43 48'2722802'282971 3-533925'9622180 12 18'2638730'273569 3-655384'9645574 42 49'2725601 *283285 3'530005'9621387 11 19'2641536'273881 3-651211'9644806 41 50'2728400'283599 3-526093'9620594 10 i 20'2644342'274194 3-647046 -9644037 40 51 -2731198'283914 3-522190 -9619800 9 O 21'2647147'274507 3-642891 *9643268 39 52 27.33997'284228 3-518294'9619005 8 22'2649952'274820 3-638744'9642497 38 53'2736794'284543 3-514407'9618210 7 23'2652757'275133 36i34606'9641726 37 54'2739592'284857 3-510527'9617413 6 24'2655561'275445 3-630477'9640954 36 55'2742390'285172 3-506655'9616616 5 25'2658366'275758 3-626356'9640181 35 56'2745187'285486 3'502791'9615818 4 26'2661170'276071 3-622244'9639407 34 57'2747984'285801 3-498935'9615019 3 27'2663973'276385 3-618141 -9638633 33 58'2750781'286115 3-495087'9614219 2 28'2666777'276698 3-614046'9637858 32 59'2753577'286430 3-491247'9613418 1 29'2669581'277011 3-609960'9637081 31 60'2756374 -286745 3-487414'9612617 0 30'2672384'277324 3-605883 -9636305 30 I COSINE. COTAN. TANNG. SINE. I I COSINE. E. COTANOG. TANG. SINE. I. DEG. 74. DEG. 74. Cj NATURAL SINES AND TANGENTS TO A RADIUS 1. s16 DEG. 16 DEG. O, SINE. TANG. COTAN, COSINE. SINE. TANG. COTANG. COSINE. I 0'2756374 *286745 3-487414'9612617 60 31'2842942 296529 3372340'9587371 29 1'2759170'287060 3483589 9611815 59 32 2845731.296846 3-368745 9586543 28 2 2761965 *287375 3-479772 *9611012 58 33 *2848520 *297163 3-365156 *9585715 27 3 2764761 *287690 3-475963 *9610208 57 34 2851308 *297479 3-361575 *95848 26 4 2767556'288005 3-472161'9609403 56 35 *2854096 *297796 3'358000 9584056 25 6 2770352 *288320 3-468367 *9608598 55 36 *2856884 *298112 3-35133 9583226 24. 6'2773147'288635 3-464581'9607792 54 37'2859671 298429 3-350872 9582394 23 m 7 *2775941 288950 3'460802'9606984 53 38'2862458 *298746 3-347319'9581562 22 9 8.2778736 -289265 3'457031 *9606177 52 39 2865246 *299063 3-343772 *9580729 21 9 *2781530 *289580 3'453267 89605368 51 40'2868032 *299380 3-340232 *9579895 20 10.2784.324 *289896 3-449512 *9604558 50 41'2870819 *299697 3-336699'9579060 19 11.2787118'290211 3445763 *9603748 49 42'2873605'300014 3'333173 *9578225 18 " 12.2789911'290526 3'442022'9602937 48 43.2876391 *300331 3'329654'9577389 17 9 13'2792704'290842 3'438289'9602125 47 44 2879177 *.300648 3.326141'9576552 16 14'2795497 *291157 3 434563'9601312 46 45'2881963 300965 3 322636'9575714 15 15.2798290'291473 3'430844'96()0499 45 46'2884748 *'301283 3'319137'9574875 14 16.2801083'291789 3'427133'9599684 44 47'2887533'301600 3-315645 *'95740.35 13 9 17.2803875'292104 3'423429'9598869 43 48'2890318'301917 3-312159'9573195 12 18'2806667'292420 3'419733'9598053 42 49'2893103'302235 3'308681'9572354 11 6 19 *2809459'292736 3'416044'9597236 41 50'2895887'302552 3-305209'9571512 10 20'2812251'29.3052 3'412362'9596418 40 51'2898671'302870 3-'301743'9570669 9 M 21'2815042'293368 3'408688'9595600 39 52'2901455'303187 3'298285'9569825 8 22'2817833'293683 3'405021'9594781 38 53'2904239'30&3505 3'294833'9568981 7 23'2820624'293999 3'401361'9593961 37 54'2907022'30382.3 3-291387'9568136 6 24'2823415'294316 3'397708'9593140 36 55' 2909805'304141 3287948'9567290 5 25'2826205 *294632 3-394063'9592318 35 56'2912588'304458.3284516'9566443 4 26 -2828995'294948 3-390424'9591496 34 57'2915871'304776 3-281090'9565595 3 27'2831785'295264 3'386793'9590672 33 58'2918153'305094 3'277671'9564747 2 28'2834575'295580 3"38&169 9589848 32 59'2920935 305412 3-274258'9563898 1 29'2837364'295897 3'3795.53'9589023 31 60'2923717 305730 3-2,0852'9563048 0 30 -28401.53'296213 3'375943'9588197 30 I COSINE. COTANG. TANG. SINE. / / COSINE. |COTANO. TANG. SINE. 1)EG. 73. DEa. 73. NATURAL SINES AND TANGENTS TO A RADIUS 1. 17 DEG. 17 DEG. SINE. TANG. COTAN.G. COSINE. I. SINE. TANG. COTANG. COSINE. 0'2923717'305730 3-270852'9563048 60 31 -3009832 -315618 3-168'480'9536294 29 1'2926499 30(6048 3-267452 *9562197 59 32 -3012606 -315938 3-185172'9535418 28 00 2'2929280'306367 3-264059'9561345 58 33 3015 380 -316258 3-161970'9534542 27 3'2932061'306685 3-260672'9560492 57 34'3018153 -31C578 3'158774'9533664 26 4'2934842'307003 3-257292'9559639 56 35'3020926'3168598 3'155584'9532786 25 5'2937623'307321 3-253918'9.558785 55 36.3023699'317218 3-152399'9531907 24 6'294040.3'307640 3-250550'9557930 54 37'3026471'317538 3.149220 -9531027 23 7'2943183'307958 3-247189'9557074 53:38 3029244'317859 3-146047'9530146 22 P 8'2945963'308277 32438&34'9556218 52 39'3032016'318179 3-142880'9529264 21 O 9'2948743'308595 3-240486'9555361 51 40'3034788'31.8499 3-139719 -9528.32 20 Q 10'2951522'308914 3-237143 -9554502 50 41'3037559'318820 3'136563 -9527499 19 W 11'2954.302'309233.3'233807'9553643 49 42'30403.31'319140 3-133414 -9526615 18 12'2957081'309551 3-230478'9552784 48 43'3043102'319461 3-130270'9525730 17 3 13'2959859'309870 3'227154'9551923 47 44'3045872'319781 3-127131'9524844 16 14'2962638'310189 32238.37'9551062 46 45'3048643'320102 3'1253999 9525958 15 C 15'2965416'310508 3-220526'9550199 45 46'3051413'320423 3'120872 9523071 14 16'2968194'310827 3-217221'9549336 44 47'305418.3'320744 3-117750'9522183 13 17'2970971'311146 3-213922 -9548473 43 48'3056953'321064 3-114655 9521294 12 18'2973749'311465 3-210630 -9547608 42 49'3059723'321385 3'111525'9520404 11 t 19'2976526'311784 3-207344'9546743 41 50.3062492'321706 3-108421'9519514 10 - 20'2979303'312103 3-204063'9545876 40 51 -3065261'322027 3-105322'9518623 9 O 21'2982079'312422 3-200789 -9545009 39 52'3068030'322348 35102229'9517731 8 22'2984856'312742 3-197521 -9544141 38 53'3070798'322670 3'099141'9516838 7 23'2987632'31.3061 3-194259'9543273 37 54'3073566'322991 3-096(059 -9515944 6 24'2990408'313381 3-191003'9542403 36 55'3076304'323312 3-092983'9515050 5 25'2993184'313700 3-187754'9541533 35 56'3079102'323633 3-089912'9514154 4 26'2995959'314020 3-184510'9540662 34 57'3081869'323955 3'086846'9513258 3 27'2998734'314,339 3-181272'9539790 33 58'3084636 -324276 308.3786 -9512361 2 28'3001509'3146i59 3-178040'95,38917 32 59 -3087403'324598 3-080732'95114C4 1 29'3004284'314979 3-174814 95.38044 31 60'3090170'324919 3-077683'9510565 0 30'3007058'315298 3-171594 -9537170 30 I COSINE. COTANG. TANG. SINE. I- I COSINE. COTANG. TANG. SINE. I DEG. 72. DEG. 72. -1O DEG. 72. DEG. 72. T NATURAL SINES AND TANGENTS TO A RADIUS 1. 18 DEG. 18 DEGc. CIO I SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. i 0 *3090170 324919 3-077683 *9510565 604 31 *3175805 *334918 2-985798 *9482313 29 1 *3092936 *325241 3-074640 9509666 59 32 *3178563 *335242 2-982916 *9481389 28 2 *3795702 *325563 3-071602 *9508766 58 33 *3181321 *335566 2-980040 *9480464 27 3 *3098468 *325884 3-068569 *9507865 57 34 *3184079 335889 2-977168 *9479538 26 4 *3101234 *326206 3-065542 9506963 56 35 *3186836 336213 2-974301 *9478612 25 5 *3103999 *326528 3-062520 *95060i61 55 36 3189593 336537 2-971439 *9477684 24 6 *3106764 *326850 3-059503 *9505157 54 37 *3192350 *336801 2-968583 *9476756 23 W 7 3109529 *327172 3 056492 *9504253 53 38 *3195106 337185 2-9 731 9475827 22 M 8 *3112294 327494 3053487 -950348 52 39 *3197863 337509 2962884 2 9474897 21 9 53115058 327816 3050486 *9502443 51 40 *320619 -337833 2 960042 *9473966 20 10 *3117822 *328138 3-047491 9501536 50 41 3203374 338157 2 957205 9473035 19 n 11 *3120586 328461 3044501 9500629 49 42 3206130 338481 2 954372 *9472103 18 M 12 312.3349 *328783 3 041517 *9499721 48 43 3208885 33W805 2 951545 9471170 17 vi 13 *3126112 *329105 3-038538.9498812 47 44 3211640 3.9129 2-948722 9470236 16 14 *3128875.329425 3-035564.9497902 46 45 3214395 339454 2945905 9469301 15 15 *3131638 329750 3-032595 *9496991 45 46 *3217149 *339778 2-943092 *9468366 14 16 *3134400 *330073 3-029632 9496080 44 47 3219903.340103 2 940284 9467430 13 M 17 *3137163 *330395 3-026673 *9495168 43 48 *3222657 *340427 2-937480 *9466493 12 t 18 *3139925 330718 3-023720 9949255 42 49 3225411 *340752 2-9342 9465555 11 Q 19 *3142686 *331041 3-020772 *9493341 41 50 *3228164 *341077 2-931888 *9464616 10 20 *3145448 *331363 3-017830 *9492426 40 61 *3230917 341401 2-929099 -9463677 9 M 21 *3148209 *331686 3-014892 9491511 39 52 32. 2 70 *341726 2-926315 *9462736 8 M 22 3150969 332509 30119GO 9490595 38 53 3236422 342051 2-92.353 9461795 7 23 *3153730 332332 3 009033 9489678 37 54 3239174 342376 2920761 9460854 6, 24 *3156490 *332655 3 006110 9488760 36 55, 3241926 *342701 2 917990'9459911 5 25 *3159250 *332978 35003193 94237842 35 56 3244678 *343026 2 915225 *9458968 4 26 *3162010 333302 35000282 *9486922 34 57 3247429 34.3351 2912464 *9458023 3 27 *3164770 *33,3625 25997375 9486)02 33 58 3250150 34.3677 21909708 *9457078 2 28 *3167529 333948 2 994473 *9485081 32 59.3252931 344002 2 906957 9460132 1 29'3170288 *334271 2 991576 *9484159 31 60 *32556832.3327 2904210 *9455186 0 30 3173047 *334595 2*988685 *9483237.30 / COSINE. COTANO. TANG. SINE. I I COSINE. COTANG. I TANG. SINE. I DEG. 71. DEG. 71. NATURAL SINES AND TANGENTS TO A RADIUS 1. 19 DEG. 19 )EGs. / SINE. TANG. COTANG. COSINE. I I STNE. TANG. COTA:-'. I. COSINE. I 0 -3255682 -344327 2-904210 194551,86 60 31. 3348)410 -3-154446 2821304 *9425444 29 1 *3258432.344G53 2-901468 *9454238 59 32 -.334:52.354770' 2 81i8700 9424471 28 32961182'344978 2'898731 *9453290 58 33 3346293'3551i01 2 816100 094213498 27 3 -32ti39332 -345304 2'895998'9452341 57 34 -3340034 3.55428 2-813'5)4 *9422525 26 4 3266;81'l 345629 2-893270 *9451391 56 35'3351775 1355756 2-810913.9421550 25 5 53269430'345955 2-890546 *9450441 55.36 3354516 -3560ut 2-'30-326 -9420575 24 6 *3272179'346281 2-887827 *9449489 54 37 3357256 135l1 2-805743'9419598 23 7 -32 7492,'34i06 2885113 -9448537 53 38' 33.59996 356739 2'803164'9418621 22 I'' 3277676'346932 2'882403 *9447.584 52 39 1 3362735 3577067 26800590 *9417644 21 O 9 3290424'317258 2'879S97'9446630 51 40 -3365475'357395 2'798019'9416ti65 20 0 1'0 32S3172'347584 2-876997 -9445675 50 41'33!;8214 3577723 279.5453'9415686 19 pi 11 32919'31347910 23874300 -9444720 49 42'3370953'358051 2-792891'9814705 18 M 12'32,;'36iT6 *348236 2-871608 9443764 48 43 373fi91 3,58380 2' 79:'3 -9413724 17 3 13'3291413'348563 2-868921 /9442807 47 44 -337649) -358708 2-7 7;7 8 *9412743 16 14'3)9411(0 3481.889 2-866238 -9441849 46 45 337916T7 350)36 2'7&594W 9411760 15 15 5:2906 9103.349215 2 863560 9440890 45 46 33581 9035 -359365 2 78,2f'3 *9410777 14 1(6'325'.465.3 934S542 2-860886 -9438931 44 47 -334i642'359693 2 780114 4 9409793 13 17'33C"21q9'349868 2-858216'9438971 43 48 3.387379'360022 2 7776V6.'9408608 12 18'33 051 44'350195 2-855551'9438010 42 49 -3390116 1 36030 2-775073 9407822 11 19'330789'350521 2-852891'9437048 41 50' 3302852 *360679 2-772544 094068f3 10 20'3310634.350848 2-850234'9436085 40 51 -3395189'361008 2-770019 -9405848 9 O 21'3313.379'351175 2-847583'9435122 39 52 -3398325 -361337 2-767499'9404860 8 22 3'.16123'351501 2-844935'9434157 38 153 -3401060 -361666 2-764982'9403871 7 23'3318867'351828 2-842292'9433192 37 54'3403796 -361994 2-762469'9402881 6 24.I21 011 *:352105 2-86986i3 -9432227 36 55 -3406531 -362324 2-7599!60'9401891 5 25 3324355 35 24-52 2-837019 -9431260 35 56 -3409261 -36263.3 2-757456 -9400899 4 26.3327098 -352809 2-834389 -9430293 34 57 -3412000 -362982 27.54955'9399907 3 27 -3329841 -'353136 2-831763'9429324 33 58 3414734 -363311 2-752458 -9398914 2 28 3.332584'353464 2-829142'9428355 32'59 3417408 -363640 2-749966'939)7921 1 29 -33.3326t'35.3791 2-826525'9427386 31 60 -3420201 -363970 2-747477'93196926 0 30 3338069.354118 2-823912 -9426415 30 - COSINE. COTAN. TAN. SINE. SINE I COSINE. - COTANG. TANG. SINE. I - DEG. 70. DEG. 70. c. NATURAL SINES AND TANGENTS TO A RADIUS 1. 20 DEG. 20 DEG0. I SINE. TANG. COTANG. COSINE. I SINE. TANG. COTANG. COSINE. I 0 *3420201 -363970 2-747477 *9396926 60 31'3504798 *374216 2-6722B1 9365703 29 1 -3422935 *36429.9 2-744992 *9395931 59 32'3507523'374547 2-66988.5 936468.3 28 2 *3425668 *364629 2-742512 *9394935 58 33'3510246' 374879 2-667522 9:933662 27 3 13428400 *364958 2-740035 9393938 57 34 *3512970.375211 2-665163 9362641 26 4 *3431133 *365288 27137562 *9392940 56 35'3515693'375543 2-662808'9361618 25 5'3433865 *365618 27.35093 *9391942 55 36' 3518416'37587 5 2660456'9360595 24 6 *34.36597 *365948 2-732628 *9390943 54 37'3521139'376207 2-658108'9359571 23 7.3439329'366277 2'730167 389.s943 53 38'3523862 -376539 2-655764'9358547 22 8 *3442060'366607 2-727710 9.388942 52 39 3526584'376871 2-653423'9357521 21 9'3444791 *366937 2-725256 9387940 51 40 3529306 *377203 2-651086'9356495 20 10 *3447521 367268 2-722807 *9386938 50 41'35,32027'377536 2-648753'9355468 19 11'34502352 *367598 2-720362 *9385934 49 42'3534748'377868 2-646423 9.354440 18 12'3452982 *367928 2-717920 9384930 48 43'3537469'378201 2-644096 8.353412 17 13'3455712 *368258 2-715482.9383925 47 44'3540190.378533 2'641774'9.35'82 16 14'3458441 *368589 2-713048 9.382920 46 45'3542910'378866 2-639454'93k51. 52 15 15 *3461171'368919 2'710618 *9381913 45 46'3545630'379198 2-637139 9&30321 14 16 *3463900' 369250 2-708192 93850906 44 47'3548350'379531 2-634827'9349289 13 17,3466628 *369580 2-705769 *9379898 43 48'3551070 *379864 2-6.32518 91348257 12 18 *3469357'369911 2-703351 *9378889 42 49'3553789'380197 2-630213'9347223 11 19 *3472085 -370242 2-700936 9377880 41 50'3556508 380530 2-627912'9346189 10 20 3474812'370572 2-698525 9376869 40 51 3559226'380863 2-625614'9345154 9 21'3477540'370903 2-696118 9375858 39 52'3561944'381196 2-62,319'9344119 8 22'3480267 *371234 2-693714 *9374846 38 53'3564662'381529 2-621028'9343082 7 23 3k482994 371565 2'691314 9.373833 37 54'3567380'381862 2-618741'9342045 6 24 *3485720'371896 2-688919'9372820 36 55'.3570097'382196 2-616457'9341007 5 25'34-88447'378227 2-686526 8 9371806 35 56 *3572814 *382529 2-614176 09339968 4 226.3491173'372559 2'684138 9370790 34 57 3575531 *382863 2-611899 *9338928 3 27'349'%98 |372890 2-681753 *9369774 33 58'3578248'383196 2-609625'9337888 2 28 f3496)624 37'3221 2679372 *936858 32 59 3580964 38.3530 2-607355 *9336846 1 29'3499.'49 373553 2-676995 9367740 31 60'3583679'383864 2'605089'9335804 0 30'3502074'37.384 2-674621 *9366722 30 I COSINE. -COT AN. TANG. SINE. I COSINE. COTANG. TANG. SINE. I DEG. 69. DEG. 69. NATURAL' SINES AND TANGENTS TO A RADIUS 1. 21 DEG. 21 DEG. SINE. TANG. CO!ANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0'3583679'383864 2'605089'9335804 60 31'3667719'394246 2-536483 9.303109 29 1'3586395'384197 2-602825'9334761 59 32'3670425 *394582 2'5S4323'9302042 28 2'3589110'384531 2'600565'9333718 58 33'3673130'394918 2-532165 93:'0974 27 3'.3591825'384865 2-598309.9332673 57 34'36752436'395255 2-530011.9299905 26 4'3594540.385199 2-596056.9331ii28 56 35.3678541.395591 2-527859 -9298&35 25 5'3597254 3855,33 2'593806.9330582 55 36.35511246'395928 2 525711'9297765 24 6 3599968'385867 2'591560 9.3295,35 54 37 836&3950'396264 2-52.3566. 9296694 23 7 -3602682'386202 2-589317.9328488 53 38.368)(654'396601 2-521424.929.5622 22 8'3605395'386536 2-587078.9327439 52 39'3689.358'3969,37 2-519286 -9294549 21 9'3608102'386870 2-584842.93263.90 51 40'3692061'397274 2-517150'9293475 20 Q 10'3610821'387205 2-582609.9325340 50 41'3694765'397611 2-515018.9292401 19 9 11'3613534'387539 2-580380'9324290 49 42'3697468 *397948 2-512889 9291 1326 18 10 12'3616246'387874 2-578153.9323238 48 43'3700170 *398285 2-510762'9290250 17 3 13'3618958'388209 2-575931 9.322186 47 44'3702872'398622 2'508639'9289173 16 14'3621699'388543 2-573711'9321133 46 45'3705574'398959 2-506519'9288096 15 0 15'3624380'388878 2-571495 -9320079 45 46'3708276'399296 2-504402 9287017 14 O 16'3627091'389213 2'569283 -9319024 44 47.3710977'399634 2-502289'92859.38 13 17'3629802'389548 2-567073'9317969 43 48'3713678'399971 2-500178'9284858 12 18'3632512'389883 2-564867 9.316912 42 49'.3716379'400308 2-498070'9283778 11 19'3635222'390218 2-562664'9315855 41 60'3719079'400646 2-4959(;6'9282696 10 n 20'3637932'390554 2-560464'9314797 40 51'3721780'400984 2.493864'9281614 9 O 21'3640641'390889 2-558268'9313739 39 52 *3724479'401321 2-491766'9280531 8' 22'3643351'391224 2-556075'9312679 38 53'3727179'401659 2-489670'9279447 7 23'3646059'391560 2-553885'9311619 37 54'3729878'401997 2-487,578'9278363 6 24'3648768'391895 2-551699 -9310558 36 55'3732577'402.335 2-485488'9277277 5 25'3651476'392231 2-549516'9309496 35 56'3735275'402673 2 483402'9276191 4 26'3654184'392567 2-547335'9308434 34 57'3737973'403011 2-481319'9275104 3 27'3656891'392902 2-545159.9307370 33 58'3740671'403349 2-479238' 9274016 2 28'3e59599'393238 2.542985'9306306 32 59'3743369'403687 2-477161 -9272928 1 29'362306 393:574 2-5180815'9305241 31 60 -3746066'404026 2'475086'9271839 0 30'3665012.'393910 2-538647'9304176 30 I OOSINE. COTANG. TANG. T SINE. I I COSINE. COTANG. TANG. SINE. I!i DEG. 68. DEG. 68. NATURAL SINES AND TANGENTS TO A RADIUS 1. 22 DEG. 22 DEG. I SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. i 0 -3746066'404026 2-475086 -9271839 60 31'3829522'414554 2'412228 -9237682 29 1'3748763'404364 2'473015 -9270748 59 32'3832209 *414895 2-410246'9236567 28 2'3751459.404703 2-470947'9269658 58 33'3834895 *415236 2'408267'9235452 27 3'3754156'405041 2-468881'9268566 57 34'3837582 -415577 2-406290'9234336 26 4'3756853.'405380 2-466819'9267474 56 35'3840268 -415918 2-404316 89233220 25 5'3759547'405719 2-464759'9266380 55 36'3842953 *416259 2-402,345 9232102 24 6'3762243 *406057 2-462703'9265286 54 37'3845639.416601 2-400377'9230984 23 7'37649'38'406396 2-460649'9264192 53 38'3848324 *416942 2'398411'9229865 22 l 8'3767632 -406735 2-458598'9263096 52 39 93851008'417284 2'396449'9228745 21 9'3770327 407074 2-456551'9262000 51 40'3853693 *417625 21394488 *9227624 20 10'3773021'407413 2-454506'9260902 50 41'3856377'417967 2'392531'9226503 19 n 11 3775714 407753 2-452464'9259805 49 42 3859060'418309 2-390576'9225381 18 F 12'3778408'408092 2'450425'9258706 48 43'3861744'418650 2'388625 -9224258 17 x 13'3781101 -408431 2-448389'9257606 47 44'3864427'418992 2-386675'9223134 16 O 14'3783794 -408771 2-446355' 9256506 46 45'3867110'419334 2-384729'9222010 15 15'3786486 *409110 2-444325'92.55405 45 46'3869792'419676 2-382785'9220884 14 16 -3789178 4()9450 2-442298'9254303 44 47'3872474'420019 2-380844'9219758 13 m 17.3791870 -409790 2'440273'9253201 43 48'3875156'420361 2'378906'9218632 12 Z 18'3794562'410129 2'438251'9252097 42 49'3877837'420703 2-376970'9217504 11 Q 19'3797253 410469 2-436233'9250993 41 50'3880518'421046 2-375037'9216375 10 20'3799944'410809 2'434217'9249688 40 51'3883199.421388 2-373106'9215246 9 21'3802(i34'411149 2-4,32204'9248782 39 52'3885880'421731 2-371179'9214116 8 tJ 22'3805324'411489 2-430193'9247676 38 53'3888560'422073 2'369254'9212986 7 M 23'3808014'411830 2-428186'9246568 37 54'3891240'422416 2-367331'9211854 6 24 -3810704'412170 2-426181'9245460 36 55'3893919'422759 2-365411'9210722 5 25'3813393 -412510 2'424180) 9244351 35 56'3896598'423102 2-363494'9209589 4 26'3816082'41251 2-422181'9243242 34 57'3899277'423445 2'361580'9208455 3 27'3818770'413191 2-420185'9242131 33 58'3901955'423788 2-359668'9207320 2 28'3821459'413532 2'418191'9241020 32 59'3904633'424131 2-357759'9206185 1 29'3824147'413872 2-4.16201'9239908 31 60'3907311'424474 2-355852'9205049 0 30'3826 834'414213 2-414213'9238795 30 I COSINE. COTANG. TANG. SINE. I/ COSE.. COTANG. TANG. SINE. DEG. 67. DEG. 67. NATURAL SINES AND TANGENTS TO A RADIUS 1. 23 DEG&. 23 LDEG SINE.S TANG. COTANG. COSINE. / Si NE. TANG. COTANG. (COSINE. 0 -3907311 *424474 2-355852'9205549 60 31.3990158 *435158 2-298014 9-169440 29 1 3909989'424818 2'353948'9203912 59 32'39928225'435504 2-296188 { 91018279 2S 2'3912660 *425161 2-352046'9202774 58 33 53995492 *435850 2'294365'9167118 27 3'3915343'425505 2-350148'92016.35 57 34'3998158'436196 2'2927r44'9165955 26 4 3918019 *425848 2'348251'9200496 56 35'4(1)0825 *436542 2'290725'9164791 25 5'3920t;95'426192 2'346358'9199.356 55 36 4003490 4G36889 2-288909 -9163627 24 6'3923371 *426536 2'344467'9198215 54 37'4006156 *437235 2'287095 *9162462 23 7'3926047 *426880 2-342578'9197073 53 38'4008821 -437582 22,85284'9161297 22 1 8'3928722'427223 2-340692 *9195931 52 39 4011486 437928 2-283475 li'91(0)10 21 O 9'3931397'427568 2-338,09'9194788 51 40'4014150 *438275 2-281609'91589963 20 Q 10'3934071 *427912 2'336928'9193644 50 41 *4016814'438622 2'279865'9157795 19 P 11'3935745'428256 2-335050'9192499 49 42'4019478'438969 2-278063'9151626 18 W 12 ~3939419'428600 2-3.33174'9191353 48 43'4022141'439316 2-2762(64 09155456 17 t3 13'39121)93 428944 2-331301'9190207 47 44'4024804'439663 2-274467'9154286 16 14'3944765 *429289 2-329431'9189060 46 45'4027467'440010 2-272672 *9153115 15 15'3947439'429633 2-327563'9187912 45 46' 4030129' 440.357 2-270890'9151943 14 16'3950111'429978 2-325697'9186763 44 47'4032791'44070.5 2-26)9090 9150770 13 17'3952783'430323 2-323834'9185614 43 48'4035453'441052 2-267303'9149597 12 18'3955455'430668 2'321974'9184464 42 49'4038114'441400 2-2(65518'9148422 11! 19'3958127'431012 2-320116'9183313 41 50'4040775'441747 2-263735' 9147247 10 20'396t)798'431.357 2-318260'9182161 40 51'4043436'442095 2-261955'9146072 9 O 21'3963468'431703 2'316407'9181009 39 52'4046096'442443 2-260177 *9144895 8 t 22'3966139'432048 2-314557'9179855 38 53'4048756'442791 2-258401'9143718 7 23'3968809'432393 2-312709'9178701 37 54 *4051416'443139 2-256628'9142540 6 24'3971479'432738 2-310863'9177546 36 55'4054075'443487 2-254857'9141361 5 25'3974148'433084 2-309020'9176391 35 56'4056734 *443835 2-253088Mi'9140181 4 26'3976818'433429 2-307180'9175234 34 57'4059393'44418.3 2'251322 89139001 3 27'3979486'433775 2-305342'9174077 33'58 4062051'444531 2-249558'9137819 2 28'3982155'434120 2'303506'9172919 32 59'4064709'444880 2-247796'9136637 1 29'3984823'434466 2-3')1673'9171760 31 60'4067366'445228 2-246036'9135i55 0 30'3987491'434812 2-299842'9170601 30 - COSINE. COTANG. TANG. SINE. I I COSINE. COTANG. TANG. SINE. I DEG. 66. DEG. 66 ce NATURAL SINES AND TANGENTS TO A RADIUS 1. 24 DEG. 24 DEG. SINE. 1 TANG. COTANGr. COSINE. I I SINE. TANG. COTANG. COSINE. 0 4067366 445228 2-246036'9135455 60 31'4149579'456077 2-192609'9098406 29 1 ~4070024 *445577 2-244279'9134271 59 32'4152226'456429 2-190921'9097199 28 2 4)072681'445926 2'242524'9133087 58 33'4154872'456780 2-189234'9095990 27 3 *tl;75337'446274 2-240772 -9131902 57 34'4157517 *457132 2-187551'9094781 26 4 )4077993 *446623 2'239021'9130716 56 35'4160163'457483 2-185869'9093572 25 5 40(80649.446972 2'237273 9129529 55 36'4162808 *457835 2'184189.9092361 24 3 6 -40)8.305'447321 2'235528 -9128342 54 37' 4165453 -458187 2-182511'9091150 23 9 7'40,85960'447670 2'233784 -9127154 5.3 38' 4168097'458539 2-180836'9089938 22 80 8 -4088615'448020 2'232043'9125965 52 39'4170741'458891 2-179163'9088725 21 9 -4091269.448369 2'230304'9124775 51 40'4173385'459243 2-177492'9087511 20 10'4093923'448718 2-228567 -9123584 50 41'4176028'459596 2-175822'9086297 19 11'4096577'449068 2-226833'9122393 49 42'4178671'459948 2-174155 -9085082 18 H 12 -4099230'449417 2-225100'9121201 48 43'4181313'460301 2-172491'9083866 17 O 13'4101883 *449767 2'223370'9120008 47 44'4183956'460653 2-170828'9082649 16 ~ 14'4104536'450117 2-221643'9118815 46 45 -4186597'461006 2-169167 -9081432 15 15 -4107189 *450467 2'219917'9117620 45 46'4189239'461359 2-167509 -9080214 14 16'4109841'450817 2'218194'9116425 44 47'4191880'461711 2-165852'9078995 13 8 17'4112492 -451167 2'216473'9115229 43 48'4194521 -4620(64 2'164198'9077775 12 18'411.51t44'451517 2'214754'9114033 42 49'4197161'462117 2'162546 -9076554 11 Q 19'4117795'451867 2-213037'9112835 41 50'4199801'462771 2-160895'9075333 10 20'4120445'452217 2-211323'9111637 40 51'4202441'463124 2-159247'9074111 9 x 21'4123096 452568 2-209611'9110438 39 52'4205080'463477 2-157601 -9072888 8 8 22'41253745'452918 2-207901'9109238 38 53'4207719'463831 2-155957'9071665 7 9 23 -4128395'453269 2-206193'9108038 37 54'4210358'464184 2'154315'9070440 6 _ 24'4131044'453620 2-204487 -9106837 36 55'4212966'464538 2-152675'9069215 5 25'41334,93.453970 2-202784'9105635 35 56.421 5634'464891 2-151037'9067989 4 26'4136342'454321 2'201083'9104432 34 57'4218272'465245 2-149402'9066762 3 27'4138990'454672 2-199384'9103228 33 58'42209)9'465599 2-147768'9065355 2 28'4141638'455023 21(97687 -9102024 32 59'4223546'465953 2-146136'9064.07 1 29 -4144285'455375 2-195992'9100819 31 60'4226183'466307 2-144506'9063078 0 30.4146932'455726 2-1(14290'9099613 30 I COSINE. COTANG. I TANG. SINE. I I COSINE. COTANG. TANNG. SINE. I DEG. 65. DEG. 65. NATURAL SINES AND TANGENTS TO A RADIUS 1. 25 DEG. 25 DEG. SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTAN(G. COSINE. I 0 -4226183'466307 2-144506'9063078 60 31 4.307736'477332 2-094975'9024600 29 1'4228S19'466661 2-142879'9061848 59 32 -4310361 -477689 2-093408'9023347 28 2'4231455'467016 2-141253'9060618 58 33 -4312986'478047 2-091843 -9022092 27 3'4234090'467370 2'139630'9059386 57 34 -4315610'478404 2-090280'9020838 26 4'42,36725'467725 2-138008'9058154 56 35'4318234'478762 2-088720'9019582 25 5'4239360'468079 2-136389 *9056922 55 36'4320857'479119 2-087161'9018.325 24 6 *4241994 *468434 2-134771'9055688 54 37 -4323481'479477 2-085603 -9017068 23 7'4244628'468789 2-133155 -9054454 53 38'4326103'479835 2-084048'9015810 22, 8'4247262'469143 2-131542'9053219 52 39 -4328726'480193 2q082495 -9014551 21 O 9'4249895'469498 2-129930 -905198.3 51 40'4331348'480551 2-081)43'9013292 20 Q 10'4252528'469853 2'128321'9050746 50 41'4333970 2480909 2-079394'9012031 19 11 *4255161. -470209 2'126713'9049509 49 42'4336591 -481267 2'077846 -9010770 18 M 12 42.57793'470564 21 25108'9048271 48 43'4339212'481625 2-076300 -9009.508 17'3 13'4260425' 470919 2'123504'9047032 47 44'4341832'481984 2-074756'9008246 16 14'4263056'471275 2-121903 *9045792 46 45 -4344453'482342 2'073214 -9006982 15 Q 15'4265687'471630 2-120303 90144551 45 46'4347072'482701 2-071674'9005718 14 O 16'42658318'471986 2-118705'9043310 44 47'4349692 -483060 2'0701. 900-45. 13 17'4270949'472342 2-117110'9042068 43 48 4352311 483418 2'06&599 9003188 12 18 *4273579'472697 2-115516'9040825 42 49'4354930 48&3777 2'06706-4 9001921 1 z 19'4276208'473053 2-113924 0.39582 41 50'4357548'484136 2`065531'9000.4 10 w 20'4278838'473409 2-112334'9038338 40 51'4360166'484495 2-0f64000 -8999386 9 O 21'4281467'473765 2-110747'9037093 39 52'4362784 4484355 2'062471'8998117 8 22'4284095'474122 2-109161'9035847 38 53'4365401'485214 2-060944'8996848 7 23'4286723 *474478 2-107577'9034600 37 54'4368018'485573 2-059418'8995578 6 24'4289351 *474834 2-105995'9033.353 36 55'4370634'485933 2-057895 -8994307 5 25'4291979'475191 2-104415'9032105 35 56'4373251'486293 2.0'56373'8993035 4 26'4294(606 4753A48 2-1028?36'9030856 34 57'4375866'486652 2-054853'8991763 3 27'4297233'475904 2-101260'9029606 33 58'4378482'487012 2-05,3334'8990489 2 28'42998359'476261 2-099686'9028356 32 59 *4.381097 *487.372 2-051218 -8989215 1 29'4302485'476618 2'098114'9027105 31 60'4383711 - 487732 2-050303 89S17940 0 30.4305111 -476975 2-096543'9025853 30 I COSINE. I COTANG. TANG. SINE. I I COSINE. COTANG. TANG. lFNE. I DEG. 64. DECG. 64. cy NATURAL SINES AND TANGENTS TO A RADIUS 1. 26 DEG. 20 DEG. C SINE. TANG. COTANG. COSINE. Iq SINE. TANG. COTANG. COSINE. I. _ I ~~ ~~ ~' 1 1- o~ _ oo_ _ 0'4383711'487732 2-050303'8987940 60 31 -4464581 *498944 2-004229 I8948045 29 1 4386326 488092 2'048791 *8986665 59 32 4467184 499308 2002771 894716 28 2'4388940 *488453 20(47280 *8985389 58 33 4469786 *499671 2-001314'8945446 27 3 4391553 [48S813 2-045770'8984112 57 34'4472388 500035 1-9998,59 8944146 26 4 4394166'489173 2-044263'8982834 56 35 4474990 1500398 1-998405 80824() 4 25 5 *4396779'489534 2-042757 8981555 55 36 4477591 500762 1-996953 8941542 24 9 3 6 4399392 489894 2'041254 8980276 54 37 4480192 *501126 1'995503 81940240 23 t 7 449'04'490255 2'039751 8978996 53 38'4482792 0501490 1'99-4055 1:1903 22 6 8 4404615 490616 2038251 *897715 52 39'4485392 -501854 1-992608''9t632 21 9 4407227- 400977 2'036753'8976433 51 40'4487992 502218 1-991163 883632 20 10 -4409838 *491338 2'035256'8975151 50 41'4490591 *502,583 1-989720 82t1i021 19 11 4412448'491699 2'033761'8973868 49 42'4493190 (502947 1-988278'8933714 18 t 12 4415059 492061 2-032268'8972584 48 43 4495789'503312 1-986838'8932400 17. 13 4417668'492422 2-030776'8971299 47 44 4498387'503676 1-985400 1892`1098 16 14 4420278 *492783 2-029287'8970014 46 45'4500984'504041 1'983963 8 1'29789 15 15'4422887 493145 2027799 8968727 45 46 4503582 *504406 1982528 21(2,S0 14 16'4425496'4980MO7 2'026313 8967440 44 47'4506179'504771 1-981095 81i27119 13 00 17 *4428104'493868 2-024828 *8966153 43 48'4508775'505136 1-979663 1F92181 8 12 t 18 *443,0712 494230 2-023346'8964864 42 49 *4511372'505501 1-978233 89241140 11 Q 19'4433319 *494592 2-021865 *8963575 41 50 4513967 *505866 1 976805 *8923234 10 20 *4435927 494954 2-020386'8962255 40 51 -4516563'506232 1-975378'8921 120 9 0 21'4438534'495317 2 018908 *8960994 39 52'4519158'506597 197 3953 *8920!106 8 0 22'4441140 495679 2-0174383'8959703 38 53'4521753'506963 1-972529'8919291 7 0 23'4443746 *486041 2 015959.895S411 37 54'4524347 507329 1-971107'8917971 6 24 *4446352 496404 2-014486'8957118 36 55 4526941'507694 1-"(9(187'8916651) 5 25'4448957'496766 2-013016'8955824 35 5' *4529535'508060 1-968268'8915342 4 26 4451562'497129 2011547'8954529 34 57'4532128'508426 1 966851'8914024 3 27'4454167'497492 2011(0080'8953234.33 58'4534721'508792 1-965436'891 075 2 28'4456771'497855 2,008615'8951938.32 59'4537313'509159 1-964022 *80113,S 29 4459575'498218 7.007151'8950011 31 60'4539905'509525 1-962610'891U06 u 30'4461978'498581 2-005689'8949344 30, COSINE. COTANG. TANG. SINE. I I COSINE. ICOTANG. TANG. SINE. DEG. 63. DEG. 63. NATURAL SINES AND TANGENTS TO A RADIUS 1. 29 DEa. 27 D,. / SINS. TANG. COTANG. COSINE. I i SINE. TANG. COTANG. COSINE. I 0'4539905'509525 1-962610'8910065 60 31 "4620066'5209335 1'919618 -8868765 29 1'4542497'509891 1-961200'8908744 59'. *4622646'521,06 1'918256i 8867420 28 2'4545088'510258 1-959791'8907423 58 3'4625225'521676 1'91(6896 8866(75 27 3'4547679 *510625 1958383'8906100 57'4 4627804'522046 -1915537 88M4730 26 4'4550269'510091 1'956978'8904777 56 35' 4630382 [522417 1'914179 -4;.'13 25 5'4552859 *511358 1'955573'8903453 55 36'4632960 J 22787 1-'912823'8862(3 24 6 *4555449 *511725 1-954171 58902128 54 37'4635538'5231.57 1-911469 886;0i688 2 7'45580,38'512093 1'952770'8900803 53 38 4638115'523528 1'910116 88)59339 22 8'4560627'512460 1-951371 8899476 52 39 464(0692 52:3899 1-908764'8857989 21 o 9'4563216'512827 1-949973'8898149 51 40'4643269 524269 1'907414 98856639 20 10'4565804'513195 1'948577'8896822 50 41'4645845'5241i40 1'06066 *8855288 19 W 11 45(68392'513552 11947182'8895493 49 42'4648420'525011 1-904719'8853936 18 M 12'4570979'513930 1-945789'8894164 48 43'4650996'525382 1-9033713'8,j2584 17 13'4573566'514298 1-944398'8892834 47 44 4fi5:571.527 54 1-9020209',851 120 16 14'4576153'514665 1-943008'8891503 46 45'4656145 *52(;125 1')900(87 -8849876 15 15'4578739'515033 1-941620'8890171 45 46 -4658719'52;496 1'99346'89483522 14 16'4581325'515401 11940233'8888839 44 47'4661293'52680i8 1'8(.)006'88471(66 13 17'4583910'515770 1-938848'8887506 43 48'4663866'527240 1'896668'8845810 12 18'4586496'516138 1-937464'8886172 42 49'4666439 527i'1 2 1-895332'8844453 11 19'4589080'516506 1-936082'8884838 41 50'4669012'527983 1-893997 1 8843095 10 H 20'4591665'51 6875 1-934702'8 350.3 40 51 *4671584 *528356 1 8926 3 &'841736 9 O 21'4594248'517244 1-933323 88216 39 52'4674156'528728 1-891331'8840377 8 t 22'4596832'517612 1-9.31945'8880830 38 53 4676727 529100 1890(1000 9839017 7 23'4599415'517981 1-'930569'8879492 37 54'4679298'529472 188;671 8837fi56 6 24'4601998'518350 1-929195'8878154.36 55 4681869'529845 1887343 8836295 5 25'4604586'518719 1-927822'8,q76815.35 56'4684439'530217 1'886017 883.'1;3 4 26'4607162'519089 1-1926451'875475 34 57 4687009'530590 1 8i94692 8,32569 3 27 4t09744'519,158 1-925081'89874134'33 58 4689578'5309&3 1'988369'88132206 2 28 4612325 519827 1-923713'8q872793 32 59'4692147'531,336 1-882047'88030841 1 26) 461490(6 -52()197 1-922347'8871451 31 60'4694716'531709 1-880726 *8829476 0 30 *4617486'2(05367 1-920982'8870108 30 i COSINr. COTANG. TANGI. SINE. COSINE. GOTANG. TANG. SINE. I DGa. 62. DIa. 62.. NATURAL SINES AND TANGENTS TO A RADIUS 1. 28 DEG. 28 DEG. 0 SINE. TANG. COTANG. COSINE. I / SINE. TANG. COTANG. COSINE. / 0 4694716'531709 1'&0726'8829476 60 31'4774144 -543332 1-840494'8786783 29 1'4697284'532082 1-879407 -8828110 59 32'4776700 -543709 1'839218 *8785394 28 2'4699852'532455 1'878089.8826743 58 33' 4779255'544086 1'837944'8784004 27 3'4702419'532829 1-876773.8825.376 57 34'4781810'544463 1-836671'8782613 26 4'4704986'533202 19875458.8824007 56 35'4784364'544840 1-835399'8781222 25 5'4707553 -533576 1-874145.8822638 55 36' 4786919 5415217 1-834129'8779830 24 p3 6'4710119'533950 1'872833.8821269 54 37'4789472 -545595 1-832861'8778437.23 7'4712685'534324 1'871523.8819898 53 38'4792026'545972 13831593'8777043 22 ~J 8 -4715250'534698 1-870214.8818527 52 39'4794579'546350 1-830327'8775649 21 9'4717815'535072 1-868906'8817155 51 40 -4797131'546728 1-829062'8774254 20 10'4720380'535446 1-867600'8815782 50 41'4799683'547106 1'827799'8772858 19 11'4.722944'535820 1-866295.8814409 49 42'4802235'547484 1-8265.37'8771462 18 12'4725508 -536195 1-864992.8813035 48 43'4804786'547862 1-825276 -8770064 17 X 13'4728071'536569 1-863690'8811660 47 44'4807337'548240 1-824017'8768666 16 O 14 -4730634'536944 19862389'8810284 46 45'4809888'548618 1'822759 -8767268 15 15'4733197'537319 1'861090'8808907 45 46'4812438'548997 1-821502'8765868 14 16'4735759 -537694 1-859792'8807530 44 47'4814987'549375 1-820247'8764468 13 M 17'4738321'538069 1-858496'8806152 43 48'4817537'549754 1'818993'8763067 12 Z4 18'4740882 -538444 1-857201'8804774 42 49'4820086'550133 1'817740'8761665 11 0 19'4743443'538819 1855908'8803394 41 50'4822634'550512 1-816489'8760263 10 20'4746004 5391935 1-854615'8802014 40 51'4825182'550891 1-815239 758859 9 21'4748,564'5.39570 1'853325'8800633 39 52'4827730'551270 1'813990'8757455 8 M 22'4751124'535946 1-852035'8799251 38 53'4830277'551650 1-812743'8756051 7 8 23'4753683'540322 1-850747'8797869 37 54'4832824'552029 1-811496'8754645 6 24'4756242'540698 1'849461'8796486 36 55'4835370'552409 1-810252'8753239 5 25'4758801'541074 1-848176'8795102 35 56'4837916'552789 1-809008'8751832 4 26'4761359'541450 1-846892'8793717 34 57'4840462'553168 1-807766'8750-125 3 27'4763917'541826 1'845609'-8792332 33 58'4843007'553548 1'806525'8749016 2 28'4766474'542202 1-844328'8790946 32 59'4845552'553928 1-805286'8747607 1 29'4769031'542579 1'843049'8789559 31 60'4848096'554309 1'804047'8746197 0 30'4771588'542955 1-841770'8788171 30 COSINE. CSINE. COTAN. TANG SINE. I I COSINE. COTANG. TANG. SINE. I DEG. 61. DEG. 61. NATURAL SINES AND TANGENTS TO A RADIUS 1. 29 DEa. 29 DEGa. SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0 -4848096 -554309 1-804047.8746197 60 31 49026767.566156 1-766295 *8702124 29 1 *4850640 1554689 1-802810 98744786 59 32'4929298 *566541 1-765097'8700691 28 2'4853184'555069 1-801575 *8743375 58 33'4931829'566925 1-763900' 8699256 27 3 *4855727 555450 1 800340 98741963 57 34'4934359'567309 1-762705 *8697821 26 4 *4858270'555831 1'799107'8740550 56 35'4936889'567694 1'761511 -8696386 25 5 *4860812 556211 1-797875'8739137 55 36'4939419'568079 1-760.318'8694949 24 6'4863354'556592 1-796645'8737722 54 37'4941948'568463 1-759126 -8693512 23 7'4865895'556973 1'795416 98736307 53 38'4944476'568848 1-757936'8692074 22 ] 8 48684.36 *557355 1'794188'8734891 52 39'4947005'569233 1'756747'86906.36 21 O 9 *4870977'557736 1-792961 98733475 51 40' 4949532'569619 1-755559'8689196 20 0 10'4873517'558117 1-791736'8732058 50 41'4952060'570004 1-754372'8687756 19 1 9 11'4876057'558499 1-790512'8730640 49 42 *4954587'570389 1'753186'8686315 18 M 12 *4978597 *55W81 1*789289'8729221 48 43'4957113 *570775 1'752002 98684874 17 S 13'4881136'559262 1-788067'8727801 47 44'4959639 *571161 1-750819'8683431 16 14 *48&3674'559644 1-786847'8726381 46 45'4962165 *571547 1-749637'8681988 15 15'4886212 *5f60026 1-785628'8724960 45 46 -4964690 5719.33 1'748456.8680544 14 16'4888750'560409 1'784410'8723538 44 47'4967215 *572319 1'747276'8679100 13 17 4891288 -560791 1-783194'8722116 43 48'4969740 *572705 1-746098 *8677655 12 18'4893825'561173 1-781979'8720693 42 49 4972264 *573091 1-744921.8676209 11 19'4896361'561556 1-780765'8719269 41 50'4974787 *573478 1'743745'8674762 10' 20'4898897'561939 1-779552'8717844 40 51'4977310 *573864 1-742570'8673314 9 O 21'4901433'562321 1-778340'8716419 39 52'4979833 *574251 1-741396'8671866 8 j 22'4903968 8562704 1-777130'8714993 38 53 4982.355 *574638 1-740224'8670417 7 23'4906503'563087 1-775921'8713566 37 54.4984877 *575025 1-730053'8668967 6 24 4909038'563471 1-774714'8712138 36 55'4987399 *575412 1-737883'8667517 5 25'4911572'563854 1'773507'8710710 35 56'4989920 *575799 1-736714'8666066 4 26'4914105'564237 1-772302'8709281 34 57'4992441 *576187 1-73.5546'8664614 3 27'49166.3 8564621 1-771098'8707851 33 58'4994961 *576574 1-734.380 -8663161 2 28'4919171'565005 1'769895'8706420 32 59 -4997481 -576962 1-713214'8661708 1 29'4921704'565388 1'768694'8704989 31 60 5000000'577350 1'732050 *8660254 0 30.49242.365 565772 1767494'8703557 30 I COSINE-. ICOTN1'. TANG. SINE. I I CO'INE. ICOTANG. TANG. I INE. I, DEG. 60. DEG. 60. C. NATURAL SINES AND TANGENTS TO A RADIUS 1. 30 DEG. 30 DEG. O B SINE. TANG. COTAN_. COSINE. oI SINE. TANG. COTANG. COSINE. 0 -5000000 577350 1-732050.8860254 60 31.5077890 589436 10696534 8614815 29 1 *5002519 577738 1-730887.8k588799 59 32'50803956 589828 10695406.8613337 28 2 *5095037 *578126 1-729726.8857344 58 33 5082901 5(930221 1'694280.8611859 27 3 50075.56'578514 13728565.865,5887 57 34'5085406 5590ti13 10693155.8610380 26 4'5010073'578902 1'727406.8654-43 56 35'5087910'591005 1'692030 -8608901 25 5 ~5012591'579291 1'726247.8652973 55 3S6 5090414'591398 16950907'8607420 24 3 6'5015107'579679 1-725090 8651514 54 37'5092918'591791 106897,5 8fi605939 23 7'5017624'580068 l1723934 86500;55 53 38 5095421'592183 1'688664.8604457 22 8'5020140'580457 1'722779 86 15i 55 52 39'5097924'592576 1-687544.8602975 21 9'5022655'580846 1-721626 8647134 51 40'5100426' 592969 1068642ti 8601491 20 10'5025170'58125 1-720473 8645673 50 41 05102928'593363 1-68530S 8t00007 19 11'5027685'581624 1-719322 8644211 49 42'5105429'593756 1-684191'8598523 1S f 12'5030)199'582013 1-718172 8642748 48 63'5107930'5!94150 1-683076.8597037 17 9 13 -5032713'582403 1-717023 8641284 47 44'5110431'594543 1'681962.8595551 16 O 14'5035227'582793 1-715875 8639820 46 45'5112931'594"37 1680848 8594064 15 15'5037740'583182 1-714728 8638.35 45 46 *5115431'595331 1-679736.8592576 14 16'5040252'583572 1-71.3582 86,36889 44 47'5117930'595725 1-678625.8591088 13 M 17'5042765'583962 1'712438 8635423 43 48 5120429'596119 1'677515 -8589599 12 t 18'5045276'584352 1-711294 8633956 42 49 *5122927'596514 1-6764(06 8S588109 11 Q 19 *5047788 -584743 1-710152 8632488 41 50.5125425 596908 1-6752!'8.8586619 10 20'5050298 -585133 1-709011 8631019 40 51 5127923'59733 1-6742 8585127 9 21'5052809 -585524 1-707871 8629549 39 52 5130420'597697 1-6730:-'; 8583635 8 M 22'5055319'585914 1-706732 8628079 38 53 5132916'598092 1-6719:'. l 8582143 7 23 5057828 -586305 1-705595 8626608 37 54 5135413 598487 1-6 70878'8580649 6 24'5060338 586696 1-704458 8625137 36 55.5137908 598882 1-6697;3 8579155 5 25'5062846'587087 1-703323 8623664 35 56 *5140404 599278 1'668674'8577660 4 26'5065355'587478 1-702189 8622191 34 57'5142899 599673 1'667574 -8576164 3 27 -5067863'587870 1'701055 8620717 33 58'5145393 600059 1'6664';4'8574668 2 28 *5070370 588261 1G699923 8619243 32 59 5147887 600464 1-665.i 76 8573171 1 29'5072877'588653 1'698792 861]7768 31 60'5150381 600860 10664279'8571673 0 30 5075384'589045 1-697663 8616292 1 30 -I CoNE. COTANG. TANG. SINE. I I COSIN. COTANO. TANG. I SINE. I DEG. 59. DEG. 59. NATURAL SINES AND TANGENTS TO A RADIUS 1. 31 DEG. 31 DEG. / SIN1E. TANG. COTANG. COSTNE. / / SINE. TANG. COTANG. COSINE. I 0'5150381'600860 1-664279'8571673 60 31 *5227466'613201 1-630786.8524881 29 1'5152874'601256 1-663183'8570174 59 32 15229945 613601 1-629722.8523360 28 2.5155367'601652 1-662088 8568675 58 33'5232424 614001 16fi28659.8521839 27 3'5157859'602049 1'660994 8567175 57 34'5234903'614402 1'627597.8520316 26 4'5160351 5602445 1-659901 8565674 56 35'5237381'614803.1626536.8518793 25 5'5162842'602841 1-658809 8564173 55 36'52239859'615204 1-625476 -8517269 24 6'5165333'6032.38 1-657718 8562671 54 37'5242336'615605 1-624417.8515745 23 7'5167824'603635 1-656629 8561168 53 38'5244813'616006 1-623359.8514219 22' 8 5170314'604032 1-655540 8559664 52 39'5247290 -616407 1-622302.8512693 21 O 9 5172804'604429 1-654452 8558160 51 40'52497(iG 616809 1-621246 -8511167 20 ( 10 -5175293 604826 1-653366 8556655 50 61'5252241 617210 1-620192.8509639 19 P 11'5177782'605224 1-652280 8555149 49 42'5254717'617612 1-619138.8508111 18 M 12 5180270 -605621 1'651196 8553643 48 13'5257191 0618014 1 61 80%5 -8506582 17 ~ 13'5182758'606019 1-650112 8552135 47 14'5259665'618416 1-617033 -8505053 16 14 5185246 606417 1-649()30 8550627 46 65'5262139'618818 1-615982.8503522 15 15 5187733 -606814 1-647949 8549119 45 6'5264613'619221 1-614932 -8501991 14 16 5190219 607213 1-646868 8547609 44 67'5267085'619623 1-613882.8500459 13 17 5192705 0607611 1-645789 8546099 43 68'5269558'620026 1-612834.8498927 12 18 5195191 608009 1-644711 8544588 42 19 -5272030'620 29 1-611787.8497394 11 19 5197676 608408 1'643633 8543077 41 50'5274502'620832 1-610741 -8495860 10 ( 20 5200161 608806 l'642557 8541564 40 51'5276973'621235 1-609696 8,194325 9 O 21 5202646 609205 1'641482 8540051 39 52'5279443 -621638 1'608652'8492790 8 22 52051 30 609604 1-640408 8538538 38 53.5281914'622041 1-607609.8491254 7 23 5207613 610003 1'639.335 8537023 37 54'5284383 -622445 1-606567 -8489717 6 24 5210096 610402 l1638263 8535508 36 55'5286853 -622848 1-605526'8488179 5 25 5212579 610801 1'637191 8533992 35 56'5289322'623252 1'604485'8486641 4 26 5215061 611201 1-636121 8532475 34 57 -5291790'623656 1-603446 -8485102 3 27 5217543 611601 1-635052 8530958 33 58'5294258 *624060 1-602408'8483562 2 28 5220024 612()0 1'633984 8529440 32 59'5296726.624465 1-601370'8482022 1 29 5222505 612400 1'632917 8527921 31 60'5299193 -624869 1-600334'8480481 0 30 5224986 612800 1'631851 8526402 30 I COSINE. COTANG. TANG. SINE. I i COSINE. COTANG. TANG. SINE. I I DEG. 58. DEG. 58 -a NATURAL SINES AND TANGENTS TO A RADIUS 1. - 32 DEG. 32 DEG. I SINE. TANG. COTANG. COSINE. I. I SINE. TANG. COTANG. COSINE. I 0'5299193 -624869 1-600334.8480481 60 31 55375449 *637479 1-568678.8432351 29 1'5301659'625273 1'599299.8478939 59 32'537790)2'637888 1-567672'8430787 28 2'5304125'625678 1-598264.8477-397 58 33 5380(354 *638297 1-566666.8429222 27 3'5306591'626083 1-5972,31 -8475853 57 34'5382806'6G8707 1 565662 ~8427657 26 4 *5309057'626488 1-596198.8474309 56 35'5385257'639116 1-564659.8426091 25 5 *5311521'626893 1-595167.8472765 55 36' 5387708'639526 156i3656.8424524 24 sr 6'5313986'627298 1-594136.8471219 54 37'5390158'639936 1-562654 -8422956 23 m 7'5316450'627704 1-593107 98469673 53 38'5392608'640346 1-561654'8421388 22 0J 8'5318913.628109 1-592078.8468126 52 39'5395058'640756 1-560654.8419819 21 9'5321376'628515 1-591050.8466579 51 40'5397507'641167 1-559655'8418249 20 10'5323839'628921 1-590023 -8465030 50 41'5399955'641577 1-558657'8416679 19, 11'5326301'629327 1-688997.8463481 49 42'5402403'641988 1'557660.8415108 18 12'5328763'629733 1'587973.8461932 48 43'5404851'642399 1-556663 8413536 17 13'5331224'630139 1-586949 -8460381 47 44'5407298'642810 1-555668 -8411963 16 O 14'5333685'630546 1-585926.8458830 46 45'5409745'643221 1.-554674.8410390 15 15.5336145'630953 1.584904.8457278 45 46.5412191'643632 1-553680'8408816 14 16'5338605'631,359 1,5&3883 -8455726 44 47 -5414637'644044 1.552688 -8407241 13 W 17.5341065'631766 1.582862 -8454172 43 48 -5417082'644456 1-551696 -8405666 12 Z 18'5343523'632173 1-581843.8452618 42 49'5419527'644867 1-550705'8404090 11 2 19'5345982'632581 1-580825.8451064 41 50'5421971'645279 1-549715 -8402513 10 20'5348440 632988 1-579807'8449508 40 51'5424415'645691 1548726 8400936 9 M 21 5350898 633395 1-578791'8447952 39 52'5426859'646104 1-547738'8399357 8 M 22 *5353355'633803 1-577776'8446395 38 53'5429302'646516 1-546751 8397778 7 09 23 -5355812'634211 1-576761'8444838 37 54'5431744'646929 1-545764'8396199 6 24'5358268 -6346i9 1-575747'8443279 36 55'5434187'647341 1-544779 -8394618 5 25 5.360724'635027 1-574735'8441720 35 56 -5436628'647754 1-543794'8393037 4 26'5363179'635435 1'5t3723'8440161 34 57'5439069'648167 1] 542810'8391455 3 27'5365634'635844 1-572712'84.38600 33 58'5441510'648W 1-541828 -8389873 2 28'5368089'636252 1'571702'8437039 32 59'54143951'648994 1-540846'9388290 1 29'5370543'636661 1-570693'84.5477 I 31 60'5446390'649407 1-539865 -8386706 0 30'5372996'637070 1'569685'8433914 30 __ / COSINE. COTANG.1 TAN.' SINE. I I COSINE. COTANG. TANG. SINE. I aEG. o7. DEG. 57 NATURAL SINES AND TANGENTS TO A RADIUS 1. 33 DEG. 33 DEG. SINE. TANG. COTANG. COSINE. / I SINE. TANG. COTANG. COSINE. I 0 35446390.649407 1'539865'8386706 60 31 5521795'662304 1'509880.8337252 29 1 5448830'649821 1-538884 -8385121 59 32 5624220'662722 1-508927 *8135646 28 2 35451269 -650235 1-537905'8383,536 58 33.5526645'663141 1-507974.8334038 27 3 35453707'6,50649 1-536927'8381950 57 34'5529069 6i6.3G560 1-507022.8332430 26 4 *5456145 -651053 1-535949 8380363 56 35 5531492 G663979 1'506071 -8330822 25 5 35458583'651477 1-534972'8378775 55 36'5533915'664398 1'505121.8329212 24 6 *5461020'651 89l 1-533996'8377187 54 37'5536,338' 664817 1'504171 88327602 23 7 *5463456 -652306 1-533021 -8375598 53 38'5538760 *665237 1'503222 *8325991 22 8 ~5465892'652721 1-532047 -8374009 52 39'5541182'665657 1-502275 -8324380 21 O 9 *5468.328'653136 1-531074 -8372418 51 40'5543603 666076 1'501328.8322768 20 0 10'5470763.653551 1-530102'8370827 50 41'5546024'666496 1-500382 -8321155 19 m 11'5473198'653966 1-529130'8369236 49 42'5548444.666917 1-499436'8319541 18 M 12 -5475632'654381 1'528160 *8367643 48 43'5550864'667337 1-498492 *8317927 17 13 -5478066'654797 1-527190'8366050 47 44'5553283'667758 1-497548 8316312 16 14'5480499'655212 1'526221 -8364456 46 45'555.5702'668178 1'496605'8314696 15 Q 15 -5482932'655628 1'525253'8362862 45 46'5558121.668599 1'495663 -8313080 14 16'5485.365'656044 1-524286'8361266 44 47'5560539'669020 1-494722 *8311463 13 17'5487797'656460 1'523320'8.359670 43 48'5562956 *669441 1'493782 -8309845 12 18'5490228'656877 1'522354'8358074 42 49'5565.373 *669863 1'492842'8308226 11 19'5492659 *657293 1-521389'8356476 41 50 -5567790'670284 1'491903'8306607 10' 20'5495090'657710 1'520426'8354878 40 51'5570206'670706 1'490965 *8304987 9 O 21'5497520 -658127 1 519463'8353279 39 52'5572621'671128 1'490028 -8303366 8 t 22'5499950 -658544 1-518501'8351680 38 53'5575036'671550 1 489092 8301745 7 23 *5502379'658961 1-517540'8350080 37 54'5577451'671972 1-488157 8300123 6 24 *5504807'659378 1-516579'8348479 36 55'5579865'672394 1'487222 8298500 5 25 *5507236'659796 1-515620'8346877 35 56 *5582279.672816 1'486288 *8296877 4 26 ~5509663 -660213 1'514661'8345275 34 57'5584692'673239 1-485,355'8295252 3 27 *5512091'660631 1'513703'8343672 33 58 -5587105 *673662 1'48442.3'8293628 2 28'5514518 -661049 1'512746'8.342068 32 59'5589517 -674085 1-483491 8. 292002 1 29 *5516944'661467 1-511790 8.340463 31 60'5591929'674508'1482561'8290376 0 30 *5519370'661885 1-510835'8338858 30 I COSINE. COTANG. TANG. I INE. I I COSINE. COTANG. TANG. SIN. DEG. 56. DEG. 56. CO NATURAL SINES AND TAN(GENTS TO A RADIUS 1. r-.~4 DEw. 34 DEG. I INE. TANB. COTANG. s COSINE.I II / SINE. TANG;. COTAKrG. COSINE. 0 -5591929 674508 1.482561 8290.376 60 31.5666459.687709 1Y4541V2 239614 29 1 *5694340.674931 1.481,631.828749 59 132.~66~885 688137 1.453197 ~ 23,965 28 2 ~5596751 ~6753.55 1 480702 ~8287121 68 L133 ~5671252 ~688566 1-452292 8s236316 27 3 *5599162 *675779 1.479773 ~82549- 57 34 5673648.688995 1451388 8234666 26 4 -5601572'676202 1-478846.828.3864 56 35'5676043'689424 1'450485 -8233015 25 5'5603981'676626 1-477919.8282234 55 36 -5678437 *689853 1'449582.8231364 24 6 *5606390'677050 1'476993.8280603 54 37 65680832 *690283 1'448680 8229712 23 ] 7 5608798'677475 1-4,76068 -8278972 53 38 -5683225'690712 1-447779.8228059 22 8'5611 206'677899 1-475144.8277340 52 39'5685619 *691142 1-446879 -8226405 21 9'5613614'678324 1'474221 -8275708 51 40'5688011 *691572 1'445980.8224751 20 10 -5616021 -678749 1'473298 -8274074 50 41 [5690403'692002 1-445081.8223096 19 11 -5618428 -679174 1'472376.8272440 49 42 {5692795 -692432 1'444183.8221440 18 12 -5620834'679599 1'471455.8270806 48 43 *5695187'692863 1'443286 -8219784 17 13'5623239'680024 1'470535 -82fi9170 47 44 *5697577 *693293 1'442389.8218127 16 O 14 -5625645'680450 1'469615'8267534 46 45 *5699968' 693724 1'441494.82164f9 [15 15'5628049'680875 1'468696'8265897 45 46'5702.357'694155 1'44P 599.8214811 14 16'5630453'681301 1'467778' 8264260 44 47 *5704747 *694586 1'439704 -8213152 13 [ 17'5632857'681727 1-466861 -8262622 43 48 *5707136 [695018 1'4.38.811 -8211492 12 18 -5635260 -682153 1'465945'8260983 42 49 *5709524 {695449 1'437918'8209832 11 19 5637663'682580 1-465029'8259343 41 50 *5711912 {695,81 1'4.37026 -82()8170 10 w 20'5640066'683006 1-464114'825770.3 40 51 5714299 *696313 1'436! 35'820609 9 21'5642467'68&333 1-463200 -8256062 39 52 ['5716686 { 69a,7 45 1'43,5245 -8204846 8 p 22 5644869'683X60 1'42287 -8254420 38 53'5719073'6J7177 1'434355 -8203183 7 ] 23 *5647270 *684287 1-461374 -8252778 37 54 *5721459 *697609 ]' 43466 - 8201519 6 m' 24'5649670 -684714 1'460463 -8251135 36 55'5723844'691042 1'432578 -8199'54 5 25 -5652070'685141 1'459552 -8249491 35 56 [c -52622 9 *i694,474 1-431690'81981 89 4 26 *5654469 *685569 1'458642'8247 847 34 57 *572,8(SG14 *'699907 1'430X03'819G523 3 27'5656868'6&5996 1'457732'824620' 3.3 5S -57:~i)9i8 *6!)99340 1'429917'819 4,%.56 2 28'5659267'686424 1'456824'8 2-4-55G 32 59 *5733381 *699774 1'4290)32'8 93i,9 9 1 29'5661665'686852 1'4559)16'824'2 ()9 3l, 60'5735764 *,00207 1-42814S *SI )]5i 0 30'5664062'687281 1'455009'8??(i2 30 -__ _ __~ _,., _ _ / COSINE. COTANT((. TASG.. IN-E. / /i COSIB1'. CCTASG.'I A~(. SI4i~. / Dr(/l. 55 ]3~. ~. NATURAL SINES AND TANGENTS TO A RADIUS 1. 35 DEG. 35 IEG. SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTAI G. COSINE. 0 *5735764'700207 1-428148'8191520 60 31'5809397'713732 1-401086 *8139466 29 1'5738147'700641 1-427264'8189852 59 32'5811765 -714171 1'400224'8137775 28 2 -5740529'701074 1-426,381'8188182 58 33'5814132'714610 1-399363 -8136084 27 3'5742911'701508 1-425498'8186512 57 34'5816498 -715050 1-398503'8134393 26 4'5745292'701943 1-424617'8184841 56 35' 5818864'715489 1-397644'8132701 25 5 *5747672'702377' 1-423736'8183169 55 36'5821230 -715929 1-396785 -8131008 24 6'5750053'702811 1-422856'8181497 54 37'5823595'716369 1-395927.8129314 23 7'5752432'703246 1'421976'8179824 53 38'5825959 -716810 1'395069'8127620 22 _ 8'5754811 -703681 1-421097'8178151 52 39'5828323'717250 1-394213 -8125925 21 0 9'5757190'704116 1'420220'8176476 51 40'5830687 -717691 1-39.357'8124229 20 0 10'5759568'704551 1-419342'8174801 50 41'5833050'718131 1-392501'8122532 19 m 11'5761946' 704986 1-418466'8173125 49 42'5835412'718572 1-391647'8120835 18 M 12 -5764:32.3 705422 1417590 *8171449 48 43'5837774 -719014 1-390793'8119137 17 3 13 -5766700 1 705858 1-416715'8169772 47 44'5840136'719455 1-389940'8117439 16 14'57(69076 -706294 1-415840.'8168094 46 45' 5842497'719897 1-389087'8115740 15 C 15'5771452'706730 1-414967'8166416 45 46'5844857'720338 1-388235'8114040 14 16'5773827'707166 1-414094'8164736 44 47'5847217 -720780 1-387384 -8112339 13 17'5776202'707602 1-413222'8163056 43 48 58049577'721222 1-386534'8110638 12. 18'5778576 -708039 1-412350'8161376 42 49 5,851936'721665 1-385684'8108936 11 19'5780950'708476 1-411479'8159695 41 50 -5854294'722107 1-384835'8107234 10 - 20'578323'708913 1-410609'8158013 40 51'5856652'722550 1-383986'8105530 9 O 21 -5785696'709.350 1-409740'8156330 39 52 -5859010 *722993 1-383139 -8103826 8.z 22'5788069'709787 1'408871'8154647 38 53'5861367 *723436 1'382292 -8102122 7 23'5790440 -710225 1-408003'8152963 37 54'5863724 -723879 1-381445'8100416 6 24 -5792812'710ti63 1-407136'8151278 36 55'5866080 *724322 1-380600 -8098710 5 25'5795183'711100 1-406270'8149593,35 56'58684:45'724766 1-379755'8097004 4 26'5797553'711539 1-405404'8147906 34 57'5870790 -725210 1-378910'8095296 3 27'5799923'711977 1-404539'8146220 33 58'5873145 -725654 1'378067'8093588 2 28'5802292 -712415 1-403674'8144532 32 59 -5875499'726098 1-377224 -8091879 1 29'5804661'712854 1-402811'8142844 31 60 -5877853'726542 1-376381'8090170 0 30'5807030 -713293 1'401948'8141155 30 I COSINE. ICOTNG. TANG SINE. / I COSINE. COAG TA NG.. SINE. I DEG. 54. DEG. 54. > NATURAL SINES AND TAMGENTS TO A RADIUS 1.,-~ ['O 36 DrG. 36 DEa. ~.~..... t........ I SINE. TANG., C0'TA_~O. COSINE. / / SI?_ E: TAN?____L_. C0TANG. COSI_.NE_.L... i 0'5877853.726542 1'376381.8090170 60'31'5950566.740411 1'350600 -8036838 29 1'5880206.726987 1-375540.8088460 59 32'5952904'740861 1'349779 -8035107 28 2 -5882558.727431 1'374699.8086749 58 33'5955241.741312 1-348958 -8033375 27 3'5884910.727876 1'373859.8085037 57 34'5957577.741763 1'348139.8031642 26 4'5887262'728321 1'373019.80&3325 56 35'5959913.742214 1'347319.8029909 25 5'5889613.728767 1'372180,8081612 55 36'5962249'742665 1'346501'8028175 24 6'5891964'729212 1'371342.8079899 54 37'5964584'743117 1'345683 -8026440 2,3 7'5894314'729658 1'370504.8078185. 53 38'5966918.743568 1'344865 -8024705 22 8'5896663 -730104 1'369667.8076470 52 39'5969252'744020 1'344049.8022969 21 9 -5899012.730550 1'368831.8074754 51 40'5971586'744472 1'343233.8021232 20 10'5901361 -730996 1'367995'8073038 50 41'5973919'744924 1'342417.8019495 19 11'5903709'731442 1'367161 -8071321 49 42'5976251.745377 1'341602 -8017756 18 12'5906057'731889 1'366326'8069603 48 43'5978583'745829 1'340788.8016018 17 13'5908404'732336 1'365493'8067885 47 44'5980915'746282 1'339975'8014278 16 14'5910750 -732783 1'364660'8066166 46 45'5983246 -746795 1'339162.8012538 15 15'5913096'733230 1'363827'8064446 45 46'5985577'747!88 1'338:550'8010797 14 16 -5915442.733677 1'362996 -8062726 44 47'5987906'747642 1-337538'8009656 13 17'5917787'734125 1'362165'8061005 43 48'5990236'748095 1'336727'$007314 12 18'5920132'734573 1'361335'8059283 42 49'5992565'748549 1'3.35917'8005571 11 19'5922476.735021 1'360505 -8057560 41 50'5994893'749003 1'.335107'8003827 10 20'5924819.735469 1'359676'8055837 40 51'5997221'749457 1-334298 -8002083 9 21 -5927163.735917 1'358848'8054113 39 52 -5999549'749911 1'333490'8000338 8 22'5929505.736366 1'358020'8052389 38 53'6001876'750366 1'332682'7998593 7 23'5931847'736814 1'357193'8050664 37 54'6004202'750821 1'331875'7996847 6 24'5934189'737263 1'356367'8048938 36 55 -6006528'751276 1'331068'7995100 5 25'5936530.737712 1'355541'8047211 35 56 -6008854'751731 1'330262'7993352 4 26 5938871.738162 1-354716 -8045484 34 57'6011179'752186 1'329457.7991604 3 27'5941211.738611 1'353891'8043756 33 58'6013503'752642 1'328653 -7989855 2 28'5943550.739061 1'353068'8042028 32 59'6015827'753098 1'327848 -7988105 1 29'5945889'739511 1'~2244'8040299 3l 60'6018150'753554 1'327044'7986355 0 30'5945228.739961 1'351422'8038569 30 COSINE. COTANG. TANG.. I! COSINE... SINE. COTANG. TANG. SINE. I....... Dr, G. 53. DEG. 53. NATURAL SINES AND TANGENTS TO A RADIUS 1. 87 DEG. 37 DEG. SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0 6018150 *753554 1-3-27044'7986355 60 31 6089922'767789 1-302440'7931762 29 1.6020473 *754010 1-326242.7984604 59 32'6092229 *768251 13301656 *7929990 28 2 *6022795 *754466 1-325439'7982853 58 33 *6094535 *768714 1-300873 *7928218 27 3 ]6025117'754923 1-324638'7981100 57 34'6096841 769177 1'300090 *7926445 26 4'6027439 755379 1-323837'7979347 56 35'6099147'769640 1'299308 *7924671 25 5 *6029760 *755836 1-323036'7977594 55 36 6101452 *770103 1-298526'7922896 24 6'6032080 *756294 1-322237'7975839 54 37'61303756 770567 1-297745 *7921121 23 7'6034400 *756751 1-321437'7974084 53 38'6106060 771030 1-296964 *7919345 22 8'6036719 *757209 1-320639 *7972329 52 39'6108363'771494 1-296185'7917569 21 O 9'6039038'757666 1-319841'7970572 51 40'6110666'771958 1-295405'7915792 20 0 10'6041356'758124 1-319044'7968815 50 41'6112969'772423 1'294627'7914014 19 [: 11'6043674'758582 1-318247'7967058 49 42'611.5270 [772887 1-293848'7912235 18 m 12'6045991'759041 1-317451'796.5299 48 43'6117572'773352 1-293071'7910456 17 3 13'6048308'759499 11316655'796:3540 47 44 6119873'773817 1-292293'7908676 16 14'6050624'759958 1-315861'7961780 46 45'6122173'774282 1-291517 *7906896 15 15'6052940'760417 1-315066'79600)20 45 46 6124473'774748 1-290742 7905115 14 16'6055255'760876 1-314273'7958259 44 47'6126772'775213 1-289966'7903333 13 17 6057570 *761336 1-313480 *7956497 43 48 *6129071'775679 1-289192 *7901550 12 18'6059884'761795 1-312687'7954735 42 49'6131369'776145 1'288418'7899767 11 z 19'6062198'762255 13i1895 *'7952972 41 50'6133666'776611 1-287644'7897983 10 w 20'6064511'762715 1-311104'7951208 40 51'6135964'777078 1-286871'7896198 9 O 21 *6066824'76.3175 1 310314'7949444 39 52 *6138260'777544 1*286099'7894413 8 22'6069136'763636 1-309523'7947678 38 53'6140556'778011 1-285327 7892627 7 23'60714,17 764097 1-308734'7945913 37 54'6142852'778478 1-284556'7890841 6 24'6073758'764557 1-307945'7944146 36 55'6145147'778946 1-283786'7889054 5 25'6076069'765018 1307157'7942379 35 56'6147442'779413 1-283016'7887266 4 26'6078379'765480 1-306369'7940611 34 57 -6149736'779881.1-282246'7885477 3 27'6080689.765941 1-305582'79388,13 3 58 66152029 78S0349 1-281477.7883688 2 28'608299'i766403 1.304,796'7937074 32 59'6154322'780817 1-280709'7881898 1 29'6085306'766864 1304010'7935304 31 60'6156615'781285 1279941'7880108 0 30'6087614 ~767327 13303225'7933533 30 I COSINE. COTANG. TA-NG. SINE. I COSINE. COTANG. TANG. SINE. I DEGr. 52. DEG. 52. NATURAL SINES AND TANGENTS TO A RADIUS 1. 38 DEG. 38 DEG. t OD I SINE. TANG. COTANG. COSINE. / I SINE. TANG. COTANG. COSINE. I 0.6156615 781285 15279941 -7880108 60 31'6227423'795911 1-256421.7824270 29 1 *6158907'781754 1'279174.7878316 59 32'6229698'796386 1-255672 -7822459 28 2'6161198'782222 1-278407.7876524 58 33'6231974'796861 1-254922'7820646 27 3'6163489'782691 1-277641 -7874732 57 34'6234248'797337 1.254174'7818833 26 4'61635780.783161 1'276876'7872939 56 35'6236522'797813 1-253426'7817019 25 5.6168069'783630 1-276111'7871145 55 36'6238796'798289 1-252678'7815205 24 - 6'6170359'784100 1-275347'7869350 54 37'6241069'798765 1'251931'7813390 23 > 7'6172648 -784570 1-274583.7867555 53 38'6243342 *799242 1'251184 *7811574 22 M 8'6174936.785040 1.273820'7865759 52 39'6245614'799719 1'250438'7809757 21 9'6177224.785510 1'273057'7863963 51 40'6247885'8(00196 1'249693'7807940 20 10'6179511'785980 1-272295'7862165 50 41'6250156'800673 1'248948 7806123 19 11.6181798.786451 1-271534.7860367 49 42'6252427'801151 1-248204'7804304 18 - 12'6184084 *786922 1-270773'7858569 48 43'6254696'801628 1-247460 -7802485 17 9 13'6186370.787393 1'270013'7856770 47 44'6256966'802106 1'246716 -7800665 16 O 14'6188655.787864 1-269253'7854970 46 45'6259235'802584 1-245974 *7798845 15 15'6190939.788336 1-268494'7853169 45 46'6261503'803063 1-245232'7797024 14 t 16'6193224 -788808 1-267735' 7851368 44 47'6263771'803541 1-244490'7795202 13 M 17'6195507.789280 1-266977:7849566 43 48'6266038'804020 1-243749'7793380 12 t 18.6197790'789752 1-266219'7847764 42 49'6268305'804499 1-243008'7791557 11 19'6200073'790224 1*2605462'7845961 41 50'6270571'804979 1.242268'7789733 10 " 20'6202355'790697 1-264706'7844157 40 51'6272837'805458 1-241529'7787909 9 m 21'6204636'791170 1-263950'7842352 39 52 -6275102'805938 1-240790'7786084 8 22'6206917.791643 1-263195'7840547 38 53'6277366'806418 1-240051'7784258 7 23'6209198.792116 1-262440'7838741 37 54'6279631'806898 1-239313 -7782431 6 24'6211478'792590 1-261686'7836935 36 55 -6281894'807378 1'238576'7780604 5 25'6213757'793064 1-260932'7835127 35 56'6284157'807859 1-237839'7778777 4 26'6216036'793537 1-260179'7833320 34 57 -6286420'808340 1-237103'7776949 3 27 -6218314'794012 1-259426'7831511 33 58 -6288682'808821 1-2.36367'7775120 2 28'6220592'794486 1'258674'7829702 32 59'6290943'809302 1-235631'7773290 1 29'6222870'794961 1'257923'7827892 31 60'6293204 -809784 1'234897'7771460 0 30'6225146'795435 1-257172'7826082 30 I I COSINE. COTANG.7 TANG. SINE. I I COSINE. COTANGr. TANGI. SINE. J DEo. 51. DEG. 51. NATURAL SINES AND TANGENTS TO A RADIUS 1. 39 DEG. 39 DEG. SINE. TANG. COTANO. COSINE. I SINE. TANG. COTANG. COSINE. 0'6293204 -809784 1-234897 -7771460 60 31 -6363026'824825 I 212378'7714395 29 1 -6295464'810265 1-234162'7769629 59 32'6365270'825314 1'211660'7712544 28 2'6297724'810747 1-233429'7767797 58. 33'6367513'825803 1.210942 -7710692 27 3'6299983 -811230 1-232696'7765965 57 34 -63(69756'826292 1-210225'7708840 26 4 -6302242'811712 1-231963'7764132 56,35'6371998 82(i782 1-209508'7706986 25 5'6304500'812195 1-231231'7762298 55 36'6.374240'827271 1-208792 7'705132 24 6'63067568 812678 1'230499'7760464 54 37 6.37(i481'827762 1-208076'7703278 23 7'6309015'813161 1'229768'7758629 53 38 -6378721'828252 1'207361'7701423 22 8 6311272'813644 1-229038'7756794 52 39 -6380961'828742 1-206646 7699567 21 9 6313528'814128 1-2283(8'7754957 51 40 6:83201 -829233 1-205932'7697710 20 Q 10'6315784'814611 1-227578 -7753121 50 41'6385440'829724 1'205219'7695853 19 m 11'6318039'815095 1'226849'7751283 49 42'6387678.830216 1'204505'7693696 18 m 12'6320293'815580 1-226121'7749445 48 43 6:389916'830707 1'203793'7692137 17 1 13'6322547'816064 1'225393'7747606 47 44'6392153'831199 1-203081'7690278 16 Q 14'6324800'816549 1-224665'7745767 46 45'6394390 *'91691 1'202369'7688418 15 O 15 -6327053'817034 1-223938'7743926 45 46'6396626'832183 1-201658'7686558 14 16'6329306'817519 1-223212'7742086 44 47'6398862'832675 1-200947'7684697 13 17'6331557'818004 1-222486'7740244 43 48'6401097'833168 1-200237'7682835 12 18'6333809'818490 1'221761'7738402 42 49'6403332'833661 1-199527'7680973 11 19'6336059'818976 1-221036 -7736559 41' 50'6405566'834154 1-198818'7679110 10 H 20'6338310'819462 1-220(312 -7734716 40 51'6407799'834648 1-198109'7677246 9 O 21'6340559'819948 1-219588'7732872 39 52'6410032 -835141 1-197401 -7675382 8 22'6342808'820435 1-218865 -7731027 38 53'6412264'835635 1-196693'7673517 7 23'6345057'820922 1-218142'7729182 37 54'6414496'836129 1-195986'7671652 6 24 -6347305'821409 1'217419 -7727336 36 55'6416728'836624 1] 195279 -7669785 5 25'6349553'821896 1-216698'7725489 35 56'6418958'837118 1-194573'7667918 4 26'6351800'822384 15215976'7723642 34 57'6421189'837613 1'193867 -7666051 3 27'6354046'822871 1'215256'7721794 33 58'6423418'838108 1-193162'76641&3 2 28'6356292'821.359 1'214535 -7719945 32 59 -6425647'838604 1'192457'7662314 1 29'6358537 8238i47 1-213816 -7718096 31 60 -6427876'839099 1'19175.3 7660444 0 30 -6360782'824336 1'213097'7716246 30 I COSINE. COTAN AG. TAN. SINE. I I COSINE. COTANG TANG. SINE. / DEG. 50. DEG. 50.,. NATURAL SINES AND TANGENTS TO A RADIUS 1. 40 DEG. 40 DEG.O SI;\1'3. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. I 0 *642876 839099 1-191753'7660444 60 31 -6496692'854583 1-170160.7602170 29 1 6430104 839595 1.191049 *7658574 59 32 *6498903 855087 1-169471 -7600280 28 2 6412332 840091 1190346.7656704 58 33 66501114 855591 1168782 759859 27 3 6434559 840587 1'189643'7654832 57 34'6503324 -856095 1.168094'7596498 26 4 6436785 841084 1'188941 *7652960 56 35 *6505533 -56599 1.167407'7594606 25 5 -6439011 841581 1'1&8239'7651087 55 36 *6507742.857103 1-166720'7592713 24. 6'6441236 842078 1-187538 *7649214 54 37'6509951 -857608 1-166033 *7590820 23 t 7 6443461 842575 1.1868.37 {7647340 53 38 -6512158 85S113 1-165347 *7588926 22 W 8.6445685 843073 13186136 *7645465 52 39 6514366'858618 1164661 7587031 21 9 6447909 84.3570 1'185437'764.3590 51 40 *6516572 8.59124 1-163976 -7585116 20 10 *'6450132 ]844068 1'184737'7641714 50 41'6518778 -859629 1-163291'7583240 19 11i 6452355 844567 1'184038'7639838 49 42'6520984'860135 1-1626107'7581343 18 12'6454577 845065 1'183340 7637960 48 43'6523189'860641 1-161923'7579446 17 13 *6456798 845564 1'182642'7636082 47 44'6525394'861148 1-161240'7577548 16 14 -6459019 846063 1'181944'7634204 46 45'6527598'861655 1-160557'7575650 15 15 -6461240 846562 1'181247 -7632325 45 46'6529801 -862162 1'159874'7573751 14 16'6463460 847062 1'180551'7630445 44 47 *6532004 -862669 1'159192 -7571851 13 1 17 *6465679 847561 1'179855'7628564 43 48'6534206 863176 1158511 I 7569951 12 t 18'6467898 848061 1'179159'7626683 42 49'6536408 863684 1157830'7568050 11 6 19 6470116 848561 ['178464'7624802 41 50'6538609'864192 1-157149'7566148 10 20 6472334 8490(;2 1'177769'7622919 40 51 -6540810'864700 1'15640:9'7564246 9 21 6474551 849561 1-177075'7621036 39 52'6543010'865209 1 15-5789'7562343 8 M 22'6476767 850064 1'176382'7619152 38 53'6545209'865718 1'155110 1'7560439 7 23'6478'84 850565 ['175688'7617268 37 54'6547408'866227 1-154431 7558535 6 24'6481199'851066 1'174996'7615385 36 55'6549607 866736 1-153753'7556630 5 25'6483414'851568 ['-174303'7613497 35 56'6551804'867246 1-153075 *7554724 4 26'6485628'&82070 1'173612'7611611 57'6554002 867755 1152397'7552818 3 27 6487842 852572 1 172520'7009724 33 58'6556198'868265 1-151721'7550911 2 28'6490056 8&53075 1'172229'7607837 32 59'6558595'868776 1-151044'7549004 1 29 6492268 883577 1-171539'7605949 31 60'6560590'869286 1-150368 7547096 0 30 6494480'854080 1'170849'70)4060 30 I COSINE. N. TA. INE. 7I COSINE. COTANG. TANG. SINE,. DEG. 49. DEG. 49. NATURAL SINES AND TANGENTS TO A RADIUS 1. 41 DEG. 41 DEG. I SINE. TANG. COTANG. COSINE. I SINE. TANG. COTANG. COSINE. I 0 *6560590 *869286 1-150368 -7547096 60 31'6628379 -885244 1-129632'7487629 29 1 ~GC562785'869797 1-149692 -7545187 59 32'6630557.885763 1-128970.7485701 28 2 -6564980'870308 1'149017.7543278 58 33'6632734 -886282 1-128308'7483772 27 3 *6567174'870820 1 148342 -7541368 57 34'6634910'886801 1-127647'7481842 26 4 -6569367 -871331 1'147668.7539457 56 35'6637087 -887321 1.126987'7479912 25 5.6571560'871843 1'146994'7537546 55 36'6639262 8-&7841 1-126327 7'477981 24 6.6573752'872355 1-146321.7535684 54 37'6641437'88361 1'125667'7476049 23 7.6575944'872868 1'145648'7533721 53 38'6643612'888882 1'125008'7474117 22 8.6578135'873380 1'144976'7531808 52 39'6645785'889403 1-124349'7472184 21 0 9 *6580326'873893 1-144304'7529894 51 40'6647959 *889924 1-123690 *7470251 20 Q 10 -6582516.874406 1'143632'7527980 50 41 *6650131'890445 1-123032 7468317 19 W 11 -6584706'874920 1-142961'7526065 49 42'6652304'890967 1-122375'7466382 18 M 12'6586895'875433 1'142290'7524149 48 43 *6654475'891489 1'121718 7464446 17 H 13.6589083'875947 1'141620'7522233 47 44'6656646 -892011 1-121061'7462510 16 Q 14.6.591271'876462 1:140950 7520316 46 45'6658817'892534 1-120405 *7460574 15 O 15.6593458'876976 1'140281'7518398 45 46'6660987'893056 1'119749'7458636 14 16'6595645 877491 1-139612'7516480 44 47'6663156'893579 1-119094'7456699 13 e 17.6597831'878006 1-1.38944 -7514561 43 48'6665325'894103 1-118439 17454760 12 P 18.6600017'878521 1-138276'7512641 42 49'6667493 -894626 1-117784'7452821 11 Y4 19'.6602202'879037 1-137608'7510721 41 50'6669661'895150 1-117130'7450881 10 A 20 -6604386 87 9552 1-136941'7508800 40 51'6671828 -895674 1-116476'7448941 9 O 21.6606570 -8.1068 1-136274'7506879 39 52'6673994'896199 1'115823 *7446999 8 22 -6608754'880585 11.35608'7504957 38 53'6676160 -896723 1-115170'7445058 7 23'6610936'881101 1-134942'750.3034 37 54'6678326'897248 1-114518'7443115 6 24 -6613119 -881618 1'134277'7501111 36 55'6680490'897773 1-113866'7441173 5 25.6615300'882135 1'133612'7499187 35 56'66855'898299 1-113214'7439229 4 26.6617482'-&82653 1-132947'7497262 34 57'6684818'898825 1-112563'7437285 3 27 -66196G2'&83170 1-132283'74953.37 33 58'6686981.899351 1-111912'7435340 2 28'6621842'883688 1li.31620'7493411 32 59'6689144 *899877 1-111262.7433394 1 29'6624022'5S4206 11.30957'7491484 31 60'6691306'900404 1-110612'7431448 0 30.6626200'884725 1.130294'7489557 30 I COSINE. COTANG. TANG,I SINE. I I COSINE. COTANG. TANG. SINE. I DEGy. 48. DEG. 48. " NATURAL SINES AND TANGENTS TO A RADIUS 1. _ 42 DEG. 42 DEG. SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE I 0 6691306:900404 1-110612'7431448 60 31 -6758046'916866 1'090671 -7370808 29 1 *6693468'900930 1'109963'7429502 59 32 6760190'917402 1'090034'7368842 28 2'669i5628. 901458 1.109314 7427554 58 33 6762333 917937 1.089398 -7366875 27 3 66 97789 901985 1-108665 *7425606 57 34 6764476.918474 1.088762.7364908 26 4'6699948'902513 1-108017 7423658 56 35 6766618'919010 1.088126 *7362940 25 5 6702108'903041 1-107369'7421708 55 36'6768760'919547 1-087491 [7360971 24 6 -6704266 -903569 1-106721'7419758 54 37 6770901 *920084 1-086,857 7359002 23 7'6706424'904097 1'106075'7417808 53 38 6773041'920621 1'086222 -73157032 22 M 8'6708582'904626 1-105428 7415857 52 39'6775181'921159 1'08 5588'7.355061 21 9 -6710739 -905155 1-104782'7413905 51 40'677320'921696 1084955 -7353090 20 10 -6712895 -905685 1-104136 7411953 50 41'6779459 *922235 1-0845322 *7351118 19 11 *6715051 *906214 1-103491 *7410000 49 42 6781597'929773 1*083089 *7349146 18 12 6717206 906744 1-102846'7408046 48 43 *6783734 *923312 1 08.3057 *7347173 17 13 *6719361'907274 1-102201 7406092 47 44'6785871 923851 1082425'7345199 16 O 14'6721515'907805 1-101557 7404137 46 45 6788007'924390 1081793'7343225 15 15 *6723668 *90336 1-100914'7402181 45 46'6790143 *924930 10f81162'7341250 14 16'6725821 *908867 1*100270'7400225 44 47 6792278'925470 1*080532'7336)275 13 M 17'6727973'909398 1-099628'7398268 43 48'6794413'926010 1 079901 73:47299 12 tZ 18'6730125'909930 1-098985'7396311- 42 49'6798547'926550 1-079271'7315322 11 19'6732276'910461 1-09843'7394353 41 50'6798681'927091 1*078642 7333345 10 20'6734427'910994 1-097702'7392394 40 51'6900813 927632 1-078013'7331367 9 21'6736577'911526 1-097060'7390435 39 52'6802946'928173 1*077384'7329388 8 M 22'6738727'912059 1-096420'7388475 38 53'6805078'928715 1-076756'7327409 7 23'6740876'912592 1-095779'7386515 37 54'68(07209 929257 1-076128'732542) 6 24'6743024'913125 1-095139'7384553 36 55'6809339 929799 1-075500 7323449 5 25'6745172'913659 1094500'7382592.35 56'6811469'930342 1-074873'7321467 4 26'6747319'914192 1-093861'7399629 34 57'681:599'930884 1074246'7319486 3 27'6749466'914727 1-093222'7378666 33 58'6815728'931428 1-073620'7317503 2 28'6751612'915261 1-092584'7376703 32 59'6817856'931971 1072994'7315521 1 29'6753757'915796 1-091946'7374738 31 60'6819984 932515 1-072368'731.3537 0 30'6755902'916331 1-091308'7372773 30 _ COSINE. COTANG. TANG. -SINE. I I COSINE. COTANG. ITANG. ISINE. I DEG. 47. DEG. 47. NATURAL SINES AND TANGENTS TO A RADIUS 1. 43 DEa. 43 DEG. I SINE. TANG. COTANG. 1COSINE. I I SINE. TANG. COTANG. COSINE. I 0 -6819984 -932515 1-072368 -7313537,60 31.6885655'949517 1-0531G66 7251741 29 1 -6822111.933059 1-071743 -7311553 59 32'6887765 -950070 1.052553'7249738 28 2'6824237'933603 1-071118'7309568 58 33 -6S89873'950624 1-051940 *7247734 27 3 *6826363.934147 1'070494.7307583 57 34'6891981'951178 1.05.1:;27 7245729 26 4'6828489 *934692 1'069870 -7305597 56 35'6894089'951732 1-050'715'7243724 25 5'6830613 *935238 1-069246 -7303610 55 36'6896195 -952287 1-050103'7241719 24 6'6832738.935783 1-068623.7301623 54 37'6898302'952842 1-049492'7239712 23 7.6834861'936329 1-068000.7299635 53 38'6900407'95:-!,97 16048880'7237705 22 8 68&36984'936875 1-067377'7297646 52 39'6902512' 953952 1'-(48270 -7235698 21 9'6839107 -937421 1-066755'7295657 51 40'6904617'954,508 1-0470659'7233690 20 Q 10 -6841229'937968 1-06ti134 -7293668 50 41'6906721 -955064 1-047049'7231681 19 9 11 6843350 -938515 1-065512 *7291677 49 42'6908824 -955620 1-046440 -7229671 18 M 12'6845471.939062 1 064891 *7289686 48 43'6910927'956177 1'04531' 7227661 17 * 13'6847591 -939610 1-064271 *7287695 47 44'6913029 *956734 1-045222'72261 16 Q 14 -6849711'940157 1-063651'7285703 46 45'69151.31 *957291 1-044613'722340 15 15'6851830'940706 1-063031 -7283710 45 46'6917232'957849 104400)5'7221628 14 16'6853948 -941254 1-062411'7281716 44 47 -6919332 -958407 1-043397'7219615 13 17'6856066'941803 1-061792 -7279722 43 48'6921432'958965 1-042790'7217602 12 18 -6858184 -942352 1-061174'7277728 42 49'6923531 *959524 1-042183'7215589 11 Z 19'6860300'942901 1-060556'7275732 41 50'6925630' 960082 1-041576'7213574 10 H 20 -6862416'943451 1-059938'7273736 40 51'6927728 9960642 1-040970'7211559 9 O 21'6864532 -944001 1-059320 -7271740 39 52 -6929825'961201 1-040364'7209544 8 22'6868i647 *944551 1-058703'7269743 38 53'6931922 *9617(1 1-039758 *7207528 7 23 -6868761'915102 1-058086'7267745 37 54'6934018'9623291 1-099153'7205511 6 24'6870875 *945653 1 057470 *725747 36 55'6936114'962881 1-039348 -203494 5 25'6872988'946204 1-0568,54 72fi3748.35 56 69.318209 -983442 1 0 79 44 -7201476 4 26'6875101 -946755 1-056238 -7261748 34 57 -6940304 -964003 1-03734 -71994,57 3 27'6877213 -947307 1-055623'7259748 33 58 -6942398'964565 1 036(736'7197438 2 28'6879325 -947859 1-0550018 7257747 32 59'6944491 -965126 1-0361,33 -7195418 1 29 -68814.35'948411 1-054394 -7255746 31 60'6946584'965688 1-035530'7193398 0 30'6883546 -948964 1-063780 -7253744 30 COSINE. COTANG. TANG. SINE. I. COSINE. COTANG. TANG. I SINE. I DEG 46. DEG. 46. Ca NATURAL SINES AND TANGENTS TO A RADIUS 1. X 44 DEG. 44 DEG. I SINE. TANG. COTANG. COSINE. I I SINE. TANG. COTANG. COSINE. 0 *6946584'965688 1.035530'7193398 60 31'7011167'983269 1*017015 *7130465 29 1 6946676 -966251 1-0349)27'7191377 59 32'7013241 983841 1-016423'7128426 28 2 *6950767'966813 1-034325'7189.355 58 33'7015314'984414 1-015832'7126385 27 > 3 *6952858 *967376 1-033723'7187333 57 34'7017387'984987 1'015241.7124344 26 M 4'6954949'967939 1-033122 *7185310 56 35'7019459'985560 1-014651'7122303 25 5'6957039'968503 1-032520'7183287 55 36'7021531'986133 1-014061'7120260 24 6'6959128 96.9067 1'031919'7181263 54 37'7023601'986707 1-013471 -7118218 23 4 7'6961217'969631 1-031319'7179238 53 38'7025672'987282 1-012881'7116174 22 8 *6963305 *970196 1-030719'7177213 52 39'7027741'987856 1-012292'7114130 21 4 9'6965392'970761 1-030119'7175187 51 40'7029811'988431 1-011703'7112086 20 M 10'6967479'971326 1'029520 -7173161 50 41'7031879'989006 1-011115 -7110041 19 8 11'6969565'971891 1-028921'7171134 49 42'7033947'989582 1'010527 -7107995 18 td 12'6971651 *9724.57 1-028322'7169106 48 43'7036014'990158 1-009939'7105948 17 om 13'6973736'973023 1-027724'7167078 47 44'7038081'990734 1'009352 -7103901 16 14'6975821'973590 1-027126'7165049 46 45'7040147'991311 1'008764'7101854 15 ~ 15'6977905'974156 1-026528'7163019 45 46'7042213'991888 1'008178'7099806 14 c 16'6979988 *974724 1-025931'7160989 44 47'7044278 *992465 1-007591'7097757 13 I 17'6982071 597.5291 1-025334'7158959 43 48'7046342'993042 1 00005'7095707 12 M 18'6984153'975859 1-024738'7156927 42 49'7048406'993620 1-006420'7093657 11 3 19'6986234.'976427 1-024141'7154895 41 50'7050469'994199 1-005834'7091607 10 Q 20 -6988315'976995 1-023546'7152863 40 51'7052-532'994777 1-005249 *7089556 9 O 21'6990396.977564 1'022950'7150830 39 52'7054594'995356 1'004665'7087504 8 22'6992476'978133 1-022355' 7148796 38 53'7056655'995935 1'004080 -7085451 7 23 -6994555'978702 13021760'7146762 37 54'7058716 *996515 1-003496 7083398 6 I 24 *6996633'979272 1'021166'7144727 36 55.7060776'997095 1-002913'7081345 5 Z 25 *6998711'979842 1-020572'7142691 35 56 -706835'997675 1-002329'7079291 4' 26'7000789 -980412 1'019978'7140655 34 57'7064894'998256 1-001746'7077236 3 0 27 ~7002866'9809)3 1-019385'7138618 33 58'7066953 -998837 1-001164'7075180 2 28'7004942'981554 1-018792'7136581 32 59'7069011'999418 1-000581'7073124 1 29'7007018'982125 1'018199'7134543 31 60'7071068 1-00000 1-000000 7071068 0 30'7009093'982697 1-017607'7132504 30! COSINE. COTANG. TANG. SINE. I I COSINE. COTANG. TANG. SINE. I DEG. 45. DEG. i4.