Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031438587 '1fA&,.^,B,'?,.,nautical astronomy / JSTAVIGATION NAUTICAL ASTRONOMY. PMPAEED FOB THE USB OP THE V. 8. NAYAI ACADEMl. Tiu Wm. GhnuvtmjJ: # J. H,C. Coffin. FOURTH EDITION. NEW-YORK : JO. VAW 3srosTRAiirr>, iss beoad'w^y. 1868. Entered, according to Act of GongresB, in the year 1865, by D. VAN NOSTBAND, In the Clerk's Office of the BUtrict Court of the United States, for the Southern District of New-York. N^OTIOE. This Treatise was originally prepared by Prof. Chauvenet to be used in manuscript by the students of the Naval Academy. With Bowditch's Navigator, oral instruction, the use of instruments, and computation of examples, it constituted the course of instruction in Navigation and Nautical Astronomy. In its printed form, some subjects are more fully discussed, others introduced, and various suggestions given on points of practice. In this edition examples are supplied, which will serve both as illustrations, and as forms for the arrangement of computations. Those in Nautical Astronomy are mainly adapted to the Ephemeris for 1865. It has been my purpose, as I should find time from incessant ofB- , in which attention must be paid to the signs, or names.* These formulas accord with the precepts on page 60 of Bowditch's Navigator. 3. If the distance is very small, so that the curvature of the earth may be neglected, then G A may be regarded as a right line, and the triangle G A B as a right plane triangle. From this we have J r- Pig.L + when North, —when South. -I- when West. when East. (1) cos C = r or. Bin(7 = |, tan (7= I ; I I =zd cos C, p = dsin C, p = l tan C, (2) (3) • K N. and W. are regarded as positive, S. and E. are negative, and may be treated as such, without the formality of substituting the signs + and ^. PLAJfE SAILING. 9 in which p is the departure in the latitude of C or A ; indif- ferently, as their distance is very small. The Traverse Table, or Table of Right Triangles, contains I and p for different values of C and d. Table I. in Bow- ditch's Navigator contains I and p for each unit of d from 1 to 300, and for each quarter-point of G. Table 11. contains them for each unit of d and each degree of C. These quantities form a plane right tri- angle (Fig. 2), in which d is the hypothenuse, C one of the angles, I the side adjacent ) p « opposite 1 *1^-* -gl«- In the Tables, the columns of distance, difference of lati- tude, and departure, might be appropriately headed, respect- ively, hypothenuse, side adjacerU, and side opposite. 4. The first two of equations (3) afford the solution of the most common elementary problem of navigation and survey- ing, viz. : Peoblem 1. Gfiven the course and distance, to find the difference of latitude and departure, the distance being so small that the curvature of the earth may be neglected. These equations also afford solutions of all the cases of Plane Sailing. (Bowd., pp. 52-58.)* 5. Pboblem 2. Given the course and distance, to find the difference of latitude and departure, when the distance is so great that the curvature of the earth cannot be neglected. /Solution. Let the distance C A (Fig. 1) be divided into parts, each so small that the curvature of the earth may be neglected in computing its corresponding difference of lati- tude and departure. * The first and sixth are the most important. 10 NAVIGATION. For each such small distance, as c a, l=:d cos O, p = d sin 0. Representing the" several partial distances by ], we have ^1+ 4+ 4 + &c-= {di + di+di &c.) cos G, P1+P1+P1 + &c. = (di+da + d, &o.) sin G; or, [n = Wcos G, [p] = [d] sin G. Since the distance between two parallels of latituoe is the same on all meridians, the sum of the several partial differ- euces of latitude will be the whole difference of latitude ; As in Fig. 1, C B = E A = the sum of all the sides, c 5, of the small triangles ; and we shall have generally, as in Prob. 1, whatever the distance, d, 1 = d cos G. We also have ^ = c? sin (7, if we regard p as the sum of the partial departures, each being taken in the latitude of its triangle ; so that the differ- ence of latitude and departure are calculated by the same formulas, when the curvature of the earth is taken into ac- count, as when the distance is so small that the curvature may be disregarded ; or, in other words, as if the earth were a plane. But the sum of these partial departures, b a of Fig. 1, is evidently less than C E, the distance between the meridians left and arrived at on the parallel C E, which is nearest the equator ; and greater than B A, the distance of these meri- dians on the parallel B A, which is farthest from the equa- TKAVEESE SAILING. 11 tor. But it is nearly equal to F G-, the distance of these meridians on a middle parallel between C'and A ; and exactly equal to the distance on a parallel a little nearer the pole, and whose precise position will be subsequently determined. (See Problem 10, Mercator's Sailing.) We take then Z>o — i (I'+L'), or, more exactly, as the latitude for the departure, p. 6. Middle Zatitude Sailing regards the departure, p, as the distance between the meridian left and that arrived at on the middle parallel of latitude ; or takes i^, = J (Z+Z"). TBAVBBSE SAILING. "l. If the ship sail on several courses, instead of a single course, she describes an irregular track, which iB called a Traverse. PsoBLEU 3. To reduce several courses and distances to a ■single course and distance, and find the corresponding dif- ferences of latitude and departure. Solution. If in Fig. 1 we regard C as different for each partial triangle, and represent the several courses by Ci, Cj, C„ &c., we evidently have i, = <^ cos <7i, pi = di sin C„ 4 = <4 cos C„ Pi = d, sin C„ 4 = <^ cos Ca, p3 — d^ sin C„ &c. &c. and \l] = li+ 4+ 4, &c., [p] =Pi + p,+p3, &c, ; or, as in the more simple case of a single course, ITie wTiole difference of latitttde is equai to the sum of the partial diffierenoes of latitude ; The whole departure is equioi to the sum of the partial de- partures. 12 NAVIGATION. This applies to all cases, if we use the word sum in its general or algebraic sense. If we represent by i. the sum of the northern diffs. of latitude, « " L, " " southern " " " " P„ " " western departures, " « P, « " eastern " we have as the arithmetical formulas, [ ^ ] = i„ ~ i, of the same name as the greater, lp] = P„~P. " which accord with the usual rules. (Bowd., p. 59 and p. 264.) The Traverse Form (p. 60 apd pp. 266 to 286) facilitates the computation. The course, G, and distance, \d\, corresponding to \l\ and [/>], may be found nearly by Plane Sailing.* 8. The departure may be regarded as measured on the middle parallel, either between the extreme parallels of the traverse, or between that of the latitude left and that ar- rived at. In a very irregular traverse it is difficult to deter- mine the precise parallel ; but, except near the pole, and for \ * C and [rf] are not accurately found, because [jo], the sum of the partial departures of the traverse, is not the same as p, the departure of the loxo- dromic curve connectli^ the extremities of the traverse. Thus suppose a ship to sail from C to A by the traverse C B, B A, her departure will be by tra- verse sailing d e + mn ; whereas, if the ship sail directly from to A, the depart- ure will be o p, which is greater or less than (i c + m », according as it is nearer to, or farther from the equator. Thus we should obtain in the two cases a different course and distance between the same two points. In ordinary practice, how- ever, such difference is immaterial. PARALLEL SAILING. 13 a distance exceeding an ordinary day's run, the middle lati- tude suffices. (Bowd., p. 59, note.) It is easy, however, to sepai-ate a traverse into " two or more portions, and compute for each separately. FABALLEL SAILING. 9. The relations of the quantities C, d, I, and p are ex- pressed in equations (3). When the difference of longitude also enters, then some further considerations are necessary. Pkoblem 4. To find the relations between i, the latitude of a paraUd, p, the departure of two meridians on that parallel, and D, the corresponding difference of longitude. Solution. In Fig. 3, let P A A', P C C be two meridians. AC = p, their departure on the parallel A C, whose latitude is A O A' = O A B = i, and whose radius is B A = r. A' C = J>, the measure of A P C, the difference of longitude of the same meridians, on the equator A' C, whose radius is O A'=0 A = i?. A C, A' C are similar arcs of two circles, and are there- fore to each other as the radii of those circles ; that is, AC:A'C' = BA:OA', or p:I> = r:Ii. In the right triangle O B A, B A = O A X cos O A B, or r= JR cos Z; (4) that is, t?ie radius of a parallel of latitude is equal to the radius of the equator multiplied hy ths cosine of the latitude. Substituting (4) in the preceding proportion, we obtain 14 NAVIGATION, p : J) = cos Z, : 1, or p = D cos Z, D = p sec Z,, (5) which express the relations required. (Bowd., p. 63.) These relations may be graphically represented by a right plane triangle (Fig. 4), of which D i« the hypothenuse, i, one of the angles, p, the side adjacent that angle. -p^ mg.4. The 2i-averse Table, or Table of Right Triangles, may therefore be used for the computation (Bowd., p. 65, " by inspection "). MIDDLE LATlTtJDa SAILING. 10. Peoblbm 5, Given the course and distance and the latitude left, to find the difference of longitude. Solution. By plane sailing, 1= d cos C, p— d sin C; (3) by Arts. 2 and 6, .L' = Z+l, X„ = i(i'+JC) = i+i?; (6) and by equation (5), D=p&eaZe, ' (7) or Z> = dam C sec Z^. (8) Equations (3), (6), and (T) or (8) afford the solution re- quired. The assumption of X„ = -J- (X' + i), or the middle latitude, suffices for the ordinary distance of a day's run ; but for larger distances, and where precision is required, we must take (Art. 6) MIDDLE LATITDBE SAILING. 15 in which ^d X is a small correction to be added numerically to the middle latitude. A formula for computing it is given in Prob. 10, under "Mercator's Sailing." Its value in the most common cases is given in Bowd., p. 76, and in Stanley's Tables, p. 338. 11. Strictly, the middle latitude should be used only when both latitudes, L and L\ are of the same name, as is evi- dent from Fig. 1. If these latitudes are of different names, and the distance is small, \ (i+i'); numerically, may be used; or we may even take /> = Z>, since the meridians near the equator are sensibly parallel. If the distance is great, the two portions of the track on different sides of the equator may be treated separately. Thus, in Fig. 5, the track C A is separated by the equator into two parts, C E and E A. For C E, we have Pi = —L tan G, C'E = A=iOisecfi, = —L tan G sec \ L nearly. For E A, we have A'A = ;.=:i', p^ =: U tan C, E A' = i)s =Pi sec \ L', ■=U tan G sec \ U nearly. Whence we obtain C A' or Z> = Z», + A- Instead of the middle latitudes \ L and f U, we may use more rigidly i^L+AL) and (|i'-l-^ U). When several courses and distances are sailed, as is ordi- narily the case in a day's run, p and I are found as in trav- Kg-A 16 NAVIGATION. Pig. 6. evse sailing, and then I) by regarding p as on some parallel midway between the extremes of the traverse. (Art. 8.) (Bowd., p. 264.) 12. The relations of the quantities involved in middle latitude sailing, namely, C, d, p, I, Z„ and J), are represented graphically by combin- ■ ing the two triangles of Plane Sailing and Parallel Sailing, as in Fig. 6, in which C = A C B, d=CA, i» = BA, l=CB, X„ = BAE, Z> = A E. • By these two right triangles, all the common cases classed under Middle Latitude Sailing (Bowd., p. 68) may be solved, if we add the formulas, 13. Other problems may be stated, which never occur in practice ; as, for example, — Problem 6. Given the cowse and distance, and the differ* ence of longitude, to find both ICttitttdes. . Solution. We have, c, d, and J) being given, p = dBia O, l—d cos C, T P COS i„ = ^, X=(Z„— JX)— ^7, L' = {L,~A L) + il, d L being taken from the table (Bowd., p. 76) correspond- ing to IJa- MIDDLE LATITUDE SAILING. 17 EzAiCFLKS m Middle Latitude Sailino. L and i. represent the latitude and longitude of tJie place sailed from. II and %', the latitude and longitude of the place arrived at. L % L' a OODKSE. DIST. 1. Sgs'oa 74 20E. 4128S 170 so K aW.byW. 210 2. 46 24 N. 4T IB W. 49 15 N. 42 21 W. N. E. i E. 270 S. 51 10 S. 168 87 E. 48 31 S. 158 42 E 4. 22 18 S. 57 28 K . B. by S. 317 5. 23 15 S. IS 36 W E. 255 6. 20 6 N. 154 17 W. 18 28 S. E. S E. } E. ... 7. 56N. 29 34W. S. 47° E. 168 8. 45 16 S. 8 46 E. 43 10 S. 6 22 "W. 9. 67 1QN. 178 51 W. N. 6fpts. E. 290 .10. Required the bearing and distance of Cape Race from Cape Hatteras. [tan C7 = J cos (io+ ^ L) d=lBeo C] 01 o ; Cape Hatteras, 35 15 K 75 31 W. Tab.LIV. Cape Race, 46 39 « 53 5 «« I =11 24 « = 684 , ig =40 57 2) = 22 26 E. = 1346 /iZ =+17 Zo+/i Z =41 14 log D 3.1290 leoa{Za+AZ) 9.8762 ar CO log I 7.1649 log I 2.8351 C=N.55°57'E.l.tan(7 0.1701 I. sec C 0.2518 =jp sec (Zb+^Z)] Cape Frio, 2°3 1 S. 41 59 W. . i' = 34 30 " jB = 3173 E. log^ 3.5015 /= 11 29 " logi 2.8382 Z„ = 28 45 " C=S. 11 45 E. 1. tan C 0,6633 /iZ= 11 " 1. sec C 0.6733 «?= 3247 log rf 3.5115 Zo + ^Z = 29 2" ----- 1. sec 0.0583 i> = 61 54 E. log J) 3.5698 A' = 19 55 E. 12. A ship in latitude 39° 8' N., longitude 33° 45' W., Bails N. 51° 5' E. 1014 miles ; required her position. i' = 49°'45'N". A' =15° 16' W. 13. A ship in latitude 56° 46' S., longitude 170° O'E., sails E. N. E. until she is in latitude 50° 10' S.; what is the dis- tance sailed, and what is her longitude ? i 3.1469 X' = 19° 60' S. 1. tan C 0.0868 I L+ A L = 9° 26'+l°31' = 10° 66' N. log L' 3.0766 i i' + J i' = 9° 66'+ 1°3'7' = ; 11° 32' S. 1. sec(ii'+ J i') 0.0089 A = 23° 22' E. D = : 24° 40' E. : 48° 2'E. log Di 3.110S A = 153° 46' W". A' = 106° 43' W. If A L and A U are neglected, the resulting value of U will be 105° 57' W". If the computations are made with the middle latitude, 0° 30 S., A' will be 106° 39' W., or in error nearly 1°. 17. Find the latitudes of two place's, whose longitudes are 12° 49' W. and 24° 51' W., their distance 645 miles, and the course from the first to the second N. 50° 46' W. (Problem 6.) (7 = N. 50 46 W. 1 . cos 9.8010 1. sin 9.8891 d- 645 log 9.8096 log 9.8096 l- 6 48 N. log 9.6106 log^ 9.6987 Z> = 12 2W. 46 12 N. or S. • log 9.8686 1. COS 9.8402 -£.= A i = — 6 • 46 6N. or 46° 6' s. \l = 3 24 N. 3 24 N. L = 42 42 N. or 49 30 a L = 49 30 N. or 42 42 s. Examples in Tbatbrsb Sailing. . A ship from the position, given at the head of each of the following traverse forms sails the courses and distances 20 NAVIGATION. stated in the first two columns ; required her latitude and longitude. 1. August 8, noon— Lat. by Obs., 35° 35' N, Long, by Chro. 18° 38' W. ConnsBS. DiST. N. 8. E. W. N. N. E. i E. S. iW. S. by E. i K S. S. W. i w. 50 46.2 16.8 33 41.8 44.1 26.9 / 45.7 16.8 37.8 23.6 4.8 26.9 65.3 30.7 , 38 = / 6.7 17.9 S.4iK 192.5 41.5 71.0 99.3 2^.3 24.6 = D. August 9, noon — Lat by Acct., 35° 1' 'N. Long. " 18° 0' W. 2. September 25, noon— Lat. by Obs., 49° 53' S. Long, by Acct., 158° 27' E. OOOEBEg. DlST. N. S. E. W. - Fts. ' ' / ' ' / S. 4i E. 46.3 * 28.7 35.0 , S. 6iE. 19.6 10.0 16.7 S. 7 W. 38 7.4 37.3 S. 6i W. 25.7 8.7 24.2 8.3 W. 51.2 42.6 28.4 N. 7i E. 13 1.9 12.9 N.6iE. 10 4.3 9.0 202.7 6.2 97.4 73.6 89.9 S. 1 w. • 93 91.2 16.3 D = = 26 September 26, 8 a.m. — Lat. by Acct., 51° 24' S. Long. " 158° I'W. In this example the courses are expressed in ^o»/ite, which is the preferable method. When the reductions are the eame for all the compass courses, we may find the difiierence of latitude and depar- MERCATORS SAILING. 21 tnre for these compass courses, and the course and distance made good. The traverse is thus referred to the mag- netic meridian instead of the true. The course made good may then be corrected for variation, etc. ; and with this cor- rected course and the distance made good the proper differ- ence of latitude and departure may be found. 3. September 16, 6 p.m.— Lat by Obs., 50° 16' S. Long, by Chro., 76° 10' W. Coup. Coobsb. BiST. N. S. E. w. S. W. i s. 25 1 19.3 / 15.9 &s. w. 80 21.1 - 11.5 S. by W. 18 11.1 3.5 S. 43 43 S. by E. i E. 255 24.7 6.2 c S. E. f S. 33 26.6 -19.7 l'?4.5 26.9 30.9 {maff.) S. 2° W. 159 158.9 5.0 Var-n, &c., 18° W. {trite) S. 16° E. 169 162.8 43.8 or S. by B. i E. 70 = = D September 17, noon;— 5Lat. by Acct., 52° 49' S. Long. " 75° O'W. MBBCATOB' SAILING. 14. Middle Latitude Sailing suffices for the comraon pur- poses of navigation ; but a more rigorous solution of pro- blems relating to the loxodromic curve is needed. These solutions come under " Mercator's Sailing." Peoblbm 5. A ship sails from the equcUor on a given course, G, till she arrives in a given latitude, L, to Jind^the difference of longitude, D. Solution. Let the sphere (Fig. 7) be projected upon the plane of the equator stereographically. The primitive circle A B C ... .M is the equator. 22 NAVIGATION, P, its centre, is the pole (the ey6 or projecting point being at the other pole).* The radii, P A, P B, P C, &o,, are meridians making the same angle with each other in the projection as on the sur- face of the sphere.* The distance P m, of any point m from the centre of the projec- tion, =tan^(90°— i), the tangent of -J- the polar distance of the point on the surface which m re- presents, the radius of the Sphere being 1.* This curve in projection makes the same angle with each merid- ian, as the loxodromio curve with each meridian on the surface.* Hg. 7. A M is the whole difference of longitude D. If we suppose tbis divided into an indefinite number of equal parts, A B, B C, C D, &c., each indefinitely small, and the meridians PA, P B, PC, &c., drawn, the intercepted small arcs of the curve A6c....m may be regarded as straight lines, making the angles PAS, P 5 c, P c w, If then D = the whole difference of longitude, d = one of the equal parts of D, -J will be the number of parts, and -}■ + 1 the number of meridians P A, P 5 . . . ,P »i, (Jb ■- -— .■■■■-■- * Principles of stereographic projection. mebcatob's sailing. ' 23 or the number of terms of the geometrical progression : and, employing the usual notation, the first term a ^ P A = 1, the last term Z = P m = tan i(90'— 2^, the ratio r = ^r-; . FA To find this ratio, we have in the indefiuitely small right triangle A B J, , tan B AJ = eot PA6 = 5-T, cot<7=— ^ ' whence P A— P J = = sec Zdp = tan C sec Zd L, in which C is constant. By integrating the last equation between the limits X = and Z= L, wd shall have i) = tau CjJ'secidZ, the whole difference of longitude required in Problem 6. To effect the integration, put sin i = X, . d X cos L' d X d X di sec ZdX sec ZdZ = then by differentiating and multiplying by sec L oos'Z 1— sin' i' d X and l-x» Besolving into partial fractions, we obt.ain BecZdZ = i\ + I H + X 1 — xj j\ecidi = i[log (1+ i)-log (1 -i] , . . /l+aini ' 1— sm Z = log tan (45° + i Z) Trig. (154). Whence we have J) = log tan (46° + i i) tan C. But in this the logarithm is Kaperian, and D is expressed in terms of the radius of the sphere. To reduce to common logarithins, we divide by m =.434294482, and to minutes by multiplying by r'= 343'7'.746Y7, and obtain D = '7915'.'7044'7 log tan (45° + i i) tan C as in (13). 15. To facilitate the practical application of the formula jagt obtained, put M=1915'.70U1 log tan (45°+^ Z) ; (14) 26 UrAVIGATION. and let M be computed for each minute of i fi'om up- ward, and its values given in a table. We shall thus form the Table of Meridional Parts or of Augmented Zatitudes, such as Bowditch's Table III. This formula accords with that given in the Preface. (Bowd., Pref. p. iv.) In practice, then, we have only to take the value of M corresponding to i, and j> is then found by the formula, D=M tan G. (15) Jf has the same name, or sign, as Xi. Example. To find tne meridional parts, or augmented latitudes, for each minute, from 30° to 33° ; log 7915'.7044'7— 3.898490. X. 46°+ ill, lag tan. I. log tan. log M. M. o / O i ' 30 60 0.2385606. 9.3'77599 8.276089 1888.37 23.14 30 20 60 10 ,2414830 .882887 .281377 1911.51 23.21 7 30 40 60 20 .2444154 .388129 .286619 1934.72 23.29 8 81 60 80 .24'73680 .393326 .291816 1958.01 23.38 9 8120 60 40 .2603108 .398480 .296969 . 1981.39 •23.45 7 3140 60 50 .2532'741 .403591 .302081 2004.84 23.54 9 82 61 .2562480 .408660 .307150 2028.38 23.63 9 32 20 61 10 .2592328 .413690 .312180 2052.01 23.72 9 82 40 61 20 .2622286 .418680 .317170 2075.73 23.80 8 83 61 30 .2652356 .423632 .322122 2099.63 The second differences afford a check of the work. By interpolating into the middle, M can be found for each 10' ; and then, by simple interpolation, for each 1'. In the first step, one eighth of the second difference is to be sub- tracted. The following is an example : MEKCATOS'S SAILING. 27 M L M 80 1888.37 80 11 1901.09 1 1889.53 12 1902.24 2 1890.68 13 1903.40 S 1891.84 14 1904.56 4 1893.09 15 1905.72 6 1894.15 16 1906.87 6 1895.31 17 1908.03 1 1896.46 18 1909.19 8 1897.62 19 1910.35 9 1898.T7 20 1911.61 10 1899.93 &C. 16. Problem 6. A ship sails from a latitude, X, to an- other latitude, U, upon a given course, O ; find the differ- ence of longitude, D. Solution. Let M be the augmented latitude corresponding to X, M' " " " " L'. The difference of longitude fi'om the point, A, where the track crosses the equator to the 1st position, whose latitude is L, will be I>=Mts.n G; and to the second position, whose latitude is U, D,-M' tan C I and we shall have J)=J),-D,={M'-M) tan C; (16) or, when M'<^M, D=I>-I),^(M-M') tan C ; since the sign of D is determined by the course. K L and U are of different names, so also are M and M\ and we have numerically JD={M+M') tan C. 17. The difference, M'—M, is called the meridional, or NAVIGATION. augmented, difference of latitvde. Representing this by m, we have D^=m, tan G. The relation between these quantities is represented by a plane right triangle- (Fig. 8), in which C is one of the angles, m =: C E, the side adjacent, Z> = E F, the side opposite. The triangle of " Plane Sailing" has the same angle (7, with ^ = C B, the adjacent side, and ^ = B A, the opposite side. Fig. s. Fig. 8 represents these two triangles combined. By them, all the common cases under Mercator's Sailing can be solved, either by computation or by the Traverse Table. (Bowd., p. 79.) The relations between the several parts' involved are l = dc.osC, L'=L+l, p — dsinC, m —M'—M, B^mtwC, X'=X->rD; \ (18) and since p^l tan (7, I : m =zp : D. 18: Problem 7. Given the latitudes and longitudes of two places, find the course, distance, and departure. (Bowd., p. 79, Case I.) Solution. L and L' being given, we take from Table III. ilfandJIf'. We have l-L'-L, m=M'-M, D=zX-k by Mercator's sailings and by Plane sailing, tan G = — ; m d = I sec G, p =:.l tan C; mehcatoe's sailing. 29 I, «i,;and C are north or som^A according as.i' is north or south of i. -O, jo. and O are eas< or wesf, according as A' is eas^ or west of A. If the two places are on opposite sides of the equator, we have numerically l = Z'+Z, m=M'+M. Mercator's sailing is rarely used except in this case, and when the diffei'ences of latitude and Ipngitude are consider- able. There are two limits of its accuracy : — 1. Table III. contains the augmented latitude .only to the nearest minute or mile.* 2. It is computed on the supposition that the earth is a sphere. Some works on Navigation, as Mendoza Rios and Riddle, contain a table of augmented latitudes, in which the true form is taken into consideration.! ExAilPLES. 1. Required the course and distance from Cape Frio to 34°30'S., 18° 30' E. Cape Frio, 23 1 S. 41 69 "W. M = 1420 a Z' = 34 30 S. r = 18 30 E. iT = 2208 S. log D 3.6598 I = 11 29 S. Z> = 60 29 E. M = T88 S. log™ 2.8965 =S. 11 45 E. Isec log I 0.6733 2.8382 1. tan C 0.6633 d = 324t' logrf 3.6116 * The most convenient unit for nautical distances is the geographieal, nautical, or sea rmle, which is 1' of the earth's equator, or B086.43 feet. Regarding the earth as a sphere, this is also 1' of any great circle, f The formula for the terrestrial spheroid is M= 7915'.70447 log tan (45° +i L) — 22'.98308 sin i;+0'.01276 sin 3 L+ &c. Delambre has shown that a table of meridional parts constructed for the sphere may be used for the spheroid by using as the argument the geoeen- /-.'« lafi^iHo inatonH Af thft true latitude. 30 NAVISATIOK 2. Required the course and distance from Cape Frio to Lizard Point, England. Cape Trio, 23 1 S. 41 69 W. M r= 1420 S. Lizard Pt., 49 68 N. 6 12 W. M' = 3471 N. log J) 3.3438 1= 72 59 N. D- = 36 47 E. M = 4891 N. log m 3.6894 C=N.24 1'?B. 1. sec Q 0.0402 log I 3.6414 1. tan a 9.6544 rf= 4804' log d 3.6816 3. A ship in latitude 18° 50' N"., longitude 153° 45' W., sails S. 4^ points E., 3656 miles ; what is her position ? , , „ , d — 3656 log d 3.5630 Z = 18 60 N. 7i= 153 46 W. M = 1161 N. 1 cos C 9.8024" ? = 38 40 S. , log I 3.3654 i' = 19 50a ' J/' = 1215a 1.1811(7 0.0858 M - 2366 S. log m 3.3740 log JD 3.4598 J) = 48 3 E. X' = 105 42 W. 19. Other problems might be stated than those com- monly given ; as, for example, — Pboblem 8. Given the latitude left, the course and both longitudes, to find tJie latitude arrived in. Solution. "We have Z) = A'— A, by Mercator's sailing m =Z> cot G (N". or S. as is (7), by Table HI. M corresponding to Ii, IT = M+m, and again by Table III, i' corresponding to JfT. Problem 9.- Given the difference of longitude and differ- ence of latitude of two places, and- the course between t/iem, find both latitudes. Solution. We have m = M'—M= D cot, C. But W=^ '7915'.70447 log tan \ (90° +i') Jf = 7915'.'70447 log tan \ (90°+i), mercatoe's sailing. 31 consequently, log tan i (90° +i') - log tan i (90° +Z) =if^^^^- (19) ■r. ■. . , ^ cot • /„-> Put ^»g"»*^= 791S'.70447 ' ^^°) then equation (19) gives tani(90° + i') . , __^__ = COt^. 'By PL Trig. (109) tan i(x + y) sin g + sin y tan i (x — y) sin a — sin y' In this, if we take 85 + y = 90° .+ Z' we have - or putting jB — y = 90° + Z, x = 90°+i{Z'+Z), Zo = i {Z'+Z) the middle latitude, aj = 90°+i„, y^i{Z'-Z) = il, and tan i (90° + X') _ cos L„ + sin j ; — • t <6 , tan i (90° + L) ~ cos X, — sin ^ Z ~ **** "' whence ^ cot ^ + 1 . , J ^"^-^o^ cot^-l "^"^^' which, by PI. Trig. (151), reduces to cos Za = tan (45° +0) sin i I. (21) We have also Z — Zo — il) ' /22\ The solution is effected by equations (20), (21,) (22). 32 NAVIGATION. Example. o / The difference of longitude of two places is 5 10 E. the difference of latitude, 3 28 N". the course ' N. 32 59 E ; find the latitudes. (Constant) 7915'.70447 ar. co. log 6.10151 B = 310' log 2.49136 C— 32° 59' log cot 0.18776 log cot = 0.06034 log 8.78063 (/) = 41° 2' 46°+ ^ = 86° 2' log tan 1.15900 \l= 1° 44' log sija 8.48069 i„ = 64° 8' N". or 64° 8' S. log cos 9.63969 L = 62° 24' N. or 65° 52' S. Z' = 65° 52' ISr. or 62° 24' S. This problem cannot be solved with precision when Z„ is near 0. 20. Peoblj!m 10. Ih find the correction of the middle latitude in Mid. Ziat. Sailing. (Tab. Bowd., p. 76 ; Stan- ley, p. 338.) Solution. In Mid. Lat. Sailing we have C08i„ = |. (23) in which precision requires that we take i„ = i(i'+i) + Ji; A L being a correction of the middle latitude, which it is now proposed to find. In plane sailing p =z I tan G, in Mercator's sailing D = m tan 0, which substituted in (23) give C0sA=- (24) whence 1—2 sin' | L^ mekcatoe's sailing. 33 I m' and sin i A = i/^HI. (25) ^ 2 TO Now for different values of i and i' we may find (in minutes) I =: U — i, the middle latitude, i„ = \ {U +L), m='?915'. '70447 [log tan (45° +J i') —log tan (45° +i i)],* and then L„ by (24), or if small by (25), from which sub- tracting Zi„ we have A i, which is required. In computing m, logarithms to 7 places should be used when the difference of latitude is less than 12°- The correction of the Middle latitude computed for differ- ent middle latitudes, and differences of latitude maiy be given in a table, as on page 76 (Bowd.) It becomes too large to be conveniently tabulated, when the latitudes are of differ- ent names, or the middle latitude is very small. Example. Find the correction of the Middle Latitude, when the lati- tudes are L — 12°, L' = 18°. o ; (45°+i I/) = 54:0 I. tan 0.1387390 (45° + ^ i) = 51 I. tan 0.0916308 const, log 3.89849 \ {L' + L) = 15 0.0471082 log 8.67310 ^ = 6 = 360' ar. CO. log 7.44370 La = 15 7 I. sec 0.01529 AL = 1 21. The loxodroraic curve on the surface of the earth and its stereographic projection (Fig. 7) present a peculiarity * log 7915.70447 = 3.898*896. Another formula, requiring only 5 place logaritbma, is m = 6875'.493 [j + ^ j> + • j" + > 5;' + - .] in which j = ain i Z sec ^ {L'+L). 34 NAVIGATION. worthy of notice. Excepting a meridian .and parallel of latitude, a line which makes the same angle with all the me- ridians which it crosses would continually approach the pole, until, after an indefinite number of revolutions, the distance of the spiral from the pole would become less than any assignable quantity. It is usual to say that such a curve meets the pole after an infinite number of revolutions. StiU, however, it is limited in length. For we have for the length of any portion, by plane sailing, d = {L'—L) sec G. If i=0 ' andi' = 90°=|, the whole spiral from the equator to the pole will be, with radius = 1, d =5- sec O. If Z = - 90° = -|, and X' = 90° =^, we have, as the entire length from pole to pole, d — n sec G. If also (7=0, or the loxodromic curve is a meridian, =p sec L. It can be shown that very small portions of the meridians are expanded in the same proportion. This is apparent from the table of the length of' 1° in Art. 22 ; as for example, a degree whose middle latitude is 60° is 120', or, 60' of the equator X sec 60°. But the two half degrees are unequally expanded ; for from 59^° to 60° is represented by 59', " 60° to 604° " " " 61', nearly, A small circle on the surface of the earth of 1° diameter, A mercatok's chakt. 37 at the equator is then represented by a circle, whose diame- ter is 1° ; in lat. 30° nearly by a circle, -whose diameter is l°Xsec 30°, " 60° " " " " l°Xsec60°, " L " " " " l°Xs«ci/ but not exactly by a circle, since the meridians are aug- mented more rapidly as the latitude is greater. Such a chart, then, while representing a narrow belt at the equator in proper proportions, presents a view of the earth's surface expanded at each point, both in latitude and longi- tude, proportionally to the secant of its latitude. 25. K we take any two points, C F, on this chart, and join them by a straight line, and form a right triangle by a meridian through one, and a parallel of latitude through the other, we shall have the triangle of JVftrcator's sailing (Fig. 8) : for, the intercepted portion of the meridian, C E, is the augmented dif- ference of latitude ; and of the parallel of latitude, E F, is the difference of longi- tude. Hence the angle E C F is the course. (Art. 17.) Moreover, the loxodromic curve is represented by the straight line C F ; for if we take any intermediate point of this curve, and let d be its posi- tion on the chart, d must be in the line C P, otherwise when we construct the triangle of Mercator's sailing, we shall have an angle at C different from EOF, the course ; which for every point of the loxodromic curve is the same. Thus a Mercator's chart presents two decided advantages for nautical purposes, viz. : 1. The ship's track is represented by a right line. 2. The angle, which this line makes with each meridian, is the course. Kg. 8. 88 NAVIGATION. To find the course from one point to another on the chart, all that is necessary is to draw a line, or lay down the edge of a ruler, through the two points, and measure its angle with any meridian. A convenient mode is to refer such line by m^ns of parallel rulers to the centre of one of the com- pass diagrams, which usually will be found on the chart, and reading the course from the diagram. Another mode of transferring the line to the compass diagram is described on page 88. (Bowd.) As such diagrams, except on some charts of limited ex- tent, are constructed with reference to the true meridian, the course obtained is the t/rue course, and not the compass course. ' 26. The distance, C F, however, is an augmented distance, which we may measure nearly by the augmented scale on the meridians of the chart (the middle latitude of the scale used being the same as that of the line CE). (Bowd., p. 88.) Or we may construct the proper distance, C A, by construct- ing the triangle, C B A, of plane sailing, in which C B is the proper difference of latitude, the scale for which is on the equator. The distance here spoken of, though represented on this chart by a straight line, is not the shortest distance between the two points, — for on the surface of a sphere, the shortest distance between two points ie the arc of a great circle, which joins them. To find this belongs to great-circle sailing. GBB AT-CIB CLB SAILING. 27. The rhumb-line, or spiral curve, which cuts all the meridians at the same angle, has been used mostly by navi- gators in passing from point to point on account of the sim- plicity of the calculations required in practice. But, as has been stated, it is a longer line than the great circle between the same points, and therefore the intelligent navigators of GKEAT-CIECLE SAILING. 89 the present day are substituting the latter wherever practi- cable On the Mercator chart, however, the arc of a great circle joining two points, not on the equator or on the same meri- dian, will not be projected into a strai^t line, but into a curve longer than the Mercator distance, and still greater than the distance on a rhumb-line. Hence it is an objection to the Mercator chart, that the shortest route from point to point appeHirs on it as a circuitous one ; and this is, doubt- less, one main reason why merely practical men have made so little use of the great circle. Many of those unacquainted with the mathematical piinciples of the subject are unable to comprehend how the apparently circuitous path on their chart should actually be the line of shortest dista,nce. 28. Peoblem 11. To project on a chart the arc of a great circle joining two given points on the globe. Solution. It will be necessary to project a number of points of the arc, and trace through these points the curve by hand. To project a point on the chart, we must know its latitude and longitude. The two given points, A and B (Fig. 9), and the pole, P, are the three angular points of a spheri- cal triangle, formed by the arcs joining these points with each other and with the pole. If from P we draw P €„ perpendicular to A B, the point C,, is nearer the pole than any other point of AB;. that is, it is the point of maximum latitude. This point of greatest latitude is called the vertex of the great circle. 1st. To find the latitude and longitude of this vertex. This may be done by a direct application of the rules of spherical trigonometry, first finding the angles A and B by iO NAVIGATION. Case I. of Sph. Trig., and then solving one of the right tri- angles A P Ga or B P Co. But in practice the following method is preferable. Let Xi=(90°— P A), the latitude of A, or the less latitude, Z2= (90°— P B), the latitude of B, or the greater latit^de, X=A P B, the difference of longitude of B from A, i„=:(90°— B Co), the latitude of the vertex. Ai=A P Co, the longitude of the vertex from P A. A2=B P Co, the longitude of the vertex from P B. The right triangle A P Co, gives , tan P Oo cot L„ tan L^ , , and the triangle B P Co, . tan P Co cot L„ tan L, tan P B cot L^ tan L, ' Whence by division cos \ tan Li cos \ tan ij ' and by composition and division, cos \ — cos \ tan L, — tan Z, cos 7^ + cos Ai tan Zj + tan Li By PI. Trig. (110) and (126), this equation becomes tanHA.+A.)tan^(A.-A.)^ ;!"^j^;g . But A, = (a, + A,,), and if we put J A = •J (A, — Aj), we have tan J A = ^!°^f'7f'^ cot J X, (27) sin (Xj + i,) ^ » ^ ' X^ = ^X+AX, (28) X^ = lx—AX. (29) By (27) we find A A, and then A; by (28), which, applied to the. longitude of A, or the point whose latitude is the smallest, and in the direction toward B, gives the longitude of the vertex. GEEAT-OIRCLE SAILING. 41 For finding the latitude of the vertex, equations (26) give tan L, = tan i, sec A„ ) ,^^^ tan Zi, = tan ij sec Aj, j either of which may be used. In using (2'!) attention must be paid to the signs of X, and ij. If the gi-ea,ter latitude, ij, is regarded as positive, Zi,, when of a different name is negative ; and in this case (is— ii) wiU.be nunaerically the sum, and .{L^+Zi) the dif- ference of the two latitudes. In this case we shall find Ai > 90°. When /I A > ^ A, Aj is negative, ^nd the vertex, €„, is be'- yond B, as in Fig. 10, instead of being between A and B, as in Fig. 9. In (30) we have X„ positive, ,or of the same name with the greater latitude, since numerically A^ < 90°. The vertex, wjhich is here used, is that which is nearest the point B, or the place whose latitude is numerically the greater. For th^s in {21) /I A < 90°. There are, hoyever, two vertices, which are diametrically opposite, as Co and C'o of the great circle C'o ECq in Fig. 10. For the vertex C, we have in (27) J A> 180°, or in the third quadrant, and in (30), i„ of a different name from i^. 2d. To find any number of points, C, C", C", &c., Cj, Cj, jCj, &c., we may assume at pleasure the differences of longi- tude from the vertex C„ P C, C„ P C", C,, P C", &c. It is best to assume them at equal intervals of 5° or 10°. Let A' = C„ P C, II = (90°- P C), the lat. of C, A" = C„PC", i"=::(90°-,PC"), " C", A"'=C„PP'", i"'= (90°- P C"), « C", (35)' tan J0"= tan A siu A'" &c. J The great circle intersects tha equator at two opposite 44 NAVIOATIOlir. points. The intersection, E, given by these formulas is that which is nearest A, the place whose latitude is the smallest. This method is preferable to that of Art. 28, only when the two places are on different sides of the equator, and tho intersection with the equator is between them. In this case, Aj and Aj, as well as Li and Z^, will have opposite signs. 30. Peoblem 12. To find the great-circle distance of two given points. Solution. Let A and B (Fig. 9) be the two given points. In the triangle P A B are given, as in Problem 11, P A = 90° -^i,, P B = 90° -Za A P B = A, to find A B = (36) and d=di + d^. J d reduced to minutes will be the distance in geographical miles. When Aj is negative, which happens when the vertex is beyond B (Fig. 10), d^ is also negative, and dis numerically the difference of di and d^. 2d Method. By (33) and (34) we may find Aj and A^ the longitudes of the two given places from E (Fig. 10), the in- tersection, and i„ the angle which the circle makes with the equator. Then from the right triangles, E a A, E 5 B, we have, putting d= E A, d"= E B, tan 90°, when the vertex is beyond B (Fig. 10). Id Method. Haring found the intersection and angle with the equator by (83) and (34), we have from the right tri- angles E a A, E 6 B (Fig. 10), cos A = sin i„ cos X^ \ cos B = — sin L, cos A3 ) ^ ' Zd Method. By Napier's Analogies we have 46 NAVKJATIOW. tan i (B+A) _ cos i {L^—L^ ~ sin i- (ij+Zi) tan ^ (B— A) = sin ^ (Xj — ii) cot J A, cot J A (41) ■ cos J (ij + 4) A = i(B+A)-.i(B-A) B=i(B + A)+i(B-A) When A and B are on opposite sides of the equator, J (ij— ii) is numerically half the sum, and ^ (ij+i,)- is half the difference of the two latitudes. 31. When the courses are found by this last method, the distance may be found by or. (42) The 1st is preferable, when \ (ij+Z,) is near 0, and conse- quently ^ (B+A) is near 90° ; the second when ^ (i,— i,) and consequently -J^ (B — A) are near 0. (Sph. Trig. "74.) 32. Example. To find the great circle from San Francisco to Jedo. (Formulas 27, 28, 29, 30.) San Francisco, Lat. L^= 37 48 N. Jedo, Xi= 35 40N. Za+i,= 78 28 Li—L^= 2 8 A = 97 88 iX= 48 49 JA= 1 67 \i=\7^A ;i. = 50 46 Ji^=ii^M= 46 52 Long 122 22 W. 140 E. L cosee 0.0183 1. sin 8.5708 1. cot 9.9420 1. tan 8.5311 L sec \ 0.1990 1. sec X, 0.1651 1. tan i, 9.8559 1. tan i, 9.8897 rw*m,Z,=48''87'N.Long.l69 14W. 1. tan i, 0.0549 0.0548 GBEAT-CIKCLE SAILING. 47 Locg. from Vertex. L cos A, 1. tan X. Lat. Longitudes. o o / o / o J 0.0000 0.0549 48 37 N. 169 14 W. 169 14 W. (Vertex). ± 5 9.9983 .0532 48 30 164 14 174 14 ±10 .9933 .0482 48 10 169 14 179 14 W. ±15 .9849 .0398 47 37 164 14 175 46 E. ±20 .9729 .0278 46 50 149 14 170 46 ±25 .9573 .0122 45 48 144 14 165 46 ±30 .9375 9.9924 44 30 139 14 160 46 ±36 .9134 .9683 42 55 134 14 155 46 ±40 .8842 .9391 41 129 14 150 46 ±45 .8494 .9043 38 44 124 14 145 46 ±50 .8081 .8630 36 7 119 14 W. 140 46 E. (36). 1. tan l^ 0.0880 (39) 1. sin A, 9.8891 1. tan A, 0.0283 1. sin A, 9.8632 1. cos i, 9.8202 1. sin i, 9.8753 1. tan di 9.9082 1. cos Ci 9.7444 1. tan di 9.8485 1. cos C, 9.7385 di = 39° 0' Course, Cj N". 54° 27' E. from Jedo. di=35 13 C, IsT. 56 48 W. from San Francisco. Distance, d = 74° 13' = 4,453 miles. Distance by Mercator's sailing, 4,689 mUes. 33. To follow a great circle rigorously requires a con- tinual change of the course. As this is diiScult, and indeed in many cases is practically impossible, on account of currents, adverse winds, &c., it is usual to sail from point to point by compass, thus making rhumb-lines between these points. When the ship has deviated from the great circle which it was intended to pursue, it is necessary to make out a new one from the point reached to the place of destination. It is a waste of time to attempt to get back to an old line. 34. As the course, in order to follow a great circle, is practically the most important element to be determined, mechanical means of doing it have been devised. Towson's Chart and Table is used much by English navigators. 48 NAVIGATION. Chativenet's Great-Circle Protractor renders it as easy as taking the rhutnb-line course from a Mercator chart. Charts have been constructed by a gnomonic projection, on whicli great circles are represented by straight lines ; but by these computation is necessary to find the course. 35. A great circle between twd points near the equator or near the same meridian differs little from a loxOdromic curve. But when the differences both of latitude and of longitude are large, the divergence is very sensible. It is then that the great circle, as the line of shortest distance, is preferred. But it is to be noted, that in either hemisphere the great- circle route lies nearer the pole, and passes into a higher latitude, than the loxodromic curve. Should it reach too high a latitude, it is usually recommended to follow it to the highest latitude to which it is prudent to go, then follow that parallel until it intersects the great circle again. 36. A knowledge of great-circle sailing will often enable the navigator to shape his course to better advantage. Let A B (Fig. 11) be the loxodromic curve on a Mercator's chart, A C B, the projected arc of a great circle. The length on the globe of the great circle A C B is less than that of the rhumb-line A B, or of any other line, as A D B, between the two. But A C B is also less than lines that may be drawn from A to B on the' other side of it, that is, nearer the pole ; and there will be some line, as A D' B, nearer the pole than the great circle, and equal in length to the rhurab-line. Betweeli this and the rhumb-line may be drawn curves from A to B, all less than the rhumb-line. If the wind should prevent the ship from sail- rig. ii. SHAPING THE COUESfi. 49 ing on the great circle, a course as near it as practicable should be selected. If she cannot sail between A B and A C, there is the choice of sailing nearer the equator than A B, or nearer the pole than A C. The ship may be Hear- ing the place B better by the second than by the first, al- though on the chart it would appear to be very &r off from the direct course. This may be stiikingly illnstrated by the extreme case of a ship from a point in a high latitude to another on the same parallel 180° distant in longitude. The great-circle route is across the pole, while the rhumb-line is along the small circle, the parallel of latitude, east or west ; the two courses differing 90°- Any arc of a small circle drawn between the two points, and lying between the pole and the parallel of latitude, will be less than the arc of the parallel. Hence the ship may sail on one of these small circles nearly west, and make a less distance than on the Mercator rhumb, or parallel due east. This is, indeed, an impossible case in practice, but it gives an idea of the advantage to be gained in any case by a knowledge of the great-circle route. It is possible in high latitudes that a ship may have such a wind as to sail close-hauled on. one tack on the rhumb- line, and yet be approaching her port better by sailing on the othesr tack, or twelve points from the rhumb-liae course. 37. The routes between a number of prominent ports recommended by Captain Maury are mainly great-cu-cle- routes, modified in some cases .by his conclusions respecting the prevailing winds. SHAPIITG THE OOUSSB. 38. The intelligent navigator, in selecting his course to a destined port, will not only have regard to the directness of the route, but will take into consideration obstructions and dangers which may be in his way ; prevailing winds 50 NAVIGATION. and currents ; and, in ease of a threatening storm, the course to be taken to avoid its greatest violence, or being driven on a lee shore. Good charts and books of sailing directions afford all requisite information respecting obstructions and dangers in the most frequented seas. Exploring expeditions from England, France, and the United States have of late years added greatly to this branch of nautical knowledge. The labors of Maury, and his recent colaborers in Eng- land and France, have greatly increased our knowledge of prevailing winds in large portions of the ocean. The care- ful observations of intelligent navigators are much needed still further to develop it. A few of the stronger currents, such as the Gulf Stream off the coast of the TJnited_ States, are well known. But more extended observations are wanted. Currents are often indicated by the difference in the temperature of their wa- ters from that of those surrrounding them ; so that the thermometer, as well as barometer, has become an import- ant instrument to the navigator. The works of Redfield, and especially of Reid and Pid- dington, afford much information respecting storms and tor- nadoes. That class of storms called cyclones is particularly deserving of attention. These branches of physical geography are well worthy of study by those engaged in navigating the ocean. CHAI'TEI?, II. REFRACTION.— DIP OF THE HORIZON.— PARALLAX. — SEMIDIAMETERS. EEPEACTION. 89. It is a fundamental law of optics, that a ray of light passing from one medium into another of different density- is refracted, or bent, from a rectilinear course. If it passes from a lighter to a denser medium, it is bent toward the perpendicular to the surface, which separates the two me- dia ; if it passes from a denser to a lighter medium, it is bent from that perpendicular. Let M and N" (Fig. 12) represent two media each of uniform density, but the density, or refracting power, of N" being the greater ; a b c, the path of the ray of light through them ; P b, the normal line, or perpendicu- lar, to the separating surface at 6. If a J is the incident ray, & c is the refracted ray ; P^ a is the angle of incidence / P 6 a' is the angle of refraction. If c 6 is- the incident ray, J a is the refracted ray, and V b a' and V h a are respectively the angles of incidence and refraction. Moreover, these angles are in the same plane, which, as it passes through P b, is perpen'dicular to the surface at which the refraction takes place ; and we have for the refraction Fig. 12. 52 NAVIGATION. a' ba — 'Pba-'Pba', or the difference of direction of the incident and refracted rays. A more complete statement of the law for the same two media is, that -: — „ , ■ = m, a constant for these media ; or, the sines of the angles of incidence and refraction are in. a constant ratio. This law is also true when the surface is curved as well as when it is a plane. 40. If the medium N instead of being of uniform density is composed of parallel strata, each uniform but varying from each other, the refracted ray b c will be a broken line ; and if, as in Fig. 18, the thickness of these strata is indefinitely small, and the density gradually increases in proceeding frqm the surface b,bc will become a I jf curved line. But we shall still have for any point c of this curve, c a' be- ing a tangent to it, Kg. 18. sin P J a -7 = m, sin P' c a' a constant for the particular stratum in which c is situated. This law, which is true for strata in parallel planes, ex- tends also to parallel spherical strata, except that the nor- mals P 5, P' c are no longer parallel, but will meet at the centre of the sphere. But the refraction takes place in the common plane of these two normals. 41. The earth's atmosphere presents such a series of pa- rallel spherical strata, denser at the surface of the earth, and decreasing in density, until at the height of fifty miles the re&acting power is inappreciable. REFRACTION. 63 Hg. 14. In Fig. ] 4, the concentric circles M H" represent sections of these parallel strata, formed, by the vertical plane passing through the star S and the zenith of an observer at A. The jiormals C A Z at A, and C B E at B, are in this vertical plane. S B, a ray of light from the star S, passes through the atmosphere in the curve B A, and is re- ceived by the observer at A. Let A S' be a tangent to this curve at A ; then the apparent direction of the star is that of the line A S' ; and the astronomical re- fraction is the difference of directions of the two lines B S and A S'. This difference of directions is the difference of the angles E B S, E D S', which the lines S B, S' A make with any right line C B E, which intersects thein. If, then, r represent the refraction, we have r 3= E B S-E D S'. Also, E B S is the angle of incidence, and Z A S', the appa- rent zenith distance, is the angle of refraction; and we have sjn E B S _ sinZAS'"'^' a constant ratio for a ^ven condition of the atmosphere and a given position of A ; but varying with the density of the atmosphere, and for different elevations of A aboVe the sur- face. For a mean state of the atmosphere and at the sur- face of the earth, experiments give m — 1.000294. The principles of Arts. 39 and 40, applied to this case, 64 NAVIGATION. show that astronomical refraction takes place in vertical planes, so as to increase the altitude of each star without affecting its azimuth. The refraction must therefore be subtracted from an observed altitude to reduce it to a true altitude; or « A = K—r, in which h is the true altitude, A', the apparent altitude, r, the refraction. These laws are here assumed. The facts and reasoning on which they depend belong to works on optics. (Bowd., p. 153 ; Hei-schel's Astronomy, p. 37 ; Lardner's Optics.) 42. Problem 14. To find the refraction of a star. Solution. In Fig. 14, let a = Z A S', the apparent zenith distance of the star, »• = E B S - E D S', the refraction, M=ZCE; then EDS' = ADC = ZAS'-ZCE = s-M, EBS =EDS' + r = a-M + r, , sin E B S sin (z — m + '') ana -: — „ . „, = : ■=■ m, sin Z A S sm s or sin [s— («—»•)] == m sin s ; (43) which is of the same form as (309) of Plane Trig., sin (z+a) = m sin s, the solution of which gives tan (z+^ a) — — — -- tan \ a. Putting « = — («—»•), its value in (43), we have tan \z—\ («—»•)] = ^_^ tan \ («—>•) ; whence tan \ (m— r) = YVm *^° ^~^ ("-'")]• (44) EEFEACTION. 55 In this u and r are both unknown, but we may note that each is a very small angle, being when the zenith distance is 0, and increasing with the zenith distance. As it is ne- cessary to make some supposition respecting them, let us assume that they vary proportionally, and that 7 = !?' a constant, reserving it for observations to test the rigor of this assumption. Equation (44) then becomes tan i (q-l) r = j=^ tan [z-^ {3-I) r], or, since i (y— 1) '* is quite small, i (2-1) r sin 1' = j-^ tan [z-i (g-l) r] ; whence r = 7 ,. ■ .,, • T— ; — tan (z— i (g— 1) r). (j — 1) sm 1' 1 + TO ^ ■* ^^ ' ' Since observation is to determine q, we may as well con- sider that it determines the whole of the constant factor into which q enters. ^ , 2 1 — m Put then n = ^_-,^^^^. • y+^, /> = *(?-!), and the formula reduces to r = w tan {e—p r), which is known as Bradley's formula. Suppose at two given zenith distances z' and s' the refrac- tions / and r^ are found by observations in a mean state of the atmosphere, then we have the two equations, r' =n tan {z'—p r'), * r* = « tan {z"—p r*) ; 66 NAVIGATIOK. and the two nnknowTi quantities n and p may be fonnd ly proper transformations, or by successive approximations. By comparing pairs of observations in this way at various zenith distances, the values of n and p come out very nearly the same, except at very low altitudes ; so that the hypothe- sis that q, and therefore n and p are constant, is found to be nearly, though not rigorously exact. The values that have been found are, with the barometer at 29.6 inches, and the thermometer at 50°, n = 57''.036, > = 3 ; and the formula by which Tab. XII. (Bowd.) has been com- puted is r = ST'.OSe tan (a — 3 r). (46) In computing by this formula, we must find an approximate value of" r, by assuming first r = 5'7'.036 tan s, and substitute the value thus obtained in the second member of the proper equation. Example. Find the refraction for the altitude 30°. log 57'.036 = 1.75615 1.75615 g=60° l.tanO.23856 3— 3r= 59°55'4' l.tan 0.23712 r= 98'.8 log 1.99471 »• = 98" 46 log 1.99327 3 r = 4' 56'.4 r — l'38''.46 43. Laplace, from a more profound investigation of this problem, obtained a more complicated formula, which agrees better with observations. Bessel has modified and improved Laplace's formula. His tables of refractions are now considered the most reliable. - They are found in a convenient form for nautical problems in Chauvenet's Method of Equal Altitudes, Table HI. The mean refractions m this Table are for the height of the REFKACTIOK 57 barometer, 30 uiches, and tbe temperature SO** of Fah renheit.* 44. The mean values of n and p iA Art. 42 correspond to' the height of the barometer, b = 29.6 inches, the thermometer, t = 50° Fahrenheit. Now, the refraction in different conditions of the atmos- phere is nearly proportional to the density of the air ; and this density, the temperature being the constant, is propor- tional to its elasticity ; that is, to the height of the baro- meter. Hence, if b is the noted height of the barometer (ia inches), r, the mean refraction of Tab. XII., 4 r, the correction for the barometer, then -^^ = ^^, . J— 39.6 ,,.. or j^=____^. (47) By this formula the correction for the barometer in Tab. XXXVI. is computed. Again, the elastic force being constant, the density in- creases by jfj part for each depression of 1° Fahrenheit. Hence, if ^ r = the correction for the thermometer, t = the temperature in degrees of Fahrenheit, ^''• = ^^('-+^''-) (48) which reduces to ji' r- 400° 50° ^ t 350° + «' • Chanvenet's Astronomy, Vol. I. pp. 12'7-1'72, contains a thorough in restagatlon of the problem of refraction, especiaHy of Bessel's formulas. 58 NAVIGATIOM'. by which the correction for the thermometer in Table XXXVI. is conjputed. Beesel's formulas are more rigid, but more complex. 45. Peoblem 15. To find the radius of curvature of the paih of a ray in the earth! s atmospJiere. Solution. By the radius of curyature for any point of a curve, is meant the radius of the circular arc, which most nearly coincides with the curve at that point. If we consider the curvature of the path of a ray to be uniform from B to A (Fig. 14), it is the same as considering the curve B A itself to be a circular arc, and the problem is reduced to finding the radius of this arc. Let C be the centre of the arc A B, R' = C A, the radius of curvature, J2 = C A, the radius of the earth. Since S B and S' A are tangents to the curve at B and A respectively, they are perpendicular respectively to the radii C B, C A ; hence, A C B = r, the difference of direc- tions of S B and S' A. As A B is a very small arc, we may put I A B = JR' sin r; and, since they are very nearly equal, we may also put AD=AB = JB'sinr. REFBACTIOIT. 69 or In the triangle ADC, A D _ sin A C D AC" sin ADC B' sin r _ sin « It sin (z — «) ' whence ^' = ^_5 ^^Ji (49) sin (a — u) sin r '^ ' or nearly enough, since .m and r are small, R'=^^. ^L, (50) sin z r ^ ' But in the preceding problem whence g' = 7, w = 7 ?• ; so that i2'= -t^, (51) sin z '^ ' which is the required formula, nearly. 46. When z = 0, or the star is in the zenith, that is, the path is a straight line. When 3 = 90°, or the object is in the horizon, that is, near the earth's surface a ray of light nearly horizon- tal moves in a curve, which is nearly the arc of a circle whose radius is seven times the radius of the earth. This, however, is in a mean condition of the atmosphere. The curve is greatly varied in extraordinary states of the atmosphere, or by passing near the earth's surface of differ- ent temperatures ; in very rare cases even to the extent of becoming convex to the surface a short distance. NAVIGATION. DIP O] THB HOEIZON. 47. Pboblem 16. Tofind the dip of the. horizon. Solution. Let A (Fig. 15) be the position of the observer at the height B A = A, above the level of th ; ssa ; A H, perpendic- ular to the vertical line, C Aj re- presents the true horizon. The most distant point of the" ^orizon visible from A is that at which the visual ray, H' A, is tangent to the earth's surface. The apparent direction of H' is A H', the tangent to the curve A H" at A. J H =P p A H' is the horizon to be the dip found. Let of Kg. 15. C be the centre of the earth, C, the centre of the arc H" A. H", C, C, are in the same straight Kne, since the ares H' B, H" A are tangent to each other at H', C A, C A, are perpendicular respectively to AH, A H' ; C A C'= H A H'= A H, the dip. Let Ji = CB, the ra,dius of the earth ; then i? + A = C A, 7 i2 = C A = C H", the radius of curvature of H'A, 6iJ = CC'. "We have, then, in the triangle C A C, by PI. Trig. (268), R—ih)(ih)^ 7 £.(.£ + A) ' and, since h is comparatively very small ^ai may therefore be omitted alongside of Ji, DIP OF THE HORIZON. 61 or, pnttiiig Bin i J ^ = ^ J ^ sin l', ^ sinT' r IS - sin 1" y IB Taking B = 20923596 feet (Herschel, p. 126), we find the ponstant factor ^ i/lL = 59''.040, AM= 59".Q40 \% (53) and log A H== 1.7V115 + i log A, h being expressed in feet, which is nearly the formula for Tab. Xin. (Bowd.) 2 / 3 Since —. — T77 y =r^ is constant, depending only upon the radius of the earth, A B is proportional to Vh, or the dip is proportional to the square root of the height of the observer above the level of the sea. 48. "Were the path of the ray, H' A, a straight line, we should have J'5^=HAH'=H'C A, and in the triangle H' C A whence, 2 sin* ^ A'H = ;gT^= ;g» nearly, or with h in feet, 4' S- &Z".11\ Vh. (54) Comparing this with A H= 59'.040 Vh, we find J^= A'H- 4:'.%Z\ Vh = A'H-.Q1i. 4' -H; 62 NAVIGATION. or that the dip is decreased by refraction by .074, or nearly ^ of it. But from the irregularity of the refraction of horizontal rays (Art. 46), the dip varies considerably, so that the tabu- lated dip for the height of 16 feet can be relied on ordinarily only within 2'. When the temperatures of the air and water differ greatly, variations of the dip from its mean value as great as 4' may be experienced. In some rare cases, varia- tions of 8' have been found. The dip may be directly measured by a dip-sector. A series of such measurements carefully made, and under different - circumstances, both as to the height of the eye, temperature a^d pressure of the atmosphere, and temperature of the water, is greatly needed. Prof. Chauvenet (Astron. I., p. 176) has deduced the fol- lowing formula, which it is desirable to test by observations : in seconds, ^JB- A'B-2i021" ^^, in minutes, J M= A'M—6'.6l t- in which t is the temperature of the air, t„ that of the water, by a Fahrenheit thermometer. When the sea is warmer than the air, the -^isible horizon is found to be below its mean position, or the dip is greater than the tabulated value ; when the sea is colder than the air, the dip is less than its tabulated value. (Raper's Nav., p. 61.) This uncertainty of the dip affects to the same extent aU altitudes observed with the sea horizon. 49. Nearthe shore, or in a harbor, the horizon maybe ob- structed by the land. (Bowd., p. 155.) The shore-line may then be used for altitudes instead of the proper horizon. Tab. DIP OF THE HOKIZOK. 63 XVI. (Bowd.) contains the dip of such water-line, or of any object on the water, for different heights in feet and dis- tances in sea miles. It is computed by the formula 7 ' a (55) in which h is the height in feet ; d, the distance of the object in sea miles ; D, the dip in minutes. 50. Pkoblem 17. To find the distance of an object of knovm, height, which is just visible in the horison. Solution. K the observer is at the surface of the earth at the point H' (Fig. 15), a point A appears in the horizon, or is just visible, when the visual ray A H" just touches the earth at H". Let A = B A, the height of A, (Z = H" A, the distance of A. As this arc is very small, we Kg. 16. liave (?= H' C A sinl' X C A = 7 J? X H' C A sin 1', smce by (51) C A —.1 B. From the three sides of the triangle C C A by PI. Trig, (268), sin i U O A - y ^-gi , or nearly i H' C A sin 1'= y g^g. and H'C'Asinl' = y^ 64 NAVIGATION-, Tlvisj substituted in, the expression for d, gives In this, Z C S. Thus we see. that this parallax takes place in a vertical plane, and increases the zenith distance, or decreases the altitude, of a heavenly body without affecting its azimuth. 55. This sufiices for all nautical problems except the com- plete reduction of lunar distances. For these and the more refined ob- servations at observatories, the spher- oidal form of the earth must be con- sidered. Then, as in Fig. 18, the radius C A does not coincide with the normal or vertical line C A Z, but meets the celestial sphei'e at a point Z', in the celestial meridian, nearer the equator than the zenith, Z. We may remark here that A C E, the angle which the vertical line makes with the equator, is the latitude of A ; and PABALLAX. 67 ACE, the angle -which the radius makes with the equator, is its ffeooentric latitude. 56. PeoSlbm 18. To Jind the parallax of a heavenly body for a given altitude. Solution. In Fig. 1^, let ^ = S, the parallax in altitude ; « = Z A S, the apparent zenith distance of S, corrected for refraction ; JR—GA, the radius of the earth ; c? = C S, the distance of the body, S, from the centre of the earth. Then from the triangle C A S, we have A sin C S A = g-H sin C A S, H sin z ,_ „. or sm^ = — ^ — , (58) . If the object is in the horizon as at H, the angle A H C is called its horizontal parallax ; and denoting it by n, we have from (58), or from the right triangle C A H, sin 7r = -=-, (59) which substituted in (58) gives sin ^ = sin 77 sin s. (60) If A = 90°— s, the apparent altitude of the object, we "have sin^ = sin n cos h/ (61) or, nearly, since p and tt are small angles, p = TT cos h. (62) 57. The horizontal parallax, n, is given in the Ifautical Almanac for the sun, moon, and planets. From Fig. 17 it is obviously the semidiameter of the earth, as viewed from the body. As the equatorial semidiameter is larger than any other, so also will be the equatorial horizontal pa/raUax. 68 NAVIGATION. This is what is given in the Almanac for the moon. Strictly it requires reduction for the latitude of the observer, and such reduction is made at observatories, and in the higher order of astronomical observations. 58. Tables X. A and XIV. (Bowd.) are computed by for- mula (62). Table XXIX. contains the correction of the moon's alti- tude for parallax and refraction corresponding to a mean value of the horizontal parallax, 57' 30". It should be used, however, only, for very rough observations or a coarse ap- proximation. Tab. XIX. contains the diflference of 59' 42' and the com- bined correction of the moon's altitude for parallax and re- fraction. The numbers taken from this table subtracted from 59' 42', give the correction of an apparent altitude for pftrallax and refraction. To this may be applied the reduo- tion of the refraction to the actual condition of the atmos- phere (Art. 42). If, instead of the equatorial hor. pa;rallax, we enter the table with the augmented parallax of Chau- venet's Lunar Method (Tab. HI.), we shall obtain the re- duction, not to the centre of the earth, C (Fig. 18), but to the point, C, where the normal line through A intersects the axis of the earth. Tabje XIX. of Bowdttch was arranged especially for one of the Lunar methods in that work, so that the redu(StiOtis of the distance should all be additive. AFPABENT S S M I D I AM S T S B S . 59. The apparent dia&ieter of a body is the angle which ts disk subtends at the {dace of the observer. Peoblbm 19. To jmi the apparent semidiameter of a heavenly body. Solution. In Fig. 19, let M be the body ; <2 = C M, its distance from ^e centre of the earth ; APPAHENT SEMIDIAMETEES. 69 TO NAVIGATION'. 8 = 0.2729 TT, and logs = 9.43600+ log By this formula the moon's horizontal semidiameter may be found from her horizontal parallax. The Nautical Almanac contains the semidiameters as well as the horizontal parallaxes of the sun, moon, and planets. 2d. For finding «', the apparent semidiameter as viewed by an observer at A on the surface of the earth, the right triangle A B' M gives . , r ,„^, sm 8' = ^ . (67 In the triangle CM A, sinMAO CM BinMCA" AM' or, putting and h = 90°— Z A M, the apparent, A' = 90°— Z C M, the true altitude of M, cosh _d . . co^h' - d" ^^^ whence, cos A which, substituted in (67), and by (63), gives r cos h . cos h sm s = -3 77 = sm s d cos h' cos A" or approximately, s' = s ■ — 77, (69) by which a' may be found when s and h are known. Since h < A', cos h > cos A', and consequently s' > sf that is, the semidiameter increases with the altitude of the body. The excess J s = s'— s, is called the augmentation. The moon is the only bofly for which this augmentation^ is sensible. ' APPARENT SfiMIDIAMETEES. 7-1 # 60. Peoblem 20. To find the augmentcctiwi of the momCs horizontal semidiameter. /Solution. From (69) we find J, t cos h — cos A' cos A ' which, by PI. Trig. (108), becomes, ^ J ^ ^ 2siDi(A' + A)smi(A'-A) ^ cos h' h' — h=:p, the parallax ; since it is small, we may put 2 sin i (A' — h) — 2 sin ^p =p sin 1" = tt cos h sin 1' ; and, in computing so small a quantity as A s, we may take A for i (h'+h), and cos h for cos A'/ and then A s = s TT sin 1' sin A, or, since (65) s = ^it, J « = ■= tt' sin 1' sin A, For the moon -5 = 0.2729 ; then J « = .000001323 n' sin A. (70) If we take tt = 57' 20", which is nearly its mean value, we have J s = 15''.65 sin A, (71) which agrees nearly with the formula for Tab. XV. (Bowd.) The augmentation may differ 2' from this mean value. Tab. II. of Chauvenet's Lunar Method contains this aug- mentation for different values of s, as well as of A, computed by a more precise formula. CHAPTER III. TIME. 91 . Transit. The instant when any point c^ the celestial sphere is on a giyen meridian is designated as the transit of the point over that meridian, 62. Hour-angle. The hour-angle of any point of the sphere is the angle at the pole, which the circle of declination pass- ing through the point makes with the meridian. It is pro- perly reckoned from the upper branch of the meridian, and positively toward the west. It is usually expressed in hours, minutes, and seconds of time. The iiitercepted arc of the equator is the measure of this angle. 63. Sidereal Ti-me. The intervals between the successive transits of any fixed point of the sphere (as, for instance, of a star which has no proper motion) over the same meridian would be perfectly equal, were it not for the variable effect pf nutation. (Hersch, Ast., Art, 82?.) This correction, arising from a change in the position of the earth's axis, i^ most perceptible in its effect upon the transit of stars near the vanishing point of that axis, i. e. near the poles of the heavens. Hence, for the exact measurement of time, we use the transits of some point of the equator, as the vernal equinox. This point is often called the Jlrst point of Aries. Its usual symbol is °P . 64. The interval between two successive transits of the vernal equinox is a sid^eal day; and such a day is regarded as commencing at the instant of the transit of that point. TIME. 73 The sidereal time is then 0'' 0" 0^ This instant is sometimes called sidereal noon. The effect of nutation and precession in changing the time of the transit of the vernal equinox is so nearly the same at two successive transits, ikaX the sidereal days thus defined are sensibly equal. It is unnecessary, then, except in refined discussions, to discriminate between mean and apparent sidereal time. 65. Ths sidereal, time at any instant is the hour-angle of the vernal equinox at that instant, and is reckoned on the equator from the meridian westward around the entire cir- cle, that is, from to 24''. It is equal to the right ascension of the meridian at the same instant. 66. Solar Ihne. The interval between two successive transits of the sun over a given meridian is a solar day, and the hour-angle of the sun at any instant is the solar time of that instant. In consequence of the motion of the earth about the sun from west to east, the sun appears to have a like motion among the stars at such a rate that it increases its right as- cension daily nearly 1°, or 4™ of time. With reference to the fixed stars, it therefore an-ives at the meridian each day about 4™ later than on the previous day; consequently, solar days are about 4" longer than sidereal days. 67. Apparent and Mean Solar Time. Jf the sun changed its right ascension uniformly each day, solar d^ys would be exactly equal. But the sun's motion in right ascension is not uniform,, varying from 3" 35" to 4'" 26« in a solar, day. There are two reasons for this,-^ 1st. The sun does not move in the equator, but in the ecliptic. 2d. Its motion in the ecliptic is not uniform, being most rapid at the time of the earth's perihelion, about January 1, and slowest at the time of the aphelion, about July 2. 74 NAVIGATION. To obtain a uniform measure of time depending on the sun's motion, the following method is adopted. A fictitious sun, called a mean sun, is supposed to move uniformly in the ecliptic at such a rate as to return to the-perigee and apogee at the same time with the true sun. A second mean sun ia also supposed to move uniformly in the eqteator at the same rate that the first moves in the ecliptic, and to return to each equinox at the same time wijh the first mean sun. The time which is measured by the motion of this second mean sun is uniform in its increase, and is called mean time. That which is denoted by the true sun is called true or apparent time. The difierence between mean and apparent time is called the equation of tims. It is also the diflference of the right ascensions of the true and mean suns. The instant of transit of the true sun over a given meridian is called apparent noon. The instant of transit of the second mean sun is called mean noon. The mean time is then Q^ 0™ 0». Mean noon occurs, then, sometimes before and sometimes after apparent noon, the greatest difference being about 16", early in November. 68. Astronom,ical Thne. The solar day (apparent or mean) is regarded by astronomers as commencing at noon (apparent or mean), and is divided into 24 hours, numbered successively from to 24. Astronomical time (apparent or mean) is, then, the hour- angle of the sun (true or mean) reckoned on the equator westward throughout the entire circle from O'' to 24''. 69. Civil Mme. For the common purposes of life, it is more convenient to begin the day at midnight, that is, when the sun is on the meridian below the horizon, or at the sun's lower transit. The civil day begins 12'^ before the astro- nomical day of the same date; and is divided into two TIME. 76 periods of 12'' each, namely, from midnight to noon, marked A.M. (ante-meridian), and from noon to midnight, marked P.M. (post-meridian). Both apparent and mean time are nsed. The affixes A.M. and P.M. distinguish civil time from as- tronomical time. During the P.M. period, this is the only distinction, — the day,' hoursj &c. being the same in both. 70. SeorTime. Formerly, in sea-usage, the day was sup- posed to commence at noon, 12'' before the civil day, and 24'' before the astronomical day of the same date ; and was di- vided into two periods, the same as the civil day. Sea-time is now rarely used. '11. To convert civil into astronomical time, it is only necessary to drop the A.M. or P.M., and when the civil time is A.M., deduct \^ from the day and increase the hours by 12i>. To convert astronomical into civil time, if the hours are less than 12'', simply affix P.M. ; if the hours are 12'' or more than 12'', deduct 12'', add l^, and affix A.M. Examples. Ast. Time. Civil Time. 1860 May 10 14 15 ]0 = 1860 May 11 2 15 10 A.M. 1862 Sept. 8 9 19 20 =: 1862 Sept.' 8 9 19 20 P.M. 1863 Jan. 3 23 22 16 == 1863 Jan. 4 11 22 16 A.M. 1863 Jan. 4 3 30 = 3 863 Jan. 4 3 30 P.M. 72. The hour-angle of the sun (true or mean), at any me- ridian, is called the local (apparent or mean) solar time. The hour-angle of the sun (true or mean) at Greenwich at the same instant is the corresponding Gheenwich time. So also the hour-angle of T at any meridian, and its hour- angle at Greenwich at the same instant, are corresponding local and Greenwich sidereal times. 76 NAVIGATION-. V3. The di^erence of the local times of ant/ two meridiam is equal to the difference of longitude of those meridians. jDem,onstration. In Pig. 20, let p P M, P M' be the celestial merid- iaiis of two places ; A y^ I \ \. ^M" 1* S, the declination circle through W\ the sun (true or mean) ; MPS, the hour-angle of the sun "^ M at all places whose meridian is ^'^- ^• P M, will be the local time (apparent or mean) at those places ; «o also M' P S will be the corresponding local time at all places whose meridian is P M' ; and M P M'= M P S - M' P S will be the difference of longitude of the two meridians. If P T is the equinoctial colure, M P T and M'P f will be the corresponding sidereal times at the two meridians ; still, however, MPM'=MPT-M'P A = A,-t' A. i ^^^' Thus, for using hourly differences, we wish the hours, 82 NAVIGATION. minutes, &c., of the Greenwich time expressed in hours and parts of an hour ; for using the differences for 1™, we wish the minutes and seconds of Greenwich time expressed in ininutes and parts of a minute. Decimal parts are usually most convenient; though some computers prefer aliquot parts. 84. The quantities in the Almanac, as commonly in other mathematical tables, are approximate numbers, that is, each is given only to the nearest unit of the lowest retained or- der ; and no refinement of interpolation can give a result to a higher degree of precision. In interpolating, more than- one lower order in any case is superfluous. Thus, the sun's declination is given to the nearest O'.l, and 'in no way can we by interpolation obtain a value which will be reliable within a narrower limit. Moreover, the Greenwich times are uncertain to a greater or less extent ; and ]S first differences only are used, the in- terpolated result can be regarded as true only within much wider limits than the approximation of the Ephemeris. In interpolating, then, it is well to consider the degree of approximation which is wanted in any particular case ; and if the nearest 1', or 10", or 1" suffices, contract the interpo- lation so as to retain at the most one lower order ; or else, consider the degree of approximation attainable in any par- ticular case, and contract the work so as to retain only the reliable figures. All lower orders are superfluous, and are deceptive, as- giving the appearance of a higher degree of accuracy than has actually been obtained ; as, for instance, using tenths and hundredths of seconds, when the data will give a result reliable within 2' or 3' only. A convenient method of contracting the multiplication and division of decimals is given in a pamphlet on the sub- ject. 85. Should it be desirable to interpolate more accurately THE NAUTICAL ALMANAC. 83 than can he done by first differences alone, the reduction for second differences may be introduced by a simple process. Let idj be the change of A^ in the time T\ then instead of ^1, as found in the Almanac for the nearest Greenwich datCj we may substitute that is, the value of 4,, interpolated for -J t, or to the middle instant between the Almanac date and the given time. This is simply using the mean rate of change during the in- terval. If J, is a "Diff. for l*"" given for the Almanac for each day, T = 24"; if A^ is a "Diff. for l"" given in the Al- manac for each hour, T' — 60"°. The interpolation of A^ to the middle instant may often be performed mentally. . , Example. If the sun's right ascension for 1865, Jan. 30, S"" 9"" time be required, we find in the Almanac, for Jan. 30 0" 4 = 10'.246 J, =— 0".035 31 0" 4= 10.211 and by interpolation for Jan. 30 i*", the middle instant be- tween Jan. 30 O"" and Jan. 80 8", J, = 10''.246 — O'.OOe = 10'.240, which is the mean hourly change in the interval from 0* to 8". 86. Formula (76), however, applies only to the American Ephemeris, where the differences for 1' or for l", which are designated by ^„ are given for the same instants of Green- wich time as the functions. A, to which they belong.* In the British Almanac before 1863 they are given for the * The " Prop. Logs, of Diff." of the ^unax Distances are given for the middle iiistaut. 84 NAVIGATION. middle instant between Wo dates.* For instance, the "Diff, for l"" " given in each as if for noon Jan. l"*, is in the Amer- ican Ephenieris the change per hour at Jan. 1^ 0'' ; in the British Almanae,' the change per hour, Jan, l"* 12^, or mid- night. * For the British Almanac (76) becomes A + ^ 4 (.77) Ai being taken from the same line or for the same d'ate as A„. This is the date preceding that of A^. 87. Problem 21. To find from the Almanac a required quantity for a given mean time at a given place. Solution. The preceding considerations lead to the follow- ing rule : — 1. Express the given m,ean time astronomically, stating the day as well as the hours, &c.,*and reduce it to Green- wich mean time by adding the longitude if west, subtracting if east. 2. Take from the Almanac for the nearest />rece'=i* — 2.1 = — 7°.56x. 2729 16 28 16 27.6 In Art. (59) we have for the moon, s = .2729 n ; whence ids = .2729 An: so that the reduction of the semidiameter may be readily found by multiplying that of the horizontal parallax by .2729, as in the above example. This coefficient .admits of a con- venient set of aliquot parts ; for .2730 = .25 + .025 — .002," so that As={i + -J^ — si-g) A n nearly. 83 NAVIGATIOir. 9". "il^gHght cuaensidn^ (Page 230.) hhrne s hms ■ Jan. go m 22 24.4 + \.9li 11 22 24.40 + l.9t\ (jit 4'') ( 15.8 in 8'' fl6.768 in S"* + 16.0 \ .2 .12 +16.01 1 .197 .1 1 89 .02 17 22 40 17 22 40.41 I 8 .004 10. 2f 's declinatim. (Page 230.) II . ii 82:-* ^IM_ ( 14.8 in 9" { 14.72 in 8'' iS.Q \ .2 .1 — l^.fl \ .18 .: -. ^ y^ o I II . 1/ a I II II Jan. SO -22 39 82:-* --1.86 —22 39 S2.4 --1.84 ,1 .02 — 22 39 47 — 22 39 47.3 11. 2f's semidiarmkr and horizontal parallax. (Page a84.) Jan. 30, Vert. sem. diam.,, 16'.29, Hor. Par., 1'.46. this is the vertical semi'diameter when thS plaAet is oa the meridian, or tlie semidiamete^ in the direction of the decUnaltion circle of the planet. The polar, or minor, semidiameter of the elliptic disk is ^ven on page 230. 12. The right ascension and declination of a Scorpii. (Antares.) Thf Washington mean time is Jan. 30 2' SO", or Jan. 30.12. On page 258, which serves as an index, the mean R. A. is le*" 21". The apparent R. A. and I3ec. are for Jan. 30.8 m. t, Washington, R. A. le"" 21" 8'.67'+ 0".33 Dec— 26» 7' 37M — 0"8. Change in — 0''.7 — .02 + .i 16 21 8.6S -^ 26 7 37.0 91. Problem 22. To find from the Almanac the smw's right ascension and declinatiouj and the equation of time for a given apparent time at a given place. Solution. This diflfers from the preceding problem simply in nsing the apparent instead of the mean time, and in tak- ing the quantities from page I. for the month, where they are given for apparent noouj instead of from page II., where they are given for mean noon. THE NAUTICAIi ALMANAC. ExAltPLBS. Find the Q's R. A. and Dec. and the equation of time for 1865 Jan. 30 9^ O" 48' A.M. apparent time in Long. 163° 14' W. Ast app. time 1865 Jan. 2i9' 21'' 0* 4^' EOn^l + I'O ^2' 56' Gj%pp. time 30 7 53 44 1 53.13 7.896 S'aE. A.2d62 37.1 + 10.'246 O's dec— 17 S3 47.2 +41.44 + 1 20.9 i'i .7' r 290.1 20 53 58 ■ ' + ^ ^''■^ \ . S.I — 17 28 20 ' [ .3- ma fl Equation- of time + 13 38.9 + 0.390 + 3.1 + 13 42.0 ' ^ 92. Problem 23. To find the right ascension and decli- nation of the sun, and the equation of time at apparent noon of a pivenr place, or' when the sun is onithe Meridian. Sohition. The local apparent time is O"" 0"° 0". The Green- wich appafelif time is then equal to the longiliud'e if west, that is, it is after the noon of the same date by a number of hours, &c., equal to the longitude. If the longitude is east, the Greenwich iipparestit time is before the noon of the same date by a number of hours, &c., equal to the longitude. Hende, take these quantities from the AlrHaUac for Green- wich appai'Sn-t noon (page I.) of the same day as the local (civil) day, and apply a correction equal to the hourly differ- ence multiplied by the hours and parts of an hour of the longitude ; observing" to add- or subtraict the correction,' ac- cording as th'e' niitabers in the Almanac may require, for a time after noon, if the longitude is west/ for a time b^ore noon, if the longitude is east. 90 NAVIGATION". The hourly differences from the British Almanac should be taken as given for the preceding instead of the same day in vast longitude. Eyamflbs. 1. Find the sun's right ascension and declination, and the equation of time for apparent noon, 1865, Jan. 30, in Long, 163° 14' W. Long. + 10 52 56' G'a R A. 20 32 37.14 + lo'.2S8 102.38 inlOh 5.129 30°> + 1 61.86.} 8.413 20 .427 2 30» 20 34 28.60 1 25 O^sdec.— 1?3S47".2 +41.62 '416.2 + 1 32.6 20.81 • 13.87 — 17 26 14.6 1.73 8 Eq. of T. + 13 38.90 + 0.382 '3.82 in 10* +4.16 J .191 30" .127 20 +13 43.06 (. .016 2i 2. For apparent noon, 1865, March 21, in Long. 163° 14' E. LoDg. —10 62 56 —10 62.93 —10.882 h m B I ©'a E. A. —0 3 20.43 +9.099 , „„„„ r 90.99 — 139.02, Y.279 ■ -.728 1 41.41 18 ©'s dec. +0° 21 44.9 + 69.21 '592.1 — 10 44.3 47.37 i 4.74 +0 11 0.6 [ .12 Eq. of T. +7 16.59 —0.767 r7.67 +8.24 1 606 61 +7 23.83 2 3. For apparent noon, 1865 , March 20, in Long. 150° 35' W. IiOng.+10 2 20 O's B. A. 23 5^9 42!o7+ 9'.101 91.01 in 10* + 1 31.36 .803 i" 61 20* 1 13.43 THE NAUTICAL ALMAJTAC. 91 I M 0's dec. —0 1 56.2 + 69.^3 Eq. of T. +1 S3.12—MSS C 592.3 in 10'' (75.6 +5 54.6 i 1.91 2°' — T.58 ■{ 25 ( .33 20" ( 4 +0 3 58.4 +1 26.14 In the 1st and 2d examples, the Diff's. for l' have been in- terpolated for S'.S or half the longitude, forward in the first, back in the second: in the third they have been interpolated forward for 5\ 93. Peoblem 24. To find the right ascension of the mean sun for a given time and place. Solution. At the instant of mean noon, or when the mean sun is on the meridian, at any place, the right ascension of the mean sun is equal to the sidereal time. The quantity on page n. of each month, in the Almanac, called " sidereal time," is also the right ascension of the mean sun at Green- wich mean noon, and may be interpolated for a given local time in the same way as the right ascension of the true sun. (Prob. 21.) The constant "Diff. for 1"" is 9'.856. A table for converting mean time into sidereal time intervals (Tab. n.) facilitates the interpolation. We have also the right ascension of the mean sun equal to that of the true sun + the equation of time, using for the equation of time the sign of its application to mean time. 94. Problem 25. To find the mean' time of the moon^s' transit over a given meridian on a given day. Solution. The Almanac contains the mean time of each transit of the moon over the meridian of Greenwich (page IV.). This mean time is the hour-angle of the mean sun (Art. Y2) when the moon is on the meridian ; and is there- fore the difference of right ascensioh of the moon and the mean sun. As this difference is constantly increasing, in consequence of the moon's more rapid increase of right y^ NAVIGATIOlir. ascension, tbe mean time of each transit is later than that of the one preceding by a number of minutes, varying, ao- cbrding to the rate of the moon's motion, from 40" to 66°- If, then, Ti and T, denote the mean times of two succe* sive transits of the moon over the Greenwich meridian, Tj — Ti is the retardation of the moon in passing; over 24'' of longitude'; so that for aily longitude X- (expressed in^ hours), the retardation is nearly The mean time of a transit is; then, reduced from" the Greenwich to any other meridian by interpolating for the longitude ; forwcvrd, if the longitude is west; baokward, if the Ibngitude is east^ since east longitudes are regarded as negative. The American Ephemerls gives also the hourly differeilces, ■which facilitate the interpolation. For greater exactness, these differences may be interpolated for/ta{/'the longitude. The practical rule will be : — Take from the Almanac the mean time of meridian passage . for the given astronomical* day, and add to it the product of the " i)iff. for 1''" by the longitude in hours, if the longi- tude is west; subtract that product if the longitude is east. Prom the firitish Almanac the daily retardation may be found by taking the difference for two successive transits ; and the reduction by multiplying it by the longitude in parts of a day ; or it may be taken from Tab. XXYIII. (Bowd.) The mean time of meridian passage for the given day, and that for the day following in west longitude, or for the day preceding in east longitude, are those which are commonly * It is important to notice whetSer the mean time of transit is more or less tlian 12''. In the former case,, the astronomical day is li less than the civil day THE NA,UTICAL A,LMANAC. , 93 used. ,(Bowd.7p. 1.70.) But it is more exact to use half tlie difference of the times of meridian passage for the day preceding and the day following the given day : Jj of this is the "Diff. for l""" of the American Ephemeris. The times of transit are given only to tenths of a minute, which suffices the purposes of the navigator. They may be found more exactly for any meridian by the method herp- after given in Prob. 33. 95. Eroblbm 26. To find on a given day the mean time of transit of a planet over a given meridian. tSolution. The mean time of each meridian passage at Greenwich is given, in the Alniaijac, for each planet. It may be reduced to any meridian in the same way as for the rnoon ; except ithat, in the case qf an acceleration, thesiga of the reduction is reversed. Examples. 1. In Long. 100°. 15' W., find the times of meridian pass- age of the moon and Jupiter for 1865, June 6 (civil day). Long. + e** $1" 0' =. 6''.683. M. T. of mer. pass. Junes 9 69.6 +1.99 June 6.12 41.8 ^4.46 in 1'' {1^94 ( 1.11 in 6'' 1.19 — 1.2 i .11 .6 .16 ( ,1 .08 1 June 6 12 40.6 , or June 6 40.6 A.M. 2. In Long. 100° 15' E. for 1865, June 6 (civil day), finji ,the times of meridian passage of the moon and Jupiter. Long. -^ 6'' 41° 0' = — 6\683. J U fas B h m m M. T. of mer. pass. June 6 9 69.6 + 1,97 June 6 12 41.8 — 4.45 in 1* flL82 (17llin6'' -13.2 I 1.18 + 1.2 i .11 .6 1 ..16 ( 1 J)8 June 6 9 46.4 [ 1 June 6 12 48.0 or June 6 43.0 A.M. 94 NAVIGATION. In the case of the moon the hourly differences have been interpolated for half the longitude. 96. Peoblem 27. To find the right ascension or declina- tion of the moon, or a planet, at the time of its transit over a given meridian on a given day. Solution. Find the local mean time of transit, as in Pro- blem 25 ; deduce the corresponding Greenwich time by ap- plying the longitude ; and for this Greenwich time take out the right ascension or declination, as in Problem 21. If the time of transit has been noted by a clock or chrono- meter, regulated to either local or Greenwich time, it should be used in preference to the time of transit computed from the Almanac. 97. Pkoblem 28. To find the Greenwich mean time of a given lunar distance. /Solution. The angular distances of the moon from the sun, the principal planets, and several selected stars, are given in the Almanac for each S' of Greenwich mean time. If d represent the given distance ; da, the nearest distance of the same body in the Almanac preceding in time the given distance ; Ji, the change of distance in S"" ; t, the required time (in hours) from the date of d^ ; by (75) we have approximately, using 1st differences only, whence, for the inverse interpolation, « = |(c?-<4), (79) or, with t in seconds of time, which is better for computation, 10800" , , , v t = —^{d-da), (80) , n which it is most convenient to express J, and {d—d^ va, seconds. THE NAUTICAL ALMAKAO. 95 Then by logarithms : / ■, t\ , losao ,„,. log t = log {d - do) + log— 3-, (81 ) ' : is the change of distance in 1' ; hence log — -j — is the 10800 " „ e^ V.X ^.^.^ V.V, , ^ V, s> j^ ar. complement of the " log diff. for 1'." It is given in the Almanac for the middl^ instant between the tabulated distances under the head " P. L.* of Diff." ; the index, which is 0,'and the separatrix being omitted. In the same way, if meris, previously referred, to. THE NAUTICAL ALMANAC. 97 Dividing both members of (80) by 10800, and inverting, we have 10800 _ _J,__ 10800 t ~ 10800 ^ d — d^ and, P. log ft = P. log id—d^-V. log Ji, (83) ■which accords with the rule on page 231. (Bowd.) 100. Examples. 1865, Oct. 31, the distance of Fomalhaut from the moon's centre is 42° 3' 35", what is the Greenwich mean time ? d = 42° 3' 35' Oct, 3115'' 2i\ Mi, may be taken out for 1'' later than stated in the previous precept, and interpolated for the excess of (A +(5) over 24'' ; and when (/L + iS) is negative, to retain its negative character, or else take out M^ for one day earlier. 3d Solution. From (89) we baTe t = iS-[/S'o + .0027379 («+A)], (94) so that, when the Gneenwidi mean time (t+X) is sufficiently known, we may find for it the right ascension of the mean Bun, (Ai-t. 107) >% +.00273 79 (t+l), and subtract it from the given sidereal time : or, the mean tim^ is equal to the sidereal time — the right ascension &f the mean sun. So also we have from Art. 107 the precept: — the apparent time is equcd to the sid&real time — the right ascension of the true sun. 106 NAVIGATION. Examples. 1. 1865, Jan. 30 (ast. day), in long lO"- O" 52'.T W., the eidereal time is G*" 57"° 42.'4 ; find the mean time. h m « h m B L. sid. t. 6 57 42.4 L. sid. t. 6 57 42.4 — iSo (Jan. 30) —20 38 56.00 M^ (Jan. 30) 3 20 31.06 — Red. for long. — 1 38.71 Red. for long. — 1 38.44 Sid. int. 10 17 7.69 Red. of sid. t. — 1 8.43 Red. of sid. int. — 1 41.10 L. m. t. Jan. 30 10 15 26.6 L. m. t. Jan. 30 10 15 26.6 2. 1865, Jan. 30, (ast. day,) in long. lO* 0" 52'.7 E., the sidereal time is G"" 54°' 25'.0 ; what is the mean time ? h m a h m a L. sid. t. 6 54 25.0 L. sid. t. , 6 54 25.0 — S„ (Jan. 30) —20 38 56.00 M^ (Jan 30) 3 20 31.06 — Red. for long. + 1 38.71 Red. for long. + 1 38.44 Sid. int. 10 17 7.71 Red. of sid. t. — 1 7.89 Red. of sid. int. — 1 41.10 L. m. t. Jan. 30 10 15 26.6 L. m. t. Jan. 30 10 15 26.6 3. 1865, Sept. 26, 9" A.M., ii) long. 4" O" 52'W., the sidereal time is g' 37°' 40M ; find the mean time. h m a h m B L. sid. t. 9 37 40.1 L. sid. t. 9 37 40.1 — So (Sept. 25) —12 17 15.89 M^ (Sept. 25) 11 40 48.98 — Red. for long. — 39.57 Red. for long. — 39.46 Sid. int. 21 19 44.64 Red. of si^. t. —1 34.64 Red. of sid. int. — 3 29.65 L. m. t. Sept. 25 21 16 15.0 L. m. t. Sept. «5 21 16 15.0 4. 1865, Sept. 25, 3' P.M., in long. 8'' 16*° 25'.3 E., the side^ real time is 15'' 32" 41'. 6 ; find the mean time. EELATIOW OF HOUB-ANGLES AND TIME. 107 h m s h m a L. sid. t. 15 32 41.6 L. sid. t. 15 32 41.6 — S„ (Sept. 26) —12 17 15.89 M^ (Sept. 24) 11 44 44.89 — Red. for long. + 1 21.55 Red. for long. + 1 21.33 Sid. int. 3 16 47.26 Red. of sid. t. — 2 32.80 Red. of sid. int. — 32.24 L. m. t. Sept. 25 3 16 15.0 L. ui. t. Sept. 25 3 16 15.0 EELATIOlf OF HOTTE-ANGLBS AND TIME. 109. Pboblem 33. To find tlie mean time of meridian transit of a celestial body, the longitude of the place or the Crreenwich time being known. Solution. In the case of the sun the instant of meridian transit is apparent noon of the place ; for which we have (84) T^^=E, the equation of time, 'which can be taken from page I. of the Almanac, and inter- polated for the longitude, which in this case is also the Greenwich apparent time; or from page II., and interpo- lated for the Greenwich mean time. When jE' is subtractive, the subtraction from the number of days can be performed. The apparent right ascension of any body at the instant of its meridian transit is also the right ascension of the me- ridian, or sidereal time. (Art. 65.) It suffices therefore to find the right ascension of the body, and, regarding it as the sidereal time, reduce it to mean time by Problem 31. The American Ephemeris contains the apparent right as- censions of two hundred principal stars for the upper cul- minations at Washington ; the British Almanac contains the positions of one hundred for the upper culminations at Greenwich. They are reduced to any other meridian, when necessary, by interpolating for the longitude. The right ascensions of the moon are given for each hour, and of the planets for each noon, of Greenwich mean time, 108 NAVIGATION. and may be found for a given Greenwich mean time by Problem 21. If, however, the longitude of the place is given, the local mean time of transit of the moon, or a planet, may first be found from the Almanac to the nearest minute or tenth (Probs. 25, 26) ; then for this mean time the right as- censions of the moon, or of the planet (Prob. 21), and of the mean sun (Prob. 24), may be computed. Subtracting the right ascension of the mean sun from the right ascen- sion of the moon, or planet, will give the mean time of transit (Pi-ob. 32, 3d solution). If it differ sensibly fi-bm that previously obtained, the process may be repeated with this new approximation. If the time of transit has been noted by a clocks or chro- nometer, tegulated eith'er to local or Greenwich time, it should be used in preference to the approximate time of transit found fronl the Almanac irl computing the right ascensions. The American Ephemeris contains also the right ascen- sions of the moon* and principal planets at their transits of the upper meridian at Washington. They can be reduced to any other meridian by interpolating for the longitude from Washington. This solution will give the time of the upper culmination of a heavenly body. To find the time of a lower culmi- nation, 12' may be added to the right ascension of the body, if sufficiently well known ; or, as is generally preferable, 12'' may be added to the longitude of the place. The in- stant of a lower culmination on any meridian will be that of an upper culmination on the opposite meridian. EXAMPLBS. 1. Find thetimes of meridian passage of the moon and Jupiter for 1865, June 6 (civil day), in long. 100° 15' W. ^Example 1, Art. 95, p. 93.) * The mooa's R. A. has been omitted since 1864. RELATION OF ttOUE-ANGLES AND TIME. 109 D U h ID b m Approx. in. t. June 6 10 12.9 June 6 12 40.6 Lung. +6 41.0 +6 41.0 G. m. t. June 6 16 53.9 June,6 19 21.'6 li m e fl h m fl S R. As'n. 15 13 28i54+2.1088 If 39 52.80 - - 1.340 : 105.440 f 13.40 Red. for G. m. t. + 1 53.66 ■ 6.326 ' 1.898 — 25.64 11.760 .402 i 80 R. As'h at trahSt i5 iB 22.20 17 39 27.16 «. 4 69 38.32 4 66 41.76 Red. for 6. m. t. + 2 46.56 + 3 10.82 ^i 6 2 24:88 4 68 52.58 M. t. of transit, June 6 10 12 57.32 June 6 12 40 34.58 Diff. from approx. t. + 3.32 — 1.42 .117 .009 M. t. of tranfei*. Jutie 6 10 12 57.43 in 3»32l^''-°^^-^- 110. Problem 34. To find the hour-angle of the sun for a given place and time. Solution. Tlie hour-angle of the siih, reckoned from the upper meridian toward the iifest, is the d/fyparetit timfe reckoned astvonomically (Art. 72). It& hour-aigle .east of the meridian is negative, and numerically equal to 24''=^the apparent time. A given rthean or sidereal time must then be converted into apparent time ; for this, the longitiide, or the Green- wich time, must be known apJ)rOximately. 111. Peoblem 35. ih find the hdur-dngh. of the moan, a planet, or a fixed star, for a given place and time. Solution. In Fig. 21, as descriljed in Art. 104, Y M is the right ascension of tiie meridian^ and measures M P Y, the sidereal time. Let liO NAVIGATION. P S be the deolinatioii-circle of the mean sun, then V S is the right ascension of the mean sun, and M P S is the mean time, and is measured by the arc of the equator, S M. Let P M' be the declination-circle of some other celestial body ; then "V M' is its right ascension, and M P M' is its hour-angle, and is measured by the arc M' M. From the figure, M' M =Y M -Y M'=Y S + S M -Y M'. (95) K "V S is the right ascension of the true sun, S M will measure the apparent time. From (95), then, we have the following rule : — To a given apparent time add the right ascension of the true sun ■; or to a given msan time add the right ascension of the mean sun, to find the corresponding sidereal time. Then from the sidereal time subtract the body's right ascen- sion ; tlie difference is the hour-angle west from the meri- dian. If it is more than 12', it maybe subtracted from 24': the hour-angle, then, is — , or east of the meridian. It is necessary to know the longitude, or the Greenwich time, sufficiently near to find the right ascensions of the sun and body. 112. Pkoblem 36. To find the local time, given the hour- angle of the sun and the Greenwich time. Solution. The hour-angle reckoned westward is itself the local apparent time, which may be reduced to mean or side- real time (Probs. 29, 30), as may be required. The Green- RELATION OF HOUR-ANGLES AND TIME. Ill ■wich time, or thte longitude of the place, is needed only for this reduction. 113. Peoblem 37. To find the local time, given the hour- angle of some celestial body and the Greenwich time. Solution. Find from the Almanac for the Greenwich time (Prob. 21) the right ascension of the body. Then, from (95), we have YM^Y M'+M'M, from which, and Arts. 105, 107, we have the following rule, regarding hour-angles to the east as negative : — To the right ascension of the body add its hour-angle, the result is the sidereal time. From this subtracting the right ascension of the true sun gives the apparent time ; or the right ascension of the m,ean sun gives the mean time. The Greenwich time is needed for finding the required right ascensions. If the longitude of the place is given, but not the Green- wich time, we may first use an estimated Greenwich time, and then revise the computations with a corrected value, nntU the assumed and computed values sufficiently agree. Examples. 1. 1865, Jan. 16, 12'' 15" 17' 6, mean time in long.. 150° 13' 10" W., find the hom--angle of the moon. h m a Ii m 8 L. m. t. Jan. 16 12 15 17.6 L. m. t. Jan. 16 12 16 17.6 Long. -1-10 52.7 S„ 19 43 44.22 G. m. t. ' Jan. 16 22 16 10.3 Red. for long. + 138.71 ))'BR.A.(Jan. 16 22'-) 11 48 31.61 - H'.8684 Red. of L. m. t. -|- 2 0.79 il8 .584 11.150 L. sid.t 8 2 41.32 .186 .133 J's R. A- at date 11 49 1.66 D's hour-angle — 3 46 20.34 112 NAVIGATIOl*. 2. 186Si Jaa. Id 22'" 16° lO'.S, G. ttiean time, the moon'a hour-angle is. —3'' 46° 20".3 ; find the local mean time. h m ■ i)'B houf-angle -^ 3 46 20J j)'s R. A. (Jan. 16 28'') +11 48 31.61 + 1'.8684 {IS .684 11 .160 .186 .133 —S^ (Jan. 16) —19 4S 44.22 —Red. for G. m. t. — 3 39.50 L. m. t. Jan. 16 It 15 17.6 ' Subtracting this from the G. m. t. gives for the longitxide lO"- 0° 52".7 W. 3, 1865, Jan. 16, 12'' (nearly) in long. 150'^ IS' lO' W., the moon's hour-angle is —3'' 46° 20".3 ; find the local meau tinale. Long. 10 sVt S's mer. pass. Jan. 16 15 50.7+1.74 4 'B h. ang. ^ 3' 46'».3 Bed. for long. +17.4 j^ i„{ch.ofR.A. — 7 .0 Jan. 16 16 8.1 '■ (-ch.of;S, + .6 .' . . . .-352.7 Ist ipprox. L. m. t. Jan. 16 12 16.4 Long. +10 0.8 1st approx. G. m. t. Jan. 16 22 16.2 5)'s h. ang. —3 4°6 ^'o.3 t'a R. A. (Jan. 16 22>')+ll 48 31.61 +K8684 . ( 1,8 .5S4 Bed. for G. m. t. +30.11-^11.150 ch. in— 1».6 —,048 ( .372 L. Bid. t. 8 2 41.42 —S„ (Jan. 16) —19 43 44.22 -Bed. for G. m. t. -3 39.60 — ch. in -1>.6 +^0Q4 2d L. m. t. Jan. 16 12 16 17.70 cor. for --1».6 —.04 Long. 10 5&.7 2d G. m. t. Jan. 22 22 16 10.4 Diff. from 1st G. m. t. —1.6 8d L. m. t. Jan. 16 12 15 17.7 CHAPTER VI, NAUTICAL ASTRONOMT. AliTITTTDES. AZIMUTHS. HOUE-AJSTGLES AHD TIME. 115. NAUTiCAii Astronomy comprises those prolslems of Sphedcal Astronomy which are used in determining geo- graphical positions, or in finding ithe -corrections of the in- struments employed. In general, they admit of a much more refined application on shore, where more ddiicate and stable instruments can be used, than is possible at sea, where the instability of the waves apd the uncertainty of the sea-horizon present practical obstacles, both to precision in observations and to the accuracy of the results, which can- not be obviated. 1 16. In the problems which are here discussed the jfoUow- ing notation will be employed : — * Zi = the latitude of the place -of observation ; h = the true altitude of a celestial body ; « = 90° — A, its zenith distance ; d = its declination ; p = its polar distance ; •t = its hour-angle ; Z =■ its azimuth. Let the diagram (Fig. 22) represent the projection of the celestial sphere on the plane of the horizon of a place : — Z, the zenith of the place ; N Z S, its meridian ; 114 NAVIGATION. P, the elevated pole, or that whose name is the same as that of the latitude ; M, the position of a celestial body; Z M H, a vertical circle ; and P M, a declination-circle, through M. Then, in the spherical triangle PMZ, P Z = 90°— Zi, the co-latitude of the place ; P M = jt> = 90°— c?, the polar distance of M ; Z M = 90°— A, the complement of its altitude, or its zenith distance ; Z P M = «, its hour-angle ; P Z M = Z, its azimuth. The angle P M Z is rarely used, but is sometimes called the position angle of the body. This triangle, from its involving so many of the quantities which enter into astronomical problems, is called the astro- nomical triangle. As three of its parts are sufficient to de- termine the rest, if three of the five quantities i, d, h, t, and Z are known, the other two may be found by the usual formulas of spherical trigonometry. These admit, however, of modifications which better adapt them for practical use. The following articles point out how i, d, h, and t may be obtained. 117. The latitudes and longitudes of places on shore are given upon charts, but more accurately in tables of geogra- phical positions, such as are found in books of sailing-direc- tions, and in Tab. LIV. (Bowd.) At sea it is sometimes necessary to assume them from the dead reckoning brought forward from preceding, or carried back from subsequent, determinations. (Bowd., p. 264.) NAUTICAL ASTRONOMY. 115 118. The altitude of an object may be directly measured at sea above the sea-horizon with a quadrant or sextant ; on shore, with a sextant and- artificial horizon, or with an alti- tude circle. All measurements with instruments require cor- rection for the errors of the instrument. Observed altitudes require reduction for refraction and parallax ; for semidia- meter, when a limb of the object is observed ; and at sea, for the dip of the horizon. The reductions for dip and re- fraction are subtraetive/ for parallax, additive. Strictly, the- reductions should be made in the following order : for instrumental errors, dip, refraction, parallax, semidiameter. In ordinary nautical practice it is unnecessary to observe this order. Following it we should have, — 1st. The reading of the instrument with which an altitude is measured ; 2d. The corrected reading or observed altitude of a limb ; 3d. The apparent altitude of the limb ; 4th. When corrected for refraction and parallax, the true altitude of the limb ; 5th. The true altitude of the centre. Except with the sea-horizon, the observed and apparent altitudes are the same. For the fixed stars, and for the planets when their semidiameters are not taken into account, the altitudes of the limb and the centre are the same. Unless otherwise stated, the true altitude of the centre is the altitude which enters into the following problems, and is denoted by A. 119. The hour-angle of a body can be found, when the local time and longitude, or the Greenwich time, are given. (Probs. 34, 35.) For noting the time of an observation, a clock, chronometer, or watch is used ; at sea, only the last two ; but it will be necessary to know how much it is too fast or too slow of the particular time required. 116 NAVIGATION. J20. The declination, of ^a body «an be found whe» tJio Grree^jyich time-is known, (Prob. 21.) iJhe polar distance -of a heayenly body is the aro of the d«clinationrcircle between the body and the .elevated pole of the place ; that is, (the north pole, when ,the place ig in north latitude ; the so\itjk pole, when it is in smith latitude. If P P' (Fig. 23) is the pi;ojec- ttion of (the declination-circle -through an -objeot, M ; P, the north pole-; P', the south pole ; E Q, the equator ; then the polar, distances,- m' A \ v / \v j V] Jy N P M = P Q - QM = 90°- = tan h cos A, cos a = COg(c— :0)c6S& eotB: ecrs f ' sin (c — f) cot A sin Fig. 24 Hg. 86i ■which, by substituting the fcbrrespdnding parts of the triaa gle P Z M, giye tan ib = cot d cos t. sin A : cotZ = sin (#+i) sin d cos f ' cos(0+£)cot i sin ^ (96) 118 NAVIGATION. If we put ^ = 90°—^', these become tan (j)' = tan d sec t. sin h = cos Qp' — L) sin d sin 0' ' , _ sin (d)' — L) cot * cot Z = — :f, , cos (p (97) which afford the convenient precept, 0' has tJie same name, or sign, as the declination, and is numerically in the same quadrant as t. 122. When t — &\ '= 90°, and the 3d of (97) assumes an indeterminate form. But from the 1st we have tan d cot t = which, substituted, gives tan 0' sin t' ^ „ sin ((*' — i) tan d cot Z = — . ^, . . — , sin f sin t (98) which may be used when t is near 6''. 123. h is the tn-ue altitude of M. If the apparent altitude is required, the parallax (Art. 54) must be subtracted, and the refraction (Art. 41) added. Z is the true bearing, or azimuth, of the body, reckoned from the N. point of the horizon in north latitude, and from the S. point in south latitude. It is generally most conve- nient to reckon it as positive toward the ea^t, which will re- quire in the above formulas — ^for Z, since t is positive when west. Restricting, however, Z numerically to 180°, it may be marked E. or W., like the hour-angle. 124. In Fig. 24, if M m be drawn perpen'&icular to the meridian, then 'Pm = = 90°—', Zm=: (0+i)— 90°= i— ^'; or, is the polar distance of m,, ', its declination, AITITUDE AND AZIMUTH. 119 Zi — 0, its zenith distance, positive, or of tlie same name as the latitude, toward the equator. A convenient precept is to mark it N. or S., according as the zenith is N. or S. of the point m. m falls on the same side of the zenith as the equator when Z > 90° ; at the zenith when Z = 90° ; and on the same side as the elevated pole when Z < 90°. It falls between P and Z only when t and Z are both less than 90°. 125. In the case of o Ursse Minoris {Polaris), whose polar distance is 1° 25', the more convenient formulas derived from (96) wiU be, since p and ^ are small,

90° ; (101), when ^< 90° If the body is in the visible horizon, then nearly h— — (33'+ the dip). 128. If the bearing of the body is observed with a com- pass at the same time that its altitude is measured, or if the beariflg is observed and the local time noted, the cMination, or variation, of the compass can be found. For, the true azimuth, or bearing, of the body can be found from its alti- tude (Prob. 40), or from the local time (Prob. 38) ; and the magnetic declination is simply the difference of the true and 122 NAVIGATION. magnetic bearings of the same object, determined simulta- neously if the object is in motion. It is marked £!. when the true bearing is to the right of the magnetic bearing, W. when the true bearing is to the left of the magnetic ? bearing. (Bo wd., p. 161.) 129. The amplitude of a star when in the true horizon is its distance from the east or the west point, and is marked N. or S., according as it is north or south of that point. It is, therefore, the complenjent of the azimuth. Problem 41. To find the amplitude of a heavenly body . when in the horizon of a given place. Solution. Let the body be in the horizon at M (Fig. 27), -4 =W M, its amplitude. The triangle P M N is right angled at N, and there are given PN = i, PM = 90°-rf, to find S 'S'M.=Z=9)0°—A. Kg.2T. We have cos P M = cos P N" cos N M, or sin d = cos Xi cos Z, whence cos Z = sin ^ = sin =49' 32" log 8.4730 ref. =—148 Par. and ref. =47 47 By Tab. XIX 47' 48' 5. Find the altitude aad aziintitli of Polaris, 1865, Sept. 25 eh 15™ P.M., in lat. 49° 16' N., long. 85° 16' W., (Art 125). h m ■ n. L. m. t 8 16 Long. 6' 41" 4' s„ 12 n 16 p= 1 24 27 log 3.T048 1.C08 9.99987 Ked. for long. + 66 < = 69 8 1. cos 9.5517 Red. of L. m. t. + 1 21 ^= 30 6 log 3.2565 I. sec 0.00002 L. aid. t 20 34 33 i+0 = 49 46 B 1. sec 0.1898 l.siu 9.88277 R. A. of * 1 11 6 A = 49 45 log^. 3.7048 1. sin A 9.88266 H. ang. of * — 4 36 32 1. sin t 9.9705 Z=:TH. 2 2 10B .log 3.8651 6. Required the greatest elongation of Polaris, 1865, Sept. 25, in lat 49° 16' N. p= 1 24 27 Z = 49 16 Z= 2 9 26 log 3.7048 1. sec 0.1854 log 3.8902 1. sin 8.3903 1. sec 0.1854 L sin 8.5767 1. At sea, 1865, May 20, IS"" 23™ 16= mean time Green- wich, in lat. 40° 15' S., long. 107° 15' W., the observed alti- tude of the sun's lower limb 10° 15' 20", index correction of sextant + 3' 20', height of eye 18 feet, bearing of snn by compass N. 41° 45' E. ; required the sun's azimuth and the magnetic declination or variation'. (100.) G. m. t May 20 IB' 23"' 16 ' = 16\888 O's dee. 20° 2' 28".0 N.+30".79 Q 10° 16' 20' ( + 14 59 ■ In. cor. + 3' 20" Dip. — 4 11 S. diam. + 15 50 -t- 20 7 10 63.7 22 N. '307.9 153.9 9.2 2.7 h = 10 30 19 1. see 0.00734 z = 40 16 1. sec 0.11734 p = 110 10 22 2» = 160 56 41 « = 80 27 60 1. cos 9.21925 i—p= - -29 42 32 1. cos 9.93880 19.28273 JZ = 64 1 48. 1. cos 9.64137 true^=S. 128 8 36 E. N. 51° 66' 24" E. Mag. N. 41 4B E. Variation 10 11 K. HOUR-ANGLE AND LOCAL TIME. 127 8. 1865, Sept. 20, in lat. 30° 25' N., long. 50° 16' W., the compass bearing of the sun, when one of its diameters above the horizon, was S. 19° 30' W. ; required its true bearing and the variation. (102.) L. ap. t. Sept. 20 6^ 0" (Tab. IX.) 0'sdee. 0° 69' 32" N. — 68'.4 Long. + 3 21 G. ap. t. Sept. 20 9 21 = gi.se — 9 50 6 N. ( 626. 6 \ 17.6 ( -2. 9 1. sin 8.1627 L = 30 26 N. 1. sec 0.0643 (Tab. yn.) .4 = W. 0° 68' N. true Z=S. 89 2 W. L cos 8.2270 mag. N. 100 30 W. Tar. 11 28 w. 9. On the same day at the same place, when the sun's centre was in the visible horizon, its compass bearing was S. 79° 30' W. ; height of eye 20 feet. p = 89° 10 X = 30 26 1. sec 0.0643 A = ^- 37 1. sec 2s= 118 68 » = 69 29 I. cos 9.7067 t—p=z— 29 41 1. cos 9.9389 19.7089 iZ= 44 20 L cos 9.8546 true2=N. 88 40 W. mag. N. 100 30 W. var. 11 50 W. HOUE-ANGLE AND LOCAL TIMB. 130. Peoblbm 42. To find the hour-angle of a heavenly body in the horizon. Solution. In the diagram of the last problem, M P Z = «, the hour-angle ; and in the triangle P M KT are given ^S"^:Jo ^ ftofind MPN = 180°-fc PM = 90°— 6', and consequently that 2 t, or the time that the body is above the true horizon, > 12** ; and when the latitude and dedinatioQ are of diSerent names, t < e"" 9,nd 2 « < 12'', 2 < is an interval of sidereal time for a fixed star, of ap- parent time for the sun. In the case of the sun, t would be the apparent time of sunset, were the refraction and dip nothing, and (24''— t) would be the apparent time of sunrise. Tab. IX. (Bowd.) contains t for different values of JL and d, 132. Problem 43. To find the hour-angle of a heavenly body at a given place, and thence the local time, when the altitude of the body and the Greenwich time are known. Solution. Find the declination of the body for the Greenwich time, and reduce the observed al- titude to the true altitude. Then in the triangle P Z M (Fig. 29) are given P Z = 90°— Z, PM=p, Z M = 90"-*, to find ZPM = «. For the triangle ABC (Fig. 29), we have sin i A = /( sin (» — V) sin (s— sin b sin o in which, putting A = « a = 90°— h, b=p, c = 90°— Z, lig. 29. (104) HOUB-ANGLE AJSTD LOCAL TIME. 129 we have s—b = 90° —^ {Z+p+A), s~o = i {L+p—h), and sin * « = . // g°s i (J+P+^) gin j {L+p—KA , r \ coB'i/ sin /> / * or, ff we put s' = i(X+jo+A), • 1 ^ //cos «' sin (■«' — ^)\ Sin * < =/l/ T^. —U ^ r \ cosZsmjp y which is Bowditch's rule, p. 209. Piom Tab. XXVII. (Bowd.) we may take t directly from column P. M., corresponding to the log sin \ t. ^ is — when the body is east of the meridian. When the object is the sun west of the meridian, t is the apparent solar time; when the sun east of the meridian, (24'' — «) is numerically the apparent time. When the object is the moon, a planet, or a star, we have (Prob. 37), denoting its R. A. by a, the sidereal time ^ a-Vt, and ^^ mean time =a— jiS'o+'j in which &„ is the "right ascension of the miean snn." (Art. 93.) Or the sidereal time may be converted into mean time by one of the other methods of Problem 32. 133. By the formula , , //sin « sin (» — a)\ cos J A = V ^^-^^ ^I> ' T \ sm S sm c y we may obtain for the triangle P Z'M (« being the zenith distance), .^ //sini(coi+^+z)sin ^(co X+^ — 8)\ cos * t ^ A/ \ ^i^-. , ' Y \ cosLsmp y or putting s = i (co L+p+z), , . //Mnssin/* — a)\ cos lt = A/[ jA ^ , ■* T \ cos Z sin p p which is the rule in Bowditch's 2d Method, p. 210, (105) 180 NAVIGATION. (105) is preferable to (104) when t considerably exceeds el", which may be the case in high latitudes. If i = 90°, the horizon and equator coincide, and p-\-h — QQ° and p = e; so that both (104) and (105) be^ come indeterminate. In very high latitudes, then, -these equations approach the indeterminate form, and it is imprac- ticable to find with precision the local time from an observed altitude. So also if ' 20™ 12» ; 80 that for J i = 10' S, J « =— 16«. 2. 1865, Jan. 1, 21'> at the Navy-Yard, Havana, in lat. 23° 8' 39' N., long. S^ 29" 27" W., tJie following altitudes of the sun were observed with an artificial horizon; re- quired the local mean time. T.hy Chro. B ® 8^57.5 58 lb Chro. &st of Q. m. t. 42 37.T S129.S K Bar. 8a4S ' 85 2.3 30 Tber. 75* 35 8a8 86 4.7 40 60 m'oflloTn J off 'be are +82'18'.8 ® '*'*™-1 on theare-Sa 36.0 86 ST.O 60 Index cor. — SL8 T. byChro. SS51T.85 58 35 ©'»<*«& Chro. cor. —48 37.7 &• = 2917 80'* In. cor.— *■— 22 •63'43.2'+14'.04 e.lii.t. Jw.a 3 5239.6 = 8b.S78 +1442 adiam.+ie'lS ret —1 40 +64.4. [42.12 11.23 »= »8812 par. +8 — 2 ! S2 48 .98 ■11 Eq.oft. m B i= 23 829 Lsec 0.0361304 +4 27.81+1.165 p= 112 5? 48 Lcoaec 0.4102711 8.49 S<= 1653829 +4.51 .93 A «= 824645 Lcos 9.0998144 +482.82 f «— A= 531488 I sin 9.903T2T6 19.4ffl7436 i«=-8S 816.7 Lsln 9.7248718 L ap. V Jan- 1 19 S 58.7 t=-M 633.4 =+396°63'26-.6 £q. or t. +4 38.3 L.m.t. Jan.1 19 46 26.0 We have also by subtracting the chro. time from the local ■ mean time, Chro. cor. (L. m. t.) -'6* 12™ 4».8 Long. +5 29 27 .0 Chro. cor. (G. m. t.) —0 42 37 .8 As tiie Chro. is fast, the correction is subtractive. 134 NAVIGATION-. By comparing the first and last altitudes and the corre spending times, we find that for 2 id A = +50', J « = + 2" 39".5 ; or.for 2Ah= + l',d «= +3M9; that is, an error of 1' in the double, altitude will produce an error of 3' in the resulting time. 3. At sea, 1865, Sept. 7, 8'^ 4" 16», G. mean time, in lat. 46° 16' N., long. 153° O' E., the observed altitude of the moon's upper limb, W. of the meridian, was 21° 19'; index cor. of octant, —3'; height of eye 20 feet; required the local mean time. G, m. t Sept. 7 8 4 16 In. cor. 21 19 — 3 J'sdee. + 6° 69' 34' +11'.2 dip. S. diam. h' = — 4 — 17 20 66 +48 44 .8 +7 22 2.8 S.diam. 16'36'+4' par.& ref. h = + 64 2149 H. par. 60 49 L = 46 16 L sec 0.16033 P = 83 L cosec 0.00326 h m . • 2» = J'sRA. 1 1 12.2 + 2.40 » = + 10.2 9.6 l—p = 1 1 22 .6 161 6 76 32J 53 43^ 1. cos 9.39738 L sin 9.90644 19.46740 J's H. A. 4 22 21 . L. sid. t. 6 23 43 ■ • • • . L an i 9.73370 —S, —11 6 18 — Red.forG.m.t. —1 20 L. m. t. Sept. 7 18 16 6 Long. — 10 11 49 = 1B2° 67' E. 4. 1865, Sept. 30, in lat 30° 27' N., the Chro. time of the setting of the sun's centre was ll"" 16"" 6«; the Chro. cor., + 15" 25"; height of eye 16 feet; required the local time. HOUR-ANGLE AND LOCAL TIME, 135 T. by Chro. 11 16 6* Chro. cor. + 16 25 O's. dee. — 2 64 31.9 Eq'n of i. —58.30 —10 4.40- -0.801 G. m. t. Sept 30 11 31 31 1 — 11 11.9 H [ 641.3 29.1 -9.23- : or .40 11.625 — 3 544 1 h = — 0S1 [ 1-8 —10 13.6 1 [ 2 Z= 30 27 1. sec 0.06446 p= 93 6 L cosec 0.00064 2 s = 122 56 s= 6128 1. cos 9.67913 ( — A= 62 5 1. sin 9.94627 L. ap. t. Sept. SO 5 "e 34 19.69050 1. sin } 9.84525 Eq. of t — 10 14 • L. m. t Sept 30 5 45 20 Long. + 6'' 30" 46" = 82° 41'i W. 139. Problem 44. To find ike hour-angle of a heavenly body when nearest to, or on, the prime vertical of a given place. Solution. If X, and with the same name, as for the body whose diarnal path is n n' (Fig. 30), P Z n will be • greatest, or nearest to 90°, when Z nia tangent to n n', aiid consequently Z»jt) = 90°. We then have If 30' N. ~ « = 14 42 N. £= 34 12 N. Examples 6 and 7 are identical, the Greenwich apparent time being May 17 12'' for both. They illnstrate the neces- sity as well as propriety of the rule for navigators near the meridian of 180°, to add 1^ to the date, when they pass from west longitude to east ; ta subtract 1* from the date, when they pass from east longitude to west. For instance. May 18 5' in long. 180° 15' E., is identical with May 17 5" in long. 179° 45' W. 141. The common mode at sea of measuring a meridian altitude of the sun, is to commence Observing the altitude 20 or 30 minutes before noon, repeating the operation until 142 NAVIGATION. the highest altitude is attained ; soon after which ifte sun, as seen through the sight-tube of the instrument, begins to dip^ or descend below the line of the horizon. It is preferable, however, to find, from A.M. observations . for time and by allowing for the run of the ship in the inter- val, the time of apparent noon by a watch, and observing the altitude at that time within l" or 2°". A meridian altitude of the moon, or a star, can be much more conveniently observed by finding beforehand the watch time of its culmination, and measuring the altitude at or very near that time. When the sea is heavy, it is recommended to observe three or four altitudes in quick succession, within 2°" of the time of culmination. 142. If the body is changing its declination, or the ob- server his latitude, the maximum altitude is not at the in- stant of meridian passage ; but after^ if the body and zenith are approaching ; before, if they are separating. Let t be the hour-angie of this culminating point, in minutes; A d, the combined change* of declination and latitude in 1"", if it is expressed in seconds; or in l*", if it is expressed in m^inutes; A„h, the change of altitude in 1" from the meridian passage due solely to the diurnal rotation, (from Tab. XXXII.) ; A A, the reduction of the maximum altitude ; both expressed in seconds. Now in the time t tAd will be the excess of altitude produced by the change of declination and latitude ; f A„h (as will be shown in Art. 150), the diminution of alti- tude due to diurnal rotation ; * Thdr sum, if the; both tend to elevate or both to depress ; otherwise their difference. LATITUDE. 143 and we shall have But at a point whose hour-angle is 2 1, the altitude will be the same as the meridian altitude, or = 2 « J —d, Z M = 90°— h, to find PZ=:90°-Z. LATITUDE. 145 By Sph. Trig. (146), if in the triangle ABC (Fig. 33) are ^ven a, b, and A, we find e by the formulas tan = tan b cos A, " cos 0'= cos cos a CDS b which, applied to the triangle PZM, give tan = cot d cos t, ' COS

'= 90°-i = sin d (112) These may be changed into a more convenient form for practice, if we put (j> = 90°— (pi' ; then tan '^= tan d sec t, sin 0' sin % cos © = — , '^ sin a (113) Here, observing that + and — may be rendered by N. and S. respectively, we mark <^' N. or S. like the declina- tion, and " and ' when of the same name, their difference when of differervt names, is the latitude, of the same name as the greater. There are two values of Ii corresponding to the same altitude and hour-angle, but which, unless <^' is very smaU, will differ largely from each other ; so that we may take that value which agrees best with the supposed latitude (at eea the latitude by account). When t > &\ ^" > 90°, as in (97). 146. In Fig. 32, if Mw4 be di'awn perpendicular to the meridian, we shall have <^ = P w, the polar distance of m, ifi'= 90°— P m, the declination " '= Z m, the zenith distance " 14:6 NAVIGATION. "When 0' is very small, (that is, when M m nearly coin cides with M Z), 0' cannot be found with precision from its cosine. If not greater than 12°, it can be found only to the nearest minute with 5-pIace tables ; if only 2°, it can be found only within 3'. The more nearly, then, that M m co- incides with Z m, or, in other words, the nearer the body is to the prime vertical, the less accurate is the determination of the latitude. If the body is on the prime vertical, cos ^'='1, and 0' cannot be found within 30'. 147. To find the efiect of an error in the altitude, let ^ A = a small change of altitude ; A 0'= the corresponding change of 0' ; it will also be numer- ically the change of latitude, as 0' does not depend on h; Then from the 2d of (113) cos(0'-l-J0')=£|sin(A + JA); or, since A h and generally A '= i, sin A 0'= J 0' sin l", cos A'— A 0'. sin 0' sin 1'= ^^^ (sin h + Ah.aosh sin 1*). Subtracting this from the second of (113), and reducing, we have , ,, sin «' cos A . , A 0'= ■ J ■ ^, A h: sin d sm f ' or, since cos 0' sin 0' sin h sin d ' A 0'= — Ah. cot 0' cot A. (114) But in the triangle M Z w, Zm = — cosMZP = 7 — \rn > tan M Z' that is, Z being the azimuth, LATITXJDB. . 147 — COS Z= — — =-, or sec -Z = — cot *' cot A, cot A ' ^ ^ and therefore J <^'= J h. sec Z. (115) If the body is on the meridian, Z= or 180°, and numer- ically A(j)'^Ah. The nearer Z is to 90°, the greater is ^ 0'. If Z= 90°, or the body is on the prime vertical, sec Z= oo, and 4 ^' is incalculable. If Zis near 90°, (115) is inaccurate; since ^ ^' becomes too large for the assumptions cos A '=l, sin A '=A '. sin 1' ; so, also, in (114) if ^' is very small, A (p' may become large. A star which transits the meridian near the zenith, changes its azimuth very rapidly. Unless observed on the meridian, it cannot be depended on for latitude. 148. To find the effect of an error in the time, and con- sequently in the hour-angle, we may take the formula in Prob. 39. sin A = sin i sin X, sin (L—d) = sin 2, is negative ; that is, s,, will have a different name or sign from Z (Art. 140). Properly h, ho, and /)„ h would also become negative to correspond. Still, however, we shall have nunlerically ho = h+A h. We may therefore disregard the sign of IJ—d in (121) and consider h and h„ as always positive. K the star is observed at its lower culmination, then t will be the hour-atigle from the lower branch of the meri- dian, and for d we may use 180°— c? (Art. 140). A^ h and A h are then numerically subtractive. LATITUDE. 151 ExAMPLBS. (Prob. 46.) 1. At sea, 1865, July 17 !•» P. M., in lat. 36° 38' S., long. 105° 18' E., by account; time by Chro., 5^ 47" 14=; O, 30° 15' ; N. Wy ; index cor. + 2' 30' ; height of eye, 17 feet ; Chro. cor. (G. m. t.) + 14™ 3» ; required the latitude. By (H3) T. by Ch^o.^-12^ 17 47 U ©'« dec. Eq'n of t. Chro. cor. + 14 3 +21 20 29 —25.19 —5 43.8 —0.230 6. m. t July 16 18 117 = 18.021 —7 34(262 —4.1 (2F~ —Long. + 7 1 12 +21 12 65 1 202 _6-47.9 ( 1-8 L. m. t July 17 1 2 29 ^ 30' 15' ( In. cor.+ 2'.6 dip. —4.0 Eq. of L — 5 48 +13 J S.-> P. M., in lat. 50° 36' N"., long. 135° 25' W. (by account), time by Chro. 3i> 10™ 15' ; Chro. cor. (G. m. t.)— 18" 56»; Obs'd alt. of Mars, 45° 18'; S. E'y; index cor. — 3'; height of eye, 19 feet; required the latitude. (113) T. by Chro.+12'', 16 10 15 b m • Chro. cor. —18 56 Mars' B. A. 3 66 26.1 + 0.03? G. m. t Jaa. 6 14 51 19 = 14».866 + 0.5 .37 .14 3 + 0'.50 8, 19 22.1 3 55 25.6 Red. for G. m. t. + 2 26.5 Hars' dec. + 23" C 80' G. Sid. t. 9 54 7.6 + 1 —Long. —9 1 40 + 23 37 L. sid. t 62 27.6 h' = 46" 18' (i„. - 8 } dip. cor. — 3' Mars' R. A. 3 65 25.6 , & ref. — 5 « = — 3 2 58 h = 46 10 1. sin 9.86074 or 45° 44' 30" 1. sec 0.15621 d= 23 37 N. L tan 9.62807 1. cosec 0.40793 f = 31 19.3 N. 1. tan 9.78428 1. Bin 9.71688 f = 19 26.5 N. L cos 9.974SS X= 60 45 N. LATITUDE. 153 IfJA = + 6'andJA =+ 16', J t = —15' ; and by (115).and(118) L cot (—A) 9.9975 n 1. cot 0' 0.4527 Z=^. 110° 46' E. 1. sec Z 0.4502 n J h=+ 5' log 0.699 Ji= — 14'.1 log 1.149 n 1. COS i 9.801 —At= + 15', log 1.176 1. tan 2 0.421 n Ji=— 25'.0, log l.S98n 3. 1865, Feb. 17, near noon, at the light-honse, W. end of St. George's Island, Apalachicola Bay, long. 65° 5' 15' W. ; 5 observations with sextant No. 1, art. hor'n No. 3, A end towaid observer : . T. by Chro. O"- 16" 21'.6 ; 2© 96° 14' 44', (S.) ; in. cor. + 2' 30'; Chro. cor. (L. m. t.) —18" 30'.4 ; Bar. 30.48, Ther. 43°. By (11^) T. by chro. C 16"' 21».6 Chro. cor. —18 SO .4 L. m. t. Feb. 16 23 57 51 .2 Long. +6 40 21 6.m.t.Feb.l7 5 38 12 =5''.637 Eq. oft. —14 14.1 L. ap. t ( 23 43 37 .1 0's dee. JEq. oft. —11° 51' 52'.8 + 52*.71 — 14' 16'.22+ 0'.204 +4 57.1 f 263 .55 (1.020 122 6 —11 46 55 .7-^ 263 .55 31 .63 +1 .15 1 .68 .37- -14 14 .07 r i% 16 22 .9 48° 7' 22' f i In. cor. + 1' 15' J S..diain. +16 30 +16 67 ) Ke£ — 54 (.Par. + 6 A=48 24 19 *= 4° 6'43'.5 1. sec 0.0011104 L sin A 9.8738198 ' can be greatly simplified in the ease of the Pole-star, since its polar distance is only 1° 25'. Putting d—90°—p and ^'= 90°— <^', we have tan = tan j> cos t or, =p cos f (within 0'.5)_ . ,, . , cos ^ sm rf> = sm h cosp (125) the 2d value of Z,, or (180°— . Let S represent the sidereal time, and a the right ascen- sion of the star, then t=S—a and ^=p cos (5— a). If we consider the right ascension and polar distance of the star to be constant,

. may be found approximately in the traverse table (Tab. II.) in the Xat. col., by entering the table with ^ as a course, and jo as a distance. 161. Formulas (126) may be de- rived from Fig. 34, by regarding P M w as a plane triangle, and Z OT = Z M. The first produces no error greater than O'.S. The error of the second is evidently greater the greater the altitude, or the latitude. This error, however, will not be more than 0'.5 in lati- tudes less than 20°, nor more than 2' in latitudes less than 60°. 162. We may use (125) with more exactness, but these formulas may be modified so as to facilitate computation. Put '=h+Ah then, changing the 2d of (125) to a logarithmio form, we have log sin {h+A h) = log sin h + log cos 4> — log cos^, or log sin (A + ^ h) — log sin h = log sec p — log sec 0. But A h being very small, representing by D^, the change of log sin h for 1', we have, with ^ A in seconds, log sin (A + /I A) — log sin A =.J A X D^; whence, by substituting in the preceding, we obtain . , log sec p — log sec ^ log cos ^ — log cos p Fig. 84 I>. A (127) LATITUDE BY CIKOUM-MEBIDIAN ALTITUDES. 163 The difference of the log secants, or log cosines, of jt> and <^ is readily taken from the table by inspection. D^ for log sin k is usually given in tables of '1 decimal places,- and hence ^ A is readily found. We have then If D, is the change of log sin h for 1', then in minutes ^^^ logsecp-logsec^ ^ (129) 163. The British Nautical Almanac contains three tables for the reduction of altitudes of Polaris, from which they may be found to the nearest second. 164. Altitudes of Polaris may often be observed at sea, with some degree of precision, during twilight, when the horizon is well defined, and the latitude found from them within 3' or 4'. Examples. (Prob. 48.) 1. At sea, 1S65, March 31, 1^ 15™ 19», mean time in long. 160° 15' E.; obs'd alt. oi Polaris -^9,° 18'; index cor. +3'j height of eye 17 feet : what is the latitude ? (128) 'k m ■ fa m ■ L. m t March 31 7 16 19 Long. —10 41 8^ 35 29 Red. for long. — 1 46 Ktd of L. m. t. +111*'= 38° 18' / j^ cor.+8 h. sid. L 7 60 14 — 2 ] Dip —4 ♦ 'sK. A. 1 9 14 A = 38 16 ( Ref. —1 <= 6 41 t= 100° 16' L cos 9.250 « ~—p=— 1 24.5 log 1.927» L sec .00018 — #=+ 15.0 log 1.177 l.sec 1. secj9 — L Bee ^ 16 Ah=+ 0.8 = - h= 38 16 L=+ 38 32 n. 164 NAVIGATION. 2. 1865, May 22, Qh* altitudes of Polaris, at light-house on St. George's Island, Apalachicola Bay : lat. 29° 37' N. ; long. 85° 5' 15" W. ; sextant No. 1, index cor. —3' 0" ; Art. Hor. No. 1 ; Bar. 30.04, Ther. 73° ; Chro. cor, (L. m. t.) + 5'' 35"' 33'. hms o I II hms o i ii 8 17 % 56 82 30 4 2 18 66 34 19 26 82 30 6 8 84 20 21 39 82 SO 8 27 84 50 80 35 82 80 10 16 84 60 14 40 85 40 T. by Chro. 8 22 16 2^=66 82 82 T. by Chro. 4 8 21 2V=66 84 44 Chro. oor. . +6 85 88 A'=28 16 16 Chro. cor. +6 85 88 ^=28 17 22 L. m. t. 8 67 49 tin. cor.— 1 80 Ii. m. t 9 43 64 i In. cor.— 1 SO Sa 4 30 Ref. -1 43 Se 4 80 Ret -1 43 Red. for i, +56 A=28 18 8 Red. for ;i +56 A=28 14 9 Red. for m. t. +1 28 Red. for m. t 1 86 L. Bid. t. IS 48 L. sid. t. 18 46 66 Hf'stLA. 1 9 82 *'aS.A. 1 9 82 I 11 61 11 ( 12 87 24 '-117T-47'46" 1. cos 9.99968 » '-■il89'21' 0' 1. cos 9.99419 » -p=- 1 24 44 log 8.70621 n —p^— 1 24 44 log 8.70621 n -#=+ 124 40 h>g a. 70598 — ^=+ 128 87 log 8.70040 ^h= 1. seep 1819 1 JA=+ 1 1. sec j> 1819 «.= 28 18 8 L sec^ 1817 !■ A= 28 14 9 1. sec 1284 L=+ 29 87 48 (^,=89.2) 2 J Z= + 29 87 47 (J,=89.2) 85 1. sec^ and 1. sec ^ are e.$preBsed in units of the 7th placa of decimals. CHAPTER VIII. THE CHRONOMETER.— LONGITUDE. 165. AsTEONOMicAixT the longitude of a place is the diP ference of the local and Greenwich times of the same instant. It is west or east, according as the Greenwich time is greater or less than the local time. (Art. 13.) The mean solar, the apparent, or the sidereal times of the two places may be thus compared. 166. A chronometer is simply a correct time-measurer, but the name is technically applied to instruments adapted to use on board ship. It is here used more generally, as in- cluding clocks which are compensated for changes of tem- pei'ature. - A mean time chronometer is one regulated to mean time ; that is, so as to gain or lose daily but a few seconds on mean time. A sidereal chronometer is one regulated to sidereal time. 167. A chronometer is said to be regulated to the local time of any place, when it is known how much it is too fast, or too slow, of that local time, and how much it gains or loses daily. The first is the error (on local time) ; the second is the daily rate. Both are + if the chronometer is fast. and gaining. It is preferable, however, to use the cxyrrection of the chro- nometer, which is the quantity to be applied to the chrono- meter time to reduce it to the true time, and its daily change. Both are -f- when the chronometer is slow and losing. 166 NAVIGATION. They will be designated by c and A a. A chronometer is said to be regulated to Greenwich time, when its correction on Greenwich time and its daily change are known. If Co is the chro. cor. to reduce to Greenwich time, and c^ the chro. cor. to reduce to the time of a place whose longi tude is A (+ if west). c„=e+A, or c = Co — X', (130) so that the one can readily be converted into the other. 168. If the correction of the chronometer at a given date, and its daily change, are known, the correction at another date can easily be found. For let c be the given correction at the date T, c', the required correction at the date T ', t = T'— T, expressed in days, J c, the daily change ; then c'=c + t.Ac. (131) t is negative if the date for which the correction is re- quired is before that for which it is given. If Ac is large, t must include the parts of a day in the elapsed time. J c may be given for two different dates, and vary in value. It may then be interpolated for the middle date be- tween the two of this problem. Thus, if A'c be a second value determined n days after the first, the daily variation of A c, regarded as uniform, will be A'c-Ae n (132) Representing this by A^ c, we have for the mean daily change of the chronometer correction during the period t, or that at the middle date, Ac + it. A^c, THE CHRONOMETER. 167 and the required chronometer correction, e'=c + t. Ac + if. A3 c. (133) When the chronometer is in daily use, it is convenient to form a table of its correction for each day at a particular hour. For a stationary chronometer, the most convenient hour is 0' of local time ; for a Greenwich chronometer, O"" of Greenwich time. Examples. 1. Chro. 1675, regulated to Greenwich mean time ; 1865, Jan. 15, 0^; correction +1'" 1„ 25'.0 ; daily change —1'So; required the correction, Jan. 26, 6''. Jan. 15, 0^, Chro. cor. +1"' 16" 25'.0 —1K65 X 11.25 = — 1 26 .1 Jan. 26, 6'' Chro. cor. + 1 14 68.9 This chronometer is slow and gaining. 2. To find the chro. cor. to reduce to local time, Jan. 26, 0\ in long. 85° 16' E. C!hro. cor. (Jan. 26 e"- 6. t) +1"' 14" 68>.9 —Long. +6 +5 41 4 Bed. for —12 +3 .8 Chro. cor. (Jan. 26 L. t) +6 56 6 .7 or — S* 3° 53".S 3. To form a table of chronometer correction for each day from Jan. 26, 6" to Feb. 6, 6''. ff. m. t. Chro. cor. Feb. 1 e"" +1'' 14"° IS'.O 2 6 14 5 .4 3 6 IS 57 .1 4 6 13 50 .1 6 6 IS 42 .4 6 6 +1 18 34 .8 169. To find -the raie, or daily change, of a chronometer, it is necessary to find the correction of the chronometer on two difierent days, either from observations, or by compari- a. m. t. Chro. cor. Jan. 26 6'' +1^ 14°' 58».9 27 6 14 51.3 28 6 14 43 .6 29 6 14 36 .0 SO 6 14 28.3 31 6 + 1 14 20.7 168 NAVIGATION. son with a chronometer, whose correction is known. Let Cj and Cj be the two corrections, t the interval expressed in days ; then we have for the daily change, Jc = ^j (134) that is, the daily change is equal to the difference of the two chronometer corrections divided by the number of days and parts in the interval. If attention is paid to the signs, + "will indicate that the chronometer is losing, — that it is Examples. Chro. 1616 b h m ■ Chro. cor. April 16 +0 18 16.2 Chro. 4872 —1 16 2''?.5 Chro. 799 h tn s +0 16.6 " " " 21 8 +0 18 29.6 —1 14 68,6 —00 6.3 Change in 12.3 days, +13.4 +28.9 —21.9 Daily change of cor. +1.09 + 2.35 —2.71 At fixed observatories an interval of one day may suffice. For rating sea-chronometers by observations made with a sextant and artificial horizon, an interval of from 5 to 15 days is desirable. The sea-rate of a chronometer is sometimes different from its rate on shore, or even from its rate while on board ship in port. Some chronometers are affected by magnetic in- fluences, so that their rates are varied by changing the di- rection of the XII. hour mark to different points of the ho- rizon. All are slightly affected by changes of temperature, as perfect compensation is rarely attainable. The excellence of a chronometer depends upon the permanence of its rate. The rate may be large, but if its variations are small, the chronometer is good. 1 70. A watch is often used for noting the time of an ob- servation. It is compared with the chronometer by noting THE CHKONOMETER. 169 the time of each at the same instant. The most favorable instant is when the watch shows no 0*. Let C and W be these noted times ; then A W^= {C — W) is the reduction of the watch time to the chronometer time : for C=W+{0-W). "Comparisons should be made before and after the observa- tion, and the results interpolated to the time of observation. A practised observer may, by looking at the watch and counting the beats of the chronometer, make the comparL son to the nearest 0^.25. It is better to take the mean of several comparisoas than to trust to a single one. A mean "time and a sidereal chronometer may be com- pared within 0^.03 by watching for the coincidence of beats, which occurs at intervals of 3", for chronometers, which ■beat half-seconds. ExATWPT.WS. - Chro. 476 Chro. 4 "e 56.2 Chro. 4072 h m s 3 16 17.5 Chro. 1976 11 48 18.2 Chro. 1976 1 °0 28.6 Watch 16 7 36 30 3 16 4 28 C—W.+S 11 66.2 • —4 20 12.5 —3 27 41.8 —3 27 31.5 The last two are comparisons of the watch with the same chronometer. Suppose the time of an observation as noted by the watch to be S' 37™ 17^ ; for finding the corresponding time by the chronometer we have, The change of C— W in 1''.2, + 10».3 ; 'whence the change in l"" is + 8 .6, and the change in 21™.3 = 0''.35, the interval between the 1st comparison and the observation, +3».0 ; or, by proportion, we have 72°' : 21".3 = + 10^.3 : -|-3».0 Then, Time by Watch = 3" 37" 17» C—W = — 3 27 38.8 Time by chi-o. = 9 38.2 170 NAVIGATION. 171. Problem 49. To find the correction of a chronomA- ter at a place whose latitude and lonffittcde are given. 1st Method. (By single altitudes.) Observe an altitude, or set of altitudes, of the sun or a star, noting the time by the chronometer, or a watch com- pared with it. Find from the altitude (Prob. 43) the local mean, or sidereal, time, as may be required. The " local time " — the " chronometer time," or c = T-G (art. 135), is the correction of the chronometer on, local time. Applying to this the known longitude of the place of observation, gives the correction on Greenwich time. 172. If an artificial horizon is used, as it should be when practicable, it is best to make two sets of observations with the roof in reversed positions. In A. M. observations of the sun with a sextant and artificial horizon, the lower limb of the sun and the upper limb of its image in the horizon are made to lap, and the instant of separation is watched for ; while in P. M. observations the limbs are separated and ap- proaching, and the instant of contact is noted. In observa- tions of the upper limb this is reversed. Even a good ob- server may estimate the contact of two disks differently when they are separating and when they are approaching. Both limbs, then, should be observed. In observing altitudes which change rapidly it is better, when circumstances permit, to set the instrument so as to ead exact divisions at regular intervals, and watch the in- tant of contact. A good observer, with a sextant and ar- tificial horizon, can observe the double altitudes at regular intervals of 10'. 1 73. On a subsequent day repeat this observation, and find again the correction of the chronometer. The differ THE CHEONOMETEE. 171 ence between these two corrections divided by the number of days and parts in the interval is the daily change, as in Art. 169. It is important that both the observations thus compared should be at nearly the same altitude and on the same side of the meridian (when the sun is observed, both in the fore- noon, or both in the afternoon), and in general, that they should be made with the same instruments, and as nearly as practicable under the same circumstances. Thus, an error in the assumed latitude and constant errors of the instruments or the observer will affect the two chronometer corrections nearly alike, but will very slightly affect their difference, and, consequently, the rate determined from it will be nearly exact. The chronometer correction, derived from single al- titudes, may be erroneous a few seconds. But for sea chronometers this is of less importance than an erroneous determination of the rate'. For instance, suppose the deter- mined chronometer correction in error 4", and the daily change in error 1^ ; in 20 days (Art. 168) the computed change of the correction will be in error 20=, and in 30 days will be in error 30=. 1 74. 2d Method. (By double altitudes.) It is better to observe altitudes of the body on both sides of the meridian, and as nearly at the same altitude as prac- ticable, either on the same day Or on two consecutive days. Altitudes of two stars also may be used, one east, the other west of the meiidian. The mean of the two results is better than a determination from either alone ; for constant errors of the latitude, the instrument, or the observer, affect the two results in oppo- site directions ; that is, if one result is too large, the other is too small, and by nearly the same amount. 172 NAVIGATION. Examples. (Prob. 49.) I. Chronometer Correction. Pensacola Kavy-Tard, 30° 20' 30' N., 87° 15' 21' W, 1865, May 30 2lh; Chro. 1876. T. hy Chro. Sextant No. 2. AH. JSor. No. 1. 8141 , 2 99 5bXen 22.S Daily change — 3.8 f 24 26.5 100 30 23.0 G's diam. off arc +32 12.5 ( 24 48.5 100 20 on arc — SO 59.2 ) 24.0 25 12.6 100 10 In. cor. + 36.6 22.3 26 34.8 100 23.4 26 68.2 99 60 28 33.5 2 97 40 a end. Bar. 30.14 23.6 ' 28 57 97 30 Ther. 76° 23.5 29 20.5 97 20 22.6 29 43 97 10 23.0 30 6 97 23.6 30 29.5 96 50 9 26 0.37 2 100 16 9 29 31.68 2 97 16 • Computation. T. by Ohro. 9 25 0.37 0'« dee. Eq^n oft. C 1 1 // m s R Chro. cor. - 42 27 +21 57 40.3 +20.92 +2 36.36 —0.362 167.36 2.896 O. m. t. May 31 8 42 33 +3 2.2.^ 14.64 —3.15 • .263 8.709 +22 42.5 ( 19 +2 33.20 8 ' ■© "50 7 30 —16 16.3- i In. cor. /, „ + 18.3 ref.— 61.6 S. diam. h= 49 51 13.7 +16' 48.4 par. +6.4 L= SO 20 30 L sec 0.0639749 h m ■ P= 67 59 17.5 L coaec 0.0328703 L. ap. t. May 31 2 56 12.04 2»= 148 11 1.2 — Eq. of t. —2 33.20 «= 74 5 30.6 1. C03 9.4379032 L. m. t. May 31 2 53 38.84 s-h= 24 14 1619 L sin 9.6133431 T. by Chro. 9 26 0.37 o 1 a 19.1480915 ©,Chro.oor.(L.m.t.)— 6 31 21.53 i<= 22 1 30.3 1. sin 9.5740458 THE CHfiONOMETEK. 175 T. byChro. 9 29 31.58 Q's dee. Eg'no/t. CJhro. cor. —42 21 4-22 42.5 + 20.92 +2 33.20 — 0.362 G. m. t May 31* 8 4Y 5 in0''.076 +1.6 —.03 8.785 +22 44.1 +2 33.17 L. ap. t May 31 — Eq. of t L. nt t. May 31 T. by Chro. Sj Chro. cor.{L.in.t)- 3 42.96 —2 33.17 48 37 30 + 15 17.8 h= 48 52 47.8 Z= 30 20 30 p= 67 59 15.9 Jin. cor. // // +18.3 ref.— 64.6 S. diam. +15' 48.4 par. +6.6 1. sec 0.0639749 L cosec 0.0328717 2 58 9.78 2 s =147 12 33.7 9 29 31.88 73 36 16.8 —6 31 21.80 s-h = 24 43 29.0 6 31 21.66 it= 22 35 22.1 +.48 « =r 45 10 44.2 (mean) Red for — S^O Chro.eor.(L.in.t.)— 6 31 21.18, May 31 O^ 1. cos L sin L sin 9.4506644 9.6214453 19.1689463 9.6844732 May 31 O"- Chro. cor. (L. m. t.) —6'' 31" 22'.24 by A.M. and P.M. obs. Long. +6 49 1 .4 May 31 6' Chro. cor. (G. m. t) — 42 20 .84 3. Table of Chro. Corrections. Chro. 1876 ; fast of Greenwich mean time and gaining. a.m. t. Chro. cor. Baily Ch. h m s —0 40 20.5 8 —4.14 41 26.8 3.88 41 68.3 3.76 42 20.8 Jiemarix. 1865, May 1 3 17 3 25 6 31 6 0, A.M. Key West Light-House. 0,A.M. " » " " O, A.M. & P.M. Pensacola Navy-Yard. ©,A.M.&P.M. ' " " " Long.* of Key West Light-House, 81° 48' 40' W. Long, of Pensacola Navy-Yard, 87 15 21 W. * The assumed longitudes of places, where the clironometer is rated, should be stated. 176 NATIGATION. 4. Comparisons and Corrections of Chronometers. 1865, May 31, 6'', G. mean time. Chro. 43T6 Chro. 9163 Chro. 789 Chro. 5166 h m a h m ■ fa m a h m a Chro. 6 50 16.3 5 3 29.7 2 15 27.5 11 69 16.8 (1876) 6 30 6 31 6 32 10 6 S3 30 (1876)— Chro. —0 20 16.3 + 1 27 30.3 +4 16 42.5 — 5 25 46.8 Cor. of (1876) —42 20.8 —42 20.S -42 20.8 —42 20.8 Chro. cor. —1 2 37.1 -0 45 9.5 +3 34 21.7 —6 18 7.6 , or +5 41 62.4 175. Zd Method. (By eg'MaZ altitudes.) A heavenly body, which does not change its declination, is at the same altitude east and west of the meridian at the same interval of time from its meridian passage. If, then, such equal altitudes are observed and the times noted by the chronometer, or by a watch and reduced to the chronometer (Art. 1'70), the mean of these times, or the middle time, is the chronometer time of the star's meridian transit. The corresponding sidereal time is the star's right ascen- sion, when the first observation is east of the meridian ; 12'' + the right ascension when the first observation is west of the meridian. This, for a mean time chronometer, may be converted into local mean time (Prob. 32) ; and for a Greenwich chrono- meter into the corresponding Greenwich time. Subtracting the chronometer time, we have the correction of the chronometer. Example. 1865, Jan. l4, at Washington, in longitude 1^1° 2' 48* W., equal altitudes of a Canis Minoris were observed, and the times noted by a chronometer regulated to Greenwich mean time ; from which were obtained : THE CHEONOMETEB. 177 Mean of chro. times ( % east) 2'' 16° ' 35».65 " " " " (-X- west) 7 59 16.38 Chro. time of %'s transit 5 1 66 .01 Ii.sid.t.= *'sR A. 1 32 16.26 . Long. +5 8 11 .2 6. sid. t 12 40 27.46 -S, (Jan. 14)- -19 35 51 11 Sd. int. from Jan. 14 0^ 17 4 36.35 Red. to m. t. inL —2 47 .86 G. mean time Jan. 14 17 1 48.49 Chro. time 17 7 56.01 Chro. cor. —6 7.52 176. If equal altitudes of the sun are observed in the fore- noon and afternoon of the same day, the mean ,of the noted times would be the chronometer time of apparent noon, were it not for the change of the sun's declination between the observations. Peoblem 50. In equal altitudes of the sun, to find the correction of the middle time for the chatige of the sun's declination in the interval between the obsisrvations. Solution. Let h == the sun's true altitude at each observation, t := half the elapsed apparent time between the observa- tions, T„=^ the mean of the chronometer times of the two ob- servations, or the middle chronometer time, ATa^= the correction of this mean to reduce to the chrono- meter time of apparent noon ; X = the latitude of the place, d = the sun's declination at local apparent noon, Ad^= the change of this declination in the time t ; -then, when both observations are on the same day, t +ATa will be numerically the hour-angle at the A. M. ob- servation, t—ATf, the hoar-angle at the P. M. observation. 178 NAVIGATION. d — dd, the declination* at the A. M. observation, d +Ad, the declination* at the P. M. observation. By (116), we have for the two observations, smh=smLsm(d— J d) + cos L cos (d— J d) cos (t + dT„) ) ,^^^^ smh=smLsm(d+Ad) + cosLcos(d+Jd)cos{t—AT,)\ But sin {d±dd) = sin dcoB dd± cos d sm A e''. The computation of the two parts of J 7J is facilitated by tables of log A and log B. Such tables are given in Chau- venet's " Method of finding the error and rate of a chrono- meter," in the American Ephemeris and Nautical Almanac for 1856, and reprinted in a pamphlet with his "New method of correcting Lunar distances." The argument of these tables is 2 t, or the elapsed time. The signs of A and B are given. Apply the two parts oi A T^ according to their signs, to the Middle Chronometer Time ; the result is the Chrorunne- ter 'n.me of Apparent Noon. Apply to this the equation of time {adding, when the equation of time is additive, to mean time ; otherwise sub- tracting) ; the result is the Chronometer Time of Mean Noon at the place. ■» Applying to this the longitude (in time), subtracting if west, adding if east, gives the Chronometer lime of Mean Noon at Greenwich. 180 NAVIGATION. *12''— Chro. T. at local Mean JVbon, will be the Chro. corree- Hon if the chronometer is regulated to local time. *12''— Chro. T. at Greenwich Mean Woon, will be the Chro. correction, if the chroaometer is regulated to Greenwich time. 111. If a set of altitudes is observed in the afternoon oi one day, and a set of equal altitudes in the forenoon of the next day, the middle time would correspond nearly to the instant of apparent midnight ; and half the elapsed time i, would be nearly the hour-angle from the lower branch of the meridian, or the supplement of the proper hour-angle. In this case l80°—{t + ATo) will be the hour-angle at the P.M. observation. l80°-{t-AT„) " " " " " " A.M. " d—^d, the declination at the P.M. " d+Ad, " " " " A.M. " and we have for the two observations, as in (135) sin A=sin L sin (d — A d) — eos L cos (d — A d) cos {t+ A ?i) ( \{iaq\ sin A=sin L sin (d+A d)— cos £ cos {d-\-A d) cos (t — A To) > Treating these in the same way as (135) we shall have = A d. sm L cos d-'r A d. cos L mi d cos t — \b A Tg cos L cos d sin t; whence * I'his is better noted as O*. f These may be written — sin k=— sin L sin (d — A d)+aoi L cos (d — A d) cos (<+ A T^ — sin h = — sin i sin {d+ A d)+eQS L coa (d + Ad) cos {t — A T^- They differ from (13fi) in the signs of h and L and in reckoning the hour- angles from the lower, instead of the upper, branch of the meridian. This would be the case, if we suppose the observations to be referred to the lati- ^de and meridian of the antipode. The only effect in (136) is to change the sign of tan L, or of the first term in the equation of equal altitudes. THE CHKONOMETER. 181 . _, Ai. tan L dd. tan d 15 siq t 15 tan t or, patting as before ^ d = \d. t t B = ~ 15 sin i ' ~ 15 tan « ' AT„=—A.^^d.t2MZ+B.^^d.ta.nd, (141) which diSevs from (139) only in the sign of A. This is the reduction of the middle time to the Chro. Time of appar- ent midnigtU : applying the eqviation of time reduces it to the Chro. Time of mean midnighi. 1 78. d, A' d, and the equation of time are to be taken from the Almanac for the instaat of apparent noon, or of apparent midnight, according as the observations are made on the same day, or on consecutive days. 2 t is propei-ly the dapsed. apparent time. The elapsed time by chronometer requires, then, not only a correction for the rate, which is Q t gjj-J c, (+ when the chronometer loses) ; but also a reduction to an apparent time interval, which, for a mean time chronometer, is the change* of the equation of time in the time, 2 *, additive when the equation of time is additive to m^an time and increasing, or subtractive from mean time and decreasing. For a sidereal chronometer, it is the change in the sun's right^scension in the time 2 t, and subtractive. 1 79. Equal altitudes of the moon or a planet may be ob- served ; but in the case of the moon admit of less precision than of the sun, and moreover require correction for the inequality produced by change of parallax. If 2 J a is the increase of right ascension in the interval, * The ma^mam daily change is 30^. The elapsed time by Chronometer iu Udually regarded as sufficiently accurate. 182 NAVIGATION. the body -will arrive at its second position later than would a fixed star, supposed coincident with it at the first posi- tion ; and the elapsed sidereal time will be greater than the dcable hour-angle of the body by the quantity 1 A a. Jf 2 « = the elapsed sidereal time, then in (137) we must take 2 « = 2 s— 2 A a,or t = s—A a. (142) K <„ == half the elapsed mean time (expressed in hours when used as a coefficient), and ^h <* = the increase of right ascension in 1^ of mean time, by (87) s = C + 9^8565 C and < = «„+<» (9^8565- J^ a), (143) by which t and 2 t may be found from 2 <„ the elapsed mean time. In this expression the last two terms are in seconds. Se- ducing to hours we have , t ii , 9".8565 —A AM. P.M. a / h m s .h m 8 8 20 60 OA. md. 8 39 26.T 8 55 69.7 43.2 '« diam. +82' 25".D +82' 26'.T 10 89 58.8 65 27.0 42.9 -82 41 .1 —82 48.8 20 40 81.5 54 66.3 43.4 In. cor. —8 .8 —8.3 80 41 4.0 64 22.0 48.0 40 41 8S.7 63 60.3 48.0 Bar. 80.22 80.18 EO 42 9.0 63 16.6 42.8 Ther. 77* 80- 2060 42S8.5 62 26.7 42.6 "" 10 43 31.3 61 68.5 42.4 Kot — 1'84'.9 —1' 34'.4 20 44 3.8 61 21.0 42.4 Diff. of alt. Ah = +0.5 80 44 87.6 50 48.0 42.7 '40 4S 10.0 60 156 42;8 For 2 ZJ A =10*, id« = 32..8 50 40 48.7 49 42.0 42.? a4»= r. At= 0066 60 25 a 42 84.21 8 52 51.46 THE CHRONOMETER. 185 Elapsed Chro. t. 5 10 17.25 Long. +5k 2T» 14s.5 Bq. of t+T" S6a.96 tO».993 Mid. Chro. t 6 17 42.SS Sli.456 +5. 45 j 4.990 RoUror Ja, 03.065 X as = +. OS 0d.22T +8 2. 41 1 *56 IstpartofEq. —2.89 © 's dec^gl" W 48' J^r8JA A'=S8S8 7.6 2JA =10', A «=8S'.S = 1', /l*=2.sr G. ap. t., June 19, Vl^ 56" 218.2 = 19, 1'7»'.939 = 19*.r4r ■I-2S26 2S +2.01 +29 — .7T +1SS6 57 +1.24 —1.03 +0 59.99 + 0.548 &4S 3.801 .489 U (.73 1 S 5 +9.74 +1 9.TS 186 NAVIGATION. Middle Gtiro. t* 18 39 S.T5 Elapsed 1. by Chro.* 15 23 46 Red. for zjA,—6'.62x2i.83=— 18.09 .... +26 1st part of Eq. +.39 Oh. of Bq. of t — 8 2d " " " — .18 Elapsed ap. t 15 24 4 Chro. t. of ap. 12h 18 38 B2.93 L= + 29' T'.l 1. tan 9.7459 d= +28'2T'l. tan 9.6378 — Eq'noft. —1 9.T3/Ihtf=+1'.24 log 0.0984 .... log 0.0934 Chro. t of mean 12h 18 8T 43.19 A log 9.7550 B log 9.3891 n —Long. —5 56 21.2 +0'.39 log 9.5943 —QMS log 9.1198 » Chro. t of G. mean 121i 12 41 21.99 Chro. oor. (G. m. t.) —41 21.99 June 19 18h 184. ith method of finding the correction of a chrono- meter. (By transits.) On shore, the most accurate method of finding the correc- tion of a chronometer is by noting the times of transit of the sun or a star across the threads of a well-adjusted transit in- strument. The mean of these times is taken and corrected for the errors of the instrument, or reduced to the meridian. In the case of the sun, the transits of both limbs may be ob- served ; or only one, and the " sidereal time of the semi- diameter passing the meridian," found on page I. of each month in the almanac, added for the Umb, which transits first ; subtracted for the second limb. At the instant of a star's transit of the meridian, the right ascension of the star is the sidereal time. The instant of transit of the sun's centre is apparent noon. From either of these, the local sidereal or mean time, as may be required, can be found ; and thence the chronometer correction by subtracting the chronometer time of transit. The moon should not be used for finding the time, when precision is required. Stars are preferred to the sun, either when transits are observed, or equal altitudes with the arti- ficial horizon ; chiefly because many stars may be observed * To obtain these, 24'' was added to the P. M. chro. time. Twice the reduction of the middle time for the diif. of alts, is to be added to the elapsed time when the P. M. obserration is last ; subtracted when the P. M. observation is first. This may be neglected unless the diff. of altitudes is quite large. LONGITUDE. 187 during the same night, and the instrument is not exposed to the rays of the sun. 185. By repeating the transits on a subsequent day, the chronometer correction can be again found, and from the two corrections, the rate as in Art. 1 69. If the transit instru- ment is not well adjusted, or the instrumental corrections are imperfectly known, the rate of the chronometer can still be quite well determined from transits of the same star, or the same set of stars, on different days, provided the position of the instrument, or its adjustments, have not been disturbed in the interval. 186. A rough substitute for a transit instrument is a ver- tical comer of a building, and a position for the eye in its meridian. The instant of the appearance or disappearance of a star, or a limb of the sun, may be noted by a chrono- meter, and the chronometer correction obtained as with a transit instrument ; but with much less accuracy, since the mode of observing is rough, and the position for the eye can- not be adjusted to the meridian with much precision : stiU the rate may be found with tolerable accuracy from the transits of the same body on different days. LONGITUDE. 187. To find the longitude of a place by astronomical ob- servations, it is generally necessary to determine independ- ently the local and Greenwich times of the same instant. The difference of these times is the longitude, which is west when the Greenwich time is the greater, and east when the Greenwich is the less (Art. 165). This is expressed by (72) in which 7J is the Greenwich time, and 7', the corresponding local time of the same Mnd. These times may be apparerU, mean, or sidereal. 388 NAVIGATION, The apparent time is the hour-angle of the true sun ; the mean time, that of the mean sun ; the sidereal time, that of the vernal equinox. In the same way we may use the local and Greenwich hour-angles of any other body or point of the heavens, regarded as + toward the west. This is evident from Fig. 35 ; for if P M is the meridian of Greenwich, P M', the local meridian, P S, the declination circle of a heavenly body ; M P M' will be the longitude of the place, MPS, the hour-angle of the body at Greenwich, M' P S, the local hour-angle ; and we shall have, as in Art. 74, M P M'= M P S - M' P S. The several methods of finding the longitude diflfe*- in th© modes of finding and comparing the two times, or the two hour-angles. 188. Pkoblem 51. To find the longitude of a place by a portable chronometer regulated to Greenwich time. /Solution. The correction and rate of the chronometer are supposed to have been found by suitable observations at a place whose longitude is known. Let the chronometer be transported to the place whose longitude is required ; and let an observation suitable for finding the hour-angle of a heavenly body, or the local time, be made and the time noted by the chronometer, or by a watch compared with it. There are then two parts of the process to be pursued : 1st, from the noted tune to find the Greenwich time (mean, apparent, or sidereal), or the hour-angle of the body, as may be deemed most convenient. 2d, from the observations, to find the corresponding local time, or hour-angle. Subtract- LONGITUDE. 189 ing the local time, or hour-angle, from the Greenwich time, or hour-angle, will give the longitude. 189. 1st. To find the Greenwich time, or hour-angle, of the body observed, apply to the noted time the reduction of the watch time to chronometer time, G — TF^ (if a watch has been used), and the chronometer correction, c', reduced to the date of observation (Art. 168). The result is, the Greenwich time ; and will be mean or sidereal, according as the chronometer is regulated to mean or sidereal time.* If it is sidereal time, it will be necessary to reduce it to mean time (Prob. 32), except when a fixed star has been ob- served, so as to take from the almanac the quantities which will be required. If, now, the Greenwich hour-angle of the body observed is desired : In the case of the sun, reduce the Greenwich mean time to apparent time, by applying the equation of time. If some other body has been observed, reduce the Green- wich mean time to sidereal time by adding the right ascen- sion of the mean sun ; and thence find the hour-angle of the body, by subtracting its right ascension. Or, if a sidereal chronometer has been used, from the Greenwich sidereal time subtract the right ascension of the body. ' Attention to the signs will giv« the hour-angle thus ob- tained, -f- if toward the west, — if toward the east. 190. The Greenwich time or hour-angle is affected by the error of the chronometer correction, which consists, 1st, of the error in its original determination, which includes any error of the assumed longitude of the place of rating ; 2d, of the error arising from an erroneous rate. This last error is cumulative, increasing vnth the number of days from the * For observations of stars, a sidereal chronometer is most convenient. 190 NAVIGATION. date, when the correction of the chronometer -was found from observations. 191. The chronometer correction for the date of observa- tion can be derived from subsequent as v^ell as from prior determinations of it and its daily change. In finding the longitude of a place on shore, or of a shoal, both values should be obtained, when practicable, and copibined by giv- ing weights to each inversely proportional to its interval of time from the original determination. Thus, if c' and c'' are two such chronometer corrections, the first' brought forward t' days, the second carried back t" days, we may take as the mean value* f g'+ t! c' t'+f ' or, in a form more convenient for computation, *'+ H+t" • For example, suppose that on Jan. 17, the chronometer correction brought forward from Jan. 1, is —IS" 56».5, and reduced back from Jan. 25, is —19" 3^.4 ; the value by the above formula will be —IS™ 568.5 + \, =—19'° l«.l. 24 Two longitudes may be combined in a similar way. 192. Reports of longitudes by chronometer are regarded as of but little value, unless, the number of chronometers, the assumed longitude of the place, where the chronometer * This assumes that > 16" 10«; obs'd alt. _L, 20° 16' 0", W. of meridian; Index, cor. + 2' 20"; height of eye 15 feet; required the longitude. h T. by Ohro. 12 + 10 3*6 25 22 40 4*1.9 +2.370 —4 39 5 + ll".62 Chro. cor. —1 16 10 G. m. t. Sept 4 21 20 16 +48.0 47.4 +3 65 ( 232 22 41 29.9 ( -6 _4 36 10 1 3 s. + 10 54 28.3 Bed. for G. m. t. + 3 30.3 A 20 16 1 In. cor. + 2 20 H. par. 60 39 G. Sid. t. S,'bB.A. 8 18 13.6 22 41 29.9 + 15 14 i Dip -3 49 A'= 20 31 4(S.diam. + 16 33 +10' Ji's G. h. ang. + 9 36 44 + 5419 Par. and ret h= 20 25 23 ' L= 20 16 1. sec 0.02776 p= 85 24 50 1. cosec 0.00139 2»=126 6 13 «= 63 3 7 1. cos 9.66627 s- -k= 42 37 44 L sin 9.83074 19.51616 D'sL. h. ang. +4 39 37 1. sini 9.76808 Long. +•4 67 7 or 74° 17' W. Note. — ^The examples under Problem 30 can be adapted to this by regard- ing the chronometer correction given, instead of the longitude. 200. 3d Method, (Littrow's. By double altitudes of the same body.) LONGITUDE. 197 When two altitudes of a body have been observed, and the times noted by the chronometer or watch, the hour- angles and local times can be found from each separately ; and thence the longitude for each. But we may also com- bine them and find the hour-angle for the middle instant be- tween- them. Pboblem: 52. From two altitudes of a heavenly iody, sup- posing the declinoition to he the same for both, to find the mean of the two hour-angles, the latitude of the place and the Oreenwieh time being given. Solution. Take the mean of the two noted times and re- duce it to Greenwich mean time ; and find for it the declinar tion of the body. Reduce the observed altitudes to true altitudes. Let h and h' be the two altitudes, 7* and T', the corresponding hour-angles ; then we have, by (116), sin h = sin Xi sin d + cos L cos d cos T, sin A'= sin i sin 23' O's true alt., 81° 1' O', « « " 2 53 11 " " " 88 40 30; Chronometer coiTection on G. mean time + 40"° 51'; re- quired the longitude. The distance sailed in the interval is 2'.8. The sun's azimuth at the 1st observation is found to be N. 131° E., which differs 86° from the course. The reduction of the 1st altitude to the place of the 2d is (Prob. 58) ' 2'.8 X cos 86' ' = + 0'.2 - + 12'. 1st chro. t. 2 3°2 23 O's (fee. :Eg.of t. 2d " " Elapsed cIito. t Mid. " " Chro. cor. 2 53 11 < = 20 48 2 42 47 + 40 51 o / " /; +19 10 4 +34.3 +1 67 . 103. + 19 12 1 ( 14- o ; ir -=-3 5*2.41+0.06 +.17 —3 62.2 G. m. t. May 16 3 23 38 h= 81 1 12 — Eq. of t. + 3 62 h' = 83 40 30 G. ap. t 8 27 SO i {h—h') =— 1 19 39 1. sin 8.3649 n L. ap. t 23 4144 ff„= 82 21 L cos 9.1242 Long, at 2d obs. ( +3 46 46 ) 66°26'.2W. Z = 25 16 d= 19 12 L sec 0.0436 L sec 0.0249 Ked. for O'.l 0.6 E. t= Oi'20»'48' Lcoseci^ 1.3433 Iiong. at noon 56 27 W. 3;=— 18 16 1. sin 8.9009 n 202 KAYIGATION. By (1 50), if A {h-h') =+1', d T, = + 6».9 ;i£AH,= + V, A T, = +2<>A; if Ji: = +10', d 7;= + P.5. 2. At sea, 1865, June 29, lat. at noon by mer. alt. of O, 33° 25' N., long, by account 147° 10' E. ; nearllA.M. T.byChro.,P55" 54- ),^^^.^^ ^1^^4021,30, « 1 P. M. " " " 3 45 ) ^ Chro. cor. on G. m. t. —36°" 28' ; the ship run from 1st observation, to noon, N. 3 pts. W. 9'.9 } " noon to 2d observation, N. 2 " W. 1 .2;) required the longitude at noon. N. 3W. N. 2W. 9'.9 7.2, 8'.2 K 6.6 6'.5 W. 2.8 J 2. = 6'.6 W. 3.4 N. 30° W. 17.0 14.8 8.3 10.0 The Btin's aramuth vas found to be N. 127" E. at the Ist observation ; N. 127° W. at the 2d obserration. The difference of N. 30° W. and N. 127° E. is 157° j the' difference of S. 30° E. and N. 127° W. is 83°. It will be better, therefore, to reduce the second aldtnde to the position of the first. By Prob. 58, (or Bowd, p. 183,) this reduction is 17'.0 x cos 83° = + 2'.1. The latitude at the time of the 1st observation was 33° 16'.8 N. b m a A. M. chro. t 13 56 54 0'» dec. Eq. of t. P. M. " •" 15 45 +23° 16' 59"— 7'.3 +2" 65».6+0».607 Elapsed" "■ t= 149 6 — 145 +7.2(7.10 +23 16 14 +3 2.8 ( .11 Mid. " " 14 50 27 Chro. cor. — 36 28 h = 74° 21' 30' L cosec i t 0.627 Hid. G. m. t. June 28.14 13 59 A' = 74 23 86 log Jr 8.824 — Eq. of t. — 3 3 J(A— A')= —18 log 1.799 » Mid. G. ap. t. 14 10 66 Zr. = 74 22 33 Lcos 9.430 " L. ap. t 23 69 64 i = 33 17 L sec 0.078 Long, at 1st obs, ( —9 48 68 i 147° 14'.6 E. , the iipparent distance ; m' S'=i>', the true distance ; and in the triangle m Z S, Zm=90°— h 5- being given, Hg.85. ZS=:90°-^) to find the angle Z, we have by Sph. Trig. (32),* , , „ _ cos i (h+H +D) cos i (h+S—J)) cos -If £i ^ ^:^ "^ • cos a cos M Then in the triangle m' Z S', Z »»'= 90°- A' and Z S'= 90°-^' being given, m' S' may be found by Sph. Trig. (I7),t sin' i iy= cos' i {h'+S') - cos h' cos S' cos' i Z, or by substituting the value of cos' -j Z, and putting s = i.{h+S+I)), - (155), sin' i 2>'=co8' i {h'+ m) - ""^ \ ""' ^ cos s cos (s-D). ^ ' cos A cos f \ ■ I To adapt this for logarithmic computation put . . , cos ^' cos S' I -r^-^ , ^ sm' * »» = - — =-- = cos s cos (s— x)), . (156) ' cos A cos jH ^ " ^ ' • , sin -i (a+6-l-c) ain \ (6+c— a) COB ■» ,0. — ; 5 — ^ : ^ . aino sm e f an' i a = sin' i (i-l-e) — sin i sin c cos* ^ ^ 212 NAVIGATIOir. then sin' i D'= cob' ^ {fi'+H') - sin' J m, which by PI. Trig. (134), becomes sin' J D'= cos i {h'-\-H'+ m) cos \ {h'+H'- m), or, if we put s'=i{h'+S'+m), (157) we have srni J)' = y'[cos «' cos («'— >w)]. (158) The solution is effected by formulas (155), (156), (157), and (158). This is only one of several direct trigonometric solutions. It is easUy remembered, involving only cosines in the second members. But in all such methods 7-place logarithms are required for the computations. 212. If the moon's augmented parallax has been used, the distance obtained, H', is not the true distance as seen fi"om the centre of the earth, but from the point O' (Fig. 36), where the vertical line of the place intersects the earth's axis. A reduction to the centre, (7, is still required, for which we have the foi'mula — * /sin i, sin S„\ ' ^iiiro' ~" tanZ»7' • in which , 6, is the sun's declination, d„, the moon's declination, ir, the moon's equatorial horizontal parallax, whose mean value is 51' 30', A, a coefficient depending on the eccentricity of the terres- /iiy=AirsmZ\ (159) • Chauvenet's ABtrouomy, Vol. I., p. 399. LONGITUDE. — LUNAK DISTANCES. 213 trial meridian, the mean value of which, for latitude 45°, is .0066855, or of log A, 7.8251, ■A sin i, the distance C C, with C E = 1. The mean values of ^tt = 23"'.07, or log ^7r= 1.3630, may be used, unless great precision is required. The signs of the declinations and latitude are + when north, and J D' is to be added algebraically to ly. If the augmentation of the parallax has been neglected, the distance has been reduced to appoint on the vertical line between C and C" and at a distance from A equal to the equatorial radius C E. 213. To find the corrections needed for the contraction by refraction of the semi-diameters of the sun and moon in the direction in which the distance is measured, let q = the angle ZSm (Fig. 35), at the sun or star, Q = the angle Z m S, at the moon, A s and A's, the contractions of the sun's semi-diameter respectively in the vertical direction S Z, and in the directicta of the distance S m / A S and A'S, the contractions of the moon's semi-diameter respectively in the vertical direction m Z, and in the direction of the distance m S. To find q and Q from the three sides of the triangle ZSm, putting, as in (155), s = i{h+S+D) we have • 1 ^ //cos s sin («— 5')\ . , //cos s sin (s—h)\ (160) for which it wUl suffice to use a rough approximation of D, and for the computation logarithms to four places ; as q and Q are required only within 30'. The contractions, 4 s and J iS, of the vertical semi-diame- 214 NAVIGATION. ters may each be fbn]i4 from the refi'q.ction table, bytakipg the difference of refractions for the limb and centre. Then, for the required corrections, we have the formulas,* J's = J s cos" q, A'S=4S cos' Q. (161) This contraction for either body is less than l', if the alti- tude is greater than 40°. For a very low altitude, it is best to subtract it from the semi-diameter in the preparation of the data, so that D will be corrected for it. But, unless quite large, it will suffice to compute it subsequently, and subtract it from X>' when the nearest limb is used, or add it to D' when the farthest limb is used. 214. Let A J) ^ the reduction of the apparent distance to the true, ov 1^=1) + AB. A ^reat variety of methods have been given for finding A D, requiring 4 or, at the most, 5-place logarithms ; but also needing special tables. Four such methods are con- tained in Bowditch's Navigator. They generally neglect to take into account the spheroidal form of the earth, the cor- rection of refraction for the barometer and thermometer, and the contraction of the semi-diameters of the sun ^.nd moon. These together, at very low altitudes and in extreme cases, may produce an error of 3™ in the calculated Greenwich time, and do actually, in the average of cases, produce errors from, 108 to 1".. Prof. Chauvenet has given in the American Ephemeris for 1855,j- a new form to the problem, with convenient tables, by which all these corrections are readily introduced. It is but little longer than the other approximative methods, in which they are neglected. * Chauvenet's Astronomy, Vol. I., p. 186. \ Reprinted in a pamphlet witli Iiis method of equal altitudes. LONGITUDE.— L0NAE DISTANCES. 215 215. The moon's mean change of longitude is 13°. 1*7640 in a day (Herschel's Ast., p. 22.2), or 33" in l"" of time. An error, then, of 33" in the distance will, in the average, produce an error of 1"" in the Greenwich time, or 1 5' in the longitude ; or an error of 10" in the distance will produce an error of about 20« in the Greenwich time, or 5' in the longitude. We may, however, readily find the effect of an error of 1", and thence amy number of seconds, in the distance, by taking the number corresponding in a table of common log- arithms to the " Prop. Log. of Diff." in the Almanac ; for .this prop. log. is simply the logarithm of the change of time in seconds for a change of 1" in the distance, (p. 95.) 216. Errors of observation are diminisbed by making a number of measurements of the distance. But even with a skilful observer a single set of distances is liable to a possi- ble error" pf 10" or even 20". Errors of the instrument are diminished by combining re- sults from distances of different magnitudes, especially from those measured on opposite sides of the moon. This cannot usually be done with longitudes at sea, but may be with de- terminations of the chronometer correction. The error pe- culiar to the observer, that is, in making the contacts always too close, or always too open, is not eliminated in this way, but will remain as a constant error of his results. The accuracy of the reductions of the observed to the true distance, depends more upon the precision with which the differences of the apparent and true altitudes — that is, the parallax and refraction — have been introduced, than upon the accuracy of the altitudes themselves. 217. Lunar distapces ard used at the present day, not so much for finding the longitude, as for finding the Greenwich mean time, with which to -compare the chronometer. They may thus serve as checks upon it, which in protracted' 216 NAVIGATIOIT. voyages may be much needed. If the chronometer cor- rection thus determined agrees with that derived from the original correction and rate, the chronometer has run well, and its rate is confirmed ; if otherwise, more or less doubt is thrown upon the chronometer, according to the degree of accuracy of the lunar observation itself. If the discordance is not more than 20^, it is well still to trust the chronometer, as the best observed single set of distances may give a result in error to that extent. If it is large, then by repeated measurements of lunar distances, differing in magnitude, and especially on both sides of the moon, and cai-efully re- duced, the chronometer correction can be found quite satis- factorily. By taking the rate into consideration, observa- tiohs running through a number of days can be combined. Example. At sea, 1855, Sept. 7, about 6^ A. M., in lat. 35° 30' K, long. 30° W. by account ; Time by chro. 8" 29" -5'7'.5 ; app. chro. cor. (G.m.t.)— 21° l^.S ; Observed distance of O and D 43° 52' 30", index cor, —20"; Observed altitude of ^ 49'' 31 ' 50", index cor, + 1' 0" ; Observed altitude of O 5° 27' 10", index cor, 0"; Bar. 29.10 inches ; ther. 75° ; height of eye 20 feet; Required from these observations the chronometer correction on Greenwich time. • Preparation, h m 6 / // u T.bychro. 12''+8 29 57.5 B'sH-par. 6419.4 3)'aS.(Jiam. 14 60.0 Chro. cor. —21 1;6 Aug. +8.6 Aug. +11.2 6. m. t. Sept. 6 20 8 66 D 'b Aug. H. par. 64 23.0 D 'b Aug. S. diajn. l6 1.2 _B * 49° 81' 60" O. 6° 27' 10" • G's H. par. 8^5 In. cor. +10 In. cor. Dip —4 28 Dip —4 23 ©'b S. diam. 18' Se'.l Aug. S. diam. -|-1S 1 V. S. diam. +16 84 V. cont. —21.6 V. S. diam. 16 83 .5 LONGITUDE. — ^LUNAR DISTANCES. 217 A = 49»43'28' J5'=6»38'21' Eef. —46 Eef. —8 12 Par. +85 10 Par. -j-8 ft'= SO 11 62 a" =6 30 17 Obs'd dist. i O 43° 52' 10' A = 49° 43' B'a Aug. S. diam. +16 1.2 ff= 5 38 I. sec 0.002 1, „ a- Tib «ei i) = 44 22 L cosec 0.155 ®^ ^^ <^^- +1« ""^ 2« = 99 43 Cont J'« = —21.6 s = 49 52 1. cos 9.809 i) = 44 22 46 i - A = 9 L sin Y.418 Cont. ©'« 5. diam. (161). ' . . 17.384 2 log COS q 9.999 g= 2_49 Lsini 8.692 loi 21.6 1.334 log J's 1.333 Computation of IVue Distance. (155-158) O I II O I u ft = 49 43 28 S= 6 38 21 2) = 44 22 45 2 > = 99 44 34 » = 49 62 17 s—D= 5 29 32 Compression. (159) * (*'+,^^- 0'sdee.=+ 6°.3 Lsiii9.040 Z)' 1. cosec 0.150 »= 0.165 log 9.190 D'sdec.= +25°.3 tsiu 9.631 D' 1. cot 9.999 n'= .427 log 9.630 n—n'=— .272 log 9.435 n Air log 1.363 L 1. sin 9.764 Comp.— 3'.6 log 0.662 ro Finding the Greenwich msan tims and chronom^er cor- rection. (82) True distance 2)"= 46° 4' 43" Distance at 18''. !>,= 46 3 17 P. L. 0.3438 Diffi +6 J}"—D^= 58 34 log 3.6468 < <= "^ 9" 6' log 3.8891 L sec ft 0.1894654 LsecZr 0.0021069 K= 60 17 62 1. cos h' 9.8053633 S'= 6 30 17 IcobM' 9.9979926 I cos « 9.8092266 o 1 a Lcos(«-D)9.9980017 h'+B')=: 27 64 4.6 19.8021464 im= 62 46 39.4 Lsin^m 9.9010732 «'= 80 40 44 ni=105 33 19 L cos s' 9.2094277 m-»'= 24 52 36 1. cos (m-s') 9.9577114 o 1 a 19.1671391 iD'= 22 32 23.7 L sin 9.6835696 D'= 45 4 47 Cor. for Comp. —i D'= 45 4 43 true distance. 218 NAVIGATION. 6. m. t. of 2), 18* 0» 0* Bed. for 2d diff. —2 6. mean time, Sept. 6 20 9 4 T. by chro. 20 29 67 Chro. cor. —20 53 by lunar. —21 1 by previous cor. and rate. Difference +8 This example is taken from the pamphlet of Frof. Chauvenet, where it ig reduced by his method with far less labor of computation. The true distance by that method is 45° i' 46' ; by Bowditch's 1st method, in which the small corrections are omitted, it is 45° 6' 44", dETering very nearly 1' from the correct value. This would produce an error of 2"" 10* in the Greenwich time. 2 1 8. Other lunar methods for finding the longitude, be- side that of lunar distances, are — 1. By moon culminations, or observing the meridian transits of the moon and several selected stars near its path, whose right ascensions are considered well determined. 2. By occuUations, or noting the instant that a star dis- appears by being- eclipsed by the moon, or that it reappears from behind the moon. The first is called an immersion, the second an emersion. 3. By altitudes of the moon near the prime vertical. 4. By asim,uths of the moon and stars observed near the meridian. These methods, except occasionally the second, are avail- able only on shore. They require good instruments, careful observations and determinations of the instrument correc- tions, and scrupulous exactness in the reductions, especially those which involve the moon's parallax. By each may be found the moon's right ascension, and thence, by inverse interpolation in the Almanac, the corre- sponding Greenwich mean time. Subtracting from it the local mean time, which must also be found from good ob- servations, gives the longitude. 219. If corresponding observations are made at two dif- LONGITUDE. — LUNAES. 219 ferent places, their difference of longitude can be found with much less dependence on the accuracy of the Ephemeris. When the two local times of the occultation of the same star have been noted, they can each be reduced to the in- stant of the geocentric conjunction of the moon's centre and the star in rightascension ; and the difference of the reduced times will be the longitude. By the other methods, the change of the right ascension of the moon, in passing from one meridian to the other, may be found. This, divided by the mean change in a unit of time, as !•> or 1™, computed from the Ephemeris, will give the difference of longitude in the same unit. CHAPTER IX. SUMNER'S METHOD: LATITUDE AND LONGITUDE BY DOUBLE ALTITUDES. CIRCLES OF EQUAL ALTITUDE.^ — (sUMNEe's METHOD.) Fig. 37. 219. Suppose that at a given in- stant the sun, or any other heavenly body, is in the zenith of the place M (Fig. 37), on the earth ; and let A A' A' be a small circle described from M as a pole. The zenith distance of the body will be the same at all places on this small circle, namely, the arc M A ; for if the representation is transferred to the celestial sphere, or projected on the celestial sphere from the centre as the projecting point, M will be the place of the sun, or other body, and the circle A A' A" will pass through the zeniths of all places on the terrestrial circle, and M A, M A', &c., will be equal zenith distances. The altitude of the body will also be the same at all places on the terrestrial circle A A' A"; hence such a circle is called a circle of equal altitude. It is evident that this circle will be smaller the greater the altitude of the body. 220. The latitude of M is equal to the declination of the body, and its longitude is the Greenwich hour-angle of the CIHCLES OF EQUAL ALTITUDES. 221 body ; which, in the case of the sun, is the Greenwich appar- ent time, or 24'' — ^that apparent time, according as the time is less or greater than 12''. This is evident from the dia- gram, in which, regarded as on the celestial sphere, P M is the celestial meridian of the place, whose zenith is M, and its co-latitude ; and also the declination circle, and co-declination, of the body M ; and if P G is the celestial meridian of Greenwich, G P M is, at the same time, the longitude, of the place, and the Green- wich hour-angle of the body. If, then, the Greenwich time is known, the position of M may be found and marked on an artificial globe. 221. IF, moreover, the altitude gf the body is measured, and a small circle is described on the globe about M as a pole, with the complement of the altitude as the polar radius, the position of the observer will be at some point of this cir- cle. His position, then, is just as well determined as if he knew his latitude alone, or his longitude alone ; since a know- ledge of only one of these elements simply determines his position to be on a particular circle, without fixing upon any point of that circle. As, however, he may be presumed to know his latitude and longitude approximately, he will know that his position is within a limited portion of this circle. Such portion only he need consider. It is commonly called a line of posi- tion.* 222. The direction of this line at any point is at right angles with the direction of the body ; for the polar radius M A is perpendicular to the circle A A' A" at A, A', A', and every other point of the circle. 223. Artificial globes are constructed on so small a scale that the projection of a circle of equal altitude on a globe ' * Inappropriately termed a line of bearing. 222 NAVIGATION. ■would give only a rough determination. But the projection of a limited portion may be made upon a chart by finding as many points of the curve as may be necessary, and, having plotted them upon the chart, tracing the curve through them. The portion required is usually so limited that, when the altitude of the body is not very great, it may be regarded as a straight line ; and hence two points suifice. Wiih high altitudes, three points, or if the body is veiy near the zenith, four may be necessary, and even the entire circle may be required. 224. Pkoblem 55. From an altitude of a heavenly body to find the line of position of the observer, the Oreenwioh, time of the observation being hnown. Solution. From the given altitude, and assumed latitudes Xi, ij, is, &e., differing but little from the supposed latir tude, find the corresponding local times (frob. 43), and thence, by the Greenwich time, the longitudes A,, Aj, A,, &c. Thus we shall have the several points, whose positions are conveniently designated as (i„ Ai,), (ij,, A^,), (i,, A3,), &c. It facilitates the computation to assume latitudes difiering 10' or 20', as the \ sums and i-emainders diflfer 6' or 10', and only one of each need be written. Or, from the Greenwich time and assumed longitudes, Aj, Aj, A,,. &c., find the corresponding local times (Art. "TT), and thence the hour-angles of the body (Probs. 34, 35). With these and the obsterved altitude, find the corresponding lati;- tudes , i„ ij, Ls, &o. (Prob. 46). This is more convenient than the preceding method, when the body is near the meridian. In either mode the computation may be arranged so that the like quantities in the several sets shall be in the same Une, and taken out at the same opening of the tables. The several points may then be plotted on a chart, each by its latitude and longitude, and a line traced through CIRCLES OF EQUAL ALTITUDES. 223 them, which will be the required line of position. Two points connected by a straight line are sufficient, unless the altitude is very great, or the points widely distant. Thus in (Fig. 38), let A and B be two 5* Ii such points plotted respectively on the /^ parallels of latitude L„ La, and each in its proper longitude ; A B is the line of position, and the place of observation is .^ , at some point of A B, or A B produced. ^- *8- This is aU which can be determined from an observed alti- tude, unless either the latitude, or the longitude, is definitely known. And as these are both uncertain at sea, except at the time when found directly by observation, the position of the ship found from a single altitude, or set of altitudes, is a li9ie, of greater or less extent as the latitude, or the longitude, is more or less accurately known. In imcertain currents, or when no observations have been had for several days, the extent of this line may be very great. Yet, if it is parallel to the coast, it assures the na- vigator of his distance from land ; if directed toward some point of the coast, it gives the bearing of that point. 225. If there is uncertainty in the altitude, for instance of 3', the line of position having been computed and plotted, parallels to it on each side may be drawn at the distance of 3'. So, also, if there is uncertainty in the Greenwich time, parallels may be drawn at a distance in longitude equal to the amount of uncertainty. In either case, the position of the ship is within the in- closed belt. In Fig. 38, a J is such a parallel to the line of position A B, its perpendicular distance from it measuring a diflFer- ence of altitude; the distance A a on a parallel of latitude measuring a difference of longitude. 224: NAVIGATION. 226. Since the line of position is at right angles with the direction of the body (Art. 222), the nearer the body is to the meridian in azimuth, the more nearly the line of position coincides with a parallel of latitude ; and thus a position of the body near the meridian is favorable for finding the lati- tude from an observed altitude, and not the longitude. So also, the nearer the body is to the prime vertical, the more nearly the line of position coincides with a meridian, and the less does any error in the assumed latitude affect the longitude computed from an observed altitude. So that, if the body is on the prime vertical, a very large error in the assumed latitude will not sensibly affect the result. Such a position of the body is, then, the most favorable for finding the longitude from an observed altitude. These conclusions have been previously stated, drawn from analytical considerations. 227. Two or more points of a line of position as (X, ^), (-£j, Aj) etc., having been determined by Prob. 55, if the true latitude, X, be subsequently found, ■p -o the corresponding longitude, X, may be ' "^ ^ obtained by interpolation. Or, the place of the ship may be found graphically upon the chart, by / i i j . drawing a parallel in the latitude, i, -A- EC and taking its intersection P, with the ^' ' line of position A B. So also, if the true longitude, X, is subsequently found, the corresponding latitude, X, may be obtained by interpo- lation ; or, a meridian E F may be drawn in the longitude, A, which will intersect the line of position in P, the place of the ship. If there is uncertainty in either of these elements, two parallels of latitude (as in Fig. 38), or two meridians, may be drawn at a distance apart equal to the uncertainty. CiScLES OF EQUAL -ALTITUDE. 225 As altitudes, latitudes, and longitudes are never found at sea with much precision, and may under unfavorable circum- stances be largely in error, the position of the ship on the chart is not properly a point, but a belt, more or less limited according to the accuracy of the elements from which it has been formed. 228. In Fig. 39, if A is the position (-£„ ^i), B, the position (Zi„ A^), both near P, the true position, whose latitude ia i, and longitude is A ; the right triangles* A C B, A E P, being formed, C B = ij— i„ the difference of the two latitudes, A C = A3— A,, the difference of the corresponding longi- tudes, E P = J X = i— ii, the correction of i„ A E = ^ A = A— Ai, the correction of Aj ; then CB:EP = AC : AE or, (i,— ii) :JL^ (As— Ai) : J A, whence we have, "^^''^^^ \- (162) and A = Ai + J A as the formulas for finding A, the longitude of the true posi- tion, when its latitude, i, is known. Or, we have '^^='^^^=XI (163) and Z = Zi + AZ ) as the formulas for finding i, when A is given. They are the same formulas as for an interpolation. The several differences are most conveniently expressed in minutes of arc, or, in the case of longitudes, in seconds of time. The • This is different from the projection on a Mereator's chart, where 6 B and £ F would be augmented differences of latitude. 226 NAVIGATION. local times may be used instead of the longitudes and in^ terpolated in the same way. From the first of (162) we may readily determine how much a supposed error in an assumed latitude affects the resulting local time, or longitude. 229. Problem 56. To find from a line of position the azimuth of the body observed. Solution. We have the positions (L^, A-i), (ij, Aj), or the latitudes and longitudes of two points, from which the azi- muth, or course of the line of position, can be found by middle latitude sailing. Adding or subtracting 90°, according as the azimuth of the body is greater or less, gives the azimuth required. Or, a perpendicular to the line of position may be drawn upon the chart, and the angle which it makes with a meri- dian may be measured with a protractor. The azimuth may thus be found to the nearest 1°. Example. At sea, 1 865, Nov. 23, lOj A. M., by account in lat. 36° 50'N., long. 65° 20' W. ; Greenwich mean time 2'' 40™ 47= ; the sun's correct central altitude 29° 6' 25' ; to find the line of position. G. m. t. Nov. 23 2'' 40"" 4'7"=2\68 0's dee. Bg'n oft. Eq. oft. +13 18 —20° 26' 33'— 80".9 — 13" 20'.4 + 0'.72 6. ap. t. 2 S4 5 ("iTTs +1-9 —1 23 i 18 .7 —13 18 6( -6 —20 26 66 2 .4 1. With assumed Latitudes. (Prob. 43.) h = 29° 6' 26" Li = 36° 30', Zs : = 36° 50', is = : 37° 10', ii = 36 30 1. sec 0.09482 .09670 .09861 J5=110 26 66 I. cosec 0.0282T .02827 .02827 2»=176 3 21 8=: 88 1 40 1. cos 8.63674 .49841 .46636 »-h= 68 66 16 1. sin 9.93271 .93346 .93422 G. ap. t." 2'' 64°' 6» 18.69264 .66684 .61746 CIRCLES OF EQUAL ALTITUDE. 227 t. ap. t. i ( (( 66° 22'.0 W. — ST'.S 66 24.6 Kg' g }is= i 17 43 = 64 25.7 i, = 4 26 28 Aj = 4 21 38 • (1) 22 28 37 L sin 9.29677 .27842 .26873 ] (2) 22 32 27 ; (3) 22 36 22 Li = 36° 30' S: Zs = 36 80 Xs = 37 10 For AL — +4:0', JA,=— 116'.3; or a change of 40' in latitude produces a change of — 116' in longitude. From A.Z = + 40', /d A = — 116'.3, we find, by middle lati- tude sailing, the dep. 93'.0, and then the bearing of the line of position, regarded as a rhumb line, which it nearly is, N. 66°.7 E. ; the sun's azimuth therefore is N. 156°.7 E. Suppose the correct latitude to be 36° 57' N., to find the corresponding longitude on the line of position, we have i=36°67'N. Xj = 36° 50' N. ■K,= : 66° 24'.6 W. Ji= +7' JA = — 2'.9 X 7 = -- — 20'.6 - = —2 94 20 X=i 65° 4' W. 2. With assumed Longitudes. (Prob. 46.) 6. ap. time 2'' 64" 5" Xi= 4 19 ■^ - 4h 21m 0. ^5 = 4''23'» 0«' L. ap. time t = —\ 24 66 - 1 26 66 — 1 28 65 L sec t 0.03062 .03201 .03354 (164) by which the latitude, JO, and the longitude, X, of the inter- section can be found. 231. Either assumed latitude may be designated as J,, and either observation by the accent ' , or be called the first latitude and xh& first observation ; but the several differences of latitude and longitude must be marked with their appro- priate names, or signs. If the differences of longitude A",— A'„ X",—X\, on the two parallels have the same name, their difference is taken in finding m, which will be -I-, when X'j—X\ > X°^ — A-V, or the difference of longitude on the first parallel is the greater. In this case m > 1, 4 i > Xj— i, and AX'> X'^—X\. The point P is then, as in Fig. 40, in the same direction as B from A, and beyond B. But m will be — , when X"i—X\ and X°^—X'^ have the same name, and A',— A'l < A'j— A',, or the difference of longitude on the first parallel, is the less. J Ij and A X will then have different names respectively from ij — 2/1 and A'j — A-',. In this case P and B are in opposite directions from A. A negative value of m may be avoid- ed, so that P and B will fall always on the same side of A, or P and D always on the same side of C, (Fig. 40), if we take as X, the latitude of the parallel on which is the greatest difference of longitude. If the differences of longitude, A,",— A.',, X'^—X\ , on the two parallels have different names, their sum is taken numerically in finding m y in that case m is + and less than 1 , J i < ij — III S'ld 4 A < A', — A', , with the same names CIKCr.ES OF EQUAL ALTITUDE. 231 respectively ; and, as in Fig. 41, P is between A and B, and between C and D. When three or more latitudes are used in the compnia- tions, those for which the differences of longitude are small- est shonld be taken as 2^ and L^. 232. The more nearly perpendicular the lines of position are to each other, the better is the determination of their intersection. Hence, the nearer the difference of azimuths of the body or bodies at the two observations is to 90°, the better is the determination of jJDsition from double altitudes. K the azimuths are the same, or differ 180°, the two lines of position coincide in direction, and there is no intersection. In this case the great circle joining the two bodies, or the two positions of the same body, is an azimuth circle, and passes through the zenith. An approach to this condition is generally to be avoided. (Bowd., pp. 181, 195, notes.) StiU, however, if the two bodies, or positions of the same body, are near the meridian, the lines of position nearly coincide with a parallel of latitude. The latitude is then well determined, but not the longitude. If the two bodies, or positions of the same body, are near the prime vertical, the lines of position more nearly coincide with a meridian and the longitude is well determined, but not the latitude. When the difference of azimuths is small, the intersection of the two lines may be computed with tolerable accuracy, while it cannot be definitely found by the projection of the lines upon a chart. 233. The operations indicated in (164) are to subtroMt, 1. The first assumed latitude from the second, {L^—Z,^ ; 2. for the first observation, the longitude corresponding to the first latitude from that corresponding to the second latitude, (^'j— ^'i); 232 NAVIGATION. 3. for each latitude, the longitude deduced from the first observation from the longitude deduced from the second, (V,— A',) and {X\—X\) ; 4. the difference of longitude for the second latitude from that from ihe first, [(A"i— A',) — (A'j— r,)], (or add numerically these differences of longitude when they are of different names.) Then 5. To divide by this last result the difference of longi- tude, (^"i— *',), for the first latitude, to obtain the coefficient m, (which will be — only when the dif- ference of longitude, (A'j— A'j), for the second lati- tude has the same name as and is greater than the difference of longitude, (A",— A',), for the first lati- tude), 6. To multiply m by the difference, (ij— ij), of the two assumed latitudes to obtain the correction of the first latitude ii ; and by the difference, A'j— A'l), of the two longitudes derived from the first ob- servation, to obtain the correction of the first of these longitudes, X\. These corrections have the same name as the differences from which they are derived, when «i is + ; but contrary names when to is — ; and are to be applied accordingly. 234. The lines of position may be found from two as- sumed longitudes A, and A,, instead of two latitudes (Art. 224). The formulas for finding their intersection will differ from (164) only by an interchange of the 'letters L and A. We shall have, then, L\-L\ nC = {L\-IJ,)~{L\-IJii ^ X=^m! (Aj— Ai), A= Ai + J A JL = m' (i',-i',), L = L\-\-AL (165) CIECLES OF EQUAL ALTITUDE. 233 Examples. 1. With -C =: 30« '28' N., Z,= 30< '8' N. by 1st alt. X, = 69 16 W., x\ = 69 W. b; 2d alt. A'l = 58 43 32 w., E. 7\ = 69 8 W. Siffereoces, 8 w. m = 82 _ 40 ~ .8 Diff. 20' S. " 16 E. " 25 W. = 40 K. 4 i = — 20' X .8 = — 16', L = 30° 28' — 16' = 30» 12' N. A A = — 16' X .8 = — 12', A = 69 16 — 12' = 69 8 W. The differences of longitude on the two parallels, 32' E. and 8' W., being in opposite directions, the intersection is between the two parallels, or L is between Li and -£,. 2. With Ji = 48» 10' a, Za = 48° 30' S. Diff. 20' S. by 1st alt. X'l = 88 16 K, X's = 88 24 E. " 8 E. by 2d alt. Vx = 88 30 E., A'a = 88 55 E. « 25 E. Differences, 14 E., — 81 E. - 11 W. m = — ^*- 82 .---.82 AL = - -.82 X 20' = — 16', X=48''10' — 16' = 47° 64' S. A ;i = - -.82 X 8' = — i; X = 88 16 — r = 88 9 E. In this example it is convenient to regard south latitudes and east longitudes as + . The differences of longitude.'on the two parallels, 14' E. and 31' E., being in the same direc- tion, the intersection is outside of the parallels and nearer the first, for which we have the smallest difference. 3. With Xi = 165° 60' W., ?9 = 166° 20' W. Diff. 30' W. by 1st alt. X', = 36 16 S., L'3 = .36 25 S. " 9 S. by 2d alt. i', = 36 38 S., Z't= 36 29 S. " 9 N. Differences, 22 S., — 4 S. = 18 S. •"■ = ?? = l.n J i = 1.1'7 X 9' = + 10'.5, X = 36° 16' + 10' = 36° 26' S. ^ X= 1.17x30' = + 36'.1, A = 166 50 + 35' = 166 26 W. The differences of latitude on the two meridians, 22' S. and 4' S., are in the same direction ; and the intersection is 234 NAVISATION. outside of the meridians and nearer the second, on whieh the difference of latitude is least. 235. Problem 57 supposes the two altitudes observed at the same place. This at sea is rarely the case. Peoblem 58. To reduce an observed oMitude for a change of position of the observer. Solution. Let Z (Fig. 42) be the zenith of the place of observation ; h = 90° — Zw, the observed altitude; Z', the zenith of the new position ; h' = 90°— Z' m, the altitude reduced to the new position, Z. 90°. (Bowd., p. 183.) It is smaller, and can, CIECLES OF EQUAL AliTITUDE. 2S5 therefore, be more accurately computed the nearer G—Z approaches 90°. It is, therefore, better to reduce that alti- tude for which the difference of the course and azimuth is nearest 90°. If the second is the one reduced, then C is the opposite of the course. In practice Z Z' does not usually exceed 30', so that al- though an arc of a great circle of the celestial sphere, it may be regarded as representing the distance, d, of the two places on the earth ; or, at sea, the distance mn. The azi- muth, or bearing, of the body can be observed with a com- pass, or be computed to the nearest degree, or half-degree, from the altitude. The assumption, Z' »w = O 9W, is more nearly correct, the greater Z' »i or Z m, that is, the smaller the altitude. If we treat Z Z' »w as a spherical triangle, 82°. Example. The two altitudes of the sun are 36° 16' 20', 58° 15' 20', the compasa bearings of the sun respectively S. E. by E. -J E. and W. S. W. ; the ship's compass course, and distance made good in the interval N. N. W. i W. 25 miles ; S. 5^ E. differs from N. 2^ W. 13 points, so that the re. duction of the 1st altitude to the position of the 2d is 25' X cos 13 pts. = — 25' cos 3 pts. = — 20'.8 = — 20' 48'. S. 6 W. differs from S. 2^ E. 8^ points, and the reduction of the 2d altitude to the position of the 1st is 236 NAVIGATION. 26' COS 8i pts. = — 25' COS 1^ pts. = — 2' 30'; c- —2' 39", if the last term of (167) is included. 236. By (166) or (167) we may reduce one of the two alti- tudes for the change of the ship's position in the interval. But instead of this we may put down the line of position for each observation, and afterwards move one of them to a parallel position determined by the course and distanc^ sailed in the interval. Thus in Fig. 43, let A B be the line of position for the first observation, and A a represent in direction and length the course and distance sailed jjg 43;, in the interval ; then a b, drawn parallel to A B, is . the line of position which would have been found had the first altitude been observed at the place of the second. If the second observation is to be reduced to the place of the first, then A a in direction must be the opposite of the course. The perpendicular distance of A B and a 6 is the reduc- tion of the altitude for the change of position : for that dis- tance is A a X cos (B A a — 90°). LATITUDB BT TWO ALTITtTDES. 237. In Sumner's method the latitude and longitude are both found by two altitudes, either of the same or difierent bodies. It is sometimes desirable to find the latitude only, or at least to make this the chief object of combining the two observations. 238. Pboblbm 59. To find the latitude from two altitudes of the sun, or other boch/, supposing the declinaiion to be the LATITUDE BY TWO AliTITUDES. 237 same at both observations, and the Greenwich time to be known cqjproxinwtely. Solution. Let two altitudes, or sets of altitudes, be ob served and the times noted by a chronometer, or a watch compared with it ; reduce the altitudes to true altitudes, and at sea one of them for the change of the ship's position in the interval by Prob. 58. Find also the difference of the chronometer times of the two observations, and correct it for the rate in the interval. " ( + when the chronometer loses), This correction is t being the interval in hours of chro. time, and J c the daily change. The result is the elapsed mean time for a mean time chro- nometer; the elapsed sidereal time for a sidereal chrono- meter. The Greenwich mean time of the greater altitude, or of the middle instant, should also be obtained from, the chro- nometer times, sufficiently near for finding the declination of the body. In Fig. 44, let M and M' be the two positions of the body, A = 90°— Z M, the/rs« altitude, h'= 90°— Z M', the second altitude, .%hioht^. (170) b. For the moon or a planet, if a and a + ^a represent the right ascensions of the body at the two times, we have (Art. Ill), T=S — a, T'-S'-a-Aa, and t = T-T=s—Aa, (171) that is, the elapsed sidereal time diminished by the increase of the right ascension of the body in the interval. A^a, the change of right ascension in l'' of mean time, may be obtained from the Almanac for the middle Green- wich time. The change in l*" of sidereal time will be, by (86), '(1— .00273) Jtfl, which can readily fee found by regarding JtO as a sidereal interval, and reducing it to its equivalent mean time inter- val. Expressing t„ and s in hours, when used as coefficients, we have Aa = t„. 4o = s. d^a (1 -.00273),; (172) and, for an elapsed sidereal time, t = 8-s. A^a (1 -.00273) ; (173) LATITUDE BY TWO ALTITUDES. 239 for an elapsed mean time, by (i 70), t = «.+ *. (9».8565 — Ao). (174) c. For the sun, the angle.MPM', or t, is the elapsed ap- parent time. If, then, i^E is the hourly change of the equa- tion of time (+ -when the equation of time is additive to mean time and increasing, or subtractive from mean time and decreasing ; that is, when apparent time is gaining on mean time), t = t,+ t^.d^B, (175) by which t may be fonnd from a mean time interval. If the sidereal interval is given, we have, as in (173), for a planet, t = s — s. JiO (1— .00273). The reduction of the elapsed mean time to an apparent time interval, is commonly neglected by navigators ; but on December 21, 4^E^^ 1^.25, and during a large part of the year exceeds 0^.5. 239. We have given A = 90°-ZM, ■ (181) Bin i = cos C sin {B+Z), ) LATITUDE BY TWO ALTITUDEa 241 which determine (7, Z, and L. If, however, we add the squares of (180), we shall have xsos' G = cos' H. cos' q + sin' H^ or 1— sin' C = cos' jET (1 — sin' g) + sin'^y whence sin C= cos -ffsin q. (182) Substituting this in (1Y9), and the 2d of (180) in (178), we have . ~ sin i {h — A') cos \ (A+ A') sin A y_ cos i (ft— ft') sin \ (A+A') cos iLi — . ™ , COS A cos 6 and (181) sin i = cos C sin (B^-Z). (183) Z, being found from its cosine, may have two values numerically equal with contrary signs. Representing, these, we have sin 2/ = cos C sin {£±Z), which gives two values of i. The value which accords .most nearly with the latitude by account may be taken. We shall see presently how the admissible value of Z may be selected. 241. To avoid using both the sine and cosecant of ^ and the cosine and secant of C, we may take the reciprocals of (176) and the 2d of (183) ; we shall then have, as in the 1st method of Bowditch (p. 180), cosec A = sec d cosec* ^ t, "| cosec J5 = cosec d cos A, I I sin C= sin i^ (A— A') cos ^ {h+h') cosec A, - (184) I sec Z—BBci {h—h') cosec ^ {h+h') cos A cos C, sin i = cos C sin (-&±Z). * log sec 4 < and log cosec -J t may be taken from Table XXVII. coto- qranding to < in the column F M (Art. 126). 242 NAVIGATION. (185) It is wnnecessary to find A and (7, as log cosiae A can be taken from the tables corresjtonding to log cosecant A, and log cosine C corresponding to log sine C. Indeed, we may dispense with A entirely by substituting the Ist of the pre- ceding equations in the 3d, and the 2d in the 4th, and em- ploying (1V7). We shskll then have, for the solution of the problem, — tan i?=tan d sec* \ i, sin <7=8in \ {h—h!) cos \ {h+h') sec d cosec* ^ t, sec Z=aeci (^— A') coseci {h+h') sine? cosec ^ cos C, sin X=co8 Csin {J3'±Z). A, JB, C, Z, SLT^d Z, are^each numerically less than 90°;, A is in the 1st quadrant; C is -f- when the 1st altitude is the greatear, — when it is the smaller ; JB has the same sign, or name, as the declination ; and i the same as {JS'+Z) or (^— ^),fl:om which it is obtained. 242. If Z O (Figs. 44 and 45), be drawn perpendicular to P T, we shall find from (182), C=±Z O, + when P T is west, — when it is east, of the meri- dian ; Z=T O, ^+Z=90°-PO in Fig. 44, ^-Z=96°-P O in Fig. 45 ; Z, or T O, being -I- or — according as P and^Z are on the same side of M M', as in Fig. 44, or on opposite side^, as in Fig. 45. This may also be shown in an- other way: for, in the first ease, the * log see i t and log cosec ^ t may be takea from Table XXYIL corre- epondiDg to £ in the column P M (Art. 126). , LATITUDE BY TWO ALTITUDES. 243 Hg. 45. angle q =" P T Z < 90°, cos g' is + ; and since, ia (180), cos C and cos ^ are +, sin Z, and therefore Z, will also be + . In the second case, when M M' produced passes between P and Z, ^ = P T Z > 90°, cos q and, consequently, sin Z and Z are — . Instead of marking Z + or — , we may use the symbols N and S, as for d. By and i. "We shall then have the rule (Bowd., p. 181) : — Mark Z north, or south, according as the zenith and north pole, or the zenith and south pole, are on the same side of the great circle, which joins the two positions of the body. By thus noting the position of this circle, the ambiguity of ^ Z is removed. We may, however, remove the ambiguity by noting the azimuths of the two points M and M'. In Fig. 44, P Z M > P Z M'; in Fig. 45, P Z M < P Z M'; which would be the case also if one or both points were on the other side of the meridian. Hence we have the rule : — Z has the same name as the latitude when the azimuth of the body is numerically the greater at the greater altitude ; but a different name from the latitude when the azimuth at the greater altitude is the less. The azimuths are to be reckoned both east and west from to 180°, and from the N. point in north latitude ; but fi'om the S. point in south latitude. 243. If Z is very small, it cannot b6 accurately found fi-om its cosine, or secant ; ijs sign may be doubtful ; and the lati- tude cannot be determined with precision. This will be the case^ when the altitudes are very great j when M and M' are near the -prime vertical ; or, in general, when. M and M' are remote from the meridian and the difference of azimuilis, 244 NAVIGATION. M Z M', is either very small, or near 180°. In each of these cases M M' intersects the meridian very near the zenith. It has been seen, with regard to lines of position derived from two altitudes (Art. 232), that the most favorable con- dition is when M Z M'=90° ; but that the latitude alone can be well determined when M and M' are quite near the meri- dian in azimuth and M Z M' quite small. Indeed, if both azimuths are 0, or 180°, the two altitudes become a meridian altitude. These conditions belong to all methods of finding the lati- tude from two altitudes. 244. The latitude having been found, we may proceed to find the hour-angle of the body from one of the altitudes (Prob. 43), if it is sufficiently near the prime vertical, and thence the longitude, if the times have been carefully noted by a Greenwich chronometer (Art. 188). Instead of this, by putting T„-i {T+T') = Z V T, the middle hour-angle, we have the formula (146), sin T, ^sinJi^. ^')cosH^ +^ (183) " cos d cos L sin i t ' ^ ' and from (185) . ^ sin -J^ (h—h') cos i (h+h') ^ Sin O — 7 : z 7 « COS a sin i^ * ' whence sin*r, = ^. (187) ° cos i ^ ' This could also have been obtained from the right triangle P O Z, (Figs. 44, 45) ; from which we have also J. rmn tan Z tan Z P O = -; — 5-j:r, sin P 0' or * If we enter Table XXVII. (Bowd.) with log sin T„ or log tan T„ we shall find 2 T, corresponding in the F. M. column. LATirUDE BY TWO ALTITUDES. 245 Thus, by a brief additional computation, :2J can be found by (186) or. (187). We shall have also T=To-it, ,T'=T, + it <189) for the hour-angles at the times of the observations. The longitude can be found from either T„ T,or T\ and the cor- responding chronometer time. (186) is the formula of LittroVs method* (Art. 200). The favorable conditions, as there stated, for finding 7J, are - a small value of T^ and high altitudes near the meridian, or altitudes on each side of the meridian near the prime verti- cal. But such altitudes are unfavorable for finding the lati- tude. When both latitude and longitude are to be found from two altitudes, the nearer the difference of azimuths is to 90° the better will be the determination. The most favorable conditions for combining them will be equal azimuths of 45°, or 135°, on each side of the meridian. , If one of the altitudes is very near the prime vertical, and the other very near the meridian, it will generally be better to find the time and longitude fi-om the first by Prob. 43, and the latitude from the second by Prob. 46 or 47. 245. In this problem the declinations are supposed to be the same at both observations. This will be the case with the sun only at the solstices ; with a planet, or the moon, only when 90° from its node, and with the latter body for a very brief period. Navigators usually neglect the change of declination of the sun, or a planet, and use the mean de- clination, or that for the middle instant. It is better, how- ever, when the change is neglected, to employ the declina- tion at the time of the greater altitude,f except when the * The novelty of Littrow'a method consists in finding T^ from very high altitudes near the meridian. (146), or (186), is by no means a new formula, f Chauvenet's Astronomy, Vol. I, pp. 276, 315. 246 NAVIUATIOW. Lour-angle of this altitude is greater than the middle hour- angle. This can be the case only when the altitudes are on different sides of the meridian. When the middle declina- tion is used, we may, with little additional labor, find, the correction of the computed latitude by the following formula from Chauvenet's Astronomy (Vol. I., p. 267). cos 2/ sm i t ^ ' , or, by substituting (187) Ji = _l^.^PL (191) sin i t ^ ' in which A d is half the change of declination in the inter- val of the observations. Noting whether it is toward the north or the south, we can apply it with the same name to the computed latitude, when the lesser altitude was ob. served ^rst; but with a different name when the lesser alti- tude was observed '2°>2». 1. cosec ! i 0.68001 ^(h+h')= 40O 16' 57 1. cos 9.88256 i(A-A')=- 1 69 32 1. Bin 8.54113 » log 2 0.30108 tne change of altitude m 1™ " sm(L — a) ' ° from the meridian, (Tab. XXXII., Bowd.,) and as before, 252 NAVIGATIOlir. h and h', the true altitudes, 7' and I", the corresponding hour-angles, (in minutes of time,) t = T—T, the difference of the hour-angles, T„ — \{T'+T),t\\& middle hour-angle. By (120) h, = h'+A,h.r\ J ^^°^^ The mean of these equations is A„ = i (A-i-A') -|-^(r '+2"). A, h. (202) But i(r'-t-r')=(^y-i-(^) = (i«)' -Hii' ■which substituted in (202) gives h, = \ (h+h) + [i t ' + T,'] d,h. (203) The difference of the two equations of (201) gives h-h' = {T'-T'). Jo ft = 2 Tot. Jo h. Substituting this in (203) we have ' The reduction to the meridian, then, is effected " by add- ing to the mean of the two altitudes two corrections; Ist, the quantity (| t)'. Jo h, which is nothing more than the common reduction to the meridian (122) computed with the half-elapsed time as the hour-angle ; 2d, the square of one fourth the difference of the altitudes divided by the first correction." Several pairs of altitudes can be thus com- bined, and the mean of the meridian altitudes taken, from which the latitude can be obtained as from an observed me- ridian altitude. ■ LATITUDE BY TWO ALTITUDES. 253 251. The restriction of the method corresponds with that of circum-meridian altitudes (Art. 150).* Quite accurate results can be obtained with hour-angles limited to 5™ when the altitude is 80°, to 25" when the altitude is only 10°.- K the interval t, however, exceed 10™, ^Ji should be computed to two or three places of decimals, as it is given in Table XXXn. (Bowd.) only to the nearest O'.l. , The accuracy of the method depends mainly upon the ac- curacy of the 2d correction, and therefore upon the preci- sion with which the difiFerence of altitudes has been obtained. The altitudes, then, should be obsei-ved with great care- Errors of the tabulated dip and refraction, and a constant error of the instrument will affect both altitudes nearly alike. * Note to Art. 150 (omitted in its proper place). From (119) we have with more exactness, cos T. cos d 2 sin' \ t Ah-- or putting sin(i— d)' sinl'-' 2 Bin' it , , cos Z cos (f and '• "*- sinl" "^^ ^- aii{Z-dy jjh = Am and A, = A +A m. Delambre's formula, obtained by developing in series the preceding equa- tion, (118) is Aj = A +A m —B n, in which 2 sin* it „ Aii IT j\ n — — ■ ,; , £ = A'taii{Z—d) sm 1 Table V., of Chauvenet's Astronomy, contains m and n, and Table VL contains log m acd log n for different values of t from to 30". Table VIL A gtves the limiting hour-angle at which the error resulting from neglecting the 2d reduction, £n, amounts to 1'. It varies from in the zenith to 36" in latitude 40°, or to 67" m latitudes 0° and 80°, for an altitude of 10°. Table XXXIL (Bowd.) gives J„A only to the nearest O'.l ; if, then, it is taken from this table, dji. ^ may be in error 1", if <> 4". If, however, J,A is computed to the nearest O'.OOl, the error of usmg AJ>~ ^, instead of 4 n>, ?rai not exceed 1", unless <> 20" and A > 60°, 254 NAVIGATION. If the altitudes are equal, this second correction becomes 0. The most favorable condition is, therefore, that of equal alti- tudes observed on each side of the meridian. At sea, the method is especially useful for altitudes of the sun observed with a clear, distinct horizon. A long interval between the observations is to be avoided on account of the uncertainty of the reduction of one of the altitudes for the run of the ship. 252. The hour-angle of either altitude may also be ob- tained approximately ; for we have from (204), in minutes, and (206) Example. 1865, March 14, near noon, in lat. 45° 30' S., long. 1 20° 10' E., by account, two altitudes were observed for latitude,- T. by Chro. 4'' 15™ 20^ ; Obs'd alt. 46° 45' 30', (North) " " « 4 26 16 " " " 46 54 40; Index cor. of sextant +5' 20"; height of eye 18 feet. The sun's declination at noon —2° 31' 57', H. ch. -|-59'. By preliminary computation AJi = 2".02. log A^h = 0.306. « = 10°" 56- A- A" = — 9' 10" it= 5 28 = 5"°.4'7 i(h—h')=— U1 (i if = 29.9 Afi = 2'.02 2 log i (h-h'] 4.2T4 log (1st cor.) 1.781 log (2d cor.) 2.498 logi(A-A') 2.137 « ar. CO. log i t 9.262 ar. CO. log JjA 9.694 logy. 1.098 T,= - 12°'.4 T =- 17 .9 2" = - 6 .9 i(A+A') = 46»60' 5" 1st cor. 29.9x2' .02= + 1 o., 13'?' 2d cor. — -— 60.4 = +5 11 46 66 16 In. cor. + 6 20 S. diam. + 16 7 Dip — 4 U Bef. and par. —48 K = 47 12 44 ^0 = 42 47 1« S. (12» before 0'') — ^ - I sin M' ^ TV cot < sin (Jf ' — d') cot P* = ^Tv cos Jf (212) • Cob a = coa b coa c+ain 6 sin c cos ^, (4) sin A cot £ = Bin e cot b — cos c cos A (10) LATITUDE BY TWO ALTITUDES. 261 We may take -3f ' in the same quadrant as t and give to it the sign of d. —. — ^p is then positive, and S will be in the 1st or 2d quadrant as the numerical value of -3f — 180°, or is negative. If the proper value of Q ', and therefore g', has not been LATrrUDE BY TWO ALTITUDES. 265 previously determined, we shall have two values of T\ but may ordinarily take that which agrees best with its known approximate value. 261. The preceding formulas employ the angles at M', and the triangle P M' Z. We may also use the angles at M and the triangle P M Z ; and, as M and M' are similarly situated with regard to the triangles, except that the angles at each are estimated in opposite directions, we shall obtain, by interchanging accented and unaccented letters in the prer ceding formulas, a set similar in form, but with this diflfer- ence of interpretation, that t is posi- tive in the opposite direction of the diurnal rotation, and q is less than 180° east of the meridian and great- er than 180°, or negative, west of the meridian. This difference is shown in Fig. 49, in which the primitive position of the triangles is east of the meridian, instead of west as in Fig. 47. We have, then, tan Jf^ tan 6S sec t. cos -B = cot P — sin i' cos (J/" — d) sin if ' cat t sin {M — i) em cos M. ' cos « sin (s — A*) i« = |/f cos A sin £ tan i\r= cot A cos y, ) sin A sin {N-\- d) > >l Bin i = ■ cot7'=- cos If J cot g cos (jr+ d) (212') (213') (Z14') (215') (216') 266 NAVIG-ATION. sin T-. T_ _ ain q cos h,^ {'iVl'S cos L ' * ' y = P— $, when M bears to tlfe left of M', y = P + §, when M bears to the right of M', * 262'. Either set of formulas may be used ; but, in general^ the latitude can be best found from the altitnde, which is nearest the meridian; the hour wangle, from the altitude which is nearest the prime vertical. The distinction made with regard to M and M' (Art. 254)> is important- only so far as it may aid in determining the hour-angles and selecting the proper value of q or q'. So that it is su:Mcient practically to find t numerically less than VH^ without regai'd to its sign. 263. The most favorable condition is, as stated in Article (243), when the difference of azimuths is 90°. But -altitudes near the meridian will give a good determination of the lati- tude, and altitudes near the prime vertical, a good deter- mination of the hour-4ngle8, when the difference of azimuths is stAall, or taear 1 80° ; especially if the altitudes have been carefully observed, and their difference is nearly exact. 264. If we put in (212' &c.) M--A, JS = C, P=90°-F, Q=Z, qz=90°-G, N=I, -we sh^U have tan A=-— tan d' sec t, cos'C : cbt ii^=- sin d' cos (J. + TER X. ; AZIMUTH OF A TEREESTRHX OBJECT. 271. In conducting a trigonometric garvey, it is necessary to find the azimuth, or true hearing, of one or more of its lines, or of one station from another. Thence, hy means of the measured horizontal angles, the azimuths of other lines or stations can be found 5 and, still further, a meridian line can be marked out ppon the ground, or drawn upon the chart. For example, suppose at a station, A, the angles reckonejl to the right are BXaC, 48° 15' 35"; C to i>, 73° 37' 16'; 1> to E, 59° 45' 20"; and that the azimuth of 2> is N. 35° 16' 15' E.; the azimuths of the several lines are A B, N. 86° 36' 36' W. A 2>, N. 35° 16' 15' E. A C, N. 38 21 1 W. A E, N. 95 1 35 E. If upon the chart a line be drawn, making with ^ ^ aa angle of 86° 36' 36' to the rigM, or with ^ 2> an angle of 35° 16' 15' to the left, it will be a meridian line. Or, if a theodolite or compass be placed at A in the field, and its line of sight, through the telescope or si^t-vanes, be directed to Z>, and the readings noted ; and then the line of sight be revolved to the left until the readings differ 35° 16' 16' from those noted, it will be directed north. If a stake or mark be placed in that direction, i^will be a meru dian mark north from A. 276 NAVIGATION. 272. If the azimuth of a terrestrial object is known, it may be conveniently used in finding the magnetic declina- tion, or variation of the •compass. For, let the bearing of the object be observed with the compass, — the difference of this magnetic bearing and the true bearing is the magnetic declination, or variation, required. It is east if the true bearing is to the right of the magnetic bearing ; but west if the true bearing is to the left of the magnetic bearing.* 2^3. Peoblbm 62. To find the azimuth, or true hearing, of a terrestrial ol^ect. Solution. Let Z (Fig. 50) be the zenith, or place, of the observer ; O, the terrestrial object ; M, the apparent place of the sun, or some other celestial body ; Z = N Z O, the azimuth of O ; s = N Z M, the azimuth of M ; ^ = Z— s = MZO, the azimuth an- gle between the two objects, or the diflference of azimuth of M and O. The problem requires that z and ^ be found ; then we have Z=s + ^. Or, numerically, * Thia haa reference to the two readings. The actual direction of the object is the same ; but the true and magnetic meridians, from which the angles are estimated, are different When the magnetic declination is east, the magnetic meridian is to the right of the true meridian ; when the mag- netic declination is west, the magnetic meridian is to the left of the true meridian. It is sometimes necessary to distinguish between the m^netic bearing and the compass bearing. The latter is affected by the errors of the instrument employed and by local disturbances ; the former is fi:ee from them. AZIMUTH OF A TERRESTRIAL OBJECT. 277 Z=. 3 + f , when the azimuth of' the terrestrial object is greater than that of the celestial, Z=z — ^, when it is less. The sign of ^ should be noted in the observations. 274. 2 = N Z M, the azimuth of the celestial body, may be found from an observed altitude (Prob. 40), or from the local time (Prob. 38). In- the first case, the most favorable position is on or nearest the prime vertical ; for then the azimuth changes most slowly with the altitude. In thi latter, positions near the meridian may also be successfull; nsed. 275. ^ = M Z O, the azimuth angle between the two ob- jects, may be found in one of the following -ways : — Is* Method. (By direct measurement.) M Z O, being a horizontal angle, may be measured direct- ly by a theodolite or a compass, by directing the line of sight of the instrument first to one of the objects and reading the - horizontal circle, then to the other and reading again. The difference of the two readings is the angle required. Or, the telescope or sight-vanes of a plane table may be directed successively to the objects, and lines drawn upon the paper along the edge of the ruler in its two positions, and the an- gle which they form measured by a protracter. At the instant when the observation is made of the celes- tial object, either its altitude should be measured, or the time noted, so as to find its azimuth simultaneously. The instrument should be carefully adjusted and levelled. With the compass or plane table, it is not well to observe objects whose altitudes are greater than- 15°. A theodolite can be used with greater precision than the other instruments ; but the greater the altitude of the object, the more carefully must the cross-threads be adjusted to the axis of collimation, and the telescope be directed to the object. 278 NAVIGATION. The error of cottimation is eliminated by making two ob- servations with the telescope reversed either in is Vs, or by rotation on its axis. Low altitudes are generally best. 2T6. If the sun is uspd, each limb may be observed alter- nately ; or a separate set of observations may be made for each. To find the azimuth reduction for semi-diameter, when but one limb is observed ; Let A— 90°— Zs (Fig. 51), the altitude of the sun, « = S s, its semi-diameter, «'= S Z s, the reduction of the azimuth for the semi-diameter. We have . _ „ sin S s sm S Z s = -. — 5— » sin Z s or, since s and «' are small, "8- *^ «'= « sec h, (229) which is the reduction required. The sign, with which it is to be applied, depends upon the limb observed. 277. If the observations are made at night, and the ter- restrial object is invisible, a temporary station in a conve- nient position may be used, and its azimuth found. The horizontal angle between this and the terrestrial object may be measured by daylight, and added to, or subtra'cted fi-om. this azimuth. A board, with a vertical slit and a light behind it, forms a convenient mark for night observations. The place of the theodolite should be marked, that the instrument may be replaced in the same position. But in doing this, and selecting the temporary station, it should be kept in mind that a change of tiie position of the instru- AZIMUTH OF A TEKRESTEIAL OBJECT. 279 ment of j^Vj of the distance of the object may change the azimuth 1' ; or of to oVoo of t^ie distance may change the azimuth more than 1'. 278. 2d Method. Finding the difference of azimuths of a celestial and a terrestrial object by a sextant • sometimes called an " astronomical bearing.''^ Measure with a sextant the angular distance M O (Fig. 52) of the two objects, and either note the time by a watch regulated to local time, or measure simultaneously the alti- tude of the celestial object. Measure, also, the altitude of the terrestrial object (if it is not in the horizon), either with a theodolite which is furnished with a vertical circle, or with a sextant above the water-line at the base of the ob- ject, when there is one. Correct the readings of the instru- ments for index errors, and when only one limb of the sun is observed, for semidiameter.* Observed altitudes of either object above the water-line are also to be corrected for the dip by (53) or Tab. XIII. (Bowd.), if the horizon is free ; but by (55) or Tab. XVI. (Bowd.), if the horizon is obstructed. The altitude of the celestial object, when not observed simultaneously, may be interpolated from altitudes before and after, by means of the noted tunes. (Bowd, p. 246.) Or the true altitude may be computed for the local time (Prob. 38 or 39), and the reiractipn added and the parallax subtracted to obtain the apparent altitude. Kg. * It is best in measuring the distance of the sun from the" terrestrial obi ject to use each limb alternately. 280 NAVIGATION. Let h' = 90°— Z O (Fig. 50), the apparent altitude of 0, H = 90°— Z M, the appcvren^ altitude of M. 2) = M O, the corrected distance.^ We have then in the triangle M Z O, the three sides fi-om which ^ = M Z O, may be found by one of the follow- ing formulas : — 1. By Sph. Trig. (164) we have . ,, /sini(i) + 5'' — ^'IsinifD— ir' + V) ■* r cos S^ cos K or, letting d = S'—h', ' sin i (2) + d) sin i (B—dy 8m^< = j/- (230) cos 3' cos A' 2. By Sph. Trig. (165), ' cos i (£■' + h' + D)c6si(S'+h'—J)) COS H = /-^ cos S' cos A' or, putting , > / cos « cos (« — B) > (231) COS i ^ = i/ ffr^ — rj-^ ' y cos S^ cos A J (230) is preferable when ^ < 90° ; (231), when ^ > 90°. 279. If O is in the true horizon, or its measured altitude above the water line equals the dip, A = 0, and the right triangle M m O gives cos ^ = cos m O = cos D sec S' ; (232) or more accurately when ^ is small (Sph. Trig., 105), tan 1 < = 4/ (tan i {B+H') tan i {B-H') ). (233) If the terrestrial object is in the water-line, h' is negative, and equals the dip. * The true altitude of M ia uBe4 in finding z, its azimuth. AZIMUTH OF A TERRESTRIAL OBJECT. 281 280. If both objects are in the horizon, or H and h are equal and very small, we have simply ^ = -O. . (234) In general the result is more reliable the smaller the in- clination of M O to the horizon. If M O is perpendicular to the horizon, the problem is indeterminate by this method. 281. If the terrestrial object presents a vertical line to which the sun's disk is made tangent, the reduction of the observed distance for semidiameter is s' = s sin M O Z (235) and not s, the semidiameter itself. This follows from the sun's diameter through the point of contact, O, being per- pendicular to. the vertical circle Z O and not in the direc- tion of the distance O M. As the altitude of the terrestrial object is always very small, we may find M O Z by the formula •\i r\ rw sin V cos M O Z = -: — ^ , sm Zr ' jy being the unreduced distance. 282. When precision is requisite, the axis of the sextant with which the angular distance is measured must be placed at the station Z ; and if the object seen direct is sufficiently near, the parallactic correction must be added to the sex- tant reading. If J represent the distance of the object, d, the distance of the axis from the line of sight or axis of the telescope, this correction is p = -T cosec 1' = 206265' — . (236) It is 1', when J = 3437.75 d. 283. If the distance of the terrestrial object and the dif- ference of level above or below the level of the instrument 282 NAVIGATION. are known, we may find its angle of elevation, nearly, by the formula tan K = — , ^ being the distance of the object, and E, its elevation above the horizontal plane of the instru- ment. If the object is below that plane, E and K will have the negative sign. Note. — The horizontal angle between two terrestrial objects may also be found by measuring their angular distance with a sextant, and employing the same formulas (230 to 234) as for a celestial and terrestrial object ; H' and K represendng their apparent angles of elevation. Each of these may be found by direct measurement, or from the known distance and the elevation, or depression, from the horizontal plane of the observer. If the two objects are oq the same levd as the observer, we have simply as in (234) ExiMPLB. 1865, May 16, 5| A. M. ifflat 38° 15' N., long. 76° 16' W.; the angular distance of the smi's centre from the top of" a light-house measured by a sextant (Q to the right of L. H.), 75° 16' 25", index cor. —1' 15"; altitude of O above the sea-horizon observed at the same time, 10° 18' 20', index cor. -f-2' 10'; observed altitude of the top of light-house above the water-line, distant 7300 feet, 1° 15' 20', index cor., + 2' 10'; height of eye, 20 feet; required the true bearing of the light-house. From the data we find \ 0'sap. alt. fl^' = 10°31'67'; ap. alt. of L. H. A'= 1° r 34" O'strue" iSr =10 27 7; ang. dist, J)=.1h 15 10 e's dec -I- 19 9 30. AZIMUTH OF A TERKESTKIAL OBJECT. 283 Computation (100) ani (230). B= 10° ar 1" L sec 0.00726 S' — 10° 31' 57" 1. sec 0.00738 L = 38 15 L sec 0.10496 A' = 1 7 34 L sec 0.00008 P- 10 50 30 d = 9 24 23 %i — 119 32 37 Z> = 75 16 10 « = 59 46 18 L COS 9.70196 i{D+d) = i2 19 46 L sin 9.82827 p-t = 11 4 12 L COS 9.99184 HD—d) = 32 56 24 L sin 9.73521 19.80602 19.57094 iZ = 36° 53'.0 L COS 9.90301 if=37°36'.2 f=75 12.4 1. sin 9.78547 O's aamnih Z=}S.iZ 46 .OE. Troe beadng of Ii. House (2-0 = N. 1 S6.4W. D. VAN NOSTRAND S PUBUCATIONS. A Treatise on Ordnance and Naval Gunnery. Compiled and arranged as a Text-Book for the U. S. Naval Acade- my, by Lieutenant Edward Simpson, U. S. N. Third edi- tion, revised and enlarged, l voL, 8vo, plates and cuts, clo. *^As the compiler has cKaigeof the Instraction in l^av^ Ganneryat the Naval Acad- emy, his work, in the compilation of which he has consnlted a large nmnber of eminent aathorities, is probably well suited for the purpose designed by it — namely, the circulation of information which many ofEicers, owing to constant service afloat, may not have been able to collect In simple and plain language it gives instruction as to cannon, gun-car- riages, gmipowder, projectiles, fuzes, locks, and primers ; the theory of pointing guns, rifleS| the practice of gunnery, and a great variety of other similar matters, interesting to fight- ing meu on sea and land.'" — Washington I>a4ly Globe^ G unnery Catechism, As Applied to the Service of Naval Ordnaiice. 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