VlBRARf OF THE UNIVERSITY OF C^LiFOR^h fa< -< 3P" 22 c^> /s. NAVIGATION AND NAUTICAL ASTRONOMY BY PROF. J. H. C. COFFIN Lai e Professor of Astronomy, Navigation, and Surveying at the U.S. Naval Academy REVISED BY COMMANDER CHARLES BELKNAP U. S. NAVY PREPARED FOR THE USE OF THE U.S. NAVAL ACADEMY Seventh Edition, Revised NEW YORK D. VAN NOSTRAND COMPANY 1898 Copyright, 1898, By D. Van Nostrand Company. Typography by C. J. Peters ] = p x + p 2 +p 3 , etc. ; or, as in the more simple case of a single course, The whole difference of latitude is equal to the sum of the partial differences of latitude ; The whole departure is equal to the sum of the partial de- partures. tt u a (t a a 6 NAVIGATION. This applies to all cases, if we use the word sum in its general or algebraic sense. If we represent by L n the sum of the northern diffs. of latitude* L s " southern " " P w " western departures, JP e " eastern " we have as the arithmetical formulas, [ I ] = L n ^^ L s of the same name as the greater, which accord with the usual rules. (Bowd., Arts. 115, 155.) The Traverse Form (pp. 10 and 11) facilitates the compu- tation. The course, (7, and distance [d~\, corresponding to [£] and [p], 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 arrived at. In a very irregular traverse it is difficult to determine the * C and [d] are not accurately found, because [p], the sum of the partial departures of the traverse, is not the same as p, the departure of the loxodromic curve connecting 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 traverse sailing de + mn; whereas, if the ship sail directly from C to A, the departure will be op, which is greater or less than de + m n, accord- ing 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, however, such dif- ference is immaterial. PARALLEL SAILING. precise parallel ; but, except near the pole, and for a distance exceeding an ordinary day's run, the middle latitude suffices. It is easy, however, to separate a traverse into two or more portions, and compute for each separately. PARALLEL 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, since the earth's surface must now be regarded, not as plane, but spherical. Problem 4. To find the relations between — L, the latitude of a parallel, p, the departure of two meridians on that parallel, and D, the corresponding difference of longitude. Solution. In Fig. 3, let P A A', PCC be two meridians. A C = p, their departure on the paral- lel AC, whose latitude is AOA' = O A B = Z, and whose radius is BA = n A' C = D, the measure of A P C, the difference of longitude of the same meridians, on the equator A' C, whose radius is A' = A = R. A C, A' C are similar arcs of two circles, and are therefore to each other as the radii of those circles ; that is, A C : A' C = B A : A', or p : D = r : B. In the right triangle B A, B A = A x cos O A B, or r = R cos L ; (4) 8 NAVIGATION. that is, the radius of a parallel of latitude is equal to the radius of the equator multiplied by the cosine of the latitude. Substituting (4) in the preceding proportion, we obtain p : D = cos Z : 1, or p = D cos Z, D =p sec Z, (5) which express the relations required. (Bowd., Art. 118.) These relations may be graphically represented by a right plane triangle (Fig. 4), of which D is the hypothenuse, Z, one of the angles, p, the side adjacent that angle. The Traverse Table, or Table of Bight Triangles, may therefore be used for the computation. MIDDLE LATITUDE SAILING. 10. Problem 5. Given the course and distance and the latitude left, to find the difference of longitude. Solution. By Plane Sailing, . I = d cos C, p = d sin C\ (3) by Arts. 2 and 6, Z' = Z + l, Z = t(Z' + Z) = Z + $l; (6) and by equation (5), D=p sec Z , (7) or Z = d sin C sec Z . (8) Equations (3), (6), and (7) or (8) afford the solution required. The assumption of Z = | (Z' -f- Z), or the middle latitude, suffices for the ordinary distance of a day's run ; but for larger distances, and where precision is required, we should use " Mercator's Sailing " (Art. 14). v\'i>u MIDDLE LATITUDE SAILING. 11. Strictly, the middle latitude should be used only when both latitudes, L and J/ 9 are of the same name, as is evident from Fig. 1. If these latitudes are of different names, and the distance is small, \ (Z + _Z/), numerically, may be used ; or we may even take p = D, 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. (Art. 18.) 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- erse Sailing, and then D by regarding p as on some parallel midway between the extremes of the traverse. (Art. 8.) (Bowd., Art. 155.) 12. The relations of the quantities involved in Middle Latitude Sailing, namely, O y d, p, I) X , and D, are represented graphically by combining the two triangles of Plane Sailing and Parallel Sailing, as in Fig. 5, in which C = A C B, d = CA, p =BA, Z = BAE i)=AE, By these two right triangles, all the Fig. 5. common cases classed under Middle Lat- itude Sailing (Bowd., Art. 121) may be solved, if we add the formulas, Z' = Z + l A' = \ + D. 10 NAVIGATION. Example in Middle Latitude Sailing 1. Required the course and distance from San Francisco to Yokohama. San Francisco, 37° 48' N. Yokohama, 35° 26' N. 122° 28' W. (Table 49,) 139° 39' E. ( Bowd. ( 1= 2° 22' = 142' X = 36° 27' 1. cos logD ar. co. log I G — S. 88° 17' W. 1. tan C 11.52203 d = 4726' D = 97° 53' = 5873' 9.90546 3.76886 7.84771 log I 2.15229 1. sec C 11.52222 log d 3.67451 Examples in Traverse Sailing. 13. A ship from the position given at the head of each of the following traverse forms sails the courses and dis- tances 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. Courses. DlST. N. s. E. W. / / / / / N. N. E. i E. 50 44.1 23.6 S.|W. 46.2 45.7 6.7 S. by E. h E. 16.5 15.8 4.8 N. E. 38 26.9 26.9 S. S. W. i w. 41.8 37.8 17.9 192.5 71.0 99.3 55.3 24.6 S. E. i E. 41.5 28.3 30.7 38 = D. August 9, noon — Lat. by Acct., 35° 7' N. Long, by Acct., 18° 0' W. MERCATOR'S SAILING. 11 2. Apr. 23, noon — Lat. by Obs., 41° 31' N. Long, by Chro., 178° 25' W. Courses. DlST. N. S. E. W. S. 21° W. 29 27.1 10.4 S. 37° W. 20.6 16.5 12.4 S. 56° W. 72 40.3 59.7 S. 71° W. 16.4 5.3 15.5 S. 82° W. 23.7 3.3 23.5 N. 88° W. 45 1.6 45 1.6 92.5 90.9 166.5 D = 219.8 Apr. 24, 7 a.m., Lat. by Acct., 40° OO'.l N. Long, by Acct., 177° 55'.2 E. In this example the courses are expressed in degrees, which is the preferable method. MERCATOR'S SAILING. 14. Middle Latitude Sailing suffices for the common pur- poses of navigation ; but a more rigorous solution of problems relating to the loxodromic curve is needed. These solutions come under " Mercator's Sailing." Problem 6. A ship sails from the equator on a given course, C, till she arrives in a given latitude, L ; to find the difference of longitude, D. Solution. Let the sphere (Fig. 6) be projected upon the plane of the equator stereographically. The primitive circle ABC. . . . M is the equator. P, its centre, is the pole (the eye or projecting point being at the other pole).* The radii, P A, P B, P C, etc., are meridians making the * Principles of stereographic projection. 12 NAVIGATION. 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° — L), the tan- gent of J the polar distance of the point on the surface which m rep- resents, the radius of the sphere being 1.* This curve in projection makes the same angle with each merid- ian as the loxodromic curve with F1 *- 6 - each meridian on the surface.* A M is the whole difference of Longitude D. If we suppose this divided into an indefinite number of equal parts, A B, B C, C D, etc., each indefinitely small, and the meridians P A, P B, P C, etc., drawn, the intercepted small arcs of the curve Abe. . . . m may be regarded as straight lines, making the angles ~P A b, ¥ b c, I* c d, etc., each equal to the course (7; and consequently the triangles PAi, P b c, P c d, etc., similar. We have then PA:P£ = P£:Pc=Pc:Ptf, etc., or the geometrical progression, T , ,, P A : P b : P c : . . . . P m. If then D = the whole difference of longitude, d = one of the equal parts of D, -= will be the number of parts, and a -z + 1 the number of meridians PA, P b. . . . P m, * Principles of stereographic projection. MERCATOR'S SAILING. 13 or the number of terms of the geometrical progression : and, employing the usual notation, the first term a = P A = 1, the last term I = P m = tan \ (90° — L), 71 — 1 == — > d the ratio r = 7—7 • PA To find this ratio, we have in the indefinitely small right triangle ABb, tan B A b = cot P A b = ^ > B A or cot C = PA- d Yb > whence PA -Yb = d cot 0; Yb = :PA - d cot C, and, since PA = 1, r = P^ : PA~ 1 — d cot a Then by the formula for a geometrical progression, 1= ar"- 1 , (Algebra, p. 240) we have tan I (90° - Z) = (l-d cot C)*. Taking the logarithm of each member, we have log tan J (90° - L) = - log (1 - d cot C). (9) But we have in the theory of logarithms (JVaperian) log (1 + n) = n -r — -f — — — -f etc and 2 8 4 (Common) log(l-f») = ww-| + |-j + etc. . . .1. (10) in which the modulus m = .434294482. 14 NAVIGATION. Hence, putting n= — d cot C, log (1 - d cot C) = m [- d cot C - J tf 2 cot 2 6 r — J 6? 3 cot 3 (?— etc ], and substituting in (9) and reducing, log tan \ (90° - Z) = - m X 2> [cot 6"+ J (7 cot 2 C + J tf 3 cot 3 6' + etc....]. (11) This equation is the more accurate the smaller d is taken, so that if we pass to the limit and take d = 0, it becomes per- fectly exact. The broken line Abe .... m then becomes a continuous curve, and our equation (11) becomes log tan I (90° - L) = - m X D cot 6 y ; whence J= _ logtan U 90°-J) tanC , } But in this equation D is expressed in the same unit as tan C, that is, in terms of radius. (Pl. Trig., Art. 11.) To reduce it to minutes we must multiply it by the radius in minutes, or / = 3437 / .74677. Substituting the value of m, we shall have (in minutes), 3437' 74677 D " - .434294482 ^ tan * ^ 90 ° " Z > ta " C To avoid the negative sign, we observe that 1 1 tan J (90° - Z) cot £ (90° - Z) tan J (90° + Z) ' or that - log tan J (90° - Z) = log tan J (90° + Z). Hence we have, by reducing, D = 7915 / .70447 long tan (45° + J X) tan (7. (13) Note. — Problem 6 may be more readily solved, and equation (13) obtained by aid of the Calculus. MEE CA TOR ' S SA I LING. 15 In Fig. 1, suppose c a to be an ele- ment of the loxodromic curve C A : cb will be the corresponding element of the meridian, and b a x sec L, the element of the equator; L being the latitude of the indefinitely small triangle cab. By Articles 5 and 10, using the no- tations of the Calculus, we have d.L = cosCdd dp = tanCdZ d D — sec L d p = tan C sec IdX, in which C is constant. By integrating the last equation between the limits L = and L = X, we shall have D = tan C I sec L d i, Jo the whole difference of longitude required in Problem 6. To effect the integration, put sin L = x, then by differentiating, dx and multiplying by sec i, di = sec L d L = sec i d X = dx cosL dx cos 2 £ 1 — sin 2 1/ dx , or 1— x 2 Resolving into partial fractions, we obtain dx , dx sec and Jo sec L d L = | [log (1 + x) - log (1 - x)] = log = .o gV /£ Whence we have sin L = log tan (45° + h L) Pl. Trig. (154). D = log tan (45° -J- i L) tan C. 16 NAVIGATION. But in this the logarithm is Naperian, and D is expressed in terms of the radius of the sphere. To reduce to common logarithms, we divide by m = .434294482, and to minutes by multiplying by r' = 3437'. 74677, and obtain T) = 7915'. 70447 log tan (45° + $ L) tan C, as in (13). 15. Formula (13) is based upon the assumption that the earth is a sphere. Allowing for the meridional eccentricity of the earth, which according to Bessel is \— = 0.003342773 = c, 299.1528 ' the formula used for computing the table of meridional parts 1S M = 7915 , .7045 log tan (45° + \ L) — a {f sin X+ J e 4 sin 2 L) (14) in which e = V2 c - c 2 = 0.0816968, and ae 2 = 22 r .9448, \ae* = 0'. 051047. The values of M computed by (14) for each minute of L from 0° to 86° form the Table of Meridional Parts or Aug- mented Latitudes. (Bowd., Table 3.) In practice, then, we have only to take the value of M cor- responding to X, and D is then found by the formula, D = M tan C. (15) M has the same name, or sign, as L. 16. Problem 7. A ship sails from a latitude, L, to another latitude, U , upon a given course, C; find the difference of longitude, D. Solution. Let M be the augmental latitude corresponding to X, M' " " " " L f . MERCATOR'S SAILING. 17 The difference of longitude from the point, A, where the track crosses the equator to the first position, whose latitude is L. will be D J = M tan C; and to the second position, whose latitude is X', D tt — M' tan 0\ and we shall have D = D u - D 4 = (M' - M) tan C\ (16) or, when M ' < M, B = 2> rf - 2> y/ = (Jf- IT) tan C; since the sign of D is determined by the course. If L and Z' are of different names, so also are M and M' , and we have numerically &=(M+M') tan £ 17. The difference, J!/' — J/, is called the meridional, or augmented, difference of latitude. Representing this by m, we have D = m tan C. The relation between these quantities is represented by a plane right triangle (Fig. 7), in which C is one of the angles, m = C E, the side adjacent, i)=EF, the side opposite. The triangle of " Plane Sailing " has the same angle C, with I = C B, the adjacent side, and p = B A, the opposite side. (17) Pig. 7. 18 NAVIGATION. Fig. 7 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., Art. 128.) The relations between the several parts involved are I = d cos C, L' = L+ I, p — d sin C, m = M' — M, D = m tan C, a' = A + D ; (18) and since p = I tan C, I : m = p : D. 18. Problem 8. Given the latitudes and longitudes of two places, find the course, distance, and departure. (Bowd., Art. 128, Case I.) Solution. L and U being given, we take from Table 3 M and M '. We have I = U - Z, m = M' - M, D = \' - X ; by Mercator's Sailing, tan C = — ; and by Plane Sailing, d= I sec C, p = 1 tan C; ?, m, and C are north or south according as U is north or south of JL. D, jo, and C are east or west, according as \ r is east or west of A. If the two places are on opposite sides of the equator, we have numerically l=Z'+Z, m = M' + M. Examples. 1. Kequired the course and distance from Cape Frio to Lizard Point, England. MERCATOR'S SAILING. 19 Cape Frio, L = 23° Or S. A. = 42° 00' W. M = 1410.7 S. Lizard Point, 2/ = 49° 58' N. X'= 5°12'W. Jf' = 3453.8 N. log- 2) = 3.34400 J = 72° 59' N. D = 36°48 / E. m = 4864.5 N. log m = 3.68704 C = N. 24° 24' 48" E. 1. sec O = 0.04068 1. tan C = 9.65696 logZ = 3.64137 d = 4809'. log d = 3.6S205 2. Required the course aud distance from San Francisco to Yokohama. San Francisco, 37° 48' N. 122° 28' W. M = 2439 Yokohama, 35° 26' N. 139° 39' E. M' = 2262.8 log D = 3.76886 I = 2° 22' S. B = 97° 53' E. = 5873' m = 176.2 co. log m = 7.75399 C = N. 91° 43' W. 1. sec = 11.52308 1. tan C = 11.52285 or, S. 88°17 / W. log I = 2.15229 d = 4735.5 log d = 3.67537 19. The loxodromic curve on the surface of the earth and its stereographic projection (Fig. 6) present a peculiarity worthy of notice. Excepting a meridian and parallel of lati- tude, a line which makes the same angle with all the merid- ians 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 as- signable quantity. It is usual to say that such a curve meets the pole after an infinite number of revolutions. Still, how- ever, it is limited in length. For we have for the length of any portion, by Plane Sailing, d = (L' — X) sec C. If L = and If = 90° = -, 2' the whole spiral from the equator to the pole will be, with radius = 1, 20 NAVIGATION. d = \ sec <7. If i = - 90° = - * and L' = 90° = £, we have, as the entire length from pole to pole, d—TT sec C. If also (7=0, or the loxodromic curve is a meridian, d — *, a semicircumference, as it should be. So also the length of the projected spiral Abe... (Fig. 6) from A to m can readily be shown to be (calling this length 8) ; 8 = Jf m sec (7= [1 — tan (45° — j-X)] sec (7, /-r. m -.,--. x * 2 tan 4 L sec (7 or (Pi. Tbio., 151), 8- 1+ * tanii - ; and its length from the equator to the pole, taking X = 90°, 8 = sec C. A MERCATOR'S CHART. 20. On a Mercator's chart, the equator and parallels of latitude are represented by parallel straight lines ; and the meridians also by parallel straight lines at right angles with the equator. Two parallels of latitude, usually those which bound the chart, are divided into equal parts, commencing at some meridian and using some convenient scale to represent degrees, and subdivided to 10', 2', V, or some other convenient part of a degree, according to the scale employed. Two meridians, usually the extremes, are also divided into degrees, and subdivided like the parallels of latitude, but by a scale increasing constantly with the latitude : so that any degree of latitude on such meridian, instead of being equal to a degree of the equator, is the augmented degree, or augmented A MERCATOR'S CHART. 21 difference of 1° of latitude, derived from a table of " meridi- onal parts." (Bowd., Table 3.) The meridian is graduated most conveniently by laying off from the equator the aug- mented latitudes ; or from some parallel, the augmented differ- ence of latitude for each degree and part of a degree, — using the same scale of equal parts as for the equator. 21. As on other maps and charts, parallels of latitude and meridians are drawn at convenient intervals ; places, shore lines of continents and islands, harbors and rivers, etc., are plotted, each point in its proper position ; and such configu- rations of the land represented as the purpose of the map requires. 22. To plot on a chart a point whose latitude and longitude are given. By means of the scales at the sides, draw a paral- lel of latitude in the latitude, and by means of the scales at the top or bottom, a meridian in the longitude of the point ; or so much of each as suffices to find their intersection. 23. In nautical charts the soundings in shoal water are put down, and even the character of the bottom ; and on those of a large scale, also, the contour lines of the bottom, or lines of equal depth. The variation of the compass at convenient intervals, and lines of equal variation, are valuable additions. 24. The meridians on this chart being parallel, arcs of par- allels of latitude are represented as equal to the corresponding arcs of the equator : thus each is expanded in the proportion of the secant of its latitude to 1 ; as is evident from the formula J) = p sec L. It can be shown that very small portions of the meridians 22 NAVIGATION. are expanded in the same proportion ; as for example, a de- gree 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 60£° " " 61', nearly. A small circle on the surface of the earth of 1° diameter at the equator is then represented by a circle, whose diameter is 1°; in lat. 30° nearly by a circle, whose diameter is 1° X sec 30°, " 60° " " " " 1° X sec 60°, " Z " " « " 1° X sec X; but not exactly by a circle, since the meridians are augmented 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. If we take any two points, C F, on this chart, and join them by a straight line, and form a right triangle by a merid- ian through one, and a parallel of latitude through the other, we shall have the tri- angle of Mercator's sailing (Fig. 8) : for the intercepted portion of the meridian, C E, is the augmented difference of latitude ; and of the parallel of latitude, E F, is the difference of longitude. Hence the angle E C F is the course. (Art. 17.) Moreover, Fig. 8. A MERCATOR'S CHART. 23 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 position on the chart, d must be in the line C F, other- wise when we construct the triangle of Mercator's Sailing we shall have an angle at C different from E C F, 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. 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 means 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. When such diagrams are constructed with reference to the true meridian, the course obtained is the true course, and not the magnetic 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 C E). Or we may con- struct the proper distance, CA, by constructing 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 24 NAVIGATION. distance between two points is the arc of a great circle which joins them. To find this belongs to great-circle sailing. 27. In Polyconic Projection each parallel of latitude is developed upon its own cone, the vertex of which is on the axis at its intersection with the tangent to the meridian at the parallel. The advantage of a chart so constructed is that those portions lying near the central meridian will be but little distorted. The method of construction, together with 'tables for the Polyconic as well as Mercator's Projection, are given in Pro- jection Tables for the use of the United States Navy. (Bur. Navgn.). great-circle sailing. 28. The rhumb-line, or spiral curve, which cuts all the meridians at the same angle, was used formerly by navi- gators in passing from point to point on account of the simplicity 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 navi- gators of the present day are substituting the latter wherever practicable. On the Mercator chart, however, the arc of a great circle joining two points, not on the equator or on the same me- ridian, will not be projected into a straight 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 appears on it as a circuitous one ; and this is, doubtless, one main reason why merely practical men have made so little use of the great circle. Many of those unacquainted GREAT-CIRCLE SAILING. 25 with the mathematical principles of the subject are unable to comprehend how the apparently circuitous path on then- chart should actually be the line of shortest distance. 29. Problem 9. 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 spher- ical triangle, formed by the arcs joining these points with each other and with the pole. If from P we draw P C perpendicular to A B, the point C is nearer the pole than any other point of A B ; that is, it is the point of maximum latitude. This point of greatest latitude is called the vertex of the great circle. 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 Case I. of Sph. Trig., and then solving one of the right tri- angles A P O or B P C . But in practice the following method is preferable. Let Z x = (90° — P A), and \ x be the latitude and longitude of A, the point left. X 2 = (90° — P B), and X 2 , be the same of B, the point arrived at. Fig. 9. 26 NAVIGATION. L v = (90° — P C ), and \ v be the same of the vertex, C . A = (A x — A 2 ), is the difference of longitudes of A and B. A B = d, is the distance between A and B. Draw K per- pendicular to P A, dividing it into Y D = and A D = 90° - (A + +). Then in the triangles A B D, and B D P, by Napier's Rules (Sph. Trig., Art. 46) we have cos A. = tan tan Z 2 or, tan = cos A cot Z 2 (19) sin as cot A tan K (20) cos (L x + ) = cot A tan K or, cot A = cos (X x -f- ) cot A cosec . (21) A is the course from A. In the right triangle P C A, we have cos L v — cos L x sin A (22) sin L x = cot A cot (A x — X v ) or, cot (A x — X c ) = sin X x tan ^1. (23) sin d sin ^4 = cos Z 2 sin A (check). (24) 30. To find any number of points, C, C", C", etc., C 1? C 2 , C 8 , etc., we may assume at pleasure the differences of longi- tude from the vertex C P C , C P C", C P C", etc. It is best to assume them at equal intervals of 5° or 10°. Let A' = C P C, U = (90° - P C), the lat. of C, X" = C P C", X" = (90° - P C"), « C", X'" = C P C", 2/" = (90° - P C"), " C" etc. etc. then the right triangles C P C, C P C", C P C", etc., give tan U = tan Z„ cos A', \ tan Z" == tan L v cos A", I (25) tan If" = tan Z r cos A'", etc.] GREAT-CIRCLE SAILING. 27 Or we may assume values of L', L" ', L f ", etc., and find the corresponding values of a', a", a'", etc., by the formulas cos A/ = tan U cot Z v , "I cos A" = tan L" cot L v , \ (26) cos X ffr = tan U" cot L v , etc. J from which we shall have two values of X for each value of L. Having thus found as many points as may be deemed sufficient, we may plot them upon the chart, and through them trace the required curve. 31. Problem 10. To find the great-circle distance and course between two given points. Solution. Let d be the distance between the two points A and B (Fig. 9). Then in the triangles BDP and ADB, by Napier's Rules, we have, sin X 2 = cos <£ cos K, (27) cos d = sin (L x -\- ) cos K, (28) cos d = sin (B 1 + <£) sin X 2 sec <£, (29) cot d = cos A tan (L x -\- ). (Check.) (30) d, reduced to minutes, will be the distance in geographic miles. The course from A is found by (21). The course from B may be found from the right triangle B &***> cos B = sin Z v sin (X, — A 2 ). (31) The vertex lies between A and B, unless either A or B is >90°. 32. Example. To find the great circle from San Fran- cisco to Yokohama. (Formulas 19, 21, 22, 23, 25, 29.) 28 NAVIGATION. San Francisco, Lat. L x — Yokohama, 37 48 N. Long. 122 28 W. 35 26 N. 139 39 E. 97 53 cos — 9.13722 cot — 9.14134 L 2 = 35 26 i cot 0.14870 sin 9.76324 L x = 37 48 ' = 169° 05' 21 " tan 9.28502 cosec 0.72290 sec - 0.00792 £i + =206° 53' 21 cos — 9.95031 sin- 9.65540 C = N. 56°52 / 40 /, W. cot 9.81455 d = 74 o 30'44" = 4470'.75 cos 9.42656 Lj = 37 48 cos 9.89771 sin 9.78739 C = 56 52 40 sin 9.92299 tan 0.18545 L v = 48 33 55 L - X v = 46 47 26 cos 9.82070 \ cot 9.97284 K = 169 15 26 W. Long from Vebte l.COS /. 1. TAN L. X. Latitude. Longitudes. 0.00000 0.05421 48 34 " N. 169 15 W. 169 15 W. (Vertex.) ± 5 9.99834 0.05255 48 27 30 164 15 174 15 ±10 9.99335 0.04756 48 08 159 15 179 15 W. ±15 9.98494 0.03915 47 35 154 15 175 45 E. ±20 9.97299 0.02720 46 48 149 15 170 45 ±25 9.95728 0.01149 45 45 30 144 15 165 45 ±30 9.93753 9.99174 44 27 30 139 15 160 45 ± 35 9.91336 9.96757 42 52 134 15 155 45 ±40 9.88425 9.93846 40 57 129 15 150 45 ±45 9.84949 9.90370 38 42 124 15 145 45 ±50 9.80807 9.86228 36 04 119 15 W. 140 45 E. Course N. 56° 52' 40" W. from San Francisco. Distance = 44 70| miles. Distance by Mercator's Sailing = 4735J miles. 33. To follow a great circle rigorously requires a contin- ual change of the course. As this is difficult, and indeed in many cases is practically impossible, on account of currents, GREAT-CIRCLE SAILING. 29 adverse winds, etc., 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 Tables and Bergen's Tables are used by English navigators. Charts are constructed by a gnomonic projection, on which great circles are represented by straight lines ; but by these, computation is necessary to find the course. 35. A great circle between two 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. 10) 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 30 NAVIGATION. than that of the rhumb-liue A B, or of any other line, as ADB, 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 rhumb-line. Between 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 sailing 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 nearing the place B better by the second than by the first, although on the chart it would appear to be very far off from the direct course. Fig. 10. 37. This may be strikingly illustrated 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 be- tween 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. GREAT-CIRCLE SAILING. 31 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 other tacky or twelve points from the rhumb-line course. 38. The routes between a number of prominent ports rec- ommended by Captain Maury are mainly great-circle routes, modified in some cases by his conclusions respecting the prevailing winds. 32 NAVIGATION. CHAPTER II. REFRACTION. — DIP OF THE HORIZON.— PARALLAX. — SEMIDIAMETERS. REFRACTION. 39. 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 per- pendicular to the surface which separates the two media ; if it passes from a denser to a lighter medium, it is bent from that perpendicular. Let p • M and N (Fig. 11) represent two me- dia 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 #, the normal line, or perpendicu- lar, to the separating surface at b. If a b is the incident ray, b c is the refracted ray ; P ba is the angle of incidence ; P b a r is the angle of refraction. If c b is the incident ray, b a is the refracted ray, and P b a! and Y ba are respectively the angles of incidence and refraction. Moreover, these angles are in the same plane, which, as it M N c Fig. 11. BEFRACTION. 33 passes through P b, is perpendicular to the surface at which the refraction takes place ; and we have for the refraction 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 : sin P b a' or, the sines of the angles of iyicideyice ayid 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. 12, the thickness of these strata is inde- finitely small, and the density gradu- ally increases in proceeding from the surface b, b c will become a curved line. But we shall still have for any point c of this curve, c a! being a tangent to it, . ._, 7 sin P b a m, Fig 12. 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 normals P b, 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. 34 NAVIGATION. 41. The earth's atmosphere presents such a series of par- allel spherical strata, denser at the surface of the earth, and decreasing in density, until at the height of fifty miles the refracting power is inappreciable. . In Fig. 13, the concentric circles M N 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 normals 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 atmo- sphere in the curve B A, and is received by the observer at A. Let AS' be a tangent to this curve at A ; then the apparent direction of the star is that of the line A S' j and the astronomical refraction 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 them. If, then, r represent the refraction, we have r=EBS-EDS'. Fig. 13, Also, E B S is the angle of incidence, and Z A S', the appar- ent zenith distance, is the angle of refraction j and we have sin E B S sin Z AS' m a constant ratio for a given condition of the atmosphere and REFRACTION, 35 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 surface of the earth, experiments give m = 1.000294. The principles of Arts. 39 and 40, applied to this case, 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 sub- tracted from an observed altitude to reduce it to a true alti- tude ; or h = h — r, in which h is the true altitude, h', 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., Art. 248.) 42. After a profound investigation of the problem, Laplace obtained a complicated formula for determining the refrac- tion. Bessel has modified and improved Laplace's formula. His tables of refraction are now considered the most reliable. They are found in a convenient form for nautical problems in Table 20, Bowditch. The mean refractions in this table are for the height of the barometer 30 inches, and the temperature 50° Fahrenheit. 2 * 43. Tables 21 and 22, Bowditch, contain corrections to be applied to the normal refraction for changes in temperature and barometric height, deduced also from Bessel's Tables. * Chauvenet's Astronomy, I, 127-172, contains a thorough investi- gation of the problem of refraction, especially of Bessel's formulas. 36 NAVIGATION. 44. When h = 90°, or the object is in the zenith, r — ; that is, the path is a straight line. When h = 0, or the object is in the horizon, the ray of light, nearly horizontal, describes near the earth's surface a curve which is approximately the arc of a circle whose radius is seven times the radius of the earth ; or, E' = 7R. This, however, is in a mean condition of the atmosphere. The curve is greatly varied in extraordinary states of the at- mosphere, or by passing near the earth's surface of different temperatures ; in very rare cases even to the extent of becom- ing convex to the surface a short distance. DIP OF THE HORIZON. 45. Problem 11. To find the dip of the horizon. Solution. Let A (Fig. 14) be the position of the observer at the height BA = A, above the level of the sea ; A H, perpendicu- lar to the vertical line, C A, repre- sents the true horizon. The most distant point of the horizon visible from A is that at which the visual ray, H" A, is tan- gent to the earth's surface. The apparent direction of H" is AH', the tangent to the curve A H" at A. AH = HAH' is the dip of the horizon to be found. Let C be the centre of the earth, C, the centre of the arc H" A. H", C, C, are in the same straight line, since the arcs H" B, Fig. 14. DIP OF THE HORIZON. 37 H" A are tangent to each other at H", C A, C' A, are perpendicular respectively to A H, A H' ; hence C A C = H A H' = A H, the dip. Let R = C B, the radius of the earth ; then R + h = C A, 7J?-CA.C H", the radius of curvature of H" A, 6 R = C C. We have, then, in the triangle C A C, by Pl. Trio. (268), f V 7R(R+h) ' and, since A is comparatively very small, and may therefore be omitted alongside of R, I O I i a tt a ion sin * A H = 1/ ; * \ 7 R or, putting sin J A ZT= J A ZTsin 1", A ^ = sTnT>V7^ = sin-rV7^ V ^ ( 32 > 46. Taking R = 20902433 feet, we find the constant factor 2 pVnr 69 '' 071 ' sin A#=59".071 y/h, (33) and log A IT= 1.77137 -f } log h, h being expressed in feet, which is nearly the formula for Table 14 (Bowd.). 2 /~3 Since - — — , y — — is constant, depending only upon the radius of the earth, A H is proportional to >/h, or the dip is proportional to the square root of the height of the ob- server above the level of the sea. 38 NAVIGATION. 47. Were the path of the ray, H" A, a straight line, we >uld have \ r H — R 1 and in the triangle H" C A, should have A 'jy =H AH"=H"CA cosA'ZT= R + h' whence, 2 sin 2 £ A' H = — = — , nearly, R -\- h R or with h in feet, A' H= 63".803 VA. (34) Comparing this with A £T= 59".07 Vh, we find A JI= A' #- 4".733 Vh = A' ^T- .074 A' H, or that the dip is decreased by refraction by .074, or nearly tV of it- But from the irregularity of the refraction of horizontal rays (Art. 44), 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 dif- ferent circumstances, both as to the height of the eye, tem- perature and pressure of the atmosphere, and temperature of the water, is greatly needed. 48. Professor Chauvenet (Astron., 1, 176) has deduced the following formula, which it is desirable to test by observa- tions . — DIP OF THE HORIZON. 39 in seconds, A# = M H - 24021" t a' a' or in minutes, A H= A' H- 6'.67 Lzih ; A'^ ' 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 visible 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 all altitudes observed with the sea horizon. 49. Near the shore, or in a harbor, the horizon may be obstructed by the land. (Bowd., Art. 253.) The shore-line may then be used for altitudes instead of the proper horizon. Table 15 (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 in which D = |^+0.56514^ (35) h is the height in feet ; d, the distance of the object in sea miles ; D, the dip in minutes. 50. Problem 12. To find the distance of an object of known height, which is just visible in the horizon. Solution. If 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 40 NAVIGATION. Tig, 14 visible, when the visual ray A H" just touches the earth at H". Let h = B A, the height of A, d = H" A, the distance of A. As this arc is very small, we have d=H"C'Asinl" x C'A = 7 J RXH"C'A sin 1", since C A = 7 R. From the three sides of the tri- angle CC'Aby Pl. Trig. (268), or nearly \ H" C A sin 1" = \J~ , and H" C A sin V vO 21 B This, substituted in the expression for d, gives d = 1R \hE=^i Rh ) < 36 > In this, d, h, and It, are expressed in the same denomination. But if h and H are in feet, in statute miles, d 5280 in geographical miles, Taking i? == 20902433 feet as before, we find in stat. miles d = 1.323 VA, or log d = 0.12156 + \ log h, | in geog. " d = 1.148 Vh, or log d = 0.05994 + J log A. J ^ The first of these is nearly the formula given in Bowditch for computing Table 6. PARALLAX. 41 51. Were the visual ray, H" A, a straight line, we should have from the right triangle C H" A, H" A = V ( c A2 ~ H " c *)> or d ' = V (2X + A) A ; or nearly oT = VOJ X VA. Introducing the same numerical values as before, we have in statute miles d' = 1.225 VA. Comparing this with the expression above, we see that the distance is increased about T \ part by refraction. This, how- ever, is subject to great uncertainty. 52. If the observer is also elevated at the height of B' A' (Fig. 16), and sees the object A in his horizon, then its dis- ta " 0e iS A' H" + H" A, or the sum of the distances of each from the common horizon, H". By entering Table 6 with the heights of the observer and the object respect- ively, the sum of the corresponding distances is the distance of the object from the observer. The distances in ^ 98n this table are in statute miles. Multi- plying them by 6qqa~o = .86839, reduces them to geographical miles. PARALLAX. 53. The change of the direction of an object, arising from a change of the point from which it is viewed, is called paral- lax ; and it is always expressed by the angle at the object, which is subtended by the line joining the two points of view. 42 NAVIGATION. Thus in Fig. 17, the object S would be seen from A in the direction A S ; and from C in the direction C S. The angle at S, subtended by A C, is the difference of these directions, or the parallax for the two points of view, C and A. 54. In astronomical ob- servations, the observer is on the surface of the earth ; the conventional point to which it is most convenient to reduce them, wherever they may be made, is the earth's centre. In those problems of practical as- tronomy which are used by the navigator, we have only to consider this geocentric parallax, which is the dif- ference of the direction of a body seen from the surface • and from the centre of the earth. It may also be defined to be the angle at the body subtended by that radius of the earth which passes through the place of the observer. Thus, in Fig. 17, if C is the centre of the earth, and A the place of the observer, the geocentric parallax of a body, S, will be the angle S = ZAS-ZCS, at the body subtended by the radius C A. If the earth is regarded as a sphere, C A Z will be the vertical line through A, and will pass through the zenith, Z. Then will the plane of C A S be a vertical plane ; Fig. 17. PARALLAX. 43 ZAS, the apparent zenith distance of S as observed at A ; ZCS, its geocentric or true zenith distance ; and ZAS> ZCS. 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 suffices for all nautical problems except the com- plete reduction of lunar distances. For these and the more refined obser- vations at observatories, the spheroidal form of the earth must be considered. 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 sphere 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 ACE, the angle which the radius makes with the equator is its geocentric latitude. 56. Problem 13. To find the parallax of a heavenly- body for a given altitude. Solution. In Fig. 17 let p = S, the parallax in altitude ; z = Z A S, the apparent zenith distance of S, corrected for refraction ; R = C A, the radius of the earth ; c?=CS, the distance of the body S, from the centre of the earth. 44 NAVIGATION. Then from the triangle C A S, we have C A sin C S A = ^-^ sin CAS, B sin z or, sm^= — — , (38) If the object is in the horizon as at H, the angle A H C is called its horizontal parallax ; and denoting it by P, we have from (38), or from the right triangle C A H, sinP = — , (39) which, substituted in (38), gives sin p = sin P sin z. (40) If h = 90° — z, the apparent altitude of the object, we have — sin p = sin P cos h ; (41) or nearly, since p and P are small angles, p — P cos h. (42) 57. The horizontal parallax P, is given in the Nautical 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 parallax. 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. It is given in Table 19 (Bowd.). 58. Tables 16 and 17 (Bowd.) are computed by formula (42). Table 23 contains the correction of the moon's altitude for APPARENT SEMIDIAMETEBS. 45 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 approximation. 59. Table 24 contains, to each minute of horizontal paral- lax, and every 10' of altitude from 5°, the combined correction for parallax and refraction of the apparent altitude of the moon's centre : barom., 30" ; therm., 50° F. Before using this table, the observed altitude of the moon's limb should be corrected for instrumental errors, dip, and semidiameter. APPARENT SEMIDIAMEIEEB. 60. The apparent diameter of a body is the angle which its disk subtends at the place of the observer. Problem 14. To find the apparent semidiameter of a heavenly body. Solution. In Fig. 19, let M be the body ; d = C M, its distance from the centre of the earth ; d! = A M, its distance from A ; r = M B, its linear radius or semidiameter ; s = M C B, its apparent semidi- ameter, as viewed from C; s' = M A B', its apparent semi- diameter, as viewed from A (B and B' are too near each other to be distin- guished in the diagram) ; R = C A, the earth's radius. Fisr. 19. 46 NAVIGATION. 1. For finding s, the right triangle C B M, gives sin* = -. (43) Were the body M in the horizon of A, or Z A M = 90°, its distance from A and C would be sensibly the same, so that the angle s is called the horizontal semidiameter. In (39) we have for the horizontal parallax, . " B R sin P — — , or(/ = a sin? which, substituted in (43), gives r sin s = — sin P } (44) R or nearly, since s and P are small, * = ^P, (45) T — is constant for any particular body, as it is simply the ratio of its linear diameter to that of the earth. For the moon, ^=0.272, 5= 0.272 P, ) and log s = 9.43457 + log P. ) By this formula the moon's horizontal semidiameter may be found from its horizontal parallax. (Naut. Alm., p. 506.) The Nautical Almanac contains the semidiameters as well as the horizontal parallaxes of the sun, moon, and planets. 2. For finding /, the apparent semidiameter as viewed by an observer at A on the surface of the earth, the right tri- angle AB'M gives r sin • as ^ . (47) APPARENT SEM1DIAMETERS. 47 In the triangle C M A, sin M A C CM sin M C A AM' or, putting h = 90° — Z A M, the apparent, and ti = 90° — Z C M, the true altitude of M, cos h _ d cos ti d -, 7/ j cos ti whence, a — d , cos h which, substituted in (47), and by (43), gives , r cos h cos h sin 6' = = sin s (48) dcosh' cos h' 9 cos h cos h! , i / COS « ,,-v or approximately, s = s , (49) by which / may be found when s and h are known. Since h < N, cos A > cos ti, and consequently s' > s ; that is, the semidiameter increases with the altitude of the body. The excess A s = / — s, is called the augmentation. The moon is the only body for which this augmentation is sensible. It is given in Table 18 (Bowd.). 48 NAVIGATION. CHAPTER III. TIME. 61. Transit. The instant when any point of the celestial sphere is on a given 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 prop- erly 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 intercepted arc of the equator is the measure of this angle. 63. Sidereal Time. 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 of nutation. This correction, arising from a change in the position of the earth's axis, is 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 first point of Aries. Its usual symbol is °f . TIME. 49 64. The interval between two successive transits of the vernal equinox is a sidereal day ; and such a day is regarded as commencing at the instant of the transit of that point. The sidereal time is then h m s . 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, that the sidereal days thus denned are sensibly equal. It is unnecessary, then, except in refined discussions, to discriminate between mean and apparent sidereal time. 65. The 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 24A It is equal to the right ascension of the meridian at the same instant. 66. Solar Time. 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 ascension daily nearly 1°, or 4 TO of time. With reference to the fixed stars, it therefore arrives at the meridian each day about 4 OT later than on the previous day ; consequently, solar days are about 4 m longer than sidereal days. 67. Apparent and Mean Solar Time. If the sun changed its right ascension uniformly each day, solar days would be exactly equal. But the sun's motion in right ascension is not 50 NAVIGATION. uniform, varying from S m 35 s to 4 m 26 5 in a solar day. There are two reasons for this, — 1. The sun does not move in the equator, but in the ecliptic. 2. 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. 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 is also supposed to move uniformly in the equator at the same rate that the first moves in the ecliptic, and to return to each equinox at the same time with the first mean sun. The time which is measured by the motion of this sec- ond 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 difference between mean and apparent time is called the equation of time. It is also the difference of the right ascensions of the true and mean suns. The instant of transit of the true sun over a given merid- ian is called apparent noon. . The instant of transit of the second mean sun is called mean noon. The mean time is then h m s . Mean noon occurs, then, sometimes before and sometimes after apparent noon, the greatest difference being about 16 m , early in November. 68. Astronomical Time. The solar day (apparent or TIME. 51 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 0* to 24*. 69. Civil Time. 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 astronomi- cal day of the same date ; and is divided into two 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 used. The affixes a.m. and p.m. distinguish civil time from astro- nomical time. During the p.m. period, this is the only distinc- tion, — the day, hours, etc., being the same in both. 70. Sea- Time. 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 divided into two periods, the same as the civil day. Sea-time is now rarely used. 71. To convert civil into astronomical time, it is only neces- sary to drop the a.m. or p.m., and when the civil time is a.m., deduct l d from the day, and increase the hours by 12*. 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 d , and affix a.m. 52 NAVIGATION. Examples. 1IC AL TIME. CIVIL TIME. d h m a d h m 8 1860 May 10 14 15 10 = 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. 1868 Jan. 4 3 30 = 1863 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 Greenwich time. So also the hour-angle of *f at any meridian, and its hour- angle at Greenwich at the same instant, are corresponding local and Greenwich sidereal times. 73. The difference of the local times of any two meridians is equal to the difference of longitude of those meridians. Demonstration. In Fig. 20, let PM,PM' be the celestial meridians of two places ; P S, the declination circle through the sun (true or mean) ; MPS, the hour-angle of the sun at all places whose meridian is P M, will be the local time (apparent or mean) at those places ; so also M'PS will be the corresponding local time at all places whose meridian is P M' ; and MPM' = MPS-M'PS will be the difference of longitude of the two meridians. If P V is the equinoctial colure, TIME. 53 MPV and M'PT will be the corresponding sidereal times at the two meridians ; still, however, MPM^MPT-M'Pf. The proposition is true, then, whether the times compared are apparent, mean, or sidereal The difference of longitude is here expressed in time. It is readily reduced to arc by observing that 24* = 360 c 1° _ £ m 1* = 15° - or - V =4 S 1* = 15' 1" = tV 1* = 15" In comparing corresponding times of different meridians, the most easterly meridian is that at which the time is great- est. 74. If (Fig. 20) P M is the meridian of Greenwich, M P S is the Greenwich solar time, and MPM' the longitude of the meridian P M'. MPM' = MP8-M'PS; so also MPM' = MPT-M'PT; or, the longitude of any meridian is equal to the difference be- tween the local time of that meridian and the corresponding Greenwich time. 75. If we put T = M P S, the Greenwich time, T = M'P S, the corresponding local time, X = M P M', the longitude of the meridian, P M', we have \ = T — T, and T = T + \ in which \ is + for west longitudes, and T and T are sup- posed to be reckoned always westward from their respective r ;\ ^ 54 NAVIGATION. meridians from h to 24* ; that is, r l\ and T are the astronom- ical times, which should always be used in all astronomical computations. 76. Usually the first operation, in most computations of nautical astronomy is to convert the local civil time into the corresponding astronomical time (Art. 71). The Greenwich time should never be otherwise expressed than astronomically. On this account it would be convenient to have chronometers intended for nautical or astronomical purposes marked from h to 24*, instead of h to 12 A as is now customary with sea-chronometers. 77. The second operation often required is to convert the local astronomical time into Greenwich time. For this we have (50), which numerically is t 77 | J + when the longitude is west, ■Iq = J. -4- K \ . ( — when it is east, and, in words, gives the following Kule. Having expressed the local time astronomically, add the longitude, if west ; subtract it, if east ; the result is the corresponding Greenwich time. TIME. Examples. 1. In Long. 76° 32' W., the local time being 1898, April l rf 9 A 3 m 10 s a.m., what is the Greenwich time ? Local Ast. T. = March 31*21*3™ 10 s Longitude = +568 G. T. = April 1 2 9 18 TIME. 55 2. In Long. 30° E., the local time being March 20 rf 6'' S m a.m., what is the G. T. ? Loc. Ast. T. = March 19 d 18 A 3 m Long. = — 2 G. T. = March 19 16 3 3. In Long. 105° 15' E., the local time being August 21^4'' 3™ p.m., what is the G. T. ? Loc. Ast. T. = August 21<* 4 A 3 m Long. = — 7 1 G. T. = August 20 21 2 By reversing this process, that is, by subtracting the longi- tude if taest, or adding it if east, we may reduce the Green- wich time to the corresponding local time. When observations are noted by a chronometer regulated to Greenwich time, an approximate knowledge of the longi- tude and local time is necessary in order to determine whether the chronometer time is a.m., or p.m., and thus fix the true Greenwich date. If the time is a.m., the hours must be in- creased by 12*. Examples. 1. In Long. 5 A W., about S n p.m., on August 3 d , the Green- wich chronometer shows 8* ll" 1 7 5 , and is fast of G. T. 6 m 10*. What is the Greenwich time ? Approx. Loc. T. Aug. S d S h G. Chro. 8 h ll m 7 s Long. -f- 5 Correction — 6 10 Approx. G. T. Aug. 3^8* G. T.Aug/ 3 d 8* i m 57 s 2. In Long. 10* E., about 1* a.m., on December 7 d , the G. Chro. shows 3* U m 13 s .5, and is fast 25 m 18 s .7. Find the G. T, Approx. Loc. T. Dec. 6 d 13 A G. Chro. 3M4™13 S .5 Long. — 10 Correction — 25 18 s . 7 Approx. G. T. Dec. 6 d 3* G. T. Dec. 6<* 2 A 48 m 54*.8 56 NAVIGATION. 3. In Long. 9* 12 m W., about 2 h a.m., on February 13 rt , the G. Chro. shows 11* 21 m 13 A .3, and is fast 30™ 3(K3. Find the G. T. Approx. Loc. T. Feb. 12 d 14 A m G. Chro. ll*27 m 13 8 .3 Long. + 9 12 Correction — 30 30*.3 Approx. G. T. Feb. 12 d 23 A 12"' G. T. Feb. 12^22^56 m 43 s .0 The operations on the approximate times may be per- formed mentally. 78. Standard Time. By this system, introduced origi- nally for the convenience of railways and now adopted by the United States and other countries, the civil mean time of cer- tain standard meridians is used throughout the adjacent dis- tricts. The standard meridians are one hour (15°) apart, and those in use in North America are the 60th, 75th, 90th, 105th and 120th meridians west from Greenwich; the times are designated respectively Intercolonial, Eastern, Central, Moun- tain, and Pacific. The belts of territory for 7^° on each side of a standard use as far as possible the time of that meridian. To reduce Local Mean Time to Standard Time. If the local meridian is E. of the standard, subtract the difference of longitude between the two meridians from the 1. m. t., and if W., add it. THE NAUTICAL ALMANAC. CHAPTER IV. THE NAUTICAL ALMANAC. 79. The American Ephemeris and Nautical Almanac " is divided into two distinct parts. One part is designed for the special use of navigators, and is adapted to the meridian of Greenwich. The other is suited to the convenience of astron- omers, on this continent particularly, and is adapted to the meridian of Washington." 80. The Nautical part of this Ephemeris and the British Nautical Almanac give at regular intervals of Greenwich time the apparent right ascensions and declinations of the sun, moon, planets, and principal fixed stars, the equation of time, the horizontal parallaxes and semidiameters of the sun, moon, and planets, and other quantities, some of which little concern the navigator, but are needed by astronomers. 81. Before we can find the value of any of these quanti- ties for a given local time, we must first find the correspond- ing Greenwich time (Art. 77). When this time is exactly one of the instants for which the required quantity is put down in the Almanac, it is only necessary to transcribe the quan- tity as it is there given. When, as is mostly the case, the time falls between two Almanac dates, the required quantity is to be obtained by interpolation. And generally, except 58 NAVIGATION. when great precision is desired, it is sufficient to use first differences only ; that is, regard the changes of the quantity as proportional to the small intervals of time which are em- ployed. Thus, for a day, the change of the sun's right ascension may be regarded as uniform, so that for l h it is J ? of the daily change ; for 2 h , ^ ; and in general for any part of a day it will be the same part of the daily change. Generally, then, if A d represent the quantity in the Almanac for a date preced- ing the given Greenwich time ; A 1? its change in the time, T-, t, the time after the Almanac date for which the value of the quantity is required, expressed in the same unit as T, and A, the required value ; we have, A-A. + 1*. (61) When A is increasing, A : has the same sign as A ; but when A is decreasing, A x has the opposite sign. 82. If the given time is nearer the subsequent than the preceding Almanac date, it may be convenient to interpolate backward. If, then, A x represent the quantity in the Almanac for a subsequent Greenwich date, and t' the time before the Almanac date, we have A = A-l^. (52) 83. The Almanac contains the rate of change, or difference of each of the principal quantities for some unit of time. Thus, in the Ephemeris of the sun and planets, the " Diff. for l h ," in part of that of the moon, the " Diff. for l m ," are given. THE NAUTICAL ALMANAC. 59 : 2 +;.::} If t or t' is expressed in the same unit of time as that for which the "Diff.," A x , is given, formulas. (51) and (52) become A = A + t A 1? A Thus, for using hourly differences, we wish the hours, minutes, etc., of the Greenwich time expressed in hours and parts of an hour ; for using the differences for l m , we wish the minutes and seconds of Greenwich time expressed in min- utes 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 order ; 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 dec- lination is given to the nearest 0".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 if 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 60 NA VIGA TION. 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. 85. Should it be desirable to interpolate more accurately than can be done by first differences alone, the reduction for second differences may be introduced by a simple process. Let A 2 be the change of A x in the time T'. . Then, instead of A x , as found in the Almanac for the nearest Greenwich date, we may substitute Ai + ^|r,A 2 ; (54) that is, the value of A : , interpolated for \ 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 A x , is a " Diff. for l h " given for the Almanac for each day, T = 24*; if Aj is a "Diff. for l m " given in the Al- manac for each hour, T' = 60 m . The interpolation of A x to the middle instant may often be performed mentally. Example. If the sun's right ascension for 1898, Jan. 30, 8* 9"* time be required, we find in the Almanac, for Jan. 30 0* A x = 10".244 A 2 = _ 0".035 31 0* A, = 10".209 and by interpolation for Jan. 30 4*, the middle instant be- tween Jan. 30 0* and Jan. 30 8*, Aj = 10 ,, .244 - 0".006 = 10".238, THE NAUTICAL ALMANAC. 61 which is the mean hourly change in the interval from h to 8*. 86. Formula (54), however, applies only to an Ephemeris where the differences for l h or for l m , which are designated by A 1} are given for the same instants of Greenwich time as the functions, A, to which they belong.* For instance, the "Diff. for l h " given for noon Jan. l d , is in the Ameri- can Ephemeris the change per hour at Jan. l d 0*; and the same in the British Almanac. 87. Problem 15. To find from the Almanac a required quantity for a given mean time at a given place. Solution. The preceding considerations lead to the fol- lowing rule : — 1. Express the given mean time astronomically, stating the day as well as the hours, etc., 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 preceding mean time date the required quantity and the corresponding " Diff. for l h " or " Diff. for l m ," noting the name or sign of each ; multiply the "Din , for 1 A " by the hours and parts of an hour, or the " Diff. for l m " by the minutes and parts of a minute, of the remaining Greenwich time ; and add the product algebraically. Or, take out for the nearest subsequent date the required quantity and its difference ; multiply the " Din ." by the hours and parts of an hour, or the minutes and parts of a * The u Prop. Logs, of Diff." of the Lunar Distances are given for the middle instant. 62 NAVIGATION. minute, of the interval from the given Greenwich date to the Almanac date ; and subtract the product algebraically. When greater precision is required, interpolate the differ- ence to the middle instant between the given Greenwich date and the Almanac date, and use the result instead of the dif- ference given in the Almanac. This rule is applicable to all those quantities which are given at regular intervals of Greenwich mean time, except the moon's meridian passage and age and lunar distances. For the " Sidereal Time at Greenwich Mean Noon," on p. II of each month, the "Diff. for 1*" is 9 S .8565 ; Table 3 of the American Ephemeris, for converting a mean solar into a sidereal interval, may be used for the interpolation. The " Mean Time of Sidereal h " on p. Ill, is given at in- tervals of 24* of sidereal time. The " Diff. for l h » is — 9 5 .8296 ; and Table 2, for converting a sidereal into a mean solar inter- val, may be used. 88. The quantities given in the American Ephemeris for Washington mean time may be interpolated in the same way, by reducing the local time to Washington time instead of to Greenwich time. 89. The apparent places of the fixed stars are given in the British Almanac for the upper transit over the meridian of Greenwich ; in the American, for the upper transit over the Meridian of Washington. In the latter, the Washington mean time is given. The sidereal time at either place for the instant of transit is the right ascension of the star (Art. 65). Generally, the position given for the nearest day suffices. But if greater precision is required, it is necessary to reduce the local mean time to the sidereal time of the prime merid- ian, and interpolate for it. THE NAUTICAL ALMANAC. 63 90. In the following examples the required quantities are taken from the American Ephemeris, and interpolated to the nearest second by first differences (53), and to the highest precision attainable by 2d differences (54). [Ordinarily, in- terpolation to the nearest second by (53) suflices for the practical purposes of navigation.] Examples. For the local mean time, 1898, Jan. 30 d 9 h 14 m 30 s a.m. in Long. 163° 14' W., find the following quantities from the Nautical Almanac : — The equation of time. O's right ascension, j)'s declination, O's declination, 3)'s horizontal parallax, J) 's right ascension, J) 's semidiameter ; The R. ascension and declination of a Scorpii (Antares). Ast. mean time, 1865, Jan. 29<* 21 A 14 m 30 s Long. +10 52 56 G. mean time, 1865, Jan. 30 8 7 26 8 7.433 8.1239 1. The Equatio7i of Time (Page II). m s to s s Jan. 30, 0*. 13 34.14 + 0.387 13 34.14 + 0.387 \ = - .035 8.124 .035 3.! 24 X4=-006 -3.15 .04 +0.381 (at 4 A ) .01 8.124 ( 3.048 038 ;.15 \ .( + 3.10 Subtractive from mean time, 13 37.24 2. The O's right ascension (Page II). I: 008 001 64 NAVIGATION. - — X Jan. 30, 0* 20 52 32.2 + 10.244 20 52 32.23 + 10.244 \ = — .035 (-82. + 1 23.2 j 1.0 24 I .2 + 10.238 (at 4 A ) 20 53 55.4 f 81.904 + 123.17 «g 20 53 55.40 ^ * 041 3. The O's declination (Page II). Jan. 30, - 17 34 04.0 + 4L42 - 17 34 04.0 + 41.42 A 2 = + .77 331 ' 4 ^X4 = + .1S 4.1 24 + 5 36 ' 5 I .8 41.55 (at 4*) .2 ( 332.40 -17 23 27.5 +5375 J 4.16 -17 28 26.44 ^ * 1 ' 7 4. The 1> 's right ascension (Page XII). h m 8 h m s Jan. 30, 8\ 3 23 30.8 + 2.1083 3 23 30.77 + 2.1083 A2 = + -0026 7 ' 433 ^x 3.7= +.0002 ( 14.8 - 60 + 15.7 J .8 + 2.1085 (at 3 m .7) I .'1 7.433 3 23 46.5 5. The 3>'s declination (Page XII). Jan. 30, 3 A , + 23 26 05.9 + 6*24 +23 26 05.9 +6.240 A 2 = -.109 43.7 _^x3.7=-.007 + 46.4 {2.5 60 {43.1 .2 + 6.233 (at 3 m .7) + 23 26 52.3 c 43 63 -46.3 J 2.40 .02 THE NAUTICAL ALMANAC. 65 6. The D's horizontal parallax (Page IV). Jan. 30, 0*, 54 24.7 - 0.78 54 24.7 - 0.78 A 2 = + 0.21 ( 6.2 -6.3 | j •■|x4= + .07 54 18.4 -0.71 (at 4*) i 5.68 ~ 5 ' 8 | .09 54 18.9 7. TAe J> 9 g semidiameter (Page IV). Jan. 30, 0* 14 51.4 - 2.2 in 12* 14 51.4 5.77 -1.5 in 8* .272 14 49.9 r .115 -1.6 \ .040 (-.001 14 49.8 In Art. (60) we have for the moon, s = .272 P\ whence As = .272 AP: so that the reduction of the semidiameter may be readily found by multiplying that of the horizontal parallax by .272, as in the above example. (Naut. Alm., p. 506.) The right ascension and declination of a Scorpii (Antares). The Washington (long. + 5* 08 m 12 s ) mean time is Jan. 30, 2 h 59 m 14 s , or Jan. 30.124. On page 299, which serves as an index, the mean R. A. is 16* 23 m . The apparent R. A. and Dec. (p. 346) are for Jan. 29.8 m. t. Washington. h m s s o / // // R. A. 16 23 09.92 + 0.35 Dec. - 26 12 24.6 - 0.7 (lOd) change in + 0.325d + .01 — .02 16 23 09.43 — 26 12 24.62 91. Problem 16. To find from the Almanac the sun's right ascension and declination, and the equation of time for a given apparent time at a given place. 66 NAVIGATION. Solution. This differs from the preceding problem simply in using the apparent instead of the mean time, and in taking the quantities from page I for the month, where they are given for apparent noon, instead of from page II, where they are given for mean noon. 92. Problem 17. To find the right ascension and dec- lination of the sun, and the equation of time at apparent noon of a given place, or when the sun is on the meridian. Solution. The local apparent time is h m s . The Green- wich apparent time is then equal to the longitude if west ; that is, it is after the noon of the same date by a number of hours, etc., equal to the longitude. If the longitude is east, the Greenwich apparent time is before the noon of the same date by a number of hours, etc., equal to the longitude. Hence, take these quantities from the Almanac for Green- wich apparent noon (p. I) of the same day as the local (civil) day, and apply a correction equal to the hourly difference mul- tiplied by the hours and parts of an hour of the longitude ; observing to add or subtract the correction, according as the numbers in the Almanac may require, for a time after noon, if the longitude is west ; for a time before noon, if the longi- tude is east. Examples. 1. Mnd the sun's right ascension and declination, and the equation of time for apparent noon, 1898, Jan. 30, in Long. 163° 14' W. h m s h m s 1. Long. + 10 52 56 O 's R. A. 20 52 34.55 + 10.237 h = + 10.882 f 102.37 in 10 + 1 51.40 J 8 ' 19 in ' 8 .82 in .08 20 54 25.95 I .02 in .002 THE NAUTICAL ALMANAC. 67 m a s O 's dec. - 17° 33' 54". 6 + 41".61 Eq. of t. + 13 34.23 + 0.379 ''416'U ^3.79 + r8sr J 33.29 +4.12 .30 1 3 .33 I .03 .08 - 17 26'21".8 +13 38.35 2. For apparent hood, 1898, March 21, in Long. 163° 14' E. h m s h m 8 s 2. Long. - 10 52 56 Q's R. A. 03 20.43 + 9.103 = — 10.882 f 91.03 -139.06^ 7 ' 28 .73 I .02 01 41.37 O's dec. + 0°21'44".8 + 59".23 Eq. of t. + 7 13.44 - 0.752 {592".3 (7.52 47-38 +8.18 .60 4".74 I .06 .12 + 0°ir00 // .3 . + 7 21.62 In the first and second examples the Diffs. for l h have been interpolated for 5^.5 or half the longitude, forward in the first, back in the second ; ordinarily such precision is un- necessary. 93. Problem 18. 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 II of each month, in the Almanac, called " sidereal time/' is also the right ascension of the mean sun at Greenwich mean noon, and may be interpolated for a given local time in the 68 NAVIGATION. same way as the right ascension of the true sun. (Pkob. 15.) The constant " Diff. for l h " is 9 S .8565. A table for convert- ing mean time into sidereal time intervals (Table III) facili- tates 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 19. 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 (p. IV). This mean time is the hour-angle of the mean sun (Art. 72) when the moon is on the meridian ; and is therefore the dif- ference of right ascension of the moon and the mean sun. As this difference is constantly increasing, in consequence of the moon's more rapid increase of right ascension, the mean time of each transit is later than that of the one preceding by a number of minutes, varying, according to the rate of the moon's motion from 40 m to 66 m . If, then, T x and T 2 denote the mean times of two successive transits of the moon over the Greenwich meridian, T 2 — T 1 is the retardation of the moon in passing over 24J 1 of longitude ; so that for any longitude A. (expressed in hours) the retarda- tion is nearly The mean time of a transit is, then, reduced from the Greenwich to any other meridian by interpolating for the longitude; forward, if the longitude is west; backward, if the longitude is east, since east longitudes are regarded as negative. THE NAUTICAL ALMANAC. 69 The American Ephemeris gives also the hourly differences, which facilitate the interpolation. For greater exactness, these differences may be interpolated for half 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 " Diff. for l h " by the longitude in hours, if the longitude is west; subtract that product if the longitude is east; or it may be taken from Table 2 (Bowd.). The mean time of merid- ian 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 used. But it is more exact to use half the difference of the times of meridian passage for the day preceding and the day following the given day : ^ ¥ of this is the " Diff. for l h " 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 here- after given in Problem 27. 95. Problem 20. To find on a given day the mean time of transit of a planet over a given meridian. Solution. The mean time of each meridian passage at Greenwich is given, in the Almanac, for each planet. It may be reduced to any meridian in the same way as for the moon ; except that, in the case of an acceleration, the sign of the reduction is reversed. * It is important to notice whether the mean time of transit is more or less than 12 A . In the former case, the astronomical day is l d less than the civil day. 70 NAVIGATION. Examples. 1. In Long. 100° 15' W., find the times of meridian passage of the moon and Jupiter for 1898, June 7 (civil day). Long. + 6*41-0' = + 6\683. J> % h 71% Wl H ftl 771 M. T. of mer. pass., June 6, 14 32.2 + 2^49 June 7, 6 59.4 —3^ in 1 d. 114.94 r 0.95 in 6* . 1.49 -1.1 \ .10 in .6 .20 I .01 in .08 .01 June 7, 2 48.8, a.m. June 7, 6 58.3, p.m. 2. In Long. 100° 15' E., for 1898, June 7 (civil day), find the times of meridian passage for the moon and Jupiter. Long. -6*41-0* = - 6 A .683. D % h m to Am m M. T. of mer. pass., June 6, 14 32.2 + 2^55 June 7, 6 59.4 - 3.8 in 1 d. {15.30 r 0.95 in Q h 1.53 + 1-1 \ .10 in .6 .20 I .01 in .08 .01 June 7, 2 15.2, a.m. June 7, 7 00.5, p.m. In the case of the moon the hourly differences have been interpolated for half the longitude. (Ordinarily this pre- cision is unnecessary.) 96. Problem 21. To find the right ascension or dec- lination 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 Prob- lem 19; 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 15. THE NAUTICAL ALMANAC. 71 If the time of transit has been noted by a clock or chro- nometer, regulated to either local or Greenwich time, it should be used in preference to the time of transit computed from the Almanac. 97. Problem 22. 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 h of Greenwich mean time. If d represent the given distance ; d , the nearest distance of the same body in the Almanac preceding in time the given distance ; Aj. , the change of distance in 3 A ; t, the required time (in hours) from the date of d ; by (51) we have approximately, using 1st differences only, whence, for the inverse interpolation, or, with t in seconds of time, which is better for computation, t = —£—(d-d Q ), (57) in which it is most convenient to express A x and (d — d ) in seconds. Then by logarithms : log t = log (d - cl,) + log ^00 f (58) 10800 ence log ar. complement of the " log diff. for I s ." S— . is the change of distance in 1* ; hence log is the 10800 8 A x 72 NAVIGATION. It is given in the Almanac for the middle 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 di represent the distance in the Almanac following the given distance ; and tf, the interval before the date of d lf we shall have by (52) and tf = — (di - d), or with if in seconds, and by logarithms, log t'= log (<£ - d) + log i^OO _ (59) The computation is simplified by using a table of " loga- rithms of small arcs in space or time." f It differs from the common table of logarithms only in having the argument in sexagesimal instead of natural numbers. With such a table it is unnecessary to reduce differences of distance to seconds, or to first find the intervals of time in seconds. From (58) and (59) we have the following rule : Find in the Almanac the two distances between which the given distance falls ; take out the nearest of these, the hours of Greenwich time over it, and the " P. L. of Diff." between them. Find the difference between the distance taken from the Almanac and the given distance ; and to the log. of this difference add the " P. L. of Diff." from the Almanac. The sum is the log. of an interval of time to be added to the hours of Greenwich time taken from the Almanac, when the earlier * Proportional Logarithm. t Table 34 (Bowd.). THE NAUTICAL ALMANAC. 73 Almanac distance is used ; to be subtracted from the hours of Greenwich time when the later Almanac distance is used. (Chauvenet's " Lunar Method/' p. 8.) 98. The result, however, may not be sufficiently approxi- mate, owing to the neglect of 2d differences. To correct it for 2d differences, Table 10 of Chauvenet's Method, Table I of the Almanac, or Table 35 (Bowd.) may be used. For either, take the difference between the two Prop. Logs., which pre- cede and follow the one taken from the Almanac. With half this difference, and the interval of time just found, enter the table and take out the seconds, which are to be added to the approximate Greenwich time when the Prop. Logs, are decreasing, but subtracted when they are increasing. Second differences may also be introduced by first finding, or estimating, the Greenwich mean time to the nearest 10 m , and interpolating the Prop. Log. in the Almanac to the middle instant between that time and the Almanac hour used, as in Art. 88 for direct interpolation. 99. Maskelyne, the author of the present arrangement of lunar distances, to facilitate their interpolation, devised what he chose to call proportional logarithms. If n represent any number of seconds, either of space or time, the proportional logarithm of n is the log. of Table 45 (Bowd.) contains these proportional logarithms for each second of n from to 3°, or to 3^, the argument being in ° ' " or in h m s . But such a table is less useful for other purposes than Table I of the American Ephemeris, previously referred to. Dividing both members of (57) by 10800, and inverting, we have 74 NAVIGATION. 10800 _ A x 10800 « 10800 tf - <% ' and P. log £ = P. log (d — c? ) — P. log A x , which accords with the rule in Art. 310 (Bowd.). 100. Example. 1898, Oct. 27, the distance of Fomalhaut from the moon's centre is 52° 3' 35", what is the Greenwich mean time ? d = 52 3 35 Oct. 27, 15*, d = 51 41 15 P. log 0.3323 diff. — 22 d — d = 22 20 log 3.1271 t = + 48 00 log 3.4594 Red for 2d diff. + 05 Table 35 (Bowd.). G. m. t., Oct. 27, 15 48 05 or, by back interpolation, d = 52 3 35 Oct. 27, 18*, d x = 53 5 P. log 0.3323 diff. — 22 d 1 — d= 1 125 log 3.5664 t =-2M2™ log 3.8987 Red for 2d diff. + 05 G. m. t., Oct. 27, 15 48 05 CONVERSION OF TIME. 75 CHAPTER V. CONVERSION OF THE SEVERAL KINDS OF TIME.— RELATION OF TIME AND HOUR-ANGLES. CONVERSION OF TIME. 101. Problem 23. To convert apparent into mean time, or mean into apparent time. Solution. For the same instant, let T m represent the local mean time ; T a , the local apparent time ; and JSf the equation of time with the sign of its application to apparent time. Then, since the equation of time is the difference of mean and apparent times (Art. 67), ^m = T a + JEJ, \ } (61) T a =T m - E. The reduction, then, is made by finding from the Almanac the equation of time for a given apparent time, from page I of the month (Prob. 16), or for a given mean time from page II (Prob. 15), and applying it to the given time according to the precept at the head of the column where it is found. 102. The equation of time on page I is sometimes called the mean time of apparent noon ; and on page II the apparent 76 NAVIGATION. time of mean ?won. Eegarding it, as in (61), as the reduction of apparent to mean time, it indicates, when additive and in- creasing, or subtractive and decreasing, that mean time is gaming on apparent time. 103. Problem 24. To convert a mean into a sidereal time interval, or a sidereal into a mean time interval. Solution. The sidereal year is 365.25636 mean solar days, or 366.25636 sidereal days ; so that the same interval of time which is measured by 365^.25636 reckoned in mean time, is measured by 366 d .25636 if reckoned in sidereal time. Since both are uniform measures of time, if we represent any inter- val by t, if expressed in mean time, s, if expressed in sidereal time, then g = 366.25636 = 10Q27379 t 365.25636 ' whence 8 = 1.0027379 t = t + .0027379 t, (62) t = 0.9972696 s = s - .0027304 s, (63) by which the reduction from one to the other may be made. The computation is facilitated by Table II of the American Ephemeris, for converting sidereal into mean solar time, which contains for each second of s the value of .0027304 s ; and by Table III, for converting mean solar into sidereal time, which contains for each second of t the value of .0027379 t. Tables 8 and 9 (Bowd.) contain the same quantities. 104. If in (62) t = 24* ; 8 = 24* S m 56 s .5553 ; or in a mean solar day sidereal time gains on mean time 3 m 56 s .5553. In l h of mean time the gain is 9 S .8565. If in (63) 8 = 24* ; t = 24* — S m 55 s .9094 ; or in a sidereal CONVERSION OF TIME. 77 day mean time loses on sidereal time S m 55*.9094. In l h of sidereal time the loss is 9 5 .8296. If t and s in the last term are expressed in hours (62), and (63) become . . ft _ OKaK . v J s = t + 9 S .8565 t, ) t = s-9 s .S296s; \ by which the reductions may be more readily calculated, when the tables are not at hand. 105. Problem 25. To convert mean time at a given place into sidereal time. Solution. Let A. represent the longitude of the place, expressed in time, + when west, T, the local mean time, S, the corresponding sidereal time, t, the interval from mean noon in mean time (differing from T only by omitting the day), s, the same interval in sidereal time, S , the sidereal time of mean noon at Greenwich, S r , the sidereal time of mean noon at the place ; then, since X expresses the Greenwich time of local noon, (Art. 92), $ Q ' = # + .0027379 X A evidently #=« + #/ I (65) and by (62) s = t+ .0027379 t;j whence we have S== t + # + -0027379 (A + t). (66) The Almanac (page II) contains S for each Greenwich mean noon, under the head " Sidereal Time." It should be taken out for the given astronomical day of the place; 78 NAVIGATION. .0027379 A. is then the reduction for longitude, additive in west longitude, subtractive in east. It, as well as .0027379 t, the reduction to a sidereal interval, may be taken from Table III of the Almanac, or from Table 9 (Bowd.) ; or either may be computed by (62), or first of (64). From (66), then, we have the following rule : To the local mean time add the sidereal time of Greenwich mean noon of the given astronomical day, the reduction of this sidereal time for the longitude of the place, and the reduction of the hours, minutes, etc., of the mean time to a sidereal interval. The astronomical (solar) day is usually retained. But if it be desirable to state the sidereal day, as well as the hours, etc., of the sidereal time, we prefix to S the sidereal day at the instant of mean noon, which is the same as the astronom- ical day after the vernal equinox of each year ; one day less before that date. At the instant of the vernal equinox the sidereal time and mean solar time coincide. Before that time the mean sun transits before the vernal equinox ; after that time it transits after the vernal equinox. 106. T + X is the Greenwich mean time. When this is given, or found in the course of computation, it will be more convenient to take out /S for the Greenwich day, and the combined reduction, .0027379 (t + X), for the hours, minutes, etc., of Greenwich mean time, instead of for t and X separately. It should be noted, however, that in the first method (Art. 105), S is taken out for the local day ; in this, it is taken out for the Greenwich day, provided X + t, as used, expresses properly the Greenwich time. 107. S + .0027379 (t -f X) is the " sidereal time " of the Almanac interpolated for the Greenwich mean time. It is CONVERSION OF TIME. 79 more convenient to term it the right ascension of the mean sun (Art. 93) ; and then the translation of (66) will be, the sidereal time is equal to the right ascension of the mean sun -|- the mean time. This is also evident from Fig. 21, in which P is the pole ; P M, the meridian ; V, the vernal equinox ; T M, the equator. T M is also the right ascension of the meridian, and measures MP T, the hour-angle of T, or the sidereal time (Art. 65). If P S is the declination-circle passing through the mean sun, T Sis the right ascension of the mean sun, and M P S is its hour-angle or the mean time (Art. 72), and is measured by the arc of the equator, S M. Evidently TM=tS + SM. (67) The hour-angles MP f , MPS, are reckoned from the meridian toward the west ; hour-angles east from the meridian are then regarded as negative. If P S is the declination-circle of the true sun, then will 12 17 18.26 Long., + 4 CL52 Red. for long., + 39.57 G. m.t., Sept. 26, 1 17 07 Red. of L. m. t., + 3 29.66 So> 12 21 14.82 Sid. time, 9 37 42.49 Red. G. m. t., + 12.67 Sid. time, 9 37 42.49 4. Find the sidereal time of 1898, Sept. 25, 3* 16 w 15 s .0, in long. 8* 16 m 25 s .3 E. L.m.t., Sept. 25, 3 16 15 L. m. t., Sept. 25, S , 12 17 18.26 Long., Red. for long., — 1 21.55 G.m.t., Sept. 24, Red. of L. m. t., + 32.24 So i Sid. time, 15 32 43.95 Red. for G. m. t., Sid. time, 3 16 15 - 8 16 25.3 18 59 49.7 12 13 21.71 + 3 07.25 15 32 43.96 CONVERSION OF TIME. 81 108. Problem 26. To convert sidereal time at any- place into mean time. 1st Solution. The sidereal time at mean noon at the place is from (65) Si = Sq + . 0027379 X; the sidereal interval from mean noon, s= S- S '= S- S - .0027379 X ; (68) and from (63) the corresponding mean time interval, t = s- .0027304 s. (69) The mean time T is completed by prefixing to t the astronom- ical day. From (68) and (69) we have the following rule : From the local sidereal time subtract the sidereal time of Greenwich mean moon of the given astronomical day and the reduction of this sidereal time for the longitude of the place ; and from the sidereal interval thus obtained subtract the reduc- tion to a mean time interval ; and to the result prefix the given astronomical day. The local sidereal time may be increased by 24 A if neces- sary. The reduction for longitude, .0027379 A, may be taken from Table III of the Almanac, or from Table 9 (Bowd.) ; nu- merically, it is subtractive in west longitude, additive in east, as applied to the given sidereal time. The reduction of the sidereal interval, .0027304 s, may be taken from Table II, or from Table 8 (Bowd.), and is always subtractive. 2d Solution. Let M represent the " mean time of the preceding sidereal h " at Greenwich ; M ' , the " mean time of the preceding sidereal h " at the place ; 82 NAVIGATION. £, the interval from h in sidereal time ; t, the same interval in mean time : then, since A. will be the sidereal interval between the Green- wich and local sidereal h (Art. 92), Mi = M - .0027304 A, evidently, T = t •+- Mi , and by (63) * = # - .0027304 #; whence we have T = £ + M* - .0027304 (A. + J8 ). (70) The Almanac (page III) contains, J^ for the Greenwich sidereal h on each mean day. The Almanac date of the pre- ceding sidereal 0* is generally the same as the local astronom- ical date when the sidereal time is less than the " sidereal time at mean noon" (page II), but l d less when the sidereal time is greater than that at mean noon. The doubtful case is when the mean time is within 4 m of noon: the comparison must then be made with the sidereal time at the nearest local mean noon. The reduction of M to the local meridian is — .0027304 A., which may be taken from Table II, or from Table 8 (Bowd.). It is subtr active in west longitude, additive in east. The reduction of the sidereal interval, .0027304 S, may be taken from the same tables ; it is always subtractive. The combined reduction, .0027304 (X + >S), may be taken out for the Greenwich sidereal time, (X + $)> instead of for X and S separately; but with these precautions, that when X + /S > 24 h , M may be taken out for l d later than stated in the previous precept, and interpolated for the excess of (A. -f S) over 24^ ; and when (A. + /S) is negative, to retain its negative character, or else take out M for one day earlier. CONVERSION OF TIME. 83 3d Solution. From (66) we have t = S - [# + .0027379 (t + A)], (71) so that, when the Greenwich mean time (t -f- A) is sufficient^ known, we may find for it the right ascension of the mean sun (Art. 107), S + .0027379 (t + A), and subtract it from the given sidereal time: or, the mean time is equal to the sidereal time — the right ascension of the mean sun. So also we have from Art. 107 the precept : the apparent time is equal to the sidereal time — the right ascension of the true sun. Examples. 1. 1898, Jan. 30 (ast. day), in long. 10" m 52*.7 W., the sidereal time is 6* 57 m 44 8 .5 ; find the mean time. h m 8 h m s L.sid.t., 6 57 44.5 L.sid.t., 6 57 44.5 S (Jan. 30), — 20 38 58.09 M (Jan. 30), 3 20 28.98 Red. for X, — 1 38.71 Red. for A, — 1 38.44 Sid. int., 10 17 07.7 Red. of sid. t., - 1 08.44 Red. of sid. int., — 1 41.1 L. m. t., Jan. 30, 10 15 26.6 L. m. t., Jan. 30, 10 15 26.6 2. 1898, Jan. 30 (ast. day), in long. 10* m 52*.7 E., the sidereal time is 6* 54 m 27 8 .08 ; what is the mean time ? h m s h m s L. sid. t. 6 54 27.08 L. sid. t. 6 54 27.08 S (Jan. 30), - 20 38 58.09 M (Jan. 30), 3 20 28.98 Red. for A , +1 38.71 Red. for A, +1 38.44 Sid. int., 10 17 17.7 Red. for sid. t., - 1 07.9 Red. of sid. int., - 1 41.1 L. m. t., Jan. 30, 10 15 26.6 L. m. t., Jan. 30, 10 15 26.6 3. 1898, Sept. 26, 9\ a.m., in long. 4 A W 52 8 W., the side real time is 9* 37 w 42 8 .49 ; find the mean time. 84 NAVIGATION. h m s h m 8 L. sid. t., 9 37 42.49 L. sid. t., 9 37 42.49 S (Sept. 25), - 12 17 18.26 M Q (Sept. 25), 11 40 46.62 Red. for X , — 39.57 Red. for A , - 39.46 Sid. int., 21 19 44.66 Red. for sid. t., - 1 34.65 Red. of sid. int., - 3 29.66 L. m. t., Sept. 25, 21 16 15 L. m. t., Sept. 25, 21 16 15 4. 1898, Sept. 25, 3 A , p. m., in long. 8* 16 m 25'.3 E., the sidereal time is 15* 32 m 43*.95 ; find the mean time. h m s h m 8 L. sid. t., 15 32 43.95 L. sid. t., 15 32 43.95 S (Sept. 25), — 12 17 18.26 M (Sept. 24), 11 44 47.52 Red. for A., +1 21.55 Red. for A, +1 21.33 Sid. int., 3 16 47.24 Red. of sid. t., - 2 32.80 Red. of sid. int., - 32.24 L. m. t., Sept. 25, 3 16 15 L. m. t., Sept. 25, 3 16 15 RELATION OF HOUR-ANGLES AND TIME. 109. Problem 27. To find the mean time of meridian transit of a celestial body, the longitude of the place or the Greenwich 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 (61) T m = E, the equation of time, which can be taken from page I of the Almanac, and interpo- lated for the longitude, which in this case is also the Green- wich apparent time ; or from page II, and interpolated for the Greenwich mean time. When E is subtractive, the sub- traction 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 merid- ian, or sidereal time. (Art. 65.) It suffices therefore to find the right ascension of the body, and, regarding it as the side- real time, reduce it to meantime by Problem 26. RELATION OF HOUR-ANGLES AND TIME. 85 The American Ephemeris contains the apparent right as- censions of two hundred principal stars for the upper culmi- nations at Washington ; the British Almanac contains the positions for the upper culminations at Greenwich. They are reduced to any other meridian, when necessary, by interpolat- ing 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, and may be found for a given Greenwich mean time by Prob- lem 15. 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. 19, 20) ; then for this mean time the right ascensions of the moon, or of the planet (Prob. 15), and of the mean sun (Prob. 18), may be computed. Subtracting the right ascen- sion of the mean sun from the right ascension of the moon or planet, will give the mean time of transit (Prob. 26, 3d Solu- tion.) If it differ sensibly from that previously obtained, the process may be repeated with this new approximation. If the time of transit has been noted by a clock, or chro- nometer, regulated either to local or Greenwich time, it should be used in preference to the approximate time of transit found from the Almanac in 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 culmination, 12* may be added to the right ascension of the body, if suffi- ciently well known; or, as is generally preferable, 12* may 86 NAVIGATION. be added to the longitude of the place. The instant of a lower culmination on any meridian will be that of an upper culmi- nation on the opposite meridian. Examples. 1. Find the times of meridian passage of the moon and Jupiter for 1898, June 7 (civil day), in long. 100° 15' W. (Example 1, Art. 95, p. 70.) D h m h m Approx. m. t., June 6, 14 48.8 June 7, 6 58.3 Long. + 6 41.0 6 41.0 G.m. t., June 6, 21 29.8 June 7, 13 39.3 = 13*.665 3)'sR. A., June 6, 21 A , h m 8 a a h m a 8 19 50 50.93 + 2.5350 A 2 _ .0079 12 04 08.50 0.281 A 2 + .028 •° 079 x 15= .002 ^X 6.8=. 008 60 24 4- 2.533 29.8 0.289 13.665 h m 8 ( 50.66 (3.76 Red. for G. m. t, 4- 1 15.49 \ 22.80 + 3.95J .17 I 2.03 I .02 R. A. at transit, 19 52 06.42 12 04 12.45 S , 4 59 40.55 5 03 37.11 Red.forG.m.t., 3 31.88 2 14.59 6V, 5 03 12.43 5 05 51.70 M. t. of transit, June 7. , 6 58 20.75 June 6, 14 48 53.99 + 2.75 Diff. f . appr. t. + 5.99 In5s99i Ch - ofRA " + - 253 ln5 - yy j_Ch.ofS ,-.016 M. t. of transit, June 6, 14 48 54.13 110. Problem 28. To find the hour-angle of the sun for a given place and time. RELATION OF HOUR-ANGLES AND TIME. 87 Solution. The hour-angle of the sun, reckoned from the upper meridian toward the west, is the apparent time reckoned astronomically (Art. 72). Its hour-angle east of the meridian is negative, and numerically equal to 24* — the apparent time. A given mean or sidereal time must then be converted into apparent time : for this, the longtitude, or the Greenwich time, must be known approximately. 111. Problem 29. To find the hour- angle of the moon, a planet, or a fixed star, for a given place and time. Solution. In Fig. 21, as described in Art. 104, °f M is the right ascension of the meridian, and measures MPf, the sidereal time. Let P S be the declination-circle of the mean sun, then T S is the right ascension of the mean sun, and MPS 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 T 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=rM-M'. (72) If f S is the right ascension of the true sun, S M will measure the apparent time. From (72), then, we have the following rule : 88 NAVIGATION. To a given apparent time add the right ascension of the true sun; or to a given mean 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; the difference is the hour-angle west from the merid- ian. If it is more than 12* , it may be subtracted from 24 A : 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. Problem 30. 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 sidereal time (Probs. 23, 24), as may be required. The Greenwich time, or the longitude of the place, is needed only for this reduction. 113. Problem 31. 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. 15), the right ascension of the body. Then, from (72), We haVC T M = T 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. Prom this subtracting the right ascension of the true sun gives the apyparent time ; or the right ascension of the mean sun gives the mean time. RELATION OF HOUR-ANGLES AND TIME. 89 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, until the assumed and computed values sufficiently agree. 114. Examples. 1. 1898, Jan. 12, 12* 15 w 17 s .6, mean time in long. 150° 13' 10" W., find the hour-angle of the moon. h m a A to « L. m. t., Jan. 12, 12 15 17.6 L. m. t., Jan. 12, 12 15 17.6 Long., +10 00 52.7 S , 19 28 00.06 G. m.t., Jan. 12, 22 16 10.3 = 16 m .17 Red. for G. m. t., +3 39.5 5'sR.A. (Jan. 12, 22*), 11 36 04.83 + 1.9622 i 19.62 11.77 L. sid. t., 7 46 57.16 20 .14 D 's R. A. at date 1136 36.56 D 's hour angle, — 3 49 39.4 2. 1898, Jan. 12, 22* 16 m 10 8 .3, G. m. t., the moon's hour angle is — 3* 49 w 39 s .4 ; find the L. m. t. h m s D 's hour angle, — 3 49 39.4 D »a R. A., Jan. 12, 22*, 11 36 04.83 + 1.9622 16.17 ( 19.62 Red. for G. m. t., I .14 L. sid. t., 7 46 57.16 £ , Jan. 12, 19 28 00.06 Red. for G. m. t., 3 39.50 L. m. t., Jan. 12, 12 15 17.6 90 NAVIGATION. 3. 1898, Jan. 12 (12» nearly), in long. 150° 13' 10" W., the moon's hour-angle is — 3 h 49 m 39 8 .4 ; find the L. m. t. h m s h m m Long., 10 00 52.7 J)'s mer. pass., Jan. 12, 15 53.5+1.84 3) 's h. a., — 3 49.7 Red. for long., + 18.4 ch.ofR.A., -7.5 Jan. 12, 16 11.9 ^-ch.of So, +0.6 -356.6 1st approx. L.m.t., Jan. 12, 12 15.3 Long., + 10 00.8 1st. approx. G.m.t., Jan. 12, 22 16.1 h m s 3>'sh. a., —3 49 39.4 D 's R. A., Jan. 12, 22*, 11 36 04.83 + 1.9622 16.1 ( - 19.62 Red. for G. m. t., + 31.59 j 11.77 .20 ch. in + 4M7 + .136 L. sid. t., 7 46 57.02 — S , Jan. 12, 19 28 00.06 — Red. for G. m. t., 3 39.49 — ch. in +4M7 — .012 2dL. m. t., 12 15 17.47 cor. for 4 s . 17 + .124 Long., 10 00 52.7 2dG. m.t., 22 16 10.17 Diff. from 1st G. m. t., + 4.17 3d L. m. t., Jan. 12, 12 15 17.6 NAUTICAL ASTRONOMY. 91 CHAPTER VI. NAUTICAL ASTRONOMY. ALTITUDES.-AZIMUTHS. — HOUR-ANGLES AND TIME. 115. Nautical Astronomy comprises those problems of Spherical Astronomy which are used in determining geograph- ical positions, or in finding the corrections of the instruments employed. In general they admit of a much more refined application on shore, where more delicate and stable instru- ments can be used, than is possible at sea, where the insta- bility of the waves and the uncertainty of the sea-horizon present practical obstacles, both to precision in observations and to the accuracy of the results, which cannot be obviated. 116. In the problems which are here discussed, the follow- ing notation will be employed : L = the latitude of the place of observation h = the true altitude of a celestial body ; z = 90° — A, its zenith distance ; d = its declination ; p = its polar distance ; t = its hour angle ; Z= its azimuth. Let the diagram (Tig. 22) represent the projection of the celestial sphere on the plane of the horizon of a place : 92 NAVIGATION. Z, the zenith of the place ; NZS, its meridian ; 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 PM,a declination-circle, through M. Then, in the spherical triangle PMZ, P Z = 90° — X, the co-latitude of the place ; P M =p = 90° — d, the polar distance of M; Z M = 90° — A, the complement of its altitude or its zenith distance ; Z P M = t, 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 whioh enter into astronomical problems, is called the astro- nomical triangle. As three of its parts are sufficient to deter- mine the rest, if three of the five quantities X, d, h, % and Z are known, the other two may be found by the usual formulas of spherical trigonometry. These admit, however, of modifi- cations which better adapt them for practical use. The fol- lowing articles point out how X, d, A, and t may be obtained. 117. The latitudes and longitudes of places on shore are given upon charts, but more accurately in tables of geograph- ical positions, such as are found in books of sailing directions, and in Table 49 (Bowd.). At sea it is sometimes necessary NAUTICAL ASTRONOMY. 93 to assume them from the dead reckoning brought forward from preceding, or carried back from subsequent, determina- tions. (Bowd., Art. 155.) 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 altitude circle. All measurements with instruments require correction for the errors of the instrument. Observed altitudes require reduction for refraction and parallax ; for semidiameter, when a limb of the object is observed ; and at sea, for the dip of the horizon. The reductions for dip and refraction are subtractive ; 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, 1. The reading of the instrument with which an altitude is measured ; 2. The corrected reading or observed altitude of a limb ; 3. The apparent altitude of the limb ; 4. When corrected for refraction and parallax, the true altitude of the limb ; 5. 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 plan- ets when their semidiameters are not taken into account, the altitudes of the limb and the centre are the same. For the moon, see Art. 59. Unless otherwise stated, the true altitude of the centre is the altitude which enters into the following problems, and is denoted by A. 94 NA VIGA TION. 119. The hour-angle of a body can be found when the local time and longitude, or the Greenwich time, are given. (Probs. 28, 29.) 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. 120. The declination of a body can be found when the Greenwich time is known. (Prob. 15.) The polar distance of a heavenly body is the arc of the declination-circle between the body and the elevated pole of the place ; that is, the north pole, when the place is in north latitude ; the south pole, when it is in south latitude. If P P' (Fig. 23) is the projection of the declination-circle through an object, M; P, the north pole ; P', the south pole ; E Q, the equator ; then the polar distances, P M = P Q-QM = 90° -d. F M = P' Q + Q M = 90° + d. Fig. 23. That is, the polar distance is 90° — d or 90° -f d, according as the pole from which it is reckoned is N. or S. This, how- ever, is regarding declination, like the latitude, as positive when N., negative when S. To avoid, however, the double sign in the investigation of the formulas of Nautical Astronomy, we shall in most cases consider the declination, which is of the same name as the latitude, as positive, and that which is of a different name ALTITUDE AND AZIMUTH. 95 from the latitude, as negative; hence the polar distance will be represented by * J p = 90° - d. When the declination is of a different name from the lati- tude, we have numerically p = 90° + d. ALTITUDE AND AZIMUTH. 121. Problem 32. To find the altitude and azimuth of a heavenly body at a given place and time. (Time- Azimuth.) Solution, Find the declination of the body and its hour- angle at the given time. (Probs. 15, 28, and 29.) Then in the spherical triangle PMZ (Fig. 24), we have N given to find PZ = 90° - Z, P M = 90° - d, ZPM = £, Z M = 90° - A, P Z M = Z. By Sph. Trig. (122), (123), if in the triangle ABC (Fig. 25), we have given b, c, and A to find a and B, we have tan ft — tan b cos A, cos (c — ft) cos b cos ft cos a = cot B = sin (c — ft) cot A sin ft Fig. 25. which, by substituting the corresponding parts of the triangle P Z M, give 96 NAVIGATION. tan = cot d cos t, sm^ 8in ^+ Z ) sillrf , COS cot Z = cos ( = 90° — r = tan c£ sec £, cos (d/ — X) sin c? sin A = ^—. — -f 9 sin ' I (74) „ sin ( r — L) cot £ cot Z = ^ -r^ cos <£ which afford the convenient precept, has the same name, or sign, as the declination, and is numerically in the same quad- rant as t. 122. When t = 6 h , ' = 90°, and the 3d of (74) assumes an indeterminate form. But from the 1st we have tan d cot t = tan ' sin t 9 which, substituted, gives cot Z = sin

— X) tan d (75) 123. h is the true altitude of M. If the apparent altitude is required, the parallax (Art. 54) must be subtracted, and the refraction (Art. 41) added. It is sometimes necessary to compute 'the altitude of one or both bodies, to use in connection with an observed lunar distance. The rules for this purpose in Art. 313 (Bowd.) are derived from the above formulas. The result is evidently more accurate, the smaller the hour-angle t, especially if the ALTITUDE AND AZIMUTH. 97 altitude is near 90°. In these rules it is best to find the " si- dereal time," or " right ascension of the meridian," from the mean local time, instead of the apparent (Art. 105). 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 east, which will re- quire in the above formulas — Z 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 perpendicular to the meridian, then , P m = = 90° - 4>' 9 Zm= (4> + Z) -90° = Z-<£'; or, <£ is the polar distance of m, , its zenith distance, positive, or of the same name as the latitude, toward the equator. A convenient precept is to mark it N. or S., according as the zenith is J5T. 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 a Ursae Minoris (Polaris), whose polar distance is 1° 25', the more convenient formulas derived from (73) will be, since p and <£ are small, =p cos t, (which gives within 0".5) sin h = sin (Z + $)52^£, 98 NAVIGATION. tan 7 — tan P si" t cos cos (X -f <£) ; or approximately, A = X + <£, Z = /> sin t sec (X -f- <£). Z is a maximum, or the star is at its greatest elongation, when the angle Z M P (Fig. 24), or Z n P (Fig. 30), is 90°. We then have sin Z = sin p sec X, or nearly Z = p sec X. 126. Problem 33. To find the altitude of a heavenly- body at a given place and time, when its azimuth is not required. Solution. The 1st and 2d of (73) or (74) may be used; or, by Sph. Trig. (4), cos a = cos b cos c -f- sin b sin c cos A, we have sin h — sin X sin d -\- cos X cos d cos t. (76) which, since cos t = 1 — 2 sin 2 J t, reduces to sin h = cos (JO — d) — 2 cos X cos e? sin 2 \ t. (77) (L — d) becomes numerically (X + <#) when X and d are of different names. Table 44 contains for the argument t in column p.m. the log sin i t in the column of sines; which, doubled, is log sin 2 \ t. It is well to note this ; for mistakes are often made by regarding the logarithms in this table as log sin, log cos, etc., of t instead of J t. 127. Problem 34. To find the azimuth of a heavenly body from its observed altitude at a given place. (Alti- tude-Azimuth.) ALTITUDE AND AZIMUTH. 99 Solution. In this the Greenwich time of the observation must be known sufficiently near for finding the declination of the body. The observed altitude must be reduced to the true alti- tude. Then in the triangle P Z M we have given the three sides to find the angle P Z M. In the triangle ABC, putting s = \ (a + b + c), we have, Sph. Trig. (33), 00Si v = J( 8ins8in ( s - b) ). * V \ sin a sin c J For the triangle P Z M, B = Z, a = 90° — h, h being the true altitude, b = p, the polar distance, c = 90° — X, the co-latitude, S = 90°-J(X + A-^), 5 _^ = 90° -* (Z+ h +p), and the formula becomes P)\ /c os % (X + h -f p) cos \(L -\-h cos X cos A cos \ Z== d\ or, if we put s' = £ (X + A + jt>), . ~ // cos / cos (&' cos 4 Z =i/ =-^- V \ cos X cos A which accords with Bowditch's rule, Art. 334. In a similar way we may find from the formula p) (78) sin J B /( sin (s- a) sin (s-c )\ V \ sin a sin c / sin \Z=J[ CQS £ ( coZ + * ± ji sil] £ ( CQ L + h - d) ^ V \ cos X cos A 100 NAVIGATION. in which coi= 90° — L ; or, if we put s" = £ (co L + A -f J), S iniZ = v /f coss " si °( s/, -^) \ l V \ cos L cos A / (78) is preferred when Z > 90° ; (79) when Z < 90°. If the body is in the visible horizon, then nearly h = - (36' 30" + 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 bearing is observed and the local time noted, the error of the compass can be found. For the true azimuth, or bearing, of the body can be found from its altitude (Prob. 34), or from the local time (Prob. 32) ; and the compass error is simply the difference of the true and compass bearings of the same object, determined simultaneously if the object is in motion. It is marked JEJ. when the true bearing is to the right of the compass bearing, W. when the true bearing is to the left of the compass bearing. (Bowd., Art. 323.) [In the triangle P M Z (Fig. 26), representing the positive angle, P M Z, by M, we have by Napier's " Analogies " tan J (Z — m) = cot \ t sin \ (L — d) sec J (L -\- d) ] tan J (Z + m) = cot J t cos \ (X — d) cosec J (L + d) J ^ ' The Azimuth Tables (Hydrographic Office, No. 71), issued by the Bureau of Navigation, were computed by means of (80). From them the azimuth of any heavenly body whose declina- tion does not exceed 23° may be found, its hour-angle, decli- nation, and the latitude being known. These tables afford a very simple as well as accurate method of ascertaining the azimuth, and are therefore spe- cially valuable for the usual compass-work on board ship.] ALTITUDE AND AZIMUTH. 101 129. The amplitude of a heavenly body 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, then, the complement of the azimuth. Problem 35. 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), A = W M, its amplitude. The trian- gle P M N is right angled at JT, and there are given PN = X, P M = 90° - d, to find N M = Z = 90° - A. We have cos P M = cos P N cos N M, or sin d = cos L cos Z, whence cos Z — sin A = sin d sec Z, (81) as in Bowditch, Art. 326. By (81) A is N. or S. like the declination. As the equator intersects the horizon of any place in the east or west points, it is plain that the star will rise and set north or south of these points, according as its declination is N. or S. Table 39 (Bowd.) contains the amplitude, A, for each 1° of latitude up to 70°, and each J° of declination to 30° computed by (80). The convenience of this table, in the case of the sun ? is the only reason for introducing amplitudes. It is generally best to express the bearing of an object by its azimuth. 102 NAVIGATION. In this problem the body is supposed to be in the true horizon, or about (36' + the dip) above the visible horizon. Hence the rule to " observe the bearing of the sun, when its centre is about one of its diameters above the visible horizon." Or, the body may be observed in the visible horizon and a correction (Table 40, Bowd.) applied for the vertical displace- ment. Art. 326 (Bowd.). Examples. (Probs. 32-35.) 1. 1898, Jan. 25, 2* 33 w 13 s local mean time in lat. 49° 30' S., long. 102° 39' 15" E. ; required the sun's true altitude and azimuth. (74) h m • Jan. 25, O's dec. Eq.oft.m « s L. m. t., Jan. 25, 2 33 13 - 18° 52' 54".3 +37". 34 -12 37.22-0.561 Long. —6 50 37 c 149.4 r 2.24 7.5 +2.40J .11 3.4 I .05 G-. m. t. Jan. 24, 19 42 36 I 3.4 I .05 or Jan. 25, — 4.29 18° 55' 34" —12 34.8 Eq. of t. — 12 34.8 L. ap. t. 2 20 38.2 t = 35 09 33 1. sec. 0.08748 1. cot. 0.15221 d= 18 55 34 1. tan. 9.53516 1. sin. 9.51101 f = 22 45 14 1. tan. 9.62264 1. cosec. 0.41254 1. sec. 0.03518 L= 49 30S <£'- L = -26 44 46 N 1. cos. 9.95085 1. sin. 9.65325 n. h= 48 29 30 1. sin. 9.87440 Z = S 124 43 W 1. cot. 9.84064 n. The reduction for refraction and parallax of h = 48°.5 is + 45" ; and the apparent altitude is h'= 48° 30' 15". If the compass bearing of the sun at the same instant had been N. 34° 20' W. = S. 145° 40' W., the compass error would have been 20° 57' W. ALTITUDE AND AZIMUTH. 108 2. 1898, July 20, 5* 5S m 20 s a.m., mean time in lat. 38° 19' 20" N., long. 150° 15' 30" E. ; required the sun's azimuth (75). a m s July 19, 0' s dec. Eq.ojt. „ , , L. m. t. July 19, 17 58 20 +20° 48' 47" — 27 // .48 -6 02.72 —0.169 Long. —10 0102 f 192.4 ,1.18 G. m. t. July 19, 7 57 18 J 24.7 -1.34 ) .15 = 7.955 _3 39 ' 6 1 1.4 ( .01 Eq. oft. - 6 04.1 +20 45 08.4 -6 04.06 ,. ap. t. July 19, 17 52 15.9 t = 91° 56' 02'' E. 1. sec. 1.47178 n 1. cosec. 0.00025 d = 20 45 08 N, 1. tan. 9.57854 1. tan. 9.57854 *'=" 95 05 23 N. 1. tan. 1.05032 n 1. cosec. 0.00171 L = 38 19 20 X. '-L = 56 46 03 N. 1. sin. 9.92244 Z = N 72 20 23 E. 1. cot. 9.50294 Entering Azimuth Tables (p. 89) with lat. 38, dec. 20° 45', we find by interpolation for 1. ap. t. 5* 52 m Az. = K 72° 14' E. In the same way with lat. 39° (p. 91), we find Az. = N. 72° 26' E. .-. The true azimuth for 38 J° = N. 72° 18' E. 3. At sea, 1898, May 20, 15* 23™ 16 s 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 -f- 3' 20", height of eye 18 feet, bearing of sun by compass 1ST., 41° 45' E. ; required the sun's azimuth and the compass error (78). G. m. t., May 20, 15* 23 m 16* = May 21, -8.6 O 10 15 20 rl. c + 3 20 58 -J S.d. + 15 50 I Par. + 9 Dip — 4 09 Ref . - 5 12 104 NAVIGATION, o h = 10 25 18 1. sec. 0.00723 O's dec. + 20 14 55 + 30.21 L = 40 15 1. sec. 0.11734 —4 20 (241.7 p= 110 10 35 +201035 | 18.1 2s= 160 50 53 s = 80 25 27 1. cos. 9.22103 8 — p = -29 45 10 1. cos. 9.93861 9.28421 h Z= 63° 59' 1. cos. \ 9.64211 True Z= S 127° 58' E. = N. 52° 02' E. Comp. bearing N. 41 45 E. Coinp. error 10 17 E. The 1. ap. t. is 8* 18 m nearly. Entering Azimuth Tables (p. 178) with lat. 40° and dec. 20°, by interpolation we find Z = S. 127° 58' E. In same manner with lat. 41° (p. 179) we find, at 8.18 a.m., Z = S. 128° 06' E. The true azimuth for lat. 40J° S. is then, by Table, S. 128° E. or N. 52° E. 4. 1898, Sept. 20, in lat. 30° 25' N., long. 50° 16' W., the compass bearing of the when its centre was in the visible horizon was S. 79° 30' W. Kequired the true bearing and the compass error (81). The 1. ap. t. of sunset for lat. 30 N. and dec. 1° K is 6* 02™. Azimuth Tables (Bur. Nav.), Table 10 (Bowd.). h m L. ap. t., Sept. 20, 6 02 Long. + 3 21 O's dec. Sept. 20 (Page I) + 0° 59' 06".3 - 58' '.35 G. ap. t., Sept. 20, 9 23 =r9.38 ,525 - 9 07 .4 ] 17 + 49 59 ' 4 .2 .5 .7 a = 50 1. sin. 8.16268 L = 30 25 1. sec. 0.06431 True azimuth = N. 89 02 W. 1. cos. 8.22699 Comp. bearing N. 100 48 W. Comp. error 11 46 E. HOUR-ANGLE AND LOCAL TIME. 105 Entering Azimuth Tables (p. 72) with lat. 30°, by interpo- lation for d. 50', we get the same result, Az. == N. 89° 02' W. , with lat. 31°, the Az. = N. 89° 03' W. Very nearly the same result is obtained from Table 39 (Bowd.), by interpolation. HOUR-ANGLE AND LOCAL TIME. 130. Problem 36. To find the hour-angle of a heavenly body in the horizon. Solution. In the diagram of the last problem, M P Z = t, the hour-angle ; and in the triangle PMN are given PM=5°-tfJ tofind MPN = 180 °-'' We have cos M P N = tJ tan P M ' whence cos t = — tan d tan L. (82) 131. Prom this it is apparent that when the latitude and declination have the same name, t > 6 A , and consequently that 2 t, or the time that the body is above the true horizon, > 12 A ; and when the latitude and declination are of different names, t< 6 h and 2 t < 12*. 2 t is an interval of sidereal time for a fixed star, of appar- ent time for the sun. In the case of the sun, t would be the apparent time of sun- set, were the refraction and dip nothing, and (24 ft — t ) would be the apparent time of sunrise. Table 10 (Bowd.) contains t, for different values of L and d. 106 NAVIGATION. 132. Problem 37. 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 Txreenwich 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. 28) are given P Z = 90° - Z, PM=jt>, Z M = 90° - A, to find ZPM = *. For the triangle ABC (Fig. 29), we have, (Sph. Trig., 31), sin i A = // sin(.-6)sin(.-c) \ y \ Bin o Bin c / in which, putting A = t a = 90° - A, b=p, c = 90° - X, we have s - b = 90° - J (Z + jt> + A), •— c = i ( Z +P~ h )> and sin J t = y/7 or, if we put / cos J (X + p -f A) sin i (Z +p - h) \ cos L sin j». / ' 8' = i(X+/>+^ . . . //cos / sin (s' — A)\ sin i « = 4 / —\ L , V \ cos L sm p J which is Bowditch's rule, Art. 262. (83) HOUB-ANGLE AND LOCAL TIME. 107 From Table 44 (Bowd.) we may take t directly from col- umn p. m., corresponding to the log sin -J- t. 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 is east of the meridian, (24 A — t) is numerically the apparent time. When the object is the moon, a planet, or a star, we have (Prob. 31), denoting its R. A. by a, the sidereal time = a -f i, and the mean time = a — S ' -f- t, in which SJ is the "right ascension of the mean sun." (Art. 93.) Or the sidereal time may be converted into mean time by one of the other methods of Problem 26. [The Sunrise and Sunset Tables (Hyd. Office, No. Ill) were computed by applying the equation of time to the local apparent times found by (83), assuming h = — 56' 08", (ref. - 36' 29" ; S.D. - 16' ; parallax, + 9" ; dip - 3' 48") for lati- tudes between 60° K and S. From them the local mean time of sunrise and sunset may be found, the declination and latitude being known.] 133. By the formula //sin 5 sin (s — a)\ a m , . COS I A = 1 /[ r—r\ L ), SPH. TRIG. (32), 2 V y smb sin c J v J we may obtain for the triangle P Z M (z being the zenith distance), \[ cosXsinjt? J or, putting s = \ (co L + p + z) (84) cos^ = v/f sin * sin(s - z) \ Y \ cos L sin p J 108 NAVIGATION. (84) is preferable to (83) when t considerably exceeds 6 h , which may be the case in high latitudes. If L = 90 °, the horizon and equator coincide, and p -\- h — 90° and p = z ; so that both (83) and (84) become indeter- minate. In very high latitudes, then, these equations approach the indeterminate form, and it is impracticable to find with precision the local time from an observed altitude. So also if d = 90°, the star is at the pole and L = h ; and the problem is indeterminate. A great declination is there- fore unfavorable. 134. If the object is in the visible horizon (rising or set- ting), h = — (36' -f dip) nearly. With the sun, the instants when its upper and lower limbs are in the horizon may be noted, and the mean of the two times taken as the time of rising or setting of its centre. The irregularities of refraction would affect nearly alike the dip and the apparent position of the sun. 135. If the time at which the altitude is observed is noted by a watch, clock, or chronometer, we may readily find how much the watch or chronometer is fast or slow of the local time. (Prob. 45.) For, let G be the time noted, T, the local time deduced from the observation : c = T — C will be the correction of the watch or chronometer to reduce it to apparent time, when T is the local appar- ent time ; to mean time, when T is the local mean time ; or to sidereal time, when T is the local sidereal time. 136. The observed altitude is affected by errors of obser- vation, errors of the instrument, and errors arising from the circumstances in which the observation is made ; such as irreg- HOUR-ANGLE AND LOCAL TIME. 109 ularities of refraction affecting both the position of the body and the dip of the horizon. Errors of the first class are dimin- ished by taking a number of observations. Thus several alti- tudes may be observed, and the time of each noted ; and the mean of the altitudes taken as corresponding to the mean of the times, so far as the rate at which the body is rising or falling can be regarded as uniform during the period of ob- servation. This period should then be brief. 137. We may easily find how much a supposed error of 1' in the altitude will affect the resulting hour-angle, by dividing the difference of two of the noted times by the difference in minutes of the two corresponding altitudes. The effect will evidently be least when the body is rising or falling most rapidly. This will be the case when its diurnal circle makes the smallest angle with the vertical cir- cle. An inspection of the diagram (Fig. 30) shows that this is the case when the object is nearest the prime vertical, or bears most nearly east or west. Thus Zn being tangent to the diurnal circle nn f 9 the angle which it makes with it is ; and is there- fore less than the angle which any other vertical circle, as Znf, makes with nn r . The diurnal circle m m r makes a smaller angle with Z m, the prime vertical, than with any other vertical circle, as Z m r . 110 NAVIGATION. The diurnal circle oo' makes a smaller angle with Zo than with Z d . The diurnal circles make right angles with the meridian ; so that at the instant of transit, the change of altitude is 0. 138. At sea, and to a less extent on the land, the latitude is uncertain. To ascertain the effect of an error of V in the assumed latitude, the hour-angles may be found for two lati- tudes separately, differing, say, 10' ; and the difference of these hour-angles divided by 10. This is an essential feature of Sumner's method, which will be explained hereafter. This method will also show that an error in latitude least affects the deduced hour-angle when the body is nearest the prime vertical. Examples. (Prob. 37.) 1. At sea, 1898, March 20, 10* 15 Hi 20 s G. mean time, in lat. 41° 15' S., long. 86° 45' W. (by account) ; observed p.m. altitude of the sun's lower limb 18° 20' ; index, cor. of sextant — 8' 20", height of eye 18 feet ; required the local mean time (82). G. m. t., Mar 20, 10* 15™ 20 s O's dec. Eq. of t. = 10.256 O / // // - 02 04.5 + 59.28 C 592.8 + 10 08.1 | U s l m s s 7 31.56 — 0.749 (7.49 -7.68 ] .15 ' .04 4- 08 04 7 23.9 Q = 18° 2(Y 00" 4-50 r S. d. + 16' 05" < Par. + 08 I. c. Dip. Kef. - 8' 20" -4 09 — 2 54 • 4- 16 13 -15 23 HOUR-ANGLE AND LOCAL TIME. Ill h = 18° 20' 50" L = 41 15 1. sec 0.12387 p = 90 08 04 1. cosec 0.00000 2 s = 149 43 54 s = 74 51 57 1. cos 9.41677 S-h= 56 31 07 1. sin 9.92120 9.46184 L. ap. t., Mar. 20, 4 h 20™ 28 s .3 1. sin £ 9.73092 Eq. of t. +7 23 .9 L. m. t., Mar. 20, 4 27 52.2 Subtracting the local mean time from the G. mean time gives the long. + 5 A 47" 27 s .8 = 86° 52' W. If we take L = 41° 25' S., we shall find the 1. ap. t. 4* 20" 1 12-.3 ; so that for A L = 10' S, A t = - 16 s . At sea, 1898, Sept. 2, S h 4 m 16 s , G. mean time, in lat. 46° 16' K, long. 152° 0' 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. ft m s o / // G. m. t., Sept. 2, 8 04 16 J 21 19 00 D 's dec. = 8 04.27 I. c. - 3 +8° 47' 01".8 +13".737 h m s r54 .9 Dip. - 4 23 + 58 .6 \ 2 .7 I 1 .0 S. D. - 15 55 +8 48 00 ft' = 20 55 42 S. diam. 15' 49" 4- 6' Par. and Ref . = + 51 37 H. P. 57 55 h = 21 47 19 L = 46 16 1. sec 0.16033 D 's R. A. 30 09.5 + 2.0957 p = 81 12 1. cosec 0.00514 r8.38 + 8.9 | .42 2 a = 149 15 19 ' .15 8 = 74 37 40 1. cos 9.42339 30 18.4 § - ft ^ 52 50 20 1. sin 9.90142 9.49028 112 NAVIGATION. D 's H. A. 4* 30'" 17 s . 1. sin ft 9.74514 L. sid. t. 5 00 35.4 - S 10 46 37.5 — Red. G. m. t. 1 19.6 L. m. t., Sept. 2, 18 12 38.3 Long. — 10 08 22.3 = 152° 05' 35" E. 139. Problem 38. To find the hour- angle of a heavenly body when nearest to, or on, the prime vertical of a given place. Solution. If d > Z, and with the same name, as for the body whose diurnal path is n n' (Fig. 30), PZw will be greatest, or nearest to 90°, when Z n is tangent to n n', and consequently Z up = 90°. We then have tan p tan L . > cos t = £- = . (85) cot L tan d If d < X, and with the same name, as for the body whose diurnal path is mm', the body will be on the prime vertical at m, and P Z m = 90° ; whence we have tan d /0 _x cos t = . (86) tan L If d and L are of different names, the diurnal circle inter- sects the prime vertical below the horizon, if at all, and the visible point nearest the prime vertical is in the horizon. The hour-angle of this point can be found by (82), omitting the effect of refraction, cos t = — tan d tan L. Altitudes less than 8°, however, are to be avoided. If d = L, the diurnal circle passes through the zenith, and the body would be on the meridian and prime vertical at the same instant ; so that ? when d and 1, are nearly equal, HOUR-ANGLE AJSfl) LOCAL TIME. 113 latitudes observed within a few minutes of the meridian passage of the body may be used for finding the time. It is only necessary that the change of altitude shall be suffi- ciently rapid. But when the body is very near the meridian in azimuth the change of altitude is proportional, not to the intervals of time, but to the squares of the hour-angles. (Art. 150.) Hence, when the body is in such a position, the mean of several times does not correspond to the mean of the alti- tudes. From the hour-angle the local time may be found by Problems 30 and 31. When the declination of the body does not exceed 23°, t may be taken from Azimuth Tables (Bur. Nat.) with suffi- cient accuracy for ordinary purposes. (See Art. 128.) Example. (Prob. 38.) 1898, June 25, in lat. 40° 15' N., long. 65° 17' W., re- quired the times when a Lyras and a Aquilse are on the prime vertical. a Lyrce. a Aquilce. L = +40° 15' 1. cot 0.07234 L = + 40° 15' 1. cot 0.07234 d=+3841.3 1. tan 9.90354 d=+ 836 1. tan 9.17965 t = q= l h 15™. 7 1. cos 9.97588 t = ^ 5 h 18 m .8 1. cos 9.25199 R.A. = 18 33 .5 R.A. = 19 45 .9 L. sid.t. = 17 17 .8 or 19* 49 m .2 14 27 .1 or l h 04™.7 — S ' 6 15 .3 6 15 .3 6 15 .3 6 15 .3 Sid. int. 11 02 .5 13 33 .9 8 11 .8 18 49 .4 red. -1.8 -2.2 - 1 .3 -3.1 L. m. t., June 25, 11 00 .7 13 31 .7 June 25, 8 10.5 18 46 .3 114 NAVIGATION. CHAPTER VII. LATITUDE. 140. Problem 39. To find the latitude from an ob- served altitude of a heavenly body on the meridian. Solution. Let the diagram (Fig. 31) be a projection of the sphere on the plane of the meridian NZ S: Z, the zenith; N S, the horizon ; P, the elevated pole ; PF, the axis of the sphere; EQ, the equator; Q Z, the declination of the ze- nith, and NP, the altitude of the pole, are each equal to the lati- tude, L. Fig. 31. Let M be the position of the body; QM = d, its declination ; M Z = z = 90° — h, its zenith distance, which it is conve- nient to mark N. or S., according as the zenith is north or south of the body. From the diagram, we have QZ = QM-fMZ LATITUDE. 115 or, L = z + d, (87) which is the general formula. If the body is at M', numerically L == z — d\ if at M", Z=d—z; or " the latitude is equal to the sum of the zenith distance and declination, when they are of the same name ; to their difference, when of different names ; and is of the same name as the greater." If the body is at M"', or below the pole, Q M'" = 180° - d, and L = 180° - d - 2, numerically ; or (87) is the correct formula, provided we use 180° — d, or the supplement of the declination, instead of the declination. But in this case we have also from the diagram L=p + h, (88) as in Bowditch, Art. 274. The declination of the body must be found from the Alma- nac for the time of meridian passage. (Probs. 17, 21.) The observed altitude must be corrected for dip, refraction, etc., and the true altitude derived. From (87) we see that an error of V in the altitude will produce an error of V in the resulting latitude. Examples. 1. At sea, 1898, June 30, in lat. 2\° N., long. 105° 18 ; W., the observed meridian altitude of the sun's lower limb was 69° 15' 20", sun bearing N. ; index cor. + 3 ; 20" ; height of eye, 20 feet ; required the latitude. 116 NAVIGATION. Long. -M rhlm 12 s = 7 A .02 69° 15' 20" fin. cor. + 3' 20' + 14 24 J S. diam. + 15 46 |Dip — 4 23 I Ref. & p. — 19 O's dec. 23° 1(Y 11".6 N. - 9". 32 - 1 5 .2 65".a d = 23 09 06 W. 29 44 2 = 20 30 16 S. L= 2 38 50 N. 2. At sea, 1898, June 30, in lat. 43£° N., long. 150° 15' E. O = 69° 15' 20'' ; on meridian bearing S. ; index cor. + '3' 20" ; height of eye, 20 feet ; required the latitude. Long. — 10 A l m s = — 10 A .02 © 69° 15' 20" / In. cor. + 3' 20" Q's dec - I S. diam. +15 46 23° 10' 11 ".6 N. - 9".32 + 14 24 -, D . p _ 4 23 +1 33 .2 93".2 [ Ref. & p. - 19 d = 23 11 45 N. A = 69 29 44 z = 20 30 16 N. £ = 43 42 01 N. 3. At sea, 1898, Aug. 9, about 5 a.m., in lat. 17° 40' S., long. 85° 15' W., obs'd mer. alt. of >>s upper limb, 50° 18 ; moon north ; index cor. — 2' ; height of eye, 16 feet ; required the latitude. Long. + 5 h 41™ = 5 A .68 D 's mer. pass. Aug. 8, 17* 38 m .9 + 2™. 03 D 's S. diam. 15' 03"+ 11" f 10"U5 Red. for long. +11 .5-j 1.22 p'sH.P. 55 08 .16 L. m. t., Aug. 8, 17 50 .4 G. m. t., Aug. 8, 23 31 .4 F = 50° 18' ( I.e. - 2' 00' D's dec. + 21° 47' 11"9 + 7".165 - 21.09 )s.d. -15 14 r 214". 9 (Dip - 3 55 + 03 45 ] 7 .2 •' 2 .9 Zi , =49 56' 51" d = 21° 50' 57" N. + 34 40 Par and Ref. z = 39 28 29 S. „ h =50 31 31 L = 17 37 32 S. LATITUDE. 117 4. At sea, 1898, Oct. 9, 5 p.m., in lat. 65%° K, long. 150° E. ; obs'd mer. alt. of a Lyrae 63° 17', bearing S. ; index cor. -J- 3' 30" ; height of eye, 17 feet ; required the latitude. *'s alt. 63° 17' f In. cor. + 3'.5 _ 1 \ Dip - 4 .0 h= 63 16 l ** - .5 z =26 44 N. d = 38 42 N". L = 65 26 N. If the star bore K, the latitude would be 11° 58' N. 5. At sea, 1898, June 18, in lat. 23J° K, long. 163° 0' E. ; obs'd mer. alt. of O's N. limb from N. point of the horizon, 89° 50' ; index cor. + V 20" ; height of eye, 21 feet ; required the latitude. Long. - 10* 52™ s = - 10*.87 O's dec. 23° 25' 25" N". + 3".02 89° 50' ( In cor. + 1'.3 - 33 + 12 .6 ] S. diam. + 15 .8 (Dip -4.5 a = 23 24.9 N. h = 90 2 .6 z = 2.6 N. L = 23 27.5 N. In this example, the corrected altitude of the O's centre is more than 90° ; this changes the sign of z. 6. At sea, 1898, May 18, in long. 180° 0' E., the true mer. alt. of the sun was 75° 18' ; sun bearing S. ; required the latitude. Long. - 12* / 0" d = 19° 30'. 5 N. z = U 42 N. £ = 34 ly.SN. 7. At sea, 1898, May 17, in long. 180° 0' W. ; the true mer. alt. of the sun was 75° 18' ; sun bearing S. ; required the latitude. Long. + 12*0*0' d = 19° 30'. 5 N". z = U 42 N. £ = 34 12'.5N. 118 NAVIGATION. Examples 6 and 7 are identical, the Greenwich apparent time being May 17, 12 h for both. They illustrate the neces- sity as well as propriety of the rule for navigators near the meridian of 180°, to add l d to the date when they pass from west longitude to east; to subtract l d from the date when they pass from east longitude to west. For instance, May 18, 5 h in long. 180° 15' E., is identical with May 17, 5 h 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 the highest altitude is attained ; soon after which the 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 m or 2 m . 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 m 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 instant of meridian passage; but after, if the body and zenith are approaching; before, if they are separating. Let t be the hour-angle of this culminating point, in minutes ; LATITUDE. 119 A d, the combined change * of declination and latitude in l m , if it is expressed in seconds ; or in l h if it is expressed in minutes / A h, the change of altitude in l m from the meridian passage due solely to the diurnal rotation (from Table 26) ; A A, the reduction of the maximum altitude ; both expressed in seconds. Now in the time t t A d will be the excess of altitude produced by the change of declination and latitude ; f A A (as will be shown in Art. 150), the diminution of alti- tude due to diurnal rotation ; and we shall have A A = t A d — t 2 A A. But at a point whose hour-angle is 2 t, the altitude will be the same as the meridian altitude, or = 2t&d- (2*) 2 A A; Ad whence t = 2A A' (89) which accord with the rule in Bowditch, Art. 277. and Ah= htAd 2 4 A A Example. A ship in lat. 62 K, on March 21, sails south 14 miles per hour. A d = 14' + r = 15' per hour, or 15" per minute ; A A = 1".0; * Their sum, if they both tend to elevate or both to depress ; other- wise their difference. 120 NAVIGATION. = 7^; AA = "X 15" = 56.' .-■If -T|P, A * -Ml The uncertainty of altitudes at sea makes such a correc- tion of little practical importance; but it is generally ne- glected by those navigators who work out their latitudes to seconds, supposing that they have attained that degree of accuracy. In the above example, the maximum altitude of the sun would have been greater than the meridian altitude, and the latitude obtained from it in error, by nearly 1'. The sun would not have sensibly dipped until 9 or 10 minutes after noon. 143. A difficulty occurs at sea in measuring the meridian altitude of the sun when it passes near the zenith, on account of its very rapid change of azimuth ; the change being made from east to west, 180°, in a very few minutes. What is wanted is the angular distance of the sun from the N". or S. points of the horizon. One of these points may be sufficiently fixed by means of the compass, and then the angular distance from this point observed within l m or 2 m of the meridian passage as determined by a watch regulated to apparent time. 144. From (87) we have z = Z — d, (90) by which the zenith distance may be found when the latitude and declination are given. Also d = L — z, which may be used at sea for estimating the declination of a bright star from its estimated meridian altitude. If the time when it is near the meridian be also noted, and converted into sidereal time, we have the right ascension and declination of the star sufficiently near for determining what star it is. LATITUDE. 121 Example. July 16, S h 45 m , in lat. 11° N., a bright star is seen near the meridian S., at an estimated altitude of 55°. L. m. t. July 16 S h 45™ i = ll°N. S 7 37 z = 35 N. L. sid. t. 16 22 d = 24 S. The R. A. of a Scorpii (Antares) is 16* 23 m , and its declina- tion 26° 13' S. 145. Problem 40. To find the latitude from an alti- tude of a heavenly body observed at any time, the local time of the observation and the longitude of the place being given. 1st Solution. Reduce the observed altitude to the true al- titude, and from the local time and longitude find the declination and hour-angle of the body. (Probs. 21, 28, 29.) Then in the triangle P Z M (Fig. 32) there are given ZPM = «, P M = 90° - d, Z M = 90° - h, to find P Z = 90° - L. By Sph. Trig. (146), if in the triangle ABC (Fig. 33) are given a, b, and A, we find c by the formulas tan = tan b cos A, ,, cos d> cos a cos = - — - — , cos b which, applied to the triangle P Z M, give 122 NAVIGATION. tan = cot d cos t, ., cos sin h cos = - — , sin d 90° - L mm ± <£'. (91) These may be changed into a more convenient form for practice, if we put = 90° — <£" ; then tan " = tan d sec £, ., sin " sin A cos = ■? , sin « Z = 4," =f *'. (92) Here, observing that -f an d — may be rendered by N. and S. respectively, we mark " N. or S. like the declination, and either N. or S. ; then the sum of " and ' when of the same name, their difference when of different names, is the latitude, of the same name as the greater. There are two values of L corresponding to the same altitude and hour- angle, but which, unless tf is very small, will differ largely from each other ; so that we may take that value which agrees best with the supposed latitude (at sea the latitude by ac- count). When t > 6*, $>" > 90°, as in (74). is positive if Z > 90°, and negative if Z < 90° ; the sign of may therefore be determined by the bearing of the body. 146. In Fig. 32, if M m be drawn perpendicular to the meridian, we shall have =P»i, the polar distance of ra, 4>" = 90° - P ra, the declination " ' = Z m, the zenith distance " When is very small (that is, when M m nearly coin- cides with M Z), ' cannot be found with precision from its LATITUDE. 123 cosine. If not greater than 12% it can be found only to the nearest minute with 5-place tables ; if only 2°, it can be found only within 3'. The more nearly, then, that M m coin- cides 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, ' cannot be found within 30'. 147. To find the effect of an error in the altitude, differen- tiate the second equation of (91), regarding and h as varia- bles ; and we have d n = - sin *" coah dh sin d sin or, since cos _ sin f/ sin h sin d ' d f = - d A. cot $ cot h. (93) But in the triangle MZm, iv/r rj M7T) tan m Z cos MZm=- cos M Z P tan M Z ' that is, Z being the azimuth, — cos Z = ?-, or sec Z = — cot d> cot h, (94) cot h ,< N ' and therefore, d <£' = d A sec Z. (95) If the body is on the meridian, Z = or 180°, and numer- ically d '= d A. The nearer Z is to 90°, the greater is d '. If Z = 90°, or the body is on the prime vertical, sec Z = oo, and d when the body is on the meridian ; and the effect is incalculable when Z = 90° or 270°, or the body is on the prime vertical. Moreover, the effect is opposite on different sides of the meridian, and would be eliminated by two observations of the same body, or of different bodies, at the same azimuth E. and W. of the meridian. LATITUDE. 125 149. 2d Solution. If the latitude is already approxi- mately known, we have (76) sin A = sin L sin d -j- cos L cos d cos t ; whence cos (X — d) = sin A + 2 cos L cos 6? sin 2 £ t ; or since (L — d) is the meridian zenith distance of the body, (87), denoting it by z , and the meridian altitude by A , we have cos z = sin A = sin A + 2 cos i cos rf sin 2 J £, (97) in which we may use the approximate value of L in comput- ing the term 2 cos L cos d sin 2 £ £, which term is smaller the nearer the observation is taken to the meridian. Having found the meridian zenith distance, we may find the latitudes as in Problem 39. If the computed value of L differs largely from the assumed value, the computation should be repeated, using this new value. 150. 3d Solution. Reduction to the Meridian. When the observation is taken very near the meridian, we may find the correction to be applied to the observed altitude to reduce it to the meridian altitude, thus : From (97) we have sin h — sin h = 2 cos L cos d sin 2 \ t, whence, by Pl. Trig. (106), cos J (A + A) sin \ (A — A) = cos L cos d sin 2 -J t. But A and A differing very little, we may put cos i (A -J- A) = cos A = sin z = sin (Z — d), so that , ,, _. cos L cos d sin 2 4- t , „ 3 m H *o-A)= gin(Z _ rf) * («) 126 NAVIGATION. Put A h = h — h, the reduction of the observed to the meridian altitude, or, as it is usually called, The reduction to the meridian ; and, since A h and t are quite small, put sin J A h — -J A h sin 1" (A h being expressed in seconds of arc), sin \t = \t X 15 sin 1" (t " " " of time), then (98) reduces to . , 112.5 sin 1" cos L cos d sin (L — d) or, since sin 1" = 0.000004848, A . /r .000545 cos Z cos d ... A A = : — 7-= — — r;^ X t* (t w seconds). sin (L — d) v J In this formula t is in seconds of time ; but if, as is usual, t is expressed in minutes, we must put (60 t) 2 for t 2 , so that we have A . 1".96349 cos L cos d Ah= :— ry ^r X« 2 (99) sin (L — d) K J If ^ = l m , the formula expresses the change of altitude in one minute from the meridian. Representing this by A h, we have A . 1".96349 cos L cos d , \ \ h = — Tf T\ ( 10 °) sin (L — d) Ah = t 2 \h, and (101) h Q — h + A h, the meridian altitude. J Whence the latitude is found as by a meridian altitude (Prob. 39). Art. 278 (Bowd.). Bowditch's Table 26 contains the values of A h for each 1° Of declination from to 24°, and each 1° of latitude from LATITUDE. 127 to 70° ; except when L — d < 4°, for then A A is so large that (99) and (100) become inaccurate. In this case the body is near the zenith, and altitudes out of the meridian do not afford a reliable determination of the latitude. Bowditch's Table 27 contains t 2 for each l 5 of t from to 13™. When h is small, the reduction to the meridian may be found by this method quite accurately, even when t is as great as 12™. If h is near 90°, t must be taken within much nar- rower limits. Indeed, in this case z , or its equal (i — d), is very small, and consequently A h becomes large. Unless then t is sufficiently small, A h will be too great for the assumption sin J A h = A h sin 1". If d > X, sin {L — d) = sin z is negative ; that is, z will have a different name or sign from L (Art. 140). Properly h, h , and A A would also become negative to correspond. Still, however, we shall have numerically h = h + A h. We may therefore disregard the sign of L — d in (100), and consider h and h as always positive. If the star is observed at its lower culmination, then t will be the hour-angle from the lower branch of the meridian, and for d we may use 180° — d (Art. 140). A h and A h are then numerically subtractive. Examples. (Prob. 40.) 1. At sea, 1898, July 17, l h p.m., in lat. 36° 38' S., long. 105° 18' E., by account ; time by Chro., 5" 47 TO 14 s ; Q , 30° 15' ; N*. W'y ; index cor. + 2' 30" ; height of eye, 17 feet ; Chro. cor. (G. m. t.) -f 14 m 3 5 ; required the latitude. (By 128) - 128 NAVIGATION. T.byChro. + 12*,17 47 14 Chro. cor. + 14 3 G. m. t. July 16 18 117=18.021 - Long. +7 1 12 L. m. t. July 17 1 2 29 &sdec. 11 tk Eq. oft. VI th + 2°1 10 03.6 —25.7 -5 53.51 +6.214 - 6 -6 -1.28 + 2 34.2 154.2 + 2112 37.8 -5 54.79 Eq. of t. -5 54.8 30 15 ( I. c. + 2 30 Dip -4 02 L. ap. t. 56 34.2 +12 43 Js. d. + 15 47 Ref. & Par -1 32 h =30 27 43 1. sin 9.70497 t* = 14 08 33 1. sec 0.01337 d = 21 12 38 1. tan 9.58892 1. cosec 0.44154 f ' = 21 48 42 N. 1. tan 9.60229 1. sin 9.57002 f = 58 37 32 S. 1. cos 9.71653 L = 364850 S. If we suppose an uncertainty of 3' in the altitude and 20' in the longitude, by (94), (95), and (96) 1. cot (-h) 0.2305 n I. cos L 9.903 1. cot # 9.7852 — d t = —20' log 1.301 n Z = S. 164° 40' W. 1. sec Z 0.0157 n 1. tan Z 9.438 n d/i^+3' log 0.477 — di=+4 , .4 1og 0.642 di=-3 , .l log 0.493 n That is, an increase of 3' in the altitude will numerically decrease the latitude 3'.1 ; and a numerical increase of 20' in the assumed longitude will increase the latitude 4'.4. This may be conveniently expressed in the following way : Long. 105° 18' ± 20' E. ; Q, 30° 15' i & L = 36° 48'.8 i 4'.4 =f 3M S. * Instead of changing t into arc, we may enter col. p.m. of Table 44 with 2 t = 1* 53 m 22 s . LATITUDE. 129 2. At sea, '1898, Dec. 6, about 5 a.m., in lat. 50° 30' K, long. 135° 25' W. (by account), time by Chro. 2 h 00™ 52 s ; Chro. cor. (G. m. t.) - index cor. — 3' tude. (92.) 12™ 34 s ; Obs'd alt. of Mars, 58° 10' S. W'y ; height of eye, 19 feet ; required the lati- T. by chro. Chro. cor. G. m.t.,Dec. 6, S Ked. for G. m. t. G. sid. t. Long. L. sid. t. Mars' K. A. t = 2 00 52 - 12 34 1 48 18 = 17 01 10.2 17.8 18 49 46 9 0140 9 48 06 8 47 17.3 100 48.7 1 A .8 h m s s Mars' R.A. 8 47 16.3 + 0.578 + 1.0 ( .58 8 47 17.3 ) .46 t a 4? 4>' = 15° 12' 11" = 20 49 38 N. = 21 30 53 N. = 28 56 35 N. = 50 27 28 N. 1. sec 0.01548 1. tan 9.58025 1. tan 9.59573 Mars' dec. + 20 49 29.3 + 4.66 + 8.4 1 4.7 + 20 49 37.7 I 3.7 tf= 58° 10' ( I. c. -3.00 - 7. 48 J Dip —4.16 (P. and R.-. 32 h = 58 02.12 1. sin 9.92860 1. cosec 0.44910 1. sin 9.56436 1. cos 9.94206 If d h = + 5' and d A = + 15', d t = - 15'; and by (94), (95), and (96), 1. cot (-h) = 9.79491 n 1. cor <£' = 0.25722 Z = N. 152° 29' W. 1. sec Z = 0.05213 n dh = + 5' log. 0.69897 d^ = dl= - 5'.5 0.74110 n 1. cos L 9.80388 -dt = +15' log 1.17609 1. tanZ 9.71679 n dl = - 5' log 0.69676 n 3. 1898, May 15 ; in lat. 41° 30' K (approx.) ; long. 4* 47™ 30 s W.j obs'd alt. O 67° 18', bearing S'ly. j Chro. T. 5* 38™ 28* ; Chro. cor. (G. m. t.) - 51- 48 5 .5 ; i.e., - V 50" ; height of eye, 25 ft. Eequired the latitude. (101.) 130 NAVIGATION. h m s Chro. T. 5 38 28 O'.s dec. (Page 1L). Ch. cor. — 51 48.5 + lb° 56' 20".4 +35". 2 G. m. t. 4 46 39.5, May 15, 4*.78 r 140".8 Long. 4 47 30 +2 48 .2] 24 .6 L.m.t 23 59 09.5, May 14 + 18 59 08 .6 I 2.8 Eq. t. +3 51 O 67°18 / 00 // < S. D. +15' 51" L. ap. t 03 00.5, May 15 + 8 47 ] Par. + 4 t = 3 00.5 ^=67 26 47 ' £ 2 =9 (Table 27.) t* A h +32.4 A h = 3.6 (Table 26.) h = 67° 27' 19".4 j .10 z = 22 32 40 .6 1ST. .02 ffgX+ 3 51.15 -0.025 N". ~ d = 18 59 08 .6 N. +3 51 i = 41 31 49 N. I. c. - V 50" Dip —4 54 Kef. - 24 151. Problem 41. To find the latitude from a num- ber of altitudes observed very near the meridian, the local times being known. Solution. By (101) we see that very near the meridian the altitude of a body varies very nearly as the square of its hour- angle. Hence we cannot regard the mean of several altitudes as corresponding to the mean of the times, since this is assum- ing that the altitude varies as the hour-angle. Let, h x , h 2 , A 3 , etc., be the several altitudes; h > h > h> etc., the corresponding hour-angles expressed in minutes / and we have as the reduction of each altitude to the meridian, and the deduced meridian altitude, & 1 h=t'i.A h h = h 1 + b 1 Ji\ A 2 A=«1.A A A = A 2 + A 2 M etc. (102) A 3 h = t\ . A h etc. h = h 3 + A 8 h) Thus the meridian altitude may be derived from each alti- LATITUDE. 131 tude, and the mean of all these meridian altitudes taken as the correct meridian altitude. But the following is a more expeditious method : — If n is the number of observations, the mean value of h will be - = At -f A 2 + A 3 -I h n A t A + A 2 A -j - A 3 A j A„A n ^ or, X ._ ^1 ~T~ "2 H~ '*3 4~ • • • jjjl _|_ ^1 H~ ^2 H~ ^3 ~j~ • • • tn £ fo (10$) n n Whence the rule : Take the mean of the squares of the hour-angles in min- utes (Table 27, Bowd.) ; multiply it by the change of altitude in l m from the meridian (Table 26) ; and add the product to the mean of the altitudes. The result is the mean meridian altitude required. (Bowd., Art. 278.) From the meridian altitude thus found, deduce the latitude as from any other meridian altitude. (Prob. 39.) Strictly, however, the dec- lination to be used is that which corresponds to the mean of the times, and the hour-angles, t, are intervals of apparent time for the sun, and of sidereal time for a fixed star. 152. It is unnecessary to reduce each observed altitude separately to a true altitude ; as the reductions, excepting slight changes of refraction and parallax, are the same for all, and may be computed for the mean of the observed altitudes, and applied to this mean with the reduction to the meridian. 153. Should it be desirable to compare the several obser- vations with each other, and test their agreement, it will be sufficient to compute the several reductions to the meridian, 132 NAVIGATION. Ai A, A 2 A, A 3 h, etc., and apply them separately to the read- ings of the instrument; or to the half-readings when the altitudes are observed with an artificial horizon : applying, also, the semidiameter when both limbs of the body are observed. 154. If the altitudes are taken on both sides of the me- ridian, and at nearly corresponding intervals, a small error in the local time will but slightly affect the result ; for such error will make the estimated hour-angles and the corre- sponding reductions on one side of the meridian too large, and on the other side too small. 155. This method is rarely used at sea, as a single alti- tude on or near the meridian suffices. No increase of the number of observations will diminish at all the error of the dip, which affects alike each observation and the mean of all.* But on land it is preferable to measure a number of altitudes at the same culmination of the body, and thus diminish the "error of observation." Altitudes of the sun are used, but the best determinations are from the altitudes of a bright star. To facilitate the operations, and avoid mistaking one star for another, it is well to compute the altitude approximately beforehand. (Art. 144.) If an artificial horizon is employed, the error of the roof is partially eliminated by making two sets of observations with the roof in reversed positions. 156. If two stars are observed which culminate at nearly the same altitude, one north, the other south of the zenith, * Such an error is called constant ; those which affect the several observations differently are called variable. LATITUDE. 133 the error of the instrument is nearly eliminated ; for such error (except accidental error of graduation) will make the latitude from one of the stars too great, and that from the other too small by very nearly the same amount; the more nearly, the less the difference of the altitudes. The error peculiar to the observer is also eliminated. If the observations are made with an artificial horizon, the error of the roof is eliminated if the same end is toward the observer in both sets of observations. 157. Bowditch's Table 26 extends only to d = 24°. If a star is used whose declination is beyond this limit, or if greater precision than the table affords is required, A A may be computed for the star and place by (100). 1".9635 cos L cos d A h = sin (L — d) 158. If the observations are made at the lower. culmina- tion of the star, we have only to use in the formulas 180° — d instead of d. (Art. 140.) The altitudes observed at the same culmination are very nearly the same. To render the measurements independent, after each observation move slightly the tangent screw of the instrument. With the sextant, it is best to make the final motion of the tangent screw at each observation always in the same direction ; for example, in advance. Example. (Prob. 41.) 1898, May 22, 9 A , circum-meridian altitudes of a Virginis (jSpica) at lighthouse on St. George's Island, Apalachicola Bay, Florida, lat. 29° 37' K, long. 85° 5' 15" W. 134 NAVIGATION. T. BY CH. Sext. No. 1. Art. Hor.No. 1. h m 8 o / // 3 28 56 2 alt. 99 22 50 A. end. In. cor. — 3' 0" 3124 26 50 Bar. 30.04, Ther. 73° 33 36 30 00 Chro. cor. (L.m.t.) + b h 37 m 14».55 34 56 32 40 Long. + 5 40 21 37 8 32 40 38 58 35 00 B. end. 42 45 34 30 *'sR. A. 13* 19™ 52*. 23 44 33 31 10 - which, substituted in (105), gives A = i (A + A') + [J t 2 + ^o 2 ] A A. (106) The difference of the two equations of (104) gives A - A' = (I 7 ' 2 - T 72 ) A A = 2^ A A. Hence, Substituting this in (106), we have K = i(h + h') + (i ty a h + ^r^ 3 ' - (los) 136 NAVIGATION. The reduction to the meridian, then, is effected " by adding to the mean of the two altitudes two corrections; 1st, the quantity (££) 2 A A, which is nothing more than the, common reduction to the meridian (101), 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 combined, and the mean of the meridian altitudes taken, from which the latitude can be obtained as from an observed meridian altitude. 160. The restriction of the method corresponds with that of reduction to the meridian (Art. 150). * Quite accurate re- sults can be obtained with hour-angles limited to 5 m when the altitude is 80°, to 25 m when the altitude is only 10°. If the interval t, however, exceed 10 w , A h should be computed to two or three places of decimals, as it is given in Table 26 (Bowd.) only to the nearest 0".l. The accuracy of the method depends mainly upon the ac- curacy of the second correction, and therefore upon the pre- cision with which the difference of altitudes has been obtained. The altitudes, then, should be observed with great care. Errors of the tabulated dip and refraction, and a constant error of the instrument, will affect both altitudes nearly alike. If the altitudes are equal, this second correction becomes 0. The most favorable condition is, therefore, that of equal altitudes 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 * Table 26 (Bowd.) gives A h only to the nearest 0".l; if, then, it is taken from this table, A h t 2 may be in error 1", if t>4 m . If, however, Ao^ is computed to the nearest 0".001, the error of using A h t 2 will not exceed 1", unless t > 20 m and /i>60°. LATITUDE. 137 uncertainty of the reduction of one of the altitudes for the run of the ship. 161. The hour-angle of either altitude may also be ob- tained approximately ; for we have from (107), in minutes, T _ \Jh - W) °~ HA.* and (Art. 299, Bowd.) (109) Example. Sept. 3, 1898, in lat. 37° 30' N., long. 5* W., by account, observed two altitudes, near noon, for latitude. C. time 10* 30™ 21 s ; observed alt. 59° 43' 40" (South). M u 1Q h 35m 36 s. u u M 590 33/ 40 // I. c. — / 30 // ; ht. eye, 18 ft. o / // h m 8 O's dec. h = 59 43 40 C. t x 10 30 21 B s „ h' = 59 38 40 C. U 10 35 26 + 7 27 23.8 - 55.15 4 1 15 t = 05 15 —04 35.8 + 5 = 59 41 10 ~ = 02 37.5 + 7 22 48 2 2 1. c. - 30 S. D. + 15 54 (U» 6.89 Dip - 4 09 A h = 3.06 log 0.48572 Par. and Kef. -0 30 /A 2 h = 59 51 55 = 6.89 log 0.83822 1st cor. =21.08 log 1.32394 1st cor. 21.1 [ L ~a L ) ~ 56,25 log 3.75012 2d cor. 4 26.8 2d cor. =266.80 log 2.42618 h 59 56 43 z 30 03 17 N. d 7 22 48 1ST. Lat. 37 26 05 N. 138 NAVIGATION. 162. Problem 43. To find the latitude from an ob- served altitude of Polaris or the North Pole-star. Solution. The formulas (91) of Prob. 40. tan = cot d cos t ., cos <£> sin h cos = £ sin d 90° - L = cf> -|- ' can be greatly simplified in the case of the Pole-star, since its polar distance is only 1° 25'. Putting d = 90° - p and tf = 90° - <£", we have tan = tan p cos t or <\> = p cos t (within 0".5) j ft • z. cos <£ sin <£ = sin h - cosp L = " — , the 2d value of X, or (180° — " — ), being excluded, as it exceeds 90°. p and are so small, that the cosine of each is nearly 1, and consequently sin <£" = sin h and " = A, nearly. Thus we have =p cos t (110) L = h-cf> l If £ is more than 6* or less than 18 ft , cos £ is negative, and we have numerically Let S represent the sidereal time, and a the right ascen- sion of the star, then t=S—a and =pcos (S — a). If we consider the right ascension and polar distance of the star to be constant, may be computed and tabulated for dif- LATITUDE. 139 ferent hour-angles of the star, as in Table 4, Appendix, in the Nautical Almanac. Owing to the change of right ascension and declination, such a table requires correction for each year. It will furnish an approximate value of the latitude ; but it is more accurate to take the apparent right ascension and decli- nation from the Almanac, and compute t and . may be found approximately in the traverse table (Table 2) in the Lot. col., by entering the table with t as a course, and p as a distance. Formulas (111) may be derived from Fig. 34, by regarding P M m as a plane triangle, and Z m = Z M. The first produces no error greater than 0"5. The error of the sec- ond is evidently greater the greater the altitude, or the latitude. This error, however, will not be more than 0'.5 in latitudes less than 20°, nor more than 2' in latitudes less than 60°. 163. We may use (110) with more exactness, but these formulas may be modified so as to facilitate computation. Put <£" = h -f A h then, changing the 2d of (110) to a logarithmic form, we have log sin (A -\- A h) = log sin h -f log cos — log cos p, log sin (h + A h) — log sin h = log sec p — log sec <£. But A h being very small, representing by D /y the change of log sin h for V , we have, with A A in seconds, log sin (A -j- A h) -- log sin h = A h X D tl ; whence, by substituting in the preceding, we obtain 140 NAVIGATION. A h = log sec P — lo g sec j = lQ g cos j — lQ g cos ff - / 112 ) ss a The difference of the log secants, or log cosines, of p and <£ is readily taken from the table by inspection. D /y for log sin h is usually given in tables of 7 decimal places, and hence A h is readily found. We have then =p cos t Z = h+Ah- v If D y is the change of log sin h for 1', then in minutes T) AA Jo g sec^-logsec f 164. Bowditch * contains four tables (28 A, B, C, D) for the reduction of altitudes of Polaris, from which they may be found to the nearest second. (Art. 287, Bowd.) 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'. Example. (Prob. 43.) 1. At sea, 1898, March 31, 7* 15 m 19*, mean time in long. 160° 15' E. ; obs'd alt. of Polaris 38° 18' ; index cor. + 3' ; height of eye, 17 feet : what is the latitude ? (113.) L. m. t., Mar. 31, 1 h 15 m 19* Long. — 10 A 41™ S 35 31.3 h'= 38° 18' 00" I.e. + 3' Red. for long. — 1 45.3 - 2' 16" Dip - 4' 02" Red. of L. m. t. +1 11.5 h = 38 15 44 Ref. - 1 14 L. sid. t. 7 50 16.5 *'sK.A. 1 20 47.8 t = 6 29 28.7 t = 97° 22' 11" 1. cos 9.10813 n p = 73 .9 log 1.86864 i = ^- < ^ + 38°25 , .2 <£ = — 9 .5 log 0.97677 n * Chauvenet's Astronomy, I, 256. LATITUDE. 141 By Table IV, Naut. Alm., App. L. sid. t. l h 5CM h = 38° 15'.7 Less 1 21 .8 Cor. per Table IV. + 9 .9 H. a. 6 28 .5 L = + 38 25 .6 ^ X 3.5 = 1.1 8.8 + 1.1 = 9.9 142 NAVIGATION. CHAPTER VIII. THE CHRONOMETER. — LONGITUDE. 165. Astronomically the longitude of a place is the dif- 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. 73.) 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 including clocks which are compensated for changes of temperature. 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 -f if the chronometer is fast and gaining. It is preferable, however, to use the correction of the chro- nometer, which is the quantity to be applied to the chronom- THE CHRONOMETER — LONGITUDE. 143 eter time to reduce it to the true time, and its daily change. Both are -f- when the chronometer is slow and losing. They will be designated by c and A c. A chronometer is said to be regulated to Greenwich time when its corrections on Greenwich time and its daily change are known. If c is the chro. cor. to reduce to Greenwich time, and c the chro. cor. to reduce to the time of a place whose longitude is A. ( + if west). c =c|A, or c = c — X ; (115) 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 r t = T' — T, expressed in days, A c, the daily change ; then 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 gain- ing. Examples. Chbo. 1615. Chbo. 4872. Chbo. 796. A h m s h m s h m s Chro. cor. April 15 + 18 16.2 — 1 15 27.5 + 00 16.6 " " " 27 8 +0 18 29.6 - 1 14 58.6 -0 5.3 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 sex- tant and artificial horizon, an interval of from 5 to 15 days is desirable. The sea-rate of a chronometer is sometimes different from 146 NAVIGATION. its rate on shore, or even from its rate while on board ship in port. Some chronometers are affected by magnetic influences, so that their rates are varied by changing the direction of the XII. hour-mark to different points of the horizon. All are slightly affected by changes of temperature, as perfect com- pensation is rarely attainable. The excellence of a chronom- eter depends upon the permanence of its rate. The rate may be large, but if its variations are small the chronometer is good. 170. A watch is often used for noting the time of an ob- servation. It is compared with the chronometer by noting the time of each at the same instant. The most favorable instant is when the watch shows 0*. Let O 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+(C-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 comparison to the nearest 8 .25. It is better to take the mean of several comparisons than to trust to a single one. A mean time and a sidereal chronometer may be compared within 8 .03 by watching for the coincidence of beats, which occurs at intervals of 3 W , for chronometers, which beat half- seconds. Examples. - Chro. 476. Chro. 4072. Chro. 1976. Chro. 1976. h TO s h m s h m s h TO s Chro. 4 16 56.2 3 15 17.5 11 48 18.2 1 28.5 Watch 1 5 7 35 30 3 16 4 28 C-W. + 3 11 56.2 r-4 20 12.5 -3 27 41.8 - 3 27 31.6 THE CHRONOMETER — LONGITUDE. 147 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 3* 37 w 17 s ; for finding the corresponding time by the chronometer we have, The change of O - W in 1\2, + 10 8 .3 ; whence the change in l h is + 8 .6, and the change in 21 m .3 = A .35, the interval between the 1st comparison and the observation, -\- 3 s .O ; or, by proportion, we have 72 m : 21"\3 = + 10 5 .3 : + 3 5 .0 Then, Time by watch = 3 ft 37 m 17 s C- W= - 3 27 38.8 Time by chro. = 9 38.2 171. Problem 44. To find the correction of a chronom- eter at a place whose latitude and longitude 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 compared with it. Find from the altitude (Prob. 37) the local mean, or si- dereal, time, as may be required. The " local time " — the " chronometer time," or c= T- C (Art. 135), is the correction of the chronometer on local time. Applying to this the known longitude of the place of observa- tion, 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 148 NAVIGATION. 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 observer 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 read exact divisions at regular intervals, and watch the in- stant of contact. A good observer, with a sextant and arti- ficial horizon, can observe the double altitudes at regular intervals of 10'. 173. On a subsequent day repeat this observation, and find again the correction of the chronometer. The difference 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 forenoon, 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 chronome- ter correction, derived from single altitudes, may be erroneous a few seconds. But for sea chronometers this is of less im- THE CHRONOMETER— LONGITUDE. 149 portance than an erroneous determination of the rate. For instance, suppose the determined chronometer correction in error 4 s , and the daily change in error I s ; in 20 days (Art. 168) the computed change of the correction will be in error 20 s , and in 30 days will be in error 30 s . 174. 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 practi- cable, 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 meridian. The mean of the two results is better than a determination from either alone ; for constant errors of the latitude, the in- strument, or the observer, affect the two results in opposite directions ; that is, if one result is too large, the other is too small, and by nearly the same amount. Examples (Prob. 44.) 1. Chronometer Correction. Pensa cola Navy- Yard, 30° 20' 30" K, 87° 15' 21" W. 1898, May 30, 21* ; Chro. ] L876. T. BY Chro. Sextant No. 2 Art. Hor. No. 1. m s 3141 22.7 2 99 50 A . end. m s Chro. cor. (G. m. t. ) - 42 26 ) Daily change — 3.8 ) 32 3.7 23.3 100 32 27 24 100 10 32 51 23 100 20 O's diam. off arc + 32 8.3 \ on arc — 30 59.2 J 33 14 23.7 100 30 33 37.7 100 40 34 7.5 23 2 99 50B . end. In cor. -f- 34.5 34 30.5 23 100 34 53.5 23.3 100 10 35 16.8 23 100 20 Bar. 30.14 35 39.8 23.2 100 30 36 3 100 40 Ther. 76° 3 32 39.07 2 100 15 3 35 5.18 2 100 15 150 NAVIGATION. Computation. h TO » T. by Chro. 3 32 39.07 O's dec. Eq. of t. Chro. cor. - 42 26 + 21 57 47.1 + 21.02 + 2 33.34 - 0.360 '42.04 r .720 G. m. t. May 31, 2 50 11 f 59.6 16.82 — 1.02 J .288 .63 1 .011 .15 +2 32.32 [ .002 2.837 + 21 58 46.7 Q 50° 07' 30" { I. c. + ir.3 Ref. -46 - 16 11 j S. D. - W^'.^ Par. 4-06 h m s O / // L. ap. t., May 30, 21 03 51.75 h = 49 51 19 — Eq. t. — 2 32.32 L = 30 20 30 1. sec 0.06398 L. in. t., May 30, 21 01 19.43 p = 68 01 13 1. cosec 0.03277 T. by Chro. 3 32 39.07 2 s = 148 13 02 0Ch. cor. (L.m.t ) - 6 31 19.64 s = 74 06 31 1. cos 9.43745 s - h = 24 15 12 1. sin 9.61360 h m 8 9.14780 t =9 03 51.75 1. sin 9.57390 h to a ©\s dec. Eq. of t. T. by Chro. 3 35 05.18 O / // // m $ s Chro. cor. — 42 26 + 21 58 46.7 + 21.02 +2 32.32-0.360 G.m.t., May 31, 2 52 41 in 0&.041 f 0.9 -.01 2.878 + 21 58 47.6 + 2 32.31 50° 07' 30" I. c. + 17".3 Ref. - 46 + 15 26 S.D. + 15 48 .5 Par. + 06 L. ap. t., May 30, h m s 21 06 18 O / // h = 50 22 56 — Eq. t. -2 32.31 L = 30 20 30 1. sec 0.06398 L. m. t., May 30, 21 03 45.69 p = 68 01 12 1. cosec 0.03277 T. by Chro. 3 35 05.18 2 s =148 44 38 Chro. cor. (L.m.t.) -6 31 19.49 s = 74 22 19 1. cos 9.43039 s-h s= 23 59 23 1. sin 9.60914 Mean —6 31 19.57 9.13628 Red. for 3 A -.48 t = 9 h 0Q m 18 s 1. sin 9.56814 Chro. cor. (L. m. t.) —6 31 20.05 May 31, 0*. THE CHRONOMETER — LONGITUDE. 151 2. Chronometer Correction. Pensacola Navy- Yard, 30° 20' 30" N., 87° 15' 21" W. 1898, May 31, S h . T. BY Chro. Sextant No 2. Art. Hor. ! No. 1. la j 9 24 2.7 s 22 8 20 100 40 A end. Chro. cor. (G. in. m s t.)-42 27 1 24 25.5 23.0 100 30 Daily change - 3.8 i 24 48.5 25 12.5 25 34.8 25 58.2 24.0 22.3 23.4 100 20 100 10 100 00 99 50 0's diam. off arc on arc In cor. + 32 12.5 I - 30 59.2 f + 36.6 28 33.5 23.5 20 97 40 B end. Bar. 30.14 28 57 23.5 97 30 Ther. 76° 29 20.5 29 43 30 6 30 29.5 22.5 23.0 23.5 97 20 97 10 97 00 96 50 9 25 0.37 2 100 15 9 29 31.58 2 97 15 Computation. h m s O's dec. Eq. oft. T. hy Chro. 9 25 00.37 , „ „ m $ s Chro. cor. - 42 27 + 21 57 47.1 +20.91 +2 33.34 - 0.362 (■ 167.28 r 2.896 G.m.t.,May31, 8 42 33 +3 02.1 \ 14.64 -3.15 \ .253 8.709 + 22 00 49.2 I .19 +2 30.19 I .003 50 07 30 I.e. + 18.3 Ref.-46 — 16 10 S.D.-1548.5 Par. + C6 h = 49 51 20 L = 30 20 30 1. sec. 0.06398 h m s p = 67 59 11 1. cosec 0.03288 L. ap. t., May 31, 2 56 11.75 2s = US 11 01 - Eq. t. - 2 30.19 3 = 74 05 30 1. cos 9.43790 L.m. t., May 31, 2 53 41.56 s-h = 24 14 10 1. sin 9.61331 T. by Chro. 9 25 00.37 9.14807 Ch. c, L.m.t.-6 31 18.81 t = 2» 56™ 11*. 75 9.57403 152 NAVIGATION. h m s 0\s- dec. T. by Chro. 9 29 31.58 , „ Chro. cor. — 42.27 + 22 00 49.2 + 20.91 + G. m. t., May 31, 8 47 05 in 0^.076 + 1.0 8.785 4- 22 00 50 .8 4- 18".3 CD 48° 37' 30' 4- 15' 25' I.e. 4- S.d. +15'48".5 Eq. oft. m 8 s 2 30.19 -0.302 - .03 2 30 16 Ref . - 48" Par. 4- 06". L. ap. t., May 31, — Eq. t. L. m. t. T. by Chro. Q Ch. cor. L. m. t. 3 00 42.67 - 2 30.16 2 58 12.51 9 29 31.58 6 31 19.07 Mean - 6 31 18.94 Red. for — 3 A + .48 Chro. cor. L. m. t. — 6 31 18.46 May 31, 0*, Chro. cor. (L.m t.) Long. May 31, 6 A , Chro. cor. (G. m. t.) h = 48 52 55 L = 30 20 30 1. p = 67 59 11 1. 2 s = 147 12 36 8 = 73 36 18 1. s -k = 24 43 23 1. t = 3 A 00™ 42 s . 67 1. May 31, h . — 6 h 31™ 19*.25, by A. m. 4- 5 49 01 .4 - 42 17.85 sec 0.06398 cosec. 0.03288 cos sin 9.45064 9.62142 9.16892 sin 9.58446 and p. m. obs. 3. Table of Chronometer Corrections. Chro. of 1876 ; fast of Greenwich mean time and gaining. G M.T. Chro. Cor. Daily Ch. Remarks. h h m s 1898, May 13 -0 40 20.5 -4.14 0, a. if., Key West Light- House. 17 3 41 26.8 3.88 o, a. m., Key West Light- House. 25 6 41 58.3 3.75 O, A. m. & p. m., Pensacola Navy- Yard. 316 42 17.8 o, a. m. & p. m., Pensacola Navy- Yard. Long.* of Key West Light-House, 81° 48' 40" W. Long, of Pensacola Navy- Yard, 87° 15' 21" W. * The assumed longitude of places where the chronometer is rated should be stated. THE CHRONOMETER — LONGITUDE. 153 4. Comparisons and Corrections of Chronometers. 1898, May 31, 6 A , G. mean time. Chro. 4375 Chro. 9163. Chro. 789. Chro. 5165. h m s h m s h m 8 A TO s Chro. 6 50 16.3 5 3 29.7 2 15 27.5 11 59 16.8 (1876) 6 30 6 31 6 32 10 6 33 30 (1876)— Chro. — 20 16.3 + 1 27 30.3 + 4 16 42.5 -5 25 46.8 Cor. of (1876) -42 17.8 —42 17.8 -42 17.8 —42 17.8 Chro. cor. -1 2 34.1 -0 45 12.5 +3 34 24.7 -6 18 4.6 or + 5 41 55.4 175. 3d Method. (By equal 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. 170), 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* -j- 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. 26) ; and for a Greenwich chronome- ter into the corresponding Greenwich time. Subtracting the chronometer time, we have the correction of the chronometer. Example. 1898, Jan. 14, at Washington, in longitude 77° 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 : 7* 34'" 00 s . 34 + 5 08 11 .2 12 42 11 .54 -19 35 53.18 17 06 18.36 2 48.14 17 03 30 .22 17 07 56.01 -4 25 .79 154 NAVIGATION. Mean of Chro. times (* east) 2 h 16 m 35 s . 65 Mean of Chro. times ^* west) 7 59 16 .38 Chro. time of *'s transit 5 07 56 .01 L. sid. t. = *'s R. A. Long. G. sid. t. - S (Jan. 14) Sid. int. from Jan. 14 0* Red. to m. t. int. G. mean time (Jan. 14) Chro. time Chro. cor. 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. Problem 45. In equal altitudes of the sun, to find the correction of the middle time for the change of the sun's declination in the interval between the observa- tions. 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 obser- vations, or the middle chronometer time, A T = the correction of this mean to reduce to the chronom- eter time of apparent noon ; L — the latitude of the place, d = the sun's declination at local apparent noon, A(?= the change of this declination in the time t\ THE CHRONOMETER — LONGITUDE. 155 then, when both observations are on the same day, t -f- A T will be numerically the hour-angle at the a.m. ob- servation, t — AT , the hour-angle at the p.m. observation, d — A d, the declination * at the a.m. observation, d-\- Ad, the declination* at the p.m. observation. By (76), we have for the two observations, sin A=sinX sin(d—Ad)-\-GOsLcos (d—Ad) cos (<+AT ) ) sinA=sinXsin((i+A^) + cosXcos (<#-f AJ)cos (t—AT ) ) But sin (d ^ Ad) = sin d cos A^^ cos d sin A d, cos (<# i A <#) = cos d cos A c? =p sin d sin A df, cos (d i A 7JJ = cos £ cos A 2J =f sin t sin A T . Since Ac?, and therefore AT , are very small, we may put cos Ac? =1, sin Ad = A d sin 1", cos A TJ = 1, sin A T = 15 A T sin 1" ; A d being expressed in seconds of arc, and A T in seconds of time ; we shall then have sin (d ^ A d) = sin d -\- A d sin 1" cos J, cos (d ^ Ad) = cos c? =p A d sin 1" sin d, cos ( 6*. The computation of the two parts of A T 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 Chro- nometer," and in Bowditch, Table 37. The argument of these tables is 2 t, or the elapsed time. The signs of A and B are given. THE CHRONOMETER — LONGITUDE. 157 Apply the two parts of A T , according to their signs, to the Middle Chronometer Time; the result is the Chronometer Time of Apparent Noon. Apply to this the equation of time {adding, when the equa- tion of time is additive, to mean time ; otherwise subtracting) ; 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 Time of Mean Noon at Greenwich. * 12* — Chro. T at local Mean Noon will be the Chro. correc- tion, if the chronometer is regulated to local time. 12* — Chro. T at Greenwich Mean Noon will be the Chro. correction, if the chronometer is regulated to Greenwich time. 177. If a set of altitudes is observed in the afternoon of one day, and a set of equal altitudes in the forenoon of the next day, the middle time would correspond nearly to the in- stant of apparent midnight ; and half the elapsed time t would be nearly the hour-angle from the lower branch of the merid- ian, or the supplement of the proper hour-angle. In this case 180° — (t -f A T ) will be the hour-angle at the p.m. observation. 180° — (t — A T ) " " « « " " a.m. " d — Ad, the declination at the p.m. d + A d, " " " " a.m. and we have for the two observations, as in (120) * This is better noted as h . ti sin A=sinZ, s'm(d—&d) —cosLcos(d—Ad) cos(£-f A T ) 158 NAVIGATION. smh=smLsm(d-\-Ad)—cosLcos(d-\-Ad)cos(t—AT )) v ' Treating these in the same way as (120) we shall have = A d sin L cos d -f- A <#cos L sin d cos t — 15 A T cos L cos c? sin £ ; whence . yr A(? tan L £±d tan ^ " 15 sin t 15 tan t or, putting as before A d = A h d t A = - , B 15 sin t' 15 tan t 9 & T Q = — A & h d ten L + JB & h d tan d, (126) which differs from (124) only in the sign of A. This is the reduction of the middle time to the Chro. Time of apparent midnight: applying the equation of time reduces it to the Chro. Time of mean midnight. 178. d, A d y and the equation of time, are to be taken from page I of the Almanac, and interpolated as in Art. 90 for the instant of apparent noon, or of apparent midnight, according as the observations are made on the same day, or on consecutive days. 2 t is properly the elapsed apparent time. The elapsed * These may be written — sin h = — sin L sin (d — A d) + cos L cos (d — A d) cos (t + A ^o) — sin/i = — sin L sin (d + \d) + cos Lcos (d + A^)cos(£ — A T ). They differ from (120) in the signs of h and X, 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 latitude and meridian of the antipode. The only effect in (121) is to change the sign of tan L, or of the first term in the equation of equal altitudes. THE CHRONOMETER — LONGITUDE. 159 time by chronometer requires, then, not only a correction for the rate, which is 2 t A c (+ when the chronometer loses) ; (41) 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 t, additive when the equation of time is additive to mean time and increasing, or subtractive from mean time and decreasing. For a sidereal chronometer, it is the change in the sun's right ascension in the time 2 1, and subtractive. 179. 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 A a is the increase of right ascension in the interval, 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 double hour-angle of the body by the quantity 2 A a. If 2 s = the elapsed sidereal time, then in (122) we must take 2t = 2s — 2Aa,OYt = s — Aa. (127) If t m = half the elapsed mean time (expressed in hours when used as a coefficient), and A A a = the increase of right ascension in l h of mean time, by (64) s = t m + 9 S .8565 t m and t = t m + t m (9 S .8565 — A* a), (128) * The maximum daily change is 30 s . The elapsed time by Chro- nometer is usually regarded as sufficiently accurate. 160 NAVIGATION. by which t and 2 t may be found from 2 t m the elapsed mean time. In this expression the last two terms are in seconds. Reducing to hours we have t = tm (l + 9 *- 8565 -**« ) = tj 1.002738 - ^) (129) W V 3600 J m \ 3600 J V ; If A A d = the change of declination in l h of mean time, then in (121) Adf= t m A h d or, substituting for t m its value from (129), A d = t A h d ■s- f 1.002738 - -^Y V 3600/ Equations (123) and (124) may then be used for other bodies than the sun, provided we give t its proper value from (127) or (128), and for A h d substitute A', d=A h d+( 1.002738 - **1*\ 3600 y' or, which will be sufficiently exact, w 7 k j . A h a — 9 8 .856 A 7 /--u _v A'* df = A* <* + h 36Q() A A £ (130) 180. Observing the double altitudes at regular intervals of 10', or 20', especially facilitates the method of equal alti- tudes; for, if the first set is observed at equal intervals, in the second the observer, having set the instrument for the last reading of the first and observed the contact, for the subsequent observations has only to move back successively the same intervals. 181. It is not requisite that the instrument should give the true altitude ; it is sufficient if the altitude is the same THE CHRONOMETER — LONGITUDE. 161 at the two corresponding observations. Hence the two obser- vations should be made with the same instruments, without change of adjustment, and in general as nearly as practicable under the same circumstances. This purpose is promoted by making the final movement of the tangent screw in both sets always in the same direc- tion. Thus, in reversing the movement, the screw may be turned a little too far, and then the final contact made by a motion in the same direction as before. If the sun is used, both limbs should be observed. The error arising from want of parallelism of the surfaces of the roof-glasses of the horizon is eliminated by having the same end of the roof toward the observer. The roof may be tested by observing sets of altitudes with it in reversed positions. 182. Although the readings of the instrument may be the same in the two sets of observations, the altitudes may be slightly different, 1st, from changes in the instrument in the interval ; 2d, from difference of refraction at the two times. A change in the index correction may be detected by observation ; but there may be expansion or contraction of various parts of the instrument which may affect the readings of the altitudes without altering the index correc- tion. The change of refraction may be found by noting the barometer and thermometer at each set, and finding the re- fraction for both sets of altitudes. 183. To correct the middle time for any small difference of the altitudes, whether from refraction or actual change of 162 NAVIGATION. readings, we may find, from the difference between two read- ings, and the difference of the corresponding times, the change of time for a change of 1', or 1", of altitude. This multiplied by half the inequality of altitudes, expressed in minutes, or seconds, will give the correction of the middle time, to be added when the p.m. altitude is the greater ; to be subtracted when the p.m. altitude is the less. If twice the altitude is observed with an artificial horizon, we may find the change of time for a change of 1', or 1", of the double altitude, and multiply it by the whole inequality of the altitudes. Example. (Prob. 45.) 1. 1898, Jan. 10, 9fc* a.m. and 2j* p.m. Equal altitudes of O at the Custom House, Key West, Florida ; 24° 33' 20" K, 81° 48' 37" W. Chro. 1085 ; Chro. cor. (G. m. t.) 42 m 18 s .O ; daily change + 8 3. Sex. No. 1. Art. Hor. No. 2. o / 2 GO A. end. 10 20 30 40 50 2060 10 20 30 40 50 60 25 T. by Chro. a.m. h m s 3 39 26.7 39 58.8 40 31.5 41 4.0 41 35.7 42 9.0 42 58.5 43 31.3 44 3.8 44 37.5 45 10.0 45 43.7 3 42 34.21 8 52 51.46 P.M. h m 8 8 55 59.7 55 27.0 54 55.3 54 22.0 53 50.3 53 16.5 52 26.7 51 53.5 51 21.0 50 48.0 50 15.5 49 42.0 Mid. Time. 6 A 17 OT 8 43.2 42.9 43.4 43.0 43.0 42.8 42.6 42.4 42.4 42.7 42.8 42.8 Elapsed Chro. t. Mid. Chro. t. 1st part of Eq. 2d part of Eq. Chro. t. of ap. noon Eq. t. 5 10 17.25 6 17 42.83 — 2.89 — 1.99 6 17 37.95 — 7 57.18 Long. O's diam. H In cor. Bar. Ther. Ref. - 32' 25".0 32 41 .7 -8.3 30.22 77° 1' 36" p.m. + 32'26".7 -32 43 .3 -8.3 30.18 -1' h TO 8 m 8 8 + 5 27 14.5 Eq. t. —7 51.74—0.997 5.456 z-4.985 d . 227 —5.44 J .399 J .050 — 7 57.18 I .006 THE CHRONOMETER — LONGITUDE. 163 Chro. t. of mean noon 6 09 40.77 0's dec. —21 54 46.4 A h d +22.78 Ch. i nld+1.06 Long. (W.) —5 27 14.5 +.24 ,.212 Chro. t. of G. m. noon 42 26.27 + 23.02 J .021 Chro. cor. (G. m. t.) — 42 26.27 22.90 ' .007 Jan. 10, 5^.5 fl 14.50 + 2 05.0 J 9.26 1 1.15 — 2152 41.4 I .14 L - + 24° 33' 20" 1. tan 9.6598 d = - 21° 54' 46" 1. tan 9.6045 n Ahd= +23" .02 log 1.3621 log 1.3621 A log 9.4396 m B log 9.3315 — 2s.89 log 0.4615 n — ls.99 log 0.2981 n (The elapsed apparent time is 5 h 10™ 12 s .) 2. 1898, June 19, 4J* p.m., and 20, 1\ a.m. ; nearly equal altitude of Q; at Belize, S. E. pass of Mississippi River, 29° V 8" K, 89° 5' 18" W., chro. 1085; chro. cor. (G. m. t.) — 41 m 28 s ; daily change -f 1 5 .0 ; sextant No. 2 ; art. hor. No. 1; (A. end toward observer). P. M A.M. P.M. A.M. Sex. Read. T. by Chr. T. by Chr. Sex. Read. In. cor. + 41 // .8 + 40" .4 o / h m s ft m 8 > 2 ©65 10 10 55 48 2 22 44.5 2Q65 30 Bar. 30 .09 30 .12 65 56 10.8 56 34.3 22 21 2134 65 20 65 Ther. 81° 80° 64 50 J In cor. + ' 20".9 + ' 20". 2 64 40 56 57.5 21 11.2 64 50 Ref. -1 28 -1 28 57 27.5 20 41.5 2 65 40 - 1 07 .1 -1 07 .8 2 065 30 (P.M. — A.M.) 65 20 57 50.5 20 18 65 30 Diff. ref., etc. + / 0".7 65 10 58 14.3 19 54 65 20 Diff. obs. alts. —5 37 .5 65 58 37.3 19 7.5 65 A/i = -5 36 .8 2 65 5 10 57 12.53 2 20 58.96 2 65 16 15 For 2 A h = 10', A t = 23s.3 /i'=32 32 30 fc'=32 38 7.5 2 A h = 1' At= 2s.38 G. ap. t., June 19, 17* 56 m 21 s .2 = 19, 17 A .939 = 19*747 O's Dec. Ah d Ch. in Id. Eq. of T. + 23° 26' 25".4 + 1".99 - 1".04 1.60 — l m 04 s .39 — 0».546 f -73 r 5.46 + 28 .7 -.78 \ .04 I .01 -9.80 - 3.82 .50 -f- 23 26 54 + 1.21 - 1 14.19 { .02 164 naviga: riON. h m 8 h m 8 Middle chro. t. 18 39 05.75 Elap. t. by chro.* 15 23 46 Red. for A h,— 5'. 61 x 2 s . 33 - 13.07 -26 1st part of Eq. + .38 Ch. of Eq. t. - 8 2d part of Eq. - .13 Elap. ap. t. 15 23 12 Chro. t. of ap. 12* 18 38 52.93 O / // Eq. of t. - 1 14.19 L= +29 07 08 1. tan 9.7459 Chro. t. of mean 12* 18 37 38.74 & h d = + l".21 log 0.0828 Long. — 5 56 21.2 A log 9.7542 Chro. t. of G. mean 12* 12 41 17.54 +0 S .38 log 9.5829 Chro. cor. (G. m. t.) - 41 17.54 June 19, 18*. d = - 23° 26' 54" 1. tan 9.6372 log 0.0828 B log 9.3862 n — s . 13 log 9.1062 n 184. 4th Method of finding the correction of a chronom- eter. (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 instrument. 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 observed ; 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 limb, 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. * Twice the reduction of the middle time for the diff. of alts, is to be added to the elapsed time when the p.m. observation is last ; subtracted when the p.m. observation is first. This may be neglected unless the diff. of altitudes is quite large. THE CHRONOMETER — LONGITUDE. 165 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 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. 169. 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. Fifth Method. (By time signals) These signals can be obtained at the W. U. Telegraph Office in any part of the United States, without delay, in any weather, and with abso- lute certainty as to comparisons. Since well-adjusted transit instruments are not generally available, the electric signal from Washington furnishes the best means of rating chro- nometers, with the utmost simplicity of method, and a high degree of accuracy. But if time signals, or a time-ball, be employed, the chronometer error should be found also by astronomical observations, as a check ; for in so important a matter, the navigator ought not to accept the unsupported or uncorroborated work of another person. (" Notes on Naviga- tion." Nav. Academy, 1872.) 166 NAVIGATION. LONGITUDE. 187. To find the longitude of a place by astronomical ob- servations, it is generally necessary to determine indepen- dently 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 Green- wich time is the less (Art. 165). This is expressed by (50) x = T„ - T, in which T is the Greenwich time, and T y the corresponding local time of the same kind. These times may be apparent, mean, or sidereal. 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 ; MPM' 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, MPM' = MPS-M'PS. The several methods of finding the longitude differ in the modes of finding and comparing the two times, or the two hour-angles. LONGITUDE. 167 188. Problem 46. 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 time 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. Subtracting 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, C — W (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. 26), except when a fixed star has been observed, 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 : * For observations of stars, a sidereal chronometer is most conve- nient. 168 NAVIGATION. 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 Greenwich mean time to sidereal time by adding the right ascension 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 give 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 cu- mulative, increasing with the number of days from the date, when the correction of the chronometer was found from obser- vations. 191. The chronometer correction for the date of observation can be derived from subsequent as well as from prior deter- minations of it and its daily change. In finding the longitude of a place on shore, or of a shoal, both values should be ob- tained, when practicable, and combined by giving weights to each inversely proportional to its interval of time from the original determination. Thus, if cf and c" are two such chro- nometer corrections, the first brought forward t' days, the sec- ond carried back if' days, we may take as the mean value * * This assumes that c' and c" are derived from two chronometer corrections of equal weight, and consequently that the longitudes used in finding them are equally reliable. This may not be the case if the LONGITUDE. 169 f d + 1 c" a + 1" ' or, in a form more convenient for computation, For example, suppose that on Jan. 17, the chronometer correc- tion brought forward from Jan. 1, is — 18 m 56 s .5, and reduced back from Jan. 25, is — 19 m 3 5 .4 ; the value by the above formula will be - 18- 56 s .5 + 16 X ~ 68,9 = - 19 m 1M. 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 as- sumed longitude of the place where the chronometer is rated, and the age of the rates, are stated. Strictly, the chronometer merely determines the difference of longitude between the two places where the observations are made. This may be ob- tained by using the chronometer correction on the time of the place of rating, instead of the Greenwich time. It is prefera- ble to report such differences rather than absolute longitudes. 193. 2d. To find the hour-angle of the body, and thence the local time. 1st Method. (Prob. 37. By single altitudes.) Observe in quick succession several altitudes of the heavenly body, chronometer corrections were found from observations at two different places. The student is referred to Chauvenet's " Astronomy," I., 317, etc., for the methods of allowing for changes in the rates and combining the results of several chronometers. 170 NAVIGATION. noting the time of each by the chronometer, or by a watch compared with it. Take the mean of the noted times, and from it find the Greenwich mean time; for which take from the Almanac the declination of the body, its semidiameter and horizontal parallax when sensible, as well as the quantities required for finding the Greenwich hour-angle. (Art. 189.) Take the mean of the readings of the instrument, with which the altitudes were measured, and from it find the true altitude of the centre of the body. (Art. 118.) With this and the known, or assumed, latitude of the place find the local hour-angle of the body by Problem 37. This hour angle, which for the sun is the local apparent time, subtracted from the corresponding Greenwich hour- angle already found, will give the longitude. Or, the local mean time may be found from it, for the sun, by applying the equation of time; for other bodies, by add- ing the right ascension of the body, which will give the local sidereal time, and subtracting the right ascension of the mean sun (Prob. 31) : and the local time subtracted from the corresponding Greenwich time will give the longitude. 194. On shore it is best to use an artificial horizon, even when a sea-horizon can be had, and for precise observations, stars in preference to the sun. At sea the sun is most conveniently used ; but altitudes of the moon and bright stars can be employed when the sun is not available. The chief difficulty is the obscurity of the sea-horizon at night. During twilight, however, or in a bright moonlight, it is often distinct and well defined. 195. The most favorable position of the body for finding LONGITUDE. 171 its hour-angle from its altitude is, as previously stated, when it is nearest the prime vertical ; provided its altitude is not so small as to involve to too great an extent the uncertainty of refraction ; and, observed on shore, is within the limits * of the instruments employed. • On shore the time and circumstances most favorable for observations can generally be selected. At sea long con- tinuance of bad weather may render poor observations, made under unfavorable circumstances, the only ones available. While, then, it is not well to use for finding the time an altitude less than 10°, or of an object whose azimuth is less than 45° or more than 135°, it may sometimes be neces- sary to exceed these limits. 196. When the declination and latitude are nearly the same, the body is nearest the prime vertical but a short time before and after its meridian passage, so that a very great altitude may be used. Thus in lat. 20° N., the sun, when its declination is 19° 55' N. or 20° 5' N., is nearest the prime vertical within 22 m of noon at an altitude of nearly 85° ; and the local time can be as accurately obtained from an altitude of 89°, 4 TO from noon, and about 5° in azimuth from the prime vertical, as from an altitude of 30°, provided the assumed latitude can be depended on within 2'. Nearer noon, the rapid change of the sun's azimuth, averaging 10° in l m , would make it difficult to observe the altitude with sufficient precision. 197. The local time or hour-angle is affected by errors in the altitude and in the assumed latitude. (Arts. 136, 138.) When several observations have been made in rapid succession, * For a sextant and artificial horizon, between 20° and 60°. 172 NAVIGATION. the effect of a supposed error of V in the altitude* may- be found by dividing the difference of two of the noted times by the difference, in minutes, of the corresponding altitudes. In a similar way we may find the change of altitude in l 9 * of time by dividing the difference of two altitudes by the difference in minutes of the corresponding times. The maximum change of altitude in l m is 15'; when L = and d = 0. The more rapid the change of altitude, the less will errors of altitude affect the result. To ascertain the effect of an error of 1' in the assumed latitude, f the local times or hour-angles may be computed separately for two latitudes differing 10', or 20', from each other, and the difference of these times divided by 10', or 20'. At sea the latitude by account is used, either brought forward to the time of observation from a preceding, or car- ried back from a subsequent, determination. It may be very largely in error, especially in uncertain currents, or after run- ning several days without observations. A small error may also result from the assumption that * Differentiating equation (76) sin h — sin L sin d + cos L cos d cos £, regarding h and t as variables, we have cos h d h = — cos L cos d sin t d t but cos d sin t = cos h sin Z Sph. Trig. (114) whence d t = — — — ; — - » 15 cos L sin Z which is a minimum when Z = ± 90°, and incalculable when Z = 0° or 180°. t From (96) we find &t= - 15 cos L tan Z which is 0, when Z— ± 90°, and also incalculable when Z = or 180°. LONGITUDE. 173 the mean of the instrumental readings corresponds to the mean of the noted times. The reduction of the mean of the altitudes to the mean of the times can be found,* but it can be avoided by limiting the series of observations, which are combined together, to so brief a period that the error becomes insensible ; or, when the body is near the meridian in azimuth, by reducing each observation by itself. This last case, how- ever, should be avoided in this problem. 198. At sea it is usual to reduce longitudes obtained from day observations to noon by allowing for the run of the ship in the interval, and for currents when known. Those from night observations are recorded for the time of observation. 199. 2d Method. Altitudes in the forenoon and in the af- ternoon, or on different sides of the meridian, are preferable to single altitudes for finding the local time, for the reasons already stated in Article 174. The longitudes can be found from each set separately, and then combined. At sea the longitudes derived from each can be reduced to noon, and the mean of the two taken as the true longitude ; or, if the difference can be regarded as due to currents, the longitude at noon can be found by interpolating for the elapsed time. It is desirable that the observations should be made at nearly equal intervals from noon. Longitudes by a.m. and p.m. observations are enjoined in the directions of the Navy Department whenever practicable. Example. (Prob. 46.) 1. At sea, May 17, 1898, 9* 45™ a.m. ; 24° 50' K., 82° 18' W. by reckoning from preceding noon ; * Qiauvenet's " Astronomy," I., 214. 174 NAVIGATION. T. by Watch 9 h 30™ 15 s ; obs'd altitude of 58° 17'; Chro. - Watch + 5* 12- 26 s ; Chro. cor. + 25™ 15 5 ; Index cor. of sextant -f- 3' 20" ; height of eye 18 feet ; required the longitude. h m s Q's dec, Eq. of t. W. T. 9 30 15 o - II m s C.-W. 5 12 26 + 19 23 51.6 + 33 /, .59 + 3 48.81 C. C. + 25 15 r 100".8 -.23 G. m. t., May 17, + 1 45.2 - 3 .4 3 07 56 = :3U3 + 19 25 37 I 1 .0 + 3 48.6 Eq. of t. +3 48.6 O / // / // / // G. ap. t., May 17, 58 17 (I.e. + 14 30 (S.D. + 3 20 Dip — 4 09 3 11 44.6 + 15 51 R. &P.— 32 h= 58 3130 L = 24 50 1. sec 0.04214 p= 70 34 23 1. cosec 0.02545 2 s = 153 55 53 s= 76 57 57 1. cos 9.35321 s — h= 18 26 26 1. sin 9.50012 L. ap. t., May 16, 8.92092 21 45 45.3 9 45 45.3 1. sin %t 9.46046 Long. + 5 25 59.3 or 81° 29' 50" W. May 17, noon, lat. by mer. alt. of O, 25° 8' K ; run of the ship from 9 J* a.m. E. N. E. (true) 18 miles. For E. N. E. 18', I = 6'.9 N., p = 16'.6 E., D = 18'.4 E. At the time of the a.m. observations, then, the latitude carried back from noon was 25° 1' N. Using this in the com- putation of the time, we find the L. ap. t. May 16, 21* 45™ 48 5 .2, and the long. 81° 29'.1 W. Applying D = 18'.4 E., we have for the longitude, May 17, noon, 81° 10'.7 W., from observations made at 9.45 a.m. By p.m. observations, and reduced to noon, the longitude was found to be, LONGTTTJDE. 175 May 17, noon, 80° 44' W. from observations at 3.45 p.m. As the position is in the Gulf Stream, where there is a strong easterly current, the difference of the two longitudes is attributed to that cause. We take, then, as the longitude at noon, 81° 10'.7 - 2 - 25 * 27 ' = 81° 00'.5 W. o Note. — The examples under Problem 45 can be adapted to this by regarding the chronometer correction given, instead of the longitude. 200. 3d Method. (Littrow's. By double altitudes of the same body.) 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 combine them, and find the hour-angle for the middle instant between them. Problem 47. From two altitudes of a heavenly body, supposing the declination to be the same for both, to find the mean of the two hour-angles, the latitude ox the place and the Greenwich 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 declina- tion of the body. Eeduce the observed altitudes to true altitudes. Let h and h' be the two altitudes, T and T', the corresponding hour-angles ; then we have, by (76), sin h = sin L sin d -J- cos L cos d cos T, sin N ** sin L sin d -J- cos L cos d cos T T ; 176 NAVIGATION. and by subtracting the first from the second, sin N — sin h = cos L cos d (cos T r — cos T). By Pl. Trig. (106) and (108), this reduces to sin J (h! — h) cos i (h r -f A) = -cosicostfsin^r' + T) sin i (T 7 '-:? 7 ); whence 2V y sin J ( Z" — T) cos L cos J v ' which is the formula used in Art. 300 (Bowd.). (T' — T) for the sun is the elapsed apparent time ; for a star, the elapsed sidereal time ; and for the moon or a planet, the elapsed sidereal time — the increase of right ascension in the interval ; and can be found from the difference of the two chronometer times. Then, by (131), J (T r + T) can be found, and, as any other local hour-angle, subtracted from the corresponding Greenwich hour-angle, which in this case is to be derived from the mean of the noted times. \ {T r -\- T) is + or — according as the second altitude is less or greater than the first ; so that it is on the same side of the meridian as the body at the time of its less altitude. 201. The method presents no special advantages for ob- servations on shore, except in the case of two nearly equal altitudes of a fixed star on opposite sides of the meridian. In the case of the sun and planets, it is necessary to take the change of declination into consideration to obtain precise results. The special case for which the method provides is at sea, within the tropics, when the sun passes the meridian at a high altitude. In that case, when by reason of clouds observations LONGITUDE. 177 near noon only can be made, or it is desired to obtain the longitude as near noon as practicable, let a pair of altitudes, or several pairs, be measured, and the times noted with all the precision practicable. The altitudes should be reduced to true altitudes, and one of each pair for the run of the ship in the interval * by the method given in Prob. 53, and in Bowd., Art. 288. From each pair the middle apparent time can be found by (131), and the mean of these times subtracted from the mean of the Greenwich apparent times for the longitude. 202. If the altitude changes uniformly with the time, or nearly so, the mean of several altitudes observed in quick suc- cession can be taken for a single altitude. If the observations have been made with care, the errors of instrument, refraction, and dip will affect the two altitudes of each pair nearly alike ; and if the reduction for the run of the ship is carefully made, the difference of altitudes in com- parison with the difference of times will be nearly exact. 203. This method was proposed by M. Littrow, Director of the Vienna Observatory. It should be used cautiously, and the errors to which the result is liable in any case carefully computed. A table showing the error of time which may correspond to an error of one minute in each of the observed altitudes, when t = 30 inin., is given in Art. 301 (Bowd.). Altitudes greater than 80° and an interval of more than half an hour are recommended, but an intelligent navigator can readily determine when he can safely depart from these limits. This will be especially the case when the altitudes are on both sides of the meridian. * This may be avoided, if the course of the ship is at right angles to the bearing of the sun. 178 NAVIGATION. Example. 1. 1898, May 16, 11.30 a.m., in lat. 25° 15' N., long. 56° 20' W., by account ; the ship running N. E. (true) 8 knots an hour. T. by Chro. 2 h 32™ 23 s O's true alt., 81° Y 0", " " " 2 53 11 " " " 83 40 30; Chronometer correction on G. mean time -|- 40 w 51 s ; required 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. 53), 2'.8 x cos 86° = + 0'.2 = + 12". h m s 1st chro. t. 2 32 23 O's dec. Eq. of t. o / // // m s $ 2d " " 2 5311 + 19 10 15.7 +34.4 +3 50.28— 0.049 Elapsed chr.t. (r-T)= 2048 +157 (103.2 -17 (.15 Mid. " " 2 42 47 + 19 12 12.7 / 13.8+3 50.1 (.02 o Chro. cor. + 40 51 G. m. t. May 16 3 23 38 h =81 01 12 Eq. t. + 3 50.1 /i'=83 4030 G. ap. t. 3 27 28.1 h (h'-h)= 1 19 39 1. sin 8.36489 L. ap. t. 23 41 44 \ (h+h')=82 20 51 1. cos 9.12439 •. . (+3 45 44.1 i=25 15 1. sec 0.04361 Long.at2dobs. | 5 6 26 02W. d=1912 13 1. sec 0.02486 T— T = 20 48 1. cosec £ t 1.34330 I (T'+T) = -01816 1. sin 8.90105 204. 4th Method. (By equal altitudes.) Let equal altitudes of a heavenly body be observed east and west of the meridian (Art. 175) and the times noted as in other observations ; and the mean of the watch-times in each set. if a watch is used, reduced to chronometer time. If both sets have been observed LONGITUDE. 179 at the same place, and the declination of the body has not changed, the mean of the two times will be the chronometer time of its meridian transit. If the declination has changed in the interval, as is ordi- narily the case with the sun, moon, or a planet, the correction for such change, found by the methods of Problem 46, should be applied. Applying then the chronometer correction, we have the corresponding Greenwich time, which will be mean or sidereal as the time to which the chronometer is regulated. Finding from this, by the method in Art. 189, the Green- wich hour-angle of the body (which in the case of the sun is the Greenwich apparent time), we have the longitude, if the first observation was east of the meridian, as the correspond- ing local hour-angle is then 0. But if the first observation was west of the meridian, the local hour-angle is 12*, and must be subtracted. This method should be used on shore, when practicable, in preference to either of the preceding. 205. Equal altitudes of the sun can be conveniently used at sea when the sun passes the meridian near the zenith ; that is, when its declination and the latitude are nearly the same. Altitudes very near noon are then available for finding the time (Art. 196), and equal altitudes can be observed with only a short interval. In the example of Art. 196, an interval of eight minutes would have been sufficient. If the ship does not change her position in the interval, the middle time corresponds to apparent noon ; as the change of declination may be neglected, unless the interval between the observations is so great as to require it. 180 NAVIGATION. 206. If the longitude only has changed, the middle time corresponds to apparent noon at the middle meridian, and will give the longitude of that meridian. This will be the longitude at noon, if the speed of the ship has been uniform. But if it has not, subtracting half the change of longitude, when the true course is west, or adding it when the course is east, will give the longitude of the place where the first alti- tude was observed. This can then be reduced to noon by allowing for the run of the ship. If the change of longitude is west, the sun arrives at the corresponding altitude of the afternoon later than it would do if observed at the same place as in the forenoon ; if the change is east, it arrives earlier; and the difference is the time of the sun's passing from the one meridian to the other ; that is, the difference of longitude expressed in time. If, then, 2 t is the elapsed apparent time, A A, the change of longitude (-J- when west), the hour-angle of the sun at each observation is t — % A\; and (122) becomes A T = A„ dttsuiZ A„ d nan d 15sin(«-JAX)' + "l5tan(«-lAX) { } But even when the elapsed time is so great that it is thought necessary to correct for the change of declination, A A is never large enough to produce a change of I s . If the latitude only has changed, the middle time requires correction for such a change, which can be deduced in a simi- lar way to that for a change of declination in Prob. 46. But, as in the fundamental formula (76), sin h = sin L sin d -f- cos L cos d cos t, LONGITUDE. 181 L and d enter with the same functions, the} are interchange- able. If, then, A A L is the hourly change of latitude (+ toward the north and expressed in seconds), and A' T , the required correction, we have from (122) and (124), = _ ^Lttmd A h L t tan L 15 sin t T 15 tan t K } and A' T = A A h L tan d + B A* L tan X, (134) for which Chauvenet's tables can be used. If both latitude and longitude have changed, for t in the denominators of (133), we may substitute t — J A A: but this at sea is a needless refinement. The restriction of this method to a short interval between the observations depends upon the uncertainty of the run of the ship, and consequent imperfect determination of A h X, the mean hourly change of latitude in the interval. If its error is supposed to be -^ A A X, the consequent error in A' T is \ A' T . When equal altitudes near noon are practicable, a meridian altitude of the sun can ordinarily be taken for latitude, so that L will be sufficiently exact. Moreover, the latitude and longitude are both found for noon. Examples. 1. At sea, 1898, March 17, noon, lat. by mer. alt. of the sun 3° 16' S., long, by account 84° 58' W. ; equal altitudes of the sun were observed at 5 h 34 w 18 s and 6 A 3 m 24 s G. mean time ; the ship running S. S. E. (true) 10 knots an hour ; re- quired the longitude. For S. S. E., 10 r , A» L = - 9 r .2, A A A = - 3'.8. 182 NAVIGATION. h m s O's dec. Eq. of t. 1st G. m. t., 5 34 18 O / // // m s s 2d G. m.t. 6 3 24 — 1 13 14 +59.29 - 8 24.86 + 0.73 Elapjed t. 29 6 + 5 44 j 296.5 + 4.23 ( 3.65 - 8 20.6 ( .58 Mid. G. m. t. 5 48 51 - 1 07 30 j 47.4 — Eq. of t. - 8 20.6 A h L=— 552" log 2.742 n log 2.742 n Mid. G. ap. t. 5 40 30.4 L =-3°16 / 1. tan 8.756 n Red. for A L -I- 5.2 d = — 1 07.5 1. tan 8.293 n ( G. ap. t. of \ or Long. noon 5 A 40" 1 35 s .6 log A 9.406 n log B 9.405 85° 09' W. j - 2«.76 0.441 n \ + 8 .00 log 0.903 In this example the sun's azimuth was 120°, and in l m the altitude changed 13'. An inequality of 30" in the altitudes would therefore affect the result only ^ of l m , or 1 5 .2. An error of V in the hourly change of latitude would affect the 5 s result — , or (K6. 2. At sea, 1898, June 29, 0^ ; lat. by mer. alt. of 0, 33° 25' K, long, by account, 147° 10' E. ; near 11 a.m., T. by Chro. 1* 55 m 54 8 obs'dalt. of o 74° 9' 10"; 1 p.m., " " " 3 45 ) u - Chro. cor. on G. m. t. - 36 m 28 s ; In. cor. of sex't + 0' 50" ; height of eye, 18 feet. The ship run from 11 a.m. to noon 1ST. 3 p'ts W. 11' ) from noon to 1 p.m. N. 2 " W. 8') required the longitude at noon. For N. 3 W. 11' N. 2 W. 8 whence A A A, L L L = + 9'.1 A X = + 7 .4 A a = + 8 .25 = 495" = + TA = +3.7 A. m. Chro. t. 4- 12 A p. if. Chro. t. h m s 13 55 54 15 45 O's dec. O / // // + 23 16 50 - 7.3 Eq. oft. m s s - 2 59.8 —0.51 Elapsed time Mid. Chro. t. Chro. cor. (G. m. t.) 149 6 14 50 27 -36 28 - 1 44 + 23 15 05 - 7.2 (7.14 - 307 | .11 Middle long. LONGITUDE. 183 h m s Mid. G. m. t., June 28, 14 13 59 log A & L 2.695 log A A L 2.695 — Eq. of t. — 3 07 log A 9.410 n log B 9.398 Mid. G. ap. t. 14 10 52 1 tan d 9.633 1. tan L 9.819 Red. for A L + 27 log 1.738 n log 1.912 G. ap. t. of noon 14 11 19 - 54 s . 7 4- 81 s . 7 = + 27 s - 9 48 41 or 147° 10'.2 E. Red. to noon 1 .8 W. Long, at noon 147 8 .4 E. The sun's azimuth was 127° ; for A t = l m , A h = 10", and an inequality of 1/ in the altitudes will affect the result ?^ of 27 s l m , or 3 s . An error of V in A h L will affect the result tt^z? or 3 S .3. 207. 5th Method. (By transits.) Observe the transits of the sun or a star across the threads of a well-adjusted transit instrument, noting the times. Re- duce the mean of the noted times for semidiameter and errors of the instrument as in Art. 184 ; and thence find the Green- wich hour-angle of the body in the way described in Art. 189. This will be the longitude, if the upper culmination has been observed, as the local hour-angle is 0. If the lower culmina- tion has been observed, the local hour-angle is 12*. This method can be used only on shore. Example. 1898, May 17, 17* 16 m 20 s .5 G. mean time, the meridian transit of a Bootis (Arcturus) was observed ; required the longitude of the place of observation. G. mean time May 17 17 A W m 20 s . 5 80 3 40 49.40 Red. for G. m. t. + 2 50.24 G. sid. t. 21 00 00.14 *'s R. A. 14 11 03.73 *'s H. angle or Long. + 6 48 56 .4 or 102° 14' 06' w. 184 NAVIGATION. LONGITUDE. — LUNAR DISTANCES. 208. Problem 48. To find the longitude by the distance of the moon from some other celestial object. Solution. If we have given the local mean time and the true distance of the moon from some celestial object as seen from the centre of the earth, we may find, by interpolating the Nautical Almanac lunar distances (Prob. 22), the Green- wich mean time corresponding to this distance. The differ- ence of this from the local time is the longitude. The local time may be found for the instant of observation, either from an altitude of a celestial object observed at the same time, or by a chronometer regulated to the local time. At sea the correction of the chronometer on local time can be found from altitudes observed near the time of measuring the lunar distance, and reduced for the change of longitude in the interval by the formula (Art. 167), d = c + A X, A X being in time and -j- when the change is west. In practice the apparent distance of the moon's bright limb from the sun or a star is observed, and the true distance derived by calculation, as in the next problem. 209. Problem 49. Given the apparent distance of the moon's bright limb from a star, the centre of a planet, or the sun's nearest limb, to find the true distance of the moon's centre from the star, or the centre of the planet or the sun. Solution. It is necessary that the altitudes of the two bodies should be known, either directly from observations at the same time, or from observations before and after, and longitude: — lunar distances. 185 interpolated to the time of observation (Bowd., Art. 312) ; or computed from the local time (Prob. 32), (Bowd., Art. 313). The Greenwich time is also supposed to be known approxi- mately, either from the local time and approximate longitude, or, as is preferable, from the time noted by a Greenwich chronometer. A complete record of the observations will include the ap- proximate latitude and longitude of the place, the local time and chronometer correction, the index corrections of the in- struments used, the height of the barometer and thermome- ter, and at sea, the height of the eye above the water, as well as the noted times of observation and the observed distances and altitudes. Several observations may be made at brief intervals, and the means taken. 210. The preparation of the data embraces : 1. Finding the Greenwich mean time approximately from the chronometer time, or from the local time. 2. Taking from the Almanac for this time the semi-diame- ter and horizontal parallax of the moon, and of the other body * when they are of sensible magnitude ; adding to the moon's semi-diameter its augmentation. (Art. 59.) At low altitudes the contractions produced by refractions should be subtracted from the semi-diameters of the sun and moon. Formulas for finding these are given in Art. 213. When the spheroidal form of the earth is taken into con- sideration, to the moon's equatorial horizontal parallax (Art. 57), as taken from the Almanac, should be added the augmen- tation to reduce to the latitude of the place, which is found * The sun's horizontal parallax may be taken as 8". 5. 186 NAVIGATION. in Table 19 (Bowd.). The decimations of the two bodies to the nearest degree are required from the Almanac for this purpose. 3. Applying to the observed distance the index correction of the instrument, and, when the sun is used, adding the moon's augmented semi-diameter and the sun's semi-diame- ter ; when a planet or star is used, adding the moon's aug- mented semi-diameter if its nearest limb is observed, but subtracting it if the farthest limb is observed. 4. Applying to the observed altitude of each body the index correction, dip, and semi-diameter (when necessary), so as to find the apparent altitude of its centre. If the true altitude is computed, the parallax must be subtracted and the refraction added. In the following direct method it is necessary also to find the true altitudes. 211. To find the true distance, let D =s the apparent distance of the centres, iy= the approximate true distance, h = the apparent altitude ) „ . ;■ . , >of 3>'s centre, h = the true altitude J H= the apparent altitude ) . _ , , \ of O s centre, planet, or star. H'= the true altitude ) In Fig. 35, let m and S be the apparent places of the moon and other body ; m f and S', their true places. The true and apparent places of each are on the same ver- tical circle, Z m, Z S respectively, since they differ only by refraction and parallax, which act only in vertical circles, except so far as a small term of the moon's parallax is con- cerned, which will be subsequently considered. LONGITUDE. — L UNA R DISTANCES. 187 == 90° — h ) being given, Fig. 35. Then mS = D> the apparent distance ; mf S' = 17, the true distance ; and in the triangle m Z S, m$ = D Zm Z S = 90° - H to find the angle Z, we have by Sph. Trig. (32),* 2 _ cos \ (h + JJ+ />) cos } (A + H- J>) cos h cos H Then in the triangle mf Z S', Zm' = 90° - A' and ZS' = 90° - J5P being given, mf S' may be found by Sph. Trig. (17),t sin 2 i J7 = cos 2 i (// + H') - cos A' cos H' cos 2 } Z, or by substituting the value of cos 2 \ Z, and putting s = i(h + ir+D), (135) sin 2 J- jy = cos 2 J (A' + H') - cos A ' cos H ' cos « cos (s - D). cos A cos If To adapt this for logarithmic computation, put then • 2 . cos N cos H' , -r^ sir \ m = cos s cos (s — Za), 1 cos h cos JI y J ' sin 2 £ 2/ = cos 2 i (*T + J5T ) - sin 2 £ m, which by Pl. Trig. (134), becomes * cos 2 hA = sin H<* + ft + c) sin £ (6 + c - a) # sin 6 sin c t sin 2 ^ a = sin 2 £ (6 + c) — sin 6 sin c cos 2 ^ A. (136) 188 NAVIGATION. sin 2 i jy = cos i (// + H' + m) cos i (h! + H' - m), or, if we put /= i (h' + H' + m), (137) we have sin £ jy = V[ cos «' cos (/ — m)]. (138) The solution is effected by formulas (135), (136), (137), and (138). This is only one of several direct trigonometric solutions. It is easily 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, Z^, is not the true distance as seen from the centre of the 'Z earth, but from the point C (Fig. 36), where the vertical line of the place in- tersects the earth's axis. A reduction to the centre, C, is still required, for which we have the for- mula,* Fig. 36. A Jy = A P sin L ( sin 8 5 sin 8, ■> (139) \sin jy tan D r in which 8 S is the sun's declination, 8 TO , the moon's declination, P, the moon's equatorial horizontal parallax, whose mean value is 57' 30", A, a coefficient depending on the eccentricity of the terres- trial meridian, the mean value of which, for latitude 45°, is .0066855, or of log A, 7.8251, * Chauvenet's "Astronomy," I., 399. LONGITUDE. — LUNAR DISTANCES. 189 A sin X, the distance C C, with CE = 1. The mean values of A P = 23".07, or log ^ P = 1.3630 may be used, unless great precision is required. The signs of the declinations and latitude are + when, north, and A & is to be added algebraically to D\ If the augmentation of the parallax has been neglected, the distance has been reduced to a point 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 direction of the distance S m ; A aS 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 (135), 8 — i (h + H+ D) We haVG *1A../ / coB,Bin(,-iry siM^y/f sin | q = //cos s sin (s — h)\ = V V sin D cos H J (140) for which it will 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 r . 190 NAVIGATION. The contractions, A s and A S, of the vertical semi-diameters, may each be found from the refraction table, by taking the difference of refractions for the limb and centre. Then, for the required corrections, we have the formulas,* A's = As cos 2 q, A'# = A S cos 2 Q. (141) This contraction for either body is less than 1", 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 D' when the nearest limb is used, or add it to 1/ when the farthest limb is used. 214. Let A D = the reduction of the apparent distance to the true, or J7 = D + A D. A great variety of methods have been given for finding A D, requiring 4- or, at the most, 5-place logarithms ; but also needing special tables. They generally neglect to take into account the spheroidal form of the earth, the correction of refraction for the barometer and thermometer, and the con- traction of the semi-diameters of the sun and moon. These together, at very low altitudes and in extreme cases, may produce an error of 3 m in the calculated Greenwich time, and do actually, in the average of cases, produce errors from 10' to 1*. In 1855, Professor Chauvenet gave a new form to the prob- lem, with convenient tables, by which all these corrections are readily introduced. It is reprinted in a pamphlet with his method of equal altitudes, and it is also given in Bowditch, Arts. 306 et seq. * Chauvenet's "Astronomy," I., 186. LONGITUDE. — LUNAR DISTANCES. 191 215. The moon's mean change of longitude is 13°. 17640 in a day, or 33" in l m of time. An error, then, of 33" in the distance will, in the average, produce an error of l m in the Greenwich time, or 15' in the longitude ; or an error of 10" in the distance will produce an error of about 20 s in the Greenwich time, or 5' in the longitude. We may, however, readily find the effect of an error of 1", and thence any 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. 216. Errors of observation are diminished by making a number of measurements of the distance. But even with a skilful observer a single set of distances is liable to a pos- sible error of 10" or even 20". Errors of the instrument are diminished by combining results 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 determinations of the chronometer correction. The error peculiar to the observer, that is, in making the con- tacts 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. 192 NAVIGATION. 217. Lunar distances are rarely used at the present day. They are given, however, in the Nautical Almanac, and might possibly be used for rinding the Greenwich mean time, with which to compare the chronometer. They may thus serve as checks upon it, which in protracted voyages might be much needed. If the chronometer correction 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 chro- nometer, 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 carefully reduced, the chro- nometer correction can be found quite satisfactorily. By taking the rate into consideration, observations running through a number of days can be combined. 218. Other lunar methods for finding the longitude, be- sides that of lunar distances, are, 1. By moon culminations, or observing the meridian tran- sits of the moon and several selected stars near its path, whose right ascensions are considered well determined. 2. By occultations, 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 azimuths of the moon and stars observed near the meridian. LONGITUDE. — LUNAB DISTANCES. 193 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. If corresponding observations are made at two different 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 right ascension ; 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 1* or l m , computed from the Ephemeris, will give the difference of longitude in the same unit. 194 NAVIGATION. CHAPTER IX. LATITUDE AND LONGITUDE BY SUMNER'S METHOD. Fig. 37. CIRCLES OF EQUAL ALTITUDE. -( SUMN EB' S METHOD.) 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', etc., 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. CIRCLES OF EQUAL ALTITUDE. 195 220. The latitude of M is equal to the declination of the body, and its longitude is the Greenwich hour-angle of the body ; which, in the case of the sun, is the Greenwich appar- ent time, or 24 A — 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 of 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 circle. His position, then, is just as well determined as if he knew his latitude alone, or his longitude alone ; since a knowledge 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 position. 222. The direction of this line at any point is at right angles with the direction of the body, or the line of bearing, as it is called ; for the polar radius M A is perpendicular to the 196 NAVIGATION. 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 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 suffice. With high alti- tudes, three points, or if the body is very near the zenith, four may be necessary, and even the entire circle may be required. 224. Problem 50. From an altitude of a heavenly body to find the line of position of the observer, the Green- wich time of the observation being known. Solution. From the given altitude, and assumed latitudes L 19 X 2 , X 8 , etc., differing but little from the supposed lati- tude, find the corresponding local times (Prob. 37), and thence, by the Greenwich time, the longitudes \ 1} A 2 , A 3 , etc. Thus we shall have the several points, whose positions are conveniently designated as (Zj, A*,), (X 2 ,A 2 ,), (A>^3>)> etc - It facilitates the computation to assume latitudes differing 10' or 20', as the £ sums and remainders differ 5' or 10', and only one of each need be written. Or, from the Greenwich time and assumed longitudes, A 1? A 2 ,A 3 ,etc, find the corresponding local times (Art. 77), and thence the hour-angles of the body (Probs. 28, 29). With these and the observed altitude, find the corresponding lati- tudes, ZjjZgjZ^etc. (Prob. 40). CIRCLES OF EQUAL ALTITUDE. 197 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 line, 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 them, which will be the required line of position. Two points con- nected 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 Bfl t, 2 such points plotted respectively on the P// parallels of latitude L : , L 2 , and each in its ~7~/ proper longitude ; A B is the line of posi- // tion. and the place of observation is at Att _ rIG. oo. some point of A B, or A B produced. This is all which can be determined from an observed altitude, 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 line, of greater or less extent as the latitude, or the longitude, is more or less accurately known. In uncertain 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 navigator 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'. 1 98 NA VIGA TION. 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 en- closed belt. In Fig. 38, a b is such a parallel to the line of position A B, its perpendicular distance from it measuring a difference of altitude ; the distance A a on a parallel of latitude measuring a difference of longitude. 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 po- sition coincides with a parallel of latitude ; and thus a posi- tion of the body near the meridian is favorable for finding the latitude from an observed altitude, and not the longi- tude. 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 find- ing 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 (L x , A. x ), (X 2 , A 2 ), etc., having been determined by Prob. 50, if the true latitude, Z, be subsequently found, the corresponding longitude, A, may be obtained by interpolation. CIRCLES 01 EQUAL ALTITUDE. 199 Or, the place of the ship may be found graphically upon the chart, by drawing a parallel in the latitude, X, and tak- ing its intersection P, with the line of position AB. So also, if the true longitude, A, 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 F B ±2 E C the line of position in P, the place of ■* . . Fig. 39. 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. 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 (X x , A x ), B, the position (X 2 , A 2 ), both near P, the true position, whose latitude is X, and longitude is A; we have, by interpolation A 2 — Ai and as the formulas for finding A, the longitude of the true posi- tion, when its latitude, X, is known. Or, we have 200 NAVIGATION. AX = AX X2 ~ Xl A 2 - *! (143) and X = X, -)- A Z as the formulas for finding Z, when X is given. The several differences are most conveniently expressed in minutes of arc, or, in the case of longitudes, in seconds of time. The local times may be used instead of the longitudes and in- terpolated in the same way. From the first of (142) we may readily determine how much a supposed error in an assumed latitude affects the resulting local time, or longitude. 229. Problem 51. To find from a line of position the azimuth of the body observed. Solution. We have the positions (Z x , X{), (Z 2 ,A. 2 ), 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 meridian may be measured with a protractor. The azimuth may thus be found to the nearest 1°. 230. Problem 52. To find the position of the observer from two altitudes of the same or different bodies, the Greenwich time being; known. Solution. Find the line of position from each. If the lines are plotted on the chart, their intersection gives the position required. CIRCLES OF EQUAL ALTITUDE. 201 This intersection may also be readily found by computa- tion, when the lines are regarded as straight. Let (L\ \\) (X' 2 X. r 2 ) De the position of two points of first line, (L'\ k'\) (Z"a k" 9 ) " " « " " " " second line. A L and A A be the run in lat. and long, between the two observations (LX) be the position at the time of the second observation, the upper accents distinguishing the observations, the lower accents distinguishing the latitude used for each point. Then by Plane Co-ordinate Geometry, assuming (L\ X\) as the origin, we have, y-AZ = Z '*-^ (x-AX) (144) y - (L\ - z\) = L "?rJ;> (« - ex", - vo) (145) L = L\ + y \ = \\ +x } ("«) (144) is the equation to the first line, moved for the run. (145) is the equation to the second line. (146) is the intersection of the first line, moved for the run, with the second line^ or the position at the time of the second observation. 231. Directions N. and E. are to be marked -f , and those S. and W. with the negative sign. If both lines have been ob- tained by simultaneous observations of two bodies, A L and A a become 0, and if the same assumed latitudes are used in both observations, of course L'\ — L' = 0. 232. The more nearly perpendicular the lines of position are to each other, the better is the determination of their 202 NA VIGA TION. 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 position from double alti- tudes. If 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., Art. 292, note.) Still, 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. Examples. 1. 1898, Nov. 24, about 6 a.m. in lat. 38° 40' 1ST., long. 125° W. (approx.), obs'd alt. Sirius 15° 27' (West) ; chro. t. 2 h ±7 m 43 s . Ran thence S. 58°.2 (true), 28.3 km when obs'd alt. O 14° 15', chro. t. 5 h 11 m 24 s . For both observations, the chro. cor. is — 25 m 09 s , i. c. + 2' 30", and height of eye 18 feet. Required the latitude and longitude by Sumner's Method at the time of the second observation. CIRCLES OF EQUAL ALTITUDE. 203 Solution. First line of Position. Chro. t. Chro. cor. h m s 2 47 43 — 25 09 *'sR. A h m s , 6 40 43.6 ft =15° 27' G. m. t. Nov. 24 2 22 34 ^'sdec. -16° 34' 30" I.e. + 2 30' h h m s 2.38 S 16 13 51.54 Dip. - 4 09 Red. G. m. t. + 23 42 Ref. - 3 28 S' 16 14 14.96 h = 15 21 53 1. sec h= 15 21 53 p = 106 34 30 1. cosec 2 S = 160 26 23 S = 80 13 12 1. cos S — h= 64 51 19 1. sin h m s >jc's t. = 3 35 22.5 1. sin | t 9.65588 $ R. A. = 6 40 43.6 L. s. t. = 10 16 06.1 S'o = 16 14 15 L. m. t. 18 01 51.1 Nov. 23 Long. 8 20 42.9 = 125° 10'.7 W. C / 0.10646 L\ = 38 50 1. sec 0.10848 0.01843 160 46 23 0.01843 9.23010 80 23 12 9.22271 9.95676 65 01 19 9.95735 9.31175 9.30697 9.65588 3 34 05.5 6 40 43.6 10 14 49.1 16 14 15 18 00 34.1 9.65348 8 21 59.9 = 125° 30' W. /Second line of Position. Ean S. 58°, E. 28 r .3 ; A lat. 15 S. ; dep. = 24 E. ; A long. = 31 E. Lat. left, 38° 40 r N. ; lat. in, 38° 25' N. Chro. t. 5* 11™ 24 s Q 14° 15' I. c. + 2' 30" Chro. cor. — 25 09 + 10 .59 S. D. 16 15 G. m.t.,Nov. 24, 4 46 15 = 4*.77 h = 14 25 .59 Par. 09 Eq. t. + 13 02 .1 Dip — 4 09 G. ap. t. 4 59 17.1 Ref. — 3 46 Q'sdec. — 20°37'39".8 — 2 23 .2 - 20 40 03 • 30.02 Eq. t. + 13™ 05 s . 67 - s . 739 120.1 (2 .96 21.0 ~ 3 - 53 J .52 2.1 + 13 02.14 I .05 204 4 NAVIGATION. £"i 38° 15' 1. sec 0.10496 L" 2 = 38° 35' 0.10696 ft 14 25 59'' p 110 40 03 1. cosec 0.02889 0.02889 2s 163 21 02 163° 41' 02" s 81 40 31 1. cos 9.16072 81 50 31 9.15201 -h 67 14 32 1. sin 9.96481 9.25938 67 24 32 9.96533 9.25319 ap. t. 20* 38 w 09 s d sin it 9.62969 20* 39™ 40 s .7 9.62660 Long. 8 21 08.1= 125° 17' W. 8 A 19™ 36.4 = 124° 54'. 1 W. To compute the lat. and long, of the intersection of the first line moved for the run, with the second line. L\ = 38° 30'N. L' 2 -L\ = +20' l\ = 125° 10/7 W. V 2 -l\ = -19'.3 L' 2 = 38 50 L" 2 -L\ = +20 2' 9 = 125 30 l" 2 -l'\ = +22 .9 L'\ = 38 15 L'\ -L\ = -15 X'\ = 125 17 l'\ -l\ = - 6 .3 Z" 2 = 38 35 AX =-15 *" 2 =124 54.1 A* =+31 Then by substitution in (144) and (145) we have V + 15 = from which 20 _ (x _ 31) and y + 15 = — (x + 6.3) 19.3 v 7 " 22.9 x = + 13.9 and y = + 2.7. Therefore the position at the time of the second observation is by (146) L = 38° 30' + 2'.7 N. = 38° 32'.7 N. \ = 125° 10'.7 W. + 13'.9 E. = 124° 56'.8 W. 2. 1898, Aug. 19, at sea, making passage from Honolulu to San Francisco: position at noon, lat. 33° 15' N., long. 135° 40' W. Thence ran N. 62° E. (true) 202 kn. until about 8 a.m. Aug. 20, when obs'd alt. O 37° 01', chro. t., 5 h 37™ 37 5 ; chro. cor. - 18™ 17 5 ; i. c. - 2'~30" ; height eye, 25 ft. Ran thence N. 62° E. (true) 28 kn. until about 10.45 a.m., when obs'd alt. O 64° 25', chro. t. l h 16 m 40 s ; c. c, i. c, and dip as before. CIRCLES OF EQUAL ALTITUDE. 205 Made noon, Aug. 20, after running 9 kn. farther on the same course. No meridian observation. Kequired the position at noon, Aug. 20, and the set and drift of the current. SOLUTION. N. Noon — 8 a.m. N. 62° E. 202' Noon, Aug. 19, Lat. 33° 15' N. Al 1 34 .8 N. 8 a.m., Aug. 20, Lat. 34 49 .8 N. E A X = 215'.3 Long. 135° 40' W. A A. 3 35.3 E. Long. 132 04 .7 W. by D. R. First line of position. Chro. t. 5* 37 m 37 s Q. 31 °01' fS.D. + 15' 51" C. c. - 18 17 + 7 17" Dip — 4 54 G. m. t. 5 19 20 = 5 A .32, h = S h t ' 08 17 J Par. 07 Eq. t. — 3 08 .5 Aug. 20 Ref . - 1 17 G. ap. t. 5 16 11 .5 [i. c. - 2 30 ©'s dec. + 12° 22' 01".7 - 49".64 Eq. t — 3 W ll s .63 + 0*. 59 r 248".2 r2.95 -4 24 - 1 \ 14 .9 + 3.14 >18 + 12 17 37 .6 I 1 - 3 08 .5 I .01 L\ = 34° 30' L sec 0.08401 L' 2 =35° 1. sec 0.08664 h = 37 08 17" p = 77 42 22 1. cosec 0.01008 0.01008 2 s = 149 20 39 149° 50' 39" s = 74 40 20 1. cos 9.42217 74 55 20 9.41520 s- h = 37 32 03 1. sin 9.78479 37 47 03 9.78724 9.30105 9.29916 1. ap. t. 20* 27 m 28 s .3 1. sin It 9.65053 20* 27™ 58 s . 3 9.64958 G.ap.t. 5 16 11.5 5 16 11.5 8 48 43.2 8 48 13.2 A'i = 132°10'.8) L\= 34 30 j A/ 2 =132°03'.3 / L' 2 = 35 00 j Second line of Position. Ean 8 — 10.45 a.m. N. 62° E. 28'; A L = 13.1 Dep. = 24.7, A a = 30'.1 E. 206 NAVIGATION. Mean long, from 1st line 132° 07' W. at 8 a.m. Approx. long, at 10.45 a.m. 131° 37 W. = S h 46™ 28 s . By D. R. at 10.45 a.m. lat. 35° 02'.9 N., long. 131° 34'.6 W. Chro. t. 8 16 40 Q 64 25 00 f S. D. + 15 51 Dip -4 54 C. c. - 18 17 +8 03 \ Par. + 4 Ref . - 28 G. m. t. 7 58 23 = 7^.97, h = 64 33 03 I I.e. -2 30 Eq. t. — 3 06.9 Aug. 20 G.ap.t. 7 55 16.1 Q's dec. Eq. of t. Mean \ 8 46 28 + 12° 22' 01".7 - . 49". 64 - 3" n 11 s . 63 + 0*. 59 Meant. 51 11.9 j 347".5 c 4 s . 13 - 6 35 .7 \ 44 .7 + 4.70 | .53 + 12 15 26 t 3 .5 - 3 06.9 I .04 h' = 64° 33' 03" 1. sin 9.95567 t x = 12 33 00 1. sec 0.01050 d = 12 15 26 1. tan 9.33696 1. cosec 0.67305 <£" = 12 32 51 1. tan 9.34746 1. sin 9.33695 <£' = 22 29 1. cos. 9.96567 l/\- 35°01'.9) X"i = 131 22 ) t 2 = 13° 03' 00" 1. 1. sin 9.95567 sec 0.01136 1. tan 9.33696 1. cosec i 0.67305 <£" = 12 34 18 1. tan 9.34832 1. sin 9.33778 $ = 22 13 1. cos 9.96650 L" 2 = 34°47 , .3) A"* =131 52 J To compute the position at 10.45 A. m. L'\ =34° 30' N. L'2 =35 00 N. L"\ =35 01 .9 N. L" 2 =34 47 .3 N. A L = + 13M A'i = 132° 10'.8 W. A'a = 132 03 .3 W. A"i = 131 22 W. A" 2 = 131 52 W. AA= +30M L'2 -L'\ = L"2 - L"\ = L"\ -L'x = 30' 14.6 31.9 A' 2 — A'i = + 7'.5 A" 2 — A"i = — 30 A"! - a'x = + 48 .8 Then by substitution in (144) and (145) y - 13.1 = y - 31.9 - -f30 (x - 30.1) = 4 (x - 30.1) + 7.5 - 30 {X ^'* } 15 whence x = + 32.9 and y = + 24.3. CIRCLES OF EQUAL ALTITUDE. 207 by (146) L = 34° 54.3 K X =131° 37.9 W. at 10.45 a.m. 10.45 a.m. lat. 35° 02.9 N. long. 131° 34.6 W. by D. R. " " 34 54.3 N. M 131 37.9 " " obs. A L 8.6 S. A A 3.3 W. dep. = 2.7 Current for 22f hrs. 9 kn. S. 17° W. = .4 kn. per hour. N. S. E. W. 10.45 — noon N. 62° E. 9. Current to noon S. 17 W. 0.5 10.45 a.m. lat. 34° 54.3 H". Run to noon A L 3.7 N. Noon Aug. 20 lat. 34 58 N. 4.2 7.9 _ as _ 02 3.7 N. 7.7 AX = 9.4E. long. 131° 37.9 W. by obs. A X 9.4 E. long. 131 28.5 W. 233. The correction for the run of the ship between two observations may be determined as follows. (Art. 288, Bowd.) Problem. 53. To reduce an observed altitude for a change of position of the observer. Solution. Let Z (Fig. 40) be the zenith of the place of observation ; h = 90° — Zm, the observed altitude ; Z', the zenith of the new position ; h' = 90° — 7/ m, the altitude reduced to the new position, Z r . d = ZZ', the distance of the two places, here referred to the celestial sphere ; C= P Z Z', the course; Z=PZ»i, the azimuth of m ; Z — 0= m Z Z', the difference of the course and azimuth. Z Z', being small, may be regarded as a right line, Z Z' O as a plane right triangle, and 208 NAVIGATION. m, without material error, as equal to 71 m ; so that we shall have ZO = ZZ'cosZ'Zm Z'wi=Zm-ZO or putting A h = Z O, Ah=dcos(C-Z) X h' = h+Ah I { V A A = d cos ( (7 — Z) is, then, the reduction of the observed altitude to the new position of the observer: it is additive when C — Z < 90° numerically ; subtractive when C — Z> 90°. It is smaller, and can, therefore, be more accurately computed the nearer C — Z approaches 90°. It is, therefore, better to reduce that altitude for which the difference of the course and azimuth is nearest 90°. If the second is the one reduced, then (7 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 run. The azimuth, or bearing, of the body can be observed with a compass, or be com- puted to the nearest degree, or half-degree, from the altitude. The assumption, 7J m = O m, is more nearly correct, the greater Z'morZ m, that is, the smaller the altitude. If we treat ZZ'wasa spherical triangle, d = Z Z' being expressed in minutes and still very small, we shall find A h = d cos (O - Z) - \ d 2 sin V tan h sin 2 (O - Z) ; (148) but the last term is inconsiderable unless d and h are both large. For instance, if d = 30', it will not exceed V unless h > 82°. CIRCLES OF EQUAL ALTITUDE. 209 Example. The two altitudes of the sun are 36° 16' 20", 58 6 15' 20", the compass bearings of the sun respectively S. E. by E. £ E. and W. S. W. ; the ship's compass course, and distance made good in the interval N. K W. £ W. 25 miles ; S. &i 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 25' cos 8£ pts. = - 25' cos 1\ pts. = - 2' 30" or — 2' 39", if the last term of (148) is included. 234. By (147) or (148) we may reduce one of the two altitudes 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 par- allel position determined by the course and distance sailed in the interval. Thus in Fig. 41, let A B be the line of position for the first observation, and Aft represent in direction and length the course and distance sailed 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 AB and a b is the reduc- 210 NAVIGATION. tion of the altitude for the change of position : for that dis- tance is A a X cos (B A a — 90°). 235. There are several other methods of finding the lati- tude by two altitudes either of the same or of different bodies ; but, with the exception of the one following, their methods are so intricate and Sumner's Method has proved so valuable a substitute for them, that they are rarely if ever used at the present time. Four methods are given in Bow- ditch, Arts. 288 to 292 ; a full discussion of the principles upon which they are based and of a method by three alti- tudes may be found in Chauvenet's Astronomy. 236. Problem 54. To find the latitude by the rate of change of altitude near the prime vertical (PresteVs Meihoa)* In the note to Art. 197 we have, for a very brief interval of time, and a small change of altitude, or, T'-T=t 15 cos L sin Z h'-h 15 cos L sin Z whence cos L = cosec Z; (149) ±o t in which H — h is expressed in seconds of arc and t in seconds of time, and, Z being -f- when east, — when west, cos L is always positive. If Z is near 90°, its cosecant varies slowly. When Z = 90° we have, , , h CO s.L= — — . (150) If, then, two altitudes are carefully observed near the prime vertical, and the times noted with great precision, the interval * Chauvenet's Astronomy, I., 303, 311. CIRCLES OF EQUAL ALTITUDE. 211 not exceeding 8 or 10 minutes, an approximate latitude may be found by (150), when the altitudes are within 2° or 3° of the prime vertical ; or by (149) when they are at a greater distance, and Z is approximately known. The time of passing the prime vertical can be found by (86). Z may be roughly computed from the altitudes, or found within 2° from the bearing observed by a compass, which will suffice, if the observations are made within 10° of the prime vertical. As, near the prime vertical, the altitude changes uniformly with the time, several altitudes may be observed in quick suc- cession, and the mean taken as a single altitude. The larger h! — h and t, consistent with the supposition of uniformity of change and the condition by which they are sub- stituted for their trigonometric functions, the more accurate in general will be the result. Still the method does not admit of much precision. It is entirely unavailable near the equator, and in latitude 45° may give a result in error from 5 to 10 minutes, even when the greatest care has been bestowed on the observations. It may, however, be useful to the navigator in high latitudes, as it can be used for altitudes of the sun, when almost exactly east or west, and it will restrict the position of the ship to a limited portion of the line of position found by Sumner's Method. There are occasions at sea, when to find the latitude only with- in 10' is very desirable. Examples. 1. 1898, June 15, 7 h a. m., in lat. 60° N., long. 60° W. ; T. by Chro. 11* 13™ 25*.3, obs'd alt. O 27° 00' 23") O's Az. " " " 11 19 51.0, " " " 27 48 42 j N. 88° E.; required the latitude. 212 NAVIGATION. tV log 8.8239 h'-h = 48' 19" log 3.4622 t = 6 m 25 s . 7 ar. co. log 7.4137 Z - 88° 1. cosec 0.0003 L = 59° 55' N. 1. cos 9.7001 If A (H - h) = 10", A log (N - h) = A 1. cos L = .0015, and A X = 6'. If the difference of altitudes can be depended on within 5", the latitude is correct within 3'. 2. 1898, July 13, 5 h p.m., in lat. 54° 20' N., long. 113° W., by account ; the altitude of the sun's lower limb was observed at 0* 23 m 34* by the chronometer, which was slow of G. mean time 10 w 18 s ; and the sextant remaining clamped, the upper limb arrived at the same altitude at h 27 m 8 8 .5 ; the true alti- tude of both limbs was 27° 18' 20" ; required the latitude. The sun's diameter, 31' 33", is the difference of altitudes in this case. The sun's azimuth computed from the altitude and supposed latitude is N. 88 %° W. tV log 8.8239 ti = 31' 33" log 3.2772 t = 3 m 34*.5 ar. co. log 7.6686 Z = 88£° 1. cosec 0.0002 L = 53° 56' N. 1. cos 9.7699 If we suppose t to be in error l 8 , 1. cos L will be in error .0020 and L 11'. If the elapsed time can be depended on within 0*.5, the latitude is correct within 6'. The longitude obtained from the same observations is 113° 5 r W. This method of observing the successive contacts of the two limbs of the sum with the horizon with the sextant clamped is recommended. CIRCLES OF EQUAL ALTITUDE. 213 CHAPTER X. AZIMUTH OF A TERRESTRIAL OBJECT. 237. In conducting a trigonometric survey, it is necessary to find the azimuth, or true bearing, of one or more of its lines, or of one station from another. Thence, by means of the measured horizontal angles, the azimuths of other lines or stations can be found ; and, still further, a meridian line can be marked out upon the ground, or drawn upon the chart. For example, suppose at a station, A, the angles reckoned to the right are B to C, 48° 15' 35" ; CtoD, 73° 37' 16" ; D to E, 59° 45' 20" ; and that the azimuth of D is N. 35° 16' 15" E. ; the azimuths of the several lines are A B, K 86° 36' 36" W. A B, K 35° 16' 15" E. A O, K 38 21 1W. A E, K 95 1 35 E. If upon the chart a line be drawn, making with A B an angle of 86° 36' 36" to the right, or with A B 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 sight-vanes, be directed to B, and the readings noted, and then the line of sight be revolved to the left until the readings differ 35° 16' 214 NAVIGATION. 15" from those noted, it will be directed north. If a stake or mark be placed in that direction, it will be a meridian mark north from A. 238. 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 ashore — the difference of this magnetic bearing and the true bearing is the magnetic declination, or variation, required. It is east if the true bear- ing is to the right of the magnetic bearing ; but west if the true bearing is to the left of the magnetic bearing.* 239. Problem 55. To find the azimuth, or true bear- ing of a terrestrial object. Solution. Let Z (Fig. 42) 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 0, the azimuth of ; z = 1ST Z M, the azimuth of M ; f = Z-2=MZ0, the azimuth angle between the two objects, or the difference of azimuth of M and 0. The problem requires that z and £ be found ; then we have * This has reference to the two readings. The actual direction of the object is the same; hut the true and magnetic meridians, from which the angles are estimated, are different. When the variation is east, the magnetic meridian is to the right of the true meridian ; when AZIMUTH OF A TERRESTRIAL OBJECT. 215 Or, numerically, Z — z -j- £, 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. 240. 2=NZ M, the azimuth of the celestial body, may be found from an observed altitude (Prob. 34), or from the local time (Prob. 32). In the first case, the most favorable posi- tion is on or nearest the prime vertical ; for then the azimuth changes most slowly with the altitude. In the latter, positions near the meridian may also be successfully used. 241. £ = M Z 0, the azimuth angle between the two ob- jects, may be found in one of the following ways : — 1st Method. (By direct measurement.) M Z 0, being a horizontal angle, may be measured directly 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 suc- cessively to the objects, and lines drawn upon the paper along the edge of the ruler in its two positions, and the angle which they form measured by a protracter. At the instant when the observation is made of the celestial the variation is ivest, the magnetic meridian is to the left of the true meridian. It is necessary to distinguish between the magnetic bearing and the compass bearing. The latter is affected by the errors of the instrument employed and by local disturbances ; the former is free from them. 216 * NAVIGATION. 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. The error of collimation 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. 242. If the sun is used, 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 h = 90° — Z s (Fig 51), the altitude of the sun, s = S s, its semi-diameter, Z s' = S Z s, the reduction of the azimuth for the semi-diameter. We have ~ „ sin S s sin S Z s = — — , sin Z s or, since s and it are small, s' = s sec A, (151) which is the reduction required. The sign with which it is to be applied depends upon the limb observed. Fig. 43. AZIMUTH OF A TERRESTRIAL OBJECT. 217 243. 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 subtracted from, 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 the position of the instrument of siVs °f the distance of the object may change the azimuth 1' ; or of 2 o oW(T °f the distance may change the azimuth more than 1". 244. 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 (Fig. 44) of the two objects, and either note the time by a watch reg- ulated to local time, or measure simultaneously the altitude 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 object, when there is one. Correct the readings of the instruments for index errors, and when only one limb of the sun is ob- served, for semidiameter.* Observed altitudes of either object above the water-line * It is best in measuring the distance of the sun from the terrestrial object to use each limb alternately. 218 NAVIGATION. are also to be corrected for the dip by (33) or Table 14 (Bo wd.), if the horizon is free ; but by (35) or Table 15 (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 times. (Bowd., Art. 312.) Or the true altitude may be computed for the local time (Prob. 32 or 33) and the refraction added and the parallax subtracted to obtain the ap- parent altitude. Z (Fig. 42), the apparent altitude of 0, JET » 90° - Z M, the apparent* altitude of M. - i) = MO, the corrected distance. Let N = 90 c We have then in the triangle M Z the three sides from which ( = MZO may be found by one of the following formulas : — 1. By Sph. Trig. (164) we have un * * -y — cos Rr co — r or, letting d = H' — ti, "j sin l I _ 4 / fanj(l> + d)smi(J)-d) \ 2 4 " V COS JSP cos h! J (152) 2. By Sph. Trig. (165), 1508 **-y- cos ir COS h' * The true altitude of M is used in finding z, its azimuth. AZIMUTH OF A TERRESTRIAL OBJECT. 219 or, putting s = i(ir + h' + D) , j. /cos s cos (s — D) C ° SH = V cob IT cos A' (153) (152) is preferable when £ < 90° ; (153), when £ > 90°. 245. If is in the true horizon, or its measured altitude above the water line equals the dip, h = 0, and the right triangle M m gives cos £ = cos mO = cos D sec IF ; ( 154 ) or, more accurately, when £ is small (Sph. Trig., 105), tan J £ = V (tan £(2> + ^0 tan \ B ~ ir ) • ( 155 ) If the terrestrial object is in the water-line, h! is negative, and equals the dip. 246. If both objects are in the horizon, or H and h are equal and very small, we have simply £ '—■!>. (156) 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. 247. If the terrestrial object presents a vertical line to which the sun's disk is made tangent, the reduction of the observed distance for semi-diameter is tf = s sin M O Z (157) and not 5, the semi-diameter 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 direction of the distance O M. /^ 220 NAVIGATION. As the altitude of the terrestrial object is always very small, we may find M Z by the formula ,^ ^ ~ sin A' cos M Z = — : — =j sin Ir jy being the unreduced distance. 248. 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 sextant reading. If A represent the distance of the object, d y the distance of the axis from the line of sight or axis of the telescope, this correction is p = -^cosec 1" = 206265" — (158) It is 1', when A = 3437.75 d. 249. If the distance of the terrestrial object and the dif- ference of level above or below the level of the instrument are known, we may find its angle of elevation, nearly, by the formula -^ tan h' = — , A A being the distance of the object, and E, its elevation above the horizontal plane of the instrument. If the object is below that plane, E and h! 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 ; IT and h' representing their apparent angles of elevation. Each of these may be found by direct measurement, or from the known dis- tance and the elevation, or depression, from the horizontal plane of the AZIMUTH OF A TERRESTRIAL OBJECT. 221 observer. If the two objects are on the same level as the observer, we have simply as in (234) J = D. Example. 1898, May 16, 5 h 45™ a.m. in lat. 38° 15' K, long. 76° 16' W. ; the angular distance of the sun's centre from the top of a light-house measured by a sextant (O to the right of L. H.), 75° 16' 25", index cor. - V 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 7,300 feet, 1° 15' 20", index cor., + 2' 10"; height of eye, 20 feet ; required the true bearing of the light- house. G. m. t., May 15, 22* 50 m 04 s = May 16 - 1M7 Obs'dalt.Q 10° 18' 20" O's dec, May 16. I. c. +2 10 + 19° 10' 15".7 + 34".4 Dip - 4 23 r34".4 Ap. alt. 10 16 07 -40 .2 j 3.4 S. D. + 15 51 + 19 09 35 .5 I 2.4 Ref. and Par. — 5 03 Ap. alt. -e, H'= 10 31 58 Obs'dalt. L. H. 1° 15' 20" Tr. alt. -e, H = 10 26 55 I.C. + 2 10 Ang. dist. = 75 15 10 Dip - 9 56 by (55) App. alt. L. H. 1 07 34 = h! Computation by (78) and (152). H= 10° 26' 55" 1. sec 0.00726 H'= 10° 31' 58" 1. sec 0.00738 L = 38 15 1. sec 0.10496 h> = 1 07 34 1. sec 0.00008 p = 70 50 25 d = 9 24 24 2 s =119 32 20 i (D-M) = 42 19 47 1. sin 9.82827 s = 59 46 10 1. cos 9.70198 h (D-d) = 32 55 24 1. sin 9.73521 9.57094 p—s = 11 04 15 1. cos 9.99184 if =37° 36'.2 1. sin 9.78547 9.80604 J = 75 12 .4 i Z = 36° 53' 1. cos 9.90302 O's Azimuth Z = N. 73 46 .0 E. True bearing of L. H. (Z-£)=N. 1°24'.4W. BOOKS ON NAVIGATION ^SEAMANSHIP. Text-Book of Steamatiship* The Equipping and Handling of Vessels under Sail or Steam. For the use of the U. S. Naval Academy. By Rear Admiral S. B. Luce. Revised and enlarged by Lieutenant Wm, S. Benson, U.S.N. 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