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the same as M. Delambre's formula already given at page 94, which formula is there illustrated by an example.

(647) In a trigonometrical survey, where a formula of the same nature as that given above is usually applied, ZP and ZB do not materially differ from 90°, or the altitudes or depressions Ps and Bo do not exceed 2° or 3°; in such cases, the problem will admit of a convenient and useful approximation, such as that given by Legendre at page 413, of his Eléments de Géométrie, sixiéme èdition, and also by other authors.

Let Ps and Bo be represented by н and h, and let D be the observed angle BDP and x the correction, viz. let D+x be the required angle ODS, for the ODS will always be greater than the BDP (page 91). Now cos z=

COS PB-COS PZ COS BZ

(420) the radius being 1, or cos (D+x) =

COS D-Sin H. sin h

COS H. cos h

and cos h=1

; but cos H=1

h2 h+

+

2 2.3.4

H2

+

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sin PZ sin BZ

H4

2 2.3.4

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&c.

--&c. (271), and in very small arcs,

the sines do not differ essentially from the arcs; hence

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COS D-Hh (1—§н2) . (1— h2) 1 – 1⁄2í2 — ¦2 rejecting all the powers of н and h above the second, as being small. But cos (D+x)=cos D. cos x-sin D sin x, the radius being 1 (237), and as x is very small, by supposition, cos x=rad=1, and sin x= X,

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cos D-x sin D=COS D+COS D. (н2+h2) —нh, by rejecting the smaller terms after multiplication;

hence x=

2

нh-cos D. (H2+h2), the correction of the ob

sin D

served angle D, the same as Legendre's formula.

(648) The preceding formula may be simplified, by following the method of Legendre; thus, let (H+h)=p, and (H− h)=q; then p2-q2 Hнh, p2+q2={(x2+h2), and therefore x=

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(p2 -q2)-(p2+q2). cos D_p2-p2.cos D-q-q.cos D

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1 + cos D sin D

the radius being 1 (2 and cot D=

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; therefore r=p2 . tan §D—q2. cot D. In applying this formula to practice, p and q are generally given in seconds, a should therefore be expressed in seconds; hence if r be the number of seconds contained in the radius, the number of seconds contained in x=P. tan D

SCHOLIUM.

02

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cot D.

The five preceding propositions, viz. from the 3d to the 7th inclusive, comprehend the principal rules and formulæ used in the calculation of a trigonometrical survey, considering the earth as a sphere. The spheroidal form of the earth will occasion a small correction, and require formulæ somewhat different, though of no great difficulty, but they are here omitted because they do not properly belong to spherical trigonometry.

THE END OF SPHERICAL TRIGONOMETRY.

377

BOOK IV.

THE THEORY OF NAVIGATION.

CHAPTER I.

DEFINITIONS AND PLANE SAILING.

(649) NAVIGATION is the art of finding the latitude and longitude of a ship at sea, and her course and distance from that place to any other given place.

(650) The Earth is considered as a perfect sphere or globe, revolving on an imaginary line called its axis, from west to east, in twenty-four hours. This rotation towards the east causes all the heavenly bodies to have an apparent motion from east to west.

(651) The equator, generally called the line by seamen, divides the globe into two equal parts, called the northern and southern hemispheres.

(652) Meridians are great circles cutting the equator at right angles, and passing through its poles. Every point upon the surface of the earth is supposed to have a meridian passing through it. That meridian passing through Greenwich is called the first * meridian.

(653) Longitude of places on the earth is reckoned on the equator from the first meridian. If they be situated eastward of the first meridian, they are said to be in east longitude; if westward, they are in west longitude. The greatest longitude on the earth is 180 degrees.

(654) The difference of longitude between two places is an arc of the equator, contained between the two meridians passing through these places.

(655) The latitude of a place on the earth is reckoned from the equator, upon a meridian passing through the place. The greatest latitude a place can have is 90 degrees.

(656) Parallels of latitude are small circles parallel to the

* It is necessary for the purposes of Geography and Navigation, to call the meridian of some remarkable place the first meridian, and to estimate the lungitudes of all other places from that meridian. And, as all the tables in the Nautical Almanac, and other English astronomical tables, are adapted to the meridian passing over the Royal Observatory at Greenwich, our seamen always reckon their longitude from that meridian.

equator. Every place upon the surface of the earth is supposed to have a parallel of latitude passing through it.

(657) The difference of latitude between two places, is an arc of a meridian contained between the parallels of latitude which pass through these places.

(658) Meridional distance is the distance between the meridian sailed from and that arrived at, and is reckoned on that parallel of latitude which the ship is in.

(659) The Mariner's compass is a representation of the horizon; and is divided into 32 points, each point 11° 15'.

(660) The variation of the compass is the deviation of its points from the corresponding points of the horizon. When the north point of the compass is to the east of the true north point of the horizon, the variation is east; if it be to the west, the variation is west.

(661) If a ship be steered due north or due south, her distance sailed is equal to her difference of latitude; and her track will be on some meridian.

(662) If a ship be steered due east or west, her track will be either on the equator or some parallel of latitude; and the distance sailed will be equal to her departure, or meridional

distance.

(663) If a ship be steered towards any point of the horizon between the north and east, north and west, south and east, or south and west, the track she describes will be a Rhumb line.

(664) A rhumb line is a curve upon the surface of the sphere, cutting all the meridians in equal angles.

(665) The course of a ship is the angle in which the track she

describes cuts the meridians.

(666) The bearing between two places on the same parallel of latitude is east and west, on the same meridian north and south; in all other situations it is a rhumb line, continually approaching the pole.

(667) The departure is the whole easting, or westing, the ship makes in any single course.

PROPOSITION I. (Plate III. Fig. 2.)

(668) In sailing upon a rhumb line the differences of latitudes are proportional to the distances sailed.

Let p represent the pole, woQE a portion of the equator, AbezeuL a rhumb line, or the track described by a ship sailing from A to L; AP, dr, fp, gp, qp, &c. meridians; ib, hc, kz, &c. parallels of latitude; and let the elementary triangles aib, bhc, ckz, zte, &c. be conceived so indefinitely small as to differ insensibly from plane triangles.

Then, the angles iab, hbc, kcz, &c. are equal (664); and the angles aib, bhc, ckz, &c. are right angles, for the parallels of latitude cut the meridians at right angles. Therefore all the elementary triangles Aib, bhc, ckz, zte, &c. are equiangular and similar.

Hence, ab: ai :: bc: bh :: cz: ck :: ze: zt, &c. (4 Euclid VI). Therefore Ab: ai :: ab + bc +cz+ze, &c. : ai+bh+ck+zt, &c. (12 Euclid V). That is,

ab: ai :: al: al, where ab and AL are distances, and ai and al corresponding differences of latitude.

PROPOSITION II. (Plate III. Fig. 2.)

(669) In sailing upon a rhumb line, the departure corresponding to any course and distance is equal to the sum of all the intermediate departures.

For, as in the preceding proposition,

Ab: ib:: bc: hc :: cz: kz:: ze: te, &c. (4 Euclid VI); therefore ab : ib :: ab + bc + cz + ze, &c. : ib + hc + kz + te, &c. (12 Euclid V). But the whole distance AL is equal to the sum of all the intermediate distances ab+be+cz, &c.; hence Ab ib AL: ib+hc+kz + te, &c.

A

(670) SCHOLIUM. Hence it appears that the meridional distance, departure, and difference of longitude, are essentially different. Let a ship sail from a to L, when she arrives at L her meridional distance will be Ll, her departure ib+hc+kz +te, &c. and her difference of longitude wE. But the meridional distance is evidently less than the departure (which is equal to the sum of all the arcs ib+hc+kz, &c.); because the several meridians converge towards the pole; and for the same reason the difference of longitude WE is greater than the departure. Again, let the ship return from L to A along the rhumb line LA, her meridional distance will then be Aa, and her departure xu+we+qz+gc, &c. the same as before; for the elementary triangles are equal, an equal portion of the ship's track being the hypothenuse of each. Here the meridional distance aa is greater than the departure; hence in the same course, or track, backward and forward, the departure and difference of longitude remain the same, but the meridional distance is variable.

Aa

(671) While the course remains the same, it has been shown that the departure is greater than the meridional distance Ll, and less than the meridional distance aa; yet it is very nearly equal to the meridional distance MN, in the middle latitude, between the latitude sailed from, and the latitude arrived at. This was probably a casual discovery; and when the places

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