Or, the sides a, b, or c, respectively, may be found by the formulæ in the fifth set of equations, page 214.

OF THE AMBIGUITY OF THE DIFFERENT CASES.

(U) When two sides, and an angle opposite to one of them, are given to find the rest (see Case Ist, IId, and Hid) the values of these required parts are sometimes ambiguous.

(W) Rules for determining whether the quantities sought are acute, obtuse, or ambiguous, have already been given in the solutions of the different cases.

The two following tables are the same as those given by Legendre at pages 400 and 401 of the 6th edition of his Geometry, and are deduced from Prop. XVII, page 145, and Prop. XVIII, page 146 of this treatise.

TABLE I. Let A, a and b be the given parts. Then,

90° ab one solution.

(ab two solutions.

90° (a+b180° one solution.

a+b180° two solutions.

a+b180° one solution.

90° (a+b180° two solutions.

If A=90°, a=b, or a+b=180, the cases will not be ambiguous, but if b=90°, there will be two solutions.

When

When two angles, and a side opposite to one of them, are given to find the rest (see Case Iivth, vth, and vith), the values of the required parts are subject to ambiguity; this triangle being supplemental to that wherein two sides and an angle opposite to one of them are given.

TABLE II. Let A, B, and a be the given parts. Then,

If a=90°, A=B, or A+B=180°, there will be but one solution, but if B=90°, there will be two solutions.

(X) We may likewise remark, that, since any side or angle of a spherical triangle is less than 180°, the half of any angle, or half the difference between any two sides, or half the difference between any two angles must be acute.

2

Hence in the equation, where cot c. cos(a-b)=tang (A+B). cos (a+b) (0.218) it is plain that cot c and cos (a−b) are both positive (K. 94), and therefore tang (A+B) and cos (a+b) must be both positive; consequently, half the sum of any two sides of a spherical triangle is of the same species as half the sum of their opposite angles. This rule has been applied in the practical solutions of the different cases, and will frequently remove the ambiguity which would otherwise arise, where a quantity sought is to be determined by means of a sine.

BOOK IV.

THE THEORY OF NAVIGATION,

CHAPTER I.

DEFINITIONS AND PLANE SAILING.

(A) 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.

(B) 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.

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

(D) 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.

(E) 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.

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

(G) 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.

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 lon gitudes 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 through the Royal Observatory at Greenwich, our seamen always reckon their longitude from that meridian.

(H) Parallels

(H) Parallels of latitude are small circles parallel to the equator. Every place upon the surface of the earth is supposed to have a parallel of latitude passing through it.

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

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

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

(M) The variation of the compass is the deviation of its points from the correspondent 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.

(N) 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.

(0) 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.

(P) 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.

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

(R) The course of a ship is the angle in which the track she describes cuts the meridians.

(S) 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.

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

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

(U) 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, Abczeul a rhumb line, or the track described by a ship sailing from A to L; AP, dr, fr, gr, qp, &c. meridians; ib, hc, kz, &c.

z z 2

parallels

parallels of latitude; and let the elementary triangles aib, bhc, ckz, zte, &c. be conceived so indefinitely small as to differ insensibly from plane or rectilinear triangles.

Then, the angles iab, hbc, kcz, &c. are equal (Q. 355); and the anglès 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. (Euclid. VI. and 4.) Therefore, ab: ai :: ab+be+cz+ze, &c. : ai+bh+ ck+zt, &c. (Euclid. V. and 12). That is,

ab : ai :: al : al; where ab and AL are distances, and ai and al correspondent differences of latitude. Q. E. D.

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

(W) In sailing upon a rhumb line, the departure correspondent 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. (Euclid. VI. and 4.) Therefore, ab ib:: Ab+be+cz+ze, &c. : ib+hc+kz+ te, &c. (Euclid V. and 12.) But the whole distance AL is equal to the sum of all the intermediate distances Ab+bc+ cz, &c.; hence, Ab: ib:: AL: ib+hc+kz + te, &c. Q. E. D.

(X) 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 diagonal 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.

(Y) While the course remains the same, it has been shewn 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, be

tween