sine .0002556634; but at the end of the 11th division the arc is 1'.45".28".7"".30", and its sine.0005113269; therefore the sine of the 12th division is just half the sine of the eleventh, in the same manner as the twelfth arc is half the eleventh. Now in indefinitely small arcs, the arcs will be to each other as their corresponding sines, hence 52".44" .3 .45.:1':: *0002556634, &c. ; •0002908882 the sine of one minute. The cosine of l'=the square-root of the radius 1, diminished by the square of the sine of 1', viz. cosine l'=•9999999577. And it is shewn (G. 122.) that 2 x cos.l'x sinel'-sinel' =.0005817764=sine2',or cos.890.58' 2 x cos.l'x sine2 --sinel'=.0008726645=sine3',or cos.89o.57' 2 x cos.l'x sine31 - sine2 =:0011635526=sine4',orcos.89o.56' 2 x cos.l'x sinet! - sine3'= .0014544406 =sine5', &c. &c. This method may likewise be extended to the cosines, by beginning at the other end of the arc, for the cosine is only the sine of the complemental arc. 2 x cos.l'x cos.1'cos.O'=.9999998308 =cos.2,or sine89°.58' 2 x cos.l' x cos.2' - cos.l'=.9999996192 =cos.3',orsine899.57 2 x cos.l'x cos.S' - cos.2'=.9999993231=cos.4', or sine890.56' 2 x cos.l'x cos.4'-cos.3'=:9999989423=cos.5',or sine890.55' Proceed thus to find the sine and cosine for every minute of the arc as far as 30°. sine A.COS B (K) Then, if A and B be any two arcs, =isine rad (A+B) + isine (A - B) E. 115.; let a=30°, then sine a=į radius (K. 37.); and cosB = sine (30° +B) + sine(30°—B); hence, sine (30° + B)=cos B-sine (90°-B). If, therefore, b=1.2.3.4, &c. successively, we shall have Sine 300.1'=cos l'-sine 29o.59' Sine 30.3'=cos 3'-sine 29o.57', &c. And in this manner find all the sines, and thence all the cosines (M. 104). as far as 45°. By these means all the sines and cosines from 0° to 90° will be obtained; for sine (45° +A)= cos (45° — A) and cos (45° + A)=sine (45° — A). (L) The sines and cosines being constructed. Cosine : sine:: radius : tangent (U.103); hence the tangents are found. Tangents: radius:: radius : co-tangent(Z. 103); hence the cotangents are found. Cosine : radius:: radius : secant(X. 103); hence the secants are found. Sine : radius:: radius : co-secant (Y. 103); hence the cosecants are found. The versed sines are found by subtracting the cosine from radius, or adding the cosine to the radius, if the arc be greater than a quadrant. (M) ARTIFICIAL, or LOGARITHMICAL, SINES, &c. are only the logarithms of the natural sines, &c. The natural sines are generally calculated to the radius 1, and being in all cases, except when the arc is 90°, less than radius or 1, they must of course be decimals. Now the logarithm of a whole number and the logarithm of a decimal is the same, only the index or whole number prefixed to the former is affirmative, and that to the latter negative. Hence if logarithmical sines, &c. were immediately formed from natural sines which are calculated to the radius 1, their indices would all be negative. To avoid this, the logarithmical radius instead of being taken 1, as in the natural sines, is generally considered as ten thousand millions. (N) To find the logarithmical sine of 1' to 7 places of figures, without the index. Rad. 1 : rad. 10,000,000,000 :: 0002908882 (the natural sine of 1', the radius being 1): 2908882 (the natural sine of 1', the radius being 10,000,000,000.) The logarithm of 2908882 = 6.4637261 the logarithmical sine of 1'. Hence the indices to the logarithmical sines from 0 to 4', will be 6; from 4' to 35' the indices will be 7; from 35' to 5o. 45' the indices will be 8; from thence upwards to 89 they will be 9; and at 90° the index will be 10=the logarithm of the radius 10,000,000,000. (0) Hence if we take the natural sine of any arc from a table of natural sines (where the radius is unity), and multiply it by 10,000,000,000; the logarithm of the product will give the logarithm sine of that arc to as many places of figures as the natural sines are carried to. It may be proper to inform the learner, that this method will not be exactly true for the first five degrecs ; because the natural sines in the tables are not carried to a sufficient number of places. (P) In logarithms the operation of multiplication is performed by addition, and division by subtraction. The logarithm sines being constructed, the tangents, &c. are formed thus : Sine+10-cosine= tangent (U. 103.) =co-tangent (Z. 103.) 20-sine =co-secant (Y. 103.) If you double the logarithmical sine of half an arc, and subtract 9.6989700 from the product; the remainder will be the logarithmical versed sine of that arc. For radius x } versed sine=square sine } arc (I. 107.) or, ž radius x versed sine-square sine i arc, or, log. 5,000,000,000 tlog, versed sine = 2 x log, sine arc, or, 9.6989700 + log. versed sine=2 x log. sine į arc. Therefore, log. versed sine= 2 x log. sine arc-96989700. (Q) The following logarithmical formulæ for the sine, cosine, &c. of any arc, may in some cases be useful, viz. Sine=cos + tang~10=tang + 10 – sec=cos + 10-cot= 20- cosec. Cos=sine +10 - tang=20-sec=sine + cot-10=cot+ 10~cosec. Tang=sine+10-cos=20-cot. Sec=tang+10-sine= 30-sine-cotscosec +10-cot= 20-cos. Cosec=cot+10-cos= 30-cos-tang=sec+ 10-tang= 20-sine. BOOK III. SPHERICAL TRIGONOMETRY. CHAPTER I. DEFINITIONS, &c. OF SPHERICAL ANGLES, ARCS, AND TRI ANGLES. (A) SPHERICAL TRIGONOMETRY treats on the properties of spherical triangles, or the position and magnitudes of arcs of circles described upon the surface of a sphere or globe. (B) A sphere, or globe, is a round solid body, each part of its surface being equidistant from its centre. This centre is common to every circle described on the surface of the sphere, wherein spherical trigonometry is concerned, and such circles are called great circles. (C) All great circles divide the globe into two equal parts, and consequently bisect each other at the distance of a semicircle or 180°. Hence two arcs cannot enclose a space, unless they are both semi-circles. (D) A spherical angle is the inclination of the planes of two great circles to each other, which circles intersect or meet each other on the surface of the sphere, in a point called the angular point. The inclination of these planes must always be ineasured on the arc of a great circle, 90° from the angular point. (E) A spherical triangle is formed on the surface of the sphere by the intersection of three great circles, and consists of three sides and three angles; any three of which parts being given the rest may be found. (F) All the sides of a spherical triangle are arcs of equal circles, and the angles of a spherical triangle are measured by arcs of circles, having the same radii as the sides. Hence the sides and angles of spherical triangles are always expressed in degrees, and parts of degrees. (G) If one angle of a triangle be 90°, it is called a rightangled triangle.—If one side be 90°, a quadrantal triangle. -If no angle or side be 90°, it is called an oblique-angled triangle. (H) The pole of a circle is a point on the surface of the sphere equidistant from every part of that circle of which it is the pole. Consequently every circle has two poles diametrically opposite to each other, and the arc of a circle comprehended between each of these poles, and the circumference of such a circle, is a quadrant. No two circles can have the same, or a common pole. If a straight line be drawn from the pole of any circle to the centre of the sphere, it will cut the diameter of that circle at right angles. (I) All great circles passing through the pole of another great circle, cut that circle at right-angles; and if two circles cut each other at right-angles, in the poles of a third circle, the four points of intersection with that third circle, will be the four poles of the cutting circles; viz. the two opposite points will be the poles of that circle which is described between them. (K) Sides and angles of spherical triangles are said to be of the same species, kind, or affection, when by comparing any two sides, any two angles, or an angle and a side together, you discover each to be greater or less than a right-angle, or equal to a right-angle. But when by comparing a side with a side, an angle with an angle, or a side with an angle, you discover one to be less and another greater than a right-angle; such sides and angles are said to be of different species. (L) Spherical triangles are equilateral, isosceles, or scalene, according as they have three equal sides, two equal sides, or three unequal sides. N.B. In any of the following propositions, wherever the word circle, or arc of a circle occurs, it must always be understood to be a great circle, or the arc of a great circle. And, that all circles concerned in spherical trigonometry are equal to each other. PROPOSITION I.' (M) If one great circle intersect another great circle in any point A, all the angles described from the point A, on the same side of any arc CAD, or EAB, are equal to two rightangles; and all the angles made about the L point A, are equal to four right-angles. E D DEMONSTRATION. Let ca be perpendicular to EB, then will the angles Eac and caB be each of them a right-angle. If al be drawn, then LAB and EAL are equal to two right-angles, for the angle cab is increased by the angle Lac, and cae is diminished by the same, therefore LAB+EAL=EAC+ CAB. In the same manner it may be proved that the angles EAD and DAB are together equal to two right-angles, for if ac be |