TRIGONOMETRY..

761

between the centre and the sine, and is equal to the sine of thecomplement of that arc. Thus CF is the co-sine of the arc AB,and is equal to Bl, the sine of its complement HB.

The tangent of an arc, is a right-line touching (he circle in oneextremity of that arc, confined from thence to meet a line drawnfrom the'centre through the other extremity; which line is calledthe secant of the same arc; thus AG is the tangent, and CG thesecant of the arc AB.

The co-tangent and co-secant of an arc, are the tangent and se-cant of the complement of that arc ; thus I1K and CK are theco-tangent and co-secant of the arc AB.

The lines here described belong equally to an angle as to thearc by which it is measured ; and, except ttie chords and versedsines, they are all common to two arcs or angles which are thesupplements of each other. So that if the sine, tangent, See. ofany arc or angle above 90° are required, it is the same thing as tofind the sine, tangent, Sic. of its supplement, or vvliat it wants of180°. They are also called the natural sines, tangents, Sec. of thearcs or angles to which they belong; and the logarithms of thenumbers by which they are represented, are the logarithmic sines,tangents, &c. And as one or other of these lines make a part ofevery trigonometrical operation, they have been calculated to agiven radius, for every degree, minute, &c. of the quadrant, andranged in tables for use. Whence, by the help of such a table,the sine, tangent, &c. of any arc or angle, may be found by in-spection ; and, vice versa, the arc, or'Angle , to which any sine,tangent, Sec. belongs. Upon this table also, ami the doctrine ofsimilar triangles, depends the solution of the several cases of planetrigonometry, which may be performed either by the natural orlogarithmic sines, tangents, &c. as occasion requires. But thelogarithmic sines, tangents, &c. are those most commonly used ;as the calculations, in this case, are all performed by adding andsubtracting only, instead of multiplying and dividing, as is re-quired by the natural sines, &c. The sine, tangent, &c. ot anyarc or angle being of the same magnitude as the sine, tangent, Sec.of its supplement, it is plain that a table of these lines made forevery degree, minute, &c. of the quadrant, or 90°, will serve forthe whole circle. It is also to he observed that, in every suchtable, the natural sines, tangents, Sec. are usually calculated toradius 1 ; but in order that the logarithmic sines, tangents, Sec.may be all positive, they arc- calculated to radius 1.0000(100000,,or 1 with 10 cyphers, the logarithm of which is 10, so that the'latter are the logarithms of the former, with 10 added (o the index.And, as the natural situs, tangents, &c. of any angles or arcs ofdifferent circles, are proportional to the radii of those circles, theirv alues may be readily found, or made to correspond to any radius'vhatever. In every plane triangle, three things must be given tofind the rest; and of these three one at least must be a side, be-cause the same angles are common to an infinite number of tri-angles. It is also to be observed, that all the varieties that canpossibly happen in the solution of plane triangles, are comprisedUnder the three following eases: viz. 1. When two of the threegiven tilings are a side and its opposite angle. 2. When two sitlesa «d their included angle are given. 3. When the three sides aregiven. Each of which cases may be resolved, either by geo-metrical construction, by arithmetical computation, or instrumenl-a My. In the first of these methods, the triangle is constructed,'T laying down the sides from a scale of equal parts, anti thea »glesfroin a scale of chords, cr a protractor, and then measuring'he unknown parts by the same scale or instrument from whichme others were taken. In the second method, having stated theproportion, according to the proper rule, multiply the secondat 'd third terms together, and divide the product by the first, andt*'e quotient will be the fourth term required, tor the naturalj'ymbers. Or, in working by logarithms, add the logarithms ofjhe second anti third terms together, and from the sum take the■mgarithin of the first, and the number answering to the remainderU| h he the fourth term sought. In the third method, or instru-mcntaliy, a, suppose by the logarithmic lines on one side of theuoininon two-foot scales, extend the compasses from the hrst termto the second or third as they happen to be of the same kind ;an< l tha; extent will reach from the other term to the fourth termWquired, taking both extents towards the same end of the scale.* “ e second of these methods, however, or that in which the ope-fation is peiformed by logarithms, is the one generally employed ;

VOL, IV.—NO. 209.

I he other two being chiefly of use as checks i,n the calculations,or, in certain simple cases,'where a near approximate value of thequantities to be determined is thought sufficient.

Properties of Plane Triangles, required in the prac-tical Part of this Science.

T lie sum of all the three angles of any plane triangle is equal totwo right angles, or 180°.

The greater side is opposite to the greater angle : and the lessside to the less angle.

'l'lie sum of any two sides is greater than the third ; and thedifference of any two sides is less than the third.

The triangle is equilateral, isosceles, or scalene, according as itsthree angles are all equal, or only two of them equal, or all threeunequal.

The angles opposite to the two least sides are acute ; and ifthere is an obtuse angle, it is opposite to the greatest side.

A perpendicular drawn from the oppo-ite angle to the longestside will fall within the triangle; and the greater and less segmentwill be next the greater and less side.

In an isosceles triangle, a perpendicular drawn from the vertexwill bisect both the base and (he vertical angle.

In a right-angled triangle tile Inpothenuse is equal to the squareroot ot the sum of the squares of the other two sides ; and eitherof the sides is equal to the square root of the difference of thesquares of the liypothenuse and the other side.

Note, also, that if the half-diirerence of any two quantities isadded to their half-sum, it will give the greater of those quantities,and if it is subtracted from the half-sum, it will o-ive die less.Thus DC, (fig. 12,) half the difference, added to A1 S half the sumof AC and CB, gives AC the greater, and subtracted gives CBthe less.

When two of the three given things are a side and its oppositeangle, to find the rest.

Axiom 1. 'I he sides of any plane triangle are to each other asthe sines of their opposite angles, and vice versa;—That is,

As any side is to the sine of its opposite angle, so is any otherside to the sine of its opposite angle.

Or, As the sine of any angle is to its opposite side, so is the sineof any other angle to its opposite side.

lienee, to find an angle, begin the proportion with a side oppo-site to a given angle; and to find a side, begin with an angle op-posite to a given side.

Demonstration. Let the triangle ABC, (fig. 2,) be inscribedin a circle ; then, from the property of the circle, half of each sideis the sine of its opposite angle ; but the sines of these angles, intabular parts, are proportional to the sines of the same in anyother measure ; therefore, in the triangle ABC, the sines of theangles will be as the halves of their opposite sides; and since thehalves are as the w holes, it follows, that the sines of the angles areas their opposite sides; i. e. S, C : S, A : : AB : BC, &c.

Case II, When two sides and their included angle are given,to find the rest.

Axiom 2. In any plane triangle, as ABC, (fig. 3,) the sum ofthe sides, AB and BC, is to the difference of these sides, as thetangent of half the sum of the angles BAC , ACB, at the base, is tothe tangent of half the difference of these angles.

Demonstration. Produce AB; and make BI1 equal to BC ;join HC, and from B let fall the perpendicular BE; through Bdraw BD parallel to AC, and make HE equal to CD, and joinBE; also take BI equal to BA, and draw IG parallel to BD orAC. Then it is evident that AH will be the sum, and HI thedifference, of the sides AB and BC ; and since HB is equal toBC, and BE perpendicular to HC, therefore I1E is equal to EC ;and BD being parallel to AC and IG, and AB equal to BI,therefore CD'or HF is equal to GD, and consequently HG isequal to FD, and half HG is equal to half ED or ED.' Ao a i n ,since IiB is equal to BC, and BE perpendicular to IIC, thereforethe angle EBC is half Uie angle HBC ; hut the angle ElBC isequal to the sum of the angles A and C, consequently the angleEBC is equal to half the sum,of the angles A and C. Also sinceHB is equal to BC?, and HF equal to CD, and the included an-gles BHF, BCD, equal, it follows that the angle DBF is equal tothe angle DBC, which is equal to BCA ; and since HBD isequal to the angle A, and HBF equal to BCA, therefore FBD is9 H the