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the arcs, of their sum, and their difference, be what they may. This might be rendered evident by a suitable modification of the diagram; and will still farther appear in the fourth chapter of this work.

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PRoP. XI.

11. As the sum of the sines of two unequal arcs, is to their difference, so is the tangent of half the sum of those two arcs, to the tangent of half their difference.

Let AE and AB be two unequal arcs, of which Ek and BG, are the sines; and let EK be produced to cut the circle in D, and BI be drawn parallel to the diameter . AA’. Draw ID, IE, from the centre I with the radius ca of the assumed circle, describe an arc dhe, and through b draw LöM parallel to DE. Then, it is evident that NE = KE + Bo, is the sum of the sines of AE and AB, and ND = KE – BG = KD — KN, is their difference. Also, since BIE (at the circumference) = }Bce (at the centre); and BID = #Bed (Euc. iii. 20); bai is equal to the tangent of (AE-FAB), and bla to the tangent of #(AE-AB), that is, of 4(AD — AB.) But, by reason of the parallels DE, LM, we have EN: DN :: bM: bl; which is evidently the theorem enunciated above.

Cor. The sum of the cosines of two arcs, is to their difference, as the cotangent of half the sum of those two arcs, is to the cotangent of half their difference.

For, the cosines being the sines of the complements, it follows from the proposition that the sum of the cosines, is to their difference, as the tangent of half the sum of the complements, is to the tangent of half their difference. But half the sum of the complements of two arcs is the complement of half the sum of those two arcs, and half the difference of the complements is the same as the complement of half the difference; whence the truth of the corollary. .

PRoP. XII.

12. Of any three equidifferent arcs, it will be, as l

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radius, to the cosine of their common difference, so is the sine of the mean arc, to half the sum of the sines of the extremes; and, as radius, to the sine of the common difference, so is the cosine of the mean arc to half the difference of the sines of the two extremes. Let AD", AB, AD, (in the figure to prop. 8, of this chapter), be the three equidifferent arcs. Then DF = DF", is the sine of their common difference, and CF its cosine. Also FG, being an arithmetical mean between the sines DK, D'k', of the two extreme arcs, is equal to half their sum, and DE equal to half their difference. By reason of the similar triangles cBH, cFe, DFE, we have, cB ; cf:: BH; FG, and CB : D F :: ch: DE; which are the analogies in the proposition. Cor. 1. From the preceding proportions, we have

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Cor. 2. Hence, if the mean arc AB be one of 60°, its cosine ch, will (prop. 5) be equal to #cb, and DK — d'k’ = DF; consequently DK will in that case equal DF + D'K'. From this conjointly with the preceding corollary result these two theorems: .

(A). If the sine of the mean of three equidifferent arcs (radius being unity) be multiplied into twice the cosine of the common difference, and the sine of either extreme be deducted from the product, the remainder will be the sine of the other extreme.

(B). The sine of any arc above 60°, is equal to the sine of another arc as much below 60°, together with the sine of its excess above 60°.

Remark. From this latter corollary, the sines below 60° being known, those of arcs above 60° are determinable by addition only.

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13. In any right angled triangle, the hypothenuse is to one of the legs, as the radius to the sine of the angle opposite to that leg; and one of the legs is to the other, as the radius to the tangent of the angle opposite to the latter.

Let ABC be a triangle, right-angled at AC B, and let AR on the leg AB, be the radius of the tables. With centre A and T radius AR, describe an arc to cut the hy- D pothenuse in D, and draw DH, TR, perpendicular to AB. Then DH is the sine, TR the tangent, and At the secant, of the ATH RTB arc DR, or angle A: and the similar triangles ABC, ART, AHD, give

Ac: CB :: AD : DH:: rad: sin A; and AB : Bc :: AR ; RT :: rad: tan A.

Cor. To the hypothen use as a radius, each leg is the sine of its opposite angle; and to one of the legs as a radius, the other leg is the tangent of its opposite angle, and the hypothemuse is the secant of the same angle.

Prop. XIV.

14. In any plane triangle, as one of the sides, is to another, so is the sine of the angle opposite to the former, to the sine of the angle opposite to the latter.

Let ABC be a plane triangle, 'C " ' in which it is required to determine the relation which subsists between the sides Ac and Bc. 7 *~

With the angular point A as a of A & HT TH centre, and the distance Ac, equal to the radius of the tables, describe the semicircle Gech. From c draw parallel to CB, and cd p. to AB : draw also eA parallel to cB, and from the point of intersection e (of that line with the circle) demit the perpendicular ef on BA produced. Then cd is the sine of the angle cAB, to the radius Ac, and ef is the sine of the angle eag = angle CBA, to the same radius. * . But Ac: Bc :: Ac: be, because of the parallels Bc and be, :: Ae: b.c., because Ae = Ac, , :: £f: ca, because of the similar triangles Aff, bed. That is, Ac: Bc :: sin B : sin A. In a similar manner it may be shewn that Ac: AB :: sin B : sinc, and AB : Bc :: sin c : sin.A. And by drawing a figure for each case, it will be seen that the circumstance of any one of the angles being obtuse will make no difference in the demonstration.

Otherwise, From c the vertical angle of the . C. triangle let fall the perpendicular cD upon AB, or As produced, according as the angles A and s are both acute, B or one obtuse. A D

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15. In any plane triangle it will be, as the sum of the sides about the vertical angle, is to their difference, so is the tangent of half the sum of the angles at the base, to the tangent of half their difference. - By the preceding prop. Ac: Bc: sin B : sin A, . ... by comp. and div. Ac.4 bc. Ac – Bc is sin B + sin a : Sin B - $111 A.

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