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and make the angle NED equal to the angle GFL; and from the centre A at the distance AN describe a circle, and let its circumference meet ED in D, and draw DB perpendicular to AN, and DM, making the angle BDM equal to the angle GLH. Lastly, produce BM to C, so that MC be equal to HK; then is AB the first, BC the second, and BD the third of the straight lines that were to be found.

For the triangles EBD, FGL, as also DBM, LGH being equiangular, as EB to BD, so is FG to GL; and as DB to BM, so is LG to GH; therefore, ex æquali, as EB to BM, so is (FG to GH, and so is) AE to HK or MC; wherefore (12. 5.) AB is to BC, as AE to HK, that is, as FG to GH, that is, in the given ratio; and from the straight line BC taking MC, which is equal to the given straight line HK, the remainder BM has to BD the given ratio of HG to GL; and the sum of the squares of AB, BD is equal (47. 1.) to the square of AD or AN, which is the given space. Q. E. D.

I believe it would be in vain to try to deduce the preceding construction from an algebraical solution of the problem.

FINIS.

THE

ELEMENTS

OF

PLANE AND SPHERICAL

TRIGONOMETRY.

PLANE TRIGONOMETRY.

LEMMA I. FIG. 1.

LET ABC be a rectilineal angle; if about the point B as a centre, and with any distance BA, a circle be described, meeting BA, BC, the straight lines including the angle ABC in A, C; the angle ABC will be to four right angles, as the arch AC to the whole circumference.

Produce AB till it meet the circle again in F, and through B draw DE perpendicular to AB, meeting the circle in D, E.

By 33. 6. Elem. the angle ABC is to a right angle ABD, as the arch AC to the arch AD; and quadrupling the consequents, the angle ABC will be to four right angles, as the arch AC to four times the arch AD, or to the whole circumference.

LEMMA II. FIG. 2.

LET ABC be a plane rectilineal angle as before: about B as a centre, with any two distances BD, BA, let two circles be described meeting BA, BC in D, E, A, C; the arch AC will be to the whole circumference of which it is an arch, as the arch DE is to the whole circumference of which it is an arch.

By Lemma 1, the arch AC is to the whole circumference of which it is an arch, as the angle ABC is to four right angles; and by the same Lemma 1, the arch DE is to the whole circumference of which it is an arch, as the angle ABC is to four right angles; therefore the arch AC is to the whole circumference of which it is an arch, as the arch DE to the whole circumference of which it is an arch.

DEFINITIONS. FIG. 3.

I.

LET ABC be a plane rectilineal angle; if about B as a centre, with BA any distance, a circle ACF be described, meeting BA, BC in A, C; the arch AC is called the measure of the angle ABC.

II.

The circumference of a circle is supposed to be divided into 360 equal parts called degrees; and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, &c. And as many degrees, minutes, seconds, &c, as are contained in any arch, of so many degrees, minutes, seconds, &c. is the angle, of which that arch is the measure, said to be. COR. Whatever be the radius of the circle of which the measure of a given angle is an arch, that arch will contain the same number of degrees, minutes, seconds, &c. as is manifest from Lemma 2.

III.

Let AB be produced till it meet the circle again in F; the angle CBF, which, together with ABC, is equal to two right angles, is called the Supplement of the angle ABC.

IV.

A straight line CD drawn through C, one of the extremities of the arch AC perpendicular upon the diameter passing through the other extremity A, is called the Sine of the arch AC, or of the angle ABC, of which it is the measure.

COR. The Sine of a quadrant, or of a right angle, is equal to the radius.

V.

The segment DA of the diameter passing through A, one extremity of the arch AC, between the sine CD and that extremity, is called the Versed Sine of the arch AC, or angle ABC.

VI.

A straight line AE, touching the circle at A, one extremity of the arch AC, and meeting the diameter BC passing through the other extremity C in E, is called the Tangent of the arch AC; or of the angle ABC.

VII.

The straight line BE, between the centre and the extremity of the tangent AE, is called the Secant of the arch AC, or angle ABC.

COR. to def. 4, 6, 7. The sine, tangent, and secant of any angle ABC, are likewise the sine, tangent, and secant of its supplement CBF.

It is manifest from def. 4, that CD is the sine of the angle CBF. Let CB be produced till it meet the circle again in G; and it is manifest that AE is the tangent, and BE the secant, of the angle ABG or EBF, from def. 6, 7.

COR. to def. 4, 5, 6, 7. The sine, versed sine, tangent, and secant, of any arch which is the measure of any given angle ABC, is to the sine, versed sine, tangent and secant, of any other arch which is the measure of the same angle, as the radius of the first is to the radius of the second.

Let AC, MN be measures of the angle ABC, according to def. 1, CD the sine, DA the versed line, AE the tangent, and BE the secant of the arch AC, according to def. 4, 5, 6, 7, and NO the sine, OM the versed line, MP the tangent, and BP the secant of the arch MN, according to the same definitions. Since CD, NO, AE, MP are parallel, CD is to NO as the radius CB to the radius NB, and AE to MP as AB to BM, and BC or BA to BD as BN or BM to BO; and, by conversion, DA to MO as AB to MB. Hence the corollary is manifest; therefore, if the radius be supposed to be divided into any given number of equal parts, the sine, versed sine, tangent, and secant, of any

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