THE

ELEMENT S

O F

PLANE AND SPHERICAL

TRIGONOMETRY.

EDINBURGH:

Printed for J. NoURSE, London; and J. BALFOUR, Edinburgh. M,DCC.LXXV.

L

LEMMA I. FIG. I.

ET ABC be a rectilineal angle, if about the point B as a centre, and with any diftance BA, a circle be defcribed, meeting BA, BC, the ftraight 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 confequents, 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.

LE ET ABC be a plane rectilineal angle as before: About B as a centre with any two distances BD, BA, let two circles be defcribed 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 fame 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 defcribed meeting BA, BC, in A, C; the arch AC is called the measure of the angle ABC.

II.

The circumference of a circle is fuppofed to be divided into

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360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called feconds, &c. And as many degrees, minutes, feconds, &c. as are contained in any arch, of so many degrees, minutes, feconds, &c. is the angle, of which that arch is the measure, faid to be.

COR. Whatever be the radius of the circle of which the meafure of a given angle is an arch, that arch will contain the fame number of degrees, minutes, feconds, &c. as is manifeft 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 ftraight line CD drawn through C, one of the extremities of the arch AC, perpendicular upon the diameter paffing 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 fegment DA of the diameter paffing through A, one extremity of the arch AC between the fine CD, and that extremity is called the Verfed Sine of the arch AC, or angle ABC.

VI.

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

VII.

The ftraight 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 fine, tangent, and fecant of any angle ABC, are likewife the fine, tangent, and fecant of its fupplement CBF.

It is manifeft from Def. 4. that CD is the fine of the angle CBF. Let CB be produced till it meet the circle again in G; and it is manifeft that AE is the tangent, and BE the fecant, of the angle ABG or EEF, from def. 6. 7.

COR.