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LEMMA I. FIG. 1.

ET ABC be a rectilineal angle, if about the point B as a center, 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.

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ET ABC be

a center 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 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.

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DEFINITIONS. FIG. 3.

I.

ET ABC be a plane rectilineal angle; if about B as a center, 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 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 feconds, &c. And as many degrees, minutes, seconds, &c. as are contained in any arch, of so many degrees, mi

a

Fig. 4.

nutes, seconds, &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, 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 ftraight 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 fine CD, and that extremity is called the Verfed 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 center 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 secant of its supplement 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 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 fine, versed sine, tangent, and secant, of any arch which is the measure of any given angle ABC, is to the fine, versed fine, tangent, and secant, of any

other arch which is the measure of the fame angle, as the radius of the first is to the radius of the second.

Let AC, MN be measures of the angles ABC, according to Def. 1. CD the fine, DA the versed fine, AE the tangent, and BE the secant of the arch! AC, according to Def. 4. 5. 6. 7. and NO the fine, OM the versed fine, MP the tangent, and BP the fecant of the arch MN, according to the fame 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 converfion, 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 fine, versed fine, tangent, and secant of any given angle, will each contain a given number of these parts; and, by trigonometrical tables, the length of the fine, versed sfine, tangent, and fecant of any angle may be found in parts of which the radius contains a given number: and, vice versa, a number expressing the length of the fine, versed sine, tangent, and secant being given, the angle of which it is the fine, versed fine, tangent, and secant may be found.

VIII.

The difference of an angle from a right angle is called, the Complement of that angle. Thus, if BH be drawn perpendicular to AB, the angle CBH will be the complement of the angle ABC, or of CBF.

IX.

Let HK be the tangent, CL or DB, which is equal to it, the fine, and BK the secant of CBH, the complement of ABC, according to Def. 4. 6. 7, HK is called the co-tangent, BD the co-fine, and BK the co-fecant of the angle ABC.

Cor. 1. The radius is a mean proportional between the tangent and co-tangent.

For, fince HK, BA are parallel, the angles HKB, ABC will be equal, and the angles KHB, BAE are right; therefore the triangles BAE, KHB are fimilar, and therefore AE is to AB, as BH or BA to HK.

Cor. 2. The radius is a mean proportional between the co-fine and fecant of any angle ABC.

Since CD, AE are parallel, BD is to BC or BA, as BA to BE.

Fig. 3.

PROP. I. FIG. 5.

Na right angled plain triangle, if the hypothenuse

be made radius, the fides become the fines of the angles opposite to them; and if either fide be made radius, the remaining fide is the tangent of the angle opposite to it, and the hypothenuse the fecant of the fame angle.

Let ABC be a right angled triangle; if the hypothenuse BC be made radius, either of the fides AC will be the fine of the angle ABC opposite to it; and if either side BA be made radius, the other fide AC will be the tangent of the angle ABC oppofite to it, and the hypothenuse BC the fecant of the fame angle.

About B as a center, with BC, BA for distances, let two circles CD, EA be described, meeting BA, BC in D, E: fince CAB is a right angle, BC being radius, AC is the fine of the angle ABC by def. 4. and BA being radius, AC is the tangent, and BC the secant of the angle ABC, by def. 6. 7.

COR. 1. Of the hypothenuse a fide and an angle of a right angled triangle, any two being given, the third is also given. COR. 2. Of the two fides and an angle of a right angled triangle, any two being given, the third is also given.

PROP. II. FIG. 6. 7.

HE fides of

THE

a plane triangle are to one another,

as the fines of the angles opposite to them. In right angled triangles this prop. is manifest from prop. r. for if the hypothenuse be made radius, the fides are the fines of the angles opposite to them, and the radius is the fine of a right angle (cor. to def. 4.) which is opposite to the hypothenuse.

In any oblique angled triangle ABC, any two fides AB, AC will be to one another as the fines of the angles ACB, ABC which are opposite to them.

From C, B draw CE, BD perpendicular upon the oppofite fides AB, AC produced, if need be. Since CEB, CDB are right angles, BC being radius, CE is the fine of the angle CBA, and BD the fine of the angle ACB; but the two triangles CAE, DAB have each a right angle at D and E; and likewife the common angle CAB; therefore they are similar, and confequently,