LEMMA I. Fig. 1.

L

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

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.

LET
ET ABC be a plane rectilineal angle as before : about B as

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 archi 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 circumfer rence of which it is an arch, as the arch DE to the whole cird cumference of which it is an arch.

DEFINITIONS. Fig. 3.

I. LET ABC be a plane rectilineal angle ; if about B as a center;

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.

It. The circumference of a circle is supposed to be divided into 360

equal parts called degrees, and each degree into 69 equat

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, said to be. COR. Whatever be the radius of the circle of which the mea

sure of a given angle is an arch, that arch will contain the
fame number of degrees, minutes, seconds, &c. as is manifat
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 sight 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 meafure.
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 extre-

mity 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 sine, tangent, and secant of any

angle ABC, are likewise the fine, tangent, and secant of its

fupplement CBF. It is manifest from def. 4. that CD is the sine of the angle CBF.

Ler CB be produced till it meet the circle again in G; and it is manifest that AE is the tangent, and BE the fecant, of the

angle ABG or EBF, from def. 6. 7. Tig. 4. Cor. to def. 4. 5. 6. 7. The fine, versed fine, tangent, and

fecant, of any arch which is the measure of any given angle
ABC, is to the fine, verfed fine, tangent, and fecant, of any
other arch which is the measure of the same angle, as the re-

dius 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 fecant of the arch AC, according to def. 4. 5. 6. 7.
and NO the sine, OM the versed fine, MP the tangent, and
BP the fecant of the arch MN, according to the fame defia
nitions. Since CD, NO, AE, MP are parallel, CD is to NO
as the fadius CB to the rådiùs 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 AL to MB. Hence the corollary
is manifeft; therefore, if the radius be supposed to be divided
into any given number of equal parts, the sine, versed sine,
tangent, and fecant of any given angle, will each contain a
given number of these parts; and, by trigonometrical tables,
the length of the fine, versed sine, tangent, and fecant of any
angle may be found in parts of which the radius contains a
given number : and, vice verfa, a number expressing the length
of the sine, versed sine, tangent, and secant being given, the
angle of which it is the fine, versed fine, tangent, and secant

may be found.

Fig. 3.

VIII.
The difference of an angle from a right angle is called the Com-

plement 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 CEH, the complement of ABC, ac
cording 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, since HK, BA are parallel, the angles HKB, ABC will be

equal, and the angles KHB, BAE are right; therefore the tri-
angles BAE, KHB are similar, and therefore AE is to AB, as

BH or BA to HK..
COR. 2. The radius is a mean proportional between the co-line

and secant of any angle ABC.
Since CD, AL are parallel, BD is to BC or BA, as BA to LF.

PROP. I. FIG. 5.

IN

a right angled plain triangle, if the hypothenuse

be made radius, the sides become the fines of the angles opposite to them; and if either side 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 sides AC will be the fine of the angle ABC opposite to it, and if either side BA be made radias

, the other side AC will be the tangent of the angle ABC oppaík to it, and the hypothenuse BC the secant of the same angle.

About B as a center, with BC, BA for diftances, let nog circles CD, EA be described, meeting BA, BC in D, E: fitue 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 tangers and BC the fecant of the angle ABC, by def. 6. 7.

Cor. 1. Of the hypothenuse a fide and an angle of a rigt: angled triangle, any two being given, the third is also given.

Cor. 2. Of the two sides and an angle of a right angled tiangle, any two being given, the third is also given.

THE

HE sides of a plane triangle are to one another,

as the lines of the angles opposite to them. In right angled triangles this prop. is manifeft from prop ! for if the hypothenuse be made radius, the sides are the fines the angles opposite to them, and the radius is the fine of a riga 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, A50 which are opposite to them.

From C, B draw CE, BD perpendicular upon the oppost: fides AB, AC produced, if need be. Since CEB, CDB are rice: 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 likewise ebe common angle CAB; therefore they are similar, and consequenc's,

CA is to AB, as CE to DB; that is, the sides are as the fines of the angles opposite to them.

Cor. Hence of two sides, and two angles opposite to them, in a plain triangle, any three being given, the fourth is also given.

PROP. III. Fig. 8.

IN
N a plain triangle, the sum of any two sides is to

their difference, as the tangent of half the sum of the angles at the base, to the tangent of half their difference.

Let ABC be a plain triangle, the sum of any two fides AB, AC will be to their difference as the tangent of half the sum of the angles at the base ABC, ACB to the tangent of half their difference.

About A as a center, with AB the greater side for a distance, let a circle be described, meeting AC produced in E, F, and BC in D; join DA, EB, FB; and draw FG parallel to BC, meeting EB in G.

The angle EAB (3 2. 1.) is equal to the sum of the angles at the base, and the angle EFB at the circumference is equal to the half of EAB at the center (20. 3.); therefore EFB is half the sum of the angles at the base; but the angle ACB (3 2. 1.) is equal to the angles CAD and ADC, or ABC together ; therefore FAD is the difference of the angles at the base, and FBD at the circumference, or BFG, on account of the parallels FG, BD, is the half of that difference; but since the angle EBF in a femicircle is a right angle (1. of this) FB being radius, BE, BG, are the tangents of the angles EFB, BFG; but it is manifest that EC is the sum of the sides BA, AC, and CF their difference ; and since BC, FG are parallel (2. 6.) EC is to CF, as EB to BG; that is, the sum of the sides is to their difference, as the tangent of half the sum of the angles at the base to the tangent of half their difference.

PROP. IV. FIG. 18.

IN any plain triangle BAC, whose two sides are BA,,

AC and base BC, the less of the two sides, which let be BA, is to the greater AC as the radius is to the tangent of an angle; and the radius is to the