a LEM. 1. as

N unequal circles, the arches, upon which equal

angles at the centre stand, have the fame ratio to the circumferences of the circles.

Let ABC, GBH be equal angles at the centre of the
circles ACD, GHK, standing upon the arches AC, GH: and
because ABC is an angle at the
centre, it is to four right angles,
a the arch AC to the circumfe-

H
rence ACD. For the same reason,
the angle GBH or ABC is to four
right angles, as the arch GH to

D K

B

G the circumference GHK: There

A fore the arch AC is to the circumb11. s. ference ACD, as the arch GH to

the circumference GHK

Note. The arches AC, GH are
said to be similar, when they have the same ratio to the circum-
ferences ACD, GHK.

I

F two magnitudes consist of parts each equal to the

same magnitude; they have to one another the fame ratio which the numbers denoting the parts of which they confift, have to one another.

Let A, B consist of parts each equal to C: and let I be unity,
and M the number of parts equal to C, in A; and
N the number of parts equal to c, in B! A is to B, ACB
as M to N.

MIN
Becaufe there are as many parts equal to C, in A,

as there are units I in the number M; A,
a C. s. equimultiples of C, I; therefore A is to c, as a M to I: and
b Cor.4.5. B, N are equimultiples of C, I; therefore A is to B, as M

to N.

M

are

LEMMA IV.
ALF the sum of two unequal magnitudes, to-

gether with half their difference, is equal to
the greater; and half the difference taken from half
the sum is equal to the lefs.

Let

HAL

Let AB be the greater of two unequal magnitudes, and BCPI. Trig. the less ; therefore AC is their sum: make AD equal to BC; therefore BD is their difference: and if BD be bifected in E, BE or ED is half their difference: and because AD is equal to BC, and DE to EB; the whole AE is equal to EC; therefore AE A D E B с or EC is the half of the sum AC. But the greater magnitude AB is equal to AE, EB together; that is, to half the fum added to half the difference; and the less BC is equal to the excess of EC half the fum, above EB half the difference.

DEFINITIONS.

AN

I.
N arch of a circle is called the measure of the angle at
the centre standing on that arch. Thus, the arch AC is the
measure of the angle ABC.

II.
The circumference of every circle is supposed to consist of 360

equal parts, each of which is called a degree: and a degree is
supposed to consist of 60 equal parts, called minutes; and a
minute of 60 equal parts, called seconds; and so on. Also, an
angle at the centre is said to be of as many degrees, minutes,
&c. as there are in the arch, which is the measure of that

angle. Cor. 1. A right angle is an angle of 90 degrees. For the

measure of it is the fourth part of the circumference. Cor. 2. Arches of different circles, which measure the same

angle, contain the same number of degrees, and parts of a degree. For the number of degrees and parts of a degree in each of them is to 360, as each of the arches to its circumference &; that is, as the angle which they measure to four a LEM. 3; right angles b.

b LEM, 1. III. The fine of an arch is a straight line drawn from one extremity

of the arch perpendicular to the diameter passing through the other extremity of it. Thus, CD drawn from C perpendicu.

lar to AF, is the fine of the arch AC. Cor. 1. The fine of AH, the fourth part of the circumference, is

BH equal to the radius. Cor. 2. The fine of an arch is the half of the chord of twice that arch. For CM is bifected in DC.

C 3. 3 IV. The versed fine of an arch is the segment of the diameter, to

which the fine is perpendicular, between the fine and the arch. Thus, AD is the versed fine of the arch AC.

V.

Pl. Trig.

V.
The tangent of an arch is a straight line touching the circle at

one extremity, and meeting the diameter that passes through
the other extremity. Thus, AE is the tangent of the arch

AC.
Cor. The tangent of the eighth

H
part of the circumference is
equal to the radius.

L
VI.

2
The secant of an arch is the

Atraight line drawn from the F
centre to the farthest extre-

A

B D
mity of the tangent of that
arch. Thus, BE is the se.

G
cant of the arch AC.

VII.
The fine, verfed fine, tangent,

and secant of an arch, are also called the fine, versed fine,
tangent, and secant of the angle, of which that arch is the
measure. Thus, CD is the fine of the angle ABC; AD is

its versed sine; AE is its tangent; and BE its secant.
Cor. The radius is equal to the fine, or versed fine of a right
angle; or to the tangent of half a right angle.

VIII.
The difference between an arch and the half of the circumference,

or between an angle and two right angles, is called the sup-
plement of that arch, or of that angle. Thus, the arch CF
is the supplement of the arch AC; and the angle CBF is the

supplement of the angle ABC. Cor. The fine, tangent, and fecant of any angle ABC, are also

the fine, tangent, and secant of its fupplement: For, by Def.
4th, CD is the fine of the arch CF, that is, of the angle
CBF, by Def. gth: and if CB be produced to G, it is mani.
feft, from Def. 5th and 6th, that AE is the tangent, and
BE the fecant of the arch AG, that is, of the angle ABG or
CBF, by Def. 7th.

IX.
The difference between an arch and the fourth

part

of the cir. cumference, or between an angle and a right angle, is called the complement of that arch, or of that angle. Thus, if BH be at right angles to AB, the arch CH is the complement of the arch CA, and the angle CBH is the complement of the angle ABC, or of CBF.

The fine, tangent, or fecant of the complement of any angle, is called the cofine, cotangent, or cosecant of that angle. Thus,

if CL, HK be perpendiculars to BH; CL is the cofine of the Pl. Trig.

angle ABC, and HK its cotangent, and BK its cosecant. Cor. 1. The segment of the diameter to which the fine is per

pendicular, between the fine and the centre, is equal to the
cofise. Thus, BD is equal to CL“, the cofine of the angle d 34. 1.

ABC.
Cor, 2. The cofine of any arch is to the fine, as the radius to

the tangent of the same arch. For the triangles BDC, BAE
are equiangular, because the angle ABE is common, and e 32. 1.
BDC, BAĚ are right angles; therefore BD is to DC, as BA
to AE f.

f 4. 6. Cor.

3. The radius is a mean proportional between the cosine and the secant of any angle. For BD is to BC or BA, as

BA to BE.
COR. The radius is a mean proportional between the tangent

and cotangent of any angle. For, the alternate angles ABE,
BKH are equal & ; and BAE, BHK are right angles; therefore g 29. 1.
the triangles BAE, BHK are equiangular; wherefore EA is

to AB, as BH or BA to HK. Cor. 5. The tangent and cotangent of any angle are reciprocally

proportional to the tangent and cotangent of any other angle. For the tangent of the first angle is to the radius, as the radius to its cotangent; and the radius is to the tangent of the other angle, as the cotangent of that other angle to the radius: Therefore, by perturbate equality, the tangent of h 12. S. the first angle is to the tangent of the other, as the cotangent of the other to the cotangent of the first.

N a right angled triangle, if the hypothenuse, or

fide opposite to the right angle, be made the radius of a circle ; the other fides are the fines of the angles opposite to them, or the colines of the angles adjacent to them. And if either of the sides about the right angle be made the radius, the other side is the tangent of the angle oppolite to it, and the hypothenuse is the secant of the same angle.

Let ABC be a triangle, having the right angle BAC: and from the centre B at the distance BC, describe a circle meeting BA produced in D: and because AC is drawn from C the extremity of the arch CD, perpendicular to the radius BD, which passes through the other extremity D; CA is the fine of

the

2

3.

Pl. Trig. the arch CD ", or of the angle CBD opposite to CA; and WBA the segment of the diameter, to which CA is perpendicular,

Def. between the fine CA and the centre 17. Det. B, is the cofine of the angle ABC CI. Cor.

F
In the same manner, if from the centre
jo. Def.

C, at the distance CB, the arch BE be
described, it may be proved, that BA,
is the fine, and AC the cofine of the

В.
angle ACB.

A D Again, from the centre B, at the distance BA, describe the arch AF.

And because BAC is a right angle, Cor.16.3. AC touches the circle at Afand it

meets the diameter pafling through F, in C; therefore AC is d 5. Def. the tangent of the arch AF, or of the angle ABC: and BC,

which is drawn from the centre to the extremity C of the tan6. Def. gent, is the secant of the same angle ABC.

E

3. Def.

PRO P. II. (F two unequal arches measure the same angle, the

fine, versed fine, tangent, and secant of one of them, have to the radius of that arch, the fame ratios, which the fine, versed fine, tangent, and secant of the other arch, have to the radius of it.

Let AB, DE be two unequal arches, which measure the angle a ri.or ACB: and draw · BF, AG, EH, DK at right angles to AC:

Then, BF is the fine , AF the versed fine ", AG the tangent b

and CG the fecant of the arch AB : and EH is the fine 6, HD C4. Def.

the versed fine , DK the tangenta, and CK the secant of the S.

Def.
e 6. Def. arch DE.

And because CFB,
CHE are right angles, and FCB

B
common to the triangles CFB,
f 32. 1. EHE ;- they are equiangular f;

and therefore FB is to BC, as 8
HE to EC; that is, the fine of
the arch AB is to the radius BC,
as the fine of the arch ED to the
radius CE.

Also CB or CA is
to CF, as CE or CD to CH ;

C

HD h E. 5. and, by conversion \, CA is to

AF, as CD to DH; that is, the radius CA is to the versed fine AF, as the radius CD to the versed fine DH. In the same manner, it inay be proved, that the tangent AG is to the radius

AC,

84. 6.

E