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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 * to the square of AD or AN, which is the given • 47. 1. 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.

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THE

ELEMENTS

OF

PLANE AND SPHERICAL

TRIGONOMETRY.

PLANE TRIGONOMETRY.

LEMMA I.

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 circum

D ference.

Produce AB till it meet the circle again in F
F, and through B, draw DE perpendicular to Fig. 1.
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.

B В

LEMMA II.

E

A

D

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.

B By Lemma 1. the arch AC is to the whole Fig.2, 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.

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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. COROLLARY. 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 See fig. 3.

angle CBF, which together with ABC is equal to two right angles, is called the Supplement of the angle ABC.

ly. A straight line CD drawn through C, one of the extremi- See fig. 3.

ties 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.

See fig. 3.

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.

See fig. 3.

See fig. 3.

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 end

tremity 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

Fig.4.
which is the measure of the same angle,
as the radius of the first is to the radius
of the second.

BA
Let AC, MN be measures of the angle

OMD
ABC, according to Def. 1.; CD the sine,
DA the versed sine, 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 sine, 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

E

A А

number of equal parts, the sine, versed sine, tangent, and secant of any given angle will each contain a given number of these parts; and, by trigonometrical tables, the length of the sine, versed sine, tangent, and secant of any angle may be found in parts of which the radius contains a given number; and, vice versâ, a number expressing the length of the sine, versed sine, tangent, and secant, being given, the angle of which it is the sine, versed sine, tan

gent, and secant, may be found. VIII. The difference of an angle from a right angle, is called See fig. 3.

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, See fig. 3.

the sine ; and BK the secant of CBH, the complement of
ABC, according to Def. 4. 6. 7. HK is called the cotangent,

BD the cosine, and BK the cosecant, of the angle ABC.
Cor. 1. The radius is a mean proportional between the tan-

gent and cotangent.
For, since HK, BA are parallel, the angles HKB, ABC will be

equal, and the angles KHB, BAE are right; therefore the triangles 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 cosine

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

PROPOSITION I.

If a right-angled plane triangle, if the hypothenu se be made

radius, the sides become the sines of the angles opposite to them: and if either side be made radius, the remaining side is the tangent of the angle opposite to it, and the hypothenuse the secant of the same angle.

Let ABC be a right-angled triangle: if the hypothemuse BC be made radius, either of

Fig.5. the sides AC will be the sine of the angle ABC opposite to it; and if either side BA be made radius, the other side AC will be the

A D tangent of the angle ABC opposite to it, and the hypothenuse BC the secant of the same angle.

E

B

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