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PLANE TRIGONOMETRY.

SECT. I.

PRINCIPLES OF PLANE TRIGONOMETRY.

LEMMA I. Fig. 1.

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LET ABC be a rectilineal angle; if about the point B as a centre, and at any distance BA, a circle be described, meeting BA and BC, (the straight lines including the angle ABC,) in A and 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 and 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.

LEMMA II. Fig. 2.

Let ABC be a plane rectilineal angle as before: About B as a centre, at any two distances BD and BA, let two circles be described, meeting BA and 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 same Lemma, 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.

• Any part of the circumference of a circle, as AC, is called an arch or arc.

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