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1.TYLANE Trigonometry is the art whereby,

|T having given any thfee parts of a plane

T triangle, (except the three angles) the rest are determined. In order to which, it is not only ..". the peripheries of circles, but also certain right-lines in, and about, the circle, be supposed divided into some assigned number of equal parts. . . . . . . . . . . . . . . . , a. The periphery of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 6o equal parts, called minutes; and each minute into 60 equal parts, called

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3. Any part AB of the periphery of the circle is called an arch, and is said

TYH. f C. to be the measure of the angle ACB at . the center, which it subtends. - - - -• - - - - * :

Note, The degrees, minutes, seconds, &c. contained in any arch, or angle, are wrote in this manner, 50° 18' 35", which signifies that the given arch, or angle, contains 50 degrees, 18 minutes, and 35 seconds.

. The difference of any arch from 90 (or a

uadrant) is called its complement; and its dif

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from one extremity of an arch to the other: thus the right line BE is the chord, or subtense, of the arch BAE or BDE. - - - 6. The fine, or right-fine, of an arch, is a right line drawn from one extremity of the arch, perpendicular to the diameter passing through the other extremity. Thus BF is the fine of the arch AB or DB. . . *. 7. The versed fine of an arch is the part of the diameter intercepted between the fine and the periphery. Thus AF is the versed fine of AB; and DF of DB. - ... " 8. The

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8. The co-fine of an arch is the part of the diameter intercepted between the center and fine; and is equal to the fine of the complement of that arch. Thus CF is the co-sine of the arch AB, and is equal to BI, the fine of its complement HB. - t 9. The tangent of an arch is a right line touching the circle in one extremity of that arch, produced from thence till it meets a right-line pańing through the center and the other extremity. Thus AG is the tangent of the arch AB. Io. The secant of an arch is a right-line reaching, without the circle, from the center to the extremity of the tangent. Thus CG is the secant of AB. 1 1. The co-tangent, and co-secant, of an arch are the tangent, and secant, of the complement of that arch. Thus HK and CK are the cotangent and co-secant of AB, ** 12. A trigonometrical canon is a table exhibiting the length of the fine, tangent, and secant, to every degree and minute of the quadrant, with respect to the radius; which is supposed unity, and conceived to be divided into 1oooo, or more, decimal parts. By the help of this table, and the doćtrine of similar triangles, the whole business of trigonometry is performed; which I shall now proceed to shew. But, first of all, it will be proper to observe, that the fine of any arch Ab greater than 90°. is equal to the fine of another arch AB as much below 90’; and that, its co-sine Cf. tangent Ag, and secant Cy, are also respectively equal to the co-fine, tangent, and secant of its supplement AB; but only are negative, or fall on contrary sides of the points C and A, from whence they have their origin. All which is manifest from the definitions.

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