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Printed for F. WINGRAVE, Successor to

Mr. NOURSE, in the Strand;
By LUKE HANSARD, No 6, Great Turnstile, Lincoln's Inn Fields.

1799*

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Plane Trigonometry.

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DEFINITION S.

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LANE Trigonometry is the art whereby,

having given any tKfee parts of a plane

triangle, (except the three angles) the rest are determined. In order to which, it is not only requisite, that the peripheries of circles, but also certain right-lines in, and about, the circle, be supposed divided into fome assigned number of equal parts.

2. The periphery of every 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, or second minutes, &c.

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

4. The difference of any arch' from 90° (or a quadrant) is called its complement; and its difference from 180° for a femicircle) its supplement.

5. A chord, or subtense, is a right-line drawn from one extremity of an arch to the other: thus the right line BE is the chord, or subrense, 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 sine 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 co-line of an arch is the part of the diameter intercepted between the center and sine; and is equal to the fine of the complement of that arch. Thus CF is the co-line of the arch AB, and is equal to BI, the fine of its complement HB.

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 pahing through the center and the other extremity. Thus AG is the tangent of the arch AB.

10. 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.

11. 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 co. tangent and co-secant of AB,

12. A trigonometrical canon is a table exhibiting the length of the fine, tangent, and fecant, to every degree and minute of the quadrant, with respect to the radius; which is supposed unity, and conceived to be divided into 10000, or more, decimal parts. By the help of this table, and the doctrine of fimilar triangles, the whole business of trigonometry is performed; which I shall now proceed to fhew. But, first of all, it will be proper to observe, that the fine of any arch Ab greater than 90°. is equal to the line of another arch AB as much below 90'; and that, its co-line Cf; tangent Ag, and secant Cg, are also respectively equal to the co-fine, tangent, and fecant 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 manifeft from the definitions.

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