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P L A NE

AND

SPHERICAL;

WITH THE

CONSTRUCTION and APPLICATION

OF

LOGARITHM S.

By THOMAS SIMPSON, F.R.S.

The FIFTH EDITION.

LONDON:

Printed for F. WINGRAVE, Succeffor to
Mr. NOURSE, in the Strand;

By LUKE HANSARD, No 6, Great Turnftile, Lincoln's Inn Fields.

1799.

RB 23. b. 3891

BRITISH

IBRAR

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LANE Trigonometry is the art whereby, having given any three parts of a plane triangle, (except the three angles) the reft are determined. In order to which, it is not only requifite that the peripheries of circles, but allo certain right-lines in, and about, the circle, be fuppofed divided into fome affigned number of equal parts.

2. The periphery of every circle is fuppofed 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 feconds, or fecond minutes, &c.

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Note, The degrees, minutes, feconds, &c. contained in any arck, 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 feconds.

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

ment.

5. A chord, or fubtenfe, is a right-line drawn from one extremity of an arch to the other: thus the right line BE is the chord, or fubtenfe, 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 paffing through the other extremity. Thus BF is the fine of the arch AB or DB.

7. The verfed fine of an arch is the part of the diameter intercepted between the fine and the periphery. Thus AF is the verfed fine of AB; and DF of DB.

8. The

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-fine of the arch AB, and is equal to BI, the fine of its complement HB.

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

10. The fecant of an arch is a night-line reaching, without the circle, from the center to the extremity of the tangent. Thus CG is the fecant of AB.

11. The co-tangent, and co-fecant, of an arch are the tangent, and fecant, of the complement of that arch. Thus HK and CK are the cotangent and co-fecant of AB,

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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 fuppofed 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 bufinefs of trigonometry is performed; which I fhall now proceed to fhew. But, firft of all, it will be proper to obferve, 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-fine Cf, tangent Ag, and fecant Cg, are alfo refpectively equal to the co-fine, tangent, and fecant of its fupplement AB; but only are negative, or fall on contrary fides of the points C and A, from whence they have their origin. All which is manifeft from the definitions.

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