Note, The degrees, minutes, feconds, &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 fe

conds.

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.

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

10. The fecant of an arch is a right-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-fècant of AB,

1

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 refpect 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, first 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 also 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.

THEOREM I.

In any right-angled plane triangle ABC, it will be as the bypotbenufe is to the perpendicular, fo is the radius (of the table) to the fine of the angle at the

base.

A

For, let AE or AF be the radius to which the table of fines, &c. is adapted, and ED the fine of the angle A or arch EF (Vid. Def. 3. and 6,); then,

B D F because of the fimilar triangles ACB and AED, it will be AC: BC:: AE: ED (by 14. 4.) 2. E. D.

Thus, if AC,75, and BC,45; then it will be, 75,45 :: 1 (radius): the fine of A = 6; which, in the table, answers to 36° 52', the measure, or value of A.

THEOREM II.

In any right-angled plane triangle ABC, it will be, as the bafe AB is to the perpendicular BC, fo is the radius of the table) to the tangent of the angle at the bafe.

For, let AE or AF be the radius of the table, or canon (fee the preceding figure), and FG the tangent of the angle A, or arch EF (Vid. Def. 3. and 9.); then, by reafon of the fimilarity of the triangles ABC, AFG, it will be, AB: BC :: AF: FG. 2. E. D.

Note, In the quotations where you meet with two numbers, (as 14. 4) without any mention of Prop Theor. Sc. reference is made to the Elements of Geometry published by the fame author; to swhich this little tract is defigned as an Appendix.

Thus

Thus let AB,8, and BC=,5; then we shall have,8,5 I (radius): tangent A=,625; whence A itself is found, by the canon, to be 32° 00'.

THEOREM III.

In every plane triangle ABC, it will be, as any one fide is to the fine of its oppofite angle, jo is any other fide to the fine of its oppofite angle.

Now the triangles

CBD, CFE being fimilar, we have CB: BD (fin. A) :: CF (AB): FE (sin. C). Q. E. D.

THEOREM IV.

As the bafe of any plane triangle ABC, is to the fum of the two fides, fo is the difference of the fidesş to twice the distance DE. of the perpendicular from the middle of the bafe.

THEOREM V.

In any plane triangle, it will be, as the fum of any two fides is to their difference, fo is the tangent of half the fum of the two oppofite angles, to the tangent of half their difference.

For, let ABC be the triangle, and AB and AC the two propofed fides; and from the center A, with the radius AB, let a circle be described, in• terfecting CA produced, in D and F; fo that CF may ex

prefs the fum, and CD the difference, of the fides AC and AB: join F, B and B, D, and draw DE parallel to FB, meeting BC in E.

Then, because 2ADB ADB + ABD (by 12. 1.) = C + ABC (by 9. 1.) it is plain that ADB is equal to half the fum of the angles oppofite to the fides propofed. Moreover, fince ABC = ABD (ADB) + DBC, and C =‚ADBDBC (by 9. 1.) it is plain that ABC-C is = 2. 2DBC; or that DBC is equal to half the difference of the fame angles.

Now, because of the parallel lines BF and ED, it will be CF: CD:: BF: DE; but BF and DE, because DBF and BDE are right-angles (by 13.3. and 7. 1.) will be tangents of the forefaid angles FDB (ADB) and DBE (DBC) to the radius BD. 2, E. D.

COROL