Cor. 1 to these definitions.-The sine, tangent or secant, of any arch or angle, is the sine, tangent or secant of its supplement, or complement to a semicircle or two right angles.

For it is manifest from these definitions, that the same right lines are the sine, tangent and secant of the arches BD and AGD, and of the angles BCD and ACD.

Cor. 2-The sine of a quadrant or right angle, is equal to the radius.

Cor. 3. The tangent of half a quadrant or half a right angle is equal to the radius; for if the angle BCF were half a right angle, the angle CBF being right (18. 3 Eu.), the angle CFB would be half a right angle (32. 1 Eu.), whence, the angles BCF, BFC being equal, the tangent BF would be equal to CB (6. 1 Eu.), or to radius.

Cor. 4-The radius is a mean proportional between the tangent (F) and cotangent (GH) of any arch (BD).

The triangles CBF, HGC, having the angles at B and G right, and the angles at C and H equal, being alternate angles formed by CH meeting the parallels CB, GH (29. 1 Eu.), are equiangular (32. 1 Eu.), therefore BF is to CB, as CG or CB to GH (4. 6 Eu).

Cor. 5.-The tangents of any two arches of the same circle, and, of course, of any two angles, are reciprocally as their cotangents.

For the rectangle under the tangent and cotangent of the arch BD, is equal to the rectangle under the tangent and cotangent of any other arch of the circle, each of these rectangles being equal to the square of radius [preced. cor. and 17. 6 Eu.], and therefore the tangent of D is to the tangent of that other arch, as the cotangent of the same arch is to the cotangent of BD (16. 6 Eu.)

Cor. 6.-The radius is a mean proportional, between the cosine (CE) and secant (CF), or between the sine (DE) and cosecant (CH), of any arch (BD).

Since ED and BF are parallel, CE is to CB, as CD or CB is to CF (2. 6 and Schol. 18. 5 Eu).

And since KD is parallel to GH, CK or DE is to CG, as CD or CG is to CH. [2. 6. and Schol. 18. 5. Eu].

Cor. 7.-The sines of any two arches of the same circle, are reciprocally as their cosecants; and the cosines, reciprocally as the secants.

Since the rectangle under the sine and cosecant of BD, is equal to the rectangle under the sine and cosecant of any other

arch of the circle, each of these rectangles being equal to the square of radius [preced. cor. and 17. € Eu.], the sine of BD is to the sine of that other arch, as the cosecant of the same arch is to the cosecant of BD [16. 6 Eu].

In like manner it may be shewn, from cor. 6 above, and 16 & 17.6 Eu. that the cosines of any two arches, are reciprocally as their secants.

Cor. 8. The ratio of the radius, to the sine tangent or secant of any angle (as A), is the same, whatever be the dimension or magnitude of the radius.

CE G

K

DB HF

On either leg, as AB, of the angle A, from the point A, take any unequal parts as AB, AF, and from the centre A, at the distances AB, AF, let arches of cir cles be described meeting the other leg of the angle A, in C and G; A from the points C, G, let fall the perpendiculars CD, GH on AF; and at the points B, F, raise the perpendiculars BE, FK meeting AC produced in E and K; CD and GH are the sines of the arches BC, FG (Def. 6. Pl. Tr.), BE and FK the tangents [Def. 8. Pl. Tr.], and AE and AK the secants of the same arches (Def. 9. Pl. Tr.); which arches are the measures of the angle CAB. The radius has in both cases the same ratio to the corresponding sine, tangent or

secant.

The triangles ADC, AHG are, because of the right angles at D and H, and the common angle at A, equiangular (32. 1 Eu.), therefore AC is to CD, as AG is to GH [4. 6 Eu.]; but : AC, AG are the radiuses, and CD, GH the sines of the arches CB, GF [Def. 6. Pl. Tr.], which are measures of the angle A.

And the triangles ABE, AFK, being right angled at B and F, and having a common angle at A, are equiangular; therefore AB is to BE, as AF to FK, and AB to AE as AF to AK; but AB, AF are radiuses, and BE, FK tangents (Def. 8. Pl. Tr.), and AE, AK secants [Def. 9. Pl. Tr.], of the arches CB, CF, which are measures of the angle A.

Scholium. The reader should beware, when the sines, tangents or secants of angles are mentioned, that he do not consider them, as right lines of a given length; but rather, as terms, denoting the ratios, which the right lines representing them have to radius, their length varying according to the radius to which they are referred, but the ratios between them are definite and

fixed, which is sufficient for the purposes of trigonometry. They may be considered as the quotients, which arise from dividing the right lines which represent them, by the radius, which quotient would express their ratio to the radius, and would be a fixed quantity; but this way of conceiving the thing would rather belong to arithmetick than to geometry, would not be so easily understood, and would lead to nothing, which is not attainable in the usual way, being that which is in this tract pursued.

PROPOSITION I. THEOREM.

In a right angled plain triangle (ABC), the hypothenuse (AC) is to either of the legs, or sides including the right angle (as BC), as radius is to the sine of the angle (BAC) opposite that leg.

From the centre A, at the distance AC, let an arch of a circle CD be described, meeting AB produced in D; CB is the sine of the arch CD, or angle CAD (Def. 6. Pl. Tr.), and the ratio of radius to the sine of a given angle is invariable [Cor. 8. Def. Pl. Tr.]; therefore AC is to CB, as radius to the sine of the angle CAB.

E

B D

In like manner, if a circle be described from the centre C, at the distance CA, meeting CB produced; it may be proved, that AC is to AB, as radius to the sine of the angle ACB.

PROP. II. THEOR.

In a right angled triangle (ABC, see fig. to preced. prop.), either of the legs (as AB) is to the other (BC), as radius is to the tangent of the angle (BAC) opposite that other, and to the hypothenuse (AC), as radius to the secant of the same angle.

From the centre A, at the distance AB, describe a circle, meeting AC in E; BC is the tangent, and AC the secant, of the arch BE, or angle EAB [Def. 8 and 9. Pl. Tr.], and the ratio of radius to the tangent or secant of a given angle is inva riable [Cor. 8. Def. Pl. Tr.], therefore AB is to BC, as radius to the tangent, and to AC, as radius to the secant, of the angle CAB, opposite to BC.

In like manner, if a circle be described from the centre C, at the distance CB meeting CA; it may be proved, that BC is to AB, as radius to the tangent, and to AC, as radius to the secant, of the angle ACB, opposite to AB.

Scholium. From this proposition and the preceding, arises that usual mode of speaking among mathematicians, that in a right angled plain triangle, if the hypothenuse be made radius, the legs become the sines of the opposite angles, and if either of the legs be made radius, the other leg becomes the tangent of the angle opposite to it, and the hypothenuse, the secant of the same angle.

PROP. III. THEOR.

The sides of plain triangles (as ABC, see figure 1, 2 and 3) are as the sines of the opposite angles.

About the triangle ABC describe a circle, which is done as in 5. 4 Eu. by bisecting two of its sides AC and BC by perpendiculars DG and EG meeting each other in the centre G; and if, in the cases of fig. 1 and 3, GF be drawn perpendicular to AB, and AG be joined, AD becomes, in all the figures, the sine of the angle AGD, AG being radius [Def. 6. Pl. Tr.]; but AD is the half of AC (3. 3 Eu.), and the angle AGD is equal to ABC, being each of them half of the angle at the centre on the arch AC (20: 3 and 4. 1 Eu.); therefore the half of the side AC is equal to the sine of the angle ABC, AG being radius. In like manner it may be shewn, that the half of BC is the sine of the angle BAC, and, in fig. 1, that the half of AB is the sine. of the angle C, to the same radius.

In a right angled triangle, fig. 2, AG the half of AB is radius, and therefore equal to the sine of the right angle ACB, [Cor. 2. Def. Pl. Tr.].

In an obtuse angled triangle fig. 3, join GB, AH and HB; AF the half of AB is the sine of the angle AGF [Def. 6. Pl. Tr.], and the angle AGF is the half of the angle AGB (4. 1 Eu.), and therefore equal to the angle H (20. 3. Eu.) but the angle H is the complement of the angle C to the two right angles (22. 3 Eu.), and the sine of an angle and of its complement to two right angles is the same [Cor. 1. Def. Pl. Tr.], therefore AF is the sine of the angle C. Thus, in every case, the halves of the sides becomes sines of the opposite angles, to the radius of the circumscribing circle, and the sides are as their halves (15. 5. Eu.), therefore, in every case, the sides are to each other, as the sines of the opposite angles.

triangle ADC, AC is to CD, as radius to the sine of A [1 Pl. Tr.], and CD is to CB, as the sine of the angle CBD to the radíus (By the same); therefore, by perturbate equality, AC is to CB, as the sine of CBD to the sine of A (23. 5. Eu.) and therefore, in the case of fig. 2, the sines of the angles CBA and CBD being equal [Cor. 1. Def. Pl. Tr.], as the sine of the angle CBA to the sine of A. In like manner it may be proved, that any other two sides are to each other, as the sines of the opposite angles.

Otherwise.

The same construction remaining, the perpendicular CD being radius, the sides AC and CB are to each other, as the consecants of the adjacent angles at the base AB [2. Pl. Tr. and 22.5 Eu.], or, the consecants of any two angles being inversely as their sines [Cor. 7. Def. Pl. Tr.], as the sines of the remote angles at the base, or, of the opposite angles.