LEMMA I. Fig. 1. L ET ABC be a rectilineal angle, if about the point B as it center, and with any distance BA, a circle be described, meeting BA, BC, the straight lines including the angle ABC in A, C; the angle ABC will be to four right angles, as the arch AC to the whole circumference. Produce AB till it meet the circle again in F, and through B draw DE perpendicular to AB, meeting the circle in D, E. By 33. 6. Elem. the angle ABC is to a right angle ABD, as the arch AC to the arch AD, and quadrupling the consequents; the angle ABC will be to four right angles, as the arch AC to four times the arch AD, or to the whole circumference. LET a center with any two distances BD, BA, let two circles' be described meeting BA, BC in D, E, A, C; the arch AC will be to the whole circumference of which it is an arch, as the archi DE is to the whole circumference of which it is an arch. By Lemma 1. the arch AC is to the whole circumference of which it is an arch, as the angle ABC is to four right angles ; and by the same Lemma 1. the arch DE is to the whole circumference of which it is an arch, as the angle ABC is to four right angles; therefore the arch AC is to the whole circumfer rence of which it is an arch, as the arch DE to the whole cird cumference of which it is an arch. DEFINITIONS. Fig. 3. I. LET ABC be a plane rectilineal angle ; if about B as a center; with BA any distance, a circle ACF be described meeting BA, BC, in A, C; the arch AC is called the measure of the angle ABC. It. The circumference of a circle is supposed to be divided into 360 equal parts called degrees, and each degree into 69 equat parts called minutes, and each minute into 60 equal parts called feconds, &c. And as many degrees, minutes, feconds, &c. as are contained in any arch, of so many degrees, minutes, feconds, &c. is the angle, of which that arch is the measure, said to be. COR. Whatever be the radius of the circle of which the mea sure of a given angle is an arch, that arch will contain the III. CBF, which, together with ABC, is equal to two sight angles, IV. the arch AC, perpendicular upon the diameter passing through of the angle ABC, of which it is the meafure. V. mity of the arch AC between the fine CD, and that extremity VI. the arch AC, and meeting the diameter BC passing through VII. tangent AE, is called the Secant of the arch AC, or angle ABC. angle ABC, are likewise the fine, tangent, and secant of its fupplement CBF. It is manifest from def. 4. that CD is the sine of the angle CBF. Ler CB be produced till it meet the circle again in G; and it is manifest that AE is the tangent, and BE the fecant, of the angle ABG or EBF, from def. 6. 7. Tig. 4. Cor. to def. 4. 5. 6. 7. The fine, versed fine, tangent, and fecant, of any arch which is the measure of any given angle dius of the first is to the radius of the second. def. 1. CD the fine, DA the versed fine, AE the tangent, and may be found. Fig. 3. VIII. plement of that angle. Thus, if BH be drawn perpendicular IX. and BK the secant of CEH, the complement of ABC, ac co-fine, and BK the co-fecant of the angle ABC. and co-tangent. equal, and the angles KHB, BAE are right; therefore the tri- BH or BA to HK.. and secant of any angle ABC. PROP. I. FIG. 5. IN a right angled plain triangle, if the hypothenuse be made radius, the sides become the fines of the angles opposite to them; and if either side be made radius, the remaining fide is the tangent of the angle opposite to it, and the hypothenuse the fecant of the fame angle. Let ABC be a right angled triangle ; if the hypothenuse BC be made radius, either of the sides AC will be the fine of the angle ABC opposite to it, and if either side BA be made radias , the other side AC will be the tangent of the angle ABC oppaík to it, and the hypothenuse BC the secant of the same angle. About B as a center, with BC, BA for diftances, let nog circles CD, EA be described, meeting BA, BC in D, E: fitue CAB is a right angle, BC being radius, AC is the fine of the angle ABC by def. 4. and BA being radius, AC is the tangers and BC the fecant of the angle ABC, by def. 6. 7. Cor. 1. Of the hypothenuse a fide and an angle of a rigt: angled triangle, any two being given, the third is also given. Cor. 2. Of the two sides and an angle of a right angled tiangle, any two being given, the third is also given. THE HE sides of a plane triangle are to one another, as the lines of the angles opposite to them. In right angled triangles this prop. is manifeft from prop ! for if the hypothenuse be made radius, the sides are the fines the angles opposite to them, and the radius is the fine of a riga angle (cor. to def. 4.) which is opposite to the hypothenuse. In any oblique angled triangle ABC, any two fides AB, AC will be to one another as the fines of the angles ACB, A50 which are opposite to them. From C, B draw CE, BD perpendicular upon the oppost: fides AB, AC produced, if need be. Since CEB, CDB are rice: angles, BC being radius, CE is the fine of the angle CBA, and BD the fine of the angle ACB ; but the two triangles CAE. DAB have each a right angle at D and E ; and likewise ebe common angle CAB; therefore they are similar, and consequenc's, CA is to AB, as CE to DB; that is, the sides are as the fines of the angles opposite to them. Cor. Hence of two sides, and two angles opposite to them, in a plain triangle, any three being given, the fourth is also given. PROP. III. Fig. 8. IN their difference, as the tangent of half the sum of the angles at the base, to the tangent of half their difference. Let ABC be a plain triangle, the sum of any two fides AB, AC will be to their difference as the tangent of half the sum of the angles at the base ABC, ACB to the tangent of half their difference. About A as a center, with AB the greater side for a distance, let a circle be described, meeting AC produced in E, F, and BC in D; join DA, EB, FB; and draw FG parallel to BC, meeting EB in G. The angle EAB (3 2. 1.) is equal to the sum of the angles at the base, and the angle EFB at the circumference is equal to the half of EAB at the center (20. 3.); therefore EFB is half the sum of the angles at the base; but the angle ACB (3 2. 1.) is equal to the angles CAD and ADC, or ABC together ; therefore FAD is the difference of the angles at the base, and FBD at the circumference, or BFG, on account of the parallels FG, BD, is the half of that difference; but since the angle EBF in a femicircle is a right angle (1. of this) FB being radius, BE, BG, are the tangents of the angles EFB, BFG; but it is manifest that EC is the sum of the sides BA, AC, and CF their difference ; and since BC, FG are parallel (2. 6.) EC is to CF, as EB to BG; that is, the sum of the sides is to their difference, as the tangent of half the sum of the angles at the base to the tangent of half their difference. PROP. IV. FIG. 18. IN any plain triangle BAC, whose two sides are BA,, AC and base BC, the less of the two sides, which let be BA, is to the greater AC as the radius is to the tangent of an angle; and the radius is to the |