f. 28. Dat pofition cut one another is given f. and the ftraight line DB which g. 33 Dat is at right angles to AB is given & in position, and AB is given in pofition, therefore f the point B is given. and the points A, D are h. 29. Dat. given, wherefore 1 the ftraight lines AB, BD are given. and the i. 2. Dat. ratio of AB to BC is given, and therefore i BC is given. The Compofition. Let the given ratio of FG to GH be that which AB is required to have to BC, and let HK be the given ftraight line which is to be taken from BC, and let the ratio which the remainder is required to have to BD be the given ratio of HG to GL, and place GL at right angles to FH, and join LF, LH. Next, as HG is to A E D BNM A C F G. K Η GF, fo make HK to AE; produce AE to N fo that AN be the ftraight line to the square of which the fum of the squares of AB, BD is required to be equal; and make the angle NED equal to the angle GFL. from the center A at the distance AN defcribe a circle, and let its circumference meet ED in D, and draw DB perpendicular to AN, and DM making the angle BDM equal to the angle GLH. laftly, produce BM to C fo that MC be equal to HK. then is AB the firft, BC the second and BD the third of the ftraight lines that were to be found. k For the triangles EBD, FGL, as alfo DBM, LGH being equiangular, as EB to BD, fo is FG to GL; and as DB to BM, fo is LG to GH; therefore, ex aequali, as EB to BM, so is (FG to k. 12. 5. GH, and fo is) AE to HK or MC; wherefore * AB is to BC, as AE to HK, that is, as FG to GH, that is, in the given ratio. and from the straight line BC taking MC which is equal to the given ftraight line HK, the remainder BM has to BD the given ratio of d. 47. 1. HG to GL. and the fum of the fquares of AB, BD is equal a to the fquare of AD or AN which is the given space. Q. E. D. I believe it would be in vain to try to deduce the preceding Construction from an Algebraical Solution of the Problem. d THE ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY. GLASGOW: PRINTED BY J. & M. ROBERTSON, SALTMARKET. LEMMA I. FIG. 1. ET ABC be a rectilineal angle, if about the point B as a L' center, and with any distance BA, a circle be described, meeting BA, BC, the ftraight 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 confequents, the angle ABC will be to four right angles, as the arch AC to four times the arch AD, or to the whole circumference. LEMMA II. FIG. 2. ET ABC be a plane rectilineal angle as before; about B as 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 arch 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 fame 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 circumference of which it is an arch, as the arch DE to the whole circumference of which it is an arch. L DEFINITIONS. FIG. 3. I. ET ABC be a plane rectilineal angle; if about B as a center, with BA any distance, a circle ACF be defcribed meeting BA, BC, in A, C; the arch AC is called the measure of the angle ABC. II. The circumference of a 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, &c. And as many degrees, minutes, feconds, &c. as are contained in any arch, of fo many degrees, mi a Fig. 4. nutes, feconds, &c. is the angle, of which that arch is the measure, faid to be. COR. Whatever be the radius of the circle of which the meafure of a given angle is an arch, that arch will contain the fame number of degrees, minutes, feconds, &c. as is manifeft from Lemma 2. III. Let AB be produced till it meet the circle again in F, the angle CBF, which, together with ABC, is equal to two right angles, is called the Supplement of the angle ABC. IV. A ftraight line CD drawn through C, one of the extremities of the arch AC, perpendicular upon the diameter paffing through the other extremity A, is called the Sine of the arch AC, or of the angle ABC, of which it is the measure. COR. The Sine of a quadrant, or of a right angle, is equal to the radius. V. The fegment DA of the diameter paffing through A, one extremity of the arch AC between the fine CD, and that extremity is called the Verfed Sine of the arch AC, or angle ABC. VI. A ftraight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC paffing through the other extremity C in E, is called the Tangent of the arch AC, or of the angle ABC.. VII. The straight line BE between the center and the extremity of the tangent AE, is called the Secant of the arch AC, or angle ABC. COR. to Def. 4. 6. 7. The fine, tangent, and fecant of any angle ABC, are likewife the fine, tangent, and fecant of its fupplement CBF. It is manifeft from Def. 4. that CD is the fine of the angle CBF. Let CB be produced till it meet the circle again in G; and it is manifeft that AE is the tangent, and BE the fecant, of the angle ABG or EBF, from Def. 6. 7. COR. to Def. 4. 5. 6. 7. The fine, verfed fine, tangent, and fecant, of any arch which is the measure of any given angle ABC, is to the fine, verfed fine, tangent, and fecant, of any |