A Redilincal figure (ABCD) is faid to be inscribed in another_rectilineal figure (EFGH), when all the angles (A, B, C, D) of the infcribed figure, are upon the fides of the figure in which it is inscribed (fig. 1). II. In like manner a rectilineal figure (EFGH) is faid to be defcribed about another rectilineal figure (ABCD); when all the fides (EF, FG, GH, HE) of the circumfcribed figure pafs thro' the angular points (A, B, C, D) of the figure about which it is described, each thro' each (Fig. 1), III. A rectilineal figure (ABCD) is faid to be infcribed in a circle, when all the angles (A, B, C, D) of the infcribed figure are upon the circumference of the cricle (ABCDE) in which it is infcribed (Fig. 2). IV. A rectilineal figure (ABCDE) is faid to be defcribed about a circle, when each of the fides AB, BC, CD, DE, EA) touches the circumference of the circle (Fig. 3). A Circle (ABCD) is faid to be inscribed in a rectilineal figure (EFGH), when the circumference of the circle touches each of the fides (EF, FG, GH, HE) of the figure in which it is infcribed (Fig. 1). VI. A circle (ABCD) is described about a rectilineal figure (ABC), when the circumference of the circle paffes thro' all the angular points (A, B, C) of the figure about which it is described (Fig. 2). VII. A ftraight line (AB) is faid to be placed in a circle (ADBE), when the extremities of it (A & B) are in the circumference of the circle (fig. 3). PROPOSITION I. PROBLEM I IN Na given circle (ABD), to place a ftraight line (AB) equal to a given ftraight line (N), not greater than the diameter of the circle (ABD). THER CASE I. If AD is to N. HERE has been placed in the given ○ ABD a straight line to the given N. Pof. I. D. 7. B. 4. 2. From the center A at the distance AE describe the O EBF, BECAUSE DEMONSTRATION. ECAUSE AB is to AE (D. 15. B. 1), and the straight line N is to AE (Ref. 1.) 1. The ftraight line AB, placed in the O ABD, will be alfo = { to N. P. 3. B. 1. Pof. 3. Ax. 1. B. 1. Which was to be done. IN PROPOSITION II. PROBLEM II. Na given circle (ABHC), to inscribe a triangle (ABC) equiangular to a given triangle (DFE). Sought. The ABC infcribed in the O ABHC, equiangular to the ▲ DFE. Given. AO ABHC together with the ▲ Refolution. 1. From the point M, to the O ABHC draw the tangent MN. BECAUS DEMONSTRATION. P. 17. B. 3. BAM P. 23. B. 1. ECAUSE the V BCA is to the V BAM (P. 32. B. 3), and the V FED is to the fame V BAM (Ref. 2); alfo the V CBA is = to the V CAN (P. 32. B. 3.) and V FDE is to V CAN (Ref. 2. = to the 1. It follows that V BCA is to V FED, and CBA to FDE. Ax. 1. B. 1. 2. Confequently, the third V BAC, of the ▲ ABC, is also third DFE of the ADFE, and this ▲ ABC is infcribed in the SP. 32. B. I. O ABHC. D. 3. B. 4. Which was to be done. ABOUT D E PROBLEM III. BOUT a given circle (EFG) to describe a triangle (ABD), equiangular to a given triangle (HKL). Given. Sought. The A ABD defcribed about the O 1. Produce the fide HL, of the ▲ HKL, both ways. 3. At the point C in CE, make the ECF to the Pof. 2. KHM, P. 23. B. 1. 4. Upon CE, CF, CG, erect the L AD, AB, DB produced. BECAUSE ECAUSE the V CEA, CFA are L (Ref. 4.) Pof. 1. Ax. 8, B. 1. 1. V FEA + EFA are < 2L, & AD, AB meet fome where in A. {..... It may be demonftrated after the fame manner, that. 2. AD, DB & AB, DB meet fomewhere in D & B. And fince AD, AB, DB are at the extremities E, F, G of the rays 3. Thefe ftraight lines touch the O EFG; and the A ABD formed 8. Therefore the ▲ ABD described about the O EFG is equiangular to the given ▲ HKL. Which was to be done, |