AD,CB,make the internal angles on the fame fide, BAD,ABC,less than two right angles, those two right lines produced fhall meet on that fide, where the angles are lefs than two right angles. 14. Two right lines do not contain a space. 15. If to equal things,you add things unequal, the excess of the wholes fhall be equal to the excess of the additions. 16. If to unequal things equal be added, the excess of the wholes fhall be equal to the excess of those which were at firft. 17. If from equal things, unequal, things be taken away, the excefs of the remainders fhall be equal to the excess of the wholes. 18. If from things unequal, things equal be taken away, the excefs of the remainders fhall be equal to the excess of the wholes. 19. Every whole is equal to all its parts taken together. 20. If one whole be double to another, and that which is taken away from the first to that which is taken away from the fecond, the remainder of the firft thall be double to the remainder of the fecond. 1 The Citations are to be understood in this manner; When you meet with two numbers, the first fhews the Propofition, the fecond the Book; as by 4. 1. you are to underftand the fourth Propofition of the firft Book; and fo of the reft. Moreover, ax. denotes Axiome, poft. Poftulate, def. Definition, fch. Scholium, cor. Corollary, PRO From the centers A and point C; from whence b draw two right lines CA, CB. Then is AC cb 1. poft. ABC BC d AC. e Wherefore the Tri-c 15.def. angle ACB is equilateral. Which was to be done, d 1. ax. Scholium. After the fame manner upon the line AB may be described an Ifofceles triangle, if the diftances of the equal circles be taken greater or less than the line AB. e 23. def. At a point given A, to make a right line AG equal to a right line given BC. From the center Cat the diftance of CB,a de-a 3.poft. Tcribe the circle CBE. b join AC; upon which b 1. poft. raife the equilateral triangle ADC. d Produce c 1. I. DC to E. From the center D, at the diftanced 2. post. of e z. poft. of DE, defcribe the circle DEH; and let DA e be produced to the point G in the circumference thereof. Then AG- CB. f 15. def. gconftr. h 3. ax. kis.def. 1 1. ax. a 2. I. For DG f DE, and DAg=DC. Wherefore AG ↳ — CĘ k — BC 1=AG. Which was to be done. The putting of the point A within or without the line BC varies the cafes; but the construction, and the demonftration, are every where alike. Schol. The line AG might be taken with a pair of compaffes; but the fo doing anfwers to no Poftulate, as Proclus well intimates. A B PROP. III. Two right lines, A and BC, being given, from the greater BC to take away the right line BE equal to the leffer A. At the point B a draw the right line BD = A. The circle described from the center B at the distance of BD fhall cut off b 15. def. BE b BD c A d=BE. Which was to be done. c conftr. dr.ax. PROP. IV. A ·D B CE If two triangles BAC, EDF, have two fides of the one BA, AC equal to two fides of the other ED, DF, each to its correfpondent fide (that is, BA-ED, and and AC-DF) and have the angle A equal to the angle D contained under the equal right lines; they fhall have the bafe BC equal to the bafe EF; and the triangle BAC fhall be equal to the triangle EDF; and the remaining angles B, C, fhall be equal to the remaining angles E, F, each to each, under which the equal fides are fubtended. DF. b 14. as. If the point D'be applied to the point A, and the right line DE plac'd upon the right line AB, the point E fhall fall upon B,because DE a AB, a hyp. alfo the right line DF fhall fall upon AC,because the angle A a D. moreover the point F fhall fall upon the point C, becaufe AC a Therefore the right lines EF, BC fhall agree, because they have the fame Terms, and confequently are equal. Wherefore the triangles BAC, DEF, and the angles B, E, as also the angles C, F, do agree, and are equal. Which was to be Demonftrated. B PROP. V. The angles ABC, ACB, at the bafe of an Ifofceles triangle ABC, are equalone to the other: And if the equal fides AB, AC be produc'd, the angles CBD, BCE, under the bafe, fhall be equal one to the other. a Take AE-AD; and b join a 3. r. CD, and BE. b 1. poft. Because, inthe triangles ACD, E ABE, are AB c AC, and AE c hyp. d-AD, and the angle A common to them both, d conftr. etherefore is the angle ABE ACD,and the angle e 4. 1. AEB e ADC, and the base BE e-CD; also ECf DB. Therefore in the triangles BEC, f 3. ax. BDC g fhall be the angle ECB DBC. Which g 4. I. was to be Dem. Alfo therefore the angle EBC= DCB. but the angle ABE ACD; therefore h before. the angle ABC k—ACB, Which was to be Dem. k 3. ax. Coroll. Coroll. Hence, every equilateral triangle is also equiangular. PROP. VI. If two angles ABC, ACB of a triangle ABC be equal the one to the other, the fides AC, AB fubtended under the equal angles, shall also be equal one to the other. If the fides be not equal, let one be bigger than the other, fuppofe BACA. a Make BD-CA, and b draw the line CD. a 3. 1. br. post. In the triangles DEC, ACB, because BD c cfuppof. CA, and the fide BC is common, and the angle DBC d ACB, the triangles DBC, ACB e fhall be equal the one to the other, a part to the whole. f Which is impoffible. Coroll. d hyp. € 4. I. f 9. ax. Hence, Every equiangular triangle is also equilateral. BA BA ་ B Upon the fame right line AB two right lines being drawn AC, BC, two other right lines equal to the former, AD, BD, each to each (viz. AD AC, and BD BC) cannot be drawn from the fame points A, B, on the fame fide C, to feveral points, as C and D, but only to C. 1. Cafe. If the point D be fet in the line AC, a 9. ax. it is plain that AD is a not equal to AC. 2. Cafe. If the point D be placed within the triangle ACB,then draw the line CD,and produce BDF, and BCE. Now you would have ADAC. then |