III.-Line BF nearer diam. > CF more remote & CF > GF. IV. The lines FB, FI, making with AD BFA = IFA, are equal. C. 2 4, I. 3 Def.15,I.Ax.I. but EI EC.. EC EL. = the whole, which is absurd. .. Neither FB nor FI the gr.; V.-Also only two equal lines, FG, FH, from F to the Oce. E in EF make FEH = and join FH. 23. I. Pst. 1. | At D.1 Def. 15, I. C... GE 2 4, I. 4 Sup. 5 C. Ax. 1. 6 Remk. 7 Case III. 8 Rec. = HE, EF com. and ZHEF, .. FG = FH, FI, FEG, GEF = And from F to Oce. no other line = FG, •· FK = FGFH.. FK = FH, Therefore, If any point which is not the centre, SCH-If from a point, within a circle, not the centre, as F, a st. line of indef. length, as FX, revolve so as in each part of its revolution to be terminated or cut off by the circumference, as in A, B, C, G, D; its maximum length FA, will be attained when it coincides with that part of the diam. AD, in which E the centre is; and its minimum FD, when it coincides with the other part of the diam.; and the nearer F X, is to the maximum the greater it is, as FB, and the nearer to the minimum, the less, as FG. USE. THEODOSIUS, mentioned p. 11, by aid of this proposition proves that, if from the pole of the world, which is not the pole of the horizon, (for the zenith is its pole) several arcs of great circles be drawn to the circumference of the horizon, the greatest arc shall be that part of the meridian which passes through the zenith. By this proposition we may also prove that the Earth being in Aphelion is at the greatest distance from the Sun, in Perihelion, at the least; and so for all the other planets. PROP. 8.-THEOR. If any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one passes through the centre; 1st, those which make equal angles with the line passing through the centre are equal; 2nd, of those which fall upon the concave circumference, the greatest is that which passes through the centre; and of the rest, that which is nearer to the one passing through the centre is always greater than one more remote; but, 3rd, of those which fall upon the convex circumference, the least is that between the given point without the circle and the diameter, and of the rest, that which is nearer to the least is always less than one more remote; and, 4th, only two equal lines can be drawn from the same point to the circumference, one upon each side of the line which passes through the centre. CON. 1, III., Pst. 1. 3, I. 23, I. DEM. 4, I. Def. 15, I. Ax. 1 & 2. 20, I. 21. I. If from the ends of the side of a to a Ax. 9. 24, I. Ax. 5. A there be drawn two st. lines within the A, these lines shall be less than the other two sides of the A, but shall contain a gr. 2. E.1 Hyp. 1. Let ABC be a O and D any without it; 2. from D let st. lines DA, 3 3. 4 DE, DF, DC, and DO be Of these let D A pass through 4. and DE, DO make with DA, ZADE = ▲ ADO; 5 Conc. 1. the line DE = line DO; 1st. Of lines incident on AEFC the concave Oce, 2. the greatest is DA passing through cen. M; 3. and any line nearer DA is > a line more remote, i. e. DE > DF and DF > DC: 2nd. But of lines incident on HLKG the convex Oce, Also only two equal lines, DB and D K, from D to Oce, can be drawn, one upon each side of DA. Take M the cen. of O ABC and join MO, ME, MF, MC, MH, ML, and MG. I. The st. lines DE and DO are equal, making equal ▲s with DA. C. 3, I. Pst. 1. D.1 C. and H. 2 4, I. 4 Ax. 9. 5 Conc. Make DP DE, and join ME, MP. In As MPD, MED MD com. DP=DE II. Of lines incident on AEFC the concave Oce, DA through the cen. is greatest, and DE, nearer DA is > DF more remote, and DF > DC: D.1 Def. 15,I.Add.] AM ME, to each add MD; 2 Ax. 2. 320, I. ·· = .. AM + MD, i. e. AD = EM + MD; but EM + MD > ED .. AD > ED. 4 Def.15,I. Ax.9 Again ME MF, MD com. & EMD 524, I. 6 Sim. 7 Conc. >< FMD, .. ED > FD. = In like manner FD > CD. .. DA through cen. M is the greatest, DE> DF & DF > DC. III. Of lines incident on HLKG the convex Oce, DG, between D and the diam. AG, is the least, DK < DL & DL < DH. D.120, I. Def. 15, I.¡ ·.· MK + KD > MD, & MK = MG; 2 Ax. 5. 3 C. .. rem. KD > rem. GD; i. e. GD < KD, & MLD is a A, and from s M, D, extrs. of one side MD, MK & DK are drawn to D.4, 21, I. .. MK + KD < ML + LD; ML .. rem. KD < rem. LD. 5 Def. 15,I. Ax.5 but MK 6 Sim. 7 Conc. In like manner DL < DH: .. DG the least, DK < DL & DL < DH. IV. Also only two equal st. lines can be drawn from the D to the Oce, one on each side of AD the line which passes through cen. M. At M in MD make DMB = ▲ DMK, & join DB C. 23, I. Pst. 1. D.1 C. 24, I. 4 SUP. in As KMD, BMD, MK = com. & KMD = ▲ BMD; .. DK = DB. MB, MD But besides DB no other line from D to the If there can, let it be DN; 5 D.4 & 3, Ax. 1. ·· DK = - DN & also = DB.. DB = DN 6 Remk. 7 D. II. 7. i.e. a line nearer the least one more remote; but this has been proved to be impossible; .. No line but DB = DK. Therefore, If any point be taken, &c. Q. E.D. SCH. 1.-The concave and convex parts of a circumference are determined by the tangents from a point external to the circumference. All the parts of the circumference which are farther from the external point than the tangent points of the circle, are concave with respect to that point; and those nearer to the external point than the tangent points are convex. SCH. 2.-A proposition, analagous to Prop. 7 & 8, explains why in Def. 4, III. and in 14 & 15, III. the perp. from a point out of a st. line to the line is called the distance of the point from or to the line.—Hose, p. 302. If any C be taken out of a st. line AB; then of all st. lines, CD, CE, CF, CG, from that to the st. line, the least, CD, shall be that which is perp. to the st. line; and of the rest, CE, CF, CG, that which is nearer, as CE, to the perp., shall always be less than that, CF or CG, which is more remote; and from this C there can be drawn to the st. line AB, only one st. line a given st. line, CE, drawn from the same C, to the st. line, which shall be on the opp. A side of the perp. I. The perp. CD is the least line from C to AB; CE > CD, CF > CE and CG > CF. D.1 H 2 H. & Sup. 317, I. 4 D. & 19, I. 5 D. 4. 6 Sim. 719, I. 8 Sim. 9 Conc. II. From on the LCDE is a rt. 2, .. / CED < art. .. ▲ CDE > CED.. CE > CD. CED<a rt... CEF > a rt. L. and in like manner CFD < a rt. . So also CG > CF; Hence the perp. CD is the least; CE > CD, CF > CE, & C only one st. line, CK CE a given st. line from C to AB opp. side of the perp. CD. C.11 23, I. 2 Pst. 2. 3 Conc. D.1 Ax.11, 2 26, I. 3 Remk. SUP. C.1.rt. / CDK= rt. / CDE, DCK= LDCE, and 5 H.D.2Ax.1 6 Case I. Conc. .. CK CE. Also no line from C to A B, but CK = CE; If there can be, let it be CL; Then.. CL =CE & CK CE.. CL = CK; Hence from C only one st. line to AB, namely CK CE on SCH. 3.-Thys Proposition," says the Translator of 1570, "is called commonly in old bookes amongest the barbarous Cauda Pauonis, that is, the Peacockes taile."-Fol. 88. USE AND APP.-I. The Proposition 8 Bk. III. is employed to show, that i a tangent and secant be drawn to the same point, the tang. <sec., but > external part of the secant: Thus in fig. to Prop. 8, let D H represent a tang. and MD a sec. to arc HG, or HMG, then D H< DM but > DG. II. By aid of Props. 7 & 8, and Ax. A, bk. III., we may demonstrate; 1st. When one circle A is contained within another, B, without touching it, the distance between the centres, DE < (DF EG), the difference of the radii: and conversely, when the distance between the B centres, DE (DF~ EG), the difference of the radii, the lesser circle will be within the greater without meeting it. DE G |