. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. A2: AG2 :: CA2—CD2 : AG2—DH2 ;consequently, D> 2 = AG2 — DH2 = Ca2— FD=CF —CD,andFD2^=CF2 —2CFCD+CD2;therefore FE2 = CF2 + Ca2 — 2CF-CD + CD2 — DH3. But, by Prop. IV. G,2 -|-CF2 = CA3 ;and by supposition, 2CF • CD=2C A • CI ;therefore, CA2—2CA • Cl-f CD2— DH2. But, by supp. CA2 : CD2 :: CF2 or CA2—AG3 : CI3 ;and, by sim. tri.^CA* : CD* :: CA2—AG2 : CD2—DH* ;therefore CI2 = CD2—DH2 ; consequently, FE2 = C
. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. A2: AG2 :: CA2—CD2 : AG2—DH2 ;consequently, D> 2 = AG2 — DH2 = Ca2— FD=CF —CD,andFD2^=CF2 —2CFCD+CD2;therefore FE2 = CF2 + Ca2 — 2CF-CD + CD2 — DH3. But, by Prop. IV. G,2 -|-CF2 = CA3 ;and by supposition, 2CF • CD=2C A • CI ;therefore, CA2—2CA • Cl-f CD2— DH2. But, by supp. CA2 : CD2 :: CF2 or CA2—AG3 : CI3 ;and, by sim. tri.^CA* : CD* :: CA2—AG2 : CD2—DH* ;therefore CI2 = CD2—DH2 ; consequently, FE2 = CA2 — 2CA • CI-f-CI2. And the root or side of this square is FE=CA—CI=AI. In the same manner is found fE = CA -f- CI =BI. Q. E. D. Cor. 1. Hence CI or CA—FE is a fourth proportionalto CA, CF, CD. Cor. 2. And /E—FE=2CI ; that is, the differencebetween two lines, drawn from the foci to any point in th«curve, is double the fourth proportional to CA, CF, CD. -288 MATHEMATICS. PROPOSITION VI. The sum of two lines, drawn from the foci to meet in d*«%y point of the curve, is equal to the transverse axis. That is, FE+/E=AB. E &. For, by the last Prop. FE = C A — CI = AI,And, by the same, /E = CA + CI = BI jTherefore, by addition, FE-f/E=AB. Cor. Hence is derived the common method of the curve mechanically by points, or with a thread,thus :
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