Book V. A T K A K C H HT And because AG is the fame multiple of E as CH is of F; and GB is equal to E; and CK equal to F; therefore (by 2. 5.) AB is the fame multiple of E which KH is of F; but AB is fuppofed to be the fame multiple of E, which CD is G-of F; therefore KH is the fame multiple of F, which CD is of F; wherefore because each of the lines KH, CD is the fame multiple of F; therefore (by com. not. 6.) KH is equal; to CD; let CH which is common be taken away; therefore the, remainder KC is equal to the remainder HD: but KC is equal to F; therefore HD is alfo equal to F; fo that when GB is equal to E, HD will also be equal to F. B DEF BDEF Certainly in the fame manner we shall demonftrate, that when GB is a multiple of E, that HD will be the fame multiple of F. Wherefore if two magnitudes be equimultiples of two magnitudes; and some magnitudes equimultiples of the same be taken away; alfo the remainders are either equal to thofe magnitudes, or equimultiples of them. Which was to be demonstrated. Equal magnitudes have the fame ratio to the fame magnitude; and the fame magnitude has the fame ratio to equal magnitudes. Let A and B be equal magnitudes; and C any other magnitude which may accidentally happen; I fay that each of the magnitudes A, B has the fame ratio to C; and that C has the fame ratio to each of the magnitudes A, B. . For let D, E be taken equimultiples of A, B ; and let F be taken any other multiple of C which may accidentally happen. And because D is the fame multiple of A, which E is of B; and A is (by fupp.) equal to B; therefore D is equal to E; but F is any other multiple of C which may accidentally happen; wherefore if D exceed F, E alfo exceeds F; and if equal, equal; and if lefs, lefs and D, E are lequimultiples of A, B; and Fany other multiple of C which may accidentally happen; therefore (by (by 5 def. 5.) it is as A is to C fo is B to C. I fay also that C has the fame ratio to each of the magnitudes A, B, D For the fame things being conftructed; in the fame manner we shall demonstrate that D is equal to E; and that F is any other magnitude; therefore if F exceed D; it alfo exceeds E: and if E equal, equal and if lefs, lefs: and F is a multiple of C and D, E any other equimultiples of A and B which may accidentally happen; therefore (by 5. def. 5.) it is, as C to A fo is C to B. Therefore equal magnitudes have the same ratio : F to the fame magnitude; and the fame magnitude has the fame ratio to equal magnitudes. Which was to be demonstrated. Of unequal magnitudes; the greater has a greater ratio to the fame magnitude, than the lefs has and the fame magnitude has a greater ratio to the lefs, than it has to the greater. Let AB and C be unequal magnitudes; and let AB be greater than C; and let D be any other magnitude which may accidentally happen: I say that AB has a greater ratio to D than C has to D : And D has a greater ratio to C, than it has to AB. For because AB is greater than C; make BE equal to C: certainly (by 4. def. 5.) the leffer of the two AE, EB being multiplied will at length be greater than D. First let AE be lefs than EB; and let AE be multiplied until what is produced shall be greater than D: and let FG be the multiple of it, which is greater than D and whatsoever multiple FG is of AE, let GH be the fame multiple of EB; and K the fame multiple of C: and let L be taken the double of D, and M triple; and more by one in order, until the multiple of D taken becomes the first greater than K: let it be taken: and let it be N four times D, the first greater than K. Wherefore because K is the multiple of D greater than K]; VOL. I. firft less than N or [N is the first * B and Book V. -E Book V. and because FG is the fame multiple of AE F which GH is of EB; therefore (by 1. 5.) FG is the fame multiple of AE which FH is of AB; and FG is the fame multiple of AE which K is of C; therefore FH is the fame multiple of AB, which K is of C; therefore FH and K are equimultiples of AB and C: Again becaufe GH is the fame multiple of EB which K is of C; and EB is equal to C; therefore GH is equal to K: but K is not lefs than M; therefore neither is GH less than M; but (by conft.) FG is greater than D; therefore the whole FH is greater than both D and M together; but D and M together are equal to N ; therefore FH exceeds N, but K does not exceed N; and FH and K are equimultiples of AB and C ; and N is any other multiple of D which may accidentally happen; therefore (by 7. def. 5.) AB has a greater ratio to D, than C has to D. KH C D L M N 1 fay alfo, that D has a greater ratio to C, than D has to AB. For the fame things being conftructed; in like manner we fhall demonftrate, that N exceeds K; but does not exceed FH; and N is a multiple of D; and FH and K [any other] equimultiples of AB and C [which may accidentally happen]; therefore (by 7. def. 5) D has a greater ratio to C, than D has to AB. E But let AE be greater than EB; certainly EB the lefs being multiplied will at length (by 4. def. 5.) be greater than D: let it be multiplied; and let GH be the multiple of EB, greater than D; and whatsoever multiple GH is of EB; let FG be made the fame multiple of AE; and K of C: Certainly in the fame manner we shall demonftrate, that FH and K are equimultiples of AB and C and in like manner let N be taken, a multiple of D, the first greater than FG; fo that again FG be not lefs than M; but GH is greater than D; therefore the whole FH exceeds D and M together, that is N; but K does not exceed N; fince FG B HCDLMN being greater than GH; that is K, does not exceed N; and in like Book V. manner following the fteps above we finish the demonstration. Wherefore of unequal magnitudes, the greater has a greater ratio to the fame magnitude, than the less has and the fame magnitude has a greater ratio to the lefs, than it has to the greater. Which was to be demonftrated. PROP. IX. Magnitudes having the fame ratio to the fame magnitude are equal to one another; and thofe are also equal to one another, to which the fame magnitude has the fame ratio. For let each of the magnitudes A, B have the fame ratio to the fame magnitude C: I fay that A is equal to B. For if not, (by 8. 5.) each of the magnitudes A, B could not have the fame ratio to C: but it has (by supp.); therefore A is equal to B. Again let C have the fame ratio to each of the magnitudes A, B; I fay that A is equal to B. B For if not ; C could not (by 8. 5.) have the fame ratio A to each of the magnitudes A, B; but it has (by fupp.); therefore A is equal to B. Wherefore, magnitudes having the fame ratio to the C fame magnitude are equal to one another; and those are equal to one another to which the fame magnitude has the same ratio. Which was to be demonftrated. Of magnitudes having ratio to the fame magnitude, that is the greater which has the greater ratio: and that is the less to which the fame has a greater ratio. For let A have a greater ratio to C than B has to C: I fay that A is greater than B. For if not; either A is equal to B, or less; but A is not equal to B; for then each of the magnitudes A, B (by 7. 5.) has the same ratio to C; but (by fupp.) each of them has not therefore A is not equal to B; neither is A lefs than B; for (by 8. 5.) A would Book V. have a less ratio to C than B would have to C; but (by Again let C have a greater ratio to B than C has to A : AT C For if not; it is either equal, or greater: but B is not equal to A, for (by 7. 5,) C would have the fame ratio to each of the magnitudes A, B: but (by supp.) it has not; therefore A is not equal to B: Neither is B greater than A; for (by 8. 5.) C would have a less ratio to B than to A; but (by supp.) it has not; therefore B is not greater than A; but it has been demonstrated that neither is it equal; therefore B is less than A. Wherefore of magnitudes having ratio to the fame magnitude, that is the greater, which has the greater ratio: and that is the lefs to which the fame has a greater ratio. Which was to be demonftrated. PRO P. XI.. Ratios which are the fame to the fame ratio, are the fame to one another. For let the ratios be; as A to B fo is C to D; and as C to D fo is E to F; I say that it is, as A is to B fo is E to F. For let G, H, K be taken equimultiples of A, C, E; and let L, M, N be taken any other equimultiples of B, D, F which may accidentally happen. And because it is (by fupp.) as A is to B fo is C to D; and G and H have been and if equal, equal; and if lefs, lefs: GABL HCDM KEFN |