tudes the antecedent to the confequent; but as in the first magni- Book V. tudes the confequent is to fome other, fo in the second magnitudes is fome other to the antecedent. PROP. I. If there be any number of magnitudes, equimultiples of an equal number of magnitudes, each of each; whatsoever multiple one of the magnitudes is of one, the fame multiple will all be of all. Let AB, CD be any number of magnitudes, equimultiples of an equal number of magnitudes E, F ; each of each: I say that whatsoever multiple AB is of E, the fame multiple will AB, CD together be of E, F together. A G B C E For because AB is the fame multiple of E that CD is of F; as many magnitudes as there are in AB equal to E, fo many are there in CD equal to F; let AB be divided (by 3. 1.) into the magnitudes AG, GB equal to E; and CD into the magnitudes CH, HD equal to F: certainly the number of parts CH, HD will be equal to the number of parts AG, GB: and because AG is equal to E, and CH equal to F, therefore AG, CH together are equal to E and F together; for the fame reafon GB is equal to E, and HD to F; therefore alfo GB, HD together are equal to E, F together therefore as many magnitudes as there are in AB equal to E ; so many many are there in AB, CD together equal to E, F together: wherefore whatsoever multiple AB is of E, the fame multiple will AB, CD together be of E, F together. H D F Wherefore if there be any number of magnitudes, equimultiples of an equal number of magnitudes, each of each; whatsoever multiple one of the magnitudes is of one, the fame multiple will all be of all. Which was to be demonftrated. PROP. II. If the first be the fame multiple of the fecond as the third is of the fourth; and if the fifth be the fame multiple of the second as * A 2 the Book V. the fixth is of the fourth: alfo the first and fifth together will be the fame multiple of the fecond, as the third and fixth together is of the fourth. A For let the first AB be the fame multiple of the second C as the third DE is of the fourth F; and let the fifth BG be the fame multiple of the fecond C as the fixth EH is of the fourth F; I fay that AG the first together with the fifth will be the fame multiple of thesecond C as DH the third and fixth is of F the fourth. For because AB is the fame multiple of C that DE is of F; as many magnitudes as there are in AB equal to C, fo many will there be in DE equal to F: certainly for the fame reafon alfo, as many magnitudes as there are in BG equal to B C, fo many are there alfo in EH equal to F; therefore as many as there are in the whole AG equal to C fo many are there in the whole DH equal to F wherefore whatsoever multiple AG is of C, the fame multiple will DH be of F; therefore AG the first together with the fifth will be the fame multiple of C the fecond, that DA the third and fixth is of F the fourth. то E C F H Wherefore if the first be the fame multiple of the fecond as the third is of the fourth; and if the fifth be the fame multiple of the second as the fixth is of the fourth; alfo the first and fifth together will be the fame multiple of the second, as the third and fixth together is of the fourth. Which was to be demonstrated. If the first be the fame multiple of the fecond as the third is of the fourth; and equimultiples be taken of the first and third; also by equality, each of thofe taken will be an equimultiple of each, the one of the fecond, and the other of the fourth.. For let the firft A be the fame multiple of the fecond B, as the third C is of the fourth D; and let EF, GH be taken equimultiples of A and C ; I say that EF is the fame multiple of B, which GH is of D. For Book V. F H L For because EF is the fame multiple of A (by conft.) which GH is of C: therefore as many magnitudes as there are in EF equal to A, fo many are there in GH equal to C: let EF be divided into magnitudes equal to A, viz. EK, KF; and GH into magnitudes K equal to C, viz. GL, LH; the number of parts EK, KF will be equal to the number of parts GL, LH: And because A is the fame multiple of B, as C is of D; and EK is equal to A, and GL equal to C; therefore EK is the fame multiple of B, which GL is of D. Certainly for the same reason alfo KF is the fame multiple of B, which LH is of D: wherefore because the first EK is the fame multiple of the fecond B, as the third GL is of the fourth D; and also the fifth KF is the fame multiple of the fecond B, which the fixth LH is of the fourth D; therefore (by 2. 5.) EF the firft together with the fifth is the fame multiple of the fecond B, which GH the third and fixth is of the fourth D. EA B D C G Wherefore if the first be the fame multiple of the fecond, as the third is of the fourth; and equimultiples be taken of the first and third alfo by equality, each of those taken will be an equimultiple of each, the one of the fecond, and the other of the fourth. Which was to be demonftrated.. If the first have the fame ratio to the second as the third to the fourth; alfo the equimultiples of the first and third, will have the fame ratio to the equimultiples of the second and fourth, according to any multiplication whatfoever; taken fo as to answer each other. For let the first A have the fame ratio to the fecond B, which the third C has to the fourth D; and let E and F be taken equimultiples of A and C ; but let G and H be taken any equimultiples of B and D which may accidentally happen; I say that as E is to G fo is F to H. For let K and L be taken equimultiples of E and F ; and M, N any other equimultiples of G and H which may happen.. And Book V. And because E is the fame multiple of A and M and N [any other] KEABG dentally happen]; therefore (by 5 def. 5.) if equimultiples of E and F; and K and L are and M and N any other equimultiples, which may accidentally LFC DHN M Wherefore if the first have the fame ratio to the fecond as the third to the fourth; alfo the equimultiples of the first and third, will have the same ratio to the equimultiples of the second and fourth ; according to any multiplication whatsoever; taken fo as to answer each other. Which was to be demonstrated. : Cor. Wherefore because it has been demonftrated, that if K exceed M; L exceeds N, and if equal, equal: and if lefs, lefs. Certainly also if M exceed K, N exceeds L: and if equal, equal; and if lefs, lefs and on this account it will be; as G is to E, fo is H to F. Certainly from this it is manifest that if four magnitudes be proportionals, they will be inversly proportionals; (by def. 14. 5.). If a magnitude be, the fame multiple of a magnitude, which a magnitude taken from the one is of a magnitude taken from the other; whatsoever multiple the whole is of the whole, also the remainder will be the fame multiple of the remainder. For let the magnitude AB be the fame multiple of the magnitude CD, which AE the magnitude taken from the one is of the magnitude magnitude CF taken from the other: I fay that whatsoever multiple Book V. the whole AB is of the whole CD, alfo EB the remainder will be the fame multiple of FD the remainder, For whatsoever multiple AE is of CF; let EB be the fame multiple of CG. G F And because AE is the fame multiple of CF (by the conft. and 1. 5.) which AB is of GF; and AE is supposed to be the fame multiple of CF which AB is of CD: therefore AB is the fame multiple of either of E the lines GF, CD; therefore (by com. not. 7.) GF is equal to CD; let CF which is common be taken away; therefore the remainder GC is equal to the remainder DF; and because AE is the fame multiple of CF which EB is of GC; and GC is equal to DF; therefore AE is the fame multiple of CF which EB is of FD; but AE is supposed to be the fame multiple of CF, which AB is of CD: therefore EB is the fame multiple of FD which AB is of CD ; therefore whatsoever multiple the whole AB is of the whole CD; alfo the remainder EB is the fame multiple of the remainder FD. B D Wherefore if a magnitude be the fame multiple of a magnitude, which a magnitude taken from the one is of a magnitude taken from the other; whatsoever multiple the whole is of the whole, also the remainder will be the fame multiple of the remainder. Which was to be demonstrated. PROP. VI. 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. For let the two magnitudes AB, CD be equimultiples of the two magnitudes E, F ; and let AG, CH the magnitudes taken away be equimultiples of the fame E, F: I fay that the remainders GB, HD are either equal to E, F or equimultiples of them. For first let GB be equal to E; I say that HD is also equal to F: for put CK equal to F. And |