A is to B, as C to D, and of A and C, than E has to F. (7. Def. 5.) WhereM and G are equim ultiples : and of fore, if the first, &c. Q. E. D. B and D, N and K are equimultiples; if M be greater than N, G is greater Cor. And if the first have a greater than K; and if equal, equal; and if ratio to the second, than the third has less, less ; (5. Def. 5.) but G is great- to the fourth, but the third the same er than K, therefore M is greater than ratio to the fourth, which the fifth has N: But H is not greater than L; to the sixth; it may be demonstrated, and M, H are equimultiples of A, E; in like manner, that the first has a and N, L equimultiples of B, F: greater ratio to the second, than the Therefore A has a greater ratio to B, fifth has to the sixth. PROP. XIV. THEOR. If the first has to the second, the same ratio which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less. Let the first A, have to the second fore also C has to D a greater ratio B, the same ratio which the third C, than C has to B. (13. 5.) But of two has to the fourth D; if A be greater magnitudes, that to which the same than C, B is greater than D. has the greater ratio is the lesser. (10. Becasue A is greater than C, and B 5.) Wherefore D is less than B; that is any other magnitude, A has to B a is, B is greater than D. greater ratio than C to B: (8. 5.) Secondly, if A be equal to C, B is equal to D : For A is to B, as C, that is A, to D; B therefore is equal to D. 2 3 (9. 5.) Thirdly, if A be less than C, B shall be less than D: For C is greater than A, and because C is to D, as A is to A BOD ABCD ABCD B, D is greater than B, by the first case; wherefore B is less than D. But, as A is to B so is C to D; there. Therefore, if the first, &, Q. E. D. PROP. XV. THEOR. Magnitudes have the same ratio to one another which their equimultiples have. Let AB be the same multiple of C, ber of the first AG, GH, HB, shall that DE is of F; C is to F, as AB to be equal to the number of the last DE. DK, KL, LE: and because AG, GH, Because AB is the same multiple HB are all equal, and A of C, that DE is of F; there are as that DK, KL, LE, are D many magnitudes in AB equal to C, also equal to one anoas there are in DE equal to F: Let ther; therefore AG is K AB be divided into magnitudes, each to DK, as GH to KL, u! equal to C, viz. AG, GH, HB; and and as HB to LE : DE into magnitudes, each equal to (.7. 5.) And as one of F, viz. DK, KL, LE: Then the num- the antecedents to its B C E consequent, so are all the antecedents equal to C, and DK to F: Therefore, together to all the consequents toge- as C is to F, so is AB to DE. There ther; (12. 5.) wherefore, as AG is to fore magnitudes, &c. Q. E. D DK, so is AB to DE: But AG is PROP. XVI. THEOR. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately. Let the four magnitudes A, B, C, D, E is to F: But as A is to B, so is C be proportionals, viz. as A to B, so Ċ to D: Wherefore as C is to D, so (11. to Do They shall also be proportion- 5.) is E to F: Again, because G, H als when taken alternately; that is, are equimultiples of C, D, as C is to A is to C, as B to D. D, so is G to H; (15. 5.) but as C is Take of A and B any equimultiples to D, so is E to F. Wherefore, as E whatever of E and F; and of Cand is to F, so is G to H. (11. 5.) But, D take any equimultiples whatever G when four magnitudes are propor. and H: and because E is the same tionals, if the first be greater than the multiple of A,, that F is of B, and that the third, the second shall be greater magnitudes have the same ratio to than the fourth; and if equal, equal; one another which their equimultiples if less, less. (14.5.) Wherefore, if É have ; (15. 5.) therefore A is to B, as be greater than G, F likewise is greater than H; and if equal, equal; i less, less; and E, F are any equimulE tiples whatever of AB; and G, H any A C whatever of C, D. T'herefore A is to B D- C, as B to D. (Def. 5.) If then four magnitudes, &c. Q. É. D. F PROP. XVII. THEOR. If magnitudes, taken jointly, be proportionals, they shall also be propor. tionals when taken separately ; that is, if two magnitudes together have to one of them the same ratio which two others huve to one oj these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD, DF be the mag- tiple of AE, that HK is of EB, where. nitudes taken jointly which are pro- fore GH is the same multiple (1. 5.) portionals ; that is, as AB to BE, so of AE, that GK is of AB: But GH is is CD to DF; they shall also be pro- the same multiple of AE, that LM is portionals taken separately, viz. as of CF; wherefore GK is the same AE to EB, so CF to FD. multiple of AB, that LM is of CF: Take of AE, EB, CF, FD any equi. Again, because LM is the same mulmultiples whatever GH, HK, LM, tiple of CF, that MN is of FD); MN, and again, of EB, FD take any therefore LM is the same multiple equimultiples whatever KX, NP: (1. 5.) of CF, that LN is of CD: But And because GH is the same mul- LM was shown to be the same mul tiple of CF, that GK is of AB; GK MP are equimultiples ; if GK be therefore is the same multiple of AB, greater than HX, then LN is greater that LN is of than MP; and if equal, equal; and CD; that is, Х if less, less : (5. Def. 5.) But if GH GK, LN, are be greater than KX, by adding the equimultiples P the common part HK to both, GK is of AB, CD. greater than HX; wherefore also LN Next, because is greater than MP; and by taking HK is the same F away MN from both, LM is greater B multiple of EB than NP: Therefore, if GH be greatthat MN is of E er than KX, LM is greater than FD, and that NP. In like manner it may be deKX is also the monstrated, that if GH be equal to same multiple G A CL KX, LM likewise is equal to NP; of EB, that and if less, less : And GH, LM are NP is of FD; therefore HX is the any equimultiples whatever of AE, same multiple (2. 5.) of EB, that CF, and KX, NP are any whatever MP is of FD. And because AB is to of EB, FD. Therefore, (5. Def. 5.) BE, as CD is to DF, and that of AB as AE is to EB, so is CF to FD. If and CD, GK and LN are equimul- then magnitudes, &c. Q. E. D. tiples, and of EB and FD, HX and PROP. XVIII. THEOR. If magnitudes taken separately be proportionals, they shall also be pro portionals when taken jointly, that is, if the first be to the second, as he third to the fourth, the first and second together shall be to the second, as the third and fourth together to the fourth. Let AE, EB, CF, FD be propor- equimultiples of AB, BE, and that tionals ; that is, as AE to EB, so is AB is greater than BE, therefore GH CF to FD: they shall also be propor- is greater (3. Ax. 5.) ihan HK; but tionals when taken jointly ; that is, KO is not greater than KH, whereas AB to BE so CD to DF. fore GH is greater than KO. Iu like Take of AB, BE, CD, DF any manner it may H equimultiples whatever GH, HK, LM, beshewn, that MN: and again of BE, DF, take any LM is greatwhatever equimultiples KO, NP: er than NP. And because KO, NP are equimul- Therefore, if K tiples of BE, DF; and that KH, NM KO be not B D, N are e quimultiples likewise of BE, DF, greater than if KO, the multiple of BE, be greater KH, then GH, E B than KH, which is a multiple of the the multiple same BE, NP, likewise the multiple of AB, is al C'A' C L of DF shall be greater than MN, the ways greater than Ko, the multiple multiple of the same DF; and if KO of BE; and likewise LM, the mulbe equal to KH, NP shall be equal tiple of CD, greater than NP, the to NM ; and if less, less. multiple of DF, First, let KO not be greater than Next, let KO be greater than KH : KH, therefore NP is not greater than therefore, as has been shown, NP is NM: And because GĂ, HK are greater than MN: and because the M whole GH is the same multiple of the than HO, LN is greater than MP; whole AB, that HK is of BÈ, the re- and if equal, equal ; and if less, (A. 5.) mainder GK is the same multiple of less. the remaind But let HO, MP be equimultiples er AE that of EB, FD; and because AE is to GH is of AB: EB, as CF to FD, and that of AE, H (5. 5.) which P CF are taken equimultiples GK, LN; is the same and of EB, FD, the equimultiples Mt that LM is of Кв. HO, MP; if GK be greater than HO, CD. In like LN is greater than MP; and if equal, manner, beE equal; and if less, less ; (5. Def. 5.) cause LM is which was likewise shown in the prethe same ceding case. If therefore GH be multiple of greater than KO, taking KH from CD, that MN is of DF, the remaind. Both, GK is greater than HO; whereer LN is the same multiple of the re- fore alse LN is greater than MP; mainder CF, that the whole LM is and consequently, adding NM to both, of the whole CD: (5. 5.) But it was LM is greater than NP: Therefore, shown that LM is the same multiple if GH be great Or of CD, that GK is of AE; therefore er than KO, LM GK is the same multiple of AE, that is greater than H н LN is of CF; that is, GK, LN are NP. In like man. M equimultiples of AE, CF: And bea ner it may be KI B cause KO, NP are equimultiples of shewn, that if BE, DF, if from KO, NP there be GH be equal to taken KH, NM, which are likewise KO, LM is equal equimultiples of BE, DF, the remain. to NP; and it O ACL ders HO, MP are either equal to BE, less, less. And DF, or equimultiples of them. (6. 5.) in the case in which KO is not greatFirst, let HO, MP, be equal to BE, er than KH, it has been showu that DF; and because AE is to EB, as GH is always greater than KO, and CF to FD, and that GK, LN are likewise LM than NP: But GH, equimultiples of AE, CF; GK shall LM are any equimultiples of AB, CD, be to EB, as LN to FD: (Cor. 4. 5.) and KO, NP, are any whatever of But HO is equal to EB, and MP tú BE, DF; therefore, (5. Def. 5.) as FD; wherefore GK is to HO, as LN AB is to BE, so is CD to DF. If to MP. If therefore GK be greater then, magnitudes, &c. Q. E. D. PROP. XIX. THEOR. If a whole magnitude be to a whole, as a magnitude taken from the first, is to a magnitude taken from the other; the remainder shall be to the remainder, as the whole to the whole. Let the whole AB be to the whole is to AE, as DC to CD as AE, a magnitude taken from CF: And because, AB, to CF, a magnitude taken from if magnitudes, ta E cr CD; the remainder EB shall be to ken jointly, be prothe remainder FD, as the whole AB portionals, they are to the whole CD. also proportionals E Because AB is to CD, as AE to (17. 5.) when taCF ; likewise, alternately, (16. 5.) ken separately ; therefore, as BE is to DF, so is EA to FC; and alter Cor. If the whole be to the whole, nately, as BE is to EA, so is DF to as a magnitude taken from the first, FC: But, as AE to CF, so by the hy- is to a magnitude taken from the pothesis, is AB to CD; therefore also other ; the remainder likewise is to BE, the remainder, shall be to the re the remainder, as the magnitude taken mainder DF, as the whole AB to the from the first to that taken from the whole CD: Wherefore, if the whole, other : The demonstration is contain. &c. Q. E. D. ed in the preceding. PROP. E. THEOR. If four magnitudes be proportionals, they are also proportionals by corversion, that is, the first is to its excess above the second, as the third to its excess above the fourth. Let AB be to and by inversion, (B. 5.) BE is to BE, as CD to DF; EA, as DF to FC. Wherefore, by then BA is to AE, AT composition, (18. 5.) BA is to AE, as as DC to CF. DC is to CF: If therefore, four, &c. Because AB is Q. E. D. to BE, as CD to DF, by division, B (17. 5.) AE is to EB, as CF to FD; PROP. XX. THEOR. If there be three magnitudes, and other three, which, taken iwo and two, have the same ratio ; if the first be greater than the third, the fourth shall be greater than the sixth ; and if equal, equal ; and i fless, less. Let A. B, C be three magnitudes, F is to E; and D was shewn to have and D, E, F other three, which, ta- to E a greater ratio than C to B; ken two and two, have the same ra- therefore D has to E tio, viz. as A is to B, so is D to E; a greater ratio than and as B to C, so is E to F. If Á F to E: (Cor. 13. 5.) be greater than C, D shall be greater But the magnitude than F; and, if equal, equal ; and, if which has a greater less, less. ratio than another to Авс Because A is greater than C, and the same magnitude, D E F B is any other rnagnitude, and that is the greater of the the greater has to the same magni. two: (10. 5.) Dis tude a greater ratio than the less has therefore greater than to it; (8. 5.) therefore A has to B a F. greater ratio than C has to B: But Secondly, let A be equal to C; D as D is to E, so is A to B ; therefore shall be equal to F: Because A and (13. 5.) D has to E a greater ratio Care equal to one another, A is to B, than C'to B; and because B is to C, as C is to B : (7. 5.) But A is to B, as E to F, by inversion, C is to B, as as D to E; and C is to B, as F to E; |