this. lefs. But G does exceed K: Therefore M will alfo exceed N. But H does not exceed L. And M and H are Equimultiples of A and E; and N and L any *Df. 7. ofothers of B and F. Therefore A has a greater Proportion to B, than E has to F. Wherefore, if the firft bas the fame Proportion to the second, as the third to the fourth; and if the third has a greater Proportion to the fourth, than the fifth to the fixth; then, aljo, fhall the firft have a greater Proportion to the fecond, than the fifth bas to the fixth; which was to be demonftrated. PROPOSITION XIV. If the first has the fame Proportion to the second, L ET the firft A have the fame Proportion to the fecond B, as the third C has to the fourth D'; and let A be greater than C. I fay, B is also greater than D. For, becaufe A is greater than C, and B is any other Magnitude; A will have a 8 of it. * greater Proportion to B, than C has to B: But as A is to B, fo is C to D; there13 of bis fore, alfo, C fhall I have a greater Proportion to D, than Chath to B But that Magnitude to which the fame bears 10 of this a greater Proportion, is + the leffer Magnitude. Wherefore D is lefs than B; and confequently B will be greater than D. In like manner we demonftrate, if A be equal to C, that B will be equal to D ; and if A be lefs than C, that B will be lefs than D. Therefore, if the firft has the fame Proportion to the fecond, as the third has to the fourth; and if the first be greater than the third; then will the fecond be greater tban the fourth. But if the first be equal to the third, AB then then the fecond fhall be equal to the fourth; and if the firft be less than the third, then the fecond will be less than the fourth; which was to be demonftrated. PROPOSITION XV. THEOREM. Parts have the fame Proportion as their like L A G+ D ET AB be the fame Multiple of C, as D E is of F. I fay, as C is to F, fo is A B to DE. For, because A B and D E are Equimultiples of C and F, there fhall be as many Magnitudes equal to C in A B, as there are Magnitudes equal to F in DE. Now, let A B be divided into the Magnitudes AG, GH, HB, each equal to C; and ED into the Magnitudes DK, KL, LE, each equal to F; then the Number of the Magnitudes A G, GH, HB, will be equal to the Number of the Magnitudes D K, KL, LE. Now, because AG, K+ H+ L+ BC EF GH, H B, are equal, as likewife DK, KL, LE; it fhall be *, as AG is to DK, fo is GH to KL, and 7 of this. fo is HB to LE. But as one of the Antecedents is to one of the Confequents, fo + all the Antecedents tot 12 of this. all the Confequents. Therefore, as A G is to DK, fo is A B to D E. But A G is equal to C, and D K to F. Whence, as C is to F, fo fhall A B be to D E. Therefore, Parts have the fame Proportion as their like Multiples, if taken correfpondently; which was to be demonftrated, PRO PROPOSITION XVI. THEOREM. If four Magnitudes of the fame Kind are proportional, they shall also be alternately proportional. LE ET four Magnitudes A, B, C, D, be proportional; whereof A is to B, as C is to D. I fay, likewife, that they will be alternately proportional; viz. as A is to C, fo is B to D: For take E and F, Equimultiples of A and B; and G and H, E B Then, because F Eis the fame Mul G C D H tiple of A, as F is of B, and Parts have the fame Pro15 of this. portion to their like Multiples, if taken correfpondently; it fhall be, as A is to B, fo is E to F. But as A is to B, fo is C to D. Therefore, alfo, as C is to 11 of this. D, fot is E to F. Again, because G and H are Equimultiples of C and D, and Parts have the fame Proportion with their like Multiples, if taken correfpondently, it will be, as C is to D, fo is G to H; but as C is to D, fo is E to F. Therefore, alfo, as E is to F, fo is G to H; and if four Magnitudes be proportional, and the first greater than the third, then the tecond 14 of this. will be greater than the fourth; and if the first be equal to the third, the fecond will be equal to the fourth; and if lefs, lefs. Therefore, if E exceeds G, F will exceed H; and if E be equal to G, F will be equal to H; and if lefs, lefs. But E and F are any Equimultiples of A and B ; and G and H, any Equimultiples of C and D. Whence as A is to C, fo fhall * Def. 5. B be* to D. Therefore, if four Magnitudes of the fame Kind are proportional, they shall also be alternately proportional; which was to be demonftrated. PRO. PROPOSITION XVII. THEOREM. If Magnitudes compounded are proportional, they fball also be proportional when divided. L ET the compounded Magnitudes A B, BE, CD, DF, be proportional; that is, let AB be to BE, as CD is to DF. I fay, thefe Magnitudes divided are proportional; viz. as A E is to EB, fo is C F to F D. For let GH, HK, LM, and MN, be Equimultiples of A E, EB, CF, and FD; and KX, and N P, any Equimultiples of EB and FD. Becaufe GH is the fame Multiple of A E, as HK is of EB; therefore GH* is the fame Multiple of A E, as G K is of AB. But GH is the fame Multiple of A E, as L M is of CF. Wherefore G K is the fame Multiple of A B, as X K+ P B N +1 of this D H+ E F+ M+ A L A-+ LM is of CF. Again, becaufe LM is the fame Multiple of CF, as M N is of FD, LM will be the fame Multiple of CF, as LN is of CD. Therefore GK is the fame Multiple of A B, as LN is of CD. And fo GK and LN will be Equimultiples of AB and CD. Again, because H K is the fame Multiple of EB, as MÑ is of FD; as likewife K X the fame Multiple of E B, as NP is of FD; the compounded Magnitude H X is + also the fame Multiple of E B, as † 2 of this. MP is of FD. Wherefore, fince it is, as A B is to BE, fo is CD to DF; and G K and LN are Equimultiples of A B and C D; and alfo H X and MP any Equimultiples of EB and FD; if G K exceeds HX, then LN will exceed MP; and if GK bet Def. 5. equal to H X, then LN will be equal to MP; if lefs, lefs. Now let G K exceed H X ; then, if H K, which is common, be taken away, G H fhall exceed K X. But when G K exceeds H X, then L N exceeds MP; therefore L N does exceed MP. If MN, which is common, common, be taken away, then L M will exceed N P. And fo, if G H exceeds K X, then LM will exceed NP. In like manner we demonftrate, if GH be equal to K X, that L M will be equal to N P ; and if lefs, lefs. But GH and L M are Equimultiples of AE and CF; and K X and N P are any EquimulDef. 5. tiples of E B and FD. Whence, as AE is to E B, fo is CF to FD. Therefore, if Magnitudes compounded are proportional, they shall also be proportional when divided; which was to be demonftrated. PROPOSITION XVIII. THEOREM. If Magnitudes divided be proportional, the fame alfo being compounded, fhall be proportional. LET the divided proportional Magnitudes be A E EB, CF, FD; that is, as A E is to E B, fo is For, if A B be not to B E, as CD is to Firft, Let it be to a leffer, viz. to GD. 17 of this. portional; and confequently they will A C E+ F+ G B D be proportional when divided. Therefore A E is to EB, as CG is to GD. But (by the Hyp.) as A E is to EB, fo is CF to FD. Wherefore, alfo, as 11 of this. CG is to GD, fo + is CF to FD. But the firft CG is greater than the third CF; therefore the second 14 of this. DG fhall be † greater than the fourth DF. But it is lefs, which is abfurd. Therefore A B is not to BE, as CD is to DG. We demonftrate in the fame manner, that AB to BE is not as CD to a greater than D F. Therefore A B to BE muft neceflarily be as CD is to DF. And fo, if Magnitudes divided be proportional, they will also be proportional when com◄ pounded; which was to be demonstrated. PRO |