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BOOK. V.

a I.

b 11.

da.

c Def. 5.

Let the compounded magnitudes AB, BE, CD, DF, be proportional; that is, let AB be to BE as CD is to DF; thefe magnitudes fhall be proportional when divided; that is, AE fhall be to EB as CF is to FD.

For, take GH, HK, LM, MN, equimultiples of AE, EB, CF, FD, and KX NP, any equimultiples of EB, FD; now, be caufe GH is the fame multiple of AE that HK is of EB; and GH the same multiple of AE that LM is of CF; and LM the fame multiple of CF that MN is of FD; therefore GK is the fame multiple of AB that GH is of AE. But GH is the fame multiple of AE that LM is of CF; therefore GK is the fame multiple of AB that LM is of CF. But LM is the fame multiple of CF, that MN is of FD; therefore LN is the same multiple of CD, that LM is of CF2; therefore GK is the fame multiple of AB, that LN is of CD. But HK, MN, are the fame multiples of EB, FD; and KX, NP any other equimultiples of EB, FD; wherefore HX is the fame multiple of EB that MP is of FDd. But GK, LN are equimultiples of AB, CD; and XH, MP, any other equimultiples of EB, FD; if GK be equal to HX, LN will be equal to MP; take HK, MN, from both; then, if GH be equal to KX, LM will be equal to NP; if greater, greater; and, if lefs, lefs; wherefore AE is to EB as CF is to FD. Wherefore, &c.

a 17.

b Hyp.

C II.

d 14.

PROP. XVIII. T HEOR.

IF magnitudes divided be proportional, they shall also be proportional when compounded.

Let AE, EB, CF, FD, be the divided magnitudes, viz. as AE is to EB, fo is CF to FD; they fhall likewife be proportional when compounded, viz. as AB is to BE, fo is CD to DF; if not, let AB be to BE as CD is to fome magnitude, either greater or less than FD; firft, to a lefs, as DG, viz. AB to BE, as CD to DG; therefore AE is to EB as CG is to GD; but AE is to EB as CF is to FD; therefore CG is to GD as CF is to FD; but CG is greater than CF; therefore DG is greater than FDd; but it is alfo lefs, which is impoffible; therefore AB is not to BE as CD to DG. In the fame manner it is proved, that AB is not to BE as CD to one greater than DF; therefore AB is to BE as CD is to DF. Wherefore, &c.

Book V,

THEOR.

PRO P. XIX. THE OR.

IF the whole be to the whole, as a part taken from the one is to a part, taken from the other, then shall the refidue of the one be to the refidue of the other, as the whole is to the whole; and if four magnitudes be proportional, they shall be conversely proportional.

17.

Let the whole AB be to the whole CD, as a part taken away AE, is to the part taken away CF; then the refidue EB is to the refidue FD as the whole AB is to the whole CD; for alternately as AB is to AE, fo is CD to CF2; then BE is to AE as a 16. DF is to FC; and BE is to DF, as AE to CF; but as AE is to CF, fo is AB to CD; therefore EB is to the refidue FD as c Hyp. the whole AB is to the whole CDd: Again, if AB be to BE as d 11. CD to DF, then they fhall be converfely proportional; for AE is to BE as CF is to FD; and BE is to AE as DF is to CFf; f Cor. 4. therefore as AB is to AE, fo is CD to DF8; therefore the first AB is to AE, its excefs above the fecond, as CD, the third, is to DF, its excefs above the fourth. Wherefore, &c.

e 17.

g 18.

h Def. 17.

PRO P. XX. THE OR.

[F there be three magnitudes, and others equal to them in num

fame ratio; and if the first magnitude be equal to the third, then the fourth will be equal to the fixth; and, if the first be greater than the third, then the fourth will be greater than the fixth; and, if the first be less than the third, then the fourth will be less than the fixth.

Let A, B, C be three magnitudes, and D, E, F, others equal to them in number; which being two in two in each order, are in the fame proportion, viz. A to B as D to E, and B to C as E to F; and if the firft A be equal to the third C, then the fourth D fhall be equal to the fixth F; if greater, greater; and, if lefs, lefs; for if A is equal to C, and B fome other magnitude, A has the fame proportion to B that C hath to B; but A is to B a 7. as D is to Eb; therefore D hath the fame proportion to E that b Hyp. A has to B; but B is to C as E to F; and, inversely, C is to Bas Fisto E; therefore F has to E the fame proportion that C

has

Book V. has to B; but A has the fame proportion to B that C has to B; therefore D has the fame proportion to E that F has to E; therefore D is equal to F; if greater, greater; and, if lefs, less. Wherefore, &c.

€ 9.

IG.

PROP. XXI. THEO R.

TF there be three magnitudes, and others equal to them in number, which, taken two and two in each order, are in the fame ratio; and, if the proportion be perturbate; if the first magnitude be greater than the third, then the fourth will be greater than the fixth; and if the first be equal to the third, then the fourth will be equal to the fixth; if lefs, less.

Let the three magnitudes A, B, C, and others D, E, F, equal to them in number, be taken two and two in the fame ratio, and if their analogy be perturbate, viz. as A is to B, fo is E to F, and B to Cas D to E; and if the firft A be greater than the third C, then the fourth D will be greater than the fixth F; if equal, equal; and, if less, less.

For, if A is greater than C, A has a greater ratio to B than C has to B; but A is to B as E is to F; therefore E has to F a greater ratio than C hath to B; and inversely as C is to B, fo is E to D; therefore E has to F a greater ratio than E to D. But that magnitude to which the fame has a greater ratio, is the leffer magnitude b; therefore F is less than D; that is, D is greater than F; if equal, equal; and, if lefs, lefs. Wherefore, &c.

PRO P. XXII. THE OR.

TF there be any number of magnitudes, and others equal to them in number, which, taken two and two, are in the fame ratio; then they fhall be in the fame proportion by equality.

Let there be any number of magnitudes A, B, C, and others, D, E, F, equal to them in number, which, taken two and two in the fame ratio, viz. A to B as D to E, and B to C as E to F; then they fhall be in the fame proportion by equality; that is, A to C, as D to F.

For, let G, H be equimultiples A, D, and K, L any equimultiples of B, E, and M, N any equimultiples of C, F; then,

because A is to B as D to E, G is to K as H is to L ; but B is Book V. to C as E is to F; therefore K is to M as L to N'; wherefore, if G is equal to M, H will be equal to N; if greater, greater; a 4. and, if lefs, lefs; but G, M are equimultiples of A, C and H, b 20. N of D, F; wherefore A is to C as D to F. Wherefore, &c. c Def. 5.

IFt

PROP. XXIII. THEOR.

there be three magnitudes, and others equal to them in number, which, taken two and two, are in the fame ratio; and if their analogy be perturbate, they shall be in the fame proportion by equa lity.

Let there be three magnitudes A, B, C, and others D, E, F, equal to them in number, which, taken two and two, are in the fame ratio; and if their analogy be perturbate, that is, as A is to B, fo is E to F, and as B is to C, fo is D to E; then they fhall be in the fame proportion by equality; that is, A is to C as D to F.

For, let G, H, L be equimultiples of A, B, D; and K, M, N, any equimultiples of C, E, F; then as A is to B, fo is G to H; and as E to F, fo is M to N; but A is to B as E is to F ;a 15. therefore G is to Has M to N; and, because B is to C as Db 11. to E, His to K as L to M; therefore, if G is equal to K, L is equal to N; but G, K are equimultiples of A, C; and L, Nc 21. of D, F; therefore A is to C as D to F. Wherefore, &c.

IF

PRO P. XXIV. T HEOR.

the first magnitude has the fame proportion to the second that the third has to the fourth; and if the fifth has the fame proportion to the fecond that the fixth has to the fourth; then the first, compounded with the fifth, fhall have the fame proportion to the fecond, that the third, compounded with the fixth, has to the fourth.

Let the first magnitude AB, have the fame proportion to the fecond C, that the third DE has to the fourth F, and the fifth BG have the fame proportion to the fecond C, that the fixth EH has to the fourth F.

d Def. s.

For, because BG is to C as EH is to F; inversely, C is to BG, as F is to EH; but AB is to C as DE is to F; therefore AB is to BG, as DE is to EH; and AG is to GB as DH is to a 22? HE; but as GB is to C, fo is EH to F; therefore AG is to b 18. C as DH is to F. Wherefore, &c.

с Нур.

PROP.

BOOK V.

a 19.

PRO P. XXV. THE OR.

IF four magnitudes be proportional, the greates, and least will be greater than the other two.

Let four magnitudes AB, CD, E, F, be proportional, viz. AB to CD as E to F; of which let AB be the greatest, and F the leaft; then AB and F together, will be greater than CD and E; for, cut off AG equal to E, and CH to F; then AB is to CD as AG is to CH; therefore the remainder BG, will be to the remainder DH, as the whole AB is to the whole DC; but AB is greater than CD; therefore GB is greater than HD; and, because AG is equal to E, and CH to F, then AG and F are equal to CH and E; but BG is greater than HD; therefore AB and F are greater than DC and E. Wherefore, &c.

THE

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