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d Cor. 4. 5.

that GK, LN are equimultiples of AE, CF; GK shall be Book V. to EB, as LN to FDd: But HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP. If therefore GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less.

P

MH

N

D

But let HO, MP be equimultiples of EB, FD; and because AE is to EB, as CF to FD, and that of AE, CF are taken equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LN is greater than MP; and if equal, O equal; and if less, less f; which was likewise shown in the preceding case. If therefore GH be greater than KO, taking KH from both, H GK is greater than HO; wherefore also LN is greater than MP; and consequently adding NM to both, LM is greater than NP: K Therefore, if GH be greater than KO, LM is greater than NP. In like manner it may be shown, that if GH be equal to KO, LM is equal to NP; and if less, less. And in the case in which KO is not greater than KH, it has been shown that GH is always greater than KO, and likewise LM than NP: But GH, LM are any equimultiples of AB, CD, and KO, NP are any whatever of BE, DF; therefore, as AB is to BE, so is CD to DF. If then magnitudes, &c. Q. E. D.

G

B

E

Al

• A. 5.

f 5 Def. 5.

PROP. XIX. THEOR.

If a whole magnitude be to a whole, as a magni- see N. tude 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 CD, as AE, a magnitude taken from AB, to CF, a magnitude taken from CD ; the remainder EB shall be to the remainder ED, as the whole AB to the whole CD.

Because AB is to CD, as AE to CF; likewise, alternate

b

17.5.

BOOK V. lya, BA is to AE, as DC is to CF; and because
if magnitudes, taken jointly, be proportionals,
a 16. 5. they are also proportionals when taken sepa-
rately; therefore, as BE is to EA, so is DF to
FC, and alternately, as BE is to DF, so is EA to E
FC: But, as AE to CF, so by the hypothesis, is
AB to CD: therefore also BE, the remainder,
shall be to the remainder DF, as the whole AB
to the whole CD: Wherefore, if the whole, &c.
Q. E. D.

A

C

F

COR. If the whole be to the whole, as a mag- B D nitude taken from the first, is to a magnitude taken from the other; the remainder likewise is to the remainder; as the magnitude taken from the first to that taken from the other: The demonstration is contained in the preceding.

PROP. E. THEOR.

Ir four magnitudes be proportionals, they are also proportionals by conversion: that is, the first is to its excess above the second, as the third to its excess above the fourth.

Let AB be to BE, as CD to DF; then BA is to AE, as DC to CF.

Because AB is to BE, as CD to DF, by di* 17.5. visiona, AE is to EB, as CF to FD; and by inB. 5. version, BE is to EA, as DF to FC. Where18. 5. fore, by composition, BA is to AE, as DC is to CF: If, therefore, four, &c. Q. E. D.

A

C

F

B D

PROP. XX. THEOR.

See N. Ir there be three magnitudes, and other three, which, taken two 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 if less, less.

Let A, B, C, be three magnitudes, and D, E, Fother three, Book V. which, taken two and two, have the same ratio, viz. as A is to B, so is D to E; and as B to C, so is E to F. If A be greater than C, Dshall be greater than F; and if equal, equal; and if less, less.

A B

E

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C

b 13. 5.

F

Because A is greater than C, and B is any
other magnitude,'and that the greater has to
the same magnitude a greater ratio than the
less has to it; therefore A has to B a greater
ratio than C has to B: But as D is to E, so
is A to B; therefore b D has to E a greater
ratio than C to B; and because B is to C, as J)
E to F, by inversion, C is to B, as F is to E;
and D was shown to have to E a greater ra-
tio than C to B; therefore D has to E a greater
ratio than F to E. But the magnitude which
has a greater ratio than another to the same magnitude, is
the greater of the twod; D is therefore greater than F.
Secondly, Let A be equal to C; D shall be equal to F:

Because A and C are equal to
one another, A is to B, as C is to
Be: But A is to B, as D to E;
and C is to B, as F to E;
wherefore D is to E, as F to Ef;
and therefore D is equal to F5. A
Next, Let A be less than C;
D shall be less than F: For C is D
greater than A, and, as was

shown in the first case, C is to
B, as F to E, and in like man-
ner, B is to A, as E to D; there-
fore F is greater than D, by the

< Cor. 13.5.

d 10.5.

7.5.

f11. 5.

B C

89.3

A B C

E F

DEF

first case; and therefore D is less than F. Therefore, if there be three, &c. Q. E.D.

PROP. XXI. THEOR.

If there be three magnitudes, and other three, See N. which have the same ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

Book V.

Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio, taken two and two, but in a cross order, viz. as A is to B, so is E to F, and as B is to C, so is D to E. If A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

Because A is greater than C, and B is any 28. 5. other magnitude, A has to B a greater ratioa than C has to B: But as E to F, so is A to B : 13. 5. therefore E has to F a greater ratio than C to B: And because B is to C, as D to E, by A B C inversion, C is to B, as E to D: And E was shown to have to F a greater ratio than C to DEF B; therefore E has to F a greater ratio than

Cor. 13. 5. E to Dc; but the magnitude to which the same has a greater ratio than it has to another, is 10. 5. the lesser of the twod: F therefore is less than D; that is, D is greater than F.

Secondly, Let A be equal to C; D shall be equal to F. 7. 5. Because A and C are equal, A ise to B, as C is to B: But A is to B, as E to F; and C is

to B, as E to D; wherefore E 11. 5. is to F, as E to Df; and 8 9. 5. therefore D is equal to F5.

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See N. IF there be any number of magnitudes, and as many other, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words "ex æquali," or " ex æquo."

First, Let there be three magnitudes A, B, C, and as Book V. many others D, E, F, which, taken two and two, have the same ratio: that is, such that A is to B as D to E; and as B is to C, so is E to F; A shall be to C, as D to F.

Take of A and D any equimultiples whatever G and H; and of B and E any equimultiples whatever K and L; and of C and F any whatever M and N: Then because A is to B, as

D to E, and that G, H are equimultiples of A, D, and K, LA equimultiples of B, E; as G is

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HLN 4. 5.

to K, so is a H to L: For the GK M
same reason, K is to M, as L to
N; and because there are three
magnitudes G, K, M, and other
three H, L, N, which, two and
two, have the same ratio; if G
be greater than M, H is greater
than N; and if equal, equal;
and if less, less; and G, H are

b 20.5.

с

any equimultiples whatever of A, D, and M, N are any equimultiples whatever of C, F: Therefore, as A is to C, 5 Def. 5. so is D to F.

A. B. C. D.

Next let there be four magnitudes, A, B, C, D, and other four E, F, G, H, which, two and two, have the same ratio, viz. as A is to B, so is E to F; and as B to C, so F to G; and as C to D, so G to H: A shall be to D, as E to H.'

E. F. G. H.

Because A, B, C, are three magnitudes, and E, F, G, other three, which, taken two and two, have the same ratio; by the foregoing case, A is to C, as E to G: But Cis to D, as G is to H; wherefore again, by the first case, A is to D, as E to H; and so on, whatever be the number of magnitudes. Therefore, if there be any number, &c. Q. E. D.

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