that GK, LN are equimultiples of AE, CF; GK fhall be to Book V. EB, as LN to FD: But HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP. If therefore GK bed Cor. 4. s. greater than HO, LN is greater than MP; and if equal, equal; and if lefs, lefs.

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 e quimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LN is greater than MP; and if equal, equal; and if lefs, lefs f; which was likewife fhewn in the preceeding cafe. If therefore GH be greater than KO, H taking KH from both, GK is greater than HO; wherefore alfo LN is greater than MP; and confequently, adding NM to both, LM is greater K than NP: Therefore, if GH be greater than KO, LM is greater than NP. In like manner it may be fhewn,

P

M

B

N

D

E

A

L

that if GH be equal to KO, LM is
equal to NP; and if lefs, lefs. And
in the cafe in which KO is not great-
er than KH, it has been fhewn that
GH is always greater than KO, and likewife LM than NP:
But GH, LM are any equimultiples of AB, CD, and KO,
NP are any whatever of BE, DF; therefore f as AB is to BE,
fo is CD to DF. If then magnitudes, &c. Q. E. D.

e Ax. 5v

f 5. def. 5

PROP. XIX. THE OR.

I a whole magnitude be to a whole, as a magnitude Se N taken from the firft is to a magnitude taken from the other; the remainder fhall 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 fhall be to the remainder FD, as the whole AB to the whole CD.

Because AB is to CD, as AE to CF; likewife, alternately, a 16.

BA

b 17.5.

Book V. BA is to AE, as DC to CF: And because, if mag-
W nitudes taken jointly be proportionals, they are
alfo proportionals b when taken feparately; there-
fore as BE is to EA, fo is DF to FC; and alter-
nately, as BE is to DF, fo is EA to FC: But as E
AE to CF, fo, by the hypothefis, is AB to CD;
therefore alfo BE the remainder fhall be to the re-
mainder DF, as the whole AB to the whole CD:
Wherefore, if the whole, &c. Q. E. D.

F

C

B D

COR. If the whole be to the whole, as a magnitude taken from the firft is to a magnitude taken from the other; the remainder likewife is to the remainder, as the magnitude taken from the first to that taken from the other: The demonftration is contained in the preceeding.

a 17. 5. b B. 5.

c 18. 5.

PROP. E. THEOR.

IF four magnitudes be proportionals, they are alfo proportionals by converfion, that is, the firft is to its excefs above the second, as the third to its excefs above the fourth.

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

Becaufe AB is to BE, as CD to DF, by divi- E
fion, AE is to EB, as CF to FD; and by in-
verfion, BE is to EA, as DF to FC. Wherefore,
by compofition, BA is to AE, as DC is to CF:
If, therefore, four, &c. Q. E. D.

F

Se N.

B D

PROP. XX. THEOR.

IF there be three magnitudes, and other three, which

taken two and two have the fame ratio; if the firfl be greater than the third, the fourth fhall be greater than the fixth; and if equal, equal; and if lefs, lels.

Let

Let A, B, C be three magnitudes, and D, E, F other Book V. three, which taken two and two have the fame ratio, viz. as

A is to B, fo is D to E; and as B to C, fo is
E to F. If A be greater than C, D fhall be
greater than F; and if equal, equal; and if lefs,
lefs.

D

E

a 8.56

C

b 13.5.

F

Because A is greater than C, and B is any
other magnitude, and that the greater has to
the fame magnitude a greater ratio than the less
has to it; therefore A has to B a greater ra-
tio than Chas to B: But as D is to E, fo is A A B
to B; therefore D has to E a greater ratio
than C to B: And because B is to C, as E to F,
by inverfion, C is to B, as F is to E; and D
was fhewn to have to E a greater ratio 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 fame magnitude, is the greater of the
two: D is therefore greater than F.

Secondly, Let A be equal to C; D fhall be equal to F: Be

cause A and C are equal to one an-
other, 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 Fe.

A B C

Next, Let A be lefs than C; D
fhall be lefs than F: For C is great-DEF

er than A, and, as was fhewn in
the first cafe, C is to B, as F to E,
and in like manner B is to A, as E

to D; therefore F is greater than
D, by the firft cafe; and therefore

A B C
DEF

Dis lefs than F. Therefore, if there be three, &c. Q. E. D.

c Cor. 13.5.

d 10. 5.

e 7.5.

f II. 5.

8 9. 5.

1

I'

there be three magnitudes, and other three, which have the fame ratio taken two and two, but in a crofs order; if the firft magnitude be greater than the third, the fourth fhall be greater than the fixth; and if equal, equal; and if lefs, lefs.

Book V.

a 8. 5.

b 13. 5.

Let A, B, C be three magnitudes, and D, E, F other three, which have the fame ratio, taken two and two, but in a cross order, viz, as A is to B, fo is E to F,

and as B is to C, fo is D to E. If A be great-
er than C, D fhall be greater than F; and if e-
qual, equal; and if lefs, lefs.

a

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio 2 than C has to B: But as E to F, fo is A to B ; therefore E has to F a greater ratio than C to B: And becaufe B is to C, as D to E, by inverfion, C is to B, as E to D: And E was fhewn to have to F a greater ratio than C to B; therec Cor. 13.5. fore E has to F a greater ratio than E to Dc; but the magnitude to which the fame has a greater ratio than it has to another, is the leffer of the two; F therefore is less than D; that is, D is greater than F.

d 10. 5.

7. 5.

f II. 5.

8 9.5.

See N.

e

A B C
DEF

Secondly, Let A be equal to C; D fhall be equal to F. Because A and C are equal, A is 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 is to F as E
to Df; and therefore D is equal
to F 8.

Next, Let A be less than C;

D fhall be lefs than F: For Cis A B C
greater than A, and, as was

fhewn, C is to B, as E to D, DEF

and in like manner B is to A,
as F to E; therefore F is greater
than D, by cafe firft; and
therefore D is lefs than F.
Therefore, if there be three,
&c. Q. E. D.

PROP. XXII.

A B

DE F

THEOR.'

IF there be any number of magnitudes, and as many

others, which taken two and two in order have the fame ratio; the firft fhall have to the last of the first magnitudes the fame ratio which the firft of the others, has to the laft. N. B. This is ufually cited by the words, "ex aequali," or "ex aequo.""

First, Let there be three magnitudes A, B, C, and as ma- Book V. ny others D, E, F, which taken two and two have the fame ratio, that is, fuch that A is to B, as D to E; and as B is to

C, fo is E to F; A fhall be to C, as D to F.

A B C

G KM

D E F
HL N 4.5.

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
Fany 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, Lequimultiples of B, E;
as G is to K, fo is H to L: For
the fame reafon 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 fame ratio; if G be
greater than M, H is greater than
Ñ; and if equal, equal; and if
lefs, lefs: And G, H are any e-
quimultiples whatever of A, D,
and M, N are any equimultiples

whatever of C, F: Therefore as A is to C, fo is D to F.
Next, Let there be four magnitudes A, B, C, D, and other
four E, F, G, H, which two and two have the

fame ratio, viz. as A is to B, fo is E to F;
and as B to C, fo F to G; and as C to D,
fo G to A: H fhall be to D, as E to H.

A. B. C. D.

E. F. G. H.

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

b 20. 5.

c 5. def. 5.