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ber, which being taken two and two in each Order, * 20 of tbis. are in the same Proportion. If G exceeds M, *H

will exceed N; if G be equal to M, then H shall be equal to N; and if G be less than M, H shall be lefs than N. But G, H, are Equimultiples of A, D, and

M, N, any other Equimultiples of C and F. Whence Def . 5. as A is to C, so shall + D be to F. Therefore, if

there be any Number of Magnitudes, and others equal to them in Number, which taken two and two, are in the same Proportion, then they shall be in the same Proportion by Equality; which was to be demonstrated.

of ibis.

PROPOSITION XXIII.

THE O'R EM
If there be three Magnitudes, and others equal to

them in Number, which, taken two and two, are
in the same Proportion ; and if their Analogy be
perturbate, then all they be also in the same

Proportion by Equality.
LET there be three Magis

nitudes A, B, C, and o-
thers equal to them in Num-
ber D, E, F, which, taken
two and two, are in the same
Proportion, and their Analo-
gy be perturbate, that is, as A
is to B, so is E to F; and as A B C D E F
B is to C, so is D to E. I say,
as A is to C, fo is D to F. G H K L M N

For let G, H, L, be Equimultiples of A, B, D, and K, M, N, any Equimultiples of C, E, F.

Then because G, H, are Equimultiples of A and B, and fince Parts have the same

Proportion as their like Multiples when taken corre* 15 of töis. fpondently, it shall be * as A is to B, fo is G to H;

and by the fame Reason, as E is to F, so is M to N. † 11 of rbis. But A is to B as E is to F. Therefore, t as G is to H, fo is M to N. Again, because B is to C, as D

is

2

is to E, and H, L are Equimultiples of B and D, as likewise K, M any Equimultiples of C, E; it Ihall be as H to K, so is L to M. But it has been also proved, that as G is to H, so is M to N. Therefore, because three Magnitudes, G, H, K, and others, L, M, N, equal to them, in Number, which taken two and two are in the same Proportion, and their Analogy is perturbate; then if G exceeds K, also L * will exceed * 21 of tbis,

N; and if G be equal to K, then L will be equal to - N; and if G be less than K, L will likewise be less

than N. But G, L are Equimultiples of A, D; and K, N Equimultiples of C, F. Therefore, as A is to C, so shall D be to F. Wherefore, if there be three Magnitudes

, and others equal to them in Number, which taken two and two are in the fame Proportion; and if their Analogy be perturbate, then all they be also in the same Proportion by Equality; which was to be demonstrated. PROPOSITION XXIV.

THEOREM.
If the first Magnitude has the same Proportion to

the second, as the third to the fourth; and if the
fifth has the same Proportion to the second, as the
fixtb bas to the fourth, then shall the first, com-
1. pounded with the fiftb,
bave the same Proportion

G
to the second, as the third

H
compounded with the sixth
kas to the fourth.

ET the first Magnitude AB

E

the second C, as the third DE has B

to the fourth F. Let also the fifth - BG have the fame Proportion to

the second C as the fixth EH has
to the fourth F. I fay, AG the
first compounded with the fifth,
has the same Proportion to the
second C, as DH the third com-
pounded with the fixth, has to A
the fourth F.

1

F

For because BG is to C, as EH is to F, it shall be (inversely) as C is to BG, fo is F to EH. Then fince AB

is to C; as DE is to F, and as C is to BG, fo is F to 22 of this. EH; it shall be * by Equality as A B is to BG, fo is

DE to EH. And because Magnitudes, being divided, † 18 of this are proportional, they shall also bet proportional when

compounded. Therefore, as AG is to GB, fo is DH Hyp to HE: But as GB is I to C, so also is HE to F.

Wherefore, by Equality *, it shall be as AG is to C, fo is DH to F. Therefore, if the first Magnitude has the fame Proportion to the fécond, as the third to the fourth; and if the fifth has the same Proportion to the second, as the sixth has to the fourth; then shall the first

, compounded with the fifth, have the same Proportion to the second, as the third compounded with the sixth has to the fourth; which was to be demonstrated.

PROPOSITION XXV.

THE ORE M. If four Magnitudes be proportional, the greatest and the least of them, will be greater than the other two.

be Le

propor* tional, whereof AB is to CD, as E is to F; let AB be the greatest of them, and F the least. I say AB, and B F, are greater than CD and E.

For let AG be equal to E,
and CH to F. Then because

G
A B is to CD, as E is to F;
and fince AG, and CH, are D
each equal to E and F, it shall
be as AB is to DCi, fo is AG

H
to CH. And because the whole
AB is to the whole CD, as the
Part taken away AG, is to the

Part taken away CH; it shall * 19 of this. also be * as the Residue GB to

the Residue HD; fo is the A C E F
whole AB to the whole CD.
But AB is greater than CD, therefore also GB shall
be greater than HD. And since AG is equal to E

and

and CH to F, AG and F will be equal to CH and E. But if equal things are added to unequal things, the wholes shall be unequal. Therefore GB, HD being unequal, for GB is the greater: If A G, and F, are added to GB, and CH, and E, to HD; AB and F will necessarily be greater than CD and E. Wherefore, if four Magnitudes be proportional, the greatest, and the least of them,' will be greater than the other two; which was to be demonstrated.

The End of the FIFTH Book,

L

EUCLI D's

148

E U C L I D's

ELEMENTS

BOOK VI.

DÉFINITION S.

'S

1. IMILAR Right-lined Figures, are fuck as have each of their several

Angles equal to one another, and the Sides about the

equal Angles proportional to each other. II. Figures are said to be reciprocal, when the An

tecedent and Consequent Terms of the Ratio's are

in each Figure. III. A Right Line is said to be cut into mean and

extreme Proportion, when the whole is to the greater Segment, as the greater Segment is to the

leser. IV. The Altitude of any Figure, is a perpendicular

Line drawn from the Top, or Vertex to the

Base.
V. Å Ratio is said to be compounded of Ratio's,

when the quantities of the Ratio's being multi-
plied into one another, do produce a Ratio.

PRO

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