* 1.

X.

A magnitude is said to be less than another by a given magnitude, when this given magnitude being added to it, the whole is equal to the other magnitude.

PROPOSITION I.

See N. THE ratio of given magnitudes to one another is given.

Let A, B, be two given magnitudes, the ratio of A to B is given.

Because A is a given magnitude, there *1 Def. may be found one equal to it; let this be Dat. C: And because B is given, one equal to

it may be found; let it be D: And since 7.5. A is equal to C, and B to D: therefore b A is to B, as C to D; and consequently the ratio of A to B is given, because the ratio of the given magnitudes C, D, which is the same with it, has been found.

A B C D

See N. IF a given magnitude has a given ratio to another magnitude," and if unto the two magnitudes by "which the given ratio is exhibited, and the given "magnitude, a fourth proportional can be found;" the other magnitude is given.

Let the given magnitude A have a given ratio to the magnitude B: if a fourth proportional can be found to the three magnitudes above-named, B is given in magnitude. Because A is given, a magnitude may be

1 Def. found equal to ita; let this be C: And be

E F

cause the ratio of A to B is given, a ratio which is the same with it may be found; A B C D let this be the ratio of the given magnitude E to the given magnitude F: -Unto the magnitudes E, F, C, find a fourth propor-. tional D, which, by the hypothesis, can be donc. Wherefore, because A is to B, as E 11. 5. to F; and as E to F, so is C to D; A is b

*The figures in the margin show the number of propositions in the other

editions.

с

to B, as C to D. But A is equal to C: therefore B is 14. 5. equal to D. The magnitude B is therefore given", because * 1 Def. a magnitude D equal to it has been found.

The limitation within the inverted commas is not in the Greek text, but is now necessarily added; and the same must be understood in all the propositions of the book which depend upon this second proposition, where it is not expressly mentioned. See the note upon it.

PROP. III.

If any given magnitudes be added together, their sum shall be given.

Let any given magnitudes AB, BC, be added together, their sum AC is given.

3.

Because AB is given, a magnitude equal to it maya be *1 Def.

found; let this be DE: And because

BC is given, one equal to it may be

D

E

F

found; let this be EF: Wherefore
because AB is equal to DE, and BC
equal to EF; the whole AC is equal to the whole DF; AC
is therefore given, because DF has been found which is
equal to it.

PROP. IV.

Ir a given magnitude be taken from a given magnitude; the remaining magnitude shall be given.

From the given magnitude AB, let the given magnitude AC be taken the remaining magnitude CB is given. Because AB is given, a magnitude equal to it may be "1 Def.

a

found; let this be DF: And

because AC is given, one

equal to it may be found; let

this be DE: Wherefore, be- D

E

F

cause AB is equal to DF, and

AC to DE; the remainder CB is equal to the remainder FE. CB is therefore given, because FE which is equal to it has been found.

See N. IF of three magnitudes, the first together with the second be given, and also the second together with the third; either the first is equal to the third, or one of them is greater than the other by a given magnitude.

Let AB, BC, CD, be three magnitudes, of which AB together with BC, that is, AC, is given; and also BC together with CD, that is, BD, is given. Either AB is equal to CD, or one of them is greater than the other by a given magnitude.

Because AC, BD, are each of them given, they are either equal to one another, or not

equal. First, let them be

A B

C D

equal, and because AC is equal to BD, take away the com-
mon part BC; therefore the remainder AB is equal to the
remainder CD.

But if they be unequal, let AC be greater than BD, and
make CE equal to BD. Therefore CE is given, because
BD is given. And the whole
A E BC

4 Dat. AC is given; therefore a AE

D

the remainder is given. And because EC is equal to BD, by taking BC from both, the remainder EB is equal to the remainder CD. And AE is given; wherefore AB exceeds EB, that is, CD, by the given magnitude AE.

PROP. VI.

See N. IF a magnitude has a given ratio to a part of it, it shall also have a given ratio to the remaining part of it.

Let the magnitude AB have a given ratio to AC a part of it; it has also a given ratio to the remainder BC.

Because the ratio of AB to AC is given, a ratio may be

* 2 Def. founda which is the same to it: Let this be the ratio of DE, a given magnitude to the given magnitude DF. And because DE,

с

4 Dat. DF, are given, the remainder FE is b

given And because AB is to AC, as

:

A

D

C

B

E. 5. DE to DF, by conversion AB is to BC, as DE to EF. Therefore the ratio of AB to BC is given, because the ratio of the given magnitudes DE, EF, which is the same with it, has been found,

COR. From this it follows, that the parts AC, CB, have a given ratio to one another: Because as AB to BC, so is DE

to EF; by division, AC is tó CB, as DF to FE; and DF, 17. 5. FE, are given; therefore the ratio of AC to CB is given. * 2 Def.

a

a

If two magnitudes which have a given ratio to one See N. another be added together; the whole magnitude shall have to each of them a given ratio.

Let the magnitudes AB, BC, which have a given ratio. to one another, be added together: the whole AC has to each of the magnitudes AB, BC, a given ratio.

Because the ratio of AB to BC is given, a ratio may be founda which is the same with it; let this be the ratio ofa 2 Dei.

B C

b3 Dat.

E F

the given magnitudes DE, EF:
And because DE, EF, are given,
the whole DF is given: And
D
because as AB to BC, so is DE
to EF; by composition AC is to CB as DF to FE; and,
by conversion, AC is to AB, as DF to DE: Wherefore
because AC is to each of the magnitudes AB, BC, as DF
to each of the others DE, EF; the ratio of AC to each of
the magnitudes AB, BC, is givena.

Ir a given magnitude be divided into two parts See N. which have a given ratio to one another, and if a fourth proportional can be found to the sum of the two magnitudes by which the given ratio is exhibited, one of them, and the given magnitude; each of the parts is given.

Let the given magnitude AB be divided into the parts AC, CB, which have a given ratio to one another; if a fourth proportional can be found to the above-named magnitudes; AC D

and CB are each of them given.

FE

Because the ratio of AC to CB is given, the ratio of AB

to BC is given, therefore a ratio, which is the same with a 7 Dat.

2 Def. it can be found; let this be the ratio of the given magni

tudes, DE, EF: And because

Ç B

BC the given ratio of DE to

EF, if unto DE, EF, AB, a
fourth proportional can be

2 Dat. found, this which is BC is given; and because AB is à 4 Dat. given, the other part AC is given.

In the same manner, and with the like limitation, if the difference AC of two magnitudes AB, BC, which have a given ratio be given; each of the magnitudes AB, BC, is given.

MAGNITUDES which have given ratios to the same
magnitude, have also a given ratio to one another.

Let A, C, have each of them a given ratio to B; A has
a given ratio to C.

Because the ratio of A to B is given, a ratio which is the 2 Def. same to it may be founda; let this be the ratio of the given magnitudes D, E: And because the ratio of B to C is given, a ratio which is the same with it may be found a: let this be the ratio of the given magnitudes F, G: To F, G, E, find a fourth proportional H, if it can be done; and because as A is to B, so is D to E; and as B to C, so is (F to G, and so is) E

to H; ex æquali, as A to C, so is A B C DE H

D to H: Therefore the ratio of
A to C is given", because the ra-
tio of the given magnitudes D
and H, which is the same with
it, has been found: But if a

F G

19

fourth proportional to F, G, E, cannot be found, then it
can only be said that the ratio of A to C is compounded of
the ratios of A to B, and B to C, that is, of the given ra-
tios of D to E, and F to G.