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fo is C to D; A is to B, as C to D. but A is equal to C, therefore b 11. 5. B is equal to D. the magnitude B is therefore given, because a magnitude D equal to it has been found.

The limitation within the inverted commas is not in the Greek text, but is now neceffarily added; and the fame must be understood in all the Propofitions of the Book which depend upon this fecond Propofition, where it is not exprefly mentioned. See the Note upon it.

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IF

F any given magnitudes be added together, their fum fhall be given.

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

Because AB is given, a magnitude equal to it may be found; a. 1. Def. let this be DE. and because BC is given, one equal to it may be found; let this be

A

EF. wherefore because AB is equal to DE, D

and BC equal to EF; the whole AC is e

B

C

E F

qual to the whole DF. AC is therefore given, because DF has been found, which is equal to it.

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IF

a given magnitude be taken from a given magnitude;
the remaining magnitude fhall be given.

From the given magnitude AB let the given magnitude AC be taken; the remaining magnitude CB is given.

a

Because AB is given, a magnitude equal to it may be found; a. 1. Def.

let this be DE. and becaufe AC is given,
one equal to it may be found; let this be
DF. wherefore because AB is equal to DE,
and AC to DF; the remainder CB is equal
to the remainder FE. CB is therefore given, becaufe FE which is

A

C

B

D

F

E

equal to it has been found.

PROP.

12. See N.

IF

PROP. V.

F of three magnitudes, the first together with the se: cond be given, and alfo the fecond 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 alfo 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 equal, and because AC is

A

B

equal to BD, take away the com

C D

2.

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

4. Dat. the whole AC is given, therefore a

See N.

5.

AE the remainder is given. and be-
caufe EC is equal to BD, by taking

AE

B

CD

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.

IF

PRO P. VI.

Fa magnitude has a given ratio to a part of it; it fhall alfo 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.

2. 2. Def.

b. 4. Dat.

c. E. s.

Because the ratio of AB to AC is given, a ratio may be found which is the fame to it. let this be the ratio of DE a given magnitude to the given magnitude DF. and because DE, DF are given, the remainder FE is given. and because AB is to D AC, as DE to DF, by converfion AB

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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 fame with it has been found.

COR.

COR. From this it follows, that the parts AC, CB have a given ratio to one another. because as AB to BC, fo is DE to EF; by divifion, AC is to CB, as DF to FE; and DF, FE are given; there- d. 17. 5.

a

fore the ratio of AC to CB is given.

PROP. VII.

a. 2. Def.

6.

F two magnitudes which have a given ratio to one ano- See N. ther, be added together; the whole magnitude fhall

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 found a. a. Def. which is the fame with it; let this be the ratio of the given magnitudes

DE, EF. and because DE, DF are gi- A

ven, the whole DF is given b. and be

caufe as AB to BC, fo is DE to EF; by D compofition, AC is to CB, as DF to

B

C

b. 3. Dat.

E F

c. 18. 5.

FE; and by converfion, AC is to AB, as DF to DE. wherefore d. E. s. 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 given a.

IF

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a given magnitude be divided into two parts which See N. have a given ratio to one another, and if a fourth proportional can be found to the fum 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 mag- A

nitudes; AC and CB are each of them

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с в

FE

ven, the ratio of AB to BC is given; therefore a ratio which is a. 7.

the

Dat.

A

C

B

b. 2. Def. the fame with it can be found b, let this be the ratio of the given magnitudes DE, EF. and because the given magnitude AB has to BC the given ratio of DE to EF, if unto DED

EF, AB a fourth proportional can be

FE

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

8.

In the fame 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.

PROP. IX.

MAGNITUDES which have given ratios to the fame

magnitude, have alfo 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 fame 2. Def. to it may be found; 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 fame with it may be found; 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, fo is (F to G,
and fo is) E to H; ex aequali, as A to

C, fo is D to H. therefore the ratio of A B C D E H

A to C is given, because the ratio of
the given magnitudes D and H, which
is the fame with it, has been found. but

if a fourth proportional to F, G, E

FG

cannot be found, then it can only be faid 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 ratios of D to E, and F to G.

PROP.

PROP. X:

F two or more magnitudes have given ratios to one another, and if they have given ratios, tho' they be not he fame, to fome other magnitudes; thefe other magniudes fhall also have given ratios to one another.

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Let two or more magnitudes A, B, C have given ratios to one nother; and let them have given ratios, tho' they be not the fame, › fome other magnitudes D, E, F. the magnitudes D, E, F have ven ratios to one another.

A

D

Because the ratio of A to B is given, and likewise the ratio of
to D; therefore the ratio of
to B is given; but the ratio
B to E is given, therefore B
e ratio of D to E is given. and
cause the ratio of B to C is

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C

E

F

ven, and alfo the ratio of B to E; the ratio of E to C is given
d the ratio of C to F is given; wherefore the ratio of E to F is
ven. D, E, F have therefore given ratios to one another.

PRO P. XI.

F two magnitudes have each of them a given ratio to another magnitude; both of them together fhall have given ratio to that other.

A

Let the magnitudes AB, BC have a given ratio to the magnitude
; AC has a given ratio to the fame D.
Because AB, BC have each of them
given ratio to D, the ratio of AB to
is given". and by compofition, the
io of AC to CB is given b. but the
io of BC to D is given; therefore the ratio of AC to D is

a

D

9.

a. 9. Dat.

22.

в с

a. 9. Dat.

b. 7. Dat.

ren.

PROP.

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