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Co is C to D; A is o to B, as C to D. but A is equal to C, therefore b 11. $. B is equal to D. the magnitude B is therefore given", because a c. 14. 5. 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 fame must be understood in all the Propofitions of the Book which depend upon this second Proposition, where it is not expresiy mentioned. See the Note

pon it.

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PRO P. III.
1 F any given magnitudes be added together, their sum

shall be given.

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

Because AB is given, a magnitude equal to it may be found * ; 4. 1. Def. let this be DE. and because BC is given,

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pre equal to it may be found; let this be
EF. wherefore because AB is equal to DE, D E F F
and BC equal to EF; the whole AC is e-
qual to the whole DF. AC is therefore given, because DF has been
found, which is equal to it.

А

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IF 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 found ; a. 1. Def. let this be DE. and because AC is given,

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A

B
one equal to it may be found; let this be
DF, wherefore because AB is equal to DE,

D

F E and AC to DF; the remainder CB is equal to the remainder FE, CB is therefore given, because FE which is equal to it has been found.

PROP.

12. See N.

PRO P. V.
IF of three magnitudes, the first together with the se:

cond 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 cqual 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 common 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 BR is given, and a. 4. Dat. the whole À C is given, therefore a AE the reinainder is given. and be. AĘ B C

오D caufe 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.

C D

5. See N.

2. 2. Def,

PRO P. VI.
IF
Fa 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 found. which is the same to it. let this be the ratio of DE a given magnitude to the given magnitude DF. and

A

C B because DE, DF are given, the remainb. 4. Dat. der FE is b given. and because AB is to

D F E c. E. s. AC, as 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.

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 divilion ", AC is to CB, as DF to FE; and DF, FE are given; there- d. 17. s. fore the ratio of AC to CB is given.

a, 2. Def.

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IF.
F two magnitudes which have a given ratio to one ano. See N.

ther, 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 found a. 2. Def. which is the same 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

b. 3. Dat. caufe as AB to BC, so is DE to EF; by D composition “, AC is to CB, as DF to

C, 18. s. FE; and by conversion 4, 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 cach of the magnitudes AB, BC is given’. .

PRO P. VIII.

7. IF F a given magnitude be divided into two parts which Sec N.

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 mag

A c B nitudes; AC and CB are each of them given.

F E Because the ratio of AC to CB is given, the ratio of AB to BC is given; therefore a ratio which is a. 7. Dat.

the

B

b. 2. Def. the same with it can be found b, let this be the ratio of the girea magnitudes DE, EF. and because the

A
given magnitude AB has to BC the gi-
ven ratio of DE to EF, if unto DE,D F E

EF, AB a fourth proportional can be C6 2. Dat. found, this which is BC is given; and because AB is given the d. 4. Dat. other part AC is given d.

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

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MA

AGNITUDEs which have given ratios to the same

magnitude, have also a given rațio to one another.

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

Because the ratio of A to B is given, a ratio which is the face a 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 same with it may be founda; let this be the ratio of the girta
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 so is) E to H; ex aequali, as A to
C, so is D to H. therefore the ratio of A B C D E H
A to C is given“, because the ratio of

F G
the given magnitudes D and H, which
is the same with it, has been found. but
if a 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 ratios of D to E, and F to G.

PROP.

9
.

PROP. X.
Ftwo or more magnitudes have given ratios to one ano-

ther, and if they have given ratios, tho' they be not he same, to some other magnitudes; these other magniodes shall also have given ratios to one another.

Let two or more magnitudes A, B, C have given ratios to one lother; and let them have given ratios, tho' they be not the same, , some other magnitudes D, E, F. the magnitudes D, E, F have ven ratios to one another. Because the ratio of A to B is given, and likewise the ratio of to D; therefore the ratio of

A

D

a. g. Dat. to B is given"; but the ratio B to E is given, therefore * B

Ee ratio of D to E is given. and

С

F cause the ratio of B to C is ven, and also 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 Eto F is ren. D, E, F have therefore given ratios to one another.

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F two magnitudes have each of them a given ratio to

another magnitude; both of them together shall have ziven ratio to that other.

B

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

A given ratio to D, the ratio of AB to is given". and by composition, the

D o of AC to CB is given b. but the o of BC to D is given ; therefore the ratio of AC to D is en.

a. 9. Dat. b. 7. Dat.

PROP.

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