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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 to CB, as DF to FE; and DF, FE are given; d. 17. 5. therefore a the ratio of AC to CB is given.

a. 2. Def.

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

another, be added together; the wholé 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 a. 2. Defe which is the same with it; let this be the ratio of the given magnitudes DE, EF. and because DE, EF

А

B С are given, the whole DF is given 6.

b. 3. Dat. and because as AB to BC, so is DE to EF; by composition , AC is to CB,

D E F 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 given a.

2. 18. 5.

d. E. S.

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F 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 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 diyided into the parts AC, CB which have a given rațio to one ano

A

с B ther; if a fourth proportional can be found to the above-named magnitudes; AC and CB are each of them given.

D

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

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

A

CB ven magnitude AB has to BC the given ratio of DE to EF, if unto DE, EF, AB

a fourth proportional can be found, this D F E C. 2. Dat. which is BC is given “; and because AB d. 4. Dat. is given, the other part AC is given d.

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

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AGNITUDES which have given ratios to the

fame magnitude, have also a given ratio to one another.

MA

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 same är 2. Def. to it may be found a; 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 *; 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, fo is
D to E; and as B to C, so is (F to G,
and so is) E to H; ex aequali, as A to
C, fo is D to H. therefore the ratio of A B C D È H
A to C is given a, 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.

11

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F two or more magnitudes have given ratios to one

another, and if they have given ratios, tho they be not the same, to some other magnitudes: these other magnitudes shall also have given ratios to one another.

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

Because the ratio of A to B is given, and likewise the ratio of A to D; therefore the ratio of D to B is given ~; but the ratio of

A

D

a. g. Dat. B to E is given, therefore a the B

Eratio of D to E is given. and be- C cause the ratio of B to C is given, and also the ratio of B to E; the ratio of E to C is given a. and the ratio of C to F is given ; wherefore the ratio of Eto F is given. 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 a given ratio to that other.

I

Let the magnitudes AB, BC have a given ratio to the magnitude D; AC has a given ratio to the fame D.

A Because AB, BC have each of them

B C a given ratio to D, the ratio of AB to BC is given a, and by composition, the

D

a. 9. Dat. ratio of AC to CB is given b. but the

b. 7. Dat. ratio of BC to D is given; therefore the ratio of AC to D is given.

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See N.

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F the whole have to the whole a given ratio, and the

parts have to the parts given, but not the same, ratios, every one of them, whole or part, shall have to every one a given ratio.

b.

Let the whole AB have a given ratio to the whole CD, and the parts AE, EB have given, but not the same, ratios to the parts CF, FD; every one shall have to every one, whole or part, a given ratio.

Because the ratio of AE to CF is given, as AE to CF, so make AB to CG; the ratio therefore of AB to CG is given; wherefore

the ratio of the remainder EB to the remainder FG is given, bea. 19. 5.

cause it is the same with the ratio of AB to CG. and the ratio of EB to FD is given, wherefore the ratio of A E

B FD to FG is given b; and by converc. 6. Dat. fion, the ratio of FD to DG is given c. C F G D and because AB has to each of the

magnitudes CD, CG a given ratio, the ratio of CD to CG is given b; and therefore " the ratio of CD to DG is given. but the ratio of GD

to DF is given, wherefore b the ratio of CD to DF is given, and d. Cor. 6. consequently d the ratio of CF to FD is given; but the ratio of CF

to AE is given, as also the ratio of FD to EB; wherefore e the

ratio of AE to EB is given; as also the ratio of AB to each of f. 7. Dat. them f. the ratio therefore of every one to every one is given.

Dat.

Dat. e. Io. Dat.

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Let A, B, C be three proportional straight lines, that is as A to B, so is B to C; if A has to C a given ratio, A shall also have to B a given ratio.

Because the ratio of A to C is given, a ratio which is the same a. 2. Def. with it may be found a ; let this be the ratio of the given straight d. 13. 6. lines D, E; and between D and E find a mean proportional F;

therefore the rectangle contained by D and E is equal to the square of F, and the rectangle D, E is given because its fides D, E are given; wherefore the square of F, and the straight line F is given. and because as A is to C, fo is D to E; but as A to C, so is the square of A to the square of B; and as D to E, so is the square of D to the square of F; therefore the square d of A is to the square À B C d. 11. 5. of B, as the square of D to the square of F.

DF F E as therefore e the straight line A to the straight line B, so is the straight line D to the straight linę F. therefore the ratio of A to B is given, because the ratio of the given straight lines D, F which is the same with it has been found.

C. 2. Cor.

20. 6.

e. 22. 6.

a. 2. Def.

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F a magnitude together with a given magnitude has a See N.

given ratio to another magnitude; the excess of this other magnitude above a given magnitude has a given ratio to the first magnitude. and if the excess of a magni. tude above a given magnitude has a given ratio to another magnitude; this other magnitude together with a given magnitude has a given ratio to the first magnitude.

Let the magnitude AB together with the given magnitude BE, that is AE, have a given ratio to the magnitude CD; the excess of CD above a given magnitude has a given ratio to AB.

Because the ratio of AE to CD is given, as AE to CD, so make BE to FD; therefore the ratio of BE to FD is given, and BE is given, wherefore FD is given a. and A

B

E a. 2. Dat. because as AE to CD, fo is BE to FD, the remainder AB is o to the C F D remainder CF, as AE to CD. but the ratio of AE to CD is given, therefore the ratio of AB to CF is given; that is, CF the excess of CD above the given magnitude FD has a given ratio to AB.

Next, Let the excess of the magnitude AB above the given magnitude BE, that is, let AE have a given ratio to the magni

b. 19. 5.

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