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PRO P. X.

If two or more magnitudes have given ratios to one

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

a 9. dat.

Let two or more magnitudes A, B, C have given ratios to one another ; and let ihem have given ratios, though they be not the same, 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 10 D; therefore the ra

A

-D tio of D :o B is given a ; but the ratio of B to E is given, B therefore a the ratio of Dio E is given : And because the C

F ratio of B to C is given, and also the ratio of Bio E ; the ratio of Eto C is given a : And the ratio of C to F is given ; where. fore the ratio of Eco F is given ; D, E, F have therefore given ratios to one another.

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If two magnitudes have each of them a given ratio

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

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

Because AB, BC have each of them a given ratio to D, the ratio A в с of AB to BC is given a : And by com

a g. dat. position, the ratio of AC to CB is D given b: But the ratio of BC to D

bo 7. dat. is given ; therefore a the ratio of AC to D is given.

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PRO P. XII.

See N.

TF 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.

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

Becaule 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 a 19.5. FG is given, because it is the same a with the ratio of AB 10 CG: And the ratio of EB to FD is

A E

B given, wherefore the ratio of FD b dat. to FG is given b; and by conver

fion, the ratio of FD to DG C F G D € 6. dat. given

c: 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 c the ratio of CD 10 DG is

given : But the ratio of GD 10 DF is given, wherefore b the d cor. 6. ratio of CD to DF is given, and confequently d the ratio of CF

to FD is given, but the ratio of CF to AE is given, as also the 4 10. dat. ratio of FD to EB; wherefore e the ratio of AE to EB is given; $7. dat. as also the ratio of AB to each of them f: The ratio therefore

of every one to every one is given.

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See N.

If the first of three proportional straight lines has a

given ratio to the third, the first shall also have a given ratio to the second.

Let A, B, C be three proportional fraight lines, that is, as A to B, to is B to C; if A has to C a given ratio, A Mall also have to B a given racio.

Becauíe the ratio of A to C is given, a ratio which is the 9 .. dcf. same with it may be found a; let this be the ratio of the gi13. $. yen staight lines D, E; and between D and E find a b mean

proportional

+ 2 cor. 20. 6.

proportional F; therefore the rectangle contained by D and
E is equal to the square of F, and the rect-
angle 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, so is D to E; but as A (oC, so is c the
square of A to the square of B; and as D to
E fo is c the square of D to the Aquare of F; A B C
therefore the square o of A is to the square of
B, as the square of D to the square of F: D F E
As therefore e the itraight line A to the
Straight line B, fo is the firaight line D to the
Atraight line F: Therefore the ratio of A to B
is given a, because the ratio of the given
Straight lines D, F which is the same with it
has been found.

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PRO P. XIV.

A.

Sce N.

Fa

a magnitude together with a given magnitude las

a given ratio to another magnitude; the excess of this other magnitude above a given inagnitude has a given ratio to the first magnitude: And if the excess of a magnitude 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 2:

A And because as AE to CD, so is BE

Β Ε

2 2. dat. to FD, the remainder AB is b to the remainder CF, as AE to CD : But the C F D 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 mag

gitude

b 19.5.

A 24

nitude CD; CD together with a given magnitude has a given ratio 1o AB.

Because the ratio of AB to CD is given, as AE 10 CD, fo make BE to FD ; therefore the ratio of A

E B BE to FD given, and BE is give! 1 2. dat. wherefore FD is given a : And becaufe

as E 1o CD, 10 is BĘ to FD, AB C D F € 19. 5. to CF, asc AE 10 CD): But the ratio

of AŁ to CD is given, theretore the ratio of AB !o UF is gi. yen; that is, CF which is equal to CD together with the given magnitude DF has a given ratio to AB.

B.

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PRO P. XV. See N. IF a magnitude together with that to which another

magnitude has a given ratio, be given; the sum of this other, and that to which the first magnitude has a given ratio is given.

Let AB, CD be two magnitudes of which AB together with BE to which CD has a given ratio, is given; CD is given 103e: ther with that magnitude to which AB has a given ratio.

Becaule the ratio of CD to BE is given, as PE to CD, fo

make AE FD; therefore the ratio of AE to FD is given, ? 7. dat. and AE is given, wherefore a FD

А

B
is given : And because as BŁ to
Cor. 19. CD, fo is AE to FD: AB is b to
FC, as BE to CD: And the ratio

F Ç_D
of BE to CD is given, wherefore
the ratio of AB to FC is given: And FD is given, that is, CD
together with FC to which AB has a given ratio is given.

PRO P. XVI.
See N.

IF
F the excess of a magnitude above a given magni.

tide, has a given ratio to another magnitude; the excess of both together above a given magnitude Muall have to that other a given ratio : And if the excels of two magnitudes together above a given magnitude, has to one of them a given ratio ; either the excels of the other above a given magnitude has to that area given ratio; or tlie other is given together with The magnitude to which that one has a given ra.

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Les

Let the excess of the magnitude AB above a given magni. tude, have a given ratio to the magnitude BC; the excess of AC, both of them together, above the given magnitude, has a given

rario to BC. Let AD be the given magnitude, the excess of AB above which, viz. DB has a given ratio to BC: And because DB has a gi- A DB С ven ratio to BC, the ratio of DC 10. CB is given a, and AD is given; therefore DC, the excets of a 7. dat. AC above the given magnitude AD, has a given ratio to BC.

Next, let the excess of two magnitudes AB, BC together, above a given magnitude, have to

A one of them BC a given ratio ; ei.

D BEC ther the excess of the other of them AB above the given magnitude shall have to BC a given ratio; or AB is given, together with the magnitude to which BC has a given ratio.

Let AD be the given magnitude, and first let it be lefs than AB; and because DC the excess of AC above AD has a given ratio to BC, DB has b a given ratio to BC; that is, DB, b Cor. 6. the excess of AB above the given magnitude AD, has a given dat. ratio to BC.

But let the given magnitude be greater than AB, and make AE equal to it; and because EC, the excess of AC above AE, has to BC a given ratio, BC has ç a given ratio to BL; and be-c 6. dat cause AE is given, AB together with Be to which BC has a given ratio, is given.

PRO P. XVII.

IF the excess of a magnitude above a given magnitude Sec N.

has a given ratio to another magnitude; the excess of the fame first magnitude above a given magnitude, fhall have a given ratio to both the magnitudes together : And if the excess of either of two magnitudes above a given magnitude has a given ratio to both magnitudes together; the excess of the same above a given magnitude shall have a given ratio to the other.

Let the excess of the magnitude AB above a given magnitude have a given ratio to the magnitude BC; the excels of AB above a givep magnitude has a given ratio to AC.

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