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14. Inverse ratio is the assumption of the con'eq ients, as the antecedent to the antecedent taken as th: conse.

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15. Composition of ratio is an assumption o' the ante. cedent, together with the consequent as one to the fame consequent !

16. Division of ratio is the assumption of the excess, whereby the antecedent exceeds the consequent, to the fame consequent m.

17 Converse ratio, is the assumption of the antecedent to the excess whereby the antecedent exceeds the consequent

18. Ratio of equality is, when there are several magnitudes in one rank or order, and as many others in another rank or order, comparing two to two, being in the fame ratio, it shall be in the first rank as the first nignitude is to the last, so in the second rank shall the first magnitude be to the last. OR/ ELSE it is the assumption of the extremes by taking away the intermediate terms

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between a line and a number, as is evident from the fifth definition of this fifth book. But in the next following ways of arguing, viz. by inverse ratio, composition, division, and conversion, the two first magnitudes may be of one kind, and the two last of another, as is evident by the demonftrations of this book.

k Let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude D. Then inversely [by cor. 4. 5.] as B is to A, fo is D to c.

As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude D. Then by composition [18. 5.] as A and B together is to B, so is c and D together to d.

m As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude D. Then by divifion (17.5.] as the difference between A and B is to B, so is the difference between c and D to D.

n As let the magnitude a be to the magnitude B, as the magnitude c is to the magnitude d. Then conversely [by cor. 19. 5.) as A is to the difference between A and B, so is c to the difference between c and D.

• As let there be three magnitudes A, B, C in one rank or order, and three magnitudes D, E, F in another order. And let a be' to B, as p is to E, and B be to c, as e is to F. Then by equality (by 22. 5.) will it be as a is to c, fo is a to F.

A

E

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19. Ordi.

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19. Ordinate proportion is, when it shall be as an antecedent to a consequent, fo is an antecedent to a consequent; and as a consequent is to some other magnitude, so is a consequent to some other magnitude P.

20. Inordinate or perturbate proportion is, when there are three magnitudes, and the same number of others, it Thall be in the first magnitudes as an antecedent is to a consequent ; so in the fecond magnitudes is an antecedent to a consequent: But as in the first magnitudes a consequent is to some other magnitude, so in the second magnitudes is some other magnitude to the antecedent %

P As let the magnitude A be to the magnitude B, as the magnitude c is to the magnitude D : and again let one of the consequents B, be to some other magnitude c, as the other consequent e is to some other magnitude f. Then is this or. dinate proportionality; and it shall be true [by 22. 5.) that a is to c, as D is to f. I take this definition to be almost useless ; it being in a manner contained in the last.

9 As let the magnitude a be to the magnitude B, as the magnitude e is to the magnitude F. And again, as in the first inagnitudes, the consequent s is to some other magnitude c; fo in the fecond magnitudes is fome other magnitude to the antecedent magnitude E This sort of proportion is called inordinate or perturbate, because the same order is not kept in the proportion of the magnitudes. And it will be [by 23. 5.] as A is to c, so is D to F.

PROPOSITIONI. THEOREM. If there be bow many soever magnitudes equimultiples of as many other magnitudes, each of each ; the

; jame multiple one magnitude is of cne, all shall be

of oll.

Let any number of magnitudes A P., co be equimultiples of the fame number of magnitudes E, F, each of each : I fay, A B and C D is the same multiple of e and F, as A B is of E.

For because A B is the same multiple of E, as 6 D is of T; as many magnitudes as there are in A B equal to B, lo

many

G

LE

E, F.

many will there be in c d equal to F: Divide A B into parts equal to E, viz. A G, GB, and cd into parts equal to F, viz. C H, HD: Then the multitude

A of the parts c H, HD of the one is equal to the multitude of the parts AG, GB of the other. And because AG is equal to E, and c H to F; AG, C H will be equal to

By the same reason G B is equal to B E, and HD to F: Therefore GB, HD are equal to E, F: Wherefore there are as с many parts in A B equal to E, as there are in A B, C D equal to E, f: Therefore A B, CD will be the same multiple of E, F, as H AB is of e.

Therefore if there be how many soever magnitudes equimultiples of as many other DI magnitudes, each of each; the same multiple one magnitude is of one, all shall be of all. Which was to be de monstrated.

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• The fifth and fixth propofitions of the seventh book of Euclid answer to this in numbers. And what is here pronounced of multiple ratio only, is afterwards dumonstrated universally at prop. 12. of any proportionals rational, or irrational whatsoever. And this proposition was first put down to prove others, which were necessary towards the demonstration of the twelfth proposition,

PROP. II. THEOR.

If the first of four magnitudes be the same multiple of

the second, as the third is of the fourth, and there be a fi; tb the jame multiple of the second, as a sixth magnitiide is of th fourth; then the first and fifth taken together will be the same multiple of the second, as the third and fixth are of the fourth .

For let the first of four magnitudes A B be the same multiple of the fecond c, as the third D E is of the fourth F. And let a fifth magnitude B G be the same multiple of the second 4, as a fixth Eh is of the fourth r: I say, A G the first:

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and

IC

IT and fifth taken together will be

the same multiple of the fe

cond c, as D H the third and AT

sixth taken together is of the fourth F.

For because A B is the same Ε:

multiple of c, as D E is of F, B

F there will be as many magni

tudes in AB equal to c, as there are in D e equal to F. And by the same reason as many as are

in B G equal to c, fo many G

there are in E H equal to F. Therefore as

many

as there are in the whole A G, equal to c, H

so many there will be in the whole D H equal to F: Therefore D H is the same multiple of F, as a G is of c: Wherefore a G the first and fifth taken together, is the same multiple of the second c, as DH the third and sixth, is of the fourth F.

Wherefore, if the first of four magnitudes be the same multiple of the second, as the third is of the fourth, and a fifth magnitude be the same multiple of the second, as a fixth is of the fourth, then the first and fifth taken together will be the same multiple of the second, as the third and fixth is of the fourth.

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• This proposition fuprofes the fifth and fixth magnitudes to be equimultiples of the second and fourth. But the propofition would hold the fame, if the fifth were only equal to the second, and the fixth to the fourth,

PRO P. III. THEOR. If the first of four magnitudes be the same multiple of

the second as the third is of the fourih, and any equimultiples of the first and third be taken ; tben, by equality, erch of the assumed equimultiples will be an equimultiple of rach, the one of the Second, and the other of the fourth magnitude .

For let the first of four magnitudes a be the same multiple of the second B, as the third c is of the fourth D; and

take

A

F

ELIT

take E F, G H equimultiples of a, c: I say, E F is the same multiple of B, as G H is of D.

For because E F is the same multiple of A, as GH is of c; there will be as many magnitudes in gh equal to co as there are in E F equal to A. Divide E F into the magnitudes EF

HT EK, K F equal to A, and G H into the magnitudes GL, LH equal to c; then will the multitude of EK, KF bc equal to the multitude of

L GL, LH. And because a is the same multiple of B, as c is of D,

K and Ek is equal to A, and GL equal to C; E K will be the same multiple of b, as G L is of D. By the same reason KF is the same

A B GCD multiple of B, as L H is of D. Then because the first ek is the same multiple of the second B, as the third G L is of the fourth D; and the fifth K F is the same multiple of the second b, as the fixth L H is of the fourth D: the magnitude E F made up of the first and fifth will [by 2. 5.) be the same multiple of the second B, as G H the third and fixth is of the fourth D.

If therefore the first of four magnitudes be the same multiple of the second, as the third is of the fourth, and any equimultiples of the first and third be taken; then, by equality, each of the assumed magnitudes will be an equimultiple of each, the one of the second, and the other of the fourth magnitude. Which was to be demonstrated.

* This is demonstrated nniversally at prop. 22. it being here only proposed in multiple ratio; and that for the sake of demonftrating the fourth proposition.

PROP. IV. THEOR. If the first of four magnitudes has the same ratio to the

second, as the third bas to the fourth; then will any equimultiples of the first and third bave the same ratio to any equimultiples of the second and fourth, when compared to one anotker.

For let the first magnitude a have the same ratio to the second B, as the third c has to the fourth D. Take E, F

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