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VII.

When, of equimultiples, the multiple of the firft exceeds the multiple of the fecond, but the multiple of the third does not exceed the multiple of the fourth; the first to the fecond is faid to have a greater ratio than the third to the fourth.

VIII.

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When three magnitudes are proportionals, the first has to the third a duplicate ratio of what it has to the fecond.

XI.

When four magnitudes are proportional, the firft has to the fourth a triplicate ratio of what it has to the fecond; and always one more in order as the proportionals fhall be extend

ed.

XII.

Homologous magnitudes, or magnitudes of a like ratio, are fuch whose antecedents are to the antecedents and confequents to the confequents in the fame ratio.

XIII.

Alternate ratio is the comparing the antecedent with the antecedent, and confequent with the confequent.

XIV.

Inverfe ratio is, when the confequent is taken as the antecedent, and compared with the antecedent as a confequent.

XV.

Compounded ratio is, when the antecedent and confequent, taken as one, are compared with the confequent itfelf.

XVI.

Divided ratio is, when the excefs, by which the antecedent exceeds the confequent, is compared with the confequent.

XVII.

Converse ratio is, when the antecedent is compared with the excefs by which the antecedent exceeds the confequent.

XVIII.

Ratio of equality is when there are taken more than two magnitudes in one order, and a like number of magnitudes in ano❤ ther order, comparing two to two, being in the fame ratio; it fhall be in the firft order of magnitudes, as the first is to the laft; fo, in the second order of magnitudes, is the first to the laft.

XIX.

Ordinate proportion is, the ratio being, as in the last, as the antecedent is to the confequent, in the first order of magnitudes;

Lo

Book V.

Book V.

fo is the antecedent to the confequent in the fecond order of magnitudes; and as the confequent is to any other, so is the confequent to any other.

XX.

Perturbate proportion is, when there are three or more magnitudes, and others equal to them in number, taken two and two in the fame ratio; in the first order of magnitudes, as the antecedent is to the confequent; fo, in the fecond order of magnitudes, is the antecedent to the confequent; and, as in the first order, the confequent is to fome other, fo, in the second order, is fome other to the antecedent.

A X I OM S.

I.

Etudes, are equal to each of the fame, or of equal magni.

II.

Thefe magnitudes that have the fame equimultiples, or whofe equimultiples are equal, are equal to each other.

I

PRO P. I. THE OR.
THEOR.

F there be any number of magnitudes, equimultiples of a like number of magnitudes, each, of each, whatever multiple any one of the former magnitudes is of its correfpondent one, the fame multiple are all the former magnitudes of all the latter.

Let AB, CD, be magnitudes, equimultiples of E, F, whatever multiple AB is of E, and CD of F, the fame multiple AB, CD, together, is of E, F, together.

For, let the magnitudes in AB, equal to E, be AG, GB ; and the magnitudes in CD, equal to F, be CH, HD; then AG, CH, are equal to E, F ; and BG, HD, likewife equal to E, F; therefore, as often as AB contains E, and CD, F, fo often AB, CD, contains E, F Wherefore, if there are, &c.

I'

PROP. II. THE OR.

F the first be the fame multiple of the fecond, as the third is of the fourth; and if the fifth be the fame multiple of the fecond, that the fixth is of the fourth; then shall the first, added to the fifth, be the fame multiple of the fecond, that the third, added to the fixth, is of the fourth.

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Let the first AB be the fame multiple of the fecond C, that Book V. the third DE is of the fourth F; and let the fifth BG be the fame multiple of the fecond C, that the fixth EH is of the fourth F; then AG will be the fame multiple of C that DH is of F.

For, becaufe AB is the fame multiple of C that DE is of F, there are as many magnitudes in AB equal to C, as in DE, equal to F. For the fame reason, there are as many magnitudes in BG equal to C, as there are in EH, equal to F; therefore there are as many magnitudes in AG equal to C, as there are in DH equal to F. Wherefore, &c.

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PRO P. III. THE OR.

TF the first be the fame multiple of the fecond, that the third is of the fourth, and there be taken equimultiples of the first and third, then will the magnitudes fo taken be equimultiples of the fecond and fourth.

Let the firft A be the fame multiple of the fecond B that the third C is of the fourth D; and let EF, GH, be equimultiples of A, C ;then EF is the fame multiple of B, that GH is of D. For, let the magnitudes in EF, equal to A, be FK, KE; and the magnitudes in GH, equal to C, be HL, LG; then there are as many magnitudes equal to B in FE, as there are magnitudes equal to D in GH; wherefore FE is the fame multiple of B, a 2. that GH is of D. Wherefore, &c.

PRO P. IV. THE OR.

IF the first have the fame ratio to the fecond that the third has to the fourth, then shall alfo the equimultiples of the first have the fame ratio to the equimultiple of the fecond that the equimultiple of the third has to that of the fourth.

Let there be four magnitudes, A, B, C, D, fuch, that A is to Bas C to D. Let E, F, be taken the fame multiples of A, C ; and G, H, the fame multiples of B, D; then E is to G as F is to H. For, take K, L, any equimultiples of E, F, and M, N, any equimultiples of G, H, then K is the fame multiple of A that L is of C. For the fame reafon, M is the a 3 fame multiple of B that N is of D ; but, because A is to B as Cis to D, if K be equal to M, L will be equal to N; if great

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b def, 5.

Book V. er, greater, and, if lefs, lefs; but K, L, are equimultiples of E, F, and M, N, of G, H; wherefore E is to Gas Fis to H; but it is proved, that, if K be equal to M, L is equal to N; but, if K is equal to M, M is equal to K, and N to L; if greater, greater, and, if lefs, lefs; wherefore G is to E as H is to F. Wherefore, if four magnitudes be proportional, they will also be inverfely proportional. Wherefore, &c.

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PRO P. V. THE OR.

IF F one magnitude be the fame multiple of another magnitude, that a part taken from the one is of a part taken from the other? then the refidue of the one shall be the fame multiple of the refidue of the other that the whole is of the whole.

Let AB be the fame multiple of CD, that a part taken away AE is of a part taken away CF; then the refidue EB fhall be the fame multiple of the refidue FD, that the whole AB is of the whole CD.

For, let BE be the fame multiple of CG that AE is of CF; then, because AE is the fame multiple of CF that BE is of CG, AE will be the fame multiple of CF that AB is of GF2; but AE is the fame multiple of CF that AB is of CD, and AB is the fame multiple of GF that it is of CD; therefore GF is eb Ax. 2. qual to CD b; take CF, which is common, from both, there remains GC equal to FD; therefore AE is the fame multiple of CF that EB is of FD. Wherefore, &c.

IF

PROP. VI. THE OR.

TF two magnitudes be equimultiples of two magnitudes, and fome magnitudes, equimultiples of the fame, be taken away, the refidue fball either be equal to thefe magnitudes or equimultiples of them.

For, firft, let GB be equal to E; if HD is not equal to F, let KC be equal to F; then AB is the fame multiple of E that KH is of F ; but AB, CD, are put equimultiples of E, F; therefore HK is the fame multiple of F that CD is of F; therefore KH is Axequal to CD. Take CH, which is common, from both, there remains KC equal to HD; but KC was put equal to F; therefore HD is equal to F; after the fame manner it may be demonftrated that, if GB is any equimultiple of E, HD will be the like equimultiple of F. Wherefore, &c.

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