cumference; if BC, CE, &c. be joined, the equilateral and Book IV. equiangular quindecagon will be infcribed: Which was requi

red.

COR. If, from what has been faid of the pentagon, right lines be drawn through the divifions of the circle, tangents to the fame, an equilateral and equiangular quindecagon will be defcribed about the circle; or a circle may be infcribed or des fcribed about the quindecagon.

THE

}

Book V.

A

PART is a magnitude of a magnitude; the less of the greater, when the lefs measures the greater.

II.

A multiple is a magnitude of a magnitude; the greater of the lefs, when the less measures the greater.

HI.

Ratio is a certain mutual habitude of magnitudes of the fame kind, according to quantity.

IV.

Magnitudes have proportion to each other; which, being multiplied, can exceed one another.

V.

Magnitudes have the fame ratio to each other, viz. the first to the fecond, and third to the fourth, when there are taken any equimultiples of the first and third, and likewife any equimultiples of the fecond and fourth; if the multiple of the firft, be equal to the multiple of the fecond, then the multiple of the third will be equal to the multiple of the fourth; if greater, greater; and, if less, less.

VI.

Magnitudes which have the fame proportion are called Proportionals.

VII

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.

When three magnitudes are proportionals, the firft 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 extended.

XII.

Homologous magnitudes, or magnitudes of a like ratio, are such whofe 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.

Inverse ratio is, when the confequentis 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 itself.

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 another order, comparing two to two, being in the fame ratio ; it fhall be in the firft order of magnitudes, as the firft is to the laft; fo, in the fecond 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;

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

Perturbate proportion is, when there are three or more magni-
tudes, 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 fe-
cond order, is fome other to the antecedent.

AXIOM S.

I.

QUIMULTIPLES of or

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

other.

II.

These magnitudes that have the fame equimultiples, or whose equimultiples are equal, are equal to each other,

PRO P. I. THE OR.

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 lattér.

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.

PRO P. 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.

Let ·

I

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, because 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 reafon, 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.

a. I.

PRO P. III. THE OR.

IF 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 first 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, that GH is of D. Wherefore, &c.

PROP. IV. THE OR.

IF
F the first have the fame ratio to the fecond that the third has to
the fourth, then shall also 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 Da; but, becaufe A is to B as Ç is to D2 if K be equal to M, L will be equal to N; if great.