Dividendo, by division: when there are four proportionals, and it is inferred, that the excess of the first above the second is to the second, as the excess of the third above the fourth is to the fourth. 17th prop. book 5.

XVIII.

Convertendo, by conversion: when there are four proportionals, and it is inferred, that the first is to its excess above the second, as the third to its excess above the fourth. Prop. D. book 5.

XIX.

Ex æquali (sc. distantia), or ex æquo, from equality of distance; when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the last of the first rank of magnitudes, as the first is to the last of the others. Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken two and two.

XX.

Ex æquali, from equality; this term is used simply by itself, when the first magnitude is to the second of the first rank as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order; and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in the 22d prop. book 5.

XXI.

Ex æquali, in proportione perturbata, seu inordinata; from equality, in perturbate, or disorderly, proportion; this term is used when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank;

and as the third is to the fourth of the first rank, so is the Book V. third from the last to the last but two of the second rank; and so on in a cross, or inverse, order; and the inference is as in the 19th definition. It is demonstrated in the 23d prop. of book 5.

AXIOMS.

I.

EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one another.

II.

Those magnitudes of which the same, or equal, magnitudes are equimultiples, are equal to one another.

III.

A multiple of a greater magnitude is greater than the same multiple of a less.

IV.

That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

Book V.

a Ax 2.1.

PROP. I. THEOR.

IF any number of magnitudes be equimultiples of as many others, each of each, what multiple soever any one of the first is of its part, the same multiple is the sum of all the first of the sum of all the rest.

Let any number of magnitudes A, B, and C be equimultiples of as many others, D, E, and F, each of each; A+B +C is the same multiple of D+E+F that A is of D.

Let A contain D, B contain E, and C contain F, each the same number of times, as, for instance, three times. Then because A contains D three times, A=D+D+D.

For the same reason B=E+E+E;

And also

C=F+F+F.

Therefore, adding equals to equals, A+B+C is equal to D+E+F, taken three times. In the same manner, if A, B, and C were each any other equimultiple of D, E, and F, it might be shown that A+B+C was the same multiple of D+E+F. Therefore, &c. Q. E. D.

COR. Hence, if m be any number, mD+mE+mF =m.D+E+F. For mD, mE, and mF are multiples of D, E, and F by m, therefore their sum is also a multiple of D+E+F by m.

PROP. II. THEOR.

IF to a multiple of a magnitude by any number a multiple of the same magnitude by any number be added, the sum will be the same multiple of that magnitude that the sum of the two numbers is of unity.

Let A=mC, and B-nC; then A+B=m+n.C.

For, since A-mC, A=C+C+C+ &c. C being repeated m times. For the same reason, B=C+C+ &c. C being re

peated n times. Therefore, if equals be added to equals, Book V. A+B is equal to C taken m+n times; that is, A+B=m+n.C. Therefore A+B contains C as often as there are units in m+n. Q. E. D.

COR. 1. If there be any number of multiples whatever, as A=mE, B=nE, C=pE; it may be demonstrated in the same manner that A+B+C=m+n+p.E.

COR. 2. A+B+C=m+n+p.E, and A=mE, B=nE, and C-pE; hence mE+nE+pE=m+n+p.E.

PROP. III. THEOR.

IF the first of three magnitudes contain the second as often as there are units in a certain number, and if the second contain the third as often as there are units in a certain number, the first will contain the third as often as there are units in the product of these two numbers.

Let A=mB, and B=nC; then A=mnC.

Since B=nC, mB=nC+nC+ &c. repeated m times. But nC+nC &c. repeated m times is equal to Ca multiplied by a2 Cor.2.5. n+n+ &c. n being added to itself m times; and n added to itself m times is n multiplied by m, or mn. Therefore nC+ nC+ &c. repeated m times =mnC; whence also mB=mnC. But by hypothesis A=mB, therefore A=mnC. Therefore, &c. Q. E. D.

PROP. IV. THEOR.

IF the first of four magnitudes have the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and fourth; the multiple of the first will have the same ratio to the multiple of the second that the multiple of the third has to the multiple of the fourth.

S

Book V.

a 3. 5.

Let A: B:: C: D, and let m and n be any two numbers; mA: nB:: mC : nD.

Take of mA and mC equimultiples by any number p, and of nB and nD equimultiples by any number 9. The equimultiples of mA and mC by p are equimultiples also of A and C, for they contain A and C as often as there are units in pma, and are equal to pmA and pmC. For the same reason the multiples of nB and nD by q are qnB, qnD. Since A: B:: C: D, and of A and C there are taken any equimultiples pmA and pmC, and of B and D any equimultiples qnB, qnD, if pmA be greater than qnB, pmC must be greater b Def. 5. 5. than qnDb; if equal, equal; and if less, less. But pmA, pmC are also equimultiples of mA and mC, and qnB, qnD are equimultiples of nB and nD; therefore mA: nB:: mC : nD3. Therefore, &c. Q. E. D.

COR. If A: B:: C: D, and of A and C equimultiples be taken by any number m, it may be demonstrated in the same manner that mA: B:: mC: D. This may also be considered as included in the proposition, and as being the case when n=1.

a 1. 5.

PROP. V. THEOR.

IF one magnitude be the same multiple of another which a magnitude taken from the first is of a magnitude taken from the other, the remainder is the same multiple of the remainder that the whole is of the whole.

Let mA and mB be any equimultiples of the two magnitudes A and B, of which A is greater than B; mA-mB is the same multiple of A-B that mA is of A, that is, mA-mB-m.A-B.

Let D be the excess of A above B, then A-B=D; therefore by adding B to both, A=D+B; thereforea mA=mD+ mB; take mB from both, then mA-mB=mD. But D-A-B, therefore mA-mB-m.A-B. Therefore, &c. Q. E. D.