19. 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.

20. 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.

21. 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 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.

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

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

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

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

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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 to 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,

For the same reason,
And also,

A=D+D+D.
B=E+E+E;

C=F+F+F.

Therefore, adding equals to equals (Ax. 2. 1.), 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 would be shown that A+ B+C was the same multiple of D+E+F.

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.

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; 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 repeated n times. Therefore, adding equals to equals, A+B is equal to C taken m+n times; that is, A+B=(m+n\C. Therefore A+B contains C as oft as there are units in m+n.

COR. 1. In the same way, if there be any number of multiples whatsoever, as A=mE, B=nE, C=pE, it is shown, that A+B+C=(m+n +p)E.

COR. 2. Hence also, since A+B+C=(m+n+p)E, and since A=mE, B=nE, and CpE, 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 also, 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 C (2. Cor. 2. 5.), multiplied by n+n+ &c. n being added to itself m times; but 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, and by hypothesis A=mB, therefore A=mnC.

PROP. IV. THEOR.

If the first of four magnitudes has the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be taken of the first and never third, and any whatever of the second and fourth; the multiple of the first shall have the same ratio to the multiple of the second, that the multiple of the third has to the multiple of the fourth.

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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 q. Then the equimultiples of mA, and mC by p, are equimultiples also of A and C, for they contain A and C as oft as there are units in pm (3. 5.), and are equal to pmA and pmC. For the same reason the multiples of nB and nD by q, are qnB, qnD. Since, therefore, A B C D, and of A and C there are taken any equimultiples, viz. pmA and pmC, and of B and D, any equimultiples qnB, qnD, if pmA be greater than qnB, pmC must be greater than qnD (def. 5. 5.); 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 nĎ, therefore (def. 5. 5.), mA : nB :: mC : nD.

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

=1.

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, and adding B to both, A=D+B. Therefore (1. 5.) mA=mD+mB; take mB from both, and mA-mB=mD; but D=A-B, therefore mA-mB=m(A—B).

PROP. VI. THEOR.

If from a multiple of a magnitude by any number a multiple of the same magnitude by a less number be taken away, the remainder will be the same multiple of that magnitude that the difference of the numbers is of unity.

Let mA and nA be multiples of the magnitude A, by the numbers m and n, and let m be greater than n; mA—nĂ contains A as oft as m―n contains unity, or m×nA=(m-n)A.

Let m-n-q; then m=n+q. Therefore (2. 5.) mA=nA+qA; take nA from both, and mA-nA=qA. Therefore mA-nA contains A as oft as there are units in q, that is, in m—n, or mA—nA=(m—n)A.

COR. When the difference of the two numbers is equal to unity or mn=1, then mA—nA=A.

PROP. A. THEOR.

If four magnitudes be proportionals, they are proportionals also when taken inversely.

If A : B :: C : D, then also B: A :: D: C.

Let mA and mC be any equimultiples of A and C ; nB and nD any equimultiples of B and D. Then, because A: B:: C: D, if mA be less than nB, mC will be less than nD (def. 5. 5.), that is, if nB be greater than mA, nD will be greater than mC. For the same reason, if nB=mA, nl)=mC, and if nB/mA, nD/mC. But nB, nD are any equimultiples of B and D, and mA, mC any equimultiples of A and C, therefore (def. 5. 5.), B : A :: D: C.

PROP. B. THEOR.

If the first be the same multiple of the second, or the same part of it, that the third is of the fourth; the first is to the second as the third to the fourth.

First, if mA, mB be equimultiples of the magnitudes A and B, mA: A:: mB: B.

Take of mA and mB equimultiples by any number n; and of A and B equimultiples by any number p; these will be nmA (3. 5.), pA, nmB (3. 5.), pB. Now, if nmA be greater than pA, nm is also greater than p; and if nm is greater than p, nmB is greater than pB, therefore, when nmA is greater than pA, nmB is greater than pB. In the same manner, if nmA=pA, nmB=pB, and if nmA ▴pA, nmB/pB. Now, nmA, nmB are any equimultiples of mA and mB; and pA, pB are any equimultiples of A and B, therefore mA A:: mB: B (def. 5. 5.).

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Next, Let C be the same part of A that D is of B; then A is the same multiple of C that B is of D, and therefore, as has been demonstrated, A : C:: B: D and inversely (A. 5.) C: A :: D: B.

PROP. C. THEOR.

If the first be to the second as the third to the fourth; and if the first be a multiple or a part of the second, the third is the same multiple or the same part of the fourth.

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Let A B C : D, and first, let A be a multiple of B, C is the same multiple of D, that is, if A=mB, C=mD.

Take of A and C equimultiples by any number as 2, viz. 2A and 2C and of B and D, take equimultiples by the number 2m, viz. 2mB, 2mD (3.

5.); then, because A=mB, 2A=2mB; and since A: B:: C: D, and since 2A=2mB, therefore 2C=2mD (def. 5. 5.), and C=mD, that is, C contains D, m times, or as often as A contains B.

Next, Let A be a part of B, C is the same part of D. For, since A: B :: C: D, inversely (A. 5.), B : A :: D: C. But A being a part of B, Bis a multiple of A; and therefore, as is shewn above, D is the same multiple of C, and therefore C is the same part of D that A is of B.

PROP. VII. THEOR.

Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other; A: C:: B: C. Let mA, mB, be any equimultiples of A and B; and nC any multiple of C.

Because A=B, mA=mB (Ax. 1. 5.); wherefore, if mA be greater than nC, mB is greater than nC; and if mA=nC, mB=nC; or, if mA /nC, mB ZnC. But mA and mB are any equimultiples of A and B, and nC is any multiple of C, therefore (def. 5. 5.) A: C:: B: C.

Again, if A=B, C : À :: C: B; for, as has been proved, A : C::B; C, and inversely (A. 5.), C : A :: C : B.

PROP. VIII. THEOR.

Of unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less than it has to the greater.

Let A + B be a magnitude greater than A, and C a third magnitude, A+B has to C a greater ratio than A has to C; and C has a greater ratio to A than it has to A+B.

Let m be such a number that mA and mB are each of them greater than C; and let nC be the least multiple of C that exceeds mA+mB; then nC -C, that is (n-1)C (1. 5.) will be less than mA+mB, or mA+mB, that is, m(A+B) is greater than (n-1)C. But because nC is greater than mA+mB, and C less than mB, nC-C is greater than mA, or mA is less than nC-C, that is, than (n-1)C. Therefore the multiple of A+B by m exceeds the multiple of C by n-1, but the multiple of A by m does not exceed the multiple of C by n-1; therefore A+B has a greater ratio to C than A has to C (def. 7. 5.).

Again, because the multiple of C by n-1, exceeds the multiple of A by m, but does not exceed the multiple of A+B by m, C has a greater ratio to A than it has to A+B (def. 7. 5.).

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