Let A B

B:C::C: D:: D: E; then if AB,

A-BA: A-E: A+B+C+D.

For (V. 19.), A:B::A+B+C+D: B+C+D+E, and consequently (V. 11. cor.), A-B: A:: (A+B+C+D) ~(B+C+D+E) : A+B+C+D; that is, omitting B+C+Din the third term, A-B:A::A-E:A+B+C+D.

If AB, then B-A: A: (B+C+D+E)-(A+B +C+D): A+B+C+D, that is, B-A: A :: E-A: A +B+C+D.

The same reasoning, it is evident, will hold for any number of terms.

Scholium. Hence the summation of continued progressions, whether ascending or descending, is easily derived.

PROP. XXII. THEOR.

The products of the like terms of any numerical proportions, are themselves proportional.

Let A: B::C: D

E: F:: G: H

I: K:: L: M;

then AEI: BFK:: CGL : DHM.

For (V. 6.), from the first analogy AD=BC, from the second analogy EH=FG, and from the third analogy IM =KL; whence the compound product AD.EH.IM≤ BC.FG KL. But AD.EH.IM=AEI.DHM (V. 2.), and BC.FG.KL BFK.CGL; wherefore AEI.DHM = BFK.CGL, and consequently (V. 6.), AEI : BFK :: CGL: DHM.

The same reason, it is obvious, will apply to any number of proportionals.

Cor. 1. Hence the powers of the successive terms of numerical proportions, are likewise proportional. For, if A: B:: C:D, and, repeating the analogy, A : B :: C : D; then, by multiplication, AA: BB:: CC: DD, or A': B2:: C: D2.

Again, let A: B:: C: D, and, repeating the analogy, A: B:: C: D,

and A: B :: C: D; whence, by multiplying the corresponding terms,

And so the induction may be pursued generally.

Cor. 2. Hence also the roots of the terms of a numerical proportion, are proportional. If A: B:: C: D, then WA: WB:: ✔C: ✔D. For let ✔A: ✔B:: √C: √E, and, by the last corollary, A: B::C: E; but A: B :: CD, whence C E C: D, and consequently E=D, or ✔A: √B:: √C: ✔D.-In the same manner, it may be shewn in general that, if A: B:: C: D, VA: VB:: VC: VD.

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PROP. XXIII. THEOR.

The ratio which is conceived to be compounded of other ratios, is the same as that of the products of their corresponding numerical expressions.

Suppose the ratio of A: D is compounded of A: B, of B: C, and of C: D, and let A: B:: K: L, B: C::

M N, and C: D:: 0 : P; then will A : D: : KMO: LNP.

For, since A: B:: K: L,

B: C:: M: N,

and CD:: 0 :P,

the products of the similar terms are proportional (V. 22.), or ABC: BCD:: KMO: LNP. But A: D::ABC: BCB (V. 3.), and consequently A: D:: KMO: LNP.

The same mode of reasoning is applicable to any number of component ratios.

PROP. XXIV. THEOR.

A duplicate ratio is the same as the ratio of the second powers of the terms of its numerical expression, and a triplicate ratio is the same as that of the third powers of those terms.

Let A: B:: B: C :: C: D; then A2: B2 : : A : C, and A3: B3 :: A : D. ́

For, since A: B:: B: C,

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and A B A: B, the products of the corresponding terms are proportional (V. 22.), or A2 : B2 : : BA: CB. Whence (V. 3.) A: B2 : : A : C.

Again, since A: B:: B: C,

and A: B:: C: D,

and A: B: A: B, as before, (V.

22.), A3: B3 :: BCA: CDB. And consequently (V. 3.)

A3: B3 A: D.

PROP. XXV. THEOR.

The product of the numbers expressing the sides of a rectangle, will represent its quantity of surface, as measured by a square described on the linear unit.

Let ABCD be a rectangle and OP the linear measure; and suppose the side AB to contain OP, m times, and the side BC to contain it, n times. Divide these sides accordingly (1.36.), and, through the points of section, draw straight lines (I. 23.) parallel to AD and DC: the whole rectangle will thus be divided into cells, each

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of them equal to the square of OP. It is evident, that there stand on BC, n columns, and that each of these columns contains, m cells; consequently the entire space includes, m.n cells, or is equal to the square of OP repeated mn times.

Cor. 1. If m=n, then AB=BC, and the rectangle becomes a square; but mn is in that case equal to nn, or n2. Whence the surface of a square is expressed by the second power of the number denoting its side.

Cor. 2. Rectangles which have the same altitude m are as their bases n and p; for (V. 3.) mn: mp :: n: p. And triangles having the same altitude, being (II. 5. cor.) the halves of these rectangles, must likewise be as their bases.

Cor. 3. If two rectangles be equal, their respective sides are reciprocally proportional, or form the extremes and means of an analogy. For if mn=pq, then (V. 6.) m:p::

g: n.

PROP. XXVI. PROB.

Given two homogeneous quantities, to find, if possible, their greatest common measure.

Let it be required to find the greatest common measure, which two quantities A and B, of the same kind, will admit.

Supposing A to be greater than B, take B out of A, till the remainder C be less than it; again, take C out of B, till there remain only D; and continue this alternate operation, till the last divisor, suppose E, leave no remainder whatever; E is the greatest common measure of the quantities proposed.

For, the quantity sought, as it measures B, will measure its multiple; and since it also measures A, it must measure the difference between the multiple of B and A (V. 1. cor. 1.), that is, C; the required measure, therefore, measures the multiple of C, and consequently the difference of this multiple and B, which it measured,—that is D: And lastly, this measure, as it measures the multiple of D, must consequently measure the difference of this from C, or it must measure E. Supposing the decomposition to terminate here, the common measure of A and B, since it measures E, must be E itself; and it is also the greatest possible measure, for nothing greater than E can be contained in this quantity.

By retracing the steps likewise, it might be shown, that E actually measures, in succession, all the preceding terms D, C, B, and A.

If the process of decomposition should never terminate, the quantities A and B do not admit of a common mea