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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 equal to 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 mn: mp :: n :p (V. 3.)

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 m:n::p: q (V. 6.)


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

Let it be required to find the greatest common measure, that 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, that which measures B will measure its multiple; and being a common measure, it also measures A, and measures, therefore, the difference between the multiple of B and A (V. 1. cor. 1.), that is, C; the required measure, hence, 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. Here the decomposition is supposed to terminate. Wherefore, the common measure of A and B, since it measures E, may 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 measures, in succession, all the preceding terms D, C, B, and A.

If the process of decomposition should never come to a close, the quantities A and B do not admit a common measure, or they are incommensurable. But, as the residue of the subdivision is necessarily diminished at each step of this operation, it is evident that an element may be always discovered, which will measure A and B nearer than any assignable difference whatever.


To express by numbers, either exactly or approximately, the ratio of two given homogeneous quantities.

Let A and B be two quantities of the same kind, whose numerical ratio it is required to discover.

Find, by the last Proposition, the greatest common measure E of the two quantities; and let A contain this measure K times, and B contain it L times: Then will the ratio K: L express the ratio of A: B.

For the numbers K and L severally consist of as many units, as the quantities A and B contain their measure E. It is also manifest, since E is the greatest possible divisor, that K and L are the smallest numbers capable of expressing the ratio of A to B.

If A and B be incommensurable quantities, their decomposition is capable at least of being pushed to an unlimited extent; and, consequently, a divisor can always be found so extremely minute, as to measure them both to any degree of precision,

Otherwise thus.

But the numerical expression of the ratio A: B, may be deduced indirectly, from the series of quotients obtained in the operation for discovering their common measure.

Let A contain B, m times, with a remainder C; C contain B, n times, with a remainder D; and, lastly, suppose D to contain C, p times, with a remainder E, and which is contained in D, q times exactly. Then D=qE, C=pD+E, B=nC+D, and A≈mB+C; whence the terms D, C, B, and A, are successively computed, as multiples of E;—A and B will, therefore, be found to contain their common measure K and L times, or the numerical expression for the ratio of those quantities, is K: L.

Or thus.

It is more convenient, however, to derive the numerical ratio, from the quotients of subdivision in their natural order; and this method has besides the peculiar advantage of exhibiting a succession of elegant approximations.

The quantities A, B, C, D, &c. are determined, as before, by these conditions: A=mB+C, B=nC+D, C=pD+E, D=qE+F, &c. But other expressions will arise from substitution: For,

1.A=mB+C=m(nC+D)+C=(mn+1)C+mD, or, putting m.n+m', A=m'C+mD.

2. A=m'C+mD=m'(pD+E)+mD=(m'p+m)D+m'E, or, putting m'.p+m=m", A=m"D+m'E.

3 A=m"D+m'E=m"(qE+F)+m'E=(m"q+m')E+m"F, or, putting m"q+m'=m", A=m"E+m"F.

Again, the successive values of B are developed in the

same manner:

1. B=nC+D=n(pD+E)+D=(np+1)D+nE, or, putting n.p+n', B=n'D+n.E.

2. B=n'D+nE=n'(qE+F)+nE=(n'q+n)E+n'F, or, putting n'.q+n=n", B=n′′E+n'F.

These results will be more apparent in a tabular form:

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Whence the law of the formation of the successive quantities, is easily perceived.

But, to find the ratio of A to B, it is not requisite to know the values of the remainders C, D, E, &c. Suppose the subdivision to terminate at B; then AmB, and consequently A: B, as mB: B, or m: 1. If the subdivision ex, tend to C, then A=m'C, and B-nC; whence A: B, as m':n. In general, therefore, the second term, in the expressions for A and B, may be rejected, and the letter which precedes it considered as the ultimate measure, and corresponding to the arithmetical unit. Hence, resuming the substitutions and combining the whole in one view, it follows that, the ratio of A to B may thus be successively represented :

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The formation of these numbers will evidently stop, when the corresponding subdivision terminates. But even though the successive decomposition should never terminate, as in the case of incommensurable quantities,-yet the expression thus obtained must constantly approach to the ratio of A: B, since they suppose only the omission of the remainder of the last division, and which is perpetually diminishing.


A straight line is incommensurable with its segments formed by medial section.

If the straight line AB be cut in C, such that the rectangle AB, BC is equal to

the square of AC; no part
of AB, however small, will
measure the segments AC, BC.




For (V. 25.) take AC out of AB, and again the remainder BC out of AC. But AD, being made equal to BC, the straight line AC is likewise divided in D, by a medial section (II. 26. cor. 1.); and, for the same reason, taking away the successive remainders CD, or AE, from AD, and DE or AF from AE, the subordinate lines AD and AE are also divided medially in the points E and F. This operation produces, therefore, a series of decreasing lines, all of them divided by medial section: Nor can the process of decomposition ever terminate; for though the remainders BC, CD, DE, and EF thus continually diminish, they still must constitute the segments of a similar division. Consequently there exists no final quantity which would measure both AB and AC.

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