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ON INCOMMENSURABLE MAGNITUDES.
1. A magnitude that measures two other magnitudes, measures also their sum and difference.
Let M be a magnitude that measures each of the magnitudes A and B; and let M measure A, m times, and B, n times; then A is equal to m times M, and B is equal to n times M; therefore A and B together are equal to m times M, together with n times M; that is, to as many times M as there are units contained in the sum of m and n; hence M measures the sum of A and B. It likewise measures their difference; for the difference between A and B is equal to the difference between m times M and n times M; that is to as many times M as there are units in the difference between m and n; hence M measures the difference of A and B.
Hence if M measures B, and either the sum of A and B, or the excess of A above B, it will measure A. For if M measure B and the excess of A above B, it will measure A, which is the sum of B, and the excess of it above B. For the same reason, if M measures B and the sum of A and B, it will measure their difference ; that is, M measures A.
2. Two magnitudes of the same kind being given, to find their greatest common measure.
Let A and B be the given magnitudes; it is required to find the greatest magnitude that will measure each of them. Let A be the greater of the two magnitudes, and find the multiple of B that is nearest to A, either greater or less than it, and let the difference between A and this multiple be C, which must be either equal to the half of B, or less than its half. In like manner find the multiple of C nearest to B,
B) A (P and let the difference between B
pB and this multiple be D, a mag
C) B (9 nitude either less than the half of
qC C, or exactly equal to its half: pro
D) C (9 ceed in this manner till no re
rd mainder be left, or till the last difference measures the preceding one; then shall the last difference be the greatest common measure required.
Let D be the last difference; then D measures C, and therefore it measures any multiple of C; hence D measures both the sum and difference of D, and a multiple of C; but the sum or difference of D and a multiple of C is equal to B; hence D measures B, and therefore D measures any multiple of B; but the difference of A and a multiple of B is C, which is measured by D.; hence D measures the sum or difference of C and a multiple of B; that is, D measures A; consequently D measures both A and B.
Also, D is the greatest common measure of A and B; for since the magnitude required ineasures A, B, and also any multiple of B; therefore it measures the difference between A and a multiple of B; that is, it measures C. Again, since it must measure any multiple of C, it must also measure the difference between B and a multiple of C; that is, it measures D.
But a magnitude cannot be measured by a magnitude greater than itself; hence the last difference is the greatest common measure of A and B.
If the preceding process is interminable, the given magnitudes cannot have a common measure, and are consequently incommensurable.