the parts of the second is greater, equal, or less than the first, so is the whole quantity formed by the parts of the fourth greater, equal, or less than the third. The first quantity is then said to be to the second as the third is to the fourth.

If we assume as we legitimately may that the meaning of equality of whole numbers is known (Introduction) we can put the definition in a shorter form. The criterion of proportionality will then be as follows.

Divide the second quantity into equal parts and the fourth quantity into the same number of equal parts; take any number of the parts of the second and the same number of the parts of the fourth, then according as the whole formed by the parts of the second is greater, equal, or less than the first, so the whole formed by the parts of the fourth must be greater, equal, or less than the third. We must next define what is meant by the sum of two numbers.

The sum of the numbers by which two quantities are expressed in terms of a third quantity is the number by which the sum of the quantities is expressed in terms of the third quantity.

But is the sum of two numbers independent of the unit of measure by means of which the quantities are expressed? If we take a fourth and fifth quantity expressed in terms of the sixth by the same numbers as the first and second are in terms of the third, will the sum of the fourth and fifth quantities be expressed in terms of the sixth by the same number as the sum of the first and second is in terms of the third?

To prove this is the same as to prove the following theorem :

If we have six quantities, such that the first, third, fourth, sixth and also the second, third, fifth, sixth are proportional, then the sum of the first and second will be to the third as the sum of the fourth and fifth is to the sixth.

We assume what has been shewn in the introduction, that the sum of two whole numbers is independent of the special objects to which those numbers apply.

We represent the quantities by lines A, B, C, D, E, F. We divide C and F each into the same number of equal parts, say into 10 equal parts, and take, suppose, 13 of the parts of C and also 13 of the parts of F. And in the first place let the whole line formed by A and B be greater than the 13 parts of C, then we must shew that the whole line formed by D and E is greater than the 13 parts of F.

Let PQ be one of the parts of C. Then the line formed by A and B exceeds 13 times PQ by some quantity, and this quantity however small must be greater than some part of PQ. Let it be greater than the third

triangles and each of the small squares will contain two of these triangles. Therefore the large square is equal to the sum of the small squares and if the small squares be each a square inch, the large square will contain two square inches. Hence the square root of 2 which is commonly written √2 may be considered to be the length, expressed in inches of the longest side of a right-angled triangle, when each of the shorter sides is one inch; or in other words 2 may be considered to be the length of a diagonal of a square when each of the sides is I.

Now it is obvious geometrically that according as a line is greater or less than another line, its square is greater or less than the square of that line; and conversely according as the square of a line is greater or less than the square of another line so is the first line greater or less than the second line. A line √2 inches in length must then be greater than any line whose square is less than 2 square inches, and less than any line whose square is

part of PQ. Divide C into 30 equal parts, each equal to PR, then PQ will contain 3 of these parts; and the line formed by A and B exceeds 13 times PQ or 39 times PR by a quantity greater than PR. Therefore this line is greater than 40 times PR. Now if we take successively 1, 2, 3, 4, &c. times PR we shall arrive at a multiple of PR just greater than A. Suppose then A is greater than 27 times PR and less than 28 times PR. Since A is less than 28 times PR, and the whole line is greater than 40 times PR it will follow that B is greater than 12 times PR. Now divide F into 30 equal parts, then since A is greater than 27 times PR, D will be greater than 27 of these parts, and since B is greater than 12 times PR, E will be greater than 12 of these parts. Hence the whole line formed by D and E is greater than 39 of the 30th parts of F and therefore greater than 13 of the 10th parts of F, since each 10th part contains three 30th parts.

Next let the line formed by A and B be less than 13 times the 10th part of C, then we must shew that the line formed by D and E is less than 13 times the 10th part of F.

Making the same suppositions as before, we shall have the line formed by A and B less than 38 times PR. Also if A is greater than 27 times PR and less than 28 times PR, B will be less than II times PR. Hence D is less than 28 times the 30th part of F, and E is less than II times the 30th part of F. Therefore the whole line formed by D and E is less than 39 times the 30th part of F, or than 13 times the 10th part of F.