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 1.

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

greater than 2 square inches. For example √2 inches is greater than I inch since I square inch is less than 2 square inches, and less than 3 inch, since & square inches are greater than 2 square inches. Since this relation is independent of the special unit chosen whether inch, foot, yard, &c., we may say simply √2 is greater than I and less than 2. And we have seen that the number of square inches, feet, &c., in a square is the arithmetical square of the number of inches, feet, &c., in the side of the square, whether that number be whole or fractional. We come then to the following arithmetical definition of √2.

The square root of 2 is not equal to any number, whole or fractional, but is always greater than any number whose square is less than 2, and less than any number whose square is greater than 2.

Thus like the whole number 6 or the fraction or any other symbol, √2 is defined by its fundamental relations to other symbols.

Though √2 cannot be expressed exactly by any fraction we can find fractions which will differ from it by less than any number we please.

For instance, we want to express √2 approximately as so many fifths. We multiply 2 by 25 and find the square root of the square number nearest to 50. In this case it is 7, and since 2 is greater than the square of 7 or 1%, is greater than . But since 2 is less than 2, √√2 is less than & and therefore √2 must differ from by less than }.

So if wanted to find a fraction differing from √/2 by less than a tenth, a hundredth, a thousandth, a millionth, &c., we should multiply 2 by the square of ten, one hundred, one thousand, one million, &c. and find the square root of the nearest square number. We can thus find fractions approaching √2 as closely as we please.

The general method of carrying out such calculations will be explained in the next section, but a special mode of approximating to √/2 deserves mention from its early date and from being a simple case of a general theory discovered long afterwards. It is given by Theon of Smyrna, an arithmetician who lived in the beginning of the 2nd century of our era.

We form two rows of numbers each beginning with 1, and write them one above the other. The pair of numbers in each vertical line is formed from the preceding pair thus we add the two numbers to make the next lower number; we add the upper number to twice the lower to make the next upper number. The rows will be,

since I and I make 2, I and twice I make 3, 3 and 2 make 5, 3 and twice 2 make 7, &c. The numbers in the vertical lines are respectively the numerators and denominators of fractions which continually approach √2 and which are alternately greater and less than √2. Thus is greater than √2, is less than √2, is greater than √/2, &c.

Since no whole number can have a square double the square of another whole number, it is impossible that the side and diagonal of a square can be expressed at once as whole numbers in terms of any unit of measure. In other words, the side and diagonal of a square cannot both be measured exactly by the same straight line; they can have no common measure. Two lines so related are called incommensurable. Of course such a relation is only a conception; it cannot be shewn to exist between the lines we meet with in actual experience. Any real physical line can be expressed by some whole number or fraction in terms of any other. For instance, if the hundredth of an inch were the utmost length we could distinguish by the eye we should when measuring with no other help express every line as so many hundredths of an inch. The diagonal of a square whose side was an inch would appear to be 14 inches in length. We can only create a difference by thought, and we do so by assuming as true beyond the limits of the senses properties which we perceive to be true within those limits.

A line which is incommensurable with the unit of measure is said to contain an incommensurable number of these units. It follows that the essential property and therefore definition of an incommensurable number is that it can be approximated to indefinitely but never expressed exactly by means of fractions. The square roots of 2, 3, 5, are examples of such numbers.

The cube root of a given number is a number whose cube is equal to the given number. Thus 2 is the cube root of 8, 3 is the cube root of 27, 4 of 64, &c.

The fourth root of a given number is a number whose fourth power is equal to the given number, and similar definitions apply to all other roots. Thus 5 is the 4th root of 625, 6 is the 4th root of 1296, 4 is the 5th root of 1024, 3 is the 6th root of 729, 2 is the 7th root of 128, &c.

The cube roots of 2, 3, 4, 5, 6, 7 are clearly not whole numbers, and it may be proved that they cannot be fractions. The cube root of 2 can however be expressed as nearly as we please by fractions if we assume that it is greater than any fraction whose cube is less than 2 and less than any fraction whose cube is greater than 2, and this also applies to the other cube roots. These cube roots are therefore what we have called incommensurable numbers. So likewise are the fourth roots of all the whole numbers from 2 to 15, the fifth roots of all the whole numbers between 2 and 63, &c.

Besides the roots of whole numbers and fractions there are other incommensurable numbers. The number which expresses the circumference of a circle in terms of its diameter is an example. This number is greater than 3 and less than 4, greater than 3'1 and less than 3*2, greater than 314 and less than 3'15, and we can continue the approximation as far as we wish. But we can never express the number exactly by means of any fraction.

When the length of a line is expressed in terms of some unit of measure by means of an incommensurable number, that length is defined exactly. There cannot be two lengths expressed by the same incommensurable number. For if it were possible these lines must differ by