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We can, however, find a number whose square shall be as nearly equal to 5 as we please, and it is this number that we treat arithmetically as 1 15.

00Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms.

00if B and C are expressions involving x which are different in form but are arithmetically equal for all values of x), then the statement A = C is an equation which is true for the same value of x for which A = B is true.

00We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.

00- The following are expressions for the areas of some simple figures; the expressions in (i) and (ii) are obtained arithmetically, while those in (iii) - (v) involve dissection and rearrangement.

00Gunter's Line, a logarithmic line, usually laid down upon scales, sectors, &c. It is also called the line of lines and the line of numbers, being only the logarithms graduated upon a ruler, which therefore serves to solve problems instrumentally in the same manner as logarithms do arithmetically.

00We can, however, find a number whose square shall be as nearly equal to 5 as we please, and it is this number that we treat arithmetically as 1 15.

00Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms.

00if either or both of the expressions A and B involves x, and A is arithmetically equal to B for the particular value of x which we require), and if B = C is an identity (i.e.

00if B and C are expressions involving x which are different in form but are arithmetically equal for all values of x), then the statement A = C is an equation which is true for the same value of x for which A = B is true.

00We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as the limit of x+ On the other hand, if f(y) has a definite and finite value for y = x, it must not be supposed that this is necessarily the same as the limit which f (y) approaches when y approaches the value x, though this is the case with the functions with which we are usually concerned.

00- The following are expressions for the areas of some simple figures; the expressions in (i) and (ii) are obtained arithmetically, while those in (iii) - (v) involve dissection and rearrangement.

00Gunter's Line, a logarithmic line, usually laid down upon scales, sectors, &c. It is also called the line of lines and the line of numbers, being only the logarithms graduated upon a ruler, which therefore serves to solve problems instrumentally in the same manner as logarithms do arithmetically.

00

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