580 VIII. ON THE STRUCTURE OF MATHEMATICS |
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In the limit as
Ax approaches zero, the value of the above ratio, or the rate of change of our function, would be 2x, as Ax would disappear. If our x = 100, the above ratio would be 200, as determined above in the case of the numerical example. Another way of symbolizing the derivative is but thisrequires a short explanation.
In Chapter XV we have already discussed the problem of the 'infinitesimal' and we have seen that 'infinitesimal' is a misnomer and that there is no such thing at all. Yet this word is very often uncritically used by mathematicians and is therefore often confusing. By an 'infinitesimal' mathematicians mean a
variable which approaches zero as a limit. The condition that it should be a variable is essential. It would probably be better to call an 'infinitesimal' an indefinitely small quantity or 'indefinitesimal', and that is what the reader should understand when he sees anywhere the word 'infinitesimal' or 'infinitely small quantity'. |
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