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74 THE NATURE OF MATHEMATICS than any term of the sequence and is not a term of the sequence. A sequence like 1, l + , ! + + , l++J+| . . ., has an analogous upper limit 2. A function /(a), as the 9 independent variable x approaches a certain value, like x as x approaches 0, may have a value (in this case 2, though at 0, is indeterminate). The question of the limits of a cc function in general is somewhat complicated, but the most fix + Ax) f (x) important limit is J - -'- fv ' as Ax approaches ; this, if y =/(*), is *. Ax That the infinitesimal calculus, with its rather obscure " in- finitesimals" treated like finite numbers when we write dy 1 dx 3 dx = dy and dp = 3~> and then, on occasion, neglected ax ay leads so often to correct results is a most remarkable fact, and a fact of which the true explanation only appeared when Gauchy, Gauss (1777-1855), Kiemann (1826-1866), and Weierstrass had developed the theory of an extensive and much used class of functions. These functions happen to have properties which make them especially easy to be worked with, and nearly all the functions we habitually use in mathematical physics are of this class. A notable thing is that the complex numbers spoken of in the second chapter make this theory to a great extent. Large tracts of mathematics have, of course, not been mentioned here. Thus, there is an elaborate theory of integer numbers to be referred to in a note to the seventh chapter, and a geometry using the conceptions of the ancient Greeks and methods of modern mathematical thought ; and very many men still regard space-perception as something mathematics deals with. We will return to this soon. Again, algebra has developed and branched off ; the study of functions in general and in particular has grown ; and soon a list of some of the many great men who have helped in all this would not be very
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