CALCULUS, The Infinitesimal. The In finitesimal, or Differential and Integral, Cal culus is not so much a branch of mathematics as a method or instrument of mathematical in vestigation, of indefinite applicability. The masters now seldom try to treat it in less than a thousand large pages; here we may hope no more than to expose its basic principles, to illustrate its characteristic processes and to ex hibit some of its more immediate applications, with their results. Even so little will require the utmost condensation and self-explaining abbreviations.
We might define the Calculus as the Theory and Application of Limits, so central and dominant is this latter concept. We must, then, clear the ground for its full presentment.
Saceisive addition of the unit 1, continued without end, gives rise to the Assemblage of positive integers, in which all additions, multi plications and involutions are possible. This assemblage is ordered: i.e., of two different elements, a and b, either ab; and if a
For a precise definition of the processes of annexation here involved, see ALGEBRA, DEFI NITIONS AND FUNDAMENTAL CONCEPTS.
The operation of involution is direct, a special case of multiplication, but is not com mutative like addition and multiplication: thus a + b= b a, ab ba, but in general ab ba. Hence the direct operation ab, yielding c, has two inverses: Given b and c, to find a, and given a and c, to find b. The former gives rise to roots or surds, the tatter to logarithms. But neither of these can in general be found in the universe of rationals; to make such in versions always possible, we must still further enlarge the domain of number by annexing Irrational:. These demand exact definition.
Divide the assemblage of rationals into two classes, A and B, any member a of the first being < any member b of the second. Three
possibilities present themselves : 1. A may contain and be closed by a number a > any other a but < any b.
2. B may contain and be closed by a number 3 < any other b but > any a, 3. Neither A may contain a largest a, nor B a smallest b. Thus we may form (1) A of 2 and all rationals .< 2; or (2) B of 2 and all rationals > 2,—in either case 2 is a border (frontiere) number ; or (3) A of all negatives and all positive rationals whose squares are < 2 and B of all positive rationals whose squares are > 2. Here there is no border number among rationals. But a border does exist, de fined as > any a but < any b. We name it second root of 2 and denote it by VY or 2i. All such common borders are called Irrational:. The assemblage of irrationals is determined by all such possible partitions of rationals (A,B). The assemblage of all rationals and all such irrationals is the assemblage of Reals. It remains and is possible to extend the opera tions of arithmetic to all teals. In particular, the assemblage of rationals is dense; i.e., be tween every two there is an infinity of others; in the same sense the assemblage of seals also has density. Again, always on dividing all seals into A and B, in such a manner that each mem ber of A < each member of B, there will be a border y, the greatest in A or the least in B, all less numbers being in A, all greater in B. Hence, and in this sense, the assemblage of reals is named continuous.
In this continuum, admitting no further in troductions, suppose a magnitude to assume successively an infinity of values: v., v., Vn+k, . . .; it is then called a variable, V, and its values in order form a sequence, S. It often happens that V will approach some con stant L, so that by enlarging n we may make and keep the modulus or absolute worth (i.e., regardless of sign) of the difference V—L