By Menger K.

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Substitution will be assumed to satisfy the following lawst I. Associative Law. ,ha )]. For some purposes it is convenient to denote the a± functions substituted into gA by \v***±a. ,r). ,j) - f. III. Ljw^of jpep£esslon. • In the classical notation, a function admitting the suppression of the Index 1 is one which does not depend upon its 1-th variable, as f(x,y,z) = 4*x + 5*log z does not depend upon y. *gr) no matter which functions g^> •••jgp we substitute, then we can suppress any r-1 of the indices and thus arrive at a constant function.

Tan(f+g)- tanf+tang 1-tan f «tan g 2. D tan 0 * 1. For this function we have Q D tan • 1 4- tan m Q rec cos . In the classical analysis, for each constant a the function tan (a-x) satisfies postulate 1. The function associating tan x with z is the only one which satisfies postulates 1 and 2. In a paper "e and ir In Elementary Calculus11 (to appear in the near future ) we describe how the postulates D tan 0*1 and D exp 0 * 1 in conjunction with the functional equations for the tangential and exponential functions yield an intuitive introduction of ir and e, as well as a simple development of the ^natural* tangential and exponential functions ex and tan x (x measured in radians).

We shall now postulate that there Is a tangential function tan for which D tan 0 = 1. From now on we shall reserve the sym^ bol tan for this function given by the postulates 1. tan(f+g)- tanf+tang 1-tan f «tan g 2. D tan 0 * 1. For this function we have Q D tan • 1 4- tan m Q rec cos . In the classical analysis, for each constant a the function tan (a-x) satisfies postulate 1. The function associating tan x with z is the only one which satisfies postulates 1 and 2. In a paper "e and ir In Elementary Calculus11 (to appear in the near future ) we describe how the postulates D tan 0*1 and D exp 0 * 1 in conjunction with the functional equations for the tangential and exponential functions yield an intuitive introduction of ir and e, as well as a simple development of the ^natural* tangential and exponential functions ex and tan x (x measured in radians).

### Algebra of Analysis by Menger K.

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