In this chapter the classical transcendental functions, already treated by Euler in his Introductio [E], will be discussed. At the center stands the exponential function, which is determined (§1) both by its differential equation and its addition theorem. In Section 2 we will prove directly, using differences and the logarithmic series and without borrowing any facts from real analysis, that the exponential function defines a homomorphism of the additive group ℂ onto the multiplicative group ℂ×. This epimorphism theorem is basic for everything else; for example, it leads immediately to the realization that there is a uniquely determined positive real number π such that exp z = 1 precisely when z is one of the numbers 2nπi, n ∈ ℤ. This famous constant thereby “occurs naturally among the complex numbers”.