"I have finally discovered the true solution: in the same way that to one sine there correspond an infinite number of different angles I have found that it is the same with logarithms, andeach number has an infinity of different logarithms, all of them imaginary unless the number is real and positive; there is only one logarithm which is real, and we regard it as its unique logarithm. (Leonhard Euler, [letter to Cramer] 1746)
"Right away, if somebody wishes to introduce a new function into analysis, I will ask himto make clear if he simply wishes to use it for real quantities (real values of the argument of the function), and at the same time will regard the imaginary values of the argument as an appendage [Gauss here spoke of a ganglion, Uberbein], or if he accedes to my principle ¨that in the domain of quantities the imaginary a + b √−1 = a + bi must be regarded as enjoying equal rights with the real. This is not a matter of utility, rather to me analysis is an independent science which, by slighting each imaginary quantity, loses exceptionally in beauty and roundness, and in a moment all truths that otherwise would hold generally, must necessarily suffer highly tiresome restrictions." (Carl FriedrichGauss, [letter to Bessel] 1821)
"The integral ∫ϕx.dx will always have the same value along two different paths if it is never the case that ϕx = ∞ in the space between the curves representing the paths. This is abeautiful theorem whose not-too-difficult proof I will give at a suitable opportunity [. . . ]. In any case this makes it immediately clear why a function arising from an integral ϕx.dx can have many values for a single value of x, for one can go round a point where ϕx = ∞ either not at all, or once, or several times. For example, if one defines logx by dx x , starting from x = 1, one comes to logx either without enclosing the point x = 0 or by going around it once or several times; each time the constant +2πi or −2πi enters; so the multiple of logarithms of any number are quite clear." (Carl FriedrichGauss, [letter to Bessel] 1821)
"We will only announce to geometers a dozen memoirs by M. Cauchy on various questions in mathematical analysis. Without doubt the scholarly and industrious author of these writings does not count on many readers; a general desertion seems to condemn his works to disuse. He might as well content himself with knowing, and write nothing at all. The coldness in the mathematical public, which is not however the subject of a whim, is not a matter of indifference: if it was possible to know the motive, whatever it might be, one would know something more about the methods of the sciences or the profession of the scholar. It is not perhaps the first time that a remarkable and ever-active talent has entirely wasted its forces and its time – a strange phenomenon that one notices with regret." (Claude–Joseph Ferry, Revue encyclopedique 35, 1827)
"A theory of those functions [algebraic, circular or exponential, elliptical and Abelian] on the basis of the foundations here established would determine the configuration of the function (i.e., its value for each value of the argument) independently of any definition by means of operations [analytical expressions]. Therefore one would add, to the general notion of a function of a complex variable, only those characteristics that are necessary for determining the function, and only then would one go over to the different expressions that the function can be given." (Bernhard Riemann, "Theorie der Abel'schen Functionen", Journal für die reine und angewandte Mathematik 54, 1857)
"[...] gradually and unwittingly mathematicians began to introduce concepts that had little or no direct physical meaning. Of these, negative and complex numbers were most troublesome. It was because these two types of numbers had no 'reality' in nature that they were still suspect at the beginning of the nineteenth century, even though freely utilized by then. The geometrical representation of negative numbers as points or vectors in the complex plane, which, as Gauss remarked of the latter, gave them intuitive meaning and so made them admissible, may have delayed the realization that mathematics deals with man-made concepts. But then the introduction of quaternions, non-Euclidean geometry, complex elements in geometry, n-dimensional geometry, bizarre functions, and transfinite numbers forced the recognition of the artificiality of mathematics." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)
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