"[…] although the symbol √-1 be beyond the power of arithmetical computation, the operations in which it is introduced are intelligible, and deserve, if any operations do, the name of reasoning." (Robert Woodhouse,"On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"By use of the symbol √-1 and of the forms proved to obtain in the combination of real quantities, a mode of notation is obtained, by which we may express sines and cosines, relatively to their arc." (Robert Woodhouse,"On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
“Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers” (Carl F Gauss, “Disquisitiones arithmeticae” [“Arithmetical Researches”], 1801)
"The application of imaginary quantities to the theory of equations, has perhaps been made more extensively than to any other part of analysis. To consider the propriety of this application on the grounds of perspicuity and conciseness, a long discussion would be necessary. I may, however, be here permited merely to state my opinion, that impossible quantities must be employed in the theory of equations, in order to obtain general rules and compendious methods." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The introduction of impossible quantities, is assigned as a great and primary cause of the evils under which mathematical science labours. During the operation of these quantities, it is said, all just reasoning is suspended, and the mind is bewildered by exhibitions that resemble the juggling tricks of mechanical dexterity." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The operations performed with imaginary characters, though destitute of meaning themselves, are yet notes of reference to others which are significant. They, point out indirectly a method of demonstrating a certain property of the hyperbola, and then leave us to conclude from analogy, that the same property belongs also to the circle. All that we are assured of by the imaginary investigation is, that its conclusion may, with all the strictness of mathematical reasoning, be proved of the hyperbola; but if from thence we would transfer that conclusion to the circle, it must be in consequence of the principle just now mentioned. The investigation therefore resolves itself ultimately into an argument from analogy; and, after the strictest examination, will be found without any other claim to the evidence of demonstration." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The symbol √-1 might arise from translating questions of which the statement involved a contradiction of ideas into algebraic language, and reasoning on them, as if they really admitted a solution." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"Whether or not I have found a logic, by the role of which operations with imaginary quantities are conducted, is not now the question. but surely this is evident that since they lead to right conclusions they must have a logic! […] Till the doctrines of negative and imaginary quantities are better taught than they are at present taught in the University of Cambridge, I agree with you that they had better not be taught [...]" (Robert Woodhouse, [letter to Baron Meseres] 1801)
"√-1 is therefore not the sign of an arithmetic operation, nor of an arithmetic-geometric operation, but of a purely geometric operation. It is a sign of perpendicularity." (Adrien Q Buée, "Memoire sur les Quantités Imaginaires", Philosophical Transactions of the Royal Society, 1806)
“At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes (real values of the argument) and regard the imaginary values as just vestigial –or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications. “ (Carl F Gauss, [letter to Bessel,] 1811)
"What should one understand by ∫ ϕx · dx for x = a + bi? Obviously, if we want to start from clear concepts, we have to assume that x passes from the value for which the integral has to be 0 to x = a + bi through infinitely small increments (each of the form x = a + bi), and then to sum all the ϕx · dx. Thereby the meaning is completely determined. However, the passage can take placein infinitely many ways: Just like the realm of all real magnitudes can be conceived as an infinite straight line, so can the realm of all magnitudes, real and imaginary, be made meaningful by an infinite plane, in which every point, determined by abscissa = a and ordinate = b, represents the quantity a+bi. The continuous passage from one value of x to another a+bi then happens along a curve and is therefore possible in infinitely many ways. I claim now that after two different passages the integral ∫ ϕx · dx acquires the same value when ϕx never becomes equal to ∞ in the region enclosed by the two curves representing the two passages." (Carl F Gauss, [letter to Bessel] 1811)
"Now as far as the arithmetical signs for addition, multiplication, etc. are concerned, I believe we shall have to take the domain of common complex numbers as our basis; for after including these complex numbers we reach the natural end of the domain of numbers." (Gottlob Frege, [letter to Peano])
"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813)
"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)
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