On Numbers: On Complex Numbers (1750-1799)

"Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such numbers, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in the imagination." (Leonhard Euler, "Vollständige Anleitung zur Algebra", 1768-69)

“All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.” (Leohnard Euler, "Algebra" , 1770)

"I consider it as one of the most important steps made by Analysis in the last period, that of not being bothered any more by imaginary quantities, and to be able to submit them to calculus, in the same way as the real ones." (Joseph-Louis de Lagrange, [letter to Antonio Lorgna] 1777)

"In the following I shall denote the expression √-1 by the letter i so that i*i =-1.” (Leohnard Euler, "De formulis differentialibus angularibus" Vol. IV, 1794)

"Certain authors who seem to have perceived the weakness of this method assume virtually as an axiom that an equation has indeed roots, if not possible ones, then impossible roots. What they want to be understood under possible and impossible quantities, does not seem to be set forth sufficiently clearly at all. If possible quantities are to denote the same as real quantities, impossible ones the same as imaginaries: then that axiom can on no account be admitted but needs a proof necessarily." (Carl F Gauss, "New proof of the theorem that every algebraic rational integral function in one variable can be resolved into real factors of the first or the second degree", 1799)

"How these magnitudes of which we can form no idea whatsoever - these shadows of shadows - are to be added or multiplied cannot be understood with the kind of clarity required by mathematics." (Carl F Gauss, 1799) [critisizing Euler's proof in effect made the assumption that "every equation can be satisfied by a real value of the unknown, or by an imaginary value of the form a + b√-1, or by a value that is not subsumed under any form." (see David A Cox' "Galois Theory" 2nd Ed., 2012)]