"From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised." (Gottfried W Leibniz, "Specimen novum analyses pro Scientia infinity circa summas et quadraturas", 1700)
"[…] even if someone refuses to admit infinite and infinitesimal lines in a rigorous metaphysical sense and as real things, he can still use them with confidence as ideal concepts (notions ideales) which shorten his reasoning, similar to what we call imaginary roots in the ordinary algebra, for example, √-2." (Gottfried W Leibniz, [letter to Varignon], 1702)
"Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity." (Gottfried W Leibniz, [letter to Varignon], 1702)
“Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being.” (Gottfried Leibniz, 1702)
“But it is just that the Roots of Equations should be often impossible (complex), lest they should exhibit the cases of Problems that are impossible as if they were possible." (Isaac Newton, “Universal Mathematic” 2nd Ed., 1728)
“[…] such numbers, which by their natures are impossible, are ordinarily called imaginary or fanciful numbers, because they exist only in the imagination.” (Leohnard Euler, 1732)
“For it ought to be considered that both –b and –c , as they stand alone, are, in some Sense, as much impossible Quantities as √(-b) and √(-c) ; since the Sign –, according to the established Rules of Notation, shews the Quantity, to which it is prefixed, is to be subtracted, but to subtract something from nothing is impossible, and the Notion or Supposition of a Quantity actually less than Nothing, absurd and shocking to the Imagination.” (Thomas Simpson, “A Treatise of Algebra”, 1745)
"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, and each 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)
“After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential." (Leonhard Euler, ”Introductio in analysin infinitorum”, 1748
“Even zero and complex numbers are not excluded from the signification of a variable quantity.” (Leonhard Euler, “Introductio in Analysin Infinitorum” Vol. I, 1748)
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