“The famous sixteenth-century algebraist Jerome Cardan called negative roots fictitious, and the founder of modern symbolic algebra, François Viète, discarded negative roots entirely. Descartes, called them false on the ground that they represented numbers less than nothing and so were meaningless.”

—  Morris Kline

Source: Mathematics and the Physical World (1959), p. 52.

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American mathematician 1908–1992

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“As for negative numbers… most mathematicians of the sixteenth and seventeenth centuries did not accept them… In the fifteenth century Nicolas Chuquet and, in the sixteenth, Stifel both spoke of negative numbers as absurd numbers. …Descartes accepted them, in part. …he had shown that, given an equation, one can obtain another whose roots are larger than the original one by any given quantity. Thus an equation with negative roots could be transformed into one with positive roots. Since we can turn false roots into real roots, Descartes was willing to accept negative numbers. Pascal regarded the subtraction of 4 from zero as utter nonsense.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 252.

“By 1700 all of the familiar members of the [number] system… were known. However, opposition to the newer types of numbers was expressed throughout the century. Typical are the objections of… Baron Francis Masères… in 1759 his Dissertation on the Use of the Negative Sign in Algebra… shows how to avoid negative numbers… and especially negative roots, by carefully segregating the types of quadratic equations so that those with negative roots are considered separately; and… the negative roots are to be rejected.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 592.

“One of the first algebraists to accept negative numbers was… placed negative numbers on a par with positive numbers and gave both roots of a quadratic equation, even when both were negative. Both Girard and Harriot used the minus sign for the operation of subtraction and for negative numbers.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), pp. 252-253.

“When an equation…clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. …Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 143.

David Eugene Smith photo

“The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, p.469

“The unnaturalness of mathematical symbolism is attested to by history. The algebra of the Egyptians, the Babylonians, the Greeks, the Hindus, and the Arabs was what is commonly called rhetorical algebra. …on the whole they used ordinary rhetoric to describe their mathematical work. Symbolism is a relatively modern invention of the sixteenth and seventeenth centuries…”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 59

“The Hindus introduced negative numbers… The first known use is Brahmagupta about 628; he also states the rules for the four operations with negative numbers. Bhāskara points out that the square root of a positive number is twofold, positive and negative. He brings up the matter of the square root of a negative number but says that there is no square root because a negative number is not a square. No definitions, axioms, or theorems are given.
The Hindus did not unreservedly accept negative numbers. Even Bhāskara, while giving 50 and -5 as two solutions of a problem, says, "The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions."”

Morris Kline (1908–1992) American mathematician

However, negative numbers gained acceptance slowly.
Source: Mathematical Thought from Ancient to Modern Times (1972), p. 185.

“The Hindus saw clearly that if the arithmetic operations… were properly defined for negative numbers, these numbers could be employed to as good advantage as people had previously derived from positive numbers. …To people to whom the word number had always meant positive whole numbers and positive fractions, the very idea that there could be other numbers came hard. For many centuries negative numbers were either rejected or treated as second-class citizens.
What was especially difficult for mathematicians to swallow was that negative numbers could be acceptable roots of equations.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 51.

“The chief innovator of symbolism in algebra was François Viète… an amateur in the sense that his professional life was devoted to the law… John Wallis… says that Viète, in denoting a class of numbers by a letter, followed the custom of lawyers who discussed legal cases by using arbitrary names [for the litigants]… and later the abbreviations… and still more briefly A, B, and C. Actually, letters had been used occasionally by the Greek Diophantus and by the Hindus. However, in these cases letters were confined to designating a fixed unknown number, powers of that number, and some operations. Viète recognized that a more extensive use of letters, and, in particular, the use of letters to denote classes of numbers, would permit the development of a new kind of mathematics; this he called logistica speciosa in distinction from logistica numerosa.”

Morris Kline (1908–1992) American mathematician

...the growth of symbolism was slow. Even simple ideas take hold slowly. Only in the last few centuries has the use of symbolism become widespread and effective.
Source: Mathematics and the Physical World (1959), p. 60

David Eugene Smith photo

“Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type x^3 + bx = c; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. …With respect to the irreducible case… we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic… before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2</sup”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, pp.461-464

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