Paul J. Nahin Quotes

• “The next, magnificent step would of course have been to write , but the Stereometria records it as , and so Heron missed being the earliest known scholar to have derived the square root of a negative number in a mathematical analysis of a physical problem.”

  Paul J. Nahin

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• “French mathematician Jacques Hadamard (1865–1963): “The shortest path between two truths in the real domain passes through the complex domain.” I”

  Paul J. Nahin

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• “In particular, the three roots to a cubic polynomial either must all be real, or there must be one real root and one conjugate pair.”

  Paul J. Nahin

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• “The quadrature of a circle, the construction by straightedge and compass alone of the square equal in area to the circle,”

  Paul J. Nahin

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• “the quadrature problem is a measure of the greatest difficulty, since it was shown in 1882 to be impossible.”

  Paul J. Nahin

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• “mean proportional, i.e., if b and c are two given positive numbers, then x is the mean proportional of b and c if it satisfies the statement “b is to x as x is to c.” Or, in algebra which”

  Paul J. Nahin

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• “The conclusion is that the original assumption of ordering leads us into contradiction, and so that assumption must be false.”

  Paul J. Nahin

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• “That is, he would have started over from the beginning to solve with, again, both p and q non-negative. This is totally unnecessary, however, as at no place in the solution to x3 + px = q did he ever actually use the non-negativity of p and q. That is, such assumptions have no importance, and were explicitly made simply because of an unwarranted aversion by early mathematicians to negative numbers. This”

  Paul J. Nahin

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• “the tangent function is periodic with period 180°,”

  Paul J. Nahin

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• “for the irreducible case with all three roots real, there is just one positive root; that is, the root given by the Cardan formula”

  Paul J. Nahin

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• “The breakthrough for came not from quadratic equations, but rather from cubics which clearly had real solutions but for which the Cardan formula produced formal answers with imaginary components.”

  Paul J. Nahin

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• “the product of two sums of two squares of integers is always expressible, in two different ways, as the sum of two squares of integers.”

  Paul J. Nahin

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