“The remark which you make concerning roots that can not be extracted, and containing imaginary quantities which when added together give none the less a real quantity, is surprising and entirely new. One would never have believed that √(1 + √-3) + √(1 - √- 3) would make √6, and there is something hidden in this which is incomprehensible.” (Christaan Huygens, [letter to Gottfried W Leibniz] cca. 1670)
"And just as the advantage of decimals consists in this, that when all fractions and roots have been reduced to them they take on in a certain measure the nature of integers, so it is the advantage of infinite variable-sequences that classes of more complicated terms (such as fractions whose denominators are complex quantities, the roots of complex quantities and the roots of affected equations) may be reduced to the class of simple ones: that is, to infinite series of fractions having simple numerators and denominators and without the all but insuperable encumbrances which beset the others." (Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)
"But it is just that the Roots of Equation should be impossible, lest they should exhibit the cases of Problems that are impossible as if they were possible." [Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)
“We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative) […].These ‘Imaginary’ Quantities (as they are commonly called) arising from ‘Supposed’ Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible.” (John Wallis, "A Treatise of Algebra, Both Historical and Practical", 1673)
"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)
"For this evil I have found a remedy and obtained a method, by which without experimentation the roots of such binomials can be extracted, imaginaries being no hindrance, and not only in the case of cubics but also in higher equations. This invention rests upon a certain peculiarity which I will explain later. Now I will add certain rules derived from the consideration of irrationals (although no mention is made of irrationals), by which a rational root can easily be extracted from them." (Gottfried W Leibniz, cca. 1675)
"The nature, mother of the eternal diversities, or the divine spirit, are zaelous of her variety by accepting one and only one pattern for all things, By these reasons she has invented this elegant and admirable proceeding. This wonder of Analysis, prodigy of the universe of ideas, a kind of hermaphrodite between existence and non-existence, which we have named imaginary root?" (Gottfried W Leibniz, "De Bisectione Latereum", 1675)
"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, in "Treatise of Algebra", 1685)
“Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality.” (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)
No comments:
Post a Comment