Harriott. 17. Harriott, the companion of Sir Walter Raleigh in Virginia, and the friend of the Earl of Northumberland, in whose house he spent the latter part of his life, was destined to make the last great discovery in the pure science of algebra. Though he is mentioned here after Girard, since the Artis Analyticæ Praxis was not published till 1631, this was ten years after the author’s death. Harriott arrived at a complete theory of the genesis of equations, which Cardan and Vieta had but partially conceived. By bringing all the terms on one side, so as to make them equal to zero, he found out that every unknown quantity in an equation has as many values as the index of its powers in the first term denotes; and that these values, in a necessary sequence of combinations, from the co-efficients of the succeeding terms into which the decreasing powers of the unknown quantity enter, as they do also, by their united product, the last or known term of the equation. This discovery facilitated the solution of equations, by the necessary composition of their terms which it displayed. It was evident, for example, that each root of an equation must be a factor, and consequently a divisor, of the last term.[608]
[608] Harriott’s book is a thin folio of 180 pages, with very little besides examples; for his principles are shortly and obscurely laid down. Whoever is the author of the preface to this work cannot be said to have suppressed or extenuated the merits of Vieta, or to have claimed anything for Harriott but what he is allowed to have deserved. Montucla justly observes, that Harriott very rarely makes an equation equal to zero, by bringing all the quantities to one side of the equation.
18. Harriott introduced the use of small letters instead of capitals in algebra; he employed vowels for unknown, consonants for known quantities, and joined them to express their product.[609] There is certainly not much in this; but its evident convenience renders it wonderful that it should have been reserved for so late an era. Wallis, in his History of Algebra, ascribes to Harriott a long list of discoveries, which have been reclaimed for Cardan and Vieta, the great founders of the higher algebra, by Cossali and Montucla.[610] The latter of these writers has been charged, even by foreigners, with similar injustice towards our countryman; and that he has been provoked by what he thought the unfairness of Wallis to something like a depreciation of Harriott, seems as clear as that he has himself robbed Cardan of part of his due credit in swelling the account of Vieta’s discoveries. From the general integrity, however, of Montucla’s writings, I am much inclined to acquit him of any wilful partiality.
[609] Oughtred, in his Clavis Mathematica, published in 1631, abbreviated the rules of Vieta, though he still used capital letters. He also gives succinctly the praxis of algebra, or the elementary rules we find in our common books, which, though what are now first learned, were, from the singular course of algebraical history, discovered late. They are, however, given also by Harriott. Wallisii Algebra.
[610] These may be found in the article Harriott of the Biographia Britannica. Wallis, however, does not suppress the honour due to Vieta quite as much as is intimated by Montucla.
Descartes. 19. Harriott had shown what were the hidden laws of algebra, as the science of symbolical notation. But one man, the pride of France and wonder of his contemporaries, was destined to flash light upon the labours of the analyst, and to point out what those symbols, so darkly and painfully traced, and resulting commonly in irrational or even impossible forms, might represent and explain. The use of numbers, or of letters denoting numbers, for lines and rectangles capable of division into aliquot parts, had long been too obvious to be overlooked, and is only a compendious abbreviation of geometrical proof. The next step made was the perceiving that irrational numbers, as they are called, represent incommensurable quantities; that is, if unity be taken for the side of a square, the square-root of two will represent its diagonal. Gradually the application of numerical and algebraical calculation to the solution of problems respecting magnitude became more frequent and refined.[611] It is certain, however, that no one before Descartes had employed algebraic formulæ in the construction of curves; that is, had taught the inverse process, not only how to express diagrams by algebra, but how to turn algebra into diagrams. The ancient geometers, he observes, were scrupulous about using the language of arithmetic in geometry, which could only proceed from their not perceiving the relation between the two; and this has produced a great deal of obscurity and embarrassment in some of their demonstrations.[612]
[611] See note in Vol. II., p. 445.
[612] Œuvres de Descartes, v. 323.
His application of algebra to curves. 20. The principle which Descartes establishes is that every curve, of those which are called geometrical, has its fundamental equation expressing the constant relation between the absciss and the ordinate. Thus, the rectangle under the abscisses of a diameter of the circle is equal to the square of the ordinate, and the other conic sections, as well as higher curves, have each their leading property, which determines their nature, and shows how they may be generated. A simple equation can only express the relation of straight lines; the solution of a quadratic must be found in one of the four conic sections; and the higher powers of an unknown quantity lead to curves of a superior order. The beautiful and extensive theory developed by Descartes in this short treatise displays a most consummate felicity of genius. That such a man, endowed with faculties so original, should have encroached on the just rights of others, is what we can only believe with reluctance.
Suspected plagiarism from Harriott. 21. It must, however, be owned that independently of the suspicions of an unacknowledged appropriation of what others had thought before him, which unfortunately hang over all the writings of Descartes, he has taken to himself the whole theory of Harriott on the nature of equations in a manner which, if it is not a remarkable case of simultaneous invention, can only be reckoned a very unwarrantable plagiarism. For not only he does not name Harriott, but he evidently introduces the subject as an important discovery of his own, and in one of his letters asserts his originality in the most positive language.[613] Still it is quite possible that, prepared as the way had been by Vieta, and gifted as Descartes was with a wonderfully intuitive acuteness in all mathematical reasoning, he may in this, as in other instances, have struck out the whole theory by himself. Montucla extols the algebra of Descartes, that is, so much of it as can be fairly claimed for him without any precursor, very highly; and some of his inventions in the treatment of equations have long been current in books on that science. He was the first who showed what were called impossible or imaginary roots, though he never assigns them, deeming them no quantities at all. He was also perhaps the first who fully understood negative roots, though he still retains the appellation, false roots, which is not so good as Harriott’s epithet, privative. According to his panegyrist, he first pointed out that in every equation (the terms being all on one side) which has no imaginary roots, there are as many changes of signs as positive roots, as many continuations of them as negative.