Science, however, was the gainer; and this theorem, known by the name of "Cardan's rule," at this day marks a point beyond which no efforts of subsequent algebraists have been able to carry them in the solution of cubic equations; and, indeed, but little success has attended their researches in the general exact solution of equations of higher degrees. Cardan pointed out the approximation to the roots of those cubic equations which his rule did not solve. He also clearly understood some of the leading principles of the roots, and the formation of the co-efficients, when we examine the forms of equations in general synthetically. This must have been more difficult than we can readily conceive, owing to the imperfection of the algebraic language. The usual practice of the day was to put the rules into verse. Cardan delivered his in a poetical dress. The unavoidable obscurity of such a mode far more than counterbalances any advantage it may give to the memory. Indeed, even in this respect, a symbolical formula must far surpass any versification.

Algebra was at the same time cultivated in Germany. Stiphelius, in a work on that science published at Nuremburg in 1544, employed integer numeral exponents of powers, both positive and negative; and introduced the same characters for plus and minus which we now use. In equations he did not go beyond the second degree.

Robert Recorde, an Englishman, of about the same date, published the earliest treatise on algebra in the English language, in which the sign of equality first appears.

The properties of algebraic equations were discovered very slowly. Pelitarius, a French mathematician, in a treatise bearing the date of 1558, first observed that the root of an equation is a divisor of the last term. He also remarked the curious property of numbers, that the sum of the cubes of the natural numbers is the square of the sum of the numbers themselves.

Bombelli, in Italy, wrote on algebra; and pointed out that the problems involving the irreducible case of Cardan's rule, admit of geometrical construction by the

trisection of a circular arc. He also mentions a manuscript of Diophantus in the Vatican, in which the invention of algebra is ascribed to the Indians. Nothing of this, however, appears in the works of Diophantus as subsequently printed. This point remains without explanation.

Among the mathematicians of the 16th century, none, perhaps, has secured a more lasting reputation than Vieta, a native of Fontenoy, who is entitled by Riccioli the great ornament of French science. He was equally remarkable for industry and originality. He advanced the algebraic notation by the introduction of letters to stand for the known as well as the unknown quantities, so that it was in his hands that the language of algebra first became capable of expressing general truths. He delivered a rule for solving some cases of biquadratic equations. And in the case of an equation of any degree, complete with all its terms (that is, containing in separate terms all the powers of the unknown quantity up to the highest, by which the degree of the equation is designated); and when the roots are all positive, he discovered the relation between the roots and the coefficients of the terms. Thus was another step slowly taken towards the complete theory.

Vieta was not less celebrated for the improvements he introduced into trigonometry: and in his treatise on angular sections, made a most important application or algebra to the theorems and problems of geometry. He restored some of the books of Apollonius in a manner highly creditable to his own ingenuity, though not precisely accordant with the taste and style of his author. He also directed his attention to astronomical subjects, and submitted to Pope Clement VIII. a plan for reforming the Calendar. He is said also to have composed an important astronomical work, called "Harmonicon Caleste," which being lent to a friend, was, by some unprincipled rival, surreptitiously taken from him, and destroyed or suppressed. His mathematical treatises

were first published about 1590, and afterwards collected by Schooten in one volume, in 1646.

About the same period algebra became greatly indebted to Albert Girard, a Flemish mathematician; though his principal work, "Invention Nouvelle en Algebre," was not printed till 1669. This ingenious author perceived to a greater extent than Vieta had done, but still not in all its generality, the principle of the successive formation of the co-efficients of an equation from the sum of the roots; the sum of their products taken two and two, the same taken three and three, &c., whether the roots be positive or negative. He also conceived the notion of imaginary roots; and showed that the number of the roots of an equation cannot exceed its dimensions. To these speculations he added some others connected with the use of negative roots in geometrical constructions, &c. He also first introduced the phrase of "a quantity less than nothing," which has been so severely censured by some mathematicians, who cavil at terms without considering that they only become fair subjects of question when the wrong use of the term is likely to involve a mistake in the thing. This phrase, of course, never implies any thing more than the very conception by which negative quantities take their origin; the supposed increase of the quantity subtracted till it becomes equal to, and then greater than, that from which it is to be taken. The variable value of the whole is thus said to pass through zero, and then become negative; or, in this figurative sense, less than zero. That is, it becomes a quantity at variance with the first hypothesis; but which may still enter into our equation, if we change the signs of all the terms; or, which comes to the same thing, if the signs remain, and we allow the fiction of a quantity essentially negative. This is, in fact, one of those simple but refined artifices arising out of the symbolical language of algebra, which has been so mystified by persons of obscure conceptions, and ignorantly

copied from one treatise to another, as to have become involved in real obscurity, and to have given ground for serious accusation against the science.

The greatest advance in this part of the science of quantity was effected by Thomas Harriot, who was born at Oxford in 1560. Having graduated in the university in 1579, he was employed afterwards in the second expedition sent out by Sir W. Raleigh to Virginia; and on his return published an account of that country. He afterwards devoted himself entirely to the study of mathematics and astronomy. His principal work is entitled “ Artis Analyticæ Praxis," which was not published till after his death, in 1631. In this work we find, for the first time, the complete developement of the apparently very simple truth, to which Vieta, Girard, and others we have named, had been so long making their successive approximations; viz. the formation of general equations of all degrees, by the multiplication together of as many simple equations as amount to their dimension, or the highest power of the unknown quantity involved. This, of course, includes the case where one or more of such factors is an equation of a higher degree than the first, but where, in general, the number of factors is such that the sum of their dimensions is the dimension of the resulting equation. The slow progress made in the developement of this principle, which, when understood, does not appear to be of any very abstruse or difficult kind, is a remarkable fact in the history of discovery. It would seem as if these labourers in science had been working with, and gradually improving upon, an instrument whose full powers they did not yet understand. It is to the further extension of those powers, and the full confidence with which the analyst may trust himself to be passively borne along by the apparatus he has thus set in motion, that the great modern discoveries are owing.

Harriot introduced a minor improvement in the facility of the notation, by using the small Italic letters

instead of Roman capitals, which had hitherto been employed; thus bringing the notation almost exactly into its present form. His labours were encouraged, and even a maintenance afforded him by that liberal patron of science and letters, Henry Percy, earl of Northumberland.*

Progress of Optics, Mechanics, &c.

Maurolycus, already mentioned as a mathematician, was also distinguished in optics. He conceived the use of the crystalline lens in the eye, though not the office of the retina; and the application of lenses for remedying both long and short sight. In his work, "Theoremata de Lumine et Umbrâ” (1575), he gives an explanation of the fact noticed by Aristotle, that the light of the sun passing through a small hole of whatever shape, always gives a circular illuminated space on a screen at a little distance. The rays from the different parts of the sun's disk cross at the aperture (which we will suppose to be, for example, triangular), and each ray gives a small triangular bright spot on the screen; these being partially superposed, but arranged in the form of the sun's disk, will give an image sensibly circular; and the more accurately so as the hole is smaller, or the screen more distant.

This principle is precisely that of the camera obscura in its simplest form, which was invented about this time (1560) by Baptista Porta, who described it in a work entitled "Magia Naturalis." The subsequent addition of a lens at the aperture, only substitutes the artificial crossing of the rays at the centre of the lens, and at the same time increases the quantity of light. He perceived, in general, the resemblance between this construction and that of the eye; but extraordinary as it may seem, failed in tracing the place of the image,

Some highly interesting particulars of his analytical labours will be found in Prof. Rigaud's "Supplement to the Works of Bradley." Oxf. 1833, pp. 43. 52, &c. To his astronomical pursuits we shall refer in another place.