Cardan, who was born in 1501, and died in 1576, relates in his book De Arte magna, published in 1545, that Scipio Ferrei, professor of mathematics at Bologna, was the first who gave the formula for resolving equations of the third order: that about thirty years afterward, Florido, a venetian, informed of this discovery by his master Ferrei, proposed to Nicholas Tartaglia, a celebrated mathematician of Brescia, several problems, the solution of which depended on this formula: and that Tartaglia discovered it, by meditating on these problems. In another place Cardan confesses, that, at his urgent intreaty, Tartaglia communicated to him this formula, but without adding the demonstration: and that, having discovered this demonstration, with the assistance of his scholar Lewis Ferrari, a young man of great penetration, he thought it his duty, to give the whole to the public. But Tartaglia was much displeased with the conduct of Cardan. He claimed the sole invention of the formula; maintained, that Florido himself was unacquainted with it; and asserted, that Cardan was guilty both of treachery and plagiarism, in having made public a rule entrusted to him under a promise of secrecy, and to which he had not the least claim.

The resolution of equations of the fourth order closely followed that of equations of the third. And here too, we learn from Cardan, that Lewis Ferrari made this new discovery. His method, known at present to all analysts by the name of the italian method, consisted in disposing the terms of the equation of the fourth order in such a manner, that, by adding to each side

2

one

one and the same quantity, the two sides might be resolved by the method of the second order. By ac complishing this condition we are brought to an equation of the third order, so that the complete resolution of the fourth order is connected with that of the third, the difficulties of which equally affect the other. I say the difficulties; for in fact there is one case in the third order, which has embarrassed all algebraists, and for this reason is called the irreducible case. This case embraces equations of which the three roots are real, unequal, and incommensurable to each other: also the formule, which represent them, include imagiuary terms; and we should at first be induced to believe, that these expressions were imaginary, if an attentive examination of their nature did not prevent us from being too hasty in our judgment. Tartaglia and Cardan did not venture to decide upon the subject. The latter only applied himself to the resolution of some particular equations, which appeared referrible to it, and in which the difficulty fortuitously disappeared.

Raphael Bombelli of Bologna, who was a little posteriour to Cardan, first showed in his algebra, printed in 1579, that the terms of the formula which represents each root in the irreducible case, form when taken together a real result in all cases. This proposition was at that time an actual paradox: but the paradox disappeared, when Bombelli had demonstrated by geometrical constructions, nearly of the same nature as those which Plato had employed to find the two mean proportionals in the problem of the duplication of the cube, that the imaginary quan

tities included in the two sides of the formula must necessarily destroy each other by the opposite nature of their signs. In support of this general demonstration the author produced several particular examples, in which, by extracting the cube roots of the two binomials, which compose the value of the unknown quantity, according to the usual methods for known quantities, and then adding the two roots, real results were obtained. Mathematicians have since arrived at the same conclusion by other means more direct and simple; but this first effort of Bombelli was a great step in the analysis of equations for the time.

It was natural to suppose, that the methods for the third and fourth orders should be extended farther, or at least give rise to new views respecting the forms of roots in still higher orders. But if we except those equations, which by transformations in their calculation are reducible, in their ultimate analysis, to the first four orders, the art of resolving equations in general, and with the utmost rigour, has made no progress since the labours of the italians just mentioned.

Maurolicus, abbot of Santa Maria del Porto in Sicily, who was born in 1494, and died in 1575, was profoundly skilled in all parts of the mathematics. He applied himself to another branch of analytical calculation, at that time almost unknown. This was the summation of several series of numbers, as the series of natural numbers, that of their squares, that of triangular numbers, &c. On this subject he gave

theorems remarkable for subtilty of invention and simplicity of result.

The reader will perceive, that I feel pleasure in doing justice to learned foreigners: but the same equity requires me, to ascribe to one of my illustrious countrymen the glory of having generalized the algorithm of algebra, and made several important discoveries in the science. This was Vieta, born in

1540, and who died in 1603. Before him equations of the kind called numerical were alone resolved: the unknown quantity was represented by a particular character, or by a letter of the alphabet; and the other quantities were absolute numbers. It is true, that the method applied to one equation could afterward be applied equally to another similar equation. But it was desirable, that all the quantities indifferently, should be represented by general characters, and that all the particular equations of the same order should be only simple translations of one general formula. This advantage Vieta conferred on algebra, by introducing into it the letters of the alphabet, to represent all sorts of quantities, known or unknown: an easy and commodious method of notation, both because the use of letters is very familiar to us, and because a letter may express indifferently a weight, a distance, a velocity, &c. He himself applied this new algorithm very happily in several cases. He taught us to make various transformations in equations of every order, without knowing their roots; to deprive them of the second term; to free them from fractional coefficients; to increase or diminish the roots by a given

a given quantity; and to multiply or divide the roots by any number whatever. He also gave a new and ingenious method for resolving equations of the third and fourth order. Lastly, in default of a rigorous resolution of equations of all orders, he came to an approximate resolution; which is founded on this principle, that any given equation is but an imperfect power of the unknown quantity: and he employs nearly. the same modes of proceeding, as for finding by approximation the roots of numbers which are not perfect powers. If we at present possess more simple and commodious means of attaining the same end, we ought not the less to admire these first efforts of genius.

About the same time several algebraists published very useful treatises for diffusing a knowledge of the science, though they contained no new views, that have any thing in them remarkable.

The commencement of the seventeenth century is marked by the noble discovery of logarithms, which has rendered, and will never cease to render the most important services to all the practical parts of science, and particularly to astronomy, by it's abridgment of numerical calculations, without which the most unconquerable patience must have been obliged to relinquish a number of useful researches. For this invention we are indebted to baron Napier of Merchiston in Scotland, who was born in 1550, and died in 1617.

Every one knows, that of the four fundamental rules of arithmetic, addition, subtraction, multiplication, and division, the former two are easily performed with