tuting, in the place of the unknown quantity, one of the values given by these component equations, the totality of the terms of the equation proposed becomes equal to nought. And these theorems have greatly facilitated the complete resolution of some particular equations, as well as other researches. Harriot was born in 1560, and died in 1621.

To the general advancement of the science of analysis no one has contributed more than our illustrious des Cartes, who was born in 1596, and died in 1650. Nature had bestowed on him the genius and energy necessary to extend all the boundaries of human knowledge. In his Method, he taught mankind the art of seeking truth; and in his mathematical works he illustrated his precepts by his example. The glory he has acquired by his writings will never perish, because the truths he discovered are truths for all ages; though it must be avowed, that most of his philosophical systems, begotten by Imagination, and contradicted by Nature, have already disappeared, and produced no advantage, except that of having abolished the tyranny of the peripatetic philosophy.

Algebra stands indebted to him for several important discoveries. He introduced, instead of the repeated multiplications of one and the same letter, the notation of the powers by exponents, which simplifies the calculation, and has been the germe of the method for developing the radical quantities in series. The analysts who preceded him were unacquainted with the use of negative roots in equations, and rejected them as useless he showed, that they are all as real, and as proper for resolving a question, as positive roots;

the

the distinction that ought to be made between them having no other foundation, than the different manner of considering the quantities, of which they are the symbols. He taught how to distinguish, in an equation containing real roots only, the number of positive roots, and that of negative roots, by the combination of the signs which precede the terms of the equation. The method of indeterminates, of which Vieta had a glimpse, was developed by des Cartes, who made a clear and distinct application of it to equations of the fourth order: he assumes, that the general equation of this order is the product of two equations of the second, which he affects with indeterminate coefficients; and by the comparison of the terms of this product with those of the equation proposed, he arrives at an equation reducible to the third order, which gives the unknown coefficients. This method is applicable to an infinite number of problems in all parts of the mathematics.

I shall not here mention several learned algebraists, who, soon after the death of des Cartes, studied and even improved his methods. One, however, deserves particular notice, the celebrated Hudde, a burgomaster of Amsterdam, who died at a very advanced age in 1704. He published in 1658, in Schooten's Commentary on the Geometry of des Cartes, a very ingenious method of discovering whether an equation of any order contain several equal roots, and of de-. termining these roots.

Pascal, who died in 1662 at the age of thirty-nine, opened to himself a new path in analysis by his wellknown Arithmetical Triangle. This is a kind of ge

nealogical

nealogical tree, in which, by means of an arbitrary number written at the point of the triangle, he forms suc cessively, and in the most general manner, all the figurate numbers; and determines the ratios, which the numbers of any two cases have to each other; with the different sums, which must result from the addition of all the numbers in one rank, taken in whatever direction you please. He afterward makes many interesting applications of these principles.

That in which he determines the odds, to be laid between two persons playing at various games, deserves particularly to be noticed, since it gave rise to the calculation of probabilities in the theory of games of chance. Some authors have ascribed the elements of this calculation to Huygens, who published in 1657 an excellent treatise, entitled, De Ratiociniis in Ludo Alea: but Huygens himself informs us, with a modesty worthy of so great a man, that this subject had already been handled by the greatest geometricians in France, and that he laid no claim to the honour of the invention. In fact we see by the letters betweeen Pascal and Fermat, printed in Fermat's Works, that the principles of the arithmetical triangle were spread through France as early as 1654, though the tracts in which Pascal explains them at large were not published till after his death.

While Pascal was thoroughly investigating the nature of figurate numbers at Paris, Fermat discovered several elegant properties of them at Toulouse, by pursuing a different method. These two great men frequently hit on the same things in the course of their researches: and this was so far from altering the friendship, to which the similarity of their stu

dies had given birth without their having ever seen each other, that each rendered the other that disinterested justice, to which mediocrity is a stranger.

The predilection of Fermat for numerical researches led him particularly to the theory of prime numbers, which had not yet been examined, and in which he made profound discoveries. It is known, that every number is no more than a ratio of the unit: but it is often difficult to perceive, whether this ratio be simple, or produced by the multiplication of several others. Fermat established general and distinguishing characters, calculated to discriminate, on a variety of occasions, those numbers which have divisors, from those which have none. The Analysis of Diophantus equally exercised his genius. Bachet de Meziriac, editor and commentator of the greek algebraist, had already resolved several new problems dependent on the doctrine of his author; but Fermat carried the same subject still farther. Since which time, all these researches have been extended and improved by other great geometricians.

In 1655, Wallis, an english mathematician, whom I have already quoted, published his Arithmetic of Infinites: a work full of genius, and the object of which, like that of the arithmetical triangle, was to determine the sum of different series of numbers. By this method curves are squared, when the ordinates are expressed by a single term: as may likewise curves with complex ordinates, by resolving these ordinates into series of which each term is a monomial. The author's dispute with Pascal on the subject of the cycloid shall be mentioned hereafter. Wallis was a profound analyst:

analyst to him we are indebted for the notation of radicals by fractional exponents; as we are for that of negative exponents; des Cartes having employed exponents for whole and positive powers only.

As the path of truth is beset with difficulties, at which the feebleness of the human understanding is perpetually liable to stumble, the means of avoiding them, or of approaching our object when it is impossible to reach it completely, cannot be rendered too numerous. Such is the advantage procured by the theory of continued fractions, when an irreducible fraction is expressed by numbers too great to be employed in practice, in it's direct form. Instead of a complicated expression it substitutes, one that is simple, and nearly equivalent. William lord Brounker, viscount of Castle Lyons in Ireland, who was born about 1620, and died in 1684, gave the elements of this theory; which was afterward enlarged, improved, and applied to various important uses, by Huygens and other celebrated geometricians.

All these particular branches of analysis did not induce mathematicians to lose sight of the problem of the general resolution of equations. Newton, who was then young, was a long while engaged in this research he did not find it; but in other respects he considerably enlarged the bounds of algebra. He gave a method for decomposing an equation into commensurable factors, whenever it is possible; a method extending to all the orders, and as simple as can be desired he summed up the powers of the roots of an equation, be they what they might: he taught the