till Leibnitz, long after this period, proved, by developing these expressions by means of the binomial theorem, that in every case these results can be converted into real roots.

Soon afterwards, Scipio Ferrei, and Louis Ferrari, a disciple of Tartaglia, discovered similar methods for the solution of Biquadratic Equations. Methods depending on the same principle, which may be considered to be the preceding method generalised, were given by Simpson and Waring. Descartes and Euler also discovered different methods for the solutions of these equations. All these methods, however, depend on the solution of an equation of the third degree, called the "reduced equation;" and, consequently, the difficulty arising from the irreducible case is common to equations of the third and fourth degree.

The science now began to be cultivated in Germany, and its notation was considerably improved by the ingenuity of Stifel, a Protestant clergyman, who published an excellent treatise of Algebra, entitled Arithmetica Integra (1544). He first introduced numeral exponents to denote the integral powers of quantities, and negative exponents to represent their reciprocals. Simon Stevin, of Bruges (1605), extended the improvements in the notation of powers by introducing fractional exponents. Stifel invented the positive and negative signs, and, and the radical sign, √, and employed the capital letters A, B, C, to denote unknown quantities. He also discovered the properties of figurate numbers, and showed how the coefficients for the different powers of a binomial can be obtained from the arithmetical triangle; a method of arranging the figurate numbers, the invention of which is improperly ascribed to Pascal.

...

At this time it appears the science was known in England, a treatise having been published by Robert Recorde, a Welshman, quaintly entitled "The Whetstone of Witte" (1557). The only improvements contained in it are the method of extracting roots of compound quantities, the introduction of the terms binomial and residual, and the sign of equality, = In a work published nearly at the same time (1558), by James Pelletier, a native of Mons, in France, it is proved, for the first time, that every rational

root of an equation is a divisor of its last term; and he also gives methods for reducing binomial and trinomial radicals to their simplest form, by multiplying them by proper factors. Vieta, a celebrated French mathematician of Fontenoy (about 1600), who was by profession a lawyer, possessed of great learning, and of profound talents and originality, first generalised the notation of Algebra, by introducing capital letters to denote known as well as unknown quantities, the consonants being appropriated to the former, and the vowels to the latter, quantities. Previous to this improvement, algebraists considered only numerical equations, and could have no formulas for the values of their roots, although the method of solving one numerical equation might be easily applied to the solution of similar equations; but the coefficients of equations being now represented by letters, thus forming literal equations, the solutions were now capable of being expressed in a general manner by proper formulas. This notation of Vieta, therefore, produced an entire change in the character of Algebra, by generalising its results. He also invented methods for performing certain preliminary operations on equations, to facilitate their solution, as the method of removing the second term, and of multiplying or dividing their roots by any number; and also invented new methods for the solution of equations of the third and fourth degree. He also introduced the terms affirmative and negative, pure and adfected, and coefficient. Simon Stevin, of Bruges, an eminent engineer, invented fractional exponents, as before stated, and a general method for the solution of Numerical Equations (1605). His works were edited by Albert Girard, also a Belgian, who published several improvements and discoveries in Algebra (1629). He discovered the composition of the coefficients of an equation in terms of the roots; that the number of roots of an equation is the same as the exponent of its degree; and he resolved the irreducible case of cubics by means of a table of sines; and also determined the sums of the powers of the roots of an equation, in terms of its coefficients, which was also effected afterwards by Newton. Girard also introduced the terms less than nothing and impossible guantities.

The science was now advancing in England. Harriot made several important improvements in the notation, and discoveries in the theory of equations, which were published by his friend Walter Warner (1631). He employed small letters instead of capitals to represent quantities, the vowels as formerly representing the unknown quantities; but he did not use the concise notation for powers invented by Stifel nearly a century before. Harriot also invented the signs of majority and minority, > and <. He was the first to arrange all the terms of an equation on one side, and to equate their sum with zero, = = 0. By this mode of arrangement he was enabled to discover the important theorem, that an equation is the product of as many simple binomial factors as there are units in the exponent of its degree, and also the composition of its coefficients, as was before done by Girard, and that the last term of an equation is the product of its roots with their signs changed. By means of Harriot's theorems has been effected the complete resolution of some particular equations. Harriot also improved the approximate methods of solving numerical equations.

In a work published by William Oughtred, another English mathematician, the first instance is afforded of employing geometry in the investigation of properties of geometrical figures, and also of the use of the sign of multiplication, X. He used the sign for continued proportion, and also the signs for common proportion, except that he placed only one dot instead of two between the first and second, and the third and fourth terms thus, a.b::c.d. By the radical sign, with q placed over it, he denoted the square root; with c over it, the cube root; and with qq, the fourth root. He is also the first known to express decimal fractions without their denominators, by using a parenthesis instead of the decimal point [thus, 34(28 would denote 34-287

No mathematician contributed more to the advancement of this now rapidly progressive science than Descartes, by the publication of his geometry (1637).* Besides improv

See the History of Geometry prefixed to the Plane Geometry of Chambers's Educational Course.

ing considerably the methods of solution and management of equations previously known, he treated more particularly of the negative and imaginary roots of equations. He discovered the important theorem, called "the rule of signs," that in an equation whose roots are all real, the number of positive roots is equal to the number of variations of the signs of its terms taken in succession, and the number of the negative roots to that of the permanencies of signs. The truth of this theorem was questioned, till a satisfactory demonstration of it was given by De Gua. He also invented the method of Indeterminate or rather Undetermined Coefficients, of such extensive use in the higher mathematics; and he applied this method successfully to the solution of equations of the fourth degree. He also effected some improvements in the notation, by appropriating the first letters of the alphabet to known, and the last to unknown, quantities. Descartes was also devoted to the study of metaphysics; and although his philosophical systems were mere creatures of the imagination, inconsistent with nature, and soon vanished from the creed of orthodox philosophy, yet his speculations were most efficient in undermining that tyrannical influence of the mere authority of a system, by which, with the aid of ignorance and superstition, the human mind was enchained during the long period of the intellectual despotism of the Peripatetic philosophy.

An edition of the work of Descartes was published (1659) by Francis Schooten, who was mathematical professor in the University of Leyden, in which are inserted some posthumous tracts of De Beaune, an eminent mathematician, and friend of Descartes, on the Theory of Equations, containing methods for determining the limits of the greatest and least roots. To this edition were also appended some articles by Hudde, a burgomaster of Amsterdam, relating to the method of drawing tangents to curves, and to the theory of maxima and minima, and the resolution of equations having equal roots, which he effected by means of the relations that he discovered between an equation of this kind, and what is termed its limiting equation.

Fermat, a contemporary of Descartes, is understood to have invented the method of representing curve lines by

algebraical equations, and thence deducing their geometrical properties, before the publication of the work of the latter, and he thus has a rival claim for the honour of inventing the method of Analytical Geometry.

The analogous theories of indivisibles, and of the arithmetic of infinities, were published by their respective inventors, Cavalieri and Wallis, in 1635 and 1655. The latter method was supported by more satisfactory reasoning than the former.* By his method of infinities, Wallis greatly extended the calculation of the more difficult kinds of series, and was the first to discover a series expressing the area of a circle. Wallis also perfected the hitherto incomplete notation by fractional exponents.

Abraham De Moivre, of Champagne, published a tract on the solution of several particular equations of odd degrees (1707); and some years before, he published a method of determining a root of an infinite equation, and of developing any powers, or extracting any roots, of polynomials or infinite series; and he thus laid the foundation of the Combinatorial Analysis.

Notwithstanding all the zealous and vigorous efforts of analysts, there still remained a formidable stumblingblock to obstruct their progress-the general solution of equations. Newton applied all the resources of his profound and comprehensive mind to the subject, but his attempt was unsuccessful, though his labour was not lost; for he extended considerably the theory and solution of equations. He invented a method by which the roots of numerical equations of all degrees may be found to any degree of approximation, which is effected by successive substitution; and showed how to decompose, when it is possible, an equation into rational factors; and he found also the sums of any powers of the roots of an equation, which was, however, effected before by Girard; and the theory of elimination is indebted to him for its origin and greatest progress; a method afterwards extended by Bezout, who reduced it to a system. The binomial theorem was partially understood before the time of Newton, in the case of positive integral powers, although the coefficients were found by a tedious

* See the History of Geometry formerly referred to.