method, except that of Briggs, who knew how to derive any term of a power independently of preceding powers; but Newton extended it to the case of fractional and negative powers, and discovered the concise rule for determining the coefficients, but he left no demonstration of this theorem, and he had probably arrived at it by a process of induction. The subject of Series was cultivated by several eminent mathematicians. Halley laid the foundation of logarithmic series, a subject that was afterwards developed in a superior manner by Lagrange, who also improved its notation. De Moivre first discovered recurring series, and the method of summing them, and Newton invented the reversion of series. James Bernouilli, Nicole, Taylor, Stirling, Maclaurin, Lambert, Landen, and Waring, contributed to these researches. But the subject was chiefly extended and generalised, with the aid of common Algebra, by Riccati, D. Bernouilli, and Euler. Montmort and James Bernouilli applied series to the investigation of the doctrines of chance; and this method was very much extended by De Moivre in his researches in probabilities and contingent annuities. The theory of numbers advanced with the progress of Algebra. Lord Napier discovered the singular relation between two corresponding series, the one an equidifferent, and the other an equirational, series, on which he founded his method of Logarithms, which was adapted to the decimal system of numeration by Briggs. Lord Brounker, who was originally a clergyman, and obtained his living through his interest with the Long Parliament, was the inventor of Continued Fractions (1670). He discovered a fraction of this kind that expressed the ratio of the square of the diameter of a circle to its area. This subject was further extended by Wallis; but Euler has the merit of laying the foundation of the theory of this subject in a published memoir (1737). He also contributed some of the most important theorems on this subject. He converted a series for the quadrature of the circle into the continued fraction given by Lord Brounker for this purpose; and a series expressing the length of a quadrantal are, discovered by Gregory and Leibnitz, was converted by Lambert into a continued fraction, by means of which he proved the incommensurability of the diameter and circumference of a circle, and also of their squares (1761). Lagrange also applied the principle of continued fractions to the solution of equations; the value of the root, by his method of solution, being exhibited in the form of a continued fraction. Gauss published a work on the theory of numbers, which also contained the complete theory and solution of binomial equations of different orders. By means of an ingenious theory, founded on the indeterminate analysis, Gauss was enabled, without using equations higher than quadratics, to discover a method of inscribing a regular polygon of seventeen sides in a circle; a problem previously considered impossible. He also established, by a more simple process, the theorem of Cotes, which had before been generalised by De Moivre. Legendre has also published important researches on this subject (1808). The theory of Indeterminate Equations of the second degree, and particularly a branch of this subject sometimes called the Diophantine Analysis, was greatly advanced by Euler in his Algebra (1770); and in an edition of it published in France, the same subject was extended by Lagrange, who also added to it an excellent article on Continued Fractions. The analytical branch of trigonometry, which was founded by Girard, was improved by De Moivre and John Bernouilli, who contributed the singular exponential formulas for sines and cosines. But the subject was completed chiefly by Euler and Lagrange, the former of whom invented its convenient notation. The general solution of numerical equations, including the determination of the real roots, and of the number of imaginary roots, was first completely effected by Lagrange. His method is one of successive substitution, and it exhibits the values of the roots in the form of a continued fraction. He determined the number of real roots by means of an auxiliary equation, called "the equation of the squares of the differences of the roots." Newton and Waring had, however, previously made observations that suggested this equation; as the former had remarked, that the separation of the real roots depended on finding a number less than the least of the differences between every two roots; and Waring had stated in a work of his, that this number could be determined by means of the above auxiliary equation. But Lagrange was not acquainted with this work of Waring when he published his method. This method of Lagrange, though perfectly rigorous, is so excessively tedious for equations higher than the fourth degree, that it is practically useless. Methods for calculating the real roots, but not for determining the number of imaginary roots, have been published by Fourier, Holdred, and Atkinson; but another method has also been published by Horner, which, from its superior simplicity and conciseness, supersedes every other. Budan discovered a method for determining the number of real and imaginary roots; but a method more compendious in practice, and more interesting in a theoretical point of view, has been discovered by Sturm. The problem, therefore, of the general solution of numerical equations, is now satisfactorily solved by the method of Horner, combined with the theorem of Sturm. The still more difficult problem of the general solution of literal equations yet remains to engage the labours of analysts, for the purpose either of obtaining a complete solution of it, or else of proving its impossibility. All the important steps in the progress of the science, and the names of the distinguished algebraists who have contributed to its advancement, have been enumerated in this outline of its history; but many names have been omitted of authors who have been eminent for their extensive knowledge of Algebra. It would be tedious, in the present state of the student's knowledge of the science, to enter into more minute details, as he will be better qualified, after he has made greater progress in its study, to appreciate a more complete history of its progressive advancement. |