To the Arabians we owe our knowledge of the rudiments of algebra; we owe to them the very name under which this branch of mathematics passes.  They had carefully added, to the remains of the Alexandrian School, improvements obtained in India, and had communicated to the subject a certain consistency and form.  The knowledge of algebra, as they possessed it, was first brought into Italy about the beginning of the thirteenth century.  It attracted so little attention, that nearly three hundred years elapsed before any European work on the subject appeared.  In 1496 Paccioli published his book entitled “Arte Maggiore,” or “Alghebra.”  In 1501, Cardan, of Milan, gave a method for the solution of cubic equations; other improvements were contributed by Scipio Ferreo, 1508, by Tartalea, by Vieta.  The Germans now took up the subject.  At this time the notation was in an imperfect state.

The publication of the Geometry of Descartes, which contains the application of algebra to the definition and investigation of curve lines (1637), constitutes an epoch in the history of the mathematical sciences.  Two years previously, Cavalieri’s work on Indivisibles had appeared.  This method was improved by Torricelli and others.  The way was now open, for the development of the Infinitesimal Calculus, the method of Fluxions of Newton, and the Differential and Integral Calculus of Leibnitz.  Though in his possession many years previously, Newton published nothing on Fluxions until 1704; the imperfect notation he employed retarded very much the application of his method.  Meantime, on the Continent, very largely through the brilliant solutions of some of the higher problems, accomplished by the Bernouillis, the Calculus of Leibnitz was universally accepted, and improved by many mathematicians.  An extraordinary development of the science now took place, and continued throughout the century.  To the Binomial theorem, previously discovered by Newton, Taylor now added, in his “Method of Increments,” the celebrated theorem that bears his name.  This was in 1715.  The Calculus of Partial Differences was introduced by Euler in 1734.  It was extended by D’Alembert, and was followed by that of Variations, by Euler and Lagrange, and by the method of Derivative Functions, by Lagrange, in 1772.

But it was not only in Italy, in Germany, in England, in France, that this great movement in mathematics was witnessed; Scotland had added a new gem to the intellectual diadem with which her brow is encircled, by the grand invention of Logarithms, by Napier of Merchiston.  It is impossible to give any adequate conception of the scientific importance of this incomparable invention.  The modern physicist and astronomer will most cordially agree with Briggs, the Professor of Mathematics in Gresham College, in his exclamation:  “I never saw a book that pleased me better, and that made I me more wonder!” Not without reason did the immortal Kepler regard Napier “to be the greatest man of his age, in the department to which he had applied his abilities.”  Napier died in 1617.  It is no exaggeration to say that this invention, by shortening the labors, doubled the life of the astronomer.