266 HISTORY OF SCIENCE. 



who published a treatise QT\ Fluxions, in which the demonstrations in- 

 cluded no supposition of infinitely small quantities. D'Alembert showed 

 the real application, of the principle of limits ; and, finally, Lagrange 

 was able to discard limits also, by reducing the whole theory of the dif- 

 ferential calculus into an elementary, but protracted, algebraical in- 

 vestigation by developing functions in series. 



It would be impossible, within our limits, to convey an idea of the 

 great improvements which were effected during the eighteenth century 

 in every branch of mathematical analysis. The theory of algebraical 

 equations received great development by the labours of several eminent 

 mathematicians, more particularly of the indefatigable EULER (1707 

 1 783)5 whose life presents one of the most remarkable examples of 

 scientific labour. He composed, it is said, more than one-half of the 

 papers on mathematical subjects which occupy forty-six quarto volumes, 

 published by the Academy of Sciences of St. Petersburg, and at his death 

 he left nearly one hundred memoirs ready for the press. He contri- 

 buted during his lifetime to the journals of various scientific societies, 

 besides publishing twenty-nine quarto and two octavo volumes in Latin, 

 six octavo volumes in German, and five in French. He lost his sight, 

 it is said, by his continually writing and calculating ; but after this 

 misfortune, which occurred in his fifty-ninth year, he continued to cal- 

 culate as actively as before, and dictated his books instead of writing 

 them. His " Elements of Algebra" was produced in this way; and 

 Euler's amanuensis, who was only a tailor's apprentice, ignorant of 

 algebra when he began his task, is said to have acquired a competent 

 knowledge of algebra by merely taking clown Euler's words, such was 

 the clearness and simplicity of the book. These qualities caused it to 

 become widely known, and there is no European language into which 

 it has not been translated. 



The application of algebra to geometry led to the study of a great 

 number of different curves, as we have already seen, and this branch 

 of mathematics was largely cultivated in the eighteenth century; almost 

 new fields of research were now opened out by the extension of the 

 principle of co-ordinates to lines of double curvature, and to various 

 kinds of surfaces. The annals of mathematics would furnish a long 

 series of valuable labours in every branch of the science ; but as these 

 could not be made intelligible to a general reader by much less than 

 a complete treatise on mathematics, they must here be passed over. 

 The names of the most distinguished mathematicians of the eighteenth 

 century include Euler, the two brothers John and James Bernouilli, 

 Laplace, Lagrange, Clairaut, Brook Taylor, Maciaurin, Stirling, Cotes, 

 Saurin, Robins, Demoivre, D'Alembert, Daniel and Nicolas Bernouilli, 

 etc. The last-named published some essays on the " Doctrine of Chances 

 or Probabilities," and the theory received most important development 

 from Demoivre. The theory of probabilities is not a mere mathema- 

 tical curiosity, but one of great utility, not only in the determination 



