INTRODUCTION 401 It will be helpful,in appreciating the work and arguments of the eighteenth-century thinkers,to keep in mind that they did not distinguish between algebra and analysis.Because they did not appreciate the need for the limit concept and because they failed to recognize the problems intro- duced by the use of infinite series,they naively regarded the calculus as an extension of algebra. The key figure in eighteenth-century mathematics and the dominant theoretical physicist of the century,the man who should be ranked with Archimedes,Newton,and Gauss,is Leonhard Euler (1707-83).Born near Basel to a preacher,who wanted him to study theology,Leonhard entered the university at Basel and completed his work at the age of fifteen.While at Basel he learned mathematics from John Bernoulli.He decided to pursue the subject and began to publish papers at eighteen.At nineteen he won a prize from the French Academie des Sciences for a work on the masting of ships.Through the younger Bernoullis,Nicholas (1695-1726)and Daniel (1700-82),sons of John,Euler in 1733 secured an appointment at the St Petersburg Academy in Russia.He started as an assistant to Daniel Bernoulli but soon succeeded him as a professor.Though Euler passed some painful years (1733-41)under the autocratic government,he did an amazing amount of research,the results of which appeared in papers published by the Petersburg Academy.He also helped the Russian government on many physical problems.In 1741,at the call of Frederick the Great,he went to Berlin,where he remained until 1766.During this period,he gave lessons to the princess of Anhalt-Dessau,niece of the king of Prussia.These lessons, on a variety of subjects -mathematics,astronomy,physics,philosophy,and religion- -were later published as the Letters to a German Princess and are still read with pleasure.At the request of Frederick the Great,Euler worked on state problems of insurance and the design of canals and waterworks.Even during his twenty-five years in Berlin he sent hundreds of papers to the St. Petersburg Academy and advised it on its affairs In 1766,at the request of Catherine the Great,Euler returned to Russia, although fearing the effect on his weakened sight (he had lost the sight of one eye in 1735)of the rigors of the climate there.He became,in effect, blind shortly after returning to Russia,and during the last seventeen years of his life was totally blind.Nevertheless,these years were no less fruitful than the preceding ones.Euler had a prodigious memory and knew by heart the formulas of trigonometry and analysis and the first six powers of the first 100 prime numbers,to say nothing of innumerable pocms and the entire Eneid.His memory was so phenomenal that he could carry out in his head numerical calculations that competent mathematicians had difficulty doing on paper. Euler's mathematical productivity is incredible.His major mathematical fields were the calculus,differential equations,analytic and differential