392 MATHEMATICS AS OF 1700 became not only an effective methodology for its own ends but also the superior approach to the solution of geometric problems.The greater effec- tiveness of analytical methods in the calculus decided the competition,and algebra became the dominant substance of mathematics. It was Wallis and Newton who saw clearly that algebra provided the superior methodology.Unlike Descartes,who regarded algebra as just tech nique,Wallis and Newton realized that it was vital subject matter.The work of Desargues,Pascal,and La Hire was depreciated and forgotten,and the geometric methods of Cavalieri,Gregory of Saint Vincent,Huygens,and Barrow were superseded.Pure gcometry was eclipsed for about a hundred years,becoming at best an interpretation of algebra and a guide to algebraic thinking through coordinate geometry.It is true that excessive reverence for Newton's geometrical work in the Principia,reinforced by the enmity against the Continental mathematicians engendered by the dispute between Newton to what the Continentals were able to achieve using the analytical approach. What was evident by 1700 was explicitly stated by no less an autho prity than Euler,who,in his Introductio in Analysin Infinitorum (1748),praises algebra as far superior to the synthetic methods of the Greeks It was with great reluctance that mathematicians abandoned the geo- metric approach.According to Henry Pemberton (1694-1771),who edited the third edition of Newton's Principia,Newton not only constantly expressed great admiration for the geometers of Greece but censured himself for not ollowing them more closely than he did.In a letter to David Gregory (1661-1708),a nephew of James Gregory,Newton remarked that"algebra is the analysis of the bunglers in mathematics."But his own Arithmetica Universalis of 1707 did as much as any single work to establish the supremacy of algebra.Here he set up arithmetic and algebra as the basic science, allowing geometry only where it made demonstrations easier.Leibniz,too. noted the growing dominance of algebra and felt obliged to say,in an un- published essay,1 "Often the geometers could demonstrate in a few words what is very lengthy in the calculus.the view of algebra is assured,but it is not better.' Another,more subtle,change in the nature of mathematics had beer unconsciously accepted by the masters.Up to 1550 the concepts of mathe- matics were immediate idealizations of or abstractions from experience.By that time negative and irrational numbers had made their appearance and were gradually gaining acceptance.When,in addition,complex numbers, an extensive algebra employing literal coefficients,and the notions of deriva tive and integral entered mathematics,the subject became dominated by 1.Couturat,L.:Opuscules et fragments inddits de Leibniz,1903,reprinted by Georg Olms, 1961,P.181