IQ Calculus in the Eighteenth Century And therefore,whether the mathematicians of the present age act like men of science,in taking so much more pair their principles than to understand them. s to apply BISHOP BERKELEY 1.Introduction The greatest achievement of the seventeenth century was the calculus.From this source there stemmed major new branches of mathematics:differential equations,infinite series,differential geometry,the calculus of variations functions of complex variables,and many others.Indeed,the beginnings of sme of these subjects were already present in the works of Newton and Leibniz.The eighteenth century was devoted largely to the development of some of these branches of analysis.But before this could be accomplished,the calculus itself had to be extended.Newton and Leibniz had created basic methods,but much remained to be done.Many new functions of one variable and functions of two or more variables had either to be recognized explicitly or created;the techniques of differentiation and integration had to be extended to some of the existing functions and to others yet to be intro- duced;and the logical foundation of the calculus was still missing.The first goal was to expand the subject matter of the calculus and this is the subject of the present chapter and the next one The eighteenth-century men did extend the calculus and founded new branches of analysis,though encountering in the process all the pangs errors,incompleteness,and confusion of the creative process.The mathe maticians produced a purely formal treatment of calculus and the ensuing branches of analysis.Their technical skill was unsurpassed;it was guided, however,not by sharp mathematical thinking but by intuitive and physical insights.These formal efforts withstood the test of subsequent critical examination and produced great lines of thought.The conquest of new domains of mathematics proceeds somewhat as do military conquests.Bold dashes into enemy territory capture strongholds.These incursions must then be followed up and supported by broader,more thorough and more cautious operations to secure what has been only tentatively and insecurely grasped. 400