11.9. SOME HISTORICAL OBSERVATIONS 259

11.9 Some Historical ObservationsAs mentioned earlier, one of the ill defined notions in calculus was the infinitesimal, dx.What is it? No one knew what exactly it was. It wasn’t any nonzero real number and itwasn’t 0 either. However, people thought in terms of dy

dx and this was the derivative so theywished to understand the quotient of these unknown things. Gradually it became clear thatwhatever meaning the quotient had, it was closely connected to the methods for finding itand these methods eventually became the definition of its meaning, being formalized as theconcept of limit. This was done by Bolzano early in 1800’s.

Even though the notion of dx was not very well defined, the notation turned out to bevery useful as in the methods presented above for changing variables in an integral.

The concept of an integral also developed gradually. It was possible to consider mostof the physical applications in terms of an initial value problem for an unknown functiony satisfying y′ (x) = f (x) ,y(0) = y0 and this is essentially what was done in the 1700’s,but this did not resolve fundamental questions concerning the existence of the integral. Ofcourse this was impossible without a careful definition of what was meant by the integralwhich did not exist at that time. These kinds of questions were not considered very muchin the 1700’s and were first addressed by Cauchy around 1823 who considered what wecall one sided Riemann sums for continuous functions. Since such a definition gives theintegral for continuous functions, Cauchy’s proof of the fundamental theorem of calculuswas the first one which was complete although it is not clear whether he had all the detailsregarding uniform continuity, a concept developed later by Weierstrass. It is unsatisfactoryto prove a theorem about something you have not defined precisely and before Cauchy, thiswas the state of the fundamental theorem of calculus. Riemann’s improved description ofthe integral dates from around 1854 and was completed later by Darboux who proved thetheorem about his integral and the Riemann integral being equivalent.

Newton discovered the binomial theorem for (1+ x)α in 1665. It is certainly a mar-velous thing, but the importance of this and other power series tended to be over empha-sized for much of the 1700’s. Power series became much more understandable with theinvention and development of complex analysis. This subject was continually expandedduring the 1800’s starting with Cauchy and continuing with most of the other mathemati-cians of that century.

What we now refer to as real analysis began early in the 1800’s with the work ofBolzano. It was an effort to make calculus rigorous by removing intuitive geometric rea-soning. Later on Weierstrass found nowhere differentiable continuous functions and Peanofound examples of space filling continuous curves. Weierstrass also showed the impor-tance of uniform convergence and uniform continuity. Eventually calculus was brought toits present form through his efforts. Of course the entire subject is built on completenessof R. Dedekind and Cantor constructed R from the rational numbers in 1872 althoughDedekind did it earlier in 1858, but before this time, the mathematicians of that centuryused the essential characteristics of R in their development of calculus. Dedekind, Cantor,and Weierstrass completed the removal of geometry from the foundations of calculus.