Mathematicians of the early seventeenth century provided a wealth of analysis infinite and infinitesimal methods, in algebraic methods, and in functional concepts upon which the invention and development of calculus was based. Among the major players were Christian Huygens, Bonaventura Cavalieri, and John Wallis. Remarkably, once again we see Pierre Fermat making his usual profound contributions, this time to early analysis.

Most readers of mathematical history are surprised by how much calculus was actually invented before Newton and Leibnitz took their turn. Let us note that by this time symbolism is well developed and its implications are well under development. With symbols, many, many expressions take on a compact form that reveal structure such as symmetries and formulas. The easiest formula to imagine is the exponent rule for powers. Moreover, the addition, multiplication and division of polynomials becomes transparently simple. We see at this time missing formal structures of proof, such as induction, are still missing --- but applied nonetheless.

What Newton and Leibnitz accomplished, both nearly the same, essentially independently, and at the most extreme level, must remain one of the truly remarkable achievements of all time. Though we hardly do justice to their work here, subsequent chapters build upon their foundation.


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