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A Euclidean Approach to the FTC - Some Conclusions

Andrew Leahy (Knox College)

So why hasn't Gregory received the same sort of accolades usually given to Newton and Leibniz? Probably the most persuasive argument against Gregory's role in the discovery of the FTC is that he didn't realize what he had. For Gregory, the FTC is essentially a technical lemma which is used without comment in the course of proving a result which most today would find uninteresting. It isn't even stated explicitly. However, instead of arguing this point, I'll let you decide for yourself. Follow this link  for a translation of the proposition where the FTC occurs and this link  for the Latin original.

Also, as noted above, Gregory's proof of the FTC is not without its problems, so some may point to a lack of rigor. But given the standards of rigor of the day, Gregory's omissions in the proof were probably not considered serious. In Gregory's time Euclid was the model of mathematical rigor, although he makes the same sorts of oversights even in the first proof he presents in the Elements . Because it makes use of these Euclidean techniques, the geometrical work of Gregory was undoubtedly considered very rigorous, especially when compared to the proofs offered by Newton and the other analysts. It would be 150 years before the analytic methods they used were given the rigorous "d-e" foundation known today.

On the other hand, it's likely that one reason Gregory's work is largely forgotten is precisely because he avoided the limit. Euclidean geometry is a mathematics for describing static relations between unchanging objects. But calculus today is often called "the language of change". Newton, Leibniz, and the other analysts are celebrated for introducing an entirely new and much more powerful way of looking at the same problems. It was this way of thinking later codified in the limit we learn today that was paramount, because it gave mathematics a new-found power to model the constantly changing world we live in.

But this also points to a reason why Gregory's work should be interesting for educators. For many students, Euclidean geometry is simply a stepping stone to the vastly more powerful tools of analytic geometry, useful only for studying simple relations between points, lines, and circles. But the work of Gregory shows just how rich the elementary language of geometrical similarity really is. Indeed, Gregory's geometrical arguments are interesting precisely because they push the language of Euclidean geometry further than most would have believed possible.

Andrew Leahy (Knox College), "A Euclidean Approach to the FTC - Some Conclusions," Loci (August 2010), DOI:10.4169/loci002156