Are there any 'lost' theorems of calculus that could be used to 'simplify' it? For example, are there ways to calculate derivatives without using limits, maybe by some forgotten methods in calculus?
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7$\begingroup$ "There is no royal road to geometry." [Euclid] $\endgroup$– Gerald EdgarCommented Feb 14, 2015 at 13:30
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1$\begingroup$ I'm more of a visual learner even though I love math yet why is it a lot of math books and those used as textbooks use a lot of boring black and white printed pages. I have a somewhat hyperactive mind which makes it hard to focus even when I like to study, also a soporific presentation found in many math books ( where they don't really care about any visual or 'informal' presentations) makes in hard to focus. There may be no 'easy' way to learn math but there may be ways with more visual and artistic approaches. $\endgroup$– 201044Commented Feb 15, 2015 at 16:20
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3$\begingroup$ @201044 Arnold quipped in a popular book "Bourbaki writes, somewhat derisively, that Barrow's book had 90 figures on twice as many pages. Bourbaki's own books have no figures on thousands of pages, and I don't know which is worse." books.google.com/books/about/… The irony is that the founding works of conventional calculus by Fermat, Newton and Leibniz are very visual and geometric, the analytization of teaching it happened exactly because that was considered the "royal road" to calculus. $\endgroup$– ConifoldCommented Feb 16, 2015 at 1:29
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2$\begingroup$ @201044 One of the best textbooks I have ever used was, "Calculus with Analytic Geometry; Second Edition" by Howard Anton. In my opinion, it was extremely well and painstakingly written. $\endgroup$– InquisitiveCommented Feb 16, 2015 at 19:26
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3$\begingroup$ @201044 Archimedes used limits implicitly. If you're familiar with how he calculated $\pi$, it was like that but more sophisticated and applied to volumes. $\endgroup$– Stella BidermanCommented Feb 7, 2017 at 4:18
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2 Answers
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As a matter of fact, there was something now called "lost calculus" or "algebraic calculus" in the 17th century, that avoided concepts like limits or infinitesimals, which where problematic at the time. It was developed by Descartes, Hudde and others, and is described in Suzuki's award winning paper The Lost Calculus (1637-1670): Tangency and Optimization without Limits. However, it only applied to algebraic functions, and it "simplifies" calculus only in the sense of avoiding more abstract concepts, rather than necessarily making computations simpler. So it was abandoned in favor of more general calculus of Newton and Leibniz based on infinitesimals, and later formalized using limits.
Descartes first introduced the idea in La Geometrie in 1637, and later simplified it to "the method of tangents" in 1638. Suppose we want to find the slope of the tangent to $y=x^2$ at $x=1$. The general equation of a line passing through $(1,1)$ is $y-1=m(x-1)$, and $m$ is the slope we are looking for. Since nearby lines intersect the graph at two points and the tangent only touches it at one algebraically the system
$$y=x^2,\ \ y-1=m(x-1)$$ must have a double root at $x=1$. Eliminating $y$ and factoring we get $$x^2-1-m(x-1)=(x-1)(x+1-m)=0,$$ so for $x=1$ to be a double root we must have $1-m=-1$ or $m=2$, which is the sought slope of the tangent.
This approach works for any polynomial $y=p(x)$, and more generally for rational and even algebraic functions given implicitly. For example, to find slope of the tangent at $x=a$ we write $y-p(a)=m(x-a)$ and look for the value of $m$ that makes $a$ a double root of $p(x)-p(a)=m(x-a)$. This only requires long division of $p(x)-p(a)$ by $x-a$, no limits or infinitesimals, and if the quotient is $q(x)$ then $m=q(a)$. To make the method computationally viable an algorithm more efficient than long division is needed for detecting double roots. Such method was provided by Jan Hudde, a talented Dutch mathematician who had to abandon mathematics for politics to save Netherlands from Spanish invasion, in two letters included into 1659 edition of Descartes' La Geometrie. It involves a clever modular reduction of polynomials that anticipates methods of modern algebraic geometry.
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1$\begingroup$ This would totally be worth teaching in a Calculus class.. $\endgroup$– hjhjhj57Commented Feb 16, 2015 at 2:33
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$\begingroup$ Could the 'lost calculus be adapted to apply to any type of function? Could it be made to 'cover' functions used in Lebesgue Integration? $\endgroup$– 201044Commented Oct 22, 2015 at 4:24
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1$\begingroup$ @201044 I don't think so. The notion of multiplicity can be stretched at most to analytic functions, and even there it becomes computationally unwieldy. $\endgroup$– ConifoldCommented Oct 25, 2015 at 0:25
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$\begingroup$ This is rather brilliant, and I feel enlightened for learning of it. $\endgroup$ Commented Feb 7, 2017 at 4:20
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$\begingroup$ I find this is a nice different way to come up with the formula, but in the end what you compute is $\displaystyle \lim_{x\to a} \frac{p(x)-p(a)}{x-a}$, just doing it "by hand" by performing long division of the polynomials and then substituting $x=a$, which after all is the standard method to compute such a limit taught early in a calculus course; the whole point of derivative rules is to avoid that cumbersome calculation. It's cool that people came up with fast algorithms for polynomial division, but for "real" applications it's obviously faster to compute these limits with the power rule. $\endgroup$ Commented Feb 28, 2019 at 22:05
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Fermat developed his method(s) of adequality in late 1620s. Fermat's manuscript was deposited in the library of d'Espagnet at Bordeaux. One of the methods involved the double-root idea for finding tangent lines, minima/maxima, and other applications. The method was communicated to Mercenne's circle in Paris around 1637. Jean Itard mentioned in his article from the 1970s that Fermat's method was superior to that of Descartes because the latter used normal directions to solve the same type of problems, whereas in truth the problems are affine in nature and should not involve finding the normal direction. Descartes' methods were therefore unnecessarily complicated. Conifold's answer is inaccurate on this point.
