Ramble, Part 22: One Graph Is Worth...
Apr. 24th, 2007 10:45 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI'll return to Descartes in the next post, but this is the appropriate place to mention another of Pierre de Fermat's contributions.
Fermat's role in the development of analytic geometry is not as well known as Descartes', but it deserves some attention on a couple of counts.
First: although Descartes did some significant work in algebra, his approach to analytic geometry came from the geometric side; his interest was in applying algebra to solve geometric problems. I alluded to this in the last Ramble post, in connection with his choice of a coordinate system. The familiar system of rectangular coordinates that goes by his name was not, in fact, his, and is in some respects contrary to his own geometric concerns.
Fermat arrived at roughly the same place as Descartes, but from the opposite side. His interest was in gaining insight into algebraic problems by geometric means. Since there was no pre-existing geometric content, he was free to impose whatever coordinates he found convenient, and the right-angled coordinate system was thus his innovation, not Descartes'.
Second: coming from the algebraic side as he did, Fermat was the one who developed the notion of graphing. He was not, I should mention, graphing functions - that notion had not yet appeared - but rather the relation between two variables, one independent, the other dependent on the first.
This notion paid an immediate dividend. After experimenting with graphing polynomial relations for a time, Fermat noticed that, whenever the dependent variable achieved a maximum or minimum value, the tangent to the graph was horizontal; phrased algebraically, this meant that, at such a value, a small change in the independent variable produces little or no change in the dependent variable. This led him to the following procedure for locating a maximum or minimum value.
Suppose, for example, that you wish to divide a quantity B into two parts, so that their product is a maximum. (Fermat used Viète's notation, with consonants for constants and vowels for variables.) Let one of the parts be A; the other is B-A. Now reduce A by a small amount E; the product changes from A(B-A) to (A-E)(B-(A-E)). Equating these, expanding, and cancelling, we obtain the equation 2AE-BE-E2=0. We divide by E: 2A-B-E=0. Finally, set E=0 to obtain 2A=B, and thus the maximum is achieved by dividing B into two equal parts.
Note that this argument is logically flawed; at one step, Fermat divides by E, but in the next he sets E to 0. This problem, in various guises, was to recur repeatedly over the next couple of centuries, until the notion of "limit" was finally clarified by Cauchy. Rigorous or not, though, the method worked, and in the springtime of the new era, this was enough.
It should be mentioned that what Fermat had found was, essentially, the First Derivative Test, well-known to calculus students. Although calculus had not been unified and formalized yet, elements of the subject were beginning to appear, which Newton and Leibniz were to draw upon in the next generation.
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Fermat's role in the development of analytic geometry is not as well known as Descartes', but it deserves some attention on a couple of counts.
First: although Descartes did some significant work in algebra, his approach to analytic geometry came from the geometric side; his interest was in applying algebra to solve geometric problems. I alluded to this in the last Ramble post, in connection with his choice of a coordinate system. The familiar system of rectangular coordinates that goes by his name was not, in fact, his, and is in some respects contrary to his own geometric concerns.
Fermat arrived at roughly the same place as Descartes, but from the opposite side. His interest was in gaining insight into algebraic problems by geometric means. Since there was no pre-existing geometric content, he was free to impose whatever coordinates he found convenient, and the right-angled coordinate system was thus his innovation, not Descartes'.
Second: coming from the algebraic side as he did, Fermat was the one who developed the notion of graphing. He was not, I should mention, graphing functions - that notion had not yet appeared - but rather the relation between two variables, one independent, the other dependent on the first.
This notion paid an immediate dividend. After experimenting with graphing polynomial relations for a time, Fermat noticed that, whenever the dependent variable achieved a maximum or minimum value, the tangent to the graph was horizontal; phrased algebraically, this meant that, at such a value, a small change in the independent variable produces little or no change in the dependent variable. This led him to the following procedure for locating a maximum or minimum value.
Suppose, for example, that you wish to divide a quantity B into two parts, so that their product is a maximum. (Fermat used Viète's notation, with consonants for constants and vowels for variables.) Let one of the parts be A; the other is B-A. Now reduce A by a small amount E; the product changes from A(B-A) to (A-E)(B-(A-E)). Equating these, expanding, and cancelling, we obtain the equation 2AE-BE-E2=0. We divide by E: 2A-B-E=0. Finally, set E=0 to obtain 2A=B, and thus the maximum is achieved by dividing B into two equal parts.
Note that this argument is logically flawed; at one step, Fermat divides by E, but in the next he sets E to 0. This problem, in various guises, was to recur repeatedly over the next couple of centuries, until the notion of "limit" was finally clarified by Cauchy. Rigorous or not, though, the method worked, and in the springtime of the new era, this was enough.
It should be mentioned that what Fermat had found was, essentially, the First Derivative Test, well-known to calculus students. Although calculus had not been unified and formalized yet, elements of the subject were beginning to appear, which Newton and Leibniz were to draw upon in the next generation.
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no subject
Date: 2007-04-29 12:49 am (UTC)Pity Fermat didn't get credit for the standard coordinate system. Moral lesson: don't deface books.
no subject
Date: 2007-04-29 10:16 am (UTC)no subject
Date: 2007-04-29 11:05 am (UTC)*acknowledges distinction and retreats to meditate*
no subject
Date: 2007-04-29 11:06 am (UTC)no subject
Date: 2007-04-29 11:09 am (UTC)*wanders off to make coffee*