Ramble, Part 28: Symbols of Change
Jul. 20th, 2007 12:06 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowHaving devoted something like two and a half posts to Newton, I should probably give Gottfried Leibniz his turn. Of Leibniz' contributions to calculus, I'm not going to say much; philosophically, his work suffered from the same general defects as Newton's, and there's not much point in rehashing that. I do, however, want to say a few words about the one respect in which Leibniz' formulation is clearly superior to Newton's, under the cut.
Calculus is the mathematics of change. Newton, in introducing the notions of fluent and fluxion, tacitly restricted this to change with respect to time. The fluxion of a fluent x, denoted by placing a dot over the variable, was its time rate of change; other rates of change could be computed, by taking ratios of fluxions, but the notation gave primacy to the time rate.
Leibniz chose a different tack. He introduced "differentials" as being infinitesimal changes in his variables, and derivatives as ratios of differentials; this is the familiar dy/dx notation. Though philosophically suspect, this symbolism was considerably more flexible than Newton's. Furthermore, it was suggestive: the chain rule, in Leibniz' notation, takes the form dz/dx = dz/dy dy/dx. Though we do not, at present, regard derivatives as ratios, still, having the chain rule encoded in a natural-looking form is worthwhile.
In Newton's notation, higher-order fluxions were denoted by placing additional dots over the fluent; in Leibniz', the second derivative, for example, takes a form like d2y/dx2. The advantage is rather like that of Hindu-Arabic notation over tally marks. Additionally, it becomes possible (using Leibniz' notation) to write, say, the n-th derivative without any special difficulty; how such a concept could be written using Newton's method is not at all obvious.
Unfortunately for the British mathematical community, the controversy between Newton and Leibniz over priority took on a nationalistic character; for the next century or more, British mathematicians confined themselves to Newton's inferior notation, while their counterparts on the Continent used that of Leibniz (and, later, that of Lagrange). In the nineteenth century, the Analytical Society of Cambridge, founded by (among others) Charles Babbage, finally broke with tradition, advocating "the principles of pure d-ism as opposed to the dot-age of the university".
Today, there is a profusion of different ways of representing the derivative. Some - mostly engineers and scientists - still use Newton's notation, and Leibniz' notation is still popular. The "prime" notation introduced by Lagrange is perhaps the most widespread, but I know of at least two other forms which have their proponents. This lack of consensus suggests to me that a truly satisfactory notation has yet to be devised, more than three centuries after the development of calculus!
Previous Next
Ramble Contents
Calculus is the mathematics of change. Newton, in introducing the notions of fluent and fluxion, tacitly restricted this to change with respect to time. The fluxion of a fluent x, denoted by placing a dot over the variable, was its time rate of change; other rates of change could be computed, by taking ratios of fluxions, but the notation gave primacy to the time rate.
Leibniz chose a different tack. He introduced "differentials" as being infinitesimal changes in his variables, and derivatives as ratios of differentials; this is the familiar dy/dx notation. Though philosophically suspect, this symbolism was considerably more flexible than Newton's. Furthermore, it was suggestive: the chain rule, in Leibniz' notation, takes the form dz/dx = dz/dy dy/dx. Though we do not, at present, regard derivatives as ratios, still, having the chain rule encoded in a natural-looking form is worthwhile.
In Newton's notation, higher-order fluxions were denoted by placing additional dots over the fluent; in Leibniz', the second derivative, for example, takes a form like d2y/dx2. The advantage is rather like that of Hindu-Arabic notation over tally marks. Additionally, it becomes possible (using Leibniz' notation) to write, say, the n-th derivative without any special difficulty; how such a concept could be written using Newton's method is not at all obvious.
Unfortunately for the British mathematical community, the controversy between Newton and Leibniz over priority took on a nationalistic character; for the next century or more, British mathematicians confined themselves to Newton's inferior notation, while their counterparts on the Continent used that of Leibniz (and, later, that of Lagrange). In the nineteenth century, the Analytical Society of Cambridge, founded by (among others) Charles Babbage, finally broke with tradition, advocating "the principles of pure d-ism as opposed to the dot-age of the university".
Today, there is a profusion of different ways of representing the derivative. Some - mostly engineers and scientists - still use Newton's notation, and Leibniz' notation is still popular. The "prime" notation introduced by Lagrange is perhaps the most widespread, but I know of at least two other forms which have their proponents. This lack of consensus suggests to me that a truly satisfactory notation has yet to be devised, more than three centuries after the development of calculus!
Previous Next
Ramble Contents
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2007-07-21 06:21 pm (UTC)no subject
Date: 2007-07-21 09:43 pm (UTC)no subject
Date: 2007-07-23 11:52 am (UTC)I seem to remember having seen both notations, but mostly remember having used dy/dx. What's the "prime" Lagrange notation look like?
I'm really enjoying these when I get the chance to read them!
PS love the deism vs dotage ;-)
no subject
Date: 2007-07-23 01:23 pm (UTC)I'm glad you enjoy these!
no subject
Date: 2007-07-23 01:31 pm (UTC)Ah, right. I'm pretty sure we used that too. :-D