Understanding Leibniz Notation: A Physics History Lesson

[clarkie_49]

Hi all,

While reading through physics history book, i en counted an attempt to show the very basics of Leibniz notation; the following is shown:

If
s = (1/2)gt^2, then
s + ds = (1/2)g(t + dt)^2
s + ds = (1/2)gt^2 + gtdt + (1/2)gdt^2, then because of the first line
ds = gtdt + (1/2)gdt^2

It then goes on to say (1/2)gdt^2 "can be ignored because it is so small". Obviously this is not really correct, and i was looking for a way to explain using limits. I came up with something, but it seems incorrect in some way?

If
ds = gtdt + (1/2)gdt^2, then
(ds/dt) = gt + (1/2)gdt . Let (ds/dt) = f, then
lim f {dt -> 0} = gt

However, the confusion is that this does not seem to imply
(ds/dt) = gt as is normally written; it seems to imply
lim f_{dt -> 0} = lim (ds/dt)_{dt -> 0} = gt

Can somebody please explain how Leibniz would have came to (ds/dt) = gt without "ignoring (1/2)gdt^2" ? Was the notion of a limit used back then?

Thanks in advance,

Brendan

 

[HallsofIvy]

Hi all,

While reading through physics history book, i en counted an attempt to show the very basics of Leibniz notation; the following is shown:

If
s = (1/2)gt^2, then
s + ds = (1/2)g(t + dt)^2
s + ds = (1/2)gt^2 + gtdt + (1/2)gdt^2, then because of the first line
ds = gtdt + (1/2)gdt^2

It then goes on to say (1/2)gdt^2 "can be ignored because it is so small". Obviously this is not really correct, and i was looking for a way to explain using limits. I came up with something, but it seems incorrect in some way?

If
ds = gtdt + (1/2)gdt^2, then
(ds/dt) = gt + (1/2)gdt . Let (ds/dt) = f, then
lim f {dt -> 0} = gt

However, the confusion is that this does not seem to imply
(ds/dt) = gt as is normally written; it seems to imply
lim f_{dt -> 0} = lim (ds/dt)_{dt -> 0} = gt

Can somebody please explain how Leibniz would have came to (ds/dt) = gt without "ignoring (1/2)gdt^2" ? Was the notion of a limit used back then?

Thanks in advance,

Brendan

Two important points: this was a physics book and a history book. Neither one of those implies you are going to get a rigorous, modern derivation. Both Leibniz and Newton used what they called "infinitesmals" which had certain properties (if I recall correctly Leibniz and Newton gave their "infinitesmals" slightly different properties). Essentially infinitesmals (Bishop Berkeley famously referred to infinitesmals as "ghosts of vanished quantities") were very, very small quantities that could be "ignored" in respect to "normal quantities" but no in comparison to other infinitesmals. Squares of infinitesmals were another order of "smallness" (this is a hierachichal system) that could be dropped in comparison to infinitesmals. That is why x+ dx= x but (x+ dx)- x= dx is not 0.

No, limits were not used back then. It was conceptual problems with "infinitesmals" that led to the use of limits instead. Interestingly, recent work in symbolic logic has allowed mathematicians to define "infinitesmals" so that we can restore that method- that's "non-standard" analysis. We don't teach it in college courses because it requires, as I said, deep results from symbolic logic and "model theory" and it gives exactly the same results as calculus based on limits.

 

[clarkie_49]

Thanks for your help HallsofIvy,

Once again this forum has been a great help! Your explanation was very good, particularly the part on powers being a "hierarchical system".

It's funny how things pop up; i have had Henley and Kleinberg's infinitesimal calculus on my amazon wish-list for a while now, though i did not plan to get the book or touch on the subject until i had completed multivariable calculus (semester 1, 2010). The approach seems very interesting, and i can't wait to read more about it.

On a side note, the physics history book in question is Great Physicists by William Cropper. I have not finished it yet, but so far it is very informative with regards to the physicists lives and achievements. It has plenty of maths in it (as opposed to some other similar books) and could be the physics equivalent of Journey through genius.