Stephen Loewenthal – Leibniz vs. Newton

This is a generally good summary of the contributions of Newton and Leibniz to the development of calculus and the “priority dispute” that erupted over the question of who should get primary credit for the discovery of these ideas. Your writing is good although you need to be a bit more careful about mechanics or proofreading at some points (plurals vs. possessives, etc.)

One point I think you underestimate is the extent to which Newton’s work on the calculus actually came from his studies of physics – in particular his derivation of Kepler’s laws for the motions of planets from the inverse square law for gravitation. It is true that he did not explain what he did using the language of calculus in his Principia Mathematica, but I think most historians believe that he worked out his derivations using fluxions and fluents, then translated them into a more traditional, purely geometric language for publication. He may have done that to avoid the criticism that would probably have resulted from the unfamiliarity of the methods. And, it is probably also true that Newton’s avoidance of publishing these ideas in his own work until much later probably was a major contributing factor in the priority dispute – the continental Europeans did not want to give Newton credit because they didn’t know it, or because they considered that Newton had never subjected his work to the public scrutiny Leibniz had.

Newton also applied those methods to things like the derivation of the binomial series (which is the same as our Taylor series:

(1 + x)^{p} = 1 + px + \frac{p(p − 1)}{2}x^{2} + \frac{p(p − 1)(p − 2)}{3!}x^{3} + ⋯

by the way – it would have been good to point that out!) But I don’t think it is really accurate to focus on that as the main impetus for the development of Newton’s ideas on calculus.

Some specific comments

1) I’m not sure what you mean by saying that as a result of Newton’s and Leibniz’s studies on series, they were no longer just “approximating tools”. This is not very clear. Do you mean to say that their work focused peoples’ attention on infinite series “for their own sakes,” as algebraic objects in their own right? I think that might be a rather anachronistic point of view (the focus on series as a primary way to represent functions really only came into its own through work of later mathematics like Euler, Lagrange, Cauchy, and others.)

2) Like many of the other people in this story, Huygens was not only a mathematician. He was also a physicist who studied optics, made lenses for telescopes and microscopes, etc. Interestingly, Leibniz himself was probably atypical in that regard. His “outside” interests were mainly philosophical rather than what we would call scientific.

3) Your description of the after-effects of the Newton-Leibniz controversy on British mathematics is good, but you should also be aware that the isolation that British mathematicians imposed on themselves in time led to a very original approach to algebra, logic, and other subjects by later British mathematicians in the 19th century. Isolation from prevailing fashions is not always all bad!

4) It would have been good to say more about this to tie it into what we did in this course. Mention the \frac{dy}{dx} notation for derivatives, the ∫ f(x) dx notation for integrals, etc. in more detail. Ironically, the one place where Newton’s fluxion notation survives is in some physics books and papers where the ẋ notation is still used for the derivative of x with respect to time.

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Micki Rupon – The Newton-Leibniz Debate

This is a good summary of the Newton-Leibniz debate and the contributions that both men made to the development of calculus. The main place for improvement, I think, would be to say a bit more explicitly about how the each of their ideas influenced things we did in this course. For instance, you could have mentioned that we used the \frac{dy}{dx} notation for derivatives, the ∫ f(x) dx notation for integrals, etc. from Leibniz. Ironically, the one place where Newton’s fluxion notation survives is in some physics books and papers where the ẋ notation is still used for the derivative of x with respect to time.

A technical comment: usually when you make use of a direct quotation in a paper like this, you should give not only the bibliography item you took the quotation from, but also a page number or some other more specific reference.

Some specific comments

1) What you are referring to here is Newton’s development of the general binomial series, which is the same as our Taylor series:

(1 + x)^{p} = 1 + px + \frac{p(p − 1)}{2}x^{2} + \frac{p(p − 1)(p − 2)}{3!}x^{3} + ⋯

It would have been good to point out that connection. In the case where p is a positive integer, the series terminates after a finite number of terms. That form of the binomial formula had been known long before Newton. The new thing here was the inclusion of the cases where p was not a positive integer.

2) I think you mean to say Leibniz’s proof of the indefinite integral formulas for

∫ \sqrt{a^{2} ± x^{2}} dx

(like the entries in our table of integrals!). What you said seems like an incomplete thought.

3) It was “unfair” primarily because Newton himself was an officer of the Royal Society and picked the investigative committee!

4) Your description of the after-effects of the Newton-Leibniz controversy on British mathematics is good, but you should also be aware that the isolation that British mathematicians imposed on themselves in time led to a very original approach to algebra, logic, and other subjects by later British mathematicians in the 19th century. Isolation from prevailing fashions is not always all bad!

5) This sort of priority dispute has happened many other times as well in science (for instance there’s one in full swing right now over who should get credit for the development of the idea of MRI medical imaging, and who should have received the Nobel Prize given for that this very year!). Maybe these disputes do look “trivial” to outsiders, but when the development of new ideas is the primary way that scientists’ careers are evaluated, giving credit where credit is due becomes a very personal and emotional thing to the people involved.

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Lauren Stillwell – Newton vs. Leibniz

This is a good summary of the Newton-Leibniz debate and the contributions that both men made to the development of calculus. The main place for improvement, I think, would be to say a bit more explicitly about how the each of their ideas influenced things we did in this course. For instance, you could have mentioned that we used the \frac{dy}{dx} notation for derivatives, the ∫ f(x) dx notation for integrals, etc. from Leibniz. Ironically, the one place where Newton’s fluxion notation survives is in some physics books and papers where the ẋ notation is still used for the derivative of x with respect to time.

Some specific comments

1) This is a good quotation! But you should identify the source.

2) What you are referring to here is Newton’s development of the general binomial series, which is the same as our Taylor series:

(1 + x)^{p} = 1 + px + \frac{p(p − 1)}{2}x^{2} + \frac{p(p − 1)(p − 2)}{3!}x^{3} + ⋯

It would have been good to point out that connection. In the case where p is a positive integer, the series terminates after a finite number of terms. That form of the binomial formula had been known long before Newton. The new thing here was the inclusion of the cases where p was not a positive integer.

3) Since we never discussed “quadratures” it would have been good to explain this term in more detail. In our terms, “quadrature” is essentially an old-fashioned synonym for definite integration to find an area – the actual area between the graph and the xaxis. The connection with rectangles is that you can think of the result of integration ∫_{a}^{b} f(x) dx to find an area as equivalent to the determination of a rectangle that has the same area as the region under the graph y = f(x) – this is the same as the idea of the formula

\overline{y} = \frac{1}{b − a} ∫_{a}^{b} f(x) dx

for the average value of f(x) on the interval a ≤ x ≤ b that we talked about in class.

4) It would have been good to say more about this, since it is a nice development of things we have used a lot in this course. The “previous formula” is just the product rule for derivatives:

\frac{d}{dx}(y_{1}y_{2}) = y_{1} \frac{dy_{2}}{dx} + y_{2} \frac{dy_{1}}{dx}

The general product rule comes from differentiating this repeatedly. For instance for the second derivatives:

\frac{d^{2}}{dx^{2}} (y_{1}y_{2}) = y_{1} \frac{d^{2}y_{2}}{dx^{2}} + 2\frac{dy_{1}}{dx} \frac{dy_{2}}{dx} + y_{2} \frac{d^{2}y_{1}}{dx^{2}}

and then similarly for all higher-order derivatives. The coefficients that appear here are the same as the binomial coefficients appearing in the expansion of (a + b)^{n} for positive integer n.

5) Your description of the after-effects of the Newton-Leibniz controversy on British mathematics is good, but you should also be aware that the isolation that British mathematicians imposed on themselves in time led to a very original approach to algebra, logic, geometry, and other subjects by later British mathematicians in the 19th century. Isolation from prevailing fashions is not always all bad!

6) I’m not sure exactly what you mean by “analytical” here. If you mean, “more mechanical, hence easier to use” then I think you’re onto something interesting. Leibniz’s notation for calculus was far more successful than Newton’s ultimately because it facilitated understanding, even for those who were not Newtons or Leibnizes themselves(!)

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Kristie Giannetto – Newton vs. Leibniz

Your paper on the Newton-Leibniz controversy contains a lot of good information and your conclusions are correct, I think. But I think the organization of the paper is not ideal – at several points you tend to introduce a good idea briefly, then talk about something else, then return to the original idea at greater length. In a paper of this length, that’s not a very good strategy because it tends to dilute the points you are making. I also think you could have done more to talk about specific connections between the mathematics that Newton and Leibniz did and topics we discussed in this course. For instance, you might have noted Newton’s work on infinite series and the general binomial series he developed:

(1 + x)^{p} = 1 + px + \frac{p(p − 1)}{2}x^{2} + \frac{p(p − 1)(p − 2)}{3!}x^{3} + ⋯

which is the same as one of our examples of Taylor series(!) The discussion of Newton’s fluxions and fluents could also have made a connection with ideas about parametric curves as we discussed them briefly. (Indeed, I think much of the ultimate explanation for the way Newton approached calculus really came from the fact that he was thinking of applications to description of motion in physics.)

Some specific comments

1) This analysis of the after-effects of the Newton-Leibniz controversy on British mathematics is good, but you should also be aware that the isolation that British mathematicians imposed on themselves in time led in time to a very original approach to algebra, logic, parts of geometry, and other subjects by British mathematicians later in the 19th century. In time, this became a big part of 20th century mathematics. Isolation from prevailing fashions is not always all bad! (Organizational point: You continue this discussion later on page 5 of the paper – it might have been better to consolidate those two sections; as it is, your paper seems to be rather repetitive and not efficiently organized.)

2) I think another (maybe the most important?) reason that Leibniz’s approach eventually “won out” was that it was just a lot easier to use. You didn’t have to be a Newton or Leibniz yourself to learn how to compute derivatives and integrals using Leibniz’s formulation and notation! You get to this point in the next paragraph, but again, the organization of your paragraphs blunts what could have been a very good discussion.

3) Interestingly enough, Newton himself was an officer of the Royal Society and picked the investigative committee! Does that sound like a fair investigation?

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Matthew Frigault – Newton vs. Leibniz: Plagiarism or Coincidence?

This is a good summary of the Newton-Leibniz debate about the origins of calculus. I agree with your differentiation between Newton’s physical approach and Leibniz’s more general mathematical approach. The major place for improvement I see is that it would have been good to say more about specific relations between things we did in this course and the contributions of the two men. See comment 4 below for one suggestion. Other possibilities would have been to mention some of Newton’s work on what we now call Taylor series (in particular the binomial series we studied just recently), some of Leibniz’s integration formulas which closely parallel entries in our table of integrals, etc.

Some specific comments

1) I’m not sure why you say this is “unfortunate”. Part of the reason for publication is so that a review of the ideas and methods being proposed in the new work can be made by other scientists. Subjecting your work to that kind of scrutiny is a way of checking its accuracy and establishing that it is of interest to others. Newton’s reluctance to publish can be seen as selfish or even egotistical. For instance, there is evidence he thought his ideas would be misunderstood and he didn’t want to be bothered with trying to explain them to others he considered his scientific inferiors. I don’t find that very admirable personally!

2) The main subject of Principia Mathematica was actually Newton’s derivation of Kepler’s Laws of planetary motion from the inverse square law of gravitation. Moreover, although most people think he developed all that using calculus, he did not present it that way – he relied mostly on traditional Euclidean geometry proofs.

3) It’s not surprising in a sense that neither Newton nor Leibniz specialized exclusively in mathematics. There was far less specialization in general at this time and there were only a handful of university professorships in existence, so the time when one could make a career solely as a mathematician was still largely in the future.

4) Note how much this sounds like the idea of parametric curves as we discussed them in this course(!) It would have been good to point out that connection.

5) This is not very clear. I think what you want to say is that the process of integration for Newton is the “inverse process” of taking differences.

6) Don’t forget that “our experience” (I presume you mean by this the way we discussed the derivative as the limit

f′(x) = lim_{h→0} \frac{f(x + h) − f(x)}{h})

profits from the later developments of calculus by many generations of people after Newton and Leibniz! It is typical in the history of mathematics for the intuition of how something should work (and even methods for using it effectively) to come before the precise understanding of how and why the idea works!

7) I think another (maybe the most important?) reason that Leibniz’s approach and notation eventually “won out” was that it was just a lot easier to use. You didn’t have to be a Newton or Leibniz yourself to learn how to compute derivatives and integrals using Leibniz’s formulation of calculus and the \frac{dy}{dx} and ∫ y dx symbolism! It’s true that British mathematicians tended to isolate themselves out of a sense of loyalty to Newton, but you should also be aware that that self-imposed isolation in time led in time to a very original approach to algebra, logic, parts of geometry, and other subjects by British mathematicians later in the 19th century. In time, this became a big part of 20th century mathematics. Isolation from prevailing fashions is not always all bad!

8) This is overstated. I think most historians now would agree with what you say elsewhere: Newton had these ideas first but did not publish them. In the meantime, Leibniz also developed many very similar ideas and did publish them. Leibniz’s name is not “tarnished” now.

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P.J. Brennan – Bernhard Riemann

This is a reasonably good summary of Riemann’s life and of some of his mathematical work most closely related to things we have discussed in this course. There is one major organizational problem though – see comment 4 below.

Some specific comments

1) Be careful – you are looking at this from a present-day point of view. At Riemann’s time, this was probably within 5 years of the average life expectancy in Europe. Also, there were a number of other very famous and influential mathematicians in the first half of the 19th century who died much earlier than Riemann. The French mathematician Évariste Galois, and the Norwegian mathematician Niels Abel are two examples.

2) It’s good that you tried to include a discussion of exactly what Riemann’s ideas in geometry were, but what’s here doesn’t really do them justice. Here’s a more complete description of what’s involved. For a parametric curve (x(t), y(t)) in the plane, there is an arc-length formula that is similar to what we did for graphs of functions:

s = ∫_{a}^{b} \sqrt{(\frac{dx}{dt})^{2} + (\frac{dy}{dt} )^{2}} dt

We can write

ds^{2} = dx^{2} + dy^{2} = (\frac{dx}{dt})^{2} + (\frac{dy}{dt} )^{2}

for the term under the square root. Then the arc-length is s = ∫_{t=a}^{t=b} ds. From earlier work of Gauss, Riemann knew that there were similar formulas for arclengths of curves on other surfaces, with

ds^{2} = Adx^{2} + Bdxdy + Cdy^{2}

where A, B, C are functions of the coordinates x, y on the surface whose form depends on what the surface is. Riemann’s idea was to base general geometry in any dimension on the form of this so-called “metric” – the ds^{2} involved in the arclength integral. The general form of a Riemannian metric is

ds^{2} = ∑_{i,j=1}^{n} g_{ij}dx_{i}dx_{j}

Einstein then used this idea to describe the geometry of the 4-dimensional space-time in his theory of general relativity.

3) It would have been good to mention exactly what the Riemann hypothesis says – namely that all the “non-trivial” zeroes of the ζ(s) function occur at points of the form s = \frac{1}{2} + it. This is probably the most famous unsolved problem in mathematics at present. It is one of the Clay Mathematics Institute’s prize problems (anyone coming up with a solution that is published and not challenged for a period of 2 years will will a million-dollar prize!)

4) It would have been better from an organizational point of view to combine these two portions of the discussion of Riemann’s theory of integration. Splitting it this way makes the whole paper seem disjointed. Also, integrals of functions which have vertical asymptotes are rather subtle. Some of them do “converge” to a finite value, and that can be seen by using Riemann integrals but taking a limit:

∫_{0}^{1} \frac{1}{\sqrt{sin(x)}} dx = lim_{a→0} ∫_{a}^{1} \frac{1}{\sqrt{sin(x)}} dx ≐ 2.035

Others do not converge. Also, much more severely discontinuous functions are interesting as well, and Riemann’s integral is not “the last word” on those. Other mathematicians have developed even more sophisticated integration theories that can handle functions like

f(x) = \{ \begin{matrix}1 & if x is a rational number \\ 0 & if x is an irrational number \end{matrix}

that are not continuous anywhere. According to Riemann’s theory this function is not integrable over any interval, but according to later integration theories like Lebègue integration, it is.

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Robert Molt – The Power of Newton’s Mathematics (in) Science: Justification of Kepler’s Laws

This is a very good presentation of (a modern version of) Newton’s derivation of Kepler’s Laws from the inverse square law for the gravitational force. It is interesting to note that Newton also showed the converse – namely that if we assume Kepler’s Laws hold, then the only pure power gravitational formula that can be the correct one is the inverse square law. The only criticism I have is that the long string of formulas at the top of page 2 of the text should still be broken up and more explanation of each step should be included. For instance, one standard method is to number previous equations and refer back to them as you use those results to substitute in later, etc. But it is clear that you have worked out each step and understand what is going on. Good work!

Some specific comments

1) You don’t need this line I think. Namely, from the line before, since u_{r} and u_{θ} are perpendicular, it follows from vector algebra that the coefficients of u_{r} and u_{θ} on both sides must be the same, i.e.

F_{θ} = m (2\frac{dr}{dt} \frac{dθ}{dt} + r \frac{d^{2}θ}{dt^{2}} )

and

F_{r} = m (2\frac{d^{2}r}{dt^{2}} − r (\frac{dθ}{dt} )^{2}) ,

which is what you want.

2) Since presumably m ≠ 0, it’s not really necessary to put in all three of these equations. It would be OK to skip a step and just write the last one.

3) It would be better to say: “Now we will choose our coordinate system to make θ = 0 correspond to the perihelion point (i.e. minimum of r, or maximum of z = 1/r).

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