An open problem of a different sort

I have been anxiously awaiting Dr. Lovric’s email all weekend.  Since our first meeting, I have discovered a lot about Newton’s notation, how it compares with that of Leibniz, and also a little bit about Newton’s contributions to calculus, the conflict between Newton and Leibniz, and finally, that notation truly is important to the development of mathematics. 

But I have been unable to find any more information as to whether Newton’s d notation has anything to do with differentiation, and thus anything to do with Leibniz.  Late Sunday night, I received a reply from Dr. Lovric.  It seems as though there is no clear resolution to the mystery.  It might be that the dz’s which abound at the end of Opticks represents the derivative, but this is not certain given some of the equations in the text.  As Dr. Lovric put it in his email:

[this] definitely needs more work!

 So, the question of whether or not the notation I found is something new (or not new) with Newton remains undecided!

The power of notation

After doing some research on Leibniz, I have discovered that, just like Dr. Lovric said, the development of calculus was not as straightforward as many suggest. Although Newton did develop his method of fluxions before Leibniz developed his “differential” method, Leibniz published his work first.  A bitter battle ensued between the two (and between the British and the Continental scientific communities) as to who was the real “creator” of calculus. 

In fact, the way in which Leibniz and Newton approached calculus was quite different, and Newton’s method (no pun intended) has been judged to be much closer to our modern ideas of the subject.  But whereas Leibniz was obsessed with notation, Newton was not.  Quite ironically, Leibniz’ superior notation allowed Continental mathematicians to make great progress in the subject, and the British mathematical community was left behind because of their unwillingess to adopt Leibniz’ style.  All this despite the fact that Newton might have been conceptually more sound than Leibniz.  But, as the French mathematician Jacques Hadamard once stated:  

The creation of a word or a notation for a class of ideas may be, and often is, a scientific fact of very great importance, because it means connecting these ideas together in our subsequent thought.  

Thus, one cannot underestimate the power of notation!

Newton was probably the first to formulate the Fundamental Theorem of Calculus (although Leibniz was aware of it as well). His integration of rational functions was also almost as sophisticated as our modern methods. Here, we have Newton's proof of the quotient rule.

 

 

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My own “echoes” of Fermat

I searched the library catalogue for relevant works by Newton and Leibniz, in hopes that I could shed some light on the “mystery” of the interloping d notation.  It was a difficult search.  I was forced to face my fear of being crushed to death between the automated bookshelves.

Although I could not find much on the notation of Newton, I was able to find some works of Leibniz which provided some insight.  When I was examining Leibniz’ works in the archives, I was hoping to come across some of his notation with which I could compare the strange symbols I had found in Opticks.  After all, one of the reasons that the notation in Newton’s work is so interesting is that it looks like modern Leibniz notation.  But, I have yet to see any of Leibniz’ notation from Newton’s era!

Notice the integrals at the top left and bottom right, as well as the "dt" and "dr" at the bottom left.

Leibniz is credited with developing not only the modern d notation, but also with the elongated “s,” ∫,  which is the modern integral sign.   I was able to find a book which contained pictures of the notes Leibniz himself wrote in the margins of his copy of Newton’s Principia Mathematica!  I lucked out.  Not only did I get to see Leibniz’ notation (which is eerily similar to that which we use today), I got to see it written in his own hand! If the d notation Newton used really was referring to differentiation, then it certainly was not his own invention!

A page out of "Principia," with Leibniz' notes in the margins. Is everything interesting in the history of mathematics written by a mathematician in the margins of a work by another mathematician?

Further investigations

After my final Friday class, I returned to the archives to do some more searching.  I took out a few more books about Newton, but I did not find any interesting notation.  I even took out whatever books by Leibniz that were in the archives.  Unfortunately, they were all printed in German and not one of them had anything to do with mathematics.  (But, one of the archives staff, clearly impressed by my fierce concentration whilst “reading” books in Latin, 18th Century English, and German, asked me what languages I research in, to which I replied: “Oh, I’m just looking for pictures”).

The elements with dots above them are what we would call derivatives. For Newton, they were "fluxions." Notice the higher-order derivatives, those elements with multiple dots.

During our conversation, Dr. Lovric started to explain Newton’s system of notation to me.  Newton used dots above his letters to indicate derivatives (as some physicists continue to do).  If you took the second derivative of something you would place two dots above it, the third derivative three dots, etc.  This is very similar to how we notate higher-order derivatives with primes.

Newton, however, called his derivatives “fluxions.”  (He was the first to do this.  His approach to calculus is known as the “method of fluxions”).  Fluxions are the rates of change of a quantity called a “fluent.”  The fluent was either denoted as a letter with no symbol above it or as a letter with a small tick mark in place of a dot.   

After my unsuccessful search in the archives, I have decided to go search for some more books in the rest of Mills to see what I can find out about Newton’s notation.

An element is the fluxion of the one preceding it, and the fluent of the one after it. This is analogous to our notion of derivatives and antiderivatives.

You mean it’s new?

I was sitting on the benches in the halls of Hamilton Hall, waiting for my meeting with Dr. Lovric.  Right before we were supposed to meet he leaves his office and tells me he’ll be back in a minute.  He also asks me something along the lines of: “I need to know; did you see those dz’s with your own eyes?”  I am very excited.

I tell Dr. Lovric about what I had noticed in Opticks, and I even show him the pictures I took earlier that day in the archives.  Dr. Lovric tells me that he can’t quite explain what I’ve seen.  In his words, “it [the Leibniz notation] shouldn’t be there!”  He shows me some pictures of Newton’s “usual” notation, and we discuss Newton as we head to the archives for a closer look.

A few curves.

Dr. Lovric tells me that the development of calculus between Leibniz and Newton is not as clear-cut as most people tend to think.  I had erroneously believed that Newton invented the f (x) notation that we use today.  But Dr. Lovric told me that Newton did not consider functions as we do, and instead dealt with equations and implicitly defined curves.  There are some beautiful pictures of these curves in Opticks.

I show Dr. Lovric the two places where I found the suspect notation.  On one page, Newton seems to be dealing with what we would call differentiation and antidifferentiation.  The d beside the z to the exponent n-1 seems like it might be playing the role of a constant.  For a moment, my hopes that I found something interesting are dashed. 

If the equation on the left is differentiation, then the equation next to it is its antiderivative.

But, Dr. Lovric is not convinced that what I had found is simply a lowly constant.  When I show him the second place where I found this notation, the purpose of the d becomes less clear. The page on which I had originally found the notation is the first of a series of formulae, which reminded me of the tables of integration that appear in every first-year calculus textbook. Here, Dr. Lovric cannot determine the purpose of the d, since one would expect to find an x in the numerator in the third equation from the left of row (1), if we are dealing with derivatives and antiderivatives.  (See the picture below).  But, that x is no where in sight.  Dr. Lovric told me he is going to think about this and get back to me sometime this weekend.  

This is a very exciting turn of events.  Could it be that what I had thought was evidence of something being not new with Newton is itself something new? 

The third equation from the left should have an "x" in the numerator.

The whole page of "confusing" equations.

Ça commence!

Dr. Garay suggested I email Dr. Lovric about what I had found, so I sent him an email and am going to meet with him today.  I thought that before my meeting I would head back down to the archives to take a closer look at Opticks.

I have always been interested in the history of mathematics, but had never really had the chance to look at the notation used a few hundred years ago.  In some respects, I was surprised at how modern Newton’s notation looked, especially in terms of some of the algebraic notation.  Take a look at some of his equations:

The square roots and exponents are similar to modern notation. But, notice that he sometimes omits the exponent and will simply write the variables out multiple times.

His equation of a third degree polynomial looks very modern.

But again, there a few dz‘s popping up.  I am excited to see what Dr. Lovric has to say.

A glimpse of a future project…

On Wednesday, October 1st, we had our field trip to the archives.  There were many wonderful volumes to peruse, but the one that really caught my eye was the first edition of Newton’s Opticks.  It felt surreal to hold such an old book, written by one of the most important mathematicians of all time.  The wannabe-mathematician within me was stirred with feelings of excitement, knowing that the book I had in front of me was an original, and that someone had held it and read it 300 years ago, and moreover, that this person had read it as new mathematics.

As I was flipping through the pages, admiring the complex diagrams of conic sections and curves, I came upon something which I found strange.  I had always thought that Newton used a “prime” notation to denote the derivative, something similar to our modern  f ‘ (x) that every calculus student is familiar with.  But, at the very back of Opticks Newton had used the dz notation on a few of the final pages of the section “Tractatus De Quadratura Curvarum.”  But this notation is Leibniz notation!  This seemed highly irregular, since Leibniz (as every calculus student knows) was, along with Newton, the “creator” of calculus, and also had the distinction (as did many) of being Newton’s rival.  Thus, I could not understand why Newton, known for his arrogance and pride, would choose to use his adversary’s notation.

The offending notation!

It was then that I came up with a vague notion of what my Newton assignment would consider: Newton’s notation.  I have never lost the somewhat childish excitement that comes with learning new notation.  It is like learning a new language, but a language endowed with a kind of “magic” that allows you to consider ideas which you could never have dreamed of without these new symbols.  And, with my little “discovery,” I realized that this topic fit in well with the question “What is new/not new with Newton?”   Was Newton using some unoriginal (i.e., “old”) notation?  What were some of his contributions to mathematical notation?  Had someone else noticed what I noticed in “Quadratura” (i.e., was what I saw in itself new?).  These are the questions that I intend to investigate!