Newton and the Case of the Missing Calculus

Posted on October 3, 2011 by Kirsten Walsh

Kirsten Walsh writes…

The case of the missing calculus is well-known.  Newton (co-)invented calculus in the late 1660s, and he wrote Principia in the late 1680s.  It would be natural to expect that Newton used the calculus in Principia.  But it seems that he didn’t.  Instead, Newton wrote Principia in the style of Euclid’s Elements, that is, using Classical Greek geometry.  This is surprising indeed, given the powerful new tool he had at his disposal.  What should we make of this?

Almost thirty years after the publication of Principia, Newton explained that he had used algebraic calculus to discover the propositions of Principia, but used classical geometry to demonstrate them:

“By the help of the new Analysis [i.e. algebraic calculus] Mr. Newton found out most of the Propositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry.  And this makes it now difficult for unskilful men to see the Analysis by which those Propositions were found out.”

But Newton was lying.  Scholars have found no evidence that he wrote or developed Principia in any other way than the published form.  Moreover, few, if any, of the propositions in Principia can even be presented in the form of algebraic calculus.

This raises two questions:

1. Why did Newton lie?
2. Why did Newton eschew modern algebraic calculus in favour of classical geometry?

These questions have been discussed by numerous scholars including A. Rupert Hall and I. Bernard Cohen.  The answer to (1) can be found in Newton’s priority dispute with Leibniz.  The answer to (2) was summarised neatly by Thony Christie last year:

“Put simply Newton had serious doubts about the reliability of the new analytical mathematics and that is why he didn’t use it for his magnum opus.”

But what caused these doubts?

In 1714, Newton wrote that the algebraic calculus is “arithmetic applied to geometrical matters… Its operations are complicated and excessively susceptible to errors, and can be understood by the learned in algebra alone”.  Whereas geometry “may be appreciated by the great majority and thus most impress the mind with [its] clarity”.  One might wonder why Newton bothered to invent algebraic calculus at all!

Well it seems that Newton wasn’t always so anti-algebra, nor was he always so interested in classical geometry.  In fact, as an undergraduate, Newton didn’t read the ancients.  Rather, he read a few modern summaries of the ancient texts, building his own mathematics on the algebraic work of mathematicians such as Descartes, Wallis and Barrow.

Newton seems to have become interested in classical geometry in the late 1670s, after re-reading Descartes’ La Géométrie.  La Géométrie was an attempt to unite algebra and geometry – Descartes aimed to show how symbolic algebra could be applied to the study of plane curves.  Guiccardini writes:

“[Descartes’] tract could be read as a deliberate proof of the superiority of the new analytical method, uniting symbolic algebra and geometry, over the purely geometrical ones of the ancients.”

Newton was very critical of Descartes’ text, writing comments such as “error” and “I hardly approve” in the margins.  He even drafted a paper entitled ‘Errors in Descartes’ Geometry’. To find support for his position, Newton began to read the ancient texts, including Pappus.

Newton wrote:

“To be sure, [the ancients’] method is more elegant by far than the Cartesian one.  For [Descartes] achieved the result by an algebraic calculus which, when transposed into words (following the practice of the Ancients in their writings), would prove to be so tedious and entangled as to provoke nausea, nor might it be understood.  But they accomplished it by certain simple propositions, judging that nothing written in a different style was worthy to be read, and in consequence concealing the analysis by which they found their constructions.”

Newton was neither the first, nor the only, philosopher to equate algebra and geometry with the ancient methods of analysis and synthesis respectively.  But he was the first to reject modern algebraic calculus in favour of ancient geometry.  (If only because he was the first to invent it!)  Does Newton’s rejection of algebraic calculus stem from his anti-Cartesian stance?  What if Newton had never re-read Descartes’ Géométrie?  Could his priority dispute with Leibniz have been avoided?

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