Ich bin ein Gastblogger II: The wrong question.

I’m an alien

I’m a legal alien

I’m an Englishman in Nürnberg1

Being an English historian of mathematics resident in Germany I have been often asked, over the years, by people who know a little about the history of mathematics, “Who invented the calculus, Newton or Leibniz?” This is probably the most famous argument about priority of discovery and possible plagiarism in the history of science and still able to provoke nationalist sensibilities 300 years after the fact. Now as I mentioned in my first post this was the first theme in the history of mathematics that caught my attention and over the years I have devoted a considerable amount of time and effort to investigating the subject. There are two possible answers to the question. The short semi-correct answer is, both of them. The much longer and much more correct answer is nobody, calculus wasn’t invented by a single person but evolved piece by piece over more than two thousand years. What follows is not a history of calculus but a very bare and incomplete skeleton naming some of the important stations between the first appearance of concepts considered central to the calculus and the work of Newton and Leibniz.

The fundamental idea behind the infinitesimal integral calculus is first recorded in the so-called method of exhaustion of the Greek mathematician Eudoxus of Cnidus who flourished at the beginning of the fourth century BCE and is used for a handful of proofs by Euclid in his Elements. Refined by possibly the greatest of all Greek mathematicians, Archimedes, it became a powerful tool for the determination of areas and volumes as well as centres of gravity and most famously for his, for the time, highly accurate determination of the value of P, the relation between the circumference and diameter of a circle. The Greeks were also nominally aware of the problem of determining tangents to given curves, the fundamental concept of the differential calculus, but it did not play a significant role in their mathematical considerations. No further progress was made in antiquity before the general decline in learning beginning in the 2nd century CE and it was first in the High Middle Ages that integration returned to European mathematics.

However earlier than that there were interesting developments in Kerala in West India. At its core calculus is about summing infinite converging series, diverging series can’t be summed, and in the 17th century several important series representing important geometrical constants such as P and trigonometrical functions such as sine and cosine were analysed and discussed by European mathematicians and named after their supposed discoverers such as Gregory, Leibniz and Newton. The series had however already been discovered and analysed by the so-called Madhava or Kerala school of mathematics founded by Madhava who flourished in the second half of the 14th century. The same mathematicians also made extensive use of the method of Archimedes to determine areas and volumes. Attempts have been made to prove the hypothesis that the further development of the calculus in the 17th century was stimulated by Jesuit missionaries bringing knowledge of the work of the Kerala School to Europe, however despite extensive research no evidence of transition has been found up to now. In the Early Middle Ages Islamic mathematicians were also aware of and used Archimedean methods.

In the 14th century the Oxford Calculatores proved the mean speed theorem, which is usually attributed to Galileo, and in the next century Oresme proved it graphically (drawing graphs two hundred years before Fermat and Descartes!) and integrating the area under the graph. In the 16th century the works of Archimedes experienced a renaissance in Europe and many of the leading mathematicians devoted themselves to determining centres of gravity using his methods. The 17th century sees an acceleration in the application of what would become the calculus. Kepler used integration to prove his second law of planetary motion, the areas law, basically summing segment of the ellipse and letting them become smaller and smaller until infinitesimal. However as he had no concept of limits even he was aware of the fact that he was claiming to be able to add areas after they had ceased to exist! This piece of highly dubious mathematics contributed to the fact that the second law was still rejected long after the first and third laws had been accepted. In fact the second law was only finally accepted in 1672 when Nicolas Mercator provided a new more reliable proof. Kepler also used a form of integral calculus in his small pamphlet on determining the volume of wine barrels, a work that is often mentioned in a mocking tone but is actually an important milestone in the history of the calculus. The developments now come thick and fast with Galileo, Cavalieri (a pupil of Galileo’s), Grégoire de Saint-Vincent (a Jesuit mathematician who first gave the method of exhaustion its name), the Frenchmen Roberval, Fermat, Pascal and Descartes, the Dutchman van Schooten and in Britain John Wallis, Isaac Barrow and James Gregory all making significant contributions. It was also in the 17th century with the development of the science of mechanics that the differential calculus came to the fore with the problem of finding tangents to curves in order to determine rates of change. Many people in the list above made major contributions to the solution to this problem. Fermat is sometimes referred to as the “father of calculus” because he was the first mathematician to use what we now call the h-method (a method that I have to explain regularly to my private maths pupils) to determine first derivatives of functions. However like Kepler he has no real concept of a limit and just lets his ‘h’ (in his case its actually an ‘e’) disappear at the appropriate moment without explanation!

I hope I have said enough to make it clear that there was an awful lot of calculus around before Newton and Leibniz even considered the subject, so what did they do? It is often claimed that their major contribution was the discovery of the fundamental theorem of the calculus, i.e. that integration and differentiation are inverse operations but even this is not true. The theorem first appears in an implied form in the work of James Gregory and more explicitly in that of Isaac Barrow both of which are explicitly cited by both Leibniz and Newton in their own work. Newton and Leibniz collected up the strands scattered throughout the work of the mathematicians listed above and collating, sorting and standardising create a coherent body of work that we now call infinitesimal calculus but even their effort where actually only a milestone along the route. Finding sums of numerous infinite series and determining integrals and derivatives of many functions proved a very difficult process and many 18th century mathematicians won their spurs by solving a particularly difficult problem in the now developing analysis, most notably Leonard Euler. However one central and absolutely fundamental problem still remained, neither Leibniz nor Newton had a limit concept and their rather cavalier attitude to elimination of infinitesimals led to Bishop George Berkeley’s famous and very justified retort about ghosts of departed quantities. This problem was not really solved until the German mathematician Karl Weierstraß came along in the 19th century.

I have entitled my post “The wrong question” because I personally thing that in any area of science the question as to who discovered/invented a particular discipline, method, theory etc is almost always displaced. We shouldn’t be asking who invented the calculus Leibniz or Newton but rather what did Leibniz and Newton contribute to the on going evolution of that branch of mathematics that we now call the calculus? All branches of science (and I consider mathematics to be a science, see my last guest post here next week), all theories all discoveries have long evolutionary histories and individuals only make contributions to those histories they don’t write the whole history alone.

Let’s take a very brief look at another example where people tend to express themselves as if one individual had produced a major scientific theory complete in one go, like Athena springing fully armed from the head of Zeus, the theory of relativity. If one were to take the popular accounts literally then Einstein dreamt up the whole affair whilst travelling to his work at the Patent Office in Bern on the tram. However the theory of relativity also has a long history. The principle of the relativity of motion to a frame of reference can be found in the works of Galileo, to whom it is oft falsely attributed, but it can also be found in Copernicus’ De revolutionibus and two thousand years earlier in the works of Euclid. The central discussion as to whether time and space are absolute or relative can be found in the Leibniz Clarke correspondence at the beginning of the 18th century with Samuel Clarke basically fronting for Newton. Einstein own work was largely prompted by the incompatibility of the theories of Newton and James Clerk Maxwell, a problem much discussed and analysed in the 19th century. Einstein famous discussion of synchronicity of clocks is foreshadowed by a similar discussion in the 19th century by the operators of railway networks.  Moving from special to general relativity we have the contributions of Minkowski, Hilbert and others.

To close I have made much use of the concept of evolution in this post and anybody who regularly reads John Wilkins at Evolving Thoughts will know that the biological theory of evolution has a long history before Darwin published that book 150 plus years ago and readers of Larry Moran or the fearsome P Z Myers will know that modern evolutionary theorists object to being called Darwinians because the theory of evolution has evolved since Charles’ day. To recap, it is wrong to ask who invented or discovered a scientific discipline or theory, one should instead ask what did a given individual contribute to the theory or discipline in question?

For those who wish to know more about such things as the method of exhaustion or the fundamental theory of calculus then the articles at Wikipedia are mostly OK. On the individual mathematicians and their contributions to the history of calculus a visit to MacTutor is recommended.

For those who prefer books, you can read about the details of the priority dispute between Leibniz and Newton in definitive form in Rupert Hall’s “Philosophers at War” or in more popular form in Jason Bardi’s “The Calculus Wars”. A very general popular account of the history of infinite in mathematics is Ian Stewart’s “Taming the Infinite” a much more challenging book on the history of the infinite in mathematics is David Foster Wallace’s “Everything or More”.

On the history of calculus the standard works are, in ascending order of technical difficulty, Carl B. Boyer’s “The History of the Calculus”, Margaret E. Baron’s “The Origins of the Infinitesimal Calculus” and C. H. Edwards Jr.’s “The Historical Development of the Calculus”.

There is a chapter on the Kerala School in George Gheverghese Joseph’s “The Crest of the Peacock”. Joseph has also written a complete book on the subject his “Passage to Infinity”. For a corrective to some of Joseph’s more exaggerated claims I recommend reading the relevant parts of Kim Plofker’s “Mathematics of India”.

“The Leibniz-Clarke Correspondence” has been edited and annotated by H.G. Alexander and anybody interested in the connections between 19th century train time tables and Einsteins Theory of Relativity should read Peter Galison’s excellent “Einstein’s Clocks and Poincare’s Maps”

If you actually read and digest all of the above then you can start writing your own blog posts on the history of calculus.

1) With apologies to Sting!

• Kepler used integration to prove his second law of planetary motion, the areas law, basically summing segment of the ellipse and letting them become smaller and smaller until infinitesimal.

I'm confused by this. I thought that Kepler's understanding of the laws of planetary motion was purely empirical- it fit the data. Proof waited until Newton showed that an inverse square law for gravity gave the Keplerian orbits first approximation. What am I missing here?

• Thony C. says:

Kepler's second law states that a ray connecting the planet to the focus of its eliptical orbit sweeps out equal areas in equal times. Kepler determined the areas swept out by using what we would now regard as integration. He divided the area to be found into triangular segments (the bases of which are actually curved) which he summed reducing the apex angle of the triangles to zero so he was basically summing straight lines without breadth. Question, how could he determine area by summing lines that have no breadth? Answer, he coudn't! Kepler was well aware of the fact that his "proof" was not mathematically legitimate and even ridiculed it.

Newton's proof of the second law consists of deducing it as a consequence of the laws of motion and gravity and or the third law.

• I remember being in math camp as a nerd having to research fermat's last theorem for a paper that I was writing. Thanks for bringing back painful memories.

• Thony C. says:

In my case they would have been fond memories 😉

As far as I'm concerned Pierre F. rocks but we all have our own perversions!

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• […] Although not one of his most successful works Kepler’s Nova stereometria doliorum is historically important for two different reasons. It was the first book to present a systematic study of the volumes of barrels based on geometrical principles and it also plays an important role in the history of infinitesimal calculus. […]

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