Montaigne on Calculus, Natural Selection, and the History of Science


Is there anything more curious than a coincidence? Just ask King Umberto I of Italy who met a bizarre looking guy in a restaurant in Monza.

The story goes that the king was feeling a bit hungry while on the road, so he settled in a nearby restaurant. It wasn’t a fancy place worthy of his highness by any measure. His entourage stopped just outside the establishment, the door of his carriage opened, and out he came. Dressed in full regalia – most probably – he went inside, found himself a comfy chair, then asked for the menu. Knowing the unusual opportunity brought by this random royal reservation, the owner himself went out to address the king. As their eyes met for the first time, bewilderment filled both of them. They stared aghast at each other. Then silence. To the king and owner’s surprise, they both looked  exactly the same! But that’s just where things started to get really interesting, albeit a little too weird. They talked and got to know each other. The king’s wife was named Margherita. And so was the owner’s wife. The owner was born on March 14, 1844, the very same day the king was born. Mamma mia! On his way back to his royal abode, the king was still reeling from the chance encounter with his doppelganger that he thought of  inviting the owner over to the royal palace for a second scrutiny. This wasn’t something that the king could easily forget, because the owner’s name was also Umberto!

There is no evidence proving that the above event did happen. Its existence solely rests on the vitality of its form, that of urban legends. It has found new life and revived interest thanks to the internet. But so do many things without foundation. Question is, how does it resist the erosion of memory?

The power of the story is in its portrayal of a concurrence. Two or more seemingly unrelated objects converge on a single point. Further simplified, it shows the arrival of two or more separate lines at a vertex (or a single point): lines A and B both ending up at point F. What matters is the event that they did meet, they did chance upon each other: in short, a coincidence.

Little is known about the paths A took to arrive at F. Did A pass through d or e? How about B? Such considerations are often neglected thanks to the singular importance placed on F. But this was not the case when a rich French aristocrat decided to put his thoughts on paper in a form that would later be the archetype of an art. Montaigne was thinking about coincidences and he was going to write about it.

Portrait of Montaigne by Daniel Dumonstier in 1578

Michel Eyquem de Montaigne (1533-1592) decided to quit public life in 1571 and settle in seclusion surrounded by his books. Locked in his citadel, Montaigne would embark on a writing spree that would later revolutionize a literary genre. From this solitary sojourn, he came up with a tome modestly titled Essais. (Montaigne’s journals and letters show that he was actually quite active during this period and traveled a lot.)

Essais is a book of essays: of attempts, of trials, of weighing the gravity and scope of various subjects. The text is an assemblage of assorted articles that each talk about different things like thumbs, cannibalism, children’s education, getting drunk, prayers, pedantry, and many more. The first topic that Montaigne tackled in this collection (at least in my copy of his essays) deals with a subject that we have discussed earlier.

In the essay That Men by Various Ways Arrive at the Same End, Montaigne is concerned with coincidences: two unrelated lines, A and B, meeting at point F. But Montaigne’s approach is novel. He knows that F is the important point as it allows the two disparate objects to have a common subject. What Montaigne does is to go further by going backwards, altogether shifting his attention to another important question. He asks, at which points did A and B pass through before they reached F? Montaigne, like a nosy buddy who just heard of your recent romance, is thus less interested in the question where did you meet? but is more focused on the question how did you meet?

Montaigne begins the essay with a scenario:

The most usual way of appeasing the indignation of such as we have any way offended, when we see them in possession of the power of revenge, and find that we absolutely lie at their mercy, is by submission, to move them to commiseration and pity; and yet bravery, constancy, and resolution, however quite contrary means, have sometimes served to produce the same effect.

He proceeds to the crux of the matter right off the bat: one can either submit or resist to have oneself remitted of an offence – A and B can both reach F by taking different routes. A can pass through c, and B can go by way of e, and both can arrive at F. While clear in its skeletal framework, he realized that this scenario was simply an exercise in idle speculation and it wasn’t convincing enough to forward his argument. Not settling for hypotheticals, Montaigne’s next move was to provide examples from history.

During his French campaign, Edward the Black Prince of House Plantagenet, was feeling a bit cross “having been highly incensed by the Limousins” after successfully taking their city.  His rancor found translation in remorselessness when his resolve to severely punish its inhabitants was not to be dampened “either by the cries of the people, or the prayers and tears of the women and children.” Edward’s fury was only soothed when he encountered three brave French soldiers resisting the onslaught of his army. This resolute defiance to submission “first stopped the torrent of his fury, and that his clemency, beginning with these three cavaliers, was afterwards extended to all the remaining inhabitants of the city.” Now, who was it again that said that the French had a penchant for surrendering all too often?

From Montaigne’s example, we can devise an illustration. Let us assume that F is the peaceful resolution. The French inhabitants, who we will refer to as A, arrived at F not through c, which is resignation to submission, but through d which is resistance to it.

Having traced A’s passage through d to arrive at F, Montaigne explores another example that exhibits the contrary.

Montaigne tells the story of the great Roman military general Gnaeus Pompeius Magnus, or Pompey the Great, and how he pardoned the Mamertines despite being “furiously incensed against it”. The Mamertines didn’t put up a strong fight, neither did they exhibit a courageous stand – so, how did they achieve a blood-less resolution? The simple cause was, according to Montaigne, “the virtue and magnanimity of one citizen, Zeno”, who retaliated through submission. But this was not just any run-of-the-mill submission. Zeno quickly turned from zero to hero when he did the unthinkable by taking the “the fault of the public wholly upon himself; neither entreated other favour, but alone to undergo the punishment for all”. Zeno’s act was submission writ large.

Assuming that B is the Mamertines, Montaigne writes that B arrived at F not through d, which is where the French Limousins passed through, but they went by way of c. The two examples now complete Montaigne’s argument: A and B have both arrived at F through different routes. Or, if we are allowed the convenience of a more modern but polite example, we can say that Allan and Bob both became fascists: Allan by being a dick, and Bob by being a cock.

Since the point of this blog is to draw lessons from history about men and women doing science, it is in this area where Montaigne’s light will be directed. And since the essay discussed is replete with examples, it is only apt to follow this line of exploration – i.e. to investigate examples of coincidences in the history of science. Perhaps there are no better examples than the two major coincidences in the two major revolutions in modern science: the revolutions instigated by Isaac Newton in the 17th century and Charles Darwin in the 19th century.

But what can a French man of letters say about two British men of science?

A little historical background on both is necessary, starting with the older Isaac Newton. What else is there to say, though? He is almost universally known thanks to the powerful mnemonic of the apocryphal apple falling on his head which then germinated into his theory of gravity. (The truth, however, is far from it. According to his biographer William Stukeley, Newton was only “occasion’d by the fall of an apple, as he sat in contemplative mood” – an apple did fall, but not on his head.) To the dismay of many high-school and college students taking up advanced maths, the fall of the apple can be blamed for the fall of their grades, because one of Newton’s greatest contributions is a staple part of modern advanced mathematics courses: calculus.

Newton was the first to “develop” it around the 1665 (he called it fluxions). He wrote his first paper on it titled De analysi per aequationes numero terminorum infinitas in 1669. But he didn’t want it to be published for reasons unknown – what he did was share it to his friends and colleagues. In 1671 he finished another text on calculus, now known as Methods of Fluxions, but again, he didn’t publish it. Decades would pass until these two texts saw the light of publication: the first one in 1711 and the second one posthumously in 1736 (Newton died in 1727).

Newton was having his moments of doubt and reluctance. Even with Principia, which is what he is most famous for, it took the goading of Edmond Halley for him to start working on it. It was only when the journal Acta Eruditorum published a curious article titled A New Method for Maxima and Minima, as Well Tangents, Which is not Obstructed by Fractional or Irrational Quantities in 1684 that Newton seriously reconsidered his reluctance to publish. There were some things that were interesting in this paper, like the product and quotient rules that are also found in today’s calculus lessons. Wait a minute, does that mean that this paper talked about Newton’s “invention” the calculus? It did. And it beat Newton to the punch in publication. The author of the paper was the German polymath Gottfried Wilhelm Liebniz.

A boat had just arrived back in England in 1836. It was filled with natural history oddities, collected from a five-year long survey of South America, a brief cruise in Australia, and a stop in South Africa. Fossils of unknown wonders were first welcomed by British eyes; specimens heretofore unfamiliar were now available for everyone to inspect. Suffice to say the trip was a major success. And it brought fame and fortune to its crew and forever immortalized the name of the ship, the HMS Beagle. And of course, one of those that alighted the boat to set foot in England once again was Charles Darwin.

Darwin never went out of Britain ever again after his Beagle escapade. Why would he? There were no reasons to. The success of the expedition launched him into scientific super-stardom, and the publication of his The Voyage of the Beagle in 1839 made him a literary sensation. Everything seemed to go Darwin’s way. Scientific societies opened their doors to welcome him, and they sat him among the elites of Victorian science. It is no wonder that many in the field wanted to consult him on matters regarding natural history. And it so happened that such a letter arrived on his doorstep on the 8th of June 1858. It came all the way from Indonesia. The envelope contained an essay titled On the Tendency of Varieties to Depart Indefinitely From the Original Type. Included in the parcel was a little request from the sender, asking Darwin to review the essay and if found worthy, endorsed to Charles Lyell, the foremost authority in geology at that time. After reading the contents of the essay, Darwin made no hesitation to forward it to Lyell – in fact, he did it in haste. Alfred Russel Wallace, a British naturalist based in the Indonesian island of Ternate, had just formulated a similar idea to that of Darwin after recovering from malaria – the very same idea that Darwin had been working on for 20 years.

Word spread around that Leibniz had plagiarized Newton and the commotion reached its boiling point in 1712. Actually, it was not the first time that Leibniz got accused of plagiarism. Leibniz visited some colleagues in London in 1673, and he happened to mention his original mathematical work on the method of differences for series. What he didn’t know was that this was already done by the French mathematician François Regnaud. So his host, John Pell, with a hint of malice, advised him to read up on it. He went to the local library, picked up the book suggested by Pell, and was perturbed at finding out that Pell had told the truth. The agitated Leibniz had to show his notes to prove that his work was in no way copied (plagiarized) from Regnaud.

The calculus affair was messy. Made all the much worse by each’s epigones. When John Keill, an avowed Newtonian, launched a series of scathing accusations against Leibniz, the much maligned Leibniz had no other recourse but to forward the matter to the Royal Society for mediation. He was adamant to clear his name from any wrongdoings. He had plenty of reasons to. Aside from having his integrity inconsiderately tarnished by this indictment, Leibniz was also concerned with the persecution of posterity. Leibniz believed that it was important and “most useful that the true origins of memorable inventions be known.” This was a belief in the historical duty and function of men and women of prestige as icons and beacons of inspiration, and properly attributing things to their originators may lead to the realization “that the art of discovery may be promoted and its method become known through brilliant examples.”

Charles Lyell and Joseph Dalton Hooker, one of the most eminent British botanists of the 19th century and Darwin’s closest friend, came up with a plan. This decision was made in consideration of Darwin’s priority as the originator of natural selection – after all, Darwin already discussed the germs of his theory to Hooker as early as 1844. You have to publish soon, they told Darwin. And so the date was set: July 1, 1858. Venue: the halls of the Linnean Society. The agenda: a joint reading of Darwin and Wallace’s essays to introduce the basics of natural selection. But Darwin was in a domestic dilemma and couldn’t attend. His infant son had just died and he was in no mood nor propensity to set himself straight for a presentation. Wallace, steeped in the damp of South East Asian forests, had no idea that his paper was to be read alongside Darwin’s. No one informed him,  let alone asked for his opinion. He would later remark in 1869 that his essay “was printed without my knowledge, and of course without any correction of proofs”. In fact, everything was all too hurried for both Darwin and Wallace. The reading happened barely a month after Wallace’s letter reached Darwin.

The Royal Society finally released its verdict in 1712. The report cited several of Newton’s letters to and from his colleagues, setting up an insurmountable body of evidence in Newton’s favor that placed everything “past all dispute that he had invented the method of fluxions”. The decision not only highlighted Newton’s invention of it, but it also added that Newton “brought it to great perfection, and made it exceeding general.” It was clear to the Royal Society that Newton invented the calculus and he was the master of it. There was no point in acknowledging Leibniz’s part in the birth of calculus, as Newton boldly and harshly declared that, “second inventors have no right”. It is interesting to note that Newton’s strict stance found consonance with the Royal Society report. More so with the then president of the society, who was the one who mainly penned the document. Thing was, the president of the Royal Society at that time was no other than Newton himself.

Around thirty gentlemen of the Linnean society were present at the event. None were particularly looking forward to listen to the joint publication of Darwin’s and Wallace’s papers as there were other business matters that required more attention. And it is almost certain that the busy gentlemen never did pay close attention to the proceedings, as even the society’s president Thomas Bell noted in his summary of 1858 that the year “has not, indeed, been marked by any of those striking discoveries which at once revolutionize, so as to speak, the department of science on which they bear.” They would have to wait another year for that revolution to arrive. And when it did, it blasted out of the printers in 1859 in an explosion that would shake Victorian sensibilities. It was a revolution that did not depose kings and tyrants off their thrones, but it did kick humans off the divine pedestal. Darwin’s On the Origins of Species deported Homo sapiens from God’s kingdom and sent them back to their rightful place in kingdom Animalia. Central in that deportation was the power of natural selection, a theory that Darwin successfully elaborated with profound clarity and established with undeniable force. The body of thought it would spawn would then be eventually named Darwinism, without any nod or reference to Wallace.

And the rest, especially for Newton and Darwin, is history.

History rife with scandal and opportunity for profit, that is. Modern academics and historians have made an industry from these two coincidences. There are some that say Leibniz should be honored as the real originator of calculus and smear malice on Newton; there are some that claim Darwin and his colleagues conspired to rob Wallace of his deserved part in the spotlight. While informative and rather interesting, most of these claims border on tabloid journalism. It is best that we steer our essay on course with Montaigne’s direction.

Newton and Leibniz devised calculus through different paths; same with Darwin and Wallace when they formulated natural selection. Newton did poorly in school when he was young, to a point where his mother wanted him to put down his books and pick up the plow. It was only thanks to the intercession of the school principal that Newton was sent back to the classroom. He did well enough to earn a place in Trinity College, Cambridge when he was 19. The outbreak of plague in 1665 forced the authorities to close down the university and wait until the plague subsided. Newton and the other students were sent back home, and it was at that time when Newton fully devoted himself to the study of mathematics. Over in Germany, the young Leibniz immersed himself in Latin poetry, law, and philosophy culminating in a doctorate in law at the age of 20. His first formal foray into mathematics started at the age of 26 under the tutelage of the prominent Dutch mathematician Christiaan Huygens. His fast progression in maths as a newcomer was mostly attributed to his prodigious intellect, for he was widely considered as a young prodigy. On the other side of the scientific fence, and on the other side of the hemisphere was Wallace, who grew up in less privileged means compared to Darwin. Prestige and connections placed Darwin in financial security, and made it possible for him to pursue his projects without any monetary constraints. While Wallace, after a failed investment in construction, had to work for various projects, scientific and otherwise, just to make ends meet.

The death of Leibniz and Newton’s old age in the last years of the squabble made sure that their relationship never received any repair. The vociferous insistence of their followers also ensured that their conflict would reach posterity. Whereas Wallace and Darwin found a working relationship towards the end of their lives, almost friendship: knowing that Wallace was in dire straights, Darwin lobbied and petitioned to secure a government science pension for Wallace; and when Darwinism was facing rabid opposition from clerics and other institutions, Wallace was at the forefront of defending natural selection, also acknowledging that the idea was rightly named after Darwin.

So where does Montaigne come into this?

As a model, we can use Montaigne’s approach to draw lessons from the two cases. But we can even go further than Montaigne by merging the contents of the two coincidences to arrive at a new arena of knowledge. Thus we ask the question: what can calculus and natural selection teach us about history in general and the history of science in particular?

One of the basic ideas of calculus is the instantaneous rate of change. It asks what is the rate of change at a particular moment in time. For example, how fast is the apple falling to the ground after falling for a second? How fast is it after two seconds? And so on. Applying it to Montaigne’s agenda, we can ask, how fast was A when it passed d? The point to remember is the importance of quantity with respect to time.

Darwin writes in On the Origins that natural selection is the “principle, by which each slight variation, if useful, is preserved”. It is a simple statement that describes the power of his theory in full: traits that are beneficial to the survival of species are preserved and passed on, like the insulating feature of feathers in birds or the allocation of proteins for the production of venom in certain snakes. It is also able to explain the phenomena of extinctions. Species unable to adapt to new conditions, i.e. fail to preserve or come up with useful traits for survival, lose the battle for survival. The point to remember is the importance of the quality with respect to time.

Taking both into account, the quality and quantity of things happening at a particular moment in history should always be considered, since the two are powerful tools in making correct conclusions and revisions – especially in the history of science. For example, it is still prevalent in popular history to call the medieval period as “the dark ages” for it was believed that science regressed during this period. However, a brief review of the number (quantity) of contributions from this period – from Boethius, the Venerable Bede, Ibn Zuhr, the Parisian masters, the translation movement in Andalusia, and etc. – show that there were vibrant scientific communities in this “dark” period that helped advance nascent science well towards the enlightenment. Furthermore, an analysis of the contents of their works (quality) also show that they made great strides in various fields, like in mathematics and physics as uncovered by Pierre Duhem. Quality and quantity should serve as two eyes that give us the best possible overall view of historical events, periods, and scenarios; preferring one over the other not only deprives us of historical riches, but also leads us astray towards scientific myopia and intellectual poverty. Hit two notes that are in key and chances are you end up with a beautiful sounding harmony.

What this underlines is rather important. It simply asserts that the object of history is man (I take this to mean the totality of mankind, while also recognizing its sexist overtone). What man undertakes, be it art, culture, science, and others, are subjects of history. Thus, if we are to follow Montaigne and analyze a particular moment in history, that analysis is an inquiry into the total humanity of that specific period. For instance, we can only begin to fully understand Newton’s arrival at a theory of gravity if we also understand and recognize the body of knowledge that had been built up to that time by earlier and older thinkers for Newton’s perusal. Moreover, this also shows how ideology and environment play a direct role in science – Darwin’s journey in the Beagle was preceded by previous projects of British scientists, initiated by Joseph Banks, to catalogue everything in the natural world. This project in turn, was made possible by advancements in cartography which was highly encouraged and fostered to assist the goals of empire.

Scientists are nothing but human creatures with all the foibles that come along with being human. They cannot be separated from the immediate contexts from where they belong. Science as practiced by these scientists is but a collective effort to make nature intelligible to add to the total body of knowledge accumulated across history. In this respect, modern science, under capitalism, has become a specialized and unique division of labour where one’s “commodity” serves the research agenda of specific science bodies. Modern science is informed by history and enforced by the politics it finds itself in. The eugenic science of Nazi Germany served the racist ideology of the Third Reich. Pharmaceutical companies employ thousands and thousands of scientists to discover the newest drug to secure patent, and thus, profits.

Corollary to this is the concern for the state of science in today’s societies. The widespread revival of right-wing beliefs has sparked renewed interest in the false science of eugenics. Creationism is on the rise. Silly conspiracies such as the flat-earth theory have been steadily gaining traction. Moreover, general mistrust of science has been growing, with people resorting to mysticism and pseudo-science for alternatives. While we have seen episodes in history where societies have suffered for abandoning the fruits of knowledge to rot, who is to say that we are in a far better position to avoid an intellectual famine? Didn’t Montaigne tell us that there is more than one way to reach a certain point?

The general condition of people all over the world is also of concern. With all the problems of the world compounding into one massive pile, the world needs to pool enough people to look for solutions. However, both the quality and quantity of conditions and institutions that enable people to do so have been scant and wanting. Wallace arrived at the correct conclusions despite having little support. I am inclined to believe that people from all walks of life are also capable of doing so. More so in our modern digital age. But most solutions are caged in minds punished daily by terrible work conditions, whether it be in sweatshops or in mines. Words are locked inside mouths wanting of food. Gifted hands are used to beg for scraps. Children of immense talent are fettered by famine and drought. If Montaigne were alive, he might have viewed the depressing situation as a flagrant fodder for an essay called That Men and Women by Various Ways are Wasted to No End.

Tasks gargantuan lay dreadful ahead. I find no reason to believe that Homo sapiens are incapable of hurdling this mission – the species that launched its members to the moon deserve no less than due honor. But humanity needs all the Zenos and cavalier Frenchmen that it can marshal to overcome the quagmire of challenges that blockade its journey to a more rational tomorrow. They need not be such, though, neither do they have to be the modern day Darwin or Newton. They just need to be themselves, but without the stifling chains that impede their potential to come up with the next big idea. Who knows, the cure for cancer might be in a pan in your local diner, sautéed to perfection by an undiscovered science royalty.



Bardi, Jason Socrates. The calculus wars: Newton, Leibniz, and the greatest mathematical clash of all time. New York: Basic Books. 2007

Desmond, Adrian & Moore, James. Darwin. London: Penguin. 1992

Mayer, Ernst. The growth of biological thought: diversity, evolution, and inheritance. Harvard: Belknap Press. 1982

de Montaigne, Michel. Michel de Montaigne – the complete essays (Screech, M.A. trans). London: Penguin Classics. 1993

Westfall, Richard. Never at rest: a biography of Isaac Newton. Cambridge: Cambridge University Press. 1983


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