Daniel almost writes like the integration symbol used in the calculus. But he suppresses that, since that symbol, and indeed the term calculus, were invented by Leibniz.
I made some waggish student-like remark about this curve, as curves had been much on our minds the previous year, and Newton began to speak with confidence and enthusiasm-demonstrating that the ideas he spoke of were not extemporaneous speculation but a fully developed theory on which he had been working for some time.
“Yes, and suppose we were on one of those punts,” Newton said, pointing to one of the narrow, flat-bottomed boats that idle students used to mess about on the Cam. “And suppose that the Bridge was the Origin of a system of Cartesian coordinates covering Jesus Green and the other land surrounding the river’s course.”
No, no, no, no. Daniel dips his quill and scratches that bit out. It is an anachronism. Worse, it’s a Leibnizism. Natural Philosophers may talk that way in 1713, but they didn’t fifty years ago. He has to translate it back into the sort of language that Descartes would have used.
“And suppose,” Newton continued, “that we had a rope with regularly spaced knots, such as mariners use to log their speed, and we anchored one end of it on the Bridge-for the Bridge is a fixed point in absolute space. If that rope were stretched tight it would be akin to one of the numbered lines employed by Monsieur Descartes in his Geometry. By stretching it between the Bridge and the punt, we could measure how far the punt had drifted down-river, and in which direction.”
Actually, this is not the way Isaac ever would have said it. But Daniel’s writing this for princes and parliamentarians, not Natural Philosophers, and so he has to put long explanations in Isaac’s mouth.
“And lastly suppose that the Cam flowed always at the same speed, and that our punt matched it. That is what I call a fluxion-a flowing movement along the curve over time. I think you can see that as we rounded the first limb of the S-curve around Jesus College, where the river bends southward, our fluxion in the north-south direction would be steadily changing. At the moment we passed under the Bridge, we’d be pointed northeast, and so we would have a large northwards fluxion. A minute later, when we reached the point just above Jesus College, we’d be going due east, and so our north-south fluxion would be zero. A minute after that, after we’d curved round and drawn alongside Midsummer Commons, we’d be headed southeast, meaning that we would have developed a large southward fluxion-but even that would reduce and tend back towards zero as the stream curved round northwards again towards Stourbridge Fair.”
He can stop here. For those who know how to read between the lines, this is sufficient to prove Newton had the calculus-or Fluxions, as he called it-in ’65, most likely ’64. No point in beating them over the head with it…
Yes, beating someone over the head is the entire point.
Almost five thousand years agone, there were pilgrims walking to the Celestial City, as these two honest persons are; and Beelzebub, Apollyon, and Legion, with their companions, perceiving by the path that the pilgrims made that their way to the City lay through this town of Vanity, they contrived here to set up a fair; a fair wherein should be sold all sorts of vanity, and that it should last all the year long. Therefore at this Fair are all such merchandise sold, as houses, lands, trades, places, honours, preferments, titles, countries, kingdoms, lusts, pleasures, and delights of all sorts, as whores, bawds, wives, husbands, children, masters, servants, lives, blood, bodies, souls, silver, gold, pearls, precious stones, and what not.
And moreover, at this Fair there is at all times to be seen jugglings, cheats, games, plays, fools, apes, knaves, and rogues, and that of all sorts.
-JOHNBUNYAN,The Pilgrim’s Progress
IT WAS LESS THAN ANhour’s walk to the Fair, strolling along gently sloped green banks with weeping-willows, beneath whose canopies were hidden various prostrate students. Black cattle mowed the grass unevenly and strewed cow-pies along their way. At first the river was shallow enough to wade across, and its bottom was carpeted with slender fronds that, near the top, were bent slightly downstream by the mild current. “Now, there is a curve whose fluxion in the downstream direction is nil at the point where it is rooted in the bottom-that is to say, it rises vertically from the mud-but increases as it rises.”
Here Daniel was a bit lost. “Fluxion seems to mean a flowing over time-so it makes perfect sense when you apply the word to the position of a punt on a river, which is, as a matter of fact, flowing over time. But now you seem to be applying it to the shape of a weed, which is not flowing-it’s just standing there sort of bent.”
“But Daniel, the virtue of this approach is that it doesn’t matter what the actual physical situation is, a curve is ever a curve, and whatever you can do to the curve of a river you can do just as rightly to the curve of a weed -we are free from all that old nonsense now.” Meaning the Aristotelian approach, in which such easy mixing of things with obviously different natures would be abhorrent. All that mattered henceforth, apparently, was what form they adopted when translated into the language of analysis. “Translating a thing into the analytical language is akin to what the alchemist does when he extracts, from some crude ore, a pure spirit, or virtue, or pneuma. The f?ces-the gross external forms of things-which only mislead and confuse us-are cast off to reveal the underlying spirit. And when this is done we may learn that some things that are superficially different are, in their real nature, the same.”
Very soon, as they left the colleges behind, the Cam became broader and deeper and instantly was crowded with much larger boats. Still, they were not boats for the ocean-they were long, narrow, and flat-bottomed, made for rivers and canals, but with far greater displacement than the little punts. Stourbridge Fair was already audible: the murmur of thousands of haggling buyers and sellers, barking of dogs, wild strains from bagpipes and shawms whipping over their heads like twists of bright ribbon unwinding in the breeze. They looked at the boat-people: Independent traders in black hats and white neck-cloths, waterborne Gypsies, ruddy Irish and Scottish men, and simply Englishmen with complicated personal stories, negotiating with sure-footed boat-dogs, throwing buckets of mysterious fluids overboard, pursuing domestic arguments with unseen persons in the tents or shacks pitched on their decks.
Then they rounded a bend, and there was the Fair, spread out in a vast wedge of land, bigger than Cambridge, even more noisy, much more crowded. It was mostly tents and tent-people, who were not their kind of people-Daniel watched Isaac gain a couple of inches in height as he remembered the erect posture that Puritans used to set a better example. In some secluded parts of the Fair (Daniel knew) serious merchants were trading cattle, timber, iron, barrelled oysters-anything that could be brought upriver this far on a boat, or transported overland in a wagon. But this wholesale trade wanted to be invisible, and was. What Isaac saw was a retail fair whose size and gaudiness was all out of proportion to its importance, at least if you went by the amount of money that changed hands. The larger avenues (which meant sluices of mud with planks and logs strewn around for people to step on, or at least push off against) were lined with tents of rope-dancers, jugglers, play-actors, puppet shows, wrestling-champions, dancing-girls, and of course the speciality prostitutes who made the Fair such an important resource for University students. But going up into the smaller byways, they found the tables and stalls and the cleverly fashioned unfolding wagons of traders who’d brought goods from all over Europe, up the Ouse and the Cam to this place to sell them to England.