Chocolate, Mint And Notation

“But calculus, Mr Moire. Why do they insist we learn calculus? You said that Newton and Leibniz started it but why did they do it?”

“Scoop me a double-dip chocolate-mint gelato, Jeremy, and I’ll tell you about an infamous quarrel. You named Newton first. I expect most Europeans would name Leibniz first.”

“Here’s your gelato. What does geography have to do with it?”

“Thanks. Mmm, love this combination. Part of the geography thing is international history, part of it is personality and part of it is convenience. England and continental Europe have a history of rivalry in everything from the arts to trade to outright warfare. Each naturally tends to favor its own residents and institutions. Some people say that the British Royal Society was founded to compete with the French philosophical clubs. Maybe England’s king appointed Newton as the society’s President to upgrade the rivalry. Dicey choice. From what I’ve read, Newton’s didn’t hate everybody, he just didn’t like anybody. But somehow he ran that group effectively despite his tendency to go full‑tilt against anyone who disagreed with his views.”

“Leibniz did the same thing on the European side?”

“No, quite the opposite. There was no pre‑existing group for him to head up and he didn’t start one. Instead, he served as a sort of Information Central while working as diplomat and counselor for a series of rulers of various countries. He carried on a lively correspondence with pretty much everyone doing science or philosophy. He kept the world up to date and in the process inserted his own ideas and proposals into the conversation. Unlike Newton, Leibniz was a friendly soul, constantly looking for compromise. Their separate calculus notations are a great example.”

“Huh? Didn’t everyone use the same letters and stuff?”

“The letters, yeah mostly, but the stuff part was a long time coming. What’s calculus about?”

“All I’ve seen so far is proofs and recipes for integrating different function types. Nothing about what it’s about.”

Newton approximates arc ABCDEF.

<sigh> “That’s because you’re being taught by a mathematician. Calculus is about change and how to handle it mathematically. That was a hot topic back in the 1600s and it’s still central to Physics. Newton’s momentum‑acceleration‑force perspective led him to visualize things flowing with time. His Laws of Motion made it easy to calculate straight‑line flows but what to do about curves? His solution was to break the curve into tiny segments he called fluxions. He considered each fluxion to be a microscopic straight line that existed for an infinitesimal time interval. A fluxion’s length was its time interval multiplied by the velocity along it. His algebraic shorthand for ‘per time‘ was to put a dot over whatever letter he was using for distance. Velocity along x was . Acceleration is velocity change per time so he wrote that with a double dot like . His version of calculus amounted to summing fluxion lengths across the total travel time.”

“But that only does time stuff. What about how, say, potential energy adds up across a distance?”

“Excellent question. Newton’s notation wasn’t up to that challenge, but Leibniz developed something better.”

“He copied what Newton was doing and generalized it somehow.”

“Uh, no. Newton claimed Leibniz had done that but Leibniz swore he’d been working entirely independently. Two lines of evidence. First, Newton was notoriously secretive about his work. He held onto his planetary orbit calculations for years before Halley convinced him to publish. Second, Leibniz and other European thinkers came to the problem with a different strategy. Descartes invented Cartesian coordinates a half‑century before. That invention naturally led the Europeans to plot anything against anything. Newton’s fluxions combined tiny amounts of distance and time; Leibniz and company split the two dimensions, one increment along each component. Leibniz tried out a dozen different notations for the increment. After much discussion he finally settled on a simple d. The increment along x is dx, but x could be anything quantitative. dy/dx quantifies y‘s change with x.”

“Ah, the increments are the differentials we see in class. But those all come from limit processes.”

“Leibniz’ d symbol and its powerful multi‑dimensional extensions carry that implication. More poetry.”

~~ Rich Olcott