Chapter 1
Two problems, and why they turned out to be one
Calculus exists because two questions refused to go away for two thousand years.
The first question is about steepness at an instant. A car's speedometer says 50 miles an hour. It has not traveled 50 miles, and it will not travel for an hour. So what is that number describing? Judith Grabiner opens her article on the origins of rigor with exactly this exchange between a student and a teacher, and then the student's follow-up: "How in the world did anybody ever think of such an answer?" (Source 360, Grabiner, p. 185). That question is the whole of differential calculus.
The second question is about totals of things that keep changing. How much wine is in a barrel whose sides curve? How far has something traveled if its speed changed the whole time? What is the area under a curve? That is integral calculus.
For most of recorded history these looked like separate crafts. Finding tangents was one trade. Finding areas was another. People were good at both, in Syracuse and in Kerala and in Baghdad and in Oxford, centuries before anyone had a symbol for either.
Guess before you read on
Tangents were being found and areas were being found, in several countries, for centuries before 1670. So the field already worked when Newton and Leibniz walked into it. Name the one thing they added that nobody had.
I have a guess
The algorithm. A set of rules you can hand to a competent person who does not have to be a genius, and they get the answer. Seeing that tangents and areas are inverse operations, and then building a symbolic machine that made the connection routine, is the part that was new.
If you guessed "the derivative" or "the area under a curve", you gave the answer most textbooks give. Both had been done, and the next four chapters are the people who did them.
The thing Newton and Leibniz did, and the reason their names are on the subject, is that they saw the two problems are inverse operations, and then built a symbolic machine that made the connection routine. Not "discovered the area under a curve". Not "invented the derivative". Both of those had been done. What was new was the algorithm: a set of rules you could hand to a competent person who did not have to be a genius, and they would get the answer.
That distinction matters for teaching, because it changes who the story is about. It stops being two men and becomes about thirty people across a dozen centuries, most of whom never met, several of whom got no credit, and a few of whom got someone else's.
The word itself
Calculus is Latin for pebble. Specifically a small limestone counter, from calx, limestone. Roman reckoning boards used them, and a teacher of arithmetic in Rome had two different job titles depending on his legal status: calculo if he was a slave, calculator if he was freeborn (Source 453, Miller and Aldrich). Quintilian records that Demosthenes practiced speaking with pebbles in his mouth, using the same word (Source 455, Lewis and Short, at Quintilian 11.3.54).
The chain runs: limestone, then little stone, then counter on a reckoning board, then the act of reckoning, then a method of reckoning, then this method of reckoning. The subject you are taking is named after a slave's pebble.
And the kidney stone is the same word, arriving by a separate route from the same source. There is no metaphor connecting them beyond the shape (Source 455, Lewis and Short).
One question before you go
A Roman could have held a calculus in his hand. What was it?
Show the answer
A pebble. A small limestone counter, from calx, limestone, pushed around a reckoning board (Source 453, Miller and Aldrich). The chain runs limestone, little stone, counter, the act of reckoning, a method of reckoning, this method of reckoning.
That is the word sitting at the top of Derivatives 1.0, and it is worth keeping there: the subject with the frightening name is named after the cheapest object in the room.