Thursday, 19 December 2013

Calculus - Or the Parable of the Pebbles

An abacus
(photo by Steven Depolo)


Noun. Plural caculi, calculuses. Mid-17th century.
[Latin = small stone (used in reckoning with an abacus).]

1(a) A particular method or system of calculation or reasoning;
especially (MATHEMATICS) infinitesimal calculus (see below). M17

1(b) obsolete. generally. Computation; calculation. L17-L19

2 MEDICINE. A stone. M18

Differential calculus the part of the infinitesimal calculus that deals with derivatives and differentiation.

Infinitesimal calculus the branch of mathematics that deals with the finding and properties of derivatives and
integrals of functions, by methods originally based on the summation of infinitesimal differences.

Integral calculus the part of the infinitesimal calculus that deals with integrals and integration. 

Predicate calculus see PREDICATE.

Calculus elicits many feelings, but I'm pretty sure fondness is rarely among them. And calculus teachers? Well, that fondness might just be ... asymptotically rare.
But I can cite the exception.
As a student, I was always able to do mathematics. I could perform its procedures and processes. But I seldom felt that deeper intuition of its ways, of its whys and wherefores. Trigonometric ratios, polynomial series, imaginary numbers – eek, logarithms. These were all meaningless to me unless the method in their all-too-methodical madness could be made plain. Unless they "clicked." And a very loud click it had to be indeed.
So, I resisted taking calculus in secondary school not because I did not want to do differentials or integrals – OK, surely there was some motivation in the mix – but because I knew the very principles of differentiation and integration would test the limits of my comprehension.
But it was precisely this – that more intimate, more intuitive comprehension of math's way of understanding the world – that my calculus teacher worked so hard to inspire within me. And she succeeded in the most ... well ... etymological way.
At some point early in the course, I, Latin geek that I was, brought to her that calculus means "little stone" in Latin. She well knew this, and when I asked her why, she asked me to imagine a chessboard. On this chessboard I was placing pebbles, trying to cover its whole, flat surface. But no matter how many pebbles I placed on the board, no matter how small the pebbles were, I could never completely cover every last little bit of space. There would always be some unimaginably tiny speck uncovered. "Calculus is about studying the infinitely small and the infinitely big," she said.
See? If you look really closely, you can still just about see a little teensy bit of chessboard
(photo by Alan Stokes)

The Etymology of Calculus

Calculus (plural calculi) indeed comes from the Latin for "little stone" or "pebble," but not for the reasons my teacher claimed above. The word is the diminutive of calx, meaning "limestone," and related to our words chalk and calcium. In Ancient Rome, calculi were used to, well, calculate, especially on tools like the abacus. According to the Oxford Classical Dictionary, an abacus was a "counting-board, the usual aid to reckoning in antiquity ... The number might be marked in writing or by pebbles, counters, or pegs." In Latin, to make a calculation was, quite literally, calculos ponere, "to place pebbles."
If you're like me, abacus may evoke the Chinese iteration, sliding counters on a wooden frame, but a variety indeed existed, such as a Roman version comprising pebbles that reckoned numbers along grooves of assigned value on a tablet. The origin of abacus itself points to yet another method: sand or dust. Hebrew has abaq, meaning "dust," likely from a Semitic root, a-b-q, "to fly off," as in drawing numbers into sand or dust spread across a board or table.
These pebbly numbers weren't so black and white, though. Or were they? Roman judges would issue their opinions in the form of a calculus ater (a black stone) for a condemnation and a calculus albus (a white stone) for an acquittal, and thus these calculi came to stand for sentencing and voting. But they had a lighter side, too, used as they were as pieces in games.
Calculus takes on its more formally mathematical sense – that is, signifying a system of calculation – in the Philosophical Transactions of the Royal Society of London in 1672: "I cannot yet reduce my Observations to a calculus." Think Isaac Newton. Think Leibniz. Think calculus differentialis (read differentials) or calculus integralis (read integrals), the shortening of which yields calculus as we think of it now, naming that branch of mathematics dealing with change. With the infinitely big or small.
If all this feels like you're passing a kidney stone, well, how apt, if coincidental. Calculus, medically, is the name for those kinds of concretions as well.
My teacher's anecdote was not true, historically speaking. Calculus tells a simpler story of increasingly sophisticated substitution: stones, counting, formal systems of counting, calculus. But I need perform no ethical or moral calculus to value the truth of her account. Sometimes in a good etymology we hunger not for a fact, but for a parable – for that eureka, for that a-ha, for that more intuitive understanding.
Has the origin of a word ever given you new understanding of a difficult concept?
What are your memories of taking calculus?
Did you pass calculus or did you pass a calculus?

John Kelly
Mashed Radish

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