19 Calculus
A calculus is a way of calculating, so mathematicians sometimes talk about the ‘calculus
of logic’, the ‘calculus of probability’, and so on. But all are agreed there is really only
one Calculus, pure and simple, and this is spelled with a capital C.
Calculus is a central plank of mathematics. It would now be rare for a
scientist, engineer or a quantitative economist not to have come across Calculus,
so wide are its applications. Historically it is associated with Isaac Newton and
Gottfried Leibniz who pioneered it in the 17th century. Their similar theories
resulted in a priority dispute over who was the discoverer of Calculus. In fact,
both men came to their conclusions independently and their methods were quite
different.
Since then Calculus has become a huge subject. Each generation bolts on
techniques they think should be learned by the younger generation, and these
days textbooks run beyond a thousand pages and involve many extras. For all
these add-ons, what is absolutely essential is differentiation and integration, the
twin peaks of Calculus as set up by Newton and Leibniz. The words are derived
from Leibniz’s differentialis (taking differences or ‘taking apart’) and integralis
(the sum of parts, or ‘bringing together’).
In technical language, differentiation is concerned with measuring change and
integration with measuring area, but the jewel in the crown of Calculus is the
‘star result’ that they are two sides of the same coin – differentiation and
integration are the inverses of each other. Calculus is really one subject, and you
need to know about both sides. No wonder that Gilbert and Sullivan’s ‘very
model of a modern Major General’ in The Pirates of Penzance proudly proclaimed
them both:
With many cheerful facts about the square of the hypotenuse.
I’m very good at integral and differential calculus.
Differentiation
Scientists are fond of conducting ‘thought experiments’ – Einstein especially
liked them. Imagine we are standing on a bridge high above a gorge and are