# Beginnings

Was it like that for you as well, when you were at school? Words lived. They had histories, back-trails to ancient Greek rhetoric or Roman sensibility, to mediaeval French farms, soggy lowland water-meadows, absurd colonial rituals. They sang, they danced, with their own characteristic rhythm and energy. You could play with words, write stories, compose poems, tell jokes, formulate riddles, act them out, set them to music. And you could read them, voraciously.

Numbers, by contrast, just sat solemnly upon the page. They had no music, no stories, no personality. They obeyed rigid rules. 6 + 7 was always equal to 13 whether you liked it or not – or so I thought at the time.

Maths at school was relentless drudgery. Tables. Fractions. Geometry. The horrors of trying to construct a triangle using only pencil, ruler and compass. Equations that seemed to be as meaningless to my teacher as they were to me. The hopeless muddle of percentages.

As exams loomed my mother, normally so laid back about my education, must have had a little moment of panic because she engaged a mathematics tutor for me. Bob was a PhD student, with the long sideburns that were fashionable in those days, and an endearing Dutch accent. He was diffident and charming. Remarkably, mathematics seemed to live for him in the way that words lived for me.

We began with the gradient of the straight line. Gradient is a hard word, like pedalling your bicycle uphill. With the skill of a super fit pacesetter, Bob led me through a few examples. Perhaps he sensed that my attention was flagging, because unexpectedly he turned to me and asked, ‘Have you ever thought about how we determine the gradient of a curve that’s always changing? A parabola, for example?’

Parabola. Here was a word I liked, even though parabolas were beasts to draw by hand. Many miserable hours had been spent plotting their points on an uncompromising grid of graph paper and trying to connect them with a wavering line.

‘No,’ I replied. ‘That’s not possible, surely?’

‘Let’s think about it.’ He sketched, freehand, the graceful swoop of a parabola. ‘We’ll take two points close together on the curve – here and here, see – and draw a little right-angled triangle to connect them, like this. We can approximate the gradient by the vertical change over the horizontal change.’

So far, so good. At least a parabola was more interesting to look at than a straight line, and there wasn’t a number anywhere in sight.

‘Now suppose you take points closer and closer together, until your triangle becomes infinitesimally small.’

Infinitesimally. This was a word that sang and danced, that opened up stories, questions, endless possibilities.

I looked at him. ‘But how can you do that? How can you make a triangle infinitesimally small? How can you measure it?’

Bob smiled, as though he was about to share an extraordinary secret with me that only he and I would know. ‘Let’s call the change in x – the length of this horizontal line in our triangle – delta x, and the change in y, the vertical line, delta y.’

Greek letters! Delta x and delta y: δx and δy. A whole new world was opening, one where words and mathematics held hands, sang and danced together. As Bob continued to explain the principles of differential calculus I fell passionately, delightedly, irrevocably in love.

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