I have a visitor this evening. She's sitting on the edge of the bed.
I'm sitting at the side of the room, next to a small table stuffed against the wall. This is where I used to prepare lessons; now I write programs.
But today it's an autumn evening and I'm teaching her maths.
She sits on my bed and looks at me. She's curvaceous, petite with a round face and fine black hair curling onto her shoulders. She's wearing a cardigan over a patterned blouse, pleasantly distorted by her ample breasts. I track down to her short green skirt and black tights…
Still, to business.
I don't really know why she's here. I've given up teaching maths myself, I can make more money with much less stress by programming computers. It's also much more interesting.
But I know she’s taught in the past - both junior school, and secondary maths - so I suppose it's prudent.
If she ever returns to teaching, that is.
Today we're doing the basics of differential calculus.
I beckon her across, she stands next to me as I sketch the parabola of y = x2 in the exercise book.
"We're going to work out the gradient of the curve at some arbitrary point x," I say, remembering when I did this in sixth form with Dr Dickinson, our irascible pure maths teacher. Then it was all epsilon-deltas and rigorous definitions of limits and convergence. Hell, he even had us constructively defining the real numbers - Dedekind’s sections, I vaguely recall.
None of that is going to be mentioned today.
“I hate to see you standing,” I say, “this would work a lot better if you sat here.”
I point to my lap.
Unaccountably she seems to agree: I pull my chair back a bit and she sits across my knees, looking very intently at the diagram.
Her left arm is draped around my neck - for stability of course; the same reason my right arm is curled around her waist.
“So: the concept of gradient,” I say.
I put my left hand against her shin, halfway between her ankle and knee, feeling smooth, glossy fabric under my fingers.
Here the gradient is steep, I say, running my hand slowly up towards her knees, pushing them slightly up.
My fingers slide over her left knee.
“So here we have what's called a local maximum. See how, just here, my hand is flat, the gradient is zero. But as we move further, we're now descending rather than ascending - the gradient’s turned negative.”
I collide with the hem of her skirt; I push forward so that my palm and fingers curve upon her warm thigh.
“Ok," I say, "That's the basic idea. Differential calculus is just doing the same thing with maths.”
And so we continue.
She seems to find the lesson useful.
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Another lesson, this time we're doing Taylor series. Again, this is something we did ‘properly’ at school. We cared about error terms and convergence; we looked at pathological functions which lack a Taylor expansion. I'm only recently appreciating how weird it is that infinite sums of polynomials converge to functions which really do look nothing like polynomials. The power of calculus, etc.
As usual, nothing of such deep reality will go into this evening's lesson.
She understands some trig now: sine and cosine. I just about managed to convey the differentiation of sin and cos from first principles. So now we have enough to derive the Taylor series for sine. Of course, I have an idea as to how to make this memorable: the sums of successive terms cumulatively approach the target function - tonight sine seems very appropriate.
She's lying flat on her back on my bed. I'm standing by my table, looking down at her. The room is already darkening, the room illuminated by a bedside table light.
Tonight she's wearing a white tee shirt, faded blue jeans, and socks. Her head is lying on my pillow, hair curling and sprawling on the pillowcase. She seems confident, relaxed.
I say, “Three terms give us a pretty accurate approximation for sine, up to say 70 degrees; although of course we actually measure the angle in radians.”
I lean over and grasp the fabric of her left sock, at the toes.
“Call this the first term of the Taylor expansion of sin(x),” I say in a professorial manner, “So that would be x itself, a straight line through the origin.”
With a flourish I pull the sock off.
“But apart from here at the origin," - I tap her now-bare instep - "this is not a terribly good approximation to the whole you.”
I now take the second sock between finger and thumb, and slide it off.
“The second term,” I say, “x cubed over three factorial, with a minus sign. It's a bit better but still not great.”
I straighten, still in my best tutorial manner, and add:
“A third term will dramatically improve things. We add back x to the fifth over five factorial.”
I look mock-puzzled: "How are we going to do that?"
She is looking towards the ceiling, listening quietly, not a care in the world. My gaze is drawn by a quick movement at her waist as the stud is released and the zip loosened. She subsides into stillness again.
“Ok,” I say, pulling at the fabric at her ankles, “now we're getting closer to approximating the real you.”
She helps by raising herself as her jeans join her socks in an untidy pile on the floor.
She's very pretty, there on my bed.
“But we only got to seventy degrees,” she says, “Surely we owe it to Mr Taylor to… approximate even more accurately?”
To the faint rustle of fabric, I turn to my notebook, to work out how many extra degrees we get by including the fourth term. It's interesting; we'd go past the right angle to 106 degrees... I turn back.
Her moving hands now enfold the hem of her tee shirt: “Close enough, d'you think?” she murmurs, "Or should we go even further?"
