They were now (the headmaster glanced at his watch) thirty-five minutes into the maths section. As the boy sat back yet again, his arms dangling and his eyes half-closed behind the lenses of his spectacles, so Harmon quietly stood up and approached him from the rear. Outside, the rain was blowing in gusts against the windowpanes; in here, an old clock ticked on the wall, pacing the head's breathing. He glanced over Keogh's shoulder, not really knowing what he expected to see.
His glance became a fixed stare. He blinked, blinked again, and his eyes opened wide. His eyebrows drew together as he craned his neck the better to see. If Keogh heard his gasp of astonishment he made no sign, remained seated, continued to gaze blearily at the rain rivering the windows.
Harmon took a step backwards away from the boy, turned and went back to his desk. He seated himself, slid open a drawer, held his breath and took out the answers to the maths section. Keogh had not only answered the questions, he'd got them right! All of them! That last frenzied burst of work had been him working on the sixth and last. Moreover he'd accomplished it with the very minimum of rough work and hardly any use at all of the familiar and accepted formulae.
Finally the head allowed himself a deep, deep breath, gawped again at the printed answer sheets in his hand — the masses of complicated workings and neatly resolved solutions — then carefully placed them back in the drawer and slid it shut. He could hardly credit it. If he hadn't been sitting here through the entire examination, he'd swear the boy must have cheated. But quite obviously, that was not the case. So… what did Harmon have here?
'Intuitive,' Howard Jamieson had called the boy, an intuitive mathematician'. Very well, Harmon would see how well (if at all) this intuition of his worked with the next paper. Meanwhile –
The headmaster rubbed his chin and stared thoughtfully at the back of Keogh's head. He must speak to both Jamieson and young George Hannant (who'd first brought the boy to Jamieson's attention, apparently) at greater length. These were early days, of course, but… intuition? It seemed to Harmon that there just might be another word for what Keogh was, one which the teachers it Harden simply hadn't thought to apply. Harmon could well understand that, for he too was reluctant. > The word in Harmon's mind was 'genius', and if this was so then certainly there was a place for Keogh at the Tech. Harmon would soon discover if he was right. And of course he was. It was only in his application that he was wrong. Keogh's 'genius' lay in an entirely different direction.
Jack Harmon was short, fat, hirsute and generally apish. He would be quite ugly except that he exuded a friendlin ess and an aura of well-being that cut right through his outer guise to show the man inside for what he really was: one of Nature's truest gentlemen. He also had a quite brilliant mind.
In Harmon's younger days he had known George Hannant's father. That was when J. G. Hannant had been head at Harden and Harmon had taught elementary Maths and Science at a tiny school in Morton, another colliery village. On and off over the intervening years he'd met the younger Hannant and so watched him grow up. It had come as no great surprise to him to learn that George Hannant, too, had finally come into 'the business' — teaching must be as much a part of him as it had been of his father.
'Young Hannant', Harmon had always thought of him. Ridiculous — for of course George had been a teacher now for almost twenty years!
Harmon had called the Maths teacher down from his own school to Hartlepool in order to talk to him about Harry Keogh. It was the Monday following Keogh's 'examination' and they had met at the Tech. Harmon lived close by and had taken the younger man home with him for a lunch of cold meats and pickles. His wife, knowing it was business, served the food then went shopping while the two men ate and talked. Harmon opened with an apology:
'I hope it isn't inconvenient for you, George, to be called away like this? I know Howard keeps you pretty busy up there.'
Hannant nodded. 'No problem at all. "Himself is standing in for me this afternoon. He likes to take a crack at it now and then. Says he "misses" the classroom. I'm sure he'd swap that study of his — and the admin that goes with it — for a classroom full of boys any time!'
'Oh, he would, he would! Wouldn't we all?' Harmon grinned. 'But it's the money, George, it's the money!
And I suppose the prestige has a little to do with it, too. You'll know what I mean when you're a "head" in your 'own right. Now then, tell me about Keogh. You're the one who discovered him, aren't you?'
'I think it's truer to say he discovered himself,' Hannant answered. 'It's as if he's only recently woken up to his town potential. A late starter, so to speak.' 'But one who's all set to overtake the rest of the runners in a flash, eh?'
'Ah!' said Hannant. Since Harmon hadn't yet said: anything about the results of Keogh's tests, he had half-feared that the boy had failed. Being called down here had reassured him a little, and now Harmon's remark about Keogh 'overtaking the rest' had clinched it. 'He passed then?
'Hannant smiled.
'No,' Harmon shook his head. 'He failed — miserably! ' The English paper let him down. He tried hard, I believe, but-'
Hannant's smile faded. His shoulders slumped a little. ' — but I'm taking him anyway,' Harmon finished, grinning again as Hannant's wide eyes came up once more to meet his. 'On the strength of what he did with the other papers.' 'What he did with them?'
Harmon nodded. 'I admit that I gave him the most difficult questions I could find — and he made mincemeat of them! If he has any fault at all, I'd say it was his unorthodox approach — if that in itself is a fault. It's just that he seems to dispense with all the customary ' formulae.'
Hannant nodded, made no comment, thought: / know exactly what you mean! And when he saw that Harmon was waiting, he said out loud, 'Oh, yes — he does that.'
'I thought it might just be Maths,' said the other, 'but it was just the same with the other paper. Call it "IQ" or "spatial" or whatever, it's mainly designed to test the potential of the intellect. I found his answer to one of the questions especially interesting; not the answer itself, you understand, which was absolutely correct anyway, but the way he arrived at it. It concerned a triangle.'
'Oh, yes?' Ah! Trig, Hannant thought, forking a piece of chicken into his mouth. / wondered how he'd do with that.
'Of course, it could have been solved with simple trigonometry,' (Harmon had almost read his mind,) 'or even visually — it was that simple. Indeed it was the only simple question in the batch. Here, let me show you:'
He pushed his plate aside, took out a pen and sketched on a paper napkin:
'Where AD is half AC, and AE is half AB, how much greater is the larger triangle than the smaller?' Hannant dotted the diagram so:
and said: four times greater. Visual, as you said.'
'Right. But Keogh simply wrote down the answer. No dotted lines, just the answer. I stopped him and asked: "How did you do that?" He shrugged and said: "A half times a half is a quarter — the smaller triangle is one quarter as great as the big one.'"
Hannant smiled, shrugged. 'That's typical of Keogh,' he said. 'It's what first attracted me to him. He ignores formulae, jumps gaps in the normal reasoning process, leaps from terminal to terminal.'
Harmon's expression hadn't changed. It was a very serious expression. 'What formulae?' he asked. 'Has he done Trig yet?'
Hannant's smile slipped. He frowned, paused with his fork half-way to his mouth. 'No, we were just starting.'
'So he wouldn't have known this formula anyway?'
'No, that's true,' Hannant's frown deepened.
'But he does now — and so do we!'
'Sorry?' Hannant had been left behind somewhere.
Harmon went on: 'I said to him, "Keogh, that's all very well, but what if it wasn't a right-angled triangle? What if it was like… this?"'
Again he sketched
‘ And I said to him,' Harmon continued, '"this time AD is half AB, but BE is only a quarter of BC." Well, Keogh just looked at it and said: "One eighth. Quarter times a half.' And then he did this
'What point are you trying to make?' Hannant found himself fascinated by the other's tense expression, if not by his subject. What was Harmon getting at?
'But isn't it obvious? This is a formula, and he'd figured it out for himself. And he'd done it during the examination!'
'It may not be as clever or inexplicable as you think,' Hannant shook his head. 'As I said, we were going to be starting on Trig in the near future. Keogh knew that. He may have done some reading in advance, that's all.'
'Oh?' said Harmon, and now he beamed, reached across the table and punched the other on the shoulder. 'Then do me a favour, George, and send me a copy of the text-book he's been swotting from, will you? I'd very much like to see it. You see, in all my years of teaching, that's a formula I never came across. Archimedes might well have known it, Euclid or Pythagorus, but I certainly didn't!'
'What?' Hannant stared again at the diagram, stared harder. 'But surely I know this? I mean, I understand Keogh's principle. Surely I've seen it before? I must have — Christ, I've been teaching Trig for twenty years!'
'My young friend,' said Harmon, 'so have I, and longer. Listen: I know all about sines, cosines, tangents — I fully understand trigonometrical ratios — I am as familiar with all the common or garden mathematical formulae as you yourself are. Probably more familiar. But I never saw a principle so clearly set forth, so brilliantly logical, so expertly… exposed! Exposed, yes, that's it! You can't say Keogh invented this because he didn't — no more than Newton invented gravity — or "discovered" it, as they say. No, for it's as constant as pi: it has always been there. But it took Keogh to show us it was there!' He shrugged defeatedly. 'How might I explain what I mean?'