Chapter 1: A Matter of Interest

Proceedings in the Grimshaw versus Pepper case began at ten-thirty a.m., Judge Embert Wimple presiding. The plaintiff alleged that the defendant was guilty of failure to repay a loan, the defendant responding hat he had not done so because the plaintiff had deceived him. Neil Grimshaw was represented by the redoubtable Desmond Oddley-Staggers, while Gordon Pepper was in the hands of the no less distinguished Rodney Melliflewes.

Judge Wimple, aged eighty-three and officially long-retired, still appeared frequently in court to assist hard-pressed colleagues. The hearing was to be held in camera, which pleased the judge, who foresaw some stimulating entertainment, which he was ill-disposed to share with a courtroom full of jurors and other undesirables. In fact, from this case onward until his retirement, the judge was to hold all his hearings with the public excluded, though he would continue to avail himself of a courtroom, as he considered his chambers inviolate.

After satisfying himself that he was clear as to the respective charges, Judge Wimple addressed prosecuting counsel. “Very well, Mr Taggart, you may begin.”

Oddley-Staggers was accustomed to the judge’s habit of addressing advocates by whatever names occurred to him, irrespective of true identities. His honour’s view was that he was dealing with a pool of lawyers and a like-sized array of names, and which of the latter he applied to any of the former was not important to him. On the occasions when his attention wavered more than average, he had been known to refer to barristers long dead. It had become a convention that, in the interest of reaching the end of a case within a tolerable period, nobody corrected Judge Wimple. The litigants were usually primed and almost always obliged.

Thumbs in waistcoat pockets, Oddley-Staggers inclined his head towards the bench. “May it please Your Honour,” he said, “the facts in this case are quite simple. Some fourteen months ago, my client loaned the sum of one pound to the defendant, the arrangement being that Mr Pepper would, at the end of one year from the date of the loan, repay the principal, plus interest. Mr Pepper did not honour this obligation and still has not done so. My client’s purpose here is to recover the amount owed. He appreciates that the sum concerned is a modest one, but contends that there is a principle involved, which must be upheld. Subject to a satisfactory outcome, he is willing to ignore any interest incurred after the one-year period.”

“Seemingly most generous,” said the judge inspecting his papers. “We must never ignore matters of principle, or indeed in this case also principal. Now, it says here that the amount owed is two pounds, seventy-two pence. I am puzzled as to how we get from the one figure to the other. Presumably this arises from either administrative charges or the interest rate. Were there any charges and what was that rate?”

Oddley-Staggers reddened slightly. “No charges, Your Honour. The interest rate was” – his voice fell to a mumble – “one hundred per cent, nominal.”

“Speak up, Mr Olliphant. I’m not sure I heard that correctly.”

“No charges. The interest rate was one hundred per cent, nominal.”

“Bless my soul,” said the judge. “I am not au fait with current trends, but if your client makes a habit of this, he must be in a lucrative business. Furthermore, my arithmetic, though possibly defective, suggests an amount of two pounds owing. I still don’t see how we get to two pounds, seventy-two pence. Please explain.”

“Your Honour, my client does not operate in the financial world, but was merely doing a favour. As to the sum involved, we are dealing here with a factor known as the exponential constant, which is the base of the natural logarithm. It is usually referred to by its initial letter, e.”

“Most interesting. Would you care to regale us further?”

“Yes. This kind of situation occurs frequently in certain areas of mathematics, physics and commerce, where two interacting elements are involved, one rising as the other falls. In this case, it concerns multiple compounding of interest. At the time the loan was made, there was no discussion between the parties as to the number of periods my client was to employ.”

The judge was enthralled. “And this makes such a large difference, does it?”

“Indeed it does. My client realised that, as the rate was nominal – or at least that there had been no agreement that it was not – there was no impediment to his compounding interest at periods of less than one year. He found that by doing so at increasingly frequent intervals, the sum owed became ever higher. He considered half-yearly periods, then quarterly ones, then monthly, then weekly and so on, until he reached the point at which no further meaningful increment could be achieved. It is a question of an arithmetical series, leading to the exponential constant I mentioned. This series consists of one, plus one, plus one divided by two, factorial, plus one divided three, factorial, and so –”

“One moment” said the judge. “You say ‘factorial’. Perhaps you would expand?”

“Willingly. The expression is mathematical shorthand. Any number factorial means that number multiplied by the one immediately below it, then the result by the next lower one and so on until unity is reached. For example, five factorial means five, times four, times three, times two, times one, the last operation being of course academic, since it does not change the total. In this case, no matter how often the compounding occurs, there is an effective limit, which to two decimal places – the practical level in financial matters – is two, point seven two. We are concerned here with an irrational number.”

“We certainly are,” said the judge. “I never heard of anything less rational.”

“If I may explain, Your Honour, an irrational number is one which has no precise value, but which can be calculated to any desired degree of accuracy, the digits following the decimal point proceeding to infinity, with no repeated pattern and always with a remainder.”

The judge was well aware of the meaning of the term concerned, but was not inclined to miss an opportunity to allow any counsel, especially one of his regulars, to demonstrate a grasp of whatever was at issue. The more experienced ones enjoyed these diversions as much as he did. His honour’s view was that all of this added colour to the proceedings. “Fascinating!” he said. “Please continue.”

“I was about to say that mathematicians usually consider the result to five decimal places as satisfactory, this being two, point seven one eight two eight.”

“Well, well,” said the judge. “I imagine that financiers find the idea even more agreeable. Was there no obligation on Mr Grimshaw to reveal this multiple compounding to Mr Pepper as it proceeded?”

“Nothing to that effect was specified at the outset, Your Honour. The technique is widely used in commercial transactions.”

“Thank you, Mr Oddment,” said the judge, turning his attention to defending counsel, “Now, Mr Mildew,” he said. “What have you to say?”