Chapter XIV: "Supposed Impossibilities"

It was only to be expected that the announcement of a lecture with such an alluring title by such a distinguished scholar and scientist as Professor Franklin Marmion should fill the theatre of the Royal Society, as the reporters said tritely but truly, “to its utmost capacity.”

The mere words, “An Examination of Some Supposed Mathematical Impossibilities,” were just so many bomb-shells tossed into the middle of the scientific arena. The circle-squarers, the triangle-trisectors, the cube-doublers, the flat-worlders, and all the other would-be workers of miracles plainly impossible in a world of three dimensions jumped--not incorrectly--to the conclusion that their favourite impossibility would be selected for examination, and, perhaps--blissful thought!--demonstration by one of the foremost thinkers of the day, to the lasting confusion of the scoffers. Learned pundits of the old school, who were firmly convinced that Mathematics had long ago said their last word, and that to talk about “supposed impossibilities” was blasphemy of the rankest sort, came with note-books and a grim determination to explode Franklin Marmion’s heresies for good and all. Dreamers of Fourth Dimensional dreams came hoping against hope, for the Professor was known to be something of a dreamer himself; and added to all these there assembled a distinguished company of ladies and gentlemen who looked upon the lecture as a “function” which their social positions made it necessary for them to patronise. The reader’s personal friends and acquaintances, including Prince Oscarovitch and Phadrig, were naturally among the most anxiously interested of the Professor’s audience.

It is almost needless to say that Hoskins van Huysman had donned all his panoply of scientific war, and had armed himself with what he believed his keenest weapons; and that Professor Hartley looked with amused confidence to a veritable battle royal of wits when the lecture was over and the discussion began. The Prince and Phadrig were keenly anticipative, and the latter not a little nervous.

A verbatim report of that famous lecture would, of course, be out of place in these pages. If Professor Marmion’s words of wonder are not already written in the archives of the Royal Society, no doubt they will be in the fullness of time when the minds of men shall have become prepared to receive them. Here we are mainly concerned with the results which they produced upon his audience. Certain portions may, however, be properly reproduced here.

When the decorous murmur of applause which greeted the President’s closing sentences had died away, and Franklin Marmion went to the reading-desk and unfolded his notes, there was a tense silence of anticipation, and hundreds of pairs of eyes, which had some of the keenest brains in Europe behind them, were converged upon his spare, erect figure and his refined, clear-cut, somewhat sternly-moulded face.

“Mr President, my lords, ladies, and gentlemen,” he began, in his quiet, but far-reaching tones. “The somewhat peculiar title which I have chosen for my lecture was not, I hope I need scarcely say, selected with a view of arousing any but that intelligent curiosity which is always characteristic of such a distinguished audience as that which I have the honour of addressing to-night. I chose it after somewhat anxious consideration, because I am aware that the bulk of opinion in the world of science strongly insists upon the finality of the axioms of mathematics, and therefore it was with no little hesitancy that I approached such a subject as this. I am well aware that, in the estimation of most of my learned confrères and fellow-seekers after scientific truth, to suggest those axioms may not embody final and universal truth is, if I may put it so, to lay sacrilegious hands on the Ark of the Scientific Covenant.”

A low murmur, prelude of the coming storm, ran through the theatre, and Professor van Huysman permitted himself to snort distinctively, for which he was very promptly, though quietly, called to order by his daughter, who was sitting in front of the platform between him and Lord Leighton. Franklin Marmion paused for a moment and smiled ever so faintly. Nitocris looked round at the now eager audience a trifle anxiously, for she had a fairly clear idea of the trouble that might possibly be ahead. Her father went on as quietly as before:

“Of course, every one here is aware that the great Napoleon once said that the word ‘impossible’ was not French. I need not remind such an audience as this that more than one distinguished student and investigator has suggested that it also may not be scientific.”

The murmur broke out again, and Hoskins van Huysman blew his nose somewhat aggressively. His scientific bile was beginning to rise. He disapproved very strongly of the tone which his rival had begun. Its quiet confidence was somewhat ominous. The lecturer continued without this time noticing the interruption, and proceeded to give a lengthy and learned but singularly lucid resumé of the more recent progress in the higher mathematics and the deeply interesting speculations to which it had given rise. This, with certain demonstrations which he made on the great black-board beside him, occupied nearly an hour. When he had finished there was another murmur, which this time was wholly of applause, for this part of the lecture had not only been masterly but entirely orthodox. Then silence fell again, the silence of expectant waiting, for every one felt that the “Examination” was coming now. He began again in a slightly altered voice.

“What I have just been saying was necessary to my subject as far as it went, but for all that it was chiefly introductory to what I am now going to bring to your notice. But this is a matter rather for illustration and discussion than for mere disquisition. Therefore, to save your time as much as possible, I will proceed at once to the illustration, and then we will have the discussion.”

Professor van Huysman snorted again, even as a war-horse that snuffs the fray. This time Franklin Marmion seemed to recognise the implied challenge, for he looked round the crowded theatre with a curious smile, which seemed to say: “Yes, gentlemen, I see that some of you are getting ready for a tussle. I am in hopes of being able to oblige you.”

“Now,” he continued, “it is generally conceded that an ounce of practice is worth a good many pounds of precept, so I will get to the practice. I need hardly remind you that ever since mathematics became an exact science, three problems have been recognised as impossible of solution--trisecting the triangle, squaring the circle, and doubling the cube. I have now the pleasure of announcing that I have had the great good fortune to discover certain formulæ which, so far, at least, as I can see, make the solution of those problems not only possible, but comparatively easy--to those who know how to use them.”

As he said this, Franklin Marmion looked directly at Hoskins van Huysman. He was the challenger now, and there was a glint in his eyes and a smile on his lips which showed that he meant business. The American writhed, and had it not been for Brenda’s gently but firmly restraining hand, he might have jumped to his feet and precipitated matters in a somewhat embarrassing fashion. The chairman looked up at the lecturer with elevated eyelids which had a note of interrogation under each of them, and then there came that sound of shifting in seats and breathing in many low keys which denotes that an audience has been wound up to a very tense pitch of expectation. If a smaller man had said such words to such hearers some one would have laughed, and then would have burst forth a storm of derision. But the keenest critic had never found Franklin Marmion wrong yet, and he had far too great a reputation to permit himself to say in such a place that which he did not seriously mean. So the hum died down as he went to the black-board, and Nitocris looked at Merrill with something like fear in her eyes.

“If he does that,” whispered Phadrig to the Prince in Russian, “the story that Pent-Ah and Neb-Anat told will be true--which the High Gods forbid!”

“As the trisection of the triangle is, perhaps, the simplest of the three problems,” said the lecturer, with almost judicial calmness, “we will, if you please, begin with that. I hope that gentlemen who have brought note-books with them will be kind enough to follow my calculations and check any error that I may make.”

But a good threescore note-books, pencils, and stylographic pens were out already, and hundreds of eyes were eagerly fastening their gaze on the black-board, their owners desperately anxious to detect the first slip in the demonstration. The demonstrator drew an isosceles triangle rapidly, and without speaking filled the remainder of the board with formulæ. The almost breathless silence was broken only by the click of the chalk on the board and the scratching of pencils and pens on paper. When he had finished he ran through the calculations aloud, and said in the most commonplace voice: