"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:
"Now, gentlemen, if, as I hope, you have found my working correct, I may
draw the two lines which will trisect the triangle."
He drew them, and then, as calmly as though he had done nothing more
than cross the much-trodden _pons asinorum_, he told two attendants to
take the board down and put it in front of the platform; then, while
they were lifting another on to the easel, he said:
"As those who have followed me would no doubt like a little time to
revise the figures, I will go on with the next problem, which will be
our old friend, or enemy, the squaring of the circle."
The second board was filled with diagrams and formulæ as rapidly as the
first.
"There is the demonstration, gentlemen," he said, as the attendants
placed it beside the other in full view of everybody. "Now, as time is
shortening, I will get on with the third problem."
The chalk began to click again, and the pens and pencils scratched on to
the accompaniment of murmurs and whispers and occasional grunts and
snorts of incredulity. By a master-stroke of strategy Franklin Marmion
had, in placing the three demonstrations of the long-supposed impossible
before them in quick succession, kept the learned, but now utterly
bewildered mathematicians so busy that they literally had not time to
begin "the trouble" which Brenda was now actually dreading. Her father's
face, bent down over his note-book, was getting more terrible to look
upon every moment. The mere fact that he had not uttered a sound since
the demonstrations had begun was sufficiently ominous, for it meant that
he was puzzled--perhaps even beaten--and if that was so, she dreaded to
even imagine what might happen. On the other hand, Nitocris felt her
spirits rising as she looked round and saw the many learned heads
bending and shaking over the note-books, each owner of them working at
high pressure to win the honour of first finding the error which all
firmly believed must exist, and which none of them could detect.
When he had finished his third demonstration, Franklin Marmion, without
interrupting the hard thinking that was going on, took a chair by the
side of the President, poured out a glass of water, and waited for
results.
"Marmion, what is this white magic that you have been springing upon
us?" whispered the presiding genius of the learned assembly, looking up
from several sheets of paper which he had been rapidly covering with
formulæ. "These things are impossible, you know--unless, of course, you
have got a good deal farther than any of us. And yet the calculations
are correct as far as I can follow them, and no one else seems to have
hit on any error yet. I must confess, though, that these progressives of
yours are too deep for me. I can follow them, and yet I can't. At a
certain point they seem to elude me, and yet the calculations are
rigidly right. It's almost enough to make one think you had done what
Cayley once told us in this room some one might do some day."
"My Lord," replied Franklin Marmion, almost inaudibly, "I began my
address by remarking, as you will remember, that perhaps, after all, the
word 'impossible' might not be scientific."
Their eyes met, and the President, than whose there was no greater name
in the higher realm of learning, saw something in Marmion's which sent a
little chill through him, and that something told him that he was in the
presence of a superior being.
"Dear me!" he murmured, looking down at his papers again, "the age of
miracles is not past, after all--in fact, it is only just beginning."
"It is re-beginning, my Lord--for us," came the reply, in a voice which
seemed to come from very far away.
The President did not reply. As a matter of fact, he had no reply ready,
and he had something else to do. He rose, and said in a somewhat
constrained voice:
"Ladies and gentlemen, Professor Marmion has shown us some very strange
demonstrations which have certainly amply justified the title which he
selected. A good many gentlemen, and some ladies as well, I am glad to
see, have followed his calculations very carefully. I have done the same
myself, but I am bound to confess that I have not been able to find any
error. I think I shall be right in saying that no one will be more
pleased than the learned and--er--gifted lecturer to hear that some one
else has been able to do so."
Franklin Marmion bowed his assent, and a faint smile flickered across
his clean-shaven lips. The next instant Professor van Huysman was on his
legs, note-book in one hand and stylo in the other. All the fresh colour
had gone out of his face; his eyes were burning, and his lips were
twitching with uncontrollable excitement.
"My Lord," he began, in a voice that even Brenda hardly recognised,
"like yourself, I have been unable to find any actual error in the
lecturer's demonstrations of which I will take permission to call the
possibility of the impossible; in other words, that a contradiction in
terms can be true and false at one and the same time. That, my Lord, and
ladies, and gentlemen," he went on, raising his voice almost to a
shout, "is still, and, I hope, in the interests of true science, and not
adroit jugglery with figures and formulæ, will ever remain, another
impossibility. Professor Marmion has apparently trisected the triangle,
squared the circle, and doubled the cube. It may be that he has
persuaded some present that he really has done so; but, again, in the
interests of science, I desire to protest against the way in which these
demonstrations have been sprung upon us. Calculations which he has
doubtless taken months to elaborate, he has asked us to test in a few
minutes. For myself, I decline to accept them as true, and I hope that
others will do the same until we have had time to satisfy ourselves that
the hitherto impossible has been made possible."
He sat down, breathing hard and white with anger and excitement, and
then the trouble began. The trisectors, the circle-squarers, and the
cube-doublers, had seen their long-flouted theories proved to
demonstration by one of the most learned and responsible men of science
in the world, and one of their most sarcastic and hitherto successful
flouters had been compelled to confess that he could find no flaw in the
calculations of this mathematical Daniel so unexpectedly come to
judgment. They did not understand his proofs, but that was no reason why
they should reject them, and so they rose as one man in support of their
champion to demand that Professor van Huysman should withdraw his
imputations of jugglery. He sat still, and shook his head. He was too
disgusted and bewildered to do or say anything more until he had made a
searching analysis of these diabolical formulæ.
But there were others who wanted to have their say in defence of
scientific orthodoxy, and they had it--and the rest was a chaos of
intellectual conflict until, at the end of nearly an hour, the
President, who now saw with clearer eyes than any of the disputants,
rose and put an end to the discussion by remarking that they had not the
whole night before them, and that all that Professor Marmion had said
and done would be published in the scientific papers; further, that such
a controversy would perhaps be more profitably conducted in print than
by word of mouth. Such a course would give every one ample leisure to
work out the problems in the light of the new demonstrations, and also
give a much better prospect of reaching a logical, and therefore just,
conclusion than a discussion in which haste, and possibly pre-conceived
opinions, from the influence of which no human being was really free,
could possibly promise.
This, of course, put an end to the matter for the time being, and, after
the usual votes of thanks and acknowledgments, the distinguished company
dispersed--amused, mystified, gratified, bewildered, and exasperated:
but, saving only four of its members, with no idea of the effect which
that evening's proceedings were destined to have upon the fate of
Europe, perhaps of the whole human race.