Vance made an insignificant gesture.
"But these are element'ry figures—the every-day facts of journeyman calculators. The higher mathematician goes vastly further. He deals in abstruse and apparently contradict'ry speculations which the average mind can not even grasp. He lives in a realm where time, as we know it, is without meaning save as a fiction of the brain, and becomes a fourth co-ordinate of three-dimnensional space; where distance also is meaningless except for neighboring points, since there are an infinite number of shortest routes between any two given points; where the language of cause and effect becomes merely a convenient shorthand for explanat'ry purposes; where straight lines are non-existent and insusceptible of definition; where mass grows infinitely great when it reaches the velocity of light; where space itself is characterized by curvatures; where there are lower and higher orders of infinities; where the law of gravitation is abolished as an acting force and replaced by a characteristic of space—a conception that says, in effect, that the apple does not fall because it is attracted by the earth, but because it follows a geodesic, or world-line. . . .
"In this realm of the modern mathematician, curves exist without tangents. Neither Newton nor Leibnitz nor Bernoulli even dreamed of a continuous curve without a tangent—that is, a continuous function without a differential co-efficient. Indeed, no one is able to picture such a contradiction,—it lies beyond the power of imagination. And yet it is a commonplace of modern mathematics to work with curves that have no tangents.—moreover, pi—that old friend of our school-days, which we regarded as immutable—is no longer a constant; and the ratio between diameter and circumference now varies according to whether one is measuring a circle at rest or a rotating circle. . . . Do I bore you?"
"Unquestionably," retorted Markham. "But pray continue, provided your observations have an earthly direction."
Vance sighed and shook his head hopelessly, but at once became serious again.
"The concepts of modern mathematics project the individual out of the world of reality into a pure fiction of thought, and lead to what Einstein calls the most degenerate form of imagination—pathological individualism. Silberstein, for instance, argues the possibility of five- and six-dimensional space, and speculates on one's ability to see an event before it happens. The conclusions contingent on the conception of Flammarion's Lumen—a fictive person who travels faster than the velocity of light and is therefore able to experience time extending in a reverse direction—are in themselves enough to distort any natural and sane point of view.[32] But there is another conceptual Homunculus even weirder than Lumen from the standpoint of rational thinking. This hypothetical creature can traverse all worlds at once with infinite velocity, so that he is able to behold all human history at a glance. From Alpha Centauri he can see the earth as it was four years ago; from the Milky Way he can see it as it was 4,000 years ago; and he can also choose a point in space where he can witness the ice-age and the present day simultaneously! . . ."
Vance settled himself more deeply in his chair.
"Toying with the simple idea of infinity is enough to unhinge the average man's mind. But what of the well-known proposition of modern physics that we cannot take a straight and ever-advancing path into space without returning to our point of departure? This proposition holds, in brief, that we may go straight to Sirius and a million times further without changing direction, but we can never leave the universe: we at last return to our starting-point from the opposite direction! Would you say, Markham, that this idea is conducive to what we quaintly call normal thinking? But however paradoxical and incomprehensible it may seem, it is almost rudiment'ry when compared with other theorems advanced by mathematical physics. Consider, for example, what is called the problem of the twins. One of two twins starts to Arcturus at birth—that is, with accelerated motion in a gravitational field—and, on returning, discovers that he is much younger than his brother. If, on the other hand, we assume that the motion of the twins is Galilean and that they are therefore travelling with uniform motion relative to each other, then each twin will find that his brother is younger than himself! . . .
"These are not paradoxes of logic, Markham: they're only paradoxes of feeling. Mathematics accounts for them logically and scientifically.[33] The point I'm trying to make is that things which seem inconsistent and even absurd to the lay mind, are commonplaces to the mathematical intelligence. A mathematico-physicist like Einstein announces that the diameter of space—of space, mind you—is 100,000,000 light-years, or 700 trillion miles; and considers the calculation abecedarian. When we ask what is beyond this diameter, the answer is: 'There is no beyond: these limitations include everything.' To wit, infinity is finite! Or, as the scientist would say, space is unbounded but finite.—Let your mind meditate on this idea for half an hour, Markham, and you'll have a sensation that you're going mad."
He paused to light a cigarette.
"Space and matter—that's the mathematician's speculative territ'ry. Eddington conceives matter as a characteristic of space—a bump in nothingness; whereas Weil conceives space as a characteristic of matter,—to him empty space is meaningless. Thus Kant's noumenon and phenomenon become interchangeable; and even philosophy loses all significance. But when we come to the mathematical conceptions of finite space all rational laws are abrogated. De Sitter's conception of the shape of space is globular, or spherical. Einstein's space is cylindrical; and matter approaches zero at the periphery, or 'border condition.' Weyl's space, based on Mach's mechanics, is saddle-shaped. . . . Now, what becomes of nature, of the world we live in, of human existence, when we weigh them against such conceptions? Eddington suggests the conclusion that there are no natural laws—namely, that nature is not amenable to the law of sufficient reason. Alas, poor Schopenhauer![34] And Bertrand Russell sums up the inevitable results of modern physics by suggesting that matter is to be interpreted merely as a group of occurrences, and that matter itself need not be existent! . . . Do you see what it all leads to? If the world is non-causative and non-existent, what is a mere human life?—or the life of a nation?—or, for that matter, existence itself? . . ."
Vance looked up, and Markham nodded dubiously.
"So far I follow you, of course," he said. "But your point seems vague—not to say esoteric."
"Is it surprising," asked Vance, "that a man dealing in such colossal, incommensurable concepts, wherein the individuals of human society are infinitesimal, might in time lose all sense of relative values on earth, and come to have an enormous contempt for human life? The comparatively insignificant affairs of this world would then become mere petty intrusions on the macrocosmos of his mental consciousness. Inevitably such a man's attitude would become cynical. In his heart he would scoff at all human values, and sneer at the littleness of the visual things about him. Perhaps there would be a sadistic element in his attitude; for cynicism is a form of sadism. . . ."
"But deliberate, planned murder!" objected Markham.
"Consider the psychological aspects of the case. With the normal person, who takes his recreations daily, a balance is maintained between the conscious and the unconscious activities: the emotions, being constantly dispersed, are not allowed to accumulate. But with the abnormal person, who spends his entire time in intense mental concentration and who rigorously suppresses all his emotions, the loosening of the subconscious is apt to result in a violent manifestation. This long inhibition and protracted mental application, without recreation or outlet of any kind, causes an explosion which often assumes the form of deeds of unspeakable horror. No human being, however intellectual, can escape the results. The mathematician who repudiates nature's laws is nevertheless amenable to those laws. Indeed, his rapt absorption in hyperphysical problems merely increases the pressure of his denied emotions. And outraged nature, in order to maintain her balance, produces the most grotesque fulminations—reactions which, in their terrible humor and perverted gaiety, are the exact reverse of the grim seriousness of abstruse mathem
atical theories. The fact that Sir William Crookes and Sir Oliver Lodge—both great mathematical physicists—became confirmed spiritists, constitutes a similar psychological phenomenon."
Vance took several deep inhalations on his cigarette.
"Markham, there's no escaping the fact: these fantastic and seemingly incredible murders were planned by a mathematician as forced outlets to a life of tense abstract speculation and emotional repression. They fulfil all the indicated requirements: they are neat and precise, beautifully worked out, with every minute factor fitting snugly in place. No loose ends, no remainders, apparently no motive. And aside from their highly imaginative precision, all their indications point unmistakably to an abstrusely conceptive intelligence on the loose—a devotee of pure science having his fling."
"But why their grisly humor?" asked Markham. "How do you reconcile the Mother-Goose phase of them with your theory?"
"The existence of inhibited impulses," explained Vance, "always produces a state favorable to humor. Dugas designates humor as a 'détente'—a release from tension; and Bain, following Spencer, calls humor a relief from restraint. The most fertile field for a manifestation of humor lies in accumulated potential energy—what Freud calls Besetzungsenergie—which in time demands a free discharge. In these Mother-Goose crimes we have the mathematician reacting to the most fantastic of frivolous acts in order to balance his superserious logical speculations. It's as if he were saying cynically: 'Behold! This is the world that you take so seriously because you know nothing of the infinitely larger abstract world. Life on earth is a child's game—hardly important enough to make a joke about.' . . . And such an attitude would be wholly consistent with psychology; for after any great prolonged mental strain one's reactions will take the form of reversals—that is to say, the most serious and dignified will seek an outlet in the most childish games. Here, incidentally, you have the explanation for the practical joker with his sadistic instincts. . . .
"Moreover, all sadists have an infantile complex. And the child is totally amoral. A man, therefore, who experiences these infantile psychological reversals is beyond good and evil. Many modern mathematicians even hold that all convention, duty, morality, good, and the like, could not exist except for the fiction of free will. To them the science of ethics is a field haunted by conceptual ghosts; and they even arrive at the disintegrating doubt as to whether truth itself is not merely a figment of the imagination. . . . Add to these considerations the sense of earthly distortion and the contempt for human life which might easily result from the speculations of higher mathematics, and you have a perfect set of conditions for the type of crimes with which we are dealing."
When Vance had finished speaking Markham sat silent for a long time. Finally he moved restively.
"I can understand," he said, "how these crimes might fit almost any of the persons involved. But, on the basis of your argument, how do you account for the notes to the press?"
"Humor must be imparted," returned Vance. "'A jest's prosperity lies in the ear of him who hears it.' Also, the impulse toward exhibitionism enters into the present case."
"But the 'Bishop' alias?"
"Ah! That's a most vital point. The raison d'être of this terrible orgy of humor lies in that cryptic signature."
Markham turned slowly.
"Does the chess player and the astronomer fulfil the conditions of your theory as well as the mathematical physicist?"
"Yes," Vance replied. "Since the days of Philidor, Staunton and Kieseritzki, when chess was something of a fine art, the game has degenerated almost into an exact science; and during Capablanca's régimeit has become largely a matter of abstract mathematical speculation. Indeed, Maroczy, Doctor Lasker and Vidmar are all well-known mathematicians. . . . And the astronomer, who actually views the universe, may get an even more intense impression of the unimportance of this earth than the speculative physicist. Imagination runs riot through a telescope. The mere theory of existing life on distant planets tends to reduce earthly life to second'ry consideration. For hours after one has looked at Mars, for instance, and dallied with the notion that its inhabitants outnumber and surpass in intelligence our own population, one has difficulty in readjusting oneself to the petty affairs of life here on earth. Even a reading of Percival Lowell's romantic book[35] temporarily takes away from the imaginative person all consciousness of the significance of any single planet'ry existence."
There was a long silence. Then Markham asked:
"Why should Pardee have taken Arnesson's black bishop that night instead of one from the club where it would not have been missed?"
"We don't know enough of the motive to say. He may have taken it with some deliberate purpose in view.—But what evidence have you of his guilt? All the suspicions in the world would not permit you to take any step against him. Even if we knew indubitably who the murderer was, we'd be helpless. . . . I tell you, Markham, we're facing a shrewd brain—one that figures out every move, and calculates all the possibilities. Our only hope is to create our own evidence by finding a weakness in the murderer's combination."
"The first thing in the morning," declared Markham grimly, "I'm going to put Heath to work on Pardee's alibi that night. There'll be twenty men checking it up by noon, questioning every spectator at that chess game, and making a door-to-door canvass between the Manhattan Chess Club and the Drukker house. If we can find some one who actually saw Pardee in the vicinity of the Drukkers' around midnight, then we'll have a very suspicious piece of circumstantial evidence against him."
"Yes," agreed Vance; "that would give us a definite starting-point. Pardee would have considerable difficulty in explaining why he was six blocks away from the club during his set-to with Rubinstein at the exact hour that a black bishop was being left at Mrs. Drukker's door. . . . Yes, yes. By all means have Heath and his minions tackle the problem. It may lead us forward."
But the Sergeant was never called upon to check the alibi. Before nine o'clock on the following morning Markham called at Vance's house to inform him that Pardee had committed suicide.
22. THE HOUSE OF CARDS
(Sunday, April 17; 9 a.m.)
The astounding news of Pardee's death had a curiously disturbing effect on Vance. He stared at Markham unbelievingly. Then he rang hastily for Currie and ordered his clothes and a cup of coffee. There was an eager impatience in his movements as he dressed.
"My word, Markham!" he exclaimed. "This is most extr'ordin'ry. . . . How did you hear of it?"
"Professor Dillard phoned me at my apartment less than half an hour ago. Pardee killed himself in the archery-room of the Dillard home some time last night. Pyne discovered the body this morning and informed the professor. I relayed the news to Sergeant Heath, and then came here. In the circumstances I thought we ought to be on hand." Markham paused to light his cigar. "It looks as if the Bishop case was over. . . . Not an entirely satisfactory ending, but perhaps the best for every one concerned."
Vance made no immediate comment. He sipped his coffee abstractedly, and at length got up and took his hat and stick.
"Suicide. . . ," he murmured, as we went down the stairs. "Yes, that would be wholly consistent. But, as you say, unsatisfact'ry—dashed unsatisfact'ry. . . ."
We rode to the Dillard house, and were admitted by Pyne. Professor Dillard had no more than joined us in the drawing-room when the door-bell rang, and Heath, pugnacious and dynamic, bustled in.
"This'll clean things up, sir," he exulted to Markham, after the usual ritualistic handshake. "Those quiet birds . . . you never can tell. Yet, who'd've thought. . . ?"
"Oh, I say, Sergeant," Vance drawled; "let's not think. Much too wearin'. An open mind—arid like a desert—is indicated."
Professor Dillard led the way to the archery-room. The shades at all the windows were drawn, and the electric lights were still burning. I noticed, too, that the windows were closed.
"I left everything exactly as it was," explained the professor.
Mark
ham walked to the large wicker centre-table. Pardee's body was slumped in a chair facing the range door. His head and shoulders had fallen forward over the table; and his right arm hung at his side, the fingers still clutching an automatic pistol. There was an ugly wound in his right temple; and on the table beneath his head was a pool of coagulated blood.
Our eyes rested but a moment on the body, for a startling and incongruous thing diverted our attention. The magazines on the table had been pushed aside, leaving an open space in front of the body; and in this cleared area rose a tall and beautifully constructed house of playing cards. Four arrows marked the boundaries of the yard, and matches had been laid side by side to represent the garden walks. It was a reproduction that would have delighted a child's heart; and I recalled what Vance had said the night before about serious minds seeking recreation in children's games. There was something unutterably horrible in the juxtaposition of this juvenile card structure and violent death.
Vance stood looking down at the scene with sad, troubled eyes.
"Hic jacet John Pardee," he murmured, with a sort of reverence. "And this is the house that Jack built . . . a house of cards. . . ."
He stepped forward as if to inspect it more closely; but as his body struck the edge of the table there was a slight jar, and the flimsy edifice of cards toppled over.
Markham drew himself up and turned to Heath.
"Have you notified the Medical Examiner?"
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