CHAPTER IX.
The Tensor Formula
(Monday, April 11; 11.30 a. m.)
Markham sat staring at Vance like a man hypnotized. Heath stood rigid, his mouth partly open, his cigar held a few inches from his lips. There was something almost comic in the Sergeant’s attitude, and I had a nervous inclination to laugh; but for the moment my blood seemed frozen, and all muscular movement was impossible.
Markham was the first to speak. Jerking his head backward, he brought his hand down violently on the desk-top.
“What new lunacy of yours is this?” He was fighting desperately against Vance’s dumbfounding suggestion. “I’m beginning to think the Robin case has affected your mind. Can’t a man with the commonplace name of Sprigg be shot without your trying to turn it into some grotesque hocus-pocus?”
“Still, you must admit, Markham old dear,” returned Vance mildly, “this particular Johnny Sprigg was shot with ‘a little gun’, through ‘the middle of his wig’, so to speak.”
“What if he was?” A dull flush had crept into Markham’s face. “Is that any reason for your going about babbling Mother-Goose rhymes?”
“Oh, I say! I never babble, don’t y’ know.” Vance had dropped into a chair facing the District Attorney’s desk. “I may not be a thrillin’ elocutionist; but really, now, I don’t babble.” He gave Heath an ingratiating smile. “Do I, Sergeant?”
But Heath had no opinion to express. He still held his astonished pose, though his eyes had now become mere slits in his broad, pugnacious face.
“Are you seriously suggesting——?” began Markham; but Vance interrupted him.
“Yes! I’m seriously suggesting that the person who killed Cock Robin with an arrow has vented his grim humor upon the hapless Sprigg. Coincidence is out of the question. Such repetitive parallels would knock the entire foundation out from all sanity and reason. ’Pon my soul, the world is mad enough; but such madness would dissipate all science and rational thinking. Sprigg’s death is rather hideous; but it must be faced. And however much you may force yourself to protest against its incredible implications you’ll eventually have to accept them.”
Markham had risen, and was pacing nervously up and down.
“I’ll grant there are inexplicable elements in this new crime.” His combativeness had gone, and his tone had moderated. “But if we assume, even tentatively, that some maniac is at large reconstructing the rhymes of his nursery days, I can’t see how it will help us. It would practically close all routine lines of investigation.”
“I shouldn’t say that, don’t y’ know.” Vance was smoking meditatively. “I’m inclined to think that such an assumption would supply us with a definite basis of inquiry.”
“Sure!” snapped Heath with ponderous sarcasm. “All we gotta do is to go out and find one bug among six million people. A cinch!”
“Don’t let the fumes of discouragement overcome you, Sergeant. Our elusive jester is a rather distinctive entomological specimen. Moreover, we have certain clews as to his exact habitat. . . .”
Markham swung round. “What do you mean by that?”
“Merely that this second crime is related to the first not only psychologically, but geographically. Both murders were committed within a few blocks of each other,—our destructive demon at least has a weakness for the neighborhood in which the Dillard house is situated. Furthermore, the very factors of the two murders preclude the possibility of his having come from afar to give rein to his distorted humor in unfamiliar surroundings. As I learnedly pointed out to you, Robin was translated into the Beyond by some one who knew all the conditions obtaining at the Dillard house at the exact hour the grisly drama was performed; and surely it’s obvious that this second crime could not have been so tidily staged had not the impresario been acquainted with Sprigg’s ambulat’ry intentions this morning. Indeed, the entire mechanism of these weird playlets proves that the operator was intimately cognizant of all the circumstances surrounding his victims.”
The heavy silence that followed was broken by Heath.
“If you’re right, Mr. Vance, then that lets Sperling out.” The Sergeant made even this qualified admission reluctantly; but it showed that Vance’s argument had not been without its effect on him. He turned desperately to the District Attorney. “What do you think we’d better do, sir?”
Markham was still battling against the acceptance of Vance’s theory, and he did not answer. Presently, however, he reseated himself at his desk and drummed with his fingers upon the blotter. Then, without looking up, he asked:
“Who’s in charge of the Sprigg case, Sergeant?”
“Captain Pitts. The local men at the 68th-Street Station grabbed it first; but when the news was relayed to the Bureau, Pitts and a couple of our boys went up to look into it. Pitts got back just before I came over here. Says it’s a washout. But Inspector Moran15 told him to stay with it.”
Markham pressed the buzzer beneath the edge of his desk, and Swacker, his youthful secretary, appeared at the swinging door that led to the clerical room between the District Attorney’s private office and the main waiting-room.
“Get Inspector Moran on the wire,” he ordered.
When the connection had been made he drew the telephone toward him and spoke for several minutes. When he had replaced the receiver, he gave Heath a weary smile.
“You’re now officially handling the Sprigg case, Sergeant. Captain Pitts will be here presently, and then we’ll know where we stand.” He began looking through a pile of papers before him. “I’ve got to be convinced,” he added half-heartedly, “that Sprigg and Robin are tied up in the same sack.”
Pitts, a short, stocky man, with a lean, hard face and a black tooth-brush moustache, arrived ten minutes later. He was, I learned afterwards, one of the most competent men in the Detective Division. His specialty was “white-collar” gangsters. He shook hands with Markham and gave Heath a companionable leer. When introduced to Vance and me he focussed suspicious eyes on us and bowed grudgingly. But as he was about to turn away his expression suddenly changed.
“Mr. Philo Vance, is it?” he asked.
“Alas! So it seems, Captain,” Vance sighed.
Pitts grinned and, stepping forward, held out his hand.
“Glad to meet you, sir. Heard the Sergeant speak of you often.”
“Mr. Vance is helping us unofficially with the Robin case, Captain,” explained Markham; “and since this man Sprigg was killed in the same neighborhood we thought we’d like to hear your preliminary report on the affair.” He took out a box of Corona Perfectos, and pushed it across the desk.
“You needn’t put the request that way, sir.” The Captain smiled, and selecting a cigar held it to his nose with a kind of voluptuous satisfaction. “The Inspector told me you had some ideas about this new case, and wanted to take it on. To tell you the truth, I’m glad to get rid of it.” He sat down leisurely, and lighted his cigar. “What would you like to know, sir?”
“Let us have the whole story,” said Markham.
Pitts settled himself comfortably.
“Well, I happened to be on hand when the case came through—a little after eight this morning—and I took a couple of the boys and beat it up-town. The local men were on the job, and an assistant Medical Examiner arrived the same time I did. . . .”
“Did you hear his report, Captain?” asked Vance.
“Sure. Sprigg was shot through the top of the head with a .32. No signs of a struggle—no bruises or anything. Nothing fancy. Just a straight shooting.”
“Was he lying on his back when found?”
“That’s right. Stretched out nice and pretty, right in the middle of the walk.”
“And wasn’t his skull fractured where he’d fallen on the asphalt?” The question was put negligently.
Pitts took his cigar from his mouth and gave Vance a sly look.
“I guess maybe you fellows over here do know something about this case.” He nodded his head sagaciously. “Yes, the back of the guy’s skull was all bashed in. He sure had a tough fall. But I guess he didn’t feel it—not with that bullet in his brain. . . .”
“Speaking of the shot, Captain, didn’t anything about it strike you as peculiar?”
“Well . . . yes,” Pitts admitted, rolling his cigar meditatively between his thumb and forefinger. “The top of a guy’s head isn’t where I’d ordinarily look for a bullet-hole. And his hat wasn’t touched,—it must have fallen off before he was potted. You might call those facts peculiar, Mr. Vance.”
“Yes, Captain, they’re dashed peculiar. . . . And I take it the pistol was held at close range.”
“Not more’n a couple of inches away. The hair was singed round the hole.” He made a broad gesture of inconsequence. “Still and all, the guy might have seen the other fellow draw the gun, and ducked forward, spilling his hat. That would account for his getting the shot at close range in the top of the head.”
“Quite, quite. Except that, in that case, he wouldn’t have fallen over back, but would have pitched forward on his face. . . . But go on with the story, Captain.”
Pitts gave Vance a look of crafty agreement, and continued.
“The first thing I did was to go through the fellow’s pockets. He had a good gold watch on him and about fifteen dollars in bills and silver. So it didn’t look like a robbery—unless the guy that shot him got panicky and beat it. But that didn’t seem likely, for there’s never any one round that part of the park early in the morning; and the walk there dips under a stone bluff, so that the view is cut off. The bird that did the job certainly picked a swell place for it. . . . Anyhow, I left a couple of men to guard the body till the wagon came for it, and went up to Sprigg’s house in 93rd Street,—I’d got his name and address from a couple of letters in his pocket. I found out he was a student at Columbia, living with his parents, and that it was his habit to take a walk in the park after breakfast. He left home this morning about half past seven. . . .”
“Ah! It was his habit to promenade in the park each morning,” murmured Vance. “Most interestin’.”
“Even so, that don’t get us anywheres,” returned Pitts. “Plenty of fellows take an early constitutional. And there was nothing unusual about Sprigg this morning. He wasn’t worried about anything, his folks told me; and was cheerful enough when he said good-bye to ’em.—After that I hopped up to the university and made inquiries; talked to a couple of the students that knew him, and also to one of the instructors. Sprigg was a quiet sort of chap. Didn’t make friends and kept pretty much to himself. Serious bird—always working at his studies. Stood high in his classes, and was never seen going around with Janes. Didn’t like women, in fact. Wasn’t what you’d call sociable. From all reports he was the last man to get in a mess of any kind. That’s why I can’t see anything special in his getting shot. It must have been an accident of some kind. Might have been taken for somebody else.”
“And he was found at what time?”
“About quarter of eight. A bricklayer on the new 79th-Street dock was cutting across the embankment toward the railway tracks, and saw him. He notified one of the post officers on the Drive, who phoned in to the local station.”
“And Sprigg left his home in 93rd Street at half past seven.” Vance gazed at the ceiling meditatively. “Therefore he would have had just enough time to reach this point in the park before being killed. It looks as if some one who knew his habits was waiting for him. Neatness and dispatch, what? . . . It doesn’t appear exactly fortuitous, does it, Markham?”
Ignoring the jibe Markham addressed Pitts.
“Was there nothing found that could possibly be used as a lead?”
“No, sir. My men combed the spot pretty thoroughly, but nothing showed up.”
“And in Sprigg’s pockets—among his papers. . . ?”
“Not a thing. I’ve got all the stuff at the Bureau—a couple of ordinary letters, a few odds and ends of the usual kind. . . .” He paused as if suddenly remembering something, and pulled out a dog-eared note-book. “There was this,” he said unenthusiastically, handing a torn, triangular scrap of paper to Markham. “It was found under the fellow’s body. It don’t mean anything, but I stuck it in my pocket—force of habit.”
The paper was not more than four inches long, and appeared to have been torn from the corner of an ordinary sheet of unruled stationery. It contained part of a typewritten mathematical formula, with the lambda, the equals and the infinity sign marked in with pencil. I reproduce the paper here, for, despite its seeming irrelevancy, it was to play a sinister and amazing part in the investigation of Sprigg’s death.
Vance glanced only casually at the exhibit, but Markham held it in his hand frowning at it for several moments. He was about to make some comment when he caught Vance’s eye; and, instead, he tossed the paper to the desk carelessly with a slight shrug.
“Is this everything you found?”
“That’s all, sir.”
Markham rose.
“We’re very grateful to you, Captain. I don’t know what we’ll be able to make out of this Sprigg case, but we’ll look into it.” He pointed to the box of Perfectos. “Put a couple in your pocket before you go.”
“Much obliged, sir.” Pitts selected the cigars, and placing them tenderly in his waistcoat pocket, shook hands with all of us.
When he had gone Vance got up with alacrity, and bent over the scrap of paper on Markham’s desk.
“My word!” He took out his monocle and studied the symbols for several moments. “Most allurin’. Now where have I seen that formula recently? . . . Ah! The Riemann-Christoffel tensor—of course! Drukker uses it in his book for determining the Gaussian curvature of spherical and homaloidal space. . . . But what was Sprigg doing with it? The formula is considerably beyond the college curricula. . . .” He held the paper up to the light. “It’s the same stock as that on which the Bishop notes are written. And you probably observed that the typing is also similar.”
Heath had stepped forward, and now scrutinized the paper.
“It’s the same, all right.” The fact seemed to nonplus him. “That’s a link anyway between the two crimes.”
Vance’s eyes took on a puzzled look.
“A link—yes. But the presence of the formula under Sprigg’s body appears as irrational as the murder itself. . . .”
Markham moved uneasily.
“You say it is a formula that Drukker uses in his book?”
“Yes. But the fact doesn’t necessarily involve him. The tensor is known to all advanced mathematicians. It is one of the technical expressions used in non-Euclidean geometry; and though it was discovered by Riemann in connection with a concrete problem in physics,16 it has now become of widespread importance in the mathematics of relativity. It’s highly scientific in the abstract sense, and can have no direct bearing on Sprigg’s murder.” He sat down again. “Arnesson will be delighted with the find. He may be able to work out some astonishing conclusion from it.”
“I see no reason,” protested Markham, “to inform Arnesson of this new case. My idea would be to keep it under cover as much as possible.”
“The Bishop won’t let you, I fear,” returned Vance.
Markham set his jaw.
“Good God!” he burst out. “What damnable sort of thing are we facing? I expect every minute to wake up and discover I’ve been living a nightmare.”
“No such luck, sir,” growled Heath. He took a resolute breath like a man preparing for combat. “What’s on the cards? Where do we go from here? I need action.”
Markham appealed to Vance.
“You seem to have some idea about this affair. What’s your suggestion? I frankly admit I’m floundering about in a black chaos.”
Vance inhaled deeply on his cigarette. Then he leaned forward as if to give emphasis to his words.
“Markham old man, there’s only one conclusion to be drawn. These two murders were engineered by the same brain: both sprang from the same grotesque impulse; and since the first of them was committed by some one intimately familiar with conditions inside the Dillard house, it follows that we must now look for a person who, in addition to that knowledge, had definite information that a man named John Sprigg was in the habit of taking a walk each morning in a certain part of Riverside Park. Having found such a person, we must check up on the points of time, place, opportunity, and possible motive. There’s some interrelation between Sprigg and the Dillards. What it is I don’t know. But our first move should be to find out. What better starting-point than the Dillard house itself?”
“We’ll get some lunch first,” said Markham wearily. “Then we’ll run out there.”