ceptions: the line at infinity and its relation to the three types of conics he had constructed, cross ratio, correlations -the wonderful duality between line and point geometry. He amused himself one whole afternoon writing corresponding theorems in parallel colums-theorems of Carnot, Ceva, Pascal, etc.

These courses gradually prepared him for more advanced work, which would ultimately lead to a thesis and a degree perhaps. Geometry was his stronghold-the favorite branch to which he clung. Later he became fascinated with the cubic surfaces. How often he stood before the case which contained the famous Brill Collection of plaster models, viewing them from all angles, counting the lines, the nodes and the binodes! He loved to hold them in his hands just as a child loves to hold a pretty kitten in its lap.

The ruled surfaces, illustrated by the thread models, seemed to fascinate him most. He wanted to have them near all the time, that he might see and enjoy them. It was for this reason that he undertook to construct some of his own. He purchased cardboard, went to the planing mill and ordered a load of lumber-just a small load-a bundle of small square sticks, which he sandpapered and then painted with drawing ink. He also bought brass tacks, a small hammer and a saw. He went to the department store and selected several spools of colored silk, some needles, a package of black beads and a package of white ones; the girl behind the counter snickered at him, but he was too much absorbed to notice it. He hurried home, worked out the equations of the traces of the surfaces made by parallel planes; plotted these traces on the cardboards, which were held apart and in place by the small wooden sticks. Then he threaded the needles with various shades of silk and strung it through the perforations in the cardboard; and where two threads intersected, he placed a bead, just as the original model had instructed him to do. His work was very successful. The models with their glossy silk threads sparkled in the sunshine, which

poured through the high windows and flooded his desk with light.

His room had now become a workshop and a study; for he did no small amount of hammering and sawing and needlework. One of his professors gave him several prints of famous mathematicians-Klein, Cayley, Kummer, Lobatchefsky, Riemann, Sylvester, Darboux; he tacked them on the walls where they served him as inspiration. He was happy-extremely happy—as he sat and mused and planned and worked and thought there alone in his cozy little study among his pictures, his models and his books.

What had become of his violin? Alas! he had forgotten it. The case was under his bed, and in it lay the mute instrument, which he had once loved so well. It had ceased to vibrate, ceased to sing to him. Cayley, Salmon, Grassmann and Clebsch had put Moskowski, Schubert, Wagner and Beethoven to flight; Barcarolles, sonatas, operas and Spanish dances had been superseded by congruences, cyclides, matrices and integrals.

Miss Jones listened in vain for her favorite aria from Saint-Saëns, but never a note came from the room overhead. Had it not been that Milton went out for his meals, she would have all reason to believe that he had passed away. "The boy has gone insane," she would murmur to herself as she glanced at the thread models when moving them to dust his desk. "All he does is build and study these mouse traps." Once she opened a volume of his notes, trying to read what he had written. To her it was like so much Greek and all these mysterious figures he drew in the text! “The boy is mad." Then she saw an hyperbolic paraboloid drawn with the Z axis horizontal and the X axis vertical: "A pair of corsets!" she exclaimed. "Ah," she sighed, "there's still hope for him-his thoughts are human once in a while."

Miss Jones was quite right: Milton was insane. He had become a monomaniac, and his mania was geometry. He thought geometry all the time he was walking, talking, eating, sleeping. He was interested in anything geometrical.

He avoided everything else, unless his imagination could transform it into some configuration with which he had already met in his work. He always walked alone, selecting the least frequented streets and allies; these streets to his mind formed a system of orthogonal trajectories. The telegraph wires became a pencil of lines with its vertex at infinity; the buildings along the streets, polyhedrons; the church steeple became an octagonal pyramid, and its large colored glass window became a system of polar coördinates; the trunks of trees became cylinders; the limbs became skew curves of higher order, on which he sought to locate cusps and nodes; the outlines of the leaves became cardioids and other well-known loci; the birds in the air described lemniscates and conchoids; in the flowers he saw roulettes; the fountain in the park became a paraboloid of revolution. At dinner, the plates became circles; the peas became spheres; the butter became cubes; the doughnuts became cyclides; the crullers became helices; the buns became surfaces with a double line; the piece of pie became a sixtydegree sector. The boy was geometry-mad. He saw geometry in everything-in everything except humanity, and in humanity he saw nothing.

CHAPTER XII

WHEN ONE STUDIES HARD ENOUGH ONE BEGINS

TO SEE THINGS

There was nothing extraordinary about Paul Milton's insanity. We are all more or less insane at times; we have some mad hobby or other in which we become so completely absorbed as to forget about those persons in whose very midst we live. We continue existing in this state until something occurs-something which suddenly awakens us from our futile dream and brings us back to our senses.

Paul Milton had even forgotten about his mother. He walked by her letters on the table in the hall. Miss Jones had to bring them to his room, where they would sometimes remain on his desk for several days unread. But the widow was never disturbed because he did not write frequently. Her husband had often become so enwrapped working over his musical manuscript that at times he seemed to ignore her, but love her the more ardently after he came out of his trance. She knew well the trend of genius; and her son's seeming negligence was, after all, natural and necessary.

Paul knew there were other students rooming in the same house, but he had never spoken with any of them. Why should he seek their acquaintance? Their conversation would be absolutely ungeometrical.

Furthermore, why should they care to know him? They would pass on the stairs and in that way were occasionally reminded of the fact that some graduate student did his "grinding" up there in the attic. As long as that "grinding" did not interfere with their affairs, why take notice of him? Thus Miss Jones was the only one who spoke to him as he went to or came from his lectures.

Aside from his professors, there were only two persons. with whom Milton cared to converse: the two graduate students who were taking the same courses as he. All three of them ate at the same table in the boarding house and talked of nothing but geometry across their plates. The other students were bored to distraction, but the three geometers appeared totally unaware of the disgust and the ridicule they were provoking. More than once Milton left the table without having tasted his food. The old colored waiter wondered how the boy existed and used to stand dumbfounded by the figures which he formed with knives, forks and spoons to illustrate the theorems he was proving to his fellow-maniacs.

Milton remained in his room every evening save one— one in every two weeks. Then he would attend the fortnightly meetings of the Mathematical Club. The papers presented at these meetings were usually beyond him, but he seldom left the lecture without having acquired a half dozen or so new conceptions, and he used to walk home alone, often repeating to himself the theorems he had learned.

One night, returning from the Club, he was ascending the stairs to his room when he saw a white form glide across the hall. The hall was only dimly lighted by the reflection from the lamp on the lower floor. The white form had come out of the landlady's room. Aside from his geometry, this vision was the only thing in several months which had attracted his attention. He was still thinking about it when he reached his room. He concluded at first that it was merely an hallucination, for he had been working very hard over his books that afternoon. But even if it were not an hallucination, he decided not to be disturbed, for it might have been none other than Miss Jones herself, however much it resembled one of her students.

But, two weeks later when he was returning from the meeting of the Club, he met with the same vision and observed quite clearly that it was a student in white pajamasand not Miss Jones-who had come out of her room.