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DIGGING INTO MATHEMATICS

473

reason or prove? How does demonstration differ from other proof?" He looked in Noah Webster's dictionary and learned that demonstration is "proof beyond the possibility of doubt."

The definition didn't satisfy him; he went to all the dictionaries and books of reference he could find for the meaning of the word "demonstrate" and in the end said to himself that their definitions meant about as much to him as the color blue when explained to a blind man. He said to himself, "Lincoln, you can never make a lawyer of yourself until you understand what 'demonstrate' means."

He bought "The Elements of Euclid," a book twenty-three centuries old. It began with definitions, such as: (1) A point is that which has no parts and which has no magnitude; (2) a line is length without breadth; (15) a circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another; (16) and this point is called the centre of the circle.

Also it began with Axioms or Common Notions: (1) Things which are equal to the same thing are equal to one another; (2) if equals be added to equals the wholes are equal; (3) if equals be taken from equals the remainders are equal; (4) if equals be added to unequals the wholes are unequal; (5) if equals be taken from unequals the remainders are unequal; (6) things which are double of the same thing are equal to one another; (7) things which are halves of the same thing are equal to one another; (8) magnitudes which coincide with one another are equal to one another; (9) the whole is greater than its part; (10) two straight lines cannot enclose a space.

Quietly, by himself, he worked with these definitions and axioms. The book, "The Elements of Euclid," went into his carpetbag as he went out on the circuit. At night, when with other lawyers, two in a bed, eight and ten in a hotel room, he read Euclid by the light of a candle after others had dropped off to sleep.

Herndon and Lincoln had the same bed one night, and Herndon

noticed his partner's legs pushing their feet out beyond the footboard of the bed, as he held Euclid close to the candlelight and learned to demonstrate such propositions as: "In equal circles, equal angles stand on equal arcs, whether they be at the centres or circumferences," and "Equal parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and parallelograms which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another."

"In this troublesome world we are never quite satisfied,” he remarked once. Dreaminess filtered through him. He planned a speech in Congress; a week passed without his getting a chance to make the speech; and he commented, "Now my interest in the subject has passed too."

One night in Danville at the McCormick House, the ladies' parlor was turned into a bedroom for Judge David Davis, who had a bed to himself, and Lincoln and his fellow lawyer, Henry C. Whitney, who slept two in a bed. In the morning a thing happened that Whitney later told in this way:

"I was awakened early, before daylight, by my companion sitting up in bed, his figure dimly visible by the ghostly firelight, and talking the wildest and most incoherent nonsense all to himself. A stranger to Lincoln would have supposed he had suddenly gone insane. Of course, I knew Lincoln and his idiosyncrasies, and felt no alarm, so I listened and laughed. After he had gone on in this way for, say, five minutes, while I was awake, and I know not how long before I was awake, he sprang out of bed, hurriedly washed, and jumped into his clothes, put some wood on the fire and then sat in front of it, moodily, dejectedly, in a most sombre and gloomy spell, till the breakfast bell rang, when he started as if from sleep, and went with us to breakfast. Neither Davis nor I spoke to him; we knew this trait; it was not remarkable for Lincoln, although this time to which I refer was a radical manifestation of it, a proof that 'true wit to madness, sure, is oft allied.'”

John T. Stuart had remarked to Whitney that Lincoln was a

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hopeless victim of melancholy. "Look at him now," said Stuart, in a McLean County courthouse. "I turned a little," said Whitney, "and saw Lincoln sitting alone in a corner of the bar, remote from any one, wrapped in abstraction and gloom. . I watched him for some time. He seemed to be pursuing in his mind some specific painful subject, regularly and systematically through various sinuosities, and his sad face would assume, at times, deeper phases of grief. No relief came till he was roused by the adjournment of court, when he emerged from his cave of gloom, like one awakened from sleep."

He was spending more and more time by himself. Books, newspapers, his own thoughts, kept him alone in his room on evenings when the other lawyers on the circuit had all gone to a party and returned to find Lincoln asleep. If he went to a concert, lecture, or negro minstrel show, he would as soon go alone.

The habit stuck to him of reading out loud to himself whatever he wanted particularly to remember, and of reading out loud as he wrote. The proverb about "wits gone a-wool-gathering," he applied to some of his own moods. Whitney noticed him often during a court session "with his mind completely withdrawn from the busy scene before his eyes, as abstracted as if he were in absolute and unbroken solitude." Whitney noticed also: "Lincoln had no method, system, or order in his exterior affairs; no library, clerk, no index rerum, no diary. When he wanted to preserve a memorandum, he noted it down on a card and stuck it in a drawer or in his vest pocket or his hat. While outside of his mind all was anarchy and confusion, inside all was symmetry and method. His mind was his workshop; he needed no office, no pen, ink and paper; he could perform his chief labor by selfintrospection." For his important business matters he had an envelope marked, "When you can't find it anywhere else, look in this."

The branches and crotches of trees interested him more in the winter-time, stripped of leaves and naked in design, than in the summer, when covered; he searched for basic anatomy of structure.

Finding Herndon reading a new book, "The Annual of Science," he glanced through it and commented that the book was on the right track because it took account of failures as well as successes in its field. "Too often we read only of successful experiments in science and philosophy, whereas if the history of failure and defeat was included there would be a saving of brain-work as well as time. The evidence of defeat, the recital of what was not as well as what cannot be done serves to put the scientist or philosopher on his guard-sets him to thinking on the right line."

These remarks were prophetic, in their way, for Herndon found Lincoln had arrived earlier than usual one morning at the office. Spread before him on his desk were sheets of paper covered with figures and equations, plenty of blank paper, a compass, rule, pencils, bottles of ink of different colors. He hardly turned his head as Herndon came in. He covered sheet after sheet of paper with more figures, signs, symbols. As he left for the courthouse later in the day he told Herndon he was trying to square the circle.

He was gone only a short time, came back and spent the rest of the day trying to square the circle, and the next day again toiled on the famous problem that has immemorially baffled mathematicians. After a two days' struggle, worn down physically and mentally, he gave up trying to square the circle.

He was trying to organize his mind and life so that he could not accuse himself, as he had accused President Polk, of being "a bewildered, confounded, and miserably perplexed man." He wanted to be simple as the alphabet, definite as the numbers used in arithmetic, sure as the axioms or common notions that are the starting-points of Euclid. Had he trusted too much to his feelings, and not reasoned, proved, and demonstrated his propositions clearly enough in his own mind before speaking them during his term in Congress? He wasn't sure. Hadn't he made a sort of fool of himself, and made his friends sorry for him, when he spoke before the Scott Club in Springfield in reply to the Richmond speech of Judge Douglas? He wasn't sure.

He would see if he could be as simple as the alphabet, as

FROM A LINCOLN NOTEBOOK

477

definite as numbers, as sure as a demonstrated proposition in Euclid. He scribbled notes trying to be as absolute as mathematics.

own.

"If A can prove, however conclusively, that he may, of right, enslave B, why may not B snatch the same argument, and prove equally that he may enslave A? . . . You say A is white, and B is black. It is color, then: the lighter, having the right to enslave the darker? Take care. By this rule, you are to be slave to the first man you meet with a fairer skin than your You do not mean color exactly? You mean the whites are intellectually the superiors of the blacks, and therefore have the right to enslave them? . . . Take care again. By this rule, you are to be slave to the first man you meet, with an intellect superior to your own. . . . But, say you, it is a question of interest; and, if you can make it your interest, you have the right to enslave another? Very well. And if he can make it his interest, he has the right to enslave you.

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Into these notes he put the high human hopes spoken by the men who made the American Revolution. One note read:

"The ant who has toiled and dragged a crumb to his nest will furiously defend the fruit of his labor against whatever robber assails him. So plain that the most dumb and stupid slave that ever toiled for a master does constantly know that he is wronged. So plain that no one, high or low, ever does mistake it, except in a plainly selfish way; for although volume upon volume is written to prove slavery a very good thing, we never hear of the man who wishes to take the good of it by being a slave himself.

"Most governments have been based, practically, on the denial of the equal rights of men, as I have, in part, stated them; ours began by affirming those rights. They said, some men are too ignorant and vicious to share in government. Possibly so, said we; and, by your system, you would always keep them ignorant and vicious. We proposed to give all a chance; and we expected the weak to grow stronger, the ignorant wiser, and all better and happier together. We made the experiment, and the fruit is before us. Look at it, think of it. Look at it in its aggregate

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