People From Pennsylvania

"It should be mentioned also that the Hindus, unlike the Greeks, regarded irrational roots of numbers as numbers. This was of enormous help in algebra, and Indian mathematicians have been much praised for taking this step; but one must remember that the Hindu contribution in this case was the result of logical innocence rather than of mathematical insight. We have seen the lack of nice distinction on the part of Hindu mathematicians between exact and inexact results, and it was only natural that they should not have taken seriously the difference between commensurable and incommensurable magnitudes. For them there was no impediment to the acceptance of irrational numbers, and later generations followed their lead uncritically until in the nineteenth century mathematicians established the real number system on a sound basis."

"A Hebrew belief asserted that if Yahweh lays aside his bow and hangs it in the clouds, this is a sign that his anger has subsided. Other peoples have had similar ideas, based upon the tradition that an archer carries his bow with the ends pointing downward when he wishes to indicate his peaceful intentions."

"The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery."

"Carl B. Boyer … is more or less the Gibbon of math history."

"But what, after all, are the integers? Everyone thinks that he or she knows, for example, what the number three is — until he or she tries to define or explain it."

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. … Fermat applied his method … and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"Descartes maintained his confidence in the instantaneity of light. … Yet in his derivation of the law of refraction, Descartes reasoned that light travelled faster in a dense medium than in one less dense. He seems to have had no qualms about comparing infinite magnitudes!"

"Android and the DROID X are, warts and all, already neck and neck with the iPhone 4. It's scary to think how one-sided this would be if Google just put a handful of UI experts on the [Android app] marketplace. Game over, Apple. Game over."

"Make no mistake, this transition is happening. ... The world we're heading toward belongs to Windows RT."

"Apple's fans are more interested in spending money than they are with facts. … That the lackluster iPhone 4S can sell so well in a market dominated by more capable Android handsets (not to mention Windows Phones) only bolsters that notion."

"Apple is a hugely successful company and its Mac business, even though it trails the wider PC market by a wide margin, is a great business, a very, very successful and desirable business. For Apple. Why anyone would care about that, other than employees of Apple, is unclear to me."

"These early [Windows Phone sales] reports don't provide any credible figures. But even if sales are as bad as all get-out, you're forgetting one thing: It almost doesn't matter, because Microsoft is in this for the long haul. They're going to continue pushing this system ahead, and pushing it to developers and users."

"Is this thing even worth reviewing? Right off the bat, I'm glad to see that my initial reactions to this thing were accurate. Anyone who believes this thing is a game changer is a tool. I'm sorry, but that's just the way it is."

"[T]he facial scanner in the iPhone X is based on the technology that Microsoft first used, disastrously, in its Xbox Kinect sensor. This probably explains why [Apple's Face ID] works so poorly: If Microsoft could never perfect this in a relatively huge device, how could Apple's component makers ever fit the technology into 'a space a few centimeters across and millimeters deep'?"

"[Apple] events are tough for non-sycophants such as myself. I don't take everything Apple says at face value, as they would prefer. And I certainly don't buy into the theory that the world's most profitable company is in any way out to help humanity or make the world a better place."

"Apple Watch is superfluous, an unnecessary accessory that literally no one needs. ... it just seems so pointless. ... I understand, even accept, that Apple hardware purchases involve most of the points made above and that most fans of the company simply don’t care or are essentially pushing their fingers into their ears and going "na-na-na-na" so they can’t hear the logic of the argument against their buying decisions. I am at peace with this."

"There are three [Apple Watch] lineups that range in price from "just" $350 for an Apple Sport stripper model with low-end materials to an astonishing $17,000 for an 18 karat gold silly version. As I noted on Twitter, this isn't consumer electronics anymore. It's consumerism run amok."

"And yet scientists continue to study the brain in hopes of understanding whether a sense of the gendered self can, at least in part, be the result of neurology. A study described by author Francine Russo in Scientific American examined the brains of 39 prepubertal and 41 adolescent boys and girls with gender dysphoria. The experiment examined how these children responded to androstadienone, a pungent substance similar to pheromones, that is known to cause a different response in the brains of men and women. The study found that adolescent boys and girls who described themselves as trans responded like the peers of their perceived gender. (The results were less clear with prepubescent children.)"

"Gender is many things, but one thing it is surely not is a hobby."

"But what does that mean, a male brain, or a female brain, or even a transgender one? It’s a fraught topic, because brains are a collection of characteristics, rather than a binary classification of either/or."

"In the past decade, there has been some fascinating research on the brains of transgender people. What is most remarkable about this work is not that trans women’s brains have been found to resemble those of cisgender women, or that trans men’s brains resemble those of cis men. What the research has found is that the brains of trans people are unique: neither female nor male, exactly, but something distinct."

"You know, don’t you, that no amount of wishing that this were not the case can make it not the case. No amount of praying that you are not transgendered will make you something other than what you are. No amount of love from anyone will make you fit inside a body that does not match your spirit."

"The results of a battle pained him as much as if he was receiving the wounds himself, for I have often heard him express himself in pained accents while talking over some of the scenes of the war; he was not the man to assume a character for feelings he did not possess ; he was as guileless in some respects as a child. How could one avoid liking such a man?"

"I think every man should be obliged to take that oath, for I have seen more treason in the last ten days than I ever supposed could exist in the United States Navy."

"As to the navy, it is said the several commanding officers grounded on the beef-bones thrown overboard from their flag-ships, but this I do not believe."

"I was much disappointed at seeing those who had once belonged to the United States navy excelled in matters of honor and propriety by the officers of another corps."

"By breaking up the great wooden navies of Europe, Ericsson helped to place us more on an equality with them as a naval power, and the distance is not so great between us but that we may hope to overtake them when the people of this country demand a navy commensurate with our national importance, and when the exigencies of politics can no longer prevent proper measures being taken for the defense of the nation, which should at all times be in a position to protect its citizens at home and abroad."

"The people fairly went wild; they set fire to the cotton along the levees, and seemed determined that nothing valuable should fall into our hands. They did not apparently remember that, so far, our navy had respected private rights and protected those made homeless by the actions of wild mobs."

"There was a time not long ago when Vicksburg could have been easily captured, but it is now a second Gibraltar, and the navy alone could do nothing toward capturing it."

"Why, Mister President, the general impression is that Grant won the battle of Shiloh; as he commanded the army, he would seem entitled to the credit."

"Well, Mister President, with all due deference to you, I don't believe in natural-born generals except where they have had proper military training, and it seems to me the siege of Vicksburg is too important a matter to trust to anybody except a scientific military man; besides, if you take troops from Grant and Sherman to give them to McClernand, you will weaken the army."

"Great complaints were made by both sides as to whose fault it was that there was a failure, but I told the navy I didn't want to hear anything about it; they did not get through, and didn't get the fort, and the less said about it the better."

"Regulations of the Navy provide that medical officers shall exercise no military authority. If I give you a flag, the line officers will think I have gone crazy."

"I never encouraged officers to discuss politics at all, and, as a rule, officers of the navy were exempt from political bias, and considered that it was their duty to heartily support the Government in any measures which might be taken to preserve the Union. This was my view of the subject, and I tried to impress it upon others, and succeeded in excluding politics from the mess-table."

"[T]he navy must be an adjunct to the army, yet the officers and men of the navy should always have full credit for the service they perform."

"[T]he navy performed its part of the operations."

"'Good wine needs no bush' is an old and good saying, and I think that the navy had very little cause to exculpate itself on any occasion when it co-operated with the army, and never entertained a difference of opinion when it came in contact with regular officers."

"I then made some observations to the general on the beautiful weather we were having, and the satisfaction I experienced at seeing him in a position requiring so much judgment and forbearance, and that our co-operation so far had been of such a pleasant nature that I should always look back to this time with the most delightful recollections, as there could not by any possibility be any mis-understanding between the army and navy, their duties being so distinct from each other, and the only chance of their clashing would be through the stupid blunder of an irresponsible officer."

"The army and navy had plenty of bad powder and worthless vessels in fact, material for half a dozen powder-boats if necessary. I don't know whether the general claimed the powder-boat as an original idea, but there is nothing new under the sun, and such a means of attack has been employed before."

"The navy and the powder-boat would be all-sufficient, and I rather liked the notion, as the expedition would be entirely a naval affair."

"It looked queer to me to see boxes labeled 'His Excellency, Jefferson Davis, President of the Confederate States of America'. The packages so labeled contained Bass ale or Cognac brandy, which cost 'His Excellency' less than we Yankees had to pay for it. Think of the President drinking imported liquors while his soldiers were living on pop-corn and water!"

"Lincoln had a wonderful faculty for understanding the topography of a country, and he was quite familiar with the one in which the army was about to operate; he carried a small chart in his pocket, on which were marked all the rivers and hills about Richmond, with the city itself, and the different points where General Lee had his forces posted, the lines of defense, and, in fact, all the information that a general of an army wanted. During our rides which were always within the lines he would stop and spread out his chart on his knees and point out to me what he would do if he were the general commanding, taking good care, at the same time, never to interfere in any way with General Grant, whom, I rather think, he considered the better strategist of the two."

"I had often heard of the wonderful power of the President in telling anecdotes, but no one could form an adequate idea of his ability in this line unless he had been alone with him for ten days as I was. He had an illustration for everything, and if anything particular attracted his attention he would say, 'That reminds me of something that occurred when I was a lawyer in Illinois,' or 'when I was a boatman on the Mississippi.' He was not at all ashamed of any business he had ever been engaged in, because it was honest business, and he made an honest living by it; and he told me many stories of his earlier life, which were as creditable to him as anything he was engaged in while occupying a higher sphere."

"Lincoln seemed to me to be familiar with the name, character, and reputation of every officer of rank in the army and navy, and appeared to understand them better than some whose business it was to do so; he had many a good story to tell of nearly all, and if he could have lived to write the anecdotes of the war, I am sure he would have furnished the most readable book of the century. To me he was one of the most interesting men I ever met; he had an originality about him which was peculiarly his own, and one felt, when with him, as if he could confide his dearest secret to him with absolute security against its betrayal. There, it might be said, was 'God's noblest work an honest man,' and such he was, all through. I have not a particle of the bump of veneration on my head, but I saw more to admire in this man, more to reverence, than I had believed possible; he had a load to bear that few men could carry, yet he traveled on with it, foot-sore and weary, but without complaint; rather; on the contrary, cheering those who would faint on the roadside. He was not a demonstrative man, so no one will ever know, amid all the trials he underwent, how much he had to contend with, and how often he was called upon to sacrifice his own opinions to those of others, who, he felt, did not know as much about matters at issue as he did himself. When he did surrender, it was always with a pleasant manner, winding up with a characteristic story."

"In the strife between the North and the South there was no bitterness in Mr. Lincoln's composition; he seemed to think only that he had an unpleasant duty to perform, and endeavored to perform it as smoothly as possible."

"I do think if I had given him two fence-rails to sleep on he would not have found fault. That was Abraham Lincoln in all things relating to his own comfort. He would never permit people to put themselves out for him under any circumstances."

"There was a small house on this landing, and behind it were some twelve negroes digging with spades. The leader of them was an old man sixty years of age. He raised himself to an upright position as we landed, and put his hands up to his eyes. Then he dropped his spade and sprang forward. "Bress de Lord," he said, "dere is de great Messiah! I knowed him as soon as I seed him. He's bin in my heart fo' long yeahs, an' he's cum at las' to free his chillun from deir bondage! Glory, Hallelujah!" And he fell upon his knees before the President and kissed his feet. The others followed his example, and in a minute Mr. Lincoln was surrounded by these people, who had treasured up the recollection of him caught from a photograph, and had looked up to him for four years as the one who was to lead them out of captivity."

"It was a touching sight that aged negro kneeling at the feet of the tall, gaunt-looking man who seemed in himself to be bearing all the grief of the nation, and whose sad face seemed to say, "I suffer for you all, but will do all I can to help you.""

"Mr. Lincoln looked down on the poor creatures at his feet; he was much embarrassed at his position. "Don't kneel to me," he said. "That is not right. You must kneel to God only, and thank him for the liberty you will hereafter enjoy. I am but God's humble instrument; but you may rest assured that as long as I live no one shall put a shackle on your limbs, and you shall have all the rights which God has given to every other free citizen of this Republic.""

"His face was lit up with a divine look as lie uttered these words. Though not a handsome man, and ungainly in his person, yet in his enthusiasm he seemed the personification of manly beauty, and that sad face of his looked down in kindness upon these ignorant blacks with a grace that could not be excelled. He really seemed of another world."