Editors From The United States

"Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type x^3 + bx = c; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. ...With respect to the irreducible case... we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic... before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2"

"It is difficult to say who it is who first recognized the advantage of always equating to zero in the study of the general equation. It may very likely have been Napier, for he wrote his De Arte Logistica before 1594, and in this there is evidence that he understood the advantage of this procedure. Bürgi also recognized the value of making the second member zero, Harriot may have done the same, and the influence of Descartes was such that the usage became fairly general."

"Of the contemporaries of Newton one of the most prominent was John Wallis. ...Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics... He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xn, y=x1/n, and y=x0 + x1 + x2 +... He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of ds = \!dx \sqrt{1+(\frac{dy}{dx})^2} for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature."

"In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science. This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning."

"Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates."

"Grégoire de Saint-Vincent... was a Jesuit, taught mathematics in Rome and Prag (1629-1631), and was afterwards called to Spain by Phillip IV as tutor to his son... He wrote two works on geometry [Principia Matheseos Univerales (1651); Exercitationum Mathematicarum Libri quinque (1657)], giving in one of them the quadrature of the hyperbola referred to its asymptotes, and showing that as the area increased in arithmetic series the abscissas increased in geometric series."

"1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves. 2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra. 3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry. Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics."

"The field of mathematics is now so extensive that no one can [any] longer pretend to cover it, least of all the specialist in any one department. Furthermore it takes a century or more to weigh men and their discoveries, thus making the judgment of contemporaries often quite worthless."

"More than any of his predecessors Plato appreciated the scientific possibilities of geometry. .. By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is... clear. ...That Plato should hold the view... is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought."

"His writings include works on mechanics, sound, astronomy, the tides, the laws of motion, the Torricellian tube, botany, physiology, music, the calendar (in opposition to the Gregorian reform), geology, and the compass,—a range too wide to allow of the greatest success in any of the lines of his activity. He was also an ingenious cryptologist and assisted the government in deciphering diplomatic messages."

"The fact that arithmetic and geometry took such a notable step forward... was due in no small measure to the introduction of Egyptian papyrus into Greece. This event occurred about 650 B.C., and the invention of printing in the 15th century did not more surely effect a revolution in thought than did this introduction of writing material on the northern shores of the Mediterranean Sea just before the time of Thales."

"Among his [John Wallis'] interesting discoveries was the relation \frac{4}{\pi} = \frac32\cdot\frac34\cdot\frac54\cdot\frac56\cdot\frac76\cdot\frac78\cdots one of the early values of π involving infinite products."

"Aside from Cauchy, the greatest contributory to the theory [of determinants] was Carl Gustav Jacob Jacobi. With him the word "determinant" received its final acceptance. He early used the functional determinant which Sylvester has called the Jacobian, and in his famous memoirs in Crelle's Journal for 1841 he considered these forms as well as that class of alternating functions which Sylvester has called alternants."

"The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts."

"The Arabs contributed nothing new to the theory, but al-Khowârizmî (c. 825) states the usual rules, and the same is true of his successors."

"Johannes Tropfke... described the history of those individual parts of mathematics that he believed were most important for mathematics as taught in secondary schools. He intended his history to inform teachers about the origin of special problems, terms, and methods in school mathematics. ...Tropfke's approach to the history of mathematics at this time was new and even now is not yet out of date. The only comparable work is the second volume of D.E Smith's History of Mathematics... which gives far less detailed information."

"In every state and nearly every civilized nation in the developed world, readers know where to go for action and reaction of news – at least one day ahead... Free from any corporate concerns, there are simply too many to thank since the site's inception in 1994. This new attempt at the old American experiment of full freedom in reporting is ever exciting. Those in power have everything to lose by individuals who march to their own rules."

"I am a conservative. I'm very much pro-life. If you go down the list of what makes up a conservative, I'm there almost all the way."

"To produce 1 lb. of feedlot beef requires 7 lbs. of feed grain, which takes 7,000 lbs. of water to grow. Pass up one hamburger, and you'll save as much water as you save by taking 40 showers with a low-flow nozzle. Yet in the U.S., 70% of all the wheat, corn and other grain produced goes to feeding herds of livestock. Around the world, as more water is diverted to raising pigs and chickens instead of producing crops for direct consumption, millions of wells are going dry. … In the U.S., livestock now produce 130 times as much waste as people do. Just one hog farm in Utah, for example, produces more sewage than the city of Los Angeles. These megafarms are proliferating, and in populous areas their waste is tainting drinking water. In more pristine regions, from Indonesia to the Amazon, tropical rain forest is being burned down to make room for more and more cattle. … We, at least, have the flexibility—the omnivorous stomach and creative brain—to adapt. We can do it by moving down the food chain: eating foods that use less water and land, and that pollute far less, than cows and pigs do. In the long run, we can lose our memory of eating animals, and we will discover the intrinsic satisfactions of a diverse plant-based diet, as millions of people already have."

"[[Tucker Carlson|[Tucker] Carlson]] has never been a stickler for the truth, as he proved in the run-up to this interview, when he claimed that he was the only western media figure who cared enough to get [[Vladimir Putin|[Vladimir] Putin]] on the record. That's absurd. Many American reporters have tried unsuccessfully to sit down with Putin, especially since the invasion of Ukraine. But the Russian president was waiting for the right stooge. With Carlson, he got just that."

""Fair and balanced" was the original Fox News lie, one of the rotten planks that built the foundation for Wednesday's democratic disaster. Over decades, with that false promise accepted as gospel by millions of devotees, Fox News radicalized a nation and spawned more extreme successors such as Newsmax and One America News. Day after day, hour after hour, Fox gave its viewers something that looked like news or commentary but far too often lacked sufficient adherence to a necessary ingredient: truth. Birtherism. The caravan invasion. Covid denialism. Rampant election fraud. All of these found a comfortable home at Fox. In the Trump era, the network — now out of favor for not being quite as shameless as the president demands — was his best friend and promoter. So to put it bluntly: The mob that stormed and desecrated the Capitol on Wednesday could not have existed in a country that hadn’t been radicalized by the likes of Sean Hannity, Tucker Carlson and Laura Ingraham, and swayed by biased news coverage."

"[O]ne key to running Twitter is the tricky matter of "managing up". Anyone who's ever worked in a corporation or big agency, especially as a manager, knows that you have to handle the boss. You have to keep them informed, hold off their worst instincts, tactfully set boundaries and, most of all, somehow convince them that every move you make is really their brilliant idea – or at least a fulfillment of their underlying vision. And there's the rub. Twitter’s problems are solvable. But the volatile and narcissistic Elon Musk|Musk]] may be the boss that can’t be managed."

"Often it has been that reporter who has most skillfully played the access game — the one who has curried just enough favor with the powerful newsmaker to be smiled upon, without giving up basic credibility and integrity. That’s access journalism. Accountability journalism, by contrast, is often performed off to the side, by those who don’t have to deal with the news provider on a regular basis."

"Trump is, of course, a master of distraction and . It’s possible to resist being his chump, but it takes continued self-regulation."

"If news organizations learned anything after the campaign, they should have learned that groupthink has a tendency to miss the point and journalistic myopia requires some extra-strength corrective lenses. Do something different. Represent the interests of a broader, more ideologically diverse population. Figure out what they’re thinking and feeling — and why."

"We — the traditional, the legacy, the mainstream media — have to change."

"In the wacky new world of fake news, conspiracy theories, es — and social media’s unthinking participation in spreading all of that — facts and truth get lost in the noise. A responsible media needs to be especially careful not to unwittingly spread lies by amplifying them. Some early coverage of Trump’s recent unwarranted, evidence-free blasts about the illegality of some of the popular vote fell into that trap. It’s depressing but a fact of life that a lot of people don’t know the difference between fake news and conspiracy bilge and verified fact. Nor do they seem to care."

"He wouldn’t drain the swamp, but merely feed different alligators."

"Defeat seems to be the brand for these boys. In any culture war, Rich Lowry and the gang have always been the first to stand athwart history, crying: "We surrender first!” They’ve been so weak and defeatist during the Trump years that a year's subscription to the magazine [National Review] could be marketed as an estrogen supplement."

"After spending the first three decades of my career at one of Buffett’s papers, the Buffalo News, I’m not willing to accept that. Even now, my former newsroom — down by about half from its peak — is doing critically important work, not just crucial watchdog journalism (insider trading by a congressman) but cultural coverage (memories of a concert venue) that knits the community together. Amid this nightmare financial scenario, what can be done? [Philip] Napoli, for one, thinks that American citizens and our big thinkers need to buckle down — fast — about substantial policy changes that could involve both direct and indirect public funding for local journalism. It "would take us in a more European direction," he said. That notion, once radioactive in journalism because it seems to threaten the independence of news organizations, must now be taken seriously."

"Some other attendees of the National Conservatism Conference, such as National Review's Rich Lowry, took pains to distance themselves from the president's brand of nationalism. "We have to push back against Donald Trump when he does things to increase that breach between the right and African Americans," Lowry said. But in the fall of 2017, when Trump attempted to silence black athletes protesting police brutality, Lowry praised his "gut-level political savvy," writing, "This kind of thing is why he's president.""

"Trump obviously never meant to impose a wealth test on his administration, or he would have failed it himself. He is proof that a fortune isn’t necessarily an obstacle to being a champion of an agenda of populist reform."

"The fact is that in a country with an enormous federal government and a First Amendment that guarantees the right to petition the government, the swamp is always going to be extensive and miasmic. As long as there is so much power and money in D.C., the lobbyists, the consultants, the associations, the media pooh-bahs, the contractors and the courtiers will gather and jockey for influence here. There is no neutron bomb that can be set off to vaporize them."

"A proper anti-swamp agenda should consist of two things. First, and most fundamentally, it should seek to reduce the size of the federal government, and cut regulations and make them as simple as possible. The more government does, the more incentive every special interest has to hire swamp creatures, for both protection and advantage. And the more complex government is, the more opportunity those creatures have to thrive in niches unknown or poorly understood by everyone except insiders. Second, and more specifically, the federal government should be wrenched out of its cozy relationship with large, established businesses and institutions in areas ranging from health care to finance to education."

"It wasn’t quite “build the wall” or “lock her up,” but “” was a signature Donald Trump slogan. ... This was always a fantasy. The oldest story in Washington is a new president elected on a pledge to clean up Washington, who then turns to old Washington hands and well-connected financiers to help shepherd his administration. It was true of George W. Bush and Barack Obama and now will be true of the even more and anti-Washington Donald Trump. The swamp will endure; it always does. This doesn’t mean that a Trump administration can’t make the swamp less important."

"Viewed through any conventional lens, President-elect Donald Trump’s candidacy was improbable from start to finish. Today, two things about his victory seem to be in sharper focus: one, that Trump’s victory might best be understood as the success of the country’s first independent president, and second, that the Trump coalition may be even more uniquely his than President Obama’s has turned out to be."

"Trump owes his success in part to the fact that he ran for president in an environment that favored change over the status quo. But his luck or genius goes beyond that. It has long been noted that the conditions have existed for an independent candidate to run a serious campaign for president."

"What has stood in the way of people running as an independent is that winning the presidency in a system that so clearly favors the two major parties is something of a hopeless cause."

"Trump took the elements of an independent candidacy — the lack of clear ideology, the name recognition of a national celebrity and the personal fortune needed to fund a presidential campaign — and then did what no one seemed to have thought of before. He staged a hostile takeover of an existing major party. He had the best of both worlds, an outsider candidacy with crosscutting ideological appeal and the platform of a major party to wage the general election. By the time he had finished, he had taken down two political dynasties: the Bush dynasty in the primaries and the Clinton dynasty in the general election."

"“There’re all kinds of people in this world. But they break down into two main groups, one big and one smaller. There’s the people who get moved out of the way or into line, and then there’s the people who do the moving. It’s safer and a lot more comfortable to go where you’re pushed. You don’t take any of the responsibility, and if you do what you’re told, every once in a while you get thrown a fish. “Being a mover isn’t safe, because you may be heading for a hole, and it isn’t comfortable because you do a lot of jostling back and forth, and what’s more, it’s up to you to get your own fish. But it’s a hell of a lot of fun.” He looked into Hawks’ eyes. “Isn’t it?”"

"Claire threw her head and laughed. “There are all kinds of men. The only kind that’re worth anyone’s time are the ones I can’t mangle the first time out.”"

"There are the people in this world who act, and the people who scheme. The ones who act get things done, and the ones who scheme try to take credit for it. You must know that as well as I do."

"Of course, you have to remind yourself that you might be seeing things that were never there. You might be maneuvering your memories to bring them into line with what you’d want them to be. You can’t be sure you’re not just daydreaming."

"That was the way things worked out: you got a certain fair share of good breaks from life, and you had no right to expect things your way every time."

"By the time of his adolescence, he had discovered an absolute lack of involvement with the human race. He studied it, because it was the salient feature of his environment. He did not live with it. It said nothing to him that was of personal value; its motivations, morals, manners and morale did not find responsive reactions in him. And his, of course, made absolutely no impression on it."

"The idea was to hang on to reality. It was the one crutch everybody used."

"“You look like any other brainless jackanapes,” he mused, “but apparently there’s some gray matter left in your artfully coiffed skull after all.”"

"I do what seems best—I live, I let live; so far I’ve done well. I don’t see why I should act like a predator, simply because I’m in a predatory society. I’m a human being, I’m myself—if I make my life something vicious, who’ll give me another if, too late, I regret what I’ve done?"

"Charging back and forth isn’t progress. Small but steady forward motion is."

"Only gods and television audiences see the pattern of human events."