People From New York State

"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"

"He states that the root of x^3 + 6x = 20 isx = \sqrt[3]{\sqrt{108} + 10} - \sqrt[3]{\sqrt{108} - 10}."

"In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation x^2 + ax + b = 0 he placed u + z for x. He then hadu^2 + (2z + a)u +(z^2 + az + b) = 0.He now let 2z + a = 0, whence z = -\frac{1}{2}a,and this gaveu^2 - \frac{1}{4}(a^2 - 4b) = 0. u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.andx = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}."

"He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam."

"He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns."

"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."

"Algebra in the Renaissance period received its first serious consideration in Pacioli's Sūma (1494)... which characterized in a careless way the knowledge... thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations. The noteworthy work... and the first to be devoted entirely to the subject, was Rudolff's Coss (1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence. The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). The next great work... to appear in print was the General Trattato of Tartaglia..."

"The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra."

"The first noteworthy attempt to write an algebra in England was made by , whose Whetstone of witte (1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde. The first Italian textbook to bear the title of algebra was Bombelli's work of 1572. By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. ...this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675). So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century."

"With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra..."

"The first writer on algebra whose works have come down to us is . He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical."

"There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose Āryabhatiyam (c. 510) included problems in series, permutations, and linear and quadratic equations; , whose Brahmasiddhānta (c. 628) contains a satisfactory rule for solving the quadratic... Mahāvīra, whose Ganita-Sāra Sangraha (c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose Bija Ganita (c. 1150)... extends the work through quadratic equations."

"Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due."

"It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B.C. In 's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position... and an arrangement of terms in a kind of notation. The rules given by Liu Hui form a kind of rhetorical algebra. The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving s. ...Sun-tzï solved such problems by analysis and was content with a single result... The Chinese certainly knew how to solve quadratics as early as the 1st century B.C., and rules given even as early as the K'iu-ch'ang Suan-shu... involve the solution of such equations. Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should... By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625). The culmination of Chinese is found in the 13th century. ...numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner (1819)."

"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."

"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."

"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."

"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."

"Vieta: 1QC - 15QQ + 85C - 225Q + 274N, aequator 120. Modern form:x^6 - 15x^4 + 85x^3 - 225x^2 + 274x = 120"

"When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to solve the equation ax^2 + bx + c = 0, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the science was known about 1800 B.C., and probably still earlier.<"

"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."

"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."

"Art must unquestionably have a social value; that is, as a potential means of communication it must be addressed, and in comprehensible terms, to the understanding of mankind."

"While reasonable men will ungrudgingly submit to such curtailment of their liberties as will promote at least the greater liberty of all, to the exact degree to which such curtailment may become unreasonable will the enforcement of it have to rest on force. Force against reason: reason, because it has the power of enlisting force to fight for it, will win. From the recognition of that truth has come democracy."

"June 29, 2009, is the two-year anniversary of the first shipment of the iPhone. Not one of those people [whose 2-year service contracts expire] will still be using an iPhone a month later. ... I'm on a 10-year plan here. They are going to run out of gas way before we are."

"We do, to what extent of freedom we have earned or are allowed by others and ourselves, what we most want to do. We say - speak, paint, carve, write, express ourselves...as we damn please. And in both the doing of things and the talking about them - which together seem to me to sum up life - we so crave freedom or liberty or whatever one may choose to call it as to justify our Declaration's romantically terming it an unalienable right of Man, bestowed on Man by his Creator."

"Euryalus, is it the gods who put this fire in our minds, or is it that each man's relentless longing becomes a god to him?"

"I sing of arms and of a man: his fate had made him fugitive; he was the first to journey from the coasts of Troy as far as Italy and the Lavinian shores."

"For other peoples will, I do not doubt, still cast their bronze to breathe with softer features, or draw out of the marble living lines, plead causes better, trace the ways of heaven with wands and tell the rising constellations; but yours will be the rulership of nations, remember Roman, these will be your arts: to teach the ways of peace to those you conquer, to spare defeated peoples, tame the proud."

"I want to also thank my beautiful wife"

"I go home, I tell the first lady, ‘I spoke to Vladimir today, we had a wonderful conversation.’ And she says, ‘Oh really, another city was just hit’"

"She hates him."

"The slogan 'My Body, My Choice' is typically associated with women activists and those who align with the pro-choice side of the debate. [...] But if you really think about it, 'My Body, My Choice' applies to both sides – a woman's right to make an independent decision involving her own body, including the right to choose life. Personal freedom."

"What I like about Trump is his wife — her beauty, her style and her charisma. I haven't met her, but I hear many good things about her. Even the Democrats have nothing bad to say about her."

"I could say that I'm the most bullied person on the world"

"We are responsible for empowering our next generation with values."

"But we need to vet them. We need to know who they are, Chain migration, he doesn't want to just cut it off completely. We need to vet the people, and we need to make sure that they believe in our system"

"Total honesty is what we as citizens deserve from our president"

"You can see from the tape, the cameras were not on—it was only a mic. And I wonder if they even knew that the mic was on. Because they were kind of, ah, boy talk. And he was led on. Like egg on from the host to say, uh, dirty and bad stuff."

"Had I been fully informed of all the details, naturally, I would have immediately denounced the violence that occurred at the Capitol Building."

"I'm working my ass off on the Christmas stuff that, you know, who gives a fuck about Christmas stuff and decorations but I need to do it, right?"

"Sometimes I say I have two boys at home — I have my young son and I have my husband."

"A lot of people say I am using all the procedures for my face. I didn't do anything. I live a healthy life, I take care of my skin and my body. I'm against Botox, I'm against injections; I think it's damaging your face, damaging your nerves. It's all me. I will age gracefully, as my mom does."

"On July 28th 2006, I was very proud to become a citizen of the United States — the greatest privilege on planet Earth."

"From a young age, my parents impressed on me the values that you work hard for what you want in life: that your word is your bond and you do what you say and keep your promise; that you treat people with respect. They taught and showed me values and morals in their daily life. That is a lesson that I continue to pass along to our son, and we need to pass those lessons on to the many generation to follow because we want our children in this nation to know that the only limit to your achievements is the strength of your dreams and your willingness to work for them."

"You judge a society by how it treats its citizens. We must do our best to ensure that every child can live in comfort and security, with the best possible education."

"It is imperative to guarantee that women have autonomy in deciding their preference of having children, based on their own convictions, free from any intervention or pressure from the government. [...] Why should anyone other than the woman herself have the power to determine what she does with her own body? A woman’s fundamental right of individual liberty, to her own life, grants her the authority to terminate her pregnancy if she wishes. Restricting a woman's right to choose whether to terminate an unwanted pregnancy is the same as denying her control over her own body. I have carried this belief with me throughout my entire adult life."

"...when he was tested again and it came up positive. Luckily he is a strong teenager and exhibited no symptoms... He has since tested negative."

"And then I do it, and I say I'm working on Christmas and planning for Christmas. And they say, 'What about the children that are separated?' Give me a fucking break."