Educators From The United States

"The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes."

"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}."

"[Zuanne de Tonini] da Coi... impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but... without any explanation. At any rate, the two cubics x^3 + ax^2 = c and x^3 + bx = c could now be solved. The reduction of the general cubic x^3 + ax^2 + bx = c to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types x^3 = ax^2 + c and x^3 + ax^2 = c by substituting x = y + \frac{1}{3}a and x = y - \frac{1}{3}a respectively, and transformed the type x^3 + c = ax^2 by the substitution x = \sqrt[3]{c^2/y}, thus freeing the equations of the term x^2. This completed the general solution, and he applied the method to the complete cubic in his later problems."

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

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

"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"

"The medium of printed scientific text is first of all a visual one."

"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."

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

"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)."

"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."

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

"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 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."

"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.<"

"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."

"Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression... [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations."

"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."

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

"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."

"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."

"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."

"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."

"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 &pi; 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."

"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."

"We should know that only replacing the economics of competition and greed with the economics of equitable cooperation will guarantee a globalization that takes advantage of potential efficiency gains in ways that also promote environmental protection, international equity, economic democracy, and variety."

"And knowing that in every specific battle, what we are fighting for is merely the substitution of the human agenda for the corporate agenda is what can guide and sustain us."

"What truth is not, according to Kuhn, is an accurate representation of the world as it is in itself. Scientific theories represent a world, but one partially constituted by the cognitive activities of the scientists themselves. This is not a commonsensical view, but it has a distinguished philosophical pedigree, associated most strongly with Kant. The Kantian view is that the truths we can know are truths about a ‘phenomenal’ world that is the joint product of the ‘things in themselves’ and the organising, conceptual activity of the human mind. Kuhn, however, is Kant on wheels. Where Kant held that the human contribution to the phenomenal world is invariant, Kuhn’s view is that it changes fundamentally across a scientific revolution. This is what he means by his notorious statement that, after a scientific revolution, ‘the world changes’. This is neither the trivial claim that scientists’ beliefs about the world change, nor the crazy claim that scientists can change the things in themselves simply by changing their beliefs. It is the claim that the phenomenal world changes because the human contribution to it changes."

"The fall of communism confirms century-old libertarian claims that equity and justice cannot be imposed by force, that interpreting "to each according to work" as "to each according to the marginal revenue product of one's labor" rationalizes privilege, and that central planning stifles workers' creative potentials. Clearly, enterprises whose inputs and outputs have been determined by a central planning procedure exclude workers and consumers from decision making, separate conceptual and manual tasks, and offer unequal consumption and work opportunities. For the Soviet, East German, Polish, Czechoslovakian, and Hungarian people to reject these injustices is encouraging. But it is dishonest to say this demonstrates that capitalism is optimal. It only bespeaks a lack of alternatives."

"Under central planning neither planners, managers, nor workers had incentives to promote the social economic interest. Nor did impeding markets for final goods to the planning system enfranchise consumers in meaningful ways. But central planning would have been incompatible with economic democracy even if it had overcome its information and incentive liabilities. And the truth is that it survived as long as it did only because it was propped up by unprecedented totalitarian political power."

"Recognizing that the current form of globalization is nothing more than a generalized downward leveling in which global corporations are extracting more and more of the wealth, power, and productive energies from communities and the environment is the right approach. Recognizing that our response must be nothing less than upward leveling that does entail the transfer of resources, power, wealth, and knowledge from the world's haves to the world's have-nots is the right approach."

"For Kuhn, scientific development is neither cumulative nor teleological: science is not moving towards the goal of a theory of everything. His favoured analogy is with biological evolution by natural selection. Organisms develop under selective pressures from the environment, but development, though tending to an increase in complexity and differentiation, has no fixed goal. Moreover, the selective environments that determine which organisms survive and which die off themselves change and are partially constituted by the organisms’ own activities. Biological evolution is not moving towards an ideal organism, and scientific evolution is not moving towards the Truth. Both processes are pushed from behind, not pulled from the front."

"Students in the Department would often attempt to help one another by ‘Liptonating’ (or ‘Liptonising’): expressing the content of one’s interlocutor’s garbled thoughts in far superior terms of clarity, precision, and concision."

"His ability charitably to “summarise” the inchoate and/or confused thoughts of his interlocutors was legendary: Lipton invariably said what you wished you had meant."

"Lipton once gave me some advice that I like quite a lot: if a footnote says something important, put it in the main text where people will read it. If it says nothing important, delete it."

"The structure known, but not yet accessible by synthesis, is to the chemist what the unclimbed mountain, the uncharted sea, the untilled field, the unreached planet, are to other men. The achievement of the objective in itself cannot but thrill all chemists, who even before they know the details of the journey can apprehend from their own experience the joys and elations, the disappointments and false hopes, the obstacles overcome, the frustrations subdued, which they experienced who traversed a road to the goal. The unique challenge which chemical synthesis provides for the creative imagination and the skilled hand ensures that it will endure as long as men write books, paint pictures, and fashion things which are beautiful, or practical, or both."

"By the time they first encounter porn, most men have internalized the sexist ideology of our culture, and porn, rather than being an aberration, actually cements and consolidates their ideas about sexuality. And it does this in a way that gives them intense sexual pleasure. This framing of sexist ideology as sexy and hot gives porn a pass to deliver messages about women that in any other form would be seen as completely unacceptable. Imagine what would happen if suddenly we saw a slew of dramas and sitcoms on television where, say, blacks or Jews were repeatedly referred to in a racist or anti-Semitic way, where they got their hair pulled, faces slapped, and choked by white men pushing foreign objects into their mouths. My guess is that there would be an outcry and the images would not be defended on the grounds that they were just fantasy but rather would be seen for what they are: depictions of cruel acts that one group is perpetrating against another group. By wrapping the violence in a sexual cloak, porn renders it invisible, and those of us who protest the violence are consequently defined as anti-sex, not anti-violence."

"No anti-porn feminist I know has suggested that there is one image, or even a few, that could lead a non-rapist to rape; the argument, rather, is that taken together, pornographic images create a world that is at best inhospitable to women, and at worst dangerous to their physical and emotional well-being. In an unfair and inaccurate article that is emblematic of how anti-porn feminist work is misrepresented, Daniel Bernardi claims that Andrea Dworkin and Catharine MacKinnon believed that “watching pornography leads men to rape women.” Neither Dworkin nor MacKinnon, pioneers in developing a radical feminist critique of pornography, saw porn in such simplistic terms. Rather, both argued that porn has a complicated and multilayered effect on male sexuality, and that rape, rather than simply being caused by porn, is a cultural practice that has been woven into the fabric of a male-dominated society. Pornography, they argued, is one important agent of such a society since it so perfectly encodes woman-hating ideology, but to see it as simplistically and unquestionably leading to rape is to ignore how porn operates within the wider context of a society that is brimming with sexist imagery and ideology. If, then, we replace the “Does porn cause rape?” question with more nuanced questions that ask how porn messages shape our reality and our culture, we avoid falling into the images-lead-to-rape discussion. What this reformulation does is highlight the ways that the stories in pornography, by virtue of their consistency and coherence, create a worldview that the user integrates into his reservoir of beliefs that form his ways of understanding, seeing, and interpreting what goes on around him."

"I literally couldn't believe the images. I couldn't believe that men created such images, and that other men wanted to watch them."

"When most of Europe was still illiterate, the Hutterites had established a system of primary schools. Among them education was compulsory. They believed that their movement depended on an educated people who could practice discipleship in light of New Testament teachings. Of course, the state churches felt no such need. Their religion was primarily in the hands of religious professionals. The layman's chief function in such ecclesiastical systems was to obey. For this purpose ignorance served as well as, if not better than, knowledge."

"The initial press commentary in Moscow on the formation of the first Mussolini government was not overwhelmingly anti-Fascist, despite the Duce’s talk of a ‘revolutionary rivalry’ with Lenin. Fascism was sometimes perceived not inaccurately as more of a heresy from, rather than a moral challenge to, revolutionary Marxism."

"At the Twelfth Party Congress in Moscow in 1923, Nikolia Bukharin stressed that the Nazi Party had ‘inherited Bolshevik political culture exactly as Italian Fascism had done.’ On June 20, 1923, Karl Radek gave a speech before the Comintern Executive Committee proposing a common front with the Nazis in Germany. That summer several Nazis addressed Communist meetings and vice versa, as the German Communist Party took a strong stand for ‘national liberation’ against the Treaty of Versailles and inveighed against ‘Jewish capitalists.’ It is said that a few of the more radical Nazis even told German Communists that if the latter go rid of their Jewish leaders, the Nazis would support them."

"Some of the similarities and parallels include: Frequent recognition by Hitler and various Nazi leaders (and also Mussolini) that their only revolutionary and ideological counterparts were to be found in the Soviet Union . . . [and the] espousal of the have-not, proletarian-nation theory, which Lenin adopted only after it had been introduced in Italy . . . Hitlerian National Socialism more nearly paralleled than has any other non-Communist system."

"There was the populist extreme left, whose main spokesman was the journalist Curzio Malaparte, who wanted to see Fascism make a ‘revolution of the people’ that would reflect what the populist left considered true Italian popular culture, both intellectually and socially. There were small sectors of a dissident extreme left or ‘free Fascism’ that promoted a progressivist and leftist revolution of ‘liberty’ under the Fascist banner."

"Fascism was created by the nationalization of certain sectors of the revolutionary left, and the central role in its conceptual orientation was played by revolutionary syndicalists who embraced extreme nationalism."