History of mathematics

"My object has been to notice particularly several points in the principles of algebra and geometry, which have not obtained their due importance in our elementary works... The perusal of the opinions of an individual, offered simply as such, may excite many to become inquirers, who would otherwise have been workers of rules and followers of dogmas. ...It has been my endeavor to avoid entering into the purely metaphysical part of the difficulties of algebra. The student is, in my opinion, little the better for such discussions, though he may derive such conviction of the truth of results by deduction from particular cases, as no à priori reasoning can give to a beginner. In treating, therefore, on the negative sign, on impossible quantities, and on fractions of the form \frac{0}{0}, etc., I have followed the method adopted by several of the most esteemed continental writers, of referring the explanation to some particular problem, and showing how to gain the same from any other. Those who admit such expressions as -a, \sqrt{- a}, \frac{0}{0}, etc., have never produced any clearer method; while those who call them absurdities, and would reject them altogether, must, I think, be forced to admit the fact that in algebra the different species of contradictions in problems are attended with distinct absurdities, resulting from them as necessarily as different numerical results from different numerical data. ...[D]ifferent misconceptions... give rise to the various expressions above alluded to."

"The late Professor Leslie... [i]n his Philosophy of Arithmetic... entered... into much of its history. ...[O]ne principal, thing to be cautious of is, his almost monomaniac antipathy to every thing Hindoo—a most unfortunate turn... Leslie... generalises... fearfully every now and then. He informs us that it was the practice throughout Europe to reduce the rules of arithmetic to memorial verses, and that [William] Buckley's Arithmetica Memorativa appears at one period to have gained possession of the schools and colleges of England. Now the truth... the verses attributed to Sacrobosco had never... been printed when Leslie wrote; and Buckley... was printed only once... and two or three times as an appendix to a work on logic. Dr. Peacock expresses the truth in saying... before the invention of printing, the practice of writing memorial verses was common, as appears by manuscript libraries. ...[H]ad the practice of using them been common, the presses of the fifteenth and sixteenth centuries would have given them forth in great numbers. But I cannot learn that any metrical work was printed in the fifteenth century, except the Compotus of [Magister] Anianus, and that only once."

"If the people at large be not already convinced that a sufficient general case has been made out for Administrative Reform, I think they never can be, and they never will be. ...Ages ago a savage mode of keeping accounts on notched sticks was introduced into the Court of Exchequer, and the accounts were kept, much as Robinson Crusoe kept his calendar on the desert island. In the course of considerable revolutions of time, the celebrated Cocker was born, and died; Walkinghame, of the Tutor's Assistant, and well versed in figures, was also born, and died; a multitude of accountants, book-keepers and actuaries, were born, and died. Still official routine inclined to these notched sticks, as if they were pillars of the constitution, and still the Exchequer accounts continued to be kept on certain splints of elm wood called "tallies." In the reign of George III an inquiry was made by some revolutionary spirit, whether pens, ink, and paper, slates and pencils, being in existence, this obstinate adherence to an obsolete custom ought to be continued, and whether a change ought not to be effected. All the red tape in the country grew redder at the bare mention of this bold and original conception, and it took till 1826 to get these sticks abolished. In 1834 it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? I dare say there was a vast amount of minuting, memoranduming, and despatch-boxing on this mighty subject. The sticks were housed at Westminster, and it would naturally occur to any intelligent person that nothing could be easier than to allow them to be carried away for fire-wood by the miserable people who live in that neighbourhood. However, they never had been useful, and official routine required that they never should be, and so the order went forth that they were to be privately and confidentially burnt. It came to pass that they were burnt in a stove in the House of Lords. The stove, overgorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Lords; the House of Lords set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; we are now in the second million of the cost thereof, the national pig is not nearly over the stile yet; and the little old woman, Britannia, hasn't got home to-night. ...The great, broad, and true cause that our public progress is far behind our private progress, and that we are not more remarkable for our private wisdom and success in matters of business than we are for our public folly and failure, I take to be as clearly established as the sun, moon, and stars."

"Like anything else, mathematics is created within the context of history, and it of interest to place Cardano's solution of the cubic two years after the publication of Copernicus's heliocentric theory and two years before the death of England's Henry VIII, or to emphasize the impact of the Restoration upon Cambridge University when a young scholar named Isaac Newton entered it in 1666."

"Mathematics is the product of real, flesh-and-blood human beings whose lives may reflect the inspirational, the tragic, or the bizarre. ...Understanding something of the lives of these diverse individuals can only enhance an appreciation of their work."

"Abel did not deny that we might solve quintics using techniques other than algebraic ones of adding, subtracting, multiplying, dividing, and extracting roots. ...the general quintic can be solved by introducing... "elliptic functions," but these require operations considerably more complicated than those of elementary algebra. In addition, Abel's result did not preclude our approximating solutions... as accurately as we... wish. What Abel did do was prove that there exists no algebraic formula... The analogue of the quadratic formula for second-degree equations and Cardano's formula for cubics simply does not exist... This situation is reminiscent of that encountered when trying to square the circle, for in both cases mathematicians are limited by the tools they can employ. ...the restriction to "solution by radicals"... hampers mathematicians... what Abel actually demonstrated was that algebra does have... limits, and for no obvious reason, these limits appear precisely as we move from the fourth to the fifth degree."

"In Greek theoretical mathematics (as distinguished from practical or commercial arithmetic) a fraction that we would write as a/b was not regarded as a number, as a single entity, but as a relationship or a : b between the whole numbers a and b. Thus the ratio a : b was, in modern terms, simply an ordered pair, rather than a rational number. ... More formally, a : b = c : d provided [a/b and c/d are both integral multiples of some p/q, i.e.,] there exist integers p, q, m, n such that a = mp, b = mq, c = np, d = nq."

"The seventeenth century is outstandingly conspicuous in the history of mathematics. Napier revealed his invention of logarithms, Harriot and Oughtred contributed to the notion and codification of algebra, Galileo founded the science of dynamics, and Kepler announced his laws of planetary motion. Later in the century, Desargues and Pascal opened a new field of pure geometry, Descartes launched modern analytic geometry, Fermat laid the foundations of modern number theory, and Huygens made distinguished contributions to the theory of probability and other fields. Then, toward the end of the century, after a host of seventeenth-century mathematicians had prepared the way, the epoch-making creation of the calculus was made by Newton and Leibniz. ...Thus, we see that... many new and vast fields were opened up for mathematical investigation."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of descriptive geometry by Monge in 1795. The s of Sir William Rowan Hamilton the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period."