mathematics

"[W]hat else... the reason of doubts arising in solutions... of plane and spherical triangles, but the want of accurate determinations and explanations?"

"But in Book II. i.e. in Spherical Trigonometry, the greatest pains were to be taken, and the greatest difficulties to be overcome. For though... Spherical Trigonometry is not so easy as the Plane, as it wants those previous helps and that determination, which the Plane... has; yet, when out of three parts of a proposition one only or two are laid down; when a proposition is demonstrated only in one case out of several; when, on account of bad definitions, several things, wanting demonstration, are passed by, or dignified with the name of axioms; when an argument turns in circulo... when whole steps are omitted; when, instead of a direct way, we go round about; when things are scattered about without order; when a whole set of triangles is neglected, &c. &c. surely, all this is not the fault of Spherical Trigonometry."

"It is from the vagueness of the proposition... and from misunderstanding the terms supplement and complement, that disputes have arisen in spherics: these may be seen at the end of [Samuel] Cunns Euclid, in his remarks and the appendix. Whatever be the mistakes of Mr. Heynes... the respectable names of Dr. Keil, Mr. Caswell, and Dr. Harris, whom Mr. Cunn joins in company with Heynes, are treated by him... injuriously; especially as he himself had not examined his subject with sufficient attention. His own rule... is indeed true... but it is more troublesome to the memory. Mr. Ham... awards his own rule, which, notwithstanding, is much more unmanageable... it using subtraction of natural versed sines, to whose difference therefore (and every one knows the thing is not easy) logarithms are to be accommodated. But it were time, long ago, to bury these worthless disputes in oblivion, that learners of spherics should not be discouraged by seeing them printed and reprinted so often."

"As in the rest of mathematical sciences, so in trigonometry, were the Arabs pupils of the Hindus…"

"[I]n 1575 Western Europe had recovered most of the major mathematical works of antiquity now extant. Arabic algebra had been... mastered and improved... through the solution of the cubic and quartic and through... partial... symbolism; and trigonometry had become an independent discipline. The time was almost ripe for rapid strides... The transition from the Renaissance to the modern world was... made through... intermediate figures, a few of the more important... Galileo Galilei... and ... from Italy; several... as .., Thomas Harriot.., and ... were English; two... Simon Stevin... and ... were Flemish; others came from varied lands—John Napier... from Scotland, Jobst Bürgi... from Switzerland, and Johann Kepler... from Germany."

"Ptolemy made observations at Alexandria from the years 125 to 150... .but an indifferent practical astronomer, and the observations of Hipparchus are... more accurate..."

"The most conspicuous authors of both Trigonometries amongst the British, who have been consulted by us, are Caswell, in Wallis's Works, Keil, Simpson, Robertson, Mr. , Emerson, and [Benjamin] Martin; and of those who have treated of them occasionally, Oughtred, in his Clavis Mathematica, and Circles of Proportion, Wallis, Jones, Wingate, [Henry] Sherwin, and Gardiner, in their Tables of Logarithms; Sir Isaac Newton, in his Univ. Arithm. Geometrical Problem II. Harris and Chambers in their dictionaries. Plane Trigonometry alone has been treated by [Philip] Ronayne, Mr. Thomas Simpson, Maseres, and Muller. However, the merits of some foreigners also cannot, without injustice, be suppressed. Such are Copernicus, in his Astronomia Instaurata; Balanus, a modern Greek; Simon Stevin, commented upon by '; Clavius; M. de la Caille; M. de la Lande, in his Astronomie, tom. III. de Chales; Ozanam; Segnerus; and the labours of Schottus, Tacquet, and others, are commendable. We need not mention the parents of these sciences, Theodosius, Ptolemy, Menelaus, and ."

"Ptolemy... produced his great work on astronomy, which will preserve his name as long as the history of science endures. This... is... the '...founded on the writings of Hipparchus, and, though it did not... advance the theory... it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise."

"Isaac Newton... went to school at Grantham and in 1661 came up as a subsizar to Trinity. ...He had not read any mathematics before coming into residence but was acquainted with Sanderson's Logic, which was then frequently read as preliminary to mathematics. At the beginning of his first October term he... picked up a book on astrology, but could not understand it on account of the geometry and trigonometry. He therefore bought a Euclid, and was surprised to find how obvious the propositions seemed. He thereupon read Oughtred's Clavis and Descartes's Geometry, the latter of which he managed to master by himself though with some difficulty. The interest he felt in the subject led him to take up mathematics rather than chemistry as a serious study. His subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures. At a later time on reading Euclid more carefully he formed a very high opinion of it as an instrument of education, and he often expressed his regret that he had not applied himself to geometry before proceeding to algebraic analysis. ...He was elected to a scholarship in 1663."

"Prop. 14. and its corollaries deserve... examination. It is hard to say, by whom they were invented, though... probably by the English; and perhaps corr. 3 & 2. of prop. 29. in spherics, have given rise to them all, as they are to be found in most books of the last age. They are all to be seen in Caswell... Wallace, Newton Univ. Arithm. Geom. Probl. 11. Thos. Simpsons Algebra, Geom. Probl. 15. Dr. Robertsons Navigation, Emerson, and [Benjamin] Martin. The analogy of the prop. in particular, is to be met with in Trigonometria Britannica, [Henry] Sherwins Tables, de la Caille, Dr. Simpson, and Ward."

"In continuation of... the most brilliant period of ancient geometry, the century of Euclid, Archimedes and Apollonius, recourse must again be had to the Collectio of the much later writer Pappus, for information about the lost three books of s of Euclid. ...In the Sphœrica of Menelaus, a geometer and astronomer of the first century A.D., is found the theorem (lib. III. lemma 1 p. 83, Oxon. 1758): If the sides ag, gd, da of a plane triangle be met by any transversal in the points erb respectively, thenge : ea=gr.db : rd.ba,or the product of three non-adjacent segments of the sides of the triangle by any transversal is equal to the product of the remaining three. This was... extended to spherical triangles... as a basis for the spherical trigonometry of the ancients."

"We shall... find similar Algebraical derivations of formulæ from two fundamental expressions for the cosine of an angle. The principle of the derivation... is not new; it originated with Euler, who inserted in the Acta Acad. Petrop. for 1779, a Memoir entitled Trigonometria Spherica Universa, ex primis principiis breviter et dilucide derivata. Gua next, in the Memoirs of the Academy of Sciences for 1783, p. 291, deduced... Spherical Trigonometry "from the Algebraical solution of the simplest of its Problems." In 1786, Cagnoli... derived fundamental expressions for the sine and cosine of the sum of two arcs. And lastly, Lagrange and Legendre, the one in the Journal de L Ecole Polytechnique, the other in his Elemens de Geometrie, have followed and simplified Euler’s method, and instead of three fundamental expressions, have shewn one to be sufficient."

": A. K. Dewdney from the famous April Fools' Day article in the computer recreations column of the Scientific American, April 1989."

": {{citation"

"The Axiom of Choice is necessary to select a set from an infinite number of pairs of socks, but not an infinite number of pairs of shoes [...] Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair. [...] There is no difficulty in doing this with the boots. The pairs are given as forming an ℵ0, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pair unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible."

"The axiom gets its name not because mathematicians prefer it to other axioms."

"The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

"One, two, buckle my shoe; Three, four, knock at the door; Five, six, pick up sticks; Seven, eight, lay them straight; Nine, ten, a big fat hen; Eleven, twelve, dig and delve; Thirteen, fourteen, maids a-courting; Fifteen, sixteen, maids in the kitchen; Seventeen, eighteen, maids in waiting; Nineteen, twenty, my plate's empty."

"The king was in his counting-house, Counting out his money;"

"Can there be any day but this, Though many sunnes to shine endeavour? We count three hundred, but we misse: There is but one, and that one ever."

"Ford carried on counting quietly. This is about the most aggressive thing you can do to a computer, the equivalent of going up to a human being and saying “Blood...blood...blood...blood...”"

"Whoe’er the number would define Of sports and joys that shall be thine, He first must count the grains of sand That spread the Erythræan strand, And every star and twinkling light That stud the glistening arch of night."

"If thou dost the number know Of the leaves on every bough, If thou can’st the reckoning keep Of the sands within the deep; Thee of all men will I take, And my Love’s accomptant make."

"In Riemann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our asymptotes no longer out of phase, We shall encounter, counting, face to face."

"Count the bees that on Hybla are playing, Count the flow’rs that enamel its fields, Count the flocks that on Tempe are straying, Or the grain that rich Sicily yields; Go number the stars in the heaven, Count how many sands on the shore, When so many kisses you’ve given I still shall be craving for more."

"Let’s number out the hours by blisses, And count the minutes by our kisses;"

"When I do count the clock that tells the time,"

"Sooner may you count the starres, And number hayle downe pouring, Tell the Osiers of the Temmes, Or Goodwins Sands devouring, Then the thicke-showr’d kisses here Which now thy tyred lips must beare."

"Those who argue that the concept of set is not sufficiently clear to fix the truth-value of CH have a position which is at present difficult to assail. As long as no new axiom is found which decides CH, their case will continue to grow stronger, and our assertion that the meaning of CH is clear will sound more and more empty."

"... there is Brouwer's , which is utterly destructive in its results. The whole theory of the \,\aleph\,'s greater than \,\aleph_1\, is rejected as meaningless (Brouwer 1907, 569). Cantor's conjecture itself receives several different meanings, all of which, though very interesting in themselves, are quite different from the original problem. They lead partly to affirmative, partly to negative answers (Brouwer, 1907, I: 9; III: 2). Not everything in this field, however, has been sufficiently clarified. The “semi-intuitionistic” standpoint along the lines of H. Poincaré and H. Weyl ... would hardly preserve substantially more of set theory."

"In the , one wholeheartedly accepts traditional mathematics at face value. All questions such as the Continuum Hypothesis are either true or false in the real world despite their independence from the various axiom systems. The Realist position is probably the one that most mathematicians would prefer to take."

"...attributes may be maintained because of deformations in fields. Such conservation laws are called topological. Thus, it may happen that a knot in a set of field lines, called a soliton, cannot be smoothed out. As a result, the soliton is prevented from dissipating and behaves much like a particle. A classic example is a magnetic monopole, which has not been found in nature but shows up as twisted configurations in some field theories. In the traditional view, then, particles such as electrons and quarks (which carry Noether charges) are seen as fundamental, whereas particles such as magnetic monopoles (which carry topological charge) are derivative. In 1977, however, Claus Montonen, now at the Helsinki Institute of Physics in Finland, and David I. Olive, now at the University of Wales at Swansea, made a bold conjecture. Might there exist an alternative formulation of physics in which the roles of Noether charges (like electrical charge) and topological charges (like magnetic charge) are reversed? In such a “dual” picture, the magnetic monopoles would be the elementary objects, whereas the familiar particles—quarks, electrons and so on—would arise as solitons."

"A method is proposed to calculate quantum numbers on solitons in quantum field theory. The method is checked on previously known examples and, in a special model, by other methods. It is found, for example, that the fermion number on kinks in one dimension or on magnetic monopoles in three dimensions is, in general, a transcendental function of the coupling constant of the theories."

"There are three essentially different types of lunar theory — that of de Pontécoulant, that of Delaunay, and that first developed by Hill, to which may perhaps be added that of Hansen as containing many features of more or less importance different from the others. That of de Pontécoulant and most of his predecessors consists in developing certain coordinates in periodic series of assumed form with the time or true anomaly as argument and determining the coefficients step by step as powers of the small parameters involved ; that of Delaunay consists in applying the method of the variation of parameters in the canonical form over and over in such a way as to remove the most important parts of the perturbative function ; that of Hill consists in finding very accurate particular solutions of the differential equations after the parts depending on the parallax of the sun, the eccentricity of the earth's orbit, and the latitude of the moon have been neglected, and then finding the deviations from this orbit due to general initial conditions and the neglected part of the perturbative function."

"In Book I, Prop. LXVI of the first edition of Philosophiae naturalis principia mathematica, Newton (1687) discussed the dynamical problem of three bodies in a general way, and then in Book III he asserted that the vagaries of the Moon's motion could be accounted for by the gravitational attraction of the Sun. He recognized that he needed to develop the theory further, and summarized his later results in The theory of the Moon's motion of 1702 (Cohen 1975). He continued to refine his treatment up to the publication of the second edition of Principia (Newton 1712), some sections of which differ greatly from the first edition. He made almost no further changes of his own in the third edition, but added a scholium by Machin (1726) on the motion of the nodes. The published account of the rotation of the apse line, much the same in all versions, was seriously wrong, but even before 1690 Newton had developed a somewhat more satisfactory treatment, with which, however, he remained dissatisfied and never published (Whiteside 1976). (Since this article was prepared, the new English translation of the Principia by Cohen and Whitman (1999) has appeared. It is a translation of the third edition of 1726, which differs significantly in a few places from the first and second editions, as will be indicated.)"

"While J. Scott Russell first observed solitons in water waves in Augst 1834, a full-fledged theory of solitons has only come of age in the last decade. This advance is due primarily in the discovery of a generalization of the , the . While this method can be used to solve exactly only a certain number of nonlinear equations, many of these are relevant to broad areas in physics."

"The idea that in some sense the ordinary proton and neutron might be solitons in a non-linear sigma model has a long history. The first suggestion was made by Skyrme more than twenty years ago ... David Finkelstein and Rubinstein showed that such objects could in principle be fermions ... in a paper that probably represented the first use of what would now be θ vacua in quantum field theory. A gauge invariant version was attempted by Faddeev ... Some relevant miracles are known to occur in two space-time dimensions ... ; there also exists a different mechanism by which solitons can be fermions ..."

"The general problem of Celestial Mechanics consists in the determination of the relative motions of p bodies attracting one another according to the Newtonian law. This problem is not able to be solved directly: in order to deal with it, certain limitations must be made. ... Again, owing to the conditions under which the bodies of our solar system move, we are further able to divide the problem of p bodies into several others, each of which may be treated as a case of the problem of three particles, or, as it is generally called, the Problem of Three Bodies. The greater part of the Lunar Theory is a particular case of the Problem of Three Bodies; it involves the determination of the motion of the Moon relative to the Earth, when the mutual attraction of the Earth, Moon and Sun, considered as particles, are the only forces under consideration. When this has been found, the effects produced by the actions of the planets, the non-spherical forms of the bodies etc., can be be exhibited as small corrections to the coordinates."

"Inflation has attracted cosmologists because of its potential to free the standard big bang model from its worst flaw, the need for special initial conditions and, in particular, the requirement of initial acausal homogeneity. Naturally one must check whether inflation itself depends critically on initial conditions. Several “no hair” theorems and perturbation calculations have indicated that inflation is stable, and that it will take place when the initial conditions are perturbed. This has led to the belief that inflation will start in any generic universe."

"In the Green-function treatment of particle motion, a unit impulse is represented by a force F(t) = δ(t–t′) and is the analogue of the unit point source in spatial problems. The initial conditions play the role of boundary conditions."

"The surprising discovery of Newton’s age is just the clear separation of laws of nature on the one hand and initial conditions on the other. The former are precise beyond anything reasonable; we know virtually nothing about the latter. ,,, … how can we ascertain that we know all the laws of nature relevant to a set of phenomena? If we do not, we would determine unnecessarily many initial conditions in order to specify the behavior of the object."

"'(6). In this elementary example, if the length of the given base AB be taken as the standard of length, and be on that account called unity, or one, then the length of the side BC (or AC) of the triangle must also be denoted by the same number, ONE; and these TWO NUMBERS, one, and sixty, serve in this view to define, or to describe, the length and direction of the new or constructed line BC; at least if the latter number (sixty) be combined with the consideration of a certain hand, or direction of rotation, towards which the old line BA may be conceived to turn, in the plane of the triangle (or of the paper)..."

"(5). As early as the first book of Euclid's Elements, an attentive student is (or may be) led to consider the relative length, and also the relative direction, of one straight line as compared with another. Thus when Euclid shows, in his very first proposition, how to construct on a given base AB an equilateral triangle ABC, he virtually teaches how, when one line AB is proposed or given, to draw a new line BC (or AC), which shall in length be equal to the given one, and in direction shall make with it an angle of sixty degrees, namely, the angle ABC (or BAC), which is the third part of 180 degrees, or of two right angles."

"'(7). The foregoing view, although not precisely the same with that adopted by Euclid himself, in his exposition of the elements of geometry, is at least consistent therewith; and has been made the basis of an important and modern method of calculation, respecting directed lines in one plane, which seems to have been first introduced about the commencement of the present century, by Argand in France, and for which Professor De Morgan... has lately proposed the name of Double Algebra: because it recognises and employs two numerical elements (such as the numbers 1 and 60 in the foregoing example), as required for the joint determination of the length and direction of a straight line. And it is now to be shown what is the nature of the passage that has been made, by the author of the Lectures on Quaternions, from such a double system of algebraic geometry, to what may be called, by analogy and contrast, a quadruple system of calculations respecting directed lines, or a system of QUADRUPLE ALGEBRA."

"'(3). One general form of answer... is... that in the mathematical quaternion is involved a peculiar synthesis, or combination, of the conceptions of space and time; and that while TIME is usually pictured or represented by metaphysicians under the figure of a line—a single stream with its ONE current—an unique axis of progression, SPACE is, on the contrary, imagined or conceived in connexion with THREE distinct axes, three lines at right angles to each other... height, length, and breadth. In time, we have only the forward and the backward, looking before and after. In space, there is not merely the contrast between the directions of upward and downward, but also between those of southward and northward, and again between westward and eastward. Time is said to have only one dimension, and space to have three dimesions. The former is an unidimensional, the latter a tridimensional progression. The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space," or "space plus time": and in this sense it has, or at least it involves a reference to, four dimensions. In an unpublished sonnet to Sir John Herschel, entitled "The "(...Greek ...equivalent to the Latin Quaternio), the author of the Lectures introduced the two following lines... an expression of the view... in the foregoing remarks..:"And how the One of Time, of Space the Three, Might in the Chain of Symbol girdled be.""

"'(4). Those who are entirely unacquainted with mathematical science may yet derive, from what has been above remarked, a sufficient preliminary insight into the nature of the speculations and inquiries to which the "Lectures on Quaternions" relate. A philosophical, if not a technically scientific, knowledge of the author's general aim, and of the idea which has guided him, may in this way be easily attained. But a very moderate acquaintance with the conceptions of geometry will suffice to render intelligible, from another point of view, the importance which the author attaches to the number Four in mathematics."

"'(8). This passage from the one system to the other may be said to consist mainly in the consideration of the variable plane of an angle. If, after tracing the equilateral triangle ABC on a card, which at first rests on a horizontal table, we then lift up that card, with the figure traced thereon, and lay it on a sloping desk, the triangle in its new position takes also a new aspect; it faces a different region of space, and may be conceived to look at, or be looked at by, a new point of the heavens, which is not now the vertical point (or ), as before. This new aspect of the figure, or of the plane (or desk) on which it is now situated, is the new circumstance introduced, in the transition from Double to Quadruple Algebra. And in fact it is easy to see that this new circumstance, of the varied position of the figure, namely, of the triangle, or simply (if we choose) of the ANGLE ABC, requires the consideration of two new numerical elements. For we have now two new questions to answer, or two new things to determine: namely, 1st, the slope of the desk (or inclination of the plane), suppose forty-five degrees, conducting to a first new number, 45 ; and 2nd, the direction of the edge (or, technically speaking, the line of the nodes), where that slope meets the table, and which may deviate from the line of north and south by any other number of degrees, suppose seventy, giving thus a second new number, in this case 70.'"

"I first became personally acquainted with Tait a short time before he was elected Professor in Edinburgh… It must have been either before his election or very soon after it that we entered on the project of a joint treatise on Natural Philosophy. He was then strongly impressed with the fundamental importance of Joule’s work… We incessantly talked over the mode of dealing with energy which we adopted in the book, and we went most cordially together in the whole affair. … We have had a thirty-eight years’ war over quaternions. He had been captivated by the originality and extraordinary beauty of Hamilton’s genius in this respect; and had accepted, I believe, definitely from Hamilton to take charge of quaternions after his death, which he has most loyally executed. Times without number I offered to let quaternions into Thomson and Tait if he could only show that in any case our work would be helped by their use. You will see that from beginning to end they were never introduced."

"I see no possible objection to your now publishing the deferred "scrap" if you yourself approve of what is said in it in favour of quaternions. I think you are right in your use of the word "Volapuk," but I don't think you should confine it to the vector part of quaternions. The whole affair has in respect to mathematics a value not inferior to that of "Volapuk" in respect to language."

"Yet, though few, if any—Clerk-Maxwell perhaps only excepted—ever possessed the same almost magical quality of physical insight, none could be more strict than Lord Kelvin in requiring demonstration freed from untenable assumptions or undemonstrable hypotheses. Daring as he was, at least in his earlier days, in the application of analytical methods to the phenomena of nature, he was in several ways very conservative. For example, he never would countenance the use in physics of the method of quaternions. At the British Association Meeting at Cambridge in 1845, he had met Hamilton, who there read his first paper on Quaternions. One might have thought that the young enthusiast would have readily welcomed a new and ingenious method of symbolic analysis: but it was not so. He would not use quaternion notation or quaternion methods himself, nor did he admit the into his work."