User:Mdd/Memorabilia mathematica

Memorabilia mathematica or, The philomath's quotation-book by Robert Edouard Moritz. Published 1914 (at gutenberg.org)

CHAPTER XVI : ARITHMETIC

 * There is no problem in all mathematics that cannot be solved by direct counting. But with the present implements of mathematics many operations can be performed in a few minutes which without mathematical methods would take a lifetime.
 * Ernst Mach, Popular Scientific Lectures, [McCormack] (Chicago, 1898), p. 197.


 * There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations.
 * Auguste Comte, Positive Philosophy, [Martineau], Bk. 1, chap. 1.


 * Pythagoras says that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe. But the law of number possesses an immanent order, which is at first sight mystifying, but on a more intimate acquaintance we easily understand it to be intrinsically necessary; and this law of number explains the wondrous consistency of the laws of nature.
 * Paul Carus, Reflections on Magic Squares; Monist, Vol. 16 (1906), p. 139.


 * It is number which regulates everything and it is measure which establishes universal order.... A quiet peace, an inviolable order, an inflexible security amidst all change and turmoil characterize the world which mathematics discloses and whose depths it unlocks.
 * Christian von Dillmann, Die Mathematik die Fackelträgerin einer neuen Zeit. (Stuttgart, 1889), p. 12.


 * Number, the inducer of philosophies, The synthesis of letters,....
 * Aeschylus. Quoted in, Thomson, J. A., Introduction to Science, chap. 1 (London).


 * Amongst all the ideas we have, as there is none suggested to the mind by more ways, so there is none more simple, than that of unity, or one: it has no shadow of variety or composition in it; every object our senses are employed about; every idea in our understanding; every thought of our minds, brings this idea along with it. And therefore it is the most intimate to our thoughts, as well as it is, in its agreement to all other things, the most universal idea we have.
 * John Locke, An Essay concerning Human Understanding, Bk. 2, chap. 16, sect. 1.


 * The simple modes of number are of all other the most distinct; every the least variation, which is an unit, making each combination as clearly different from that which approacheth nearest to it, as the most remote; two being as distinct from one, as two hundred; and the idea of two as distinct from the idea of three, as the magnitude of the whole earth is from that of a mite.
 * John Locke, An Essay concerning Human Understanding, Bk. 2, chap. 16, sect. 3.


 * The number of a class is the class of all classes similar to the given class.
 * Bertrand Russell, Principles of Mathematics, (Cambridge, 1903), p. 115.


 * Number is that property of a group of distinct things which remains unchanged during any change to which the 263 group may be subjected which does not destroy the distinctness of the individual things.
 * Henry Burchard Fine Number-system of Algebra, (Boston and New York, 1890), p. 3.


 * The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions and Logarithms.
 * Florian Cajori, History of Mathematics, (New York, 1897), p. 161.


 * The grandest achievement of the Hindoos and the one which, of all mathematical investigations, has contributed most 264 to the general progress of intelligence, is the invention of the principle of position in writing numbers.
 * Florian Cajori, History of Mathematics, (New York, 1897), p. 87.


 * The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio.
 * James Whitbread Lee Glaisher, Encyclopedia Britannica, 9th Edition; Article “Logarithms.”


 * All minds are equally capable of attaining the science of numbers: yet we find a prodigious difference in the powers of different men, in that respect, after they have grown up, because their minds have been more or less exercised in it.
 * Samuel Johnson, Boswell’s Life of Johnson, Harper’s Edition (1871), Vol. 2, p. 33.


 * What a benefite that onely thyng is, to haue the witte whetted and sharpened, I neade not trauell to declare, sith all men confesse it to be as greate as maie be. Excepte any witlesse persone thinke he maie bee to wise. But he that most feareth that, is leaste in daunger of it. Wherefore to conclude, I see moare menne to acknowledge the benefite of nomber, than I can espie willying to studie, to attaine the benefites of it. Many praise it, but fewe dooe greatly practise it: onlesse it bee for the vulgare practice, concernying Merchaundes trade. Wherein the desire and hope of gain, maketh many willying to sustaine some trauell. For aide of whom, I did sette forth the first parte of Arithmetike. But if thei knewe how faree this seconde parte, doeeth excell the firste parte, thei would not accoumpte 265 any tyme loste, that were emploied in it. Yea thei would not thinke any tyme well bestowed till thei had gotten soche habilitie by it, that it might be their aide in al other studies.
 * Recorde, Robert. Whetstone of Witte, (London, 1557).


 * Why are wise few, fools numerous in the excesse? ’Cause, wanting number, they are numberlesse. Lovelace.
 * Noah Bridges, Vulgar Arithmetike, (London, 1659), p. 127.


 * The clearness and distinctness of each mode of number from all others, even those that approach nearest, makes me apt to think that demonstrations in numbers, if they are not more evident and exact than in extension, yet they are more general in their use, and more determinate in their application. Because the ideas of numbers are more precise and distinguishable than in extension; where every equality and excess are not so easy to be observed or measured; because our thoughts cannot in space arrive at any determined smallness beyond which it cannot go, as an unit; and therefore the quantity or proportion of any the least excess cannot be discovered.
 * John Locke. An Essay concerning Human Understanding, Bk. 2, chap. 16, sect. 4.


 * Battalions of figures are like battalions of men, not always as strong as is supposed
 * Michel Sage. Mrs. Piper and the Society for Psychical Research, [Robertson] (New York, 1909), p. 151.


 * Number was born in superstition and reared in mystery,... numbers were once made the foundation of religion and philosophy, and the tricks of figures have had a marvellous effect on a credulous people.
 * Francis Wayland Parker, Talks on Pedagogics (New York, 1894), P. 64.


 * God made integers, all else is the work of man.
 * Leopold Kronecker, Jahresberichte der Deutschen Mathematiker Vereinigung, Bd. 2, p. 19.


 * Plato said “ἀεὶ ὁ θεὸς γεωμέτρε.” Jacobi changed this to “ἀεὶ ὁ θεὸς ἀριθμητίζει.” Then came Kronecker 270 and created the memorable expression “Die ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk”
 * Felix Klein. Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 6, p. 136.


 * Integral numbers are the fountainhead of all mathematics.
 * Hermann Minkowski, Diophantische Approximationen, (Leipzig, 1907), Vorrede.


 * Many of the greatest masters of the mathematical sciences were first attracted to mathematical inquiry by problems relating to numbers, and no one can glance at the periodicals of the present day which contain questions for solution without noticing how singular a charm such problems still continue to exert. The interest in numbers seems implanted in the human mind, and it is a pity that it should not have freer scope in this country. The methods of the theory of numbers 271 are peculiar to itself, and are not readily acquired by a student whose mind has for years been familiarized with the very different treatment which is appropriate to the theory of continuous magnitude; it is therefore extremely desirable that some portion of the theory should be included in the ordinary course of mathematical instruction at our University. From the moment that Gauss, in his wonderful treatise of 1801, laid down the true lines of the theory, it entered upon a new day, and no one is likely to be able to do useful work in any part of the subject who is unacquainted with the principles and conceptions with which he endowed it.
 * James Whitbread Lee Glaisher. Presidential Address British Association for the Advancement of Science, (1890); Nature, Vol. 42, p. 467.


 * Let us look for a moment at the general significance of the fact that calculating machines actually exist, which relieve mathematicians of the purely mechanical part of numerical computations, and which accomplish the work more quickly and with a greater degree of accuracy; for the machine is not subject to the slips of the human calculator. The existence of such a machine proves that computation is not concerned with the significance of numbers, but that it is concerned essentially only with the formal laws of operation; for it is only these that the machine can obey—having been thus constructed—an intuitive perception of the significance of numbers being out of the question.
 * Felix Klein, Elementarmathematik vom höheren Standpunkte aus. (Leipzig, 1908), Bd. 1, p. 53.


 * Zu Archimedes kam ein wissbegieriger Jüngling,
 * Weihe mich, sprach er zu ihm, ein in die göttliche Kunst, 272
 * Die so herrliche Dienste der Sternenkunde geleistet,
 * Hinter dem Uranos noch einen Planeten entdeckt.
 * Göttlich nennst Du die Kunst, sie ist’s, versetzte der Weise,
 * Aber sie war es, bevor noch sie den Kosmos erforscht,
 * Ehe sie herrliche Dienste der Sternenkunde geleistet,
 * Hinter dem Uranos noch einen Planeten entdeckt.
 * Was Du im Kosmos erblickst, ist nur der Göttlichen Abglanz,
 * In der Olympier Schaar thronet die ewige Zahl.


 * Carl Gustav Jacob Jacobi,  Journal für Mathematik, Bd. 101 (1887), p. 338.


 * To Archimedes came a youth intent upon knowledge,
 * Quoth he, “Initiate me into the science divine
 * Which to astronomy, lo! such excellent service has rendered,
 * And beyond Uranus’ orb a hidden planet revealed.”
 * “Call’st thou the science divine? So it is,” the wise man responded,
 * “But so it was long before its light on the Cosmos it shed,
 * Ere in astronomy’s realm such excellent service it rendered,
 * And beyond Uranus’ orb a hidden planet revealed.
 * Only reflection divine is that which Cosmos discloses,
 * Number herself sits enthroned among Olympia’s hosts.”


 * The Theory of Numbers has acquired a great and increasing claim to the attention of mathematicians. It is equally remarkable for the number and importance of its results, for the precision and rigorousness of its demonstrations, for the variety of its methods, for the intimate relations between truths apparently isolated which it sometimes discloses, and for the numerous applications of which it is susceptible in other parts of analysis.
 * Henry John Stephen Smith, Report on the Theory of Numbers, British Association, 1859; Collected Mathematical Papers, Vol. 1, p. 38.


 * I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, 274 which like space may be in its essence poly-dimensional, and that this and such sort of truths would become self-evident to a being whose mode of perception is according to superficially as distinguished from our own limitation to linearly extended time.
 * James Joseph Sylvester, Collected Mathematical Papers, Vol. 4, p. 600, footnote.