1601. 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.—Mach, Ernst.

Popular Scientific Lectures [McCormack] (Chicago, 1898), p. 197.

1602. 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.—Comte, A.

Positive Philosophy [Martineau], Bk. 1, chap. 1.

1603. 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.—Carus, Paul.

Reflections on Magic Squares; Monist, Vol. 16 (1906), p. 139.

1604. An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of arithmetic, to translate it into lines necessitates the use of geometry.—Lagrange.

LeÇons ElÉmentaires sur les MathÉmatiques, LeÇon seconde.

1605. 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.—Dillmann, E.

Die Mathematik die FackeltrÄgerin einer neuen Zeit (Stuttgart, 1889), p. 12.

1606.

—Aeschylus.

Quoted in, Thomson, J. A., Introduction to Science, chap. 1 (London).

1607. 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.—Locke, John.

An Essay concerning Human Understanding, Bk. 2, chap. 16, sect. 1.

1608. 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.—Locke, John.

An Essay concerning Human Understanding, Bk. 2, chap. 16, sect. 3.

1609. The number of a class is the class of all classes similar to the given class.—Russell, Bertrand.

Principles of Mathematics (Cambridge, 1903), p. 115.

1610. Number is that property of a group of distinct things which remains unchanged during any change to which the group may be subjected which does not destroy the distinctness of the individual things.—Fine, H. B.

Number-system of Algebra (Boston and New York, 1890), p. 3.

1611. The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time.—Parker, F. W.

Talks on Pedagogics (New York, 1894), p. 64.

1612.

—Howison, G. H.

Journal of Speculative Philosophy, Vol. 5, p. 175.

1613. That arithmetic rests on pure intuition of time is not so obvious as that geometry is based on pure intuition of space, but it may be readily proved as follows. All counting consists in the repeated positing of unity; only in order to know how often it has been posited, we mark it each time with a different word: these are the numerals. Now repetition is possible only through succession: but succession rests on the immediate intuition of time, it is intelligible only by means of this latter concept: hence counting is possible only by means of time.—This dependence of counting on time is evidenced by the fact that in all languages multiplication is expressed by “times” [mal], that is, by a concept of time; sexies, ??a???, six fois, six times.—Schopenhauer, A.

Die Welt als Vorstellung und Wille; Werke (Frauenstaedt) (Leipzig, 1877), Bd. 3, p. 39.

1614. The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions and Logarithms.—Cajori, F.

History of Mathematics (New York, 1897), p. 161.

1615. The grandest achievement of the Hindoos and the one which, of all mathematical investigations, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers.—Cajori, F.

History of Mathematics (New York, 1897), p. 87.

1616. 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.—Glaisher, J. W. L.

Encyclopedia Britannica, 9th Edition; Article “Logarithms.”

1617. 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.—Johnson, Samuel.

Boswell’s Life of Johnson, Harper’s Edition (1871), Vol. 2, p. 33.

1618. The method of arithmetical teaching is perhaps the best understood of any of the methods concerned with elementary studies.—Bain, Alexander.

Education as a Science (New York, 1898), p. 288.

1619. 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 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).

1620. You see then, my friend, I observed, that our real need of this branch of science [arithmetic] is probably because it seems to compel the soul to use our intelligence in the search after pure truth.

Aye, remarked he, it does this to a remarkable extent.

Have you ever noticed that those who have a turn for arithmetic are, with scarcely an exception, naturally quick in all sciences; and that men of slow intellect, if they be trained and exercised in this study ... become invariably quicker than they were before?

Exactly so, he replied.

And, moreover, I think you will not easily find that many things give the learner and student more trouble than this.

Of course not.

On all these accounts, then, we must not omit this branch of science, but those with the best of talents should be instructed therein.—Plato.

Republic [Davis], Bk. 7, chap. 8.

1621. Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and if visible or tangible objects are obtruding upon the argument, refusing to be satisfied.—Plato.

Republic [Jowett], Bk. 7, p. 525.

1622. Good arithmetic contributes powerfully to purposive effort, to concentration, to tenacity of purpose, to generalship, to faith in right, and to the joy of achievement, which are the elements that make up efficient citizenship.... Good arithmetic exalts thinking, furnishes intellectual pleasure, adds appreciably to love of right, and subordinates pure memory.—Myers, George.

Monograph on Arithmetic in Public Education (Chicago), p. 21.

1623. On the one side we may say that the purpose of number work is to put a child in possession of the machinery of calculation; on the other side it is to give him a better mastery of the world through a clear (mathematical) insight into the varied physical objects and activities. The whole world, from one point of view, can be definitely interpreted and appreciated by mathematical measurements and estimates. Arithmetic in the common school should give a child this point of view, the ability to see and estimate things with a mathematical eye.—McMurray, C. A.

Special Method in Arithmetic (New York, 1906), p. 18.

1624. We are so accustomed to hear arithmetic spoken of as one of the three fundamental ingredients in all schemes of instruction, that it seems like inquiring too curiously to ask why this should be. Reading, Writing, and Arithmetic—these three are assumed to be of co-ordinate rank. Are they indeed co-ordinate, and if so on what grounds?

In this modern “trivium” the art of reading is put first. Well, there is no doubt as to its right to the foremost place. For reading is the instrument of all our acquisition. It is indispensable. There is not an hour in our lives in which it does not make a great difference to us whether we can read or not. And the art of Writing, too; that is the instrument of all communication, and it becomes, in one form or other, useful to us every day. But Counting—doing sums,—how often in life does this accomplishment come into exercise? Beyond the simplest additions, and the power to check the items of a bill, the arithmetical knowledge required of any well-informed person in private life is very limited. For all practical purposes, whatever I may have learned at school of fractions, or proportion, or decimals, is, unless I happen to be in business, far less available to me in life than a knowledge, say, of history of my own country, or the elementary truths of physics. The truth is, that regarded as practical arts, reading, writing, and arithmetic have no right to be classed together as co-ordinate elements of education; for the last of these is considerably less useful to the average man or woman not only than the other two, but than many others that might be named. But reading, writing, and such mathematical or logical exercise as may be gained in connection with the manifestation of numbers, have a right to constitute the primary elements of instruction. And I believe that arithmetic, if it deserves the high place that it conventionally holds in our educational system, deserves it mainly on the ground that it is to be treated as a logical exercise. It is the only branch of mathematics which has found its way into primary and early education; other departments of pure science being reserved for what is called higher or university instruction. But all the arguments in favor of teaching algebra and trigonometry to advanced students, apply equally to the teaching of the principles or theory of arithmetic to schoolboys. It is calculated to do for them exactly the same kind of service, to educate one side of their minds, to bring into play one set of faculties which cannot be so severely or properly exercised in any other department of learning. In short, relatively to the needs of a beginner, Arithmetic, as a science, is just as valuable—it is certainly quite as intelligible—as the higher mathematics to a university student.—Fitch, J. G.

Lectures on Teaching (New York, 1906), pp. 267-268.

1625. What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking—consecutively, closely, logically—can be effectually taught.—Fitch, J. G.

Lectures on Teaching (New York, 1906), pp. 292-293.

1626. Arithmetic and geometry, those wings on which the astronomer soars as high as heaven.—Boyle, Robert.

Usefulness of Mathematics to Natural Philosophy; Works (London, 1772), Vol. 3, p. 429.

1627. Arithmetical symbols are written diagrams and geometrical figures are graphic formulas.—Hilbert, D.

Mathematical Problems; Bulletin American Mathematical Society, Vol. 8 (1902), p. 443.

1628. Arithmetic and geometry are much more certain than the other sciences, because the objects of them are in themselves so simple and so clear that they need not suppose anything which experience can call in question, and both proceed by a chain of consequences which reason deduces one from another. They are also the easiest and clearest of all the sciences, and their object is such as we desire; for, except for want of attention, it is hardly supposable that a man should go astray in them. We must not be surprised, however, that many minds apply themselves by preference to other studies, or to philosophy. Indeed everyone allows himself more freely the right to make his guess if the matter be dark than if it be clear, and it is much easier to have on any question some vague ideas than to arrive at the truth itself on the simplest of all.—Descartes.

Rules for the Direction of the Mind; Torrey’s Philosophy of Descartes (New York, 1892), p. 63.

1629.

—Lovelace.

Noah Bridges: Vulgar Arithmetike (London, 1659), p. 127.

1630. 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.—Locke, John.

An Essay concerning Human Understanding, Bk. 2, chap. 16, sect. 4.

1631. Battalions of figures are like battalions of men, not always as strong as is supposed.—Sage, M.

Mrs. Piper and the Society for Psychical Research [Robertson] (New York, 1909), p. 151.

1632. 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.—Parker, F. W.

Talks on Pedagogics (New York, 1894), P. 64.

1633. A rule to trick th’ arithmetic.—Kipling, R.

To the True Romance.

1634. God made integers, all else is the work of man.—Kronecker, L.

Jahresberichte der Deutschen Mathematiker Vereinigung, Bd. 2, p. 19.

1635. Plato said “?e? ? ?e?? ?e??t?e.” Jacobi changed this to “?e? ? ?e?? ?????t??e?.” Then came Kronecker and created the memorable expression “Die ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk”—Klein, F.

Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 6, p. 136.

1636. Integral numbers are the fountainhead of all mathematics.—Minkowski, H.

Diophantische Approximationen (Leipzig, 1907), Vorrede.

1637. The “Disquisitiones Arithmeticae” that great book with seven seals.—Merz, J. T.

A History of European Thought in the Nineteenth Century (Edinburgh and London, 1908), p. 721.

1638. It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory.—Mathews, G. B.

Theory of Numbers (Cambridge, 1892), Part 1, sect. 48.

1639. Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories.—Mathews, G. B.

Theory of Numbers (Cambridge, 1892), Part 1, sect. 1.

1640. 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 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.—Glaisher, J. W. L.

Presidential Address British Association for the Advancement of Science (1890); Nature, Vol. 42, p. 467.

1641. 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.—Klein, F.

Elementarmathematik vom hÖheren Standpunkte aus. (Leipzig, 1908), Bd. 1, p. 53.

1642. Mathematics is the queen of the sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.—Gauss.

Sartorius von Waltershausen: Gauss zum GedÄchtniss. (Leipzig, 1866), p. 79.

1643.

—Jacobi, C. G. J.

Journal fÜr Mathematik, Bd. 101 (1887), p. 338.

1644. The higher arithmetic presents us with an inexhaustible store of interesting truths,—of truths too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed.—Gauss, C. F.

Preface to Eisenstein’s Mathematische Abhandlungen (Berlin, 1847), [H. J. S. Smith].

1645. 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.—Smith, H. J. S.

Report on the Theory of Numbers, British Association, 1859; Collected Mathematical Papers, Vol. 1, p. 38.

1646. The invention of the symbol = by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic.—Mathews, G. B.

Theory of Numbers (Cambridge, 1892), Part 1, sect. 29.

1647. As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious.—Mathews, G. B.

Theory of Numbers (Cambridge, 1892), Part 1, sect. 167.

1648. 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, 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.—Sylvester, J. J.

Collected Mathematical Papers, Vol. 4, p. 600, footnote.