MEMORABILIA MATHEMATICA THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS ATLANTA SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO Ex Libris C. K. OGDEN MEMORABILIA MATHEMATICA OR THE PHILOMATH'S QUOTATION-BOOK BY ROBERT EDOUARD MORITZ, PH. D., PH. N. D. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF WASHINGTON THE MACMILLAN COMPANY 1914 All rights reserved COPYRIGHT, 1914, BY ROBERT EDOUARD MORITZ (5 A LIBRARY -, UNIVERSITY OF CALIFORNIA SANTA BARBARA PREFACE EVERY one knows that the fine phrase "God geometrizes " is attributed to Plato, but few know where this famous passage is foun '. or the exact words in which it was first expressed. Those who, like the author, have spent hours and even days in the search of the exact statements, or the exact references, of similar famous passages, will not question the timeliness and usefulness of a book whose distinct purpose it is to bring together into a single volume exact quotations, with their exact references, bearing on one of the most time-honored, and even today the most active and most fruitful of all the sciences, the queen- mother of all the sciences, that is, mathematics. It is hoped that the present volume will prove indispensable to every teacher of mathematics, to every writer on mathe- matics, and that the student of mathematics and the related sciences will find its perusal not only a source of pleasure but of encouragement and inspiration as well. The layman will find it a repository of useful information covering a field of knowledge which, owing to the unfamiliar and hence repellant character of the language employed by mathematicians, is peculiarly in- accessible to the general reader. No technical processes or technical facility is required to understand and appreciate the wealth of ideas here set forth in the words of the world's great thinkers. No labor has been spared to make the present volume worthy of a place among collections of a like kind in other fields. Ten years have been devoted to its preparation, years, which if they could have been more profitably, could scarcely have been more pleasurably employed. As a result there have been brought together over one thousand more or less familiar passages pertaining to mathematics, by poets, philosophers, historians, statesmen, scientists, and mathematicians. These have been gathered from over three hundred authors, and have been v VI PREFACE grouped under twenty heads, and cross indexed under nearly seven hundred topics. The author's original plan was to give foreign quotations both in the original and in translation, but with the growth of mate- rial this plan was abandoned as infeasible. It was thought to serve the best interest of the greater number of English readers to give translations only, while preserving the references to the original sources, so that the student or critical reader may readily consult the original of any given extract. In cases where the translation is borrowed the translator's name is inserted in brackets [ ] immediately after the author's name. Brackets are also used to indicate inserted words or phrases made necessary to bring out the context. The absence of similar English works has made the author's work largely that of the pioneer. Rebie're's " Mathematiques et Mathematiciens" and Ahrens' "Scherz und Ernst in der Mathematik" have indeed been frequently consulted but rather with a view to avoid overlapping than to receive aid. Thus certain topics as the correspondence of German and French mathematicians, so excellently treated by Ahrens, have pur- posely been omitted. The repetitions are limited to a small number of famous utterances whose absence from a work of this kind could scarcely be defended on any grounds. No one can be more keenly aware of the shortcomings of a work than its author, for none can have so intimate an acquaint- ance with it. Among those of the present work is its incom- pleteness, but it should be borne in mind that incompleteness is a necessary concomitant of every collection of whatever kind. Much less can completeness be expected in a first collection, made by a single individual, in his leisure hours, and in a field which is already boundless and is yet expanding day by day. A collection of great thoughts, even if complete today, would be incomplete tomorrow. Again, if some authors are quoted more frequently than others of greater fame and authority, the reason may be sought not only in the fact that the writings of some authors peculiarly lent themselves to quotation, a quality singularly absent in other writers of the greatest merit and authority, but also in this, that the greatest freedom has been exercised in the choice of selections. The author has followed PREFACE Vll the bent of his own fancy in collecting whatever seemed to him sufficiently valuable because of its content, its beauty, its origi- nality, or its terseness, to deserve a place in a "Memorabilia." Great pains has been taken to furnish exact readings and references. In some cases where a passage could not be traced to its first source, the secondary source has been given rather than the reputed source. For the same reason many references are to later editions rather than to inaccessible first editions. The author feels confident that this work will be of assistance to his co-workers in the field of mathematics and allied fields. If in addition it should aid in a better appreciation of mathe- maticians and their work on the part of laymen and students in other fields, the author's foremost aim in the preparation of this work will have been achieved. ROBERT EDOUARD MORITZ, September, 1913. CONTENTS CHAPTER PAGE I. DEFINITIONS AND OBJECT OF MATHEMATICS . . 1 II. THE NATURE OF MATHEMATICS 10 III. ESTIMATES OF MATHEMATICS 39 IV. THE VALUE OF MATHEMATICS 49. V. THE TEACHING OF MATHEMATICS 72 VI. STUDY AND RESEARCH IN MATHEMATICS ... 86 VII. MODERN MATHEMATICS 108 VIII. THE MATHEMATICIAN 121 IX. PERSONS AND ANECDOTES (A-M) 135 X. PERSONS AND ANECDOTES (N-Z) 166 XI. MATHEMATICS AS A FINE ART 181 XII. MATHEMATICS AS A LANGUAGE 194 XIII. MATHEMATICS AND LOGIC 201 XIV. MATHEMATICS AND PHILOSOPHY 209 XV. MATHEMATICS AND SCIENCE 224 XVI. ARITHMETIC 261 XVII. ALGEBRA 275 XVIII. GEOMETRY 292 XIX. THE CALCULUS AND ALLIED TOPICS .... 323 XX. THE FUNDAMENTAL CONCEPTS OF TIME AND SPACE 345 XXI. PARADOXES AND CURIOSITIES 364 INDEX . 385 Alles Gescheite ist schon gedacht worden; man muss nur versuchen, es noch einmaljzu denken. -GOETHE. Spruche in Prosa, Ethisches, I. 1. A great man quotes bravely, and will not draw on his invention when his memory serves him with a word as good. EMERSON. Letters and Social Aims, Quotation and Originality. MEMORABILIA MATHEMATICA MEMORABILIA MATHEMATICA CHAPTER I DEFINITIONS AND OBJECT OF MATHEMATICS 101. I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music. SYLVESTER, J. J. Presidential Address to the British Association, Exeter British Association Report (1869); Collected Mathematical Papers, Vol. 2, p. 659. 102. ... all the sciences which have for their end investiga- tions concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, ac- cordingly, there ought to exist a general science which should explain all that can be known about order and measure, con- sidered independently of any application to a particular subject, and that, indeed, this science has its own proper name, con- secrated by long usage, to wit, mathematics. And a proof that it far surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides; . . . DESCARTES. Rules for the Direction of the Mind, Philosophy of D. [Torrey] (New York, 18918), p. 72. 103. [Mathematics] has for its object the indirect measure- ment of magnitudes, and it purposes to determine magnitudes by each other, according to the precise relations which exist between them. COMTE. Positive Philosophy [Martineau], Bk.l, chap. 1. 2 MEMORABILIA MATHEMATICA 104. The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenom- enon under consideration; and these equations are the starting- point of the calculus; which must obtain from them certain quantities by means of others. COMTE. Positive Philosophy [Martineau], Bk. 1, chap. 2. 105. Mathematics is the science of the connection of magni- tudes. Magnitude is anything that can be put equal or unequal to another thing. Two things are equal when in every assertion each may be replaced by the other. GRASSMANN, HERMANN. Stiicke aus dem Lehrbuche der Arithmetik, Werke (Leipzig, 1904), Bd. 2, p. 298. 106. Mathematic is either Pure or Mixed: To Pure Mathe- matic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered. . . . Mixed Mathe- matic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to ex- plain, demonstrate and actuate these. BACON, FRANCIS. De Augmentis, Bk. 3; Advancement of Learning, Bk. 2. 107. The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the considera- tion of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, com- bined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics. WHEWELL, WILLIAM. The Philosophy of the Inductive Sciences, Part 1, Bk. 2, chap. I, sect. 4. (London, 1858). DEFINITIONS AND OBJECTS OF MATHEMATICS 3 108. Higher Mathematics is the art of reasoning about numerical relations between natural phenomena; and the sev- eral sections of Higher Mathematics are different modes of viewing these relations. MELLOR, J. W. Higher Mathematics for Students of Chemistry and Physics (New York, 1902), Prologue. 109. Number, place, and combination . . . the three inter- secting but distinct spheres of thought to which all mathemati- cal ideas admit of being referred. SYLVESTER, J. J. Philosophical Magazine, Vol. 24. (1844), p. 285; Collected Mathematical Papers, Vol. 1, p. 91. 110. There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all three of them combined, every mathematical truth admits of be- ing referred; these are the three cardinal notions, of Number, Space and Order. Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space. SYLVESTER, J. J. A Probationary Lecture on Geometry, York British Association Report (1844), Part 2; Collected Mathematical Papers, Vol. 2, p. 5. 111. The object of pure mathematics is those relations which may be conceptually established among any conceived elements whatsoever by assuming them contained in some ordered mani- fold; the law of order of this manifold must be subject to our choice; the latter is the case in both of the only conceivable kinds of manifolds, in the discrete as well as in the continuous. PAPPERITZ, E. Uber das System der rein mathematischen Wissenschaften, Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 1, p. 86. 4 MEMORABILIA MATHEMATICA 112. Pure mathematics is not concerned with magnitude. It is merely the doctrine of notation of relatively ordered thought operations which have become mechanical. NOVALIS. Schriften (Berlin, 1901), Zweiter Teil, p. 282. 113. Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of math- ematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics. CHRYSTAL, GEORGE. Encyclopedia Britannica (9th edition), Article 11 Mathematics." 114. The purely formal sciences, logic and mathematics, deal with those relations which are, or can be, independent of the particular content or the substance of objects. To mathematics in particular fall those relations between objects which involve the concepts of magnitude, of measure and of number. HANKEL, HERMANN. Theorie der Complexen Zahlensysteme, (Leipzig, 1867), p. 1. 115. Quantity is that which is operated with according to fixed mutually consistent laws. Both operator and operand must derive their meaning from the laws of operation. In the case of ordinary algebra these are the three laws already indicated [the commutative, associative, and distributive laws], in the algebra of quaternions the same save the law of commutation for multiplication and division, and so on. It may be questioned whether this definition is sufficient, and it may be objected that it is vague; but the reader will do well to reflect that any defini- tion must include the linear algebras of Peirce, the algebra of logic, and others that may be easily imagined, although they have not yet been developed. This general definition of quan- DEFINITIONS AND OBJECTS OF MATHEMATICS 5 tity enables us to see how operators may be treated as quanti- ties, and thus to understand the rationale of the so called sym- bolical methods. CHRYSTAL, GEORGE. Encyclopedia Britannica (9th edition), Article " Mathematics." 116. Mathematics in a strict sense is the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations. MURRAY, J. A. H. A New English Dictionary. 117. Everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science, mathematics. HERBART, J. F. Pestalozzi's Idee eines ABC der Anschauung, Werke [Kehrbach], (Langensalza, 1890), Bd. 1, p. 163. 118. Perhaps the least inadequate description of the general scope of modern Pure Mathematics I will not call it a defini- tion would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations. HOBSON, E. W. Presidential Address British Association for the Advancement of Science (1910); Nature, Vol. 84, p. 287. 119. The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and pre- cisely stated. So that all serious thought which is not philos- ophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus. WHITEHEAD, A. N. Universal Algebra (Cambridge, 1898), Preface. 6 MEMORABILIA MATHEMATICA 120. Mathematics is the science which draws necessary con- clusions. PEIRCE, BENJAMIN. Linear Associative Algebra, American Journal of Mathematics, Vol. 4 (1881), p. 97. 121. Mathematics is the universal art apodictic. SMITH, W. B. Quoted by Keyser, C. J. in Lectures on Science, Philosophy and Art (New York, 1908), p. 18. 122. Mathematics in its widest signification is the develop- ment of all types of formal, necessary, deductive reasoning. WHITEHEAD, A. N. Universal Algebra (Cambridge, 1898), Preface, p. vi. 123. Mathematics in general is fundamentally the science of self-evident things. KLEIN, FELIX. Anwendung der Differential-und Integral- rechnung auf Geometric (Leipzig, 1902}, p. 26. 124. A mathematical science is any body of propositions which is capable of an abstract formulation and arrangement in such a way that every proposition of the set after a certain one is a formal logical consequence of some or all the preceding propositions. Mathematics consists of all such mathematical sciences. YOUNG, CHARLES WESLEY. Fundamental Concepts of Algebra and Geome- try (New York, 1911), p. 222. 125. Pure mathematics is a collection of hypothetical, deduc- tive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition. FITCH, G. D. The Fourth Dimension simply Explained (New York, 1910), p. 58. 126. The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning. WHITEHEAD, A. N. Universal Algebra (Cambridge, 1898), p. 12. DEFINITIONS AND OBJECTS OF MATHEMATICS 7 127. Pure mathematics consists entirely of such assevera- tions as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. ... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. RUSSELL, BERTBAND. Recent Work on the Principles of Mathematics, International Monthly, Vol. 4 (1901), P- $4- 128. Pure Mathematics is the class of all propositions of the form "p implies q," where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, Mathematics uses a notion which is not a constituent of the propositions which it considers namely, the notion of truth. RUSSELL, BERTRAND. Principles of Mathematics (Cambridge, 1903), p.L 129. The object of pure Physic is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence. SYLVESTER, J. J. On a theorem connected with Newton's Rule, etc., Collected Mathematical Papers, Vol. 3, p. 424- 130. First of all, we ought to observe, that mathematical propositions, properly so called, are always judgments a priori, and not empirical, because they carry along with them necessity, which can never be deduced from experience. If people should 8 MEMORABILIA MATHEMATICA object to this, I am quite willing to confine my statements to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure knowledge a priori. KANT, IMMANUEL. Critique of Pure Reason [Muller], (New York, 1900), p. 720. 131. Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex. . . . WHITE, WILLIAM F. A Scrap-book of Elementary Mathematics, (Chicago, 1908), p. 215. 132. The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to dis- cover such systems. Any such system is a branch of mathe- matics. KEYSER, C. J. Science, New Series, Vol. 35, p. 107. 133. [Mathematics is] the study of ideal constructions (often applicable to real problems), and the discovery thereby of rela- tions between the parts of these constructions, before unknown. PEIRCE, C. S. Century Dictionary, Article "Mathematics" DEFINITIONS AND OBJECTS OF MATHEMATICS 9 134. Mathematics is that form of intelligence in which we bring the objects of the phenomenal world under the control of the conception of quantity. [Provisional definition.] HOWISON, G. H. The Departments of Mathematics, and their Mutual Relations; Journal of Speculative Philosophy, Vol. S, p. 164. 136. Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number. HOWISON, G. H. The Departments of Mathematics, and their Mutual Relations; Journal of Speculative Philosophy, Vol. 5, p. 170. CHAPTER II THE NATURE OF MATHEMATICS 201. Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the con- trary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be as- cribed to a revolution, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us. ... A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was Thales or any other name), for he found that he had not to investigate what he saw hi the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts a priori, placed into that figure and represented in it, so that, in order to know anything with certainty a priori, he must not attribute to that figure anything beyond what neces- sarily follows from what he has himself placed into it, in accord- ance with the concept. KANT, IMMANUEL. Critique of Pure Reason, Preface to the Second Edition [Mutter], (New York, 1900), p. 690. 202. [When followed in the proper spirit], there is no study in the world which brings into more harmonious action all the faculties of the mind than the one [mathematics] of which I 10 THE NATURE OF MATHEMATICS 11 stand here as the humble representative and advocate. There is none other which prepares so many agreeable surprises for its followers, more wonderful than the transformation scene of a pantomime, or, like this, seems to raise them, by successive steps of initiation to higher and higher states of conscious intellectual being. SYLVESTER, J. J. A Plea for the Mathematician, Nature, Vol. 1, p. 261. 203. Thought-economy is most highly developed in mathe- matics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathe- matics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ deter- minants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar, we see in all this but a faint reflection of the intellectual activity of a La- grange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one. MACH, E. . Popular-wissenschafliche Vorlesungen (1908), pp. 224-225. 204. Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols. DlLLMANN, E. Die Mathematik die Fackeltragerin einer neuen Zeit (Stuttgart, 1889), p. 94. 12 MEMORABILIA MATHEMATICA 206. The essence of mathematics lies in its freedom. CANTOR, GEORGE. Mathematische Annalen, Bd. 21, p. 564- 206. Mathematics pursues its own course unrestrained, not indeed with an unbridled licence which submits to no laws, but rather with the freedom which is determined by its own nature and in conformity with its own being. HANKEL, HERMANN. Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tubingen, 1884), P- 16. 207. Mathematics is perfectly free in its development and is subject only to the obvious consideration, that its concepts must be free from contradictions in themselves, as well as definitely and orderly related by means of definitions to the previously existing and established concepts. CANTOR, GEORGE. Grundlagen einer attgemeinen Manigfaltigkeits- lehre (Leipzig, 1883), Sect. 8. 208. Mathematicians assume the right to choose, within the limits of logical contradiction, what path they please in reaching their results. ADAMS, HENRY. A Letter to American Teachers of History (Washington, 1910), Introduction, p. v. 209. Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself, will some day find it employed against himself. HERBART, J. F. Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 105. 210. Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hy- pothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics. PEIRCE, BENJAMIN. Linear Associative Algebra, American Journal of Mathematics, Vol. 4 (1881), p. 97. THE NATURE OF MATHEMATICS 13 211. Mathematics is a science continually expanding; and its growth, unlike some political and industrial events, is attended by universal acclamation. WHITE, H. S. Congress of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. 455. 212. Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, dec- lination and right ascension, longitude and latitude are re- ferred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those 'trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that every- thing is accomplished by natural means; one is rather impressed by the multitude of skilful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures. HERBART, J. F. Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 101. 213. They [mathematicians] only take those things into consideration, of which they have clear and distinct ideas, designating them by proper, adequate, and invariable names, and premising only a few axioms which are most noted and certain to investigate their affections and draw conclusions from them, and agreeably laying down a very few hypotheses, such as are in the highest degree consonant with reason and not to be denied by anyone in his right mind. In like manner they assign generations or causes easy to be understood and readily ad- mitted by all, they preserve a most accurate order, every proposition immediately following from what is supposed and proved before, and reject all things howsoever specious and probable which can not be inferred and deduced after the same manner. BARROW, ISAAC. Mathematical Lectures (London, 1734), P- 66. 14 MEMORABILIA MATHEMATICA 214. The dexterous management of terms and being able to fend and prove with them, I know has and does pass in the world for a great part of learning; but it is learning distinct from knowledge, for knowledge consists only in perceiving the habitudes and relations of ideas one to another, which is done without words; the intervention of sounds helps nothing to it. And hence we see that there is least use of distinction where there is most knowledge: I mean in mathematics, where men have determined ideas with known names to them; and so, there being no room for equivocations, there is no need of dis- tinctions. LOCKE, JOHN. Conduct of the Understanding, Sect. 81. 215. In mathematics it [sophistry] had no place from the beginning: Mathematicians having had the wisdom to define accurately the terms they use, and to lay down, as axioms, the first principles on which their reasoning is grounded. Accord- ingly we find no parties among mathematicians, and hardly any disputes. REID, THOMAS. Essays on the Intellectual Powers of Man, Essay 1, chap. 1. 216. In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to the old structure. HANKEL, HERMANN. Die Entwickelung der Mathematik in den letzten J ahrhunderten (Tubingen, 1884), P- %5- 217. Mathematics, the priestess of definiteness and clear- ness. HERBART, J. F. Werke [Kehrbach] (Langensaha, 1890), Bd. 1, p. 171. 218. . . . mathematical analysis is co-extensive with nature itself, it defines all perceivable relations, measures times, spaces, forces, temperatures; it is a difficult science which forms but slowly, but preserves carefully every principle once acquired; it increases and becomes stronger incessantly amidst all the changes and errors of the human mind. THE NATURE OF MATHEMATICS 15 Its chief attribute is clearness; it has no means for expressing confused ideas. It compares the most diverse phenomena and discovers the secret analogies which unite them. If matter escapes us, as that of air and light because of its extreme tenuity, if bodies are placed far from us in the immensity of space, if man wishes to know the aspect of the heavens at successive periods separated by many centuries, if gravity and heat act in the interior of the solid earth at depths which will forever be inaccessible, mathematical analysis is still able to trace the laws of these phenomena. It renders them present and measur- able, and appears to be the faculty of the human mind destined to supplement the brevity of life and the imperfection of the senses, and what is even more remarkable, it follows the same course in the study of all phenomena; it explains them in the same language, as if in witness to the unity and simplicity of the plan of the universe, and to make more manifest the unchange- able order which presides over all natural causes. FOURIER, J. Theorie Analytique de la Chaleur, Discours Preliminaire. 219. Let us now declare the means whereby our understand- ing can rise to knowledge without fear of error. There are two such means: intuition and deduction. By intuition I mean not the varying testimony of the senses, nor the deductive judgment of imagination naturally extravagant, but the conception of an attentive mind so distinct and so clear that no doubt remains to it with regard to that which it comprehends; or, what amounts to the same thing, the self-evidencing conception of a sound and attentive mind, a conception which springs from the light of reason alone, and is more certain, because more simple, than deduction itself. . . . It may perhaps be asked why to intuition we add this other mode of knowing, by deduction, that is to say, the process which, from something of which we have certain knowledge, draws consequences which necessarily follow therefrom. But we are obliged to admit this second step; for there are a great many things which, without being evident of themselves, never- theless bear the marks of certainty if only they are deduced from true and incontestable principles by a continuous and uninter- 16 MEMORABILIA MATHEMATICA rupted movement of thought, with distinct intuition of each thing; just as we know that the last link of a long chain holds to the first, although we can not take in with one glance of the eye the intermediate links, provided that, after having run over them in succession, we can recall them all, each as being joined to its fellows, from the first up to the last. Thus we distinguish intuition from deduction, inasmuch as in the latter case there is conceived a certain progress or succession, while it is not so in the former; . . . whence it follows that primary propositions, derived immediately from principles, may be said to be known, according to the way we view them, now by intuition, now by deduction; although the principles themselves can be known only by intuition, the remote consequences only by deduction. DESCARTES. Rules for the Direction of the Mind, Philosophy of D. [Torrey] (New York, 1892), pp. 64, 65. 220. Analysis and natural philosophy owe their most impor- tant discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity. LAPLACE. A Philosophical Essay on Probabilities [Tru- scott and Emory] (New York 1902), p. 176. 221. There is in every step of an arithmetical or algebraical calculation a real induction, a real inference from facts to facts, and what disguises the induction is simply its comprehensive nature, and the consequent extreme generality of its language. MILL, J. S. System of Logic, Bk. 2, chap. 6, 2. 222. It would appear that Deductive and Demonstrative Sciences are all, without exception, Inductive Sciences: that their evidence is that of experience, but that they are also, in virtue of the peculiar character of one indispensable portion of the general formulae according to which their inductions are made, Hypothetical Sciences. Their conclusions are true only upon certain suppositions, which are, or ought to be, approxi- mations to the truth, but are seldom, if ever, exactly true; and THE NATURE OF MATHEMATICS 17 to this hypothetical character is to be ascribed the peculiar certainty, which is supposed to be inherent in demonstration. MILL, J. S. System of Logic, Bk. 2, chap. 6, 1. 223. The peculiar character of mathematical truth is, that it is necessarily and inevitably true; and one of the most im- portant lessons which we learn from our mathematical studies is a knowledge that there are such truths, and a familiarity with their form and character. This lesson is not only lost, but read backward, if the student is taught that there is no such difference, and that mathematical truths themselves are learned by experience. WHEWELL, W. Thoughts on the Study of Mathematics. Prin- ciples of English University Education (London, 1838). 224. These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assum- ing a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects. WHEWELL, W. The Philosophy of the Inductive Sciences, Part 1, Bk. 2, chap. 1, sect. 2 (London, 1858). 225. The apodictic quality of mathematical thought, the certainty and correctness of its conclusions, are due, not to a special mode of ratiocination, but to the character of the con- cepts with which it deals. What is that distinctive charac- teristic? I answer: precision, sharpness, completeness* of definition. But how comes your mathematician by such com- pleteness? There is no mysterious trick involved; some ideas admit of such precision, others do not; and the mathematician is one who deals with those that do. KEYSER, C. J. The Universe and Beyond; Hibbert Journal, Vol. 3 (1904-1905), p. 809. 226. The reasoning of mathematicians is founded on certain and infallible principles. Every word they use conveys a deter- * i. e., in terms of the absolutely clear and indefinable. 18 MEMORABILIA MATHEMATICA minate idea, and by accurate definitions they excite the same ideas in the mind of the reader that were in the mind of the writer. When they have defined the terms they intend to make use of, they premise a few axioms, or self-evident principles, that every one must assent to as soon as proposed. They then take for granted certain postulates, that no one can deny them, such as, that a right line may be drawn from any given point to another, and from these plain, simple principles they have raised most astonishing speculations, and proved the extent of the human mind to be more spacious and capacious than any other science. ADAMS, JOHN. Diary, Works (Boston, 1850), Vol. 2, p. 21. 227. It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a court of justice, without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel. BARROW, ISAAC. Mathematical Lectures (London, 1734), p. 64- 228. What is exact about mathematics but exactness? And is not this a consequence of the inner sense of truth? GOETHE. Spruche in Prosa, Natur, 6, 948. 229. . . . the three positive characteristics that distinguish mathematical knowledge from other knowledge . . . may be briefly expressed as follows: first, mathematical knowledge bears more distinctly the imprint of truth on all its results than any other kind of knowledge; secondly, it is always a sure prelimi- THE NATURE OF MATHEMATICS 19 nary step to the attainment of other correct knowledge; thirdly, it has no need of other knowledge. SCHUBERT, H. Mathematical Essays and Recreations (Chicago, 1898), p. 35, 230. It is now necessary to indicate more definitely the reason why mathematics not only carries conviction in itself, but also transmits conviction to the objects to which it is applied. The reason is found, first of all, in the perfect precision with which the elementary mathematical concepts are determined; in this respect each science must look to its own salvation .... But this is not all. As soon as human thought attempts long chains of conclusions, or difficult matters generally, there arises not only the danger of error but also the suspicion of error, because since all details cannot be surveyed with clearness at the same instant one must in the end be satisfied with a belief that nothing has been overlooked from the beginning. Every one knows how much this is the case even in arithmetic, the most elmenetary use of mathematics. No one would imagine that the higher parts of mathematics fare better in this respect; on the con- trary, in more complicated conclusions the uncertainty and suspicion of hidden errors increases in rapid progression. How does mathematics manage to rid itself of this inconvenience which attaches to it in the highest degree? By making proofs more rigorous? By giving new rules according to which the old rules shall be applied? Not in the least. A very great un- certainty continues to attach to the result of each single com- putation. But there are checks. In the realm of mathematics each point may be reached by a hundred different ways; and if each of a hundred ways leads to the same point, one may be sure that the right point has been reached. A calculation without a check is as good as none. Just so it is with every isolated proof in any speculative science whatever; the proof may be ever so ingenious, and ever so perfectly true and correct, it will still fail to convince permanently. He will therefore be much deceived, who, in metaphysics, or in psychology which depends on meta- physics, hopes to see his greatest care in the precise determina- tion of the concepts and in the logical conclusions rewarded by conviction, much less by success in transmitting conviction to 20 MEMORABILIA MATHEMATICA others. Not only must the conclusions support each other, without coercion or suspicion of subreption, but in all matters originating in experience, or judging concerning experience, the results of speculation must be verified by experience, not only superficially, but in countless special cases. HERBAHT, J. F. Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 105. 231. [In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general proposi- tions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity. HELMHOLTZ, H. Ueber das Verhaltniss der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vortrdge und Reden, Bd. 1 (1896), p. 176. 232. It is true that mathematics, owing to the fact that its whole content is built up by means of purely logical deduction from a small number of universally comprehended principles, has not unfittingly been designated as the science of the self- evident [Selbstverstandlichen]. Experience however, shows that for the majority of the cultured, even of scientists, mathematics remains the science of the incomprehensible [Unversta'ndlichen]. PRINGSHEIM, ALFRED. Ueber Wert und angeblichen Unwert der Mathematik, Jahresbericht der Deutschen Mathematiker Vereinigung (1904), P- 357. 233. Mathematical reasoning is deductive in the sense that it is based upon definitions which, as far as the validity of the reasoning is concerned (apart from any existential import), needs only the test of self-consistency. Thus no external verification of definitions is required in mathematics, as long as it is considered merely as mathematics. WHITEHEAD, A. N. Universal Algebra (Cambridge, 1898), Pref- ace, p. vi. THE NATURE OF MATHEMATICS 21 234. The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demon- strative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowl- edge. REID, THOMAS. Essays on the Intellectual Powers of Man, Essay 1, chap. 8. 235. ... for the saving the long progression of the thoughts to remote and first principles in every case, the mind should provide itself several stages; that is to say, intermediate prin- ciples, which it might have recourse to in the examining those positions that come in its way. These, though they are not self-evident principles, yet, if they have been made out from them by a wary and unquestionable deduction, may be de- pended on as certain and infallible truths, and serve as unques- tionable truths to prove other points depending upon them, by a nearer and shorter view than remote and general maxims. . . . And thus mathematicians do, who do not in every new problem run it back to the first axioms through all the whole train of intermediate propositions. Certain theorems that they have settled to themselves upon sure demonstration, serve to resolve to them multitudes of propositions which depend on them, and are as firmly made out from thence as if the mind went afresh over every link of the whole chain that tie them to first self- evident principles. LOCKE, JOHN. The Conduct of the Understanding, Sect. 21 . 236. Those intervening ideas, which serve to show the agree- ment of any two others, are called proofs; and where the agree- ment or disagreement is by this means plainly and clearly perceived, it is called demonstration; it being shown to the understanding, and the mind made to see that it is so. A quick- ness in the mind to find out these intermediate ideas, (that shall discover the agreement or disagreement of any other) and to apply them right, is, I suppose, that which is called sagacity. LOCKE, JOHN. An Essay concerning Human Understanding, Bk. 6, chaps. 2, 3. 22 MEMORABILIA MATHEMATICA 237. . . . the speculative propositions of mathematics do not relate to facts; ... all that we are convinced of by any demon- stration in the science, is of a necessary connection subsisting between certain suppositions and certain conclusions. When we find these suppositions actually take place in a particular instance, the demonstration forces us to apply the conclusion. Thus, if I could form a triangle, the three sides of which were accurately mathematical lines, I might affirm of this individual figure, that its three angles are equal to two right angles; but, as the imperfection of my senses puts it out of my power to be, in any case, certain of the exact correspondence of the diagram which I delineate, with the definitions given in the elements of geometry, I never can apply with confidence to a particular figure, a mathematical theorem. On the other hand, it appears from the daily testimony of our senses that the speculative truths of geometry may be applied to material objects with a degree of accuracy sufficient for the purposes of life; and from such applications of them, advantages of the most important kind have been gained to society. STEWART, DUGALD. Elements of the Philosophy of the Human Mind, Part 8, chap. 1, sect. 8. 238. No process of sound reasoning can establish a result not contained in the premises. MELLOR, J. W. Higher Mathematics for Students of Chemistry and Physics (New York, 1902), p. 2. 239. ... we cannot get more out of the mathematical mill than we put into it, though we may get it in a form infinitely more useful for our purpose. HOPKINSON, JOHN. James Forrest Lecture, 1894- 240. The iron labor of conscious logical reasoning demands great perseverance and great caution; it moves on but slowly, and is rarely illuminated by brilliant flashes of genius. It knows little of that facility with which the most varied instances come thronging into the memory of the philologist or historian. Rather is it an essential condition of the methodical progress of mathematical reasoning that the mind should remain concen- THE NATURE OF MATHEMATICS 23 trated on a single point, undisturbed alike by collateral ideas on the one hand, and by wishes and hopes on the other, and moving on steadily in the direction it has deliberately chosen. HELMHOLTZ, H. Ueber das Verhdltniss der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vortrdge und Reden, Bd. 1 (1896), p. 178. 241. If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the inter- vening propositions. And there is art also in this; for as the mediate truths (which are called lemmas, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the understanding and memory, to choose of them those which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another obstacle, viz. : that it is not easy to demonstrate all the axioms, and to reduce demonstrations wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry. LEIBNITZ, G. W. New Essay on Human Understanding [Lang- ley], Bk. 4, chaps. 2, 8. 242. In Pure Mathematics, where all the various truths are necessarily connected with each other, (being all necessarily connected with those hypotheses which are the principles of the science), an arrangement is beautiful in proportion as the principles are few; and what we admire perhaps chiefly in the science, is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises. STEWART, DUGALD. The Elements of the Philosophy of the Human Mind, Part 3, chap. 1, sect. 3. 243. Whenever ... a controversy arises in mathematics, the issue is not whether a thing is true or not, but whether the proof might not be conducted more simply in some other way, or whether the proposition demonstrated is sufficiently important 24 MEMORABILIA MATHEMATICA for the advancement of the science as to deserve especial enunciation and emphasis, or finally, whether the proposition is not a special case of some other and more general truth which is as easily discovered. SCHUBERT, H. Mathematical Essays and Recreations (Chicago, 1898), p. 28. 244. . . . just as the astronomer, the physicist, the geologist, or other student of objective science looks about in the world of sense, so, not metaphorically speaking but literally, the mind of the mathematician goes forth in the universe of logic in quest of the things that are there; exploring the heights and depths for facts ideas, classes, relationships, implications, and the rest; observing the minute and elusive with the powerful microscope of his Infinitesimal Analysis; observing the elusive and vast with the limitless telescope of his Calculus of the Infinite; making guesses regarding the order and internal harmony of the data observed and collocated; testing the hypotheses, not merely by the complete induction peculiar to mathematics, but, like his colleagues of the outer world, resort- ing also to experimental tests and incomplete induction; fre- quently finding it necessary, in view of unforeseen disclosures, to abandon one hopeful hypothesis or to transform it by retrench- ment or by enlargement: thus, in his own domain, matching, point for point, the processes, methods and experience familiar to the devotee of natural science. KEYSER, CASSIUS, J. Lectures on Science, Philosophy and Art (New York, 1908), p. 26. 245. That mathematics "do not cultivate the power of generalization," . . . will be admitted by no person of compe- tent knowledge, except in a very qualified sense. The generaliza- tions of mathematics, are, no doubt, a different thing from the generalizations of physical science; but in the difficulty of seizing them, and the mental tension they require, they are no con- temptible preparation for the most arduous efforts of the scientific mind. Even the fundamental notions of the higher mathematics, from those of the differential calculus upwards are products of a very high abstraction. ... To perceive the mathematical laws common to the results of many mathematical THE NATURE OF MATHEMATICS 25 operations, even in so simple a case as that of the binomial theorem, involves a vigorous exercise of the same faculty which gave us Kepler's laws, and rose through those laws to the theory of universal gravitation. Every process of what has been called Universal Geometry the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diver- sity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics. MILL, JOHN STUART. An Examination of Sir William Hamilton's Philosophy (London, 1878), pp. 612, 613. 246. When the greatest of American logicians, speaking of the powers that constitute the born geometrician, had named Conception, Imagination, and Generalization, he paused. Thereupon from one of the audience there came the challenge, "What of reason?" The instant response, not less just than brilliant, was: "Ratiocination that is but the smooth pave- ment on which the chariot rolls." KEYSER, C. J. Lectures on Science, Philosophy and Art (New York, 1908), p. 31. 247. . . . the reasoning process [employed in mathematics] is not different from that of any other branch of knowledge, . . . but there is required, and in a great degree, that attention of mind which is in some part necessary for the acquisition of all knowledge, and in this branch is indispensably necessary. This must be given in its fullest intensity; . . . the other elements especially characteristic of a mathematical mind are quickness 26 MEMORABILIA MATHEMATICA in perceiving logical sequence, love of order, methodical arrange- ment and harmony, distinctness of conception. PRICE, B. Treatise on Infinitesimal Calculus (Oxford, 1868), Vol. 3, p. 6. 248. Histories make men wise; poets, witty; the mathematics, subtile; natural philosophy, deep; moral, grave; logic and rheto- ric, able to contend. BACON, FRANCIS. Essays, Of Studies. 249. The Mathematician deals with two properties of objects only, number and extension, and all the inductions he wants have been formed and finished ages ago. He is now occupied with nothing but deduction and verification. HUXLEY, T. H. On the Educational Value of the Natural His- tory Sciences; Lay Sermons, Addresses and Reviews; (New York, 1872), p. 87. 250. [Mathematics] is that [subject] which knows nothing of observation, nothing of experiment, nothing of induction, noth- ing of causation. HUXLEY, T. H. The Scientific Aspects of Positivism, Fortnightly Review (1898); Lay Sermons, Addresses and Reviews, (New York, 1872), p. 169. 251. We are told that "Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation." I think no statement could have been made more opposite to the facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas, and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activities of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close atten- tion to discern as those of the outer physical world (to which the inner one in each individual man may, I think, be conceived to stand somewhat in the same relation of correspondence as a shadow to the object from which it is projected, or as the hollow palm of one hand to the closed fist which it grasps of the other), that it is unceasingly calling forth the faculties of observation THE NATURE OF MATHEMATICS 27 and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verifica- tion, and that it affords a boundless scope for the exercise of the highest efforts of the imagination and invention. SYLVESTER, J. J. Presidential Address to British Association, Exeter British Association Report (1869), pp. 1-9.; Collected Mathematical Papers, Vol. 2, p. 654. 252. The actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences; obser- vation, comparison, classification, trial, and generalisation are essential in both cases. Not only are special results, obtained independently of one another, frequently seen to be really in- cluded in some generalisation, but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthesised in one single body of doctrine. The essential nature of mathematical thought manifests itself in the dis- cernment of fundamental identity in the mathematical aspects of what are superficially very different domains. A striking example of this species of immanent identity of mathematical form was exhibited by the discovery of that distinguished mathematician . . . Major MacMahon, that all possible Latin squares are capable of enumeration by the consideration of certain differential operators. Here we have a case in which an enumeration, which appears to be not amenable to direct treat- ment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares. HOBSON, E. W. Presidential Address British Association for the Advancement of Science (1910); Nature, Vol. 84, p. 290. 253. It has been asserted . . . that the power of observation is not developed by mathematical studies; while the truth is, 28 MEMORABILIA MATHEMATICA that; from the most elementary mathematical notion that arises in the mind of a child to the farthest verge to which mathematical investigation has been pushed and applied, this power is in constant exercise. By observation, as here used, can only be meant the fixing of the attention upon objects (physical or mental) so as to note distinctive peculiarities to recognize resemblances, differences, and other relations. Now the first mental act of the child recognizing the distinction between one and more than one, between one and two, two and three, etc., is exactly this. So, again, the first geometrical notions are as pure an exercise of this power as can be given. To know a straight line, to distinguish it from a curve; to recognize a triangle and distinguish the several forms what are these, and all perception of form, but a series of observations? Nor is it alone in securing these fundamental conceptions of number and form that observation plays so important a part. The very genius of the common geometry as a method of reasoning a system of investigation is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds. So characteristic of common geometry is this method of investigation, that Comte, perhaps the ablest of all writers upon the philosophy of mathematics, is disposed to class geometry, as to its method, with the natural sciences, being based upon observation. Moreover, when we consider applied mathematics, we need only to notice that the exercise of this faculty is so essential, that the basis of all such reasoning, the very material with which we build, have received the name observations. Thus we might proceed to consider the whole range of the human faculties, and find for the most of them ample scope for exercise in mathematical studies. Certainly, the memory will not be found to be neglected. The very first steps in number counting, the multiplication table, etc., make heavy demands on this power; while the higher branches require the memorizing of formulas which are simply appalling to the uninitiated. So the imagination, the creative faculty of the THE NATURE OF MATHEMATICS 29 mind, has constant exercise in all original mathematical in- vestigations, from the solution of the simplest problems to the discovery of the most recondite principle; for it is not by sure, consecutive steps, as many suppose, that we advance from the known to the unknown. The imagination, not the logical faculty, leads in this advance. In fact, practical observation is often in advance of logical exposition. Thus, in the discovery of truth, the imagination habitually presents hypotheses, and observation supplies facts, which it may require ages for the tardy reason to connect logically with the known. Of this truth, mathematics, as well as all other sciences, affords abun- dant illustrations. So remarkably true is this, that today it is seriously questioned by the majority of thinkers, whether the sublimest branch of mathematics, the infinitesimal calculus has anything more than an empirical foundation, mathemati- cians themselves not being agreed as to its logical basis. That the imagination, and not the logical faculty, leads in all original investigation, no one who has ever succeeded in producing an original demonstration of one of the simpler propositions of geometry, can have any doubt. Nor are induction, analogy, the scrutinization of premises or the search for them, or the balancing of probabilities, spheres of mental operations foreign to mathe- matics. No one, indeed, can claim pre-eminence for mathe- matical studies in all these departments of intellectual culture, but it may, perhaps, be claimed that scarcely any department of science affords discipline to so great a number of faculties, and that none presents so complete a gradation in the exercise of these faculties, from the first principles of the science to the farthest extent of its applications, as mathematics. OLNEY, EDWARD. Kiddle and Schem's Encyclopedia of Education, (New York, 1877), Article "Mathematics." 254. The opinion appears to be gaining ground that this very general conception of functionality, born on mathematical ground, is destined to supersede the narrower notion of causa- tion, traditional in connection with the natural sciences. As an abstract formulation of the idea of determination in its most general sense, the notion of functionality includes and tran- 30 MEMORABILIA MATHEMATICA scends the more special notion of causation as a one-sided determination of future phenomena by means of present condi- tions; it can be used to express the fact of the subsumption under a general law of past, present, and future alike, in a sequence of phenomena. From this point of view the remark of Huxley that Mathematics "knows nothing of causation" could only be taken to express the whole truth, if by the term "causation" is understood "efficient causation." The latter notion has, how- ever, in recent times been to an increasing extent regarded as just as irrelevant in the natural sciences as it is in Mathematics; the idea of thorough-going determinancy, in accordance with formal law, being thought to be alone significant in either domain. HOBSON, E. W. Presidential Address British Association for the Advancement of Science (1910); Nature, Vol. 84, p. 290. 255. Most, if not all, of the great ideas of modern mathe- matics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss's transcendent genius; or the doctrine of double periodi- city, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre's law of reci- procity; or Sturm's theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendu- lums; or Huyghen's method of continued fractions, charac- terized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, "that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, alge- THE NATURE OF MATHEMATICS 31 braic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics" (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) "have all felt its influence." SYLVESTER, J. J. A Plea for the Mathematician, Nature, Vol. 1 , p. 238; Collected Mathematical Papers, Vol. 2, pp. 655, 656. 256. The ability to imagine relations is one of the most indispensable conditions of all precise thinking. No subject can be named, in the investigation of which it is not impera- tively needed; but it can be nowhere else so thoroughly acquired as in the study of mathematics. FISKE, JOHN. Darwinism and other Essays (Boston, 1893), p. 296. 257. The great science [mathematics] occupies itself at least just as much with the power of imagination as with the power of logical conclusion. HERBART, F. J. Pestalozzi's Idee eines ABC der Anschauung. Werke [Kehrbach] (Langensaltza, 1890), Bd.l, p. 174. 258. The moving power of mathematical invention is not reasoning but imagination. DE MORGAN, A. Quoted in Graves' Life of Sir W. R. Hamilton, Vol. 3 (1889), p. 219. 259. There is an astonishing imagination, even in the science of mathematics. . . . We repeat, there was far more imagina- tion in the head of Archimedes than in that of Homer. VOLTAIRE. A Philosophical Dictionary (Boston, 1881), Vol. 3, p. 40' Article "Imagination." 260. As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention. SYLVESTER, J. J. A Plea for the Mathematician, Nature, Vol. 1 , p. 261; Collected Mathematical Papers, Vol. 2 (Cambridge, 1908), p. 717. 32 MEMORABILIA MATHEMATICA 261. A marvellous newtrality have these things mathe- maticall, and also a strange participation between things supernaturall, immortall, intellectuall, simple and indivisible, and things naturall, mortall, sensible, componded and divisible. DEE, JOHN. Euclid (1570), Preface. 262. Mathematics stands forth as that which unites, mediates between Man and Nature, inner and outer world, thought and perception, as no other subject does. FROEBEL. [Herford translation] (London, 1893), Vol. 1, p. 84. 263. The intrinsic character of mathematical research and knowledge is based essentially on three properties: first, on its conservative attitude towards the old truths and discoveries of mathematics; secondly, on its progressive mode of development, due to the incessant acquisition of new knowledge on the basis of the old; and thirdly, on its self-sufficiency and its consequent absolute independence. SCHUBERT, H. Mathematical Essays and Recreations (Chicago, 1898), p. 27. 264. Our science, hi contrast with others, is not founded on a single period of human history, but has accompanied the devel- opment of culture through all its stages. Mathematics is as much interwoven with Greek culture as with the most modern problems in Engineering. She not only lends a hand to the progressive natural sciences but participates at the same time in the abstract investigations of logicians and philosophers. KLEIN, F. Klein und Riecke: Ueber angewandte Mathe- matik und Physik (1900), p. 228. 265. There is probably no other science which presents such different appearances to one who cultivates it and to one who does not, as mathematics. To this person it is ancient, venera- ble, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere THE NATURE OF MATHEMATICS 33 stretching out after the "attainable but unattained" and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other and branch off into innumerable by-paths that end in a wilderness. CHAPMAN, C. H. Bulletin American Mathematical Society, Vol. 2 (First series), p. 61. 266. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathe- matical knowledge, we are still clearly conscious of the simi- larity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected rela- tions are disclosed between hitherto separated branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but manifests itself the more clearly. HILBERT, D. Mathematical Problems, Bulletin American Mathematical Society, Vol. 8, p. 418. 267. The mathematics have always been the implacable enemies of scientific romances. ARAGO. Oeuvres (1855), t. 8, p. 498. - 268. Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression. COURNOT, AUGUSTIN. Mathematical Theory of the Principles of Wealth [Bacon, N. T.], (New York, 1897), p. 3. 269. Coterminous with space and coeval with time is the Kingdom of Mathematics; within this range her dominion is supreme; otherwise than according to her order nothing can exist; in contradiction to her laws nothing takes place. On her 34 MEMORABILIA MATHEMATICA mysterious scroll is to be found written for those who can read it that which has been, that which is, and that which is to come. Everything material which is the subject of knowledge has number, order, or position; and these are her first outlines for a sketch of the universe. If our feeble hands cannot follow out the details, still her part has been drawn with an unerring pen, and her work cannot be gainsaid. So wide is the range of mathe- matical sciences, so indefinitely may it extend beyond our actual powers of manipulation that at some moments we are inclined to fall down with even more than reverence before her majestic presence. But so strictly limited are her promises and powers, about so much that we might wish to know does she offer no information whatever, that at other moments we are fain to call her results but a vain thing, and to reject them as a stone where we had asked for bread. If one aspect of the sub- ject encourages our hopes, so does the other tend to chasten our desires, and he is perhaps the wisest, and in the long run the happiest, among his fellows, who has learned not only this science, but also the larger lesson which it directly teaches, namely, to temper our aspirations to that which is possible, to moderate our desires to that which is attainable, to restrict our hopes to that of which accomplishment, if not immediately practicable, is at least distinctly within the range of conception. SPOTTISWOODE, W. Quoted in Sonnenschein's Encyclopedia of Education (London, 1906), p. 208. 270. But it is precisely mathematics, and the pure science generally, from which the general educated public and independ- ent students have been debarred, and into which they have only rarely attained more than a very meagre insight. The reason of this is twofold. In the first place, the ascendant and consecu- tive character of mathematical knowledge renders its results absolutely insusceptible of presentation to persons who are unacquainted with what has gone before, and so necessitates on the part of its devotees a thorough and patient exploration of the field from the very beginning, as distinguished from those sciences which may, so to speak, be begun at the end, and which are consequently cultivated with the greatest zeal. The second THE NATURE OF MATHEMATICS 35 reason is that, partly through the exigencies of academic instruc- tion, but mainly through the martinet traditions of antiquity and the influence of mediaeval logic-mongers, the great bulk of the elementary text-books of mathematics have unconsciously assumed a very repellant form, something similar to what is termed in the theory of protective mimicry in biology "the terrifying form." And it is mainly to this formidableness and touch-me-not character of exterior, concealing withal a harmless body, that the undue neglect of typical mathematical studies is to be attributed. McCoRMACK, T. J. Preface to De Morgan's Elementary Illustra- tions of the Differential and Integral Calculus (Chicago, 1899). 271. Mathematics in gross, it is plain, are a grievance in natural philosophy, and with reason: for mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning. Mathematical proofs are out of the reach of topical arguments; and are not to be attacked by the equivocal use of words or declaration, that make so great a part of other discourses, nay, even of controversies. LOCKE, JOHN. Second Reply to the Bishop of Worcester. 272. The belief that mathematics, because it is abstract, be- cause it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. That such is the case a few indications will suffice to show. The mathematical concepts of constant and variable are represented familiarly in life by the notions of fixedness and change. The concept of equation or that of an equational 36 MEMORABILIA MATHEMATICA system, imposing restriction upon variability, is matched in life by the concept of natural and spiritual law, giving order to what were else chaotic change and providing partial freedom in lieu of none at all. What is known in mathematics under the name of limit is everywhere present in life in the guise of some ideal, some excellence high-dwelling among the rocks, an "ever flying perfect" as Emerson calls it, unto which we may approxi- mate nearer and nearer, but which we can never quite attain, save in aspiration. The supreme concept of functionality finds its correlate in life in the all-pervasive sense of interdependence and mutual determination among the elements of the world. What is known in mathematics as transformation that is, lawful transfer of attention, serving to match in orderly fashion the things of one system with those of another is conceived in life as a process of transmutation by which, in the flux of the world, .the content of the present has come out of the past and in its turn, in ceasing to be, gives birth to its successor, as the boy is father to the man and as things, in general, become what they are not. The mathematical concept of invariance and that of infinitude, especially the imposing doctrines that explain their meanings and bear their names What are they but mathematicizations of that which has ever been the chief of life's hopes and dreams, of that which has ever been the object of its deepest passion and of its dominant enterprise, I mean the finding of the worth that abides, the finding of permanence in the midst of change, and the discovery of a presence, in what has seemed to be a finite world, of being that is infinite? It is need- less further to multiply examples of a correlation that is so abounding and complete as indeed to suggest a doubt whether it be juster to view mathematics as the abstract idealization of life than to regard life as the concrete realization of mathematics. KEYSER, C. J. The Humanization of the Teaching of Mathe- matics; Science, Neiv Series, Vol. 85, pp. 643- 646. 273. Mathematics, like dialectics, is an organ of the inner higher sense; hi its execution it is an art like eloquence. Both alike care nothing for the content, to both nothing is of value but the form. It is immaterial to mathematics whether it THE NATURE OF MATHEMATICS 37 computes pennies or guineas, to rhetoric whether it defends truth or error. GOETHE. Wilhelm Meisters Wanderjahre, Zweites Buck. 274. The genuine spirit of Mathesis is devout. No intellec- tual pursuit more truly leads to profound impressions of the existence and attributes of a Creator, and to a deep sense of our filial relations to him, than the study of these abstract sciences. Who can understand so well how feeble are our conceptions of Almighty Power, as he who has calculated the attraction of the sun and the planets, and weighed in his balance the irresistible force of the lightning? Who can so well under- stand how confused is our estimate of the Eternal Wisdom, as he who has traced out the secret laws which guide the hosts of heaven, and combine the atoms on earth? Who can so well understand that man is made in the image of his Creator, as he who has sought to frame new laws and conditions to govern imaginary worlds, and found his own thoughts similar to those on which his Creator has acted? HILL, THOMAS. The Imagination in Mathematics; North American Review, Vol. 85, p. 226. 275. . . . what is physical is subject to the laws of mathe- matics, and what is spiritual to the laws of God, and the laws of mathematics are but the expression of the thoughts of God. HILL, THOMAS. The Uses of Mathesis; Bibliotheca Sacra, Vol. 82, p. 523. 276. It is in the inner world of pure thought, where all entia dwell, where is every type of order and manner of correlation and variety of relationship, it is in this infinite ensemble of eternal verities whence, if there be one cosmos or many of them, each derives its character and mode of being, it is there that the spirit of Diathesis has its home and its life. Is it a restricted home, a narrow life, static and cold and grey with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of solar light, not clad in the colours that liven and glorify the things of sense, but it is an illuminated world, and over it all and every- 38 MEMORABILIA MATHEMATICA where throughout are hues and tints transcending sense, painted there by radiant pencils of psychic light, the light in which it lies. It is a silent world, and, nevertheless, in respect to the highest principle of art the interpenetration of content and form, the perfect fusion of mode and meaning it even sur- passes music. In a sense, it is a static world, but so, too, are the worlds of the sculptor and the architect. The figures, how- ever, which reason constructs and the mathematic vision be- holds, transcend the temple and the statue, alike in simplicity and in intricacy, in delicacy and in grace, in symmetry and in poise. Not only are this home and this life thus rich in aesthetic interests, really controlled and sustained by motives of a sub- limed and supersensuous art, but the religious aspiration, too, finds there, especially in the beautiful doctrine of invariants, the most perfect symbols of what it seeks the changeless in the midst of change, abiding things in a world of flux, con- figurations that remain the same despite the swirl and stress of countless hosts of curious transformations. The domain of mathematics is the sole domain of certainty. There and there alone prevail the standards by which every hypothesis respect- ing the external universe and all observation and all experiment must be finally judged. It is the realm to which all speculation and all thought must repair for chastening and sanitation the court of last resort, I say it reverently, for all intellection whatsoever, whether of demon or man or deity. It is there that mind as mind attains its highest estate, and the condition of knowledge there is the ultimate object, the tantalising goal of the aspiration, the Anders-Streben, of all other knowledge of every kind. KEYSER, C. J. The Universe and Beyond; Hibbert Journal, Vol. 3 (1904-1905), pp. 813-314. CHAPTER III ESTIMATES OF MATHEMATICS 301. The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of the truths with which mathematical science is concerned, these, and such like, are the surest grounds of its title of human regard, and would remain unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance. SYLVESTER, J. J. A Plea for the Mathematician, Nature, 1, p. 262; Collected Mathematical Papers (Cam- bridge, 1908), 2, p. 659. 302. It may well be doubted whether, in all the range of Science, there is any field so fascinating to the explorer so rich in hidden treasures so fruitful in delightful surprises as that of Pure Mathematics. The charm lies chiefly ... in the absolute certainty of its results: for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light! 'Ei> 8e