CHAPTER XIX

THE CALCULUS AND ALLIED TOPICS

=1901.= It may be said that the conceptions of differential quotient and integral, which in their origin certainly go back to Archimedes, were introduced into science by the investigations of Kepler, Descartes, Cavalieri, Fermat and Wallis.... The capital discovery that differentiation and integration are _inverse_ operations belongs to Newton and Leibnitz.--LIE, SOPHUS.

_Leipziger Berichte, 47 (1895), Math.-phys. Classe, p. 53._

=1902.= It appears that Fermat, the true inventor of the differential calculus, considered that calculus as derived from the calculus of finite differences by neglecting infinitesimals of higher orders as compared with those of a lower order.... Newton, through his method of fluxions, has since rendered the calculus more analytical, he also simplified and generalized the method by the invention of his binomial theorem. Leibnitz has enriched the differential calculus by a very happy notation.

--LAPLACE.

_Lés Intégrales Définies, etc.; Oeuvres, t. 12 (Paris, 1898), p. 359._

=1903.= Professor Peacock’s Algebra, and Mr. Whewell’s Doctrine of Limits should be studied by every one who desires to comprehend the evidence of mathematical truths, and the meaning of the obscure processes of the calculus; while, even after mastering these treatises, the student will have much to learn on the subject from M. Comte, of whose admirable work one of the most admirable portions is that in which he may truly be said to have created the philosophy of the higher mathematics.

--MILL, J. S.

_System of Logic, Bk. 3, chap. 24, sect. 6._

=1904.= If we must confine ourselves to one system of notation then there can be no doubt that that which was invented by Leibnitz is better fitted for most of the purposes to which the infinitesimal calculus is applied than that of fluxions, and for some (such as the calculus of variations) it is indeed almost essential.--BALL, W. W. R.

_History of Mathematics (London, 1901), p. 371._

=1905.= The difference between the method of infinitesimals and that of limits (when exclusively adopted) is, that in the latter it is usual to retain evanescent quantities of higher orders until the end of the calculation and then neglect them. On the other hand, such quantities are neglected from the commencement in the infinitesimal method, from the conviction that they cannot affect the final result, as they must disappear when we proceed to the limit.--WILLIAMSON, B.

_Encyclopedia Britannica, 9th Edition; Article “Infinitesimal Calculus,” sect. 14._

=1906.= When we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs.--LAGRANGE.

_Méchanique Analytique, Preface; Oeuvres, t. 2 (Paris, 1888), p. 14._

=1907.= The essential merit, the sublimity, of the infinitesimal method lies in the fact that it is as easily performed as the simplest method of approximation, and that it is as accurate as the results of an ordinary calculation. This advantage would be lost, or at least greatly impaired, if, under the pretense of securing greater accuracy throughout the whole process, we were to substitute for the simpler method given by Leibnitz, one less convenient and less in harmony with the probable course of natural events....

The objections which have been raised against the infinitesimal method are based on the false supposition that the errors due to neglecting infinitely small quantities during the actual calculation will continue to exist in the result of the calculation.--CARNOT, L.

_Réflections sur la Métaphysique du Calcul Infinitésimal (Paris, 1813), p. 215._

=1908.= A limiting ratio is neither more nor less difficult to define than an infinitely small quantity.--CARNOT, L.

_Réflections sur la Métaphysique du Calcul Infinitésimal (Paris, 1813), p. 210._

=1909.= A limit is a peculiar and fundamental conception, the use of which in proving the propositions of Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. The axiom just noted that what is true up to the limit is true at the limit, is involved in the very conception of a limit: and this principle, with its consequences, leads to all the results which form the subject of the higher mathematics, whether proved by the consideration of evanescent triangles, by the processes of the Differential Calculus, or in any other way.

--WHEWELL, W.

_The Philosophy of the Inductive Sciences, Part 1, bk. 2, chap. 12, sect. 1, (London, 1858)._

=1910.= The differential calculus has all the exactitude of other algebraic operations.--LAPLACE.

_Théorie Analytique des Probabilités, Introduction; Oeuvres, t. 7 (Paris, 1886), p. 37._

=1911.= The method of fluxions is probably one of the greatest, most subtle, and sublime discoveries of any age: it opens a new world to our view, and extends our knowledge, as it were, to infinity; carrying us beyond the bounds that seemed to have been prescribed to the human mind, at least infinitely beyond those to which the ancient geometry was confined.--HUTTON, CHARLES.

_A Philosophical and Mathematical Dictionary (London, 1815), Vol. 1, p. 525._

=1912.= The states and conditions of matter, as they occur in nature, are in a state of perpetual flux, and these qualities may be effectively studied by the Newtonian method (Methodus fluxionem) whenever they can be referred to number or subjected to measurement (real or imaginary). By the aid of Newton’s calculus the mode of action of natural changes from moment to moment can be portrayed as faithfully as these words represent the thoughts at present in my mind. From this, the law which controls the whole process can be determined with unmistakable certainty by pure calculation.--MELLOR, J. W.

_Higher Mathematics for Students of Chemistry and Physics (London, 1902), Prologue._

=1913.= The calculus is the greatest aid we have to the appreciation of physical truth in the broadest sense of the word.

--OSGOOD, W. F.

_Bulletin American Mathematical Society, Vol. 13 (1907), p. 467._

=1914.= [Infinitesimal] analysis is the most powerful weapon of thought yet devised by the wit of man.--SMITH, W. B.

_Infinitesimal Analysis (New York, 1898), Preface, p. vii._

=1915.= The method of Fluxions is the general key by help whereof the modern mathematicians unlock the secrets of Geometry, and consequently of Nature. And, as it is that which hath enabled them so remarkably to outgo the ancients in discovering theorems and solving problems, the exercise and application thereof is become the main if not sole employment of all those who in this age pass for profound geometers.--BERKELEY, GEORGE.

_The Analyst, sect. 3._

=1916.= I have at last become fully satisfied that the language and idea of infinitesimals should be used in the most elementary instruction--under all safeguards of course.--DE MORGAN, A.

_Graves’ Life of W. R. Hamilton (New York, 1882-1889), Vol. 3, p. 479._

=1917.= Pupils should be taught how to differentiate and how to integrate simple algebraic expressions before we attempt to teach them geometry and these other complicated things. The dreadful fear of the symbols is entirely broken down in those cases where at the beginning the teaching of the calculus is adopted. Then after the pupil has mastered those symbols you may begin geometry or anything you please. I would also abolish out of the school that thing called geometrical conics. There is a great deal of superstition about conic sections. The student should be taught the symbols of the calculus and the simplest use of these symbols at the earliest age, instead of these being left over until he has gone to the College or University.

--THOMPSON, S. P.

_Perry’s Teaching of Mathematics (London, 1902), p. 49._

=1918.= Every one versed in the matter will agree that even the elements of a scientific study of nature can be understood only by those who have a knowledge of at least the elements of the differential and integral calculus, as well as of analytical geometry--i.e. the so-called lower part of the higher mathematics.... We should raise the question, whether sufficient time could not be reserved in the curricula of at least the science high schools [Realanstalten] to make room for these subjects....

The first consideration would be to entirely relieve from the mathematical requirements of the university [Hochschule] certain classes of students who can get along without extended mathematical knowledge, or to make the necessary mathematical knowledge accessible to them in a manner which, for various reasons, has not yet been adopted by the university. Among such students I would count architects, also the chemists and in general the students of the so-called descriptive natural sciences. I am moreover of the opinion--and this has been for long a favorite idea of mine--, that it would be very useful to medical students to acquire such mathematical knowledge as is indicated by the above described modest limits; for it seems impossible to understand far-reaching physiological investigations, if one is terrified as soon as a differential or integration symbol appears.--KLEIN, F.

_Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 2 (1902), p. 131._

=1919.= Common integration is only the _memory of differentiation_ ... the different artifices by which integration is effected, are changes, not from the known to the unknown, but from forms in which memory will not serve us to those in which it will.--DE MORGAN, A.

_Transactions Cambridge Philosophical Society, Vol. 8 (1844), p. 188._

=1920.= Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom: to it nothing would be uncertain, and the future as the past would be present to its eyes. The human mind offers a feeble outline of that intelligence, in the perfection which it has given to astronomy. Its discoveries in mechanics and in geometry, joined to that of universal gravity, have enabled it to comprehend in the same analytical expressions the past and future states of the world system.--LAPLACE.

_Théorie Analytique des Probabilités, Introduction; Oeuvres, t. 7 (Paris, 1886), p. 6._

=1921.= There is perhaps the same relation between the action of natural selection during one generation and the accumulated result of a hundred thousand generations, that there exists between differential and integral. How seldom are we able to follow completely this latter relation although we subject it to calculation. Do we on that account doubt the correctness of our integrations?--BOIS-REYMOND, EMIL DU.

_Reden, Bd. 1 (Leipzig, 1885), p. 228._

=1922.= It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock.--SYLVESTER, J. J.

_Notes to the Meditation on Poncelet’s Theorem; Mathematical Papers, Vol. 2, p. 214._

=1923.= The experimental verification of a theory concerning any natural phenomenon generally rests on the result of an integration.--MELLOR, J. W.

_Higher Mathematics for Students of Chemistry and Physics (New York, 1902), p. 150._

=1924.= Among all the mathematical disciplines the theory of differential equations is the most important.... It furnishes the explanation of all those elementary manifestations of nature which involve time....--LIE, SOPHUS.

_Leipziger Berichte, 47 (1895); Math.-phys. Classe, p. 262._

=1925.= If the mathematical expression of our ideas leads to equations which cannot be integrated, the working hypothesis will either have to be verified some other way, or else relegated to the great repository of unverified speculations.--MELLOR, J. W.

_Higher Mathematics for Students of Chemistry and Physics (New York, 1902), p. 157._

=1926.= It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations.--KLEIN, F.

_Lectures on Mathematics (New York, 1911), p. 8._

=1927.= Every one knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.... A curve is the totality of points, whose co-ordinates are functions of a parameter which may be differentiated as often as may be required.--KLEIN, F.

_Elementar Mathematik vom höheren Standpunkte aus. (Leipzig. 1909) Vol. 2, p. 354._

=1928.= Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along telegraph wires, and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.--THOMSON AND TAIT.

_Elements of Natural Philosophy, chap. 1._

=1929.= The principal advantage arising from the use of hyperbolic functions is that they bring to light some curious analogies between the integrals of certain irrational functions.

--BYERLY, W. E.

_Integral Calculus (Boston, 1890), p. 30._

=1930.= Hyperbolic functions are extremely useful in every branch of pure physics and in the applications of physics whether to observational and experimental sciences or to technology. Thus whenever an entity (such as light, velocity, electricity, or radio-activity) is subject to gradual absorption or extinction, the decay is represented by some form of hyperbolic functions. Mercator’s projection is likewise computed by hyperbolic functions. Whenever mechanical strains are regarded great enough to be measured they are most simply expressed in terms of hyperbolic functions. Hence geological deformations invariably lead to such expressions....--WALCOTT, C. D.

_Smithsonian Mathematical Tables, Hyperbolic Functions (Washington, 1909), Advertisement._

=1931.= Geometry may sometimes appear to take the lead over analysis, but in fact precedes it only as a servant goes before his master to clear the path and light him on the way. The interval between the two is as wide as between empiricism and science, as between the understanding and the reason, or as between the finite and the infinite.--SYLVESTER, J. J.

_Philosophic Magazine, Vol. 31 (1866), p. 521._

=1932.= Nature herself exhibits to us measurable and observable quantities in definite mathematical dependence; the conception of a function is suggested by all the processes of nature where we observe natural phenomena varying according to distance or to time. Nearly all the “known” functions have presented themselves in the attempt to solve geometrical, mechanical, or physical problems.--MERZ, J. T.

_A History of European Thought in the Nineteenth Century (Edinburgh and London, 1903), p. 696._

=1933.= That flower of modern mathematical thought--the notion of a function.--MCCORMACK, THOMAS J.

_On the Nature of Scientific Law and Scientific Explanation, Monist, Vol. 10 (1899-1900), p. 555._

=1934.=

Fuchs. Ich bin von alledem so consterniert, Als würde mir ein Kreis im Kopfe quadriert.

Meph. Nachher vor alien andern Sachen Müsst ihe euch an die Funktionen-Theorie machen. Da seht, dass ihr tiefsinnig fasst, Was sich zu integrieren nicht passt. An Theoremen wird’s euch nicht fehlen, Müsst nur die Verschwindungspunkte zählen, Umkehren, abbilden, auf der Eb’ne ’rumfahren Und mit den Theta-Produkten nicht sparen. --LASSWITZ, KURD.

_Der Faust-Tragödie (-n)ter Tiel; Zeitschrift für den math.-natur. Unterricht, Bd. 14 (1883), p. 316._

Fuchs. Your words fill me with an awful dread, Seems like a circle were squared in my head.

Meph. Next in order you certainly ought On function-theory bestow your thought, And penetrate with contemplation What resists your attempts at integration. You’ll find no dearth of theorems there-- To vanishing-points give proper care-- Enumerate, reciprocate, Nor forget to delineate, Traverse the plane from end to end, And theta-functions freely spend.

=1935.= The student should avoid _founding results_ upon divergent series, as the question of their legitimacy is disputed upon grounds to which no answer commanding anything like general assent has yet been given. But they may be used as means of discovery, provided that their results be verified by other means before they are considered as established.--DE MORGAN, A.

_Trigonometry and Double Algebra (London, 1849), p. 55._

=1936.= There is nothing now which ever gives me any thought or care in algebra except divergent series, which I cannot follow the French in rejecting.--DE MORGAN, A.

_Graves’ Life of W. R. Hamilton (New York, 1882-1889), Vol. 3, p. 249._

=1937.= It is a strange vicissitude of our science that these [divergent] series which early in the century were supposed to be banished once and for all from rigorous mathematics should at its close be knocking at the door for readmission.--PIERPONT, J.

_Congress of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. 476._

=1938.= Zeno was concerned with three problems.... These are the problem of the infinitesimal, the infinite, and continuity.... From him to our own day, the finest intellects of each generation in turn attacked these problems, but achieved broadly speaking nothing.... Weierstrass, Dedekind, and Cantor, ... have completely solved them. Their solutions ... are so clear as to leave no longer the slightest doubt of difficulty. This achievement is probably the greatest of which the age can boast.... The problem of the infinitesimal was solved by Weierstrass, the solution of the other two was begun by Dedekind and definitely accomplished by Cantor.--RUSSELL, BERTRAND.

_International Monthly, Vol. 4 (1901), p. 89._

=1939.= It was not till Leibnitz and Newton, by the discovery of the differential calculus, had dispelled the ancient darkness which enveloped the conception of the infinite, and had clearly established the conception of the continuous and continuous change, that a full and productive application of the newly-found mechanical conceptions made any progress.--HELMHOLTZ, H.

_Aim and Progress of Physical Science; Popular Lectures [Flight] (New York, 1900), p. 372._

=1940.= The idea of an infinitesimal involves no contradiction.... As a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares.--PIERCE, C. F.

_The Law of Mind; Monist, Vol. 2 (1891-1892), pp. 543, 545._

=1941.= The chief objection against all _abstract_ reasonings is derived from the ideas of space and time; ideas, which, in common life and to a careless view, are very clear and intelligible, but when they pass through the scrutiny of the profound sciences (and they are the chief object of these sciences) afford principles, which seem full of obscurity and contradiction. No priestly _dogmas_, invented on purpose to tame and subdue the rebellious reason of mankind, ever shocked common sense more than the doctrine of the infinite divisibility of extension, with its consequences; as they are pompously displayed by all geometricians and metaphysicians, with a kind of triumph and exultation. A real quantity, infinitely less than any finite quantity, containing quantities infinitely less than itself, and so on _in infinitum_; this is an edifice so bold and prodigious, that it is too weighty for any pretended demonstration to support, because it shocks the clearest and most natural principles of human reason. But what renders the matter more extraordinary, is, that these seemingly absurd opinions are supported by a chain of reasoning, the clearest and most natural; nor is it possible for us to allow the premises without admitting the consequences. Nothing can be more convincing and satisfactory than all the conclusions concerning the properties of circles and triangles; and yet, when these are once received, how can we deny, that the angle of contact between a circle and its tangent is infinitely less than any rectilineal angle, that as you may increase the diameter of the circle _in infinitum_, this angle of contact becomes still less, even _in infinitum_, and that the angle of contact between other curves and their tangents may be infinitely less than those between any circle and its tangent, and so on, _in infinitum_? The demonstration of these principles seems as unexceptionable as that which proves the three angles of a triangle to be equal to two right ones, though the latter opinion be natural and easy, and the former big with contradiction and absurdity. Reason here seems to be thrown into a kind of amazement and suspense, which, without the suggestion of any sceptic, gives her a diffidence of herself, and of the ground on which she treads. She sees a full light, which illuminates certain places; but that light borders upon the most profound darkness. And between these she is so dazzled and confounded, that she scarcely can pronounce with certainty and assurance concerning any one object.--HUME, DAVID.

_An Inquiry concerning Human Understanding, Sect. 12, part 2._

=1942.= He who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in Divinity.--BERKELEY, G.

_The Analyst, sect. 7._

=1943.= And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?--BERKELEY, G.

_The Analyst, sect. 35._

=1944.= It is said that the minutest errors are not to be neglected in mathematics; that the fluxions are celerities, not proportional to the finite increments, though ever so small; but only to the moments or nascent increments, whereof the proportion alone, and not the magnitude, is considered. And of the aforesaid fluxions there be other fluxions, which fluxions of fluxions are called second fluxions. And the fluxions of these second fluxions are called third fluxions: and so on, fourth, fifth, sixth, etc., _ad infinitum_. Now, as our Sense is strained and puzzled with the perception of objects extremely minute, even so the Imagination, which faculty derives from sense, is very much strained and puzzled to frame clear ideas of the least particle of time, or the least increment generated therein: and much more to comprehend the moments, or those increments of the flowing quantities in _status nascenti_, in their first origin or beginning to exist, before they become finite particles. And it seems still more difficult to conceive the abstracted velocities of such nascent imperfect entities. But the velocities of the velocities, the second, third, fourth, and fifth velocities, etc., exceed, if I mistake not, all human understanding. The further the mind analyseth and pursueth these fugitive ideas the more it is lost and bewildered; the objects, at first fleeting and minute, soon vanishing out of sight. Certainly, in any sense, a second or third fluxion seems an obscure Mystery. The incipient celerity of an incipient celerity, the nascent augment of a nascent augment, i.e. of a thing which hath no magnitude; take it in what light you please, the clear conception of it will, if I mistake not, be found impossible; whether it be so or no I appeal to the trial of every thinking reader. And if a second fluxion be inconceivable, what are we to think of third, fourth, fifth fluxions, and so on without end.--BERKELEY, G.

_The Analyst, sect, 4._

=1945.= The _infinite_ divisibility of _finite_ extension, though it is not expressly laid down either as an axiom or theorem in the elements of that science, yet it is throughout the same everywhere supposed and thought to have so inseparable and essential a connection with the principles and demonstrations in Geometry, that mathematicians never admit it into doubt, or make the least question of it. And, as this notion is the source whence do spring all those amusing geometrical paradoxes which have such a direct repugnancy to the plain common sense of mankind, and are admitted with so much reluctance into a mind not yet debauched by learning; so it is the principal occasion of all that nice and extreme subtility which renders the study of Mathematics so difficult and tedious.--BERKELEY, G.

_On the Principles of Human Knowledge, Sect. 123._

=1946.= To avoid misconception, it should be borne in mind that infinitesimals are not regarded as being actual quantities in the ordinary acceptation of the words, or as capable of exact representation. They are introduced for the purpose of abridgment and simplification of our reasonings, and are an ultimate phase of magnitude when it is conceived by the mind as capable of diminution below any assigned quantity, however small.... Moreover such quantities are neglected, not, as Leibnitz stated, because they are infinitely small in comparison with those that are retained, which would produce an infinitely small error, but because they must be neglected to obtain a rigorous result; since such result must be definite and determinate, and consequently independent of these _variable indefinitely small quantities_. It may be added that the precise principles of the infinitesimal calculus, like those of any other science, cannot be thoroughly apprehended except by those who have already studied the science, and made some progress in the application of its principles.

--WILLIAMSON, B.

_Encyclopedia Britannica, 9th Edition; Article “Infinitesimal Calculus,” Sect. 12, 14._

=1947.= We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

--VOLTAIRE.

_A Philosophical Dictionary; Article “Infinity.” (Boston, 1881)._

=1948.= Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!--CARUS, PAUL.

_Logical and Mathematical Thought; Monist, Vol. 20 (1909-1910), p. 69._

=1949.=

Great fleas have little fleas upon their backs to bite ’em, And little fleas have lesser fleas, and so _ad infinitum._ And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on. --DE MORGAN, A.

_Budget of Paradoxes (London, 1872), p. 377._

=1950.= We have adroitly defined the infinite in arithmetic by a loveknot, in this manner ∞; but we possess not therefore the clearer notion of it.--VOLTAIRE.

_A Philosophical Dictionary; Article “Infinity.” (Boston, 1881)._

=1951.= I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a _facon de parler_, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.--GAUSS.

_Brief an Schumacher (1831); Werke, Bd. 8 p. 216._

=1952.= In spite of the essential difference between the conceptions of the _potential_ and the _actual_ infinite, the former signifying a _variable_ finite magnitude increasing beyond all finite limits, while the latter is a _fixed_, _constant_ quantity lying beyond all finite magnitudes, it happens only too often that the one is mistaken for the other.... Owing to a justifiable aversion to such _illegitimate_ actual infinities and the influence of the modern epicuric-materialistic tendency, a certain _horror infiniti_ has grown up in extended scientific circles, which finds its classic expression and support in the letter of Gauss [see 1951], yet it seems to me that the consequent uncritical rejection of the legitimate actual infinite is no lesser violation of the nature of things, which must be taken as they are.--CANTOR, G.

_Zum Problem des actualen Unendlichen; Natur und Offenbarung, Bd. 32 (1886), p. 226._

=1953.= The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously _withdrawing_ any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.

--LEWES, G. H.

_Problems of Life and Mind (Boston, 1875), Vol. 2, p. 384._

=1954.= A great deal of misunderstanding is avoided if it be remembered that the terms _infinity_, _infinite_, _zero_, _infinitesimal_ must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined.--MATHEWS, G. B.

_Theory of Numbers (Cambridge, 1892),