📓On Literature: On Infinitesimals (From Fiction to Science-Fiction)

"Science has gone down into the mines and coal-pits, and before the safety-lamp the Gnomes and Genii of those dark regions have disappeared [...] Sirens, mermaids, shining cities glittering at the bottom of quiet seas and in deep lakes, exist no longer; but in their place, Science, their destroyer, shows us whole coasts of coral reef constructed by the labours of minute creatures; points to our own chalk cliffs and limestone rocks as made of the dust of myriads of generations of infinitesimal beings that have passed away; reduces the very element of water into its constituent airs, and re-creates it at her pleasure." (Charles Dickens, Poetry of Science, [Book review of Robert Hunt]1848)

"A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable human wills, is continuous. To understand the laws of this continuous movement is the aim of history." (Lev N Tolstoy, "War and Peace", 1867)

"Arriving at infinitesimals, mathematics, the most exact of sciences, abandons the process of analysis and enters on the new process of the integration of unknown, infinitely small, quantities." (Lev N Tolstoy, "War and Peace", 1867)

"Only by reducing this element of free will to the infinitesimal, that is, by regarding it as an infinitely small quantity, can we convince ourselves of the absolute inaccessibility of the causes, and then instead of seeking causes, history will take the discovery of laws as its problem." (Lev N Tolstoy, "War and Peace", 1867)

"Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history."  (Lev N Tolstoy, "War and Peace", 1867)

"To-day alone was real. Never was man brought into contact with reality save through the evanescent emotions and sensations of that single moment, that infinitesimal fraction of a second, which was passing now - and it was in the insignificance of this moment, precisely, that religious persons must believe. So ran the teachings of all dead and lingering faiths alike. Here was, perhaps, only another instance of mankind's abhorrence of actualities; and man's quaint dislike of facing reality was here disguised as a high moral principle. That was why all art, which strove to make the sensations of a moment soul-satisfying, was dimly felt to be irreligious. For art performed what religion only promised." (James B Cabell, "The Cream of the Jest", 1917)

"Measured objectively, what a man can wrest from Truth by passionate striving is utterly infinitesimal. But the striving frees us from the bonds of the self and makes us comrades of those who are the best and the greatest." (Albert Einstein, cca. 1920s)

"It is venturesome to think that a coordination of words (philosophies are nothing more than that) can resemble the universe very much. It is also venturesome to think that of all these illustrious coordinations, one of them - at least in an infinitesimal way - does not resemble the universe a bit more than the others." (Jorge L Borges, "Discussion", 1932)

"To be so closely caught up in the teeth of things that they kill you, no matter how infinitesimally kill you, is, truly, to be a poet: and to be a poet in fact it is additionally necessary that you should possess the tongues and instruments with which to record this series of infinitesimal deaths." (George Barker,"Therefore All Poems Are Elegies", 1940)

"An ugliness unfurled in the moonlight and soft shadow and suffused the whole world. If I were an amoeba, he thought, with an infinitesimal body, I could defeat ugliness. A man isn’t tiny or giant enough to defeat anything." (Yukio Mishima, "The Sailor Who Fell from Grace with the Sea", 1963)

"The creative element in the mind of man [...] emerges in as mysterious a fashion as those elementary particles which leap into momentary existence in great cyclotrons, only to vanish again like infinitesimal ghosts." (Loren Eiseley, "The Night Country", 1971)

"It is tempting to wonder if our present universe, large as it is and complex though it seems, might not be merely the result of a very slight random increase in order over a very small portion of an unbelievably colossal universe which is virtually entirely in heat-death. Perhaps we are merely sliding down a gentle ripple that has been set up, accidently and very temporarily, in a quiet pond, and it is only the limitation of our own infinitesimal range of viewpoint in space and time that makes it seem to ourselves that we are hurtling down a cosmic waterfall of increasing entropy, a waterfall of colossal size and duration." (Isaac Asimov, 1976)

"I wonder if being sane means disregarding the chaos that is life, pretending only an infinitesimal segment of it is reality." (Rabih Alameddine, "Koolaids: The Art of War", 1998)

∑On Analysis: On Infinitesimals (1875-1899)

"For an understanding of Nature, questions about the infinitely large are idle questions. It is different, however, with questions about the infinitely small. Our knowledge of their causal relations depends essentially on the precision with which we succeed in tracing phenomena on the infinitesimal level." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)

"To prove that, until this very day, life has never been shown to man as a product of the forces that govern matter, it could be useful the spiritual doctrine which has been very neglected elsewhere, but always finds at least a glorious refuge in your groups. Perhaps you know that in this difficult question concerning the origin of the infinitesimal, I will have brought experimental rigor that has grown weary of contradiction. Referring to the merit, however, we have inherited severe rules of the method from the great experimenters: Galileo, Pascal, Newton and their followers for two centuries." (Louis Pasteur, [Discours de réception de Louis Pasteur] 1882)

"What relation has the life of the individual to the life of the universe? [...] The former is absolutely subordinate, inconceivably infinitesimal compared with the latter. The becoming of the latter bears not the slightest apparent reference to the becoming of the former. [...] The one seems finite, limited, temporal, the other by comparison infinite, boundless, eternal. This disparity has forced itself upon the attention of man ever since his first childlike attempts at thought." (Karl Pearson, "The Ethic of Freethought", 1883) 

"During my slumber I had a dream. I thought I was once more by the side of the Sphere, whose lustrous hue betokened that he had exchanged his wrath against me for perfect placability. We were moving together towards a bright but infinitesimally small Point, to which my Master directed my attention." (Edwin Abbott, "Flatland: A Romance of Many Dimensions", 1884)

"In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"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." (George B Mathews, "Theory of Numbers", 1892)

"Accordingly, time logically supposes a continuous range of intensity of feeling. It follows then, from the definition of continuity, that when any particular kind of feeling is present, an infinitesimal continuum of all feelings differing infinitesimally from that, is present." (Charles S Peirce, "The Law of Mind", 1892)

"From these two immediate perceptions, we gain a mediate, or inferential perception of the relation of all four instants. This mediate perception is objectively, or as to the object being represented, spread over the four instants; but subjectively, or as itself the subject of duration, it is completely embraced in the second moment. (The reader will observe that I use the word instant to mean a point in time, and moment to mean an infinitesimal duration." (Charles S Peirce, "The Law of Mind", 1892)

"How can a past idea be present?… it can only be going, infinitesimally past, less past than any assignable past date. We are thus brought to the conclusion that the present is connected to the past by a series of real infinitesimal steps." (Charles S Peirce, "The Law of Mind", 1892)

"The first character of a general idea so resulting is that it is living feeling. A continuum of this feeling, infinitesimal in duration, but still embracing innumerable parts, and also, though infinitesimal, entirely unlimited, is immediately present. And in its absence of boundedness a vague possibility of more than is present is directly felt." (Charles S Peirce, "The Law of Mind", 1892)

"To ignore the subject would be an act of cowardice - an act of cowardice I feel no temptation to commit. To stop short in any research that bids fair to widen the gates of knowledge, to recoil from fear of difficulty or adverse criticism, is to bring reproach on science. There is nothing for the investigator to do but to go straight on; to explore up and down, inch by inch, with the taper his reason; to follow the light wherever it may lead, even should it at times resemble a will-o'-the-wisp. I have nothing to retract. I adhere to my already published statements. Indeed, I might add much thereto. I regret only a certain crudity in those early expositions which, no doubt justly, militated against their acceptance by the scientific world. My own knowledge at that time scarcely extended beyond the fact that certain phenomena new to science had assuredly occurred, and were attested by my own sober senses and, better still, by automatic record. I was like some two-dimensional being who might stand at the singular point of a Riemann's surface, and thus find himself in infinitesimal and inexplicable contact with a plane of existence not his own." (William Crookes, [Presidential Address to the Society for Psychical Research] 1897)

"Consciousness arises with, or out of, and accompanies, these clay compounds called creatures, but it does not cause, nor in any way interfere with, their phenomena. If it were possible to construct artificial clods, chemically as accomplished as philosophers, but without any accompanying consciousness, these soulless mechanisms, without will, feeling, or conscious intelligence, simply acting out their chemical and physical affinities, would not behave otherwise in any infinitesimal particular than the real, conscious meditators on things." (J Howard Moore," Better-World Philosophy: A Sociological Synthesis", 1899)

∑On Analysis: On Infinitesimals (1750-1799)

"Newton started out from another principle; and one can say that the metaphysics of this great mathematician on the calculus of fluxions is very exact and illuminating, even though he allowed us only an imperfect glimpse of his thoughts. He never considered the differential calculus as the study of infinitely small quantities, but as the method of first and ultimate ratios, that is to say, the method of finding the limits of ratios. Thus this famous author has never differentiated quantities but only equations; in fact, every equation involves a relation between two variables and the differentiation of equations consists merely in finding the limit of the ratio of the finite differences of the two quantitiescontained in the equation." (Jean LeRond D'Alembert, "Differentiel" ["Differentials", 1754)

"What concerns us most here is the metaphysics of the differential calculus. This metaphysics, of which so much has been written, is even more important and perhaps more difficult to explain than the rules of this calculus themselves: various mathematicians, among them Rolle, who were unable to accept the assumption concerning infinitely small quantities, have rejected it entirely, and have held that the principle was false and capable of leading to error. Yet in view of the fact that all results obtained by means of ordinary Geometry can be established similarly and much more easily by means of the differential calculus, one cannot help concluding that, since this calculus yields reliable, simple, and exact methods, the principles on which it depends must also be simple and certain." (Jean LeRond D'Alembert, "Differentiel" ["Differentials", 1754)

"Any change involves at least two conditions, one preceding and one following, which are distinct from one another in such a way that the difference between the former and the latter can be established. Now the law of continuity prohibits the thing which is being changed to transcend abruptly from the former to the latter. It must pass through an intermediate condition which is as little distinct from the previous as from the subsequent one. And because the difference between this intermediate condition and the previous condition can be established still, there must be an intermediate condition between these two as well, and this must continue in the same way, until the difference between the previous condition and the one immediately succeeding it vanishes. As long as the set of these intermediate conditions can be established, every difference between one and the next can be established as well: hence their set must become larger than any given set if these differences shall vanish, and thus we imagine infinitely many conditions where one differs from the next to an infinitely small degree." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"It is held because of this law in particular, that no change may occur suddenly, but rather that every change always passes by infinitely small stages, of which the trajectory of a point in a curved line provides a first example." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"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." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"At the time the book of Marquis de l'Hôpital had appeared, and almost all mathematicians began to turn to the new geometry of the infinite [that is, the new infinitesimal calculus], until then little known. The surprising universality of the methods, the elegant brevity of the proofs, the neatness and speed of the most difficult solutions, a singular and unexpected novelty, all attracted the mind and there was in the mathematical world a well marked revolution [une révolution bien marquée." (Bernard Le Bovier de Fontenelle, 1792) 

"The infinitely smallest part of space is always a space, something endowed with continuity, not at all a mere point or the boundary between specified places in space." (Johann G Fichte, "Grundriss des Eigenthümlichen der Wissenschaftslehre in Rücksicht auf das theoretische Vermögen", 1795)

"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." (Pierre-Simon Laplace, "Exposition du système du monde" ["Exposition of the System of the World"], 1796)

∑On Analysis: On Infinitesimals (1700-1749)

"[…] even if someone refuses to admit infinite and infinitesimal lines in a rigorous metaphysical sense and as real things, he can still use them with confidence as ideal concepts (notions ideales) which shorten his reasoning, similar to what we call imaginary roots in the ordinary algebra, for example, √-2." (Gottfried W Leibniz, [letter to Varignon], 1702)

"Of late the speculations about Infinities have run so high, and grown to such strange notions, as have occasioned no small scruples and disputes among the geometers of the present age. Some there are of great note who, not contented with holding that finite lines may be divided into an infinite number of parts, do yet further maintain that each of these infinitesimals is itself subdivisible into an infinity of other parts or infinitesimals of a second order, and so on ad infinitum. These I say assert there are infinitesimals of infinitesimals, etc., without ever coming to an end; so that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts." (George Berkeley, "The Principles of Human Knowledge', 1710)

"In fact, a similar principle of hardness cannot exist; it is a chimera which offends that general law which nature constantly observes in all its operations; I speak of that immutable and perpetual order, established since the creation of the Universe, that can be called the LAW OF CONTINUITY, by virtue of which everything that takes place, takes place by infinitely small degrees. It seems that common sense dictates that no change can take place at a jump; natura non operatur per saltion; nothing can pass from one extreme to the other without passing through all the degrees in between." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

"The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment. The sciences must proceed to first causes, above all mathematics where one cannot assume, as in physics, principles that are unknown to us. For there is in mathematics, so to speak, only what we have placed there […] If, however, mathematics always has some essential obscurity that one cannot dissipate, it will lie, uniquely, I think, in the direction of the infinite; it is in that direction that mathematics touches on physics, on the innermost nature of bodies about which we know little." (Bernard Le Bovier de Fontenelle, "Elements de la géométrie de l'infini", 1727)

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire, "Letters on the English", 1733)

"And yet in the calculus differentialis, which method serves to all the same intents and ends with that of fluxions, our modern analysts are not content to consider only the differences of finite quantities: they also consider the differences of those differences, and the differences of the differences of the first differences. And so on ad infinitum. That is, they consider quantities infinitely less than the least discernible quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding infinitesimals, and so on without end or limit. Insomuch that we are to admit an infinite succession of infinitesimals, each infinitely less than the foregoing, and infinitely greater than the following. As there are first, second, third, fourth, fifth, &c. fluxions, so there are differences, first, second, third, fourth, &c., in an infinite  progression towards nothing, which you still approach and never arrive at. And (which is most strange) although you should take a million of millions of these infinitesimals, each whereof is supposed infinitely greater than some other real magnitude, and add them to the least given quantity, it shall never be the bigger. For this is one of the modest postulata of our modern mathematicians, and is a cornerstone or ground-work of their speculations."  (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"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? [...] The method of Fluxions is the general key by help whereof the modern mathematicians unlock the secrets of Geometry, and consequently of Nature." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"The foreign mathematicians are supposed by some, even of our own, to proceed in a manner less accurate, perhaps, and geometrical, yet more intelligible. Instead of flowing quantities and their fluxons, they consider the variable finite quantities as increasing or diminishing by the continual addition or subduction of infinitely small quantities. Instead of the velocities wherewith increments are generated, they consider the increments or decrements themselves, which they call differences, and which are supposed to be infinitely small. The difference of a line is an infinitely little line; of a plane an infinitely little plane. They suppose finite quantities to consist of parts infinitely little, and curves to be polygons, whereof the sides are infinitely little, which by the angles they make one with another determine the curvity of the line. Now to conceive a quantity infinitely small, that is, infinitely less than any sensible or imaginable quantity, or than any the least finite magnitude is, I confess, above my capacity. But to conceive a part of such infinitely small quantity that shall be still infinitely less than it, and consequently though multiplied infinitely shall never equal the minutest finite quantity, is, I suspect, an infinite difficulty to any man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust." (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

∑On Analysis: On Infinitesimals (2000-)

"[…] all roads in mathematics lead to infinity. At any rate, most of the attempts to do the impossible have called upon infinity in one way or another: not necessarily the infinitely large, not necessarily the infinitely small, but certainly the infinitely many." (John Stillwell, "Yearning for the impossible: the surpnsing truths of mathematics", 2006)

"The mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Within a few decades of its discovery, the fantasy of infinitesimals had completely overpowered the honest method of exhaustion by joining forces with algebra to form an infinitesimal calculus - a symbolism for solving problems about curves by routine calculations, like those already known for the geometry of straight lines. The calculus, as we know it today, is perhaps the most powerful mathematical tool ever invented, yet it originated in the dream world of infinitesimals. We look at these strange origins in the next section." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"[...] a high degree of unpredictability is associated with erratic trajectories. This not only because they look random but mostly because infinitesimally small uncertainties on the initial state of the system grow very quickly - actually exponentially fast. In real world, this error amplification translates into our inability to predict the system behavior from the unavoidable imperfect knowledge of its initial state." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)

"Probability is often expressed using large but finite numbers: ‘one in a thousand’, ‘one in a million’. But perhaps the probability of life, intelligent life, appearing somewhere in our universe is infinitesimal. If so, a universe would need infinitely many planets to produce even a finite number of civilisations (i.e., one)." (Daniel Tammet, "Thinking in Numbers" , 2012)

"One of the remarkable features of these complex systems created by replicator dynamics is that infinitesimal differences in starting positions create vastly different patterns. This sensitive dependence on initial conditions is often called the butterfly-effect aspect of complex systems - small changes in the replicator dynamics or in the starting point can lead to enormous differences in outcome, and they change one’s view of how robust the current reality is. If it is complex, one small change could have led to a reality that is quite different." (David Colander & Roland Kupers, "Complexity and the art of public policy : solving society’s problems from the bottom up", 2014)

"In the applications of mathematics, infinitesimal quantities are often considered as variable points on the real number line. Cauchy took this viewpoint by defining an infinitesimal to be a variable quantity that becomes arbitrarily small. In modern notation, this idea can be represented as a null sequence, which is simply a sequence that tends to zero." (Ian Stewart & David Tall,"The Foundations of Mathematics" 2nd Ed., 2015)

"Newton explained naturally occurring phenomena, such as gravity and the movement of the planets, using a combination of Greek geometry and symbolic algebra to build his ideas in the calculus. Leibniz imagined quantities that could be infinitesimally small and produced a powerful symbolism for the calculus that has stood the test of time, despite widespread concerns about its logical foundation. Later giants in mathematical development focused on different aspects. Euler manipulated symbols algebraically using power series and complex numbers, and Cauchy imagined infinitesimals geometrically as variable quantities on the line or in the plane that become arbitrarily small. His approach led to major advances, using a blend of visual and symbolic methods in real and complex analysis, but it also generated significant criticism about the precise meaning. The critics had a point: the meaning had not then been fully worked out. What prevailed was more an act of faith, that everything would work out much as it always had done." (Ian Stewart & David Tall,"The Foundations of Mathematics" 2nd Ed., 2015)

"Calculus is the study of things that are changing. It is difficult to make theories about things that are always changing, and calculus accomplishes it by looking at infinitely small portions, and sticking together infinitely many of these infinitely small portions." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"But in another way infinitesimals were not like imaginaries. While it is true that imaginaries are not part of the ordinary number system, that system can easily be enlarged to accommodate them – for example, by defining imaginaries as ordered pairs of ordinary numbers, as Hamilton did in 1835. It is not nearly as easy, or convenient, to enlarge the number system to include infinitesimals. It was not even known to be possible until the twentieth century, long after mathematicians had decided that it was better to avoid infinitesimals the way Euclid and Archimedes avoided infinity." (John Stillwell, "A Concise History of Mathematics for Philosophers", 2019) 

"Infinitesimals were a bit like imaginary numbers. They seemed to contradict accepted principles – just as imaginaries contradicted the principle that squares are positive, infinitesimals contradicted the Archimedean axiom for geometric quantities, [...] yet they enabled calculations that were otherwise difficult or impossible. For mathematicians of the seventeenth and eighteenth centuries this was generally good enough reason to accept them." (John Stillwell, "A Concise History of Mathematics for Philosophers", 2019) 

∑On Analysis: On Infinitesimals (1800-1874)

"It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum." (Joseph-Louis Lagrange, "Leçons sur le calcul des fonctions", 1806)

"What should one understand by ∫ ϕx · dx for x = a + bi? Obviously, if we want to start from clear concepts, we have to assume that x passes from the value for which the integral has to be 0 to x = a + bi through infinitely small increments (each of the form x = a + bi), and then to sum all the ϕx · dx. Thereby the meaning is completely determined. However, the passage can take placein infinitely many ways: Just like the realm of all real magnitudes can be conceived as an infinite straight line, so can the realm of all magnitudes, real and imaginary, be made meaningful by an infinite plane, in which every point, determined by abscissa = a and ordinate = b, represents the quantity a+bi. The continuous passage from one value of x to another a+bi then happens along a curve and is therefore possible in infinitely many ways. I claim now that after two different passages the integral ∫ ϕx · dx acquires the same value when ϕx never becomes equal to ∞ in the region enclosed by the two curves representing the two passages." (Carl F Gauss, [letter to Bessel] 1811)

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials." (Pierre-Simon Laplace, "Essai philosophique sur le calcul des probabilities", 1812)

"[…] a function of the variable x will be continuous between two limits a and b of this variable if between two limits the function has always a value which is unique and finite, in such a way that an infinitely small increment of this variable always produces an infinnitely small increment of the function itself." (Augustin-Louis Cauchy, "Mémoire sur les fonctions continues" ["Memoir on continuous functions"], 1844)

"If they [mathematicians] find a quantity greater than any finite number of the assumed units, they call it infinitely great; if they find one so small that its every finite multiple is smaller than the unit, they call it infinitely small; nor do they recognise any other kind of infinitude than these two, together with the quantities derived from them as being infinite to a higher order of greatness or smallness, and thus based after all on the same idea." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis." (Richard Dedekind," Stetigkeit und irrationale Zahlen", 1872) 

"The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858." (Richard Dedekind," Stetigkeit und irrationale Zahlen", 1872) 

On Analysis: On Infinitesimals (1900-1924)

"[...] 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, resorting also to experimental tests and incomplete induction; frequently finding it necessary, in view of unforeseen disclosures, to abandon one hopeful hypothesis or to transform it by retrenchment or by enlargement: - thus, in his own domain, matching, point for point, the processes, methods and experience familiar to the devotee of natural science." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"What, in fact, is mathematical discovery? It does not consist in making new combinations with mathematical entities that are already known. That can be done by anyone, and the combinations that could be so  formed would be infinite in number, and the greater part of them would be absolutely devoid of interest. Discovery consists precisely in not constructing useless combinations, but in constructing those that are useful, which are an infinitely small minority. Discovery is discernment, selection." (Henri Poincaré, "Science and Method", 1908)

"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!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"I have described at some length the application of Positive Rays to chemical analysis; one of the main reasons for writing this book was the hope that it might induce others, and especially chemists, to try this method of analysis. I feel sure that there are many problems in chemistry, which could be solved with far greater ease by this than any other method. The method is surprisingly sensitive - more so than even that of spectrum analysis, requires an infinitesimal amount of material, and does not require this to be specially purified; the technique is not difficult if appliances for producing high vacua are available." (Joseph J Thomson, "Rays of Positive Electricity", 1913)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"[...] political and social and scientific values […] should be correlated in some relation of movement that could be expressed in mathematics, nor did one care in the least that all the world said it could not be done, or that one knew not enough mathematics even to figure a formula beyond the schoolboy s=(1/2)gt2. If Kepler and Newton could take liberties with the sun and moon, an obscure person ... could take liberties with Congress, and venture to multiply its attraction into the square of its time. He had only to find a value, even infinitesimal, for its attraction." (Henry B Adams, "The Education of Henry Adams: An Autobiography?", 1918)

"Eudoxes […] not only based the method [of exhaustion] on rigorous demonstration […] but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus, though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous methods of Eudoxes. Archimedes adds that we must give no small share of the credit for these theorems to Democritus […] another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, 'seems to have thought of everything' […] Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as equal or unequal [...] Democritus was already close on the track of infinitesimals." (Thomas L Heath, "Achimedes", 1920)

"The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a linear transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem 'linear' by reverting to infinitely small quantities." (Hermann Weyl, "Space - Time - Matter", 1922)

On Analysis: On Infinitesimals (1950-1999)

"No fact in the world is instant, infinitesimal and ultimate, a single mark. There are, I hold, no atomic facts. In the language of science, every fact is a field - a crisscross of implications, those that lead to it and those that lead from it. […] We condense the laws around concepts. Science takes its coherence, its intellectual and imaginative strength together, from the concepts at which its laws cross, like knots in a mesh" (Jacob Bronowski, "Science and Human Values" 1956)

"Our psyche is set up in accord with the structure of the universe, and what happens in the macrocosm likewise happens in the infinitesimal and most subjective reaches of the psyche." (Carl G Jung, "Memories, dreams, reflections", 1962)

"Thus far we have regarded integration as the inverse of the operation of differentiation, and the integral was thought of as an anti-derivative. It is possible, however, to consider integration from another point of view, namely, as a process of summation, or as the addition of many similar elements. Indeed, it is largely from this point of view that the Integral Calculus developed historically, growing out of early attempts to determine the area bounded by various curves. A given area was subdivided into many small parts, and these 'infinitesimal parts' were then added." (William L Schaaf, "The Calculus, a college course guide", 1963)

"When comparing infinitesimals, we refer to their order. This is a relative term, suggesting comparative degree of smallness. If the limit of the quotient of two infinitesimals is a constant, not zero, they are said to be of the same order; if this limit is zero, the first differential (the numerator) is said to be of higher order than the second, and the second of lower order than the first. If the limit is infinite, the first differential (the numerator) is said to be of lower order than the second, and the second of higher order than the first." (William L Schaaf, "The Calculus, a college course guide", 1963)

"Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"The world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)

"Mathematics […] is mired in a language of symbols foreign to most of us, [it] explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding." (Robert Kanigel, "The Man Who Knew Infinity", 1991)

"The inflationary period of expansion does not smooth out irregularity by entropy-producing processes like those explored by the cosmologies of the seventies. Rather it sweeps the irregularity out beyond the Horizon of our visible Universe, where we cannot see it . The entire universe of stars and galaxies on view to us. […] on this hypothesis, is but the reflection of a minute, perhaps infinitesimal, portion of the universe's initial conditions, whose ultimate extent and structure must remain forever unknowable to us. A theory of everything does not help here. The information contained in the observable part of the universe derives from the evolution of a tiny part of the initial conditions for the entire universe. The sum total of all the observations we could possibly make can only tell us about a minuscule portion of the whole." (John D Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[The black hole] teaches us that space can be crumpled like a piece of paper into an infinitesimal dot, that time can be extinguished like a blown-out flame, and that the laws of physics that we regard as 'sacred', as immutable, are anything but." (John A Wheeler & Kenneth Ford, "Geons, Black Holes & Quantum Foam: A Life in Physics", 1998)

On Analysis: On Infinitesimals (-1699)

"This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles…. for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals." (John Wallis, "Arithmetica Infinitorum", 1656)

"The same thing is confirmed through the previous proposition since, for example, an infinite collection from which units can be subtracted (not only ten but infinitely many) while the collection remains infinite is, obviously, infinitely greater before the subtraction takes place than after; thus, since the collection does not cease being infinite after the subtraction, the infinite will be, as such, infinitely smaller than it was earlier. You could say that this is in conflict with the generally accepted thesis that holds that the terms 'greater' and 'smaller' can only apply to finite quantities but not to infinite quantities; or, at least, that they can be applied to infinite quantities only in a very improper way. I reply that this idea has the following feature, namely that it is widespread. This fact notwithstanding, I would say, with permission, that it also has this other feature, namely that its ground is nothing else but a false notion of infinity. Moreover, the advantage it offers, which consists in apparently solving some difficulties that are usually put forth by denying that 'greater' and 'smaller' are properties that can be predicated of infinity, does not subsist for it ends up not resolving the difficulties." (Emmanuel Maignan, 1673)

"Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum's composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysics except he who has passed through this [labyrinth]." (Gottfried W Leibniz, "Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae", 1676)

"Where, by the way, we may observe a great difference between the proportion of Infinite to Finite, and, of Finite to Nothing. For 1/∞, that which is a part infinitely small, may, by infinite Multiplication, equal the whole: But 0/1 , that which is Nothing can by no Multiplication become equal to Something." (John Wallis, "Treatise of Algebra", 1685)

"Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality." (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)

"A quantity diminished or enlarged by an infinitely smaller quantity is neither diminished nor enlarged." (Johann Bernoulli, cca 1691-92)

"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)