"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."
"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)
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