Part 1, sect. 104.

=1955.= This further is observable in number, that it is that which the mind makes use of in measuring all things that by us are measurable, which principally are _expansion_ and _duration_; and our idea of infinity, even when applied to those, seems to be nothing but the infinity of number. For what else are our ideas of Eternity and Immensity, but the repeated additions of certain ideas of imagined parts of duration and expansion, with the infinity of number; in which we can come to no end of addition?

--LOCKE, JOHN.

_An Essay concerning Human Understanding, Bk. 2, chap. 16, sect. 8._

=1956.= But of all other ideas, it is number, which I think furnishes us with the clearest and most distinct idea of infinity we are capable of.--LOCKE, JOHN.

_An Essay concerning Human Understanding, Bk. 2, chap. 17, sect. 9._

=1957.=

Willst du ins Unendliche schreiten? Geh nur im Endlichen nach allen Seiten! Willst du dich am Ganzen erquicken, So musst du das Ganze im Kleinsten erblicken. --GOETHE.

_Gott, Gemüt und Welt (1815)._

[Would’st thou the infinite essay? The finite but traverse in every way. Would’st in the whole delight thy heart? Learn to discern the whole in its minutest part.]

=1958.=

Ich häufe ungeheure Zahlen, Gebürge Millionen auf, Ich setze Zeit auf Zeit und Welt auf Welt zu Hauf, Und wenn ich von der grausen Höh’ Mit Schwindeln wieder nach dir seh,’ Ist alle Macht der Zahl, vermehrt zu tausendmalen, Noch nicht ein Theil von dir. _Ich zieh’ sie ab, und du liegst ganz vor mir._ --HALLER, ALBR. VON.

_Quoted in Hegel: Wissenschaft der Logik, Buch 1, Abschnitt 2, Kap. 2, C, b._

[Numbers upon numbers pile, Mountains millions high, Time on time and world on world amass, Then, if from the dreadful hight, alas! Dizzy-brained, I turn thee to behold, All the power of number, increased thousandfold, Not yet may match thy part. _Subtract what I will, wholly whole thou art._]

=1959.= A collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection.--RUSSELL, BERTRAND.

_International Monthly, Vol. 4 (1901), p. 93._

=1960.= An assemblage (ensemble, collection, group, manifold) of elements (things, no matter what) is infinite or finite according as it has or has not a part to which the whole is just _equivalent_ in the sense that between the elements composing that part and those composing the whole there subsists a unique and reciprocal (one-to-one) correspondence.--KEYSER, C. J.

_The Axioms of Infinity; Hibbert Journal, Vol. 2 (1903-1904), p. 539._

=1961.= Whereas in former times the Infinite betrayed its presence not indeed to the faculties of Logic but only to the spiritual Imagination and Sensibility, mathematics has shown ... that the structure of Transfinite Being is open to exploration by the organon of Thought.--KEYSER, C. J.

_Lectures on Science, Philosophy and Art (New York, 1908), p. 42._

=1962.= The mathematical theory of probability is a science which aims at reducing to calculation, where possible, the amount of credence due to propositions or statements, or to the occurrence of events, future or past, more especially as contingent or dependent upon other propositions or events the probability of which is known.--CROFTON, M. W.

_Encyclopedia Britannica, 9th Edition; Article, “Probability.”_

=1963.= The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which ofttimes they are unable to account. If we consider the analytical methods to which this theory has given birth, the truth of the principles on which it is based, the fine and delicate logic which their employment in the solution of problems requires, the public utilities whose establishment rests upon it, the extension which it has received and which it may still receive through its application to the most important problems of natural philosophy and the moral sciences; if again we observe that, even in matters which cannot be submitted to the calculus, it gives us the surest suggestions for the guidance of our judgments, and that it teaches us to avoid the illusions which often mislead us, then we shall see that there is no science more worthy of our contemplations nor a more useful one for admission to our system of public education.--LAPLACE.

_Théorie Analytique des Probabilitiés, Introduction; Oeuvres, t. 7 (Paris, 1886), p. 153._

=1964.= It is a truth very certain that, when it is not in our power to determine what is true, we ought to follow what is most probable.--DESCARTES.

_Discourse on Method, Part 3._

=1965.= As _demonstration_ is the showing the agreement or disagreement of two ideas, by the intervention of one or more proofs, which have a constant, immutable, and visible connexion one with another; so _probability_ is nothing but the appearance of such an agreement or disagreement, by the intervention of proofs, whose connexion is not constant and immutable, or at least is not perceived to be so, and it is enough to induce the mind to judge the proposition to be true or false, rather than contrary.--LOCKE, JOHN.

_An Essay concerning Human Understanding, Bk. 4, chap. 15, sect. 1._

=1966.= The difference between necessary and contingent truths is indeed the same as that between commensurable and incommensurable numbers. For the reduction of commensurable numbers to a common measure is analogous to the demonstration of necessary truths, or their reduction to such as are identical. But as, in the case of surd ratios, the reduction involves an infinite process, and yet approaches a common measure, so that a definite but unending series is obtained, so also contingent truths require an infinite analysis, which God alone can accomplish.--LEIBNITZ.

_Philosophische Schriften [Gerhardt] Bd. 7 (Berlin, 1890), p. 200._

=1967.= The theory in question [theory of probability] affords an excellent illustration of the application of the theory of permutation and combinations which is the fundamental part of the algebra of discrete quantity; it forms in the elementary parts an excellent logical exercise in the accurate use of terms and in the nice discrimination of shades of meaning; and, above all, it enters into the regulation of some of the most important practical concerns of modern life.--CHRYSTAL, GEORGE.

_Algebra, Vol. 2 (Edinburgh, 1889), chap. 36, sect. 1._

=1968.= There is possibly no branch of mathematics at once so interesting, so bewildering, and of so great practical importance as the theory of probabilities. Its history reveals both the wonders that can be accomplished and the bounds that cannot be transcended by mathematical science. It is the link between rigid deduction and the vast field of inductive science. A complete theory of probabilities would be the complete theory of the formation of belief. It is certainly a pity then, that, to quote M. Bertrand, “one cannot well understand the calculus of probabilities without having read Laplace’s work,” and that “one cannot read Laplace’s work without having prepared oneself for it by the most profound mathematical studies.”--DAVIS, E. W.

_Bulletin American Mathematical Society, Vol. 1 (1894-1895), p. 16._

=1969.= The most important questions of life are, for the most part, really only problems of probability. Strictly speaking one may even say that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, induction and analogy, the principal means for discovering truth, are based on probabilities, so that the entire system of human knowledge is connected with this theory.--LAPLACE.

_Théorie Analytique des Probabilitiés, Introduction; Oeuvres, t. 7 (Paris, 1886), p. 5._

=1970.= There is no more remarkable feature in the mathematical theory of probability than the manner in which it has been found to harmonize with, and justify, the conclusions to which mankind have been led, not by reasoning, but by instinct and experience, both of the individual and of the race. At the same time it has corrected, extended, and invested them with a definiteness and precision of which these crude, though sound, appreciations of common sense were till then devoid.--CROFTON, M. W.

_Encyclopedia Britannica, 9th Edition; Article “Probability.”_

=1971.= It is remarkable that a science [probabilities] which began with the consideration of games of chance, should have become the most important object of human knowledge.--LAPLACE.

_Théorie Analytique des Probabilitiés, Introduction; Oeuvres, t. 7 (Paris, 1886), p. 152._

=1972.= Not much has been added to the subject [of probability] since the close of Laplace’s career. The history of science records more than one parallel to this abatement of activity. When such a genius has departed, the field of his labours seems exhausted for the time, and little left to be gleaned by his successors. It is to be regretted that so little remains to us of the inner workings of such gifted minds, and of the clue by which each of their discoveries was reached. The didactic and synthetic form in which these are presented to the world retains but faint traces of the skilful inductions, the keen and delicate perception of fitness and analogy, and the power of imagination ... which have doubtless guided such a master as Laplace or Newton in shaping out such great designs--only the minor details of which have remained over, to be supplied by the less cunning hand of commentator and disciple.--CROFTON, M. W.

_Encyclopedia Britannica, 9th Edition; Article “Probability.”_

=1973.= The theory of errors may be defined as that branch of mathematics which is concerned, first, with the expression of the resultant effect of one or more sources of error to which computed and observed quantities are subject; and, secondly, with the determination of the relation between the magnitude of an error and the probability of its occurrence.--WOODWARD, R. S.

_Probability and Theory of Errors (New York, 1906), p. 30._

=1974.= Of all the applications of the doctrine of probability none is of greater utility than the theory of errors. In astronomy, geodesy, physics, and chemistry, as in every science which attains precision in measuring, weighing, and computing, a knowledge of the theory of errors is indispensable. By the aid of this theory the exact sciences have made great progress during the nineteenth century, not only in the actual determinations of the constants of nature, but also in the fixation of clear ideas as to the possibilities of future conquests in the same direction. Nothing, for example, is more satisfactory and instructive in the history of science than the success with which the unique method of least squares has been applied to the problems presented by the earth and the other members of the solar system. So great, in fact, are the practical value and theoretical importance of least squares, that it is frequently mistaken for the whole theory of errors, and is sometimes regarded as embodying the major part of the doctrine of probability itself.--WOODWARD, R. S.

_Probability and Theory of Errors (New York, 1906), pp. 9-10._

=1975.= Direct and inverse ratios have been applied by an ingenious author to measure human affections, and the moral worth of actions. An eminent Mathematician attempted to ascertain by calculation, the ratio in which the evidence of facts must decrease in the course of time, and fixed the period when the evidence of the facts on which Christianity is founded shall become evanescent, and when in consequence no faith shall be found on the earth.--REID, THOMAS.

_Essays on the Powers of the Human Mind (Edinburgh, 1812), Vol. 2, p. 408._