CHAPTER XV

MATHEMATICS AND SCIENCE

=1501.= How comes it about that the knowledge of other sciences, which depend upon this [mathematics], is painfully sought, and that no one puts himself to the trouble of studying this science itself? I should certainly be surprised, if I did not know that everybody regarded it as being very easy, and if I had not long ago observed that the human mind, neglecting what it believes to be easy, is always in haste to run after what is novel and advanced.--DESCARTES.

_Rules for the Direction of the Mind; Philosophy of Descartes [Torrey], (New York, 1892), p. 72._

=1502.= All quantitative determinations are in the hands of mathematics, and it at once follows from this that all speculation which is heedless of mathematics, which does not enter into partnership with it, which does not seek its aid in distinguishing between the manifold modifications that must of necessity arise by a change of quantitative determinations, is either an empty play of thoughts, or at most a fruitless effort. In the field of speculation many things grow which do not start from mathematics nor give it any care, and I am far from asserting that all that thus grow are useless weeds, among them may be many noble plants, but without mathematics none will develop to complete maturity.--HERBART, J. F.

_Werke (Kehrbach), (Langensalza, 1890), Bd. 5, p. 106._

=1503.= There are few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning, and when they cannot, it’s a sign our knowledge of them is very small and confus’d; and where a mathematical reasoning can be had, it’s as great folly to make use of any other, as to grope for a thing in the dark, when you have a candle standing by you.--ARBUTHNOT.

_Quoted in Todhunter’s History of the Theory of Probability (Cambridge and London, 1865), p. 51._

=1504.= Mathematical Analysis is ... the true rational basis of the whole system of our positive knowledge.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 1, chap. 1._

=1505.= It is only through Mathematics that we can thoroughly understand what true science is. Here alone we can find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain. Any scientific education setting forth from any other point, is faulty in its basis.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 1, chap. 1._

=1506.= In the present state of our knowledge we must regard Mathematics less as a constituent part of natural philosophy than as having been, since the time of Descartes and Newton, the true basis of the whole of natural philosophy; though it is, exactly speaking, both the one and the other. To us it is of less use for the knowledge of which it consists, substantial and valuable as that knowledge is, than as being the most powerful instrument that the human mind can employ in the investigation of the laws of natural phenomena.--COMTE, A.

_Positive Philosophy [Martineau], Introduction, chap. 2._

=1507.= The concept of mathematics is the concept of science in general.--NOVALIS.

_Schriften (Berlin, 1901), Teil 2, p. 222._

=1508.= I contend, that each natural science is real science only in so far as it is mathematical.... It may be that a pure philosophy of nature in general (that is, a philosophy which concerns itself only with the general concepts of nature) is possible without mathematics, but a pure science of nature dealing with definite objects (physics or psychology), is possible only by means of mathematics, and since each natural science contains only as much real science as it contains _a priori_ knowledge, each natural science becomes real science only to the extent that it permits the application of mathematics.

--KANT, E.

_Metaphysische Anfangsgründe der Naturwissenschaft, Vorrede._

=1509.= The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers;... The modern, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things.--SAFFORD, T. H.

_Mathematical Teaching etc. (Boston, 1886), p. 9._

=1510.= It seems to me that no one science can so well serve to co-ordinate and, as it were, bind together all of the sciences as the queen of them all, mathematics.--DAVIS, E. W.

_Proceedings Nebraska Academy of Sciences for 1896 (Lincoln, 1897), p. 282._

=1511.= And as for Mixed Mathematics, I may only make this prediction, that there cannot fail to be more kinds of them, as nature grows further disclosed.--BACON, FRANCIS.

_Advancement of Learning, Bk. 2; De Augmentis, Bk. 3._

=1512.= Besides the exercise in keen comprehension and the certain discovery of truth, mathematics has another formative function, that of equipping the mind for the survey of a scientific system.--GRASSMANN, H.

_Stücke aus dem Lehrbuche der Arithmetik; Werke (Leipzig, 1904), Bd. 2, p. 298._

=1513.= Mathematicks may help the naturalists, both to frame hypotheses, and to judge of those that are proposed to them, especially such as relate to mathematical subjects in conjunction with others.--BOYLE, ROBERT.

_Works (London, 1772), Vol. 3, p. 429._

=1514.= The more progress physical sciences make, the more they tend to enter the domain of mathematics, which is a kind of centre to which they all converge. We may even judge of the degree of perfection to which a science has arrived by the facility with which it may be submitted to calculation.--QUETELET.

_Quoted in E. Mailly’s Eulogy on Quetelet; Smithsonian Report, 1874, p. 173._

=1515.= The mathematical formula is the point through which all the light gained by science passes in order to be of use to practice; it is also the point in which all knowledge gained by practice, experiment, and observation must be concentrated before it can be scientifically grasped. The more distant and marked the point, the more concentrated will be the light coming from it, the more unmistakable the insight conveyed. All scientific thought, from the simple gravitation formula of Newton, through the more complicated formulae of physics and chemistry, the vaguer so called laws of organic and animated nature, down to the uncertain statements of psychology and the data of our social and historical knowledge, alike partakes of this characteristic, that it is an attempt to gather up the scattered rays of light, the different parts of knowledge, in a focus, from whence it can be again spread out and analyzed, according to the abstract processes of the thinking mind. But only when this can be done with a mathematical precision and accuracy is the image sharp and well-defined, and the deductions clear and unmistakable. As we descend from the mechanical, through the physical, chemical, and biological, to the mental, moral, and social sciences, the process of focalization becomes less and less perfect,--the sharp point, the focus, is replaced by a larger or smaller circle, the contours of the image become less and less distinct, and with the possible light which we gain there is mingled much darkness, the sources of many mistakes and errors. But the tendency of all scientific thought is toward clearer and clearer definition; it lies in the direction of a more and more extended use of mathematical measurements, of mathematical formulae.--MERZ, J. T.

_History of European Thought in the 19th Century (Edinburgh and London, 1904), Vol. 1, p. 333._

=1516.= From the very outset of his investigations the physicist has to rely constantly on the aid of the mathematician, for even in the simplest cases, the direct results of his measuring operations are entirely without meaning until they have been submitted to more or less of mathematical discussion. And when in this way some interpretation of the experimental results has been arrived at, and it has been proved that two or more physical quantities stand in a definite relation to each other, the mathematician is very often able to infer, from the existence of this relation, that the quantities in question also fulfill some other relation, that was previously unsuspected. Thus when Coulomb, combining the functions of experimentalist and mathematician, had discovered the law of the force exerted between two particles of electricity, it became a purely mathematical problem, not requiring any further experiment, to ascertain how electricity is distributed upon a charged conductor and this problem has been solved by mathematicians in several cases.--FOSTER, G. C.

_Presidential Address British Association for the Advancement of Science, Section A (1877); Nature, Vol. 16, p. 312-313._

=1517.= Without consummate mathematical skill, on the part of some investigators at any rate, all the higher physical problems would be sealed to us; and without competent skill on the part of the ordinary student no idea can be formed of the nature and cogency of the evidence on which the solutions rest. Mathematics are not merely a gate through which we may approach if we please, but they are the only mode of approach to large and important districts of thought.--VENN, JOHN.

_Symbolic Logic (London and New York, 1894), Introduction, p. xix._

=1518.= Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought.--HOBSON, E. W.

_Presidential Address British Association for the Advancement of Science, Section A (1910); Nature, Vol. 84, pp. 285-286._

=1519.= The immense part which those laws [laws of number and extension] take in giving a deductive character to the other departments of physical science, is well known; and is not surprising, when we consider that all causes operate according to mathematical laws. The effect is always dependent upon, or in mathematical language, is a function of, the quantity of the agent; and generally of its position also. We cannot, therefore, reason respecting causation, without introducing considerations of quantity and extension at every step; and if the nature of the phenomena admits of our obtaining numerical data of sufficient accuracy, the laws of quantity become the grand instruments for calculating forward to an effect, or backward to a cause.

--MILL, J. S.

_System of Logic, Bk. 3, chap. 24, sect. 9._

=1520.= The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to use a wide range of mathematical theory, the chemist is only just beginning to apply the first derivative, i.e. the rate of change at which certain processes are going on; for second derivatives he does not seem to have found any use as yet.--KLEIN, F.

_Lectures on Mathematics (New York, 1911), p. 47._

=1521.= The bond of union among the physical sciences is the mathematical spirit and the mathematical method which pervades them.... Our knowledge of nature, as it advances, continuously resolves differences of quality into differences of quantity. All exact reasoning--indeed all reasoning--about quantity is mathematical reasoning; and thus as our knowledge increases, that portion of it which becomes mathematical increases at a still more rapid rate.--SMITH, H. J. S.

_Presidential Address British Association for the Advancement of Science, Section A (1873); Nature, Vol. 8, p. 449._

=1522.= Another way of convincing ourselves how largely this process [of assimilation of mathematics by physics] has gone on would be to try to conceive the effect of some intellectual catastrophe, supposing such a thing possible, whereby all knowledge of mathematics should be swept away from men’s minds. Would it not be that the departure of mathematics would be the destruction of physics? Objective physical phenomena would, indeed, remain as they are now, but physical science would cease to exist. We should no doubt see the same colours on looking into a spectroscope or polariscope, vibrating strings would produce the same sounds, electrical machines would give sparks, and galvanometer needles would be deflected; but all these things would have lost their meaning; they would be but as the dry bones--the _disjecta membra_--of what is now a living and growing science. To follow this conception further, and to try to image to ourselves in some detail what would be the kind of knowledge of physics which would remain possible, supposing all mathematical ideas to be blotted out, would be extremely interesting, but it would lead us directly into a dim and entangled region where the subjective seems to be always passing itself off for the objective, and where I at least could not attempt to lead the way, gladly as I would follow any one who could show where a firm footing is to be found. But without venturing to do more than to look from a safe distance over this puzzling ground, we may see clearly enough that mathematics is the connective tissue of physics, binding what would else be merely a list of detached observations into an organized body of science.--FOSTER, G. C.

_Presidential Address British Association for the Advancement of Science, Section A (1877); Nature, Vol. 16, p. 313._

=1523.= In _Plato’s_ time mathematics was purely a play of the free intellect; the mathematic-mystical reveries of a Pythagoras foreshadowed a far-reaching significance, but such a significance (except in the case of music) was as yet entirely a matter of fancy; yet even in that time mathematics was the prerequisite to all other studies! But today, when mathematics furnishes the _only_ language by means of which we may formulate the most comprehensive laws of nature, laws which the ancients scarcely dreamed of, when moreover mathematics is the _only_ means by which these laws may be understood,--how few learn today anything of the real essence of our mathematics!... In the schools of today mathematics serves only as a disciplinary study, a mental gymnastic; that it includes the highest ideal value for the comprehension of the universe, one dares scarcely to think of in view of our present day instruction.--LINDEMAN, F.

_Lehren und Lernen in der Mathematik (München, 1904), p. 14._

=1524.= All applications of mathematics consist in extending the empirical knowledge which we possess of a limited number or region of accessible phenomena into the region of the unknown and inaccessible; and much of the progress of pure analysis consists in inventing definite conceptions, marked by symbols, of complicated operations; in ascertaining their properties as independent objects of research; and in extending their meaning beyond the limits they were originally invented for,--thus opening out new and larger regions of thought.--MERZ, J. T.

_History of European Thought in the 19th Century (Edinburgh and London, 1903), Vol. 1, p. 698._

=1525.= All the effects of nature are only mathematical results of a small number of immutable laws.--LAPLACE.

_A Philosophical Essay on Probabilities [Truscott and Emory] (New York, 1902), p. 177; Oeuvres, t. 7, p. 139._

=1526.= What logarithms are to mathematics that mathematics are to the other sciences.--NOVALIS.

_Schriften (Berlin, 1901), Teil 2, p. 222._

=1527.= Any intelligent man may now, by resolutely applying himself for a few years to mathematics, learn more than the great Newton knew after half a century of study and meditation.

--MACAULAY.

_Milton; Critical and Miscellaneous Essays (New York, 1879), Vol. 1, p. 13._

=1528.= In questions of science the authority of a thousand is not worth the humble reasoning of a single individual.--GALILEO.

_Quoted in Arago’s Eulogy on Laplace; Smithsonian Report, 1874, p. 164._

=1529.= Behind the artisan is the chemist, behind the chemist a physicist, behind the physicist a mathematician.--WHITE, W. F.

_Scrap-book of Elementary Mathematics (Chicago, 1908), p. 217._

=1530.= The advance in our knowledge of physics is largely due to the application to it of mathematics, and every year it becomes more difficult for an experimenter to make any mark in the subject unless he is also a mathematician.--BALL, W. W. R.

_History of Mathematics (London, 1901), p. 503._

=1531.= In very many cases the most obvious and direct experimental method of investigating a given problem is extremely difficult, or for some reason or other untrustworthy. In such cases the mathematician can often point out some other problem more accessible to experimental treatment, the solution of which involves the solution of the former one. For example, if we try to deduce from direct experiments the law according to which one pole of a magnet attracts or repels a pole of another magnet, the observed action is so much complicated with the effects of the mutual induction of the magnets and of the forces due to the second pole of each magnet, that it is next to impossible to obtain results of any great accuracy. Gauss, however, showed how the law which applied in the case mentioned can be deduced from the deflections undergone by a small suspended magnetic needle when it is acted upon by a small fixed magnet placed successively in two determinate positions relatively to the needle; and being an experimentalist as well as a mathematician, he showed likewise how these deflections can be measured very easily and with great precision.--FOSTER, G. C.

_Presidential Address British Association for the Advancement of Science, Section A (1877); Nature, Vol. 16, p. 313._

=1532.=

Give me to learn each secret cause; Let Number’s, Figure’s, Motion’s laws Reveal’d before me stand; These to great Nature’s scenes apply, And round the globe, and through the sky, Disclose her working hand. --AKENSIDE, M.

_Hymn to Science._

=1533.= Now there are several scores, upon which skill in mathematicks may be useful to the experimental philosopher. For there are some general advantages, which mathematicks may bring to the minds of men, to whatever study they apply themselves, and consequently to the student of natural philosophy; namely, that these disciplines are wont to make men accurate, and very attentive to the employment that they are about, keeping their thoughts from wandering, and inuring them to patience in going through with tedious and intricate demonstrations; besides, that they much improve reason, by accustoming the mind to deduce successive consequences, and judge of them without easily acquiescing in anything but demonstration.--BOYLE, ROBERT.

_Works (London, 1772), Vol. 3, p. 426._

=1534.= It is not easy to anatomize the constitution and the operations of a mind [like Newton’s] which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization.--WHEWELL, W.

_History of the Inductive Sciences (New York, 1894), Vol. 1, p. 416._

=1535.= The domain of physics is no proper field for mathematical pastimes. The best security would be in giving a geometrical training to physicists, who need not then have recourse to mathematicians, whose tendency is to despise experimental science. By this method will that union between the abstract and the concrete be effected which will perfect the uses of mathematical, while extending the positive value of physical science. Meantime, the use of analysis in physics is clear enough. Without it we should have no precision, and no co-ordination; and what account could we give of our study of heat, weight, light, etc.? We should have merely series of unconnected facts, in which we could foresee nothing but by constant recourse to experiment; whereas, they now have a character of rationality which fits them for purposes of prevision.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 3, chap. 1._

=1536.= It must ever be remembered that the true positive spirit first came forth from the pure sources of mathematical science; and it is only the mind that has imbibed it there, and which has been face to face with the lucid truths of geometry and mechanics, that can bring into full action its natural positivity, and apply it in bringing the most complex studies into the reality of demonstration. No other discipline can fitly prepare the intellectual organ.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 3, chap. 1._

=1537.= During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become _mathematical_. The question now is, not whether this or that hypothesis is better or worse to the pure thought, but whether it accords with observed phenomena in those consequences which can be shown necessarily to follow from it, if it be true. Even in those sciences which are not yet under the dominion of mathematics, and perhaps never will be, a working copy of the mathematical process has been made. This is not known to the followers of those sciences who are not themselves mathematicians, and who very often exalt their horns against the mathematics in consequence. They might as well be squaring the circle, for any sense they show in this particular.--DE MORGAN, A.

_A Budget of Paradoxes (London, 1872), p. 2._

=1538.= Among the mere talkers so far as mathematics are concerned, are to be ranked three out of four of those who apply mathematics to physics, who, wanting a tool only, are very impatient of everything which is not of direct aid to the actual methods which are in their hands.--DE MORGAN, A.

_Graves’ Life of Sir William Rowan Hamilton (New York, 1882-1889), Vol. 3, p. 348._

=1539.= Something has been said about the use of mathematics in physical science, the mathematics being regarded as a weapon forged by others, and the study of the weapon being completely set aside. I can only say that there is danger of obtaining untrustworthy results in physical science, if only the results of mathematics are used; for the person so using the weapon can remain unacquainted with the conditions under which it can be rightly applied.... The results are often correct, sometimes are incorrect; the consequence of the latter class of cases is to throw doubt upon all the applications of such a worker until a result has been otherwise tested. Moreover, such a practice in the use of mathematics leads a worker to a mere repetition in the use of familiar weapons; he is unable to adapt them with any confidence when some new set of conditions arise with a demand for a new method: for want of adequate instruction in the forging of the weapon, he may find himself, sooner or later in the progress of his subject, without any weapon worth having.

--FORSYTH, A. R.

_Perry’s Teaching of Mathematics (London, 1902), p. 36._

=1540.= If in the range of human endeavor after sound knowledge there is one subject that needs to be practical, it surely is Medicine. Yet in the field of Medicine it has been found that branches such as biology and pathology must be studied for themselves and be developed by themselves with the single aim of increasing knowledge; and it is then that they can be best applied to the conduct of living processes. So also in the pursuit of mathematics, the path of practical utility is too narrow and irregular, not always leading far. The witness of history shows that, in the field of natural philosophy, mathematics will furnish the more effective assistance if, in its systematic development, its course can freely pass beyond the ever-shifting domain of use and application.--FORSYTH, A. R.

_Presidential Address British Association for the Advancement of Science, Section A; Nature, Vol. 56 (1897), p. 377._

=1541.= If the Greeks had not cultivated Conic Sections, Kepler could not have superseded Ptolemy; if the Greeks had cultivated Dynamics, Kepler might have anticipated Newton.--WHEWELL, W.

_History of the Inductive Science (New York, 1894), Vol. 1, p. 311._

=1542.= If we may use the great names of Kepler and Newton to signify stages in the progress of human discovery, it is not too much to say that without the treatises of the Greek geometers on the conic sections there could have been no Kepler, without Kepler no Newton, and without Newton no science in the modern sense of the term, or at least no such conception of nature as now lies at the basis of all our science, of nature as subject in the smallest as well as in its greatest phenomena, to exact quantitative relations, and to definite numerical laws.

--SMITH, H. J. S.

_Presidential Address British Association for the Advancement of Science, Section A; Nature, Vol. 8 (1873), p. 450._

=1543.= The silent work of the great Regiomontanus in his chamber at Nuremberg computed the Ephemerides which made possible the discovery of America by Columbus.--RUDIO, F.

_Quoted in Max Simon’s Geschichte der Mathematik im Altertum (Berlin, 1909), Einleitung, p. xi._

=1544.= The calculation of the eclipses of Jupiter’s satellites, many a man might have been disposed, originally, to regard as a most unprofitable study. But the utility of it to navigation (in the determination of longitudes) is now well known.--WHATELY, R.

_Annotations to Bacon’s Essays (Boston, 1783), p. 492._

=1545.= Who could have imagined, when Galvani observed the twitching of the frog muscles as he brought various metals in contact with them, that eighty years later Europe would be overspun with wires which transmit messages from Madrid to St. Petersburg with the rapidity of lightning, by means of the same principle whose first manifestations this anatomist then observed!...

He who seeks for immediate practical use in the pursuit of science, may be reasonably sure, that he will seek in vain. Complete knowledge and complete understanding of the action of forces of nature and of the mind, is the only thing that science can aim at. The individual investigator must find his reward in the joy of new discoveries, as new victories of thought over resisting matter, in the esthetic beauty which a well-ordered domain of knowledge affords, where all parts are intellectually related, where one thing evolves from another, and all show the marks of the mind’s supremacy; he must find his reward in the consciousness of having contributed to the growing capital of knowledge on which depends the supremacy of man over the forces hostile to the spirit.--HELMHOLTZ, H.

_Vorträge und Reden (Braunschweig, 1884), Bd. 1, p. 142._

=1546.= When the time comes that knowledge will not be sought for its own sake, and men will not press forward simply in a desire of achievement, without hope of gain, to extend the limits of human knowledge and information, then, indeed, will the race enter upon its decadence.--HUGHES, C. E.

_Quoted in D. E. Smith’s Teaching of Geometry (Boston, 1911), p. 9._

=1547.= [In the Opus Majus of Roger Bacon] there is a chapter, in which it is proved by reason, that all sciences require mathematics. And the arguments which are used to establish this doctrine, show a most just appreciation of the office of mathematics in science. They are such as follows: That other sciences use examples taken from mathematics as the most evident:--That mathematical knowledge is, as it were, innate to us, on which point he refers to the well-known dialogue of Plato, as quoted by Cicero:--That this science, being the easiest, offers the best introduction to the more difficult:--That in mathematics, things as known to us are identical with things as known to nature:--That we can here entirely avoid doubt and error, and obtain certainty and truth:--That mathematics is prior to other sciences in nature, because it takes cognizance of quantity, which is apprehended by intuition (_intuitu intellectus_). “Moreover,” he adds, “there have been found famous men, as Robert, bishop of Lincoln, and Brother Adam Marshman (de Marisco), and many others, who by the power of mathematics have been able to explain the causes of things; as may be seen in the writings of these men, for instance, concerning the Rainbow and Comets, and the generation of heat, and climates, and the celestial bodies.”--WHEWELL, W.

_History of the Inductive Sciences (New York, 1894), Vol. 1, p. 519. Bacon, Roger: Opus Majus, Part 4, Distinctia Prima, cap. 3._

=1548.= The analysis which is based upon the conception of function discloses to the astronomer and physicist not merely the formulae for the computation of whatever desired distances, times, velocities, physical constants; it moreover gives him insight into the laws of the processes of motion, teaches him to predict future occurrences from past experiences and supplies him with means to a scientific knowledge of nature, i.e. it enables him to trace back whole groups of various, sometimes extremely heterogeneous, phenomena to a minimum of simple fundamental laws.

--PRINGSHEIM, A.

_Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 13, p. 366._

=1549.= “As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).” [Riemann.]

The first of the two problems here indicated by Riemann consists in setting up the differential equation, based upon physical facts and hypotheses. The second is the integration of this differential equation and its application to each separate concrete case, this is the task of mathematics.--WEBER, HEINRICH.

_Die partiellen Differentialgleichungen der mathematischen Physik (Braunschweig, 1882), Bd. 1, Vorrede._

=1550.= Mathematics is the most powerful instrument which we possess for this purpose [to trace into their farthest results those general laws which an inductive philosophy has supplied]: in many sciences a profound knowledge of mathematics is indispensable for a successful investigation. In the most delicate researches into the theories of light, heat, and sound it is the only instrument; they have properties which no other language can express; and their argumentative processes are beyond the reach of other symbols.--PRICE, B.

_Treatise on Infinitesimal Calculus (Oxford, 1858), Vol. 3, p. 5._

=1551.= Notwithstanding the eminent difficulties of the mathematical theory of sonorous vibrations, we owe to it such progress as has yet been made in acoustics. The formation of the differential equations proper to the phenomena is, independent of their integration, a very important acquisition, on account of the approximations which mathematical analysis allows between questions, otherwise heterogeneous, which lead to similar equations. This fundamental property, whose value we have so often to recognize, applies remarkably in the present case; and especially since the creation of mathematical thermology, whose principal equations are strongly analogous to those of vibratory motion.--This means of investigation is all the more valuable on account of the difficulties in the way of direct inquiry into the phenomena of sound. We may decide the necessity of the atmospheric medium for the transmission of sonorous vibrations; and we may conceive of the possibility of determining by experiment the duration of the propagation, in the air, and then through other media; but the general laws of the vibrations of sonorous bodies escape immediate observation. We should know almost nothing of the whole case if the mathematical theory did not come in to connect the different phenomena of sound, enabling us to substitute for direct observation an equivalent examination of more favorable cases subjected to the same law. For instance, when the analysis of the problem of vibrating chords has shown us that, other things being equal, the number of oscillations is in inverse proportion to the length of the chord, we see that the most rapid vibrations of a very short chord may be counted, since the law enables us to direct our attention to very slow vibrations. The same substitution is at our command in many cases in which it is less direct.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 3, chap. 4._

=1552.= Problems relative to the uniform propagation, or to the varied movements of heat in the interior of solids, are reduced ... to problems of pure analysis, and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis. The differential equations ... contain the chief results of the theory; they express, in the most general and concise manner, the necessary relations of numerical analysis to a very extensive class of phenomena; and they connect forever with mathematical science one of the most important branches of natural philosophy.--FOURIER, J.

_Theory of Heat [Freeman], (Cambridge, 1878), Chap. 3, p. 131._

=1553.= The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe.--FOURIER, J.

_Theory of Heat [Freeman], (Cambridge, 1878), Chap. 1, p. 12._

=1554.= Dealing with any and every amount of static electricity, the mathematical mind has balanced and adjusted them with wonderful advantage, and has foretold results which the experimentalist can do no more than verify.... So in respect of the force of gravitation, it has calculated the results of the power in such a wonderful manner as to trace the known planets through their courses and perturbations, and in so doing has _discovered_ a planet before unknown.--FARADAY.

_Some Thoughts on the Conservation of Force._

=1555.= Certain branches of natural philosophy (such as physical astronomy and optics), ... are, in a great measure, inaccessible to those who have not received a regular mathematical education....

--STEWART, DUGALD.

_Philosophy of the Human Mind, Part 3, chap. 1, sect. 3._

=1556.= So intimate is the union between mathematics and physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create “for each successive class of phenomena, a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature.” Sometimes, indeed, the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armoury of the mathematician the weapons which he has needed ready made to his hand. But, much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma.--SMITH, H. J. S.

_Presidential Address British Association for the Advancement of Science, Section A; Nature, Vol. 8 (1873), p. 450._

=1557.= Of all the great subjects which belong to the province of his section, take that which at first sight is the least within the domain of mathematics--I mean meteorology. Yet the part which mathematics plays in meteorology increases every year, and seems destined to increase. Not only is the theory of the simplest instruments essentially mathematical, but the discussions of the observations--upon which, be it remembered, depend the hopes which are already entertained with increasing confidence, of reducing the most variable and complex of all known phenomena to exact laws--is a problem which not only belongs wholly to mathematics, but which taxes to the utmost the resources of the mathematics which we now possess.--SMITH, H. J. S.

_Presidential Address British Association for the Advancement of Science, Section A; Nature, Vol. 8 (1873), p. 449._

=1558.= You know that if you make a dot on a piece of paper, and then hold a piece of Iceland spar over it, you will see not one dot but two. A mineralogist, by measuring the angles of a crystal, can tell you whether or no it possesses this property without looking through it. He requires no scientific thought to do that. But Sir William Roman Hamilton ... knowing these facts and also the explanation of them which Fresnel had given, thought about the subject, and he predicted that by looking through certain crystals in a particular direction we should see not two dots but a continuous circle. Mr. Lloyd made the experiment, and saw the circle, a result which had never been even suspected. This has always been considered one of the most signal instances of scientific thought in the domain of physics.

--CLIFFORD, W. K.

_Lectures and Essays (New York, 1901), Vol. 1, p. 144._

=1559.= The discovery of this planet [Neptune] is justly reckoned as the greatest triumph of mathematical astronomy. Uranus failed to move precisely in the path which the computers predicted for it, and was misguided by some unknown influence to an extent which a keen eye might almost see without telescopic aid.... These minute discrepancies constituted the data which were found sufficient for calculating the position of a hitherto unknown planet, and bringing it to light. Leverrier wrote to Galle, in substance: “_Direct your telescope to a point on the ecliptic in the constellation of Aquarius, in longitude 326°, and you will find within a degree of that place a new planet, looking like a star of about the ninth magnitude, and having a perceptible disc._” The planet was found at Berlin on the night of Sept. 26, 1846, in exact accordance with this prediction, within half an hour after the astronomers began looking for it, and only about 52′ distant from the precise point that Leverrier had indicated.

--YOUNG, C. A.

_General Astronomy (Boston, 1891), Art. 653._

=1560.= I am convinced that the future progress of chemistry as an exact science depends very much indeed upon the alliance with mathematics.--FRANKLAND, A.

_American Journal of Mathematics, Vol. 1, p. 349._

=1561.= It is almost impossible to follow the later developments of physical or general chemistry without a working knowledge of higher mathematics.--MELLOR, J. W.

_Higher Mathematics (New York, 1902), Preface._

=1562.=

... Mount where science guides; Go measure earth, weigh air, and state the tides; Instruct the planets in what orb to run, Correct old time, and regulate the sun. --THOMSON, W.

_On the Figure of the Earth, Title page._

=1563.= Admission to its sanctuary [referring to astronomy] and to the privileges and feelings of a votary, is only to be gained by one means,--_sound and sufficient knowledge of mathematics, the great instrument of all exact inquiry, without which no man can ever make such advances in this or any other of the higher departments of science as can entitle him to form an independent opinion on any subject of discussion within their range._

--HERSCHEL, J.

_Outlines of Astronomy, Introduction, sect. 7._

=1564.= The long series of connected truths which compose the science of astronomy, have been evolved from the appearances and observations by calculation, and a process of reasoning entirely geometrical. It was not without reason that Plato called geometry and arithmetic the wings of astronomy; for it is only by means of these two sciences that we can give a rational account of any of the appearances, or connect any fact with theory, or even render a single observation available to the most common astronomical purpose. It is by geometry that we are enabled to reason our way up through the apparent motions to the real orbits of the planets, and to assign their positions, magnitudes and eccentricities. And it is by application of geometry--a sublime geometry, indeed, invented for the purpose--to the general laws of mechanics, that we demonstrate the law of gravitation, trace it through its remotest effects on the different planets, and, comparing these effects with what we observe, determine the densities and weights of the minutest bodies belonging to the system. The whole science of astronomy is in fact a tissue of geometrical reasoning, applied to the data of observation; and it is from this circumstance that it derives its peculiar character of precision and certainty. To disconnect it from geometry, therefore, and to substitute familiar illustrations and vague description for close and logical reasoning, is to deprive it of its principal advantages, and to reduce it to the condition of an ordinary province of natural history.

_Edinburgh Review, Vol. 58 (1833-1834), p. 168._

=1565.= But geometry is not only the instrument of astronomical investigation, and the bond by which the truths are enchained together,--it is also the instrument of explanation, affording, by the peculiar brevity and perspicuity of its technical processes, not only aid to the learner, but also such facilities to the teacher as he will find it very difficult to supply, if he voluntarily undertakes to forego its assistance. Few undertakings, indeed, are attended with greater difficulty than that of attempting to exhibit the connecting links of a chain of mathematical reasoning, when we lay aside the technical symbols and notation which relieve the memory, and speak at once to the eyes and the understanding:....

_Edinburgh Review, Vol. 58 (1833-1834), p. 169._

=1566.= With an ordinary acquaintance of trigonometry, and the simplest elements of algebra, one may take up any well-written treatise on plane astronomy, and work his way through it, from beginning to end, with perfect ease; and he will acquire, in the course of his progress, from the mere examples put before him, an infinitely more correct and precise idea of astronomical methods and theories, than he could obtain in a lifetime from the most eloquent general descriptions that ever were written. At the same time he will be strengthening himself for farther advances, and accustoming his mind to habits of close comparison and rigid demonstration, which are of infinitely more importance than the acquisition of stores of undigested facts.

_Edinburgh Review, Vol. 58 (1833-1834), p. 170._

=1567.= While the telescope serves as a means of penetrating space, and of bringing its remotest regions nearer us, mathematics, by inductive reasoning, have led us onwards to the remotest regions of heaven, and brought a portion of them within the range of our possibilities; nay, in our own times--so propitious to the extension of knowledge--the application of all the elements yielded by the present conditions of astronomy has even revealed to the intellectual eyes a heavenly body, and assigned to it its place, orbit, mass, before a single telescope has been directed towards it.--HUMBOLDT, A.

_Cosmos [Otte], Vol. 2, part 2, sect. 3._

=1568.= Mighty are numbers, joined with art resistless.--EURIPIDES.

_Hecuba, Line 884._

=1569.= No single instrument of youthful education has such mighty power, both as regards domestic economy and politics, and in the arts, as the study of arithmetic. Above all, arithmetic stirs up him who is by nature sleepy and dull, and makes him quick to learn, retentive, shrewd, and aided by art divine he makes progress quite beyond his natural powers.--PLATO.

_Laws [Jowett,] Bk. 5, p. 747._

=1570.= For all the higher arts of construction some acquaintance with mathematics is indispensable. The village carpenter, who, lacking rational instruction, lays out his work by empirical rules learned in his apprenticeship, equally with the builder of a Britannia Bridge, makes hourly reference to the laws of quantitative relations. The surveyor on whose survey the land is purchased; the architect in designing a mansion to be built on it; the builder in preparing his estimates; his foreman in laying out the foundations; the masons in cutting the stones; and the various artisans who put up the fittings; are all guided by geometrical truths. Railway-making is regulated from beginning to end by mathematics: alike in the preparation of plans and sections; in staking out the lines; in the mensuration of cuttings and embankments; in the designing, estimating, and building of bridges, culverts, viaducts, tunnels, stations. And similarly with the harbors, docks, piers, and various engineering and architectural works that fringe the coasts and overspread the face of the country, as well as the mines that run underneath it. Out of geometry, too, as applied to astronomy, the art of navigation has grown; and so, by this science, has been made possible that enormous foreign commerce which supports a large part of our population, and supplies us with many necessaries and most of our luxuries. And nowadays even the farmer, for the correct laying out of his drains, has recourse to the level--that is, to geometrical principles.--SPENCER, HERBERT.

_Education, chap. 1._

=1571.= [Arithmetic] is another of the great master-keys of life. With it the astronomer opens the depths of the heavens; the engineer, the gates of the mountains; the navigator, the pathways of the deep. The skillful arrangement, the rapid handling of figures, is a perfect magician’s wand. The mighty commerce of the United States, foreign and domestic, passes through the books kept by some thousands of diligent and faithful clerks. Eight hundred bookkeepers, in the Bank of England, strike the monetary balance of half the civilized world. Their skill and accuracy in applying the common rules of arithmetic are as important as the enterprise and capital of the merchant, or the industry and courage of the navigator. I look upon a well-kept ledger with something of the pleasure with which I gaze on a picture or a statue. It is a beautiful work of art.--EVERETT, EDWARD.

_Orations and Speeches (Boston, 1870), Vol. 3, p. 47._

=1572.= [Mathematics] is the fruitful Parent of, I had almost said all, Arts, the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to Human Affairs. In which last Respect, we may be said to receive from the _Mathematics_, the principal Delights of Life, Securities of Health, Increase of Fortune, and Conveniences of Labour: That we dwell elegantly and commodiously, build decent Houses for ourselves, erect stately Temples to God, and leave wonderful Monuments to Posterity: That we are protected by those Rampires from the Incursions of the Enemy; rightly use Arms, skillfully range an Army, and manage War by Art, and not by the Madness of wild Beasts: That we have safe Traffick through the deceitful Billows, pass in a direct Road through the tractless Ways of the Sea, and come to the designed Ports by the uncertain Impulse of the Winds: That we rightly cast up our Accounts, do Business expeditiously, dispose, tabulate, and calculate scattered Ranks of Numbers, and easily compute them, though expressive of huge Heaps of Sand, nay immense Hills of Atoms: That we make pacifick Separations of the Bounds of Lands, examine the Moments of Weights in an equal Balance, and distribute every one his own by a just Measure: That with a light Touch we thrust forward vast Bodies which way we will, and stop a huge Resistance with a very small Force: That we accurately delineate the Face of this Earthly Orb, and subject the Oeconomy of the Universe to our Sight: That we aptly digest the flowing Series of Time, distinguish what is acted by due Intervals, rightly account and discern the various Returns of the Seasons, the stated Periods of Years and Months, the alternate Increments of Days and Nights, the doubtful Limits of Light and Shadow, and the exact Differences of Hours and Minutes: That we derive the subtle Virtue of the Solar Rays to our Uses, infinitely extend the Sphere of Sight, enlarge the near Appearances of Things, bring to Hand Things remote, discover Things hidden, search Nature out of her Concealments, and unfold her dark Mysteries: That we delight our Eyes with beautiful Images, cunningly imitate the Devices and portray the Works of Nature; imitate did I say? nay excel, while we form to ourselves Things not in being, exhibit Things absent, and represent Things past: That we recreate our Minds and delight our Ears with melodious Sounds, attemperate the inconstant Undulations of the Air to musical Tunes, add a pleasant Voice to a sapless Log and draw a sweet Eloquence from a rigid Metal; celebrate our Maker with an harmonious Praise, and not unaptly imitate the blessed Choirs of Heaven: That we approach and examine the inaccessible Seats of the Clouds, the distant Tracts of Land, unfrequented Paths of the Sea; lofty Tops of the Mountains, low Bottoms of the Valleys, and deep Gulphs of the Ocean: That in Heart we advance to the Saints themselves above, yea draw them to us, scale the etherial Towers, freely range through the celestial Fields, measure the Magnitudes, and determine the Interstices of the Stars, prescribe inviolable Laws to the Heavens themselves, and confine the wandering Circuits of the Stars within fixed Bounds: Lastly, that we comprehend the vast Fabrick of the Universe, admire and contemplate the wonderful Beauty of the Divine Workmanship, and to learn the incredible Force and Sagacity of our own Minds, by certain Experiments, and to acknowledge the Blessings of Heaven with pious Affection.--BARROW, ISAAC.

_Mathematical Lectures (London, 1734), pp. 27-30._

=1573.= Analytical and graphical treatment of statistics is employed by the economist, the philanthropist, the business expert, the actuary, and even the physician, with the most surprisingly valuable results; while symbolic language involving mathematical methods has become a part of wellnigh every large business. The handling of pig-iron does not seem to offer any opportunity for mathematical application. Yet graphical and analytical treatment of the data from long-continued experiments with this material at Bethlehem, Pennsylvania, resulted in the discovery of the law that fatigue varied in proportion to a certain relation between the load and the periods of rest. Practical application of this law increased the amount handled by each man from twelve and a half to forty-seven tons per day. Such study would have been impossible without preliminary acquaintance with the simple invariable elements of mathematics.--KARPINSKY, L.

_High School Education (New York, 1912), chap. 6, p. 134._

=1574.= They [computation and arithmetic] belong then, it seems, to the branches of learning which we are now investigating;--for a military man must necessarily learn them with a view to the marshalling of his troops, and so must a philosopher with the view of understanding real being, after having emerged from the unstable condition of becoming, or else he can never become an apt reasoner.

That is the fact he replied.

But the guardian of ours happens to be both a military man and a philosopher.

Unquestionably so.

It would be proper then, Glaucon, to lay down laws for this branch of science and persuade those about to engage in the most important state-matters to apply themselves to computation, and study it, not in the common vulgar fashion, but with the view of arriving at the contemplation of the nature of numbers by the intellect itself,--not for the sake of buying and selling as anxious merchants and retailers, but for war also, and that the soul may acquire a facility in turning itself from what is in the course of generation to truth and real being.--PLATO.

_Republic [Davis], Bk. 7, p. 525._

=1575.= The scientific part of Arithmetic and Geometry would be of more use for regulating the thoughts and opinions of men than all the great advantage which Society receives from the general application of them: and this use cannot be spread through the Society by the practice; for the Practitioners, however dextrous, have no more knowledge of the Science than the very instruments with which they work. They have taken up the Rules as they found them delivered down to them by scientific men, without the least inquiry after the Principles from which they are derived: and the more accurate the Rules, the less occasion there is for inquiring after the Principles, and consequently, the more difficult it is to make them turn their attention to the First Principles; and, therefore, a Nation ought to have both Scientific and Practical Mathematicians.--WILLIAMSON, JAMES.

_Elements of Euclid with Dissertations (Oxford, 1781)._

=1576.= _Where there is nothing to measure there is nothing to calculate_, hence it is impossible to employ mathematics in psychological investigations. Thus runs the syllogism compounded of an adherence to usage and an apparent truth. As to the latter, it is wholly untrue that we may calculate only where we have measured. Exactly the opposite is true. Every hypothetically assumed law of quantitative combination, even such as is recognized as invalid, is subject to calculation; and in case of deeply hidden but important matters it is imperative to try on hypotheses and to subject the consequences which flow from them to precise computation until it is found which one of the various hypotheses coincides with experience. Thus the ancient astronomers _tried_ eccentric circles, and Kepler _tried_ the ellipse to account for the motion of the planets, the latter also compared the squares of the times of revolution with the cubes of the mean distances before he discovered their agreement. In like manner Newton _tried_ whether a gravitation, varying inversely as the square of the distance, sufficed to keep the moon in its orbit about the earth; if this supposition had failed him, he would have tried some other power of the distance, as the fourth or fifth, and deduced the corresponding consequences to compare them with the observations. Just this is the greatest benefit of mathematics, that it enables us to survey the possibilities whose range includes the actual, long before we have adequate definite experience; this makes it possible to employ very incomplete indications of experience to avoid at least the crudest errors. Long before the transit of Venus was employed in the determination of the sun’s parallax, it was attempted to determine the instant at which the sun illumines exactly one-half of the moon’s disk, in order to compute the sun’s distance from the known distance of the moon from the earth. This was not possible, for, owing to psychological reasons, our method of measuring time is too crude to give us the desired instant with sufficient accuracy; yet the attempt gave us the knowledge that the sun’s distance from us is at least several hundred times as great as that of the moon. This illustration shows clearly that even a very imperfect estimate of a magnitude in a case where no precise observation is possible, may become very instructive, if we know how to exploit it. Was it necessary to know the scale of our solar system in order to learn of its order in general? Or, taking an illustration from another field, was it impossible to investigate the laws of motion until it was known exactly how far a body falls in a second at some definite place? Not at all. Such determinations of _fundamental measures_ are in themselves exceedingly difficult, but fortunately, such investigations form a class of their own; our knowledge of _fundamental laws_ does not need to wait on these. To be sure, computation invites measurement, and every easily observed regularity of certain magnitudes is an incentive to mathematical investigation.

--HERBART, J. F.

_Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 97._

=1577.= Those who pass for naturalists, have, for the most part, been very little, or not at all, versed in mathematicks, if not also jealous of them.--BOYLE, ROBERT.

_Works (London, 1772), Vol. 3, p. 426._

=1578.= However hurtful may have been the incursions of the geometers, direct and indirect, into a domain which it is not for them to cultivate, the physiologists are not the less wrong in turning away from mathematics altogether. It is not only that without mathematics they could not receive their due preliminary training in the intervening sciences: it is further necessary for them to have geometrical and mechanical knowledge, to understand the structure and the play of the complex apparatus of the living, and especially the animal organism. Animal mechanics, statical and dynamical, must be unintelligible to those who are ignorant of the general laws of rational mechanics. The laws of equilibrium and motion are ... absolutely universal in their action, depending wholly on the energy, and not at all on the nature of the forces considered: and the only difficulty is in their numerical application in cases of complexity. Thus, discarding all idea of a numerical application in biology, we perceive that the general theorems of statics and dynamics must be steadily verified in the mechanism of living bodies, on the rational study of which they cast an indispensable light. The highest orders of animals act in repose and motion, like any other mechanical apparatus of a similar complexity, with the one difference of the mover, which has no power to alter the laws of motion and equilibrium. The participation of rational mechanics in positive biology is thus evident. Mechanics cannot dispense with geometry; and beside, we see how anatomical and physiological speculations involve considerations of form and position, and require a familiar knowledge of the principal geometrical laws which may cast light upon these complex relations.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 5, chap. 1._

=1579.= In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry on their researches.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 5, chap. 1._

=1580.= In this school [of mathematics] must they [biologists] learn familiarly the real characters and conditions of scientific evidence, in order to transfer it afterwards to the province of their own theories. The study of it here, in the most simple and perfect cases, is the only sound preparation for its recognition in the most complex.

The study is equally necessary for the formation of intellectual habits; for obtaining an aptitude in forming and sustaining positive abstractions, without which the comparative method cannot be used in either anatomy or physiology. The abstraction which is to be the standard of comparison must be first clearly formed, and then steadily maintained in its integrity, or the analysis becomes abortive: and this is so completely in the spirit of mathematical combinations, that practice in them is the best preparation for it. A student who cannot accomplish the process in the more simple case may be assured that he is not qualified for the higher order of biological researches, and must be satisfied with the humbler office of collecting materials for the use of minds of another order. Hence arises another use of mathematical training;--that of testing and classifying minds, as well as preparing and guiding them. Probably as much good would be done by excluding the students who only encumber the science by aimless and desultory inquiries, as by fitly instituting those who can better fulfill its conditions.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 5, chap. 1._

=1581.= There seems no sufficient reason why the use of scientific fictions, so common in the hands of geometers, should not be introduced into biology, if systematically employed, and adopted with sufficient sobriety. In mathematical studies, great advantages have arisen from imagining a series of hypothetical cases, the consideration of which, though artificial, may aid the clearing up of the real subject, or its fundamental elaboration. This art is usually confounded with that of hypotheses; but it is entirely different; inasmuch as in the latter case the solution alone is imaginary; whereas in the former, the problem itself is radically ideal. Its use can never be in biology comparable to what it is in mathematics: but it seems to me that the abstract character of the higher conceptions of comparative biology renders them susceptible of such treatment. The process will be to intercalate, among different known organisms, certain purely fictitious organisms, so imagined as to facilitate their comparison, by rendering the biological series more homogeneous and continuous: and it might be that several might hereafter meet with more or less of a realization among organisms hitherto unexplored. It may be possible, in the present state of our knowledge of living bodies, to conceive of a new organism capable of fulfilling certain given conditions of existence. However that may be, the collocation of real cases with well-imagined ones, after the manner of geometers, will doubtless be practised hereafter, to complete the general laws of comparative anatomy and physiology, and possibly to anticipate occasionally the direct exploration. Even now, the rational use of such an artifice might greatly simplify and clear up the ordinary system of biological instruction. But it is only the highest order of investigators who can be trusted with it. Whenever it is adopted, it will constitute another ground of relation between biology and mathematics.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 5, chap. 1._

=1582.= I think it may safely enough be affirmed, that he, that is not so much as indifferently skilled in mathematicks, can hardly be more than indifferently skilled in the fundamental principles of physiology.--BOYLE, ROBERT.

_Works (London, 1772), Vol. 3, p. 430._

=1583.= It is not only possible but necessary that mathematics be applied to psychology; the reason for this necessity lies briefly in this: that by no other means can be reached that which is the ultimate aim of all speculation, namely conviction.--HERBART, J. F.

_Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 104._

=1584.= All more definite knowledge must start with computation; and this is of most important consequences not only for the theory of memory, of imagination, of understanding, but as well for the doctrine of sensations, of desires, and affections.

--HERBART, J. F.

_Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 103._

=1585.= In the near future mathematics will play an important part in medicine: already there are increasing indications that physiology, descriptive anatomy, pathology and therapeutics cannot escape mathematical legitimation.--DESSOIR, MAX.

_Westermann’s Monatsberichte, Bd. 77, p. 380; Ahrens: Scherz und Ernst in der Mathematik (Leipzig, 1904), p. 395._

=1586.= The social sciences mathematically developed are to be the controlling factors in civilization.--WHITE, W. F.

_A Scrap-book of Elementary Mathematics (Chicago, 1908), p. 208._

=1587.= It is clear that this education [referring to education preparatory to the science of sociology] must rest on a basis of mathematical philosophy, even apart from the necessity of mathematics to the study of inorganic philosophy. It is only in the region of mathematics that sociologists, or anybody else, can obtain a true sense of scientific evidence, and form the habit of rational and decisive argumentation; can, in short, learn to fulfill the logical conditions of all positive speculation, by studying universal positivism at its source. This training, obtained and employed with the more care on account of the eminent difficulty of social science, is what sociologists have to seek in mathematics.--COMTE, A.

_Positive Philosophy [Martineau], Bk. 6, chap. 4._

=1588.= It is clear that the individual as a social unit and the state as a social aggregate require a certain modicum of mathematics, some arithmetic and algebra, to conduct their affairs. Under this head would fall the theory of interest, simple and compound, matters of discount and amortization, and, if lotteries hold a prominent place in raising moneys, as in some states, questions of probability must be added. As the state becomes more highly organized and more interested in the scientific analysis of its life, there appears an urgent necessity for various statistical information, and this can be properly obtained, reduced, correlated, and interpreted only when the guiding spirit in the work have the necessary mathematical training in the theory of statistics. (Figures may not lie, but statistics compiled unscientifically and analyzed incompetently are almost sure to be misleading, and when this condition is unnecessarily chronic the so-called statisticians may well be called liars.) The dependence of insurance of various kinds on statistical information and the very great place which insurance occupies in the modern state, albeit often controlled by private corporations instead of by the government, makes the theories of paramount importance to our social life.--WILSON, E. B.

_Bulletin American Mathematical Society, Vol. 18 (1912), p. 463._

=1589.= The theory of probabilities and the theory of errors now constitute a formidable body of knowledge of great mathematical interest and of great practical importance. Though developed largely through the applications to the more precise sciences of astronomy, geodesy, and physics, their range of applicability extends to all the sciences; and they are plainly destined to play an increasingly important rôle in the development and in the applications of the sciences of the future. Hence their study is not only a commendable element in a liberal education, but some knowledge of them is essential to a correct understanding of daily events.--WOODWARD, R. S.

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

=1590.= It was not to be anticipated that a new science [the science of probabilities] which took its rise in games of chance, and which had long to encounter an obloquy, hardly yet extinct, due to the prevailing idea that its only end was to facilitate and encourage the calculations of gamblers, could ever have attained its present status--that its aid should be called for in every department of natural science, both to assist in discovery, which it has repeatedly done (even in pure mathematics), to minimize the unavoidable errors of observation, and to detect the presence of causes as revealed by observed events. Nor are commercial and other practical interests of life less indebted to it: wherever the future has to be forecasted, risk to be provided against, or the true lessons to be deduced from statistics, it corrects for us the rough conjectures of common sense, and decides which course is really, according to the lights of which we are in possession, the wisest for us to pursue.--CROFTON, M. W.

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

=1591.= The calculus of probabilities, when confined within just limits, ought to interest, in an equal degree, the mathematician, the experimentalist, and the statesman. From the time when Pascal and Fermat established its first principles, it has rendered, and continues daily to render, services of the most eminent kind. It is the calculus of probabilities, which, after having suggested the best arrangements of the tables of population and mortality, teaches us to deduce from those numbers, in general so erroneously interpreted, conclusions of a precise and useful character; it is the calculus of probabilities which alone can regulate justly the premiums to be paid for assurances; the reserve funds for the disbursements of pensions, annuities, discounts, etc. It is under its influence that lotteries and other shameful snares cunningly laid for avarice and ignorance have definitely disappeared.--ARAGO.

_Eulogy on Laplace [Baden-Powell], Smithsonian Report, 1874, p. 164._

=1592.= Men were surprised to hear that not only births, deaths, and marriages, but the decisions of tribunals, the results of popular elections, the influence of punishments in checking crime, the comparative values of medical remedies, the probable limits of error in numerical results in every department of physical inquiry, the detection of causes, physical, social, and moral, nay, even the weight of evidence and the validity of logical argument, might come to be surveyed with the lynx-eyed scrutiny of a dispassionate analysis.--HERSCHEL, J.

_Quoted in Encyclopedia Britannica, 9th Edition; Article “Probability.”_

=1593.= If economists expect of the application of the mathematical method any extensive concrete numerical results, and it is to be feared that like other non-mathematicians all too many of them think of mathematics as merely an arithmetical science, they are bound to be disappointed and to find a paucity of results in the works of the few of their colleagues who use that method. But they should rather learn, as the mathematicians among them know full well, that mathematics is much broader, that it has an abstract quantitative (or even qualitative) side, that it deals with relations as well as numbers, ....--WILSON, E. B.

_Bulletin American Mathematical Society, Vol. 18 (1912), p. 464._

=1594.= The effort of the economist is to _see_, to picture the inter-play of economic elements. The more clearly cut these elements appear in his vision, the better; the more elements he can grasp and hold in his mind at once, the better. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looms up in firm, bold outlines. The old phantasmagoria disappear. We see better. We also see further.--FISHER, IRVING.

_Transactions of Connecticut Academy, Vol. 9 (1892), p. 119._

=1595.= In the great inquiries of the moral and social sciences ... mathematics (I always mean Applied Mathematics) affords the only sufficient type of deductive art. Up to this time, I may venture to say that no one ever knew what deduction is, as a means of investigating the laws of nature, who had not learned it from mathematics, nor can any one hope to understand it thoroughly, who has not, at some time in his life, known enough of mathematics to be familiar with the instrument at work.

--MILL, J. S.

_An Examination of Sir William Hamilton’s Philosophy (London, 1878), p. 622._

=1596.= Let me pass on to say a word or two about the teaching of mathematics as an academic training for general professional life. It has immense capabilities in that respect. If you consider how much of the effectiveness of an administrator depends upon the capacity for co-ordinating appropriately a number of different ideas, precise accuracy of definition, rigidity of proof, and sustained reasoning, strict in every step, and when you consider what substitutes for these things nine men out of every ten without special training have to put up with, it is clear that a man with a mathematical training has incalculable advantages.--SHAW, W. H.

_Perry’s Teaching of Mathematics (London, 1902), p. 73._

=1597.= Before you enter on the study of law a sufficient ground work must be laid.... Mathematics and natural philosophy are so useful in the most familiar occurrences of life and are so peculiarly engaging and delightful as would induce everyone to wish an acquaintance with them. Besides this, the faculties of the mind, like the members of a body, are strengthened and improved by exercise. Mathematical reasoning and deductions are, therefore, a fine preparation for investigating the abstruse speculations of the law.--JEFFERSON, THOMAS.

_Quoted in Cajori’s Teaching and History of Mathematics in the U. S. (Washington, 1890), p. 35._

=1598.= It has been observed in England of the study of law,--though the acquisition of the most difficult parts of its learning, the interpretation of laws, the comparison of authorities, and the construction of instruments, would seem to require philological and critical training; though the weighing of evidence and the investigation of probable truth belong to the province of the moral sciences, and the peculiar duties of the advocate require rhetorical skill,--yet that a large proportion of the most distinguished members of the profession has proceeded from the university (that of Cambridge) most celebrated for the cultivation of mathematical studies.--EVERETT, EDWARD.

_Orations and Speeches (Boston, 1870), Vol. 2, p. 511._

=1599.= All historic science tends to become mathematical. Mathematical power is classifying power.--NOVALIS.

_Schriften (Berlin, 1901), Teil 2, p. 192._

=1599a.= History has never regarded itself as a science of statistics. It was the Science of Vital Energy in relation with time; and of late this radiating centre of its life has been steadily tending,--together with every form of physical and mechanical energy,--toward mathematical expression.--ADAM, HENRY.

_A Letter to American Teachers of History (Washington, 1910), p. 115._

=1599b.= Mathematics can be shown to sustain a certain relation to rhetoric and may aid in determining its laws.--SHERMAN L. A.

_University [of Nebraska] Studies, Vol. 1, p. 130._