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._