QUANTITATIVE INDUCTION.
We have not yet formally considered any processes of reasoning
which have for their object to disclose laws of nature expressed in
quantitative equations. We have been inquiring into the modes by which
a phenomenon may be measured, and, if it be a composite phenomenon, may
be resolved, by the aid of several measurements, into its component
parts. We have also considered the precautions to be taken in the
performance of observations and experiments in order that we may know
what phenomena we really do measure, but we must remember that, no
number of facts and observations can by themselves constitute science.
Numerical facts, like other facts, are but the raw materials of
knowledge, upon which our reasoning faculties must be exerted in order
to draw forth the principles of nature. It is by an inverse process of
reasoning that we can alone discover the mathematical laws to which
varying quantities conform. By well-conducted experiments we gain a
series of values of a variable, and a corresponding series of values
of a variant, and we now want to know what mathematical function the
variant is as regards the variable. In the usual progress of a science
three questions will have to be answered as regards every important
quantitative phenomenon:--
(1) Is there any constant relation between a variable and a variant?
(2) What is the empirical formula expressing this relation?
(3) What is the rational formula expressing the law of nature involved?
*Probable Connection of Varying Quantities.*
We find it stated by Mill,[398] that “Whatever phenomenon varies in any
manner whenever another phenomenon varies in some particular manner, is
either a cause or an effect of that phenomenon, or is connected with
it through some fact of causation.” This assertion may be considered
true when it is interpreted with sufficient caution; but it might
otherwise lead us into error. There is nothing whatever in the nature
of things to prevent the existence of two variations which should
apparently follow the same law, and yet have no connection with each
other. One binary star might be going through a revolution which, so
far as we could tell, was of equal period with that of another binary
star, and according to the above rule the motion of one would be the
cause of the motion of the other, which would not be really the case.
Two astronomical clocks might conceivably be made so nearly perfect
that, for several years, no difference could be detected, and we might
then infer that the motion of one clock was the cause or effect of the
motion of the other. This matter requires careful discrimination. We
must bear in mind that the continuous quantities of space, time, force,
&c., which we measure, are made up of an infinite number of infinitely
small units. We may then meet with two variable phenomena which follow
laws so nearly the same, that in no part of the variations open to
our observation can any discrepancy be discovered. I grant that if
two clocks could be shown to have kept *exactly* the same time during
any finite interval, the probability would become infinitely high
that there was a connection between their motions. But we can never
absolutely prove such coincidences to exist. Allow that we may observe
a difference of one-tenth of a second in their time, yet it is possible
that they were independently regulated so as to go together within
less than that quantity of time. In short, it would require either an
infinitely long time of observation, or infinitely acute powers of
measuring discrepancy, to decide positively whether two clocks were or
were not in relation with each other.
[398] *System of Logic*, bk. iii. chap. viii § 6.
A similar question actually occurs in the case of the moon’s motion. We
have no record that any other portion of the moon was ever visible to
men than such as we now see. This fact sufficiently proves that within
the historical period the rotation of the moon on its own axis has
coincided with its revolutions round the earth. Does this coincidence
prove a relation of cause and effect to exist? The answer must be in
the negative, because there might have been so slight a discrepancy
between the motions that there has not yet been time to produce any
appreciable effect. There may nevertheless be a high probability of
connection.
The whole question of the relation of quantities thus resolves itself
into one of probability. When we can only rudely measure a quantitative
result, we can assign but slight importance to any correspondence.
Because the brightness of two stars seems to vary in the same manner,
there is no considerable probability that they have any relation with
each other. Could it be shown that their periods of variation were
the same to infinitely small quantities it would be certain, that
is infinitely probable, that they were connected, however unlikely
this might be on other grounds. The general mode of estimating such
probabilities is identical with that applied to other inductive
problems. That any two periods of variation should by chance become
*absolutely equal* is infinitely improbable; hence if, in the case of
the moon or other moving bodies, we could prove absolute coincidence
we should have certainty of connection.[399] With approximate
measurements, which alone are within our power, we must hope for
approximate certainty at the most.
[399] Laplace, *System of the World*, translated by Harte, vol. ii.
p. 366.
The principles of inference and probability, according to which we
treat causes and effects varying in amount, are exactly the same as
those by which we treated simple experiments. Continuous quantity,
however, affords us an infinitely more extensive sphere of observation,
because every different amount of cause, however little different,
ought to be followed by a different amount of effect. If we can measure
temperature to the one-hundredth part of a degree centigrade, then
between 0° and 100° we have 10,000 possible trials. If the precision
of our measurements is increased, so that the one-thousandth part of
a degree can be appreciated, our trials may be increased tenfold. The
probability of connection will be proportional to the accuracy of our
measurements.
When we can vary the quantity of a cause at will it is easy to discover
whether a certain effect is due to that cause or not. We can then
make as many irregular changes as we like, and it is quite incredible
that the supposed effect should by chance go through exactly the
corresponding series of changes except by dependence. If we have a
bell ringing *in vacuo*, the sound increases as we let in the air, and
it decreases again as we exhaust the air. Tyndall’s singing flames
evidently obeyed the directions of his own voice; and Faraday when he
discovered the relation of magnetism and light found that, by making
or breaking or reversing the current of the electro-magnet, he had
complete command over a ray of light, proving beyond all reasonable
doubt the dependence of cause and effect. In such cases it is the
perfect coincidence in time between the change in the effect and that
in the cause which raises a high improbability of casual coincidence.
It is by a simple case of variation that we infer the existence of
a material connection between two bodies moving with exactly equal
velocity, such as the locomotive engine and the train which follows
it. Elaborate observations were requisite before astronomers could all
be convinced that the red hydrogen flames seen during solar eclipses
belonged to the sun, and not to the moon’s atmosphere as Flamsteed
assumed. As early as 1706, Stannyan noticed a blood-red streak in
an eclipse which he witnessed at Berne, and he asserted that it
belonged to the sun; but his opinion was not finally established until
photographs of the eclipse in 1860, taken by Mr. De la Rue, showed that
the moon’s dark body gradually covered the red prominences on one side,
and uncovered those on the other; in short, that these prominences
moved precisely as the sun moved, and not as the moon moved.
Even when we have no means of accurately measuring the variable
quantities we may yet be convinced of their connection, if one always
varies perceptibly at the same time as the other. Fatigue increases
with exertion; hunger with abstinence from food; desire and degree of
utility decrease with the quantity of commodity consumed. We know that
the sun’s heating power depends upon his height of the sky; that the
temperature of the air falls in ascending a mountain; that the earth’s
crust is found to be perceptibly warmer as we sink mines into it; we
infer the direction in which a sound comes from the change of loudness
as we approach or recede. The facility with which we can time after
time observe the increase or decrease of one quantity with another
sufficiently shows the connection, although we may be unable to assign
any precise law of relation. The probability in such cases depends upon
frequent coincidence in time.
*Empirical Mathematical Laws.*
It is important to acquire a clear comprehension of the part which
is played in scientific investigation by empirical formulæ and laws.
If we have a table containing certain values of a variable and the
corresponding values of the variant, there are mathematical processes
by which we can infallibly discover a mathematical formula yielding
numbers in more or less exact agreement with the table. We may
generally assume that the quantities will approximately conform to a
law of the form
*y* = A + B*x* + C*x*^{2},
in which *x* is the variable and *y* the variant. We can then select
from the table three values of *y*, and the corresponding values of
*x*; inserting them in the equation, we obtain three equations by the
solution of which we gain the values of A, B, and C. It will be found
as a general rule that the formula thus obtained yields the other
numbers of the table to a considerable degree of approximation.
In many cases even the second power of the variable will be
unnecessary; Regnault found that the results of his elaborate inquiry
into the latent heat of steam at different pressures were represented
with sufficient accuracy by the empirical formula
λ = 606·5 + 0·305 *t*,
in which λ is the total heat of the steam, and *t* the
temperature.[400] In other cases it may be requisite to include the
third power of the variable. Thus physicists assume the law of the
dilatation of liquids to be of the form
δ_{t} = *at* + *bt*^{2} + *ct*^{3},
[400] *Chemical Reports and Memoirs*, Cavendish Society, p. 294.
and they calculate from results of observation the values of the
three constants *a*, *b*, *c*, which are usually small quantities
not exceeding one-hundredth part of a unit, but requiring to be
determined with great accuracy.[401] Theoretically speaking, this
process of empirical representation might be applied with any degree of
accuracy; we might include still higher powers in the formula, and with
sufficient labour obtain the values of the constants, by using an equal
number of experimental results. The method of least squares may also be
employed to obtain the most probable values of the constants.
[401] Jamin, *Cours de Physique*, vol. ii. p. 38.
In a similar manner all periodic variations may be represented with any
required degree of accuracy by formulæ involving the sines and cosines
of angles and their multiples. The form of any tidal or other wave may
thus be expressed, as Sir G. B. Airy has explained.[402] Almost all
the phenomena registered by meteorologists are periodic in character,
and when freed from disturbing causes may be embodied in empirical
formulæ. Bessel has given a rule by which from any regular series of
observations we may, on the principle of the method of least squares,
calculate out with a moderate amount of labour a formula expressing the
variation of the quantity observed, in the most probable manner. In
meteorology three or four terms are usually sufficient for representing
any periodic phenomenon, but the calculation might be carried to any
higher degree of accuracy. As the details of the process have been
described by Herschel in his treatise on Meteorology,[403] I need not
further enter into them.
[402] *On Tides and Waves*, Encyclopædia Metropolitana, p. 366*.
[403] *Encyclopædia Britannica*, art. *Meteorology*. Reprint, §§
152–156.
The reader might be tempted to think that in these processes of
calculation we have an infallible method of discovering inductive
laws, and that my previous statements (Chap. VII.) as to the purely
tentative and inverse character of the inductive process are negatived.
Were there indeed any general method of inferring laws from facts it
would overturn my statement, but it must be carefully observed that
these empirical formulæ do not coincide with natural laws. They are
only approximations to the results of natural laws founded upon the
general principles of approximation. It has already been pointed out
that however complicated be the nature of a curve, we may examine so
small a portion of it, or we may examine it with such rude means of
measurement, that its divergence from an elliptic curve will not be
apparent. As a still ruder approximation a portion of a straight line
will always serve our purpose; but if we need higher precision a curve
of the third or fourth degree will almost certainly be sufficient. Now
empirical formulæ really represent these approximate curves, but they
give us no information as to the precise nature of the curve itself to
which we are approximating. We do not learn what function the variant
is of the variable, but we obtain another function which, within the
bounds of observation, gives nearly the same values.
*Discovery of Rational Formulæ.*
Let us now proceed to consider the modes in which from numerical
results we can establish the actual relation between the quantity of
the cause and that of the effect. What we want is a *rational* formula
or function, which will exhibit the *reason* or exact nature and
origin of the law in question. There is no word more frequently used
by mathematicians than the word *function*, and yet it is difficult
to define its meaning with perfect accuracy. Originally it meant
performance or execution, being equivalent to the Greek λειτουργία
or τέλεσμα. Mathematicians at first used it to mean *any power of a
quantity*, but afterwards generalised it so as to include “any quantity
formed in any manner whatsoever from another quantity.”[404] Any
quantity, then, which depends upon and varies with another quantity may
be called a function of it, and either may be considered a function of
the other.
[404] Lagrange, *Leçons sur le Calcul des Fonctions*, 1806, p. 4.
Given the quantities, we want the function of which they are the
values. Simple inspection of the numbers cannot as a general rule
disclose the function. In an earlier chapter (p. 124) I put before the
reader certain numbers, and requested him to point out the law which
they obey, and the same question will have to be asked in every case of
quantitative induction. There are perhaps three methods, more or less
distinct, by which we may hope to obtain an answer:
(1) By purely haphazard trial.
(2) By noting the general character of the variation of the quantities,
and trying by preference functions which give a similar form of
variation.
(3) By deducing from previous knowledge the form of the function which
is most likely to suit.
Having numerical results we are always at liberty to invent any kind
of mathematical formula we like, and then try whether, by the suitable
selection of values for the unknown constant quantities, we can make it
give the required results. If ever we fall upon a formula which does
so, to a fair degree of approximation, there is a presumption in favour
of its being the true function, although there is no certainty whatever
in the matter. In this way I discovered a simple mathematical law which
closely agreed with the results of my experiments on muscular exertion.
This law was afterwards shown by Professor Haughton to be the true
rational law according to his theory of muscular action.[405]
[405] Haughton, *Principles of Animal Mechanics*, 1873, pp. 444–450.
Jevons, *Nature*, 30th of June, 1870, vol. ii. p. 158. See also the
experiments of Professor Nipher, of Washington University, St. Louis,
in *American Journal of Science*, vol. ix. p. 130, vol. x. p. 1;
*Nature*, vol. xi. pp. 256, 276.
But the chance of succeeding in this manner is small. The number of
possible functions is infinite, and even the number of comparatively
simple functions is so large that the probability of falling upon the
correct one by mere chance is very slight. Even when we obtain the law
it is by a deductive process, not by showing that the numbers give the
law, but that the law gives the numbers.
In the second way, we may, by a survey of the numbers, gain a general
notion of the kind of law they are likely to obey, and we may be much
assisted in this process by drawing them out in the form of a curve.
We can in this way ascertain with some probability whether the curve
is likely to return into itself, or whether it has infinite branches;
whether such branches are asymptotic, that is, approach infinitely
towards straight lines; whether it is logarithmic in character, or
trigonometric. This indeed we can only do if we remember the results of
previous investigations. The process is still inversely deductive, and
consists in noting what laws give particular curves, and then inferring
inversely that such curves belong to such laws. If we can in this way
discover the class of functions to which the required law belongs,
our chances of success are much increased, because our haphazard
trials are now reduced within a narrower sphere. But, unless we have
almost the whole curve before us, the identification of its character
must be a matter of great uncertainty; and if, as in most physical
investigations, we have a mere fragment of the curve, the assistance
given would be quite illusory. Curves of almost any character can be
made to approximate to each other for a limited extent, so that it is
only by a kind of *divination* that we fall upon the actual function,
unless we have theoretical knowledge of the kind of function applicable
to the case.
When we have once obtained what we believe to be the correct form of
function, the remainder of the work is mere mathematical computation to
be performed infallibly according to fixed rules,[406] which include
those employed in the determination of empirical formulæ (p. 487).
The function will involve two or three or more unknown constants, the
values of which we need to determine by our experimental results.
Selecting some of our results widely apart and nearly equidistant,
we form by means of them as many equations as there are constant
quantities to be determined. The solution of these equations will then
give us the constants required, and having now the actual function we
can try whether it gives with sufficient accuracy the remainder of
our experimental results. If not, we must either make a new selection
of results to give a new set of equations, and thus obtain a new set
of values for the constants, or we must acknowledge that our form
of function has been wrongly chosen. If it appears that the form of
function has been correctly ascertained, we may regard the constants
as only approximately accurate and may proceed by the Method of Least
Squares (p. 393) to determine the most probable values as given by the
whole of the experimental results.
[406] Jamin, *Cours de Physique*, vol. ii. p. 50.
In most cases we shall find ourselves obliged to fall back upon the
third mode, that is, anticipation of the form of the law to be expected
on the ground of previous knowledge. Theory and analogical reasoning
must be our guides. The general nature of the phenomenon will often
indicate the kind of law to be looked for. If one form of energy or one
kind of substance is being converted into another, we may expect the
law of direct simple proportion. In one distinct class of cases the
effect already produced influences the amount of the ensuing effect,
as for instance in the cooling of a heated body, when the law will be
of an exponential form. When the direction of a force influences its
action, trigonometrical functions enter. Any influence which spreads
freely through tridimensional space will be subject to the law of
the inverse square of the distance. From such considerations we may
sometimes arrive deductively and analogically at the general nature of
the mathematical law required.
*The Graphical Method.*
In endeavouring to discover the mathematical law obeyed by
experimental results it is often desirable to call in the aid of
space-representations. Every equation involving two variable quantities
corresponds to some kind of plane curve, and every plane curve may
be represented symbolically in an equation containing two unknown
quantities. Now in an experimental research we obtain a number of
values of the variant corresponding to an equal number of values of the
variable; but all the numbers are affected by more or less error, and
the values of the variable will often be irregularly disposed. Even if
the numbers were absolutely correct and disposed at regular intervals,
there is, as we have seen, no direct mode of discovering the law, but
the difficulty of discovery is much increased by the uncertainty and
irregularity of the results.
Under such circumstances, the best mode of proceeding is to prepare a
paper divided into equal rectangular spaces, a convenient size for the
spaces being one-tenth of an inch square. The values of the variable
being marked off on the lowest horizontal line, a point is marked for
each corresponding value of the variant perpendicularly above that of
the variable, and at such a height as corresponds to the value of the
variant.
The exact scale of the drawing is not of much importance, but it may
require to be adjusted according to circumstances, and different values
must often be attributed to the upright and horizontal divisions, so
as to make the variations conspicuous but not excessive. If a curved
line be drawn through all the points or ends of the ordinates, it will
probably exhibit irregular inflections, owing to the errors which
affect the numbers. But, when the results are numerous, it becomes
apparent which results are more divergent than others, and guided by a
so-called *sense of continuity*, it is possible to trace a line among
the points which will approximate to the true law more nearly than the
points themselves. The accompanying figure sufficiently explains itself.
[Illustration]
Perkins employed this graphical method with much care in exhibiting
the results of his experiments on the compression of water.[407] The
numerical results were marked upon a sheet of paper very exactly ruled
at intervals of one-tenth of an inch, and the original marks were left
in order that the reader might judge of the correctness of the curve
drawn, or choose another for himself. Regnault carried the method to
perfection by laying off the points with a screw dividing engine;[408]
and he then formed a table of results by drawing a continuous curve,
and measuring its height for equidistant values of the variable. Not
only does a curve drawn in this manner enable us to infer numerical
results more free from accidental errors than any of the numbers
obtained directly from experiment, but the form of the curve sometimes
indicates the class of functions to which our results belong.
[407] *Philosophical Transactions*, 1826, p. 544.
[408] Jamin, *Cours de Physique*, vol. ii. p. 24, &c.
Engraved sheets of paper prepared for the drawing of curves may be
obtained from Mr. Stanford at Charing Cross, Messrs. W. and A. K.
Johnston, of London and Edinburgh, Waterlow and Sons, Letts and Co.,
and probably other publishers. When we do not require great accuracy,
paper ruled by the common machine-ruler into equal squares of about
one-fifth or one-sixth of an inch square will serve well enough. I
have met with engineers’ and surveyors’ memorandum books ruled with
one-twelfth inch squares. When a number of curves have to be drawn, I
have found it best to rule a good sheet of drawing paper with lines
carefully adjusted at the most convenient distances, and then to
prick the points of the curve through it upon another sheet fixed
underneath. In this way we obtain an accurate curve upon a blank sheet,
and need only introduce such division lines as are requisite to the
understanding of the curve.
In some cases our numerical results will correspond, not to the
height of single ordinates, but to the area of the curve between two
ordinates, or the average height of ordinates between certain limits.
If we measure, for instance, the quantities of heat absorbed by water
when raised in temperature from 0° to 5°, from 5° to 10°, and so on,
these quantities will really be represented by *areas* of the curve
denoting the specific heat of water; and since the specific heat varies
continuously between every two points of temperature, we shall not get
the correct curve by simply laying off the quantities of heat at the
mean temperatures, namely 2-1/2°, and 7-1/2°, and so on. Lord Rayleigh
has shown that if we have drawn such an incorrect curve, we can with
little trouble correct it by a simple geometrical process, and obtain
to a close approximation the true ordinates instead of those denoting
areas.[409]
[409] J. W. Strutt, *On a correction sometimes required in curves
professing to represent the connexion between two physical
magnitudes*. Philosophical Magazine, 4th Series, vol. xlii. p. 441.
*Interpolation and Extrapolation.*
When we have by experiment obtained two or more numerical results,
and endeavour, without further experiment, to calculate intermediate
results, we are said to *interpolate*. If we wish to assign by
reasoning results lying beyond the limits of experiment, we may be
said, using an expression of Sir George Airy, to *extrapolate*. These
two operations are the same in principle, but differ in practicability.
It is a matter of great scientific importance to apprehend precisely
how far we can practise interpolation or extrapolation, and on what
grounds we proceed.
In the first place, if the interpolation is to be more than empirical,
we must have not only the experimental results, but the laws which they
obey--we must in fact go through the complete process of scientific
investigation. Having discovered the laws of nature applying to the
case, and verified them by showing that they agree with the experiments
in question, we are then in a position to anticipate the results of
similar experiments. Our knowledge even now is not certain, because we
cannot completely prove the truth of any assumed law, and we cannot
possibly exhaust all the circumstances which may affect the result. At
the best then our interpolations will partake of the want of certainty
and precision attaching to all our knowledge of nature. Yet, having the
supposed laws, our results will be as sure and accurate as any we can
attain to. But such a complete procedure is more than we commonly mean
by interpolation, which usually denotes some method of estimating in a
merely approximate manner the results which might have been expected
independently of a theoretical investigation.
Regarded in this light, interpolation is in reality an indeterminate
problem. From given values of a function it is impossible to determine
that function; for we can invent an infinite number of functions which
will give those values if we are not restricted by any conditions, just
as through a given series of points we can draw an infinite number of
curves, if we may diverge between or beyond the points into bends and
cusps as we think fit.[410] In interpolation we must in fact be guided
more or less by *à priori* considerations; we must know, for instance,
whether or not periodical fluctuations are to be expected. Supposing
that the phenomenon is non-periodic, we proceed to assume that the
function can be expressed in a limited series of the powers of the
variable. The number of powers which can be included depends upon the
number of experimental results available, and must be at least one less
than this number. By processes of calculation, which have been already
alluded to in the section on empirical formulæ, we then calculate the
coefficients of the powers, and obtain an empirical formula which will
give the required intermediate results. In reality, then, we return
to the methods treated under the head of approximation and empirical
formulæ; and interpolation, as commonly understood, consists in
assuming that a curve of simple character is to pass through certain
determined points. If we have, for instance, two experimental results,
and only two, we assume that the curve is a straight line; for the
parabolas which can be passed through two points are infinitely various
in magnitude, and quite indeterminate. One straight line alone can
pass through two points, and it will have an equation of the form,
*y* = *mx* + *n*, the constant quantities of which can be determined
from two results. Thus, if the two values for *x*, 7 and 11, give
the values for *y*, 35 and 53, the solution of two equations gives
*y* = 4·5 × *x* + 3·5 as the equation, and for any other value of *x*,
for instance 10, we get a value of *y*, that is 48·5. When we take a
mean value of *x*, namely 9, this process yields a simple mean result,
namely 44. Three experimental results being given, we assume that they
fall upon a portion of a parabola and algebraic calculation gives the
position of any intermediate point upon the parabola. Concerning the
process of interpolation as practised in the science of meteorology
the reader will find some directions in the French edition of Kaëmtz’s
Meteorology.[411]
[410] Herschel: Lacroix’ *Differential Calculus*, p. 551.
[411] *Cours complet de Météorologie*, Note A, p. 449.
When we have, either by direct experiment or by the use of a curve,
a series of values of the variant for equidistant values of the
variable, it is instructive to take the differences between each
value of the variant and the next, and then the differences between
those differences, and so on. If any series of differences approaches
closely to zero it is an indication that the numbers may be correctly
represented by a finite empirical formula; if the *n*th differences
are zero, then the formula will contain only the first *n* - 1 powers
of the variable. Indeed we may sometimes obtain by the calculus of
differences a correct empirical formula; for if *p* be the first term
of the series of values, and Δ*p*, Δ^{2}*p*, Δ^{3}*p*, be the first
number in each column of differences, then the *m*th term of the series
of values will be
*p* + *m*Δ*p* + *m*[(*m* - 1)/2]Δ^{2}*p* +
*m*[(*m* - 1)/2][(*m* - 2)/3]Δ^{3}*p* + &c.
A closely equivalent but more practicable formula for interpolation
by differences, as devised by Lagrange, will be found in Thomson and
Tait’s *Elements of Natural Philosophy*, p. 115.
If no column of differences shows any tendency to become zero
throughout, it is an indication that the law is of a more complicated,
for instance of an exponential character, so that it requires different
treatment. Dr. J. Hopkinson has suggested a method of arithmetical
interpolation,[412] which is intended to avoid much that is arbitrary
in the graphical method. His process will yield the same results in all
hands.
[412] *On the Calculation of Empirical Formulæ. The Messenger of
Mathematics*, New Series, No. 17, 1872.
So far as we can infer the results likely to be obtained by variations
beyond the limits of experiment, we must proceed upon the same
principles. If possible we must detect the exact laws in action,
and then trust to them as a guide when we have no experience. If
not, an empirical formula of the same character as those employed in
interpolation is our only resource. But to extend our inference far
beyond the limits of experience is exceedingly unsafe. Our knowledge is
at the best only approximate, and takes no account of small tendencies.
Now it usually happens that tendencies small within our limits of
observation become perceptible or great under extreme circumstances.
When the variable in our empirical formula is small, we are justified
in overlooking the higher powers, and taking only two or three lower
powers. But as the variable increases, the higher powers gain in
importance, and in time yield the principal part of the value of the
function.
This is no mere theoretical inference. Excepting the few primary
laws of nature, such as the law of gravity, of the conservation of
energy, &c., there is hardly any natural law which we can trust in
circumstances widely different from those with which we are practically
acquainted. From the expansion or contraction, fusion or vaporisation
of substances by heat at the surface of the earth, we can form a most
imperfect notion of what would happen near the centre of the earth,
where the pressure almost infinitely exceeds anything possible in our
experiments. The physics of the earth give us a feeble, and probably a
misleading, notion of a body like the sun, in which an inconceivably
high temperature is united with an inconceivably high pressure. If
there are in the realms of space nebulæ consisting of incandescent
and unoxidised vapours of metals and other elements, so highly heated
perhaps that chemical composition is out of the question, we are
hardly able to treat them as subjects of scientific inference. Hence
arises the great importance of experiments in which we investigate the
properties of substances under extreme circumstances of cold or heat,
density or rarity, intense electric excitation, &c. This insecurity
in extending our inferences arises from the approximate character
of our measurements. Had we the power of appreciating infinitely
small quantities, we should by the principle of continuity discover
some trace of every change which a substance could undergo under
unattainable circumstances. By observing, for instance, the tension of
aqueous vapour between 0° and 100° C., we ought theoretically to be
able to infer its tension at every other temperature; but this is out
of the question practically because we cannot really ascertain the law
precisely between those temperatures.
Many instances might be given to show that laws which appear to
represent correctly the results of experiments within certain limits
altogether fail beyond those limits. The experiments of Roscoe and
Dittmar, on the absorption of gases in water[413] afford interesting
illustrations, especially in the case of hydrochloric acid, the
quantity of which dissolved in water under different pressures
follows very closely a linear law of variation, from which however it
diverges widely at low pressures.[414] Herschel, having deduced from
observations of the double star γ Virginis an elliptic orbit for the
motion of one component round the centre of gravity of both, found that
for a time the motion of the star agreed very well with this orbit.
Nevertheless divergence began to appear and after a time became so
great that an entirely new orbit, of more than double the dimensions of
the old one, had ultimately to be adopted.[415]
[413] Watts’ *Dictionary of Chemistry*, vol. ii. p. 790.
[414] *Quarterly Journal of the Chemical Society*, vol. viii. p. 15.
[415] *Results of Observations at the Cape of Good Hope*, p. 293.
*Illustrations of Empirical Quantitative Laws.*
Although our object in quantitative inquiry is to discover the exact or
rational formulæ, expressing the laws which apply to the subject, it
is instructive to observe in how many important branches of science,
no precise laws have yet been detected. The tension of aqueous vapour
at different temperatures has been determined by a succession of
eminent experimentalists--Dalton, Kaëmtz, Dulong, Arago, Magnus, and
Regnault--and by the last mentioned the measurements were conducted
with extraordinary care. Yet no incontestable general law has been
established. Several functions have been proposed to express the
elastic force of the vapour as depending on the temperature. The first
form is that of Young, namely F = (*a* + *b t*)^{m}, in which *a*,
*b*, and *m* are unknown quantities to be determined by observation.
Roche proposed, on theoretical grounds, a complicated formula of an
exponential form, and a third form of function is that of Biot,[416] as
follows--log F = *a* + *b*α^{t} + *c*β^{t}. I mention these formulæ,
because they well illustrate the feeble powers of empirical inquiry.
None of the formulæ can be made to correspond closely with experimental
results, and the two last forms correspond almost equally well. There
is very little probability that the real law has been reached, and
it is unlikely that it will be discovered except by deduction from
mechanical theory.
[416] Jamin, *Cours de Physique*, vol. ii. p. 138.
Much ingenious labour has been spent upon the discovery of some general
law of atmospheric refraction. Tycho Brahe and Kepler commenced the
inquiry: Cassini first formed a table of refractions, calculated on
theoretical grounds: Newton entered into some profound investigations
upon the subject: Brooke Taylor, Bouguer, Simpson, Bradley, Mayer,
and Kramp successively attacked the question, which is of the highest
practical importance as regards the correction of astronomical
observations. Laplace next laboured on the subject without exhausting
it, and Brinkley and Ivory have also treated it. The true law is
yet undiscovered. A closely connected problem, that regarding the
relation between the pressure and elevation in different strata of
the atmosphere, has received the attention of a long succession of
physicists and was most carefully investigated by Laplace. Yet no
invariable and general law has been detected. The same may be said
concerning the law of human mortality; abundant statistics on this
subject are available, and many hypotheses more or less satisfactory
have been put forward as to the form of the curve of mortality, but it
seems to be impossible to discover more than an approximate law.
It may perhaps be urged that in such subjects no single invariable
law can be expected. The atmosphere may be divided into several
variable strata which by their unconnected changes frustrate the exact
calculations of astronomers. Human life may be subject at different
ages to a succession of different influences incapable of reduction
under any one law. The results observed may in fact be aggregates of an
immense number of separate results each governed by its own separate
laws, so that the subjects may be complicated beyond the possibility of
complete resolution by empirical methods. This is certainly true of the
mathematical functions which must some time or other be introduced into
the science of political economy.
*Simple Proportional Variation.*
When we first treat numerical results in any novel kind of
investigation, our impression will probably be that one quantity
varies in *simple proportion* to another, so as to obey the law
*y* = *mx* + *n*. We must learn to distinguish carefully between the
cases where this proportionality is really, and where it is only
apparently true. In considering the principles of approximation we
found that a small portion of any curve will appear to be a straight
line. When our modes of measurement are comparatively rude, we must
expect to be unable to detect the curvature. Kepler made meritorious
attempts to discover the law of refraction, and he approximated to it
when he observed that the angles of incidence and refraction *if small*
bear a constant ratio to each other. Angles when small are nearly as
their sines, so that he reached an approximate result of the true law.
Cardan assumed, probably as a mere guess, that the force required to
sustain a body on an inclined plane was simply proportional to the
angle of elevation of the plane. This is approximately the case when
the angle is small, but in reality the law is much more complicated,
the power required being proportional to the sine of the angle. The
early thermometer-makers were unaware whether the expansion of mercury
was proportional or not to the heat communicated to it, and it is only
in the present century that we have learnt it to be not so. We now
know that even gases obey the law of uniform expansion by heat only
in an approximate manner. Until reason to the contrary is shown, we
should do well to look upon every law of simple proportion as only
provisionally true.
Nevertheless many important laws of nature are in the form of simple
proportions. Wherever a cause acts in independence of its previous
effects, we may expect this relation. An accelerating force acts
equally upon a moving and a motionless body. Hence the velocity
produced is in simple proportion to the force, and to the duration of
its uniform action. As gravitating bodies never interfere with each
other’s gravity, this force is in direct simple proportion to the
mass of each of the attracting bodies, the mass being measured by, or
proportional to inertia. Similarly, in all cases of “direct unimpeded
action,” as Herschel has remarked,[417] we may expect simple proportion
to manifest itself. In such cases the equation expressing the relation
may have the simple form *y* = *mx*.
[417] *Preliminary Discourse*, &c., p. 152.
A similar relation holds true when there is conversion of one substance
or form of energy into another. The quantity of a compound is equal
to the quantity of the elements which combine. The heat produced in
friction is exactly proportional to the mechanical energy absorbed.
It was experimentally proved by Faraday that “the chemical power of
the current of electricity is in direct proportion to the quantity of
electricity which passes.” When an electric current is produced, the
quantity of electric energy is simply proportional to the weight of
metal dissolved. If electricity is turned into heat, there is again
simple proportion. Wherever, in fact, one thing is but another thing
with a new aspect, we may expect to find the law of simple proportion.
But it is only in the most elementary cases that this simple relation
will hold true. Simple conditions do not, generally speaking, produce
simple results. The planets move in approximate circles round the sun,
but the apparent motions, as seen from the earth, are very various. All
those motions, again, are summed up in the law of gravity, of no great
complexity; yet men never have been, and never will be, able to exhaust
the complications of action and reaction arising from that law, even
among a small number of planets. We should be on our guard against a
tendency to assume that the connection of cause and effect is one of
direct proportion. Bacon reminds us of the woman in Æsop’s fable, who
expected that her hen, with a double measure of barley, would lay two
eggs a day instead of one, whereas it grew fat, and ceased to lay any
eggs at all. It is a wise maxim that the half is often better than the
whole.