GENERAL RULES FOR THE CONSTRUCTION OF THE CONCEPTION.
APHORISM XXXVIII.
_The Construction of the Conception very often includes, in a great
measure, the Determination of the Magnitudes._
APHORISM XXXIX.
_When a series of_ progressive _numbers is given as the result of
observation, it may generally be reduced to law by combinations of
arithmetical and geometrical progressions._
APHORISM XL.
_A true formula for a progressive series of numbers cannot commonly
be obtained from a_ narrow range _of observations._
APHORISM XLI.
Recurrent _series of numbers must, in most cases, be expressed by
circular formulæ._
APHORISM XLII.
_The true construction of the conception is frequently suggested by
some hypothesis; and in these cases, the hypothesis may be useful,
though containing superfluous parts._
I. IN speaking of the discovery of laws of nature, those which
depend upon _quantity_, as number, space, and the like, are most
prominent and most easily conceived, and therefore in speaking of
such researches, we shall often use language which applies
peculiarly to {196} the cases in which quantities numerically
measurable are concerned, leaving it for a subsequent task to extend
our principles to ideas of other kinds.
Hence we may at present consider the Construction of a Conception
which shall include and connect the facts, as being the construction
of a Mathematical Formula, coinciding with the numerical expression
of the facts; and we have to consider how this process can be
facilitated, it being supposed that we have already before us the
numerical measures given by observation.
2. We may remark, however, that the construction of the right
Formula for any such case, and the determination of the Coefficients
of such formula, which we have spoken of as two separate steps, are
in practice almost necessarily simultaneous; for the near
coincidence of the results of the theoretical rule with the observed
facts confirms at the same time the Formula and its Coefficients. In
this case also, the mode of arriving at truth is to try various
hypotheses;--to modify the hypotheses so as to approximate to the
facts, and to multiply the facts so as to test the hypotheses.
The Independent Variable, and the Formula which we would try, being
once selected, mathematicians have devised certain special and
technical processes by which the value of the coefficients may be
determined. These we shall treat of in the next Chapter; but in the
mean time we may note, in a more general manner, the mode in which,
in physical researches, the proper formula may be obtained.
3. A person somewhat versed in mathematics, having before him a
series of numbers, will generally be able to devise a formula which
approaches near to those numbers. If, for instance, the series is
constantly progressive, he will be able to see whether it more
nearly resembles an arithmetical or a geometrical progression. For
example, MM. Dulong and Petit, in their investigation of the law of
cooling of bodies, obtained the following series of measures. A
thermometer, made hot, was placed in an enclosure of which the
temperature was 0 degrees, and the rapidity of {197} cooling of the
thermometer was noted for many temperatures. It was found that
For the temperature 240 the rapidity of cooling was 10·69
220 " 8·81
200 " 7·40
180 " 6·10
160 " 4·89
140 " 3·88
and so on. Now this series of numbers manifestly increases with
greater rapidity as we proceed from the lower to the higher parts of
the scale. The numbers do not, however, form a geometrical series,
as we may easily ascertain. But if we were to take the differences
of the successive terms we should find them to be--
1·88, 1·41, 1·30, 1·21, 1·01, &c.
and these numbers are very nearly the terms of a geometric series.
For if we divide each term by the succeeding one, we find these
numbers,
1·33, 1·09, 1·07, 1·20, 1·27,
in which there does not appear to be any constant tendency to
diminish or increase. And we shall find that a geometrical series in
which the ratio is 1·165, may be made to approach very near to this
series, the deviations from it being only such as may be accounted
for by conceiving them as errours of observation. In this manner a
certain formula[26\3] is obtained, giving results {198} which very
nearly coincide with the observed facts, as may be seen in the
margin.
[Note 26\3: The formula is _v_ = 2·037(_a^t_ - 1) where _v_ is the
velocity of cooling, _t_ the temperature of the thermometer
expressed in degrees, and _a_ is the quantity, 1·0077.
The degree of coincidence is as follows:--
Excess of temperature of Observed Calculated
the thermometer, or values values
values of _t_. of _v_. of _v_.
240 10·69 10·68
220 8·81 8·89
200 7·40 7·34
180 6·10 6·03
160 4·89 4·87
140 3·88 3·89
120 3·02 3·05
100 2·30 2·33
80 1·74 1·72 ]
The physical law expressed by the formula just spoken of is
this:--that when a body is cooling in an empty inclosure which is
kept at a constant temperature, the quickness of the cooling, for
excesses of temperature in arithmetical progression, increases as
the terms of a geometrical progression, diminished by a constant
number.
4. In the actual investigation of Dulong and Petit, however, the
formula was not obtained in precisely the manner just described. For
the quickness of cooling depends upon two elements, the temperature
of the hot body and the temperature of the inclosure; not merely
upon the _excess_ of one of these over the other. And it was found
most convenient, first, to make such experiments as should exhibit
the dependence of the velocity of cooling upon the temperature of
the enclosure; which dependence is contained in the following
law:--The quickness of cooling of a thermometer in vacuo for a
constant excess of temperature, increases in geometric progression,
when the temperature of the inclosure increases in arithmetic
progression. From this law the preceding one follows by necessary
consequence[27\3].
[Note 27\3: For if _θ_ be the temperature of the inclosure, and _t_
the excess of temperature of the hot body, it appears, by this law,
that the radiation of heat is as _a^θ_. And hence the quickness of
cooling, which is as the excess of radiation, is as _a^θ+t_ - _a^θ_;
that is, as _a^θ_(_a^t_ - 1) which agrees with the formula given in
the last note.
The whole of this series of researches of Dulong and Petit is full
of the most beautiful and instructive artifices for the construction
of the proper formulæ in physical research.]
This example may serve to show the nature of the artifices which may
be used for the construction of formulæ, when we have a constantly
progressive series of numbers to represent. We must not only
endeavour by trial to contrive a formula which will answer the
conditions, but we must vary our experiments so as to determine,
first one factor or portion of the formula, and then the other; and
we must use the most {199} probable hypothesis as means of
suggestion for our formulæ.
5. In a _progressive_ series of numbers, unless the formula which we
adopt be really that which expresses the law of nature, the
deviations of the formula from the facts will generally become
enormous, when the experiments are extended into new parts of the
scale. True formulæ for a progressive series of results can hardly
ever be obtained from a very limited range of experiments: just as
the attempt to guess the general course of a road or a river, by
knowing two or three points of it in the neighbourhood of one
another, would generally fail. In the investigation respecting the
laws of the cooling of bodies just noticed, one great advantage of
the course pursued by the experimenters was, that their experiments
included so great a range of temperatures. The attempts to assign
the law of elasticity of steam deduced from experiments made with
moderate temperatures, were found to be enormously wrong, when very
high temperatures were made the subject of experiment. It is easy to
see that this must be so: an arithmetical and a geometrical series
may nearly coincide for a few terms moderately near each other: but
if we take remote corresponding terms in the two series, one of
these will be very many times the other. And hence, from a narrow
range of experiments, we may infer one of these series when we ought
to infer the other; and thus obtain a law which is widely erroneous.
6. In Astronomy, the series of observations which we have to study
are, for the most part, not progressive, but _recurrent_. The
numbers observed do not go on constantly increasing; but after
increasing up to a certain amount they diminish; then, after a
certain space, increase again; and so on, changing constantly
through certain _cycles_. In cases in which the observed numbers are
of this kind, the formula which expresses them must be a _circular
function_, of some sort or other; involving, for instance, sines,
tangents, and other forms of calculation, which have recurring
values when the angle on which they depend goes on constantly {200}
increasing. The main business of formal astronomy consists in
resolving the celestial phenomena into a series of _terms_ of this
kind, in detecting their _arguments_, and in determining their
_coefficients_.
7. In constructing the formulæ by which laws of nature are
expressed, although the first object is to assign the Law of the
Phenomena, philosophers have, in almost all cases, not proceeded in
a purely empirical manner, to connect the observed numbers by some
expression of calculation, but have been guided, in the selection of
their formula, by some _Hypothesis_ respecting the mode of connexion
of the facts. Thus the formula of Dulong and Petit above given was
suggested by the Theory of Exchanges; the first attempts at the
resolution of the heavenly motions into circular functions were
clothed in the hypothesis of Epicycles. And this was almost
inevitable. 'We must confess,' says Copernicus[28\3], 'that the
celestial motions are circular, or compounded of several circles,
since their inequalities observe a fixed law, and recur in value at
certain intervals, which could not be except they were circular: for
a circle alone can make that quantity which has occurred recur
again.' In like manner the first publication of the _Law of the
Sines_, the true formula of optical refraction, was accompanied by
Descartes with an hypothesis, in which an explanation of the law was
pretended. In such cases, the mere comparison of observations may
long fail in suggesting the true formulæ. The fringes of shadows and
other diffracted colours were studied in vain by Newton, Grimaldi,
Comparetti, the elder Herschel, and Mr. Brougham, so long as these
inquirers attempted merely to trace the laws of the facts as they
appeared in themselves; while Young, Fresnel, Fraunhofer, Schwerdt,
and others, determined these laws in the most rigorous manner, when
they applied to the observations the Hypothesis of Interferences.
[Note 28\3: _De Rev._ l. i. c. iv.]
8. But with all the aid that Hypotheses and Calculation can afford,
the construction of true formulæ, in {201} those cardinal
discoveries by which the progress of science has mainly been caused,
has been a matter of great labour and difficulty, and of good
fortune added to sagacity. In the _History of Science_, we have seen
how long and how hard Kepler laboured, before he converted the
formula for the planetary motions, from an _epicyclical_
combination, to a simple _ellipse_. The same philosopher, labouring
with equal zeal and perseverance to discover the formula of optical
refraction, which now appears to us so simple, was utterly foiled.
Malus sought in vain the formula determining the Angle at which a
transparent surface polarizes light: Sir D. Brewster[29\3], with a
happy sagacity, discovered the formula to be simply this, that the
_index_ of refraction is the _tangent_ of the angle of polarization.
[Note 29\3: _Hist. Ind. Sc._ b. ix. c. vi.]
Though we cannot give rules which will be of much service when we
have thus to divine the general form of the relation by which
phenomena are connected, there are certain methods by which, in a
narrower field, our investigations may be materially
promoted;--certain special methods of obtaining laws from
Observations. Of these we shall now proceed to treat.
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