SPECIAL METHODS OF INDUCTION APPLICABLE TO QUANTITY.
APHORISM XLIII.
_There are special Methods of Induction applicable to Quantity; of
which the principal are, the_ Method of Curves, _the_ Method of
Means, _the_ Method of Least Squares, _and the_ Method of Residues.
APHORISM XLIV.
The Method of Curves _consists in drawing a curve of which the
observed quantities are the Ordinates, the quantity on which the
change of these quantities depends being the Abscissa. The efficacy
of this Method depends upon the faculty which the eye possesses, of
readily detecting regularity and irregularity in forms. The Method
may be used to detect the Laws which the observed quantities follow:
and also, when the Observations are inexact, it may be used to
correct these Observations, so as to obtain data more true than the
observed facts themselves._
APHORISM XLV.
The Method of Means _gets rid of irregularities by taking the
arithmetical mean of a great number of observed quantities. Its
efficacy depends upon this; that in cases in which observed
quantities are affected by other inequalities, besides that of which
we wish to determine the law, the excesses_ above _and defects_
below _the quantities which the law in question would produce, will,
in a collection of_ many _observations_, balance _each other._ {203}
APHORISM XLVI.
The Method of Least Squares _is a Method of Means, in which the mean
is taken according to the condition, that the sum of the squares of
the errours of observation shall be the least possible which the law
of the facts allows. It appears, by the Doctrine of Chances, that
this is the_ most probable _mean._
APHORISM XLVII.
The Method of Residues _consists in subtracting, from the quantities
given by Observation, the quantity given by any Law already
discovered; and then examining the remainder, or_ Residue, _in order
to discover the leading Law which it follows. When this second Law
has been discovered, the quantity given by it may be subtracted from
the first Residue; thus giving a_ Second Residue, _which may be
examined in the same manner; and so on. The efficacy of this method
depends principally upon the circumstance of the Laws of variation
being successively smaller and smaller in amount (or at least in
their mean effect); so that the ulterior undiscovered Laws do not
prevent the Law in question from being_ prominent _in the
observations._
APHORISM XLVIII.
_The Method of Means and the Method of Least Squares cannot be
applied without our_ knowing the Arguments _of the Inequalities
which we seek. The Method of Curves and the Method of Residues, when
the Arguments of the principal Inequalities are known, often make it
easy to find the others._
IN cases where the phenomena admit of numerical measurement and
expression, certain mathematical methods may be employed to
facilitate and give accuracy to the determination of the formula by
which the observations are connected into laws. Among the most usual
and important of these Methods are the following:--{204}
I. The Method of Curves.
II. The Method of Means.
III. The Method of Least Squares.
IV. The Method of Residues.
SECT. I.--_The Method of Curves._
1. THE Method of Curves proceeds upon this basis; that when one
quantity undergoes a series of changes depending on the progress of
another quantity, (as, for instance, the Deviation of the Moon from
her equable place depends upon the progress of Time,) this
dependence may be expressed by means of a _curve_. In the language
of mathematicians, the variable quantity, whose changes we would
consider, is made the _ordinate_ of the curve, and the quantity on
which the changes depend is made the _abscissa_. In this manner, the
curve will exhibit in its form a series of undulations, rising and
falling so as to correspond with the alternate Increase and
Diminution of the quantity represented, at intervals of Space which
correspond to the intervals of Time, or other quantity by which the
changes are regulated. Thus, to take another example, if we set up,
at equal intervals, a series of ordinates representing the Height of
all the successive High Waters brought by the tides at a given
place, for a year, the curve which connects the summits of all these
ordinates will exhibit a series of undulations, ascending and
descending once in about each Fortnight; since, in that interval, we
have, in succession, the high spring tides and the low neap tides.
The curve thus drawn offers to the eye a picture of the order and
magnitude of the changes to which the quantity under contemplation,
(the height of high water,) is subject.
2. Now the peculiar facility and efficacy of the Method of Curves
depends upon this circumstance;--that order and regularity are more
readily and clearly recognized, when thus exhibited to the eye in a
picture, than they are when presented to the mind in any other
manner. To detect the relations of Number considered directly as
Number, is not easy: and we might {205} contemplate for a long time
a Table of recorded Numbers without perceiving the order of their
increase and diminution, even if the law were moderately simple; as
any one may satisfy himself by looking at a Tide Table. But if these
Numbers are expressed by the magnitude of _Lines_, and if these Lines
are arranged in regular order, the eye readily discovers the rule of
their changes: it follows the curve which runs along their
extremities, and takes note of the order in which its convexities
and concavities succeed each other, if any order be readily
discoverable. The separate observations are in this manner compared
and generalized and reduced to rule by the eye alone. And the eye,
so employed, detects relations of order and succession with a
peculiar celerity and evidence. If, for example, we thus arrive as
ordinates the prices of corn in each year for a series of years, we
shall see the order, rapidity, and amount of the increase and
decrease of price, far more clearly than in any other manner. And if
there were any recurrence of increase and decrease at stated
intervals of years, we should in this manner perceive it. The eye,
constantly active and busy, and employed in making into shapes the
hints and traces of form which it contemplates, runs along the curve
thus offered to it; and as it travels backwards and forwards, is
ever on the watch to detect some resemblance or contrast between one
part and another. And these resemblances and contrasts, when
discovered, are the images of Laws of Phenomena; which are made
manifest at once by this artifice, although the mind could not
easily catch the indications of their existence, if they were not
thus reflected to her in the clear mirror of Space.
Thus when we have a series of good Observations, and know the
argument upon which their change of magnitude depends, the Method of
Curves enables us to ascertain, almost at a glance, the law of the
change; and by further attention, may be made to give us a formula
with great accuracy. The Method enables us to perceive, among our
observations, an order, which without the method, is concealed in
obscurity and perplexity. {206}
3. But the Method of Curves not only enables us to obtain laws of
nature from _good_ Observations, but also, in a great degree, from
observations which are very _imperfect_. For the imperfection of
observations may in part be corrected by this consideration;--that
though they may appear irregular, the correct facts which they
imperfectly represent, are really regular. And the Method of Curves
enables us to remedy this apparent irregularity, at least in part.
For when Observations thus imperfect are laid down as Ordinates, and
their extremities connected by a line, we obtain, not a smooth and
flowing curve, such as we should have if the observations contained
only the rigorous results of regular laws; but a broken and
irregular line, full of sudden and capricious twistings, and bearing
on its face marks of irregularities dependent, not upon law, but
upon chance. Yet these irregular and abrupt deviations in the curve
are, in most cases, but small in extent, when compared with those
bendings which denote the effects of regular law. And this
circumstance is one of the great grounds of advantage in the Method
of Curves. For when the observations thus laid down present to the
eye such a broken and irregular line, we can still see, often with
great ease and certainty, what twistings of the line are probably
due to the irregular errours of observation; and can at once reject
these, by drawing a more regular curve, cutting off all such small
and irregular sinuosities, leaving some to the right and some to the
left; and then proceeding as if this regular curve, and not the
irregular one, expressed the observations. In this manner, we
suppose the errours of observation to balance each other; some of
our corrected measures being too great and others too small, but
with no great preponderance either way. We draw our main regular
curve, not _through_ the points given by our observations, but
_among_ them: drawing it, as has been said by one of the
philosophers[30\3] who first systematically used this method, 'with
a bold but careful hand.' {207} The regular curve which we thus
obtain, thus freed from the casual errours of observation, is that
in which we endeavour to discover the laws of change and succession.
[Note 30\3: Sir J. Herschel, _Ast. Soc. Trans._ vol. v. p. 1.]
4. By this method, thus getting rid at once, in a great measure, of
errours of observation, we obtain data which are _more true than
the_ individual _facts themselves_. The philosopher's business is to
compare his hypotheses with facts, as we have often said. But if we
make the comparison with separate special facts, we are liable to be
perplexed or misled, to an unknown amount, by the errours of
observation; which may cause the hypothetical and the observed
result to agree, or to disagree, when otherwise they would not do
so. If, however, we thus take the _whole mass of the facts_, and
remove the errours of actual observation[31\3], by making the curve
which expresses the supposed observation regular and smooth, we have
the separate facts corrected by their general tendency. We are put
in possession, as we have said, of something more true than any fact
by itself is.
[Note 31\3: _Ib._ vol. v. p. 4.]
One of the most admirable examples of the use of this Method of
Curves is found in Sir John Herschel's _Investigation of the Orbits
of Double Stars_[32\3]. The author there shows how far inferior the
direct observations of the angle of position are, to the
observations corrected by a curve in the manner above stated. 'This
curve once drawn,' he says, 'must represent, it is evident, the law
of variation of the angle of position, with the time, not only for
instants intermediate between the dates of observations, but even at
the moments of observation themselves, much better than the
individual _raw_ observations can possibly (on an average) do. It is
only requisite to try a case or two, to be satisfied that by
substituting the curve for the points, we have made a nearer
approach to nature, and in a great measure eliminated errours of
observation.' 'In following the graphical process,' he adds, 'we
have a conviction almost approaching to moral certainty that {208}
we cannot be greatly misled.' Again, having thus corrected the raw
observations, he makes another use of the graphical method, by
trying whether an ellipse can be drawn 'if not _through_, at least
_among_ the points, so as to approach tolerably near them all; and
thus approaching to the orbit which is the subject of
investigation.'
[Note 32\3: _Ib._]
5. The _Obstacles_ which principally impede the application of the
Method of Curves are (I.) our _ignorance of the arguments_ of the
changes, and (II.) the _complication of several laws_ with one
another.
(I.) If we do not know on what quantity those changes depend which
we are studying, we may fail entirely in detecting the law of the
changes, although we throw the observations into curves. For the
true _argument_ of the change should, in fact, be made the
_abscissa_ of the curve. If we were to express, by a series of
ordinates, the _hour_ of high water on successive days, we should
not obtain, or should obtain very imperfectly, the law which these
times follow; for the real argument of this change is not the _solar
hour_, but the _hour_ at which the _moon_ passes the meridian. But
if we are supposed to be aware that _this_ is the _argument_, (which
theory suggests and trial instantly confirms) we then do immediately
obtain the primary Rules of the Time of High Water, by throwing a
series of observations into a Curve, with the Hour of the Moon's
Transit for the abscissa.
In like manner, when we have obtained the first great or
Semi-mensual Inequality of the tides, if we endeavour to discover
the laws of other Inequalities by means of curves, we must take from
theory the suggestion that the Arguments of such inequalities will
probably be the _parallax_ and the _declination_ of the moon. This
suggestion again is confirmed by trial; but if we were supposed to
be entirely ignorant of the dependence of the changes of the tide on
the Distance and Declination of the moon, the curves would exhibit
unintelligible and seemingly capricious changes. For by the effect
of the Inequality arising from the Parallax, the convexities of the
curves which belong to the {209} spring tides, are in some years
made alternately greater and less all the year through; while in
other years they are made all nearly equal. This difference does not
betray its origin, till we refer it to the Parallax; and the same
difficulty in proceeding would arise if we were ignorant that the
moon's Declination is one of the Arguments of tidal changes.
In like manner, if we try to reduce to law any meteorological
changes, those of the Height of the Barometer for instance, we find
that we can make little progress in the investigation, precisely
because we do not know the Argument on which these changes depend.
That there is a certain regular _diurnal_ change of small amount, we
know; but when we have abstracted this Inequality, (of which the
Argument is the _time of day_,) we find far greater Changes left
behind, from day to day and from hour to hour; and we express these
in curves, but we cannot reduce them to Rule, because we cannot
discover on what numerical quantity they depend. The assiduous study
of barometrical observations, thrown into curves, may perhaps
hereafter point out to us what are the relations of time and space
by which these variations are determined; but in the mean time, this
subject exemplifies to us our remark, that the method of curves is
of comparatively small use, so long as we are in ignorance of the
real Arguments of the Inequalities.
6. (II.) In the next place, I remark that a difficulty is thrown in
the way of the Method of Curves by _the Combination of several laws_
one with another. It will readily be seen that such a cause will
produce a complexity in the curves which exhibit the succession of
facts. If, for example, we take the case of the Tides, the Height of
high water increases and diminishes with the Approach of the sun to,
and its Recess from, the syzygies of the moon. Again, this Height
increases and diminishes as the moon's Parallax increases and
diminishes; and again, the Height diminishes when the Declination
increases, and _vice versa_; and all these Arguments of change, the
Distance from Syzygy, the Parallax, the Declination, complete their
circuit and {210} return into themselves in different periods. Hence
the curve which represents the Height of high water has not any
periodical interval in which it completes its changes and commences
a new cycle. The sinuosity which would arise from each Inequality
separately considered, interferes with, disguises, and conceals the
others; and when we first cast our eyes on the curve of observation,
it is very far from offering any obvious regularity in its form. And
it is to be observed that we have not yet enumerated _all_ the
elements of this complexity: for there are changes of the tide
depending upon the Parallax and Declination of the Sun as well as of
the Moon. Again; besides these changes, of which the Arguments are
obvious, there are others, as those depending upon the Barometer and
the Wind, which follow no known regular law, and which constantly
affect and disturb the results produced by other laws.
In the Tides, and in like manner in the motions of the Moon, we have
very eminent examples of the way in which the discovery of laws may
be rendered difficult by the number of laws which operate to affect
the same quantity. In such cases, the Inequalities are generally
picked out in succession, nearly in the order of their magnitudes.
In this way there were successively collected, from the study of the
Moon's motions by a series of astronomers, those Inequalities which
we term the _Equation of the Center_, the _Evection_, the
_Variation_, and the _Annual Equation_. These Inequalities were not,
in fact, obtained by the application of the Method of Curves; but
the Method of Curves might have been applied to such a case with
great advantage. The Method has been applied with great industry and
with remarkable success to the investigation of the laws of the
Tides; and by the use of it, a series of Inequalities both of the
Times and of the Heights of high water has been detected, which
explain all the main features of the observed facts. {211}
SECT. II.--_The Method of Means._
7. The Method of Curves, as we have endeavoured to explain above,
frees us from the casual and extraneous irregularities which arise
from the imperfection of observation; and thus lays bare the results
of the laws which really operate, and enables us to proceed in
search of those laws. But the Method of Curves is not the only one
which effects such a purpose. The errours arising from detached
observations may be got rid of, and the additional accuracy which
multiplied observations give may be obtained, by operations upon the
observed numbers, without expressing them by spaces. The process of
curves assumes that the errours of observation balance each
other;--that the accidental excesses and defects are nearly equal in
amount;--that the true quantities which would have been observed if
all accidental causes of irregularity were removed, are obtained,
exactly or nearly, by selecting quantities, upon the whole, equally
distant from the extremes of great and small, which our imperfect
observations offer to us. But when, among a number of unequal
quantities, we take a quantity equally distant from the greater and
the smaller, this quantity is termed the _Mean_ of the unequal
quantities. Hence the correction of our observations by the method
of curves consists in taking the Mean of the observations.
8. Now without employing curves, we may proceed arithmetically to
take the Mean of all the observed numbers of each class. Thus, if we
wished to know the Height of the spring tide at a given place, and
if we found that four different spring tides were measured as being
of the height of ten, thirteen, eleven, and fourteen feet, we should
conclude that the true height of the tide was the _Mean_ of these
numbers,--namely, twelve feet; and we should suppose that the
deviation from this height, in the individual cases, arose from the
accidents of weather, the imperfections of observation, or the
operation of other laws, besides the alternation of spring and neap
tides. {212}
This process of finding the Mean of an assemblage of observed
numbers is much practised in discovering, and still more in
confirming and correcting, laws of phenomena. We shall notice a few
of its peculiarities.
9. The Method of Means requires a knowledge of the _Argument_ of the
changes which we would study; for the numbers must be arranged in
certain Classes, before we find the Mean of each Class; and the
principle on which this arrangement depends is the Argument. This
knowledge of the Argument is more indispensably necessary in the
Method of Means than in the Method of Curves; for when Curves are
drawn, the eye often spontaneously detects the law of recurrence in
their sinuosities; but when we have collections of Numbers, we must
divide them into classes by a selection of our own. Thus, in order
to discover the law which the heights of the tide follow, in the
progress from spring to neap, we arrange the observed tides
according to the _day of the moon's age_; and we then take the mean
of all those which thus happen at the _same period_ of the Moon's
Revolution. In this manner we obtain the law which we seek; and the
process is very nearly the same in all other applications of this
Method of Means. In all cases, we begin by assuming the Classes of
measures which we wish to compare, the Law which we could confirm or
correct, the Formula of which we would determine the coefficients.
10. The Argument being thus assumed, the Method of Means is very
efficacious in ridding our inquiry of errours and irregularities
which would impede and perplex it. Irregularities which are
altogether accidental, or at least accidental with reference to some
law which we have under consideration, compensate each other in a
very remarkable way, when we take the Means of _many_ observations.
If we have before us a collection of observed tides, some of them
may be elevated, some depressed by the wind, some noted too high and
some too low by the observer, some augmented and some diminished by
uncontemplated changes in the moon's distance or motion: but in the
course of a year or two at the longest, all these causes of
irregularity balance {213} each other; and the law of succession,
which runs through the observations, comes out as precisely as if
those disturbing influences did not exist. In any particular case,
there appears to be no possible reason why the deviation should be
in one way, or of one moderate amount, rather than another. But
taking the mass of observations together, the deviations in opposite
ways will be of equal amount, with a degree of exactness very
striking. This is found to be the case in all inquiries where we
have to deal with observed numbers upon a large scale. In the
progress of the population of a country, for instance, what can
appear more inconstant, in detail, than the causes which produce
births and deaths? yet in each country, and even in each province of
a country, the proportions of the whole numbers of births and deaths
remain nearly constant. What can be more seemingly beyond the reach
of rule than the occasions which produce letters that cannot find
their destination? yet it appears that the number of 'dead letters'
is nearly the same from year to year. And the same is the result
when the deviations arise, not from mere accident, but from laws
perfectly regular, though not contemplated in our
investigation[33\3]. Thus the effects of the Moon's Parallax upon
the Tides, sometimes operating one way and sometimes another,
according to certain rules, are quite eliminated by taking the Means
of a long series of observations; the excesses and defects
neutralizing each other, so far as concerns the effect upon any law
of the tides which we would investigate.
[Note 33\3: Provided the argument of the law which we neglect have
no coincidence with the argument of the law which we would
determine.]
11. In order to obtain very great accuracy, very large masses of
observations are often employed by philosophers, and the accuracy of
the result increases with the multitude of observations. The immense
collections of astronomical observations which have in this manner
been employed in order to form and correct the Tables of the
celestial motions are perhaps the most signal instances of the
attempts to obtain {214} accuracy by this accumulation of
observations. Delambre's Tables of the Sun are founded upon nearly
3000 observations; Burg's Tables of the Moon upon above 4000.
But there are other instances hardly less remarkable. Mr. Lubbock's
first investigations of the laws of the tides of London[34\3],
included above 13,000 observations, extending through nineteen
years; it being considered that this large number was necessary to
remove the effects of accidental causes[35\3]. And the attempts to
discover the laws of change in the barometer have led to the
performance of labours of equal amount: Laplace and Bouvard examined
this question by means of observations made at the Observatory of
Paris, four times every day for eight years.
[Note 34\3: _Phil. Trans._ 1831.]
[Note 35\3: This period of nineteen years was also selected for a
reason which is alluded to in a former note. It was thought that
this period secured the inquirer from the errours which might be
produced by the partial coincidence of the Arguments of different
irregularities; for example, those due to the moon's Parallax and to
the moon's Declination. It has since been found (_Phil. Tr._ 1838.
_On the Determination of the Laws of the Tides from Short Series of
Observations_), that with regard to Parallax at least, the Means of
one year give sufficient accuracy.]
12. We may remark one striking evidence of the accuracy thus
obtained by employing large masses of observations. In this way we
may often detect inequalities much smaller than the errours by which
they are encumbered and concealed. Thus the Diurnal Oscillations of
the Barometer were discovered by the comparison of observations of
many days, classified according to the hours of the day; and the
result was a clear and incontestable proof of the existence of such
oscillations although the differences which these oscillations
produce at different hours of the day are far smaller than the
casual changes, hitherto reduced to no law, which go on from hour to
hour and from day to day. The effect of law, operating incessantly
and steadily, makes itself more and more felt as we give it a longer
range; while the effect of accident, followed out in the {215} same
manner, is to annihilate itself, and to disappear altogether from
the result.
SECT. III.--_The Method of Least Squares._
13. The Method of Least Squares is in fact a method of means, but
with some peculiar characters. Its object is to determine the _best
Mean_ of a number of observed quantities; or the _most probable Law_
derived from a number of observations, of which some, or all, are
allowed to be more or less imperfect. And the method proceeds upon
this supposition;--that all errours are not _equally_ probable, but
that small errours are more probable than large ones. By reasoning
mathematically upon this ground, we find that the best result is
obtained (since we cannot obtain a result in which the errours
vanish) by making, not the _Errours_ themselves, but the _Sum of
their Squares_, of the _smallest_ possible amount.
14. An example may illustrate this. Let a quantity which is known to
increase uniformly, (as the distance of a star from the meridian at
successive instants,) be measured at equal intervals of time, and be
found to be successively 4, 12, 14. It is plain, upon the face of
these observations, that they are erroneous; for they ought to form
an arithmetical progression, but they deviate widely from such a
progression. But the question then occurs, what arithmetical
progression do they _most probably_ represent: for we may assume
several arithmetical progressions which more or less approach the
observed series; as for instance, these three; 4, 9, 14; 6, 10, 14;
5, 10, 15. Now in order to see the claims of each of these to the
truth, we may tabulate them thus.
Sums of Sums of Squares
Observation 4, 12, 14 Errours Errours. of Errours.
Series (1) 4, 9, 14 0, 3, 0 3 9
" (2) 6, 10, 14 2, 2, 0 4 8
" (3) 5, 10, 15 1, 2, 1 4 6
Here, although the first series gives the sum of the {216} errours
less than the others, the third series gives the sum of the squares
of the errours least; and is therefore, by the proposition on which
this Method depends, the _most probable_ series of the three.
This Method, in more extensive and complex cases, is a great aid to
the calculator in his inferences from facts, and removes much that
is arbitrary in the Method of Means.
SECT. IV.--_The Method of Residues._
15. By either of the preceding Methods we obtain, from observed
facts, such Laws as readily offer themselves; and by the Laws thus
discovered, the most prominent changes of the observed quantities
are accounted for. But in many cases we have, as we have noticed
already, _several_ Laws of nature operating at the same time, and
combining their influences to modify those quantities which are the
subjects of observation. In these cases we may, by successive
applications of the Methods already pointed out, detect such Laws
one after another: but this successive process, though only a
repetition of what we have already described, offers some peculiar
features which make it convenient to consider it in a separate
Section, as the Method of Residues.
16. When we have, in a series of changes of a variable quantity,
discovered _one_ Law which the changes follow, detected its
Argument, and determined its Magnitude, so as to explain most
clearly the course of observed facts, we may still find that the
observed changes are not fully accounted for. When we compare the
results of our Law with the observations, there may be a difference,
or as we may term it, a _Residue_, still unexplained. But this
Residue being thus detached from the rest, may be examined and
scrutinized in the same manner as the whole observed quantity was
treated at first: and we may in this way detect in _it_ also a Law
of change. If we can do this, we must accommodate this new found Law
as nearly as possible to the Residue to which it belongs; and {217}
this being done, the difference of our Rule and of the Residue
itself, forms a _Second Residue_. This Second Residue we may again
bring under our consideration; and may perhaps in _it_ also discover
some Law of change by which its alterations may be in some measure
accounted for. If this can be done, so as to account for a large
portion of this Residue, the remaining unexplained part forms a
_Third Residue_; and so on.
17. This course has really been followed in various inquiries,
especially in those of Astronomy and Tidology. The _Equation of the
Center_, for the Moon, was obtained out of the _Residue_ of the
Longitude, which remained when the _Mean Anomaly_ was taken away.
This Equation being applied and disposed of, the _Second Residue_
thus obtained, gave to Ptolemy the _Evection_. The _Third Residue_,
left by the Equation of the Center and the Evection, supplied to
Tycho the _Variation_ and the _Annual Equation_. And the Residue,
remaining from these, has been exhausted by other Equations, of
various arguments, suggested by theory or by observation. In this
case, the successive generations of astronomers have gone on, each
in its turn executing some step in this Method of Residues. In the
examination of the Tides, on the other hand, this method has been
applied systematically and at once. The observations readily gave
the _Semimensual Inequality_; the _Residue_ of this supplied the
corrections due to the Moon's _Parallax_ and _Declination_; and when
these were determined, the _remaining Residue_ was explored for the
law of the Solar Correction.
18. In a certain degree, the Method of Residues and the Method of
Means are _opposite_ to each other. For the Method of Residues
extricates Laws from their combination, _bringing them into view in
succession_; while the Method of Means discovers each Law, not by
bringing the others into view, but by _destroying their effect_
through an accumulation of observations. By the Method of Residues
we should _first_ extract the Law of the Parallax Correction of the
Tides, and _then_, from the Residue left by this, obtain the
Declination Correction. But we might at once employ the Method {218}
of Means, and put together all the cases in which the Declination
was the same; not allowing for the Parallax in each case, but taking
for granted that the Parallaxes belonging to the same Declination
would neutralize each other; as many falling above as below the mean
Parallax. In cases like this, where the Method of Means is not
impeded by a partial coincidence of the Arguments of different
unknown Inequalities, it may be employed with almost as much success
as the Method of Residues. But still, when the Arguments of the Laws
are clearly known, as in this instance, the Method of Residues is
more clear and direct, and is the rather to be recommended.
19. If for example, we wish to learn whether the Height of the
Barometer exerts any sensible influence on the Height of the Sea's
Surface, it would appear that the most satisfactory mode of
proceeding, must be to subtract, in the first place, what we know to
be the effects of the Moon's Age, Parallax and Declination, and
other ascertained causes of change; and to search in the
_unexplained Residue_ for the effects of barometrical pressure. The
contrary course has, however, been adopted, and the effect of the
Barometer on the ocean has been investigated by the direct
application of the Method of Means, classing the observed heights of
the water according to the corresponding heights of the Barometer
without any previous reduction. In this manner, the suspicion that
the tide of the sea is affected by the pressure of the atmosphere,
has been confirmed. This investigation must be looked upon as a
remarkable instance of the efficacy of the Method of Means, since
the amount of the barometrical effect is much smaller than the other
changes from among which it was by this process extricated. But an
application of the Method of Residues would still be desirable on a
subject of such extent and difficulty.
20. Sir John Herschel, in his _Discourse on the Study of Natural
Philosophy_ (Articles 158-161), has pointed out the mode of making
discoveries by studying Residual Phenomena; and has given several
illustrations of the process. In some of these, he has also {219}
considered this method in a wider sense than we have done; treating
it as not applicable to quantity only, but to properties and
relations of different kinds.
We likewise shall proceed to offer a few remarks on Methods of
Induction applicable to other relations than those of quantity.
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