Part III. BIOLOGY.--_Organosophy_, _Phytogeny_, _Phyto-physiology_,

_Phytology_, _Zoogeny_, _Physiology_, _Zoology_,

_Psychology_.

A glance over this confused scheme shows that it is an attempt to

classify knowledge, not after the order in which it has been, or may be,

built up in the human consciousness; but after an assumed order of

creation. It is a pseudo-scientific cosmogony, akin to those which men

have enunciated from the earliest times downwards; and only a little

more respectable. As such it will not be thought worthy of much

consideration by those who, like ourselves, hold that experience is the

sole origin of knowledge. Otherwise, it might have been needful to dwell

on the incongruities of the arrangements--to ask how motion can be

treated of before space? how there can be rotation without matter to

rotate? how polarity can be dealt with without involving points and

lines? But it will serve our present purpose just to point out a few of

the extreme absurdities resulting from the doctrine which Oken seems to

hold in common with Hegel, that "to philosophise on Nature is to

re-think the great thought of Creation." Here is a sample:--

"Mathematics is the universal science; so also is Physio-philosophy,

although it is only a part, or rather but a condition of the universe;

both are one, or mutually congruent.

"Mathematics is, however, a science of mere forms without substance.

Physio-philosophy is, therefore, _mathematics endowed with substance_."

From the English point of view it is sufficiently amusing to find such a

dogma not only gravely stated, but stated as an unquestionable truth.

Here we see the experiences of quantitative relations which men have

gathered from surrounding bodies and generalised (experiences which had

been scarcely at all generalised at the beginning of the historic

period)--we find these generalised experiences, these intellectual

abstractions, elevated into concrete actualities, projected back into

Nature, and considered as the internal framework of things--the skeleton

by which matter is sustained. But this new form of the old realism is by

no means the most startling of the physio-philosophic principles. We

presently read that,

"The highest mathematical idea, or the fundamental principle of all

mathematics is the zero = 0."....

"Zero is in itself nothing. Mathematics is based upon nothing, and,

_consequently_, arises out of nothing.

"Out of nothing, _therefore_, it is possible for something to arise; for

mathematics, consisting of propositions, is something, in relation to

0."

By such "consequentlys" and "therefores" it is, that men philosophise

when they "re-think the great thought of Creation." By dogmas that

pretend to be reasons, nothing is made to generate mathematics; and by

clothing mathematics with matter, we have the universe! If now we deny,

as we _do_ deny, that the highest mathematical idea is the zero;--if, on

the other hand, we assert, as we _do_ assert, that the fundamental idea

underlying all mathematics, is that of equality; the whole of Oken's

cosmogony disappears. And here, indeed, we may see illustrated, the

distinctive peculiarity of the German method of procedure in these

matters--the bastard _à priori_ method, as it may be termed. The

legitimate _à priori_ method sets out with propositions of which the

negation is inconceivable; the _à priori_ method as illegitimately

applied, sets out either with propositions of which the negation is

_not_ inconceivable, or with propositions like Oken's, of which the

_affirmation_ is inconceivable.

It is needless to proceed further with the analysis; else might we

detail the steps by which Oken arrives at the conclusions that "the

planets are coagulated colours, for they are coagulated light; that the

sphere is the expanded nothing;" that gravity is "a weighty nothing, a

heavy essence, striving towards a centre;" that "the earth is the

identical, water the indifferent, air the different; or the first the

centre, the second the radius, the last the periphery of the general

globe or of fire." To comment on them would be nearly as absurd as are

the propositions themselves. Let us pass on to another of the German

systems of knowledge--that of Hegel.

The simple fact that Hegel puts Jacob Boehme on a par with Bacon,

suffices alone to show that his standpoint is far remote from the one

usually regarded as scientific: so far remote, indeed, that it is not

easy to find any common basis on which to found a criticism. Those who

hold that the mind is moulded into conformity with surrounding things by

the agency of surrounding things, are necessarily at a loss how to deal

with those, who, like Schelling and Hegel, assert that surrounding

things are solidified mind--that Nature is "petrified intelligence."

However, let us briefly glance at Hegel's classification. He divides

philosophy into three parts:--

1. _Logic_, or the science of the idea in itself, the pure idea.

2. _The Philosophy of Nature_, or the science of the idea considered

under its other form--of the idea as Nature.

3. _The Philosophy of the Mind_, or the science of the idea in its

return to itself.

Of these, the second is divided into the natural sciences, commonly so

called; so that in its more detailed form the series runs thus:--Logic,

Mechanics, Physics, Organic Physics, Psychology.

Now, if we believe with Hegel, first, that thought is the true essence

of man; second, that thought is the essence of the world; and that,

therefore, there is nothing but thought; his classification, beginning

with the science of pure thought, may be acceptable. But otherwise, it

is an obvious objection to his arrangement, that thought implies things

thought of--that there can be no logical forms without the substance of

experience--that the science of ideas and the science of things must

have a simultaneous origin. Hegel, however, anticipates this objection,

and, in his obstinate idealism, replies, that the contrary is true; that

all contained in the forms, to become something, requires to be thought:

and that logical forms are the foundations of all things.

It is not surprising that, starting from such premises, and reasoning

after this fashion, Hegel finds his way to strange conclusions. Out of

_space_ and _time_ he proceeds to build up _motion_, _matter_,

_repulsion_, _attraction_, _weight_, and _inertia_. He then goes on to

logically evolve the solar system. In doing this he widely diverges

from the Newtonian theory; reaches by syllogism the conviction that the

planets are the most perfect celestial bodies; and, not being able to

bring the stars within his theory, says that they are mere formal

existences and not living matter, and that as compared with the solar

system they are as little admirable as a cutaneous eruption or a swarm

of flies.[2]

Results so outrageous might be left as self-disproved, were it not that

speculators of this class are not alarmed by any amount of incongruity

with established beliefs. The only efficient mode of treating systems

like this of Hegel, is to show that they are self-destructive--that by

their first steps they ignore that authority on which all their

subsequent steps depend. If Hegel professes, as he manifestly does, to

develop his scheme by reasoning--if he presents successive inferences as

_necessarily following_ from certain premises; he implies the postulate

that a belief which necessarily follows after certain antecedents is a

true belief: and, did an opponent reply to one of his inferences, that,

though it was impossible to think the opposite, yet the opposite was

true, he would consider the reply irrational. The procedure, however,

which he would thus condemn as destructive of all thinking whatever, is

just the procedure exhibited in the enunciation of his own first

principles.

Mankind find themselves unable to conceive that there can be thought

without things thought of. Hegel, however, asserts that there _can_ be

thought without things thought of. That ultimate test of a true

proposition--the inability of the human mind to conceive the negation of

it--which in all other cases he considers valid, he considers invalid

where it suits his convenience to do so; and yet at the same time denies

the right of an opponent to follow his example. If it is competent for

him to posit dogmas, which are the direct negations of what human

consciousness recognises; then is it also competent for his antagonists

to stop him at every step in his argument by saying, that though the

particular inference he is drawing seems to his mind, and to all minds,

necessarily to follow from the premises, yet it is not true, but the

contrary inference is true. Or, to state the dilemma in another

form:--If he sets out with inconceivable propositions, then may he with

equal propriety make all his succeeding propositions inconceivable

ones--may at every step throughout his reasoning draw exactly the

opposite conclusion to that which seems involved.

Hegel's mode of procedure being thus essentially suicidal, the Hegelian

classification which depends upon it falls to the ground. Let us

consider next that of M. Comte.

As all his readers must admit, M. Comte presents us with a scheme of the

sciences which, unlike the foregoing ones, demands respectful

consideration. Widely as we differ from him, we cheerfully bear witness

to the largeness of his views, the clearness of his reasoning, and the

value of his speculations as contributing to intellectual progress. Did

we believe a serial arrangement of the sciences to be possible, that of

M. Comte would certainly be the one we should adopt. His fundamental

propositions are thoroughly intelligible; and if not true, have a great

semblance of truth. His successive steps are logically co-ordinated; and

he supports his conclusions by a considerable amount of

evidence--evidence which, so long as it is not critically examined, or

not met by counter evidence, seems to substantiate his positions. But it

only needs to assume that antagonistic attitude which _ought_ to be

assumed towards new doctrines, in the belief that, if true, they will

prosper by conquering objectors--it needs but to test his leading

doctrines either by other facts than those he cites, or by his own facts

differently applied, to at once show that they will not stand. We will

proceed thus to deal with the general principle on which he bases his

hierarchy of the sciences.

In the second chapter of his _Cours de Philosophic Positive_, M. Comte

says:--"Our problem is, then, to find the one _rational_ order, amongst

a host of possible systems." ... "This order is determined by the degree

of simplicity, or, what comes to the same thing, of generality of their

phenomena." And the arrangement he deduces runs thus: _Mathematics_,

_Astronomy_, _Physics_, _Chemistry_, _Physiology_, _Social Physics_.

This he asserts to be "the true _filiation_ of the sciences." He asserts

further, that the principle of progression from a greater to a less

degree of generality, "which gives this order to the whole body of

science, arranges the parts of each science." And, finally, he asserts

that the gradations thus established _à priori_ among the sciences, and

the parts of each science, "is in essential conformity with the order

which has spontaneously taken place among the branches of natural

philosophy;" or, in other words--corresponds with the order of historic

development.

Let us compare these assertions with the facts. That there may be

perfect fairness, let us make no choice, but take as the field for our

comparison, the succeeding section treating of the first

science--Mathematics; and let us use none but M. Comte's own facts, and

his own admissions. Confining ourselves to this one science, of course

our comparisons must be between its several parts. M. Comte says, that

the parts of each science must be arranged in the order of their

decreasing generality; and that this order of decreasing generality

agrees with the order of historical development. Our inquiry must be,

then, whether the history of mathematics confirms this statement.

Carrying out his principle, M. Comte divides Mathematics into "Abstract

Mathematics, or the Calculus (taking the word in its most extended

sense) and Concrete Mathematics, which is composed of General Geometry

and of Rational Mechanics." The subject-matter of the first of these is

_number_; the subject-matter of the second includes _space_, _time_,

_motion_, _force_. The one possesses the highest possible degree of

generality; for all things whatever admit of enumeration. The others are

less general; seeing that there are endless phenomena that are not

cognisable either by general geometry or rational mechanics. In

conformity with the alleged law, therefore, the evolution of the

calculus must throughout have preceded the evolution of the concrete

sub-sciences. Now somewhat awkwardly for him, the first remark M. Comte

makes bearing upon this point is, that "from an historical point of

view, mathematical analysis _appears to have risen out of_ the

contemplation of geometrical and mechanical facts." True, he goes on to

say that, "it is not the less independent of these sciences logically

speaking;" for that "analytical ideas are, above all others, universal,

abstract, and simple; and geometrical conceptions are necessarily

founded on them."

We will not take advantage of this last passage to charge M. Comte with

teaching, after the fashion of Hegel, that there can be thought without

things thought of. We are content simply to compare the two assertions,

that analysis arose out of the contemplation of geometrical and

mechanical facts, and that geometrical conceptions are founded upon

analytical ones. Literally interpreted they exactly cancel each other.

Interpreted, however, in a liberal sense, they imply, what we believe to

be demonstrable, that the two had _a simultaneous origin_. The passage

is either nonsense, or it is an admission that abstract and concrete

mathematics are coeval. Thus, at the very first step, the alleged

congruity between the order of generality and the order of evolution

does not hold good.

But may it not be that though abstract and concrete mathematics took

their rise at the same time, the one afterwards developed more rapidly

than the other; and has ever since remained in advance of it? No: and

again we call M. Comte himself as witness. Fortunately for his argument

he has said nothing respecting the early stages of the concrete and

abstract divisions after their divergence from a common root; otherwise

the advent of Algebra long after the Greek geometry had reached a high

development, would have been an inconvenient fact for him to deal with.

But passing over this, and limiting ourselves to his own statements, we

find, at the opening of the next chapter, the admission, that "the

historical development of the abstract portion of mathematical science

has, since the time of Descartes, been for the most part _determined_ by

that of the concrete." Further on we read respecting algebraic functions

that "most functions were concrete in their origin--even those which are

at present the most purely abstract; and the ancients discovered only

through geometrical definitions elementary algebraic properties of

functions to which a numerical value was not attached till long

afterwards, rendering abstract to us what was concrete to the old

geometers." How do these statements tally with his doctrine? Again,

having divided the calculus into algebraic and arithmetical, M. Comte

admits, as perforce he must, that the algebraic is more general than the

arithmetical; yet he will not say that algebra preceded arithmetic in

point of time. And again, having divided the calculus of functions into

the calculus of direct functions (common algebra) and the calculus of

indirect functions (transcendental analysis), he is obliged to speak of

this last as possessing a higher generality than the first; yet it is

far more modern. Indeed, by implication, M. Comte himself confesses this

incongruity; for he says:--"It might seem that the transcendental

analysis ought to be studied before the ordinary, as it provides the

equations which the other has to resolve; but though the transcendental

_is logically independent of the ordinary_, it is best to follow the

usual method of study, taking the ordinary first." In all these cases,

then, as well as at the close of the section where he predicts that

mathematicians will in time "create procedures of _a wider generality_",

M. Comte makes admissions that are diametrically opposed to the alleged

law.

In the succeeding chapters treating of the concrete department of

mathematics, we find similar contradictions M. Comte himself names the

geometry of the ancients _special_ geometry, and that of moderns the

_general_ geometry. He admits that while "the ancients studied geometry

with reference to the bodies under notice, or specially; the moderns

study it with reference to the _phenomena_ to be considered, or

generally." He admits that while "the ancients extracted all they could

out of one line or surface before passing to another," "the moderns,

since Descartes, employ themselves on questions which relate to any

figure whatever." These facts are the reverse of what, according to his

theory, they should be. So, too, in mechanics. Before dividing it into

statics and dynamics, M. Comte treats of the three laws of _motion_, and

is obliged to do so; for statics, the more _general_ of the two

divisions, though it does not involve motion, is impossible as a science

until the laws of motion are ascertained. Yet the laws of motion pertain

to dynamics, the more _special_ of the divisions. Further on he points

out that after Archimedes, who discovered the law of equilibrium of the

lever, statics made no progress until the establishment of dynamics

enabled us to seek "the conditions of equilibrium through the laws of

the composition of forces." And he adds--"At this day _this is the

method universally employed_. At the first glance it does not appear the

most rational--dynamics being more complicated than statics, and

precedence being natural to the simpler. It would, in fact, be more

philosophical to refer dynamics to statics, as has since been done."

Sundry discoveries are afterwards detailed, showing how completely the

development of statics has been achieved by considering its problems

dynamically; and before the close of the section M. Comte remarks that

"before hydrostatics could be comprehended under statics, it was

necessary that the abstract theory of equilibrium should be made so

general as to apply directly to fluids as well as solids. This was

accomplished when Lagrange supplied, as the basis of the whole of

rational mechanics, the single principle of virtual velocities." In

which statement we have two facts directly at variance: with M. Comte's

doctrine; first, that the simpler science, statics, reached its present

development only by the aid of the principle of virtual velocities,

which belongs to the more complex science, dynamics; and that this

"single principle" underlying all rational mechanics--this _most

general form_ which includes alike the relations of statical,

hydro-statical, and dynamical forces--was reached so late as the time of

Lagrange.

Thus it is _not_ true that the historical succession of the divisions of

mathematics has corresponded with the order of decreasing generality. It

is _not_ true that abstract mathematics was evolved antecedently to,

and independently of concrete mathematics. It is _not_ true that of the

subdivisions of abstract mathematics, the more general came before the

more special. And it is _not_ true that concrete mathematics, in either

of its two sections, began with the most abstract and advanced to the

less abstract truths.

It may be well to mention, parenthetically, that in defending his

alleged law of progression from the general to the special, M. Comte

somewhere comments upon the two meanings of the word _general_, and the

resulting liability to confusion. Without now discussing whether the

asserted distinction can be maintained in other cases, it is manifest

that it does not exist here. In sundry of the instances above quoted,

the endeavours made by M. Comte himself to disguise, or to explain away,

the precedence of the special over the general, clearly indicate that

the generality spoken of is of the kind meant by his formula. And it

needs but a brief consideration of the matter to show that, even did he

attempt it, he could not distinguish this generality, which, as above

proved, frequently comes last, from the generality which he says always

comes first. For what is the nature of that mental process by which

objects, dimensions, weights, times, and the rest, are found capable of

having their relations expressed numerically? It is the formation of

certain abstract conceptions of unity, duality and multiplicity, which

are applicable to all things alike. It is the invention of general

symbols serving to express the numerical relations of entities, whatever

be their special characters. And what is the nature of the mental

process by which numbers are found capable of having their relations

expressed algebraically? It is just the same. It is the formation of

certain abstract conceptions of numerical functions which are the same

whatever be the magnitudes of the numbers. It is the invention of

general symbols serving to express the relations between numbers, as

numbers express the relations between things. And transcendental

analysis stands to algebra in the same position that algebra stands in

to arithmetic.

To briefly illustrate their respective powers--arithmetic can express in

one formula the value of a _particular_ tangent to a _particular_ curve;

algebra can express in one formula the values of _all_ tangents to a

_particular_ curve; transcendental analysis can express in one formula

the values of _all_ tangents to _all_ curves. Just as arithmetic deals

with the common properties of lines, areas, bulks, forces, periods; so

does algebra deal with the common properties of the numbers which

arithmetic presents; so does transcendental analysis deal with the

common properties of the equations exhibited by algebra. Thus, the

generality of the higher branches of the calculus, when compared with

the lower, is the same kind of generality as that of the lower branches

when compared with geometry or mechanics. And on examination it will be

found that the like relation exists in the various other cases above

given.

Having shown that M. Comte's alleged law of progression does not hold

among the several parts of the same science, let us see how it agrees

with the facts when applied to separate sciences. "Astronomy," says M.

Comte, at the opening of Book III., "was a positive science, in its

geometrical aspect, from the earliest days of the school of Alexandria;

but Physics, which we are now to consider, had no positive character at

all till Galileo made his great discoveries on the fall of heavy

bodies." On this, our comment is simply that it is a misrepresentation

based upon an arbitrary misuse of words--a mere verbal artifice. By

choosing to exclude from terrestrial physics those laws of magnitude,

motion, and position, which he includes in celestial physics, M. Comte

makes it appear that the one owes nothing to the other. Not only is this

altogether unwarrantable, but it is radically inconsistent with his own

scheme of divisions. At the outset he says--and as the point is

important we quote from the original--"Pour la _physique inorganique_

nous voyons d'abord, en nous conformant toujours a l'ordre de généralité

et de dépendance des phénomènes, qu'elle doit être partagée en deux

sections distinctes, suivant qu'elle considère les phénomènes généraux

de l'univers, ou, en particulier, ceux que présentent les corps

terrestres. D'où la physique céleste, ou l'astronomie, soit géométrique,

soit mechanique; et la physique terrestre."

Here then we have _inorganic physics_ clearly divided into _celestial

physics_ and _terrestrial physics_--the phenomena presented by the

universe, and the phenomena presented by earthly bodies. If now

celestial bodies and terrestrial bodies exhibit sundry leading phenomena

in common, as they do, how can the generalisation of these common

phenomena be considered as pertaining to the one class rather than to

the other? If inorganic physics includes geometry (which M. Comte has

made it do by comprehending _geometrical_ astronomy in its

sub-section--celestial physics); and if its sub-section--terrestrial

physics, treats of things having geometrical properties; how can the

laws of geometrical relations be excluded from terrestrial physics?

Clearly if celestial physics includes the geometry of objects in the

heavens, terrestrial physics includes the geometry of objects on the

earth. And if terrestrial physics includes terrestrial geometry, while

celestial physics includes celestial geometry, then the geometrical part

of terrestrial physics precedes the geometrical part of celestial

physics; seeing that geometry gained its first ideas from surrounding

objects. Until men had learnt geometrical relations from bodies on the

earth, it was impossible for them to understand the geometrical

relations of bodies in the heavens.

So, too, with celestial mechanics, which had terrestrial mechanics for

its parent. The very conception of _force_, which underlies the whole of

mechanical astronomy, is borrowed from our earthly experiences; and the

leading laws of mechanical action as exhibited in scales, levers,

projectiles, etc., had to be ascertained before the dynamics of the

solar system could be entered upon. What were the laws made use of by

Newton in working out his grand discovery? The law of falling bodies

disclosed by Galileo; that of the composition of forces also disclosed

by Galileo; and that of centrifugal force found out by Huyghens--all of

them generalisations of terrestrial physics. Yet, with facts like these

before him, M. Comte places astronomy before physics in order of

evolution! He does not compare the geometrical parts of the two

together, and the mechanical parts of the two together; for this would

by no means suit his hypothesis. But he compares the geometrical part of

the one with the mechanical part of the other, and so gives a semblance

of truth to his position. He is led away by a verbal delusion. Had he

confined his attention to the things and disregarded the words, he would

have seen that before mankind scientifically co-ordinated _any one class

of phenomena_ displayed in the heavens, they had previously co-ordinated

_a parallel class of phenomena_ displayed upon the surface of the earth.

Were it needful we could fill a score pages with the incongruities of M.

Comte's scheme. But the foregoing samples will suffice. So far is his

law of evolution of the sciences from being tenable, that, by following

his example, and arbitrarily ignoring one class of facts, it would be

possible to present, with great plausibility, just the opposite

generalisation to that which he enunciates. While he asserts that the

rational order of the sciences, like the order of their historic

development, "is determined by the degree of simplicity, or, what comes

to the same thing, of generality of their phenomena;" it might

contrariwise be asserted, that, commencing with the complex and the

special, mankind have progressed step by step to a knowledge of greater

simplicity and wider generality. So much evidence is there of this as to

have drawn from Whewell, in his _History of the Inductive Sciences_, the

general remark that "the reader has already seen repeatedly in the

course of this history, complex and derivative principles presenting

themselves to men's minds before simple and elementary ones."

Even from M. Comte's own work, numerous facts, admissions, and

arguments, might be picked out, tending to show this. We have already

quoted his words in proof that both abstract and concrete mathematics

have progressed towards a higher degree of generality, and that he looks

forward to a higher generality still. Just to strengthen this adverse

hypothesis, let us take a further instance. From the _particular_ case

of the scales, the law of equilibrium of which was familiar to the

earliest nations known, Archimedes advanced to the more _general_ case

of the unequal lever with unequal weights; the law of equilibrium of

which _includes_ that of the scales. By the help of Galileo's discovery

concerning the composition of forces, D'Alembert "established, for the

first time, the equations of equilibrium of _any_ system of forces

applied to the different points of a solid body"--equations which

include all cases of levers and an infinity of cases besides. Clearly

this is progress towards a higher generality--towards a knowledge more

independent of special circumstances--towards a study of phenomena "the

most disengaged from the incidents of particular cases;" which is M.

Comte's definition of "the most simple phenomena." Does it not indeed

follow from the familiarly admitted fact, that mental advance is from

the concrete to the abstract, from the particular to the general, that

the universal and therefore most simple truths are the last to be

discovered? Is not the government of the solar system by a force varying

inversely as the square of the distance, a simpler conception than any

that preceded it? Should we ever succeed in reducing all orders of

phenomena to some single law--say of atomic action, as M. Comte

suggests--must not that law answer to his test of being _independent_ of

all others, and therefore most simple? And would not such a law

generalise the phenomena of gravity, cohesion, atomic affinity, and

electric repulsion, just as the laws of number generalise the

quantitative phenomena of space, time, and force?

The possibility of saying so much in support of an hypothesis the very

reverse of M. Comte's, at once proves that his generalisation is only a

half-truth. The fact is, that neither proposition is correct by itself;

and the actuality is expressed only by putting the two together. The

progress of science is duplex: it is at once from the special to the

general, and from the general to the special: it is analytical and

synthetical at the same time.

M. Comte himself observes that the evolution of science has been

accomplished by the division of labour; but he quite misstates the mode

in which this division of labour has operated. As he describes it, it

has simply been an arrangement of phenomena into classes, and the study

of each class by itself. He does not recognise the constant effect of

progress in each class upon _all_ other classes; but only on the class

succeeding it in his hierarchical scale. Or if he occasionally admits

collateral influences and intercommunications, he does it so grudgingly,

and so quickly puts the admissions out of sight and forgets them, as to

leave the impression that, with but trifling exceptions, the sciences

aid each other only in the order of their alleged succession. The fact

is, however, that the division of labour in science, like the division

of labour in society, and like the "physiological division of labour" in

individual organisms, has been not only a specialisation of functions,

but a continuous helping of each division by all the others, and of all

by each. Every particular class of inquirers has, as it were, secreted

its own particular order of truths from the general mass of material

which observation accumulates; and all other classes of inquirers have

made use of these truths as fast as they were elaborated, with the

effect of enabling them the better to elaborate each its own order of

truths.

It was thus in sundry of the cases we have quoted as at variance with M.

Comte's doctrine. It was thus with the application of Huyghens's optical

discovery to astronomical observation by Galileo. It was thus with the

application of the isochronism of the pendulum to the making of

instruments for measuring intervals, astronomical and other. It was thus

when the discovery that the refraction and dispersion of light did not

follow the same law of variation, affected both astronomy and physiology

by giving us achromatic telescopes and microscopes. It was thus when

Bradley's discovery of the aberration of light enabled him to make the

first step towards ascertaining the motions of the stars. It was thus

when Cavendish's torsion-balance experiment determined the specific

gravity of the earth, and so gave a datum for calculating the specific

gravities of the sun and planets. It was thus when tables of

atmospheric refraction enabled observers to write down the real places

of the heavenly bodies instead of their apparent places. It was thus

when the discovery of the different expansibilities of metals by heat,

gave us the means of correcting our chronometrical measurements of

astronomical periods. It was thus when the lines of the prismatic

spectrum were used to distinguish the heavenly bodies that are of like

nature with the sun from those which are not. It was thus when, as

recently, an electro-telegraphic instrument was invented for the more

accurate registration of meridional transits. It was thus when the

difference in the rates of a clock at the equator, and nearer the poles,

gave data for calculating the oblateness of the earth, and accounting

for the precession of the equinoxes. It was thus--but it is needless to

continue.

Here, within our own limited knowledge of its history, we have named ten

additional cases in which the single science of astronomy has owed its

advance to sciences coming _after_ it in M. Comte's series. Not only its

secondary steps, but its greatest revolutions have been thus determined.

Kepler could not have discovered his celebrated laws had it not been for

Tycho Brahe's accurate observations; and it was only after some progress

in physical and chemical science that the improved instruments with

which those observations were made, became possible. The heliocentric

theory of the solar system had to wait until the invention of the

telescope before it could be finally established. Nay, even the grand

discovery of all--the law of gravitation--depended for its proof upon an

operation of physical science, the measurement of a degree on the

Earth's surface. So completely indeed did it thus depend, that Newton

_had actually abandoned his hypothesis_ because the length of a degree,

as then stated, brought out wrong results; and it was only after

Picart's more exact measurement was published, that he returned to his

calculations and proved his great generalisation. Now this constant

intercommunion, which, for brevity's sake, we have illustrated in the

case of one science only, has been taking place with all the sciences.

Throughout the whole course of their evolution there has been a

continuous _consensus_ of the sciences--a _consensus_ exhibiting a

general correspondence with the _consensus_ of faculties in each phase

of mental development; the one being an objective registry of the

subjective state of the other.

From our present point of view, then, it becomes obvious that the

conception of a _serial_ arrangement of the sciences is a vicious one.

It is not simply that the schemes we have examined are untenable; but it

is that the sciences cannot be rightly placed in any linear order

whatever. It is not simply that, as M. Comte admits, a classification

"will always involve something, if not arbitrary, at least artificial;"

it is not, as he would have us believe, that, neglecting minor

imperfections a classification may be substantially true; but it is that

any grouping of the sciences in a succession gives a radically erroneous

idea of their genesis and their dependencies. There is no "one

_rational_ order among a host of possible systems." There is no "true

_filiation_ of the sciences." The whole hypothesis is fundamentally

false. Indeed, it needs but a glance at its origin to see at once how

baseless it is. Why a _series_? What reason have we to suppose that the

sciences admit of a _linear_ arrangement? Where is our warrant for

assuming that there is some _succession_ in which they can be placed?

There is no reason; no warrant. Whence then has arisen the supposition?

To use M. Comte's own phraseology, we should say, it is a metaphysical

conception. It adds another to the cases constantly occurring, of the

human mind being made the measure of Nature. We are obliged to think in

sequence; it is the law of our minds that we must consider subjects

separately, one after another: _therefore_ Nature must be

serial--_therefore_ the sciences must be classifiable in a succession.

See here the birth of the notion, and the sole evidence of its truth.

Men have been obliged when arranging in books their schemes of education

and systems of knowledge, to choose _some_ order or other. And from

inquiring what is the best order, have naturally fallen into the belief

that there is an order which truly represents the facts--have persevered

in seeking such an order; quite overlooking the previous question

whether it is likely that Nature has consulted the convenience of

book-making.

For German philosophers, who hold that Nature is "petrified

intelligence," and that logical forms are the foundations of all things,

it is a consistent hypothesis that as thought is serial, Nature is

serial; but that M. Comte, who is so bitter an opponent of all

anthropomorphism, even in its most evanescent shapes, should have

committed the mistake of imposing upon the external world an arrangement

which so obviously springs from a limitation of the human consciousness,

is somewhat strange. And it is the more strange when we call to mind

how, at the outset, M. Comte remarks that in the beginning "_toutes les

sciences sont cultivées simultanément par les mêmes esprits_;" that

this is "_inevitable et même indispensable_;" and how he further remarks

that the different sciences are "_comme les diverses branches d'un tronc

unique_." Were it not accounted for by the distorting influence of a

cherished hypothesis, it would be scarcely possible to understand how,

after recognising truths like these, M. Comte should have persisted in

attempting to construct "_une échelle encyclopédique_."

The metaphor which M. Comte has here so inconsistently used to express

the relations of the sciences--branches of one trunk--is an

approximation to the truth, though not the truth itself. It suggests the

facts that the sciences had a common origin; that they have been

developing simultaneously; and that they have been from time to time

dividing and subdividing. But it does not suggest the yet more important

fact, that the divisions and subdivisions thus arising do not remain

separate, but now and again reunite in direct and indirect ways. They

inosculate; they severally send off and receive connecting growths; and

the intercommunion has been ever becoming more frequent, more intricate,

more widely ramified. There has all along been higher specialisation,

that there might be a larger generalisation; and a deeper analysis, that

there might be a better synthesis. Each larger generalisation has lifted

sundry specialisations still higher; and each better synthesis has

prepared the way for still deeper analysis.

And here we may fitly enter upon the task awhile since indicated--a

sketch of the Genesis of Science, regarded as a gradual outgrowth from

common knowledge--an extension of the perceptions by the aid of the

reason. We propose to treat it as a psychological process historically

displayed; tracing at the same time the advance from qualitative to

quantitative prevision; the progress from concrete facts to abstract

facts, and the application of such abstract facts to the analysis of new

orders of concrete facts; the simultaneous advance in generalisation and

specialisation; the continually increasing subdivision and reunion of

the sciences; and their constantly improving _consensus_.

To trace out scientific evolution from its deepest roots would, of

course, involve a complete analysis of the mind. For as science is a

development of that common knowledge acquired by the unaided senses and

uncultured reason, so is that common knowledge itself gradually built up

out of the simplest perceptions. We must, therefore, begin somewhere

abruptly; and the most appropriate stage to take for our point of

departure will be the adult mind of the savage.

Commencing thus, without a proper preliminary analysis, we are naturally

somewhat at a loss how to present, in a satisfactory manner, those

fundamental processes of thought out of which science ultimately

originates. Perhaps our argument may be best initiated by the

proposition, that all intelligent action whatever depends upon the

discerning of distinctions among surrounding things. The condition under

which only it is possible for any creature to obtain food and avoid

danger is, that it shall be differently affected by different

objects--that it shall be led to act in one way by one object, and in

another way by another. In the lower orders of creatures this condition

is fulfilled by means of an apparatus which acts automatically. In the

higher orders the actions are partly automatic, partly conscious. And in

man they are almost wholly conscious.

Throughout, however, there must necessarily exist a certain

classification of things according to their properties--a classification

which is either organically registered in the system, as in the inferior

creation, or is formed by experience, as in ourselves. And it may be

further remarked, that the extent to which this classification is

carried, roughly indicates the height of intelligence--that while the

lowest organisms are able to do little more than discriminate organic

from inorganic matter; while the generality of animals carry their

classifications no further than to a limited number of plants or

creatures serving for food, a limited number of beasts of prey, and a

limited number of places and materials; the most degraded of the human

race possess a knowledge of the distinctive natures of a great variety

of substances, plants, animals, tools, persons, etc., not only as

classes but as individuals.

What now is the mental process by which classification is effected?

Manifestly it is a recognition of the _likeness_ or _unlikeness_ of

things, either in respect of their sizes, colours, forms, weights,

textures, tastes, etc., or in respect of their modes of action. By some

special mark, sound, or motion, the savage identifies a certain

four-legged creature he sees, as one that is good for food, and to be

caught in a particular way; or as one that is dangerous; and acts

accordingly. He has classed together all the creatures that are _alike_

in this particular. And manifestly in choosing the wood out of which to

form his bow, the plant with which to poison his arrows, the bone from

which to make his fish-hooks, he identifies them through their chief

sensible properties as belonging to the general classes, wood, plant,

and bone, but distinguishes them as belonging to sub-classes by virtue

of certain properties in which they are _unlike_ the rest of the general

classes they belong to; and so forms genera and species.

And here it becomes manifest that not only is classification carried on

by grouping together in the mind things that are _like_; but that

classes and sub-classes are formed and arranged according to the

_degrees of unlikeness_. Things widely contrasted are alone

distinguished in the lower stages of mental evolution; as may be any day

observed in an infant. And gradually as the powers of discrimination

increase, the widely contrasted classes at first distinguished, come to

be each divided into sub-classes, differing from each other less than

the classes differ; and these sub-classes are again divided after the

same manner. By the continuance of which process, things are gradually

arranged into groups, the members of which are less and less _unlike_;

ending, finally, in groups whose members differ only as individuals, and

not specifically. And thus there tends ultimately to arise the notion of

_complete likeness_. For, manifestly, it is impossible that groups

should continue to be subdivided in virtue of smaller and smaller

differences, without there being a simultaneous approximation to the

notion of _no difference_.

Let us next notice that the recognition of likeness and unlikeness,

which underlies classification, and out of which continued

classification evolves the idea of complete likeness--let us next notice

that it also underlies the process of _naming_, and by consequence

_language_. For all language consists, at the beginning, of symbols

which are as _like_ to the things symbolised as it is practicable to

make them. The language of signs is a means of conveying ideas by

mimicking the actions or peculiarities of the things referred to. Verbal

language is also, at the beginning, a mode of suggesting objects or acts

by imitating the sounds which the objects make, or with which the acts

are accompanied. Originally these two languages were used

simultaneously. It needs but to watch the gesticulations with which the

savage accompanies his speech--to see a Bushman or a Kaffir dramatising

before an audience his mode of catching game--or to note the extreme

paucity of words in all primitive vocabularies; to infer that at first,

attitudes, gestures, and sounds, were all combined to produce as good a

_likeness_ as possible, of the things, animals, persons, or events

described; and that as the sounds came to be understood by themselves

the gestures fell into disuse: leaving traces, however, in the manners

of the more excitable civilised races. But be this as it may, it

suffices simply to observe, how many of the words current among

barbarous peoples are like the sounds appertaining to the things

signified; how many of our own oldest and simplest words have the same

peculiarity; how children tend to invent imitative words; and how the

sign-language spontaneously formed by deaf mutes is invariably based

upon imitative actions--to at once see that the nation of _likeness_ is

that from which the nomenclature of objects takes its rise.

Were there space we might go on to point out how this law of life is

traceable, not only in the origin but in the development of language;

how in primitive tongues the plural is made by a duplication of the

singular, which is a multiplication of the word to make it _like_ the

multiplicity of the things; how the use of metaphor--that prolific

source of new words--is a suggesting of ideas that are _like_ the ideas

to be conveyed in some respect or other; and how, in the copious use of

simile, fable, and allegory among uncivilised races, we see that complex

conceptions, which there is yet no direct language for, are rendered, by

presenting known conceptions more or less _like_ them.

This view is further confirmed, and the predominance of this notion of

likeness in primitive times further illustrated, by the fact that our

system of presenting ideas to the eye originated after the same fashion.

Writing and printing have descended from picture-language. The earliest

mode of permanently registering a fact was by depicting it on a wall;

that is--by exhibiting something as _like_ to the thing to be remembered

as it could be made. Gradually as the practice grew habitual and

extensive, the most frequently repeated forms became fixed, and

presently abbreviated; and, passing through the hieroglyphic and

ideographic phases, the symbols lost all apparent relations to the

things signified: just as the majority of our spoken words have done.

Observe again, that the same thing is true respecting the genesis of

reasoning. The _likeness_ that is perceived to exist between cases, is

the essence of all early reasoning and of much of our present reasoning.

The savage, having by experience discovered a relation between a certain

object and a certain act, infers that the _like_ relation will be found

in future cases. And the expressions we constantly use in our

arguments--"_analogy_ implies," "the cases are not _parallel_," "by

_parity_ of reasoning," "there is no _similarity_,"--show how constantly

the idea of likeness underlies our ratiocinative processes.

Still more clearly will this be seen on recognising the fact that there

is a certain parallelism between reasoning and classification; that the

two have a common root; and that neither can go on without the other.

For on the one hand, it is a familiar truth that the attributing to a

body in consequence of some of its properties, all those other

properties in virtue of which it is referred to a particular class, is

an act of inference. And, on the other hand, the forming of a

generalisation is the putting together in one class all those cases

which present like relations; while the drawing a deduction is

essentially the perception that a particular case belongs to a certain

class of cases previously generalised. So that as classification is a

grouping together of _like things_; reasoning is a grouping together of

_like relations_ among things. Add to which, that while the perfection

gradually achieved in classification consists in the formation of groups

of _objects_ which are _completely alike_; the perfection gradually

achieved in reasoning consists in the formation of groups of _cases_

which are _completely alike_.

Once more we may contemplate this dominant idea of likeness as exhibited

in art. All art, civilised as well as savage, consists almost wholly in

the making of objects _like_ other objects; either as found in Nature,

or as produced by previous art. If we trace back the varied art-products

now existing, we find that at each stage the divergence from previous

patterns is but small when compared with the agreement; and in the

earliest art the persistency of imitation is yet more conspicuous. The

old forms and ornaments and symbols were held sacred, and perpetually

copied. Indeed, the strong imitative tendency notoriously displayed by

the lowest human races, ensures among them a constant reproducing of

likeness of things, forms, signs, sounds, actions, and whatever else is

imitable; and we may even suspect that this aboriginal peculiarity is in

some way connected with the culture and development of this general

conception, which we have found so deep and widespread in its

applications.

And now let us go on to consider how, by a further unfolding of this

same fundamental notion, there is a gradual formation of the first germs

of science. This idea of likeness which underlies classification,

nomenclature, language spoken and written, reasoning, and art; and which

plays so important a part because all acts of intelligence are made

possible only by distinguishing among surrounding things, or grouping

them into like and unlike;--this idea we shall find to be the one of

which science is the especial product. Already during the stage we have

been describing, there has existed _qualitative_ prevision in respect to

the commoner phenomena with which savage life is familiar; and we have

now to inquire how the elements of _quantitative_ prevision are evolved.

We shall find that they originate by the perfecting of this same idea of

likeness; that they have their rise in that conception of _complete

likeness_ which, as we have seen, necessarily results from the continued

process of classification.

For when the process of classification has been carried as far as it is

possible for the uncivilised to carry it--when the animal kingdom has

been grouped not merely into quadrupeds, birds, fishes, and insects, but

each of these divided into kinds--when there come to be sub-classes, in

each of which the members differ only as individuals, and not

specifically; it is clear that there must occur a frequent observation

of objects which differ so little as to be indistinguishable. Among

several creatures which the savage has killed and carried home, it must

often happen that some one, which he wished to identify, is so exactly

like another that he cannot tell which is which. Thus, then, there

originates the notion of _equality_. The things which among ourselves

are called _equal_--whether lines, angles, weights, temperatures, sounds

or colours--are things which produce in us sensations that cannot be

distinguished from each other. It is true we now apply the word _equal_

chiefly to the separate phenomena which objects exhibit, and not to

groups of phenomena; but this limitation of the idea has evidently

arisen by subsequent analysis. And that the notion of equality did thus

originate, will, we think, become obvious on remembering that as there

were no artificial objects from which it could have been abstracted, it

must have been abstracted from natural objects; and that the various

families of the animal kingdom chiefly furnish those natural objects

which display the requisite exactitude of likeness.

The same order of experiences out of which this general idea of equality

is evolved, gives birth at the same time to a more complex idea of

equality; or, rather, the process just described generates an idea of

equality which further experience separates into two ideas--_equality of

things_ and _equality of relations_. While organic, and more especially

animal forms, occasionally exhibit this perfection of likeness out of

which the notion of simple equality arises, they more frequently

exhibit only that kind of likeness which we call _similarity_; and which

is really compound equality. For the similarity of two creatures of the

same species but of different sizes, is of the same nature as the

similarity of two geometrical figures. In either case, any two parts of

the one bear the same ratio to one another as the homologous parts of

the other. Given in any species, the proportions found to exist among

the bones, and we may, and zoologists do, predict from any one, the

dimensions of the rest; just as, when knowing the proportions subsisting

among the parts of a geometrical figure, we may, from the length of one,

calculate the others. And if, in the case of similar geometrical

figures, the similarity can be established only by proving exactness of

proportion among the homologous parts; if we express this relation

between two parts in the one, and the corresponding parts in the other,

by the formula A is to B as _a_ is to _b_; if we otherwise write this, A

to B = _a_ to _b_; if, consequently, the fact we prove is that the

relation of A to B _equals_ the relation of _a_ to _b_; then it is

manifest that the fundamental conception of similarity is _equality of

relations_.

With this explanation we shall be understood when we say that the notion

of equality of relations is the basis of all exact reasoning. Already it

has been shown that reasoning in general is a recognition of _likeness_

of relations; and here we further find that while the notion of likeness

of things ultimately evolves the idea of simple equality, the notion of

likeness of relations evolves the idea of equality of relations: of

which the one is the concrete germ of exact science, while the other is

its abstract germ.

Those who cannot understand how the recognition of similarity in

creatures of the same kind can have any alliance with reasoning, will

get over the difficulty on remembering that the phenomena among which

equality of relations is thus perceived, are phenomena of the same order

and are present to the senses at the same time; while those among which

developed reason perceives relations, are generally neither of the same

order, nor simultaneously present. And if further, they will call to

mind how Cuvier and Owen, from a single part of a creature, as a tooth,

construct the rest by a process of reasoning based on this equality of

relations, they will see that the two things are intimately connected,

remote as they at first seem. But we anticipate. What it concerns us

here to observe is, that from familiarity with organic forms there

simultaneously arose the ideas of _simple equality_, and _equality of

relations_.

At the same time, too, and out of the same mental processes, came the

first distinct ideas of _number_. In the earliest stages, the

presentation of several like objects produced merely an indefinite

conception of multiplicity; as it still does among Australians, and

Bushmen, and Damaras, when the number presented exceeds three or four.

With such a fact before us we may safely infer that the first clear

numerical conception was that of duality as contrasted with unity. And

this notion of duality must necessarily have grown up side by side with

those of likeness and equality; seeing that it is impossible to

recognise the likeness of two things without also perceiving that there

are two. From the very beginning the conception of number must have been

as it is still, associated with the likeness or equality of the things

numbered. If we analyse it, we find that simple enumeration is a

registration of repeated impressions of any kind. That these may be

capable of enumeration it is needful that they be more or less alike;

and before any _absolutely true_ numerical results can be reached, it is

requisite that the units be _absolutely equal_. The only way in which we

can establish a numerical relationship between things that do not yield

us like impressions, is to divide them into parts that _do_ yield us

like impressions. Two unlike magnitudes of extension, force, time,

weight, or what not, can have their relative amounts estimated only by

means of some small unit that is contained many times in both; and even

if we finally write down the greater one as a unit and the other as a

fraction of it, we state, in the denominator of the fraction, the number

of parts into which the unit must be divided to be comparable with the

fraction.

It is, indeed, true, that by an evidently modern process of abstraction,

we occasionally apply numbers to unequal units, as the furniture at a

sale or the various animals on a farm, simply as so many separate

entities; but no true result can be brought out by calculation with

units of this order. And, indeed, it is the distinctive peculiarity of

the calculus in general, that it proceeds on the hypothesis of that

absolute equality of its abstract units, which no real units possess;

and that the exactness of its results holds only in virtue of this

hypothesis. The first ideas of number must necessarily then have been

derived from like or equal magnitudes as seen chiefly in organic

objects; and as the like magnitudes most frequently observed magnitudes

of extension, it follows that geometry and arithmetic had a

simultaneous origin.

Not only are the first distinct ideas of number co-ordinate with ideas

of likeness and equality, but the first efforts at numeration displayed

the same relationship. On reading the accounts of various savage tribes,

we find that the method of counting by the fingers, still followed by

many children, is the aboriginal method. Neglecting the several cases in

which the ability to enumerate does not reach even to the number of

fingers on one hand, there are many cases in which it does not extend

beyond ten--the limit of the simple finger notation. The fact that in so

many instances, remote, and seemingly unrelated nations, have adopted

_ten_ as their basic number; together with the fact that in the

remaining instances the basic number is either _five_ (the fingers of

one hand) or _twenty_ (the fingers and toes); almost of themselves show

that the fingers were the original units of numeration. The still

surviving use of the word _digit_, as the general name for a figure in

arithmetic, is significant; and it is even said that our word _ten_

(Sax. _tyn_; Dutch, _tien_; German, _zehn_) means in its primitive

expanded form _two hands_. So that originally, to say there were ten

things, was to say there were two hands of them.

From all which evidence it is tolerably clear that the earliest mode of

conveying the idea of any number of things, was by holding up as many

fingers as there were things; that is--using a symbol which was _equal_,

in respect of multiplicity, to the group symbolised. For which inference

there is, indeed, strong confirmation in the recent statement that our

own soldiers are even now spontaneously adopting this device in their

dealings with the Turks. And here it should be remarked that in this

recombination of the notion of equality with that of multiplicity, by

which the first steps in numeration are effected, we may see one of the

earliest of those inosculations between the diverging branches of

science, which are afterwards of perpetual occurrence.

Indeed, as this observation suggests, it will be well, before tracing

the mode in which exact science finally emerges from the merely

approximate judgments of the senses, and showing the non-serial

evolution of its divisions, to note the non-serial character of those

preliminary processes of which all after development is a continuation.

On reconsidering them it will be seen that not only are they divergent

growths from a common root, not only are they simultaneous in their

progress; but that they are mutual aids; and that none can advance

without the rest. That completeness of classification for which the

unfolding of the perceptions paves the way, is impossible without a

corresponding progress in language, by which greater varieties of

objects are thinkable and expressible. On the one hand it is impossible

to carry classification far without names by which to designate the

classes; and on the other hand it is impossible to make language faster

than things are classified.

Again, the multiplication of classes and the consequent narrowing of

each class, itself involves a greater likeness among the things classed

together; and the consequent approach towards the notion of complete

likeness itself allows classification to be carried higher. Moreover,

classification necessarily advances _pari passu_ with rationality--the

classification of _things_ with the classification of _relations_. For

things that belong to the same class are, by implication, things of

which the properties and modes of behaviour--the co-existences and

sequences--are more or less the same; and the recognition of this

sameness of co-existences and sequences is reasoning. Whence it follows

that the advance of classification is necessarily proportionate to the

advance of generalisations. Yet further, the notion of _likeness_, both

in things and relations, simultaneously evolves by one process of

culture the ideas of _equality_ of things and _equality_ of relations;

which are the respective bases of exact concrete reasoning and exact

abstract reasoning--Mathematics and Logic. And once more, this idea of

equality, in the very process of being formed, necessarily gives origin

to two series of relations--those of magnitude and those of number: from

which arise geometry and the calculus. Thus the process throughout is

one of perpetual subdivision and perpetual intercommunication of the

divisions. From the very first there has been that _consensus_ of

different kinds of knowledge, answering to the _consensus_ of the

intellectual faculties, which, as already said, must exist among the

sciences.

Let us now go on to observe how, out of the notions of _equality_ and

_number_, as arrived at in the manner described, there gradually arose

the elements of quantitative prevision.

Equality, once having come to be definitely conceived, was readily

applicable to other phenomena than those of magnitude. Being predicable

of all things producing indistinguishable impressions, there naturally

grew up ideas of equality in weights, sounds, colours, etc.; and indeed

it can scarcely be doubted that the occasional experience of equal

weights, sounds, and colours, had a share in developing the abstract

conception of equality--that the ideas of equality in size, relations,

forces, resistances, and sensible properties in general, were evolved

during the same period. But however this may be, it is clear that as

fast as the notion of equality gained definiteness, so fast did that

lowest kind of quantitative prevision which is achieved without any

instrumental aid, become possible.

The ability to estimate, however roughly, the amount of a foreseen

result, implies the conception that it will be _equal to_ a certain

imagined quantity; and the correctness of the estimate will manifestly

depend upon the accuracy at which the perceptions of sensible equality

have arrived. A savage with a piece of stone in his hand, and another

piece lying before him of greater bulk of the same kind (a fact which he

infers from the _equality_ of the two in colour and texture) knows about

what effort he must put forth to raise this other piece; and he judges

accurately in proportion to the accuracy with which he perceives that

the one is twice, three times, four times, etc., as large as the other;

that is--in proportion to the precision of his ideas of equality and

number. And here let us not omit to notice that even in these vaguest of

quantitative previsions, the conception of _equality of relations_ is

also involved. For it is only in virtue of an undefined perception that

the relation between bulk and weight in the one stone is _equal_ to the

relation between bulk and weight in the other, that even the roughest

approximation can be made.

But how came the transition from those uncertain perceptions of equality

which the unaided senses give, to the certain ones with which science

deals? It came by placing the things compared in juxtaposition. Equality

being predicated of things which give us indistinguishable impressions,

and no accurate comparison of impressions being possible unless they

occur in immediate succession, it results that exactness of equality is

ascertainable in proportion to the closeness of the compared things.

Hence the fact that when we wish to judge of two shades of colour

whether they are alike or not, we place them side by side; hence the

fact that we cannot, with any precision, say which of two allied sounds

is the louder, or the higher in pitch, unless we hear the one

immediately after the other; hence the fact that to estimate the ratio

of weights, we take one in each hand, that we may compare their

pressures by rapidly alternating in thought from the one to the other;

hence the fact, that in a piece of music we can continue to make equal

beats when the first beat has been given, but cannot ensure commencing

with the same length of beat on a future occasion; and hence, lastly,

the fact, that of all magnitudes, those of _linear extension_ are those

of which the equality is most accurately ascertainable, and those to

which by consequence all others have to be reduced. For it is the

peculiarity of linear extension that it alone allows its magnitudes to

be placed in _absolute_ juxtaposition, or, rather, in coincident

position; it alone can test the equality of two magnitudes by observing

whether they will coalesce, as two equal mathematical lines do, when

placed between the same points; it alone can test _equality_ by trying

whether it will become _identity_. Hence, then, the fact, that all exact

science is reducible, by an ultimate analysis, to results measured in

equal units of linear extension.

Still it remains to be noticed in what manner this determination of

equality by comparison of linear magnitudes originated. Once more may we

perceive that surrounding natural objects supplied the needful lessons.

From the beginning there must have been a constant experience of like

things placed side by side--men standing and walking together; animals

from the same herd; fish from the same shoal. And the ceaseless

repetition of these experiences could not fail to suggest the

observation, that the nearer together any objects were, the more visible

became any inequality between them. Hence the obvious device of putting

in apposition things of which it was desired to ascertain the relative

magnitudes. Hence the idea of _measure_. And here we suddenly come upon

a group of facts which afford a solid basis to the remainder of our

argument; while they also furnish strong evidence in support of the

foregoing speculations. Those who look sceptically on this attempted

rehabilitation of the earliest epochs of mental development, and who

more especially think that the derivation of so many primary notions

from organic forms is somewhat strained, will perhaps see more

probability in the several hypotheses that have been ventured, on

discovering that all measures of _extension_ and _force_ originated from

the lengths and weights of organic bodies; and all measures of _time_

from the periodic phenomena of either organic or inorganic bodies.

Thus, among linear measures, the cubit of the Hebrews was the _length of

the forearm_ from the elbow to the end of the middle finger; and the

smaller scriptural dimensions are expressed in _hand-breadths_ and

_spans_. The Egyptian cubit, which was similarly derived, was divided

into digits, which were _finger-breadths_; and each finger-breadth was

more definitely expressed as being equal to four _grains of barley_

placed breadthwise. Other ancient measures were the orgyia or _stretch

of the arms_, the _pace_, and the _palm_. So persistent has been the use

of these natural units of length in the East, that even now some of the

Arabs mete out cloth by the forearm. So, too, is it with European

measures. The _foot_ prevails as a dimension throughout Europe, and has

done since the time of the Romans, by whom, also, it was used: its

lengths in different places varying not much more than men's feet vary.

The heights of horses are still expressed in _hands_. The inch is the

length of the terminal joint of _the thumb_; as is clearly shown in

France, where _pouce_ means both thumb and inch. Then we have the inch

divided into three _barley-corns_.

So completely, indeed, have these organic dimensions served as the

substrata of all mensuration, that it is only by means of them that we

can form any estimate of some of the ancient distances. For example, the

length of a degree on the Earth's surface, as determined by the Arabian

astronomers shortly after the death of Haroun-al-Raschid, was fifty-six

of their miles. We know nothing of their mile further than that it was

4000 cubits; and whether these were sacred cubits or common cubits,

would remain doubtful, but that the length of the cubit is given as

twenty-seven inches, and each inch defined as the thickness of six

barley-grains. Thus one of the earliest measurements of a degree comes

down to us in barley-grains. Not only did organic lengths furnish those

approximate measures which satisfied men's needs in ruder ages, but they

furnished also the standard measures required in later times. One

instance occurs in our own history. To remedy the irregularities then

prevailing, Henry I. commanded that the ulna, or ancient ell, which

answers to the modern yard, should be made of the exact length of _his

own arm_.

Measures of weight again had a like derivation. Seeds seem commonly to

have supplied the unit. The original of the carat used for weighing in

India is _a small bean_. Our own systems, both troy and avoirdupois, are

derived primarily from wheat-corns. Our smallest weight, the grain, is

_a grain of wheat_. This is not a speculation; it is an historically

registered fact. Henry III. enacted that an ounce should be the weight

of 640 dry grains of wheat from the middle of the ear. And as all the

other weights are multiples or sub-multiples of this, it follows that

the grain of wheat is the basis of our scale. So natural is it to use

organic bodies as weights, before artificial weights have been

established, or where they are not to be had, that in some of the

remoter parts of Ireland the people are said to be in the habit, even

now, of putting a man into the scales to serve as a measure for heavy

commodities.

Similarly with time. Astronomical periodicity, and the periodicity of

animal and vegetable life, are simultaneously used in the first stages

of progress for estimating epochs. The simplest unit of time, the day,

nature supplies ready made. The next simplest period, the mooneth or

month, is also thrust upon men's notice by the conspicuous changes

constituting a lunation. For larger divisions than these, the phenomena

of the seasons, and the chief events from time to time occurring, have

been used by early and uncivilised races. Among the Egyptians the rising

of the Nile served as a mark. The New Zealanders were found to begin

their year from the reappearance of the Pleiades above the sea. One of

the uses ascribed to birds, by the Greeks, was to indicate the seasons

by their migrations. Barrow describes the aboriginal Hottentot as

denoting periods by the number of moons before or after the ripening of

one of his chief articles of food. He further states that the Kaffir

chronology is kept by the moon, and is registered by notches on

sticks--the death of a favourite chief, or the gaining of a victory,

serving for a new era. By which last fact, we are at once reminded that

in early history, events are commonly recorded as occurring in certain

reigns, and in certain years of certain reigns: a proceeding which

practically made a king's reign a measure of duration.

And, as further illustrating the tendency to divide time by natural

phenomena and natural events, it may be noticed that even by our own

peasantry the definite divisions of months and years are but little

used; and that they habitually refer to occurrences as "before

sheep-shearing," or "after harvest," or "about the time when the squire

died." It is manifest, therefore, that the more or less equal periods

perceived in Nature gave the first units of measure for time; as did

Nature's more or less equal lengths and weights give the first units of

measure for space and force.

It remains only to observe, as further illustrating the evolution of

quantitative ideas after this manner, that measures of value were

similarly derived. Barter, in one form or other, is found among all but

the very lowest human races. It is obviously based upon the notion of

_equality of worth_. And as it gradually merges into trade by the

introduction of some kind of currency, we find that the _measures of

worth_, constituting this currency, are organic bodies; in some cases

_cowries_, in others _cocoa-nuts_, in others _cattle_, in others _pigs_;

among the American Indians peltry or _skins_, and in Iceland _dried

fish_.

Notions of exact equality and of measure having been reached, there came

to be definite ideas of relative magnitudes as being multiples one of

another; whence the practice of measurement by direct apposition of a

measure. The determination of linear extensions by this process can

scarcely be called science, though it is a step towards it; but the

determination of lengths of time by an analogous process may be

considered as one of the earliest samples of quantitative prevision. For

when it is first ascertained that the moon completes the cycle of her

changes in about thirty days--a fact known to most uncivilised tribes

that can count beyond the number of their fingers--it is manifest that

it becomes possible to say in what number of days any specified phase of

the moon will recur; and it is also manifest that this prevision is

effected by an opposition of two times, after the same manner that

linear space is measured by the opposition of two lines. For to express

the moon's period in days, is to say how many of these units of measure

are contained in the period to be measured--is to ascertain the distance

between two points in time by means of a _scale of days_, just as we

ascertain the distance between two points in space by a scale of feet or

inches: and in each case the scale coincides with the thing

measured--mentally in the one; visibly in the other. So that in this

simplest, and perhaps earliest case of quantitative prevision, the

phenomena are not only thrust daily upon men's notice, but Nature is, as

it were, perpetually repeating that process of measurement by observing

which the prevision is effected. And thus there may be significance in

the remark which some have made, that alike in Hebrew, Greek, and Latin,

there is an affinity between the word meaning moon, and that meaning

measure.

This fact, that in very early stages of social progress it is known that

the moon goes through her changes in about thirty days, and that in

about twelve moons the seasons return--this fact that chronological

astronomy assumes a certain scientific character even before geometry

does; while it is partly due to the circumstance that the astronomical

divisions, day, month, and year, are ready made for us, is partly due to

the further circumstances that agricultural and other operations were at

first regulated astronomically, and that from the supposed divine

nature of the heavenly bodies their motions determined the periodical

religious festivals. As instances of the one we have the observation of

the Egyptians, that the rising of the Nile corresponded with the

heliacal rising of Sirius; the directions given by Hesiod for reaping

and ploughing, according to the positions of the Pleiades; and his maxim

that "fifty days after the turning of the sun is a seasonable time for

beginning a voyage." As instances of the other, we have the naming of

the days after the sun, moon, and planets; the early attempts among

Eastern nations to regulate the calendar so that the gods might not be

offended by the displacement of their sacrifices; and the fixing of the

great annual festival of the Peruvians by the position of the sun. In

all which facts we see that, at first, science was simply an appliance

of religion and industry.

After the discoveries that a lunation occupies nearly thirty days, and

that some twelve lunations occupy a year--discoveries of which there is

no historical account, but which may be inferred as the earliest, from

the fact that existing uncivilised races have made them--we come to the

first known astronomical records, which are those of eclipses. The

Chaldeans were able to predict these. "This they did, probably," says

Dr. Whewell in his useful history, from which most of the materials we

are about to use will be drawn, "by means of their cycle of 223 months,

or about eighteen years; for at the end of this time, the eclipses of

the moon begin to return, at the same intervals and in the same order as

at the beginning." Now this method of calculating eclipses by means of a

recurring cycle,--the _Saros_ as they called it--is a more complex case

of prevision by means of coincidence of measures. For by what

observations must the Chaldeans have discovered this cycle? Obviously,

as Delambre infers, by inspecting their registers; by comparing the

successive intervals; by finding that some of the intervals were alike;

by seeing that these equal intervals were eighteen years apart; by

discovering that _all_ the intervals that were eighteen years apart were

equal; by ascertaining that the intervals formed a series which repeated

itself, so that if one of the cycles of intervals were superposed on

another the divisions would fit. This once perceived, and it manifestly

became possible to use the cycle as a scale of time by which to measure

out future periods. Seeing thus that the process of so predicting

eclipses is in essence the same as that of predicting the moon's monthly

changes, by observing the number of days after which they repeat--seeing

that the two differ only in the extent and irregularity of the

intervals, it is not difficult to understand how such an amount of

knowledge should so early have been reached. And we shall be less

surprised, on remembering that the only things involved in these

previsions were _time_ and _number_; and that the time was in a manner

self-numbered.

Still, the ability to predict events recurring only after so long a

period as eighteen years, implies a considerable advance in

civilisation--a considerable development of general knowledge; and we

have now to inquire what progress in other sciences accompanied, and was

necessary to, these astronomical previsions. In the first place, there

must clearly have been a tolerably efficient system of calculation. Mere

finger-counting, mere head-reckoning, even with the aid of a regular

decimal notation, could not have sufficed for numbering the days in a

year; much less the years, months, and days between eclipses.

Consequently there must have been a mode of registering numbers;

probably even a system of numerals. The earliest numerical records, if

we may judge by the practices of the less civilised races now existing,

were probably kept by notches cut on sticks, or strokes marked on walls;

much as public-house scores are kept now. And there seems reason to

believe that the first numerals used were simply groups of straight

strokes, as some of the still-extant Roman ones are; leading us to

suspect that these groups of strokes were used to represent groups of

fingers, as the groups of fingers had been used to represent groups of

objects--a supposition quite in conformity with the aboriginal system of

picture writing and its subsequent modifications. Be this so or not,

however, it is manifest that before the Chaldeans discovered their

_Saros_, there must have been both a set of written symbols serving for

an extensive numeration, and a familiarity with the simpler rules of

arithmetic.

Not only must abstract mathematics have made some progress, but concrete

mathematics also. It is scarcely possible that the buildings belonging

to this era should have been laid out and erected without any knowledge

of geometry. At any rate, there must have existed that elementary

geometry which deals with direct measurement--with the apposition of

lines; and it seems that only after the discovery of those simple

proceedings, by which right angles are drawn, and relative positions

fixed, could so regular an architecture be executed. In the case of the

other division of concrete mathematics--mechanics, we have definite

evidence of progress. We know that the lever and the inclined plane were

employed during this period: implying that there was a qualitative

prevision of their effects, though not a quantitative one. But we know

more. We read of weights in the earliest records; and we find weights in

ruins of the highest antiquity. Weights imply scales, of which we have

also mention; and scales involve the primary theorem of mechanics in its

least complicated form--involve not a qualitative but a quantitative

prevision of mechanical effects. And here we may notice how mechanics,

in common with the other exact sciences, took its rise from the simplest

application of the idea of _equality_. For the mechanical proposition

which the scales involve, is, that if a lever with _equal_ arms, have

_equal_ weights suspended from them, the weights will remain at _equal_

altitudes. And we may further notice how, in this first step of rational

mechanics, we see illustrated that truth awhile since referred to, that

as magnitudes of linear extension are the only ones of which the

equality is exactly ascertainable, the equalities of other magnitudes

have at the outset to be determined by means of them. For the equality

of the weights which balance each other in scales, wholly depends upon

the equality of the arms: we can know that the weights are equal only by

proving that the arms are equal. And when by this means we have obtained

a system of weights,--a set of equal units of force, then does a science

of mechanics become possible. Whence, indeed, it follows, that rational

mechanics could not possibly have any other starting-point than the

scales.

Let us further remember, that during this same period there was a

limited knowledge of chemistry. The many arts which we know to have been

carried on must have been impossible without a generalised experience of

the modes in which certain bodies affect each other under special

conditions. In metallurgy, which was extensively practised, this is

abundantly illustrated. And we even have evidence that in some cases the

knowledge possessed was, in a sense, quantitative. For, as we find by

analysis that the hard alloy of which the Egyptians made their cutting

tools, was composed of copper and tin in fixed proportions, there must

have been an established prevision that such an alloy was to be obtained

only by mixing them in these proportions. It is true, this was but a

simple empirical generalisation; but so was the generalisation

respecting the recurrence of eclipses; so are the first generalisations

of every science.

Respecting the simultaneous advance of the sciences during this early

epoch, it only remains to remark that even the most complex of them

must have made some progress--perhaps even a greater relative progress

than any of the rest. For under what conditions only were the foregoing

developments possible? There first required an established and organised

social system. A long continued registry of eclipses; the building of

palaces; the use of scales; the practice of metallurgy--alike imply a

fixed and populous nation. The existence of such a nation not only

presupposes laws, and some administration of justice, which we know

existed, but it presupposes successful laws--laws conforming in some

degree to the conditions of social stability--laws enacted because it

was seen that the actions forbidden by them were dangerous to the State.

We do not by any means say that all, or even the greater part, of the

laws were of this nature; but we do say, that the fundamental ones were.

It cannot be denied that the laws affecting life and property were such.

It cannot be denied that, however little these were enforced between

class and class, they were to a considerable extent enforced between

members of the same class. It can scarcely be questioned, that the

administration of them between members of the same class was seen by

rulers to be necessary for keeping their subjects together. And knowing,

as we do, that, other things equal, nations prosper in proportion to the

justness of their arrangements, we may fairly infer that the very cause

of the advance of these earliest nations out of aboriginal barbarism was

the greater recognition among them of the claims to life and property.

But supposition aside, it is clear that the habitual recognition of

these claims in their laws implied some prevision of social phenomena.

Even thus early there was a certain amount of social science. Nay, it

may even be shown that there was a vague recognition of that fundamental

principle on which all the true social science is based--the equal

rights of all to the free exercise of their faculties. That same idea of

_equality_ which, as we have seen, underlies all other science,

underlies also morals and sociology. The conception of justice, which is

the primary one in morals; and the administration of justice, which is

the vital condition of social existence; are impossible without the

recognition of a certain likeness in men's claims in virtue of their

common humanity. _Equity_ literally means _equalness_; and if it be

admitted that there were even the vaguest ideas of equity in these

primitive eras, it must be admitted that there was some appreciation of

the equalness of men's liberties to pursue the objects of life--some

appreciation, therefore, of the essential principle of national

equilibrium.

Thus in this initial stage of the positive sciences, before geometry had

yet done more than evolve a few empirical rules--before mechanics had

passed beyond its first theorem--before astronomy had advanced from its

merely chronological phase into the geometrical; the most involved of

the sciences had reached a certain degree of development--a development

without which no progress in other sciences was possible.

Only noting as we pass, how, thus early, we may see that the progress of

exact science was not only towards an increasing number of previsions,

but towards previsions more accurately quantitative--how, in astronomy,

the recurring period of the moon's motions was by and by more correctly

ascertained to be nineteen years, or two hundred and thirty-five

lunations; how Callipus further corrected this Metonic cycle, by leaving

out a day at the end of every seventy-six years; and how these

successive advances implied a longer continued registry of observations,

and the co-ordination of a greater number of facts--let us go on to

inquire how geometrical astronomy took its rise.

The first astronomical instrument was the gnomon. This was not only

early in use in the East, but it was found also among the Mexicans; the

sole astronomical observations of the Peruvians were made by it; and we

read that 1100 B.C., the Chinese found that, at a certain place, the

length of the sun's shadow, at the summer solstice, was to the height of

the gnomon as one and a half to eight. Here again it is observable, not

only that the instrument is found ready made, but that Nature is

perpetually performing the process of measurement. Any fixed, erect

object--a column, a dead palm, a pole, the angle of a building--serves

for a gnomon; and it needs but to notice the changing position of the

shadow it daily throws to make the first step in geometrical astronomy.

How small this first step was, may be seen in the fact that the only

things ascertained at the outset were the periods of the summer and

winter solstices, which corresponded with the least and greatest lengths

of the mid-shadow; and to fix which, it was needful merely to mark the

point to which each day's shadow reached.

And now let it not be overlooked that in the observing at what time

during the next year this extreme limit of the shadow was again reached,

and in the inference that the sun had then arrived at the same turning

point in his annual course, we have one of the simplest instances of

that combined use of _equal magnitudes_ and _equal relations_, by which

all exact science, all quantitative prevision, is reached. For the

relation observed was between the length of the sun's shadow and his

position in the heavens; and the inference drawn was that when, next

year, the extremity of his shadow came to the same point, he occupied

the same place. That is, the ideas involved were, the equality of the

shadows, and the equality of the relations between shadow and sun in

successive years. As in the case of the scales, the equality of

relations here recognised is of the simplest order. It is not as those

habitually dealt with in the higher kinds of scientific reasoning, which

answer to the general type--the relation between two and three equals

the relation between six and nine; but it follows the type--the relation

between two and three, equals the relation between two and three; it is

a case of not simply _equal_ relations, but _coinciding_ relations. And

here, indeed, we may see beautifully illustrated how the idea of equal

relations takes its rise after the same manner that that of equal

magnitude does. As already shown, the idea of equal magnitudes arose

from the observed coincidence of two lengths placed together; and in

this case we have not only two coincident lengths of shadows, but two

coincident relations between sun and shadows.

From the use of the gnomon there naturally grew up the conception of

angular measurements; and with the advance of geometrical conceptions

there came the hemisphere of Berosus, the equinoctial armil, the

solstitial armil, and the quadrant of Ptolemy--all of them employing

shadows as indices of the sun's position, but in combination with

angular divisions. It is obviously out of the question for us here to

trace these details of progress. It must suffice to remark that in all

of them we may see that notion of equality of relations of a more

complex kind, which is best illustrated in the astrolabe, an instrument

which consisted "of circular rims, movable one within the other, or

about poles, and contained circles which were to be brought into the

position of the ecliptic, and of a plane passing through the sun and the

poles of the ecliptic"--an instrument, therefore, which represented, as

by a model, the relative positions of certain imaginary lines and planes

in the heavens; which was adjusted by putting these representative lines

and planes into parallelism and coincidence with the celestial ones; and

which depended for its use upon the perception that the relations

between these representative lines and planes were _equal_ to the

relations between those represented.

Were there space, we might go on to point out how the conception of the

heavens as a revolving hollow sphere, the discovery of the globular form

of the earth, the explanation of the moon's phases, and indeed all the

successive steps taken, involved this same mental process. But we must

content ourselves with referring to the theory of eccentrics and

epicycles, as a further marked illustration of it. As first suggested,

and as proved by Hipparchus to afford an explanation of the leading

irregularities in the celestial motions, this theory involved the

perception that the progressions, retrogressions, and variations of

velocity seen in the heavenly bodies, might be reconciled with their

assumed uniform movement in circles, by supposing that the earth was not

in the centre of their orbits; or by supposing that they revolved in

circles whose centres revolved round the earth; or by both. The

discovery that this would account for the appearances, was the discovery

that in certain geometrical diagrams the relations were such, that the

uniform motion of a point would, when looked at from a particular

position, present analogous irregularities; and the calculations of

Hipparchus involved the belief that the relations subsisting among these

geometrical curves were _equal_ to the relations subsisting among the

celestial orbits.

Leaving here these details of astronomical progress, and the philosophy

of it, let us observe how the relatively concrete science of geometrical

astronomy, having been thus far helped forward by the development of

geometry in general, reacted upon geometry, caused it also to advance,

and was again assisted by it. Hipparchus, before making his solar and

lunar tables, had to discover rules for calculating the relations

between the sides and angles of triangles--_trigonometry_ a subdivision

of pure mathematics. Further, the reduction of the doctrine of the

sphere to the quantitative form needed for astronomical purposes,

required the formation of a _spherical trigonometry_, which was also

achieved by Hipparchus. Thus both plane and spherical trigonometry,

which are parts of the highly abstract and simple science of extension,

remained undeveloped until the less abstract and more complex science of

the celestial motions had need of them. The fact admitted by M. Comte,

that since Descartes the progress of the abstract division of

mathematics has been determined by that of the concrete division, is

paralleled by the still more significant fact that even thus early the

progress of mathematics was determined by that of astronomy.

And here, indeed, we may see exemplified the truth, which the subsequent

history of science frequently illustrates, that before any more

abstract division makes a further advance, some more concrete division

must suggest the necessity for that advance--must present the new order

of questions to be solved. Before astronomy presented Hipparchus with

the problem of solar tables, there was nothing to raise the question of

the relations between lines and angles; the subject-matter of

trigonometry had not been conceived. And as there must be subject-matter

before there can be investigation, it follows that the progress of the

concrete divisions is as necessary to that of the abstract, as the

progress of the abstract to that of the concrete.

Just incidentally noticing the circumstance that the epoch we are

describing witnessed the evolution of algebra, a comparatively abstract

division of mathematics, by the union of its less abstract divisions,

geometry and arithmetic--a fact proved by the earliest extant samples of

algebra, which are half algebraic, half geometric--we go on to observe

that during the era in which mathematics and astronomy were thus

advancing, rational mechanics made its second step; and something was

done towards giving a quantitative form to hydrostatics, optics, and

harmonics. In each case we shall see, as before, how the idea of

equality underlies all quantitative prevision; and in what simple forms

this idea is first applied.

As already shown, the first theorem established in mechanics was, that

equal weights suspended from a lever with equal arms would remain in

equilibrium. Archimedes discovered that a lever with unequal arms was in

equilibrium when one weight was to its arm as the other arm to its

weight; that is--when the numerical relation between one weight and its

arm was _equal_ to the numerical relation between the other arm and its

weight.

The first advance made in hydrostatics, which we also owe to Archimedes,

was the discovery that fluids press _equally_ in all directions; and

from this followed the solution of the problem of floating bodies:

namely, that they are in equilibrium when the upward and downward

pressures are _equal_.

In optics, again, the Greeks found that the angle of incidence is

_equal_ to the angle of reflection; and their knowledge reached no

further than to such simple deductions from this as their geometry

sufficed for. In harmonics they ascertained the fact that three strings

of _equal_ lengths would yield the octave, fifth and fourth, when

strained by weights having certain definite ratios; and they did not

progress much beyond this. In the one of which cases we see geometry

used in elucidation of the laws of light; and in the other, geometry and

arithmetic made to measure the phenomena of sound.

Did space permit, it would be desirable here to describe the state of

the less advanced sciences--to point out how, while a few had thus

reached the first stages of quantitative prevision, the rest were

progressing in qualitative prevision--how some small generalisations

were made respecting evaporation, and heat, and electricity, and

magnetism, which, empirical as they were, did not in that respect differ

from the first generalisations of every science--how the Greek

physicians had made advances in physiology and pathology, which,

considering the great imperfection of our present knowledge, are by no

means to be despised--how zoology had been so far systematised by

Aristotle, as, to some extent, enabled him from the presence of certain

organs to predict the presence of others--how in Aristotle's _Politics_

there is some progress towards a scientific conception of social

phenomena, and sundry previsions respecting them--and how in the state

of the Greek societies, as well as in the writings of Greek

philosophers, we may recognise not only an increasing clearness in that

conception of equity on which the social science is based, but also some

appreciation of the fact that social stability depends upon the

maintenance of equitable regulations. We might dwell at length upon the

causes which retarded the development of some of the sciences, as, for

example, chemistry; showing that relative complexity had nothing to do

with it--that the oxidation of a piece of iron is a simpler phenomenon

than the recurrence of eclipses, and the discovery of carbonic acid less

difficult than that of the precession of the equinoxes--but that the

relatively slow advance of chemical knowledge was due, partly to the

fact that its phenomena were not daily thrust on men's notice as those

of astronomy were; partly to the fact that Nature does not habitually

supply the means, and suggest the modes of investigation, as in the

sciences dealing with time, extension, and force; and partly to the fact

that the great majority of the materials with which chemistry deals,

instead of being ready to hand, are made known only by the arts in their

slow growth; and partly to the fact that even when known, their chemical

properties are not self-exhibited, but have to be sought out by

experiment.

Merely indicating all these considerations, however, let us go on to

contemplate the progress and mutual influence of the sciences in modern

days; only parenthetically noticing how, on the revival of the

scientific spirit, the successive stages achieved exhibit the dominance

of the same law hitherto traced--how the primary idea in dynamics, a

uniform force, was defined by Galileo to be a force which generates

_equal_ velocities in _equal_ successive times--how the uniform action

of gravity was first experimentally determined by showing that the time

elapsing before a body thrown up, stopped, was _equal_ to the time it

took to fall--how the first fact in compound motion which Galileo

ascertained was, that a body projected horizontally will have a uniform

motion onwards and a uniformly accelerated motion downwards; that is,

will describe _equal_ horizontal spaces in _equal_ times, compounded

with _equal_ vertical increments in _equal_ times--how his discovery

respecting the pendulum was, that its oscillations occupy _equal_

intervals of time whatever their length--how the principle of virtual

velocities which he established is, that in any machine the weights that

balance each other are reciprocally as their virtual velocities; that

is, the relation of one set of weights to their velocities _equals_ the

relation of the other set of velocities to their weights; and how thus

his achievements consisted in showing the equalities of certain

magnitudes and relations, whose equalities had not been previously

recognised.

When mechanics had reached the point to which Galileo brought it--when

the simple laws of force had been disentangled from the friction and

atmospheric resistance by which all their earthly manifestations are

disguised--when progressing knowledge of _physics_ had given a due

insight into these disturbing causes--when, by an effort of abstraction,

it was perceived that all motion would be uniform and rectilinear unless

interfered with by external forces--and when the various consequences of

this perception had been worked out; then it became possible, by the

union of geometry and mechanics, to initiate physical astronomy.

Geometry and mechanics having diverged from a common root in men's

sensible experiences; having, with occasional inosculations, been

separately developed, the one partly in connection with astronomy, the

other solely by analysing terrestrial movements; now join in the

investigations of Newton to create a true theory of the celestial

motions. And here, also, we have to notice the important fact that, in

the very process of being brought jointly to bear upon astronomical

problems, they are themselves raised to a higher phase of development.

For it was in dealing with the questions raised by celestial dynamics

that the then incipient infinitesimal calculus was unfolded by Newton

and his continental successors; and it was from inquiries into the

mechanics of the solar system that the general theorems of mechanics

contained in the _Principia_,--many of them of purely terrestrial

application--took their rise. Thus, as in the case of Hipparchus, the

presentation of a new order of concrete facts to be analysed, led to the

discovery of new abstract facts; and these abstract facts having been

laid hold of, gave means of access to endless groups of concrete facts

before incapable of quantitative treatment.

Meanwhile, physics had been carrying further that progress without

which, as just shown, rational mechanics could not be disentangled. In

hydrostatics, Stevinus had extended and applied the discovery of

Archimedes. Torricelli had proved atmospheric pressure, "by showing that

this pressure sustained different liquids at heights inversely

proportional to their densities;" and Pascal "established the necessary

diminution of this pressure at increasing heights in the atmosphere:"

discoveries which in part reduced this branch of science to a

quantitative form. Something had been done by Daniel Bernouilli towards

the dynamics of fluids. The thermometer had been invented; and a number

of small generalisations reached by it. Huyghens and Newton had made

considerable progress in optics; Newton had approximately calculated the

rate of transmission of sound; and the continental mathematicians had

succeeded in determining some of the laws of sonorous vibrations.

Magnetism and electricity had been considerably advanced by Gilbert.

Chemistry had got as far as the mutual neutralisation of acids and

alkalies. And Leonardo da Vinci had advanced in geology to the

conception of the deposition of marine strata as the origin of fossils.

Our present purpose does not require that we should give particulars.

All that it here concerns us to do is to illustrate the _consensus_

subsisting in this stage of growth, and afterwards. Let us look at a few

cases.

The theoretic law of the velocity of sound enunciated by Newton on

purely mechanical considerations, was found wrong by one-sixth. The

error remained unaccounted for until the time of Laplace, who,

suspecting that the heat disengaged by the compression of the undulating

strata of the air, gave additional elasticity, and so produced the

difference, made the needful calculations and found he was right. Thus

acoustics was arrested until thermology overtook and aided it. When

Boyle and Marriot had discovered the relation between the density of

gases and the pressures they are subject to; and when it thus became

possible to calculate the rate of decreasing density in the upper parts

of the atmosphere, it also became possible to make approximate tables of

the atmospheric refraction of light. Thus optics, and with it astronomy,

advanced with barology. After the discovery of atmospheric pressure had

led to the invention of the air-pump by Otto Guericke; and after it had

become known that evaporation increases in rapidity as atmospheric

pressure decreases; it became possible for Leslie, by evaporation in a

vacuum, to produce the greatest cold known; and so to extend our

knowledge of thermology by showing that there is no zero within reach of

our researches. When Fourier had determined the laws of conduction of

heat, and when the Earth's temperature had been found to increase below

the surface one degree in every forty yards, there were data for

inferring the past condition of our globe; the vast period it has taken

to cool down to its present state; and the immense age of the solar

system--a purely astronomical consideration.

Chemistry having advanced sufficiently to supply the needful materials,

and a physiological experiment having furnished the requisite hint,

there came the discovery of galvanic electricity. Galvanism reacting on

chemistry disclosed the metallic bases of the alkalies, and inaugurated

the electro-chemical theory; in the hands of Oersted and Ampère it led

to the laws of magnetic action; and by its aid Faraday has detected

significant facts relative to the constitution of light. Brewster's

discoveries respecting double refraction and dipolarisation proved the

essential truth of the classification of crystalline forms according to

the number of axes, by showing that the molecular constitution depends

upon the axes. In these and in numerous other cases, the mutual

influence of the sciences has been quite independent of any supposed

hierarchical order. Often, too, their inter-actions are more complex

than as thus instanced--involve more sciences than two. One illustration

of this must suffice. We quote it in full from the _History of the

Inductive Sciences_. In book xi., chap, ii., on "The Progress of the

Electrical Theory," Dr. Whewell writes:--