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

philosophize 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 generalized (experiences which had been

scarcely at all generalized at the beginning of the historic period)--we

find these generalized experiences, these intellectual abstractions,

elevated into concrete actualities, projected back into Nature, and

considered as the internal frame-work 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 philosophize 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

_a priori_ method, as it may be termed. The legitimate _a priori_ method

sets out with propositions of which the negation is inconceivable; the _a

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 B[oe]hme on a par with Bacon,

suffices alone to show that his stand-point 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.[F]

[F] It is somewhat curious that the author of "The Plurality of

Worlds," with quite other aims, should have persuaded himself

into similar conclusions.

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 Philosophie 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 _a 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 historic 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 cognizable 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,

hydrostatical, 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 endeavors 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

generalite et de dependance des phenomenes, qu'elle doit etre partagee en

deux sections distinctes, suivant qu'elle considere les phenomenes generaux

de l'univers, ou, en particulier, ceux que presentent les corps terrestres.

D'ou la physique celeste, ou l'astronomie, soit geometrique, 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 generalization 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, &c., 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

generalizations 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 generalization 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 generalize the phenomena of gravity, cohesion, atomic affinity, and

electric repulsion, just as the laws of number generalize 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 generalization 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 specialization 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 Picard's more exact measurement was

published, that he returned to his calculations and proved his great

generalization. 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 cultivees simultanement par les

memes esprits_;" that this is "_inevitable et meme 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 echelle encyclopedique_."

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

sub-dividing. But it does not suggest the yet more important fact, that the

divisions and sub-divisions thus arising do not remain separate, but now

and again re-unite 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 specialization, that there might be a

larger generalization; and a deeper analysis, that there might be a better

synthesis. Each larger generalization has lifted sundry specializations

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 generalization and

specialization; 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, &c., 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, &c., 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 sub-divided 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

symbolized 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 dramatizing

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 civilized 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

notion 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 uncivilized 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 generalization 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 generalized. 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, civilized 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 likenesses 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 uncivilized 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 that 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 analyze 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 were 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 symbolized. 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 re-considering 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

generalizations. 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, &c.; 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 but 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, &c. 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 uncivilized 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 uncivilized 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 uncivilized 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 civilization--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 civilized

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 generalized 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

generalization; but so was the generalization respecting the recurrence of

eclipses; so are the first generalizations 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 organized 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-day 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 recognized 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, moveable 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 generalizations were made respecting

evaporation, and heat, and electricity, and magnetism, which, empirical as

they were, did not in that respect differ from the first generalizations 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

systematized 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 connexion with astronomy, the other solely by analyzing

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 analyzed, 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 Bernoulli towards the dynamics of fluids. The

thermometer had been invented; and a number of small generalizations

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

neutralization 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 as 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 Ampere 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 dipolarization 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:--