THE METHODS OF SCIENCE 91

such facts and to our consciousness. But these are not, in

any true sense, " measurements." On the other hand, all the

problems solved by careful measurements in physical science

are in every case ascertained and solved by the attainment of

a correct appreciation of relations existing between different

objects and activities. And, indeed, Metaphysics may also

be said to be occupied about metaphysical relations. Thus

all science is one vast process of ascertaining, as correctly as

possible, relations (e.g., co-existence, succession, and causation)

of very different orders of things.

But owing to our organization, every such inquiry must be

carried on, and every conclusion arrived at, through either our

sense -perceptions* or by the aid of sensuous imaginations,

however supersensuous the essential nature of the object of

our inquiry may be.

The imaginations we make use of need not, of course, be

mental pictures of concrete, extended things ; they may be

the merest symbols, and such symbols are not only of the

greatest utility, but are absolutely necessary for the very

simplest kinds of science.

Spoken and written words are such audible and visible

symbols, and so are numerals and all algebraic signs. By

means of symbols we can work out the most complicated

results without any need of thinking, meanwhile, what it is such

symbols represent. But in the end, to arrive at any practical

or complete result, the symbols must be retranslated into the

things they symbolized, and thus the correspondence of pro-

cesses gone through (simple or complex) may be tested by

our direct or indirect sense-perceptions. Thus, in matters so

elementary as the simple addition of numerals, the result may ,

be tested by taking parcels of things, e.g., marbles, each

corresponding in number with one of the (symbols) numbers

to be added together, and, having mixed the whole, then

* See ante, p. 9.

9 2

counting them, and so seeing that the senses of sight and

touch confirm the previous result of the addition of the

numerical symbols. It is the same as regards the process

of subtraction ; its correspondence with the real relations

which exist between the substantial things may be similarly

tested.

The symbolism of science may be very well exemplified

by the simplest facts of algebra, which, as our readers know,

is a branch of science replete with the most beautiful,

complex, ingenious, and far-reaching processes, whereby alone

many calculations are made possible, or the labours of

investigation lessened, while the results arrived at have

complete accuracy. This is the case even when we find

need to employ symbols which express not only unreal, but

even impossible, quantities, by means of which we may arrive

at otherwise unattainable truths concerning real or possible

existences. Such is the case, because they express abstract

truths which have real applications, or would have them

could the impossible conditions, sometimes supposed, really

exist. Thus even the absurd and impossible quantities ex-

pressed by the symbol \/-x has its relations with reality.

It is, of course, really impossible in itself, since there is no

quantity which, being multiplied by itself, gives a negative

product. Yet it has its relation with reality, inasmuch as

it can be used as if it were a real quantity, and all the

laws and relations relating to real quantities can be applied

to it.

The truths and processes of algebra may be tested by

our direct sense-experience (as those of arithmetic may)

by making use of definite numbers as representatives of

algebraic symbols, and so translating algebra into arithmetic

in order to be practically tested. The truths of geometry

may be tested by being made evident to the eye and by

reasoning.

such facts and to our consciousness. But these are not, in

any true sense, " measurements." On the other hand, all the

problems solved by careful measurements in physical science

are in every case ascertained and solved by the attainment of

a correct appreciation of relations existing between different

objects and activities. And, indeed, Metaphysics may also

be said to be occupied about metaphysical relations. Thus

all science is one vast process of ascertaining, as correctly as

possible, relations (e.g., co-existence, succession, and causation)

of very different orders of things.

But owing to our organization, every such inquiry must be

carried on, and every conclusion arrived at, through either our

sense -perceptions* or by the aid of sensuous imaginations,

however supersensuous the essential nature of the object of

our inquiry may be.

The imaginations we make use of need not, of course, be

mental pictures of concrete, extended things ; they may be

the merest symbols, and such symbols are not only of the

greatest utility, but are absolutely necessary for the very

simplest kinds of science.

Spoken and written words are such audible and visible

symbols, and so are numerals and all algebraic signs. By

means of symbols we can work out the most complicated

results without any need of thinking, meanwhile, what it is such

symbols represent. But in the end, to arrive at any practical

or complete result, the symbols must be retranslated into the

things they symbolized, and thus the correspondence of pro-

cesses gone through (simple or complex) may be tested by

our direct or indirect sense-perceptions. Thus, in matters so

elementary as the simple addition of numerals, the result may ,

be tested by taking parcels of things, e.g., marbles, each

corresponding in number with one of the (symbols) numbers

to be added together, and, having mixed the whole, then

* See ante, p. 9.

9 2

counting them, and so seeing that the senses of sight and

touch confirm the previous result of the addition of the

numerical symbols. It is the same as regards the process

of subtraction ; its correspondence with the real relations

which exist between the substantial things may be similarly

tested.

The symbolism of science may be very well exemplified

by the simplest facts of algebra, which, as our readers know,

is a branch of science replete with the most beautiful,

complex, ingenious, and far-reaching processes, whereby alone

many calculations are made possible, or the labours of

investigation lessened, while the results arrived at have

complete accuracy. This is the case even when we find

need to employ symbols which express not only unreal, but

even impossible, quantities, by means of which we may arrive

at otherwise unattainable truths concerning real or possible

existences. Such is the case, because they express abstract

truths which have real applications, or would have them

could the impossible conditions, sometimes supposed, really

exist. Thus even the absurd and impossible quantities ex-

pressed by the symbol \/-x has its relations with reality.

It is, of course, really impossible in itself, since there is no

quantity which, being multiplied by itself, gives a negative

product. Yet it has its relation with reality, inasmuch as

it can be used as if it were a real quantity, and all the

laws and relations relating to real quantities can be applied

to it.

The truths and processes of algebra may be tested by

our direct sense-experience (as those of arithmetic may)

by making use of definite numbers as representatives of

algebraic symbols, and so translating algebra into arithmetic

in order to be practically tested. The truths of geometry

may be tested by being made evident to the eye and by

reasoning.