William Kingdon Clifford.

Lectures and essays (Volume v. 2) online

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VOL. 11.

Digitized by the Internet Arciiive

in 2010 witii funding from

Open Knowledge Commons and Harvard Medical School




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' Z« vii-ite est toute pour tous'—PAVL-homa Courier



[The nght of translation is reserved]



thp: second volume.



Instruments used in Measurement 3

Body and Mind . . . . . . . . 31

On the Nature of Things-in-themselves . . . .71

On the Types of Compound Statement involving Four

Classes 89

On the Scientific Basis of Morals 106

Right and Wrong : the Scientific Ground of their Dis-
tinction . . 124

The Ethics of Belief 177

The Ethics of Religion 212

The Influence upon Morality of a Decline in Religious

Belief 244

Cosmic Emotion 253

ViRCHow ON the Teaching of Science 286





By Measurement^ for scientific purposes, is meant the
measurement of quantities. In each special subject
there are quantities to be measured ; and these are very-
various, as may be seen from the following list of those
belonging to geometry and dynamics.

Geometrical Quantities.




Angles (plane and soHd)

Curvatures (plane and solid)

Strains (elongation, torsion, shear).

Circumstances of Motion. Properties of Bodies.

Time Mass

Velocity Weight

Momentum Density

Acceleration Specific gravity

Force Elasticity (of form and
Work volume)

Horse-power Viscosity

Temperature Diffusion
Heat. Surface tension

Specific heat.

^ [' Handbook to Loan Collection of Scientific Apparatus, 1876].

B 2


Notwithstanding the very different characters of these
quantities, they are all measured by reducing them to
the same kind of quantity, and estimating that in the
same way. Every quantity is measured by finding a
length proportional to the quantity, and then measuring
this length. This will, perhaps, be better understood if
we consider one or two examples.

The measurement of angles occurs in a very large
majority of scientific instruments. It is always effected
by measuring the length of an arc upon a graduated
circle ; the circumference of this circle being divided
not into inches or centimetres, but into degrees and
parts of a degree — that is, into aliquot parts of the
whole circumference.

As a step towards their final measurement, some
quantities, of which work is a good instance, are repre-
sented in the form of areas ; and there seems reason
to beheve that this method is hkely to be extended.
Instruments for measuring areas are called Planimeters ;
and one of the simplest of these is Amsler's, consisting
of two rods jointed together, the end of one being fixed
and that of the other being made to run round the area
which is to be measured. The second rod rests on a
wheel, which turns as the rod moves ; and it is proved
by geometry that the area is proportional to the distance
through which the wheel turns. Thus the measure-
ment of an area is reduced to the measurement of a

Volumes are measured in various ways, but all
depending on the same principle. Quantities of earth
excavated for engineering purposes are estimated by a
rough determination of the shape of the cavity, and the


measurement of its dimensions, namely, certain lengths
belonging to it. The contents of a vessel are some-
times gauged in the same way ; but the more accurate
method is to fill it with liquid and then pour the liquid
into a cyhnder of known section, when the quantity
is measured by the height of the liquid in the cylinder,
that is, by a length. The volumes of irregular solids
are also measured by immersing them in liquid con-
tained in a uniform cylinder, and observing the height
.to which the liquid rises ; that is, by measuring a length.
An apparatus for this purpose is called a Stereometer.
The liquid must be so chosen that no chemical action
takes place between it and the solid immersed, and that
it wets the sohd, so that no air bubbles adhere to the
surface. Thus mercury is used in the case of metals
by the Standards Department.

Time is measured for ordinary purposes by the length
of the arc traced out by a moving hand on a circular
clock-face. For astronomical purposes it is sometimes
measured by counting the ticks of a clock which beats
seconds, and estimating mentally the fractions of a
second ; and in cases where the period of an oscillation
has to be found, it is determined by counting the
number of oscillations in a time sufiicient to make the
number considerable, and then dividing that time by
the number. But by far the most accurate way of
measuring time is by means of the line traced by a
pencil on a sheet of paper rolled round a revolving
cylinder, or a spot of light moving on a sensitive surface.
If the pencil is made to move along the length of the
cyhnder so as to indicate what is happening as time goes
along, the time of each event will be found when the


cylinder is unrolled by measuring the distance of the
mark recording it from the end of the unrolled sheet,
provided that the rate at which the cyhnder goes round
is known. In this way Helmholtz measured the rate of
transmission of nerve-disturbance.

A very common case of the measurement of force
is the barometer, which measures the pressure of the
atmosphere per square inch of surface. This is deter-
mined by finding the height of the column of mercury
which it will support (mercurial barometer), or the
strain which it causes in a box from wliich the air has
been taken out (aneroid barometer). The height in the
former case may be measured directly, or it may first
be converted into the quantity of turning of a needle,
and then read off as length of arc on a graduated circle ;
in the latter case the strain is always indicated by a
needle turning on a graduated circle.

The mass^ and (what is proportional to it) the weight,
of different bodies at the same place, are measured by
means of a balance ; and at first sight this mode of
measurement seems different from those which we
have hitherto considered. For we put the body to be
weighed in one scale, and then put known weights into
the other until equihbrium is obtained or the scale
turns, and then we count the weights. But in a steel-
yard the weight is determined directly by means of a
length ; and in a balance which is accurate enough for
scientific purposes, both methods are employed. We get
as near as we can with the weights, and then the remain-
der is measured by a small rider of wire which is moved
along the beam, and which determines the weight by
its position ; that is, by the measurement of a length.


For the measurement of weight in different places a
spring-balance has to be used, and the weight is deter-
mined by the alteration it produces in the length of the
spring ; or else the length of the seconds pendulum is
measured, from which the force of gravity on a given
mass can be calculated. This last is an example of a
very common and useful mode of measuring forces
called into play by displacement or strain ; namely, by
measuring the period of the oscillations which they

It seems unnecessary to consider any further
examples, as all other quantities are measured by means
of some simple geometrical or dynamical quantity which
is proportional to them ; as temperature by the height
of mercury in a thermometer, heat by the quantity of
ice it will melt (the volume of the resulting water),
electric resistance by the length of a standard wire
which has an equivalent resistance. It only remains to
show how, when a length has been found proportional
to the quantity to be measured, this length itself is

For rough purposes, as for example in measuring
the length of a room with a foot-rule, we apply the rule
end on end, and count the number of times. For the
piece left, we should apply the rule to it and count the
number of inches. Or if we wanted a length expressed
roughly for scientific purposes, we should describe it in
metres or centimetres. But if it has to be expressed
with greater accuracy, it must be described in
hundredth, or thousandth, or milhonth parts of a milli-
metre ; and this is still done by comparing it with a


But in order to estimate a length in terms of these
very small quantities, it must be magnified ; and this is
done in three ways. First, geometrically, by what is
called a vernier scale. This is a movable scale, which
gains on the fixed one by one-tenth in each division.
To measure any part of a division, we find how many
divisions it takes the vernier to gain so much as that
part ; this is how many tenths the part is. The quantity
to be measured is here geometrically multiplied by ten.
Next, optically, by looking at the length and scale with
a microscope or telescope. Third, mechanically, by a
screw with a disc on its head, on which there is a
graduated rim, called a micrometer screw. If the pitch
of the screw is one-tenth and the radius of the disc ten
times that of the screw, the motion is multiplied by one
hundred. The two latter modes are combined together
in an instrument called a micrometer-microscope.
Another mechanical multiplier is a mirror which turns
round and reflects light on a screen at some distance, as
in Thomson's reflecting galvanometer.

Properly speaking, however, any description of a
length by counting of standard lengths is imperfect and
merely approximate. The true way of indicating a
length is to draw a straight hue which represents it on
a fixed scale. And this is done by means of self-record-
ing instruments, which measure lengths from time to
time on a cyhnder in the manner described above. It
is only by this graphical representation of quantities
that the laws of their variation become manifest, and
that higher branch of measurement becomes possible
which determines the nature of the connexion between
two simultaneously varying quantities.



Science of Motion.

Geometry teaches us about the sizes, the shapes, and
the distances of things ; to know sizes and distances
we have to measure lengths., and to know shapes we
have to measure angles. The science of Motion, which
is the subject of the present sketch, tells us about the
changes in these sizes, shapes, and distances which take
place from time to time. A body is said to move when
it changes its place or position ; that is to say, when
it changes its distance from surrounding objects. And
when the parts of a body move relatively to one another,
i.e. when they alter their distance from one another,
the body changes jn size, or shape, or both. All these
changes are considered in the science of motion.


The science of motion is divided into two parts : the
accurate description of motion, and the investigation of
the circumstances under which particular motions take
place. The description of motion may again be divided
into two parts, namely, that which tells us what changes
of position take place, and that which tells us when
and how fast they take place. We might, for example,
describe the motion of the hands of a clock, and say
that they turn round on their axes at the centre of the
clock-face in such a way" that the minute-hand always
moves twelve times as much as the hour-hand ; this is


the first part of the description of the motion. We
might go on to say that when the clock is going
correctly, this motion takes place uniformly, so that the
minute-hand goes round once in each hour ; and this
would be the second part of the description. The first
part is what was called Kinematics by Ampere : it tells
us how the motions of the dijQferent parts of a machine
depend on each other in consequence of the machinery
which connects them. This is clearly an apphcation of
geometry alone, and requires no more measurements
than those which belong to geometry, namely, measure-
ments of lines and angles. But the name Kinematics is
now conveniently made to include the second part also
of the description of motion — when and how fast it
takes place. This requires in addition the measurement
of time, with which geometry has nothing to do. The
word Kinematic is derived from the Greek kinema,
' motion ; ' and will therefore serve equally well to bear
the restricted sense given it by Ampere, and the more
comprehensive sense in which it is now used. And since
the principles of this science are those which guide the
construction not only of scientific apparatus, but of
all instruments and machines, it may be advisable to
describe in some detail the chief topics with which it


That part of the science which tells us about the
circumstances under which particular motions take
place is called Dynamics. It is found that the change
of motion in a body depends on the position and state
of surrounding bodies, according to certain simple laws ;


when considered as so depending on surrounding bodies,
the rate of change in the quantity of motion is called
force. Hence the name Dynamic, from the Greek
dynamis, ' force.' The word force is here used in a
technical sense, pecuHar to the science of motion ; the
connexion of this meaning with the meaning which the
word has in ordinary discourse will be explained further

Statics and Kinetics.

Dynamics are again divided into two branches : the
study of those circumstances in which it is possible for
a body to remain at rest is called Statics, and the study
of the circumstances of actual motion is called Kinetics.
The simplest part of Statics, the doctrine of the Lever,
was successfully studied before any other part of the
science of motion, namely by Archimedes, who proved
that when a lever with unequal arms is balanced by
weights at the ends of it, these weights are inversely
proportional to the arms. But no real progress could
be made in determining the conditions of rest, until the
laws of actual motion had been studied.

. Translation of Rigid Bodies.

Eeturning, then, to the description of motion, or
Kinematics, we must first of all classify the different
changes of position, of size, and of shape, with which
we have to deal. We call a body rigid when it changes
only its position, and not its size or shape, during the
time in which we consider it. It is probable that every
actual body is constantly undergoing slight changes of


size and shape, even when we cannot perceive them ; but
in Kinematics, as in most other matters, there is a great
convenience in talking about only one thing at a time.
So we first of all investigate changes of position on the
assumption that there are no changes of size and shape ;
or, in technical phrase, we treat of the motion of rigid
bodies. Here an important distinction is made between
motion in which the body merely travels from one place
to another, and motion in which it also turns round.
Thus the wheels of a locomotive engine not only travel
along the hne, but are constantly turning round ; while
the coupling-bar which joins two wheels on the same
side remains always horizontal, though its changes of
position are considerably comphcated. A change of
place in which there is no rotation is called a translation.
In a rotation the different parts of the body are moving
different ways, but in a translation all parts move in
the same way. Consequently, in describing a translation
we need only specify the motion of any one particle of
the moving body ; where by a particle is meant a piece
of matter so small that there is no need to take account
of the differences between its parts, which may therefore
be treated for purposes of calculation as a point.

We are thus brought down to the very simple
problem of describing the motion of a point. Of this
there are certain cases which have received a great deal
of attention on account of their frequent occurrence in
nature ; such as Parabolic Motion, Simple Harmonic
Motion, Elliptic Motion. We propose to say a few
words in explanation of each of these.


Parabolic Motion.

The motion of a projectile, that is to say, of a body-
thrown in any direction and falhng under the influence
of gravity, was investigated by Gahleo ; and this is the
first problem of Kinetics that was ever solved. We
must confine ourselves here to a description of the
motion, without considering the way in which it depends
on the circumstance of the presence of the earth at a
certain distance from the moving body. Gahleo found
that the .path of such a body, or the curve which it
traces out, is a parabola ; a curve which may be
described as the shadow of a circle cast on a horizon-
tal table by a candle which is just level with the highest
point of the circle.

It is convenient to consider separately the vertical
and the horizontal motion, for in accordance with a law
subsequently stated in a general form by Newton, these
two take place in complete independence of one another.
So far as its horizontal motion is concerned, the projec-
tile moves uniformly, as if it were sliding on perfectly
smooth ice ; and, so far as its vertical motion is con-
cerned, it moves as if it were falhng down straight.
The nature of this vertical motion may be described in
two ways, each of which implies the other. Eirst, a
falhng body moves faster and faster as it goes down ;
and the rate at which it is going at any moment is
strictly proportional to the number of seconds which
has elapsed since it started. Thus its downward velo-
city is continually being added to at a uniform rate.
Secondly, the whole distance fallen from the starting-
point is proportional to the square of the number of


seconds elapsed ; thus, in three seconds a body will fall
nine times as far as it will fall in one second. The
latter of these statements was experinientally proved by
Galileo ; not, however, in the case of bodies falhng ver-
tically, which move too quickly for the time to be con-
veniently measured, but in the case of bodies falling
down inchned planes, the law of which he at first as-
sumed, and afterwards proved to be identical with that
of the other. The former statement, that the velocity
increases uniformly, is directly tested by an apparatus
known as Attwood's machine, consisting essentially of a
pulley, over which a string is hung with equal weights
attached to its ends. A small bar of metal is laid on
one of the weights, which begins to descend and pull
the other one up ; after a measured time the bar is
lifted off, and then, both sides pulhng equally, the
motion goes on at the rate which had been acquired at
that instant. The distance travelled in one second is
then measured, and gives the velocity ; this is found to
be proportional to the time of falling with the bar on.

The second statement, that the space passed over is
proportional to the square of the number of seconds
elapsed, is verified by Morin's machine, which consists
of a vertical cyhnder which revolves uniformly while a
body falhng down at the side marks it with a pencil.
The curve thus described is a record of the distance the
body had fallen at every moment of time.


This investigation of Galileo's was in more than one
aspect the foundation of dynamical science ; but not the
least important of these aspects is the proof that either


of the two ways of stating the law of faUing bodies in-
volves the other. Given that the distance fallen is pro-
portional to the square of the time, to show that the
velocity is proportional to the time itself ; this is a par-
ticular case of the problem. Given where a body is at
every instant, to find how fast it is going at every in-
stant. The solution of this problem was given by New-
ton's Method of Fluxions. When a quantity changes
from time to time, its rate of change is called the
fluxion of the quantity. In the case of a moving body the
quantity to be considered is the distance which the body
has travelled ; the fluxion of this distance is the rate at
which- the body is going. Newton's method solves the
problem, Given how big a quantity is at any time, to
find its fluxion at any time. The method has been
called on the Continent, and lately also in England, the
Difierential Calculus ; because the diflerence between
two values of the varying quantity is mentioned in one
of the processes that may be used for calculating its
fluxion. The inverse problem, Given that the velocity
is proportional to the time elapsed, to find the distance
fallen, is a particular case of the general problem. Given
how fast a body is going at every instant, to find where
it is at any instant ; or, Given the fluxion of a quantity,
to find the quantity itself. The answer to this is given
by Newton's Inverse Method of Fluxions ; which is also
called the Integral Calculus, because in one of the pro-
cesses which may be used for calculating the quantity,
it is regarded as a whole (integer) made up of a number
of small parts. The method of Fluxions, then, or Dif-
ferential and Integral Calculus, takes its start from
Gahleo's study of parabohc motion.


Harmonic Motion.

The ancients, regarding the circle-as the most perfect
of figures, beheved that circular motion was not only
simple, that is, not made up by putting together other
motions, but also perfect., in the sense that when once set
up in perfect bodies it would maintain itself without
external interference. The moderns, who know nothing
about perfection except as something to be aimed at,
but never reached, in practical work, have been forced
to reject both of these doctrines. The second of them,
indeed, belongs to Kinetics, and will again be mentioned
under that head. But as a matter of Kinematics it has
been found necessary to treat the uniform motion of a
point round a circle as compounded of two oscillations.
To take again the example of a clock, the extreme point
of the minute-hand describes a circle uniformly ; but
if we consider separately its vertical position and its
horizontal position, we shall see that it not only oscil-
lates up and down, but at the same time swings from
side to side, each in the same period of one hour. If
we suppose a button to move up and down in a slit
between the figures XII and VI, in such a way as to be
always at the same height as the end of the minute-hand,
this button will have only one of the two oscillations
which are combined in the motion of that point ; and
the other oscillation would be exhibited by a button con-
strained to move in a similar manner between the figures
in and IX, so as always to be either vertically above or
vertically below the extreme point of the minute-hand.
The laws of these two motions are identical, but they are
so timed that each is at its extreme position when the


other is crossing the centre. An oscillation of this kind
is called a simple harmonic motion : the name is due to
Sir William Thomson, and was given on account of the
intimate connexion between the laws of such motions
and the theory of vibrating strings. Indeed, the har-
monic motion, simple or compound, is the most univer-
sal of all forms ; it is exemphfied not only in the motion
of every particle of a vibrating solid, such as the string
of a piano or viohn, a tuning-fork, or the membrane of
a drum, but in those minute excursions of particles of

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Online LibraryWilliam Kingdon CliffordLectures and essays (Volume v. 2) → online text (page 1 of 22)