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LECTURES AND ESSAYS

VOL. 11.

Digitized by the Internet Arciiive

in 2010 witii funding from

Open Knowledge Commons and Harvard Medical School

http://www.archive.org/details/lecturesessays02clif

C/

yj^r^^^^

A^-C^>^

[y/C^^^VT^l-'^U^ ^ ^

LECTURES AM) ESSAYS

BY THE LATE

WILLIAM KINGDON CLIFFOED, F.E.S.

LATE PROFESSOR OP APPLIED MATHEMATICS AND MECHAIJICS IN UNIVERSITY COLLEGE, LONDON

AND SOJIETIJIE FELLOW OP TRINITY COLLEGE, CAMBRIDGE

EDITED BY

LESLIE STEPHEN and FREDERICK POLLOCK

WITH an INTRODUCTION by F. POLLOCK

' ZÂ« vii-ite est toute pour tous'â€”PAVL-homa Courier

IN TWO VOLUMES

VOL. XL

MACMILLAN AND CO.

1879

[The nght of translation is reserved]

CONTENTS

OF

thp: second volume.

LECTURES AND ESSAYS -continued.

PAGE

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

LECTUEES AND ESSAYS

{continued)

VOL. II.

INSTRUMENTS USED IN MEASUREMENT.^

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.

Lengths

Areas

Volumes

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

4 INSTRU^IENTS USED IN MEASUREMENT.

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 â€” 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

length.

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

INSTEUMENTS USED IN MEASUREMENT. 5

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

6 INSTRUMENTS USED IN MEASUREMENT.

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.

INSTEUMENTS USED IN MEASUREMENT. 7

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

produce.

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

measured.

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

scale.

8 INSTRUMENTS USED IN MEASUREMENT.

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.

INSTRUMENTS USED IN MEASUREMENT.

INSTRUMENTS ILLUSTRATING KINEMATICS, STATICS,

AND DYNAMICS.

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.

Kinematics.

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

10 INSTKUMENTS USED IN MEASUREMENT.

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 â€” 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

deals.

Dynamics.

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 ;

INSTRUMENTS USED IN MEASUREMENT. 11

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

on.

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

12 INSTRUMENTS USED IN MEASUREMENT.

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.

INSTRUMENTS USED IN MEASUREMENT. 13

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

14 INSTRUftlENTS USED IN MEASUREMENT.

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.

Fluxions.

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

INSTRUMENTS USED IN IMEASUREMENT. 15

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.

16 INSTRUMENTS USED IN MEASUREMENT.

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

INSTRIBIENTS USED IN MEASUREMENT. 17

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

l-\

' ./ â™¦-

". ,'^^^''

Boston

Medical Library

8 The Fenway

fift-?: â€¢ s..

:h^

â– 1 K

.,;/.>

w

. ^ -.;â– ;â– , :-' ','. â€¢.; . â– . :"

â– l-""

i.^^.ci â– , '4;- .":;;", V. ' . â– ; â– '

^'- ^^%

LECTURES AND ESSAYS

VOL. 11.

Digitized by the Internet Arciiive

in 2010 witii funding from

Open Knowledge Commons and Harvard Medical School

http://www.archive.org/details/lecturesessays02clif

C/

yj^r^^^^

A^-C^>^

[y/C^^^VT^l-'^U^ ^ ^

LECTURES AM) ESSAYS

BY THE LATE

WILLIAM KINGDON CLIFFOED, F.E.S.

LATE PROFESSOR OP APPLIED MATHEMATICS AND MECHAIJICS IN UNIVERSITY COLLEGE, LONDON

AND SOJIETIJIE FELLOW OP TRINITY COLLEGE, CAMBRIDGE

EDITED BY

LESLIE STEPHEN and FREDERICK POLLOCK

WITH an INTRODUCTION by F. POLLOCK

' ZÂ« vii-ite est toute pour tous'â€”PAVL-homa Courier

IN TWO VOLUMES

VOL. XL

MACMILLAN AND CO.

1879

[The nght of translation is reserved]

CONTENTS

OF

thp: second volume.

LECTURES AND ESSAYS -continued.

PAGE

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

LECTUEES AND ESSAYS

{continued)

VOL. II.

INSTRUMENTS USED IN MEASUREMENT.^

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.

Lengths

Areas

Volumes

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

4 INSTRU^IENTS USED IN MEASUREMENT.

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 â€” 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

length.

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

INSTEUMENTS USED IN MEASUREMENT. 5

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

6 INSTRUMENTS USED IN MEASUREMENT.

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.

INSTEUMENTS USED IN MEASUREMENT. 7

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

produce.

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

measured.

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

scale.

8 INSTRUMENTS USED IN MEASUREMENT.

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.

INSTRUMENTS USED IN MEASUREMENT.

INSTRUMENTS ILLUSTRATING KINEMATICS, STATICS,

AND DYNAMICS.

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.

Kinematics.

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

10 INSTKUMENTS USED IN MEASUREMENT.

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 â€” 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

deals.

Dynamics.

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 ;

INSTRUMENTS USED IN MEASUREMENT. 11

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

on.

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

12 INSTRUMENTS USED IN MEASUREMENT.

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.

INSTRUMENTS USED IN MEASUREMENT. 13

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

14 INSTRUftlENTS USED IN MEASUREMENT.

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.

Fluxions.

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

INSTRUMENTS USED IN IMEASUREMENT. 15

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.

16 INSTRUMENTS USED IN MEASUREMENT.

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

INSTRIBIENTS USED IN MEASUREMENT. 17

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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