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of a piano or violin, a tuning-fork, or the membrane of

a drum, but in those minute excursions of particles of

air which carry sound from one place to another, in the

waves and tides of the sea, and in the amazingly rapid

tremor of the luminiferous ether which, in its varying

action on different bodies, makes itself known as light

or radiant heat or chemical action. Simple harmonic

motions differ from one another in three respects ; in

the extent or amplitude of the swing, which is measured

by the distance from the middle point to either extreme ;

in the period or interval of time between two successive

passages through an extreme position ; and in the time

of starting, or epoch, as it is called, which is named

by saying what particular stage of the vibration was

being executed at a certain instant of time. One of the

most astonishing and fruitful theorems of mathematical

science is this ; that every periodic motion whatever,

that is to say, every motion which exactly repeats itself

again and again at definite intervals of time, is a com-

pound of simple harmonic motions, whose periods are

successively smaller and smaller ah quo t parts of the

original period, and whose amplitudes (after a certain

number of them) are less and less as their periods are

more rapid. The ' harmonic ' tones of a string, which

VOL. n. c

18 INSTRUMENTS USED IN MEASUREMENT.

are always heard along with the fundamental tone, are

a particular case of these constituents. The theorem

was given by Fourier in connexion with the flow of

heat, but its applications are innumerable, and extend

over the whole range of physical science.

The laws of combination of harmonic motions have

been illustrated by some ingenious apparatus of Messrs.

Tisley and Spiller, and by a machine invented by Mr.

Donkin ; but the most important practical application

of these laws is to be found in Sir W. Thomson's Tidal

Clock, and in a more elaborate machine which draws

curves predicting the height of the tide at a given port

for all times of the day and night with as much

accuracy as can be obtained by direct observation.

One special combination is worthy of notice. The

union of a vertical vibration with a horizontal one of

half the period gives rise to that figure of 8 which M.

Marey has observed by his beautiful methods in the

motion of the tip of a bird's or insect's wing.

Elliptic Motion.

The motion of the sun and moon relative to the

earth was at first described by a combination of circular

motions ; and this was the immortal achievement of the

Greek astronomers Hipparchus and Ptolemy. Indeed,

in so far as these motions are periodic, it follows from

Fourier's theorem mentioned above that this mode

of description is mathematically sufficient to represent

them ; and astronomical tables are to this day calcu-

lated by a method which practically comes to the

same thing. But this representation is not the simplest

that can be found ; it requires theoretically an infinite

INSTRUMENTS USED IN MEASUREMENT. 19

number of component motions, and gives no informa-

tion about the way in which these are connected with

one another. We owe to Kepler the accurate and com-

plete description of planetary or elliptic motion. Hi a

investigation applied in the first instance to the orbit

of the planet Mars about the sun, but it was found true

of the orbits of all planets about the sun, and of the

moon about the earth. The path of the moving body in

each of these motions is an ellipse, or oval shadow of a

circle, a curve having various properties in relation to

two internal points or foci, which replace as it were

the one centre of a circle. In the case of the ellipse

described by a planet, the sun is in one of these foci ;

in the case of the moon, the earth is in one focus. So

much for the geometrical description of the motion.

Kepler further observed that a line drawn from the sun

to a planet, or from the earth to the moon, and sup-

posed to move round with the moving body, would

sweep out equal areas in equal times. These two laws,

called Kepler's first and second laws, complete the

kinematic description of elliptic motion ; but to obtain

formulae fit for computation, it was necessary to cal-

culate from these laws the various harmonic compo-

nents of the motion to and from the sun, and round it ;

this calculation has much occupied the attention of

mathematicians.

The laws of rotatory motion of rigid bodies are

somewhat difficult to describe without mathematical

symbols, but they are thoroughly known. Examples

of them are given by the apparatus called a gyroscope,

aad the motion of the earth ; and an application of the

former to prove the nature of the latter, made by

c 2

20 INSTRUMENTS USED IN MEASUREMENT.

Foucault, is one of the most beautiful experiments

belonging entirely to dynamics.

Rotation.

Next in simplicity after the translation of a rigid

body, come two kinds of motion which are at first sight

very different, but between which a closer observation

discovers very striking analogies. These are the motion

of rotation about a fixed point, and the motion of slid-

ing on a fixed plane. The first of these is most easily

produced in practice by what is well known as a ball-

and-socket joint ; that is to say, a body ending in a

portion of a spherical surface which can move about in

a spherical cavity of the same size. The centre of the

spherical surface is then a fixed point, and the motion

is reduced to the sliding of one sphere inside another.

In the same way, if we consider, for instance, the

motion of a flat-iron on an ironing-board, we may see

that this is not a pure translation, for the iron is

frequently turned round as well as carried about ; but

the motion may be described as the sliding of one plane

upon another. Thus in each case the matter to be

studied is the sliding of one surface on another which it

exactly fits. For two surfaces to fit one another exactly,

in all positions, they must be either both spheres of the

same size, or both planes ; and the latter case is really in-

cluded under the former, for a plane may be regarded

as a sphere whose radius has increased without limit.

Thus, if a piece of ice be made to slide about on the

frozen surface of a perfectly smooth pond, it is really

rotating about a fixed point at the centre of the earth ;

for the frozen surface may be regarded as part of an

INSTRUMENTS USED IN MEASUREMENT. 21

enormous sphere, having that point for centre. And

yet the motion cannot be practically distinguished from

that of sliding on a plane.

In this latter case it is found that, excepting in the

case of a pure translation, there is at every instant a

certain point which is at rest, and about which as a

centre the body is turning. This point is called the in-

stantaneous centre of rotation ; it travels about as the

motion goes on, but at any instant its position is per-

fectly definite. From this fact follows a very important

consequence ; namely that every possible motion of a

plane sliding on a plane may be produced by the rolling

of a curve in one plane upon a curve in the other. The

point of contact of the two curves at any instant is the

instantaneous centre at that instant. The problems to

be considered in this subject are thus of two kinds:

Given the curves of rolling to find the path described

by any point of the moving plane ; and, Given the

paths described by two points of the moving plane

(enough to determine the motion) to find the curves of

rolling and the paths of all other points. An important

case of the first problem is that in which one circle rolls

on another, either inside or outside ; the curves de-

scribed by points in the moving plane are used for the

teeth of wheels. To the second problem belongs the

valuable and now rapidly increasing theory of link-work,

which, starting from the wonderful discovery of an

exact parallel motion by M. Peaucellier, has received an

immense and most unexpected development at the

hands of Professor Sylvester, Mr. Hart, and- Mr. A. B.

Kempe.

Passing now to the spherical form of this motion,

22 INSTRUMENTS USED IN MEASUREMENT.

we find that the instantaneous centre of rotation (which

is clearly equivalent to an instantaneous axis perpen-

dicular to the plane) is replaced by an instantaneous

axis passing through the common centre of the moving

spheres. In the same way the rolling of one curve on

another in the plane is replaced by the rolling of one

cone upon another, the two cones having a common

vertex at the same centre.

Analogous theorems have been proved for the most

general motion of a rigid body. It was shown by M.

Chasles that this is always similar to the motion of a

corkscrew desending into a cork ; that is to say, there is

always a rotation about a certain instantaneous axis,

combined with translation along this axis. The amount

of translation per unit of rotation is called the pitch of

the screw. The instantaneous screw moves about as

the motion goes on, but at any given instant it is per-

fectly definite in position and pitch. And any motion

whatever of a rigid body may be produced by the

rolling and sliding of one surface on another, both

surfaces being produced by the motion of straight

lines. This crowning theorem in the geometry 01

motion is due to Professor Cayley. The laws of combina-

tion of screw motions have been investigated by Dr. Ball.

Thus, proceeding gradually from the more simple to

the more complex, we have been able to describe every

change in the position of a body. It remains only to

describe changes of size and shape. Of these there are

three kinds, but they are all included under the same

name strains. We may have, first, a change 01 size

without any change of shape, a uniform dilatation or

contraction of the whole body in all directions, such as

INSTRUMENTS USED IN MEASUREMENT. 23

happens to a sphere of metal when it is heated or cooled.

Next, we may have an elongation or contraction in one

direction only, all lines of this body pointing in this

direction being increased or diminished in the same

ratio ; such as would happen to a rod six feet long and

an inch square, if it were stretched to seven feet long,

still remaining an inch square. Thirdly, we may have a

change of shape produced by the sliding of layers over

one another, a mode of deformation which is easily pro-

duced in a pack of cards ; this is called a shear. By

appropriate combinations of these three, every change

of size and shape may be produced ; or we may even

leave out the second element, and produce any strain

whatever by a dilatation or contraction, and two

shears.

Dynamics.

We have already said- that the change of motion

of a body depends upon the position and state of sur-

rounding bodies. To make this intelligible it will be

necessary to notice a certain property of the three kinds

of motion of a point which we described.

The combination of velocities may be understood

from the case of a body carried in any sort of cart or

vehicle in which it moves about. The whole velocity

of the body is then compounded of the velocity of the

vehicle and of its velocity relative to the vehicle.

Thus, if a man walks across a railway carriage his

whole velocity is compounded of the velocity of the

railway carriage and of the velocity with which he

walks across.

When the velocity of a body is changed by adding

24 INSTRUMENTS USED IN MEASUREMENT.

to it a velocity in the same direction or in the opposite

direction, it is only altered in amount ; but when a

transverse velocity is compounded with it, a change of

direction is produced. Thus, if a man walks fore and

aft on a steamboat, he only travels a little faster or

slower ; but if he walks across from one side to the

other, he slightly changes the direction in which he is

moving.

Now, in the parabolic motion of a projectile, we

found that while the horizontal velocity continues

unchanged, the vertical velocity increases at a uniform

rate. Such a body is having a downwards velocity

continually poured into it, as it were. This gradual

change of the velocity is called acceleration : we may

say that the acceleration of a projectile is always the

same, and is directed vertically downwards.

In a simple harmonic motion it is found that the

acceleration is directed towards the centre, and is

always proportional to the distance from it. In the

case of elliptic motion it was proved by Newton that

the acceleration is directed towards the focus, and is

inversely proportional to the square of the distance

from it.

Let us now consider the circumstances under

which these motions take place. To produce a simple

harmonic motion we may take a piece of elastic string,

whose length is equal to the height of a smooth table ;

then fasten one end of the string to a bullet and the

other end to the floor, having passed it through a hole

in the table, so that the bullet just rests on the top of

the hole when the string is unstretched. If the bullet

be now pulled away from the hole so that the string is

INSTRUMENTS USED IN MEASUREMENT. 25

stretched, and then let it go, it will oscillate to and fro

on either side of the hole with a simple harmonic

motion. The acceleration (or rate of change of velo-

city) is here proportional to the distance from the hole;

that is, to the amount of elongation of the string. It is

directed towards the hole ; that is, in the direction of

this elongation. In the case of the moon moving round

the earth, the acceleration is directed towards the

earth, and is inversely proportional to the square of

the distance from the earth.

In both these cases, then, the change of velocity

depends upon surrounding circumstances ; but in the

case of the bullet, this circumstance is the strained con-

dition of an adjoining body, namely, the elastic string ;

while in the case of the moon the circumstance is the

position of a distant body, namely, the earth. The

motion of a projectile turns out to be only a special

case of the motion of the moon ; for the parabola

which it describes may be regarded as one end of a

very long ellipse, whose other end goes round the

earth's centre.

There is a remarkable difference between the two

cases. The swing of the bullet depends upon its size ;

a large bullet will oscillate more slowly than a small

one. .This leads us to modify the rule. If a large

bullet is equivalent to two small ones, then when it is

going at the same rate it must contain twice as much

motion as one of the small ones ; or, as we now say,

with the same velocity it has twice the momentum.

Now the change of momentum is found to be the same

for all bullets, when the momentum is reckoned as pro-

portional to the quantity of matter in the bullet as well

26 INSTRUMENTS USED IN MEASUREMENT.

as to the velocity. The quantity of matter in a body is

called its mass : for bodies of the same substance it is, of

course, simply the quantity of that substance ; but for

bodies of different substances it is so reckoned as to

make the rule hold good. The rule for this case may

then be stated thus ; the change of momentum of a

body (that is, the change of velocity multiplied by the

mass), depends on the state of strain of adjoining bodies.

Eegarded as so depending, this change of momentum is

called the pressure or tension of the adjoining body,

according to the nature of the strain ; both of these

are included in the name stress, introduced by Eankine.

But in the case of projectiles, the acceleration is

found to be the same for all bodies at the same place ;

and this rule holds good in all cases of planetary

motion. So that it seems as if the change of velocity,

and not the change of momentum, depended upon the

position of distant bodies. But this case is brought

under the same rule as the other by supposing that the

mass of the moving body is to be reckoned among the

' circumstances.' The change of momentum is in this

case called the attraction of gravitation, and we say

that the attraction is proportional to the mass of the

attracted body. And this way of representing the facts

is borne out by the electrical and magnetic attractions

and repulsions, where the change of momentum depends

on the position and state of the attracting thing, and

upon the electric charge or the induced magnetism of

the attracted thing.

Force, then, is of two kinds ; the stress of a strained

adjoining body, and the attraction or repulsion of a

distant body. Attempts have been made with more or

INSTRUMENTS USED IN MEASUREMENT. 27

less success to explain each of these by means of the

other. In common discourse the word ' force ' means

muscular effort exerted by the human frame. In this

case the part of the human body which is in contact

with the object to be moved is in a state of strain, and

the force, dynamically considered, is of the first kind.

But this state of strain is preceded and followed by

nervous discharges, which are accompanied by the

sensations of effort and of muscular strain ; a complica-

tion of circumstances which does not occur in the

action of inanimate bodies. What is common to the

two cases is, that the change of momentum depends on

the strain.

Having thus explained the law of Force, which is

the foundation of Dynamics, we may consider the

remaining laws of motion. It is convenient to state

them first for particles, or bodies so small that we need

take account only of their position. Every particle,

then, has a rate of change of momentum due to the

position or state of every other particle, whether

adjoining it or distant from it. These are compounded

together by the law of composition of velocities, and

the result of the whole is the actual change of momen-

tum of the particle. This statement, and the law of

Force stated above, amount together to Newton's first

and second laws of motion. His third law is, that the

change of momentum in one particle, due to the posi-

tion or state of another, is equal and opposite to the

change of momentum in the other, due to the position

or state of the first.

By the help of these laws D'Alembert showed how

the motion of rigid bodies, or systems of particles, might

28 INSTRUMENTS USED IN MEASUREMENT.

be dealt with. It appears from his method that two

stresses, acting on a rigid body, may be equivalent, in

their effect on the body as a whole, to a single stress,

whose direction and position will be totally independent

of the shape and nature of the body considered. The

law of combination of stresses acting on a system of

particles is, in fact, the same as the law of combination

of velocities, so far as regards the motion of the system

as a whole. This beautiful but somewhat complex

result of Dynamics has been used in some text-books as

the independent foundation of Statics, under the name of

the parallelogram of forces ; a singular inversion of the

historical order and of the methods of the great writers.

When the result of all the circumstances surround-

ing a body is that there is no change of momentum, the

body is said to be in equilibrium. In this case, if the

body is at rest, it will remain so ; and on this ^ account

the study of such conditions is called Statics. In deal-

ing with the statics of rigid bodies, we have only to

examine those cases in which the resultant of the

external stresses and attractions acting on the body

amounts to nothing. But the most important part of

statics is that which finds the stresses acting in the

interior of bodies between contiguous parts of them ;

for upon this depends the determination of the requisite

strength of structures which have to bear given loads.

It is found that the way in which the stress due to a

given strain depends on the strain varies according to

the physical nature of the body ; for bodies, however,

which are not crystalline or fibrous, but which have the

same properties in all directions, there are two quanti-

ties which, if known, will enable us always to calculate

INSTRUMENTS USED IN MEASUREMENT. 29

the stress due to a given strain. These are, the

elasticity of volume, or resistance to change of size ; and

the rigidity, elasticity of figure, or resistance to change

of shape. Problems relating to the interior state of

bodies are far more difficult than those which regard

them as rigid. Thus, if a beam is supported at its two

ends, it is very easy to find the portion of its weight

which is borne by each support ; but the determination

of the state of stress in the interior is a problem of

great complexity.

There is one theorem of kinetics which must be

mentioned here. If we multiply half the momentum of

every particle of a body by its velocity, and add all the

results together, we shall get what is called the kinetic

energy of the body. When the body is moved from

one position to another, if we multiply each force acting

on it whether attraction or stress by the distance

moved in the direction opposite to the force, and add

the results, we shall get what is called the work done

against the forces during the change of position. It

does not at all depend on the rate at which the change

is made, but only on the two positions. If a body

moves, and loses kinetic energy, it does an amount of

work equal to the kinetic energy lost. If it gains kinetic

energy, an amount of work equal to this gain must be

done to take it back from the new position to the old

one. The amount of work which must be done to take

a body from a certain standard position to the position

which it has at present is called the potential energy of

the body. The theorem may be stated in this form ;

the sum of the potential and kinetic energies is always

the same, provided the surrounding circumstances do

30 INSTRUMENTS USED IN MEASUREMENT.

not alter. Hence the theorem is called the Conservation

of Energy. It is one fact out of many that may be

deduced from the equations of motion ; it is not suffi-

cient to determine the motion of a body, but it is ex-

ceedingly useful as giving a general result in cases

where it might be difficult or undesirable to investigate

all the particulars ; and it is especially applicable to

machines, the important question in regard to which is

the amount of work which they can do.

It will have been seen that the science of motion

depends on a few fundamental principles which are

easily verified, and consists almost entirely of mathema-

tical deductions and calculations based on those princi-

ples. It is no longer therefore an experimental science

in the same sense as those are in which the fundamental

facts are still being discovered. The apparatus con-

nected with it may be conveniently classified under

three heads :

(a) Apparatus for illustrating theorems or solving

problems of kinematics, such as those mentioned

above for compounding harmonic motions.

There is reason to hope for great extension of

our powers in this direction.

(b) Apparatus for measuring the dynamical

quantities, such as weight, work, and the

elasticities of different substances. These are

more fully classified under Measurements.

(c) Apparatus designed for purposes belonging to

other sciences, but illustrating by its structure

and functions the results of kinematics or

dynamics. In this class the remainder of the

collection is included.

31

UNIVERSITY

TWtt.1

BODY AND MIND. 1

THE subject of this Lecture is one in regard to which

a great change has recently taken place in the public

mind. Some time ago it was the custom to look with

suspicion upon all questions of a metaphysical nature

as being questions that could not be discussed with any

good result, and which, leading inquirers round and

round in the same circle, never came to an end. But

quite of late years there is an indication that a large

number of people are waking up to the fact that Science

has something to say upon these subjects ; and the

English people have always been very ready to hear

what Science can say understanding by Science what

we shall now understand by it, that is, organized com-

mon sense.

When I say Science, I do not mean what some

people are pleased to call Philosophy. The word ' phi-

losopher,' which meant originally ' lover of wisdom/

has come in some strange way to mean a man who

thinks it his business to explain everything in a certain

number of large books. It will be found, I think, that

in proportion to his colossal ignorance is the perfection

and symmetry of the system which he sets up ; because

it is so much easier to put an empty room tidy than a

1 Sunday Lecture Society, November 1, 1874 ; * Fortnightly Keview/

a drum, but in those minute excursions of particles of

air which carry sound from one place to another, in the

waves and tides of the sea, and in the amazingly rapid

tremor of the luminiferous ether which, in its varying

action on different bodies, makes itself known as light

or radiant heat or chemical action. Simple harmonic

motions differ from one another in three respects ; in

the extent or amplitude of the swing, which is measured

by the distance from the middle point to either extreme ;

in the period or interval of time between two successive

passages through an extreme position ; and in the time

of starting, or epoch, as it is called, which is named

by saying what particular stage of the vibration was

being executed at a certain instant of time. One of the

most astonishing and fruitful theorems of mathematical

science is this ; that every periodic motion whatever,

that is to say, every motion which exactly repeats itself

again and again at definite intervals of time, is a com-

pound of simple harmonic motions, whose periods are

successively smaller and smaller ah quo t parts of the

original period, and whose amplitudes (after a certain

number of them) are less and less as their periods are

more rapid. The ' harmonic ' tones of a string, which

VOL. n. c

18 INSTRUMENTS USED IN MEASUREMENT.

are always heard along with the fundamental tone, are

a particular case of these constituents. The theorem

was given by Fourier in connexion with the flow of

heat, but its applications are innumerable, and extend

over the whole range of physical science.

The laws of combination of harmonic motions have

been illustrated by some ingenious apparatus of Messrs.

Tisley and Spiller, and by a machine invented by Mr.

Donkin ; but the most important practical application

of these laws is to be found in Sir W. Thomson's Tidal

Clock, and in a more elaborate machine which draws

curves predicting the height of the tide at a given port

for all times of the day and night with as much

accuracy as can be obtained by direct observation.

One special combination is worthy of notice. The

union of a vertical vibration with a horizontal one of

half the period gives rise to that figure of 8 which M.

Marey has observed by his beautiful methods in the

motion of the tip of a bird's or insect's wing.

Elliptic Motion.

The motion of the sun and moon relative to the

earth was at first described by a combination of circular

motions ; and this was the immortal achievement of the

Greek astronomers Hipparchus and Ptolemy. Indeed,

in so far as these motions are periodic, it follows from

Fourier's theorem mentioned above that this mode

of description is mathematically sufficient to represent

them ; and astronomical tables are to this day calcu-

lated by a method which practically comes to the

same thing. But this representation is not the simplest

that can be found ; it requires theoretically an infinite

INSTRUMENTS USED IN MEASUREMENT. 19

number of component motions, and gives no informa-

tion about the way in which these are connected with

one another. We owe to Kepler the accurate and com-

plete description of planetary or elliptic motion. Hi a

investigation applied in the first instance to the orbit

of the planet Mars about the sun, but it was found true

of the orbits of all planets about the sun, and of the

moon about the earth. The path of the moving body in

each of these motions is an ellipse, or oval shadow of a

circle, a curve having various properties in relation to

two internal points or foci, which replace as it were

the one centre of a circle. In the case of the ellipse

described by a planet, the sun is in one of these foci ;

in the case of the moon, the earth is in one focus. So

much for the geometrical description of the motion.

Kepler further observed that a line drawn from the sun

to a planet, or from the earth to the moon, and sup-

posed to move round with the moving body, would

sweep out equal areas in equal times. These two laws,

called Kepler's first and second laws, complete the

kinematic description of elliptic motion ; but to obtain

formulae fit for computation, it was necessary to cal-

culate from these laws the various harmonic compo-

nents of the motion to and from the sun, and round it ;

this calculation has much occupied the attention of

mathematicians.

The laws of rotatory motion of rigid bodies are

somewhat difficult to describe without mathematical

symbols, but they are thoroughly known. Examples

of them are given by the apparatus called a gyroscope,

aad the motion of the earth ; and an application of the

former to prove the nature of the latter, made by

c 2

20 INSTRUMENTS USED IN MEASUREMENT.

Foucault, is one of the most beautiful experiments

belonging entirely to dynamics.

Rotation.

Next in simplicity after the translation of a rigid

body, come two kinds of motion which are at first sight

very different, but between which a closer observation

discovers very striking analogies. These are the motion

of rotation about a fixed point, and the motion of slid-

ing on a fixed plane. The first of these is most easily

produced in practice by what is well known as a ball-

and-socket joint ; that is to say, a body ending in a

portion of a spherical surface which can move about in

a spherical cavity of the same size. The centre of the

spherical surface is then a fixed point, and the motion

is reduced to the sliding of one sphere inside another.

In the same way, if we consider, for instance, the

motion of a flat-iron on an ironing-board, we may see

that this is not a pure translation, for the iron is

frequently turned round as well as carried about ; but

the motion may be described as the sliding of one plane

upon another. Thus in each case the matter to be

studied is the sliding of one surface on another which it

exactly fits. For two surfaces to fit one another exactly,

in all positions, they must be either both spheres of the

same size, or both planes ; and the latter case is really in-

cluded under the former, for a plane may be regarded

as a sphere whose radius has increased without limit.

Thus, if a piece of ice be made to slide about on the

frozen surface of a perfectly smooth pond, it is really

rotating about a fixed point at the centre of the earth ;

for the frozen surface may be regarded as part of an

INSTRUMENTS USED IN MEASUREMENT. 21

enormous sphere, having that point for centre. And

yet the motion cannot be practically distinguished from

that of sliding on a plane.

In this latter case it is found that, excepting in the

case of a pure translation, there is at every instant a

certain point which is at rest, and about which as a

centre the body is turning. This point is called the in-

stantaneous centre of rotation ; it travels about as the

motion goes on, but at any instant its position is per-

fectly definite. From this fact follows a very important

consequence ; namely that every possible motion of a

plane sliding on a plane may be produced by the rolling

of a curve in one plane upon a curve in the other. The

point of contact of the two curves at any instant is the

instantaneous centre at that instant. The problems to

be considered in this subject are thus of two kinds:

Given the curves of rolling to find the path described

by any point of the moving plane ; and, Given the

paths described by two points of the moving plane

(enough to determine the motion) to find the curves of

rolling and the paths of all other points. An important

case of the first problem is that in which one circle rolls

on another, either inside or outside ; the curves de-

scribed by points in the moving plane are used for the

teeth of wheels. To the second problem belongs the

valuable and now rapidly increasing theory of link-work,

which, starting from the wonderful discovery of an

exact parallel motion by M. Peaucellier, has received an

immense and most unexpected development at the

hands of Professor Sylvester, Mr. Hart, and- Mr. A. B.

Kempe.

Passing now to the spherical form of this motion,

22 INSTRUMENTS USED IN MEASUREMENT.

we find that the instantaneous centre of rotation (which

is clearly equivalent to an instantaneous axis perpen-

dicular to the plane) is replaced by an instantaneous

axis passing through the common centre of the moving

spheres. In the same way the rolling of one curve on

another in the plane is replaced by the rolling of one

cone upon another, the two cones having a common

vertex at the same centre.

Analogous theorems have been proved for the most

general motion of a rigid body. It was shown by M.

Chasles that this is always similar to the motion of a

corkscrew desending into a cork ; that is to say, there is

always a rotation about a certain instantaneous axis,

combined with translation along this axis. The amount

of translation per unit of rotation is called the pitch of

the screw. The instantaneous screw moves about as

the motion goes on, but at any given instant it is per-

fectly definite in position and pitch. And any motion

whatever of a rigid body may be produced by the

rolling and sliding of one surface on another, both

surfaces being produced by the motion of straight

lines. This crowning theorem in the geometry 01

motion is due to Professor Cayley. The laws of combina-

tion of screw motions have been investigated by Dr. Ball.

Thus, proceeding gradually from the more simple to

the more complex, we have been able to describe every

change in the position of a body. It remains only to

describe changes of size and shape. Of these there are

three kinds, but they are all included under the same

name strains. We may have, first, a change 01 size

without any change of shape, a uniform dilatation or

contraction of the whole body in all directions, such as

INSTRUMENTS USED IN MEASUREMENT. 23

happens to a sphere of metal when it is heated or cooled.

Next, we may have an elongation or contraction in one

direction only, all lines of this body pointing in this

direction being increased or diminished in the same

ratio ; such as would happen to a rod six feet long and

an inch square, if it were stretched to seven feet long,

still remaining an inch square. Thirdly, we may have a

change of shape produced by the sliding of layers over

one another, a mode of deformation which is easily pro-

duced in a pack of cards ; this is called a shear. By

appropriate combinations of these three, every change

of size and shape may be produced ; or we may even

leave out the second element, and produce any strain

whatever by a dilatation or contraction, and two

shears.

Dynamics.

We have already said- that the change of motion

of a body depends upon the position and state of sur-

rounding bodies. To make this intelligible it will be

necessary to notice a certain property of the three kinds

of motion of a point which we described.

The combination of velocities may be understood

from the case of a body carried in any sort of cart or

vehicle in which it moves about. The whole velocity

of the body is then compounded of the velocity of the

vehicle and of its velocity relative to the vehicle.

Thus, if a man walks across a railway carriage his

whole velocity is compounded of the velocity of the

railway carriage and of the velocity with which he

walks across.

When the velocity of a body is changed by adding

24 INSTRUMENTS USED IN MEASUREMENT.

to it a velocity in the same direction or in the opposite

direction, it is only altered in amount ; but when a

transverse velocity is compounded with it, a change of

direction is produced. Thus, if a man walks fore and

aft on a steamboat, he only travels a little faster or

slower ; but if he walks across from one side to the

other, he slightly changes the direction in which he is

moving.

Now, in the parabolic motion of a projectile, we

found that while the horizontal velocity continues

unchanged, the vertical velocity increases at a uniform

rate. Such a body is having a downwards velocity

continually poured into it, as it were. This gradual

change of the velocity is called acceleration : we may

say that the acceleration of a projectile is always the

same, and is directed vertically downwards.

In a simple harmonic motion it is found that the

acceleration is directed towards the centre, and is

always proportional to the distance from it. In the

case of elliptic motion it was proved by Newton that

the acceleration is directed towards the focus, and is

inversely proportional to the square of the distance

from it.

Let us now consider the circumstances under

which these motions take place. To produce a simple

harmonic motion we may take a piece of elastic string,

whose length is equal to the height of a smooth table ;

then fasten one end of the string to a bullet and the

other end to the floor, having passed it through a hole

in the table, so that the bullet just rests on the top of

the hole when the string is unstretched. If the bullet

be now pulled away from the hole so that the string is

INSTRUMENTS USED IN MEASUREMENT. 25

stretched, and then let it go, it will oscillate to and fro

on either side of the hole with a simple harmonic

motion. The acceleration (or rate of change of velo-

city) is here proportional to the distance from the hole;

that is, to the amount of elongation of the string. It is

directed towards the hole ; that is, in the direction of

this elongation. In the case of the moon moving round

the earth, the acceleration is directed towards the

earth, and is inversely proportional to the square of

the distance from the earth.

In both these cases, then, the change of velocity

depends upon surrounding circumstances ; but in the

case of the bullet, this circumstance is the strained con-

dition of an adjoining body, namely, the elastic string ;

while in the case of the moon the circumstance is the

position of a distant body, namely, the earth. The

motion of a projectile turns out to be only a special

case of the motion of the moon ; for the parabola

which it describes may be regarded as one end of a

very long ellipse, whose other end goes round the

earth's centre.

There is a remarkable difference between the two

cases. The swing of the bullet depends upon its size ;

a large bullet will oscillate more slowly than a small

one. .This leads us to modify the rule. If a large

bullet is equivalent to two small ones, then when it is

going at the same rate it must contain twice as much

motion as one of the small ones ; or, as we now say,

with the same velocity it has twice the momentum.

Now the change of momentum is found to be the same

for all bullets, when the momentum is reckoned as pro-

portional to the quantity of matter in the bullet as well

26 INSTRUMENTS USED IN MEASUREMENT.

as to the velocity. The quantity of matter in a body is

called its mass : for bodies of the same substance it is, of

course, simply the quantity of that substance ; but for

bodies of different substances it is so reckoned as to

make the rule hold good. The rule for this case may

then be stated thus ; the change of momentum of a

body (that is, the change of velocity multiplied by the

mass), depends on the state of strain of adjoining bodies.

Eegarded as so depending, this change of momentum is

called the pressure or tension of the adjoining body,

according to the nature of the strain ; both of these

are included in the name stress, introduced by Eankine.

But in the case of projectiles, the acceleration is

found to be the same for all bodies at the same place ;

and this rule holds good in all cases of planetary

motion. So that it seems as if the change of velocity,

and not the change of momentum, depended upon the

position of distant bodies. But this case is brought

under the same rule as the other by supposing that the

mass of the moving body is to be reckoned among the

' circumstances.' The change of momentum is in this

case called the attraction of gravitation, and we say

that the attraction is proportional to the mass of the

attracted body. And this way of representing the facts

is borne out by the electrical and magnetic attractions

and repulsions, where the change of momentum depends

on the position and state of the attracting thing, and

upon the electric charge or the induced magnetism of

the attracted thing.

Force, then, is of two kinds ; the stress of a strained

adjoining body, and the attraction or repulsion of a

distant body. Attempts have been made with more or

INSTRUMENTS USED IN MEASUREMENT. 27

less success to explain each of these by means of the

other. In common discourse the word ' force ' means

muscular effort exerted by the human frame. In this

case the part of the human body which is in contact

with the object to be moved is in a state of strain, and

the force, dynamically considered, is of the first kind.

But this state of strain is preceded and followed by

nervous discharges, which are accompanied by the

sensations of effort and of muscular strain ; a complica-

tion of circumstances which does not occur in the

action of inanimate bodies. What is common to the

two cases is, that the change of momentum depends on

the strain.

Having thus explained the law of Force, which is

the foundation of Dynamics, we may consider the

remaining laws of motion. It is convenient to state

them first for particles, or bodies so small that we need

take account only of their position. Every particle,

then, has a rate of change of momentum due to the

position or state of every other particle, whether

adjoining it or distant from it. These are compounded

together by the law of composition of velocities, and

the result of the whole is the actual change of momen-

tum of the particle. This statement, and the law of

Force stated above, amount together to Newton's first

and second laws of motion. His third law is, that the

change of momentum in one particle, due to the posi-

tion or state of another, is equal and opposite to the

change of momentum in the other, due to the position

or state of the first.

By the help of these laws D'Alembert showed how

the motion of rigid bodies, or systems of particles, might

28 INSTRUMENTS USED IN MEASUREMENT.

be dealt with. It appears from his method that two

stresses, acting on a rigid body, may be equivalent, in

their effect on the body as a whole, to a single stress,

whose direction and position will be totally independent

of the shape and nature of the body considered. The

law of combination of stresses acting on a system of

particles is, in fact, the same as the law of combination

of velocities, so far as regards the motion of the system

as a whole. This beautiful but somewhat complex

result of Dynamics has been used in some text-books as

the independent foundation of Statics, under the name of

the parallelogram of forces ; a singular inversion of the

historical order and of the methods of the great writers.

When the result of all the circumstances surround-

ing a body is that there is no change of momentum, the

body is said to be in equilibrium. In this case, if the

body is at rest, it will remain so ; and on this ^ account

the study of such conditions is called Statics. In deal-

ing with the statics of rigid bodies, we have only to

examine those cases in which the resultant of the

external stresses and attractions acting on the body

amounts to nothing. But the most important part of

statics is that which finds the stresses acting in the

interior of bodies between contiguous parts of them ;

for upon this depends the determination of the requisite

strength of structures which have to bear given loads.

It is found that the way in which the stress due to a

given strain depends on the strain varies according to

the physical nature of the body ; for bodies, however,

which are not crystalline or fibrous, but which have the

same properties in all directions, there are two quanti-

ties which, if known, will enable us always to calculate

INSTRUMENTS USED IN MEASUREMENT. 29

the stress due to a given strain. These are, the

elasticity of volume, or resistance to change of size ; and

the rigidity, elasticity of figure, or resistance to change

of shape. Problems relating to the interior state of

bodies are far more difficult than those which regard

them as rigid. Thus, if a beam is supported at its two

ends, it is very easy to find the portion of its weight

which is borne by each support ; but the determination

of the state of stress in the interior is a problem of

great complexity.

There is one theorem of kinetics which must be

mentioned here. If we multiply half the momentum of

every particle of a body by its velocity, and add all the

results together, we shall get what is called the kinetic

energy of the body. When the body is moved from

one position to another, if we multiply each force acting

on it whether attraction or stress by the distance

moved in the direction opposite to the force, and add

the results, we shall get what is called the work done

against the forces during the change of position. It

does not at all depend on the rate at which the change

is made, but only on the two positions. If a body

moves, and loses kinetic energy, it does an amount of

work equal to the kinetic energy lost. If it gains kinetic

energy, an amount of work equal to this gain must be

done to take it back from the new position to the old

one. The amount of work which must be done to take

a body from a certain standard position to the position

which it has at present is called the potential energy of

the body. The theorem may be stated in this form ;

the sum of the potential and kinetic energies is always

the same, provided the surrounding circumstances do

30 INSTRUMENTS USED IN MEASUREMENT.

not alter. Hence the theorem is called the Conservation

of Energy. It is one fact out of many that may be

deduced from the equations of motion ; it is not suffi-

cient to determine the motion of a body, but it is ex-

ceedingly useful as giving a general result in cases

where it might be difficult or undesirable to investigate

all the particulars ; and it is especially applicable to

machines, the important question in regard to which is

the amount of work which they can do.

It will have been seen that the science of motion

depends on a few fundamental principles which are

easily verified, and consists almost entirely of mathema-

tical deductions and calculations based on those princi-

ples. It is no longer therefore an experimental science

in the same sense as those are in which the fundamental

facts are still being discovered. The apparatus con-

nected with it may be conveniently classified under

three heads :

(a) Apparatus for illustrating theorems or solving

problems of kinematics, such as those mentioned

above for compounding harmonic motions.

There is reason to hope for great extension of

our powers in this direction.

(b) Apparatus for measuring the dynamical

quantities, such as weight, work, and the

elasticities of different substances. These are

more fully classified under Measurements.

(c) Apparatus designed for purposes belonging to

other sciences, but illustrating by its structure

and functions the results of kinematics or

dynamics. In this class the remainder of the

collection is included.

31

UNIVERSITY

TWtt.1

BODY AND MIND. 1

THE subject of this Lecture is one in regard to which

a great change has recently taken place in the public

mind. Some time ago it was the custom to look with

suspicion upon all questions of a metaphysical nature

as being questions that could not be discussed with any

good result, and which, leading inquirers round and

round in the same circle, never came to an end. But

quite of late years there is an indication that a large

number of people are waking up to the fact that Science

has something to say upon these subjects ; and the

English people have always been very ready to hear

what Science can say understanding by Science what

we shall now understand by it, that is, organized com-

mon sense.

When I say Science, I do not mean what some

people are pleased to call Philosophy. The word ' phi-

losopher,' which meant originally ' lover of wisdom/

has come in some strange way to mean a man who

thinks it his business to explain everything in a certain

number of large books. It will be found, I think, that

in proportion to his colossal ignorance is the perfection

and symmetry of the system which he sets up ; because

it is so much easier to put an empty room tidy than a

1 Sunday Lecture Society, November 1, 1874 ; * Fortnightly Keview/

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