Chapter 1.
Introduction
Article 9. System of three
particles
The position of B with respect to A is indicated
by the vector AB, and that of C with respect to B
by the vector BC.
Article 10. Addition of
vectors
Then, by Article 9, the position of a point P of
the second system, with respect to the first origin, O, is represented
by the sum of the vector OP of that point with respect to the second
origin, O´ and the vector OO´ of the second origin,
O´
with respect to the first, O.
Physical science is that department of knowledge
which relates to the order of nature, or, in other words, to the regular
succession of events.
The name of physical science, however, is
often applied in a more or less restricted manner to those branches of
science in which the phenomena considered are of the simplest and most
abstract kind, excluding the consideration of the more complex phenomena,
such as those observed in living beings.
The simplest case of all is that in which
an event or phenomenon can be described as a change in the arrangement
of certain bodies. Thus the motion of the moon may be described by stating
the changes in her position relative to the earth in the order in which
they follow one another.
In other cases we may know that some change
of arrangement has taken place, but we may not be able to ascertain what
that change is.
Thus when water freezes we know that the
molecules or smallest parts of the substance must be arranged differently
in ice and in water. We also know that this arrangement in ice must have
a certain kind of symmetry, because the ice is in the form of symmetrical
crystals, but we have as yet no precise knowledge of the actual arrangement
of the molecules in ice. But whenever we can completely describe the change
of arrangement we have a knowledge, perfect so far as it extends, of what
has taken place, though we may still have to learn the necessary conditions
under which a similar event will always take place.
Hence the first part of physical science relates
to the relative position and motion of bodies.
In all scientific procedure we begin by marking
out a certain region or subject as the field of our investigations. To
this we must confine our attention, leaving the rest of the universe out
of account till we have completed the investigation in which we are engaged.
In physical science, therefore, the first step is to define clearly the
material system which we make the subject of our statements. This system
may be of any degree of complexity. It may be a single material particle,
a body of finite size, or any number of such bodies, and it may even be
extended so as to include the whole material universe.
All relations or actions between one part of
this system and another are called internal relations or actions.
Those between the whole or any part of the system and
bodies not included in the system are called external relations
or actions. These we study only so far as they affect the system itself,
leaving their effect on external bodies out of consideration. Relations
and actions between bodies not included in the system are to be left out
of consideration. We cannot investigate them except by making our system
include these other bodies.
When a material system is considered with respect
to the relative position of its parts, the assemblage of relative positions
is called the configuration of the system.
A knowledge of the configuration of the system at a given
instant implies a knowledge of the position of every point of the system
with respect to every other point at that instant.
The configuration of material systems may be
represented in models, plans, or diagrams. The model or diagram is supposed
to resemble the material system only in form, not necessarily in any other
respect.
A plan or a map represents on paper in two
dimensions what may really be in three dimensions and can only be completely
represented by a model. We shall use the term diagram to signify
any geometrical figure, whether plane or not, by means of which we study
the properties of a material system. Thus, when we speak of the configuration
of a system, the image which we form in our minds is that of a diagram,
which completely represents the configuration, but which has none of the
other properties of the material system. Besides diagrams of configuration
we may have diagrams of velocity, of stress, etc., which do not represent
the form of the system, but by means of which its relative velocities or
its internal forces may be studied.
A body so small that, for the purposes of
our investigation, the distances between its different parts may be
neglected, is called a material particle.
Thus in certain astronomical investigations
the planets, and even the sun, may be regarded each as a material particle,
because the difference of the actions of different parts of these bodies
does not come under our notice. But we cannot treat them as material particles
when we investigate their rotation. Even an atom, when we consider it as
capable of rotation, must be regarded as consisting of many material particles.
The diagram of a material particle is of
course a mathematical point, which has no configuration.
The position of B relative to A is indicated
by the direction and length of the straight line AB drawn from
A to B. If you start from A and travel in the direction indicated
by the line AB and for a distance equal to the length of that line,
you will get to
B. This direction and distance may be indicated
equally well by any other line, such as ab, which is parallel and
equal to AB. The position of A with respect to B is
indicated by the direction and length of the line BA, drawn from
B to A, or the line ba, equal and parallel to BA.
It is evident that BA = —AB.
In naming a line by the letters at its extremities, the
order of the letters is always that in which the line is to be drawn.
The expression AB, in geometry, is merely
the name of a line. Here it indicates the operation by which the line is
drawn, that of carrying a tracing point in a certain direction for a certain
distance. As indicating an operation, AB is called a vector,
and the operation is completely defined by the direction and distance of
the transference. The starting point, which is called the origin of the
vector, may be anywhere.
To define a finite straight line we must state its origin
as well as its direction and length. All vectors, however, are regarded
as equal which are parallel (and drawn toward the same parts) and of the
same magnitude.
Any quantity, such, for instance, as a velocity or a
force, which has a definite direction and a definite magnitude may be treated
as a vector and may be indicated in a diagram by a straight line whose
direction is parallel to the vector, and whose length represents, according
to a determinate scale, the magnitude of the vector.
Let us next consider a system of three particles. Its
configuration is represented by a diagram of three points, A,
B,
C.
Figure 1
It is manifest that from these data, when
A is known we can find B and then C, so that the configuration
of the three points is completely determined.
The position of C with respect to
A is indicated by the vector AC, and by the last remark the
value of
AC must be deducible from those of AB and BC.
The result of the operation AC is
to carry the tracing point from A to C. But the result is
the same if the tracing point is carried first from A to B
and then from
B to C, and this is the sum of the operations
AB +
BC.
Hence the rule for the addition of vectors may be stated
thus: From any point as origin draw the successive vectors in series, so
that each vector begins at the end of the preceding one. The straight line
from the origin to the extremity of the series represents the vector which
is the sum of the vectors.
The order of addition is indifferent;
for if we write
BC + AB the operation indicated may be performed
by drawing AD parallel and equal to BC, and then joining
DC, which, by Euclid, I. 33, is parallel and equal
to AB, so that by these two operations we arrive at the point C
in whichever order we perform them.
The same is true for any number of vectors,
take them in what order we please.
To express the position of C with respect
to B in terms of the positions of B and C with respect
to A, we observe that we can get from B to C either
by passing along the straight line BC or by passing from B
to A and then from
A to C. Hence
BC = BA + AC.
= AC + BA since the order of addition
is different.
Or the vector BC, which expresses the position of
C with respect to B, is found by subtracting the vector of
B from the vector of C, these vectors being drawn to B
and C respectively from any common origin A.
= AC — AB since
AB is equal and
opposite to BA.
The positions of any number of particles belonging
to a material system may be defined by means of the vectors drawn to each
of these particles from some one point. This point is called the origin
of the vectors, or, more briefly, the origin.
This system of vectors determines the configuration of
the whole system; for if we wish to know the position of any point B
with respect to any other point A, it may be found from the vectors
OA
and OB by the equation
If the configurations of two different systems
are known, each system having its own origin, and if we then wish to include
both systems in a larger system, having, say, the same origin as the first
of the two systems, we must ascertain the position of the origin of the
second system with respect to that of the first, and we must be able to
draw lines in the second system parallel to those in the first.
Figure 2
We have an instance of this formation of a large
system out of two or more smaller systems, when two neighbouring nations,
having each surveyed and mapped its own territory, agree to connect their
surveys so as to include both countries in one system. For this purpose
three things are necessary.
1st. A comparison of the origin selected by the one country
with that selected by the other.
2nd. A comparison of the directions of reference used
in the two countries.
3rd. A comparison of the standards of length used in
the two countries.
1. In civilized countries latitude is always
reckoned from the equator, but longitude is reckoned from an arbitrary
point, as Greenwich or Paris. Therefore, to make the map of Britain fit
that of France, we must ascertain the difference of longitude between the
Observatory of Greenwich and that of Paris.
2. When a survey has been made without astronomical
instruments, the directions of reference have sometimes been those given
by the magnetic compass. This was, I believe, the case in the original
surveys of some of the West India islands. The results of this survey,
though giving correctly the local configuration of the island, could not
be made to fit properly into a general map of the world till the deviation
of the magnet from the true north at the time of the survey was ascertained.
3. To compare the survey of France with
that of Britain, the metre, which is the French standard of length, must
be compared with the yard, which is the British standard of length. The
yard is defined by Act of Parliament 18 and 19 Vict. c. 72, July 30, 1855,
which enacts ‘‘that the straight line or distance between the centres of
the transverse lines in the two gold plugs in the bronze bar deposited
in the office of the Exchequer shall be the genuine standard yard at 62
Fahrenheit, and if lost, it shall be replaced by means of its copies.’’
The metre derives its authority from a law
of the French Republic in 1795. It is defined to be the distance between
the ends of a certain rod of platinum made by Borda, the rod being at the
temperature of melting ice. It has been found by the measurements of Captain
Clarke that the metre is equal to 39.37043 British inches.
We have now gone through most of the things to
be attended to with respect to the configuration of a material system.
There remain, however, a few points relating to the metaphysics of the
subject, which have a very important bearing on physics.
We have described the method of combining several configurations
into one system which includes them all. In this way we add to the small
region which we can explore by stretching our limbs the more distant regions
which we can reach by walking or by being carried. To these we add those
of which we learn by the reports of others, and those inaccessible regions
whose position we ascertain only by a process of calculation, till at last
we recognize that every place has a definite position with respect to every
other place, whether the one place is accessible from the other or not.
Thus from measurements made on the earth’s
surface we deduce the position of the centre of the earth relative to known
objects, and we calculate the number of cubic miles in the earth’s volume
quite independently of any hypothesis as to what may exist at the centre
of the earth, or in any other place beneath that thin layer of the crust
of the earth which alone we can directly explore.
It appears, then, that the distance between one
thing and another does not depend on any material thing between them, as
Descartes seems to assert when he says (Princip. Phil., II. 18)
that if that which is in a hollow vessel were taken out of it without anything
entering to fill its place, the sides of the vessel, having nothing between
them, would be in contact.
This assertion is grounded on the dogma
of Descartes, that the extension in length, breadth, and depth which constitute
space is the sole essential property of matter. ‘‘The nature of matter,’’
he tells us, ‘‘or of body considered generally, does not consist in a thing
being hard, or heavy, or coloured, but only in its being extended in length,
breadth, and depth’’ (Princip., II. 4). By thus confounding the
properties of matter with those of space, he arrives at the logical conclusion
that if the matter within a vessel could be entirely removed, the space
within the vessel would no longer exist. In fact he assumes that all space
must be always full of matter.
I have referred to this opinion of Descartes
in order to show the importance of sound views in elementary dynamics.
The primary property of matter was indeed distinctly announced by Descartes
in what he calls the ‘‘First Law of Nature’’ (Princip., II. 37):
‘‘That every individual thing, so far as in it lies, perseveres in the
same state, whether of motion or of rest.’’
We shall see when we come to Newton’s laws
of motion that in the words ‘‘so far as in it lies,’’ properly understood,
is to be found the true primary definition of matter, and the true measure
of its quantity. Descartes, however, never attained to a full understanding
of his own words (quantum in se est), and so fell back on his original
confusion of matter with space—space being, according to him, the only
form of substance, and all existing things but affections of space. This
error runs through every part of Descartes’ great work, and it forms one
of the ultimate foundations of the system of Spinoza. I shall not attempt
to trace it down to more modern times, but I would advise those who study
any system of metaphysics to examine carefully that part of it which deals
with physical ideas.
We shall find it more conducive to scientific progress
to recognize, with Newton, the ideas of time and space as distinct, at
least in thought, from that of the material system whose relations these
ideas serve to coordinate.
The idea of time in its most primitive form is
probably the recognition of an order of sequence in our states of consciousness.
If my memory were perfect, I might be able to refer every event within
my own experience to its proper place in a chronological series. But it
would be difficult, if not impossible, for me to compare the interval between
one pair of events and that between another pair—to ascertain, for instance,
whether the time during which I can work without feeling tired is greater
or less now than when I first began to study. By our intercourse with other
persons, and by our experience of natural processes which go on in a uniform
or a rhythmical manner, we come to recognize the possibility of arranging
a system of chronology in which all events whatever, whether relating to
ourselves or to others, must find their place. Of any two events, say the
actual disturbance at the star in Corona Borealis, which caused the luminous
effects examined spectroscopically by Mr. Huggins on the 16th May, 1866,
and the mental suggestion which first led Professor Adams or M. Leverrier
to begin the researches which led to the discovery, by Dr. Galle, on the
23rd September, 1846, of the planet Neptune, the first named must have
occurred either before or after the other, or else at the same time.
Absolute, true, and mathematical time is conceived by
Newton as flowing at a constant rate, unaffected by the speed or slowness
of the motions of material things. It is also called duration. Relative,
apparent, and common time is duration as estimated by the motion of bodies,
as by days, months, and years. These measures of time may be regarded as
provisional, for the progress of astronomy has taught us to measure the
inequality in the lengths of days, months, and years, and thereby to reduce
the apparent time to a more uniform scale, called mean solar time.
Absolute space is conceived as remaining always
similar to itself and immovable. The arrangement of the parts of space
can no more be altered than the order of the portions of time. To conceive
them to move from their places is to conceive a place to move away from
itself.
But as there is nothing to distinguish one
portion of time from another except the different events which occur in
them, so there is nothing to distinguish one part of space from another
except its relation to the place of material bodies. We cannot describe
the time of an event except by reference to some other event, or the place
of a body except by reference to some other body. All our knowledge, both
of time and place, is essentially relative. When a man has acquired the
habit of putting words together, without troubling himself to form the
thoughts which ought to correspond to them, it is easy for him to frame
an antithesis between this relative knowledge and a so-called absolute
knowledge, and to point out our ignorance of the absolute position of a
point as an instance of the limitation of our faculties. Anyone, however,
who will try to imagine the state of a mind conscious of knowing the absolute
position of a point will ever after be content with our relative knowledge.
There is a maxim which is often quoted, that
‘‘The same causes will always produce the same effects.’’
To make this maxim intelligible we must
define what we mean by the same causes and the same effects, since it is
manifest that no event ever happens more than once, so that the causes
and effects cannot be the same in all respects. What is really meant is
that if the causes differ only as regards the absolute time or the absolute
place at which the event occurs, so likewise will the effects.
The following statement, which is equivalent
to the above maxim, appears to be more definite, more explicitly connected
with the ideas of space and time, and more capable of application to particular
cases: ‘‘The difference between one event and another does not depend on
the mere difference of the times or the places at which they occur, but
only on differences in the nature, configuration, or motion of the bodies
concerned.’’
It follows from this, that if an event has occurred at
a given time and place, it is possible for an event exactly similar to
occur at any other time and place.
There is another maxim which must not be
confounded with that quoted at the beginning of this article, which asserts
that ‘‘Like causes produce like effects.’’
This is only true when small variations
in the initial circumstances produce only small variations in the final
state of the system. In a great many physical phenomena this condition
is satisfied; but there are other cases in which a small initial variation
may produce a very great change in the final state of the system, as when
the displacement of the points causes a railway train to run into another
instead of keeping its proper course.
© IDEA YAYINEVI, IDEA PUBLISHING HOUSE, ISTANBUL 1998