... from the Manuals of Elementary Science ...
J. CLERK MAXWELL
Chapter 1 - INTRODUCTION
Chapter 1 ... INTRODUCTION
Article 1 ... Nature of Physical Science
Article 2 ... Definition of a Material System
Article 3 ... Definition of Internal and External
Article 4 ... Definition of Configuration
Article 5 ... Diagrams
Article 6 ... A Material Particle
Article 7 ... Relative Positions of Two Material Particles
Article 8 ... Vectors
Article 9 ... System of Three Particles
Article 10 ... Addition of Vectors
Article 11 ... Subtraction of one Vector from another
Article 12 ... Origin of Vectors
Article 13 ... Relative Postions of Two Systems
Article 14 ... Three Data for the Comparison of Two Systems
Article 15 ... On the Idea of Space
Article 16 ... Error of Descartes
Article 17 ... On the Idea of Time
Article 18 ... Absolute Space
Article 19 ... Statement of the General Maxim of Physical Science
[INDEX to all Chapters][Contents]
Article 1 - Nature of Physical Science
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 brances 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 ist 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.
Article 2 - Definition of a Material System
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.
Article 3 - Definition of Internal and External
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 ot 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.
Article 4 - Definition Of Configuration
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 configuation 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, &c., which do not represent the form of the system, but by means of which its relative velocities or its internal forces may be studied.
Article 6 - A Material Particle
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 not configuration.
Article 7 - Relative Position of Two Material Particles
The diagram of two material particles consists of two points, as, for instance, A and B. 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 indicting 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 equl which are parallel (and drawn towards the same parts) and of the same magnitude. Any quantity, such, for instance, as a veloecity 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.
Article 9 - System of Three Particle
Let us next consider a system of three particles. Its configuration is represented by a diagram of three points, A, B, C. 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. It is manifest that from these data, when A is know we can find B and then C, so that the configuation of the three points is completely determined. The position of C with respect of 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.
Article 10 - Addition of Vectors
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.
Article 11 - Subtraction of One Vector From Another
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 indifferent = AC - AB since AB is equal and opposite to BA. 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.
The positions of any number of particles belonging to a material system 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 AB = OB - OA. We may choose any point whatever for the origin, and there is for the present no reason why we should choose one point rather than another. The configuration of the system - that is to say, the position of its parts with respect to each other - remains the same, whatever point is chosen as origin. Many enquiries, however, are simplified by a proper selection of the origin.
Article 15 - On the Idea of Space
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 mthod of combining several configurations into one system which includes them all. In this way we add to the small region we can explore by stretching our limbs the more distant regions which we can reach be walking or by being carried. Th these we add those which we learn by the reports of others, and those inacessible regions whose position we ascertain only be a process of calculation, till at last we recognise that every place has a definite postion with respect to every other place, whether the one place is accessible from the other place or not.
This from measurements made on the earth's surface we deduce the position of the center of the earth relative to known objects, and we calculate the number of cubic miles in the earth's volume quite independently of any other hypothesis as to what may exist at the center of the earth, or in any other place beneath that thin layer of the crust of the earth which alone we can directly explore.
Article 16 - Error of Descartes
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 in 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 is 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 Newtons'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, 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 recognise, 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.
Article 17 - On the Idea of Time
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 to 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 own experience of natural processes which go on in a uniform or a rhythmical manner, we come to recognise 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 efects examined spectroscopically by Mr. Huggins on the 16th May, 1866, and the mental suggestion which first led Professor Adams or M. Leverrier to being the researches which led to the discovery, by Dr. Galle, on 23rd September 1946, 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 in 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 with 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. Any one, however, who will try to imagine the state of a mind conscious of knowling the absolute position of a point will ever after be content with our relative knowledge.
Article 19 - Statement of the General Maxim of Physical Science
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 efects 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 the 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 tue 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 cause.
[Index to Chapters][Contents - This Chapter][Next Chapter]
... from the Manuals of Elementary Science ...
J. CLERK MAXWELL
Chapter 1 - INTRODUCTION