Grau theurer Freund ist alle Theorie
Und gruen des Lebens goldener Baum.
GOETHE
Everyone realizes the constraint that is placed upon the lives of
men by the physical necessities of the world in which we live, and
although in one way this constraint is more and more relieved with the
progress of the applied sciences, in another way it becomes more and
more exacting. It is indeed easier to cross the Atlantic Ocean now
than it was in Leif Ericsson's time, but consider the discipline of
the shop, and above all consider the rules of machine design! Could
even the hardy Norsemen have known anything as uncompromisingly
exacting as these? To do things becomes easier and easier, but to
learn how to do things becomes more and more difficult.
Every person I have ever talked with, old or young, theorist or
practician, student-in-general or specialist in whatever line, has
exhibited more or less distinctly a certain attitude of impatience
towards the exactions of this or that phase of the precise modes of
thought of the physical sciences.
"Da wird der Geist Euch wohl dressiert
In spanische Stiefeln eingeschnuert."
In a recent article[D] on the distinction between the liberal and
technical in education, my friend and colleague, Professor Percy
Hughes, says that in speaking of an education as liberal we thereby
associate it with liberalism in politics, in philosophy and theology,
and in men's personal relations with each other. In each case
liberalism seems fundamentally, to denote freedom, and liberalism in
education is the freedom of development in each individual of that
character and personality which is his true nature. All this I accept
in the spirit of an optimist, assuming men's true natures to be good,
but I do not, and I am sure that Professor Hughes does not, consider
that technical education, unless it be inexcusably harsh and narrow,
is illiberal; nor that liberal education, unless it be inexcusably
soft and vague, is wholly non-technical. The liberal and the technical
are not two kinds of education, each complete in itself. Indeed,
Professor Hughes speaks of liberal education, not as a category, but
as a condition which makes for freedom of development of personality
and character.
It seems to me, however, that there are phases of education which have
but little to do with personality, and I call to your attention this
definition of liberalism in education, in order that I may turn sharply
away from it as a partial definition which, to a great extent, excludes
the physical sciences. Indeed, I wish to speak of a condition in education
which is the antithesis of freedom. I wish to explain the teaching of
elementary physical science as a mode of constraint, as an impressed
constructive discipline without which no freedom is possible in our
dealings with physical things. I wish to characterize the study of
elementary physical science as a reorganization of the workaday mind
of a young man as complete as the pupation of an insect; and I wish
to emphasize the necessity of exacting constraint as the essential
condition of this reorganization.
There is a kind of salamander, the axolotl, which lives a tadpole-like
youth and never changes to the adult form unless a stress of dry
weather annihilates his watery world; but he lives always and
reproduces his kind as a tadpole, and a very funny-looking tadpole he
is, with his lungs hanging like feathery tassels from the sides of
his head. When the aquatic home of the axolotl dries up, he quickly
develops a pair of internal lungs, lops off his tassels and embarks on a
new mode of life on land. So it is with our young men who are to develop
beyond the tadpole stage, they must meet with quick and responsive inward
growth that new and increasing "stress of dryness," as many are wont to
call our modern age of science and organized industry.
Stress of dryness! Indeed no flow of humor is to be found in the
detached impersonalities of the sciences, and if we are to understand
the characteristics of physical science we must turn our attention
to things which lead inevitably to an exacting and rigid mathematical
philosophy. It certainly is presumptive to tell a reader that he must
turn his attention to such a thing, but there is no other way; the
best we can do is to choose the simplest path. Let us therefore
consider the familiar phenomena of motion.
The most prominent aspect of all phenomena is motion. In that realm
of nature which is not of man's devising[E] motion is universal. In the
other realm of nature, the realm of things devised, motion is no less
prominent. Every purpose of our practical life is accomplished
by movements of the body and by directed movements of tools and
mechanisms, such as the swing of scythe and flail, and the studied
movements of planer and lathe from which are evolved the strong-armed
steam shovel and the deft-fingered loom.
The laws of motion. Every one has a sense of the absurdity of the idea
of reducing the more complicated phenomena of nature to an orderly
system of mechanical law. To speak of motion is to call to mind
first of all the phenomena that are associated with the excessively
complicated, incessantly changing, turbulent and tumbling motion of
wind and water. These phenomena have always had the most insistent
appeal to us, they have confronted us everywhere and always, and life
is an unending contest with their fortuitous diversity, which rises
only too often to irresistible sweeps of destruction in fire and
flood, and in irresistible crash of collision and collapse where
all things mingle in one dread fluid confusion! The laws of motion!
Consider the awful complexity of a disastrous tornado or the dreadful
confusion of a railway wreck, and understand that what we call the
laws of motion, although they have a great deal to do with the ways in
which we think, have very little to do with the phenomena of nature. The
laws of motion! There is indeed a touch of arrogance in such a phrase
with its unwarranted suggestion of completeness and universality, and
yet the ideas which constitute the laws of motion have an almost unlimited
extent of legitimate range, _and these ideas must be possessed with a
perfect precision if one is to acquire any solid knowledge whatever of
the phenomena of motion._ The necessity of precise ideas. Herein lies
the impossibility of compromise and the necessity of coercion and
constraint; one must think so and so, there is no other way. And
yet there is always a conflict in the mind of even the most willing
student because of the constraint which precise ideas place upon our
vivid and primitively adequate sense of physical things; and this
conflict is perennial but it is by no means a one-sided conflict
between mere crudity and refinement, for refinement ignores many
things. Indeed, precise ideas not only help to form[F] our sense of
the world in which we live but they inhibit sense as well, and their
rigid and unchallenged rule would be indeed a stress of dryness.
The laws of motion. We return again and yet again to the subject, for
one is not to be deterred therefrom by any concession of inadequacy,
no, nor by any degree of respect for the vivid youthful sense of those
things which to suit our narrow purpose must be stripped completely
bare. It is unfortunate, however, that the most familiar type
of motion, the flowing of water or the blowing of the wind, is
bewilderingly useless as a basis for the establishment of the simple
and precise ideas which are called the "laws of motion," and which
are the most important of the fundamental principles of physics. These
ideas have in fact grown out of the study of the simple phenomena
which are associated with the motion of bodies in bulk without
perceptible change of form, the motion of rigid bodies, so called.
Before narrowing down the scope of the discussion, however, let us
illustrate a very general application of the simplest idea of motion,
the idea of velocity. Every one has, no doubt, an idea of what is
meant by the velocity of the wind; and a sailor, having what he calls
a ten-knot wind, knows that he can manage his boat with a certain spread
of canvas and that he can accomplish a certain portion of his voyage in
a given time; but an experienced sailor, although he speaks glibly of
a ten-knot wind, belies his speech by taking wise precaution against
every conceivable emergency. He knows that a ten-knot wind is by no
means a sure or a simple thing with its incessant blasts and whirls; and
a sensitive anemometer, having more regard for minutiae than any sailor,
usually registers in every wind a number of almost complete but
excessively irregular stops and starts every minute and variations of
direction that sweep around half the horizon!
Wer will was Lebendig's erkennen und beschreiben
Sucht erst den Geist heraus zu treiben.
GOETHE.
We must evidently direct our attention to something simpler than
the wind. Let us, therefore, consider the drawing of a wagon or the
propulsion of a boat. It is a familiar experience that effort is
required to start a body moving and that continued effort is required
to maintain the motion. Certain very simple facts as to the nature and
effects of this effort were discovered by Sir Isaac Newton, and on the
basis of these facts Newton formulated the laws of motion.
The effort required to start a body or to keep it moving is
called force. Thus, if one starts a box sliding along a table
one is said to exert a force on the box. The same effect might
be accomplished by interposing a stick between the hand and
the box, in which case one would exert a force on the stick
and the stick in its turn would exert a force on the box. We
thus arrive at the notion of force action between inanimate
bodies, between the stick and the box in this case, and Newton
pointed out that the force action between the two bodies _A_
and _B_ always consists of two equal and opposite forces, that
is to say, if body _A_ exerts a force on _B,_ then _B_ exerts
an equal and opposite force on _A,_ or, to use Newton's words,
action is equal to reaction and in a contrary direction.
In leading up to this statement one might consider the force with
which a person pushes on the box and the equal and opposite force with
which the box pushes back on the person, but if one does not wish to
introduce the stick as an intermediary, it is better to speak of
the force with which the hand pushes on the box, and the equal and
opposite force with which the box pushes back on the hand, because in
discussing physical phenomena it is of the utmost importance to pay
attention only to impersonal [42] things. Indeed our modern industrial
life, in bringing men face to face with an entirely unprecedented
array of intricate mechanical and physical problems, demands of every
one a great and increasing amount of impersonal thinking, and the
precise and rigorous modes of thought of the physical sciences
are being forced upon widening circles of men with a relentless
insistence--all of which it was intended to imply by referring to
the "stress of dryness" which overtakes the little axolotl in his
contented existence as a tadpole.
When we examine into the conditions under which a body starts
to move and the conditions under which a body once started is
kept in motion, we come across a very remarkable fact, if we
are careful to consider every force which acts upon the body,
and this remarkable fact is that the forces which act upon _a
body at rest_ are related to each other in precisely the same
way as the forces which act upon _a body moving steadily along
a straight path._ Therefore it is convenient to consider,
_first_ the relation between the forces which act upon a body
at rest, or upon a body in uniform motion, and _second_ the
relation between the forces which act upon a body which is
starting or stopping or changing the direction of its motion.
Suppose a person _A_ were to hold a box in mid-air. To do so
it would of course be necessary for him to push upwards on the
box so as to balance the downward pull of the earth, the weight
of the box as it is called. If another person _B_ were to take
hold of the box and pull upon it in any direction, _A_ would
have to exert an equal pull on the box in the opposite direction
to keep it stationary. _The forces which act upon a stationary
body are always balanced._
Every one, perhaps, realizes that what is here said about the
balanced relation of the forces which act upon a stationary
box, is equally true of the forces which act on a box
similarly held in a steadily moving railway car or boat.
Therefore, _the forces which act upon a body which moves
steadily along a straight path are balanced._
This is evidently true when the moving body is surrounded on
all sides by things which are moving along with it, as in a
car or a boat; but how about a body which moves steadily along
a straight path but which is surrounded by bodies which do not
move along with it? Everyone knows that some active agent such
as a horse or a steam engine must pull steadily upon such a
body to keep it in motion. If left to itself such a moving
body quickly comes to rest. Many have, no doubt, reached this
further inference from experience, namely, that the tendency
of moving bodies to come to rest is due to the dragging
forces, or friction, with which surrounding bodies act upon a
body in motion. Thus a moving boat is brought to rest by the
drag of the water when the propelling force ceases to act; a
train of cars is brought to rest because of the drag due to
friction when the pull of the locomotive ceases; a box which
is moving across a table comes to rest when left to itself,
because of the drag due to friction between the box and the
table.
We must, therefore, always consider two distinct forces when
we are concerned with a body which is kept in motion, namely,
the _propelling force_ due to some active agent such as
a horse or an engine, and the _dragging force_ due to
surrounding bodies. Newton pointed out that when a body is
moving steadily along a straight path, the propelling force
is always equal and opposite to the dragging force. Therefore,
_The forces which act upon a body which is stationary, or
which is moving uniformly along a straight path, are balanced
forces._
Many hesitate to accept as a fact the complete and exact
balance of propelling and dragging forces on a body which is
moving steadily along a straight path in the open, but
direct experiment shows it to be true, and the most elaborate
calculations and inferences based upon this idea of the
complete balance of propelling and dragging forces on a body
in uniform motion are verified by experiment. One may ask, why
a canal boat, for example, should continue to move if the pull
of the mule does not exceed the drag of the water; but
why should it stop if the drag does not exceed the pull?
Understand that we are not considering the starting of the
boat. The fact is that the conscious effort which one must
exert to drive a mule, the cost of the mule, and the expense
of his keep, are what most people think of, however hard one
tries to direct their attention solely to the state of tension
in the rope that hitches the mule to the boat after the
boat is in full motion; and most people consider that if the
function of the mule is simply to balance the drag of the
water so as to keep the boat from stopping, then why should
there not be some way to avoid the cost of so insignificant
an operation? There is, indeed, an extremely important matter
involved here, but it has no bearing on the question as to the
balance of propulsion and drag on a body which moves steadily
along a straight path.
Let us now consider the relation between the forces which act
upon a body which is changing its speed, upon a body which is
being started or stopped, for example. Everyone has noticed
how a mule strains at his rope when starting a canal boat,
especially if the boat is heavily loaded, and how the boat
continues to move for a long time after the mule ceases to
pull. In the first case, the pull of the mule greatly exceeds
the drag of the water, and the speed of the boat increases; in
the second case, the drag of the water of course exceeds the
pull of the mule, for the mule is not pulling at all, and
the speed of the boat decreases. When the speed of a body is
changing, the forces which act on the body are unbalanced. We
may conclude therefore that _the effect of an unbalanced force
acting on a body is to change the velocity of the body,_ and
it is evident that the longer the unbalanced force continues
to act the greater the change of velocity. Thus if the mule
ceases to pull on a canal boat for one second the velocity of
the boat will be but slightly reduced by the unbalanced drag
of the water, whereas if the mule ceases to pull for two
seconds the decrease of velocity will be much greater. _In
fact the change of velocity due to a given unbalanced force
is proportional to the time that the force continues to act._
This is exemplified by a body falling under the action of the
unbalanced pull of the earth; after one second it will have
gained a certain amount of velocity (about 32 feet per second),
after two seconds it will have made a total gain of twice as
much velocity (about 64 feet per second), and so on.
Since the velocity produced by an unbalanced force is
proportional to the time that the force continues to act, it
is evident that the effect of the force should be specified as
so-much-velocity-produced-per-second, exactly as in the case
of earning money, the amount one earns is proportional to
the length of time that one continues to work, and we
always specify one's earning capacity as
so-much-money-earned-per-day.
Everyone knows what it means to give an easy pull or a hard
pull on a body. That is to say, we all have the ideas of
greater and less as applied to forces. Everybody knows also
that if a mule pulls hard on a canal boat, the boat will get
under way more quickly than if the pull is easy, that is, the
boat will gain more velocity per unit of time under the
action of a hard pull than under the action of an easy
pull. Therefore, any precise statement of the effect of an
unbalanced force on a given body must correlate the precise
value of the force and the exact amount of velocity produced
per unit of time by the force. This seems a very difficult
thing, but its apparent difficulty is very largely due to
the fact that we have not as yet agreed as to what we are to
understand by the statement that one force is precisely three,
or four, or any number of times as great as another. Suppose,
therefore, that _we agree to call one force twice as large as
another when it will_ as _produce in a given body twice as
much velocity in a given time_ (remembering of course that we
are now talking about unbalanced forces, or that we are assuming
for the sake of simplicity of statement, that no dragging forces
exist). As a result of this definition we may state that _the
amount of velocity produced per second in a given body by an
unbalanced force is proportional to the force._
Of course we know no more about the matter in hand than we did before
we adopted the definition, but we do have a good illustration of how
important a part is played in the study of physical science, by what
we may call making-up one's mind, in the sense of putting one's
mind in order. This kind of thing is very prominent in the study of
elementary physics, and the rather indefinite reference (in the story
of the little tasseled tadpole) to an inward growth as needful
before one can hope for any measure of success in our modern world of
scientific industry was an allusion to this thing, the "making-up" of
one's mind. Nothing is so essential in the acquirement of exact and
solid knowledge as the possession of precise ideas, not indeed that
a perfect precision is necessary as a means for retaining knowledge,
_but that nothing else so effectually opens the mind for the
perception even of the simplest evidences of a subject_[G].
We have now settled the question as to the effect of different
unbalanced forces on a given body on the basis of very general
experience, and by an agreement as to the precise meaning to
be attached to the statement that one force is so many times
as great as another; but how about the effect of the same
force upon different bodies, and how may we identify the force
so as to be sure that it is the same? It is required, for
example, to exert a given force on body _A_ and then exert the
same force on another body _B._ This can be done by causing
a third body _C_ (a coiled spring, for example) to exert the
force; then the forces exerted on _A_ and _B_ are the same if
the reaction in each case produces the same effect on body
_C_ (the same degree of stretch, for example). Concerning the
effects of the same unbalanced force on different bodies three
things have to be settled by experiment as follows:
(a) In the first place let us suppose that a certain force
_F_ is twice as large as a certain other force _G,_ according
to our agreement, because the force _F_ produces twice as much
velocity every second as force _G_ when the one and then the
other of these forces is caused to act upon a given body, a
piece of lead for example. Then, does the force _F_ produce
twice as much velocity every second as the force _G_ whatever
the nature and size of the given body, whether it be wood, or
ice, or sugar? Experiment shows that it does.
(b) In the second place, suppose that we have such amounts
of lead, or iron, or wood, etc., that a certain given force
produces the same amount of velocity per second when it is
made to act, as an unbalanced force, upon one or another of
these various bodies. Then what is the relation between the
amounts of these various substances? Experiment shows that
they all have the same mass in grams, or pounds, as determined
by a balance. That is, a given force produces the same amount
of velocity per second in a given number of grams of any kind
of substance. Thus the earth pulls with a certain definite
force (in a given locality) upon _M_ grams of any substance
and, aside from the dragging forces due to air friction, all
kinds of bodies gain the same amount of velocity per second
when they fall under action of the unbalanced pull of the
earth.
(c) In the third place, what is the relation between the
velocity per second produced by a given force and the mass in
grams (or pounds) of the body upon which it acts. Experiment
shows that _the velocity per second produced by a given force
is inversely proportional to the mass of the body upon which
the force acts._ In speaking of the mass of the body in grams
(or pounds) we here refer to the result which is obtained by
weighing the body on a balance scale, and the experimental
fact which is here referred to constitutes a very important
discovery: namely, when one body has twice the mass of
another, according to the balance method of measuring mass, it
is accelerated half as fast by a given unbalanced force.
The effect of an unbalanced force in producing velocity may
therefore be summed up as follows: _The velocity per second
produced by an unbalanced force is proportional to the force
and inversely proportional to the mass of the body upon which
the force acts, and the velocity produced by an unbalanced
force is always in the direction of the force._
* * * * *
"We advise all men," says Bacon, "to think of the true ends of
knowledge, and that they endeavor not after it for curiosity,
contention, or the sake of despising others, nor yet for reputation
or power or any other such inferior consideration, but solely for
the occasions and uses of life." It is difficult to imagine any other
basis upon which the study of physics can be justified than for the
occasions and uses of life; in a certain broad sense, indeed, there
is no other justification. But the great majority of men must needs be
practical in the narrow sense, and physics, as the great majority
of men study it, relates chiefly to the conditions which have been
elaborated through the devices of industry as exemplified in our mills
and factories, in our machinery of transportation, in optical and
musical instruments, in the means for the supply of power, heat,
light, and water for general and domestic use, and so on.
From this narrow practical point of view it may seem that there can
be nothing very exacting in the study of the physical sciences; but
what is physics? That is the question. One definition at least is to
be repudiated; it is not "The science of masses, molecules and the
ether." Bodies have mass and railways have length, and to speak
of physics as the _science of masses_ is as silly as to define
railroading as the _practice of lengths,_ and nothing as reasonable as
this can be said in favor of the conception of physics as the science
of molecules and the ether; it is the sickliest possible notion of
physics, whereas the healthiest notion, even if a student does not
wholly grasp it, is that physics is the science of the ways of taking
hold of things and pushing them!
Bacon long ago listed in his quaint way the things which seemed to
him most needful for the advancement of learning. Among other things
he mentioned "A New Engine or a Help to the mind corresponding to
Tools for the hand," and the most remarkable aspect of present-day
physical science is that aspect in which it constitutes a realization
of this New Engine of Bacon{6}. We continually force upon the extremely
meager data obtained directly through our senses, an interpretation
which, in its complexity and penetration, would seem to be entirely
incommensurate with the data themselves, and we exercise over physical
things a kind of rational control which greatly transcends the native
cunning of the hand. The possibility of this forced interpretation and
of this rational control depends upon the use of two complexes:
(a) A _logical structure,_ that is to say, a body of mathematical and
conceptual theory which is brought to bear upon the immediate materials
of sense, and (b) a _mechanical structure,_ that is to say, either (1)
a carefully planned _arrangement of apparatus,_ such as is always
necessary in making physical measurements, or (2) a carefully planned
_order of operations,_ such as the successive operations of solution,
reaction, precipitation, filtration, and weighing in chemistry.
These two complexes do indeed constitute a New Engine which helps
the mind as tools help the hand; it is through the enrichment of
the materials of sense by the operation of this New Engine that the
elaborate interpretations of the physical sciences are made possible,
and the study of elementary physics is intended to lead to the
realization of this New Engine: (a) By the building up in the mind,
of the logical structure of the physical sciences; (b) by training
in the making of measurements and in the performance of ordered
operations, and (c) by exercises in the application of these things
to the actual phenomena of physics and chemistry at every step
and all of the time with every possible variation.
That, surely, is a sufficiently exacting program; and the only
alternative is to place the student under the instruction of Jules
Verne where he need not trouble himself about foundations but may
follow his teacher pleasantly on a care-free trip to the moon or with
easy improvidence embark on a voyage of twenty-thousand leagues under
the sea.
What it means to study physical science may be explained further by
mentioning the chief difficulties encountered in the teaching of
that subject. One difficulty is that the native sense of most men is
woefully inadequate without stimulation and direction for supplying
the sense material upon which the logical structure of the science is
intended to operate. A second difficulty is that the human mind is so
in the habit of considering the practical affairs of life that it can
hardly be turned to that minute consideration of apparently
insignificant details which is so necessary in the scientific analysis
even of the most practical things. Everyone knows the capacity of the
Indian for long continued and serious effort in his primitive mode of
life, and yet it is difficult to persuade an Indian "farmer" to plow.
Everyone knows also that the typical college student is not stupid,
and yet it is difficult to persuade the young men of practical and
business ideals in our colleges and technical schools to study the
abstract elements of science. Indeed it is as difficult to get the
average young man to hold abstract things in mind as to get a young
Indian to plow, and for almost exactly the same reason. The scientific
details of any problem are in themselves devoid of human value, and
this quality of detachment is the most serious obstacle to young
people in their study of the sciences.
A third difficulty which indeed runs through the entire
front-of-progress of the human understanding is that the primitive
mind-stuff of a young man must be rehabilitated in entirely new
relations in fitting the young man for the conditions of modern life.
Every science teacher knows how much coercion is required for so
little of this rehabilitation; but the bare possibility of the process
is a remarkable fact, and that it is possible to the extent of bringing
a Newton or a Pasteur out of a hunting and fishing ancestry is indeed
wonderful. Everyone is familiar with the life history of a butterfly,
how it lives first as a caterpillar and then undergoes a complete
transformation into a winged insect. It is, of course, evident that
the bodily organs of a caterpillar are not at all suited to the needs
of a butterfly, the very food (of those species which take food) being
entirely different. As a matter of fact almost every portion of the
bodily structure of the caterpillar is dissolved as it were, into
a formless pulp at the beginning of the transformation, and the
organization of a flying insect then grows out from a central nucleus
very much as a chicken grows in the food-stuff of an egg. So it is in
the development of a young man. In early childhood the individual, if
he has been favored by fortune, exercises and develops more or less
extensively the primitive instincts and modes of the race in a free
outdoor life, and the result is so much mind-stuff to be dissolved
and transformed with more or less coercion and under more or less
constraint into an effective mind of the twentieth-century type.
A fourth difficulty is that the possibility of the rehabilitation of
mind-stuff has grown up as a human faculty almost solely on the
basis of language, and the essence of this rehabilitation lies in the
formation of ideas; whereas _a very large part of physical science is
a correlation in mechanisms._
The best way of meeting this quadruply difficult situation in the
teaching of elementary physics is to relate the teaching as much as
possible to the immediately practical and intimate things of life,
and to go in for suggestiveness as the only way to avoid a total
inhibition of the sense that is born with a young man. Such a method
is certainly calculated to limber up our theories and put them all at
work, the pragmatic method, our friends the philosophers call it, a
method which pretends to a conquering destiny.