Storm's Journal

--| Light Course > The Parallelogram of Forces |--- 

Increasingly in modern time, the mathematical way of studying the phenomena
of Nature — i.e. the study of natural phenomena in terms of mathematical
formulae — has grown to be the determining factor in the way we think even
of Nature herself.

Concerning these things we really must reach clarity. You see, dear Friends,
along the accustomed way of approach to Nature we have three things to begin
with — things that are really exercised by man before he actually reaches
Nature. The first is common or garden Arithmetic. In studying Nature
nowadays we do a lot of arithmetic — counting and calculating. Arithmetic —
we must be clear on this — is something man understands on its own ground,
in and by itself. When we are counting it makes no difference what we count.
Learning arithmetic, we receive something which, to begin with, has no
reference to the outer world. We may count peas as well as electrons. The
way we recognize that our methods of counting and calculating are correct is
altogether different from the way we contemplate and form conclusions about
the outer processes to which our arithmetic is then applied.

The second of the three to which I have referred is again a thing we do
before we come to outer Nature. I mean Geometry, — all that is known by
means of pure Geometry. What a cube or an octahedron is, and the relations
of their angles, — all these are things which we determine without looking
into outer Nature. We spin and weave them out of ourselves. We may make
outer drawings on them, but this is only to serve mental convenience, not to
say inertia. Whatever we may illustrate by outer drawings, we might equally
well imagine purely in the mind. Indeed it is very good for us to imagine
more of these things purely in the mind, using the crutches of outer
illustration rather less. Thus, what we have to say concerning geometrical
form is derived from a realm which, to begin with, is quite away from outer
Nature. We know what we have to say about a cube without first having had to
read it in a cube of rock-salt. Yet in the latter we must find it. So we
ourselves do something quite apart from Nature and then apply it to the

And then there is the third thing which we do, still before reaching outer
Nature. I am referring to what we do in “Phoronomy” so-called, or
Kinematics, i.e. the science of Movement. Now it is very important for you
to be clear on this point, — to realize that Kinematics too is,
fundamentally speaking, still remote from what we call the “real” phenomena
of Nature. Say I imagine an object to be moving from the point a to the
point b (Figure 1a). I am not looking at any moving object; I just imagine
it. Then I can always imagine this movement from a to b, indicated by an
arrow in the figure, to be compounded of two distinct movements. Think of it
thus: the point a is ultimately to get to b, but we suppose it does not go
there at once. It sets out in this other direction and reaches c. If it then
subsequently moves from c to b, it does eventually get to b. Thus I can also
imagine the movement from a to b so that it does not go along the line a — b
but along the line, or the two lines, a — c — b. The movement ab is then
compounded of the movements ac and cb, i.e. of two distinct movements. You
need not observe any process in outer Nature; you can simply think it —
picture it to yourself in thought — how that the movement from a to b is
composed of the two other movements. That is to say, in place of the one
movement the two other movements might be carried out with the same ultimate
effect. And when in thinking I picture this. The thought — the mental
picture — is spun out of myself. I need have made no outer drawing; I could
simply have instructed you in thought to form the mental picture; you could
not but have found it valid. Yet if in outer Nature there is really
something like the point a — perhaps a little ball, a grain of shot — which
in one instance moves from a to b and in another from a to c and then from c
to b, what I have pictured to myself in thought will really happen. So then
it is in kinematics, in the science of movement also; I think the movements
to myself, yet what I think proves applicable to the phenomena of Nature and
must indeed hold good among them.

Thus we may truly say: In Arithmetic, in Geometry and in Phoronomy or
Kinematics we have the three preliminary steps that go before the actual
study of Nature. Spun as they are purely out of ourselves, the concepts
which we gain in all these three are none the less valid for what takes
place in real Nature.

And now I beg you to remember the so-called Parallelogram of forces, (Figure
1b). This time, the point a will signify a material thing — some little
grain of material substance. I exert a force to draw it on from a to b. Mark
the difference between the way I am now speaking and the way I spoke before.
Before, I spoke of movement as such; now I am saying that a force draws the
little ball from a to b. Suppose the measure of this force, pulling from a
to b, to be five grammes; you can denote it by a corresponding length in
this direction. With a force of five grammes I am pulling the little ball
from a to b. Now I might also do it differently. Namely I might first pull
with a certain force from a to c. Pulling from a to c (with a force denoted
by this length) I need a different force than when I pulled direct from a to
b. Then I might add a second pull, in the direction of the line from c to b,
and with a force denoted by the length of this line. Having pulled in the
first instance from a towards b with a force of five grammes, I should have
to calculate from this figure, how big the pull a — c and also how big the
pull c — d would have to be. Then if I pulled simultaneously with forces
represented by the lines a — c and a — d of the parallelogram, I should be
pulling the object along in such a way that it eventually got to b; thus I
can calculate how strongly I must pull towards c and d respectively. Yet I
cannot calculate this in the same way as I did the displacements in our
previous example. What I found previously (as to the movement pure and
simple), that I could calculate, purely in thought. Not so when a real pull,
a real force is exercised. Here I must somehow measure the force; I must
approach Nature herself; I must go on from thought to the world of facts.

If once you realize this difference between the Parallelogram of Movements
and that of Forces, you have a clear and sharp formulation of the essential
difference between all those things that can he determined within the realm
of thought, and those that lie beyond the range of thoughts and mental
pictures. You can reach movements but not forces with your mental activity.
Forces you have to measure in the outer world. The fact that when two pulls
come into play — the one from a to c, the other from a to d, — the thing is
actually pulled from a to b according to the Parallelogram of Forces, this
you cannot make sure of in any other way than by an outer experiment. There
is no proof by dint of thought, as for the Parallelogram of Movements. It
must be measured and ascertained externally. Thus in conclusion we may say:
while we derive the parallelogram of movements by pure reasoning, the
parallelogram of forces must be derived empirically, by dint of outer
experience. Distinguishing the parallelogram of movements and that of
forces, you have the difference — clear and keen — between Phoronomy and
Mechanics, or Kinematics and Mechanics. Mechanics has to do with forces, no
mere movements; it is already a Natural Science. Mechanics is concerned with
the way forces work in space and time. Arithmetic, Geometry and Kinematics
are not yet Natural Sciences in the proper sense. To reach the first of the
Natural Sciences, which is Mechanics, we have to go beyond the life of ideas
and mental pictures.

Even at this stage our contemporaries fail to think clearly enough. I will
explain by an example, how great is the leap from kinematics into mechanics.
The kinematical phenomena can still take place entirely within a space of
our own thinking; mechanical phenomena on the other hand must first be tried
and tested by us in the outer world. Our scientists however do not envisage
the distinction clearly. They always tend rather to confuse what can still
be seen in purely mathematical ways, and what involves realities of the
outer world. What, in effect, must be there, before we can speak of a
parallelogram of forces? So long as we are only speaking of the
parallelogram of movements, no actual body need be there; we need only have
one in our thought. For the parallelogram of forces on the other hand there
must be a mass — a mass, that possesses weight among other things. This you
must not forget. There must be a mass at the point a, to begin with. Now we
may well feel driven to enquire: What then is a mass? What is it really? And
we shall have to admit: Here we already get stuck! The moment we take leave
of things which we can settle purely in the world of thought so that they
then hold good in outer Nature, we get into difficult and uncertain regions.
You are of course aware how scientists proceed. Equipped with arithmetic,
geometry and kinematics, to which they also add a little dose of mechanics,
they try to work out a mechanics of molecules and atoms; for they imagine
what is called matter to be thus sub-divided, In terms of this molecular
mechanics they then try to conceive the phenomena of Nature, which, in the
form in which they first present themselves, they regard as our own
subjective experience.

We take hold of a warm object, for example. The scientist will tell us: What
you are calling the heat or warmth is the effect on your own nerves.
Objectively, there is the movement of molecules and atoms. These you can
study, after the laws of mechanics. So then they study the laws of
mechanics, of atoms and molecules; indeed, for a long time they imagined
that by so doing they would at last contrive to explain all the phenomena of
Nature. Today, of course, this hope is rather shaken. But even if we do
press forward to the atom with our thinking, even then we shall have to ask
— and seek the answer by experiment — How are the forces in the atom? How
does the mass reveal itself in its effects, — how does it work? And if you
put this question, you must ask again: How will you recognize it? You can
only recognize the mass by its effects.

The customary way is to recognize the smallest unit bearer of mechanical
force by its effect, in answering this question: If such a particle brings
another minute particle — say, a minute particle of matter weighing one
gramme — into movement, there must he some force proceeding from the matter
in the one, which brings the other into movement. If then the given mass
brings the other mass, weighing one gramme, into movement in such a way that
the latter goes a centimetre a second faster in each successive second, the
former mass will have exerted a certain force. This force we are accustomed
to regard as a kind of universal unit. If we are then able to say of some
force that it is so many times greater than the force needed to make a
gramme go a centimetre a second quicker every second, we know the ratio
between the force in question and the chosen universal unit. If we express
it as a weight, it is 0.001019 grammes' weight. Indeed, to express what this
kind of force involves, we must have recourse to the balance — the
weighing-machine. The unit force is equivalent to the downward thrust that
comes into play when 0.001019 grammes are being weighed. So then I have to
express myself in terms of something very outwardly real if I want to
approach what is called “mass” in this Universe. Howsoever I may think it
out, I can only express the concept “mass” by introducing what I get to know
in quite external ways, namely a weight. In the last resort, it is by a
weight that I express the mass, and even if I then go on to atomize it, I
still express it by a weight.

I have reminded you of all this, in order clearly to describe the point at
which we pass, from what can still be determined “a priori”, into the realm
of real Nature. We need to be very clear on this point. The truths of
arithmetic, geometry and kinematics, — these we undoubtedly determine apart
from external Nature. But we must also be clear, to what extent these truths
are applicable to that which meets us, in effect, from quite another side —
and, to begin with, in mechanics. Not till we get to mechanics, have we the
content of what we call “phenomenon of Nature”.

All this was clear to Goethe. Only where we pass on from kinematics to
mechanics can we begin to speak at all of natural phenomena. Aware as he was
of this, he knew what is the only possible relation of Mathematics to
Natural Science, though Mathematics be ever so idolized even for this domain
of knowledge.

To bring this home, I will adduce one more example. Even as we may think of
the unit element, for the effects of Force in Nature, as a minute atom-like
body which would be able to impart an acceleration of a centimetre per
second per second to a gramme-weight, so too with every manifestation of
Force, we shall be able to say that the force proceeds from one direction
and works towards another. Thus we may well grow accustomed — for all the
workings of Nature — always to look for the points from which the forces
proceed. Precisely this has grown habitual, nay dominant, in Science. Indeed
in many instances we really find it so. There are whole fields of phenomena
which we can thus refer to the points from which the forces, dominating the
phenomena, proceed. We therefore call such forces “centric forces”, inasmuch
as they always issue from point-centres. It is indeed right to think of
centric forces wherever we can find so many single points from which quite
definite forces, dominating a given field of phenomena, proceed. Now need
the forces always come into play. It may well be that the point-centre in
question only bears in it the possibility, the potentiality as it were, for
such a play of forces to arise, whereas the forces do not actually come into
play until the requisite conditions are fulfilled in the surrounding sphere.
We shall have instances of this during the next few days. It is as though
forces were concentrated at the points in question, — forces however that
are not yet in action. Only when we bring about the necessary conditions,
will they call forth actual phenomena in their surroundings. Yet we must
recognize that in such point or space forces are concentrated, able
potentially to work on their environment.

This in effect is what we always look for, when speaking of the World in
terms of Physics. All physical research amounts to this: we follow up the
centric forces to their centres; we try to find the points from which
effects can issue, For this kind of effect in Nature, we ate obliged to
assume that there are centres, charged as it were with possibilities of
action in certain directions. And we have sundry means of measuring these
possibilities of action; we can express in stated measures, how strongly
such a point or centre has the potentiality of working. Speaking in general
terms, we call the measure of a force thus centred and concentrated a
“potential” or “potential force”. In studying these effects of Nature we
then have to trace the potentials of the centric forces, — so we may
formulate it. We look for centres which we then investigate as sources of
potential forces.

Such, in effect, is the line taken by that school of Science which is at
pains to express everything in mechanical terms. It looks for centric forces
and their potentials. In this respect our need will be to take one essential
step — out into actual Nature — whereby we shall grow fully conscious of the
fact: You cannot possibly understand any phenomenon in which Life plays a
part if you restrict yourself to this method, looking only for the
potentials of centric forces. Say you were studying the play of forces in an
animal or vegetable embryo or germ-cell; with this method you would never
find your way. No doubt it seems an ultimate ideal to the Science of today,
to understand even organic phenomena in terms of potentials, of centric
forces of some kind. It will be the dawn of a new world-conception in this
realm when it is recognized that the thing cannot be done in this way,
Phenomena in which Life is working can never be understood in terms of
centric forces. Why, in effect, — why not? Diagrammatically, let us here
imagine that we are setting out to study transient, living phenomena of
Nature in terms of Physics. We look for centres, — to study the potential
effects that may go out from such centres. Suppose we find the effect. If I
now calculate the potentials, say for the three points a, b and c, I find
that a will work thus and thus on A, B and C, or c on A', B' and C'; and so
on. I should thus get a notion of how the integral effects will be, in a
certain sphere, subject to the potentials of such and such centric forces.
Yet in this way I could never explain any process involving Life. In effect,
the forces that are essential to a living thing have no potential; they are
not centric forces. If at a given point d you tried to trace the physical
effects due to the influences of a, b and c, you would indeed be referring
to the effects to centric forces, and you could do so. But if you want to
study the effects of Life you can never do this. For these effects, there
are no centres such as a or b or c. Here you will only take the right
direction with your thinking when you speak thus: Say that at d there is
something alive. I look for the forces to which the life is subject. I shall
not find them in a, nor in b, nor in c, nor when I go still farther out. I
only find them when as it were I go to the very ends of the world — and,
what is more, to the entire circumference at once. Taking my start from d, I
should have to go to the outermost ends of the Universe and imagine forces
to the working inward from the spherical circumference from all sides,
forces which in their interplay unite in d. It is the very opposite of the
centric forces with their potentials. How to calculate a potential for what
works inward from all sides, from the infinitudes of space? In the attempt,
I should have to dismember the forces; one total force would have to be
divided into ever smaller portions. Then I should get nearer and nearer the
edge of the World: — the force would be completely sundered, and so would
all my calculation. Here in effect it is not centric forces; it is cosmic,
universal forces that are at work. Here, calculation ceases.

Once more, you have the leap — the leap, this time, from that in Nature
which is not alive to that which is. In the investigation of Nature we shall
only find our way aright if we know what the leap is from Kinematics to
Mechanics, and again what the leap is from external, inorganic Nature into
those realms that are no longer accessible to calculation, — where every
attempted calculation breaks asunder and every potential is dissolved away.
This second leap will take us from external inorganic Nature into living
Nature, and we must realize that calculation ceases where we want to
understand what is alive.

Now in this explanation I have been neatly dividing all that refers to
potentials and centric forces and on the other hand all that leads out into
the cosmic forces. Yet in the Nature that surrounds us they are not thus
apart. You may put the question: Where can I find an object where only
centric forces work with their potentials, and on the other hand where is
the realm where cosmic forces work, which do not let you calculate
potentials? An answer can indeed be given, and it is such as to reveal the
very great importance of what is here involved. For we may truly say: All
that Man makes by way of machines — all that is pieced together by Man from
elements supplied by Nature — herein we find the purely centric forces
working, working according to their potentials. What is existing in Nature
outside us on the other hand — even in inorganic Nature — can never be
referred exclusively to centric forces. In Nature there is no such thing; it
never works completely in that way: Save in the things made artificially by
Man, the workings of centric forces and cosmic are always flowing together
in their effects. In the whole realm of so-called Nature there is nothing in
the proper sense un-living. The one exception is what Man makes artificially;
man-made machines and mechanical devices.

Rudolf Steiner, Light Course, Lecture 1, Stuttgart, 23rd December 1919.


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