ON
FORCE OF FALLING BODIES
AND
DYNAMICS OF MATTER,
CLASSIFIED WITH PRECISION TO THE MEANING
OF DYNAMICAL TERMS.
BY
JOHN W. NYSTROM, C.E.
PHILADELPHIA:
J. B. LIPPINCOTT & CO.
187 3.




Entered according to Act of Congress, in the year 1872, by
JOHN W. NYSTROM,
In the Office of the Librarian of Congress at Washington.
IIPPINCOTT'S PRESS,
PHILADELPHIA.




FORCE OF FALLING BODIES.
A DISCUSSION on this subject has originated in the Scientijic
American, from a question published in that journal, June 8th,
I872, as follows:
" Force of Fallinfg Bodies.-We have a steam-hammer weighing exactly three tons, including piston and rod; the stroke is four
feet, and the hammer falls by its own gravity. What will be
the force of the blow, making no allowance for friction? What
is the formula for the calculation?
"(Sioned)                                  "J. E."
The following answers were given:
[Frorm tSe Scientfic American, 7uly 6, I872.]
Force of Falling Bodies.
To J. E., query I2, June 8.-The hammer will strike with a
momentum of I60, I64.5472 pounds. The formula is
the square root of (4 X 64.33)-  I6.03I2 velocity.
Then
4.426 X 6ooo X I60312  I60I64.5472.
Or, multiply the fall in feet by 64.33; the square root of the
sum is the velocity; and multiply the weight in pounds by 4.426
and that by the velocity, and you have the momentum.
E. E. W., of W. Va.




4
[From the Scien/ific American, 7uly 13, I872.]
Force of Falling Bodies.
If J. E., query I8, page 385, last volume, will multiply the
weight of any falling body, in pounds, by the height of the fall
in feet, he will have the force of the blow in foot pounds.
Leaving friction out of the question, the force of the blow of
his hammer is precisely equal to the force expended in raising
it, namely, 6000 X 4-  24,000 foot pounds. Converted into
heat, this force would be competent to raise the temperature of
one pound of water a little more than 3 I ~, thus: 24,000 divided
by 772 equals 31.09 units of heat.
W. H. P., of Iowa.
[Fronm the Scientific American, August Io, 1872.]
Force of Falling Bodies.
In view of the difference between the two answers to J. E.,
query I 2, June 8, and of my own ideas, somewhat different from
either, I would say: The striking force of a moving body, in
whatever direction it moves, is its momentum. Its momentum
is the joint result of its quantity of matter and its velocity. The
ratio of this momentum to that of other moving bodies is compounded of the ratio of its quantity of matter, which is indicated by its weight, and of its velocity at the instant in question.
Its momentum, therefore, is not weight any more than it is
space or time, and it cannot be expressed by pounds, in the
ordinary sense of that word, any more than by feet or by
seconds, nor is it expressed by any two of those terms. To
obtain a statement of the momentum of a body for the purpose
of comparison: Multiply its weight by its velocity-its number
of pounds, for instance, by the number of feet it would move
in a second if it should proceed for a second at the rate for the
instant in question. The velocity of a falling body is continually accelerated, and it increases not as the space fallen over
but as the square root ('tqaery-"E) of that space. Therefore,




to multiply the weight by the space fallen over, will not give
the momentum. The velocity of a falling body at the end of
one second of its fall is 321 feet per second, and it has fallen
one-half that distance. It will fall 4:T feet in half a second,
and its velocity is then 8 1 feet in half a second. The velocity
at four feet descent is nearly the same, but more exactly is
I6.0312 feet per second. This multiplied by the weight in
pounds gives the momentum. The general formula is: The
square root of (64.33 multiplied by the distance fallen) -- the
velocity, and the velocity multiplied by the weight - the momentum.  So much for determining the momentum.  The
extent of change produced by the blow of a hammer has a
compound relation to the force of the blow and the ability of
that which it strikes to resist. Some obstacles resist in proportion not only to intrinsic power, but also to the time during,
which they exert their resistance, and their resistance to a blow'
is less as the velocity of the blow is greater. Such are the
different attractive, repulsive, and expansive forces, and such is,
substantially the case where springs are to be bent and where
many forms of cohesion are to be overcome. In such cases,
the change produced is as the weight multiplied by the square
of the velocity, and in case of a falling body is as the weight
multiplied by the distance fallen. Other resistances are independent of time, and are in proportion to the space over which
the resistance operates. Such is substantially the case of friction.  Here the change is as the momentum  of the blow. It
is so in the case of bodies resisted by the momentum or inertia
of other bodies, or, as in greater or less degree is the case of a
body moving through liquids, of the particles of bodies. The
case of forging with a hanimer presents a compound of both
these kinds of resistance, varying in their proportions with the
nature of materials, degree of heat, and other considerations.
G. M. T.




6
[Fronm the Scienltfc Antercan, Auggst Io, I872.]
Force of Falling Bodies.
TO THE EDITOR OF THE SCIENTIFIC AMERICAN:
Since you are publishing a series of articles on " Weight,
Pressure, Power, Force," etc., it would be useful to so explain
the acting force of a body in motion, its momentum or striking
force, that, if such a thing be possible, your readers may
understand by what means, by what it is measured, and how
determined.
While this is one of the simplest problems in physics, as well
as one of the most essentially practical, it is one of those of
which the majority of the people are most profoundly ignorant,
as is shown by the frequent questions on the subject in your
valuable paper, and by the replies, no two of which are alike
and which indicate that the correspondents are hopelessly befogged.
In your number of July 6, page IO, a correspondent-misled
by Haswell, probably-estimates the force of the hammer,
weighing three tons and falling four feet, at over I60,0oo
pounds. But what does he mean? What is a pound of force?
To what is it equal? What work will it do? He does not say
foot pounds, and if he means that, he is wide of the mark in his
estimate. A blow cannot be compared with weight or pressure
alone.
It should be universally known, if possible, that force is estimated by the measure of the work it is competent to perform,
the number of pounds it will raise one foot high. The force
which will lift one pound one foot is called a foot pound,
and is the unit used to express the amount of a force. Gravitation, being a constant quantity, is a convenient standard, and
force measured by the amount of gravitation it will overcome
affords a statement quite intelligible to any intelligent person.
Next, it should be known that this same one pound, in falling
freely one foot, will accumulate the same amount of force, that
is, gravity will impart to it in its descent the same amount of
force which it took from it in its ascent, and therefore the force
of the blow will be just one foot pound; and, if converted into




7
heat, would produce exactly the amount of heat which would
be required to lift the one pound one foot high again.
In general, the force with which any falling body will strike
is precisely the amount required to lift the same body to the
height from which it fell. When, therefore, the weight and
height are given, their product is the force of the blow in foot
pounds, and in the case of this hammer, would be 6,ooo X 4
24,000 foot pounds. The force of a " weight of one pound
falling two feet" would be I X 2-2 foot pounds, while Haswell's "Engineer's and Mechanic's Pocket Book," page 4I9,
gives it at I 1.34 pounds, whatever that may mean.
If the velocity is given, we find the height as follows: Dividing the velocity by 321 (the velocity acquired in each second)
gives the time of fall in seconds, and multiplying the square of
the time by I6-j1, we have the height from which the body
must have fallen to acquire the given velocity, which, of course,
is also the height to which the body would ascend, if projected
upward with the same initial velocity before its force would
be expended in overcoming gravitation. Obviously, the force
of the blow will be the same, with the same velocity, whether
the motion be downward, upward, or horizontal; hence, to find
the force with which it is moving, we only require to find the
height from which a body must fall to acquire the given velocity, and said height, multiplied by the weight, gives the striking force in foot pounds, or the amount of work the body would
perform, the resistance it would overcome, the weight it would
lift one foot, or the heat it would produce; and also, what is
the same thing, we have the amount of force expended in imparting to the body the given velocity.
The general confusion of ideas upon this subject is probably
largely due to the fact that the text-books differ widely, and
the majority of them are entirely wrong, as they almost all
teach that the striking force is proportional to the velocity,
whereas it is, in fact, proportional to the square of the velocity,
as is readily shown by the law of falling bodies enunciated in
the very same books.
The formula above given is far more simple than the various
arbitrary and fantastic ones so often presented by your correspondents, and has the peculiarity of being correct, and con



8
sequently consistent with all the laws of motion; and if you
will give me space for a few examples, I believe its application
will be perfectly plain to your readers. Instead of dividing the
velocity by 32.I6 and multiplying the square of the quotient by
I6.o8, we may, of course, obtain the same result by the shorter
process of dividing the velocity by 8.02, and squaring the
quotient.
I. A one pound ball moves IOOO feet per second; (IOOO -
8.02)2-= I5,545. Its force then is I5,545 foot pounds, and as it
weighs one pound, if its motion were directly upward it would
mount to the height of I5,545 feet, and on returning would
acquire in its descent the same velocity of IOOO feet. The force
expended, then, in imparting this velocity was equivalent to
that required to raise i5,545 pounds one foot.
2. A  twenty-four pound ball has a velocity of 50 feet
per second; (50 - 8.02)2 X 24- 931.44 foot pounds.  If this
twenty-four pound weight were a hammer with a stroke of
38.81 feet, it would acquire a velocity of 50 feet, and would strike
with a force of 38.8i X 24=93I.44 foot pounds, and this
amount of force, in any available form or mode of manifestation,
would be sufficient to impart a velocity of 50 feet to a mass of
24 pounds, or to lift 24 pounds 38.8  feet, or to lift or throw one
pound 931.44 feet high, or 931.44 pounds one foot high. In
these calculations, there is no allowance made for atmospheric
resistance.
W. H. PRATT.
DAVENPORT, IOWA.
Neither of those four communications answers the question
of Mr. J. E., which is, What will be the force of the blow?
They state, how many foot-pounds, how much work, and
how much momentum there is in the hammer, neither of which
is force, as is required by the question. As I considered it difficult for the inquirer to make out anything from those answers,
I volunteered to answer the question in my own way, as
follows:




9
[From the Scientific American, August 31, I872.]
(otrmpoaldfrr~.
The Editors are not responsiblefor the olinions expressed by their Correspondents.
Force of Falling Bodies.
To THE EDITOR OF THE SCIENTIFIC AMERICAN:
The question "With what force does a falling body strike?"
has been frequently repeated in the Scientific American for the
last twenty-five years, and has generally been answered by the
batch of dynamical terms used in colleges and styled " scientific." The answers have invariably made the problem more
obscure. Each one generally says that " the problem is very
simple," and he pretends to understand the subject perfectly. I
am one of those pretenders, and propose to answer the question
in my own way, reference being made to the accompanying
figu re.
Let us assume the case of driving a nail into a piece of wood
by the aid of a lever whose fulcrum is at C. The applied force
is represented by the weight W, acting on the lever L.  Let
R denote the force of resistance in the wood, expressed by the
same unit of weight as that of W, say pounds.
The weight W, acts on the long lever L, and the resistance
R, on the short lever 1. Then




WZL
R: W=-L: /, and R=That is to say, the force of resistance in the wood is to the
weight or force W, as the long lever L, is to the short lever 1.
Let S represent the vertical height which the weight WV4
moved, and s the distance which the nail was driven into the
wood. Then
R: W=S: s, and R-W
That is to say: the force of resistance in the wood is to the
force or weight /V, as the height S, is to the distance s.
Now let the same weight W, fall from an equal height S,
directly upon the head of the nail, and the latter will be driven
into the wood the same distance as by the aid of the lever.
Therefore: the force with which the falling body acted upon
the nail is to the weight of the falling body as the height of
fall is to the distance the nail is driven into the wood. The
force of the falling body is equal to its weight multiplied by its
height of fall, and the product divided by the distance which
the nail is driven into the wood.
JOHN W. NYSTROM.
PHILADELPHIA, PA.
My explanation on force of falling bodies was not well appreciated by an employee of the Scientific Amevrcan, Dr. Vander
Weyde, who made the following remarks on the subject:
[From tIe Scienific American, September 14, I872.]
The Editors are not responsiblefor the opinions expressed by their Correstondents.
Force of Falling Bodies.
To THE EDITOR OF THE SCIENTIFIC AMERICAN:
I see, in an article on page I31 of your paper, that Mr. John
W. Nystrom acknowledges himself to be " one of those pretenders" who think that they " understand perfectly the subject" of measuring the force of a falling body by taking, as




II
unit of measurement, the mere weight of matter without motion.
I desire here to say to him and to all those interested in the
important question of measuring forces, in consideration that
force must be distinguished from mere weight, that weight is
merely a measure for an amount of matter, for a mere mass, and
for nothing else; such weight, of course, is caused by gravitation, and thus can exert pressure, but as long as the weight
does not produce motion, there is no force generated; therefore
strictly speaking gravitation is no force, notwithstanding the
conventional way of speaking of the force of gravitation; however, gravitation can beget force, and only does so in case it is
allowed to produce motion. According to the modern conception of force, it is not something immaterial, independent of
matter, but absolutely nothing but matter in motion. This
motion may be hidden, molecular, when the force manifests
itself as heat, electricity, etc., or the motion may be in the
masses, when the force is directly measurable by two elements,
the mass and the velocity. Accepting the customary symbols
for these two different elements, the different degrees of force
are expressed by the formulae v X in and v2 X in, which are both
correct according to circumstances. In the case of the effect
of a blow produced by a falling body, for instance, the driving
in of a nail, the identical case represented on page I3I, the
latter formula, corresponding with the theory of the vis viva
(see any text-book on mechanics), must be applied. This is the
first point in which the formulae of Mr. Nystrom are faulty, as
they are based on the lever, and thus not on the square of the
velocity or space, but on the simple velocity: v X in.
The result of this law of the vis viva is that, where gravitation
increases or decreases, and with it the velocity of the falling
body, the force of the blow will increase or decrease as the
square of the gravitation, while the weight of the body will
only increase or decrease in the simple ratio of the gravitation.
Mr. Nystrom's figure and formulae fail to take any account of
this whatsoever.
But let us consider the expressions v X en and v2 X oZ theoretically. It is evident that they have no value at all as soon as
either of the quantities v or m becomes immeasurably small or
disappears.  Let, for instance, in the function v X in or v2 X vZ,




12
n become = o; then we have v X o  o and v2 X O  o, which
conform to practical experience, because a blow with a mass
equivalent to nothing must necessarily amount to nothing.
Let, inversely, v be = o, and we have
o x m- o and o2 X  - o,
again equivalent to nothing; a mathematical proof that a mere
mass without velocity (motion) cannot possibly be reckoned
equivalent to any force; and we see here the great mistake,
thus far made by the authors of many text-books, in speaking
of a force of, say, Ioo pounds, or a ton.
The cause of this error is mainly to be found in the fact that
a mere weight by its pressure will in some cases produce results similar to that of a force or blow. If, however, we attempt
to measure force (matter in motion) by mere weight (matter in
rest), we must continually fail and obtain incongruous results,
as they are two incomparable quantities. This confounding of
an actual force produced by a moving mass with mere weight
or pressure produced by a stationary body, is the cause of fifty
per cent. of the attempts, continually being made by the half
educated, to find perpetual motion.
Now for a few practical illustrations: With a comparatively
light hammer, we may easily drive a nail into a brick wall; if
we try to do it by mere pressure, we shall crush the nail, or, to
take Mr. Nystrom's own illustration, we can drive a nail into a
board by the blow produced by dropping the head of a hammer
on it from a suitable height, directed by guiding pieces, as in a
pile driver; but take a similar nail, place it on the same board,
attach the lever of proper length and hang the hammer head at
the end of the lever, following practically the figure on page
I3I, and see if the nail will penetrate at all. If Mr. Nystrom
had tried the experiment, he surely would never have taken the
trouble to illustrate and publish his explanation.
The blow or percussion gives to a mass a shock, transmitted
through it with the same velocity as a wave of sound would
travel in that same mass; when the blow is violent and there is
somewhere a want of continuity, or lack of strength, which prevents the wave from pursuing its course, its power will be expended there in crushing the material. This is the case in
driving a nail. The motion will not be communicated to the




I3
board, but the force will be expended in crushing and cutting
the fibres of the wood under the nail, so as to allow it to enter,
while a weight or pressure placed on the nail will have plenty
of time to communicate itself to the whole board. A striking
illustration of this may be had when balancing a heavy board
on its centre; it is then possible to drive a well-pointed nail
with a smart blow deep in the board without moving the latter,
while the same nail with a weight on top will scarcely make a
mark on its surface, but will move the whole board. A pistol
ball may be fired through a door without moving it on its
hinges, which latter may be done by the slightest pressure of
the finger. Scores of other familiar examples may be adduced,
all proving the immense difference between force and mere
pressure, and it is only to be wondered at and at the same time
deplored that still so much confusion prevails in regard to this
all-important subject.
P. H. VANDER WEYDE.
NEW YORK CITY.
TIhe inquirer Mr. J. E., and other readers of the Scientific
American, will no doubt be well informed on force of falling
bodies, after having read Dr. Vander Weyde's philosophy on
that subject. When I read the article I could not credit the
possibility that Dr. Vander Weyde has ever graduated as
doctor of philosophy in any creditable college; which, however, would make no difference to me if he has or not, as long
as he keeps himself within his due limits. When an individual
bears the title of Doctor or Professor we expect him to issue
authority; but when he imposes upon the public with quack
philosophy, as Dr. Vander Weyde has done, we have reason
to suspect that there may be something wrong about his title,
and we are justified in asking the doctor to show his diploma,
and in holding him responsible for such offences. With this
motive I addressed the editor of the Scientzfic American on the
subject, and received the following answer:




I4
[From the Scientific Anterican, Seltember 28, 1872.]
Who is Dr. Vander Weyde?
During the past few weeks, an esteemed correspondent, J.
W. Nystrom, Esq., C. E., of Philadelphia, has furnished to our
readers several interesting communications, some of which
have been answered and criticised by another of our valued
correspondents, Dr. P. H. Vander Weyde, of this city. From
the tenor of the following letter it would seem that our Philadelphia correspondent is a little suspicious of the respectability
of his antagonist. But we can assure him that, in Dr. Vander
Weyde, he has a foeman worthy of his lance.
TO THE EDITOR OF THE SCIENTIFIC AMERICAN:
SIR:-Will you be kind enough to inform me, through the
Scientific American, if Mr. P. H. Vander Weyde, of New
York, has a doctor's diploma, and if so, from which college he
has received that title? And what kind of a doctor is he?
The answer to these questions will greatly oblige yours very
respectfully,
JOHN W. NYSTROM.
Ioo1 SPRUCE STREET, PHILADELPHIA, September 7, 1872.
We would inform our correspondent that Dr. Vander Weyde
is a physician of the strictest orthodox sect; that he is an
honored graduate of the New York University Medical College, of which John W. Draper, LL.D., is President; that he
holds the regular diploma of that institution; that he enjoys
the fellowship and esteem of many of our leading physicians
and prominent men of science; that he is a native of Holland,
where he received a university education; took the degree of
Doctor of Philosophy in I840; was the editor of a scientific
periodical; in I845, at Amsterdam, he received the honorary
prize, consisting of the gold medal of the Society of Sciences,
for his essays upon natural philosophy.
Dr. Vander Weyde is now a citizen of the United States.
From 1859 to I864, he was Professor of Physics, Higher Mathematics and Mechanics at Cooper Institute in this city. During nearly the same period, he was also Professor of Chemistry
in the New York Medical College. From I864 to 1866 he was




I5
Professor of Industrial Science in Girard College, Philadelphia,
Pa. His contributions to the scientific literature of the day
have been very extensive, and are widely known.
These are only a few of the items of Dr. Vander Weyde's
public record. But they are sufficient, we trust, to satisfy the
inquiries of our correspondent, and remove from his mind any
adverse prejudices that he may have formed concerning the
qualifications of the distinguished gentleman whose public
standing he has questioned.
Upon the authority of the Scientzfic American it appears that
the said Vander Weyde has really graduated as doctor of philosophy in some college or university in Holland, which, however, I shall not feel convinced of until I have seen his diploma,
or am informed of the fact direct from that university.
Holland stands very high in sciences, in fact, on the level
with other nations, and its colleges are very strict in their
studies and examinations, in which it is doubtful whether Dr.
Vander Weyde's philosophy on dynamics of matter could have
passed for good.
I cannot enter into any discussions in the Scientz)fc American
with such a philosopher as Dr. Vander Weyde has proved
himself to be, for it is the duty of the university in Holland
from which he bears his diploma (if he has any such) to have
instructed him in the sciences in which he has proven himself
deficient.
As soon as I can find out the name of that university, I shall
endeavor to have the subject attended to, and if that university
cannot endorse and sustain its doctor's philosophy, he ought
to be called back and made to study over again, or be requested
to return his doctor's diploma.
I have learned that he has been professor of natural philosophy in Girard College, Philadelphia; and if he has advanced
such philosophy there as that he has done on force of falling
bodies, his students are simply deceived.
If Dr. Vander Weyde had limited himself within the confusion which really exists in the want of precision to the meaning




of dynamical terms, I should have had forbearance with him;
but he dictates a doctrine which is contrary to established facts
in physics.
It may be considered that I am rather severe upon Dr. V.
W., but there are so many of this kind of philosophers that
it will do no harm to tell the truth to one of them every now
and then; and what has been said about Dr. V. W. is equally
applicable to all the rest of the high authorities who have invariably attacked me in the style of quack der Weyde. When
my ideas differ from what is written in their books, they
blindly suppose that I am wrong, and they attack me with
irrelevant philosophy, by which the public has been juggled,
and I have been embarrassed all my life by quack opinions
of high authorities. In many cases I have doubted whether the
high authorities have themselves believed in their own statements, but they evidently expected me to have faith in their
profound reasonings of perfect nonsense.
The object of this writing is, however, not to attack Dr. V.
W., nor to prove that I am right and that he is wrong, all of which
is of secondary importance to me and to the public; but my
principal object is to call the serious attention of scientific institutions to the confusion in dynamics, and to the annoyance
which that confusion causes to the public, and between
individuals.
For the last ten years I have had repeated discussions on the
confusion of dynamical terms, in which my opponents have not
been able to sustain themselves. Some of these discussions
are published in the 7ouzrnal of the Franklin Institute, and also
in the Scientific American for the year 1865. The high authorities generally maintain that work is independent of time!!!
which is contrary to the opinion of manufacturers in regard to
the eight or ten hours' struggle for a day's work.
We have no good text-books on dynamics, and, even in colleges, the subject is confused, not so much in substance as in
terms, and the result is that graduated students, even from the
same college, differ as to which is which in dynamics. The confusion is, however, acknowledged by high authorities, who
have not been able to sustain themselves against me in open
contests, and, when such authorities are professors in colleges,




17
the public cannot thus receive proper information, and the subject remains obscure, as proven in the case before us. Under
such circumstances I have a right to claim the issue of authority, which is as follows:
CLASSIFICATION  OF  DYNAMICAL  TERMS.
I do not acknowledge the existence of more than one kind
of force in physics, and that is, that action which can be expressed simply by weight, without regard to motion, time,
power, or work. Force is derived from a great variety of
sources; but when it is simply force, it can always be expressed by weight.
Force, motion, and time are simple physical elements. Space,
power, and work are functions of those elements.
Elem/eents.                 Fu nctionis.
Force      F.... I.Space S-VT.... 4.
Motion.... 2. Power P FV.... 5.
Time...  3. Work K= FVT..    6.
The weight of a body is the force of attraction between that
body and the earth; and as the force of attraction varies inversely as the square of the distance between the centres of the
attracting bodies, the weight of a defined body is not a
constant quantity.
Mass means the real quantity of matter in a defined body,
and it is a constant quantity which cannot be increased or
diminished by force. Mass is generally denoted by the letter
M, and is one of the four elements which constitute the dynamics of matter.
The relation between these four elements is well known and
accepted as an established fact, namely,
M: FT: V.....                 7.
This is the fundamental analogy in the dynamics of matter,
of which the two functions,
Momentum of motion, ffMV- FT, momentum of time, 8.
The function MV has been termed force, which is a great
error, for it only denotes the product of the force and time consumed in giving the mass Mthe velocity X, and which is equal
2




to the product of another force and time required in bringing
the mass from motion to rest. There is no relation between
the two forces, which are entirely governed by their respective
times of action.
The function 6 expresses the
Work K- FVT.6.
Multiply the momentums function 8, by the velocity V; and
we have the
Work K-= MV2 =  FVT...
We see here that the function MV2 means work, but it has
been termed force and vis viva, which are also erroneous.
MV2 only denotes the product of force, motion, and time,
which is the work consumed in giving the mass Mthe velocity
V, and which is equal to the product of another force, motion,
and time, or the work required or executed in bringing the
moving mass to rest.
The function 4 expresses the
Space S= VT.4,
which, inserted in function 9, will be the
Work K- MV2: FS.                  Io,
in which the function MV2 denotes the product of the force
and space in which the mass M attained the velocity T, and
which is equal to the product of another force and space in
which the moving body is brought to rest, and which is the
same as the primitive function FVT.
None of these functions should be called force.
The present confusion in dynamics consists in that the functions are called force, for which we can make neither head nor
tail of it.
Dynamics is now in the same condition as geometry would
be if there was no distinction between line, surface, and solid,
and it will remain in that confusion as long as the colleges
teach that MV and MV2 are forces.
The erroneous idea that MVand MV2 are forces has stuck
so tightly into the heads of philosophers as to cause an epidemic
in dynamics.
Dr. Vander Weyde says: "Accepting the customary symbols for these two different elements, the different degrees of
force are expressed by the formulae VX M and V2 X M, which




I9
are both correct according to circumstances."  I would ask
Dr. V. W. if the circumstances depend upon the weather.
The Scientific Amenrican of the 22d of June, I872, gives the
following ideas of dynamics:
Weight, Pressure, Force, Power, Work.
The fact that the above words are often confounded together,
for the simple reason that their true meaning is not well understood, has been the cause of many fruitless attempts at
mechanical inventions and improvements. Most searchers for
perpetual motion make no distinction between pressure and
force, and are under the delusion that mere pressure can produce work, and we have seen writers on mechanics and we
have even heard lecturers on scientific subjects speak of a force
of, say, two tons weight. Weight alone is not force, neither
is pressure equivalent to work; and it may therefore be useful
to attempt some clear definitions of the above terms, in order
to protect inventive minds against mistakes in mechanical
reasoning.
Weight is simply the measure of an amount of matter referred to a certain standard accepted as a unit. This unit may
be a gramme, a pound, a ton, or our whole earth, which the astronomers use; but, in either case, it conveys to the mind
nothing but the conception of an inert mass, or a certain
amount of matter, for the determination of which gravitation
gives us the means of measuring and comparing. Therefore
we may say: To have "a mass of two tons," but not "a force
of two tons."
Pressure is a result of this gravitation, and a mass of two
tons will exert a pressure of two tons; in this way we may
estimate the effect of a spring, hydraulic press, or other similar
contrivance, by saying its pressure (not its power) is equal to
two tons, meaning thereby that it has the effect, on the material to be pressed, as if two tons weight were placed upon it;
but we have in pressure neither force nor power. These con



20
ceptions of the latter require other elements, as we shall soon
see.
Force is matter in motion, nothing more, nothing less; the
abstract idea of force without matter is a nonentity. All the
modern discoveries in science tend to prove this more and
more plainly. Without matter, force would have no existence,
but it may be hidden in matter as molecular invisible motion
in the form of heat, electricity, etc. The steam engine, electromagnetic engine, etc., are there to prove how this molecular
motion, or hidden force, may be changed into visible force
or motion of matter. Inversely, the caloric friction machine
changes motion into heat; the ordinary and also the Holtz
electric machine change motion into electricity. In any case,
we are driven to the conclusion that all force proceeds from
motion of matter, and is finally resolved into motion of matter,
either of masses, or into molecular motion, generating one of
the so-called imponderable forces.
Chemistry has proved since the last century that the amount
of matter in the universe is a constant invariable quantity, and
that we cannot create or destroy a single material atom, but
can only change its form solid to liquid, or gaseous, or vice
vcrsa. So the modern philosophy of mechanics proves that'
the amount of force (that is, motion of matter) in the universe
is a constant quantity, and that we cannot create or destroy
the slightest amount of this force, but can only change it from
mass motion to molecular motion, that is, heat, electricity, etc.,
or vice versa'.
The measure of force is thus the product of the mass with
the distance through which it moves; and as the unit of measure of ordinary masses is the pound, and of distances, the
foot, we have adopted the foot-pound as the standard unit of
force, meaning " one pound ltzjrd against gravitation one foot,"
not " one pound moved one foot," as we have seen and heard
it stated, which of course gave rise to the most absurd calculations in regard to the immense power obtained to drive a
steamship or railroad train.
If one pound weight is raised one foot, one unit of force is
expended; if, inversely, we cause one pound to descend one
foot, we obtain a unit of force back, and may transform this




21
into other mass motion, or into molecular motion. We may
cause this mass of one pound to be raised slowly if we have
little power to apply, or rapidly if we have greater power; and,
inversely, we may cause it to descend slowly, as is done in the
weight of a clock, and spend itself gradually during a long
period of time, producing slight effects throughout that time;
or we may cause it to descend quickly, as is the case with the
blow of a hammer, and spend itself during a very short period
of time, almost instantaneous, producing a powerful effect for
that short time. So the driving in of a nail, which often the
pressure of a ton weight would not accomplish, the blow of a
hammer of one pound, lasting a small fraction of a second,
will accomplish easily. This remark points out forcibly the
difference between the weight of masses at rest and of masses
in motion; in other words, the immense difference between
mere pressure and force.
We may understand from the Scientific American that modern discoverers have undiscovered the discoveries of Sir Isaac
Newton.
The law of force of universal attraction, which was established by Newton, is as follows: The force of attraction between
any two masses is proportionate to the product of the masses, and
inversely as the square of their distance apart.
There is no motion in this definition of force, and it makes
no difference whether either one or both the masses are at rest
or in motion, the force will always hold good with the definition. Call F- force of attraction, between the masses /and
z, and D  - their distance apart.
Then F= i... II.
ED2      m..... I2.
F: M-I:' Di..       I13.
Call V= the mean velocity in the time T, in which the
masses Mand im would be drawn together by their own attraction, and we have D -  VT and D2 _=  PT2.
M in
The work K= FVT=                     I4.
VT




22
The object of this treatise does not involve the development of this very interesting subject, which must be deferred to
a more appropriate occasion.
In the application of the physical elements to practice, we
must assume units of measures for each of them, namely:
F- force in pounds, acting on the mass M.
V- velocity in feet per second.
T- time in seconds, in which the force Facts on the mass
M, and produces the velocity VF
W — weight in pounds of a body, or of the mass MA
When the body falls freely under the action of gravity, then
F - W.
M=  mass, which is proportionate to weight when compared
in one or the same locality.
The mass of a body is said to be its weight divided by the
acceleratrix g, or 32.I7, for which the unit of mass should be a
quantity of matter weighing 32.I7 pounds.
There is no name adopted for this unit of mass, the want of
which makes the subject obscure.  If we are told that the
weight of a body is 20, we naturally ask what twenty? and we
cannot conceive the magnitude of the weight until we know
the name and magnitude of its unit. Although we may know
that thirty-two pounds of matter is a unit of mass, it does not
make the clear impression as if the unit had a name, for which
reason I would propose to christen this unit with the name
mtat., from the word matter.
Then, the unit mat. means a quantity of matter weighing
32. I 7 pounds, and there will be 69.63 mats in a ton weight of 2240
pounds. Now, we can apply our reasonings to practice.
The actual force Fin pounds of attraction between any two
masses Al and m, expressed in mats, and D - their distance
apart in feet, will be closely approximated by the formulaF      M                            I5
5274460D. 
ILLUSTRATION OF DYNAMICS OF MATTER.
Let a force F be applied on a mass Al at rest, but free to
move; the mass will then be set in motion with an accelerated
velocity as long as the force acts. If the force Fis constant,




23
the acceleration of velocity will also be constant, and the
velocity V, attained in the time T, will be from the fundamental
analogy of dynamics of matter 7.
FT
V —T...       I6.
That is to say, the force Facting upon the mass M, in the
time T, will generate the velocity V.
Exalnple. —A force F — 8 pounds, is acting upon the mass
M_  4 mats, for a time of T- 3 seconds.  Required the
velocity V.
8 X 3- 6 feet per second.
4
In the functions 4, 5, and 6, V means the mean velocity in
the time Tor space S, and when V means the actual velocity
of a moving body accelerated from rest to;, or retarded from
Vto rest, the mean velocity will be    V.. Then we have the
real work expended on the moving mass, or generated in
bringing the mass to rest, to be,
K- MV2  FVT
2     2
VT
The space S- —, which inserted in formula 17 will be the
2V/Fz
Work             FS...   8.
2
Referring again to the illustration, the force Facted on the
mass Monly in the space S, after which the body will continue
with uniform velocity and generate the space VT, when another
force F', independent of F, is applied in opposite direction to
stop the motion of the mass M, which is accomplished in the
space s, independent of the first space S, but the work consumed or generated will be alike in both cases, namely,
MV2
2
Of which F: F'- s: S.        20.




24
In the case of a falling body, the force F is equal to the
weight of the body, and when it strikes a blow, the mean force
of resistance is F', acting in the space s, but as the force F or
weight of the body is constant even through the space s, the
falling body is acted upon by two opposite forces Fand F', for
which the space S of the fall must also include the space s of
the blow.
Then we have the force of a falling body equal to its weight
multiplied by the whole height of fall, and the product divided
by the space of the blow.
When the force of resistance is irregular, the mean force will
still be as the above rule.
In the case of the problem of driving a nail into a piece of
wood, as before illustrated, whether accomplished by the aid of
a lever or by the weight falling the same space directly upon
the nail, the rule will hold good in both cases.
In the case of a steam hammer weighing WV  3 tons, and
falling S: 4 feet, including the compression of the iron by the
blow, say, s = o.I of a foot, required the force of the blow.
F =__ WS  _ 3      X  4    120 tons.
S     O.I
This is the answer to the original question of Mr. J. E.
The force of resistance F' is not uniform in the space s of
compression of the iron forged. It is generally smallest when
the hammer first touches the iron, and greatest at the moment
the hammer stops, but the mean force throughout the compression s, will be according to the formula.
The Scielzzf'c American attacked me on the subject of dynamics some seven years ago, as follows:
[From the Scientific American, 7une 29, 1865.]1
Work and Power.
In the pages of the _7ournal of the Franklin Institute, a discussion is going on between De Volson Wood, Professor of
Civil Engineering in the University of Michigan, and J. W.
Nystrom, Acting Chief Engineer, U. S. N., on the subject of
work, force and power. The main purpose of Mr. Nystrom
seems to be to deny the position that work is independent of




25
time, and he succeeds in involving the question in considerable
confusion. The facts of the case are simple and plain enough.
Work is the overcoming of mechanical resistance of any
kind, either by raising a weight, dragging a body along, turning off a shaving, or by any other action. The question
whether it is independent of the time depends entirely upon
the meaning of the language employed. A foot-pound of
work is the raising of one pound of matter one foot in vertical
height, and this foot-pound is precisely the same quantity
whether one second or one thousand years be consumed in
the operation.
We may say that a machine is doing the work of raising
one foot at the rate of one inch per second; then the work
done by the machine will depend upon the time that it is in
operation; it will take it twelve seconds to do one foot-pound
of work, and twenty-four seconds to do two foot-pounds.
In this case, however, we have attached to the word "work"
a meaning for which the word "power" is employed by the
standard writers on mechanical philosophy. To keep our ideas
clear, it is better to regard the machine as exerting a power of
one inch-pound per second, and to confine the word " work" to
the aggregate resistance overcome.
One writer argues that 2 and 2 do not always make 4, sometimes making 22. By analogous tricks of language we may
confuse our minds in regard to any problem whatever; but a
more useful aim of discussion is to free our minds from confusion, and to accomplish this one of the most important steps is
to use words always in their exact signification.
Regarding work as the overcoming of physical resistance, it
is plain that the aggregate amount of any given quantity is
independent of the time required for its performance.
There is probably no higher authority on the philosophy of
mechanics than Arthur Morin, and from his "Legons de
MRecanique Pratique," translated by Bennett, we take the following extract:
" The Idea of Work is Independent of Time.-We see from
what precedes that in the measure of work we have only regarded the effort exerted, and the space described in the direction peculiar to this effort. It is, therefore, independent of
time.




26
"Thus, in raising goods the effect is not measured by the
duration of labor, but by the product of the load into the height
of its elevation."
The editor of the Scientjfic American declined at first to publish niy reply to the above criticism, and he wrote me a letter
advising me to consult some authority on the subject, indicating that my views on dynamics were all visionary; whereupon
I called at the office of the editor and insisted upon it that my
reply should be published, which was at last consented to,
and which is as follows:
[From the Scientific American, September 9, I865.]
Force, Power, and Work.
(For the Scientific American.)
FORCE is a mutual tendency of bodies to attract or repel each
other. Its physical constitution is not yet known. We only
know its action, which is recognized as pressure and measured
by weight. The unit of weight being assumed from the attraction of the earth upon a determined volume of any specific
substance; for example, the force of attraction between the
earth and 27.7 cubic inches of distilled water, at the temperature of 39.8~ Fahr., in an atmosphere balancing 30 inches of
mercury, at the level of the sea, which is called one pound
avoirdupois. Force is the first element of Power and Work,
and can be likened to length, which is a primary element in
geometry. Force will here be denoted by the letter F, expressed in pounds.
VELOCITY is the second element of Powert and Work, and
may be likened to breadth in geometry. It is that continuous
change of position recognized as motion, and is here denoted
by the letter V, expressed in feet per second. Velocity is a
simple element, although it appears to be dependent on time
and space, but the space is divided by the time, and therefore
both relieved from the velocity.
TIME is the third element of work, and may be likened to
thickness in geometry. It implies a continuous action recog



27
nized as duration.  Time is here denoted by the letter T,
expressed in seconds.
POWER is a function of the two first elements-force F, and
velocity V-as area in geometry is a function of length and
breadth. Power is here denoted by P=FV, which means
that the power P, is the product of the force F, multiplied by
the velocity V. The power so obtained is expressed in footpounds, and called dynamic effect, of which there are 550 in
a horse-power; or if the velocity is measured in feet per
minute, there will be 33,000 foot-pounds in a horse-power.
Power is independent of space and time, but it has often been
confounded with work, which essentially depends on time and
space.
SPACE is a function of the second and third elementsvelocity 7, and time T-and may be likened to a cross section of a solid, which is a function of breadth and thickness.
Space is here denoted by S= VT, which means that the space
S, is the product of the velocity Vg and the time T, expressed
in linear feet.
WORK is a function of the three elements,-force F, velocity
V, and time T. It maybe likened to a solid in geometry, which
has the three dimensions,-length, breadth, and thickness.
Work is here denoted by K - FVT, which means that the work
K, is the product obtained by multiplying together the three
elements-force F. velocity V, and time T.
Work may also be denoted by K - FS, or the product of the
force Fmultiplied by the space S, where it appears as if the
work was independent of time, but the time is included in the
space S- VT.
Work may also be denoted by K PT, which means the
power P, multiplied by the time T. Either of the three cases
expresses the work in foot-pounds.
Force, velocity, and time are simple physical elements.
Power, space, and work are functions or products of those
elements.
The Scientific American is read by most mechanics in this
country, and it may be further said that that journal is met with
in most parts of the world. It evinces a habitual and sincere
desire to furnish its readers with correct and instructive articles




28
on scientific subjects, in consideration of which it would be a
neglect of duty on my part to pass over in silence its articles
on "Work and Power," published on page 71 of the present
volume. In that article you say, " The main purpose of Mr.
Nystrom seems to be to deny the position that work is independent of time, and he has succeeded in involving the question in considerable confusion." And you think "the facts of
the case are simple and plain enough." You proceed to give
an antithetical description of what work is, and say, "In this
case, however, we have attached to the word work a meaning for
which the word power is employed by. the standard writers on
mechanical philosophy." Now you will allow me to remark
that, in this expression, you have, together with the standard
writers, confounded work with power. You have thus not followed your own good advice, namely, "to free our minds from
confusion" by taking "most important steps to use words
always in their exact signification." You then go on to say,
" Regarding work as the overcoming of physical resistance, it
is plain that the aggregate amount of any given quantity is
independent of the time required for its performance." Do you
not here convey the idea that work is independent of w/lat it
requires, namely, the time? You evidently mean to say that a
given quantity of work may be performed in any desired length
of time; but you do not seem to conceive that the work is
dependent on whatever time is required for its completion.
Referring to a geometrical figure, you may be able to comprehend the position of your argument about work, which is
substantially this-tlhe cubic content of a pancake is independent
of t1he thickness required to make it zup! You say, " The question whether it (work) is independent of the time depends
entirely upon the meaning of the language employed." To
this I respectfully object, inasmuch as I recognize only one
meaning in the language I have employed; and, indeed, the
entire controversy upon this subject appears to have sprung
from a rejection or misappreciation of the specific meaning I
have struggled to establish to the terms force, power, and work.
But supposing, for the sake of argument, that my generaf language is not sufficiently clear, if you can read my algebraical
formulas you will not misunderstand me. Why, therefore, do




29
you not exemplify your argument upon my formulas and
thereby show its effect in practice? To say that "work is
indeperident of time," is to say that work is dependent of no
time, or that any amount of work may be performed in no
time-a proposition which is not yet realized.
Work, as before stated, is the product of the three elements,
-force, velocity, and time. For a given quantity of work,
either one or two of these elements can vary ad libitum, but
only at the expense of the remaining two or one. Work is
thus not confined to any specific relation or ratio to either of
those elements, but independent of either one of them it ceases
to be work.
I am well aware that the standard authors have to this day
considered work independent of time, and they have also confounded force, power, and work with each other, so that we are
yet thrown upon our individual authority to decide which is
right. Thus, when you and Professor Wood cannot defend
your position, it may be very convenient to assert that I use
"cant phrases" when I exemplify your arguments, and that
"by analogous tricks of language we may confuse our minds
in regard to any problem whatever." I nevertheless trust that
I have given both perspicuity and precision to my language
and meaning in this article, and sincerely hope that it may be
tributary to " the consummation-devoutly to be wished for"of reducing to certainty and system the future reasonings of
the scientific world on this subject.
JOHN W. NYSTROM.
[From the Scientzfic American, September 9, I865.]
Nystrom on Work and Power.
We have a kindly feeling towards Mr. Nystrom, having received from him several valuable contributions. It has seemed
to us, however, that his method of explaining the difference
between work and power was calculated rather to confuse than
to elucidate the subject. In reply, he forwards us a communication containing his explanation, with a request that we would




30
lay it before our readers and let them judge for themselves.
We comply with his request with pleasure, and the communication will be found on another-page.
The raising of one pound of matter one foot in vertical
height is one foot-pound of "work." The raising of 33,000
pounds one foot is 33,000 foot-pounds of work, whether one
minute or one hundred years be consumed in the operation.
The power-either of a steam-engine, waterfall, or animalthat can raise 33,000 pounds one foot in each minute of time
is one-horse power; the power that can raise 33,000 pounds
in half a minute is two-horse power; and the power that can
raise 63,000 pounds in one-tenth of each minute is ten-horse
power.
Morin and other writers, therefore, say that the idea of work
is independent of time, but that time is an element in the
measure of power. It seems to us that these writers are correct. It seems to us, also, that the matter is extremely plain
and simple.
Mr. Nystrom, on the other hand, while accepting, if we understand him, the above illustrations of both work and power,
denies that work is independent of time, or that time is an element of power, and asserts that the subject is not generally
understood even by educated engineers. We have criticised
his arguments upon it as calculated rather to confuse than elucidate it. From this criticism he wishes to appeal to the judgment of our readers,-an appeal in which we cheerfully concur.
I do not doubt the kindly feeling of the editors of the Scientific American towards me, which has been proven in their very
kind advice; and I am convinced that they have no improper
motives in attacking me on dynamics; but the misfortune is
that they do not understand the subject.
In regard to work being dependent or independent of time,
we have only to refer to the operation of a steam-engine of a
definite power. Whatever time the engine is running, say one
hour or one year, the power is constant, independent of time;




3 I
but the work executed by that engine depends entirely upon
the time of operation.
The best joke of all is that the editors of the Scientific American are not ashamed of themselves for undertaking to give me
advice on the subject. I have lately received a letter, signed
Munn & Co., praising Dr. Vander Weyde and advising me to
drop the subject, evidently with the intention of frightening me.
The publication of this pamphlet is the result of their last
advice. I have heretofore never entertained the slightest idea
of giving advice to the editors of the Scientific American, but
now I feel disposed to do so, namely:
I will hereby advise the editors of the Scientific American to
inform themselves well on the subject of dynamics, before they
give me advice and attack me on that subject.
I am informed on good authority that Dr. Vander Weyde is
engaged for the Scientiflc American to write on scientific subjects or as an acting editor, and he writes articles which are
published under the head of correspondence, stating that " the
editors are not responsible for the opinions expressed by their
correspondents."
It will be noticed in this pamphlet that the Scientific American, as well as myself, is attempting to clear up the subject of
dynamics in regard to definite meanings of its terms, and we are
of different opinions about the true meaning of the term force.
In the year I865 I submitted a communication on the subject
of dynamical terms to the National Academy of Sciences, which
met at Washington that year; but the papers were handed back
to me immediately, with information that "the subject is all
clear."
In Conclusion, I most respectfully invite the attention of
scientific institutions to my classifications of dynamical terms,
that a decision may be given, whether it be adopted or rejected.
A prompt action on this subject, in aid of establishing precision to the meaning of dynamical terms, would render a great
service to the public.