Chapter XII (1)

Will Perpetual Motion Ever Be Accomplished?

The antiquity of the problem of Perpetual Motion, and the countless attempts by clever and ingenious minds to accomplish its solution, and the uniform failure of such attempts is no proof at all, scientifically speaking, that Perpetual Motion is an impossibility. If there be scientific proof that Perpetual Motion is unattainable, that proof must be found elsewhere than in the number of attempts and the universality of failures, or in the number or eminence of the people who believe it to be impossible.

Dircks in his work printed in 1861, being "A History of the Search for Self-Motive Power, During the 17th 18th and 19th Centuries," says on the subject:

"The subject of Perpetual Motion opposes paradox to paradox. It is viewed both as being most simple and most difficult to find. The learned justify both its possibility and impossibility. Many mechanics believe it possible * * * Its pursuit always commences in confidence, only to end in doubt. * * *

We think a careful perusal of all that has been gathered respecting Perpetual Motion clearly establishes that much remains to be done to prove the impossibility of practically solving this knotty problem; and that a full demonstration of the difficulties that environ it is worthy of being attempted, even by the most exalted mathematicians. It is not requisite that they should descend to the level of the most ordinary minds, but leave it for others to reduce their elaborated reasonings on the subject to some generally comprehensible form. We fear the proposal partakes too much of the difficulty of proving a negative; but still, as the attempt has been made by celebrated savants, and is generally considered insufficient; and as data may have been wanting, which we conceive a collection of the chief known examples will supply; we recommend the consideration of this matter to all geometers. * * *

In a mathematical point of view, we think this subject is far from being exhausted; and, after what has been advanced, may very properly be considered as claiming grave considerations. And that, scientifically examined, it is a mark of mere shallowness and querulousness to attempt the substitution of ridicule and satire for the more difficult, but consistent course of sound, close reason and argument, such as the wonted sobriety and severity of scientific criticism accords to its investigations generally."

At the time of the publication of Dircks's work from which the above quotation is taken (1861), the doctrine of Conservation of Energy had not been announced and accepted as an established generalization of a scientific fact, and it is apparent was not understood by him. Dircks's statement "as data may have been wanting, which we conceive a collection of the chief known examples will supply," shows that he misconceived the nature of the problem of proving the impossibility of Perpetual Motion. If, however, the principle of Conservation of Energy is a true scientific fact, the impossibility of self-motive power follows as an inevitable scientific corollary, and the ignis fatuus hope of attaining Perpetual Motion which has deluded so many bright minds is forever destroyed and demolished.

A perusal of the arguments against Perpetual Motion made by thinking men with scientific minds even though long before the thorough establishment of the doctrine of transmutation and Conservation of Energy, discloses the fact that those arguments in fact depend finally on the principle now known and designated Conservation of Energy.

It is amusing to note in reading the arguments on the subject by our greatest philosophers, Newton, Gallileo, Huyghens, and Descartes, that while they lived and labored long before Conservation of Energy in its generalized form was known, or announced, they seemed to have a perception that energy could not be created; that energy must produce an effect commensurate with its own activity; that the existence of energy in one body is proof positive that some agency furnished and lost an exact equivalent of that energy. In other words, these men in reasoning on specific problems presented to them, and on the problem of Perpetual Motion in particular, appear to have appreciated and applied in their reasonings, the principle of Conservation of Energy.

Men who have worked at the problem of Perpetual Motion before the establishment of the doctrine of Conservation of Energy, and men who still work at the problem, who, through lack of opportunity have not become familiar with that doctrine, are not to be blamed or thought stupid because of that folly, but those who knowing that principle, or being in a situation to know it, must be mechanically and mathematically stupid not to realize that Perpetual Motion and Conservation of Energy are irreconcilable, and that both cannot be possibilities. In this day when the principle of Conservation of Energy is taught in the High Schools of the United States, and in every other civilized country in the world, it is not surprising that fewer people work on Perpetual Motion than formerly, and that public interest in the subject is waning, as waning it surely is.

A generation ago, however, this principle was not known and taught, and the state of the world's learning was at such a stage that many even scientific minds thought Perpetual Motion possible, and worked for its attainment.

The principle of Conservation of Energy as applied to all Perpetual Motion devices can be stated as follows: There can be no mechanical effect without an equal mechanical cause. Energy--i. e., the capacity to do work, can only be imparted by an equal amount of work done. It therefore follows axiomatically that Perpetual Motion is possible only if and when a machine be produced that runs absolutely without friction and absolutely without atmospheric resistance, or the resistance of bending of cords, or other like mechanical resistance. If there be such resistance, then the energy imparted to the machine will be diminished by that resistance, with the result that the machine can only yield the amount of energy imparted, less the energy required to overcome such resistance. That no machine can be built free of such resistance is patent to even a tyro in mechanics.

It will be interesting here, and perhaps more interesting than useful, to add some of the arguments quoted by Dircks and reproduced in his work for and against the possibility of Perpetual Motion. They have little scientific value at this time, as they were all made by men who were unfamiliar with the decisive principle of Conservation of Energy. Nevertheless, for their historical interest we offer a few:

The Possibility of Perpetual Motion Denied Remarks of Dr. Papin on a French Contrivance

In 1665, Dr. Papin, Fellow of the Royal Society, brought before the Royal Society of London, a paper concerning a French contrivance for Perpetual Motion. The following excerpt will illustrate and explain the contrivance:

The paper printed in French, and containing contrivance for
perpetual motion, being set down in such a manner that can hardly
be understood but by those that are much acquainted with such
descriptions, I have endeavored to explain it as follows:

Let D E F be a pair of bellows forty inches long, that may be
opened by removing the part F from E; let them be exactly shut
everywhere but at the aperture E; and let a pipe E G, twenty
or twenty-two inches long, be soldered to the said aperture E,
having its other end in a vessel G, full of mercury, and placed
near the middle of the bellows.

A is an axis for the bellows to turn upon.

B, a counterpoise fastened to the lower end of the bellows.

C, a weight with a clasp to keep the bellows upright.

Now, if we suppose the bellows opened only to one-third or
one-fourth, standing upright, and full of mercury, it is plain
that the said mercury, being forty inches high, must fall, as in
the Torricellian experiment, to the height of about twenty-seven
inches, and, consequently, the bellows must open towards F, and
leave a vacuity there. This vacuity must be filled with the
mercury ascending from G through the pipe G E, the said pipe
being but twenty-two inches long; by this means the bellows must
be opened more and more, till the mercury continuing to ascend
makes the upper part of the bellows so heavy that the lower part
must get loose from the clasp C, and the bellows should turn
quite upside down; but the vessel G being set in a convenient
place, keeps them horizontal, and the part F engageth there in
another clasp C; then the mercury, by its weight, runs out from
the bellows into the vessel G through the pipe E G, and the
bellows must shut closer and closer until the part E F comes to
be so light that the counterpoise B is able to make the part F
get loose from the clasp C; then the bellows come to be upright
again; the mercury left in them falls again to the height of
twenty-seven inches, and, consequently, all the other effects
will follow as we have already seen, and the motion will continue
forever. Thus much for the French author.

Upon this it is to be observed, that the bellows can never be
opened by the internal pressure, unless the said pressure be
stronger then the external; now, in this case, the weight of the
atmosphere doth freely press up the outward part of the bellows,
but it cannot come at the inward part but through the pipe G E,
which, containing twenty-two perpendicular inches of mercury,
does counterpoise so much of the weight of the atmosphere, so
that this being supposed to be twenty-seven inches of mercury,
it cannot press the inward part of the bellows but with weight
equivalent to five perpendicular inches of mercury. From this
we may conclude, that the pressure of the atmosphere, being
weakened within the bellows more then it can be helped by the
mercury contained in the same, as may easily be computed, the
said bellows standing upright must rather shut then open. Thus,
without losing any labor and charges in trying, people may be
sure that the thing can never do.

Two "Certain" Plans for (Not) Producing Perpetual Motion

In 1834, the following article was contributed to "Mechanics' Magazine." The contributor was very frank, and presents some splendid suggestions for Perpetual Motion workers. His article is as follows:

Very few young mechanicians escape being seduced into an attempt
to produce a perpetual movement, by making gravitation counteract
itself. They are not contented with being told by older men,
that a cause can never be made to exceed its own power; yet
gravitation is expected by them to lift up on one side more
weight than sinks on the other, with some percentage of friction
into the bargain. Nature, however, is too true to itself to be
so taken in by all or any of the multitudes of various ways
the inventive genius of man has contrived, and still keeps
contriving, to circumvent her immutable laws, with no other
effect than to render the case so complicated as to puzzle the
judgment of the inventors, which ends usually in their firm
belief that they have outwitted nature instead of themselves. I
acknowledge that in my youth I was one of this class, and, for
the benefit of the young, I beg to present you with two _certain_
plans for producing perpetual motion, and compelling gravity to
be frolicsome, and do more work than she ought.

Let A (Fig. 1) be a cistern full of oil or water, above 4 feet
deep. Let B be a wheel; freely suspended within it, on its axle,
let there be four wide glass tubes, 40 inches long, _c c c c_,
having large bulbs, holding, say a pint, blown at the closed
end. Fill these tubes with mercury, fix on an Indian-rubber ball
or bladder, that will hold a pint, to each of them at the open
end, and let them be attached round the wheel, as exhibited in
the figure. As the pressure of 40 inches of mercury will exceed
the atmospheric pressure, and also that of the four-foot column
of water, when the Indian-rubber bottle is lowest, and the tube
erect, at D, the mercury will fill it, leaving a vacuum in the
glass bulb above. On the opposite side the mercury will fill the
glass bulb, and the Indian-rubber bottle will be pressed flat,
as will also be the case in the two horizontal tubes. Now, it is
evident that the two horizontal tubes exactly balance each other;
but the tube D, with its bulb swelled out, displaces a pint of
water more than its opposite tube, and hence will attempt to rise
with the force of about one pound; and each tube, when it arrives
at the same position, must produce the same result, the wheel
must have a continual power, equal to about one pound, with a
radius of two feet.--Q. E. D.

Let Fig. 2 represent a light drum of wood--one-half of which is
inserted into a cleft in a water-cistern A, which fits it, and
from which the water is prevented from escaping by a strip of
leather, which the water presses against the drum, and which thus
operates as a valve, without much friction (especially if oil be
substituted for water in the cistern). Now, as this drum is much
lighter than water, it must ever attempt to swim, and thus, in
perpetually rising, cause the drum to revolve forcibly round its
axle.--Q. E. D.

I tried this last method thirty years ago, but it was so
obstinate as not to move one inch at my bidding, though it
obviously is proved, to demonstration, that it ought to have
gone on swimmingly. I have just heard that an Italian gentleman
has hit upon the same plan; so it seems that the mania is not
confined to England.

The article above quoted elicited a varied correspondence on the subject of self-motive power. The editor finally made the following apt and happy remark concerning the two "Certain" plans:

We think our correspondent, S. F., has entirely misconceived the
scope of the playful account, given in our last number, of two
plans of perpetual motion. The object of the writer seems to have
been, to impress on the minds of young mechanicians the folly of
wasting their time in vain endeavors to render the effects of
causes greater than the causes themselves; or, in other words,
to gain power out of nothing--a process without limit or value,
were it not cut short by the want of all limit to its folly; and
this he could not, perhaps, have done in any way so well, as by
exhibiting a couple of infallible perpetual movers that would
not stir at all, though they bade as fair for it as any of their
kindred.

Article by Rev. John Wilkins

Rev. John Wilkins of England, born 1614; died 1672, published a work called "Mathematical Magic," in which he discoursed scientifically and technically on efforts that had been made up to that time to attain Perpetual Motion. His work shows great scholarship, diligent search, and a thorough knowledge of mathematics and mechanics. Considering the state of scientific knowledge at the time when he lived and worked, his insight into scientific subjects is truly remarkable.

Considering the state of scientific learning in his day, his observations on the subject of Perpetual Motion show him to have possessed really a great scientific and analytical mind. Of all those who wrote or thought extensively on the subject in that century we regard what he had to say as being the most worthy of reproduction. The following excerpt from "Mathematical Magic," will give the reader an idea of his course of reasoning and conclusions on the subject of self-motive power:

CHAP. IX.--_Of a Perpetual Motion--The seeming facility and real
difficulty of any such contrivance--The several ways whereby it
hath been attempted, particularly by Chemistry._

It is the chief inconvenience of all the automata
before-mentioned, that they need a frequent repair of new
strength, the causes whence their motion does proceed being
subject to fail, and come to a period; and, therefore, it
would be worth our enquiry to examine whether or no there may
be made any such artificial contrivance, which might have the
principle of moving from itself so that the present motion should
constantly be the cause of that which succeeds.

This is that great secret in art which, like the Philosopher's
Stone in Nature, has been the business and study of many more
refined wits for divers ages together; and it may well be
questioned whether either of them as yet have ever been found
out; though if this have, yet like the other, it is not plainly
treated of by any author.

Not but there are sundry discourses concerning this subject, but
they are rather _conjectures_ than _experiments_. And though many
inventions in this kind may at first view bear a great show of
probability, yet they will fail, being brought to trial, and will
not answer in practice what they promised in speculation. Any one
who has been versed in these experiments must needs acknowledge
that he has been often deceived in his strongest confidence;
when the imagination has contrived the whole frame of such an
instrument, and conceives that the event must infallibly answer
its hopes, yet then does it strangely deceive in the proof and
discovers to us some defect which we did not before take notice
of.

Hence it is that you shall scarce talk with any one who has never
so little smattering in these arts, but he will instantly promise
such a motion as being but an easy achievement, till further
trial and experience has taught him the difficulty of it. There
being no enquiry that does more entice with the _probability_ and
deceive with the _subtilty_.

I shall briefly recite the several ways whereby this has been
attempted, or seems most likely to be effected, thereby to
contract and facilitate the enquiries of those who are addicted
to these kind of experiments; for when they know the defects of
other inventions, they may the more easily avoid the same or the
like in their own.

The ways whereby this has been attempted may be generally reduced
to these three kinds:

1. By Chemical Extractions. 2. By Magnetical Virtues. 3. By the
Natural Affection of Gravity.

1. The discovery of this has been attempted by chemistry.
Paracelsus and his followers have bragged that by their
separations and extractions they can make a little world which
shall have the same perpetual motions with this microcosm, with
the representation of all meteors, thunder, snow, rain, the
courses of the sea in its ebbs and flows, and the like. But these
miraculous promises would require as great a faith to believe
them as a power to perform them; and though they often talk of
such great matters:

At nusquam totos inter qui talia curant, Apparet ullus, qui re
miracula tanta Comprobet--

yet we can never see them confirmed by any real experiment; and
then, besides, every particular author in that art has such
a distinct language of his own (all of them being so full of
allegories and affected obscurities), that 'tis very hard for any
one (unless he be thoroughly versed amongst them) to find out
what they mean, much more to try it.

One of these ways (as I find it set down) is this: Mix five
ounces of ☿ with an equal weight of ♃; grind them together with
ten ounces of sublimate; dissolve them in a cellar upon some
marble for the space of four days, till they become like oil
olive; distil this with fire of chaff, or driving fire, and it
will sublime into a dry substance; and so, by repeating of these
dissolvings and distillings, there will be at length produced
divers small atoms, which, being put into a glass well luted and
kept dry, will have a perpetual motion.

I cannot say anything from experience against this; but I think
it does not seem very probable, because things that are forced
up to such vigorousness and activity as these ingredients seem to
be by their frequent sublimings and distillings, are not likely
to be of any duration. The more any thing is stretched beyond
its usual nature, the less does it last; violence and perpetuity
being no companions. And then, besides, suppose it is true, yet
such a motion could not well be applied to any use, which will
needs take much from the delight of it.

Amongst the chemical experiments to this purpose may be reckoned
up that famous motion invented by Cornelius Dreble, and made for
King James; wherein was represented the constant revolutions of
the sun and moon, and that without the help either of springs
or weights. Marcellus Vranckhein, speaking of the means whereby
it was performed, he calls it _Scintillula animae magneticae
mundi, seu astralis et insensibilis spiritus_; being that grand
secret for the discovery of which those dictators of philosophy,
Democritus, Pythagoras, Plato, did travel unto the Gymnosophists
and Indian Priests. The author himself, in his discourse upon
it, does not at all reveal the way how it was performed. But
there is one Thomas Tymme who was a familiar acquaintance of his,
and did often pry into his works (as he professes himself), who
affirms it to be done thus: By extracting a fiery spirit out of
the mineral matter, joining the same with his proper air, which
included in the axletree (of the first moving wheel), being
hollow, carried the other wheels, making a continual rotation,
except issue or vent be given in this hollow axletree, whereby
the imprisoned spirit may get forth.

What strange things may be done by such extractions I know not,
and, therefore, dare not condemn this relation as impossible;
but I think it sounds rather like a chemical dream than a
philosophical truth. It seems this imprisoned spirit is now
set at liberty, or else is grown weary, for the instrument
(as I have heard) has stood still for many years. It is here
considerable that any force is weakest near the center of a
wheel; and therefore, though such a spirit might of itself have
an agitation, yet 'tis not easily conceivable how it should
have strength enough to carry the wheels about with it. And
then, the absurdity of the author's citing this, would make one
mistrust his mistake. He urges it as a strong argument against
Copernicus; as if, because Dreble did thus contrive in an engine
the revolution of the heavens and the immovableness of the earth,
therefore it must needs follow that 'tis the heavens which are
moved, and not the earth. If his relation were no truer than his
consequence, it had not been worth the citing.

CHAP. XIII.--_Concerning several attempts of contriving a
Perpetual Motion, by Magnetical Virtues._

The second way whereby the making of a perpetual motion has
been attempted, is by Magnetical Virtues, which are not without
some strong probabilities of proving effectual to this purpose;
especially when we consider that the heavenly revolutions (being
as the first pattern imitated and aimed at in these attempts)
are all of them performed by the help of these qualities. This
great orb of earth, and all the other planets, being but as so
many magnetical globes, endowed with such various and continual
motions as may be most agreeable to the purposes for which they
were intended. And, therefore, most of the authors who treat
concerning this invention, do agree that the likeliest way to
effect it, is by these kind of qualities.

It was the opinion of Pet. Peregrinus, and there is an example
pretended for it in Bettinus (apiar. 9, progym. 5, pro. 11) that
a magnetical globe, or terella, being rightly placed upon its
poles, would of itself have a constant rotation, like the diurnal
motion of the earth. But this is commonly exploded as being
against all experience.

Others think it possible so to contrive several pieces of steel
and loadstone that, by their continual attraction and expulsion
of one another, they may cause a perpetual revolution of a wheel.
Of this opinion were Taisner, Pet. Peregrinus, and Cardan, out
of Antonius de Fantis. But D. Gilbert, who was more especially
versed in magnetical experiments, concludes it to be a vain and
groundless fancy.

But amongst all these kinds of inventions, that is most likely,
wherein a loadstone is so disposed that it shall draw unto it on
a reclined plane a bullet of steel, which steel, as it ascends
near to the loadstone, may be contrived to fall down through
some hole in the plane, and so to return unto the place from
whence at first it began to move; and, being there, the loadstone
will again attract it upwards till coming to this hole, it will
fall down again; and so the motion shall be perpetual, as may be
more easily conceivable by this figure:

Suppose the loadstone to be represented at A B, which, though it
have not strength enough to attract the bullet C directly from
the ground, yet may do it by the help of the plane E F. Now, when
the bullet is come to the top of this plane, its own gravity
(which is supposed to exceed the strength of the loadstone) will
make it fall into that hole at E; and the force it receives in
this fall will carry it with such a violence unto the other end
of this arch, that it will open the passage which is there made
for it, and by its return will again shut it; so that the bullet
(as at the first) is in the same place whence it was attracted,
and, consequently, must move perpetually.

But, however, this invention may seem to be of such strong
probability, yet there are sundry particulars which may prove it
insufficient; for--

1. This bullet of steel must first be touched, and have its
several poles, or else there can be little or no attraction
of it. Suppose C in the steel to be answerable unto A in the
stone, and to B; in the attraction C D must always be directed
answerable to A B, and so the motion will be more difficult; by
reason there can be no rotation or turning round of the bullet,
but it must slide up with the line C D, answerable to the axis
A B.

2. In its fall from E to G, which is _motus elementaris_, and
proceeds from its gravity, there must needs be a rotation of it;
and so 'tis odds but it happens wrong in the rise, the poles
in the bullet being not in the same direction to those in the
magnet; and if in this reflux it should so fall out, that D
should be directed towards B, there should be rather a flight
than an attraction, since those two ends do repel, and not draw
one another.

3. If the loadstone A B have so much strength, that it can
attract the bullet in F, when it is not turned round, but does
only slide upon the plane, whereas its own gravity would rowl
it downwards; then it is evident the sphere of its activity
and strength would be so increased when it approaches much
nearer, that it would not need the assistance of the plane,
but would draw it immediately to itself without that help; and
so the bullet would not fall down through the hole, but ascend
to the stone, and, consequently, cease its motion: for, if the
loadstone be of force enough to draw the bullet on the plane, at
the distance F B, then must the strength of it be sufficient to
attract it immediately unto itself, when it is so much nearer
as E B. And if the gravity of the bullet be supposed so much
to exceed the strength of the magnet, that it cannot draw it
directly when it is so near, then will it not be able to attract
the bullet up the plane, when it is so much further off.

So that none of all these magnetical experiments, which have
been as yet discovered, are sufficient for the effecting of
a perpetual motion, though these kind of qualities seem most
conducible unto it; and perhaps, hereafter, it may be contrived
from them.

CHAP. XIV.--_The seeming probability of effecting a Continual
Motion by Solid Weights in a Hollow Wheel or Sphere._

The third way whereby the making of a perpetual motion has been
attempted is by the Natural Affection of Gravity; when the
heaviness of several bodies is so contrived, that the same motion
which they give in their descent, may be able to carry them up
again.

But (against the possibility of any such invention) it is thus
objected by Cardan:--All sub-lunary bodies have a direct motion
either of ascent or descent; which, because it does not refer to
some term, therefore cannot be perpetual, but must needs cease
when it is arrived at the place unto which it naturally tends.

I answer, though this may prove that there is no natural motion
of any particular heavy body which is perpetual, yet it does not
hinder, but that it is possible from them to contrive such an
artificial revolution as shall constantly be the cause of itself.

Those bodies which may be serviceable to this purpose are
distinguishable into two kinds:

1. Solid and consistent; as weights of metal, or the like.

2. Fluid or sliding; as water, sand, etc.

Both these ways have been attempted by many, though with very
little or no success. Other men's conjectures in this kind you
may see set down by divers authors. It would be too tedious to
repeat them over, or set forth their draughts.

I shall only mention two new ones, which (if I am not
over-partial) seem altogether as probable as any of these kinds
that have been yet invented; and, till experience had discovered
their defect and insufficiency, I did certainly conclude them to
be infallible.

The first of these contrivances was by solid weights being placed
in some hollow wheel or sphere, unto which they should give a
perpetual revolution; for, as the philosopher has largely proved,
only a circular motion can properly be perpetual.

But, for the better conceiving of this invention, it is requisite
that we rightly understand some principles in Trochilicks, or the
art of wheel instruments; as, chiefly, the relation betwixt the
parts of a wheel and those of a balance; the several proportions
in the semi-diameter of a wheel being answerable to the sides
in a balance, where the weight is multiplied according to its
distance from the center.

Thus, suppose the center to be at A, and the diameter of
the wheel, D C, to be divided into equal parts (as is here
expressed), it is evident, according to the former ground, that
one pound at C will equiponderate to five pound at B, because
there is such a proportion betwixt their several distances from
the center. And it is not material whether or no these several
weights be placed horizontally; for though B do hang lower than
C, yet this does not at all concern the heaviness; or though the
plummet C were placed much higher than it is at E, or lower at
F, yet would it still retain the same weight which it had at C;
because these plummets (as in the nature of all heavy bodies),
do tend downwards by a straight line; so that their several
gravities are to be measured by that part of the horizontal
semi-diameter, which is directly either below or above them.
Thus, when the plummet C shall be moved either to G or H, it will
lose one-third of its former heaviness, and be equally ponderous
as if it were placed in the balance at No. 3; and if we suppose
it to be situated at I or K, then the weight of it will lie
wholly upon the center, and not at all conduce to the motion of
the wheel on either side; so that the straight lines which pass
through the divisions of the diameter may serve to measure the
heaviness of any weight in its several situations.

These things thoroughly considered, it seems very possible and
easy for a man to contrive the plummets of a wheel, that they may
be always heavier in their fall, than in their ascent; and so,
consequently, that they should give a perpetual motion to the
wheel itself; since it is impossible for that to remain unmoved
as long as one side in it is heavier than the other.

For the performance of this, the weights must be so ordered: 1.
That in their descent they may fall from the center, and in their
ascent may rise nearer to it. 2. That the fall of each plummet
may begin the motion of that which should succeed it, as in the
following diagram:

Where there are sixteen plummets, eight in the inward circle,
and as many in the outward. (The inequality being to arise from
their situation, it is therefore most convenient that the number
of them be even.) The eight inward plummets are supposed to be
in themselves so much heavier than the other, that in the wheel
they may be of equal weight with those above them, and then the
fall of these will be of sufficient force to bring down the
other. For example, if the outward be each of them four ounces,
then the inward must be five; because the outward is distant from
the center five of those parts whereof the inward is but four.
Each pair of these weights should be joined together by a little
string or chain, which must be fastened about the middle, betwixt
the bullet and the center of that plummet which is to fall first,
and at the top of the other.

When these bullets, in their descent, are at their farthest
distance from the center of the wheel, then shall they be
stopped, and rest on the pins placed to that purpose; and
so, in their rising, there must be other pins to keep them
in a convenient posture and distance from the center, lest,
approaching too near unto it, they thereby become unfit to fall
when they shall come to the top of the descending side.

This may be otherwise contrived with some different
circumstances, but they will all redound to the same effect. By
such an engine it seems very probable that a man may produce
perpetual motion; the distance of the plummets from the center
increasing with weight on one side, and their being tied to one
another, causing a constant succession in their falling.

But now, upon experience, I have found this to be fallacious;
and the reason may sufficiently appear by a calculation of the
heaviness of each plummet, according to its several situation;
which may easily be done by those perpendiculars that cut
the diameter (as was before explained, and is here expressed
in five of the plummets on the descending side). From such a
calculation it will be evident, that both the sides of this
wheel will equiponderate; and so consequently, that the supposed
inequality whence the motion should proceed, is but imaginary
and groundless. On the descending side, the heaviness of each
plummet may be measured according to these numbers (supposing the
diameter of the wheel to be divided into twenty parts, and each
of those sub-divided into four):

_The Outward Plummets._ _The Inward Plummets._

7.0 } 1.0 }
10.0 } The sum 24. 7.2 } The sum 19.
7.0 } 7.2 }
3.0 }

On the ascending side, the weights are to be reckoned according
to these degrees:

_The Outward._ _The Inward._

1.3 } 4.1 }
7.2 } 7.0 } The sum 19.
9.0 } The sum 24. 5.2 }
5.3 } 2.1 }
0.0 }

The sum of which last numbers is equal with the former, and
therefore both the sides of such a wheel in this situation will
equiponderate.

If it be objected, that the plummet A should be contrived to pull
down the other at B, and then the descending side will be heavier
than the other; for answer to this, it is considerable--

1. That these bullets towards the top of the wheel, cannot
descend till they come to a certain kind of inclination.

2. That any lower bullet hanging upon the other above it, to pull
it down, must be conceived, as if the weight of it were in that
point where its string touches the upper; at which point this
bullet will be of less heaviness in respect of the wheel, than if
it did rest in its own place; so that both the sides of it, in
any kind of situation, may equiponderate.

CHAP. XV.--_Of composing, a Perpetual Motion by Fluid
Weights--Concerning Archimedes his Water Screw--The great
probability of accomplishing this enquiry by the help of that,
with the fallibleness of it upon experiment._

That which I shall mention as the last way, for the trial of
this experiment, is by contriving it in some Water Instrument;
which may seem altogether as probable and easy as any of the
rest; because that element, by reason of its fluid and subtle
nature (whereby, of its own accord, it searches out the lower
and more narrow passages), may be most pliable to the mind of
the artificer. Now, the usual means for the ascent of water is
either by suckers or forces, or something equivalent thereunto;
neither of which may be conveniently applied unto such a work as
this, because there is required unto each of them so much or more
strength, as may be answerable to the full weight of the water
that is to be drawn up; and then, besides, they move for the most
part by fits and snatches, so that it is not easily conceivable,
how they should conduce unto such a motion, which, by reason of
its perpetuity, must be regular and equal.

But, amongst all other ways to this purpose, that invention of
Archimedes is incomparably the best, which is usually called
_Cochlea_, or the Water Screw; being framed by the helical
revolution of a cavity about a cylinder. We have not any
discourse from the author himself concerning it, nor is it
certain whether he ever writ anything to this purpose; but if he
did, yet, as the injury of time hath deprived us of many other of
his excellents works, so likewise of this amongst the rest.

[Near five pages are occupied in describing the use of this screw, and the form and manner of making it; then follows:]

The true inclination of the screw being found, together with
the certain quantity of water which every helix does contain;
it is further considerable, that the water by this instrument
does ascend naturally of itself, without any violence or labor;
and that the heaviness of it does lie chiefly upon the centers
or axis of the cylinder, both its sides being of equal weight
(said Ubaldus); so that, it should seem, though we suppose each
revolution to have an equal quantity of water, yet the screw
will remain with any part upwards, according as it shall be
set, without turning itself either way; and, therefore, the
least strength being added to either of its sides should make
it descend, according to that common maxim of Archimedes--any
addition will make that which equiponderates with another to tend
downwards.

But now, because the weight of this instrument and the water
in it does lean wholly upon the axis, hence is it (said
Ubaldus) that the grating and rubbing of these axes against the
sockets wherein they are placed, will cause some ineptitude
and resistency to that rotation of the cylinder; which would
otherwise ensue upon the addition of the least weight to any one
side; but (said the same author) any power that is greater than
this resistency which does arise from the axis, will serve for
the turning of it round.

These things considered together, it will hence appear how a
perpetual motion may seem easily contrivable. For, if there were
but such a water-wheel made on this instrument, upon which the
stream that is carried up may fall in its descent, it would turn
the screw round, and by that means convey as much water up as is
required to move it; so that the motion must needs be continual,
since the same weight which in its fall does turn the wheel is,
by the turning of the wheel, carried up again.

Or, if the water, falling upon one wheel, would not be forcible
enough for this effect, why then there might be two or three, or
more, according as the length and elevation of the instrument
will admit; by which means the weight of it may be so multiplied
in the fall that it shall be equivalent to twice or thrice that
quantity of water which ascends; as may be more plainly discerned
by the following diagram:

Where the figure L M, at the bottom, does represent a wooden
cylinder with helical cavities cut in it, which at A B is
supposed to be covered over with tin plates, and three
water-wheels upon it, H I K; the lower cistern, which contains
the water, being C D. Now, this cylinder being turned round, all
the water which from the cistern ascends through it, will fall
into the vessel at E, and from that vessel being conveyed upon
the water-wheel H, shall consequently give a circular motion to
the whole screw. Or, if this alone should be too weak for the
turning of it, then the same water which falls from the wheel H,
being received into the other vessel F, may from thence again
descend on the wheel I, by which means the force of it will be
doubled. And if this be yet unsufficient, then may the water
which falls on the second wheel I, be received into the other
vessel G, and from thence again descend on the third wheel at
K; and so for as many other wheels as the instrument is capable
of. So that, besides the greater distance of these three streams
from the center or axis by which they are made so much heavier,
and besides that the fall of this outward water is forcible and
violent, whereas the ascent of that within is natural--besides
all this, there is thrice as much water to turn the screw as is
carried up by it.

But, on the other side, if all the water falling upon one wheel
would be able to turn it round, then half of it would serve with
two wheels, and the rest may be so disposed of in the fall as to
serve unto some other useful delightful ends.

When I first thought of this invention, I could scarce forbear,
with Archimedes, to cry out εὕρηκα, εὕρηκα {heurêka, heurêka};
it seeming so infallible a way for the effecting of a perpetual
motion that nothing could be so much as probably objected
against it; but, upon trial and experience, I find it altogether
insufficient for any such purpose, and that for these two reasons:

1. The water that ascends will not make any considerable stream
in the fall.

2. This stream, though multiplied, will not be of force enough to
turn about the screw.

1. The water ascends gently, and by intermissions; but it falls
continually, and with force; each of the three vessels being
supposed full at the first, that so the weight of the water in
them might add the greater strength and swiftness to the streams
that descend from them. Now, this swiftness of motion will cause
so great a difference betwixt them that one of these little
streams may spend more water in the fall than a stream six times
bigger in the ascent, though we should suppose both of them to
be continuate; how much more, then, when as the ascending water
is vented by fits and intermissions, every circumvolution voiding
so much as is contained in one helix; and, in this particular,
one that is not versed in these kind of experiments may be easily
deceived.

But, secondly, though there were so great a disproportion, yet,
notwithstanding, the force of these outward streams might well
enough serve for the turning of the screw, if it were so that
both its sides would equiponderate the water being in them (as
Ubaldus had affirmed). But now, upon farther examination, we
shall find this assertion of his to be utterly against both
reason and experience. And herein does consist the chief mistake
of this contrivance; for the ascending side of the screw is made,
by the water contained in it, so much heavier than the descending
side, that these outward streams, thus applied, will not be of
force enough to make them equiponderate, much less to move the
whole, as may be more easily discerned by this figure:

Where A B represents a screw covered over, C D E one helix or
revolution of it, C D the ascending side, E D the descending
side, the point D the middle; the horizontal line C F showing how
much of the helix is filled with water, viz., of the ascending
side, from C the beginning of the helix, to D the middle of it;
and on the descending side, from D the middle, to the point G,
where the horizontal does cut the helix. Now, it is evident that
this latter part, D G, is nothing near so much, and consequently
not so heavy as the other, D C; and thus is it in all the other
revolutions, which, as they are either more or larger, so will
the difficulty of this motion be increased. Whence it will appear
that the outward streams which descend must be of so much force
as to countervail all that weight whereby the ascending side in
every one of these revolutions does exceed the other. And though
this may be effected by making the water-wheels larger, yet then
the motion will be so slow that the screw will not be able to
supply the outward streams.

There is another contrivance to this purpose, mentioned by
Kircher de Magnete, 1, 2, p. 4, depending upon the heat of the
sun and the force of winds; but it is liable to such abundance of
exceptions that it is scarce worth the mentioning, and does by no
means deserve the confidence of any ingenious artist.

Thus have I briefly explained the probabilities and defects of
those subtle contrivances whereby the making of a perpetual
motion has been attempted. I would be loath to discourage the
enquiry of any ingenious artificer by denying the possibility
of effecting it with any of these mechanical helps; but yet (I
conceive) if those principles which concern the slowness of the
power in comparison to the greatness of the weight were rightly
understood and thoroughly considered, they would make this
experiment to seem, if not altogether impossible, yet much more
difficult than otherwise, perhaps, it will appear. However, the
inquiring after it cannot but deserve our endeavors, as being one
of the most noble amongst all these mechanical subtilties. And,
as it is in the fable of him who dug the vineyard for a hidden
treasure, though he did not find the money, yet he thereby made
the ground more fruitful, so, though we do not attain to the
effecting of this particular, yet our searching after it may
discover so many other excellent subtilties as shall abundantly
recompense the labor of our inquiry.

And then, besides, it may be another encouragement to consider
the pleasure of such speculations, which do ravish and sublime
the thoughts with more clear angelical contentments. Archimedes
was generally so taken up in the delight of these mathematical
studies of this familiar siren (as Plutarch styles them) that he
forgot both his meat and drink, and other necessities of nature;
nay, that he neglected the saving of his life, when that rude
soldier, in the pride and haste of victory, would not give him
leisure to finish his demonstration. What a ravishment was that,
when, having found out the way to measure Hiero's crown, he
leaped out of the bath, and (as if he were suddenly possessed)
ran naked up and down, crying εὕρηκα, εὕρηκα {Greek: heurêka,
heurêka}! It is storied of Thales that, in his joy and gratitude
for one of these mathematical inventions, he went presently
to the Temple, and there offered up a solemn sacrifice; and
Pythagoras, upon the like occasion, is related to have sacrificed
a hundred oxen; the justice of Providence having so contrived
it, that the pleasure which there is in the success of such
inventions should be proportioned to the great difficulty and
labor of their inquiry.

The Paradoxical Hydrostatic Balance

The following was contributed to an English scientific journal in 1831, the name of the author of the article is unknown to us, but here is what he wrote:

This hydrostatic balance, like the compound balance of
Desaguliers, may be introduced to illustrate the impossibility of
perpetual motion by a weight removed from the centre of a wheel.

Take the hollow-rimmed wheel A B; let it be air-tight and half
filled with water. Let C be the axle; at B place a hollow ball
loaded to near sinking. Such a wheel, however fine its axle
may be, or however well lubricated, will not make a single
revolution, though the weight B occupies that part at which every
deluded perpetual-motionist is desirous it should be placed;
concluding that, by such an arrangement, the production of
another Orffyrean wheel must be inevitable.

Discussion by P. Gregorio Fontana

P. Gregorio Fontana was professor of higher mathematics at the Royal University of Pavia, in the Province of Lombardy, Italy. In 1786 he published what he designated "Examination of a New Argument in Favor of Perpetual Motion." In part he says:

1. A vertical wheel (Fig. 2) divided in two halves by a vertical
plane which passes through its diameter F O, has the half F P O
immersed in water under the level M N, and the other half wholly
out of the water, being cut off in F O by a peculiar mechanism
from all communication with the reservoir, the exterior half
of the wheel being F Q O; this turns freely round on an axle
passing through the centre C. Now the wheel being specifically
lighter than the water, the immersed part F P O comes with a
continual rotation to the top with a force equal to the excess
of the weight of a volume of water corresponding to the immersed
portion, over the weight of the immersed portion; which rotation
passing through the centre of gravity of the exterior part, and
consequently out of the centre C, obliges the wheel to turn
around C.

Such being the case, the question to be asked is whether the
wheel has itself a perpetual motion, as may be judged at first
sight.

2. To reply adequately, it is at first necessary to know what
effect is produced on the wheel by the horizontal pressure which
the water exercises on the semi-circumference F L O.

Having taken for this purpose, a part P _p_, and having drawn to
the diameter the ordinate P. R, _p r_, and marked the radius P C,
and from it P G perpendicular to the radius C L, which determines
the quadrant O L, the distance of the lowest point O from the
level of the water will be = _b_, the semi-diameter of the wheel
= _a_, C R = _x_, and the specific gravity of the water = 1; the
perpendicular pressure against the part P _p_ = P _p_ . R D,
which resolved in two, one horizontal P R, the other vertical
P G, gives the proportion

PG : PR :: P_p_ . RD : (P_p_ . PR . RD) / (PG).

Thence the horizontal pressure against P _p_, and = (P _p_ .
P R . R D .) / (P G), that is to say P _p_ . P R = R _r_ . P G,
the given horizontal pressure is found to be = R _r_ . R D =
(_b_ - _x_) _d_ _x_, and which, multiplied by R D, giving _b_
- _x_, becomes the momentum of the pressure relatively to M N
= (_b_ - _x_)² _d_ _x_, and the sum of the momenta of pressure
exercised upon the indefinite arc, O P = _f_ (_b_ - _x_)² _d_ _x_
= -(1/3)(_b_ - _x_)³ + the side. And since acting together such
momenta equal _x_, there comes the side = (1/3)_b_³; and as the
already-given sum of the momenta = (1/3)(_b_³ - (_b_ - _x_)³) =
_b² x_ - _b x_² + (1/3)_x_³. Whence, taking _x_ = 2_a_, the sum
of all the momenta of the horizontal pressure exercised on the
whole semi-circumference O L F of the wheel, will be = 2_b_²_a_
- 4_b_ _a_² + (8/3)_a_³, and dividing that sum by the whole
horizontal pressure, that is to say by _f_(_b_ - _x_)_d_ _x_ =
(1/2)(_b_² - (_b_ - _x_)²) = _b_ _x_ - (1/2)_x_² = 2_b_ _a_ -
2_a_², gives _x_ = 2_a_, we have the formula