attempts of so many men, and the persistence with which they cling to
obviously erroneous results in the face not only of mathematical
demonstration, but of practical mechanical measurements. For even when
working in wood it is easy to measure to the half or even the one-fourth
of the hundredth of an inch, and on a ten-inch circle this will bring
the circumference to 3.1416 inches, which is a corroboration of the
orthodox ratio (3.14159) sufficient to show that any value which is
greater than 3.142 or less than 3.141 cannot possibly be correct.
And in regard to the area the proof is quite as simple. It is easy to
cut out of sheet metal a circle 10 inches in diameter, and a square of
7.85 on the side, or even one-thousandth of an inch closer to the
standard 7.854. Now if the work be done with anything like the accuracy
with which good machinists work, it will be found that the circle and
the square will exactly balance each other in weight, thus proving in
another way the correctness of the accepted ratio.
But although even as early as before the end of the eighteenth century,
the value of the ratio had been accurately determined to 152 places of
decimals, the nineteenth century abounded in circle-squarers who brought
forward the most absurd arguments in favor of other values. In 1836, a
French well-sinker named Lacomme, applied to a professor of mathematics
for information in regard to the amount of stone required to pave the
circular bottom of a well, and was told that it was impossible "to give
a correct answer, because the exact ratio of the diameter of a circle to
its circumference had never been determined"! This absolutely true but
very unpractical statement by the professor, set the well-sinker to
thinking; he studied mathematics after a fashion, and announced that he
had discovered that the circumference was exactly 3-1/8 times the length
of the diameter! For this discovery (?) he was honored by several medals
of the first class, bestowed by Parisian societies.
Even as late as the year 1860, a Mr. James Smith of Liverpool, took up
this ratio 3-1/8 to 1, and published several books and pamphlets in
which he tried to argue for its accuracy. He even sought to bring it
before the British Association for the Advancement of Science.
Professors De Morgan and Whewell, and even the famous mathematician, Sir
William Rowan Hamilton, tried to convince him of his error, but without
success. Professor Whewell's demonstration is so neat and so simple that
I make no apology for giving it here. It is in the form of a letter to
Mr. Smith: "You may do this: calculate the side of a polygon of 24 sides
inscribed in a circle. I think you are mathematician enough to do this.
You will find that if the radius of the circle be one, the side of the
polygon is .264, etc. Now the arc which this side subtends is, according
to your proposition, 3.125/12 = .2604, and, therefore, the chord is
greater than its arc, which, you will allow, is impossible."
This must seem, even to a school-boy, to be unanswerable, but it did not
faze Mr. Smith, and I doubt if even the method which I have suggested
previously, viz., that of cutting a circle and a square out of the same
piece of sheet metal and weighing them, would have done so. And yet by
this method even a common pair of grocer's scales will show to any
common-sense person the error of Mr. Smith's value and the correctness
of the accepted ratio.
Even a still later instance is found in a writer who, in 1892, contended
in the New York "Tribune" for 3.2 instead of 3.1416, as the value of the
ratio. He announces it as the re-discovery of a long lost secret, which
consists in the knowledge of a certain line called "the Nicomedean
line." This announcement gave rise to considerable discussion, and even
towards the dawn of the twentieth century 3.2 had its advocates as
against the accepted ratio 3.1416.
Verily the slaves of the mighty wizard, Michael Scott, have not yet
ceased from their labors!
FOOTNOTES:
[1] What follows is an exceedingly forcible illustration of an important
mathematical truth, but at the same time it may be worth noting that the
size of the blood-globules or corpuscles has no relation to the size of
the animal from which they are taken. The blood corpuscle of the tiny
mouse is larger than that of the huge ox. The smallest blood corpuscle
known is that of a species of small deer, and the largest is that of a
lizard like reptile found in our southern waters--the amphiuma.
These facts do not at all affect the force or value of De Morgan's
mathematical illustration, but I have thought it well to call the
attention of the reader to this point, lest he should receive an
erroneous physiological idea.
II
THE DUPLICATION OF THE CUBE
This problem became famous because of the halo of mythological romance
with which it was surrounded. The story is as follows:
About the year 430 B.C. the Athenians were afflicted by a terrible
plague, and as no ordinary means seemed to assuage its virulence, they
sent a deputation of the citizens to consult the oracle of Apollo at
Delos, in the hope that the god might show them how to get rid of it.
The answer was that the plague would cease when they had doubled the
size of the altar of Apollo in the temple at Athens. This seemed quite
an easy task; the altar was a cube, and they placed beside it another
cube of exactly the same size. But this did not satisfy the conditions
prescribed by the oracle, and the people were told that the altar must
consist of one cube, the size of which must be exactly twice the size of
the original altar. They then constructed a cubic altar of which the
side or edge was twice that of the original, but they were told that the
new altar was eight times and not twice the size of the original, and
the god was so enraged that the plague became worse than before.
According to another legend, the reason given for the affliction was
that the people had devoted themselves to pleasure and to sensual
enjoyments and pursuits, and had neglected the study of philosophy, of
which geometry is one of the higher departments--certainly a very sound
reason, whatever we may think of the details of the story. The people
then applied to the mathematicians, and it is supposed that their
solution was sufficiently near the truth to satisfy Apollo, who
relented, and the plague disappeared.
In other words, the leading citizens probably applied themselves to the
study of sewerage and hygienic conditions, and Apollo (the Sun) instead
of causing disease by the festering corruption of the usual filth of
cities, especially in the East, dried up the superfluous moisture, and
promoted the health of the inhabitants.
It is well known that the relation of the area and the cubical contents
of any figure to the linear dimensions of that figure are not so
generally understood as we should expect in these days when the
schoolmaster is supposed to be "abroad in the land." At an examination
of candidates for the position of fireman in one of our cities, several
of the applicants made the mistake of supposing that a two-inch pipe and
a five-inch pipe were equal to a seven-inch pipe, whereas the combined
capacities of the two small pipes are to the capacity of the large one
as 29 to 49.
This reminds us of a story which Sir Frederick Bramwell, the engineer,
used to tell of a water company using water from a stream flowing
through a pipe of a certain diameter. The company required more water,
and after certain negotiations with the owner of the stream, offered
double the sum if they were allowed a supply through a pipe of double
the diameter of the one then in use. This was accepted by the owner, who
evidently was not aware of the fact that a pipe of double the diameter
would carry _four_ times the supply.
A square whose side is twice the length of another, and a circle whose
diameter is twice that of another will each have an area four times that
of the original. And in the case of solids: A ball of twice the diameter
will weigh eight times as much as the original, and a ball of three
times the diameter will weigh twenty-seven times as much as the
original.
In attempting to calculate the side of a cube which shall have twice the
volume of a given cube, we meet the old difficulty of incommensurability,
and the solution cannot be effected geometrically, as it requires the
construction of two mean proportionals between two given lines.
III
THE TRISECTION OF AN ANGLE
This problem is not so generally known as that of squaring the circle,
and consequently it has not received so much attention from amateur
mathematicians, though even within little more than a year a small book,
in which an attempted solution is given, has been published. When it is
first presented to an uneducated reader, whose mind has a mathematical
turn, and especially to a skilful mechanic, who has not studied
theoretical geometry, it is apt to create a smile, because at first
sight most persons are impressed with an idea of its simplicity, and the
ease with which it may be solved. And this is true, even of many persons
who have had a fair general education. Those who have studied only what
is known as "practical geometry" think at once of the ease and accuracy
with which a right angle, for example, may be divided into three equal
parts. Thus taking the right angle ACB, Fig. 4, which may be set off
more easily and accurately than any other angle except, perhaps, that of
60 deg., and knowing that it contains 90 deg., describe an arc ADEB, with C for
the center and any convenient radius. Now every school-boy who has
played with a pair of compasses knows that the radius of a circle will
"step" round the circumference exactly six times; it will therefore
divide the 360 deg. into six equal parts of 60 deg. each. This being the case,
with the radius CB, and B for a center, describe a short arc crossing
the arc ADEB in D, and join CD. The angle DCB will be 60 deg., and as the
angle ACB is 90 deg., the angle ACD must be 30 deg., or one-third part of the
whole. In the same way lay off the angle ACE of 60 deg., and ECB must be
30 deg., and the remainder DCE must also be 30 deg.. The angle ACB is therefore
easily divided into three equal parts, or in other words, it is
trisected. And with a slight modification of the method, the same may be
done with an angle of 45 deg., and with some others. These however are only
special cases, and the very essence of a geometrical solution of any
problem is that it shall be applicable to _all_ cases so that we require
a method by which _any_ angle may be divided into three equal parts by a
pure Euclidean construction. The ablest mathematicians declare that the
problem cannot be solved by such means, and De Morgan gives the
following reasons for this conclusion: "The trisector of an angle, if he
demand attention from any mathematician, is bound to produce from his
construction, an expression for the sine or cosine of the third part of
any angle, in terms of the sine or cosine of the angle itself, obtained
by the help of no higher than the square root. The mathematician knows
that such a thing cannot be; but the trisector virtually says it can be,
and is bound to produce it to save time. This is the misfortune of most
of the solvers of the celebrated problems, that they have not knowledge
enough to present those consequences of their results by which they can
be easily judged."
[Illustration: Fig. 4.]
De Morgan gives an account of a "terrific" construction by a friend of
Dr. Wallich, which he says is "so nearly true, that unless the angle be
very obtuse, common drawing, applied to the construction, will not
detect the error." But geometry requires _absolute_ accuracy, not a mere
approximation.
IV
PERPETUAL MOTION
It is probable that more time, effort, and money have been wasted in the
search for a perpetual-motion machine than have been devoted to attempts
to square the circle or even to find the philosopher's stone. And while
it has been claimed in favor of this delusion that the pursuit of it has
given rise to valuable discoveries in mechanics and physics, some even
going so far as to urge that we owe the discovery of the great law of
the conservation of energy to the suggestions made by the
perpetual-motion seekers, we certainly have no evidence to show anything
of the kind. Perpetual motion was declared to be an impossibility upon
purely mechanical and mathematical grounds long before the law of the
conservation of energy was thought of, and it is very certain that this
delusion had no place in the thoughts of Rumford, Black, Davy, Young,
Joule, Grove, and others when they devoted their attention to the laws
governing the transformation of energy. Those who pursued such a
will-o'-the-wisp, were not the men to point the way to any scientific
discovery.
The search for a perpetual-motion machine seems to be of comparatively
modern origin; we have no record of the labors of ancient inventors in
this direction, but this may be as much because the records have been
lost, as because attempts were never made. The works of a mechanical
inventor rarely attracted much attention in ancient times, while the
mathematical problems were regarded as amongst the highest branches of
philosophy, and the search for the philosopher's stone and the elixir of
life appealed alike to priest and layman. We have records of attempts
made 4000 years ago to square the circle, and the history of the
philosopher's stone is lost in the mists of antiquity; but it is not
until the eleventh or twelfth century that we find any reference to
perpetual motion, and it was not until the close of the sixteenth and
the beginning of the seventeenth century that this problem found a
prominent place in the writings of the day.
By perpetual motion is meant a machine which, without assistance from
any external source except gravity, shall continue to go on moving until
the parts of which it is made are worn out. Some insist that in order to
be properly entitled to the name of a perpetual-motion machine, it must
evolve more power than that which is merely required to run it, and it
is true that almost all those who have attempted to solve this problem
have avowed this to be their object, many going so far as to claim for
their contrivances the ability to supply unlimited power at no cost
whatever, except the interest on a small investment, and the trifling
amount of oil required for lubrication. But it is evident that a machine
which would of itself maintain a regular and constant motion would be of
great value, even if it did nothing more than move itself. And this
seems to have been the idea upon which those men worked, who had in view
the supposed reward offered for such an invention as a means for finding
the longitude. And it is well known that it was the hope of attaining
such a reward that spurred on very many of those who devoted their time
and substance to the subject.
There are several legitimate and successful methods of obtaining a
practically perpetual motion, provided we are allowed to call to our aid
some one of the various natural sources of power. For example, there are
numerous mountain streams which have never been known to fail, and which
by means of the simplest kind of a water-wheel would give constant
motion to any light machinery. Even the wind, the emblem of fickleness
and inconstancy, may be harnessed so that it will furnish power, and it
does not require very much mechanical ingenuity to provide means whereby
the surplus power of a strong gale may be stored up and kept in reserve
for a time of calm. Indeed this has frequently been done by the raising
of weights, the winding up of springs, the pumping of water into storage
reservoirs and other simple contrivances.
The variations which are constantly occurring in the temperature and the
pressure of the atmosphere have also been forced into this service. A
clock which required no winding was exhibited in London towards the
latter part of the eighteenth century. It was called a perpetual motion,
and the working power was derived from variations in the quantity, and
consequently in the weight of the mercury, which was forced up into a
glass tube closed at the upper end and having the lower end immersed in
a cistern of mercury after the manner of a barometer. It was fully
described by James Ferguson, whose lectures on Mechanics and Natural
Philosophy were edited by Sir David Brewster. It ran for years without
requiring winding, and is said to have kept very good time. A similar
contrivance was employed in a clock which was possessed by the Academy
of Painting at Paris. It is described in Ozanam's work, Vol. II, page
105, of the edition of 1803.
The changes which are constantly taking place in the temperature of all
bodies, and the expansion and contraction which these variations
produce, afford a very efficient power for clocks and small machines.
Professor W. W. R. Ball tells us that "there was at Paris in the latter
half of last century a clock which was an ingenious illustration of such
perpetual motion. The energy, which was stored up in it to maintain the
motion of the pendulum, was provided by the expansion of a silver rod.
This expansion was caused by the daily rise of temperature, and by means
of a train of levers it wound up the clock. There was a disconnecting
apparatus, so that the contraction due to a fall of temperature produced
no effect, and there was a similar arrangement to prevent overwinding. I
believe that a rise of eight or nine degrees Fahrenheit was sufficient
to wind up the clock for twenty-four hours."
Another indirect method of winding a watch is thus described by
Professor Ball:
"I have in my possession a watch, known as the Lohr patent, which
produces the same effect by somewhat different means. Inside the
case is a steel weight, and if the watch is carried in a pocket this
weight rises and falls at every step one takes, somewhat after the
manner of a pedometer. The weight is moved up by the action of the
person who has it in his pocket, and in falling the weight winds up
the spring of the watch. On the face is a small dial showing the
number of hours for which the watch is wound up. As soon as the hand
of this dial points to fifty-six hours, the train of levers which
wind up the watch disconnects automatically, so as to prevent
overwinding the spring, and it reconnects again as soon as the watch
has run down eight hours. The watch is an excellent time-keeper, and
a walk of about a couple of miles is sufficient to wind it up for
twenty-four hours."
Dr. Hooper, in his "Rational Recreations," has described a method of
driving a clock by the motion of the tides, and it would not be
difficult to contrive a very simple arrangement which would obtain from
that source much more power than is required for that purpose. Indeed
the probability is that many persons now living will see the time when
all our railroads, factories, and lighting plants will be operated by
the tides of the ocean. It is only a question of return for capital, and
it is well known that that has been falling steadily for years. When the
interest on investments falls to a point sufficiently low, the tides
will be harnessed and the greater part of the heat, light, and power
that we require will be obtained from the immense amount of energy that
now goes to waste along our coasts.
Another contrivance by which a seemingly perpetual motion may be
obtained is the dry pile or column of De Luc. The pile consists of a
series of disks of gilt and silvered paper placed back to back and
alternating, all the gilt sides facing one way and all the silver sides
the other. The so-called gilding is really Dutch metal or copper, and
the silver is tin or zinc, so that the two actually form a voltaic
couple. Sometimes the paper is slightly moistened with a weak solution
of molasses to insure a certain degree of dampness; this increases the
action, for if the paper be artificially dried and kept in a perfectly
dry atmosphere, the apparatus will not work. A pair of these piles, each
containing two or three thousand disks the size of a quarter of a
dollar, may be arranged side by side, vertically, and two or three
inches apart. At the lower ends they are connected by a brass plate, and
the upper ends are each surmounted by a small metal bell and between
these bells a gilt ball, suspended by a silk thread, keeps vibrating
perpetually. Many years ago I made a pair of these columns which kept a
ball in motion for nearly two years, and Professor Silliman tells us
that "a set of these bells rang in Yale College laboratory for six or
eight years unceasingly." How much longer the columns would have
continued to furnish energy sufficient to cause the balls to vibrate, it
might be difficult to determine. The amount of energy required is
exceedingly small, but since the columns are really nothing but a
voltaic pile, it is very evident that after a time they would become
exhausted.
Such a pair of columns, covered with a tall glass shade, form a very
interesting piece of bric-a-brac, especially if the bells have a sweet
tone, but the contrivance is of no practical use except as embodied in
Bohnenberger's electroscope.
Inventions of this kind might be multiplied indefinitely, but none of
these devices can be called a perpetual motion because they all depend
for their action upon energy derived from external sources other than
gravity. But the authors of these inventions are not to be classed with
the regular perpetual-motion-mongers. The purposes for which these
arrangements were invented were legitimate, and the contrivances
answered fully the ends for which they were intended. The real
perpetual-motion-seekers are men of a different stamp, and their schemes
readily fall into one of these three classes: 1. ABSURDITIES, 2.
FALLACIES, 3. FRAUDS. The following is a description of the most
characteristic machines and apparatus of which accounts have been
published.
1. ABSURDITIES
In this class may be included those inventions which have been made or
suggested by honest but ignorant persons in direct violation of the
fundamental principles of mechanics and physics. Such inventions if
presented to any expert mechanic or student of science, would be at once
condemned as impracticable, but as a general rule, the inventors of
these absurd contrivances have been so confident of success, that they
have published descriptions and sketches of them, and even gone so far
as to take out patents before they have tested their inventions by
constructing a working machine. It is said, that at one time the United
States Patent Office issued a circular refusal to all applicants for
patents of this kind, but at present instead of sending such a circular,
the applicant is quietly requested to furnish a _working_ model of his
invention and that usually ends the matter. While I have no direct
information on the subject, I suspect that the circular was withdrawn
because of the amount of useless correspondence, in the shape of foolish
replies and arguments, which it drew forth. To require a working model
is a reasonable request and one for which the law duly provides, and
when a successful model is forthcoming, a patent will no doubt be
granted; but until that is presented the officials of the Patent Office
can have no positive information in regard to the practicability of the
invention.
The earliest mechanical device intended to produce perpetual motion is
that known as the overbalancing wheel. This is described in a sketch
book of the thirteenth century by Wilars de Honecourt, an architect of
the period, and since then it has been re-invented hundreds of times. In
its simplest forms it is thus described and figured by Ozanam: