THE SEVEN FOLLIES
OF SCIENCE

A
POPULAR ACCOUNT OF THE MOST FAMOUS
SCIENTIFIC IMPOSSIBILITIES
AND THE ATTEMPTS WHICH HAVE BEEN MADE TO SOLVE THEM.

To which is Added a Small Budget of Interesting
Paradoxes, Illusions, and Marvels.

WITH NUMEROUS ILLUSTRATIONS

BY
JOHN PHIN

Author of "How to Use the Microscope"; "The Workshop Companion";
"The Shakespeare Cyclopedia"; Editor Marquis of Worcester's,
"Century of Inventions"; Etc.

SECOND EDITION

NEW YORK
D. VAN NOSTRAND COMPANY
23 Murray and 27 Warren Sts.
1906

Copyright, 1906,
By D. Van Nostrand Company.

PREFACE

In the following pages I have endeavored to give a simple account of problems which have occupied the attention of the human mind ever since the dawn of civilization, and which can never lose their interest until time shall be no more. While to most persons these subjects will have but an historical interest, yet even from this point of view they are of more value than the history of empires, for they are the intellectual battlefields upon which much of our progress in science has been won. To a few, however, some of them may be of actual practical importance, for although the schoolmaster has been abroad for these many years, it is an unfortunate fact that the circle-squarer and the perpetual-motion-seeker have not ceased out of the land.

In these days of almost miraculous progress it is difficult to realize that there may be such a thing as a scientific impossibility. I have therefore endeavored to point out where the line must be drawn, and by way of illustration I have added a few curious paradoxes and marvels, some of which show apparent contradictions to known laws of nature, but which are all simply and easily explained when we understand the fundamental principles which govern each case.

In presenting the various subjects which are here discussed, I have endeavored to use the simplest language and to avoid entirely the use of mathematical formulae, for I know by large experience that these are the bugbear of the ordinary reader, for whom this volume is specially intended. Therefore I have endeavored to state everything in such a simple manner that any one with a mere common school education can understand it. This, I trust, will explain the absence of everything which requires the use of anything higher than the simple rules of arithmetic and the most elementary propositions of geometry. And even this I have found to be enough for many lawyers, physicians, and clergymen who, in the ardent pursuit of their professions, have forgotten much that they learned at college. And as I hope to find many readers amongst intelligent mechanics, I have in some cases suggested mechanical proofs which any expert handler of tools can easily carry out.

As a matter of course, very little originality is claimed for anything in the book,—the only points that are new being a few illustrations of well-known principles, some of which had already appeared in "The Young Scientist" and "Self-education for Mechanics." Whenever the exact words of an author have been used, credit has always been given; but in regard to general statements and ideas, I must rest content with naming the books from which I have derived the greatest assistance. Ozanam's "Recreations in Science and Natural Philosophy," in the editions of Hutton (1803) and Riddle (1854), has been a storehouse of matter. Much has been gleaned from the "Budget of Paradoxes" by Professor De Morgan and also from Professor W. W. R. Ball's "Mathematical Recreations and Problems." Those who wish to inform themselves in regard to what has been done by the perpetual-motion-mongers must consult Mr. Dirck's two volumes entitled "Perpetuum Mobile" and I have made free use of his labors. To these and one or two others I acknowledge unlimited credit.

Some of the marvels which are here described, although very old, are not generally known, and as they are easily put in practice they may afford a pleasant hour's amusement to the reader and his friends.

John Phin

Paterson, N. J., July, 1905.

CONTENTS

THE SEVEN FOLLIES OF SCIENCE

he difficult, the dangerous, and the impossible have always had a strange fascination for the human mind. We see this every day in the acts of boys who risk life and limb in the performance of useless but dangerous feats, and amongst children of larger growth we find loop-the-loopers, bridge-jumpers, and all sorts of venture-seekers to whom much of the attraction of these performances is undoubtedly the mere risk that is involved, although, perhaps, to some extent, notoriety and money-making may contribute their share. Many of our readers will doubtless remember the words of James Fitz-James, in "The Lady of the Lake":

Or, if a path be dangerous known
The danger's self is lure alone.

And in commenting on the old-time game laws of England, Froude, the historian, says: "Although the old forest laws were terrible, they served only to enhance the excitement by danger."

That which is true of physical dangers holds equally true in regard to intellectual difficulties. Professor De Morgan tells us, in his "Budget of Paradoxes," that he once gave a lecture on "Squaring the Circle" and that a gentleman who was introduced to it by what he said, remarked loud enough to be heard by all around: "Only prove to me that it is impossible and I will set about it this very evening."

Therefore it is not to be wondered at that certain very difficult, or perhaps impossible problems have in all ages had a powerful fascination for certain minds. In that curious olla podrida of fact and fiction, "The Curiosities of Literature," D'Israeli gives a list of six of these problems, which he calls "The Six Follies of Science." I do not know whether the phrase "Follies of Science" originated with him or not, but he enumerates the Quadrature of the Circle; the Duplication, or, as he calls it, the Multiplication of the Cube; the Perpetual Motion; the Philosophical Stone; Magic, and Judicial Astrology, as those known to him. This list, however, has no classical standing such as pertains to the "Seven Wonders of the World," the "Seven Wise Men of Greece," the "Seven Champions of Christendom," and others. There are some well-known follies that are omitted, while some authorities would peremptorily reject Magic and Judicial Astrology as being attempts at fraud rather than earnest efforts to discover and utilize the secrets of nature. The generally accepted list is as follows:

1. The Quadrature of the Circle or, as it is called in the vernacular, "Squaring the Circle."
2. The Duplication of the Cube.
3. The Trisection of an Angle.
4. Perpetual Motion.
5. The Transmutation of the Metals.
6. The Fixation of Mercury.
7. The Elixir of Life.

The Transmutation of the Metals, the Fixation of Mercury, and the Elixir of Life might perhaps be properly classed as one, under the head of the Philosopher's Stone, and then Astrology and Magic might come in to make up the mystic number Seven.

The expression "Follies of Science" does not seem a very appropriate one. Real science has no follies. Neither can these vain attempts be called scientific follies because their very essence is that they are unscientific. Each one is really a veritable "Will-o'-the-Wisp" for unscientific thinkers, and there are many more of them than those that we have here named. But the expression has been adopted in literature and it is just as well to accept it. Those on the list that we have given are the ones that have become famous in history and they still engage the attention of a certain class of minds. It is only a few months since a man who claims to be a professional architect and technical writer put forth an alleged method of "squaring the circle," which he claims to be "exact"; and the results of an attempt to make liquid air a pathway to perpetual motion are still in evidence, as a minus quantity, in the pockets of many who believed that all things are possible to modern science. And indeed it is this false idea of the possibility of the impossible that leads astray the followers of these false lights. Inventive science has accomplished so much—many of her achievements being so astounding that they would certainly have seemed miracles to the most intelligent men of a few generations ago—that the ordinary mind cannot see the difference between unknown possibilities and those things which well-established science pronounces to be impossible, because they contradict fundamental laws which are thoroughly established and well understood.

Thus any one who would claim that he could make a plane triangle in which the three angles would measure more than two right angles, would show by this very claim that he was entirely ignorant of the first principles of geometry. The same would be true of the man who would claim that he could give, in exact figures, the diagonal of a square of which the side is exactly one foot or one yard, and it is also true of the man who claims that he can give the exact area of a circle of which either the circumference or the diameter is known with precision. That they cannot both be known exactly is very well understood by all who have studied the subject, but that the area, the circumference, and the diameter of a circle may all be known with an exactitude which is far in excess of anything of which the human mind can form the least conception, is quite true, as we shall show when we come to consider the subject in its proper place.

These problems are not only interesting historically but they are valuable as illustrating the vagaries of the human mind and the difficulties with which the early investigators had to contend. They also show us the barriers over which we cannot pass, and they enforce the immutable character of the natural laws which govern the world around us. We hear much of the progress of science and of the changes which this progress has brought about, but these changes never affect the fundamental facts and principles upon which all true science is based. Theories and explanations and even practical applications change or pass away, so that we know them no more, but nature remains the same throughout the ages. No new theory of electricity can ever take away from the voltaic battery its power, or change it in any respect, and no new discovery in regard to the constitution of matter can ever lessen the eagerness with which carbon and oxygen combine together. Every little while we hear of some discovery that is going to upset all our preconceived notions and entirely change those laws which long experience has proved to be invariable, but in every case these alleged discoveries have turned out to be fallacies. For example, the wonderful properties of radium have led some enthusiasts to adopt the idea that many of our old notions about the conservation of energy must be abandoned, but when all the facts are carefully examined it is found that there is no rational basis for such views. Upon this point Sir Oliver Lodge says:

"There is absolutely no ground for the popular and gratuitous surmise that radium emits energy without loss or waste of any kind, and that it is competent to go on forever. The idea, at one time irresponsibly mooted, that it contradicted the principle of the conservation of energy, and was troubling physicists with the idea that they must overhaul their theories—a thing which they ought always to be delighted to do on good evidence—this idea was a gratuitous absurdity, and never had the slightest foundation. It is reasonable to suppose, however, that radium and the other like substances are drawing upon their own stores of internal atomic energy, and thereby gradually disintegrating and falling into other and ultimately more stable forms of matter."

One would naturally suppose that the extensive diffusion of sound scientific knowledge which has taken place during the century just past, would have placed these problems amongst the lumber of past ages; but it seems that some of them, particularly the squaring of the circle and perpetual motion, still occupy considerable space in the attention of the world, and even the futile chase after the "Elixir of Life" has not been entirely abandoned. Indeed certain professors who occupy prominent official positions, assert that they have made great progress towards its attainment. In view of such facts one is almost driven to accept the humorous explanation which De Morgan has offered and which he bases on an old legend relating to the famous wizard, Michael Scott. The generally accepted tradition, as related by Sir Walter Scott in his notes to the "Lay of the Last Minstrel," is as follows:

"Michael Scott was, once upon a time, much embarrassed by a spirit for whom he was under the necessity of finding constant employment. He commanded him to build a 'cauld,' or dam head across the Tweed at Kelso; it was accomplished in one night, and still does honor to the infernal architect. Michael next ordered that Eildon Hill, which was then a uniform cone, should be divided into three. Another night was sufficient to part its summit into the three picturesque peaks which it now bears. At length the enchanter conquered this indefatigable demon, by employing him in the hopeless task of making ropes out of sea-sand."

Whereupon De Morgan offers the following exceedingly interesting continuation of the legend:

"The recorded story is that Michael Scott, being bound by contract to procure perpetual employment for a number of young demons, was worried out of his life in inventing jobs for them, until at last he set them to make ropes out of sea-sand, which they never could do. We have obtained a very curious correspondence between the wizard Michael and his demon slaves; but we do not feel at liberty to say how it came into our hands. We much regret that we did not receive it in time for the British Association. It appears that the story, true as far as it goes, was never finished. The demons easily conquered the rope difficulty, by the simple process of making the sand into glass, and spinning the glass into thread which they twisted. Michael, thoroughly disconcerted, hit upon the plan of setting some to square the circle, others to find the perpetual motion, etc. He commanded each of them to transmigrate from one human body into another, until their tasks were done. This explains the whole succession of cyclometers and all the heroes of the Budget. Some of this correspondence is very recent; it is much blotted, and we are not quite sure of its meaning. It is full of figurative allusions to driving something illegible down a steep into the sea. It looks like a humble petition to be allowed some diversion in the intervals of transmigration; and the answer is:

"'Rumpat et serpens iter institutum'

"a line of Horace, which the demons interpret as a direction to come athwart the proceedings of the Institute by a sly trick."

And really those who have followed carefully the history of the men who have claimed that they had solved these famous problems, will be almost inclined to accept De Morgan's ingenious explanation as something more than a mere "skit." The whole history of the philosopher's stone, of machines and contrivances for obtaining perpetual motion, and of circle-squaring, is permeated with accounts of the most gross and obvious frauds. That ignorance played an important part in the conduct of many who have put forth schemes based upon these pretended solutions is no doubt true, but that a deliberate attempt at absolute fraud was the mainspring in many cases cannot be denied. Like Dousterswivel in "The Antiquary," many of the men who advocated these delusions may have had a sneaking suspicion that there might be some truth in the doctrines which they promulgated; but most of them knew that their particular claims were groundless, and that they were put forward for the purpose of deceiving some confiding patron from whom they expected either money or the credit and glory of having done that which had been hitherto considered impossible.

Some of the questions here discussed have been called "scientific impossibilities"—an epithet which many have considered entirely inapplicable to any problem, on the ground that all things are possible to science. And in view of the wonderful things that have been accomplished in the past, some of my readers may well ask: "Who shall decide when doctors disagree?"

Perhaps the best answer to this question is that given by Ozanam, the old historian of these and many other scientific puzzles. He claimed that "it was the business of the Doctors of the Sorbonne to discuss, of the Pope to decide, and of a mathematician to go straight to heaven in a perpendicular line!"

In this connection the words of De Morgan have a deep significance. Alluding to the difficulty of preventing men of no authority from setting up false pretensions and the impossibility of destroying the assertions of fancy speculation, he says: "Many an error of thought and learning has fallen before a gradual growth of thoughtful and learned opposition. But such things as the quadrature of the circle, etc., are never put down. And why? Because thought can influence thought, but thought cannot influence self-conceit; learning can annihilate learning; but learning cannot annihilate ignorance. A sword may cut through an iron bar, and the severed ends will not reunite; let it go through the air, and the yielding substance is whole again in a moment."

I.
SQUARING THE CIRCLE

ndoubtedly one of the reasons why this problem has received so much attention from those whose minds certainly have no special leaning towards mathematics, lies in the fact that there is a general impression abroad that the governments of Great Britain and France have offered large rewards for its solution. De Morgan tells of a Jesuit who came all the way from South America, bringing with him a quadrature of the circle and a newspaper cutting announcing that a reward was ready for the discovery in England. As a matter of fact his method of solving the problem was worthless, and even if it had been valuable, there would have been no reward.

Another case was that of an agricultural laborer who spent his hard-earned savings on a journey to London, carrying with him an alleged solution of the problem, and who demanded from the Lord Chancellor the sum of one hundred thousand pounds, which he claimed to be the amount of the reward offered and which he desired should be handed over forthwith. When he failed to get the money he and his friends were highly indignant and insisted that the influence of the clergy had deprived the poor man of his just deserts!

And it is related that in the year 1788, one of these deluded individuals, a M. de Vausenville, actually brought an action against the French Academy of Sciences to recover a reward to which he felt himself entitled. It ought to be needless to say that there never was a reward offered for the solution of this or any other of the problems which are discussed in this volume. Upon this point De Morgan has the following remarks:

"Montucla says, speaking of France, that he finds three notions prevalent among the cyclometers [or circle-squarers]: 1. That there is a large reward offered for success; 2. That the longitude problem depends on that success; 3. That the solution is the great end and object of geometry. The same three notions are equally prevalent among the same class in England. No reward has ever been offered by the government of either country. The longitude problem in no way depends upon perfect solution; existing approximations are sufficient to a point of accuracy far beyond what can be wanted. And geometry, content with what exists, has long pressed on to other matters. Sometimes a cyclometer persuades a skipper, who has made land in the wrong place, that the astronomers are in fault for using a wrong measure of the circle; and the skipper thinks it a very comfortable solution! And this is the utmost that the problem ever has to do with longitude."

In the year 1775 the Royal Academy of Sciences of Paris passed a resolution not to entertain communications which claimed to give solutions of any of the following problems: The duplication of the cube, the trisection of an angle, the quadrature of a circle, or any machine announced as showing perpetual motion. And we have heard that the Royal Society of London passed similar resolutions, but of course in the case of neither society did these resolutions exclude legitimate mathematical investigations—the famous computations of Mr. Shanks, to which we shall have occasion to refer hereafter, were submitted to the Royal Society of London and published in their Transactions. Attempts to "square the circle," when made intelligently, were not only commendable but have been productive of the most valuable results. At the same time there is no problem, with the possible exception of that of perpetual motion, that has caused more waste of time and effort on the part of those who have attempted its solution, and who have in almost all cases been ignorant both of the nature of the problem and of the results which have been already attained. From Archimedes down to the present time some of the ablest mathematicians have occupied themselves with the quadrature, or, as it is called in common language, "the squaring of the circle"; but these men are not to be placed in the same class with those to whom the term "circle-squarers" is generally applied.

As already noted, the great difficulty with most circle-squarers is that they are ignorant both of the nature of the problem to be solved and of the results which have been already attained. Sometimes we see it explained as the drawing of a square inside a circle and at other times as the drawing of a square around a circle, but both these problems are amongst the very simplest in practical geometry, the solutions being given in the sixth and seventh propositions of the Fourth Book of Euclid. Other definitions have been given, some of them quite absurd. Thus in France, in 1753, M. de Causans, of the Guards, cut a circular piece of turf, squared it, and from the result deduced original sin and the Trinity. He found out that the circle was equal to the square in which it is inscribed, and he offered a reward for the detection of any error, and actually deposited 10,000 francs as earnest of 300,000. But the courts would not allow any one to recover.

In the last number of the Athenæum for 1855 a correspondent says "the thing is no longer a problem but an axiom." He makes the square equal to a circle by making each side equal to a quarter of the circumference. As De Morgan says, he does not know that the area of the circle is greater than that of any other figure of the same circuit.

Such ideas are evidently akin to the poetic notion of the quadrature. Aristophanes, in the "Birds," introduces a geometer, who announces his intention to make a square circle. And Pope in the "Dunciad" delivers himself as follows:

Mad Mathesis alone was unconfined,
Too mad for mere material chains to bind,—
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square.

The author's note explains that this "regards the wild and fruitless attempts of squaring the circle." The poetic idea seems to be that the geometers try to make a square circle.

As stated by all recognized authorities, the problem is this: To describe a square which shall be exactly equal in area to a given circle.

The solution of this problem may be given in two ways: (1) the arithmetical method, by which the area of a circle is found and expressed numerically in square measure, and (2) the geometrical quadrature, by which a square, equal in area to a given circle, is described by means of rule and compasses alone.

Of course, if we know the area of the circle, it is easy to find the side of a square of equal area; this can be done by simply extracting the square root of the area, provided the number is one of which it is possible to extract the square root. Thus, if we have a circle which contains 100 square feet, a square with sides of 10 feet would be exactly equal to it. But the ascertaining of the area of the circle is the very point where the difficulty comes in; the dimensions of circles are usually stated in the lengths of the diameters, and when this is the case, the problem resolves itself into another, which is: To find the area of a circle when the diameter is given.

Now Archimedes proved that the area of any circle is equal to that of a triangle whose base has the same length as the circumference and whose altitude or height is equal to the radius. Therefore if we can find the length of the circumference when the diameter is given, we are in possession of all the points needed to enable us to "square the circle."

In this form the problem is known to mathematicians as that of the rectification of the curve.

In a practical form this problem must have presented itself to intelligent workmen at a very early stage in the progress of operative mechanics. Architects, builders, blacksmiths, and the makers of chariot wheels and vessels of various kinds must have had occasion to compare the diameters and circumferences of round articles. Thus in I Kings, vii, 23, it is said of Hiram of Tyre that "he made a molten sea, ten cubits from the one brim to the other; it was round all about * * * and a line of thirty cubits did compass it round about," from which it has been inferred that among the Jews, at that time, the accepted ratio was 3 to 1, and perhaps, with the crude measuring instruments of that age, this was as near as could be expected. And this ratio seems to have been accepted by the Babylonians, the Chinese, and probably also by the Greeks, in the earliest times. At the same time we must not forget that these statements in regard to the ratio come to us through historians and prophets, and may not have been the figures used by trained mechanics. An error of one foot in a hoop made to go round a tub or cistern of seven feet in diameter, would hardly be tolerated even in an apprentice.

The Egyptians seem to have reached a closer approximation, for from a calculation in the Rhind papyrus, the ratio of 3.16 to 1 seems to have been at one time in use. It is probable, however, that in these early times the ratio accepted by mechanics in general was determined by actual measurement, and this, as we shall see hereafter, is quite capable of giving results accurate to the second fractional place, even with very common apparatus.

To Archimedes, however, is generally accorded the credit of the first attempt to solve the problem in a scientific manner; he took the circumference of the circle as intermediate between the perimeters of the inscribed and the circumscribed polygons, and reached the conclusion that the ratio lay between 31⁄7 and 310⁄71, or between 3.1428 and 3.1408.

This ratio, in its more accurate form of 3.141592.. is now known by the Greek letter π (pronounced like the common word pie), a symbol which was introduced by Euler, between 1737 and 1748, and which is now adopted all over the world. I have, however, used the term ratio, or value of the ratio instead, throughout this chapter, as probably being more familiar to my readers.

Professor Muir justly says of this achievement of Archimedes, that it is "a most notable piece of work; the immature condition of arithmetic, at the time, was the only real obstacle preventing the evaluation of the ratio to any degree of accuracy whatever."

And when we remember that neither the numerals now in use nor the Arabic numerals, as they are usually called, nor any system equivalent to our decimal system, was known to these early mathematicians, such a calculation as that made by Archimedes was a wonderful feat.

If any of my readers, who are familiar with the Hebrew or Greek numbers, and the mode of representing them by letters, will try to do any of those more elaborate sums which, when worked out by modern methods, are mere child's play in the hands of any of the bright scholars in our common schools, they will fully appreciate the difficulties under which Archimedes labored.

Or, if ignorant of Greek and Hebrew, let them try it with the Roman numerals, and multiply XCVIII by MDLVII, without using Arabic or common numerals. Professor McArthur, in his article on "Arithmetic" in the Encyclopædia Britannica, makes the following statement on this point:

"The methods that preceded the adoption of the Arabic numerals were all comparatively unwieldy, and very simple processes involved great labor. The notation of the Romans, in particular, could adapt itself so ill to arithmetical operations, that nearly all their calculations had to be made by the abacus. One of the best and most manageable of the ancient systems is the Greek, though that, too, is very clumsy."

After Archimedes, the most notable result was that given by Ptolemy, in the "Great Syntaxis." He made the ratio 3.141552, which was a very close approximation.

For several centuries there was little progress towards a more accurate determination of the ratio. Among the Hindoos, as early as the sixth century, the now well-known value, 3.1416, had been obtained by Arya-Bhata, and a little later another of their mathematicians came to the conclusion that the square root of 10 was the true value of the ratio. He was led to this by calculating the perimeters of the successive inscribed polygons of 12, 24, 48, and 96 sides, and finding that the greater the number of sides the nearer the perimeter of the polygon approached the square root of 10. He therefore thought that the perimeter or circumference of the circle itself would be the square root of exactly 10. It is too great, however, being 3.1622 instead of 3.14159... The same idea is attributed to Bovillus, by Montucla.

By calculating the perimeters of the inscribed and circumscribed polygons, Vieta (1579) carried his approximation to ten fractional places, and in 1585 Peter Metius, the father of Adrian, by a lucky step reached the now famous fraction 355⁄113, or 3.14159292, which is correct to the sixth fractional place. The error does not exceed one part in thirteen millions.

At the beginning of the seventeenth century, Ludolph Van Ceulen reached 35 places. This result, which "in his life he found by much labor," was engraved upon his tombstone in St. Peter's Church, Leyden. The monument has now unfortunately disappeared.

From this time on, various mathematicians succeeded, by improved methods, in increasing the approximation. Thus in 1705, Abraham Sharp carried it to 72 places; Machin (1706) to 100 places; Rutherford (1841) to 208 places, and Mr. Shanks in 1853, to 607 places. The same computer in 1873 reached the enormous number of 707 places.

Printed in type of the same size as that used on this page, these figures would form a line nearly six feet long.

As a matter of interest I give here the value of the ratio of the circumference to the diameter, to 127 places:

3.14159 26535 89793 23846 26433 83279 50288 41971
69399 37510 58209 74944 59230 78164 06286 20899
86280 34825 34211 70679 82148 08651 32723 06647
09384 46+

The degree of accuracy which may be attained by using a ratio carried to only ten fractional places, far exceeds anything that can be required in even the finest work, and indeed it is beyond anything attainable by means of our present tools and instruments. For example: If the length of a curve of 100 feet radius were determined by a value of ten fractional places, the result would not err by the one-millionth part of an inch, a quantity which is quite invisible under the best microscopes of the present day. This shows us that in any calculations relating to the dimensions of the earth, such as longitude, etc., we have at our command, in the 127 places of figures given above, an exactness which for all practical purposes may be regarded as absolute. This will be best appreciated by a consideration of the fact that if the earth were a perfect sphere and if we knew its exact diameter, we could calculate so exactly the length of an iron hoop which would go round it, that the difference produced by a change of temperature equal to the millionth of a millionth part of a degree Fahrenheit, would far exceed the error arising from the difference between the true ratio and the result thus reached.

Such minute quantities are far beyond the powers of conception of even the most thoroughly trained human mind, but when we come to use six and seven hundred places the results are simply astounding. Professor De Morgan, in his "Budget of Paradoxes," gives the following illustration of the extreme accuracy which might be attained by the use of 607 fractional places, the highest number which had been reached when he wrote:

"Say that the blood-globule of one of our animalcules is a millionth of an inch in diameter.[1] Fashion in thought a globe like our own, but so much larger that our globe is but a blood-globule in one of its animalcules; never mind the microscope which shows the creature being rather a bulky instrument. Call this the first globule above us. Let the first globe above us be but a blood-globule, as to size, in the animalcule of a still larger globe, which call the second globe above us. Go on in this way to the twentieth globe above us. Now, go down just as far on the other side. Let the blood-globule with which we started be a globe peopled with animals like ours, but rather smaller, and call this the first globe below us. This is a fine stretch of progression both ways. Now, give the giant of the twentieth globe above us the 607 decimal places, and, when he has measured the diameter of his globe with accuracy worthy of his size, let him calculate the circumference of his equator from the 607 places. Bring the little philosopher from the twentieth globe below us with his very best microscope, and set him to see the small error which the giant must make. He will not succeed, unless his microscopes be much better for his size than ours are for ours."

It would of course be impossible for any human mind to grasp the range of such an illustration as that just given. At the same time these illustrations do serve in some measure to give us an impression, if not an idea, of the vastness on the one hand and the minuteness on the other of the measurements with which we are dealing. I therefore offer no apology for giving another example of the nearness to absolute accuracy with which the circle has been "squared."

It is common knowledge that light travels with a velocity of about 185,000 miles per second. In other words, light would go completely round the earth in a little more than one-eighth of a second, or, as Herschel puts it, in less time than it would take a swift runner to make a single stride. Taking this distance of 185,000 miles per second as our unit of measurement, let us apply it as follows:

It is generally believed that our solar system is but an individual unit in a stellar system which may include hundreds of thousands of suns like our own, with all their attendant planets and moons. This stellar system again may be to some higher system what our solar system is to our own stellar system, and there may be several such gradations of systems, all going to form one complete whole which, for want of a better name, I shall call a universe. Now this universe, complete in itself, may be finite and separated from all other systems of a similar kind by an empty space, across which even gravitation cannot exert its influence. Let us suppose that the imaginary boundary of this great universe is a perfect circle, the extent of which is such that light, traveling at the rate we have named (185,000 miles per second), would take millions of millions of years to pass across it, and let us further suppose that we know the diameter of this mighty space with perfect accuracy; then, using Mr. Shanks' 707 places of decimal fractions, we could calculate the circumference to such a degree of accuracy that the error would not be visible under any microscope now made.

An illustration which may impress some minds even more forcibly than either of those which we have just given, is as follows:

Let us suppose that in some titanic iron-works a steel armor-plate had been forged, perfectly circular in shape and having a diameter of exactly 185,000,000 miles, or very nearly that of the orbit of the earth, and a thickness of 8000 miles, or about that of the diameter of the earth. Let us further assume that, owing to the attraction of some immense stellar body, this huge mass has what we would call a weight corresponding to that which a plate of the same material would have at the surface of the earth, and let it be required to calculate the length of the side of a square plate of the same material and thickness and which shall be exactly equal to the circular plate.

Using the 707 places of figures of Mr. Shanks, the length of the required side could be calculated so accurately that the difference in weight between the two plates (the circle and the square) would not be sufficient to turn the scale of the most delicate chemical balance ever constructed.

Of course in assuming the necessary conditions, we are obliged to leave out of consideration all those more refined details which would embarrass us in similar calculations on the small scale and confine ourselves to the purely mathematical aspect of the case; but the stretch of imagination required is not greater than that demanded by many illustrations of the kind.

So much, then, for what is claimed by the mathematicians; and the certainty that their results are correct, as far as they go, is shown by the predictions made by astronomers in regard to the moon's place in the heavens at any given time. The error is less than a second of time in twenty-seven days, and upon this the sailor depends for a knowledge of his position upon the trackless deep. This is a practical test upon which merchants are willing to stake, and do stake, billions of dollars every day.

It is now well established that, like the diagonal and side of a square, the diameter and circumference of any circle are incommensurable quantities. But, as De Morgan says, "most of the quadrators are not aware that it has been fully demonstrated that no two numbers whatsoever can represent the ratio of the diameter to the circumference, with perfect accuracy. When, therefore, we are told that either 8 to 25 or 64 to 201 is the true ratio, we know that it is no such thing, without the necessity of examination. The point that is left open, as not fully demonstrated to be impossible, is the geometrical quadrature, the determination of the circumference by the straight line and circle, used as in Euclid."

But since De Morgan wrote, it has been shown that a Euclidean construction is actually impossible. Those who desire to examine the question more fully, will find a very clear discussion of the subject in Klein's "Famous Problems in Elementary Geometry." (Boston, Ginn & Co.)

There are various geometrical constructions which give approximate results that are sufficiently accurate for most practical purposes. One of the oldest of these makes the ratio 31⁄7 to 1. Using this ratio we can ascertain the circumference of a circle of which the diameter is given by the following method: Divide the diameter into 7 equal parts by the usual method. Then, having drawn a straight line, set off on it three times the diameter and one of the sevenths; the result will give the circumference with an error of less than the one twenty-five-hundredth part or one twenty-fifth of one per cent.

If the circumference had been given, the diameter might have been found by dividing the circumference into twenty-two parts and setting off seven of them. This would give the diameter. A more accurate method is as follows:

Given a circle, of which it is desired to find the length of the circumference: Inscribe in the given circle a square, and to three times the diameter of the circle add a fifth of the side of the square; the result will differ from the circumference of the circle by less than one-seventeen-thousandth part of it. Another method which gives a result accurate to the one-seventeen-thousandth part is as follows:

Fig. 1.

Let AD, Fig. 1, be the diameter of the circle, C the center, and CB the radius perpendicular to AD. Continue AD and make DE equal to the radius; then draw BE, and in AE, continued, make EF equal to it; if to this line EF, its fifth part FG be added, the whole line AG will be equal to the circumference described with the radius CA, within one-seventeen-thousandth part.

The following construction gives even still closer results: Given the semi-circle ABC, Fig. 2; from the extremities A and C of its diameter raise two perpendiculars, one of them CE, equal to the tangent of 30°, and the other AF, equal to three times the radius. If the line FE be then drawn, it will be equal to the semi-circumference of the circle, within one-hundred-thousandth part nearly. This is an error of one-thousandth of one per cent, an accuracy far greater than any mechanic can attain with the tools now in use.

Fig. 2.

When we have the length of the circumference and the length of the diameter, we can describe a square which shall be equal to the area of the circle. The following is the method:

Draw a line ACB, Fig. 3, equal to half the circumference and half the diameter together. Bisect this line in O, and with O as a center and AO as radius, describe the semi-circle ADB. Erect a perpendicular CD, at C, cutting the arc in D; CD is the side of the required square which can then be constructed in the usual manner. The explanation of this is that CD is a mean proportional between AC and CB.

Fig. 3.

De Morgan says: "The following method of finding the circumference of a circle (taken from a paper by Mr. S. Drach in the 'Philosophical Magazine,' January, 1863, Suppl.), is as accurate as the use of eight fractional places: From three diameters deduct eight-thousandths and seven-millionths of a diameter; to the result, add five per cent. We have then not quite enough; but the shortcoming is at the rate of about an inch and a sixtieth of an inch in 14,000 miles."

For obtaining the side of a square which shall be equal in area to a given circle, the empirical method, given by Ahmes in the Rhind papyrus 4000 years ago, is very simple and sufficiently accurate for many practical purposes. The rule is: Cut off one-ninth of the diameter and construct a square upon the remainder.

This makes the ratio 3.16.. and the error does not exceed one-third of one per cent.

There are various mechanical methods of measuring and comparing the diameter and the circumference of a circle, and some of them give tolerably accurate results. The most obvious device and that which was probably the oldest, is the use of a cord or ribbon for the curved surface and the usual measuring rule for the diameter. With an accurately divided rule and a thin metallic ribbon which does not stretch, it is possible to determine the ratio to the second fractional place, and with a little care and skill the third place may be determined quite closely.

An improvement which was no doubt introduced at a very early day is the measuring wheel or circumferentor. This is used extensively at the present day by country wheelwrights for measuring tires. It consists of a wheel fixed in a frame so that it may be rolled along or over any surface of which the measurement is desired.

This may of course be used for measuring the circumference of any circle and comparing it with the diameter. De Morgan gives the following instance of its use: A squarer, having read that the circular ratio was undetermined, advertised in a country paper as follows: "I thought it very strange that so many great scholars in all ages should have failed in finding the true ratio and have been determined to try myself." He kept his method secret, expecting "to secure the benefit of the discovery," but it leaked out that he did it by rolling a twelve-inch disk along a straight rail, and his ratio was 64 to 201 or 3.140625 exactly. As De Morgan says, this is a very creditable piece of work; it is not wrong by 1 in 3000.

Skilful machinists are able to measure to the one-five-thousandth of an inch; this, on a two-inch cylinder, would give the ratio correct to five places, provided we could measure the curved line as accurately as we can the straight diameter, but it is difficult to do this by the usual methods. Perhaps the most accurate plan would be to use a fine wire and wrap it round the cylinder a number of times, after which its length could be measured. The result would of course require correction for the angle which the wire would necessarily make if the ends did not meet squarely and also for the diameter of the wire. Very accurate results have been obtained by this method in measuring the diameters of small rods.

A somewhat original way of finding the area of a circle was adopted by one squarer. He took a carefully turned metal cylinder and having measured its length with great accuracy he adopted the Archimedean method of finding its cubical contents, that is to say, he immersed it in water and found out how much it displaced. He then had all the data required to enable him to calculate the area of the circle upon which the cylinder stood.

Since the straight diameter is easily measured with great accuracy, when he had the area he could readily have found the circumference by working backward the rule announced by Archimedes, viz.: that the area of a circle is equal to that of a triangle whose base has the same length as the circumference and whose altitude is equal to the radius.

One would almost fancy that amongst circle-squarers there prevails an idea that some kind of ban or magical prohibition has been laid upon this problem; that like the hidden treasures of the pirates of old it is protected from the attacks of ordinary mortals by some spirit or demoniac influence, which paralyses the mind of the would-be solver and frustrates his efforts.

It is only on such an hypothesis that we can account for the wild 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 31⁄8 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 31⁄8 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

his 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

his 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°, and knowing that it contains 90°, 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° into six equal parts of 60° 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°, and as the angle ACB is 90°, the angle ACD must be 30°, or one-third part of the whole. In the same way lay off the angle ACE of 60°, and ECB must be 30°, and the remainder DCE must also be 30°. 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°, 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."

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

t 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:

"Fig. 5 represents a large wheel, the circumference of which is furnished, at equal distances, with levers, each bearing at its extremity a weight, and movable on a hinge so that in one direction they can rest upon the circumference, while on the opposite side, being carried away by the weight at the extremity, they are obliged to arrange themselves in the direction of the radius continued. This being supposed, it is evident that when the wheel turns in the direction ABC, the weights A, B, and C will recede from the center; consequently, as they act with more force, they will carry the wheel towards that side; and as a new lever will be thrown out, in proportion as the wheel revolves, it thence follows, say they, that the wheel will continue to move in the same direction. But notwithstanding the specious appearance of this reasoning, experience has proved that the machine will not go; and it may indeed be demonstrated that there is a certain position in which the center of gravity of all these weights is in the vertical plane passing through the point of suspension, and that therefore it must stop."

Fig. 5.

Fig. 6.

Another invention of a similar kind is thus described by the same author:

"In a cylindric drum, in perfect equilibrium on its axis, are formed channels as seen in Fig. 6, which contain balls of lead or a certain quantity of quicksilver. In consequence of this disposition, the balls or quicksilver must, on the one side, ascend by approaching the center, and on the other must roll towards the circumference. The machine ought, therefore, to turn incessantly towards that side."

In his "Course of Lectures on Natural Philosophy," Dr. Thomas Young speaks of these contrivances as follows:

"One of the most common fallacies, by which the superficial projectors of machines for obtaining perpetual motion have been deluded, has arisen from imagining that any number of weights ascending by a certain path, on one side of the center of motion and descending on the other at a greater distance, must cause a constant preponderance on the side of the descent: for this purpose the weights have either been fixed on hinges, which allow them to fall over at a certain point, so as to become more distant from the center, or made to slide or roll along grooves or planes which lead them to a more remote part of the wheel, from whence they return as they ascend; but it will appear on the inspection of such a machine, that although some of the weights are more distant from the center than others, yet there is always a proportionately smaller number of them on that side on which they have the greatest power, so that these circumstances precisely counterbalance each other."

Fig. 7.

He then gives the illustration (Fig. 7), shown on the preceding page, of "a wheel supposed to be capable of producing a perpetual motion; the descending balls acting at a greater distance from the center, but being fewer in number than the ascending. In the model, the balls may be kept in their places by a plate of glass covering the wheel."

Fig. 8.

A more elaborate arrangement embodying the same idea is figured and described by Ozanam. The machine, which is shown in Fig. 8, consists of "a kind of wheel formed of six or eight arms, proceeding from a center where the axis of motion is placed. Each of these arms is furnished with a receptacle in the form of a pair of bellows: but those on the opposite arms stand in contrary directions, as seen in the figure. The movable top of each receptacle has affixed to it a weight, which shuts it in one situation and opens it in the other. In the last place, the bellows of the opposite arms have a communication by means of a canal, and one of them is filled with quicksilver.

"These things being supposed, it is visible that the bellows on the one side must open, and those on the other must shut; consequently, the mercury will pass from the latter into the former, while the contrary will be the case on the opposite side."

Ozanam naïvely adds: "It might be difficult to point out the deficiency of this reasoning; but those acquainted with the true principles of mechanics will not hesitate to bet a hundred to one, that the machine, when constructed, will not answer the intended purpose."

That this bet would have been a perfectly safe one must be quite evident to any person who has the slightest knowledge of practical mechanics, and yet the fundamental idea which is embodied in this and the other examples which we have just given, forms the basis of almost all the attempts which have been made to produce a perpetual motion by purely mechanical means.

The hydrostatic paradox by which a few ounces of liquid may apparently balance many pounds, or even tons, has frequently suggested a form of apparatus designed to secure a perpetual motion. Dr. Arnott, in his "Elements of Physics," relates the following anecdote: "A projector thought that the vessel of his contrivance, represented here (Fig. 9), was to solve the renowned problem of the perpetual motion. It was goblet-shaped, lessening gradually towards the bottom until it became a tube, bent upwards at c and pointing with an open extremity into the goblet again. He reasoned thus: A pint of water in the goblet a must more than counterbalance an ounce which the tube b will contain, and must, therefore, be constantly pushing the ounce forward into the vessel again at a, and keeping up a stream or circulation, which will cease only when the water dries up. He was confounded when a trial showed him the same level in a and in b."

Fig. 9.

This suggestion has been adopted over and over again by sanguine inventors. Dircks, in his "Perpetuum Mobile," tells us that a contrivance, on precisely the same principle, was proposed by the Abbé de la Roque, in "Le Journal des Sçavans," Paris, 1686. The instrument was a U tube, one leg longer than the other and bent over, so that any liquid might drop into the top end of the short leg, which he proposed to be made of wax, and the long one of iron. Presuming the liquid to be more condensed in the metal than the wax tube, it would flow from the end into the wax tube and so continue.

This is a typical case. A man of learning and of high position is so confident that his theory is right that he does not think it worth while to test it experimentally, but rushes into print and immortalizes himself as the author of a blunder. It is safe to say that this absurd invention will do more to perpetuate his name than all his learning and real achievements. And there are others in the same predicament—circle-squarers who, a quarter of a century hence, will be remembered for their errors when all else connected with them will be forgotten.

To every miller whose mill ceased working for want of water, the idea has no doubt occurred that if he could only pump the water back again and use it a second or a third time he might be independent of dry or wet seasons. Of course no practical miller was ever so far deluded as to attempt to put such a suggestion into practice, but innumerable machines of this kind, and of the most crude arrangement, have been sketched and described in magazines and papers. Figures of wheels driving an ordinary pump, which returns to an elevated reservoir the water which has driven the wheel, are so common that it is not worth while to reproduce any of them. In the following attempt, however, which is copied from Bishop Wilkins' famous book, "Mathematical Magic" (1648), the well-known Archimedean screw is employed instead of a pump, and the naïveté of the good bishop's description and conclusion are well worth the space they will occupy.

After an elaborate description of the screw, he says: "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 (Fig. 10):

"Where the figure LM at the bottom does represent a wooden cylinder with helical cavities cut in it, which at AB is supposed to be covered over with tin plates, and three waterwheels, upon it, HIK; the lower cistern, which contains the water, being CD. 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 insufficient, 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 twice as much water to turn the screw as is carried up by it.

Fig. 10.

"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 'Eureka! Eureka!' 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."

How well it would have been for many of those inventors, who supposed that they had discovered a successful perpetual motion, if they had only given their contrivances a fair and unprejudiced test as did the good old bishop!

A modification of this device, in which mercury is used instead of water, is thus described by a correspondent of "The Mechanic's Magazine." (London.)

"In Fig. 11, A is the screw turning on its two pivots GG; B is a cistern to be filled above the level of the lower aperture of the screw with mercury, which I conceive to be preferable to water on many accounts, and principally because it does not adhere or evaporate like water; C is a reservoir, which, when the screw is turned round, receives the mercury which falls from the top; there is a pipe, which, by the force of gravity, conveys the mercury from the reservoir C on to (what for want of a better term may be called) the float-board E, fixed at right angles to the center [axis] of the screw, and furnished at its circumference with ridges or floats to intercept the mercury, the moment and weight of which will cause the float-board and screw to revolve, until, by the proper inclination of the floats, the mercury falls into the receiver F, from whence it again falls by its spout into the cistern G, where the constant revolution of the screw takes it up again as before."

He then suggests some difficulties which the ball, seen just under the letter E, is intended to overcome, but he confesses that he has never tried it, and to any practical mechanic it is very obvious that the machine will not work. But we give the description in the language of the inventor, as a fair type of this class of perpetual-motion machines.

Fig. 11.

In the year 1790 a Doctor Schweirs took out a patent for a machine in which small metal balls were used instead of a liquid, and they were raised by a sort of chain pump which delivered them upon the circumference of a large wheel, which was thus caused to revolve. It was claimed for this invention that it kept going for some months, but any mechanic who will examine the Doctor's drawing must see that it could not have continued in motion after the initial impulse had been expended.

That property of liquids known as capillary attraction has been frequently called to the aid of perpetual-motion seekers, and the fact that although water will, in capillary tubes and sponges, rise several inches above the general level, it will not overflow, has been a startling surprise to the would-be inventors. Perhaps the most notable instance of a mistake of this kind occurred in the case of the famous Sir William Congreve, the inventor of the military rockets that bore his name, and the author of certain improvements in matches which were called after him. It was thus described and figured in an article which appeared in the "Atlas" (London) and was copied into "The Mechanic's Magazine" (London) for 1827:

"The celebrated Boyle entertained an idea that perpetual motion might be obtained by means of capillary attraction; and, indeed, there seems but little doubt that nature has employed this force in many instances to produce this effect.

"There are many situations in which there is every reason to believe that the sources of springs on the tops and sides of mountains depend on the accumulation of water created at certain elevations by the operation of capillary attraction, acting in large masses of porous material, or through laminated substances. These masses being saturated, in process of time become the sources of springs and the heads of rivers; and thus by an endless round of ascending and descending waters, form, on the great scale of nature, an incessant cause of perpetual motion, in the purest acceptance of the term, and precisely on the principle that was contemplated by Boyle. It is probable, however, that any imitation of this process on the limited scale practicable by human art would not be of sufficient magnitude to be effective. Nature, by the immensity of her operations, is able to allow for a slowness of process which would baffle the attempts of man in any direct and simple imitation of her works. Working, therefore, upon the same causes, he finds himself obliged to take a more complicated mode to produce the same effect.

"To amuse the hours of a long confinement from illness, Sir William Congreve has recently contrived a scheme of perpetual motion, founded on this principle of capillary attraction, which, it is apprehended, will not be subject to the general refutation applicable to those plans in which the power is supposed to be derived from gravity only. Sir William's perpetual motion is as follows:

Fig. 12.

"Let ABC, Fig. 12, be three horizontal rollers fixed in a frame; aaa, etc., is an endless band of sponge, running round these rollers; and bbb, etc., is an endless chain of weights, surrounding the band of sponge, and attached to it, so that they must move together; every part of this band and chain being so accurately uniform in weight that the perpendicular side AB will, in all positions of the band and chain, be in equilibrium with the hypothenuse AC, on the principle of the inclined plane. Now, if the frame in which these rollers are fixed be placed in a cistern of water, having its lower part immersed therein, so that the water's edge cuts the upper part of the rollers BC, then, if the weight and quantity of the endless chain be duly proportioned to the thickness and breadth of the band of sponge, the band and chain will, on the water in the cistern being brought to the proper level, begin to move round the rollers in the direction AB, by the force of capillary attraction, and will continue so to move. The process is as follows:

"On the side AB of the triangle, the weights bbb, etc., hanging perpendicularly alongside the band of sponge, the band is not compressed by them, and its pores being left open, the water at the point x, at which the band meets its surface, will rise to a certain height y, above its level, and thereby create a load, which load will not exist on the ascending side CA, because on this side the chain of weights compresses the band at the water's edge, and squeezes out any water that may have previously accumulated in it; so that the band rises in a dry state, the weight of the chain having been so proportioned to the breadth and thickness of the band as to be sufficient to produce this effect. The load, therefore, on the descending side AB, not being opposed by any similar load on the ascending side, and the equilibrium of the other parts not being disturbed by the alternate expansion and compression of the sponge, the band will begin to move in the direction AB; and as it moves downwards, the accumulation of water will continue to rise, and thereby carry on a constant motion, provided the load at xy be sufficient to overcome the friction on the rollers ABC.

"Now to ascertain the quantity of this load in any particular machine, it must be stated that it is found by experiment that the water will rise in a fine sponge about an inch above its level; if, therefore, the band and sponge be one foot thick and six feet broad, the area of its horizontal section in contact with the water would be 864 square inches, and the weight of the accumulation of water raised by the capillary attraction being one inch rise upon 864 square inches, would be 30 lb., which, it is conceived, would be much more than equivalent to the friction of the rollers."

The article, inspired no doubt by Sir William, then goes on to give elaborate reasons for the success of the device, but all these are met by the damning fact that the machine never worked. Some time afterwards Sir William, at considerable expense, published a pamphlet in which he explained and defended his views. If he had only had a working model made and the thing had continued in motion for a few hours, he would have silenced all objectors far more quickly and forcibly than he ever could have done by any amount of argument.

And in his case there could have been no excuse for his not making a small machine after the plans that he published and even patented. He was wealthy and could have commanded the services of the best mechanics in London, but no working model was ever made. Many inventors of perpetual-motion machines offer their poverty as an excuse for not making a model or working machine. Thus Dircks, in his "Perpetuum Mobile" gives an account of "a mechanic, a model maker, who had a neat brass model of a time-piece, in which were two steel balls A and B;—B to fall into a semicircular gallery C, and be carried to the end D of a straight trough DE; while A in its turn rolls to E, and so on continuously; only the gallery C not being screwed in its place, we are desired to take the will for the deed, until twenty shillings be raised to complete this part of the work!"

And Mr. Dircks also quotes from the "Builder" of June, 1847: "This vain delusion, if not still in force, is at least as standing a fallacy as ever. Joseph Hutt, a frame-work knitter, in the neighborhood of the enlightened town of Hinckley, professes to have discovered it [perpetual motion] and only wants twenty pounds, as usual, to set it agoing."

The following rather curious arrangement was described in "The Mechanic's Magazine" for 1825.

"I beg leave to offer the prefixed device. The point at which, like all the rest, it fails, I confess I did not (as I do now) plainly perceive at once, although it is certainly very obvious. The original idea was this—to enable a body which would float in a heavy medium and sink in a lighter one, to pass successively through the one to the other, the continuation of which would be the end in view. To say that valves cannot be made to act as proposed will not be to show the rationale (if I may so say) upon which the idea is fallacious."

The figure is supposed to be tubular, and made of glass, for the purpose of seeing the action of the balls inside, which float or fall as they travel from air through water and from water through air. The foot is supposed to be placed in water, but it would answer the same purpose if the bottom were closed.

Description of the Engraving, Fig. 13. No. 1, the left leg, filled with water from B to A. 2 and 3, valves, having in their centers very small projecting valves; they all open upwards. 4, the right leg, containing air from A to F. 5 and 6, valves, having very small ones in their centers; they all open downwards. The whole apparatus is supposed to be air and water-tight. The round figures represent hollow balls, which will sink one-fourth of their bulk in water (of course will fall in air); the weight therefore of three balls resting upon one ball in water, as at E, will just bring its top even with the water's edge; the weight of four balls will sink it under the surface until the ball immediately over it is one-fourth its bulk in water, when the under ball will escape round the corner at C, and begin to ascend.

"The machine is supposed (in the figure) to be in action, and No. 8 (one of the balls) to have just escaped round the corner at C, and to be, by its buoyancy, rising up to valve No. 3, striking first the small projecting valve in the center, which when opened, the large one will be raised by the buoyancy of the ball; because the moment the small valve in the center is opened (although only the size of a pin's head), No. 2 valve will have taken upon itself to sustain the whole column of water from A to B. The said ball (No. 8) having passed through the valve No. 3, will, by appropriate weights or springs, close; the ball will proceed upwards to the next valve (No. 2), and perform the same operation there. Having arrived at A, it will float upon the surface three-fourths of its bulk out of water. Upon another ball in due course arriving under it, it will be lifted quite out of the water, and fall over the point D, pass into the right leg (containing air), and fall to valve No. 5, strike and open the small valve in its center, then open the large one, and pass through; this valve will then, by appropriate weights or springs, close; the ball will roll on through the bent tube (which is made in that form to gain time as well as to exhibit motion) to the next valve (No. 6), where it will perform the same operation, and then, falling upon the four balls at E, force the bottom one round the corner at C. This ball will proceed as did No. 8, and the rest in the same manner successively."

Fig. 13.

That an ordinary amateur mechanic should be misled by such arguments is perhaps not so surprising, when we remember that the famous John Bernoulli claimed to have invented a perpetual motion based on the difference between the specific gravities of two liquids. A translation of the original Latin may be found in the Encyclopædia Britannica, Vol. XVIII, page 555. Some of the premises on which he depends are, however, impossibilities, and Professor Chrystal concludes his notice of the invention thus: "One really is at a loss with Bernoulli's wonderful theory, whether to admire most the conscientious statement of the hypothesis, the prim logic of the demonstration—so carefully cut according to the pattern of the ancients—or the weighty superstructure built on so frail a foundation. Most of our perpetual motions were clearly the result of too little learning; surely this one was the product of too much."

A more simple device was suggested recently by a correspondent of "Power." He describes it thus:

The J-shaped tube A, Fig. 14, is open at both ends, but tapers at the lower end, as shown. A well-greased cotton rope C passes over the wheel B and through the small opening of the tube with practically little or no friction, and also without leakage. The tube is then filled with water. The rope above the line WX balances over the pulley, and so does that below the line YZ. The rope in the tube between these lines is lifted by the water, while the rope on the other side of the pulley between these lines is pulled downward by gravity.

Fig. 14.

The inventor offers the above suggestion rather as a kind of puzzle than as a sober attempt to solve the famous problem, and he concludes by asking why it will not work?

In addition to the usual resistance or friction offered by the air to all motion, there are four drawbacks:

1. The friction in its bearings of the axle of the wheel B.

2. The power required to bend and unbend the rope.

3. The friction of the rope in passing through the water from z to x and its tendency to raise a portion of the water above the level of the water at x.

4. The friction at the point y, this last being the most serious of all. An "opening of the tube with practically little or no friction, and also without leakage" is a mechanical impossibility. In order to have the joint water-tight, the tube must hug the rope very tightly and this would make friction enough to prevent any motion. And the longer the column of water xz, the greater will be the tendency to leak, and consequently the tighter must be the joint and the greater the friction thereby created.

A favorite idea with perpetual-motion seekers is the utilization of the force of magnetism. Some time prior to the year 1579, Joannes Taisnierus wrote a book which is now in the British Museum and in which considerable space is devoted to "Continual Motions" and to the solving of this problem by magnetism. Bishop Wilkins in his "Mathematical Magick" describes one of the many devices which have been invented with this end in view. He says: "But amongst all these kinds of invention, 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 (Fig. 15):

"Suppose the loadstone to be represented at AB, 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 EF. 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."

Fig. 15.

Notwithstanding the positiveness of the "must" at the close of his description, it is very obvious to any practical mechanic that the machine will not move at all, far less move perpetually, and the bishop himself, after carefully and conscientiously discussing the objections, comes to the same conclusion. He ends by saying: "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."

It has occurred to several would-be inventors of perpetual motion that if some substance could be found which would prevent the passage of the magnetic force, then by interposing a plate of this material at the proper moment, between the magnet and the piece of iron to be attracted, a perpetual motion might be obtained. Several inventors have claimed that they had discovered such a non-conducting substance, but it is needless to say that their claims had no foundation in fact, and if they had discovered anything of the kind, it would have required just as much force to interpose it as would have been gained by the interposition. It has been fully proved that in every case where a machine was made to work apparently by the interposition of such a material, a fraud was perpetrated and the machine was really made to move by means of some concealed springs or weights.

A correspondent of the "Mechanic's Magazine" (Vol. xii, London, 1829), gives the following curious design for a "Self-moving Railway Carriage." He describes it as a machine which, were it possible to make its parts hold together unimpaired by rotation or the ravages of time, and to give it a path encircling the earth, would assuredly continue to roll along in one undeviating course until time shall be no more.

A series of inclined planes are to be erected in such a manner that a cone will ascend one (its sides forming an acute angle), and being raised to the summit, descend on the next (having parallel sides), at the foot of which it must rise on a third and fall on a fourth, and so continue to do alternately throughout.

The diagram, Fig. 16, is the section of a carriage A, with broad conical wheels a, a, resting on the inclined plane b. The entrance to the carriage is from above, and there are ample accommodations for goods and passengers. "The most singular property of this contrivance is, that its speed increases the more it is laden; and when checked on any part of the road, it will, when the cause of stoppage is removed, proceed on its journey by mere power of gravity. Its path may be a circular road formed of the inclined planes. But to avoid a circuitous route, a double road ought to be made. The carriage not having a retrograde motion on the inclined planes, a road to set out upon, and another to return by, are indispensable."

Fig. 16.

How any one could ever imagine that such a contrivance would ever continue in motion for even a short time, except, perhaps, on the famous descensus averni, must be a puzzle to every sane mechanic. I therefore give it as a climax to the absurdities which have been proposed in sober earnest. As a fitting close, however, to this chapter of human folly, I give the following joke from the "Penny Magazine," published by the Society for the Diffusion of Useful Knowledge.

"'Father, I have invented a perpetual motion!' said a little fellow of eight years old. 'It is thus: I would make a great wheel, and fix it up like a water-wheel; at the top I would hang a great weight, and at the bottom I would hang a number of little weights; then the great weight would turn the wheel half round and sink to the bottom, because it is so heavy: and when the little weights reach the top they would sink down, because they are so many; and thus the wheel would turn round for ever.'"

The child's fallacy is a type of all the blunders which are made on this subject. Follow a projector in his description, and if it be not perfectly unintelligible, which it often is, it always proves that he expects to find certain of his movements alternately strong and weak—not according to the laws of nature—but according to the wants of his mechanism.

2. FALLACIES

Fallacies are distinguished from absurdities on the one hand and from frauds on the other, by the fact that without any intentionally fraudulent contrivances on the part of the inventor, they seem to produce results which have a tendency to afford to certain enthusiasts a basis of hope in the direction of perpetual motion, although usually not under that name, for that is always explicitly disclaimed by the promoters.

The most notable instance of this class in recent times was the application of liquid air as a source of power, the claim having been actually made by some of the advocates of this fallacy that a steamship starting from New York with 1000 gallons of liquid air, could not only cross the Atlantic at full speed but could reach the other side with more than 1000 gallons of liquid air on board—the power required to drive the vessel and to liquefy the surplus air being all obtained during the passage by utilizing the original quantity of liquid air that had been furnished in the first place.

That this was equivalent to perpetual motion, pure and simple, was obvious even to those who were least familiar with such subjects, though the idea of calling it perpetual motion was sternly repudiated by all concerned—the term "perpetual motion" having become thoroughly offensive to the ears of common-sense people, and consequently tending to cast doubt over any enterprise to which it might be applied.

That liquid air is a real and wonderful discovery, and that for a certain small range of purposes it will prove highly useful, cannot be doubted by those who have seen and handled it and are familiar with its properties, but that it will ever be successfully used as an economical source of mechanical power is, to say the least, very improbable. That a small quantity of the liquid is capable of doing an enormous amount of work, and that under some conditions there is apparently more power developed than was originally required to liquefy the air, is undoubtedly true, but when a careful quantitative examination is made of the outgo and the income of energy, it will be found in this, as in every similar case, that instead of a gain there is a very decided and serious loss. The correct explanation of the fallacy was published in the "Scientific American," by the late Dr. Henry Morton, president of the Stevens Institute, and the same explanation and exposure were made by the writer, nearly fifty years ago, in the case of a very similar enterprise. The form of the fallacy in both cases is so similar and so interesting that I shall make no apology for giving the details.

About the year 1853 or 1854, two ingenious mechanics of Rochester, N. Y., conceived the idea that by using some liquid more volatile than water, a great saving might be effected in the cost of running an engine. At that time gasolene and benzine were unknown in commerce, and the same was true in regard to bisulphide of carbon, but as the process of manufacturing the latter was simple and the sources of supply were cheap and apparently unlimited, they adopted that liquid. The name of one of these inventors was Hughes and that of the other was Hill, and it would seem that each had made the invention independently of the other. They had a fierce conflict over the patent, but this does not concern us except to this extent, that the records of the case may therefore be found in the archives of the Patent Office at Washington, D.C. Hughes was backed by the wealth of a well-known lawyer of Rochester, whose son subsequently occupied a high office in the state of New York, and he constructed a beautiful little steam-engine and boiler, made of the very finest materials and with such skill and accuracy that it gave out a very considerable amount of power in proportion to its size. The source of heat was a series of lamps, fed, I think, with lard oil (this was before the days of kerosene), and the exhibition test consisted in first filling the boiler with water, and noting the time that it took to get up a certain steam pressure as shown by the gage. After this test, bisulphide of carbon was added to the water, and the time and pressure were noted. The difference was of course remarkable, and altogether in favor of the new liquid. The exhaust was carried into a vessel of cold water and as bisulphide of carbon is very easily condensed and very heavy, almost the entire quantity used was recovered and used over and over again.

But to the uninstructed onlooker, the most remarkable part of the exhibition was when the steam pressure was so far lowered that the engine revolved very slowly, and then, on a little bisulphide being injected into the boiler, the pressure would at once rise, and the engine would work with great rapidity. This seemed almost like magic.

The same experiment was tried on an engine of twelve horse-power, and with a like result. When the steam pressure had fallen so far that the engine began to move quite slowly, a quantity of the bisulphide would be injected into the boiler and the pressure would at once rise, the engine would move with renewed vigor, and the fly-wheel would revolve with startling velocity. All this was seen over and over again by myself and others. At that time the writer, then quite a young man, had just recovered from a very severe illness and was making a living by teaching mechanical drawing and making drawings for inventors and others, and in the course of business he was brought into contact with some parties who thought of investing in the new and apparently wonderful invention. They employed him to examine it and give an opinion as to its value. After careful consideration and as thorough a calculation as the data then at command would allow, he showed his clients that the tests which had been exhibited to them proved nothing, and that if a clear proof of the value of the invention was to be given, it must be after a run of many hours and not of a few minutes, and against a properly adjusted load, the amount of which had been carefully ascertained. This test was never made, or if made the results were not communicated to the prospective purchasers; the negotiations fell through, and the invention which was to have revolutionized our mechanical industries fell into "innocuous desuetude."

That the inventors were honest I have no doubt. They were themselves deceived when they saw the engine start off with tremendous velocity as soon as a little bisulphide of carbon was injected into the boiler, and they failed to see that this spurt, if I may use the expression, was simply due to a draft upon capital previously stored up. The capacity of bisulphide of carbon for heat is quite low, when compared with that of water; its vaporizing point is also much lower and consequently, an ordinary boiler full of hot water contains enough heat to vaporize a considerable quantity of bisulphide of carbon at a pretty high pressure.

In even a still greater measure the same is true of liquid air, and this was the underlying fallacy in the case of the tests made with liquid-air motors.

3. FRAUDS

But while the inventors of these schemes may have been honest, there is another class who deliberately set out to perpetrate a fraud. Their machines work, and work well, but there is always some concealed source of power, which causes them to move. As a general rule, such inventors form a company or corporation of unlimited "lie-ability," as De Morgan phrases it, and then they proceed by means of flaring prospectuses and liberal advertising, to gather in the dupes who are attracted by their seductive promises of enormous returns for a very small outlay. Perhaps the most widely known of these fraudulent schemes of recent years was the notorious Keeley motor, the originator of which managed to hoodwink a respectable old lady, and to draw from her enormous supplies of cash. At his death, however, the absolutely fraudulent nature of his contrivances was fully disclosed, and nothing more has been heard of his alleged discovery. But, while he lived and was able to put forward claims based upon some apparent results, he found plenty of fools who accepted the idea that there is nothing impossible to science.

It is true that the Keeley motor was examined by several committees and some very respectable gentlemen acted in such a way as to give a seeming endorsement of the scheme, but it must not be supposed for an instant that any well-educated engineers and scientific men were deceived by Mr. Keeley's nonsense. The very fact that he refused to allow a complete examination of his machine by intelligent practical men, ought to have been enough to condemn his scheme, for if he had really made the discovery which he claimed there would have been no difficulty in proving it practically and thoroughly, and then he might have formed company after company that would have rewarded him with "wealth beyond the dreams of avarice."

The Keeley motor was not put forward as a perpetual motion; in these days none of these schemes is admitted to be a perpetual motion, for that term has now become exceedingly offensive and would condemn any invention; but the result is the same in the end, and the whole history of perpetual motion is permeated with frauds of this kind, some of them having been so simple that they were obvious to even the most unskilled observer, while others were exceedingly complicated and most ingeniously concealed. Many years ago a number of these fraudulent perpetual-motion machines were manufactured in America and sent over to Great Britain for exhibition, and quite a lucrative business was done by showing them in various towns. But the fraud was soon detected and the British police then made it too warm for these swindlers.

Mr. Dircks, in his "Perpetuum Mobile," has given accounts of quite a number of these impostures. The following are some of the most notable:

M. Poppe of Tübingen tells of a clock made by M. Geiser, which was an admirable piece of mechanism and seemed to have solved this great problem in an ingenious and simple manner, but it deceived only for a time. When thoroughly examined inwardly and outwardly, some time after his death, it was found that the center props supporting its cylinders contained cleverly constructed, hidden clock-work, wound up by inserting a key in a small hole under the second-hand.

Another case was that of a man named Adams who exhibited, for eight or nine days, his pretended perpetual motion in a town in England and took in the natives for fifty or sixty pounds. Accident, however, led to a discovery of the imposture. A gentleman, viewing the machine took hold of the wheel or trundle and lifted it up a little, which probably disengaged the wheels that connected the hidden machinery in the plinth, and immediately he heard a sound similar to that of a watch when the spring is running down. The owner was in great anger and directly put the wheel into its proper position, and the machine again went around as before. The circumstance was mentioned to an intelligent person who determined to find out and expose the imposture. He took with him a friend to view the machine and they seated themselves one on each side of the table upon which the machine was placed. They then took hold of the wheel and trundle and lifted them up, there being some play in the pivots. Immediately the hidden spring began to run down and they continued to hold the machine in spite of the endeavors of the owner to prevent them. When the spring had run down, they placed the machine again on the table and offered the owner fifty pounds if it could then set itself going, but notwithstanding his fingering and pushing, it remained motionless. A constable was sent for, the impostor went before a magistrate and there signed a paper confessing his perpetual motion to be a cheat.

In the "Mechanic's Magazine," Vol. 46, is an account of a perpetual motion, constructed by one Redhoeffer of Pennsylvania, which obtained sufficient notoriety to induce the Legislature to appoint a committee to enquire into its merits. The attention of Mr. Lukens was turned to the subject, and although the actual moving cause was not discovered, yet the deception was so ingeniously imitated in a machine of similar appearance made by him and moved by a spring so well concealed, that the deceiver himself was deceived and Redhoeffer was induced to believe that Mr. Lukens had been successful in obtaining a moving power in some way in which he himself had failed, when he had produced a machine so plausible in appearance as to deceive the public.

Instances of a similar kind might be multiplied indefinitely.

The experienced mechanic who reads the descriptions here given of the various devices which have been proposed for the construction of a perpetual-motion machine must be struck with the childish simplicity of the plans which have been offered; and those who will search the pages of the mechanical journals of the last century or who will examine the two closely printed volumes in which Mr. Dircks has collected almost everything of the kind, will be astonished at the sameness which prevails amongst the offerings of these would-be inventors. Amongst the hundreds, or, perhaps, thousands, of contrivances which have been described, there is probably not more than a dozen kinds which differ radically from each other; the same arrangement having been invented and re-invented over and over again. And one of the strange features of the case is that successive inventors seem to take no note of the failure of those predecessors who have brought forward precisely the same combination of parts under a very slightly different form.

It is true that we occasionally find a very elaborate and apparently complicated machine, but in such cases it will be found, on close examination, to owe its apparent complexity to a mere multiplication of parts; no real inventive ingenuity is exhibited in any case.

Another singular characteristic of almost all those who have devoted themselves to the search for a perpetual motion is their absolute confidence in the success of the plans which they have brought forth. So confident are they in the soundness of their views and so sure of the success of their schemes that they do not even take the trouble to test their plans but announce them as accomplished facts, and publish their sketches and descriptions as if the machine was already working without a hitch. Indeed, so far was one inventor carried away with this feeling of confidence in the success of his machine that he no longer allowed himself to be troubled with any doubts as to the machine's going but was greatly puzzled as to what means he should take to stop it after it had been set in motion!

These facts, which are well known to all who have been brought into contact with this class of minds, explain many otherwise puzzling circumstances and enable us to place a proper value on assertions which, if not made so positively and by such apparently good authority, would be at once condemned as deliberate falsehoods. That falsehood, pure and simple, has formed the basis of a good many claims of this kind, there can be no doubt, but at the same time, it is probable that some of the claimants really deceived themselves and attributed to causes other than radical errors of theory, the fact that their machines would not continue to move.

While many have claimed the actual invention of a perpetual motion it is very certain that not one has ever succeeded. How, then, are we to explain the statements which have been made in regard to Orffyreus and the claims of the Marquis of Worcester? For both of these men it is claimed that they constructed wheels which were capable of moving perpetually and apparently strong testimony is offered in support of these assertions.

In the famous "Century of Inventions," published by the Marquis in 1663, four years before his death, the celebrated 56th article reads as follows (verbatim et literatim):

"To provide and make that all the Weights of the descending side of a Wheel shall be perpetually further from the Centre, then those of the mounting side, and yet equal in number and heft to the one side as the other. A most incredible thing, if not seen, but tried before the late king (of blessed memory) in the Tower, by my directions, two Extraordinary Embassadors accompanying His Majesty, and the Duke of Richmond and Duke Hamilton, with most of the Court, attending Him. The Wheel was 14. Foot over, and 40. Weights of 50. pounds apiece. Sir William Balfore, then Lieutenant of the Tower, can justifie it, with several others. They all saw, that no sooner these great Weights passed the Diameter-line of the lower side, but they hung a foot further from the Centre, nor no sooner passed the Diameter-line of the upper side, but they hung a foot nearer. Be pleased to judge the consequence."

Such is the account given by the Marquis himself, and that he exhibited such a wheel at the time and place which he names, I have not the least doubt. And that some of the weights on one side hung a foot further from the center than did weights on the other side is also no doubt true, but, as the judging of the "consequence" is left to ourselves we know that after the first impulse given to it had been expended, the wheel would simply stand still unless kept in motion by some external force.

Fig. 17.

Mr. Dircks in his "Life, Times and Scientific Labours of the Second Marquis of Worcester," gives an engraving of a wheel which complies with all the conditions laid down by the Marquis and which is thus described:

"Let the annexed diagram, Fig. 17, represent a wheel of 14 feet in diameter, having 40 spokes, seven feet each, and with an inner rim coinciding with the periphery, at one foot distance all round. Next provide 40 balls or weights, hanging in the center of cords or chains two feet long. Now, fasten one end of this cord at the top of the center spoke C, and the other end of the cord to the next right-hand spoke one foot below the upper end, or on the inner ring; proceed in like manner with every other spoke in succession; and it will be found that, at A, the cord will have the position shown outside the wheel; while at B, C, and D, it will also take the respective positions, as shown on the outside. The result in this case will be, that all the weights on the side A, C, D, hang to the great or outer circle, while on the side B, C, D, all the weights are suspended from the lesser or inner circle. And if we reverse the motion of the wheel, turning it from the right to the left hand, we shall reverse these positions also (the lower end of the cord sliding in a groove towards a left-hand spoke), but without the wheel having any tendency to move of itself."

But it is quite as likely that the wheel constructed by the Marquis was like one of the "overbalancing" wheels described at the beginning of this article.

It is upon this "scantling" that has been based the claim that the Marquis really invented a perpetual motion, but to those who have seen much of inventors of this kind, the discrepancy between the suggested claim made by the Marquis and what we know must have been the actual results, is easily explained. The Marquis felt sure that the thing ought to work, and the excuse for its not doing so was probably the imperfect manner in which the wheel was made. Only put a little better work on it, says the inventor, and it will go.

Caspar Kaltoff, mechanician to the Marquis, probably got the wheel up in a hurry so as to exhibit it on the occasion of the king's visit to the tower. If he only had had a little more time he would have made a machine that would have worked. (?) I have heard the same excuse under almost the same circumstances, scores of times.

The case of Orffyreus was very different. The real name of this inventor was Jean Ernest Elie-Bessler, and he is said to have manufactured the name Orffyreus by placing his own name between two lines of letters, and picking out alternate letters above and below. He was educated for the church, but turned his attention to mechanics and became an expert clock maker. His character, as given by his contemporaries was fickle, tricky, and irascible. Having devised a scheme for perpetual motion he constructed several wheels which he claimed to be self-moving. The last one which he made was 12 feet in diameter and 14 inches deep, the material being light pine boards, covered with waxed cloth to conceal the mechanism. The axle was 8 inches thick, thus affording abundant space for concealed machinery.

This wheel was submitted to the Landgrave of Hesse who had it placed in a room which was then locked, and the lock secured with the Landgrave's own seal. At the end of forty days it was found to be still running.

Professor Gravesande having been employed by the Landgrave to make an examination and pronounce upon its merits, he endeavored to perform his work thoroughly; this so irritated Orffyreus that the latter broke the machine in pieces, and left on the wall a writing stating that he had been driven to do this by the impertinent curiosity of the Professor!

I have no doubt that this was a clear case of fraud, and that the wheel was driven by some mechanism concealed in the huge axle. As already stated, Orffyreus was at one time a clock maker; now clocks have been made to go for a whole year without having to be rewound, so that forty days was not a very long time for the apparatus to keep in motion.

Professor Gravesande seems to have had some faith in the invention, but then we must remember that it would not have been very difficult to deceive an honest old professor whose confidence in humanity was probably unbounded. The crowning argument against the genuineness of the motion was the fact that the inventor refused to allow a thorough examination, although a wealthy patron stood ready with a large reward if the machine could be proved to be what was claimed.

And now comes up the question which has arisen in regard to other problems, and will recur again and again to the end of the chapter: Is a perpetual motion machine one of the scientific impossibilities?

The answer to this question lies in the fact that there is no principle more thoroughly established than that no combination of machinery can create energy. So far as our present knowledge of nature goes we might as well try to create matter as to create energy, and the creation of energy is essential to the successful working of a perpetual-motion machine because some power must always be lost through friction and other resistances and must be supplied from some source if the machine is to keep on moving. And since the law of the conservation of energy makes it positive that no more power can be given out by a machine than was originally supplied to it, it seems as certain as anything can be that the construction of a perpetual-motion machine is one of the impossibilities.