FOOTNOTES:

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 whichshall 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 verysimple 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 3.140625exactly. 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 thehidden 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 awell, 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 31⁄8times 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 31⁄8to 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 ofcutting 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.

[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.

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.

hisproblem 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 isone 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 carryfourtimes the supply.

A square whose side is twice the length of another, anda 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.

hisproblem 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 toallcases so that we require a method by whichanyangle 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 thesquare 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 requiresabsoluteaccuracy, not a mere approximation.

tis 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 mechanicalinventor 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 vibratingperpetually. 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.

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 aworkingmodel 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.

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 othermust 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 inthe 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 atcand pointing with an open extremity into the goblet again. Hereasoned thus: A pint of water in the gobletamust more than counterbalance an ounce which the tubebwill contain, and must, therefore, be constantly pushing the ounce forward into the vessel again ata, 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 inaand inb."

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 upmay 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 bywhich 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.

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