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.