Part 8

See Translation, by Admiral W. H. Smith, and Robert Grant, M.A., in 2 vols. 8vo. 1855, Vol. I., page 10.

"The sect of squarers then," Arago adds,--"are searching after a solution which is proved to be impossible, and which, moreover, would be of no practical use, even if their foolish hopes were crowned with success."

In the "Birds" of Aristophanes, the character is introduced of a geometer, who is going to make a square circle, showing how early this chimerical performance became an object of ridicule.

Thales, Anaxagoras, Pythagoras, Hippocrates, Plato, Apollonius, Ptolemy, with other ancient mathematicians, have given methods for approximating to the area of the circle; and many also among the moderns. In 1775, the Paris Academy of Science determined to discourage papers devoted to this subject, and their course in this respect was soon after adopted also by The Royal Society, it being found that there was among certain geometers a complete mania for settling this and similar problems, the solution of which was either unattainable, or if attained of very questionable value.

DUPLICATION OF THE CUBE.

The Duplication of the Cube it is asserted can readily be demonstrated. It is usually called the Delian Problem, from its having been suggested by the oracle of Apollo at Delphos, requiring that Apollo's cubical altar should be doubled.

It is something in its favour to say that the enquiry has had the attention of Newton and of Huygens.

TRISECTION OF AN ANGLE.

Lastly, we shall notice among problems of this class--the Trisection of an Angle, which it is asserted can only be accomplished by means of the conic sections and some other curves.

A rule for the cubic equation by which the problem of trisection is solved has been given by Cardan.

The difficulty only arises when we attempt the trisection of any other than a right angle, its trisection being easily effected with a pair of compasses.

On this subject it has been observed that, "there is no more trouble in trisecting an angle, not a right angle, than in finding a cube root."

* * * * *

These three celebrated problems have received the attention of mathematicians in every age and country, and led to many learned discussions, and controversial writings. But in point of litigiousness the Squarers of the Circle most decidedly carry off the palm, having frequently laid and lost heavy wagers, and even appeared in a Court of Justice to settle their monetary disputes. They are renowned for their pamphlets, in which philosophers of every class are charged with prejudice, conceit, and ignorance, and denounced for their want of candour and consistency in not giving audience to the projector of the last best demonstration.

PERPETUUM MOBILE.

To conclude this Lecture we shall offer a few remarks on Perpetuum Mobile, or the search for a means of obtaining a mechanical perpetual motion. As a mathematical problem it dates back some 2000 years or more, but we know nothing of any actual attempt earlier than the 14th century to construct a machine intended to be self motive, by containing within itself the means of continually overbalancing. External motive agency such as the tides, magnetism, and the like are not included; the only admitted agent being gravity.

If we considered wear and tear the question would be settled at once, but this is allowed as the single exception, and therefore any machine constantly renewing the means that first moved it might be deservedly called a perpetual motion.

Until a history of the schemes invented by numerous ingenious mechanics was published in 1861, inventors of this class were continually though unconsciously reproducing obsolete contrivances, from taking up the ordinary idea that a wheel may be kept constantly over-weighted on one side, so as to raise the next weight which is to perform the same miracle of art. It is singular to observe this particular coincidence of the inventive faculty of man, and it shows next to a demonstration, that if all mechanical inventions were swept from the face of the earth they would be reproduced in some remote age.

A common error with those who toil at perpetual motion machinery is their aiming to produce a bottled-up power; or to apply the principles of the ordinary scale or balance to a wheel, overlooking the simple facts of friction on one side acting against their most ingenious contrivances, and of non-production on the other. Sooner or later, however, they discover the inertia of matter, that a pound will not raise a pound, and that they cannot invent mechanism to move independently of the laws of action and reaction.

A ball descending a semicircular path, as suggested by Dr. Henderson, will only rise to the same height as that from which it fell; and will afterwards gradually diminish in velocity until it rests at the centre. If it would ascend to a height greater than that from which it descended, then indeed an inclined path might return the ball to repeat such evolutions until quite worn out.

And as regards the weighted wheels, it is always overlooked that they come to rest from the same fact, that the vertical line of descent and that of ascent are equal, however much the weights may on one side recede from the centre, while on the other side the weights are approaching the centre. (See Plate 6, Fig. 1.)

The most famous perpetual motive schemes were those of the Marquis of Worcester made 1630-41; (See Plate 6, Fig. 2,) and of Bessler, better known as Orfyreus, between 1712-19.

The Marquis gives a brief notice of his plan, in his "Century of Inventions," a curious catalogue of his several ingenious schemes.

But of Orfyreus's wheel we know nothing more than was communicated by the eminent mathematician, 'S Gravesande, to Sir Isaac Newton, after an external view of it, while it was rotating in a chamber of the residence of the Prince of Hesse Cassel.

The most singular part of this strange delusion is the fact of its strong hold on the minds of its infatuated votaries. Once bewitched with the idea of at last succeeding in the attainment of his grand design, fortune, health, and reputation, are resolutely set at nought, in the delirium of delight that follows; and more unreasonable creatures can scarcely be found than such self-deluded individuals, for they cannot, or will not, be convinced that their utmost efforts can at best but produce an amazingly curious toy; and nothing can be more futile than to expect any higher application, assuming such a discovery were possible.

The best proof of the sincerity and earnestness of those who seek the attainment of a mechanical perpetual motion, is afforded by the variety and number of their patented schemes; the patentees having among them divines, doctors, lawyers, civil engineers, carpenters, draughtsmen, jewellers, watchmakers, shoemakers, confectioners, and all classes of professions and trades. It is not, as is generally supposed, only the wholly ignorant and designing who can be cajoled by these chimeras; there is in them a spice of mystery, of wonder, of singularity, and of simplicity combined with much subtle difficulty, which, being once fully imbibed, acts like an opiate draught.

We have thus reviewed summarily, chimeras which are mainly associated with Astronomy, Chemistry, Mathematics, and Mechanics, and which have swayed the human mind more or less from a period anterior to the Christian era. The list of this species of deceitful systems of pseudo-philosophy, and of profitless problems, might have been enlarged; but what has been advanced may suffice as a warning to the uninitiated to beware of blind guides and of visionary pursuits. Science has lost nothing by its professors exercising that degree of caution, which all classes of superficially learned men, affecting to possess original and valuable views on certain matters, call prejudice: which, in such cases, generally means no more than the natural aversion which the learned have for all attempts to place specious dogmas on a level with sound science. Such enthusiasts are generally men of no research or depth of thought, who obtain an imperfect acquaintance with subjects with which they are incompetent to grapple; and with whom it is, therefore, hopeless to contend. Delusion will have its day, and will as certainly decay, if not die out. Chimeras constantly spring up, and find ardent professors and crowds of easily led proselytes, even up to this very present time; so that although, undoubtedly with many--Knowledge is power: yet it is to be feared that far too large a proportion of mankind favour the delusion that--Ignorance is bliss.

EXPLANATIONS OF THE PLATES.