Although the present lecture seems to require some introductory remarks, they must necessarily be brief—our time being limited and this discourse rather discursive; yet it is sufficiently condensed to suit the present occasion, and illustrates fully the truthfulness of the axiom that—A little learning is a dangerous thing: from its tendency to inspire its possessors with vanity rather than with the humility which always accompanies profound knowledge.

You are no doubt all, or most of you, well acquainted with the use made of Astrology and Alchemy in the dramas of Shakspeare—"The Antiquary" of the "Wizard of the North"—the "Strange Story" of Lord Lytton—the "Faust" of GoËthe; and are probably familiar with the more instructive works of Scott on Demonology, and of Brewster on Natural Magic. Now we always find that fiction is more suitable than truth for romantic writings; truth is circumscribed, but the fictions whether of Astrology, Alchemy, or any other pseudo-philosophy are erratic, the delight of poets and romance writers, being the comets and ignes fatui of many popular compositions in our light literature.

There is no end of fabulous writings of the class we call novels and romances, and no end of deceptions which we patronize as tricks of legerdemain; the one gratifies our imagination and fancy, the other takes our common sense by surprise; but all these are harmless because only presented to us for our amusement.

Delusion, however, assumes a startling character when romance in the form of mystic writings, and jugglery in the form of pretended communication with the spirit-world demand our respect and serious attention, by claiming to have a divine origin. But hallucination of the human intellect, as we shall see, is not confined to such remote visionary speculations, and it is not unimportant to remark that in mathematics, as in physics, and in other branches of investigation, there is a singular persistency in upholding errors.

A contemporary astrologer, assuming the pseudonym of Zadkiel, tauntingly observes in his preface to a recent publication: "This is the age of inquiry; and yet prejudice continues to press down her leaden foot upon the neck of examination in this matter"—that is, Astrology. Now with this reproof before us we hope to discuss the subject with becoming propriety.

ASTROLOGY.

The splendour of the sun by day, the glories of the firmament by night, together with the sublimity of all celestial phenomena, attract alike the attention of the most simple and the most intellectual among mankind. The distance, the magnitude, and the grandeur of the entire planetary system while exciting emotions of awe, reverence, and devotion among the mass of the human race, have at the same time been studied from the earliest period of man's history to the present time superstitiously by one class of observers, and scientifically by another.

As the telescope was not invented before the 17th century, it is evident that the study of Astronomy without that instrument must previously have been pursued under amazing difficulties; and we might have expected that when first used by Galileo at Venice in 1609, its introduction would have been hailed without a dissentient voice. Such, however, was not the fact, according to Sir David Brewster,[8] who says:—

"The principal Professor of Philosophy at Padua resisted Galileo's repeated and urgent entreaties to look at the moon and planets through his telescope; and he even laboured to convince Cosmo de Medici, the Grand Duke of Tuscany, that the satellites of Jupiter could not possibly exist. Sizzi, an astronomer of Florence, maintained that as there were only seven apertures in the head—two eyes, two ears, two nostrils, and one mouth—and as there were only seven metals, and seven days in the week, so there could only be seven planets. He seems (eventually), however, to have admitted the visibility of the four satellites through the telescope; but he argues, that as they are invisible to the naked eye, they can exercise no influence on the earth; and being useless they do not exist."

Such being the crude state of astronomical science in the 17th century, it must have been comparatively imperfect throughout all preceding centuries; and open to mystical appropriation and abuse by Egyptians, Chaldeans, Hindus, Chinese, and European and other ancient astrologers. Among that motley group the most learned were found strangely associated with ignorant impostors, and their activity in writing and travelling served to spread their different systems over the entire civilized world. It was not until late in the 17th century that Astrology could be absolutely declared to be in its decline. In England, William Lilly, the Sidrophel of Hudibras, and the most famous astrologer of his time, died in 1681, leaving behind him his Introduction to Astrology, together with many other works of the same character.

Astrology is merely a philosophism, being empirical, wholly visionary, a mere fanciful system compounded of incongruous mixtures of astronomical with human events, of mythology with theology, and of facts with pure fiction. It has been variously designated Judicial, Hororary, Atmospherical, and Mundane, Astrology. It has also many off-shoots subservient to Magic or the black art, Sorcery, Witchcraft, and other pretended mysticisms ostentatiously styled occult philosophy.

We may first observe that Astrology lays no claim to inspiration, but affects a very ancient unknown origin, tracing back to a dark, heathenish, and superstitious age, in the very infancy of traditional knowledge, when the boldest assertions of the seer were received as the authority of an oracle, no one daring to question their validity. Whatever is remotely possible the Astrologer accepts as a fact; while ignorant of much around him, he assumes with the utmost complacency an intimate acquaintance with the sun and planets thousands upon thousands of miles off; yea with the sun 969,272 miles in diameter, while he himself inhabits a globe only 7,916 miles in diameter; from which the moon is 237,000 miles distant, and the sun 400 times that distance.[9] And these immense bodies revolving millions on millions of miles away in immeasurable space are described by him as fashioning an infant's nose, directing the fortunes or misfortunes of lovers, ordering the property of traders, meting out diseases, and improving or deranging man's mental faculties. And as if such puerile influences were not sufficiently preposterous we are informed by the modern seer, Zadkiel, that the 12 signs of the Zodiac not only rule the several parts of the human frame, but also those of a ship, as Aries, the bows; Taurus, the cutwater; Gemini, the rudder; Cancer, the bottom; Leo, the upper works; Virgo, the hold; Libra, parts above the water's edge; Scorpio, the seamen's berths; Sagittarius, the seamen; Capricorn, the ends of the vessel; Aquarius, the Captain; Pisces, the oars in galleys, the wheels in steam vessels, and the sails in others; but these latter being above water, we are left in doubt about the ruler of the submerged screw propeller.

To show what a modicum of learning, and how trifling an acquaintance with matters of natural philosophy will serve the Astrologer, we will turn to a modern treatise published in the year 1801, by Francis Barrett, (styling himself a student of Natural and Occult Philosophy) a quarto volume of upwards of 370 pages, entitled, "The Magus, or Celestial Intelligencer," which affords a pretty clear insight into the nature of the superstitions which from an ancient period even to that date obtained credence and were popular with the multitude.

Treating of the wonders of Natural Magic previous to entering on the main topic of his treatise, he adduces a few of what he conceives to be ordinary matters of fact, assuring us that:—

If any one shall, with an entire new knife, cut asunder a lemon, using words expressive of hatred, contumely, or dislike, against any individual, the absent party, though at an unlimited distance, feels a certain inexpressible and cutting anguish of the heart, together with a cold chilliness, and failure throughout the body;—likewise of living animals, if a live pigeon be cut through the heart, it causes the heart of the party intended, to be affected with a sudden failure;—likewise fear is induced by suspending the magical image of a man by a single thread;—also death and destruction by means similar to these; and all these from a fatal and magical sympathy.

The loadstone, (he observes), possesses an eminent medical faculty against many violent and implacable disorders;—the back of the loadstone, as it repulses iron, so also it removes gout, swellings, rheum, &c. that is of the nature or quality of iron. Likewise the wearing the loadstone eases and prevents the cramp, and such like disorders and pains.

The influences of the stars appear to be as intimately known to Astrologers as though they had walked among, and carefully examined and fully realized their occult properties, for example:—

In every work observe Mercury, for he is a messenger between the higher gods and the infernal gods; when he goes to the good, he increases their goodness—when to the bad, he hath influence on their wickedness. It is an unfortunate sign or planet, when it is by the aspect of Saturn or Mars especially, apposite or quadrant, for these are the aspects of enmity; but a conjunction, a trine, and a sextile aspect, are of friendship; but yet if you do already behold it through a trine, and the planet be received, it is accounted as already conjoined. Now all planets are afraid of the conjunction of the sun, rejoicing in the trine, and sextile aspect thereof.

They say of the Sun and Moon:—

The Sun is the lord of all elementary virtues;—it disposes even the very spirit and mind of men.

The Moon (says Barrett) measures the whole space of the Zodiac in the time of 28 days, hence it is that the wise men of the Indians, and most of the ancient astrologers have granted 28 mansions to the Moon, which, being fixed in the eighth sphere, do enjoy divers names and properties, from the various signs and stars which are contained in them; through which, while the Moon wanders, it obtains many other powers and virtues; but every one of these mansions, according to the opinion of Abraham, contained twelve degrees, and fifty-one minutes, and almost twenty-six seconds. In the first quarter of these mansions the 1st conduces to discords and journies; the second to the finding of treasures, and to the retaining of captives; the 3rd to benefit sailors, huntsmen, and alchymists; the 4th the destruction and hindrances of buildings, fountains, mills, gold mines, the flight of creeping things, and begets discord; the 5th to help the return from a journey, the instruction of scholars, and confirms edifices, gives good health and good will; the 6th to hunting and besieging towns, and revenge of princes, destroying harvests and fruits, and hinders the operation of the physician; the 7th to confirm gain and friendship; is profitable to lovers, and destroys magistracies.

In a similar manner the remaining three quarters have the characters of their several mansions allotted to them with equal exactness, and of course indisputable veracity also.

We have here a fair example of the arrogant assumptions of ancient and indeed of all astrologers, magicians, and sorcerers, men who are incompetent to elucidate the ordinary phenomena of nature in the animal or vegetable creation, and yet with unbounded effrontery affect to build up an empirical system, delivered in a language of their own invention, a pompous parade of jargon made up of the most incomprehensible materials—which if wholly due to antiquity partakes of ancient simplicity, credulity, deceit, and superstition; and if somewhat polished and refined to suit the advances of literature and science, has never been able to prove the correctness of its groundwork, or afford a solitary instance of its possessing any meritorious quality beneficial to mankind; while on the other hand its evil consequences have been many, by destroying the peace and happiness of thousands, encouraging deceit, and misapplying in its ignoble pursuit the time and labour and property of its ardent but deluded admirers.

In Judicial Astrology it is not thought requisite to consider more than a certain number of the planets, after a method simplified by antient astrologers or astronomers, which is found to be so compact and so complete in governing the destinies of the human race that modern intelligence has failed to enlarge the field of heavenly influences. Varley notes that:—the antients discovered that the circle of the Zodiac, about 16 degrees in width, and through the middle of which runs the Ecliptic, or sun's path through the 12 signs, contains the heavenly bodies, named planets, and the principal fixed stars, and nearly the whole of the materials or significators, from which predictions are obtained.

He remarks that:—In forming a horoscope, this circle is divided into 12 equal parts, corresponding with the spaces containing the 12 hours. These 12 divisions are called houses; and they always remain fixed, while the Zodiac with the 12 signs, and all the heavenly bodies belonging to it, are considered to be moving through them all, every 24 hours. The lord of the ascendant is the planet which rules the signs rising at birth. In drawing horoscopes it is usual to make the figure square instead of round. (See Plate 1, Fig. 1.)

The various significations arising from the aspects of the starry heavens at the time of birth are so exceedingly numerous, that we must refer the curious in such matters to the works themselves, in which all these pretended revelations are minutely recorded.

Mankind rank astrologically as being of four temperaments.

1. One class is said to answer to the fiery trigon, also called diurnal, masculine, and choleric, consisting of Aries, Leo, and Sagittarius, which contains the spirited, generous, magnanimous, and princely natures.

2. We have next the earthy trigon, being nocturnal, feminine, and melancholic, consisting of Taurus, Virgo, and Capricorn, containing the careful, sordid, and penurious qualities.

3. Thirdly, the aËrial trigon, which is diurnal, masculine, and sanguine, consisting of Gemini, Libra, and Aquarius, contains the humane harmonies, and courteous principles. And—

4. Fourthly, the watery trigon, which is nocturnal, feminine, and phlegmatic; namely, Cancer, Scorpio, and Pisces, including the cold, prolific, cautious and severe qualities.

Take as a brief illustration of the manner in which Astrologers presumptuously assign to the planets their several offices relating to human nature the following:—

Those born when Aries ascends are born under the sign Aries and planet Mars. This is the diurnal, fair, and masculine house of Mars, and partakes also largely of the nature of the magnanimous Sun, and the benevolent and moral Jupiter, who rule the fiery trigon, of which Aries is the first sign.

As affecting physiognomy we are assured that:

The Scorpio noses are more aquiline than those of Aries, and are more frequently conspicuous for a sort of bracket shape beneath, which prevents the under part of the nose from forming a right angle with the upper lip; while the under lip, both being usually small, recedes in a greater degree, as if drawn tightly against the teeth; so that the mouth appears in the act of pronouncing the word SEVERE.

When we meet in volume after volume with page after page of such composition as this, when we reflect on the sublimity of the heavens and the paltriness of such combinations as are here given of the planets with mundane affairs, we ask the reasons for arriving at such judgments. To be told that it is so because it is so; or because it was an ancient belief, and is to be found in the writings of Ptolemy, Nostradamus, Dr. John Dee, William Lilly, or Zadkiel; or because it has often proved as true in its predictions as the telling fortunes by means of a pack of cards, is no evidence whatever; yet the Astrologer boasts of his very paralogisms.

Zadkiel, in prefacing a work by Lilly, says:—If a proposition of any nature be made to any individual, about the result of which he is anxious, and, therefore, uncertain whether to accede to it or not, let him but note the hour and minute when it was first made, and erect a figure of the heavens, (See Plate 1, Fig. 1,)—and his doubts will be instantly resolved. He may thus, in five minutes, learn infallibly whether the affair will succeed or not; and, consequently, whether it is prudent to adopt the offer made or not.

Such is the belief of this sound, intelligent man, as we fully believe him to be in other respects. But we say it is not given to man to assign special influences to the stars, to select one portion and discard all the rest, or to be more intimately acquainted with the starry heavens above him, than with the stony earth he inhabits, and with his fellow creatures around him.

The works claiming to expound this pretended Occult Philosophy prescribe such childish processes that one naturally wonders how in the midst of so much impudent imposture Astrology and its kindred pursuits ever found or retained any honest partizans.

Take, for example, the use of fumigations, such as of frankincense, &c. to Saturn; of cloves, &c. to Jupiter; of odoriferous woods to Mars; of all gums to the Sun; of roses, violets, &c. to Venus; of cinnamon, &c. to Mercury; of the leaves of vegetables to the Moon; of all or any of which there must be a good perfume, odoriferous, and precious, in good matters; but in evil ones quite the contrary.

The Zodiac is also favourably affected by proper suffumigations.

Astrologers in their Demonology profess to be able to ascertain the characters and seals of spirits,[10] and according to the Cabalists, tables are given of many of these in their books, in the so-called Theban Alphabet; in characters of Celestial Writing; in that called Mallachim; or in the writing called Passing the River.

They affect to have suitable bonds by which spirits can be bound, invoked, or cast out.

Of Necromancy they pretend to two kinds, one of which is raising the body of a deceased person, which it is said cannot be done without blood;—the other sciomancy, which is the production of a mere shade or shadow.

The exorcisms and conjurations of Magicians are so audaciously profane and blasphemous as to be unworthy of even a passing notice.

ALCHEMY.

We shall now proceed to consider Alchemy, another but very different chimerical pursuit, which was early cultivated in the East, and is generally ascribed to Hermes Trismegistus, although its more enthusiastic admirers pretend to trace a knowledge of it to Adam. From the earliest periods of history man was acquainted with gold, silver, and other metals, with bitumen, sulphur, sea salt, sal-ammoniac, gums, and resins, together with other varieties of substances and liquids common to modern chemistry. For the compounding and heating of certain of these materials a multiplicity of means were adopted requiring furnaces, crucibles, and distillatory apparatus. The first workers in these experimental operations formed a body of investigators into the nature and properties of all manner of substances, whether animal, vegetable or mineral, the members of which were distinguished as adepts, alchemists, and later in their career as common chemists. The most esteemed branch of the art however was Alchemy, a pseudo-science which ultimately took three forms. First, the Hermetic Art for the discovery of the Philosopher's stone; and the Alkahest, or universal solvent;—Second, a Medical Alchemy;—and Third, a Theological Alchemy pretending to conceal divine mysteries under an allegorical form, treating of the spiritual while apparently describing alchemical discoveries.

Our principal business, however, is with the so-called Hermetic Philosophy, treating of vaunted methods of transmuting the base metals into gold. It is doubtful whether this particular delusion of the adepts can be referred to a date earlier than the 8th century, but even then we cannot refrain from surprise at the fallibility of the human intellect, which could be swayed by a belief in the pretended lapis philosophorum for upwards of ten centuries. It was believed to be so secret and rare that its possession was never ascribed at any time to more than two or three favourite adepts, who transmitted it to some single favoured individual after his taking the sacrament, and going through certain prescribed religious ceremonies, preparatory to being entrusted with a verbal recipe for the composition of a peculiar smelling red powder, of which it was affirmed that when projected on heated mercury or any solid metal, it would at once change it into pure gold. Ashmole gravely assures us that Dr. John Dee and his associate Kelly, having in some way procured this precious substance, Kelly—to use Ashmole's own words—"made projection with one small grain thereof, in proportion no bigger than the least grain of sand, upon one ounce and a quarter of common mercury, and it produced almost an ounce of pure gold." With equal simplicity and earnestness, Ashmole asserts that this same Kelly was often seen to make these extraordinary transmutations,—"and in particular (he adds) upon a piece of metal cut out of a warming-pan, and without touching or handling it, or melting the metal, only warming it in the fire, the elixir being put thereon, it was transmuted into pure silver. The warming-pan and this piece of it, was sent to Queen Elizabeth by her Ambassador who then lay at Prague, that by fitting the piece into the place whence it was cut, it might exactly appear to be once part of the warming-pan."

Among the adepts there were no doubt a select few who employed themselves in their prolonged labours in all sincerity, and who were not unfrequently repaid with remarkable, and unexpected results. Brass, being the result of copper combined with zinc, would appear a singular transformation. Many stones, or more properly, ores, would yield sulphur and metals; sulphur would be found apparently to dissolve iron; and certain salts, when distilled, would yield corrosive acids. Alchemy thus presented to the ancient adepts many of the ordinary wonders of modern chemistry; in short, the latest adept of the present century is no other than an unlettered chemist. It was peculiar to the Alchemists to treat all their operations as secrets; which, when recorded, were described partly by symbols and partly in a novel nomenclature, invented to conceal their mysteries from vulgar gaze or imitation. Thus, to prepare the philosopher's stone, we have merely to—"Take of moisture, an ounce and a half; of meridional redness, that is the soul of the sun, a fourth part, that is, half an ounce; of yellow seyr, likewise half an ounce; and of auripigmentum, a half ounce; making in all three ounces. Know that the vine of wise men is extracted in threes, and its wine at last is completed in thirty." To the incredulous in these matters, Ashmole offers the admonition that, he knows "Incredulity is given to the world as a punishment!" However, when the Alkahest, or pretended Universal Solvent, was alluded to by the modern chemist Kunckel, he could not refrain from incredulously enquiring—"If it dissolves all substances, in what vessel can it be contained?"

Alchemical writings are very numerous, it might be impossible to procure a complete bibliographical list of them, but they may be estimated at from 3000 to 4000 works, and an astonishing number of manuscripts. Their authors indulge in such terms as the Ph[oe]nix, to indicate the quintessence of Fire; Realgar, for the fume of minerals; Guma, also Luna Compacta, for quicksilver; Hadid, for iron; Aurum potabile, for liquor of gold; Anathron, for saltpetre; Malek, for salt; Terra fidelis, for silver; Tinkar, for borax; and in a similar strain for all matters and operations; so that Dr. Johnson was justified in deriving the word Gibberish from the mysterious jargon employed by Geber, a celebrated Alchemist; who has, nevertheless, been appropriately styled the Pliny of the 8th century.

Weidenfeld, in an Alchemical Treatise, published in 1685, addressing students, says:—

"Under heaven is not such an art, more promoting the honour of God, more conducing to mankind, and more narrowly searching into the most profound secrets of nature, than is our true and more than laudable Chymy."

And at the conclusion of his address he observes:

"Nothing remains but upon our bended knees to return most humble thanks to the Father of Lights, in vouchsafing us this art by the writings of his servants, and the high priests of Nature; without which, it would be beyond the power of man to arrive at so great a degree of knowledge."

Some notion of the extravagance of the language employed may be obtained from his description of a Philosophical Wine, literally, rectified spirits of wine, or alcohol. He assures us that, on opening a vessel of it, "a wonderful scent" should arise: "so as that no fragrancy of the world can be compared to it; inasmuch as putting the vessel to a corner of the house, it can by an invisible miracle draw all that pass in to it; or, the vessel being put upon a tower, draws all birds within the reach of its scent, so as to cause them to stand about it. Then will you have, my son, our quintessence, which is otherwise called Vegetable Mercury, at your will, to apply in Magistery of the transmutation of metals."

How ardent an adept this Alchemist was may be gathered from his exclamation:—"May the God of Heaven put prudence in the heart of evangelical men, for whom I compose this book, not to communicate this venerable secret of God to the reprobates."

Among the remarkable discoveries made by Alchemists, due to the carefully noted and carefully examined failures and accidents, as well as successes, of their endless combinations of matter, under the treatment of fire and water, the most distinguished is that of gunpowder, noted in a recipe left on record by Roger Bacon, who died in the year 1284. He clearly names the mixture of Saltpetre with Sulphur, but the third ingredient, Carbon, is concealed in the form of an anagram.

Lord Bacon, Luther, Spinoza, Leibnitz, and many eminent moderns, were impressed with a belief in the possibility of transmuting lead, tin, copper, or other metals, into gold; in short, as it was supposed there were only four elements, fire, water, earth, and air, it was probably assumed that a fifth might be found in the Philosopher's stone.

But if ever any pursuit was more open to fraudulent practices than another, surely the pretended possession of a transmuting powder or elixir afforded a grand arena for their exercise. In this enlightened age, although we cannot fail to look with charity on the arduous labours of those adepts who honestly mixed devotional exercises with laborious experimental operations, selecting times and seasons for their alchemical work, and noting with accuracy the hours and days of fusions, sublimations, distillations, lixiviations, and so forth; still, it is scarcely possible to refrain from smiling at the docile simplicity of Ashmole in denouncing a certain class of Alchemists, as pretended masters and adepts, seeing "they are mere practisers of legerdemain," while he himself gave credence to the story of the warming-pan, already named as being shown to Queen Elizabeth, which was clearly a flagrant piece of fraud practised by Kelly, a common adventurer, and from his youth remarkable only for his indifferent character.

An easily performed trick was effected by means of nails, or other light articles, made half of gold and half iron, but disguised, so as to appear to be of one metal and colour. Sometimes these knaves employed crucibles, having an interior false bottom, below which a small quantity of gold was placed, which, being reproduced, as was pretended from base materials, was offered as an example of success. Or, by having the gold in a hollow rod, stopped at one end with wax, used to stir up the materials, the gold would naturally enough appear in the crucible. Or, their materials being conveyed into charcoal, a similar result would be obtained on heating the crucible in a furnace. At other times, by the employment of amalgams, or solutions in acids, they could perform a species of electro-plating on common metals. The extent to which these nefarious practices were carried might appear incredible, considering the evident inconsistency of the owner of the pretended golden key to countless wealth, being in such comparative poverty as to be indebted to any one of moderate means for pecuniary assistance. But, it is some apology for such credulity when we call to mind the state of public morals, of education, of political institutions, and the prevalent superstition, not only among common people, but also the higher classes of all countries and creeds, down to the seventeenth century: representing a phase of the human mind, liable to be overawed by impostors, who boldly claimed supernatural aid in abetting their impositions. And the trickery of the designing was further aided by the close secrecy adopted by the adepts in their processes, their conversations, and their writings. Ashmole freely admits that—"Their chief study was to wrap up their secrets in fables, and spin out their fancies in 'vailes' and shadows, whose radii seem to extend every way, yet so that all meet in a common centre, and point only to one thing." It was this very secrecy, this continual mystery from beginning to end, that favoured deceptions of the grossest and most bungling character, as viewed by the light of modern chemistry.

Alchemy no doubt tended to improve Medical science, by the introduction of many new mineral and vegetable preparations, but the healing art treated after the manner of the Hermetic Art, was laid open to every description of quackery. It is not our intention, however, to enlarge on this department, which has steadily advanced at every stage of improvement in chemical science.

SQUARING THE CIRCLE.

Of Mathematical Problems, the most perplexing to ancient and modern mathematicians, although of late years said to be satisfactorily demonstrated, and no longer desiderata of Geometry, are—

1. The Quadrature or Squaring of the Circle;—2. The Duplication, or doubling of the Cube;—and 3. The Trisection of the Angle.

In his "Popular Astronomy,"[11] Professor Arago, treating on the surface of a circle, observes that,—

It is mathematically equal to the product of the length of the circumference, multiplied by half the radius. To square a circle of a given diameter in mÈtres, is the same as giving the number of squares, of a mÈtre in each side, of which the surface is the equivalent. If, the diameter being given, the exact circumference were known by a sort of inspiration, the superficial extent of the circular space would be deducible from the two numbers, by the mere multiplication of the numerical length of the circumference by the fourth of the diameter, or half the radius. But, the circumference being deducible from the diameter only by approximation, the surface alluded to cannot be computed with mathematical rigour; yet the result can be obtained with all desirable precision by the aid of the ratios usually given for such purpose; for instance, the area of the space included within a circle of thirty-eight millions of leagues radius, may be determined within such a degree of precision that the probable error shall not exceed the space of a mite.

"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.

[8] See his "Martyrs of Science."[9] This portion of the subject was illustrated by means of a Diagram exhibiting the Diameters and Magnitudes of Planets, thus:—

The Moon's diameter is 2,160 miles; and its distance from the Earth is 237,000 miles.

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[10] See the Table, Plate I. Fig. 2, for distinguished names of their angels, spirits, or demons.[11] See Translation, by Admiral W. H. Smith, and Robert Grant, M.A., in 2 vols. 8vo. 1855, Vol. I., page 10.