arose, the first attempt was always to solve it geometrically, and only when that failed, were mechanical means resorted to, or were other curves constructed, the con to a narrow neck, and that again gracefully opening | this circle so nearly, that the error committed should be towards the lip, like an unfolding flower. It is sup- less than the length of that atom. The problem which ported by two handles, inserted at the concave or nar- has puzzled so many generations, is the finding what is row part. The material is a dark but transparent called a geometrical quadrature of the circle. What this blue substance, undoubtedly a sort of vitrified paste, is we proceed to explain. It is the object of geometry to or glass, although long supposed to be some species find out, by reasoning, all truths which relate to the of stone. Upon this the figures, formed of a deli- various figures which a draughtsman can construct. cate opaque white substance, are laid in bas-relief; and Not that these figures are precisely the objects of geomeso firmly are they united to the ground upon which they trical reasoning; for a geometrical line has no breadth are thus fixed, that they seem rather to have grown out or thickness, but only length, while the line we draw of it, and to be a part of itself, than to be fastened on by with a pencil has a small degree both of breadth and art. It is difficult, indeed, to conceive by what process thickness. It is agreed by geometers to take as little for the union between the two substances was effected. granted as possible, and to make all their propositions They must of course have been first brought into con- arise out of the smallest number of simple truths. It is tact when both were in a soft state, and then apparently also agreed to imagine the existence of as few figures as they were run together by heat. If the action of possible. It was therefore the practice of the ancient fire, however, was employed for this purpose, it has geometers to assume no problems except the following: not injured the finest line in any of the figures.-1. A straight line can be drawn from one point to Every stroke is as sharp and unbroken as in the most another. 2. A straight line, when finished, can afterfinished delineations that were ever drawn by the pencil, wards be made longer. 3. A circle can be drawn with or cut by the graver, or struck from the die. Of the scene any point as a centre, and any line as a radius. All represented no satisfactory explanation has yet been lines, except the straight line and circle, and afterwards given, and therefore any description of the figures would the conic sections, were called mechanical, as distinbe little better than a catalogue of unconnected particu-guished from geometrical lines; and if any problem lars. But we may say in general that they are fashioned with admirable grace and animation, and are full of expression in every look and attitude. It is impossible not to feel that there is great dramatic force and pathos instruction of which, once granted, solved the problem. the sketch, even without being able to interpret it completely. The Portland vase was discovered about the middle of the sixteenth century, enclosed in a sarcophagus, within the monument of the Emperor Alexander Severus and his mother Julia Mamæa, commonly called the Monte del Grano, about two miles and a half from Rome, on the Frascati road. The sarcophagus itself, which is a very noble work of art, is still in Rome, and the vase also remained for more than two centuries in the palace of the Barberini family in the same city, of which it was considered to form one of the chief ornaments. At last it fell into the hands of Sir William Hamilton, from whom it was purchased nearly forty years ago by the Duke of Portland. A mould of the Barberiai vase was taken at Rome before it came into the possession of Sir William Hamilton, by the gem-engraver Pechler; and from this the late Mr. Tassie, the celebrated modeller, took sixty casts in plaster of Paris, and then broke the mould. Some very beautiful imitations of it have also been fabricated by the Wedgewoods, in which not only the shape, but the colour of the original has been attempted to be preserved. Modern art, however, cannot imitate the vitrified appearance of the material in the ancient vase. ON THE QUADRATURE OF THE CIRCLE. We have undertaken this article, because we have reason to know, that there are still some persons who, from ignorance of the real question, are turning their attention to this useless and exploded problem, and deceiving themselves and their neighbours into the belief that they have succeeded in doing that which has been repeatedly shown to be impossible. By the rectification of a circle is meant the finding of a straight line which shall be equal in length to the circumference of a given circle, or the two ends of which when bent round it should exactly meet; by the quadrature of a circle is meant the finding a square which shall be equal in surface to a given circle, that is, to the whole space contained inside its circumference. There is no difficulty in doing these problems in a manner more than sufficiently correct for all useful purposes. If, for example, we take a circle as large as the earth's orbit, and an atom as small as the smallest insect which a microscope ever showed in a drop of water, nothing would be more easy than to find the circumference of The names geometrical and mechanical, as applied to distinguish one sort of solution from another, may be improper; but that is not the question. When a man asserts that he has found a geometrical quadrature of the circle, he either does or does not use the word in the sense of the ancient geometers. If he does, and his solution is correct, he has certainly solved the problem; but that no one has yet done this, is universally admitted. If he does not use the word geometrical in the ancient signification, his solution has nothing to do with the problem which has hitherto remained unsolved. Many ways have been discovered of finding the area of a circle, which take something more for granted than the use of the ruler and compasses only, and any person, with a reasonable knowledge of mathematics, might add a dozen to the number in a couple of hours. So much for the geometrical solution of the problem. It was proved long before the Christian era, that the circumferences of two different circles are to one another as the radii, that is, whatever number of times one circumference contains its radius, the other circumference contains its radius as many times, or whatever fraction one radius is of its circumference, the same fraction is the other radius of its circumference. From this it follows, that if any number of circles were taken, having for radii a foot, a yard, a mile, &c., whatever number of feet and parts of feet would go round the first, the same number of miles and similar parts of miles would go round the third, and so on. Hence it became a question of importance to discover what was the number of units and parts of units contained in the circumference of a circle whose radius was the unit. Again, it was proved that the number of square feet and parts of square feet in a circle of one foot radius, was the same as the number of square miles and similar parts of square miles contained in the circle of one mile radius. Archimedes showed that a circle of one foot in radius contained nearly 3 square feet and of a square foot, which does not differ from the truth by so much as of a square inch, and gives the circle too great by about its three-thousandth part. An ancient measure of the Hindoos makes it 3 square feet and 177 parts out of 1250 of a square foot. This is much nearer to the truth, differing from it by about one-thousandth part of a square inch, but still a little too much. Metius, who flou shed in the beginning of the seventeenth century, found that if a square foot be divided into 113 parts, the | circle of one foot radius contains about 355 of these parts; a result of surprising accuracy when the simplicity of the numbers is considered; it is too great by about the fifty-thousandth part of a square inch. These numbers may be very easily recollected, since, when put together, they give the first three odd numbers, each repeated twice; thus, 113355. The following simple rules will enable every one of our readers to find the circumference of a circle. If any one of them would go direct round the world, he would by means of them, if the earth were a perfect sphere, be able to tell the length of his journey within less than four yards. From them the word inch may be taken out, and any other unit substituted. To find the length of the circumference of a circle, multiply the number of inches in the diameter (or twice the radius) by 355, and divide the product by 113. The result is the number of inches in the circumference. To find the area, or surface of a circle, multiply the number of inches in the radius by itself, and that product by 355; divide the result by 113, the quotient of which is the number of square inches in the area. We might go on to describe still more accurate methods; it will be sufficient to say, that the latest of them gives the area of a circle true to 127 decimal places, as it is called; that is, if the radius be 1000, &c. feet, the ciphers being 127 in number, the area of the circle will be obtained without an error of a square foot. Still, this is only an approximation, and however nearly the circumference of a circle has been obtained, it has never been obtained exactly. Numberless attempts have been made to find the exact ratio of the circumference to the diameter, but without success; the reason being, as was afterwards proved, that the thing is impossible. We can now demonstrate that the ratio of the circumference to the diameter cannot be accurately expressed in numbers; all we can do with numbers is, to express it as nearly as we please. Thus, using decimals, we may say that the circumference of a circle whose diameter is 1 is greater than 3 and less than 4; greater than 3.1 and less than 32; greater than 314 and less than 3.15, and so on, assigning nearer and nearer fractions between which it must lie, but never coming to an exact result. Almost every projector who imagines he can solve this problem, is sure to produce some number or fraction as the exact ratio of the circumference to the diameter; and it is observable that the less his knowledge of geometry, the more easily does he overcome the difficulty, and the more obstinately does he believe himself in the right. Some have been found hardy enough to deny the common propositions of geometry, in order to establish their own conclusion on this point. Others, totally ignorant of geometry, hearing that a circle could not be exactly measured, have imagined that the word exact was used in the sense in which a carpenter would take it, who, very properly for his purpose, considers two rods to be of exactly the same length, when they do not differ to the naked eye. These usually cut a circle out in wood, measure it with a bit of string, pronounce their result to be perfectly accurate, and are very much surprised that an ungrateful world does not perceive their claim to one of the first places in the ranks of science. We shall give some anecdotes connected with this subject, principally extracted from Montucla's History of the Mathematics. The author of it was a modest man, and ascribed all the honour to the Virgin Mary. Another Knight of the Round Shield found out by his method that the first book of Euclid was all a mistake. About the same time a merchant of Rochelle discovered not only the quadrature of the circle, but with it, and depending upon it, a method of converting Jews, Pagans, and Mahometans to Christianity. In 1671, an anonymous writer published a treatise with the following title: 'Demonstration of the Divine Theorem of the quadrature of the circle, of the trisection of the angle, and of the perpetual motion, and the connexion of this theorem with the Vision of Ezekiel and the Revelation of St. John.' A certain Cluver found out that this problem depended upon another, which he expressed thus: "Construere mundum divinæ menti analogum." The literal translation of this (the sense is unknown) is, "To build a world resembling the divine mind." But the most singular person was one Richard White, an English Jesuit, who, having once undertaken to square the circle, was afterwards convinced by argument that he was in the wrong, which never happened to any other of this class of speculators, except perhaps to one Mathulon, a Frenchman of Lyons. This man offered to give a thousand crowns to any one who would detect an error in his solution. It was done to his satisfaction, but he refused to pay the money, and a court of justice decided that it should be given to the poor. So late as 1750 an Englishman, Henry Sullamar, found out the area of the circle by means of the number 666 mentioned in the Revelations. But in 1753, a Captain in the French Guards did more, for in squaring the circle, which he did with a piece of turf, he hit upon what he thought was a most obvious connexion between this and the doctrines of original sin and the Trinity. He offered to bet three hundred thousand francs that he was right, and actually deposited ten thousand of them. A young lady and several other persons easily won the wager, and brought actions for the money; but the courts declared that the bet was void. Such are a few of the most remarkable aberrations of the human mind on this problem. They show, in the most convincing manner, that presumption is rarely confined to one subject in the same mind, and that a man who, without studying a science, conceives himself to be more knowing than those who have passed their lives in the pursuit of it, must previously have brought himself to believe that he is almost a God, and is but one step removed from taking the government of the universe out of the hands of its Creator, and arranging it according to his own improved notions. With regard to a geometrical solution, and the possibility or impossibility of it, we shall now say a few words. We have already observed that an arithmetical solution is certainly impossible, that is, there is no number or fraction which exactly represents the circumference of the circle where the radius is a unit. In 1668, James Gregory, a well-known name in geometry, asserted that a geometrical quadrature was impossible, that is, no use of the ruler and compasses could give a square of exactly the same dimensions as a given circle. Of this he published his demonstration, which was attacked by Huyghens, another geometer of the same time. The dispute has interested mathematicians so little for the last century and a half, that few of them seem to have cared which was right. The historians of mathematics have, of course, been obliged to give an opinion, and yet Montucla and Dr. Hutton both forbear to decide In 1585, a Spanish friar published his quadrature of the question, each being apparently somewhat inclined the circle. His preface is a dialogue between himself to believe that J. Gregory was right. The demonand the circle, who thanks him in most affectionate stration of the latter appears to us to render it extremely terms for having solved the problem. The circle, how-probable that the geometrical quadrature is imposever, did not in this case attend to the maxim, "Know sible; but we will not venture a positive opinion thyself," any more than some of its squarers have since where such respectable authorities have declined to done, for the pretended quadrature was utterly wrong. give one. But we would recommend any one who imagines he can give this solution, to learn geometry, to | for without this he will never out-herod Herod so far as examine the demonstration of J. Gregory, which he will find in the library of the British Museum, and find out the error; and it deserves some attention, since neither Montucla nor Hutton, both very well informed mathematicians, would positively say it was false. We would not have entered upon this subject at such length, if it were not that there appear, from time to time, pretended solutions of this problem. To any one who is ignorant of geometry, we would recommend to be sure of two things before he undertakes it; first, that he has an imagination which will set common sense at defiance to produce anything worthy of notice, after the instances which we have mentioned: secondly, that he has his own good opinion to a very great degree, for otherwise his peace of mind will be disturbed, either by the neglect or ridicule which it will be his fate to meet with. To one who understands geometry, and who imagines himself to be the person destined by Providence to work this wonder, we have not a word to say: if the study of Euclid has not been sufficient to teach him more sense, or at least to induce him to wait until he knows more, we should almost rival him in absurdity, if we thought him a proper subject for the language of common sense. [The Banana, or Plantain Tree, with Cocoa-Nut Trees in the back-ground.] THE banana, or plantain, forms a principal article of food to a great portion of mankind within and near the tropics, offering its produce indifferently to the inhabitants of equinoctial Asia and America, of tropical Africa, and of the islands of the Atlantic and Pacific Oceans. Wherever the mean heat of the year exceeds 75° of Fahrenheit, the banana is one of the most important and interesting objects for the cultivation of man. All hot countries appear equally to favour the growth of its fruit; and it has even been cultivated in Cuba, in situations where the thermometer descends to 45° of Fahrenheit. The tree which bears this useful fruit is of considerable size: it rises with an herbaceous stalk, about five or six inches in diameter at the surface of the ground, but tapering upwards to the height of fifteen or twenty feet. The leaves are in a cluster at the top; they are very large, being about six feet long and two feet broad: the middle rib is strong, but the rest of the leaf is tender, and apt to be torn by the wind. The leaves grow with great rapidity after the stalk has attained its proper height. The spike of flowers rises from the centre of the leaves to the height of about four feet. At first the flowers are inclosed in a sheath, but, as they come to maturity, that drops off. The fruit is about an inch in diameter, eight or nine inches long, and bent a little on one side. As it ripens it turns yellow; and when ripe, it is filled with a pulp of a luscious sweet taste. The banana is not known in an uncultivated state. The wildest tribes of South America, who depend upon this fruit for their subsistence, propagate the plant by suckers. Eight or nine months after the sucker has been planted, the banana begins to form its clusters; and the fruit may be collected in the tenth and eleventh months. When the stalk is cut, the fruit of which has ripened, a sprout is put forth, which again bears fruit in three months. The whole labour of cultivation which is required for a plantation of bananas is to cut the stalks laden with ripe fruit, and to give the plants a slight nourishment, once or twice a year, by digging round the roots. A spot of a little more than a thousand square feet will contain from thirty to forty banana plants. A cluster of bananas, produced on a single plant, often contains from one hundred and sixty to one hundred and eighty fruits, and weighs from seventy to eighty pounds. But reckoning the weight of a cluster only at forty pounds, such a plantation would produce more than four thousand pounds of nutritive substance. M. Humboldt calculates that as thirty-three pounds of wheat and ninety-nine pounds of potatoes require the same space as that in which four thousand pounds of bananas are grown, the produce of bananas is consequently to that of wheat as 133: 1. and to that of potatoes as 44: 1. The facility with which the banana can be cultivated has doubtless contributed to arrest the progress of improvement in tropical regions. In the new continent civilization first commenced on the mountains, in a soil of inferior fertility. Necessity awakens industry, and industry calls forth the intellectual powers of the human race. When these are developed, man does not sit in a cabin, gathering the fruits of his little patch of bananas, asking no greater luxuries, and proposing no higher ends of life than to eat and to sleep. He subdues to his use all the treasures of the earth by his labour and his skill; and he carries his industry forward to its utmost limits, by the consideration that he has active duties to perform. The idleness of the poor Indian keeps him, where he has been for ages, little elevated above the inferior animal;-the industry of the European, under his colder skies, and with a less fertile soil, has surrounded him with all the blessings of society—its comforts, its affections, its virtues, and its intellectua riches. 66 "Duncan sighed baith out and in, THE river Clyde in the neighbourhood of the town of perplexity and despair of the rejected suitor, he saysLanark presents, according to the testimony of all travellers, some of the most romantic and picturesque scenery in the world. We shall confine ourselves at present to a short notice of the Linns or Falls which have been so much celebrated. The word Linn, we may Spak o' loupin' owre a linn," writes one of his cor remark, is the Gaelic Leum, and signifies merely a fall or respondents, the Honourable A. Erskine, to the poet, leap. Its application to a cataract, or fall of water, is" is a line of itself that should make you immortal." general throughout Scotland. Burns has introduced the But to return to the linns on the Clyde. The first preword with very happy effect in his humorous and well- cipice over which the river rushes, on its way from the known song of Duncan Grey, where, in describing the hills, is situated about two miles above Lanark—and is The word has also been derived from the Welsh Linn, signifying known by the name of Bonnington Linn. It is a per"a lake" or "water." This root is likewise found in the Greek pendicular rock of about twenty, or, as some authorities Language, and its proper signification seems to be, "water," state, thirty feet in height, over which the water, after |