[The Portland or Barberini Vase.] One of the proudest ornaments of the Museum is the art. It is undoubtedly a work of Grecian genius, and is beautiful and celebrated Portland, or, as it used to be fortunately still as perfect as when it left the hands of called, Barberini Vase. It stands on a table in the its fabricator. Its dimensions are small, its height being middle of the small ante-room at the head of the stairs only about ten, and its diameter at the broadest part leading to the gallery of antiquities. This vase is in oniy six inches. But its shape is very elegant, the swell every respect among the most exquisite productions of of the lower and central portion diminishing gradually VOL. I. 2 K 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 geome: so 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 arose, the first attempt was always to solve it geometriwith admirable grace and animation, and are full of ex-cally, and only when that failed, were mechanical means pression in every look and attitude. It is impossible not resorted to, or were other curves constructed, the conto feel that there is great dramatic force and pathos in struction of which, once granted, solved the problem. the sketch, even without being able to interpret it com- The names geometrical and mechanical, as applied to pletely. The Portland vase was discovered about the distinguish one sort of solution from another, may be middle of the sixteenth century, enclosed in a sarco- improper; but that is not the question. When a man phagus, within the monument of the Emperor Alex- asserts that he has found a geomeirical quadrature of ander Severus and his mother Julia Mamæa, commonly the circle, he either does or does not use the word in the called the Monte del Grano, about two miles and a halt sense of the ancient geometers. If he does, and his sofrom Rome, on the Frascati road. The sarcophagus lution is correct, he has certainly solved the problem ; itself, which is a very noble work of art, is still in Rome, but that no one has yet done this, is universally adand the vase also remained for more than two centuries mitted. If he does not use the word geometrical in the in the palace of the Barberini family in the same city, of ancient signification, his solution has nothing to do which it was considered to form one of the chief orna- with the problem which has hitherto remained unsolved. ments. At last it fell into the hands of Sir William Hamil- Many ways have been discovered of finding the area of ton, from whom it was purchased nearly forty years ago a circle, which take something more for granted than by the Duke of Portland. A mould of the Barberini the use of the ruler and compasses only, and any person, vase was taken at Rome before it came into the possession with a reasonable knowledge of mathematics, might add of Sir William Hamilton, by the gem-engraver Pechler; a dozen to the number in a couple of hours. So much and from this the late Mr. Tassie, the celebrated mo- for the geometrical solution of the problem. deller, took sixty casts in plaster of Paris, and then broke It was proved long before the Christian era, that the the mould. Some very beautiful imitations of it have circumferences of two different circles are to one another also been fabricated by the Wedgewoods, in which not as the radii, that is, whatever number of times one cironly the shape, but the colour of the original has been cunference contains its radius, the other circumference attempted to be preserved. Modern art, however, cannot contains its radius as many times, or whatever fraction imitate the vitrified appearance of the material in the one radius is of its circumference, the same fraction is ancient vase. the other radius of its circumference, From this it follows, that if any number of circles were taken, having ON THE QUADRATURE OF THE CIRCLE. for radii a foot, a yard, a mile, &c., whatever number of feet and parts of feet would go round the first, the We have undertaken this article, because we have reason same number of miles and similar parts of miles would to know, that there are still some persons who, from igno- go round the third, and so on. Hence it became a rance of the real question, are turning their attention to question of importance to discover what was the number this useless and exploded problem, and deceiving them of units and parts of units contained in the circumselves and their neighbours into the belief that they have ference of a circle whose radius was the unit. Again, it succeeded in doing that which has been repeatedly shown was proved that the number of square feet and parts of to be impossible. By the rectification of a circle is meant square feet in a circle of one foot radius, was the same the finding of a straight line which shall be equal in as the number of square miles and similar parts of length to the circumference of a given circle, or the two square miles contained in the circle of one mile radius. ends of which when bent round it should exactly meet; by Archimedes showed that a circle of one foot in radius the quadrature of a circle is meant the finding a square contained nearly 3 square feet and of a square foot, which shall be equal in surface to a given circle, that is, which does not differ from the truth by so much as of to the whole space contained inside its eircumference. a square inch, and gives the circle too great by about its There is no difficulty in doing these problems in a man- three-thousandth part. An ancient measure of the ner more than sufficiently correct for all useful purposes. Hindoos makes it 3 square feet and. 177 parts out of If, for example, we take a circle as large as the earth's 1250 of a square foot. This is much nearer to the orbit, and an atom as small as the smallest insect which truth, differing from it by about one-thousandth part of a microscope ever showed in a drop of water, nothing a square inch, but still a little too much. Metiys, who would be înore easy than to find the circunference of Nou ished in the beginning of the seventeenth century, found that if a square foot be divided into 113 parts, the | The author of it was a modest man, and ascribed all the circle of one foot radius contains about 355 of these honour to the Virgin Mary. Another Knight of the parts; a result of surprising accuracy when the simpli- Round Shield found out by his method that the first city of the numbers is considered; it is too great by book of Euclid was all a mistake. About the same time about the fifty-thousandth part of a square inch. These a merchant of Rochelle discovered not only the quadranumbers may be very easily recollected, since, when put ture of the circle, but with it, and depending upon it, á together, they give the first three odd numbers, each method of converting Jews, Pagans, and Mahometans to repeated twice; thus, 113355. Christianity. In 1671, an anonymous writer published The following simple rules will enable every one of a treatise with the following title: Demonstration of the our readers to find the circumference of a circle. If any Divine Theorem of the quadrature of the circle, of the one of them would go direct round the world, he would trisection of the angle, and of the perpetual motion, and by means of them, if the earth were a perfect sphere, be the connexion of this theorem with the Vision of Ezekiel able to tell the length of his journey within less than and the Revelation of St. John.' A certain Cluver four yards. From them the word inch may be taken found out that this problem depended upon another, out, and any other unit substituted. which he expressed thus: “ Construere mundum divinæ To find the length of the circumference of a circle, menti analogum.” The literal translation of this (the multiply the number of inches in the diameter (or twice sense is unknown) is, “ To build a world resembling the the radius) by 355, and divide the product by 113. divine mind.” But the most singular person was one The result is the number of inches in the circumference. Richard White, an English Jesuit, who, having once To find the area, or surface of a circle, multiply the undertaken to square the circle, was afterwards convinced number of inches in the radius by itself, and that pro- by argument that he was in the wrong, which never duct by 355; divide the result by 113, the quotient of happened to any other of this class of speculators, except which is the number of square inches in the area. perhaps to one Mathulon, a Frenchman of Lyons. This We might go on to describe still more accurate man offered to give a thousand crowns to any one who methods; it will be sufficient to say, that the latest of would detect an error in his solution. It was done to them gives the area of a circle true to 127 decimal | his satisfaction, but he refused to pay the money, and a places, as it is called ; that is, if the radius be 1000, &c. court of justice decided that it should be given to the feet, the ciphers being 127 in number, the area of the poor. So late as 1750 an Englishman, Henry Sullamar, circle will be obtained without an error of a square foot. found out the area of the circle by means of the number Still, this is only an approximation, and however | 666 mentioned in the Revelations. But in 1753, a nearly the circumference of a circle has been obtained, Captain in the French Guards did more, for in squaring it has never been obtained exactly. Numberless attempts the circle, which he did with a piece of turf, he hit upon have been made to find the exact ratio of the circum- what he thought was a most obvious connexion between ference to the diameter, but without success; the reason this and the doctrines of original sin and the Trinity. being, as was afterwards proved, that the thing is impos- He offered to bet three hundred thousand francs that he sible. We can now demonstrate that the ratio of the was right, and actually deposited ten thousand of them. circumference to the diameter cannot be accurately A young lady and several other persons easily won the expressed in numbers; all we can do with numbers is, wager, and brought actions for the money; but the to express it as nearly as we please. Thus, using courts declared that the bet was void. decimals, we may say that the circumference of a Such are a few of the most remarkable aberrations of circle whose diameter is 1 is greater than 3 and less the human mind on this problem. They show, in the than 4; greater than 3:1 and less than 3.2; greater most convincing manner, that presumption is rarely conthan 3.14 and less than 3•15, and so on, assigning fined to one subject in the same mind, and that a man nearer and nearer fractions between which it must lie, who, without studying a science, conceives himself to be but never coming to an exact result. Almost every more knowing than those who have passed their lives in projector who imagines he can solve this problem, is the pursuit of it, must previously have brought himself sure to produce some number or fraction as the eract to believe that he is almost a God, and is but one step ratio of the circumference to the diameter; and it is removed from taking the government of the universe out observable that the less his knowledge of geometry, the of the hands of its Creator, and arranging it according to more easily does he overcome the disficulty, and the his own improved notions. more obstinately does he believe himself in the right. With regard to a geometrical solution, and the possiSome have been found hardy enough to deny the common bility or impossibility of it, we shall now say a few words. propositions of geometry, in order to establish their own We have already observed that an arithmetical solution conclusion on this point. Others, totally ignorant of is certainly impossible, that is, there is no number or geometry, hearing that a circle could not be eractly fraction which eractly represents the circumference of measured, have imagined that the word exact was used the circle where the radius is a unit. In 1668, James in the sense in which a carpenter would take it, who, Gregory, a well-known name in geometry, asserted that very properly for his purpose, considers two rods to be a geometrical quadrature was impossible, that is, no use of exactly the same length, when they do not differ to of the ruler and compasses could give a square of exactly the naked eye. These usually cut a circle out in wood, the same dimensions as a given circle. Of this he measure it with a bit of string, pronounce their result to published his demonstration, which was attacked by be perfectly accurate, and are very much surprised that Huyghens, another geometer of the same time. The an ungrateful world does not perceive their claim to dispute has interested mathematicians so little for the one of the first places in the ranks of science. We last century and a half, that few of them seem to have shall give some anecdotes connected with this subject, cared which was right. The historians of mathematics principally extracted from Montucla’s History of the have, of course, been obliged to give an opinion, and Mathematics. 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 to produce anything worthy of notice, after the instances will find in the library of the British Museum, and which we have mentioned : secondly, that he has his find out the error ; and it deserves some attention, own good opinion to a very great degree, for otherwise since neither Montucla nor Hutton, both very well his peace of mind will be disturbed, either by the neglect informed mathematicians, would positively say it was or ridicule which it will be his fate to meet with. To one false. who understands geometry, and who imagines himself to We would not have entered upon this subject at such be the person destined by Providence to work this wonlength, if it were not that there appear, from time to time, der, we have not a word to say: if the study of Euclid pretended solutions of this problem. To any one who has not been sufficient to teach him more sense, or at is ignorant of geometry, we would recommend to be sure least to induce him to wait until he knows more, we of two things before he undertakes it; first, that he has should almost rival him in absurdity, if we thought him an imagination which will set common sense at defiance 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 size: it rises with an herbaceous stalk, about five or six food to a great portion of mankind within and near the inches in diameter at the surface of the ground, but tropics, offering its produce indifferently to the inhabit- tapering upwards to the height of fifteen or twenty feet. ants of equinoctial Asia and America, of tropical Africa, The leaves are in a cluster at the top; they are very and of the islands of the Atlantic and Pacific Oceans. large, being about six feet long and two feet broad: the Wherever the mean heat of the year exceeds 75° of middle rib is strong, but the rest of the leaf is tender, Fahrenheit, the banana is one of the most important and and apt to be torn by the wind. The leaves grow with interesting objects for the cultivation of man. All hot great rapidity after the stalk has attained its proper countries appear equally to favour the growth of its fruit; height. The spike of flowers rises from the centre of and it has even been cultivated in Cuba, in situations the leaves to the height of about four feet. At first the where the thermometer descends to 45° of Fahrenheit. flowers are inclosed in a sheath, but, as they come to Thę tree which bears this useful fruit is of considerable maturity, that drops off. The fruit is about an inch in diameter, eight or nine inches long, and bent a little on ninety-nine pounds of potatoes require the same space as one side. As it ripens it turns yellow; and when ripe, that in which four thousand pounds of bananas are it is filled with a pulp of a luscious sweet taste. grown, the produce of bananas is consequently to that of The banana is not known in an uncultivated state. wheat as 133: 1. and to that of potatoes as 44:1. The wildest tribes of South America, who depend upon The facility with which the banana can be cultivated this fruit for their subsistence, propagate the plant by has doubtless contributed to arrest the progress of imsuckers. Eight or nine months after the sucker has been provement in tropical regions. In the new continent planted, the banana begins to form its clusters; and the civilization first commenced on the mountains, in a soil fruit may be collected in the tenth and eleventh months. of inferior fertility. Necessity awakens industry, and When the stalk is cut, the fruit of which has ripened, a industry calls forth the intellectual powers of the human sprout is put forth, which again bears fruit in three race. When these are developed, man does not sit in a months. The whole labour of cultivation which is re- cabin, gathering the fruits of his little patch of bananas, quired for a plantation of bananas is to cut the stalks asking no greater luxuries, and proposing no higher laden with ripe fruit, and to give the plants a slight nou- ends of life than to eat and to sleep. He subdues to his rishment, once or twice a year, by digging round the use all the treasures of the earth by his labour and his roots. A spot of a little more than a thousand square skill; and he carries his industry forward to its utmost feet will contain from thirty to forty banana plants. A limits, by the consideration that he has active duties to cluster of bananas, produced on a single plant, often perform. The idleness of the poor Indian keeps him, contains from one hundred and sixty to one hundred | where he has been for ages, little elevated above the and eighty fruits, and weighs from seventy to eighty inferior animal ;-the industry of the European, under pounds. But reckoning the weight of a cluster only at his colder skies, and with a less fertile soil, has surforty pounds, such a plantation would produce more than rounded him with all the blessings of society-its four thousand pounds of nutritive substance. M. Hum- comforts, its affections, its virtues, and its intellectua: boldt calculates that as thirty-three pounds of wheat and I riches. (View of Corra Linn.) 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 tra “ Duncan sighed baith out and in, vellers, some of the most romantic and picturesque Grat his een baith bleer'd and blin', scenery in the world. We shall confine ourselves at Spak o' loupin' owre a linn; present to a short notice of the Linns or Falls which Ha, ha, the wooin' o't.” 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 |