ideal of Hercules completed this class of forms. His Heifer and his Sea-monster, are famous among his animal forms. But one thing was wanting to this great sculptor-grace of expression; in this he was surpassed by a rival sculptor, who adopted the undulating line of beauty, and first expressed the sinews and veins with accuracy. He created the ideal of Apollo in the position of an archer, who has just shot the serpent Python-the figure indicating in its expression a placid satisfaction and assurance of victory. This splendid work of art was found at Antium, the modern Capo d'Anzo, at the end of the fifteenth century. It was purchased by Pope Julius II., then a cardinal, and placed in that part of the Vatican called the Belvidere, whence it has been commonly named the Apollo Belvidere. The fore part of the right arm and the left hand, which had been destroyed, were restored by Angelo du Montorsoli, a pupil of Michael Angelo. The ease of the attitude and excellent proportions of the figure are universally admired. Our small outline engraving affords but an imperfect idea of the majestic original. After the ideal style of Phidias and his disciples, succeeded the period in Grecian art distinguished for the beautiful. Praxiteles and Scopas were the great leaders of this improved style, in which beauty was united with grace. The most celebrated works of Scopas are his furious Bacchante the head bending backwards, uniting the highest beauty with Bacchanalan frenzy; his Cupid, his Venus, and his Achilles, who is placed in a mournful attitude, contemplating as if lamenting the loss of his friend Patroclus. Praxiteles, the most feeling of all sculptors, created the perfect ideals of Diana and of Bacchus; the latter being designed by him as a contrast to the satyrs and fauns, whose figures express rudeness and licentiousDess. The figure of Bacchus was soft and tender, without being effeminate, and expressed perpetual gaiety and sport. He effected, also, the admired statue of a Satyr, and the ideal of Eros, or Cupid, which was that of a playful boy. Praxiteles was the first to represent Tenus entirely naked, thus giving to the world a new ideal of the goddess. His most celebrated works are his Venus of Cos and of Cnidos; the former covered from the hip downwards, the latter entirely naked, holding her garment with her left hand over the bath. The group of Niobe is also ascribed to this master. To the epoch which followed that of Praxiteles is Deally ascribed the statue of Venus, styled the Venus de Medicis, from having been placed in the gallery of the Medici at Florence, after its discovery at Tivoli in 1695. It is of pure white marble, and measures, ac cording to one authority, 4 feet 11 inches, and according to another, 5 feet 2 inches, in stature. Some small portions have been restored. It is not ascertained who was the sculptor of this exquisitely designed figure. The pedestal exhibits the name of Cleomenes; but the inscription is modern. It has been the object of the artist to represent Venus either as just coming from the bath, on the point of dressing herself, taken by surprise, and full of modesty, or as appearing before Paris for his judgment in the contest with Juno and Minerva for the prize of beauty. By many, the countenance is thought unintellectual, or at least unexpressive. The graceful waving lines of the body have never been surpassed; and, taken in different points of view, they offer a most advantageous study for ideal grace and beauty. In this latter age, appeared Lysippus of Sicyon, Euthycrates, Apollodorus, and others, among whom stand pre-eminent Polydorus, father and sons, of Rhodes, who formed the celebrated group of Laocoon. A number of great works of this era were executed by Greek sculptors at Rome, to whom we are indebted for many busts of distinguished Romans. Among the numerous ancient sculptures, which, like those already mentioned, have survived until modern times, and been preserved in museums, may be mentioned the following, as useful studies: The Dying Gladiator, a naked manly figure, reclining on his shield, his weapon broken, and expiring from a mortal wound in the side. The Three Graces, a group of female figures in different attitudes, calculated to show the symmetry of the form in various positions. Antinous, the figure of a graceful youth, in a simple attitude, expressive of melancholy, and executed with great correctness of proportion. Adonis, a figure somewhat more slender and elegant. Venus Genetrix, a figure draped, and possessing a noble simplicity of expression; the drapery hanging in the most elegant folds, is in itself a study. Cleopatra, meditating in a reclining position. Laocoon, a group of figures, above referred to, consisting of a father and his two sons, struggling in the folds of serpents, and strongly characteristic of the distraction and suffering which may be conceived to be endured in that dreadful situation. Hercules, a figure expressive of robust muscular strength. After a lapse of nearly ten centuries, the art of sculpture was revived in Italy, and thence spread to France and other modern nations. Germany has latterly produced various sculptors of eminence; among these may be mentioned Dannecker of Stuttgard, and Tieck of Berlin. Dannecker has executed, in pure white marble, a figure of Ariadne seated on a tigress, in an attitude of inexpressible elegance, and equal to the works of ancient art. It is contained in a private museum at Frankfort on the Maine. Thorwaldsen, a Danish sculptor, has likewise produced several works of striking grandeur and beauty; not the least imposing of his designs is the colossal figure of a lion carved in the solid rock at Lucerne in Switzerland. The animal is supposed to be dying from the effects of a wound from a spear, and reclining over a shield emblazoned with fleurs de lis: it is a monument emblematic of the fidelity of the Swiss Guards who perished in defending their master Louis XVI., on the 10th of August 1792, from the brutality of the Parisian mob. Canova, an Italian (born 1757, died 1822), has been justly esteemed the restorer of the graceful and lovely in sculpture. All his works display a surprising degree of softness and delicacy, and will form admirable studies to the young artist. His Cupid; Psyche, standing half dressed, with a butterfly; the repentant Magdalene; Hebe, smiling and animated; a Venus, partially draped; Beneficence (several figures); Graces rising from the bath-are only a few of the works by which Canova gained his great reputation. Modern sculpture has likewise been distinguished by various works executed in England, by Flaxman, Chantrey, Westmacott, Baily, and others. Modern art has almost reached the ancient in the figure of Eve at the Fountain, which is conceived in a style of pure simplicity and grace, with somewhat | eighteenth century, was celebrated for his sea-pieces more intellectuality in the features than is generally to be found in the Grecian sculpture of female figures. Eve at the Fountain. and figures. In the course of the seventeenth century, Holland produced, but cannot be said to have encou raged, many distinguished painters. Among these are included David Teniers, celebrated for his representations of domestic and familiar scenes: Paul Potter, renowned for his cattle pieces, the most remarkable of which is his picture of a bull; it is contained in the royal museum at the Hague, and valued at £5000: Philip Wouvermans, noted for his landscapes, beautiful skies, and scenes with groups of figures hunting, or otherwise engaged in field sports: Berghem, also noted for his landscapes, his foliage, cattle finely drawn and coloured; the woodland scenes of this painter are exquisitely finished and true to nature: Vandervelde, a painter of naval victories and sea-pieces, all remarkable for richness of composition and effect: Gerard Douw, like Teniers, famed for his domestic scenes. Jacob Ruysdael, who painted in the style of Berghem, but gained great celebrity for his representations of water. a high degree of refined taste. If unable to accom plish this desirable object, we advise you to pay occa sional visits to any private collection to which you can gain admission, and also to exhibitions and museums open to the public. Among recent works of English art, generally accessible, the historical pieces of Hayden and Hilton, the historical and pathetic pieces of Allan, the grand architectural idealities of Martin, the church and other architecture of Roberts, the landscapes of Gainsborough, Calcott, Mulready, Thomson, and the Nasmyths, the animals of Landseer, the sea-views of Stanfield, Turner, and Williams, and the faithful delineations of humble life by Wilkie, and many other works of art which might be mentioned, will afford much pure pleasure and instruction, and show what can be accomplished by a cultivated observation, and All the eminent works of art, whether in sculpture or painting, are now contained in a few great national museums, or in the private collections of men of taste Besides studying the manifold graces of ancient and or opulence. The principal museums are those of the modern sculpture, with the view of improving the taste Vatican at Rome, the Gallery of the Medici at Floin reference to figures, you are recommended to study rence, the royal galleries at Munich (now the head the styles and compositions of the more celebrated school of painting and other fine arts in Germany), painters, as nothing could be more suitable for impart the Louvre at Paris, the British Museum and National ing correct ideas respecting drawing, disposal of groups Gallery in London, and the Royal Museum at the of objects, and colouring. Another great benefit will Hague. If at all within the means of young men deconsist in making you feel your own deficiency, and sirous of pursuing professions in which taste in draw how much you require to attain by diligent study. ing is requisite, we strongly recommend them to visit Painting, you will learn, has, since the revival of art, the museum of the Louvre, which is rich in ancient taken the character of schools, or peculiar styles, each sculpture and modern paintings; the sight of the many of which has had its leaders and followers; for ex-fine works of art in Paris would scarcely fail to inspire ample, there is the Florentine school, commenced by Michael Angelo Buonarotti (born 1474, died 1564), who delighted in representations of the grand and terrible. The Roman school, of whom Raphael (14831520) was the head. The great characteristics of this school are, truthful representations of nature, a just expression of the passions, a chaste nobleness of design, and correctness of drawing. The Venetian school, headed by Titian (1477-1576), the characteristics of which were the harmony of colours, delicacy of tints, and a judicious contrast of light and shade. This school was improved by Corregio and Tintoretto. The German school, led by Albert Durer (1471-1528) and Holbein. The second Lombard school, distinguished by the works of the three Caracci (1555-1609). The French school, founded by Nicholas Poussin (15941665), Vouet, and Charles Le Brun. The Flemisha persevering desire to excel. school was founded by Peter Paul Rubens (1577-1640), In the course of your observations you will learn, whose design is dignified, his drawing of anatomy and that in the delineation of human figures great care reperspective correct, and his colouring brilliant. The quires to be paid to historical costume and the fashion only objection to some of his figures is, that they are of artificial objects represented. A person who lived too heavy; and certainly they want the grace of those in the eighteenth century, for instance, should not be of Raphael. The Dutch school, of which the most pro- dressed as an ancient Roman; nor should the interior minent painter is Rembrandt (1606-1668), is less distin- of a house of the fifteenth be decorated like one of the guished for taste than the faithful adherence to nature. sixteenth century. On this account, every student of The works of the eminent masters in these various the fine arts requires to be well instructed in history, schools were chiefly scriptural and historical; and archaiology, and other branches of learning. With scenes of a miscellaneous kind, embracing landscapes, respect to statues, it is so important to give an easy figures, animals, sea-pieces, architecture, and other and graceful effect, that a departure from exact costume subjects, were painted by contemporary artists, who is allowable, so far as to place a loose garment over of followed no particular school. Among these great about the person. We have only to add, that a dus masters may be named Claude Gelee of Lorraine (1600-perception of the beautiful and truthful in pictorial 1682) whose landscapes are exceedingly beautiful, his colouring delicate, his tints tender, and his lights and shades unrivalled; Salvator Rosa (1615-1673), whose taste was for the wild, rugged, and romantic aspects of nature; Gaspar Poussin (1613-1675), whose pictures are grand, and remarkably true to nature. The sixteenth and seventeenth centuries produced the following masters, all of whose works are esteemed :-Paul Veronese, Guido, Carlo Muratti, and Spagniolletto historical; Murillo (Spanish)-figures; Hobbimalandscape; Canaletti-buildings; and De Witt-the teriors of churches. Vernet, a Frenchman of the | delineation must be in all cases a work of time. A first, the unpractised eye, or, properly speaking, the untutored mind, will perhaps be most charmed with gaudy daub, and see in the finest work of art only a dull and valueless scene. But the repeated contemplation of pictures, the comparison of one with another, and the constant reference to actual nature, will remove such impressions, and the work of true merit standing apparent, will receive the highest meed of approbation. Printed and published by W. and R. CHAMBERS, Edinburgi CHAMBERS'S INFORMATION FOR THE PEOPLE. CONDUCTED BY WILLIAM AND ROBERT CHAMBERS, EDITORS OF CHAMBERS'S NUMBER 67. NEW AND IMPROVED SERIES. ARITHMETIC-ALGEBRA. In the present and succeeding sheet, an attempt is made to convey to the comparatively unlearned mind some knowledge of Mathematical science, both as regards measurement by numbers (ARITHMETIC) and meacrement of dimensions (GEOMETRY.) The sketch we fer of each is necessarily brief and imperfect; but ar end will be gained if we afford that amount of information on the subject which is generally possessed ty persons of moderately well-cultivated intellect.—Ed. A recognition of the value of numbers is coeval with the dawn of mental cultivation in every community; but considerable progress must be made before methods of reckoning are reduced to a regular system, and a rotation adopted to express large or complex quantities. Au inability to reckon beyond a few numbers is always A proof of mental obscurity; and in this state various vage nations have been discovered by travellers. Some are found to be able to count as far as five, the its of the hand most likely familiarising them with that number; but any further quantity is either said ta consist of so many fives, or is expressed by the more convenient phrase, "a great many." Among the North American Indians, any great number which the mind incapable of distinctly recognising and naming is guratively described by comparing it to the leaves of the forest; and in the same manner, the untutored Negro of Africa would define any quantity of vast ount by pointing to a handful of sand of the desert. On the first advance of any early people towards vilisation, it would be found impossible to give a parate name to each separate number which they had occasion to describe. It would therefore be necessary to consider large numbers as only multiplications of Sertain smaller ones, and to name them accordingly. This is, no doubt, what gave rise to classes of numbers, toch are different in different countries. For instance, the Chinese count by twos; the ancient Mexicans reckoned by fours. Some counted by fives, a number which the fingers would always be ready to suggest. The Hebrews, from an early period, reckoned by tens, which would also be an obvious mode, from the number of the fingers of the two hands, as well as of the toes of the two feet. The Greeks adopted this plan; from the Greeks it came to the Romans, and by them was read over a large part of the world. NOTATION. The representation of numbers by written signs is an art generally believed to have taken its rise after the formation of alphabets. One of the earliest sets of written signs of numbers of which we have any notice, is certainly the scries of letters of the Hebrew alphabet which was used by that people-Aleph, beth, gimel, daleth, he, vau, zain, cheth, teth, standing respectively for the numbers one, two, three, four, five, six, seven, eight, nine. The Greeks directly adopted this plan PRICE 1ąd. from the Hebrews, forming their numbers thus :1 alpha, 2 beta, 3 gamma, 4 delta, 5 epsilon-here, having no letter corresponding with the Hebrew vau, they put in the words or Bau to denote six; after which they proceeded with 7 zeta, 8 eta, &c. Before adopting this plan, they had indicated one by iota, probably because it was the smallest of their letters, five by II (P) being the first letter of pente, five; ten by ▲ (D) being the initial of deka, ten. After having for some time adopted the Hebrew plan, they divided their alphabet into three classes; the first ten letters expressing the numbers from one to ten, while twenty, thirty, forty, and so on up to a hundred, were signified by the next nine, ninety being expressed by a figure formed on purpose, and resembling the Arabic 5 inverted. The remaining seven letters expressed 200, 300, 400, 500, 600, 700, 800; and for 900 there was another inverted figure. Larger numbers were represented by letters accented in various ways. The Romans, from an early period, had a plan of expressing numbers, which seems to have been at first independent of the alphabet. The following clear account of it was given a few years ago by Professor Playfair :-" To denote one, a simple upright stroke was assumed ; and the repetition of this expressed two, three, &c. Two cross strokes X marked the next step in the scale of numeration, or ten; and that symbol was repeated to signify twenty, thirty, &c. Three strokes, or an open square, were employed to denote the hundred, or the third stage of numeration; and four interwoven strokes M, sometimes incurved M, or even divided CIO, expressed a thousand. Such are all the characters absolutely required in a very limited system of numeration. The necessary repetition of them, however, as often occasionally as nine times, was soon found to be tedious and perplexing. Reduced or curtailed marks were therefore employed to express the intermediate multiples of five; and this improvement must have taken place at a very early period. Thus, five itself was denoted by the upper half V, and sometimes the under half_, of the character for ten; L, or the half of, the mark for a hundred, came to represent fifty; and the incurved symbol M, or CIO, for a thousand, was split into 15, to express five hundred. These important contractions having been adopted, another convenient abbreviation was introduced. To avoid the frequent repetition of a mark, it was prefixed to the principal character, and denoted the effect by counting backwards. Thus, instead of four strokes, it seemed preferable to write IV; for eight and nine, the symbols were XII and IX; and ninety was expressed by XC. This mode of reckoning by the defect was peculiar to the Romans, and has evidently affected the composition of their numerical terms. Instead of octodecem [eight and ten-for eighteen], and novemdecem [nine and ten-for nineteen], it was held more elegant, in the Latin language, to use undeviginti [one | designation of Sylvester II., travelled into Spain, and from twenty], and duodeviginti [two from twenty]. But the alphabetic characters now lent their aid to numeration. The uniform broad strokes were dismissed, and those letters which most resembled the several combinations were adopted in their place. The marks for one, five, ten, and fifty, were respectively supplied by the letters I, V, X, and L. The symbol for a hundred was aptly denoted by C, which had originally a square shape, and happened, besides, to be the initial of the very word centum. The letter D was very generally assumed as a near approximation to the symbol for five hundred; and M not only represented the angular character for a thousand, but was likewise, though perhaps accidentally, the first letter of the word mille."-Edin. Rev. xviii. 193. studied for several years the sciences there cultivated by the Moors. Among other acquisitions, he gained from that singular people a knowledge of what are now called the Arabic numerals, and of the mode of arithmetic founded on them, which he forthwith disclosed to the Christian world, by whom at first his learning caused him to be accused of an alliance with evil spirits. The knowledge of this new arithmetic was about the same time extended, in consequence of the intercourse which the Crusaders opened between Europe and the East. For a long time, however, it made a very slow and obscure progress. The characters themselves appear to have been long considered in Europe as dark and mysterious. Deriving their whole efficacy from the use made of the cipher, so called from the Arabic word tsaphara, denoting empty or void, this term came afterwards to express, in general, any secret mark. Hence, in more troublous times than the present, a mode of writing was practised, by means of marks previously concerted, and called writing in cipher. The Arabic characters occur in some arithmetical tracts composed in England during the thirteenth and fourteenth centuries, particularly in a work by John of Halifax, or De Sacrobosco; but another century elapsed before they were generally adopted. They do not ap pear to have settled into their present forms till about the time of the invention of printing. It would be impossible to calculate, even by their own transcendent powers, the service which the Arabic numerals have rendered to mankind. NUMERATION. The Arabic numerals take the following well-known forms :- 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine of these, called digits or digital numbers, represent, each, one of the numbers between one and nine, and when thus employed to represent single numbers, they are considered as units. The last (0), called a nought, nothing, or cipher, is in reality, taken by itself, expressive of an absence of number, or nothing; but, in connexion with other numbers, it becomes expressive of number in a very remarkable manner. The Hebrew, improved Grecian, and Roman numerals, were perhaps sufficient to express any single number with tolerable precision; but it is easy to see that they must have been nearly unfitted for use in the processes of arithmetic. The Greeks certainly contrived to overcome many obstacles in the business of calculation, and even could express fractions-though, from a practice of adding from left to right, and ignorance of the plan of carrying tens to the higher places, their problems were at all times awkward and complicated. The Romans, however, careless of old inconveniences, were still more awkwardly situated than the Greeks. Let any reader just suppose, for instance, even so simple a question as the amount of XLVIII added to XXXIV! It is evident that placing the figures below each other, as we do with the Arabic numerals, would serve little to facilitate such a calculation. In fact, the Romans were obliged, where mental calculation would not serve, to resort to a mechanical process for performing problems in arithmetic. A box of pebbles called loculus, and a board called abacus, constituted their means of calculation; and of these every schoolboy, and many other persons, possessed a set. The word calculation claims no higher descent than from calculus, a stone or pebble. The board was divided from the right to the left hand by upright columns, on which the pebbles were placed, to denote units, tens, hundreds, thousands, &c. The labour The valuable peculiarity of the Arabic notation is the of counting and arranging the pebbles was afterwards enlargement and variety of values which can be given sensibly abridged by drawing across the board a hori- to the figures by associating them. The number ten is zontal line, above which each single pebble had the expressed by the 1 and 0 put together-thus, 10; and power of five. In the progress of luxury, tali, or dies all the numbers from this up to a hundred can be exmade of ivory, were used instead of pebbles; and after-pressed in like manner by the association of two figures wards the whole system was made more convenient by substituting beads strung on parallel threads, or pegs stuck along grooves; methods of calculation still used in Russia and China, and found convenient in certain departments of Roman Catholic devotion, and in several familiar games in more civilised countries. With such instruments, problems in addition and subtraction would not be very difficult; but those in multiplication and division, not to speak of the more compound rules, must have been extremely tedious and irksome. So disagreeable, indeed, was the whole labour, that the Romans generally left it to slaves and professional calculators. The numerals now in use, with the mode of causing them by peculiar situation to express any number, and whereby the processes of arithmetic have been rendered so highly convenient, have heretofore been supposed to be of Indian origin, transmitted through the Persians to the Arabs, and by them introduced into Europe in the tenth century, when the Moors invaded and became masters of Spain. Such in reality appears to have been in a great measure the true his tory of the transmission of these numerals; but as it has been lately found that the ancient hieroglyphical inscriptions of Egypt contain several of them, learned men are now agreed that they originated in that early seat of knowledge, between which and India there exist more points of resemblance, and more traces of intercourse, than is generally supposed. In the eleventh century, Gerbert, a Benedictine monk of Fleury, and o afterwards ascended the papal throne under the thus, twenty, 20; thirty, 30; eighty-five, 85; ninetynine, 99. These are called decimal numbers, from decem, Latin for ten. The numbers between a hundred and nine hundred and ninety-nine inclusive, are in like manner expressed by three figures-thus, a hundred, 100; five hundred, 500; eight hundred and eighty-five, 885; nine hundred and ninety-nine, 999. Four figures express thousands; five, tens of thousands; six, hundreds of thousands; seven, millions; and so forth. Each figure, in short, put to the left hand of another, or of severs! others, multiplies that one or more numbers by ten. Or if to any set of figures a nought (0) be added towards the right hand, that addition multiplies the number by ten thus, 999, with 0 added, becomes 9990, nine thousand nine hundred and ninety. Thus it will be seen that, in notation, the rank or place of any figure in a number is what determines the value which it bears. The figure third from the right hand is always one of the hundreds that which stands seventh always expresses millions; and so on. And whenever a new figure is added towards 1, 2 3 4, 5 6 7,8 9 0 the right, each of the former set obtains, as it were, a promotion, or is made to express ten times its former value. A large number is thus expressed in the Arabic nume rals, every set of three from the right to the left hand being divided by a comma for the sake of distinctness. Thousands of millions. Hundreds of millions. ➡Millions. co Hundreds. Tens of millions. Thousands. Tens of thousands. Tens. Units |