MATHEMATICAL QUESTIONS, SOLUTIONS, &c.

EDITED BY W. J. REYNOLDS, ESQ. B.A.

Solutions to Questions on Pp. 152, 182, & 212. 23. We may conduct our investigations, in reference to this question, solely with regard to the time of coincidence of the hour hands, as that of the minute hands will evidently follow as a necessary consequence.

Now, as the minute hand of the first clock traverses 318 of the circumference of the face in three hours, or 1 of that space in one hour, its hour hand describes of the circumference in the same

time.

In a similar way, it will be found that the portions of the circumference described in one hour by the hour hands of the second and third clocks are expressed by the fractions 247 and 143 respectively. These fractions, expressed with their least common denominator, are 7, 640 640 respectively.

792 741 715

Hence, at the end of 8640 hours the extremity of the hour hand of the first clock will have traversed the circumference of the face 792 times; that of the second, 741 times; and that of the third, 715 times exactly.

Now, 8640 contains 24 precisely 360 times; consequently, when the. clocks in question again show 11 o'clock together, they will indicate the true time in the forenoon of the 360th day following that on which they were first wound up.

24. Let x = the number of shillings. Then 21a = the number of sixpences, represented by the same number of halfguineas.

So that, by the question, we have =20x

21x 12 =

... x == 12.

Consequently he had 12 shillings.

25. In order to solve this problem, we will determine-1. The rate of the special

train: 2. The distance between C and B: 3. The distance between the trains when the special train leaves the intermediate station D.

Now the express travels 1 mile in of an hour, and consequently 123 miles in 23, or 43, of an hour-that is, in 2h. 44m., which is therefore the time that elapses between 2 P.M., and the moment of its arrival at D. There is a delay here of 5 minutes, and the train therefore leaves D 2h. 49m. after starting; consequently at 4h. 49m. P.M.

The special arrived at D 44 minutes after this, or at 5h. 33m. P.M.; and as it started at 3h. 30m. P.M., it follows that this train travelled 123 miles in 2h. 3m., or 123 minutes. Therefore, the rate of the special was a mile in a minute, or 60 miles an hour.

Again, as we learn from the question that a train at B was due at A in 12 minutes, and that the distance BA is traversed at the rate of 20 miles an hour, or 1 mile in 3 minutes, it follows that B is 4 miles from A. Now C is 61 miles from A. Therefore the distance between B and C is 21⁄2 miles.

Lastly, as the express, travelling at the rate of 45 miles an hour, or of a mile in a minute, left D 47 minutes before the special, therefore it was of 47 miles from D when the special started thence. Also, when the express was at B, the special was at C, 211 miles behind it. Therefore, in travelling from D to C, the latter train gained on the former the difference between

of 47 miles and 22 miles, which will be found to be 27 miles. But the respective rates being 65 and 45 miles per hour, it follows that 20 miles is gained every hour by the latter train, and therefore 27 miles will be gained in

179-Weight of Audiences. J. J.-The pressure of an ordinary crowd at a public meeting, or in the pit of a theatre, does not exceed sixty pounds to the square foot. 180-Pronunciation. B.-It is customary to emphasize the name Saladin on the second syllable, as if spelled -as so many similar names are-Sa-ladd-in.

181-Pronunciation. Beaumaris. J. L. K.-The first syllable is pronounced like bow in "rainbow;" the second as if spelled "maw." The accent falls on the second syllable.

182-Lost M.S. C. R.-It is stated, upon apparently good authority, that some of the lost works of the old Greek authors have recently been discovered in a cave at the foot of Mount Athos.

183-Jordan. T. F.-The waters of the Jordan flow into the Dead Sea, from which they rise by evaporation, leaving the saline particles, which they have dissolved in the course of the river, in the dense waters of the lake. 184-British Insects. Y. Z.-The total amount of known British insects is 10,012, which is equal to twice the number of ascertained birds, and to more than ten times the number of ascertained quadrupeds, throughout the whole world.

185-"Mash Allah." R. S.-The Turks mingle in their opium cakes a quantity of syrup made from native fruits; these are much prized on great festival days, and are stamped with the words "Mash Allah," which signifies, the work of God.

186-The first Opera. N. O. A.-The first composer who set an opera to music was Francesco Barbarini, an Italian artist; and the piece to which he gave the garb of harmony was, "The Conversion of St. Paul." It was brought out in Rome in 1160.

187-A Second Deluge. L. D.-Albers computed that after a lapse of 83,000 years, a comet will approach the earth, in the same proximity as the moon; after 4,000,000 years, it will reach within 7,700 miles; and then, if its attraction equals that of the earth, the waters of the earth will be elevated 13,000 feet, and cause a second deluge. 188-The Original Bank Charter. F. G.-The original charter was, in 1708, extended to August 1732, and was five years after extended to 1742: it was then renewed to 1764; it was in this manner extended from time to time, in five years' grants, until 1800, when it was extended 33 years, the final extent of the original charter.

189-Goethe's early Essays. W. M.-The first attempt in publishing was in 1773 (his 24th year), when the Gotz with the Iron Hand appeared. His Sorrows of Werter made its appearance at the close of the year following. He was educated at the university of Leipsic. He was born at Frankfort, and died at Weimar.

190-The Feudal System. G. D.-Every lord having authority over his tenant in capite-in chief, or direct from the king-was bound by oath to see that all under his tenure were of good behaviour to the sovereign. In making the laws, the common people were not consulted, but as stepping Letween such and the crown. The barons were called concilium regni.

191-Fossils of unknown Animals. M. P. S.-Various specimens of the footprints of unknown birds and beasts have been discovered of late years, chiefly in sandstone. Among the most perfect afe those of the Cheirotherium, found in the quarries of Hildburghunsen in Saxony, and the Ornithichnites, brought from the sandstone beds of Greenfield, Massachusets.

192-Botany. W. L. R.-An excellent microscope suited for almost every botanical purpose, can be purchased for twelve guineas. We know of no better work to guide the amateur botanist to discover the genera, species and English names of plants, than Lindley's Synopsis of the British Flora, price 8s. Its classification and arrangement is founded on the natural system of Jussieu, and De Candolle.

193-Melanchthon Library. R. Y.-The amiable Melanchthon possessed in his library only four authors, Pliny, Plato, Plutarch, and Ptolemy the geographer. From these, however, he derived more information than is acquired by students who have facilities a thousand-fold as great. Hence it is not an uncommon compliment to the possessor of a small but well-worn collection of books, to compare it to Melanchthon's library.

194-Diamond. J. H. C.-A French chemist is said to have produced crystalline black diamonds, which had all the qualities of the true gem, except its pure colour. Diamonds increase in value in proportion to the purity of their transparency, as well as in proportion to their size. In many cases the colour seems to be derived from charcoal not perfectly crystalline; but to test the impurities in the stone is almost impossible, as it undergoes no change except under a high degree of heat.

195-Origin of the Commons. J. B.-In the twentysecond year of Henry II. (A. D. 1176), Benedict Abbas, one of our old monkish annalists, relates, that about the Feast of St. Paul the king came to Northampton, and there held a great council concerning the statutes of his realm, in the presence of the bishops, earls, and barons of his dominions, and with the advice of his knights and men. This is the first record which has the appearance of including the Commons in the national councils.

196--Gold Mines. M. F.-The gold veins which were at first supposed to be confined to North Carolina, extended in very close succession from the vicinity of the Potomac, into Alabama and Tennessee. There have been instances of immense gain from the mines, and even factories erected for the extraction of the ore. The United States Mint have coined as much as 460,000 dollars' worth of gold in one year, the produce of the Carolinas, Virginia, and Georgia.

197-Shakspere's Contemporaries. A. S. H.-The most valuable contributor to authentic notices of the life and times of the immortal Will. Shakspere, is Francis Meres. Ile published a book, called Palladis Tamia, or Wit's Treasury. Sir Philip Sidney's Defence of Poesy, was not published till 1595, though written some years before. Meres was a master of arts at Cambridge, and subsequently entered the church. Sir P. Sidney was killed at the battle of Zutphen, in the year 1586.

198-Et. And. J. L. K.-The figure "&" is sometimes called amperzand. It really stands for the Latin word et, and arose, perhaps, from the habit of the old manuscript writers of writing "et" in one stroke, beginning at the bottom of the t, and after forming an upright stroke, returning to make the e ia like manner from below upwards, finishing by crossing the stroke of the t before taking the pen from the paper. The upstroke of the t then was omnitted in quick writing, till at last the word et came to be written "&."

199-Coal. E. F. F.-There can be no doubt that coal is vegetable matter, which has undergone a peculiar change consequent upon its exposure to heat, pressure, and moisture, for an immense period of time. The coal strata are not all derived from forests probably, but from peat bogs upon which trees grew. On the Irish and Scottish moor lands we may actually see the process of change of vegetable matter into carbonaceous material going on. [See Grandfather Whitehead's Lecture on Coal, Family Friend, vol. ii., p. 6.

200-Large Steamers. F. J. C.-The largest steamship ever yet built, we believe, is the United States' mail steamer, named the Atlantic. It is 2,860 tons burthen. The following are her dimensions:-Length between perpendiculars 276 feet; breadth of beam 45 feet; breadth across paddles 75 feet; depth of hold 31 feet 7 inches; saloon 67 feet long by 20 feet wide; dining saloon 60 feet long by 42 feet wide. The steam power is equal to 1000 horse. Having no bowsprit or jib-boom, her appearance is rather peculiar.

201-Bibliomania. A. L. L.-This folly is by no means a matter of ancient history. We are familiar with prettily-nay, luxuriously-furnished rooms, in which valuable books are shut up, in wood, and wire, and glass, like criminals forbidden to hold intercourse with mankind. We plead guilty to a desire to see our friends in the bookshelves clad in good binding, and we love well-printed editions" a rivulet of print running through a broad meadow of margin," but to so adorn and bedizen books as to make them too costly for use, is as foolish an act as that of the man who, for very love, drowned his nightingale in nectar.

202-Silk. G. C. D.-Two Nestorian monks of Persia had travelled to Serindi (China), and lived there long enough to become acquainted with the history and treatment of the silk, as well as the processes of manufacture. Happening to be at Constantinople after their return, they stated their information to the emperor, who engaged them to return to Serindi, and bring away some of the eggs of the silkworm. They accordingly went back, and secured a quantity of the eggs, which they deposited in a hollow cane, and brought in safety to Constantinople. They succeeded in hatching the eggs by the heat of a dunghill, after which they fed the worms with mulberry leaves, furnishing an abundant supply for manufacturing.

203-Camera Obscura. J. H. J.-The dimensions of the box for a camera obscura must depend upon the size and focus of the lens. The most perfect camera is formed by placing an inclined mirror in a revolving frame on the top of a building, by which the rays are thrown down on a convex lens in the root, and the images of all the surrounding objects received on a table. The images being accompanied by the motions belonging to the objects, a

very pleasing picture is formed. Very few amateurs possess sufficient ingenuity to construct a camera calculated for daguerreotype, and they may now be purchased at a very moderate rate. Lenses vary greatly in price. The distance of the object to be portrayed from the camera must depend upon the size it is intended to make the portraiture, and the focal distances of the lens. Such instruments cannot be constructed accurately without a knowledge of the fundamental principles of optics.

204-Chiltern Hundreds. S. C.-This name is given to a portion of the high lands of Buckinghamshire, known by the name of the Chiltern Hills. Formerly these hills were covered with timber, and are said to have afforded shelter to numerous banditti. To put these down, and to protect the inhabitants of the neighbouring parts from their depredations, an officer was appointed under the crown, called "the steward of the Chiltern Hundreds." The duties have long since ceased, but the nominal office is retained for a convenient purpose. As the law now stands a member of the House of Commons, who is not in any way disqualified, cannot resign his seat. A member, therefore, who wishes to resign, accomplishes his object by applying for the office of steward to the Chiltern Hundreds of Stoke, Desborough, and Bodenham," which being held a place of honour and profit under the crown, vacates the seat, and a new writ is in consequence ordered. As soon as the office is obtained, it is resigned. It is in the gift of the Chancellor of the Exchequer, who exercised the right of refusal in 1842.

205-Papuas. R. G.-They are the aboriginal inhabitants of the great islands of the Malayan archipelago, and to this day occupy all the inland and mountainous parts of New Guinea, spreading themselves in scattered hordes over the southern coast of New Holland. Their migration to the Land of Dieman was by passing along the group of the New Hebrides, and from New Caledonia. They distinguish themselves by the term Arfuki, or mountaineers. That a relationship exists between these woolly-haired natives of New Guinea and the Madecusses (natives of Madagascar) may be traced in the unquestionable affinities in the dialects of the two races; and whether allied in lineage or not, it is certain that they belong to the same variety of the human species, since they resemble in form and colour, and more particularly in the texture of their hair, which in general differs from that of the Hottentot, and from the lank hair of other nations. There is also a distinct race of the Papuas, distinguished by their coarse lank hair, and they are denominated the Endamênes, as inhabitants of the interior; these are the people termed by some writers the Alfourns, Alfoërs, or Halaforas. 206-Locks. E. D. A.-The most ancient locks, of whose form and construction there is any certain knowledge, is the Egyptian, which has been in use for upwards of 4,000 years. In Chubb's lock, patented in 1818, and since improved, there are six separate and distinct tumblers, capable of being elevated to different heights, and which lie by the side of each other like the blades of a knife. It appears that with an average sized key having six steps, there could be, by different combinations, 86,400 changes; or if the seventh step, which throws the boit, were taken into account, with a capability of reduction of ten times, the number of changes would be 864,000-that is to say, there would be 864,000 chances to one that any key except the right one would not fit the lock. In Chubb's lock, moreover, there is a detector, which indicates when any attempt at pic ing has been made. A detector lock has also been patented by Mr. Cotteril of Birmingham, in which the nicety of construction is such that it is impossible for the maker to produce a key precisely to fit the same lock. In this there is also a detector, which, on the application of a false key, may be said to "lock the lock," making all its parts fast.

207-Hydraulic Presses. V. V. Mr. Clark, in his essay on the Tubular Bridges, states, that in most cases hydraulic presses fail by splitting down the centre, the fracture being generally quiet and gradual, owing to the incompressibility of water. It appears that cylinders can be made constantly to bear a pressure of four tons to the circular inch; where the cylinders fail, they are almost invariably proved to be defective castings. The pressure in the large press used at the Britannia Bridge, was equal nearly to four tons per square inch, which would raise a column of water about five and a half miles, and throw a jet over the highest mountains on the globe. Hydraulic presses are used extensively in almost every branch of trade where steady and slow compression is required, and it is necessary to regulate the pressure precisely. Among other purposes to which hydrostatic power is applied, we find it used to press veneered wood, to compress wool, hay, cloth, or ca.icoes for packing, to squeeze oil from

seeds, to draw lead and pewter pipes, in the manufacture of gutta percha and leathern fancy articles, for glazing or hot-pressing paper, for raising vessels in dry docks, to press metals into moulds, &c. &c.

209-Reviews.

208-Gold and Silver Coin. M. R.-In 1717 gold was made to bear, for the first time, a fixed price in silver coin, the guinea heing ordered to pass current at, and for no more than 218. By this regulation, a guinea and 21s. in silver became equally legal tenders, and so they continued from 1717 to 1774, except that from 1750 guineas were not allowed that privilege if they were deficient in weight more than five grains. The standard fineness of our silver coin was, from the earliest time, three grains alloy in forty grains. Fine gold is worth 2 and one-eighth of a penny per grain; crown gold, 2d. per grain. A Tower pound, of 5.400 grains, at 2 and one-eighth per grain, is equal to a Troy pound of 5,760 grains, at 2d. per grain. The standard fineness of our gold coin was, till 1527, half a grain alloy in 96 grains. The old standard is called fine gold, the new, crown gold. The Tower pound was in use till 1327; since that period the Troy pound has been the standard at the Mint. The seven-shilling pieces of gold have not been in general issue since 1797. From 1774 to 1797, silver coins to an amount not exceeding £25 at one payment, and silver bullion to any amount at 58. 2d. per ounce, were made a legal tender, with guineas which had not lost more than one grain of their full weight. Dollars (value 5s. 6d.) were issued in 1811. B.E.-The origin of literary journals was the project of Denis de Sallo, a counsellor in the parliament of Paris. In 1665 appeared his Journal des Scavans, which he published in the name of his footman. This was so successful that it was imitated throughout Europe, and translated into many languages. Yet the criticism of Sallo was full of asperity and malignant wit, and thus excited murmurs on all sides from authors, so that at the conclusion of his third volume, Sallo was compelled to cast down his biting pen. Sallo was followed by the Abbé Gallois, who was as insipidly mild as his predecessor was waspisaly severe he confined himself to extracts from the works which he noticed. Bayle, in 1684, undertook his Nouvelles de la Republique des Lettres. He possessed the happy art of presenting the reader with the main features of bocks which came under review. It is said of him that "he wreathed the rod of criticism with roses;" but yet he failed to satisfy, and in his later volumes he forsook the path in which he had set out. He gave to the wor'd thirty-six small volumes of criticism, the last pub.ished in 1687. The work was continued by Bernard, and afterwards with more success by Basnage. Le Clerc was the contemporary of Bayle, and his antagonist. He gave to the world eighty-two volumes, comprising three Bibliotheques-Universelle et Historique, Choisie, and Ancienne et Moderne. Gibbon referred to Le Clerc's volumes "as an inexhaustible source of amusement and instruction." Beausobre and L'Enfant wrote a Bibliotheque Germanique, from 1720 to 1740, in fifty volumes. The Bibliotheque Britannique contains an account of English books, ficm 1373 to 1747. It contains twenty-three volumes, published at the Hague. The Journal Britannique exhibits a view of English literature, from 1750 to 1755; it was edited by Dr. Maty, a foreign physician residing in London. Our own early journals notice but few pulications. The original Monthly Review commenced in 1749, and was the mother of our reviews. The labour of reviewing is enormous; and the science requires mental endowment of a high order in those who would extensively practise it. It has been justly remarked, that "the most noble criticism is that in which the critic is not the antagonist so much as the rival of the author."

QUESTIONS ANSWERED.

13-Duke Humphrey. T.-In Old St. Paul's in London, one of the aisles was known by the name of Humphrey's Walk, because, says Stowe, ignorant people mistook the monument of Sir John Beauchamp for that of Humphrey, Duke of Gloucester. St. Paul's, about 1590, was used as a promenade, and even as a place of business; and unfortunate gallants who did not know where to find a dinner, spent the dinner hour in walking up and down the aisle. which gave rise to the expression-"to dine with Duke Humphrey."-T. B. J.

14-Druids. J. H.-For information on the relizion and customs of the Druids, see Cæsar, Galtic War, book vi., chap. 13; and also Brande's Popular Antiquities, and Fosbroke's Encyclopædia of Antiquities, where various works are mentioned containing valuable information es the subject.-T. B. J.