MATHEMATICAL QUESTIONS, SOLUTIONS, &c. EDITED BY W. J. REYNOLDS, ESQ. B.A. Solutions to Questions on pages 62 and 91. 15. Let AOB be a circular arc, which can be moved so as always to pass through the fixed point O, while one extremity of the arc is always on the fixed straight the other, and has one of its extremities hinged at the middle point C thereof, the other being capable of turning in the same plane about a fixed point, O, and one extremity, A, of the longer rod be moved along a straight line OA, then the other extremity, B, must also describe a straight line. This principle has been ingeniously applied as a parallel motion. (The word motion is frequently used by mechanicians to imply a system of machinery by which motion is modified or directed.) Many of our readers are, no doubt, aware that in a steam-engine the motion of the piston in the cylinder is communicated to the rest of the machinery by means of a piston rod, which is attached to the piston, and is line OA. Let this circular arc be moved into the position aOb; then, since the angles in the same segment of a circle are equal to one another (Euclid, B. iii. Prop. 21), therefore the straight line 60 makes the same angle with the fixed straight line Oa that BO does; consequently the points O, B, and b, must lie in one straight line. Remark. As the angle in a semicircle is a right angle, it is plain that if the arc in question be semicircular, the lines described by its extremities will be at right angles to one another: and it is imB A mediately evident, that if C (see Fig.) be the centre of the arc in any position, the line OC is equal to CA, the radius of arc. Consequently, the centre C of the arc describes, during the motion, a circular arc of equal radius about O as centre. It follows from this, that if OC and BA be two rods, one of which is half the length of connecting rod drawn up and down by it. The piston rod acts on the connecting rod, on which it exerts alternately a pulling and a thrusting force in a direction generally oblique to the line of its own motion. Thus it will be seen that there will exist an oblique reaction on the piston rod, tending to alter the line of its motion; and to counteract the effect of this reaction, the parallel-motion machinery is requisite. is A glance at the wood-cut will show how the principle we have been considering may be applied for this purpose. In that Fig. the "radius rod" corresponds to the line OC in the diagram; the "beam" the diameter AB; the groove in which one end of the "beam" moves represents so much of the line OA as is requisite. The machinery which is moved by the connecting rod" is not shown. As the piston rod is (by the pressure of the steam acting on the piston) thrust out of the cylinder, it is plain that the oblique reaction of the connecting rod will tend to pull its end inwards, which tendency is neutralized by the parallel motion; and so the piston rod travels in a straight line towards the point a. 66 16. It is only necessary to find the weights of hydrogen and nitrogen in the mixture, taking a cubic inch as the unit of volume. Now the volume of nitrogen in the mixture is 5 cubic inches, and that of the hydrogen is 15 cubic inches. ... the weight of the nitrogen is 971 X 5, or 4855; and the weight of the hydrogen is 0694 X 1.5, or 1041. .. the weight of the mixture is 4855 +1041, or 5896; and as this is the weight of a cubic inch of ammonia at the given temperature and pressure, it is the specific gravity of that gas. QUESTION FOR SOLUTION. 26. The other day three scientific friends who reside in the country came with their wives to see me. Their names are Arthur, Frederick, and Herbert; and the ladies' names are Gertrude, Margaret, and Rosamond. They had been purchasing at Knight's a number of botanical and horticultural specimens, and each of my friends had spent three guineas more than his wife. On looking over the list of their purchases which they showed me, I found that each person had bought as many specimens as the number of shillings which he or she had given on an average for each specimen; and that Arthur had bought 23 specimens more than Margaret, and Frederick 11 more than Gertrude. What was the name of each gentleman's wife? ELOCUTIONARY EXERCISES. VI. O WINDING pathways, o'er whose scanty blades My thoughts grew humorous, look terrible, Where fly? I see the myrtle-cradled spot Upon that earth, as on a mother's breast, VII. My lord, we women swim not with our hearts, That scorn (which masks your fear) of what is wise, From "Philip Van Artevelde." 118-Gutta Percha. H. J. H.-Tubing of this material may be safely used for conducting oxygen gas; but the best material for flexible gas-tubes is vulcanized caoutchouc. 119-Stock Buying and Selling. J. T. B.-The system of buying and selling stock will be fully expressed in future pages of the Family Tutor, when we shall present our readers with a treatise on Arithmetic, in which we shall endeavour to simplify that science as much as possible. 120-Diagrams. W. B., X. C., &c.-The diagrams in mathematical manuscript should be drawn clearly with ink, and lettered distinctly; but it is not necessary that they should be drawn with compass and ruler. A welldrawn diagram with a pen, neatly executed, will answer all purposes. 121-St. Swithin. T. B.-Saint Swithin was Bishop of Winchester, and when he was dying requested to be buried in the open churchyard; the monks, however, determined to remove him to the chancel, and appointed the 15th of July for that purpose: but it rained so heavily on that day, and for six weeks after, that they abandoned the design. 122-Galvanic Battery. A. S.-Daniel's battery, and that bearing the name of Mr. Smee, are useful forms of the apparatus. The subject will be treated in our papers on Chemistry. In the meantime we may refer our pupil to the Treatise on Galvanism in the Library of Useful Knowledge, or to Fownes's Elements of Chemistry, pp. 87, 205, third edition. 123-Hard Words. J. C.-A young student-in a great measure self-taught-the type of a large class, desires to be informed, how he can best obtain a knowledge of the true use of what he calls "hard words, " by which he means long terms of classical derivation. The best plan, undoubtedly, is to learn the meanings of the classic words of which the "hard words" are composed. A little work, entitled, Black's Dictionary of Words derived from the Latin, vol. 12mo., 1832, will be useful to our cor respondent. 124-Arsenic. C. A.-Since the discovery of arsenic in the deposits in certain chalybeate springs, it has been supposed that the poisonous properties of this substance are neutralized by the state of combination in which it is found. From a series of experiments on the subject, M. Lassaigne concluded that the poisonous property of arsenic is destroyed by its combination with the peroxide of iron, and thus confirms what has been before asserted-that peroxide of iron by combining with arsenious and arsenic acid, destroys their pernicious qualities, and consequently becomes an antidote for them. 125-J. C., S. F. K., J. S. M.-For a beginner in Arithmetic who cannot obtain the advantage of private tuition, we believe Tate's Arithmetic will be found most serviceable, price 1s. 6d. The best modern work on Arithmetic, showing the reason of each rule, and giving copious examples for practice, is that by the Rev. J. W. Coleuso, 48. 6d. After having read Euclid, and learned the mode of solving simple equations, it will be well to proceed to quadratics. Mr. Tate's Algebra made Easy, will be found very useful to a private student, 1s. 6d. At this state of his progress, however, our correspondent would do well to procure Bland's Algebraical Problems, Ed. 1849, 10s. 6d. 126-Common-place Book. W. B.-An Inder Rerum such as that suggested by the Rev. John Todd (author of Todd's Student's Guide) is to be preferred to a Commonplace Book. We have had an Index Rerum for many years, and can speak most decidedly of the advantages of using such a book. The following extract from the "directions for using the Index" will explain the character of the book briefly: "Read nothing which is not worth remembering; have your Index at hand, and when you meet with any thing of interest, note down the subject, the book, and the page. There are a few slight improvements upon Mr. Todd's arrangement which have been suggested by our own experience; to these we may refer hereafter. We strongly recommend the use of the "Index Rerum" to all classes of our pupils. 127-Captivity of Bruce's Wife. L. M. O.-She was the Countess of Carrick. In 1306 Bruce was defeated at Methven, near Perth, by the English under Aymer de Valence, Earl of Pembroke. Bruce fled into the mountains, his queen and several ladies sharing his misfortunes and dangers. At length, taking refuge at the Castle of Kildrummie, and retreating from thence for greater security, they were captured by the Earl of Ross. The queen was confined at Brustwick. A writ of the 6th Feb. 1312, directed her to be removed to Windsor. In October of the same year she was removed to Shaston (now Shaftesbury), and afterwards to the Abbey of Barking; finally to Rochester, until the battle of Bannockburn, so fatal to the English, when the Constable of Rochester Castle was commanded to conduct the wife, sister, and daughter of Robert Bruce, to Carlisle (usque Kariolum), where an exchange of prisoners was made. She was Bruce's second wife. 128-Soap-Test. E. C.-Water containing sulphate of magnesia alone acts towards the soap-test in producing with it a perfect lather, similarly, or nearly so, as does water containing a lime-salt alone, but only when the equivalent of magnesia salt does not exceed six grains of carbonate of lime in a gallon of water. The degrees of hardness of an ordinary water cannot be inferred by this test. Compute the grains of lime, magnesia, oxides of iron, and alumina, in a gallon of water, each into its equivalent of chalk; the sum of these equivalents will be the hardness of the water. The degrees of hardness of a water containing magnesia and lime salts, as shown by the soap-test, cannot be relied upon as an indication of the amount of these salts in the water, nor of the amount of lime in a water when magnesia is present. Water may show by the soap-test a small degree of hardness, in comparison to the quantities of salts of magnesia and lime which it may contain, and thus by trusting to that test in the selection of water for ordinary uses or steam purposes, a very unfortunate selection might be made. 129-Astronomy. T. W. G.-An excellent globe may be purchased at Wylde's, Charing Cross, for about two pounds. The stars in the six maps, on the Gnomic Projection, published by Charles Knight, are very excellent; the price is for the set 3s. plain, or 6s. coloured. The following course of reading has been recommended by a popular lecturer on Astronomy. Dick's Solar System, is. 3d.; Chauntler's Elements of Astronomy, 2s. 6d.; Reid's Elements of Astronomy, 2s. 6d.; Chambers' Information for the People, No. 1, 1d.; Popular Lectures on Astronomy, by M. Arago, 23., published by Whittaker. Olmstead's Mechanism of the Heavens, 58.; Dick's Celestial Scenery, 58. 6d.; Dick's Sidereal Heavens, 58. 6d.; Herschel's Astronomy, in Lardner's Cabinet Cyclopædia, 6s.; Nichol's Solar System, 10s. 6d.; Nichol's Planetary System, 68. 6d. ; Nichol's Architecture of the Heavens, 10s. 6d.; Nichol's System of the World, 10s. 6d.; Nichol's Planet Neptune, 58. 6d.; Chambers' Papers for the People, No. 21, 1d. Mitchell's Planetary and Stellar Worlds, 28.; Herschel's Outlines of Astronomy, 188.; Milner's Descriptive Atlas of Astronomy, &c. £1 11s. 6d. These works are arranged in the order in which they ought to be read. 130-Stephenson. W. Ö.-Robert Stephenson, F.R.S., was the only son of George Stephenson, Esq., of Tapton House, Derbyshire. He was born at Wilmington, near Newcastle-upon-Tyne, Nov. 16, 1803. He was educated at the academy of Mr. John Bruce of Newcastle, afterwards became a pupil of Mr. Riddle (now of Greenwich), and was there apprenticed to a coal-viewer. In 1821 he entered the University of Edinburgh; in 1822 returned to Newcastle, and commenced his studies in engineering, &c., under his father, who had just established a steamengine manufactory. After two years, his health failing, he took charge of a mining expedition to South America, and remained nearly four years in Columbia. In 1826 the Liverpool and Manchester line was commenced by George Stephenson, and was nearly completed. For three years Robert Stephenson devoted the greater part of his time to his father's steam engine manufactory, but worked also as an engineer. In 1829, the directors of the Manchester and Liverpool Railway having offered a premium for the best locomotive which should consume its own smoke, and fulfil certain other conditions, four engines were entered for competition; of these the best was that constructed by Robert Stephenson. This was, in effect, the commencement of our present system of locomotion on railways. 131-Mathematics. J. J.-The question, "Whose are the most complete works in the highest branches of mathematics?" strikes us as, perhaps, verging slightly on the indefinite. The works of many authors must be digested by the student who would obtain high mathematical proficiency. He will, however, find in Pratt's Mechanical Philosophy, a spirited and successful attempt to exhibit the train of mathematical reasoning, which leads from the most elementary mechanical principles to the highest investigations of physical astronomy, (Parker, London, £1 18.) We would also recommend a work which is unique of its kind-De Morgan's Differential and Integral Calcutus, Library of Useful Knowledge. The object of the talented author has been "to contain within the limits of his work the whole of the student's course from the confines of elementary algebra and trigonometry, to the entrance of the highest work on mathematical physics." After our correspondent has carefully studied these two works, he will find himself quite able to enjoy the works of Laplace, Fourier, and Gauss; and if he have a chance of obtaining the perusal of the Journal de l'Ecole Polytechnique, Crelle's Journal, and the Cambridge Mathematical Journal, he will find many entertaining mathematical novelties of a delightfully intricate character. 132-Chemistry. M. H.-The letter of our correspondent is written in a tone which makes us somewhat afraid that the writer prefers amusement to instruction. Unless he seeks knowledge for its own sake, as well as for the pleasures it secures, he will never gain any knowledge worth possessing. M. H. complains that the lectures on chemistry are too hard, and expresses a fear that they will increase in difficulty as they go on." We might, perhaps, make our papers on chemistry more easy by making them more superficial, and confining our instructions to a few showy experiments, but this would defeat the object which we have had in view, viz:-to give a complete elementary treatise on the subject. To those who find these lectures too abstruse we recommend the Lectures by Grandfather Whitehead, in the Family Friend. To master the rudiments of any science requires attention and perseverance, and those who are not willing to give these in exchange, can never purchase wisdom-for they are the only coins that pass current in the market of knowledge. Our correspondent complains likewise, that "the apparatus described is too expensive." To this we reply, that the apparatus described in our Familiar Lectures, &c., are those which have been found by the lecturer to be most economical, and the best adapted to illustrate the subjects in question. It would be inconsistent with our plan to describe and recommend defective apparatus. The pupil should exercise his ingenuity in devising the best substitute in his power; we should especially observe, moreover, that without a considerable exercise of such ingenuity, no great progress in the practical part of chemistry can be made. 133-Dr. & Cr. in Book-keeping. H. C. N., XIII.-The left-hand side of all accounts is Dr., and the right-hand side Cr. No reason can be assigned for this except that of custom. It might have been reversed. The cash account, whether kept in the cash-book or the ledger, forms no exception to this rule, as our correspondent seems to suppose. Money received is entered on the left-hand, or Dr. side of the cash-book, thus making the cash account Dr., and money paid is placed on the right-hand, or Cr. side, making in like manner the cash account Cr. It is true that when these sums are transferred, or posted-as it is technically called-to other accounts in the ledger, that which stands Dr. in the Cash becomes Cr. in the ledger, and that which stands Cr. in the cash becomes Dr. in the ledger. For instance, suppose we sell to a Bookseller, named John Thompson, 1,000 copies of the Family Tutor, value £8 68. 8d. On the left-hand side of John Thompson's account in the ledger, is placed £8 6s. 8d., signifying that he stands Dr. to us for that sum. When the amount is paid to us, it is entered on the left-hand or Dr. side of the cash, and transferred to his account in the ledger, on the right-hand or Cr. side, and so balancing the transaction. The word "To" is placed at the commencement of entries on the Dr. side, and the word "By" on the Cr. side; but these words have no very definite meaning, except where books are kept by double entry. In this system every step in a transaction, whether a sale of goods, payment of cash or otherwise, is twice entered in the ledger, once on the Dr. side, and again on the Cr. side. The word "To" on the Dr. side pointing to a corresponding credit, and the word "By" to a corresponding debit; so that when an entry is made on the left-hand side of the cash book in this form-" To John Thompson-Cash received from him," the cash is Dr., and John Thompson is Cr. If the entry be made on the right hand-side-"By John Thompson-Cush paid to him," the cash is Cr., and John Thompson Dr. 134-Degerando on Simplicity of Character.-G. B. writes:-"Dear Sir,-As you have kindly expressed your willingness to be questioned by your pupils upon any difficulties or obscurities with which they have met in their studies, I shall feel obliged if you will assist me in my efforts to understand the true and correct meaning of the opening part of an article entitled 'Simplicity of Character,' in your second Number of the Family Tutor. The part to which I allude is this- Simplicity of character is never acquired. It may be preserved, and may be lost, but, like innocence, once lost it can never be recovered. My own views are more in accordance with the following quotation from the same article-That whilst simplicity of character is one of the fundamental essential conditions of moral advancement, it is accessible to all men, whatever may be their rank or circumstances, and even more easily attainable by those least favoured of fortune.' If you will help me to reconcile, or to draw a right conclusion from, these apparently contradictory assertions, you will confer a lasting favour upon one who is anxiously striving to attain to that simplicity of character which is advocated in the article to which I have aliuded."-Our translation is almost a literal one. The original is:-"On n'atteint pas à la simplicité; on lui reste fidèle; elle est comme l'innocence-elle se conserve, perdue; elle ne se retrouve plus." Again: "En même temps qu'elle est une des conditions essentielles et fondamentales de notre éducation morale, elle est aussi une condition accessible au plus grand nombre des hommes, et précisément aux classes les moins favorisées; les situations les plus obscures sont redevables de cet avantage à leur obscurité même." It might probably be questioned whether, as Degerando asserts, simplicity once lost can never be recovered. We believe not; but one thing is very certain, that the difficulties in the way of preserving the original simplicity of childhood, though of infinite value if preserved, are very great amidst the artificial usages of civilized life. Degerando's meaning, therefore, is-that because the great mass of mankind, by the obscurity of their condition, are less exposed to the influence of these artificial usages, the maintenance or preservation of this gift of nature, as a moral fact, is more easily accessible or attainable amongst them than amongst the more favoured of fortune. It is not a matter of much consequence whether our correspondent's "anxious strife" be for the preservation or the attainment of simplicity. If honest with himself, he will succeed. 135-Cyclopædias-J. C.-The British Cyclopædia was published in six vols. 8vo., in 1807-9, and we do not remember to have seen any new edition. A large portion of the work would, therefore, be out of date. Our correspondent wishes to know whether he had better complete his set of this work, or Knight's National Cyclopædia, of which he has twenty-three Parts. Though the National Cyclopædia is not one that we can strongly recommend, it is yet very far superior to a work so much out of date as that published in 1809. Dr. Rees' Cyclopædia, in 45 vols. 4to., was published during the years 1802 and 1819, being an enlargement of Chambers' Cyclopædia of 1785. This may be had for a few pounds; but the money could be better spent. The Encyclopædia Brittanica, the seventh edition of which is still considered a standard work, and is a high authority, was published in 1832-1840. It consists of twenty-one vols. 4to.; but its price is beyond the means of the majority of our readers. The Encyclopaedia Metropolitana is also a work of the highest value; it is republishing in a neat form; but the cost of the whole is such, as to put it out of the power of any but persons of considerable means. Of the Orford Cyclopædia, published in 1828, we have not sufficient knowledge to write. We have heard some commendations of Partington's Cyclopædia, published in 10 vols. in 1836, but do not remember to have seen the work. Of all the Cyclopædias with which we are acquainted, the one that is the most worthy of the attention of those who wish to possess a good book of reference of modern date, is The Penny Cyclopædia, which was published between the years 1833 and 1843, and to which a Supplement was added in 1846. It is advertised to sell new at about £10, but can be purchased bound in cloth (with the Supplement included), at various prices, ranging from £6 10s. to £8. Second-hand copies may be purchased in London, half-bound, for £9 108. The Imperial Cyclopedia, now in course of publication, is intended to be an extension and improvement of the Penny Cyclopædia, but will be very much more expensive, and will be of little use as a book of reference till it is completed. The National Cyclopædia is too abbreviated, but is a useful work. The series of Dictionaries published by Longman & Co., and which includes Loudon's Encyclopædia of Agriculture, Ure's Dictionary of Arts and Manufactures, Brande's Dictionary of Science, &c., and McCullock's Dictionary of Commerce, is well worthy of attention as a comprehensive series, of which the student can purchase a detached part complete. As dictionaries of a less comprehensive character, but of great value, and well adapted for young students, we know of none more worthy of commendation than Maunder's Treasuries, published at 10s. 6d. each. The titles of them are as follows:-Treasury of Knowledge; The Scientific and Literary Treasury; Biographical Treasury; The Treasury of History; and the Treasury of Natural History. The Cabinet Cyclopædia, known as Dr. Lardner's, consists of complete works on History, Biography, Philosophy, Literature, Arts, Sciences, &c. The series is complete in one hundred and thirty-three vols.; advertised price, £39 185. The works may be had separately, at 6s. a volume. In this series is that delightful book, Herschet's Preliminary Discourse on the Study of Natural Philosophy. |