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in addition to my own, to make a free use of examples which have passed the test of years of experience in the best schools of Italy and Germany. Iam more anxious to serve the interests of my pupils than gratify a literary vanity; and even were I to make an effort at originality, by the preparation of exclusively new exercises, one man could hardly hope to excel the united labours of many grammarians in this direction.

The exercises ought to be read over frequently, and always aloud; and if committed to memory, so much the better for the knowledge of the student.

only to make every country a different colour from those that are around it, taking care to keep the colour within the dotted boundary line, and to lay it on very lightly indeed. Some taste may of course be shown in the selec tion and the arrangement of the colours of adjoining countries.-ANXIOUS, MASTER (Wolverhampton), should apply to Henry Dunn, Esq. Secretary to must free himself of his incog. if he wishes us to answer him.—A SCHOOLthe British and Foreign School Society, Borough Road, London, for the Pamphlet entitled "The Normal Schools, &c.," which will give him all the information he wants.-J. R. SMITH (Stoke-Newington): We cannot promise to publish any letter till we see it, and can judge of its contents. The regulations relating to the degrees at the University of London, are contained in vol. ii. p. 213, and p. 137; and for the rest, he should at once apply to the University Almanac. We give the same advice to ANNI SEPTENDE CIM: Doncaster.-J. MARCH (Marrick): See p. 60, col. 1, vol. iv.-UN GARGON GALLOIS (Aberystwith): His French letter to us is very well done; but it is too flattering to be inserted; besides if we inserted his letter we should be completely deluged with French letters from all parts of the Empire.-T. PoWELL, will find a key to the Latin Exercises in the P. E., and a correction of the errata in various parts of the subsequent volumes.-N. T. N. (Beech Lane): His suggestion is good and will be conto the "Law of the Assation of Ideas," and says that in the study of Latin, he learned the voc bulary prefixed to each exercise, and that in the very next book which he ook up for casual reading, he often found several words derived from those that he had thus learned. He adds that in studying the French in connection with the Latin, he recalls to mind the words of the latter, which have the same meaning as those of the former, and thus fixes both in his memory.

paste As to colouring maps, it is the simplest thing in the world; you have

As I have so very fully explained the elementary principles of pronunciation, even at a length which may have damped the ardour of more impatient readers, it will not henceforth be necessary to give the pronunciation of each Italian word used. Should any doubt occur, the student can always refer to the pronouncing lessons or to the general table which pre-sidered. SEPTENDECIM (Louth), in a very well written letter to us, adverts cedes these remarks. As it is, however, most desirable that the reader should have as much assistance as possible, I shall aid him by a new, and, I believe, a most effective method, namely, by dividing each Italian word used, into syllables, for the most part, as the words are divided in Italian spelling and writing. I shall not omit to mark the accent of tone with the acute sign or with the circumflex sign over the e and o; signs, be it remembered, not used in Italian writing or printing, with the exception of the words commented on in my remarks on the use of the accent. The grave accent will, henceforth, always be placed where the usage of writing requires it, and in such cases it will serve, likewise, to denote the accent of tone. I am induced, by three reasons, to adopt this method of dividing words into syllables :—

First, to correct the great fault of Englishmen in pronouncing Italian by slurring over words, the component sounds of which are unfamiliar to the ear. By this means, the learner will be in some measure compelled to do justice to each syllable Secondly, it will be a practical aid to the memory. This dwelling on the ingredients of the word will impress the word itself better on the memory.

Thirdly, it will be useful in the case of compound words, in indicating at once the elementary constitution of the words.

ANSWERS TO CORRESPONDENTS. CHEMISTRY.-F. M., S. J. R., a Diligent Pupil, a Young Chemist, and a Dunce, have experienced difficulty in generating sulphuret of iron according to directions given.-The iron bar must be white hot; a piece of iron as large as a kitchen poker cannot be heated to whiteness in a common open fire. Those who can have access to a smith's forge, may avail themselves of it.

H. DUNKLEY: Hydrosulphate of ammonia, and hydrosulphuret of ammonia, are terms commonly employed to indicate one and the same substance, nor can any ambiguity arise from their indiscriminate use. H. Dunkley, H. Dunkley, however, is right in assuming that, viewed in relation to their analogies, these two expressions should indicate two different bodies. The most recent term for the liquid in question is sulphuret of ammonium; but ammonium is a hypothetical compound-it may exist or it may not. It has never been separately obtained.-Pharmacien, H. Hud, and a Novice-will receive answers to their questions next week.-P. S. (Trafalgar-road): Is respectfully informed that we cannot find room for his article, which will be returned on application for it.

J. JONES (Royal Marines): We should have been happy to insert his poetical communication on the question of Autodidactos, but it is too late. We love to encourage the Welsh.-W. N. BARKER (Islington) and another friend have apprised us that a list of eleven or twelve students passed in Classics at the last matriculation examination of the University of London. The omission of the three names is our fault; we concluded too hastily that there were none, the paper sent, us not containing them. We must atone for this another time.-H. WARDINGLEY (Leeds): The wolf and the tiger did not eat so long as twenty minutes TOGETHER. RATTLER (Liverpool): It is not legal to acknowledge the payment of a debt by a bill by post without the ld. receipt stamp.-E. A. SUTER (Portsmouth): The "History of England," by Dr. Ferguson, at 3s., 3s. 6d., or 4s.

Our correspondents give us more credit for knowledge of their affairs, their mental capacities, their physical capabilities, and their general habitudes and respective positions in life, than they themselves possess; they must take us for the greatest conjurer that ever was known. Thus: M. A. C. (Huddersfield), asks us "what are the best studies he should pursue to be a really practical mau, like Franklin !!"-LEO (Brompton), asks us "how many numbers of the Popular Educator make a volume !!" SOCIUS asks us for "a universal rule for placing Latin words in a sentence !!" C. D. REDDING asks us" whether the students of Greek are to do their best, without knowing whether they are right or not!!" J. THOMAS (Halifax), asks us "if the phrase it is cold, be grammatical!!" IN Loco (Birmingham), earnestly asks us "where Cain's wife came from !!" C. Y. PARTRIDGE (North Molton), asks us to give the Chemical analysis of the North Molton W. Townlime-stone, as its properties are unknown to the residents !!" SEND asks us if a wife can be put in a Lunatic Asylum because she doubts the fidelity of her husband!!!" A CONSTANT SUBSCRIBER (Spilsby), asks us "whether the letter h is silent or not in the word Chobham," see p. 28, line 26 from the bottom!! J. EDWARDS (Lancaster), asks us for "Mr. Bell's address," which was given before; viz. 13, Hope-street, Charlotte-square, Edinburgh!! G. JACKSON (Leicester), accuses us of not fulfilling our engagement as to Music. and asks us for the name of a work teaching it by Mr. Curwen's system;" see Mr. Curwen's "Grammar of Vocal Music," price 2s. 6d. "The Pupil's Manual of the Tonic Sol-Fa Method of Singing, and Sol-Fa School Music," price ls.-" The School Course of Sol-Fa-Exercises," price 4d." The Sol-Fa edition of the People's service of song,' price 1s. 6d. &c., and especially the "Tonic Sol-Fa Reporter, and Magazine of Vocal Music for the People, price 1d. each number!!! J. S. M. (Norfolk), asks us to give the Analysis of Rice and Wheat!! J.T. (Napier-street), asks us to "inform him of the cheapest class where our English Lessons are studied!!" F. B. (Buckley), asks us," what is the cure for disease brought on by hard study!!" H. Jean (Norwich). asks us when we think of introducing the Hebrew Language in the P. E.!! And lastly, SELF-TAUGHT, with a thousand others, asks us if his style of writing or penmanship will do for a clerk's situation !!!!

T. T. is right.-R. TORKINGTON (Bolton-le-Moors): We'request him to exercise a little patience, the agent is not in fault; no one can help the illness of an editor.-J. DoWELL (Birmingham): The word peer comes from the French pair, thus defined in Boniface's Dictionary; "anciennement titre de dignité; l'un des ducs ou comtes qui avaient séance au parlement de Paris; membre de la chambre des seigneurs d'Angleterre; vassal qui-a droit de juger avec le seigneur du lieu."-T. T. KIELY; pocket compasses may be had from 3s. to 3gs. according to mounting, at Knight and Sons, Foster Lane, Cheapside. Back numbers of both editions of vols. 1, 2 and 3, of the P. E., may be had onemand.-E. J. HOSKINS (Winchester): An English sovereign can neither legally marry a subject nor a foreigner who is not a Protestant. The rain falls because the pressure of the atmosphere is diminished, and consequently the barometer sinks. Dr. Black's oalance, which you have described in his own words as follows, may be useful to our chemical students:-" A thin piece of fir-wood of the thickness of a shilling is divided into 20 parts, i.e. 10 on each side of the middle; being altogether a foot long, and half an inch broad. These are the principal divisions, and these are subdivided into halves and quarters. Across the axis is fixed a very small needle, which is fitted to its place by sealing-wax. The fulcrum is a piece of brass plate, the middle of which lies flat upon the table; the two ends are bent at right angles so as to stand upright. These two ends are ground at the same time on a flat hone. A grain weight is placed on one division of the balance, and the object to J. H. EASTWOOD (Middleton): Keceived.-D. A. R. (Forfar): The trans- be weighed on another; the position of the two will indicate the weight of lation of "Non sum ita hebes ut isthuc dicam," is I am not so dull as I say the latter." The mode of calculating the weight by this balance is this: in that matter; or Scotticè, I am not so blind as I am blear-eyed.-J. E. H, suppose for instance that half a grain weight on division 10 of one end of (Kidderminster): Many thanks for his corrections.-G. WILLIAMS (Bristol): the beam, was balanced by an object on division 6 of the opposite end, Prop. VII. Book II. might be made a Corollary to Prop. IV, but the advan- | what is the weight? It is shown in Mechanics, that the distances of the tage would be too small to compensate for the disarrangement of the proposi-weights from the fulcrum are to one another, inversely as the weights them tions. As to the exercise appended to Prop. XV. Book III., the term chord selves; therefore, we have 6: 10:::; hence, the weight of the object as defined in Cassell's Euclid is opposed to the term diameter: and of course is 24 of a grain.-A TEACHER (Torquay): The publication of the treatises canuot pass through the centre. By joining G L in the figure to Prop. II. from the P. E. is only a question of time and demand.-T. MACKEN (Dublin) : Book I., it can be proved that D G L is an equilateral triangle, but not till triangle, but not till His suggestion has been often made, but it would be a serious undertaking. you come to Prop. XXXII. The vendor can legally refuse to exchange old numbers for new ones even on H. GARRETT (Derby): French is the easiest language to learn. With paying the difference.-ISRAEL. (Glasgow): No one requires to study Dr.ordinary care, the maps will not wear out; but they may be strengthened Stoddard and Andrews' Latin Grammar unless he likes; but the more at the folds with narrow strips of thin paper of strong texture and a little knowledge he can get the better.

20

MATHEMATICAL ILLUSTRATIONS.-No. I.

ASYMPTOTES TO CURVES.

Mathematical Lines which continually approach each other, and yet never mest.—The numerous inquiries which have been made on this subject, have induced us to give a short account of certain mathematical lines, which, from their nature, and by definition, are such that they continually approach each other, and yet never meet. The last communication which we received, informs us of the pleasing fact that a class of young men, at Clitheroe, frequently meet to compare notes on their studies in English Grammar, and that (according to an early recommendation of ours) they select one of their number to be president for the occasion. Having assembled one evening lately for this purpose, they met with the following passage, in a paper on the "Immortality of the Soul," which was contained in the book they were reading and parsing, viz. "Enfield's Speaker;"—" The soul considered in relation to its Creator, is like one of those mathematical lines that may draw nearer to another for all eternity, without a possibility of touching it; and can there be any thought so transporting as to consider ourselves in these perpetual approaches to HIM, who is not only the standard of perfection, but of happi

ness ?"

The very natural inquiry arose, What mathematical lines are those here referred to? The first thought was that they were parallel straight lines; but this was self-contradictory, because parallels, though they never meet, do not approach each other, but keep always at the same distance from one another. Appli

on each side of P A, making any angles APE, A PE', &c. with the straight line A P. In BP, take any point C, and make BA equal to B c; then, make D E and D F equal to B A or B C ; also D'E′ and D'F'; &a. Now, draw a line through all the points E, E', B", &c. be they ever so near one another, and a line through all the points F, F, F", &c., in like manner; then the curve of Nicomedes; and the curve CFFF" will be the second AE BE" will be first or superior conchoid, called the conchoid or inferior conchoid.

From the construction of this curve, it is plain that as the angle APE increases, the conchoid continually approaches on both sides, on both sides, the straight line BD D'D', which is called the asymptoté of the curve; that the curve has four infinits branches, two on the right and two on the left of the axis AP; or two on one side and two on the other side of the asymptoté BD; and that the curve, though continually approaching the asymptoté, can never meet it, however far both may be continued, because the straight lines DE, DE', DE", &c. and the straight lines D F, D F, DF", &c., can never coincide with the asymptoté B D. That the curve is continually approaching the asymptoté is manifest, from the consideration that the equal angles E D D' and B D F, &c., are continually diminishing in magnitude, as the angles APE, A PE, &c., increase; and the points E, E', E", &c., as well as the points F, F', F", &c., of the curve, are perpetually drawing nearer and nearer to the straight line BDD'.

In the preceding curve, at a and c, are points corresponding to what are called Maximum and Minimum ordinates, and are therefore the points remotest from the straight line BD D' On

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ation was made to various quarters for a solution of the difficulty, but in vain; and as we have been requested to unfold the secret, on behalf of the class, as well as of many other readers, we proceed to give as clear an account of the matter as we can, to our non-mathematical readers.

The term Asymptote, which very aptly conveys the meaning implied in the description of the lines in question, is derived from the Greek word aovμπTwos, which signifies incoincident, or that which does not fall in with or meet another. The asymptote of a plane curve is generally defined as a straight line to which the curve continually approaches without ever being able to meet it; or, in other words, it is a tangent to the curve of which the point of contact is placed at an infinite distance. In order, however, to give our readers a more definite and simple idea of an asymptoté, we shall draw a curve according to its mechanical construction by the help of certain lines and angles, instead of by that of points referred to coordinates, which is the usual mathematical method.

each side of it; and the points in the vicinity of E” and r'are called points of comrary flexure, because at these points the curve bends in the contrary direction, and instead of being concave towards the straight line e D D', it becomes conver, and yet it still continues to approach the asymptoté. When the straight line B A or B C is less than BP, the curve always assumes a form similar to fig. 1, but when BA or B C is equal to BP, the curve assumes the form represented in fig. 2; and when BA or B C is greater than BP, the curve assumes the form represented in fig. 3. The point c in fig. 2, is, in the mathe

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The Conchoid (shell-like) curve, which was invented by Nicomedes, about the end of the second century B.C., and was much used by the ancients in the solution of difficult problems, is constructed in the following manner: Draw two straight lines AP and BD intersecting each other at right angles in the point B, fig. 1. In Br, take any point P, and from it matical language, called a cusp (from Lat. cuspis, a pont_of draw any number of indefinite straight lines PA, PE, PE', &c. | anything that has a point); and the loop P c, in fig. 3, is called

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a node (from Lat, nodus, a knot) or an oval (from Lat. ovum, an egg). For the sake of some mathematical readers we may just add the equation belonging to the conchoid, viz.

x2y2(b2—y2) (a±y)3,

where the double sign indicates that plus is to be taken when the properties of the superior conchoid are investigated; and that minus is to be taken when those of the inferior conchoid are under consideration. This equation shows that the conchoid is a curve of the fourth order, because, when expanded, it becomes an equation of the fourth degree. Of the figures here shown, the first was drawn exactly according to the method above described, but the second and third figures were copied, merely to give an idea of their form, and their perfect accuracy is doubtful. We would therefore advise our students to draw these figures for themselves; the process will form an interesting and amusing exercise.

The common Hyperbola, one of the conic sections, described in the Lessons in Drawing, fig. 36, p. 226, vol. II, is another curve which has asymptotés, as above defined. In order to explain the nature of these asymptotés in the easiest manner, suppose that, in fig. 4, between the two axes az and Ay of the

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hyperbola, two straight lines н H' and кK' are drawn through the centre A, in such a manner that AH and AK', the hypotenuses of the two right-angled triangles ABH and ACK', are at distances from the centre each equal respectively to the distances AF and A F' from that point to either of the foci F and F of the curve; then, these two straight lines, indefinitely produced, are the asymptotes of the hyperbola, and possess the property of continually approaching the four branches of the curve in as many different directions, without ever meeting them, though both were continued to infinity. In like manner, if DE be the transverse axis of another hyperbola, having the two foci f and f', the same asymptotés HH' and KK' are the asymptotes of this hyperbola. In speaking thus of these hyperbolas, we have considered the two opposite curves passing through the vertices в and c, as one hyperbola; and the two opposite curves, which pass through the vertices D and E, as another hyperbola; but, the former are frequently called opposite hyperbolas, and the latter conjugate hyperbolas.

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In reference to the passage above quoted, which gave rise to our preceding remarks, we may join in the language of Zophar the Naamathite, and say, "Canst thou by searching find out God? Canst thou find out the Almighty_unto perfection? High as heaven, what canst thou do? Deeper than hell, what canst thou know?" And to this we may add the words of Elihu the Buzite: "Touching the Almighty, we cannot find him out; he is excellent in power and in judgment.' Nevertheless we are permitted to illustrate spiritual and eternal things by material and visible objects. Hence, in Scripture, we find that God is spoken of as having eyes and hands, which are with him, and to us, the emblems of knowledge and power. With this example before us, we may reverently compare the existence of the Eternal, which is "from everlasting to everlasting," to a straight line which had no beginning, and which has no end; and we may, in the same spirit, liken the infinite and unbending rectitude, truth and justice of the great Creator, to the directrix or axis of a curve which extends both ways to infinity, without ever deviating to one side or to the the other; for with him "one day is as a thousand years, and a thousand years as one day;" and with him, there is "no variableness, neither shadow of turning."

How different from all this is the condition of man! Truly may he be compared to the curve line, of which the direction is continually changing; the curve line, which depends for its direction at every point, upon its relation to the great and invariable axis; the curve line which, even in a state of comparative approach to that axis, can only become parallel to it at an infinite distance. Hence, it is that even when we borrow a simile from the curve and its asymptoté, the metaphorical comparison between God and man falls infinitely short of the real state of the case; for, although it may be admitted that, during the endless ages of eternity, the purified and saved soul of man shall be continually drawing nearer and nearer unto God, through the contemplation of his visible and ineffable glory, yet there will be such an inconceivably great distance between the Creator and the created, that the comparison dwindles down to that of continually approximating parallels, as infinite in their mutual distance as they are endless in their mutual approach, and everlasting in their asymptotic relation to one another. Yet the apostle John, speaking to believers, says, "we shall be like him, for we shall see him as he is;" true, we shall be like him in kind, but not in

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The equation belonging to the hyperbola, is of the following degree. His greatness, his goodness and his holiness shall fill form, viz.,

b2x2—a2y2—a2b2;

the universe; ours, although similar in kind, shall be an infinitesimal only in magnitude, and shall fill only that sphere to which he shall appoint us. Our respective spheres may

a form which is very similar to the equation belonging to the enlarge as eternal ages roll on, but the mighty sphere of the ellipse, another conic section, viz.,

b2x2+a2y2—a2b2.

Eternal is as unlimited as his duration, and as comprehensive as that infinite snace which he ever continues to fill with his own glory.

LESSONS IN BOOKKEEPING.-No. IX.
(Continued from page 146).

In the following Day-Book, the entries of the Purchases and
Sales of Cotton, detailed in the Memoranda of Transactions, are
here entered in the proper DR. and CR. form, and in business they
would constitute the original record of these transactions. The
original documents relating to these transactions would, of
course, be found in the Invoice Book; those relating to Pur-

chases, in the Invoice Book inward; and those relating to Sales, in the Invoice Book outward; the former consisting of the actual invoices sent in with the good, which are usually posted in a Blue Paper Book; the latter consisting of exact copies of the actual invoices sent out with goods.

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DAY BOOK.

February 25th.

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