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in addition to my own, to make a free use of examples which only to make every country a different colour trom those that are around it

,

As to colouring maps, it is the simplest thing in the world; you have have passed the test of years of experience in the best schools taking care to keep the colour within the dorted boundary line, and to lay it of Italy and Germany. Iam more anxious to serve the interests on very lightly indeed. Some taste may of course be shown in the selecof my pupils than gratify a literary vanity; and even were I to tion and the arrangement of the colours of adjoining countries.-ANXIOUS, make an effyrt at originality, by the preparation of exclusively MASTER (Wolverhampton), should apply to Henry Dunn, Esq. Secretary to new exercises, one man could hardly hope to excel the the British and Foreign School Society, Borough Road, London, for united labours of many grammarians in this direction.

the Pamphlet entitled “The Normal Schools, &c.," which will give him all

the information he wants.-J. R. SMITH ( Stokie-Newington): We cannot The exercises ought to be read over frequently, and always promise to publish any letter till we see it, and can judge of its contents. aloud ; and if committed to memory, so much the better for The regulations relating to the degrees at the University of London, are the knowledge of the student.

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 As I have so very fully explained the elementary principles of SEPTENDECIM: Doncaster.-J. MARCH (Marrick): See p. 60, col. 1, vol. pronunciation, even at a length which may have damped the iv. UN GARÇON GALLOIS (Aberystwith): His French letter to us is very ardour of more impatient readers, it will not henceforth 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 be necessary to give the pronunciation of each Italian word of the Empire.-T. Powell, will find a key to the Latin Exercises in the used. Should any doubt occur, the student can always refer 2. E., and a correction of the errata in various parts of the subsequent to the pronouncing lessons or to the general table which pre-volumes.- N. T. N. (Beech Lane): His suggestion is good and will be concedes these remarks. As it is, however, most desirable that to the Law of the Assi lation of Ideas," and says that in the study of the reader should have as much assistance as possible, I shall Latin, he learned the roa bulary prefixed to each exercise, and that in the aid him by a new, and, I believe, a most effective method, very next book which he sook up for casual reading, he often found several Damely, by dividing each Italian word used, into syllables, foring the French ic connection with the Latin, he recalls to mind the words of the most part, as the words are divided in Italian spelling and the latter, which have the same ineaning as those of the former, and thus writing. I shall not omit to mark the accent of tone with the fixes both in his memory. acute sign or with the circumflex sign over the e and o; signs, Our correspondents give us more credit for knowledge of their affairs, be it remembered, not used in Italian writing or printing, with their mental capacities, their physical capabilities, and their general habithe exception of the words commented on in my remarks on must take us for the greatest conjurer that ever was known. Thus: M.A.C. the use of the accent. The grave accent will, henceforth, (Huddersfield), asks us “ what are the best studies he should pursue to be

a always be placed where the usage of writing requires it, and really practical man, like Franklin !!”—Leo (Brompton), asks us " how in such cases it will serve, likewise, to denote the accent of many numbers of the Popular Educator make a volume I!" Socius asks us

for * a universal rule for placing Latin words in a sentence !!” C. D. REDI am induced, by three reasons, to adopt this method Ding asks us " whether the students of Greek are to do their best, without of dividing words into syllables :

knowing whether they are right or not!!” J. THOMAS (Halifax), asks us

“ if the phrase it is cold, be granımatical !!” IN Loco (Birminghain), First, to correct the great fault of Englishmen in pronouncing earnestly asks us “where Cain's wife came from !!" C. Y. PARTRIDGE Italian by slurring over words, the component sounds of which (North Molton), asks us to give the Chemical analysis of the North Molton are unfamiliar to the ear. By this means, the learner will be lime-stone, as its properties are upknown to tie residents !!”

" W. Townin some measure compelled to do justice to each syllable

SEND asks us "if a wife can be put in a Lunatic Asylum because she doubts

the fidelity of her husband !!!" A CONSTANT SUBSCRIBER (Npilsby), asks Secondly, it will be a practical aid to the memory. This us "whether the letter h is silent or not in the word Chobham," see p. 28, dwelling on the ingredients of the word will impress the word line 26 from the bottom !! J. EDWARDS (Lancaster), asks us for Mr. Bell's itself better on the memory,

address," which was given before ; viz. 13, Hope-street, Charlotte-square,

Edinburgh!! G. JACKSON (Leicester), accuses us of not fulfilling our Thirdly, it will be useful in the case of compound words, in engagement as to Music. and asks us for the name of a work teac.ing it by

Mr. Curwen's system;" see Dir. Curwen's “ Giammar of Vocal Music, is indicating at once the elementary constitution of the words.

price 2s. 6d.-" 'ihe Pupil's Manual of the Tonic Sol-Fa Method of Singing,
and Sol-F'a School Music," price ls.-" The School Course of Sol-Fa.Exerc
cises," price 4d.--" The Sol-Fa edition of the People's service of song,

price ls. 60. &c., and especially the “ Tonic Sol-Fa Reporter, and Magazine
ANSWERS TO CORRESPONDENTS.

of Vocal Music for the People, price ld. each number!!! J.S.M. (Norfolk),

asks us to give the Analysis of Rice and Wheat1! J.T. (Nupier-street), asks CHENISTRY.-F.M., S. J. Ru, a Diligent Pupil, a Young Chemist, and a us to " inform himn of the cheapest class where our English Lessons are Dunce, have experienced difficulty in generating sulphuret of iron according studied ! !" F. B. (Buckley), asks us," what is the cure for disease brought to directions given.--The iron bar must be white hot; a piece of iron as large on by hard study !!" H. Jean (Norwich). asks us when we think of introducas a kitchen poker cannot be heated to whiteness in a common open Are.ing ihe Hebrew Language in the P. E.!.! And lastly, SELF-TAUGHT, with Those who can have access to a smith's forge, may avail themselves of it. a thousand others, asks us if his style of writing or penmanship will do for a

clerk's situation !!!! H. DUNKLEY: Hydrosulphate of ammonia, and hydrosulphuret of ammonia, are terms commonly employed to indicate one and the same substance,

T. T. is right.-R. TORKINGTON (Bolton-le-Moors): We'request him to por can any ambiguity arise from their indiscriminate use. H. Dunkley, exercise a little patience, the agent is not in fault; no one can help the however, is right in assuming that, viewed in relation to their analogies, illness of an editor.-J. DowELL (Birmingham): The word peer comes from

"anciennement these two expressions should indicate two different bodies. The most recent the French pair, thus defined in Boniface's Dictionary; term for the liquid in question is sulphuret of ammonium; but ammonium titre de dignité; l'un des ducs ou comtes qui avaient séance au parlement is a hypothetical compound-it may exist or it may not. It has never been de Paris; membre de la chambre des seigneurs d'Angleterre; vassal qui a separately obtained.-Pharmacien, H. Hud, and a Novice will receive droit de juger avec le seigneur du lieu."-T. T. KIELY; pocket compasses answers to their questions next week.-P. s. (Trafalgar-road): Is respect- may be had from 3s. to 3gs. according to mounting, at Knight and Sons, tully informed that we cannot find room for his article, which will be Foster Lane, Cheapside. Back numbers of both editions of vols. 1, 2 and returned on application for it.

3, of the P. E., may be had on emand.-C. J. HOSKINS (Winchester): An J. Jones (Royal Marines) : We should have been happy to insert his English sovereign can either legally marry a subject nor a foreigner who is

not a Protestant. The rain falls because the pressure of the atmosphere is poetical communication on the question of Autodidactos, but it is too diminished, and consequently the barometer siuks. Dr. Black's oalance, late. We love to encourage the Welsh.-W. N. BARKER (Islington) and which you have described in his own words as follows, may be useful to our another friend have apprised us that a list of eleven or twelve students chemical students:-"A thin piece of fir-wood ol' the thickness of a shilling passed in Classics at the last matriculation examination of the University is divided into 20 parts, i.e. 10 on each'side of the middle; being altogether of London. The omission of the three names is our fault; we concluded too

foot long, and half au inch broad. These are the principal divisions, and hastily that there were none, the paper sent, us not containing them.

these are subdivided into halves and quarters. Across the axis is fixed must atone for this another time.-H. WARDINGLEY (Leeds): The wolf

& very small needle, which is fitted to its place by sealing-wax. The and the tiger did not eat so long as twenty minutes.TOGETHER, RATTLER fulcrum is a piece of brass plate, the middle of which lies flat upon (Liverpool): It is not legal to acknowledge the payment of a debt by a bill the table; the two ends are bent at right angles so as to stand upright. by post without the ld. receipt stainp.-E. A. SUTER (Portsmouth): The These two ends are ground at the same time on a flat hone. A grain "History of England," by Dr. Ferguson, at 3s., 38, 6d., or 4s.

weight is placed on one division of the balance, and the object to J. H. EASTWOOD (Middleton): keceived.-D. A. R. (Forsar): 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 dull as I say the latter." The mode of calculating the weight by this balance is this: in that matter ; or ecottice, 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, inversoly as the weights them tions. As to the exercise appended to Prop. XV. Book III., the term chord selves; therefore, we have 68:10 :: 5:11; hence, the weight of the object as defined in Cassell's Euclid is opposed to the term diameter: and of course is an of a grain. A TEACHER (Torquay): The publication of the treatises canuot pass through the centre. By joining GL in the figure to Prop. II. from the. 1. E. is only a question of time and demand.-T. MACKEN (Dublin): Book 1., it can be proved that D G L is an equilateral triangle, but not till His suggestion has been often made, but it would be a secious undertaking. you come to Prop. XXXII.

The vendor can legally refuse to exchange old numbers for new ones even on H. CARRETT (Derbys: 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 lixos; but the more at the folds rith narrow strips of thin paper of strong texture and a little I knowledge he can get the better.

a

We

on each side of PA, making any angles A PE, A PE', &c. with MATHEMATICAL ILLUSTRATIONS.-No. I.

the straight line Ap. In B P, take any point C, and make Ba ASYMPTOTES TO CURVES.

equal to BC; then, make de and dp equal to B A Or BC; also D'e' Mathematical Lines which continually approach each other, and and d'E';&a. Now, draw a line through all the points 5,E', B", yet never mest.—The numerous inquiries which have been made ! &c. be they ever so near one another, and a line through all on this subject, have induced us to give a short account of the points 1, P, F", &c., in like manner; then the curve certain mathematical lines, which, from their nature, and by

A E B'2" will be first or superior conchoid, called the conchoid

1 definition, are such that they continually approach each other, of Nicomedes ; and the curve crrr" will be the second

or inferior conchoid. and yet never meet. The last communication which we received, informs us of the pleasing fact that a class of young From the construction of this curve, it is plain that as the men, at Clitheroe, frequently meet to compare notes on their angle APE increases, the conchoid continually approaches studies in English Grammar, and that (according to an early on both sides, the straigut line BDD'D', which is called recommendation of ours) they select one of their number to the asymptoté of the curve; that the curve bas four infinite be president for the occasion. Having assembled one evening branckes, two on the right and two on the left of the axis AP; lately for this purpose, they met with the following passage, or two on one side and two on the other side of the asymptoté in a paper on the “Immortality of the Soul,” which was BD; and that the curve, though continually approaching the contained in the book they were reading and parsing, viz. asymptoté, can never meet it, however far both may be con“ Enfield's Speaker;"_" The soul considered in relation to its tinued, because they straight lines DE, DE', Ds", &c. and the Creator, is like one of those mathematical lines that may straight lines D F, D F', DF", &c., can never coincide with the draw nearer to another for all eternity, without a possibility asymptoté BD. That the curve is continually approaching the of touching it; and can there be any thought so transporting asymptoté is manifest, from the consideration that the equal as to consider ourselves in these perpetual approaches to Him, angles E D D' and BDF, &c., are continually diminishing in magwho is not only the standard of perfection, but of happi- nitude, as the angles ÁP E, A PE', &c., increase; and the points

E, E', E", &c., as well as the points F, F, F", &c., of the curve, The very natural inquiry grose, What mathematical lines are

are perpetually drawing nearer and nearer to the straight line those here referred to?

BDD'. The first thought was that they were parallel straight lines; but this was self-contradictory, because In the preceding curve, at A and c, are points corresponding parallels, though they never meet, do not approach each other, to what are called Boximum and Mininuin ordinates, and use but keep always at the same distance from one another. Appli therefore the points remotest from the straight line BD D'OR

ness?"

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ation was made to various quarters for a solution of the, each side of it; and the points in the vicinity of E" and f' are difficulty, but in vain; and as we have been requested to called points of contrary flexure, because at these points the unfold the secret, on behalf of the class, as well as of many curve bends in the contrary direction, and instead of being other readers, we proured to give as clear an account of the concave towards the straight line EDD', it becomes convess matter as we can, to our non-mathematical readers.

and yet it still continues to approach the asymptoté. When The term Asymptoté, which rery aptly conveys the meaning the straight line B A or Bc is less than BP, the curve always implied in the description of the lines in question, is derived assumes a form similar to fig. 1, but when BA or Bc is equal to from the Greek word ao vpli Twos, which signifies incoinci- BP, the curve assumes the form represented in fig. 2; and dent, or that which does not fall in with or meet another. The when BA or Bc is grealer than BP, the curve assumes the form asymptoté of a plane curve is generally defined as a straight represented in fig. 3. The point c in fig. 2, is, in the mathe line to which the curve continually approaches without ever

Fig. 2.

Fig. 3. 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 asymp.oté, we shall draw a curre 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 inathematical method.

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 A P and B D 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 point out draw any number of indefinite straight lines Pa, PE, PE', &c. ) anything that has a point); and the loop pc, in fig. 3, is called

А

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& node (from Lat, nodus, a krot) or an oval (from Lat, orum, If in the latter equation, 6 becomes equal to a, the equation on 699). For the sake of some mathematical readers we is changed into that belonging to the circle, viz., may just add the equation belonging to the conchoid, viz.

#*+y=a x+y=(62_7°) (a+y)”,

These three equations may also be exhibited in the following where the double sign indicates that plus is to be taken when curious forms, by which they are most likely to be more easily the properties of the superior conchoid are investigated ; and remembered, viz., that mini is to be taken when those of the inferior conchoid are under consideration. This equation shows that the conchoid

(1.

=l, equation to the circle, 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,

+

- l, equation to the ellipse. 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 interest

(3.)

- 1, equation to the hyperbola. ing and ainusing exercise,

The common Hyperbola, one of the conic sections, described The equation to the conic sections, generally, is of the folu in the Lessons in Drawing, fig. 36, p. 226, vol. II, is another lowing form :eurve which has asymptotés, as above defined. In order to

yoanz fmnx? ; explain the nature of these asymptotés in the easiest manner, suppose that, in fig. 4, between the two axes Ax and ay of the and this equation includes all the curves known by this name,

viz., the parabola, the ellipse, and the hyperbola; but we

must stop here till we take up the subject in our mathematical Fig. 4.

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 Scrip

ture, we find that God is spoken of as having eyes and hands, F

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 directioi hyperbola, two straight lines u u' and KK' are drawn through is continually changing; the curve line, which depends for its the centre a, in such a manner that ah and AK', the hypotenuses direction at every point, upon its relation to the great and of the two right-angled triangles ABH and A CK', are at dis-invariable axis ; the curve line which, even in a state of comtances from the centre each equal respectively to the distances parative approach to that axis, can only become parallel to it AF and A F' from that point to either of the foci F and F of the at an infinite distance. Hence, it is that even when we eurve; then, these two straight lines, indefinitely produced, borrow a simile from the curve and its asymptoté, the metaare the asymptotés of the hyperbola, and possess the property phorical comparison between God and man falls infinitely of continually approaching the four branches of the curve in short of the real state of the case; for, although it may be as many different directions, without ever meeting them, though admitted that, during the endless agee of eternity, the purified both were continued to infinity. In like manner, if D E be and saved soul of man shall be continually drawing nearer the transverse axis of another hyperbola, having the two focif and nearer unto God, through the contemplation of his visible and f', the same asymptotés - Ħ' and « K' are the asymptotes and ineffable glory, yet there will be such an inconceivably of this hyperbola. In speaking thus of these hyperbolas, we great distance between the Creator and the created, that the have considered the two opposite curves passing through the comparison dwindles down to that of continually approximatvertices B and c, as one hyperbola; and the two opposite ing parallels, as infinite in their mutual distance as they are curves, which pass through the vertices D and E, as another endless in their mutual approach, and everlasting in their hyperbola; but, the former are frequently called opposite asymptotic relation to one another. Yet the apostle John, hyperbolas, and the latter conjugate hyperbolas.

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

the universe ; ours, although similar in kind, shall be an 021? 6 -a?y'=a282 ;

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.,

Eternal is as unlimited as his duration, and as comprehensive

as that infinite snace which he ever continues to fill with his logo+ay?cao 63.

own glory.

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LESSONS IN BOOK KEEPING.–No. IX.

(Continued froin-page 146).

IN the following Day-Book, the entries of the Purchases and chases, in the Invoice Book inward; and those relating to Sales, Sales of Cotton, detailed in the Memoranda of Transactions, are in the Invoice Book outward; the former consisting of the here enteredin the proper DR. and CR. form, and in business they actual invoices -sentin with the good.com, which are usually posted would constitute the original record of these transactions. The in a Blue Paper Book; the latter consisting of exact copies original documents relating to these transactions would, of of the actual invoices sent out with goods. course, be found in the Invoice Book; those relating to Por

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