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The stones taken from quarries, were cut into irregular polygons, and placed one upon another in such a manner as to make the different faces of the geometrical figures which they employed, coincide, the salient angles filling up the re-entrant angles formed by two adjoining stones. (Fig. 5.) This was the Fig. 5. ordinary manner of building under this system of construction. It is met with from the lake Van, on the frontiers of Armenia, to the west of Italy, Sardinia, and the Balearic Isles; and it is found in temples and in tombs, in pubPelasgic wall. lic and private buildings, and in innumerable military constructions. At last, a third method presents itself in the walls of these early buildings; namely, that in which the stones are fashioned in the square form; and the buildings themselves, assuming the same form, exhibit a greater degree of civilisation; and the invention and application of more exact instruments. The walls of the ancient Mycenae were built in this manner. (Fig. 6.) Fig. 6.
the union of these materials architectural forms, without going through that initiatory process which characterises the origin of all human inventions. Yet the plains of Chaldea soon exhibited constructions which had a great influence over primitive art in the east, and formed the basis of a system which extended its branches even to the west. The want of stones in Mesopotamia soon taught the inhabitants to mould bricks, and their most ancient temple mentioned in the Bible, called the tower of Babel, was an immense pyramid built of bricks piled on one another, and forming, according to report, eight stories or rows, gradually receding from each other. At the top of this building they sacrificed to Baal; at a later period the Chaldean kings placed his statue there, when their artists had made some progress in the art of sculpture. It is probable that this pyramidal-formed temple owed its origin to their remembrance of the practices of those Caucasian_countries whence the Shemitic tribes derived their origin. Herodotus gave a glimpse of the truth, when he said that the Scythians made their temples or altars with a great quantity of wood heaped in the form of a pyramid. However the case may be, this very simple form, which appears to have come naturally to the mind of those men who were the first to raise large constructions, spread itself over all Asia; the ancient pagodas of India are built in this form; the most ancient monuments of Lower Egypt and Ethiopia, where the Shemitic tribes settled in Africa, are all of them pyramids. (Fig. 7.) In Asia, Fig. 7.
The continued and progressive order of these Pelasgic constructions, is one of the most interesting facts in the history of the art of building-particularly when we refer them to an antiquity which goes back to the heroic time of Greece. Doubtless the gradual improvement which is to be seen in the walls constructed by this original people, does not reveal all the revolutions of this art in early antiquity; but it enables us to perceive the progress of the greater part of the civilised world, a progress which it must necessarily follow, because it is the nature of all human inventions to pass from early and rude attempts, to successive periods of improvement and perfection, The Pelasgic monuments, sketched and studied at the present day, extend over a zone, which, comprising the breadth of Western Asia, stretches over Greece and Central Italy; and this is not the whole of the ancient world, as we have already said, in which early monuments composed of rocks in their natural state, have been seen by ancients and moderns; but they have been discovered in all the northern countries, and in Africa, from Egypt to the neighbourhood of Carthage; and we have reason to believe that in these countries, to the primitive constructions, a second period succeeded, more refined in its productions, and forming a step from the first attempts, to the more perfect examples, of which we behold the numerous ruins in India, in Central Asia, in the valley of the Nile, and in the oases of the desert. These monuments of transition, so to speak, have disappeared under early and actual civilisation, and have even escaped the investigation of travellers.
FIRST REGULAR CONSTRUCTIONS, PYRAMIDS, &c. The Pelasgi, proceeding from the Asiatic plateaus (tablelands), directed their steps towards the west; other Shemitic tribes marched towards the south and east, and peopled India, Persia, Assyria, and Arabia, as well as Ethiopia and Egypt. The art of these tribes, like that of the western branch, passed through a rude and primitive state, as we have shownthrough the BETH-EL style, or constructions in unhewn stones. It cannot be supposed that these tribes were more privileged than others, and were able, without previous attempts, to hew stones regularly, to mould and cement bricks, and to give to
Pyramids of Memphis.
whole cities, Ecbatana for example, presented numerous concentric enclosures rising one above another in such a way as to exhibit the pyramidal form. The celebrated gardens of Babylon, formed of numerous terraces, one above another, had also the same configuration. In short, this must be considered as the progress of architecture, when we see that the most ancient religious edifices of the Mexicans are immense pyramids, simple at first like those of Chaldea, and of Lower and Upper Egypt; but at a later period, ornamented with sculpture like the pagodas of India. Ancient public buildings were also found in Mexico of a pyramidal form. It is evident that these first regular constructions were thus generally established; and the greater part of the primitive world adopted them, with the exception of those countries where great political events interrupted the first movements of civilisation and suspended the march of the arts; with the exception also of those whose inhabitants, less endowed by nature, necessarily remained in the rear of civilisation, and only received a movement of this kind from their neighbours, or from an invasion of some
to follow the road in which they were placed. The want of experience, the absence of instruments and machines prevented them from raising, at first, great edifices with vertical fagades or fronts, such as they were enabled to construct at a later period. To form large foundations, and to raise above them materials with gradual and numerous recesses such as would prevent the fall of the upper parts of the building, was the first law of construction and of statics to which they were obliged to submit. This is so true that after having made their great steps in the art of building, the Indians, the Chaldeans, the Ethiopians, and the Egyptians, and become able builders, they still continued in the path of which the pyramid was starting-point, by raising their edifices in such a manner as to give to their façades a great inclination in order to obtain greater stability; a wise system, which was adopted by the Etruscans when they left Asia, where these principles were long established. They were also spread over a part of Italy, and traces of them are found at Norchia. The same ideas exerted their influence over the early edifices of the Greeks, and they are found in a modified form among the finest specimens of their later architecture. They are recognised, for instance, in the Parthenon, where the inclination of the jambs of the doors and windows still exists. Mexico also bears witness to this, as may be seen in our remarks on the first regular constructions of that country.
QUESTIONS ON THE PRECEDING LESSONS. Among what early tribes had the Pelasgi style of architecture its origin?
Who were the Pelasgi; and how far back are their monuments traced?
In what countries are the Pelasgic monuments found; and in what parts of America were similar constructions discovered? How many centuries ago were the walls of Tyrins built, and what was the size of the stones? Where are examples of temples built like these walls to be found?
Where did the square form of stones first appear, and where did brick-building first commence?
utility, and cannot be depended upon as a satisfactory check upon
By casting the nines out of the multiplicand, as shown at page 266, we obtain the remainder 3; by casting the nines out of the multiplier, we obtain the remainder 8; and by multiplying these remainders together, we obtain the product 24. By casting the nines out of this product, we obtain the remainder 6. Now, by casting the nines out of the product of the factors in the above operation, we obtain also the remainder 6. Hence, it is presumed that the operation is right.
If the remainder, arising from casting the nines out of either of the factors, or of both, should be 0, the remainder arising from casting the nines out of the product should also be 0; because the product of 0 by any figure or number is 0.
The proof of division by casting out the nines, proceeds on the tient is one of the factors, the divisor is the other factor, and the same principle as that referred to in multiplication; for the quodividend is their product. To this principle must be added the very simple one, that if the same thing be added to equals, the wholes are equal. Accordingly, when there happens to be a remainder over in the division of one number by another, the remainder arising from the casting out of the nines from this remainder must be added to the product of the remainders of both the factors, in order to get the true remainders of the nines; because the remainder of the division is included in the dividend, or else there would have been no remainder at all. The rule for proving division by casting out the nines, then, is the following:-Cast the nines out of the divisor and the quotient, and multiply these remainders together; cast the nines out of this product, and note the remainder; cast the nines, also, out of the remainder of the noted, and cast the nines out of their sum; the remainder of this casting out should be the same as that arising from casting the nines out of the dividend, if the work be right; this rectitude or accuracy being subject to the same limitations and restrictions, as those above explained under the rule for proving multiplication. EXAMPLE. Divide 7543210 by 476, and prove the operation by casting out the nines.
Divisor. Dividend. Quotient.
LESSONS IN ARITHMETIC.-No. XIV. THE proof of multiplication, by casting out the nines, proceeds on the following principle: that if two factors and their product by divided by any number, the product of the remainders of the factors, when divided by that number, will give the same remainder as the product of the factors. Thus, if 24 and 20 be the factors, 480 is their product; and if each of these factors be divided by a number, say 7 for instance, the remainders are 3 and 6, and their product 18, divided by 7, gives the remainder 4; but the product 480 divided by 7, gives the same remainder 4, as we have said. Com-division, and note this remainder: add it to the former remainder bining this principle with what was explained in our last lesson on the subject of finding the remainder of a number divided by 9, the rule for proving multiplication, "by casting out the nines," is as follows:-Cast the nines out of the factors, i.e., the multiplicand and the multiplier, and multiply the two remainders together, cast the nines out of this product, and also out of the product of the factors; if the remainders be the same in both cases, it may be presumed that the work is right. The accuracy of the operation, as we said in our last number, cannot be depended on, for the reasons there stated, and others that might be added: thus, the arrangement of the partial products might be all wrong by the displacement of the products, of the units, or of the tens, or of the hundreds, &c., i.e., of any one by itself or of all of them together; and yet the casting out of the nines would not indicate the fact, but would give the same result as if they were all arranged in a manner perfectly correct. Hence the propriety of giving other methods of proof, as we have done in past numbers,-methods which are sure to detect the slightest inaccuracy, and put the calculator fully on his guard. For doing this, we have been found fault with by a number of individuals, who fancy that if they know any thing at all they must know arithmetic, and that they can teach us, forsooth, better than we can teach them! But true knowledge is always humble, and ready to acknowledge its own deficiencies or mistakes; it is also anxious to gain useful information from every quarter, and more willing to learn than to find fault. We hope that such persons will take these hints in good part. We assert, without fear of their displeasure, that the method of proving multiplication, or, indeed, any rale in arithmetic, by "casting out the nines," is of no practical
METHOD OF PROOF.
The principle of most of these operations was very well ex. plained by some of our correspondents, among whom we must mention the following:-Robert Boland, Great Shelford, Cambridgeshire; J. L. N., Dublin (who applied it most ingeniously to other scales of notation); Charles Currie, Glasgow (who added some curious and ingenious rules for dividing by a number consisting of nines, extracted from old authors); P. G. Anderson, Birmingham; Joseph Webster, Bramley (who applied the proof of casting out the nines to involution and evolution, and gave rules for the same); J. Carey, Clapham; and others. From the review of all the correspondence we have received, we are convinced that many of the observations made on the properties of the number 9, especially in their application to the methods of proving arithmetical operations, are more curious than useful; and that their place may be fairly taken by something of much greater value.
numbers put down by the pupils.
by the teacher.
The following is a method of interesting young pupils in Casting the nines out of the divisor gives remainder 8, and out of the irksome rule of addition. Tell the members of a the quotient gives remainder 7. The product of these remainders class to put down on their slates a sum in addition, a certain 34765 is 56, and casting the nines out of it gives remainder 2; also, cast- number of lines deep, and a certain number of figures broad, 27184 ing the nines out of the remainder of the division gives remainder as in the margin; and to leave room enough below for you 36958 2, and the sum of these remainders is 4. Lastly, casting the nines to add as many lines. This being done, you proceed to 47366 out of the dividend gives the same remainder 4; hence it is pre-add as many lines to each in the following manner :-Looksumed that the work is right; but, as we have said before, this is ing at the first line, write down rapidly the difference between each figure in it and 9; do the same only a presumption. with the second line, then with the third, and so on. You then tell them to add up all the lines, and find the amount of the whole. But before they do this, you may just as well tell them the answer or sum You do this by which it will make. giving them the product of a number consisting of as many nines as there are figures in the breadth, by the figure which indicates the number of lines in the original depth of the sum before you added your own lines to it. This operation will be seen at once in the example placed in the margin. In this operation it is to be observed, that, however different the numbers may be which are put down to be added, it is plain that, by the addition of your figures to them, they must all come to the same sum. Here you multiply 99999, the number composed of as many nines as there are figures in the breadth, by 4, the number of lines in the original breadth, that is, the depth of the numbers put down by the pupils. The reason of this process is so obvious, that we shall leave it for the consideration of those who may be interested in the subject, satisfied that they are sure to make the discovery for themselves.
We were reminded of a curious property of 9 and its multiples by a correspondent, Joseph Bowman, Preston, namely, that the number 12345679 being made the multiplicand, and the multiples of 9 as far as 81 the multipliers, the products will consist of numbers all composed of the same figure, and that figure will be the number of times 9 which constitutes the multiple employed;
63041 52633 399996 Sum.
98765432 12345679 222222222
Although we have seen these curious results many years ago, we give our correspondent full credit for his individual rediscovery. Another correspondent, J. L. N. (Dublin), suggests the following curious operation in subtraction as an amusement for the junior classes in a school. Tell the members of a class to put down upon their slates individually any number they please, as in the margin; then, to put down under this number, the same figures written in any other order whatever, taking care that its first figure, or the figure of highest value, shall be less than the one above it; next, to subtract this number, or row of figures, from the one above, and mark out one of the figures of the remainder. Lastly, tell them to add together all the rest of the figures, and be prepared to tell you the result. Then, beginning at one end of the class, ask the number required, subtract it in your mind from the next multiple of 9 above it, and tell the remainder aloud; this remainder will be the number which each individual marked out. Thus ; if in the above example 6 were marked out, the sum of the rest of the figures would be 12; now the next multiple of 9 above 12, is 18; and subtracting 12 from 18, you have 6, the number marked out. The principle of this apparent trick is fully explained in our last lesson on Arithmetic. By telling them to cast the nines out of the rest of the figures, and tell you the remainder, the process would be easier, for you would only have to subtract it from 9, and this remainder would be the figure marked out. But this process would more readily reveal your secret. It is to be noted, that when the sum of the rest of the figures is itself 9, or a multiple of 9, the figure marked out may be either 9 or 0. This renders two cases dubious, and diminishes the completeness of the artifice. Pupils rejoice to find their master in a dilemma.
HAVING been frequently requested by our correspondents to recommend to the students of French, some book for reading, in order to extend their knowledge of that language, and to give them an interest in some useful study besides, we have selected for their special use, a collection of the most valuable MAXIMS and MORAL SENTIMENTS which the French language affords. This collection is arranged alphabetically, and will be continued in several numbers of the POPULAR EDUCATOR, until complete. When this is done, our readers will be in possession of as fine a body of social morality and experience of the world, as the language of any country
can afford. The names of the authors are added to the sentiments
or maxims, to give them zest and authority; those of La Rochefoucauld, La Bruyère, Bacon (our own Francis), Pascal, Montaigne, of the wisdom of their sayings, and the excellence of their STYLE, Fenelon, Montesquieu, and many others, will be a sufficient guarantee
PENSEES MORALES ET MAXIMES.
Dans l'esprit de l'ambitieux, le succès couvre la hont des moyens. Massillon.
Le lâche a moins d'affronts à dévorer que l'ambitieux.Vauvenargues.
Lorsque mon ami rit, c'est à lui à m'apprendre le sujet de sa joie; lorsqu'il pleure, c'est à moi à découvrir la cause de son chagrin.-Desmalis.
Vous ne chercherez pas vos amis dans un rang trop au-dessus ni trop au-dessous du vôtre.-Barthélemy.
Celui qui compte dix amis n'en a pas un.-Malesherbes. Les épanchements de l'amitié se retiennent devant un témoin quel qu'il soit; il y a mille secrets que trois amis doivent savoir, et qu'ils ne peuvent se dire que deux à deux.-J. J. Roussecu.
Si nos amis nous rendent des services, nous pensons qu'à titre d'amis ils nous les doivent; et nous ne pensons pas du tout qu'ils ne nous doivent point leur amitié.-Vauvenargues."
Il est plus honteux de se défier de ses amis que d'en être trompé. -La Rochefoucauld,.
Il y a un goût dans la pure amitié où ne peuvent atteindre ceux qui sont nés médiocres.-La Bruyère.
Il ne faut pas regarder quel bien nous fait un ami, mais seulement le désir qu'il a de nous en faire.-La Rochefoucauld.
Il en est de l'amour comme de ces montagnes en forme de pic, dont le sommet n'offre point de lieu de repos; à peine monté, il faut descendre.-Lévis.
Le commencement et le déclin de l'amour se font sentir par l'embarras où l'on est de se trouver seuls ensemble.-La Bruyère. Vouloir oublier quelqu'un, c'est y penser.-La Bruyère.
L'amour-propre des sots excuse celui des gens d'esprit, mais ne le justifie pas.-Lévis.
SOLUTIONS OF PROBLEMS AND QUERIES.
4. Draw two straight lines making a right angle between them; and on one leg of this angle cut off from its vertex, a part equal to the side of the smaller square; from the point where this part is cut off, as a centre, with radius equal to the side of the larger square, describe a circle, cutting the other leg of the angle; the distance from the point where it is cut to the vertex of the angle is the side of the square equal to the difference between the two and of the part cut off the other leg, are equal to the square of the given squares. For by the 47th E. I., the squares of this distance radius of the circle. There the square of the said distance is equal to the difference of the squares of the radius and the part cut off, which straight lines are equal to the sides of the given squares re spectively. 5. SHOW how the squares B G and c H must be cut so that the pieces may be laid upon the square B E, and made exactly to cover it.
The following solution contains fewer pieces than that inserted in No. 16, but it is not so easy to slide the pieces into their position, as their sides do not possess the same property of making the same angle with the base of the largest square.
In the figure, the pieces marked 1, 2, 3, 4, 5, of the two squares BG and C H, will exactly cover the parts marked 1, 2, 3, 4, 5, of the square BE. We have received a
L'amour propre est l'amour de soi-même et de toutes choses pour soi; il rend les hommes idolâtres d'eux-mêmes, et les rendrait les geometrical demonstration of this construction, or rather dissectyrans des autres si la fortune leur en donnait les moyens.-tion of the parts, from the author. La Rochefoucauld.
Aimer les animaux, avoir de la charité pour eux est la marque d'un bon naturel.-Christine.
Rien dans la vie ne doit être stationnaire, et l'art est pétrifié quand'il ne change plus.-Madame de Stael.
La pauvreté manque de beaucoup de choses, l'avarice manque de tout.-La Bruyère.
Ce que l'on prodigue, on l'ôte à son héritier; ce que l'on épargne sordidement, on se l'ôte à soi-même : le milieu est justice pour soi et pours les autres.- La Bruyère.
L'illusion des avares est de prendre l'or et l'argent pour des moyens pour en avoir.-La Rochefoucauld.
L'avarice des pères ou des mères envers leurs enfants est un vice inexcusable. Elle les décourage, les avilit, les excite à tromper, les porte à fréquenter de mauvaises compagnies; puis, quand ils sont une fois maîtres de leur bien, ils donnent dans la crapule ou dans un luxe outré, et se jettent dans des dépenses excessives qui les ruinent en peu du temps. La conduite la plus judicieuse que les pères et les mères puissent tenir à cet égard envers leurs enfants, c'est de retenir avec plus de soin leur autorité naturelle que leu, bourse.-Bacon.
Il y a beaucoup de gens dont la facilité de parler ne vient que d'une impuissance de se taire.-Cyrano de Bergerac. Les hommes ont sur les bêtes l'avantage de la parole; mais les bêtes sont préférables aux hommes, si les paroles ne sont de bon sens.-Marimes des Orientaux.
Personne ne fait plus paraître sa bêtise que celui qui commence de parler avant que celui qui parle ait achevé. Maximes des
We have great pleasure in directing the attention of the readers of the POPULAR EDUCATOR to the following announcement of the publication of EUCLID'S ELEMENTS, at One Shilling. As nothing less than the most extensive circulation can possibly remunerate us for the necessary outlay in producing such a work at such a price, we trust that the large body of our correspondents, at whose suggestions we engaged in the publication of it, will now do their part, by making it as widely known as possible among their friends and acquaintances. We have no fear, if this be done, that the demand will fully equal our most sanguine expectations, which, if realised, will inspire us with confidence to continue the series of valuable educational works, of which this may be considered the pioneer.
CASSELL'S SHILLING EDITION OF EUCLID. THE ELEMENTS OF Books of Euclid, from the text of Robert Simson, M.D., Emeritus GEOMETRY, Containing the First Six, and the Eleventh and Twelfth Professor of Mathematics in the University of Glasgow; with Corrections, Annotations, and Exercises, by Robert Wallace, A.M., of the same university, and Collegiate Tutor of the University of London, is now ready, price 18. in stiff covers, or 1s. 6d. neat cloth.
THE ILLUSTRATED EXHIBITOR AND MAGAZINE OF ART.-The
First Volume of this splendidly embellished work, handsomely bound, price.Cs. 6d., or extra cloth gilt edges, 78. 6d., is now ready, and contains upwards of Two Hundred principal Engravings and an equal number of minor Engravings, Diagrams, &c.
HISTORY OF HUNGARY, WITH UPWARDS OF EIGHTY ILLUSTRATIONS. -The First Volume of the New Series of the WORKING MAN'S FRIEND, neatly bound in cloth, price 38. 6d., contains the completest History of Hungary ever published; also, a History of China and the Chinese, with Forty-six Illustrations of the Manners, Customs, Public Buildings, Domestic Scenes, &c., of this most remarkable people; together with numerous instructive Tales and Narratives; Biographics, with Portraits; Scientific and Miscellaneous Articles, &c. - The WORKING MAN'S FRIEND is regularly issued in weekly Numbers, 1d. each, and monthly Parts at 5d. or 6d., according to the number of weeks Il est des vices dangereux, il en est de déplaisants, il en est de in each month. Part IV. of Vol. II. (for August), price 5d. is now ready. ridicules; le babil réunit tous ces inconvénients. En disant des CASSELL'S EMIGRANT'S HANDBOOK, a Guide to the Various Fields choses ordinaires, le babillard est ridicule; en disant des méchan- of Emigration in all Parts of the Globe, Second Edition, with considercetés, il est odieux; en ne sachant pas taire un secret, il se met enable Additions, and a Map of Australia, with the Gold Regions clearly péril.-Plutarque. marked, is now ready, price 9d.
Un discours à contre-temps est comme une musique pendant le deuil-Ecclésiastique.
THE PATHWAY, a Monthly Religious Magazine, is published on the Ce que j'ai dépensé, je l'ai perdu; ce que je possédais, je l'ai 1st of every month, price twopence-32 pages enclosed in a neat laissé à d'autres: mais ce que j'ai donné est encore à moi.-wrapper. No. 82, for August, is now ready, and Vols. I. and 11., Epitaphe. neatly bound in cloth and lettered, price 2s. 3d. each, may be obtained by order of any Bookseller.
(To be continued.)
ANSWERS TO CORRESPONDENTS.
E. A. He writes a legible dand we recommend him to practise as much as possible-Maisy. We shall have an article on German pronunciation in our next or the blowing Number, when, we trust, his dithculties will be remoVOJ. 1. The wame of the vowel e is ey ey in prey; but on being pronounced, is➡e in met, bed, get, etc.; the same takes place in all 'anguague, the names of the letters seldom agree with thea prva olation, o g. the English letter w. 2. It should be ae.J.C. With It is sounded --J. H. BARKER (Islington): German nouns are distinguished by gender to a much greater extent than is the case with bugish nouns, the waggon, the hat, etc., are nouns of the masculine gender m German. We refer our correspondent to the vocabularies, in which the gender of nouns is indicated by being followed by an m, for u, respectively, the first standing for masculine, the second for feminwc, and the last for neuter.-GUILLAUME'S (London): With diligence and carefulness, he may accomplish much in a short period; there is no objection to his studying the two languages together. Respecting pronunciation, we refer him to "Stanley." Flügel's Dictionary is the best.--JUVENIS (Leeds): "Becker" is a good book, and well adapted for his purpose; though Schiller's historical works may be more justly pointed out as models of elegance. ADMIRATOR wishes to know whether the human body changes every seven years. Physiology deals with facts and not with theories. It is A fact that there is a constant waste, and as constant a reparation of all the tissues of the body, but whether every individual particle has been renewed within the space of seven years, is more than we can venture to affirm.-JULIANA QUICK will find that a PACE measures five feet; and a stadium, as used in the third lesson, is equal to one hundred and twenty-five paces, or six hundred and twenty-five feet. Suppose a young lady had to go this latter distance to see her lover, how many miles would she have to walk?-E. D. (Tynemouth): The questions which he suggests as to the origin and early progress of Architecture, are the very points in which the most learned men are divided. origin has been claimed by Egypt, India, and Persia. And if the primitive history of Meroë was involved in less obscurity, might be brought within the range of an easier solution.
ISAAC WALTON,,jun. (Birmingham): Dialling will be treated of. THAUMASTES (London): The philosopher of San Souci was Frederic of Prussia, and Voltaire was the author of the "Philosophical Dictionary." you will soon observe that it is his by the sneers he throws out against revelation. The cloven foot soon appears. The king and the wit were a dread
ful pair! To multiply 1+x+x2+; by 1-x, observe that the deno
minator of the fraction in the multiplicand is the same as the multiplier; now if you multiply a fraction by a quantity which is the same as its denominator, the product is the numerator; therefore you have only to multiply 1+x+x by 1-x, and to the product add the numerator of the fraction; this will give the result 1.-J. M. H.: The British Colonies will come in their order in the geography, and being of so much importance to this country, and especially to our surplus population, shall receive our best attention.-A. B. (Rochdale): We entirely sympathise with him on the subject of the hedgehog and the toad; the charges made against them are vulgar errors. We shall call the attention of our natural historian to the subject.-RICHD. JACKSON (Manchester): If the sentence quoted had been wrong, through inadvertence or any other cause, we should have thanked him, and acknowledged our error; but we are right.-A. M. WHITE (City): We rather think that the query about the nine rows of nine digits, is insoluble to the full extent of the requirement; but we shall not be too positive. Poetry, good poetry, shall find a place with us; but our standard is very high. We thank him for his philanthropic intentions regarding the POPULAR EDUCATOR.-LYCO (Wortley): We prefer Young's Algebra to Chambers's, and Riddle's Latin Dictionary to any he has mentioned; thanks for his recommendation.-J. B. (St. NEOTS): We recommend Cassell's" History of the Steam-Engine," price 7d. as the plainest we know.-S. B. (Bridlington): Read both lessons together.-R. C. A. (Gourock): Our worthy friend at the " Saut Watter," asks too many questions at once; we cannot cool our "kutes," and take matters as easily as ne does. To rise early get an alarm-clock, or get a friend to put a wet sponge to your nose. The sentences" Alfred rides a horse," and "If it rains to-morrow," are both grammatical. A tank 12 feet deep, 41 feet long, and 9 feet broad, contains 4,428 cubic, or solid feet, for 12×41X9 4428. Now multiply 4428 by 1728 the number of solid inches in a solid foot, and divide by 277.274 the number of solid inches in an imperial gallon, and you have the content of the tank in gallons; the answer will be about 27,575 gallons.
T. B. S. (Leeds): Messrs. Horne and Thornthwaite, Newgate-street, London, have published a cheap Manual of Photography.- J. P. (Norwich) shall be gratified by the mysteries of the number 9.-W. S. should of course improve his penmanship; but bad writing is not the reason of his getting no reply to letters in answer to advertisements.-SELAW: His suggestions as to mechanical arts are followed up to a considerable extent in the "Illustrated Exhibitor."-AMINTUS (Glas
gow) will get Dr. Beard's "Latin Made Easy," by applying to Messrs. Simpkin and Marshall, London. A list of errata will be given with the volume.
RICHARD RIGBY (Brades) wishes us to tell him more about our proceedings than we know ourselves. Now, curiosity is very praiseworthy when properly directed. What we have said in former numbers of this work, we are making every effort to fulfil. Let this be enough. The law of gravitation holds good in respect to all bodies. Two bodies of very unequal weight let fall from a high tower, would reach the ground in the same time. This is nicely illustrated by the Guinea and Feather experiment. In the exhausted receiver of an air-pump they both fall to the bottom of the receiver at one and the same instant.-WILLIAM SIMPSON (Yorkshire) may acquire a pretty good knowledge of French in three months.-UN JEUNE MONTAGNARD (Glasgow): We advise him to remain in his present employ, to try to improve his handwriting, and do everything to please his employers. His reward will come sooner or later. Perseverantia vincit omnia.-J. O. N. (Liverpool) has been received and, will be attended to with pleasure.-The CLASS OF WORKING YOUTHS (Leicester), which wrote to us by the hands of the chairman, is advised to get a book on ETIQUETTE, written by Ayayos; it is published by Longmans, Paternoster-row; we believe the price is 18.-H. M. F. (Cheltenham): His solution of query 3, p. 223, is wrong; those of queries 3 and 4, are right.-R. S. (Piccadilly): Proceed with Geometry, by all means. Get Cassell's Shilling Euclid.-JOHN WRIGHT (Astley): The solutions will be given.-J. C. G. M. (Walworth): His solution of query 3, p. 223, is wrong.-HORATIO BLYTH (Norwich): His first essay at composition is passable. Heralded is not a good word to use in speaking of the beginning of time. The Almighty is the best word to use for God, when speaking of creation. He was indeed ALL-MIGHTY, and the Hebrew word is best rendered by this English word, or the Latin term Omnipotent.-ISLINGTONIAN, if he wishes to be a man, should set his face against theatricals and novels. History and biography are good. Let him study Latin and English well, and stick to Blackstone. -C. BOLDERO (Shrewsbury): He has now got the Latin key; let him go on and prosper; fortune favours the brave.-A WORKING MAN (Aberdeenshire): Received.-W. S. G-N: His solution of query 3, p. 223, is wrong; those of queries 4, 5, and 6, are right.-DAVID CRAWFORD (Largs), has also sent right solutions of these three.DAWSON (Knaresborough): His solutions of queries 1 and 3 are incorrect; of queries 2 and 4, right.-JOHN DENNIS (Droylsden): We are pleased with his communication, and shall take his hints in good part. We shall be down on Naper's bones soon, and the circle too.ÕPIFEX: We shall make some use of his interesting communication. V. A. (Richmond): His solution of the cup question is right.-A. T. (Coventry) should consult the Key to the Latin Lessons in the POPULAR EDUCATOR.-A LAD who works in a factory should consult the Regulations for Degrees in the London University Almanack; any college connected with that university is open to him.-JAMES HALE will find all that is required for the German in the POPULAR EDUCATOR.-J. D. (Torpoint): Professor Liebig, of Giessen, is reckoned the most famous Chemist of the day; but he is a German.-NIL DESPERANDUM (Bristol): An is used as well as ne, but the distinction will be pointed out in the lessons.-A LABOURER (St. Osyth) will find the answer in the Key to the Latin Lessons.-W. S. WAKEFIELD (Waltham Abbey): We fear his benevolent plan would not succeed.-MECENAS (London) wishes a solution to the following:-If two circles touch each other externally. and parallel diameters be drawn; the straight lines joining opposite extremities of the diameter shall pass through the point of contact. See "Bland's Geometrical Problems," sect. 2, No. 85.-F. W. M. (London) should study Geometry, beginning with the lectures on Euclid.-R. M. (Aberdeen) will obtain all he wants in succeeding lessons.-R. W. (Brighton): Both.-R. BOWEN (Hayes): His suggestions are received.— H. H. (Farnham): England allowed an institution for the training of civil engineers to die. There is now no remedy but apprenticeship.I. H. (Tyldesley): As the lines of small angles are nearly proportional to the angles themselves, you will find the sine of 865 by multiplying the sine of 1' by S-65 and dividing by, 60-DEXTER (York) will find a geometrical solution to the following question, in Dr. Thomson's Euclid, prop. 26, book 3 of the appendix,
"A gentleman a garden had,
Five score feet long, and four score broad;
Tell us how wide the walk must be."
Construct the rectangle according to its dimensions, draw the diagonal, and find the centre of the circle inscribed in the triangle. Parallels to the sides drawn through this point will divide the rectangle as required.-A. E. should walk and converse with a friend, as often as hie can; one wiser and older than himself.
Printed and Published by JOHN CASSELL, La Belle Sauvage Yard, Ludgatehill, London.-August 7, 1852.