The use of the plates is explained in the Soy at the ond of the book. Several examples in each section are perforined in the Key, to show the method of solving them No answers are given in the book, except where it is necessary to oxplais something to the pupil. Most of the explanations are given in the Key ; because pupils generally will not understand an explanation given in a book, especially at sa early an age The instructer must, therefore, give the explan:tion vina toca These, however, will sccupy the instructer but a very show time. The first section contains addition and subtrartion, the sec ond multiplication. Tlie third section contains division. di this section the pupil learns the first principles of fractions ang the terms which are applied to them. This is done by making bin observe that one is the half of two, the third of three the fourth of four, &c. and that two is two thirds of three, iwa fourths of four, two fifths of five, &c. The fourth section commences with multiplication. In this the pupil is taught to repeat a number a certain number o times, and a part of another time. In the second part of th: Rection the pupil is tanght to change a certain number !wos into threes, threes into fours, &c. In the fifth section the pupil is taught find , , !, &c and 3, 4, 5, &c. of numbers, which are exactly divisible intu these parts. This is only an extension of the principle of fractions, which is contained in the third section. In the sixth section the pupil learns to teil of what number any numhor, as 2, 3, 4, &c. is one half, one third, one fourth, &c.; and also, knowing 1, 3, 5, &c. of a number, 10 find that number. These combinations contain all the most common and most useful operations of vulgar fractions. But being applied only to number which are exactly divisible into these fractional, parts, the pupil will observe no principles but multiplication und division, unless he is told of it. In fact, fractions contain no other principle. The examples are so arranged, that almost any child of six or seven years old will readily compre. bend them. And the questions are asked in such a manner, that, if the instructer pursues the method explained in the Key, it will be almost impossible for the pupii to perform any example without understanding the reason of it. Indeed, in formed by addition, serves both for multiplication and division, La This treatise the same plate serves for the four operations. This reinack shows the necessity of making the pupil attend to hig manter of performing the examples and of esplaining in hin the dif Perence between liem. every example which he performs, he is obligod to go throydo a complete com outration of the principle by which is duch it; ud at 10 same time he dock it in the simplest way possibly These whoservations apply to the remaining part of the link These principia are sufficient to enable the pupil to per fornia imosi il kiteli of exükles that ever occur. He will 310i, however, the insie: to solve questions in which it is necessary to takie iracional parts of unity, thougla ile principles are the swie. Afier seciion sixili, there is a collection of miscellaneoak examples, in which are contained imese all the kinds that usually occur. Ture are llone, however, which the prin ciples expolimedamot sitlicient to solve. in section cight in the fullowing, fractions of unity are explaincu, ind, it is believed, so simply as to be intelligiblo to mosi pupils of seven or cight years viage. The operations do noi diler materially from those in the preceding sections There are sorne esperations, however, peculiar 10 fractiune. the two last piles are used to illustrate fractions. When the pupil is made funiiliar with all the principles contained in this book, he will be able to perform all exango pies, in which the numbers are so small, that the operations may be pertormed in the mind. Afterwards he has only us earn the application of figures to these operations, and his unowledge orarithmetic will be complete. The Rule of Three, and all the other rules which are usually contained in our arithmetics, will be found useless. The examples under these rules will be performed upon general principles with much greater facility, and with a greater de. gree of certainty. Thie following are some of the principal diffioulties which a cliild has to encounter in learning arithmetic, in the usual way, and which are seldom overcome. First, the examples are so large, that the pupil can form no conception of The numbers theinselves ; therefore it is impossible for him to comprehend the reasoning upon them. Secondly, the first examples are usually abstract nunbers. This increases the difficulty very much, for even if the numbers were so small, that the pupil could comprehend them, he would discover but very little connexion between them, and practical examples. Abstract numbers, and the operations upon them, must be searned from practical examples; there is no such thing as deriving practical examples from those which are abstract, unless the abstract have been first derived from stuse which are practical. Thirdly, the numbers are ex. pressed by figures, which, if thex ker nised only as a con. tracted way of writing numbers, would be much more dif ficult to be understood ai first, than the numbers written at length in worde, But they are not used merely as words, they require operations peculiar to thomselves. They are, in fact, a now language, which the pupil has to learn. The pupil, therefore, when he commences arithmetic is present. eit with a set of abstract numbers, written with figures, and šis large that he has not the least conceprion of them even when expressed in words. From these he is expected to earn what the figures signify, and what is meant by addition, subtraction, multiplication, and division; and at the same time how to perform these operations with figures. The consequence is, that he learns only one of all these things, and that is, how to perform these opera wons on figures. He can perhaps translate the figures into words, but this is useless since he does not understand the words themselves. Of tho effect produced by the four fundamental operations he has not the least conception. After the abstract examples a few practical examples are usually given, but these again are so large that the pupil cannot reason upon them, and consequently he could not tell whether he must add, subtract, "multiply, or divide even if he had an adequate idea of what these operations The common muthod, therefore, entirely reverses the natural process; for the pupil is expected to learn general principles, before he has obtained ihe particular ideas of which they are composed. The usual mode of proceeding is as follows. The pupil bearns a rule, which, to the man that made it, was a general principle; but with respect to him, and often times to the instructer himself, it is so far from it, that it hardly deserves to bo called even a mechanical principle. He performs the ex. amples, and makes the answers agree with those in the book, and so presumes they are right. He is soon able to do this with considerable facility, and is then supposed to be master of the rule. He is next to apply his rule to practical examples, but if he did not find the examples under the rule, he would never so much as mistrust they belonged to it. But finding them there, he applies his rule to them, and obtains the an. swers, which are in the book, and this satisfies him that they are right. In this manner he proceeds from rule to rulo through the book. When an example is proposed to liim, which is not in the brok, his sagacity is exercised, not in discovering the opera wong necessary to solve it ; buit in comparing it with the exam ples which he has verfurmed barere and endeavouring to ati. are. Sover some analogy between it and them, either in the sound »r in sometiing elit. If he is fortunate enough to discover any such analogy, lie finds what rute to apply, and if he las not been deceived in tracing the analogy, he will probably solve tbe question. His knowledge of the principles of his rules, is so imperfect, that he would never discover to which of them the example belongs if he did not trace it by some analogy, to the examples which he had found under it. These observations do not apply equally to all; for some will find the right course themselves, whatever obstacles bo thrown in their way. But they apply to the greater purt; and it is probable that there are very few who have not ex: perienced more or less inconvenience from this mode of proceeding Almost all, who have ever fully understood arillo retic, have been obliged to learn it over again in their own wi, And it is not too bold an assertion to say, that no inan ever actually learned mathematics in any other method, than by analytic induction; that is, by learning the principles by the examples he performs; and not by learning principloa, first, and then discovering by them how the exaınples are des be performed. In forming and arranging the several combinations tho an thor has received considerable assistance from the system of Pestalozzi. He has not however had an opportunity of seeing Pestalozzi's own tvork on this subject, but only a brief outline of it by another. The plates also are from Pestalozzi. In selecting and arranging the examples to illustrate these com binations, and in the manner of solving questions generally he has received no assistance from Pestalozzi. THE BOY WITHOUT A GENIUS. Mr. Wiseman, the schoolmaster, at the end of his summer vacation, received a new schular with the following etter: Sir,- This will be delivered to you by my son Samuel, whom I beg leave to commit to your care, hoping that by your well-known skill and atteirtion you will be able to make something of lim; which, I am sorry to say, none of his inasiers have hitherto done. He is now elevon, and yet can do nothing but read his mother tongue, and wat but ndifferently. We sent bim at seven io a grammar school in our neighbourhood; but his master soon and that his genius was not viriied to learning languages. He was then put to writing, but he Bet about it so awkwardly inal he male nothing of it. He was tried A accounts, but it appeared that he had 10 genius for that eitleg He could do minilag ili geography or wand of memory in short of tio nas any genius at alt, does not yet show Itsell. But I trnt four experience in cases of this nature to discover what he is fit for, and to instruct him accordingly. I beg to be favoured shortly with tour opinion about himn, and remain, sir, Your most obedient servant, HUMPHREY ACRES. When Mr. Wiseman had read this letter he shook his head, and said to his assistant, a pretty subject they have sent us here! a lad that has a great genius for nothing at all. But perhaps my friend Mr. Acres expects that a boy should show a genius for a thing before he knows any thing about it-no unconmon error! Let us see, however, what 'he youth looks like. I suppose he is a human creature at least. Master Samuel Acres was now called in. He came hanging down his head, and looking as if he was going to be fingged. Come hither, my dear! said Mr. Wiseman-Stand by me, and do Dot be afraid. Nobody will hurt you. How old are you? Eleven last May, sir. No! Why, how do you think other boys do ? Have they more fingu than you ? No, sir. I see nothing here to hinder you from writing as well as any boy in the school. You can read, I suppose ? Yes, sir. Samuel with some hesitation reail, WHATEVER MAN HAS DONE MAN MAY DO, Pray how did you learu to read ?-Was it not with taking paine ! Yes, sir. Well-aking more pains will enable you to read better. Do you know any thing of the Latin Grammar ? No, sir. Why, you caramy soing things by heart. I dare say you can tell mo die apos aí ve days of the week in their order |