together are worth 16 dollars, but the money is worth 7 times as much as the purse; how much money was there in the purse ? and what is the value of the purse ? 168. A man being asked the price of his horse, answered, that his horse and saddle together were worth 100 dollars, but the horse was worth 9 times as much as the saddle. What was each worth? 169. A man having a horse, a cow, and a sheep, was asked what was the value of each. He answered, that the cow was worth twice as much as the sheep, and the horse 3 times as much as the sheep, and that all together were worth 60 dollars. What was the value of each ? 170. A man bought an apple, an orange, and a melon, for 21 cents; for the orange he gave twice as much as for the apple, and for the melon he gave twice as much as for the orange. How much did he give for each ? 171. If 80 dollars worth of provision will serve 20 men 24 days, how many days will 100 dollars' worth of provision serve 30 men ? 172. There is a pole, and į under water, and 10 feet out; how long is the pole ? 173. In an orchard of fruit trees, į of them bear apples, $ of them bear plums, į of them pears, 7 of them peaches, and 3 of them cherries ; how many trees are there in the whole and how many of each sort ? 174. A farmer being asked how many sheep he had, answered, that he had them in 4 pastures; ir the first he had of his flock; in the second į ; in the third }; and in the fourth 15; how many sheer had he? 175. A man driving his geese to market, wiis met by another, who said, good morrow, master, with your hundred geese ; says he, I have not a hundred; but if I had half as many more as I now have, and two geese and a half, I should have a hundred; how many had he ? 176. What number is that, to which if its half be added the sum will be 60 ? 177. What number is that, to which if its third be atided the sum will be 48 ? 178. What number is that, to which if its fifth be added the sum will be 54 ? 179. What number is that, to which if its half and its third be added the sum will be 55 ? 180. A man being asked his age, answered, that if its half and its third were added to it, the sum would be 77; what was his age ? 181. What number is that, which being increased by its half, its fourth, and eighteen more, will be doubled ? 182. A boy being asked his age, answered, that if į and of his age, and 20 more were added to his age, the sum would be 3 times his age. What age ? 183. A man being asked how inany sheep be had, answered, that if he had as many more, s as many more, and 2 sheep, he should have 100. How many had he? was his ARITHMETIC. PART II. KEY. Tar Key contains remarks on the principles employed and illustrations of the manner of solving the examples in each section. All the most difficult of the practical examples are solved in such a manner as to show the principles by which they are performed. Care has been taken to select examples for solution, that will explain those which are not solved. Many remarks with regard to the manner of illustrating the principles to the pupils are inserted in their proper places. Instructers who may never have attended to fractions need not be afraid to undertake to teach this book. The author flatters himself that the principles are so illustrated, and the processes are made so simpie, that any one, who shall undertake to teach it, will find himself familiar with frạctions before he is aware of it, although he knew nothing of them before ; and that every one will acquire a facility in solving questions, which he never before possessed. The reasoning used in performing these small examples is precisely the same as that used upon large ones. And when any one finds a difficulty in solving a question, he will remove it much sooner, and much more effectually, by taking a very small example of the same kind, and observing how he does it, than by recurring to a rule. The practical examples at the commencement of each section and article are generally such as to show the pupil what the combination is, and how he is to perform it. This will learn the pupil gradually to reason upon ab stract numbers. In each combination, there are a few abstract examples without practical ones, to exercise the learner in the combinations, after he knows what these combinations are. It would be an excellent exercise for the pupil to put these into a practical form when he is reciting. For instance when the question is, how many are 5 and 3 ? Let him make a question in this way; if an orange cost 5 cents, and an apple 3 cents, what would they both come to? This may be done in all cases. The examples are often so arranged, that several de pend on each other, so that the preceding explains the following one. Sometimes also, in the same example, there are several questions asked, so as to lead the pupil gradually from the simple to the more difficult. It would be well for the pupil to acquire the habit of doing this for himself, when difficult questions occur. The operations can be illustrated by counters, or marks on the blackboard, according to the necessity of the pupils. These illustrations will be les necessary as the pupils advance in the work; but a frequent reference to them throughout most of the book will be useful in fixing more clearly in mind the principles involved in the operations. The book may be used in classes where it is conveniente The pupil may answer the questions with the book before him or not, as the instructer thinks proper. A very useful mode of recitation is for the instructer to read the ex. ample to the whole class, and then, allowing sufficient time for them to perform the question, call upon some one to answer it. In this manner every pupil will be obliged to perform the example, because they do not know who is to answer it. In this way it will be best for them to answer without the book. It will often be well to let the elder pupils hear the younger. This will be a useful exercise for them, and an assistance to the instructer. SECTION 1. A. This section contains addition and subtraction The first example may be solved by means of beans, peas &c., or by means of the blackboard. The former method is preferable, if the pupil be very young, not only by the examples in the first part of this section, but by the first examples in all the sections. The pupil will probably solve the first examples without any instruction. B & C. The articles B and C contain the common addition table as far as the first 10 numbers. In the first the numbers are placed in order; and in the second, out of order. The pupil should study these until he can find the answers readily, and then he should commit the answers to memory. D. In this article the numbers are larger than in the preceding; and, in some instances, three or more numbers are added together. In the abstract examples, the numbers from one to ten are to be added to the numbers from ten to twenty. E. This article contains subtraction. F. This article is intended to make the pupil familiar with adding the nine first numbers to all others. The pupil should study it until he can answer the questions very readily. G. In this article all the preceding are combined to gether, and the numbers from 1 to 10* are added to all numbers from 20 to 100, and subtracted in the same manner. 18. 57 and 6 are 63, and 3 arə 66, and 5 are 71, and 2 are 73, less 8 are 65. H. This article contains practical questions which show the application of all ths preceding articles. 6. 37 less 5 are 32, less 8 are 24, less 6 (which he kept himself) are 18; consequently he gave 18 to the third boy. Y • Figures are used in the key, because the Instructor is sapposed to be acquainted with them. They are not used in the filrst part of the book, bar cause tho papil would not understand tham so well as he will the words |