wilh the most important aritbmetical operations performed with whote "The third part is mostly practical, and composed of such rules and NOTE. In his progress through the second part the pupil should be 11 Sec. II. Formation of SEC. VII. Characters ex- See. III. Espression of Sec. VIII. Miscellaneous Sec. IV. Fractions 24 Sec. IX. Geometrical det- Sec. V. Tables of Com. initions Sect. V. Per cent Interest 112 Sect II Ground Rules 65 Sect. VII. Vulgar Fractions 143 Sect. VIII. Powers & Roots 161 Sect. IV. Compound Num. Articles PART 111. Page Sect. I. Exchange 184 | Sect. IV. Miscellaneous. 199 bers ARITHMETIC. PART I. INTELLECTUAL ARITHMETIC. PRELIMINARY OBSERVATIONS. That frequent exercises in mental computations, have a salutary influ. ence upon the mind, by inducing habits of attention, by strengthening the memory, and by producing a promptness of recollection, is, at pres ent, very generally adınitted. And, that exercises of this kind should be more extensively introduced into our primary schools, is acknowl. edged, and even urged, by our most experienced and successful teachers. The suocess, which has, in most cases, attended the introduction of Intellectual Arithmetic into schools, has been such, as would doubt. less appear incredible to those unaccustomed to witness it; but expe. rience has shown that children may be made acquainted with the first principles of Arithmetic at as early an age as they can be taught the Alphabet and its most siinple combinations. We bave seen, says Dr. Griscom, a class of girls, whose ages everage not more than nine years, by the force of memory, and a few plain rules, inultiply sevea or eight figures by an equal number, enumerate and announce accurately ihe product and then extract the square root of this targe product, and state the root and the remainder, without varying a ligure from the truth. In the ordinary course of instruction, Arithmetic has been studied only by the boys; and by them it has usually been defered to the very last portion of their attendance at school. The consequence has been, that few have become familiar with its first principles, before they have been obliged to quit school and enter upon the business of life. Commencing the study of Arithinetic at this advanced period, the scholar is sensible that he bas but little time to devote to it, bolt being determined to cipher through his book, he applies himself with diligence, yet he hurries on from rule to rule with such rapidity, that he learns nothing as he ought. He may iodeed reach the end and thus accomplish his principal purpose; but, of what he has gone over, scarcely a trace remains upon his mind. He bas not even made him self thoroughly acquainted with the elements of the science, nor bas he made himself so familiar with the rules as to derive from them any considerable advantage in the transaction of business. It is asserted with confidence, that children, after baving learned to talk, cannot too soon be made acquainted with numbers, and exercis. ed in mental computations. But great care should be taken that these exercises be adapted to the age and capacity of the child-that the questions proposed, be such as the child can fully comprebend. And as young children are incapable of the exercise of abstraction, the instructer will find it necessary to begin by employing sensible objects. These should be placed before the child, and the first questions pro. posed should relate tc the objects themselves, and be solved by them. Questions may then be asked respecting things which are not present; and the child may soon be led to conceive the objects before bim to represent men, cents, or any other things you please. In performing these exercises the child will at length discover that numbers are not inherent qualities of the objects ther.selves, but tbat they merely de. note a succession of similar quentities, and may be applied as well to one kind of quantity as another. After this discovery the child will find but little difficulty in forming a conception of abstract numbers, that is, of numbers, or successions, without applying them to any spe. cific objects. By repeating and varying these simple operations, children will soon become familiar with the fundamental principles of Arithmetic and their application to practical purposes. They will at the same time be acquiring habits of attention, and a promptness of computation, which will be of inestimable value to them in after life. And this may be done in our primary schools, as an amusement and relaxation to the scholars, without interrupting, in the least, their other pursuits. The proper place to commence these instructions is in our summer schools. These, it is true, are usually taught by females, many of whom have not had the advantages of much arithmetical instruction. But this defect in their qualifications, is not owing to a want of capa. city to learn, but to a fault of the times when the study of Arithmetic was regarded as projer only for boys. But those times are pass. ing by and with them this defect will vanish. A moderate share of attention to the subject would enable every young lady, who en. gages in teaching, to give instruction in the Intellectual Arithmetic contained in this work, and it is believed that tbey would find themselves amply repaid for this attention by the improvement of their own minds. By beginning with children at the commencement of their going to school, every boy and girl of ordinary capacity may be made more thoroughly acquainted with the principles of Arithme. tic before they arrive at the age of ten years, than most of our scholars are on leaving school, after having plodded through all the rules of Arithmetic in the ordinary way. Some knowledge of Arithmetic is no less necessary to the female sex than to our own; and experience has proved, that, if the course, bere recommended, be pursued, they will not be found less capable of proficiency in this science. It is hoped that our instrncters, both male and female, will take this subject into consideration and unite their efforts in bringing about a reforma: fion so desirable in the course of arithmetical instruction. 1 Jn commencing a course of instruction in Intellectual Arithmetic with very young children, it should be the teacher's first object to learn them to count. For tbis purpose beans, small blocks of wood, marks on a slate or paper, or some other sensible objects must be employed. It would perhaps be advisable to use no more than five counters at first, and in selecting these, care should be taken that they resemble each other as nearly as possible, that the child may not be led to suppose that the names used in counting denote a difference among the ob. jects employed. Having called the little class around him, the in. structer should begin by laying down one of the counters, which he has provided, and wbich we shall here suppose to be beans, and say. ing, there is one, require the children'to repeat after him, one. Then, putting down another, he should say, one and one are two. Another bean inay then be laid down, and the children taught in like manner to count ihree; and so on to five. After the children have learned to count five with facility, five'more beans may be taken and ibe chil. dren taught in the same way to count ten; after which they may be taught, by the help of the beans, to answer the following questions : 2. 1. How many beans are one beans and two beans ? bean and one bean more? | 15. How many beans are seven 2. How many beans are two beans and two beans ? beans and one bean ? | 16. How many beans are eight 3. How many beans are three beans and two beans ? beans and one bean ? 17. How many beans are two 4. How many beans are four beans and three beaus ? beans and one bean ? 18. How many beans are three 5. How many beans are fise beans and three beans ? beans and ope bean ? . | 19. How many beans are four 6. How many beans are six beans and three beans ? beans and one bean ? | 20. How many beans are five 7. How many beans are seven beans and three beans ? beans and one bean ? | 21. How many beans are six 8. How many beans are eight beans and three beans ? beans and one bean? | 22. How many beans are seven 9, How many beans are nine beans and three beans ? beans and one bean ? | 23. How many beans are two IJ. How many beads are two beans and four beans ? beans and two beans? 24. How many beans are three 11. How many beans are three beans and four beans ? beans and two beans? | 25. How many beans are four 12. How many beans are four beans and four beans ? beans and two beans? | 26. How many beans are five 13. How many beans are five beans and four beans ? beans and two beans ? | 27. How many beans are six 14. flow many beans are six bears and four beans ? * This Seelion is designed for rery young children ; older ones may conta merce at Section II, 28. How many beans are twol 38. How many pents are i-wo beans and five beans ? fcents and two cents ? 29. How many beans are three 39. How many cents are three beans and five beans? cents and two cents ? 30. How many beans are four 40. How many plumbs are three beans and five beans ? plumbs and three pluinbs? 31. How many beans are five 41. How many nuts are four beans and live beans ? nuts and three nuts ? 32. How many beans are two 42. How many feet has one beans and six beads ? " horse ? 33. How many beans are tbree 43. How many feet have two beans and six beans ? horses? 34. How many beans are four 44. How many bands have two beals and six beaus? boys? 33. How many beans are two 45. How many hands have foor beaus and seven beans? boys ? 36. How many beans are three 46. How many hands have five beans and seven bears? boys ? 37. How many beans are two 47. How many legs are there to beans and seven beans ? a chair? 3. 1. How many beans are two times two beans ? times one bean? 17. How many beans are three 2. How many beans are three times three bean's ? times one bean? 18 How many beans are four 3 How many beans are fourltimes two beans ? lines one bean? 19 How inany beans are five 4. How many beans are five times two beans ? times one bean: 20. If I give two boystwo plumbs 5 How many beans are six timesla piece, how many plumbs will one bean? boili bave ? 6. How inany beans are seven! 21: At one cent a piece bow times one bean? Imuch do four apples cost? 7. How many beans are eight 22. At two cents a piece how times one bean? Touch do three pears cost ? 8. How many beans are nine 23. What animal has as many times one bean ? · again feet as you bave and how 9. How many beans are ten many feet bas it ? . times one bean? 24. How many eyes bave two 10. How many beans are one boys? time one bean ? | 25. How many ears have three 11. How many beans are one boys ? time two beans ? 26. How many eyes and ears 12. How many beans are two have two boys ? times turo beans ? | 27. How many gloves do two • 13. How many beans are two pair of hands require ? times three beans ? | 28. How inany ear rings must I 14. How many beans are twoget for tbree pair of ears? times four beans? 29. I gave a boy three cents and 15. How many beans are two he gave me twice as many apples, times five beans ? How many did he give me? 16. How many beans are threel 30. What do three oranges cost |