BELLOWS-FALLS, VT. BOSTON: DISTRICT OP BASSACHL'SETI, TO WIT District Clerk's Often BE # remembered that on the twenty-third day of March, A. D. 18304 la tho Anleth year of the Independence of the United Stales of American Cummings, Hilliard, and Company, of the said District, have deposited is this office the title of a book, the right wberoof" they claim as proprietors, in the words following, to wit: "Intellectual Arithmetic, upon the Inductivo Method of Instruction. By Warren Colburn, A. M." In conformity to the act of the Congress of the United States, entitlech # An Act for the encouragement of learning, by securing the copies of maps, eharts, and books, to the authors and proprietpro of such sopics, during the times therein mentioned and also to an act, ontitlod, An Act, supplementary to an act, entitled, An Act for the encouragoment of learning, by securing the copies of maps, charts, and books, to the anthops and proprietors of such copies during the times therein nientioned; and extending the benefits lbereof to the acts of designing, engraving, and etching, historical, and other print." JOHN W. DAVIS, RECOMMENDATIONS. Sir, Boston, 18 November, 1824. I have made use of the Arithmetic and Tables, which you sometime race prepared, on the system of Pestalozzi; and bave been much grate fied with the improved edition of it, which you have shown me. I are walisfied from experiment, that it is the most effectual and interesting mode of teaching the science of mumbers with which I am acquaistedo Respectfully, HENRY COLMAN. Having been made acquainted with Mr. Calburn's treatise on Arith maalic, and having attended an examination of his scholars, who had been caught according to this system, I am well satisfied that it is the most Basy, simple, and natural way of introducing young persons to the first principles in the seience of numbers. The method here proposed is the Fruit of much study and reflection. The author has had considerable arperience as a teacher, added to a strong interest in the subject, and a thorough knowledge not only of this but of inany of the higher branches of mathematics. This little work is therefore carnestly recara pended to he notice of those who are employed in this branch of early instruction, with the belief that it czy reznures a fair trial in order to be fully approx and adopled. I FARRAR, from Murth Strand Valuersitet Lamtridge, Ne fi sai, la-15-30 23095 PREFACE. As soon as a child begins to use his senses, nature creto ually presents to his eyes a variety of objects; and one of the first properties which he discovers, is the relation of number He intuitively fixes upon unity as a measure, and from thia he forms the idea of 'more and less; which is the idea of quantity. The names of a few of the first numbers are usually learn. od very early; and children frequently learn to count as far as a hundred before they learn their letters. As soon as children have the idea of more and less, and the Dames of a few of the first numbers, they are able to make small calculations. And this we see them do every day about their playthings, and about the little affairs which they are called apon to attend to. The idea of more and less implies addition; hence they will cften perform these operations without any previous instruction. if, for example, one child has three ap ples, and another five, they will readily tell how many they both have; and how many one has more than the other. Ira child be requested to bring three apples for each person in the room, he will calculate very readily how many to bring, if the number does not exceed those he has learnt Again, if a child be requested to divide a number of apples anong a certain number of persons, he will contrive a way to do it, and will tell how many each must have. The method wbick children take to do these things, though always correct, is not always the most expeditious. The fondness which children usually manifest for these exercises, and the facility with which they perform them, seem to indicate that the science of numbers, to a certain ex. tent, should be among the first lessous taught to them.* To succeed in this, however, it is necessary rather to fure nish occasions for them to exercise their own skill in performing examples, than to give them rules. They should be allowed to pursue their own method first, and then they should be made to observe and explain it, and if it was not * See on this suhject two essays, entitled haenil. Saudtes, in the Prize Book of the Latin school, Nar doud Il Publishod bs Cuna. mungs as Williarul 1920 and in the best some improroment should be uggested By lowing this code, and making the examples grailually in crease in difficulty ; experience proves, that, at an early age, children may be taught a great variety of the most uselil combinations of nuinbers Few exercise strengthen and mature the mind so much and arithmetical calculations, if the examples are nade sufficient ly simple to be understood by the pupil; because a regular, ihongh simple process of reasoning is requisite to perfori: them, and the results are attended with certainty. The idea of number is first acquired by observing sensible objects. Having observed that this quality is common to ali things with which we are acquainted, we obtain an abstract idea of numben We first make calculations ahout sensible ohjects ; and we soon observe, that the same calculations will apply to things very dissimilar; and finally, that they may be made without reference to any particular things Hence from particulars, we 'estabilish general principles which serve as the basis of our reasonings, and enable us to proceed step by step, from the most simple to the more com plex operations. It appears, therefore, that inathematica reasoning proceeds ax nmoh upon the principle of analytie induction, as that of any other science. Examples of any kinci upon abstract numbers, are of vers little use, until t'e lourner hen disevered the principle fror practical examples. They are more difficult in themselves for the learner does not use their use; and therefore does not so readily understand the quortion But questions of a practical kind, if judiciously chosen, slow at once what the combination is, and what is to be effected by it. ' Hence the pupil will much more readily discover the ineans by which ihe result is to be obtained. The mind is also greatly assisted in the operations by reference to sensible objects. When the pupil learns a new combination by means of abstract examples, it very seldom happens that he understande practical examples more easily for it, because he does not discover the zonnexion, until he has performed several practical exam ples and begins to generalize thein. Alter the pupil cor.prehends an operation, abstract examples are usefil, to exercise him, and make him familiar with it. And they serve better to fix the principle, because they each the late to generalize. Front the above observations, and from his own expert ence, the author has been induced to publish this treatise ; im which he has pursued the following plan, which seemed to him the most agreeable 10 literatural progress of the mind, GENERAL VIEW OF THE PLAN Every combination commences with practical examples Care has been taken to select such as will aptly illustrate thens combination, and assist the imagination of the pupil in per forming it. In most instances, immediately after the prac. tical, abstract examples are placed, containing the same numbers and the same operations, that the pupil may the more easily observe the connexion. The instructer should be careful to make the pupil observe the connexion. After these are a few abstract examples, and then practical ques. tions again. The numbers are small, and the questions so simple, that almost any child of five or six years old is capable of understadding more than half the book, and those of seven or aight years old can understand the whole of it, The examples are to be performed in the mind, or by means of sensible objecte, such as beans, nuts, &c. or by means of the plate at the end of the book. "The pupil should first perform the examples in his own way, and then be made to observe and tell how he did them, and why he did them 1 * It is remarkable, that a cnild, althougn ne is able to perform a va riety of examples which involve addition, subtraction, multiplication, and division, recognises no operation but addition. Indeed, it we ana. lyze these operations when we perform them in our minds, we shall find that they all reduce themselves to addition. They are only different ways of applying the same principle. And it is only when w z use an artificial method of performing them, that they take a different form. If the following questions were proposed to a child, his answers would be, in substance, like those annexed to the questions. How much is five less than eight? Ans. Three. 'Why? hecause tive and three are eight. What is the difference between five and eight? Ans. Three. Why? because five and three are eight. If you divide eight into two parts, such that one of the parts may be five, what will the other be! Ans. Three. Why? because five and three are eight. How much must you give for four apples at two cents apiece ? Ans. Cight cents. Why? because two and two are four, and iwo are six, and two are eight. How many apples, at two cents apiece, can you buy for eight cents ? Ans. Four. Why? because two and two are four, and two are six, and two are eight. We shall be further convinced of this if we observe that the same mable series for addition ang bawacuon ; AAA aber icla varian in |