THE ORIGINAL PREFACE. As soon as a child begins to use his senses, nature continually 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 this 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 learned 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 names 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 upon to attend to. The idea of more and less implies addition; hence they will often perform these operations without any previous instruction. If, for example, one child has three apples, and another five, they will readily tell how many they both have; and how many one has more than the other. If a 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 among a certain number of persons, he will contrive a way to do it, and will tell how many each must have. The method which chil dren 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 extent, should be among the first lessons taught to them. To succeed in this, however, it is necessary rather to furnish 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 the best, some improvement should be suggested. By following this mode, and making the examples gradually increase in difficulty, experience proves, that at an early age, children may be taught a great variety of the most useful combinations of numbers. Few exercises strengthen and mature the mine so much as arithmetical calculations, if the examples are made sufficiently simple to be understood by the pupil; because a regular though simple process of reasoning is requisite to perform 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 all things with which we are acquainted, we obtain an abstract idea of number. We first make calculations about sensible objects; 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 establish 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 complex operations. It appears, therefore, that mathematical reasoning proceeds as much upon the prin ciple of analytic induction, as that of any other science. From the above observations, and from his own experi. ence, the author has been induced to publish this treatise; in which he has pursued the following plan, which seemed to him the most agreeable to the natural 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 the combination, and assist the imagination of the pupil in performing it. In most instances, immediately. after the practical, abstract examples are placed, containing the same numbers and the same operations, that the pupil may the more easily observe the connection. The instructor should be careful to make the pupil observe the connection. After these are a few abstract examples, and then practical questions again. The examples are to be performed in the mind, or by means of sensible objects, such as beans, nuts, &c. 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 so.* * It is remarkable that a child, although he is able to perform a variety of examples which involve addition, subtraction, multiplication, and division, recognizes no operation but addition. In deed, if we analyze these operations when we perform them in our minds, we shall find that they all reduce themselves to addi tion. They are only different ways of applying the same princi ple. And it is only when we use an artificial method of perform. ing them, that they take a different forin. The first section contains addition and subtraction, the second multiplication. The third section contains division In this section the pupil learns the first principles of frac tions and the terms which are applied to them. This is done by making him 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, two 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 of times, and a part of another time. In the second part of this section the pupil is taught to change a certain number of twos into threes, threes into fours, &c. 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? Because five and three are eight. What is the difference between five and eight? Ans.: Three. Why? Because tive 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. Eight cents. Why? Because two and two are four, and two 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 table serves for addition and subtraction; and another table, which is formed by addition, serves both for multiplication and division. This remark shows the necessity of making the pupil attend tr his manner of performing the examples and of explaining to him Che difference between them. In the fifth section the pupil is taught to find,}, 4, &c., and 3, 1, 3, &c., of numbers which are exactly divisible into 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 tell of what num ber any number, as 2, 3, 4, &c., is one half, one third, one fourth, &c.; and also knowing, 2, t, &c. of a number, to find that number. These combinations contain all the most common and most useful operations of vulgar fractions. But being applied only to numbers which are exactly divisible into these fractional parts, the pupil will observe no principles but multiplication and 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 comprehend them. And the questions are asked in such a manner that, if the instructor pursues the method explained in the Key, it will be almost impossible for the pupil to perform any example without understanding the reason of it. Indeed, in every example which be performs, he is obliged to go through a complete demonstration of the principle by which he does it; and at the same time he does it in the simplest way possible. These observations apply to the remaining part of the book. These principles are sufficient to enable the pupil "to perform almost all kinds of examples that ever occur. He will not, however, be able to solve questions in which it is necessary to take fractional parts of unity, though the principles are the same. After section sixth, there is a collection of miscellaneous *xamples, in which are contained almost all the kinds that |