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METHOD OF USING THIS BOOK. BEGINNERS should commence by daily practice on the Improved Numeral Frame, described in p. 11, followed by the exercises in Oral Arithmetic, Chap. I., pp. 15—66. When the learner, or class, has become familiar with a few sections in this chapter, he may commence the study of the first chapter in Written Arithmetic, pp. 108—124. The oral and written Exercises should now proceed simultaneously ; the former clearing the way and facilitating the operations in the latter. Chaps. II. and III. Oral Arithmetic, should be fully mastered before the pupil commences Chap. III. Written Arithmetic.

Those who have already made some progress in Arithmetic should pursue pretty much the same course, omitting, or passing rapidly over such parts as may seem familiar to them, if any such there be. It is believed, however, that this will seldom be found to be the case.

See the Preface for the general OBJECT of the work.

The following paragraphs on “ Exchange" have been omitted by mis

take in their proper place at p. 248. STERLING may be changed to Canada money by adding je to its amount; Why? [See Table of Provincial Currencies, p. 227.] It may be changed to New England by adding } ; Why? To New York by adding ] ; Why? To Pennsylvania by adding } ; Why? To South Carolina by adding z'; ; Why?

Canada may be changed to New England money by adding } ; to New York by adding ; to Pennsylvania by adding d.

New ENGLAND may be changed to New York by adding } ; to Pennsylvania,

PENNSYLVANIA may be changed to New York by adding is.

SOUTH CAROLINA may be changed to Canada by adding ; to New England by adding & ; to New York by adding ; to Pennsylvania by

adding ?

Every one of these operations may be reversed by subtracting irstead of adding the proportionate part, after changing the respective fraction by adding the numerator to the denominator for a new denominator, and allowing the numerator to remain as before. Thus in changing Canada to Sterling money, the fraction f must be changed to 1o ; while New York to Sterling requires to be changed to 16. Why is this so ? The reason will be readily discovered by attentively operating with a fraction that has 1 for numerator, and then enlarging it to 2, 3, &c.

The same principles are applicable to Foreign Exchange.

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ARITHMETIC is the science of numbers, or that branch of mathematics which teaches us to combine and separate numbers with ease and rapidity. The propriety of this definition appears evident, when it is considered that there are only two operations in arithmetic, increase and decrease. A number may be increased by one or more additions. It may be diminished by one or more subtractions. Such is the whole sum and substance of arithmetic. Now both these operations may be performed by NUMERATION, that is, by the successive addition or subtraction of a unit at a time. If, for instance, a child wishes to know the amount of 4 and 3, he enumerates five, six, seven; if he wants to know how many of 7 apples will remain after parting with three, he also enumerates, six, five, four.

But practice makes perfect. These slow though sure movements seem every day more tedious. The child's numeration, therefore, is transformed into ADDITION and SUBTRACTION, by the omission of superfluous steps. Four and 3 at one step make 7; 3 from 7, in the same manner, make 4.

One more progressive movement completes the system. When several equal numbers are to be added or subtracted, the same desire for economy in time and space prompts to a similar omission of superfluous steps. In place of three steps to reckon 3, and 3, and 3, and 3, one is made to suffice, by saying 4 times 3; and the operation is reversed, by finding at once how many times 3 is contained in 12, in place of subtracting at three different times. Thus another important improvement is introduced, by substituting, when the numbers are identical, MULTIPLICATION for addition, Division for subtraction; and the whole secret lies, as before, in the omission of superfluous steps.

INVOLUTION and EVOLUTION are merely slight modifications of multiplication and division. The former differs in no other respect from multiplication, save that of the factors being identical. In the latter, the divisor and quotient being also equal factors, both are required to be found by analysis of the dividend.

Such, in all its beautiful simplicity, is our system of Decimal Arithmetic. The whole science may be considered a mere enumeration of numbers, with wider and more rapid steps, as the subject becomes more and more familiar. We shall endeavor to follow up this natural mode of improving the science, by suggesting to the pupil from time to time very many other cases in which both time and labor may be saved by the omission of superfluous steps.

For the sake of convenience, arithmetic is commonly arranged under two heads, viz., Oral Arithmetic and Written Arithmetic.

PART I.

ORAL ARITHMETIC.

INTRODUCTION,

ADDRESSED TO TEACHERS.

ALTHOUGH oral and written arithmetic occupy separate places in this book, it is intended that they should be taught simultaneously. By this means, while the pupil is acquiring a knowledge of Notation and Numeration in written arithmetic, he is mentally practising the elementary operations, preparatory to the larger processes on the slate; and thus, if care be taken to proceed with equal steps in both parts of the science, the oral operations precede, and smooth, and facilitate the progress of the pupil in uritten arithmetic, and enable him wholly to dispense with artificial rules, or, should these be thought necessary, to form them for himself.

Arithmetic is suited to the capacity of the youngest child that attends school. At a very early age children understand. and delight to operate with, abstract as well as concrete numbers. Indeed, within certain limits, the effort is less in young than in older children. Experience shows that, of two boys of equal capacity, the one of seven, the other of nine or ten years of age, when commencing arithmetic, their progress is more nearly in an inverse than in a direct ratio to their respective ages. Nor is this all. The faculty of attention, one of the most important powers of the mind, is thus surely, yet easily and gradually, developed by oral arithmetic when properly taught. Indeed, children in the babitual practice of such operations, acquire the art of reading and orthography much more rapidly than those who delay the study of arithmetic to a later period.

This work commences with the most simple elements. The very first lessons may not, therefore, be necessary to all. The teacher should use a proper discretion in this respect. His judgment, indeed, must be relied on throughout the oral division of the subject, as well in the supply of additional questions for a dull class, as in the omission of such as may be superfluous for brighter scholars. No treatise can be formed exactly to suit every degree of mental capacity. It is believed, however, that it will be beneficial for all to proceed regularly through the book, and more especially through the fundamental exercises on the frame given below. Probably no other means would convey to the mind of childhood such clear and exact conceptions of the nature of our decimal arithmetic.

Description of the Improved Numeral Frame. A numeral frame is an indispensable tool for a good teacher of arithmetic. No contrivance could be better adapted to convey clear ideas of the first principles of that science. But, as commonly arranged, with twelve wires, and twelve beads on each wire, its worth, if it has any, must be exceedingly limited. To fit it for our decimal system, it should be pierced for eleven wires only, ten of which should be at equal distances, the eleventh farther apart. In one already formed, the eleventh wire may be taken out, which will leave the others properly arranged. The beads, as in the old-fashioned frame, should be of two dissimilar colors, such as black and white, or blue and yellow. There should be ten beads on each wire, arranged as follows: three dark, two light, and two dark, three light. By this disposition of the beads, any number, not exceeding eleven hundred,

can be read from the frame at a glance. Of the upper 100 beads, each stands for a single unit, while each of the 10 beads on the lower wire represents 100. By this simple arrangement into units, tens, and hundreds, our decimal system of arithmetic is distinctly presented to the eye, and is readily comprehended by the youngest pupil, and read off as easily and rapidly as by the use of figures.

The first lesson for beginners should be as follows: Let the teacher hold the frame so that all the beads fall to one side, and, passing one of those on the upper wire across, say, “ There is one bead. Repeat after me, one bead [passing another across), two beads,&c., till the first ten beads are all passed across and named.

The chief object of the second lesson is to qualify the pupil to read off any number, from one to ten, upon the frame at a glance, for which the arrangement of the beads upon the wires (3 and 2, and 2 and 3) renders it eminently qualified. The teacher should commence by a repetition of the first lesson, and, when that is perfectly known, proceed further, thus: Pass three beads across separately, and name as before, adding, “ Now try to recollect three.” Then pass those across at once on a different wire, and ask the number. If the child does not know, let this part of the lesson be repeated till the number three on the frame becomes familiar to the eye. Use four, five, six, and seven beads, till these numbers also can be readily named on the frame, without counting them. For eight, nine, and ten, direct attention to the opposite end of the wire; eight being known at first by two beads opposite, nine by one, ten by

When the pupil or class has become familiar with the first ten numbers, and able to name them on the frame at a glance, · the difficulty of the nomenclature is nearly surmounted ; the names of the others being chiefly a repetition of the first ten. It will be found most convenient and useful to give these larger numbers their original appellation before introducing their common or contracted names, as the former explains and simplifies the whole system of arithmetic. Let the class, therefore, be informed that ten has three different names, viz. I. Ten, standing by itself, is called

ten. II. Ten, added to another number,

teen. III. In the plural, i. e., more than one ten, ty. Applying this to the frame, when the beads on the first wire

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