Lands (211, 248) as time poorly spent. These articles we regard as being "for the greatest good of the greatest number"; since their position, somewhat early in the work, serves the purpose of giving to many who would never see them if placed on the last pages with Mensuration, an idea of what they shall have to labor at in after years. Besides, the rules may be laid by for future use and reference.

In addition to the points named, we wish to call attention to our Outlines. It will be observed that we have a General Outline and for each chapter a Sub-outline. And although these may not be all that the heart could desire, we do think that every teacher may make them a valuable aid in the prosecution of his labors. We like them better than synopses placed at the close of the chapter; for the mind, like the eye, takes in an object as a whole, then separates it into parts, then sub-divides each part, and so proceeds, until all the minutiæ are, in turn, examined. This is the principal object of the Outlines; though they also serve the purpose of a ready index.

In the arrangements of subjects, deviation has been made from the "beaten track"; but only with great caution. The placing of Decimals immediately after Division of Integers was decided upon only after careful experiment by many of our most experienced educators.

And last, though, perhaps, to some, not least, is the fact that although the publishers have spared neither pains nor expense (as the most casual examination of the books themselves shows) to make this series, consisting of the FIRST BOOK IN ARITHMETIC and THE COMPLETE ARITHMETIC, Second to none, they propose to offer to their customers in these two volumes, what is usually found in from three to five, and at correspondingly low prices; and thus relieve our "humble poor," as well as those of moderate means, of much of the expense they can so ill afford in the procuring of text-books for their children. They do not expect, however, to change in a day the habits strengthened by years. There will be men who will differ from them and us in opinion, and who will think, despite what has been said, that two books are not enough; that Intellectual and Written Arithmetic should be separate works. With these the publishers have no quarrel, but, having acted in accordance with their best judgment, now submit to an intelligent public for their decision this question so important to all interested in the subject of education.

One word to our fellow-teachers. We have made an earnest effort to present such a work as will meet with your approval and suit your wants. From comments of some of you on portions of the manuscript and proof-sheets that you have seen, we have reason to believe that you will be pleased with the result of our labors. But we cannot, and do not, expect you all to agree with us in all points. You have minds of your own; and we are glad to know that you have not only the independence to think for yourselves, but that you possess that liberality of view, that grants others the privileges you claim for yourselves. Therefore, while we do not hope to escape your criticism, we do look for that honest, straightforward expression of opinion that becomes those who are engaged in a profession which, when properly pursued, develops the noblest qualities of mind and heart.

In the preparation of this work, we have been greatly aided and encouraged by many friends, whose kind suggestions have been thankfully received and freely used. To all we return our best thanks, as well as to our co-laborers, Messrs. J. M. LOGAN and H. 1. GOURLEY, of the Pittsburgh Schools, to whose superior taste we are indebted for the neat arrangement of headings, plates, and outlines, and by whose care and vigilance many errors and crudities have been avoided. M. B. G.

WESTERN UNIVERSITY OF PA., June, 1876.

Suggestions

Art. 1-19. The teacher must exercise discretion in use of definitions. Those in Section I. need not all be committed at once, as some of them are given in the body of the work as needed.

55. Addends, although a comparatively new term, is not used without authority.

61. In Mental Exercises, use the model best adapted to each pupil. If deemed best, give the younger pupils the mental problems as dictation exercises on the slate.

73. Refer to definitions (4 and 5), or explain fully concrete and abstract.

78. As early as possible, it is well to show that the placing of the subtrahend under the minuend, is a matter of convenience.

89. When multiplying we may regard both terms as abstract, and then attach to the product such name as the nature of the question demands. Thus, since 1 bbl. flour costs $8, 9 bbls. cost 9 times as much, or $72. 9 × 8 = 8 × 9 = 72. The answer must be dollars. Therefore, $72, answer.

93. Prob. 10 may be contracted thus:

374781 1402

749562 5246934 525442962

First multiply by 2; then multiply this product by 7, placing the right-hand figure under 4, and the remaining figures in order to the left; for 2 × 7 = 14. This is given merely as a sample. Once show the pupil how to lighten his labor, and generally he is not slow to take advantage of any short process; and will soon reap great benefit. Try Prob. 21.

Many operations in multiplication may be solved by such devices as this:

95. When ciphers are on the right of significant figures, either in the multiplicand or multiplier, or both, they should not be considered until the significant fig ures are multiplied together, when the ciphers must be annexed to the product

102. EXAMPLE SECOND. This may also be solved thus: If each of two persons receive $1, to divide $10 equally between them, each must receive as many times $1 as $2 is contained times in $10, or 5 times $1 = $5. This relieves us of the inconsistency of calling the divisor an abstract and the dividend a concrete number.

Or, since taking one-half of $10 is the same as multiplying $10 by, we have the multiplicand and product of the same kind (89).

106-108. Show that the divisor may be written in any other convenient place as well as at the left or right of the dividend.

PROBLEMS. The earnest teacher will not fail to supply the student with abundant examples. Pp. 73-78 are thought to afford a fair variety of such examples as the pupil needs to make him thoroughly familiar with the principles and operations already discussed.

112-119. Show the strong resemblance of a whole number, or Integer, and a decimal; and exhibit in the strongest light the importance of the decimal point. 120. The Second Method of Numeration and Notation possesses such great advantages over the first method that we wonder at the limited use of the former.

129-138. Too much care cannot be exercised in teaching "Multiplication of Decimals" and "Division of Decimals "; and no part of the Arithmetic will better repay this care; for, pupils once thorough in these, will move along easily and rapidly.

138. Reducing dividend and divisor to the same denomination before dividing has the advantage of clearness, but is sometimes inconvenient in practice. Thus: Divide 1728 by 1.2. Annexing .0 to 1728, we have 1728.0, dividend and diviboth tenths, and the quotient a whole number, 1440. Dividing .0001728 by 1.2 by this method, however plain it may be, is a clumsy performance.

sor,

167. In finding the factors of a number, the facts in this article may be greatly extended, as for example: FOURTH. Any even number, the sum of whose digits is divisible by 3, has 6 for an exact divisor; FIFTH. Any number, whose two righthand figures are divisible by 4, has 4 for an exact divisor, etc., etc.

183. Special Rules on p. 148 are given, that teacher and pupil may have both variety and choice. We prefer, however, the GENERAL Rule.

184. The same remarks apply to RULES on pp. 154, 155.

185. The pupil should be able at once to change a decimal into a common fraction, or a common fraction into a decimal.

186, 187, are eminently practical and should be thoroughly mastered.

192-232 embrace the tables used in Denominate Numbers, and are placed together as a matter of convenience for easy reference. The exercises which follow are placed under their appropriate headings, so that a table, or a convenient number of tables, with the exercises belonging to each, may be readily assigned as a lesson. 193. The Table of Federal Money, together with definitions, are here inserted to preserve the uniformity of the system.

The representations of the coins of the United States are all that are coined at the present time. Of the coins of Great Britain, Germany, and France, only a lim