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Late Paine and Burgess,




"A special examination of this valuable work will show that its author has compiled it, as all books for school use ought to be compiled, from the results of actual experiment and observation in the school-room. It is entire. ly a practical work, combining the merits of Colburn's system with copious practice on the slate.

"Two circumstances enhance very much the value of this book. It is very comprehensive, containing twice the usual quantity of matter in works of this class; while, by judicious attention to arrangement and printing, it is rendered, perhaps, the cheapest book in this department of education. The brief system of Book-Keeping, attached to the Arithmetic, will be a valuable aid to more complete instruction in common schools, to which the work is, in other respects, so peculiarly adapted.

"There are several very valuable peculiarities in this work, for which we cannot, in a notice, find sufficient space. We would recommend a careful examination of the book to all teachers who are desirous of combining good theory with copiouts and rigid practice."



The utility, and even necessity, of a work of this description, will scarcely be questioned by those who have had any experience in teaching Arithme


Most young persons, after having been persuaded, again and again, to review a long arithmetical process, feel, or affect to feel, certain that they have performed it correctly, although the result, by the book, is erroneous. They then apply to their instructor; and, unless he points out their mistake, or performs the operation for them, they become discouraged, think it useless "to try" longer, and the foundation for a habit of idleness is thus im perceptibly established. Now, in a large school, it is always inconvenient, and sometimes impossible, for the instructor to devote the time necessary to overlook or perform a very simple, much more a complex, question in Arithmetic. This is at once obviated by having at hand a Key, to which reference can be easily and speedily made. The time of the teacher will thus be saved, and the pupil will not have his ardor damped by being told that "his sum is wrong," without learning where or how.

This work is not designed for, and can scarcely become a help to lazi. ness; its object is to lighten the burden of teachers, and facilitate the progress of scholars. To promote both of these important purposes it is now presented to the public.

Entered according to Act of Congress, in the year 1835, by

In the Clerk's Office of the District Court of Massachusetts.

W. BENEDIT & CO., Stereotypers

No 16 Spruce street, New York.



When a new work is offered to the public, especially on a subject abounding with treatises fike this, the inquiry is very naturally made, Does this work contain anything new "Are there not a hundred others as good as this? To the first inquiry it is replied, that there are many things which are believed to be new; and, as to the second, a candid public, after a careful examination of its contents, and not till then, it is hoped, must decide. Another inquiry may still be made: "Is this edition different from the preceding?" The answer is, Yes, in many respects. The present edition professes to be strictly on the Pestalozzian, or inductive, plan of teaching. This, however, is not claimed as a novelty. In this respect, it resembles many other systems. The novelty of this work will be found to consist in adhering more closely to the true spirit of the Pestalozzian plan; consequently, in differing from other systems, it differs less from the Pestalozzian. This similarity will now be shown.

1. The Pestalozzian professes to unite a complete system of Mental with Written Arithmetic. So does this.

2. That rejects no rules, but simply illustrates them by mental questions. So does this.

3. That cominences with examples for children as simple as this, is as extensive, and ends with questions adapted to minds as mature.

Here it may be asked, "In what respect, then, is this different from that?" To this question it is answered, In the execution of our common plan.

The following are a few of the prominent characteristics of this work, in which it is thought to differ from all others.

1. The interrogative system is generally adopted throughout this work. 2. The common rules of Arithmetic are exhibited so as to correspond with the occurrences in actual business. Under this head is reckoned the application of Ratio to practical purposes, Fellowship, &c.

3. There is a constant recapitulation of the subject attended to, styled "Questions on the foregoing."

4. The mode of giving the individual results without points, then the aggregate of these results, with points, for an answer by which the relative value of the whole is determined, thus furnis. 'ng a complete test of the knowledge of the pupil. This is a characteristic a. Terence between this and the former editions.

5. A new rule for calculating interest for days with months.

6. The mode of introducing and conducting the subject of Proportion. 7. The adoption of the Federal Coin, to the exclusion of Sterling Money, except by itself.

8. The Arithmetical Tables are practically illustrated, previously and subsequently to their insertion.

9. As this mode of teaching recognizes no authority but that of reason, it was found necessary to illustrate the rule for the extraction of the Cube Root, by means of blocks, which accompany this work.

These are some of the predominant traits of this work. Others might be mentioned, but, by the examination of these, the reader will be qualified to decide on their comparative value.

As, in this work, the common rules of Arithmetic are retained, perhaps the readerás ready to propose a question frequently asked, "What is the use of so many rules Why not proscribe them The reader must here be reminded, that these rules are taught differ ently, in this system, from the common method. The pupil is first to satisfy himself of the truth of several distinct mathematical principles. These deductions, or truths, are then

generalized; that is, briefly summed in the form of a rule, which, for convenience' sake, m named. Is there any impropriety in this? Ou the contrary, is there not a great convenience in it? Should the pupil be left to form his own rules, it is more than probable he might mistake the most concise and practical one. Besides, different minds view things dif ferently, and draw different conclusions. Is there no benefit, then, in helping the pupil to the most concise and practical method of solving the various problems incident to a business life?

Some have even gone so far as to condemu the Rule of Three, or Proportion, and almost all the successive rules growing out of it. With more reason, they might condemn Long Division, and even Short Division; and, in fact, all the common and fundamental 1ales of Arithmetic, except Addition; for these may all be traced to that. The only question, then, is, "To what extent shall we go?" To this it is replied, As far as convenience requires. As the Rule of Three is generally taught, it must be confessed, that almost anything else, provided the mind of the pupil be exercised, would be a good substitute. But when taught as it should be, and the scholar is led on in the same train of thought that originated the rule, and thus effectually made to see, that it is simply a couvenient method of arriving at the result of both Multiplication and Division combined, its necessity may be advocated with as much reason as any fundamental rule. As taught in this work, it actually saves more figures than short, compared with Long Division. Here, then, on the ground of cou venience, it would be reasonable to infer, that its retention was more necessary than eit er, But, waiving its utility in this respect, there is another view to be taken of this sub eet, ani that not the least in importance, viz., the ideas of beauty arising from viewing the harmoni ous relations of numbers. Here is a delightful field for an inquisitive mind. It here imbibes truths as lasting as life. When the utility and convenience of this rule are once conceded, all the other rules growing out of this will demand a place, and for the same reason.

It may, perhaps, be asked by many, "Why not take the principle without the name?" To this it is again replied, Convenience forbids. The name, the pupil will see, is only an aggregate term, give. to a process embodying several distinct principles. And is there no convenience in this? hall the pupil, when in actual business, ie obliged to call off his mind from all other puits, to trace a train of deductions arising from abstract reasoning, when his attention is most needed on other subjects? With as much propriety the name of captain may be dispensed with; for, although the general, by merely summoning his captain, may summon 100 men, still he might call on each separately, although not quite so conveniently. With these remarks the subject will be dismissed, merely adding, by way of request, that the reader will defer his decision till he has examined the doctrine of Proportion, Fellowship, &c., as taught in this work.

In this work, the author has endeavored to make every part conform to this maxim, viz.. that names should succeed ideas. This method of communicating knowledge is diametrically opposed to that which obtains, in many places, at the present day. The former, by first giving ideas, allures the pupil into a luminous comprehension of the subject, while the latter astounds him, at first, with a pompous name, to which he seldom atfixes any definite ideas. and it is exceedingly problematical whether he ever will. In addition to this is the fact, that. by the last-mentioned method, when the name is given and the process shown, not a single reason of any operation is acduced; but the pupil is dogmatically told he must proceed thus and so, and he will come out so and so. This mode of teaching is very much as if a mer chant of this city should direct his clerk, without intrusting in with any business, first to go to South Boston, then to the state-house, afterwards to the market, and then to return, leav ing him to surmise, if he can, the cause of all this peregrination. Many are fools enough to take this jaunt pleasantly; others are restive, and some fractions. This sentiment is fully sustained by an article in Miss Edgeworth's works, from which the following extract is made: "A child's seeming stupidity, in learning arithmetic, may, perhaps, be a proof of intelligence and good sense. is easy to make a boy, who does not reason, repeat, by rote, any technical rules, which a common writing master, with magisterial solemnity, may lay down for him; but a child who reasons will not be thus easily managed; he stops, frowns, hesitates, questions his master, is wretched and refractory, until he can discover why he is to proceed in such and such a manner; he is not content with seeing his preceptor make figures and lines on the slate, and perform wondrous operations with the self-complace dexterity of a conjuror; he is not content to be led to the treasures of science blindfold; he would tear the banda e from his eyes, that he might know the way to them again."

In confirmation of the preceding remarks, and as fully expressive of the author's views on this subject, the following quotation is taken from the preface to Pestalozzi's system: "The Pestalozzian plan of teaching arithmetic, as one of the great branches of the mathe matics, when communicated to children upon the principles detailed in the following pages, needs not fear a comparison with her more favored sister, Geometry, either in precision of ideas, in clearness and certainty of demonstration, in practical utility, or in the sublime de ductions of the most interesting truths.

"In the regular order of instruction, arithmetic ought to take precedence of geometry, as it has a more immediate connection with it than some are willing to admit. It is the science which the mind makes use of in measuring all things that are capable of augmentation or dimination; and, when rationally taught, affords to the youthful mind the most advanta geous exercise of its reasoning powers, and that for which the human intellect becomes early ripe, while the more advanced parts of it may try the energies of the most vigoron and matured understanding"


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Practice in Federal Money

Numeration........................ .............

Simple Subtraction.................

Simple Multiplication...........................

Simple Division....

Questions on the foregoing.


Reduction of Federal Money

Federal Money-different Denominations

Addition of Federal Money.

Subtraction of Federal Money.

Multiplication of Federal Money
















Questions on the foregoing-Bills of goods sold.....

Reduction-Tables of Money, Weight, Measure, &c. .... ..................

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Fractions arise from Division................................................................................. .106 Proper, Improper, &c....


To change an Improper Fraction to a Whole or Mixed Number..... ..109 To change a Mixed Number to an Improper Fraction....


To change a Fraction to its lowest Terms"


To multiply a Fraction by a Whole Number.............. ..........


To multiply a Whole Number by a Fraction


To multiply one Fraction by another....


To find the Least Common Multiple of two or more Numbers..


To find the Greatest Common Divisor of two or more Numbers (reference) 120 To reduce Fractions of Different Denominations to a Common Denominator 122 Addition of Fractions.

Subtraction of Fractions...



Division of Fractions-To divide a Fraction by a Whole Number


To divide a Whole Number by a Fraction


To divide one Fraction by another...


To reduce Whole Numbers to the Fraction of a greater Denomination... 13)
To reduce Fractions to Whole Numbers of less Denominations
To reduce Fractions of a higher Denomination to a lower............... 134
To reduce Fractions of a lower Denomination into a higher ·
........... 135


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