mill by a mill, would produce a dollar, and though they are aware of the absurdity, cannot tell how to avoid the conclusion.

The above difficulties arise chiefly from not making a proper distinction between abstract and concrete numbers. Not one of these cases can ever occur in the manner here proposed. They are imperfect examples. When a perfect example is proposed, which involves one of the above cases, the difficulty is entirely removed.

It is not proper to speak of dollars being multiplied c divided by dollars or gallons.

At 5 dollars per barrel, what cost 3 barrels of flour?

Instead of saying that 5 dollars is to be multiplied by 3 Darrels, say 3 barrels will cost three times as much as I barrel, that is three times 5 dollars.

If I dollar will buy 7 lbs. of raisins, how many pounds may be bought for 4 dollars?

Say 4 dollars will buy 4 times as many pounds as I dollar. In these two examples there is no doubt what the answer should be. In one it is dollars, and in the other it is pounds.

In a piece of cloth 5 feet long and 3 feet wide, how many square feet?

If it were 5 feet long and 1 foot wide, it would contain 5 square feet, but being 3 feet wide it will contain three times as many, or three times 5 feet.

In a certain town a tax was laid of 1 dollar upon every $150; how much did a man possess whose tax was 3 dollars?

It is evident that he possessed three times $150.

At 1 cent each, how many apples may be bought for 1 cent ?

Here the divisor is 1 cent and the dividend is I cent, and the result is an apple instead of a dollar.

How many gallons of wine at 2 dollars per gal., may be bought for 6 dollars?

As many times as 2 dollars are contained in 6 dollars, so many gallons may be bought.

The truth is, the numbers are always used as abstract numbers, but a reference to particular objects is kept in view, and the nature of the question will always show to what the result must be applied.

It may however be established as a general principle, that

the multiplier and multiplicand are never applied to the same object, and in precisely the same way; and the product will be applied to the object which is mentioned in one denomination, as being the value of a unit in the other.

In division there are two numbers given to find a third, two of which will always be of the same denomination, and the other different, or differently applied.

If the divisor and dividend are of the same denomination and applied in the same way, the question is, to find how many times the one is contained in the other, and the quotient will be applied differently.

If the divisor and the dividend are of diferent denominations, or differently applied to the same denomination, the question is to divide the dividend into parts, and the quotient will be applied in the same manner as the dividend.

When any difficulty occurs in solving a question, it is best to supply very small numbers, and solve it first with them, and then with the numbers given. If the question is in an abstract form, endeavour to form a practical one, which shall require the same operation, and the difficulty is generally very much diminished.

In all cases reason from many to one, or from a part to one; and then from one to many or to a part. If several parts be given, always reason from them to one part, and then to many parts, or to the whole.

IMPROVED

SCHOOL BOOKS.

Colburn's First Lessons, or, Intellectual Arithmetic

THE merits of this little work are so well known, and so highly appreciated in Boston and its vicinity, that any recommendation of it is unnecessary, except to those parents and teachers in the country, to whom it has not been introduced. To such it may be interesting and important to be informed, that the system of which this work gives the elementary principles, is founded on this simple maxim; that, children should be instructed in every science, just so fast as they can understand it. In conformity with this principle, the book commences with examples so simple, that they can be perfectly comprehended and performed mentally by children of four or five years of age; having performed these, the scholar will be enabled to answer the more difficult questions which follow. He will find, at every stage of his progress, that what he has already done has perfectly prepared him for what is at present required. This will encourage him to proceed, and will afford him a satisfaction in his study, which can never be enjoyed while performing the merely mechanical operation of ciphering according to artificial rules.

This method entirely supersedes the necessity of any rules, and the book contains none. The scholar learns to reason correctly respecting all combinations of numbers; and if he reasons correctly, he must obtain the desired result. The scholar. who can be made to un

derstand how a sum should be done, needs neither book nor instructer to dictate how it must be done.

This admirable elementary Arithmetic introduces the scholar at once to that simple, practical system, which accords with the natural operations of the human mind. All that is learned in this way is precisely what will be found essential in transacting the ordinary business of life, and it prepares the way, in the best possible manner, for the more abstruse investigations which belong to maturer age. Children of five or six years of age will be able to make considerable progress in the science of numbers by pursuing this simple method of studying it; and it will uniformly be found that this is one of the most useful and interesting sciences upon which their minds can be occupied. By using this work children may be farther advanced at the age of nine or ten, than they can be at the age of fourteen or fifteen by the common method. Those who have used it, and are regarded as competent judges, have uniformly dec'ded that more can be learned from it in one year, than can be acquired in two years from any other treatise ever published in America. Those who regard economy in time and money, cannot fail of holding a work in high estimation which will afford these important advantages.

Colburn's First Lessons are accompanied with such instructions as to the proper mode of using them, as will relieve parents and teachers from any embarrassinent. The sale of the work has been so extensive, that the publishers have been enabled so to reduce its price, that it is, at once, the cheapest and the best Arithmetic in the country.

Colburn's Sequel.

THIS work consists of two parts, in the first of which the author has given a great variety of questions, ar

ranged according to the method pursued in the First Lessons; the second part consists of a few questions, with the solution of them, and such copious illustrations of the principles involved in the examples in the first part of the work, that the whole is rendered perfectly intelligible. The two parts are designed to be studied together. The answers to the questions in the first part are given in a Key, which is published separately for the use of instructers. If the scholar find any sum difficult, he must turn to the principles and illustrations, given in the second part, and these will furnish all the assistance that is needed.

The design of this arrangement is to make the scholar understand his subject thoroughly, instead of performing his sums by rule.

The First Lessons contain only examples of nunbers so small, that they can be solved without the use of a slate. The Sequel commences with small and simple combinations, and proceeds gradually to the more extensive and varied, and the scholar will rarely have occasion for a principle in arithmetic, which is not fully illustrated in this work.

Colburn's Introduction to Algebra.

THOSE who are competent to decide on the merits of this work, consider it equal, at least, to either of the others composed by the same author.

The publishers cannot desire that it should have a higher commendation. The science of Algebra is so much simplified, that children may proceed with ease and advantage to the study of it, as soon as they have finished the preceding treatises on arithmetic. The same method is pursued in this as in the author's other works; every thing is made plain as he proceeds with his subject.

The uses which are performed by this science, give it a high claim to more general attention. Few of the