When they have become familiar with the nomenclature, which will be in three or four lessons, the abbreviations, which are few and simple, may be explained. The most difficult are the two first, oneteen, twoteen, which are changed to eleven, twelve; the others are very simple, namely, threeteen and threety are shortened to thirteen and thirty; fiveteen and fivety, to fifteen and fifty; lastly, twoty is twenty; and fourty is forty.

This explanation is principally intended for those who know nothing of arithmetic. But it would be profitable for the whole school to go over the frame once or twice, as there are few who have clear notions of the meaning of ty and teen.

Our little pupils, having thus acquired the nomenclature of numbers, the fundamental processes of addition, subtraction, multiplication, and division, may now be commenced. The first two should be taught simultaneously on the frame thus, passing two beads and two beads, the class will see they make four; and, if two be taken from four, two will remain. If this be practised a very few minutes every day, in a week or two the class will add or subtract, instantly, any two numbers, not exceeding one thousand. Multiplication and division should also proceed simultaneously. Thus, taking eight beads, ask, how many twos it contains; and, if one of the class separate them on the wire into twos, all will see there are four; consequently, four twos make eight, and eight contains four twos. It will not be necessary to go further than the fifth line in multiplication and division; as the higher numbers will be more readily taught from Colburn's 'First Lessons,' of which anon. The frame need now be no longer used as a regular exercise, but should always be convenient to the teacher's desk; as, if properly used, it will be of much advantage to the class.

For very small children, Fowle's Mental Arithmetic' should precede Colburn's Lessons ;' but those of seven or eight years of age may pass, at once, into Colburn. Of these books, there should be only one copy in school. Any intelligent teacher can use them, even though unpractised in mental arithmetic. When this is the case,

however, she should work out every question, mentally, along with the class. The main advantage of mental arithmetic is, the wonderful manner in which it disciplines some of the most important faculties of the mind, particularly those of attention, abstraction, and reasoning. But to gain these advantages, in any considerable degree, the pupils should distinctly know, that the questions are never to be repeated. They must give their whole attention, while the question is reading, and they must retain the whole in their minds, until they have found the answer, and explained the process by which it was discovered. The books ought to-be used thus: The teacher reads, "Your brother William gave you nineteen cents, your brother John, ten, and your cousin Mary, two. many have been given to you, in all ?”

C. (after consideration.) Thirty-one.

T. How do you know?

How

C. Because brother William gave me nineteen, brother John, ten; now ten and nineteen make twenty-nine; and cousin Mary gave me two; twenty-nine and two make thirty-one.

T. Very well. Twelve men are to have ninety-six dollars for performing a piece of work. How much is due to each ?

C. Eight dollars.

T. Why?

C. Because, as the twelve men were to have ninetysix for their work, and as there are eight twelves in ninety-six, of course each man would have eight.

I now give a question in a more advanced stage.

T. A cistern has two cocks; the first will fill it in three hours, the second in six hours; how long would it take both to fill it?

C. Two hours.

T. Why?

C. Because if the first can fill it in three hours, it will fill one third of it in one hour; and if the second will fill it in six hours, it will fill one sixth in one hour; but one third is equal to two sixths; therefore, both will fill three sixths, or one half, in one hour; or the whole, in two hours.

Some of the exercises in addition, in Colburn's 'First Lessons,' are so easy, though not the less important, that there is some danger of the class allowing their minds. to wander, and yet answering correctly. This may be checked, by varying the questions, as follows: Instead of, Nine and four? Nineteen and four? Twenty-nine and four? Thirty-nine and four? regularly increasing the number of ty, let them be varied, thus: Twenty-nine and four? Forty-nine and four? Thirty-nine and four? Fifty-nine and four? &c.

once.

It requires some tact, to gain the utmost advantage from mental arithmetic; but it is easily acquired. The main point is, that the attention of the teacher be kept wide awake. The dull and slow must be allowed time; the bright must not be suffered to monopolize the answers. At the same time, it will not do for the answers to be received, in the order in which the pupils stand in the class; for, in this case, only one child would be occupied at Each pupil would attend only to his own question; whereas all should be occupied, and should actually solve every question put to the class. The best plan, then, is, for each to hold up a finger, when ready to answer, leaving the teacher to select whose turn it shall be. Thus, every one might have an equal chance. The dull and the bright, however, ought not to be together, but in different classes. In fact, it would be well to have the classes differently arranged, for each separate study. Some are bright at reading, and dull in arithmetic, and vice versa. To chain the dull to the bright has bad

effects on both.

The Pestalozzian plates, at the end of Colburn's book, may or may not be studied, at the option of the teacher. They are explained in the 'Key,' page 141. At all events, they should be clearly understood by the teacher.

Abbreviations in Mental Arithmetic.-The following abbreviations have never before been published. They may probably not only be useful to the student, but lead to the invention of others, equally profitable.

To multiply by 5. Take half the number, and multi

ply by 10.

=

We take half, because multiplying by 10 gives double of multiplying by 5. Thus, 5 × 64 = 64 X 10 32ty, or 320. When the number is odd, halving leaves a remainder of 1, which, of course, is one 5. Thus, 73 X5 X 10=36ty and five, or 365. Let us next proceed to 15, 20, 25, 30, &c., and afterwards take up the intervening numbers.

Fifteen is 10 and half of 10; therefore, increasing any number a half, and multiplying by 10, is the same as multiplying by 15. Thus, as 64 and half of 64 make 96, 64 X 1596ty, or 960. When the number is odd, proceed as above, in speaking of 5. Thus, 75 X 15= 112ty and five, or 1125, and the square of 15 is 22ty and five, or 225.

Twenty being two tens, to multiply by 20, double the number, and multiply by 10. Thus, 20 X 45 90ty,

or 900.

=

Twenty-five is one fourth of 100; therefore, to mul tiply by 25, take of the number for hundreds : every unit in the remainder is one twenty-five. Thus :

27 X 2527 X 100=675, &c.

Fifty is half of 100; therefore, to multiply by 50, take the number for hundreds. Thus, 24 X 50-24 x 100 1200.

=

Thirty is thrice ten; therefore, to multiply by 30, take thrice the number, and multiply by 10. Thus, 24 × 30 =72ty, or 720.

Let us now examine the intermediate numbers, which are all done on one principle. Fourteen times any number is 15 times that number, less once the number; and 13 times any number is 15 times the number, less twice the number. Thus, 14 X 24: 15 X 24, less once 24; and 13 X 24=15 X 24, less twice 24. Again, 16 X 24= 15 X 24, more once 24; and 17 X 24 = 15 X 24, more twice 24. Thus, by connecting two numbers less,

and two numbers more, with our 15, 20, 25, 30, &c., we have all the intermediate numbers.

Division is performed by reversing these processes; that is, multiplying, where division is shown above; and dividing, where multiplication is indicated. Though not so easy as multiplication, some practice in it will be useful.

This system of abbreviations may seem obscure or difficult, perhaps, to those who have never practised mental arithmetic. But nothing is hazarded in the assertion, that, where Colburn's Arithmetic is used, as pointed out above, the class will understand and apply it with ease and rapidity, before they have gone half through that work. The teacher may exemplify the abbreviations for himself, on the slate; but they should be performed by the school, exclusively in the mind.

It is a matter of the first importance, that the teacher should have a distinct idea of the objects to be gained by the practice of mental arithmetic; as, otherwise, the main advantages that might result from it will assuredly be lost. Let it be constantly borne in mind, then, by the teacher, that the knowledge of arithmetic is not the chief benefit to be derived from it, but one of secondary importance. It is the mental discipline, the power of abstraction, the habit of attention and of reasoning which it developes, that constitutes its chief value. But all these advantages are lost, if the child is allowed to study the book; more especially by working out the questions on the slate. They can only be completely attained, by calling on the class to solve each question mentally, merely from hearing it once read, and then to give a clear account of his mental operations. And, so beautifully are the questions arranged, so completely does the knowledge gained in each question, come into requisition in those that follow, that, if the plan of study be commenced right, and strictly followed, the most intricate and difficult questions will give no trouble to the class.

It may, perhaps, be incredible to some, but it is not the less true, that Colburn's book may be gone through, and correct notions be attained of the principles of arith