tutes for the inch rod another rod 14 inches in diameter; will it sustain the required weight?

4. There are two ropes of the same material; one, 14 inches in diameter; the other, 2 inches; what is the ratio of their strength?

Case Second.-The strength of beams to resist fracture crosswise. In beams of the same material, length and width, but of different depth, the strength varies, as the square of the depth.

1. There are two beams of equal length, but the depth of one is 10 inches; of the other, 12 inches; what is the ratio of their strength?

2. There is a stick of timber 4 inches thick and 12 inches deep; if sawed into three 4 inch joists, what part of the former strength of the whole stick, when placed edgewise, will each part possess, allowing nothing for waste in sawing?

3. There is a stick of timber 10 inches in depth; if 4 inches of its depth be removed, what will be its strength compared to what it was before?

4. There are two sticks of timber, equal in length and width; one, 7 inches deep; the other, 5; what is the ratio of their strength?

5. If a stick of timber 6 inches deep have 2 inches of the depth removed, will it be weakened more than one half?

What is the exact ratio of its present, compared with its former strength?

6. A builder went to a lumber-yard, wishing to obtain an oak beam 5 inches wide and 10 inches deep; the lumbermerchant said, “I have not such a stick; but I have two oak sticks of the right length and width, and 7 inches deep; they will both, placed side by side, be stronger than one beam 10 inches deep." "Not so strong," said the builder.

Which was right? and what is the ratio of strength in the two cases

?

10*

NOTES TO PART FIRST.

NOTE 1.-PAGE 15.

This exercise should be often reviewed till the pupils can go through it with ease, and without mistake. No exercise can be devised that will more rapidly increase the learner's powers in Addition.

The word complement means, something to fill up. In arithmetic, the complement of a number, strictly speaking, is that number which must be added to it, to make it up to the next higher order. The complement of a number consisting of units only, as 8, 7, 9, is the number that must be added to make it up to 10, and consists of units only. If the number consist of tens, as 20, 50, its, complement is the number that must be added to make a hundred, and consist of tens. If the number is hundreds, its complement is so many hundreds as will make up a thousand.

If the number consist of several orders, its full complement will consist of the same orders, of such an amount as to raise the sum to the next order above the highest named in it. The complement of 745 is 255, for 745+255-1000, which is the order next above the highest named in the given sum.

The more restricted use of the word, as employed in the text, is sufficient for the purposes here had in view.

A few suggestions will here be made in reference to the best mode of conducting the accompanying recitation. The object of the lesson is to cultivate the power of instantly associating a number and its complement together. In conducting the recitation, the answer to each question as it is given out, should be required simultaneously by the whole class. The teacher should stand before them, and require that every eye be fixed on him. The questions should not be hurried, but the class should be encouraged to answer instantly on hearing the question. This will be easy in the first class of numbers given, which are even tens. In regard to

the remaining numbers, however, which are not even tens, something more will be necessary. Suppose the question is, what is the complement of 37? it may be conducted as follows:

Teacher. What is the complement of — 30?

Class. 70.

Teacher. Now listen to me without speaking; what is the complement of 30 -? you observe, I am going to say something more; what will it be?

Class. Something between 30 and 40.

Teacher. Well then, whereabouts will the complement be found?

Class. Between 60 and 70.

Teacher. Very good! Now when I say 30, and keep my voice suspended, showing that that is not all, what number can you think of, that you know will be a part of the complement ?

Class. 60.

Teacher. Very well. Now listen; what is the complement of 30- ? what have you now in your mind?

Class. 60.

Teacher. Well, now once more listen, and all answer as soon as you hear the question; what is the complement of 37?

Class.

63.

In the following questions, let the teacher always make a short pause between pronouncing the tens, and the units; and if the class hesitate or disagree in their answer, let the question be resolved into its elements, and each one presented separately. Thus, if 64 is the number, and the class have not answered promptly and alike, say thus, what is the complement of 60 ?

Class. 40.

Teacher. What is the complement of 60

think of? Class. 30.

? what do you

Teacher. Now answer all together; what is the complement of 64? Class. 36.

In the examples of addition that follow, the teacher should make a pause between the two numbers, and see that every member of the class is intent and eager to catch the second number, and answer instantly. A few questions answered by the whole class in this way, will benefit them more than whole pages recited in an indolent and listless manner.

NOTE 3. PAGE 17.

In these and all other examples, the large numbers should be taken first. If the pupil begins with the units, as in written arithmetic, he should be checked at once. Such a method would

only lead to a laborious imitation of the process of written arithmetic, which is not the natural one, and could give no new power to the pupil, nor awaken any new interest in the study. Only a small portion of these questions should be recited at one lesson.

NOTE 4.-PAGE 23.

Care must be taken here that the pupil does not imitate the process of written arithmetic, but be required to regard every number in its true value. Thus in the question, what is one fifth of 250? he must not say 5 in 25 will go 5 times; and 5 in 0, no times; but one fifth of 25 is 5; therefore, one fifth of 250 is 50.

NOTE 5.- PAGE 25.

In the higher as well as the lower numbers, let the pupil grapple at once with the number as it stands. In this way his interest will be very much increased. He will see, throughout, the progress he is making; whereas, in written arithmetic as usually studied, the pupil has no sooner begun an operation than he loses sight of the process, and goes on in blind bondage to his rule, till he comes out at the end, and then looks to the book, as to an oracle, for the

answer.

Let the oldest class in arithmetic in a school be called up, and one of them be required to perform on the board the question, "what is one sixth of 43,248?" and when he has obtained the first quotient figure, stop him, and ask him, what he has now done; he will most likely be unable to tell. The answer he will give will probably be, that he has divided 43 by 6; and no one of his class will probably have a better answer to offer. If he says he has divided 43 thousand, he is still wrong; for he has divided only 42 thousand, leaving one thousand undivided.

In some of the examples given in this section, the large numbers may be separated in different ways preparatory to division. Thus, in the last example, 92,648 may be divided 80,000, 12,000, 600, 48; or 88,000, 4000, 640, 8, and in still other ways.

Pupils should be encouraged to exhibit more methods than one for obtaining the answer. If a scholar has two methods he should be allowed to give them both, and if another has a different one still, it should be brought forward, and the most lucid and easy one should receive the commendation of the teacher.

PART SECOND:

CONTAINING

RULES AND EXAMPLES FOR PRACTICE

IN

WRITTEN ARITHMETIC.

NUMERATION OF WHOLE NUMBERS.

IN common Arithmetic there are 9 figures used for the expression of numbers. 1, one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine. When one of these figures stands alone, it signifies so many units, or ones; when two figures stand side by side, the left hand figure signifies so many tens; when three stand side by side, the left hand figure signifies so many hundreds; and universally, as you advance to the left, the figures increase in value tenfold at each step, as will be seen in the table on the next page.

The right hand place is always that of units. When there are tens, and no units, a cipher, 0, must stand in the unit's place, thus, 20; this merely serves to occupy the unit's place, and shows that the figure, 2, is in the place of tens. When there are hundreds, and no tens nor units, two ciphers are wanted; one in the unit's place, and one in the place of tens; as, 200; and so of all higher numbers.

To annex a cipher to a figure, therefore, is the same as to multiply the number by ten, for it removes the figure from the unit's place to the place of tens. To annex two ciphers is the same as to multiply the number by a hundred, for it removes the figure from the unit's place to that of hundreds.