Plan of Conducting the Exercise.

Let the teacher, or one of the pupils, write a few of the following ex-
amples on the black-board, and number them, and then let each member
of the class at once proceed to the computation, in perfect silence, writing
the complete result on his slate with the appropriate number, as below, as
soon as it is ascertained.

The teacher, however, should be careful to see that no pupil write
down a partial result, nor have any of them ready written on the slate. As
the object is to cause the class to go through the whole of each exercise
mentally without a pause, it is easy to see that either of these practices
would render the whole proceeding nugatory.

1. Addition by Inspection.

All these exercises should be thoroughly practised on the slate before
recitation on the black-board, the pupils being cautioned never to use the
pencil until they are prepared to write the result in full. All the les-
sons should be short. They may be practised while the rest of the book
is reviewed.

Find the sums or amounts of the following columns of figures, not
writing them on the slate till the whole result of each exercise is ascer-
tained. Prove by addition in the ordinary manner.

The class may now take the first four exercises, and form them into
one, with 6 figures in width and 6 in depth, and proceed in like manner
with the remainder, thus increasing the exercises both in width and depth
to the end, always remembering that the result of each addition is not to
be written till the student has it complete in his mind.

Repeat all the above exercises in addition, commencing each at the
left instead of the right. For example, in No. 5, say: sixteen hundred ;
a hundred and ten, seventeen hundred and ten; twenty-six, seventeen
hundred and thirty-six.

2. Subtraction by Inspection.

Find the differences of the following numbers, observing that the sub-
trahend is placed sometimes above and sometimes below. The whole
result should be ascertained before any part of it is written. Prove by
subtraction in the ordinary method.

12. 796346
149287

13. 9712645
2793487

14. 287638472. 15. 8296384
639724576
5149427

Repeat the above exercises, by throwing two into one, proving as be-
fore; and repeat them once more, performing the subtraction by adding
the complement of the subtrahend.

Find the differences of the following pairs of numbers, by the addi-
tion of the complement, proving by adding the result to the double sub-
trahend.

Repeat the first ten exercises immediately above, first by doubling two
horizontally, and again by doubling each pair vertically, placing the
four numbers of the minuend together, and also the four of the subtra-
hend, connecting each set by a brace.

3. Multiplication by Inspection.

Find the products of the following factors, not writing them till every
figure of the result is attained. Prove by multiplication in the ordinary

Method of Operation.-[To be read slowly by the teacher, the class
keeping their eye on the figures.] No. 1. Beginning on the left, twelve
thousand; sixteen hundred, thirteen thousand, six hundred; two hun-
dred, thirteen thousand, eight hundred; thirteen thousand, eight hun-
dred and eight.

No. 2. Thirty thousand; fifteen hundred, thirty-one thousand, five
hundred; two hundred and fifty, thirty-one thousand, seven hundred
and fifty; thirty-one thousand, seven hundred and seventy.

1. Repeat the above eighteen exercises, omitting the words in italics : that is, throwing the partial product at once into the general result without naming it.

2. Repeat the same eighteen exercises, with an additional figure to each multiplier: that is, 1 ten to each, then 2, 3, 4, and 5 tens to each.

Method of operating with two figures in the multiplier. No. 1, with 14. First by 10, then by 4. 10. Thirty-four thousand, five hundred and twenty. 4. Twelve thousand, forty-six thousand, five hundred and twenty; sixteen hundred, forty-eight thousand, one hundred and twenty; two hundred and eight, forty-eight thousand, three hundred and twentyeight.

No. 2, with 35. First by 30, then by 5 [5]. 30. A hundred and eighty-nine thousand; sixteen hundred and twenty, a hundred and ninety thousand, six hundred and twenty. 5. Thirty-one thousand, seven hundred, and seventy, two hundred and twenty-two thousand, three hundred, and ninety.

No. 3, with 26. First by 20, then by 6. 20. A hundred and sixtynine thousand, eight hundred and forty. 6. Fifty thousand, four hundred, two hundred and twenty thousand, two hundred and forty; five hundred and forty; two hundred and twenty thousand, seven hundred and eighty; twelve; two hundred and twenty thousand, seven hundred and ninety-two.

No. 5, with 38. 3. A hundred and forty-seven thousand; a thousand and eighty, a hundred and forty-eight thousand and eighty. 8. Thirtytwo thousand; a hundred and eighty thousand and eighty; seventy-two hundred; a hundred and eighty-seven thousand, two hundred and eighty; two hundred and forty; a hundred and eighty-seven thousand, five hundred and twenty; forty-eight; a hundred and eighty-seven thousand, five hundred and sixty-eight.

3. Repeat the 18 exercises with the figures of the multiplicand in reverse order, and with 6, and 7, and 8, and 9, in the ten's place of the multiplier.

The teacher may extend these exercises as far as he may find it profitable to the class. Some pupils have accomplished the multiplication of 9 figures in the one factor, and 5 in the other, after a very short practice. To others they come hard. But all will be highly benefited by their use.

4. Division by Inspection.

Find the quotients in the following exercises by inspection, beginning at the left, not writing them till the whole quotient is attained. Prove by multiplication by inspection.

1. 2)653,492

2. 4)134,684

3. 7)179,426 4. 5)286,942

Repeat each of the above 11 exercises, with an additional figure for tens in the divisor, namely, 1, 2, 3, 4. Then repeat again, with the figures in each dividend reversed, with an additional figure, 6, 7, 8, 9 tens in the divisor. Prove as above.

The teacher can extend this practice by new exercises, increasing the number of figures, both in the divisor and the dividend, till the class has acquired sufficient dexterity.

5. Evolution by Inspection.

a. Extraction of the Square Root by Inspection.

Find the nearest square root of each of the following numbers: 576; 1296; 1458; 3975; 2482; 9176; 7056; 2209; 20736; 53824. Prove by involution by inspection.

b. Extraction of the Cube Root by Inspection.

Find the nearest cube root of the following numbers by inspection; and prove by involution by inspection: 13824; 46656; 53824; 592,704; 638,576; 2,985,984; 12,487,168.

When a perfect cube does not exceed 1,000,000, its root may be found almost at a glance, as follows: Fill up the table of squares and cubes in p. 164, if not already done; and then take notice that the unit's figure in the cubes of 2 and of 3, and also in those of their complements (8 and 7), is the same as the complements of their respective roots. Observe, also, that the unit's figure in the cubes of all the other digits (1, 4, 5, 6, 9) is the same as that of their root. Thus :

From the above table it is evident that it only requires a knowledge of the cube of each of the digits, to determine the root of any perfect cube not exceeding 1,000,000, by a mere glance at the second period and at the unit's figure of the power.