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seventeen and five hundred and thirty-eight? Five hundred and thirty-seven and three hundred and forty-eight ? Seven from fifteen ? Eighteen from thirty-five ? Seventeen from sixty-five? Fifty-eight from eighty-five? Twenty-seven from eighty-five? Eighteen from a hundred and forty-five? Two hundred and seventeen from seven hundred and fifty-five ? Five hundred and thirty-seven from eight hundred and eightyfive? Three hundred and forty-eight from eight hundred and eighty-five ?

6. Nine and four, how many? Nineteen and fourteen? Nineteen and twenty-four? Thirty-nine and thirty-four? A hundred and twenty-nine and a hundred and fourteen? Two hundred and nineteen and three hundred and thirty-four ? Nine from thirteen? Four from thirteen? Nineteen from thirtythree? Twenty-four from forty-three? Thirty-nine from seventy-three? Thirty-four from seventy-three ? A hundred and fourteen from two hundred and forty-three? Two hundred and nineteen from five hundred and fifty-three? Three hundred and thirty-four from five hundred and fifty-three ?

7. Six and eight, how many ? Sixteen and eighteen? Sixteen and thirty-eight? Seventy-six and eighteen? A hundred and sixteen and nine hundred and eighteen? Two hundred and thirty-six and two hundred and thirty-eight ? Six hundred and sixteen and a hundred and twenty-eight ? Six from fourteen ? Eight from fourteen? Sixteen from thirty-four ? Sixteen from fifty-four ? Eighteen from ninety-four? Seventy-six from ninety-four ? A hundred and sixteen from a thousand and thirty-four? Two hundred and thirty-eight from four hundred and seventy-four ? Six hundred and sixteen from seven hundred and forty-four ? A hundred and twenty-eight from seven hundred and forty-four ?

8. Nine and seven, how many? Nineteen and seventeen ? Forty-nine and seventeen ? Sixty-nine and twenty-seven? Thirty-nine and forty-seven? Two hundred and nineteen and a hundred and forty-seven? Four hundred and nineteen and five hundred and seventeen? Nine from sixteen? Seven from sixteen? Nineteen from thirty-six ? Seventeen from thirty-six ? Seventeen from sixty-six ? Sixty-nine from ninety-six ? Forty-seven from eighty-six ? A hundred and fortyseven from three hundred and sixty-six ? Four hundred and nineteen from nine hundred and thirty-six ? Five hundred and seventeen from nine hundred and thirty-six ?

9. Two and nine, how many? Twelve and nineteen? Thirty-two and nineteen? Sixty-two and twenty-nine? A hundred and twelve and nineteen? Two hundred and fifty-two and two hundred and thirty-nine ? Three hundred and thirty-two and three hundred and thirty-nine ? A hundred and twelve and nine hundred and nineteen? Two from eleven? Ninę from eleven? Nineteen from thirty-one? Thirty-two from fifty-one? Sixty-two from ninety-one? A hundred and twelve from a hundred and thirty-one ? Two hundred and fifty-two from four hundred and ninety-one ? Three hundred and thirtynine from six hundred and seventy-one ? · A hundred and twelve from a thousand and thirty-one? Nine hundred and nineteen from a thousand and thirty-one ?

10. Five and nine, how many ? Fifteen and nineteen? Twenty-five and thirty-nine? Sixty-five and twenty-nine? A hundred and five and three hundred and nineteen? Six huudred and fifteen and two hundred and fifty-nine? Five from fourteen? Nine from fourteen? Fifteen from thirty-four ? Thirty-nine from sixty-four ? Twenty-nine from ninety-four ? A hundred and five from four hundred and twenty-four ? Six hundred and fifteen from eight hundred and seventy-four ? Two hundred and fifty-nine from eight hundred and seventyfour ?

SECTION VIII. — Addition Circles. [EXPLANATION.- Addition circles are formed by adding a number continually to itself, dropping the hundreds, till we return to the number with which we commenced. Take, for instance, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, then omit the hundreds, 17, 26, &c. Each circle should be recited again and again, till it can be repeated as fast as the words can be spoken. The circles for 7 and 3 may be formed in the same manner as for 9. The even numbers and 5 must be managed somewhat differently, to insure sufficient variety. As the repetition of 5 will only give fives and ciphers in the unit's place, after repeating the circle sufficiently with that number, commence the circle again with the other digits. Thus, commencing with 1 gives 6, 11, 16, &c., with 2 gives 7, 12, 17, &c., and so with the other digits. In forming circles with the even numbers, in order that there may be sufficient variety, it will be necessary to commence at least twice, namely, once with an odd figure, and once with an even one. The exercises that follow increase in difficulty, nearly in the order in which they are arranged. They will be formed with more ease, if the pupil observes that the addition of 9, 19, 29, &c., to any number diminishes the number of units by 1; the addition of 8, 18, &c., by 2, and 7, &c., by 3. It may also be noticed that the addition of 1, 11, 21, &c., 2, 12, 22, &c., 3, 13, 23, &c., increase the units by 1, 2, and 3, respectively.]

1. Form an addition circle with 9; with 7; with 3; dropping the hundreds; and continuing the process till the pupil again arrives at the number 9, or 7, or 3, with which he commenced.

2. Form a circle with 5, commencing with that number. Form it again, commencing with 6; with 7; with 8; with 9.

3. Form addition circles with 2, 4, 6, 8; commencing, the first time, with these numbers severally; afterwards, with an additional 1, 3, 5, or 7.

4. Form addition circles with 11, 12, 13, 19, 18, 17, 16, 15, in a similar manner as with the single digits.

5. Form circles with the numbers 21 to 29, varying the commencement by the addition of other numbers as above, especially the even numbers and 25.

[These exercises ought to be repeated till the circles can be recited rapidly and without hesitation.]

SECTION IX. — Increase and Decrease by Large Numbers, the

Tens causing a change in the Hundreds. 1. SEVENTY and seventy, how many ? [A hundred and forty. Why? Because seventy and seventy make fourteenty, and as ten ty make a hundred, fourteen ty are a hundred and forty. Repeat this idea in the following questions, till the principle is sufficiently familiar to the class, when the questions may be answered in the usual terms.] Ninety and forty? [Ans. A hundred and thirty. Why, &c. ? Some pupils will add numbers like these more rapidly by subtracting a sufficient number of ty from the smaller number to increase the larger number to 100. Thus, 80+70=170—20=150; 90+40=140-10=

130; 84+42=146420=126; 365+253=668—40=618. Observe, however, that one of the two steps in each of the above operations should be dropped as superfluous after a little practice.] Sixty and fifty ? Eighty and ninety? Forty-two and eighty-four ? Thirty-six and eighty-two ? Fifty-four and sixty-two ? Seventy-eight and eighty-one? A hundred and fifty and seventy ? [Explain again, if any hesitation here.] Two hundred and eighty and a hundred and sixty ? Three hundred and sixty-five and two hundred and fifty-three ?

2. Seventy from a hundred and forty ? [A hundred and forty being the same as fourteen ty, this question is, in fact, seventy from fourteenty, which is seventy.] Ninety from a hundred and thirty ? Forty-two from a hundred and twentysix ? Fifty-four from a hundred and sixteen? Seventy-eight from a hundred and fifty-nine? A hundred and sixty from four hundred and forty? Three hundred and sixty-four from six hundred and eighteen ?

3. Sixty and eighty, how many ? Seventy-three and fortyfive? Seventy-two and seventy-six ? Thirty-six and ninetythree ? Five hundred and forty-six and two hundred and ninety-one? Five hundred and forty-six and two hundred and ninety-two? [Seven hundred and thirteenty-eight, or, &c.] Three hundred and eighty-eight and two hundred and fortyone? [Five hundred and twelvety-nine, or, &c.] Sixty from a hundred and forty ? [or fourteenty.] Forty-five from a hundred and eighteen? [or eleventy-eight.] Seventy-two from a hundred and forty-eight? Thirty-six from a hundred and twenty-nine ? [twelvety-nine.] Two hundred and ninety-two from eight hundred and thirty-eight? [seven hundred and thirteenty-eight.] Three hundred and eighty-eight from six hundred and twenty-nine ? [five hundred and twelvety-nine.]

4. Ninety and eighty, how many ? Seventy-three and sixtytwo? Eighty-three and ninety-six? A hundred and fifty-four and two hundred and eighty-two? Three hundred and twentynine and two hundred and ninety? Four hundred and twentytwo and five hundred and thirty-three ? Ninety from a hundred and seventy ? [seventeenty.] Eighty from a hundred and seventy? Sixty-two from a hundred and thirty-five? [thirteenty-five.) Eighty-three from a hundred and seventynine? [seventeenty-nine.] Two hundred and eighty-two from four hundred and thirty-six ? [three hundred and thirteentysix.] Two hundred and ninety from six hundred and nine

teen? [five hundred and eleventy-nine.] Four hundred and twenty-two from a thousand and fifty-five ? [nine hundred and fifteenty-five.]

5. Seventy and sixty, how many ? Sixty-two and eighty-two? Two hundred and thirty-six and a hundred and eighty-two? Two hundred and seventy-one and two hundred and eighty-four? Six hundred and ninety-five and three hundred and seventythree? Seventy from a hundred and thirty? (thirteenty.] Sixty-two from a hundred and forty-four? Two hundred and eighty-two from four hundred and eighteen? (three hundred and eleventy-eight.] Two hundred and seventy-one from five hundred and fifty-five? Six hundred and ninety-five from a thousand and sixty-eight? [nine hundred and sixteentyeight.]

6. Sixty-three and seventy-two, how many ? Two hundred and fifty-four and three hundred and sixty-five ? Four hundred and seventy-two and two hundred and fifty-three? Eight hundred and fifteen and a hundred and ninety-four ? Sixtythree from a hundred and thirty-five? Two hundred and fifty-four from six hundred and nineteen? Four hundred and seventy-two from seven hundred and twenty-five? A hundred and ninety-five from a thousand and nine? Eight hundred and fifteen from a thousand and nine ?

7. Forty-two and ninety-three, how many? Two hundred and sixty-seven and three hundred and sixty-one ? Four hundred and forty-three and three hundred and ninety-four ? Five hundred and seventy-eight and three hundred and sixty-one ? Forty-two from a hundred and thirty-five? Two hundred and sixty-seven from six hundred and twenty-eight? Three hundred and ninety-four from eight hundred and thirty-seven? Five hundred and seventy-eight from nine hundred and thirtynine ?

8. Seventy-six and forty-three, how many ? Two hundred and fifty-six and six hundred and eighty-three? Four hundred and ninety-six and two hundred and sixty-three? Eight hundred and seventy-six and ninety-three? Two hundred and forty-six and six hundred and eighty-three? Seventy-six from a hundred and nineteen? Forty-three from a hundred and nineteen? Six hundred and eighty-three from nine hundred and thirty-nine ? Four hundred and ninety-six from seven hundred and fifty-nine? Ninety-three from nine hundred and sixty-nine? Six hundred and eighty-three from nine hun

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