9. In working out this Example, you would prepare by putting under the horizontal line certain ciphers. Where would you put them? How many would you put?

10. Why are those ciphers necessary?

11. What is meant by "The Back Figure"?

12. With what multipliers may the "Back Figure Contraction" be employed?

13. What is "A Mixed Number"? Give instances of Mixed Numbers.

14. What is every Mixed Number equivalent to?

15. How may an Integer be expressed as a Fraction? (p. 76')

16. How is a Mixed Number reduced to its equivalent Improper Fraction?

17. What is meant by "A Common Denominator?"

18. When do Fractions require to be reduced to a Common Denominator?

19. Repeat Principle XII.

20. What does this principle enable us to do?

21. How are Fractions reduced to a Common Denominator?

22. How are Fractions prepared for Addition?

23. Repeat Principle v.

24. What is Subtraction? Describe its Arithmetical Symbol.

25. How is the following expression to be read, and what is its meaning?

26. Explain the words Minuend, Subtrahend, Remainder, and Dif

ference.

27 Name each of those in the foregoing example.

28. When borrowing is necessary in Subtraction, whence do we borrow?

29. What is done with the quantity borrowed?

30. Why is the borrowed quantity never returned?

31. When we borrow from the next place to the left, what alterations are made in the value of the two Minuend Figures?

32. When one or more ciphers stand between the Minuend Figure for which we borrow and the Minuend Figure from which we borrow, what alterations are made in the values of the Minuend Figures, and of the cipher or ciphers between them? 33. As practised on the New System of working Subtraction, does "Borrowing" increase or decrease Minuend or Subtrahend? 34. On the New System, when is carrying required in Subtraction? 35. Why is "Convert" a more appropriate word than "Borrow" in Subtraction?

36. How many modes are there of proving Subtraction?

37. If I subtract 64 from 20000 I find the Remainder or Difference to be 19935. I thus find at once the answers to seven different questions. Write out those seven questions.

38. Write out the seven answers to them.

39. Of two Fractions having equal Numerators which is the Greater

Fraction?

40. Which is the Greater of two Fractions having a Common De

nominator?

41 Compare and . Two features distinguish the Greater of those two Fractions. What are they?

42. What is Continued Subtraction?

Give an instance of it. 43. What is the object of Continued Subtraction?

44. How many things are ascertained at once by a Continued Subtraction?

45. By what other method, beside Continued Subtraction, can you find how many nines there are in sixty-seven?

46. Of what two steps does this last method consist?

47. Which of the two methods is most natural for Experiment, and which for Calculation?

DIVISION.

Division is the distribution of any Number into lots, each containing some stated Number, or a certain portion of that stated Number.

Arithmetical Symbol. The sign of Division is a short horizontal line, having one dot immediately over its centre and other immediately under it: thus, . It is called in reading, "Divided by."

Hence, the expression 306 is read, "Thirty divided by six," and means that thirty units are to be distributed into lots of six units each.

Four words employed in Division must be clearly understood. They are these: Dividend, Divisor, Quotient, and Measure.

Take thirty units. Separate them into lots of six units each. There will be five lots each containing six units.

Again, take thirty-three units. Distribute these into lots of six units each. There will be five lots of six units each, and there will also be one half of such a lot.

... 306. 5 and 336.

In the first of these two Examples,

= 51

30 is the Dividend, the Number to be distributed.

6 is the Divisor, the Number to be comprised in each lot.

5 is the Quotient, shewing the Number of lots when distributed.

And, in this example, the Divisor 6 is also called a Measure of 30, because it exactly divides 30 into lots, leaving no remainder or part of a lot.

In the second Example,

33 is the Dividend.

Quotient.

6 is the Divisor. 5 is the

But, here, the Divisor 6 is not a Measure of the Divi

dend 33, because 6 does not divide 33 into an exact number of lots, but into five lots and a half of a lot.

The Dividend is the Number to be divided.

The Divisor is the Number to be comprised in each of the lots into which the Dividend is to be distributed.

The Quotient is the Number of lots into which the Dividend is distributed.

A Measure is any exact Divisor of the Dividend.

THE FIRST RESULT OF DIVISION.

"Divide forty by five," means, as already explained, "Distribute_forty units into groups of five units each." This being done, there will be eight such groups.

The first object attained by this Division is, then, the same as that of Continued Subtraction; namely, To find how many fives there are in forty.

And, it is now shewn that there are three ways of arriving at the answer to the question, "How many fives in forty?"

First. By Continually Subtracting five, and counting the subtractions.

Second. By Distributing into lots of five each, and counting the lots.

Third. By Subtracting at once the highest possible Multiple of five.

The second method, which is Pure Division, can only be employed experimentally, that is to say, when we have the real units actually before us, and can effect their distribution by hand.

The first method, Continued Subtraction, is adapted either for experiment or calculation.

The third method, Multiplication and Subtraction combined, is the best for calculation. It is, therefore, the one always used.

THE DIVISION TABLE.

In order to construct the Division Table, it is only necessary to alter the arrangement of the Multiplication Table, so that each line shall shew, not the same multiple of each of the first twelve numbers, but the first twelve multiples of the same number.

For example, the third line of the Multiplication Table shews the third multiples of 1 of 2 of 3 &c. up to 12; but the third line of the Division Table shews the first twelve multiples of three.

To exhibit more clearly this difference of arrangement, we print, below, the fourth line of each Table, side by side.

Write out and learn by heart the whole Division Table.

SIMPLE EXAMPLES IN DIVISION OF INTEGERS, TO BE WORKED OUT EITHER BY EXPERIMENT OR MENTALLY. ABSTRACT NUMBERS.

1. How many twos in 24, 28, 12, 32, 66, 42, 100', 88.?

2. Into how many lots of four each can you distribute 16, 32°, 96, 48, 64, 32, 80, 200, 28, 36′, 44′, 88′, 76° ?

3. What are the Quotients of 3311, 427, 63·9· =? 4. What are the Quotients of 84 12, 663, 91.7. =? 5. How many nines can be obtained from 27', 81', 36, 18', 63.? 6. Name four Measures of 12, six of 24, and seven of 36? 7. How many of the following Numbers is 7 a Measure of: 35', 42, 24, 53', 29′, 92, 49', 94°, 14°, 41•?

CONCRETE OR APPLICATE NUMBERS.

8. How many crowns in 45's? How many in 39's?

9. How many sixpenny loaves will 90 pence pay for?

10. A windlass took up, at each revolution, 5 feet of a well-rope. How many times must it turn to raise a bucket from the bottom to the top of a pit 140 feet deep?

11. How many paces of 3 feet must one make in walking a distance of 117 feet?

12. How many lb. of beef at 4 pence per lb. could I pay for with half-a-crown, so as to receive two pence in change?

13. Out of a bag of apples containing eleven dozen, how many boys may receive 5 each, supposing 7 to be assigned to the distributor?

14. How many gallons of water will weigh 1 cwt?

15. From a piece of whip-cord measuring 45′ feet, how many topstrings, each 4-5 feet long, can be cut?

16. Out of a crown I received ten-pence change, after paying for

K

the same number of three-penny balls and two-penny tops. How many of each did I purchase?

17 Supposing a loiterer to arrive at school, daily, 3 minutes too late in the forenoon, and the same in the afternoon, in how many days will he have lost one hour of his school-time? 18. At 9 pence per day, how long will a man be earning 6 shillings? 19. For how many weeks must one lay by 5 pence a week, in order to accumulate 10 shillings?

20 Tea is weighed by the Avoirdupois lb. of 16 ounces.

What

would 1 ounce cost at 4's. per lb.? What at 1's. per lb.? 21. How many lb. of tea would weigh 64 ounces?

22. How many ounces of tea at three farthings per ounce can be be bought for sixpence?

23. In moving over a distance of 300 feet, a coach-wheel made 40 revolutions. What was the circumference of the wheel? miles in 5 hours, how long shall I be in

24. If I can walk 20

in walking 8 miles at the same rate?

25 Exeter being 40 miles from Tavistock, how many miles per hour must one travel to make the journey between the two places in 300 minutes?

SECOND RESULT OF DIVISION.

To express one Number "in terms of" another, simply means to state what Multiple or Fraction the first Number is of the second.

Division enables us to do this. For, in order to express 90 in terms of a score, we have only to state the number of scores into which 90 can be distributed. And this number is the Quotient of 9020; that is to say, 41.

...

then, 90 contains 4 twenties and of twenty;

90, expressed in terms of a score, is 4 (scores).

In the same way:

The number of crowns in a guinea = 215· — 41.
For, a guinea = 21's.; and a crown = 5's.

.. One guinea "in terms of a crown" is 43 (crowns).

The second result of Division, then, is, that the Quotient expresses the Dividend in terms of the Divisor.

Now, because in the expression 3284, the Quotient 4represents 4 groups each containing 8. such units as the Dividend is made up of, it is evident that the Quotient 4 has the same absolute value as the Dividend.

This is the case with every Quotient, and arises from the nature of Pure Division, which neither increases nor diminishes the Dividend, but merely rearranges it.*

*NOTE TO TEACHER. This, the essential characteristic of Pure experimental Division, is too frequently lost sight of, and the consequence is utter confusion of identity with coincidence, the very broad and important distinction between which, we, by starting with the true idea of Division, are enabled to maintain with perfect precision. See page 103.