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It will at once suggest itself to the mind, that to withdraw one number from another, the number we withdraw must be less than that from which it is withdrawn; therefore, in this rule we have three terms, the greater, the lesser, and the remainder.*

There are two numbers given, the greater number we write down first, and under it we write the lesser; under the lesser number draw a line, and below the line write the difference there is between the greater and the lesser numbers, which shews or yields the remainder. Example 1.-From 8 take 4. 8 greater

4 lesser

4 remainder.

Example 2.---From 270 take 191.

270 greater
191 lesser

79 remainder.

Here you perceive that although the bottom number is considerably the smallest, yet the figures in the first and second place are more than the corresponding figures in the larger number, and in subtracting them you will perceive the great simplicity and beauty of this rule. Beginning at the right, or units place, we say 1 from 0 we cannot; but here we lend ten to the top line and say, 1 from 10, nine remains; then you repay the ten you lent the top line, by adding it to the next figure of the bottom line, saying 1 and 9 make ten, 10 from 7 I cannot, but 10 from 17, seven remains,

* There are two terms used by some Authors : Minuend, to be diminished ; and Subtrahend, to be subtracted; but as greater and lesser are plainer terms, I prefer using them.

which I write under the 9, or in the second place; again repaying the ten I added to the top line, I say, carry

I to 1 making two, and 2 from 2 leaves nothing, which I do not write, for a cipher on the left of a series of figures expresses nothing.

Having in the last example shewn how a larger figure in the lesser or bottom sum, is subtracted from a smaller figure in the greater or top, viz., by adding ten, first to the top figure, and then to the next figure on the left in the bottom line, it now remains but to shew, that these additions of ten, when thus added to each line, do not alter the proportion between the lesser and the greater sums. In the first example we find the difference between 8 and 4 is 4, now let us add ten to each of these figures and they will stand thus- 18 Here you see although both numbers are in- 14 creased, yet the remainder remains the same.

I have now but to point out, in the following example, the reason that ten is added to the top line and apparently but one to the bottom.

I begin by saying 9 from 6 I cannot, but from 16, and 7 remains; I write the 7 in the place of units,

389 lesser and say 1 to 8 makes 9, from 9 and

107 remainder. nothing. (You perceive that 8 from its place represents 8 tens, and by adding 1 to it, it represents nine tens; and we do in fact add 10 to the bottom line.) Write a cipher under the 8, and say 3 from 4 and 1 remains, which we write under the 3. The remainder is one hundred and seven.

When reading these directions, you will work the examples carefully on your slate, after which do for practice the following sums.

Minus, the sign of Subtraction.

4

496 greater

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(10.) The Battle of the Boyne was fought in 1690 ; how many years is it from that time to the

year 1847 ? (11.) What is the difference between seventy-five hundred and forty-seven, and six thousand three hundred and one ?

(12.) Magna Charta was signed in 1915; how many years is it from that time to the present?

(13.) The Battle of Cressy was fought on the 26th August, 1346 ; how many years is it from that time to the 26th August 1850 ?

(14.) What is the difference between the diameters of the sun and earth, the sun's diameter being 893246 miles and the earth's 7912 miles ?

(15.) Charles II. was restored in the year 1660; how long is it from that time till the year 1848?

(16.) The Battle of Genappe was fought in the year 1815, the Battle of Bunker's Hill was fought 40 years previous to Genappe ; how many years is it since the Battle of Genappe, and in what year was the Battle of Bunker's Hill fought?

* Diameter, from the Greek dove letgov, a measure through.

(17.) The Spanish Armada was dispersed in 1588; how many years is it from that to the year 1847 ?

(18.) The Battle of Blenheim was fought in 1704, and the Battle of Roleia in 1808; how many years was Blenheim fought before Roleia?

(19.) The art of printing was discovered in the year 1449; how long is it from that time to the year 1847?

(20.) The Battle of Hastings was fought in 1066, the Prince of Orange landed in the year 1688; how many years were there between the Battle of Hastings and the landing of the Prince of Orange; and also how many years from the landing of the Prince of Orange to the year

1848?

MIXED QUESTIONS IN ADDITION AND SUBTRACTION.

EXAMPLES.

(1.) A man had to walk a journey of 467 miles; the first day he walked 30 miles, the second day 27 miles; the third day 40 miles, and the fourth day 36 miles ; how many miles did he walk, and how much farther had

he to go?

(2.) Received on 2nd April £17, paid on 3rd April £5, received on 4th April £70, paid on 5th April £79, received on 6th April £132, and paid on 7th £113; how much did I receive, how much did I pay, and what is the difference?

(3.) It was 166 years from the death of King Alfred to the Battle of Hastings, from the Battle of Hastings until the landing of the Prince of Orange it was 622 years, from the landing of the Prince of Orange to the birth of Queen Victoria was 131 years; Queen Victoria was born in 1819; how many years had King Alfred been dead before Queen Victoria was born, and in what year did he die?

ANSWERS TO SUBTRACTION.

EXAMPLES. (1.) 214.

(9.) 62109. (2.) 133.

(10.) 157. (3.) 504.

(11.) 1246. (4.) 16175.

(12.) 632. (5.) 898757.

(13.) 504. (6.) 273590.

(14.) 875334. (7.) 39666.

(15.) 188. (8.) 19929. (16.) 32 years since Genappe; in 1775, Bunker's

Hill fought. (17.) 259. (18.) 104. (19) 398. (20.) From the Battle of Hastings to the landing of the Prince of Orange 622 years.

Since the land. ing of the Prince of Orange 160 years.

ANSWERS TO MIXED QUESTIONS. (1.) Walked 133 miles ; had to walk 334 miles. (2.) Received £219. Paid £197. Difference £22. (3.) 919 years. Died in the year 900.

PROOF OF ADDITION AND SUBTRACTION.

There are few things in which man is more liable to error than in arithmetical calculations. A sudden thought passes over the mind, a momentary dulness, and you may set down a wrong figure, which if not discovered at once and rectified, would often cause much trouble and serious loss.

Such being the case, every sum after it is done ought to be carefully proved. The examples given for practice in this book being so very few, I trust the learner, as soon as he has made himself master of the mode of proof adapted to each rule, will turn to the first sum in that rule and prove every sum therein set down.

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