We have given in the first part different ways for making the proof of multiplication, but there is another very short, which I will give here, though not infallible, it is called the proof of multiplication by the cross, or the proof of multiplication by 9.

Make a cross thus X, then add all the figures in the multiplicand together, as in Addition, and cast away the nines, as often as they arise, and bear the remainder to the next figure; when you come to the end of the line, note what remains after the nines are cast away, and set such remainder on the left side of the cross,

Then do the same by the multiplier, and note what remains there also, setting that on the right of the

The proof may be found right, and the fum be wrong, when, in the product, a cypher is put for a nine, or when the places of the figures are changed.

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cross; multiply these two figures together, and cast the nines out of the product, setting the remainder on the top of the cross.

Lastly, cast away the nines out of the product, and if the remainder be like the figure on the top of the cross, you may hope that the work is right.

I will give one example, to make the foregoing directions the more intelligible.

Here, after having made the cross, I begin at the multiplicand, saying, 5 and 6 is 11, I cast away 9, and there remains 2; now 2 and 7 is 9, and there rests nothing; then 1 and 5 is 6 (I skip the 9) and 2 is 8, which I set on the left hand of the cross, as it is

seen.

Going to the multiplier, I say, 1 and 2 is 3 and 3 is 6, which I place on the right of the cross.

Then

I multiply 8 by 6, and of the product 48, casting away the nines there remains 3, which I write on the top of the cross.

After the same manner, I cast away the nines out of the product, and at the last there remains 3 likewise, and so the proof is done.

CONTRACTIONS.

1st. When the multiplier is 10 or 100, or 1000, &c. The rule has been given in the 1st part (page 17.)

2nd. When there are cyphers at the right hand, in both the multiplicand and multiplier.

These two cases have been explained in the first part, and are not in need of examples.

3d. When the multiplier is 11 or 12.

Multiply at once each figure of the multiplicand by the two figures of the multiplier, so that the product may be in one line only.

4th. When the multiplier is 5.

Annex a cypher to the multiplicand, then halve it because 5 is the of 10.

When the multiplier exceeds 12, and is less than 20. Write under the multiplicand, its product by the unit figure of the multiplier, and set the first figure of this product one place forward to the right hand.

Or this way, that the product may be in one line only:

Multiply by the unit figure, and add to the product of each figure, that which is next on the right hand.

QUESTIONS.

First. The sum of two numbers is 360; the less is 114. What is the product of these two numbers? Ans. 28044.

Second. There are two numbers; the greater of them is 73 times 109, and their difference 17 times 28. We demand the product of these two numbers? Ans. 59526317.

Third. I bought 10 bales of cloth; in each bale 59 pieces; in each piece 25 yards; and the price of each yard is 300 cents. What sum must I pay Ans. 4425000 cents.

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Fourth. A Merchant bought 11 bales of cloth; in each bale 1475 yards; for each yard he paid 120 eents; this merchant sold 13928 yards at 135 cents each. How much do his receipts fall short of his expenditure? Ans. 66720 cents.

Fifth. A Vintner bought 5 hogsheads of brandy, for the sum of 157 dollars; he sold the quart 18 cents; it is known, that one hogshead contains 252 quarts, and that one dollar is worth 100 cents. What was the gain of the Vintner?

cents.

Ans. 6980