The general rule therefore for multiplying any number of figures may be expressed thus, Maltiply each figure of the multiplicand by each figure of the multiplier separately, taking care when multiplying by units to make the first figure of the result stand in the unit's place; and when multiplying by tens, to make the first figure stand in the tens' place; and when multiplying by hundreds, to make the first figure stand in the hundreds place, &c. and then add the several products together. Note. It is generally the best way to set the first figure of each partial product directly under the figure by which you are multiplying. Proof. The proper proof of multiplication is by division, consequently it cannot be explained here. There is also a method of proof by casting out the nines, as it is called. But the nature of this cannot be understood, until the pupil is acquainted with division. It will be explained in its proper place. The instructer, if he chooses, may explain the use of it here. Subtraction. VIII. A man having ten dollars, paid away three of them; how many had he left ? We have seen that all numbers are formed by the successive addition of units, and that they may also be formed by adding together two or more numbers smaller than themselves, but all together containing the same nuinber of units as the number to be formed. The number 10, for example, may be formed by adding 3 to 7,7 + 3 = 10. It is easy to see therefore that any number may be decomposed into two or more numbers, which taken together, shall be equal to that number. Since 7 + 3 10, it is evident that if 3 be taken from 10, there will remain 7. The following examples, though apparently different, all require the same operation, as will be immediately perceived. A man having 10 sheep sold 3 of them; how many had he left ? That is, if 3 be taken from 10, what number will remain ? A man gave 3 dollars to one son, and 10 to another ; how much more did he give to the one than to the other? That is, how much greater is the number 10 than the number :3 ? A man owing 10 dollars, paid 3 dollars at one time, and the rest at another ; how much did he pay the last time? That is, how much must be added to 3 to make 10? From Boston to Dedham it is 10 miles, and from Boston to Roxbury it is only 3 miles ; what is the difference in the tuo distances from Boston ? A boy divided 10 apples between two other boys ; to one he gave 3, how many did he give to the other? s'hat is, if io be divided into two parts so that one of the parts may be 3, what will the other part be? It is evideot that the above five questions are all answered by taking 3 from 10, and finding the difference. This operation is called subtraction. It is the reverse of addition. Addition puts numbers together, subtraction separates a number into two parts. i A man paid 29 dollars for a coat and 7 dollars for a hat, how much more did he pay for his coat than for his hat? In this example we have to take the 7 from the 29; we know from addition, that 7 and 2 are 9, and consequently that 22 and 7 are 29; it is evident therefore that if 7 be taken from 29 the remainder will be 22. A man bougkt an ox for 47 dollars ; to pay for it he gave a cow worth e3 dollars, and the rest in money; how much money did he pay? Operation. It will be best to perform this example by parts. It is plain that we must take the twenty from the forty, and the three from the seven; that is, the tens from the tens, and the units from the units. I take twenty from forty and there remains twediy. I then take three from seven and there remains four, and the whole remainder is twenty four. Ans. 24 dollars. It is generally most convenient to write the numbers under each other. The smaller number is usually written under the larger. Since units are to be taken from units, and tens from tens, it will be best to write units under units, tens under tens, &c. as in addition. It is also most convenient, and, in fact, frequently necessary, to begin with the units as in addition and multiplication. Operation. Ox 47 dollars I say first, 3 from 7, and there Cow 23 dollars will remain 4. Then 2 (tens) from 4 (tens) and there will re24 difference. main 2 (tens), and the whole remainder is 24. A man having 62 sheep in his flock, sold 17 of them; how many had he then ? Operation. He had 62 sheep In this example a difficulty imSold 17 sheep mediately presents itself, if we at tempt to perform the operation Had left 45 sheep as before; for we cannot take 7 from 2. We can, however, take 7 from 62, and there remains 55; and 10 from 55, and there remains 45, which is the answer. The same operation may be performed in another way, which is generally more convenieni. I first observe, that 62 is the same as 50 and 12; and 17 is the same as 10 and 7. They may be written thus : 62 = 50 + 12 That is, I take one ten from the 17 = 10 + 7 six tens, and write it with the two units. But the 17 I separate simply 45 = 40 + 5 into units and tens as they stand. Now I can take 7 from 12, and there remains 5. Then 10 from 50, and there remains 40, and these put together make 45.* * Let the pupil perform a large number of examples by separating tbe:n this way, when he first commences subtractiun. This separation may be made in the mind as well as to write it down. Operation. 62 Here I suppose 1 ten taken from the 6 tens, 17 and written with the 2, which makes 12. I say 7 from 12, 5 remains, then setting down 45 the 5, I say, 1 ten from 5 tens, or simply 1 from 5, and there remains 4 (tens), which written down shows the remainder, 45. The taking of the ten out of 6 tens and joining it with the 2 units, is called borrowing ten. Sir Isaac Newton was born in the year 1612, and he died in 1727; how old was he at the time of his decease? It is evident that the difference between these two numbers must give his age. Operation. 1600 + 40 + 2 = 1642 80 + 5 = 85 years old. In this example I take 2 from 7 and there remains 5, which I write down. But since I cannot take 4 (tens) from 2 (tens), 1 borrow 1 (hundred) or 10 teos from the 7 (hundreds), which joined with 2 (tens) makes 12 (tens), then 4 (tens) from 12_(tens) there remains 8 (tens), which I write down. Then 6 (hundreds) from Ô (hundreds) there remains nothing. Also I (thousand) from 1 (thousand) nothing remains. The answer is 85 years. A man bought a quantity of flour for 15,265 dollars, and sold it again for 23,007 dollars, how much did he gain by the bargain ? Operation. 23,007 Here I take 5 from 7 and there re15,265 mains 2 ; but it is impossible to take 6 (tens) from 0, and it does not immedia 2 ately appear where I shall borrow the 10 (tens), since there is nothing in the hundreds' place. This will be evident, however, if I decompose the num bers into parts. Operation. 7,000 + 700 + 40 + 2 = 7,742 The 23,000 is equal to 10,000 and 13,000; tbis last is equal to 12,000 and 1,000; and 1,000 is equal to 900 and 100. Now I take 5 from 7, and there remains 2; 60 from 100, or 6 tens from 10 tens, and there remains 40, or 4 tens; 2 hundreds from 9 hundreds, and there remains 7 hundreds ; 5 thousands from 12 thousands, and there remains 7 thousands; and 1 ten-thousand from 1 ten-thousand, and nothing remains. The answer is 7,742 dollars. This example may be performed in the same manner as the others, without separating it into parts except in the mind. I say 5 from 7, there remains 2; then borrowing 10 (which must in fact come from the 3 (thousand) I say, 6 (tens) from 10 (tens) there remains 4 (tens); then I borrow ten again, but since I have already used one of these, I say, 2 (hundreds) from 9 (hundreds) there remains 7 (hundreds); then I borrow ten again, and having borrowed one out of the 3 (thousand), I say, 5 (thousand) from 12 (thousand) there remains 7 (thousand); then i (ten-thousand) from 1 (ten-thousand) nothing remains. The answer is 7,742 as before. The general rule for subtraction may be expressed thus ; The less number is always to be subtracted from the larger. Begin at the right hand and take successively each figure of the lesser number from the corresponding figure of the larger number, that is, units from units, tens from tens, &c. If it happens that any figure of the lesser number cannot be taken from the corresponding figure of the larger, borrow ten and join it with the figure from which the subtraction is to be made and then subtract ; before the next figure is subtracted take care to diminish by one the figure from which the subtraction is to be made. N. B. When two or more zeros intervene in the number from which the subtraction is to be made, all, except |