V. There are four places on the same line in the order A, B, C, and D. From A to D it is 1463 miles; from A to C it is 728 miles; and from B to D it is 1317 miles. How far is it from A to B, from B to C, and from C to D?--_Answer._ From A to B 146, from B to C 582, and from C to D 735 miles. VI. In the following table subtract B from A, and B from the remainder, and so on until B can be no longer subtracted. Find how many times B can be subtracted from A, and what is the last remainder. No. of A B times. Remainder. 23604 9999 2 3606 209961 37173 5 24096 74712 6792 11 0 4802469 654321 7 222222 18849747 3141592 6 195 987654321 123456789 8 9 SECTION III. MULTIPLICATION. 47. I have said that all questions in arithmetic require nothing but addition and subtraction. I do not mean by this that no rule should ever be used except those given in the last section, but that all other rules only shew shorter ways of finding what might be found, if we pleased, by the methods there deduced. Even the last two rules themselves are only short and convenient ways of doing what may be done with a number of pebbles or counters. 48. I want to know the sum of five seventeens, or I ask the following question: There are five heaps of pebbles, and seventeen pebbles in each heap; how many are there in all? Write five seventeens in a column, and make the addition, which gives 85. In this case 85 is called the _product_ of 5 and 17, and the process of finding the product is called MULTIPLICATION, which gives nothing more than the addition of a number of the same quantities. Here 17 is called the _multiplicand_, and 5 is called the _multiplier_. 17 17 17 17 17 ---- 85 49. If no question harder than this were ever proposed, there would be no occasion for a shorter way than the one here followed. But if there were 1367 heaps of pebbles, and 429 in each heap, the whole number is then 1367 times 429, or 429 multiplied by 1367. I should have to write 429 1367 times, and then to make an addition of enormous length. To avoid this, a shorter rule is necessary, which I now proceed to explain. 50. The student must first make himself acquainted with the products of all numbers as far as 10 times 10 by means of the following table,[8] which must be committed to memory.