_ab_ _b_ ------ = _a_ × ---- (88) _c_ _c_ _ac_ _a_ ----- = ------ (89) _b_ _b_ ---- _c_ It must be recollected, however, that these have only been proved in the case where all the divisions are without remainder. 91. When one number divides another without leaving any remainder, or is contained an exact number of times in it, it is said to be a _measure_ of that number, or to _measure_ it. Thus, 4 is a measure of 136, or measures 136; but it does not measure 137. The reason for using the word measure is this: Suppose you have a rod 4 feet long, with nothing marked upon it, with which you want to measure some length; for example, the length of a street. If that street should happen to be 136 feet in length, you will be able to _measure_ it with the rod, because, since 136 contains 4 34 times, you will find that the street is exactly 34 times the length of the rod. But if the street should happen to be 137 feet long, you cannot measure it with the rod; for when you have measured 34 of the rods, you will find a remainder, whose length you cannot tell without some shorter measure. Hence 4 is said to measure 136, but not to measure 137. A measure, then, is a divisor which leaves no remainder. 92. When one number is a measure of two others, it is called a _common measure_ of the two. Thus, 15 is a common measure of 180 and 75. Two numbers may have several common measures. For example, 360 and 168 have the common measures 2, 3, 4, 6, 24, and several others. Now, this question maybe asked: Of all the common measures of 360 and 168, which is the greatest? The answer to this question is derived from a rule of arithmetic, called the rule for finding the GREATEST COMMON MEASURE, which we proceed to consider. 93. If one quantity measure two others, it measures their sum and difference. Thus, 7 measures 21 and 56. It therefore measures 56 + 21 and 56-21, or 77 and 35. This is only another way of saying what was said in (74). 94. If one number measure a second, it measures every number which the second measures. Thus, 5 measures 15, and 15 measures 30, 45, 60, 75, &c.; all which numbers are measured by 5. It is plain that if 15 contains 5 3 times, 30, or 15 + 15 contains 5 3 + 3 times, or 6 times, 45, or 15 + 15 + 15 contains 5 3 + 3 + 3 or 9 times; and so on. 95. Every number which measures both the dividend and divisor measures the remainder also. To shew this, divide 360 by 112. The quotient is 3, and the remainder 24, that is (72) 360 is three times 112 and 24, or 360 = 112 × 3 + 24. From this it follows, that 24 is the difference between 360 and 3 times 112, or 24 = 360-112 × 3. Take any number which measures both 360 and 112; for example, 4. Then 4 measures 360, 4 measures 112, and therefore (94) measures 112 × 3, or 112 + 112 + 112.