This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ...( 180). Still if CD is the unit of measure, and----= V2, we shall call V2 the numerical measure of AB. 183. It is evident from the above that the ratio of two magnitudes of the same kind, whether commensurable or incommensurable, is equal to the ratio of their numerical measures when referred to a common unit. THE ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ...( 180). Still if CD is the unit of measure, and----= V2, we shall call V2 the numerical measure of AB. 183. It is evident from the above that the ratio of two magnitudes of the same kind, whether commensurable or incommensurable, is equal to the ratio of their numerical measures when referred to a common unit. THE METHOD OF LIMITS 184. We define a variable as a quantity which, under the conditions imposed upon it, may have an indefinitely great number of different values. We define a constant as a quantity which remains unchanged throughout the same discussion. 185. A limit of a variable is a constant quantity, the difference between which and the variable may be made less than any assigned quantity, however small, In other words, a limit of a variable is a fixed quantity to which the variable approaches indefinitely near. 186. Suppose, for example, that a point moves from A towards B under the condition that it O D E B shall move, during successive equal inter-1 1 1--!--I vals of time, first from A to C, halfway between A and B; then to D, halfway between C and B; then to E, halfway between D and B; and so on indefinitely. In this case, the distance between the moving point and B can be made less than any assigned distance, however small. Then, the distance from A to the moving point is a variable which approaches the constant distance AB as a limit. Again, the distance from the moving point to B is a variable which approaches the limit 0. As another illustration, consider the series-"-2, 4 H IB"'.' where each term after the first is one-half the preceding. In this case, by taking terms enough, the last term may be made less than any assigned number, however small. Then, the last term of the series is a variable which approaches the limit 0...
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