ON MEASUREMENT. 161. Ratio is the relation with respect to magnitude which one quantity bears to another of the same kind, and is expressed by writing the first quantity as the numerator and the second as the denominator of a fraction. a Thus the ratio of a to b is it is also expressed a : b. b' The numerical value of a ratio is the quotient obtained by dividing the numerator by the denominator. 162. To measure a quantity is to find its ratio to another quantity of the same kind called the unit of measure. 163. The number which expresses how many times a quantity contains the unit, prefixed to the name of the unit, is called the numerical measure of that quantity; as 5 yards, etc. 164. Two quantities are commensurable when they have a common measure; that is, when there is some third quantity of the same kind which is contained an exact number of times in each. Thus if EF is contained in AB 3 times and in CD 2 times, then AB and CD are commensurable, and EF is a common measure. 165. Two quantities are incommensurable when they have no common measure. The ratio of such quantities is called an incommensurable ratio. This ratio cannot be exactly expressed in figures; but its numerical value can be obtained approximately as near as we please, Thus, suppose G and H are two lines whose ratio is √2. We cannot find any fraction which is exactly equal to √2 but by taking a sufficient number of decimals we may find √2 to any required degree of approximation. Thus and therefore √2=1.4142135..., √2> 1.414213 and 1.414214. 1000000 That is, the ratio of G to H lies between 1414213 and 1414214 and therefore differs from either of these ratios by less than one-millionth. And since the decimals may be continued without end in extracting the square root of 2, it is evident that this ratio can be expressed as a fraction with an error less than any assignable quantity. 166. And in general, if the approximate numerical value of the ratio of two incommensurable quantities is desired within n let the second quantity be divided into n equal parts, and suppose that one of these parts is contained between m and m + 1 times in the first quantity. Then the numerical value of the ratio of the first quantity m+1 to the second is between and 1; that is, the approxi m n mate numerical value of the ratio is correct within 9 n 1 1 n And since n can be taken as great as we please, is made n correspondingly small, or until it becomes less than any assignable value, though it can never reach zero, or absolute nothing. THE METHOD OF LIMITS. 167. A Variable Quantity, or simply a Variable, is a quantity, which under the conditions imposed upon it, may assume an indefinite number of values. 168. A Constant is a quantity which remains unchanged throughout the same discussion. 169. The Limit of a variable is a constant quantity which the variable may approach indefinitely near, but never reach. A M M' M" B 170. Suppose a point to move from A toward B, under the conditions that the first second it shall move one-half the distance from A to B; that is, to M; the next second, one-half the remaining distance; that is, to M'; the next second, one-half the remaining distance; that is, to M"; and so on indefinitely. Then it is evident that the moving point may approach as near to B as we please, but will never arrive at B; that is, the distance AB is the limit of the space passed over by the point. 171. THEOREM. If two variables are always equal and each approaches a limit, then the two limits are equal. Let AM and A'M' be two equal variables which approach indefinitely the limits AB and A'B' respectively. To prove that AB= A'B'. If possible, suppose AB > A'B', and lay off AC = A'B'. Then the variable AM may assume values between AC and AB, while the variable A'M' is restricted (by 169) to values less than AC; which is contrary to the hypothesis that the variables should always be equal. Hence AB cannot be > A'B', and in like manner it may be proved that AB cannot be < A'B'; therefore AB = A'B'. 172. Cor. If two variables are in a constant ratio, their limits are in the same ratio. To prove that their limits have the same ratio. Now let a approach the limit x', and y the limit y'. Then since the variables x and my are always equal (by 171), their limits are equal; that is, x': = my'. Therefore = m. y' MEASUREMENT OF ANGLES. PROPOSITION VIII. THEOREM. 173. In the same circle, or in equal circles, angles at the centre are in the same ratio as their intercepted arcs. CASE I. When the arcs are commensurable. In the circle O, let AOB and BOC be two angles at the centre intercepting the commensurable arcs AB and BC. B C D A Let AD be the common measure of the arcs AB and BC, and by applying it to the arcs it is found that AB contains it 3 times, and BC 4 times. If radii be drawn from the several points of division, they will divide the angle AOB into 3 parts which (by 145) are equal, and BOC into 4 parts which are equal. CASE II. When the arcs are incommensurable. If the arcs AB and BC are incommensur B C"C' C COC" arc CC Now CB is the limit of the arc, and COB is the limit of the angle, therefore since the ratio of the angles is equal to the ratio of the arcs at different stages of their variation, they will have (by 171) the same ratio when they reach their limits; that is, 174. SCHOLIUM. Since the angle at the centre of a circle and its intercepted arc increase and decrease in the same ratio, it is said that an angle at the centre is measured by its intercepted arc. PROPOSITION IX. THEOREM. 175. An inscribed angle is measured by one-half the arc intercepted between its sides. In the circle 0, let BAC be an inscribed angle. Since OB and OA are radii, the triangle OBA (by 68) is isosceles, and (by 93) B=L BAO. |