CHAPTER III. THE MEASUREMENT OF STRAIGHT LINE SEGMENTS. 232. A straight line-segment is said to be exactly measured when we find how many times it contains a certain other segment which is taken as a unit. The number thus found is called the numerical measure, or the length of the segment. E.g. a line-segment is 9 in. long if a segment 1 in. long can be laid off on it 9 times in succession. Thus, 9 is the numerical measure, or the length of the segment, when 1 in. is taken as a unit. 233. In selecting a unit of measure it may happen that it is not contained an integral number of times in the segment to be measured. Thus, in measuring a line-segment the meter is often a convenient unit. Suppose it has been applied five times to the segment AB and that the last time the end falls on A1, AB being less than one meter. Then, taking a decimeter (one tenth of a meter) as a new unit, suppose this is contained three times in A1B with a remainder A,B less than a decimeter. Finally, using as a unit a centimeter (one tenth of a decimeter), suppose this is contained exactly six times in A,B. Then, the length of AB is 5 meters, 3 decimeters, and 6 centimeters, or 5.36 meters. The process of measuring considered here is ideal. In practice we cannot say that a given segment is contained exactly an integral number of times in another segment. See § 235. 234. It may also happen that, in continuing this ideal process of measuring as just described, no subdivided unit can be found which exactly measures the last interval, that is, such that the final division point falls exactly on B. E.g. it is known that in a square whose sides are each one unit the diagonal is √2, and that this cannot be exactly expressed as an integer or a fraction whose numerator and denomi nator are both integers. By the ordinary process of extracting square root we find √2 = 1.4142 each added decimal mak ing a nearer approximation. But this process never terminates. V2-1.4142 1 Hence, in attempting to measure the diagonal of a square whose side is one meter, we find 1 meter, 4 decimeters, 1 centimeter, 4 millimeters, etc., or 1.414 meters approximately. It should be noticed, however, that 1.415 is greater than the diagonal and hence the approximation given is correct within one millimeter. 235. Evidently any line-segment can be measured either exactly or to a degree of approximation, depending upon the fineness of the instruments and the skill of the operator. The word measure is commonly used to include both exact and approximate measurement. For practical purposes, a line-segment is measured as soon as the last remainder is smaller than the smallest unit available. It should be noticed that all practical measurements are in reality only approximations, since it is quite impossible to say that a given distance is, for instance, exactly 25 ft. It may be a fraction of an inch more or less. E.g. in the above example 1.414 meters gives the length of the diagonal for practical purposes if the millimeter is the smallest unit available. The error in this case is less than one millimeter. 236. Definition. Two straight line-segments are commensurable if they have a common unit of measure. Otherwise they are incommensurable. E.g. two line-segments whose lengths are exactly 5.27 and 3.42 meters respectively have one centimeter as a common unit of measure, it being contained 527 times in the first segment and 342 times in the second. But the side and the diagonal of a square have no common unit of measure. In the example of § 234, the millimeter is contained 1000 times in the side and 1414 times in the diagonal, plus a remainder less than one millimeter. A similar statement holds for any unit of measure, however small. 237. For the purposes of practical measurement any two line-segments may be considered as commensurable, but for theoretical purposes it is necessary to take account of incommensurable segments also. The theorems in this chapter are here proved for commensurable segments only. They are proved for incommensurable segments also in Chapter VII. RATIOS OF LINE-SEGMENTS. 238. The ratio of two commensurable line-segments is the quotient of their numerical measures taken with respect to the same unit. E.g. if two segments are respectively 3 ft. and 4 ft. in length, the ratio of the first segment to the second is and the ratio of the second to the first is 239. The ratio of two commensurable segments is the same, no matter what common unit of measure is used. E.g. two segments whose numerical measures are 3 and 4 if one foot is the common unit, have 36 and 48 as their numerical measures if one inch is the common unit. But the ratio is the same in both cases, namely: ;; = 1. 240. The approximate ratio of two incommensurable linesegments is the quotient of their approximate numerical measures. It will be seen that this approximate ratio depends upon the length of the smallest measuring unit available, and that the approximation can be made as close as we please by taking the measuring unit small enough. is E.g. an approximate ratio of the side of a square to its diagonal 1 100 1 1000 Another and closer approximation is 1.41 141 1.414 1414 = In this case the numerical measure of one of the segments is exact. An approximate ratio of √2 to √3, in which neither has an exact 241. It should be clearly understood that the numerical measure of a line-segment is a number, as is also the ratio of two such segments. Hence they are subject to the same laws of operation as other arithmetic numbers. For example, the following are axioms pertaining to such numbers: (1) Numbers which are equal to the same number are equal to each other. (2) If equal numbers are added to or subtracted from equal numbers, the results are equal numbers. (3) If equal numbers are multiplied by or divided by equal numbers, the results are equal numbers. It is understood, however, that all the numbers here considered are positive. For a more complete consideration of axioms pertaining to numbers, see Chapter I of the Advanced Course of the authors' High School Algebra. 242. A proportion is an equality, each member of which is a ratio. Four numbers, a, b, c, and d, are said to be in proportion, in the order given, if the ratios and are b d α с equal. In this case a and c are called the antecedents and b and d the consequents. Also a and d are called the extremes and b and c the means. The proportion α с b d is sometimes written a : b = c : d, and in either case may be read a is to b as c is to d. If D and E are points on the sides of the triangle ABC, and if m, n, p, and q, the numerical measures respectively of AD, DB, AE, and EC, are such that m n q m D E n then the points D and E are said to divide the sides AB and AC proportionally, that is, in the same ratio. For convenience it is common to let AD, DB, AE, and EC stand for the numerical measures of these segments, and thus to B write the above proportion, AD AE = or AD: DB=AE: EC. THEOREMS ON PROPORTIONAL SEGMENTS. 243. THEOREM. If a line is parallel to one side of a triangle and cuts the other two sides, then it divides these Proof: Choose some common measure of AD and DB, as AK. Suppose it is contained 3 times in AD and 5 times in DB. |