499. In like manner it may be shown that if a bbc cd, then a da3: b3. NOTE. Here the four quantities are said to be in continued proportion (§481). 500. It will be evident to the student from the preceding articles that, if four quantities are proportional, many other theorems than those given may be derived. Thus, for example, a: b = c : d, if pa+gb pc + qd' ma + nb: pa + ql = me+nd: pc + qd. 501. It has been assumed in our definition of a proportion that one quantity is a definite multiple of another, or, what is equivaleut to the same thing, that the fraction formed by making one of the numbers the numerator, and the other the denominator, is a determinate fraction. This will be the case when the numbers have a common measure. Let the common measure of a and b be x; then a = mx, and b = nx, and where m and n are integers. a mx m nx n 502. Incommensurable Numbers.-But it sometimes happens that the two quantities do not have a common unit of measure; that is, both can not be expressed as integers in terms of a common unit. They are then said to be incommensurable. For example, the ratio of the diagonal of a square to its side is the irrational number 1 2. A b C Now 21.41421356 which is greater than 1.414213 and less than 1.414214. If a millionth part of b is taken as the unit of length, then the value of the ratio is 1 therefore differs from either of these fractions by less than b 1000000 Similarly, if the decimal is carried to the nth place, the corresponding fraction will differ from the true value of the ratio by less than 1 [2341] and this fraction can be made as small as one chooses, if n is taken as large as may be desired, i. e., by carrying the decimal as far as may be desired. Hence, in case two quantities are incommensurable, there is no fraction which will exactly express the value of the ratio of the given quantities; but it is possible, by taking the unit of measure small enough, to find a fraction that will differ from the true value of the ratio by as small a quantity as is desired. 503. THEOREM XIV.-In case a and b are incommensurable quantities, a fraction can be found which will differ from the true value of the ratio by as small a quantity as is desired. Let b and suppose that if x diminishes, then n increases (b being constant) and, therefore, n 1 1 diminishes. Hence if x is made as small as may be desired, n can be made as large as may be desired, and therefore can be made less than any assigned fraction. Therefore the difference n can be made less than any assigned fraction (2341). 504. THEOREM XV.-If c and d, as well as a and b, are incommensurable; and if, when m a m + - " n If and are not equal, their difference. must be some assignable b d n and this difference must be less than n as large as is desired, can be made less than any assigned fraction, however small; therefore the difference between and can be made as small as is desired, which can only be true if Hence all the propositions respecting proportionals are true of the four quantities a, b, c, d. 505. The property involved in Euclid's definition follows from the algebraic definition. Euclid's definition of a proportion is: "The first of four magnitudes is said to have the same ratio to the second that the third has to the fourth, when any equimultiples whatever of the first and the third being taken, and any equimultiples whatever of the second and the fourth, if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth, and if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth, and if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth." (Euclid, Book V.) Hence pc is greater than, equal to, or less than qd, according as pa is greater than, equal to, or less than qb. 506. Conversely, the property involved in the algebraic definition follows from Euclid's. Let a, b, c, d, be four quantities which are proportional according to Euclid's definition; prove that. For, if is not equal to then one of them must be greater than the other. Suppose that b then it must be possible to choose some fraction, 2, which lies between them. Then is greater than and is greater than b Then qa>pb, and qc <pd, hence, a, b, c, d are not proportionals according to Euclid's definition; which is contrary to the supposition. Therefore and can not be unequal. d 507. Euclid's definition of the ratio and proportion is the preferable one. Straight lines can be represented geometrically, but the abstract number which expresses how often one straight line is contained in another, can not be represented geometrically. Hence the common algebraic definition of proportion can not be used in Geometry. The algebraic definition is, strictly speaking, applicable to commensurable quantities only; but it should be noticed that Euclid's definition is applicable to incommensurable quantities as well. This consideration alone is sufficient reason for the definition which is given in Euclid. 508. EXAMPLES.-1. Solve the equation, 5x-3a: 5x + 3 a = 7 a−5: 13 a — 5. 2. If x: y = [2492] (x — z)3: (y —-z); prove that z is a mean propor yx2 - xy2 = xz2 — yz2, xy (x − y) = (x − y) z2, xyz2. z is a mean proportional between x and y (2483). 3. If prove [1482] that aac+c2 : a2 — ac + c2 = b2 + bd + d2 : b2 — bd + ď2. By Theorems XIII and IX, --- EXERCISE LXXXI Find the ratio compounded of Ans. 1: 2. 1. The ratio 32: 27 and the triplicate ratio of 3: 4. 3. The triplicate ratio of x:y and the ratio 2y2:3x2. 4. Find a fourth proportional to x3, xy, 5 x2y2. Ans. 1:5. Ans. 2x: 3 y. Ans. 5 y3. 5. Find a mean proportional between 4 ax2 and 16 a3. Ans. 8 a2x. 6. Find a third proportional to 6.3 and 5 x2. 7. Find a mean proportional between 25 Ans. x. 6 13. 2a5b: 4a-3b2c+5d: 4c-3 d. 14. If a, b, and c are proportional, and a the greatest, show that 16. 3x-2a: 3x+2a5a-3: 15 a +5. 17. 3 x 16 x 77 x 109 x + 10. 2 = 18. y 16 y 25 y 2 y 24 y2-3y-10. 19. 2-√1-x: 3 + VI − x = Va-Va−b: Va+Va−b. x +4 y when 21. Find x and (3x-5y: 5x+3y=-16: 15 xy = 33.58. 22. Find x when x2-2x+3x2 -3x+5=2x-3: 3x-5. 23. Find xy, if given x2 + 6 y2 = 5 xy. Ans. 2 or 3. |
