1 1000000 therefore differs from either of these fractions by less than b 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 ; 10n [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. if x diminishes, then n increases (b being constant) and, therefore, 1 diminishes. Hence if x is made as small as may be desired, n can n n be made as large as may be desired, and therefore can be made less than any assigned fraction. Therefore the difference a m b n can be made less than any assigned fraction (341). 504. THEOREM XV.-If c and d, as well as a and b, are incommensurable; and if, when m m If and are not equal, their difference must be some assignable quantity, since each lies between and this difference must be less than 1. Now, since n may be made n a as large as is desired, — can be made less than any assigned frac tion, however small; therefore the difference between and can be made as small as is desired, which can only be true if d 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 pe 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 d then it must be possible to choose some fraction, 2, which lies between them. Then is greater than and is greater than . с 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. 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+3a= = 7a-5:13a - 5. tional between x and y. or or and z)2: (y —- z)2; prove that z is a mean propor - dividing by xy, .. = xy2 2 xyz +xz2, z is a mean proportional between x and y (3483). [1482] + c2= b2 + bd + d2: b2 — bd + d2. EXERCISE LXXXI Find the ratio compounded of 1. The ratio 32: 27 and the triplicate ratio of 3: 4. 3. The triplicate ratio of xy and the ratio 2y2:3x2. 4. Find a fourth proportional to x3, xy, 5 x2y2. Ans. 1: 2. 36. Ans. 5 y3. 5. Find a mean proportional between 4 ax2 and 16 a3. Ans. 8 a2x. 6. Find a third proportional to 6 x3 and 5 x2. 7. Find a mean proportional between 25 Ans. x. 13. 2a5b4a-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. 3x16x-7= 7 x 109 x 10. 18. y 16 y2 - 25 y2 - 2 y 24: y2 - 3y-10. : = 19. 2−√1-x: 3 + VI — x = √ a − Va−b: Va+Va — b. 1 22. Find when x-2x+3x2-3x+5=2x-3: 3x-5. 23. Find xy, if given x2 + 6 y2 = 5 xy. Ans. 2 or 3. 4 24. Find two numbers in the ratio 3: 4 (suggestion, 3 x and 4 x) of which their sum is to the sum of their squares as 7: 50. 25. Find two numbers in the ratio of 5: 4, such that their sum has to the difference of their squares the ratio of 1: 18. 26. Find two numbers such that if 7 is added to each they will be in the ratio of 4: 3; and if 11 is added to the greater and subtracted from the smaller the results will be in the ratio of 5: 2. 27. If 7 x 4 z: 8x-3z4y-7z:3y8z, prove that z is a mean proportional between x and y. 29. If mx + ny : px + qy = my + nz : py + qz, show that n: q = m : p. 30. If 2a+3b: 3 a 4b2c3d: 3 c abc: d. 4d, prove that 31. If 2 men working 9 hours a day can do a piece of work in 32 days, in how many days can x men working y hours a day do the work? If a b c d, prove that: 32. aa + c = a + b: a+b+c+d. 33. a2c + ac2 : b3d + bd2 = (a + c)3 : (b + d)3. C3 35. (a+b+c+d) (a−b−c+d) = (a−b+c−d) (a+b—c—d). 36. Show that, when four quantities of the same kind are proportional, the sum of the greatest and the least is greater than the sum of the other two. 38. Each of two vessels contains a mixture of wine and water; a mixture consisting of equal measures from the two vessels contains as much wine as water, and another mixture consisting of four measures from the first vessel and one from the second is composed of wine and water in the ratio of 2: 3. Find the proportion of wine and water in each of the two vessels. Ans. In the first the wine is, in the second . 39. If the increase in the number of male and female criminals is 1.8%, while the decrease in the number of males alone is 4.6% and the increase in the number of females is 9.8%, compare the number of male and female criminals respectively. Ans. Number of female criminals four-fifths the number of male criminals. |