α For if and are not equal, they must have some assignable Ъ m difference, and because each of them lies between and 1 n this difference must be less than But since n may, by sup position, be increased without limit, n limit; that is, it may be made less than any assigned magnitude; a therefore and have no assignable difference, so that we may say that b a с = d d Hence all the propositions respecting propor tionals are true of the four magnitudes a, b, c, d. 408. It will be useful to compare the definition of proportion which has been given in this chapter with that which is given in the fifth book of Euclid. The latter definition may be stated thus; four quantities are proportionals when if any equimultiples be taken of the first and third, and also any equimultiples of the second and fourth, the multiple of the third is greater than, equal to, or less than, the multiple of the fourth, according as the multiple of the first is greater than, equal to, or less than, the multiple of the second. We will first shew that the property involved in this definition follows from the algebraical definition. For suppose ab: cd; then Hence pc is greater than, equal to, or less than qd, according as pa is greater than, equal to, or less than qb. 409. Next we may deduce the algebraical definition of proportion from Euclid's. Let a, b, c, d be four quantities, such that pc is greater than, equal to, or less than qd, according as pa is greater than, equal to, or less than qb, then shall α = с suppose c and d are commensurable; then we can take such that pc qd; hence, by hypothesis, pa qb. Thus, and, = First Ρ and 9 Next suppose c and d are incommensurable; then we can not find whole numbers p and q such that pc = qd. In this case take any multiple of c as pc; then since this quantity must lie between some two consecutive multiples of d, suppose it to lie between qd and (q+1) d. Thus 2 is greater than unity, and qd рс is less than unity; therefore is greater than and less (q+1) d than 9+1 Ρ And, by hypothesis, pa is also greater than unity, ρα qb α is less than unity, so that Ъ is greater than and less than ?+ 1 Ρ p and q may be, it follows, by Art. 407, that = q. 410. It is usually stated that the common algebraical definition of proportion cannot be used in Geometry, because there is no method of representing geometrically the result of the operation of division. Lines can be represented geometrically, but not the abstract number which expresses how often one line is contained in another. But it should also be noticed that Euclid's definition is rigorous and can be applied to incommensurable as well as to commensurable quantities, while the algebraical definition is, strictly speaking, confined to the latter quantities. Hence this consideration alone would furnish a sufficient reason for the definition adopted by Euclid. EXAMPLES OF PROPORTION. 1. The last three terms of a proportion being 4, 6, 8, what is the first term? 2. Find a third proportional to 25 and 400. 3. If 3, x, 1083 are in continued proportion, find x. 4. If 2 men working 8 hours a day can copy a manuscript in 32 days, in how many days can x men working y hours a day copy it? 5. If x have to y the duplicate ratio of x + ≈ to y + z, prove that z is a mean proportional between x and y. 7. If four quantities are proportionals, and the second is a mean proportional between the third and fourth, the third will be a mean proportional between the first and second. 8. If (a+b+c+d) (a-b-c+d) = (a−b+c-d) (a+b-c-d), prove that a, b, c, d are proportionals. 9. Shew that when four quantities of the same kind are proportional, the greatest and least of them together are greater than the other two together. 10. 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 vessels. 11. A and B have made a bet, each staking a sum of money proportional to all the money he has. If A wins he will have double what B will have, but if he loses, B will have three times what A will have. All the money between them being £168, determine the circumstances. 12. If the increase in the number of male and female criminals be 18 per cent., while the decrease in the number of males alone is 4.6 per cent., and the increase in the number of females is 9.8; compare the number of male and female criminals respectively. XXVIII. VARIATION. 411. The present chapter consists of a series of propositions connected with the definitions of ratio and proportion stated in a new phraseology, which is convenient for some purposes. 412. One quantity is said to vary directly as another when the two quantities depend upon each other, and in such a manner that if one be changed the other is changed in the same proportion. Sometimes for shortness we omit the word directly, and say simply that one quantity varies as another. 413. Thus, for example, if the altitude of a triangle be invariable, the area varies as the base; for if the base be increased or diminished, we know from Euclid that the area is increased or diminished in the same proportion. We may express this result by Algebraical symbols thus; let A and a be numbers which represent the areas of two triangles having a common altitude, and let B and b be numbers which represent the bases of these triangles respectively; then A a B b And from this we deduce Α α В (Art. 392). If there be a third triangle having the same 'altitude as the two already considered, then the ratio of the number which represents its area to the number which represents its Here A may represent the area of any one of a series of triangles which have a common altitude, and B the corresponding base, and m remains constant. Hence the statement that the area varies as the base may also be expressed thus; the area has a constant ratio to the base; by which we mean, in accordance with Article 392, that the number which represents the area bears a constant ratio to the number which represents the base. We have made these remarks for the purpose of explaining the notation and language which will be used in the present chapter. When we say that A varies as B, we mean that A represents the numerical value of any one of a certain series of quantities, and B the numerical value of the corresponding quantity in a certain other series, and that A = mB, where m is some number which remains constant for every corresponding pair of quantities. We will give a formal proof of the equation A = mB_deduced from the definition of Art. 412. 414. If A vary as B then A is equal to B multiplied by some constant quantity. Α ― = B Let a and b denote one pair of corresponding values of two quantities, and let A and B denote any other pair; then by definition. a a b Hence A =B = mB, where m is equal to the constant 415. The symbol is used to express variation; thus A ∞ B stands for A varies as B. |
