4. Required approximate expressions in small numbers for the ratio 7853981633: 10000000000, being that of the area of a circle, to the square of its diameter, nearly? 5. If the side of a square be 1234000, its diagonal will be 1745139, nearly; required approximations to this ratio in smaller numbers ? OF PROPORTION " 53. Four quantities are said to be proportionals, when the first has to the second the same ratio which the third has to the fourth; that is, when the first is the same multiple, part, or parts of the second that the third is of the fourth. d Ratio is the comparison of magnitudes or quantities; proportion is the equality of ratios; hence there must be two ratios to constitute that equality which is called proportion; that is, there must be three terms at least to express the two ratios necessary to a comparison. Some authors have, with the most unaccountable negligence, confounded and perplexed their inexperienced readers with the definitions they have given of ratio and proportion. Dr. Hutton, to whose useful labours almost every branch of the mathematics is indebted for elucidation or improvement, in his system of Elementary Mathematics for the use of the Royal Military Academy, thus defines them: "Ratio is the proportion which one magnitude bears to another magnitude of the same kind, with respect to quantity ;" and immediately after, " Proportion is the equality of ratios." Now it has always been held as a necessary maxim in logic, that "in every definition the ideas implied by the terms of the definition, should be more obvious to the mind than the idea of the thing defined," otherwise the definition fails of its purpose; it leaves us just as wise as it found us. Wherefore, supposing the above definitions of ratio and proportion to be adequate and perspicuous, as they ought to be, if we apply this doctrine to them, it will follow from the former, that the idea of proportion is more obvious than that of ratio; and from the latter, that the idea of ratio is more obvious than that of proportion; but the supposition that both these conclusions are true, implies a manifest absurdity, and consequently, that one or both of these definitions must be faulty. It is but justice to suppose, that the learned Doctor must have used the term proportion, in the former definition, 54. This proportion, or equality of ratios, is usually expressed by four dots, thus :: interposed between the two ratios. Thus, ab:: cd, shews that a has to b the same ratio that c has to d, or that the four quantities, a, b, c, and d, are proportionals, and are usually read, a is to b, as c to d. 55. The first and last terms of the proportion (viz. a and d) are called the extremes, and the two middle terms (b and c) the means. 56. Since it has been shewn, (Art. 27.) that any ratio is truly expressed by placing its terms in the form of a fraction; therefore, when four quantities are proportionals, that is, when the first has to the second the same ratio which the third has to the fourth, it follows, that the fraction constituted by the terms of the first ratio, will be equal to the fraction constituted by the terms of the other ratio placed in the same order. α с b d ΟΤ , b d a с Thus, if a:b::c:d, then will == 57. If four quantities are proportionals, the product of the extremes is equal to the product of the means. Let ab::c:d, then by the preceding article, a a с ; muld C tiply the terms of this equation by bd, and (—× bd=—=—=xbd, or) ad bc. Euclid 16, 6. 58. Hence, if three quantities are proportionals, the product of the extremes is equal to the square of the mean. a с Let acc: d, then tiply both sides by ed, and (=xcd=&xcd, or) ad=c3, Euclid 17, 6. according to its vulgar acceptation,. (namely, the comparison of one thing with another,) and in the latter, according to its mathematical import. The learner ought to be cautioned to study not to be imposed on by the double meaning of words, and especially to scorn the mean artifice of availing himself on any occasion of the ambiguity of language. A wrangler may confound his opponent by using the same word in two or three different senses; but truth (which is the grand object of science) is discovered only when our reasoning proceeds by means of terms which are strictly limited in their signification. 59. Hence, if three terms of any proportion be given, the For since bc, if a, d, and b, are given, then · 60. If the product of two quantities be equal to the product of two others, then if the terms of one product be made the means, and the terms of the other product the extremes, the four quantities will be proportionals. Thus, if ad=bc, divide both sides by bd, and ad bc bd bd' or) 61. If the first term be to the second, as the third to the fourth, and the third to the fourth as the fifth to the sixth, then will the first be to the second as the fifth to the sixth. Let ab:: cd, and c:d::e:f, then will a:b:: e:f; for 62. Hence, if the same ratio subsists between every two adjacent terms of any rank of quantities, that is, if the terms are in continued proportion, the first term will be to the second as the last but one to the last. = For, let a, b, c, d, e, f, g, h, k, l, &c. be such, then с d = d e 63. If four quantities are proportionals, they are also proportionals when taken inversely. e This article furnishes a demonstration of the Rule of Three, except that part of it which respects the reducing of the terms: but the latter is obvious; since in order to compare quantities, it is plain we must bring them to a simple form, and likewise the quantities compared must be of the same denomination, otherwise a comparison cannot be made. a = с et a : b::c:d, then will b: a::d: c; for since & b d let be divided by each of these equal fractions, and the quotients this operation and property is usually cited under the name TENDO. Euclid pr. B. Book 5. If four quantities be proportionals, they are also propors when taken alternately. Let a:b::c:d, then will a:c::b:d; for: = where that is, a: c::b: d; this is named ALTER 5. If four quantities be proportionals, the sum of the first second is to the second, as the sum of the third and fourth he fourth. Let ab:: cd, then will a+b:b::c+d: d. Because a let unity be added to each, and (~— b a b a+b b wherefore a+b:b::c+d:d; this is named COMPO Ndo. Euclid 18, 5. 66. In like manner, the first is to the sum of the first and cond, as the third to the sum of the third and fourth. 67. If four quantities be proportionals, the excess of the first bove the second is to the second, as the excess of the third above the fourth is to the fourth. Let a:b::c:d, then will a-b: b::c-d: d. Because VOL. 11. F α b 59. Hence, if three terms of any proportion be given, th fourth may be found: ad For since bc, if a, d, and b, are given, then =c; if b 60. If the product of two quantities be equal to the pro of two others, then if the terms of one product be made means, and the terms of the other product the extremes four quantities will be proportionals. Thus, if adbc, divide both sides by bd, and ad bd -1 61. If the first term be to the second, as the third fourth, and the third to the fourth as the fifth to the six will the first be to the second as the fifth to the sixth. Let ab::c:d, and cd::e:f, then will a: :b:: 62. Hence, if the same ratio subsists between ever jacent terms of any rank of quantities, that is, if the in continued proportion, the first term will be to the the last but one to the last. = For, let a, b, c, d, e, f, g, h, k, l, &c. be such, tỉ C d e = d e b::k: l. |