E B49 A A A X or A is to E in the quadruplicate ratio of A to B, DEFF. XIV., XV., XVI., XVII., XVIII., XIX., XX., XXI. 193. The various rules laid down by these definitions for the changes that may take place in the terms of a proportion may be exhibited in one view by the following ADDITIONAL FORMS USED IN ARITHMETIC AND ALGEBRA. A+ BAB:: C+D: C - D A. P.W: B. Q. X:: C.R.Y: D.S.Z &c. &c. 194. In Newton's "Principia," and other works on Natural Philosophy, treated Geometrically, the following different kinds of ratios are to be found, as well as those specified in Euclid, viz. RATIO OF EQUALITÝ, CONSTANT OR GIVEN RATIO, DIRECT RATIO, INVERSE OR RECIPROCAL RATIO, SUBDUPLICATE RATIO, SUBTRIPLICATE RATIO, SESQUIPLICATE RATIO. Now Euclid being merely a preliminary step to Philosophy, it may not be amiss to explain the meaning of these several technical terms. 195. What is meant by "Ratio of Equality?" A ratio of equality is that in which the antecedent and consequent are equal. Thus A: A is a ratio of equality. 196. What is a "Constant or Given Ratio ?" Two magnitudes, mA and mB, which for all values of m constitute the same ratio, are said to be in a given ratio, viz. that of A: B. 197. What is the distinction between Direct and Inverse, or Reciprocal Quantities? Taking the words "magnitude" and "Quantity," in the sense of Euclid (see Def. III., Book V.), a magnitude (A) may be any thing whatever having extension; but if this thing or magnitude be considered as to quantity, it must be compared with the unit of its species; that is, the ratio A: 1 or denotes the quantity of the magnitude A. A Hence the magnitude A being compared with its unit, gives A the quantity, or, as it is written for the sake of brevity, A. But if we make the unit the antecedent, and A the conse quent, or compare i with A, the resulting ratio is 1 Α' ; and these quantities being formed in an inverse or reciprocal order, the one is called the inverse or reciprocal of the other. From the above reasoning it also appears that all ratios are quantities; and that magnitudes are nothing but certain extensions. Example. A certain unmeasured area (A) is proposed for consideration. This is a Magnitude. Let it be measured, and suppose that it contain 1000 square inches. Then the unit of surface being supposed 1 square inch, we have 198. What is a Direct Ratio? That in which the antecedent and consequent are either both direct quantities, or both inverse quantities. 199. What is an Inverse, or Reciprocal Ratio? That in which the antecedent and consequent are neither both direct quantities, nor both inverse quantities. are inverse or reciprocal ratios. 200. What is a Subduplicate Ratio? If three magnitudes A, B, C, be proportionals, then A is said to have to C the duplicate ratio of that which A has to B (see Def. X.); and conversely, A is said to have to B the subduplicate ratio of that which A has to C. Algebraically exhibited. If A, B, C be proportionals, then (192) AC A A2 A2: B2 or = and extracting the roots of these equals, Hence then A B = A or A : B :: 4a : c A subduplicate ratio is such, that, of three proportionals, the first is said to have to the second the subduplicate ratio of that which the first has to the third. 201. What is a Subtriplicate Ratio? As before, the reasons may be manifested why it should be defined, Of four proportionals the first is said to have to the second the subtriplicate ratio of that which the first has to the fourth. Thus, in algebraic notation, if A, B, C, D be proportionals; then A : B :: A3 ; c3. |