Proportions Of Numbers And Magnitudes :: essays research papers

Proportions of Numbers and MagnitudesIn the Elements, Euclid devotes a book to magnitudes (Five), and he devotes abook to numbers racket (Seven). Both magnitudes and numbers represent quantity, up to now magnitude is continuous while number is discrete. That is, numbers atomic number 18composed of units which can be used to divide the whole, while magnitudes cannot be distinguished as parts from a whole, therefore numbers can be moreaccurately comp ared because there is a standard unit representing matchless ofsomething. Numbers allow for measurement and degrees of ordinal positionthrough which one can better compare quantity. In short, magnitudes tell youhow much there is, and numbers tell you how many there are. This is cause fordifferences in comparison among them.Euclids definition five in Book Five of the Elements states that " Magnitudesare said to be in the equivalent ratio, the first to the second and the third to thefourth, when, if any equimultiples whatever be inte rpreted of the first and third, andany equimultiples whatever of the second and fourth, the former equimultiplesalike exceed, are alike equal to, or alike fall short of, the latterequimultiples respectively taken in corresponding order." From this it followsthat magnitudes in the same ratio are proportional. Thus, we can use thefollowing algebraic proportion to represent definition 5.5(m)a (n)b (m)c (n)d.However, it is necessary to be more specific because of the way in which thedefinition was worded with the diction "the former equimultiples alike exceed,are alike equal to, or alike fall short of.". Thus, if we take any fourmagnitudes a, b, c, d, it is defined that if equimultiple m is taken of a and c,and equimultiple n is taken of c and d, then a and b are in same ratio with cand d, that is, a b c d, only if(m)a > (n)b and (m)c > (n)d, or(m)a = (n)b and (m)c = (n)d, or(m)a < (n)b and (m)c < (n)d.Though, because magnitudes are continuous quantities, and an exact measurementof magnitudes is impossible, it is not possible to say by how much one exceedsthe other, nor is it possible to contain if a > b by the same amount that c >d.Now, it is important to realize that taking equimultiples is not a test to seeif magnitudes are in the same ratio, but rather it is a condition that definesit. And because of the phrase "any equimultiples whatever," it would be correctto say that if a and b are in same ratio with c and d, then any one of the three