Proportions Of Numbers And Magnitudes :: essays research papers

Proportions of Numbers and MagnitudesIn the Elements, Euclid devotes a handwriting to magnitudes (Five), and he devotes abook to numbers (Seven). Both magnitudes and numbers salute quantity,however magnitude is continuous while number is discrete. That is, numbers becomposed of units which stinkpot be used to divide the whole, while magnitudes can non be distinguished as parts from a whole, consequentlyce numbers can be moreaccurately comp ared because on that point is a standard unit representing one 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 interpretation five in keep Five of the Elements states that " Magnitudesare said to be in the same balance, the initiatory to the second and the third to thefourth, when, if any equimultiples whatever be taken of the stolon and third, andany equimultiples whatever of the second and fourth, the former equimultiples besides exceed, are alike equal to, or alike fall short of, the latterequimultiples on an individual basis taken in corresponding order." From this it followsthat magnitudes in the same ratio are proportional. Thus, we can use thefollowing algebraic proportion to represent description 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 phrase "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 define 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 continuou s 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 determine 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, only when rather it is a condition that definesit. And because of the phrase "any equimultiples whatever," it would be square upto say that if a and b are in same ratio with c and d, then any one of the three