11 It appears that the Greeks overlooked a simple point that

11. It appears that the Greeks overlooked a simple point that might have led them to break out of the confining circle of Euclidean methods. If only they had realized that composite ratios represent multiplication, they would have been freed from the need for dimensional consistency, since their ratios were dimensionless. They could, for example, multiply and number of ratios, wheras interpreting the product of two lines as a rectangle precluded the possibility of any geometric interpretation of product containing more than three factors. Could they have developed analytic geometry if they had made this realization? What else would they have needed?

Solution

No.

They would have developed analytic geometry even though they had made the realization.

Greeks overlooked found out of the confining circle of Euclidean methods,

As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes\' numerical definitions and representations.

Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates.

hence analytical geometry has its own importance other than euclidian geometry.