La Géométrie

“Any problem in geometry can be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.”

“Let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA; then BE is the product of BD and BC.If it be required to divide BE by BD, I join E and D, and draw AC parallel to DE; then BC is the result of division.”

“If the square root of GH is desired, I add, along the same straight line, FG equal to unity; then, bisecting FH at K, I describe the circle FIH about K as the center, and draw from G a perpendicular and extend it to I, and GI is the required root.”

“Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter. ...it must be observed that by a, and similar expressions, I ordinarily mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of the terms employed in algebra.”

“This is also evident from what Pappus has done in the beginning of his seventh book, where... he refers to a question which he says that neither Euclid nor Apollonius nor any one else had been able to solve completely...”

“The considerations that forced ancient writers to use arithmetical terms in geometry, thus making it impossible for them to proceed beyond a point where they could see clearly the relation between two subjects, caused much obscurity and embarrassment, in their attempts at explanation.”