Excerpt from Replies by Prof. Lawrence S. Benson, Author of Benson's Geometry, to Prof. E. T. Quimby, Dartmouth College, Hanover, N. H., Prof. Wm. Chauvenet, LL. D., Washington University, St. Louis, Mo., Prof. Robert D. Allen, and Prof. A. T. Bledsoe, LL. DIn giving the direct demonstrations in all cases, I had to consider the Reductio ad Absurdum of euclid, whereby he endeavored to sustain the proposition that the area of the circle is the rectangle on the semi-circumference and radius. See benson's geometry, pp. 139, 140, 141, 142, 143. Now, if this proposition be true, it can be directly demonstrated, which all true propositions can be, and the fact that it is impossible of being directly demonstrated, is a positive proof that it is not true. And, in giving the direct demonstration for the problem of the circle, I was unavoidably led to a different result from the false proposition of euclid.I will give a sketch of my first demonstration, which is totally different in method from those I published in benson's geometry, but they all give the same conclusion.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.