Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/346

322 38. We may mention, in particular, that a consequence would contradict results already established, if we could shew that it would lead to the solution of a problem already given up as impossible. There are three famous problems which are now admitted to be beyond the power of Geometry; namely, to find a straight line equal in length to the circumference of a given circle, to trisect any given angle, and to find two mean proportionals between two given straight lines. The grounds on which the geometrical solution of these problems is admitted to be impossible cannot be explained without a knowledge of the higher parts of mathematics; the student of the Elements may however be content with the fact that innumerable attempts have been made to obtain solutions, and that these attempts have been made in vain.

The first of these problems is usually referred to as the Quadrature of the Circle. For the history of it the student should consult the article in the English Cyclopædia under that head, and also a series of papers in the Athenæum for 1863 and subsequent years, entitled a Budget of Paradoxes, by Professor De Morgan.

For approximate solutions of the problem we may refer to Davies's edition of Hutton's Course of Mathematics Vol. . page 400, the Lady's and Gentleman's Diary for 1855, page 86, and the Philosophical Magazine for April, 1862.

The third of the three problems is often referred to as the Duplication of the Cube. See the note on VI. 13 in Lardner's Euclid, and a dissertation by C. H. Biering entitled Historia Problematis Cubi Duplicandi... Hauniæ, 1844.

We will now give some examples of Geometrical analysis.

39. From two given points it is required to draw to the same point in a given straight line, two straight lines equally inclined to the given straight line.

Let A and B be the given points, and CD the given straight line.

Suppose AE and EB to be the two straight lines equally inclined to CD. Draw BF perpendicular to CD, and produce AE and BF to meet at G. Then the angle