A little problem that always has some interest for pupils is one that Napoleon is said to have suggested to his staff on his voyage to Egypt: To divide a circle into four equal parts by the use of circles alone. [Illustration] Here the circles _B_ are tangent to the circle _A_ at the points of division. Furthermore, considering areas, and taking _r_ as the radius of _A_, we have _A_ = [pi]_r_^2, and _B_ = [pi](_r_/2)^2. Hence _B_ = 1/4_A_, or the sum of the areas of the four circles _B_ equals the area of _A_. Hence the four _D'_s must equal the four _C'_s, and _D_ = _C_. The rest of the argument is evident. The problem has some interest to pupils aside from the original question suggested by Napoleon. At the close of plane geometry teachers may find it helpful to have the class make a list of the propositions that are actually used in proving other propositions, and to have it appear what ones are proved by them. This forms a kind of genealogical tree that serves to fix the parent propositions in mind. Such a work may also be carried on at the close of each book, if desired. It should be understood, however, that certain propositions are used in the exercises, even though they are not referred to in subsequent propositions, so that their omission must not be construed to mean that they are not important. An exercise of distinctly less value is the classification of the definitions. For example, the classification of polygons or of quadrilaterals, once so popular in textbook making, has generally been abandoned as tending to create or perpetuate unnecessary terms. Such work is therefore not recommended. FOOTNOTES: [83] Bosanquet and Sayre, "The Babylonian Astronomy," _Monthly Notices of the Royal Asiatic Society_, Vol. XL, p. 108. [84] This and the three illustrations following are from Kolb, loc. cit. [85] This was in five colors of marble. [86] The proof is too involved to be given here. The writer has set it forth in a chapter on the transcendency of [pi] in a work soon to be published by Professor Young of The University of Chicago. CHAPTER XIX THE LEADING PROPOSITIONS OF BOOK VI