At the close of Book IV, also, the geometric equivalents of the algebraic formulas for (_a_ + _b_)^2, (_a_ - _b_)^2, and (_a_ + _b_)(_a_ - _b_) are given. The class may like to know that Euclid had no algebra and was compelled to prove such relations as these by geometry, while we do it now much more easily by algebraic multiplication. FOOTNOTES: [79] See, for example, G. B. Kaye, "The Source of Hindu Mathematics," in the _Journal of the Royal Asiatic Society_, July, 1910. [80] An interesting Japanese proof of this general character may be seen in Y. Mikami, "Mathematical Papers from the Far East," p. 127, Leipzig, 1910. [81] Special recognition of indebtedness to H. A. Naber's "Das Theorem des Pythagoras" (Haarlem, 1908), Heath's "Euclid," Gow's "History of Greek Mathematics," and Cantor's "Geschichte" is due in connection with the Pythagorean Theorem. [82] The rule was so ill understood that Bhaskara (twelfth century) said that Brahmagupta was a "blundering devil" for giving it ("Lilavati," Sec. 172). CHAPTER XVIII THE LEADING PROPOSITIONS OF BOOK V [Illustration] Book V treats of regular polygons and circles, and includes the computation of the approximate value of [pi]. It opens with a definition of a regular polygon as one that is both equilateral and equiangular. While in elementary geometry the only regular polygons studied are convex, it is interesting to a class to see that there are also regular cross polygons. Indeed, the regular cross pentagon was the badge of the Pythagoreans, as Lucian (_ca._ 100 B.C.) and an unknown commentator on Aristophanes (_ca._ 400 B.C.) tell us. At the vertices of this polygon the Pythagoreans placed the Greek letters signifying "health." Euclid was not interested in the measure of the circle, and there is nothing in his "Elements" on the value of [pi]. Indeed, he expressly avoided numerical measures of all kinds in his geometry, wishing the science to be kept distinct from that form of arithmetic known to the Greeks as logistic, or calculation. His Book IV is devoted to the construction of certain regular polygons, and his propositions on this subject are now embodied in Book V as it is usually taught in America. If we consider Book V as a whole, we are struck by three features. Of these the first is the pure geometry involved, and this is the essential feature to be emphasized. The second is the mensuration of the circle, a relatively unimportant piece of theory in view of the fact that the pupil is not ready for incommensurables, and a feature that imparts no information that the pupil did not find in arithmetic. The third is the somewhat interesting but mathematically unimportant application of the regular polygons to geometric design.