The distinctive feature of Euclid's "Elements," compared with the modern American textbook, is perhaps this: Euclid begins a book with what seems to him the easiest proposition, be it theorem or problem; upon this he builds another; upon these a third, and so on, concerning himself but little with the classification of propositions. Furthermore, he arranges his propositions so as to construct his figures before using them. We, on the other hand, make some little attempt to classify our propositions within each book, and we make no attempt to construct our figures before using them, or at least to prove that the constructions are correct. Indeed, we go so far as to study the properties of figures that we cannot construct, as when we ask for the size of the angle of a regular heptagon. Thus Euclid begins Book I by a problem, to construct an equilateral triangle on a given line. His object is to follow this by problems on drawing a straight line equal to a given straight line, and cutting off from the greater of two straight lines a line equal to the less. He now introduces a theorem, which might equally well have been his first proposition, namely, the case of the congruence of two triangles, having given two sides and the included angle. By means of his third and fourth propositions he is now able to prove the _pons asinorum_, that the angles at the base of an isosceles triangle are equal. We, on the other hand, seek to group our propositions where this can conveniently be done, putting the congruent propositions together, those about inequalities by themselves, and the propositions about parallels in one set. The results of the two arrangements are not radically different, and the effect of either upon the pupil's mind does not seem particularly better than that of the other. Teachers who have used both plans quite commonly feel that, apart from Books II and V, Euclid is nearly as easily understood as our modern texts, if presented in as satisfactory dress. The topics treated and the number of propositions in the plane geometry of the "Elements" are as follows: Book I. Rectilinear figures 48 Book II. Geometric algebra 14 Book III. Circles 37 Book IV. Problems about circles 16 Book V. Proportion 25 Book VI. Applications of proportion 33 ---- 173 Of these we now omit Euclid's Book II, because we have an algebraic symbolism that was unknown in his time, although he would not have used it in geometry even had it been known. Thus his first proposition in Book II is as follows: If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.