By George David Birkhoff, Ralph Beatley

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**Example text**

It is essen- z. 31. The point of ID[e. 20-36 to stated after Exs. 34 and 36; namely to provide justification - aptly phrased for asserting any proportion that can be derived from a given proportion, and to ahoy the applicability of this to situations involving triangles. ,a5 +3+f+--b+d+f+- - 38. An imoeediate application of PS[. 37. Page 67: Summary. The summaries at the and of Chapters 2, 3, 4, 5, and 7 give the logical plan of BASIC GROl1ETRY. as a rough index of its chapter. Zach summary serves also These eummaries will help teachers who are familiar with other systems of geometry to keep the pattern of this geometry in mind.

Definition of half-line demands that it have an end-point. The The phrase "divides the straight line into two half-lines" Is not to be understood in the sense that two halves make a whole, because the point P must do double duty, serving no end-point of each half-line. The pupil may be puzzled also by the statement that the point P may be selected anywhere on the endless straight line. Be must abandon any idea he may have had that P is the mid-point of this line; for the term "mid-point" is defined or.

The proof given of Principle 7 to unusual in that it applies Principle 5, which was worded so as to refer to two triangles, to a situation Involving only a single isosceles triangle. It is clear from the text that an isosceles triangle can be considered to be similar to itself, regardless of the order In which the equal sides are read. Had we mentioned under Principle 5 that we intended to apply that Principle occasionally in this special manner, the remark would have conveyed no meaning. The authors deem it to be good teaching not to refer to this special application of Principle 5 until the need arises, as here in Principle 7.