Euclid and His Modern Rivals/Act III. Scene I. § 2.
ACT III.
Scene I.
§ 2. Chauvenet.
'Where Washington hath left
His awful memory
A light for after times!'
Nie. I lay before you 'A Treatise on Elementary Geometry,' by W. Chauvenet, LL.D., Professor of Mathematics and Astronomy in Washington University, published in 1876.
Min. I read in the Preface (p. 4) 'I have endeavoured to set forth the elements with all the rigour and completeness demanded by the present state of the general science, without seriously departing from the established order of the Propositions.' So there would be little difficulty, I fancy, in introducing into Euclid's own Manual all the improvements which Mr. Chauvenet can suggest.
P. 14. Pr. i, and p. 18. Pr. v, taken together, tell us that only one perpendicular can be drawn to a Line from a point. And various additions, about obliques, are made in subsequent Propositions. All these may well be embodied in a new Proposition, which we might interpolate as Euc. I. 12. B.P. 26. Pr. xv, asserts the equidistance of Parallels. This might be interpolated as Euc. I. 34. B.
Another new Theorem, that angles whose sides are parallel, each to each, are equal (which I observe is a great favourite with the Modern Rivals), seems to me a rather clumsy and uninteresting extension of Euc. I. 29.
I see several Propositions which might well be inserted as exercises on Euclid (e.g. Pr. xxxix, 'Every point in the bisector of an angle is equally distant from the sides'), but which are hardly of sufficient importance to be included as Propositions: and others (e.g. Pr. xl, 'The bisectors of the three angles of a Triangle meet in the same point) which seem to belong more properly to Euc. III or IV. I have no other remarks to make on this book, which seems well and clearly written.