Only one or two extracts need be given, which, it is hoped, will suffice to illustrate the character and style of the book:
Act II., Scene v.—Niemand and Minos are arguing for and against Henrici's "Elementary Geometry."
Minos.—I haven't quite done with points yet. I find an assertion that they never jump. Do you think that arises from their having "position," which they feel might be compromised by such conduct?
Niemand.—I cannot tell without hearing the passage read.
Minos.—It is this: "A point, in changing its position on a curve, passes in moving from one position to another through all intermediate positions. It does not move by jumps."
Niemand.—That is quite true.
Minos.—Tell me then—is every centre of gravity a point?
Niemand.—Certainly.
Minos.—Let us now consider the centre of gravity of a flea. Does it—
Niemand (indignantly).—Another word, and I shall vanish! I cannot waste a night on such trivialities.
Minos.—I can't resist giving you just one more tit-bit—the definition of a square at page 123: "A quadrilateral which is a kite, a symmetrical trapezium, and a parallelogram is a square!" And now, farewell, Henrici: "Euclid, with all thy faults, I love thee still!"
Again, from Act II., Scene vi.:—
Niemand.—He (Pierce, another "Modern Rival,") has a definition of direction which will, I think, be new to you. (Reads.)
"The direction of a line in any part is the direction of a point at that part from the next preceding point of the line!"
Minos.—That sounds mysterious. Which way along a line are "preceding" points to be found?
Niemand.—Both ways. He adds, directly afterwards, "A line has two different directions," &c.
Minos.—So your definition needs a postscript.... But there is yet another difficulty. How far from a point is the "next" point?
Niemand.—At an infinitely small distance, of course. You will find the matter fully discussed in my work on the Infinitesimal Calculus.
Minos.—A most satisfactory answer for a teacher to make to a pupil just beginning Geometry!
In Act IV. Euclid reappears to Minos, "followed by the ghosts of Archimedes, Pythagoras, &c., who have come to see fair play." Euclid thus sums up his case:—
"'The cock doth craw, the day doth daw,' and all respectable ghosts ought to be going home. Let me carry with me the hope that I have convinced you of the necessity of retaining my order and numbering, and my method of treating Straight Lines, Angles, Right Angles, and (most especially) Parallels. Leave me these untouched, and I shall look on with great contentment while other changes are made—while my proofs are abridged and improved—while alternative proofs are appended to mine—and while new Problems and Theorems are interpolated. In all these matters my Manual is capable of almost unlimited improvement."
In Appendices I. and II. Mr. Dodgson quotes the opinions of two eminent mathematical teachers, Mr. Todhunter and Professor De Morgan, in support of his argument.
Before leaving this subject I should like to refer to a very novel use of Mr. Dodgson's book—its employment in a school. Mr. G. Hopkins, Mathematical Master in the High School at Manchester, U.S., and himself the author of a "Manual of Plane Geometry," has so employed it in a class of boys aged from fourteen or fifteen upwards. He first called their attention to some of the more prominent difficulties relating to the question of Parallels, put a copy of Euclid in their hands, and let them see his treatment of them, and after some discussion placed before them Mr. Dodgson's "Euclid and His Modern Rivals" and "New Theory of Parallels."
Perhaps it is the fact that American boys are sharper than English, but at any rate the youngsters are reported to have read the two books with an earnestness and a persistency that were as gratifying to their instructor as they were complimentary to Mr. Dodgson.