Euclid.—­At that rate there would probably be within
    the limit of my First Book—­how many?

    Minos.—­A thousand at least.

    Euclid.—­What a popular school-book it will be!  How
    boys will bless the name of the writer who first brings out
    the complete thousand!

With a view to discussing and criticising his various modern rivals, Euclid promises to send to Minos the ghost of a German Professor (Herr Niemand) who “has read all books, and is ready to defend any thesis, true or untrue.”

“A charming companion!” as Minos drily remarks.

This brings us to Act II., in which the Manuals which reject Euclid’s treatment of Parallels are dealt with one by one.  Those Manuals which adopt it are reserved for Act III., Scene i.; while in Scene ii., “The Syllabus of the Association for the Improvement of Geometrical Teaching,” and Wilson’s “Syllabus,” come under review.

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?