that a man could have all those three maladies, and survive? And yet the thing is possible!

Let me now read you a statement (at p. 112) about incommensurables :—

'When one of the magnitudes can be represented only by an interminable decimal, while the other is a finite whole number, or finite decimal, no finite common submultiple can exist; for, though a unit be selected in the last place of the whole number or finite decimal, yet the decimal represented by all the figures which follow the corresponding place in the interminable decimal, being less than that unit in that place and unknown in quantity, cannot be a common measure of the two magnitudes, and is only a remainder.'

Now can you lay your hand upon your heart and declare, on the word of an honest man, that you understand this sentence-beginning at the words ' yet the decimal'?

Nie. (vehemently) I cannot!

Min. Of the two reasons which are mentioned, to explain why it cannot be a common measure of the two magnitudes,' does the first-that it is less than that unit in that place-carry conviction to your mind? And does the second-that it is unknown in quantity' ripen that conviction into certainty?

6

Nie. (wildly) Not in the least!

Min. Well, I will not 'slay the slain' any longer. You may consider Dr. Willock's book as rejected. And I think we may say that the whole theory of 'direction' has collapsed under our examination.

Nie. I greatly fear so.

ACT III.

SCENE I.

§ 1. THE OTHER MODERN RIVALS.

'But mice, and rats, and such small deer,
Have been Tom's food for seven long year.'

Min. I consider the question, as to whether Euclid's system and numeration should be abandoned or retained, to be now set at rest: the subject of Parallels being disposed of, no minor points of difference can possibly justify the abandonment of our old friend in favour of any Modern Rival. Still it will be worth while to examine the other writers, whose works you have brought with you, as they may furnish some valuable suggestions for the improvement of Euclid's Manual.

Nie. The other writers are CHAUVENET, LOOMIS, MORELL, REYNOLDS, and WRIGHT.

Min. There are a few matters, as to which we may consider them all at once. How do they define a straight Line?

Nie. All but Mr. Reynolds define it as the shortest distance between two points, or more accurately, to use the

words of Mr. Chauvenet, 'a Line of which every portion is the shortest Line between the points limiting that portion. Min. We discussed that Definition in M. Legendre's book.

How does Mr. Reynolds define it?

Nie. Not at all.

Min. Very cautious. What of angles?

Nie. Some of them allow larger limits than Euclid does. Mr. Wright talks about 'angles of continuation' and 'angles of rotation.'

Min. Good for Trigonometry: not so suitable to early Geometry. How do they define Parallels?

Nie. As in Euclid, all of them.

Min. And which Proposition of Tab. II. do they assume? Nie. Playfair's, or else its equivalent, 'only one Line can be drawn, parallel to a given Line, through a given point outside it.'

Min. Now let us take them one by one.

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. 1, 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.

M