Euclid and His Modern Rivals/Act II. Scene III.
ACT II.
Scene III.
Treatment of Parallels by angles made with transversals.
Cooley.
'The verbal solemnity of a hollow logic.'
Cooley, Pref. p. 20.
Nie. I have now the honour to lay before you 'The Elements of Geometry, simplified and explained,' by W. D. Cooley, A.B., published in 1860.
Min. Please to hand me the book for a moment. I wish to read you a few passages from the Preface. It is always satisfactory—is it not?—to know that a writer, who attempts to 'simplify' Euclid, begins his task in a becoming spirit of humility, and with some reverence for a name that the world has accepted as an authority for two thousand years.
Nie. Truly. Minos reads.
'The Elements of Plane Geometry … are here presented in the reduced compass of 36 Propositions, perfectly coherent, fully demonstrated, and reaching quite as far as the 173 Propositions contained in the first six books of Euclid.' Modest, is it not?
Nie. A little high-flown, perhaps. Still, you know, if they really are 'fully demonstrated'
Min. If! In page 4 of the Preface he talks of 'Euclid's circumlocutory shifts': in the same page he tells us that 'the doctrine of proportion, as propounded by Euclid, runs into prolixity though wanting in clearness': and again, in the same page, he states that most of Euclid's ex absurdo proofs 'though containing little,' yet 'generally puzzle the young student, who can hardly comprehend why gratuitous absurdities should be so formally and solemnly dealt with. These Propositions therefore are omitted from our Book of Elements, and the Problems also, for the science of Geometry lies wholly in the Theorems. Thus simplified and freed from obstructions, the truths of Geometry may, it is hoped, be easily learned, even by the youngest.' But perhaps the grandest sentence is at the end of the Preface. 'Then as to those Propositions (the first and last of the 6th Book), in which, according to the same authority' (he is alluding to the Manual of Euclid by Galbraith and Haughton), 'Euclid so beautifully illustrates his celebrated Definition, they appear to our eyes to exhibit only the verbal solemnity of a hollow logic, and to exemplify nothing but the formal application of a nugatory principle.' Now let us see, mein Herr, whether Mr. Cooley has done anything worthy of the writer of such 'brave 'orts' (as Shakespeare has it): and first let me ask how you define Parallel Lines.
Niemand reads.
'Right Lines are said to be parallel when they are equally and similarly inclined to the same right Line, or make equal angles with it towards the same side.'
Min. That is to say, if we see a Pair of Lines cut by a certain transversal, and are told that they make equal angles with it, we say 'these Lines are parallel'; and conversely, if we are told that a Pair of Lines are parallel, we say 'then there is a transversal, somewhere, which makes equal angles with them'?
Nie. Surely, surely.
Min. But we have no means of finding it? We have no right to draw a transversal at random and say 'this is the one which makes equal angles with the Pair'?
Nie. Ahem! Ahem! Ahem!
Min. You seem to have a bad cough.
Nie. Let us go to the next subject.
Min. Not till you have answered my question. Have we any means of finding the particular transversal which makes the equal angles?
Nie. I am sorry for my client, but, since you are so exigeant, I fear I must confess that we have no means of finding it.
Min. Now for your proof of Euc. I. 32.
Nie. You will allow us a preliminary Theorem?Min. As many as you like.
Nie. Well, here is our Theorem ii. 'When two parallel straight Lines AB, CD, are cut by a third straight Line EF, they make with it the alternate angles AGH, GHD, equal; and also the two internal angles at the same side BGH, GHD equal to two right angles.
For AGH and EGB are equal because vertically opposite, and EGB is also equal to GHD (Definition); therefore—'
Min. There I must interrupt you. How do you know that EGB is equal to GHD? I grant you that, by the Definition, AB and CD make equal angles with a certain transversal: but have you any ground for saying that EF is the transversal in question?
Nie. We have not. We surrender at discretion. You will permit us to march out with the honours of war?
Min. We grant it you of our royal grace. March him off the table, and bring on the next Rival.