Euclid and His Modern Rivals/Act II. Scene IV.
ACT II.
Scene IV.
Treatment of Parallels by equidistances.
Cuthbertson.
'Thou art so near, and yet so far.'
Modern Song.
Nie. I now lay before you 'Euclidian Geometry,' by Francis Cuthbertson, M.A., late Fellow of C. C. C, Cambridge; Head Mathematical Master of the City of London School; published in 1874.
Min. It will not be necessary to discuss with you all the innovations of Mr. Cuthbertson's book. The questions of the separation of Problems and Theorems, the use of superposition, and the omission of the diagonals in Book II, are general questions which I have considered by themselves. The only points, which you and I need consider, are the methods adopted in treating Right Lines, Angles, and Parallels, wherever those methods differ from Euclid's.
The first subject, then, is the Right Line. How do you define and test it?Nie. As in Euclid. But we prove what Euclid has assumed as an Axiom, namely, that two right Lines cannot have a common segment.
Min. I am glad to hear you assert that Euclid has assumed it 'as an Axiom,' for the interpolated and illogical corollary to Euc. I. 11 has caused many to overlook the fact that he has assumed it as early as Prop. 4, if not in Prop. 1. What is your proof?
Niemand reads.
'Two straight Lines cannot have a common segment.'
'For if two straight Lines ABC, ABH could have a common segment AB; then the straight Line ABC might be turned about its extremity A, towards the side on which BH is, so as to cut BH; and thus two straight Lines would enclose a space, which is impossible.' Min. You assume that, before C crosses BH, the portions coinciding along AB will diverge. But, if ABH is a right Line, this will not happen till C has passed H.
Nie. But you would then have one portion of the revolving Line in motion, and another portion at rest.
Min. Well, why not ?
Nie. We may assume that to be impossible ; and that, if a Line revolves about its extremity, it all moves at once.
Min. Which, I take the liberty to think, is quite as great an assumption as Euclid's. I think the Axiom quite plain enough without any proof.
Your treatment of angles, and right angles, is the same as Euclid's, I think?
Nie. Yes, except that we prove that 'all right angles are equal.'
Min. Well, it is capable of proof, and therefore had better not be retained as an Axiom.
I must now ask you to give me your proof of Euc. I. 32.
Nie. We prove as far as I. 28 as in Euclid. In order to prove I. 29, we first prove, as a Corollary to Euc. I. 20, that 'the shortest distance between two points is a straight Line.'
Min. What is your next step?
Nie. A Problem (Pr. F. p. 52) in which we prove the Theorem that, of all right Lines drawn from a point to a Line, the perpendicular is the least.
Min. We will take that as proved.
Nie. We then deduce that the perpendicular is the shortest path from a point to a Line.
Next comes a Definition. 'By the distance of a point from a straight Line is meant the shortest path from the point to the Line.'
Min. Have you anywhere defined the distance of one point from another?
Nie. No.Min. We had better have that first.
Nie. Very well. 'The distance of one point from another is the shortest path from one to the other.'
Min. Might we not say 'is the length of the right Line joining them?'
Nie. Yes, that is the same thing.
Min. And similarly we may modify the Definition you gave just now.
Nie. Certainly. 'The distance of a point from a right Line is the length of the perpendicular let fall upon it from the given point.'
Min. What is your next step?
Niemand reads.
P. 33. Ded. G. 'If points be taken along one of the arms of an angle farther and farther from the vertex, their distances from the other arm will at length be greater than any given straight line.'
In proving this we assume as an Axiom that the lesser of two magnitudes of the same kind can be multiplied so as to exceed the greater.
Min. I accept the Axiom and the proof.
Niemand reads.
P. 34. Ax. 'If one right Line be drawn in the same Plane as another, it cannot first recede from and then approach to the other, neither can it first approach to and then recede from the other on the same side of it.'
Min. Here, then, you assume, as axiomatic, one of the Propositions of Table II. After this, you ought to have no further difficulty in proving Euc. I. 32 and all other properties of Parallels. How do you proceed?
Nie. We prove (p. 34. Lemma) that, if two Lines have a common perpendicular, each is equidistant from the other.
Min. What then?
Nie. Next, that any Line intersecting one of these will intersect the other (p. 35).
Min. That, I think, depends on Deduction G, at p. 33?
Nie. Yes.
Min. A short, but not very easy, Theorem; and one containing a somewhat intricate diagram. However, it proves the point. What is your next step?
Niemand reads.
p. 34. Lemma. 'Through a given point without a given straight Line one and only one straight Line can be drawn in the same Plane with the former, which shall never meet it. Also all the points in each of these straight Lines are equidistant from the other.'
Min. I accept all that.
Nie. We then introduce Euclid's definition of 'Parallels. It is of course now obvious that parallel Lines are equidistant, and that equidistant Lines are parallel.
Min. Certainly.
Nie. We can now, with the help of Euc. I. 27, prove I. 29, and thence I. 32.
Min. No doubt. We see, then, that you propose, as a substitute for Euclid's 12th Axiom, a new Definition, two new Axioms, and what virtually amounts to five new Theorems. In point of 'axiomaticity' I do not think there is much to choose between the two methods. But in point of brevity, clearness, and suitability to a beginner, I give the preference altogether to Euclid's axiom.
The next subject to consider is your practical test, if any, for two given Lines meeting when produced.
Nie. One test is that one of the Lines should meet a Line parallel to the other.
Min. Certainly: and that will suffice in such a case as Euc. I. 44 (Pr. M. p. 60, in this book) though you omit to point out why the Lines may be assumed to meet. But what if the diagram does not contain 'a Line parallel to the other'? Look at Pr. (h) p. 69, where we are told to make, at the ends of a Line, two angles which are together less than two right angles, and where it is assumed that the Lines, so drawn, will meet. That is, you assume the truth of Euclid's 12th Axiom. And you do the same thing at pp. 70, 123, 143, and 185.
Nie. Euclid's 12th Axiom is easily proved from our Theorems.
Min. No doubt: but you have not done it, and the omission makes a very serious hiatus in your argument. It is not a thing that beginners are at all likely to be able to supply for themselves.
I have no adverse criticisms to make on the general style of the book, which seems clear and well written. Nor is it necessary to discuss the claims of the book to supersede Euclid, since the writer makes no such claim, but has been careful (as he states in his preface) to avoid any arrangement incompatible with Euclid's order. The chief novelty in the book is the introduction of the principle of 'equidistance,' which does not seem to me a desirable feature in a book meant for beginners: otherwise it is little else than a modified version of Euclid.