Euclid and His Modern Rivals/Act III. Scene I. § 6.
ACT III.
Scene I.
§ 6. Wright.
'Defects of execution unquestionably exist.'
Wright, Pref. p. 10
Nie. I lay before you 'The Elements of Plane Geometry', by R. P. Wright, Teacher of Mathematics in University College School, London; the second edition, 1871.
Min. Some of the changes in Euclid's method, made in this book, are defended in the Preface.
First, he claims credit for having more Axioms than Euclid, whom he blames for having demonstrated 'much that is obvious.' I need hardly pause to remind you that 'obviousness' is not an invariable property: to a perfect intellect the whole of Euclid, to the end of Book XII, would be 'obvious' as soon as the Definitions had been mastered: but Geometricians must write for imperfect intellects, and it cannot be settled on general principles where Axioms should end and Theorems begin. Let us look at a few of these new Axioms. In p. viii of the Preface, I read 'with the conception of straightness in a Line we naturally associate that of the utmost possible shortness of path between any two of its points; allow this to be assumed, &c.' This I consider a most objectionable Axiom, obliging us, as it does, to contemplate the lengths of curved lines. This matter I have already discussed with M. Legendre (p. 56).
Secondly, for the host of new Axioms with which we are threatened in the Preface, I have searched the book in vain: possibly I have overlooked some, as he never uses the heading 'Axiom,' but really I can only find one new one, at p. 5. 'Every angle has one, and only one bisector,' which is hardly worth stating. Perhaps the writer means that his proofs are not so full as those in Euclid, but take more for granted. I do not think this any improvement in a book meant for beginners.
Another change, claimed in the Preface as an improvement, is the more constant use of superposition. I have considered that point already (p. 47) and have come to the conclusion that Euclid's method of constructing a new figure has all the advantages, without the obscurity, of the method of superposition.
I see little to remark on in the general style of the book. At p. 21 I read 'the straight Line AI satisfies the four following conditions: it passes through the vertex A, through the middle point I of the base, is a perpendicular on that base, and is the bisector of the vertical angle. Now, two of these four conditions suffice to determine the straight Line AI,… Hence a straight Line fulfilling any two of these four conditions necessarily fulfils the other two.' All this is strangely inaccurate: the fourth condition is sufficient by itself to determine the line AI.
At p. 40 I notice the startling announcement that 'the simplest of all Polygons is the Triangle'! This is surely a new use for 'many'? I wonder if the writer is prepared to accept the statement that 'many people have swum across the Bosphorus' on the strength of Byron's
'As once (a feat on which ourselves we prided)
Leander, Mr. Ekenhead, and I did.'
As a specimen of the wordy and unscientific style of the writer, take the following:—
'From any point O, one, and only one, perpendicular can be drawn to a given straight Line AB.
'Let O’ be the point on which O would fall if, the paper being folded along AB, the upper portion of the figure were turned down upon the lower portion. If from the points O, O’ straight Lines be drawn to any point whatever I on the line AB, the adjacent angles OIB, O’IB will be equal; for folding the paper again along AB and turning the upper portion down upon the lower, O falls on O’, I remains fixed, and the angle OIB exactly coincides with O’IB. Now in order that the Line OI may be perpendicular to AB, or, in other words, that OIB may be a right angle, the sum of the two adjacent angles OIB, O’IB must be equal to two right angles, and consequently their sides IO, IO’ in the same straight Line. But since we can always draw one, and only one, straight Line between two points O and O’, it follows that from a point O we can always draw one, and only one, perpendicular to the line AB.'
Do you think you could make a more awkward or more obscure proof of this almost axiomatic Theorem?
Nie. (cautiously) I would not undertake it.
Min. All that about folding and re-folding the paper is more like a child's book of puzzles than a scientific treatise. I should be very sorry to be the school-boy who is expected to learn this precious demonstration! In such a case, I could not better express my feelings than by quoting three words of this very Theorem:—'I remains fixed'!
In conclusion, I may say as to all five of these authors, that they do not seem to me to contain any desirable novelty which could not easily be introduced into an amended edition of Euclid.
Nie. It is a position I cannot dispute.