Euclid and His Modern Rivals/Act III. Scene II. § 2.
ACT III.
Scene II.
§ 2. Wilson's 'Syllabus'-Manual.
'No followers allowed.'
Times' advertisement-sheet, passim.
Nie. I lay before you 'Elementary Geometry, following the Syllabus prepared by the Geometrical Association,' by J. M. Wilson, M.A., 1878.'
Min. In what respects is this book a 'Rival' of Euclid?
Nie. Well, it separates Problems from Theorems
Min. Already discussed (see p. 18).
Nie. It adopts Playfair's Axiom
Min. Discussed (see p. 40).
Nie. It abandons diagonals in Book II
Min. Discussed (see p. 50).
Nie. And it adopts a new sequence and numeration.
Min. That, of course, prevents us from taking it as merely a new edition of Euclid. It will need very strong evidence indeed to justify its claim to set aside the sequence and numeration of our old friend. We must now examine the book seriatim. When we come to matters that have been already condemned, either in Mr. Wilson's book, or in the 'Syllabus,' I shall simply note the fact. We need have no new discussion, except as to new matter.
Nie. Quite so.
Min. In the 'Introduction,' at p. 2, I read 'A Theorem is the formal statement of a Proposition,' &c. Discussed at p. 189.
At p. 3 we have the 'Rule of Conversion,' which I have already endeavoured to understand (see p. 190).
At p. 6 is a really remarkable assertion. 'Every Theorem may he shewn to be a means of indirectly measuring some magnitude.' Kindly illustrate this on Euc. I. 14.
Nie. (hastily) Oh, if you pick out one single accidental excep
Min. Well, then, take 16, if you like: or 17, or 18
Nie. Enough, enough!
Min. (raising his voice)—or 19, or 20, or 21, or 24, or 25, or 27, or 28, or 30!
Nie. We abandon 'every.'
Min. Good. At p. 8 we have the Definitions of 'major conjugate' and 'minor conjugate' (discussed at p. 185).
At p. 9 is our old friend the 'straight angle' (see p. 101).
In the same page we have that wonderful triad of Lines, one of which is 'regarded as lying between the other two' (see p. 185).
And also the extraordinary result that follows when one straight Line 'stands upon another' (see p. 186).At p. 27, Theorem 14, is a new proof of Euc. I. 24, apparently an amended version of Mr. Wilson's five-case proof, which I discussed at p. 137. He has now reduced it to three cases, but I still think the 'bisector of the angle' a superfluity.
At p. 37 we have those curious specimens of 'Theorems of Equality,' which I discussed at p. 139.
At p. 53 is the Theorem which asserts, in its conclusion, part of its own data (see p. 192).
At p. 54 we are told that 'parallel Lines, which have equal projections on another Line, are equal' (see p. 193).
At p. 55 we have the inconceivable triad of 'equal intercepts' made by a Line cutting three Parallels (see p. 193).
At p. 161 I am surprised to see him fall into a trap in which I have often seen unwary students caught, while trying to say Euc. III. 30 ('To bisect a given arc') After proving two chords equal, they at once conclude that certain arcs, cut off by them, are equal; forgetting to prove that the arcs in question are both minor arcs.
But I must go no further: I have already wandered beyond the limits of Euc. I, II. The one great merit of this book
Nie. You have mentioned all the faults, then?
Min. By no means. You are too impatient. The one great merit, as I was saying, of Mr. Wilson's new book (and a most blessed change it is!) is that it ignores the whole theory of 'direction.' That he has finally abandoned that night-mare of Elementary Geometry, I dare not hope: so all I have said about it had better stand, lest in some future fit of inspiration he should bring out a yet more agonising version of it.
But it has the usual hiatus of a system which replaces Euclid's Axiom by Playfair's: it provides no means of proving that the Lines contemplated by Euclid will meet if produced. (This I have discussed at p. 187.)
Its proposed changes in the sequence of Euclid I have discussed at p. 188.
It has a few other faults, which I have already discussed in Mr. Wilson's own book, and a few peculiar to the Syllabus; but I spare you such minute criticisms.
But what I have now to ask you is simply this. What possible pretext have you left for suggesting that Euclid's Manual, and specially his sequence and numeration, should be abandoned in favour of this far from satisfactory infant?
Nie. There are some new Theorems
Min. Those constitute no reason: you might easily interpolate them.
Nie. I fear there are no other grounds to urge. But I should like to consult the doppelgänger of the Association before I throw up my brief.
Min. By all means.
[For a minute or two there is heard a rustling and a whispering, as of ghosts. Then Niemand speaks again.]
Nie. They think that, considering that this book is but just published, and that it is definitely put forward as the Manual to supersede Euclid, it ought to be examined more in detail, with reference to what is new in it—that is, new proofs of Euclid's Propositions, and new Propositions.
Min. (with a weary sigh) Very well. It will perhaps be more satisfactory to do this, if only to ascertain exactly how much this new Manual contains that is really new and really worthy of adoption. But I shall limit my examination to the subject-matter of Euc. I, II.
Nie. That is all we ask.
Min. We begin, then, at p. 12.
Theorem 1. 'All right angles are equals.' This is provedby their being halves of a 'straight angle,' a phrase which I have already criticised. There is a rather important omission in the proof, no distinction being drawn between the 'straight angle' on one side of a Line, and the other (of course named by the same letters) which lies on the other side and completes the four right angles. This Theorem, if proved without 'straight angles,' might be worth adding to a new edition of Euclid.
Th. 2 (p. 13) is Euc. I. 13, proved as in Euclid.
Th. 3 (p. 14) is Euc. I. 14, where, unfortunately, a new proof is attempted, which involves a fallacy. It is deduced from an 'Observation' in p. 9, that 'a straight Line makes with its continuation at any point an angle of two right angles,' which deduction can be effected only by the process of converting a universal affirmative 'simpliciter' instead of 'per accidens.'
Th. 4 (p. 14) is Euc. I. 15, proved as in Euclid.
At p. 17 I find a 'Question.' 'State the fact that "all geese have two legs" in the form of a Theorem.' This I would not mind attempting; but, when I read the additional request, to 'write down its converse theorem,' it is so powerfully borne in upon me that the writer of the Question is probably himself a biped, that I feel I must, however reluctantly, decline the task.
Th. 5 (p. 18) is Euc. I. 4, proved as in Euclid.
Th. 6 (p. 20) is Euc. I. 5, proved by supposing the vertical angle to be bisected, thus introducing a 'hypothetical construction' (see p. 20).
Th. 7 (p. 21) is Euc. I. 26 (1st part), proved by superposition. Euclid's proof, by making a new Triangle, is quite as good, I think. The areas are here proved to be equal, a point omitted by Euclid: I think it a desirable addition to the Theorem.
Th. 8 (p. 22) is Euc. I. 5, proved by reversing the Triangle and then placing it on itself (or on the trace it has left behind), a most objectionable method (see p. 48).
Theorems 9 to 13 (pp. 22 to 26) are Euc. I. 16, 18, 19, 20, 21, with Euclid's proofs.
Th. 14 (p. 27) is Euc. I. 24, proved by supposing an angle to be bisected: another 'hypothetical construction.'
Th. 15 (p. 28) is Euc. I. 8, for which two proofs are offered:—one by Euc. I. 24 (which seems to be reversing the natural order)—the other by an application of Euc. I. 5, a method involving three cases, of which only one is given. All this is to save the introduction of Euc. I. 7, a Theorem which I think should by no means be omitted. (See p. 220.) Here, as in Th. 7, the equality of the areas is, I think, a desirable addition to Euclid's Theorem.
Th. 16 (p. 29) is Euc. I. 25, with old proof.Th. 17 (p. 30) is Euc. I. 26 (2nd part) proved by superposition instead of Euclid's method (which I prefer) of constructing a new Triangle.
Th. 18 (p. 32) is Euc. I. 17, with old proof.
Th. 19 (p. 33) is new. 'Of all the straight Lines that can he drawn from a given point to meet a given straight Line, the perpendicular is the shortest; and of the others, those making equal angles with the perpendicular are equal; and that which makes a greater angle with the perpendicular is greater than that which makes a less.' This I think deserves to be interpolated.
Th. 20 (p. 34) is new. 'If two Triangles have two sides of the one equal to two sides of the other, each to each, and the angles opposite to two equal sides equal, the angles opposite to the other two equal sides are either equal or supplementary, and in the former case the Triangles are equal in all respects.' I do not think it worth while to trouble a beginner with this rather obscure Theorem, which is of no practical use till he enters on Trigonometry.
Th. 21 (p. 43) is Euc. I. 27: old proof.
Th. 22 (p. 44) is Euc. I. 29 (1st part), proved by Euc. I. 27 and Playfair's Axiom (see p. 40).
Th. 23 (p. 45) is new. 'If a straight Line intersects two other straight Lines and makes either a pair of alternate angles equals or a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary; then, in each case, the two pairs of alternate angles are equal, and the four pairs of corresponding angles are equal, and the two pairs of interior angles on the same side are supplementary.' This most formidable enunciation melts down into the mildest proportions when superfluities areomitted, and only so much of it proved as is really necessary to include the whole. Euclid proves all that is valuable in it in the course of I. 29, and I do not see any sufficient reason for stating and proving it as a separate Theorem.
Th. 23, Cor. (p. 46) is the rest of Euc. I. 29: old proof.
Th. 24 (p. 46) is Euc. I. 30, proved as a Contranominal of Playfair's Axiom.
Th. 25, 26, and Cor. (pp. 47, 48) are Euc. I. 32 and Corollaries: old proof.
Th. 27, 1st part (p. 50), is a needless repetition of part of the Corollary to Th. 23.
Th. 27, 2nd part (p. 50), is part of Euc. I. 34: old proof.
Th. 28 (p. 51) is the rest of Euc. I. 34, proved as in Euclid.
Th. 29 (p. 52) is new. 'If two Parallelograms have two adjacent sides of the one respectively equal to two adjacent sides of the other, and likewise an angle of the one equal to an angle of the other; the Parallelograms are identically equal.' This might be a useful exercise to set; but really it does not seem of sufficient importance to be selected for a Manual.
Th. 30 (p. 53) is Euc. I. 33,: old proof.
Th. 31 (p. 54) is new. 'Straight Lines which are equal and parallel have equal projections on any other straight Line; conversely, parallel straight Lines which have equal projections on another straight Line are equal; and equal straight Lines, which have equal projections on another straight Line, are equally inclined to that Line.' The first and third clauses might be interpolated, though I think their value doubtful. The second is false. (See p. 193.)
Th. 32 (p. 55) is new. 'If there are three parallel straight Lines, and the intercepts made by them on any straight Line that cuts them are equal, then the intercepts on any other straight Line that cuts them are equal.' This is awkwardly worded (in fact, as it stands, its subject, as I pointed out in p. 193, is inconceivable), and does not seem at all worth stating as a Theorem.
At p. 57 I see an 'Exercise' (No. 5). 'Shew that the angles of an equiangular Triangle are equal to two-thirds of a right angle.' In this attempt I feel sure I should fail. In early life I was taught to believe them equal to two right angles—an antiquated prejudice, no doubt; but it is difficult to eradicate these childish instincts.
Problem 1 (p. 61) is Euc. I. 9: old proof. It provides no means of finding a radius 'greater than half AB,' which would seem to require the previous bisection of AB. Thus the proof involves the fallacy 'Petitio Principii.'
Pr. 2 (p. 62) is Euc. I. 11, proved nearly as in Euclid.
Pr. 3 (p. 62) is Euc. I. 12, proved nearly as in Euclid. It omits to say how a 'sufficient radius' can be secured, a point not neglected by Euclid.
Pr. 4 (p. 63) is Euc. I. 10, proved nearly as in Euclid. This also, like Pr. 1, involves the fallacy 'Petitio Principii.’
Pr. 5 (p. 64) is Euc. I. 32, proved nearly as in Euclid, but claims to use compasses to transfer distances, a Postulate which Euclid has (properly, I think) treated as a Problem. (See p. 212.)
Pr. 6, 7 (pp. 65, 66) are Euc. I. 23, 31: old proofs.Problems 8 to 11 (pp. 66 to 69) are new. Their object is to construct Triangles with various data: viz. A, B, and c; A, B, and a; a, b, and C; a, b, and A. They are good exercises, I think, but hardly worth interpolating as Theorems. The first of them is remarkable as one of the instances where Mr. Wilson assumes Euc. Ax. 12, without giving, or even suggesting, any proof. If he intends to assume it as an Axiom, he makes Playfair's Axiom superfluous. No Manual ought to assume both of them.
Theorem 1 (p. 82) is Euc. I. 35, proved as in Euclid, but incompletely, as it only treats of one out of three possible cases.
Th. 2 (p. 83) is new. 'The area of a Triangle is half the area of a rectangle whose base and altitude are equal to those of the Triangle.' This is merely a particular case of Euc. I. 41, and may fairly be reserved till we enter on Trigonometry, where it first begins to have any practical value.
Th. 2, Cor. 1 (p. 84) is Euc. I. 37, 38: old proofs.
Th. 2, Cor. 2 (p. 84) is new. 'Equal Triangles on the same or equal bases have equal altitudes.' No proof is offered. It is an easy deduction, of questionable value.
Th. 2, Cor. 3 (p. 84) is Euc. I. 39, 40. No proof given.
Th. 3 (p. 84) is new. 'The area of a trapezium [by which Mr. Wilson means 'a quadrilateral that has only one pair of opposite sides parallel'] is equal to the area of a rectangle whose base is half the sum of the two parallel sides, and whose altitude is the perpendicular distance between them.' I have no hesitation in pronouncing this to be a mere 'fancy' Proposition, of no practical value whatever.
Th. 4 (p. 86) is Euc. I. 43: old proof.Th. 5 (p. 87) is Euc. II. 1: old proof.
Th. 6, 7, 8 (p. 88, &c.) are Euc. II. 4, 7, 5. The sequence of Euc. II. 5, and its Corollary, is here inverted. Also the diagonals are omitted, and nearly every detail is left unproved, thus attaining a charming brevity—of appearance!
Th. 9 (p. 91) is Euc. I. 47: old proof.
Th. 10, 11 (pp. 94, 95) are Euc. 12, 13: old proof.
Th. 12 (p. 95) is new. 'The sum of the squares on two sides of a Triangle is double the sum of the squares on half the base and on the line joining the vertex to the middle point of the base.' This, Mr. Wilson tells us, is 'Apollonius' Theorem': but, even with that mighty name to recommend it, I cannot help thinking it rather more curious than useful.
Th. 13 (p. 96) is Euc. II. 9, 10. Proved algebraically, and thus degraded from the position of a (fairly useful) geometrical Theorem to a mere addition-sum, of no more value than millions of others like it.
In the next proposition we suddenly transfer our allegiance, for no obvious reason, from Arabic to Latin numerals.
Problem i (p. 99) is Euc. I. 42: old proof.
Pr. ii. (p. 100) is Euc. I. 44: proved nearly as in Euclid, but labours under the same defect as Pr. 8 (p. 66) in that it assumes, without proof, Euc. Ax. 12.
Pr. iii (p. 100) is Euc. I. 45: old proof.
Pr. iv (p. 101) is Euc. II. 14: old proof.
Pr. v (p. 103) is new. 'To construct a rectilineal Figure equal to a given rectilineal Figure and having the number of its sides one less than that of the given figure; and thence to construct a Triangle equal to a given rectilineal Figure.' This I have already noticed (see p. 193). It really is not worth interpolating as a new Proposition. And its concluding clause is, if I may venture on so harsh an expression, childish: it reminds me of nothing so much as the Irish patent process for making cheap shoes—by taking boots and cutting off the tops!
Pr. vi (p. 103) is 'To divide a straight Line, either internally or externally, into two segments such that the rectangle contained by the given Line and one of the segments may be equal to the square on the other segment.' The case of internal section is Euc. II. 11, with the old proof. The other case is new, and worth interpolating.
I have now discussed, with as much care and patience as the lateness of the hour will permit, so much of this new Manual as corresponds to Euc. I, II, and I hope your friends are satisfied.
[A gentle cooing, as of satisfied ghosts, is heard in the air.]
I will now give you in a few words the net result of it all, and will show you how miserably small is the basis on which Mr. Wilson and his coadjutors of the 'Association' rest their claim to supersede the Manual of Euclid.
[An angry moaning, as of ghosts suffering from neuralgia, surges round the room, till it dies away in the chimney.]
By breaking up certain of the Propositions of Euc. I, II, and including some of the Corollaries, we get 73 Propositions in all—57 Theorems and 16 Problems. Of these 73, this Manual omits 14 (10 Theorems and 4 Problems); it proves 43 (32 Theorems and 11 Problems) by methods almost identical with Euclid's; for 10 of them (9 Theorems and a Problem) it offers new proofs, against which I have recorded my protest, one being illogical, 2 (needlessly) employing 'superposition,' 2 deserting Geometry for Algebra, and the remaining 4 omitting the diagonals in Euc. II; and finally it offers 6 new proofs, which I think may fairly be introduced as alternatives for those of Euclid.
In all this, and in all the matters previously discussed, I fail to see one atom of reason for abandoning Euclid. Have you any yet-unconsidered objections to urge against my proposal 'that the sequence and numeration of Euclid be kept unaltered'?
[Dead silence is the only reply.]
Carried, nemine contragemente! And now, Prisoner at the Bar (I beg your pardon, I should say 'Professor on the Sofa'), have you, and your attendant phantoms, any other reasons to urge for regarding this Manual as in any sense a substitute for Euclid's—as in any sense anything else than a revised edition of Euclid?
Nie. We have nothing more to say.
Min. Then I can but repeat with regard to this newborn 'follower' of the Syllabus, what I said of the Syllabus itself. Restore the Problems (which are also Theorems) to their proper places; keep to Euclid's numbering (interpolating your new Propositions where you please); and your new book may yet prove a valuable addition to the literature of Elementary Geometry.
[A tremulous movement is seen amid the ghostly throng. They waver fitfully to and fro, and finally drift off in the direction of one corner of the ceiling. When the procession has got well under way, Niemand himself becomes hazy, and floats off to join them. The whole procession gradually melts away into vacancy, Diamond going last, nibbling at the heels of Nero, for which a pair of gorgeous Roman sandals seem to afford but scanty protection.]