P. 12, Def. 34. Is not 'identically equal' tautology? Things that are 'identical' must surely be Again, every part of one being equal,' &c. mean by every part' of a rectilineal figure? Nos. Its sides and angles, of course. equal' also. What do you Min. Then what do you mean by Ax (b) in p. 3. 'The whole is equal to the sum of its parts'? This time, I think I need not 'pause for a reply '! P. 15, Def. 38. When a straight Line intersects two other straight Lines it makes with them eight angles etc.' Let us count the angles at G. They are, the major and 'minor' angles which bear the name EGA; do. for EGB; do. for AGH; and do. for BGH. That is, eight angles at G alone. There are sixteen altogether. P. 17, Th. 30. If a quadrilateral has two opposite sides equal and parallel, it is a parallelogram.' This re-asserts part of its own data. P. 17, Th. 31. 'Straight Lines that are equal and parallel have equal projections on any other straight Line; conversely, parallel straight Lines that have equal projections on another straight Line are equal.' , The first clause omits the case of Lines that are equal and in one and the same straight Line. The second clause is not true: if the parallel Lines are at right angles to the other Line, their projections are equal, both being zero, whether the Lines are equal or not. P. 18, Th. 32. 'If there are three parallel straight Lines, and the intercepts made by them on any straight Line that cuts them are equal, then etc.' The subject of this proposition is inconceivable: there are three intercepts, and by no possibility can these three be equal. P. 25, Prob. 5. 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.' May I ask you to furnish me with the solution of this problem, taking, as your 'given rectilineal figure,' a triangle? Nos. (indignantly) I decline to attempt it! Min. I will now sum up the conclusions I have come to with respect to your Syllabus. In the subjects of Lines, Angles, and Parallels, the changes you propose are as follows: You give a very unsatisfactory definition of a 'Right Line,' and then most illogically re-state it as an axiom. You extend the definition of Angle-a most disastrous innovation. Your definition of Right Angle' is a failure. You substitute Playfair's axiom for Euclid's 12th. All these things are very poor compensation indeed for the vital changes you propose the separation of Problems and Theorems, and the abandonment of Euclid's order and numeration. 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 Syllabus may yet prove to be a valuable addition to the literature of Elementary Geometry. 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 Nie. It abandons diagonals in Book II Min. Discussed (see p. 49). 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. evidence indeed to justify its It will need very strong 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. 6 Min. In the 'Introduction,' at p. 2, I read A Theorem is the formal statement of a Proposition,' &c. Discussed at p. 163. At p. 3 we have the Rule of Conversion,' which I have already endeavoured to understand (see p. 163). 6 At p. 6 is a really remarkable assertion. Every Theorem may be 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 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. 159). At p. 9 is our old friend the straight angle' (see p. 75). In the same page we have that wonderful triad of Lines, one of which is 'regarded as lying between the other two' (see p. 159). And also the extraordinary result that follows when one straight Line stands upon another' (see p. 160). |