July lo, 1879] 



NATURE 



241 



We pass over many passages we had marked, with 

 saying that in many cases the objections are sound but 

 trivial. Objection is taken to Mr. Wilson's remark, 

 " Every theorem may be shown to be a means of in- 

 directly measuring some magnitude," and Niemand 

 abandons "every." We think, however, that Niemand 

 might have made a better fight of it and suggested that 

 what is intended is that, for instance, all the theorems of 

 the first book are directly or indirectly required for the 

 proof of the 47th Proposition, which is surely a proposi- 

 tion concerned with the measurement of magnitude. 



On p. 177 Minos says of the exercise, "Show 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 a/igks—an antiquated prejudice, no 

 doubt ; but it is difficult to eradicate these childish 

 instincts." Mr. Dodgson was taught that the t/trei 

 angles were equal to this magnitude ; the question says 

 '■' angles " surely in the plain sense of each angle being 

 equal, &c. Again, in the construction for proposition 

 corresponding to Eucl. i. 9 objection is taken to "finding 

 a radius greater than half AB" (it should be A Q: "it 

 would seem to require the previous bisection of A B'' 

 {AC). Thus the proof involves the fallacy " Petitio 

 PrincipiiP Surely one can take a line greater than or 

 equal to A C; where, then, is the fallacy ? Exception is 

 taken to the proposition " the area of a trapezium 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 " as being " a mere 

 'fancy' proposition of no practical value whatever." We 

 have met with it in works on co-ordinate geometry and 

 elsewhere. Then again the theorem (ApoUonius's) on 

 Mr. Wilson's p. 95 is branded "new," "but even with 

 that mighty name to recommend it, I cannot help think- 

 ing it rather more curious than useful." It is our own 

 impression that it is one of the most important "riders" 

 from the second book, and if Mr. Dodgson has been 

 teaching geometry for nearly five-and-twenty years, so 

 have we— but we do not confine our teaching to the text- 

 book only, we devote a great part of our geometrical 

 teaching time to the working of exercises. 



Our conclusion from the examination of Mr. Dodgson's 

 objections to Mr. Wilson's last book is that the majority 

 of them can be easily met; indeed, many of them are 

 lere verbal quibbles ; the rest arise from the very dif- 

 ferent standpoints taken up by the two writers, and here 

 there is likely to be "war to the knife." 



A word or two on Morell's (J. R.) "Euclid Simplified." 

 It is very easy work to pick this little book to pieces, but 

 we cannot understand a statement of Mr. Dodgson's on 

 p. 148. Of the proposition " Every convex closed line 

 .1 B CD enveloped by any other closed Xme PQRSTxs 

 less than it," he says the method used fails, " as of course 

 all methods must, the thing not being capable of proof." 

 We cannot call to mind any English text-book in which 

 the proposition is proved, but there is what we have 

 thought was a proof in Sannia and D'Ovidio's " Elementi 

 di Geometria," p. 32. 



We are bound to say that " Euclid and his Modem 

 Rivals " is not all amusing reading. It alternates 

 " From grave to gay," 



and more than a third part is devoted to appendices, the 

 third to the sixth of which (73 pages) must have cost the 

 author a great deal of thought and labour. We fear, 

 however, it will not get the attention it deserves. It is 

 hard reading, and one has hardly been led up to it by the 

 amusement provided in the four Acts of the Drama. 

 Some little trouble is involved in mastering the symbols 

 and their significance. 



The fourth act considers the objections brought by Mr. 

 Wilson ("EucUd as a Text-Book," iScc.) and others 

 against the use of Euchd for junior pupils on the score of 

 unsuggestiveness and want of simplicity of style, the 

 exclusion of hypothetical constructions, &c. We need not 

 consider them here, but refer to two articles by the Rev. 

 Dr. Jones (" On the Unsuitableness of Euclid as a Text- 

 Book of Geometry," Trans, of Liverpool Lit. and Phil. 

 Society, published in a separate form ; and " Review of 

 Mr. Todhunter's Essay on Elementary Geometry," 

 {Monthly Journal of Education, 1875, pp. 97-112, 150- 

 160), neither of which is referred to by our author, though 

 he quotes largely in the appendix from Mr. Todhunter's 

 Essay and also from a review of Mr. Wilson's first 

 Geometry in the Athcnaum for July 18, 1868, written by 

 Prof. De Morgan. We could instance other geometries 

 which have an equal claim to be considered with any of 

 those criticised by Mr. Dodgson, and we should rather 

 have written " Euclid and some of his Modern Rivals." 



LETTERS TO THE EDITOR 



\The Editor does not liold himself responsible for opinions expressed 

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The Papau or Papaye 



In Nature, vol. xix. p. 447, is a paragraph relative to the 

 singiilar qualities of the Carica papaya. I cannot but think that 

 some of the properties attributed to this vegetal in British 

 Guiana by the natives of that colony are exaggerated somewhat, 

 e.g., the tempering of steel by its sap, &c. 



Sir Wyville Thomson, in the first volume of " The Voyage of 

 the Challenger," gives a capital representation of a group of 

 these papaw-trees in the garden of the Admiral commanding on 

 the North American station at Clarence Hill, Bermudas, where 

 they seem to abound ; I do not know if these dicecious plants are 

 indigenous to these islands or introduced from the West Indies 

 and tropical America. From the cut above mentioned can be 

 seen the quaint grow th of these paradoxical trees, which must 

 have- been esteemed by the early voy.igers, as they have been 

 introduced into all parts of the tropics. The singular-looking 

 straight stems (not unlike the gigantesque tree-cabbage stalks of 

 the Channel Islands) are crow ned with a tuft of digitate leaves, 

 somewhat at a distance resembling those of the Aralia fapy- 

 rijera, under which the clusters of black purple fruit protrude. 

 In the islands of Bourbon and Mauritius they make a passable 

 compote of these fruit, which are pulpy and full of black seeds 

 when ripe, and the Creole children eat them raw, with what 

 effect on their insides I know not ; the birds, however, will not 

 touch them, and as they fall they rot on the ground beneath. 

 In Mauritius, where we lived principally on ration beef cut from 

 the tough flesh of Malagasy oxen, we were in the habit of hanging 

 the ration under the leaves of the tree itself, and if we were in a 

 hurry for a very tender piece of filet, our cook would wrap up 

 the undercut of the sirloin in the leaves, when the newly-killed 

 meat would be as tender as if it had been hung for a considerable 

 time. Whence arc these deleterious effects causing rapid de- 

 composition of animal fibre ? and are there any other trees which 

 possess similar properties ? . 



