November 3, 1892] 



NA TURE 



1 1 



and " Newton " are generally considered to yield a more valuable 

 mental training than such subjects as analytical geometry is that 

 the older authors, perhaps because they were a bit afraid of 

 purely symbolic argument, tried constantly to keep real pictures 

 and ideas before the minds of their readers. But even so our con- 

 viction of the truth of any but the simplest theorems of geometry 

 depends chiefly on the symbolic argument, not on the realization 

 in succession of the actuality of the relations and operations 

 discussed in the course of the proof This is perhaps sufficiently 

 obvious in the higher branches of even Euclidian geometry, but 

 it becomes absolutely indisputable when we reach such theorems 

 as " Any two conies in one plane intersect in four points." Not 

 only may some of the-e points be at an infinite distance, but 

 some, or all, may be what is called, on the luctts a non lucendo 

 principle, "imaginary"; that is, they may be such that they 

 cannot be imagined by anybody, much less actually drawn. 



Accordingly I cannot admit that the theorems of geometry 

 are established by induction at all. If they are interpreted in 

 either of the first two ways I have described, they are only 

 particular propositions, and the inference from them to a general 

 proposition would no more yield a "mathematical certainty " in 

 this case than in any other. And though the third way of 

 looking at the proposition may be paraphrased into a form 

 which appears general {e.g., anything which may fairly be called 

 "an isosceles triangle" may also be said " to have two equal 

 angles"), it is really only a particular proposition about the 

 words " isosceles triangle," and so on. Its wide applicability 

 and usefulness depends on the fact that we can, and do, often 

 find things which can fairly be called isosceles triangles ; but it 

 must be admitted that the assertion that, on any given occasion, 

 we have found such a thing,- — -is not a mathematical certainty. If 

 the triangle in question is an objective one, we can only say that 

 it is probably, or approximately, isosceles ; and though perhaps 

 we may subjectively conceive j erfectly isosceles triangles, and 

 so regard the/^;w asinorum as a subjective necessary truth, it 

 must be doubtful whether we could do so in the case of a more 

 complex proposition such as Pascal's Theorem, and it is quite 

 certain that we could not d > so in the case of such theorems as 

 that about the intersections of two conies. 



It is to be hoped, therefore, that logicians will come to 

 recognize the importance of symbolic reasoning, as mathe- 

 maticians have already done. And when they do so we may 

 hope for this further advantage, that they in turn will teach 

 mathematicians and others not to confuse a purely symbolic with 

 a real conclusion — not to assume that, because they have 

 correctly proved a conclusion symbolically, that it therefore 

 necessarily gives any information about real things, or even real 

 concepts. Edward T. Dixon. 



Trin. Coll., Cambs., October 22. 



Bell's Idea of a new Anatomy of the Brain. 



In Nature of October 27 the writer of the review of Mr. 

 Horsley's "Structure and Functions of the Brain," speakingof the 

 rarity of the above book, states that he only knows of one copy 

 in London, viz., that in the British Museum. It may be useful 

 to some of your readers to know that there is a very interesting 

 copy ill the library of the Royal College of Surgeons, It is the 

 presentation copy to Dr. Roget " from Mr. C. Bell, 34, Soho 

 Square " : by Dr. Roget it was given to Lady Bell, who pre- 

 sented it to the Royal College of Surgeons through Mr. Alex- 

 ander Shaw. 



Mr. Shaw has added in MS. a copy of the letter received 

 from the printers fixing the original date of publication, and also 

 the list of persons to whom presentation copies were sent. The 

 letter and the list are both published in Mr. Shaw's reprint of the 

 Tract in the Journal of Anatomy, \o\. iii., 1869. 



Jas. B. Bailey, 



October 27. Librarian Roy. Coll. Surgeons. 



Photographic Dry Plates. 



In reference to "Prevention's" note on Photographic Dry 

 Plates, one cannot but agree with him that packets should be 

 dated when issued from the factory. 



I would venture, however, to suggest that good makers' plates 

 do not deteriorate within a reasonable length of time. 



As an illustration of my experience I may mention that in 

 April this year I opened a box of plates {\ plate Extra Rapid) 

 which I bought in July 1886. 



NO. I 201, VOL. 47] 



I had carried them on a three months' tour in the Mediter- 

 ranean in i888. and had taken no special care of them since. 



They proved in every way as good as new, both in sensitive- 

 ness, and perfection and evenness of film. 



Arthur E. Brown. 



THE GENUS SPHEIVOPHYLLUM. 

 VrOTWITHSTANDING the small size and compara- 

 ■'-^ tive scarcity of the plants belonging to this Palaeo- 

 zoic genus, they have long attracted a rather unusual 

 amount of attention. This has been partly due to their 

 peculiar external forms, which suggested even to the 

 earliest observers the idea of resemblances to the Marsi- 

 liae ; but the interest they have excited has been further 

 increased of late years by discoveries respecting the 

 peculiar organizations of their stems. In 1822 Adolph 

 Brongniart assigned to them the name of " Spenophyl- 

 lites," and in 1823 Sternberg figured some of them 

 under the generic title of " Rotularia.''^ Sternberg's 

 figures appeared in his " Versuch einer Geognostisch- 

 Botanischen Darstellung der Flora der Vorwelt," 

 which work is now best known through the French 

 translation of it by Comte de Bray. To the first of his 

 specimens figured {loc. cit., tab. xxvi., figs, ^a and b)y 

 Sternberg gave the name of Rotularia pusi/la, and the 

 example so designated is very characteristic of the 

 simpler type of the group, in which we have a somewhat 

 branched stem, with verticils of wedge-shaped leaves 

 at each node. A second form was figured on a later plate 

 of the same work. It is interesting to note that Stern- 

 berg associated with these figures the observation, 

 " Piantse organisatione foliorum Marsileis, forma caulis 

 Hippuri Maritimas." The generic name thus given by 

 this author represents the rotate arrangements of the 

 leaves in each verticil, as the wedge-shaped contour of 

 each separate leaf is further indicated by Brongniart's 

 generic term, " Sphenophyllites." In 1820 Von Schlot- 

 heim had also included similar examples in his too com- 

 prehensive genus, " Palmacites.'-* " 



In 1828 Brongniart published his classic " Prodrome 

 d'une Histoire des Vdgetaux Fossiles," in which work 

 we find the generic name of these plants changed to 

 Sphenophyllum, which name they have retained to the 

 present time. In this work Brongniart examines in some 

 detail the probable affinities of these plants, which even 

 in 1822 he inclined to regard as having some affiaities 

 with the Marsileae. He defines them as having six, 

 eight, ten, or twelve leaves in each nodal verticil, each 

 leaf being wedge-shaped ; sometimes entire, truncated at 

 its apex, which is denticulate. In some others these 

 leaves are bilobed, and in other species they are not only 

 profoundly bifid, but each of these lobes is either divided 

 into two, or their ends are laciniated. Lastly, in some 

 cases the lobes become narrow and linear. Brongniart 

 here compares these leaves with those of Ceratophyllum 

 and Marsilea, concluding with the statement, " We 

 cannot for the moment decide between these two rela- 

 tionships." At this date the fructification was wholly 

 unknown. 



In his introduction toithe" Natural System of Botany," 

 p. 37, Brongniart again reverts to the idea that Sphe- 

 nophyllum had Marsileaceous affinities. 



In 1831 the authors of the "Fossil Flora of Great 

 Britain" conmenced their publication of that work, 

 and in one of its early numbers they figured and 

 described under the name of Spenophylltim crosum 

 what appears to be identical with the first 

 figure published by Sternberg. When discussing 

 the relationships of this plant, Lindley and Hutton 



' These figures were preceded in 1709 by a still earlier one by Scheuchier 

 in his " Herbanum Diluvianum." (Cocmans and Kickz, " Monographie des 

 Sphenophyllum d'Europe "). ^^ 



2 "Die Petrefactenkunde auf ihrem jelziien Standpuncte, 



