296 



NATURE 



[July 30, 1903 



Spirals in Nature and Art. 



HAVE to thank you for a very kind notice of my little 

 ly on spirals, and I venture to trouble you further on the 

 subject, because your last paragraph, criticising my attribu- 

 tion of spiral curves in flight to Leonardo, gives me an 

 opportunity of making a correction to which, I feel sure, 

 vour courtesy to a distinguished scientific writer will enable 

 me to give publicity. It appears that, in pp, 153 to 155 

 of my study of spirals, and in the figures 45 and 46 therein 

 included, I have unconsciously done an injustice to the 

 original researches on flight published by Dr. J. Bell 

 Pettigrew, M.D., LL.D., F.R.S., Chandos professor of 

 medicine and anatomy at the University of St. Andrew's, 

 who,_ I now find, has been steadily engaged on the problem 

 of flight since 1867, and has apparently published many 

 papers and memoirs on the subject in the Proceedings of 

 the Royal Institution of Great Britain, the Transactions 

 of the Linnean Society and of the Royal Society of Edin- 

 burgh, and elsewhere. 



My figure 45, which you acutely ascribe to its right 

 author, is of very little importance to my argument, and 

 only a side-issue in my essay, but it is right to say that 

 it is Dr. Pettigrew 's original figure, and should have been 

 acknowledged as such in my pages. Had I known of this, 

 I think I need hardly assure you that this acknowledgment 

 would have been inserted, and that Dr. Pettigrew 's own 

 explanation of the figure would have been substituted for 

 what he would justly stigmatise as the incorrect explan- 

 ation given in my text. I have also to add that Prof. 

 Marey's photograph of a flying pigeon, which I attributed 

 to the only source I knew, was really an illustration of the 

 alternate and opposite rise and fail of the body and the 

 wings of a bird in flight, a principle first described and 

 figured by Dr. Pettigrew in his memoir on " The Physio- 

 logy of Wings " (Trans. Roy. Soc. Edin., 1870), and 

 acknowledged by Prof. Marey as a previous discovery. 

 Theodore Andrea Cook. 



Distribution of Calostoma. 



In December, 1891, I found in a pit near Port Katsura, 

 a few miles off this place, a species of Calostoma in abund- 

 ance, and this year I see the same fungus now and then 

 occurring here. I send you some specimens of it herewith, 

 in the hope that some mycologist of your acquaintance may 

 determme it in my behalf. Of all the species given in Mr. 

 Massee's monograph of the genus in the Annals of Botany, 

 vol. li. 1888, it seems most near C. Ravenelii, Mass. 



If my memory deceives me not, Mr. Massee, in the same 

 paper, divided the genus Calostoma into two groups, the 

 so-called eastern group, growing in Asia and the adjacent 

 islands, with globose spores, and the western group, the 

 habitats of which are America and Australia, with elliptical 

 spores. Now the Japanese species in question has its spores 

 oblong-elhptical, which fact would seem to necessitate such a 

 naming of the groups as eastern and western to be modified 



"^^? O"" l^fS. KUMAGUSU MiNAKATA. 



Mount Nachi, Kii, Japan, June 5. 



The specimens of fungi from Japan belong to Calostoma 

 Naveneht, Mass., agreeing in every essential point with 

 the type of that species preserved in the herbarium at 

 Kew. 



In the monograph referred to in the letter accompanying 

 the specimens, the form of the spores was not made a basis 

 of classification, but the fact was simplv pointed out that 

 eastern species possessed globose spores, whereas in all 

 known western species the spores were elliptical. 



The fact of a North American species occurring in Japan 

 while very interesting, will not cause surprise to botanists 

 considering the intimate relationship between the phanero- 

 gamic flora of the two countries. Geo. Massee. 



School Geometry Reform. 

 In your issue of June 25, Mr. R. W. H. T. Hudson 

 criticises the fact that, in my "Elementary Geometry," 



NO. 1 76 1, VOL. 68] 



I give three meanings of the word angle, the third 

 being what may be called the " sector of plane 

 space " meaning. 



He considers that, even if not wrong, it is undesirable 

 in a school book. It seems to me that the one essential 

 point which requires attention in introducing a new subject 

 to boys and girls is to attach a clear, definite meaning to 

 the terms employed, and that, if there be any terms such 

 as this word " angle," of which many people have confused 

 notions owing to the bringing together and blurring of 

 two or three distinct meanings, then those meanings should 

 be carefully dissected. 



Mr. Hudson quotes with approval the French writers 

 who, while stating that an angle is a simple undefinable 

 idea, incidentally give " inclinaison mutuelle " as a 

 synonym ; personally, I am adverse to the word " inclin- 

 ation, " it seems to mean a "leaning towards one 

 another," whereas an angle is a "leaning away from 

 one another," if it be a leaning at all. I have endeavoured 

 to express this idea in my second meaning, viz. the " wide- 

 ness " of the opening between two radii drawn from a 

 point. 



That the space-sector meaning is implied in nineteenth 

 century Euclids is indisputable, e.g. in iii. 20 we have 

 " Case i., when the centre is within the angle " — how could 

 the centre lie within a " mutual inclination " or within " an 

 amount of turning "? Again, " a solid angle is . . . made 

 by . . . plane angles . . . meeting at one point " — how 

 can " mutual inclinations " meet? I doubt even if a 

 " mutual inclination " is more capable of being bisected 

 than is any other abstract quality, say, for example, 

 gratitude. 



Mr. Hudson speaks of the axiom, " whole is greater than 

 its part " : surely this is no axiom at all ; it is a definition, 

 whether of " a part " or of " greater than " I would not 

 venture to say. 



Whether my position be right or wrong, it is surely 

 preferable to the attitude which makes geometry the 

 " science of the undefinable." 



I am grateful to your reviewer for the suggestion that 

 angles should be quoted in decimals of a degree rather than 

 to the nearest ten minutes, and will adopt the suggestion 

 as soon as possible. 



Frank R. Barrell. 



University College, Bristol, July 6. 



The Moon's Phases and Thunderstorms. 



In connection with the note in Nature (July 9, p. 232), 

 it is interesting to compare the results of Prof. W. H. 

 Pickering with those obtained by Schiaparelli in 1868, from 

 the discussion of observations made in Vigevano (north 

 Italy) for thirty-eight years (1827-1864) by Dr. Siro 

 Serafini. 



" Sebbene i numeri della seconda colonna presentino delle 

 grandi irregolarita nel loro andamento, sembra tuttavia 

 indubitato, che nella prima met^ della lunazione i temporali 

 debbano in generale essere meno frequenti che nella 

 seconda. Facendo la somma di 5 in 5 per veder meglio la 

 legge di progressione, si vede che il minimum cade verso 

 il 5° giorno della lunazione, il maximum verso il 24°. E la 

 proporzione della frequenza minima alia massima fe quella 

 di loi : 153, ci6 h quasi esattamente di 2:3." 



Translated into English, the quotation reads as follows : — 

 " Although the figures of the second column show great 

 irregularities in their proceeding, it seems nevertheless 

 undoubted that in the first half of a lunation thunder- 

 storms may be, generally speaking, less frequent than 

 in the second. Adding 5 by 5 in order to see better 

 the law of progression, one remarks that the mini- 

 mum falls towards the 5th day of the lunation and the 

 maximum towards the 24th. The ratio of the least 

 frequency to the greatest is that of loi : 153, or almost 

 exactly of 2:3." (Clima di Vigevano: Milano Vallardi, 

 1868, p. 81.) 



The conclusion is thus exactly the reverse of what Prof. 

 W. H. Pickering has found. 



Ottavio Zanotti Bianco. 



