NA TURE 



[Dec. 2, 1886 



LETTERS TO THE EDITOR 

 [The Editor Joes not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake to 

 return, or to correspond with the writers of, refected manu- 

 scripts. No notice is taken of anonymous comtnunications. 

 [ The Editor urgently reqtiests correspondents to keep their letters 

 as short as possible. The pressure on his space is so great 

 that it is impossible otherwise to insure the appearance even 

 of communications cotitaining interestijig and novel facts.~\ 



Longitudes in Brazil 



Le numero du 18 novembre de Nature publie un article du 

 professeur Young sur les progres de I'aslronomie depuis dix aiis, 

 dans leqiiel il est dit que les observations de longitude tele- 

 graphiques des officiers aniericains ont corrige une erreur de 

 8 '545. sur Lisbonne, et une bien plus etonnante encore de 35s. 

 sur Rio. 



II y a la une grosse erreur inexplicable de la part du profes?eur 

 Young, contre laquelle je dois protester comme auteur des cartes 

 hydrographiques du Bresil encore employes aujourd'hui, et 

 auteur de toutes les determinations geographiques relatives et 

 alKolues faites douze ou quinze ans avant la mission americaine 

 de MM. Green et Davis pour les longitudes telegraphiques entre 

 le Bresil et I'Europe. 



Sur les mille lieues de cote du Bresil la mission americaine a 

 determine six longitudes entre le Para et Buenos Ayres. Voici 

 la comparaison des resultats obtenus par MM. Davis et Green, 

 ^ I'aide du telegraphe, et par moi, a I'aide de chronometres et 

 d'observations astronomiques directes. Les observations ameri- 

 caines sont publiees dans le numero 59 (1880, je crois) " Hydro- 

 graphic Notice," et les miennes dans les " Annales hydro- 

 graphiques, 1866." 



Para Pernatnbouco Bahia 



li. m. s. Lh. m. s. h. m. s. 



Long. telegrapTiique. 3 23 20'94 2 2S 48'6 2 43 296 



Long. Moiichez ... 3 23 i8'67 2 28 47'5 2 43 26'9 



Erreur 



-2 •27s 



Montevidei 



-27s. 

 5 Ayres 



4 2 49-9 

 4 2 49-9 



Long, telegraphique. 322-3 3 54 9-9 

 Long. Moucbez ... 3 2 o'l 3 54 9-4 



Erreur -2'2s. -0'5s. o'os. 



II resulte de ce tableau que la plus grande erreur que j'ai 

 commise est -2'7s. sur Bahia. A Rio I'erreur est de -2'2s., 

 et non de 35s. comrae le pretend M. Young. Dans le Rio de 

 la Plata I'erreur a ete trouvee nulle. 



Je ne crois pas qu'aucune etendue de cote de mille lieues eut 

 jamais presente moins d'erreur absolue ou relative que la cote 

 du Bresil apres la publication de mes cartes et de mes observa- 

 tions. 



Quant a I'erreur sur Lisbonne je I'avais signalee depuis plus 

 lie trente ans, elle etait connue. 



Je vous serais tres oblige de vouloir bien publier au moins le 

 tableau comparatif des longitudes que j'ai I'honneur de vous 

 envoyer aujourd'hui, pour protester contre I'erreur qui m'est 

 indirectement imputee. 



Veuillez agreer I'assurance de ma parfaite consideration. 



E. MOUCHEZ 



Cooke's "Chemical Physics" 

 I AM told that I have been the object of severe strictures in 

 your journal for republishing my old " Chemical Physics " as if it 

 were a new book. It is a sufficient answer to say that the book 

 was stereotyped when first issued in i860, and that there has 

 never been any pretence on my pai t that it has been revised 

 since. I find, on inquiry, that the American publishers have 

 made, since the first edition, three reprints from the plates, and 

 have called these reprints second, third, and fourth editions, 

 changing, with each issue, the date on the title-page ; a usa^e 

 which I regard myself as reprehensible, but which must be 

 sanctioned by the trade since it is so universally followed. All 

 this lime, however, the date accompanying my signature after 

 the preface, and the date of the copyright, have remained 

 unaltered. I had supposed the book entirely out of print ; and 

 the last reprint of a very few copies to meet a small demand 

 still existing, chiefly in England, was made entirely without my 

 knowledge or consent. On its very face the whole aspect of the 



book is antiquated ; but in it there was brought together certain 

 positive knowledge in connection with the weighing and measur- 

 ing of aeriform matter, derived chiefly from the classical researches 

 of Regnault, which is still of great importance and not readily 

 found elsewhere ; and this is, unquestionably, the reason of the 

 continued demand for a compilation made more than twenty- 

 five years ago. I have, until within a few years, had the expec- 

 tation of revising the book and presenting the old facts in their 

 new dress, but the failure of my sight has obliged me to give up 

 the plan, and younger men must do the work. 



JosiAH Parsons Cooke 

 Cambridge, U.S.A., November 16 



Note on Mr. Budden's Proof that only One Parallel can 

 be drawn from a Given Point to a Given Straight Line 



Mr. Budden's paper in the last number of Nature (p. 92) 

 is full of inaccuracies of a more or less serious character. With- 

 out pointing out these, I wish to show that the essential idea 

 which underlies his reasoning is altogether wrong, as it is based 

 on the *' infinite," which be introduces in the most innocent 

 manner by letting his figure grow without limit, and about which 

 he then calmly reasons as if he still dealt with a finite figure. 

 If we let a quantity " increase without limit," we get a quantity 

 which has increased beyond our comprehension, and no one in 

 his senses will wittingly and seriously draw conclusions from 

 wh.-it he does not comprehend. Here we might stop, were it 

 not that the constant use in modern mathematics of the infinite 

 (both the small and the great) has made us so familiar with it 

 that an attempt to base an elementary proof on it might seem 

 to many a very natural thing. 



In algebra, the infinite number is shown to have one property 

 which we can comprehend, viz. that its reciprocal is zero ; and 

 with this property alone we work safely. 



In modern geometry, on the other hand, the infinite is used 

 as a kind of shorthand, which enables us to make long state- 

 ments short, and, at the same time, general. Taking the axiom 

 about parallels for granted, it is shown that all points at an in- 

 finite distance in a line may be taken to be one point as far as 

 constructions at a finite distance are concerned. For all lines 

 joining a fixed point, P, to any point at infinity in a line may be 

 taken as parallel to this line, and therefore as coincident. To 

 express this more shortly, it is said that the whole indefinite and 

 infinite part of a line which is out of the reach of our compre- 

 hension plays for us only the part of a single point, and accord- 

 ingly it is called a " point," viz. the point at infinity of the line. 

 Similarly it is shown that all points in a plane which are at an 

 infinite distance may be considered as lying in one line, which 

 is then spoken of as the line at infinity in the plane, and which 

 is freely and safely used in deducing theorems and solving 

 problems. 



If, then, a line in a plane be moved to infinity, making always 

 a given angle with a fixed line, it will ultimately become 

 coincident with — which here means indistinguishable from — the 

 line at infinity. The latter then makes wiih the fixed line a 

 given angle. But this angle may be anything. Hence the "line 

 at infinity " makes any angle we like with any given finite line ; 

 in other words, it makes no definite angle at all with it. 



It follows, if we take a property of a figure which depends 

 upon the magnitude of an angle, that this property will not neces- 

 sarily any longer hold if one of the limits of the angle be moved 

 to an infinite distance ; for then this angle has not any longer a 

 definite magnitude. To base any reasoning on that properly 

 after the figure has been indefinitely increased must therefore 

 necessarily be fallacious. Eut this is exactly what Mr. Budden 

 does. His proof is based on the implied assumption that if a 

 figure in a plane be increased indefinitely, we can still reason 

 upon it as if it were finite. He may take this as an axiom, but 

 then he has replaced Euclid's axiom by another, and has not 

 proved it ; and the question would arise, Which form of the axiom 

 is preferable ? I prefer Euclid's. O. Henrici 



Lunar Glaciation 



I TRUST you will allow me a small space to explain regarding 

 this theory of lunar glaciation, referred to by Mr. Darwin in 

 Nature (vol. xxxiv. p. 264). 



First, I must thank him for the remarks made, and say that I 

 certainly was not aware that Capt. Ericsson had been at work 

 in the same direction some ten years or more before me. 



