26 



NATURE 



\Nov. 9, 1 871 



as all astronomers know, by the direction of the line of nodes, 

 not by its Uitiial place. 



By-lhe-bye, Sir John Herschel is sometimes very careful to 

 use the words " actual place " where my criiics contend that the 

 word " position " would be sulTiciently definitive. 



It seems overlooked that I pointed out in the beginning that 

 "position" was often but erroneously used as synonymous with 

 " place." It is not my fault if this error appears in the technical 

 use of the word "position" in some mathematical treatises. I 

 say again with Colonel Mannering, Abusiis iwt: tellit iisitm — 

 " The abuse of anything doth not abrogate the lawful use thereof." 

 It was a /ii/sus calami of mine to say that " position " could not 

 be niisundcrsiood. It could be, for it has been misused. 



Frof. Ilir.st is quite right in saying I should be unable to de- 

 scribe the aspect of a horizontal plane. I should not think of 

 trying to. lie says, however, that Mr. Wilson would unhesiia- 

 tirgiy pronounce its aspect veilical. (Does it look vertically /// 

 or vertically i/tni'ii .') What would Mr. Wilson assign — un- 

 hesitatingly or otherwise— as the aspect of the "prime vertical " ? 



lias a true plane {as distinguished from a plane face of a solid) 

 one aspect or two ? It lias one posilioii or situation, and one 

 place or localioit, but I conceive that it has two aspects. 



Mr. Laughton seems quite unaware of Sir J. llerschel's re- 

 peated use of the word "tilt." 



His comment on my remark about the books which I have 

 written is unworthy. He must surely perceive that I only sought 

 to indicate how much occasion I had had to consider the subject 

 of plane-position ; more occasion, I think, than any of my critics, 

 save Prof. Hirst, the weight of whose opinion I recognise fully, 

 though 1 cannot agree with him. But 1 have net felt free to use 

 the word "position" so systematically as I should wish, pre- 

 cisely because of its misuse to indicate place. 1 have only been 

 able to use it where there could be no fear ol tliat wrong meaning 

 being assigned to it. 



As I claim no credit for the invention of any word for indi- 

 cating plane-position, and as 1 could not take from Mr. Laughton 

 that which is not his — the credit for Hamilton's word " aspecc " 

 — perhaps I may be permitted to say that if I am " pertinacious " 

 (as Mr. l-p.nghton asserts) there is nothing personal in my per- 

 tinacity. It is not my custom to admit that I am wrong when I 

 consider that 1 am right. 



[My objections to the word "aspect" are confirmed by Mr. 

 Wdson's letter. I wrote that the word could not be used 

 in the sense indicated, "unless a new and artificial meaning 

 were assigned to it." Mr. Wilson obligingly proves this by 

 assigning to it just such a meaning. "Tne aspect of a plane is 

 the direction of its normal," it would seem. Now no special 

 objection need be urged against this definition, if it is to be con- 

 fined ligidly within the limits of mathematical text-books. The 

 definition is strange and arlihcial no doubt ; but it is nothing 

 new to see the familiar and natural banished from such works. 

 As a writer on astronomy, however, 1 must decline to accept the 

 proposed usage, whicli seems to me altogether objectionable. If 

 I wiite respecting ihe ccUstial equator-plane that **its position is 

 at right angles to the polar axis of the heavens," I find that I am 

 undc stood ; but 1 am sure my readers would be very much per- 

 plexed if I wrote that " the aspect of the equator-plane is the 

 direction of the polar axis." Again, I shouhl be undci stood, I 

 think, if 1 said that " the positions of two hour-planes determine 

 the direction of the polar axis," or ihat "the diieciions of the 

 polar axis and the vertical determine the position of the meridian- 

 plane." But if I wrote "aspect" where I have here written 

 "position," I scarcely know what my readers would think. 



By the way, what would be the "aspect" of the meridian- 

 plane according to the proposed usage ? Would it be " east" 

 or " west " ? The normal to that plane would lie east and west ; 

 but we could not hear of an " east-and-west " aspect without 

 thinking of certain "clearstories towards the south-noith, lustrous 

 as ebony." 



I am bound to point out, however, though I may seem to 

 weaken my position by doing .so, that a very eminent .luihoriiy 

 long since used the word " aspect " in the sense suggested by Mr. 

 Laughton. In one of his well-known " Leiteis to a Lady," on 

 quatcmions. Sir W. R. Hamilton uses the words "position, ' 

 "slope," "ledge," and ".aspect," to express the relations which 

 1 have called respectively "place," "slope," "aspeet," and 

 "position." (See r^ichol's "Cyclop.tdia of the Phy.'-ical 

 Sciences," 2nd ediiior, p. 70S.) 1 app ehind, however, ihat he 

 lays no special stu-ss on tins verbiage. H hid used the word 

 " position " lor " p.ace," and thib left him wi.hout any word to 

 indicate position. Besides, liis illustrative plane is the surface of 



a desk, and a surface may be conceived to have an aspect de- 

 finable by the direction of its normal, but a geometrical plane is 

 two-faced. ] * 



This is my last letter on the present subject— unless one of 

 your correspondents should employ arguments showing me to be 

 in error, in which case I shall crave two lines of your space to 

 admit as much. RicHD. A. PitocroR 



Brighton, Nov. 3 



P.S. — Let it be noticed that the question is not how the word 

 "position " has been used by some, but how it ought to be used 

 by all. 



I CANNOT agree with Mr. Wilson that "aspect" is exactly 

 the word wanted. The same wall has te'o aspects ; if a southern, 

 then also a northern aspect on the o'her side. In fact the word 

 seems adapted, according to its common usage, to express the 

 "sense" (sens), as well as the direction of the plane's normal, 

 whereas I t.ike it that the word sought for should express the di- 

 reciion only without connoting the " sense." 



1 think a word sometimes used by geologists would be, if we 

 dare use it, exactly the word. As they speak of the he of strata, 

 defined (with resi-iect to the horizon) by its two elements, strike 

 an<l (lip, so geometers might well speak of the Iteoi a plane ; but 

 would our English language permit us to say that "two lies de- 

 termine one direction," and " two directions determine one lie "? 

 I (ear the moral connotation of the word, although an etymo- 

 logical accident, is too ugly. 



If we are reduced to coin a new word, I would suggest that 

 the Latin root "pand" (spread), would afford for a plane the 

 fitting analogue of the root "reg" (rule, make straighi), for a 

 line', and so the word "dispansion" would be the an.alogue of 

 "direction." "Parallel planes h.ave the same dispansion." 

 "Two dispansions determine one direction, and two directions 

 determine one dispansion." Will not the neatness of this mode 

 of expressing Mr. Wdson's test propositions atone for the strange- 

 ness of the word ? 



The word "aspect," however, is too good to be rejected from 

 geometrical science, th.iugh I believe its chief use will be found 

 beyond the domain of pure geometry. Should it not be appro- 

 priated to cases where the plane presents different aspects to the 

 portions of space on either side of it ? For instance, if two 

 bodies revolve in the same or parallel planes, their orbits might 

 be said to have the same or contrary aspects, according as ihe 

 bodies revolve in the same or contrary directions, and so the posi- 

 tive aspect of a planet's orbit would determine, not only the 

 "lie" or " dispansion " of the plane of ihe orbit, but also the 

 direction of revolution in that orbit. So, too, the statement that 

 all the planetary orbits have nearly the same aspect, would 

 imply not only that their planes nearly coincide, but also that 

 they all revolve in the same direction. I cannot help thinking 

 that Mr. Proctor would find his account in adopting this sense of 

 the word " aspect " in his astronomical wriings, especially since 

 he might, as Dr. Hirst suggests, retain the word where he has 

 hitherto employed it, by simply qualilying it with an appropriate 

 adjective. (Would the adjective "azimuti.al" satisfy him? 



May I conclude with a question which I have ofien wished to 

 propound ? What is the proper English equivalent for the French 

 ^'siiis"'! English mathematicians generally seem shy of using 

 the word "sense,'' while, to use theword "direction" as well for 

 the " sens" as the " direction" of a line, is very awkward and 

 inconvenient. The difficulty, I imagine, is the same as appears 

 to me almost fatal to the word "lie " proposed above, namely, 

 that the proposed technical use diverges too widely from the 

 familiar use of the word. Is not the superior flexibility of the 

 German language in the formation of new terms in part due to a 

 less degree of fastidiousness in this respect ? 



Harrow, Nov. 6 Robt. B. H.-\YWARn 



After all, I fear the word " aspect " is not quite the right thing. 

 What is wanted is a word to express " plane-direction ;" some- 

 thing in the plane, and not looking out from it. And I am not 

 sure that the compound word 'plane-direction," whii.h is not 

 ambiguous nor colloquial, will not be better even than "aspect." 



We should then have axioms on planes analogiius to those on 

 straight lines : that planes may have the same or different plane- 

 directions : that intersecting planes have diflerent plane-directions ; 

 and conversely. 



Par.ibel p anes will be defined as those which have the same 

 plane-direction. 



* The matter between brackets was written on October 37.— Ed. 



