456 



NATURE 



[June 9, 1921 



Letters to the Editor. 



\The Editor does not hold himself responsible for opinions 

 expressed by his correspondents. Neither can he undertake to 

 return, or to correspond with the writers of. rejected manu- 

 scripts intended for this or any other part of NATURE, ^o 

 notice is taken of anonymous communications.] 



Phenomena of " Intelligence " in the Protozoa. 



I REGRET to observe the spirit of the letter in which 

 Mr. Dunkerly (Nature, May 26, p. 395) replies to 

 Mr. Ludford, though, as being primarily responsible 

 for the statement of the theory suggested, and 

 periodically referred to of late years, 1 fully realise 

 how extremely careful one should be in the choice ol 

 words in conducting the discussion. There is prob 

 ably no theory occupying the attention of zoologists 

 in connection with which the motto of the Royal 

 Society, " NuUius in verba," applies with greater 

 force. Unfortunately, the "journalistic instinct" of 

 many writers on scientific subjects has led them to 

 credit observers with views which they have — to put 

 it mildly — not yet reached, and to saddle them with 

 responsibilities which they have never assumed. For 

 instance, in Prof. Boycott's letter on the same page he 

 credits my friend Earland with my views on "the 

 selective intelligence of the Foraminifera," which is 

 the one subject upon which my esteemed collaborator 

 does not entirely agree with me. 



The term "gregarious instinct" used by Mr. Lud- 

 ford is an unfortunate one. The " grouping " of 

 Protozoa to which he refers must be considered with 

 a cautious appreciation of the elements of (a) fear, 

 (b) reflex action, and (c) surface tension, but the most 

 indignant opponent of my views will scarcely deny 

 that the sense of fear is perhaps the most elementary 

 phenomenon dependent upon a sensory system. It is, 

 no doubt, related to, but it must not be confounded 

 with, the " intelligence " displayed by many arenaceous 

 Foraminifera in building their tests of adventitious 

 material, and in using that material in such a manner 

 as to protect the surface of the test from naturally 

 incidental dangers of damage, and to protect the 

 apertures of the tests against the entrance of preda- 

 torv parasites. 



The "grouping" to which Mr. Ludford directs 

 attention must not be confounded with the associations 

 of marine Rhizopoda, which gain protection against 

 sufTocation in soft muds by the co-operative use of 

 spicules, arranged as catamaran spars to maintain 

 them upon the surface (as in Psammosphaera rustica, 

 H.-A. and E.), or with the aggregation of simple 

 arenaceous tests for purposes of strength and protec- 

 tion, which, unfortunately, has led some of the earlier 

 rhizopodists to treat such associations as new genera 

 or species. It is as if they were to describe a litter 

 of little pigs huddled together for warmth (which 

 is an elementary, phenomenon of intelligence) as a 

 new and " polythalamous " genus of pig. 



Edward Heron-Allen. 



Large Acres, Selsey, May 31. 



An Algebraical Identity 4X = Y'-37Z'. 



The following is a well-known theorem derived 

 from the theory of numbers. Let p be any ordinary 

 odd prime, and let X = (xP— i)l(x-i); then there is 

 an algebraical identity 



4K = Y'±pZ\ 

 where Y, Z are polynomials of degree ^(p—i) and 

 hip-i) respectively; and the sign of the ambiguity 

 is -t- or — according as p is of the form 4n+3 or 

 4n-M. The cases up to /» = 3i inclusive have been 

 published; the result for /) = 37 has just been com- 

 municated to me by Pundit Oudh Upadhyaya, 

 NO. 2693, VOL. 107] 



research scholar of the University of Calcutta. He 

 finds that 



4X=V-37Z» 

 with 



+ ii:!c"-4x'+ii«'-8jc'+I7:ic'-5«''+is.ic*-4x' 



+ IOAC' + X + 2. 



Z = x"+ox'*+2x"-x'*+3x"-x''' + 2x''-x'' 



+ 2X*-X' + 2X'-X'' + 2X^-X* + 2X* + '0X' + X. 



I have tested this result in various 4^ays, and have 

 no reason to doubt its correctness. 



It should be noted that Y may be obtained by 

 expanding 2(x— i)", and reducing the coefficients to 

 their absolutely least residues mod. 37. It would be 

 interesting to know the least value of p for which 

 this rule does not apply. It must be less than 61. 



G. B. Mathews. 



7 Menai View, Bangor, May 29. 



Atmospheric Refraction. 



The following proposition regarding the effects of 

 refraction may be known, but I do not remember to 

 have seen it stated. It is: "The course of a nearly 

 horizontal ray of light in the lower part of the 

 atmosphere is a circular, arc having a radius of 14,900 

 geographical miles." 



The velocity of light in that lower part of the atmo- 

 sphere for which the decrease of pressure with the 

 increase of heig'ht is nearly linear is given by the 

 relation 





where v^ is the velocity in vacuo, Vtt the velocity at 

 the height h above the ground, and H the height of 



the homogeneous atmosphere (0=00029 nearly). At 

 ground level the velocity is Va(i — a). 



Let a plane vertical wave surface start from P as 

 in Fig. I. After the lapse of the time t it will have 

 advanced v^t at the height H, and Vo{i-a)t at the 

 surface of the ground. (This assumes the linearity 

 of the relations between v^ and h to hold up to 

 H, and though this is not true, the conclusions drawn 

 from the assumption are correct, at any rate up to a 

 few thousand feet.) 



Thus at the time t the wave surface will be in- 

 clined forward, making an angle 





H 



or vJ 



H' 



