May 6, 1922] 



NATURE 



579 



constant may involve a very different constant from 

 that which, with small modification, applies to the 

 three known radioactive families. Still more funda- 

 mental would be the discovery of some other than a 

 radioactive origin for these haloes. I have considered 

 many alternatives. One naturally thinks of a 

 chemical influence emanating from the nucleus. 

 Apart from the difficulty of accounting for the 

 consistent measurements, the existence of bleached 

 haloes in this mica, which possess the characteristic 

 dimensions of uranium and emanation haloes, seems 

 a formidable difficulty. The relation between the 

 radioactive staining and the bleaching is everywhere 

 such as to suggest that the latter is a modification of 

 the former. A quantitative difficulty also exists. 

 The volume of the nucleus varies from the -^-^-^ to the 

 TjJff,T part of the halo- volume which it has affected, 

 ^et the nucleus may be a limpid particle revealing no 

 trace of loss or decomposition. Only radioactivity 

 can confer on one atom the energy requisite to ionise 

 many hundreds. 



Of course some one of the known elements may be 

 responsible for these haloes. Geiger and Nutall long 

 ago pointed out the difficulty which would attend the 

 discovery of radioactivity in elements having a 

 radioactive constant proportional to such ranges. 

 But here we have an integration such as far transcends 

 the resources of the laboratory. Until this point is 

 settled — if it ever will be possible to settle it — a 

 distinguishing name seems desirable. To this name 

 the addition of numerals would suffice to deal with 

 such halo-developments as may be ascertainable. 



J. JOLY. 



Trinity College, Dublin, April 25. 



Pythagoras 's Theorem as a Repeating Pattern. 



The interesting communication from Major 

 MacMahon on the above subject reminds me of a 

 proof which I discovered over the chessboard a few 

 years ago, of the well-known fact, that if the lengths of 

 the sides of a right-angled triangle are 3 and 4, the 

 length of the hypotenuse will be 5. 



Placing pawns at A, B, and C (Fig. i), we require to 

 prove that A is equidistant from B and C. We put 



^^ 



^^':^ 



Fig 



B 



^ R^^ 



^ 



two more pawns at D and E, when it will be readily 

 seen, even by a person unacquainted with Euclid, 

 that A, D, E are in line and that CEBD is a square. 

 Since any point on a diagonal of a square must, by 

 symmetry, be equidistant from the extremities of the 

 other diagonal, AB =AC. 



The corresponding general proof of Pythagoras's 

 theorem is as follows. Given x^-\-y^ = z^, we get 



z-\-x-y _ x-\-y-\-z 



x+y-2~z+y-x ' ' • vij 



by algebra. Taking horizontal and vertical axes of 

 reference through an origin O (Fig. 2), we mark down a 

 point A the co-ordinates of which are h{z + x-y), 

 \{x+y-z), and a point B the co-ordinates of" which are 

 l{x+y + z),^(z + y-x). By the given relation ( i ) O AB 

 is a straight Hne. Through A and B draw ordinates PQ, 



NO. 2740, VOL. 109] 



SR, each equal to PS, and in the square PQRS inscribe 

 the square ADBC by marking off RD = PC = SB. 

 Then, drawing the ordinate DN, we have 

 I)]<i = PS = i{x+y + z)-^iz + x-y)=y, 

 01<! = 0P + PA = ^{z + x-y) + ^{x+y-z)=x, 

 OD = OC = OP + SB = ^{z + x-y) + ^{z+y-x)=z, 



which proves the theorem. 



It is to be noted that in any special case where 

 X, y, z are given integers (as in the case given above), 

 it can be easily shown that OAB is a straight line 



without knowmg anything about proportion. Thus 

 in such cases Pythagoras's theorem is proved without 

 introducing areas. It has, I believe, been suggested 

 that the ancient Egyptians must have been acquainted 

 with Pythagoras's theorem, since they knew that a 

 triangle with sides 3, 4, 5, is right-angled. But they 

 may possibly have known only the special proof here 

 given. 



Lastly, it may be noticed that Euclid's axiom about 

 parallels is tacitly assumed when we allow that a 

 repeated pattern of squares can be constructed. 



J. R. Cotter. 



Trinity College, Dublin, April 15. 



Man. 



Man is a social animal through habit, not instinct. 

 Religions, morals, -taboos, customs, conventions, 

 which he learns through imitation, supply him with 

 rules of thought and conduct. Without them 

 human society could not exist. But there are two 

 sorts of rules. The one kind binds the body, limits 

 action, supplies rules of conduct, and impels men to 

 " play the game " fairly. The other sort binds the 

 mind, limits thought, impels men never to question 

 the rules. When the rules that hind the mind are many 

 and strait, men tend to regard lightly the rules that 

 bind to conduct. All this may seem far-fetched, 

 but consider universal history. Is it not the fact that 

 communities have been inefficient, stagnant, and tur- 

 bulent in proportion as their minds have been bound ? 



I may be afflicted with racial prejudice, but 

 to me it seems that the men of English speech 

 owe their predominant position in the world to the 

 fact that they more than others " play the game " 

 scrupulously, and yet have been freest of all in their 

 thoughts, and so, while obeying their existing rules, 

 have most readily altered the rules both of conduct 

 and of thought. We English may have only one 

 sauce, but, fortunately, we have a hundred heresies. 

 Modem English history tells of continuous evolution, 

 but of only one revolution. Compare the histories 

 of more orthodox countries. Men cannot get away 

 from habit, and mental habits depend not so much 

 on the things that are learned as on the way in which 

 they are learned. Through imitation we get 

 emotional convictions and closed habits of mind ; 

 through curiosity, intellectual convictions and open, 

 reflective habits of mind. When a man is mentally 

 " too old at forty " his mind has been artificially 

 closed. It can no longer profit from experience. 



