54^ 



NA TURE 



[October 21, 1922 



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 manuscripts intended for 

 this or any oilier part of Nature. No notice is 

 taken of anonymous communications.'] 



Mersenne's Numbers. 



In my presidential address to Section A of the 

 British Association, reprinted in Nature (September 

 16), I stated that 137 was the least value of n for 

 which the prime or composite character of 2" - 1 

 was still undecided. Mr. W. W. Rouse Ball has 

 pointed out to me that this is incorrect, as 2 13 ' - 1 

 has been shown to be composite by M. A. Gerardin 

 (Comptes rendus du Congres des Societes Savantes, 

 1920, pp. 53-55). The result is quoted in The 

 .hue 1 noii Mathematical Monthly, vol. 28, 1921, p. 380. 

 The number 139 should therefore be substituted for 

 137 wherever it occurs in my address. 



The authorities on which I relied were Prof. L. E. 

 Dickson's "History of the Theory of Numbers" 

 (vol. 1, Washington, 1919) and the seventh edition 

 of Mr. Rouse Ball's " Mathematical Recreations " 

 (1917, now superseded by the tenth). My quotation 

 from Mr. Rouse Ball was taken, as I stated, from a 

 pamphk-t written thirty years ago, and is, of course, 

 not to be interpreted as an expression of his present 

 view. G. H. Hardy. 



New College, Oxford, October 4. 



Animal Mechanism. 



The notion that the legs of animals behave as 

 pendulums is ascribed to the brothers Weber. I can 

 find no indication that the notion was more than a 

 general one, and, in the general sense, when pointed 

 out, it is obvious to a student of dynamics, for legs 

 have inertia and weight and dynamics is reasonably 

 near to the truth. 



A better view may arise from the supposition that 

 animals may be regarded as dynamical systems with 

 many natural modes and frequencies, and that animals 

 adapt their methods of locomotion and other actions 

 to suit these fundamental characteristics. As 

 examples, we have the lounging gait of very tall men 

 and the apparently energetic step of short men. The 

 tripping, half running step of women and children is 

 also in point. Apart from mere legs the moment of 

 inertia about the feet must be important, as may be 

 seen in the stately carriage of quite short women in 

 the East when carrying water vessels on the head. 

 Sir George Greenhill has given several examples of 

 this in his notes on dynamics, among them being 

 interesting examples of the carrying of soldiers' kit. 



A further point of interest arises in regarding legs 

 a little closely, for they are not simple but multiple 

 pendulums with more than one natural mode. When 

 a horse or man is walking the leg appears to vibrate 

 in the slowest mode of the pendulum and the joints 

 are or appear to be on one side of the vertical. In the 

 running gait, however, the thigh points forward while 

 the lower parts point backwards. Probably the 

 " reason " why a horse's forelegs are more flexible 

 than the hind legs is to make him nimble in balance 

 and steering ; a horse could not stumble with his 

 stiff hind legs. 



The dynamics of locomotion is of interest to the 

 student of engine balancing, for in the natural gaits of 



NO. 2764, VOL. I IO] 



man and horse there is a utilisation of balancing 

 principles. In man the right leg moves forward 

 while the right arm moves backward, in the horse the 

 right legs are always moving in opposition and 

 similarly the left legs, of course. This holds for the 

 walk, the trot, and the gallop, all natural modes. In 

 the amble, an artificial stride due to the trainer, the 

 legs on either side are in phase and an ungainly motion 

 results, though it is comfortable for the rider. This 

 amble stride is natural to the giraffe, but the latter 

 has a long neck to give it poise. The balancing view 

 of animal locomotion may be realised at once by 

 any one who will try to run with stiff arms or will 

 try to walk with his arms tucked up in the running 

 posture. The runner is compelled by dynamics to 

 move his legs in a quicker mode than when walking. 

 His arms are so jointed that he cannot alter their 

 type of vibration, and he is therefore compelled to 

 reduce their inertia in order that they may oscillate 

 in time with his legs. The balance from the engineer's 

 point of view is imperfect, and thus stresses are imposed 

 in the trunk. Hence sprinters are well-bodied men 

 and horses need girth for speed. As a final example 

 of these facts, let any one try to run to the station 

 with a heavy suit-case in his hand. Porters usually 

 carry such things on their shoulders and stride 

 rather slowly. 



There is a further point of interest in connexion 

 with the viscera. If dynamics is true, the various 

 internal organs have inertia and their attachments 

 have elasticity ; thus they must possess natural 

 frequencies. This being so, they must be subject more 

 or less to the phenomena of resonance. Is sea- 

 sickness, subjective agencies apart, to be explained in 

 this way ? Some people before embarking have a 

 copious meal, others pin their faith and hope to a 

 single bottle of stout, while yet others proceed fasting. 

 Is this a phase of dynamical tuning ? In 1914 I read 

 a short paper on the dynamics of the human foot at 

 the British Medical Association's summer meeting. 

 The outcome of the discussion was that tonicity was 

 more potent than mechanics, or, in other words, living 

 tissue may vary in its properties on account of tone 

 or debility to a degree which will exceed the influence 

 of configuration. The contention would be that 

 while astringents or food may alter the effects of a 

 sea voyage, the action is due to dynamical effects ; 

 the inertia of the stomach or the stiffness of its 

 suspension is varied — opiates and such like are here 

 excluded. 



A medical writer of some eminence recently advo- 

 cated walking because " Nature has ordained that the 

 finest exercise of all is that which she bestows." 

 While this is reminiscent of Heine's " Harzreise," the 

 greenness of grass and the length of a donkey's ears, 

 there is in it matter for reflection. Motor car designers, 

 led by Dr. Lanchester, have found that the most 

 comfortable predominant natural frequency of a 

 motor car is between 80 and 100 per minute ; it is a 

 curious coincidence that this is also the frequency of 

 the ordinary walking step. Has the human system, 

 enforced by dynamics to walk in a certain rhythm, 

 acquired an internal system and a nervous organisa- 

 tion to meet this rhythm ? It is worthy of note that 

 in certain cars several dogs and children have been 

 actually and violently sick and in other cars sprung to 

 vibrate with a different natural period they are immune. 

 Shall we, disagreeing with Shylock, say, " It is not 

 their humour, but their natural frequency " ? 

 H. S. Rowell, 

 Director of Research, 

 Research Association of British Motor 

 and Allied Manufacturers. 



15 Bolton Road, W.4, September 20. 



