Dec. 9, 1881.] 



KNOWLEDGE 



115 



satisfactory evidence to prove the conclusion advocated by Sir John 

 Labbock. Let ns take a single instance of Mr. Darwin's " method " 

 of treating facts and inferences. He tells us that the rattle of tlie 

 rattlesnake was probaby evolved by the desire of the creature " to 

 frighten its enemies." [Darwin says nothing abont the rattle- 

 snake's desire. — Ed.] Xow the " enemies " of any animal seek it 

 out in order to attack and destroy it ; and, therefore, tlie rattle, so 

 far from being a source of alarm, would act rather as an invitation 

 to the snake's " enemies " to pursue and overcome it. [How if 

 thev dislike the noise ? — Ed.] Further, if the desire to frighten an 

 enemy is a creative cause of such an organ as a rattle, why does 

 not the same desire develop the same product in other creatures ? 

 Asimilar cause ought to produce a similar effect under similar 

 conditions. 



Sir John also alluded to those kno\m transformations which occur 

 in a short time, and whicli have been observed in many objects of 

 creation; and he cited them as collateral testimony of those greater 

 and more profound transmutations which have been supposed to be 

 wrought in higher organisations during remote ages. I subn\it that 

 this argument is altogether illusory. Those obvious minor transfor- 

 mations, to which Sir John referred, take a certain course — pursue 

 a certain round — in obedience to regular processes and laws of 

 being, and then terminate their career under known conditions. 



Tlie attention of the audience was also invited to the circumstance 

 that the stripes of the tiger correspond with the long grass in which 

 he makes his habitat ; and that the spots of the leopard resemble 

 the speckled appearance of the light falling through the leaves of 

 trees. Now, if this eminent teacher really means that we should 

 believe that in these instances the long grass and the specks of light 

 are in the remotest degree the caxifes of the stripes of the tiger and 

 the spots of the leopard, then I must say that the impression he 

 wished to create is the most astounding and intolerable tax upon 

 our credtjity ever levied by the greatest scientific fanatic. 



London, Nov. 19, 1881. Newton Croslaxp. 



[Mr. Crosland entirely misapprehends the Danvinian theory and 

 Sir J. Lubbock's remarks on it. We feel justified in excluding ob- 

 jections based on mere misinterpretation of the theory attacked ; 

 though inquiries suggested by such misinterpretations will always 

 find a place here. — Ed.] 



PROBLEMS GEOMETRICALLY INSOLUBLE. 



[90] — Would the geometrical solution of one or two problems, 

 hitherto unsolved by geometry, bo suitable for the pages of 

 Knowledge ? One is to determine the centre of gravity of a semi- 

 circle, and a computation of its distance from the centre of the 

 circle by trigonometry. This, of course, is analogous to what is 

 called squaring the circle. 



Another is to determine the diameter of a sphere equal in volume 

 to a given parallelopipedon. I should esteem it a favour if you 

 would let me know what you think about them. — Tours, etc., 



J. G. Moore. 



[The trigonometrical computation of the centre of grarity of a 

 circular arc is well-known. If our correspondent knows of any 

 simpler form of it, we shall, of course, be glad to have his demon- 

 stration. If, as a preliminary to either of his problems, he proposes 

 to " square the circle " geometrically, we shall be content to wait 

 awhile. If he has squared the circle geometrically, we shall never 

 succeed in showing ** where tlie error comes in." — Ed.] 



A REMARKABLE RAINBOW.— LOGIC versus MATHEMATICS. 



[91] — About three o'clock in the afternoon, one day last week, I 

 observed a very bright rainbow, accompanied by its secondary 

 external arc. "To my great surprise, I noticed that the primary 

 bow consisted of a triple series of colours, the red of the second 

 band being in close contact >vith the violet of the first, and so with 

 the third. The colours of the third band were very faint, and it 

 was only distinctly visible for a short time at the stmimit of the 

 arc ; but the second band was visible over almost the whole length 

 of the bow for three or four minutes. I should be glad to know 

 whether this phenomenon has been noticed before, and how it may 

 be explained. 



I am rather sorry to see that, not^vithstanding the warning of 

 " F.R.A.S.," in No. 1, your admirable j»per is being seized upon 

 by the crotchet-mongers to air their remaikable notions, and I trust 

 you will not think that I wish to dispute accepted and infallible 

 laws of mathematics or of logic because I send you a paradox in 

 which they appear to confute one another. Jly paradox is as 

 follows : — 



For every whole number there is a square, which is also a whole 

 number. No two whole numbers have the same square. 



Tlierefore there are as many whole numbers which are squares 

 as there are whole numbers. 



But there are many whole numbers which are not squares to 

 other whole numbers. 



It follows that there arc whole numbers which are not whole 

 numbers. 



I prefer, however, to infer that the series of numbers being 

 infinite, and the series of their squares being therefore also infinite, 

 the latter infinity includes the former. I should like, however, to 

 see a more satisfactory explanation. — Faithfully yours, Theta. 



[We do not see how there can be a more satisfactorj- explanation 

 than the one" Theta " has himself supplied. If we considerany cor- 

 responding parts of the two series, 1, 2, 3, i, &c., 1, 4, 9, 16, &c. ; we 

 see that, taking them together towards infinity, the latter will run on 

 to a higher infinity, as it were, the highest number in the latter 

 series being always the square of the highest number in the former. 



Or, algebraically :" — 1' -t- 2' + 3' -f -)- it^_ J)i(n -f l)(2>i + 1) ^ 2)1 + 1 



3 



1-1-2 -H 3 -I- + n 



or is infinite when n is infinite. — Ed.] 



i"( + l) 



STONE ON WHEELS. 



[92] — The results obtained by "Queensland'' and his mathe- 

 matical friend (Querj- 2S, p. 80), with regard to the stone rolled on 

 wheels, probably differ through their not understanding each other. 

 It " Queensland " means the wheels to be supported by and revolve 

 on fixed axles, then the stone will move, as he supposes, through a 

 distance of 75 in. If, however, he intends the wheels to rest on the 

 ground, they will themselves move along a distance of 75 in. for 

 each revolution, carrying the stone with them, and at the same time 

 they project the stone forward (with regard to their own position) 

 an equal distance; it will, therefore, move a total distance of 75 in. 

 + 75 in. = 150 in. G. M. 



POLARITY versus GRAVITATION (Ahs'rr.ct). 



[93] — I AM too old a stager to take umbrage at any usage, however 

 rough, which I may experience in the arena of debate ; when, 

 therefore, you tell your readers that I am " a paradoxer," who 

 hardly knows what he is about, I accept the imputation in the par- 

 liamentary and controversial sense in which it is meant. Iprestune 

 it is your mode of saj-ing that you differ from me in opinion. 



Mv ideas must natm-ally suffer some loss of cogency by the 

 necessity which exists of compressing their exposition within tie 

 space which you have kindly allotted to me. 



Permit me to submit a few words of reply to your remarks 

 on my letters. With regard to my objections to the Newtonian 

 theory of the tides, as you merely content yourself with reiterating 

 that theory, and asserting its correctness, I can, of course, say 

 nothing more on this subject. Discussion becomes profitless when 

 one disputant sets up what the other knocks down. 



Touching my criticism on the centrifugal and centripetal forces 

 as regulators of the motions of the universe, I beg leave to say that 

 " the ill-informed writers" who used the term " centrifugal force" 

 in the sense which I condemned are Joyce, in " Scientific Dia- 

 logues ;" Milner, in " The Gallery of Nature ;" Ferguson, in his 

 " Lectures edited by Brewster ;" Dr. Lardner, in his " Astronomy ;" 

 and " Keith on the Globes " — all well-known expounders of the 

 Newtonian system — and Sir Isaac Newton himself. When you say 

 that " centrifugal force is only another way of viewing the centri- 

 petal force," I know what you mean, but I fancy that the general 

 reader will require further explanation. The revolution of a 

 planet round the sun is supposed to be effected by the attraction of 

 gravitation or centripetal force of the sun drawing the planet out 

 of the straight line on which it was first impelled by its Maker. 



In one part of its orbit the sun draws the planet nearer to itself, 

 and thereby accelerates its speed. This increase of speed is sup- 

 posed to generate " a centrifugal force " which has a repellent effect, 

 and thus sends the planet off again on its proper course. Now, here 

 comes in the ticklish part of this theory. When once the attrac- 

 tion of gravitation overcomes a rival force, nothing can stay its 

 career of conquest, except the intervention of a third power of 

 equal potency and independent jurisdiction. As the accelerated 

 speed above-mentioned has no such independent origin, but 

 proceeds directly from the centripetal force which draws 

 the planet towards the sun, the planet cannot, by any straggle 

 of a centrifugal force, escape from the catastrophe of ulti- 

 mately being precipitated upon the face of its ruler, and there ter- 

 minating its blundering career. Fortunately for us, the planet 

 know.ii better, and obeys the lay of polarity, not that of gravitation. 

 I submit that my theory of polarity— attraction and repulsion — 

 gets rid of the difficulty here so patent, and enables us to arrive at 

 a sounder idea of the laws of revolution. 



I now leave my ideas to their fate. If thej- are good for any- 



