406 



KNOWLICDGE. 



OCTOBKK. 1912. 



representing the distance of llir Kartli from tlif Siiii. whilst he 

 would have rcaclu'd quite diflVrent conchislons if he had 

 chniiiialed the factor represeiitiny the velocity of our planet. 

 In short, having only the ri^ht to atlirni the constancy of one 

 product — that of the sijuare of the angular velocity by the 

 cube of the distance — the author iniplicilly introduces into his 

 calculatiou.s (p.ages 194 and 195 ibiii) the arbitrary supposition 

 which allows him, later on, to affirm that the Sun cannot 

 have received the necessary amount of cosmic matter to 

 conserve its heat, without the Karth suffering an acceleration 

 sulTieiently great to have been detected. 



It was Poincare who wrote in his celebrated "La Science 

 et I'Hypothese " this disconcerting phrase : " Les axiomes de 

 la geometric nc sont que des definitions dcguisces" (the 

 italics arc Poincar6's), and this in order to deprive the most 

 beautiful of the pure sciences of its majestic prestige, and to 

 concede belligerence to other so-called geometries. And then 

 he affirms (page 661 that one geometry cannot be truer than 

 another, but only more convenient (" Une geometrie ne pent 

 pas ctre phis vraie qu'une autre, elle peut seulement etre plus 

 commode ") — again, the italics are Poincare's. So that if we 

 believe that one side of a triangle is less than the sum of the 

 other two and greater than their difference, or that the sum of 

 the three angles of a triangle is equal to two right angles, it 

 would not be, according to this way of reasoning, because it is 

 absolutely true, but because it would be more convenient to 

 have it so than to believe that the said sum is equal to more 

 than two right angles (geometry of Riemann), or less than two 

 right angles (geometry of Lobatchewsky). 



Now, if we measure the angles in millions of triangles, we 

 shall always see that their sum is eijual to two right angles, 

 and the same verification can be made with any other con- 

 clusion of pure Euclidian geometry, while we fail to verify the 

 conclusions of the pseudo-geometries. Of course, the purpose 

 of those who overlook such facts, is to find a pretext for 

 asserting that it is impossible to find an absolute truth, even 

 in geometry. 



We should not forget, by the way, that the school of the 

 inipossibilists has been proved wrong on many occasions, and 

 looks like having to sustain a few shocks yet. Pasteur 

 declared that it was impossible to produce by synthesis the 

 substances endowed with the property of polarising light, and 

 yet this synthesis is now a daily affair in lat)oratories. Auguste 

 Com^e was quite certain that man would never know the 

 chemical composition of the stars, and notwithstanding 

 spectrum analysis allows us to analyse a star to-day as 

 easily as we analyse a piece of a mineral. 



Louis Favre has rightly said, in his "Histoire Generale des 

 Sciences," that a very interesting " History of Errors" might 

 be written by simply enumerating the supposed impossibilities 

 asserted as such by great men which have turned out in the 

 end to be cjuite feasible. 



There is a " convenience " argument now fashionable 

 amongst many intellectual men, which deals with the rotation 

 of the Earth, and which appears at first sight quite reasonable. 

 It admits the rotation of our planet for the sake of con- 

 venience, because there are many facts that support it, and 

 because it is much easier to grant that the Earth revolves 

 aroimd its axis every twenty-four hours than to assinne that 

 all the stars revolve around us in one day ; but, after all, it is 



contended, many millions of people, after observing the 

 .ipparent movement of the Sim, mistakenly aflirmcd that it 

 was our great luminary which effected its revolution every 

 twenty-four hours ; so that this is not a question of pure 

 mathematics, and our belief cannot be called an absolute 

 truth. 



To say this, is to overlook the fact that this truth, in common 

 with many others, can be proved mathematically by the appli- 

 cation of certain principles of pure mathematics, not to non- 

 exact sciences, but to other purely mathematical principles. 

 For instance, astronomical trigonometry enables us to calculate 

 the distance separating us from Mars, and Kinetics tell us that, 

 in order to revolve round the Earth in twenty-four hours, the 

 red planet would need to possess an orbital velocity of about 

 nine thousand miles a second. Besides, Dynamics tell us that 

 if a body revolves around another, at the mean Martian 

 distanceof one hundred and thirty million miles with a velocity 

 of nine thousand miles a second, the central body requires a 

 mass nearly one hundred thousand million times greater than 

 that possessed by the Earth. .And since a thing cannot be 

 millions of times greater than itself, we see, by pure 

 mathematical reasoning, that Mars cannot revolve around our 

 planet. And the same reasoning applies to every celestial 

 body except the Moon. For instance, were Alpha Centauri, 

 the nearest star to the solar .system, to revolve around us, we 

 should require for our poor little abode a mass many miUions of 

 times greater than that calculated by Lord Kelvin for the 

 whole of the visible Universe. So, if we are compelled to 

 recognise that it is not the stars which gravitate around us in 

 twenty-four hours, but that it is our Earth which performs its 

 rotation in one day, we have been forced to accept this con- 

 clusion, not only from convenience, but also as an irresistible 

 consecjuence of a purely mathematical reasoning. 



It is not only in such complex matters that mathematics 

 show us their power and their beauty. The simplest problem 

 affords them boundless opportunities. For example, if we ask 

 algebra for two numbers of which the sum shall be four and 

 the product twenty, it gives us as a result two imaginary 

 quantities: the first is two plus the imaginary unit multiplied 

 unto four, and the second is two minus the same imaginary 

 unit multiplied also unto four. And yet, though none of these 

 numbers be real, their algebraic sum is four, and their algebraic 

 product is twenty, as required. It is as if this noble and 

 infallible science, while reminding us gently — by giving us 

 imaginary quantities — that we were talking nonsense in asking 

 for such an unreal thing, wants at the same time to show that 

 even things which are physically impossible to us are not 

 outside her powerful grasp. Do they not give us the weight 

 of a double star as easily as scales give us the weight of a 

 loaf? 



It is they, the majestic and poetical mathematics, the 

 charming friends who always tell the truth, who never 

 deceive, flatter or discourage, who have pointed out to 

 Professor Bickerton the secrets of the partial impact in the 

 birth of the new stars, they who ga\e Adams and Le \'errier 

 news of the existence of Neptune before Galle discovered the 

 most remote of the known planets, they, in short, who have 

 whispered to Sir Joshua Thomson the romance of the 

 electrons, those ultimate particles of matter which perhaps 

 the eye of man will never see ! 



REVIEWS. 



ASTKONOMV. 



Catalonitc of 9SI2 Naked-eye Stars for the Epoch 1900.— 



By T. W. Backhouse, F.R.A.S. 



(Sunderland: Hills & Co. Price 10 6.) 



This catalogue was drawn up with a view to the preparation 



of fourteen large star maps on the gnomonic projection, 



designed for use in meteoric observations. The author, 



however, has go[ie far beyond what was strictly necessary for 



this purpose, and thu vdliime will bo useful to many besides 

 meteoric observers. It contains all the stars that can be 

 readily seen with an unaided eye of normal acuteness in 

 the entire heavens from pole to pole : several are also 

 included which, though individually too faint, might be seen 

 in conjunction with near neighbours. They arc arranged by 

 constellations in alphabetical order, and this is not a very 

 convenient plan, for the boundaries of the constellations are 

 irregular and differ in ditTercnt atlases, besides which the 



