132 SCIENCE PROGRESS 



(3) If the cubic has 1 real positive root and 2 real negative roots, 



the quartic has either 2 real and 2 unreal roots, or 4 unreal roots. 

 If a 2 be incommensurable, a result can be obtained which will be accurate to 

 any number of decimal places required. In the second example, cr = 2, so a is 

 a surd. 

 Ex. (1) x i - 252? + 6ar - 36 = o = f{x). 



As the term involving x 3 is absent, we can immediately write down the 

 auxiliary cubic <p(z 2 ). 



z 6 - 50s 4 + 7692-' — 3600 = o. 

 or (s* - 5 3 ) Csr* - 25^ + 144). 



The cubic has here 3 real positive roots, the quartic must by (1) have 4 real 

 roots. 



/(*) = {* 2 +5* + ^(25 -25 - 2v/l44)}{* 2 - 5^ + ^(25 -25 + 2 yi44)} 



.*. x = - 2'5± a/6'25 + 6 or 2*5 ± ^'25 - 6. 

 • • X\ = 1 j X2 = — o j x$ = 3 l x$ == 2. 



Ex. (2) x* + 4X - 1 = o =f{x) 



Auxiliary cubic ^{z 2 ) = z 6 + 4s- 2 - 16 = o. 



<l>(z') = (z 3 - 2) («* + 2£ 2 + 8). 

 f{x) = (x 2 + V2 jt + 1 - / J 2 )(x i - sjl x + 1 + V2). 



.'. ^= -i V2± a/^2 - I or 1^2 ± a/- ^2--. 



2 V 2 2 V 2 



The 2 real roots are consequently -249038376 . . . and — 1*663251938 . . . 

 and the 2 unreal roots are 707106781 . . . ± v - I'9i42i3562 . . . 



SOME OF THE EVOLUTIONARY CONSEQUENCES OF WAR 



(Ronald Campbell Macfie, M.A., M.B., CM., LL.D.) 



THOUGH war has been in the world since the time of the trilobites, and though its 

 importance in the evolution of animal types has long been a cardinal article in the 

 creed of biologists ; yet the sociological and biological significance of human war- 

 fare with reference to the evolution of man's body and mind has never been quite 

 adequately studied, either from the standpoint of sociology or of biology. 



In view of the universality of war, there naturally arises the question : What 

 physical and spiritual types of Man does war select, and what types does it 

 eliminate ? No question, indeed, would seem more to invite and more to merit 

 thorough investigation. But the problem is not only difficult : it is readily 

 obscured by associated sentiments and passiong, and, so far, it has been more 

 often exploited by military or pacific partisans than elucidated by patient and 

 impartial investigators. Even the wary spirit of scientific inquiry seems readily 

 corrupted by the emotional psychology of war, and even in scientific papers we 

 find doubtful data leading to undoubting generalisations, and strong prejudices 

 drawing, from very weak premises, very wild conclusions. 



Thus, we find competent biologists, such as the eminent pacifist Prof. David 

 Starr Jordan, stating, with reference to the dysgenics of war, that war caused 

 degeneracy in the Romans, and that the Napoleonic wars lopped inches off the 

 stature of the Frenchmen. 



Yet both statements, though widely current, have never been proved, and are 

 probably erroneous. 



