MATHEMATICS IN EVOLUTION. 207 



sideration of the laws which have been stated, and which are closely- 

 borne out by observation, would lead us to expect just what we find, 

 namely, in the processes of development intermediate links would 

 drop out after comparatively brief existence between planes of life 

 increasingly separated, so that the last difference of power and intelli- 

 gence Avould be the greatest of all. 



And, furthermore, the same laws make intelligible the vast gulfs 

 we fiud fixed between our intellectual giants and the rest of mankind, 

 so that they form a small solitary band above us all, leaving a mere 

 understanding of their mighty works the test of our highest powers. 

 A single English dramatist and a single English mathematician have 

 probably equaled in scope and excellence of original work, in their 

 several fields, all the like labors of their countrymen put together. 



Two other mathematical laws, abstractedly of great power and 

 generality, may be noticed in the many phases of evolution, namely, 

 those treating of the relations between areas and solids of the same 

 form, varying in size. In like plane figures, boundaries increase di- 

 rectly as like dimensions and areas, as the square; in similar solids, 

 surfaces increase as the square and contents, as the cube of like dimen- 

 sions. These laws state in an abstract way the economy of aggrega- 

 tion, whether domestic, industrial, social, or political. The farmer 

 profits by them when he takes down costly fences in enlarging his 

 domain ; the ship-builder avails himself of them when he models his 

 monster craft which shall carry the cargo of half a dozen small vessels 

 at half the expense ; the Broadway architect embodies them in 

 his lofty designs, rivaling in a business structure the height of a 

 common church-steeple, putting two ordinary buildings on one lot of 

 ground. 



From the time when animals first noticed that two together were 

 stronger than two singly, the gregarious instinct has been assisted in 

 taking a firm hold on many species from its usefulness in attack and 

 defense ; where it is not exhibited, exceptional circumstances prevent : 

 for instance, a spider would have nothing to gain by going into part- 

 nership, for it preys on flies much weaker than itself, and no company 

 of spiders, however large, could do battle with a swallow, or a house- 

 maid armed with a broom. 



Speaking in a general way, such savage tribes of men as have had 

 the strongest social feeling, and the largest mutual confidence, have, 

 other things equal, had an advantage over less coherent neighbors, 

 and so on, until now modern history deals with national groups fewer 

 than ever before, and becoming fewer still. 



In commerce, also, the largest banks, mercantile firms, and facto- 

 ries, grow continually larger by virtue of the less expense attending 

 the management of extensive groups. The costly competition of 

 many small manufactories and merchants is passing away before the 

 more economical methods of a few strong concerns. Cooperation in 



