December 23, 1898.] 



SCIENCE. 



911 



by one magnitude, that one magnitude to be 

 always interpreted as a number. This radical 

 Innovation, the creatiou of this epoch- marking 

 paradox, is due to Newton. Newton takes this 

 vast step explicitly and consciously. The lec- 

 tures which he delivered as Lucasiau professor 

 at Cambridge were published under the title, 

 'Arithmetica Universalis.' At the beginning 

 of his Arithmetica Universalis he says, page 2 : 

 " Per Numerum uon tarn multitudiuem uuita- 

 tum quam abstractam quantitatis cujusvis ad 

 aliam ejusdem generis quantitatem quae pro 

 unitate habetur rationem intelligimus." [In 

 quoting this, Pringsheim, p. 51, misses the first 

 word. He omits the Per.'] 



As "Wolf puts it (1710): "Number is that 

 which is to unity as a piece of a straight line 

 [a sect] is to a certain other sect." Thus the 

 length of any sect is a real number, and the 

 length of any possible sect incommensurable 

 ■with the ulait sect is an irrational number. 



Says Hayward in his Vector Algebra (1892), 

 page 5 : " Number is essentially discrete or dis- 

 continuous, proceeding from one value to the 

 next by a finite increment or jump, and so can- 

 not, excejjt in the way of a limit, represent, 

 relatively to a given unit, a continuous magni- 

 tude for which the passage from one value to 

 another may always be conceived as a growth 

 through every intermediate value." 



But the moment we accept Newton's defini- 

 tion of number it takes on whatever continuity 

 is possessed by the sect. However, from this 

 alone does not follow that for every irrational 

 there is a sect whose length would give that ir- 

 rational. G. Cantor was the first to bring out 

 sharply that this is neither self-evident nor de- 

 monstrable, but involves an essential pure geo- 

 metric assumption. 



To free the foundations of general arithmetic 

 from such geometric assumption, G. Cantor and 

 Dedekind each developed his pure arithmetic 

 theory of the irrational. 



Professor Fine, in his ' Number- System of 

 Algebra,' seems to miss this point completely. 

 He gives, page 42, what purports to be a de- 

 monstration that " corresponding to every real 

 number is a point on the line, the distance of 

 which from the null-point is represented by the 

 number," without any mention of the geometric 



assumption necessary, and then proceeds, page 

 43, to borrow the continuity of his number sys- 

 tem from the naively supposed continuity of the 

 line, the very thing for the avoidance of which 

 G. Cantor and Dedekink made their systems. 



Says Dedekind. " Um so schoner erscheint 

 es mir, dass der Mensch ohne jede Vorstellung 

 von messbaren Grossen, und zwar durch ein 

 endliches System einfacher Denkschritte sich 

 sur Schopfung des reinen, stetigen Zahlen- 

 reiches aufschwingen kann ; und erst mit diesem 

 Hiilfsmittel wird es ihm nach meiner Ansicht 

 moglich, die Vorstellung von stetigen Raume 

 zu einer deutlichen auszubilden." 



George Bruce Halsted. 



Austin, Texas. 



The Tides and Kindred Phenomena hi the Solar 



System. By George Howard Darwin. 



Boston and New York, Houghton, Mifilin & 



Co. 1898. Pp. xviii + 378. 



During October and November, 1897, Profes- 

 sor Darwin delivered a course of semi-popular 

 lectures on tidal phenomena at the Lowell 

 Institute, Boston, Massachusetts. Since then 

 the author has prepared these lectures for the 

 press, and they are now, through the enterprise 

 of Messrs. Houghton, Mifflin & Co., placed be- 

 fore the reading public in attractive book form. 



The salient features of oceanic tides are more 

 or less familiar to most people in these days. 

 Indeed, some intelligent people will tell us that 

 it is only necessary to read the daily papers of 

 the seaboard towns, or to look in ' The Farmers' 

 Almanac,' to learn when high and low water 

 will occur. The educated public was not al- 

 ways so well informed, however. When, for 

 example, Alexander the Great attempted to 

 make a landing at the mouth of the Indus his 

 fleet was nearly overwhelmed by the inrush of 

 the tide. " The nature of the ocean," accord- 

 ing to his biographer, Curtius, "was unknown 

 to the multitude, and grave portents and evi- 

 dences of the wrath of the gods were seen in 

 what happened.!' The admirals of the present 

 day know more about tides than the admiral of 

 Alexander, and the wrath of a court of en- 

 quiry, rather than the wrath of the gods, hangs 

 over the head of any commander who exposes 

 his fleet to tidal dangers. But whence comes 



