2 10 THE POPULAR SCIENCE MONTHLY. 



" Whether the multiplication of fractions is an increasing process ? " 

 In order to prove that to multiply means to increase, he bases his ar- 

 gument on Scripture, and clinches the whole by quoting the promise 

 to Abraham, " I will multiply thy seed like the stars of the firma- 

 ment," To this devout logician there would be no joke in the com- 

 mon conundrum that proves Abraham to have been a mathematician 

 because he increased and multiplied on the " face of the earth." But 

 how is this to be reconciled with the numerical result in the cases 

 under consideration ? He supposes the units of the product to be of 

 greater virtue and significancy than those of the factors : thus, if ^ 

 and ^ represent the sides of a square, their product will represent the 

 area of the square. 



The first actual mention of real decimal fractions is in a Flemish 

 work published in 1590. There the mixed number 2T'847 is written 



2 V' ^J ^7 7 To the present advocates of the metric system it may 

 afford encouragement to know that Stevinus, in this work, enumerates 

 the advantages which would result from the decimal subdivision of 

 the units of length, area, capacity, value, etc. 



In 1619 the contents of the Flemish book were embodied in an 

 English work — " The Art of Tens, or Decimal Arithmetike, exercised 

 by Henry Lyte, Gent., and by him set forth for his Countries Good." 

 After enlarging upon the value of his system to all classes, he adds : 

 "If God spare me life, I. will spend some time in most cities of this 

 land for my countries good to teach this art. I hold the lively voice 

 of a mcane sj^eculator somewhat practised, furthereth ten fold more 

 in my judgement than the finest writer that is." Rather severe on those 

 " meane speculators," his contemporaries, Francis Bacon and William 

 Shakespeare. 



SPENCER'S DATA OF ETHICS. 



By ALEXANDER BAIN, LL. D., 



PROFESSOR OF LOGIC IN THE UNIVERSITY OF ABERDEEN. 



IN the immense abundance of literary production a great deal of 

 criticism is avowedly calculated to supersede the jDerusal of the 

 works themselves. Such a book as the present, however, is among the 

 rarest ; and being on the most interesting of all themes, and withal 

 lucid and short, the critic would be much mistaken in assuming that it 

 will not be read by his own readers and many besides. 



The field of ethics has been crossed and recrossed in many direc- 

 tions ; and we are now called to follow a new and unbeaten track. 

 Our interest and expectation are awakened, not simply on account of 

 the general philosophic ability of the writer, which disposes us to 



