62 Notices respecting New Books. 



mean 



hence 



(i+2)+3+ *4i-*)=(i+3)+2+ \<i-y); 



\*Q.-z)=\w(l-y), 



which appears to mean 



4=4. 



x 

 The most general meaning of - cannot be considered to be a class 



which, on restriction by z, produces x ; that is its meaning only 

 when x{\ — z)=Q. It appears that we are entitled to equate the 

 sum of the first and fourth terms on the one side to the sum of the 

 first and fourth terms on the other side, and the second term to the 

 second. But we are certainly not eutitled to equate, as Mr. Yenn 

 thinks we are, the first to the first, and the fourth to the fourth. 

 Eor let x(l —z)=0 and tt'(l —y) = 0, then the state of affairs is still 

 represented by (1 + 2) + 3= (1+3) + 2. 



Another excellent feature of the work, and the one which appears 

 to me to display the greatest amount of originality, is the invention 

 of appropriate diagrams to represent the classes into which the 

 whole is broken up when a given number of attributes are taken into 

 consideration. Intersecting circles suffice in the case of two or 

 three attributes ; but when the number is increased to four, we 

 encounter the difficulty of making four closed plane figures inter- 

 sect so that each of the resulting classes is continuous. Mr. Yenn 

 succeeds, by means of ellipses, in drawing figures which make the 

 final classes continuous, though some of the intermediate classes 

 are detached. He has also 

 shown how to apply these 

 diagrams to express data, by 

 shading the classes which 

 are given to be non-existent. 

 A paper describing this me- 

 thod appeared in the Philo- 

 sophical Magazine for Julv 

 1880. To investigate the 

 above problem, the method 

 gives us the four-ellipse dia- 

 gram (fig. 2); and the condi- 

 tion expressed in the equa- 

 tion xy — zw is expressed by the shading out represented. It indi- 

 cates the same conclusion as the other figure. 



A difficulty arises in the application of this process of shading- 

 out in the case of the contrary class (pp. 271, 281), owing to the 

 fact that Mr. Yenn does not represent the whole class of things 

 considered by an enclosure such as the square in fig. 1. Not only 

 does the shading-out process require this, but the logical- diagram 

 machine insists upon it (p. 122). This point is intimately con- 



