370 



MEMOIRS OF THE AMERICAN ACADEMY. 



i. Transitives are further divisible into those whose products by themselves are 



qual to themselves, and those whose products by the 



than them 



selves ; the former may be termed continuous, the latter discontinuous. An example 



of the second is found in the pure mathematics of a continuum, where if a is greater 



than b it is greater than something greater than b ; and as long as a and b are not of 



the same magnitude, an intervening magnitude always exists. All concurrents are 

 continuous. 



/. Intransitives may be divided 



products) of which 



those the number of the powers (repeated 



contained in the first is infinite, and those some power of 



which is contained in the first. 

 Infinite inexhaustibles are cyclic 



The former may be called infinites, the latter, finites 



In addition to these 



old divisions of 



of the latter into aptitudinal and actual, and of 

 are often useful * 



of reason and real 

 into extrinsic and 



i 



" Not" 



We have already seen that "not," or "other than," is denoted by Q-r. It is often 

 more convenient to write it, n. The fundamental property of this relative has been 

 given above (111). It is that, 



6 



X 



1 



x . 



4 



Two other properties are expressed by the principles of contradiction and excluded 



middle 



They 



x,Q 

 x-^Q 



X 







I* 



X 



1. 



The following deduced properties are of frequent application 

 (167.) • 



6-( 



6 



-fc6-f 



i 



(168.) 



Q-xy 



6 



X 



y 



z x , zv. The 



The former of these is the counterpart of the general formula, & + y 



* "Duplex est relatio: scilicet rationis et realis. Unde relatio • rationis est qua, fit per actum comparativum intel- 

 lect, «t sunt secunda, intentiones; sed relatio realis est duplex, scilicet aptitudinalis et actualis. Aptitudinalis est 

 qu* non requmt terminum actu existere sed solum in aptitudine ; cujusmodi sunt omnes propria, passiones, omnes 

 aputudmes, et omnes inclinationes ; et tales sunt in illo pr*dicamento reductive in quo sunt ilia quorum sunt propria, 

 passes. Sed relatio actualis est duplex, scilicet, intrinsecus adveniens, et extrinsecus adveniens. Intrinsecus adveni- 

 ens est qua, necessario ponitur positis extremis in quacunque etiam distantia ponantur, ut similitudo, paternitas, equali- 



extref n ""• T"" * ^ neCeSSari ° ™ ^^ P ° siti9 extremis > sed «■***« deWta approximate 



praedicamenta 



Tartaretus 



