394 



Mr. A. B. Kempe. 



[June 18, 



§§ 3 — 13. Fundamental Principles. 



(3.) Whatever may be the true nature of things and of the con- 

 ceptions which we have of them (as to which points we are not 

 concerned in the memoir to inquire), in the operations of reasoning 

 they are dealt with as a number of distinct entities or units. 



(4.) These units come under consideration in a variety of guises — 

 as points, lines, statements, relationships, arrangements, intervals or 

 periods of time, algebraical expressions, &c, &c. — occupy various 

 positions, and are otherwise variously circumstanced. Thus while 

 some units are undistinguished from each other, others are by these 

 peculiarities rendered distinguishable. For example, the angular 

 points of a square are distinguishable from the sides, but are not dis- 

 tinguishable from each other. In some instances where distinctions 

 exist they are ignored as not material. Both cases are included in 

 the general statement that some units are distinguished from each 

 other and some are not. 



(5.) In like manner some pairs of units are distinguished from 

 each other while others are not. Pairs may be distinguished even 

 though the units composing them are not. Thus the angular points 

 of a square are un distinguish able from each other, and a pair of such 

 points lying at the extremities of a side are undistinguishable from 

 the three other like pairs, but are distinguishable from the two pairs 

 formed by taking angular points at the extremities of a diagonal, 

 which pairs again are undistinguishable from each other. Further, a 

 pair, ab, may sometimes be distinguished from a pair, ha, though 

 the units a and b are undistinguished. Thus if a, b, c, be the angular 

 points of an equilateral triangle, and barbs be drawn on the ' sides 

 pointing from a to b, from b to c, and from c to a respectively, the 

 angular points a, b, c will be undistinguished from each other, each 

 has an arrow proceeding from it and to it ; but the pair ab is dis- 

 tinguished from the pair ba, for an arrow proceeds from a to b, but 

 none from b to a. 



(6.) It will be convenient to speak of ab and ba as different aspects of 

 the collection of two units, a, b. Here the terms " aspect " and 

 " collection " are each to be understood as referring to two separate 

 units, and not to those units regarded in the aggregate as a single 

 unit. 



(7.) Again we have also distinguished and undistinguished triads, 

 tetrads, .... m,-ads .... n-ads . . . . ; every m-ad being, of course* 

 distinguished from every w-ad. Just as we may have ab distinguished 

 from ba, so we may have an n-&d pqrst . . . . uv distinguished from 

 qusvt .... rp. Here pqrst .... uv and qusvt .... rp will be 

 termed, as in the case of pairs, different aspects of the collection 

 jp, 2, r, 5, t, .... u, v ; the term " collection " being understood to 



