170 Prof. De Morgan on Anharmonic Hatio, 



it by CC", and AA'BB' is complete, while ABA'B', which 

 was complete, takes its place among the incomplete ones. As 

 follows : 



AA'BB' has diagonals AB, A'B' wants CC" and dispenses with CC. 

 BB'CC ... BC, B'C ... A A" ... A A' 



CC'AA' ... CA, CA' ... BB" ... BB' 



CC 



AA' 

 BB'. 



(16.) That each diagonal is harmonically divided by the 

 other two is well-known ; but this is only a very small part of 

 the following. Take any two points and their opposites, say 

 A, C, A', C. Any triangle made from three of these, say 

 ACC, has BB'B" for an harmonic inscript, as proved by the 

 lines which meet in the fourth point, A'. This gives twelve 

 cases of harmonic inscription. (17.) Let A"B"C" be called 

 the diagonal triangle', it has four harmonic inscripts. Of the 

 remaining six letters take any three which are in the same 

 straight line, say A'BC. The other three AB'C, show an 

 harmonic inscript of the diagonal triangle, A'BC, first named, 

 being the evanescent common inscript. 



(18.) Hence it follows that A"A, B"B, C"C, &c. meet in 

 one point. Let them meet in ; A"A, B"B', C'C, in 1 ; 

 A" A', B"B, O'O, in 2; A"A', B"B', C'C in 3. There are 

 then six new lines 01, 12, 23, 30, 02, 31. If C'A" be taken 

 as a seventh line, then 0123 is complete ; if B"C", then 1203 ; 

 if A"B", then 2013. 



The above will be sufficient for my purpose, and will show, 

 I think, the value of the anharmonic ratio. It will be observed 

 that I adhere closely to the language of geometry, and do not 

 admit that of algebra: but I go further, and, while thinking 

 of the subject, do not admit the Jiotions of arithmetic. From 

 various writers I gather that they think, in compounding the 

 ratio, say of AB to CD and PQ to RS, there is no choice 

 except either to compare the rectangles under AB and PQ 

 and under CD and RS, or else the products AB x Pd and 

 CD X RS, the linear symbols being interpreted numerically: 

 so that the composition of three linear ratios in plane geometry 

 is resisted as involving the use of solids. But if any one will 

 accustom himself to fix his mind upon alteration i?i a ratio as 

 an operation which can be conceived independently of number, 

 and executed independently of rectangles, he will find that the 

 propositions of the geometry of transversals, &c. have a vitality 

 which the algebraic forms cannot give them. The following 

 process would not only help the confirmed algebraist to receive 



