Prof. De Morgan on Anharmonic Ratio. 167 



in one line, in any order, let (PQRS) denote the ratio com- 

 pounded of PQ: QR and RS: SP, or (PQ: QR)(RS: SP). 

 This is what M. Chasles has styled the anharmonic ration by 

 an extension of its well-known case in which (PQRS) is 1 : 1, 

 in which case PR is said to be harmonically* divided in Q 

 and S. Since the letters P, Q, R, S can be written in twenty- 

 four different orders, there are as many forms in which an 

 anharmonic ratio may be presented. But all these forms are 

 only three ratios and their inverses : nor are these three di- 

 stinct, for they are always of the connected forms A : B, 

 A : A + B, B : A + B. And the rules for separation and col- 

 lection are as follows. 



To find the distinct anharmonic ratios. Take any one, say 

 (QSRP), advance the first letter one interval, and two inter- 

 vals, giving (SQRP), (SRQP): theee three are distinct; and 

 if the first be A : B, the second is A: C, the third is B : C, and 

 of A, B, C, some one is equal to the sum of the other two. 



To Jind forms of equivalent ratio. Invert all the letters, or 

 pairs, or one after the other ; thus(RSPQ), (QPSR), (PQRS), 

 (SRQP), are the same. 



To find the forms iiohich have ratios inverse to that in any 

 given form. Change either extreme into the other, or inter- 

 change the first and third, or second and fourth: thus if 

 (SQRP) = A:B, either of these four (QRPS), (PSQR), 

 (RQSP), (SPRQ), isB:A. 



To express the ratios circuitously^ or in the order A:B, B:C, 

 C : A. Make two transferences of the first letter and of the 

 second: thus if (PQRS) be A:B, (QRPS) is B : C, and 

 (PRSQ) is C:A. 



Tofnd convenient values of A, B, C. Look for the order 

 in which the letters occur on the line, say P, Q, R, S. Take 

 the rectangle under PQ and RS, with wholly unconnected 

 sides; under PS and QR, one of which contains the other; 

 and under PR and QS, in which each side is partly in and 

 partly out of the other. Call these A, B, C ; then C = A + B, 

 and (PQRS) = A : B. Call this, and its inverse, the primary 

 ratio. 



(5.) If two systems be anharmonically equivalent, the six 

 ratios (counting inverses) of one being severally equal to those 

 of the other, then the primary ratios are equal, each to each. 



(6.) To what is usually said on harmonic division, I should 

 add the following. If C, D, and C, D' be conjugate pairs of 

 points to AB, it is known that of the two lines CD, CD', the 



* This mode of speaking is not exactly consistent with the derivation of 

 the phrase ; it is PR which is an harmonic mean between PQ and QS, ac- 

 cording to the received use of the word harmonic. 



