Olservations on Equation of Payments. 269 



Mr, Alsager exhibited various new and unexpected results, 

 of chords intly tuned on this Instrument, such as the marked 

 and pleasing character of the IX, very closely associating it with 

 the less grateful of the Concords ; and he said that this interval 

 could be almost as certainly adjusted by the ear at once in 

 tuning, as one of the Concords : the same with regard to the 

 *VI : and the b! 0,(726) ; the perfectly smooth and agreeable 

 effect of the discords XIV and XVI, when heard in the common 

 Chord on tlic Twelfth and Diapason Stops, and others, which it 

 might be tedious to mention. 



It having been hinted by some of the Profession, to whom 

 Messrs. Flight and Robson had handed the printed Paper copied 

 into your last, that a more detailed Ta])le of two Octaves of tiie 

 Scale of the Instrument would be desirable ; these gentlemen in 

 their laudable zeal to CKplain the principles of the Instrument 

 most fully, have since printetl and circulated such a Table, v.'hich 

 I cannot doubt but your Readers will approve of seeing preserved 

 ill your Magazine. See the preceding page. 



I hope the zeal at present beginning to be manifested for en- 

 couraging Euharmonic Organs, will be productive of their spread 

 amongst us, and I am 



Your obedient servant, 

 April 6, 1817. PhILO-MusICUS. 



LXVII. Observations on Equation of Payments; tending to prove 

 that the generally -received Riile of Malcolm is not correct. 

 By Mr. J. B. Ben well. 



To'Mr. Tilloch, 



Sjr, — In the cessation from other engagements, I send you 

 the following observations for insertion in the Philosophical 

 Magazine. 



it has been asserted, and with evident reason too, by rnatlie- 

 matical writers of the present day, that no rule in arithmetic has 

 been subject to more dispute than the rule for equating of pay- 

 ments. But as in the ca-'c of the binomial theorem mathema- 

 ticians have long been seelchig a general demonstration from 

 principles too refined and foreign to its object, so in this rule 

 called equation of payments, a principle has been assumed which 

 (although accurate in itself) will not apply as the basis of a legi- 

 timate datum, excej)ling by virtue of an express convention prc- 

 viouslv' ;u'd purpo.^eiy made. 



But it would really sr-em that the controversy had been effec- 

 tually decided by the unanimous adoption of Malcolm's rule, as 

 aimplified since his time, and whicii may be found in the few 



trealisca 



