&f Sciences of the Inslllute of France. 3-15 



MATHEMATICAL PART. 



By M, Le Chevalier DiiLAMBRE, Perpetual Secretary, 

 " Never, perliaps," says the learned analyzer of this dei.avt- 

 Hient, " was the zeal of 'geometricians better sustained— -ne%-er 

 have they devoted thciuselves with more constancy to their ac- 

 cnstomeri labours, to the development of their first ideas, to tlw 

 completion of works already published ; and yet never have v/c . 

 found greater difficulty iu giving the history of this branch of thn 

 annual labours of the Academy." After a statement of ^some 

 oljstacles arising from the private arrangements of the Academ.y, 

 " the more," it is added, " mathematics have advanced, the more 

 diificult their ulterior progress has become, the more impracli- 

 cable it is found to render sensible and striking the results newly 

 obtained." It will not therefore, it is hoped, excite surprise, that 

 in the present analysis little more than the titles of some valu- 

 able works should be indicated, nor (if we may be allowed to 

 supply the parallel implied, but not expressed, by the learned se- 

 cretary) that in entering more at large into other works, a suc- 

 cessive exposition of the contents of each should be preferred to any 

 attempt to give a condensed and systematic view of the whole. 



The length, however, to winch on these accounts the ana- 

 lysis of this branch of th.e kibours of the Academy extends (eighty 

 pages quarto), is a disadvantage which, we regret, precludes the 

 practicability of transferring it to our pages, as we have done that 

 of the preceding department. Not a few of the articles indeed, 

 particularly those of M.Laplace, Mr.Farey, &c. have already been 

 fullv noticed in the Philosophical Magazine ; and opportunities 

 will' afterwards be taken to make our readers acquainted from 

 time to time with such others as merit, but have not yet been 

 under, their observation. 



The article of Mr. Farey's to which we have just alluded, and 

 to which the ingenious secretary refers with no more than the 

 respect which it merits, appeared originally in the Philosophical 

 Magazine, vol. xlvii. p. ."^85, and refers to a curious property of 

 vulgar fractions, which Mr. Farey was the first to discover and 

 communicate to the world *. 



* My injienious correspondent Mr. Farey, referring to this subject in a 

 letter vvhicli I lately received from him, wrileb as trjlluws: 



" I still entertain hopes, that some Englisli Mathc-mutician will turn h\s 

 attention to the demonstration of the properties of complete fractional 

 Series: your Correspondent S. A. in p. 204 of vol. xlviii. has failed entuelv, 

 and reasoned in a circle: he fust (onus nril/nnclical progressions, which 

 he could not find in any Series, and then shows, that the mscparahic 

 properly of aiitlimetical progressions attends such scrirs, of his own 

 forming. 1 wish likewise, that some of those fond of contemplating Mubical 

 Ratios and Intervals, wouhl consider the complete Hurmoiuc Scries X\ II, 

 :■: V, XII, X, Vllf, VI, V, 4, in and 1, whose ratio;; arc eiveii in p. ^85. 

 What would he the effect on the Ear, of a Chord thus ..Jinilculy (.•Had, on 

 -Mr. Liiton's Euhurraonic Organ .\"^£i>;ruJi. 



