1820,] Royal Academy of Sciences. 219 



equation of the fourth degree, which is solved in the same way 

 as those of the second. M. Peyrard begins by giving the alge- 

 braic calculation of it ; afterwards, by translating the Greek 

 demonstration into the modern style, he renders its strict and 

 accurate progress more forcibly evident. 



"Amongst all the propositions contained in this! last volume, not 

 one occasioned the etlitor so much trouble as the 17th of the 

 12th Book of the Elements. In all manuscripts and editions 

 whatever, the figure was so defective, that most of the reason- 

 ings of Euclid were quite inapplicable to it. The translator has, 

 by means of adding some hnes, rendered the demonstration 

 exact in every respect. 



" There is, in every other part, as in the preceding volumes, the 

 most perfect fidelity in the translation, and the same care is 

 taken to correct the text, and to collect the various readings, 

 which here form 84 pages. The editor had asserted that the 

 fine Oxford edition was not more correct than that of Basil, 

 since, besides republishing all the errors, even the most palpable, 

 of the latter, the former contained a considerable number of 

 other faults, from which the Basil edition was free. This state- 

 ment excited astonishment, and was probably but httle credited; 

 yet we cannot perceive what can be objected to the eight pages 

 in which M. Peyrard has exhibited a comparative view of the 

 two editions. 



" M. Peyrard has now brought a long and laborious work tea 

 successful conclusion. We propose to tile Academy to extend 

 the same approbation it has been pleased to bestow on the two 

 orhers, to the third volume, in the hopes that this merited 

 approbation may facilitate the publication of the author's Apol- 

 lonius, the manuscript of which has long since been finished." 



We have much pleasure in announcing that this new edition 

 is begun, and that we have already seen several sheets of it. 



A Treatise on Descriptive Geometry; by M. Vallee, Civil 

 Engineer. — Committee, Messrs. de Prony, Fourier, and Arago, 

 Secretary. 



The lectures on descriptive geometry of the celebrated 

 author of that doctrine, M. Mongc, contain an explanation of 

 the principles of the science which will always be cited as a 

 perfect model of perspicuity. It is to be regretted that the work 

 is not more extensive, for artists who have not made a particular 

 study of mathematics cannot familiarize themselves with the 

 methods of projection, without varying the data of the ques- 

 tions, and practising upon a great number of examples. M. 

 Hachettc has partly filled up this chasm by a supplement, 

 which obtained the approbation of the Academy. By following 

 the footsteps of these two scientific men, M. Vallee has com- 

 piled his treatise, which he divides into six books, forming 

 more than 60U pages in quarto. " The Committee, who devoted 

 their attention chiefly to the most difficult parts, feel pleasure in 



