40 



MATHEMATICAL PAPEPvS. 



Remarks on Mr. Winthrop's paper on the duplication of the Cube 



part ijl of this voh 



By GEORGE BARON, late Ma} 



ter of the Mathematical Academy at South Shields^ in the 

 County of Durham^ in England. 



Mr 



"Wintlirop has attempted the dupl 



r 



th 



e 



Cube, in a paper inferted at page 25, part I. yo\. 2, of the 

 Memoirs o£ the Amei-ican Academy of Arts and Sciences. 



_ J 



This paper is entitled " Geometrical methods of finding any- 

 required feries of mean proportionals between given extremes." 

 It confiflis of four problenis. In problem i. is fliewn the well 

 known method of findhig one mean proportional between two 



" To find two mean propor- 



givcu extremes. Problem 





IS 



between two given extremes 



5> 



On 



truth of 



problem the two remaining problems, together with Mr. \\ 



th top's dupl 



of th^ Cube, entirely depend. I fliall 



therefore here, fij^i defcribe Mr. Winthrop's method of find 



ing two mean prppoi 



between two 



given 



Secondly, ihew that the demonftration which he has attempt 

 ed to give of that m^ 



It method is not tn^e. And thirdly, I fliall 

 dcmonflrate that that method is univerfally falfe. And hence 



I 



it will follow, that Mr. Winthrop's duplication of the Cube is 

 univerfally falfe, 



Firft then, I am to defcribe Mr. Winthrop's method of 

 finding tw^o mean proportionals between two given extremes. 

 The fubflance of it is as follow-s. 



> 



Let 



