42 ASTRONOMICAL and 



I am tKIrdly to demonftrate that Mr. Wiiitarop*s method 

 of performing this problem is univerfally falfe. 



Becaufe Mr. Winthrop has not proved that I K is parallel to 

 E D, it does not follow that I K is not really parallel to E D. 



J* 



Imagine therefore that I K is parallel to E D, Then 



triangles A I E, K B D, G E C, E D C, DEC, AI C, I KG and 

 K B C be all iimilar to one another. Therefore IE: K D : ;. 

 CE: CD; andalfoIE :KD ;: AI: BO. But A I is to BD 

 In the complicated ratio of A I to KB and of K B to B E), and 



—2 



AIiKB t: AC: KC::CE:CD; alfa KB : B D : : CE 



'•^ 



C D. Therefore A I : B D : : C E ; C B. .But it has jnft been 

 proved that AI : BD :: IE i KD ;: CE ; CD ; and hence 



C E : CD : : C E : CD, which is abfurd. I K cannot there- 



* _- 



fore be parallel to E D ; and confequently this method of 

 performing the problem is uftiverfally falie. And as Mr. 

 Winthrop's mode of duplicating the Cube depends upon thd. 

 truth of this problem, it is alfo univerfally falfe. r Q. E. D.' 



n 



The fame may be demonftrated by a variety of other metli 

 ods. Eor inftance by the addition of finesj tangents, &c. or 

 by the logarithmic fpiral, 



n I 



I muft at the fame timte acknowledge that Mr, Winthi^op's 

 ixiiRfikc on this fubJ€(ffe was fo natural, that I at lirft followed 

 Him ia k. A fecond inipe<5lion convinced me of my miflake. 

 Hallowell^ in the DizrRicr of Main Ey June 2^ 1798, 



N. B. Soon after the paper, to which Mr. Barents obrcrvatioas relate^ was.com- 



r 



L 



Tnunkated to the Academy, it was examined by fevcral of the members, fkilled in 

 mathen^tical fcicnce, \v*ha were of ooinion* that tlie a>irhnr \\:\A not demonftrated 



ihtOTLXXiM E^i 



the ccmmlttee foip- 



dert 



-? 



._ J-' 



2 



Mn 



