A'o. IX. 



OBSERVATIONS 



On the Duplication of the Cube, and the Trtsection of 



an Aiigle. 



BY COL, JARED MANSFIELD, 



SURVEYOR-GENERAL OF THE UNITED STATES. 



IN the memoirs of the American Academy of Art» 

 and Sciences, Vol. II. are two papers (No. 1 and 2) 

 by James Winthrop, Esq. which from the importance 

 of their subjects, cannot fail to excite the attention of the 

 mathematical reader. In the first of those papers, the 

 author proposes to solve the ancient and difficult pro- 

 blem of the duplication of the cube, or of finding-, geome- 

 trically, two mean proportionals between any two ex- 

 tremes of a geometric series. This problem, in arith- 

 metic, amounts only to the extraction of the cube root, 

 and may very easily be solved by numbers, or loga- 

 rithms. A solution, however, by the strict principles of 

 geometry, is not so easily effected, and by no means do 

 I apprehend that any propositions of Euclid, or of a right 

 line and circle, are sufficient for this purpose. Newton 

 and others, who have used a circle, have either produ- 

 ced a mechanical solution of this problem, or have intro- 

 duced principles of the higher geometry. Mr. Win- 

 throp appears to be the only person who has ever attempt^ 



