MATHEMATICAL PAPERS. 



41 



Let it be required to find two mean proportionals between 

 any two given extremes Y and Z. (Plate i , fig. 1 3tli. ) 



Take any ftraight line A C=Z, (PI 



fig 



4) and from 



tlie point G draw C I making any angle A C I, lefs than one 

 third of two right angles. Make the angles I C K and KGB 



r . ■ ■ 



each eqnal A C I, and make C B=Y. Join A B interfecling 

 i C and K C in E and D. From E and D draw E G and 

 D F, making the angles AEG and B D F each eqnal A C I, 

 and interfering A C and C B in G and F. Through A and B 



■ > _ 



draw A I parallel to G E, and B K parallel D F, meetmg C E 



r 



and C D produced in I and K. Then are C K and G I the 



mean proportionals required. 



Secondly, I am to fhew that the demonftr 



which 



Mr, Winthrop has attempted to give of the above method 



not true. 



The fubftance of it is as follows. 



The angle B D G=D E C+E C D 



r- 



b 



B D F-E C D 



Hence the 



gles F D C and D E G are fimilar 



e 



Alfo 



hence the 



angle A E C =E D C+D C E, but A E G-D C E : 



I 



triangles G E C, E D G and D F C are all fimilar to each other. 

 And therefore G F .: G D : G E : C G. Again becaufe A I is 



■ 



parallel to G E, and K B parallel to D F, join I K, and I K is 

 parallel to E D. Gonfequently the triangles A C I, I C K and 



KGB are all fimilar one to another ; and therefore C B (Y) : 

 C K : C I : C A (Z) as was required. 



But although A I is parallel to G E, and K B is parallel to 

 B F, it does not follow that I K is parallel to E D ; and there- 



fore Mr. Winthrop's demonftration i§ not true. 



G 



I am 



