368 GEOMETRY. 



be divided into triangles. Upon this the practice 

 of land-surveying and making plans of estates is 

 founded. 



Prob. 21. To make a square equal to two given 

 squares. Make the sides D E and D F of the two 

 given squares A and B ; from the sides of a right- 

 angled triangle F D E ; draw the hypothenuse 

 F E ; on it describe the square E F G H, and it 

 will be the square required. 



Prob. 22. Two right lines A B, C D, being 

 given, to find a third proportional. Make an angle 

 H E I at pleasure ; from E make E F equal to 

 A B, and E G equal to C D : join F G. Make 

 E I equal to E F, and draw H I parallel to FG ; 

 then E H will be the third proportional required ; 

 that is, EF : E G : : E H : E I, or A B : CD : : 

 C D : E I. 



Prob. 23. Three lines being given, to find a 

 fourth proportional. Make the angle H G I at 

 pleasure ; from G make G H equal to A B, G I 

 equal to C D, and join H I. Make G K equal to 

 E F ; draw K L through K, parallel to H I ; then 

 G L will be the fourth proportional required ; that 

 is, G H : G I : : G K : G L, or A B : C D : : E F : 

 GL. 



Prob. 24^. To divide a given line A B in the 

 same proportion as another C D is divided. 



Make any angle K H I, and make H I equal to 

 A B ; then apply the several divisions of C D from 

 H to K, and join K L Draw the lines he, if, kg, 

 parallel to I K ; and the line H I will be divided in 

 e>f> g> as was required. 



Prob. 25. Between two given lines A B and C D, 

 to find a mean proportional. 



Draw the right line E G, in which make E F 

 equal to A B, and F G equal to C D. Bisect E G 



