XII. Demonstrations offome of Dr Ma tthew S te w- 

 ARTs General Theorems. By Rorert 

 Small, D. D. F. R. S. Edin. 



[Read by the Jutbor, Feb. 7. 1785.] 



THAT excellent Geometer the late Dr Matthew Stew- 

 art, as long ago as the year 1746, publiflied his book 

 of General Theorems, all of them, except the firll five, without 

 the demonftrations. As 1 do not find that any demonftration 

 of them has ever been made public, I may perhaps flatter my- 

 felf that what I now communicate to this Society will not be 

 wholly unacceptable. The demonftrations given are of the 

 propofitions relating to the fums of the fquares, and of the 

 fourth powers of lines drawn in a certain manner, and are 

 felecSed from the reft, as moft connedled with one another. The 

 theorems that refpe(5l the cubes and other higher powers, may 

 afford materials for another paper, fhould this meet with the 

 approbation of the Society. 



A FEW lemmas and corollaries have been introduced that are 

 not among Dr Stewart's Theorems, and which are therefore di- 

 ftinguiftied by afteriflcs. The references are to the edition of 

 thofe Theorems publifhed at Edinburgh, 1746, and the propofi- 

 tions are numbered as in that edition, beginning with the fixth 

 Theorem. 



THE- 



