That excellent Geometer the late Dr Matthew Stewart, as long ago as the year 1746, published his book of General Theorems, all of them, except the first five, without the demonstrations. As I do not find that any demonstration of them has ever been made public, I may perhaps flatter myself that what I now communicate to this Society will not be wholly unacceptable. The demonstrations given are of the propositions relating to the sums of the squares, and of the fourth powers of lines drawn in a certain manner, and are selecled from the rest, as most connected with one another. The theorems that respect the cubes and other higher powers, may afford materials for another paper, should this meet with the approbation of the Society.