This paper professes to investigate certain properties of the series of whole numbers whose ultimate differences are constant, and incidentally to treat of Fermat’s theorem of the polygonal numbers, and some other properties of numbers. Its object is to show that the same (or an analogous) property which Fermat discovered in the polygonal numbers belongs to other series of the same order, also to all series of the first order, and probably to all series of all orders. It also proposes to prove the first case of Fermat’s theorem (that is of the triangular numbers) from the second case of the squares (which had not before been done), and to dispense with the elaborate proof of Legendre (Théorie des Nombres), finally, to prove all the cases by a method different from that either of Lagrange, Euler, or Legendre.