theory of these numbers was born on the Nile at a remote epoch, and was developed by Diophantes, the father of arithmetic, at the school of Alexandria. In his treatise occurs the proposition, giving, as an 
Fig. 3.
Fig. 4. essential condition of such a number, that the octuple of a triangular number, augmented by unity, is a perfect square. This fact is made evident by the examination of the diagram (Fig. 3). An arithmetical progression is a series of numbers in which each member is equal to the preceding member, plus a constant number which is denominated the common difference. Thus the odd numbers one, three, five, seven, nine, eleven, form an arithmetical progression, the common difference of which is two. We can demonstrate, as was done in the preceding case, that the sum of the terms in such a procession is equal to the product of the number of terms by the half-sum of the extremes; and, in the same way, the area of a trapeze is half the area of a parallelogram of the same height, the base 
Fig. 5 The Square of Cabbages. of which is equal to the sum of the bases of the trapeze (Fig. 4).
Our second example is borrowed from Plato. Fig. 5 represents a square of cabbages. To get the number of cabbages contained in the square, we multiply by itself the number on one of the sides.
We have marked lines bounding the successive squares that contain one, two, three, four, five, or six cabbages to the side. Now observe the difference between the numbers of cabbages in one square and the next one. We find that the numbers included in the successive inclosures bordered by our lines are one, three, five, seven, nine, eleven; and we perceive, by reference to the short dotted lines, that the number from one inclosure to another increases by two. We come at once to the propo-
