The parallelogram which formed the subject of this communication is the rectangle whose sides are to each other in the ratio of the diagonal of a square to its side,—a figure well known to architects, sculptors, and painters, from its beauty, and frequently adopted in the practical arts.

The author shewed that if the given rectangle be bisected by a line parallel to its shortest side, each segment will be a figure similar in all respects to the original rectangle; and if either of these halves be itself bisected in the same manner, their halves will be rectangles similar to the original rectangle; and so on ad infinitum. The sides of the primary figure and its halves are continual proportionals, represented by the series

The author endeavours to trace an analogy between the properties of this parallelogram and the logarithmic spiral.