The general properties of numbers, considered without reference to the notation in which they are expressed, have been very fully investigated by several of the most distinguished mathematicians. Little attention, however, has been paid to the particular properties resulting from the principle of the modern notation, which is the expression of every number in a series, a + bn + cn2, &c. where a, b, c, are the digits, and n the local value or root of the notation. Having been led to examine some of these results, and to account for them, I am now desirous of laying them before the Society. I do not flatter myself that they possess any great practical importance; but as I have reason to believe that they are new, I trust the Society will not think them entirely unworthy of their attention.