“Having been frequently admitted to the honor of hearing his late most excellent Majesty, Charles XII. discourse on mathematical subjects, I presume an account of a new arithmetic invented by him, may merit the attention of my readers. “His Majesty observed then, that the denary arithmetic, universally received and practiced, was most probably derived from the original method of counting on the fingers; that illiterate people of old, when they had run through the fingers of both hands, repeated new periods over and over again, and every time spread open both hands; which being done ten times, they distinguished each step by proper marks, as by joining two, three, or four fingers. Afterwards, when this method of numeration on the fingers came to be expressed by proper characters, it soon became firmly and universally established, and so the denary calculus has been retained to this day. But surely, were a solid geometrician, thoroughly versed in the abstract nature and fundamentals of numbers, to set his mind upon introducing a still more useful calculus into the world, instead of ten, he would select such a perfect square, or cube number, as by continual bisection, or halving, would at length terminate in unity, and be better adapted to the subdivisions of measures, weights, coins, etc. “Thus intent on a new arithmetic, the hero pitched upon the number eight, as most fit for the purpose, since it could not only be halved continually down to unity, without a fraction, but contained within it the square of 2, and was itself the cube thereof, and was also applicable to the received denomination of several sorts of weights and coins, rising to 16 and 32, the double and quadruple of 8. Upon these first considerations, he was pleased to command me to draw up an essay on an octonary calculus, which I completed in a few days, with its application to the received divisions, coins, measures, and weights, a disquisition on cubes and squares, and a new and easy way of extracting roots, all illustrated with examples. “His Majesty having cast his eye twice or thrice over it, and observing, perhaps from some hints in the essay, that the denary calculus had several advantages not always attended to, he did not at that time seem absolutely to approve of the octonary: or, it is likely he might conceive, that though it seemed easy in theory, yet it might prove difficult to introduce it to practice. Be this as it may, he insisted on fixing upon some other that was both a cube and a square number, referrible to 8, and divisible down to unity by bisection. This could be no other than 64, the cube of 4, and square of 8, divisible down to unity without a fraction.