JF Ptak Science Books Quick Post
I found this very interesting history of calculating machines in the February 1885 issue of The Popular Science Monthly (Volume XXVI, No IV)--it is a wonderful piece, nicely illustrated, too. The entire article is reproduced below and the original is offered for sale on our website.
When I was a little boy, I sometimes went for the bread to a short distance from the house. The baker would take my tally-stick, put it alongside of his, and cut a notch in both. Then I would go away with my bread and the baker's account on the tally-stick. At the end of a fortnight or a month the tally-notches were reckoned up and the account was settled. The number of notches represented the number of loaves of bread bought, and this number, multiplied by the price per loaf, gave the amount of money I had to take to the baker. [Lucas, 1842-1891, above.]
Although in our present article we shall make use of systems of numeration, and particularly of the decimal system, it is proper to observe that the most important properties of numbers are independent of such systems, and that they are used by the arithmetician in his calculations only for aids, as the chemist uses bottles and retorts. We give two specimens of properties of numbers, which we see illustrated in the problems called the flight of cranes and the square of the cabbages. Cranes in their flight dispose themselves regularly in triangles. We wish to get a rule for finding the number of the birds when we know the number of files; or, supposing that we have arranged the files with increasing numbers from unity to a determined limit, we seek to find the total of the unities contained in the collection. To make the matter plainer, let us seek the sum of the first six numbers, or the number of units represented to the left of the broken line in Fig. 1, by the black pawns. We will represent the same numbers, in an inverse order, by white pawns, to the right of the same line. We shall see at once that each horizontal line contains six units plus one; and, since there are six lines, the number of units in the whole square is six times seven. The number we are seeking, then, or the number in the half-square, is half of forty-two, or twenty-one. The same reasoning may be applied