This problem may be resolved by the means of the geometrical progression, 1,3,9, 27, 81, etc. the property of which is such, that the last, sum is twice the number of all the rest, and one more; so that the number of pounds being forty, which is also the sum of 1, 3, 9, 27, these four weights will answer the purpose required. Suppose it was required, for example, to weigh eleven pounds by them: you must put into one scale the one-pound weight, and into the other the three and nine-pound weights, which, in this case, will weigh only eleven pounds, in consequence of the one-pound weight being in the other scale; and therefore, if you put any substance into the first scale, along with the one-pound weight, and it stands in equilibrio with the three and nine in the other scale, you may conclude it weighs eleven pounds.

In like manner, to find a fourteen-pound weight, put into one of the scales the one, three, and nine-pound weights, and into the other that of twenty-seven pounds, and it will evidently outweigh the other three by fourteen pounds; and so on for any other weight.