De Boot makes the carat (four grains) his unit of comparison, increasing his base value by one third for pearls weighing eleven carats (forty-four grains) or over. In Pio Naldi’s treatise, published in Bologna in 1791, the unit is the grain, the base being the fourth part of the value of four pearls weighing together one carat. Naldi, also, increases his base value making it 1½ lire ($.30) for pearls weighing less than ten grains, and 2½ lire ($.50) for those weighing twenty grains and upward. A curious method of valuing pearls by their weight is shown in a treatise by Buteo, published in 1554.[382] The writer states that a pearl weighing two carats was valued at 5 gold crowns; one of four carats at 25 crowns; and so on, the price increasing fivefold when the weight was doubled. The intermediate figures were obtained by computing the proportional mean of any two known weights and values. For example: 8 × 4 = 32, the square root of which is 5.656. Now, the value of a four-carat pearl is 25 and that of an eight-carat pearl 125 crowns, and 125 × 25 = 3125, the square root being 55.9; hence a pearl weighing 5.656 carats was worth 55.9 crowns. The base value of a necklace can be determined in the following way. Should the center pearl weigh 25 grains, multiply 25 by 25; the result is 625; then, take the next two, three, or four pearls, as many as are of approximately the same weight, add their weights together, multiply the resulting figure by itself and divide the product by the number of pearls in the group. Proceed in exactly the same way with the remainder of the necklace, always grouping the pearls so that there shall not be a considerable difference in weight between the smallest and the largest pearl, and then add together the figures obtained for the center pearl and for the various groups and divide the price of the necklace by this total; the quotient will represent the multiple or base. As may be seen by comparison of the first with the second and third of the accompanying tables, the result arrived at in this way will, if there is any difference in the weight of the pearls in the various groups, vary slightly from that obtained by calculating the weight of each pearl separately, but it represents a satisfactory approximation. NECKLACE OF 41 GRADUATED PEARLS ON A $10 BASE