The Book of Curiosities

by John Platts

This is the same thing as to find a number, which being divided by 2, 3, 4, 5, and 6, there shall remain 1, but being divided by 7, there shall remain nothing; and the least number, which will answer the conditions of the question, is found to be 301, which was therefore the number of eggs the old woman had in her basket.

To find the least Number of Weights, that will weigh, from One Pound to Forty.

This problem may be resolved by the means of the geometrical progression, 1, 3, 9, 27, 81, &c. 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.

To break a Stick which rests upon two Wine Glasses, without injuring the Glasses.

Take a stick, (see Plate,) AB. fig. 1, of about the size of a common broomstick, and lay its two ends, AB, which ought to be pointed, upon the edges of two glasses placed upon two tables of equal height, so that it may rest lightly on the edge of each glass. Then take a kitchen poker, or a large stick, and give the other a smart blow, near the middle point c, and the stick AB will be broken, without in the least injuring the glasses: and even if the glasses be filled with wine, not a drop of it will be spilt, if the operation be properly performed. But on the contrary, if the stick were struck on the underside, so as to drive it up into the air, the glasses would be infallibly broken.

A Number of Metals being mixed together in one Mass, to find the Quantity of each of them.

Vitruvius, in his Architecture, reports, that Hiero, king of Sicily, having employed an artist to make a crown of pure gold, which was designed to be dedicated to the gods, suspected that the goldsmith had stolen part of the gold, and substituted silver in its place: being desirous of discovering the cheat, he proposed the question to Archimedes, desiring to know if he could, by his art, discover whether any other metal were mixed with the gold. This celebrated mathematician being soon afterwards bathing himself, observed, that as he entered the bath, the water ascended, and flowed out of it; and as he came out of it, the water descended in like manner: from which he inferred, that if a mass of pure gold, silver, or any other metal, were thrown into a vessel of water, the water would ascend in proportion to the bulk of the metal. Being intensely occupied with the invention, he leaped out of the bath, and ran naked through the streets, crying, “I have found it, I have found it!”

The way in which he applied this circumstance to the solution of the question proposed was this: he procured two masses, the one of pure gold, and the other of pure silver, each equal in weight to the crown, and consequently of unequal magnitudes; then immersing the three bodies separately in a vessel of water, and collecting the quantity of water expelled by each, he was presently enabled to detect the fraud, it being obvious, that if the crown expelled more water than the mass of gold, it must be mixed with silver or some baser metal. Suppose, for instance, in order to apply it to the question, that each of the three masses weighed eighteen pounds; and that the mass of gold displaced one pound of water, that of silver a pound and a half, and the crown one pound and a quarter only: then, since the mass of silver displaced half a pound of water more than the same weight of gold, and the crown a quarter of a pound more than the gold, it appears, from the rule of proportion, that half a pound is to eighteen pounds, as a quarter is to nine pounds; which was, therefore, the quantity of silver mixed in the crown.

Since the time of Archimedes, several other methods have been devised for solving this problem; but the most natural and easy is, that of weighing the crown both in air and water, and observing the difference.