Mathematicians

A square is a two-dimensional shape that has four sides (a quadrilateral), where all of the sides are of equal length, and all of the corners form 90 degree angles. A square can be described as an equilateral rectangle, or conversely as an equiangular rhombus.

An architectural square surrounded by buildings can be called a quadrangle or courtyard.

Two of his major contributions to physics are his understanding of the principle of buoyancy, and his analysis of the lever. He also invented many ingenious technological devices, many for war, but also the Archimedean screw, a pumping device for irrigation systems.

Heiro, after gaining the royal power in Syracuse, resolved, as a consequence of his successful exploits, to place in a certain temple a golden crown which he had vowed to the immortal gods. He contracted for its making at a fixed price and weighed out a precise amount of gold to the contractor. At the appointed time the latter delivered to the king's satisfaction an exquisitely finished piece of handiwork, and it appeared that in weight the crown corresponded precisely to what the gold had weighed.

But afterwards a charge was made that gold had been abstracted and an equivalent weight of silver had been added in the manufacture of the crown. Heiro, thinking it an outrage that he had been tricked, and yet not knowing how to detect the theft, requested Archimedes to consider the matter. The latter, while the case was still on his mind, happened to go to the bath, and on getting into a tub observed that the more his body sank into it the more water ran out over the tub. As this pointed out the way to explain the case in question, without a moments delay and transported with joy, he jumped out of the tub and rushed home naked, crying in a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, "Eureka, Eureka."

Taking this as the beginning of his discovery, it is said that he made two masses of the same weight as the crown, one of gold and the other of silver. After making them, he filled a large vessel with water to the very brim and dropped the mass of silver into it. As much water ran out as was equal in bulk to that of the silver sunk in the vessel. Then, taking out the mass, he poured back the lost quantity of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quantity of water.

After this experiment, he likewise dropped the mass of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quantity: namely, as much less as a mass of gold lacks in bulk compared to a mass of silver of the same weight. Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over for the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold and made the theft of the contractor perfectly clear.

What is going on here is simply a measurement of the density-the mass per unit volume-of silver, gold and the crown. To measure the masses some kind of scale is used, note that at the beginning a precise amount of gold is weighed out to the contractor. Of course, if you had a nice rectangular brick of gold, and knew its weight, you wouldn't need to mess with water to determine its density, you could just figure out its volume by multiplying together length, breadth and height, and divide the mass, or weight, by the volume to find the density in, say, pounds per cubic foot or whatever units are convenient. (Actually, the units most often used are the metric ones, grams per cubic centimeter. These have the nice feature that water has a density of 1, because that's how the gram was defined. In these units, silver has a density of 10.5, and gold of 19.3. To go from these units to pounds per cubic foot, we would multiply by the weight in pounds of a cubic foot of water, which is 62.)

The problem with just trying to find the density by figuring out the volume of the crown is that it is a very complicated shape, and although one could no doubt find its volume by measuring each tiny piece and calculating a lot of small volumes which are then added together, it would take a long time and be hard to be sure of the accuracy, whereas lowering the crown into a filled bucket of water and measuring how much water overflows is obviously a pretty simple procedure. (You do have to allow for the volume of the string!). Anyway, the bottom line is that if the crown displaces more water than a block of gold of the same weight, the crown isn't pure gold.

Actually, there is one slightly surprising aspect of the story as recounted above by Vitruvius. Note that they had a weighing scale available, and a bucket suitable for immersing the crown. Given these, there was really no need to measure the amount of water slopping over. All that was necessary was first, to weigh the crown when it was fully immersed in the water, then, second, to dry it off and weigh it out of the water. The difference in these two weighings is just the buoyancy support force from the water. Archimedes' Principle states that the buoyancy support force is exactly equal to the weight of the water displaced by the crown, that is, it is equal to the weight of a volume of water equal to the volume of the crown.

This is definitely a less messy procedure-there is no need to fill the bucket to the brim in the first place, all that is necessary is to be sure that the crown is fully immersed, and not resting on the bottom or caught on the side of the bucket, during the weighing.

Of course, maybe Archimedes had not figured out his Principle when the king began to worry about the crown, perhaps the above experiment led him to it. There seems to be some confusion on this point of history.