agoras showed that a right angle can be formed
without the contrivances of the artisan. Thus, the result which
carpenters reach very laboriously, but scarcely to exactness, with their
squares, can be demonstrated to perfection from the reasoning and
methods of his teaching. If we take three rules, one three feet, the
second four feet, and the third five feet in length, and join these
rules together with their tips touching each other so as to make a
triangular figure, they will form a right angle. Now if a square be
described on the length of each one of these rules, the square on the
side of three feet in length will have an area of nine feet; of four
feet, sixteen; of five, twenty-five.

7. Thus the area in number of feet made up of the two squares on the
sides three and four feet in length is equalled by that of the one
square described on the side of five. When Pythagoras discovered this
fact, he had no doubt that the Muses had guided him in the discovery,
and it is said that he very gratefully offered sacrifice to them.

This theorem affords a useful means of measuring many things, and it is
particularly serviceable in the building of staircases in buildings, so
that the steps may be at the proper levels.

8. Suppose the height of the story, from the flooring above to the
ground below, to be divided into three parts. Five of these will give
the right length for the stringers of the stairway. Let four parts, each
equal to one of the three composing the height between the upper story
and the ground, be set off from the perpendicular, and there fix the
lower ends of the stringers. In this manner the steps and the stairway
itself will be properly placed. A figure of this also will be found
appended below.

9. In the case of Archimedes, although he made many wonderful
discoveries of diverse kinds, yet of them all, the following, which I
shall relate, seems to have been the result of a boundless ingenuity.
Hiero, 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.

10. But afte