204
to those contained in the square, whose side is five feet.
When Pythagoras discovered this property, convinced
that the Muses had assisted him in the discovery, he
evinced his gratitude to them by sacrifice. This propo¬
sition is serviceable on many occasions, particularly in
measuring, no less than in setting out the staircases of
buildings, so that each step may have its proper height.
For if the height from the pavement to the floor above
be divided into three parts, five of those parts will be the
exact length of the inclined line which regulates the
blocks of which the steps are formed. Four parts, each
equal to one of the three into which the height from the
pavement to the floor was divided, are set off from the
perpendicular, for the position of the first or lower step.
Thus the arrangement and ease of the flight of stairs
will be obtained, as the figure will show.
CHAPTER III.
OF THE METHOD OF DETECTING SILVER WHEN
MIXED WITH GOLD.
THOUGH Archimedes discovered many curious matters
which evince great intelligence, that which I am about
to mention is the most extraordinary. Hiero, when he
obtained the regal power in Syracuse, having, on the
fortunate turn of his affairs, decreed a votive crown of gold
to be placed in a certain temple to the immortal gods,
commanded it to be made of great value, and assigned
an appropriate weight of gold to the manufacturer. He,
in due time, presented the work to the king, beautifully
wrought, and the weight appeared to correspond with
that of the gold which had been assigned for it. But a
report having been circulated, that some of the gold had