VITRUVIUS. 
the discovery, he had been asfisted by the Muses, to render them the greater thanks, it is 
said he offered to them sacrifices. 
This principle is useful in measuring, and in many other cases; it is so likewise in build¬ 
ings, for the erection of the staircases, that the ascent of the seps may have a moderate incli¬ 
nation; for if the height (AB) of the story be divided into three parts, five of 
Fig. LXVI. 
these will be the just length of the inclined fhafts (40) of the stairs; for the 
height of the story, whatever it may be, being divided into three parts, four (B0) is set off 
from the perpendicular (B.A), and there (at C) the interior footings of the shafts are fixed; 
by this means the disposition of the stairs, and the ascent of the steps, will be moderate. 
(1*) This problem of Pythagoras, as well as that of 
Plato foregoing, is founded on the 47th proposition of 
the first book of Euclid, by which it is demonstrated, 
that, the square of the hypothenuse of any right angled 
triangle is equal to the sum of the squares of the two 
sides. The discovery of this proposition is generally 
ascribed to Pythagoras, who, it is said, usually sacrificed 
an ox every time he made any discovery in geometry, 
but that on this occasion he sacrificed an hecatomb. 
(2*) Galiani observes that the word interiores should 
be read inferiores, or anteriores, which appears to me 
highly probable. I should prefer the former, but have 
not chosen to differ from the text. 
The interior would probably mean the under side of 
the scapi; and in that case their upper ends could not 
abut against the floor, but would rise wholly above it, 
or at least that part of it from whence the measures ori- 
ginate. 
(3*) By this means the tread of the steps will be to 
the rise as 4 to 3, so that if the tread be a foot broad, 
the rise will be nine inches, which the moderns in 
general think too much. We now universally fix half 
a foot, or thereabout, as the standard for the rising of 
the steps in convenient staircases; although we often 
deviate therefrom in particular cases; and a foot is as 
generally considered as the proper breadth of the tread; 
so that the breadth or tread of our steps are to their height 
or rise, as 2 to 1, and are therefore much less steep than 
those of the antients.