VITRUVIUS.
202
quantity of water it had caused to overflow was not so great; but was as much less as the
magnitude of the mass of gold was less than that of the same weight of silver: lastly, filling
again the same vase with water, he put therein the crown itself, and found that more water
was displaced by the crown, than by the mass of gold of the same weight: so that from
the water displaced by the crown, more than that by the mass, he discovered by calcu-
lation the quantity of silver mixed with the gold; and thus detected the fraud of the work-
man.
Let us now transfer our attention to the inventions of Architas the Tarentine, and of
Eratosthenes of Cyrene, who have by their mathematical knowledge made many discoveries
useful to mankind: and although for other inventions they may be applauded, for the solution
of the following problem they are chiefly celebrated. Each undertook to solve, by different
methods, the response uttered by Apollo in Delos, to make an altar like his, but containing double
the number of cubic feet; and that, thereafter, those who might be in that island should be
freed by the religion. This Architas solved by the description of the hemicylinder, and Era¬
tosthenes by the mechanism of the mesolabium.
(2*) The altar of Apollo at Delos was a cube, and
the proposition was to find the measure of another cube,
whose quantity should be exactly double that of the
former.
If a cube be formed, whose side is double that of the
given cube, it will contain eight times the cubical quan¬
tity, being the fourth number of a geometrical series in-
creasing in a duplicate ratio, as 1. 2. 4. 8, and of which
the cube required is the second number of the series. It
is said, that Hippocrates, reflecting upon this principle,
reduced the proposition to the finding two geometric mean
proportionals between two given lines, one equal to the
side of the given cube, and the other, double thereof; the
leaft of which mean proportionals will be the measure of
the side of a cube that fhall be exacty double in quantity
to the given cube.
To find these two mean proportions, several methods
were discovered by the antients. Architas, Eratosthenes,
Plato, Nicomedes, Apollonius, and others, have invented
different methods; but they are all tentative: these may
be met with in books of geometry, to which I refer the
feader. I hall, however, here describe the principle on
which the mechanical contrivance of Eratosthenes was
founded.
Fig. LXVIII.
Let AB and CD be the two given lines, drawn parallel
to each other; join AC and BD, draw the line AF to any
point F, and from the point F raise FE parallel to AB;
then from E draw the line EH parallel to the former line
AF, and from H raise HG parallel also to AB: lastly, from
G draw GD parallel to the former lines AF and EH, and
if the line GD intersects the line CD in D, the operation
is right; if not, the direction of the lines AF, EH, and
GD, must be altered till it so happens ; when EF, and
GH, will be the two mean proportionals sought. This
method is founded upon the 4th proposition of the 6th
book of Euclid, which demonstrates that the correspond-
ing sides of equiangular triangles are proportionals.
By the same process any number of mean proportionals
may be found between the two given lines AB and CD.
As every addition to the several inventions left us by the
antients for finding two mean proportionals, may not be
without its use, I shall here annex one that I have dis¬
covered.
BD, Fig. LXVIII. is the longest given line, and DE is
the shortest, perpendicular to the former: continue BD
toward C indefinitely; also draw the perpendicular BA