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