goras,
Democritus, Plato, and Aristotle, and the daily life of other learned
men, spent in constant industry, yield fresh and rich fruit, not only to
their own countrymen, but also to all nations. And they who from their
tender years are filled with the plenteous learning which this fruit
affords, attain to the highest capacity of knowledge, and can introduce
into their states civilized ways, impartial justice, and laws, things
without which no state can be sound.

3. Since, therefore, these great benefits to individuals and to
communities are due to the wisdom of authors, I think that not only
should palms and crowns be bestowed upon them, but that they should even
be granted triumphs, and judged worthy of being consecrated in the
dwellings of the gods.

Of their many discoveries which have been useful for the development of
human life, I will cite a few examples. On reviewing these, people will
admit that honours ought of necessity to be bestowed upon them.

4. First of all, among the many very useful theorems of Plato, I will
cite one as demonstrated by him. Suppose there is a place or a field in
the form of a square and we are required to double it. This has to be
effected by means of lines correctly drawn, for it will take a kind of
calculation not to be made by means of mere multiplication. The
following is the demonstration. A square place ten feet long and ten
feet wide gives an area of one hundred feet. Now if it is required to
double the square, and to make one of two hundred feet, we must ask how
long will be the side of that square so as to get from this the two
hundred feet corresponding to the doubling of the area. Nobody can find
this by means of arithmetic. For if we take fourteen, multiplication
will give one hundred and ninety-six feet; if fifteen, two hundred and
twenty-five feet.

5. Therefore, since this is inexplicable by arithmetic, let a diagonal
line be drawn from angle to angle of that square of ten feet in length
and width, dividing it into two triangles of equal size, each fifty feet
in area. Taking this diagonal line as the length, describe another
square. Thus we shall have in the larger square four triangles of the
same size and the same number of feet as the two of fifty feet each
which were formed by the diagonal line in the smaller square. In this
way Plato demonstrated the doubling by means of lines, as the figure
appended at the bottom of the page will show.

6. Then again, Pyth