In this part we propose to investigate the extension of circular lines, and we make this imaginary
extension because a circular line, be it natural or artificially drawn with a compass, cannot be
extended across a surface.
And now let us see how this investigation is done: in the first part, where we proved the quadrature
and triangulature of the circle, we said that the capacity of the square is halfway between the
capacity of the circle and that of the triangle; and thus we proved that line x.y., which is worth 4
units of the circle in the master figure, is worth half a unit more than the circular line of the circle,
and that line a.b. is worth one half unit more than x.y.
Thus, we see that if the line of the circle in the plenary figure could be extended, it would only be
worth three and a half units of line x.y. in which the square is extended, and line a.b. would be
worth one unit of the square more than the circle, so that in the mind's eye, the extended circular
line is half a unit shorter than the line of the square, and one unit shorter than the line of the
triangle, while all the figures are equal in capacity.
Following what we said about the extended line of the whole circle which is worth 8 half-units of the
square, we can discuss the parts of the lunules in the master figure, like circular line e. which has
the same value as straight line ik, and as much more as the fourth part of 7 and a half units, or as
a quarter unit of the straight line i.k.
We showed how circular lines are extended and how their extended value compares with that of
straight lines, and following the example we gave with the master figure, examples can be given
with figures of five, six or more angles, following the natural measurements of the capacity of the
circle, square and triangle, and we just discussed this capacity here, in Part 2 of this book.
At this point we can consider how science originates in imaginary quantities drawn from quantities
that are sensed, as when we make an imaginary measurement of a circular line by straightening it,
and mentally measuring it with the straight lines of the square and the triangle, and these
considerations expand the imagination's virtue as it feeds and verifies itself with imaginary species;
in this way, one attains natural secrets inherent to the composition of figures, and we will give
examples of this in the third part.
