In the first part of this book, we describe the usefulness of this science as follows. It is clear to the
wise that science begins in the sensitive and imaginative powers where the human intellect
receives species which are likenesses of sense objects, and then makes these species intelligible
in its essence, and by reason of this intelligibility, the intellect attains the natural secrets of
corporeal substances, and as the intellect cannot do this without help from the imagination, this
science also nurtures the imagination in imagining imaginary mensurations based on sense data.
And so this science is good for strengthening the imagination's imagining and the intellect's
understanding, and likewise with memory, for when the imagination has a more virtuous disposition
in imagining, the memory also has a more virtuous disposition in retaining the species transmitted
to it by the intellect and acquired through the imagination.
Through the master figure, this science provides a doctrine for penetrating the quadrangulature and
triangulature of the circle, and a doctrine for the other figures as well. Thus, the imagination
develops its virtue inasmuch as it attains new species in the equality of circular and straight lines
and their surfaces, following the doctrine given in the first book. And by multiplying species attained
by the imagination through diverse investigative methods, the intellect and memory acquire much
material that enables them to reach a greater understanding of corporeal substances and
For instance, we extend the line of the circle to consider it as a straight line, and likewise, in the
minor square, we extend a 5th line measured in the master figure by the intellect with the
imagination, taking this 5th line along with the 4 other lines to produce the middle square of the
master figure, as we proved in the second figure where the circle is squared, and likewise with the
6th mathematical line of the pentagon in third figure, and so with the other figures used in this
science to imagine mensurations that are not, and cannot be sensed. Through this, the imagination
has a loftier and more virtuous act than with lines and figures that can be sensed.
For instance, the 5th line of the square potentially exists in the circle of the master figure, and is
more useful to imagine than the 4 lines of the minor square in the master figure. This kind of
imagining raises the imagination aloft and helps the intellect and memory to better attain
imaginable objects which cannot be imagined in reality, but which can be understood and loved
spiritually, like God, angels as well as prime general and abstract principles like general goodness
and other such things, following the comments given in the first book on each particular figure and
things deduced from them.
And this science is useful, as we said, and as it is understandable per se, it is very dear to those
who desire to know it and who desire to have an intellect that understands well and a memory that