On the Principles of Arithmetic and On the Principles of Music

...is the grasping of truth. We say that those things "exist" which
neither increase by expansion nor diminish by contraction nor are
changed by variations, but always preserve themselves by the power of
their own nature, supported by their own foundations. These are
5 qualities, quantities, forms, sizes, smallnesses,
equalities, relations, acts, dispositions, places, times, and
whatever is found in some way united to bodies; for while these
things are themselves incorporeal by nature and thrive by the reason of
immutable substance, they are changed by participation in the
10 body, and by the touch of a variable thing, they
pass into a changeable inconstancy. Therefore, since these things
have—as was said—received a nature of immutable substance and power,
they are truly and properly said to "exist." Wisdom, then, professes the
knowledge of these things—that is, of those things which properly
15 exist and which are called "essences" by their own
name. There are two parts to essence: one is continuous and joined by
its own parts, not distributed by any boundaries—such as a tree, a
stone, and all bodies of this world, which are properly called
"magnitudes." The other is truly disjoined from itself and determined
20 by parts, brought together as if in a gathered
assembly—such as a flock, a people, a chorus, a heap, and whatever else
whose parts are limited by their own boundaries and are distinct from
the limit of another. To these the proper name is "multitude."
Furthermore, within multitude, some things exist through themselves,
25 such as three or four or a square number
original: tetragonus. A number resulting from the product of a number multiplied by itself, e.g., 2x2=4., or any number which requires nothing else in order to exist.
But others do not consist through themselves, but are referred to
something else, such as the double, the half, the sesquialter
original: sesqualterum. A ratio of 1.5 to 1, or 3:2., or
the sesquitertian original: sesquitertium. A ratio of 1.33 to 1, or 4:3.,
and whatever is of such a kind that it could not exist unless it were
related to something else. Within magnitude, moreover, some things are
30 remaining and lacking in motion, while others are
always turned by a mobile rotation and do not rest at any time...

3 by the power of their own original: se vi suae. Variant in manuscript a.. 8 of immutable original: inmutabilis. Variants in d, l; s corrected.. 10 untraceable original: investibilem. Variant in a.. 12 they obtain (above the line: or "have received") original: sortiuntur (ł sortita sunt). Variant in s.. 15 above "knowledge": also original: quoque. Variant in a.. 17 tree original: arbos. Variant in b.. 18 of this world original: huius mundi. Variant in f.. 20 in a heaped manner original: coacervatim. Variants in f, s; "acervatim" with "co" syllable added in a.. 21 above "whatever": it is of those original: est eorum. Variant in a; "est horum" in b, d.. 23 distinct original: discreta. Variant in c.. 27 after "half": or original: vel. Variant in f.. 31 by reason original: ratione. Variants in a, c, f, s. Above "reason": others say "rotation" (rotatione) b, r. || let them be turned original: vertantur. Variants in d, r.. let them rest original: adquiescant. Variants in d, r..