PROPOSITION 93.
If an area be contained by a rational straight line and a third apotome, the “side”
of the area is a second apotome of a medial straight line.
For let the area
AB be contained by the rational straight line
AC and the third apotome
AD; I say that the “side”
of the area
AB is a second apotome of a medial straight line.
For let
DG be the annex to
AD; therefore
AG,
GD are rational straight lines commensurable in square only, and neither of the straight lines
AG,
GD is commensurable in length with the rational straight line
AC set out, while the square on the whole
AG is greater than the square on the annex
DG by the square on a straight line commensurable with
AG. [
X. Deff. III. 3]
Since then the square on
AG is greater than the square on
GD by the square on a straight line commensurable with
AG, therefore, if there be applied to
AG a parallelogram equal to the fourth part of the square on
DG and deficient by a square figure, it will divide it into commensurable parts. [
X. 17]
Let then
DG be bisected at
E, let there be applied to
AG a parallelogram equal to the square on
EG and deficient by a square figure, and let it be the rectangle
AF,
FG.
Let
EH,
FI,
GK be drawn through the points
E,
F,
G parallel to
AC.
Therefore
AF,
FG are commensurable; therefore
AI is also commensurable with
FK. [
VI. 1,
X. 11]
And, since
AF,
FG are commensurable in length, therefore
AG is also commensurable in length with each of the straight lines
AF,
FG. [
X. 15]
But
AG is rational and incommensurable in length with
AC; so that
AF,
FG are so also. [
X. 13]
Therefore each of the rectangles
AI,
FK is medial. [
X. 21]
Again, since
DE is commensurable in length with
EG, therefore
DG is also commensurable in length with each of the straight lines
DE,
EG. [
X. 15]
But
GD is rational and incommensurable in length with
AC; therefore each of the straight lines
DE,
EG is also rational and incommensurable in length with
AC; [
X. 13] therefore each of the rectangles
DH,
EK is medial. [
X. 21]
And, since
AG,
GD are commensurable in square only, therefore
AG is incommensurable in length with
GD.
But
AG is commensurable in length with
AF, and
DG with
EG; therefore
AF is incommensurable in length with
EG. [
X. 13]
But, as
AF is to
EG, so is
AI to
EK; [
VI. 1] therefore
AI is incommensurable with
EK. [
X. 11]
Now let the square
LM be constructed equal to
AI, and let there be subtracted
NO equal to
FK and being about the same angle with
LM; therefore
LM,
NO are about the same diameter. [
VI. 26]
Let
PR be their diameter, and let the figure be drawn.
Now, since the rectangle
AF,
FG is equal to the square on
EG, therefore, as
AF is to
EG, so is
EG to
FG. [
VI. 17]
But, as
AF is to
EG, so is
AI to
EK, and, as
EG is to
FG, so is
EK to
FK; [
VI. 1] therefore also, as
AI is to
EK, so is
EK to
FK; [
V. 11] therefore
EK is a mean proportional between
AI,
FK.
But
MN is also a mean proportional between the squares
LM,
NO, and
AI is equal to
LM, and
FK to
NO; therefore
EK is also equal to
MN.
But
MN is equal to
LO, and
EK equal to
DH; therefore the whole
DK is also equal to the gnomon
UVW and
NO.
But
AK is also equal to
LM,
NO; therefore the remainder
AB is equal to
ST, that is, to the square on
LN; therefore
LN is the “side”
of the area
AB.
I say that
LN is a second apotome of a medial straight line.
For, since
AI,
FK were proved medial, and are equal to the squares on
LP,
PN, therefore each of the squares on
LP,
PN is also medial; therefore each of the straight lines
LP,
PN is medial.
And, since
AI is commensurable with
FK, [
VI. 1,
X. 11] therefore the square on
LP is also commensurable with the square on
PN.
Again, since
AI was proved incommensurable with
EK, therefore
LM is also incommensurable with
MN, that is, the square on
LP with the rectangle
LP,
PN; so that
LP is also incommensurable in length with
PN; [
VI. 1,
X. 11] therefore
LP,
PN are medial straight lines commensurable in square only.
I say next that they also contain a medial rectangle.
For, since
EK was proved medial, and is equal to the rectangle
LP,
PN, therefore the rectangle
LP,
PN is also medial, so that
LP,
PN are medial straight lines commensurable in square only which contain a medial rectangle.
Therefore
LN is a second apotome of a medial straight line; [
X. 75] and it is the “side”
of the area
AB.
Therefore the “side”
of the area
AB is a second apotome of a medial straight line. Q. E. D.
