73[viii]
tion is at the ſecond Epitagma;
and farther, the Limiting Ratio is
therein a maximum, as it ought. Again, the Problem, wherein
it is propoſed to make the ſquare on AO in a given ratio to the rect-
angle contained by EO and P, has its limiting ratio a minimum
when the required point is ſought beyond (E) that of the given ones
which bounds the ſegment concerned in the conſequent term of
the ratio; which, therefore, I apprehend muſt have been the third
Epitagma, and if ſo, this of courſe muſt have been the third Pro-
blem: and as there remains only one wherein the number of given
pointsare two, I make that the fourth. With reſpect to the fifth and
ſixth Problems, in which three points are given, it ſhould ſeem
that that would be the firſt in order, wherein there is a given ex-
ternal line concerned.
therein a maximum, as it ought. Again, the Problem, wherein
it is propoſed to make the ſquare on AO in a given ratio to the rect-
angle contained by EO and P, has its limiting ratio a minimum
when the required point is ſought beyond (E) that of the given ones
which bounds the ſegment concerned in the conſequent term of
the ratio; which, therefore, I apprehend muſt have been the third
Epitagma, and if ſo, this of courſe muſt have been the third Pro-
blem: and as there remains only one wherein the number of given
pointsare two, I make that the fourth. With reſpect to the fifth and
ſixth Problems, in which three points are given, it ſhould ſeem
that that would be the firſt in order, wherein there is a given ex-
ternal line concerned.
But it ſhould, by no means, be diſſembled that objections may be
brought againſt the identity, and arrangment of ſome of theſe Pro-
blems. For firſt, Pappus no where expreſsly ſays that Apollo-
NIUS compared together two ſquares, wherefore, if this cannot be
implied, the identity of the fourth Problem is deeply ſtruck at:
and moreover, this fourth Problem perhaps cannot with propriety,
be ſaid to have its limiting ratio either maximum or minimum, un-
leſs the ratio of equality, can be admitted as ſuch. Laſtly, in the
fifth Problem, the ſaid limiting ratio is a minimum, and not a maxi-
mum as it is ſaid to have been by Pappus: either, therefore,
brought againſt the identity, and arrangment of ſome of theſe Pro-
blems. For firſt, Pappus no where expreſsly ſays that Apollo-
NIUS compared together two ſquares, wherefore, if this cannot be
implied, the identity of the fourth Problem is deeply ſtruck at:
and moreover, this fourth Problem perhaps cannot with propriety,
be ſaid to have its limiting ratio either maximum or minimum, un-
leſs the ratio of equality, can be admitted as ſuch. Laſtly, in the
fifth Problem, the ſaid limiting ratio is a minimum, and not a maxi-
mum as it is ſaid to have been by Pappus: either, therefore,