The former four are all conſtructed by making B to fall between A and

I, C between U and E, and drawing DH parallel to BC; and it will ap-

pear by reaſonings ſimilar to thoſe uſed for the like purpoſe in Caſe III. of

Epitagma I. that O muſt fall between E and I as was propoſed. See Fig. 54, 55, 56 and 57.

Limitation
. In theſe four Caſes, the given ratio of R to S muſt not

be leſs than that which the ſquare on the ſum of two mean proportionals

between AE and IU, AI and EU bears to the ſquare on EI. For it has

been proved (Lem. II.) that when the ratio of the rectangle contained by

AO and UO to that contained by EO and IO; or, which is the ſame thing,

the given ratio of R to S is the leaſt poſſible, the ſquare on EO will be to

the ſquare on IO as the rectangle contained by AE and UE is to that con-

tained by AI and UI; and (
Lem
. V. Fig. 31.) that FG will then be the

fum of two mean proportionals between AE and UI, AI and UE: it

therefore only remains to prove that the rectangle contained by AO and

UO is to that contained by EO and IO as the ſquare on FG is to the

fquare on EI. Now it has been proved in demonſtrating Lem. V. that the

triangles EOG and IOF are ſimilar, and that the angle at V is right,

whence it follows that the triangles AOG and FOU are alſo ſimilar, and

conſequently that AO is to OG as OF is to UO; therefore the rectangle

contained by AO and UO is equal to that contained by GO and OF. More-

over GO is to OF as EO is to IO, and ſo by compoſition and permutation,

FG is to EI as OG is to EO, and as OF is to IO: hence by compound

ratio the ſquare on FG is to the ſquare on EI as the rectangle contained

by (OG and OF) AO and UO is to that contained by EO and IO.

Q. E. D.

Scholium
. In the four Caſes, wherein the given ratio is of a leſs to a

greater, and wherein the point O muſt fall between thoſe given ones which

bound the antecedent rectangle, the limiting ratio will be a maximum, and

the ſame with that which the ſquare on AU bears to the ſquare on FG.