rectangle contained by AB and AC; but the ſquare on HO is equal to

the rectangle contained by AB and EC: now EC is, by ſuppoſition,

greater than AC, therefore the rectangle AB, EC is greater than the

rectangle AB, AC, and the ſquare on HO greater than the ſquare on AG,

conſequently HO is itſelf greater than AG; but this could not be the

Caſe unleſs O fell beyond A. In the ſame manner my it be proved that O

will fall beyond U in Fig. 59 and 60.

Limitation
. In the above four Caſes the given ratio of R to S muſt

not exceed that which the ſquare on AU bears to the ſquare on the ſum of

two mean proportionals between AI and UE, AE and UI. For (Fig. 30.) demit from A, on KO produced, the perpendicular AH. Now it has been

proved (Lem. III.) that the ratio of the rectangle continued 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 greateſt poſſible; and (Lem. IV.) that KF is a mean

proportional between AI and UE, alſo that YF is a mean proportional

between AE and UI: but HK is equal to YF, therefore HF is equal to

the ſum of two mean proportionals between AI and UE, AE and UI; it only then remains to prove, that the rectangle contained by AO and UO

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

on HF. The triangles OEK, OHA, OIY and OUF are all ſimilar; con-

ſequently OK is to OE as OA is to OH, as OY is to OI, and therefore

by compound ratio, the rectangle contained by AO and UO (OK and

OY) is to that contained by EO and IO as the ſquare on AO is to the

ſquare on OH; but alſo AO is to UO as HO is to EO, and by compoſi-

tion and permutation, AU is to HF as AO is to HO, or (Eu. VI. 22.) the ſquare on AU is to the ſquare on HF as the ſquare on AO is to the

ſquare on HO, and ſo by equality of ratios, the rectangle contained by AO

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

ſquare on HF.

Q. E. D.

Scholium
. In the four Caſes wherein the points A and U are means,

the limiting ratio will be a minimum, and the ſame with that which the

ſquare on HF bears to the ſquare on EI.