and ſecondly when it is ſought beyond both the other given points: in the

firſt, the ratio of R to S muſt be of a greater to a leſs, and in the latter of

a leſs to a greater. The former is conſtructed in Fig. 17. at the ſame time

with Caſe I of Epitagma I. and is repreſented by the ſmall letters h and o: the latter in Fig. 18, and pointed out by the ſame letters. That O will

fall in both as is required needs not inſiſting on.

Epitagma
III. In which there are ſix Caſes, viz. I being the middle

point, when O is ſought beyond A, or beyond E; and that whether the

given ratio be of a leſs to a greater, or of a greater to a leſs; and again, I

being an extreme point, when O is ſought between A and E, and that let

the order of the points A and E be what it will.

Cases
I and II. Are when I is a mean point and the given ratio of a

leſs to a greater; and theſe are both conſtructed at once by Fig. 20,

wherein B is made to fall beyond A, and C beyond E with reſpect to the

middle point I, and DH is drawn through the center of the circle on BC.

Cases
III and IV. Here, the points remaining as before, the given

ratio is of a greater to a leſs; and the conſtruction will be effected by

making B fall beyond E, and C beyond A, and drawing DH parallel to

BC, as in Fig. 21 and 22.

Case
V. Wherein I is one extreme point and A the other, and O is

ſought between A and E: in conſtructing this Caſe, B muſt be made to

fall between A and I, C between E and I, and DH drawn parallel to BC,

as is done in Fig. 23. The directions for Conſtructing Caſe VI. are exactly

the ſame, as will appear by barely inſpecting Fig. 24.

Limitation
. It is plain that in the four laſt Caſes, the ratio which the

rectangle contained by AO and EO bears to the ſquare on IO, or which is

the ſame thing, the given ratio of R to S cannot exceed a certain limit; and it is farther obvious that the ſaid limit will be when the ſtraight line

DH becomes a tangent to the circle on BC, as in Fig. 25. 26, for after

that the problem is manifeſtly impoſſible. Now when DH is a tangent to

the circle on BC, HO will be equal to half BC; but the ſquare on HO

is equal to the rectangle contained by IB and EC, wherefore the ſquare on

half BC will then be equal to the rectangle contained by IB and EC. Moreover, by the conſtruction, R is to S as AB is to IB, and as EC is to

IC; therefore by compoſition or diviſion, the ſum or difference of R and

S is to R as EI to EC, and the ſaid ſum or difference is to S as AI is to