168 On the ORIGIN and 
13. We are therefore to enquire, in what circumftances the 
point L may coincide with the point A, and the point M with 
with the point C, and of confequence the circumference L F M 
with the circumference AB C. 
On the fuppofition that they coincide, EA is to AD, and 
alfo EC toC D, as «to 8; and therefore EA is to EC as AD 
to CD, or, by converfion, EA to AC as AD to the excefs of 
CD above AD, or to twice DO, O being the centre of the cir- 
cle ABC. Therefore alfo, E A is to AO, or the half of AC, 
as AD to DO, and EA together with AO, to AO, as AD 
together with DO, toDO; that is, EO to AO as AO to DO, 
and fo the relangle EO. O D equal to the fquare of AO. 
Hence, if the fituation of the given points E and D, (fig. 4.) 
in. refpect of the circle A BC, be fuch, that the rectangle _ 
EO. OD is equal to the fquare of AO, the femidiameter of 
the circle; and if, at the fame time, the given ratio of «to 6 
be the fame with that of EA to AD, or of EC to CD, the 
problem admits of innumerable folutions ; and as it is mani- 
feft, that if the circle ABC, and one of the points D or E be 
given, the other point, and alfo the ratio which is required to 
render the problem indefinite, may be found, therefore we have 
this Porifm: “ A circle ABC being given, and alfo a point D, 
a point E may be found, fuch, that the two lines inflected 
from thefe points to any point whatever ‘in the circumference 
ABC, fhall have to one another a given ratio, which ratio is 
alfo to be found.” 
_ Tuts Porifm is the fecond in the treatife De Pori/imatibus, 
where Dr SIMSON gives it, not as one of Euciip’s propofitions, 
but as an illuftration of his own definition. It anfwers equally well 
for the purpofe 1 have here in view, the explaining the origin 
of Porifms ; and I have been the more willing to introduce it, 
that it has afforded me an opportunity of giving what feems 
to be the fimpleft inveftigation of the fecond propofition in the 
fecond book of the Laci Plani, by proving, as has been done 
above, 
