74[ix]
miſtake muſt be admitted in this Author, or the fifth Problem is
wrong placed. I am not prepared farther to obviate theſe objec-
tions, and only mention them to ſhew that although I ſaw them
in their full force, I could by no means agree, that they are pow-
erful enough to overturn thoſe already advanced for what I have
11[Handwritten note 1]22[Handwritten note 2]done.
wrong placed. I am not prepared farther to obviate theſe objec-
tions, and only mention them to ſhew that although I ſaw them
in their full force, I could by no means agree, that they are pow-
erful enough to overturn thoſe already advanced for what I have
11[Handwritten note 1]22[Handwritten note 2]done.
I come now to Book II, which if I am not much miſtaken, was
entirely employed about what Snellius makes his fourth Problem.
In this I am confirmed not only by the account which Pappus gives
in his Preface, but much more by the Lemmas of Apollonius
11[Handwritten note 1]22[Handwritten note 2] which he hath left us. For we there find that Lemma 21, where-
in is aſſigned the leaſt ratio which the rectangle contained by AO
and UO can bear to that contained by EO and IO, when O is ſought
between the two mean points of the four given ones, is ſaid to be
concerned in determining the μοναχὴ, or ſingle Caſe , of Problem 1. This Problem therefore of Apollonius contained only thoſe
Caſes of the general one, where O is ſought between the two mean
points. In like manner, we gather from Lemma 22, that his ſe-
cond Problem was concerned in determining the point O when ſought
between a mean point, and an extreme one. And laſtly, from
Lemma 24, that the third Problem of Book II. determined
the point O when required without all the given ones.
1
entirely employed about what Snellius makes his fourth Problem.
In this I am confirmed not only by the account which Pappus gives
in his Preface, but much more by the Lemmas of Apollonius
11[Handwritten note 1]22[Handwritten note 2] which he hath left us. For we there find that Lemma 21, where-
in is aſſigned the leaſt ratio which the rectangle contained by AO
and UO can bear to that contained by EO and IO, when O is ſought
between the two mean points of the four given ones, is ſaid to be
concerned in determining the μοναχὴ, or ſingle Caſe , of Problem 1. This Problem therefore of Apollonius contained only thoſe
Caſes of the general one, where O is ſought between the two mean
points. In like manner, we gather from Lemma 22, that his ſe-
cond Problem was concerned in determining the point O when ſought
between a mean point, and an extreme one. And laſtly, from
Lemma 24, that the third Problem of Book II. determined
the point O when required without all the given ones.
1
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