1Apollonius, at which you made no ſeruple.
SAGR. It may be either that I knew them by chance, or that I
might for once gueſſe at, and take for granted ſo much as ſerved my
turn in that Tractate: but here where I imagine that we are to
hear all the Demonſtrations that concern thoſe Lines, it is not con
venient, as we ſay, to ſwallow things whole, loſing our time and
pains.
might for once gueſſe at, and take for granted ſo much as ſerved my
turn in that Tractate: but here where I imagine that we are to
hear all the Demonſtrations that concern thoſe Lines, it is not con
venient, as we ſay, to ſwallow things whole, loſing our time and
pains.
SIMP. But as to what concerns me, although Sagredus were,
as I believe he is, well provided for his occaſions, the very firſt
Terms already are new to me: for though our Philoſophers have
handled this Argument of the Motion of Projects, I do not remem
ber that they have confined themſelves to deſine what the Lines
are which they deſcribe, ſave only in general that they are alwaies
Curved Lines, except it be in Projections Perpendicularly upwards.
Therefore in caſe that little Geometry that I have learnt from Eu
clid ſince the Time that we have had other Conferences, be not ſuf
ficient to render me capable of the Notions requiſite for the under
ſtanding of the following Demonſtrations, I muſt content my ſelf
with bare Propoſitions believed, but not underſtood.
as I believe he is, well provided for his occaſions, the very firſt
Terms already are new to me: for though our Philoſophers have
handled this Argument of the Motion of Projects, I do not remem
ber that they have confined themſelves to deſine what the Lines
are which they deſcribe, ſave only in general that they are alwaies
Curved Lines, except it be in Projections Perpendicularly upwards.
Therefore in caſe that little Geometry that I have learnt from Eu
clid ſince the Time that we have had other Conferences, be not ſuf
ficient to render me capable of the Notions requiſite for the under
ſtanding of the following Demonſtrations, I muſt content my ſelf
with bare Propoſitions believed, but not underſtood.
SALV. But I will have you to know them by help of the Au
thor of this Book himſelf, who when he heretofore granted me a
ſight of this his Work, becauſe I alſo at that time was not perfect
in the Books of Apollonius, took the pains to demonſtrate to me
two moſt principal Paſſions of the Parabola without any other Pre
cognition, of which two, and no more, we ſhall ſtand in need in
the preſent Treatiſe; which are both likewiſe proved by Apollonius,
but after many others, which it would take up a long time to look
over, and I am deſirous that we may much ſhorten the Journey, ta
king the firſt immediately from the pure and ſimple generation of
the ſaid Parabola, and from this alſo immediately ſhall be deduced
the Demonſtration of the ſecond. Coming therefore to the firſt;
thor of this Book himſelf, who when he heretofore granted me a
ſight of this his Work, becauſe I alſo at that time was not perfect
in the Books of Apollonius, took the pains to demonſtrate to me
two moſt principal Paſſions of the Parabola without any other Pre
cognition, of which two, and no more, we ſhall ſtand in need in
the preſent Treatiſe; which are both likewiſe proved by Apollonius,
but after many others, which it would take up a long time to look
over, and I am deſirous that we may much ſhorten the Journey, ta
king the firſt immediately from the pure and ſimple generation of
the ſaid Parabola, and from this alſo immediately ſhall be deduced
the Demonſtration of the ſecond. Coming therefore to the firſt;
Deſcribe the Right Cone, whoſe Baſe let be the Circle I B K C,
and Vertex the point L, in which, cut by a Plane parallel to the
144[Figure 144]
Side L K, ariſeth the Section B A C
called a Parabola; and let its Baſe
B C cut the Diameter I K of the
Circle I B K C at Right-Angles;
and let the Axis of the Parabola
A D be Parallel to the ſide L K;
and taking any point F in the Line
B F A, draw the Right-Line F E
parallel to B D. I ſay, that the Square
of B D hath to the Square of F E
the ſame proportion that the Axis
D A hath to the part A E. Let a Plane parallel to the Circle I B K C
and Vertex the point L, in which, cut by a Plane parallel to the
144[Figure 144]Side L K, ariſeth the Section B A C
called a Parabola; and let its Baſe
B C cut the Diameter I K of the
Circle I B K C at Right-Angles;
and let the Axis of the Parabola
A D be Parallel to the ſide L K;
and taking any point F in the Line
B F A, draw the Right-Line F E
parallel to B D. I ſay, that the Square
of B D hath to the Square of F E
the ſame proportion that the Axis
D A hath to the part A E. Let a Plane parallel to the Circle I B K C