1meter C F, perpendicular to one of the two Lines A B, or D E
and immoveable; we ſuppoſe all this Figure to turn round about
that Perpendicular: It is manifeſt, that there will be deſcribed by
the Parallelogram A D E B, a Cylinder; by the Semi-circle A F B,
an Hemi-Sphære; and by the Triangle C D E a Cone. This pre
ſuppoſed, I would have you imagine the Hemiſphære to be taken
away, leaving behind the Cone, and that which ſhall remain of
the Cylinder; which for the Figure, which it ſhall retain like to a
Diſh, we will hereafter call a Diſh: touching which, and the
Cone, we will ſirſt demonſtrate that they are equal; and next
a Plain being drawn parallel to the Circle, which is the foot or
Baſe of the Diſh, whoſe Diameter is the Line D E, and its Center
F; we will demonſtrate, that ſhould the ſaid Plain paſs, v. gr. by
the Line G H, cutting the Diſh in the points G I, and O N; and
the Cone in the points H and L; it would cut the part of the
Cone C H L, equal alwaies to the part of the Diſh, whoſe Profile
is repreſented to us by the Triangles G A I, and B O N: and more
over we will prove the Baſe alſo of the ſame Cone, (that is the
Circle, whoſe Diameter is H L) to be equal to that circular Su
perficies, which is Baſe of the part of the Diſh; which is, as we
may ſay, a Rimme as broad as G I; (note here by the way what
Mathematical Definitions are: they be an impoſition of names, or,
we may ſay, abreviations of ſpeech, ordain'd and introduced to
prevent the trouble and pains, which you and I meet with, at pre
ſent, in that we have not agreed together to call v. gr. this Super
ficies a circular Rimme, and that very ſharp Solid of the Diſh a
round Razor:) now howſoever you pleaſe to call them, it ſufficeth
you to know, that the Plain produced to any diſtance at pleaſure,
ſo that it be parallel to the Baſe, viz. to the Circle whoſe Diame
ter D E cuts alwaies the two Solids, namely, the part of the Cone
C H L, and the upper part of the Diſh equal to one another: and
likewiſe the two Superficies, Baſis of the ſaid Solids, viz. the ſaid
Rimme, and the Circle H L, equal alſo to one another. Whence
followeth the forementioned Wonder; namely, that if we ſhould
ſuppoſe the cutting-plain to be
ſucceſſively raiſed towards the
57[Figure 57]
Line A B, the parts of the Solid
cut are alwaies equall, as alſo the
Superficies, that are their Baſes,
are evermore equal; and, in
fine, raiſing the ſaid Plain higher
and higher, the two Solids (ever
equal) as alſo their Baſes, (Su
perficies ever equal) ſhall one couple of them terminate in a Cir
cumference of a Circle, and the other couple in one ſole point;
and immoveable; we ſuppoſe all this Figure to turn round about
that Perpendicular: It is manifeſt, that there will be deſcribed by
the Parallelogram A D E B, a Cylinder; by the Semi-circle A F B,
an Hemi-Sphære; and by the Triangle C D E a Cone. This pre
ſuppoſed, I would have you imagine the Hemiſphære to be taken
away, leaving behind the Cone, and that which ſhall remain of
the Cylinder; which for the Figure, which it ſhall retain like to a
Diſh, we will hereafter call a Diſh: touching which, and the
Cone, we will ſirſt demonſtrate that they are equal; and next
a Plain being drawn parallel to the Circle, which is the foot or
Baſe of the Diſh, whoſe Diameter is the Line D E, and its Center
F; we will demonſtrate, that ſhould the ſaid Plain paſs, v. gr. by
the Line G H, cutting the Diſh in the points G I, and O N; and
the Cone in the points H and L; it would cut the part of the
Cone C H L, equal alwaies to the part of the Diſh, whoſe Profile
is repreſented to us by the Triangles G A I, and B O N: and more
over we will prove the Baſe alſo of the ſame Cone, (that is the
Circle, whoſe Diameter is H L) to be equal to that circular Su
perficies, which is Baſe of the part of the Diſh; which is, as we
may ſay, a Rimme as broad as G I; (note here by the way what
Mathematical Definitions are: they be an impoſition of names, or,
we may ſay, abreviations of ſpeech, ordain'd and introduced to
prevent the trouble and pains, which you and I meet with, at pre
ſent, in that we have not agreed together to call v. gr. this Super
ficies a circular Rimme, and that very ſharp Solid of the Diſh a
round Razor:) now howſoever you pleaſe to call them, it ſufficeth
you to know, that the Plain produced to any diſtance at pleaſure,
ſo that it be parallel to the Baſe, viz. to the Circle whoſe Diame
ter D E cuts alwaies the two Solids, namely, the part of the Cone
C H L, and the upper part of the Diſh equal to one another: and
likewiſe the two Superficies, Baſis of the ſaid Solids, viz. the ſaid
Rimme, and the Circle H L, equal alſo to one another. Whence
followeth the forementioned Wonder; namely, that if we ſhould
ſuppoſe the cutting-plain to be
ſucceſſively raiſed towards the
57[Figure 57]Line A B, the parts of the Solid
cut are alwaies equall, as alſo the
Superficies, that are their Baſes,
are evermore equal; and, in
fine, raiſing the ſaid Plain higher
and higher, the two Solids (ever
equal) as alſo their Baſes, (Su
perficies ever equal) ſhall one couple of them terminate in a Cir
cumference of a Circle, and the other couple in one ſole point;