1is 2, and between the two Squares 9 and 4, 6 is the Mean. The
property of Cubes is to have neceſſarily between them two Num
bers Mean-proportional. Suppoſe 8, and 27, the Means between
them are 12 and 18; and between the Unite and 8 the Means
are 2 and 4; betwixt the Unite and 27 there are 3, and 9. We
therefore conclude, That there is no other Infinite Number but the
Vnite. And theſe be ſome of thoſe Wonders, that ſurmount the
comprehenſion of our Imagination, and that advertize us how ex
ceedingly they err, who diſcourſe about Infinites with thoſe very
Attributes, that are uſed about Finites; the Natures of which have
no congruity with each other. In which affair I will not conceal
from you an admirable accident, that I met with ſome time ſince,
explaining the vaſt difference, yea, repugnance and contrariety of
Nature, that a terminate quantity would incur by changing or paſ
ſing into Infinite. We aſſign this Right Line A B, of any length at
pleaſure, and any point in the ſame, as C being taken, dividing it
into two unequal parts: I ſay, that many couples Lines, (hold
ing the ſame proportion between themſelves as have the parts
A C, and B C,) departing from the terms A and B to meet with
one another; the points of their Interſection ſhall all fall in the
Circumference of one and the ſame Circle: as for example, A L
and B L departing [or being drawn] from the Points A and B, and
having between themſelves the ſame proportion, as have the parts
A C and B C, and concurring in the point L: and the ſame pro
portion being between two others A K, and B K, concurring in K,
alſo others as A I, and B I; A H, and B H; A G, and B G; A F,
and B F; A E, and B E: I ſay, that the points of their Interſecti
on L, K, I, H, G, F, E, do all fall in the Circumference of one
and the ſame Semi-circle: ſo that we ſhould imagine the point
C to mve conti
58[Figure 58]
nuallyafter ſuch
a ſort, that the
Lines produced
from it to the fix
ed terms A and
B retain alwaies
the ſame propor
tion that is be
tween the firſt
parts A C and C B, that point C ſhall decribe the Circumference
of a Circle, as we ſhall ſhew you preſently. And the Circle in ſuch
ſort deſcribed ſhall be alwaies greater and greater ſucceſſively,
according as the point C is taken nearer to the middle point
which is O; and the Circle ſhall be leſſer which ſhall be deſcribed
from a point nearer to the extremity B, inſomuch, that from the
property of Cubes is to have neceſſarily between them two Num
bers Mean-proportional. Suppoſe 8, and 27, the Means between
them are 12 and 18; and between the Unite and 8 the Means
are 2 and 4; betwixt the Unite and 27 there are 3, and 9. We
therefore conclude, That there is no other Infinite Number but the
Vnite. And theſe be ſome of thoſe Wonders, that ſurmount the
comprehenſion of our Imagination, and that advertize us how ex
ceedingly they err, who diſcourſe about Infinites with thoſe very
Attributes, that are uſed about Finites; the Natures of which have
no congruity with each other. In which affair I will not conceal
from you an admirable accident, that I met with ſome time ſince,
explaining the vaſt difference, yea, repugnance and contrariety of
Nature, that a terminate quantity would incur by changing or paſ
ſing into Infinite. We aſſign this Right Line A B, of any length at
pleaſure, and any point in the ſame, as C being taken, dividing it
into two unequal parts: I ſay, that many couples Lines, (hold
ing the ſame proportion between themſelves as have the parts
A C, and B C,) departing from the terms A and B to meet with
one another; the points of their Interſection ſhall all fall in the
Circumference of one and the ſame Circle: as for example, A L
and B L departing [or being drawn] from the Points A and B, and
having between themſelves the ſame proportion, as have the parts
A C and B C, and concurring in the point L: and the ſame pro
portion being between two others A K, and B K, concurring in K,
alſo others as A I, and B I; A H, and B H; A G, and B G; A F,
and B F; A E, and B E: I ſay, that the points of their Interſecti
on L, K, I, H, G, F, E, do all fall in the Circumference of one
and the ſame Semi-circle: ſo that we ſhould imagine the point
C to mve conti
58[Figure 58]nuallyafter ſuch
a ſort, that the
Lines produced
from it to the fix
ed terms A and
B retain alwaies
the ſame propor
tion that is be
tween the firſt
parts A C and C B, that point C ſhall decribe the Circumference
of a Circle, as we ſhall ſhew you preſently. And the Circle in ſuch
ſort deſcribed ſhall be alwaies greater and greater ſucceſſively,
according as the point C is taken nearer to the middle point
which is O; and the Circle ſhall be leſſer which ſhall be deſcribed
from a point nearer to the extremity B, inſomuch, that from the