1would require to divide it into forty, you will be content more
willingly to admit of it in our future Diſcourſe?
willingly to admit of it in our future Diſcourſe?
SIMP. I am pleaſed with your way of arguing, as you now do
mix it with ſome pleaſantneſs: and to your queſtion I reply, that
the facility would ſeem more than ſufficient, if the reſolving it into
Points were but as eaſie, as to divide it into a thouſand parts.
mix it with ſome pleaſantneſs: and to your queſtion I reply, that
the facility would ſeem more than ſufficient, if the reſolving it into
Points were but as eaſie, as to divide it into a thouſand parts.
SALV. Here I will tell you a thing, which haply will make you
wonder in this matter of going about, or being able to reſolve the
Line into its Infinites, keeping that order which others obſerve in
dividing it into forty, ſixty, or an hundred parts; namely, by di
viding it firſt into two, then into four: in which order he that
ſhould think to find its infinite Points would groſly delude himſelf;
for by that progreſſion, though continued to eternity, he ſhould
never arrive to the diviſion of all its quantitative parts: yea, he is
in that way ſo far from being able to arrive at the intended term
of Indiviſibility, that he rather goeth farther from it; and whilſt
he thinks by continuing the diviſion, and multiplying the multi
tudes of the parts, to approach to Infinite, I am of opinion, that he
more and more removes from it: and my reaſon is this; In the
Diſcourſe, we had even now, we concluded, that, in an infinite
Number, there was, of neceſſity, as many Square, or Cube Num
bers, as there were Numbers; ſince that thoſe and theſe were as ma
ny as their Roots, and Roots comprehend all Numbers: Next we
did ſee, that the greater the Numbers were that were taken, the
ſeldomer are their Squares to be found in them, and ſeldomer yet
their Cubes: Therefore it is manifeſt, that the greater the Number
is to which you paſs, the farther you remove from Infinite Num
ber: from whence it followeth, that turning backwards, (ſeeing
that ſuch a progreſſion more removes us from the deſired term) if
any number may be ſaid to be infinite it is the Unite: and, indeed,
there are in it thoſe conditions, and neceſſary qualities of the Infi
nite Number, I mean, of containing in it as many Squares as Cubes,
and as Numbers.
wonder in this matter of going about, or being able to reſolve the
Line into its Infinites, keeping that order which others obſerve in
dividing it into forty, ſixty, or an hundred parts; namely, by di
viding it firſt into two, then into four: in which order he that
ſhould think to find its infinite Points would groſly delude himſelf;
for by that progreſſion, though continued to eternity, he ſhould
never arrive to the diviſion of all its quantitative parts: yea, he is
in that way ſo far from being able to arrive at the intended term
of Indiviſibility, that he rather goeth farther from it; and whilſt
he thinks by continuing the diviſion, and multiplying the multi
tudes of the parts, to approach to Infinite, I am of opinion, that he
more and more removes from it: and my reaſon is this; In the
Diſcourſe, we had even now, we concluded, that, in an infinite
Number, there was, of neceſſity, as many Square, or Cube Num
bers, as there were Numbers; ſince that thoſe and theſe were as ma
ny as their Roots, and Roots comprehend all Numbers: Next we
did ſee, that the greater the Numbers were that were taken, the
ſeldomer are their Squares to be found in them, and ſeldomer yet
their Cubes: Therefore it is manifeſt, that the greater the Number
is to which you paſs, the farther you remove from Infinite Num
ber: from whence it followeth, that turning backwards, (ſeeing
that ſuch a progreſſion more removes us from the deſired term) if
any number may be ſaid to be infinite it is the Unite: and, indeed,
there are in it thoſe conditions, and neceſſary qualities of the Infi
nite Number, I mean, of containing in it as many Squares as Cubes,
and as Numbers.
The Unite of all
Numbers may
moſt properly be
ſaid to be Infinite.
Numbers may
moſt properly be
ſaid to be Infinite.
SIMP. I do not apprehend very well, how this buſineſs ſhould
be underſtood.
be underſtood.
SALV. The thing hath no difficulty at all in it, for the Unite
is a Square, a Cube, a Squared Square, and all other Powers; nor
is there any particular whatſoever eſſential to the Square, or to the
Cube, which doth not agree with the Unite; as v. gr. one proper
ty of two Square-numbers is to have between them a Number
mean-proportional; take any Square number for one of the terms,
and the Unite for the other, and you ſhall likewiſe ever find be
tween them a Number Mean-proportional. Let the two Square
Numbers be 9 and 4, you ſee that between 9 and 1 the Mean
proportional is 3, and between 4 and 1 the Mean-proportional
is a Square, a Cube, a Squared Square, and all other Powers; nor
is there any particular whatſoever eſſential to the Square, or to the
Cube, which doth not agree with the Unite; as v. gr. one proper
ty of two Square-numbers is to have between them a Number
mean-proportional; take any Square number for one of the terms,
and the Unite for the other, and you ſhall likewiſe ever find be
tween them a Number Mean-proportional. Let the two Square
Numbers be 9 and 4, you ſee that between 9 and 1 the Mean
proportional is 3, and between 4 and 1 the Mean-proportional
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