Add MS 6782
*362*
*De infinitis. De continuo.*
Aristotle in the beginning of his 6th booke of his physicks, & in the 26th treatise of the 5th booke, defineth those thinges to be And in the 22nd treatise of the said 5th booke that: . .
Now for the of the definitions as also of their truth. Let us understand first two cubes A & B to be separate, that is, to be in diverse planes, extremes & all.

*363*
*De Infinitis progressionibus*
In progressions that be infinite be they increasing or decreasing.
There are these passes.
First to a quantity that haveth no rate to the first quantity given, or rather because betwixt positive quantityes there is a positive rate, I may call that rate infinite either in great- ness or litles according to the , in respect of the first quantity given.
Yet in respecte of the progression following it is divisible or mul- tiplicable till the progression being infinite hath for his second passe also a quantity infinite rate.
Which is not only infinite in respecte of the first quantity of the last progression; but infinitely infinite in respect of the first in the first progresse.
And also the summe of the second pro- gression is infinite in respect of the first summe of the first pro- gression, or the first quantity of all.
And so a third, fourth & infinite other progressions and passes; of which any quantity or the summe of all infinitely all, is of an infinite quantity in greatness of litleness in respect, of the summe or first quantity of the first progression.
And yet last in decreasing progressions we must needes under- stand a quantity absolutely indivisible; but multiplicable infinitely infinite till a quantity absolutely immultiplicable be produced which I may call universally infinite.
And in increasing progressions we must needes understand that last there must be a quantity immultiplicable absolute, but divisible infinitely infinite till that quantity be issued that is absolutely indivisble.
That such quantity which I call universally infinite: hath not only act rationall, by supposition, or by consequence from supposition: but also act reall, or existence: in an instant, having actuall being, or in time, passed by motion both finite & infinite: with many reall consequences or properties consequent; & accidents adioyning: shalbe declared in the papers following.

*364*
*De Infinitis.*
Seing that any finite line will subtend an angle at summe distance; as let $ bc $ subtend the angle $ bac $.
Then a line double to $ bc $, which let be $ de $, will subtend the same angle at a double distance, so that $ bd $ will be aequall to $ ab $.
In those subtensions I understand that the poynt $ a $ be perpendicular to the middle of the subtendent lines.
as also in all the others which follow.
Now I suppose $ bc $ to be removed to a further distance from the poynt $ a $.
Then the angle $ bac $ subtended must be lesse than before.
And $ de $. shall subtend the same angle at a double distance as before.
And this is true continually that the further $ bc $ is removed the lesse angle it subtendeth & $ de $ always must subtend the same angle at a double distance.
Then I suppose $ bc $ to be removed to an infinite distance; at which distance the supposition altereth not the quantity of $ bc $. but the consequence is of the angle.
Which wilbe, that the angle then subtended to be of an infinite quantity in litleness in respecte of the former angles.
Yet it cannot be sayd to be no angle negatively because it is positive. & it must also follow that the line $ de $ must subtend the same positive angle at a double distance.
Which is Double to the former infinite distance.
Also, let the distance of the subtendents be nearer infinite, it cannot be otherwise inferred but that the lines $ af $ & $ ag $ though infinite, be , because $ bc $ & $ de $ are betweene them, & have agreement or concurrence but only in the poynt $ a $, in no distance out of the poynt $ a $.
And yet the nearness of there congruence & conrrence in all other partes at the utmost is such, that although they be remote; the angle is of no proportion explicable by nomber finite, but infinite , to any other angle which we call finite.
The like inexplicable proportion is of the lines $ de $ & $ bc $, to there infinite distance position from $ a $.
And yet the sayd lines $ de $ & $ bc $. as also that infinite litle or improportio- nable angle is divisible still . & still, although improportionable yet in an other respect, that is to say of his owne partes, is proportionable.

*365*
*De Infinitis.*
That in a finite time an infinite space may be moved
It is now convenient that $ cf $ be in this line.
Suppose the line $ cef $ to be infinite, & the line $ ab $ suppose to revolve & describe a circle in a finite time, fro $ b $ towards $ g $. $ ab $ doth first respect $ c $, then $ d $, after $ e $, & so forth successively no poynt in the infinite line is unrespected by that time the line $ ab $ cometh to $ ag $ where then the line is parallel & cutteth not the former line infinite.
Now seing that a motion may be of any thing according the continuall succession of a poynt, as well in respect of .
Whatsoever may be or not be in respect of the moment, it maketh no matter: the purpose is manifest.
Consequentia Accidentis quædam huius motus.
The line $ ab $ having moved till he comes to be $ ah $ that is parallel to $ cf $. & so that continuing his motion of revolution:
The lines are parallel but in one instant.
They never cut at an infinite distance but at that instant they are parallel.
And if they cut then, they must cut & then being right lines there must be no space betwixte them, but there distance by supposition is more than the line $ ab $.
Which implies contradiction.
And yet there must be a cutting at an infinite distance or else all the poyntes of the infinite line could not have been respected. &
if that be not some part of the infinite line, that is some quantity finite is only cut; & that is at a finite distance; & then it maketh an angle at the greatest distance of such cutting: from that cutting the line by motion came to be parallel: That motion is made in an instant or in time.
If in time, then in half the time the cutting must be further than the supposed furthest;
If in an instant, our line wilbe in two places in one instant; .
The lines therefore must cut at an infinite distance before they come to be parallel.
And that must be in time before or in an instant before.
If in time, then in half the time they cut at greater distance than infinite or are parallel before they are parallel.
Which both do imply contradiction.
If in an instant before; the two instants are one or different.
If one, .
If two there must be no other betwixt them.
And then there be a time greater than an instant & lesse than any time of quantity that is indivisible, that is agayne, indivisible into partes of quantity. & so also like of poyntes &c.

*366*
*De Infinitis.*
line $ ab $ by his revolution cometh at length to be parallel to the infinite line $ bf $.
Which motion being from $ b $ to $ g $ suppose to have been æqually.
The degree of the motion let be $ mn $. the time $ op $.
The beginning of the time or first instant $ o $.
The last instant wherein the line is parallel, $ p $. Now seing that $ ab $ must cut at an infinite distance & his last cutting must be before the instant $ p $.
Which suppose $ q $.
That $ q $ as it is argued by the premises must differe from $ p $ by an indivisible time, so that $ q $ must be the next instant to $ p $. & no other between.
In which instant $ q $, $ ab $ must not be parallel but make his last cutting at an infinite distance.
And therefore it must have a certayne at that instant out of the point $ g $ towards $ b $, which let be $ af $, as it maketh his last section.
In which situation the motion ordering it hath the sayd degree $ mn $, as in all other situations.
From the which situation to the situation of being parallel it must be moved unto (as it is sayd) in the next instant.
Now suppose (as it may be) that the motion from $ b $ to $ g $ be in half the time of $ op $.
Then doth it follow necessarily that the degree of motion or velo- city be double to $ mn $. And therefore, what space or parte of a space, (be it finite or infinite, so it be positive,) it moved before according to the degree of $ mn $. it moveth the same now, in half the time.
Therefore in this second motion when $ ab $ cometh to have his situation at $ af $ to make the sayd last section; seing that then it hath double degree of velocity; it must afterward be parallel in half an instant that is to say, in half that time which was sayd to be indivisible.
Which doth imply contradiction.
Agayne if it be sayd that $ af $ at that instant (when & where it maketh his last section with $ bf $ before it be parallel) be to $ ah $. or that the poynts $ f $ & $ h $ be at an infinite distance so that no point can be between.
Yet from the poynt $ k $ to $ f $ may be interposed a line $ kf $. and also from $ l $ to $ f $. & by the doctrine of Elements the angle $ fkh $, or $ flh $ must be than $ fah $. & therefore lesse than that which was sayd to be least or indivisible. & therefore the lines $ af $ & $ ah $, or the poynts $ f $ & $ h $ be not .

*367*
*De Infinitis. Ratio Achilles*
There is a reason in Aristotle (in the 6th booke of his phisickes. text. 78.) which for the sorce it seemeth to carry is called Achilles.
And for that cause, no doubt, is name also Achilles used in the example to expresse the reason.
The which because it is against Aristotles doctrine & for that it compryseth matter of greater consequence concerning the doctrine of infinites, it being there but briefly & obscurely set downe with an answere uncertayne: I thinke good to set downe more & largely: with Aristotles Answere as he hath it in the place allwayes, as also at full according to his owne doctrine in other places.
To the end that comparing one with the other, the truth may appear, & perhaps otherwise to be, then yet hath been by the peripateticles either noted or observed.
The proposition of Zeno is.
The swift runner (runne he never so swiftly) shall never overtake the slow runner (runne he never so slowly.
That there may be doubte of the meaning of the proposition we will declare what thinges are therein supposed.

ffirst, (as it ought to be, else the proposition were ridiculous) The motion of the runner & slow mover are understood to be both one way & in one right line.
Secondly The

*368*
*Ratio Achilles.*
Let Achilles be $ A $.
Testudo $ B $.
The Motion of Achilles from $ A $ to $ B $ in the time $ ef $. of Testudo from $ B $ to $ C $ in the time .
.
Which space of $ BC $ let be the tenth parte of $ AB $.
Now the quaestion is, both these motions being continued in the same proportion as 10 to 1. where & when shall $ A $ overtake $ B $. .
At some point or other it must really be.
Suppose that $ d $.
There must be $ A $ & $ B $, at the same instant of time.
And therefore the time wherein $ A $ hath moved to $ d $ must be the same wherein $ B $ hath moved to $ d $.
But the space $ Ad $ to $ Bd $ must be as 10 to 1.
Now by the supposition it must follow (because these motions be proportionall ) * As $ AB $ to $ BC $. so: $ Ad $ to $ Bd $. which same termes proportionall call by these letters & in the same order.
As $ \\beta $ is known to be 1. $ \\gamma $ is $ \\frac{1}{10} $. $ \\beta + \\alpha $ is unknown. & so is $ \\alpha $. yet this is known that.
* Now what other proportion is this than if a man should say as the first to the second so all the antecedents to all the consequents which in this be infinite in nomber.
X To find that poynt geometrically is set downe in my other papers .

*369*
*De Infinitis.*
Now will I propound some dfficultyes to be considered of.
Seing that every line is compounded of atomes, & therefore the periphery of a circle. one is succeeding one an other infinitely in such manner as the perifery is at last compounded and made.
Now also seing that the whole is compounded of about the poynt $ a $. so many times infinitely, & to that number of them infinitely, till the circle supposed be accomplished.
I demand what wilbe the nomber of that are about the point $ a $.
Infinite they must needes be, or else infinite lines could not be supposed actually from the point $ a $ to the perifery.
And infinite also are in the perifery.
But now I demande whether they are aequally infinite or not.
If about the center are lesse infinite then there cannot from the center $ a $ to every poynt in the perifery be understood a right line but we must understand those that we supposed indivisible, divisble .
and if they be æqually infinite: then in a great place, (where the nomber, although infinite, yet in them selves definite; because they being supposed to have acte there is not one more nor lesse.
Neither can there be more because they being one more cannot be between there being no distance: & if there one lesse; there lacketh of the supposed actaull, & definite & positive number although infinite.
Then I say in a greate place where there could be no more or lesse, in a lesse place there are an æquall nomber; which seemeth to imply.
An other difficulty riseth from the square.
If a line be compounded of , the diametrall line wilbe found to be aæquall to the side.
ffor suppose the line $ ab $ to be drawne from the point $ a $ to the point $ b $, of the line $ bc $. Then from the next point $ e $, which is to $ a $ in the line $ ad $, draw a line to $ f $ the next point to $ b $ in the line $ bc $.
So likewise from every next point in the line $ ad $, to every next point in the line $ bc $.
Now the lines so drawne must needs be the least & most that may be, because they are & all. & they all cut the line $ ac $ & of the line $ ac $ there can be no point betwixt two of the former lines .
And therefore the nomber of the poynts of the line $ ac $, aequally infinite to the poynts of $ ab $ & consequence the lines $ ab $ & $ ac $ aequall.
But this difficulty wilbe made more playne by the next following, which wilbe found the meanes for the solution of all.
An other question is. where two are . whether an other (the not disioyned) may either passe or have situation betwixt them.

*370*
*De Infinitis. Notanda.*
De tactu duorum corporum per superficies. an duæ superficies sint realiter distantes in corporum contactu.
Because is negative to in respect of that thing may be sayd to be either.
If yet which is is not & that which is is not . therefore the one being knowne the other cannot be unknowne what it is.
Now although there be great controversy of the essence & quality of . yet there is no such of . we will therefore lay downe what is manifest of it, that the ratio & essence of may appeare.

*370v*
William Sprat a wolle draper at the sign of the rope in Watlin street at Soper Lane corner. serveth for a for his wifes brother for . there are 4.

*371*
*De Infinitis.*
That there may be two magnitudes given, of which the one shalbe infinite in respect of the other, & yet in respect of two other magnitudes they shalbe finite.
That a line finite, cannot have his partes, of a finite magnitude; but they must be of a finite nomber.
That a finite line may have an infinite nomber of partes, & if the partes be in continuall proportion: the nomber must be compounded of an infinite nomber of finite partes; & an infinite nomber of infinite partes.
If a line be understood to be compounded of poyntes: the nomber of them is infinite of the first passe, second or any nomber of passes finite or infinite.

*372*
*De Infinitis. Ratio Clava Herculis.*

*373*
*De Infinitis.*
Suppose the line $ bc $ doth touch the the circle in the point $ b $. & touching in that point it only & in no other point toucheth, as Euclide suffi- ciently demonstrateth. Now I say there is (a point ) a next poynt that doth not touch the line $ bc $.

*374*
*De Infinitis.*
Minimum. That will kill men by piercing & running through.
Maximum. That which will presse men to death.
Unitas. Numeris unitatum. finitis infinitis
Finites finitorum. Infinites finitorum. finites Infinitorum. Infinites Infinitorum. Infiniti infinitorum infinitum. Infiniti infinitorum finitum.
finitorum minimum. Infinitorum minimum. finitus minimorum. Infinites minimorum. finites finiti minimorum. Infinites finiti minimorum. Infinites finiti maximorum. Infinites infiniti maximorum. finiti. finitorum maximum .1. Infinitorum Infinitorum maximum. Infiniti.
Ratio Achilles
All the mistery of infinites lieth in which is only respective, & from where the knowledge & import of of quantity doth spring.
A finite space may be moved in infinite time.
There is a motion that a finite space cannot be moved but in an infinite time.
Also: that a finite space given cannot be moved in an infinite time.
Also: that an infinite space may be moved in a finite time.
Also: that an infinite space may be moved not in a finite time but in an infinite time.
Also: that an infinite space given, may not be moved either in an infinite time nor finite.
Of contradictions that spring from diverse suppositions it cannot truly sayd that the one parte or other is false, for they are true consequently from there suppositions & in that respect are both true. but that which followeth is, that one of the suppositions is necessarily false, from where one of the partes of the contradiction was inferred.
As in the reason Achilles & other reasons of Zeno &c.

Add MS 6782
*362*
*De infinitis. De continuo.*
Aristotle in the beginning of his 6th booke of his physicks, & in the 26th treatise of the 5th booke, defineth those thinges to be And in the 22nd treatise of the said 5th booke that: . .
Now for the of the definitions as also of their truth. Let us understand first two cubes A & B to be separate, that is, to be in diverse planes, extremes & all.

*363*
*De Infinitis progressionibus*
In progressions that be infinite be they increasing or decreasing.
There are these passes.
First to a quantity that haveth no rate to the first quantity given, or rather because betwixt positive quantityes there is a positive rate, I may call that rate infinite either in great- ness or litles according to the , in respect of the first quantity given.
Yet in respecte of the progression following it is divisible or mul- tiplicable till the progression being infinite hath for his second passe also a quantity infinite rate.
Which is not only infinite in respecte of the first quantity of the last progression; but infinitely infinite in respect of the first in the first progresse.
And also the summe of the second pro- gression is infinite in respect of the first summe of the first pro- gression, or the first quantity of all.
And so a third, fourth & infinite other progressions and passes; of which any quantity or the summe of all infinitely all, is of an infinite quantity in greatness of litleness in respect, of the summe or first quantity of the first progression.
And yet last in decreasing progressions we must needes under- stand a quantity absolutely indivisible; but multiplicable infinitely infinite till a quantity absolutely immultiplicable be produced which I may call universally infinite.
And in increasing progressions we must needes understand that last there must be a quantity immultiplicable absolute, but divisible infinitely infinite till that quantity be issued that is absolutely indivisble.
That such quantity which I call universally infinite: hath not only act rationall, by supposition, or by consequence from supposition: but also act reall, or existence: in an instant, having actuall being, or in time, passed by motion both finite & infinite: with many reall consequences or properties consequent; & accidents adioyning: shalbe declared in the papers following.

*364*
*De Infinitis.*
Seing that any finite line will subtend an angle at summe distance; as let $ bc $ subtend the angle $ bac $.
Then a line double to $ bc $, which let be $ de $, will subtend the same angle at a double distance, so that $ bd $ will be aequall to $ ab $.
In those subtensions I understand that the poynt $ a $ be perpendicular to the middle of the subtendent lines.
as also in all the others which follow.
Now I suppose $ bc $ to be removed to a further distance from the poynt $ a $.
Then the angle $ bac $ subtended must be lesse than before.
And $ de $. shall subtend the same angle at a double distance as before.
And this is true continually that the further $ bc $ is removed the lesse angle it subtendeth & $ de $ always must subtend the same angle at a double distance.
Then I suppose $ bc $ to be removed to an infinite distance; at which distance the supposition altereth not the quantity of $ bc $. but the consequence is of the angle.
Which wilbe, that the angle then subtended to be of an infinite quantity in litleness in respecte of the former angles.
Yet it cannot be sayd to be no angle negatively because it is positive. & it must also follow that the line $ de $ must subtend the same positive angle at a double distance.
Which is Double to the former infinite distance.
Also, let the distance of the subtendents be nearer infinite, it cannot be otherwise inferred but that the lines $ af $ & $ ag $ though infinite, be , because $ bc $ & $ de $ are betweene them, & have agreement or concurrence but only in the poynt $ a $, in no distance out of the poynt $ a $.
And yet the nearness of there congruence & conrrence in all other partes at the utmost is such, that although they be remote; the angle is of no proportion explicable by nomber finite, but infinite , to any other angle which we call finite.
The like inexplicable proportion is of the lines $ de $ & $ bc $, to there infinite distance position from $ a $.
And yet the sayd lines $ de $ & $ bc $. as also that infinite litle or improportio- nable angle is divisible still . & still, although improportionable yet in an other respect, that is to say of his owne partes, is proportionable.

*365*
*De Infinitis.*
That in a finite time an infinite space may be moved
It is now convenient that $ cf $ be in this line.
Suppose the line $ cef $ to be infinite, & the line $ ab $ suppose to revolve & describe a circle in a finite time, fro $ b $ towards $ g $. $ ab $ doth first respect $ c $, then $ d $, after $ e $, & so forth successively no poynt in the infinite line is unrespected by that time the line $ ab $ cometh to $ ag $ where then the line is parallel & cutteth not the former line infinite.
Now seing that a motion may be of any thing according the continuall succession of a poynt, as well in respect of .
Whatsoever may be or not be in respect of the moment, it maketh no matter: the purpose is manifest.
Consequentia Accidentis quædam huius motus.
The line $ ab $ having moved till he comes to be $ ah $ that is parallel to $ cf $. & so that continuing his motion of revolution:
The lines are parallel but in one instant.
They never cut at an infinite distance but at that instant they are parallel.
And if they cut then, they must cut & then being right lines there must be no space betwixte them, but there distance by supposition is more than the line $ ab $.
Which implies contradiction.
And yet there must be a cutting at an infinite distance or else all the poyntes of the infinite line could not have been respected. &
if that be not some part of the infinite line, that is some quantity finite is only cut; & that is at a finite distance; & then it maketh an angle at the greatest distance of such cutting: from that cutting the line by motion came to be parallel: That motion is made in an instant or in time.
If in time, then in half the time the cutting must be further than the supposed furthest;
If in an instant, our line wilbe in two places in one instant; .
The lines therefore must cut at an infinite distance before they come to be parallel.
And that must be in time before or in an instant before.
If in time, then in half the time they cut at greater distance than infinite or are parallel before they are parallel.
Which both do imply contradiction.
If in an instant before; the two instants are one or different.
If one, .
If two there must be no other betwixt them.
And then there be a time greater than an instant & lesse than any time of quantity that is indivisible, that is agayne, indivisible into partes of quantity. & so also like of poyntes &c.

*366*
*De Infinitis.*
line $ ab $ by his revolution cometh at length to be parallel to the infinite line $ bf $.
Which motion being from $ b $ to $ g $ suppose to have been æqually.
The degree of the motion let be $ mn $. the time $ op $.
The beginning of the time or first instant $ o $.
The last instant wherein the line is parallel, $ p $. Now seing that $ ab $ must cut at an infinite distance & his last cutting must be before the instant $ p $.
Which suppose $ q $.
That $ q $ as it is argued by the premises must differe from $ p $ by an indivisible time, so that $ q $ must be the next instant to $ p $. & no other between.
In which instant $ q $, $ ab $ must not be parallel but make his last cutting at an infinite distance.
And therefore it must have a certayne at that instant out of the point $ g $ towards $ b $, which let be $ af $, as it maketh his last section.
In which situation the motion ordering it hath the sayd degree $ mn $, as in all other situations.
From the which situation to the situation of being parallel it must be moved unto (as it is sayd) in the next instant.
Now suppose (as it may be) that the motion from $ b $ to $ g $ be in half the time of $ op $.
Then doth it follow necessarily that the degree of motion or velo- city be double to $ mn $. And therefore, what space or parte of a space, (be it finite or infinite, so it be positive,) it moved before according to the degree of $ mn $. it moveth the same now, in half the time.
Therefore in this second motion when $ ab $ cometh to have his situation at $ af $ to make the sayd last section; seing that then it hath double degree of velocity; it must afterward be parallel in half an instant that is to say, in half that time which was sayd to be indivisible.
Which doth imply contradiction.
Agayne if it be sayd that $ af $ at that instant (when & where it maketh his last section with $ bf $ before it be parallel) be to $ ah $. or that the poynts $ f $ & $ h $ be at an infinite distance so that no point can be between.
Yet from the poynt $ k $ to $ f $ may be interposed a line $ kf $. and also from $ l $ to $ f $. & by the doctrine of Elements the angle $ fkh $, or $ flh $ must be than $ fah $. & therefore lesse than that which was sayd to be least or indivisible. & therefore the lines $ af $ & $ ah $, or the poynts $ f $ & $ h $ be not .

*367*
*De Infinitis. Ratio Achilles*
There is a reason in Aristotle (in the 6th booke of his phisickes. text. 78.) which for the sorce it seemeth to carry is called Achilles.
And for that cause, no doubt, is name also Achilles used in the example to expresse the reason.
The which because it is against Aristotles doctrine & for that it compryseth matter of greater consequence concerning the doctrine of infinites, it being there but briefly & obscurely set downe with an answere uncertayne: I thinke good to set downe more & largely: with Aristotles Answere as he hath it in the place allwayes, as also at full according to his owne doctrine in other places.
To the end that comparing one with the other, the truth may appear, & perhaps otherwise to be, then yet hath been by the peripateticles either noted or observed.
The proposition of Zeno is.
The swift runner (runne he never so swiftly) shall never overtake the slow runner (runne he never so slowly.
That there may be doubte of the meaning of the proposition we will declare what thinges are therein supposed.

ffirst, (as it ought to be, else the proposition were ridiculous) The motion of the runner & slow mover are understood to be both one way & in one right line.
Secondly The

*368*
*Ratio Achilles.*
Let Achilles be $ A $.
Testudo $ B $.
The Motion of Achilles from $ A $ to $ B $ in the time $ ef $. of Testudo from $ B $ to $ C $ in the time .
.
Which space of $ BC $ let be the tenth parte of $ AB $.
Now the quaestion is, both these motions being continued in the same proportion as 10 to 1. where & when shall $ A $ overtake $ B $. .
At some point or other it must really be.
Suppose that $ d $.
There must be $ A $ & $ B $, at the same instant of time.
And therefore the time wherein $ A $ hath moved to $ d $ must be the same wherein $ B $ hath moved to $ d $.
But the space $ Ad $ to $ Bd $ must be as 10 to 1.
Now by the supposition it must follow (because these motions be proportionall ) * As $ AB $ to $ BC $. so: $ Ad $ to $ Bd $. which same termes proportionall call by these letters & in the same order.
As $ \\beta $ is known to be 1. $ \\gamma $ is $ \\frac{1}{10} $. $ \\beta + \\alpha $ is unknown. & so is $ \\alpha $. yet this is known that.
* Now what other proportion is this than if a man should say as the first to the second so all the antecedents to all the consequents which in this be infinite in nomber.
X To find that poynt geometrically is set downe in my other papers .

*369*
*De Infinitis.*
Now will I propound some dfficultyes to be considered of.
Seing that every line is compounded of atomes, & therefore the periphery of a circle. one is succeeding one an other infinitely in such manner as the perifery is at last compounded and made.
Now also seing that the whole is compounded of about the poynt $ a $. so many times infinitely, & to that number of them infinitely, till the circle supposed be accomplished.
I demand what wilbe the nomber of that are about the point $ a $.
Infinite they must needes be, or else infinite lines could not be supposed actually from the point $ a $ to the perifery.
And infinite also are in the perifery.
But now I demande whether they are aequally infinite or not.
If about the center are lesse infinite then there cannot from the center $ a $ to every poynt in the perifery be understood a right line but we must understand those that we supposed indivisible, divisble .
and if they be æqually infinite: then in a great place, (where the nomber, although infinite, yet in them selves definite; because they being supposed to have acte there is not one more nor lesse.
Neither can there be more because they being one more cannot be between there being no distance: & if there one lesse; there lacketh of the supposed actaull, & definite & positive number although infinite.
Then I say in a greate place where there could be no more or lesse, in a lesse place there are an æquall nomber; which seemeth to imply.
An other difficulty riseth from the square.
If a line be compounded of , the diametrall line wilbe found to be aæquall to the side.
ffor suppose the line $ ab $ to be drawne from the point $ a $ to the point $ b $, of the line $ bc $. Then from the next point $ e $, which is to $ a $ in the line $ ad $, draw a line to $ f $ the next point to $ b $ in the line $ bc $.
So likewise from every next point in the line $ ad $, to every next point in the line $ bc $.
Now the lines so drawne must needs be the least & most that may be, because they are & all. & they all cut the line $ ac $ & of the line $ ac $ there can be no point betwixt two of the former lines .
And therefore the nomber of the poynts of the line $ ac $, aequally infinite to the poynts of $ ab $ & consequence the lines $ ab $ & $ ac $ aequall.
But this difficulty wilbe made more playne by the next following, which wilbe found the meanes for the solution of all.
An other question is. where two are . whether an other (the not disioyned) may either passe or have situation betwixt them.

*370*
*De Infinitis. Notanda.*
De tactu duorum corporum per superficies. an duæ superficies sint realiter distantes in corporum contactu.
Because is negative to in respect of that thing may be sayd to be either.
If yet which is is not & that which is is not . therefore the one being knowne the other cannot be unknowne what it is.
Now although there be great controversy of the essence & quality of . yet there is no such of . we will therefore lay downe what is manifest of it, that the ratio & essence of may appeare.

*370v*
William Sprat a wolle draper at the sign of the rope in Watlin street at Soper Lane corner. serveth for a for his wifes brother for . there are 4.

*371*
*De Infinitis.*
That there may be two magnitudes given, of which the one shalbe infinite in respect of the other, & yet in respect of two other magnitudes they shalbe finite.
That a line finite, cannot have his partes, of a finite magnitude; but they must be of a finite nomber.
That a finite line may have an infinite nomber of partes, & if the partes be in continuall proportion: the nomber must be compounded of an infinite nomber of finite partes; & an infinite nomber of infinite partes.
If a line be understood to be compounded of poyntes: the nomber of them is infinite of the first passe, second or any nomber of passes finite or infinite.

*372*
*De Infinitis. Ratio Clava Herculis.*

*373*
*De Infinitis.*
Suppose the line $ bc $ doth touch the the circle in the point $ b $. & touching in that point it only & in no other point toucheth, as Euclide suffi- ciently demonstrateth. Now I say there is (a point ) a next poynt that doth not touch the line $ bc $.

*374*
*De Infinitis.*
Minimum. That will kill men by piercing & running through.
Maximum. That which will presse men to death.
Unitas. Numeris unitatum. finitis infinitis
Finites finitorum. Infinites finitorum. finites Infinitorum. Infinites Infinitorum. Infiniti infinitorum infinitum. Infiniti infinitorum finitum.
finitorum minimum. Infinitorum minimum. finitus minimorum. Infinites minimorum. finites finiti minimorum. Infinites finiti minimorum. Infinites finiti maximorum. Infinites infiniti maximorum. finiti. finitorum maximum .1. Infinitorum Infinitorum maximum. Infiniti.
Ratio Achilles
All the mistery of infinites lieth in which is only respective, & from where the knowledge & import of of quantity doth spring.
A finite space may be moved in infinite time.
There is a motion that a finite space cannot be moved but in an infinite time.
Also: that a finite space given cannot be moved in an infinite time.
Also: that an infinite space may be moved in a finite time.
Also: that an infinite space may be moved not in a finite time but in an infinite time.
Also: that an infinite space given, may not be moved either in an infinite time nor finite.
Of contradictions that spring from diverse suppositions it cannot truly sayd that the one parte or other is false, for they are true consequently from there suppositions & in that respect are both true. but that which followeth is, that one of the suppositions is necessarily false, from where one of the partes of the contradiction was inferred.
As in the reason Achilles & other reasons of Zeno &c.