9«. ill. 
r 2242 ) 
aUmlly Be, or be p§Jftble U he performed ; but only, that they 
fuppofed^ (Itbeingufua! with theiBj u^on fuppojitten of things 
im/^^^^/^/Vj to infer ufeful Truths.) kndFMthde (in his fecond 
Po/Zz^te^) requiring, the producing a (ireiyht line Infinitely^ either 
way did not rnean5that it fliould he afiuallyperfotmed. (for it is 
noc poffible for any man to produce a ftreight line lafinitelyy) 
but.th^tiih^/uppofed. And if AB * be fuppofed fo 
^gesr^^.i. produced,thoughbiitone way ; its length muft be 
^ Juppofedio become Infinite (or more than any Finite 
length ajjt^nable-^^ V ox ^xihxii Finite^ a Finite production would 
ferve. But^if 10 produced both ways ; it will be ycc Greater^ 
that is. Greater than that Infimtey ox Greater than was neceff^iry 
tonaakeitmorethan any Finite length afsignable. (And who- 
ever doth thus fuppo(e Infinites *^ mvi^ Qonk(yxQm\y fuppofe^One 
Infinite greater than amtber.) Again, when (by ^z/^-//^^/ tenth 
Propofition) the fame AB*, mzy b^BifeHei'mM. 
and each of the halves in and onwards. In- 
finitely I it is not his meaning (when fuch continual 
feftion is propofed) that it fliould be aUually done^ (for, who 
can do it?) but that it be fuppofed. And upon fuch Q'uppofed) 
ic&.\on infinitely continued^ the parts muft be (fuppsfed) infinitely 
mmy\ for no Finite number of parts would fuffice iot Infinite - 
fediioas. And if further , the fame AB fo divided 5 be fuppofed 
the fide of a Triangle ABC ''^ 5 and,fromeach point 
ofdivifion^ /^^/?fl/e^hnes (as rac, Mc, &:cj parallel 
* toBC; thefe parallels (reckoniiig downward from 
A toBC) muftconfequently be (fuppofed) infinitely many and 
t\\ok^\n Arithmetical progreffnn^ as i, 2^ 33 8cc* each exceeding 
its Antecedent as much as that exceeds the next before it /) 
tindyVphereof the lafi (^^C) is given : (and their Squares, as i, 4^ 9, 
&c. their Cubes^as i,§52 7,&c.) And this I fay, to {hew that the 
fuppoftion of Infinites ("with thefe attendants) is not fo nevi^, or 
io Peculiar to Cavallerius or Dr.Wallis^ but that Euclide admits 
itj and all Mathematicians with him 5 as at leaft fuppofahle^y^hf- 
ther Pofihle Gv not. 
In particular, therefore^ to his ^^r^V^ I anfwer^, i. There 
ra^y b^ fuppofed a row of Quantities Infinitely mmy ^ and con^ 
tinually increafing, fas the fuppofed parallels in the Triangle 
ABC,; reckoning downwards from A to BCJ r^htreof the lafl 
(tC^is givefi^ 2y A Finite Qiiantity (as AB) may, be fuppofed 
(by. 
