" ( S 7 5 ) 
td-nen cdMcqucnfur^dtii) (LCMijJl.ni Cavalerii; f >?rlndlvi fibiHum doclrinam 
nobis amicijpmam. 
And when thus carefully to apply it, of that fee Lalovera'sE/ententaTe- 
tragonifinica Tolofa 1 65 1. where more Archlmedeo he demonftrates the 
truth of this Method ; which Book if Kngeli had feen 5 he would certainly 
have quoted it, and admired the Author 
For want of this Method , it was , faith Angeli 3 by way of complaint, of Tacquct > that 
he omitted fome Theorems , which by aid the eof lie might eafily have found out. See him 
in his Pre c acc to his Infinite Spirals 3 but efpecially'at SchoL 3. Prop. 1 $. 1. 1. 
Si ergo Tacquet rccepifset doctrinam Cavalerii, potuiffct non folum Cubare portio- 
fiem Cylindrici Parxbolici fuper quicunque Infinitarum Parabolarum per Bafin Parabo~ 
l# (? Punftum in latere; fed ttiam exiis 3 qux in Exercitat. 4. Cavalerii tradunt ipfe 
& Beaugrand, potuiffet Cubare Segments portions cu'yufcunque Cylindrici Parabolkire- 
ft ft* plank fcclioti maxima parallelis : Imo ex doclrina Cavalerii potuiffet etiam Gw 
bare , & portiomm Cylindrici upgr Hyperbola per bxfin Hyperbola & Puntt^m in latere, 
& fegmenia bujm portions refe'cl a plants feclionl mixima parallelis {juppofita tamen Hy- 
perbolae Jguadratura.) 
Angeli finds afterwards another defer vedlyjfamoiis Msn , vi%. Dr. John JJ'aflii , own- 
ing an. i ufing the Method of Indivifibles , and advancing it to admiration inhh Aritfone- 
ticalnfinitorum ; who in his Book de Cycloide at Oxford 16^ , faith thus, Pa r. 9. 
Supponimus en'm ( qutdct facile 3 ft opus eft, probability ) Planum quodvis t ant tin - m 
dem hufufmodi Converfione ( feu Rotatione) producere 3 quantum eft quod Jit exeodem 
Piano in lineam ipfius Centro grav 2 tat is defer ip lam duclo h quod & delinea quavh five 
refta five etirva > in co Piano defcripta , pariter intelligendum eft : ^uod quidem enim 
ipfeolimme primum invenijj'e putaverim , monitus moxeram 3 nonnihilapud Guldinum 
exjiare qu&d bucfftHeu Id autem fi animadvertifset Tacqu.nus , dum de Cylindricis & 
Aiinu'aribus acutum Opus confer ip fit , nonpirum UHfuifset adjumcnto , midtaqu&qiue 
Wc extant , turn Univerfalius turn contractus forte fuifstnt edita. 
All which is not recited here 3 todifparage our Author 5 but to take off the prejudicc 3 
which he may beget in his Readers againft the Method of Indivifibles , which hath been 
o*ne<fby other famoua Men, befides thofe already recked ; vi\. by Mcngolut, who from 
the Excellencies of this Method,' Archimel's Method, and Meta's Specious Algebra 3 
composed his Geometric Speciofa\ by Antimo farby , alias fas 'tisfuggeikdj Hon. Fabri 
in Tra& Ve Linea sinuum iff Cycloide j by Pafcal, alias Bettonville ^ by Ves Cartes 
himfelf Vol. 5. of Letters, who faith, that by it he fquared the Cycloid , and iately by the 
excellent Slufius, (pc* 2. To remove the other prejudice that may be againfl this Author as 
defective: for the f th Book CylMricorum & Annularium (now printed with the reft} the 
Prefictr affertsto be firft extant in 16 ?o. And becaufe we prefume, the reft of thefe Books 
are already known and common, and that this hath not formerly been expos'd to fale in 
England > and becaufe alfo it lupplies and compenfates thofe defecV,we think fit to acquaint 
tfrc Reader with the Argument thereof. The Author divides this Fifth Book into fis 
parts : 
1 * In the flrft he demonftrates (in 6 Lemma's and 9 Proportions) That, if any Plain 
Surface have a Rotation about its Axis in any Situation whatfoever , and at anydiftanee 
whitfoever , or none , it produceth a Round Solid equal to an Upright Solid , *hofe Bafe 
Is the begetting Figure, and Height is equal to the Cireumference defcribed by its Center 
of Gravity. ( this Univcrjal Rule was invented fc/Guldin , and is the Bafts of moil of 
hk Decfrine 5 but hs could not demonflrate the fame, though 'twas much defired.) 
z. Inlikemsnner , If any Perimeter have a Rotation about its Axis in any Situation 
whatfoever , it begets around Surface , equal to aright Surface , made by the fame Peri- 
meter as a Bafe (which may be evolv'd and made a Plain Surface ) whofe height is the way 
or circumfejenee defcribed by its Center of Gravity. This by 5 lemmas and ivVropo- 
fiiions. 
rule 
