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ELEMENTORUM 

UNIVERSyfi MATHESEOS 

^ U C T O TL ^ 

P. ROGERIO JOSEPHO 

BOSCQVICH 

Societalis Jesu 

PUBLICO MATHESEOS PROFESSORE 

T O M U S III. 

coNTinBns 

SECTIONUM CONICARUM ELEMENTA 
nov3 quadam methodo coadRnaa Sc Diflcrtaiionciil 
de TaANsroiLMATioHE LocoiLUM Geomitricoruu, 
ubi de Contiauitatjs kgCj ac de quibuidam loiiiuti 
Myfteiiis, 

E 1> 1 T 1 O P S. r M A VENMTAi 
fmfmalahoriilcdUisentUiiierrBribfuexluriiaat 



VENETIIS, MDCCLVII. 

A P U D A N T O M 1 D M P E & L 1 N I. 






V 



m 




AtrCTORlS 

PR ^ I? ATlOi 



EcHontSn tonicArum , Elements prii 
miferAm }am a plurihus Annis , 4^ 
flHrihus in lecis , ffova quaddm me. 
thbdp 'ekien&aUiefinitidHe dedntU 
nlt in komdhi^ titterdtifrum Uiario 
ad annum 174(5. extat Jche^diafma 
hretdj/iinum , quo ex eadem illa de^ 

, - — TT -. finitione demonfiratur in primis ra* 

Uo confians inter Una re£liniuU feimento^Um hindrum 
thordmm Seaidnis ConiU Cujufvis hahehtium Incliha^ 
tiomm conQihtefh^ \& fe ihvicem /ecdhtium • tUmpar^ 
hm ej6 eo theoremate , partim iterum ex ipfa definitio^ 
he pracipui hmnia i iu^ ad ejufmodi cwrvas pertinent * 
dmvdnt0i 

Id quideih driumehhhi plus quam deciei ergo fane 
^ditut fum in Auditorum meoT^um gratiam ^ neque »w- 
iumnmpitrare potui d mi ipfo , ut ordinim ^ quemji^ 
mel fufceperam , tenerem , at^ porro perierem 5 fed no. 
V4M qudnd4m,i;i4m^ ikdmvii ab iddem definitioni dU 
ire/fHti irtii fempe^ , & fape etiani fere Ufque ad exu 
tm tenia i Ex dlierd inim pam Mdmirdbilis quiddm 
^nter geimetricds veritate} nt:iiui * ut in intricaHjfmo 

\ ^*?^ Idl^riht^ , mille ad eUndem ixitum diverfos 
mrtbdt trdhtiies i eii altetd vel hrehiorit j vel expe^ 

i t^T^ uineHs ohorta fpes tadium iuoddam jamiolerd. 

. ti laboins irtdu^erai i 

\^^^'f^pe hd0em diHiius i hifi fupinoti 4nno irdvif 
^pmumJtbeMei a& inaturandam editionem incitamen^ 
sjm . fonfcripferdm ego quidiin latin» fermone jam ah 
mno I7l7jn^ufum nohilis ddolefcentis 4 quem geoHne^ 
tncis fiudiis znitidndum fufcejfetdm , hreve qudddam 
^tmttnit plana cmpendiolum , qudm dd 14 pnfpofitio' 

425906 



V 



\ 



fV P R ifi F A T I o: 

tium fmm tfeluti capirs redegerAtn , adjeSlis cor^llamf 
n$nmlliSi & fcholiis ita^ ut propofitiambks quidem^ & 
corollariis a^erte contineretur , vel fere ^onte indeflue* 
ret^ ac facHlime dednci pofet , quidquid ad cateras fifi> 
4:idtares mathematicas.vel ^hyficas ex ipfa Gecmetris 
requiritur, in fcholiis autem ufus haberentur nonnulii 
torum^y qu^ pertrailatajam fiifranty quibus Tyronisani^ 
fnus incitaretur , & eorum fiuEluum , q^os olim ex ipfa 
Ceometria percepturus ejfet , jam ali^uam voluptatem 
pricfperet . Haud ita muLto pofi Italico formone hrev4 
itidem Arithmetices compendiotum exaraveram inalia* 
rum quorundam ufum^ uhi primo capite pnteepta , quds 
ad computationem pertinerent , indicaveram tantumma^ 
do ^ fecundo proportionts , ac argumentanS modos , ter* 
tlQ pTQgreJfiones ^ ac logarithmos aliquanto diligentius^ 
perfecmus eram , demonjirationibus ^ in iis , qua ad com» 
putationem pertihebant , pierumque omiffts , in reliquis 
fummo femper cum rigore deduElis . Compendiofa itidem 
Trigonometria fpherica elemenh^ Romana Taquetiana^ 
rum Elementorum editioni inferueramy qudt fimplicitatt. 
quadam , & ordine fe commendabant , nec omnium im4 
prohabantur. 

Dum Gcoiraphif corrigend^ caufa^ & Meridiani stc- 

curatius per Tontificiam ditionem traduSti menfuranda^ 

rumgraduum ^ ditionem ipfampercHrrerem itinereperfum^ 

mos montes Uboriofijftmo\ & abamicis^ & ab iisy qui^ 

kus kt paream j mihi ob ipfam inftituti mei rationem re^ 

tigio eji^per litteras induElus fum^ ut eorum editionem 

permitterem tanquam exordium quoddam Elementorum 

univerfa Mathefeos^ adje^tis iis , ^ua necejfaria vide^ 

rentur . Jjiffum autem Geometria plana compendium illud, 

meum > ejus djfcejfuy in cujus gratiam confcriptum fue^ 

rat , ]am tnm amiferam , quod extabat ab alio Italicti 

redditum^ , unde iterum ab Editore latinitate donatum 

fu^rati & idem Arithmetiae quooke compendium in l^^^ 

tinum ftrmonem converterat , quibus in verfionibus mt^ 

tationes etiam extiterant nonnulUy uti fit , aliis etiani 

quandeque adjeStis^ omij/is aliis arbitrio interpretuni ^ 

/nterea vero Editor mihi aPnicijfimus Geometria SolidQ^. 

rum comvendim s dr Planm Trigonometriam , ac ^g^ 

gehr^ ^ 



yw^fmAMmptt^ trmeigo ntgefuijtllnhWeiit ex* 
ftfcntitx " . - . 

■^timipfi, elemenia Midorum inwedio itikere con. 
prtfff lUmMn tfanjmfi ^ fatit , ni fdlor , & expeM- 
t4, CTferfstcHA & vero fimul etiam copiofa, Trigono, 
mtru Mitem ffhertt* itligarum adtnodum mutata pU. 
mm adjen^qua »« umm eum e4 veluti eorpus coale- 
fieree» ^t Janie^utriufque eiementa adeo paucis innitm^ 
t»r trineifus. & tttm exyedita methodo , 4c tam conti- 
m,a, er necejfma deduibom funt concinnata, HtirliU 
fi xternm etimn edtnda. ejfent , . nihit fere fit , nuod^mm 
tatm vehm , five ordinem ffe£iem j five demonfiratio. 
nmtextum , Atque e» quidem omnia cum ad Vrhm 
vemffem , redeundum enim erdt identidem , imweffa. 
tnveni , qutlna emnino addendam cenfui appendicem 
quand4m Miqmnto fufiorem ad calcem , quaquadam . 
quaad Geometrtam illam, & Arithmeticam necefraria 
cenjetam t eonttnerentur , In ea demonfirationeL qua^ 
tUerant , fupflentHr fajfm , ac nbernma theorematum 
cmmum elementanum feges coUigitur, indicaturqne , quo 
tnr^ne, qna ratione ex iit filisi^ Geometrit propofitio^ 
mbtuvel lifottus , ( »4«» Hna ad frefortiones fecunde 
jtrubmetica cafite uhrius pertralfatas pertinent ) , 
guidqutdjd Elementarem Geometriam requiritur, dedu. 
ctndtim fit, acflura innuumur^blemata Tironi exer^ 
cendo aftiffima. \\ •' 

In hac apfendive cmintmuY' ea , qu* tneis igoquidet/t 
lyronitus ,vtva voce infinuate confueveram, velinqui. 
bus eofdem exercebam , qua fane ad Geometriam addi^ 
ieendam cum fruElu fumma arhitror utilitatit , Obrul.' 
turfUrumque Tyronis animus rirum tUfparatdruM mul* 
tttudine, dum ea omnia, qu* ad elemema pertinere pof- 
iunt, umco veluthiatufercutrity ac licet fingula feriuatn 
fanlt amfiat , rerum futnmam , ac admirahiiem quem^ 
dam nexiti» non tenet , Hinc utilijfmum fore arbitratut 
fum,Ji ad fraeifua qutdam cafita tota h*e tam ampta 
matenes redigeretur, qua fine aliorum adjumento fufli. ' 
nerentfefe , ex quibusautem, ut ex frimariis quibufdai» 
fontibus, catera omniafaeile dedwerentur . Ubi it^Ty. 
n ferffcxerit , & totiut adificii tfuodddm velMjJ^^ _ 

^ 3 hiyui 



ti P R ^ F A T I O, 

^ihs cMfaStum fnicim^mum habHcrif^ tim reliqiU i^4 

fkm longe majore frk^ii^ ^dji^icf ^ in quibus dcdftcendit 

Ji vircs primum fu4s exfcri^tHr , tumuhi impAres fen*' 

fcrit , Pv^ceftoris, opem implqxct , ne iUc c^ ad inven^ 

tioncmy ne(;cjfmam fane ^fed rar^tm ad^d^m , viam, 

fihi Iternet cxpe^tijfimof^ , IUud enii^, omninci, niihiper" 

fkafnm cJH y idcirca t^m p4^cos prQdire Qcojnctras , qui 

novOf inv^nirc pojftnt , vcl propaJitorHm theorcmatum de^ 

monfirationcs, ftpplere , licet tammulti Qcomctricis, fiu^. 

diis, opcram ^avcnt^ & m^l^i ijH^m M ali^rum inver^. 

ta percipicnda dcvcni^nt; ; (jimd, uhi primum fe ad, Ge^r. 

mctri^m addifccndam apj^licueru.nt y expJicatfi omjtiaj ac: 

difcrtc ded^Sl^ repcrerinf^ nkllo mt ifiventiq^i^ atttdf^ 

dufHoni reliilq locq, , quo acncrctur indufiria , ^- exercirt 

tatio mcntcm cxcolcrct , (^crum 4d eam hujus difcipHna^, 

rationem du£iore cfl opus, e^crcitatp , qui noperit ejufmor 

di infinkarc notitias > quas ad invcntionem pro Tyronij^ 

C^ptu fytii fore ccnfkcrit i qu,a Ji^ adhucipfum ^ quo tcn^. 

dit^ ncq^^iH^ P^^df*^.erint\ it^ipji rcliquafaul^^imad-i, 

^ae y ut fempcr itid^m relinquat aliquid > quod djfjmumi 

per fe ipfe i^fcr^t » quo^ nimirHrn ili^. tam(^m inpent% 

fuo^ gratulakitur Jihiy dn ingontem inde voiuftatcm pcr-r 

cipiett £P hac quidem, dc appendicc illa 9 4eqHain,ip^, 

fd prim^ lihrifrontCy ac editoris pr^atione^ qu^ nimi-^ 

rumimpre^4]amfi(ranti fMjA ^*» i^idem i^JJ^fk^. </f 
^entiq^ 

* f)^um hac ederentur jilgcbr^ Etcment^ expofcchantur ^ 
£4 ex Urhc iterum digteffo cqf^ffcrihenda fuerune fartint^ 
in itiiniirc^ \ p4r(im Jirimini , uhi diutius oh plures qh-- 
fcpvation^s ihidem infiitutas fum cmmqrdtusyunde ipfi^ 
q^tmJtntA « HP effiuih^nt q qalmA j ijA RomAm tranfmit* 
tchantur cdqnda , ir^ ^uiku^ 04 qm»i4 ?«4 44 aquatiqpumi 
froprietA^ct^ generales pertinent , ac ad tcrtium , CT quar^ 
fHm gr^du/n in primM j. €«« ^d variahilej, fojrmularnm 
yalore^ ^ ad idrunfUm increment^ ^ dir dccrcmcma ^ ad 
msiximqrujm ^ 4C minimorusn detcrminatjoncm fpe^ 
{lant , aliqu^ntOL acckratiujf: , dr fij^u^ fum perfccHtus , ac^ 
imagirta,riarum quamiiaj:unk ^f^^m in radici^us, aquation^ 
numgradus tertii , ex ^ad^m ^nica formula erucndijt 
wotuli ncc inutiUmt ut arktrqr^^ ncc indcgdntcm. ^t 



P R iE F A T I O. vfi 

f «94 ind illum s quem aigorithmum vocant % fivt ad prsi 
fifuds cmpatsndi rstiancs ftrtineti compindiofiore me^ 
thodo y infiitHtionum mtre » ^««e msximt necejfaria vi^ 
debdntwr^ innui tantummadoy 4C demonflntvi ^ exemptie 
uHiue 4idjeSKs, fed ddmodnm pAucis, plura Pr^ceptoris 
drkitfio relinqHensy qui ea pra Tyronis capttt fuggerat^ 
& fM i^ppertHna videaneur Oid Hkeritrem rerHm intelli'' 
gentiam t fippltat viva voce • 

Ea omnia jam prodierant fine meo nomine 9 cum de^» 
mum oifervationitus omnibus confe£Hs Romam regrejfus^ 
ac meo Mathefeos tradenda mnneri refiitutus , ad Co» 
nic4fHm SeSiionum elementa perficienda animum ap* 
fHeare caaftms fum , & i^fornm editionem matnrare . Ita 
autem applicui^ ut veteribus illis lahorihus omnibMS pra>* 
termijfis navam rurfum ratianem inierim^ ({r ab ii/^^mm 
nino diverfam veritatum feriem adornarim . /d autem 
aliqu4nto plus atii naStus in hoc mihi opere prafiandum 
in primis dnxt , ut fingula quam dilucide fieri poffet , ex* 
foneremy mhil non acuratijpme demonflrarem per fini^ 
tam Ge^metriam^ quam unam htc mihi adhibendamconm 
fiitidj its^ Ht quod ad ^lgebra ufam in Conicis pertU 
neret , eo referzHirem , ubi dt ipfius Algebra applica-» 
catiane ad Geametri^ agendum erit • Nexum autem 
qHemd^m in primis j & d^dnSHonis ordinem ita rerum 
ffotura canfentaneum perfecutus fum i ^t inde manifefla 
0fpartre pajfet,^ ipfa Geometria duce ex ajfumpta defini* 
tiane ad prcfrieta^es omnes neceffario deveniri » qua la* 
tw nequeant inquirantem , licet earum amnino ignarus 
4d hnnc ardinem cantemplationis acced^t. Atque idqui* 
dem itd' me affecutnm effe arbitrar , ut quicumque fatit 
in Qeametria ^eritus ad htc elementa percurrenda ani^ 
ntum ^licHent^ per fe ipfe fine datjore ulta & theore- 
mmum demanftratianes amnes admodum facile ajfequipof* 
jfit i& ordinem ipfum , ac nexum perfpicere ^ cujus vi 
per ftfe iterum eodtm ingrejfus caUe eadem pojjit vetpra* 
hiematafiH falvere ^ vel de^wtjharetheoiremdta^ &eam-^ 
idem pracipuarHm veritatum feriem cantaxere • Eam ab 
caufam iUHm ipfum ardinem , quem in ea fchefdiafmafa 
fropofueramy immutaviplurimum^&quodibi ex ipfa dt» 
finieiatnt theorem^ deduxerAm primum ^ hsc Md jixtam 

A 4 



ftr.ofojitionem reje^Hm ejl^ ut generalis cijafddni conjim^^ 
Hionis fruU^us pracipttHs quidem , fed qui alios ame fe 
flurimos , ex eadem itidem profluentes , haberet , qui 
frdtermittendiy ac differendi non ejfent . Ordinen\^ au* 
$em hunc ipfum novum ^ quem reJiquis prafireffdum een^ 
fui,^.prdeeipua j qua congeffi , quds fcholiis interjeStis 
adjeH y qua in fufiore differtatione ad calcem adiita' 
fertraiia^i , hic quam breviffime fieri poterit , perflrin^^ 
gam, 

In primis' curvas hafce conjiderandas mihi duxi non^ 
in coTki ipfoy a quo nomen hahent^ cum folidorum conjt^ 
deratiomulto complicatior fie , & mulio majorem vim\ 
imaginationis requirat^ fedin plano pofitai^ quod ^ 
aiii prafiiterunt fane multi . Definita autem earumfor'»' 
ma 9 & proprietatiius plurimis in ipfo plano deduEtiir^ 
tum demum ad Coifi^ Cylindri^ Conadum feiHonesgr^t^ 
dumfcciy qua itA multo expeditiores evadunt. 

Eam igieur Se£Honis Conica perimetrum appellavi{m^ 
hil enim rcfert ^fi interea quamcumque nominis, adar*' 
hitrium ajjumfti , definitienem ufurpes , undecumque id 
Ttomen finxerit 9 ac nominis ipfius derivationem alio re^ 
ferves ) , in quapun£li cujufvis difiantia a dato quodanf 
fundpy quod Focus dicitur , ad difiantiam a data qua^ 
dam reltaj quam Dire6):riccm e^ellavi 9 fit in ratione. 
data^ quant efiet minoris inaqualitatis y aqualitatii^ vel 
maj^risinaqualitatisj adEllipfim^ Paraholatn^ velHy^ 
ferbolani perimeter pertineret ^ ubi illud accidit fatis ad 
rem oppofiti, utdef€^us\^aqualitas& exceffusqui iisipfis 
curvis Graco vocabulo id nomen jam olim dederant iH 
tommuni methodoex Unge alia proprietaufetitum^ in hac 
mea tx ipfa definitiwe pend$rint\ — 

Mira fane atque incredihilis efl ejus definitionisuher^ 

fas , atque facunditas , qua , ut in adjeQa dijfertatione 

demonfiravi num. 766 9 cmnis hac traSatio ad unicum 

/ frobiema reducitUr , quo ex datis foco , direQriee , rs^ 

tione illa data , dat^ reEla concurfus cum perimetroin^ 

quiritur ; cujus problematis folutio rite ad cafus omnes 

applicata , vel immediate per fefe ^ vel ex iis , qua inde 

frimo deduEta funt^ omnss exhihet hartm curvarum pro" 

triet4tes , ^iw/ ki $. fropofitiones redegi *) Priores tres 

- ■ ' -' tria 



ffiafroH^^^ comimnt ( num. 34, 1^8, 140) &deS'^ 
jfiniHmconsHrfumjferimetn eum reSa qusvij direElrick 
f^dUU^ cum trMfcHntt ptr focum , cnm hAhente dire^ 
Siomm qHamcHmque , cujus tenii frohlematis conftru^ 
Sio gentralis fatis elegans , & fecHndijfima , tum i» 
frioribus binis cafibus fallerei , bina iUa e^gie fra^\ 
minerc fingularia froblemata , minus iila qtddefifrfacm'^ 
da^ ae ad naturam ^ & varias^ triun^ turvarmn formas 
determinandas , fifiendas anima , & verd etiam deline*' 
andasj atque oculis frofonendas afti/fma • Reliqua funt 
theoremata e tertio froblemate derivata . Quarta frofth 
fitio {tnum. 181 ) focorum frofYietatem e^rt , qua iis 
nmendedit^ ab ^qualitate angulorttm cum tangente pe* 
iittmy quinta(num.2o6) diametros exhibet fecantes ii^ 
fariamorcUnatas fuas. Sexta (num. 299) enHnciat\theo^ 
rcma iilud generale confiantis reHangulotum rationis *• 
de qua mentio fuferius injeEla efi . Et ea qnidem tria 
thecremasa ah iffo tertio froblemate fingulafer fefe im^ 
meiiate deducuntur . Exeorum autem fojiremo foti/fimHm 
alia bina froveniunt . Seftima nimirum frofofitio com^ 
fleSitttr (num. JJi* ) relationem qnadrati femiordinats 
cujufvis diametri ad reStangulum fib abfcijfis , vel ad 
Mbjcifiam unicam in Parkbola i & latus reElum . OHava 
( num» 3^. ) fr^ortionem quandam armonicam reSa e 
binarum tangentium concurfu duSla^ & occurrentis fcri^ 
metro his , ac reSdt contaSus jungenti femel ; cuJHs qni* 
dem tkeorematis mira efi fane j atque incredibilis^ fecun^ 
ditas, Ex feftima frofofiti$ne nona deducitur ( num»49$*) 
quit itidem ex illa fexta deduci immediate fojfet , & il* 
lam exfomt quadrati femiordinata relationem ad latus 
reSlum , & abfcijfam , qudt afud f^eteres Elliffeos > Pa^ 
raboUt ^ Uyferbola nomen dedit ; qua quidem frofofitio 
tdamftravit ad demonftranda accuratifijme ferfinitam 
Cemeiriamj quecumque ad circHlosConicarum Seitionum 
ofculasores fertinent , quos fluribus corollariis diligentif^ 
Jime fum ferfecutus. 

Et' quidem frtfofitionibus omnibus coroUaria adjeUa 
funt flurima , ^ufbus Jingulis fafe ir^ens theorematum nu^ 
merusy & ex frofofitionibus iffir % & ex fe mutuo con^ 
fertim frormfentiHm continetur • Uabet d^nitio. iffa» 

* Coroz ^ 



f PR 4 P ATI O. 

CorolUria 5 9 propofitio prima ^ fHd^& cum nsvis dejim 
flitionibus conjkfjfla alia zo: fecnnda propofitio tantum^ 
mod0 29 mult^ enim ex iisy qua frima exMbuity fx e^ 
itiiem deduci pojfent i tertia 9« nam reliqua omniaytiu^ 
eonffquutntur ad finem ufque pra ffus c^rollariis hakeri 
foffunts primum autem ejufdem C(frollarium t4m multafi^ 
mul theoremata centinet <v diverfis cafuum eonditiom^ 
tks derixmt^ pariter ( num. 49) ut fili enunciationi wx: 
infegra pagtna fuffecent ; quartaj: quinta 22, quibutin* 
frimisea omniacontinentuTy qua M. ffyperbolarum afym- 
ftotos (piSfant: fexta i^i feptima 12: oSfava reliquis fie^ 
cundior 28 .* nana den^unk |o ^4 circulos ofcul^fores potijp' 
Jimum pertinentia. 

Corollarii immixta funt fcholia fane multa funt enin^ 
num^o ^% qtdius vMi qu^ maxime nefatu erant dignk^ 
ddnotarentury vttl qua cum earumdem curvarum proprie* 
tatibus copulat^ ad Geometriam generatiter pertinerent^ 
fertra^^r^ntury vel quibus ordo deduEHonis indicaretur » 
fuad intanta facundhate^ in t^ntoveritatum nexu necef^ 
farinm mninoextitit^, Jllud enim res ita ar^^ inter fe^ 
cepulatas % &pendentes 4fe invicem litteris C4nfignatur% ^r- 
tidit firquam incommwkum^ quod^ licet diutius meditOt^ 
tus ^ fkrraginem omnem totvcritatum, & deduElionum 
unico etiam demum intuitu compleEhatur^ & quacumqu^ 
velitfihi 4nima feorfum fi/fat ; non nifi fingula enun'^ 
Hare poffit , atque confcribere , dum alia deducit ipfa 
itemm aque facunda^aliis interea omnibut pratermiffis% 
ad qua regrteU debeat memotry & ad omnes derivatianest 
delatus iterumy ramos jam peragratos emittere int^Slor 
^ggredi^ ac nullo fnrcula nutla pretermiJfafronde,,]pari 
circuit pertu/irare ^ 

Finge tibi denfis confertum frondibus arbu/fum ; t&t 
rmttot exurgentes, eitrunco, tot minores eramis ramufcu* 
iosf e ramufcutis jurculos prorumpentes , e Jurculis fron^ 
des t e ffondibus fiores, & fotia. Haec tum rnnnia unica 
intuitu^ Contemplaris , quot e quibus prorumpant j| vides , 
^od libuerit fi'ium , quemcumque fiorem animo elegeris , 
$ i$nta acie | ^dmota m^nu per aerem^ carfis, nuttora^ 
mo^ nutto fkrcutio attatlo . At fi formicam quampiamdo^ 
ure debeas^ qua iter difponendum ratione pt^ ut adfin^ 

guld. 



^Afili^t dd finguloi fiqres , nuLlo demum pf^tirf^ijfi ^ 
fjerHn^4t ^ quMta tibi arte §pHs erit , nt qf^iJL omittns « 
ti^ qtufidirn itexHM l^ore^ itti^a^ formiam ttum red^AS \ 
^ftt/fitndHm fer ttun^am : ubi frimum, derivantur rsh 
m^ nojt^ndus: dHigtnter- eorum mmtrns , 4f f^teris i W« 
reA frdttrmijps r arri^ifndf^ mim^ t^r ^Htnk sffenddu,^ 
foji faueoji ireffuj^ plwrei p^^uruntur rnmHfcuUi uniicusjte^^ 
tHm feligendus ^ fefofitis. (^teris : idem in fi^culorum ^ 
idem infrondiuiny idem, in folior^i & fUrumy etuptu^» 
nc multiflici frafi^n4um femfiery don^tiin s^icumfiorem,^ 
vel foliujsi , exig^uum Hlim vistorem tnum inveJKtrk • /n^ 
dt M froximi^ bivinm > vel^ etiam trivium redenndfim « 
&d4aliafi^index at^f^^aliay donecfroniet^tafersgratoi 
defcendas ^ iterum ad frond^um. ipfarum derivationemL , 
(um. ad divifiojtes; Jln^MUs, furculorum onmium , ramufetk!^ 
lorwmy ramMftfn^ initere p^iefiijpmo fane , ^ awkbiguig^ 
Wit idfiquf plenij^ma^ 

H^c ^ffi4tm imago, qu^dam efl: ineujidi laboris % rei 
adiwbrahda utgumque far , fatis exf^endds omnino iinn 
far ^ ^eque enim ibi , ubi fe fyrculi^^ frondcf%ue dfvifip^, 
rint^ iterum, coeunty nqivo. ambigHitofM f»nte ^ &- erroria 
fericulo ; quod iyfnm Ji forte aHquand^ accidat^ » foperii 
fane sd f^remum fiottm afcend^re unic^ , & eontinuoi 
via » licet ad novam fluxiwn furculorum coniunlli^neH 
deveneris unicofetagrato,^ reliquisadhHcintaHii^ A,tlne% 
ubi ( dcfydtiorie CQ^firuilionej; quafdam, ermtris » ex iia 
tl^eoretnata d^m^njir^rii flura % qua^^jutrfim facunda ^ 

fere femfer ab novamquamdamveritatemeducendamx W 
illis, veluti ramis , & furculis flura Jimul necejfaaniafunt^ 
C<r quibus ea ita fendet , ut nifi omnia ferlufiraris > C^ 
mente adhu^ retineaSi illo veluti fiore fotiri om$tin^noj% 

Sn igftMr incredibilem fanet difficulta^m % quam 
t^o^fckoliis idtntidem, interjeSliSf moJUre f^ltem conam, 
fftt.fm^ quorum efe quid omitt^m intere^i unde recef^ 
f^ri^x q^. re^rediar, Tyro^nemad^^cineo^ /IM enim.min 
Ihi^ in hifce elementis, qojncimAndii frofofui^ ut dfidffltiom 
ffi^i fateret ordoyd^Qe^metrid^ mira indoJes%atqHe arStijp-f 
mus oMniHm veritatum nejfus franjfieeretur utcumque ; 
Wjn tum, demus^ is fatete omnino fojfet ^ cum ^Jiis^ atque^ 

^ alUi 



mliis dtjimtionihH} aJf-ATKptis ^ alio\ atque alw ordiHe i 
veritAtes e^detn dedt^cerentur^ qtdod irtnumeris fane , CT 
a Je invicevi in inimenftim difcre^atttibus rationibus fne'*^ 
fiari foffet. /llud in mentem venerat , ut hujus mea me-^ 
thodi quamdam , quemadmodum in familiarum derivatio" 
tiibut fierifolety arboretH dejignarem^ inqua ttuncumte-* 
tteret definitio ^ tria prima problemata ternos ramos i 
theoremata reliqkis propojitionibus y corolUriis ^ fcholiij 
edntenta abirent in ramufculos^ furculos^ frondes , ac fo'^ 
Ua ita, ut a quovis theoremate cUrvdi quadam line<jt ad 
definitionem ufque traducerentur per illa omnia theorema^' 
ta numeris paragraphorurh^ i quibus continentur , deligna^ 
ta » quibus ad ejus demonfirationefn efi opus 5 ubi et-^ 
iamfignis quibufdam denotari p&teraty qu<t omnibus tribuj^ 
communia ejfent Conicis SeElionibus^ qua adfingulasy vel 
hinas pertinerem . Verum arbor ejufmodi ita excrefcit^! 
ut tanta amplitudo exigui voluminis mole continerinonpof^ 
fit. Eam parieti affigendam facile fibi quifque effortHare 
foterit i fi velity regreffu e Jingnlis theorematis faEloy ufi\^ 
que ad definitionem ipfam ^ ac adnotatis diiigenteir iis^^^ 
qua ad abfolutam ejtis demonfirationem ajfumuntur jamde'* 
vtonflratai 

Ethdtc quidem ad ea fcholia pertinent^ quibusdeduEHom/ 
ftries identidem denotatur . In reliquis continentur fane^ 
Multa adnotatu dignijfma. jiliud { num. 18. ) proportionis' 
armonicdt proprietates perfequitur \ kliud ( num, 1 1 1 . ) figu»' 
tarum Jimilitudinem contemplatur ^quarum complementum[ 
quoddam efi detenninatio fatis etegans punifi communis^ 
homologi, quod nifi in infinitum recedat in binis quibuf- 
quefiguris h^betur femper , & nnicum , ac re^arum homo^'' 
Ugdrum etmmuniimy quarum in inverfa Jimilitudine fem-f^ 
ftr habetur unica per id punStum traduSa y in direltoi^ 
vero vel omnes ejufmodi.funty vel nulla ; & ea quiden% 
in adjeSla differtatione habetur a num, ^zS.aliud {nuirH^ 
13, toi} Cbnicarum SeStionum trahsformationem in re-'] 
Ras^ in eitculum^ infe invicemperftquitur: aliud {num^ 
102, 280, j88, 43?, 44.2) ipfarumconfiruEliones multi'-l 
pliceSj ae deterthinationes exponit : aliud [ num. 280^ 
curvaturas determinat y & plt&es tangentium , ac fecan^ 
fpnm prt^rijftates fro divfrfa pofittone jfuniti , per quod 

' ^ ■ '" ' 4 f4J5r* -- 



P R iE F A T I Oi xiU 

ducMur, definit. uiiind {nnm.zyo) docet inventionem 
binarHm mediarum cominue frofortioHMlium inter hin^i- 
re£lar datas » & arcuj circularis » fii$e anguli irifcStion 
nem^ qnam omnino haberi non foffc per Eudideam Gioif 
fnctriam fatis ibi quidem accurate , ni fallor , evinco i 
& ifjius refugnatia fontem aperio : aliud ( »»». 337 ) 
h»ze alium ordinem exhibet^ quo elemeuta hac ipfadii^, 
geri potuiffent : aliud ( nHm. tM; ) fimilium EUipfium i 
& Hyperbolarum 9 ac aq^ualium^^arabolarMn proprietatem 
tvolviti qua aliiC refpeilu aliaru^funganturvicibusafymi» 
ftotorum: aliud{num.^i6 ) vatias circuli Sefliancm Co-» 
nicam contingentis mutationes confiderat , donec demum 
is in ofculat^rem definat. Accedit Itis^ ut alia brcviors 
fcholia pr^ttqrmittam , unicum generalciiomctricum lom^ 
ma ( num., 204 ) de tribus re£Hs ad jfunUum quoddam 
convergentibusj & parallelas rcllasintercipientibusy quod 
mihijumma fluribus itt locis cxtitit utUitatis . 

Hifce omnibus abfolutis , qua pertincnt ad harum cur^ 
varum confiderationem in plano > ad folidorum fciHones 
Zrdidum fcci. Definitis Cono (num. 546)9 Cylinaro{num^ 
550), Comide (num. 516), fe£Honum formas evolvi co^ 
rolUfiis quibusdam , ac fcholia fuo loco difpofui^ quibua 
mira in frimis trasformationum gcomctricarum indoles 
cominetur, Ibi autcm notatuomnino digniffimafunt% qua 
cccurrunt (num. 653^ in folido genito convcrfione Hyper* 
hla circa axcm con]ugatumy in quo (num.666) quadam 
etiam punSi cujufdam adefi veluti difcijpo , & crurum 
permutatioy pofi recejfum Hyperboldt in rcStas lineas, mi^ 
ra fanc y & ad continuitatis legem illufirandam aptif- 
fim4 1 f^erum > quod ad ejufmodi transformationes perti^ 
net , in^ adjcita dijfcrtationo multo cfl ubcrius pertra* 
ilatum. . 

Lijfertatio autem iffa aliquanto longior , quam initio 
arhitrarery evafit \ at td in Geometridt arcana intimio* 
ra irrumpere meditanti facem^ prafcret^ & viam fterne^ 
mirum in modum^ Aiulta autemcontinet^ qudt licctfcitn 
fanc dignijp^^a^ cgo quidem nufquam alibi cffendi^ mul-' 
ta , qua licet alibi etiam occurrant fdCpe » nufquam ego 
quidem ad certo^ rcfcri redaita canones , & geometric^ 
methodo ^crtraStata . Ea tamcn, jpro noyis vcjtditare non 



. >j 



kiv I^RiEt Atia. 

Mhdeo ; cum mthi quidem infeitidt me€ culfa , nov4 ej^ 
fe Pojftnti licet fortAJfe Jint ^ud Litterariam Remff. ve* 
tufti^tma. ^ ^ , . . . 

DiffertatU tpfa de Locorum Geometntorum , irdnsfor^ 
inationiiui aiit i UH nimirum frohlema quodpiam gent' 
raliter folveris \ mutdta nonriihil ddtorum difpojitione ; 
fUrumut ipfa toH/fru8io mutari plUrimtim debet ^ ^Ma^ 
daM fmima in dijferintids aheutit ^ qudidam riStarum ^ 
& dngulorum direchidries rnutdntur^ quidam terrninievd* 
dunt impojftbiles ^ quiddm ih infinitumexcrtfcUnt itd^ ut 
interfelfio , qu^e dd prohlemaHs folutidhen$ hecijfarid e* 
fat^ nufiuam Jit^ ut uhi hiha reEldC convergeniei dheptni 
in paraltela^y quidam circuli j abeiknti centro in ihfini* 
tum^ ktuiantur in reSias liheds ; dcalid t]ufniodi acci^ 
dunt fane multd ; in iis dutefn Cbnfiantifftmas quafdam 
legts obfervat Geometriai qua nihil ufquam bpiratur per 
faltum. Sed in ejufmbdi Coniint^iiatt fervanda occurtunt 
ftepe iuidam progrejfus in infiniiumi &4uidam ii^dhfiiki 
fer infihitkfn^ qui fecum trahuntqudtdam i qu4S haudfudi 
an alia meliui nothine dppelldti pejfint i quam, myfteria» 
fum quorunddm injiniti $ qud idfneh^ tb €^crefcuntijttj,tk 
vefa demum abfurda videantuirrtcidh^t\ 

tioc argumentum in ea mihi dijfertatiohe evolvenduni 
tdnliituiy qub futceffu i videhiti qui legerit: nihil auteni 
ufpiami pfater eommunia Gtomettic^ i &mea Conicd^ 
rum Seuionum elefnenta , re^uititkr ad dlffolutdff^ omnim 
um intellifintiam i Vrimd quidefn hegaiivas qudhiiidiei 
in (jeometria confideirandai efte^ ui in jilgebra i geome^ 
iricd mithodo oflendtf i &uhidire£Ho^udntitatummutaiuf4 
mutdtionum numerumparem in iuahtitatihui dettn^inanti* 
tus^ evincoi ttliniuere dire£Hortem ikdhiiiatis deiermi-i 
ftatdt i impdrem vero humerum eandem mutare \ Unde mi^ 
hi imaginoHdi iuttiue ikdntitaies ptofiuknt ih lateribus 
quddfatorum , qua in negdtii/a migrarint . Eorum ver4 
^mnium plura . exempld profero i fimplici Geometrid ad- 
modum mdnifeftai 

£x thecrematis defnSnftrdtis deduco d num . 692 for^ 
inam curvarum omniumi qu^e ad fublimiores Tdraholds i 
vel tiyperbdas inter afymptms reduCuntur , in quibus 
irdinstta efl iu aliqua rkti^he raiiehali abfcijfie , ^uarttm 

fur* 



I^ R iE F A T I O- x¥ 

turv^im gemetricMn aecuratam confiru3i§nem frofer^ 

fer fuffHdy qud cum erutis ex fojitivortm ^ mgAtivorum^ 

imagindnerum leiibui mirkm in modum conferttiunt 4 

Tum a num. 714 ad contiHidiatis leiem conjiderandan^. 

Iradum facioi quam uH quantitates i fofiiivii irdnfeum 

innegativas, reUtidfe obfervaridemonflfo. Trarifiiuntas^ 

tem e]ufmodi^^ oflendd fieri;, tam per nihilkm i quam fer 

ihfinitum^ ubi in^ens quoddam infiniti myflerim fepr^ 

dii. ReSa nimirum linedi quauttimque ininfiniiumfro^ 

duOa in illis iuibufdam infiniiis difldntiis iffojitit 

t^nneaUur quoddaimdo i & in fe ipfam ftdii^ tamquam^ 

fi ajfei cifculu^ iuiddm infinitui , tM reiidm lineam aquU 

valefe demvnflfoi ac eundem nexum j & ih vfuribus «>- 

\finitis ^ufvafum evineo manifefHjfimum i plurimd exem^ 

pla pfoferens, & pracepta quidam adjiciens , iua pirti* 

neHs ad ejufmoS tranjitus. /lludautem impfifnisdfiendo 

d hum. 729, ubi deveniiur ad nihilum i vel ad idfinim 

tum, aliquando quidem trdhpiu pef eum Umiiem faHo g 

quantitatem abire in heiativdmi diiiuando veroindert^ 

ifedi fitri tse eadtfH parie, cujus ekempld frifero plura^ 

'& indi crkfum feuparaboUH j five hypfboUci generis ^ 

quorum ndtufam doceoi regreffus ex inflhite^ muteiplices r 

ac cufpidum naiuram^ & iuddah^ alia , qud ad tangete^ 

tesi & cwrvaturam pertinent , evfilvo , quee fane omnis 

funt ad continuitdtis legem , & Gemettia inddm f#- 

gnofiendam aptiffima^ 

^^fidii alius quehdam reitdt per modum ctrculi infi* 

mii ift je redeuntis confiderata ufus^ quem contemplor ai 

numi iilfVi cu)fts quantitatemf qua poft negativam\f & 

nnas pofitiva/ J$t iuartd, n^n negativam reverd effede^ 

Wf , demofiflto^ fed veluti ptufiuam infinitdfHj & datis 

bjms puneHj in re£ta ihfinita , e]us fegmentum iij pun^ 

ms interceptum , oftendff ^ effe duplex^ atteffim finitum ^ 

Mternm fer infinitufn traduitm , iucfum ptifnum bifa^ 

tiamfeectur in puntlo quodam datis interjacehte ^ fecun^ 

dtM in irfinito itU( ipfr^ in bini infiniti tra£lus reSletai 

binis illis funthis uiriniue in infifiitum produSla Conne* 

Suntur quodammodo^ & cepuldntutf qua iiddem confide^ 

rasio ingentis efl ufus in SeQioftum Conicafum analogia 

€9nfideranda. Tum^ a num. 775 migrationen^ perfequor d 

ftettfs 



xvi P R i« F A T I Ov 

fiAti* reali ad in/MginariHm > qni mmqHSP» hdtfiri fojfiti 
tiiji qHAanfds vel ad nihilim deveniat , vel ad infifii^ 
tum, & in utr$que cafu bina fnnSla collidantur quodam^ 
mtfdo y ac in fe mutuo irruant velocitate vel infiniticf 
fnofwe^ quam alihi^ vel infinities minore ^ quod analogi^ 
am etiam qhamdam exhihet haud fane inelegantem ejus 
Tnigrationis cum vero viventium interitu. Ibidem auten^ 
in cono feSo per flanum mobile quoddam $ feries curva^ 
rum nafcentesj in fe mutuo transformatas ^ ac in imag^ 
narietatem definentes ofiendo. 

> His expojitisy & tanquam materia quadam novicufuf* 
dam adificii jfraparata , ad ordin^ndam transfnrmatio' 
num theoriam progredior num. 760» quamdufiicis anal&» 
gise definitioney & 11 CanonibtucomfleElor. AnalQgadi- 
coy funEla» qua eadem confiruHione [etitaah interfeSio^ 
ttibus eorumdem Zacorum Geometricorum definiuntur: li- 
neas analogas% qua, funllis analogisy fuperficieSf quet li^ 
ffeisy folida^ quat fuperficiebus analogis terminantur * Bina 
autem diflinguo analogia genera primarium alterum , ubi 
ektiam direSiofervatury altcrum feeundariumy ubi eacon-- 
traria exifiit. Canon frimus num. 764 fertinetad quan^ 
titates , qua frimario atial(^idt genere funt analog^ ^ in 
quibus nulla mutatiofit^ niji quafiamquantitatesfer in/\-' 
nitum traduSla flufquam infinitdt cenfendafintj ubietiam 
infiniti myfleria quadam occurrunt . Secunausnum.yyxad 
eas fertinet , quit fecundario genere analmdt funt analo* 
fay ubi offenditury quando fumnuc in differentias migra- 
re debeant , & modi argumentandi mutentur . Tertiu» 
mm. 777 mutationej direElionis exfonit , qua in quavis 
frofortione utcumque comfopta nonnifi numero fari habc' 
^Kf^Jfi^f ' Qj^^^f^ num. 790 ad angulorum mutaeiones 
fertinetj qua laterum mutationem confequuntur . Qjfintus 
fium. 799 tranfitum continet anguli e fofitivo in nega- 
tivum^ mutata hiatus eUreQione , five ejufmodi mut^ 
tio fiat tranfeundo fer nihilum , five fer duos reStos . 
Sextus num. 807 , quadrati negativi latera determU 
nat imaginaria , & mediam inter . binas quantitates , 
alterius tantum dire£Hone mutata , imaginariant y bi" 
nas autem medias reales magnitudine aquales , direSio* 
ne centraritu , qui qnidem Canon in SeUionibus Coni* 

€ij 



fiend^m: .^ . 

Seftimujf fnm. 8}$ Mi qMntitatcs tranjit^ ^ut la nU 
tSUtm ^beunt y vet ifs in infivitnm , ut faltem dlter li^ 
79fef nkfquam jnm fit i qutd fi in miiqua frefonione H^ 
ms terminis m-4nentilms finitis eontingat nni e reliquisf 
tonHn^et idem ttr sUteri , nifi forte , qui manent ^ veL 
€xtremi fuerint i tftl medii , qu9 cafu abeunte altero in 
T^lHm^ dlter in infinitum ahire debet « OHsvus num^ 
V^^ efi dt relKs , qMt e convergentilfus p^ralteU fiunt ^ 
iiHerfilHone ita in infinitum reeedentei ut nufquamjam 

I fiti qHOram & angulus ex altera parte evanefcit^ ex aU 
teta d/^tdt in binos reQos. Nonus {num. ^^^) prafcri* 
t£t9 qkid agertdumfiti uH vertice trianguli aheunte im 
idtus aliquodj reEtai qua ft prius interfecalytnt ^ fuperM^ 
nunturi Decimus circuiiperifheridfn f doctt [num. 658 )« 
uhite in reRamy ubi altero radii termino extante , cefh* 
trum ita in infinitum recedatj ut nufymtmjam fit i JQe^ 
Hfum undecimus (num. 8^2) raiionem definit ^ quam ha* 
here deheant hina feSa in infinftum excre/cefrtes i qua 
ertmdie tqualitatit ratio , uhi differentia finita maneat i 
uSi mstm ea etiam in infnitum excrefcat , quavis effe 
foufi^ nulta^ infinita^ aqualitatis, vel finita inaquatita* 
tis cujustihett 

Porro finguli Canones demonfirantifr accUrafe : fingulo* 

funhexempla ex iis « qua pr^emfffk fuerant prcfiiruntur ^ 

fiugula ad Conicarum SeRionum naturam^ & analegiam 

tontemplAndam applicafftur y ac ecrum ufus in hifce meis 

earunaem elementis concinnandis ofienditur . Plura fa* 

ne aceurrunt adnotatu digtta t ut ea ^ quihsitum. 784^ 

TAtio redditur ex infiniti myfleriis quihufdam repetita f 

fur enam uhi quantitates per infinitum. tradklta abeunt 

in mgativasy adhuc fiibtrahenda fint , atque atia ejufnr^* 

JH fane multa i iltud in primis non omittendum > qued 

^lutiius in locis oftenditur , potijpmum vero i ubi fecnn" 

dkria anatogia exponitur^ & uH feecundijftmus ille fex^^ 

tus Canon nd Conicas Sefiionei applicatur . Nimirum 

ubi Etlipfis in Hyperbolam tranfit pef Parabolam , axi 

'fiftit^ EHpfeos f & eentro non Jkecedit analogus prima^ 
fio Anaioita genere axis finitus hyperhola , fed axis 

\ ' Bofc^Uh^ tomJlt. B ier 



\ 



trnm bttjm Jinitfitn, fedj^nnElHm ^uodd^m i» infinU 

tQ delitefcens ^ Di^metri dntcm fecmdan^ ^fH^^ 

/< nulh an4l(^l4 giner^ 4^/0^ xjknt tiiametrk ^itif* 

feos , feA hormt quadrata negaiive fumfta ^uaatratis il* 

Uru0 negadvis aiaaift^ ^ qHprim qimdrata idcirco /r» 

iurid^rio analogi^ genere funt analpga HUrum iHaAra^ 

fls, latera veroy qu^^ lateri^us analoga ejfem y imagina- 

ria fune^ /d i^fum manifejta ibidem evincitur. Mede ^is- 

tem deducitur , qu^ frofrietates communes effe debeam 

MUip/i\ & fiy$erhU , qua dk altera ad alteram tranf 

ferri nequeam. Inde nimivum fatet ^ cur EUiffis finite^ 

centro (avitattm > ^Myperbpljt Qonvexitatem ih/ertnt ^ 

cur axis tranfverfus » ©" ^M%ds conju^ata diameter in 

BUiffi ad ^erimetrum terminetur\.a^is coMjugatHS ffjf* 

ferbola finitus iUe , 0* Jecundaria diametri omnes /p- 

Ji ^erimetra nufquam occurrantr cur afymptotis , & tam 

multis eUgantijfmis afymptotorum fro^rietAtibus BlUpJis 

carere debeat \ 4^ alia ejufmodi evolvuntur fane mulp4k 

earum curvarum difcrimina » atque illudgeneraUter ofien' 

ditur > projrietates , qua a foUs^ diametris can^ugatis t€»» 

deant t nufquam effe dehere communes^ nee communi de* 

monfiratiane i & canfiruElione erui yojfe ; quacumque ^iiH 

4em ab earum quadratis pendeant y ea communia fare «• 

mnia^ fi quadrata diametrarum fecundariarum lAperba* 

idt habeantur fra negativis • Exempla eorum froferHntw 

jfltwimay qudc ad harum curvarum uaturam cogn^cend^m 

& meorum jtUn^ntorum commendatianem flurimum cann 

Jjtrunt • ; ^\ 

Parro ubi in fine fqftremi Cananis de rationibus agi* 
tur quantitatunfJabeuntium in infinitum , ibi \am 4em 
mum incifinfit iffa infiniti mjfierut' migrarein ahfur^ 
d^ y de quibusa num. 878 aa finem ufque ita agirur ^ 
$it infinitum iffum extenfujm fra imfaJfibiU haberi #- 
mnina debere videatur . Jlatia autem imfaJfibilUatis tfm 
fius e^ iffa Canicarum SeElianum natura demum ertU^ 
fur , qua e]ufm0di invenitur , ut infiniti iffius n^m^ 
r4 fimfiicitatem injinitam requirat , qua cum infinU' 
tit fartibus ab amni quamitatitm excrefcentium gene^ 
rt rtif^ifits mwgi mnin^ n^n fat^ i unde demum 



PKATA/rtO. ^ jSx 

dd. ^dm Wm Dei O . M\ immunem Ab mni ann: 

f9fiti4ftg fimflicitAtem immenf^tn cum inpnitate cofH 

' ptnihim C^ntemfUndam trdducimur , in quA ipfa con^ 

'timft^ii^^e fupor h^c dijferratia tandem, aliiuando a- 

' hrwnfitur. 

Hsc univerfi hu]us oferis efi Sjnoffis qujcdam , im 

' qna frditermijps quamflurimis , fracifua tantummoda 

cagita innuuntuT . ConfeqHttur aliud ^d^ens de infinitis^ 

etr infinita farvis , qu^e mihi indefinita funt , quo^ 

rum naturam exflicabo , ardines diggeram , element^ 

trddam geometrico rigore demonflrata , & ex iis ad cur^ 

varum generales frofrietates gradnm faciam , cufpides , 

' Jlexus contrarios crura infinita , contaElus , ofcula , evo^ 

lutas , maximorum ^ & minimorum thcoriam y atqne 

atia ejufmodi evotvam , ac fingulares fracifuarum , & 

maxime utilium curvarum ffofrietates deducam > ac dc* 

morrftraho^ 

/knd^unttm htc demum monendum efi . Si quis in hoc 

volnmine vei nqn foffit , vel nolit fingula ferfequi y efr 

fracifuas tantummodoy ae maxime neceffarias Settionum 

Conicarum froprietates inqtdrat; is & univerfam differ* 

tationem > & fcholia fere omnia , & plnrima etiam Co^ 

rollaria omittere foterit ftnt demonftrationis y ac de- 

' dkSionis damno • Jta entm fracifua qudtdam inter ft 

^Cifulavi eam iffam ob coHfanry ut reliquis non indige^ 

' rerft • Vix autem faginas loo requirunt fracifua ejuf 

modi frofrietates inter fe fatis arfli connexa . En nu^ 

' mtrornm feriem , quam retiquis omiffis foterit ferfe^ 

^qui , in ,fua , ubi binis numeris funtla interferuntur , 

^iud fignificatUT y intermedios numeros omnes fercurren^ 

dos eft. 

X ... 3, 6 •.. II, 18... 30, 34...47> 54, 56, J7, 

«62... 84 , 87 > 93 > 228 ... i:$7 9 140 ... 144 , 149 ••• 

^S9 » X64 .•. 171 > 173 ...f 183 , 189 ... 19% > 198 ... 

201 » 204 ... 213 > 221 ..• 231 , 242 ... 247 , 256 ... 

25« , 260 , z6l , 299 , 300 , 3of ... 308 f 328 , 331 , 

3n ..• %S1 *3^7>358> 363, 364, 397» 3^8, 402... 

•407 » 411 ••• 4H » 43* ••• 441 » 457 ••• 461 > 495 > 
497 * 503 ... 508 , $46 , J jo . . . J68 , J90 • . . 60% » 

ii5 • . « ^^ • 

♦;< B % Jftut 



' » PK£F ATIO. 

m* canJid€rMt*s vtlin fUx» , m/ » c»t:m,Si Cylimjri-, 
& CtndJnm StOimu tiddtn tibemt > mammt 590 ... 
te5 > 61 ¥ . .. 643, ftftrimku mddst , ^ «i«s fndtmt 
cm^ fict. 



SE? 



SECTlONUKf CONICARpM 

E L E M E N T A,'" '- 
' DE FI Ni rjp' l. , 

X reltam AB indcfinitar» pojitioti^ 
i d/itant, & Aiia' rta^^ PF W^- 
I Hnm f d^tuiri extr* i^fam AB ,; 
I fiteritfim^ex FP ^ PD /« w- 
1 /MOT data ; linea.m iitdfn dica' 
' «, , , "'Sc^^oncm Conicam , Ellipflm , 

I- t^tof^Iara, vtl Hypcrbplam, ^out ilU naio fHerifitti- 
'^_.t»*qiM[iiatis, aqnMitatis , vel majarij ifj^^ualita. 
I '»: n^Mm AU DiK&ritJcm, fufiSfym F j FdoJni^ r*. ' 
««««1 iJiim dofam, Ratidnemdeterminantcnl; reilam 
IT) , Ordlaarafti direarici ad anaulos reaos , reSi^ 
rr> Foci radium . ' ' " "" ■ ■ • .... 

Coroll. j. 

■ m/' ^ ^*^^ "^^" ""^'^" '^" '^^i^tnti^ iireShL 
" VH,fmper rati^ cnjufvis radU fi>ci ad fiam ordi- 
>utm^ fft, angkto Uio dato erit confiafts & data : nimi. 
rm ciT^fita ex ratione Aeterminamt , ^ ratione yf. 
I "'"indinationts adradium.' ', 

I. Nam ratio FP. ad PH cornpornemr ex rationeFP 
M PD; que cft ratio detcrminans, & rationc PD ad 
, ,.^^J^ °^ anguhim PDH re>aam e(t ratio finusah- 
I guU PHD ad radiura (Qum.88. Trig.) 

Bi>fch.Tom.II/. B 3 , Ce~ 



% SfeCTlONUM CONICArUM 

C0T6lL 3. ^ 

iiiUA rmo a^itfvii V^'pSifintlc ■ 
Tum dato pitnBo F 114 fiikm Wi 
''m data rtSla mn tranfettnti per " 
4 trit iiU iiftea ■' S<!iio Vfftrica i 
it 'tott^ianetHr tie iHa rathmeott- ' 
dii ad finifm inclin^ionis . 

5. Daaa mini, ^pcppCTidiflilb :PD ; taiibFP^l^Pl» 

^ompbrietur ex ratioiie FP ad Pf-J ■> & PH \ ad PD , 
quaiTtitR prinia datur ^ h^otcA » -fecilHdz cft ratio ra^ 
dii ad illum fitiuin. 

, -, ^orBU. 3i , _ , 

^. Mini radii foci rrmt ad jfe invicem , ut ftnt ordi~ 
t]M»,ix quof/is d/tifito cg,tftiMVii . ■^ '"^- — :~^ " , 

7. Cum enim. fit FP ad PH,, ut F/ ad fh , iiit .iihcr-. 
riando FP ad Fj>, iic Pt^ adfA. 

Corell.^. , ;, ■ 

■.j.a 8i Si re^d, quavis 'occmrtns .fiico iti F, dtre^rici iti 
Q^ etcMrat Seaiofii Cpnica in vinis punSis P, pj kltera 
ex iis jacente inter ipjfa fmUfi F , Q, altero ad fartes 
KtriHslihef^ ertt 'dtvifa in punlHs p, F ,- P; (X,in pro^ 
fbriianehahioiiic^i . , 

ji. £rit enini Fp ad FP, ur /'Qi adPQ;».Sunt afl- 
tetii in fig. 3. in tritus rcftis Ki.» FQ, » PQ, reftaS 
fO^ PQcxtvein^ re(S? Vero Fi> , FP diffetcftti» ex- 
trcmarum ? .mcdia j at iri figoTa ^.triuRi F^j FQ^ FP 
teft? Fp, FP 'extrcm^, reiaa: pdj PQ^exrfcfhatQmdif-' 
fercniix a medla . Iginir uitobiquc erunt extrenuc ad 
ic inviccm , ut pacitet ad ff: inyiccm di&renttie ex- 
trcmarum a ftlectiai que el^ ipia notio pippoitiiairis' 
harihdnicre. ■ , 

, .CoroU. 5. , V 

16. .ifitr^ r^^zi J^» ^i^ ordinatam itl qnevis /tugida 
vbliqm erit in EUipJi , & ParahoU rutio minoris ina- 
qmlitatis , in Hyperhola minoris inaqmlitatis , aqnali- 
tatlf, V el tnajorij ifjaqkalitatis , prout raditu ad fitium 
inclinatiotiis hdhuerit rationem majorem , a^lualetit > vel 
minerem ratifne dtterminante , quam qnidem inctitto' 
tiwum 



thfhan fjtm 9 qnic rationtm txhihtt ^equaliiatii $ Jidi^ 
VffHs Indinatiotiem sequalitaus , & ijux angultan kcuhM 
am JiinSrm ^yangulufn Ratiol&is asqualis $ /Ivt anii^ 
ium ^ualiutiiSi 

ti. Nam in trianguld reftangulo PDH femper ba* Kt 
fis PH ttiajor eft lacere PD ^ Adeoque ctim PD in £1- 2« 
bpfi fit major , quarn ^F (niij^. i.) > ^c in ParaboU 
aDquaiis^ crit fempe^ in iis PH mafor ) quam PF. Ai 
in Hypcrbola^ iu qUa PD eft minor ^ qiiam PF» ptro^ 
ut ratip PH ad PD ftierit major 5 aequalis , vel mi*^ 
6or lefpedlu tationis PF ^d PD, erit qtioque PH ma» 
)or^ ^Ualis, vel minoir reipe&u PF* 

SCHOLIUM t 

t^. T intf^ hufuflnodi appdlavimus Seltiones Coht^ 
JL> cas 5 quia deinde dcmdnft^abitur » coho uu 
eumque fefto lioti per verticein i obvenii^e hujufmodi 
lineas, ttt pariter Eilipfis 9 l^arabola , St Hyperbola 
liomen accipiutlt Gtxc^ origine in commutii tnctho* 
do tra&andi ^eftidnes Coiiicas , a quodam defcftu » 
xqoalitate y Vcl eJcccfiu 5 ^ui deinde demonftramr • 
C^axumqoe fit tiOmittis r;itlo 3 fnodo femper in ea fi-^ 
gnificatlone accipiatur i^in qua ih dcfitiitione ofurpa*' 
tum eft, nihil intereft * EUipfeos ^utcm defeftus ille ^ 
Par^lar afqualitas, 6c Hypcrbolas redutidatitia in hac 
noftra methodo etiam ex ip{a definitione cofnftit; cum 
tatio determinanS in prima fit miiiotis tna^qualitatis » 
iti leeunda skqualitatis » in Htti^ majoris inacqualita^ 

i;. Potris( mitum fatle » qtiam immediate ex faatf 
propHecaoe^ quam aflfumpfimus prodcfitiitioncj &quam 
mIm^ in £Ui[i(fi faltem 5 & Hyperbdla poftremam fefc 
demotiftrare foleiit ( nam pro Pardbola hanc ipfam af« 
fnmpfit etiani Hofpitaliusj praedpuac Sedtioniim Ceni"* 
carum proprietatcs fiuant > & quidcm , qua: iis cohh 
mcmcs funt y communi femper dcmonftratione eniun^ 
V vinculo qaodam » ac tniro nexu , quo Geometri^i 

B 4 indQ« 



. It SECTIONUM CONIGARUM 

itidoles > Sc vis Tane iacredibilis fponte tncurr an t inocntMt 
T4-Pr£tcrca multoexpedictorharumKnearumcofifido- 
rcitio Tyrpni evadit , fi eas in plano coofiderentiir « 
iquod & ipfe Horpitalius prseftitit, & alii multi» quam 
fi folidoruni Geometria opus iit ) & variis planormn 
-^ in cono interfcifHonibus. 

. 15. Xkmum ba:c definitio ita Conicis Se&ionibus 
jtft propria» ut eas quodammbdo & a- circulo diftii^ 
guat f qui cacteroquin inter fiUipfes ehumerari debet ^, 
& i C6no Sedio, ut infra videbimtis , ipfe etiamprok. 
dit) iive is fecetur fedtione bafi parallela > iSve aK« 
quadam , qua: dicitur fubcontrarta . Si enim EUipfis ia 
circulum abeat , dirc<9xix y ut patebit paullo inferius » 
abit in infinitum, nec ufquam j^m eft. 

16. At fi diret^rix tranfiret per ipfum pundum da* 

tum pro fbco , nuUunt aliud pun<^m invcniri poflet ^^ 

cujus diftantia ab ipfo foco ad perpendiculiHn in dir^ 

dxicem duclutTi baberet rationem datam > ubi ea rada 

^ cft nrinods insqualitatis . Sed fi«%tto eflct aequaUta- 

tis, fatisfacerent quaeftioni'pun£ta omnia re<^ diie&rir 

Ci pcrpetidicularis du^a^ cx ipfo pun^daio ifmtram^ 

vis plagam 'y Sc fi racio effet majoris inasqualitatis ^ 

fatisfacetent punda omaia binarum re&arum lunc in* 

de inclinacarum j ut radius ad i^um incUnationis ef* 

iec in ratione determinante * 

F. J. , 17. Nam fi pund:um datum m dirciaricc AB fit F^ 

quodvis aliud pundutti vel jacet in reda hfH perpea« 

4iculari ipfi AB du(3:a per F , ut R , «c cft FR tam 

dillantia. a pun Ao F , quam perpendiculutn in dirc6fari« 

cem demiflum , adeoque ea duo aequantur ; vel jacec 

extta^ ut CX» & dutto perpendiculo QZ in diredUi-* 

cem, femper erit ipfo major diftantia Qt bafis triaa«* 

guU rc^languli QZF. Qiare nufquam haberi poieftia 

co cafu ratio minoris inasquaUtatis . Ratio autem ae- 

qualitatis babetur in ipfa reda perpeodiculari HFi&, ia 

qua fumptis ubicumque pun<3;is R , & r, eft fempci: 

4ilta^n|ia FR, vel Fr, ad perpendiculum RF , vel rF 

ia fm(m isqualitatis • Ac dcmsm & rsuio fit msgoxis 



E L E M E N T A. 5 

|ii3B(|i}aliutIs fuinpto in pcrpendlculari FH fegmento £F 
ad iM^bicrium» dui^Uque pcr £ reda hEV indefinitapa^ 
rallela direi^tricL , cencro F intervallo reftas » qua: ad 
£F, i!rt m rationcf .daca determinante > inveniantur ia 
ip[:i bina pun&a m> & V» ac ducantur per ea> &per 
F redbe indcfinita: G^, Iz» & quodvis pun<5hxm utri- 
Jbudibct» ut Qj 9> fatisfaciet quxftioni • Erit enim FQ^ 
ad QZ, ut FV ad £F in ratione data , & eadem eft 
densQaftratio pro q. £ft aiitem ratio illa determinans 
FQ^ad QZ, ut radius ad finum indinationis QFZ • 
jQ^c pateat quascumqiie fuerant propoGca, 



S C H O L I U M • II. 



c » 



i2. tN CorojL^. invenimus divifionan harmonicam, 
^r'' X qii£ ia Se^^ionibus Conicispoti/nmum fa;peoc«* 
/:i3Erit, & in Geomeoria elegantiflimas proprietatcs ha* 
hct^ Pcsec^uas quafdajm » . quarum ufus nobis occurreti 
4uc tzpoaeinus. 

>- 19 Si qudtmr funila Ay B, C* D j ita diffojtta ^gy 
:fim , Ht JtifimtU AB, CB^ binorum A, C Mternduim 
fnmftmtm ah Mtero e reliquis B emiemra^i^nan habe^ 
am , ac difiantia eor»ndem AD, CD abaltero D^erunt 
i» frofortiane harmonica tres dijiantia Htriuslihet extre^ 
mi a reliqHis triimsy nimiarHm t4m/^> JHO^ COqHom 
AB, ACs AD. 

20. Primum patct : nam AD , DC crunt primiter* 
oarii cxtremx , & AB9 BC cxtremarum dtfFcrentix a 
taedia . Secundum facile deducitur . Cum nimirmn fic 
AB ad BC , ui AD ad DC i erit & alternando AB 
ad AD, ui BC ad DC . Sunt autem AB , AD extrc- 
wc &cundi ternarii , BC > DC excrcmarum differcn^ 
^ a media AC. 

3x. Pacet autem eadcm dcmonftratione , non poflfc 
projponionem harmontcam terminari ad alterum cxarc- 
jsum D» quin fimul terminetur ad alterum A. 
. zz. Si jam intervaltnm hinorum dtemorum qHornm^ 
vis AC dividatur bifariam in R , ernnt RB, RC i 
RD in ea continna ratione geometrica^ qnam habetfror 
I • forM 



« S£CTI0NUM CONlCiARUM 

partio hatmonica tritM quanmatum termir^atarum g^ 
ixtrmum A ajfumptum $fd hiJfbSiione , nimirum AB^ 
AD, vtl BC ad CD. 

ij. Aifumptts cnim Ri, Kd aqUalibus RB ^ RD i 
crunc 8c Aby kd acquaies CB> CD , adeoqiie cfit ^B 
teftanim AB, BC differenti&, AC earum fumma , ip» 
i% AC, rcaarutn AD> DC diffcrcntia ♦ dD earumfunl* 
ma. Cum igimi: fmt BC ad C5 ♦ & BA ad DA iil 
cadem ratibne, etit in eldem rationc & antecedchtium 
difiereHtia ^B ad eonfcquentium difikrcHtiatii AC , & 
illorura fumma AC ad horum fummam dD\ ( Cap. 2^ 
^ir. num. t%.)zc fumptis dimidii^, eric RB id RC» 
& RC ad RD iti cadem rationc . 

14. Conrra veto fi futrint RB , RC i RD in in^mi^ 
ttua ratione geometrita , & mdia RC ajumatuf aqu4* 
iis RA ad fattes ofpofitai , punEia^, A, B, G> D cen^ ^ 
ftituint binas fropoftiories hatmonicas quantitatum tef^ a 
minatarufn ad D & A^ & ratio illa RB , ad RC « ^ 
vel RC ad RD irit eadtm , ac ratio ftopottioriis ttr- ^ 
minorum terminatorum ad A^ ut facilc patcbic regreflii 
dcmonftracioni^ ipfius. 

25. Datis blnis funltis alternis A y C i & f^ionit 
frofortionis harmonicay habebuntfw facilei &reliiUaduoj 
medium quidem fecando AC in ea fatione in B , r^- 
tremum fecando ACbifariam in K 9 & fumendo Rj> 
tertiam frofortionalem fofiV3i KC 4 Patct autem cX 
ipfa dcmonftratione dcbcre D afliimi ad partes B fe-- 
ipcctu R , quod quidem eo recedet magis a C , quo 
ratio daca accedec magis ad rationem xqiialitatis ^ puti-> 
cco B eo pariter magis accrcdentc adR, quod quidein 
punctum abibit in ipfum R 9 ptmctum vero D ita in 
infininim recedet, ut nufquam jam iit , ubi tatio da- 
ta evafcrit rado arqualitatis. 

26. Quotiefcunque quatuar funSld A, B\ C, D ^ coh^ 
fiituunt frofortionem harmonicam , fe£ta bifariam in R 
diftantia binorum alternatorum AC , erunt geometrice 
frofortionales quatuor diftantia ab extremo D in bife^ 

^m m ^Jfjifnpv^ nimirm AD^ RP ^ i^^ BD ^ 



CD 5 d* '^^^fptor a punElo B r;//J rf/ww 9 tnmirm 4# », 
W RB3 «^ DB 4^^CB. '' -* V - 

a7. Cbm cnirti ♦( hum. %t.ynt mvcrtendo. lil) .ad 
|RC,.fiVe;M R'A in;ilia ritionebC a^tfe, frttp^ip^ , 
jrum fiitnma AD; .aci^^priAabi RD| dt p^ftenoirupi lam-? 
ima DB dil *rtiM DC, Cum vcrofir in veroRndo PCl 
ttd CB i \xi RC^, fire RA iilW; '^rlk hm^cmch^ 
»0 adCB^ut^AB-ad^RB;. . ;.. ;. ,; ^.\ ... ,. 

ytlternkim fttitn^Hfn^ defcribkiur circHtusi (^^ 4<^ W^fix 
hisy^fhefle ^punSiufh E ducknttfr ^k^ retiguis Jfi^ , 
pun^is teSi^ BE, pE,, ^eruni ei ad fi ^i^vicem^- fc^f^er 
Hn i^dem ratfont "BC^id Cp/, Jive BA ad AD ^-^^4, -; 
CE earim affgtitum J^ED fec^bit bifaridm\&' r^Eia 1^% 
iiiniuttiih hl.GV iumaliera BE^ coniinet tum olttrraDE. \ 
^oduSa^ • .. . . r V . , . « 

2'9- rJiiciis cnii?! BF , BG parall^lis .AE.* CP> «? ' ^^ 
bccurrcnubus rectae DE.in F^ & G,^ crit pj? par^k- 
lis 'DE ad ^F; utpA ad A& ^ nir^Imtn pS. pfoppr-» 
tioneni Hdtntoiiicatin lit DG ad CB^ ftvc ob . pai:aUe|a$ 
&c illa iaclem DE.ad EG|. C^4^c/aegual^5^rpnj/(jEi . 
EF, angdlus huterii. GBFv^wem cc^tttinent^rccta? GB^^ , 
BF ajqpatdf angulb,*qucm continent AE|,EQ ip^isps^J 
tallelsc qui recms . eft iri fcmicirtuio • Igitiir ^ i& f c- 
ctus critj &: circulus cbritro E diaractro GF d^fdipius 
hrtthfibitpcrBi idcdquc EBajqualii crit tan^ El^ , guajcri 
EG^ Sl habebit, ut ill2t,.ad EDcam rationcm» quani 
BA ad AD; vel BC a4 CD^ Angiili ycro BEC, FEC 
iaequalcS, illc alternd EBG, Hic infcfnd & oppofito G 
iaequdlibus ad bafim trianguK ifofcelis fiEG xquabuiu 
tiir intcr ft; & cddelm afgufticmd AEB ,' AEG »qua^ r 
les awgulis EBF , EFB ; , r 

'30i Confrd vero Jt reSla CE fecet tfifariam oniMtum 
k4 E' trianiuti BED, & EA iffi ferfendicul^ris occur;^ 
rat diametro inA^ qUatuor pinSid A, B, C, D coffji^ 
tuem jfrbf^tiinetn hdrfnonicamy cujus ratio^in ternmri^ ,. 



ter- 



>• 



g SECHPWiNfGp>IfCARUM 
ietminat0ad D erit e^demi ^tc ratU Isterum fi£> 
%y!W triangtM . Pp^a enim BG parijUna C^£ , ani^ 
EBG> £GB crunc flcquales ^qualibus BEC « p£C > 
dcoqtX^*& 4nferJie^fi<;E(^.> £^ gc' fafta?! 

iacquail ip(if .£G ) .^B ^ WguLus GEtf^> erit rc^bis i ack 
- quc BF, cpngruef c\xnx rc^a Vctftep '-^ f4^aUda-rqi 
ibidcm reftum - w^ul|im cootiacrc dd>Qt V Eric > ig*^ 
\ DA ad AB, >ut DE ad EF ^ fivc ut ipfa DE^ ^ " 
nimirum' ut pp ad )BC /ib hpc cafu e&Amrc^ 
ffcabiV^ifari^nji angujl^ra BEG> & paciier^fi i?«6li 
fecantc bifariafn atjguliim BED, rci^a £A (ecce 
rbm tnguhim BEG .', yu^^a A , B> C, D propor^. 
ilcm l}armonicanix<3^ipftitu^^ V ' ;. ' - -^, 

r.8. [^i.' jQemum fk in.ettdem cireulo dncatitr fer B «^ 
(ia ^H fer](indici4.ms ^diamtvo\ ireQ^ ^tddem DE?# 
DH contingenti circdum.in Ey ^, H ^ ^^^is. atiieM ris 
Ba in earjm angvdo du{la ex D , & occHrrent che/rJk, 
iffi in, X> ct^^gmIo UtW, &^ y^ifecabkuy in, fttnSiis M 5 L 
I, D\ in p[Qf&rtione harmofH(;aJ. * i" 

32. IHrimum pQtQC ex co., quod ( n^. az. ) crirRII! 
ad IlCV'five'^d^E!rEtr«ik.li^^ proiirie ^ 

aQgi^a^RBHjREDob^gd R' cbromune fittA 

lia crmt^^/& anguluji j^£D cc^Q RBE «quaiis : ade<f^ 
que £D' pe^pcndic^atis radib £R erit taDgeiis , & c^ 
dcm cft d^raopftracio pro rcda DH.*^ ^* 

jg.^Sccundum fic dcmonftratur ; Du^ per I ^* 
M chordiVIit K^/par^cUs £H , ac proinde perpend(2 
cularibus ad DA / & bifi^rit^ fecus , quas occurjrani 
rcdtis DE , DH in F , G , & /, ^ , patec ipfas quoqall 
F/r G^ bifariam dcbei;c iccari a^ipfaDA, adepaue fcb 
re Fz a^qu^cm I/, & Gm kcMialem gyi y ac rci^tigif 
la FI/, GM^ rc^angulis IFz\ MGw, Popro, erune FI ad| 
GM, •& 1/ ad Mf ^ ut DI ad DM: adcoguc qaadracunij 
DI ad quadracum pM ut r^angulum Fl/, fcii IFi, fi- 
vc quadramm tangencis £F ad rc4^ahgi^am GM^, fii^ 
MGi» , vcl quadratum GE : adcoque y^ qpiadratum IB 
ad quadracuot LM. Erit igicixr Dl adDM, ucJUadLNf 
uc oporccbatl * 

PRO- 



ELEMENtA. 9 

t 

\ MiOPOSITlO I. prObL 

r 

' S4. r\-^^ fi^^ * dmSrici y & rMioPie defermindrh 

LJ te 5 invetni^ amma SeSionii Gonic^ funSd.' 

3;. Dttcator ^r focum F lecta HF^ ihdtSni^ oc^ F.9« 

cdrrens directrid AB ad ongulos rector iti E , pdna^ lo, 

^torqoe Had partcs F • Capca iti directrice vtrfus paN 2X« 

,ian ucramlibet, lit verftis A , recta EK xquaU £F dci-> 

rcatur per F, & K recta indefinita Tt , polica T adi 

panes F. Ducatur per F reaa pctpendicularis ipfi EF $ 

ac in ea capianmr FV, Fm, qus fint ad F£ in rado^ 

fic determinanfie, pofito V in angulo FKE 9 quas qui* 

dem patet ('nam. i.) fofe minoires F£ , in EUipfij x- 

qoaks in Parabola, majores in Hypcrbda. JPier £> & 

9t daeanir lecu indefinita Ggy pofico G cir^a directri^ 

|«ttn «d parces F, quas ncceffario occurret tccts Tfd- 

tra directricem inoer K> &F alicubi in L^ tum perE^ 

& V recta indefinita li t pofito I citra directricem ad 

ifsnes F^ quam patet in parabola in fig« lo^debete ef< 

k parafielam ipfi T/ ( cttm nimtram EK^ VF ex ona 

parR parallels fint ^ 8c ex ^ia aBquanmr eidem FE 9 

adcoqoe & inier fe ) ac proinde ia EUipfi in Bg* 9- 

debece oGciarrere ipfi T/ alicubi ia l ci^a direco-icem 

ad partc^ T ob FV ibi minoreni » quam FE ; & con- 

sa in HTperboIa ( fig* 1 1. ) debere ipfi T^ pairiter 00 

CDcrere, led ultra directriccm aliqibi in / 4 Demumex 

pu&ctis L9 l ducannir reccae directrici parallete, oocur« 

i^ipfis Cg, Hk, U in L, M, NW, m, n. 

36. His ita femel pra^psUratisj per quodvis pnncmm 
& <tct!( Tt jac^ens in fig. 9. intra fegmentum II , in 
%> 20. ab L verfus T , in fig* ix* extra fegm6ntum . 
\U^ ^^ i^ccta parallela diriectrici» quat occurrat reais 
Cir Bh, li alicubi in O, R. Q^ centro F intervallo 
f&Q; vd RO i qus ipfi aequidis erit y inveniantur in 
ijpfa OCXbina puncta i>> P: Invcniri autemferaperpo- 
Jicrimt bina, ac bina tanmm, & omma> ac (bla ^>un^ 
^ ita invenu una cum punctis I/L9 m in EUipfi» Sc 



^ fo SECT*1(5iSft7MT.O>JlCARUM 
Hypev6oIa, &- tulii pwicto M in parabola crliftt ad 
iectioiicm Conidafti qilspfitam, - 

37* In ptimis enim fi centra F imenrallo RO,) ▼«! 
RO »1 iecta OQ; ^irectrid paitti&ist invcniattir' ptiinp 
ctotii Py vel p>^ cflb ddbet ad ^quaefitam SectioQem 
Conicant 3 8c & ^ ita in^^tiietor •. Ducta etiim -PD 
perpetidi(?ttiari ad directtieem» ad^oqae parallela ■ Rfi i 
cuiproinde ctittae?quali«,crit FPadPDv tit RQ^j-fd 
RO ad RE^ fivc ob tAfy OQ^paraHcIa»FPadPDi ui 
FV5 vcl Fwad FE'^ -liimirttm.per coaftruetionem ia 
raiione determkaifte v nndc etiam paiet ot> FV» FnraT- 
fumptas acqualcs fotc «ciatn aiqiiaUs'RO> RO« Si aii* 
teih P fuerit dd'5cctit)nem Goni(jam y ent eontra FPi 
ad PD , utrFV, vcl F» ad FEj adteoqucf FP ad PD 
nt RCL^ vel RO ad REatqoatcm PD: atdcdque opor^ 
tcbit FPefle atqualem RQj veftROi &. ponctum Piti* 
vcAior centro^ F, ra<Ko RQjm& cadcni cft ^icniQnftfa* 
^Q-piro punccb I». ; ^ *: 

38. Porro pcr tiuodvis' punctum S reciac Tt dacta 
ORQ^pat^llcla ditectrici, invcnfcntur ccntro F 'kiter- 
vaUo RQ^bina ptfnm Prf hinc inde vb Rs v«lunft- 

Tcum congrQcns cum R, vel nullumv prout RQ^ftterit 
iBa)or 9 vel' asqualis 9 vd minor^ rerpccm FR » Nani 
£FR ipfi OQ^pcrpendtcularis eft, cum ik perpendiah 
iaris dircctrici A6 r adcoqae citculus cctitf3)F ife(hft* 
ptus, cranicurrit ulcra OCX9 eamque fccat inbinispufil^ 
cfis liiac indc a perpciiidiculo FR , vel. cotitingk. ia 
R., vd ad eam non pectingit. 

39. Eft autem FR fctnper aequalis RS » cum smgkK 
lv3 FR5 fit recttt&,'& ob K£ , FE ax)aalesyac ahgu* 
lnm KEF reccum , fit femirectu^ KFE , adcoque Sc SFRv 
Ipfa vcro RS|, aflucnpto S intra limites etiunciaca^iti 
conftructione 5 erit fcmpcr pars ipfius RQ^, vcl -RO» 
adcoque minor ipfis: abeudte S in L» vcl /« ctim i^ 
ijtf congKuet : . aflfumpto veto. S extra limites enuneid^ 
10$, erit e contrario RQ^> vel RO pars ipfiiis RS j^ 
«deoque RS ipfis majcrr, qaod quidem admodunt mi^ 
idfefium erioin figuria. i:^> t3> 14« : . . .. ^ « . _ 

40. Nam 



^, .^^^ m ifg^, ja^ m EUipfi toca linca /T jaccbiiRiil 
icxtra OTgulum GEI, afteQfftQ. punctp S affumpto io/J, ij, 
bW^rftti^FW ipfius RSr. Qjiod fi S affumeretur in / r^. 
-n»W*3«R«i:ibi ^l»^ 0L> $• Tum.io.ta ih jaoct in- 
P.ji^a^ 9i»St^WPi GElj, ;^leQKiac AffumptQ S in (B ^ «ft RS 
iiff^. wAWrRQ^' .^^WW«i S 4«> Fj ^^ iEvaiiciciit ;. ;it 
, fiunpfo S.la FL cvadirRS pars ipfius R0> aboiiita 
j.^fcr^ )S. i^i: t a w6k co0vcQi«W S , O, . At loti U 
,jja£gj? i«»l^ ^Qgv^. GEI ^ .«a. aira ipfi ^ vcrticcin 
rPWpP^I»»^ J!fef jija, u^,nffi«np«> ^ Jn JJC, fit Rpfara 
►^lrtW ]^r:^b»»le 5 in l^ Yivm^X RO X afltoipta 
^i^ fe K^ jit ittrum RQ, par^ RS. 
/, r4tl!'Xt^ ig^. \Z^ H> Pacabok cadcm prQf{u$ aeciduntip 
(^Q^^o folK^^^iiG^ifnine, qopdob.Tir , U paraUelas 
-fP&49^ Jb^euwjE^.ca^um c^qarfuc /3 adco^yic tom lin^ 
^^W^I^tfcTrjfQcrintt-^ anguliiii GEI ^ tot^ U cxura 
J^^jffrp 4c nm^ iT^ ^ ad vcracom oppoGmm ? . ac 
(irdiidc pcr totam LT cft RS pars RQ^ yd RQ, pcr 
tewati^ «PWa RP3, vd RQLP^rs RS . 
-:.34?*rDcnHJTO ia fig. 14. ii^ H^thoh tota pariicr 
J^. j^ct inisra 'anigiiluin^ Gi^I> |acet autcin 4 ulua di« 
.tffSWU2Ct(ij» 4;; |9igL ^uidfni U i^cc cxtrn an^dosGEI» 
j£i^> j^ttQi^./l^ ii^tra gBi }a£ct^*.> a4$cK|ue pcr .^tam 
43?.^, RS i»m KCl? P^^ FL.pJirs Rb» pcr LK con-. 
^a* j(Q p^ RS^ pcrK/ vcrQ:RQ,pars RS * &'Pct 
JWW /^ r«rftim RS p^ra RQ^ ,, 

r* 43: Wi^ o^llMtHs |^fp^ciis.p4trO .affuwpto S in % F.^' 
9- intra L/, in %. lo. p^»toiamXT« in fig. ii* pcr 10, 
Jf^f^UTf: it invcniri in rccr^dirc$t(ici fardOicIa bina i<* 
^Wfia ad ,Scct;iQncm Conic^am qn^fit^sr eo affumpt0 
}9\ ji> vel A coc\ii|tibus ia primo ^puncas S ,» O » ia 

^jQ^^lp pu^cci^Si Q^«^^i-'i.F^ F^ ^uales ML % 
pil^ adeoqjoc qoire ibi puncta ^h^t in.unicum M, vcl 
Hfs iii*fluo niinirjjm circulus ^cmro F: r^dioML > vd. 
it^ dcTcciptu^ roctafn MN , vcl mn coatingerct, cxi« 
\ (t9(e ibidcm FM, vcl F^ ad ME, vd i^E , ut ML» 
; ^ f^l ad ipfas MEv vel nfSL^ nimirum ut E^4 vcIFV 
I ad F£ > fivc in j^atioQ^ dctcrtninmce i at affumpio S 
\ r -' ttbicunf 

f 



tibicumquc cxtra cos limices > nuUuiti invcftiti ptttl-i 
Oamt Q.E.a 

CoTolL 1. 

44, Datis fo€9 , diftHrieei & tdti&m dittmiHsinti^ 
idtitit SeElio Cdnida. 

45^ Patct » cum iis dfltis ^ kvcniaaAit: bttinia ejus 
ftjndta. Corm t^ 

46. £//4^ tdta^ehrddirearicem jacity & Pffeiffiim 
tedit * Paratold unicnm hahet i^amnm citra din/^Ehicem 
in infirtitnm exiurrentem ; Hyperhola binos ramoi Ut 
infimtum exd^ntes > alterum citra^ alterum ultra di' 
reSricem. 

47. Patct cx ipAi probkmdds eotiftniftione i tvart tiU 
fliirum cx Omnibtis rectis dirccirici parallclis omncs y 
Sc (daa rcctse ducts in fig. 9. intr^ limitc^ U odcut^ 
rant £IIipn blnc itide a rcaa Mm in binis puitctis F^ 

[ic pl ({wt deinde inM^ mCotmz i omnes atiiem, 8i 
folx kczntsi infinitam LlT in fig. lo. oc^cuitant' Pftta-^ 
bol^^ ac omncs, & fdls in f^. |i. per inftiit^ LT^ 
It Hyperbolac occurrant . 

CofolL %* 

45. ElHpfis i Pair^oU y & ratMts citiri^ tiypefho^ 
la contingunt reBas LN, Lu, NV in M , u, V ; £1^ 
lipfis autem^ &ramus ulterior^Hyperbola reSumltiiH nf^ 

49. De pun<^ M i ^ patet ; cum ibi pundta P y 
f coalefcant in unicum , 8c qCisvis directtid p^raUelS' 
€t altera parte rcctas LM, vcl Im i ducta occurfat Se« 
ctioni Conica^ in bints pimcds binc indc ab M . De 
punctis autcm V, h Cdlligitut et eo y quokt abeunte S 
in F, abeunt puneta O, Ry (^in Ui F, V , adeoque 
iit ipTa Vu invenienda funt bina puncta cencrb F inter-' 
vallo FV, qu« cnint ipfaV* u , cvaiiefdientc ftimirutir ' 
ibidem FR, & factis RP^ RQ^acqi^ibus^ inter fe^ ac 
ipfi FV. At utoamqtic parmn diftct OCXfltb uV ntraliur 
bet es pane, femper latus RP mintis cft , quam b^» j 
£P, adeoquc quam RQ^, ac (5roindc ^^ectionis Coniw^ 
cx puncmm P utrinquc circa V jacct ciora rcd^atmNV>4 
fic cadem cft dcmoiiftratio fro Ur 



50. feElunis Conict ferimtter tft Unta ci^m , mp. 
qutm tnterruffa . .-. . ' >. / 

«» cflfc non poflSt 04 Iinca . q^uam plufes-rcctx ita 
«wttngSQt M «|ico>ponci6 fiogolar , ut Ipfa utrinque 
orca contaeratn jacca^ -^id caAlein elufilem recta: pac! 

5> *^raqtMm jiiaem te<e»mippi-, ^atet cx' conftra, 
cnoK tfCi.vamfmi pateat, poftctb S excutrentc mo, 

i/ m^tas .n Parabola, ap Hypcrbolaf deberc pun 
^P^mter^exourrcre mot« continqp', Se4 fic ac, 
Oiraaas demonftnimc. . • ^ 

Jjd' w ^ alKii sitcnt M* oecttcrerct iteram in p\ 
^^ ^JI^- *' n^fq"^™' ut in, fig. 16. Prfmum fie- 
anon poteft , cum.p^«a <i.ircctPicF parcllela nonhifi 
»« nmoo punoto poflit occurrere Sectioni Conice ad 
«mfcm partem rects hfH ( num. 3». y, Sctandmn 
feiaon poteft, quia ex altero ex^emo ^ arciis M 
•all*j« <hi«ai> paraJlela direetrid ', aK« paralleii VO 
•fncro m&mx dact«' per putMrti V intetpofita punctis 
S, 1. fcc«t.m«erceptatus limiobus deffnftis , in quibus' 
f^ paralMa debet oecutrete perimetfO fcctlonis hinc 
«wfe^ ftm MH, ipfi nunj^uara ex ca partc occurre; 

D E F I N r l^ I O IT. 

9fr7*f6>rdm iihfm Va per focim dutlm dico ta, 

n^mmn ut miffi (- fig.9, ) ^ i„ HyicritU {fil 
|!^ ir ; Lanis Tranfverfam Principale , >* Attm 
jRmfvcrfiim, #/«^ fmices M, ra, acijifiMmfeaa 
£.V^. "L^' '^^ ^ Gentrunf : ereEiit aunm hincin- 
«f rtms CX i Cx terttndicfdarihus axi trMfverfi , 
nc mediis gemetrice fiu^UnalihHs inte^ FM , F«i 
-H^^m-^h, Tm,lIJ, c Hnas 



14 «fiCTlOMtfMcpWlCjARUiti 

Unai iifiMtiai foci t a kif^u vertlcitfHs axts tra»^ j 
^efj % di4;d \t AiWP^Cdfljugatiim t tiufy$f^ -^Jtrticn %i^ 
X:lti£tidti aUtm'U\i ift'(hifi^itam U PdiahoU ifisti 
.j{p; ) dicd tjus Akem ftaftfvcrftdp . i d- M ^j^/ ^kni"] 

definit/i9o \ inftlligaM totai^ i^eSlam us^if^t^ inij^s^' 
tph^^ in dM^fnni 4^^ vhficeJ ; Afi?4^ axi u^ili^^ 
ferfendiciOaiei ^ & lad SiEtionis fi^imetrmiiitriniui 'tfr^i 
minatai diro iyut Dtdjlitlatlst | Hti fkni ihoirj^ Pp vefyit^ 
£iu axis tr^Jz/i^J fegfnentm OfUm 4xis hft^frq^jthmi j 
UtiT ofdiHdtdin > W vertjj^m, , ti^ (^nfit^ f d^p A^l 
fdffaiti ak io ^tice , z^ ^X^ntr^ v i* MR 9 raR J 
/^r 4^/c/^ d l^tkiliHS U^& ta^\^ CR f^^i^ 4 \ 
€tntf04 

iJJf.lpOji Wc* dcfiaiophcs crutmus gnrno trial CcJ^ \ 
X roPad^ quac ab iU aoni pcncjedi » nili. |i^, for . j 

l^ Homintim ofurj)adoiie ^, ^ dcbiiiffcpt C9ru»ni^e fe-* a 
sicra CojfolIariort];a'pt0j^tioQis prim^ 1 ciifn fli jol^ 3 
^|us cpiiftriK^tioiilc fpdnte diiant ^ jG^d d^jM$inc$. ^i- 
cerfcrenda^ fdcriinr^ ut esl^.qibfdnji ji^ckpj^I^i^ «^unir 
^aritiir^ (uis ict (pfk enuxiciaticMe ^oraiiiibiK 4ppcU^ i 
i^cdtut w Cdnf^qilieiitiit Cpiroliaria 4^ ^ jf^ qii^<fW« 1 
^roprie Cptollacii ^fmiiidnum lateFis CQ^i ii #( &ini^ 
xis cotijui;ati ^ qui hid afliimptii^^ft ita 9 uc eju$q(|^ , 
drataiii lit arquale jtcdiangtilo diftititiatura foci a bi«» ^ 
siis vcfticibus i Tiipi CproUiUriiifn . &^ erit iittniin Co- 
rollariam {>it>pdfitioi(iis primaer ^ & COiitincbit pfcci-. 
^yxi Se^ionum Cpnic^utn prppnetat^ra ^^ qds ear.»., 
nira natiiram exbibet > U- fbetfundjifima e^ it^» tit re^ 1 
iiqqa on;;inia CoroU^ia' deinde ab ipfa p^nd^^anc » So \ 
i^voi ^o^iiUmum Cordlaria £^^ 4 Potuiflct idcircpencm-^ 
ciari per pro|^iitronera ^ C0m ob enanci^.ti tJicQrgfi^aK- 
9$ di^it^tcm » cun; ob IqKUdic^tdCein nov.^ , ^^>xij 
cidtrd ^ q^uod ^^auUo raajote arabitii inijigi^a^ a^t iVu^ 
4<;mon$ratiQpcm > /binaiijca nirair^q) rjKJ^^iiiim compo^ 



t l E U t hf r A. ff . 

nviiihfi i Vcrum cb&AiUii^. ^imas H qiftkiu^: Cof oh 
laciis unmHccrc > fum ^uii vix q^dqudm sUl iist de« 
indddra&odcin pdftolac , fxxtii Coitfttii^dAem problc^ 
m2ft& ^i:tmi 9 vXm quia piropJrieUieni Cnuticiat - ^xii 
\rj^cr& \ 4d^ (kindc iavetiihihis ^tfterakitt & a^i 
todji^gatD ) Sc iliattk^r^ timAi&us i ( qua^ iu quavis 
5eftioi& t>>nici ia&Hitz (ii&t} dt iA propdiiticiiit (^ 
tadaciabimu&; , 

f6* Jxif ^arqvhfHs Ufafiam fccaf JUai itiiM^M \ 
^ fccat tam aream , qnamr ptrifiietrum Setiienit Cdni- 
tmterminda quavir ordidata in duai fittei )^ifpis a^ 
l^tMtis ^ iir finules . ^,, a • - 

3(7. Nam ordinata j^ cflct isbori^ ckcult i^i6\^iL 
jimtrb ^, ^adidrl^Pi ade5qte|CbcolL4. p'rop;)[;GcoM) 

Lpe^didlb' #R {^ ceUtiHimr dtido feoatut bif^i^ 1 
db .aiiceut piiet > cdtam FiguraHi ^iPR ^ Vdi mRP 
fednvdi^ citca ixcni trahfvttfum debek:^ prorfus bdd-* 
liiikift figuraB Mftj^ vel iPiRf > bum quxvis lciniofiditi^* 
a KP dcfadU ^ adgUldft ad R t<^ eau^flier« iibt 
«awdl Riii. 

f ^* i^^mum fociraiiorum ihihiniHS in iltiffi eji is ^ 
fii termmatiir ad verticem axir tranfverji fropiorim ^ 
hdJHlimkrj ^p adremdtiartm tetiqui eominontsi vitlmdjo^ 
pKr,.4^ adilium-i vel hMc verticem actedkfft ma^U 
ifipia^imS^ ^ad 4f^k tMnimintitr : in Pai^ahoia > 
ict m09it JH^kda i^amo ille mftimis r ^i ad akii 
WpriiGtm terminatur in eo tami fitutn > reliiki to ma»^ 
fn^ 4^ Urmittaritur^ ad^ fuHSla- ab kodtm vertict 'te- 
m e ^kiA . ^ fke Mfi hiric iude Ifini laqUlit hjfcft 
k^Swr in efdam kiHc^ indo dnfydo ab iffo axt ttanf^ 

tPi Him fadiiis Yoci FP i ^i liabcat ad Pp ^ fi. 

Rfi. ^a:d6ii(in.cclaftailtcr c^dem ( uum. i* )sciis. 

Vcl docrcfcet» Ut if^fit HR « Pafei autcm abeunte' 

m M ^vcl m y ^lre pMtet R in eadcm punda , 

tc l^ at. M , vdi w> rtCcdcrc & R ab^ iifdcm, 

C a. 2c 




\ 



Sg SECripNUM CONICARUM 

)u: proinde ipfaruni ER inEUjpfis Parabola ^ & raniA 
citcriorc HyperbolsB minimam cffe ipfam EM, tum vc*' 
ro perpemo crefcere in his quidem in infinitum EUip- 
fi vero dencc in m cvadat maxima , ac pariter in rar 
mo ulteriore Hyperbolae in fig. ii» forc omnium FR' 
uiinimam Fw, tum cas in rcccffu puncti P ab w cre* 
fccre in infinitum. Binac vero FP, F/, <}u« fola; cora- 
niunem RE liabcnt, jaccbunt hinc inde inangulisRFP, 
KFp acqualibus ob FR commun^m^ Sc Ia^craRP> Kp^ 
iu: FP, Ff fiequalia, 

CorolL 5. 

^o. Differemia dimidii lateris reBi priwipafis , c^ 
radii foci in Elli^fi^ Parabola , a€ ramo citeriore Hy^ 
jerboU fumma in i^lteriore ad dij^amiam ordinafa a fir. 
co ejl in ra^ioTit determinant^. ^ ^ 

61. Cum cnim fit & FP, ad RE, & FV ad FE jq 
ca rarionc, erit & illarum ^iffcrenti^ vel fumma aj 
harum differeotiam > vcl funnp^ in rationc cadeth 
{ cap. 2, Arit, n. 1 3 . Porro dlftmitia FR o^^dintaae Pp % 
f oco F cft ubique diffcrcntia ipfarum ER , EF , fic itt 
famo ultcriorc Hyperfaoja? in fi^, 21 eft FR." fummaip-. 

i^xm ER' > £f, 

CorolU 4, 



6a. Dimiditm latus re£hm prmcifale ad difiantiam^ 
foci a direEbrice eji in ratione' determinanre , & inP^ 
rahqla latus r^Sumfrincipale eft duflnm ejus dijiantut ^ 
^Hodruflum tum dijian^ia foci a vqrtice ^ tum dijtantut 
"verticis a direEirice^ 

6^. Patet primum ex ipfa conftructionc prop. 1. cum 
fjt FV ad FE in rationc dcterniinanic . Porra in P^ 
rabol^ ca eft ratio aequalitatis, & pM , ME asquancuf 
Iptcr fet Pa^cnt igitur & rcliqua, 

CerolL 5. 

^4* Qyadratum femiaxis c^njugati aqttatur dijfertf^ 
fi^ q^ad^atorum femiaxis tranfverfi^ &dijiantia foci k 
ffntro I ^^fitnu i(io ^majore in Ellijfji , minore in Hy* 
t(rhk i (ff i^i^dratm difmia ^Cf ((ntro -^quatin 

in 



/ 



fi t E M E N T hr \f- 
^ i^li^ differemU quadratorum jemiaxium exiflenf% 
fimiaxje tranfverfo femjfer majore i in HyierbeU eo^ 
tmm.fumm^, . 

eji Patenc ex cd > qudcii. ti 6e&tnnont ipfa qua-* 
^^atuih femiaxis cotijugatt de&eat tBc sequale rectan*- 
{ylo M^m , ^ bb Mm fectam bifariam in C, qua- 
draiom CM /C^tf//. a & 5 prop; 13. Geom. ) «que- 
mr in EUipfi , ubi CF efr minot ouswn CM y qua- 
drato CF , & rectangulo MFm fimu! • At m Hyper^ 
TOla jriibi CF cft fempei: xnajor quam CM , quadra*» 
tum CF aDquatiit quadratGl CM , & rccfangulo MFi^ 
fimiii. 

CirolL i^ 

.]. 66. Quddtatum femiordinata axis tranfverfi ajuatur 

; pi F^ithola reQangulo fuh ahfcijfa a vertice y &, qm^ 

Jrufla di^antia foci aJb iffo vertiee , five fub eademak' 

i^fa , & latere reSio frincifali : in JE^Uitfi vero & 

if/ierhola efi ai reElangHtum fuh hinis abfci/ps\ ut qua^ 

dn^um re^atiguium fub hinis difiantiU foci a binisvef^ 

ticihus ad quadratum axis iranfvtrfi y five ut quadroi 

ium axis y vtl femiaicis tonjt^ati ad quadratum axis ^ j 

vel /emiaxis tranfverfi^ five ut tatus reSium princip^^ 

le ad Ltius tranfverfum ^ qua rationfs omn^s aquales 

funt. 

^7, Nam.pb OQ;fectani.bifiriam in R, & non bi- 
wriam in S i erit ( CofoII. 4^ $. prop. ij.Geomijqua^ 
^atum KS cum reciangulo OS(\ fimul jcquale qu^^ 
«fc«D RQi^, five quailrato FP,.fcu quadr.atls FR^, RP ; 
pim igitur, &: quadratMm RS asquecur quadrato t^.P 
^yfasJlS^ RF «quales ritiitri.^90 9 txit i€ qMdriH 
Ipi RP ^qualc rectanguio OSQ,. 
[ 6S. Eft ajitem Iri Kg. 10. S(X arquaiis FV dimidi0 
iateri recto «V , & acquf^isiiN, Uvc divl^ LM» tiU 
I fairam (cam ob angulun^XMF rectunq^ % 6c LEtH 
IJternircctum jL rium. j9v ) «qucntur irit^: k Mt y 
iTylL ) dupl^ FM . Ducta vero Ly Uprn^aii a4 05,.q^uq 
l^ditide crii fMsikli M squalis abfciSe MR » erudt 
v^ G 3 oy 



\ 




/ •»' 



1 

t«, SECTIONUM GONICAROM 

: [yS ipfi arquales . Nam tri^npl^ Sjfi^ O^L ^mtlia^ftHm 
bgulis NME, LMg <)(i fingula kttera fingiiKs Izvi^ 
?ibus paralleia:! .^4?QC)ae ut| NM', XH «qu^nnqr Mf » 
fivc ME 4 itJi & Sy ^ Oj} «(}uaqmr jjfL , Erit igftu|: I 
OS diip'^ li7 y fiyc dupla al^fciflte MR, ^ 5? rc^an^-. I 
}um 0$Q^)t five quadratum illud| fet|luoi4lQa.taE^ l^P aB« 
quale re£bangulo fub dupla abfciflfa mR: ij^^uali ch^M 
Ly, Avc toti QSi ac dimidio. Utcrc ' rcAo, FY rfquaft 
$Qj« adeoque atquale tedtlangulo fub abicida' MR > &: 
|6|Q laterc rc&p. nYy fivc re^tan^q fub, abfci^a. »/ 9c 
quadrupla dift^ntia FM f^ a Y^tjcic^ 

69kDuQ^^ autcm (Witcf L)f in % ^* & it,, quci 
fi opus^ fit , p(odu4l;a 0001$^; at reds /xr in Y » cric 
OS a4 ^f 4upl^ >^^ >. five; 4iipkmi «iF ( tiwn^ 43. j| 
. ut L5r ad y ^ fivc «e tjp ^4 ?iY ,. vcl ut MR ad 
W«! > 9c Sd M tN AffJl^m LM j^ vcl pariter duplam 
MF ,m Sllk U yf^t qt jfY a4 LY ,/vcl at R«i a4 
^fiMf . Igit^r ^oi^iHii^it raubnibij^ erit re£|angQlctm fjsfi^ 
OS * 5^ SQ^ % fiyc q^ia^^atlim |ip a4 qua^druphir» r^. 
tVangukiim fuh MF , ^ Pij»,^ iic t^Aangttlum fi^. MRi^ 
ic Km a4 quadr^m. 'MuPiy yA altcr(^afi4<^ Qaadra-. 
|um ^emfordihafisc RP ^ reftang^tuni MIM Mk ab^ 
fciflis > iit q^adniphim re&angulum MF^ fub, binis di-. 
ft^tii^ foci a vcr^Qtbus a4 (^uadratyp^ ftxii^ craxtfyei^ 

7Q, PoiKa cum CX,^ Cr fint mt^ lnt$? fMj^ Fim 
f' liam. 54, K crit qisKiPatijin CX,, yel C^ ^qaalc re- 
^«oguta MFmi & pifoin4c qiiadranim torius 2^ coa-i 
logsiti X^ aqci^e qn^plo rcft^^ngi^a MFW^ *J?oque 
jrania c|us qua4rap|t tc^aii^ 2(4 qu[ac|?4tc^ axts naii&. 
V^rfl ^ad^m ^ 9, oc ratlo qiMrad axilsi > yel femiari 
^1% ^ju^at^ a4 ;|x^m:i y?I lctniaxw fl;anfy^(«n\quan 
4wqm, 

7?!, Dmm wm ipft ?Y'«t femidr4inat^ ^ & fM, 
ft» d.(^^ffar s| ycfiicib^ tri^ qu^at^m f V a4 ?c<^anr 
wleWi Wm tfm^^ qnadfatum €X ^ «t IpfUn^ qvp- 
Iwwim CK ^ qHt*«Wn.CM: At? proMc FV^ ^, 



fi l 8 M E N T A; w 

fattti ftftmn pr^d})^» atX ^s cdQJagatmt, l^ <utk 
fcaiifKrf^ fuat condnue propontonafes ^n tefeoqufe rsi^ 
prflni ad ccnium, eit l(^<init 9lE r^tio qiudra(i (com* 

■ 75- Nsp^ qaaaiat^ ^iof^ifiiat* pefr. <;?jmrumdn^ 
«tac ad icteiat^gqltim fqb. KiC: > C^ .» <m« fen? ejq;? 
aU^as , fivfc «I qiMidr^tOia CM , «l^t <J^, itt: q^a- 
dfiim}^ fettAaxis con}ug^d QQ 9$ quat^Iratyift^ id^m 
Jem^fsd^. arafldTytrfi ^M ^^ >ic proStide iemiprdiAat^ 
pet ceutrum dtt6la aeq^at^ ipd CX > & pohiQtim X^ 
^ a4 |pciibi^ttiin[r>^ gic ^a^jcm ^ ^emotUlra^b ^ico * ^ 

Bjf^i^ > ^t rOljw^», fnb Hnfs ^fcij^ 4 vir(i^ 

7jf. &h efiim q^adtatum^ \}fiiift. drdt^att |t| P#a; 
bob 9d qoj^amm' ^terius «^ ut rc<ftwgul|un f|ib ab^ 
^ffil ^GS y^ dr bt^re r^fta principaK a4 red^^iilum 
ibb a^^iTa hu)ti^ &'codem \xivxs\,i6.)^ W id^m. ill^ 
i^ i^t^tm fion mutat,. 

^i^ IxK E%fi ^^ t ^ HyperMa cti<^ quadi^atimi 
«dos fettuordinatse a4 rc<^guiotti fub ft^$ ab!a(n$ ^ 
tt ^dadramim sdteritc^ ad reobingedilkm Aib fois^ adeo» 
qiK i^rinando Vont i|ia qttadr^a ^ |Vt itfo re(!^^^ 

Ujarbi^ 4k 4n>c* trMifb^» nai^4!«t dtr4_ fiiififtfni^ 

liU(. y MR' di-^ adiCO&t idtta qUdO^dU^iAt licrutcs > 

C 4 adw- 



id S£cf lOMuM GONf CARtrM 

Jkdtoqm Sc kmjLOvdia^OiXvm quadrata ultra qiiofciitidM^ 
quc limites .crefcant. « 

CofolL io. 

7P# Semior^i^aid axi iranfverfp dque diftanie^ aierif 

iro ^ t/H a rfffcSiivil verticihHS jHnt aqnales inier fe iJt 

Elliifi , & Hyferlpola , qm dutem centra frofiores y. 

eo majoT/^es in Ellitfi $ minoris in axe tranfverfo Hy- 

ferhoU. . . 

So. Eruat.tfiim m prd^natis «orque idiftan&ibus bin6 
abfciffe unius a^q^uaks bini» ^^cims altertus i abfcjiffit 
himiriim pnius a yertice M, a()fciflx dtcrius a vcrtice 
my^ vicevcrfa,^ adeoquc rccfangula ^b atfciflts aBqi^^ 
lia^i &c {^qualia fomiordinatarum quadrata. Ai cxjm xc^ 
ctanguiiim MKm tvi diifcrcntia quadfatQiriHn CM^CR» 
quo mitlot cHt CR ih £llip(i , co majoi: &it exceflii^ \ 
quadrati CM fiipra ejiis quadratum; & in Hypcrbola 
eo minor ejus quadrati i^x^efius fupra quadratum CM^ 
Q^arc eo ibi majus> hic sninus - rcetangulum MRm s 
6c proind« cuam quadratumi fcmiorctinats.^ 8c ipfa fe-* 
fi^oirdinata* CorolL li^ 

8i. Qf4<evis reua in Elliffiy & Hyfierbolafer centrum 
*duBay & ad ferimitrum utrinque teritnnatai in iffocen^ 
fro bijariamfecaiur 4 
p j- 82* £)ucta inim in fig, 15^ , tSf qiiavis PC^ a<l ceiw 
jg irum, ac femidrdinata PR axis tranfvcr^, mm atfuni-r 
pta Qr ^quali CR, & crecta ad partes oppofitas femi-' 
brdinata rfy ac (ducta Qf , erit rf aegualis ftP ob di- 
ftantias Cr , CR aeqtiales^ f gitiK ob angulos ad R & 
f [alterrios ^ualcs , eritint' xtk trianguUs PRC > jp^C 
jequalcs & angull ^iA C , & rect? PC , jfr'C > ac 
pfoindc cum fec'ta PC producta dcbcat cfficcre, an- 
gulum ad vcrticcm oppofitum scqualcm angulo RCP , 
dcbcbit abirc in ipfam Cf , & terminari ad p , ac iu 
ipfo centro iccari bifariara. 

CmlL 12. 

83, In Elliffiy & Hyferbola axis conjugatus omnes 
ffuts ordinatas kifariamjecat > & ejus erdknasa aque dii 
^ antes a cmro. a^titf^ funt,^qu9aut^n%rem(fti^4s a cem^ \ 

tro :, 



/ 



^^ MH iUiffi mifHffejs in HyfirioU maj&res 6 ^ ift 
Hyf€rUU'iuwis crdiff^fs axi conjui^ii^mdjar 4X$ tranfm 
^ftrfo* 

34.. 5pmpu$ tnitn 3 in fig. T^ » id j CR i Cr ihfj^ 
axr trat^vtrfo aBqcndibQi » fcniiordinaic RP ^ rjf^ ad 20 
taDdcm axis partem dncc; aequaleseriinc intet fe« Qgt* 
re ^ Vf pmgens ipfas par^Uf las » & lequalcs eric pa^ 
talleia 9 & n^^^is . H^ i c^i . cam perpendicularis fit 
fxis Xx 5 erit &' ipfi Vf perpdndicularis < quatn habe« 
J^it pco itta. ordinata > fic fec^bit in I ita, ut Pl i pl 
^qttennir ipfis CR > Cr inter fe fqualibtt$ » adeoquo 
£c ioier fe ^uales fint • Cotnplecis autem ordinatis 
PP 9 ff ^ xraafverfo , cric eodem argumcnto F/ or* 
<iinata axi aonjugato 4 P^tei , autem fore ^uales P^f % 
Pf 9 Sc eacum diftai^iias CI > CI' 4 cet^tro C ^ua<« 
l|cs squaiibui fcnuprdinatis RP > RP' axis tranfverfi * 
0ui^ aut&n diftantia CI fuerit major $ co femiordina^ 
tavRP axis traafverfi etit mftjor adeoquc. e)tts diftto-» 
tia CR a oentro eo minor in EUipfi , major in Hy«« 
jicrbolaf Sc proinde eo ibi minor 9 bie major etiam 
fcffliordmata IP axis con|ugati » & tota prdiiiata P^^ 
Cunsqoc in Hyperbola qu^vis CR abfciffa ax^s tran& 
verfi a cen^o m^jor fit femiaxe . CM j erit qo^is 
icnMordinataPIaxi conjugato majoripfo fcmiaxe» iran£. 
Verfe CM, & toca erdinsKta P^ major toto ax^ Mm^ 

Coroil. Ij. . 

t$. QuAilratum fetniordinata axi conyugata ad fum^ ^ 
fnam inHj^ioU, & differentiam in EUi^quadratorum 
femUjds conjuidti y & aifciffa a centro^ vet in ktae dd 
teSai^ldum jfuS hinis ^ihfcijfis a tinis verticibus efi y ut " 
qfiadtatim jtrniaxis , vel axis^ tranfverfi ad quadratufn fe^ 
miaxisy vel axis conjugati^ 

S94 Eft enim ( nunv 66. ) quadramm RP^ iSve O 

ad rectasgulum MRi» i ut quacbrajtiim CX ad qpadra-r 

nm CM9 adcoque ait-cmatKio quadratum Cl ad qua^ 

i4rauim C.X , uc rcctangulum MRi» qd quadratuni 

CM • P«no ob Mm fcctam bifariam in C ( CoroIL 

,2, j, propoC ijt Gecm, } in dg. 19 quadranim CM 

cit 




/ 



I» S^CTIONUW CONIGA^UM 

*|^.2&. qaadF^i^ pR ^nlfe qOadrato C^ , 8t 
^ngiilq MR/np Igicarii|idiYi4cn4o eric 4iferetittaq^^ 
4H:at»ri}m O^^ C?|^ f tcl itlS^jfdguflatq Xix\ bic eM| 
#oii^]i4q 9 eorum fiimma a4 qu^ratain QC3 ut ^ 
'4ratum CR ad qu^drato^ 04 ^ ft a^msftMei ^i^ 
Hiv€|:cefk4^ qoadratum CK i^ PI pi AKnmaiflf iQf^ 
ierbola qiKi4<^^t^f ttti). CI « CX > di|ferentiaQi in 
l^/vel id k^ M rectaagalam XU ^ uc quKdrai , 
Olttd quacfeacum OC, v^l uK^q^jitum Mni tfjf^tffi 
drafom JCff^ 

f$» aqmlwy & fimUes ; sk Hni ffyfertdlie ramfiaH 
fror^ inm fi a^natu & fiffliki y & Mm, $UifJfs ^ 
f «M^ fS^Mji kiim 'f^m%' Ak#^> » at ^reBfka^ #• 
9Mf d^mtu ^ cenero %& S Mfhtnis wrsicp^s 1 4 
h^&ites fiifdm trerfHS p^i^aiei , f^ firbtr feent % 

i9. S\ eqim f^pcr axe ^X cohvtrcamr 4imi<n# $ 
gVFa 1^9 $p. ita> uc abfcal putiaum 1». ia M > sAii|| 
iqfpa^vis^^ if^ RP> & Ip> in IP , adeoque Semiclli|il 
Mem^ inr JvMX> ac U»\ ia £Bipfi>^ ^uam tdi Hjrpdicbii 
fo «^ iq MP- 

%9. Quod fi capMi C/> Q asquallfmi! > &: oippblftb 
CF , CE , ductaquc i»# perpendiculari ^i trHni^' 
Iq , Mof l^a cofHrcn^ttir ckea ;^em cod|u^UiBa 
scCK > abibi( aeb iit locum* i£& ^ «r in locmtn M 
/ fii feeum f > & vicevei^ra^ i qu^vis ametu pefiSie 
iri fonctai aJbiac enint fci locif y in^ qoibuaf* idift f6 
rimecri puticta cr^Qt ante * Adeoque Mmiai' i qtcb ft 
fpe(:tu pnmium peritiietri puac^unf* vtrificafl^ii^^ttfr ' 
iaco f > & dirp(:aice AI^ , jam vtfrifiCabuiiQ» de £) 
f0./> & directritip 4^ • Poircr ob CM » C«f t i 
CF > C/ » & C£ , C^ a^qaafe^ inier f^ » erofl 
»ariter inter f^ s^cs ^MB? /ATi ^ MF , iki/ » fl 
Mf > «^E t 



I 
p 



i fc fi M £ H t^ Ax ♦! 

1' !^ Jir ril^i^ ^ & ^h^M^ dijhmh f^mm^ fk 

4ni^tiin9n€j(J^i!rmi»4^te^^ ' \ ■ 

fh Cijra ^nim rctf^a Fg F^ focara duft^ <>ccJi*W(| 
fgof^i Ccmica^ in punftis M , i». ; {ai^c^ce alierd Kl 
itCT ym^ P> E» qtmuor ^niifta nt\ F, M* l^. ^•^ 

^ Mlfefta %j>ifariain iAC>. ^nmt (nwaaL^P'} Cf* 

CM >, Q. condoue ^ r^porpoiiaic^. in radonc FM ad 

^^j^tmninm ki ra^otie dcterminatite : ac in ^afikiii 

7i^4^^ji^<9 p^itearpm fnnma m BU^y JlifferMi^ 

n, D?tt^ ciwn pcr P rc^a ari a«nfi?trfaj ptraBi* 

Wv^ bm^ 4*fif^?ifeu$ occurra^tt I>5^ dit <lk »*« 

!^#M ?D ^ qwcn /ft Vd P^ in Hncnt cfe^nMiialMlfi 

^ Ut Mwt ad Ee • <^^c ipf^m fipn^ ift Effi^)i 

<4tjJ (fiffiffciieia: in %pcrt)ola ?fig.?p) a<| t)4 fttoh 

fla» w ijla , diffcrcmiam i» hac ipferom M> » B* j 

|55tnrpa(ijfr^ pt M?»! ad Er.: Com ig^tiirD^» E^ «fll^ 

*s fat, ^rit & ii^anim FP, jflP fnmmaitt Ell^» 4lf. 

fepma in Hypecbola asrqu^ii axi lyanfvcrfo Mftk 

^ ' . Cefi^oit 17. 

%Si ^ ^xtf^, funait CherJU ^ trMtfetfe f4* 
TwfWe dn^antkr ad eundm fiem Hna pelta , eanm 
fmm iff JSllipfi, diferema in ffmrhtai atnatm ^ 

k?Jt Duft^ ^nim Fjpi, patet ipfimi dcWre aequaii /P» 
n conycrfa ^ra clrca axcm coQjqgamm rf)C«. Fm 
^/, & P in jpr. Quare i(to vcl ^Mfercna^ bi&aNn 
rf?> F| crit ^a^cini^ ac binarum FPj, /P, 

Cor^//, 18. 
96. J/ ai ex^ema pmSla r^< i^ mH^ duHf J 

^ 44 



/^4 SfeOTIONUMCOTSIICARllM 

!$• ad perimetrHfn utrinque terminat* ducantur in EUi 
ffii & Hyjferbola ex eodem foco hin^ reSUy eanm fm 
ma in ElUi/iy di^erentia m Hyierbola iquabitkr axi 
tranfverfo. ^ 

F.31 97. Nam In iritngulis jCF , PC/, emntlattfa CP^ 
zz C/ aequalia lateribusC^ » CF, & ^ngult ad C ad v^^ 
ticcni pppofiti acquales* Qu^re & P/> t^ xqualeseruht^^ 
Cum igitur fumnia in EUipfi» diifcrentia in Hyperbp^' 
ia rectarum PP > P/ acquetur axi oranfverfo 1 aiqualMf 
tur eidem etiam ibi fuouna i bk diilerenua redtarkai^ 
FP , F/, 

CopflL 194 

9S. piffereniisin Ellifji^^ fumind in iiyperboU late* 

rum reiii^ & tranfverji ad dijianiiam focorum funt in 

tadem ratiene determinante , in qua eji ea diftantia ad 

aasem tranfverfumy & is ad dijiantiam direStricum. . 

^flp 99* Eft cnim^ ( num. 60 ) differentia in EUipii, fum^ 

a# ma in Hyperbola red);; /P» & diinidii lateris ireai^ nU 

ptrum 8c F/x, ad /H. in eafratipne . Pdrrd abeunte P 

in V^ ^it. R in tf&c evadit /R ip4. diftantia foccfc 

^rum F/, recca vero FP abit in FV . Qiiare in £llip6 

\ diflferentia /P ^b F« evadit differentia binarum /P, KFI 

£ve ( num4 92) totius axis tranfverfi , a toto laterere-» 

cto VF« j at in Hypcrbola cum fP contineat axcm tran^ 

/vcrfi^m, & PF (nuva.92)y fiye in eo cafu kxcmtrati^ 

fverfum, &FV, cric fumma /P, Sc Vnia eocafufum;? 

tia azis tranfverfi, Sc toms Yhj 

CorolL zoi 
xoo. Si faBo centr^ in altero foco f itHjifeos in (Ig^ 
%^ , vil MyferbpU in ftg. 14 , intervallo fE., pei fi 
«fM/i 44;?i trAnfverfQ defcribatur circulu/ , ^ ejb qtidl 
vis punElo P perimetri £llipfeoSy vel Hyferbola ducani 
tut bina rella altera VF ad alterum focum F , a/te^ 
ra PD perfendicularis periph^ia ipjius circuli in £1^ 
lipfi ad partes oppojitas ejus ceritro f , in Hyperbd* 
U verfus ipfum , donec ipji peripheria occurrat ci^ 
tra ( in D ^ ut PD , vel ultra in d ^ ui pd , ^^1 
ut punHm perimftri jagmit i ut V $ in fpdm ramo 

cum 



L, . 



• E L E M E N T A. ^55 
F; tW in ^ffofito ) irf p , trum femfer te^reSs Sk 
ktudes. ^ ^ • K * 

r ' loi. Nkm pcriphaerix eirculi pcrpcndicularcs linca 
fuarradii, qui pcr ccncrum /tranfcunt> in EUipfi. au- 
\ fcm ^mac /P, FP aDquannir toti /D (num.9ij» adco- 
, ^' rcmancf FP aqualis PD . In Hypcrbola vcro /P 
hcxoe£t FP pcr diifcrcntiam «qualfcm axi tranfvcrfo 
HHUth. 9%) » adcoquc xquglcm /D . Quamobrcm crit 
L J^P «qualis rcfidua» PD , & cum F; ^xccdat ff pcr a» 
fr^tm traofvcrium aequalcm fd^ co addito > erit F^ «^ 

ftliali$/</t . / 

f 

SCHOLIUM II. 

^;?. T7 ST fatis dcgans c|us circuli analogia cum di^ 

JC/ rcctricc Parabolae . In fig. 1. fi ea Parabolan^ ' 
Itfbrat > diRantia pcrpcndicularis PD a dlrcctricc rccti- 
Hoca AB xquamr diftantiae FP a foco F.Hic infig.2;» 
^. dfftaptia pcrpcndicularis PD a pcriphcria circuli 
ci^Iinea A£B idem praeftat 3 cum squctur diftantia?* . 
FP>^ pim ipfa dircctrlx ii\ Parabola dircctioncm 
tioik itiutct, In f Ulpft cft cava vcrfus F, in Hxperbol^ 
^ohvcxa. 

T03. £x tam niultis vcro ^ quac huc ufque ex ipfa 
jriraa dcfinitiotic fcrc fponte profluxerunt , jam fainc 
fatct, quam apu fit definitio a nobis afibmpta ad pcr^ 
opicndam Sectionum Conicarum naturamj atqueiiido* 
wn. Earum autcm forinam ' niulio flbi cvidcntiu^ o- 
cplis fubjidct Tyro , fl purvas ipfas faujus problema- 
lis ope dclincavcrit, ac, 11 ducwm perpendct, namram 
&(tBig|>t. Delincabit autem admodum facilchocpactQ. 

104. Facto quovi!? angulo acuto GEI, ut in fig.. 25*^^2,5 

vd TcctOy ut in fig. 26, vel obtufo, ut tn fig. 27^ ac }$ 

bffaritoi fccto pcr rcctam EH , aflumatur iii ca P^p ^j 

fdco punctum F ad arbitrium , ducaturqtrc rccta T f » 

qtnc 'Cdm H h faciat anguliim femircctum , qusB qm- 

dcm 3lt?ri latcri a»guli aflumpti ^ ut ' EG ," occurrct 

ali* 



I 

4i:SfeCtlONUM[ CONt6ARUM, 
SllliQibi ia L» akm vctro» ut EI» .ofioiircet ia / ad 
\ Ikm parte$ id fig. 25» t)^it pardUela ih fig; i6> occar^i^ 
krd J« '/ ad ^irtes bppdficas id &g. 27vlaten,mffiiciim^ 
lE '^6ia&6 yciCiis i\ NaiJii ^Ubt iagOlus GEI eft s^.j 
iftuil ttc i^6g»^6^^an|uksH£l irit {emireftds^ Ac^i 
qiKriis. «iKCtio HFT3 adedcjtde JFt^ £1 paraileLK erudci^ 
^ v^ eft/actttui agulus. G£h bt inSgvij, deic HEllj 
.liMire^ knicior; tilbt iUe o&tufus» dt iu ng. 27 3 erit*^ 
l^ fofSiili^ld Isi^aj^^ proinde EI^ i^ ibi toiiver^ ^ 

ksm^i^ifitQ dii^geuci <o6verientei^ exj^arit bppdslt? ' 

to j[; A&mptiS atttem ih lateribus EI t EG \ vei l^i^' 
le^iieDtis £N, E« aBqualibus ipfis EL> E/» & applica* 
ta tegu^a.ih LN> ^^ ckfniiehtiir punt5br M» iv» vercioes 
ialkis trahfverii^ iciim aifumptis .prhribhs EOj EQ^ijuarA 
tibm th ipiijt kMilHis ahgul^ G£I ia&er phh^ L» it ^^ 
ScNilih Sg^ ks % ^ puhdEis L^ N Verlojt G 1 1 ih ' 
fi^. ft&> ab ip&i verfdi ar> I^ & i piind&$ /i # ycp^di.. 
^tcim q^dfit^ ^» ^ in ^. 2.7 ^ sU: iipplicati (empef 
reftda ad piiuiSi OvQ^ quc rei^ tiEh oeakrieft iii 
K io^ iil ob iiofGeliirmum aiang^ Q£C^»< 6c. idf^^ 
WA ad £ fcufaUil ^tumam: ) ipfa QO^^roetur ^ibi bttifti' 
tiiiQ ^, dc ^ a4 ^giilos redbos ; cehtro F ihtet-vallo RCU 
^l RO ihVeniaiitik bina puhifta P» p Hinc ihdcs^Plu- 
Irit^r dtii^ pilniftis ica inv^ntis doliiieafii pci ipfapciA 
tecic Sc^df Qmti^ 4Ua: determiAatis prailtiBirei puhdii 
i»»^ V per ttjQt^ ipfl &H'pikrp€ikdicularem facilihs.qiiinl 
altbJi dfJiheabitur circa pUniS:» « i> V», M^ iki^ ieqhendd: 
iju<him reftir^m &fi EV > LNj./« > quas ia iisi puhA 
Ais. diibtt cucva tojti&ihgereA 

tQ^. Coriro toUaca iia^. edhftru&idhe aim ^icts^ i 
td lif 2^ duidfoluiiQh6.probkmatiSi facile patebit rein. 
^pdtoi sediifc» Rtei$bii amem FM five LM erit. minoi: # 
wi ±diuU;^i VeLm^|0t reipedhi- M£ i prour angulu^ 
i£M Stci^ijc.femiire(3fa ihihiStj sequalis» iiel, majjor ^ td^ 
ttionn prout conis. G^T.fiii^rii! ^iouttis^ iredas.^ Vel 6b- 
tufi^4 ^ promde ia primo. cafu obveniet £Ilij^i ia 
lcpiftda parabola^ ig wdot Hyperbola», 

Qtiod 



^ 




f t 5 M 5 N t A- .. if^ 

.fiBOfi^ sc&^s BQ^ EQ i4 eoiljsm «H9ot^ li# 
94 irc^.£N> £|^.$u^ pf^ipicict i mstt^f^ 
«^cui I Sc <Mmi4 (e9)t>^ obirciMre iii9^dM)«^ 

%i ji| m8t%ij$ arigt^ GKIi ftacim ^rm iplk^^ 

d^fl^ $ ill|4 is a^d^ M ircadm i fiU^GI <4f# 

^.oDuies mdgnitiidmum grsMl^si donec:^ ^ 

^denic redoi defiaat iii ParaboUnii vei^eii^ i^ i.tajg 

riBcedcate^ iu npTqii^ J4m fijti iiP eo4iritat£»6 

qt>d|fd mtitabinir P^aM^ ii^* H}cpar^dlap9 1 ifc^^ 

c( in^g^o to iKirit oppol|(ii do^ biqi 

r#i ^t ^aodagmtodo teluu ^vtsid^SU 

l4# jmpmqvi^ i^Smt^ ^Mdii <^BoiS^ ci^ 
"^-^^ iQde ajateini pat^bit 8c ^miik qiisedaiii E^ 
ih idiih^nfiim ol^i^gati <Hii& Par^olai qiia&^ 
m, 4ftrodpinia indta$.. Coipjstaram^.iii EUip^SkuS mai^ - ^ 
luni oMo^$ hgii^^tKr ^o Patabpli|^« 1^^ qUo ^^ ^ 
\px<^ i^KgibiM. iii^ e«> 4^c» ^ 4»i eft p^ojf^inii» ^x^i. dc! 

Wttp DBAilf iwifpipito i , ^ 

I J»i Q|#^ito M(to aI» ^gi^ LKM pBiiilei iHMHi 
ViM liw i £t4S l^tio iUi dct^miiians FM «d M£ ^ 
k'4i^^kt imi pitct oinnies P^aboi^/fQCe i|ii«r fefir 
ndcs ) cum in iiA anguliis fit feippe;i( re^ftiisi SUiplcf 
^Qk i^, idjccc fe Mm^i H Wjipcrbolfts inw: fc i fi 
S^. 4rt«9nf4Qati{» fi^it eadt^i. NimirQoi i^ m % fti fjf 

W^ ^^mk t^l % <$fc^rid$ii rcci^ omMs FP isi Sm ^^ 
^ H^^Hifi iMlipitaL ad ip4^ni Si i 0V« ^ ^sleixi irai|(^ 
Kt^fil ^ ^m^ Mte yerticl^. M^ mitf^uncAr iff e^ 
Nl «(()«»» 4 Si. taim fiat bi<ii«9 €|ii/in<«li S<i}cpot)Qs^Ca« 
Ika^ Cl^ti ia mws^ EP ad; PD: fiyt RE i^ eadc^tai^ 
^ p{>^ c«]4Gtm iiatiocicm; deteiSmitiantcOtij» & Pf ad 
«& ^sdei: wgt^f; ia ttiaogulii I^RC! » adcpi(iiie U 
^Jt AiiiUMn» v4 i(}|toKlu^ M^i9At proof 

R «H 



it SECTIONUM CONlCARtJM, 
R cadac intra FE » vct cxtra , in eadem radonc cric^ 
Sc proindc etiant FP in una ad FP ia alccra conftan^ 
ter> ttc F£ in illa ad FE in hac. Quin imo cum ra^ 
cio CF ad fM tn EUipfi, S^ Hyperbbla fit eadem^ nc 
ratio detcrminans Cnum. 9o) z ea maneixte » mancbit 
eadcm rado quadrati CM ad quadramm CF, adeoque 
tc ad eorum diffieFentiam) tiimirum ad quadratum (e- 
nuaxb conjugati) (num.64) fe viceverfk- Quare fi in 
pluribus EUipfibus j vcl Hyperbolis fiierit . eadem ratio 
Kmiaxium-) vel axiumj adeoque & ratio lateris recti 
frinclpalis ad tranfverfum ^» ill$ crvin( inter fe fimiles ) 
«Uffiniiles 3 fi diverfa . 

109. Qttod & reccas liy Gi manentibus punccis F 9 
L , M > N in f^*. 25* jcvadacrc paraUelas , & punctum 
E> ac difectrix ttufquam jam fit , EUipfis mutatur in 
drculura ^ coeunttbus foco fy & .centro C cum F > ac 

f,jfi.fig. 25 abit in fig. 28> in qua cum RCV fit fcmpcr ac- 
. quaUs eidem FV, vel MN > punccum Pcft femper a4 
circulum dcfcriptum radio codcm FV , ac centro F , 
Quafnobrem circulus quidem eftquasdam velut] EUipfis , 
cujus foci coeant» fed ejus <|iretrlx ita in infinitumr^^ 
ccdit, ut nufquam fam fiCi & ejus ratio dctcrminans 
ita in infinimm dcacfcit» ut petiitus evanefc^t» & fic 
,prorfus nuUa ; adeoquc definitio a nobis afiumpta ipfi 
revcra in Geomctrica faltem 9 ac reali confidcratione 
aptari non pofiit > ut in SchoUo i. poft ipfam Dcfini^ 
tionem i. innuimus • ^ 

1 10. Atque Iioc quidem pacto Conica? Scctiones in 
fe invicem transformatKur » vcl in circulum . Poffimc 
autcm 8c ad rectas lineas » & ad punctum ita acccde- 

j rc, ut dcmum in eas definant. Nam fi potius manen-r 

f^ ^ te foco F, (fig*9« 10. iiO & directricc , adeoque pun-^ 

20 cto E, minuatur in cafu EUipfcos ratio detcrminans ia 

jj infuiitum» & penims evancfcat, a^dentibus in mfini^ 

tum punctis V, 1» ad F, ac reddcin(ibus demum in ip- 

fum j latcra lE;^*, GE^ acccderent ad axem EFH in in^ 

fiaitum, 8c in ipfum recidereot, ac interca tota EUipfis 

(pontrahcretur vQtAls iQCum f » £c ia ipjum - unicum 



t t E M E N T A V 

definec» • Si vtio ki cafo H^perbola» rado 
in immenfum> & coticipeFeeur jaiu omoe» fi« 
lutnwn iqagmcuclinum limkes tiaji^redi^ ceocdentifciis 
Mpx^ «» V ia infinictiin ita > uc nitrqiiam jam fint > 
iMeKl ip& lEi 9 GC^ acaderem ad dittctrioem» coaii- 
mm pofitionibus AEB» BEA > ac demum in ipfam 
ftckicreai> utroque Hypesboke ramo interea fe expacw 
> ac verticibtts M > 10 accedencibus ad ejus: pun- 
£ ita> ut demum iiv ipfamreciderent> Sc abeon* 
nbus tvnh ipfis in directricem. Si demum manentedi- 
i i3MBice> ic racione determinaiiKe > fo^us F ita acceda; 
ad (firccaicem> tx\ demum in eam ieoi4at in E; patec 
a namer. i6^> ElUpfim quidem debere abire in ipfum 
^ wjcam puQcmm £ > Parai)oIam in ttctam axi perpen* 
f ificnlarem £FH> Hfpcxbohm in binas rectas i^a incU^> 
I Dataa (fiicictrici > ui^ radiu& a4 finom in^iiationis Qi ia 
KradoQc decerminante • Nam fi contra focus F recedar 
i idira quofininque limiies in $ ut nufquam jam fij^ > fe« 
QW ivehec 6c recc^ Tr > 0; axium verticei> & to« 
m CMnraa m in&iiQim > quo demup obnMg Wquanv 
jp enint . Proderit aiijtem plu^roum bafoe transforma- 
1 lioQies lopoFum Geoinetricorum contemplari> quibusyis- 
^^lKdam». ^tque admiral^ CijcoipeiiriseiodQl^ imimitt& 
jji^imo per(piotor« 

SCHOllUMm. 

iiur\\Jonimi dc Seciioiiiifn Omicasupf) fimil!tudi<« 
\J[ ne mentiq injccta eft fi^periore SchoUo > notv 
911 «b^ re pattca qiiaedam de figgirarum fimilimdine hic 
^mxiltrare > fiuttra ufui tum in Secttonibus Conicis > 
I «m iQ oomi UteOeometria. Sin^ it^fil^ 2,9 y 30 > J14F29 
fH^ reS^ dat€ FG , (gx& M binas filHrss cujufam* 30 
\9ffmm< FADQ. farfb di^aisi utcun^ F£, fc> quiefa^ 31 
^0m aniutos GF£ , gfe femfer aqualcs , & vil pmfer 
W eafdem flagas , ut exhil^nt fii. 29 y dc ^oj vel ad 
• ^ofit/ij^ Ht a^, ac iiy fi( autm fmftr FE ad fc in 

D data 



5c SECtlONUM COMr(?ARl/Ki 

data ratUfti ; tjufmedi figuriu dic0 SiinUes > in prim^ 
iafu Ditectci in ficMidd Contrairie^ & reQMJt iliMsFEw 
/e Latera Hotiioldga » eai oktem i{fas\ vel ^Jvis.w^ 
lias faciinitJt tum Hsy vtl ctm t^G ^ % Mgkloi ^iegpu^ 
Us ad eafdim pariser ^dgai ^ vel ad i^jf&fiuu ^ eU- 
CQ reSias Pofidonc Hotnologas i fwr fi it^lumgfH^ 
taHter iH illd cahfiaHd rasiom i eafdai dzco fmri^ 
tet Latcrl Homdloga ^ vel teSlai Hiam MagriitiKfi* 
ne Hotnoldgas^ j^fiU vn$ illa fi f dica itidcfH Ho-* 
niologa* . 

it2. Si duElis mUmque FC^ fc magniiitdini & ^o/u 
iione homelogis > faSis mminmf angutts GFC $ ^fc ^ 
9ialikus\& captisK,^ ^ in Ula conjUnti fmias^ ^ 
erunt ^ C$ c funSia homoloiai ac rella C£ ^ -ct fm%* 
ter in iifdtm angulis duHa ad ipfas CF > cf mcttrv^Ht 
in punSa hofnologa £i e^ & erunt in tadem illa rdtioi 
ne conjianiii nimirum erunt & fofitiohey & magnitudi^ 
ne homologa^ 

ii%* Cucds enhn I^E 3 /e iti an^^is cqUalibiis wi 
^(^i fii adeoqtie 8c zd fCf fi: j tiun jEC , ic » erunc 
in ttiaDgulis FEC^ fec tam angulisad F i f iequales > 
quam Iztttz F£ i FC propof donaliir latertbus fc ^* ft ^ 
adeoqUe ipfsi tria^gdla Omilia ^ ^ ahguii F£C > fic 
stqnoltSi ac ktcrat CEy rrin eademilla ratidiie^ Qiia;- 
ttiobrem reditac ex C^ r in xqualabu^ ^gulis dc&rtae ad 
CF cf Congruet cum ipiis C£, ^^ 9 & inddent ia iUa 
puncta hdmoldga . ' 

1 14* Patet^ illd ipfa puncta £ 9 ^ fore homologa « 
cum & F£ , ^ indineiixur ad FG 9 )^ in anguUs *st^ 
qualibusy & fint itl illa conftand radone: ac dttds ia 
lingulis figuris fiQguIis puncds hotfiolo^s , cnm iieCQS 
per ea tranfeundbus^ & pofitione hotnologis, aic faciclu 
ne illa conftand pofle invtniri infinita alid ntoiefic^ 
puncta homologa 9 8c rectas , qu^ bina puncta hotholo^ 
ga binaruni figutarum conjungunt , forc p^itt lio*-/ 
cnologas 8c pofitionC) 8c mdgnimdinc , ac facile colU- ] 
ginir binas rcctas> qu; bina puncta cotijungunt intinay 
ad ali^vm, qu; in ^a coDJon^unt alia l)ina quevts > 



. E L E ]^ $ H T A. ^t 
^b^e iadinari Jn epci.eril ^gido , in qito in altera iri^ 
cfi&afiW rectai jutigchtes puncca ii^ hbmologa : ac triatl* 
j^i id fefiili' 'ipst^yk fiomologa puiictd iermitxflt^ forb 

iljr. Si aftird ifigurii pnilihu} hahHerit reSamMz* 
1«^ jffi ferimftfo » hahiiit & altera reEtam^ iffi ^fo^ 
]&i»ffe ,'# maznihtiiine hpmalogamy acfiiihd ejujfmoiti reSa 
iotkwnrani in fingUlii jiguris an^jdos aqjkales conftiiueni . 
. i 18. Sit fenim qiifmodi fccta EB vx {Sfima e figu- 
tis fi5>;j 8c;auctis fiS, 1^B*,& ad cinodvis fcjus pun- 
ttaH I tcaiivl y ddcantaf Iti fccurida ( 30 i vd 31 ) 
hdtfdijfi hdnhdldgsB ipfiifeEi F8 mmpofitidnc tum 
^a^itiiditie 3 fetuhtqdc Diincra ^ i h Ui pcHmetrd fe- 
t&nd3t Bgiifaef i ac hdmdldga ipfi^ £ > B ^ adedquc du^ 
ili i* kit' &: Bdfitidiicj & faia^ittictirie fiottiblogaEBi 
bc afigulds/lpi^.iijlialis inglild FEB. Fadd igitur an- 
Igah tfi «qudli EFI V^rfuiS * 5 doiicc tecta fi occdr- 
rat rcctsE th iii i j cfiint fimilia triaiiguld EFI , e fi 
Jrfcdqiie &:Fi dd*/i iii fci ratidhe coriftanti i adco^ 
^ & piintuim' i feirit irf pcrimcnxl fcciandac figja* 
b^rad homdiogdni I; Qtiare fecundd figiira hab^ 
WtpBo |>eitintetro rcctarti f^, & fi prima habtlcifit plii- 
fe lecUs i fediirida habebit tdtidlriri iii homoldgas j & 
fli iifdetn ^ngulis ad fe iriVicem inclinatas . 

tl7. Si frima figurd hahuerit perimetri farterh idU 

"^fiam cMrvilineaHn i tid^ihit & fecilnda > ac chordd fet 

hina fbigui^um fuhcfa homilogd iu^a ', cum re5Hs quii 

itfvis homologis toniinihuht angulos aquales , Itrunf^ 

^ & jofiiioht i & magnOAdine homologa ^ ac idk- 

fWWf/ indjefiriitit ^r punSla homaloga duilfC erunt po- 

'fakne homotoga ; ipfi vefo afcujt pun^is ^omohgis ier-^ 

Mutaei erunt in taditm illa rationi confidnti , quos 

/Mnde itidim Homolbgbs dicd i area veto gkdcunquw 

Haufa limis homoUgis five rktiis , fivt curvis erunt in 

fati^nt duplicata taterum homotogotum^ 

118. Cun^ eoim firigulis latcribui rectls alterlus £. 
^st , debeant rcfpondere latera recta altcrius , npa 

D i potcft 



33 SECTlONtJM CONICARUM 
poteft hw curvilineuiTi non refpondcre laterl curvv- 
lineo > quod nimirum fi non carvilineum ied re<2ilr 
ncum efiet» ^iUi in altcra pariter redilineura refponde- 
tct. Porro j^unfta in illi^ homologa erunc ea, in quas 
incidenc redtas bomologa; a quibufvis fingulis fingul^ 
rum homologis punftis dudle , 8c iccirco chordx ^ 
qux jungent homologa ejufmodi fxm&SL » 9c ipfa: hp- 
mologx^ierunt & poHtione » & magnitudine » quac ic« 
circo ad re(£las quafcunque homologas habebunc in* 
clinationem eandem • Sl e)ufmodi chord^ finc DE» 
dcy qu^ indefiniie producantur inM>N; m ^ n , t" 
runc ipfe MN» mn pofitione homolog^, & cumhomo- 
logis rcctis eofdem conttnebufic angulos , Coeuntibii$ 
vero pundis D» E; ^» <?» fccances MN» ivMr evar 
dunt tangentes » qu^ iccirco rcraancnc poficione ho* 
molog^ , ^ cum bomologis eofdem continenc angu- 
)os • PoiTQ cum ^rcubus in plures , ac plures pard-^"^ 
<ulas fedfcis in infinitum > chordf femper homolog^ 
iint & pQnGtton^ » & magnimdine » ac earum fum- 
me ad arcfuum ipforum magnitudinem accedant in in« 
dnitum > arcus ipfi crunt in ea ratione conftanci ^ Si 
autcm a quibufvis perimetri angulis j vcl ab t^trc* 
mis chordarum bomologarum utcumque parvanim pmi^ 
^s » ad bina pun<fta homologa aflbmpta fingula in 
iinguli^ figuris ducantur rcctf » criangula illa omnia 
jungenc tema puncta homologa » adeoquc fimilia e* 
runt , & arcjis babebunt in ratione duplicata laierum 
bomologorum « Quare omnes homologf are; figur^ 
rum fimilium fivc rcctilincf iint» iive curvUinef «' 
ad quas arcf chordaruin in infinitum accedunt » e^ 
runt in ratione duplicata latcrum homologorum • 
F.^s II9* Si ex qHQdam fHnSo F in fi^. 32, 53 ad ^<-i 
33 vis funEla E fighr^ AEB dnSit ntiis FE , fafiantw 
in iis fmf^ F.e M *FE in rationt data , vtl vtt^ 
fus E\ ut in /^-32 > vtlad partes offojitas , ut in fii^ 
33» pU0fim c defcrihit ftrimetrum fiiur^ aeb direH§\ 
fimUi4 fis^^ «ACB , hi^ii fHnilis Ikmloiis (00imtih$ 

in 



£ i t M E N T A. ja 

i^fy 9^ mtfariqftec^tnmu9f€y &fHrrlls £> c erkhr h^^ 
mJ9§^r ^ & ^^* ¥Ey Vc; ^ in iii qMvis reiid^ hd- 
mUg€ erunt imer ft fardUlt ^ qumds funSd homologd^ 
fsi^imt in diriSmn cum fttndo comnmi Fy & fi fm- 
Hnt turiMt ferUnetti ^ tdniofitci du£t^ ftr funQd. hon$o^ 
togdf Jhe for fun&d » in qtdkm forimotro occurrent ad 
.ti^dim fortesf vel-M offofitas reStM dulla fer F erunt 
faralkU^ 

IM» Pattt» eum dudti pdr F quavis jndcfiiiiu Fg» 
' & ki ea lUTumpto g ad cafdem partcs iil % ^i , ad 
oppofita^ in fig* zl^ fi:mp€r GFE>^F^ dcbcat cfTc ibi 
iiifem anguius % hk xqualis ad vcrticcm oppofitus > Sc 
toxLQ FE ad He ponatuc conRans * Kq&x vcro qua:- 
vis honudogx ad quamvis rcctam pct F ttanfcuntcm 
debebunt iia ^fuc indinari , ut ^arallclifitiUm fcr^ 
vcnt. Ex pundo F ad qiiodvi^ punAum ptimae fi- 
garz dtt^ rcCta » adfumcnda ctit In ca ipfa ad caf- 
4em parict» vcl ad oppofitas rccta ipfi homologa» quie 
pUQcnim homblogum dcfiniat $ ac tangcntcs pct pun- 
m faomologa £> e ductae debcbunt cum rcoa £r ho- 
motoga continete angulos ^quales ita % ut lcrvent pa« 
taUdifmiim • ^ 

121. Si auttm fitura Jtnt direda fimiks , & Hna 

fKuSa hmnologa cveant » ac congrHot direElio unius re- 

ctt cnm recta homotogat Ut ad eafdem fartcsy vel fro^ 

ducta (od fdrtes. offofitas, ^ recta omnes ex eo communi 

fimctm ducta ufyue ad ftrimetnA» ad eafdem fariter » 

nrf ad offofitas fartes erunt^ in ddea ratione , & homo^ 

hfftj ac habeyuntur ea omnia ^ qua fuferiore numero di- 

ttafnnt, 

^ iM. Nam (i ptinctum F fit communc , 6c coOg^ant 

biw qu«vis rcctac hbraologar FG , fg uorovis modo » 

*m quavis FE » qu« occurrat pcrimctro fccundac «• 

gurx in e, erit angulus GFE idcm ac gfe in fig. qj, 

«qualis ad vcrticcm oppofitus in fig. ;:{, adcoque FE» 

, fe debebunt cOe rect? bomolog; , & in illa rationc 

' conftanti« 

123. Sed jam redcundiimad ipfas Sedlioacs Cooicas, 

D 3 qua- 



54 SECTtONUM CQNICARUM 

gaarum eleganteiii cbn^ui^dnem per ntonn^ COfVlJh^ 
lliwn 6pe ^oiruni yidebimu$ feqnenti Sd^o&o^ 

^CHOLI U M IV. 

}[a4. T7 X proprietatc 3 qnam ^mil, 95. dcrttbntlriavt» 
X-/ JiPWS , facile eniitur mcthodus defcribendi H- 
lipfiinii ^ Hypcrbolcm moti^ coptinuQ dpc fik>rnmqiK| 
quidem paflfim utuntut fafari ligci*ii , oc mnra?|i p^ 
EUipiS . AJjHmpia ^U > cuius hngpudtf ieqmur axi fi*t 
ti&€ Elliffeos , eJHs extreiHa CAfita defyuntur fimfHj^ 

f.J^focorumV ^r f in fig. i^, tm fylo P filttm ciroumducU 
tur ita :^uifemfer exte^fum maneat y & ^^c^^x ^ 
£llifjpl defcribitHr , cUm riimirun^ bin« fP, /P fimtil 
fcmpcr sequcritur , cidcm longinidini fili • Bt vero it^ 
iam datis hinis (lliffeos axibus Mn^ , 5^ , Jhe lihtm, 
liiudine ($; latitudifte Elliffeos quajtta , Joci F , if 4tf- 
rnodum facite invenicntur ,' dufticato nxmitu^ filo , ac 
medio yus funElo , five^ iffq, fiexu fuf^fofito altett v&^ 
tici X axisconjugatiy diducantur bina cafita , dqneead 
axem tranfvfrfum dqvfn^anf , fxt^nfo, fiiq in f ^ & f^ 
Patet ciiim co$ forc focos,^ & fUipfim tranCtprani pcr 
AT, vel gconictrice fa£l:a ccntro iiii altcro vf rticc «ftaii^ 
cbnjug^tt int^ryallo CM femiaxi$ tranfvcr^ ^ inyemelH ■ 
tur in ipfq axe trarifycrfq jfoqi P , / > cqm. niixiiram ^ 
(num.6^. dcbcat cffc quadrat^m (^f difFcrenria qoa-* 
dra^orum C*if, C^;^ nimiripi bi^a quadraia Cf , CBjj 
quse sequanmr qua,4rata ^F , dcbeant a^uari quaclrac^. 
CM^ adcoque ipfi^ xE femiaxi CM. ^ •. 

125. Atfi bina fila it4 jungantur ^ ut attiri/jts cdlk 
fut tanfum excwrra^ utjxa cavut atterius ^ ^uanta ^ldH% 
gitudq axis tranfvirfi quafita Hyfetbola , & ta caftti' 

n,,^defigantur ^cis F j f , ^tim ftyto P Jimujl qvotvantur i 
** i^^i^ fii^^ Hf t^^^/^fy. np^Mnt y & aquales m *^ 
fuq far^ej in i(t^ druaficatitfnq , & exfticationf c 
Xam es^ ipfq fiyla ^ dffcribetur ramus Hyftrbota r. 

fiKS hm i f«? Ilm krv^i^. infijfim f^fr^ • ^«m 



ktm iawo, T«Pi fc «nawtjjL <34>itit>«is,. «teferifceow %^ 
im iim iimm . Fo^i «wew F , / iiM «ibi« inw- 
QiaiKF m «(«. naftCt^Q ^«^qC iQ«rY«^ Mv» cwn 
ninirmn (^ n«pa. H). %v^vm CF^ vd C/in Hypw:- 
^imiW Stmm qi^Atpmm (minii/Lm CJM « 
Cr». iKiQEivK tpO^ .CF;, MtkCfm Wt^ 
i«S^ Pwrtijbo^ib «uiein km f»^ 4e&rii^ 9o«)?it op« 

>^iirH<^4f«f 4//«rii«(» ..fiU, HPf ., nm ImUfuk 

nfmr^-mkm V-fi/m Mf£mf^i^ 4^v*tm »r- 

9i€ i»-HP. , .|4:i!f^ 4iA»pm i» FP . PaK< fer* (cat^. 
Pff W^ fqifl^m ;BP i adeqtjp: pqn^awn P «4 Pwabon 
Um fecp F» ^ii^q^Sljic^ A3,. D^im» sofsm are» 4i- 
Biid^ MP, jcKOSt convct^oi*q ^KJWn?s.afe|r 9f<m >te 

'•'''■• ■ 

V tlQHpiIlirH Tv 

Vf^i ^ft ^(mcuff% >c<l^njin ^irc^lDci par^lor^ 
?WI;j«; W* peil^adi^irfariqiTj qvu Sc(9^ioni» Conic^ 
Ma^iara ^ S<?^iignti vero pi;oycmajc detcrouqabimt» 
iXMjKfuro. tjc^ cujufek pcr focuaa ducbe , ac c|us. 
quQqae. conftrit^^ia l^atiTO ^uTYajr^m, fio^qn^xV e^QpPWV 

PRQPOSlTm tr. PRQ5L. 

'^;* ^it^i^cinio iijp^a daca t>er focani m^f^n» »^ 



V 



ts SECiTlONtri* COKlCAieUM 

F.;5 feiUek «littArid . Deltirflfa ih ipfkn dire^i^eAi in ^ 

36 35 > J^ petpeadtculQ F£ > capianturjiii ea rcAa dM| 

' FV ^ Fn ^d ip&Lttt F£ in ratioile detetminalue $ M 

(numt 4SJ eji» toncwfuf ctim perimetro Se^6ilis-Cd<« 

nitx^ erunt pun€lit u, V^ ^ 

l)o. Si aoteni fit di^ qusvis tldH parrdleUi ea-di-t 

<%rici occurret iii altqod pun^ Q^. Capiannir in tpfa^ 

4ii^drioe Qp> Q£ a^qutfes ipii QF^ pofito punao 6^ 

ad eam plagam mpeciiu Q^) ad quam jacet F re/pe^fi 

V. Du^que GV> fV» "carura .^Krarfiis, fi qui erunV^^ 

cum refta d^a Q^5 |M:odu<3;i$ ^ ipfis j & QF utravil 

ex parte 3 quslmum dpus fuerit i efbnt qUxGti tdn*> 

curfus cum Sedionis CcMicae pctimctro ^ eruntque ii> 

foU, 

iji. Nam diltaa PD pclrpehdicUhri &d ditfcdrlcfetn jl 
fimilia ernnt tfian^ula FPVj QPGi QJ^E , O^P , *•'* 
deoqui eritFP,adFV, ut QP y adQ^ , m dd Qf i 
nimirum ut PD ad FE ; adeoque aitcrniiidd fP id' 
PD, ut FV ad FE iri raiidnt diteftniUafitc , 8t cidera ^ 
cfl: demonftratip pro pundto ^, fubftitUtiS ^, ^ , rf pro : 
p3 G, D. Coritra^ero fi pundlum P fucrit ad Scdtio- ' 
ncn^ Conicam, ic ducdtur pcr V ; ac P rcaa occur- 
r^ns diredhrici in G, erit FP ad PD, ut FV ad FE in 
ratione dctcrminamc, & PD ad PQ^, ut FE ad FQ., 
6b FE, PD parallclas/ adcoquc cx ajqualitate ordidata* 
FP ad PQ, ut FV' ad FQ. Eft dUtcm FP ad PQ^, ut 
FV ad GQ^ob ipfarum FV, GQ^parallclilmum* Etgo 
crit GQ^ acgualis FQ^. Qtrfirc pUndtum < ^uckl ad Co- 
nicam Scdkioncm fit, determinati omninodcbet atfuin-'^ 
pta QG, vcl Ctf «qtiali QF, & dutti GV , Vcl ^V; 
adcoquc punda invcnta ca conftrudione funf ad ipfam' 
Scaioncm Conicam, & funt ea fola. Q^. £• D. 

Coroll. u ,j 

732* Qh£W rectd fef foctm ducta otfkrrit £lHffi in \ 
hinu funcm hinc inde s foco : qutvis fariter o^m ^ 
chtrie ParaboU hinc inde a foco , frater unicam di^ ^ 
nctrici ferfendicularem , ct^us dtera interfectio a di- 

uctri 



i ' t l E U t U r A. ff , 

MB^na nmii$t itm in infirtism nciiit^ ut fhifiium \ 

r^amk. In HyfethdU ^ntiem ^€vis HCkrHt fmel im^ / 

't€rfim j& dimtmm } sUter vtira otd&fus im inp 
^mitd recedit , ta wi^^ium jm Jk iie hinis rectii 
lanc Me directfiH Jnclik4iii im m^uh , imrn num, 
la diximns Mniulm nqmlitMiis > iH tfliiHis iHdiHM- 
fis uf Mnfuk minc¥e kaktkr eitrd directtieem fdtrsf^ 
Im Y in inclinatis ik angiUi In^^e uttrs directricem 4 
13;. Nam rca? qttidem Qf , GV fe dcdiflantes • 
acccffarto fet^l: riiii dcciirrcnt aficttbi in P imer ft>- 
cmn j tc dircAllccm i tcttx vclro GgP , ^V td crunt 
partlfclf , piracto f ita in infuiimtti ateunie * uc iiuf* 
qoamjam fil , vtl tonvcrgerti iul p^vt$ l^ » m iti 
^g- 35 » vcl ad parrcs Q/ , ut in fig» :j« , pi^oatQjri 
fivc QJ? fiierit jfequalil i m^jdr vd mtiior refpectuFV • 
Poito m ESipfi in <}ui FE cft Ih^or* qtiam FV, fcith 
pet F(li ^uevdconlruit citai FE i Vdcft ipfii mi;or * 
cnt ^tae in^or , vd nliilsti imigis ma}or quanl ipft 
FV i adcoque punctum f (cmftt habcbimr « ot iil fig^ 
|5' ^'«'i dire^tficem ad patiea oppo&u^ P ifefpectli F« 
w Parafeola j in qua FE fquamr FV , fi FQ^congttaat 
cum FE, niminim fit perpeddiciilaris direcnrtci , eric 
cqoaiii ipfi FV, & pudCtiim f iu ih itifinitimi retse^ 
^ 9 nt nuiqmim jam fir . In rdiquls verb pofitioni^ 
m anlni6us erit FQ^nlajot , quam FE, adcpquc ma* 
jpc , qiiam FV» & puncttim f bdbebitur , ut in Ellip^ 
ii . bk Hypcrbola vcro 9 in qua FE cft minor quam 
i^ I fi angulus FQE fiierit ejufinckli i ut radius ad 
^us fiftum habeat ratiteem detcAniniintem $ quam nU 
»in» habct FV ad FE , ipfa FQ habebit ad FE ra- 
iaotm candem, adcoque ^quabitur ipfi FV » fic pun« 
^ttn/ ita in infinitum teccdct» ut nufquam jam fic • 
n 4&<em is angulus flierit minot » eric FQ^ major ^ 
bam FV & habcbitur cafus figor^ 35 t ut in EUipfi » 
r Par abola -$ fi autem is anguitls fiaetit ma jor , cr ic 
IC^minor^ quam Vf 9 dccon^rfusj^abibiCf ucinfig.}^» 
fcra difcctriccxt! . 

CerelL 2« 



|t SECTKQNtJW COWICAltllM 
F-J7 i*4- f^* ^f^ PP f^ fi^ ? ^^ *» >^» 37* 58$ 

^9 f* Pr P iivf^^r kif^ii^ in R.» ifi«^« RF , RP , RQ^ 
4im ai -iftgU^^m diff^ri^ i» (Nh ^npi^ FQ£ , & fi 

#> F * P » Ct cqoftitiMMat propoiriiofict» l^armpnimaq 
( n^ni, g. > yj^Vt fecewr bifariatiii in R>. croiu REf^ 

»jp * KQ^vi wmvm ^wm W ^ PQ^* 5€c<ui*im 

. ;r|4 |>|da p«stc|^% PP p«rBei|4ifi;dvi dicod%i(;i) 8$ 

SHr <^i(lein par#:I^ y^ Qgmrat r^d^ IR.pFodtt%» 
|i o^ fit > ii H ,-€iri| RF a4 I^Q,^ Qifnirum HFa4'' 
IQ.111 di^io^a rauonc.pp .ad PQ^ » AimiruiiK ut i^U 
l^ iqu^arum ac^ hiiJQS ^«diaiwci. £ft amm ob aiH 
^Q^ IFQ^, lEF n^o% ^i^ aqgubipi «4 ^ coiKMptiEicnt 
iriaBguli^ FIQ., Elf, rcdt^ iQ^.ad IF • m IF ad IE; 
adcoqufQl ad lE, wr guJirf^ttui^QJ i4qij*Jra,wm FIj 
iH^to^ amiK^ tri^ngula rc4tat«uk Qfl j Q|>P i^ W 
qwdi^uin QP a4 qua^r^wi^ PI>i Efii igiw^ ar 
qualitate or^inata FH ad l£ \x o^M^ucg^ FP ^drqua^ 
<i«?^«Wr P^ viWiwrBin i^ rattQlW dp^e^xuift^^c d^pU^ 

' fii«>** 
. IJ7, IVMf^ c* ftciQ in JPiyrafeoI^ cft ra«ia ^c^alita- 

ib ^cMuc io iJg* j? aq^anittr FH 1 If X §^ proiadc ^ 
|H p^.raU«la^ciUc4^ar£Fv &; di^^driei perpcpidicularig. *r 
4t in Ellipfi in fig. 37, & in HypcrM^ il 6fr 3^ »• 
f^* fft ( niwn. ^<i ) ad ,CF M CM * «c. CK ad CE^ 
in raiipn^ dctcri^inantc , ^dcoqyc CF ad CE m ca4cia 
f^tiQd^ diiplicaf^. Eri^ igimr J^ utraqi;!^ FHai EK ut, 
CF 4d CE.> a^ pioindc du^s. CH , CI j triangula 
CFH, C^ Jmiifta» erwt f" C<»qJ1. 2.|i|op* l^- Gc0m.> 
dc angulus FCH^ £Q xqaaliEHts 9 pud^a I:» ;H> C ia. 
^rc<auii? jaccnt, . , 

SCHO- 



k . SCMOL ly M. 



C9 ac cacutti dilerimeii propoqit ob ocofef; com^* 

^e h Qrbera ckc^ faqxn , Parabol^ t^iafasre niiw 
tato wvzi! citra dir^ftriceni (m^enftun m in&itcu||i» 
V]F|ierMtoi tci^ ^Aiio» ^efu^nodi vxmos ^is^ ind^ai 
^SHdsict. Seeundi aiietrn Coroiiafii fqmnius iii piB^i^ 
poa q^am Se^donum Conicarum p raprifUKc 4e|ntii|h» 
fcMda ttf^ erit pauUo i^ifrn/ / 

^ IJ9. Saut ^toffiem fadlc per^icactqr ^ iUu4 ^ JXf^NTiF«J| 
g^l^ pcf G, & H dudiim ^c^re frpnfiie per j?, & j^i 
Iwipj aactam per ^, & Y^^^c tranfire pcr f, •deo* 
l^ id «Itenim e pun<£fcis V , h com tturoqiie- G > j » 
^ ^KTQm e puncm G\ g cam «oroque V » «^ Pfoiik>- 
[fui| Mvendo fttisfaa;np , " 



!•- 



FRQBQSITIO m, PI^OBL 

\ \lJ ifmt^ , i^enifr ^onevitfnm raSU dM^ ^ 

''' 141. Si fecta dara (it dir^eerici parsdfela , folyeqir 
I«WciHa pcr conftriictioncmproUematis i(n.3i|.,3^j»F4^ 
f^lJBfcat per foc^m' fo|v?tur -pec con&rud^aoop prp- 4^ 
^ffHk ^ ( numb i3f8 ).NSi fit qiKevis alia KH j.^ ^|» 4J 
j^ifa dk^^ci neccfT^i^ ali^i ocettBcet in Hr CW* 44 
jWtur probkma hoc pa6^o , ;* . ' ^J 

ifi hi figufis 41 5 41 » f ? » 4+ > +5 > q*»^ 
**" piima^ ad- Elliptini pi;rti|iet , fectiiida ad PcM^ 
, reliquce a4 Hyj^lkJam pro cafibusi > in quU 
occurrat recta data' foli rama cit^riori > vel /0- 
oltieriori , yel iitrique t Aflumpto puncto L vt>ivii| 
^ff^ directriccm dcmiifoqtte ip ipfa dirc<3xicctn pcr- 
LG , ac in co 9 fi opus (j^ , prodiH 

CtQ 



40 SECTtONUM CONlCARUM. 
60 capca LS » qux fic ad ipfum ia radone dcteftttt 
tiance » cencro L , riUUo LS defcfibjtcur ctrculas i du-> 
Aaque LO parallcla redc dacc KH j doiiec occucrat 
dice^ici in O , nim tdnjundlis punAts H 1 F , doca-^ 
tuT per O feOa «OZ ipfi HF parallela pofico in ci 
putidto Z ad eamdem diteddds partem cum centTo L , 
punfto vero x ad partetn oppoficam ) & fi ipfa OZ 
prdduda uoravis ex parce indefinite alicubi occuriric cir^ 
culo in T vcl r » duda LT vel Lr, ac ex F reAa ip. 
fi paraUda^ liujlis concurfiis Cum HK in P vel jp de« 
€etininahic pundhim quxfimm^ nec in aliis pundis ptf- 
ler hoc modo invenU tc&z, daca poteft dacas Seftioni 
Conics oceurrere^ 

I4|. Duda enim PD ^ ytl pd pe^ndiculari id di^ 
fe^lcem i ob reftas LO , GL paralklas redis PH ^ 
DP , fimilia erunc ttiangula LGO , PDH ; & ob re- 
dfcas LO, OT, TL parallehs rcdlis PH, HF, FP i fi- 
milia LTOi PFH; quare FP ad PH , uc LT ad LO« 
Sc PH ad PDi uc LO ad LG) adeoque & ex aiqttali. 
etce ordinaca FP ad PD, uc LT, five LS ad LG > ni* 
mirum in ractone dccerminance , adeoque punftum P 
eft ad dacim Sedtionem Goiiicam , Oc eadem cft dc** 
imonftracio pro punfto ^. 

144. Conora vcro fi quodiam pundhim P fic ad Se- 
^oncm Conicam dacam, & manencibus cmvctU duda- 
tur LT patalkla^ FP , doi^ec occurrat re^ OZ alicii- 
bi in T i cric LT ad LO \u FP ad PH , & LO ad 
LG ttc PH ad PO, adeoque LT ad LG uc FP ad PD 
tn racione decerminanie , in qua cum fic hS ad LG , 
eric LT asqualis LS, adeoque pun&um T ad circulum« 
Quare puncmm quodvis , in quo recca HK occur« 
rac Scilimi Conica!, def;|ec inveniri expofica conftruftio- 
ne per concurfum re^ ;cOZ cum circulo , & Ada 
pun<5la eo paao j^invcnca func ad Seccionem Coaicam 
dacam : Q. E • D , 



SCHQ^ 



E L E M t N T A . ^t 

S C H O L I U M. 

Mlrum fane quatn foccuhda eft b^ eonftra. 
cnoy quam Tyrom czerccndo apta . Phiri- 
a quidem cx ea inferri poflunt thcoremata» &plera« 
'^ uiUiffima ac iterura foecunda : curabimus autem 
vam ffcri porerit ^ ne tanta rcrum copia confiifio- 
pariat. Intcrea notaodum iUud ; pofle punctum 
^ aflbini edam ultra directriccm » quamquam noi iti 
A&c ftbcniatis ipfum fcmper ciura dircctricem affump- 
prous • Deinde poffe ipfum affumi divcrfis locis « «que 
pdto fadrtorcm conftrucuonem exhibcrcnt > fed tltir 
ius g^eralem » Sc generalibus theorematis erucfti^s 
linus :tptsm . Potiflimi c^fus , in quibiis conftructid 
Kitrahittir , funt ii > in quibus affumatiir puncmm L 
l ipfa perimctro Sectionis Conic^ , himirum in ali* 
10 puoao P jam ihvento 3 quo cafu radius cireuli ef- 
ipCz rcaa PF, quae ad perpendiculum PD rationcm 
tdeterminantcm ; vel affumamr in foco ipfo F» 
cafa radius circtili cffet dimidium latus reetum ; 
.in fig. p > 10 » II FV » cum nimirum ' fit FV » 
pctpendiculuni tE pariter in ratione determfnatite % 
pro Ellipfi, & Hyperbola in ccntro , quo cafu iH 
19) 20 radius circidi cffct fcmiaxis tranfverfus CM» 
ifnum. 90 ' ad pcrpcndiculuro CE habct pariccr 
m dctcrminantem , vcl pro quavis Scctionc 
i} in ipfa recta data , quo cafu bic punctum 
coDgracret cum puncto H > punfto nimirum L ja^ 
« in ipfa KH . Potcrit Tyro conftructionem hanc 
dcm ad hofce cafus particulares contrahere ac 
e quo paao mutata pofitione $ vel direotionc 
ctac dat? , poflint crt^i plurcs fatis diverf? & clegan* 
E conftructioncs, quibus omnia qu^fits SectionisCo^ 
itx puncta invcniantur . 
146. Et quldcm ipfa conftructione noftra g^nerali 
t^tiavtniri punctaomnia , fi nimirum mancntc dircciia* 

nc 



• ' * <• /•■• ,••*■• • ^ 



#2 SECTIONUM CbNlCARXJM 
£e rcctab datac mutetur ejus fk)fitio ^ nithirbni fi fn 
le ai^gulo ^d H . eiccqrrit jpUnctitm H per fotlm dii 
toiccto^» Vel ^T& da^m punctllm qiiodvis) tit ^ii 
^m > vel in EUipti & H)^perbdla per cehtrdm cbti^ 
iftlbr ificeta ; In iis , d^ibiis pofitkhiibtis >recl^^ iM 

jpat^clo d^li(tac>v^ I^^ 4^^^ <P^?^"^ co&ver£e I 
•ixebini^r ohniia iScttiomis Goiiicac ^uncti^ 'St Jcii 
^ikkHmen' £»:ile detegetui: > atqtie htoC ipfani fidth^ 
liebtiniis vtam id ediendis iisi qlue tatii multi fe tf" 

M <iffenilnc; / . . . i , ., » v^ 

/ 1^47. IiUerea qiiod ad ipfam coh(faructionem.peMi 
.jibk^ iUud. Sireiiax data direiiriei farAlkU.Jkij^ 
^.H» O &a in infinkHm 4iktimt ^y Mt mjfqHam 
^mi 4P ji/ca^ refStd riranfeat fer fdcim F ; conj^tthd 
iHK* I^F, cong^kunt ktiam OZ j OL; & ht i U 
iMir »5r &^; FP, F/^ in illi^^ ^uihiut in utrd^tie hoc 
fkd ienitraii /wr c^nftruEhiont- deferifnnk . Poflct 4 
dcm «X ip£ii pro ucro4iie tiftt pcciiiiaris dbnftruj' 
^dciV^i; fcd iitrique . cafui confulrum cft in.trdpi 
.^ '^. Prqindc. iis oiiiiflisi erufemiis hii primo locd 
iteralifi ibeoremata, qua: fluunc e motu paraUeld 
:bt» daiiae .eandem .fen^ indidauoiiem tetiiiehcis zA 
irecctricenl ; 

.. i43* Ac primo ^iuidem Coi^oUar»» plurima (Imullid 
ilUhgfemils oimis inter fk analoga , qux provcniune c 
imicQ eafu Hlip&ob 9 iti ^uo tecta data qiiamc 
ithofitioiieai iiabeac ; & i^ binis. Pairabolae , iri t[i 
^ink) eafic direarici liccuniqiie obliqtisi i' iii tcc 
rferpectdiailaris » ac demum e ccrnis Hyperbojias ? in 
.(um primd recta daci faeiac cttm dirccdrice ahguU 
riiiiiiQrtm angidb sequaliucis» in fccufidcji cqualeni 1 i 
lercio majorem; ' . , • 

.CoToll. u i 

i4^. £ reSii^ omn^us data. reSta farallelit iin^e , 

tir :JSi(ipfim ^otttingmt finidainfiniHlis fmctis > rei 

mnnes , qua iis intetjacent eam feCant in binis f!i 

^fmetie « ^a ixtra iUas eadunt « iffi nnfqpiam « 

r^ff^ In JPaitakola uniea continjpi in nnico vunct^ » 

liiui 




' jrtofc' M/Jito i |H«l i^r« t&f ^ fii tUrectHci iefS&^ 

f*ctuJ!hi»U , jtfMhi iMerfect&pfe itHin iS. 

^.*»fftl^ «*w iirtctrici MHgnlMm Mi^^ m. 

»«»]M»»1 *f/- <r*fr< |<ir 1*4,^^ u Si '^ectitdtttti 
f^ii»i^ *»lMit4fii. i^ uHic» -tx cmnaHi #. 

■» tr» qnk^u jd»A difimUa «^^ jferrtr i *^«» 

^i/yBjirf* r^«, «rtTM^* i ,»?/ «/,^^ . 
mervn kui^ u^ii ^Mi ^^^„, ^ ^^ ^l 

merfnthane Hd ift htfiAShiih al^mte ,• «f tttOqM^ 

M , Si.dtthum mitlMi ifitmMionh ja anguh 

fmt m^or i «mnis<reet* fectinir ks JRhvpetlni 

i ftngtdot nmimn rdmes ^tuiikt ih tmtu K»^ 

W.Hoftim ortittium (fctnofiltratid fyotih flmt fiie^^f 
wis pc^enibus dmtubus circoU rcfpcctu diitctti/^ 
tc rtctafttat LO ,- 02 pofitionc rcfpecdi circuK . t! 
JttBo ^ttidem tii EUipfii m qua ratio dettttniriatis 7; 
HBho minpris irixqtidKtiitis i ^it LS mindf , dtiam tt 
fri« m % 41 ; io Patafaola att^iiaiis i coeuritfbfli 
etpMctis G / S , rit ia %. 4*1 in Hypetbola ftra* 
Ej**'"» J^- .43 V 44 1 4J • Quare drcuius ia tU 
B*l dtffecmcem non pertiriget / in ftajeibok e^ 
•*«rt m eo ptmeto , ia quo cofeUat G, S» iri I^- 
iiltra earii tranfcuifet , qaam prolnde mmt 
lais pun<ais N , », ad ^af du^ LN ^ L» indi- 
nntf ad tpfam dhreiftrieem a an^jlo *qdaUtatij ,- 
aiimram flt radius ad -fittjmi aagftli LNG , vel 

♦ ut IN , vdi^-*! iG^ nimirum itt fttidae 
tounantc. #«. 



4+ SECTIONV.M CONICARUM 
151. Pr^^erca & Itaa <.0Z cti:ciilu ooGRnr^ ift bkik 
|UDCti$ T % $ 9 patct reaam KH d^bcre Scctiont 0>* 
nicf occurrerc pariter in Uqis puoctif 9 dcnH>to cafu.a 
quo LT I vcl Lr coiigrqac cum 4trectiooc rcct^ OL j 
recu Tcro KH uoti tr^qfcat per focum F , quo nim- 
rum cafu recta FP> vel Ff c^vadtt parallela rectat KH» 
j^uncco P vel / f iA quo d^berent concurrere ad dcm^- 
mioaniluni Sccupnis Conica; punctum j ita in iixfiai- 
|um, ^bwnte % ^t nufiiuana-^ jatu fit « Quod fi re^ 
aiOZ circulo nufquam occurrat » reoa quoque KH 
tm(quam Qccurret Scctiqni Coaica; . Facile aittem col- 
ligitu^ 9l^ iUo4 ; pun^iun P vcl / debere iaccrc cicra 
vel ulcra dir«ca'iccai> prout punctum T it vd 1 j.acuc- 
rit . aid eafd^m pariei^ direcuricts cum cenoro L » vel a4 
cfipo^t^ A cnm in figuri^ prprfu^ fimiUbui FHDP » 
TOGIL^ » & ad direCtricen) Aji fimilitcr pofitu& • dHK^ 
ct^ix ipfa debcac vel utrumque e latoribas fecarc 9 vcl 
qcutrum • Demum fi cocuntibus puctis T » ^ » recu 
ibZi ^vadat taagens circuli % evancfccn^ arcu iUo ii^- 
tjermedio Tt^ coibunt etiam puncu P « / in EUipfi % 
ic recta KH evadet tangens . 

.> IJ2. Mancnte iguur.iudin^tipnc rectae KH ad di^ 
rcct^cenij iive maoente angulo ad H » concipiatur ca 
recu.motu continuo translata iu j ut punctuni H per- 
currax j^oum direoricem % devcnicndo cx paris finu 
fira A ex diftantia quavis indefinitc m^gna vcrfus de- 
xccram B > Habebimtur eo pacto omnes rcct^ iUam 
directioncm habentes > ^ licebit contemplari quando j 
8c qua ratione in datx Scctionis Cooic; pcrimcKrunv 
incurrent. In omni eo motu punctum O manebit fem* 
pcrcummaneatpunctumL» &incliiutiQ LOad directri* 
ccm. Rccta FH pcrficiet dimldiam converfionem cir* 
ca punctum F» tcndente puncto H dextrorfuQi > adeoqoc 
Sc rectae Oz.i OZ iUi fempcr paraUelx dimidi^xn cpn* 
vcrfloncm abfolvent cpdcm ordinc } fed fi oentrumcir« 
culi L aflumpmm fuerit dura directricem 9 quod ubi«^ 
que prxflitimus, punaum z. tcndet a finiftra ad dex* 
tcram^ puncmm vci;9 ? ipfi cj^pofitum raxiiifa a dex-* 
.^ . ' *' ' ^ * ^ icra 



f ^ . , 

il loiftram . Ea' inesips CECulis difiigenter fiiteoda 
V Bt liceac imia) irehit confpeiStu cafus compleAI 
, lofo fpacio pen lineas KH> OZ iodefiniir utriaT 
piBdiK^as tan({aam p^r cverricula ^3^;^m yela( 
iifo. 
15^, Incfpiendo db ElHpft in Sg* 46 habebunrur 7 F^ffT 
fi Gafus iineas ;(iOZ re4>ondcnt€& toridein cafibus 
HK/fi>ee FH. In ppimo cafit.OZ]^ extra circit> 
cadet ex parte dextem^ tum in iecundo redaOZa 
ipfom continget alicubi pariter- ex parte dextera ia 
^^Mdoa Tfm&o Q^> deinde re^ OZ; adbuc eentrum L 
^*xieltD9iiens ad finilbam eirsuluni ipfum fbcabit in binia 
.^piiii^ Ti , /i tum 02^ tcanfiens per ipfum cen-< 
PiRiin fecabit circuhim in bini$ pundts T a » / a deindp 
^OZj relmquens pm ccntouin ad partem dexteram ipAim 
iroilum piiier fccabit in punftis T 5 , t^ ^ tiim OZ^ 
pQfcrt iterum alicubi in unico pxm^o f ex parce 
ta > at demum OZ7 extra circulum cadet pacicer 
jarK finiftra . Eodcm igitur paOru in primo «afii re-,r 
HiKi extra Ellipfim cadet ex parte Gniftra , tum 
i feoiodo recta HiKz jsLm ipfam coacinget aKcubi pa-- 
^ter cx par^ finiftra in unico puncto !> dcinde reaa 
^l adfauc fbcum F reUhquens ad dexcerani EUipfim 
^m icGahff in. binis punctisPr^ ^i) tum H4K4 msobi^. 
p^ ipfum focum fecabic Ellipfim in binis punccis 
ip9 illis nimirum.» qu£ dctierininavimus conftriH 
fecuadi problemacis num« 128 juxca num. 132 v 
H5K5 relinquens jam focum adparcem finiftram 
paricer fccabic 4n punccis P3 » jp- 3 J tum HsKfi 
i iterum alicuji^i in unico pUn(:ca i ex partc. 
a, ac demam kh^? cxcra EUipfim cadet pariter 
Iflftie dextera . Quamobrem 9 recns. smnihki dat^ 
fnoMlUlis iifia femfer Ellifjhn c$.ntinimt fingid^, 
finguli^ jpmiUs h reii^^ ofnn^ x 5«^ W int^accnt » 
ftcant in duohns j^itn^is , qn^ cxtra illas cadnnt ^ 
mifquofn ocxurru^t : quod quidw de ElJipfi gropo* 
amps. 
■154. in Parabolafi rectadata fic obliqua ad directri-F*f7 



4^ 5ECTl6SrUM CdNrCARUM 

cmi 9 qu€m cafum «xbibet fig. 47; habetutltur caful9 tan« 
tlmimodo quioquCd qui h^mifum eodem prorfos p:lcai 
procedeiltydc; JDumefp fupcriore idEUipfi. Sed qiloniam 
faic-tpra ditecaix OA cotitingit drcidum ih illo pua* 
ctO) ifl quo cocunt G, 6c S9 pbft lineath it^OZ^ qu^ 
yis iiiiea ^&jOZj^ utciimque exiguum ciuh dircdtrice an- 
gulhnl continehs ipfum circuluih fecabit ih binis pua* 
ctis Tj f /ji Quare Utcutiquc phtictum H5 recedatvcr* 
fus B^ rccist FH5 cohtinente ctun directrice angulum ut- 
cunque cxigiiuni , fctnper rccti H5K5 Pafabdlani fec»- 
bit, in binis punctis P^> pj - At fi rccta data fit pir- 
1^.48 pchdicularis direcirki j ut irt fig. 48 i jam cttara LO 
eVadchie [k:rpehdiculari ad direCtt^iccm , ipfum O coti«i» 
gruit cum G» S ih eo puncto , in qub directrix circu^. 
lum tangit^ 8c cafus deducuntuf ad tres tantiim; Qu^ 
vis ehim ;cOZ e^ illo cohtaau ducta circhlum fecabic 
in ipfo puncto Oi ih qtiodproinde abiburit ohiniapun-^ 
aa r» 6^ pr^terea in aliqud alio punctd T; Nulla igi^ 
tuf c|ufmodi recta HK^Parabolam cohtiriget; fecabitaa-* 
tcm, quaevis ti iis in aliquo ptinctd P i qhbd detdrmi^ 
nabit fecta FP pafallda fcct? LT, & in cafu.irect? Ha 
K2 trahfeuritis pef focum punchim Pi dctefminabiturcor^- 
ftructione Problcmatis feciindi , vel Pfoblcmatis pcimi ^ 
ih qud vefticem axis cujufvis Conic^ Sectionis inveni- 
mhs num. g6 & quidem ih Paf abola uhicum ; Rectsi 
ahtem ex F pafallela rectaer Lr ducta 9 quas >deber^t al« 
tcCam inteffectionem deicf minare fect? HiKi vcl H3K5 
oim Parabola ^ congruec cum ipfa FKi , qu2l ipfis pw^ 
ralleia e(i. ita , uf iaterfectio po({ iieCefTum ih infiniw 
tuiii niifquarn jam fir. Quare omnium ejufthodi recta-^ 
rtitil umca contingit irt umco punct^ , riliqM emn^ks if^ 
fam his ftcant , vit ipfi ntijquam cccwrrunt pnna fift^ 
a taniente verftts focum , vel ad fartes effofitnu , fNb^^ 
ter rectas directrici lerfendiculare^ five axi fairaUUas i 
qfiarum nulU toftgit^ fecant i/ero omnes in finj^is jHm^ 
ctisfingula^ altefainterfectioneitain infini^miidmlnfti W 
n^ifquam jam fit • Qnod dc Parabola fucrat pf^ft* 
iim. ' - ' • 

IJf^Pro 



E L. E l^ E N T A. 47 

t|j. Pr6 HypttboU feciat primo data recJla €umdi- 
Mtfllc ^illdm mlnoreitl ^^guld a^quaritdcU , ut in 
^ ^^ » ^ qdbniam t» j LN incilnaritur in ipfo ar-F4^ 
tSs.Anfufo ( iSum. 150. ) , rcda LO datac re- 





tifil p^alkfi , iiebqufe . Cotitmchi inguluui trtihbrciii 
ffSis iNh i L>/N deoeWt . difciftfici bccutrer^ m aK- 
putL&o O feitra circ^uiuni Gto ; Quarc dunfi recta 
faris diftat a foCb F iti i ht FHi fatis incline- 
ad (KrcctfifeAi , recta quideM ttOZi hoii oc- 
h&itt cittulo e^c partc 2 1 ^ fed tamcn ipfam fccabit bis cx 
||4fe bppdfita ti iA arcii dltra dirc«5h*iceni excurrentc. 
Eb ctfu patet c^ huni. 151. r^ctas FPi i F>r {)ai:al- 
l*b redis LTl i L/i debefe. bccuffei^e ipfi KiHi in 
Ifcis punctis Pl j pt iiltra difectriccm dtis , himlrdm 
fcMc pccurtcrc^f atmb bltcf lori . JHypcrbola! , atquc id ac-^ 
fidci, donectiOZicphtihgat illum ipfumarcum alicubi 
i y T^&^ HiKz iptumi ultefiorcm rairum coh tlngente 
i : tmn fcdl ZjO^j hiifqiiam circulo occufrct, & 
^a HjKj huiqudm occurfat Hypefbblaci UW autcm 
' OZ4 cbh tigcrif cifcuiuni ih (T ciuri difcdlri- 
^ fccti* IC4H4. cohtirtget jim famum dtefiorem a* 
iih l y ic deinceps cafus quihtd> i fcxtqs i &fc- 
Kmtte fe hAcbuht pforfus ut Cafus tcrtius ,; quarrus 5. 
*i}uintus ih EUIpii ; ac quocumque m imnicnfum 
'Cdat rt/ veffus 8 femper pbtiricbit idem Cafiis fcpti- 
*i . Ijgtmr ji re^a datd ijficiai cum direclrice ^ angn^ 
f^inoreM angnld ttqualitatis, , bina ex ommhHs re^ 
m jfa^allelis contingunt finztilos ramos finguU. ih 
^fi^gulis , reliqtit vero nufqkam bvcurrunt ^ vel fe- 
in hinis punSlis eundem ramum , prout iis interja^ 
iel txtra kas cadani . Qiiod prirao loco de Wy- 
xb propofuimiis; 
18. QKiod fi rccra data fatiat cura dlrcctrice a^igii- 
*qUalitatis, lit in fig. 50 , rccta 1.0 abibit in ip-^ 
LN, abeiihte puti($o O ih N . Qaamobrcmqu^- «^^ 
Kda pct O ducta fecabit Cifculum in ib^ puncto 
S vfei Nj ih qubd proihde abibunt omnla puncta t\ 
^racttrea in alib puacto T, prn^tet unipam ZzO^c^; 
* >£ i per- 






4S SECTIQNUM ^QNIGA|IUM 

{serpendicL^ar^p radio LN> qiiae circalum coQtingci». i| 

quoquc intcrfectiopc T^ ibi focunte cum t , &cumi 

'ac N . Donec igltur putictum FJi fucri;fatis iC€mot| 

a fqco, apgulq FHiBfatis ac^to , rccfa ZxO£ife< 

expartc z,iinrTi arcumcircuU j^centenpi vlvi^ div< 

cera , & recta KiHi ramum ulterioreni in Pi ^ 

uhte autem. / in O reft^ ipfi Lf paralleia ^x F d\ 

erlt parallela ipfj HiKi , ac ej^s intcrfectio ua in 

|initum recedct , i)t nufquam jam Gt facta Z2Z,2 1 

gentc circuli ,, ufai & FH2 pyaclU pwcndicularis 

ctas K2H2 , ac projnde abeunte iii Q ipfo etiam pu| 

cto T2 , recta ipfi LT^ parqfllela ducta c focQ F cji 

det parallela ipfi IC2H2 ^^ ^c pro^ndc utraq^c intei 

ctio determiiianda nimirum a ppnctis f > T ita in 

finltum abit , ut nvifqu^m jam Ijt : uhde confequia 

rectam K;iH2 nufquam occurrerc Hyperbolae . Ra 

quis autem omnibus OZ^ 9 OZ4 , OZ5 fecatitibus q| 

culum in puncto f coeunte cum » & in alio pii 

cto T? :j X4^ T5> citra directric^m fito^ feliquac o^ 

nes K3H3 , K4.H4 , H5H5 fecabijnt ramum \h€^ 

rem Hyperbola& iii unico puncto P2 > Pj , P4 fii^ 

ix , altcra interfectionc , qux nimirum in rectls K^l 

K5H5 detecminanda erat pcr panctura t ^ ita iti i 

nitum abeunce, ut nufquam jam jfit ^ quod d^ intt 

{cctionc r^etae iC^H^, coniftat ex conflructione profaL 

num. ijo ♦ Cum vero quifivis ZiQ, Z3O ^tcumqi 

parum inclinata ad illam ^20 pcrpendicularcm raij 

LO circulum ncccflario fecet iii aliquo puncta Tl 
T5 hinc, vcl indc a contactuOji paritcr quxvisKiti 
K3H3 ii|:9qnqu« proxima illi K2H2 fecabit ramut^^ 
tcriorcm , vcl ultcriorem in aliquo puncto Pi » j^ 
ac proinde rcdla illa K2H2 indefinite producta ab( 
det hinc rai"no ulteriori ^ incje citeripri indcfinitc pl 
^U(Jti$ magis ^ quam prq data quavis diftantia , q^ 
ipfis umquani occurrat , quod ipfum cxprimit afyi 
ptoti notncn . Quarc in Hyperbola , {\ rcctas , q| 
j^arallclap Vunt rec^ datae , Cum dircccricc efficiam i 




^mbus efi afymftdtus , ^«< nimirHm nufquam iffi 

uamx , fti bifios ramos iiligquit hinc inde , lieet ad 

^tiat nMgis ^ quam ^o data quavis difia ntia uf- 

"^fxrva i Yeliqua ofnhis fecant in finiHlis fHnhis 

t rahfum citeHohfH , vei ulteritfrem $ fr ou$ jacue- 

hmc inde 'ai afymftoto fibi fkralleta , alterd ek- 

» imerfe&ioffe ita in inprtitHm aheunte , ut nufl 

^)0^ fit . Quod fccuhao loco dt Hypcr bola pro- 

fuiraus. 

157- S Vcto . aetrliini r^ecta datdi faciat cam directii- 
an^uKte hiajprcm angulo a:ciualitati$,.ui in hg. ^i, 
a Id acecdct magis ad pc^pchdioilum LS , abcun-p ^| 
panqo O intira circulum . Ciuambbrcm qliacvis cc- 
tO fcf ipfuiti O ducta: feeabit clrcuiurff ift. binis 
cdi , Quorura altctilm / jaeeBit tUtra dircctriccm , 

fcrofa T citra , Qucvii igitut rccta KH fetibit pati^ 
Hypctbol^ in binis punctis , quorum altcfiim p 
bit ultra, alterum P citra ditectriceiti , quod derc- 
K2H2 ttanfeuntc pcr fociim dcriionftratuni cft cx 
!rucfioiie probfcm^tis fccundi,; Quare.fi iKle incli- 
biis togufus Cii majot angtdd JBqualitati^ , omnes 
\'fiBa fecani tiyperholofh ifi binis funSiis , nimirum 
[hUs fdmos ih fingulis : Quod Cfat pdftrcmo loc^ 

opofitum dc Hypetboli. 

CorSlL i. 

tjS: itBa Co?iica7h SeiHoneTh hk in pluribus i iptam 

Mks fuhElis fecai , nec iri pluriius , quam in uni^ 

9 mtingii .' / ' 

t)9. Patct ti Coroll. x, , cx ed nlrfiitum » quQd 

OZ Ci^ciilum ncc in plufibus, qiiam duobus pu»-» 

ii ftcat, flec in pldritys^ q[uaixi m umco, ccihftfl^k 



1 • 



E 3 



scHa 



^o sefiLTIQNUM CQNICA^UM 
5CHQI.IUM {I, 

i6o. 4 Dmirabilis faQc ac |iotam dignifCni^ ^ft fp 
J\ fywptoiorum natura , qux nimirum fi per* 
petup prpducantur , perpetuo ad lincos pariter pro^l^ 
ctas ita acccdunt» ut n^Ua fit diftantia utcunquc p^ 
va > quam aliquando oon tratifcendaat ; Jicet omninf 
nunquam coincidant » in quo cum convtrgentibus (pp 
ritbus analogiam habem Aimmarn» & plurinu f^nt eap« 
rum genera» de cfutbi^s agcmus fuo loco • Incereay ut 
evidcntior evadat Tyroni res» immeqiate ^tign^ ^o^ f^r 
cto demonftrabicjir , 

i6i.:Si J^ctji KsH^ in i|g. 50 ufpiam H)»p6rbp]| 
cccurrciret m |l., v,cl r ^ ideb^rct eflfe FR ad RH^ ^ 
vel Fr ^d rH^ iqiratione sqi;ialitax^ ci^i liaea c|ajtai0 
angulo <equalita4s potiatur indipata ad direccri^cin. |^ 
«iutem fieri omniiiQ tion potefi ob. angulos RHzF » 
fH;iF rcftos . Rccta igitur K;iH2 quautumlitei prodft- 
catur , nufquam Hyperbplas occurret • ^t fi fbmax^r ^ 
qua:vi$ ^iHu vcl KgHj iai;^ns ulrra ipfam> vel citr^ 
& ipfi utcunque prpximc/, illa Hyp e^:{)pl? occ;mn:ct , 
aique ocpurfus fafiic--<teter/m.natiir . Si. cnitn c^ occa- 
rat rcctae FH2 in i , vcl I, crit angulus FHii, FHjX 
acutus ob angulum tiHi > FIH3 rectum • Qaart fi 
fiat an^^los HiFPi^HjFP^^ ^qy^^ ipfi FH«*, PHjI» 
adeo^^jp pariter ajfvitu^ , recta FPi ^ JFP^ pccuirrct a- 
licubi-fccKB Hii 9 W?i in Pi , P2 , critqu^ Wngtt- 
lum FPiHi f J^PaH j ifofcelf s , ^icfrpinde fPi, fPz 
ad PiHi ^ P2H3 'iti f;atione «palij^ti^ , J^ PViWWVft 
Pi ^ P2 ^ Hyjpiijj<4^m , i^ifSfm priiiiwi iacgbit ju|- 
Xt;4i directricem i ut i > fecundum citra , ut I • Qu^ 
quidem demonAratiQ ^ fimpliciffima ^ Sc evidemiPt 
^ma ?ft. 

162. ^iraul ^uiem hic etiatn fint Hpirculq probl^fna 
^^lpipd^m fa^iilc folyifur mvetiifndi piinqum ad Hy- 
pi^fiiolam \n recta. iqcjinata in angulo xqualitatis , Sc 
pftM « Cqnftfijgtipn^ ipfe ^m ifl mkq puQCtp Hy- 



k 



£ t B M I. N T A.. 51. 

||S s^iittif0in KH dii;)eari<d |BctpcPdic«4.%riuiii ioterrccuo 
^}mboU faciiifis ii).V^oj^ facco. ^Qgido. HiFPl 
^mokli «QgulQ JFHiPi ,: Kcs. ^^pfiem icdii ^ cmn ibi ao^ 
ppriK mm ItqfiiaLpra x^iiJQis^QCQ. i^quirf t, >, ^ pi'oiodc 
pg^rum squ^l^atis vi^s gcrat ^ . 

k. i6j. S^qucj^^^ibos bu|u4coHiftructblus.CorolIariiserue« 
pB pnfpn^ proprictaie) quafdam liiWii^n. UQcaruii^ , 
lliMi^ IqU iiitt^ omaes iibt ^araUclas ^ecMoiiii CpAica 
i»aiMP pccurraot^ 9?}Jqius oirniitnil?. ^am lecaQCiims 
irleaKl, mm faciemo; gr«du$. ^ cas ;> q[uaruni tim t^ 
imti^, a>iiti|iguQ(^^ 

CoralL Yt. . 
F. ' -itf i^. Hyfcrk^ afymftoti funt kw^ > fi^t^f^tn^ 
^»im. TiSis s foco 4i^M ^ pmm interftEHon^ ckp§ 
^inSrkt > tranf^t fer ^ntmn , ki^os, ramos, binis ad, . 
i m^. <fifofit,is angHlis contismt > ^m^/ ^ulos, axis, 
' ^'^^^OBfui Hfaridfn fccat , ^ c^nm fegmenia intetcc^ 
j^ «00* wmjm. ^ ^cSricm, apiammr ^n^a^ ftmiom, 
i 4p^afytKfo^^ . 

^ 1^5. BMas jcffir cooft^c ex eo. , qood iiabeantor bi« 
' ^ ipdiQadoncs LN , L^ in %. ^o hinc ipde in an« 
^ {rio. cqualitads > ^. iingulac haberc debeaiit ^fympto- 
[ M fil^i paraUdb^ • f (re perpcQdic^lares reccis a^ feco. 
^ ^Mi. ad cttwn in^ecfcaiones. cum ^ircctricc > demon-* 
^t^Qn cfi Qiini.. i5^.RcHqjua fic dcmonftrantur. Gen- 
"* ^ C ifu^^ryaUo. ^Bmiaxi^: tranfvcrii CM. in fig. 52 in-^F.ji 
^lMau: ia dirciptrice punctaH, h ductifque CH^C^, 
^ *IH, ftfe, €cit CF. «d CH; ^t CH ad CjE,;iq ratio. 
\»^minanic> Oiin in ca fit CF ad CM\S & CM 
j^CE. ( num. go ) . Quorc primo. quidcra fccMC CH, 
rVv », qiiacum r^io ad QE cft eadcm ,^ ac ratio radii 
M ^tupa anguU CH£ vC^E» inclinantur dircqrici ia 
^fjjB^ ^SQUsdixajtis ( aum. 10 ) . Dcinde iimiiia: e«uac 
PigP9da Qtf, CHE ( CoroH^ 2. Prop.^ 12. Gcom. ) 
ri^ jUkgulus. CHF jcrii: a^quaiis rcc^ C£H , ad|eo- 
|W^ Ql <cric afyljQptotQs 3 & eadcm eft demonftsado 
1^ Qh , qfjgxm WBfiuc prasterca ex conftniciione )p^ - 

£ 4 ^acur 



55 5£GTIOMOMGOTIlGARtJM 

'<5tiatar fcmiaxi tranfverfo CM . Patet aiitera triMgaJ^ 
HCh ifofcclis ahguluiti HCU ab aXe CM perpcftdicuii-l 
ri bafi fecari bifariam^ lit & b^m ipfsttri, ac cttlti fiis-^ 
giils arymptoti binas ramos hitic inde relitiquatit » cvtil 
portet tami ipfi jacsafitt in bitiiscaramatirgulis od vts^ 
ticcm oppofitisrf • i 

Corplli 4* ■ ■ A 

166. DiflAntU foci Mh iTirerfeEiioni HfjmftM cfMi 

AireEhice aquatHr ftmiaxi confiigdta , ac utrique ^iquatm^ 
pgmentum tangentis per vermefn tixis dHild i & inm^ 
ceftkfn iffo 'nerti&ei, atpte afjmptoto^ a 

167. Nam ob anguUim FHC rectaftf j clf quadrji^v 
tum FH difFcrentia quadratorufiv CF , CM, cui.(niim. ^ 
^4 ) ^quatur quadtatum femiaxis cdnjugati CX, adeo«'< 
que FH, CX asqtantur intcr fe • Si autem rccra axt 
pcrpendicularis per Mdiicta, quas' ibi Hyperbolara con- < 
fingct { num. 4.8 ) y occurrat afymptotis ifl.Tj r^ aequa^ < 
lia crunt triangqla reotangula, CMT i CHF > ^otufxf 
angal^s ad, C. commtmis y ^ latera CM; CH aequaliay.' 
adeoque & MT acquatur FH , & CT acquatur GF i ac.. j 
€^cm sft detnofiftracio pro F^, Mt, Ct^ 

CoYolL 5** 

i6t* Afymffton ptnt iiametri ejus rictangnU > f ir^ , 
efficiHnt reBa ntrique axi farallela i dncta fer alteriits 
^ertices , hahentis latera iffis axihns aqualia f ac ra^ i 
dius ad tangentem anguli , quem fitrazds afymft^Hd 
continet cum utrolibet jlxc ^ efl ut ille axis ad 4/r<- 
ftttm r & Mjf^oU , qua hahent eofdem ct4» t^defm^ 
iaxe afynifiotorum angulos , funt fimiles , & viceverJa-A 

169, Si enim per altcrum axis vcrriccm in ,duca'tar»'i 
f ccta axl tranfverfo pcrpendicular is ^ occiirrcns . zfyixh^ »4 
ptoti^ CTy C^ in I , & i, ei'unt eadem demonftratio- « 
«c ml > mi aeqttalcs ipfi CX, Cjii:^^ cara & «5, &M/, i 
MT fint iis prastefea. parallclae , rcCta quoquc ^X > T* I 
parallel?^ erunt, & aeqiialcs (. Corofl. i.Prop.i.Gcom.^*! 
fciiuaxi tratifvcrib CM , 8c rccta IX > i^r feniiaxiCii», l 
ac lotum T/Ii jccuniulum habects Jatcira aEqualfa: ip^ i 

C* 



lagi >• 



tii dxilnis lim ^ Xx\ ijhi radtm M tatigeiitan angui^ 
li MQt ctt y ixtCM ad 'Ut i fivc ad GX, vcl utMwr' 
md Xxy 9C #&dius ad t^getitem anguli ^Gt , ux CX 
ndXt i rd^ CM, fi^^ ik Xk ad M;»^^ Hji^rbole 
Vero, qu^ eahdem habebtint ad e«ndem a^eth afymp^' 
totorutn, inclinationemj (iandeth habebunt raiionem a- 
iti^ trahfverfi ad ccmjttgatum 5 adfoc^iie erunt fimiks ^ 
& viceverfa^ 

170 Si altei^d i hinis HyperbdHs hahat ffo akeiranj^ 
iferf» axem conju^atim atterius j ■& vieeverfai qttai di^ 
cimus Hypetbolas Conjuga^s , tohtimnes tmbithn^ afym*' 
ptotos 9 & aqualem focarum diftantiam a communi ceni^ 
tn^\ ••',-.•' '^ '-' 

I7ij Si chihi JJia fiyperbola hab^dt 1^0 iHkit tftmf^ 
vcrfoi Xa;, pro conjugato M^, rectanguluhi iilud>At-' 
perioFis CotoIIarii efit pro litTaqtie idem *> adeoquetomi-' 
mtsnes ntriq^ue diam^^i cjus rectanguli, & diftantie f6- 
corum a centro, q[U£ ih fingulis seqdati debcnt eidem 
CT y vel O 9 communes erunii ; 

i C H 6 L I U M IIL 

iii* T T ifiC qufdcni de H^pcrbdlarum afumt>tbtJ#^ 
JrJL fcrc fpontc flilxeriint;; ex quibus fadlc fol- ' 
Vtalltur plurima problcmata, quibus qusrantur afympto^' 
fi dato foto/ ccnti^Oi & dircctrice, vcl focb » centro y 
& Vertkc Sxis tranf^crfi i vcl binis axi&us y vcl qua:- 
ratur cUrcCtti:^ datis afymptdtis > & foco, vcF alia hu^ 
jufmodi > quaB per fe quifque faciic folvet ; pcndenr 
auteh) a Gdhfibinatiohc eorum ^ qua? fn iis theorc- 
matis cbnnectuntur inter fc • Plures alix maximc' no* 
tabilcs afyptotorum proprictatrs occurrent infra . Notan- 
da^ inlercia mira ittdoks ^i^tuor r^mbi^uiii pernnehtium 
ad fcihas Hyperbola^ conjugatas , quorum crurk' in in- 
finitum ' produdta ad fe invicem accedunt- m%is , quam 
pto quavi^ data differentia y quin ufqti^ concur- 
rant ^ Povto €}u$ fij^urae 9 ^itfta ^ul condudont ^^ 



- 1 






U SECTIONUM CpNlCARUl^, 

IRi^fogi^ fl^iapa if^ ckggos (fm^ Ellipft ^q f|^D9r 
(f (^s off^cec i(Dfta • laiQnpatraiigbimMs a^ OP.^mil^ 
{^ frqprjecgtes feoarum Coaic^ni Scctio^em ^jp^aiif 

CmUf7f 

etisnwn funcpd di4cn^ cugf r^cfg franfeunte fer illHm oc^ 
dgfm^ ft' fi^m co^fmh^f ang^o^ kifff inde mMtu 
Si Amm.cmW4f » ^cta dncu 4 jSw 44 con^tnm^ 
& offtiifum fm dirccmeM r^ctm aVfdm fmini-. 

f4t X74. Nam m %4i9 42 j 43f 44 ^ quibus put^ii 

41 P9 # jaGeat ia eodepi raoiO) pofico V ia MccaJHFpro* 

45 dwcii. 44 PAT-ucs F > ia fig. Yetr045 co pofijtQ 94 pircnr 

44 Hi ^nguli HFP, VFjp , q^os r^cc^ FP , F# ooQMW 

4j ^um tcct^ VF, ^ruac ^qil^s wgiMis LT^ 9 UTy ifios 

radii JJT^ Lt iis parall^li CQacioenc wm «^E^orda TjT par- 

rallela ipfi VFH j adcoqtte c^m bi acqttcoaijr ifittcr jfe 

ob iror^relirmum triaDguli Tl/, cciam^ UU inccr fc pari- 

{cr aequaies crunt. 

175. Inde auteni jam pacec, fi cocuntibus. puactisP» 
^^ ubc ad cuqdem ramum termit^aQjW» rsfitjk^H cm- 
d^c icaagens^ foci rodiis FP^ F^ CQcwiNis^a U9i.icum> 
^ebcrc iprum buqc r^i^ cvadcce pcrp^DdU^m^ iffi- 
yFH . Sed idcm mulco magtf. nunjfeftum ^t Ift Sg. 4^» 
47, 49^ ubi ^s^us, qucm IF , ycl if ^oacii!^ cum 

J.46 JFH jfibi rcfppndcmc, 4c^Rt cflSi 9q9iii$ «ggJI«» qiwm 
4^ circuli ra^ius J^QL? '^^^ ^ P^^ paraUclus Cfinirjjf^ ^^m 
49 00, qO cangcnce circuli pflff^ff^. f9^kif'H IkIw%M 
• tcaus. 

176. Bina ia^genm dufita jrr ^mHm tm^kk iimd^ 
tr4»ffHntu fier focm (Chordam 4mm dif^ l^hm^Am 
jnniit pina q^vis ferimtn tfi9ct4, t ^^ ^ HyffrkpU 
$a fprtinea$ M rams offojitos) concjmmf if dirfctrl* 

fc, & iH fonfincm »edm .i§ JSUiffi 0cmm> in Por 

faho^ 



I L E >t f N T At n 

mSi cnim dxotdz Pf^in % 53» 5+ Wn^a|||r F. j j 

'^m F, ducta FH ipQ perpenc(iculari , dooec occurrac 54. 

ifectrici in H>, rccta? ]PH * fW ^i^% ^ngcmes f numi ' 

f7}i* Poctis aiiteni in pnino cafu i|) fig. 5:} rcais 

•J^v^'^ P«^<Ucwl?ritoi;S riiWjp^fi ^5 «rit H" « P^ 

lainfiPj in Para£>p]a ^qualis^ i^ Hyptr^pl^ laajor , 

•^IP;^. Jricogiijc ^um Pp;i P^ ^« fiaus m^l^mm 

«©^ PHF ^ wdiutn coiiur«^epf |IP ^ «,um. q^. 

J^gih m aD\guIu$ PHF iA E%ft winor;, ^^i Pa;!:^^ 

la «qijalis, HypcjrbpJa m^oi:^ qviai^ ?Iip ; ap p^wcst 

»!Vn^HF fcfpcctu ^j[^^i.^Q{?^?irt jfoxus^H^ JPMfCM- 

«wsi bini^ PHF» fHF mi^oi: m J^tpfi ^ ^g»a|^ ,|Pl 

i J^aWa^ majof in Hraarbcte ii^is PUD ^ ^i^ Ml 

^; iB?>f&, j^e rcftd^jp /id ,4uoj; Jfccjoff^ ^\m Wirjjta 

?qu»W ^nin^s ;w^i prodcvwes ^x >| Wtft^ E £- 

' ^ fwnp.iif ac pcojjti^c ipfe JRHf. r^ctp ?iWP»^ VJ %»- 

pSrSfcqaalis in PsM^abola» majoir in Hyperbola . At iti 

•4- 54> ubi Pj| f' fmt rad ^amqs fjppofitof > ob angu^ 

*QttiHFP rectuhi» acutus cii totus.FHf j adcoquc roult 

i^iff}s,.in Pi9r4iM0, chqrJ^v^ t^ &fm 4^tm fffjt^ 
-Wmmy & ejus fegmentum im£r Jlirf^tricpf^^ & skff^ 




PHF, & ,^xigMlwi.PQH* JHF^P ^ttctas^ ajEy;u)lgi^< 
»»1H? HJi^J f^q^a^s cf^ MjgulQ 1^^ <ivc ^u^ pgic- 
f^qiculari ad dircctricqn » & prpitide parallelji ^JP » 

! vigftlo PHI akcrw ipOiis JHRD , i|;ituc i^ lar. 

i> IP triaip^i PlHi^ ^tti^UH^ jtq|i4^ ogpQ^» 

nw^^ mmu ^c :?a^ 4cmoi^rau^nfi if . a;^^^ jy|I, 
ll^^u? & JP* Si wtcm. ip/a J^Cof;ciK3rat^mpjict^ 
^Yo crii f y ^qu^? VJi/ aijfpfluc *ngulH$ VFW[ «CB»-^ 

' Ks 



its VHl' . Cum igitur in triangulb rcctJingulo IfH tP 
m anguK VHF, VIF fimul scjuentur tcrtio JFH recto', 
crit ac VIF aequalis VFI « & VI «qualis VF ^ idedJ 
^ue iBe VHi 

^dHC5LltJ]^. iV: 

f 1^6. T^ CoroUario feptimo admodum facilc dediicitut' 

* - Xl/ aliud Theorefiia> quod quidem poflet hJc td 
'CoroBarioriim fcric tellodati . Vetam ciim &ritineat 
•unani i praedpuii ScCtidiiuiii Coriicatuiri^gencralibu^ 
•^opri^tiitibui , & ipfam itidefrt admddiirii facuridam , 

tatridcm fcqticnfi IPropbfltioilc fcnunciabimtls : trini. cx 
"ra piura dcduceriiias Cofoll^riaj quorum f)le^aqtic fiim- 

*)uni habetlt ufdrti l At primi raro adtriodurii iifus ad- 

venict , nec atf td aflfia pcnderit. Cmtl tamcfi ini Elc- 
' incntis dcmoriftnui fdkat, ipfiirii etiam deduccmus, & 

iu ctprimcmtis; ut genetaiitet vcrum fit, licct ab aliuf 

* m itpririii folcat, ut in aliqud cafu fit falftim; 






moPo&tno iVi THEokEMA. 



iSi. rj/ r quovis funSo ferimetn in EUifJti vel Hj- 

^perbola ducantwr bini rect^t ad hinos facos ^vel 

'in Parabola aliera ad unicHfH fdcutn^ attera dxipdrallC' 

lay ea cum iangente ftt tdifn funStumdulid kqudlescon' 

' Hnent fUtgulos hic indi . 

i$i. De Pafabola patet ex ko y quocf bfc arigulisip 
HFP in fig. 53 rectiim f num, 17:$. /^ & bafim HP 
comriiuricm, ac latera IT j PD iqualia^ «quatur aor 
gului HPF arigulo Hri) , vcl productis DP , HP ia 
©^ & Qi^ arigulo qiioqufc OPQ^ipii ad vcrticcm op* 
"-pofito. 

1S3. Iri Ellipff aritcrii, & Hypcrbda fig. 5:5 , |6 fi 

t.55tangens pcr P ducta dccurrat directtici AB peiftiricnti 

5^* *i fdcum 1!^ iri H y & dircctfici ^ pcrtincriti ad fo- 

cum^ juxta nam; 87* in h , iriclinabitur in eodcni 

* an^o ad «trafhciac i Q0ti ca^ gSmif um fint paraHdx . 

Qua- 



E L E M B N 7 A. ^ 57 

4^e crit (nuni 2, & 87/) FP ad PH, ut /P adPib 

srfboquc ob angulos j^d F 3 6^ / rccto? aoaalcs C n»* 
17;) eriaiTi (num* 25. Trig.j cofinusanguloruinFPH^ 
/Pi?, $: ipfi angiiii FpH , /PA inicr fc , ac in HypcrT 
toia, productis paritcr hP , /P in Q^, & O , ^uigu^ 
FPH, OPQ^^qualcs cruqt. Q^Ep. 

CorolL I. 

i^ ^upltm angffliy quftn cgtrninent iina tanjr$ntes y 

fqHAtur in Parsbula angnlo , quem Hna reSt^ a contOj 

HihHs 4^ focum du^dt ihi ^ontinent fi ibi continent ita ^ 

ut aiffis an^uli ffeElet concurfkm tangfntiumy in fUiffi 

vero diferenti^y ih eodem HyferhoU ramo fmmabin^ 

rm angulorum 3 quos ejufmodi r^ila ad binos focos duf 

^ Qt in iis . fonfinenf y Ji in Elli$fi b^ni hiatHs fe m^ 

• tuo fffilent , & in Hyferbola uterque . ffeSiet eandem 

fl^sm ', quod fi anguli diverfas pojftiones haheant , al^ 

ier fx iis fHhflitHi dfbft ejus comflfmenta ad qHOtuor 

reiios. . 

1^5. N;»m in fig. 57 i^ Paj^i&bla fl ^angcntcs fintF*57 
MPH, mpH , ducantur PQ , /i , H« parallelas axi 44 
pam plagam^ ad quam ipfc m in^nifum protcnditur in- 
ira Parabolam , & rccta HFN pcr focutn f , crunt bi- 
•i apguli flPHj T{H aequ?Jc$ binis in contactu MPQj 
PfP^ fivc ob paraUclas binisPH^, ftini adcoquc fimul 
toti PH/r • AQguIus autcm NJFP cxtcrnus xquatur fi» 
mul bihis FPH , FHP , & NFp binis FpH , FHp, 
adeoque totus PFf foti PH^ una cuinbiAis EPH, FfH 
ip(l zqualibus , ninairum duplo FHp • 
' .1X6. At in EUipA in fig- 58 ductis HFN, Ufn, b1^F-58> 
ni fPH , FpH aequalcs crunt binis /PM » fpm y five 
(jnatoor iqtcrnis, & oppofitis P/H, PH/, p/H, f fH^ 
pimirum toti PH^ , & toti Vff. Angulus autem Pfp. 
\ «qualis binis PFN, pFN, fivc quatuor internis FPH , 
JHP, F^rH, FH^, Y?I Wpis \V^^ W, F/Hcum aix- 
giJo VHpy adeoque angulo PHpbis, & totiPjJ feracl. 
Quare angula P/p dcmpto a PFp ^ rcmanct an^tis^ 
^Mp bis. 
I 187., pcraurn jn Hyperbolafig. 5P ductis /H^, HPN, 



|l SECitlbNUM ^6HiCAkUM • . 

;^ nhgutds Pff cbrtftaffts hmis PFN s ptN catn JCdu^hlr 
**594teiti|pr FPH , FHPj TpHy JPH/ i e^fcedit !>»> pcr bi- 
hos FPH y fpH . Siiiiili argiihicntb PHp ctttdit Vfy 
m Kiidi HP/ i Hli/yi^ribtiS dfeciuab ; tgituir PF> ,' 
Vi^ y ¥ff fiiht kt eonrirtuA ^ithniefica ^toportibric i ' 
jSC buidriini cxtreiAoi^udi (umAia ^quatur duplo hifc- 

^ itt* Quod fi artgului PFj^ ut ih ^|. 66y it^ 61 ob-^ * 

'j^trthat ihtiattim ad pAftcs bppofit^s N i pro ipfo fumcn- * 

^f dum eHt ej^i c<5riij)kmcfttUni ad qliatuoi* rectos i nimi- 

^* ttim aggfegiium tendriiiti PFNi jf^FN > ac dcmoiallra- 

fio fcadein rcdibii. 

iSj^. /;/ Ellipfi ifdrmalif tanse^ti y ^& in MyferhU 
tA^gths diviiit bifitftaf^ affjiilufH y ^ueiH ConHnent bini 
ii^drum focorUiip ^dMi hd tof^Yactum ducti , ac zpjd ncfr^ 
%Hiis j ^ Va^.iem und ciAh binii filcitj apceth ditMuni 
in ftofortione harmonica i . , . ^ 

i$&. Primiaih paterr fi ehim ih EUipfi ih fig. 6ti &t 

tft Myperbold ih fig. 64. tangehs bccurrat aki iil T , ac 

|p - Pi ipiS ndrmalis ih li FP, /P ih Hypefbda debenr fc- 

^•^^quaics ^i^uloi cdhtinerc toni tartgehttPTj quefiin El- 

*♦ lip(i prodUcatuif ittdcf&iite in H ad paftes dppofitas T y 

felrurit pariter eqilal^s anguli FiT ; fPH i adcoqua & 

fP£ \ fVlj ebruht torii^ehJchta ad tectos TPI j; HPI ; 

ijqualcs cront; ' . ^ 

!$>t. Sccundurh' dotkm dedudcuf « prithoi &er hii. 
|6,> cum hihiirtim reCtarum PT , PI altera fecet bifa- 
HSaift ahgulum FP/, alttra fit huic ipfi perpchdiculariis ; 

C(ftolh ^; 

152. iriAr^i difliimiiPP\ (P fdcorufH a contaciu ^ bi- 
na Tl , Tl /^ ^at^ conlfutaU d Aormali , ^z;^^ FTi fP iii^^;» 
c^mpuiaia a ian^ente fm^ in iadefh ratidne inter fe . 
Tres Mftaniid Cl , (^F., Ct centri C computaU in 
^i"^ ^ normali '^ a foco i d tdngente fuht in conii^ 
nud ratione geomWitd fectafum Fl s FT , ih qua fo- 
<MS dividii dijldntiam IT mrmalis a tangente . Si e - 
bipis ftcis^ & centto demiitantMr per^fidHpda FAi-CL, 
' ' i ■ ' fjL in. 



■4. CLi^a* : . 

moiiic* pro^ofiti^ aiMe Prbfi. 1* ^ htiiu. t«. Niftirdtti 
caiideiti 45& tmlmm fUlfi «: FTi T/^ acJFP$ #/; 
partifft^ eX ipfa ndtiofi^ >1-p|)drubnis hkiritioiiid» i pitf.* 
tiitt *i iki«i. 564 Redtas Ch Ct , Ct s tM daii<iHi« 
propdt^ttoliafei! ifl ritioniS I/ *i /T ^arel: eit ftiftH. i* 
ob F/iriterValiikn bidbmhl piklcfoi^uiTi alfemtti?ui# f^- 
ctum feifento iii C Siint stettrti rf id f T, ut I/adf 
/r, cx prihia hiijas p^t^t^; Dtihm ob pariileiififtiitti 4 
rea* FA* iPi CLi/^ funt}h«^r fcs lit FT^ITj CTi 
/r . His luteni CJ3b ^meorice t^bj^6tiiic^i§ g&nftit 
cx nuitl. i6i< 

CMtur ferfendiculuin in tangenttm , recta jmien^ h^ 
jtis extremum funSum cum ceniro , faralleld efl rectd 
juniinH coritddtum tufh fo^ atfefo , & aqtialis />- 
miaxi tranfver/o y i^oiue ipj! kqkalii erit ftchd i^' 
c^ntro dd fan^entitfh dkctd pd^alMd juhgiHti focM cfM 
contactH iffo ; eidem vtro dquate ejt: ttiam fhgh^htHfK 
recta trdnfeuniis per contactumy & focum utrumlihii ih^i 
tiri^timi ipfo coniactd i & ¥ecia tangenti farathtd du^ 
(td per ceniTfmi^. 

19^ i Nam d tangens TP m ^p 6%^ M ^^^ 
rac in;A i & O rcctts CA , FO paralleliS rtct* /P, 
recta veto ducta pcr C par^alfela ipfi HP tcais PF, F/i 
OFi inBi i^,Rj,ob CF, C/«quaIcs i crunt ^n^^t 
eriam PA, AO (CoroUi^ Ptopof. 1 2 Gcoin. ) intct^ccfJti^ 
iifdenfi piaraUfclis FO, GA^ /P, ac ofa candtm tadoiicitt 
CR, CJ aequalcs.crunt iflter fe, ac proindc aeqttateir 
criam VK9 fh in rriangulis RCF;^ ^C/?qualibus . Cuni 
vcrd recta FP contincat cum tangciite ejDiideni angu- 
km, <iucm /P^ adepquc curiddm., qufem FO huic pa* 
ralleia , triAiiguluWl PFO cHt ifdfc^?^ & JFO ?qua- 
U^ Fpt« 0!f^e {ttiiao qaicfetft k d^iangulis F AO ^r FAP 

ob 



4a SECTIONUHXONIC^RUM 
4k omnia lac^ra ^qualia, atiguit ad A cruac ^qualc^V 
€f rccta FA pcrpcndi^ularij tangcntit Dcindc cura RF 
^Quetur fi, Sc FO ?quetut FP in fig. 6:}, f«nmaFP, 
P^, Iffj qu? (nurn.9a) gquatur axi ^ranfycrfq» ^qualis 
crit fumm^ OF, P^, FR> fivc binis OR, P^, qaarunfi 
ineul^ cum ^qucptpr ii^tcr fc^ & cqucntur CA qb pa- 
|:^eiifmum, cric tam CA parall^Ja /P , quam.P/? g- 
qualis fepiiaxi prfinfvcrfo CMs imo cum Sc trianguluni 
BP^ (|t ifofqclcs ob aagulos ad B, & ^ ^uales angu^ 
lis alternis ad P, crit & PB ?qualis P^, adcoqiic ipf| 
ff miaxi . In Irlypcrbol^ vero in fig, 6+ cxcefllis P/ fu- 
pra FF , crit idcm , ac fumnja cxcefTuum P/ fupra ^/> 
fivc FR, (Sf ipfijis FR fupra PF > Gyc FO , qu? nimi^ 
rum, ^uabitur binis PB, OR .?qualibus intcr fc , v^l 
dupl? AC . Cum igituir iUc cxcciTus P/ fupra PF ?qu^ 
tur pariter axi tranfverfo, ^quabimr ^ejus dimidio tatn 
CA» qaaip P^ , 5f codcm , qi^o in Ellipfi , afgum^ow 
to FB , 

CordL 5f 

196. PeriendiailHm e foca in tMienum ducfum inci^ 
4U( in Elliffiy 0* Hy^erboU in cancHrfkm tmgentis if^ 
Jius cum eirctdo hahentis fro diametro axem tranfver^ 
f^m , in Parabola ifcrp in rectam axi ferfe/jdicHi^rem in 
ipfo yertice^ 

1^97* Primum conftat cx pr^cedenti , cum nimirum 
in fig. 63 64. , ob reciam CA ?qualem CM , ckculuaj 
centro C radio QA debca; tra,nfirc per A : Secunduoi 
^.65 patet in % $5 ex eo , quod fi. tangcns occurgat rect^ 
FD in A, eam ibi fccabu bifari^m, cum feget bifai;iani. 
anguium ad P trianguli ifofcelis FPD. Ac proindc , fii 
ducatur AlA , ca ob FD , FE fcctas bifariam in A , &;> 
M erit parallcla directrici ED , adcoque perpcndi^ul^^ 
ris axi* . r 

CorolL 6. 

198. In iffa Parabala id ferpendiculnm efi medium 
ieometrice frofortionaU inter quadrantem lateris recti 
frimifalisi & difiant}am (^onta^tus a fsfco , ac rn^tato^ 

tttcHm- 





^ E L E 1»^ r^ T A.^ 6t,: 

j^mitiffiHs. ' ' , 

• ■59?;|*am priangula reftaQgura FMA , f AR iiraUiA 
ImiCm hab^ant iintim angulum' fedtura ^cju^cin 9 
^ii^m PFA ?(jualis 1>DA; ob PD' > PF ^uale$ , 
'"^'^'cdam dtcirno AFM,.'ac proindc fM ad FA » 
l a3 FP. HiQ.c autcAi quacJfatum PA acquatur rc^ 
^ ftib FM, qusB (%. 69.) cft quarta?pars laterig 
fi-pimdpalis, & FI^, adedqtieobFM invamtam,u| 
Jaq* imitetur P ,id quadrajfum ctt, ut FP,nimirun:x^ 
iffe ffinfationc duplicataFA, & haec in AibdupKcata' 

m, '' " cor0ii.7P ''^ ^ : ^ 

'200. /» ParahtfU l}^a nifmalis terminat^, ai axen%^ 

i^ difU perfendickli e fico in tangentem deinijft : di^^ 

»& joci um a normaliy quam a tan^ente (omputatd^ 

iffe axej nquaHs dijfahtla contaEius a foco 5 fubtan^, 

W^dufla dfiifft\ fubnormalis dimidid lateris reEli- 

mc^Mtis ; normalis adr t^n^^ntpn , hf lam reSlHm ad^ 

l**^or; Nam riomlnc Tangentisy Norm/dis^r Subtan^ 
PWW, Subnomfalis , intelltgitur PQ^ ■itftcrccpta inicir 
Wkaftum, & axcm>"PI pcrpendiciJaris tangenfi pari-» 
% tttttiinata ad axfcm >' QR fegnietitun:^ axis' iatet^ 
pojoittm , & ordinatam 5 RI* fegmentum ejufdem in- 
floTDialcmi & ordinatam . Porroprimo Pl aequa-. 
FD ob parallelas, ad^oquc cft dupla f A. Sccurida, 
W, ut IP dupla' PA ; ita IQ^dupIa f Q^-, ^deo^ 
*";afqualts FI, fivc PD, nirairiim dlftahtfaB FP ., 
IP dupk FA, ita PQ^dupk AQ;, adco^uc&:* 
s R<X dupla MQ^, adeoqil? du^a ^tiam' abn^ 
rcfiduas RMl Q^aao in ^ian^olis FBD; iRPob, 
1 onmium par&llclifmum fivninbus , &', ob* R.P> 
cqiiales , aequalibus 3 erit fbbnormalts RI d?qualis 
diiaidio. kueri uiko principati • Quinto demum ob 
um ad I communcm tri^ngulis rcctangulis IRP> 
<rit normalis IP ad tangenterii PQ^, ut IR ad fc-^ 
inatam RP^ ad^b^ue u\ xoxjjofa, latas re<5lum a4; 
i ordinaiam . ' 

Mcmdcb.TomMI. f SOiCV 



T 



seHoiit/M. 

:&02^ T) K.op(ictas , quanl itt bac prop6fitioii6 ckmoa^ 
JL ftraviinus. eft una k potiflimi^ SciSJoaunfi Cgh 
nicanitn proprictitibus > quof iltmirum iplis foci& no«» 
men dedit» Nam radii lucis iii ifpcculum incidente^ kx, 
rcdcctuntur» iii anguluhi refilexionis faci«t aoguk) inr 
CidentiaE afqilaleni> qui ailguli>ubi rpcculi rui>erficies o& 
curva» sftimanttu: peneS tihgentem in ipfo ihcidendap^ 

T*66 Sc reflexiohis pUncto ; Nimirum ih ElUpfi la £§, 6^- 
67 ridii bttines /P egrcfli c foio / ihcidentes iii perime- 
6S triitti debcnt reficcti ab Fj & viccverfil^ in Patabolam 
ifig. 67 iradii omhes OP dclati per rectas axi paraJMaa 
debent paritcr coUigi ih F»^& radii egrdili es F debct 
abire parallcli ; In Hypcrbola ih fig. 6& fi radii OP 
defcfehtur cutn ditectione tehdchte ad / d^bibnr paritet 

^" colligi in F^ & £i egrediantur ex F, dcbetii al^ii^e tan^ 
quam fi cgfcili eflent ex / ; Atque boc paclo ighc fypi 
tis valido excrtato tn /> {5ptcli ih m^4 diflajltia ac- 
ccndi ighis in l^ > ac fpiiiCuld Pafabolico obverfi;) fQliv^ 
cujus radii advenitint ad ifcnfum pafalleli 9 excitatatr igr 
nis in ejus foco F i ibidcm vcro aocenfa candi^Ja ia. ipf# 
F> lumcn fatis validum ad magnam diftantiam trW-* 
mitti pdtcA pef radios poft rcflexionem p^trallelos; 

:20jw His perfpedbis rcgrediemuf .itef uni acj. con/lriif 
ctlonem iliani noftf am , Sc motuih lidea; paf allcl^, oiK 
de allaiH adnidduin inilghchi Secitionum Coaicflffufit 
proprieta^em cYu^mus > nimirum feauidas di^imcif^j 
quas chordasl' ottihes paf aliclas biflariam fcc^mt » ae ex 
bac jpfa alia theOfctnata tdhqiiam ex hovo qiip4ii)si 
ramo novos furculos quoquoycffunl pforump^OS: de* 
ducemus 4 Sed prarmit^cinu&JLxtmma q^oddam gttnjoralct « 

cujus ufus etiam inffa. qccuri^et^ & 'm ^emafMt W 
P^tct. 



LEM^ 



ElJEMiNTA. 6$ 
L & M M A. 

' H, i» iH»»r A^ ih^ Ht t^f^y dHcsmw Hmt faraltelaHX 
k4fBf ai idiS^aih e fiHiuhy Tt^ & hind uidemf^ 
Mttfir HR; biTy «^/ j^entesi i/el nm ^mes etirH iif 
inMr^ ifqtie md ^terdm Qj \ Wit fimfer tiA ad 
ittl^^iiib id btt yat mtMd f^cimiMe fnnEb H ^ 
M|9^l mMentikM^ reStartm UA i HR dit^ienir 
kfi nkmit eariim rMo cehfiani ; Cetnrd verd ji fac^ 
m Ifei hifaraiteid^ ititer yi v C^ H l , . hr inter fe , 
mm oimliA ai HKvta ha 4<s< hf, jacemibf^ fm- 
8ii \i\ W hy i^ vel m^ eaaidem fl^ain\ ut irt fig. 69. 
HlHii^fta}^ Ht ifi fii^ 7b;, fironi HAi ha }aet^ihp 
tdt^im^ vel dd etfof^aif > reHa Qg^ , Pp ^ Tt dttaa^^' 
l^i)cmiMfdraileU^ jfmEh H,,hi A^ a; R^ r^ 
^ ^tffua» cenis&reni ^ vii fimtd ^onckfri^ni in eedem 
jMOr P: ^ p iMieme raHiene HR ifW HA;^ idMnqua 
^iMSbir^ ^^ |iii7^4f Hi A McMrdntfer hinds reSas^ 
MMhn^ ieJbMr R ^ reiMt^y fi illa coemt \ cm/ersem 

^ Prima pars patct» qliia triaagtiial HFA , ikF^* ob 
fti^im pdiattelJiHmi ^^ates trunt fcmper fimilia , uc 
tHFRi i6£#; Ouai^ ^tit HA ad HF , ut ^ii ad ibF 

S^icfHRv' iSt/bF ddi6«^, adeocjoe cr anjuaHtate oMi^ 
hA ad HR i ut ifr4 ad Ar . Scctinda^ars dircctt 
titt itbtholkftfiai;i j^dtefti ^ fed dbducitur &i£iliw e ptima i 
iH&te cotf umibus rcctisHAi Af idP,: terta pcJrFi &f 
^" iidi* tratifirei pef Ri trinfifefc per aliiid puiidhii^ 
e Ife> & cBct HA ad HO , m A^ ad Ar> five 
tfaUi iit HA od iiRj»l&ph^isfe«a^ HR se^a.» 




PRO- 



»4 SEGTIONUM CONICARUM 

•:•-.■■;. * .4 

PROPOSITIO V. theorema; 



zo6* /r> Hardas omnes parallelas intfr fi iifitrum /^ 

K^ cat Mameter , qtut ih Ellipfi ^' & Hyferbok 

fimper per centrum tranfit , in Porahola efl direericifef^ 

pendicularisy five axi faraUela^ & data Seftione Com 

ca^ ac inplinatione ordinatarf0f$y datur:. 

zoj. De chordls par^kHs , vel perjktndiculatibus di^ 

rcctrici patet ex nmia 56 > 6c 8;, per quos bifariamie- 

cacinii: hx ab axe conjugato , iUas ab axe tranfverib J 

E^yiDe reliquis fic demonftratur . In Sg. 71, 72 , 7j , 74J 

72 quac. conftructa; funt fuxta num. i+a y & quarum prin 

73 ^Tia pertinct ad EUipfim, fecunda ad Parabqlatn) tertui 
74- ad chordas jungentes in Hypcrbola bina c)ufdcm rairf 

puncta , quarta ^ chordas fungentes in ipfa Hyperb^f 
la rairios oppofitos, agatur tV perpendiculaps ad cboef 
dam circuli Tf , quam & fecabit bifariani ^ producfti 
tur LO , qvi^ opus eft , ut circulb ipfi occui:rat in Ui 
6c mj fecetur cborda Vf bifariam in R , ducatur<]tii 
per focum F cborda P^ ipfi parallela , occunrns difCrl 
ctrici in Q^, ereccaque FI ipfi perpendicularl^-xjiiacjwij 
cefiario alicubi occurret 'directrici iii I9 ducansr IF 
ipfam pP* fecans bifariam in R , quas fnum. .134>) k 
Ellipfi , & Hyperbola tranfibit per centrum , in Pfl 
cabola erit perpcndicu^ari$ directrici , adeoqup parr' 

axi . . . ' 

208, Jam ve«p cum fit HP ad HF , ut L ad C 
& HF ad Hp,' ut O^ ad OL , crit ci aequalitatc 
turbata HR aad Hjii , ut O^ ad OT , & HR ipfa 
HP , Hp fcmifumma in pcioribus tribus figuris , 
<liffercnua i^ poftrevna, ^d priorcmHP, ut OV pari^ 
femifumma , yel femidiflfcrentia ipfaruni Q^ ^ OT 
priorcm O/. Q^arc cum ratio HR ad HA com| 
tur cx tril?us HR ad HP, HP ad HF. HF ad HA, 
p<:inia fit padem ac OV ad O^, fccunda cadcm ac 
ad OT , ae ' tertia * ob trianguloruni rcctangulon 



E i E M E N T A. 6$ 

HAt,OVL liiliilitudineni > cadem, ac OL ad OV> crii 
ipEitatb HR ad HA eadem; aC folidi fub cectis OV> 
OL,0L ad foHdmn fdfa tcctis Or, OT, OV , nimi. 
tDD) oEi VO communem ; ot qnadraturn OL ad rectan« 
gqko.TOri ii^e ad rectangalum MOm ipfi anquale 
(hif. 13; Geom.) • £a ratio eft conftans ^ utcunique 
tmasz pofitioae. cbordas Pp , dummodo e)us indinatio 
|d directriccm flt femper eadcm, mancntibus itimirum 
i^cihpct punctis O, M, LiM. Indt autem deducitur cx 
liteiuiof^ dnihla punaa R fore fSimpet iii eadeni re» 
cti. Com nimirum manedt 8c directio rcetarum HA ^ 
HR, k ratio , ac puncta H > A excurrant pcr iectas 
fflj 'F, cxcurret etiam punctum R pct recram cj^ 1 du- 
C&m> & ehefrdac bmhes parallela? *ab eadem diamctro 
iafamm feqabuntur « £a auiem diametet erit illa ipHi 
^i ^tix cbordam per fbcum tranfeilntem bifariam fe** 
^* irqoe id quidem p^tet ex eo i quod ea reaa de- 
^ fecarc bifariam cbordam quamvis utcutnque proxi^ 
ttaa^ctordac PKji tranfeunti per focufti F. Scd fic ^o- 
Ata&ffimc demonftratur y nam deoionftratio illa gene- 
l^ljs^pro chocdis oranibus non I^abet locum pro eaii 
V^ per foGum traniit , lictt facile ad candem rtduct 

/^Ratio HR ad HA cft eadem^ iC quadrati LO 
«I fcojjngulum MOw (ttura. 208 .^^ nimirum (C6roIl* 
*iA Ji I>rop, ij. Geom. ) ad dificrcntiam qnadtato-! 
*PL,LM. Qu^re eric HR ad RA diffcrcntiam in 
f^us tribusfiguris, fummam in quarta ipfarumHR, 
^9 Qt quadratum OL ad quadratum LM f qucxl pa.. 
I^ffovcnit fi in illis ^ quarfraio OL aqfcratur dif- 
PSitia quadratorum QL, LM , & jn poftrcma figura 
> nimkum mt quadratum OL ad quadratum 
^vel ut quadcatum HP ad quadracum PF , five uc 
«Iratom QP* ad quadranam P F 5 & invercendo R A 

!,KH in raciotie duplicata FP' ad P'Q^j in qua ipfa 
jone eft RT a4 WQ^ cuhx (num. 154.; RT, RP, 
Q^fiiit coQUnue in ea ratione fimplici • Reda igi- 
^ IR. dcbct tranfirc pcr K (num* 204 ) . Cum vere 

F j ipfa 



% 



. <6 SECTIONUM CONICARUM 
Ipfa IR' in Ellipfi , ^ Hjrpcrbola traafcac per refuniiq 
(num IJ4)» ih Pacafaoh fix perpciidiQilaris <Utecrici«^ 
patct cbiQrda^ amncs parallelas iiabcre fuatn^istmemm^ 
gqas eas omne^ faifaciaai feoet, & tranfcat in illis per 
oeiicrum , ih hac (it perpendicidaris dioecirici » Bc fdrdjr 
kla axi 9 adcoque dctur invenro puncco I p^r rccmi^^ 
FA perpendicularem cuiitbct ex iai^uimodi ^hauji^ \ 
E. D. 

2x0. &U4n;is reila ftt^ cgntrnm trmffi^S: in Elliffi'^ 
& ffyper^la^ ptaur falas Hyfnhda nfym^et^^r ^ f^- 
folliia a^i m Parahpla » $ft diamter fmf hatens ^frdi^. 
natas , quaj hifariam fecat y & quarum direiii» dom. 
uar 3 data Sectione Cenica ^ & iffa diametre 9 *^ 
frater axes uUa diameter- fuis ^rdinatis feffendksdai^ 

211. Rectam eniin diroctrici paraUcbm , ac pcrpcR^. 
dicuiarem^ five axcs ipfos» in EUipfi & Hypcrbola, quf 
quidcm ordifiatis fuis pcrpendicularis iity eife efufmo<tf 
conftit ex num 56. , fc 83 * Data autem quayis afiii 
recta^ quae pcr centrum Ctranfcat in fig. 71, 73) 74> 
ca dircctrici o^curret inaTiquo punctol , ex quo. doAn 
recta- ad fbcum IF , & per Frccra QF pcrpendicuAari 
ipfi IF ^ i^(k IC fcal>it bifadam c^rdas otpncft Vf 
^allelas ipfi QF> quas { num. 14.9 . fempe^ habebu»- 
tur in fig. 71 in EUipfi > ac ia Hypcjrbola habcbunttf 
lempcr , pnmr cafum , quq in fig. 73 , ^^ aecta FQ^ 
laclinetur ad direcifficcqi in angdb «q^itaci^, quo fo^ ' 
kt cafu ircaarum eam icMiinatiQnein habcniium altera> 
int^fe^io ita rcccdit ia inflninuii 9 ^ &ufqaanii JMi 
fit . At is cafiisi cft ille ipfe ia quo CI eft alt^o* 
ir^ cx afyinpiotifi j ^ ipft -QF paf aHcIa . N^ iA fij^ 
50 recta FH2 eft perpeiidi^iftlaFifi a|yi{^pio. KaHii 
franfeunti per ccnteum. y fji rectse K4FH4 habenti i%i 
cliq^tiqncni ipi^q^itatis ad dire^i^m ^ juxta numi 
]][6, lv\ Bar^ola yeKi }n lig. 7^ quatvii rccta pa- 
rgU^Ia %xi iran^rfe occ^P|tt direcii^iiei aKeubi in I » 
yndg ^HCta reer^ IF , fei^ta lCL^^Qie pcrpendieula* 



ns^ 



/ 



E t 15 M « N T A. «7 

Ipt*! Ml potesit tfle ptrpendicuhtis ^i^Qrici 9 li| 

SUo ta6( vaft^ram ipCi |)^ailel^um alrera iotcrft^ 
. i|in infinii^nr recpdit y at qul^aam jam fit; cum^ 
fft Itepfer IF fit pcrpendteol vis orlinat^ P|^ nuf(ju4« 
^^l^rpgidi^tris^ tK^inti^r IR. 

an. i^l^f^ dUmttr in £li^ niccmiit j^rWetto fk 
(Mif |ii^5v » z/r Pdrakdiu ^ukvis £h nnko > ^)? iq^ 
|^« 5ihr»i4; in duqbpii ferfinemihus ad bifios^ ramoi 
^fm/yV(l in nidh, ^ frot^ jacyerit. (n iifdm afym- 
|hwi» i$n^iU^ , jir^jr 0xis, trdnfxJ^fus bif^iriam fcca:t ^ 
Mtmi^^tiM jfft/tHk diam€tror\^ vertif^s dic6 ^ ut in 
^kro : i^ tnffi fer^ ifrw ^jfij virticts d^Ba orjlifjatis. 
pt^k tfi td^ens • forro cum d^iametri majfaitudinem 
w» Mfno^ y i^teijigo. fegmentum iffius ptterceptum^ binig 
*f*KW3<r in Hyfcrbola diamitrosj/icentes in ang^^ 
^^'^Jf^tmm^ in quibus \acet exis tranfvfffiis ^ dice^. 
K"an4$ , qjta ^u^t txtra^ dico,^ fccandariW , & h^ 
¥^,fcatrruftt innis ramis tiyferboU, cqnjugitta\ ac- 
t^.P^ilit. iferticei , dico, iiia occurjtuum^funSia , froi 
W^Jif^itu^ne affumens fegmentkt^ interceptum binis^ 
^iHfi^yJn ElUffi autem^ ^uamvis: diameipm frima^ 
^iifOf^jeBu fuiirum evd ac in utraqu^ 

^^^^^Mh faraUelam brdinatis aitefius diametrl , fen^ 
^^WKtei^ ijus vMic^s dutlts y dico ejus cpnjwacv 
t»!n\ ' " ^ ' \ .' " ■ ^ i' ■ 

' V3* Nan^ h\ priniis^ ih EMipIi, chordae Ofnan^s^ (^num^ 
^;}ji^m qupcqmc^ angulo indincntur , l^bem bi-. 
f^QQ^tfes. t>araliel2ts , quibus. daudutitur > in quam 
fj^jiilifcm fi defiftat ^prda Pp in feg. ^i> dteb^n; bina? 
f^wditutaB RP.,^ Rf , quaej^ iiimic^m femp^r ajquantur 
P^ft , fimul eyaneicere « pim^is P., jp fiitiul ciun 
Wfeitlikmetri II ^cun^ibas mjpfuni conta|Sbum, Eo— 
^ argtirtietrto. ia Par^ola, in qaadtdinat^ qiixcum-' 
V^ u^ieam tangentem, iibi ]|«raHchm habMt i diame-v; 
^ > quae cum. fi( perpendicularis dire<^ri^i , Iq *unico 
f(Qn&6 debec. occurrere curvas > iUi oociirret in iUo ip« 
4d coocaid^ « Ax^ iii H^pcrbola ordinatae omnes , mx. 

F 4 34 



6& SfiCfidNUM CdNICARljW- 
fld djredlricem inclmannir in angulo niinore ^ qttai^ 
ijtangului xqualitatis , habeiJi oinas tangentes paral- 
lelas . 9 contadti^usf pertinentibiis ad ramos oppqfito^ p 
^iias in. angulo ma)ote nuUas h^tenc « Porro in £g^ 
f*7S ^5. fi CH fit altera ex afymptotii , ^ diameter qu»«, 
dam CI accedat aH perpendiculum Cfe ma^is , quamjj 
ipfa » Ci minus, ac ipfis Fl» FH 9 F^ perpendiculare/] 
fim fO, fC^, Voy ( rium. ^34 ) qu^ num. iii.erunf^ 
][>arallel2e prdinans diamettorum CA^ CH» Ci, faitis pjt- 
tetj.FQ inclifiari ad directriccin in angulo minere ^*^ 
qiiam rQ^ quf ^P^ afymptbto parallela ^^ 00 angulurhi^ 
THQ pariter re^tum > iifidinatiit in angulo acqualitati^' 
to in angcJo majote > adeoque proii; diameter accc.fi^e^ 
rlt magis > quam iitravis ex ^rymptou^ Ctl y O ad a- 
iem CEF) vel minus, nimirum prout jacuent in ^o^ 
afymptotornm angu/o HCA; qutm axis tr^hfverfus: fe- 
cat, vel cxtra > habebit binas. tahgentes fuis ordinatij» 

fiarallelas^ & pcrtinehtes ad ramos opppfitqs , vel nut 
as (nu. 149} 9 ^ in primo cafu per iflfos ipfbs. contar' 
€lus trarifire debebit > eodcm afgtuiiehto > in feomdo'. . 
ftufquam ocdurret pefimetro , cui fi ufpiath bccurreret/ ' 
liabcretur ibi tangehs ordMatis pafallela ; deberet eniiQ i 
cjus otdinata ablre ;in tahgehtehi > coeuntibus. himi^ . 
fum binis ejus ixtremis pundlis , qu? fi non coirent ^. 
cfiameter ipfa ordinatam pef idem pundfcum tiotk ^caree 
Kfe^iam^ 

114. Diakiiter mteaM CoiiiU ^etHenis tetmnaiaik ; 
ordindtd q^Uy & toiam ih Elliffi aream hifariam yJr- / 
cat. t 

215;, Patet ex ed , quod fi cbndplatur qrdiirimta d\ 
tcrtice diametri motu continuo , & paraDefo delata, 
hiti^ femiordinatae fcmpcir fibi a^uales , & eadem cc-* 
leriiate progfcdicntes > gcncrabun^ aiccas icmpcr a»* ji 

qualcsr 



Corl/. 4. 



€cr«lL 4. 

ii6. CAar^« fer binM tktTtm^ hinatum MinMtarm^ 
y^4 Mie 3 ac tanteniis fer bina exirhma dttS^ iM^ 
dm cHrdie^ 9 jfi farallfU hon Jkni ^ ancarrmt in dU^ 
m^9 : diameter vero pejr. eoncfo/fm. tanggntimn^ dtu^ 
Ba hdit fro pfdinata okordam junientem Hnos cm^ 

i. 2i7i^iGum ciiiin m I^y^j 77, ordiMt? A.B , alj.H 
ibifariam fecentvr a diamctro ^ E^ $c e ^ cxk et ^ J^ 
fct, ut EB ad fiA, adcbcjuc Imx rcet? >! A , th dcfjfeni: 
(iw4io,4) cqncurrcrc curn diainctro te ii^co^cm cpit 
punob p *^lJbi.aut€m cocuntibiis or^inatis ai i AB ^ 
rc(2f AD, BD : defiriunt in tangcntcs Ad ^ Bd , dcbe- 
m p\incium d mancfe in ipfa oiamctro . Hiric aiitsiif 
& poftrcmum iporitc Buiti 

toM. j: 

. ^ il8, £//ie^ centroy & utriquf foco cazdtatem olver^ 
^ far^oU foca cavitatem , Ji/yperidU ramiu Hterqui 
^fmro, cohvexitatem i fico vm ramus citerior caifitatenh*^ 
^erier convexitatein . 

,, ^1$. Nam in Ellif)fi chorflg omricsi» ad^oquc bmnia 
^ pnncfa ( ntkn. 149. J jaccnt inper. binas tangct^ 
JW5 later quas & ccntnim Jacct ,^quod fitiira eft ifi 
fediointct birios contacws., & focus nterqqe , cim» 
ttotdx per cos ductse debcanr iifdcm tangcntibus cott^ 
gicdj adcoquc EUipCi & ccntro, & uniqac foco ca- .. 
^faicm obvcrtit • lii parabola focus jacct ad eas par-^. 
^> ad quas chofd« Jaccqt ^^ tangcritis ^ & ia 

^^bola ccritrum iriter binas tangentes:^ cxtra quas^ 
Wf jacierit curii arhibus » focus ad cam plagam ycr^ 
qaarii ramus citcrior pfot^nditur , ramo ultcriore^ 

fgcntc ad partes oppofitas . Patcnt igimr ^ qu? prou 

•"""ni ctiam i^ ils • * -^ 1 



^CHO. 






S C H O L 1 U M I. 

,tt|f* •S Wtd a4 curt^turam ptetmct refpeftti fbccsN 
vJf ram » poterat erai etiam immediate ex na« 
149} fcctlftuit podus bii<^ j%fervare • 4|t fitnul hab^ren^ 
Off ^tiam ea , qiise pi^netit ad cen^ • l^^.rro qao 
yjcrgat curyatur^ refpeftu fqci & ceauri., necelTftrio de«* 
nidnftraiiidum ;^0 ^^ ^ni iatci: cartera ubV in ^fechanica 
. ittquiriti»: in yircJ » ^ulbiiJ. Sc^ipnes C^onicq: ticfcribi 
^fltint y indc pendeat , utrum tit tetidere d^beant ad 
dtmm p^nt^u^^ \ an a& ipfo t t^iinircim 9i;rum, atcra^in 
fse <ife'(ictqitoc»'ati: vero r^pujtliyx* jam tero, i^iemu^ 
grad^ ad (»:oprietates qu^dam Hypc^fbol^ telatafi ad 
ifympcqtqs, q^s ab hac diatpeirorurii cbdtdas bijfariait) 
iecantium prqprietate pendent, tSt fiKundilSima; itemm 
funt, ac quxdam etiam jt q\Mf; fjypcrbola habet EUipQ 
QuoQue communia» fpqpte prpgignunt. 

Mr U9$um/fxtr^mm , &^ umm, ^fyni^tioifm ^ ^u^t&fgi^ 
numo intercepto inter altcrum p &^ ilterAfn , air ^mi^- 
t» , $Ai 4r^¥^ MfM^iam ftcat^ fetM etiam bifaritim. 
tmi^^ ji ofus efl^ froduSlarum fig^nhm imtrc^th «% 

fy^fM¥^ 

t%2i SK entm^ q^o^i chorda P/ tettniiiata ^deun^ 

jp^^gdtm Tamum in ,fig.' ^ , ad oppofitos in fig. 7J , quae 

^^ omi^ac aiympto^s m pun^is H, A . Si PH , >^ noa 

fniit aBquales, crit alicta, ut Wi,' major . Abfciffi FO 

asqunii jpi , cx C pet O ducawr rcfta, quse(nnmt3li^ 

oomttet diicubi cidem ramo in P*, aC rcAa pcl: P pAr 

raUela chorcte priori occurrat afymptotis in H' , St h\ 

tfyperbedlie itcrum jtt jp'. Di^meter quidem, quai; hujuf- 

medi.chcirdai prqordinatts/faabet, pcr ceonrunn Ctr»* 

.fibit, & ipfas chordas. fccabit bifariim in R , 6r fS < 

( num.3o6 ) . Cum igiiur scquentur & RP ^ Rp , & 

PO , j/h , crit & RO «qualis Kh 5 adcoque ( n. »4) 

^ R'F , & Kh' arqualcfs erunt , nimirum & R>, VJk 

- ♦. aquaHi- 



^ f i i J4 « 3N r^AV f I * 

w?. iT^;»^ ftlt^Af^ dfjmfit^if ttrmin^t^ > />«vi»-v 

^ff^figm»^ ^fy^0i pims dvfmttn Utter c€f$^ 

^&^ffmm* sic fy^Jtii kifmnm ffsmmm^ii^. 

mJiHkwis, ' .._«..,.-. , 

4citt AI efle aQqoalU la: qiiace duifta jp(»tccra ID! fAr 
falldiLCA « ionec ocaum O » > ccic & PC ^ 
iB>IisI)4: ac 4^ I4 ii^idiaDi Ad ^muJc ID Ss4- 

CmU,*. 
m*. Si € binis funBis^ P, p itiiiufvis ff^lfer^hoUi in 

i fiO, (^ pb 9 po /^ ^uibufvi^ ipits angdit, daiir^ 3^ 
*fwW«» BPO fiib binis inclinAds 4^- ¥^9 fun£h ^ 
^/W^ aquMc reSian^ulo bpo fub imU^afis Jm ^U*'^ 
^t^kds mutat& funllq P utatmiHt mmHm ftmf^ 

J^i. Nam ob p^r^cti^ ofit PB od /^ , iic PU «i ' 
r^bft (S»tq>cis «^^tbufi , ttt ^^ ad Pib , 111011^001» 
^NUoIar , uc f9 4ul PQ : ae proiodc rc<9tangttlini \ 
J ttiocmis PB , PQ ^ual^ rcdanjgulo, (iib ine|li% 

?ii7* Si § quffidf funOa fiyf(f!^§td P msixnmir PD> al- ; 

^ 4y^t^9 > parathU atttri , rrii^utmn fub akftif 
^ cmr^ CD, <!^ tjufm^ 0rdiMatat irit f$mfar icrn^ 
^uq4 r$iiangutum d^itur Pocentia Hypcrbol» > ^ 
nmato utci4nquf funlU , arunt ^dinata in tad^nt- . 

•^^!roca fimflici otfdfarMm ^ 

.^28. Nam fi PO, PB abeant in PD , i?R parallelas 

"'^is afymptotis , erit adbiic conftans rcctangulum fub. 



i 



'\ 



\ 



.fi SECf tOI^ItfM GOKlCAftui^ 

PD, PB^ quas abiens in PR tvadh asqpalis CO, hti^' 
n oppofico pirallclogrammi PRGD • Erit igimr con- 
(ians etiam reftangulum fob GD> DP, tc refpedhi bi^ 
aorom ptmaorom P: jp> erit PD ^ pdi ut cd adCDi 

^ CH O L I U M il; 

m 

giy; y Ti€C conftans Hyperl?oI« potentia eft rtia t 
JLl pr^cipaijf proprietatibus Hyperbolarum ,.& 
kfiiimi folet.pro decerminatione natorse ipfiu^ Hyperbd^ 
he relatx ad a^^mptotos , iii m cutvx ,* in quibiis fftm 
lUnatat funt id afiqiia ration^ mdtiplicata, vd fubmul*-' 
tiplicata reciproca abfciflarum ; ut hic funt in fimplier,, 
^pellentur Hyperbolac adtiores; £x ea plurima^ proptie- 
cates ^roflounty quariim aKquas,^ u,t mdnui eruam fe^ 
quentibds GorpUarris , tum regrediair ad eas , qixx '' erir- 
imtur e pr^cedentibus GoroUariis^ > ex quibus ctiam illa 
ipfa pocentia fpoctte [Hrdfluxit; 

«0. PeJitU iifdifn y nnatam pardlltlciirdmmiCDPRy 
^Md continent bin^ reSla ordihata ab eodem PmSi P- 
Md binas afymytotos cum iffis dfymjttofis , quam trian^ 
gHli.CjyP 9 quam continet abfcijfay crdinata afymjftoto^ 
& fimidiamiter > ac area in fig. yS AGa ,* qnam con^, 
tinit tafigens ad afymptoios terminata cum iffis afym^ 
ftotis ^ fnnt magHitmlnis femf& conflantis . 

237. Si enim PB fit afymptoto . p^pcndicularis i 
fldfauc efit cdhftans recftangalum fub PD > & PB , 
£ve 6jh CR » &: PB , nimirum fadhim ex bafi ^ & 
altitudine parallelogrammi CDPR > ackoque tam qm 
l^ .o^^ea ^ quam area ttiangdi PGD 6|us dimidii • Du* 
*^ Aa autem ID in fig. 78, paralkla^ afymptoto AC # 
erit ob Aa fedam bifariam in I f nmner; 22; ) , ^ 
tea AC4 dupla a^fese IC^ 3 adeoque » ob ^G fe<5t^ 
itidem bifariam ifi D y qoadrupla areas C D I co0« 
ftands , 



C^ 



r 

1 



E lr'E M E ftf T 4r 73 

|- CowL ii:. 

, ij>. fi in fig. iQ. € binis fHnliis P, p fjnfdem rorJ^J^ 
jp^ HjfvkoU ducantur iina ordinat^ PI^ $ pd ad alff^ 
^4^Pfft(it$my & tUfa alU P^, pr ad altfram y or 
m Dfjpd clmfa^ ^u^ ^fym^oto ,, & ^ioribnr ki^i^ (fr^ 
iimsy tqfudfitftr area RPpr elaufa eodem arcu , altor 
fA ifjmftoto , & pofterionkus hinis , ac earum fingw* 
kjmAV^es ar^/t feQorU Pij^p t^rmipati ad cen^ 

^};. Si pAim PD , fr 0bi mutuQ occprfrant in e ^ 
VcaO/rf qquabitur ( nura. ago ) are« CRPD;. Qua- 
fe topfa commijpi Cr^ D ^ & addita communi Pgp ^ 
pat ar« DPi?4 ?qa^^ arc? BJPfr . Oubniam vero & 
m trlanguli CuP aquatur are^ irianguli Caf , fi 
iW) , Q fibi iavicera ocpurrant in I , dempta com- 
IDumClD^ & addit^ ^bmmuni Plp , crit arca 
(o2Qfis Pp^ 5|?qualis mx tp£d ^ ^co<juq 5^ lU>pr, 

Gorolf. I2t 

2Jf eon^urfiis e ordinau PD i^ /^- S2 4^ ^'^^'F.St 
fPf ^j^/0/M9» , r«m ordinata rp ^ alteram , ^ r^/^^ * ^ 
fvfiv £. ordinata RP ^iii» ordinat^ dp // z;^ diamf^ 
^ tP^horia }Ci Jpalfente fKp ordinjOt^ fhodam Pp ^ ^^ 
/ ? W^f I *;V di^ffffr^i duQatur qrdinata I M 4^ 
<wr4W 4fymftotHm\eruntj^ ahfcijfa <jp ^ CM ,* 
V^ f <^ qrdinata DP , ^^I ^ dp fontinuo frofortior 

?]J. puais cnim O , CE » crlt ( ^um. 227 J CD 
?^u , uf ^^ i five P^ ad dP , fiye dB( . Quarc oB 
.^tplos GDr, C^£ ip paralleli^ (qual^s ,, iimilia erunc 

rO^a Cbe , Cif/^ , 4^ angulus DO xqu^^ ^ngu^ 
^ ^c propterca tcdtz O fiipr^ CE cadit : ipifa 
KP^m Ef, diamctcr parall^ogrammi PepE bjfariam fc« 
^ 2|It?ram cjus* diwcirum. P^ in E( , ut facilc 
W*ia? , Qjiarc QOfq te trarifcat pcr ccntriim Cj 
P» erit diamctei: habcti$ piro ordinata eatidera Pf ^ 
fp fi pccmxat pcrimepro in I , & ducarar IM 
*%W ^ ^yrapptuip ^4 ^ crii pb trian* 

gulorum ' 



^ortim fimilitiidineni GD Ht €M ^ m De y tive ii 
*M»Ry MniffaA ( iiHOT. aif. )' vac CM ^ C4, ^ctec 
qw CDi GM V C^ iit^ ^ADtii ^pra)>orcidiiii i quibul 
ttR» fiwr ife^pftlce ,p»6^i4o*aJe»* ( n*i z^f, >D,n Im^ 
Jpi^^ ttnak et iphe is oonliiillat j^d^dNbiiev 

tinM frojfortione geometrica *, & erig^ahtur ordih^k dfte^ 
j* i^finftoto pietMlHi y aruclaiifa hints ctnBkfvii jfrirxi^ 
ftj^ir ardirrarfs ; 'irctt , & atptftiti hmt ihiet fe aq^d- 
ks \ ^ ihferfe kqptafes^ drei Se^otM ftrndftktorm aJt- 
^mhthm 4, Uttis ij^ibufvis fraicimis ofdinida^httii^, vicriiciA 
$us y eonfiHtieHribus fhgrejjfohim geomtricM dfcijj^ i 
i/et ordHniiis \ arha ioinpiatx a iatk q^^kvit ofdXilfdta ^ 
vel d ddHa iHOK/is fefmdidhtetra^ fe^ ordihdti ; ;i^eriicefH 
dH&a nfque ad feqt^ntis orMnata/ ; vH finhidiametro^ 
citfcet in frogrefjione arithff^iicd ', & drea cliufa brdi^ 
najta qudvis \ arcu , & ifymftoto crefet ih ihjinitum ^ 
/? arcus & dfymftotus ih infxnitum froducaniui^ . 

%if. Narti cxifteriiitibus CD i CM j Cd in comfcirt 
pfoportioric gedmctKcai ut Sc DP^ lAl, df , ttdtx Ct 
( ni«nV2j4J fccat bifaKam chb?dam Pjf^ id Bi &: profafe 
de» Warigiila PCBy/CB haBctitia bafcs PBi jfi ^qiialfef*. 
& «lancfeth altitiidinem mChabciitdreas^qu^bSi aqfui^ 
i)tts ff licmafatiir ^e| h^pcifcolic? PIB, >IB aiqaalcs (^m 
>^ ) remaiicbanc squ^es etiam arc; fedioFtim PCl i' 
IQ, adcoqiie & arc^' HypdtboM^ H>Ul i IMdfi, qn^ 
Itos 5q«raks funt f nilin* i^i ) erunf intcr fc fqtfclldr. 
£<idchi adtem' paod fumpta Cm tatia pdft CMi Cd i 
in\^chietur arca fc<3rork fCi > vcl* qiiadf ilihci dfiffr ^ 
^iiiilis prioribus^ atque ita porto a(fumptis hovis abfcli^ 
iA coiitlhuar ptx^pprtiohe geDmetfici i ad rematnettlt^ 
kis ih eadibm reciproca drdindiis, ards {k&orixm i&ch^ 
|i«t^titisiiJr si qtiavis fcmadianlietfd CP i vei ^rei^ quddri« 
miei& indprentibus a. quavis brdin^ti PD acccdcnt no^ 
VA iacrcmentiribn^t seqaalia j. atquc arese pfoihdecrc'' 
iceat i» rikionc aritluheaca » Cumquc nun^rus ab« 

m 



i t % li ^ fi f Ai 7*- 

i^ae^w iti ^Qftwkz ^opordoat siSwafm^ aigH 
ri po(Iic in infiaitiinf k poceft.cdam in iflfiattciun AQ§t^ 
d numerus inereix^ehtotmn il|prum ^uaUum « quo- 
fttm proifldc fiimmaiqidnrKs ftiicanl hi^^oidiacid ct- 

1)8^ T T^C area» in arichnietica prog^eflioac MJil^ 
VX iccnti^ {i^Opricsas ^^ dum dMcifla^ cn&uAciil 

'^togrcflipnc gponiicalca cQ adflibdtuninfig^is & notani' 
I digna ; Inder cnini, 6iy ut area^ HypcrBoUsaCr liabe» perf!^ 

iiQC pro Ipg^LdMtiis niiinci-oruin > qiioc^/cfl^ioiun» ^.: 

!rcif£3e; (^fl4^ imo 6p6 ijidui.arcdi t^^tbdiic» ^mpK^ 
i titac. mctboda ^q^sc ope calo^ idtcgirali&facile iwtmr • 
I t^r^ 1, lo^tbmi qjuoque.computancur^^ &c., cometttatt^,, 
' fcmel ibgai^ithmiisfi arca Hy^crbbla^ 4 Wa . d4cii ocdiM^ * 
^ tis^ Sc ^bfcifljs {acile idvgnuac ; Scd bic jgcomenricasi 

&Qyct aritiimeciicas* pfopri^i^ pcfiequimuf Sc^iqnjiun 

Coniciatiim. ... 

, zpjL.. Pergam igttuf ad aliam pcbpfietatentah qpsi pi^ 

Hter cx conftaati* illa potcntia^ Hypcrbole dcdudwj ouip 

aliai ^ alii^- prorun^ntcs adjidan».. 

34 1; Si9it cnimi in tilg^ ft; jnxca nuni^ 176 itfSH^imt 
mune^ Mff»^ ^4f ^ comtouncs afKinptoci TB ,. a^ 068«l&*' 
rcpi€esT t^^ibusL por ao^ium vciiices dudll^ in Ty n » 
I Bm A • R«di. H^ p^iUeM lUjrnfptdtd: 713 iccabiniR ^ 
a4yq!>totd O iiifiiiri^tm, iti Qba uo T0 inv C . Si ancew 
gup&Tis ali^^ij^ri paraU^b IL ootfutrat O inDj eiii (vm 
tXfi ) DI ad M0> m CO ad CD ^ m DH ad^ OX^ 
adeoqqe oi» QMb OXarqoakSj acquabuncur &Dl9cD)i-« 
Indc vfr« fc reccangulk CDL^r GDJU^. ^ fttnt« biAa- 

ttttn 



7» SECTIONUM CONICARUM 
ffuni Hypecbolstfiua foajagatocum poieatue > amoalhi 
iimt* 

242. Tdngens afymftoHs intercepu ^quatur dUmetra 
f^njugata eJHs diametri ^ qua per contaSium txanfit , 
0c reQa jungens in vertue iinas diamotros conJHgatas^ 
C^ alteri afymftoto foralUia ab altera f^catwr hifa^ 
rim . ' ■^ ' 

^24;. Si entm ^a fig. 8}. fit efufmodi cangens, ote 
f num. 223 ) CA dupla DI , adcoque sequalis IL, cut 
cum parailcla fit , cnmt Sc AI » GL ?qualcs , dc pz^ 
raUel; adcoquc & corum dupk Afy U ?quaKa . Dia-r 
mcBct' aiitcm' ]LC/ eum paraRcIa fit 'tangcntl AI4 , crlt 
('nui^. 212 j coti|ugata diamctri lO, & rccia IL jun- 
gcns carum diamctrbrum vcrtias, a<yrapi;oto 'BB parat- 
Wai. ab ailyiDpioto h t>ifariam fecat^r, 

CoroU. 1$. 

344. Diametri conjugata in HyperboUs fnnt fibi in^^ 
vuem conjujgata quatuor tangentes per earim vertices du^t 
Ite toncurrum ui afyitatis ,* lAt parallelpgramum ooAJH-x 
tuttnt 4nfcripum figura claufa quatuor Hyferkolarum ra- 
mis 9 cujuf area efi fenifer confians , aqualis '«imi\ 
fum rectangulo fuh axibus \ ac farallelogrammum fe^ 
midiametrorum conjugatarush rectangulo fub femiaxibus . 

245. Ducta cmm in fig. 83. 4iLQ^pat^ck iQredt 

fegmctitiun afymptoti CQ^?quaIc IL , adeoquc clupluni 

D£;^ ac proinde 4LQ, tangcns ( num. 223 ) , & dada 

Idi^ ac fumpta dq acquali dC , patct ob Ci^ CUy as- 

qtialcs CL, CI 3 forc & U aiquaicm LI , adcoquc «- 

qualcm tam CA, quam CQ^, & proinde dl, di dimi-; 

diay GA 5 CQ^ , Quarc A/ , Qi con vcrgent ad iderti 

ptlHiJtumgita, utfitC^ dupla dq^ ScAqQ^^ fcdae bthi* 

iriam in /, i, adeoque tangentes . Eric igitur 8c dta- 

mctcr li paraJlela tangentibus ducfeis per vertices dia-. 

mctri L/, adcoque cjus con|ug^ta : & A^iQ^ crit pH^ 

rallclogrammum , quamor tangentium , cujus ai?ea con^ 

ftantcr «qualis crit arca: rtdangqliTfB^, cum fiAtqua- 

druplae 



l E t 5 M E N T. A, / 77 

llfH trisBgttlorum ACi , ^C^ ^ualium ^tium.^jo) 
K afta ClaL paraUeioigri^ainii fcraidiameirorum con^ 
|agsaanin)> v^ arca ACLI cui ea ^uamr , ^ualjs a-» 
^WCx y mm fint duplc uiangulorum ACI,TCM 
|lilp{qaalium. * ^ 

^lOmf&m diMmetrmm fnmarianim miniins ejf 
ffis, trmfvnjus , ftt^dMruirum coHjugdtus ; qMrum 
M^^ ^ m^is ^ai ai^ iff§ trantverfp vel c9n]ugat^ 
K^f^ 9 «ij ma^ores fiiht , ffjc ni^ hvna hinc inie in 
^fdih engiiUs incltn^a aquales : frimaria autem efi 
^l^ , €qit4is » z^( miwr reffeHu fua conji^ata , 4C 
0t^4w^^^^^9, ^ quUfus \acet axis tranfverfus , 
^ fijfeima^ funt acuti reSi^ vel ohtuji , froui axis 
9v^^ &^> ^ mam' , a^is x '^k ff^ip^r reffecif^ 
WWfi,. • ' • • ^ ■ • ^ ^ ; 

^H7'N2m quo niftgls fibmiprdinaca lU diftat a vcr^ 
*"* ix» M> co majgis (xefcit ( num. 79 )*& ipfa, ac 
^^^ ?bfciffa a ccntro CR , crcfqt ic ftimmaquaf 
*JWBn utriufquc , adco^uc qrcfcit fcmi^iam^tcr Cl • 
waoteiri .inagts & IL te^ecih ^ MX, adeoqvic L al>' 
?>^froin()c' co m^gisacfcit fcmidiametcr fccuqda- 
B* vL, qan ca fic pt;|maria Hyp^rbol? conjqgatae . Bi-? . 
^mn CI, jCS^I, icrminatif ad puada I , M oFdfnataQ 
¥^ in angolis RQ , RCN^um axe CM ?qualibus' ob 
3^ conunun^, fc RI, RN htera aequalia trian-. 
•^P rcaangulorum CRl , CRU , xquaJes iuQt . Porrq 
Jjij iriangulisCOM, COX latus CO fit cbramunc, 
T ^ a OX latcra xqp^ipi { n^ ^40, , prout fcmia- 
^ttinfycrfus-CM fticrit siajor , «qualis , vcl minec 
fftu qonjugati ^, , etiam.. an^uJus COM crit 
*^ «qualls , vel mirtor angulo COX , adco-»' 
I ob MX X IL patallclas , & CM inaior, xqua- 
^ vcl minor CDL , & fcmidiamctec primaria CI 
'H* > icqualis , v^thinor CL . Concra vero angu* 
^ymptotorum TCl «quaKs akcrno COX crit mi. 
' > aqualis , vel raajor OCB , qpi ^quamr MQC » 
"que is angulus TO afTipp|6(orum , in fuo jacct 
Hcoj^h.Tm.m, Q axis 



7$ ^ECTlbNUM CONlCAiltjM 
:ucis tranfyetfus 6c Hypcrbola erit acutusi fj^f^ > Vjel 
obtufui* 

24$. Differgntt^ padfaiprm binafim fmidJ^amtra^ 

fum conjtigiUrum tfi ad qtmdruflam ifotinyiam Xiyttr-^, 

hoU iffiki 3 ut coftnui an^u^ djy^tdtoirum kd tadium 

anleoiui fimpef (onfiansi f^^^mii differenii^ Vfidta- 

I tofHfn femiaxiun^ - 

F.78 " H9* Dudlaemm m ^g, 78 iV jHjrpcAdiciJ vi , ^fyi?^ . 
pjtoto O i diffcrehtia quadratotmn fcmidi^qtictt^ O | 
S^ tangcnti^ \a tju? .ta0|;ciis VqUitur ( nuni; i^i) fi?-. 
midiametro cotijugat^ diamejtri li crit ieinpcf |^^(lcnr » 
ac diilcircnti^ quadt^toi^um CV i V4 / cutxl bp '^gii*. 
Ips ad V rcdds ^ ^uadi^atUiti illiui^ fcmidiai^pfri ae- 
qllctlil: qtiadraus jCV , VI fimuii & ^iiair^t^i^ M ^ja^" 
dratis aV , VI fimuL * Pori^d qiiadratiini CV cxccdiil. 
fcina qu^drdta CPy DV pcr bini fciaartgiai^ ^DV, &: 
qtiadrattinl ^V deficit ^ bitiis c^iladrafiV P^/t5V*^j^^ 
bitia itA^guii yi)4 i fivp a lunis dU^dtatii iUis.iP- 
lii CD j DV pct binj^ iUa Jp/a^gfcingvl^ Ct)V ^lff. 
tur differeiitia quadri^Qfuixi C^i }/a sfqxi^i %?^#:* 
plo redangiilq fub CD /DV* E(t ^utetn >c4*9g:iH*P} 
fufci CD i DV ad rcdaflgulum fut CD> DI j^vf M 
po^ntiAtn Hypcrbola^ in tatibn^ DV a4 pi i mmiiriini 
ut cdfiriUs ^nguli VDli fivc iritcfiji > ^ oppofiti^CA 
ad radium^ ^dcpque c^ftans ; ''6i' cuni axcs ipfi %c 
diamctri conjugats'> cru ^quaKf^^iJff^^ 4^^?89!^ 
tum fcmiaxium^ 

scnoiiyu i^. 

250. HJIfcc jaii? cx conftanti iila Hypcrbci^ pd^ 
XTL tcntia dcdudlis rcdcundum ad n, 223 j qx. 
qiK> potentia ipfa coti^ans dcdu^a^ cft , uc aliu9i ijuf- 
culum inde fimul cutn ea pror^pf:jite;xi jf^etfpcm^i^» 
qui tamcn nunu^ (oecundus c^ • 






Co" 



l l W i N t A. f§ 

HffnMi » vil ^avii ex hif i^& iilit ^viU > 

\ iumhoii mdfHdi HreSiiM i mjtt fempi^ mm 
^^fiiMif 9 ^Mi^ ni^iitm f^^ quidiri^ 
rnV jfm4Ulil4 ij^fis chotdii } m. uH chm^ds^ 
iinku^rdmm fkminafur i imJMoid fii^m ti^€f$>^ 

\mU' icaoTii ktiJiii Hypcrhilh conji^diif ; 





•fc* ^^-\ 

151 Om eolim fiat ia fig*;78> 7^9 tquales iaoetF;^f 
(nttnu 221 ;2 HP^ pb\ 8c lip^ h? i «quaUa crunt 7^ 
^tdiatigali MPfri Hff^i PH>, PA/, & matictt- 
^tneAnmibtk PH » Pi& ad ifytOf&it^gi fc&anga-* 
HP^cfU {enipcr niagciifiUdihis cdnftanei$ ( numi^ 
;); iiUiuttibus atkem i|i g^ 98 ptu^s P i jf ia 
i^abk ircifiangulum PHf^ ih qa^atuni can^ 
tt AI i oii ikquaiii eft ( hutO; Hz } {eniidiamc^ 
imlkk ip<l> fiC' cliordls l^ : ac ii» fi^. ^9 abeucU 
^^f(mdn.H» ik iii cictoirm' C> ;ti)if re&angulii(;ii 
^io qaaieaciim rctakfiabTfefltfi Cl; Hiticaatom & ii» 
H fint fl^ li diaiietri c^n^gatas i dc ipfa chordar 
mai^i piaHffBol H^^peiiQl; cdnjiigafli lii N i » i 
, quamdi? illa twaadgdd HP* i Hiifr ; PHp , PAp ^ 
*>^!|m9orN^;9h; Khni HNi» H»^ emuic ^aiiacL» 
^foadrato rcmidiiniMrtri CLi 

CoroU, 16, , , 

^K* Si pg» 84 e Vittici p femUidineiri primarU iH 
"^ V i^infitrim ifrimarum iQ dmMw f$m(^Jtifid* 
i & r ^^/c^ D fm iMaM^f ri CD !;;<// (conjngd* 
^ D£ z» pl^ pardlliiM i «rir 4ft4dratkm GE n^ 
4 kktroteti fofierietemr d^iuUf^ feQdffgido fnt 
tl ^iffis 4 hiffis verticihdr por friorm » e^ 
v^ri4 bimnm iHsdrdforum UnatHm ahfaijfiifhm 
mr9 C£, QV^^aiHfdfitnr qHMdrMo ffmidimfifff^i C^, 

G a «s 



l9 sEcrioHuM comeARUM 

|if qudm femivrdin^tA tfi dfmijjk ; differerjtia Vir^ 
^Mdva$mm femmdinat^ pR ^ & paralUla D£ f/14- 
idrat0 femidiamftri QL conji^ata ifJiusCl, & ^dem 
if^^l^ifW^ fi <^ femi^rdinata , c^ fi«j farallela ducanm 
$m in . diametrum facundariam , /f ^ i^i quadrattm 
idfeiffa a centro fet eirdinatam aquabititr feHanguU fub 
iihfciffis a Hni$ verticihus fer faralielam , 
\} 254. Naitt fi C^) CD finc femidiametri coajugatEC ^ 
/|D crit paraHcU afympt0to AQ^( num^ 24* j , & &▼ 
ifta bifariam a Qi in V. Quarc fi R^ occurrat afym- 
ipiacts in, H» i?, & ducaTQr hD > quxL^occurrac afym? 
tCQCo HG ioH^ 3 crit ( n. aQ4 } ctiam HH* fcAa bfr 
iariam in C > & cum Hh fccetur bifariam in R (1 nu^ 
.321J erit ^H* ' parallcla CR^ adcoque ordin^ta diame-r 
iri /CL, &. afa ca feda faifariam in R*. . .' 

255. Jam vero redfcangulum hUH* ( quod cft arqua? 
ic (num.25ij quadrato CI^ una cum quadrato RO^ 
4vc C^ cquatur quadrato R'ii » (ive CR^ vcl quadrapl 
to C^I » & rectangulo |Ri ; adcoque dempto uaofaique 
quadrato CI , quadratum C£ «^quatur rc Aangulo IRi ^ 
Poritcr cum quadrata C£, CI fimul xqucntur quadran 
10 CR > crit quadratuin CI diffcrcntia quadratoruni 
<ZR > C£ ; quadratorum ycro £D , Kf > five BJb , Rji 
lliffcrcntia eft re6langulam ApH, five Y ^^um. lyi^quarr 
dratum CL* Demum ut Cp, CI funt fcmidiameari prU. 
inari?, CD, CL fecundari? rcfp^u Hypcrbol? i>Ip, ifl^ 
funt fcomdariae, h? primarije rcfpcctuHypcrbolac DL , 
Quare i>atent tam quas de primariis $ qu4m , quas dm 
f(^cm\d9p\!^ di^ni^tris affitmavcra^, ' ^ 

faroll. 21. . 

25^. ^4islr.^«jw femiofdinata ad differontiam fM^ 
dratorHm Jua femidiametri, &. ahfciffa a centto in duH 
pfffris frimariis % fumman^ in fecundariis , & ad rtQan^ 
gnlm in Ulis fuh biuiji abfcijps a tinis diametri ver^^ 
H?if^i^0 HfiHOidnatum femidiametxiyveldiam^mcojt'^ 
m4^ i^^^Hadmnm illiHs ifJtH^ fna fm^diamcm ^ vfl 

?5^ Si 



M 



^; Si teim pra^tcr^a diamettt prlmaria li ocasxtM 
i KX&aitx ia R, crit quadratimi lU i fiVc qnidta-' 
"^ R/-€ura r^etarigiao H/A/ nimininf bina quadm^ 

\ Cfc ad qoadratara lai fiye GL# ut quadratom 
fivc. quackaf^um C| cmn rdAangtdo IRi adqtf*^ 

n,GI ; ac.divicfcndd quirfratum Kg ad q^dra-» 
I CL, ot diffctctttia guadratorum CK^^ Q , fivcut 
ngiftjra.IRi 4d quadratum Q j tcI ilternandb 

caaun Rf >l diffcrciltitnfi qiiadratorum GR > CU 
ad rcoangulum IRi> ut <iuadratum Ct adquadr»- 
t O , vdnt qu«dra«m! cotius U M quadratom to- 

5A .Qood fi ditoictcr fcc^O^datia /£ . occu rrat in R* 
oKtinacc Py, afymptoti? atftscm in A^ HS critqua- 
Utimi Ki ad quartr/^tilm Li, fivc GI,,ut quadrauifii 
» M quadratam ,CL ,' & componctido quadtatuiri 
^ cam quadrato GI, fivc cura' rciSfcAngu 16 f'hP\ (fi: 
;J ninurum quadramm R>* adi quadra ttim CI , ui 
m quadratorum CR*, CL aid quadra tum CL, 6c 
«itodo quadratum R/, ^ , fgmmara ^uadratoruift 
^9 CL, ut quadrati|ra Cl ad quadratum CL , fiie 
¥»draffl(m totiii^ li ad quiadratum toiius JJ: 



./ 



se UOlilJM V; 

Hlfcc dcduais gencralitcr pro ^uavis H^per- 
Marum <pe%, addam hicf poltrcino noh- 
qu^ perjttacnt' ad Hypetboiam . aftquilatcram , a^^ 
m habtt latus rcctun? apquale ,-axi ^ traiiivcrfo * & 
& ipfos axc2f atqualcs , & . jurta ijum. 246 an^ 
afymptotdttm tccips . Pl?raaii ,; qose a \^^^ 
rbolam^cyiilateram pcrtiociit> dcducttntur ex iUi 
hic pro Hypcrbolis Jn gcoerc dtmonftravircus 
quc hic paritcr locum M Virid iam . Intcrea no* 
lum iUud : Hypcrbolam aequilateram cffc id im^I 
arbblas, qaod cft circulus int^r Ellipfes . NamEl- 
1 ^jiB ax^ azquales fint , jam in circuhim mu 



C^r^U. 



%%. 



! f» 51CTI0NUM CONiCARUM 

ttm ji^iHgMti ^ hibit ffi^ UtHs rslHifn wqn^k Msrii 

dfsuim difiantU f$d 4 tMr4f duplum quMdrnH -. ^ii, 
farinslibit : TertU Mffgiiisi MfyfHfhform r^0s •: ^um^ 
f^ti^fi^im dtqu^efn di^idU qnddtm fitmiAxU uiriusl^ 
iet : Q^into qu^fvis iis^tm con^Mtits aqHdUf i ^iJt^ 
fo qusd^attm. CUJnft^U ftmiafdi^atM vujuA^e^ dUfkifH 
frimarU aquMle riSsisfgido fiA bints a^ftijf^ 4 hiffi^ 
'taertieikus i Sfpfifn^ qud^rafm tujufvis ffmioiidinstM etik 
juslibet. diametri fecundaria ^ual^ fumma quadtMtek' 
rum, femidiametti iffiu^ vit ffimarii^ , vel ejus ca^fjugA-'. 
ta y & 4kfciffa a eentro t OSa^o iffafn fffffi41^iff4^ 
tam ad aJtwm cinifugatMffit dqualem difiantia fki eoffA 
' eurfns curn iffa axi a vtrfice 4xiji ff^anfverfi : ^ JVitfb 
€ hinUi diamitru frirfiariU , vel e H^is feennd^trHa 
aqHalibu^ 3 hdbet ffU^^m^ fdt&im cinjugdta f&ftrtdi^ff^. 
larem'. 

%6i. PriHiuh) pdtel ^ edm fi^ ( ntim. fi) latus rt^. 
{kqm t^Ftium pr6p6rf^c»iiale p.6f! bi^6$ at5^e^ • \ Secun- 
dum dciducitiir c^ fitim. 6^. Cum t^ftAs^iof^ dmunn^ 
fyd a centro «qvicmr rummc quadrator^m binprum 
femiaxiLim > adepqte fibi ii ecjQal^s; /uftt 9 duplo qua* 
drat(» tttriuslibet • T^rtium dernonftratum eft num. ^^6m 
f iSj Qjiatmm patct in fig, 8^. N^m fi afigubs TC/ fuc- 
rit in e« reao9 & COM tt MCQ fenlire^us > dc 
OC cqudis QM, ad^ue tedlangiditm fob CO ^ 8t 
bM^ <|t|oc( ( mriu 12^ ; ^i^tur pote^tia Hyperbolar^ 
, x^qqs^tnir qfututffi^Q tinriutttb^t CQ y ve) OM » nimi^ 
rum dimiittiGr <|v|iidrat0 CM^ vclCX; Quintum dcmon- 
vftratum c^ i^ ii^,^- tc cruinjir ^ti^i^ cx n. 248 t eutn 
^adraipraiii <)iCrf (eniia nuUa fit in a^ibus ^ ddeoque 
null^ if^ quibufTin dia^netris con|ugatis • Sextuin dcda- 
mtXf^ ()( ^IQintdi, Sit% i^tim. 256,, cum mmirum qua^ 
^s^tM^TI (criiio^dif^at^ ad f^dbrf^ulimr ^. m abfciflr^ 
^beat ^ ^t ^oadpat^m fbmidi^ctJl'! coirijugac; , a4 
^\ia*ate?n q« ftn^iidwm?^ ptimarhB , q^a; m; Hjl^ 

pc^ 



E t 5 M E N t A. :^ 83 ; 

. tt. tofc^ tiitim^ra i^ A^ tri di^wi* fccdnda* 

m quadratorupi qus £emidi2(n9capi 1 8c. abiciflT; 
ra parit^ it\ t^id^q a^q^ali^^tl^. ,, Q^ayun:^ Bta' 




^\k It coticipiajttir R1 . , eric efa^ 4^ddrai;atnWqaa- 
fii^ ^uadr^tis' d ; CR' adcoquc quadrata Kjf^ , 
JAk ipft R>' ipfi, Kl ^qbali^ . Non^m fd^cik 
irorfqmnto t nam io 6g. 85 fi QN/fit ^ua-p^g* 
Q tiit ^ idgulus NCR^afqualis ICR (- nu. »46 )^ ^ 
^ 8t NCG, *qualis iCD^ qui ofe Qtnnia larer^ 
. ^, itifft Cp.1, CDL a^qualia , cri^ «quali^ angu-» 
lo DCL . Quare a<idico NCD comniuai cric NCL «- 
1«*^ GCD . Sunc a^tcm-NC , IQ icmidiam&r 
*5 }^mfi4 xcfpidbi Hypcrbpte NMI > & CL! conju* 
^ potefioriif , *c ca&dcm func fccundaria: rcfp^<^hrHy*» 
^ofe tX i adcoqdt v^dkt idcra ^rd^ wirbqui} diamc^t. 
Ittapncrc^ ' '" " ' ' 

^ ^"" Corolt. ^3,^ 

^^ljfi f W^£r verticibHs, V, u i/ir^ yfj', Jt5 ctd^fvU 
WWf ifiMHrui_ HyftrbaU aqfiilaterii , ducantnr^ hin4. 
W ifdfwMr funElHm^ eJHs ftrirneiH it&fer verticeni. 
^^i^ fApf. Uffgens Vl ofCnrrehs iffi. uV in ly dJt^ 
gtyaP. i^^iufHtfir dniHlo VPR » t;c/ PVI . ^.^««««^ 
g*»»»? r*^W4J VP aqHabitur ttElan^Hlp aPI ; diftr 
^if^mfn dd bajfm Vii trianguli VPu conjlan^ 
^HiiitHf at^ulo TJiVl , 9«iw cWinet taniens VI ^wrf, 

*/^i^vu.. ; 

?^r. Dufta ctiim fcinl6rdm.at^ PR, qua? crit paral- 

^ ^^cn^ VI , cric ( o«|ri. tM ) qu^dratum ipfiiii 

wpilc rcflialngulo. «RV , adcoqi^c AR ad Plt, uc 

*1 RV, nitairam ab angulilm ad R communcm 

^ tti^glia^RP, PRir, & atlgulus R«P acqualis 

Vl^. , ^[d^dquc « aficmc^ PVI .. Quarcf ob «1- 

"^ ad P commoncm ctiwn cciangula IVP , PifV rc- 

G 4 mv 



i \ 



«4 SBCtioUxjU GoUidAkiJU 

inancnt. fimllia, & IPad' PV:, iit PVad P/», acqttt*^-^ 
tum VP xquale reftaagtuo ivp!l . Eftautem aagulus ^^^ 
^fFercntia anguli »VP ab angifld IVP » five V»P i 

^CMOLIUi^Vt. 

.164« /L Tqui boc quidcin padlo cx ccmftni<!^ioiid 
/\ jprobleniatis tertii druimus priniariam p^o-< 
prietatem dianietrorum ordinatas fuas fecantium biCa- 
riam^ ^ inde rixpetbolx-ad afymptotos relatc proprie-' 
taces dedttximus alias nihilo minus fcKCundas j ac Hy-. 
perbol; demum. (qBiIater; hatiiratii » & proprietates 
pleirafque • In ()ac poftrcma habetur , etiam aUa quf^ 
datn elegails analogia ipfius I;lyperboI; aEfquilater; cClin. 
circula, &c conftrudlrio locig^^mcfficiy cujus ttfus non- 
iiunquam occurritrf , ^ \ . . 

F#^ a65. Conftat ex primis Ce^^corias' elementis incir- 
culo fupi?a chordani quamvis V^Mn fig. 86 oA quo4- 
vis peripberise punii^ura P ad eandem a6 jpfa chorda' 

Srtem faccntis dudas ^binas redlas » continerel ingt^ 
_m VP«f femper «qualem, cujus nimiii|.m raenfura eft. 
arcus dimiditis VHi^ , cui infiftit j five qui ab eadenv 
chorda fubtendttut ad partbm oppofitam •' Quiare incir-^ 
culo ^liquorum anguk^um ^V»>Jft^ fumma clJ fem^ 
per conftans > ^ualis niminim complemenitp anguli 
VP» duos xtdtos > qui cum fit ^qii^s^, anguio mVI r 
quem tatigens' Vi ad partem opppfitam ducca cohtijaiCt 
cum ipfa chorda , erft fumma illa angulorum PV» , 
VhV squalis ahgmo mVI, qiiem ea.chorda ad eandezp* 
partera. continet chm tangena VI> duliijn HyperSoIa? 
flon furama) fcd differchtia aiiguloruni j^y«, P«V a- 
quatur angulo «VI >- quem diameter «V ,>tontinct pari- 
ter cura tangehte VI ad eandetn partcih r. 

266. Hlnc fi qhqritiir hujttimodi I^oblema » f^[ 
iMd hafi confiitMeti mahgtdtm ita , m fumma , vd 
^ifferentia angidmm ad bafim aqi/wtwr angklo dato ; \kf 
trumque Problema t^ i^dets^minatusn ^ itifinitas hi'; 
/>uir«m folutiones admittcns , quas omni^ idcm con^ 

tinuus 



I t t t M t U f A: 5*^ 

m^ iteos gcometricus coixiiple^tQr ; qul pro fuin^'* 
pftt trit irou. eirculi 9 (H-o differcQtia crus infipitum 
Hyp^cibolx « Pro utrcque Aateiii contkro^kio eft hu}u(-. 
. inodi^ Adfufifhm V ixtremwf^ i4^4, v^fa fi^ Mgu^ , 
^aVI «^ir^/ij ^»^10 fumirut y vel diffmntU. THmPraV.2^ 
i^Kuvtfii. 26 , conJtruAt^ dtrcus circuii yPvk habens 86 
M/w tgugentey Vu /r* chorJU , d* fr^; 4^ferenti4 in, 
kiLfvus Hmrid^ dquilaters VP iaJefinite frods&^. 
m mens pdruer VI fre tkngente , & Vu fra dis^ 
iuein frimsria j & 4d iuodtlis pwi£lu^ P earum dr* 
mnbiais reais VP i Pa h^eHtttr fiintio fteblemd' 

3^7« roiiro rarcttliis (min iis conditiofliibtis «dthei 

dom facOe defcribitur » Ducatiir VC perpeudicularis^ 

^Vl>x feda ta£su:iam Vm iti O i erigaoir QC pe^* 

pcQdicslaris ad Vu , d^^c occurrat ui tl prjiori per- 

rca(ficuio i ac qliitrd C intcrvaiio CV 9 vd (> ^, quas 

P^ fare aeqoales > fiajjr circulus 9 ^uerh pat^t dcbeie 

B^anfrcper V, » , & hatcrc pro tangcntc VI pcrpciiri 

<licahrem cjus radio^^ Ac eadem cdnftttitkio cdet , & 

^(iWietQr , qubd eodcm recidit, puiic^uin^ita, ,utan-> 

P^ ypH tfiec acqualis dato * Tum nimirura facicndQi 

t&t ingnlas uVi ad p^itef oppo^ti^ P aeqiialis ckto 4 

f^Pffada rcUqua confihidioiic haE>eretur ; quod qii^^ 

l^iiW : ac eodcm pariter redic Problema i quo &fii* 

tcr data yu qiiacratur fegraenmm circiili capicfil w- 

. ^ VP« atqualcm ciato • 

. .24 Hypctbola vcro jcquilatct^a/lacilc pariter dcter- 

Wo«ur data diamctrq primari;^ vu , &, taiigcnte VI • 

^cnim diamctiro ipfa Vu bi(ariam iii C» & afta 

.fv C re(9:a parallcla taiigenti 9 m qua capiantur CB^. 

w«quale$ femidiamctris CV, C« » crit B^ diamctcr 

f^jugata asqualis priraaf ias V^ 3 ac datis binis diaiae^ 

iris conjugatij datur Hypcrbola . . 

' i69i Nam in primis cx ntim- iai iniitur cs^diitif* 

ffcia mcdiodus dcfdribcndi Hypertolam per punfta da^ 

'^ poQi^ P 8c afymptons concuireiitibus in C in 

;fe'87. Cirqjmdpfta drca P ycgirfa^ qu« ipfisafynipiCffF-S; 

• ti^ 



9« 5fiCtl6NtfM CONfG^RtTM, 

ui^qtie r^ds &(^Ue deferibitur. EHtil SU±m bihH di£> 
m^tris eonj(]§atis U i U hi fij^ ^4 fydilt iiivcqiiititqf 
rfympiotij nutii 144 ) daAftdt» t)d? I , i«, /, t t^ai^ 

dMiairflint» ^fyinqjjtdtdy ^ quibYi^ ddtis ,r & dato {>iiM' 
A9 I jam datitdr pmtfii t^6|a ^er a^idfi^nii ^oA^ 

SCMQLIUM Vlt, 

• - - 

ft^, T? X i&itoi^ proprietalte Hj^peiftolae, ex qtri a/uft 
JLLff rtibdl ccirtftrariid cferivatur , & iUtfd dft^ndl 
pdWffti adnioc(um' facilc ^r doi^rftini Fjyp^rblolsp 4at* 
cmft dd^o circhld iriverti^ htfiiSf incdi^s cortdiiue pto- 
|3bi*<i6nalcs int^ bini^rtcfis; datas^^ ct^u^ ProMctftitfs 
ciiilts pai^ncul^ris eft etiam Gclcbri^^ iHd CQbi dupticido 
«b Apolline olfiti pr^cfcriptai,^ qubd PnalJcma iciciixrciVc^ 
ttres iriqud adcq tbrlit , fc tistndru fruffrA jtef plaaam 
Geomepriam, five ^ti rcd^aruin iftterfcaSon^s; inter ft^ 
vel cum <:ircuIo eft ^uaJfi^. 

ifU Cafiamf^ Pi (at^fitusf aniutt tttH HCh in fii, 
J7,g§ n. HHS feSa CR, Cr aqndtis d^is > d^. r^^ r^ 
9dniui6 RPrC ^utM^ CP , ina Affmpa. ^o d$m^ 
ieftrihMnr HttHks^ ^i 6^ angidaj^ dAKiit f^ff^ irM^ 
f^U fer iffa pmSa R, r.- fef pmBh^. Auienl^ P, dfjm- 
ftoHs HC, Ch defcrihatur Hh^Ma , ^ie uH titcnU 
ocdt&ret iterufh in p folvet frm^4i du6ld e^ifh^^ fer* 
fkndicuidri CH , i^nr jt^, Ci mdia c^ntifiui ptdpir^ 
n^e/ imer CXyCK^ 

272; pufta^im per P, f hct^j <)iWi Afj^Jtotis dc- 
<*pr^ in H, ir, eri* cx tratura Hyj^itwl^ HP »|iKdii 
fhy & Hp «qualis PA adeoqttd & <> a^iia& ^A . Er 
Il4tuf a vero dr^aili i?cc:ti C> drir pe^dndfcttfarl^ P/, ac 
triangda rc<5tangula^ Cifj fih fimilia t6tt Cph , ade#p 
que & intcr fe. Erit igitiir O^ ad Ciy ut HP rirf H/, 
flve fumptis acqaaUboy, itt iEjp id AP, fivc m zj> adrP. 

Erit 



« 1 fi il fi N f A. It 

ipilr iateitt &r hi lld If, m ip U Ci ; qiiamdbr^ ^ 

& M «f '. ip M M, a ^d f^Pt yel CR/ at 0| ijl ^ 

^i CK ddntimii prd^pi^tidf^alei^ 

^ «^ Vcfuifa ifiam fihe t6du^ HypirM^ «taftf Hf^ 

ftA&i efit de^cribto dre^d litli^tim ejfus. ^tiQct^ fi 

^Atfftimar e » circdmduGendo f (fgtilam ctrj^^ p dtmet dff 

fTetefi^ciir PH ^iii^is >i j qmti iAq etiim iiiii^ cks» 

oao feqil PG bifdriaii^ m Q fatij e^ic UgdMi (^iidtlK 

dticd^ ^ donitc A^i^t\iktat 0H fqualis 0)& / d^^dt 

eiiiin OB^ Qr pefpeu^i^lilaribus ad m dfir triatigulutil 

HO* ifofcclium cfit HB *qti^^ B/?i fi ob PQyOQ 

le^tiates i etii; PBr ftqu^^ ^f ^ ac^o^ii^ 8t PH ^^^fvni^ 

Bs i^< Sti determlitdta^ ^cfU^mitfs^ folufibn^rh dHttA^ 

fioruts locorum geometricorum circtilis , & HypiefV 

hddx eafttfiirftii^ ut» fe domifrt^i eo^ufn istc^s iuterfi^^ 

vf^ CittoR ^aricer a^ Vljpti^bfoie iiitidffkfld. e^Kib^ 
^aM a(fmo<*ham ci^peditani metE^dd^ ttifectioAk' Hi' 
^/quod l%^rnii p^ter dia st Gcfofmctfis ftt (M- 
i^m Geometi^iattt riequicqttitn ijfu^fih^m , qtiam liiiMi- 
tm cKaftfeendft pfotfos x ac el ipfa eonftrucdofitt pat<*- 
Kt^led emtiiiAofHMi ^fbflei tt^pte#circt4cun> & ctctatli 
ImtaHi iplvamr ui^mi . Satis: airtetti cdf^ftkt v ^ftyglK 
lom qBefn?is feeati iti t>afte$ «qu^ ttcu d (ttciai i<l 
ifts pitfftiis afquales ^cus cifciili babentis <feDtrani Hk 
a^oU vcrtic^ , S^ intcrceptus iiit^ auguli ipfius cfiKT^ 
lefrlkter». 

37J, Sui^tnr nreur chtHH FBVrt /f/, 8^ ficsf^ «t^j 
|«if^^ 4^»*/fi #r« , Ckard^ mF /w-rfiitr iif^riam m E . 
'^^ftir >fr E rectd Afi ii>^ ferpetidfcrti^is ^ qnt tra^ 
i^ per cerm^um C , Pe€a F , directtic^ AB , rMtione^ J^ 
t&itnfiante z nd \ fit liyp^olay qua arcui cirtMi ^- 
cmat in Pj eritqne fP pars tertia arcus FEtm itd , i»f 
<<w* PO paraUeU Fm j^ 4i«< fj^ directriH eccurfat in 
P arcus in Hni^ pitncti^ P, Q Tf^/i^i JH in tres pams 
^kales. 

-tje. Dc^naoiiftraUo tft^ $^od|nn faciKis , Quom^ 

dior» 



Mbiorda PO eft di^etfo AB .perpeJddicularis ^ ih ^ 
fecatur bifarlam,, £ft autem tp M PU ia racionc det' 
tern^nante 2t ad i«.Q^are FP eft dupla PD ,^ acieoqad 
«qualis PO, & promde arcus FP $ PO iquialcs ; €^ 
chordas autern F»*, PO parall^l^s > etiam ^ FP ctt d* 
qualis mO. C^aarc trcS partcs FP^ PO , Om Aant 'mi 
itt fc xqmks , ut bppprtebat • Q^oniani iatittra et 
& {^nf ad m^ i tit 2 ad i f pai^t m fore altcrum ^ 
xis tranfverfi vcrticein . Qpod fi ^ter vertex fit Mi 
crit FM dupla M£ ,. ^ aflumptia m y verfiis^M 
quali FM, crit &; ^V ^dupla VB , adeoque VE.i i/li 
«qualcs, & ,FM aatqualis MY> fiyc JfM, MV * V» ar* 
qitales: nimirum divifa F;^ in M> & V in partes tfta 
trunti M» m vercicsesi aiis oraafverfi , V centhm Hp 
^rbofa:.^ ^. .... ,^.- . ^ ... ... (,. , , 

277. Porro idem ramus Ffypcrbpl^ fecabic . circulataf 
etiam alicubi in f , ac ramus oppofitus alicubi( in P' i 
ic erit F/ dupla fd^ asqualis fo, ,ac tres chor^a^j^ &.ar^ 
cus Vf^ poi om aeqijales/ ac p3rim',FI^ duf^a F^^D ^ 
qaalis Pp\ qu; etiam/ob P^O'9 mi pardlc^s^ ^ritaB^ 
qualis OV» • Q^are tres .^bordas , i^ arcus FP ; P(7 ir 
P'm acqucs. Nimiram fi^t FP ^rit pars icritia atco^ 
FBw , iu FBP , crit pays tertia arcus FBPLFB^-AFB» iflvc 
ipfius Vm intcgro .firculq aucti ; & TEmAf etii pars 
lertia; areus FBi»AFBmAFBi9i!^ five arcus Pm aucti biniS 
circulis,\.& t contrarro arcus Vf ei;ir tertia pars arcuf 
^Amy FAP* cric tertia pars FA^BFAm cjufdpn FAwr 
drculo aucti FAmBP pars tertia FAmBFAj9»BFAi» ejitf* 
<iem aucti binis circulis > & cum FP fit tcrtia pors 
arcus^ FB;» , & tf tcrtia arcus tAm , crit Vtg xs> 
na tocius cjirculi : cora^ue FP fit kxxtQL VBm > 8^ 
fBP tertia FBivAFBim ^ fiVc ip^tis FB»^ circoto auoi ^ 
crit PP' pars itideni tcrtia circuli totius, & pance2 /t 
]f^, P* totum circulum divident id partes aequales arcsf 
' 278. Id autem femper contingec in qtiavis folutiooc 
geometrica» qua quaeratur pars terda arcus cujufpiam i^ 
'S^mper omniho invcdiri debebunt puncta tria^ , qu; 
locum circulum diviiiaDC ia partes {qualec tres ^ ncc 

cfffum 



E L E M E N T A. $9 

i»m punctorum inveniri umquam potcfieit unum , $i 
fie idiqurs binis • Ratio ejus cft ipfa drculi natUr 
nbjb ipAim r(^deunn$ in' infinimm ^ infihito quOf 
im qD2Funi4am yeluti fpirarum numerp » quarum 
tda prima > Qulia ultima • Semper autem ipfe cirau 
hAtz fibi fimilis crit , ut quafGumqiic proprietatesr 
Mflcrii quivis cjus arcus binis punctis intcfccpms gc^ 
Waksj ic pendcntcs unicc ab co , quod finjgula cjus 
i Toactt 9que dift^nt a ^ntro podem , cafSem liabe-i» 
\ ^ debeat tam arcus , qui ab altcro px iis punctis 
ifl^iis definat ia altcrpm in eadem fpira , quans 
^iidcfinat poft unam intcgram converfionem pera-^ 
tttffl) tam qiii poft duas , jam qui poft carum nump* 
to qDCfflcumqucj idquc t^m |ft:ogrcdicndo ab co puft- 
ctQ wr&s unam plagam , quam tendendo verfus oppa«f 
fiom f Qtiar^ ubi qu?ripit paBs -tettia arcus inci* 
pciitfj fh f y 8c dcfinentis in m , ifieri omnino noa 
V^ > m aliqua geomctrica conftrudionc dctcrminc* 
w pars icma arcus VBm , non vcro limul & arcuJ 
fftWffB»i Sc ita porro quocumquc numcro, intcgr*i 
'^ CQiivcrfioi>um affumpto . Quin imo c:^dcra fltnul 
**>Aaettoac invcnicnda crit pars tcrtia omnium o^ 
paioo arcuum , qui pergendo ab F verfus A deftnutir 
^ h five in cadcm aftiimatur fpira punctum m , fi« 
^ k quavis quotcumque infpgris conv^fionibus di<^ 
Ibca. * ' \ 

?79; Quiamobrcm l^cct tp problcmatc yio^atur tt^ 

?vi ittrica pars teriia unid arcm, RVcra reqtiimatut 

^"^^^^Jo? innumerorum arcuum ^ quod prima fronr 

j^ videretuv. focm iR^>ofli^le non fplum pcr icirculam « 

^ftaam.liB^am , fcd pcr curvas 'm immcnfum ma« 

Acpmpofitas « Scd iUud pcrquam commodc acci^ir» 

^onmium illorum numcro infinicofum arcuum tri^ 

jS^oo^ habcannir in illis ipfis tribus punctiiS P 

^J» t ^ a fe iavioem diftantibus per tertiam ciroi-» 

1l£['*^°^ • $i cniin FP tit tcrtia pars arqis FB» 5 

yctido huie intcgrum circulum , addcnda crir patt 

^ N« prioti pars circuU tcrtifli POP' , & habebituc 

. pro 



p<^ SfECf IQNUM CQNI^^RifKl. 

ffO part^ )te?fia tptius f B»?4Fa^ii ij^«fs Fl>P' : ^ add[H 
19;t arcui txlCcCmdo alio iht^grp cii[culd : adciend^ er 
'^ti terti^i iteruin pm jtertia circ^li J^f ; 8i; fffi pat 
fti^ arcus trifec^hdi $rit FPP^: addfiicio veta itenjij 
*mn cir<;uium 3 addchda erit p^rti ifirfix iffiriim jj^ 
tia circuli btius jfP '> en^iic pfirs t^tia a^us trj| 
iCfhcU FBflj^^FP» & ita porro, ho^ aidvchi^pti&iis *' 
^s iirc^i trii(Se(;iMido ; novi faipiper a^c^dcht pirt; ,^, 
qLai triehtcs qlrculii S^ tl^ifeilrtLoauh^.pMhi^ta m^P^i ^ 
fcupfcht ppr Pi'P'i t id infinitl;m. 6^!^^^^^ fHkit^ fe-' 
f/ifl^ F^ Mftc ififi^ia arc^s fMi aq noyii W*«ri? « 
j^i;^ axculis hpiici^^oncjs (UXciirrcht pcr #, F, P. iri.ij_ 
^n^ • O^imobreih tti^ rcquiruntur ad ^qq ^Pro^i^ 
Il4 ciir(;uii piihctai» S( cum recta^ vcl circu)4|jii|;{^fi^ 
a6(uu4 ih duobiji; pghqp^ fi^caire poitit » i4 J^o^lcpai') 
£^vei:c, bmhmo ti^ii potcrqht: j^o^crit Hyp^tbpl^y q^ 
^piKft ih tril^u^ puhctis Qrculo.bcciitlrft^e; in>iB6 pQyQQpc* 
CIM^ & qusiti^^r pun^ta requir^rchtur » w^ i^ applidu 
ll^hf A|se()irap a4 C^eoru^iam pl^cnd^mus bicMi^, 4^^ 
vis SdrCiQt)^^ Cohicas |>robl0matL. folvendQ &^^e ^^ 
1(1 qiij^ajyi^ cum dLi:;:ulo ;. Scd hl{<» omiqi^ iMwdl^fc ' 

4^ cft j^ a4 ii^m t^9^^ iHMt^km P^wkfliwl 



ui,* 



SCHOllUM vlii. 

< Smerali cooilrurtQM PiPopolmQB^ w^ 

tkovos^ fatis ttbcCQs capiamu^ ftuCCQs #; 

guncaim uuua Li. quod ibi afliime&ran^us ubicttQiqite^ 

!^.90|aarumamuS jam ia,fig;^a» 9I i ^i id ^pfa rectau diix£^ 

9t Jiiii oqus doiiCEirfus quaef tnir cum. Cahica Scddckic « 

^a Patet puactam. O £g. 41 debcre hic abifc id H i cufli{ 

ibi reaa IQ c^iCta fis parallcla ipli KJtf i adebquc ,|| 

ittius rcetam OZ» qvix ibidemi etar pptaUeia rectsi^Hfl 

djbifc in ipfiuti UF bujus • Qi^arc ma coniilruccid evM 

ifei multo fimplidor. Sumpto radio L^li qttt ad perpcHN 

dicidHm dcouifum cz L ia dicccti^iccm lit in raaono 

deter^ 



t 

<«?»»»» ?WM| PHWta P» p coibunt; & r^ 





^ fd «ian^ ,gK9f%p1^, dehid4ftr#i ^jflr^ 

WS W ^«nri^ ipfiii^, ap ppqd^eti la^tjwicjp^ ^ 9« 
jWty.q^pfit^s «4> % PCXsattftiir pH i pct W i P 1 
|H*W8wr 5X f ??ft?? iWeftiite «M partep H. P» Q« 
ik-wl ftciittsl ie^wriHn FPi FQjijtwi^e^ ii» d|wtri«£ 
«« % Jl4i .'W «»14« f P »fl i i «« ift 4si ay , vd 
a«m fQ^in *, li: in ^g. s4. Mw^to^ia f|f ^iipvii 
ponao L i aganif. «cta 41^. py^^li ^Ci. oficwioiiul 
gfotiri iii $i tsccti^ FHi iQLiJB A » 4>9c im p*. 
**l» ia fig. 9$. fit % oc^iitrens r««i« fii, fP «idi 

^ id VQJif& «xiMlew i »iia«8Ja, in, fiWioiK: df* 
gwwanie , ae in e«{em wtHW« fe«« .'Qs y i;^ in^ 

4i. lade Vero patet , fobun Mil % 9,4^ pui»ctiita |l 

'^iRmtB ad Sfi&^as^ Qximm ^ifkm if 9A,ph 

"^ detccnuiiaslie > (:«m ^iip^itn} io oqJIp pjsn. 

^^i {lo/fit fi ad I.S iA.c^ r^tione, ni/l id vel 

lat.cura P cqQgniet)tit>u3 A, $ cum H, vel abeafc 

'^coB^i^iibus <<, S-CHsi h . £rit igitujt pig^coibi 

Pap|/^t9 , Sc «^junqw;. Hypeftobs 

catnunij 



/ 



ramum , fi aflTumatur in fig. 9+ V y J iibictfiiique 4leit 
P, 8c m fi& 9^ inter P, jp, eric aut^ift intt^ i!lai>- ve| 
kitr^ altemm hufus ramum j fi dirumatur Iptnft' F^nitep 
ipfum & F, vcl in 6g; 516 ultra^. »• •• 
' 283. Porro cum radius circuli aflbrai debeat ad I^ 
i^ racionc dctcrmiaante, inqua fempcrcft LPad HLA , 
vel L4 patct, iinrtim fbre niajot^^mi, sequalem > tcI min 
norem rerpcctum LF, prout LS fuerit maror^ aequalis, 
vcl minor rcfpcdtu LA, vel L4 . *Patct autenv tffunv* 
pto Li ubicuraqufc imcr F, & Pi fore LtSr mftforcm^ 
^uam tiAH ^umptoM^ in P, forc LS'*aBqq4lchi LAj 
& codcm aflijmpto in fig. $6 in f forc LS aqualcHu 
Li ) aflumpto aucem Ls ubicumqtif: dtra P in %. ^q, ^ 
fc inter P ae I in rcHquis , fore LxSs minorcm quath 
i«2A2, fi L aflbracrctur fti-ipfa ditcAricc inl cvancn 
fcentcLS, evahcfcit &' circulus:; a^ in pun^m abit ^^. 

* " at aflumpto L3 ubicumquc ultraTtn fig» 95, & intci? 
I, ac p ia fig. 96, fore L^Sj minorem, quam L^^:; ^ 
ac domum aflumpto L4 ubicumquc ultra f in %. 9^ ^^ 
Ibr^ itcrum L4S4 majorcm <^uam L444. Quarc radius 
circuli erit major, aequalis, vel minor, quam diftantia 
LF a foco , prout punAum L aflttmptum fucrit intra' 
.EUipfim, Parabolam, utrundibec Hypiecbohe rs^num, veil 
in pcrimctroj, ycl ^xtra: Q^.^E . D . 

284. Indc autem facilc ^ruimr primo illud • «S^ 4^ 
ftmatur pHniStkm imra Elliyk^^ Pdrai^lant^^ vel utrum^ 
liket ramuM HyferhoUy nullam nSiam inde f^jfe duci ^ 
qua SectUnem Conieam eontingat , & qu^nrtds reclan^ 
fer ipfum ductam dehere iffdm ficaro bis , frater rectsjf 
farallelas axi in fa»»ahola > vel utrilibet afymftoto im 
Hyferhola , quarum altera intfrfeSia ita in infinitum re^ 
cedit^ ut nufquamjam fie\ 

ff9l 285. Nam in hoccafu pundum F, ut'~ih flg. ^i . 
cadet intra circulum , ncc uUa cx eo dud poterit rp^ 
Gtz FH, quas circulum tangat, quaevis ex iis, quas per 
spfum ducatur, eirciilo oceurrct bis punSiis T, # adeo- 
que fe HL Seftiptii C/6tkicx occurret in binis pundris 
fit9 nifi ^(c altcrum tx iis ita in infiaitum recedac,, 




i^ffiff^ yifn $c> quQ4 in Us; ca(it>u$ poITe Qpri psi^ 
^ num. I4p. 
.^^. {^^^ y? ]innElumAffHfimur in ferim^^ Sellionis, 
)mc^x ^^^ ^ reptis otnniJm fer ^fim ifranfiHfttitftf'^ 
^tpVft H44f^ Vf^ SeEiionem ConicdnH hHi^ omnes 
mecnmm ite^nm, fr^ter reCias farail^las axi ParA^ 
#, v^ Hyf^boU afympqtis. 

. a?7, ^am in co (?afii focijp, F iaccblt, ut. in fig. 5» 3^,91 
^cff^ peripheria > adeoquQ ^nica c rectis pcr ipfutn 
qpdbus, ut FI^^ ipfum circulivp continget x relir 
fecaatik^s iterum: imd^ CQprequitur umcam l^jH^ 
ifcds tran^untibus per P d^bere Sec^poem Conicam 
€QQ%rc ibidcm in P I rttiqvps ^x^t^ expontos c^fyis 
ilpincatibm ipfi iterum» . 

^88. ^ v^o fun^Hm affifimalur e^a Ellip/sm j Pa^ 
T4^lm ^ ^i utrHfn^at Hy^hjhola ramtfm y Mna e re* 
^ Ht ^um traftfeHn^ihts SelHonem C^nicam contiur 
Wti p^uarum omnium ody qua jacehunt in iis tangen^ 
^mgulisy in quibus focus jacet^ occurrent bisy altor^ 
.^Mi jKcurfn in rfElis axi Parabikt « vel utrUibet afynu 
\fff$ Hjierbol^^ farall^lis^ ^^unte in infinitum itA , uf 
N^fMfff jam fit X utroqu^ a^tem occurfu^ itt tiyi^rboUf 
"^pdnptte ad iundem r^mup , vel ad oj^ofitos , /wpk^ rf- 
^ificlinabitur ad dir^Ehicem in angulo minone quanj^ 
^ppoti^ ve^ majore j bini vero cofftaEhs jacebunt in 
'Ifim HyjieicbsfU ran\o » vel in offofiiis , pr^t^ punElumi^ • 
. afev jafuerit in iis afymptpt^um an^ulU , in quilmf 
H W^^t y^ z/^ ^xtra \ ^ in friore, cafu Ifrminabunr 
l^ U eum V/Omum > qni J^t in^ eod^m afymftptorun^ 
^ flpfj* cum funSo ajf^mfto , Sed c^d^nt^ functo in alr 
r.lfm afymftotum ^ alier contaElus in infiniium rece" 
^\ eo c^dentf pf c^nppum^ r^Cfdft ut^qu^x fff^ ufquan^ 
l»fr£t. 

f ,^8?. N^n i^ ^o cafq focu? F jacebit> ut iiv^g. 97^ F-^7 
|pti.99 cxpra clrculupi , adeoque bins ad ipfpm cx F y.8 
hipgeQte^ d\2(:I poficrunt FQHi , F^Hs qu; binas il > 99 
Ik Sectiotiis Cbniqe tangentes det^rniinabunt • £x rc- 
l^ vero omnibijs traQfeutvtjt^iis ^r F« c?^ omnesi^ quS 
Bofooxdcb. TmMh H tf-^n' 



j 
/ 



\ 




^4 StCTlOliXjHCoUtCA^VU 

l. dratifibonc pct quodvi^ dlrectficis pun&uni H^ jat 
intcr^^und^ Hi^ H^ » circuiiim icSabuQlt bis » 4 
Vis tr^fifictiS hind in^c t>c^ ^uncti M4. i H5 j ilu. 
qdatM eircuid oceurrct t Quard idcm J£ddc( &: iem. 
^anfeuntibii^ pcr L rci^dta Scctioni^ Cbhic^ ^^ ^ 
tct pm&um H; fore iti li^ tc£ticum Ltj U pt4^dtiC0 
rum» qua opu^ cft^ ipguli^^ in quibusi jl(%t tbaUs f§ 
iitm % ^T^r^S^ in angiild HiLHli qiii i]t ill^ cft 
ip(i ILi ad vci^ticeni dppodms i ixi hac cift ^t^rd ILi i ^ 
in fig. 99 iti aii^d ILHl , qucm <:k:^mitlcf tan$ed| 
IL5 cam tangcnte iL t^oducta; Qubd adtettl ictiifecM 
puacttim interrcctloni^ P^ vcl f xcitidt^i iti [tlShicum | 
jam tbtics yidimus ct num 149 ^ Pdiict^ vcro fonta^ 
cmum t> / jaccbunt iaramb citeriori vcl tilceH6rI> vj9 
ita in iiifinitum tcdcderit 3 iit nuTqaam \im diit 1 prbat 
pUnCta Q^s.4 ja<^aeririt rcfpccJtu dirccoricis in^ardd cir-- 
culi fecti a ditcctrice ipfa in N» & /? cbdtni cum cctU: 
tto L i Vel in oppofito $ ver incideriiie iiX iUa ipfa puil- 
cta N> ^- 

* i9o. Concipianit' dutem idctitruni circtati ti pd^iM 

tio-a dirccnriccm ^ vcl Li idtra defcrri tx patke A dt 

recnricis vcrfus B ita , ur iiiterfcctio N Ipfius circo& 

fj^tim directrice primo quicfcm iii ^g. nbd diftcc a ptii^ 

j^^ Ctd arisf £ mslgis i qaant intcrfectid H afyn^tbci CNf 

102 P^^^c^? ^P^ ^ 'f ^utri in fi^, loi abcac L iii ^/aiil 

«f)rmptotum CH , adcoqric N in H ^ ac demuttf iii 

figi lol trarifcarrat ulnra aci paftes B i. at: ^ttds qui-^ 

dem NO/^ Jaceat ad eaifdeih diriectricis^ paftcm cuitf 

ctnnrd J-i arcus tlon ad dppofilttoj, & recti VN«r ptt^ 

pctuo ta^gat ipfum circulum in N* Qudniam ^!al r^ 

cmm anl^ulum continct cam NL ^ & fH cam' H<5 

fnum* 164^ , patetj ipfum Vat forc paraMelam ipli FMi 

ac focum F rclinqucrie iri fig. lod ad. partes B ^ tt 

ipfum iricidett in figr loi i cum rcliriqa^re ^d ps^w 

tcs A in fig. io2« Qjjiare cti^m cangens fi f^cebit itt 

primo cafu in arcu N»^> abibir in fecuddo iri N y ja^ 

ccbic m cercio tri. NOir , & contadtUs Hyperbahe 1:^ 

poiutens ipfi q in primo c:^u jacebit in ramo ulitri<^^ 



in 




I L E M C N T^ A. #f 

ftewt^. 'ObUMt in . iMaitutii . t^ i iit naftfuaiti - » 



ift tari^o |abebit ts iftfoo tkeriofe i Gumq[&e ikkB 
;at packer cvtkirifc «ohntiSHii iC^; 6bi cbhttiim i^if^ 
4ev^itt>k.pkce P, veriiis A ^ )>^ieii j^dty<fliji 
_|totii HC, iS^CJn^P» i<j a6ficC t^itnctum L kiil 
imgii^ HG^; vei lilb > ^iiies contadtis iiii|rmifii^l 
,wmol raitioi opfi^ iUo.exiftbnte in arigulo HO^ 
tnfQHiiqiie coiitlNfljpi dcbere )«cere in ir$mo ciidribn i 
I 4Ib}a^^ie ih SliCD^ iitCiiBn|[iie jaeere ia .tamd uite» 
' ^ *i ilio vdro atfi^uatd in. i^ccrarti afyniptotiiln i 
<oikti€t!uSn <itpei:e abire in ixlfinUum i eIic* 



hi leibaiiler^ id eb rsunb i dd qtiem id afymptiN^ 
4 piiad^ ac^btlit i ac iUo. dcmiim iaibeutitc in, iceii- 
^\ mvia\vk cdrrtaiftuni ita irahibv^ifi i ut blsfct^^oi 

ittftttr 

, A9i*&t faifce autem omnibus ploirima Ipoate coflfr-^ 
"^^iii^» ^brtim pauca utiliora attingemu$ . £x nu&i> 
^ mhMi- JEUigm fara^^ HMin fj^irb^ h- 
^fMRftf itifHAMA, i^bvirieri iHaquaf&trjfH^ cuicfif^iA ^Ht^ 
4f itfr^ ^J^ Jtti i ipnmxiiMm uUqjii» faitetk ^tni- 
jf^ ^fiiu txira : liim H aliqusi^x patte piidi&b ifi^ 
W ftd^ 6»ivuftlaceth cSb^^tcrct arcos aliquiS i |)b0ei 
|%&d(i Veir^ii^ cum deveiiiri iA iocum i ex ^iib id . il« 
w j^geiis iduci pdffec; Uon jibteft ducibm piiilillis in» 
^m bWeit^ ^aViciH^ii nifi bbveriac convciKtiaieki 

t^i. E< iiuni a86 paist la quovis fu^» feriihkti 
^Xmi CnHica fUnnifi unkam tangeniem h/Aeri fuffe ^ 
f^ «atein 4effaonftrari pbfTct ^ ibi drckm cuHfa fOf/in^ 
¥f iircM ^oHt^SHm jacMi pn^r ad imdetH Unjtpntis 
MMi quod iattiett 8C cx riimir H9 i ^ ck liiittu 
4» 4^lc Mii i &mi nitairum ttStz ibi .mdttt f>au 
^1(5 delat4 > (lic dretunvbltitd drca pulidluhi fittim 
^tti ilei^iuiiem Coni^ath p rilhtttn iocipiat #atn c6ib« 
^if^ti x^ ici bitiis iiinc^itide i contslftu punAts &• 
^ •, Iiide auctoi GOdfequiCur ^eBiQnem ienicJtm ttei^ 
m ficAm mmm % fid tkrieim ite sadem ^agam i$t. 




96 SE€Tr01^UM CONICARUM. ^ 

S9^ ppe ipfius nuW 286 facile dcmonfhratur & 

hsd , licet rsila C^fficam ScSHotiem cemingat i» m 

funBoy nHllam aliam tectam Auci fofe in angida^ 

ta birn^ linea in ipfo contdctu confiitmnt. Nam in 

9? » in qua KPjHj cft tangens, fit qu«vis FHii^l 

PrPjFH^ utcumque parum indlnata ad tan.^cntcm cirei^ 

U FHj , fe-ei circuliim fccaJbit itcrum aliciiM in Tri 

^^ vd T5 , & recHkHiPj , vfel H^Pg Scftioncm Cont 

"^ «am in aliqufo puncto Pi , vcl P5 • Suniatur jarn putt?: 

-ctuhi quodvis P2, vej P4 tpfi Pj propius , & punctfli^ 

^ ^ T2 , vel T4 jacentc in «rcu FTi , vel FT5 , ac rc^f 

\|:ta FH2 , vel FH4 fiibeunte angolum tangentis cirqi.. 

K H;F , productas , fi opus eft , yerfus cbordam, ctun 

ipfa choida , FTi , vel FT5 , fubibit H2P3 , vel H4P3 

angulum , ()uem continet tangens Sectionis Coni«ris 

KPjHj cma iUa FHi , vc! FH5 , & jacebit iii arcq 

.P5P1 , vd P^P5 , ddcoque ut arcus circuU altqais 

'Fr^T?: , ytl FT4T5 hinc indc % contacta iubit fcm- 

per ihter tangcntcm , &-r^ctam quamvis tangettti ui- 

cumque proximam 1 iu idemi in Scf tionibus Coniob 

-cvenit • ... 

^ 294. Patci iiide iuo pacto dato puftcso in SMiomt 

Conica }emihrp duci pofftt fangens , ducefido nimi- 

rum inde ^d tocum rectam P^F , tqm hii|ic pecpendi« 

r ^ cularem FH3 ufque ad directricem , ac )ungendQ puti-' 

ct^ Hj, P?, quQd/quidera jam cx Jnijra, ifj innotutt- 

rat . Ai bic prsterea est num. 288 eruitur methodMs 

^odmodHm oxpedita ducfndi tangentem ad Sgctioncm G^ 

ttic^ e pHncttf L uhivix dato fxtra iffimtL Gentro L 

f'97m fig. 97 > 98 » 99 5 intervaUo , quod /aa\)erpendicu* 

pS lum demiflrum ex L in directricem iit ii% ip^tione de^ 

^' tcrminante', defcribatur circulus, ad quem"Hiubaiitijr re- 

. ctx FQ^, F^ tangcntes , qiKc occurrant directrici alico* 

• bi in H15 & H2. Rectae tfuctas per ea puncta^ Sc ptf 

L contingent Sectionem Oonicam , &c ponaa conta» 

cmufiii I , i invenicntur du^tis FI ., Fi perpendiculari- 

-Jbus/ad EHi , FH2 , quas femper invenicntuc , 

fftfum I mq {, <:adaf in Hyperfaote.af/mptotos . Qu< 



. i i i ii,'l i4 Tf A. — Jf 

^'fortc alteri e tangentJbus circuH FQ^i Tq tvadcki 

^ ^V^ i6i, tit nufqu&m i^iii fiti ipfa qtiocme LI * vel Id 
^adet directHd ^^tMtli, & cdntadtUs I « vel i abibit i« . 
'ierticenn axi^ tEanfverfl : Si vef 5 dtocttim cictur in di* 
Ttetfioe , dt H iil fj^: 53] 54., cittulus qukkm evan6-F*5| 
lcet, lcd ddaa a(d ipciim Hti Sc chorda Pj) pefr focutn 54 
^ pe^fldidflari > Habebiintul' lainx tangefitcf^ Hf; HP> 
-ffaoa ntrin/ iyfi r-,: < 

_ toj. Pirjctcfea fadle ilcdiicirar ^ illud , fiirrriW* , 
-.^tkji c^hcmrfu Hnkmm- Ungennim ^ fifcum dncu 
itnr^fkdire kifariain dngUum i ^utm iti cdntineni H- 
ki rddii foci dittti dd binSs teHtaettts y ijcl y pM M- 
iri conhttf4j jackn§ in hiiiis rHinis opfoftilr ; alief e^d 
-Us d^ altint prodmtLl Nain in fi^. .97 ii aAgulisF.^Y 
HiH , Hzfi t^ctJi / ilurifi ;^730 iufe^ntut angiili ^^ 
Ltflt i LFHi^iialei ob latera triAftgtfltirum FL(X» 9^' 
ti4 srqoali^ , r^Utll|Dei!tur anguli LFI » LFi azqua^ 
Its • Iti fig; 99 a tectis QFI i ^i^m^tis xquaU- 
*&f QFL > eFli teliuqudnaiit a^iiales LFI , LFi : at ifeiL 
'S* 99 producta IP iii Ol j a rectis QFO, ^Fi dettipAt 
WL y ^ ^ualibiis I [witer remanen& LFO ; LF/ 

^ j^^.^^oflet hic etiam fedfct^^diici ^inEtUifi quofh^ 
Ms rectam per cenfrum dudiain bis accrinere EUiffi hiAc ^ 
inde a ccntro » /iyfifhU aktem bis ^ vei nupniiianr » 
froHt jaceat in ilHs afyf^otorum dngtUis i quos a:tis f^ 
tat^ vel in reii^is ^ ^ixtndo primum ei num* 284» 
Com nimiram centfum intra EUipfifm ^acear, fecundum 

"fero ex lium* 288^^uorum tltriitTlque jam a npm.al2 

' *^'fcdu)Umui ii <j[uin iftiii affuni^to eentro EUipfeos Vcl Hy- 

pcrb^l^b ptb centtcr ci^U L ^ cujus radii^ effet ipfe 

^fcmiaxis nrahfverftis , lit iiiriuiiniis nura. 1^5 , dedaci 
jpoffent multa ex iis>- quoe in fig. 63 > 64 aetndaftravi« 

• 297. Scd aditioamTi elcgans eft raiio , qua hinc di- 
ifedfei "deifno&ftt^rione deducatur iH^^^i^iperbola: ad afym- 
f^G^Tcljltse propi/ietas > quam num. zit deduximus 




9« «iCTlONUMCQNieARUM 

?» «^tiira diain^t^cirum p^f t^d^ctiQOf n^ ad aJbft 

I©4 1*?^ iff L> K^ f^ Pl >. -f«*i^Pfw pli* • Si ^ni^ 
pioH o^or^ant 4irectrici in p\incM2( il). tp», & pEQ. 
«rq <^rc»li afl^s^tyr vm h\. quj^nj {/, pa^t (qi 
Wa) <kbere circvii|o$ tran$rc i^um pe? 1^^ hun? g;^i 
4f contingi ibi4cm ^|k fR> Fr asqpaftbijs iptc? fe ^ 
hk punct;i K% t im iK% intcr(t;5«Q a^^naKoti c^ 
rcctricc, quae in fig- io| c(^ in H , qij^ char<|fi^ § 
more ocQJfFat difcctrigi in H« atc rccta H? circi^is. 
T, f| TVirS crit FP paiall^a: tam LT,, quanTt 
^c F/ tam l4v W^ y«^- ap seciangwla Tf if-, 
«qnalia eruint qu^t%ti$ ^fqtialibu^ pR ^t Fr • Quan^i 
crit FP 94 Ft> fiy« pt:, ^d VU. m h a4 ?/% 4vc «ti 
^t ^4 fL» & co^ppnw<fe> ip $g. IQJ > di\?fctei|do ^H- 

i5|.. 104, crit w? M LP, « ipfi| y; stfifl,', ^ p6(^^4i)- 

da jSlt iiapp Conft^ii^iQ , i^ inult^ , 5f muJto.ysrvion 
fcwrfwt,^ af ipfa iftci^im itn fwup44 ^ w qwpqim^ 
jp mcas. n.Qvi ffjnptr jyc icpctern Yclwti trijnfio, r^ ^ n 
fai^^li§ ran^fi imm\ ^^%x %c^i ^ ftonclci <iu(|(|v^ 
Ycrfum prornmpantj. atque profiliant • Sequ^nti ^^op^ \ 
%»sm ^ismmp\ Bi^^cim^ S^ fa^cvmdiaimam Sedick 
9m Coflicafiwft firQpri^tJlBffl q: fiJ^. ^Sfmm^ , 
itedui^^mjis.. ^ i 

99 w^fmttMtc ffeeiism Seemm Cenm^ % C^ ^ne#< i 

iMf^ m(4^: iffivm ». S^^m^ pim WW i: 
H tmsmif , tH km oe^n* e«mm ^mmt ^-^ 



5 >L B H 1 N T 4- Si» 
:enic4 m^tki^ (^!^ f^S^^4 r«f« ««w nSiMZ^ 

■ ic |4^> jM TF, ua,H a4 TH, ^c ]Cp 514 ?F, mlH n 
[ •, ^t^r jtonjvwcu* raepiM^M^, ^i^t rf ctapguiajn 

'«M4r?jm iJK a4 rcctaiig\j^*Wi.THr» %c MHwi 

'^sm arf difiprintiatn ^^^dworunj IH » HM . Jm 
I f^Q {^ ad tM fivc; ad JLT cft (fadqm » ^ ra^qi 
,•'1^ W> fivc qi^api Jt^abct or^jna^^ ^ djrccmc^m 
i angglo AHL nd fo?i radianii fP, quap p^nd^t ^^ fo, 
ll f»^e 4^ym^vii^t^ fWCJt?^ JjjcrippiR Cqnjt^,^ 5c 
|iU$^<9l^ rifCt^ m, gi^Ri fii; FP ^jl PH f wjPf. ^r) 
Hl ra|pa[( ^ppfita ^x C^lionj; d^tcrmin»ns? • & r;|- 
p^onc Sai^ fjus indiin^tloois ad radiurat, pcpdfbit igi^ 
W aJt iii fqlis ^tt^m £%pq <ma4r^fl HL '^ <j!;^adrgmta 
l^ < *; gtti^Ein H|. 3^4. qpr^mquadratfprBm $iiff(?j:cft- 'i 
WW!» W«W §?. W^ rf ctWgui^ Vlf. «4 rc^qangn. 
If^QFff ^ 1^4 ii^ qw?vis ^k Htrcqdgtl ippdo o^cuir-t 
n^lll aUIf fip.^ts ?^ l! in^i)5ntcpftPffQ I.r C^do qi«^ 
— mms^) ^l^im PF4 ^4 resi^gulqiTi noy^Ptl 
it a fola rationf; dct^riJiiQitSts fpc^yn ^gcilonjf 
. ^ ,,^ ac inclipaypnc hyj^is nov^ LH t, lErgo. ^i^ 
fio unjus rcqaagyli^ PL| ^, ;qap4yjs^ agad penjjcbit; \ 
fola ration(; tlla dctcrmin^n^ ^ $<: inclinatiqn^ rccta^ 

Iftnx if^ijiin. \ Qjiafc (i; j^ iili^ pilpott|in^ pcr ^op4 
r^guc prihfcwfltj , ftii^tetiir incu^nquc , five uWqamqu^ 
,tef4^S?f '^ P^P.iPi(arn iran^^t fcQta? cum,iifd?n^^ 
|p|i^r iR(;Iin%j;i<^]M, t csi, ratiq rcg^afligii^ P^fffncorr 
M^ 4^ 1^^ ' cx ii$ recti^ ad icct^ngipiJfin p^rtincnf, 
jL ^^xa in. pmi^^^ys diycrfis pupc^i gQ^vioaibijs, «1%. 
1^1 jfQi^ftjins,^ ^, patfjC cojcunji()US pijnqti^ ^^f}^^\f.9f 

g^ Yicl 98. in I, adcoque in ipifo (jontacta fajtis LP^; g^ 
. fl^fj^j^u^ LI., rectang^Ium 'tPLp .deb^rc ^birc ia ^ 
^ad^jjh ^ftgfnnsLI, quo4 i?i rcctangulp (ijbaiitUJi 
|ot^tV iPatct ijgituf <|wdqui dcrat propofitum . 

^ H 4 SCHQ* 




• - -» 



tt» SE^CTIOMUitf tOl^K^AtVU 






SCHO^LI U M I. 

^oti. CI tatiodem ipram vcUihus ^iiareflram fioi 
v3 inciinationls , & ^dgAt^&cis AgtAsi facrilci 
tlnebimus • Si nimirum racio determindns dicatur( 
ad Q 9 fintis aatem iilciihationis dicitur idt priore 
in poheriore s» erit fatio reax FP ad PH in pri 
rado SP 4d O» &^ ihpoftcriorc ^P adQ, Qjjarc n 
primi reccanguli std feciindinn cii^ compofita tx rai 
nibus Oa ad QQj-SSPP , & (^-*/PP 4d Qf 
fivc Qf^-/iPP ad QP,-^SSf>P, qi»5 qmdem cft 
prcflio ejus ratiopb admddttm fihipleic/ 

302. Porro-<1jm Sedkiones Conicae poflihc dSiqfus 
d6 in tt&zs defihcrc ^ proprietas ratiotiisr cocilii 
reftai?||i|Iorum in re£tU daum difcftiohem fcabiikil 
6c fe imer/ecantibu^ communis cft etiam > tibi ti 
/. currant binis anguli: rediliriei lacerib^ , uc Pft 
'•*05in fig. 105 -^ Id6, vel Wnis rtftis paifaUelis » at in 
106 ^07. Id autem in iis cafibus n)ulto faciliuS' perfpicii 
^^7 Nam manebunt fempcr aftgnli triangtflorum PRP' /1 
adco(|uc & ratio rea:x RP ad RP' ,& R^ aui Rf' 
fcmper eadem z ac prolnde ratiGf ^itoqf^-tci 
> PR/ ad r^dattgulum PR/. ^ 5 

SCttOLlUM IL 

303, TXfiacAifti^atia ptopofitiohfs diai j^dtzt H 
JL^ conftfu^onc Pi!t>blcmatis tertii » non ka^' 
6et vima sbi pandtuhi de^ in directricc ipfa ^ 9^ 
caiu circidus cTane(bit 9 hec ubi reda fic dii^e&nct pi^ 
rallela » vel per focnm ttahfeat , tt ndta^Mmds in xj^ 
fa Problematis «onftruftione . Poftet qtiidem & iis c^ 
fibus aptari demonftrarlo fongiore ambi tn ; fed fatls crif 
nptarc illud : cum cz geherali cbnftrudione Tfieore^ 
m& iocum iiabeac in e^bos omnxbus » ih qujbtts^puft* 
ftum daclmi acccdlc ad dlfedricent qdahn!kmli6et » iS 
iL&x ad eas biiias poiiuones pariiei^ ad^du&r q|trah<*^ 

cura- 



/^ 



1 



t i i li-t « t t ■ ftt 

> oportet fane» ut Sc in ii$ cafibus fit ^eri L 
quos generaUs conftra^ definit > poftqaani ukrik 
hinqtie limite^ ad eps adceflTerit* 
kf Sit ed&iA fi^eret ex prdpomcte ipfa dedvtCe^» 
mi Tbedrema^ pro rcftii m_:^d afymptotflr 
Wp^ad^ iii ParsAoIa » Vel Hyper Jk>la » <;aa- 
lUuni dce^^ int6fedio ita in infinitum rece-^ 
m linfi(aam jahi fit i bonfidetando > qaid accidaf 
^ id eas ditectidnes actedentibos idtr^ qudfcum-^^ 
Mits. Scd libet per fiaitam Gebmtniam hofcd 
^ivolvtre ex ipfa Sdriftractioifi^' cum cx ptimo pd- 
tfiteQ pcndcat diametroram ortmiam nattfft in ra^ 
^iic af/mptoforum ifi Hypterbola. 

Cpt^IE I. - 
i^^ hSis Pir dkht^ fm^uf trdftjkni Jtt f^d^ 
4i tt PMrMdy & dltkri ajyfhpt0t§ in ft^^bo^ 
\k9imihm {nm. 1/^9. ) dlt^ra intirficHinif itdiii 
i ^N*» ficedente , id n^piiim jarh fity in unito pnnA 
^imat fgfimetfo i in cii fra eonflami^ratidnc re^ 
^Viifa Hnif dijhtntiis a HMi occwtfibns fiihfiitfA^ 
i^na^ulum fui diflantia ab Hnzco occt&fui (fi ri-t 
*fM^» conjianti in Parahola # vtl Uli iffi afjfnftd^ 
^ii^Paia H^irfola^ tx iffa Aa$6 JtnnSto in dngt$^ . 
I ^^^dftii , & rjiiia Bla rpnflOnf pcndeiit fraterea 4 \ 
^'^VM^e ftSa conflantis Ui Par^bla , & reSa du^ 
^ » incHnationi ad afynffhtnm in Hyftrhola • 
., }06. Nam fi iri flg.iof, 109, 1 10, qttarum ^imaSFJc* 
WDtt id Parabolamj, rcKqux 4d Hj^tboiatoi recta lo^ 
ttoccurrat bis in Vf pefimctro Sectibnis Conicac , Ilt 

^^€^0 f 1 fcmel in P, rcfeta iW^ focumtran- 
) etit tecta FP , «qaalis Vt\ cmn P»* fit c^ina* 
*fa angolo «quklitmis , & FP^ ftoallcLi Tl. Ccn'* 
fcr ihtetvallo Ff' ihirthiamr in rotta ^' pua<?niml, 
«l^oc iriangulum SfofccUs IF^* fimile ifofcelib FPIrS 
^ teBcmt t£i^um atnguluhi ad bifim conjmancm ia ^ 
^»*oique,& reliquof ^uales • Qpare eric le adf*F, 
^'? ad tY^ five ut FT^^ad^ PL $ adcoqut rectangu- \ 
« &b W^ Pt K^altf ^ rccwpgri^ ^*T' fivt 



« 



-1«; «^CTIOHUM Cb>f!CAai7M 
^cf^u|i cQfiftyiiv Hlm\ cttf .pQiiwr. WWMWr 
l^ TFf , «f ^uod ^ rc^eapgulqip FjL| ia 4m S?^ 
^one Coqi^ > In qfi^a |:iuio. diTtecmiti^as ed feq;^ 
Hpir ^^detn, faabet: T^onctfi j^d^t^tx^ z fpla incliaiu 
jj^Qrie rccu5 LH jiijcta Profu>(»tipnj$ d^monftra^Ion^f ; 
. J07: Porro iff pcp F 5*^4air rpc^ FQ in <ig. I9I 
4ireetrica parallcl^i ^ iq %• ipp , {lo (^c^f iBQdcn^ 
' R OG(Barfuiil difcGcnds qim afymptoto j^^rqJUil^i ip(| 
A P?^ ipfw frcabij: i^ Q ad smguloi; rf (?e>s , cMtR 
^!P (tc pcrpei|<jticularif dircctrici in fig. {08 <x byp9*« 
||)efi{ & F5s oicqirrait afjrrqpiQW CR^4 yigM^s, rci 
<kp5 (' nqm. ij^4 ) • Qa^e U(im f'l triangifli . ifp^cp^ 
lii /f I fccat bifari^iii in O • P WfiSP? *'P iq fig, 
X08 icmper conftans , iiiiiiirifin acqualis diftantia fod 

t .j 4i£¥cp:lj¥r tWC rP^ffl ^i ) t^ 4in^.a fes^ris fc^ 
«« PFiWKip?|is Pirafeplac 4 |4fQWc r'i ^mppr ajqualij 

Um mp mmp^f «c irfctafigjiii^jT» j^ ir^ 4? pL| 

«I rcptjngijlsjm fub p'L ^ qi»ayif. rcc{^ Qp^ift^ri b^^f 
b^it |:?tfqn^in conftantpis , quatp fe?i<cbit |^i)$ rp^ 

«ma Rr^npp»k »4 ill^ra rggiw ^ ^ ffoMc pco*h 
bV % m|gqii;45|iqe ipOys uc^. 

iqS, M m <«. ?P?i i|p ^ ^p ;|4 OR , q^? 5n 
4»«W <i«ft»»KM? peri|€ti4iculjri j^^ L al) ffyi«?iio«| 
C«5i m l?atio|if cqftHantl, i»iiBii3|m ob fimilitftdyinicai 
ffi§ngi4i r^f t^ng# ^Ot* ci|m rc(:tJ»ngj}Ip FE^ , cmu 
fluQ halift. ^WguOlitpfequalem» vcl cuqcJicrn^ »tR> %d^ 
«8? 9^^ mn&M Fcctangulo FR^C, ^n r^pqpe f R ad 
RCi Ck? ( P.^m. i^.^ 14^ ) ftmW conji}gf|ti 94 
fcgajg^^cm iriinfy^rfarp . Q^c, aiip q^^vis ?^^ iij 
MPYi? dijp ajt^gqjp iijclin^p cf l ^i afjrmptommC|i; 
«cfe^f habffe »4 r^cjapi w H>ft> l p^rp^ndicula^^iji 
«fyiDPtoto ipfi, fiifjj ad dift^ntiam ijlftm perpcndi^pl»? 
fWi «I» ^«arW OR^? raaoi»^m ijonft^intcm , .qm| 
pcij4e^it a> itidMftoc cjus r^cta?} lUa ipf;^ Pj*/S^??7 
pa qupoue eju^ ^HRI^ I^' tafepbimt $|4 qwvi? i^fi^? 
pipm c? L iji qHQvis i)ag^Io da|qt ra#c^m cojiftant 
Ifm p^|i4cni:em §[)'^jip infli^afjqtoe , cpmpoiGtaip ^5 
feipis cpi>ftapH{j«| |:Q y4 ^^ # PR| §«: QJ^ *f» ^W 

dem 



ttm at} mclimftiqDC fjvs \^m ffiWfv 

?io. Nim fi per j. d^rar qocris ^a recj» ii| ^ 
M attgi^, ^fc nifnirmn ocajErsic peiri^^tco in I > ^ 
i, anrecan^uluinl IPD, quam Lfd hjAebont (nurtt, 

^lf^nra (»« , LP ferpci, vV WH PO? ^V»P |«< W^ 
.jfwife iRdaBgntuF •» »d«!qu| ba^lwV l?»»oR«|* 

fwnili LPi>, i^^ !wi to iUwt fMsR» a^^a^ 

«liim «< Mltetvim, eontimmnt rea4»^m m^mSm* 

Ifkif GP, gg i«J.\ fp, w 2^f;.^^f 

• Jli. ^rit eniw ^i^ Qc8»4yiqi pPfpcdcftll l«l??^ 

f fcp, ^tijv, m m%sc CG »t7p . Cm g« :^- 



»84 iECtiONUM coi^rcAitiii 

4d C«, & rcaangiilum fu^ Q & Cir, ffvc fub* pi k 
>«, ^uak KCtangulo fflb-CG «c CV, fivc fiib PG & 
PV » ^uod ptoiiiac niadcbic tdiiftaritb rbignitudinii » 
utciinque mutato (>iincto P* 

313. Jara vcro proportionalium tcnrnnonira captcn- 
do (umhias, vA difFcrcntias , vclfubftiracifdo rcif^tas iis 
6iirallclas; Sc g^n^c^ i piicbit i fptc LP ad LG , dc 
I^ ad I^, a^coquc P> i Cg pafallclas; & CG *1 O», 
ixtCg ad CV,^dcoqttc Ggi V* pirallclai,' &f dHntim 
indc CG ad O, utLG ad /m, adcoquc triahgula GCE, 
iK7 fimilia , & coriim angiilos ad C arqaalcs rcdl^ 
d ^ & ^fbducatttr, qua? optts cfl, abeitnte m Li rsb^ 
fzwxt omnia^ y 

P4. ttCte qUidcm\part6 dclapfi Mdxii iH.pSici- 

JLjL tiam ifliri Hyp^rbbte cohftantcrt > quani 

*fdcmonftravimu$ riurir; lary, ciirri nimirum hia h^i^ 

ttrf ccnltans rcctangulum ctiahi fub' CG , flt GP . Indc 

«utcm fadili i^cgrclfudcraonftrar^fittir caotnn?a,' quarafd 

. afymptotos pcrtinchtia^ crhimus c proprictJtc diametro- 

i^ura chordas blfariam (fecantittm a mim. 221 ," qtue 

quidcm jdemonftrari potuiflcnt ctiani opc riufm. 2^7/, 

M quoniam ca jam dcmonftrata funt , hicprogrcdie* 

xnur ad CotoHs^ quacdam gcncfalia , qux ab ipfa 

l^ropoiCtibflt f vdt ab Mfcc ^rbUitfiisr fpente dD&ft- 

f uunrar •• ' ^ ■ \ 

i ijin Si lit ^dd/tm fHnctum trs0ifiMi Hni reSU 
fecantes SiElienis Conk^ ferimitrum 9 redanjfiSs fui 
tinis dijfantiis fhnSi iffius a hinis JingHlarum interfe-* 
tiionibus trunt inter fe i As quadrata tJtngenfiumiisfa^ 
raHelartfm y Jiqua funty a cencurfu ad cantactum, & ue^ 
] ^drata fmidiamitrorum faralleiarum . 

316/Primum patct ex i^fa cnunciatfotoc Propofitio^ 
nis i cujus eft cafus pafticuiaris • Nath fi cr uno pun^ 
&Q dkfcaatttt bins fedU» 9c ex aUo biite ii» paraJJb^. 



\/ E I. E M E N T: A; i<^ 

i.h^babcbimt ad dirc&iccm ea^dcm inclinauon^ni 
l£\ Quarc fi illo? fcccnt bis y h« laagant , iUf- 
rcdaffgula ad fc inviccm , crunt Mt Uarum qu^- 
. jecii^upi m pmm{)i2$C diamKti^is BQipfcos > & 
jttis ^rimariis HypcrMarum patct c^ co, quod 
I ad^ pcffimetrum Sedioms Conicaj tcrmin^il- 
^ fcccnn^ blfariani in.ccatro . Ddbcbunt cnim 
jgpb fpb binis (cmidtametris pc? tdem ccntrum 
l^in eadcm ratione > in qua fnm rcctangula 
iis paraDelarum aranfcimiium H^ illud alium 
punchun . Pro fpmndariis Hypcrbolc dia* 
i^ qiMc non tcrminantur ad pcrimcirttm Hyper- 
, rcjufdcm, fed ad pcrimcoiim conjugatae y fic d^ 
fflcmfeamr . Occurrai chorda I^ in fig. 1 1 } cidcm ra- 
ao JPybbli$& qtraque binis in fig* H4» prior autcmFAl| 
l^itroki^Qc alttri afymptop in G , & pcr G <}ucatur u^ 
^da I/ parallela P>'. £cit rc(%angi)Iiim VI4 ad r^ 
lum Pl^^ m rcctaogulpm PG^ ad rctlanguluni 
* '^ rec^ac\gula PGp, IG^ 

(cnv^iV^trqrBm fil» 

>i'. '^ . foroU. ff. * 

\IJ*M flHm charda , vel tangfnta jfor^t^^ ^ 
4 m^ chorda trAnfvctfim femm 3 ernnt qkadr^^ 
.kf^m ; & re£l4niHla fnb f^mentU chordor 
W^MreSaniulafui fpsmef(tis cht^da tranfverfa. 
;jl&>Si eni^n i» fig- HJ , ll6 cBord? V« occu^at^ 

»1 VL« in cadcm f atiQne , adcoque & illa inicr 

iUiJiaEj: iritcip fe, 

Qer4U 6. 

JiJfc Sihgula ejufmdi qn^drata^ v^ redangHa t^-^ 
mim , vel chordarttm farallelarmk aquantur Jift^ulis 
^leng^is f^k fezment0 ^rda ttanfverfa pit^cefta in^ 
^ 4term ei^s extremtfm , ac tangent^ » vel chor^ 

^taraUelam\ & fegmento tangmis,» vel chordapa^ 

'^ . " - re,lleU 




\ 



\ 



\H 



mHtid inH9t€pf0 intitr iffm thtntAmn trdHfvnfian , # 
iMdm MMn dmMt UuSmk f& MfihJm Wticm 

^ tmf^pti * 

3S6. Si ^nifti «X ^d^t ^dbb tticird^ iraAi 
ll iudihif R^ iU thordis , vd tMS^ntibtt pArftlldft 
^ilft fit 6d teR ui ie* i^^Uoiie dAtft ^ ih qU kft rdhfli^ 
fidllnl PIj^ ad VLh^ 8c pef if^ St S diicatitf i^eftaciid* 
f fcntiMii «idhiri^ftiis iti B ^ M tJhbhlii in D ^ i> iirlc 
MtMli|ttlbhl VLD ad recUAgttlUm VU^ \ii LD adLl»^ 
INi iit ]^JS 4i4 lUi iitita^ Utreacofiittiliub I^I# Ididett 
Vm VLi » O^are iili t«ctdngiib VLD i^ttt^bitdr ^ 
H^tk^m PLj» ; fi^ afteimiibtts L ii A v P $ / M 
tdntftetunl f f D ill Bi IhittaKtiif.rectangia^ VAB <^ 
ttatom Ali 

iiK jl jlM rW4(4 h^fmfd fnifiitaghtt iMj^i % 

Hkm ftSbe MrMitit fifttni ^ kitHmitk pNuf^dem C^^ 

fdhirit^ hmUt Ueum j dnmmUo rtBaniM fiime)^* 

idliim SttHfrM mmfiikrf^ fnJhfiiiiMttir qniirami ^dt^m 

Ut ihim^ifthr fkHiiauMi &,gd^MlMsi. 

fdlf p2. Si himiiriinpi ia fig. 117, taagens*^ V dttAi 

dourrac ^hbirdis Pjp» py paraQelis^ 8c tangetiti lA in 

l(i V i A i eriint )reftiMgtda PI^^ P^L^' i 3c ()uMra^ 

im Ai id (e iftVi^^ i iit ifiidtzti VL^ VL*^ VA » 

f tiuri)^ 317; ) St t)i pltii^td qisb^i^ k WM^tii j^ 

lliictit RS iUis paraUeia i ^a^ di ad Vl( itl rattoMi' 

ikia teemtl^i iPLi^ iid qU&dratum LV ^ fi ^diicatur VS 

dlut tecii^ paralieiis ^^iirrcnsi in D» D^j JS, «runi r& 

lArangt^a PL|>, Pi^\ <st duadtrafuih AI ^uldia 4r«K^ta& 

|uUs WWiVVO^i VAB. , 

2 1^ toHtmrtntilnu iH £ l^roimf^ tmgtns dttcid fer V inA 4 
ii ijns fizhfiHtd ^AY > aV trtM in rtaiem ctmigifits 
Ali ai, & Ei i EI. . 

Jl4^ Si eijtfii ex A ducatul: i^dca paraUeia ^afigen^ 
£s occurrcii$ pelrlltietro in P,' j^ etit qiiadratctfti VA ad 
«^uadratum V#i^ % ut rcciaDgubifii |^i)|f ad ^iiadrailim^ 

- fivc 



E l fi W £ U f Ai Uj 

littir^ { h± ^iy ) flt fecntfa^dm P^/ iitl ^ 

AI i ut ^liadratiKh £i «d ^it^ahki) Hi 

„Jratuid AV adqiiStUittnb yiVt iq r^ttaoe cdm- 

^^mtiAXL m iimiiim ^S &qittidrati £{ id 

^ feli iAeb&jk AV ^ V« tet^^Me cdt#9i 

^i$« ^i ¥)«tgr»Atl AVa^f. iibi ^ii mit tJ^^mkP-ii^ 
t4/iUelis AI J ai\ eceiarritt i.jtrit. VA *d V4 i lll Ui 
«( n } iito fiifttttifti ik tUi^fi fucHt pMtid 
jni^gti mtmiif HftHM ficid4m M mp t»*i 

3J6. Etit itM qtiiillritto VA ad «Juadr^tiirH At i 
. qnadrsitifan Va ad ^uadtatuiti /ff ^ Qjidr^ VA ^ 
il j iif.)r< ad 4^ i & alpeirnaiidd VA 44 Vm j iit AT 
' #f • Qiiod a EUipa ih fig. ii6, fufiirit A» pairalleii 

. u*^i At i *f ^iiafes i adebqtie ^quafcs 6r VAi 



fd: 



ScMdlltiM m 



pj, tJrtJc uiliue ^eduxinifas Cbroiiaria^ dt ipAf 
L^ JriL Propofitione • Hoc pofttcmum f^oute tiii 
'■^ aiiud 'thtorema udilfflmunl ^c itidcm ftecUn* 
aliorum duaifiphirium , quaf tt utdmo parfi 



-* ^^^ -|..^.., ^«... .^— ^w.w~«^ — pariier _^.. , 

^a liuc Ulque dediiCtis alia analdga i qua; % C(^^ 

io primn^ bujus propoiitidnis 6 ^ducuntu^ i pcr« 

. i qu« himlnim pei-tincnt ^d cafimi rdidohil 

it^s i iu quo ^dteira inccrfcctio in Parabd^ 

H^itdia lU in itifinituin reCcdit i ut nu£. 

- jnm idt > ac ibqutaii^ quidem duo: Cot^^Uari^ 

judetat qulflto & fei^to c Propofitionre dcducti^ $ 

\ qoarmm ttausfcrri fkoA polcft ad rcctas parallela^ 

ia imJbob^M dirc(teiciTiiU HypCitxida, qudc nullam 

caa* 



:iot SECTIONUM CONICARUM^ 

jAngcnrcm habenc libi parallelam, nec femldiamctrunii 
diametris nimirum omnibus iq Parabola in infinimtO 
producri^, ^ nuUa diametTQ exi(teat|f \a Hyperbo|apa» 
rallcl^ gfyrnptous. 

3l8. «fi //«r^/ charda % V^l t^tmes faralhta fe^ 
^ptur trAnfverfim 4» rt^a aj^ jfdraiUla i^ Parabola » 
vel alteri afymptoto in Hyffrtola^ m»t qHodrata^ taih 
temium » & r^an^ula fup ffgmontis chordarum ^ ut 
ftgmenta ejus faxaiich ^fiiffa 4^. ^jfins cpwwfH cmm 
' fcrimtro. 

1^9. Sx cnun in fi|[. 12^ » ;^3 » qu^r^m i% a4 

F^isip^abolam » |ixc ad Hype(bo|am prrtinet , rea; VI, 

)^3 parallelac ibi axi» hic alteri afymptoto » qus qutdea\ 

pccurret perim^qrQ iq unicQ puncio V ( num. 149 } , 

^ occurrant cangcntcs IA> ia inter {^^zxtil^lx^ i^ A« 4i 

& chorda Pjp» P>' id 1« > L' , oportebic ! num. 305 ) 

efle quadc^ta lA 9 ^» & reqt^qgula Vlf. V%Y adrc* 

ctans^ in Parabola q^idem fi^b (}uaris cecta coaftao* 

ti , in Hyperbola vero Aib reCca ducta a punctis A ^ 

4, L» L* in datQ qupvis angqlQ ad afymptotum il- 

lam ipfi VL parallelam , qux idcirco conftans paricer 

cric, ^. a^fciflSs VA» V4j VL, VL'iA ratioo^ conftapr 

|i^ Igitur erunt etiam LUa, quadraca^i vel rec^tangula in- 

ter fe, ut hsec rcdtangul^ inter fe, quacob re<^i ili 

iain conft^pteoi funt, uc ipfa VA, V^^VIi, VL'. 

CoroU. n. 

^l(^ $ingHla^)ufniodiquadrat^y vclreBaniula^uatP^ 

tur fingulit r^4ngulis. fub ejufmodi ahfcijfts reMa iliiuM 

. f^alltU axiy vcl afymftotoy &re^a ^uadam data. 

\ 331. Si enitn aflbixiatur quarta propprtionalis poft 

y qu^mvis VL , LP , Lp , redlangulum fub VL ,. & ipfj^ aequa* 

fcicur reft^ngulo PLf , r^d^ngula ijutem iub VA , V^ , VL*, 

8c ipfa ad reaangulum fub ipfa ^ & VL erunc , ut VA > V^ 

VL' ad VL>(ivcucqua4racaAl>-ii,&.rcdanguliun PLJp' 

^d rcd;angi4um PLjp , adcQquc quadraca AI > ^^ > & re<rtan* 

^ulum PL|^ pariter ^ualia rcdlangulis fub illa eadeai quar^ 

la proporcionali ^ 2c dbkU^s VA 1 V^ » VL^iiDguila fi^gulis • 

^ SCHQ. 



/ 



• ? L E M E N T A. XQ, 
SCHOL ! U M V. 

j?>? TT Ujus Corollarii i r. rclatto ad CoroHarium 
I 11 6 facjlius pcr/pidetur , fi affumpto paritcu 
[^ RCta VL quovts pnncib R , duoacur RS parallcla 
^{i^tibus, vcl chordis, ^'^qualis iUi c©nftaiiti quar-» 
mipropmfonali» tum pa:. S duca|ur ipfi VL parallclas 
hiQaoccurrsM: tangcntlbus in B> i, dhorctis in D, D'i 
htvmt cHim p^ilcr quadraca^AI^ /ti, &rcccangula PLp, 
PLy aBqualia rcctangulif VAB. V4^ , VLD , TLD* - 
«clg^a iij, vcl ii^ abit in ,122 vcl 125^ fi pun 
(to » in itfis ita in infinuum abcunte , uc riufquam ^ 
|ttnfit,Kct? VRj BS nulquain jam fibi occorraat i 
«faj^uc paralfcl? md^nt. ^ 
^ ' ' CoroHi 12. > pj^ 

I J|J.iri rw ^hord^ Vu> t/r/ tangefitem IB «» /^. 1^5 
j^m, I2J incHmmt fturts rolke LP^ L'P' ^£ ^ifr^/Zf^ ^ 
f» w » PimkoU , w/ 4//m afymft^to in JFiyfcrkola^ erunt 
f «ff«^f«ii VLu , yL'u fkb ptgmtntU chard^\ vtl^ qtu^ 
'^ir EL, 11} tifmgentis iH^ it ftgmjtma LP , L'P* ven 
*» lilm faralleU imerctfta imer shord^tm , vel tan^ 
*»»,■ kr ferimitrum , ^ib^. «r^ viRangHla fuit iifdom 
r&miiy p^ rectM in qtiofuis anguto dati ducta ex inn 
^^imne if&ts cum chn^da sivet t^emot^ 4uL ^ym^ 
; )f^ fmallehmi , . 

3Jf Paicnt cx ipfo GorolL i » vcl ctiam 11 • Suni 
^iibi qua<k«ta IL, IL' , vcl rccungula VL« 5 VLV 
IJftiiitoIa, ut ccaangttk fubL!P's LV, & rcctacon^» 
^mf) qti^^r^tionem ncm mutac; hicut rcctangulafub 
IP, Lt^»& rcctis cx L\. & L* ductk arf afymi- 
parallclam in quovis angulo dato., 

Coroll.t^. 
^JIJ. Segmenti Batabolici VMxk injlg. 126. stre^efiV^i^ 
^m trianguli VMnhaifefftisfro bafi chordamyai 
fftmcem in M in^ vmice diametri MKy cujus iffa 
etiinata j ut ^ iid ^ ; ad foratlelogrammum vero 
«u claufum tangente fer M dufta , &, froindc ifp 
'M^^vich.Tm.UL .\ Yy 



IW ^fiCTTlONUM CDisrlCARtrM 

Vu pof^alMa 3 ^i ad recPangHlim fut ipfd chorda Vi^ 
i^ fctpeHdicul^ in eam de^iffo ex eedet^ vtrtice ^ ut 
± ad %* 

3J6. &£ta emm bifari^m MV ia B^* agatur per It 
ted^a parallela diamctro MR> occurreD^ cbordc uV in 
L, petidietrd Parabola^ in. D • Patec forc I;.B adMR» 
ut VB ad VM» ut i ad 2, vel ut ^. ;|d 4 1» £rit au* 
tciii MR adLD (^n.33j)utrcc3tanguIuiolVRi^adre6tan- 
gulum VL« i (ive in xationc compoiita VI^ ad LV , &. 
Ku ad Lm nimirum 3 ad i^ &c z 2A % 1x\c\xi 4 ad i^ 
ac ptoitid^ BL ad LDi^ ut2 ad :;i & BL ad BD ut \ 
ad I ; Quare &: area trianguli bVL ^ dupla crit areaii 
BVD ob altitudiiiem cbmmbncm in V « fumpus BL > 
BD pro bafibus ; Area autem triabgoU VpH paritei: 
dupla cft arex trianguli VD3 ob bafiih VM duplam 
bafeos VB , Igitur area triangiili VDM erit ?qualis a- 
ttt BVL , qua^ cum fit ad aream uiangi^i fimilis 
MVR» ut quidratiim BV ad quadratum VM^ erit 9 ut 
t ad 4* Eodem vero a^timcotjQi acca trianguli iAdii 
erlt qutf ta pars area^ MR» ^ Quare toti^n triatigulum 
VMm ad bina triangula VDM> udlA fimui» yt 4 ad i^ 
£c>dein verd pa(5t6 £:d:is bifaridm chocdis VD^ DM > 
lAd i du habcrentur quattior tiriangula > ad qu£ priora 
illa dud iimul eflent > ut 4 ad i tum b€to alia > ad quse 
llla quatiioi^ eflent pariter^ ut 4 ad ii & ica pdrro, ^,i 
feric^ rcdarum fctnper magis in infiniium ac<:ederct ad 
pcrimetrum Parabols^ ^ & arca ad arcam fcgmeati pa- 
rabolici , qua Concluderciuc omnis illa progreflio in iti* 
finitum produ(3:a , cujus progreflionis ptimus. ceriiniqus 
cC^tt trianguium CMi^' , &c ^atiai primi ^cexsmtii ad fe- 
cundum^ ut 4 ad 1 . Qyx>niajni igitur m pi^ogrolfioni- 
bus geomctticc decrefccntibus eft (c. jiU. 10. Ari;h; ^dif- 
ferentiaprimi tcnnini a fibciindo ad priiiium,' ut primus 
ad totam progrefl^oms fummam i crit ut 3 di0erentia 
4 ab I ad 4, ita illuc( uriaiigulum ad aream k&oxvi 
parabolici. Pataliclogranimiim vcro. CE^^ cft duplutp( 
ejtis trianguli> & deqixale rectangdlo fub bafi Qu > & 
^auidiae MI i^ Igifur cric id parallclogrammum , & 

id 



„ . E L E M fi N T A, Hf 

Id ie^aagulmn ad iurciini ipfius fcciotiii « 4 *A 4 i 

iRat 3 ad i; 

\ 

SCH OLlti M VL 

\^. VTISI (upti dcmotiftrata fuifl€t ptdpriaasdifr^ 
1\ mietrpriim cbordas orkines bifariam fccan* 
iiiiiD admbdum facilc hic ex had ipfa ptdpdfidohc de- 
^ioci poflfet pro bmiiiBui diamctris fiUipreos ; ac Para** 
(olzi k f^o {ecuadariis Hypetbolaf : 

}3t. Si ehim CiAl binx , taiigehtes IB i ih pairallels 
wEllipii in fig. la^i vel in HyperiOli in fig; 128 , 
aciacas incidat iii Li i cKorda P/> pAraiiela li jan- 
riitj coQtaaus i debcbtnii rc^aiigula vtp^ Plp adqua** 
diaii liagijtfitiam Lt i /f taBere raHetletn candem $' 
cpm<)ue i{>ifx IL^ ^ ^qualts cflfc dcbedilt; cruht a^qiia-: 
|Iia «Dam ea re^^adgiila^ adcbque P/ ad PL , m >L ad 
r t five componcado iti EUipfii dividcndo iii Hypcrbo- 
^ U ad PL, uc ipta U ad >/ Adcdque Pli pl arquales ; 
Qprc fi f€(Jd bifariim il in G agatui? pef C refta CR 
j^t^geddbds paralldaj qus litmirum abfcindct ire-^ 
^iJLi R/aequaleisredis Clv Cr,adeoquc & intcr ft; 
" Vfi&chordam Vp feiabit bifairiini id eddem pundlro R ; 
;. J?^Et co quidcni pa6to habctctur proprictas clia- 
^ffonun omiiium iii Ellipfii fi nimirum concipiatur, 
^^tts paraUelas Bli ^* in fig.ii^convcrti circaoir- 
EUipfitn, converfa cum iis Ii> & pofitione chor- 
■iP/. Iii Hypcrbcjla vero habentur omries diame- 
S fccundatio? , quaj folaei tangciltes habent fibi parallc^ 
. &d prd primariis hoc padto progredi licerct. At- 
fiain fig. ii8 CR aequaU CRi & ducta PrR'/|^» 
"da rectc PLR/p > dcbcbuiit effc «qualcs ILi IL' , 
]uc aqualia rectanguli P'LJp'> PLp, qu2Cad ?qiia- 
quadrata IL\ IL candcm ratioiicm habent; Efferii 
sm 2X}iiales P'L* pf i Quamobrcm ob LV ^qualcm 
li W cffct major , Vcl mindr PL , ctiam L>* e£-. 
pariter rcfpcctu Lp i ad^que rectangiiiltim PL^* 
' crit eqaalo; rcctangulo PI-p; nifi PL ajquttuf Pl^ 

I 2 Ctt- 



ui SECTIONUM eONlOARUM 
Ducta igicur PP% quam QI fccct in r, caerit par^Uei 
la LL', & bifariam rectam in r, ut LL'inI,acCIcrit 
diaimcr pmncs cfaordasPP'parallclas tangfntiLLTccani 
bifariam , 6: 'cadcm cft dcn*?onhratio pro chordis pf* 
^419 340. ^t in Parabola infig. 129 fi fit quxvis chorda 
:Ppi ac e contaw i tangentts >pfi parallcIaEducanmrr^* 
Ct3L parallcla ayi, ea ip&m chordam fecabit bifariam 
in R. Duciis cnimPL, pl paritcr axi parallclis, cruni 
xjuadrata !{., il ad fe invicem , m ipfx LP , Ipy qus 
cum xqualcs cfTc dcbeant, etunt rqualia Sc ipfa qua^ 
.jlrata, 8c rqcta? it, I/, & RP, & Rp ipfis a?quales. 
i}4i. PorrQ japi CK ipfa hac dcmoiiftrarionc patct , 
inParaboIa opincs diamctros dcbcre cffe axi parallclas: 
ac ir^ EllipG, & Hyperbola omnes dcbcrc tranfire pcr 
ccntrum , dcmonflrarctur ex co , quod omncs chordac 
pcr feOitrum tranfcuntcs in ipfo ccntro bifariam fecan- 
«zr^^ii. 81 &dia|Tictri ejufmodi chordas ctiam fccaredcT 
lcant b;fariam , adtpqut per illud idcm ccntrum cranfi^ 
Xt . Atqu^ h^ec <]uid^m innuere libuit, ut pateret, quam 
facilc alio prprfus padto cx cadem definitionc feries prq- 
nricr^m dcdup pQiTct « dcdu<Sta antc alias hac con-« 
Aanti ratiope reittangplorum fub chordarum fegmcntis. 
342. Scd ii$ omiflfis contcmplabimur hic ppcius 
inirain quandam analogiam, quam habent EUipfi^s, & 
Jiypcrbohe fimilcs conimuni ccntro , & pofitione axis 
cranfveri^; ac Pafabolse squalcs communi pofiti/)ne a^ 
nisj cum afymptoti^; Hyperbolarum , quac profliiit par- 
|im cf Prop. 5, partim ex hac Prop, 6, &CoroUariii» 

$ Q^ giiu U Vlh 

f. 130 343. Olfintinfig. 130. bi9ta Ellipfes ^ & in fig. 

j?i ijiji, 13? kiM fdjpetboU fimiles , quarum 

133 comnfune centruf» Ci & fxet tranfverfi Cu, GW* pofi^ 

13 j tione cangx^^nt \ ac in fig. 13? hiri^ ParAhoU <eqHaUs^ 

fongtutonte axinm pofitionsy ordinatavum eandem inutn^ 

^ue pfififfonpn hakentium diametri pofitione congrueht > 

^ fi %Him rt/ffr^?f ordir^at^ Pp Q(curr0 ^lteri inH^ 



^ ^ i t i u t U r A: ni^ 

/JMcmilus Pj p m.fig. iji in ^adem rdmo ^ in 0J -■' 
i}2 in rm^is ,o^fitis erutn fempet aqwUid CtimeHifd 
HP> hpj C^ Hp, hP intirctpu hine indeimtt interid^ 
tm,& miriorem. 

\.\^ Si tnim ducamr in, fig* i^i ^ l^i diamctrr 
Utkjcix Pp j quas altcri EUipfl » & Hyperbote occii- 
pinE, & ^, tang«;ntes per I i &Edudla^, efuntpa- 

^ttllcl«(n.iip.) . (^are cura ofdinata Pj^debeat cffij 

Stilida tangcnti pcr vertideni fuai dlametri l , erit & 
tprallda tangcnti du<a« per vcrticcm dlametri E> 
*4«iac ipCiui ordiiiata . In fig, vcro iji fi ipfi Pj^ 
^ter para^la li occurfat alteri Hyperbols in A, de 
^Mictcr habcns pio ordinaia P/ dcber cffe f.n. iii.) 
Mlldj i^ngenti dutlac pcr verricem L cum dcbcat cf- 
^cpajugjiu.dianactri li , & parlter diamctct ordinai 
^i« HA parallcla tiinfgcnft du6tacpefil . Gu'm igiturcas 
. ttngfflttj paralicl? cflc dtbeartti candcndf iiabebunt di^ 
L^ftioacra carum crfdinatarunl diametri, & cum dc- 
bcint tranfirc pcr idcm ccntrum camnmne C> pofiticf. 
,l«coi^rac;nt. Demum iti fig. iJ3 fi cohcipiatur Paj^. 
!*o!i HVA translata pcr axem ita, ur fegraentum ^ 
^asVF abcat in fcgmcntum axis.VT*fibi acqaalci con^ 
^{^^t tota ciam ilia Parabola fibi a^quali ita s ut diaf*^ 
k^tcr£R abeat in IR* exifteme vertice I in ealdent 
^*«antla ab axc, in qua crarf, adeoque in eatdem rc- 
ki^^prioreeritdiamecerlR., & quDaiam adbuc tangeni 
y^ I iu£kz cum diamctro eundem anguium contine«« 
^i qutra tangei^s per E, erunt hujufmodi tangcn-» 
*5paraUcl3B, & pfoindc coramunls diredio ordinar 
^n utriufquc diametri , & cam.nunis ordinatarum 
"?<fctn diredfcrioncm habcntium diamcccr. 
HJ* Igitur iti omttibus ejufmodi figuris a comrath^ 
diamctro feca})untur ambaeordiaatac Pp, Hiftbifarianl 
Rj acproiadccrlcHP«quali$^i?., & Hj»arquiIisPA* 
l^^* Manmteoriinatarumeiafmodi direEtione. quAtuar 
"^^gula HPh, PHp, Hpb, Phpt/m/r;' erunt inter fe 
^w, ^ magnitudinis cortfianm , . ac femfer\ ^qn^ 
^ fii* 130, 13 1^ 133 qiiadraf:0 tangemis lA , vel 

1 3 Iz dn^ 



"IT^ SEGTIONUM C9NIGARUM' 
Jd dkcu f0r vcrnc^m l diamctri interiofis , ac dctcfu^ 
pi^afa contactti', & ffrimetrB, c^tcriori > ^a tarj^en^^ 
ff M fccta tifatia^ in I\ in fg. vcrp 132 dijferfntis 
quadratorHm fcmidiamctrorum farallciarum CI > CA . - 

347. Ducta cnim pcrP, <^ I rccia, quae altcri cur-. 
V? occ^rac in M» & ^T, crit &PM «quaJis IN, 8e 
MI afqualis PN. Q-iare rccjaQguIurn MPN crk cqua- 
le rcctangulo MIN. Eftaurcin n. 299- ) rcctangulum 
MPN ad rcctangulpra HPA> ut MIN ad rcctangulurn 
AI^. Iginit ctiara rcctangulum HVh crit ?quale rcctan- 
gulo AI4. PofrarcctanguIa^PA, PHp, Hj^, PiSf, pa* 
tct, sequ^ia cflfe ob PH, jpA, & Hp, AP a?qualcs, rc-. 
ciangulum autem Alrf crit infig. 130, 131,13; «qua- 
Jc quad"ato.AI,cumcocuntibi|spunctisP, p }n I abcanc 
HP, hf arqualcs in AI, dl^ & in fig. 132 ob| diame^ 
truni Aa fcilam bifariam in C erit rcc^angulum Al4 
difFcrentia qua4r«orum Hli Ca^ 

348. Hic autem j^i paicc atialogi^ Scftionis Co- 
mcx cxtcrnac rcfpectu ipternar cum afyniptotis . Scg- 
iiicnta recta: interccptsE hac cxtcmapcrimctro, & intcr- 
ija a?quarxtur hic inccr fc ( num. 345 ) , m fn. 221) 
fcgmenta rcctq? jritcrccpt? afymp.totls , & Hypcrfcola . 
Ex ea apquaUtatc rnfcrtur hic ( num^ 346 )' conftans 
mcnfura illorum quatuor rectangulorum , quac conti-r 
nentur fub diftantia alterius intcrfcctionis cum altcra e 
t>inis perhnetris » 8c biiiis. interfcctionibus cum altcra» 
ut in afynaptotis C numcr. 251 ^ , & ut ibi > ita ct- 
jam hxc , ubi habctun tangcns oircKnatis rcctis parat. 
Tcla , ca in ipfo coiitadhi fecatur bi&riam , ac iilarc* 
^^ngula $equantur quadrato {aagentis intcrceptc coq« 
jactu i 9c perimetro exteriorc . Ubi autchi in fig, 
1^2 non habcmr tangen» parallcia, asquantur illa re-: 
fcaagula difiercnti^ quadtatorum CI > C^ , quac jn 
.f^fypiptoti^ , ubi CA.evan^fcit , a^qtiatitur fpum.^jfi) 
q^^^t^tq tQti ipfius CI • At in ep etiani ccmycniaiit • 
§1 cnioj a3?is Vu minuatpr in inffnitum ita , ut dc- 
|nijm cYaj^^ffcit , Hypcrbola dpfinit in biqias rcctas 
Iianfcuatcs pcr C luxta nuraer. \6 .& i 10 y qav^ c- 

*^* '\ runt 



fiLEMENTA. iiy. 
tttt Iplx afj^mpcoci ^ quo cafu cvanefiiente AC » 4if« 
t»^a qaadratorvim GI 9 C A cft idem , zq ipfpm qa^ 
inma Q . Quamobrcm prppirictates jafyinp^o^orttni 
fcnt gcncralcs in Hyperbola' omnibiis HypcrboUs fi- 
piiibui comra^ni ccntro , & axium pofitiLooje,, au«j 
ue evancfccnce 5 dcfinunc demum in afymptotps. i^ 
luy in quibus. gcncrales iUs proprietates manent ^ 
iicet aliqus ex li^ ita itnmutcntur , ut remaneanc 
jncQiQmodgbe ipfis rcctis , & eyancfqcntia^ ^xis tranfi 
vcrii» ac cx natura redx lin(?ae cum iis. ipfis^ propri^^ 
lafe^s conjonpta deducanpir aiia X^iccrctnata . ' 

]49* Et qutdem in ejufir^odi fimijibus. pcrimetri^ 
Mjlogia cum afymptotis in Hyperbola , & j^arabol^ 
ftiam \ilterius prqgrcditur • Nanx in iis , ubi in in^- 
niianv pcoduaMiwi: , perira^^cr extcrior ad int^riorcni 
^it t^tra qiipfcuiiquc limites, quin (aracn iinquam 
fii occurrant . In EJIipfi quidcm p^rinjctrorum di- 
^Qa cft femper finita , & quidem minima in ipfis 
^ium conjugaiprum . vcrti^ibus^ , maxima in vcrtici- 
te ir«rfvcrforum. . At in Hyp^rba^a iti 6g.. iji , &: 
» Patibola Jn fie. iZ7 rcccdentc oirdinata Ppia in- 
PUQns , crefcit ipfa in iQfinitum , adeoquc crefcit ii| 
«Aiium & H/> ipfa major , $r cum fit Hp ad AI , 
^lAzd fh y pb rcaangtdum illud xqu^e quadrato 
^ ipfa fh deacfcit pariter ii\ infinitum . Scd, cum Hf 
■Bfiiuam abcat in iafinitum f aam omnes. ibordx pap- 
"^ alicui fccanti bis cundepi ramum incKnann^r 
•* dircctriccm ( num. X49J in anjuio miripre, quara 
^ aogQlus atqualjtatis ) , & proindc eam fecant fois ^ 
^ooBies paritcr in Parabola bis fcccnt ( n. \yi{) l^\Sfr^ 
iBam fh evantfcet . 

S C HfO LI UKf VIII. 

3J0, r Ed jam rcgrcdicndum ad (cricm. Thcorcmj^ 

tum hifce fcfaoliis interruptam ac crucmu$ 

IMpiwattrt naajdmenotabilcm, qu? licetfit quoddan;^ 

I 4 fim- 



ii6 SECTIONUM CONICARUJ^ 
fimplex Yeluti CoroUarium ipfios Propofiddfiis 6 y v 
mcn hic nova Propofitione 7 enunciibinir cum ninii< 
rum naturam ipfam Seccionum Conicatram cotittneacit 
^ ufiua habeat firequcndfiimum • 

PROfdSlf 10 Vn- THEORJEM A ; 

^51« ^^tladr^ftf» femiofdifiM^ eujkfvis dLimetri^ri^ ^ 

V^ mairue in Ellipfi & HjferhoU ad rectM^f^i > 

Itm fuh ahfcijfts a hinis verticihus eft in cenftanti rs^ 

tioHe , nimirum ut quadr^tum diametfi^ vel femidiame-- 

tri eenjtigata ad quddraium ejus diametri vel firmi^. 

diametfi five fi , ut in axe , tertis continue frof^* 

tionalis poft diafnetrum ipfam , & diaTnetrumconjugs-' 

tam dicatur pafameter , vel latus rectum^ & iffa dia* 

fneter latus tranfverfum , erit , ut latus rectum > z^l 

faram&ter ad latks trmfverfum , vel diametrum iUam^ 

iffam . In Parahala vero aquatur rectangulo fuh ahfcif^ 

fa ahunico diametri vertice^ & recta c^nftantij quane 

dico fardmetrum ; vel latus reSbim y & qu^c aquatmr 

ordinata fer ficum duEla , ac aquatur quadrufia 

diftaniia vertiiis diametri a faco' , v^i a direSrice» 

?.i34 35^* I^cr EUipfi , & rfiamcaris priitiariit Hypcrba-'. 

j^^ l£ 5 ii$ fig. 134 , 135 babtri rationem conftantem 
quadr^fl femiordinatf LP, vel hf ad rectaogulum VLm 
fub bini^ abfcii&s a binis verticibus V $ u patct 
C3C Propofitiombu^ ; , 6c 6 • Nam ex prop.* 6 rc* 
ctangulum PL/ ad rectangulom VL« habct racionem 
conftantem , manente ordinatarum directionc > Sc : 
cx Propofitione 5 reaa P> bifariain fecatur in JL y. i 
adcoque rectangulum PLj' aequatur quadrato PL , j 
vel fL • Idem pro Hypcrbola conftat ctiam ex no- 1 
^er. 256. '' - 

353* Eam rationem cflb eandem , quam pararne- 
0*1 , vel lateris recti ad diaineaum , vtl latus tx^taC* 1 
vcrfum » patebic cx definitione pcrametti y ii demon*. i 
liremr cfle candem, ac rationemquadratidiamctri>vci 

feim- 



E t B iil 6 M t A; rtf 

iet&idiamctriconjiigacs ad qaadracum diamctri , velfemr' 
itiamctri pcimarix • Id wtem pro EUipfi patrc in 6jt. 
i^, com diaroetri omnes ih ca tcrmincntu^ ad perp» 
bExrumj adeoqoc fx AO fit diameter conjiujatay cflfe 
ilfjxat in cad^m i^a rationc rcdaiigulnm ACa ad re,- 
daoguium VC^ ^ five quadratum AC ad quadra^m VQ 
«ieoque Sc quadratum Aa ad quadratum Vk . I^o Hf-« 
fcrbola demonftratum ctt num. %$6. 
. 354« ia Pataboia verQ iti ^g« 136 ctup rcdangukinii:^ 
PL|, five quadratum PL fit pcr Coroll. i. I^fop. 6 ad ' 
tcctan^um ii4> abfcifla VL» ic quavis rccta coiiftante 
in ratione conftaiiti i fi fcmel aflumaair pro rc&a illa 
c^aflti, fivc. pro paraiiictro tcrtia proportionalis po» 
iliqaam abfciflam $ Sc cjus ietniordiaatam 3 jam qua- 
draium fenuordinata: fict xqualc rcctanguio fub abfci& 
fa, & ca patamctro, adcoque ca ratio conftans in rcU« 
qms onuiibus ordinads cri^ ratie aequalicatis. 
^ 3 $5« Q$od fi ordinatd Piy tranfeat pcr fdctim Pi 
dc diamcicr LV occurtat dircdrici in H , eric ( num» 
278; W dimidia L'H, & LH diihldia F/ , ac proiri- 
de «quaKs PL*. Quarc eric LV ad VP' lit UP adPp; 
&: proinde ordiaata Pj^ pcr focum dudra cric* illa para^ 
Ineter conftans , qux erlc quadrupla VH f adcoquc Sc 
qmiiufiz VFi Q^ £. D; 



^ S C H O 1 1 tJ M 1. 

\iS6*/^ t/m cx hac quoquc Propofitione phtrk^a ^oii- 
Vw^ icccaria pro^uanc , ordinem quemdam in iia 
xndis pcrfcquar . In pdmis qiias omncs Sectionca 
ics commilnia liabenc in diamecris omnibus cum 
, quaB inicio dc axibiis iiintDcmonftrataCoroIlarioi* 
kabo > tum dcducam bina , que Parabolf foli funt 
^ria, quibiis dcraonflratis progrcdiar ad Tbcorcma- 
qacdam pcr^incntia ad £llipfiih , & Hypcrbolara gc* 
' ler : dcmum occafionc nacta comparacionis £IE« 
cnsn dvqsXof plurci cjus propriccatcs cvolvam* 






il$ SECTIONUM CONiCARUM. 

CorolL I. 
' 357- ^* dedftcta funt fro ordinatis axis tYonfver^^ 
J! in Corollariis S i lo, iik ^ 13 definit. 2 ntm. 74. , 
79, 83. $$ > f^^f^iv loctm hahnt in ordinatis diame» 
trorufH omnitm , fi fro axe conju^ato fonatftr in ^inis 
fofltemis diameter conjuf^ata. 

' J58. Denfonftratio cft fcadcm mroblquc, pctita pari- 
tcr cx r^onc conftanti \ quam habct quadramm fc- 
nriordinatac ad rc<5laagulum fub abfciflis^ & quod pcr« 
rinct ad CoToli. i y dcmonftrattim cft pro Hvpcrkolft 

nvm. 256. 

CarolL 3, 
359. Latus rectum cujufvis diametri in Parahla a^ 
quatur lateri rtcto frincifalt y & quadrupla ahfcifft a 
vertice axis per ordinatam ductam ex ejns diametri 
i/ertice. * 

F.6J 36o. Eft enim in ffg. 6$ patamctcr diamctri tracl* 
ftontis per P quadrupla (num.351) PD, adeoquc qua- 
drupla ER <;6mpofitac ex EM quarta partc latcris rc<%t^ 
principaIis/& MR cjufmotfi abfciffac a vcrticc. 

CofqlL 5. 
F.114 2^'* ^^^ quovis punito t chotdit Vu Par/AoU in 
I25 /f'^^4» vet tangtntis IL in fig, 125. ducatur LP axi 
parallela ufque ad ferimetrumy erit ihi reSlaniutttm^^lji y 
hic quadratum \\. aqptale recta^guU fuh PL > & latere 
recto ejus diametri , cujus ihi c^orda VtX efi ordin^a , 
& qua h)c tranfit fer contraSlum l . 

162. Scda cnim In fig. 124 chorda Vn btfariam m 
R, & erccta RM parallclsf dxi, qtiac crit fnum. 206, 
^213^ diamettr ejus chorda* , 'crir quadratttm; VR , 
five redtanguliim VRi^ ( num. ^$x ) afqualc rcdlangulo 
fub RM , & laferc rccto diamctri ipfrns .. Etit autcm 
rectangulum VL« ad redtangulum YRh (num. 533^ ut 
fedtangulum fub LP , & illa paJ-attJCtrb afllilTipta pto 
conftanti, ad rcctangulum fub RM, & cadcm parame- 
, tro, adeoque & rcctangiilqm yti^ crit JcquaTc *• rectan- 
gult) fub LP, & cademparahtctrio. Porro fi* eo^ttbos 
V , H fecans LV« afacat in tangentcm , quadratum 

cjus 



E L E M E N T A. |fi| 
ffas tangcntts dcbcbit acquari rc<^angulo fub L'P , Scts^ 
parametro. Sed idcm in fig. tij . patcbit, fi ift diawr 
ttum IR axi, adcoquc ipfi PL pariallelaiii ducatur f^- 
miordfnata i?R ^ qax c|:it parallela > & squalis L) ^ 
£rit CQiin quadratum RP xqualc t^ftangulo fub IR i 
ie paramctro diamctri IR , adcoquc & quadratum It 
tqaak rcctangulo fiib LP / & cadcm paramctro . 

^65. /;? Ellifji^ & ByffrboU diametri cppfjptgofffmt 
fii mvicem coTTjHg^^* 

j^ Pro Hyperbola demonftratum <rft ctiam fh,244.^ 
fcd pro utraquc fic cvlncitur communi dcmojnftratidnc . 
Sni m.fig. 1:57, i;^ binsB ordinatsB Pp, P/ cidcmF^ij^ 
diamctrb Yu apqqaliter diftantcsr a ccntro^ C pcr CL , ijg 
C/, & proindc arqualcs (^num. jyy, 8C79 ). Si dctcatur ' ' 
pcr ccnmim C dianicter ACa parallpla ordinatfs I^ « 
Py, ci fccabit chordas PP^ , jf bjfariam , cum L'P ^ 
L'P> & ffi lY debcant ajquari sequalibiK CL, C/, ad 
proindc habet ipfas chiordas PP*, />f ' pro ordinatis . Igi- 
tur Haar diamctri V« , Aa cjufmodi ftint , nt altcrius 
otdinata: fint ahcri mutuo parallclar , adcoquc ( num» 
212 j ipfa^ diametri fibi mumo conjagata: fqtit. 

Coroll. K. ' 

j6y. Si communem SAmetrum hiweanr flures Ellif- 
fltSy vel flures HyferhoU eandem frimariam diametrnm^ 
eriinatk' vero fint in quihufvis angulis inclinata ad 
iff/w diametros y femiordinata ad idem diametrifun'^ 
» Ihm fertinentes erunt in omnibus in confianti ratione 
iner fe , quam habebunt diametri con]ugata , & idem 
t^feSw Elliffium contingit femiordinata ad circtdum^ & 

eEtu Hyferbclarum tangenti ex eodem funHo diame- 

i dulhe ad circulum iffum eadem diametro defcriftum^ 

^dbita iffius circuli diametrtf fro diametro ejufdem con-^ 

ata^ cui tangenti Jemiordinata HyferboU aquitatera 
^ualis erit, 

' ^66. Si cnim in flg; 1^9, 140 cfufmodi EIiipfiun^»p|.^ 

d Hypcrbolarum femiordinataefuefinrLP, LP', crunt, *^^ 

Snvcrtcndo in proportione hujus Propofitionis 7 , qua- 7 - 

•<Jrata 



;ii6 5ECTT0MUM CO^flCJARUli^. 

drata femiordinatarum LP LP* ad quadrata fuarutn fb^ 
midiamctrorum conjungatarum in cadem rattooe qotn- 
ixiunis rcctanguli . VLm ad quadratum communis femi- 
diametri CV > adeoque & LP ad fuam femidiaixictnHn 
Conjugatam , uc LP' ad fuam i ac proindc alte^nandd 
LP ad LP', ut altera femidiameter conjugata ^d alterami 

367. Qjjiod fi in % i^9 VP«r fic circulus , iii co 
quadratum LP acquatur rectangulo VLh ^ & fi in fig. i^o. 
ducatuir LT tangens ad circulum VT» y quadracum ip- 
fius sequatur rectangulo VLh. QiX?xt etiam in iis crii 
quadratum LP figmx 1^9 ySc LT % 140. ^d quadra- 
tum fcmidiamctri circuli, ut r(ctangulum VL» ad qu^ 
dratum CV9 nimirum in farione aequalttatis , ac pra« 
inde manebic dctnionfiratio • In Hyperbola vero aequi^ 
lacera diamctri conijugata: erunt acquales, (num. 260)^ 
adeoquc ratio quadraci LP in fig. 140 ad reCianga- 
Itun VLny vcl quadracum LT racio asqcialiutis » adeo- 
que LP a^qualis LT. 

368. /« eddem cafU chorda Pp^ P^ JtuEt^ per vertU 
ces binatum erdinatarnm fernmntiim ad bina commtinia 
dumetri functa L, /> vel ta^gentes ducta per Hna ex^ 
trems punSta P > P* (nrdinatarkm pertinentium ad CQmmu- 
ne diametri punSlnm L concurrent in ipfa diametre ali* 
cubi in Q^ quod etiam in Ellipfi cum circula eompafa^ 
ta contingity in qua iccirco erit abfcijfa a cenira ad /#- 
midiametrumy ut hac ad difiantiam tarigeniis a centrm 
€$n^aratam in ipfa dianUtro^ 

, ^69. Paccc cx lemmace generali num. to^ £ric eatni 
LP td LP, ut Ip ad Ip' r adeoque rcct» Vp » L/, Vf 
ad idcm punctum C^ convcrgent • Accedcnc autetn pui^ 
cco / ad L, donec ciim ipfo congruat > crane(centibus 
fimul chordis Pp, P>*, fimul aiixbas focanccs^PCt^i^lPQ 
abibtmc in cangentts , & adhuc ipfa? tangences ii^ eo- 
dem diamecri punccoCXconcurrenc. Porro fi in fig. 139 
VPm fit circulus, &PCX,.cangcns, angulus CPQ^.critrc- 
ctus, SfC fimilia crianguia CLP , CPQ^ob angulum afl 
C communetn:, ade^que CL ad CP , live CV , uc CV 
ad CQ,. SCHo 



]B ;. £ M E N T \A. 4*1 

^ C H O L I U M II, 

• ■ • 

yo. TjLurcs hinc EUipfcos proprictatcs profluunt fant 
1 elcgantifl)ma?vtam guar ad cjus diamctros con- 
jugatas pcrtiiicnt, ^uam quac ad ipfius comparatipncm 
cum cirtulo , qux quidcm Hypcrbol? vd nuUo modo 
convcniunt, vcl non omnino communcs funt. Easali* 
quot Coroliariis pcrfcquar po ordinc , cfio ali; cx aliis 
Otiuntur . 

CorolL 7 

371. JFa Ellifjibus^ annmerato iis etiam circuk^ A*- 
imikus didmetTHm commuffem ^ fi ordinat^ ducta per 
vertices kinarum diametrorum^ quarum finguU ad fingur' 
las fertineant^ tranffant per idem fu]ufvis diametri pun* 
ctm , tranfihunt etiam ordinat^ duSta per vertices dia^ 
metrorm cpnjugatarum per aliud diametri punSlum co$n- 
mune . • . f 

372. Sint cftim in fjg. iij.i ftmidlanictri CP 3 CP 5?.^^ 
& ordinat? ad compiuncm diamctrqm V« duct? pcr 

P, P. ffanfcant pcr idcm diamctri Vu punctum L . Sit 

quoquc Q fcmidiamctcr conjugata CP> adcoquc paral- 

kra taugcnti PQ^, & ducta fcmiordinata pl , tum fc- 

miordinata /y, dcmonftrandum cfl: forc C^ femidiamc- 

trvm conjugatam CP'. Sic autcin facile dpmentoatur . 

Tangcn« ducta pcr P* tcrminatur ad idcm puncmm Qjf 

& ob fimilia triangula //C , PLCXcftC/ad/p , ut QL 

: ad LP, & (num. 3^5 ) Ip ad //, ut LP ad LP* . Qjj^ 

'■ Tjc cx cqualitatcordinataC/ ad pt ut QL ad LPS adco- 

I qoc ob angulos QLP, Q.lp' in parallclis ?qualcs, fimi- 

, fia crunt oriangula Qf-F, Cl^ , & , C^' parallda QP% 

' ^dcoquc conjugats^ fcmidiamctri CP* . 

CarolL 8. 

■ .,37,3- InElUpfi fi ad quamvis diametrum uV e veni- 

f fiius p«, p diametrorum quarumvis conjuzatarum ducan- 

f^ femiordinata VL , p1 , altevius ahfviffa a centro CL 

^it media proportionalis inter alterius abfciffas VI , ul 

' A 4dni^ vcrncihu^^ ac fumma quidem quadratorum bina* 

• rum-* 



111 SECTIO NUM CONICARUM 

rum ahffijfdrum a centro CL , CI aquahitHr quadrati ft^ ^ 
midiamttri CVi in quam e^ demijf^ funt , fumma ver$\ 
quadratorum femiordinatarum PLj p*l quadrati femidia^t 
m4tri CK conjagati ipJtM CV \ ' j 

37+. Si cnini cadem cliamctro fit circulus VP» , ^^ 
trigaHtur fcmiDrdinata: LP, /jf» erit/rium. 371 ; C^par/ 
rallcla tahgcnti QP i adeoquc angulus PCp ?qualis al- • 
tcrno CPCL^^ecto in contadm ; Quare bini anguli PCL* • 
jpC/fimul?quanturrcct6.CUmigitur5quefiturrectb&: biiu 
PCL, CPL iti triangulo rcdan^ulo. CLl?i ciit angulusj 
CPL cqualisiO, & proinde fimilia triangula CPL »^ 
iC/i qu? priftcrca ob bafes CP ; Cf ^qiialcs crunt 5- 
qualia^ adcoque CL ^qualis Ij^ medi; proportibnali in*- 
tcr V/, id ek circuli natura. Pr^tcrca yero fiimraa qua« 
dratotrum CL^ C/ ^quabimr quadrato.CP^ livc qiiadra- 
to CVj cumque fit C/ five PL ad LP', & CL, five If 
ad //, ut fcinidiameter CA ad femidiametrum CAVcon- 
jugatam CVi crit & fiimma qUadratorum C/ i Ct ad 
fummam quadratorum LP', lf*y ut quadratum CA^ fed 
CV fquale Uli prim; fumm; ad qiiadratum CA, qiiod 
proinde crit ^qualc fumme .poftcriori; 

CorolL 9. 
375. Summa quadraiofum diametrorum , feu femiMa^ 
irorum conjugatarum in Elliffi confianter aquaiur fummd 
^uadratorum axiumj. vel femiaxium) faralleUgrammum ; 
fuJMS latera femidiametri conjugata, reElangulo fub femi^ 
axihus, \ ac yarallelogrammum Elliffi circumfcriftumy qitod 
Co/ifinent iangentes ducta per diameirorum coh]ugaiarHm 
yertices reciangulo fuh axihus^ cujus farallelogramni an^ 
guiorum vertices erunt femfer in perimetro Elliffe^s alteJ. 
rius piori fimilis^ cujus latfraad^ ejuslatera homoipga e^ 
,. ^ runt in ratione fuhduflicata 2 ad i: . * 

F44i 276. Nam in fig. 142. fi Vu fucrit axis Ellipfcos' 
VVu; & diameter circuli VP«, & CP, Cjf fcmidiamc-'^ 
tri conjugat;» ductis P'LP, /// axi pcrpciidicularibus uP' 
4tie ^^ circuli peripheriam, tum CP» Q, crit quadta*^ 
tum CA ad quadratum CA , ut quadratum LP ad 
^uadratttra LP' , & (]^uadratam l^ ad quadramm // i 

adeo^ 



E t E H •£ N T a: hi 

acfeeque ut fumraa quadratorum LP' , /p ad fuounani, 
quadratorum LP, //, fcu ob PL ?qualem C/ f num; 
379 ) iumma c^adratorum PL^ pl xquatuir fumulgC^i 
/f, fivc quadrato Qi vel CA • Igituir, & fumraa qua- 
dratprum LP', ^* acquatur quadr^to CA'. Cun;i vcrp 
ctiam C/ aeqitetur LP» adebq^ue bina q^adrata C/^CL. 
^quentur binis PL, CL > fivc quacJratd CP^ vei CV, 
quatuor qiiadrata LP^, /^', CL', C/, fivc biha femidia- 
mearotum (:onjUgitarum CP' , Q' sequabuntiit binis 
quadtatis feraiaxjin CA', CV , a^coque Sc quddirat^ 
cUametrorum conjugatariim quadratis axium. 

37^* Ductis autcm i?L ; p^Ly erarit, ut CA adCAV 
tain are« triaogulorum P^, P^l, & PCt , P'CL / 
qug , cum. fiat ihtcr cafdehi j)arallela$ , funt ut b^^fe^ 
LP, LP'j quam arc? uiaiigulorum j^L/, jptly & pCi\, 
jpC/, qu«, pariicr funt iit baics jp/» fV . Sunt igituiria 
^adeni iratiohe & tota quadrilihea PL/jC>^ Pl//^', &tri« 
angiala PCLi P'CL,.ac i<7, /C/, ade6que& rcfidua 
crlangula PCjpi PC^, eft autem trianjguluiri PCj^ re-n 
ctangulum ad G ciimidiutn rectanguli fub PC^ Cf > 
five fub VC, CA, &.triangulum PC/ dimidium pa* 
rallelogfammi PC^l' ; Quare ci:it rec^anguium fub 
AC< & CV/ad fiarallclograwmum P*Cj>T paritcf, uc 
CA ad CA\ fivc ut idcm rcctangulum fub AC i & 
CV ad rectatigulum fub CA', & cadem CV , nimi- 
riim ^d r^ctanjgulum Aib femiaxibus > cui prpinde si^ 
quale, erit illud parallclogramintipl; 

37^ At iti fig. 143 fi .Qy^^ fit parallelc^ammtimp^j 
tatigcntiuni ductafumper vertic^s^P, i'-iP>!p diamctro- 
runi conjvigataruni P/ , Ff , fatis patet ob ipfacum 
taogehtii^p , parallellfmum cum. ipfis diamettis > forc 
intcr fc kqiialia quatuor parallelogramma CT , CQ> 
Cf , Cj^quoruhi proihde cufn fingula ut CT ^ g^ 
quciatur rectangulo fub f<pmiaxibus , fimul omhiai x* 
quabuntui^ rec^ao^gulo fu^ a^clbus • Diicta v^ro CQ.* 
qu2c Elli^fi occurrat in V , chordam Vf ea bifariam 
fecabit in R , & ibidem ab ea bifariam fecabitur > 

cum fini; bini diftw^tri paralWiogr^nmi » critque PP' 

ordi- 



t«4 SECTIONUM CONICARUM. 

erdinata fcmidiametri VC , adeoquc ( num. 368 ) 

CR ad CV , ut CV ad CQ. , & CQ, ad CV , in 
. rftcionc fubdupticata CQ^ad CR, fivc 2 ad i. Paritec 
fi P>', CT fibi occurrant in r , & CT Ellipfi in B , 
crit CT ad CB in rationc fubdttplicaca CT td Cr > 
five 2 ad 1 1 adeoque Q^, T ^ bufufmodi. ^llipiim per 
nura, 119. 

379. Didmetvmm •mnium in EUiffi mdxima eji 4r 
xis tTAnfverfMi minima axis cenjugatus y reliquarHm es 
mofer , qu^ dxis tranfverfo fropiar , dc time hinc ind$ 

1^442^ ^i^i^ f^ Vfi ^ifuUHttis aqudes. 

380. Nam in fige 142 (i Vn fit axis tranfvcrfus, qui 
conjugato ftrapcr cft major (num. 64), erit LP major^ 
quam LP^in eadern ratione ; adcoque quadratum CP 
squalc quadracis CL , PL crit majus quadrato CP > 
quod cft ^ualc quadr^ti» CL, LP*; ac proindc CP? 
vcl CV major quam CP', & axts uanfverfus dupl^i 
CV major qyavls diamctro dupl^ CP'. 

381. Porro quoQiam quadratum PL ad qpadratuni 
P*L eft in eonftanti rationc , ix^ cadem Fatione ac* 
fcent , & dccrefeent & ipfa » & corum dijpkrcncia . 
Crefcic autcm femper finus PL in circulo , dum P ab 
V ab A tendit, decrefcente CL, ac i» A cft maximus, 
adeoque & diffcrentia quadcatprum LP', LP qu« ca^- 
dem cft, ac difFerentia quadratorvim CP , CPS fem- 
per crefcit ab V ad A > vcl A* ; & proindc cum qus^» 
dratum CP fic fempcr idcm , dccrefccc pcrpccuo CP , 
& abeuncc P' in A*^ ficc minimum . Quare diaraetti 
quoque quo magis diftant ab axe tranfverfo co mino- 
fcs fuac, 9fC axis con)qgacus cft omnium miniiibus* 

382. Demum fi Ellipfis compleca occurrac ipfi PP" 
in I > cric jLI squ^ilis LP' , adcoque & CI seqHalrs 
CP*, & angulus LCI aqualis LCP . Quare binas fe- 
midiamecri CP*, CI hine inde in ^qualibus angulis ab 
axe cranfverfo ^quaks ^ adcoque ^qualcs & ince^ 
tdiaracari, 



E L E M E N T A. .iif 

fc^rw. II. 
I f 8;« Duimeter fer ciijfu vmkm duEi^ erdiftat^ sd 
)llKtm h^iih abfviffdn^'^ centrtf ^ufmodi ^ mA f jirx qu^^ 
^am fit ditnidium quddrati iF\ufdem femiax}s\ h^ehit 
dimetrum fenjugatdm fihi 4juslem , (ir ea y datU a^ 
4uiy fneiie dMrmitiatur. 

%i^ Si enim ^eriht CP^ C/ acgualcs^ cnint xqiKir- 

ilc$ & angtili PO,, ^Oi adcoquc & CI, ?'0 , nimi% 

*um (fi C/ fuerii conjugata CPV CL, LPi & qqadraii 

^^ CL «Cniidium qi^adra^i CP» ^ve QV. Daio autem 

iQtt kV, fi fiac angulus VCt^ flrmirectus, tum c;4>ta CP 

zquafi Cy , dttcatur PL' perpendicularis ipfiaxi» S^capi^ . 

ttri!p*aidl LP in ratibne' femid:is CA^ adfemiaxeni 

.CV, crit P'C fomidiametcr,' ocis fiise coojugat^ ^^!?^ 

Corotl. II, 
' Jlf *i tedem axf fit Elliffis , &. cmuhs \ erit ^ 
^imiipn/i^ 4^- iirrim £lliffeos , nr £^ 4Ari> 4^ alterum , 

9M r^ri» #r// ^^i^ in fegmentis comfnunes abfcijfoj 
^Mmilms; ac drea iotpis Ellipfis erit media gefimetri-^ 
^ fnfertionaiis inter are^ cirtulorm haheHtium ^^ 

^^nt kinos ejut axes , five circuli circumftrifti 0[ 
H^crmi\ ac apiatis dfe^ cvrcuii hahentis diafi^etrum 
} M^i» geometrite' frojfortienatem infer, Hffos axes. 
^ ^Si» Si enim Vii flt jam axis uterijybet , 8t drculus 
•iJittai'!^ occurrat in P,vent PP ad IP fcmp^r ut PL 
Nld IPt, five ut fi*miaxi$ GY ad CA^, fivc m ?otus a-* 

6 Yu ad aicm alterum . Qiiare & arcas' gehita eo- 

"*' mohi earundein rectariim PP, IP crunt iri ca- .. 
catione , nimirum arca fcgminti PVr. ad Ycg- 

ftnatoi IVP' , & arei totiqs circifli ^d jreani totius 

J*7. Porra ct^ni Mnc ar^a clrcuB fiabcntis pro dia- 
tto axem tran%rfum ijt ad ^reaf^ Ellip^ebs , ut' a« 
tranf?crfus ad con jiiga|mh , ik area EUipfeos ad 
^ cifcuU habentis pro dtaoi^tro a;k^ conjugatum 
itctum, ut axis tranfverius ad conjiigatum , crit ares^ 
'pTeos media inier areas iUorum circuloruin, &'cuq 
Boffovieh , TomU/Jf l^ S^^T 



- 



125 iSl^CTIONUMCONlCfRtfM 

quxvis fcmidiatncter fic mmqt fetniaxc tranrverfp^ nyt> j 
)or fcmiaxe conjugato» pateU circulum ddci^ipixiivv» af- 
funjpto pro radio illo ptiore i fore drc^mfbripjMlni y ai&ifp 
pto vcro Iioc poftcriorei forc mfcriptiinii Ci|ttt^^ ^< 
circulorum fit ih iratione dupllcata diamctrotum > pg^t 
circulum pariter habcntcm diametrummcdxamgpott^tri 
ce proportioHalcm intcr bmo^ axes > liabitpt;u(n, afe^ 
paritcr mediam intcr ^eas eorundcm illpxiim ^rbtlo^ 
rum^ & cfqualen) ircse fillipfeoSi 

5CH0 tltjivt ii|. 

^SS. A Tque hoc qiiidcm paAo nii^ta .decfaiCitDiis . 
jTjl quas Ellipfi ita propria. funt > i^ idjHxnefi. 
bolam falteni eodem pa^o transfcrri non pofljnCiiticci: 
fuas habcat Hy.pcrboIa ipfk piropdctatti » qiiar eiruni 
plcrifque rcfpQnd^atit • SiQ nonnullis eori^.y qu^ tud 
proponontuu n- 373, J jf;5 > ^79 rcfpOildcnt.,. qius p^ 
Hypcrbola propolita funt num. 253^ i^S, 144^ 

^89, £x ipfa Propofitionc facilc dcdiK^tu^^ ddti^. 14: 
tere tranfz/erfo ^ & r^Sh » ^ dirfSH$ne ^rdin^atiimf vet 
ddtis irt Blli^fi &HyierboU Unis di^e^tris X^ifi^is^i 
mainitudiney & jfofitione i ^ojfe, inveniri otnni^ ^ej^fti _ 
Cenic^ pun^a • AflTumpta cnim quavi^ abfdfia/iii lairt?*'.; 
re tranfvctfo i ic dudba fedfca id ca du^c^G^ . ^ ^wm 
habcre debctit ordinataei » qua^ nitiiitcinl iii i^pfi,» Sc \ 
Hypcrbola parallcla cft diamctrd^ cdniuS^ffi >. ^ps crit «^ 
pro Parabola afiumere in ipfa binc inde. hkv^^ fcoiiot- t 
dinatas medras proportionaxes ini^r abfciiTii^i^,,^, jaiiis ; 
redlum, in Eliipfiy ic Hyperbola aflumpta. nkdia ft^. 
portionali inter binas abfciflas a binis vcftic}l|QSi rati&. 
crir aflrumcre hinc & indc binas fcniiorc^nafasy qiiae aoj 
cam finr in ratioiie iimpUci diametr! ccinjugaisi^ ad <i\%^\ 
metrum iUam^ in.qua a^umpta cft abraiTaj^ fivQ id ra-\ 
tionc fttbduplicata lateris rcAi ad illam diaiiKiruQgi ^ 
Habcbimr cnim, ur patcCy dcbitus^ femiordiaata: yator»j 
&:. niutata uicumque abfeiila^ dcicribe^ oiQois S%Qmi 
Conica pcr puacta. . . • j 



S L E Ai E N t A. i±7 , 
f^a Sed ut pro Hjrperbola ex datis binis diatpeix!^ '. 
bqj^sads ek^ancifliinam , 8C e^fpedicidimam conftru' 
mieta babuinlQS aum« i65^ ope reguiae gycaacis citca 
\uaBfi pundfaim iiiter bitias afymptoeo^i fic bic paricer . 
i4beBm& aliam nihUo minds eipedicant i 8c elcgahcem 
^ru^Qiiem EUtpfeos ftt pixtxStt i dad:s icidem* bi- 
itts dianietris cohjugacis i idque paricer ope regube 
iiia quadattr dacar lcge gyrancis incer dacas binas red:as « 

J9i. Sin^ tinac diamecri conjugacal in fig; i44i ^45* , 
vCn^ACiP; Ex aiterius verticd A demiffo iA altetain^-*4j 
^adicald AB^ c^piacur AD ih eckleni i vel aci par- ^4J 
k^hfioda&Os iic ih fig. 144 y. vel vetfus B » iit ih 
^ 14$ i AD atqid^is femicUametrd CV4 ac pec C > <Sc 
Ddatta uidcfihita BF> prcidiidtaque indcfinite ucridque 
y» ift Gi 8c H» niciv^atuf^ linea BD ica^ tit putido B 
kxcuccentt pet! t<6t^m GH i ad piin^la P per £F abeat 
mcSi Si fm£hxt!^ A abiehs ih d defcribet tUipfim ^ 
Dlia eaim ex d redla parallek DB, qnx ocduirrac rc^ 
^ AP, Va^ in L) N^ eiU (nuni. 2164 ; ^^ ad ^N , 
litlM ad DBi five uc dd ii S i Q^c diiAi aL i 
tattt fiimlisfc. triangula sdL^ idN', ^deoqae ah paraUe- 
la &sm<^ ifOi, Erlc aucem DA ad WL ^ uc AC ad 
tCjjdcoqud qiiadrdtuih DA^ five CV ad dificrentiam 
qui&aiDrum DA» dLi live ddj dL^ nimirum ad qha- 
Jbudm aL-i ac qiiadratunl AC id di£Ferenciani quadr^- 
tea»ACf LC^ five ad feaahgiilunl fub abfciffis AL, 
"' ( Qjiard alterhandct qhadt^atum VC ad quadfatum 
i m quadcamm 4L ad fedtahguiam ALP fub abfcif- 
& pfc>ijid6 dL aequalis femiordinata?: ^ & pundunl 
a# EUtpfini f 

^i; Q^od fi iri 1%. 14^, 147 V», AI? fiidrint axes,F.i4« 

dkio evadet facilior . Sumpco enim in axc CA 147 

to AD yei ad partef oppoficas. cencri C , uc in 

1469 vcl vcrfus ipfunii ut ih fig. 147 , & ncptacis 

rcgula pundis D^ G^ Ai ipfa rcgula ica ^onVerca- 

i ttc pudtfai^i G exdvrat pcr axet?i VC?i ih c^ pun- 

► D* cxcarfer]lce per aCP ih i, a:d punccum A cran- 

in d diicribcc EUipfiiil. Ducca enim^L paraUe- 

K 2 Ja 



f 2S SECTIONUM CdNrCARUM 
U VC9 crit ^ ad ^5 ut cm td CL, adcoqix: 
tui» da^ fivc CV ad dificrcntiam quadratorum dd, dl 
(ivc quadratum ^L, ut quadramm fs^ (iv^e CA ad diM 
ferentiam quadratorum ^/i, CL, fivc GA i CL ; nii 
rum ad rcctaftgulura ALP^ adcoquc altcrnthdo quaAanj 
tum CV ad quadratum CA, ut ^uadratum Li ad rt4 
.ccangulum ALP, iit oportebat. ' 

i9i* Quoniam vcro illa puncta a^ h^ d m fig. 144 
145, vcl 4>r» d in fig. 146, 147 pofliint ctiam oflk.. 
tari fn txtrcmo rcctilinco charts marginc 3 ic cbant 
ipfa i^a cranslata , ut puncta h » d ^ yrtl e ^ d feni! 
pcr fitit in rcctis GH, £F> notari» facile pofliint quoe»^ 
Cumquc puncta a 9 6c pcr ca duci linca continua ; adf 
pbdum facilc Ellipfis dcfcribitur • Solcr autem &, in>r 
ftrumcQmin conflrui rcfpondcns (ig. 147» in quo virj^ 
dca habcat in a flylum, in r» & ^ binos pcdes infer* 
tos ita crcnis in lamina incavatis fecundun) dirccttooef 
CV, Cuj CA , CP j ut pcr ipfas cxcurrant , ac fi^ 
a motu continuo Eliipfim dcfcribat, 3c ut plurima Q'^^ 
iipfium gcncra deffxibi pofKnt, virga paratur longiot»^ 
pcr quam flylus 4, & pedcs c^ d pofHnt excurrere, ft' 
adniovcri ad fc inviccm , ac fcmoveri ica ^ ut da fiat^ 
scqualis fcmiaxi tranfvcrfb ca conjugato. 

394. Ovalem lincam , qua| rcfcrat Ellipfim , fk tt^ 

^^giam opc circini liccint dcfaibere • Fiat in fig. ifS* 
rhomlxus quivis HD&^ 3 cujus 4atcra ad partes anguloi^ 
ri|m oppoj5rorum B ,' '8c H producanmr z tum centris 
fi, £^ H, quovis, fed utrobique codem intervallo, de^ 
fcribaiarur arcus circuii FG , IL , ac centris £ , & D 
reliqui FL, GI , qui apte conncctentur cum prioriboii 
in F, G , I , L cum perpeodipnlares fint iifdidm £F 1 
Dp , DI , £L > habentibus corum ceiitra . Quin ctiaiq 
F449^ dcnitur in fig. 149. aris m^or VCn, &minor ACI^ 

* facile fic d^tgrminabitur rhombus HDB£, cujus ope 4 

' jufmodi ovalis fiat . Centro Cvradiis Ch ^ CA fiafll 

quadrantfs circuli «K> AS occurrentes in K , S ipfiii 

CA 9 Ch .. Ducatur Au occurrens arcui AS in G , al 

per quodvis punctum J arcus DG ducawr rccta CtvOC' 

ciir- 



^ l B id fe M T A; i^ 

Mttii qttadranti i»K ifji L > ducannirque rtctx uL / 
U, per qQanini Cdncurfiim F cluCta reCta ptraljela ipfi 
LC> qase occorrat rectis O > CP in B , £ afliunpufquc 
^i CD verfas V, 8c A aequaltbcis CBy C£j habebitur 
rixmibas qpatfitus EBDH « Nj^9 nrianguda fBu , FEA 
tnuit fiimlia ifofteliis LGm i IGA» adeeqiie tircus circuli 
bdio Bm abibit in F> ^ radio jEF in A. 

}95* Proqno^iil rhon^jbo^facilius in^enfetut quadra^ 
na. Sumatur AN verfus P jeqdalis nC 5 tum CM vcr* 
fasiequalis CN, ductaque MN , ac bifariam fect^ 
pR) fiunantur MB^ N£ ad partcs oppofitas C aequa* 
teilR, vd NRi &.CH, GD «qualcs.ipS^ CB, CE, 
icitabebiw intentum « Paiet cnim HDBB f^^re qua- 
^, ob aniu^iatrianpila BCD, DfCH^HCE^ £CB/ 
^a^ein RB^ R^, a^ R0 parallela !NE , obCM» 
CB eqides CN> CE patet^ MN> BE fore parallelas y 
.&pmin(fe ai^gidum RBO a»iualem alternoMRB, gve 
.^^ ob MB , M^ aBquales , vel^CBR ; Adgulus quoqiic 
aOB jqQalis cft fensirec^o NEOj' five femirecto BCR^ 
» M commuais tfiangulis BRC > BRO .' Igitur errf 
§la^uaiis CB, & duct0. aifcu tfS , crii OF ^^uali» 
«) fivc NA • Qjiare additis EO , £N > rquafifaus ei* 
^ffl NR, crit & EF arqualis EA , ac arcu^radio EF abibft 
^ A;Sed faaec coonructio locum non habet, ubi CN 
^i^ltia femiaxium fit itar niagnas ut MB evadac ma-r 
I*' ^ sequalis iAii, 

$ G H 6 £ I U M IV. 

^ pRogrcdicmur |am ad aliud ThccJirenia dedu- 
^ 1 ecndom c Prbp. e^ ^ p2uAttt focundiifimam 
nniotum ptrtinentium potiffimnm ad tangcfltes» quo» 
m nottndla etiam e Cirollariis ipfius Prop. tf , dcda* 
I^Pwetant, ut monui num. ja^.OrdiHcitJ.deductionis 
^^^ in Scholiis infierjecus^ 



K ? PRO. 



730 SECTIONUM CONICAROm 

PB.oPQSiTio vnr. theorbma, 

?^7* Q^f^ f^ncurfim (Xfi^. 150, ly», ly^ tafigm 
v3 m PQ^nrw dfmptr» QR. dfuranir r^Sfa ^ecHrrm 
perimtrs feElUnis Cpnie^ in T^ tj & frdinata Vp £| 
K , mt QK media harmenice frtfertionaUs inter ijf ^ 
Qt >/^ fig. 1^0: iji 5 w ^i^/ Ti t/iM^ iit eodemra, 
moy vfl Kr, Kt (n fi^ 152 , /« fn/i fo^m jacent #1 
ranf^is pfpefitis,, 

398. Pu^att cnUn rcda Q?» ag^ntur pcr T, $ rcfli 

Pj-^paraJkl« ipfi Vf occawcntes rc^is Q.P, C^ , Qf Ig 

i-?! '^' ^' ^* iyL\ l^ &c pcHmetro itcrum in S, ^. Q» 

j^^ niam prdi|iat« TS > // a diaxnctro Mritcr Hfartatq i^ 

■ canwr in I, i^ & f^nom, 204 j rc«^ Ht, ht a rcij^ 

QR dcbcnt fccari btfariam in I, r^ Ut rc^a Vp in Ri 

crunt & HS> ^J, jcquaks TL, ?^ & re^angula TH&x 

ths rc^angulis HTL , htl . Porro cum fit HT ad ht 

& TL ad tly yt QT ad Q/ 5 <^ri« quadlratmn QJ ad 

quadratum Qt , ut re^ratigultiiii HTL ad rcctangdoflQ 

A//, fiye ut rc(:tangulum THS M rcctangqlum r/^, nt- 

«niri^n fnum, 321 ) Pt qusidrafam PH ^d qviadratmn 

Vh , vel ut quadratum KT ad quadraf\im Kt , Qjnarc 

Oy ad Qt^ ut KT ad K/^^, Sunt autcm QT, Q/ if 

fig* 150, 151 tri^ra QTj QJC, Q£ cxfr^sc , ^ KT 

Kt difFcrcntia? extrcmarqm a media, ac in fig. 152 lu 

^xtrcma^ ^ium KT, KQ, K^, iliUe difleircntia^ earunden 

a medis^ • H^^^ur igttur ^obi^e rado barmonic: 

prqpofita* 

S C H O L I U M I, 

^9f. Q I ixGtsk QK fit parallela axi in Parabola , n 
alteri afytpptoto in Hyp^rbola , puQcro t t 
in infinitum rcccdcnte^uc nurquam }am fi( , fiet juil 
num. ^5 QJ acqua}i$ TK. Scd quoniam co c^fii ii 
P^rab<^a QK dcbcrc? congruere cum dinmetro QJfc 
ci^tn ipftim cafum , ^ui nimiroiTi afui futurus eft 

accu-? 



fe L E «1 E N T A. tji 

'«turatt |S^ hoc Scb^om pct i^nitam Ccfointtriam dc- 
monftralbo . , . -/ 

^bb. Qciod fS j)uri^m R abJr^t iii ccntrum •, Pj> in 

|%* ijb, lyi cvadcfct diimctcr ,^ ip^ ^aJ^cft^ PH <iia- 

f«»tro RQ«pi^aI&I«9 a3cociuic putt(5Him 'Q^a,l4^^ inia- 

Wtittmri qqo ^afu rcifts^Tr patitcrpattllciktangcQti H/»^ 

t&t Qrdipata diainctri Pfx&c ab ^a bi(ariaxn fecarctur 

Ift K , HlM 'paaritcr coogrclft <^m iis , qu« jAunii 15. 

^demci^afj^ fuat dc haijlnoifncae prppbhiohis ration.c in 

^cfialitatem dcfincntc , ubi ^^t^m c ^uatuor piu^.fti^ 

iaiteoB. abit in Inftiitum ^ ' 

* 4bi. In cafui vfero,' in quo C^ cv^flat diijittcter, 8c 
%riiiit cutaQR, puntjlutn T ubiquc , & fin EUi. 
1*> « ^pcrbola cvadit cfui vcrtCx , adeoquc <^vmi«- 
fcttittbm TSi //, fmt Bti W tangtintcs , fq ^cdan- 
jQ^HTLjjlfr^/ cvadunt tjAidirata tangcntium, quo ta- 
^ cafq. adhuc dcRjonftraitit» vim habct ^ $c in ca- 
'fcERipitos, ac f^ypcrbolae coipddit cum dcmoaftra- 
poocCor^ 9; PtopoQtidnis $. cxpofiqi^ num. 33) , 
^ qua ftriem j^uandani ebnftdariotum cjufdcm Pro- 
fofitiodfs abrttpitiitis, ut num. 527'. indnuii. tic nimis 
w ttrkgarcmut , Imc rcfervatis iis , q4«'lani. dcclu-^ 

^ftMlum P i«af* f« Btlipfi , & diametris Vu ifri-p,«5 



40J. Ntm iti fig^ 153 , X54 pund^t Jd, V> ft^ i^ 
%)tt^t ptiuAis CX* r» R» i fig. 156, 151. Acpro- 
^ cft VQ^ad ftQj, u> VR ad R». Idem autcm cruc- 
''^Qtctiiltt cx iUo Corolf;9,|h*op. 6; C ^riim ungcntes 
l^ Vi at k dudhe ^occurrant tangcnti ^0.,in A, & B, 
trimi paraUda?, & pcr id CoroUarium crii AV ad Bii, 
ft AP ad 4%, adcixiite VCliW Q« , ut 'VR ad Ku. 
40^ Inde autcM i^ainir pun^ta Q> & ^ debcrc fa- 

X 4 cerc 



ii^. ifiCTlONUM CbNICARUlU 
ttic ad caiKjciii ccmri partetn ,; quia ^nf' diAiCilm^ 
V(X i VR ab cbdcm vertioc V , qu? funt primus ^ 
tcrtius proportionis tejrinmus y dcbcnjc «cflc yt\ fimitf. 
majorcs ^ Ircl fimiil riiiftorcs quam binae difta^dac * 
altcro vcrtice u , adcoqu^, jaccfc ad candcrti partcal- 
ccntiri vcrticibus intcrjcescl v ^ quam. jacc^ vcridf 
piropior , ; . ; 

405. In Paraboia veio in fig. ijj; fic.piar Shmid 
Gcomctriam dcmotifttfatitf' fore Qy -> VR. «qualcs . 
Pr?tccca diariictcr duCJ^^tP occurw^ twgcqti duct? 
pcr V ih Ai 3 ordinatac in r . ac cx concurfli^ A. bi- 
<iamm.,tangcntium dtjcatur J^ diametris ^ & axipa'' 
rallcla uique aJ pcririjctriirii ,. i^runt ob parallclifmuiit 
^qualcs QV, Pr ,ScVK, MP, acjcoqic ctiam QRi 
Mr . Efit autcm ( ngm.ri^S )' QV ad.AN, wqua* 
dratum QP ad q^xtdfatum PA 5 Cvc.tit qaadratura 
QR ad quadfamiri VR^ ^ AN ad^MP, five VR ut 
qnadriturii VA ad quadjratum V&l ,- fivc ut quadratum 
rP ad quadraturii.rM, vel m quadratum C^V ad qaa- 
^atum QR ^ I^itiir cx «qtialitate pcrtqrbata ^ cric 
Qy ad. VR , ut quadratum Oy ad qua^atum^VR t 
4>io<£!a^cndit cain cffe raiioncm «ma^iatisi c,ritcnii|i 
ftcial^um^ fub Qy Sc fc ipfa , nimirum cja$ q^ai 
dratum, ad rcctangulum fub Qy , & VR , ut Iftum 
quadramm C^ ad quadratum VR > adcoque rccraiH 
gulam fub QV & VR asquak quadrata VR > ^vc 
Qy «qualis VR. ^ 

i - Corcii.i: 

"^^ 405. In illip/i & diSmetris frimariis HyftfkoUfyi^ 
mmadumim YBl* VQ» qua ^cindit ^ Mit^rozf^* 
iice V ordinfta PRp , & t^n^o^s VQ^fer idtm jMff 
Sbm V dnil^i ^ fmt nt ^nfmodi fogmentk sbfcijfd ^ 
^ltero vtfiice 5 dr es rs^io in ^lHfJi ejt mitmis > ito- 
Hyferbolfk majofis in^^4Ht4tir , m P^rsioiM €99$^ 
^iiatis. 

407. Parct primus t% prcjscdeatis Corolbrii ptopai^ 
tiooc • Nam altcrnando cil^yQ.ad: YKi st «Q^aAri^ 



^nt E::i* i> H-A.^ t^ 

hm fecaiidoiii» qiiia ordiosita.P/ iiiMsacidj^cti;^^ tA 
Ji iii EUifrfl ioicr vcrtito Y*^, in Hyprtbol» cxtra-^ 
^ggfa dc^cac C 4uij9^ :i4^ J }ac»tc ifai inicr faitias taiiH 
j^fcs fibi parallelas traafcuntcs pcr diafnpm^* vfcl^ 
Ks ( nom. 212 ) » failc cxcra • laccbic jgltur coacrk 
R.jJ^ibi cxtra, Jiic iotra, & in filljpfi.ifiR crit ininor* 
i $^ ^Q/ lA tfyperbpla major ;i In l^araliola^ velo ^ 
CordUario fapcriorc fawntur VQ.i. VR* 

408. J'* timgths VNI ^4 |r#r vtrdcem V Mamtv 
H KcwrrM in M rrffii tninftmti i§iY qj^itis Jierimt^ 
nifimElum ^ ^ & ftr altmm vtrtmm lA in EUiffi 
& H^^trhoU , ^ diamttri fmdUlfi^t mi fmab^U^ M 
JecAhuw hifMruim in A;4 tafsgtnte j^4 ftT V j 4B 
i» ParJkoU tMgentes VM ., PQ;^/?^ /«y ^*»^ W^Tj^- 
n(w diametrorHm vertices^ W^V y & t^minatit 4d ijf^ 
fdt disnietref f^ mutiio fecsnt Hfsri^m i^ A* 

fOf. Erit cnini in fig. 155 , 154 B« ad AV , ixt 
«Q^ad VC1> adcoque ( 41U. 406) iK »R ad VR^ ni- 
miiriim uc 4^ ad PM , yel deHium uc eadem Bh ad 
j&M, adeoquc AV, AM «qualcs ^ Ac in % ];55 oii 
Qy.dimidiamQR, crii VA dimidia.PR , vd VAi, 
^(M ditnidia C^ ^ acj>^oiade a^quaici & AVj AMi 
AAQi. APi 

^CHOLI UM IL 

4xa T t Adcnut Propofitionis cpnfcftaria qufdatn 

JL JL dedoxi ,j quflB a raiionis harmonicae pro^ 

(cktatibus jooa pead^t • Nunc quoniam pan&aquo^ 

9R Q^ R, Vi I» in ^jllipfi Sc H/pcrbola harmoni- 

cm proportioncm confticuuac , cujus cum priora iUk 

\4ao f cuBi liset pofieriora alterna fanfi dcducam ca , 

'^u cz proprietaci|^il9 ejufdcm harmooic; proporcionis 

conicqutlQtur , fc<%a dtQaotia binotum . alccrnorum V» 

* bilsaiam a centro .Ci quorum bina pocifiimum dc* 

iBo^aYi msfk* »A 9 z^* QjXQ4 3mm Ha ia Sg, 6. 

^ ' func 



t?4 SECTld^KUM CoMrCARUM 

.fnnt {>lm<Ska Ay R^ $. C, t) , hot hk Ih SUipfi in 

1%. 15 j. ftttrpttwaa , » , C , R, V, <I, & inHy. 

psrbok in iig. 154. «, C , Qj» V , R . I^ihniitir aa- 

-ttm pridris^dprietai^i; <potiji:Alu:i^ y tmn poftetiorii 

* Ccrath 4* 

4n. in l^Hffi y & Hyperholk dutmtns frimatUs 
^jfiftridtlgmetir Ca s t/</ C V ejf medid gtbintrice , frefer^ 
^U^slis int^ QCX^y CR diJUntias irdihata Pp , & 
Jdnfenfi^ PCL in ead^ diametra affnmftas y qnd nd 
^tHndem ^entn farHm jof^ent amU . 

41^ Paitt pri«mim tx nutn. zz. ob proportibnem 

lurmpiiicam ptmdorum Qj» R, V, i< : quorum gl^ 

^ma fimt V^», & ebriini ^iftantia fedraeft biJFariaiH 

& C Ekbere autem R» Sc Q^^acere; ^d candempar^ 

tem O^nri C patet c^x. niim. ^oi. 

Cereil. 5. 

413^ <A iifdm efi CR ii^/r^}f^ a centro sd ^ti «^. 
fiiffjtm ab Um pertice ^ mt VR ^fcijfa ab alteri ^ ad 
'KtXj^t^tanientem ^ 

4 4. Cum enim fit CR ad CV 41 ut CV iad CQ^ 
^lt in eadem rationie & RV difierentia ipfarum CR^ 
<CV a4 VQ^ difierehtiam ipfarum CV, CQ^ adcoquc 
XR, adCV , ut VR td VQ; i Jk proindc Ck ad 
R» priorum fumniam, ut VR, ad RQ^fummam pofti:^ 
riorum» CeraU.^*. 

6 415. /n Hyferhela in fig. i^i. feindiameter q$i^ 
qne fecmdfria CV , vel Cu efi media xeemetrice fre^ 
fmiePiaiif inter CR, CCLdifidntidif ^i^eUnita Pp'i & 
sangentis PQ^ in eadem dia0Uf^e offkmttMS , fed ea 
•ad fortes effofiias facent , & tdHg^Hs > ac ordifidta 
Sametrum iffam cenjugatam fecant in cadetk itdti^^^ 
fed recifreca. 

4i<6. Si enim diaftietrQ ptimartsfs Hd conj^at;^ ip- 
idSus Vn occi^rat tangens PQ^in I , femiordifi&ta fua 
PE paraUcla ipfi Yh in E,^erunt &^ EG *qualc« 
RC, RP, eritqiie RQ,ad CQ., ut RP^ fii?* CE ad 
CI j aimimm ob CI9 CD> OB [^tim» pmportio* 

- - tiona- 



t E E H E M T.ai »$f 
cloBabs ( Qum. 411/9 ut quadratum .C& vdt V^J^wk 
quadrjitum CD. Quare dividcndo K£^.^ ^^^ A 
diffierenp^ q^adratorvim'' BP, Gp ^ quadramtn CD , 
£t auienit ^^^' ?5^< -^ 4a«ic«riiin <$ 9 11A CR ad^ 
ftftangdum DEi) fivt^ di^Et^matn ^uUm^nitiiCi^ 
CE» ^ CD, I^P , m <tua4cacqfn CV «i quadracQfil 
d9^ , Me0qqr iiltdmfliido ^^wdiranim OR «ud quadkl» 
nm cV in <admi iUa tatioai 4ifimM» tiuadciHi^ 
tam ftP> Cp ad «ttffhsatito CDt Igm^ tiklkCm, 
CQ^> ut ^uadrtttifm RC M quadnsipm C V > 
RC j CV » CQ^ Awrimd wic. pvopnriiamles < 

417. Jaoebit auitift*C[Q^«d panesr -is((M>ite9 €& % 
qata eam C^, CE jaaettot ad ^fiatetii pMries ( tmt^ 
411 ) ^tiigeris {<Q;i)rliB iacidat iit ^aimmim p a iw M» 
MmCE^ c|uam fii feci»Mtariain :^^v 9^ f ^u^ |t« 
Dtliit Q^t^tra cefitrDnii (efpnAii PiVt - 

ria CQ;, <lb CRy CV 9 Cf coockiqe poopanxaoaili 
ki, |^X> jeWaJ*m CV, qobuaor p«dla «, V^ft^ 
1/ CQQdituent prtip^t^nem hdrxndQicoBi < miou «4 t) 
aiiMpK ^rit VR «4 Ra > qt Ir^f ad jgi^ ;, ifiit «t IfQ^ 
«t(]^, «c <iia»ie«sr V» fttixt ttt iQ^ 41^ 
ikpcaiio&e ^ fcd nciproc^* 

^ mdia pnfmrtUmiii imtr fmim^matam diammH 
^m^afd j &^fikm figmncm intemttam ftm 
tmrtm f sc tMM^entm fir Mtremm fmm rHi im t 

4ao^ Si enim in fig. 15} , i^j^i&tVSf {«luirdiliarpj 
t» ad- 4iamea:um CD con|ugatain CV > crit acquaUa j^| 
CR , adcoque crit ( num. 411 ^ ipfe EP ad CV-* 
ur QV ad CQ^. jK vero in £.15^9 fttmdiametep CD 
. pcodocatur iitquc ad tangenieni^ Qf^ in H > cum ftc 
-.pariier C£ a^ftatis femiordinatae PR p erit CO 190^ 
dia^ intar CE » CH ( num; ^n )^ » adeoquc inisr 
PR , CH» In figura n^o iJ4 cam CVfitinediaiits. 
* tcr 



1^6 ^ECTlONUMCONieARtiW 
yiltx CR i CQ^^ erit mcclia inter - femibrdinat» &F 4 
JJk kgtasntam CQ^ 

C§rolL 84 
4^1; /n qH4t/h StSlidne Cintca tangemis dftS^tir 
txtrem» fnnSa cujufvh ardi»at€f coiunt in dliqk^Jliii^ 
80 eJHS diametti l cujfis ea ejt mrdinata ^ ac fi finres^ 
EUiffes annkmerato iis etiam drcHlo ^ vel fiUres. t^r 
ferhla cotnmunem hah^ant diametrHm. i taijiigentes 4^ 
8a fir extrema fnn^a otdinatarHm eandink al^fbijfaiii^^ 
biAentism convergent ad ide?h ^jufdem diametri fufk^ 
Shtm • Si JtHtem. Hyferbola commHnem cnm EUipfi , ^ 
pei circHlo hahat diametrnm frimariam > & hufns saK^ '-. 
gens cum. Ulias erdinatd congtHOt in iffd dumetre gj, 
wtcmret etiam hupis o¥dinata cnm UHhs ianierite,;, 

4i2» Tangictites per txttmtt ordidatx pun^a d^ 
ftas eoncurrere ia 'diainecro i demonftracum ^ft etian 
' flum. 216 > eaftgentcs EUipfium &. drculi» yel Hjrpef^ 
bolarum commmiem faabencium diameituin» & lbfid& 
iam , concqnere in e«dent diatmecri pun^ > dei^od»* 
ftracum eft nnm. ^8 • Idem hic pateti quia tii Para« 
^bola; diftatuia concurfus cum diamecro' uttiufqite tan-« 
^ gcncis i\xdut f)cr binra cxcrema punda otdinatflb avtr-' 
',< tice ipilus diametri dcbei^ie efle asquatis eidem abfeiC 
fiCf ac in reliquts omnibus cafibus EUipfium» & Hy-» 
perbolarum exii^ente abfcifTa a centro ^ 6c femidia* 
xhetro' comrhuifi» debebic efle communis etiram diftaa* 
da concurfus tangentis cum di^^tra ab ifks centro'» 
^'^yrSc ad eandem parcem |accre / P^o circuio. aut^ eft 
itidem manifeftumi qtfia ii in fij. 157. PQ^fit taa»- 
g^ns^ PR femiordinata circuli \ in triangulis re&an-^ 
gttUs fimilibusCRPy CPCtcritOR id O?, ilt GP ad 
CQ;.> adeoquie CPj fi^ CV uiedia ietdem intec ab«' 
' ^ * fdffam. CR> & dittantiam GQ^^ tangcnic. \ 

423. Quod & fucrtt yPu vel citculus , vel EHipfis ' 
Vjp Hyperbola eadtem diametro primaria V» » &1 
•RP £?miordinaca prioris > ac R/ pofterioris tangeQ& ' 
pernneat ad idem diaraetri pun^uin R y etiafmv pcio^ ^ 
(is tangcns du^ per P> ac poftctioris fcmiordinaia 

. pcr 



E L E M E n T* A. t^f 

fcr f debcQt convcrgere td idem punftum CX. diar 
metri 9 cutn pro utraquj^ ^ebeat cBt iUa CQ^Derd^. - 
poft CR , CV, 

i 4*4, TVwjfwr/ Aa, vel Bb in fy. 15; ^ 154 , ^ IMS^j^^ 
Llcr dUfnetri vertkem V ^ vel u duHa , & termwat^ 1^4^ 
T^ ungentes VQ^, fO^ebtSas fer extrem^ ftotOa ^f- 155 
ditutit Pp in iffe verticp fecatur biferi^m , & iina . 
rcS« Ba » hA jungentes in^ EUiffi j & Hyfertels ^m^ 
lnUt effajhos qkddrilinei AabB earum qiuUuer tdtte 
gemim , tranfeunt fer cpncmfupt K ardin^t^ cufi$ 
dimttro. 

425. Patet priinuin ( num* 204 ) , cum Vf feC6-«.., ^ 
tor bifariam in R , & reA^ PQ^» R(X jfQ^ pcr idem 
punftum CX tranfeant ^ Secyndum fic dpmoi)ib:atur ^ . 
Cum fit Va aequalis VA , crit ipfa ad Bu i ut VA 
ad i^atti Bu , five ut VQt^ad Qm » vel ut VR , ad . \ 
9iK : adcoque ob anguloi RV^ , RnB in parallelis / 
sqaales, fimilia triangula RV4 , RiA , & angi£ ad 
K sqoales , ac proinde re&a ^R produfta ex p^'* 
tc R in fig. 153 , ex parte s in fig. i;4 ccm£cuet 
oan RB, 

S C H O L I U M III, . 

4^« T TiEc qiiidem profluxerunt tx illa prinui. 
JLjL proprietate proporttonis harmoaicx indi-^ . 

C2tz num. 410, & propofita numr aa; tmnc pir^gr^ 
diar ad alteram ibidem indicatam > & propofium Ht . 
26 pihUo minus foecundam. 

QoretL lo. 
■ 4*7- ^» ^fiipfi j & HjfcrMd ift fii. I53 , IJ4 
^tit geometrid frefertionMes tum qustuer difiantia 
Qy y QJK. , Qf Qfi coneurfus Q^tanientis cum eli^ 
nefro a vertife V ^ ab occurffi ordinata R r ^ centrp 
C 9 & ak altero vertice u ; tum quatuor KXX? ^^t 
ttu 9 RC , occurfus ordinata R ab. occurfu . tangentit 
O^y a vertice V » tdf altero verticf n % & a cen^ 
fra C. 428 



< ♦ 



J4*4£CtlONUlW GONldARtfSl. 
ifiS. Eft pn^tidiccas ( itttmor. 2^ ) pfopbrfiodij - 
Kanmon.ii^ qtoslior putidorum ii> V > R> C^^ qiio-'' 
tum altei^da V y ii i 8c eorum diftantia diVidiftiC/ Uh 
fariam iii C ) adeoque Q^5i R runt feliqii^ bini in 
^fi^ioa«inDfl a^nmpcal r^ 8c Q^cft extremiini itl 6^ i 
.ijti iC r-54. CareiL lU \ 

l . 42^ «fii tdf^iihfi^ JtiEhffer eMtfcfnkm oidinats pm- . 
9im P i2i /^ 15S y 15^ «rr«mi» tsg^entihks dnSit . 
fit ^filiiiees diametri V , u , i/i A ^ C^ a^ c^ fetmr 
dUmm^f miuidts CD /» H^ ^rM;»<^ fiMiRP^ CHmiu ^; 
^ • .JUt^f iidttmttwnthiM^ft^ferti^i^aiBSy qtMmfemidiamettr , 
^•*5'CD midia CQntinni ffoferiioHalis inter hinas iftmgcn^} 
*^^ ti^MA^ tiai 

43«. Naoi VA> JLP. C!H ^ 4if efunt ac( fe iavi^ 
«efd^ tic Qy» QR » Qp » Q» > qae ( num. 427 )- 
fttfR* tft geoiiitarukl propomone^ quatnobfeni hitet V^ > 
^ ^mt ioMAst RPi CH$ ^d^ue efic cfiiani taoiU . 
CD^/ qu^ ( oum^ 41^)^ ^^ media in tpfa^ P&^ HQ 
feiHiotdinJlrcat» Bimihimi^&fe|tnedmm dianMtci^ 6]»&- . 
pstitit imeocepditn cennro C , ac cangence PQ; ^ 

431. XeQdHgfilM , ^M conHnentur fib hinis Ungeri^**^^ 
tibus pmrallelis VA > ua interceptis inter centaHus i '^ 
^ quamois didm tdttsehum QP, ac fiih binis huJMS 
fegmentis PA , Pa interceptis inier illas , & contM-' 
{hm^ ^^iHdf^r qmdshMis fimidiametretum fmrdMitnm \ 
iir iffis Mnietmhu akirum aliihri i 

4^i'. Ciun enirm: (natVi 42^1 ) CD fic media mcer "« 
tatifeHM^AV yl^fi^-) ^t ejus quadratum aeqttale re^ 'i 
Aangulo fub iifdem . Qijraui &^ O fit femidiaiiieiec \ 
paf2&U cangenci QP» eric (11. 315) tam AV ad AP^ ' 
qu^nffi^ a4'ifPV ar«C1>adQ; adeoque rcd^angidum - j 
fub AVi au M re<ftaRgulum APity tic qijadranlm CD '^ 
ad quadrtfmtH CK Com igimr re<9:angulam fub AVyJ 
^ s^Udcmf quadivto CD , ecianf re^anigulamt /tfW^ 
(qvabitur ^OadMoi €1 / Gor^lLi^, i 

43;. RehangtUmm (^H fult fegmantis tangentis r#- ' 
Juf^is inHTceftif inter^J^ntalhmt > (t*- A^^^x qmulihet - 



CR. ad Rmi ccic, oi^ta ^ ajt PQ., «s HJ^ «1 P<( : 

« 

jril./^^ F^oduunt^^ 4 coafiderowi' ptoitBnea pei;«, 
^adicjuliim 4l#iW c c^tfHiK) in caage&temv i«i e piut* 
do cpoui^U9 4^. xiQgeiiteiij^ ipfam tifijus. 44 axem i^ 
trumvij;^ >. 4i|||in)m ad-^prUt^ies psrpefl^iici^ vLvm-y 
^eiiteu} p &^ Aqrfnaliu$n ^emitiaiaram ad axesitpfi»^ 
iibi cun> diaa(\erris ;^ ^xi6uf comparantur^ qoar &ele-i 
|antc; iijtit 191)!, f^ici & 6mm i«pc ulbs isi ii£b:oacr^> 

4^i4 At.iASfea 10; bini^. SchoUis ad alia ^pMOixA 
hil^ldjminus utilia, digre/liemar • In pcimis libcan-^' 
^mTlluc{;a.Pp9 hu|us i^temi CoroUarii M^midm^ 
f^fU^^iffinki. 4^.es d4tU Unii didmetrU Cfnfifdtu ^» 
Si enimiiloc iflb 6g. i4<i ^ i^t diametri Gdn)ugac0|*i^; 
Kii lCii & duftA pcr P refika Jridcfiairt HQ^pa-?^ 
rajlel^ ti^ qu^ qimirum debec effe £llip(e6f % Sc %«* 
p^i^koht, caageos » ati fumpta.PS aeqiiali dimidio i^ce* 
rl t^^ diacnficri I^ in EUipfi io Sg. i66M Gl^.pro- 
du4a ^ iit Hxperboia io % i6i veiiiis d Maqiae 
kdmwi CS in T » agaxur TiG p^rpeodicularis ipO. 
CS 1 donec occurrat HQ^ in G ^ ac ^eiuo G imenraU 
laf GC I GS 9 qux. interyaUa pacec fiKe ^nalia « in-» 
tenitociii' iil. ipfa. caagetite punda Q^ H j re^ CV, 
CH^ decermiiia|)iint pQfidoHes mmi 9 8c fumpta CV 
topdia^ gecfn^trice ptopoTcionaU iacer CQ^> CR > fic 
CD incer C^^ CH« cum fompcis Cu^ Cd ipfis ae< 
qo^ibu^ ad parces oppoficas» habcbuntur axes VM^Dd^ 

4^7« Cum enim circulus cranlfre debeac per pun«' 

^ C9 S| Q,vU> ccit r^dangulum HPCXcqu^^^^ 

ccanr* 



*t4o SECTI^NUM CONICARUM ' 

jtlAt>gulo CPS y adeoque qoadract) Clinediar nitntri 

incer CP9 & dimidium lacus redum ; ahgdm HV< 

43ectu$ jeri t » ut oporcebac > in axibus j & CD 9 Cl 

«BTunc mcdiac inter CE, CH> & CR , CQ,, qoqp 

mlrum faabcri 4ebebanc ia ejufmodi EUipfi j vel H] 

bola , ex^ftenie HPQ^tange&fe parallela diiametri 

conjugacc Pp . ' ^ i 

438. fric aiicem in fig. ifo axis cranfvtrfus is i'^ 

qui ev^et longior , in fig. 161 is , cujus occurfitfl; 

cu^ cahgente uc Q^eft propior contaftui P. InveottMI 

axibps facile ( num.' 124^ & 155) inveriinnmr fodfi 

& dacis focis ^ ac axe tranfyerfo invetmisr ( nu. 90J! ) 

4ice&rix, atquc adeo Conjca StSkio ex dciintcione iff 

qua ab inicio uQ fumus » Por«o defi:ripca iis axibor 

^ Siectione Conica , ca iiecefl^io tranfibic per punAuflj' 

P , & habebic Pf , U pro dtainearis confuga^i^ . Eiil^ 

cnim quadracum CV iivc re^angulum fub CR» CC 

ad re^angulum VRi^, (ivc dificrcntiam quadrati d 

z quadraca CV , fivc a rectangulo fub CR :« A: CQj 

nimirum rcecangulum fub RC & RCX > *i^ CQ;^ 

RQL» fivc GH ad RP, v^l ad CE, nilnirum ( imt 

^ii ) uc quadratum CD ak quadracum C% , five af 

i^ quadratum RP , adeoquc alcernando quadi^acum Cw. 

ad quadracum CD , uc rc6)sangulum VRn ad quadla(r' 

tum RPj ac proindc ipfa RP eric femtordinaca » M\ 

perimeter tranfibit per P , cufus cangeiis tiit ( txm^, 

411 ) HTQ^ob CR , CV , CQ^ continuc ptopdrt^ 

nales, & Cr, C^ icmidiam^tec conjugata , oum CM| 

gcnti parallda fi^ , & ejus quadracum -^ucciir ttJU^ 

gulo HPQl,» juxta n. 435t » *€ 

439* Hiac indc iUud cooftqnitur ; Ji w ^Hodm 

fisurs rcS^ Bb in dato ^ulo' ifKiifuits ^d doidm 

fi6t^^ Vu/w/?/«r bifmam ah iffa inK^ vel imeri 

jj$5 fiFii. V , u , «r in fig. 162 , vol extta m in fii. %i 

ac fint quadrata BK , ut reilangHla VRu- , fivo , ji 

e(tdem redit , quadratum KB ad reBangulum VKu 

data ratione. ; oa^ fiiura erit ElUfffis^ ia frimo 

Hyforhola in fecHndo . ^Otz cnim bifari^ Vi^ in 

& du- 



E L E M E N T A. * 141 
^ per C r^cGi IGMd codem illo aAguld ira , 
t]ua(irt;a CI» Gi ad qc»cir9tuin Ot fm iii illaca-* 
ratioqpi; a^ cpnltrvc^is^ El|.i|>(i Sc Hyperboia , que , 
TitipC9i y»» li pro cli^^m<U?i$, copj^gatis, ejus EI- 
k vd Hj^rbols fcmtordinata quaeyis peniocns 
^m^tp^ K dcbci>it oongrtKfrc cMcntCJI » tel K^ » 
(^b^i^t cfle paralkla li ( n. ti i)&c 4ck^Q,i (tu^$t) 
q^itum id wcctmg^um VK» tfft iii ^adem ilr 
ir^tBooe quadrad VC ^, qi^aflratum CI > adi^pque 
^ jigar; puQCta omiiita '(^opgr^ent cum punct^s cjufr 
?»iElIipfeo$, Ycl 'Hypcrfeolae ; 

«^«iBm erit , Coao non pcr. vcrriccm fcao , obv6* 
Meaiai^te tribas Conids fectioni^us initio dcflni* 
i®i<tj)^Didc lubcre 'omncs pjropti^tates ^ quas; e* 
lw^<fc|#tiQnc dediixtrnus • Scdprap^^ea ^cnduln > 

f^^^ ^hiffw VK^s^fihice didmetri PK , ci^am 
Pj^ffl^^t^my cu]us fMratfkfer tertia c^miftue jropor^ 
P^teji fHMmds eh/cijfam ^ ^.fmfn fimiordina- 
^•BK • Nijun in t% curva prod^fctiim fub par^c-^ 
2^» & q(|ai|rt9 aUa abfciflOs^, erit sequfiie quadrato fuafc 
mot^aiis;, cav^ ^oc aliud ^adratam id illud prius 
9*^ cilc , at ' rectai|igulum fub fua labfiriffa, & iUi 
*|J^.«1 lecttin^tim fab abfciffa pciore ,;^ tjcctaea- 
vAtDtua ao^^i ttiamcoro PK » ^ dirj^CtiojiQ ordi** 
lW?p B^» ac njagnitudinc unius exiis, vel paraJ. 
1^0» lertia po{( /abfcid&m PK 9 & fcmiordinatam 
F» (imrminatur Cocus , ^ dkccjprix P^s^bolsc , quir 
p *ids, «latBr Paraboja ipfa, quam dcberi^ Congruo 
poim (^modi curva, facllc demonftrattio^ ,. 
Mil^i;;^^^^ ]iiBtt|ram. diamctro KP > doncc fit 
M qi^^ paramctri^ptacs^ recta AM ipfi PM perp<ndi- 
ttis cm cfircctrix » Ducta verojpOLp^^^^^ ^^^^* 
» j & fecto angolo QPF asquoU QPM ^ afc recta 
aequaU P^, trit I focus * Si^^nim foco F , di- 
*tL AM defgribatiir. Parabola 9 ccit ( a«m l) P 
Bofiwch.remIlL L /ad 



l4i SfiCTIONUM CONICARUii 
ftd ipfam ob PF a^qualein PM : erit PQ, tang^as ( hu: 
i8i ) ob anguluth MPF feccum bifariam a PQ.; erit 
PK diameter ( 0um. %66 > & ejas parametec ( nu; 
^51 ) quadrupk PM v Quisure ejus ordihata cdngruec 
CLim B^ 8c direetione > & magnitudine .» cum dtbeac 
cflfe paraUela tangcnti P li flr femidrdihat? quadra^ 
tum asquale ( num; 351 ) rfeccangulo fiib l^, & pa^ 
ramecro) adeoque ^uale quadraco KB ^ vd K^; 

SCHOLIUM Vi 

44i. T7 Odciti picto plurima alia ProbIemat4 ti 
Xl/ demonftratis Theoremaris folvi fsicile pof-^ 
fuc^t) in quibus vel fe quifque ^ Vel Tyronem Pr^ct*^ 
pfbr exercere poterit . NonnuUa hic innuami exqui- 
bus conftent » datis 5 punElis detm^nniri S^Hi^nefk 
Gonicafni ^ frdinde hinas Seciiones nen pbjfe ocvur^ 
rere Jibi muiuo 3 vel circfdd' > qui inter Elliffit enu^ 
merMri potefi , in plurihus , quam in quMtuor funSHs • 
443. In primis ddtis hiriis chordis parMllelti i pAtet^ 
dsri direElionem nnius diametri ) fectis tiimirum ipfi^ 
chordis bifariam ^ &: pcr fbctlonum puncta ducca rc* 
cta indefinitt ; Hinc auttfm dato krcu SeSionis Conici 
facile potiffi inveniri ejus centrum . Si nimirum ducan- 
lur bina paria chordarum parallelarum 9 quarutn fin^ 
gula determinabunt fiiarum diametrorum direcrionetn^ 
quae proiiide diametri , fi concurraac i determinabtint 
cehtrum ipfius SecriQhis i quce fi diametri evaferint 
parailel£ ertt Parabola , cenrro in ea in infinicnm 
sibeunte f & ubi eae convergunc ; ac centrutn d<h 
cerrhinanc , atcus ilk Ellipfim j vel Hypcrbolam pcr<t 
tinebity ^ num» 83 ) prouc ipfum ccntrum facrit pro- 
pius longioti e bicxis chordis parallelis i vel brevioril 
qux fi forte ^quales evaferint > fatis erit aliam chor^^i 
dam ducere aliquanclo pfiorem centro , quam fit at> 
istura e ductis i & videtc , an chorda ipfa priore loa* 
Sior evaferic an brevior. Quanquam ideCD patebic e»; 

iam 



j £ i E M E N t A. )4f 

i^ar (nu.2i8)) Videndo ^ 'm arctis centro cavhatenl/ 

444« i)4ir£i: ^i^if ch§rdk f^aUelii; in^qjluUittt 4 

mm difliintibfis > ddt^Me inaqudlituf ( num. %% >-» 

^ rriTrrtf 3 f^H^ inz^niri fojfnnt tin£ - dUmeiri fih^ 

micim coni0g4t£ , tdf y^ ciHtroin infinUtan abeHntt 

m^i y Se&iowtm Cp0fcam djebere ejfe ParaboUm « 

facilt invenietur unius iiMmefrivertex , & f^ameteti 

(jjfoius dofie cum ifja ordiiidtariim fofitione > d^ ftr ( m 

e^Tfiy 438, 441 ) SeSkio Conica. , 

.445. Siht in fig^ 165 i \66y i^ii binae Chbrd; Vf^ 

t>', & ccntrum C jacclt in pttma ad partcs inajo 

ris i iii rtliquis ^ parte^ minoris y ac ii i Atcr utram-f 

yie lacejret, res eflct pcprfiis eidefh » duthmodo in ilM 

efict majori propUu > in hac minbti ^ ut ilU Ellip^ 

fcos cafum referat, hacc Hxperbolas bitios cafus > in 

.qaarani priorc chordss dat; fint. prdinatae^ ad diame- 

(Tum, primariaih , in poft^rior^ adfeciindariain; Sectis 

bifariam ipds chordis in R. R* habebitut directiodia^ 

inetri Wu eas. habentis pro fuis prdinatis i ignotis.ad'^ 

#uc eJQS Verticibus^ & ducta per ccntrum C recca iii 

farallda , ca . exbibebit, pofitionem diamefti B^ ejttsi 

coa|agate, cujus paricer vertices B, ^ adhuc ignoran-^ 

tur . Ducra vcro per P recta parallela RR' , quxoc- 

dirrat P>' in I, B^ in H^ (i fiimatur iil ea HA ae-f 

onalisj & cotitratia HP , patejt CA foirc brdinaltam 

^amcnro B^ : & proinde A ad Sictiociom Conicam i^ 

iDcbcbit autem effe (n. %;^9 ) rectangulum datum P'Ip* 

. a^ rectaiigulvtm datum PIA m fig^ 0^65, ^66 > ut re^ 

.fcangulucn PR/9 liyc quadratum PR damm » ad fc- 

.^aangglurn VKu fivc ad dif&rentiam quadracorumCR; 

i^CV, & in fig. 167 ut reetangulum BH^ , livft diffe- 

jdxuA quadratorum CH ^ GB ad quadracum HP, Ii'^' 

^ye CR datum. Dabicur igitur licrobique ea quadra- 

pomxn difFerencia i & daca prastcrea CR in fig^ 

|f65» 1^6, ac CH in %. 167 , d^bitur ibi CV , & 

^Hy hJc Cff, & C^, . . . 

4^6. Conftru<^r9 auicm erlc hu}afiii6dt « Capta in 

i. * ' 1 ■ 4 fig 



H4 SECTIONUJdCONICARlTM 

f«r AI, IP inv^nUmr quarta poft !pfa$\, & PR , cul 
«rgiii^ a4 asgulos rc^s cum CR erlganir RQ^» 9^ 
feiitFO G imcfval|o G(V, Jnvcpiptitur punAa y^ « , 
£rit CQim quadratiiin pr|nue mcdiias ad ^uadratum &- 
mnd^9 fivc rc^angalum Pip' ^d r cftangulum 4IP, ut 
iluadr^tum KR ad qtiadratiim RQ^> quod proincfe de<- 
^if cffe cquifc diflSercJ|tiaB quadratorum GR, CVc- 
xilhitte CV mifjorc , &: crlt, qim fit diffcrcndaqua- 
dratori4m'GR, CQ,. 

44.7. In idg/ 16^ inven^a codem pft<9:o quarta i8at 
trigatur GC^cx G ipfi ^u^Ks^ & ad angdos red^os 
efdem^ QK , tuni Cfntroi (JX itttefvaHo CR. ' itivctiian^ 
lur vertifr< Y^ ^% 6c dcmoqftr^tip erit pademl Spd fi 
CQ^evtlbrit squalis (3R, punfta V, h ablbunt iiii C 
rraticrcef diamcter V», 8c Hjfperbok abibit in re^am 
Uoeam, ut numcr. tio. Erit Cftim co cafu P*F/ dif' 
fercntl^ quadratorum P'R\ R?, 'fiyc P^' , PR- ad rc- 
iftangulum PIA, fivc diffcrcntisjm qua^iratorum Hl , 
HP, vcl CR^, GR, wt quadratum PR ad qaadratun| 
CR ; adeoque a^ditis propor|iojtialibu$ ettam quadra- 
tum PR ^d quadrattitn GR , ut quadratutn P^RS ad 
guadwum CR -, adeoquc (i duccrentur CP, GP*, an- 
gulus ad C in triangulif R'CP, RCP cffet idem , 8c 
punda G, P, P^in dire^um* 
' 448. Qpod fi quarta illa proportionalis obvenerit 
inajor qutm (^R in fig* 166 ', centro Q^ interyaUo 
C^ non poterunt invcniri pun&a V^ » , & tum ca- 
i\is pcrtinebin aid fig, 167 , 8c V« noii crit diameier 
pritptria, ffd fccundaria. Nimirum fa(ftisut media \n-* 
tccr PI, lA' ad mcdiam iqtccP'!, y, itaHP, fivcGW 
ad.quartum, debebit obvcnirc retfla miiior, quarnCM) 
iivePR, curn nimirum rcfta major , quam CR haP 
buerit in prioFe (:afu j^id PK e^m rationem, quamnidi 
di^ inier PI, lA ad mcdiam inter Pl j W . Eredl 
igrtur CQ^ perpcndk\ilAri ad HC scquali quartac iii 
vcntsB, ccntro Q^, intervallQ CH, vcl Rf dcte^min» 
bunmr Ycrtices B, ^diametri primaria; conjugatar ifrCiii 

^' ■■ " "• V« ,'■ 



Va, cum dcbeat quadramm iUius c|iiart£ «quari ili& 
^end^ qoadratotiirh CH » CB i , 

44^^ laveiita^ aufieiti di^ctrd ftm^iz Vh in ^g^ 
165, it6 ; 6c Bi; in figi 167 i^dmdduiti facile invt^ 
tmi. diacnetei' efui Coift|u^ata ^ Iii iUis cilihi JuthttH^ c«- 
lit CB adCVi ut PljL ad txxdiiRl iiitet Vl^ } Kii i ih 
lac CV a4 CS i ut HP ad iti^diim intet BH itU ; cum 
iumiiJiitii (^ni 351 ). qtiiitattiiii femidiamttri <f#t1fjugatte 
fit id qtiadiratutii fehiidiaiiktri ptiiti^ria!^ ul quadrftcam 
leqH<R*diiiat^ ad i^dao^dtoi Atb abTcfffts; 
, 456;. Iti p^abbla iutem ih % iiS dikla PI {>ai:a[l-Fa6^ 
fcli RR*i fi iapiaiiii? media. inter Pl i Ij/ jtuhi W- 
(ii fQ& ipCixti i ScPK i trii illi media ad foanc tei^ 
naoi ^ uc PI ad R V (umeaddiri. iii diita^fii <^m a&* 
adpatieid drdltiati^ btaoris i Erii enim PI atf RV # 
iit qoidriltism iUi«s medif ^ five icdliriguliitfi P1/' ad 
^[aadratttm Pil j fed reds&dgtdiuti PRjf i^ debet eS: pcr 

wmiii. .. , ^ .. .. ^. . -, . ^ • • 

4f^ D^ aiitem biius diametris ddn|iigatiS| dcceii^ 
io determiiiatur EUipfis, vd Hyperbola ( num; 436 j 
iiijj Sc dato vertice^ ac direCtidne diiimetri^ dc taii 
4ttavis ordinata j adboqtie 8c idw!t i!^6 fertio pott ^b- 
fctflam,' ^ feim«tdinatani ^ ac drdinatatunl directibne 
^iatur Pai:abold ihiim. 44<5. 

4^2, ^^^ fi ^^^^ ii^ordd Adfi iiiudes fint , Pr^blt^ 

n^ trit indetermifidtud <i t^t impftffliiie^ Jfrout ^qualiteri 

W indqftdlieer -4 €4ntr$ difiiitHnt « In eo Cafu piui^ 

^buri 2 Cadkt hl P, & iffirdmpta aliii cdord^ pataUeli 

biiiis iUis aeqiialibd^ ihagnit^inis eujufcuinque i per 

Am» ic pcr afaoram e datii^ determmata Secftione Co« 

i ti dctiBbic habere pro cUorda fua Ulain etiam al- 

d e, binis arqiialibus ditis » .quaef fi a cenurd ^bqud 

difticerint ^oblema im[Jo(Itbile' erit i cum qiiacvfts 

io Cohica debeac habere orditiatas»^ qux Itisquali. 

r a ccfDtro diAaxit^ iiia^qualesl » &r pcninebit cafusad 

Ufpfiin deiin^ithi iii birias re&as pai^aU^Ias i ubi axic 

jlpc^Sato inaiiiefite i 6c equali ipfis chordis datis i 

tzjs if anfyerfiis concipiatur tMcrdcerc in iafinituoi 

L 3 ita 



;f4< SECTIONUM CONIGARUM 
Uz > U( t]us vert^ces jam nufquam finr , in quas 9c 
Parabola abibit , fi binac ejqs ordinatas squaks Cmt\ 
yerti^c V in fig* i^8 i^a ip infinitum tbeunee^ utniif^ 
quani jam fit , 
fjiid 45;. Demm jsm hx fig. 169 ^ 170 qnirt^e punlfM 
* ^79A> P^ P> B, P* 9 per qna oforteMi SeHionem Gonicam, 
(^eterminare • Confunga^tur. bina quaevis paria pun£to-> 
rum rectis 4 at BA, Q^ > quae fi fuetint plarallela?, jam 
definient unius diamctri pofiticne ( nunl. 443 / , fi 
non fuerint parallelas concurrent alicubi in Q^. DuAa 
per qqintum punctnm P'* recta alteri ex ii( , ut P^ pa~ 
ralleM qccurr^nis alteri , fi opus eft productac i n I 9 
i[iat ut (inec^ia i^ter AQ^» QJB ad mediam inter Qp, 
Qj^ j ita i;nedia inter AI > Ifi ad quartum . Tnm ca^ 
piatur t^rti^ continue propprtionalis poft Pl & quai^ 
^um termiwm i^vcntum » cui in ipfa re6ia Pl prou 
duAa , fi «py$ cft, capiatur acquaMs I^^ ad partes P » 
vel ad oppofitasita, ut fi punctum Q^fiierit vcl fimtd 
in^ra u^amquc A6 > P/ > vel fimul extca utramque 
ctiam i veji (it fimul mtcr vitranique AB , V*f , ve| 
fimul ext^a ytramqpe , ft XjcrQ illud fuerit inter altr« 
j(:am , ^ cxtra utramquc > fi vero illud fuerit intra 
alteram ^ x his » &: exna alteram, critquectiam ^* ad 
candem Sectienem Conicam . Erit enim rectanguluta 
AQ§t ad rcct^gulunx PQ/ , ut rectangukim AIB atl 
rcctangulum Plp* : ac binqc chordat Vf , P>' parall^- 
lac dctermipabunt ynius diametri peficionem • Eodcttt 
modb coiijunctis ^ ,, PB » & ducta P'i parallelii Ap 
determinaLbitur /4 , 5c alterum par chordarmn parat-. 
Iclafum V*a ^ Ap ^ i^c per ipfa^ alteradianeter . Si bi* 
ixx diametd &^i:int paraUela? ^ 5ectio Conica erit Para- 
bola» 2c pcr biti^as chordas parallelas determinabiturjuiEr 
ta nq. ^50*: 4 con^urrant alicubi , determinabunt cen^ 
trum, aic pcr ipfum, &: binaschQrdas paralklas defiaie- 
tur EUipfis,/ vcl ^yperbpla juxta n. 444. Quod fi forte 
binic. ordinai»,' ut j?/, pycvafcrintacqualcs, & iequa** 
Bter acci^tro df^antcs , adearum,d^ametrum ex i;tro- 
yis i^clicjuorvun datoi: ua\ pijnctorurii A , B duaa. rccta 
" "'*' '" *""' ' ■ ' * • "* ■ ' ■'" '*"■ pa- 



E L E M E N r A; 147 

ipiniBela iii ufijne ad dkmetrum , dc producta tantum 

deoix fun iiabebitur alia chorda ia^qualiter a ctnto. 

0mSi & ^Uemaci ideternibiando pav% 

45f In flg. 169 punctum Q^crat extra uBPamqucK*^!? 

I^ A8, erat 1 intca AB » afTumen^a fuit Jp' ad par- 

«oppofitas rcfpectu IP*', ut- 1 Fctnancrct fimul| intra 

aninquc AB > P>' , ic cocfcm pacto quoniam f futt 

mz lucamque Af^ BP , & i incra A& > aflumpta cft 

W «d partcs* oppofuas ^* . Ai in iig. 170 crat Q^ in-F.i70 

tniF^, fed extra BA\ Qjiare cum I fucrit intraAfi, 

affiBncnda fuit Ip* ad partes IP\ uc 1 iaceret extr^P]^*. 

Eicam ^ fucrit intra PB^, fcd extra Af , i8c ^ cxtra 

B4 » affumenda f^t contra ia ad partcs oppo&t^ iP\ 

« » rcmancret intra aW ^ Id autem femper; neceffario 

kaUmbva pca^. oculis • Nam ubi ^itur de EUipii > & 

Picabola > fempec ooncurAis^ binarum chordaruin faa- 

Waairinter utramquc » vel extra utramquc^, prout kt 

fQBCtum jacuci^it im|:a SeCtioncm Conicam , vel t^ 

c^ • la Hypcrbola vero fi utraquc rccta vel fioiut in-. ^ 

'^tuc diuccmci in ^gulo majore' , quam fir ^ngu-i 

^ fqualitaus , vcl fimul in angulo n^inorc , utra-* 

^«c vd binos conjunget ramos oppofitos-, vel ejut 

^i lami puncta , & concurfus uirijifquc m prima 

«fii habcbitur inu:a utEamquc chordant , fi id pun- 

^ jacuerit iotcc utramque Famum , habcbitur ve-* 

^nta^j fi jacuctit intra utrumvis ramun^ i m fo- 

fWHjp vcro cafa habebitur intra utramque , % jficcar 

•»tt4 cum ramum , cxtta utramquc , fi^ cxtr-a cum ja* 

>ftt <, At fi altera indinctur. in angulo majore, altera 

C^ nuiQore.» illa conjungct utrumquc ramum, ^i, hxc 
tfdem umi biaa puncta , q«io Gafi:^ concorfus neccf^ 
h^o jaccbix fcmpei: intra altcram , & extra a^ccrani . 
r-Qpatc gcncralitcr faoc vcrum erit in Hnis^ faribus^ 
^danim , q^arum fri^rcs Hna foflerioribus hinis f$mt^ 
iff^^lfU , iieherA turji^nq^e^ c^ncHrfkm ^ w/ Jimul effd 
W^^ utramqttc » vel fimul extrA utramque , vd} Jimul 
^a alteram , & extra alterain ,. qui fi§firemus cafus 
k^ebitut in Hyferbeda , ubi ^l^erA chords debe^t 

l. 4 con^ 



14« SECTlCNUMCOMrdARt/^ 

ionjungen hin$s rMmos o^ofitos\ ^lters hindfw^iM 
den% ramit^ 

455) Infininsni efCct periequt bcnnei ca/bs) in ^l 
conitructio rcctas Itneas pro SeCtioatttus Comcis 
bebit; Ve^iim id gcneraliter liccbil etiath ante cob 
ctioneni *de{>reh€ficlcfe i ScCtio enim. Cohica nonnifi 
unam rtctam^ vcl duas abite potcft; Q^aiiihobcem 
faltcmtrid punct^nn direCtpm jaceanti tn rectds. 
incidetur , qti^ iS jacucrint ih direaiiiii ; rdtat h 
omnino habebiintur • Paritcr fl prd biois pimctis d 
tongcns cum ipfo cbntactuj, tts codbm roiibtl ; 
derato puncto dato prb dupHci^ mixpdnasi Pi jf 
rent, ic recta Qpj? abirct ih tangcntchi i ic iccirco 
d^tut tangcns cum coiitdctu i ic ttia puncta pi^setcrei^ 
vtl d^titUr bina» tahgentescumbinis coikactibijs>^ afiol 
punctum i todem. pdriter tes t^edicet : ied ifta » 6c aili 
cjufmpdi perfeqbi; ut libi ddntui^ tatlgemes ^ctxaLtti^ 
cthy infinimm edet^ (juoifuiil tidnnQUo^ cafo^ Ncvino^ 
nus elegantiflime fcdvit principtofum lib. t; . 

45^; Illiid ut)um fatis crit in^EiTc i quoc( fopra ibi 
A&imuSi SeSioftem Conicam altm StSUM ConUm ank 
fojjfi occurrere i nifi in fMtHor fMnSlis i Si cilita qaii»i 
qotf ptiiiAsi cbngruant i con|tait j^ tota Gooka ^b^ 
ctid^ cum totp Pdrrd fibihfinaerfectionescocatit tiafic^ 
rur cdfltactu^ > fl tertia iis accedat inbecnr odHtBcaMt 
acctior cxtr^' wrticei axium» qui, ut infri piateUi» 
id« quod piculum dicimus: ubi autem oitioe^ coft 
mni in tmicuhi puticaun 3 evadit ofculum odhuc ^aik 
ctius in $xiom verticibus. Sed hxC iiori Gmt hufmiA 
ci, 6c poft exchi^uni fhfiorem td folutiooeA PtobMl 
mamm peftidentiam ad< dcteffniiiacioYfehnf SeoiontBN 
Conicarum exqaibafdam datis^i regcedteoittr ad ^icril 
Corollariorum intermptam oumero 4^}; » per{e(|iiei|i 
tcs ea s quac pcrcinent ad normaiem > ac perpeiKliaftl 
lum e ceturo ia tangcn&em ac^eeta sdi^s 
iidcratis . 



C^ 




i 1 ig M £ M t i. t^ 

^f^^. figmentis Q|i ioMgintu ftr idem fHnHHm 
^ iStd r^ Hrmi9i4t€ di. Undi axts Va i Dd.ir 9K4Mi 
jlifrwrf? femiii4$mtri Cl fm^itid. iffi tmAgemi i ^ 
m^ni^td dumSttfi tr^nfiHfitie fer iiem ^nmm P. . 
4|g. S(iiu ciiim £mHisi ^a cci^guki recn^j^ 
«d^H « oim oB, iiigiuldm «d , Q^iii %. vfy 
nem cri^ngulis reccatigulis MPQ^, HCQ^> &iM 
% 171 «ilgulos ad verticem Q^ oppoficos - acqii^s fic 
ifAai MPQ^fimilis t4CQ| ac ob angtdiinai lai^ H c^m^ 
flumefli & eide» HCX^t; fiQ^ *^PH. Quare kic ME^ 
4d K2d atTH ad Piv^, & teccangiihim fub MP > « 
I^ 84b49 rcaxifisiilo fub HP > &; PQ^ j adeo^ il»^ 
itiri fauok 4331 ^ ^ii^racd CI^ 

, , , ,^ercU. iji . . 

^ 4.5^. XelUngjthm /ui perpendinin CL ^i»» # ^M^ 
«r tef^entem , ^cc mrnydi PM» tv^ ^ W aUthm e^ 
Hm Vu > z/iri Dd termindta ^uatur qHddrato femia» 
Hs 4ltirius CD i vet CV i & tini norf^ales interfi 
fittjlin ratione recif^oCU iHflkaifn dxkm i adqkn-ter'* 
pimHtr y ferfendictda i>ifo i cemro in eangentem i 
kt^i4m itttnmqne fmtS» eemaShte in rmiom jl^iffciiini/r^ 
mdit^titrimlik^m . . . , 

, 4^ tHi^i coim £:miordkatt t^, Pfi adaxes V^ 
\m tmm limilil criangula redcaa^ CI^ PI^M «b 
^a^ilos ^dC Sc P a patallcii^ conientos^^ateir, ^ 
[ftttiter llmtlia CLQ^> VnA> firk i^ GL «d CH > 
i« PK ad PM i adod^K v^imffkxvA kh KJL i U 
Mk ^qilale Msmigulo (^ CH,Si PR y iire ( mm. 
4h') ^aadraco feiiifaxis oonjugaii G£> ^ «c ptfttft 
4iL id CQ;^ uc PFad9ffi>i Mitd^i f «Miigdum ibb CL> 
A iiF aiifttate il^sii^ulo fnb €Q^ & 9£ : ^ {nU, 
\it9.) qtladtaco femiaxis CV. 

461« Hinc liufem ob CX «trique rectangulo oimr 
nanem> e»m &ti^9m^ y madctwt QD « CV > oir 



XSJ^ SECnpNUM CONICARUM . 
inagnitudincm vero conftantem rcctanguli fub CLySc 
fiaravis notmali , ipfum pcrpendicuhim CL augcbituf > 
ycl minuctur in cadem x^atipnc > ia qua contra minue*- 
lur, vcl augcbitur nonnaUs ipfa^ 

46^. Suifform^lis sd ^tfiiffdm d anm in utroqtie 
iMTf efi t nt quMdr^um alterius dxis^ ad quadrMtum 
iifiuA 9 & im Mxe tranfverfo ahftiffa efi ad dtfiantUm 
fccurfus nowmalis cum axe i^o a centro , u^ quadra- 
eum femiaxis tranfverfi ad quadraum difiantia fici m- 
iriuslibet a centro^ 

467. £ft enim in iifdem criangtdis tam fubtK^rmalis 
MR ad P£, iive kC, quam PR ad fubnormalcm £ir> 
Ut PM ad Vm.^ (ivc ( num. 4.59. ) ut quadi^adim fe- 
mi^xis CD ad quadratum femiazis CV , Hinq autem 
^rit CR ad CM diffcrcntiam in EKpfi , fummam in 
Hypcrbola ipfarum CR 3 RM ^ ut quadracum ftmiaxis 
franfverft ad quadratuin diftantif foci a ccntro , quo4 
( num, 64 ) io EUipfi arquatur difiiercntias ia Hypeir-' 
jbola bmm% qu^dratorum icmiaziumi^. 

(^oroU. 17^ 

?47J 4*4- S^ P^ verticem axh V in fig. 17J > 174 x 

174 ^75 \ '7^* ducantur re£la VO perfendicularisaxix & 

175 4equalis[dimidip lateri recto ijfiusaxis, tum CO inElli^* 
i76 in fig. 173, a€ in Uyferbola in fig. 174, 175 per cen^ 

trum y & in ParakoU in fig^ ij6 OI tardlela axi, 
^ccurrens mrdinat^ RP ^, D > erit RD aqualis fuhor^- 
nomalis KM. 

465.. Erit cnim £llipfi. ia Hypctbola RD ad abt 
jfciflam CR ^ iit dimidium latus rectum VO ad femiai^ « 
xcm CV> nijKiiirum u\ quadratum alterius axis adqua- 
dratwn ^xis V» y fivc ( num. 462 } > ut RM ad ip^ 
/am RC • In Parabola vero in fig^ 176 crit RD acqua». 
lis dirnidio latesi ii:ec(o VO« > ^eoquc ( liium* 2.00. ) 
fttbnormali RM. 

Ceroli. 18. 

4^6. Rectangtdum fub fernidiapetrp Ql conjut^ 



I L E M E> N T A. ^ ^it 

fMrMmetri CP in fig. lyi^ 172, & ferpendicuU y velFwft 
^ e vertice P diametri ejus con]ugat^ demijfo in ipfam, ite 
GL e centro C in pangentem fer P dHShtm aqHahm 
^ bfiule fuk femiaxiht/^^ (ir femidiametrty vel diametri 
^ funt in ratione recijx,oc^^ ejufmodi pejfendiculorumii, '^ 
\ 1^7. Eft ^nirn tam quadratum CL ad * rcctaogulum 
ilGL, & PM , quam rectangulum fiib CL , & Pm - 
B ftctangulutri fub PM ; & P^^ , ut CL id PM \ 65 
GLcommuoem i» utroque tertnino^ primx Fationrs, & 
f» iB utroquc fccundas . Qiiarc cum 8c nUm. 4.59^ 
rccaD^ulum fub CL, & PM iacquctur quacfrato fcTDta^^ 
»s CD, rcctangylutn ycro. fub CL > & , Pw quadrato^ 
imms CV' ; rectangulum ftib PM ^ 8c Pm ( num, 
457) qoadrato CI , crit quadratum CL ad quadratuni 
CD , ui qa^dratum €V ad; quadratom GI > sidcpquft 
GL Jjd CD, ut CV ad Ci, & rcctaguhlm fub CL, & 
a,vd fub Cl , & PO iquali ipfi CL in j^araHe; 
%ammoCLPO a?quafc rcctangulo fub fcmiaxibul 

4^^ Porro cum rcctaogulum fub co pcrpcndiculo ^' 
fc CI conftantcr scquetur cidcm rcctangtdo fub fcmiai 
fite, mutato ipfo pcrpcndiculo > iputabitur CI in rar^ 
?®c cjus reciproca, ' ' ^ ' 

SCH O t I UM VI. 

^, pGflfct hlc jara admodum ftcile communi dc^ 
*iu nioniftratione pro EIKpfi, & Hypai^bolaeruity 
*J* paraHcIogrammum circumfcriptuui EUipfi , vcl itii 
fcptum quanior ramis HyBcrboIarura conjugatarum i 
|N conrinetur rectis ducti? pct venices ahcrius c dia* 
^s conjugatis paraUelis aheri , acquari rectangulo fub 
^» ajdbus ^' quod pto Hypcrbola dcmonftravi: ntim. 
%> prd^ ElCpfi num. 575 . Nam parallclogrammum > 
|ubd potcft continerc feniidiamcter Clcum femidiamc^ 
^ GP in fuo angulo, cflct cjus pars quarta, 8c*^qua« 
^i^ectangulo fub bafi CI , &: altimdinc CL , nimr- 
^ rcciaogulo fub fcmiaxibus. Sed ad alia pergcnduny 
•^^^ntkim cmta . ' C&- 






^73. T TUC ufque perfequuti fumus pr^ipuas pro^ 
jrjL prietaces^ quc ex illa harmonica tangentis 
j^oporcione profluunt « confkkrando prius ejos unii!^ 
coniedirariai mnl introduceiido confideratioiiem ceaGTi.^ 
&: diametroruni coiljug^tarum ^ ac deinde tidrtiialesjMl: 
cuiivam) & j^rpendiculani e cenard ioi ungcncem«mnd 
tpaia focos inducemySf quotuoi relationem ad tangen^' 
lem vidimttS tium. i8x ^ cum nimirum radii foci $4 
<!fttlt^Aum du(5):t dcbeant cum ipiai tail^nce c^rititicrc 
atigulos xquales » adeoque & cum norniali ; & aliam' 
ibideni iiabuimns proporuonem harmoiiicam (nuni. i8^ 
ile&iitam a tan^nte> nortnalii &c bini^ focis* Q^o ta^ 
lacn plurcs affumuntur termiiii comparandi ^. co pliir«si 
cttami combmati«nes proveniunt > quibus animus defa-> 
tigttuTi atquc' dbruitur v Quamobrem multis omiQis i 
quas pcrlequi infiaittim ellec , j^rxdpuas caatummodaf 
ddlibabimus. 

fcW^. ao. 
.17^ 473^ Vunmti^r Mm in fii. 177, 17», 17^ ^fi mdid 
17S frtfanUfuUis intn c^rd^im Pp dtdlam ftr f^Cfm ^ & ^ 
179 ^em tr^virfumi^ 

474* 5i ^oinl ip(i Mm occurtat tangiens per P duA^ 
in A> & iemiordiaaca in O9 ac otdinflitanl Vg fua did^ 
sicca ipii PP paraU^it fecct ia J« crit (ounir 19^) CA 



i|i SECtlONUM CONfCARU^i. 

CartfUi 19» . 
4^0. Qnivis femididmittt efl ad mmalem du^dd 
jfgr verticem tjujt cpijtjidti i & ierminaiam 4d dte^ 
'dxenif Ht is fmUxiii vek djcis ad dlierum i f^ cmnu 
f^hidiimetri funti ut ejpifmodi hormales. ^ ; 

. 47X* £ft enim IC ad PM, vel Piii» ut rcft^guiuni 
jfiA lCi 8c Ct^ fS[ve,(num* 466j.fub CD i CV ^ad r©- 
ftanguliftm fub CL^ &: PM k vel Vm , nimirum ( tittiB. 
9} ad quidiratum CDi vel CV> adcoquc ut CV td ) 
3> vel ut CD ad CV: Cuiiique caratio fitcdaftattij 
Sii^untur eodem paCld ip&Cl, PMi 1?m; 

s c li 6 i I u M viL 



C t E M E N T A, ii^ 

\s frpiari tranfvcrfo CV ob faun| parallc|Hfmun| 

FP (iucca per fbcuiTi > Fl ycro dimidia P^ crit x^ 

CD. Cumigitur (imni. 411 , & 4x5 / fit CM 

intcr CA, CD , crit tota Mi» mcdit inicr Vm 

CA.ScP,, ' / 

\ Si % fy. 180 , j8( in pUpJl/j &■ IfmrM4f.iU 
% M rtirmalis rmninau Md axm trMjverfum \%X 
ifxe iffoy duc^tur ferpendiculutf^MT in feSam 
^f diflm ad punEbim V fmmetriy ^x qiw n$rmd% 
(brsfir 9 id in iffa ^k e^dem funSe abftindn fef^ 
^ PT diuke din4dio lateri recto vrincifdli « qwiA 
vfMpls l<fcum hater, ^ 

47^.Ducta cQim pcrCdlamctr© \i parallcla tangtti^, 
PQj ea a rccta PF abfeindct (num^, 194} fcgrocntURt 
' >^de fcmi^xi tranfvjcrfoj acln nojmali PM feg- 

fWflK) arquaicpcrpcndiculo^Cl- <?x cct^tro C da« 
i in tangcntcm PQ^, critquc ( num. <^5? ; Vcctangu- 
ifa(X^^^uaIe qua^rato fcmiaxis cohjugati • Erunc 
jfan fipilfa triangula rcctangula PTM, VQD , adco* 
critPD ad PO, ut iPM ad |TV & rcctan^dumi 
W*, & PD fcininc tran(vcrfo, aqualc rcctanguli 
PM, & PO, fivp quadrato fcmiaxi$ conji^ati , nl- 
^ PT prtia*ppXfcmiaxcm tranfvcrfum, & conju- 
) fivc arqualis dimidio latcri rccto principaU . In. 
h ytro in fig. iy6 ducta MT pcrpcndi^ulari adptiyf 
? «qwlia crunt mi^igul? rcctangula PTM , MRP > 
*Job latera*RP> FM «qualia (num.ioO^ fint iqua- 
«!fi£nli FPKf , FMP, & PM communis. Quarc crit 
Jj^aalis fufinormaH RM, fivc (num. 200} dimi(Iic( 
N recio principali , 

^ Cerell. 22. 

477* pimidium latus reHum frincifale ad nemalem 
^^^verfo efl^ ut ferfendicuium e centre in ta^gen^ 
^^fmiaxem ipfum tranfverfum. 
78. Eft cnim fig. 180, 181 PT ad PM,, pt PP r'F.i8d 
«s femiaxi tranfvcrfo ad PO a^qualcm pcrpcndicu- \%i 
CL, Potcrat ctiam dcduct cx num* 459, cx quo rc- 

^aR^ 



i 



i|4 scTioNUM cpNicARy\i 

tc^gultim fub i?Mj & CL acqaatur quadrato &miaxii 
conjugad» fiyc fn.71, vcl 351). rcctaiigulo fub diml- 
dio latccc rccto principali ,. & fctniax; aanfvcrfo. 
, . CorolL 2g. . , 

479* Difftrentia qHodratornm norin/dis ad axem tra/u 
fverjfnm terminata , & dimidii iateris reEli frincifalis 
mqfiotur in P^rahola ^Mdrato femiordinofa jffims axis S 
ifl in EUiffi 9 (^ Hyferbola dd iffum » ui quadratiim 
iflantia foiorum. ad quadraiHm axis trarifverji^ jfive ut 
ifferenpia ih Elli^]i\ fumfnain Hvferlfola quadratQrm 
femidxis tranfverfi^ & con]ug4ti adquadrMum femlaxis 
CQtnUgatiy five ut dijferentia in EUiffi, ifumma in, Hy- 
tferhold iotius y i>el dimidil laieris re£ii frincifali ^ & 
Sftius y vel dimidii a^is ianfverfi ad totum , vel dt 
inidium dxem irmfverfim j qua rationes omnes eadeni 

F.176 480. Patct in Parabola iti £g. 176, cani in triaqgu- 

177 (is jJli$ P*T^, I^RM scqualibus, ctiani MT dcbcat a^ 

178 ^uairi PR, ac ob an^ulum ad T rcctuni cjiis qiiadra^ 
tiitii .diifcrcntix quadratorum normalis PM« &: dimidii 
}atcri rc6ti PT ^ qiiod inimcdiatc patct in triangulo ccctao- 
gulo PRM, iii quQ PM liormalis, RM «qualis diiTxi- 
yiio latcri rccto, PR fcmiordinatap . Prb Ellipfi & Hy- 
pcrbola fic dcmonltratur in fig. 180, 181.. Ducta P/ad 
^tcrum focum* & fcmiordinata PR,fimilia.ci:unt triatt^ 

. gula rcctangida FMT , FPR ob angulum ad F ciDrama- 
icm. Quarc trit PR ad MT> iit FP ad FM, adcoque 
fetiam ( num.. 192) ut /P ^d /M , nimirum ut fumma 
in EUipfi, diffcrentia iii Hypcrbola ipfarum I^F, /P, &^ 
yi^ litrobiquc axis traofvcrAis ad riimmam .m EUxpQi 
diffcrcntiam in Hypcrbola rcctarum FWt , /M , fivc ff- 
crobique ad diftantiam /ocorum F/ . Adcoquc quadra» 

^tipi fcmiordiaatac PR ad quadratutn MT , flvc diflc- 
fiam quadratbrum normalis PM,.& dimidii lateris rc- 
iti PT , ut quadramm axis iranfvcrfi aci quadratum di- 
iiaAti; focorum , vcl fumcndo dimidiorum qiiiadra- 
(a j uc qcadratum fcmiaxis tranfverfi ad quaHratuixi 
^ftai^ii* foi^ a ccWQi mrairum (num. *4) ad* diffc- 

rcn- 



I 



. E t E M E N T A; i$| 

iiim in ElHpfi, fummam in Hypcrbdia qaadratorani 

axis tranfverii » dc conjagati i cumqi!lc fit (^nu; ^) 

atorum Tcmiaxis trMfvcrfi ad quadramm conjiiga^ 

i^ axis^ vel femiaxis uraiifvcrfus ad totum , vel di- 

lanis rccmm; eadcm illa ratio erit diiFcrentias 

, vd dimidii uis tranfverfi, 8c totius» rcl dirat« 

iaftris recti id Ellipfi , fummae in HTptrbola ad to- 

i vd dimidium axem tranfvcrfo « 

CorolL sJf. . . 

i^\J>iffi»vnU in fig. 182 in EUiffij tiAarum PF , 
it&anm a quovis fnn^ P ad biftos focos^ & /^^F.iSi 
[iiMtt/^. iS^ in Hyferkola ad CR wfciffam a cen^ \ff^ 
irsmMXf tranfverfo.tfi confianror^ ut difiantin ficofuM 
m ^ fmUxen^ tranfverfiim CV • 

4S2. Si cttim reaa Pf oceuitat in B ; Sc D recds 
TB, CD ductis e foco F i & ccniro G parailclis tan- 
fewtt QP, erit PD fnum. 194 >^ «qliaiis femiaxi tran^ 
*cr/byC, & ob-angulos PFB, PBF aequalcs iis , qui 
j^i in P cum tang^ntc ; adcoqut ( num; i^ i ) cqualcs 
iittrfcj crii Pfi «qualis PFi & /B in Ellipfi diffcircii- 
*f> in Hjppcrbola fumni» binaram P/ , PF ; qu« oB 
ifiiflam FCi fivc fC j crit dupia-/D . Erit auteni 
\pBm iila, vcl difierentia ad /F. diftamiam focorumii 
N/D ad /C > ut DP , fivc 9V ad CQ,» nimirum 
iNom 41Z) ut CR ad CV, ^ aiteraando/B a4 CR^ 
^/JadCV; 

► Ci7«f//. aj. 

483. SiOangulum fub.kinis relfis PF; Pf i)i /^^^4»F;iJ4. 
i dnSis s quovis fun&o P dA binos focos ^eqUJitUr qu^ \^^ 
*« fmidikmetri con]ug4U e]iis j qM tenSt ad P > *^* 
^tf/»^ ^z>ir nOrmMibus terminatis dd binos dxes^ 
n&amgtdo fkb fegmentis tkngentis interceflii inter 
^^SkSy ffr binos Mxes ; & iffius %reElangidi FPf, m 
"'mi ij^ta GP fmma in i,lUffis diff&entia in Hy^ 
io tquatur ibi fummk , hic differentue quadratorum 

im. 
4^- Condpiatui: ciiitri drculus circumfcriptum triatf- 
'(> Fiy, qui occurrac axi CQUJugato ia m , &c N t 

pofico 



j|f SECTIONUM CONICARUM, 
p9lko N ift «rcu FP/ ia fig. 184. itt dppofito iti ifa 
iS^, diicatut IPm occiirr^s ^iii traafv^rfo Viy m M^ 
^ NP fecatis a^rm ImY in Q^« ?f i^^ f^« Ob si 

cum F/* fe^t^m &ifariam> & ad angulo re^tos in Cj 
cKamctro Nm ; airctta FN/, im/ (ecaybantur (aif^i wn 'i| 
H ^ m . Qs^t lam r£^^ Vm in fig. 1S4. > qaam Hl 
lA ifig. 185 tccaf>it bif^iam angtiluhi FP/> cum ^mgdi 
infiftente ^qualibus arcubus Fntf jfm m Ag* 184 y Ff^ 
/N in 6g, 185 fqualcs cffe ^bcani ; recta .vcro P> 

crit ipfi Pf» perpendicularis ob angul^ l«F^ rccnxn 

in fcniicirc^lo. £rit igifur utrobiquc num. |8x ; Vn 

nocmalis / PN tangens • Angu^us autcm f^Pyeritfquj 

lil angulo M/P> cum iiucr(|ue infiftat cidcm ar^t FP 

adeoque ob angulos ^d iP j^iiale^ in triaagulis FPm 

MP/> erum: iigiilia ear triani^a, & FP nAPm^ \iv?h 

a. P/ ,/ ac f ectaQgulura FP/ ?qualc rectahgulp. MI% 

adeoqiie niim/4.57) tum quadcato. f^niicniLmetri coa 

|ngat; e}us, quf tendit ad P » tum rectafigulo NPQ^ 

Cunique fumtm in £llipfi num* 37$ » 248.) y diffeteo 

da ia Hyperbola quadratorum femi^iametrcMrum con/t' 

gatarum cquQti^r ibi: fummc y bic differaiti; quadcati 

9i)m fcmiaxium» ^quabitik ^idem ibi fuirima« hic di& 

5P«fctia rewnguli pP/, ^ 'quftjirati PC* 

43y. Rect^tdtm» FMf /«^ i^iiv// difi^ntiif canaa 
fns normalis cnm axe tranfverfo a Hnis facis aquatn 
in Elliffi differcntia , in Hyfmfola fi^mma qnadrati no\ 
mdis PM ^iffum t^inaUy & ^Mdratt femidiasm 
tri conjugata ejus, qua termjn^tum ad P» w r^ctangi 
FPf ^inarum d^^nm ^ kinos f^cos^ , fjr r^Uanguh 
FQ| fub (^is difiantiis cancurfus tangfntis a binis { 
m ^^iMfwr ^. Blliffi fumma % in Nyfifkol^ diferem 
^adrasi tangenti PQ^terminatOi ad axem ttarifvtrfm 
& quadrati eyufd^ illius femidiametri fon]i^at^ , ^ 
re£ianguli FPf ^ j 

486. Nam cx circuli natura re^tangulujn^. f Mf^ »n 
tur rectangolo nfMP , & tectangulum FQ./ ''^cinl 
lo. PvijNl* Porro rcctanguliim fub lAm, & MP, addf 

qua. 




E L E M E N T A: J57 

djtiadraco UP in % 184, dc aUato in f^. 1S5 1 cva« 
dit recrangulum fob. mPy Bc PM^ five ^uadramin iUiut 
" ainctri con)ugatx, vcl rectangulum FP/, Sc rc- 
uai fub PQ^, & C^ , ablaro in fig. liS^ qua- 
P(X& addltoin figj 185 » cvadit rcctanguluni 
\tSc PN^ fivc iliud idcm quadratura ferai^i^ 
conjugat(> vd rcct^gulum FP/, 

Hjtm$U in fig. 64. duc^ntur in tangtmem PT i^iiu 64 
firfeniicuU FA , fa farir»» zcShangiulHm ^tqudtiiur qn^ 
dtdi$ kmiaxis conyugati. 

48S. Erit cnim fnum. :i^2 } FA ad normakm IP» 
BC pctpendiculum CL e centro m tangentem ad jf4 ; ac 
proin^ rectanguKim fi^b FA > & /1 a:quabitur rectan<< 
gulo fub IP , & CL , fivc ( n. 459* > ^uadrato feniiV 
;«s conjugad. 

Corotf. 28^ 
489. Xfdius a4 finum anguli , qu^ r^Sta t fbco du^ 
84 W contaSum, continet cum tahgcnte , efi in Elliffi% 
& Hyferbola^ ut femidUmet^r faralleU tangenti adfe^ 
«ittw». conjugatum , C^ is angutus m JElliffi a reliqi 
waxime in if/ms axis conjugati verticitus dijtaty an^H^ 
U quem bina re£ia inde ad foatm duSla continent iM 
'fjdfiento maximo : tt^^ iJH^, 4iff^^f}^ ^ ^^^^ > ^^^ 
PfiMwr duflo hujus , eo magis minuitur > quo funQum 
'a^ujf ad verticem fjtpfioren^ axi^ tranft/erfi^ accedit 
is i in HyferhU is angulus eo ipagis rece^it a re* 
, & ille , quem ea Hn^ reSla continent 9 eo magi^ 
huiinr , q^. contaSus magis difiat. 4^ Wf^^., ^W 
tfverfi. 

4>p, Nam ob, angulos PPA , fVa mrobique «qualct 

narn..i8i ) eft FP adFA, ut fP ad j^ in cadcni ira- 

M«, ac proindc qu^dfatum FP ad quadratum FA , 

"rcctaoguham FP/, fivc !' ^,4^3 ) quadratum femidia^ 

;trr parallcte tangenu PT\ad'rcftartgulurafubFA , &j 

0,(1 ve C n* ^^7) qnadratoa^ femiaxis*^^ conjugati ; ac 

Rofc^ch^TomJfJ^, jf4 pip- 



1.5^ stttioNtiMCoUitAkijU 

P)^.oitidtfP ad jFA^ fivc cadius ^d Xiniim Mff^ fPAi 
^^ lilz i^ fcmiditfincQer act cvun rccmaxeiti; 

49 ^i Quaitiobrcni l^ iiniis cb trii mmor) & M^gi^ 
proUidc ^o magis rcccdct a redo ', qub ic^ ^jmidi^Pif 
icr hiajdt cric. Poirro cii fcrtiidiahi^tcr in ElJipfi ip» cft 
jnajoir^ qiid cjus, iconjiagaca. CP cft mlnor , ^tn fumuoA 
quadritoruiii uoriufque iit (nuin; 375) conllaptcr ae^uf- 
lis fummsc quadratbrum {emiaxiuitn > & CP co c{l mi- 
nor (num^ ^{79 i qu6 inigis P, aicccdit ad vefticcs a- 
Ijcis conjugati , & i^eccdit. a.vcrtic^ ptopioijj^ axis tran- 
iVeir/i . Quafe ^ngului FPA ieo magis reccdit a tcdoi 
qub tiiagis P aCCcdtt ad verticem ii^is coh jugaci i ubi 
itiaicime a tciko recediti Cumquc t}\xi diifcremig ^ re- 
cio API fit angulusf FPI^ Sc FP/Ct diiplus ipfiiis FPI; 
ipfe angiilus FP/ ju^it mixiinus piiiiikd P coogruentip 
cxim femiaxis cdnJLigati vcrcicc^ &: eo major ^rit> qud 
magis P.ad eUin yercicem.acQsdfCj ^ recedcc ^ vertice; 
fibi propiofc axis crdnfverfi; , ^^ .^ 

4.91^ At iti Hyperbbla in fig. 64 cuni diameter CP 
Iji rceefiii a ycrtiCc ixis con|ugaci pi:i^ctucr q^Ccat (liiim. 
246 i £<: diffefeiicia quadratorum femidiamecroriim cpn- 
jugdcarutti fit confiititef cidemy eciam (cinidii^iDctcr Con*^ 
jugata pcrpttup aug^bicur , adedque pefpetuo feccdec ^ 
re^p^ ainguliis FPA^ SC mitiaecur cam Ipfc > qfx%0 jpP/ 
fjtis duplusv 

^CH OLIUM Vill 

-4^?- T>Oftrcma h^ CJoroUari? , qi«r ad focUn» per- 
i cinentf licec noa profluxeriai: in^mediacp $|y 
|p& projpbficione hac 8, cainpn proffiiirefeni: ^ C<^ollar 
riis et ea dediidis combinaus cunif iis, qux antc^ ^tr 
fSLn% ejrut«> ^uam ob caufan? hinc divell^nida non fue-* 
raat« Poftremiam hoc cfecermin^t atfguliy quem fod.jl!si- 
dius cutn cangentc continetj nlagnitudincin f ac iocriew 
'm<?nia>^ dccrcmenta pro' EUipfi, & tfyperbola . Fr0 
I^arabola idcm deduci fkcile potcft t npm; }s8. Eft ni-' 
mirvm j;adias ad finum ^agiii^ FPA in i^g. i^ i uc FP 

ad 



_^ E L E M- E N t A. fj^ 

•^ FA, fivc ob FP, FA, FM eontinuc prjoportiohalci I 

& FP acijualcm, I^Jiuni., J5 i ) ijuartag^ parti latctts irc^fti 

f< ptmtMihtil; ad Uiamctrtim traiircuiitcnii pcrP; ^rif rddiiis 

• ttd euni finum in iriitionc Albduplic^ti diftaAtiar con- 

Uftos i^. foco iid quartani |>ar!etii^ lafcris rcdl. priri- 

Cipalisi Hve in, fubdtlpjiicata fatloti^ lacdtis tt(5U dia- 

ffletrt, dtt^af pcr, contai^um ad.Jattis rc<!luni pribcipa- 

ki ^ ijuoriiam ia rcccflu punSri P i verridc ax?S tran- 

frcrfi fempcr aiigctiir , ( iium. 58 ) diftaiitf d IrP \ fcnv 

pet an^ultti reCte FP cum tatigehte magis ^tccdct a 

r 494« Jatn yero progredia^ ad aliam prcsprietstcth Se- 
' &'ontttn Conicarum» cjUiSe ipfis nomj<: dcdit i & qun; 
ita paritci' a fexU Prdp<^^ipiu , pl^qfiuu ut fit nicruji 
j^ddil4^is cafus Tbcc^cmatts dcmoii^ (nuth«^i9)i 
V«nKQ,bic irerum dcmonftratur bpc Pcoj^: 7 i & ftcr- 
i^t iioKs viani ad dcflnieodos clrcutos dfculatbres St- 
^oaiim Cpciicaram ptr iShit;inl Cycbnictriatn ^ qui ni- 
ininmi iea ad arcunl Scftidnis Cohidx a<iccdaht> ur 
qoemadliicdum intet kredrii tirdiiti ^ 6c.i^€i:dhi tahgcn- 
tem auila alia rccta, duci poftlt l Woci iiiiiniei, numcro 
arcui pofBni diici ; ita inter ahiuni Sc6iioiiis 



- Conica^i ic afcum cjus cirduli dfcttlitorisi riullus alius 
aiadari^ arcus tranfoe poflfiti licet in iiiiicb puiidbo fe 
coatirigantl & infintti niimcro ardus Seclidhiiiri Coini- 
canuii pdffiht iiicerfeifii qua^ gccieralis <Hl propHtta^ prd 
drciiftis ofcutatoribut curvarum qiiatibncnt7K|ue : ^ ig-- 
grcdiaibur rcni ipfam; 



V 



PR0I*0Sltl6 IX. THfiORENiA. 



•^495. Q^t^ verticem V dlamctri cHJufvis in ElUpfi ^ 

^ in fig.1869 & Parahla iH .fi^. jS^> ac atjti fF.i8i 
"'m iiametri primaria HyperkaU infii.iSS dncatnrfa^' 1H7 
'pMVAa^^is lateji reSl^ ifjius; & fer ki^eBA tranfiens i88 
l^fer altirim verticeni u in EUiffii ac Hyfetbold i ac fa-- 
Vralkl^ aki ih Paraboia , qui ordinata PRp occurrat in 
\f Mf ^f^^fnm femiofdinata RP a/pndt reSlangulo 

• M i fb 



t6o SECTIONUM CONICARUM 
ftii Ahfcifa VR, & im€rc4fta RL imer diametrftm^ ac 
feSlam dtiBam fer A , qu^e intercefta erit qstarta frofar^ 
jionalis foji iatux tr^njverfam , re6bm , & abfcifam sA 
altero vertice^ cui latus reBum mn afflicatur. Idem ve^ 
ro quadratumy & re^angulum in Parabola ^equabitur rr- 
Ssngulo fuh illa abfiijfa VR, & latere rech; in Ellifji 
ab eodem defieiet\ in eo HjferboU ramo , cui iatus rr-. 
Rum efl afflicatupty exetdet iffum , fer rellanguhim fuk 
iffa abfciffa^ ^ quOrta frofortionali fofi latus tranfz/er-, 
fum^ r^um-y & iffam abfcijfam. 

^96. Ell cnim (xixm. 351) quadramra PR in Para* 
boia m fig. 187 acquak rectangula fub abfcifla VR , & 
laterc rccto VA, adeoque fub VR , & RL . A^ in El- 
Upfi, ac Hyperbola cft ipAimquadratuai PR ad rectaiiT 
gulum VKu , ut lams rcctum AV ad tranfv^rfum V«, 
£ve ut LR ad R«, vrl affumpta VR coramuni, uc re*. 
ctangulum fub VR, & RL ad idem rectangulumVRiy. 
Quare quadr^mm ipfum RP asqoale erit reccangolo IvSx 
VR, & RL, 

497. Pafit aatcm in Par^boh RL afquari latcri re- 
cto VA, i'n EUipfi eflcminorcm^fc^VA» ia Hypcrbo- 
la majotem; & fi in his ducatur VO ufque ad P^p pa- 
iralleh AL, cui & aequalis crit, & abfcindet CL acqua^* 
lem lateri rccto VA, crit Vh ad VA, ut VR ad RO, 
ac promde ipfa RO quarta poft latus tranfvcrfum Vu^ 
rcctum VA> & abfciffam VRi^ ac rcctangulum fubVR,^ 
& RL a rcctanguJo fub . VR , & OL , vel VA dcficic^ 
^i Eiljpfi , ipfum excedet in Hjpccbola pcr rectanguloizi 
fub Y|i> «f OR. Q^.^. D, 

S G H O L I U M I. 

■ 

498. /^ UM qu^atum ftmior^inatas rcctangulo ilH 
V^ fub abfciffa^ & latcre recto acquctur in Pa^ 
rabola, deficiat ab coinEUipfi, redundet inHy:pcrbola9 
hincParaboI^, ^Uipfi, Hy^cibokr nomen datum a Vct 
^ribus, qaod Grseco fcmdnc '«qualitatem , dcfecmm > 
& red^ndandani ^ifi^fu» exprimh^ ^c^ 4^ iioftra<der 



§ L E M £ N t A. m 

miuotiCj ut niiin; iz ^otavimus i babeatur ftatim ^-^ 
([(ualicas qu^daltx aliaj ddfeOuSi Si cxceffos rationis il- 
iiBS det^.rmiaaqtis ; .... 

49^;.Porro hic recta. AL dat^i idem prbrfus, t>t2ftac 
poEUjpfii & Hyftxbol^j qiiod nnfn.ji^; infig^ ii5,Fiitf 
ii( iUa BDy quae ibi etiam traniit per ui Jc.fi ipfum ii# 
V coBgruac ibi cum coiitacm I > & chorda Vm evadat 




quak r^tatigulo fub VL.^. & bL cyadci hic ipium qua- 
dnuu9i fcmiordinatx RP sequaJe fectangulo fub VR ; 

5oa TSed jam ex hac . Prop^fitione <;ompatata cum 
nqhu 464. ci:ua(n Corollairium non inutile » & fpontc 
fiiitns^ quod ad fuboprm^qati ^ j^ttinet, tuni ad olculsU 
torcs drcuias faciclTias gradi^m. 

• t I Caraih u. ... -^ ^ 

,301. Sui^narinalis in axi^ iranfper/i Jtfficit fer eUmU 
Hm Utttis nSi frincifalis iu.Parahola iiJf iffi iattri 
^tUo ftiwifaliy iH Elliffi & Hy^rbola squarta frofar- 
tUndi foft latus tranfverfum » reStnm i (^ aifciffam s 
vtnke axU r^moticre , jive ah illa reSa » csim ^M . 
Mnaet iohfciffa reSlanigulm muate iuaUrato femiordi^ 

;q2. Nam in fig. 17 j^ 1745 17^ .fl Caplatiit VA du-Payj 
^UVOj adeoque xqualis Ja;teri ftccd priucipali > tum ly^ 
rcaa ex A parallela ai^i VR in Paraboiajn fig. 17^ , i^l" 
feddens ad u \n t^Iiquis, &occcirrc;ns ordinatas PR iti 
l* firf>normalis RMj 4^?^ arquatiir IkO (> num. 4^4) , 
^det ab RL i qii» eft.illa ipfa rccta enunciata. ia 
liac Prop; p^ & ift hoc CorolL i »> pet X>L sequaleixi 
AO dimidio lateri recto principali VA# 

r. . • ,' CorolL 24 . 

;oj. Circulus iui cmmuntm in aiiiuo fnnSto iar^eH'* 
^m halfet CHmCionicaStliionUrfetimetro , & e dpametra 
Jer id funSum tranfeuntt J^fi:uikit chordam aqualem lateri 
tfQo €jus diainefri i maxim^f pmniutn acctditad arcum St- 

M 3 dio^ 



1«^ JEGTIQN^M CONliGARUM 

^^h^H ^jil^^ i^^ \ ^^ f^Uius altifius cireuli arcus. imef ' 
m^ W^VMm tr^nfappd^i:, yW 'eujufeufnque majoris m- " 
cus ali^w ^^mmu^ utrifi^uf^a cer$fj0^ extra utmmqu^:^ 
caidai inter ipfas (^ t^figemtt» , cujufcuniiue miftorif 4 ' ' 
t/ngentf rmdat magis-y quam uterlibet cjc iis\ &'ja' 
ceat fs( parte' ipforum earva ; quem circHlum oftulatorem '" 

voco , " " ' 

504, Manefati6us et\Im iii % i^9, 19o> t9X p^n- ' 
f'iS^Ms V, H, «, A,'L;0,; utinPjrDpoOaonc mfig,m^ 
J90 187,1881 f^perini^tet. autern Sc<kionis Cbjpic» noa 4o- ' 
\9\ dnir' YitandaE' conft«flqnis gratiai, e^m reAj^^VA, qa«. " 
Sc^oncra " Gotiicani Contingit jn V i t^gat ibiderai, & 
circulum^V»»', qi^^ a diamctro '^^bfcinidat chordain ali-^ 
quam, YHV ac |^*8. LR par^cla: t^gcnti bccurratln ' 
M,' »{ip4 vcro tangcnti VA occuirat" tangcns HTdih, '. 
a* pcr. H in' T', '8c re&x. MN ,' }»« pv^Bcla: tm^vl ' 
HT occuiTsui^ Vc<J». VH'in N^»V 

'505,'ln primis'crit guailraium ' M^ «qualt rc-aan- 
gulo fub' t^y Sc NH» acrquadramin >r'R ^^ogufq 
fub, VR, &;»H: Nani in flg.'i«9 ob, MN/MR p»- *. 
ranclas"tangen?ibus TH,'TV.suis;uK MRN; MNR*-. ' 
-puari^fur a,ngt^is''THV,'TVH Vqqalibus , eum eos fin-'^ 
gulos nietifurct dijpidiys. Vcuis VH,' ^cleoquc ^ ipfi »-.' 
quales c^nt; &' xqualcs MRV, MNH eorum^ compk-.* 

in?r\?a. a4 <>,ino,^ ^^9P.i «c «^. W '■ ^^. "^*^; ' 
Quarc cum ctiam ansolus; VMR aque^ur aJtcrnoTVH. 

«rl^ordo:. VM catT\ tangente Vt, <iHi (Copoll. 6PFop.9,.^ 



d?atum MR" afquale rccK\nguJa (bb. VJ» *? ^?J .-- .^ 
dem prorfus ^rgumento angi|li yR»»» H»»» «^.^ "in^ 
ic 2gual?s' VmK , '&' /fH»* adcoque ^ft- etiain yi^a« 
Rw, w »»»> five Rw ^.Up>, Mcoqac quadratuin R«'| 
sequale ■ reiaan^^uh f«^. H» , & VR , " 
' i:o<^, Cqm igiw^ «l^^f^wnJ^ f?mio,rd«\atap Scdbomt' 
Conicic^in quavis c trit>us iiguris awiuctOE (num,.+95> 
reaafwulo fub VR , & RL > patet fofie id qua4r»tum< 



ina- 



, fc L E M E N T A, i^^ 

ti4^if ^lM^i ▼^ niinus quadrato MR», VqI ff*K> ac 

jieiil Coi^iesu^, >^;^^, vH cxtf^ pro^t R^ ftwit ma-' 

'j©7. Jaravdro, ll^ h ctrculus Wcrcipiat: ck^qifd[amVH 
tejorcm laiere ^cto; VA; , ' aocJ^Uau: rcct^ HB: ycf fus 
V, *fi wterftpi^ chordaim itimprcm ," ^cipiamr pai^itcf 
««rius V rcc» Hi. a?}u^5 ipfi latcjr^ <;ccto; V Et qtio- 
nnai efaioFdat' Mi9» poteft acccdcrc aJ tangctnjpn^ YA 

5Quuuoin Kbi^rit:; ac in ca acccfsir ppAunr putiiSb^ M ^ 
l> 9 ad y, & adt fc inyiccm acccdcrc quantuiiilihct» 
A.ob.M]^, mn fen^pcr paraHtlas cidcm, rcctar HT „ 
«Mi panqa N, n pofeunt accedcrc arf R, & Vqiian* 
imiitlft, in (juo^ acccfi^ inj^ipict' ^fqnaiida F5S| i^fte 
H ^Kiisnf ^fu niajor i 8c in f^uncto^ H»* lYiiQOt , quam 
Kk 9 f^fmdf in paxabpla; iix '%• ^90: aqcidet' ftatim ad 
funcnBa N- ft^Bictit intcr B," Sc W vcl( n inter. k&cV^ 
€uin aiaiirum rccta RL ibi; acquctur. YA^ ,*^ fivt H8 iq; 
pdjao^ caiftt, l^ iix; ftctihdb. Itt"EUip(t yero, int figaS^ 
m pdmbc cajRi^ Vt^ ctiati^^ qtiam, ]S(' (f^^^ mttt B, &j 
V» HH Hictpitt ^sc mafcr^.qiiaiTiBsL",' cum^B, aequji^ 
Ss Af^ Ji^^ fit m^o?: RE , & i». Hypcrbpla^ in fccuin-: 
A^.osiA^'is\'%l isrr antcqiiara »- fiibcat inicr ^, 3r V,» 
|w M»^ ^t minpr, qqatn^RL, cumj Hi^. fit scqua^is VAoi 
a4K^ue m^fl^r. RL*. At pfoj fccai^do, cs^fq; EUipfcos ^ 
i!d priti^^/Hypcrb^^ acxcdcix^!^ R ad Y quatiti^m. libuin 
Ki^ eti^ Q; ad R' a^cdct quatnu^bct , dc projiade 
iti^k^ ^ BN' ifii!^ a^uaado majjor , quant Oft '^ &' 
^UL BUi>fi /ofrH*, 01, ac(^t?ales ctd<?m; AY',^ & in- 

Rfe V dfemptis inzmi^bus; rcljhquf tur ti^ mmoi^quam 
, -Sf i» Hjpct?^ OL inaeu 

|tet»;»f , OK majoi: ,, ^u^mj RL, P^r 

nvqauRi autem arcum Qmncn^^ \^V«i at' act^cixteadhuc^ 
li^gi^ N , Yd n , ad Y > &• adfaue magi« auicta, BN , vcl * 
kj & iniminuta^ RO^ m«iI^o magisHN fupcrabit-RL,; 
M W« fup*rabitur ak. t^fe. 

[^jt* <^^ire pcr^ tottpi. illtiin, aroum' rccta; RM. in 
■DU^ c^lw^cfritc raajei? ^ - q^iktoi- fenuordi/iata^ Sc^tionis. 

M 4 Co^ 



1*4 SfeCTlONUM CONtCAIlUM; 

Conicae, adcoquc multd knagisRiKi, & in fccufidocft^ 
iii rccta Km crit minor , quam ordinata cjufdcnr,. atf 
niulto magis RKI, adcoquc in ci.rculis intcrcipientibus 
thordam VH mtjorcm latcrc recto fempcr aliquis $x^ 
cus MVm uirinquecirca contacmm V jstcebic cxtri^i 
ctioficiti Conicam } in drculis verointercipicntibuschof- 
dam mihorcm ipfo latcre rccto^ aliquis arcus utrinqit 
circa ipfum cohtacmm jacebit intra ; Qi^pniam vero 
ininorcs circuli toti mfra maiorcs jacenc > & proinde 
xiiinorcm ctiam intcrdpiuht chordamVH > ohmcs ii 9 
qui intcrdpiant chordam maiorcm laterc recto /acc- 
bunt etiam cxtra cum, qui intcrcipicc squalcm , &(^ 
mncs > qui intercipicnt minorem > jaccbunt ctiara in^ 
ira cundcm : is circulus, qui arqualem intcrdpir» ita 
ad arcum Sectiohis Cbnicac accedit drca ipfum conta^ 
ctuhi» uc cujufvis aitcritis uccumque pauUo majoris ar^ 
cus aliquis ucrinquecircarontactumjaccattum extraeum 
circulum» tum cxcra eum arcum Scctionis Conic; , cu^ 
jiifvis vcro alterius utcumquc pauUo minqris arcus aliquis 
utrinque drca concactum jaccat , ' cum in tra cutii circul.ump 
cum intra Scctionis Cohicaf arcum; ac proindc nulliu^ 
circufi arcus potcrit duci intel' arcum Sectionis Coai-' 
cx 9 & arcum ejus circuli» qui intercipic chordaml^i 
teri rccto ^equalem > qui proinde prs: csceris omnibus 
ipfi arcui clt proximus > & idcirco jurc dicicur Ofcst^ 
/aur. CorolL 5. 

509. drcuLiS qui ConicdmSeElUnem ofcnlatujh inver^ 
tice axis utriuslihet , kahet fro diHmetro Utus reShm^ 
ejus axis i m lerin^etrum in eodem uhico funllo contin^ 
^it ita i ut qui efcuUtur Elliffim in vertice axis con^ 
jy^ati , totus extra ElUfpm jaeeat j ^ Mt minimus ex 
circunrfcriptis , in cateris omnihus totus jaceat intra^ 4^ 
fit maximus ex ivfcripis 9 nec in friore cafu uUus in^ 
fcriftprum maximus haheatur y in fofieriore tdlus mini^ 
mus circvmfcriftorum. 

510. Nam fi concipiamus VH pcrtincrc ad a^iA «e 
liqucm; cangens VA crit ipfi perpendicularis > adeoque 
ipla VH> Qjfxx qrquacur latcri rccco AY> cv^dic diame^ 



_ :; E^ E M "E K t h ttf 

J^ firculi. Cfaorda ,<}iioque Mm evadec ipfi VH peN 

peiKiiculariSf ac proinde fecabicur bifadam in .R ^ & 

j^» pfm icoqgtueo t cum MR 5 mK 1 punctis IJ 4 n ti^ 

kundbus in R^ quaiUataiji vcrp tam Itl^, iqaamRijf; 

cvadet ar^uale rectangulp fub yR ^ & RH • Qjiare H 

VH fiierit 2tquali$ lateri. irecto in fig» 19 V in. Hyperbc^ 

h 9 ^& in parabola in fig* 190 ^r eri^ HR femper mi- 

tkx 5 quam RL , cun^ debeat e£;e minor ^ qtiani HY i 

\ pik q)2^An VA., qiiae in Parabola a^quatur rL , , & iil 

. Hyperbola eft ipfa adhuc minor • At ij(i Ellipfi in fig^ 

189 cum fit VR ad OR , ut Yh ad AV , erit VR ma* 

)tf I vel minor , quamRO» proui: axis «V fueci^tna* 

jbr.vd mipof fuo laterc rcctb VA . Qjiare pb yHc* 

^oalcm VA ^ adeoque OL , .eri^ contra RH mjn.or i^ 

yd major KL i proiit axis fiaerit, major , vcl minor 

fuo latere rtcto . Axis autem tranfverfus Qiajor eft fuo 

kicre recto ^ conjugatus jninbr > cuin. axis tranfverfiis 

coojugato iiit major ; 8c lattts rcctum . utriusUbet axis 

fiif num. 351 } continue proportionale poft ipfuii^ » 

& axem aljtcrum.Igitur fi V ftierit ver^ex axis tranf- 

yah 9 ctii HR minor fci^tiper ^ quain RL > fi coiijugai* 

piDa)or • Qjiamobrem in vertice axis conjugti £1« 

Kpfeos crunt RM , Km jfcmper ma^ores ^^ quam or-^ 

dinata ejufdeiH ^llipfeos > in reliquis omnibiis axiun» 

vcrddbus (^runCvinijBioccs; & prinde circulus , qui Co- 

&icam iSectionem ofculatur in aliquo axis vertice » eani 

\fi codcni unico puncto contingit , Sc qui pfculams 

Hipfini in ytrtice axis eon^ugati totus extra ipfamy»» 

tirt, reliqiii /acctit inu^a bmnes , ac iUj^ eft circumfcrt^ 

pnis, hi otnnes infcripti* 

. jii. Porro quoniam in illo cafu otpnes circuli maM 

/bres cadunc extra & curyam,'^ ofculatprem , acmi«; 

Qores dmnes & intra ipfum ^ & per . aliquem arcutii 

. tltrii^que circa contattum etian\ intra EUipfim cadunt} 

iUt eft circuxnfcriptorum minittius.: cum vcro e con^ 

ivliario in reliquis ca(ibus omaes iHinores xadaiit. intra 

;&; curvam» ^ ofculatorcm » bmnes ^utem majores 8c 

I txffa ipfumi Sc fic aliqu^js) arcm ttfi:inquc eircaGoac 

ta<3tum 



^4 SPCT^IONUM COKlCAtlUM 

faAtta> cack^t extra curvam 3 » odt iiiAriploriHtf ffi^ 
^itaas '. PoFt^ noltps m pFirn^: ciaQ» mmop ofinitar ^^ 
^ rtKquif maiof» /ita «t euni^ ^|c<!dj^f , m aBi piH 
res ha^ri ifion^ P^^^t ^U^^f^ ^it^tiici'»' k&^, A^i 
ctntroiniin mterv^lo /ut Kbt^ic^, pc0 Qdy» cct^itoeic. 
ctiU intenifiedii , qui iftcerrne^iu^ ^vtd aUtUk>: Mfok 
crinqiie ctrcar contadom> cadbe fiy i^Ud pri^er aiTu 
pam iatra^ curvaiit , m hifbt ^i^a^ cxcr^. QuajK] 
lus habebicur ibi iofcftatoruflv maxiiDiis r bic nuAifti 
eircumfcriptoruiB^. 

yr2^. CircklHt , ^£ ScSUnem C^nicjtm €fc\ 
tftrtice^ cufufw^ ^wUis Ammt^i^ ISclft tmffimi iin tsn 
^emcm hwfat > tnfnen iiidem^ 0nm ficat ita ^ ut e. 
p^tt anguli obtujf chorcl^. itlins, iftntflis' hteri reHoi 
ping^e \ JMce^ extr^ ifptm SeSiUnem Canicmm \ 
fmte vero anguli acuti intra , ac fratma in alie- 
2?^ y quod in ee geeme/ricef^ de^niri fotc/l » iffam it€^ 
Tum jecat» 

fi^. Duc^tur enim- in Bg*^ r^o 2o* Farabola^ dtordai 
W parailela tangemi HT , St:- p^tet piin<9b m^ ^fisa^^ 
pto » uc figara indiQat, idtra e^clK>Fdam lcinpcr do« 
bere mn * ipfi F V p^raUelam. facer^ ukra ibTmih & Hirftiii^ 
majottm, quam HV, iive' in; c^ ciroiH' oj^^cbrii- » 
quam V A' > vd KL : at ipib, pun^ mt abeqnte' in Ff 
^ibir nittV^ ac fiem H^, KL asquateV : eodeM' vt^ 
ro pmdto iff> de(0cndence in^^ arcum FH » etiam; n^ ia^ 
gredieQir cbordam, YH , eriiqae H^' minof ' y qmm 
HV , adeoquc mtnoi? , <piam- ft£ . Q^^^ pcf totuin'. 
arcum VmF crit Bjn major quam femibrdiftatat ParsK. 
boke, io F a:quaKs^ , pcr arcuin f H minof :. pcr- cck 
tum autem arcum VMH/ crit HN- mlno^ quam FfV ^. 
adeoque minor > quam RL , Sc MSi minor y qinm- 
ftmiordiftata. Parabok» !. Ar<hi^ igi^ VMH' C3^ partt- 
anguti acuti lafSfrC' inoraf Paraliplm toti^ , & VmF ia^ 
angulo obtufo exora, quam Paral^plam iM:otfi4<^ ia cir^r 
cqIuS' fecar in V , tt ciim iterum' arcus FH< IseeaE* iiKr 
cnt Paraboten^ 9 oaor idtm ikculas Qtosit io 9'. 

5x4. Ac 



£ t t M B N T >^. 1*71 
'ji4, M in IBMfti in %. i99 arciw vW jftoeblt om,. 
Iimo emz ^ laltem donec ii ca^af excr^ cireuliim , com - 
defcicat H» eflfc majop. , quM» f^V > adeaqac m^r ,* 
quam VA » & "^4^Q w*)oc , c^uatn Rt» at pfo* pa»-^ . 
te oppofita fi^ vetfiis H captatur T^-t ad Tf^ i ^t cft 
laius isranfYerfiim Va a<J rcAira VA , &: duealui> VQ, 
CKCurrcn» circi^o 14 F , locus aKu» V^fF fe^eebir ia»'* 
cra, '& circul^s in ipfq piindo F iferum Eltipfim ftca* * 
(>it '. Du^a enim ad quodvis^ pund^um G itljler Tj. Sc 
Q^rtAi VG^ cjus cir^ulo oc^cutrat in M> ac p? odiiAa 
I^IMu^ii^ ad^tahgCHitem i^ I, eritNVadilVJ, nc NI 
a4 MI , flvc ( num. ao4 > m HT ad ©!r , «c eftt 
VR ad R,0 , ut V<» ad VA , iSvc ut T<^* ad TH • 
(Xj^si^ttx asqualitate perturbata ed^ VN ^d RO y «t ' 
TCX^ qd T€^ 3 Mcoque c^oacG *TG 6ierit nnniQif ^ ^vmuh 
TQ;, crit & RQ hhhof V quam VJ^ , m proiacli ofc * 
Ojp *, ' yH a^uaies, >?it RLma|or^ NH ,' te4miofflBnsh> 
taBlIIpfebs majo^ , quaoa '^LJ^ ^ ac pui;i(3;um M in^a 
^llipfim '. Abcun tc vero, G in Q, , & M in f\ cva- 
aenit' VN ,^ RO «quales , ^ pua^bim ' M crit * in ip^ 
Eliipfi y (^iSta autetR T€j ^huc naajoiae & cy«let 14«« > 
tra ElKpfim ,^ adeoque totu^ arcus ^ VMK js^^^^tnm *^ 
fiUipfim, qua» ciscukis dein^ itorum feoc^t m F^ 
' 515. Demum in'trkperbota('i^ ^ i$bx {^^ eri»' 
HN tnii^op , qu^ HV ^ adQoqqe miiiqc » qoai» VA» ^ 
Sc mutio/msaor , quam ft£, '; aq pi:osttde coto^ afiqu*^ 
VMH jacebic inira > hS^ autem TQ^ad T^ t% eii-. 
demratione laceri^ tran(Verfi ad fcttus/^^m ^ fe4 M 
pacce^ oppbfi^s, ac d^(%a rei^ (W ^ qi^aireirculo o^. 
ciHvac in'l^V tum pct <^odyis pto^m G ipfius TQl 
dudU GVmn' eodem pro^s argunieatGt ai« Vsi adRflw^ 
m tn ad »»i, us HT^ ad TG, & cric VR adRO;, 0« 
\h ad VAVat TQ,¥ T^ V*^ prcArKio »V a$| RO, 
uV T^ad TG ; mnuni^ctonecTG fueric mtnor » 
qnain TQ. f q^od fiec pcf . toi^m af cum Y£ , ecic RQ > 
ni)lnpir,'quafiv V», 6c prbtodc RL muior» quam Hv> »* 
Qimlrum ftnuor^iiuta Hyperbblap minoc quam Rivi » 
8c X» €xtra.^ipfam Hjcpetbolam , Abeunce m iot F , Sf 

G in 



m SfeCTIONUM C6mCAKXjU. . 

(b In Q^# habcbitur aKjualitas ^ & pund:unt m^ erit id, 
HypeVbolx perimetro 9 tutn per totum arcum FH , j^ 
vadcnt TG majore > quam TQ^ jaceblt m Intira Im 
pctbolam i 

Ccroll. 5i ; ^ 

516; NhUus ^arcHS fttcumque farvus hircuii ofculascri^ 
tonzruit cum atcu StBionis CoAic/t , fcd cum cm anii 
ium contimt quovis circulari^ fninortm. - i 

517. Patet ptimum ex ipfi^ ^moaftr^tiotie CorolUi^ 
lii fecundi , & tcrtii > ciim nafi:][u4m in cafu CorcJk! 
2. NM. > mn fiaiu ocqualcs femiordiaatls Sectlonh CqI| 
nicx , in cafa Coroli. ji punftum F coqgrudt ciim c^ 
|u;s peiriitietro ita remotiilx! ab ofcolb V 1 in arcu «oa^i 
tinuo eirca ipfum V fit NM femper itiinor nm^ fenK 
per maJQr . Pacet autem 8c fecundum cx Cbroll. 2 ^, 
cum aullus <irculatis arcus duci pbiSt inter atctmi Se^ 
ffitioms Cotlicxj & arcum eirculi ofcuiatorisi 

Coroll. &. 

^it. Hy]^bili» PardoUi & EUipfii idefn hsAtntif, 
Utus reElum , dr eandem incUnaiionem ordimtarmn ad . 
sUumetrum i cujus id efl latus teibm > hatent circulum'^ 
cfiulatorcm aquai^ i qsuecf^un fit dianifetri ms^nitw^ 
mo i ad quam tamen ubiarcus circtdi iacet intta Cpni^ 
cam Se3ionemi,ut ex parte angtdi acutii & arcuf VM 
in quart/is diametro , ac in vertice a^is Paratoie^i vet 
axis tranfverji iJy^trbola 3 onmium mMximie^acceelit EI^ 
litfai & eo magis > quo iiut dum^er efl minor 9 t$im 
farabolay tum omnikm minimi Hyferbola y & eo mi^ 
nus , quo minor efl tius diametir ; Contta vero Mtiar^ 
cus circuli jofet extra : ac ut ^ licet im angklo reSd^ 
tangentis cum arcu circuli nuUa alia rtSa duci fjft # 
^& is angulus fit quovis reSliUneo minor » foffunt t4med^ 
duci arcus circulorum maiorum i qui eo frofius ad tan^ 
gentef» accedunty quo diameter efl maiori fic licet inan-i 
gulo circuli ofculatoris fum arcu^ SeQionis Conica ntdluHi 
alius circulus duci fojfiti & is anguius fit minorquims 
cirtulariy ffffunt tamm d»fi mut ScliionumCvnicarum^ 

qui 



\ 



' 'E t E M E N T A: ^ ^ 

\ia fropius nd ciradtm oJculatar$m acted^ ^ 

pi ad eas paties tangeniis , ad quas cif 
mahres fiterinty vel ad effofitas minores<, 
)19. Omacs cjufmodi Scftioncs Conicas, 
e circul\im ofculatorcm patet > quia fi^ 
is , omncs ii circuli congruenc > omL 
tandf m habcbuat tangcntcm , & cx eadcm rcctti 
dpicnc chordam candem aequalcm comtnuni latctt 
^ Porro in fig. I8^. ^quo maror fiierit axis V« ^ 
V na&ente pundto A > crit minor re6ka RO quarta. 
^ Vf , VA , Vk a adcoquc co maior *KL » & raalor 
W^^nt ordtnats • Qjiamobrcm co magis ejufmodi or^ 
jbiiu fuperabit RM, at €io minus fiiperabftur' ab Rm, 
\C\o magis dtft abit apcus rpfius ab arcii VM-, vclmi- 
ab arcu Vw . Ifi Pat^aboh vcro in fig. ,190 , ia 
RL jam acquatar V^ > ca crit mafor ^ quam -ia 
a Effipfi • I>emum in Hyperbola in %. I9> ^ adbuc 
UL ed. major , quam VA ^ & eo major , quo major^ 
ft RG quarta poft «V, VA, VR , adeoquo eo m*a- 
ic, goo «V mii^or • Subibit igitur ex parte VM ar-f 
■scaJDfvis Hypcrbokc hab^ncis diamecrum mV maj(H 
tm tiiter arcutn habcnfis minorcm x Sc arcum VM » 
t intcr cos bFrincs , & VM fubibit arcus Parabolx , 
Ini inier'hunc quoque arcas cujufVis EU^fcos, &in^ 
t arcum EUip(e6^ habentis diametrum majorem , ac 
M fubibrt arcus babentis ipfam minorem . Ex parte 
sro Vn» intcr arcum Viff , & arcutii ElKpfeos faabei^ 
\ minorcm diamctrum Vu fubibit arcus habeatis mai«. 
rcm , mm inter hos omnes , & iUum arcus Parabo* 
r, mm Hjrpcrbolar^m omnium eo propi^s. , qua m.a« 
hnnfr habuerint diametrum Vu. EoctemverA argumeii^ 
^contrngct primum iliud utrinque it^ axium vcrtict» 
s^-nbi arass circuli jaccat intra^ hocicc^ndum» ubi 
ira. Rcliqua paceQt ex his. 

Cifroll. j. 
^T^q. In Elliffi & Hyferbola radius circuli ofculaP^ 
tertius contbine frofortionatis^ pofl ferfendiculmn 
V in tangentcm Mi^um 9 & femidiametrftf» coniu- 

iatam ^ 




> V7P SEeTlbNUM CONIGA kVU ,, 
i^lhim^ &y4dii drcfdorum ^culMwum intenfe fimt in 
rm$z0fff rtcifirTicd triflicAta eiufyiodi ferfenditulatum^ d6 
dirtSi trgiic4td mmalium ad uirumiiifet dxem termi^ 

^ ^li; Si eaim circuliis ofcakair. Eliipfim in fig. ipaj 

F.Vi^itl, Hy^tlKjlam ia llg. 193, in P i c diamctro P| 
4J^J ifcbfciodec { niihi. $6% ^, jchordam PH ?qualcni latcri 
IbSe eiiu .diametri : Sic.eius citculi . ceiltrbm. iii K ; 
9c recta k£ petpendicularis }^ii diorda: eam bl/ariam 
lceabii in £ i ac ducto CL pei^^biidfculb iti tatigentedi 
VQl eFontfiiiiiliaoriangalarectangulaCLPiPEKj iiam 
' «orum anfuU ftd C; &P enint,altet]ii m fig.^i^i % 
likternus \ ac externus 9 & oppofitus tii 6g. , i^;: Erit 
igiiur CL a^ CPi ut PE ad ,PK : Sed cuni, PE /it di^ 
Jiitdium latus fe^um diameiri I^ i/duda diametiocd- 
niii^nta^IC*; erit fnum: 351 ).CP iidXI; ut tl a^ 
P£ Klgitur tt ^c^licate . petttirbata^erit CL ad Q i 
ittClid r^dium circuli ofcuiatoiris PK;. ^ ... 
, ytil Hinc iiifem^eruitut; i foife radium KP aMfuar 
letn quadracb femidiametri coniugatas CI applicato ad 
|)tt#p^iciilum CLi adeb^ue in ^ rdtidii^ coifiipafita ex 
directa: d«iplicata iplius femidi^rneiEri; & recipreca fim- 
ijici ei;^ perpen Jicdti i nimirsmdum feniidiametr^ con* 
fttgacf iiai^ {tk. 46ii) r«cipr«ce ut eiufmbdi perpeiidicth 
iai erit ilie radius in radone reciprbcai triplieata ejof* 
4ktn peipeiidiictili^ qtts ( num. 4^9 ) eft eadem ; ac 
f^a direct:^ iriplifiata^aormaiis aKt atrumlibec axcnj ter- 

CoroU. 8« 
. ^^. In ^mfis SeSityne Conica radius cifculi efiu^ 
Utins efi quartus eemiHue frefertienalis fefi dindi^Mm 
Utus reSkm fHncifaie i & normeiem terminatam ad #^ 
Jetm tnii^perfum . 

ji4: Eft ,ciiim in EUipa , & Hyperbola PM ad PK 
m fecsarigulum fub PM,- & CLo fivc ( n. 459 ; qua- 
dci^tim. kitoisidi coAJMgdti CD , ad rectanguliim ^ fiib 
QL' y £f PK i ilve ( nmt 510 ) quadca«usB femidia-: 
teitfi 90g^i^,^ Ql , aii:qirit<^ ( vLmi. 45$- j ut ^ua- 

djfatum 



•y -. 



B l % U E ^ T A- V, I7i . 

, jKlcJ CV 1 ^vc ( fium. 477 . ) «t <jiwi<ii:atum ^Jiitiidii 
^^(c^ ccbi {)tmicipalis ad qj[Midicatiifli noniialis PM • 

Qu;»? 4 JJ^^. PM ; & radium Pfi^ fijiRatiiri*cta mc- 
^^ f«!o^r^ptudis j ad ^iiis quadr^laun cric <iua<lra- 
l^ikafim P]Vi^ uc ipu Pii ^PK# ilye ut quadfaoundi-' 
nidu lateris rcfti pcindpalis ad qu^iteatiim tiofmali^^r 
critipfa ctiabi nohnalis ad cam rcdain^ iit ditnidium 
litoi tectum principala ad noi^malcni; ^ ditnidium la« 
i it«%ftsm prinppaic noimalis PM» ci r^cca ^m^tz, 

L >c }| coaxiriu^ prdpbr^pnai^s . ^ ,^ 

^ > jii.M Paf abpl^^ vero^ fig. i^i^» fi t^tigens du^aperFa^^' 
i Poccparaf cao^u ifx^x pcr vcrd^m V in A» rcdia 
^H^tt/nu: 19.6) perppiidicularis ipd raii|etiri PA, SC 
i n. ;9$ ) micdia propQr.riboalis intcr FV i FP i ^ qua- ^ 
^.riijn pritDa cft Tnv 19S l quarra p.ars latcrU fcfti prind* 
t jiiis Mcbquc (A^ ioo) dimidia fulmof maHs RM ^/c^ 
^a vero ( o« :}5 x ) quaftapars laccris fci^i diainccfi trati* 
k^^WUf pcr P» adcqque rcctai VHi Sc. protnde dimidia 
l.iPf^ a^ triangula FVAiPRM^ t>EKamiIia fmitobbiii- 
^^iaiakra parallela: Quare critJ>M ad PK, ur RM ad 
hjKUyci nitnptis 4imidii$> utrV ad FPi niifii^imi itt 
^^M^wm FV ad quadratum I^A i fiVjB uc qoadratiini 
Jm (iiiDidii iateris rccci priticipalis ad qiiadrattiin P^' 
^oduc eddem> ar|umetito PK qiiafca <2on(inao prbpd^r^ 
\iom$ poft ipfum dimidium lauis. reftom ptmdpate ^ 
^k ijbfsx sv9rm«lfni PM« 

^ C H t I V M It 

k . ■ . . . / ^ ', 

iuk p6c6rat con^uni & fadliori cfemoiiftration^ 
% 4 idem ^ O^oUarioruni boc edam pa(;to <fe« 
^&ooftrari . Radius circuli ofcuiancis perimecnim id vet^ 
fst axis ti^atiiifver^ (. aum. 509 ) ^ttacur dimidio li* 
"^i cepto . priiidpali : ibid^n^ aiitem normalis PMf^^iS^ 
r fij. 173 > 474 > 17^ cYaftcfccncc PR evidit ap- i^df 
»li$ fubnormali t RM > iiye rocta? RP ^ quar ilbp^ |^^ 
W K » y eraAil ^^SaU* dimidj^ i^H W^ rcccor 



' ly^ SECTfONUM CONICARUM. 
'VO . Cum igitur { num. 520 } (inc radii ipff , m 
cut^i nQrmalium » erit dimidiam lacus re(^m prroe)- 
pale ad radium circuli ScdUpnem Gohicam orcdaoti» 
in quovis punfto in ratione triplieata ipfius dimidti la- 
tcris redll ad normaIem> ^ proinde ilfe radius qoactut 
continue proportionaUs poift ipfum diinidiupi Utos re- 
iptum » ^ Aor^ialepi • 

5^27. CiYCutuSy qui conmunem in diqtMfunth tAnge^, 
tem Met cum SeElionis CQnics ferimetrcy & iffi feri» 
metro in aliquo aliopunSb occurrit^ abitiniffum circulum 
efculatorem , uhi i4 functwn ita aicontactumiltum Mccedit^ 
nt demum in iffum aheat , ac concurfus binatum rects^ 

^ rum^ quarum alt^a fit ferimetrft ferfendicularis ip ex* 
trenio functo chorda cujuffiam^ alterd iffi chorda ferfen^ 
dicularis in ejus medio\ vel altero eictremo^ ^kit in cen^ 
trum cifTuti iffam ofculantis in friore illo functo,^ vet 
in finem diarnetry iffius citculi fer illui iiUm functum 
tranfi^ntisy ubi evanefiente chorda , congruunt ixtrem^ 
ejus functa. 

V^i9 52S. Sl cnim in fig. 189, 190, 19,1 V fit conta^hi* 

190 fUc > & M , vct m ad Conicam ScaJoncm percin^ar » 

191 cric ex natura circuli ('num. 505 j HN, vcl H» tcrcia 
poft VR , &c RM , vcl Km , ac ex natuca Sedtionis 
Conica: ( nura. 49? ) RL pariter tjertia port cafdcm . 
Quare femper HN , vd H» acqualis RL . Acccdac jam 
M, vel ifi ad V ica, uc demivn cpngruanu coibunt £• 
mul cum ipfo pundto V eciam punfta lA^ m^H ^ n ^ 

' ac puncpm L abibic in A . Quare & HV ficc acqita* 
lis HN, five RL,^ nimirum latcri recco VA, & prcri». 
4e circulus (nvm. 503 ) ^vad.ec ofculacor ; unde pacec 

primutn. 

529. Jam vero diametcr pcr ooncaccum V cranfiens 
eft pcrpendicgiaris cangcnci, ^M^coquc & perimctro Se- 
ctionis Conicx , ac rccca quidemex cenoro ducta ad 
itogulos reaos in chordam VM , vel Ym debec ipfam. 
fcear^bifariami reccavcrocxcx^emoiUiusdiamerri puo- 

cto 



E t « W B N T A: T7Y 
#0 diicia ad pnnccum M » vel m extremum ebordx y dcbee 
rnciiiere ansulum femicirculorectum. Q,uarQpatet»coa- 
caBC^im i:^ctae perpendiculari^ pcrimetro ductse pcrV cum 

Sa pcipencliculari ^prdc ducta per mediam ipfam choi:* 
ty vel e}us exqremuQi M> "vtlmy debere abireiacca-^ 

! frumdrculi ofculatoris^ yel extremum punctum ejusdi^ 

I mcori, ubi puactis Mi vel i^ » & Y CQcivitibus ^ eva^ 

I nefcic cborda. CmlLio. 

' 5}Op BjLnMrum n^frmdiun^ pen Hna Seltionu C^tAs^k 
paUd di^dtHm CQncHtfns, aiu in c^ntnmcixculi cfcnla^ 
uris > nki oi funlU 4(L jf^ Ua. 4^^<dHns a f*f dmnm 
fagnutnt « 

5JI. Concurtant enjm in fig. 195 ia Parabc4a » 
^9^ in EUipfi 9 197 ia Hyperbota binas. aormajicst PK)F.i95 
fK ia K > & fecaac.axem. traafverfnm ia M> m .^ ac jp^ 
gffnmpca VQ perpendiculari axi oratxfverfp. , ic a^u^ i^j 
dimidip iateri recto (iriacipali , recta ex Q duct4 pa« 
ralkla axi in fig. 195 j ad centi;um C in £§« 196 3 
1^7 occurrac iemiordinatis PR » fr profiucti^ in, D ^ 
^ 4 eci(que ( num. 4^4 ) fubno^malis RM , m s«« 
qnalis RD » !&fl(.« Qiorda Pp occurrat a^i tranfyecfo ia 
Qj> & rec^ ex P parallcla. ipfi axi occustat rectis pr, 
;K io. H, £. £rit u^ique PK a<i> MK> uc P£ adM«».» 
|ive in rationc comtpfita Pip a$( PH :k Si VH x vd 
Rr ad Mati. 

jja. Porra PE ad RH- eft C wwa^t ao4^ ^ > uc Qj» 
9dQr., & Rr in fi|g« IP5 a^uatur }Am 9 cum a^ 
qifemsff BJ^ y,vd ^ adcoque Sc RM y rm 9 9c x (km^^ 
nsk cpmmuiii Mr % ipfae Rr > Mm • Ac ia iig« 196 
iimpc^ QB aequali femiaxi tranfvcrfo^ CV Yerfu$.y, Sc 
VI ig^ff,^ partec oppoiJjcas:^ dugtai^ CB^ quac ip* 
&. ferruordmaGLs occurrat^ ifV Tx ^» duQ^qui^ ^I j ^A 
parallelis CV 9 CB iifque ad rectam DP y cril^ Mflf 
^DfyzaUs I^ . Erit enim Q3 ad DT , ui; OC ad DC , 
VC CV ad, CR X adcoquc Sc o^ OB » CV ae<yi4lcs , 

' cdt PT acqvialisi, CR , ac Qpdem argumcfi^io d^ acqofH. 

■ lis Qr , . qtiae; ^u^^i wm. iit a^qualis AT 9 crit Ri» 
apqiKtUs DlA .; c^mqup fi^. ^ KD, a^qualia^ B,M » 



!^4 SECtlONUk COi^iCARtjKi 
tHt RA ^qualis rM; cft vcro & m acqualis r^ , fivd 
Rli Igittir crit lAm acqiialis lA ; Indc vcrd cuni.bini 
quxvijt laterd triatiguloruni IdA i VCEi fiht p^aflllfela ^ 
crit dl ad lA^ five Rr ad Um^ \xt CVad VB. 

53^j Cocant jam ptihda P , / ^ & fccans |>PCX *■' 
bibit in tadgciitcmi toibunt punfta K ^ r ^ ^ punAa 
{^.IpSMi m ; fig- I9i i 196 i isi muiatuntur iri 198 ,v 
f 99 199 i 200 i & crit PK ad KM in ParaboW ih fig. ' 
4od 198 5' lit QM ad Qfliat iti rclicjuii iri ratioric Compo-'^ 
fita cx ipfi QM adQRi & ti altcr^ fcmiaxis trant' 
vcrfi CV ad VB diflct^ctttiam in EUipfl i fiimrriam iil 
Hypcrboli cjus,» & dimidii laterij reai prindpdis VO. 
534. Porrtf ob fimilia trianguldi QP\I ^ RPM^QRPv 
cft tam MQ^ad Q? / quam Qp ad QJl ^ at MP aa 
PR, ^deoque Q&f ad QR.j* m quAc^ranim MP adqas^ 
dratum PR . Quarc crit iri Parabola in fig.» 19^ PK 
fld KM- i ut qdadratuffi PM ad quadr^turii PK ^^ aideo- 
quc fumcndo differeritiam tcrriiiriotadi ^d atitc^edcn- 
t^m i erit quadratunf dimidii latcris re<^i MR ad qua^ 
dratuitl riormalis PM,- nt \p(i nofriralis' PM dd PK . 
At in EUipfiv &*' Hyperbola curil fit (riuiri*'47^)f€mia*.' 
xii traofvcrfus CV rfd yB- ibi difRlferitiarii / hic fam^- 
mam ipfius^ & dimidii laterisredti principjdisi utqua*^ 
<lrattun femiordinataEf RP ad differ^ntiam qusldratorum . 
Iiormalis PM ,' &i dimidii lateris re6K prindpalif VO , 
hmx lYLx r:iti<^ncs cbmp6fitd^ cruiit quad(att' VM. ad ' 
quadirdtum PR i Sc quadrati PR ad cam q[uadc^torunl 
^fferctltiamy quac reducuntur ad unicam ^U^dfdti PM 
ad" fuariJ diffcPentiam tf quadrato VO • Eri< igifciM: KP," 
ad KM , tt quidratum PM ad difitrcntilm qiuadra-' ' 
tcfiim PM i VO , adcd<iue PM diflfercritiS pridtum ^ 
tertriiriorum ad primum PKy* ut quaPdrattitri^VO adqua-* ^ 
dratum PM / . , . . 

53jr« Igimr ubique rado normalis PM ad PK eftei-' 1 
dem y ac quadrati OV , ad quadratum. PM ^ adcdiljue 
tcdcm argumetito r quo in fuperiorisT Corollarii <ic- ^ 
monftrationc , PK quart^ cbritiriue' projf>ottiari:alir pofk ^ 
dimidium laius rcdum principale VO 9 Sc libfroa- 

WCtO' 



t; ? 



rat PM » ac pifoincJe sequalii radid^ drcuU orcalac<Vi 
Hs> puaAd K abetificcl iii ipfia^ circtiti ofculatdfi^ 
Sattiaam i 

' s c H 6 L i ti M iiL 



Ji^; "17 Idcbimu^ fao locd i ubi nimlrum dt Cur- 
. ; y , vii^ ^cilctaHtet agcmiis dpc infiriitcfimorumi 
jca^cni hanc prcipnetaiem cfle cirduldratri ofculato- 
ram, ut nimitiim cofunl afaii Cum aircii dulfvai atigu- 
iDn catiuitiiat qudvis cirCulari bindrcni iu i ut licet 
m tmicd donveniant piindfco ^ di^ ixi co iiagulo infiniti 
^iflii ciirvarttitt ^^rdu^ clacr pciflSnt i adhuc tamcn 
&0Q jQlCt uUtis ciirculari^ arcd^ 5 Sc cdncur/us ultitnust 
ire^ fecanti j cHofdam ad ^giilds reddsj ic tsifarianl 
iam hormaii per alterdm ejus exffcmuni dufti i yel 
itnarum hdrinaliumi iiicidat itt ipfuni Cefitrdnt circuid 
)lfcubtori§; ubi binis pciinietri pundbis coeunttbus chor- 
ii evanercit, fed interea iibuit ca hid ex ipfa natuTai 
&.j>rdprictatitus Scdtionutri Coiiicarutn de ipfarumcir- 
edis ofciiiatoribtts accurdtiOrnie dcmonftrare pei: finitam 
iEcomctriitti *.,,;,,. 

• 537. Et quidem poftremuiTl hoc Corollarium ufus. 
iam in Phyfica magtios habet, at ubi quxritur Tcl- 
iris figuri pcr .grarfuiim dimciilidiies i Nam gradus 
«t« dicitur ejus ille arcus i per cujus cxtrcmapun- 
|la duStx binas norhiales , ubi Coiiveniiint i. anguluiii 
Boncineiit ttniiis gradui > iU^^vcro cdnvetiiunc propc 
bitram drco^ ipfum arciim ofcuUtitis iiiiiledioi cum 
h pua<Aa parum 1 fc invicem diftent , ic H csl con^ 
Iniaiij iii medid; concurfui ndrttialium in id cetitrum 
Mrc debct J Quarc prajccdcritis Cdroliarit vi aflunli fo- 
gr pco arcu oixvx arciis ekiguus circUli ofculatoris i q\ii 
fto parum admddum differfc pdteft^ Cuni ^cv^ cir^ 
iii, ofcalatdrenK dcfitiencis debcat ad ipfarri accc- 
idtra qudfcumque iimites i iatequam congruiat> 
&impcr.arcus aliquJs curvae concliidatur intcr ar- 
^ circi^ afcalatoi»i$ i &c arGum Vel majoris j vcl 

N 1 mi- 



17« SECT|ONUM CONICARUM 
sginoris cir^uli 9 dciinentis dempm in ofculatocem ip^ 
fum > ubi arcvks cijrv^ ix^ i^^^ojq;! ifnniinucus p^mtus 
eyanercit. 
J,l99 5^S< tlhi in Coroll.^. in fig. 189 Ellipfim eonfid6« 
lavimus» exprcflimus in ipfa figura caf^ro) in quolatus 
rcdlum VH eflct majus diametro Vi^> in quo cafu» ut 
ipft figi;ra cxtiibei , Amipta TQ^ ad TH in ratione Vm 
ad HV i pi^nAum Q^cadi^ in^cr H > & T . Si latus 
reAum arquarc^ur dianxtvo , abiret pundtum Q i^ H^ 
adeoque & pundum F > in quo circyl^s ofculatoi; EUip^ 
4m iteruin f^catf abiret ia H; quod G adbuc cflTetmir 
nus > 8c excederctur ab ipfa Vm » abiret Q citra H ia 
tangc^tcm TH produdkam, ^ F in accwmYwH, quo; 
cafu ad dep3onft!:aQdum eam par;em arcus VF i^ qu2& 
facer^t citr^ H ^ cOTe int^a £|lip£im , immutanda non-. 
Dibil cflfct demonftratio :i & ci syptai^da cafiii , quod far 
cile fieri po^uifii^t > fcc^ ^d id > quod propofitum fue-^ 
rat , id quidein non erat neccfiari^m » cum, nimirum 
fa;is cflTct oftendere , aliquem arcum VM jacerc iotraj^ 
^: aliqu^m Vm extra 8c alicubi ^bcreiterum EilipGni fcr 
cari a circulo ofculato^c in pundlQ , quod georuetrice 
dcfiniri pofiet, quae quidem omnia ex ipfa confi:ru«Slio>i 
nc cafus primi in figura exp^^cflS , pro cafibus omnibus 
iiiim fatis manifefta , ac cjm demon^lratio iis pmni^ 
bus , yel prorfu^ (qpim^i^is^ ^ft ^ ycl admodum facile 
^ccomodatur , 

^$9. Porrq non crit abi; re ooniid^rare % quo paAo 

clrculus aliquis Seifbionis Cqnics ofculator cvadat. Po- 

|»,:ioiteft eam circulus in qiiatuor punAis fecare ^ ut infig, 

2Q2 ^^^ ^^^ l^llipfim in punAis P , A , fii > C • Nani 

^Q-* circulum alic^ucni qjilibet Sc^ibni CoQi^ pofl^ oceur^ 

304 ^^re in quatuor pundtis , admpdum facile dcmonftr»* 

905 ^^^ ' ^^ H P^^ ^i^a extrcma p^nda qnius rc£ts axi 

' ordinat« , &* per un^m cxtrcmum altcrius ducamr ci^n 

culus ; is profcfto tranfibit etiam per alt^rum pofterio^ 

ris exoremum ^ Habcbit ^nim c^norum i^ ipfo axe 

^riorem 6rdina;am j fuam chordam « fccaate hif a« 

ti^. li ^dcQc^u^ Sc |9oft^rior;n) ordin^tan^ habebit pro 

. "' chorda 



*■ fe l I Ki ^ H ¥ A. tjf 

6Mi i quatn itidcm fecabit bifariam • Si }am CtA^ 

rilocus mutetur ira , uc biaa punda A > P conghi^ 

lot $ evinefcetitc cemtnuhi thorda PA » commiinis 

fecins £G abit iii totnmunfem cangentfem 3 ac ip(c 

tiiailtts EUipfim ednciiigit ih P » figura 26 1 abeuncc 

la 202 ) Ubi cirdtilus> & EUipfis fe thutuo tontingunt 

kt^Sc idbdc fe poflfunc ftcate ih binii aliis pun^ 

tts C 3 B 9 cehtnlm autem K jacebic in irefta PK per- 

fcnjidtiiari kangehci 8c conc^diis erii excetiot ^ arch 

«ccdi utridque circi cohtaAum P exiitente cttra £1- 

fipb . Qadd fi perpetud rainuatur radius PK 9 inter- 

ftffiD iUt C accedet ad P, donec ih ipfum P inci- 

iUy quo cafti ^adec eircultii ofdulator ', in cujiis 

^hio trii communid punda iiniuntui' ih unicum i 

^Qod fdiem cribus sequivalet intetfediioaibus , Vel uiil 

<OQtaftui , & ufli inctrfectiotii i' Interea rtro 8c al« 

«ra OU interfc^io B afcchdec , % li P fuefic Vettex 

kzisd^ufdami ciim PK tric ih ipfd axc^ 8c adinodum 

'Mt deAdnfttactur , fore eo eafli [intbrfcctiones Ci 

^ B arquc diftantcs a P r lit in Rg. 203 > nec} pote^ 

^t abite G itl P > nifi abcac & B , dfciilo in axiuih 

^crticilius aequivalcnce quatudr eonimuhibus punais ^ 

^quttiiot iilccrfeccidnibui^ Velbini^ interfcdlionibitS; 

^Qiii Cdptskcctit , Vel bttlis cohcadibus . Ac ubi P 

l^ cft in vertiCe axis alicujiis > uc ih fig; 162 > puh^ 

pC, & B lioti ^qub diftabunc a P^ &mucaca cii^- 

Mbili magaimdinc ptlus ali!etum > ut C i eo appeUet 9 

^o B adhuc ihde diftance per ajiquod intervaHum i 

*^ & , ut circulus 9 qui Conicam Seftioaem ofch- 

^ i& axiatTi terttcibus % ipfi nufqham alibi occur- 

^ > nee ibidcm fcoec , f^ vel infcripcus fit > vel cit^ 

tafttipcus ^ uc oftehdi CorolL ) f at ih vchicibtu 

Btuntm diamttrorum ibidem eam cangac > 8c fecct^ 
itenmi fccec alicubi i uc vidimus CoroU. 4. Quddl 
j^^bttc minnacur radius PK > jam iUa incerfcAio C 
^libit ad pattos oppdfitas P, uc ia fig. 204: conca- 
^ tiet interior > tc tirtim aliqnis atcus CB adtiuc 
ra £)Upfim cadcc > 4oni|o coeuntibus cciam punciis 

N ? C. B, , 



178 SEqTIQNUM GONIGARUM 
C> B^ cpDfifi^f ipfam ittrutn |nicriiis, ac. flcmatq tq- 
lius iqcipUt fadct^ in^ra fUipfim/ 

^40^ ]pt quidcjp (i p ,' I in (^. 25P fucrit ws cottr 
|ugatu$^ $c coqcipiarurt' fa<3:o ^cntrq ilifrf^ijii ipfqaxc 
in ' K V f i^fH^^^ f ^ip ^¥^ P^^o quidcm mtniii[ui$ , nim 
pcrpeciiq crcfccn$ ii^ Quidcip primo crit fo^us Jntra EI- 
Jipfirn,' fum cam continscc itcrum in f^ deindc jiic fh 
gurc^ ci^primit/ cam fccabit in |>iiiis pun^^isC» B\> qtii^ 
pcrpcci^q juiccdcnt ad P» C|iin quo congruent , ^bi ipfc 
circulus babqcru^pro di^ftro ^us fc^iufn ejps iEtici^, 
& cyafcf it* QfculatcM: } "ac is crif primus f x iis^ qpitW. 
gcnt EUipfim ckc^rius , qui quidetn rcliqui omiies croiijc 
co majorcsV & t^ti ciora' EUipfini cadetit^ 

5^1; M m fig. ?o? fi P/ fucrit s^xis tranfvcrfiis^ Sc 
concipiacur ciro^ns ptimo quidem' maxiinus , tom perr 
pecuq irofninutus ; priino quidem atpbict utiivcrfamEW 
jipfiin^^tum contingct cciain in f , dcinde fecabit ia 
binls piinctis C , 3j qiix. cum ipfo P. congruent!^, ubi 
is Jhabucf it prq ^adiq djmidiiun latiis ^&mti ejus axis;^' 
& cvafcrit qfiruUtor , *ac ^ crit primus cx iis , qui 
iangcni £llipfim ^ntcrius', gui 'quide(n\<3:iH|C reliqui 
omncs co ' mit^brcs t & foci incra £llip(|ui c^^cti^ . Et 
idein ^^cidet clrcuUs tangctitibus Parabo|ai^ ,] vel Hy- 
jpcrt)ol^ni jn ycrtice axi^ tranfycrfi,' fcd iti.iis circiilus 
utcunique maghus praet^ coiitactum in vcrtice feo^cr 
Iiabcbit binas interfc^iqncs , qux iUoc immin^Co ao^, 
ccdcntacl concaiflutn P , in iUvim recidcnc , nec uf- 
qiiaiii jam frunc codcm prorfus orcliiic /quo i|i fupe^ 
ri#rc ' iiumero , ' * ' \ **' s , ^^ 

S4^f ^xtra axes vcrq fluft^ PK, iH ^nfig.202, pcfr 
pcndicolari tangcnti ^ ^G » ^ fafflo circulo ingcnri , is 
tocus fadcc cxcra £Uipfim»*min itnniinutus iUam ^iciv^ 
bi conwgei circaP, dcindc fccabiciti biiiis 'pun<^sCt 
B, ac in ]Para()pla; uccuir»quc'fic'magnus, fccabif fcn»-' 
pcr , &; adhuc contingcc exccrius ,'aiiquq ejW ftrcu 
CPB jaccnic cxcra curvana ', (tUquo CDB intra '. Im- 
^ninucq Vcrq cciVm m^is ^ircij^q i iaterfe^ioncs iU« 
accedcnt ad conta<Jlum P 5 in quf m ita lncidet akcrat 

ut 



t L E M E N T A, 179 
Jpti2 9 aate alteram > ut iM circulus perimcirum ^ tAQ« 
gat, & fcccf, altpra itttcrfc^iqn^ P nori ?oogruQmc , 
ac ^r ^^ arcabus a P ad B r^man^it ^x^a, ut p^i^ 
nsp idicr crit inv^m txm r^dio adhuc imminuto, |ani 
utniique intcrius cpntit^ct in P, iranftnnte » ut yidi* 
mtis» C; M P^^s pppolScii, ut in ^ 204. > acifauc ta- 
men exeimfe i;ucco td^tquo CB cx^^ ^eclioncm Conio 
fsun > dofiec ppn^i$ -Cl » 9 f:o6un;t&^<^ >nutentur binse; 
bteiiecdpfies |n Qeiita<3;um ^ ac ^tqde mcipiat ^accr^ 
drculus toms tntt^ Sccttonem Coni^am^ 

S^i* Patet autem yel tx ejufmodi confidcratione de-> 

krc iiaberi cirt^upi aliquem j qui ad ^rcuni curvas 

[itqafinodi acccdat magis ^ quam quivis ^us ita , ut 

ui corutn angulo. ^iullus ^ius qrcularis arcus duct pof- 

^t j ac is vei infcriptus (it > yel circumfcriptusV in pri* 

laofcafu maximus ex infcdp^is , in fecuudo ^inimus 

f eiraifnfcri^tis ita» ut ubi bab^turminimus^ circtmi- 

fccipdS) iniBus fit maximus cx infcripfis , ^ viccver^ 

{a. Diim csiitn arcus» (^ui jacebat iii contafStii ^xtra 

furvsBin 9 iBotu contiiiuo xntitatus abit in japetifCiim in-* 

tra » onuiiepo alicuti is tranfitus haberi debet ^ & fi 

pb divcrCun cucv£ i^aturam , nullus circuli arcus con-t 

ffvit cum arcu ipfius cury-a? ^ debct ^licubi iUe tran- 

|uas fieri it^ , :ux c pirculis omnjibus aliquis ilt pro- 

I 3UI1IUS ly nec ijUus propior habcri pofllt , qui ii infcri- 

ftsii Gt\ &9C intra curvam jaccat 9 quivis minor mul^ 

' to tA^gjiS, l^^^cJ^M; intra» ouivis vcro major extra^ aliter 

illc f roximus npn e0et / fcd is alius » qui co tnajot^ 

adhuc jaoerct intra» •pfnninp tOkt propipr . Erit igitur 

iUc maximuscx infcriptis ; fed ut<;umc]ue paraiii alius 

«^iiifpijim illum cxcedat , femper ^ius haberi potcrit ^ 

mu ipfum es»:edat minus, mediusnimirum interu^rum-^ 

■^, Sc centrum inter cprum centra habcos, qui aid- 

' -hac 8c ipfi: circumfcriptus crLi^ , & curvae propior , & 

^ ^ofc circumfcripto niinor » ^dcoquc ille prior non 

h- «liOttfat effe circun^criptorum minimns ^ qupd idem de 

^ 'lioc ixoyp pariter dcmonftratur, & dc alio^minorcquo- 

fis > aim^|ttmii:ttm 4Uto i^^ aliquo pro cuculi 

! ■ N 4 ra« 



180 SrCTlOMUMCOKlCATlUii 
r&dio , nullum haberi poflfit ifitcFvallum , quod iA ipl 
fum accedat ita , ut iniiniti alii accedenties magis htf 
beri non pofHnt • Atque eadehn cft demonftrario pro 
cxdudendo maxlmo ex infcriptis» ubi isj qui cft pro- 
ximu^^ eft circumfcriptus • 

544. Atque in his quidem atiigimus tanmmmodv 

e^mpararionem Seftionum Contcaram cum circuto • 

Omnia, quas in prioribus S Propofiriontbus > & earum» 

acDcfiniribnum CoroUariis^ acScholtisrdemonftravimus» 

pertinent ad compararionem redlarum eum Sedioaibtis 

Conicis , 8c earuih Dtcurros, qui iicet in fingidis rc- 

€kis bini tanmmmodo efli: poflint ^ adfauc taraeh taii- 

:tam proprietamm mulrimdinem prodiderunt 9 quanim 

alif eriam habcntur quampiurimse , quas omifimus # 

quod minoris fint ufus^ & plcrxque longiore dcmoili- 

ftrarionum ambim indigeant > ac complicaiiores fint* 

Qjiod fi occurfus circuli 3 vel alterius Secrionis Co^ 

nicx i qui in fingulis quatcrni efle poffunt i confi- 

derarentur gcneraliteri quam muItS) quanto fuNiinio- 

tc$ proprietatcs profluerent 1» qu; qiiidem maxima fat- 

tem ex parte noftr^ menti impcrvias funt > qui iumi« 

xum rectae lines folius namram fatis evidcnter pcrdpi^ 

m&s, 9c veluri inmemur, ac idoirco ad ipfas rtctas 6- 

xiglmus eurvas quas contemplamur j & qiiartim pro-^ 

prietatcs immectiatc 9 & iii fc ipfis intueri non poffa^ 

Jxtus ? Aiie menris gcncre opus cflet ad ejufmodi Gco-l 

metriam 3 qas ifta omnia vel immedtate vidcrct » vd 

iacile €x iis » quae^ immcdiate videt , coUigcrct . Nos 

ca pcr quandam rclarioncm ad rcaas tamunsnod^f 

contemplaniur • 

. 545' Quamobretti iis ofniflis y lieet nonmilla lot^ 
giore ambitu poSeraus afiequi ^ progrediamur jam ad 
contemplandum Conum^ ejufque Sccrioncs > qu2 huHi 
jufmodi corvis nomen dcderont • Contcmplabimur au^ 
ccm fectiones Cylindfi» &Conoides genitas convcrfio^ 
ne Secrionum Conicarum cirez fe ipfasy eatumqne iu» 
dcm fcctioncsy ubi videbimus EUipfoidem non gigoeiie- 
mii circttlum Sc JBUipfcs # Parabolaidcm ^dcrc Pac^ 

bolas» 



B t lE Rt fc N t A: tti 

itbs l Hfpetboloidtm vecb ttiam Hyp^irbolas coarirf 
aeit. Sed iH iis aliquaato milius iixuiiorabitnur • 

DEFINITIO lil. 

546; Q / TilH lifNtk in fig. z66 utnnqui inde/mitd 

txtra TfUntm dati aradi AB fetfitM m§Hi fercurrat 
roLfite»^ circuli ferifherhm \ fuferjkicm » quam generatf 
^a ^iqicrficiem Conicam 9 faliJum ta inclufim ^ dic§¥.ifb6 
Qoamm V Verticem , rircslum ipfmn Bafim 3 reBam 
VC tranfiuntem per verticem » & cenmm circuli diee 
*Arem , qui fi fmit ferfendictdaris fldno bafisj Connm 
dic^ Riecmm» fecus S^denum, reitam amem iffim g^ 
nitricem Lams Goni> 

SCHOtlt/M i 

54^« OOknt {^krumqoe appellare edndni id isS^ 

O tdm, quodintet verticem^ & bafim intcrja^ 

tet refiqaum vero i^ eatidem appellaht cohum predii* 

«im» ad oppofitam conum oppdfimm« At libet potiils 

coni nomlne appellare quidquid reaa linea ^ qux ^ 

iocus grametricus fimpliciflimus > & natura- fiia uorin- 

qiK fine fineproduci ppieft> gignit motuccmtinuocirda 

Jocum g^nietrieuin itidem fimpUdflimum > himirum 

i.circuli peripberiam • Lodus jgtotaetricus ihteger ab eo- 

.niffi locormn cembinatione nafcimt , ciqus fruftum 

quoddam eft id , quod arta quadain b^ > ac vertid» 

terminatur • Sic ergo U^perbblarum ramos oppofitds 

*appeUari 9 qubs alii fere Uyperbolas oppbfitas uo* 

laienatt 

Vofcll. ts 

' 1 » 

' 5f9. OAuf teShs gmeratur > fi aUero^ anguli AVG 
reSiUnn buere VC immoti > akerum iatus V A convet^ 
drca iffim^ . 
j^49« & tsiim cir quon^ punctb A doeamr AC pcr« 

pcn* 



m SBCTtONUM CONIGARUM 
jpcndicularis in VC , ac in iUo mom gcQcrabic tke^ 
lum ( nanu 30 fiiid.) , qvi erit hafis coni habcnds 
vadcem ia V ^ cujus axis yC crit pecpendicularis ^ar; 

fi ipfi. 

C^alL a. 
5f<>. Si C^nus ijHhis fetct$a^ BMtmqni fUino fer, ver^^ 
ifk€m dkSle 9 feSU rfficiet in fnferficie ceni Hmm relta^ 
earinque idefi^iu fredkctM^ centinentes HnBS angnbs M 
wrticem effefiees , ^Msmm fegmensa infercefts 'ineer t^* 
fticem & bafim Vf cene r^to fquaiis erimt in^ fe , in 
e^no fcalenp pnaqtialia ita , ut omnium minimim , ac 
maximm jacenne in flano tranfeunte fer axeml &fer- 
fendicHlkm ^demiffum t vettiee in flaBo hafa^ mhtimmn^ 
quidmi i}fi fnfendiculQ frcfisu > maximim vero ab ^ 
dem remotiHSf 

551. Si cnim (cdio fiat plano jtranfcunte per vcrii- 
ccm y 9 §c bina puncca pcjipfaeriae Jbafis AB» ubi rccta 
gcnurix dcycniet ad puncta Ay Sc B congruet cum li- 
ncis yA(X, VBN fcctione genicis » cum ^lebc^t. f^- 
jre in fiipergcic coni, &: tr^rc iUa.per punctaV> A, 
ktec fcr V 5 fi . Qsafc ip& linic V AQ^, VBN erunt rc^ 
cta^ , & continebimc angidos <^N » qVn oppofitos ^ 
yerciccm. 

552. Ducds mitem AC» BC tadiis bafis uciqne ae« 
-^«lalibus, ipfi radii in cono ireao cootinebunt cura a- 
ate VC ^gtdos rcctos • Adeoque triangulorum yCA , 
VCB hajbentium prxterea Umis VC commune > baic$ 
VA, VB squalts ecunt. Reliqna paient ex onm 235. 
ibUdoram. " " 

CerotL 5. 
' 55?« Qff^i^ feetia bafi fanallelsi pit eircHlus ^ cufus 
€entrum in iffo occurfu axis cum eadem fectione* 

554. Si enim fectiobafi paraUcIaoccurrat axi in f cx 
uo^ayis partc ycrticis, plaiiis autcm yCA, VCB in rc- 
xtis f4, cb \ «ruttt tc(;tx CA i ca , ific <3B , p^ in- 
jeriec^ones planprum paralklorum fara%lc . ( imm. 9. 
/olidorum^. Quarc cum rectx Aa\ Cc ^ Bi traofeaat 
pcr iddn {^iacus&m V ^tm ( mm ^4^ ^ca M .ci^ nt 

CA 



' 



E t/E M E N T A. ^1«? 
CA ad CB » fitmirttiii ifi r aaqne acqualitadfi *< M^ 
Acnie mtur puncto At /k 4^ & uccumque osQtato Bi 
& ^ > fcpper r^ erit arqui^s ^cm f ^ i adeoquc ^ ad 
^culum .radio r^ dcforiptuni/ ^ , 

.555: Sec$i09ex forMltlA nuutfq^^ incHn$na uufdm c^ 
ni miMp Jaiffer wttrf^ 

' 556. Si enim ABy.M refcrah^ fectiones qualhiniqiie 
farallelns utciuique ftiam indinatas, ac inaneiitii«as re« 
itis VA* y^t plannm CVB gyret uicumque firca re- . 
ctam VC* crunt £ui]per & CA, c^» & CB,^ c^ faraL 
Idx inter fe, ac proinde zdinic cs ad^^» ut CA4id 
GB) adeoque puncta B «^ ( num. zzz« } ad figuras fi- 
piles» 

prW/ 5, 
$57.. /n Cetf9 Sfolenp oIm qufque fectio heifi wm Jd-: 
raUeUy qtue dicitMr fubcontraria^ eft circulHs.^ , * ' 

i^% Si enim ici fig. \ixyj. pcr centriiin C, ic Wti- -^^' 

' cen^ fiucatur (num. 74 folid J jdanutn AVB perpeHi* 

diculaire plano bafis > tum ad qtiodyis punc|um M itr 

ctx AV fiat angulus SMin sequ^is angulo VBA tta » 

iit lecta lAm faciat cum laiere VA cizm al»gulum« quefii 

Ui facitcum yS) unde ob angubun V ^inmupem « 

vcl at^uaJein ia qriaogulis AVB » lAym » coo^qciefiir 

cuam» ut eadem. Mn^ cum VB comineat 4Hmdm .«af 

gulumy ;f}uem AB coptinet cum VA» lum per Mp fiac 

iectio. f^pendicularis plano AVB (aun. V^iblicL) « tm 

fcctio di«itttr fubcooiraria bafi > 5c cam fotc drcBlittii 

iic facilc demonfiratur* 

55^. Pcr quodvis pun(S;um R rectaei Mm dnft^ fi> 

^io parallek bafi toccurrat plaiio A VB in «i^» ^^toni 

<iuAap per lAm in rc^a Pi» \ £a crit ciradus^ ( ttm» 

55}/vCujus\«^ ^rit diameter» ac ch^^^^ ^imer&iJSaa 

|>inorum planorum perpendicidariunaL cidi^^n ' AVB ^ jcom 

iie&eat ipfi perpendicu)arif ciTe» erit perp^dicu]!^isii0i« 

pqv^ tit 6c M^> ac a priprc^ uipote a qirculi dimetro* 

^v^^abicur ^fari^pi in R , eritque quadr^tum PR «quak 

rectangylo aKlt ( Ctx* u I^op» 1 3. Ckom. ) * P^rro in 

trian* 



«4 ^SECTIONUM CONICARUM 
triangulis 4RM j ^Kp^ anguli ad verticem oppofid lA 
K a^quales fiinc » & ob angitfum VMR dequalem tt 
hyooteli angulo VfiA» (ive V^R> eric &: ^MR «qualis 
iwR . Quare (imilia erunt ea cfiangiila , &^MR ad 
lUj uc R^ ad Km, five re(%angulum MKm arquale re- 
ftangulo ^R^s verquadraco RP» Sel9ni autem M^» bi- 
fariam in r quadracum thi xquacur redsanguld MRffif» 
& quadraco cR (imul CCordlli 2. Prop; i^.Geonl), »• 
deoque xquaEHtur binis quadratis cK % RP fimul , fiv^ 
ob angulum e^RP re<5i;um> quadraco rP; Eric igitur £em« 
per cP xqualis clA » adeoque puaftiim P ad circuluni 
xadio rM dcfcripmm . 

Cmll. $. 

^io. Pt9 hafi dfHmi potefi qfuevis feElio fivi patdlk^ 
U {ftim€ b^fi , fivi fubc^mraris ex Htrsvis gdrte m ver^ 
meV. 

561. Nam quevis ejufnaiddi ikStio circularis t^i 8c 
sefta pcr vercicem V tranfiens> ac cjus fuperficieib coxi- 
radens eundem generat conum ; 

CerelL 7. 
5024 QH4vis dHs feaie coni erit Ell^ , PMrajfoUi 
vel Hyferbola^ fi^out flanum per coni vertiGtm AuEbm 
fUno je^ionU fdrdlleiitm cddet extrs connm % vel enm 
fomingetfi, vel intra iffu^ immergetnr. 

5^3; Secetiir ehim quivi^ eooitis qdovi^ plaiitf HoA 

]>at;aUcld bafi , & planum ipfi feitioni paraileiam do^ 

^ftum per Vetticem V occorret plano bafis^ in ttStt qua^ 

!^2o8dam OSj qux vd cadec excra bafim , uc in fig. 208 j 

:to9 ^09) vel eam conciciget aiicubi» uc in fig.2io> vel in- 

110 tra ipfam immergecto: 4 uc in fig^ iii i ae fi dHcatilt 

aix per cencrum bafis C re(5ba CT ipfi OS perpeddicularis 

dcctirrens perimecro bafis in punais A , & B , caddc 

pundfum T in fig.2oS, 209 exnra diamecrum AB > iit 

fig.2io in attero cjus exoremo, ut B> in fig. 212 iocra 

«liametrumj qus nimirum fegmentnm re&« OS cireiH 

io intercepnim > cum ad angulos ttAo€ kcet i iecabit 

(C«roIL4«P£opv5<^G6onL>) bifafiam^ 

564. Dti- 



E L E\U E N T A. i»j 

5<4. DuSto jam pciTABy |dano> quod platiQalliOVS 

bccurrct^in rcfta VT , fupcrficici coni in rcdis VA t 

VB> plano fcdiofiis! in rcda quadam li paraUela (num. 

9. folid* > tcctx VT ob parallclirmum plani fct^ionis 

asra plano OVT > quc' idcirco rc<ftam VA ftcabit ali^ 

cubi in M > ac (i ponatur pundum I ab M vcrfus co^ 

rium j £ ad partcs oppoiitas > ncceflario £ccabit in fig. 

2o8, 209 ciiam latus VB aficubi in m yerflis I ^ erit 

in fig. 2ZO ipfi para^lcla > in fig. 211 fccabit verfus i 

^ partes opppiitas fupra vcrnccm V ipfum lams BV 

produiStum , cum ipfa VB in fig. 208 9 209 dedinet ' ab 

VT verf us paralldam U ad partes B in fig- 2 10 cum 

priore coogruat, in fig. 211 dcdinet verfus partem op^ 

pofitam. Quamobrem rectse li fcgmentum Mm totum^ 

& f«lum jacebit in fig.aoS, 209 inora comum, in 6g. 

211 exnaa, in €g. 210, tota MI indefinita jacebit intra» 

tota vero Mi cxura, 

)65. Afluippto in ipfa li pundo quovis R inter Mt 
ic n^ in fig' 20S, 209, extra eos Jtmiitcs in fig. 21^1 , 
ab M verf^s lin fig.210 ducatur pct id pundhim pla4 
muvfiaraUduin plano bafi^, quod pkno - AVB oceurrat 
inMrceta at , plano prioris fedtionis In I^ , & paoet fo« 
te ipfam iedionem hanc novam drculum (num. 55^) 
diamctfo dit^ ac ipfas «^, AT^ ac Pp, OS interfedlio* 
ncs planorum parallclorum qim eodem plat^o fore ( n» 
ffolid.^ parallcl^ in(;er fe, adeoque (num. 19 folid.) ut 
AT eft per conftrudioncm perpeodicularis.OS> itaerit 
dianocter 4^ perpendicularis chordx Pp , quam proinde 
(Coroll. 4. Prop. 5. Geom.) iecabit bifariam, adedquc 
& reaa lierit dia^^r quxdamprionsfcctionis, cujili 
nitnirum chordas pcr quodvis punctum R traafeuntes 
S^allclas cidcm datx re4k QS , & inter fe , fccabit 
biiariam. 

566. Dufta MD parallela 4B, quac rectx VB oocur- 
^t in p, ac in fig«. 208, 1:^09» ^ti ducta. paciter md 
parallela eidcm AB, qux occurrat in ^^ rectse VA > ja^, 
cente md in fig* 208 intra triaagulum VMD , in fig» 
^.09 ^xtta ad pjyrtcs MD , in fig. 211 cxtta ad partcs Vi 

con- 



- ■ V 



126 SECTIONUM CONICARUM 
cAcicipiatBr ctrculus rcctam AV^icacltiil^ens ia M » ac 
trtnficns pcr D (is duci ppITct^ fed vitandae confufio^ 
fiis gratui nbn ducihir^^^^ a' recca li tranfeuhtc peif 
contactum M abfcliid^t fegnTciitum MB iti > ut diidi 
DE i anguki^ MED aequemr (Cdroll. 6: Prop, 9. Gtom:) ' 
Mgdbi qu^ni chordd.MD conttncc ciim.ipfa, taiigente | 
AMV ad parte!» bp(ioiitasj adedqiie atiguld M/tR i qui 
in fig. ioS flcqiiatur adguld AMD > in reliqiiis dhguld * 
VMD cxtcrno i: & oppofito* • Cunique ctiam.EMp «- 
^iictut alternd MR^; fimilia criin^ tiriaagula iRMj 

EMDi ad dK ad RM, ut ME ad MD: , 

f6t* Eft auteni pr^tetea id fig; iid ; ob MR i Di 
j^drallelas; MD arquali^ R^; Erit.igimif ibi^iR adRM^ 
Ut M£ ad Rf j ddeoque rcdbangmum ^R^; £[ve qd^a* 
hinl fcimoirdinats; . RP. zqhale rectMguld fiib ajbfciffa 
MR> & teai. confianti ME{ idedqcie ( num. i}40 ) pun^ 
bti y ; p ad Parabolam diametrd MI parametro Mfi 

dcfixtptahiJ ,, , . -i . > 

. $6S. At in reliquis erit praQterefl R^ ad Km 9 # . 
MD ad Mm . Quare conjunirtis fattonibus , rectjtng^ 
lum «R^ j five qtiidratifm femidrdinatse RP ad rehaia» 
j^um MRm fub.binis abfciflis' a binis verticibu^ > uc 
fectangulntn ftib ME, 8c: MD dd fectangulum fubM^i, 
ic MD> five iii ^nHanct ratidne ME ad Mm i adeo- 
qiic (nydi«43^) puncta P{ p erutit in fig. 20S 9^ 209 
ad EUipfimj in fig 211 ad,%perboIam dcfcripcani dia-- 
inet^o Mm, 8c parametro ME; 

, / .. CVr^/A 8; . . . . ; - ' 

f «9. In Etlipjtf & liyperkotd diamfer C9W\fig^d dU^ 1 
itttH lAm eft mdi4 gtakietrkt profortio/ialis inter ' 
MD> md ; , , . . 

V 570. Erit entni md ad Mmi, ut Kd ad RM, fivcuc ' 
ME ad MD 5 adeoque recaangulum fub md y 8c MD 
aqualc rcctangulum wME fitb diametro & paraiuc- 
c<0,' Himkuni ( nuiii; 351-) quaklfato diamctri con- 

571. Sp pUnm AVB fHorit perpendkutare plano bs- ' 



7 ^ 



E L E M E N T A. i»7 
/ts^ qMd in eono rmo vomigit femfer i in cow fcMletid 
in uni€4L direSiont , didmetri AB i erie iMi 4Jcis > (jr 
i^tddlnn in Hjferbold .Mm tcmpcr in eo cafi$ erit tucis 
iranfveirfiis i in EU^ in (tonarecio fsiriier ftmfer ttmjf'^ 
ikrfusy in Cdno vero otUqm erit tr^uifverfui^ vH cohji^ 
gatusi frout fictio jacucrit inter fectionkm.fatailitmm Im^ 
fi ductani fer M^& fulfconttnriam i vei exttd mrmm 

afi^uiiM», '...'— i • ^ 

57a Si cmni planuni AVB fuenc per|ieiidiculare pU^ 
ho bafis^ recta OS jaceh^ in plano bafis» & pcrpetidt* 

cularis per conftrudibnem ihterfe&ioni AT, plani AVB 
^m ipfa hafi j eric ( n; 66; folid ) perpexidicularii^ illi 
^pfi pltho, \ade64tte &: reaac VT ; Quare 8c cirdina» 
Vf cric peri^ehdicularii diametrd Mm i adeoqae hbef 
(nunM aio) erii axis ^ . ^ . ,. * ^ . , , ♦ , - 
r 57^,Cum vero ih cohb ic&6 axis coni per C crtif-^ 
^etis fit petpehdiculatis plano bafis i quodvis idanum 
AVB^ cranfiens per V & C ^ adeoqhe per dcem coni j 
teitjhuth.62; folid.) perpehdidulare pUho bafis; Ai ift 
^oho fcalcno perpendicuIUm cx V tleiniiium iti plsinuni 
bafis cahef cxva C ^' adeoque ih ca unica . dtribftio^ 
he, ih qui diameter AB hanfien^ pdf C dirigahu? axt 
id pimcmm i plahum AVB trjiBfibic per fe^bim' per« 
pehdicularcm plano bafis^ adeoque ipfi pcrpendiedt*' 
rc crit; 

574. Por^o in Hyperbola axis cdnjugsims ipfiiis ft^ 
rimetro nufquam occurric (num» 112 ), adcoque ecmi i 
ipfi occurrat Mm in Mi ic m; ecit axis tFaniverfiiis; 

575« Pro Ellipfi verd fi fig« 21% cxhibcac criaAgiilttm .. 
AVB pfo^ cafu coni r^ti /f\gur; it; V 214 P^ cafuF.lli 
cohi fcaleni^ quod in illa erit (num.550) ifofttlcs^iil alj 
hac fcalehuih , circulus M£D i^ primo cafu ddhcii^gtc f ^r< 
^tiahr lahi^ VB.ih D V ih ^cundo ipftim ibi fecabic » 
fk itcrun} iccabic paritec aficubi ih L vcr^B; vel vtSk 
fiis VV prout lams VA; iri qiio \sacx, Mi iuem ^^hi^ 
htire VB, ut ih fig; 21J, vd minus» ht in &^i^^ 
%\ enim eius^ circuU centtum fic Oi ductis MOy IJO> 
aogultt^ OMD crir w^iftiix. angiUo OiAXi isb laecrt 
' . ^ ' ' OM, 



itS SECTIOKUM CONICARUM 
OM> OD aequalia, cumque & latus VM fit in Hg^ 
"ziz aequale lateri VD> in ^.213 mafos , io ^. 214 
mious^ eric angulus VDM aequalis in fig. *2ia angula 
VMD) major in fig. 2T3> minor in 6g. 2i4> ac pr»- 
jnde totus angula$ VPO xq^alis angulo rccto VMO 
in fiig. 2i2> ma|or in fig. 213, mindr in fig.. ^14 i^ 
Quamobrem leaa quoque VDB comkiget cirojJum i^ 
fig. 2129 ipfum in rdiquis fecabit alicubi itr Lj far 
cente i ad partqs anguli acuti radii OD cum rccta 
fVD » nimirum in fig. :^ii a D ycriius B » & ia % 
^if verfus V. 

576. Hinc in fig^ 2I2 ducta quavi? Mm^ qua&Iateri 
VB occurrat ab V vcrfus B> vel fupraMD» ut Mmi , 
vel infra m M1992 > fcmpcr ea. prius occurret circulo ia 
B19 vel £2, critque fempcr axis Mm nuijor Utere re-; 
cto M£> adeoque multo major ( num. 351 } altero a- 
»y Sc proindeerir axis tranfverfus. Atinfig. 21^3, 214} 
pbi m abicrit in L» BcmMmy M£ acquales abeuni^ ia 
L ettam £, quo cafu^ ^uabuntur axis , 8c cjus latus 
irectum % adcoque bini axes ^ £Ilipii abeunte in cijrculum 
juxta num. 109 y qui quidcm cafus pertincc ad fectio^ 
nem fubcoiurariam ob aogqlum MLD ^ualem angulo 
LMD in fig.2i3> 8c AMD in fig. >i4.-t^ngen|is cum 
ehorda MD refereme fectiQp/cm bafi parallelam. Quiare 
qu^vis Mmz jacens inter MD j ML occurret pjrius lan 
tcri VB.> quam circulo ultra ipfum procurrenti , eritque 
«xi Mm^ minor fuo latcre cec^o M^^^ >' adeoque & 
axe «Itero • Q^^vis autem jacens exora eos Bi^pes , 
ut Mmi y M193 > erit major fua M£ > & proindc ich 
itra eos limi^ crit M^ ws conjugat^s , ext^a eo% 

traafvcrfus t 

Or«//. lo, 
577» £x quaviscoff^^ ^Afcindi f$ufi qneifis dafOrJElti^ 
Sfih acP^iratfiU^^lM^im^ i$idm HyfethoU lii^t.pon 0- 
pmf> 41C €x (^om reeto nulU fofeft ex iis^ in quibfu /4- 
tm rectnm frincijfole ad ^a^em tranfverfum^ h^eat ra^ior 
nem maj^ntm y quar» tatfgenf dimidii 4tngHli AVh in 

prwe mftitHtiiad cmanffntem^fivef qjmdjeodm re^. 



E L E M E N T A, »8? 
'ifl^ ^ in fniius fxif conju^jitus ^d trdoifvirfum habeat 
iM^mM 9fajore^y quam tdngeris e^fdtm i^imidii an^uli 
fi raSiifi^m^ reltqu^ omnes jjioffunt . : 
.***578. Nainpnpio quidcm |n feg. 212 fcc^o cono ut- 
^tamque per ^eip plano AVB j, Ar afTumpto puhcto \| 
^ arbitriuni, capiamr Vp iquafjs VM , ducatur cir- 
ulus tangc^s AV in M / & tranflcns p?r b , capia»» 
or MF ad Mv in ea ration^^ m qua eft in data £1*- 
pii ' ^acus recmin principale ^d axem tranfverflim % 
qtiQ^ cojn femper fit minus ipfo ia^re tranfverfo ' ( n^ 
^^,64 ) crlt fcmper. MFipiiQQr , quam MV , adeo* 
qpeaha <;x F rcaa parallela VB. / ca ncQefTario' ocqir^ 
^r alicubi drculo in binis puhctis £1 3 £2 > cum ipfa 
.VB. illum tangat (num. 575 ) • Si autent ducantur re^ 
ctz \/tE\m\\ yfiL^mi^ ip6e dccerminabunt fcctioncs H-; 
tniWs datse SUipfi ; crlt ^nlm in iis hms' irai\fvcrfuin 
M» ad rccmm ME, nt MV ad'MF a nknirum ui ii; 
^ data ScctionV Conica Ktus tf anfvcrfum ad' rcctum , 
.jQuarc fi ialter ex iis '^xibus M^ cvaferit ^qualis axi 
^ir&fverfq daMB Ellipfcds , fcctio per ipfum ducta pjr^ 
^ fcndicalaris' piano AVB^ cxhibebit ^llipfim datam ; fl 
^.Ucatcr, fatis crtt 'afTumcrti in* ipfb Htcr^ AV aKan^ 
|VM, qux ad priiis afRiinptam fit,* ut cfV txis tranfvcr^ 
^fiis datar EUipfcos ad Mwi / Mw2 ^rf^i iovcntas j 
J&: fccdo pcrnovuhi punchimMparallelaductaepcr prio-^ 
J{em Mjrii, vcl M/i^i cxbibebit qusefitaih EUtpfim. Erit 
Lcnim (nuin. 555) priori fectioAifimilis, ac cjus axi^ 
fran^cr^ a4 Mi» prius iiiy^ntam | vit^ ' nova VM a4 
ii^prcra. * " ..,.>-- 

L* 57^. Qupd fl agamr ME3 paraHcIa VB , ea dctcfc 
JBiriaEh Parabo&m, ih qua fi fatusf rectum hdn obvc- 
tocrit arqnale latcri rccto datop P^rabofce ^ codcm aaifi^ 
mCIo mutata VM |n ea ratipn^, inyciuctujf Porab^I^ os-t 
.i^ualis datas. . 

I 580. Si dcmum aaa diametrQ DOF , tangens pcr ' 
kccurrat latcri VA ih H^ & d^mr Hypcrbofe , in q-vi 
ptus rccmm principafe aid isExem tranfverfum habeatra- 
Bonem^ mcumque minorcm \ quam HM ad MV , ftt% 
^ ffofim€h.Tm.III. W 0»- 

( 



\ 



\9Q 5EClrid^ltJM CONICARU^i 
tnatur M/ ad ipfam MY zd partes oppofitas V j five 
vetfuS H ih ratiqne cjus latcris rccti prinicipalis ad a- 
Xciii ttatifvci-fum $ & rcdta cx / parallcli VB codcm 
pado dctcrlTiinabit bin^l puiid^ E4 , E5 y ex quibiii do^ 
ctas binse E^ dcterinihabunt binas Scdioncs fihiilcs 
Hyperbolos datae^ in qua il iUi ratid latcris rc(Sti prin- 
cipalis ad axcm traiifvcrfum fucrit cadehi ] dc HM ad 
MV * cocutttibus punctis E4 i E5 in I i fcctio pti I , & 
M dticti exhibcbit Hypcrbelam (imilcm i fi ratio fuerit 
kdduc niajor^ patct Cmilcm exhiberi noil poffc ; Muta- 
to igitur pimcto M, ut priiis i itivcnictur quidem Hy- 
pctbola d^qualis data? in duplici inclinatidne io( primd< 
cafu i (inica ia fccimdo i it iti tcrtio invcniri nequa^ 
quam potcrif.- 

$81^ Porto quoniam ot tangcntcs^ij HM> &VMi 
VD 5 seqdalcs,' rcctae OH i OV fecant bifariam angulos 
lOM, MOD / angulus HOV erit aequalis binis lOH t 
yOD< qlii cum ipfo ddndituunt binos rcctos»' adeoque 
erit rcCtuSi & an^uS MOV i qui oK OMV rectum 4 
<ft (^omplemcritum ^nguli MVO > erit (^omplcmentlim 
MOH,^ adeoque ipfe MOH arqualis illi MVO dimidiof 
totiiis AVB, Cuin igitur firit HM,MV tarigc^tcs angu- 
lorum HQM,» MOV, crit illi tangcns ,' hacc cotan^cnS 
dimidii ariguli AVB 3 & Hypcrbolac , quae noni potc- 
Xunt fcc^ati c% datd cono rccto, criant eac/ loquibuslai» 
ttis rccturil principale ad tranfvcrfum habct ratioJ' 
ficm majorcm , quam tangens illiifs dimidii angul%^ 
ad cotarigcntcm 4 Quoniam vcro db fimilitudirfem triaa.i|^ 
gulorum rcctangnlorum HMO / OMV V cft HM a<f 
Mo i ut MO ad MV ^ & eft latus rcctum principj 
Ic ad axcm conjagatum , utf hic^ ad tranfvcrfum ; 
axis conjugatus habacrit ad tranfverfum rationem ms 
jfotcm , asqualtm i> vcl minorem tcfpeitif ejus < qu: 
HM tangcns dimidii angulr AVB jid radium MO 
habebit pariter laius rcdhim pripcipale ad latus tranj 
verfum racionem majorem , acqualcrii , vcl ri^iino^ei 
rcfpedu cjus f quam Labct tangcns HM ad cot 
gentcm • 



^ . E L E M E N T A; i^i 
5S2. In cono ^utera fcalcnd fi AVB iri fig. 213^ 2^4 
refcrat fedlioncm pcr axcnli qua^ fit pefpcndicularis bA- 
fi, cddcm prbrfus argumcnto haberi potcrlt quaevis El-* 
lipfis fernper duplici. incliriatiohe Mm i ; iAm^ ; ad li 
concipiatur Iji pairallcli latcri VBi qua; tadgat iii £ ^if* 
fcum , LD fitum cktri inguluni AVB ; & iratio , axii 
tcan/verfi ad conjugatum fucrit minor rationc Mh jldi 
MVi vd ci itqualisi pbterit cadem iila EUipfis erui tt 
fcodcm cond binis dire(5libnibus MEi, hinc &: iridc ab 
ii vd unica; qua E abeat in.i : Poterit feitiper Para- 
bdla dircdlione ME4 parallela lateri VBi ttim fuccedunt: 
brania Hypcrbblariim gehera iifquc.ad fcam J Clijus la- 
ias rediini principalc ad trarifverfum fit in. MH ad 
MV; Quod fi AVB hbn rcferat fectioiiem bafi pcrpcn- 
dicolarcm; fcd aliani quamciiniqiie i dcfiniri pariter |)6-: 
terunt limitcs i^atibnis» ^uini habebit latus tcctdm c^-* 
]rifpiaiA altcirius diametri ad fuaiii diametrum , ita ta« 
iiieti > ut Ciim ticc atigiilus V ; ned iticlinatio triangu- 
£ AVB ad baiim variari poflint, nifi iiitra cet*tos limi- 
bii femper ccrtus in quovi^ Cond habe^tur limct p3:<$ 
^j^bolis: 

CorolL ix. 

, jSj. Dat/i quavis SeSiione Conica inveniri pffuHi 
injiniti coni , ex quihHs ed ahfcindi pojjitj qui tameTt 
ai Hyperbolan^ ^qHilatetdih ahfcif^dendam iMhere debent 
in cono re£l6 dngtUum ad verticem V reSlHm 5 vel acuti 

m^orem*. 

^ 584; Nani quaevls EUigfis & Hyperbola abfcindi pof- 
£int cx quovis coiib ; Data autcm. quavis Hypetbola » 
(L fiipra quamvis ircdani AB iii fig. 212 fiant ^guli 
VABi VBA intcr fc^iqualcs^ & non miriofes coj cu- 
fti^ cdtangctis ad i^adium tA i ut cju^ Hyperbolas, axia 
ppiijugatus id tranfverfum» tum diametro AB defcriba^ 
fUr circulus in pland perpcddiculari ad planum AVB &: 
~ mpto y pro vertice ^ ac ed circulo pro bafi. j fiat 
aus i . ex co fetnper abfcindi poterit cjufipodi Hyper- 
i Cmn cniifl biril ang^Ui VAB i \BA fimul cunt 

i AVb 



/ 



i9ft SEC-TIONUM G0NIGARUM 
AVB contlneant bin ps rcctos, finguli fuht comglcmeiii 
ta dimidii anguli AVB » Sc eorum cotangcns trit ho- 
jtis dimidii tangens . Quoniam ycro tangens ango- 
ii '{erhirccti iquatur 'radiD f mim, 4.9. frigon. ) , & 
anguli minoris cft mini^r , majbris major ; ut apqutla^ 
tera eflfe poffit Hyperbola , debcbit dimidtum ahgult 
AVB non cflrc fnin^is fciairceto > adeoquc is to^ ao^ 
pffc acutus , . ^ • . . ^ . - 

S CH OLI y M H. 

'585. A Tquc hoc pacto jam habcntur pr«ctpua tOr 
jTSl rurii, quae ad conorum fcctiones pertincot » 
9c notari fecile potcft affinitas, quam habcnt int^r fc « 
& cum rccta> ac mutua transformatio in fe iaviccm , 
& ijl rccta$> ei fimilis, quam perftcuti fumus in 5ch^ 
lio 2 poft QjroU. 20 ddin. 2. a num. 107. Concipia-. 
tur in fig. 212 puncmm M imraotum , dum punctuin 
m primo abit in V, EUipfi co cafu in infininim anct. 
nuata, arca cvancfcit , ac cjus pcrimcccr abit utrrnqu^ 
in rectara MV. Indinat^ Sectioi^^ vcrfus D in Mmi ^ 
hab^tur fllipfis initio quidcRi tcnuiflima , 8f fortnai 
^mbdum oblongac cxiftcqtc ratione latcris rccti MEi . 
ad tranfvcrfuai Mwiadmorfumcxigua, tumfcnfim pin- 
guefcit , ac ubi n$ i abit in D , aequalibus latcr^ rc» 
CK), & tranfvcrfo, migrat in circuhm : mm In Mmx 
rcdit ad formam iterum oblongam , tc itcrum decre«^ 
fcit fatio latcris recti ME2 ad tranfvcrfiiin Mmz ptt. 
omnes gradus in immenfum ,, doticc abcuiitc E2 ia 
Ej , veciex m ita in infinitum reccdat , yx nufquam: 
|am fit ^ ac Ellipfis in Parahol^m migrct ," nufqaam. 
jn. fc fcdcuQtcmi Indintto autem adhuc raagis, uicans 
quc parum , plano fectionis per E4M , jam incipic vcr4 
tcx rn^. apjiatcre cx parte oj>poClia V , initio qoidcni 
in immenfe diftantia ita V ut nuUa fit diftantia ia ib 
dctcrminata cjufmodi, qu» cuipiam dctcrmiaatQ piitH 
ftg £r^ tioA xcfpondeat j qua proindc majorcs ali^ 

tca 



• I L E k k k.f A) : fpf 

i|tt non extittrint refpoadentes aliis punctis E^ acibu(^ 
pn)pioribus puncto £3 : Parabola autem jam in Hy-' 
t)ctboIftm migrat binos habentem , ramos utrinque in 
infinimtri prqdnctos, in qua r^tio lateris rectt ME4. ad 
tranfverf^ni Mw4 initio in imm^nfum exigua fenfim 
crefcii dilatata Hyperbola^ forma^ ddnec abeunte E4 ixk 
I^ fia^ maxima ill^a ratto s ^tum jterarnjqadem in E$ 
decrefdt,. ic cbmprimuntur Hyperbolx, ac demum eva-' 
nefceiite MB5 i ^ abeunte »5 in V > definit Hyperbola 
ia reaam^ ab M verfus Ai & V ad parces oppofitas ia 
ifumcnfum produaam. >. . j ^ 

j86^ Idcm contingit }h, ng.\ 213 , & 2^4 in co-' 

no fcalcno jEuih hoc fqlp difcrimine , quqd ,ubi Ellip-^ 

Ss primo oUonga per M^i perpetuo pmguefcit t ac a-- 

bit ia circulum in ipfo . aippiJfu mi in fig. 2x3 ad 

D 9 in fig.' 214 ,ad, L dilatamr adhuc magis > factoj 

&fo2 iam axe conjugato ^ mm iterum a4 fprmani 

occul^em redit abeunte ;» in ^g. 213 in L, iti 

l^ 214 in D , acdcinde oblongatur in jmmen'^ , 

^ ,. dum in ^Parabolam ddfinat , ^ ad Hyperbo^ 

lam tranfeat primo ,quiddm . fe veluti txpandentem > 

tiim iterum Gompreflam , dpnec abea^ in rectam 4 

4c in ofnnibus hifce cafibusi EIKpfis « ac Hyperbo« 

la,.ubi ia rectas definunt , id prsftant axe tranfver-^ 

£0 finito , .& ,latere reoto evaaefcente , ac perime- 

uo utrinqiie abeonte in axem , dum Sc axe excre- 

'icente in immenfumy &: latere recto finko , in Parar 

bolam. migrstnt • Poft omnes Ellipfium , ac H^perbola'^ 

rmti fpedcs a^lfttingentium form^ii ita, ut ratio latct^f 

rcBd sA tranfverfum, decrefcat ultra quofcpmq^e Itmi^ 

-cts, bim fuBt velutlimites quidam, rc^ta Iinea> Sf Pa^ 

x^hdaL , qua; quodammodo velut ejufdeiii , fuqt ultimas 

i^ddj 6^ ad aiteramdevcnitur axe tr^fverfo iinito 5 

^ laiere reao Qvapefqente , ad alteram finttb iatere 

s^cdo, & axe tranfvcrfo cicrcfccntc in infinitum • Ut- 

^ramque parum qusdam Ellipfis, & Hyperbola a re« 

^a df(!enti & formam adftrin^nt , habent fcaionem. 

^aJiam , ParaboU poritcr proximam , majorcm quU 

O 5 dcni^. 



I9if SECTIONUM eONIfiyiRUM 
dem ) icd fortna: prorfus c)afdem , atque ip(i omninoi 

fiinilem , 

587, Quod fi nianctite (direcSlionc feftionis » concir 

piatur punttum M accederc acf V, fain ElHpfis, quam 

Parabola, &Hyperbola, cafhdem rettn^nt formatn, jux- 

ta (nvim/ 555)9 ftd perpcdio decrefcunt, donec abeunr 

JF20?tc M it^ V Ellipfis ut patct in fig. 208 , 209 abcat ii^ 

-09 unicum pundhim V, Pvabola in fig. 210 in redtani 

210 VT 9 Hyperbola in fig. 211 iii binas redas YO , VS 

211 utrinquc in infinitqm produftas juxta num. 550. 

588/ Si mancntc bafi, & plano fcdionis , vcrtcx V 
Hiovcatur, pcr rcdam VT, aC tlcfinat jn T> EUipfisqni- 
dem iii fig. 208, 209, cocunabus pun<5tis M, m dcfi- 
nit in reftam pcrpcndicularem re6l;c CT confideratam 
ut duplicem interceptam tangentibus ex T duftis ad ba- 
fim , abcuntc fupcrficic coni in pmnc illud fpatiuni , 
quod c^ tangcntes uuinquc in infinimm produdbs- 
contincnt. Parabola in fig. 210 definit in unicam fim- 
plicem reftam itidem perpendicularcm C T tndcfiai^ 
tc produftam hinc , & indc , abcunfje coni fupcrficic 
in totam arcam bafis hinc indc ^ tangcntc OS indc- 
finitc productana • Hyperbplsc in fig. 211 ramus utcr- 
quc abit lix candcm unicam rcctam codcm modo la. 
infinitumprodyctam, & confidcratam ut dupliccm ita^ uc 
iti cam totam* finguli abcant rami , abcuntc paritcr vt^ 
traquc coni fupcrficic in planum bafii^ indcfinite pror 
ductum , * 

5??* Quod fi punctum V rccedat a bafi ia infinitum 
pcr eandem rectam ita , ut nufquam jam fit , conas 
quidcm definit in cylindrum , at Ellipfis formam EUi* 
pfis rctinet, Parabolac in fig. 210, ac Hypcrbote ^. 
211 vcrtex.V nufquam jam cft, perimcter vcro abit in 
binas rectas parallelas , quse funt ipfa cylindri tatera • 
Atquc eodem pacto liceret plurimas alias traasforma.^ 
tioncs (Tontcniplari , Quod vero ad cylindrum attincr » 
]zm hitic infcm potcft quamYis, fcctionem ajci parallc^ 
Hm cfticcrc in cjus fuperficies binas rectas , quannris 
paraUclam bafi^ vel in cylindro obliquo fubcontrariaxn 

cffi- 



t l t U E N T a; t^^5 
cfficcr^ prculum b^fi irqualcm , quamvi5 aUam cffi- 
ccre mfipfim . S^d ca , qt & pauca ^U, qua: ad cy- 
Imdri f^ctioneS pcrtinent, libct porius pcr finiiam Geo^ 
xnctriam accurate demonftrarc , quod utique pra-fta^ 
ri poterit fcrc ?adcn^ propfiw incthodQ , qu;^ in COHq 
pfi fomus. 

PE FI N I Tl Q IV, 

590. C / reSia Nn in flg. zi^ ntrinque indefini$4 fm- 
v3 fer faralleU dau adpi^irf^ reEi^ P^fi^^ extra 

tlMmem dati circHli AB ferpetno f^curraf ejufdem circuli 
feripheri^nt ^ fuferficiem-, quam gevfrat^ ^^V^^upcrficici^f^ji» 
Cylihdricam , felidum f^inclufum y dico Cylindrum » 
circulum^ ipfum Bafim ; re^a;m VCu ^er centrum J^ajis 
dnSdm , & dat^ illi reEl^ paraHellam dico Axcra , qui 
fi fuerie perpendicuUrU plano hafis , Cylindrum dicQ rc- 
ctam , fecus obliquuni > reSlam vero illam mebiiem du^§ 
f^^^liadri Latus»^ 

^ C H O L I y &1 I, 

591. T TIC parite Cylladrum appcUavi toftim locum 
Jl'*! geomctri^um, qui da^ura fua in infiaitum 

atrinque projducitur, licetpIerunqucCyIiiidrit\oxiiinc dc-* 
fignari folcat hujufmodi Cylindri fcgtncatum. tantum-» 
xncklo binis planis parall^^is tcrmiuatum^ 

Qorqll, x^ 

55>2, Cytindrus re^us generatur , J^ altero e Hni^ op^ 
fofiti^ feBanguli lateribus ^trinque in infinitum pr^du^ 
&o totum reSan^ulum cire^^ (atus ^lterHm. in^metum cen-^ 
vertatnr . 

593- Nam utrumvis c rcHqui$. binis Utcribii^ CUm 
latcri immotp pcrpcndicularc fir> defcribct f num, 30 
folid. ) circulum pcrpcndicularcm . ipfi latcri immoto » 
qaod prQinde ctit axis CyUadri , cujus iUc citculus cfi; 
% 9fi$ • 

p 4 Co- . 



t^S SBCTrONUM.CONICARUif 

CorolLz. 

594* si Cylindfut quivis fecctur utCHmque flano ,fif 
nxem duSlo , vel dxi parallelo ; fectio in e]us fufaficu 
generabit hinas rktas axl faraltelas utrinqui in infinU 
ium- froauctas * , . . 

595, Sedabit cnim bafim in quadam rccca ,AB j ^ac 
fi fcctio traafeat pcr axcm in ipfo plano fcctionis da- 
ci potcrunt pcr Ai & B binae fcete Q5, Nn paraUcI^ 
cidcm axi, fin minus> intcrfcctioncs planorum VCA ^ 
yCB i ium ipfo fcctionis plano crunt.bin^-rea? Qj3» 
1>in tranfcuntcs pof A» & t^i cum quibus dcbcbit ccwJ- 
griierc rc^ta mobilis» qus fupcrficicm gencrar> ubi a^ 
pellit ad puncta A > B • 

CoreiL 3. 

J96* ^avis fectio bafi farallela erit circfdus ha^t 
'^tqualisy cuius centrum in iffo occurfu axis cum eaiem 
fectione > ac Cylindri latera binis flanis faratlelis in^ 
tifcefta erunt ^aqualia inter fe. ., 

597. Si cnim fcctio bafi paraHcIa occurrat a3M ia- 
c ; plano auten^ VCB bafis in rccta CB > ei vero (t* 
ctioni in rccta cbi^ erupt CBr cb paraUela: ( num. 9. 
folid. ), adeoque CHbc parallelogrammum , cu|us latc- 
la oppofii;g ^qualia, Sc proinde cb femper as^aalis ci- 
dem radio circuli GB > ac pariter Sc fi^ fempcr scqoa^ 
lis fcidcm Cc^ 

CorblL 4« . 

$9^. ^^vis fectio faraliela bafi > Jpro iafi ajfuthi 
foterit. " '- . 

^ %99^ Patet ex ed, quod' fit circulus, & reaa mobi-^ 
Ih tam ipfiim^ quam bafim perpetuo conradat»^ 

CprolL 5. 

600. /n Cylindro obliquo alia qudque fectiq baji 
non farallela' , qua fubcontraria dicitur ^ efi circulus. 

601. Si cnim in fig. 21$ per axera VC ducaturpla-» 
^F.2i6^nam bafis f>Iano perpendiculare , fccans bafim in rc* 

cta AB > fupcrficiem Cylindri in rertis Qj? , N», an- 
gulorum f AB , /^BA alter etit aciitus , ut qA& ^ at- 
tcr obtufus 9 ut nBA . Qj^are fi c qu0Vis puncto Kt 



/ 



\ t t E M B N t Ai W 

iketz Q$ ducta in eodem plaao recta MD pHHlcU 
diamctro bafis AB 3 cui & xqualis ent> an^uIusAMi» 
i^ualijs ang^o ^DM^ occurr^tite.ea recta lateri l^n in 
i», crit ^ Mfl^D ^equalis ipfi MDw, cum «qoctur al- 
icnio AM»> & trlatigulum wMD ilbfceles . |?6rfq fi 
CyUndrus ftcetur^ pcr Mm plano ^erpendicuiari ipfl 
AMDB , ea ifccdo <iicettir fut)Coatraria> & erit orcu* 
ius bafi a^ualis^ 

^2. N^n per quodvis panftum.R recbe iAm factn 
itctioBe nP^p paraUela bafi , qu^ fectioni priori occuje:* 
i^i in Pp vplano MABiD in,4^ > crit ca ( Dum.596} 
qroirtts ^ ^ciijus ccncruin in axcy adeoque diameter If^ 
fa ^y erit^ue i^Rf iiucrfcctlq blnorum planorum per« 
pendiculafium eidem plano MABw per(>epdieularis ipfi 
todv.adeoque pcrpendicularis Mi»» & at^ ac jproinde 
chorda Pp biFariam fecta a cdametro ^in R> & quar- 
cffaojm PR, sequale rectangulo iiR^ , nimirum » cum ob 
triangula MKa , mRb (insiiia triangula DM;»i > acfeo^ 
quc ifofcelia 3 fn Sc MR aequalis R4 i 6c f^K {qua* 
ts R^ » rectahstule MKm » quibiis fi fccta Mm bifa^ 
triam in c addacur quadratum cKy efunt bina quadraft * 
xzcKy ^P acqualia^uadrato cKy 8c rcctan^o Mki»» 
ncmpe quadratum cF, qiioa ob aDgulum ad K, rectuni 
xquanir illisi asquale quadratoCM , quQd ob i4m tci 
am bifariam in e asquatur bis , & punctiim P ad cif* 
^ttlum radio ^M defcriptum. 

CorolL ^. .. 1 ^ 
^ 60^. Qukvis aiia jfecta mt Ellifjis habem centrm 
in 100 Cylindri axe. . ^ 

604. Nam ea noti tfit paralleia axi , qutm proigM 

^e fccabit alicubi in ^g. 217 ih r , ut paritcr & Qin-iF.a^; 

xua.Gylindri latcra , ac totam ejus perimctrQm alicu-* 

bi fccabit iii MPmf . Ncc erit psirallela ^afi / cujas 

pJano proinde alicuoi occurfdc m fecta qiiacfam OS j 

^ qjuaih ducto perpendieulo CT ex centro bafis > ftr 

pcr ipfiun ac pef axcm ducto plano > id bafim fecabi^ 

«licabi ia AB, fuperficiem Cvlindri in reais (XAq , 

NB^, planuin Scctionis ih Mm f I^Gente M^ intra 

- • C7- 



198 SECTlONUM <!;ONlCAR.UM. ^ 

Cylindrum . Ductis in co pUno MD , md paraBcE^ 
AB , adcoquc & ipfi > & intcr fc acqualibus , per quodr 
vis *pundkum*^rc6t? lAm fiat fcaio p^rallcla bafi^- qox 
trit circulus C num. 596 )y ac plano AMw^B. occurrc^ 
in rcfl^a ab fua diamctro , plano autcm MPwp Jn rcda 
Pp , qux crit pcrpendicularis ipfi ^^, cum rcctae Vp^ ah 
flcbcaqt cffc parailclx rcctis CT , OS intcrfcctionibu^ 
planorum parallclt?rum cum iifdcm planis , & CT , 
OS fibi inviccm perpcndicularcs fmt pcr conftmdlip." 
ncm . 

605. Erit jgitur i^ bifariam fecta iti R > fc qua- 
dratum PR «qualc rcctahgulo ^R^ . f ft autenr ^R ad 
MR , ut md\ five MD ad }Am , & R^ ad Rw , ut 
MD ad yim , adcoque rcctangulum ^R^ , fivc qu*. 
ilratum RP ad rectangulum MRw in ratione <poaftaa- 
ti quadrati MD ^d quadratum Mm .- Quamqbrcm crit 
lAVmf EUipfis , ctiius diameter altcra }Am , adcoque 
( nunf. 351 ) ejus conjugata MD , qua: EUipfi iti' eir- 
pjlum non abibit, nifi P/ fit perpendicularis ipfi^ M^V 
qupd non accidet , nifi pldinum AMmB fit perpendicist- 
lare plano aVhfy five plano bafis, & praetcrca Mw fit 
squalis DM, nimirum nifi fectio fit fubcontraria bafi ^ 
Patct autem Mm fccairi bifariam abV;f;| ut AB> ade<>^ 
que ccntrum efie in axe. 

CardlL 7. 

606. In Cylindro reElo femper Mm erit axis trOLnfi.^ 
verfus \ in eylindro vero pbliquo fi planum AMmB fiic-- 
Ht perpendicutare plano hafis , erit Mm pariter a:ds , /eJL 
trit conjugatHs^y vel tranfverfus ., frout feilio jacuerit ifi^ 
ter feEHpnem ' hafi farallelam , cr Jfuhcontrariam > wl 
fxtra eos limites. 

607. Nam quoticfcumquc fucrit planum AMwB pcr- 
pcttdicidarc plano bafis , quod in Cylindro recto fcm- 
pcr contingct ; crit OS perpcndicularis MT , adeoque 
ordinatac.pcrpcndicttiarcs diametro Mm , qux proindc 
crit axis. 

^08. Porro in Cylindro recto angulas MDm crit 
fcmpcr re<%us , & Mm major ; quam MD \ adoqae * 

axis 



E L E M E N T A. 199 
axis tranfverfus . Ih Cylindro fcaleno M»» cvadet mi- 
nima y ubi fuerit pcrpendicularis latere ^D > (um ia 
itccffu a perpfndiculo fiinc ,'^ indc ^quc petpetuo 
crcfcct , donec deveniat hinc fd MD parallelam bafi > 
indc ad fcctionemfubcontrariam, ^c deinde pcrget utriii'. 
quc aefcerc, adeoque crit minoryelmajor, quamMD, 
proat jacuerit AID , ^ fectioncra fubcpntrariam , ycl 
fxtra cos limites , CotolL 8. 

' 609. £ gi«?z;zj Cylindro potefi fecari Ellipfis CHJHfr 
cmqne ffecifi , fed in Cylindro reSo femfer ejfis /•*•» 
ConJHsatHs dehebit fjfe aqnalis diametro hjifis , ut fiiam 
in Cylindro obliquo qHotiefcHmque fuerit feiiio ferpendi* 
ciddris pUno per axem^ quod perpendicHlar^ fit pUno ba^ 
fo, & jacnerit extra binas fetHones. Hrcidares *, fi ver6 
jacuerif intra^ axis tranfverfus erit femper diametro bafi^ 
aqHolu. ■ "^ 

610. Nam fi fiat in Cylindrq rectd quaevis (eaio 
pet axem , & in obliquo fectio pcr axem pcrpendicu- 
hris bafi , qiias fit M A B D , in qua ducatur c quo* 
vis puncto M recta MD parallela diametro bafis, tCxtn 
capiatur rccta, qusc ad ipfam fit, iit cft axis tranfyer- 
fus ad conjugatum in data EUipfi , &c centro M , ed 
intervallo neccffario invcnictur in rccta BD cx utrali'' 
bet partc puncti D, punctum 7», ad quod ducta Mm^ 
tura fccto Cylindro plano pcr lAm pcrpendiculari aJ 
MABw habebitur Ellipfis , cujus axis tranfycrfus Mw ad 
iconjugatum MD crit^ ut iti data Ellipfi j adcoquc cric 
jpfi fimilis, ' 

5 II. In Cylindro ?mtera fcaleno , fi axis conjuga^ 
tus non fit ad tranfverfum in ratipne mino^i , quam 
fit ca finus anguli MAB ad radium, potc^it ctiamdat? 
EDipfi fimilis abfeindi Eilipfis ctiam planqducto. intcr 
Knas circularcs , Nam ubi lAm fit perpcndicuiaris j 
adeoque miniraa , erit ad MD , ut finus anguli MD?» 
fivc MAB oppofiti in paraPcIogrammo ad radlum, ac 
ccntro M ititcryallo rectae cujufyis minoris quam fic 
MD , fed non minoris quam fit id pcrpcndiculum, 
invcnictur vel unica Mw cum co perpendiculo con- 

graens 



:%6» SfeCTION,UMGpNICAIlU\f._ . 

ghichs > vcl duplcx hinc ; & indc 9 quae exhibebit^^' 
icm.conjugatum minorcm trarifvcrfo MD in ca raticS- 
tic> in qiia cft in data EUipfi » Vcrum fcmpcr iaprin-; 
mo pafu UH crie axis cbnjugatus i iti . fccijbcto ziii 
teanfvcrftts • 

itiHOLiuM ih 

iii: CI in Cylifuiro obliquo planum MARm ^t 6w 
j3 liquum . ad planiim baGs ; adhuc Sc axis uter^- 
.qu6 habcri potcnt tnae:c]ualis diamctro bafis ) crit criitn 
tum Mm diamctcr qued^m > Sc MD c|us conjugata » 
quarum utraqiie cum dcbcjit cfle ( num.,379.J minqr 
^c tranfverfo, mi^r ^onjugato > habebitur ixis coit^ 
jugatus minor ipfa MD, tranfvcrfus major. 
. 613. Qtiod 4 dcfcribatur circukis , q«u rcotam A\i 
tontingat .ih M, &, trahfcat pcr D, qjii quidcm oc- 
curreret diiaihetro M^ ih £ eodcm pafto , quo in cc^ 
ho demonftratum eft ( num. 5:66', 568 ) demohftrabi-- 
imr hic , fprc Mt lanis rectuhi dlametri Mm , ut 8a 
iUud patet fectionom' maxiihc inclinatam ad axem Cylindrt 
tflc maxime obloDgam » tum crcfcctitc. angulo paulatinai: 
accedcrc^d circulifjrmam» & eam afieqtti demum fcm* 
jpcr in Cylindco rec,to in. unica podtione petpcndico* - 
lari ad axem, ih.obliquo vcro fi planuni AMmB fit- 
bafi pcrpcndiculare, eam quidem primum.aflrcqui» tuni 
adhuc magis contrahi, & axem tranfvetfum mutarcia. 
conjugatum 9 recedendo a forma circulari fe;piperma« 
gis > donec pcrpendiculari^ cvadat ,• xum incipiar ite* 
rum ad eam formam accedere 3 ipfi iterum congruat^ 
ac iterum per eofdem gradus oblbngctia: in infinitum v 
614. Poflet etiam inquiri inmutationes omnes> quo. 
accidunt, ubi planum AMDB ,eft ihcUnaeum ad plaiw 
num bafis : fcd quoniam ej^fmodi perquifitio nec u.^ 
fus habet ferme uUos 8c proUxior eft aliquanto , catrv 
hic omitcendam duxi, ut 6c aliam ei ilmilcm in coocy 
icalcno .: ac potius g^radum faciam ad coafideraad^s 

fpha:'' 



fpbirroides ) 'ac conoideS) qaas Conic; fectioaes gene«r 
rant circa axem revolutx» earuraque feaiones ufui fd^* 
piras fxpc ^ ubi iUiid miram fx Elli]if9}de fecari nen 
fojfc nifi circulum , & Ellipjim non magis a circulari 
fimf, recedeniem ^ quam recedat EUipJis getiitrix \ # 
'Pariiholoide jfojfe circulum^ Ellifjim\ & Parabolamiex^ 
ffyftrhehide circulum^ Ellipjim.y Parabolam ^ & Hy^ 
iftriolam no/t magis a formaParahola rccedfntem^ quam 
iffd reccdat ityferiola genitrix . 

D E F I N I T I p V, 

«... ,1 * 

^r;. Q/ circa axem utrumvif convertat^r Eltipfis i 
i3 folidum ea converJUne qrtum dico EUipfoidetn» 
feu Sptueroidcm Oblongam , vel Oblatam , frout gjjh 
fet circa axeni^ tranjberfum > vel conjf^atum : Si con^. 
vmatw circa fuum axem Parahola , dica Paraboloi- 
dcm , vcl Conoidcm Parabolicam , fi Hyperbola circa 
axm tranfverfvm , dico Hyperboloidcm » ftr/e Conoi- 
dem Hyperbplicam ^ axem autem illum comerfionis di- 
co Aifem if^Ms Spharoidis^ vel Conotidfs > ^ a^is ver-^ 

ticejPolos. 

Cqroll. l. 

^i6^ SeSlio fpharoidis , vel Conoidis cujufvis fer 
Mxm aquatur pr^rfus fimra genitrici , & feSio axi 
terfendicularis efl circtdus haiens centrum in ipfo axe. 

617. Si enim in fig. ^18 fit Sphaarois Elliptica , inF.2S$ 
fig. 21 f CJonois Parabolica,' ia iig. 22oConois inHy- ^i^. 
pcrbolicai & fecctur planp pcr axem; ubi figura gcni- ^%<^ 
irix ad Id planum dcveniet, cym ca fcctionc congructt 
adeoqoe ei asqualis efle d^bct. 

6ri. Si autcm feeemr plano PBp pcrpendiculari ad 
axem > cui occurrat in R > & ducanmr bina quasvif ' 
plana p:r axcm MRP , MRB , quae ipfi fcctioni oo- 
ciTOuit in RP , RB t anguli MRP , MR8 cruni rc- 
oi , & proindc ubi figura genitrix ad oa plana devc- 
Uict , eaciem fcmiordinata ipfius primum congruet catn 
feP > tum cam RB , adcoquc fempcr quanris RB ci- ' 

deni 



7 



20S SECTIONUM CONICARUM 
dem RP zqusJis cft li 6c pundlunl B cft sld drculum ra- 
dio RB i 

S C H O L i U M i. 

iigi C Atis patet pcr Thcorcma cffc commun^ cui-i 
i3 vis folidogcnitorotationd figur^ planae cu;u£. 
Vis circd axcni quemvis pofituni in cddcm plaao ^ 
nam demonftratio nott pcndct i natura Scctioiiunl 
Conicarum, 

610; Ex hoc pifimo CoroUarib cruam pauca qiiaEi-'' 
dam y qua; pcrtincnt ad folidorunt cjufmodi rclacio- 
ricm ad fc invicem ^ ac ad dimenfioncm Sphairoidum 
Ellipticarum fumnio futura ufui., qu£ facilc pcffpici- 
iintur ^ & e fimplici Cavalleriaria nicthodd confc- 
^iiuntur : Rcliqu^ fud loCo aptiu^ dcmonfti^abuntur 
infinitcfimali^mcthddo » ap c^lculo integrali; Priiis. ta-* 
incti aliud Theorcma fponte fiiiens prd EUipfoidi&us 
deducani; 

, \ . CorstL i. , . . , . ^. 

. 621. Cfrcnlus 0mniHm maximus efi in Sph^oide£U 
iijitica is ^.qUi haketur feEiione per centrum duSx ^ 
ac ^que difiat ab utroque polo , qui etiani ejus aqua^ 
tor dicitur i Veliqui qm mtgis hinc ; & inde ak eo di-* 
ftatit i ^ 44 polum propiorem dcceduni , eo niinorer 
funt i ac Hhi hinc , & inde fque diflantes aquales 
fuhti 

., t^. Natri oiiimuni cjufmodi circulorum dianictrL 
funt rect; Vp ordinatse axi , qux in quavis EUipfi eo 
minorcs funt, qud a ccntrodiftant magis ( num.SjJ, 
idedquc caruni maxinii cft illa,* quac pcrcentrum tran- 
fit> & binse; qu^ hincj & mde acque afa ipfd centrd 
.diftant ?qualcs funt pcr n. 8^. 

CorolL i. . ... 

iii. Si plurei Ellipfoides , vel, plures Pjtraholoiies i 
vel plures Hyperioioides aqualem habentes axem inter ' 
fe conferantur , eoj/^um fegmetita planis aque a verti-^ 
ci difldntibus ahfcijfd i ojc EUipfoides tot^ 4 annumcra^ 

ia, 



E L E M E N f A: 20? 
U EUiffoidibus etum fph^rai erunt inttr ft ut earunt 
Uftera reEin fertinentia ad eundem nxem i five in EU 
liffoidibus > ac Hyterhilbidihus ut quadrata axium reli- 
qwruMi mmirum ip Sfharoidihus Ellifticis^ ut quadra^ 
ta diametroruni aqudiorisi 

^.,624. Nani qiiodvis planuni circulare PBp crit , uc 

quadraninl radii RP i £rit autcni id quadratum fem-. 

perid quavis Parab61oid& xqualc ^rcAanguIo fub ab^ 

fdffa MR i Sc laterc reClo ( nunu 351 ) ; at in El- 

lipfoidibiis i & Hypcrboloidibus ad rcdangulum MRw 

(num; iSi ) fcmpcr ut lams tc&iim id tranfverfum^ 

iivc ia Ellipfoidibus 9 ac Hypcrboloidibus $ ut qiiadra* 

ium axi^ alccrius ad quadratuni axis Mm ; Qjiare, G. 

^umantui: abfciflk MR aequalcs ^ ac prsterca in £1- 

JipfoidibijfS i & Hypcrboloidibus iint axes Mf» acqua- 

lcs j adeoquc asquales 8c Km 4 8c xqualia rcdtangula 

^Km ; crunt ubiquc qiiadrata RP i ut latera retSa i 

8c iii Eiiipfoidibus intcr fc comparatis ^ ac Hypcrbo'» 

Ipidibus intcr fc i ut quadrata axium reliquorum ^ cir^ 

td quos noil fit conVerfio i qui axcs id ^liaeroidibus 

£llipticis funt dkmcui ^qUatoris : cumquc ci ratid 

iiakatur i)bique > utcumquc mutato puncfco R ^ criinc 

iil eadcm conftanti rationc tota folida ab cjufmodt 

tircidaribus planis genita i dum R ^xcuririt per totum 

icgmenmm axis MR \ &c isx £llipfoidc pct totum ^^ 

iicm lAmi 

i CH OL I U m[ it 

^25. TjrOc ctiam Thcoremi gcncralc cft folidisom* 
iTjL nibusgcnitis rotatione circa eundem axcma 
figuris i quarum femiotdiiutac RP coitantcm fempci: 
xadoaon habeantj ut patet cx ipfa demonftratione • 

CorolL 4. 

626. Sph^rois Elliptica efl ad fpharam eodem axi 

defrriptam^ ut quadratum axis ipfius ad quadratum dia- 

ntitri aquatori^ » & fpharoides omnes funt inter fe in ra- 

iionc tmpofita ex fimplici axisi (^r dupHcata oqHatoris » 

627. 

. '0 



^of S^CTIONUM CONICARUM 

^27, Nam fphaer? codem axc dcfcripta; diamctCF m. 
quatoris cft axis illc idem • Si aqtem binsc fpha^roidci 
divcrTos axes iiabcant ; crit prima ad fphaeram codem 
axc defcrrptam in rationc duplicata dtamctri zquatorif 
priiTue ad cjiis axcm , haec fpha^ra ad fphseram haben^ 
icm axem communem cumfticunda in radoiie tripUcaa, 
jixts primx ad axem fecutidx 9 h^c fecunda fpharra a4} fcn 
cyQdam fph^roidem iBtratiohc duplicata axis feqmda^. 
ad diamcttum xquaTorts ejufdem . ^oUciSis radonir' 
t>m clifa ratione duplicata direda ^ ac reciproca axis 
print£ ad axem fccunds , habc^iur i:atio compofita ex 
iimplici ^xh primae ad axem fbcundae » Sc duplic^* 
diia^mctri a^quatoris iUius ^d diametrum hujus 4 

62S. Sfh^0is oHo»z^ y ac oblatd^ ak eadem JSlHffl' 
ienita Junt m^di^ ^eometrice frojfornonaks inter /^fei- 
ran^ infi^T^ptam , & circumfcriptam , 

629. Nam i/iicripta habebit pro axc ^xcm con;uga-. 
nim EUipfi^os, Cvc axem fpha:i:oidisoblata?, drcumfcri- 
pt^ axem tranlVerfum,.fivc axem oblong? . Quareerit 
fpba^ra inj^ipta ad fpharroidem oblatam ^ ut quadra- 
tiit^ QT^vcrfl, ^ pariter fph^roi^ oblonga ad fphseram. 
cireumfcriptam > ut idcm quadratum axis conjtjgati* ad^ 
quadratum tranfvct:fi • J^rit i^imt iphxtz infcripta act 
fphacroidem oblatam ^ ut oblbhga act " drcumfcriptain >- 
adcoquc alternando fph^ra infcripta ad oblongam , ug^ 
oblata ad circumfcriptam • Porrp cft c ti^m fphasrois oUonga 
adoblataminrationccompofitacx fimpUci axis tranfverfi 
ad conji}gati;im, & duplicata conjugati ad txanfvcrfum^ 
adeoquc in rationc flmf Hci coajugati ad tranArerfuni^ i 
in qjua ratrone duplicata cum fit fphasra infcripta a<t* 
fphaeiroidem oblatam , cnt oblonga mcdia intcr iofcri* 
ptam j & -oblataa:) ; adeoque ^h^ra infbripta , fphasr^Ms 
longa, fpha^rois oblata, f^h^ra circumfcripc9.. func 
CQHtinuc proportionalcs. - * 

Ceroll. 6. ' ' «j 

► 1 



6;o. Sphara fpharoidi ohlonga aqualis ha^et fro 
mePr0 frimam t Hnis mediis zeometricc continue frofav^ 

tio^ 



E l^ I M: F* N T A. • «^ 

P0t4liifis uiter AXim conjiigatm J^lliff^os zmtriQU ^ 
& prAnfverfHm , fph^erpidi vfro iAl^tit fecHffd^m , 

6ji. Si cnim CQjKipikntuT bih^ tncdic cpntinucpro- 
{>ortionales intcr axcm conjujatuii} EUipfcQ^ geniu^icis, 
frrc ctiamcoruin fph^ras iqfaiptx, <3caxcm tranfvcr/l|nj, 
tyc diamctrurti fph^r? drcumfcripti , quatuor fph^r? , 
nimirum infciripta liabcns pro dlaractro illum axcm gon». 
figatum ^ fpba^r^ habcns prq diainctro priiTiam p binis mc» 
(fus^fpb^rahab^qs prpdi^rncixafccundam r&: circmmfcrii 
pta^, crunt' Sc |pf? coi^tii^uc proportionales ^ cupi i^imi* 
tum finf in ration^' tpiplicata diamctrorum propQrtior. 
^^''••ci . Qiiarc cum ctiaiii fphacra iiifcripta ^ fphjerois 
oblopga , fJJhserois pblata ^ ' & (phxra circuirifcript^ fin^ 
isontinucj proportionalcisr^, crit fphjrois oblongai asqua- 
p fpbasrx habcntt prc/dianictro primam, oblata fccunt 
^cx ijlis b}ni$ m/diis con^inuc proportionalibusy 

S G H O l I U M III, 

f??*t T TIs dcraonftr^tis p^r^cndum jam ^ rcli|[a^j| 
XT. Sphxroidum, & ^yl^n^idum fc^lipQcsj (^oai 
«quc facile df tcrminantur. 

Coroll 7. 

. ^3J? ^^M Jf^^^ fi^ SjfhnroiduyftvMi Ca^nidisnoni 
Hrtfndicufarisj^xi fji^SiaiQConicAy in EUiyfqUftfem^ 
}er Ellifjisy in Paraholoidf EUipfis^ vel Paraiola^ ^or 
i^ feSio ffterit opUqua axfy ve^l ei faraUela\ ir^ JHlyper^ 
Moide EUiffisy Parahola^ z/</ Hyferhola , frout fe^io^ 
»is flaniem inclinabitur ad akem in angnlo majori , ^ 
muli , ^lminori r^£e^H ^jus^ quq afymftptus utr^vi^ 
M ijjHf^f: inclinatur,. 

Ijt4^ Rcfcrat cnim in flg. 12 ^ HMI ft#^m cujuf-p^. 
J« SphaeroWis, vclHypcrbaqi^s, & ia fig.^an HMU ;J^ 
gmpcrtincant ad binos ran^o^ oppofitos, , Sq pUngi 
Ifcaionis cujufvls PBp, oblicjqap ad axem ,' ducatur ptl 

Fn ipfum pcrpendicularc ( n^m, 74 folid. }. p^num 
I , quod cxcindct EUipfim , ParabQlam , vcl H/- 
olam gcnitrici fimilem f qum. 616), & occuri:^t 
fvfcoiiich.Tom/f/^' ]p " ili-, 



\t6 SeCtlONUM CONICARUW . 

Alicubi fcdllotii priori in redll aliqua Vpi qu; nufqUatS 
crit pctpcndic&laris axi; nam fi ipfa cflct axi pcrpendi- 
cularis s tottitrl plahum PBp cflTct cidctii axi pcrpcndi^ 
cularc ( iium. 66i folid. ) Ipfa aiifcm P/ ( nixixi^ 149 ) 
Ellipfi occuri^et fcinpcr in bini^ pundlis Vi p^ Parabo- 1 
1« ccedrrct fciftpcr in binisf i ptxtct cafum , qud pla- \ 
nutn fic ati pardlclum^ quo cafu altcrcf pun^o p in 1 
infinittirtl r^ccd^titCi ita ut nufquam jam Gti babcbicur 
unicu$ occurfus P . In Hypcrbola dcmurfi occurrct bis 
cidcm ramo^ vcl fcmcly altcra' int^rfeftionV it;ii in igp 
iinitum abefmtc^ ut nufquam jatn fit »* vcl occurrci! ra« 
mis op|p»Ofitis i protic inclinabitur ad dirc£h:icci1t in an« 
guld minorci^uali}' vd fnajofC rcfpcftcf an^uU xqnor 
litayV ttimiifutn, cum in ipfo angulor ^qualicatis indt- ' 
nentuf afyitiptotr ( hum« i4sf J » ptout id axcnl ipfi 
dircdrlci pcrpCAdicuIarcm indiiiabuiitur in anjuloma- 
jore ^ 'q6am afjmptoci ^ vcl sqtiali ^ vcl minore •' 
635/ Porro pcr quodvis ptmdum R rcdfac Vp duiGto 
plano parallclo bafi ^ fc<£lio cric circulus habchs cen- 
triiiti in axe ( rium.^ 616^),- adcoquc iii ipfrf Pll/'in- 
tcrfcdioric figiirat getiitrids HMI , St tjut mferfc<aio 
£R^ Cum plano prioris fc^ionis cric' pcrpbndicularis 
toci' plano HMI , adcoquc tam dfamctro drculi Py / 
qaum rtCtk P>, & proindc fc<ifa bifariam in R / & 
quadi^atuih BR arqiialc rcdtdngulo FRp* .' Ipfum autcm 
redlaiigtllum P^R/ in cafibiis 9-^ itf quibus jp non rccc* 
dir in infinitum, ad rcdiangulum PRjp babCt rationcm 
dafam ( rium. 29^^, matientc /nimiriijm Ppt^Sc dirc- 
<5tione cfiofdamm Pyj in cafibus vero , in quibus / 
nufquam jam cft ^ nimirum Ubi PR cfl parallcla zti 
in? Parabolay vcl afymptotcr norilibcc iir Hypcrbola', eric 
rcitangulum P*Rjp' , uc rc<aaPR . Qparc fcmpcr V/ 
crit diamctcr fc<iliOnis hVtp s cliordas omrtcs Bp <fan- 
<iem diredidncm habcntes i- cidcm nimlruiti planp 
HMI pdrpcndicuIaTcs fccans bifariafn , idque ica./ 
oc in poftrcmis hifcc cafibusr , quoruni' alccr ad' Para- 
bolam pcnincc ^ altcr ad Hypctbolam ,- fiiic qiia-' 
drata BK $ uc abfciflTar PR , & proindc ( num. 440 > 

{ciikcr , 



• f 



. E L E M E N T A; i6? 

iedio ipfa Parabola , in csteris omatbus quadratutti 
BRfit ad rcdangulum PRi' in <lata rattoile i adcdqutf 
( uumi 4^9 ) feftid EUif^fis ; vel Hypcirbol^ i prout R 
jacuerit^ ut]iflfi^:2li^interve]^tic^cisP, fi qudd fetdpet ac- 
ddetm EUipibide^ In Parabdldide Mtipcx ; pr^tcr cifuoi i in 
qud fedio axi fi t perjjcndiciilaris i in Hyperboldide ktop^i 
ii&i tiicliilatid ad aietii Uabebitui^ itiingulddiajdri^^ctam 
ad ipfiini afymptdti indiineiitur > vel jicueric ipfum R 
atra verticei.r» fi ut iti fig. lia i quod Cbntinget j 
tti diigului plani fedionis cum axc fuecit ' minoc • 

S C H 6 1 1 U M iv* 

6^6. TTIc addemus dtmenfioneni -fblidi parabolici; 
Fl qux idmodum factlc fimpliei CavaUftiana 

ttftkodo obtineturi 

, C^rffll. Si n 

*?^ Siimentutn Con^idis Pnatabolict PVp i/i ^f , 
^ij* Afciffvm fcr ikamvis ElHifim Pp ^^iuttitf dimU 
^cylindrdced cirCHmfcrifti^ CH]ns hdfis £Uiffis <adi$9$i 
^M icnirMS PA -fi«4/zj' > ^ pOf^slleU irecu RV ^ 
|^< e;tf centri £Ui{feos dHeifur faralleld AXi ParA^ 

f^jS. Si enitn duCatur reda V/ > tum qui^vis ttOCxo 
'^a> quae cylindraccum iecabic in EUipfiMxK^ cqUa^ 
>&fimtU EUipfi P/i & Cdrioidem Parabolicara iii 
Hlipfi N» pariter fimili i[rfj P/ ^ ad reCtas V> # VR ia 
«iqttlKui puriaisl, i O i cfitque BUipiis Vf l ^stUM 
5*Eflipfim N;?, tit quadfatUm R/ ai qdadratu^n On^ 
(aum,j5i ) uc VR ad Vdi nimirun ut R/, fiyc 
id OI . Igiwr cuni Oni fit cortftatls rcctae.Oi ,' 
cxponetit areas EUipfiu.n N« i M^ i & foUdiitil 
^^licum genitum ab EUi^fi ^N id, CyUiidraCeiitli:: 
^itiim ab EUipli Mw crit) ut arel dcfcripta ab Ol# 
ijrutti ariitiguliimRyp, ad.arcatndcfcrif^tatriabM»^ 
paf aUdlogrammuin KV^/ ^ fivc ut t aid :i « 



Co^ 



^ I 



-|o8 §fCTipN]JM jCONlCARUM 

" CorplL 9, 

^f^t^ ahfciffaYHfn VR, 

640, Erunt enim u| bafejS) &alntudine$. B^reserimt 
W quacjrat^ R/>, fiye qt VR, tdtltudin^s ^icr^i^ pc VI^'; 
QUVP Wun!: pt qu^drata ipfarum VR/ 

5 C H Q L I U M V, 

f^tf T Am perfcquamur alia cotifcilaci^ C^FoUatii 
J fcptimi, 

Qorolljt xo» 
^42. ^fr/4 RP erit femfer axis fectionis , & in EU 
UsfpMfi <iuiMm ohlat4 axis conjtigatus^ in o^longa ^ cj^ 
|k c^ffixis cmnihus fplidis, axU tr^nfverftis ^ 

645. Patct primum ex eo, quod diamctcrPReftpMCi;- 
pendicularis, fuis ordinatis fi^ , adeoque axis • Ubi au- 
tem chorcla P^ Hypcrfaolae geQit;rids terminatMr ad bi- 
y.52inQs iraniQs oppofitps , yt in %. 2.^2 , patct tpfaik fo» 
1^^ ^ ^Xfra tranfvcrfum ', cum fe^onis periracpro occuc. 
pi \n iptivis pun^tis P ,/>, At iq, fig. 221 ?rit in ca» 
^u Ellipfoidis <5c H][pc^bQioidis quadratum ?xis Pf^ ad 
quadratum axis alterius,» ut redangulum VKf ad quacr 
flrvu^n Q^ , fiyc ad rcclapgu^im PR]}* , nimiruta 
( Qum. ^15 ) ut quadratum diame^i curvai^ genitrlci^ 
parailclx Pp ad quadramm c^araetti parallcla^ chprdat; 
yp* y fiye ad qua^lr^turn axis tratifvcrfi^ ii^ fpbsroide 
pblat^, conjugap in oblonga, & Conoide Hypfrbolica. 
Pprro quaevis d?ame|cr in EUipfi cft fnum. 3.75}) minoi: 
jxe tranfv,erfp, m^jor cojijijgato, , Qyar^ in (ph?roidc 
9b}ata crit a^^is Pp mlnor al^ro axe, it\ oblopga. mat^ 
]f>ty adfoque i^> cpnjugatijSj hiVtranfvci^fus. At iti Hjr-j 
B^rbola dian^ct^^ parallcla cbofdrc Pj>crit (n. 149» aix) 
(empcc diaractcr fccHpd^ria , que ( nura,, 2546) rQa|€C 
. ^)f^^Itcroconjugato, adcoqiie & axis l^jp piajor axe alJ 
terq, ^{ ^n parabo^ re^anguium PK^ i^d rectangviluiq^ 
I^^Rjp^ five quadratum Bl(i crit (n. 361) *, ut latus rc- 
^tum diaractri babcntis pro ordinata^^^chordam Pp , ad 
' ' '* "^ " """ * . latus 



laius rcctum axi^ hdbenti§ pro otdihiu cbol;dam Py i 
cumque quodvis latus rcdum (it ( Hum. 359) maju^ 
laterc rcccd ptincipali iti Parabold 9 erit fempcr ifcctan^ 
guliim PRp niajus quadrdfto RB > & prdidcUt Pf axil 
tratilVerfus l , 

6^ B^i qu^k/ii Sph€r0ide atffcikdi fttt^H £llipp , 
tujufiimiqke ffecicii in qud rdfio sxiufn ah aqnditMii 
nen magis difttt > qfutm ift EUijfJi ienitfice i & Intrd 
MAecies ^uyifvis hainifudinis hdtentii dktni frAnf»^ 
verfum in oblkta 3 c6n]i^ntiitn iH ehlongd non mdforeffi 
Mci ihi tranfverfo , htc con)ug4to EU^ft^ iinerantisl 
Sx quMvit PdrdAoloidt qudvH EUifjis& fficit^ &ms* ^ 
gnitkdino Hatd y fkd ParaboU fdli zenitrici dquiilis : ei 
qudvii /Jypetboloidi iuavis EUipfis.& fpecie 9 & md^ 
imtudint , ac qutevis Porabold , Hyperbola vitro ca^ufi^ 
cunque /peciH > in qua axis tranfverfus dd 'toh]Hiatuni 
nm babeat rationem minoremy quam in gehitrice » vp* 
tra eas z/ero fpecies quicumque etiam fnainittUine datai 
in qud axis cofi)ugatut non fit minir axt cofiiugato ge^ 
^^ist r . i , > . i 

Ellipleca genitrids in C in fis;. ^24, quae exbibecfph^^^i 

toidem oblotigam, Vel 225 > qu; feihibet obktamf in^ 

ttnrallo quovis ned minore » nec majore utroquet fe^ 

iniaxe CM , CQ^ iitvexiiri pdttric punctilm S i &: fc-« 

^o per SO habcbit pro ^cerd oxt Ss » pto alierd 

(^ y eritqoc S> in priore tafil Axis cranfvArfus i \H 

6ciindo cbnjugacus j ac ftttiones Vp ducl^B pcr cho^^ 

das quafi^is P/ pardlelas S/<:rant fimiles imer fe» ci^tt 

in fig. 22 T> & 222 ihactedte ctireCciontt platiK fiP^ > 

ftaoear ilireccio reccae fp» Si proihde ( n{im,2^9^ ra- 

tio rectanguli PRf ad P^Rj^' , fivc ad 4ttadrat&m BR^ 

qoae (n\m. 1^1 ) cll racio duplicaca axiom ^ adeoqntf 

«runtc fimiles (eccidni diict^ pef S/ , . 5^ habebunc ra« 

.lidQcm axitun S/ ad Q^ • Sed 4xi^ Vp crit minoraxe - 

S/ ( num. 8 j ) « Daca igicur quavcs fpecic EUipfeos i 

^ 9«I4 ras^^ ^OWiif^ ^^ vdi\^ 4iftcc afa^ aq^Iitaic. 5 

P j <iuat» 



^jQ SECTIONUM CONIGARUM 
^nam in CUipQ genicric^ ,>brcindi poterit <jus ^eciti^ 
]£llip{|s , Sc inir^ eas fpcci^s habc|:i non pQCeri^ EUipih^ 
fujus ibi .1X15 tranrverfiis ^ blc cdnjugatui fit major a« 
xe C^9 ibi conjugato , hic pranrverfo EUipffpos geni- 
tricis ; qu£ arqualcm habcat , abfcindetpr p^ i: 5/ ; qoao 
minbrcm , abfcindctur. , fi facta CV $qua{i rcmiaxi 
Bllipfcos dats ibj conjugaCQ » faic traofv^rfo , dacatuc 
V£ femiordinata diametfi Ss , ^um Pf par^Uda mxi 
ipfi $sy quae a fliamj^tro. conjuga^a ipfius Ss^, k patal^ 
lela VR ^ta fcc;^bitur bifariam in R» ut fit. PR aeqiuiH^ 
lis VC » adeoque PP axis oov^ fectionis dufias CV> 
&: «qualis axi dato. 
f2z6 ^^^* Piror Paraboloide fi AB in fig. %z6 fit dutctrix 
P^abol^ gcnitricis , cui axis occucrac ^n A > & Tci^. 
matur AD ad AM \ \xt t!i quadratun^ axis tratTyccfi 
d^tz EUfpfeos ftd qt^adran^m {roiijagati i ducat^rque* BI 
perpen^iculacis axi , donec pccurfat ipfi P^raholafr iti I^ 
quxvis feqiq faaa per chordam Ss ordmacam diane* 
iro dua? pcr I e^hibebit EUipfim da^ fimilem ^' Si 
enim e4 dianietcr directrici Q$currat in 3 » ^nc ejiis 
lams rectum quadruplum IB ( num* 351 ; , ad latin 
r^aum princip^e quadryiplum AM , pimirum in Hgi 
321. rcct^ngid^m VRp ad rcctangylum PHf '• adeoque 
Jiic quadratinm 'axis iranfvcrfi $/ EUipfcos ei|fcca» ad 
^uadramm axis coziiugati, ut BI , five AD ad AM ^' 
^imirum ii\ ratione data 3 adeoquc Ellipfis ej\ifiiiQdi 
fimilis datse > Quod fi ipfa Ss fu^ diamen^o ooc^irtat 
ia ^ * & capt;^ CV vcrfys S asiqu^i fcmiaxi traQryf9> 
$a[ dac; EUipfcos, ^vt ca fit minor quam CS» five ur^. 
«Jumque majqr , agamr VP parallela CB, ^oncc occur* 
rat Parabole ii^ P ttum cbord^ PR; paraUela Si ^ eria 
ipfa dupla .PR , five VC , nirai^ura aequalis asi trati£» 
Tcrfo dat^ pUpfeos , adcoque EUipfis fcctloBe setuta 
^quaiis flat?. 

'(^47. At fi PR |n ^* Z2% evadat in Paraboloidt 
pmr^cla axi ^* abc^nte f in Jnfinimm itat pc iiufqiiam 
|am fit » cric rcctangulum PRp' , five qu^racvun "BR 
r quale rcctangulo fub RP ^ & parameoo diametri 

JUS 



E L E M' E N T A, «tr 
^ Pj^' orjin^ta^ n^Xtifi psuram^V^o axis, vellateri re« 
ITO pri|icipali Par^ol? ^eRitrifis % Qjiarc & cjufmodl 
Iccti0» qu^ Parabola crit^ habcbit idem latus rectuix^ 
pipifcip4l^« qAod Parabola genitrix, & EUipiis quidcn» 
qa(vis pp;trit ff ctioac Pfri^c^idis obtincri » fivc dctur 
^cdc tantum» fiv^ magnitudiac » f(cl p^rabpl^ omncs 
^^ cx&ct^ crunt genitrid (quales, 

64S. Prq Hypcrtioloi4c fjt Hypcrbol? gpni||:icis ax'' 
trinfVerfu^ M»» in fig. ^^7, conjugatu? ^ 4 Sf ccn *^**7 
vo C iii^rY^o rcctc, qu^ gd femias^em fon/ugamm 
CC^ fit >. ut axic tTAnfvcrfus dat^ ^llipfcos ad conjuga- 
tum f ix^YCz^iatvu: in Hyperbola cqn jugata pvmct^m $ ( quod 
fempcr pot^rU tum faiac, mm ind; <a CX.» ^um axis 
frai\fYcr(9s fit tn^jor conjMgato iq quavi^ £Uip6 » (ic. 
oi^Om fcmi.^ia^ienror^m pqiijugataFum mii^iimqsi Gt 
ia Hyperbola fcmia^^is CCXh duaa SC/, per qu^n^vi^ 
cfaordampJ^^ ipliparaUclapa^habicbitiJirE^Iip^ <jla(; itmi^ 
&99 cuju$nij;nirum^tsconjugatttsad tracUvcrfum crit»ut 
Qj ad 3^9 aca0uQ)pta CVvcrfus S ^^alifcipiaxitraQf^ 
verf(pi 4a.tf ^ipfcoAi duc^aq^^C VI^ parallela ^i^metro 
O coojugatgipfiuft SCjr , tam chjRrdA JPRi* parallcU 
$0 ti^bit^r Cllipfis squalis datQ^ Utpri^s, cuja^ nU 
4nirum axis pranfvcrfus ^uablnir rcct^ P/ . _ 

H^. Qiiod fi j^ qu«raiur ibidcm HjperbQlft da* 
tx jGrailis •, fatis.^ crit C^nti^o, C W^rYaJlQ cefto? , qua?. 
^ C(X i\c ' uc cft axis ijranfvcrfas datx Hyp^x^^M 
ad conJugaQim invtnire ii^ Hypci^bola PM Rtm,<^^m I « 
4}a(4 tpjum Dotcrjut , (i ca ^ati^i. noa Ht miaor ratiof* 
n§ M^ ai4 Q? 5 aara CM cft minima omntqinj CI • 
P^a^ ycTO qua^is Pjp* paraUcla VJ. 5 fe<ftiO pcr ipfam 
fdt finwljt^ 4at« ^JyperbQla^,, cui^ dcbem ba^icrq axem 

ir^fvc^funj ad conjuiga|um ^ m *0; U a4 Q^ ^ Pq^:^^ 

4|i;b?vis p/ cl^ mqjor^^^ quann li ( num^ ^ > 5^ 353 )t > 
^jCPquc fc<aioi\i4 pc? quajQ^xi^ Pi* " da<%at, ^if^ cqai^* 

4ii^ cwt m]ot9 qiwim Q{r. dafta? aiui^m pyr U critae* 

^3lii) a^coguc QfiUa^ Hxpcrbpla. <^f<(Cari indl$ potcrit» 
(pjijus axis^ CQjpgugai^s fy. minor ai^e coniogato Q^ Hy-- 
DartKdia; Wi^iagiii^ QjVl A arqualis fit fcdHo p^r I/> reln. 

.. \ P l^ ^- 



fibfolvct , fi majof utcumquc > capta CR in CI produw 
flequali fcmiaxi ttanfverfo datb, lum duda P/' parallcla 
liy qtiac rtit dupla CR,. adedcjuc «qualis axi dato ^ fe- 
£tio pct ipfam cfit litnilis, ic Jecjualis date Hypcrboia:. 

6jov Dcmufn pro paiabolis cx(lcandis*c:^hypcri()oloi- 
dc i ^bcuntc ili fg. i2i /? ultr^ qdbfciamguc limitci 
ita , ut nufquam jam ilt , crit latus redum JPafaboI^* 
t .^ -.tcftiuiti p6ft PR, & RB, fivcquartum poft Pk; RP*, 
"^R/ * Hidc fi in fig.liS. CD fit afympMus, adqu^Ti 
diicatur pcr focum f , tciki FD paralifcla dircftrici 
AB , occitfrcfis HypcrBoIaegenrtticr inV , V,* 4^« erit 
( nuhi, 54 ) ejus latus fcdtum prihcipali > tum fuma- 
tur DI fei ca dd DF , ut (ift wms te Aum principalc 
dat» l^af aboli ad lattis tcftum V« Hypcrfeote genitri- 
rfis y & duda pct I rcda PR afymptoto CD parallc- 
la 9 quae 6ccurrat Hypctfaote gcnitrici ift P , raftafi' 
yu ih I , 6i' dctc^minaBit Pdi^afaolam ^qualcfri datic •' 

.651* Si cnim ducatii^ ufquc ad difcctriccm FA paf. 
faHeta afymptota DC , qui ocfciifr^t petiihcrfo ih E, 
ea &: ei1:t «qiialis dmiidio latcti rectot ptmcipali tV , 
tct f^^ & cfitfcdlfa bifariam -iriJE. Nam cftiaa i^B pa- 
rallcla rtdcrtt afythpiofo* / cfit «qualis ipfi FA latcri 
oppofito parallclogtamitii AF«B i Sc ctit siqiiiJis Fu ^ 
cdm ih du(5ta^ ad' di^cctf iccm id atigufo' a^qualitatis ^ 
In quoad^ipfam indinantui: afyniptotl^, ac eandcm ofa 
fafidncm crft &f FE cqiaalis* EA , adcoquc erit EF ad 
FV, vLt F« ad tdtam V« ,' & tcdarigulum fub EF t 
5t Vh «(Jtialc fccfangulo VEf^ . Ducta autcili cfiOyis 
chotda^ 'P'R/ paifalfila V«,. qu« occdrfat rect« RR in^ 
<ra HypcfbolaiPn genitridctn 'iri*' R , alyrtiptoto iii H / 
^tit rcctanguliim VF» ^d rcctanguluxri vRf (tiVL. 305^ 
ut itectarigulurii fdb EF,;& FDadrcctariguIum iubPK, 
& RM , vel fub PR , & Dr , fivcjprd Flt), tD fufa- 
ftitutis taccfc rccto Mj^pfeAoIae: genitficts , & latcre rc- 
cto principaH Patabolae d^lse t (ptit tcctirigdlum illW 
Vf« ad P'K/ , tit rectangulum fdb tE\ 8c Vh adre- 
ctangulum fub PR , &riatcrc 4?cctb principali dat? Pa- 
rabol^ adcoqot cam» reotanguldm VF« «quctiir reCtan^ 

gjLil<y 



DJn 



E L E M fe N ^ A: ii^ 

jjiofijlb EP , 8c Vh 9 cii^m rectangulum PR/ acqui^ 
iitur rectangc^o fub PR; &: latere reccd principali ^i^ 
XX l>afabote; adcoqiifc. fcft PR ad RP, m R^' , adla- 
ms refctuin principdle ParaBol? dat? : ciiriiqlic fit c^^ 
fara PR ad Rp*, lit Rp* ad latus rectiiniprincipalt Parabblse 
provenicntis ex feccione ; bfc Farabola erit xqualis di; 
tx . Curaquc Ett ad DF affunii pdfllt ih quavis ratidi 
pe ', patet^qnamvis datltit PalTabolain ex quavis Hf^ 
ftAolcAdt faaberi poOe; 

s e H 6 L I u M vii 

55i. Tjflfce Sphcroidlbds, aci CoiioidiBis libef j^tA 
IjL adnecterc folidura geniturh convcrfioftc Hy- 
ptrbola! circa. axera coajugatum , in quo folida multi 
i^oirnint nbtatu digniilihia , & ad Geomttriae indo^ 
iem co^nof2:endam fanc aptidima j ,ut pefmbtatio qu^- 
|3im cnirdili ad oppofitos H)i^perT)olc ramos pertincii^ 
tiam fatis ctegdns. Enunciabo auteai tinicd veluc- hii^ 
^ qad^iimque pertinent ad fex di verfos cafus fectid^' 
^m huius folidi, tum fingula pld finguHs cafibttt dt^ 
inoiiftraiio accuraummc • , 



*" » . * 



CorolL 12.^ 

iji. Si jfjyjferhoU fonvertatur circd nxim c^if)i4^ 
^w , getitrabii foHdum * quod fi fmtuf flano , ctd oc** 
tkrrai fUnuin iffm HyferhoU g^tmtficis kd aniulos ri-*, 
8fl/> & confiderentH^ fex pofitibnHs hS^ty in qha fUnm^ 
fiHionif otcUirrit ei ftano MyferhoU ienitricis , ac in eo^ 
'^m ji\rifho riia i^fd f^ firfendkularis axi rotatidnis^- 
fnff axi cinjuiato fiypirbdU gltnitricis , iH fecundo. 
4d iffum inclinetur , fed in anguio ^oiM quam 4« 
fymftoH , ini ttrti^ fH iifymftis fardtl^lk , in roliqum 
^us ihclinetfir in an^uio minbrk , q^am afymftoti , 
ffd in quarto fecbt iramim utfiimlihet Hyferholt gt^ 
iitricis in ko flano ]acentis , io quinto altir^trum 
hntiffiat ^ in fexto neutri occurr4i$ - , hi^is nimirum 

faral^ 



«14. SECTIONUMGONICA^UM 
pirdUli/ fofJiemius inttrjtEla^ erit feHio in pnm^ f^ 
fu circHfuf 9 in fec^ndo £Uiffa , {n tertfo P^^ols. » 
vet fi fldnim tranfe^t per dltfram afj^ptotHf» , Jfin^ 
ffS^ fdrdlleUy in itmto Hyperbofa fertiindfns ilLud fl^ 
pum Hyperbolf ^er^itricis perpendicutare pldno feSf^nis » 
^ h^ihcns in ipfo plano 'pertices axis tranfv^ ^ i» 
gfiinto 4ngulus reEHlin^us fonfims binis relHs u$ri^qM. 
indefiitite protepfis , in fextq HyperbqU illud pL^ 
mm HyperboUgenitricis non attingens , fed fi^Z^i^^^ 
fuos ramos effmnans e bi^is cruribus refpondentibM 
iis> qua pertinebant in cafu ^uarto ad binos ramox 
^PP^fi^pi fi^g^U adf fingulos \ conjunHis , & permft^ 
fatis in tranfitu per cafum^ quintum , ac curvitate. 
in eppofitam pU^am ihidem, converfa \.^ £t interfe^ 
^ioni illi r cujus fyx cafus c^nfidera^twr j in fafu fe^^ 
fundo , ^ quarto parall^lus^ ^ a^is, tranfverfus^ fc^ 
0ionis 9 qu^ nii9%irum aquatur chqrda Hyperhftla gani^ 
tricis ,^ in fextq « ubi nuila ejufmqdi chorJUa ^ 
Ptdffm parallelus efl axis conj^atus^ ^ ac in^iUis z^. 
tie axis tranfverfi ad fqnfK^ofum , in hoc cotiJMgsui 
4d ^^^f^^rfum , & in caftt quinto ratio radii ad ta^ 
j^em 4nguli 5 qua reSa feEiipne obveniens ineiiaa^ 
tur ad pUnufn illud HyperboU genitrici^ ^efi csdcm 
ac^atiq diametri paralleU iili ipfi interfeHioni y ct^ 
)us fex cafus confiderantuf , ad axem ttanfvj^fum Hy^ 
ppbqU genitrifis y adeoque fqS^oner ow^nes cwrvilinc^ 
fUnis parailelis faiiffimiJ[es erunt inter fe^ frater Hyr^ 
t^boias cafus fexti , qua non erunt fitniles Hypa^bolic 
iafus quatti , fed earum conjugatis ; kab^un/(^ tamoj^ 
Hyperbola planis paraileiis edu^a cqmmunem ^Jymfcm^ 
to^ufn iniiinatio^m tam^ in eafu quajrto , quat^^ is^, 
P^lq, , qua trit eadem , ac reflarut^^ cafus. quinti . Is^ 
frimo, verq, c^fu haberi ^oterit qtdvit, Hrculns , Ciiiuf die^ 
m^ter nqn fit minor axe tranfverfo HypeKfpl^ jg</liffrH 
fis , in fecHM^ <i^^i^ tt^ufcumque Jpeciei iliipfil » c^m 
yns axis coniugapis. nqf^ fit minor axe (oningatq cinf^ 
dem HyperboU genitricis , in ttirtio fufvis 
k^4 > w qfutrto qn^m Hyp^bo(a . & fptcje l & M 



E L E M E N T A. %tf 
rtdiiidifre » r«i«^ ^uds trMfverfHs ad conjugatim non ku 
^at r^Utttm minorm^ quMm axis confugatUi ffjfferh&^ 
la iemifricis ad ttanfvfrfn^ , it$ qtfWo ttlfa inc(ijsf4i4 
adfiam^ fiypfrboU ginitricis in qu^is fnguto y qui 
pm n0n fuperft^ quo^ afymptoti ad a^om conyugatim in^. 
fliaaasur, in fexto quavjs Hyforhola & fffci^ , & m^ 
gmtudinf , in qua a^is coui)Hgatus ad tranfvcrfum non 
habeat ratiptem mi^remj quam in fiyferhola genitrice^ 
& in qua axif tranfverfus axem tranfverfum f^ff^H^ 
tcnitricit non fuferet, 

' ' ^54. ^am fi Hypf rbolt HMD gyrct drca axcm con*- 
j^amra Qj !n $g. aij, ^^0, 331,^31,* 233, gcnp-f."9 
rabit felidum qu^dam pgarx terctis; cujus fcdio qa^- 2j« 
vis P^l^' pcrpiendicularis ipfi 2x1 crit circulus juxta ( n. 23 1 
^I9jj pijus ifiamct<:r cnt chorda P^ Hypcrbpl« gcoi- 23» 
iricis, qua pm fcmpct dcbcat cflc major axe jranfvct- 23 J 
fe Mmy omnipm circqlorum minimus cr|t is , qui ka- 
Hbitur fcAo cjufmodi foltdo pct ipfpm axcm MCm ; 
ac proindc circtilus habcns minorcm diam^rtrum habcrt 
poq poterit; potcrit autcm habbns aequalcm, vd utcunv- 
que psajorem» ex quo patcnt» qua; dc primo cafu funf 

^55. Scccmr jara in fig. 119 idcm fplidum planeF22f 

P^^ ejufmodi , ut planqm HDdh per ^xcm traqfiens » i}^ 

9c ipCus fcftionis plaho pcrpfndicularc occurrat fc^io- 

fii ]p(i in fcfta Vf inclinata ad axcm conjugatum Qjt 

jfl aagul4at majorc > quam (it is , it^ quo ^ ipfym in- 

dindntuc afymptoti • Occgrret refta ejufmoii rami| op« 

pofitis (fium. 149) ip P/p« 6c n pcr quodvis cjus pvia« 

^m R faeens intcr P, f ^ucatur planuim P^- p!eti>fn« 

dicolare plat^o axi^> quod pIano*(fDWi& occurr^t in re- 

iStz Py> ^c priori feftioiu in rcftil KB pcrpcttdiqilarl / 

nd comm planum HD^A, adcoque ad Pjp^ & Pf\hx€ 

ipOi nova fcAio crit circulus habens pro dtamctro P^j^» 

4c quadramm RJB «quabimr fc^tangulQ PRj^» quod ^ 

fe^ans^um f R; frit'(iium. 31? /> Mt quadramm axis 

Vtnfvtrfi Mm Hypcrbolae gcnitricis ad quadramm dia- 

fottri St parallclac chords Pf . Erit igitur conftans rar- 

lio 



21^ SECTIONUMCONICARTJ\£ 

tk) quadrati JR.B ad rcdtangulum PR^, adeo^ue PBf^ 
£Itipfis y cu|us aus tranfverAis Vp , qui ad tonfugatUEfi^ 
criti ut cft diametcr.Sjt ad axcm tranfvcrfum M>* Hy-« 
pcrbolai gcnitrids. Ejufmddi EUipfini cxhibct fig: ^34 < 
&-patcl Ci dirc<fti6 rcfdx Pp iri fig. tz^ fit conftans 4 
cooitantcnv forc diaitictrum S/ ipfi par^llclam^ utcomque 
mutetur diftantia cjus chordas a CcntroC, adcoque cort^ 
ftahtcip forc rationem axium in EUipfi ^ & Qfntic^ c^ 
jafmodi EUipfcs planis paraUeUs ^fciffas fimiks fore ; 
Si autcm EUipfis cduccnda & fpccic i Sc magaitudiiitf 
fit data , ,nec, in. qa axis ' conjugatus fit min^r ,. quam 
axis tfanfvrsrfus Mm Hyperbola ^ninrici^ ; fa6fci$ , ui 
ibi N/r ad PJ^^ ita tn fig. 229 GVl ad CS applicitodam 
ccntro G %^ ufquc ad Hypcrbol^on gcnitricem HMD »• 
quacvis fedio du(9;a pcr rcdam S/ perpcndicularis pla^ 
jpio HDdh cxhibebit EUipfim. datas fimilcm ^ & .capta 
in fig. 22f-GV «quali PO fig. 334 , daftaqtic VP pa^ 
raUcla cfiamctro li conjugatae ipfius Ss > mm dudfac 
t^Op chorda pararallcla Ss , patet ^an) fore dc^Iam re^ 
fta5 CV ,. & acqualcm datd a^i Pf^ figiirae 3^4 / adcoi 
que & EUipfim ortam fedione per P^ fore a?qua-< 
lcm datas , unde patet quidquid de fccundo cafu eft 
propofitum. 

6 j^; Quod fi Jam fiat fciaio' PBT pcr fcaafti PR iti 
FiJQfig. 2jo> paral|clam afymptoto S/ , crit.Cnam. ^±8 X 
^iS rc6tangulum PR^ > iivc quadrammr BR, iit rcda' PR ^ 
adeoque fccti^Jpfa Parabola:) quam cxhibct fig. z^y »; 
<^u^ quidem fi dctur magtiiiudine » fatis crit iii rc€l!a 
' |:ier focum F du^ta pcrpendicuiari axi tranfverfo Mm ^ 
ic occurrentc Hyperbolae genitrici in V y m> dfytnptoto 
m Si aflumcre SL ad SF in ratione latcri» re£ti prin-- 
cipalis datae Parabole s^d axcm Mw tranfvcrfum* .Hy- 
pcrbolai gcnitricis > & ducere LPR parallclam afymptO' 
to $/; Nam fi ducatur Ff ufque ad pcrimttrum U^ 
pcrbolx genitricis, ca juxta /num# 65 r^ erit dicnidia: 
FV diinidii la^eris recti principalis, cumque ( timnmy^J 
rct^angulum MFi» acquetur quadrato femiaxi^ conjuga^ 
« CQ> cui (jium. 66j squatur euam rcftangulum Xuli^ 

dinf>iv 



E i £ M E N T A; ^17 
fimtiio laterc refto F V , & feituaxe tranfverro MC 9 
i^ric redanguliun MFi» xqjjale re(^DguIo fub F^ , & 
toto axe tranfverfo M{». £ft autem (nnttL^oj;) re£taa* 
gdum MF17» a4 re4%anguium PR|/ , five quadratum 
KB^ ut redtangiilum fab Ff * & FS ad re£langulum fqb 
'^Vy Sc IJSy five pro TS ^ LS pofitis proportionalibus 
axe M^') & latere rt£ko principali datas Parabolx ^ ut 
feftangulum fub Ffy & Mm ad reAanguIum fub RP.» 
i& latere reAo datx Parabolac :. cunique rc^banguluin 
MFffr jeqaetur reftangulo fub Ff 3 & Mm ; ctiam quqh 
dratum R.B asquabitur reftangulp fub RP» & latere rc- 
&o ParaE^oIas datse , quod cum arquetur re(5tanguIo fub 
RP, & latcre redo Parabolas PBT > crit hoc latiis te- 
&um arqqak lateri re^o datac Parabolx 9 ^deoque PBT 
datas P^abolse acquali^. \ 

657. At fi per ip£im afymptottim Ss aanfeat fcdiay 
efficiet 4ua5 ri&zs parallelas afympeoto ipfi , & ab ea 
diftantes hinc inde per intervallum azquale femiaxi rranr 
fverfo CM» Nam fi Pjf>' occurrat afymptotoin r, & r^ 
fit eccurfu; pla^i Pl3p' cum icdionc per jifymptotuiii 
iu&z y erit quadratum rn fempcr aequale rcdangulo 
f'r?\ adcoquc fcmpcr acqualc (num# 251) quadrato fcr 
niiaxis QH } & proindc M ad redtara ^^ parallelara 
afymppto dift^ntem ab ea pet intervaUum CN asqualc 
^mi^xi CM; unde ]vm patet» quiciquisl cti^m pro ter-^ 
tio cafu fucrat propofitum. 

65S. 5i autem refta fcAIoncm determinans inclinc* 
pir ad axem conjugatuq(ji in angulo adh.uc minore 9 
^uam afymptoti, vel eidctn ramo (nura, 149 ) bi$ oc-F-aj* 
currct 3 Mt in ^g. aji , |n duobus pundi^ P j P » vcl ^3^ 
cum contingct in P, ut rect;^ RJl in fig. 23^, vel ^JJ 
intcr qtrumquc ramum tra^fihit binis tangcntibus pa«. 
raUclis interjcda 3 & neuai fa^np occurretxs> \it ii\ 

659. Uhi pccurrit bis , qui cft cafus quartus ', p^c 

qiiodvis punAum R extra limites Pp diida fedione 

circulari, dudaquc diametro SO parallela ipfi P/9 critF.ijt 

({lum* 315 ) rcctang^lum P'Rjp% fivp quadratjun RR ad z^ 

rctan- 



^ %ti SECTIONUMCONICARUM 
trctangulom PRp» ut quadracum Mm ad quadratum Ssi 
ifedcoquc punctum B ad Hyperbolam i cujui axis traaf. 
▼etfus 1^9 2C ii ad coiijagatutni uc Si ad Mmi qazm 
ttiiibet fig^»36: Ciimquc racio axis tradfverfi ad con- 
jdlgatum maaeat eadem $ utcum4de mutata diftatidic char- 
4ib fp i (dtatroi diinimodd dtredtid madeati patetc > o- 
smes ^jufinodi Hypetbbla^ fore fimile^ inter fe ; Sed 
tum qucti^ ditoetei: fecoiidaria S/fic ^numi 344) ma^ 
^ axe eohjjigafo Qjii pateti in nnll^ et elufmddiHj- 
ferbolis axem tfatifverriim' ad (:!o'njag^tunl p6fk habere 
rationeni miifbt^m^ qu^ni habeat axis colnjugacu^ Qjf 
tiyfcrholk genitricis ad tradfvcrfunl Mjm,' quae ratio fi 
£ierit eadem f chordii paraliela^ axi coujagaco Qjf cxhi- 
btbbHt Hyperbolas fimile^ data^; fi nfajor, ceritro C id^ 
tbrvaUo rectSy qdas sCd CM habedt; rjiciotlem^ <itiamid 
4atat Hjp^p^rbola h^bct axis tranfverfus ad cohjugatum i 
Snvedianoir iK Hyperbola tonjugacflt Hypttbbht geniicri*. 
xi ptiriaa Si / / & chocdse P/r paraUela: diMetro S/ 
eihibebunt Hyperbola^ dacs fimiles. Aflrdmptai vet^ CV 
kk ipfa Ss squali femiaxi cfanfverfcy daca; Hyj^erbolac ; 
ac diftct VP fcrmtoCdidaca ipfius diamecri S/» Sc P/pa- 
irallela ipfi orditlitfa diametro U' Conjugata^ ipfias' S/; 
A: ab ca fetta bifariahl iii O,' habebicur Pf dupla^ CV 
'^afis axis tranfii^erfo ddx Hyperbol!; »- adeoqut Hy- 
boltf orca fcccidrie erit ipfi dalc^ Hyperbol^ ^ua^ 
is ; & hlnc patenc quccumque ad quarcum cafum per^ 
ciiiebant; 
6io; Ptio cafu 5 iri fig. 2ja rccca R^PR carigat H^- 
p-j2^piatt)olain geriitriccm in PV 8e coifaunt ibi puncta P,/,' 
^y^ Ij. O: crit autcm rcccangwum P^Rp* , fivc quadramm 
-^^ BR ad quairacum cangencis PR in illa eadcm racidtte 
quiidrati MfH ad qo^dratum Sx; Quare utcum^ue mu« 
rato puricto R cfit femper PR ad RB ; five ob arigu- 
lum PRB , rcccum radius ad catigcncem angoli RPB 
fllutii;25.Trigori;) in coriftarici raciorie Ss ad M/9, ac 
pifoindc angttlus idem cohftans ,' omni^ puritca B ad 
reccam cranfeUncem pcr P , & incUtf^tam ad plan^i* 
lihdD itn angulov cujus cangen^ ad radium eft; utMivgr 




: i X E k s M T A; ii,^ 

ad ^/f ^nz rttio non petcft cSe"thafor rmone Mni 
ad QjT» adeoque reaa IT non poteft mcUnari ad ph- 
Mum HWifr in angolo miqore j ^ain fit is, in quo 
afyitiptoti. inclinanmr ad aiem conjugattim i qu? incli- 
tiantur ad illum in angulo ^ cujus tafagens ad radiunr 
cfti ui a:iis ttarifvcffusad conjugatiim. Cutn yet6 idciri 
, eontingat hiiic, 8c indfc a contactu P^ fccijo e^ifinodi 
exhifccbit binas rectas TT*i t^i quas eihifcct fig. 237 , 
& cchtactns P detcrmifiati^ ftctionefni v^ qiia habetiir 
data ihdinatio rect{ ad ipfum planiiin Hi:i^i& invenia. 
tur^ inV^nto pimcto S > nt vx cafu pr^iiedentil in Hy^ 
pcrbbia conji^gata iti i iit Ct CS ad CM i ut eft tan* 
^ens dat( ind^ationis ad radium; Patent i^ttir ctiam 
ea omiiia* qu?' ad qtiihtum cafum pertitttafit; 

«i. Deinum pt& cafu fcitorccta R*R ih flfr ijj Jij^ 

fcadem difectiorie i kc prifis jaccaf imer vcniccs I., i aj J 

ilhui ijufdeiii diainetri ICi ; cui dtcurrat in O ; l^r 

^uodc^inique ejus punoum R ubicuinqcie affumptum agattur 

«ircul^s E'Bp' > habeb|tur fcmpcr aliqua RB i nimirunT 

aliqua diftaritia feetitni^ /NT ad plario HDHb i qupd 

pUnutix pioiiidi ipfa 66ri atringct . Qubd fi etiafai ^ ' 

O diieatur fcctio drculatis LN/, occurrens>Iand fefcdo- 

nis prioris in ON 3 ducatiirque pcr R ordinati GRz 

ad diametriim^ $rdoriitrgatam ipfms Ui a qua bifariam 

abcubt fecabitur in X> erit 6t qtiadratum Mm ad qtta« 

dfatum liy ita fectangulum P*R/ , fiv< quadratum RB 

ad rccfanglilum GIU > rit fcctangulum LO/ ,^ fiVc qoa* 

dfatirii 6N ad fcctan^ulum lOi ; Cufnquc fir rectan^^ 

gulum 6Ri exceffuf, qtiadrati XG fupra XR ^ & tc'. 

ctaogulum lOi tx ceitus qtiadf ati femidiametri CI^ iid« 

nbris (^nnm. Sj ) fctfeiofdinata Xg ^ fupra quadfatura 

tOi vcl latcris XR ipfi paralleli / crit fcmpcr rictatt- 

^um GRi riiajus tcctaugulo lOi, adcoquc & quodvis 

fiuadratam . BR majni quadrito ON ,^ puncto N oriwli- 

j^m ejtas fcctibriis purictbrurii fnaxime accederite ad ph- 

Mm HOdhi differeritia vero rectanguloruria GR^,*IOs 

l^ ^adem y ac differentia quadratorum XG , CI 6b il- 

fcs> XR^ CO fquaks^ adeoque ^ fuUatis^l^ropofnonafi*' 

' busf 



^i^tadratojpjptfxi 3^Gi,jCl cfij;, ut qua^ratum Mw ad qul 
^raium Ij. Eft autcrri > ut facjlc folUgi?ur cx derpoi^' 
ftratis injLim,*^ traasUtisad di^ii^tros, diffcr^atia <^u4r 
^lratorum fc tpiorc^inai^ XG > & /cmidiamctri prlman? 
f^I ad quadratum ab/ciff? Q^ iii di^metro fijcuQ^arla", 
jSvc ad quadratupJpR fibi par^^lel^ , 5f ?quali$ , ^c 
cft quadramm ji adi qus^dr|itqip Sj * Erit igitur cx ^ 
j4ualita«c pr4in^tA,dIffcr^ntia.quadratarum ^^ Q^Ta^ 

'fluadr^twn P{l,A ;ij^^a^^^rat^^ M/?» ,M qqadratum 
, ;Sjr, adcqquc guOHptum .JET.fd HyEcrboJaim., cuius O cen- 
, frutn», f^^iifxis yranjfv?r^us. OIJn,, a? axis ipfe traiifver- 
^iii^ a^ conjugapjra.,. u; M^ ^. Sjt . 5i cj\im''c]ufi 
liiQdi ,Hypcrbola^n rcferaf fig, 23$, crit & i^i differcn- 
r ^ , quadratorum ' RB , ON ad ji^adra^mn OR> , li^ 
^^ugc^ratutn axiV trs^nfvcrfi gd coryugamm , ac proiadc 
y^cjiptis dlR i^i, 5c in fig. ^^jj ?qualibus , fcniiordinats 
'..RB* igqu^i ^W5i ^ fij^PS^P9^^5. puo^cus O > R coa-« 
j jj^cnt. ^ .^ 

^,. ($62, ;Jn ^a %utcm Hypcr^ola jani non axls tranfvcr- 
, jus jaccbit ia illa rccta R*R , icd conjugatus , eritquc 
.. tranfycrfus N» in fig. ^38 » ipfi pcfpcndiculari , ratipi 
jLxis iranfvcrfi a4 cqnivigatijm crit^ cadqm H^^ ad Sir , 
[ qa^ in cafu quarto erat ratio axis coniuga4 ad tran- 
^vcrfum; a^c^que ^m Hypcrbole coa.ugat? ^xes pcr- 
^inutent,, ^rit ra^tio axis tranfycruadconiugatum in cafii 
^ ic3fto cadcmat ac i^Hypcrboli^conjug^^tis Hypcrbolarutn 
cafus quarti, & Hypcrbol^ omrics cafus^fcxti crunt fi- 
milcs intcr fq , & fimilcs non ipfis Hyperbolis cafiis 
quarti, fed carum coniugatis; afy;mptotos autcm in co^ 
dcm angulo babebunc inqlinatas ^4 ^ i^viccm , & a4 
_^ps axcs permut^itqs , cum tangens anguli quo ad al- 
^ tcram c rectis Il*R , vel N*^ ini^inantvr , debcat effc ca- 
, f(em 9 ac cf at in caf^ quar^o » 6c cadcm ac in rccts 
. (:afus quinti, Ramus autcm novus TBN^V in fig. 33S; 
. cpalefcet c binjs cri^ribus BX>.^'^'> q^orum alterum ia 
fig. 236 per^iacbat ad ramum TBiP^^, altcrum ad ratnom 
f!^'f BT*5 <?^ paritcr f amus f^;?BT'fig. 238 , c r^Uquss bmis 

" ' ' cruri- 



i J ■ V ». , 



» I, 15 M E N T A: »t 

•rari(>u$ % 2:^6.» con]vLn&h Bimlrum vemdbus P>,^ 
in caf^ quiQta in fig» 237 in 0> in quo cr^ra ipfa ia 
afyingcotos abeuiit, tun^ cruribus tranfgreflis j^fympto* 
^o$, dx&xsi6t;(\ crure TB a ^r, & conjundo cum fV* > 
^ curyan^ra iu Qppofitam pattem obverfa • 
* 66^. Data a^atcm. Hypcrbola % z^^y fl ejus. axis 
franfvcrfus, N/f, non fit ad conjugitum in ratione ma^ 
jprc» quam in %• 233 Mi«f ad Q5 » invenca CS > 
w^ ift cdu quii\to » 8c quarta , quas fit ad Mm > ul 
^^ conjugat^s data^ Hj^rbolas. id tranfverfuna > dia- 
xneter SlQs^ exbibebit dircAionem iedionis pro Hf^ 
perbpUs datae. iimi^bus « Quod fi etiam Nn iti 
£g. Z3S aon ^x^edat l/b» fig. z^ %. invenietur in hac 
pun£tum^.). per quo4 tranfire dcbeai: recta R.R^exhi-« 
^ns Hyperbolam asqi^Jem datJD « Nimitum capta Oi^ 
l>erpendiculari ^ h',\qux fit adNO. in fig. ^^Sdatam^ 
yt CI ad CM in fig. ^ 3 3 , ccnjtro Vintcrvallo C* invenicttu? 
in ipfa O pvuK^tuip Q qua^fituitii exutralibetcennri parn 
te . Brit enim in fig^ ^3), quadratum CV differenda 
quadratorum VO ,; Cp ,. fiv^ CI , CO , aequalis tectann 
gulo lOi, quodi. ad i^cctangulum LO/ > five quadratum 
ON eft, ut quadcatuiT} CI ad. quadratum CM, adeo- 
que Cy tam ad ON fig. ^33, quwi ON fig. 238 , 
habebit fation^m eandem, quatn CI ad CM , acproin- 
4e bina^ QN> fi.ve bini axes tranfverfi fectionis » &Hxn 
pcrbols data; erunt inter fe asquales , 8c acquales ipfs 
Hjperbolas. Patent igitur edana Qin^a ^as ad fcxtu,m 
c^fum Dcrtinebani: •^ 

S CHOEI-UM m 

66^1 A Pmodum utUc cft iUas. iransformationes Ibn 
xV corum Geometricocum in fe inviccm , & i« 
alia affinia confiderare > Ut innotefcat Geom6(rie> in- 
<{oIes , qu^ nibil inprdina^mp. admiccit ^ nifall abruptum 
pcr faltum . CpnSdcretuj: cnim puncto P* invnoto ia 
fig. 32^, planum fccjtionis cum rectaPR converci motu 
co atinuo circa ipfam . Circijjlo,, qui habctur, rccta Fi»' 



iii 5£CTI0NUM COl^ICARtTi^ 
petpcildiallari axi Q5 * fucccditi fc&ioncinclkiata, fcri^ 
4onnnM ornhiunl fpccictuni EUipfium,* iii quibus ratid 
Axis tritlfvcrfi ai cpnjugatuai pcrpcmd crcfdt , doflc<J 
tz pcr oroncs milgDitudinis finits gradus progrcflTa ^ jam 
EUipfi fucccdat PaTabola fig.230,' in qui vcrtei^; ccn^ 
trumi axisl conjiigatus nufqiiam jahi lunt; quae tamcni 
acqusCqtiani cflfc dcfincnt y hifi ubi pcr on^ncs finitarum 
maghitudinumi gradus reccfierintf • Adhnd magis Incli^ 
Hatai fcctionc,* jam ca habcnmr cx parce* oppofita j & 
io fig« 2:51,' ramus nafcimr Hypcrbolae ofipofims , cu- 
|us axis tranfvcrfus ad cdntugaihm rationen^ initio ha- 
bct utcumque' mdgriam ,; qua? ratid pcr omne^ magni- 
todinum finitaruhi gradus ab infimfo quodammodo rc-^ 
dux cum' ipfd verticc f y dccrcfcit decicfcehtc XlS ; do- 
Hcc ipfa CS cvadat aequalis CQ^y ubi nimirum fit ipfat 
PO parallcfa axi cbnjugato' Qa j Pergcntcf cohvetfio- 
nc (Sfca R y iterum' crefcerct ipfa ratio cfefccnte CS 
cx partc oppdfita' axis CCS^^ donec cotuniibus P » p ,• 
jam H^pcrbola abiret in reftas cafus quinti , fed mo- 
iii ipfo' adhuc ctefcentc y & pundd Pimmoca nori per- 
mutjrcflkur ramotuihcrura, vcrum vcrtex quidem j> tranfirci 
iff arcunt PD , & ratio' axis tranfvcrfi ad Conjuga- 
turh itcrum crcfccret irf infinimm , donec facta Pf 
alteri afymptdtcT parallela , iterum haberctui' Parabo- 
la , cut Ellipfim nova fcries fucccderct arf circuli for- 
inanr acc^dens, ac in ipfam definens ,' in ipfo regrciTu 
rcctae T?p ad prarccdcntehi pofitibnem ,' poft quam^ itc- 
rum codem ordine cardem fcrics cvoIVcrcntur / ac fcnl^ 
per' circulus a fe mumo' difcerncrct binas EUipfium fc- 
rics, ih quarum' alcera crefdbret in altcra dccrcfccrct 
ratio axis tranfverfi ad cdn|ugatum,* Parabola yero El- 
lipfes ab HyperboHs , inter quas Hypcrbolas in mcdio* 
Vcluti curfu rcccilineus ctianr ahgulus dccurrcrct ,* in 
qucm pluribus jam' vicibus Hypcrbolahi ihhtari poffc vi- 
dimuSy & nmtabitur fempcr r ubi axis tranfvetfus cvaF- 
pefcat , dum ejus ratio ad axcm conjugatum cx^ 
prcila aliis lincis ncc cvaocfcit ^ ncc ia infiaitum cx- 
crcfcit» 

^6y. At 



E L E M E N T A: ii? .. 

^ 66$. At fi potius manentc directionc fectionis pafal* 

lela eidem^S/) excurrat:planum jpfum motu parallelo i 

m priiiio carii . habetut fehiper circulusj dongruente qui- 

dem {ectj€ne miiiiinus , fed femp:r ejufdem fbrma^, a<f 

paricer . in fecuiido cafu habetur . Ellipfium feties prot^ 

fus fimiliuni ; quatum initlim^ iti fig. 229 9 quae per 

ip/am 5/ abfcinditur ; iiec itl iis quidquam notatu di-> 

gnum accidit ; Ai in cafu Parabolx in fig. 230, qub 

^magis recta^PR a^ :^fymptoto recedit i eo aiigemr n\i- 

gis latus tectmni > quo magis ill^ ac^cedit i eo boc de-^ 

crefeit ; & iii primo cafu eXpatictitur, in feCundo con- 

crahimt Parabola doncc recta PR abeuhte, in afympto- 

him i & cvatiefcente SL evariefcat ktus rcctum : fcd 

Vercex fimul iri iafinitum tcccdit ita i ut nufquam )am 

fic: quo cafu Parabola» quse evanefcentc laterc rccto > 

Sc^ vcrticc adhuc alicubi cxifteritc , abirct in axem 

fuum , tit iri cum abiit i iibi Coni Sectio (num. 5^^) 

^r verticcm tranfiit > ac Coniim jam contlgit noa fc- 

c!uic ; iii hoc cafii abic iii binas i^edtas parallelas axi 

fuo , qui in^fymptotum dcfinit . Planp aiitcm^feCtio- 

his adhuc progrcffb i vcrtex P ; qui per pmnes diftan- 

ciarum fiaiitarum magnitudides ita ih infinitum receilc- 

rat ; tic mifciuanli jara eireti ftatini ex oarte oppofitay 

cnafccrctur qiiodammodo > & codem ordine regrederc- 

iur tx iiifinitb , , aucto per cofdeni gradus latcrc re-. 

hd : ubi liotandum . maixme illud , quo pacto cius 

BT > quoJ priui vcrfiis.T feccdebat in irifinituin ab 

.axc ; & veirfus B recidebat iri ipfum*iii P paulatim 

ad axbni ipfum cx paftc T accefletit , & ad irectx 

paralleLe formdm, ut in trarifitu per afymptdtum dc- 

fdretet dcintim ipfum axem cx partc fe , & ci ^x par- 

fe oppdfita conjungetctur , e priorc iUa iii infinimm 

xecedens; , 

.. 666i In poftremls autem cafibus multo major fe ^ro- 
dic rcrum viciflitudo) fed conftans. qusedam Geometriae 
tnddles ubiquc rcgnat. Haberitur in fig. 231 , 136 bi- 
iii Hypctboli rami ; qui crhbrda acccdcilte ad Centrum , 
%d fe accedwt , & ad ftfymptbtos , donec conjunctis 

Q^ 2. pun- 



\ 




pundis P , p in ipTas afyniptotos recidaptV Vt m 
^^2 3 237, ac dcmuni-rpira illa crurum pctinutationc • 

Suam ridnnus in 8g. ijj > ij8 trknfiliam ad pjjritft 
fymptotomni oppofitaj 9 ntc curvaturam mutcnt'» tji^ 
in tranfitii pcr rc(5fcini,- licct pariici: ad rcftani m cafa 
ParaboLic arcjus appcllens , jHam ralncn hequaqvuiiji 
mutavcrit . Notandum" autcm V quofpadlo <:ru^^Ei 
TB , ih\ pynda R, / in Hg. ijt, paulaiim ad th-^ 
ccflcrint, ncc coicrinc in fi^« 23 j. ih unlcum ramuni.» 
riifi polteaquam fe in ipfo cchtrd O corijtinxcipint. in 
fig. 232 , 8c ibi vcluti conghitinavcrint arctis > qod- 
dammodo vcluti rcliftis fuis illis pundfis P > p * qua, 
cum natura fua iiidivifibilia in paftes dividi lion pch' 
tucrint , nec fimuL in oppofita^ direftiones abirc ^ l|-. 
Jida quoddamniodo ibi func , ac punctis N > ff]y (p&. 
pariter immihuto axc conjugatp; dcvcnerant ad cimrudx 
O , in cormn locum Tuffcdis , afcus iidcra ex centro 
ipfo. cum hifee novi^ verticibu? trahigrefll Tunf afjyyr 
ptotos, & progrertl. Nam punda Uk P, j«-delita gjr. 
i:ectain RR* nequaquam potucrunc fal|u "quodaiir m, 
Ccomecria abfurcjo' muiarr diredHoricm > & per afiiiKi^ 
rcftam priori pcrpcndicularcm pro^redi finc ulla infic^ 
^Kione , fcd per cafJJcm vcL rcgrcdi dcbuerii&r, ycl p«x^ 
gredi , cujufiiiodi regrcffuum.,'& *pro§rcffuum cVc;mp4a 
plurima occurrunt in transformattone locdfum gcoinc-( 
tricorum. Ec quidem pun(5ta N^ n 6^,238 noncffc ca- 
deiTj> ac P, p fig, 236 :;^patct etiam cx cq ^"qupci raub, 
SLxis N^ ad fuuiu cpnjugatum in illa noh cfl: eadcm > 
ac in hac ratio. axis Pp ad' fumh conjqgaaim, (cd.rcU*. 
ctis in fig. 2 38punctisT, p ih vcrticifaus axfs cdh|ivg^ 
lAi c^^dem reita RR\, habetyr ratia N^ ad P/> UQ:p& 
quc eadem* " ' ' '' *V ' "• ' -. 

66y. Scd dc hifcc transformaiionum myfteriis hio fc». 
tis . Agcmu^ de iis infira ordinatUis , 6c quidcra &•, 
^ipnum Coniciirum propricfates admirabilem fanc ^ 
jeifnaocli ^rmuiatioriunj, cvolutioniiiiaj myftcriomin *•. 
getcm ubiquc offcruntj qua? ahimum intimiiis ritnaiH. 
;i;m jucu^diflima quad^m conrcmplacioQe dcfiguht . iiktd 

ununi 




X L, E M E. N T .Ai , jxtj 
jii^uki hic addiemui) <^oA nDaniilli» ubi deConicis Sc* 
.^onibus agtint, notarc folcnt» . . 

'^itim ge^erat foUdum l tftjH» feSionihs Cohick SeUiones 
i^lpiBeMHr . 

\^i^9' U patfet.in hoftfo.caru,i cjuii fi rcfta SN fig. 
^o, 'conncxa cum axc Q3 per rcdlam CN, vd re^aF^jo 
^|T* .fig^, ^g2 per i:cdam, PC gyrct , crit fcmper ii^ co 23^ 
^iblido \ iq qi^o tffet i fi tpt^. figura convcrtcrctur circa 
'ittm .Qj?> nimitatil .in foliclo gcnito .conycrfionc Hy- 
^rbol^e ,cii:$a ^xcm cohjugatum^ cujui feftioncs vidimus 
ySp ConicasSqd-tioncSi , ^ . . , , 

! ^7o£ JDjltis aUteqi hm% rc^is utcuhiquc 3 altcra pro 
"^c, ahcra iTiQycndskjcivca ipfum^ facilc jinvcaietur Hy- 
x5oIa geQeran$. idelp folidmn; .Sit p^or rc<^a Q^ ia 
. ajQ ,^ poftcriot N^ . Ex quovis. priorjs punfto Q^ 
(24 Cif par^ela ^jita: ,N^ ;. ad. pjaniim 4Qy, ci 
Ijiiovis ipfiifls,N/r^.ptia^o. n^ dueatur nr pcrpendiculum 
Vf id^pUniim %, tiim jti ^ddem plapo redba ^S parallc^ 
j .qu£ idcii;co paralleU erii ctiam h^ date 
^ cugi f^a aoi;i .fuerit pajrallcla Qf ; aliter cnim 
^pdeni plaiio. jaduiflpnt » ipfarS fccabit. alicubi '\tk 
'daram ^Q^ quantum , ppus cD' produ^tam • . Dx^Qc^ 
pcrpendiculari ad Qjt , ^ i9qu,ali rir., tum fafto 
[kulo MCS^ arquali MCS p^r pundliim M, inter ifym* 
Ptos C^$ ,. pS'.d€feribatur Hyperbola , qua fui Coo^ 
^rfionc orca. Ct^ .gencr^bit . fpliduin idem 5 . quod rc- 
ISri ^n. Erit^tnim CM ejus Hyperbolac fcmiaxis jranC- ^ 
^'irfi^ , hc reda N« Jn plaQp N«r pcrpchdiculari 
planum Hyperj^ote,.g^nitricis parallc;Ia afymptoto 
difiabit ab eo per irn jsqUalem femiaxi tranf- 
JJCi • • ... 

671. Pofien; in&iii(£. a|i? Hyperboke invehiri ^ qux 

iiduai. idcth. gcnerai:ent : . nec diffiqile cfltt ctiam in 

;. '»32, data rejfta.Pt, &, affumptp iil ca pundo P> 

^ 'jirbitrium dctcrminare in planp pcr P^ & axem 

a .dudo. Hypcriqilam HMD ejufmbdi , ut dudto 

Jtt Pl^o.pcrj^jrtdicaliari/in iilius planum , in-. 

Ck i- terfe- 






2-6 SEprrpNUM CONIGARUM 

fprftcrio PE Hypcrbolam ejufmodi congcrct ra'R. Srf, 
prior illa detKminatio fatis N^ftctidit folidi iltius getiid 
a recta utcumque po/ita fectibncm quamcumque cum 
conicis Scctionfbus congruerc i ^ . - .. 

^ -* - • >*. • 

S C H O L I U M VIII. 

67 2> C Unt foKdppura gcncra, quonim fectioiies qnse. 
^ cumque cxhibetit pariter Sectiones Conicas 
cafdem, quas liuc ufque pcrfecuti fumus , nirnirum •• 
mnia gcnera corporum Cbnoidiforum ,' vel Cylindra- 
ceorum , quse^ oriuntur ex conycrfione rcctar radcntis 
non circulum , fcd aliquam c tribus Conicis Sectioni- 
bus 5 Ellipfim, Parabolam , Hyperbolam , & trahfeun- 
,tis per' datum punctum , ve^ dclatac mom paralld-. 
lo , five corpora Conoidica , & Cylindracca , habcn- 
tia pro bafi boo circulum , fed unam e tribus Co-. 
nici$,^cctionibus • Dcmoq/lratio autcm cft eadcm fc* 
r« ^i quae pro cono , Sc Cylindro fupcrius eft adhibi- 
F.zo^ta . Nam/in prlmij; fi in fig. 20^ »215 bafis AB fit 
215 quajvis Sectio Conica , recta vero^ Cr qiiacvis / tran- 
" fiens ibipcr iUod puhcmnf V , bic paralkla rcCt« gy- 
rantl ; c^dem dcmonftratione numcri 553 , & 59^ c- 
rit ibi' fcmpcr cb ^ ad CB , ' ut €4 od CA ^ hic ck «- 
F.2©8quaJis CB i &c Cd a^qualis CA * Quarc quajvis" fcctio^ 
209 bafi parallda crit ibi fimilis bafi (num.' iii j , hic ct- 
jio iam ipfi acqualis 1 Dcindc in fig. 208, 209,* 210, 211 
rftil fi AB fit quivis diamctcr bsSSs Ellipticae , Paraboli- 
" * caD i vcl Hypcrbolicae ,'* & pS^parallela ofdlnatis cjuf- 
dem bafis , crit fcmpcr ak diamctcr fcctionis dVbf^ 
Parallc ac baQ , ^ P/» ejus ordinata fecta bifariani in 
R I adcoquc Cnum- 305 ji rectanguIum'PR/>, fivc qua- 
dratum* PR , ad recjcangulum vrR^. in* rationc data . 
Erit ' autcm" tit in demonftrationc numeri 562 , rc-. 
ctangulum ^R^ in fig. 210 , ut MK , in rcliquis » 
lit rcctammlum MRw . * Igitur' crit qiaadfatum fcmior-. 
dinatae PK ibi ut abfciOa MR » hie ut rcctaaguluni 
.MRiKi» adcoquc puncmm P ubiquc ad Cooicim Se» 

ctio- 



l^- B t E M E n T a; «7 
|ii$K>a6in Jiut« Biin^ 4^9 * & 4+0 . Eaikni vere erit 
^moaftrsdo pro Cylindracei in Rg. 217» ubi ^ua^F.117 
dianini PI^ crit , ut rcct^ngiihim 4R.hy fivc ut' rc- 
cuogulnni MRi» . Quin immo ubicumque ia iis fo- 
lidis inier fecdtqies Iiab^i potcrit & circut^ : c£- 
dem fempcr enmt vcrus i;oaus , vel Cylindrus habens 
ipfum prcul)im pro b^fi • Sc4 longum eflet fisgu- 
$BS caias pe^qi^., Sc )ub ad irtipsbrmattoncs qii^f-' 
4ut] lo^tiB CcvQCtii^wi gtttcraliorem fuiemus 



Q. 4 



:. frBfi TRANS^FORMATlONl — -^ b, 
LO6ORUM GEOMETRICOR.imi:| 

19» dg cotttinfittMu' legi, ai M ikliufdtm "' 
Jujihiti myfitriis. '■ ' ." 

%T. Alra ^usdsini feprodlt m bnrni G^' 

I ttfttrJcommLocdhira tfSmifofnmtionft'' 
I Geomctti^ ihddlcs, mifi admodtnrfi ■ 
I Knoffris ftientibus pror^tis impciviat' 
ll tnoimnitJnocQloslDlniiti Gcomettl^.' 
ti mjftcria ■» qnf qatdem in iis cticffl /' 

qm. _ SeiUonibas a flobis dcmdnfttata fiirirj ' 

contcmplari licett qnam ipfam ob c«ufam ca hlc evt^ 
venda oobij een&iimus. nc ad fiiUimioteT eorvas j &?" 
intibitcfimotum tiKthoders breri cvDlguidas pronidi''^ 
T)Toi(i vift ftemcrctur,' ^ '^ 

#74. In primts qu?cariqae eujufcoiKfue geWneniciIa»' ' 
d' pars «andcm natutam habet , qiif ipfius dcliftiticttld '' 
cominetat , atqvc idcirco habctetiam prbpiiciatcspror:"^ 
&s eafdem cx iih ipfa naiori flutntlfs / QuamobtVi^'' 
qtiidquid clc una sSiqaa ejas parre demonnramf fltient ' 
cx illa ipfx nann^a > reliquis oinriibus panibtis ^ta- 
ri debei codem modo' > ncc quidquaid fola iUiu? aa>' 
tor^ contemplationc deiAonHrari potcrit de uni ali^^ 
qua partc ^ qutn dc patie alia quavi» eadcin paritct' 
iaiionc dcmonftrewr . Qufcumquc emm eandem rsM- '• 
ntfam sque parric^paot, ea omnla debcnt itrrfem xqoe ^ 
paFHcip^rc qtiidquid ex cjusunius naturxconfiderjAiotU^'" 
ifettticiiur. Atque id ipfum pirfpeximui flum. 178, ub?''' 
de arcus circularis trifccrionc cgimus , qtiMH ibt vi- { 
dimas obtiotri non poQe, quin fimul in^itornm AuJ ^ 
loei-o aliorun. aTcuum , estdcm conftroAtone nifeAto' ' 
obciaeirctuc . Atque banc ipfata ob eauliun « Bbiciun^ ' 
ipie' io Gsomctria vel folvunnir problemata > Vel de- " 
iDOitftiMitui dworcmau > ccrtum quoddam , & 6^ < 
teiininanim £:hun» luujiciur gndix r ^ui ioveftigacio, < 

voi 



tOCORUM GEeMETRICOkUM . y%4 ^ 

\tk daQon&two- UppKqttue i ti fapiUffm. fGketna mi^ 

tom c&film. Dculo fubftdat et. infiaitis nuaiero ip& 

fcorfitsi fiiBilibQg:, &: '«pijdquid tn eo^irontingere^yl* 

dent ocdli , inens dd. xelsquos ^tiines transfert , argu^ 

nentatu^e CQnunKmi pro omnibiis ; Sic fi re^a li* 

nea bifariam fecanda,.Ct ; conftru^io aptatur ccrtk 

coidem Jinea: , nt unius p<>Ili<:i;s , qu{ t^meii icadem 

aAkm ]iacx> m unius. poUicis.^ .^^a: ^taqacn eade^ 

iciuvis abcri loagitudini q<^ ^ts^u^^ nec longitudi* 

ttefii 4)£wi ^dcterminatam in ichematr pculis propofito 

tattk$ iiijituc|ttr> fcd foiam lin$a?i:cdC; Habcntis binoster- 

totnoi Dottonem ^ un^ pgun .o^pnc circuioriim ad 

foludonem pra^lemaiiU vec|iiifitoiMm^ 4c xc^ per to^ 

hu|^ xjitci&^on^ .^uccndfi» .. .. it . . ,• 

Mif ^ qiiidpm aVqiWUulo .fit „ ^iit folutia »11 Cdk 
fiti in /cbcn^AaMCulis proppllt^ .^plicjasa ^jfifi^ uUir 
I)efU|^ui.diiipriminp applicclpr,<afibu$f otniiibus ^ ai: 
jcbcina ipfum reihaneat ciufdem fo^mg « Multo tameii 
fg^^J^i jpfis tafi^s .pp%ip ,(^ya£^ ita fchema pe«!^ 
tDl&a^;rto ^roficiP quodam fi| x^pus » a^ fcrvamiaii^ 
Ujuojpam. t & rctinendam folutionis > ac demonftra>« 
ti0Dis vim 9 1|u( qijudcni poiitio iliod ctiam pt^iksit ^ 
VI quandiKiiic fiimmi aUqiu in diffcrcntiam abcat • 

1^7^^ £)xnapliun profcrcQaiis c Ceoinetria plana p^ 
dtxiPQu «; Sihl in ig4 a^f bin; itctq parallcl; indefinitt 
ABi.DG ^ <yiUs k£t% in C> & H > recu £F paritier 
iiidieifiaita m Sit autcm dti^da pcr damm punctmn P 
Jrccxst-Qccurrcns iifdcm u:i!bii^ rcctis A^ , DG > JEJf^ ki 
M^09 N ii* , m fumtna binarum ,MN , ON , quf 
intcEx^ipiiintur iiiter pr^aiti $ &c tcrtiain > ac inter ftn 
ctmdam » & fiertiam . fquccur^ rect^ dat;^ • Factp. cca* 
tro in quPvis pttnctp K alterius c .parallclis » ui AB . 
IniervaUo c^ufden^ ica6.4;ate inyeniat^ir, li ea fit fi^ 
& longia ^ m dacra^par^Wa DG punctupi I ^ dttc%> 
tnrquc Kl> nim ex P i;cc];a, ijpu K|. paraUd^ j qufc & : 
EF dcciirrtt in. N i , Inter C, & H r folvct > pro-. 
5 ^^erit cnim ipfarum .Ai^r^i^i. ^iNi fumma 0« 
MiQl i ad^oquc^ Sftaajfi^.%4 W# <Jppofito i» 

paralp 



.1 



^50 DE TR ANSF0RM;ATI0 W 
parallelogrfimfoo MiKIOi. Ubicamque puiycitfmPI» 
rit coHocatum inj ut Ni cadat iatcr C > & H t b? 
lutib problcmatir^ttc ptoccdct . At fi P faoeat in Pi, 
vclPj ita , ut N cadat cxtra CH, vcl ii| Na^ , ad 
partcs H , vcl iii N^ ad partes C ^ eadcm CDoftraccto poh 
ma fcont^ vidcbitur faUerc . Nam in uuroque cafa ca- 
rutidcm rcctarura MN/ NO ftou crit ioiiinia» icd difc 
fercntia MO, qu? icquatur Cl. 

677.^ Vcruth fi poOtiouis vis confidcretur f mtaeb^ 
ttiara ibi analogta, Sc patebit , idcm prorfbs prfftad 
in omiiibus cafibus, ac iUam , qu^ videcur diffcreatii 
biaarum qualitamm , rcv^ra effe fiHnmam . Nara fl? 
in quaatitate difcrcta, ut nutncris , ac algcbcicis 6tf- 
raulis ,'& it^ quaatitate contiaua\ ut ia Qconicrici^ 
iiacis, funt qu?dam quantitates , qu? dicunmc mga;*, 
tive V &qu? fi poficivis addaatttr ^ cas ininiiuat> 
vcl minuuntur ab iis . Si quts dccem nummos bv 
fecat y 8c lucretur alios trcs , babebit o^edecim : & & 
potius contrahat debitqin ^ium , habcbit 7 ; fi dcbi» 
tum fit 9 , babebit f i fi debitum fit jo , bf 
fccbit nihil •, (^d fi debiton fic 1} » jam^ habcbil 
dcbitum quidem , fed 2 , minus nitnimra , .qaani 
I? • Oebimra iilud ctt qu^dam iiegativa quantitas i^ 
quar conjunaa cum pofitiva Ula re habica , illam cai* 
tmit, vel ab iUa minuitur . Eodem paao fi qi2is> &• 
cundo flttviq remis ctiam urgencibus proniovcatur i tC 
intra fliivium progrcdiatur remorum opc fingulit mt 
nutts per paffu^ 10 > moctt autcm fluyii procfcdat pt 
paffus 5 ; coniunctis mocibus progredictar ptr ij. Ac 
fi fluyius rccro rcflectat mpmm, & rctrahat nayimpef 
paffbs 5 •, vel 9 9 vdi la, /vel i j, ptogreffu, & teg» 
lucoh:unais,habcbimtprogreffu$7, Vel ifvcl nthil» ndl 
nettam rcgreffns 3 . Rcgrcffus illc eft fiegativa qiuadi 
cas, qu^ progrcffum pofitivam quancitaiem JBinnit ji^ 
ycl ab co minuituri 

~ 67S. Porro in hoc fccundo c^a mutatto. 4kei» 
hls pofitivam quantitatem tnutat tn ncgativm , ^99 
neraliccr tti Geoasetcia^diitcttoaiit oppofido eaadeki 

mu- 



LOCORtrM <5EOMETkICORUM; zp 

liatiaitiOAem ifiducit • Pro qilavis quanritate variabifi 
plaga )>b(itivorum ad ai^jfrium aflumi poceft , qua (6- 
ind affuitipka ,*" ditccrio" oontratia ■ quar^fitates cxhibc- 
bttncgativas ^at (i in aliquo cafu habebatur fumma 
qafdam quatiritatum quarundam » & earum aliqua in 
ofa alio dit^ctionem mutct ; adhuc habebitut fumma 
Dmoiom, fi ea quantitas in' fummara tiegativo inodO| 
compwctut*, cam nimirum demendo ; vcl fi commu'* 
fiis c^fidcratio adhibeatur , qu; nimirum pofitioneni> 
'& direcrionem non curat , fed folam magnitudinem 
conamplatur,*difFcrctiria fucccdet fumiHfV * 

679. -Satis patctji in cxpofito probkmate in cafu fc* 

cuodo P2 dtrecrioncm M2N3 mancre eandem , quc 

ferat inMiNi , at dircctioNzO^ cft oppofita dirc^ 

ctioni NiOi V In tcrrio yero' cafu Pj direcrio qui* 

jilcln NjOrj \ manct eadcmV quc NiOi , fcd MjNj 

^cofitraria illi, qu^fucrat in MiNi • Hincnimi- 

itlm fumma, qu? in primo cafu crat fqualls reccaidaiw 

te iabiit in rcliquis in diffcrcntiam • Qubd fic' car- 

ft Pi \ progrcdi^ur ad Pi, tum indc^ad P3 , diffc- 

Toitia , qu? bafietur in primo cx bifce ttibus \ abit 

fcfamraam in fccufidq ob directionemaltcriustantum 

toutatam , mm fumma fccundi itmtatur in ' diffcrcn- 

ti«n tcrtii , cuin itcrum ihuicmr dircftio etiam altc^ 

^U5 . Cunique"^ comparando primum ex hifce cafibus 

^ra icrtio ,' utriufquc' quantitaris direftio mutctur j ia 

ottoque habctur diflfcrcnria; quia niiTurura Hl M2N2 , 

^ N2O2 , in cafuPi confider^ntur ambse , ut quanr 

fcatcs pofirivac , ficnt iii^ terria negativac anib» > i\nm 

l^feai reftituunt ncgarivo modo , fiyc dircctionc coq-. 

??Ha ,y Dcmendo Oihfi 9 abM^N^ rcUnqtticurM^Oz, 

« dcmendo N3O3 ab M3N3 rcmanct O3M J » ncgar 

ihrcfampta, fivc M^O^Vut prius, * 

f ^&B. In quavis cafuum divcrforuin conteftiplauonc , 

tit in quavis combinationc locoruqa geometricoruiii^ 9 

^BI«imis confidcrari dcfact cjufn^bdi pofirio. , qua: in 

pirum transfdrmationc fcmpcr cafdem propricBatcs rc- 

wuct, dummodo ubictun^uc [quanutaris} direbrid mu- 

i •■• . - . " ' fetur. 



^tidt j iUa habeatut.prD negativa», adeoqae jam j^^gsaik* 

-tiS^ iddcbaturs vcl cqntra addatuc , fi .eicOTcb^t& ; 

*^« cnim addenda.fuerat , dum dccrcfclt pcrpctii}/; 

fonpcr viinUs addct i.fi ^vadat nalla . , 5c cvancfcai^^ 

addct nihil ) fi . in coqtr^iam ^tiam , niutcnir i mut^ 

tk d^rectionc ^ contrarium itidem cffedum prcftare J^ 

Tbcbit^ nimlrum minuct id,^ qudd antca aHgebat^. ^, 

l.:,iSSu Et in Uneis quidcra , tibi mutcoir dlrcc^dj^ ^ 

• ^qus ope pbfitiva. migrcni^ ia negatiyai , faps ctit mat^ 

*!ifediuti pcr fcfe , yel rjccte Une« fint » vel curv? . S«: 

fi»*4^ia fig. 340,^ fi binjc circuU chprdae fc mutUo Ikccm 

Jhtra circulum In. C ,,me'nfura anguli ACB cft fcmlr 

JUmrua arcuum AB. » T)t a rectis^ (pfi|tn contincadbi|i 

^tercc^tbruiji . ( Cpr.^ 4. .Pr. ^ > Geom; ) l A^ fi.. pu4- 

^ctuni Cz jaceat extra circulum ; ea ipfa menfura ,aa« 

gufi AC2B cvadit diflFcretitia arcuuqi, AB , t>Ea qup^ 

tiimirum direceio arcus I^z cft conrraria dirccdohi 

D£ j qua: fi negativo modo ,mra^tui:; adhuc PCpt\i^ca-r 

^ura/l^abebitur femifumma i Immp prodcrit hiic cti^i^ 

omties muc;itionum vlces contemplai;i i ^afque cledtji. 

Ccre cx fqlo piimo cafu rcctarum J\B , Bpo &C pofi- 

iBione puncd £ percurrentis tocam circuli peripheriatnf;; 

Moncc cQ rcdeat^ undc digreffumfft. Anguli nirnitiini 

ACB inchfura cft fcfnifuraina arcuum AB,^ ED.^ Abear 

jwm(9iin £ in D, & arcus ED fict iiullus hiac rnc^ 

Vir^.anguU ADB , in qaem tutn abibic ACB ^ <^i(<^%- 

&idius ateus AB . Abeat £ in £2\ ,, Sc rhat%ta d|p 

ttctiohe arcus Dl^i, cohtraria nimirurp flirectioni I>j^jf 

nkDt anguli AC^B.menfura erit femidifferentia arcuum 

AB^; E^D V ivadat E3P acqualis ipfi AB » jatn £ci»i* 

^iffercntla^erit nuUa , quarc recta A£, cum Bp% iiu|U 

lum angulum continebit: & qutdem eocafu.patet^ i^ 

fas paralleks efic • CrefCa; adbuc DE^ , & jam evj^ 

djit ina|or> quam AB . lUius igitOr diniidio ctenip^Ar 

4lmidio A3.> femidifFcreada cvadet ncgativa . Quam 

Ih^alus babebicur A' 4B , fed ad parccs oppofitas j^ 

tebiif >' ac fpe&abit plagas oppoiitas.» ut figura ^exfti^ 

mt $ c>iif(}uc menfura crit acmuc iUli, fcniidiffcrenda « 



,-?-• f 



f^zt Es in A >, & cyadet A5G5 tangcns j wgujt 

Wb AC5H mchfuracrli femidlffcrentia afcuum DE5V 

Mruve DA , AB ^ quod ita cflc patct > iiamcoruii^ 

^cuam dlffcrcntia cft AE3 , ob EjD aequalcm ABf', 

fC aQgiili qucm tangcns 5A a produdla continctcivn 

cliorda AE.3; parallcl^ rcftac BD > qui idcicco squatiu! 

jntcrho , ^Sc bppofito AC5B , mcnfuf a cft dimidius af- 

Qi$ AEj . Abcat E6 inter A, & Bi & anguli AC6B 

'iQcnfura crit femidiflS:rcntfa DAE6, BA , qu^ 6b AE6 

jpbpimuhcm A fccfu<;ctur ad fcmidiffcrentiara DA , BES'» 

Abcat puh^kum E in B» <5^ cvancfccnte E6B , mcnfu- 

fa anguli ABD fict dimidium arcus folius DA. Abcac 

dcmum punthim E7 ultra B, 8c BE7 jam mutabit d^ 

redioncm, adcoquc metifura anguli ABD , (pectantu 

pfdcm plagaseric femifumma arcuum t>A > lE^B ^ ut 

patct omnino cffc . » ^, 

^ 68i* Et hx quidcm c|e lineis • At in fuperficiebus 

HQt^ndum crit illud « $i fumatur redlangulum binac- 

*runt rc^birum , Sc iina cx iis pofitioncra mutct » niu- 

tabitur^ Sc rcCUnguIum, ac e pofitivo migrabit in Qe^ 

gativum : fi vtro mutct utraque » adhuc crit cotifidcr 

tandum cj^fdcm gcncris , ac crac > cum ncutra pofi- 

iioncm mutaverat. Nam fi in f^, 241 GD. CA confi-p^^^ 

(ierqitur /uc quantitates pofitivac, & ear>um reCtahm- 

iam 0CAB,'ut pofitivum, mutcnir autcm CA in CFj 

facebit DCFE ad partcs oppofitas , adcoque id reccaa* 

i^^iim refpecliu prioris confiderandum crit , ut neg^- 

Itfnun. Qjiod fi iterum mutetur CD in CH , jam r^ 

«langulum FGGH , mutabtt dircAioncm rclpedu FGD5 

^coque debcbit pr^ftate cffe<5kum conurarium^ nimirun^ 

ihinuerc, quod id augebat, au^ercj quod id minuebai^ 

V proindc negativi negativum crit ^ & itcrum in pq^» 

|irivum migrabit. 

683. Hinc in Geoilietria idcm accidcti quod inAri^ 

TCtica, & Algebra contingit , lit nimirum ubi 4jh 

^ do unam quantitatem in aliam , oritur praduiitii^ 

^dodd^ni , fi altera e binis quantitatibus mutetur in 

|c^tivam, fiat ncgativum^&produdtums fiutraqaemar , 

'- ' ' ^ ' ncat. 







/ 



33+ ^ DE TRANSFORMATIONE' .. j 

ficat, fit pofitivum, quod ibi cxprimitur dicemdd)^'! 
multiplicatione tum binorum pofitivorum , tum biao* 
rum ncgatiyprum priri pofitivlim , cx multiplicadcK, 
lic pofitivi per ticgativuni , vcl viccvcrfa \ oriri nc| 
gativum^ five figna conformia in multiplicationc cihil 
bcrc pofitivum \ diflFormia ncgativiim . * , . . • J 

^84. Porro hinc illiid confeduitur , ut lincsc cuji 
Oimque quadrafbm pofitivum kmpcr maneat, licetci 
4lcni linea t pofitiva mutetur in neg^tivam ,.pofitioi 
mutata ; Quadratum cnim lincse cft ipfa linca in^ 
ipfam ducta, qu? e fupcriore canone producit.planui 
^ofitivum : Indc ,vero dcdudtur \ quadrati ncgativi 
tus impoflibilc effe j quod in Arithirietica ,* & Algcbz 
lippellatur qiiaatitas imagiriaria . Quadratum.autci 
qiiodcumquc bina fcrapcr. Habcre poteft latcra altcfam 
pbfitiviim^ alterum ncgativdm .' Atque idcirco ubicum| 

w quc problcma aliquod ad fui folutioriem requiret ,• ui 
inveniatur dati quadrati latus ,- fempcr id ipfum latu^; 
adbibcri poterit cum directionc utravis, tam pofitivuiili^ 
qUam negativuni. , , ,. , , .. 

^, 685..Id patebit fequenti cxemplo . Dcbcat invcniri 
ilitcr binas rectas media proportionalis . .Qp^cfit^c rac- 
di? quadratum dcbet.sequari dato rectangulo fub dati^ 
ircctis ; Quare binas omnino folutiories habcrc dcbc- 
bit id problcma, & bina ejus. quadrati latcra inveni-' 
_fi dcbebunt conftrudtionc cadem l Atquc id . qaidcrii 

'^•i+^omnirio contingct. Nam fi iri fig. 2+2 binjc rcctac da^ 
tac abfcindantur in AB , BD in cadem' rccta ita ,' uif 
carum fumma conftituat AD , ac ipfa AD fcctam bi- 
fariam iri G, tadia GA ducatur circulus , is rcctsc EBP 
perpcndiculari AD occurret iri . binis , punctis G ; G ^ 
critquc C3t natura circuli utriiislibct BG quadfatam 
quak cidem rcdangulo fub AB ; & BD i & titra[qi 
cx iis media quaefita ; UbiCumquc punctum B fucri< 
intcf A,' & D, folutio ritc proccdec i At fi id fuou^' 
nir igxtra ,' vd ad partcs A iri Ba i vd ad parces Did 
Bj , mutata in pfimo cafii difcctiohc ABi , ia fccua^ 
do DB^» jam rcctangulum ABD mutabiitfr ii^ negatl 

vunii 



LflC0RUM GEOMETRICQRUM . aj^ 

kbUk i adeoque ncgaovum evadet etiazn illud, quadtaf 
tom 9 & iddrco ejus latus inipoflibjle ; quam ob reni 
id ea coniOtructibnc ^invcniri nequaqu^m pqtcrit . En 
^uidem rcctae EzBifii EjBgl^j ip(i AE) pcrpendicula«i 
tes tiunqu^ bccurrent circulo ; Potcrit. quidcm alia 
confiructione detcrmiiiari mcdia intcr ABi i 8c BjD, 
Vel AB^; &c B3D indcpendcntcr ab illa mucatibri<^ d>* 
rectionis; BiirJrum ducendo binasi cangehtes B^H, Y^t 
B^Ht ad circulum ipfum, quflef erunc mbdix qu^fitx v 
Vtrum ibi.iterutti ABi j & B2D eoti^detantur , uc 
pofltiva: i & fi deiiidc J3i ; migrcc in K ; & pofiw 
mutetuc^) |am ea conltructio nos defcrct ; nfque enini 
cx B tangcntcs ad, circulum duci potcfunt ; quac pro- 
blcma eadcfti cohftructionc folvaht ; migrante verb B 
in B^i jam & ABji & DBj habent dircctioncs. cea- 
trarias dircctiohibus AB2 ; & DB2 ; adeoque rectaiv* 
gulum enrundem itetum evadit poficiviitn ^.ac iterum 
confiructio redit cum binis tangchtibus ; Atqye idcir- 
iq fi in rectis EF fumantur bin^B^L, vel bin^Bjlia 
±quales binis tangcntitus > puncta tii Lz eruoc aid oi-> 
nos tjufdcm Hyperbolas xquilaters ramosf 9 quas eft 
Locus Gebmctricus divcrfus ab ilio circulo ,.cum quo 
fseqhaquam Continuatur in A » ubi arcuum quamvis 
cbntiguorum natura , 6c pfoprictatcs funt admadumdi- 
verfa^,^ licct arcus afTuraanmr quani proximi.. £t haitc 
ipfam ob.cau£am circulus quidem ordinatas BG axi 
perpctidicularis habet refpondentes puhctls B affumptis 
ihtcr A , & D , nuUas autem haberc poidft cxtra eos 
liiTiitcs i contra vcro Hypcrbola extra eos limitcs 
habct fcmper , intra eos habere omnino noci potcft. 
^686. Idcm autem etiani in adraodum fimpliclbus 
Gcomciriae theorcmatis ntotare licct .* Eft quarta Eu- 
clidis Propofitio Libri 2^ puncto B jaccntc intcr A > 
Sc D ,* bina quadrata AB , BD cum binis rectahgu- 
lis fub AB , BD a^quari quadrato AD , ftptima vero, 
purfcto B2 jacente cxtra' A, & D, bina quadrataAB^, 
B2D acquarr quadraccx AD cum binis rc^tahgulis fub 
ABi y Sc BiD . H9 bin« pfopofitioncs cxhibcht fai-* 

tum- 




'^^6 Dfe TRANSFORMATTITOB 
iKOtittiodo binos «afus ejufdem theoremam-V ic , ^foai 
tla fponie flt^c e prima,^ dvunmodo Botecur > ^Jli^^^ltaH 
fcefe direakmem conrariaiti A 3 quam habet ABJ,^ 
re<2toncm vrro DB^ efle eandem i ac DB • Ep 
t)ado paccbk, quadraca quidem manjere ut ptivk»^ XHx 
l^ bkia redangula mutace poficionem » 8c fieri ^hn 
^a 4 Qiiamobrem nbi ante fcimma ex bmis dpi^ae^ 
tis A6 , BD, & binis rddlaiiguUs fub AB « Sclib''» 
^uabituc quadratd AD, |ani i^ aKjuabltur tum ft^Hia^ 
led* dffierencki, qtuc habemc demeado ab. ilK^ QiKbc&a>- 
tis illa 1>ina' tt<AanguIa i unde feqiumr illa binatiruar. 
draca sequiuri quadaco AD bsnis^ iili^ rc<%ati^ul£rwf 

497. Eodem ^ciattipafto tam qumca^ & fetc^^^qnam. 
no&a, Sc dectma, Sc immo eciam fecunda. , 5^^^'*- 
fia^ duodecima, & decithacercia. cjulilem libri «^^^.^^ 
-giila theoretnata reduci poflunt , habica ratione^utiT 
vorum, ac negativorum in mutatione dii€6ttotus,.MuT 
<UAte vaiorem rcAanguli , aon vero quadraci / ^f, ia 
reliquis quidem mutacio illa valoris enunciatiDnem Ip? 
fam cheoficmatts mutat , cum in iis habeantar ttAkrir 
"gula • At in nona / & decima^ quac contiited'^ fela 
quadrata 5 nulio in iis. rautato valore : cnunciati^ihar 
oec ftadem. • Sedla. AD bifariam in G , fi pua^Jta^ Sj 
iit inier A , & D , bina quadrata AB} BD xqfidbpor 
tur per nonam binis.GA^ & binis CB ^ Si auccm B^ 
fic excr;! eos limites , erunt pariter per declft^am hma 
quadraca AB2 , DB2 acquaiia binist QA, &binjs(^^* 
Mutata e(i dircftio laccris AB in AB iti AJ^Zy ^iyap. 
ior quadrari non eft mutatus. 

688. Infolidis patitec, fi una etrlbas redbis folidhm 
continentibus mutet dire^onem mutatur folidum e ^(ir 
tivo in negativumj ficnimcjQncipiamr planumarcl&jiis 
binis contenmm immobile, recu vero , qu; djtecno^i 
nem mucac , fic folidi altitudo , jacebit folidurn /iDrum 
ad partem oppofitam poft mutatiQncm diredioajsj iiji 
m akituicline ', ac proinde & ejus va|or. mutqbtt^r «^ 
Quod fi muteacur bin« , rcdibic^ itcru^i?, ad valprera 

poua- 



1 



LQCpRUM GEOMETWCQRUM* »t». 

jBcs » licrun^ valqr folicU, .ipm^t^Uwf , . 4(* gcftcca^ 
ubicuriKjuc alicjuod fivc jccig fi< ,..fiy^arcai fi^ 
r^^ToUdum I ddSdiatur d^^ yc^ fHropocnotiii))^ re^ 
. mrbm "^ quptcunquc ,' fi ' c aruni , ^ (lupcrus irap jm: - ^|c- 
.iBQncm rnutct'^ ipJTun^ product^ii) (nufablt valorcip s 
l8 numcrus carum , quc mutamniii , ijc paf.,.v^r.ina^ 
Sdbit .' Nain iiinjgularum' rni^tatiq d^t^t valprom •PCO^u-' 
J!cii inujarc, quodprolndc ^^ pofitivjp m n^sgo^ivHmv c: 
y&gativo ih pofitivum abibit pcr yice^ , aK&oquc .po^ 
Bpamerum patcm eodcm fcmpef ri^grcdicturj, ac aUainu-j 
^Ktone dcinde addita> in opppfituiT\ va^orcm migrabic % 
6Sp. Id mai^ifeftum ^rit.^ vibt d^tis trib^sjfcti&quz* 
tac 'quart^ proportionalis poft Ip(as ^ DucjaanKrbMias 
cec ^B. y pE indcfinit? ia fi^, 243^ , <mg^ (e» aiptuo 
^ctit in*G t rumanturCH 9. CF ,v^rius A «%ij|a|csF2+j 

G' ioribus biiii^ ^ Cl ycrfus ^ ^qualis tcrtiae ; diK;atur 244- 
i , tura cx F irccta ipfi parallcla , qju? ^biipia^et C15 245 
DE rcaara (^G , quarfitatn pqft CH ., CF ., Q . M^-? H^ 
|rtar jam dircdlp pruiis CH ih oppo^tatt^ ii^ fig^ 
44 , fhanehtibus diVcdlbn,ibtis CF , CI > Tfcta FQ 
^sfralfela IH folvet itidem prQyc^iJI , fcd CG jffdbii;, 
id partcs ^oppofitas dirc^tlionc mut^t^ . Mutetm; i^ Rr. 
"i 245 , ctiaih'CP, Sc jam rcfta FG paraJkla HI 
ibit dd Dofitioncm C(!} candcip ,, qu^ tuUbuit 
la ^. 243 ; Miitctur dcmum iu %«2^6 cii^iigf Cl^^ S^f 
* n CCquQquc itcrum^ m(;i(abituf ^d dirciftioncin op-s 
fit^ .. Quih injjmq fl quxjCi^mqu^ .1;^ iUi^ tribus 
^CH^9 GF , CI figurae prlmae; m^tetur ii\ connrarium» 
ftlcbic cqs cafiis dcUncanti' muta^i ^in|<r CQ • ^cd 
odcumquc binavum mutcti^r quavis cx iis rclid^ ivi 
Btiohc priorc , patebh fcmpcr ^ dirc^iqnem CGnpi^* 
ise ; ' ao fi quis ratloncs ctiarq c^oinpofitas adfa^bcciC . 
Kcuraquc, poteric fane mutatiopcs quotUbucrit ex- 
iri, & fcmper ihveniet, n^merum mutationum imri 
cm' induccrc mutationem , parcm vcrx? r^tiocr^ .ysv* 
em priftinum. 
l'^9o. Porro in cjufmodi rautatioxitbus angujiquoquc 
^*''^ Bofcojich.Tm.IIA R ^rc^ 



ij8 DJ&TRANStORMATlONE 
l-ectanifh mutabuntut it^ i ut mucat^' ditectionc uniui 
lateri^ ) totiietiii: angulus id eiiiii i ^iii ejiis comple' 
inentuhl eft ad duos re&o$ $ mutata autetti dite&ione 
utriufqii^ lateriji mutabittit^ atiglilui id alituii fibi adver^ 
ticem opi^dfittuh > qlii ipCi prorfui; a^qiialii c^ i .&r t]\jii 
vice^ ^qui;^ przdabit ini , detiiohftfitione quacujnqtie • 
Ac (i^tnchllratid^ vel ipfa etiatii tHfcoreitiati^ , ptopoii- 
ti veritaif adiiiodum facile ab und cafti tf ao$fei;6tuf ad 
alitim j fiiibialieiriu^ ^aniutnmo^o latetis iniitetuf ditetlio,' 
fubfiituatut dn^uld prioti ejiis cdmpkftientiittl 4d duosf 
reAos # ubi utriufque ; fubftituatur ^hgiilu^.ad verd- 
£em oppofitus ; Fict autetit alif uando in ejufmodi mu- 
tatiohibiii i ut ^di ahguli! ih pafallcli^ alterni ^rAit 9 
mtitentiit in externum v id ihtefhum ^ & oppbfittim , 
interhus itl extefnuht aliquahdcr migrec » 6c vicevct- 
faC 5 acf alia eiufthodl Cd&fequedtut »' quaf fponfe mcur- 
tent in ocutos »< ac fingulaf perfequi i ic exemplis il- 
luftrar^ iniinitum eifct v. Satis^ erir, in illis ipfi^ iafi- 
bus y quo^ expttfifihius in ejufihodl figUtis, tiotarevim' 
deh^ohftrationisr ^ &c mutatidhem Ihx angulis fa&sm J 
Triah^jf HCI ,. ECG ih fig,^ hJ fimitiscJunti qiiia 
tiabent angulum HCl v FCG commiinem»* nempe euu'- 
dem ac ACE ,- ahgulr auteih CHI i^ CIH crjcterhr «- 
qualc^ funr angulis^^ CFG , CGF intcfnisr i & oppofi- 
lis^ in parallelisr HI ,- FG • HinC eft CH id CF y ur 
Q ad^ CG itt figura:- 244 funt iudcni fimilia^ triaDgu- 
la HCl / FCG y fcd rdcircor fimiliar funt j qui ahguU 
HCI, FCG fiiht ad veftidcm oppdHtk arqualcsf &GHI; 
€IH «qualctf alterniV CFG/ CGF/ Miititicf latcrif CH 
hiutavit ahg^uTum AGE in ECBV & hiutario' lafetisCG 
motavit ipfom AGE in' ACD. Anguli vefo' CHP,CIH>^ 
qaif erarit cxtetni fefpecair CFG r CGI^ ihfctnotum > Sc 
oppdfitotum in^ fig. 243 i cvafctuhr altcfai itl fig. »44 * 
At dehionftratibnisf vis^ adhuc^ rcli(Sa eft / 

69t^Pzttt iridem mutairiohe' ipf^ dirctSlionis argu- 
inemarionem> quat fif coniponchdo , mutari deberc itf 
cam^ qoas' fit dividcndo , quouefcunqiic ia propos^uo- 
ne aliquii biav' tahtuhamoda tcrmini anteccdentes > vcl 

^ i^iiii^ 







j, ^v.^v..vv.... CEOKiETRICORtTM. n9 

fclni confcqufcntes mUt^nt.cUrc(5^icMiem, maaert» fimiir 
tcnl: priore^ binii vcl bini pdftrchiii vcl dmrles finial^ 
ftsUfabiint autem fcnipci!; ^ vcl niiHiii i ^VcI. bini * vel 
piiines ^ dum iS i: priocibm mbus mutei: pfimus folus» 
vcl Uf^y debbat^utarig, (juiirtus^^fi bini \ qUartlis m;a* 
iittt debfcat; imde jpdtcti fieri tl6n poffe i ut corUm y 
\vSi mut^nt rtitihiieruS fit impar l In fi^« i\% l eum fit 
CF ad CH^ ut CG aa di crit diviiitado FHid CH, 
iit GI/W.Ctl,At lAfig.^iUi ubi Cg; &.CH tnuta- 
^nkt dire^iiSncm, fiet C(Jlnpohtnd6,FH. ad CHi.ut GI 
ivi ^CI * ; lii fig. i4^ i ubi miitaiit prior^s ^ bini. i.. & , fig. 
246 > ubi butlht opiticsi^habietik itcruiii.argumcntum 
Ipohtpohendo^; R^tid c(t mahiiefta> quia, fummd primi» 
& ^ fircuridi ; vel tcirtii l Sc quakti tnlltatlii: in ,difiFcrcn- 
iiami vel dificrcntia in fuh)h)ihi> ub! alter ex iis pb-* 
fftion^m hiutat^ tnanct Vcro fupima> vel differcntia, fi 
VU ricutet mhticti.vel litcrqliCi , , 1 « . . 
^ tf^il Ex iis^ qtiae, Jcrhahflravimus , licebit fepc Lo- 
iporum GfeoiTiitricorum*dudlum5, & varios cafus,. ac trans-^ 
forhidUohi^s tontetopUri . Exctnplum dtfuhil^iiis d i:ur^ 
t.:. ^i-ii_i.rjji.. >^..v r..-u..^..^ -'n. liniv^rfa; Gcohictri^ 

crfequemuir ibi > iibi iri- 
. agcmlis dc fcUtVis gc- 
hc^aliter, ac ca ctirvardm gehcrai qus inajons func ii- 
fui perfcqucmut 4^ Intei^ca ^ earum. du^us hic definitUs 
jjlurithuiti j^rodcrit , ad, qu5Cdam. infiniti myfteria ei^bl- 
yeiiida, h Sc cbgnbfctndam intimiuscontinuitaitis gbothe- 
lcricas legcmi ac ij^fa. plurimorumfcafuum contebpktio, 
& loCbruhi. ^cneralis cdnftf iiftio Cbi , ubique rtfpbttdcns, 
ad Gcometriie ipfiuS inUolem, miram fane> percipicn- 
4im. pariiter plurimum prodcrit» 

t^^l Curvj; qiiarum^ naturam-^ 8c ^aeiim bk con-* 
^tnplabimur , crunc t^t in t^uibus ordinacae rado fim- 
|flex » vel utchmquc hiultipiicata eft cadehi ^ ac Ismo 
Implcx ; vel utcuhique mhltiplibaca > fivc reciproca , 
jVc dircda . abfcifiq^ * Si /algcbraicis fighis uti libeat ». 
^C:, tonfiderlurc laltibrcs lineatum p«ueQ:ates > qu^ex*- 
^ctmiUltiiic iad^iSnitc per littctas m ^ Sc ^ y .ao. abfcif- 

K % fa 



j+0 DE TRANSfORMATIONE 
^ dlcatur p, oixlinata vcto Q. j lihcac hiijufmotdi fuqL 

C2 , 19 fluibus P:^,.yf Qf ^, Fxpdimcj5ittb«« m j & ^ 

Xiumcrps^ quofctimquc ratiohalcs intcjtos , fiyc ji^fitivos, 
fivc hcgatfvoj, vcL qudd ccdbii rcdit, ih quibus fitPJ 

ur Q^ , cxprime^tc /s^ nflgacrum qwncu^iqp? ra^onat 

lcm lOtcgrum, yclfrajftum, pofuivutn, vcl ricgativum • 
Scd hic , ubl Gcpmctnara cotttcm[5^mut i gcometri- 
cum ctiam fei^mdncm lifijrpabiinus , adbrbcndo ratio« 
niim «qualium cbmpofitiprlerrt , tjucm ctiarri' multiplf- 
catio ratiohufh appcUatur, piotrus quam potcftates linca^ 
rum, qusB uhra fccnndam', & tcniarh , nimifum ulora 
quadratum , Sc ctibiihi i in ' Gcom^rrla * nbh a^guht ^ 
alTur^unt autein iii" Aritlimctica cbhfidcrationc ad 'grar 
dum qucihcumquc , fi' qusedam Ijnea' dicatuf uhitas , 
qua dc rc ibi aptius , ubi de Algcbrse VppHcationc a<f 
wometriam dJcendurn crit T Porrb int^ cjufmodi Lo- 
5*1 Gcometrica habetiir ctiarh recta Knca t;im axt 
inclinata, quam parallcla, 6c tam'. Parabbli ad axcm 
rclaia, quam HypcirboJ^ ad ' afyrnptotbs prb axc aflum-? 
ptos, & prajterca bmhiis qusedam , quam vocaai Pira- 
bolafum; ae I-fypcfbblarmh femHia/ ' 

694. Sit in~fig. '247'fccta iifdefinita MN, m quafu-: 
rnaniur abfciflfe a quodam puhctpd^to' V pofitivarvcr- 
F^47/us N, ut VR, adcoquc negativi vcrfiis M,.tit VR2,' 
'' ac dcducta ' pcr * V indcfinita OVCX pcrpchdicuJari adf 
MNs ordinataB ca]>i«ntur paraDcfe^ipfi V &' habcantur 
pro pofitivis directioite VO, i;t RP, adeequc pfb nc-' 
gati^i^ directioiic cpritraria VO, ut' R2 fi ." ' 

^95- Siiit autcm prini^ brdinatac in ratibhc fimpHcf 
abfcilTarum ♦ Sumpta VA' ad arbitrium tx partc pofiti- 
va/& crccta AB pafallek' VO ck paftc Itidcm pofiti^' 
longJiudinis cujufcuhquc , & ducta pcf V , 8r B rciSt» 
(T indcfihita ita , ut S jicct ad pattcs V-y ac T akf ' 
parte« B / pai^ci^ , cam forc LoCtttH Gcomctticura quefi- 
tum; ducca-chim quavi«'RP pafaHrfa VO, fcmj^fcfft 
otdimi^ PJR. iiid-^bfciifoin VR , m BA ad AV. «1 



^ * 






jldm rationc^Qnftanti adeo^ue illa miitabitur, ut-liiCCi 
live erir o^^nata ih ratioric fimpdici ditcdfaa abrdifnt j 
Porro in hof cafu.pjttpt i.aBfciflx poflidvaer VR dcbe** 
re fcmpcr refpondcr^: prdinatam * pdfitlv^m RP > nega^ 
tivx.vero VRa . ricgarivam R 1P2 * Nam dcbet cfle 
AB.ad PR > iii VA.ad YR > in ^ua proportione VA; 
& AB< CQnftaHteJ Cmt , a^dcoqtfc mmata pofitione ab^ 
fciflas VA i mutari ctiam dcbieit poficio ordinatc RP 9 
juxta tiUm».68S. Scmpcr: autenl rcfpondcbit cnivis db-» 
fciflx» dia^ drcliiuca ^tqiie da unica , cuni .hic nvili 
pccurra^jtlt qtta.driitortim iaxera i qux bihaf cfle poflunf 
pofitiQnum:oppofltarttm> vel qu; qdadratdr ncgativo fa^ 
<to cygdaftt impoffibilia . Creioetue .autem in infihi-. 
t|im abfciiffi ;, .dtbct ercfi:erc,& roordinata , ac ea^cva- 
i)efcet)ti? f evaaefceri^ . , JEt bxc quiiieth ottioia . dmni^ 
nb accidiint ia ea ipfa rcda. y qus Sc tranfir per 
V i Sc uttinqu^ ih irifiaiiliim r6c^dii ab a^cr ad par- 

•tfiS OppofitaSi ; . . ^ j ' - ' r 

,,,^^6. Debeat in fi|. 448 cflfe c(rditi«a RP iii ratfbnSP^^^ 
dnplicacai dirc<5ta. abfciflaB; VR . . Abiciflfis onjfnitm^ pofiti^j^; 
patct, dcbcrc ^rctpondcrc ordinataqi.pofmv^n , & iifti^ 
ocm', ^qux inventcturi capiendoRPad ADit! ratiofi^du- 
yUcata Vft ai yA>,five at cft quadratnra VR ad qna- 
4ramril VA. Fafta auccm» abfcifla VRa negativa* adhuc 
ordina^ RzPi debebit cfle politiva. Nam ini illi ration? 
^lic^taVR^ bis ini^reditur 8c proindc pofitio bis mutatnr; ' 
ac. quadratum abfciilW VRi quamvis ncgativa^ 1 cft pofitt'^' 
Xnm . Por^o paitct , . acfcente ^ infiHinitn abfeiffa y 
debcre a;efcer^ in iitfinitum i & twdirfatam » ac^fri-- 
ftiidts ma^is / litide colligitik r • bW Loci- Gcdmctri- 
ci aura in infioitum ^e ex piirte VD vcrfu? T , &' 
S^i tcccdcndo &mper& afi ate Mt4i & ah Va in itt-^ 
iQtiuh: at abfcifla tn infiatimhi.deerefcentc p^t^, ctiafm 
«riltnattani itifinitics magis debcre dccrefcc^e^ onde ihfcr-' 
tir: ^vafceme abfcii&f y dcbere cvancfc^tc &i ordin^tam, 
adcQque Locom Geomccricum.hanNt tranfire p'av\tcf ptv 
y i Q^oiiiain vera ordinata infidilici majis crefcit , 
9ixim-0titAifx i ^^ s(mb2e ^rcfti^t- ulira' q^ucHfcamque 

. ^3 *mii- 



24^ PBTRANSFORMATIOKE' 

)im]te$9 infintues aut^m magis decrefcit ^ ubt zttA^ 
-~ dccTcfcunt , patct ^ j^ per y , & P. flucatur rc^a' VI 
indcfit^ita , angplum yPR »n primo cafii , 3^ PVR 
ix^ fecundo d^crefcecc uitrsi qiiofcui^quc Itmi^i^s ^ ade^t*; 
^ue fi arcus VR concipiatiir coqtimiatas in inflnttuti^ 
verfus T , angulu^ OVI a|ternus ipfius ' VfR decrefoec 
in infinitttm, acc^dentc recla VI «id VO , ul^ra qnpC* 
cumquq limites, quqd nobis infra iifui erif r ubi ^ge^ 
411US dc infinito. Si vero ar(:us VP. cvanefcat abctfnie 
P iq V/c>jancfcet apgulu^ IVN:^ & rc6^a( VI, qu2ecd 
cafu ^vadc?\an|5ens, reqd^ in ip(ura axem M V>^ ^' qi^ 
proind^Iocum SYT in y conri^ngct . Patct kuiem ci 
ipfa propqrtiofic ^xpofita, SV^ debere ttk Para;boIaa^ 
c Cqni S^Aionc orta V cujus axi^ VO • Dudta <aim . 
P£ pcrpendiculari a4 c^m axem,^ft inParabol^' VBr 
abfciff;^^ t^t quadr^tum ^P , qua: ii| ^a* ^icitqr feini- 
ordlnat^ ^ adcoqi]^ 'KP,'qua^ hic ^i^ittir. prdinatti^, eft 
in*r^lion(5 dupUcat^ VR^ qu? hic dicitut abfcifla • Pof^ 
ro it\ Par%l>Qla Coi^ica pateVcrura VSi^VT cOe *il]|ius 
ipfius fpftnap, qi\ain bi? c3{.iUa poGtiyorum 9 & ncg^- 
(ivorura nQtionc dfdMxipus, * 
^97< Quod & dcbeat efle qrdiiiata it^ ranone tri^Ur 
^^f9cata, abfciffa?.^ ba^b^^nfur/^ uticifig. ^49-,^ bit^i arcu^ 
' VT, yS infinitif quorum altcr. jaccbit inangulbpyN, 
altcr in MYsJ^r N.a^A f*^ VR^r "^ncgativa , habetui: 
in ill^ r^tiqA<? triplicats^ 'numerus ^cgatiyqrum ia^pstf' 
& prqinde ocg^tiv^ V^ ctiam. ordiqatJ^ , ' Eqdem vc* 
ro argumcnto/ ^rujr^ ^ itifinittun abeupt , ac Wcus 
tranfit^per V » ub^ a Ce^^ MN fontingipir » a- qaa^ 
cum cti^ ftccmr , tiabcpir / ibi^dcm, mutatiQ dir^cdok 
nis curyatprac ^ quas appcUatur mutatio flcxus , Con-* 
tadtus autctn, ^'inxe^ic(£);io bic 'ui\iimt\]r l |it uhicic*' 
Culns 'qfciUa;qi; ^ectioneni Conicam f num, ^is^fecatu 
firaul,'{l('tangi{ inipfp qfculo. Porrobic kicust an>dU?> 
Jatur ' Para^Ma (rpbica, in gu^ (\ 'OVQ^ 'affumatuc {«cn^.t> 
|ixc, QjfBi qr^it^awrijn^ P£ fun^, ut abfciffx, V^ ' •» 
^98."*Gc?eraliter autcn^ (i ordiua^a PJR, fit tQ r«(io^' 
nc ^ciffai VK UCcuipqucLmuItiplicat^ pec anmcnitn !«'• 

' ' * ' tc- 



LOCOTITO G^QMETRlCORUMv M? 
^gmm pftfitivmtt pnjcm , qt fifi( in quac|r^pli?^ti , fexiupU- 

fata,deqq).Iicata,dcb^bif baberi or<^ina;a quoqu^ R%P:b poli^ 
^va y 8c fortn^ crtiirym , cadcm, qua? itx fig. ^48 »• Q pcr impa- 
f^mti^abt^ur iri tirg^tiYatn^ & (lab^binir forma fig- 1^9-» 
^ 699. $1 yero ratio (iuplic^t^ prdinata^ fi t ^adem ^ ac X9r 
^o ^itt^^ utqimKjae multipii^at^ p^r numerum impa-> 
fcin \ nam ' fi * fit cadtm , ?ic rapq naulppticata pgr 
puip^r^rn '4)arcm \ eri^ ra^o fimplcx oxH^^xx (padcm, 
frc ratiq ^fci(L^ tpi4^ip|icata pcr ^imii^i^iT) frjus tium^-* 
ri p^ris h^ cmuiB ^ducctur ^d ^terum.c ^iai^ pra^^ 
fedentibus) habcbunturbini priirs^ >;ius for^a^, qyain 
erl^ib^t fig. :j5o' jaccntia 'ii^ angdis, QYN \»l QVN fFzja 
Nam exiitentc YR pofitiya , iny^nlewr pQl^tivus valor 
qotdraci ordinatae \ adcoquc/ ^ina e}^$ Ia|;er^ ^^b^bm-> 

tur Y prtn. '6^4 ) RP \ Rp . Exlftchtc vcro YR^ n^ 
padva ,' valor 'quadrkti ordirta;« ncgaijivus ^t^^prQ- 
indc ordin^ta ipfa impoflibiUs, quam ob cauflfam reif^a 
LRiL Qrdinatis par^ii;:la nufquam occurr^; curva^ ^ Quo- 
mimayteni ^od?m Vgupiento dc<rpfccnic RP infitii* 
des isagis ,*quam"RV rVbuc YN ^ondngi^ ciirvami 
furva ipia tp' V i^dfpid^ iiabct admodum ac^tam , ia 
^ua re^orc^cdituf , ^ 

76o.*Idein gencralitcr ^omifligit 1 quo^icfcuQquc rar» 
^oordinat^ multiplicaca per quemvis \tiiimergum parcm 
ift ^adcm i ac iratio abfciffa^ ];nuItipUcata, pcr injparcm 
majorein • Lnfiparitas abfcifls , fi^ parit^as pr^inat^ d> 
tJt regrcflum curyse V X^^A QQ^y 6^ blaas prdinata* 
^m <tire^ionibus oppofitis ; cxccflus QUipexi ^K^^ifi^ 
(upra pumcrum 'prdinacae: , c^hi^^bit ^ontai^tiuxt K^^^ 
VN i ^ Qufj^idcni iu Y •rQj^pd fi, rai;io, cnrclinata? mi4^ 
VplicatatV'^ imrai?rum i^parQm/quQmyis^ ,^ fii; «ad^m. ^ 
ac ratip. abfciflk per qtj^meriu^ n^ajoreiu . parqiX «^ vel 
iinpnrcm i r^ibit forraa fig. t^^S, , '<c H9 », qpi f*fni; 
Qnmcf clmit^qdi cafus, nani ra^io ordinata? miiii^iplicar 
tz per psmn arqualis raiioiii abfciflie pcr pamn redun 
^tor conc^nua^ bifleaibnc a4^ ii^parcm ii\ ^cera % ^'^^' 
I nis , ^coquc ad unum c prasced$ntibu$ cafibii^ ^ £t 
i |?«c quidcin; omnia ^fadilc gcncrali demonftraxicuic cmi ^ 



,T,RA^.SJ^j6kMATiO>Ji 

fc M* ■ ■ '■' . ■ .:■-"■•■ 

;; '^,nuinc'r^..pci:, qupni miJtiplicaiur' (t»-. 

t t minor cb^t. per quem, miiltifJicatuE x*' 

fijflt habebumij( figura; jsj^ sja , .^515 ^ 

x;3 tj i; 34S> 34.9>v2Sa .fi .perauuepiur^^Qiif 

^53 f_ , „ ardiriaf? ac illaruiri rctSi OC^fuccaJat 

hjtoiiil axis MN fi eriiiii Uc Gni VR.^ & RP , quc 

ibrerki VE . &:'£> . fivc PR , & Va , h^^^biiur i 

■ ^fctcreatlem fclatio 'ordiriatatLivTi ad .adfdffas , (j^a: ibi 
abfciffariim ad.ordinat^s •. 'ri ijs omnibiis caGttus ctff 
OV<X tangcijs 't & numerus majoE par^ in otdinat* 
i . iin^aE i(i abfciffa pr^bcbit in rig. ayi , biflas o^iiwtaS 
opp&iltaS ei paftc abfdflij poGliva , &. cs iw^ n^»r 
iiv^ iinpo/nbiles ; impai: in ordinata , Sc iti, fbfciO* ■ 
- & ih rig. 252 ordina'[as fiilgulas, & ejuCiieiii legisouH ( 
kbl^nTs i iiripar in ordinata a par in abfcifTa, cvditia' . 
tas &r[ipcr fingulas pro riiigalis abfclnis , fcmpcr pofi- 
tivas, & c!ufpiden]. ^ , ;- . j -.j ' .' ■ 

701. Hi quidcm furit dmn« cafus ratioms dk^da: ; 
St Vcro Jtatio fuerit reciproca ,-noti dirc<^a r patft, fi 
^i^^fHcriti /inipIeX ,= haberi in fig.' i;^. Hypcrbolain /F , St 
intef Sfymptotos MN. , OC4. ,Nam iri ca (nnnv.3J7) 
cft-^Aingidiinf fub VR , & RP feinpcr conftaJns , ^ 
deoaac RP m rationc reciproca limplici .VR.. E» Wi 
ro' nyftfbola' Mnos.&.ibct ramos in binis atiguUs aA ■ 
verticem bj^efltis OVN , .MVCl m inliaitum e!(cut<; 
fcnter;'ic ab;ed£ntdt ad C|i anguloramcrura ^Ia:aqao&- 
cumqu^ Iimttes . Id dutetri ctiatn dctfucitur ex tradje 
tis- riegattvorurn regulis ,' & c^ paura ratjoais , leci»' 
proi^i fmipltdis .' Nain iViutata. dirc^ione ahfctdT; mttt - 
tairi dtfbet ' tftlam dlreflio' ordtriata: in cu^ub detett 
(riiiiaiibncin illa: feinel ingi^cditur' . Gtneraliter autemfi' 
f aqb qr^diftatz uEc^uiriquc muliiplicata pcr numeriitQ im- :l 
pifcm squetili' raiioniteciproc^aEifciir^mu^iplicatie pec ■ 
numcrum imparem , forma erit caiJem , ac in ^ ty^ i. ' 
Orttinittic pofitivis abfciftis' poiitiYz > ticgativU aegaa- : 
v^ Ec^^d^uht fm^ulis &i^g\il%t M cprc^caff; i^^i^i-f ■ 



^Ss^ ib&nttk n xicctcfczt ordinata , fed nunquam cvjuic-J 
fccc *, dccrcfccntc otdinata » abfciffa ccicfcct in iHfini- 
yatk i 8£ , fi^^r ha^bcbUor ^ iliqua ;: adcoque qpsicuor 
cswi-erdQt ftfym{>tdtica , habcbanc pro afymptdcis em- 
m*'(fi3moc latcra V.M, VO , VN , VQj. ^^^ 
4ccbiuib ih f>iilh atigtlli^ dd Vcrci^^cm oppoHtis QVN « 

. ryojt Ac fi numcrus ordinatac ftlcrlc }m(^ar\ fcd abr 
(fdfifc^t; orictuf formi figura: 25J . Qrdinatc ^fingu-Fijy 
HriibfctiBs rcfpondcbunt iingaiac , fed omties pofitivac: 
^mAt, iKdioquc bim>raii^i afymptbcici jaccbunt iii bidla 
laogbiifr £)VN , PVM . ; , . , 

. «7^4% Si deiTUim rium^u5 6rd!aacs fucrit par, &a<b- 
fctffi$}'impars iicgatms ..abfciin^ nulljc ordinatac rc;fpoiir 
•dekom , pofitivis Vcfpdridcbuttt binae oppoficae , CnglHp 
fci ^iJar ferma cric , quae cjchibctiir ih .^g^jijl^ «» |ar"^* 
^cftibits biiiis r^nils afytiipiocidj in angulis NVO » 

705. Uc ttnito veliit confpcftu cdntfemplari Uccai 



bimieiejufmodi caifus , Ct P^ i Vii ^ ; fivc j^ > 
iuQir ciprimente P abfdHaiii i Q, ordinatam > n ; 

& w flfkrricros quofCuraquc intfegros p^ficiros » vd xit* 

***. ... ,. > ji*- - 

gatcH)$| mx^ fe primoi it»i flt fradtiar- nonftofflc rc-^ 

aua; ad mmorcni citprcdioncra « Si Fuirlc ^ numJri 

XQspdiifivus ,.pcrfiricbit cafnJ.^d JSgura^ a 24?^ 
^5f *' fi- ncgaciviis. ad 3 rclictuayi 5c itl prjmo c^fulo^. 
€a ^cgittdat crunc ei familiaParabplartiim', in fccU^doct 
CuBiiift UypcH>olarixrJbif. Si ^ fueric niimcrus s^qualis n^ 




^^- ffitcritanicA^; penmftbic^aftis adrcdfjrtn cxprc& 

iimda%' i^T^. Si fh iit&ric mimcnis minor^ qu^^^^ 

xiiKt^ cafos ad figuras 348 , 249 , 250 , prout fucric m 

yxc^^n pir, vclm^ &/^impar, vcliwpar, /r impar . Si 

^fMltmajor, quam «r, iiabcliunmr fiugrc 251^2^2, 

j ji4i**4ifdertt flribiw l^ti&nri eju^ cafus .- Si 



\ 




34^ P^TB.^NSF0RMAT|OKi 

piutcm — fucrlt ncgativus habcbitur fi^ufa 254 » ^j 

456 prouf fuetit &c m^ ic n iiiipat, ycl w par r e^ipt 
par, ycl I» Impar, » par. Qljod fi m ^fl|fet mMl> adeti 
quc prdinata in nulla t^tiotic abfi^iflpse; tum vcro Qrdi^ 
fiata oflet fcrxipcr coaftans/acieoquc pro curvis iUis hh 
t>crctur tantummodo rc6la ip(i axi par^kla m quanj^ 
^ cafu capdcm^urvsc abcunt, ^ 

70</Quac cx nat^ra po/itiyorum> ac negatiYonii^ 
!iic dc^ufta funt ? poffun? omnia accufatc 4ctnonftr 
ti, ^ iiximediatc dcduci opc conftrudtionis I\arum 
varum ipfarum , qus copftru^Ho ritc pera<9:a ^xhibcbl 
pcr fcfc cutvarpm Varundcmotnnijpm ^uf^cii/^ g 
icmpcr gcomctric^ prarftari potcrit per punAa ixz , 
piusjiab^atur ^onftruif^iQ, ^arum , quse ^xhiben; ordi 
pats rationem fimpli^cm, vcfpondentem ^acioni abfciffi' 
snultiplic^taE per qucmvis liumcrum gradatim ab imi 
tc incipiendo » tum pergendo pcr unitatis additioae 
continu^m , Deitidc vcro traduci poteft conftru<9io 
guamvi$ rationem, multiplicatam ^tiam abfcidic* 

707. Quoniarn rcA^ linea exprimit cafum^ in quq 
ordinata elt in rationc fimplici direfta abfciffx , qua^ 
«^-Ji^atur in fig. 257. |inca, in qua fit ordinata ia/^atiojl 
* '^ V -nc dupliUtai directa cjufdjpra . Capiatur AB utcumqiir 
perpendicul^is, VA \ produ(;aturque indefinite : i^atcd 
per y> & B: reda indefini^a ducatur pcr quoctvi^PQivj 
^tum R axis MN rccta parallcla Qp ; qu; occiUTai! 
dicubi rcCt^ VB in P^ ducatur PD axi parallela n 
^urrens rectas BA in D: ducatur per V , & D reaa 
qi^a! rectac yii RP occurrct alicubi in E , & iW 
^terminabit o(dinatam loci quadSti . Bric etiiin 
rcctac lincas naturan\ PR » ut VR •, Erit auccm BA 
DA« ut BR ad ER , Quarc rectangulum fub ABco] 
ftafoti, & ER>quaIc.rcctangdo/fub DA > & PR, 
^coquc ER in rationc ^iompofit^ ipfarunft DAj &' 
nimirum, ciira ear acqucntur in. rationc duplicata^ PK 
fivc VR 3 ut op^tcbat . Patebit awtem ipfam con 
^tioncm contcmplanti ^ a punao P orit^ RE cm 

*^ mcm 



LOCQRUM QEOMEtRICORUM. ^47 
^KR? ac pQiitivdm, a puncto vcro P2 oritiK^Ei 
CQntr;M^iam, iive a ilcgativo itcmm pbfitivam , uc 

reS. Inf ci}icnda jam Rt curya , jn qua ptdifiaca fic 
raDCfoe ariplicata abfciffaB 1 Sic ip fig. ^58 i redtaFzji 
*' W prios,' & cuTYaJShVI^T )am cqnftruika cjufT '- ' 
di^ uc RI fic in rationc dupUpata VK^ Ducatut c:^ 
' ^ ID paraUeU W , occurr^ns AB in D, cilmpci: 
P fCcu/qua^rcctx Rpoccprre^ iil E, ac ^tttt-^, 
t puncmni E quaefict^m J Ericenifn BA ^dDA^fivc 
Pfl ^ ^d RE ! CJiiarejUcprius ER,itt ration? CQ^Tipoficai 
KP,ppor cfldupiic^^a VR Vpoltcrior fimplex . Qtta- 
id iUafompoficacrittripIicaW ipfitisYR» qt oporcC- 
Patet aucem cdam hic , punccuiti U jacens tt 
pqfici v^ ' dcber? ' i^erum redderd punctum E2 ci 
pegatiYa • P^cec etiam, fi firfic iii ra^dne trU 
^ VR> obvencuram RE *ia ratioflc ' quidruplica- 
ti mnc def)erc I2 |acere ex parte acgaciya^ & ^z 
firc a(J p^cempoficivam ,^ ^tque ica^ pocro quaeyjs 
'plicatict (racionis abfcifias h^bebicar pet gradus 3 
^cfcitipcr E2 cx par^e pofitiya '^ ubi deveiii^ut ^d' 
:rual 'p^em^ negaciya > ubi ad imparem ; ^tque 
fatiQne ^bentur o(niies cafus bujuftnodi ,' ii^ ^uit' 
I» prdi|aacs( (i; ^iiquavisra^onc Vhfci^;^ ttiulciplicaca' 
t j^ucfDvis humcrum ihcegrgim ^ ppfitivuiii ' } ac ' finiul . 
ki^ omoes ca&s ^ifl quibus dcbeac Cffe fubmulcipli- 
|i| f a racio > fivc ^n quibus raciQ Qrdttiac£ ,' utcuii« 
"; |iiiilcip{icaca, fic eadem, ac r^cio abfciffbc fimplcst. ^ .^ 
*i'fnim'^ft'mucare iixcm , '& abfciffam ihicarc itl \ * 
nacam, ^t ^x fi^uris 248» 2^9 conltrucds^clerlycoh ' * 

!W. Si rapo fic rcciproca firaplcx iti fig. 1^59, i dur 
i VB > qc prius ," ducacuf B! parallela iud OccurtcflSp^ 
|r RP m I , o^m per V , & I rect^ occurrcns rc- ^^- 
iAli io'H, ac deiitumTecta HE pav^cla axi Qc- 
ns j^P in E, ericqoe E- qiwcficum puticcuto . Erit 



c 



AS »4 AH, avci RE, ut RP ad RJ . Qaam.^ 

■ dbKta 



.9^ ©E'TItAKSfOR*lAtlOhr.i^ 

ifctax Tfastagabnn fub RP, & R£ aeqQabitixr 1 
n tcfXUiffaio fiib AB , & RI % cruqtK icj^ircor 
ratiQQC rtdproca. RP , Gvc VR ^ tit dporKbztJ . 
firti^o autem ipfa oftcadct Ei detenninart a ]?i 
psiricni ocganvam: . _ 

710I Si ratiD iit fcciproct doplicata ; maneny 
f26dSg^ ,266 VB » iir flBT Cttrva fain licfcripci^ ^^^ 
ja.ratidiic rcc^oca iimidici VR, daaaque VIHTj 
H^^up prips y iiai>ebitiit qoxlita RE • Erit eaim 
. ad AH> five RE^ ni RP zi RI, adcqque ob AB 
ftantcfh RI in. cadone . compofita RE . & RP^ 
tCE dircae mRi, ^rcciproceutRP. ^fl: satein raSj 
reaa RI. eacfem y ic (cciprpca VK. i 8c ratip reei 
ca W* cadcm , ac rcciproca VR * Quarir cri; R] 
ratidnc rcciproca duplicata VR > ut oportebat . 1 
autcm etiam hiCt puoctiim I2 jacens cx .parte ncj 
v^ debcrc itcrum r<(idcre pttnctwn £3 ex ^artc gc 
va. Patct ctiam» (i RI fit in ratione retciprbc^ '4l 
c^ata VR^ obTcanu^am RE i|i radone. rcciprofa n 
tata j fcd mac dcbcre la jacerc ex partc • ppfmv^ , 
£2 tranfire ^ partcm ncgativam, atque ita porrot 
yis mukiplicatio rationis rcciproC; habcbitur pef 
du$i ;a6ente fcmpcr £ ex partc pbfinva in numen 
riy ncgativa iii impari^ ... 

, 7^^» Qjiod & ratto (^dinat^muifcipiicata pcr ai^i 
numcrum tationalcm n ckbcat cflc. cadcm , ac r 
abfciCfe iivc dirccu * fivc fcciproca multiplicata pci 
]^26iirum (^acmvis »f> i<f facile pracftari potfrit q>c :^ry| 
162 rum ;am coaftnictarum . Fiat in 6g. 261 axe MNca 
^63 v'a SVT , cujus ab(cifl)B VH finc ia r^tionc ordinat 
rum IH multiplicata ptv n , ac curva S^VT*, Cuf 
ordinatac, RP ^at in ratione al^fciflacmn VR tnulti 
cata pcr m ♦. His' fcm'cl prxparatis pec quodyis ^u 
ctum R agamr rccta pcirpendiculafis ^M > donar.j 
currat curv« VX* in P, tum PD parallela NM i«~ 
ad OQ^, indc vcro Dtt ad anguluni VDH f&j^ 
ctum, quic occurrat in H rccta; VN, dcindc HI pafi^d 
Uk Q$>« dmeo dccufrat curva^ VX in U' ac dcmul 



cij3L parallcla-;.MN> qaa| ^c^^rtct RP iil By Sk 
iL$SLt qu^iimm pvinctuiH £• Natn eritob aligu^ 
fi fclnirccpiin /& D VH t&tum , VH ^qiialis 
I^A^ ICP • Erit autcm VH iirrarfbiic IH, fivc iar«-i 
\cKJ^ muitiplicata pcr «r, &PR ia rationc VR mulri 
bffAfCf w. Etgo ratip RE multif>licata pcr n crit 
^^ ac rattQ VR muhiplicata pcif rn^ m oportcbat^ 
3i PorijoC n fit numcrus imps^ i & w par^ ha-^ 
f Caftis figiirac 261 i compofiiat cx fig. 252 » &' 
,ac E2 jaccpu c% parcc poativa t figura ipfa fcfc-* 
Q9WS TcliguQS fi.jf4^, ycl 252: fi fueru &»> &. 
Il^ar V.fiafcebinir cafias ^» :^62 compbfitac • cx 249, 
>, ac ftaiiiibitur Ez c^ partc ncgatLva j fiiurai^ 
»nt€' cafiisbdiiquos ip£^um figur^rum 249*^*&:" 
h fiicrif >' par >.& ^ if^^r r, habcbiturcafiis 
r.z^j 'compolipe cx 251 , ic 2^9 > figora ipfa' 
titcpafus rcUquos f^. 250, nuUo cxiftcritc Ea ^' 
rcfponjlcat, Kz \ & rcfppridcii til^us JRl' binis ' E 5 
"QuckI ]S raiiQ prdinatac miiltipiicata pci!^ ^ ^" 
t pc ^aeqiiafi? faripni fccipro» atbifrifl? rtJula*'^ 
afe pcr m , /atis cflct arcubiis SVT pjirabolicis 
-iicrc crura hypcrbolica figurarura 2^4 , ^55. > 
;'iia . ui in 'figuia.260 * 261 , 262 Ri cffct m\ 
iC fcciproca abfciffac VR multiplicata pcr m 
atcc > cadctp pi^orfiijs co&ftrucuonc obtincri in» 

fv.Atque h«c dcmum.pacto conflitui. pofllit^t 

S'^ •or{u$ turvfc propofitae tam parabolici , ^an^ 
ci jencris , q\xx quidcm egregias ^ & utilifp 
j»ropribtates habent potifflmum circa^ubtangcn? 
.fiT arearum mcnfuraiti ^ qu^ iu pmnibus ac- 
^uidrabilcs fuiit , prajtcr' uiiicam. Hypcrbolaii^ 
iih ratiotiis fimplicis reciprocx » fcd earumip- 
ip pcc ad rcm pra^fcntetn facit;^ & multo ef^ - 
'or opcquantitatuminfinitcfinlarunl: intcrcapcr- 
Vad' bQnfidcria,tioncm traiifitus, qui fit epofitiyq 
tivym. 
;^irum fane ^ quam ijbi ubiquc coftaos ik Qto* 

**** ' . metria 

.•i>» 



.^5>. Dfi TRANSFORMATIONE 
,i^trupo(ifntiium inkge CQnttnuatlonisrervand^j cujui 
Vinihil qfpidm mtitattir j^i faltiimi aiittotum.ili^iilfexo 
xiturji aut ^vaiiefcit ;.fed a quacumquie iriagiiitu^in^ a< 
aliam qu^mcutiique femper ituir pei: int^rnledi*a$ dmnesi 



iti infinitum protehdituri ut crura hyperbb}ica^ 8c para 
l^olica; vel (pirii infihitis circuhia^iturl aut recfedeildo i 
{)Uiii3:6 qub^atn ci , alterd^ tantiitn^ part^ .iti iiifinimm j A 
cx alij^ra 4ccedetid0 fetnpei' » qiiin ad ipfiim pertingai 
hnqtjam i & quin tdmen, ufpian| .abriimpatiit: J quod 
ic illi. accidit i^ quam fpiraleni logirithtnicam apel 
l^t i &c cujuj hati^am.. altbi perfcqtietiiii^ i vcl- dehitHi 
f)inis faltem fpiirairuhi oirdinibus re^dendo iii ^iiifimtiiti:: 
<][Uod ali^ midiai fpiralej^ pr^ftajic^ Acordina;taD hdrma 
leS) fubtidrmale^,' t^ngetites^ dngtili tarigeh^itlni ciim a 
iccj vcl fcum reila dati quavis , i^el c(im.re(5ta^;utcufa) 
^ue pet eundehiLocum Geometri£uni deftnitai ^ur^atu 
rk ipfa» direAtd cxffv^ci aC cjuidvis aliiid finp uUd fal 
li} muutut f^mpet tranfetihdo pcf omne$ i|itcrjinedia 
quantUatesf ciufdcm* gehctiif; ,. . , - .; , .' . 
715; £a h% omnincf fervatut etiam » ubi e pofitivij 
^uantitatibus tranfitut ad liegativas, qtii nimirum (ran- 
fmi noci fjt pcr falcuma f^d per gradus c^htihuosi Fi 
autcni is tr^nfim^ dupltci modo; nitninim vel tranfciin- 
dopcrnihilum, veipeirinfihitum.Ac ubi pet hikildhi tran- 
iitur^res fanenuJUaniadmit^tidnem^parit^ cum id^ quod 
d^refcit; donecc^^atiefc^tp£tiitus,"adnaodum rerurhcori' 
flit{itipiii ,' & natursi^ ordifxi confentaneiim Gxi qc, ali- 
^uandd poft. iiitcrituhi miitetur in\ oppofittim i qjo^mad-- 
moduni j^auUd, iUperius ry\xai. 677 vidimus -, Contingere 
ii!i debitpy vcl in regreffu fluvii ; At trahfitus. d pofi*- 
fivd in negacivum vay^ttii quxd^ fecum' ttabitr quc 
hi^ evolv^nda; func' i. Sc qu£ ad Sedtiohum Conicariiti] 
Hs^curarh' { ^ ^nalpgiam ^ ac ad iiniverfahi Lqco^i^tm 
Geomctricorum ihdolem perfpicicti^dam mirum ini hio^ 
duh^ cpuducunc ; Pnmivn autcm pt^fcrchitti ^xethpla , 




f 



GEOMETRICORUM. ajj . .. 
u; ia quibus, e pofiriro iii ne^itivum fic tranfi«u& per ni^ 
t> hilum i ac pcr infihitiinl p^dta ciiam i valgari XJcomc- 
triai nihi alia» 4<i£ ad infinitunl pertiricht,^demus e . 
Secitibnibti^ Gohici^ demonftii^iitis^ ac ex%illi$ ii^s, cuTt 
^ ^si quas htc habuimus i & ad Hyperbolas , ad Para* 
bolas fubliinibrei ireferri dixihluf adjc6tis ctiani regu^ 
Uis quibu{dam prb Locorbm^ Gcoinctricorum transfor? 
inationibtis;, -^ . \ , . . . • • > ^ . '^- 
.. 7161 Sit in fig« 2^4 rccta indcfinita AB i ic «^cntrbp^g^ 
fc-jextra ipfam. affumpto , concipiatur eirculus. NKOQ; 
^iiovis ihtcrvailo r. pct Cujus ccntruhi tranfeat , recta 
D£ pailalleU ipfi AB ^ occuinrctis circulb in N , ad pzt4 
tc^ Ai Dj iil O ad partei^B, E ; Sit autcm CH perV 
I {)endicularis ad AB, occuri^ns ipfi ia H j &, clrcula| 
i^ ih K vcrfusHj acl in Q^ad panci oppofitas. Trahfcaii 
'^ ikmum pcr ipfum centturil C rcfta indefiriiti f6; qu? 
' circulo occiirrae iti piinctb t ad partei F ; 8c^ M ad 
pivttt Gi rcctas vcro ABJa P v*tqu<^ ca recta conci- 
piatur, mbtu coiitiiiub delata in, gyrum , circa ccntruni 

illud C imriibtum oWiiie NfcOCi . ^^*'^^^^** pf"^P 
ti?3 acccdifttfc L ad K; &, evahefcct: tum^ abcunti L 
ik afcum KO in L* ^ mutabit HP* difktioricm^ddco- 
^c pofi tr^nfituiti pet" riihilum tti H hititabiiur e p6i^ 
^: fitiVa iri hcgativahi y vcl viccvctfa ; Pcrgai FG coni* 

* t^i-ti i &. Piinctuni P' pcrpemo rccedct ab H; auctat 
|[)erpituo HP' per brarics finitarurii magnitudinuth gf-a- 
dlis in irifiriitura i donec L' abeat in.Oi qtio cafu in* 

^ iericctib P'in illo infiniti quodam' <reIutimmcrifopeld]ifo 

^aodataambdo abfof^ta nufquam jam crit. NamrefctiG*E', 

\ Cdhgriitt ciim' D£ flaralkla rtcts^ AB , idebquft cum 

' i^fa AB tlufqiiam cohchrret 4 licct ih infihttfini produ« 

\" tanif / Yctim hfcimique parum inde reitiov^atur ita / 

ixt abcat L' ixi M ^ & M' itl L ^ ftaitim P, .quod pofi 

diftefftor in infiriltiitn delitucr^t ep unico momcrito 

' f6h9poris ; quo L' erat iti O , jani inveniriir ci p^te 

>.oppbfiu in P, ac , fi finitas tantummodb quarititktes 

• ^fem^mar i mutata cft HP* y iri HP habcntcm di- 
^ ikcdonctoi cobaariam • Nimifiio» ^tiam' in ti^arifinj^ 
1 f^ncti 



tja DE TRANSFORMATiaMB 

f>uncci P j^rinfinitum abit ipfa HP • e ncgfttiva ia|irt 
fitivam 3 vcl vJccvcrfe . -- 

717* U tranfitus puncd P pcr infinitum ex uQA{)b.. 
ga in pUgam oppofitam vidttnr fkri motu prorfus con-, 
riqiuo^ tanquam ii rccta. infinica HE, in infinita ilk( 
dtftanitia connectereturv quodamraodo eum rcctain&ii. 
€a HA • NuUo ctHm teciipQre concinjuo deefl hxm 
«liquis puncns F» prartcr momcntum iilud> quo L* c^ 
in O9 ac aflSgnato qu6vi« niQmc]pto te.'mporis> utcucn^ 
quc parum diAaiite a momcnio illo quo L'^ cft in O^ 
jtffignari frmpcr. poteft locui punai P » qut idcircb eq 
iblo momenib tempocis in infinito dcliterdc . In ipib 
vero tiiom continuo . rect^ CF vertic quodammodo. to?; 
tun\ fpatium coaclu&im parallelis C£ > HB Jta 9 mi 
iiullum fit punctum utcumque proximum^ ficctse CE y' 
Qtcumque ireniotum a cccca CH. y pcr quod aliquaEi*^ 
da uon tranfcat > qupd ipfum accidcc rcctx CG r«&; 
pccm fpatii DGHA » ubi puactum M^ percurret ^uc^ 
cum NK , motu fciliccc fcmpec concinuo , dc xml»j 
quam inte^uptQ • 

. 718. Illud unicum eft difaimen tjiter cranfitum re- 
etae HP' per ni)iilum > & per infinlcum » quod niniK 
rum in primo £a,(u in ipib nranficu ipfa quidem HP^ 
}am nuUa fit, punccum verb P hai^eacur in H; ia (c-* 
cundo puoaum P in illo immfufo infinici pclago ve* 
iut demesfum nufquam jam fit ^ ipfa ^mcm HP- l^a^ 
beatur , Sc quidem infinita , nifi forte infinitum inw 
I)ofibiIc fic , qua de re paulo inferius • ULad intercs 
gcneraliter notaci poteft , niliilum , ifc infinitttm abf«b>^ 
lumm ia cxcenfipne ita inter fe conneeti > ut quocie^H 
cimique in aUqua proporcipne geometrioa biiu te 
finiti maneant , qui vel fimul medii finr , vel & 
cxtremi, fi reliquorum alter cvanefirat, debeat altcr 
vadere abiblute infinitus, & viceverfa, quod cciam 
manifcftum fic, fi ducatqr L^Z peipendicuiaris ad K 
cm enim CZ ad ZL*, ut CH ad HP\ ac abconte . 
ih O evancfcit CZ, & remaricnt finita: CH, & ZL 
fed hac de rc occurrct itcrum iccmo^ 




tCjCORUM «EpMETRICORUM^ a^? 

.; ^i^. Gi^rcfani quoci putidtum']^ mdtu -qa )dam .con* 
^VK) tranfeat per lofinitQm ; • dc illud i^m » qiiQcl 
cx altera parce deiiier{iim iucrat in inSaito y .Sc obiiX'^ 
^ ^ rcsredratur cx patte eppofica > vfdcrut ctruf ct- 
iam cx Iblatf otre problematis , qno quxratur iti*. ffgu* 
kit2^ , tcrria CP corrahae pr^pc^tionalw foii bshtL$7i6S 
CM , CO datai .' Si ciiiiii ceritro C .intdrwlld ma- 
Jmw CO dcfcribiataj: «ircultw , cat « occurrac in I Tt- 
te Ml pcrpcndicularis ad CO , ducatulFquc pcr I: tmi- 
jtns infiriita GF , occujrrcn^ rcctar AB aUciAi m P>, 
€«t cx circdli natura • CP tertia / pfoporttbtialts -qiacBfi- 
^ 7 ubi interea notctur &:»'ilhid ,' licct cjtifmddt' .rc^rta 
fe binis punals I cifculo occurrat', unicaaftamcnCP 
ftfpondcrc vunico puhcta&l y &'cana«m a5 utfoque 
t «hSbcrr idcircov quod cam cttam ob an^iilim! re<?tum 
ClPfir CM'^ MI ut MI ad MP ; ubipro MI *pdfiti- 
1^i'fumatin:'eadem negativa , mancntc dircctionc pti* 
Trf icrniini CM , & ca mutata in binis tcrminis pfr©- 
rfertionis (^uamor terminorurh CM , MI , Mf > Ml^- , 
■cbct maticre ctiam directio quarti t^rhiini CP V ^idco- 
^juc ubi'MI poft trinfitum puncti I pcr Ojrcdcat "wi 
n^gnitudihcm eandem, liccto()pofitamdirectioncm ac* 
foBTu ; debct rcdirc cadto & magmtudo ;, & pofi- 
tto rcctae* MP , Sc Ibcus puocti P cl?'c idcm. Stdhoc 
*I iranfitumpcrintfmmm ribn'peftinet'1 " ' * "^ 

720. -Pro ipfo trahfitu pcr infihitum cohfiderr.ndo,t 
^a CO utrinquc in infinituttl prdducahir in A ,-& 
B ; ac circulo itcrum occurrat ih N , tecta vcro ipG 
NO pcrpcndicolaris pcr G duaa occun-at circulo Q,X» 
JC'ptr utrcmqne Q^fit tiang«ni DE intjefinite productst. 
^ncipiatm* jam puntrara M mdtuf cohtirtiio ddamffi 

b ad N fca , ut fopcrato ccntro C , abeat in M*. 

icmni I trtmsfcrttuf per Q^ in I , tangtns GF per 
te, inGlF, punctuni v«rt)P pcr infinitum rcpcdciis cfx 
mti *B rcgredictur ex ipfo infinito ex parte A • Ife^ 
qiiidcm in ipfo appulfu' puncti M ad C , inn- 
ia veluti obtutum, ni/quam crit; tangens cnimTJE 
fclicla AB nufquam ipfi AB occurratj at utcumqti^ 
B^fc9vich. Tom./II. S pa- 



/ 454 t>? TRA]S^SFQRMATlONE 
parum diftet M a C , crit "pmnioQ alicuW^ex pagtc 
altcra B , vcl A. Porro iti ^o motu punfta M *, 8? 
I fcirpet intucrilicct, qusccHim pcr(i/&C ?'*an(cunt| 
tranfcutit illa qui^m P^Ptp coqtinuoi^ ncc bfnniqo mu-^ 
taiimr,'fpcj porropergiint/Pjinaum igitur P , auocr 
iis fcmpcr rcfppndct ,'quo4 fcmpct ^ncntis acic Tal-- 
tcm' intucri poffpmus ^xtra imicum infiniti cafum ,] 
vidctur >' in i|lo unicq in^niti^Mfu iti in^itq ip foi 
quodaiiimodo ^clituiffc ^ tion intcriiffe ^ ncc mutatfitn 
efs^ /dum in illoupafu uipicp in infinito dclimitquo^ 
damniodo , fcd fx pil^g^^fpntrariji^Vc^fse idcm , a* 
depque In illis plagis (X^htrariis yidcntprquodamtno^ 
do cqnncfl:! fcdac CB, CAncxu qubdam noftrjcmcn- 
|i ^npcrvio , fcd (]ui , nifi in^nitum rcpggnct, oti^ii- 
np habcri flcbcat. Pqrrq noy? tM*, rcfpondet ipfa ncv 
va CP' cx partc oppofii^ , qi|ia ex quamqr propqrtio^ ^ 
naj[ibus CM, CO5 ^O» CP, mu^ata dircctiqnc upip^ 
CM ,' ac m^^ii^ntibus CO , " dcbuif mutarj & poftrcmiC' " 
CP direftiq, ac fi pro COfumatur CN, & fiantCM%[' 
CN j CN» CP' propprtioi^alfs , m^tatis primis tti-' 
bus, mutan d^bct, Qc ^umvLS tcrminu^ cum (n.|688^ 
inutat^qnum numcrus impar , inducat mutationcm ' iq 
tcrmino pcr pracccdeQics dctcrminato par Vcro ipfum 

rctinc«it* "* *^ "^ - ; - ^ ,1 

'^721. Porrq ipfaf b« mira conittnuauo , in transla-r 
|ion? PUtttK pet. infioitum ad plagas prqrfus cqntra* 
rias »/4c tncn^i noftra^ impcryius^ injSniti ncxus. pluii^ 
xnisTallis ^xcmplis c G^pmctri;^' pctids ' fonfirin^tar ^ 
yw^jiimitpm, qux cum fundq in infini^um rcccdcnte. " 
ita^ ^t QUiguam 'jam^fit> connc^umpr , mutari ^m,i«^^' 
|nus mo$^ ^on|itmoy 5? pciJli^ ipfis (ubJ9<5la> ac quo-^ * 
dammodp yel^t SjIcYinct^ retincmys , ne in tranfim pcc. ** 
mhilum l^^i^t i^ ^ jn^tentur . yi^Hm cx 'hujufm^irfii"^ 
excmpU? hic prQfcrcm^$,''in'quq q^id?m ornninq vi»*. 
det^it^r dcmonft^ari imm^diatvis '^lc ^anfitus * Sc infi-^ 
s^ti Ipcxos ,af pat^bit '» redtar^ Iinfati[\ tiaberi dcM^ 
pro ^culo , ^ujus radius €1% itif^nitus ,' & <;ujus ceo^ 
<ri8i) |t| /|{if^Uta ij[la. ^iftantic^ , quodamtnodq velutob-:;^^ 

rutum "^ 



LOCORUM GEQMETRICORUM. 25^ 
tiiQitn delitefc^t > s^c ^eit\<ic ^x parte dppofita regredia* ^ 
oir f Ubt autpift ex ea plure^ frudus perceperimus^ » 
pco^ec^iemqr ad illiiftr^dam eji^ ppc cotftiquationis 
legcm > 8q iml^s qii^ ad cufpidcs > aique ad infinita 
curvarufn crura perdaeat > eyolvemusi • \ Multo autem 
]^^ ir^ ipfi^ S<:<%ioni;im Conicarum proprletatibus oc-> 
otfrent ex boc mira , ic noftr^ meati prorfus iinper* 
vio iafinii;i ^cxvt in plagis c(4>ofiti^ dcriyata i^bi etiam 
dam ear^ natura ^ & aaalogia evdvi^ur, myft^ria 
qu^am fk j^adent) qu? meatem altius defixam > ac 
{Coni^trtciA meditationibus iaitiatatt) iocr^c^ibi^ fanQ 
^lupta^ perf^adati^r, 

7x2. Coacipiatuv ia ipfa %• 26^ radio PH clrcuIusFi^f^ 
occarrens ipfi ^B prxt^rea iq R • Moyeamr jam , ut 
otiu^, poac^um t pcr arcum NKOQ,) 8c m^atisacies 
4e%icut in ^n&tatioaes omnes , qu^ .'iater^a accidenc 
ipfi circolo » mcn quod ad magnimd^^m > tum quod 
id dkeftioncm perdaet cqrvamr^ V Curyatu(;a. qui^ 
^knjlj^ circui^ & eft minpr , quo ra^ius e^ oiajor ; 
fo eaitn 'magis e}ufBem longitudiois arcus ad redtam 
acocdit lia^^am » ' quo c majore abfciaditu^ circuJo. > 
qt^ ^ ob caafi^m putei fupcrficiei. , qu; cx ingenti 
vxius Telluri!^ fphatra defumitur » ^aftbus appare^ 
fTorft^ plaaa : atque idcirco circuli curvamta 2fti<^ 
imari Co^\ Uz y ut fit ifk t^tione reciptoca itmplici k^ 
<&>mm. 

723. Di^ igiOEir puadh^a Lao^edit^ K ul(?^^. quof- 
taatqjac Umites » miauitur raclius HP^ pariter ultra li^ 
miics quofcumque » 8c ulara qapfct|tnqu^ limites aitge.* 
mr curvatura • A^c^te t! ad K, > appcUlt. 1? ad H a 
ry^j^it r^ius HP ^ cvanefcit d^cvlus .» ^ftqiiaiiii 
^^ pnvnes 'fioitioiiiTi magniti||<}iAum gra^us. dca:tve* 
it ; cuicvatara autem ipfiiis circuiyi pcr omnes pari- 
fioitamm QiagnitUflinum gr^us ai^a infimta eflfe 
rret in eo ^afu ^ ut in fig. ijSj^ i$Aa CP rcapro- 
OVf infinita eva^ere dcbuit » m ipfo yelat inscrL- 

rcdbe ^.M evaaefccQjds . Tranfcupte L. in LS jam 

um t:adias HP' » ac ^^ircalas per omxKet i^dem fini* 

S « taturn 



45« DE TRANSFORMATIOKE' 

tarum magiunidinum gradus crcfcuat , curvatnr ytta 
minu^mr ; at curvatura ip(a jam oppofitam dircAioiicai 
acqaidvit » 8c qux cavitas pdus rdpidcbat plagam A 
in iD^icum cxtenfam > )am plagam B rcfpidt. extcnr 
iam pariter in inrinimw ad parces comtarias a ^^dsc^ 
mus igitur jam > curvatucam in cranfim quodam pcr 
jnfinimm dirc^oacm mutalTc mom continuo, & poft-» 
quam cavitas quibufdam vdut hiantibus oculis [dagam 
A afpcdlivctat ^ utut mom continuo pcrgcns > ipfos o-> 
culos jam ad plagam B convcrfos habct '• Vcrain liie 
quidcm curvatura ipfa ad illam infiniti ma^itudincm 
vidctur acceiTiflc ulnra quofcumquc limitcs > at eam nc^ 
quaquam act-giifc , nifi in ipf# puni^o , quod part^ 
bus, & ilcxu carct, quandam vclut infinicam curvato^ 
ram animo confingamus. 

^ 724. Pcrgat jam movcri L* verfus O : per^cc augcri 
drculis radius , & ipfe circulus, ac pcr oiuncs magni* 
mdinum finicarmn gradus excrcfcent in infinimm • la-» 
terca vcfo cutvatu):a circuli decrcilret pariter ultra quo^ 
fcimquc limites , 8c peripheria ad reAam CH ucrin-. 
que in infinicitm produc^aiii in S , & T accedct pari* 
ter olcr^ qiiofcumque^ limites tta , ut nuUum fit pmit* 
dimi V ejufdeiti tedac in quacumque ctiftabtiA ab irk 
aflfuniptum , ad quod ea peripheria aliquaoito Aon 
Cedat ulcra quofcuraquc limiccs diitanci^ quofcumq; VI ' 
cumque parv2 . Ubiciunque cnim afiumacur pun^tum V 
cxtrarciSbam ST, fempcr inyeniri poterrt locuscentri P* 
in rcdta AB e jurmodi, uc periphcria pcr ipfum cranfcar • Sati» 
crit reclam HF duccrc , ciim ad rconfticuere angulum HI^- 
sequalcm angulo THB, & reda IX dcbchit rcdte 
occurrcre alii^ubi ob arigulos IH^ , H1'X fimul mi^ 
res duobas rtctis y ac ob eorum .angulorum acqoalits^ 
(cm debebic ;bIdeiTi ttiangulum conllicucrc ifofceliuBi ^ 
ac proinde ubi ad euin occurfum devencric P* txaaa&* 
bit periphcria per T , & co cranfgrcflTo , peripberia 
fa adhuc ad V accedcc propius . Quod quidem 
veium omnino fit dc quovis puadro I uccumque 
ximo cuivis pqndo V rcct« ST ; pccflphcria ip/a 



LOCORUM GEQMETRICORUM; 2J7 
to contifiuo vcrrct quodaiTiniodo > ac vclut pcrradct 
omne fpatium planum , quod a itdla ST In infinitum 
protcAditur ad partcs fi^ ita , ut i\ullum fit pun6lum 
cjos ififtiiti fpatii y ad quod aliquando pcriphcria non 
pertifigat; dum L' pcrcurrrt arcum KQ > nullum , ad 
qaod itcrum redcat, fed aflignatio quovis pundo cjus 
^atii , ubicumque pofito, afllgnari fcmpcr poHit locus 
ookxn P' iprfi fcQ>6iidens in rc(9a HB , & pun6ti L*lo- 
cus in arcu KO, ac utrolibiet ex his aflignato, adigna^ 
ri paritcr poflint omnia fupcrfiqiei punfta , per qua; 
tum tranfit ipfa * pcriphcrla • 

725. Abcuntc L' iii O , punc^um P' Inflnito obru- 
tnm, nufquam erit , at abeuntc L' in arcum bQ» & 
P cx pane oppofiia cmcrgcntc cx infinito, fam qrcu- 
lum habebimus cum dircctione curvaturac oppofita, ja- 
ctntem pcnims ad plagam a priorc prorfus avcrfam rc- 
fpcctu rcctae ST, Radius, & circulus dccrcfccnt pcr o- 
mncs finitarura magniradinum gradus , curvatura, crc-* 
fcct , arcus autcm codcm ordinc vcrtct , ac pcrradct 
omnc fpatium , qucki ab ipfa recta ST porrigitur in 
infinitum ad partcs A ita , ut pcr quodvis pun<5fcum I 
cjufiicm fpatii tFanfeat aliquando ^ donec, abcuntc dc- 
nmm L' In Q^, rccidat itcrum F iAH'» ac cadcmphc-^ 
nome&oputn fcries exordiamr. 

72^. Jam vcro quinam futurus eft pcripheriflc ftaius 

in ipfo appulfu L ad O » in quo punctum P' ita irx. 

infinitum rcccffit , ut nufquam |am cfser? Dcbuit fanq 

cottgnicre cum ipfa recta ST in infinitum producta ^ 

Concipiantur enira omncs infiniti ftatuspunctorum P, 

& t'' > ^ omnes paritcr infiniti ftams pcriphcrisB cir^ 

ca punctum H . Cuivis ex iHis rcfpondcrc dcbct ali^ 

quls ex his. Nullus autcm ipfiu^ ptripherix ftat.us ha.- 

bctur cxtra rcctam ST , cui non refpondcat altera ex 

partc rcctx ST fuus ftatui punctorum L' P' , ncc ul- 

I11S cft punai L ftatus in arcu KOO cxira O , cui 

iKm refpondeat aliquis ftatus puncti P m rccta infini-* 

la AB , & aliquis pcripheriE ftatus hinc , vcl inde 

t rcct^ ST • Qjiarc pco appulfu puncti L* ad 0> cui 

S 3 rcf- 



L 



\ 



158 JXETRANSFORMATiONE 
rcfpondet cafus illc puncti P' ia immfnfam iUam W- 
finin sby&nm^ atque v0r;^gidem vdot demecfi-^^nie ik 
nicus ftatus pcripberix ircliaqoitifi:^ nitnarum 
da cutn rccu STi Qtioniam petipberia ckca: H 
vetfam aream pcrradit ex pane B motit Cdndmid » & 
in iHo traniitn V pcr O abiit, ad pbgam iippofitaiii > 
profecto dcbuft ia ipfo appulfu L' ad O tcariiire pct 
rcctam ipfam 4 nec a ptinctis I^ ad paocfisi I iranfilicc 
potuit^ tiifl tranicutldo prr punda V^ , 

727, Inde auiem duplici via.ncxosille mfiaitl vi-^ 
detur erui : primo qoidem i qoia rccta ipfa ttifinia. 
ST deb^c coafidcrajci taaquam drcolos qttidam iBfihi-» 
tus i cuius centrum Qt in infinita quikiam diftaotia ^ 
£vc ex pafte B ^ five c:)c parte A 9' i^ pattc^ int ip- 
fo indnito Copulcntut quodainmodo f, dc.. ConjoQgaii^ . 
tur, lit ipfa circuli peripheria abH vcrfos T digre(saad. 
ipfum B cx patte S redcat qtiodatnmodc^ docm cobti* 
nuof ncc ufqqam abrupto.' Secundo verov quia uc pe- 
ripbcria illa eadem circa Hj ex parce SBT traiifiitjix>* 
tu quodam condnt^o ad pjartcmSAT^ ntc iu ^foiraa-^ 
fim cft mutata^ fcd fc explicavit quodamfnodo ^ & fi^. 
ne faltil ullo in rcaamabiit^ ac deiifde in "^ontrariatnr- 
partem fc ifeziti ita & illud c^ ceotrum jP' vidietur . 
idera.itidcm manfifse , nec in illo tran^QI pcr tofoi^ 
tum commutatom efse / 

728. Atque hinc quidem licetet jam ^ Conicaruiif 
Scctionam analogiam quandam conftderandam migrar^. 
fed quo plenips intcUigatur resipfa> addei^a quaDdam^ 
quac pertinem ad regrcfsutnponcti^ujufpiam motacoaw 
rinuo delad a flnids quibufdam limitibos ^ £c ab ipfo 
in/initoy quse ad con^inuitatem Locbrum Geometrico-* 
rum intimius cognofccndam corKducunc, 8c ciun his r 
de quibus agimus^ nexus habent arctiflimos •' - 

729^ Uk primis ubi quxpiam q^andtas perp^nid cfeB-*. 
crefcit 3 ac demum cvanefcit cocuntibos' binis illis fi- 
mitibus f quibus tcrminabatur ^ ut^^ ubi de lineis^ ag^ 
tur , binis ptpictis ; aIiqtia:ndo quidcm iir co&tratian 
muutur 9 8c ia negadvam abit % njQmoi fiium pfi^c^^ 

qucaie 



LbCORtjM GfiOMfeTRlCORUM. 15^ 
ijfiimt ilcero cjus licnite » vel ucroqiie i fi uterque $1^ 
mes fit mobilisy quod in . exemplis concigii: buc urquc 
ilbtbi aliqiiaddo vcro retrd regf^dituir i Sc Cum eadeni 
dicc(^otie icerum crefcit tt pifU pofitiv^» limitibns il« 
lis ipfisf vel tohim alcpro $ |i alcer imindcus manec» 
rctro Curfum reflecie^tibus i utide adVenebnc « Eodem 
Vero pacco etiam ubi quatititas exdrefcic in in^nitum » 
aliqoaado quideM ejus limcs e^ infinitd regreditur ex 
parte Oppofitl i uc pariier ia exeitipii^ huc urque alla«> 
05. comigic^ aliqdaildo vero tx <6adcm icidcm parcein^ 
fimri redit rttro, <}uo pacco quahtiutis Jpfiiii directio 
DOQ nautacur; Rem icidem exemplis c (implici Geome^ 
irk pedcis iJluftrabimus « 

73<x In fig. 164^ vidimus) HP mutare direccionem 
ifim ubi in appuUu L ad K » vei C\cvaaefcic , quam 
«bi in appulfu ad O $ vel N oranfic per , infinitum 4 
Id qui4cm fcmpcr accidcc i ubicunlque afsumatur C 
incra circulum 4 ducca per ipfum panctum C irec^ta 
DNOE i 6c excurrcnce puncco L ihotu Conticldo per 
tirculi peripiietiam «,At fi i uc iii Ag* 166 i puclccum|«^^^ 
C afsamacttt extra circulilm ica 5 uc c binis catlgien* 
trliiis et ipfo duccis ad circulum ipfum alcerd CQ^fic 
parallek AB i quac , producacur iiidefinicc in DE » alcc-> 
ti Gt CKy qux ipfam AB fccet in H i ac punCmm L 
ipfios circuli peripbcriam p^rcurrac omncm mocu con* 
Muo^ 6c recca GF per ipfunl» ac pcr C cranfeac fcm<* 
tier interfeccid illa P dfcUlabic quodammodo inter ni- 
bilmn i Sc infinicdm 9 fctt^d et uoroque limice rcgrc* 
dieiis fine diireckionis mucacione « Si cnim in arcu 
circuli incer Q^i K ad patccs C ponacur I f ad par* 
lestero cohtrarias R^ Sc pundum L per arcum][QiRK; 
dcorrac verfus K motd concinuo > minuecur HP: eo 
appellenie ad k » ipfa HP evanefcec ; eo abeoncc in 
V ia arcum KI^^, itetum P rcgredtecur ^ & HP cre- 
Gaec eadem dircctione » qua prius » ac pcr omnes ma- 

E* 'ttsdinnm finicarum graJus inccrjaccnccs inteir nihi- 
i 9 6c tnfinimm > evadet demum abfolute infinica t 
Bhi L* appcUic ad O r QttO abeunte ih L in arcum 



V 



'16^ DE TRANSFORMATTONg' 
QRKj iterurr P rerro redibk ex infinito excadcitfili^ 
ga fine tranfittr ad direftionemoppofitam, ac dccrcfctt 
pcr omncs magnitudinum carundcm gradus ab tnfinito 
ufquc ad nihililihi > & ad iprum nrbilum appellct ^ ne 
prius> in ipfo appulfo L ad K. 
• 73 !• Hajc autcm ipfa vidcrc eft in ilKs Parabol^-* 
mnn, ac Hypecbolarum gcncribus, dc quibus a na. 69^ 
egimus, ubi Parabolae oftendcnt binos hofce cafa pcrni- 
liflum; Hyperbolae vcro m tran£tu pcrinfinimm. 'Ndmd 
p j4^§in fi^. 249 , ubi punftum P per arcum TVS motucofl'^^ 
2A^ linuo cxcurrae» nBinuitut- tam abfcifla . VR , quam or- 
-^jQ dinata RP ultra quofcumque limitcs, cvanefcunt inipi 
fo appulfu ad V , tum abcunte P in P2 crcfcit utraguc 
ex parte oppofita , directioncm mutata- in ipfo tranfi- 
tu per nihilum . la fig. 248 in tr^nfitu- per i>ibiluroiii 
Vmutat quidcm dircdtioncm abfeiffaVR abicnsinVR», 
fcd ordinata RP non mutac 9 qusc nimirum (rctro rc- 
p, ^ .grcdimr in R^Pa . la fig» 250 t contrario , abcunte P- 
^y^ per V in ;> , ordinata RP mutat dircclioncm in Rp : 
Z. ^^ abfciffa VR retro regrcdituc . At in fig. 254 fi rcAa 
quaxlam parallela Q(> moveatur motu continuo dirc- 
iiionc NM, cxcrcfcit ordinata. RP in infinimm , pcr 
quod tranfit in ipfo appulfu R ad V, tum abeunte R in 
II2 ; mutat dire(iioneip, & abfciflSa VR delata in VRa 
tranffijreffa nihilum j & ordinata RP in R2P2 tranfjrcf- 
fa infinitum , ubi punctum P a crure P/ tranfit raota 
quodam continuo zd crus sPz. qiiafi illa infinita aur* 
ra in illa infinita diftantia licet vcrgente ad partcs op« 
pofitas, inter fe q^iodammodoconncdbcrentur, &conti* 
nuarentur, At in fig. aj5 abit quidcm abfciffa VR in 
VRi per nihilum directione mutat,at ordinata-RPdirc^ 
ctioncm fuamretinet inR^P^, quocafu crusPrcuraau- 
re /Pzconcinuaturquodammodo. in illoinfiniro^ cxqua 
cx eadcm plaga O regreditur. la fig. 256.. arcusPr ami 
arcu/^continuacur quodammodo in ill:) infinitadiftandft 
oppofita, &abfciffa quidcmVR fctroregredimr cnifaih^ 
ordiaata vcro RP in motu puuffti P pcr Pfsp trant^ 
gccff# infiaito mutatur » dc oppaficam, directioncm acs 
. .' quirit. 



LOCORUM GEOMETRICORUM. ^fi 
tfaitit ^ Porro in hoc cafu cemmuart arcum P^ ^mix 
jf in illis. plagis oppofitts > coUigitur cx eo » quod {% 
pcr B agatur.rc&a infinita IH occqrrens cruri P^ in 
In P , tum convcrtatur , ut anjulo ABH cvancfocnic 
congruat cum dirc6Hone BA » 6c fiat paralida tc€tm 
OQj mm pcrgat ultcrius inPH', puruStumP, pcragra-s 
to tcto arcu infinito Pr > ex p arte oppofita rcgrcdietur 
pcT jp in f 
k. 732. In his qnidem excmplis habuimus mutationem 
^^k- :^ftfiine 9 Sc ordinatse in fig« 149 , tc 254..; mu«; 
tationcm abfcifls , /& regrefium ordinatac a nibilo > 
vcl infinrto in fig. 248, & 255; rcgrcflum abfciflac> & 
mutadonera ordinatas in %• 250 , & 2j6 , NuUa ex 
I eurvis ejus gcneris exhibct cegrcirum utriufque um ab^ 
fciflic^ <]uam ordinatas » ac cu(pidcs quidcm , quac ibi^ 
•ocurrunt , habent binos arcus pofitos hinc > & inde a 
communi tangente, &c crura afymptotica, fi regrcdiun-: . 
W ex eadcm parte infiniti, jaccnc pariter hinc^ dc in? 
de ab afyniptoto • Scd facile eft , curvas invcnirc barum 
opc , in quibus uterquc arcus jaccat ad candcm tan< 
gentis partcm , ac uttumque crus ad candcm partcm 
afymptoti , rcgrediamr autcm abfcifla c nihilo , ac or- 
dinata fivc c nihiIo> fivc ex infinito. 

733. Sit in fig, 2^7 cufpis DOD cjufdem gciieris 3F267 

ac in 250, vel 253 , in qua tangcns OA jaoeat intcr 

binos arcus OD > OD' • Aflumpta AV; ad arbitrium 

ducatur rc6ka MN in quoyis angulo finito cum OA, 

qua: occunat rcftae OA in A , captoquc in eadcm re- 

ttti fcgmcnto AV ad arbitrium , ducatur VO indcfini- 

ta , ac pcr quodvis ejus pundfcum £ rccta £L parallela 

MN, qucT occurrat curvas D*OD in T, I, rcdacOAin 

F> ac in ipfa£L capiatur EP tcriia poft YE,EI, & EP' 

tcrtiapbftVE, £r in eadcm dirc(5kionc , in quajaccnrEI, 

JEIS nifi dircdtio VEmutata, cogat mutarc dirc(5Honcm 

jpfius EP, vcl EF, &pun(ara P, P crunt adnovamcu- 

%idcm TOS , cujus tangcns crit ipfa illa fcjfta VO ita, 

ui^ utcrque arcus OT , OS jaceat ad candcin partem tangcn- 

lis ipfius • Nam acccdcntc E ad O ulaa quorcumquc 

limiccs. 



, 2«i t>E TRANSFORMATrONE 
iimircsf i dccrcfcit ctiam EF » adcoquc tam EI , quaA 
^ El* ultra quofcamquc limircs : , Qjiamobrcm EP, & EF 
^ dccrcfccot ultira ((uordiimquc Hmitr^ ctidm rcfpe<5hi fp« 
ferunt EI^ EVi adfertqdc rcfpcftd EF i «r ifcfpeiftu EO 
habctitis ad EF rationctii fiiiitani > uiidc fit i lit redla 
pcr O r& P » vd P^dufta acccdat, ad OV, iiltei quo? 
fcahqtii lidiitcii qu^e idcirco pud(%is P^ P* abcuiitibu^ 
f in, O fiitial fict tangcns^ & ^ccidct iri re<5kam VO.- Ja- 
ccbit autcm tam EF; qdani EP iri difcttionc eadcm 
propc ci^pidcm i cwi BL i Er iii eadeni diredtiooe ja-^ 
ccanti , ^ ,',/'. • ' . . • ■' . ■■. . K 

fiiS 7J+* ^^ ^^ ^S- ^^^ ^^^ bina crura afymptotica IDi 
IT>' fainc> 8c inde ab cadem afymptoto AB, at in fig» 
255, & 256 ab iifdcm VO, VN. Scectipfam ABqu;^ 
"" vis MN in.A ^ & haiic in V fccfet OQ^parallcla ip(i 
tffy mptoW AB ; Du(fe vcro rcda EL , ut prius i fiat pa* 
ritet EP, vcl EP' tcftia poft VE,^ EU vcl Ei\ & pun- 
ita Pf P* erunt ad aiia bi^at crura Tr $ S/, quaf ha-' 
facfaunt pro commiini afjniptoto rcdlam VO i fcd jacc- 
fctmt ad caihdenl partchi refpe Au ipfius ; Crefcenic eiiim' 
yE irif infmitum» accedunt £1, EI* in infinitum adEF 
rfqiialete datsc VA ; Quamobfcm BP , EP tef tiae poft- 
V£ , & EF dc^cfcrint in iafiniti6n ; & crris utruimque 
r ^ccedit ad VO oltra quoTctiritquc limitesr, qiiantfjddrco 
habet pro afymptotd . Qjjoniariir vero rtdtx EI ,' £1* 
eanddn dirc^ioncm habeat ; habebunt candcm citam 
£Pv EP^ , 8c ramas ucerque jacebit ad eandem partem 
afymptotiv. ^ ,, .... 

71 S^ Porrb in littaqtle donfth](fiione &cile admoidttm^ 
inveniunmr punfta H, H% in quibus nova curva prio* 
rem fccat. Ea dcterminabuntur a re^ fccante btfariam 
angulum OVN .' Si enim hrcc tt&x occurrat rcdar , EL - 
ia L : patct ob angulum ELV «qualcm alterno LVN , 
adeoquc etiam an^o EVL , fore EL aEqualetti £V g 
adeoque EP, Vfel EP* tertiam poft EL/ EI, vel El^fote 
minorem i squalem < vd majorem rcfpcdu EI y pfouc 
fuerit EI refpechi £L. <^are ubi L congruet cum I i 
Vjdl r in H^ vel H'^ ibi oim iifdem congruet ettam* 

P, vcl 

/ 




f^y^ P** Scd hsc ad reni pra^encem nuUius Tuat 0»? 
fiqSk Illud; ai^eth k|dc pertinec i quod iii %^ 2^7 ii h^m 
bMnir. B>^o aA>fci(f4 OB i pro ordiuata £P ^ EP'^ vcl cu ^ 
iHin JEJi £r, cxeuhrcntc Pii vcT I. pcr arcmii TPOPS 4 
Vd DlOrO^ iqfi^cditui: ilmul e mbilq tam apfciflaQ£|ii 
quam (^4itiata EPj vel £1^ manjCQie jcadchi diifo^btMA 
«^iMi id EP'^ vtl £r ; ^t ixi % z6ii & dUc^tur or-^ 1 
dioact^ PRi ,P'R' pjtrallctB re<a« 6V i abeunte Ppcr crui ;| 
Te in. it^imm» ac redciuite [kr 4PS, cx in&iijco^ tatrt 
^fciSa VR.,rctra rcgi^cdieeur^ p^ VI^^ cni^QjE jqalai 
ordinata RP pcf RT' cx infinitoi 
. 7j6i At Jjujufmodi fcuryam 4 gua^ bmi crura afyiii- 
ptoticii hsdbcat ad candcm aiymptoti partcm , Sc qu«f 
idcircof /cutidcm iUiitn regrcffum exbibere . poflfl^ iitriHf> 
({ilei tumiti^ tam abfcifHr^ qilam drdinatae,^ 4cimodUi|j 
faciie cft cdnftruere ctiam iti fig;. i6.6; Satis c& ibideui;^ 
rc<aam CP prdduccre ita^ ut PO.» PO* fint asqiiaIesip-Fi<^ 
fe CLi Cti &c omiiia puniita O ,- O* eriiht ad, ctir-: 
vgm,SO'MOT, qu« comingct in M ircftam CH pro- 
do^ it9i ut J(it HM azqualis tangenti CK.j habcbit 
vao hina crura OS , OT in idfinicum tehdentia ab ea* 
dcm parce tc&x HA i qjax erit aQ^mptotus Utridfquc • 
Dbais cfiim CV ^ ON, 0'N' pcrpcadicuUs in ipfem 
AH.i cfit CP ad POi vcl PQ' * nlmirum ad CL , va 
CL', ut CV ad ONi vcl ON' ^ Qgarc au<ao in inifi^ 
nun pFimo fcrmino CP in acccflfu L^ ycI L' ad Q^ ^ 
fnaocdtibus finitis CVi CLi ycl CL',' debcbit ON, vcl 
dlN' dccrcfcere vltt^ quofcumqiie limites,' & cumCL^, 
CL' imbst .diredk>nem babeant fempcr eatxdefm.^ can^ 
dcm pariitf djre^tionem habcbunt 4ppcr.&^0^ PCf-» 
Bcroque puncao O jfccntc ad candem parftm. fcc^se Atl.. 
Abcuniibus ajutcm L, & L* in K^ patct O, & Qf dc-J 
6ere abirc in Miundc iUud confcqui patet, rc^m ni* 
miriim FG cVadcte ^ngcfJtcm <jurva5 TMS^ 

737; Hi$ fufius aUquanto e^pofitis iiccbit jaai tnde 
^ruierc continuir^tem quand^m in ipfis Scdbionibus.Co- 
iiicisi que in Hypcrbola fit cuin tranfitu per infinitum 
wi parccs oppofiicas 9 in Par$ibc^a Yccp -011» r^^ lo, 

fig.%69 



'^64 l)ETRANSFORMATrONE 
JFzi^pfie. 269 .fint intcr afymptotcs. MG» , NC» bini rawl 
cjufdcm Hyperbplae SDT , sdt^ ac rccta qua^dam infr 
iSita RB tranficns per cjus punctura D ipfi iterum oc- 
currat in P , & circa ipfum D* motu continuo con- 
vcrtatur , donec intcgram conyerfioneiji abfolvat • ja- 
ceat autcm Pi in crurq DS , pcr quod ita excurrct y 
m AiBi, evadcnte in A2B2 parallela afymptoto Mm , 
iiVifquam jam fit , fed^ crurc toto peragrato in infjni- 
10 iUo quodammodo dclitefcat : abcuntc A2B2 in Aj- 
S^^jam punctum P cmcrget in P:} exdiftanria infinita- 
oppofita in crure /, ac motu eontinuatp per A+B^pc- 
ragrabit 'P4 totum crus ^, donec ficta A5B5 parallcla 
afyraptoto N;/ , itcrura nufquam fit: pergeatc vcrore-.. 
cta in A6B6 rcgredietur cx infinito cx parte- bppofi- 
ta pcr crus T ^ quod pcrcurret totum 3 doncc rccK 
<fat in b > facta A?'^? tangente . Atque hoc quidenv 
pacto , ubi rcaa AB dimidiam converfionera abfelve- 
' rit motu continuo, Punctum itidcm P motu conrinu& 
percurret utcumque Ffypcrbola; ramum , & Hyperbola 
jpfa habcnda erit pro curva quadam oontinua , qu£ 
quodararaodo in orbera redeat etiara ipfa ^ & in infi- 
nitis illis oppofitis diftantlis quodararaodo veluti con- 
jugatur, conncctaturque, crure/ conjuncto cum T, & 
s cura S. Ductus autcra ejus coniinuus eft DPiS ^w-. 
finitum ) /P2P4/ {wfinmm} TP6D. 

758. Qnod fi punctum D aflumatur intra Hypcrbo- 
tx ramum ubicumque rccta binas fcmpcr habebit in- 
t€r(ccciones cura ejus periractro juxta num. 2S4., dcm- 
pto cafu t quo evadat afyraptotis parallela , quo ca- 
fi| altcra ex interfectionibus in infinitum abibit , & 
cufquam jam crit ; fcmper autem cx infiaito rcdibit 
cx P^i^re oppofita $ unde con&quitur etiam iHud^ mu- 
tari femper rectam DP c pofitiva in negativam io 
qqovis tranfitu puncti P cx altero Famo in alterum.- 
Sic DPi jacet dircctionc DAi , fcd DPj poft tranfi- 
tum per infinitum contrarium directioncm habet DB; 
quam habet ctiam DP4 ; at iterum fuperato infinito 
P^ jacct ad jparies Af . Quace fi qua rcctst digrefia x 



LOCORtlM .GEOMETRiCOftOM . ttfy 

•Sfatd puncto j & terniinata ad altcrum ramgfti fuMf . 
ieatur pro pdfitiva j ubi ad ramum alterum jcrminabl* ' 
tur , habenda crit pro ncgativa k Chotda quoqiic 
qu£visj 5 qua ad cundcm ramiim cerminabamr , fi 
tcrminctut ad utrumqiic , c pofitiva trjmfibit in tkt^^ 
gativam. 

719. Hinfc autem ctiam , fi cbiicipiatur Hypcrbol» 
brdinata I^ in fig» i i poft rctcffam puncti R in infi^ 
nituni rcgrcffumquc pcT R' cx partc oppofita rcgrcdictlli 
pcr Ff' , pcrmutabuntur puncta P^ p in p\ P* ita, uc 
exiftentc P in latcrc dextro, fit P" in finiftfb , dc yU 
ccverfa f c latcrc fiaiftrd cranicat in >^ in latus dcx-^ 
tcrum , mutata itidem ipfi.us. chordae Pp directione m 
concrariam in Pj?', caque ipfa e pofitiva migrantc ia 
ncgativam, vcl viceVerfak. 

, 74©. At in Paraboia longc alio modo ifc tcs ha-> 
bct • Habcmr nimirum rcgrcffus cx infinito^ in rtcta 
DP in fig. 270» Si cnim pcr punctum quddvis pcritric-ViIf 
pti D tranfcat recia AiBi , & occurrat ipfi pcriitictrp^ 
itcrum in Pi , tum movcatur ita , ut accedat ad po* 
fitioncm parallclam axi } TcccdetPi qltra quofirumqiie 
iimitcs pcr crus DS 3 & fcmper alicubi cxiftct i do^ 
ncc AB fiat in A2B2 parallcla ax,i ^ quo cafu juxta 
num. 149 ipfum P nufquam )am.etit : at pfogrcdiente 
recta ipfa in A^bj, ftatim habcbitur P3 in crurcTD^ 
^uod puncmm percurrct totum id crus, doncc inidem 
punctum D, cx quo fucrat digrcffum , rccidat ubi AB 
evaferit tangcns in A^B^, Hic igitur DP, ubi inDPt 
ia infinitum cxcrcvcrit, retro rcdibit in DP3 cum di- 
rectione cadcm . Erit autcm Parabola etiam ipfa curva 
qu2edam continua in fc quodammodo rcdicns hoc or* 
dinc DPiS ( infinmm ) TP3D . 

741. Hic autcra mirus itidcm vidctur hexus cm-«' 
rum S j & T in ingclita licet . diftantia a fc inviccm 
fe conjungcntium quodanmiodo . R.ccedudt iUa juxta 
nutn. 7J* Sc ab axc , & a fe inviccm ultca quofcum'* 
quc Umitcs : at ut in Hypcrbola binorum ramo- 
lum cruTa continuabantur ia iUa infinita oppofita di- 

ftantia 



t66 DETRAl^SFOKMATtOHE^ ' 
ftati^ia» ita hk cominuatotBr qaod^mtnpda cru» Sjf'^ 
in diftantUs oppofitis • Si nimirum p fit yetter ^xis 
DAa; & concipiaiur ordii^ata P1P3 ^ cpyt abeupte l^ 
^ infinitmn , Sc rcgrcffo intjc rcgrcdi^tur cum ipfo ; 
fM&A ipC^ Pi, P| non tcgrddicntur » fed Pi trmfi- 
bit in P^^, & Pj in Pl tranfgrcffo infinito., in quo& 
Mdih^fa ipfa in inilhitqm ciTCtefcehs eondnuatur quo? 
damihckio » *& crhra S' , T contih\|iaiitur • Hi^tq» vcro 
Wt itit€K bina cruf^ S , T licet cxcrcfCchs in infini^ 
iKlmr cctnfid^randlis; crit tanqukm pun£hvn quoddam in* ; 
fihitae pctipheriap infihiti crrcuii dcfctipti fa(9:o ccntra ^ 
ifl vcrtice D . ytciimquc eqim cxiguus angaltes fiat 
AiD-^a, fcmpcr (nym. t$€) rcda AiBl ocpirret iie- 
(om aUcubi ia Pi cruri paraholi^o, & pltrx ipfum cx* 
curret. Si nin;iirum fadto centto in Dt affumpto, fadio. 
qaov^s , de fc:ribatar circulus, occurreas: redi^ AiD 31 
AiDy A;D in Hi, Ka, H3, utcamque cxiguus iic ar« ^ 
cus HiHli^ ftmper puhdum At cxcorret ultFa F^abo^/ 
Ir ramun^, ut pariter utcumque exiguus fit ^cusHzF^; ^ 
cxcurret A3 ultra ipfum ramum*. Qiiar<( fi fiima|ur su% 
cus ^iHa 1n quavis utcutnque exigua rationc ad ton ^ 
tam lui circuli peripheriam, ioci|cuIa, qui concipiatpr ^ 
defcdpms radio PA fupe^ante chordam DP > adhuc mir 
Aore^Ti rationem hab^bit areus intcrceptus cruri^ ST^ 
cum eam haberc ^ebcat arcus interecptus vtQ;h PiAi , 
PjAj . Quare in ^rculp in^ito ea ratio deb^Y efle 
pror(us nuila , ita , ut arcus interceptu$^ ipfis cmri(>us. 
ntc habeat qnutt) gradum illi^s circuli^ nec ununi mi- 
nutuiih, nec unum fecundum, & ita porro^ fed bal^r^ 
4ebeat refpedu ipfius prorfus ut punftpm qupddam , " 
quod tUi idtx continuitai;is crurum ST magis, etiaoi fi^ 
vet, & yidetur cxdudere falttuii quemdam infiaimm e 
crure S ad T in itto continuato mom pundi P pera* . 
grantis ramum omnem Parabolx » qui quodammoclo/" 
redeat Ui fc ipfMm. 

742, Por^^o cadcip continuatio ^ & ncxus irorum» 
ac regrcffiis curvas in fe ipfam ope infiniti habetur ct- 
iamia curvis reliqois, dequibus hitc cgioiusr five para^ 

boli- 



LOCQRUM GEOMITRIGORTJM- 267 . 
jbolid gei^cus Gnx » fiye I^pefbolici . lo primis in fig» 
248 j & 24^ ,*(}^pdvis iparaboi^ivn 'gt^nu^ ii^ prb^mFa^S 
rcdif hoc Qr4i|ie , VPT ( inftnit^m } SP%V\ & in 249 
fig. 25q VPT^^ inftnitimf ) $fV ^ Id 'pajebit , fi con- f 50 
dpumir r^dSla indefinita tranfieos pcr p» &V« Sienim ' 
(pa Aiovpatur circa V > & difccdens a ' pofifione MN 
conyprtatur > dpacc deveniaj; prius ac| pofitionem QP^ 
nim ^ NM > pun^m P percurret prius tomm. cru$ ^ 
VT, ex qyp motu contifiuo tfgnfibtt ad crii^ SV^ quo4 
percurrct » aufc T conncxp quodau)iT^pdo cum prurc 

5 in il|^ ipfiaita diftantia • l^ ramis pariter Hypcrbo- 
|ici$ in fig. 254» fjj, 256, fempcr habpbimr ^opuy 
pumo jpruruni f, Sy ac T, S in infipita diltaptia, & r^ 
iu&us purv^ coixiipuus hsdbpbifpr pcr B|T C infinithm), ^'^ 
SP2/ ( infinihm) ^PB, ac in fig. 254 fan) T, & S,. t 
quani ty Sc 4. codiunguntur in diftaticia infinita 0{p(H 
fita, in ^g. 25J.T> & S conjunguntur in oppofita t^ 

6 / iq eadcm, tn fig, 256 contra T , & S in eadem 
f , & / m oppouta. ' 

74 j. Ceneralit^t aucem in figuris^ pmnil^us geome« 

trldi > fiyp guarufn ofnnia pupAa inveniri poiTiuiC; 

quocupiquf: modo ope fimpUcis Gcopieuriae > vcl op^ 

furvarup pcr fimplicem Gcometriaip conftru^Urumpcr 

poncu , fi qnod cru^ iti infinit^m abeac 9 iempcr )ui^ 

fccbitur crus ^cerum cx in(inito r^e<iicO|| vel ^x ca- . 

dem parce > vel ex fonoraria cum ipfo ia iUa jnfioita 

dift^tia ' connc:iEum quodafnin/udo , qup^ omniao ad 

continuitati^ legepi ^bique \f^ Gcomeuria fer^tam rc- 

ligipfi(^mc fft neceflariuui » 9C epe calculi algebraici 

gencraliter dcmonfirari poteft i^' A^ ubi de ^plicati^nc 

Aigebrc ad Gcometriam ^cndqta crit > oipnina de- ^ 

inon^abitur • ' Quamobrem ejufpio^i crura (emper crunc 

nuniero paria • Idcm autcn) ,' & fublitnipribus c^rvis 

Quibofdam fpntingi; ^ quas tranfcendentes vocant, pr^- 

ler (pi^alcs quafdam , quas e3( alter^^ parte 40 infinit^m 

leccd^nc, ex alcera cii^fa pui^(:tum quo^dam * vcl or- 

lem qocndain infinitis Ipiris circumvolvimmr acccden- 

prs icxnper > quin unquatii . \fi Ipfum rccidant , de qui- 

I ' bus 



4g8 6]& t!l'AKSFORMAl*IO^rfe ' 

bus agemus alibi. Crura aucQrahuigiiiibdi^ yel parjl»^ 

lici crunt gencris, vcl hypctbblici • Prirai gcncris d^ 

ra nullam habcdt fectllineam afymptotatn ; ad quam 

acc^daat ultra qilofcumquc liinites 3 fcd ultra quof- 

cumquc limites a quavis recca data rccedutit . Secundi 

generis crura habent rectilineaiii arymptdtum omnia s 

ad quim ultra quofcumque limitls acccdunt. Illafem* 

per reccdartt a fe inyiccm in infinit;umt & m diftio- 

tta infinif a copUlantur : h^c quandoljue a Te . itivicem 

rcccdunt 9 in infinitum quandpquc vero ^cedunt*, ac 

in primo . cafu fanpcr tcCcdunt Id plagas prorfUs oppo- 

GtsLS ita y ut adhuc afymptomm candem habeat fcm- 

pcr utrumqUc cru$ ^ qUod ubi iu infitiimm difce/ferif 

cx partc alterst cjiis afymptoti pdterit rcgredi vcl cze^* 

dem parte > ycl ex oppofita , ac vel ita , ut binacru^ 

ra jaccant refpcctu cjufdem -afymptoti ad cafdcm pla-^ 

^as> vcl ita , ut jaccant ad dppdfita« • Crus Pt reOc- 

dit in infinitum ad parte^ O afymptoti OQ;^ in 6gi 

^254254, 255,256» 26S> regreditu > autem in prima ex par- 

^55 tc oppofita Q^, & ad plagani oppofitam VM , rcipc- 

^66 dm aiymptoti ipflus , in fetunda cx eadcm parte O ; 

268 ftd paritcr ad plagaiii oppofitam VM , in tertia ex 

partc oppofita CX» ^^^ ^^ candem plagam VN , in 

quartal cx cadcm parte O, & ^ pkgam eandcmVN^ 

74.4.^ Sic autcra ctiim in arcubus 9 qui atd punTchim 

cfuoddam tcrminantur, ideih adcidit , ut ducta ibidcm 

tangente , & re43:a ipfi tadgeflti idclinata utcumque ^ 

qu2e nimi^um reeta produaa ctim ea ipfa tangente pa- 

ritcr producta continet 4 angulos 1 dreus curvs^ ipfius 

cbntinuari debeat , fed facerc pofljt in quovis cx illis 

Fi^Squamor angulis, five rcgrcdicns:, five progrediciis . Ar- 

249 CJas VS 9 qui cft continuatio arcus TV jacct ia iSg. 

250 248 , & 253 , in angulo OVM, jacente ad |acus re- 
151 fpecm anguli OVN , in quo /acet TVj in figw 24^1 
252 & 252 in angulo MVQ^i ad ver-tidem oppofico : ift 

i24j fig* 23d, &: 251 in angulo NV. jaCentc ad latus at- 

z6y terum : at in fig. 267 , tam areus TO , quam OS ja^ 

cent ia codem anguio_ VOA 2 Quotiefcumquc autcm 

con- 



iOCdRUM GEOMETRIGORUM . t&^ 
fOfitJnuatiQ habctur in angulo ad vcrriccm \)pp6(no , tk\ 
ia fccundo cx hlfce quatuor cafibus , habeiur mutatio 
flcxus in ipfonexabiiTorumarcuum, rccba , cjnaa arcum 
utrumquc tangit, ibidcm ipfum fecante^ju: in fig. 349 ^^ 
|k Z52 . Quotigfcumque habctur continuatJo in eodem 
angulQ, ut ia fig. \h'^. , habctur cufpis fccundl g^ncris 
duorum arcuum i qtldrum altcr convcxitatcm obverth 
altcrius cavitati • la rcliquis binis C2afibu$- habctur vel 
condnuatiQ qusdam curvatufx in cant)toi plagam ^ ue 
la fig. 348, & 251 , vcl cufpis prfim gencris duorum 
arcuum Cbi obvertenrium convcxitatcs , ut in fig. ^yo^ 
& 253, prout arcus continuatus jaeet ad eandcni tan-> 
geotis partcm, vel ad oppofitam , in quo poftremo ca- 
fu cuQ)idis primi generis tangcns curvam pariter in Jp, 
fo contaftu fecat . Cufpis autcm primi gcneris. figurjB 
250 , & 253 habens tangcntem infertam intir bino^ 
arcus refpondcr cruribus hypcrbolicis figurse 255 haben- 
tibus afymptotum mcdiam VO , in quam tang^ns dcfi- 
mt, ubi pun<%um contadbus ita in infinitum recedit> ut 
uufquam jam fit> & cufpis fig. 267, TOS fecundi gc, 
ncris jacctis utroqpe arcu ad candem tangencis parceni 
Cturibus Tt , Sjt fig. ^^^8 jacentibut paritcr ad candem 
partem afymptoti, qux cufpis in haec ipfa cviira dcfinit, 
Ut p2(tct ex ipfa confl;ru£lio;ie , fi tpanentibus pundi^ 
V , A pun<5him O ita in infinitum difccdat , ut nui^ 
qaam )am fit, quo cafu a cufpidc primi gcneris DOD' 
figars 267 gcnerantur crura afymptotica DBD^ figttrNC 
268, dc a cufpide TOS iilius crura T/, Ss hujvis. 

745. Porro in bis ipfis qifpidibus , & in iUo flcxu 
coutrario continuitads kgem obfcrvare liccret paritcr , 
fed connexam faepe cum illo tranfitu pcr infinitum , vel 
cum co^ifideracione ret^a:, tanquam in iof^aisis oppofitis 
diftantiis icontiauatx , ^ redeuntis ic^ fc ipfam^ actran* 
(itum e pofitivo in ncgativum, tam per nihilum, quam 
|)cr infinitum. Curvaturam enim, ut diximus num.722 
metitur radius circuli cucvam ofculancis in quovis pun- 
Gto 9 cui ea cenfetur rcciprocc proportionalis . Porro 
(^cnccum circoli ofculatoris fqmpcr )acct cx parte cava 
Bofcovkh. Tm.III* T in 



^f6 DE tRAbJStORMAtlpNl 
in reidia perpcfidicul^ri tangcnti^ quod idcirco in fteinl 
tonii^ario iigura^ ^52, vcl in ctjfpide priini generis ha- 
bcnte rangcntcra ^rcufeus intcrmcdiam in figuri 25^ 
dcbct in V tranfire e plaga VN , a4 plagam oppdfitani 
VM, qubd ficd onsnitio nod pdteR> hifi ti'at^fi:ac , vci 
pci: ipfum pUndlum V, Vcl pcr ?ihfinitu\n , tranlcun^ 
tc r^id [ofculi , vcl pcr nifailuin i vel pcr infini- 
jtUm) ac prqjlnde cumtura, vcl pcr infitlitum , vel pcf 
itiihiium* Et^q&idem ubl de circuloriim ofculatofum ge- 
tierali dcterminatione dgcmus>.vidcbimus in curvis quk«> 
bufcumque cain lcgcnii fande fcrvari fcmpef 9 ui huUa 
cufpis primi gcnerisi hullus cohtrarius fiexus hiEiibeaturi 
nifi ih (o puHi^o, in quo fadius bfculatoris circuli vel 
pci^ hihilUm tranfit^ vcl pcf infinitum ;. him vero cur- 
vatura^ &.fadii circuius migrant c pofitivis ifi ncgati* 
va^ licet aliquando ctiam radius ofculi, & cutvatura ; 
Vcl ad nibilum deVchiant^ vel ad infinitum y fcd itide 
tcgrcdiantUf, qu© cafu oritur-i vcl arcus potrd pcrgcosi 
ac itct fuum produccns> ut TVS Iri fig. 448 ; vcl cuf- 
pis feCundi gcnerisi ut TOS ih fig. 267 » qui quidem 
arcusj & qu£ cufpis habcri itidem pofiuht rftdto oficUw 
latoris circuli ad ccrtam maghitudincm dcVcnlente , ncc 
adnihilutn, ncc ad infinitum d«|atOi 
. 74<$* Prxtcrca fi cpnfidcrctur difedlo mbtus puodti 
P pcrcurrcntis arcum TVS , & concipiahir ungcns ea- 
dcm dircdtiohe, facile apparcbit* tam iri fig. 248 i x^r 
afcus pcrgcntis^ quam in 249^ 252 afcus mutanris fie- 
3cum> finc dircAionis mutatione cohtinuari motura per 
PVi^i j vcl PV/ j at in cufpidc t;im J>rimi gcncris iri 
^?r 250* 253, quam fccundi in fig. 267. motum rctro 
tcf i5ti> ac prolndc tangenrisdircAio in illts hianet» iii 
his mutatur, & ih cis Ipfa ianges abit e pofitiVa in he-« 
gativam. Sed mutatio ubicumque fit i fieri fcmpcr dc- 
bet in aliam ditedfioncrii prorfus oppofitam 5 tahqiiafa 
fi. plajsB M, & N in fig; 250 j vcl Q^i O ih fig. ijj 
it3finit1.es a fc invic^cm difiantcs in illa ipfa infinita di- 
fiantta connc^lerentur intcf fe i 6c tontihuarentur ; 
quorum analpga fiinc ea ^ qus ia hypcrbolicis cruribiis 



^ • -■». I-» ' ■ -5 ■• -. • ^ . ■(»•—•■•»- ^ » 



tl>C6tXJM GtOMtr^lCOKXM . .,i?i 
i , pfoffttrit , ut ubi de , airvis agemus gctierali tct ■ viA 
"hpt fblius^tieotnetrii , vcK6pe calculi ; fafius e^pont* 
«ms, ac d<;monflfatbiiiiusv Hic kutem ciJnntiirtiuJ; j ut 
tfin^tefcat hujus ifiexiis; j^ contiduationis ufu^ iA Uni^ 
Vfcrfa' Gfebttietrii lariflirtic. patenk ; . , , 
/ 747'. Porro a*ura hujiiftnodi ifi infmitutti protcaft itl 
]fiiHgali§ Gcobetricis Ldds. }am,biiifa funt tantunimc>dOi 
fftHQ ^uatuo^^ jam cnahiplura" Jta, lit qilivrs corum. ntt* 
fltertib par habcri Jpbftit'; Bini/tantum/patabolica hriben'* 
Var in . ParaboU .cotijca \ fli m bmnibus fubiiirJioribui 
Psoi-abolis fig; 248; 249; 2^0; 251, 152; 25 J i quift 
itnmo tina quodammodo funt etiam in lcftd liiiea 
jhinc *; 8t mde in itifitiitum pfotcnfa V Bina, tantum hy- 
perbt>li€a liabefitur . iri fig, 266 i Cujus pcrimetcr m ftj 
fedit ^ pct. MO*S ( inpnitm ) TOM ; Quatuor habcti:^ 
ttk t^perbolica in Hypcrbold conica i &, in Hypcrbo-PijA 
Ks omnlbus figurarum 254; 155; 15^; &: quattior iht- ^^^ 
Iberentut etiam ihfig. \66 > fi ^uiiduih C cflet propilis 25^- 
Wd^ar AB itai; ut tarigens ,QCE aliCubi . ocCiirreret Jfe- 26d, 
*dac AB ad partcs B , qui quidetii Ctim^ Itt orbehn' 
Itdirct.line aure infitlito j fi pundtum C jicefct J^citio- 
Bos", li tangcns quoquc CQ^bccurretet te(3t« BA ad 
- partcs A ; ut jfacile patcbit Curvai pro cjufmodi Cafibus' 
foaflTucnti ; & coiitehiplaiiti 'Whiiti origiuem i ac na-; 
turam *; Plura aufetti ', & quocumque liumero habcntur 
iXuXTL in aliisCurvi^ quimplurittiis* quartiin conftruitfo- 
bcs occith^cnti ubi generalitci* agcmus de curvis lincif/ 
748; IntcreS*'. anteqiiam tas i quas hic detctmliiavi* 
inus 9 curvas^ felinquamus ; notabimus tatlohem qUah** 
dam ' detcttninandi tangentcS 5 quas tmbant nonnihil 
cufpicles utiriirfqUe gtneris» Qux poflcnt aliquatido hott 
iatis fcaiitis imponere; Sole&t enim quandoquc dctehiii* 
iiSsc\ tangentes curyarum hoc pa€lo . In fig*,266 rcdia * 
CGf ftcans arciim^uendam iKk iti L ; & L' ita mo* 
Viratlir , iit deinum intcrfcctibncs L , L' cotant itt K .* / 
cvaricfcenic chdrda LL'; abibit ipfa fecahs irt tangcn» 
iefcn ; ,& binS ititcrfcdiohcs iii tohtaftum * Hacc me* 
Ihodus faUcrfe poteft aljquahdb; cum fieri pofllt^ ut br« 

T 1 »1« 



%p PE TRANS^FORMAI^IONfc 
|I2 interfe^iones coeat)t> quin habea^ur oontaftust^S 
j)^beatur conta<3:a(, q^tn binas intei^fec||iones cocant . 
Pripium accidet, quoticfcumqvie bab^Hyr ciilipi^ uQrins- 
jibjt genfri,?, fec-wdum .quoticfcumque Ctfrv^ ^cai^eQ- 
|c fimul fecatur , ut in mu|atiqQe( fle^us > & ii^ cjirpi!^ 
fie primigeneris . Re6ta| curvam tangenti^ ycra.no- 
tio eft ea^ yf {it reda^ qu^ ad ipfum arcum onuiium 
maxime accedi^ ita > ut C\im eo. CQntineat an^^ukia^ 
quoyis redtiiitieo mijnoi;em j fivt ita ^ ut tjiuUa alia re* 
^a duci pA^^t e pun^o/contadlus in eo angulQ, quem 
^i^^arcus ipfc continct qum tangcnte . Pqrro fi in fig^ 267 
cefta 5J,* mpveatur mpt» Ra|air^lo , docet abeat ia 
4rO/» v^l <;irca pun(5kum L > donec a.be9.t in LOiil.j m*. 
tierfe^iones l :, 1' » Vjcl P , P' coibunt in Q , nec ta- 
men ytrdibet ex iis r<<Stis evadec tangens.. utjciuslibet 
^jjaC^fpidi? . Con^ra vcro in iSg. 35.2, aj3>. tc(ksk parallo^, 
255 la |:e<p:5 BA quamvis pccurrens curv» in ui>ico pundo, 
2ji P motu cpn,tinuo. delata ^bibi; in tapgentcm OYQ^a 
' ' quin habeajur concmfus. binacjgim tang^ntium. At extra, 
cjufmQdi^ ca/iis » quotiefcumqye i^iimii^upi > yt in Gg^ 
Z5 1 « bini ^cu5 TiV ^ \S co9,tinyad j*icent in binii. 
^ngulis ^ quos ^pgen^ cym alia, rc&^ |>er Qonta&unji. 
duda ^ontin^t ad. eandem pl^gam , &rapec rite pcoc^^ 
ciet, metbodys 9 qyod dcyioiinrabimus , ubi ^^ corvis. 
lineis agemus in genei^e » ut & illud » hunc c^fum ge^ 
/ Dcraliter o^jpurrer^ in curyi? qijibuJTcumqyc : naip cur^, 
vx ip(x iti pun^is taqtummodp determin^ati^ poffunq 
Ijaberc ycl flcis^um contrarigm , ^el cufpidem primi » 
aut fecun^ genei:is ^, fivc continuajtionem arqis in a- 
liqijiQ ^ reliqui^. tribu^ apgulis ^ngcntis cum nor- 
JiTpali ,, non ycrQ in^ oiiinibus put\(Sbi^ c6ju;^am ar- 
^us continu;i , u^cumqu^ psnrvi > quod ip(um ibidcm, 
dfmonftrabittir dp cicculo ofcjulojtorc ^ quji itidcm ge-. 
pcralit^r baibebitur io quayis curva » ne^ tiifi puB^ 
dh ^ulbu^anj 'dcjcrnjiM^^^ ^n^m^j^odo dc<;ffc ph. 

7^9^ HJc intere^ monqijLdum iHud , quoiiiam ca dc-. 

Kn3Koatic\n^€ «?«?,^?»5 m Sedtlonibus Co^icis ufi fi^r 

^i:us 



tdCdRiTMGEQMEtklCQRUR. '27?^ 
iritb^tium. 151, cdnfidcrando in fig.46, & fequ^ntib'ns 
concarfum pundorum P , ;^ in I , idcir(io dcinde faunl. 
a^joftcnftira feffe, tahgentcni to |)actd detcrminatamF.4^ 
aocederc ita ^d ^cum cur^ae , ut in eo an^uld nuUa 
Uia daci po(m • Nam ifonf^renci cdnditionbm , qux:, 
habetQC in utrocjue tafd ; qucki ri^ctae Auctx a conciit*- 
lii tan^imBs cutu directrict > & a contattu ad jfocutH 
oontintailt i6i aiigiitUfli rebtuni Jtixt^ num. 175 i t^ate-^ 
bit utram^ue dctermitutidticm eddeni itiridcre . Q^iiii 
immo cuiTi iiidc conftfet gcn^r^itf > Kd pacto Ucfiairi 
poffe iii S;cctipiiibiis CoiHciff Wrigcriteto» patct fimttl iri 
iiis» nufqiiain habeti ctifpidem> aiit fioitim &ont^afiuiii. 
. 750^ Eodctn autem Virid, ac iri iifdclfi cafifcuS^ labd- 
rarci patct cSafai , incihodtim , i[ui tangcns dbtcbnl;^ 
iiacur d^thbnftrindo atctim titrim^ite a quodaiti pimcrd 
|acere ad candeiti parcem tujiifdairi recftafc i dc deduccii- 
do iridci eaih rtctam cffe tangcritcnis 9t illdd puix£tni\i 
feffc puhctdm contactiisi.Id atcidil in fig. 267 iii fcctI§Fi^f 
omuibus pct, O dtlctis, Hcet tini^a OV flt tangcris cut sji 
fidii {ecimdi gchcri<i , Bc Unic^ OA <!:ufpldii pfiml^ 
^iiin iinind ia ha^ accidi» dthnibtis re£:tis {^r^tcf ipraih 
foiam tangcntcm OA . In ipfa vcrd tangcnte id hec 
accidit hi£ in cufpide primi gericHs ^ ncc iii flg^ 252 
in flexu coriurairio ; cufai tKrdbicJfic bini -arcus hinc i flc 
inde jaceant ad par^es tangenti^ oppofita§ i At eo Vitid 
rion laboirat methodus , qua rcctai occurl-cris ctirva^ iii 
biriis punctis, convcftatiit cifia alt^ttirii ex lis imriid- 
tum donec cBofdi e^ariefceme , '^odcrtl fccid^t &t altS- 
tura. Sifc fi iri fig. 25^ pct V j & P dgatiir fecfa con- 
vcf taturqtie ii donec recidat P iii V; recta ipfa ablbit iil 
tangeitfetfa OQ^riccxffarid ^mnium rcctanitti pfdximdlm 
ita; lit ih.eodem aiiguld riiilla alia fccti duci pdflfit J 
iiain fi riova rccti utceitnquc pafdiii dedinct d pfima ^ 
jam ifit una tx M t ^uae Habebat alteram intirfcctid- 
nenii & ircum biiiis intcrCctioriibus intcfjaeeritcm m- 
terceptum angulo tangcntii cdri? chorda' . Idcrii autcm 
accideret etiara iri fig. 267^ iii^^a ix pef Oj dcl, velP 
iig^rdtar reCtaj a^ cirda ptinettJm.O coavertef cii^ y doriec 

T J libi-! 



^74 DRTRANSFORMATIWB 
llbircm ea pundla ia O , denn^ret eodcm rc6^ai_ in. i^^ 
gcncctn OA) OV, yeriun baec. ipfa la tra^^u de cix- 

. yis lincis in gencre pluribus pcrfeqaemur , Sc accoratiiis 

' omnia dcmonarabimW^. - . V 

751« Inrerca videbirniis k^c aliani quandain cclatiot 
nem , quam habct reda linea infinita , cnm' iofiniio, 
<:irculo^ qua^ nobis uroi iutura eft infra , Sc ad pltir^ 
tut^ analogias , mm anr^alias dc^gen^as^ condoce(«« 

f 37i5it in fi;C^.^7i re^a, infiaita MN i eiquc perpcQjdicii.' 
' laris 0(X i V^^ iffkm fecet in R • la h4c Ht oeit- 
trum circi^i P. occurremis ipfi in binis pundi^. I, I'., 
|accme l^ ad partcs ccQtri , SctciSbx Hy^.m (mihs.A , 
C. Pa^ ^ cbordam AC perpcndicularcm diaiiictro fe-! 
cari in R bifariam ' ab eadcm , & binos ,arc«»..' A^C > 
.AVC it>dcm bifariam in I, & ri/Recedcoie ccntro P, 
in infinituni ita , ut lcmper circulus uranica)^ per ^ 
dcm * illa pjurifta A v & C ^ /patet juxta nufn, 7^7 , tj- 
cum AIC dct)cre abirc in re&am lineam , adcoaue df- 
berc cong^crc', cuni ipfa tcOca AC » abcunte I in R • 
Rcliquus; arcus Al' ^ CU partim abibit in re^as AH , 
CN in iniinitum produdas j partini. cta ifl tufiaimfii 
reccdct cum ipfo pi^ndlo, I, ot tuifquam fam fit/Qj^ 
^nobrcm iicut in ipfo circulo, bini. habentur arcus AC» 
oimirum, AIC, AIC termifiatr binis pMn<5l;U A|C ^ qoi 
arcus fccantur bifariam in I» Sc r«» ita ba^cbuotur. 
bina: rcdbe AC 9 mmirum' AK^C > Mi ( infiMfim J^ 
NC, (Irc affumpto pro; carateriftica infiniti iinno qq > 
quo fcmper uicmur. in^^pofterum , AM «t NC> quo-1 
^^m priraa fecabitur bi/ariam in R » fccund^ iti 09 
i^^9 ut prloris dinudia fim AR) RC , poftcrioris A09 » 

C8^^,C# Quin iipm.Q:}. quoniam, ut in fig. 85^. yidunus 
num. ^78, arcus/Fw iurit numcro, infiniu tani diiaj- 
(Stionc Efim, qnanV dircAione ¥Am , qui niinirum bis 
arcubus integras qijiotcumquc ' peripberia;^ ^.danjt } d-. 
i^n. blc iJ[ifinHt num^rq' crynt arcus incipientes ab. A> 




tOOORUM GEOMETRICORUM . iyy 
Ifcft» ARC, ARCN oo MARC, ARCN oq MARCN 
« MARC> & coatra AML c^ KC AM •• NCRAM 
*l NCAMo^ NeR,AM 09 NCR^M p^ nc ^ 

^ tta porp , 

75^. Jam vcro omiffis rcliquis, raagfs (jompQfitis , 
1p(a reifta finita ARC » 9c iUa per in^itani traducl^ 
4M o^ NCeani iiicer fe^ flnalogiam. Habetit ,/ quam 
jft €0 Gtrculo accus AIC »^ ArC ; m li nimirqm, a;rcu$ 
t^dnEHnuae^ hah^nt propc^taces ,. fi alteri, pofitive fumn 
Mo fubftiuiatuc alcer fi;imptus ii^gatiy^, ita etiam ia,re- 
^ illa MN u^inque infinita fegrntpntum ejus. fjnitum. 
AC nfgadve nBfponcbat fegmoito AM oo- NC per in- 
^ninim/ triiduiS^o, & yicevcrfa hoc negadve fumptum illi 
iiiiapco pofitive , 

*7j^. Hfnc au^cm iti £^. %^f , ubi iirnpinuu CM iFi6j 
«Qgetur CP , dpn^c puncto M abeunte in C > abeat I^ 
ia infmitutr^ ita» ut nufquam jam. fit , ac puncto ipfo 
}A ab^ate ii\ M! ad partes^ oppoQtasy ^t etiam P a4 
pairtes oppoGtasi in P'' » cot^^Gderari poflTunt bjina^ CP^ ,. 
aitera diiectione QB, qus dir^cdq ii: afFumatur pro po- 
idv^ » adcoque oppofita CA pro negativa ^ eadem e- 
m adbuc pofi^va, Sc aliera direccioAe CA jam nega^. 
tiva. lUa mmirum^ erit COB oo^ AP- , hsec CNP ^ 
Hoc nsodo. fi res^ confidei^tar poft e;^ndem CM*s &: 
CO9 vcl CM habebuntur <}UQeiammoda binse tercias 
fomitiue proportlonaks, aliera negativa CNF , altera 
adtmc pc^ts^a, GOB. - oq, AP: ^ Nimjrqm cum juxta 
'i^mn^7i9^&t CP- pofirivji tertia^ poft CM, pofitivam ,. 
9c COv ut imminuta ipfa CM uhra quofcuitique limi* 
tes, aug^tur ultra^ qupfcumquc iimites CP > & illa e^ 
^lnicfctste, iive abcunie in. nibiiutii > hacc abit iti, infi- 
Bicum j ita facca CM'.)ani negativa» quas q^pdammo- 
do/ copicip^tur decreviAii; in&a nihilum , ip^. videtur 
quofbunmodo debere refpopdoee-ex eadem pftrte quatn 
iit»s pk^uam infinita , 8c cum refpondeat COPB 
4N0^ AP* , vidctur. hacc dicea4a, efle quodammodo 6c 
pofitWaj & plu^quam infinitav Sed, id quidem ad ^ny^ 
^eriom qupddam infiaiti^pebinet , ic ad analo^ias 

T ^ quaf- 



^* DE TRANSFORMATTOk^ . ^ 

l)Cia&tam conducit> ac ia Geomecria coincQuni ipli^C^' 
ncgacivas ncgati^a itidem illa finka CMP refpoadet S)- 
jzc ullo myAcrlo » Sc ita » iit sii Jis ^ 'qtiai iiide dcda- 
(iiancur > perfpiciU tibique evidedda habcatar i ac xna^ 
ximc maQifcfta. 

1^271 ^754. Coarideratio tamtii bin,ariiin AG in figura 27 1 

9 nunirum ARC» & AM to NC^ uTum eciam^ in Se- 

II 6Honibus Conicis contemfdancBs^ pauUo- ihferius habc- 

z69 bit ^ra^ftant^mttifi > ubi axi £Uip^os MCi» finito iit 
f^. 9 cdkndemui frorfus, Sc dtre<9:e analogum ,* non 
^m finicum Hypcrbol^e lAm In fi^. 11 5 i%d' zxtni 
MH 00 ^^- traduAum per infiiritum • Pariter in 
%• 2699 tibi reda AiBi pcf A2B1 ablc in A^^^ ; 
concipicurDPi per infinitum abire in DP) negattvam i 
Abit illa^ ia ^aldgia.fpe^cur direfta , & ab ia&iiti 
nyfteriis petica-, in DA) 00 BP^: adhuc pofittvam ; 
& per infinitum tradi^am, &: propricutes priorisque- 
curaque a dire^tione pcndent $ cum hufus direftione 
confpirant. SedcoQiiderari iblet pr6 ipfa illa alteraDP 
ilinita^ ac negativa,; quJC huic cont(anaIaga eft , fi h^c 
voce uci iicecy & eft f jus Complemenaim ad quendatn 
, Veluti infinitutn drculum, qua idea nobls infira opu^ 

^ erit ad o^ndendjim .iUud etiam , poife rationehi red->' 
di^ ccfi: in Begatiyis quancitacibus^ fubtradio addicfoni 
^ubftituenda iif cnmi i ubi obveaerim ci tranlQca ptm-- 
^iper infhucum;^ licee quanclcaci , qu^ faabebacur ante 
difceiium in iniiaintm , fit prorfus > 5c dirc<9% analo-^ 
ga noa hmz q|Uanticcs negaciva , fcd pofinva illa pcr 
infinicum ctadu€la,F qtia: |uxta iUam fuperiorem ideatkr 
plufquam.infinita dicer^cur<r 

7'5 5. Qjiomam autem buc ufquc tim niAlca vidlmus;,^ 
qui? perdncnc ad tranficum qaandcanim e pofitivis v& ' 
negadvas^ vel regrefTum .inde, libet hicadned^rcaliam 
qaandam anal(%iam>' quam habet cum hoC ipfo tran^ 
iim quancitatum e pofidvis in negativas , vel regreffif 
tranfitus, qui fit c fiatu reali,.ad f^atum imaginariuitl^ 
qui impofHbilitatem fecum fert juxta num. 684. Trao- 
ficus e pofietvo in negadviim aunquam fiai poteft pct 



, iOCORtJM GEOMETRICORUM . iy^ 

iktnUn queiidani , ubi adblic dccrementtim haberi ^bf* 
fyy vtl iacremenmm*cx e^m plaga^ fed gradatini i 
ut nifnirum tradfitus ipfe fia; vel per nihilum , vd 
|)cr infiniium . I^ ptimo ^afa limites magnitudinis. ^ 
ut ubi de linei^ agitur , extrema punda ad fe invi.p 
ebAi acoedunc» & coeunt , in fecundo a fe invicem 
reoedinii ki infinimm ; Eodem pa<5fco realis quamitds 
nua<{oam ih im^ginariam abibir pt^ faitum ; fed fem* 
per gradatim, nec unqtam is tran(itas fiet> nifi ubiek 
dev^oerit vcl ad nihilum, vel ad infinitura. Ad hofbe 
velud fcopulos allifa aliquando retro refiedlitur adhtic 
realis^ St pcr edfdem gradus decrefcit 9 aliquandb con-^ 
trariam diredionem acquirit per ipfum nihilum , vel 
infiiiitmn tradu<3:a i aliquando vero ia imagiaariam 
qooquie migrat> five impotlibilem : Regreffus, ac tran* 
iitus cxempla dtdimui jam plurima ; hujus migrattonis 
ia ftatom imaginatium exempla pliurima fe ubiquepro^ 
deot. £n aliqua irei illudrandae ftpta. 

75 df. Dum irt fig. 242 rcfta EF Imotd contitlub de- 
lata verfus E2F1 appeilit adA; bina pUnftaQ, G itikp^ii 
in fe io^icem incurriint> & quodammodd veluti colli* A^ 
duntur 3 m fe deftrdant j 8c motu cjus tcStx i^rgeii- ^53 
te» fam nUfquam (int, ereidiftamiaimaginariumtraii^ 
slatai qusf migtatib a migratione in iniinimm plurimuiti 
difiert; Migratio enim iti iafinimm dcterminationem quan- 
dam proBIematiaddit, ut ibiEUipfcosvettex iilinfinimm 
recedens Ellipfim ipfam mutat in ParaboUm ; ac abdueto 
feciim iii infinitum altero foco , Sc centro , rtn:itat in paralle- 
losjuxta n«t02 radioi illo^, qui cX altero foco egtef^ 
fi, coHvergebant in EUipfi pdft reflexionem ad fbcum 
abcrum, ac parallelas icidem reddit diametros omnes 
quas in EUipfi convergebant ad centtum ^ vel tibi dr^ 
culi eentrum recedens in infinitum ejusperipheriammu^ 
tat io ce&am liheam • At abitus iii imagiiiarietateitl 
fecum trahit impoilibilitatem ^folutam profokitiatis , 
qood ejus-ope fbivebatur ita , ut idem fit ik quavi^ 
rcfolutione devenire ad latus quadrati negativi, acpro''- 
blematis ia eo cafii impeflibilitaieni cvincere # quod St 

Gco- 



y^^ DE TRANSPORKf ATIQMqf 
^Geooi^tri^ , & Algc&ridis (bletDfic eft . lifiea i^ 
,GQ in eo cafu cvadit imaginarla* poftcQqao]^ per^. 
Infies iftoitariim magi^itudfouto g^ados de^^vir xjhj^ 
ad lUhiiutn, ar in Hg. 25 d erdiitara P/ 9 puodo R %> 
bcunre pet V in Rz^ eyadit quidem inij^tnaria , H 
pofteaqoam iptx omnes axM% magnitudimim fiait»* 
rum gradus crevit \n infi^mm y atqttc idem acctdecet 
in fig. z69. ordinaiae R^ 9 fi pim^um R con^naii* 
'ret curfum idrra V verfus M • Abiret ^rdinata «v 
iam i^ eot:a(u in io&iirttm i & d^«dc. imaginariiil 
cv^crcr . 

757. Illud autcm difcriminis intetcedit tnter cafuffl 

qup linea. p«ft difceSum in iniSnitum a(^it ii^ unagi* 

nariam > fiq cafuni , quo rcalis remanec >, ac w^Si^ 

vcl regrcdirur , quod in hoc fccut^o. cafu potcft habe^ 

- J^ progreffus, Vcl regrcfftB cciam , ubi uliiottm pu^ 

f Hichim aWr iQ infinicum , ur ubi ia ^. 354 ofcliflata 

"^^ RP abir in cphrrariam RaPi > vel itv fig^ 255 rcgtie-. 

^/5 dirur pcr R2P29 in quibus^ abir quidem in infimtum 

.^^ P>/fcd remi^nec^ R ;^ at in primo. illo. cafu nunquam 

^*^ iiaW)itur imagiQ^rictas ipfa , nifi utrumquc r^^^' 

?^®» rremum abcar in iiafinirutn &yc ad p^rcs, oppafitas» ut 

in fig/256, fivc ad casdcm, ut in, fig. 268 , adcoqiic 

nifi in illQ ipfo ihfinieo, coUifip qqa^^ ha^Wt '^?^^ 

>€iuii pugna inrcr bina piin<5ka fibj inyicciu oecucictttia, 

' ibidcitl, & fc muruo q^oda^rnmodo eliden tia . Hic auK« 

ipfe vclur ip jcriru$ quaKtitatis (fi^ hswc cridm cuna vcr^- 

aliarum reruiTiinrerim analogiam quandam per&q^^lf 

bcar. ) Titt habebitur fane, nifi illa ipfa punda velod' 

tarem, qua in fcmuruo irri^onr, infiairtcs^niajorcmha- 

bcanr ibi, quam alibi, ur facile demooftratur coDtifl* 

gerc punilts G, G fig. 242, P, f fig. ^68, &veio.€f- , 

iam P, |»flg.256, ubi pun^a P, ^ cx partc finitaafc 

invicem rccedentia ulrra quofcufiique limitcs, cx p« 

infinita ad fe inviccm. ^cedunt patircr ujlara, quofcuoH 

que limircs , & fibi inyiccm occurrunr quodamaioiiO) 

' & c6Uidun|ur :; vcl infiniries minorcm , quam alii« r 

Yclocitarem babcatn rclpedivam , quod^ accidei[ct ^-^ 

' quc 



'fcQEORUM GE0!#TIIIGORHM- ^ 

gue in cuj(piclibiut omnibus «« f^ts^ tamco ngulca piWcJM|u 

jcs (uiit ji^xu njgpie 70 v ^aati rctise Pt^' > & II* i^ 

|g. 2^7 paulo, antfquauoi CYancfcacu 9 diflGbreutias 'iiV 

|ca( in infinitmn ixui\Qr€< y <^ktn aj[ibi'i 111: jfa^Jki ^ 

,l9QlAft(4ri pofla,-^ poft imiiHtimatn in ioifinUq^i ti^» 

feduteni refpecviyi mo^f cxtrciiibvuuii pun^orur^ » %•. 

^C0ate £L' dtra ^/ » ^naginariXL fiunc : ur a^o? yi-r 

fleam^^ciain in Gcotne^ia. bk intcrifus/ bab(r^ j^oiQ|p C 

ttnittDHiipdoi^ vel e nttnljS 4^^4^"^ ^"^^ f^^^ 4 ?^.<if'*. 

,Rtv^n^a> ut teli cuj)if4aai i(^)i baberi (olec >\^^ 

bri, vel e languore qj(iodam » uc habetur ia %u(m^ 

^aado^ dccrepiaiS alca» ipfa , ^ virii^ i^nbc^ift^. 

Wt , quwquam id ipfum paritiiEir perqjgiam rar^ cof^, 

75^. Porr«i ^lgratiauJs ei^a^ r^ali ia iniagijQiariqfla 
|cr nitulum fatis ctiam /elcgan* cxcvfif^iffpk habctur 1(1 
Wfi$ Cou^ Seaioinibiis , quas a nura. 55 j p^fccuti fvf" 
?H«/Affumpto, iA laurc VA figujrap 2,08. quovis pi^i-Fzd^ / 
^o H a4 arbitri^m ', fi cot^cipiawjf. |^cci;a. MI'. con.-^^ 
.gnjto^ ini^o cum MV vcrf^s i>oft^ionem MA f?^^?wir :»io 
loluta per. puaftum M, c rcc^a.lin^^MV, iu, ^u^^plar an 
.flumij^ OyS paraUclu^, eo caf^ contiQgu ciiwm * * * 
.f&afiicur }am num.'585'Ellip^ prii^^p^q arcttiQma 9^ . ^ 
9U« pcrpetuo pingucfcit i^' coap fccto , *4onc(j. fegsi» 
J^ ipfq pars^lo baii f^iio ei(^a| circulus ».. pun- 
«0 T ^unc^ in infinitiinf) ita^y Vit; aufquam ji^m dt^ 
Hin inftiitq ipfq de^Utcfc^ • Pi?^^ntc mptu^ ojbloo- 
lattjr pecpctuq S^ionH (qrn^ , §f ^^(f^ wncs 
9«lys finitarum r^^npSB axi$ CQOt^gatit^ad. tranfycc- 
ioQ)) qua^ ^q^iri( ic), %^,^p^ iterpm ^, circalaJ:i for* 
jna reccdcns , ac puncnwn X V^^^^P^^ P^^ in^ittim 
m «egrcdiijur cx p^e 'cpRpfi|^,*q^ a^^^^ f^etiuini 
w, B, abit Scctionis.figura, in P^abolan^ fig^ra^ f^XQ, 
uiqua ycrtcx ille. »1 jam^nfit^itq ot^runi^ Ia;ct> &ixy)r- 
qu^ cft rProccdcntc ultjjrius T vcrAi? A > JW fi^ab^- 
^ in. (^jira 211 duplex ra^ausi, Hypcrbofae/c^ni vcrti^^ 
^c ^ rfigrcflb, cx in^nito cx parte pppofita, a,c Hypcr- 
feote ipfius forma mutatur iiidem pcrpcyib, dpnec ip- 



^U bitRANSFokKiATioNE 
fo piihctb Tj Sc Cuni co ctiain I abcuntibus fitnilui 
^A > abcant ipfl Hypcrbola^ rami in bijtias rcAis MA\; 
V^' Ihfinitas . Perit hic Seftioiiis Coni^x afcca ; & ad 
iiihiluhi deveait > podcaiquam c niluki cnata fucrat ak 
ilta itd:^ MV fig. 208 , qux rerpQndct buic ipfi MT 
fi£% iti t & is intcritus Hiab^tul: quodam Veliiti incar* 
, fu pcrimctri irmcntis in fc 9 6c in axcni tranRrcrfun^j 
^ iiinc , & ]i\dc ab axc ipfo ; Si motus plani qui e6 
• fcafii continglt codum , pcrg^t liUerius in caddem pld*' 
^am> jam puniftam T abibit ultra A cxtta cotium ^ 
ot pimdSd ^ fubeunte rc&arai VB i jd planilm itcruiii 
fccabit ipfiim cdhutn i ac iti^ram. nafcctur riota Elhp* 
fis, & lioVa ScdionUm Coiiicarum fcrics piriori • pror- 
fus fimiUin^a. Sed hxc non continuatur cum illaprict«~ 
irc , nec Hypferbdlacj iUae pdftlrcmae.in prliiiis hafcc El- 
lipfes mutatitur • Ill± fenitn definudt ih tc&im MA 
bp dm tradli£t!am ptr infinitum , ha: nafcuntur a re« 
^a fiiiita MV , qiias illi tradiiiSfcas ptt iiifinitara qucU 
danimddo tiotl analoga td i kd ^uodatnmddd veldt 
antianaloga > liimirutn ^jiis ne^ativ^ , 8i ad eatii 
tclata , tit iUi bihi cjdfdcnfi circiiU ar^us'^iiis di- 
tisf puri^lis interjddi , & contraria dircctidric conf- 
l^i^rficlcrati AlC , AI6 in fig. 271 fibi ihvicem anald* 
gi furit • Frima iUa igitilr ferics cxormiii hdbct in re- 
&d fifiita , ititcrituiii itf rcc^ta pet infitiiniiiifi tx'2daAi 
illius cjufdem fitiitae re£tae domplemdntd ad infinitutn 
circufum , ad illi jilia fucdedit itidetti oiftarn i &t intef« 
ritum habcits ita ^ ut in fingtilis convcrfidntbtis intc- 
grjs , bins cfurinodi fcirics oriancur, ic occidanr, qua« 
rum quaslibct (intc dtti^tfi , vd poft oCCafum iii im^- 
ginario ftatu fit • 

7J9* Porro in hdjufmddi &atisTofmattdnibus Sectio^ 
num Conicairum aliaruni in aliais faabcntur punabrutd 
fn^dpllccs &c tranfiti^ pcr nihilum , ac pcr infinitun( 
& i^egriffus indc : jpfi ailtem apj^ulfus ad infibitum » 
vel nihihihi^ fi£pc pUncta rctinent iri ftatu rcali' , vd 
alic^i confpic^a , vel infinito obruta > ibiqiie vdifc 
iklitcfcencxa ,- quandoque ^tiain ad' imaginarictateifi 

dctoif. 



L€>eORUM eEOMETRICGRUM ; aj» 
^mrbmit) adeoque lineatum , qux ipfis tecminantur ^ 
}iabctur jam perfcverantia in eadcm direqi 
tectioais mutaiio , }am impaflibtlttat , & 
li,a6, ac eyaiicrcentia , fxpc ptoduciio in ii 
pc ctiam ciccultus quidam per infiainim , 
veluii plirfquam ipf\nita cxtcnfio . HJi^c h 
&icatuni Sectionum transformatio apttflima 
claraQdos , cpnfirmandofque quofdam cano 
univerram law Cepmftricam obfcrvanrar , Sc eorun^ 
ficn^pla ex demonnratis harum (urvarum elementis 
depromenda . Et ipfis autem caaonikqs, corumque^p-, 
filicarionp ad hsc ipfa Conicarum Secuotiuni Elcmen- 
la patcbit eiiam, qax hifcc curvis comttiunia fint, & 
Gommuncm demoqftrationem fuTdpiant , qux ah alte- 
ra ad alieram transfccri non poflint , & ipfa ejus ano- 
m^ix ratio fc prodct, ac ooftrum in hiffc clementis 
adomandis confilium paiam fiet . Ejufmodi verp cano-, 
nes ex -iis, quz huc ufquc vidiraus pendentomncs , & 
func eortun. quidai^i veluii fructvs ■ prppo^iemu^ autem 
l^gtilos, ac corum raiionem ptofcremi^) c^^mpla da< 
I^iiUHS , & applicatioue^ ad Coni^as Sectione; . Oc- 
currene autem idc(itidem quxdain etiam inflniti tnyr ■ 
Ilcria , quz eo ufquc c^crefcenc > ut idfiniit extcnfi 
iiupofltbilitaiem dem^m fuadeaot , ac ad indefitiitopl 
i)im \ £ve itidcSnite parva fint , iive iode&iite ma- 
gfiz , theocianfi} quam alfOi op^ce pertraccabimus a no$ 
dcduceni . 

760. In ptiipis ^nMbg* cUcemus puncta , qux codem 
Diodo determinanmr in utroqueeiufdemgcometric^cop- 
flrucDonis ^aiu, antc nimicun transformationcm , 6c ; 
poft, qu3« nenipc de^rtninaQcpt per concurfum eorun- 
dem Locorum Geometricofum t rectarum cum aliis fc- 
CDS , cutn circ^ild, cum Scctionis ConicE perimecro , 
CDtn licutis per cjufmodl eoncurfus d^fiiiitis eadem le- 
ge . Sic aoaloga fupt in. ^, 2^9 tam punaa ^i ,£339 
M»> Mj.quam Oi, O2 , O^, & Ni, Ni, N; , 
cddem modo defiaita per concu^fum reccar^m incct y 
fc: anajogi fynt cam yertices M» qyapi «iufig.Siio, 
1 1 axiura 



iSi b£ tRANsi?bRMATlbN^\. 

Fijpii axiUm tranfvcrforum Enipfcos , Pir^lx , HTpciw 
9 te 9 qui ubiquc cadem lcgc dctcrrninantur pcr ratio- 
je hcm coriftaritcm/cx foco F affunipta ; & rccta* dirc- 
21 Ctrioe Aft \ Andoias autem diccipus lincls binis ana- 
togis punctis terminatas, fupcrficTes tcrminatas liheis 
^aiogis 9 folida tcrminata analogis fupcrficicbus.; ^Sit 
ift fig.ij^ dhaldga^ Tiint i*ccCT,Mtbi;MiOii Mgpji' 
ic in fig. ^ , ib> i i foci radii JFM intcr fc \ . thordar 
pcr focum ductJt. VF« intef fc, ac alia qufmodi; 

761. Dcinde biria hiilus ahalogi^C gcncra diftingui^ 
mus^: altcrum PrinidriHm \ ic fummtinr ; cum poft 

f tfansformitibhcra maiict dlrcctio quarititatis dcfinitic ;' 
vd miitatur hiimero iiiutatiohum ^ari; altcrutti Secun- 
d^ium y tum liircctio quaiititatis miitatuf fcmd; WI. 
ttumcro mutationum impari^ qua: ponct ctiam Amis» 
italiiia diCi ; Primario ahalogiaB gcricrc tfialo^ae , fudf 
iil fig. 23^ bmtics rcctae MO intcr fc ; rcctac MiNi ; 
& M2N2 intcr fc , ac NiOi ; & N3O3 intcT^ fe i 
j|>aritet in fig. 9> 109 it radii foci FMintcr fc; choir- 
<te yF« duct» pcr fociihi intcr fc ; ^u^c <Jtrccrioncm 
fcrvant; Hoc itidcm gcncrc prihiario dnalogiz atialo* 
jja foat quadrata fcctdirum. dircctioncm . tnutahtiiim ; 
qux cam Juxta num. J84 bis, hiutarii ; Et vcro ctiam 
pimario arialogiaji gciictc analogbs cft aiis ttanfycrfus 
EUipfcos finitus Mw^ ciim axc rHypcrbolsc M io m 
btr iiifinitum tradiicto ^ iqUa cxprcfliorie cxprimimu^ 
tincasi qu£ aquibufdam punctis \xt Ml 8c m tenden- 
fes ad pattes opfibfitas ipfis ; ut liic: ycrfui H ; & 
h i ^bncif^iailtur Cmjarict^ qubdammbdo ; ic con- 
ntxx in ipfb infinito ; )uxta c±i qUa; jam totics vidi- 
hiu$; Scdluddrib analogis getiei^e aiialdga: funt iii /ig. 
ij^ rectaj NiOirN^Oii ad MiNi; MjNj; iti iSg. 
9 i & li fodi radii Fm ixitclf (t i ixcs finiti Mw in- 
tcr fe i Sd alia cjufmodi^ qu« dirtfctioncm babentcbif^/ 
t^ariam , poft tt^nsformatibacm i ut ctiam folida fob 
Ctiiius litieis qUij^ufcdmqUe directldricm mutanclbus : 
Porro divcrfa axibm EUipfco^ i Sc Hypcrbola: dnald-. 
^i^ ac permucaiio axis finici cum axc pct ihfinitum 

tradii* 



LOCORUM GEOMETRICORUM. iiij , 
tf«4i]Cto ita> dc axi EUipfeos Knitb HCm dic^cte ttf- 
pondeat Hy^rbola^ a^is ', noii finitus MCm , fed M 
«o m pct irtflnituiii tradiiaus 9 & Viccvcrfa , patiftf 
ex eo y K^ilod dttm ratiii 'deicrrnitians pcrpetuo crcfcit > 
vel cotii. fectib [^rpctua indinatur pofl: (^arallelifmiim 
Cum bafe I & Ellipfis ^ccedit ad ; Paraboiam ^ axi^ 
MO» perpetud oblongatut^ 5( vettex m poft croitifitu]^ 
per parabbiam ita regtedkuif' ek pairte bppefita , ut p^ 
iTHBetcr carvJe retrb tion siedeat In orbQm. ab M ad «^ 
ied verfiis eaiidem plagam in infinitual ^toat Sc fup^^ 
irato veidti ihfinitb , e^dem iditectioile t^^^g^t regre* 
dieos ex parte oppofita ;- Hinc niniirum per quodvi$ 
Ikiticbunl R Bniti axis }ACm figiiras 9, & axb M 00 
j» per . iiifinitum ttaducti figuts it duaa recta Ml 
perpehdicularis bccurrit periinetrb in binis pimctis P^ 
p juxta num; 36; conttk tectas; quae tranfeiiAt perptitii^ 
ai ]^ axis M ob m EUipfeos , & MCm Hyperbolae 
hufquatti occurrunt perimetro : ufquc adeo axi MOmi 
iiiius ^ refpondet dirccte axis M c& m hujos j & vi^ 

fcevciijii. . ^ : . . ■ . • : ■ 

762. Etiam in punctis i fi ea determineniur a bl* 
his rectis tendentitnis ad eandem pkgam , dicemus ip^ 
fa analoga primo analogis generc ) fi ad oppofuas ^p/;^ 
recundario ; Punfta P ctefinita ( tium.ijo^ ift fig.35a l^ 
&: 2^ a tectis FQ^i VG tendentibus iitrpbiquc iacan- j^ 
dem plagam fuht analdga prinlario analogic genei*<e , ^o 
punccd p fecuiidario ; cum ipfum f in fig. ^5 definia- 
tur a rcctis Qf i ^V coeulitibus ad partes FV refpe- 
cm Ggi & in fig* 3^ ad ^artcs oppofitas . l?ariter ia 
iSg* J9j 8c io funt analoga fecundario analogia^ ^ 
nere puncta m i faltetn fi ipfum m in Hypietbola in 
fig; io concipiatur , ut vettcx axis finiti Mm j fi cnim 
boiicipiatur , ut vertcx axis M 60 ;» per infinitum 
li^adactii poterit concipi) tit ptimario analcgias gcntre 
m^ilogum ipfi m figurae tp. Centrun quoque C EN 
tipieos in fig; i^i cuni dentro C Hypeirbolac in fig. to 
brunt analbga fccuridario analogias geiierc 9 cuni in- 
kiiiantur id medio itrncre ab M ad /» veifus pattes 

oppofi- 



>«4 DETRANSFORMATIOrNB 

bppofitas • Ac axis Hyperbols per iafinimm . GCaApvi 
habebic in ipfo infinico aliud ccntrum 0% , qu^ ilt 
infiniti noca , m & axis EUtpfeos M oo ^ alui4oeii! 
trum oo jaxca num. 254» ericque analogum prioQo 4i 
nalogiae |^nere centrum finitum EUipfeos C, quodejaft 
axem . finitum MGxf» fecat bifartam ^ cuin oentgpHypcrbor 
jbe ingnito . 00 5 quod fcc^cbifaciam ejus axemM cc ^ 
cradutlum per infioitum, 5c cencrum 00. EUipfeoscuaji 
pentro Hyperbqls G '• Epiu permatationis ^ntrorum 
ciifcrimen manifefto k prodic ipfam EUipleos % aa Hyt 
perbols formam confideranti • Eliipfis obvectic «avica» 
ccm centro Cj convexitatetn centro oq uttinquc, &fc» 
catur a re(S:a per C du&a perpeodiculari axi ia binai 
aequales , ac fimiks SemicUipfes. ^'^«Ssa&ces hiaotibui 
veluti buccis. plagas MFC > mfC • Hypcirbola obvertLt 
convexitaiem centro C > cavitatem centro 001 utrior 
que } 8c in kxnos ^ualeS) ac iimilesL ramos qupclaiBs 
modo fecatur ininfinico» quo. rami ipfi exairrunt> qoi^ 
fpedtanc itidem hiata cavo ea£lem plaga;; , fed. cxi>tt& 
fas per MF 00 , mf^o. Ipfc ordopundtorum reinprc% 
dit. Nam in EUipfi incipiendo ab M propedimr infii(^ 
19 fic , MFC /wr pp EM >. in Hyp^rb^a ycco ia 
fig, zo ficc MF 09 fmi^ CEM, ubi C , & 00 fcda: 
pcrmutan& • Hinc nimirum in EUipfi qua:vis redaf^ 
C dudui occurric perimetro bi$, nqlla inHypecboladiitr 
(ksL in iis afymptotorum angulis > quos fecat axiscon* 
jugatus^ NuII^ in EUijifi contingit pQrimetrum pet C 
duAa: ia Hyperbola babenmr. pro t;angen^bas afym« 
pcoci > in qjuas tangentes de^iimnt juxtanum. 2^8, ubi 
conta(3:us ita in infinituniabeanc, ut nufquamjamfint^ 
Hinc Hyperbola afymptotos I^j^bet, EUipfis aon habct; 
adeoque tam multis, & elegahtiflimi^ fane afymptoto^ 
rum proprietatitus tUipfis caifet^ 

763. Expofitis hifce nominum definitipnibus > jfl0 
ad canones ipfos faciemu^ gradum , iti quavis jcff* 
metricarum confirudionum ^raosformacione ^adhiboh 
dos • 
,764. CaQQii I. Sk qH^nnt4itv , 4 qftitus^MufUfi^ 



LOGOHUM ©EOMETRICORTjM; t»f 

itmalds pemda -» vil ^tnnf^ixtU the^rmdtis , manedi^ 
mm fofi tfiMMfBmaticnmm Msdeg^ frimo • ^antdt^ut g^ 
mftj nee Mus hakedHtr tMtfiruf fer infimtHm \ 'mafte^ 
Ut tidm folHUm j etameidtia , deittcffrfirdtie^ ^ fhtlU re^, 
Miil» mhe mteMte^ « Qmt-fi aiitns ex iis f^ infirii^ 
tmtrad^fB»' » & in iffo infittiso e^futdtdy ac c^nnexd^ 
inttrfc eQttafiMtttHTi e^aattt^ W^^'extf^em0;i» iis^ 
iu 4 foU d^reBione feftt(efs^^ mat^eime itidm vtttttia^f 
k iis; qua.Jul magttieudistm femnent^ eewfefidebetea^ 
m TMtio tadm^-qua mitttf ete ea i^e y qua determi^ 
mtutt frorfus attatoia illi, quam haieret^^ fi f^ infn 
tatm nott tranfiffem. 

765, Prima canonis paw omftinopitct cx cb, onod 
«DQcs GeQmemcocum Locorum partcs dcbeam e^fdcm 
froprietates liabcr& ; & cum nuUus fiac trannms ptf 
ia&utum , vel per nihilam-) oulla mutatio tit ^ qua: 
|ertQrbet vulgsupcm geonMtrlcum fermonem ^ quantita^ 
^tim vel infinicb , aut per infinirum a:adu<5fcis ufque 
«d finitum oppofitum , vel negativis > & minuentibus 
'^mam. £t id quidem prorfuscongruit cumnn.67f[&: 
i75* In fig* ^39> quodefcumque pun^um N fucritin-Fz}^ 
jKtGi je H» ut Ni^ eonftruftio problenistls propo- 

iiti uum. ^76 invcnicndi fummam MN » NO xqualei)! 
^ifeftx datSj enunciatio fummx inventar demonftratio» 
cideoi erii: ubique > nec mutabitur nifi pundro |N e* 
ffcffo ex iUis Uaiidbtts aliquae quancitates dire^Ionem 
maieQt. 

c 

766. Idem vidcre licet etiam in noftrii ScAionum 
ConiGaram Eiemcmi^ • Nos ca ita adornavimus , ut 
in iis , qoar ad ipfam curvarum aaturam contcmplan* 
daai > & proprietates dcducendas pertincnt> reducercn'» 
lut omnia «d unicum problema geomctricum « cujus 
leneralis foludo > & applicatio ad cafus particuiares % 
IkI per ic ipfa » vel per ca j quas inde vfponte confe-» 
tuerentar > proprietates omncs harum curvarum c^ 
Kientarcs exfaiberct .'• Vidimus nimirum ca fere om* 
ftu ) quac in carum elementis circumfetri fglent^ con- 
liQcri comparadonibus re<Sb«:um , qua* ipfis occurrunt» 

Bofcovich. Tm.Il/. V vel 



i ■ 1 



a86 "DB TRANSfORMATlONfi^ 
vel earum poQuone confidecata , vel magnitudtne «* i 
qua pendent Jumma:^ difietetltis » rationesi^ ad fe in- 
vicem^ quadrat^ » rect^ngul^reotilmque felationes tdnl 
Vatias. Qi^aniobrem felegimus cjurmodi defkiitimem i 
qua^ onmibu^ hi^ce curvis generaliter conVenitet ^ ex^ 
preflam ratione cohftami^ qiiam habet difttflBtisl ptinai 
cujufvi^ perimetri a datQt^unctQ^ actdiftantiam pet-pen-^ 
dicularem ^ dat^ fecta t. tunl [inycftigAvimuii folutio'> 
nem hujufmodi get)erali$ ptoblenmtis * I^n^ foco i du 
reEbricty & rdtionc i§tt^u^m4 i invenin concurfiofi 
reiia datit cujufvis cum S^ione Conitd 4 Solatd gene-' 
raliter hoC probletnate 1 fatis patehat, in ipfa foIutio-> 
ne cotftineri debere fundamenrsi omnid omnium rela* 
tioaum^ quas rect^ ejufmodi concurfibusl intercept^ ha-^ 
bere poflent ad fe inviceni i & cum| ipfa peritUetro 
ConiCarunl Scctionum t dummodd tt generalibu^ 
Locorum GeomciriCorum transfbrmationibus rite ip- 
ia getieralis conftiHJCtio ad cafus fingulares applica-> 
rttur. 

"767. Porrd illud <k>ntigit i uf m ipfa illa gen^rjl^ 
rali conftructione qtiaedam rectarun'^ intcrfecoones s a 
quibus punctotum qusfitorum determinatio petidifbat^ 
vel iis rectis ^vadentibus parallelis , ita in iniinitum 2r 
bierint $ ut nu/qUam jam eflTetit # vet iis xecds con' 
gruentibus i haberi ndn poflTent^ fruftrata generali ipfa 
folutione ; quorum primum accidie in reais dii^ctrici 
patallelis^ fe(^undum id recti^ per focum tranfeuiitibus i^ 
Qunmobrem pro iis fubflituimus bin^ particularia pro- 
blemata, ad quorum fulutiones quo pacto illa genera- ^ 
lis folutio nos petduxeritj ii^ fequentiuin candailtTl ap-> 
pUcationcjr ubi nimirum ad e^s cjufmodi transforma- 
tiones pertinuerint^ ofteildemuS 4 Atque idcircd f^ro^ 
blema genetale ad prdpofitionem ;crtiam rcjccimus i 
reliquisi illis > qu^e ipfo generali nmi indiget6nt i pfac- 
n^Sis in pr^cedentibu^ binis propofitionibus 9 ubi ct- 
intn » quarcumque ad Conicarum Sectionum ptdpiic'' 
tates pertinentid fe ultrd offerrent i deduximus * Tum i 
ex ^tfielN^ |>tobkmate ninlto ubcriores frucms percc^ 

pimu» 



jpmius alia ex;aliis thedremata , dedlicendo « ipfa ct^ 
iani i ubeirtat^ (ane adniirabiii i ^oecuiictKiitnd quaqua-< 
verfunii. 

^ yMi jain vcto. iii dngulis inid i vcl probiematuttl 

fclutiohibus » vel theorematum chunciationibus 5 vel 

ficinMftrationtbus iitrorumque ^ patebit fane illud ea« 

dcih cotidderanti^ iibictimque hihil direCtioneth itiiitat « 

hibii abit ^ infinitunl i |)ec per infiniham tradiicitur, 

yim conftructionis 3 8c ei^uhciatioheni jprain ^ ac veif-* 

M ohihid prorfus eidem eflfe ubiqiie. i five cohndei^eh^ 

hir diverra: fixtti eiufdetri perimetri ejurdehi Sectio- 

hili Cooicd i five conferatur petinict^r uhius Sectio- 

bis Cdhicz di}ufcuihqijc duvA perithetris aliiiruhi qua^ 

^amciimque Vel, m^ghitiiciine tihtum i. vel 6c. hi^gni-i 

tikliil^ j 8c fpccie y 8t forma. difjferentiuni ; Ejufmodi 

likempk ubique occurriint i Badem e(t ih fig. pi ioi 

i t determihatid puncti M i fecta Fi^ in M ih ra^ 

ticne detet^hunahtc! > cddem phhcti V i vel ^ i capta 

FV i vei Fm id F£ in ipf^ iratioae detcrmihante jux- 

U n&im; 35; Eadem ih fig^ 35 i 8t ^6, deterhiin^tid 

cnjufVi^ puncti I^ peir totuih arcum VM^ ih qtiavis 

&cuoh6 Conicd i cdpti jiixta num; i^d Q^ adfiar- 

tcs dppd(itai FV arquali QF i pet idterfedciohem re^ 

i^arual VGi P0^ &caclem iifdcm verbis dehidnfiratio 

defunipta c (ih^ilibus triahguUs FPVi QPG 9 qua^ ubi- 

quc dempndrintiir (imilia ob ihgulos ad Verticeni l^ 

aiquales dppofiios i 8i. ^hguld^ ad bafim FV i alterho^ 

^ngplorum ad bafini QG 5 ddedqde ^^^uiales i Parlter 

thcdrcmafil cdnmiunia iifdcm Verbis cffcrchtuif « Chor- 

dl VFi^ ih fiifdcitl fighrii cru ubique l^tus i^edumprih-t 

dpale jiiicta dumj ^^^ ad eodem ubi(|u<; hiodd Accipic^ 

biti Chorddi qu^m ciicukiS pfi^uiatov ihtcrcipjct e etia- 

inc^o pe|c pun^tum ofctti tr^il^feyhte i eric ubique jux* 

H nitm.5Q3 ccqtialis lateiri rec(0 cji»fdem diamctri • Iii 

<>thnibus ejufmodi cafibus fatis cric puhctsi hdniolbgsi 

defignare littcris iifdcra ubiquc » & ca^deih prorfus de-> 

mon(irationcs ofevenient* 

7691 Sccunda piars hujus Canonis i qu* cft dc liricii 

V ^ pct 



m DETRANSFORMATIONE 

pcr infimtum Jtradudtis , pcrtinct ad infiniti myftcti;^ 
qua(dani> qua? ad analogiam quandam rctinendam hlc 
adhibemus y licet infra eo dcveniendum fit nobis ^ ut 
ipfura infinitum habcamus potius pro impofHbili. Id- 
circo adjccimus , Ji aliqu^e ex iis fer infinitHm tradf^ 
t?^ .... concifiantur . Nimirum fi cks hoc pactd 
concipimus , dcbcmus ctiam in iis gcncralcs illas ra- 
tiones admittcre , quae habenmr in omnibus aliisana- 
logis , eadem nimirum lcgc cum cadcm dircctione de- 
finitis pcr conftructiones eafdcm, ad quasanalogasGeo- 
mctria humanx mcntis extedditur . Nam fi infinitum 
extc|ifum eft poflibilc , id quidem humans mentts vi- 
res omnino cxccdit , qus in eo abfurda quscciam de* 
mum invenit, qus cum recta ratione nullo modocon* 
ciliari poffe videantur . Adjecimus autera illudy extan-^ 
te Htroqne extremo^ ut diftinguercfnus quantitatcs hafce 
pcr infinitum traductas, ac proindc quodammodovclu- 
ti plufquam iofinitas , quarum nimirum extrema funt 
alicubi, ic pofTunt perfpici , ab illis » quse fimplicitcr 
ih infinitum abcunt , altcro faltcm exuemo nufquam 
fam exiftcnte. 

770. lUtid , quod tn hac fecusda hujuipe Canools 
pane pcrtinct ad dircctionem rectac per infiaitum tra- 
ductse, manifeftum eft in iUa infigni Conicarum Sc- 
ctionum proprietate, quae earum focis] nomcn dcdit > 
quam num. 202 expofuimus. Radii ex fbco F egrefli 
in Ellipfi in fig. 66 poft reflexioaem ^n punctis P, f 
debcnt abire pef rcaas finitas Pjp > jp/C convcrgentes ad 
F*^^ punauin f ex parie finita . li in parab<Ja in fig. 67, 
^7 abeunte toco / in infinitum ita , ut Quiquam jam fit , 
^^ evadunt parallcli inter fe , quod pcrtinet ad unum e 
fequentibus Canonibus • At in Hyperbola^ in fig. 69 
abeunt per reaas P 00 /> / eo /, quat funtanalogc 
primario genere analogije finitis P/, ^ EUipfeos , & 
quodammodo vclut convergunt itidem ad ipfuni / cx 
parte infiniti . Scd quoniara in vulgari geomecrico fer- 
nione noUs adhibetur nota infiniti , nec rectse confid^ 
rantar in infinitum traductar^ apponenda fuit litteraO» 

qu2 



^ LqCORUM GEOMETRlCOktm. xtt 

^ilas Vices jpfius o© fupplerei> & convergcntla^ c%p^f 
xc infiniti Aibftitucncja ctivergentia ex patte ^niti • At^ 
que eodem pacto fi in fig, 6S polTent lucis radli ex / 
'^cgrcflS Aipcrato iiifinito deferrl adpuncta P^ p^ idqu? 
minirum advenirent pet rcctas OP > op ^ Colligerentut 
m ty ut in figura 66 radii fP^ fp in Ipfo ^oco Fj^I- 
figuntuT) 

771. tx hac hujus Caftotiii partc .debcnt in fig* io 
ifi Hyperbola diftantix focorum f 06 /, vctticum M 
to m^ direcuricum E ao. # pct infinitam tracjuctx ha- 
kcrijpro cohtinue prqporiionallbus intcf (c^&c diftan* 
das r od > M to > ]^ ^ > inter fe in' tatione deter- 
xninantC) ut in ratione detei^minante funt ip fig. 19 
Cominuc proportionales FG/, MO*,fi(>, &EC>MC, 
£C juxtanum. ^0« Videtur hoc Jingeiis quoddam infi^ 
liiri myfterium. Dcbet enim concipi arcus illius Circull 
infioiti cui reQ)oadet F M /majot arcu illius» Cuirc 
jpondet M m iff> Sf hic arcu £ od ^ in illa iratio-* 
tie, quanl habet in ipfa fig. (M adME> quae aratione 
«nsqualitatis poteft diftare Utcumque > uc poffit cflfc du- 
pia> decupla, ccdtupla> ^ ita porto 4 Qtiare fieri po- 
teft > uc ille arcus primu^ (ecundi > & hic tertii habc 
ti dcbeat duplus 3 dccuplus> ccntuplus i At id difcri-> 
lucn proveriirc npn {K)tcft afa illis EM, sm » VclMF* 
mf adjectis > qu« potius praeftarcnt primum arcum mi- 
tiorem fectiiido. , fccundum tettip . Debec igitur con-*^ 
cipi ilic circulus prinlus in iaftnito ipfo c^renfus lon-> 
ge ultra fccundum > fccundus longe ultra tcrdum ita > 
uc iilud 00 in aliis cjufmodi cirCulis iil alia diftantia 
infinica fic ^ pro conditione , & natura rcctarum, qud^ 
per ijifinitum traduCta: concipiatitijr , In fig» ip FC/ 
cft minot , quam MO» > & MO»] minor > quaiu 
£0* Obiongaca £Uipfi> dutn ratio determinans concl- 
Huo crcfcU^ crefciietiamejufmodiratio,qa2dumtllipfis 
ad Parabolam appcUit > evadcnte ratione detcrminan* 
tc racione arqualitatis, evadere 6i ipfa debet raclo 
«quaiitatis , ut infra vldebimus . Mutati Ellipfi in 
Hxpcrbolam in fig* zq » &c cradu(^is pcr infioicum 

V i pttu- 



|M DETRANSFOIIMATIONE 
pi}p6H5 ^7 V^\f\ ^^\ ^^^^^ d^tcrmin^n^ in rationcm 
inajpri? inaequaUtatis, qiisc pcrpetuoprcfcit, d|impun(ft^ 
f pfa accedun? a(4 E , M , F ex parte oppofita » ^uaxc 
dcbpnt concipi & iili yelufi arcus F W /> M q* m^ 
^ f9 ? il^ Uli^ ipiraenfis, 5f npftras mcntl imperviijf 
quii>i;fdam infiniti ipQu^ vcluti campis extcnfi per tr^^ 
iftjjs diycFfo^ refpQndcntes ratiogi ilU , ^beunte duplo^ 
d^fuplo , centqplp Iqngius ijlo 99 pcrciiicnte^ad Yf ^ 
quaip ^beai: illu^ ^ qqod pcrpncj ad lA,m , & hoc to-* 
pdepi fp^tii^ 'opgips ^ quan^ id ^ c^uod peniiiet adFr. 
Hoc infiqiti piyftf rium pfpi. nobis txit infra , & ufar 
fstiam bii^a: fc^i^p it^ infitiitum rcccduqt \ limiie falr 
«em alt^vq felidla in ipfo in^ni^Q^ patebit ipfra , d^r 
bere pariter fongipi altcrani altcra longiprcm in ratior 
pe quacumque • QuiQ ctiam fieri porfiet > ut ad an4ot 
giam fetyaqdaip itifittitum inf}nito etiam infiniticslma- 
jus:; fivc in r:^tIonc> quam faabct itiifinita quantitas zi, 
finttatii) ^nit^ ^ nihiium , habcri dcbeat« 5ed hxcdr 
primq ^anpnc faps ; j^m ^d fccut^duin^ 

772. Cianoq, z. Si aliqu^ ^antiutes n^aneant an^h 
loga fojo ftcun4ari§ analogi^ ^tncr^ , comfHtanda cnmt 
in fnH^ci^noniffus , ^ demqnflratiopihpis 'negativo m^ 
do, ea^ ^u€ direflighem mutart^nt ntim^o impare nma- 
tionHm'<i «f nimirum fi p tinis ait^a fantum mutetur 
fo paUo 9 fumma ai(^ fn diff^i^f^^^^j^ > ^^ t^o foji^ 
tiva haheatur ^ ttfl ^d, n^g^tiva ^ frout ea 9 qua pirr4- 
vit , erat m^inqryVfl major , & viceverfa : fi^ mutetnr 
Htrac[ue y fumnue ^ t^ differentia rentaneanf pariter fum'» 
ma^ &. di^erentU^ fe^ e pofifivis in nexoiiy^s ahiiffe 
(enfean^ur , ubi ad ulteriora vel thforemata , vel prqblC" 
mata ^dhihenda fint ^ fn denfpnflrofioniJhus vero (er^ 
proportiones inflitutis argufnentationi per compofitionem 
f^kiiit^i dehet ariHm^nt^iq per divifipnem^ & vicever- 
fa , t{hi c hinis terminii ratipnis tamt prim^a , quam^ fi' 
(undjfi ^kwit in tf^atimn, alt^ (antumfmodoi ; retinpk 
^JiPk ^ffii^nt^tiqnii ^(nus ^ fi i^^<l}^ tfjMtH rationis l-. 
XTfiudihit ^ 

77h Qt^ ^^ ^^^^^ P?T««?nt Cinpn?m coqfcquuntuc 

pmnla 



LOC©RUM GEOMETRICORUM. 29 x 
omnia ex iis > qa» fvvra vi4imus , Habcnda cffc pr# 
negauvis ca » quas pofitlon^m inaiant tiumcvo vicium 
iiiiparc, m^ncrc> qusB matant numero pari^ gaaftat cx 
nutt^. 688« Ncgativa mutarc fqmmam in dififerentiam, 
confkat cx iis pmnibus, qu» dcmonftravimus ^ n, 677 
M ^92. Mu^a^io modi ^gumentandi patet ^x co ipfo, 
quQd fuqftniap ii| 4iffercntias piigreiit, ^ y}cevcrra>, ubi 
alcer e binis terminis mutatur in ncgatiyiim • Ejus rc.i 
cxcmplum a4du6^am cft num. 69 1 • Alia exempla cxhi- 
beri pQfiuni plura ^ti^m iQ S^c^io^um ^oAicarum Ck^ 
^nentis.. £p aliqaa^ 

774. Ii^ Ellipfi in fig. 19 cft (num^^i) fumma blna- 
rum rccfarum, ^ixx i binis focis F, /ducunnir ad quod- 
vis pun^tum perimetri P conftaqter xqualis axi M^ • 
|a l^ypcrbola in fig. ao s^qualis cft axi Mw^arum dif-^F.if 
ferentia / quia nimirum P/ direCtioncm mutavi^, cum *^ 
punctuni / EUipfeos abierit in / Hyperbolx per iqfini- 
ium, unde fit, ut rccta P %c /Hyperbote (it an^oga 
primo analogias gencrc rcasc P/ pllipfeos . Cum vcro 
)P/]pLegativ4 iit m«i]or, qa^ PF» famma ipfarum, quae 
in vulgari fermone geomeoico f ft difFerentia , cvadl^ 
ficgaciva 9 & ideirco ^xis Mm ip^ diffcrcQcias ?cc^uali$ 
ncgativos cff refpecm ^xis Ellipiibos^ . ^ 

77 j. In dcmonftrafionc autem cjufdkm proprictatis; (a- 
fta aam. ^3 fummap ^ quae habentur pro Plipfi , mu^ 
tantur in differentiaspro Hyper{)ola. Com nimiram fic 
fP ad PDj. & /IP M P^ in ra^ionc dctcrmiaan^c , fivc 
^xca |ium fo , ut Mm zd Er , eruitur fummam FP > 
jfP in EUipfi , differch^iam in Hyperbola ad Dd fam* 
mam ibi , bic differcntiam ipfan^m PD ^ Vd effe, ut 
}Am a<;l £^^ adcoque ut Dd,he a^quantur, arquari il« 
iam fummam , yel differentiam ipfi Mm . Thcorema au- 
tcm nuiii^ji 90 ibi fuppofitum, quod F/, Mw, E< fint 
con^nuo in ^atione determinantc , quod num, 9% dc- 
monftravimus ex natura proportionis hapnoniQs , po- 
fcrat demonftrari mutando diffei^cntias ^ quqp habcntur 
pro EUipfi, in fummas pro Hyperbola, & viccverfa' hoc 
jtacta-. Eft F/ in EUipfi diffoTcntia , in Hypcrbola fum- 
( . V 4 mu 



^j^i ^DlTfCAmtXD RMATrONl 

^jraa^ipfariim FM> /M , cft Mm diffcnchtia in -Elllpfc 
iumma in Hypcrbola ipfarum ME> Mr, five mcy M^ 
'tadem M«^ fumma in Ellipfii diflfcrcntia iti Hypcrbo- 
la ipfarum FM , /M > fivc fm * /M j & Er fumtna in 
Elli pfi , differcnria in Hypcrbola ipfarum ME > Me . HuiC 
cum fit ^ FM ad ME, & /M ad M^ in ratione dc* 
tcrminantc, colligitur 6c antccedcntium fummaj, vd 
diffcrcntias ad confequentium fummas, vcl diffcrcntias> 
nimirutn F/ad Miw, & Mi» ad E^ fore in cadcm ra* 
tione *■ JMutauo dirc^onis ire Aarum /M i Mt mutatio^ 
nem induxit in fummas, & diffcrcntiaSi 

776.. Porro cx ipfis infinki myftcriis, nimiruni e ne- 
xu illojn infinita dtitantia» de quo jani todcsinjeAacft 
mcn tio , rcddi potcA ratio , cur ctiam ubi dtrciSkiones quan-» 
titatui> mutantur vi nranfitus per infinitum ^ adhuc pro 
ne^ativis haberi debcant» & fuborahi, licct illa; pofitiv^ 
nionmutcnmr inhas ncgativas» fedin illas petiafinitom 
tradudas, qu^ funt harura vcluti complcmcnta ad cir^ 
culum infinitum* Summa ipfarum FP > P/in fig- 79 cft 
cohdans, &£qualis axiM;»* In fig. to* ipfi P/cft an»* 
loga primario anaiogia^ genere rc^a pcr infinitum trs' 
dudt P do /• Q{3ai:e adbuc ipfarum PF, P 00 /fum- 
ma pro conftatui habcnda crit . Qliantttm igitur crefczc 
FP taiuum minut dcbct ipfa P 00 /^ quaicumcacon- 
^antem fummam reddit« Tantundcm igititr dcbet crcfc&^ 
irc /P complementum ipiius P 00 / ad illtim infinitum 
circulum 3 qui hic habetur proconftanti s ac proindcFPi 
/P a^que acfcent, & earum diffcrcntia fempcr mattcbic 
conftans* Abcunte P in M > ca differcntia erit cadcm^ 
AC diffcrenua/M,lFM , five/» ,/M ^nimirum Mw . Hocpa- 
,ctoab illa fumma EUipfeos fittranfitusadhanc Hypcfbo^ 
lae diffcrentiamex ipfis infiniti myftcriis. Scd remita^& 
haberc dcbere conftat ex ipfa conformitace otnnium pat'' 
lium LoCorumGeometricorum, quatcommunespropiic-* 
tates habcrc dcbeiit, dummodo fi directio contrariafiti 
fonrfario modo accipiantur , dcmendo , quod addcit* 
P^ tur» & addendo, quod demebatur • Sic in fig.S^.arais 
^iUi ^bmi 9c tAP^ jttxia Aom. 277, ooixunuQes propcic* 

y * caus 



LOCORUM (SfiOMETRICOJRUM. 19^ 
taffc^ habcnt » ncc dtcr in trds parce^ xquales fccarl 
pofcft» quin fccctur&altcf, liccraltcriusncgacivm fit:& 
idcirca fi ab FP trifccantc prinmm dcvcnicndum Ct ad 
Ff trifecantcm fccundam non gyrando pcr hmPAf » 
quo pado in f trirec^tur arcus FBmAFB;»AFB^ non 
arcttl if>fc FAiw , fcd retro regrcdicndo pcr PTf, muta- 
iur dirodio tam arcus FP > iti arcuiFf , quam diordf^ 
in chorda* 

777« Cadon. j. «Tj «^ 4//$m ftopMione termini dlu 
qhi fofi rrsnsfarmMtioffi^m fndneant analogi fecHndario d^ 
fhdogut gtnefe , mafithit frofortio : fed in frdportioni^ 
itis HtCumqKe eomfofitis nptnqtiam mutatio hahebitur^ ni- 
fi numero fariy in rectangulis^ vel folidis oqHalihus de-» 
Itebit^ vel im omnihus habe^i nmtationum numerus for » 
t^l imfOT in omnihus^ & terminus , qui invenitur fro^ 
forti^ibus quihufcumque , vel quovis ductu , cenfendus 
^rit n^ativus > vel fofitivus , frout mutationum nu^ 
merus fiterit in iis » a quibus fendet , imfar , velfar\ 

778. Proportioncm dcbcrc mancrc poft mutationcm 
riiteaioniS) qua analogia primaria in fccundatiamver'* 
otar , paiet cx co, quod ctiam num. 77^ ufi fumus^ 
qood nimirum omncs partcs corundcm Locorum Gco« 
mctricorum cafdem ptoprietatcs » 8c Velationes ad fe 
kivieetB habcte debcant , fivc aflumantur tx parte po- 
fitiva ^ (ive ex pairtc ncgativa* Sic in fig. S9 nuUusex 
arcnbiis tcndentibus ab F ad ^ pet B trifecari poteft 
uxta num. 776) quin (imul tf ifccentur cooftrudione 
kadem reliqui omncs , qut ab codcm pun<5(o F tcn* 
dunt ad m tontratia ditedionc per A. 

j^ji^. Tcrminum , qui invcnimr proporridnibul qui- 
bufcunaque > vcl duftb quovis , fbrc ncgatiyum , vd 
pofidvumj prout numcrus mutationum ruerit impar » 
vel par » demcmfttatum cft num. 68S ^ ic eonfirmatum 
deidde tam multis exemplis e Geometria pedtis • Inde 
tmein confcquitur , in proportionibus utcumque coni^ 
{>ofitis minquam mutationcm haberi pc^^e niu numerd 
pari. Nam fi prseccdcntes mutattoncs nterint numer^ 
impsiri % acccdet tnutattio .poftrcmi > qUiC complcbit nili* 

^icrum 



»94 DETRANSFORMATIONE 

mcrum parem > fi autem mucattone;s pr^ec^djcntes fii6> 
rint pares , maq^bit poftrcmus teriiiifius ^ ad^pque itc- j 
rum manebit numcms par. Ile^fc^gDla ^utem » vclfo» j 
lida jrqualia » debent baberc numcrt>m mutatiotiunci , 1 
vel limul imparem ^ Vfl fimul parcn^ i quia & al- ^ 
tcrum babcrct iraparem » al^eriara parcm; ^ceniuii^va^ | 
ilerct negativum^ alterum po(itivum rcmanerec » adco-r \ 
i^ue non ponfet rcmanerc ^cqualia. Jdcm ^utcm ex pricv 
re parte cruitur etiam ho6 pa^Ov Inre^^ngulis «equa^ ' 
libus efl: unum {atus prioris ad «unum poft^rit^is , ii 
in folidis planum fub binis lateribps priQris ad planum 
fub binis poHcrioris, ut rcliquum l^tus ppderipris a4 
reliquqm prioris. Hinc: |n c|ufmo4i proportioiic nunie- 
rus mutaponum erit fumm^ mutationpra \iu:iu£)uc rc- 
Aangulii vcl foli^i • Ut ea fic n^mcrus par , dcbel^it 
in utroquc rcdlangulo, vel foljdo cflc fim^I par , vcl 
(imul impar . Narn par pari, 6c impar impari addinif 
parcm reddit , par impari imparcm. JPa|ent igicur pin*. 
nes propofiti C^nonis partes. 

7S0. At hic in ipfa prima partc bujus Can^t |vi"t 
detur occurrere difficultas, qi^as folutioncm non ita (a- 
cilc admittac. Sex habcri poflfunt in proporcione ^- 
qua conftantc qi^atuor terminis binaria terminprinft i^. 
forum. Yel enim fumi poffunt bini rationis priniae ^ 
vel bini r^tionis' fecundas ^ vel prim^s c^im tertio ^ vel 
fccividus cum quarto , vcl bini cxtrcmi » vcl bini mc- 
dii) qui ^iutcntur » In-primo, ac fccundo. ^afu ^ri^ 
termiqi negativi ad negativum eadem racio , quip po;^ 
litivi ^d pofitivum : in quo nuUa c(l difficuU.;^ • b 
tertio» & quarto crijp negatiyus ii4 ppfitiyum » ^i ne- 
jativus ad pofitivum» ycl pofitiyus; ad negativum » uc 
pofitiv^ ad iiegatiyum , in quo par^ter diffic^tas cft 
nuUa, At inVpo(lremis binis oportc^ fit negativuc. acl 
pofitiyum» ur ppfitivus ad ne^a^iyumj yel pofitivus a^ 
negativum 9 U^ ncgativus s^i pofitiyum 1 quod eodcm 
rcddi^ permutato rationum xqualium ordinc ^ Id yero 
videtur omnino pugnarc cum analogia , & quidem cc^ 
iam cum modo 9 quo ncgativa concipimus • £a nixnw 

ruiii 



LOCORUM GEOMETRICORUM. 2^5 
irum concipiunmr in aliqua rarionc mJnora nihilo. Sl 
facultatc? confider^fttur^ debitum , qupd ^ ft ncgatlvum^ 
pejoris coqditionis Iu>mincm rfdcfit , quam (i nihil ha- 
^rec. Si confidcrcntur progrcffus , pqqris cpndittotiis 
0ft iDe? qui frcgrpditiif , quam jllc , qui ftac • Ahlatfs 
j8 a lo rclinquuntur 3, abjatis 10 rclinqiiitur mhil ^ 
ablatis X3 relinquuntiir duo pyinus , ^uaiti nihil. 5^^. 
cunda concjitip cft pejor prima 5 igitur & fcrria Cohr* 
ditiq fecunda cft pcjor , Qjaamobrem rarip qqanrit^ri^ 
negatiyx a4 pofi^yani cffc d^bct mult# minpr , quattl' 
^ihili ad pofjtivamllpfam, ratio aCutem pofiriv^ ad tici 
gatiyam niulto inajor , .quam pofitiyae ad pihilum' ^ 
Non igitur fpqualcs icfie poffun^. 

781. Rtc quidem difBcultas fumfham, i! rifc rerum; 
analogia confidcrecur , vim {iabet. M cjus foIuriQpcn- 
dec cx Jiifcc infiniti myftctii^ , qusc perfequimur ^, ^ 

px iis pocifllmum , quae ijum. 75? vidimus in fig.2t^Jfz6$ 
Ifal cnim nocavimus cerriam concinue propqrrionalen^ 
pqft CM- con|ideratam uc ncgatiyam, in quarii abierit 
pofitiva CM.poft niljilum habitum in appulfu M adC 
;ion cffc CP' fiiiitam , fed CB ^ AP' pcr inffiiitum 
^radui^am , Sc^ quodamiiipdq veluti plufqyam infini- 
tam . Hinc uc ^nita quar^ritas CM d^% in finitani 
CP recWit rcttangulum acqualc quWrato CO, ita qoor» 
idammbdcf i^ihilum in infinit^m du6lum , nbi M abit 
fn C9 & P in infinitum ita, u^ nufquam jam fit^ & 
ficgativa CM- in quanritatcm plufquam infinitam ^« 
0a, idcm produca^, 

782. Idcirco autcm illud in Gcomctria ubiqric fattr 
d:e pbferyabi.tur , uil in hifcc poftrerais binrs ca$b|is 
femper , fi al;cr c faini^ tcrpitpi? ^bcat. in fiegariyun| 
jraqfeunldo per nii^ilum , ^r afacat pranfcuttdo per Jar ^ 
finitum , dym in reliquis ycl ambo pranfibunt per ni-f 345 
liilym, ycl amb6 pcr infinitqm. Dum fig. 24 j aWt in 244 
244 , e quatuor ^erminis propQrponalibus illius CH > 245 
CF^ Q, CCprimus, & quartus abeiint in ncgativum* 244 
$ed pun^Slo H acccdcnte ad C, & dccrcftcntc angulo 254- 

QH 



2ff6 DE TRANSFORMATIONE 
OH in fig. 24 j ) adcoque crefcetite CHl , pundhini G 
rcccdit a C ita , ut congrucntc IH , cum IC , & k- 
ifta FG parallela CE ^ puncftum G in iniinico obra- 
cum delitefcat ; tum procedente H in fig. 244. redit 6{ 
cx parte D ex infihito . Pariter in dg. 254 , fi eare- 
ferat Hyperbolam conicam ^ in qua redtangulum fub 
,VR, & RP eft conftans , 'adcoquc VR ad VA , ut 
lAB ad RP, tranfeiinte VR in ncgativam per nlhilura J 
tranfitRP in R2P2 per infinitum, ut adco illisCCPR 
Cg^si 243, & 254. non refpoiidcat CG f5gur« 244 , 
& R2P2 figurx 254 , fcd illi C 00 G huiC R2 oo ] 
Vi . Gencralitcr ut redangulum fub extrcmis aequctut 
re&angulo fub mediis, fcmper mancntibus fmitis aite- 
rius latcribus > &c altero altcrius latere uanfcutitc pct 
nifailum, alterum latus altcrlus ttandbit pcr infinitum,^ 
cum» ut paullo infra patcbit , altcro evancfccntc , ^^* 
(crum debcat evadcre infinitum : adeoque quodaramo', 
d^ fiet plufquam infinitum ex ea parte 3 ex qua inin- 
tinitum reccficrat . At ubi figura 243 abeat in. 24^ ^ . 
fisK:iIe patebit tranfeunte CH pcr nihilum , vel per ino 
finimm motu rcAas IH circa I, tranfire idebcre paritcf] 
GF pcr nihilum , vcl per infinitum fimili motu redbi^ 
GF circa G , & idem acciderct , fi reita Hl traa.. 
firet raotu parallclo ad partes BD pcr C , vel per in i 
, finitum , oranfcuntibus H , 6c 1 fimul pcr C, vel pe^!, 
infinitum • 

7J3. Licet autem ubi agitur de proportlone $ tcrmi-^ 
aus quartus poft quantltatctn negativam CMl , 8c bi-> 
nas pofitivas CO fit Coo Ppcrinf!nitumtradu<5iiis idi 
^z6i6g.26Si tamen cum hacc traduSbio habcri noa po(IfCi 
nifi P" rcdeac ex parte oppofita , ic alicubi in fiaiui| 
quantitatibus cxiffat; fccumtrahit ncccfiatio diftaatiam^ 
CF finitam diredfcioni^ oppofitac , Sc conformis. dirc^^ 
Aioni CM , qux , fi pur^e magnitudines fpe(£bcQtur ». 
Vel tx confiderentur ut pofitivx , libera enim eft f^- 
ga pofitivorum , eafdcm habebunt relationcs ad fc i^* 
yiccm", 8c ad eandem CO , quam prius habebaiu fts^, 
menu CM % CP ejufdem Loci Geomettici eodem modo ; 

dcfini»; 



LOCORUM GEOMETRICORUM. i»7fc 

definita*, adeoquc adhuc erit CM' fid CO , ut CO *1> 
CP' finitam, & proportio quidcmraancbit, dircftio aa-^i 
tcm in cjufmodi finitis quanticatibus in oppofitam pla- 
gam tendentibus erit iterum cadcm priori contraria ; 
Idcirco proportio mancbit ctiam intet ejufmodi qua- 
tuor quantitates , quarum mcdias dirc(Sl;ionem non mu« 
tarunt > mutavit prima > 8c quarta quoque aflfumpta 
ex partc finita cpntraritm priori habet ; adcoque in 
fummis habcnda crit ctiam ipfa proncgativa» rcductio- 
ne kliqua fimili ei , quam num. 77^ confidcravimus ia 
complcmcnto ejufmodi ad^circulqm infinitum cjusquan- 
titatis pcr infinitum traduftar» quas analoga erat primo 
analogix gcncrc, 

784. Ubi vcro uterque tcrminus pcr nihilum tran- 

fit , nulla diflScuItas cffc poteft , cum praeccdcntes tcr- 

mini , qui habebantur antc transformationcm » ipijra- 

rint in hos ipfos negativos > ac ubi mutatio fit tran- 

feundo pcr infinitum *, facilc ratio rcdditur rationis 

niancntis ex illo infiniti myfterio » quod num. 776 

perfecuti fiimus ; licct mutationc fafta pcr infininjra , 

non fucccdant prioribus tcrminis ncgativi illi fitiit! » 

I fcd pofitivi pcr infinitum tradufti . Si in fig. 243 re-F243[ 

\ &SL FG abirct in infinitumcx partc AE, & rcgrcderctur 

ex parte coniraria DB in fy j illisCF, CGinonfuccc- 

dcrent C/, C^, fed C 00 /, C 00 ^ • At quoniam 

harum ratio fempcrobanalogiam debercteflccadcm, ct- 1 

' iam fi fg appcllerct ad C> idcircojuxtan.^^p ctiamin- 

: tcgri infiniti circuli CA 00 BC , CE 00 DC dcbcnt 

concipi ad fc inviccm in cadem ratione CH ad Q • 

; Qiiare ubicumque fit j^ ab intcgris circulis illis cxi- 

ftcntibus , ut CH, CI dcmehdo' {egmenta CA 00 B/# 

I CE 00 D^, qux funt in cadcm rationt » rclinqucn« . 

I tur Cf 3 Qg in ratione pariter cadcm . Quamobrera 

1 ctiam confidcrata analogia primi gcneris in transfor* 

! matione , cruitur adhuc quantitates fecundario gcncrd ' 

analogas, licet oriantur tranfitu limitis pcr infinimm> 

dcbere rctincre proportioncs^ quas aote transfojmatio- 

ncm habucranr. 

857, Et 



apS Dfi TRANSFORMATION^ 

785. Et harc quidcm ad; cxplicandum canooim » ac 
px JLocoruni Gcomctrlcoruni hombgcncitate in omni-^ 
bus fui^ partlbus^ vcl cx iniiiiiti ihyfteriis fctcmohftran- 
dum» ac vindicanduni dida abilnde funt^ i Caiteruni 
panon ipfc i ubi de finitis quacltitatibus agitut dertiffi- 
tnus omnino cft^ ac patei iil omhibiis tani multis c- 
;ccniplis i qu3S adduximu^ d hiinii 677 ad niim; 706 . 
Bx to dctcrnrihavimus duddm i 8c fbrmam tot diirva- 
tunl parabolici » ad byi^rbolici geheris i quas deihde 
ConflruAion^ geometrici acdurata ihvenimus ejufdcm 
forms^ t qux ex hod cahone iis applicatd obveherat j 
Fatct aiitem lati/fime ipfius ufus per univerfamj Gecv 
mctriam i Pauc^t quapdam attitigemus i qux perri- 
tient ad ejiis ufuni] ih noftris Cohicatuni Sedltbriiinl 
demchtis i 

f. t 786^ Ih primis in ipfa definitione ih <ig, i » & i 
4 taiii FP ad PD > quam Fp ad fd funt ih eadem ratio-: 
ne detcrmihante « F/^» & pd iri £g; 2 fiint analogorip^ 
fis tp i fd figuras i fecundario analogias; gehere i 8c tz^ 
men fcrvaht proportioncm cahdem FP ad PD i uc F/ 
ad pdi Ddndel iil c;i proportione abteruht iri hcgati- 
to^ hini tehnihi fccuiKlx r^tionis iil tranfitu a figiira 
i ad ii nimirum tiabctur numerus mutationiim par i 
8c htetqiie ferminus mutat tranfcundo pcr ihEnitum i 
0im arcus r^mi ultcridris 1 ^ cum eo puhftuni^ re- 
gr^dlatiir tx inHnitoj 

F.i9 787. Iri fijt^ 19 i Sc 20 cfl: ( «uni. 96 ) tam F/ ad 
20 M«9^ quam Mm ad £^ iri r^tidnedeterminahte FMad 
ME rf Martct utraque proportid i licet F/, M«i , E^ 
ixi fig* 20 fint analogas fccuhdario geriere ahalogix ip: | 
fis F/ :f Mw , Etf fig. 194 Iri iitraquc proportionc bini ' 
Ccrhaini t^tumthodo mutsinyc dirediohem ^ Sc cum ad 
^aadcm pertirieint rationci^ii ^ n^iitant arabo in irainii^ . 
tg pcr infinitiim^ 
<Fiii. 7**- te i^* 122 iri qua rci^anguium PLp aequatur 
(flkum. 3jVJ rcAau^ VLD , abcuntc VL iri VL' 
direclione mutata, & manenic L'D' , debcf mutari e 
i^ofiiivQ' iri ncgativum ciiatra rc<Slanguluin PVp' . Quar 

ttt de-< 



LOCOROM GEO^ETRICORUM. 299 
ftt *b«t altcra taniiim csc ipfii Pl* i ty dlrei^onctn 
mutarc 4 Mntat catn (ola L*Pi. ac inre^angqlii «qu:^ 
libtis Ptyi VL*D mvcnitut^nmiietus mutatidnum inro- 
bique imparj 

78$^* Hine cx hoc ipfo prlncipio iri fig. i^$, &: i^eFi^^ 

facilc d^iri potcft plagi ad quam po»i1 dcbcnt iH* 17« 

i^i Ip** q^a^ ttuTOj 453 dcterminavimus in problcma* 

te , quo 4u*ritut Se€Kd Cc^ici triiifiens pcif daU 

> quinque pun<ad PpP^AB * Cuttt ctiim debcat cffc ( nii^ 

i i9$i ) rcdlangiiium i^Qfi ad i^cAangulum AIB i m rr* 

danguliim PQp ad Pljf % pdftrcttium hoc P?>' dcbet 

iiabere miitatiortcs diredidnis riumcto piri j vcl impa- 

ri re(pe(au AI6, tit PQ^ habct rcfpc6hl AQBa C^ard 

' cum mnotcfcint reliqudS^uin ornnium lattfum dktSiio^ 

hei praetcr dircaioncm lateri^ quaefiti I>' i hzC ctiani 

i innbtcfcet. In fig. i^^ AIB refpcAu AQBitiutat folam 

' AQ^in Ali mancntibus QB , & IB . Quare 6c P*Jj* 

^ tcfpcdu PQj^ debct habcrc unanl mutatioticm * Mu- 

• tivic P^I refpefta PQ^^ maiicbit igitui? Ip* rcfpc£tuQg'»< 

I iit fcVcra manet < Similc cft argumcntunt pro If' mo^ 

fttnte in fig^ 170 > aC codcnl pado detcrmitiatur pd^ 

iinb i4'i quae raanet refpedU fp.iii fig^ 169 § inutt-> 

tut in fig* I7d* 

790J Canoii. 4^ Anguto , cujus ^iterum Cfus tdntum^ 

fHodo direSHonem mutavit , fuccedit is , qui e^us efi 

tomplementum ad duos reSio^ i five quem cominet crus 

^ Hon mutaium cum cruri muiato jfr^duEid i^anguio i c%-^ 

\ms Htfuinque mutavit direElionefn ^ ficcedit it i qui ip^ 

Ji ad verticem opponitut y & ut enunciatio maneaty m 

crwre quod dirtSiionem mutavit ^ communis aliqud litte^ 

fa opponendd efi iH binis cafibus fita ad partei eppofi^ 

'tas ita i ut atiera jaceat Ad partem punSli anatogi jfe- 

■^ndarii stnalogia gehere Jalterd ad partem oppofitam; 

in demohfirationibusvefo lut & in ehftHciationibus caven- 

sium fempet fieri pojfe i ui an^uli » qui congruebani , 

\^ant ad vtrticefn oppofiti , qui erat externus in paral^ 

ielisi evadat intermsy & oppofitus , vel alternus 5 au 

^jrt ta a numero muiationum pendebttnt y ita ^al^en ^ 

ni 



5do DE TRANSFORMATIONE 
kt in fingfdis cafihs admodum facilt d^ehtnJUfngt fJh 
fiimio facienda in dem^nflrationt , noraiis illis iinis 
fucceJftoBum regtdis • Generalittr autem uti vertex ofi^ 
tuliy qui erat intra binas parallelas » dheat extra ', an-^ 
ffdus ijife tnmeiatHs concurfn crurHm cum iis parallelis 
hinc i & inde ad verticem opfofitus y fiet commums , 
anguli vero crurum cum parallelis mutal^untur ex alter- 
nis in externos ^ a^ internos , & oppofitoSi & vicevet" 
fa fi punSbim aheai inter pardlelas . QuoA fi extra 
fuerit » & alreat extra ^ fed ad partes aiterius paraU 
lela , manebit ipfe angdus , & angfdi ad parallelas , 
q%A erant externi , fient interni , & vicozferfa. 

791. Hujus canoms catio cft/naaifqfta ; ubi enim» 
Vt^irx fig. 24^ abcuntc iii 244 , anguU cuiafpiam HCI crus 
2^ altcruin CH dire^lioneni mucet » a^gulus ipfe HCI > 

245 qui prius in fig. 24; erat AC£ , cvadit iam in fig. 

246 244 ^^ y quem continet crus mutatum CH prioris, 
(iyc CA productum in CB » cum latere non mutato 
Cl , vcl C£ • At in fig. 246 mutato & CHj & CI, 
angulus K H > qui congruiebat in fig. 243. cura AC£» 
|am coogruit cum DCB ad verticem oppofito • Qi^ 
.mam vero pundum C jacet in fig.243» 245» ^46 cx- 
tra parallelas HI, FG ad partes HI , in fig. 244 in« 
tcr eas \ angulus HCI cft in iliis idem > ae FCG, Ja 
hac ad vcrticem oppofitus , anguli vero CHI, QH in 
iilis extcrni , &c CFG , CGF ioterni , & oppofiti, ia 
bac alterni.At fi in illis Hl recederet a C ultra FG, 
fatis pacet , (latim ipfos CFG j CGF ex internis evaiii* 
ros cxternos. 

792. Porro plurimum facpe prodcrit litteras appone^ 
re a transformatione non pendentes , qux adhibexi 
pofltnt fme mutatione ulla , ut hic iitterae A > B» D» 
£ plurimum profunc ad plagas defignandas , cum ia 
fig. 243 ponitur A ad partes H, & in fig. 244 B ai 
partes H jam mutati , & A ad oppofitas. ProKcieritatf- 
tem id ipfum faspe ad habendam generalem cnuAd»- 
tionem , ut )am videbimus , in Conicarum Sc<%ioiiiin 
elementis prasftitum a nobis eflfct cum fuccelTu. MutatloQes 

vero 



LOCORUM eBOMETRICORUM. jot 
vfaro-ati^Qnm injoppofiros ad veracem > velexcernau; 
rum in akemos, vel intienios vidiinus ex parte n.^^o* 
videbii&as j^ uberius in ipfit Conicis Se6lt#niiyus . 

79:{. Angoli mntatio tam ex alterttis cruris , quam 
e utriufque raucattone in Conicarum' Se^onum ele- 
mentis occurrit plurimis vicibus , cui 8c demonftraito 
aliquando. idctrco aecooim^danda fuit • Jn folutione 
probk 2> num.,i3(;^» edcurrit in fig* 35 * & 36 dctcr-F.jJ 
ininatio pun6bi P per imerfedionem re<ftairum VG > 3 6 
FQf 5c punfti p pcr interCbftioiiem rei6terum V^,FQ> 
captis FV ad FE in rationc dctcrminante, & Qp,Qir 
acqualibus QF • In ejus . autem demonftracitme confide* 
ranmr fimilia fco pun^ P .trLmifUs' , PPV», QPG 3 
^ QPD, QFE, ac inde iruim FP ad W^, ut FV ^4 
Q^,fiu€QY, & PQ^4W< PD, «f FCt^rf FE, unde iw- 
jSr/Mr #A? dqtmlitatc ordinata FP /ei PD, ut FV, 4i!^FE 
^ ratione determinante ^ ut ^pfortebat. H^c demonftra- 
tio , fi afiumatur fimilitudo triangulonim , nuUumha- 
bec difcrimen in %uris 35 9 & 36, jum num; 764. , 
licet altera ad qi^amvis Se<ftionem Conicam pertiaeat , 
altera ad folam Hyperbolam \ quia omnia eemat^c primo 
analogi; genere analoga , auUo termino dire£bionem 
mutaatc , ne^ in infiaitum abit quidquam , nec per in- 
Snlcum traducitujr . Tra^Qs&rtur ea demonftratio ad pun-^ 
&um t iifdem prorfi|s verbis» 8c litteris ponendofolam 
prp puoftis, P, C , D paniSra f^ g^ d eorum analoga. 
Sttiit nirairum fimilia trioH^da FpV, QPg , & Qod , 
QFE, ac inde erHUnr Fp ad pQ, «f Fv ad Q^g, five 
QF> ^ PQ*^ P4> «^FQ^ad FE ; unde infwrtur ex^^^ 
qualieate erdinata Fp ad pi, ut fV Jtd F£ in ratione 
detnrminante, , njt offortelfat » NuUa autem iputatio fic 
in nomenclatura triangolorttm , 8c prop^rfionibus^ fi- 
ve conferatur punctom jr cum pundlo P e|ufdem figu- 
ISj fivrc f cum f ^terius, quia punctis, & rectis fu> 
jpeduar puncca, 8c vcctsc cum analogia vel primi, vel 
.fecuQdi generis ; quamob^em rationes redeunt e^dem 
juxu iium. 772, & cum nuUa argamencacio fiat com- 
j>oaendo f vel dividendo^ nuUui^ fit cranficusa fumm's> 
Bof(;ev^jp.Tei^.iJI/. X aj 



ad di&imoas> vel vtcevtrfn» qdat tmmtL demmift^ 
tioAis verbo aliquo immucent* 

794« At iimilltudinis triaoguloium illorun! iftimdn* 
ffTAtio turbatut nofuiihil a mutatiotie dire^ionk t^ 
rum in angulii • Angulo VtP in fig« ^j fucceditVF^i 
qucm PF mutata continet 9 fi producatuf » cum fV 
non mutat^ • At dirciffcio FP t ¥p cotnmubis ifi flfi ' 
13; 6 , cum FVcommont angulmn vFP communetti i%d- 
dit cumlangulo VI^ • Contra angulus |>Q£ idem eB^ 
flc jpQ|f in %• 35 c<> diredtionetti Ct^ , QP e^dcm j 
& Q/ utrobique candetn » fcd contrariam iUi priorf 
Q^ : at in fig. 3^ i<^ eft ad vetticem oppodtus tp* 
fius PQp> ob dicedionem C^ , Q/" utramque oppeffi- ' 
tam direAioni QP i Qp . Comparando anguloSFPVi * 
QPG4 babetur utrobiqne altet alteri ad vertic^m opfO-' 
il^us , at F/V s Qj^ idcm font angulus mutatis iiUrgi | 
35 fofis dire€Honibus FP, VPi dum abeuut in Fp r 
Vp, & mjirtcntibusdirediottibtisGP , QP, irt GpyQji: 
Acin fig.3^ mutatts contra dfre<5ktonibus GP > QPiu^f^t | 
Q,p, manentibus FP) VP in Fp, V/, undfe fit, ut al-»^ 
ter cx angulis illis binis titrobique , dum fit trahfittts^ 
a P ad / , mutetur in angulum fibi ad verti<fem dp« 
poCtum , maneat vero altcr , Sc proitldc qui foitkal ' 
ad verticem oppofid % jam congruant « Demum ahgoli ^, 
PFV, PVF funt utrobique alterni angulorum PC^ ,' 
PGQ. jacenite P intct i^parallelas FV , GQ^, at fiV , 
fVF , refpcdtu fQ^ , /i^Q^funt itt fig* 35 externt, in^ 
£g. ^6 i idtcrni, & oppofitt oum fact^lt j^ ifai ad partei' 
FV hic ad partts g(X,^ Qjioniam lamen e)ufmodi'm<^ 
catio angulQrum ex oppofitft ad yertictni in cdngitet^ ^ 
tes & ex altcrnis in extemos, lic Ifitei^nos, 9c oppofi^^ 
tos , vel ex externis in internos , aeqiialitatieiti corun! j 
non mutatj manebit demonftrationis vis 3 & ^^^'^{J^ 
nuttciatio muubimr dicendo proptt&<^o Panguhis FPI^ 
cqu^tur angoto QPG ad vcrtictm oppofito , & prof'*] 
^ulus F/V 3 eft idemi ag angdTus Q/j ; pro amni- 
lis vcro ad FV, GQ, & FV , fQ>o«ft <•«?» tantiilft' 
trojo suigaU ad ejufinodi bafes fQQ€ id>ique cquafescil 




G£OMETRIGORUM ; 
pirallelarum propnecatihus 4 Ucet , & ^ fOopii^Ki. 
inmieientur) rfiiutari (kbeat istpteind • Prorftts vero 4^ 
niflia obf^rvari pdffunt in coniparatioiic €ci«Qgiilociidgi 

^j. At dfi evitaadd incdxxuiKidi dtiieotiiotii^ hmeatai 
m aiiguloram, ^ vero 6tiam rof^um cnticiciatiofidi^ 
fctis 5 plurittium £sepe tidbls pro&iic alias adbibeCe io-^ 
teias pri^rer eas , quas miitaiitur j Hia6 illaB A « B dd 
%• I ^ 2 , & tam ii^tis foft letettta^ in direSkice ; 
hi^ ilbs GH1T> j/yir cooftoater re^mae ia figuris ap f.f 
M 14 ^ & 2$« 26) 27. Hinc in i^u& -poft 4.1 ptm- ^ 
3:a illa Xi Z^ & K» ac allij iii Idci^. Id auteoi cpro- ^j 
deft mulio etiani tnagis aliquandd i ubi pundutn ^li- ^ j 
<}uod it:a in infinitum abit > ut nufqaam )am fit« Sic 
piranfer fuperiora exemplaj in ^i^bus faaec utilitas bften- 
Ifi poteft, ubi figura 25 rautamr iri 28 f num. 169 ) 
6t^Uipfis in circulum, pdrifto £ jUlius abeunte intfi*F. ^^ 
fitiitutn ita, ut liufqu^m }dm fit, fruftr^a analdgia qa^* ^g 
''^^etur i^^urarum^ nifl iitrobique mansrent litcetx Qi^ 
H«r, li ab inteirfedionibus non pendfiams $' q^ P^^ 
tritorformationem fuperfuiu a 

75^6. Exempla litterse adjeifto? ^fnm ^i^u enuhciatia^ 
iris tnanemis habeiitur plura • Lucfileiuiilimum eft i^F^M 
tdk iitterae V i qux ih ggucis H 41 dd 45 idjefta eft . ^ 
in ufu lihctx V^ quae in figttris a 41 ad 45 adjc^a .1 
Cft fntltiii 172) ireftaB HF, in prioribus ad parces Fit» 44 
pofttcftid ad partses H i, Hac^tc obtigir ubique expa'' .^ 
t^sfllelarutti nat&ra asqualitas adgidorumrPf H , ptV cum 
tegolis Vtty Iff aequalibus inter fe i licet cx diver* 
fis paraUcl^rumproprietatiblis^prdduat^quaUtas ip(a iux« 
ta hunc ipfutn cahonem . Porro in %iir4s 41^ 4i>4^ 
tam VP inxer jfe rolata&,5 «qoam ^ i^r &f pofitionem 
fe^antf 8c proinde omnia eodenl modo fc babciUi iA 
ffguta 44 mutat direAio^em tam BP-i quam Ef ^ hm6 
adhuc V jacet ad partcs 'Coiltcdrias H . At (in fig. 45 
matatur Fp., ttianct FPi binclittccarum rcf pooden lium 
V i & H idtera refpechi alterijus maheniis n^cari dc 
b^t , ut }am dire&iont» FH ^ F V cong^if nt , 

X i 7 97^ 



.JB& 



^of DETRAN5FORMATIONE 
7^7. Hujufmodi artificio aufcrctui: ctiam appareni 
quxdam irregularltas , qux videtur occurrere in tbed^ 
remate expofito num. 176. Ibi enunciatur , bmas tan^ 
gentcs ductas ex cxtrcmis purictis chordac tranfeuntls per 
fbcum concurrere in dircctrice , ibique contincrc angu- | 
luni in Ellipfi acuram , in Parabola rcctum » in Hy« 
perbola obtufum, fi terminetur ad candem ramutn il"- 
la chorda , iterum vero acutum fi tcrminctur ad bU 
F.^o.nos ramos . Is angulus eft iii fig. 5j , & 54 PHp * 

53 Pcrro ubi puncmm p e ramo citeriore figurae 53 abic 

54 in ulicriorem 6gur« 54, non abit arigulus ille cx ob- 
tufo in acutum faltu quodam , fcd angulo VHp illim 
ibcccdit angulus , quem in hac contincrct PH futn 
fH producta ad partcs H , quae niniirum pH dircctio^ 
ncm mutAvit . Is eft adhuc obtufus , & excipiens po- 
ftremum fflum obtufum PHp figurae 53 , qui habetur 

.^ puncto p abeunte in infinitum , Sc tangente Hp in 

afymptotum H2K2 figurae 50 . Is pcr omncs conti- 

\ nuos gradus matatur , donec ad binos rcctos acce* 

dat. ultra quofcumque limites , imminuto PH^ acutq 

iia f ut abeuntibus P , p in vcttices axis tranfverfi > 

& factis tangent^ibus parallelis , evancfcat . Satis igitor 

fuifllit tn HP pirbducta in fig. 53 ad parics > , in 54 

^d partcs H apponere littcrani, V , & cnunciarc ita z 

angulus PHV crit in EUipfi acutus , in Parabola re- 

ctus, in Hyperbola fempcr obtufus . Sed quoniam ci 

nunciatioy &demoaftratio fineejufmodi production^ re-' 

ctx cvadebac firaplicior^ fimplicitati analogiam poftpo^ 

fuimus . i 

7^..At ex hifcc exctnplis jam pat€t » quam apt6 

hujufmodi artificio fcrvcmr farpe analogia , vulgari et« 

iam Gcomctris fermone adbibito . Nam fi infiniti n^- 

fieria libcrct adjiccrc , & reaas confidcrare pcr infioi** 

Tum traductas, ac alia, qusdam, quac fingula pcrfequi 

longum etTet 3 admifccre , thcoromatis quoque inde ' 

provcnicntibus in Gcomctriam invcctisj poffcni fcmpct 

ipfa intctfcctionum puncta rctincrc caractcrcs fups > 

dummodo aliqua notula gcncraliter exprimi pofTetf di-^ 

reciio 



mcoRUM G£o.vcfetRiedRirM. jof 

^S'!'"^ tettdcriris^ad pun<aanv, & mSig^Uudm . 
quae txprcffio commiinis etfct erianj- piuMais In ia6au 
to ktcnribus & iincis pcr mfioltuin traduOis;. SkTd 
g- " .^°S'^"« ^S.?"S"'° ^>" *''huc oppofitusP.M 

«nutalrunt » fed tjt parfc lllaram f 00 *< v of > , 

»T5.°«?? ^* ."^^'^'^ ofamru, cft an«ilus PH 
&infi;;ri S '^'=^^«)."'^"o«'i gcoraetricura idioma * 

Lf «2r/.^- "V'°' "' ^*f« »° "sanalogiaqus- 
daifl coafiderail poflit tawummodo , & afus ad ea , 

tm£Z Sf °' "f !. ^;'^ .P^-^qu^rfiefitlitaram ina- - 
g«ud num rclat.oncS ad fc inv ccm evidcntct pcrfpi.. 

S^ fL mt/f . '"^a '""' ^^ '" relatiOflibus dcdu- 

dTm SnmmnS * ''"'" demonftrariooi . C^tti cjii-. 
aam tantummodo canones eruumur , quod hic pKe, 
fiajmus.^cx qu.bas riteftafailitis po/iint p&que, qSi 

Sfno^tr'^^"- > ^^t.«.^^ qu^itariblrs feitil 
ST«;i.; L "^°'.'i* "?''''^'^" A.bftitueri«us alio tb,, 

bam othtia , qua: huc pertincfcnc. \ S?d dc iis 

Jm"'?^ r • . '"^" gcoraetfiei idiomaris defeduse^ 

JSdc^ """ *^"''°' * ^ '""^*" *"*'" "^»^* ^ 

7W. CiOoil. ^. t/k dnguii hiaikt n^ «/*W4 bltna. 



3©*^Dfi TRANSFORMATIONE 
^bfot tMHfnmd» fefr b ms^ reRos y dngulo mo juxtA 
^9mmim0m Gemet/U nomenclamnm debet fubjiieui ^ i 
€ mfiemenfm ad 4 teSofj qni /i s^jeltetHrl$ngulus penm 
vexHs y vel ut aliqut fite»t ffwns > fd^ maloiiA muU 
to melius fervkkitur. 

teo, EHun re^ Ct ia % ^^4 Sf^^l circa C ciM 
f ^«^redlra CK t(Rd% angalum KCL direaioiie KLN \ abe* 
unte i in K, it endit ndius : tum abeunte t in L% 
jam cvadic oegativus it^edhi KCL > bkm KCL po0 
traciGtum per mbiltim abeunte in KCL' direSione of* 
pofita KLO. Is ^efi^ty & fir reftus > u6i L* abit m 
O: tum Ci L*,pergat ultra moveri in M; anguIusKCM 
cft adhuc ciuAlem diretfHonis cum KCLS fed obturiis, 

Abemte M ki Q. , iam fir KCCJL ^^^^ ^i^^ t 

^ anguhis ilfe abit non in nihilum /^d in daos re-» 

^ <^s KCQ; ) qaorum menfura eft dimidia drcumfe^ 

^ rcntia KOQ^ • Ffet^gcnie M in M' , }ara angulus KCM* 

ixx vulgari ^ometrico fermone intelligitur i$, qoi Iiia^ 

m Cavo r^fpldtplagamKN» qui iterum c(! minor diio« 

bus rcdis , Ax is non fuecedit priori iHi KCM > ncc 

cft analogus ipli primario analogi^e genere» fcd fccaof 

dario • Mori* Aiccedrc angulas ^ ut eum appcHavimos» 

convexu5, quem KC cum CM* continct ex partc OQ» 

&? cujus mcafura eft atftjs KOM* fcmicirculo ma|or^ 

' h cukit t 8i: ilH cavus decrefcit , dum M' pergic ia 

L , & appeH^nre d^mum M* , vel X ad K > complciw 

FfSp tur quatuor H&i « Nimirum «ut in fig, tl9 bini iiiat 

arcus FBw , FAw <f<witraria direAionc <K)inpleates dt* 

cuium i immo infiniti 9] qui integrbs addunt drculo» 

dircc^iorie utraquc > it^ bini conflderari poftUnt anipw 

li , quos bina; tcQ^-Jn^m&o quavis condntnt di* 

rei^one eontr^rii « aiter convexu^r $ altcr cavus #i 

complectetli quatuor tt^ai j^ ktimo inftnici dirc^BfM % 

JUtraque. / ' ; 

9ou Porfo ubi attgulos^ dlfedionem mutai tcsflfe* 
undo per nihilum ,. nra^bad deEiet ut ncgativus. Ifl^K» 
Fi^o^^ti iingulus ACB cxt^rnus sequatur fumm^ a^o» 

mnAEfi I DBE i qui fuot incerm) ^ oppoiiti i&trU 



LOCORUM GBOMErWCORUM. 507 
aaguk> CBE • Hinc aagulas ACiB a^quarl dd)et di& 
(^Etmxm Mguloram AE91B, DBE^ ob dire&ioftem DBE 
ma€a«UB m DBE^ 1 ti^atifita fafto in D per fi^hiiuui, 
£c rcTcra cft ipfi difierentias ^quatis , cum AE2B ex^ 
lerntts flsquetur binis; DB£2 9^ AC2B internis , Sc op- 

So2. Quod Q mutatio &m trainfeundo per duos re« -^. 
dK>s > anguloi qd in 'ndgati fcrmone nafcitur cavus 
ad parnQni opFM>fitam 9 dcbet fubftitiu oonvexus ilk > 
qui cft ^os complcitiencufn ad quatuor rc6^os. Eftno^ 
liflimum Geomemas tkKOrema » iti circulo angulumad 
Ccmrmn ell^ duplum anguti eic^m arcui infiftentis ad 
Ctrcumfefcatiam « Ncto c^^it v^rum , nifl aogulus ad 
circumferctiiiftm fii acums , vcl oifi anguU huiufmo* 
di convezi coaiidcrcntor « Ih %27I atigulus APCeft^Fa^x 
dupluft angiifi AIC % mg^ autem AIC noa habetur 
«fajplus ia vulgsiri i^monc acceptus , i^cque cnim cft 
APC » icd cjii^ axnplcmenmm ad red;os quatuor, cu^ 
|us mcafura cft arcu& Al^Cx fivc cft angulosi APCcon-* 

toj^ HofUA «Mm caQonis iifus occurrit in Se<!^(> 
niHQ CooicaEum clcmcatis • £k niun. 184 faabemr, inF.57 
(Ifipfi iti Sig. ^8 doplam aaguli PBp binarum tangen- 58 
uiffa 9qa«ri .diffeiicnti« btnorum angiiAorum PI*p, Vfify 59 
m HxRe^^^ ^ % i9 ^trmx cohmdcm I^^ > P^% 
Nam i^ / ^t ia PMrabcJa in infinitum ita> ut nuf- 
^iutm jpin &t » aogiiai. Pfy dcarcjfteot in itce^ puft« 
^fiik iognitum past fit ni^Qiiif» 8c tdcicco ibidcm ia 
^S* I? iB Parafaola diy»lQm angujti Pf^ aeqoatur foU 
an^ido.Pfy. Ubi $iat€m ahfk curva ia Hj^pcibolam 6r 
fitt» j# ry&i f redk « pncic oppofita , ««ftguius P^ 
a^cquirit dircctionen\ oppofitam > qu^m Qjm aiicquifieric. 
10 tnattfim pQr aibiigra >^ cs^t n^at^vus» Bc, difi^rcn- 
^a debuh abirc in fHHmmam.. 

#04. Ibidcin autcm 4 Mgulas R^ iutn obvonat cuk 
Iptdcitt piuavfh» Hv M ut itik^ fig. 60,^ ^i 9 6ibiatam;|F.6o 
'caijindatio ibco8|iqHC& iOi ^goi^iCiftomcarico Armoacii^i 
faUa ^ita Nam noa clQt aooijpicncfais angylm PFf ca- 62 

X 4 TUS 



>o8 DE TRANSPOfiLMATrONE^ . 
VQS iiie quem vuigo GoafidecaQt>.fed tjus cdtnplcnieif « * 
tuni ad 4 t<(€lo!s» niniitxim illt, quon nos con^exuRi 
appeUavimus > qui Conftat adhuc fainis PFN ^ pFN , qnod 
ibidcm cnunciavimul;) & qui id non enunciant». theo* 
rema exbi^enc in boc cafu falfum « Nam in Geodie-* 
trico fermone vulgari fcmper anguli nomine &itelligi» 
tur cavus ille> non CQnvexus* 

S05. Hic folum poftremo loce tiotanduttl eft bofee 
binos mpdos mutandi dire<%ionem in angulis tranie^ 
undo per nibiium , Sc ptr duoi redloS) refponderc bi« 
nis modis, quibus line^ abit e pofitiva in negativam 
tranfeundo per nihilum) & per infinitum • Vt anrcm 
ibi non eft analoga primario atialogias gcnerc priori 
line; iitiea finita habens dirc&ionem oppofitatn nata 
in tranfitu per infinitum» fed illa per infmitun] tradu^' 
&9f plufquam itidmuy ica bic priori angulo oon ref> 
pbndit poft tranfimofi per duos ttStos angulus caviHi 
diredtionis contrari^ % fed ille > quem nos iuc conve* 
xim diximus plufquam obtufu^ < 

SoS. Canan. 6. Quadratum line^ tam yojitiva > quam 
negativa efi fefitivum ^ & iu^vis quddaMtum pfifitivum 
Hna hahet latera altirum pofitimm dlterum n^iativum*' 
Si autem qptoddatn iiuadratHm aquale fiterit - reSangnl^ ^ 
CHJfts latus alterkm diretlionem mutet ; iffum qsddem 
quadratHm cenfendum erit reale % Jed Jtegdtivum » &' 
quadratp frimi analcgum fecunda gmire anidegia \ af 
€\us latus fiet itn^ipforiumi & impQffikiley defidemeibi^ 
termina analogo Uttri qHadrati frioris : fi disteiHonem 
mutet utrumque reSlanguli lattUy nit realt utrumquf la* 
tus quadratLfpfitivum^ & negatiPumy'& finguiaixifis 
erunt analoga jnmo anakgia genere fingidis imerHmi 
frioris quadrati* • ri . 

807» Patet bic Calioti^ ejc iis« qu0 diximus a nunu 
682 ad 688 > Ubi & ejus dcmonftratto iiabetur« 6c A 
I^H^fipruntur etempla of4inat8rum BG> BGfiguit; s^a^qu; 
.Mmi funt ifitfia circulum,.,null? extra, ac BaL^B^JL^» 
i : qiA^ bateatur «xorft uuiaque in MxpclMiay aon auicm 



tOCORUM GEOMETRICORUM. jt^ 
inira» tt alk cxempla adduanir defuit^ia lat pofidoni- 
bus Euc^dts libri 2 • Quadratam autem » obi fit nega* 
imuD^ 8c adhuc appellatur quadratum, non crit qua- 
ikatum quantitatis realis^ fed produ&um tx rc<5b po* 
fidv; couiiderata» ^ rcda longimdinis ejufdcm^ dire* 
dioois otnttariie negativd confiderata ; adeo uc qusu* 
dranim negativum ubi ad reales quantitates rcferatur 
idem (ignifket^ ac ejufmodt prodQftttm» quodquadrato 
poCtiva9 fc vere quadrato rcfpondet ita , ut reda nega- 
iiFa pofitiv;.; erit autem quadratum lateris imagtnarii» 
iiye impoflibilis . Reftangula cjufmodi > & quadrata 
Hegativa cum pofitivis confundi> ic pro fe inviccm af- 
frnni poteruntj ubi folg magnitudiMs coniidetantur ; 
at ubi etiam pofitio confideratur , ae analogia ad traU'^ 
sformatioiiesj diligentcr funt diftittgucnda • 

8o8. Confequin^ autem cx ipfo canone hoc velut^ 
Corpllarium. Ifitir hinas niids tdm fimtd pojinvMS % 
fiMW fimid mzntivMs meiui frofortionMlis efi dufUx p 
dterd fefitvtfa > dtera ne^asiva « qua Ungihtdine fun^ 
^equales » direOione contraria . /nter binas alteram fo>* 
fokfom ^ negativam atteram media frofortionalis rea^ 
Hs non katetur > fed in imfojfikilem , & imn^inariam 
mraque tranfit r hahm autem fojfmt hina ntedialom 
fimdine aqnales » fed fofitione conharia altera fofiti* 
ya » altera negativa • Patec corollarium cx eo , quoil 
^nadramm mediae sequari debeat rec^angulo fub extre^ 
^i Sc dcraonliratum eft num. 685. Bina: autera iUe 
medi; iaabebuntur , ubi datarum alteta eft poiitiva ». 
dtcra ncgativa , fi carundem datarum utraque pofitivc 
sonfiderctur » & tAvcnianrar binc mcdi^» quod ibidcm 

gr^ftitimus > invcntis binis BzL , xiatdih intct ABa* 
2O s Nam' fi h; confidercnmr ut pofitiv; ambap f 
irit ABa ad uoramvis BiL, ut cadftm B2L adBaD» at 
E aliera cx iis confideretur negativo\modo , ut AB^^ ^ 
ffit ABa ad alterum e htnis BL^ ut aliera» BL,iionit«. . 
ft cadenr, ad B2D» rautata nimii^um confiideratione u« 
Kitt^t^tcrmioieiufdem prim^ratioois.Atquefapceritdi-^ 

iiwci- 



j 






ZU> DKTRANSFOI^MATIONK 
fcdaieii iatcr B CQa^aramm dfcuto,. St Bz conpM» 
cuiii Hyf^Aoix • Erit tfat AB ad al^ecmram fiG > ot 
c^Mcm BG ad BD » hk ABa ad abetam BaL > ot qm 
ca, fed ^ra tL pacitcr ad Hxperbolam tcrminata ac) 
B^J^t Atqur iioc pacto rebtioncs qaamloqiic hahrhnn^ 
tor m»i ind^ntcs hicer EUipfim » & Hypciholam » 
idnmtcs..qiiaEdam probkmata» qus vidosciitiif opc po- 
iiciTorttm « & nf gadraripii ad ntiicnni ptoidoma rodc^ 
ci pofle , S( gommuQcm habcrt cnundattOQfm » ttbi 
nimirum planis pofitiris ncgachra fucxBdant > acm li"* 
t^x lineis tanwhimodo , uc in fine eorum , qoa» ad 
hanc CaiioQem percifieiiE » paKcIuc • 

8o5r. Hujuf Canonis» & CoroUarii fiimmaa cft i^ 
ii| «Sectionum Conicarum elcmencts,» & ejci!i c^ nu^ 
rum in modum raoo redditur q^arundam y qim ^ridea^ 
tot anomaliat evertcote$ oniaemanalogiamii Sc fdaaio^ 
ncm harum corvarum ad fe invicem -^ Olud l^ foftz 
notavimus fium, 7^1, ubioflcndimus aicn( tqfpcibobQ 
per infinitqm traductum > non vero axcm finitom ic^ 
pofidccc finito axi EliipfeoS) quod nimicttm per qiiod- 
via punctum axis finid EUipGcos > ^ per nutlum fini» 
tii fed per quodvis iUiiis» qni tradudtur p^ infitiiiiua 
ifi Hypnbola) ducut ttciae ipfi axi pcrpendisiilacc^ 00« 
curruAc pcrunciro* id vero hinc fanc manif^fta pcft« 
det » & ad omnes diameiros ftimartas Hjrperboljc triir 
T.9 ducicur* Nimkum in fig* ^. in Ellipfi eft ( tnm; ifiti 
ccmilanicr asis Mm ad chordam VF», ut tcaaogulaak 
MRm ad quadr jcum fcmiordinacacRP. Jam Wx> obt- 
cutnque aflumauir punccum R in EUipfi ;in |uce fiaiB» 
Mm^ ambsMRy mR rctinenc poOtionein fuam» adco^: 
que habcncqr ocdinata& - PRf iis < irdpondentes • Ai 4 
^ puocium affnmamr cxora ad panei M » wl 4Pi j^ tm^ 
cur in ncgativam MR $ vtl mtL » mancnti; asRt ^ 
MK ^ Quatc nmcacur in ncgativnm ciiam recia&gubfli ! 
MRm Zriinc qi^cus tenninitts propqniottalia poft M* 
VHy <c ccdangQlum MRii» i qaod crac giiadracupti»" 
^ miordtnaiae , vertitur in ncfativura , & proindc ^»** 
•ordinafa refpQndcnspunAocttiUbct axis EUipfcosMM"^ 



LOCORUM CEOMET^teMUNl, jit 
per ifififlHum tracbifti eft inaagTnarU ^ Hcet qus quft-' 
drac^ rede maneat» (a9 ncgati?iim, 

Sio. Comparat^Jim f^^rbola figi^a? II cuin ^Vf^f 
lipfi: %• 9 > n R aiBamatiir m eps^ris pmSto axis in-' i z 
idm6 MH I £rcAion^m habCtMR candem, ^cftiti% 
wti eonttariam; 9c ^^i^n^tvir K* iii axe ptky^^ttit^ 
m Nfifl' > retinet mR\ Q^e in ucroqne esUU ikA^n* 
guktm MRm cvadit iKgativum . Remanet autert' Vh 
ppfitiya quantitas, Mmi ncgatlvai ^ 4ireAic^fs nhtiitym 
coQffarias , Qpare mutadi primo > ac ttttio tbrfnino 
pcoponionis» Ac mttttite ^cundo^ debet maft^requirT 
ta$ 9 adcoque qoadratiBii femrordfnaese habcmr; pofitt 
vuvi, te icmiordifRmiiitraquc rcafis per totqm ax^m * 
M QO ft tradcRmm per ihfimfum « Contra vero ii£ 
quom pvmtt;o R nffiiinpto inttr M, ^ mr retinetut di* 
re^o iteriufque MR j tffR, rcfpedh Cffipfeo^ ; adeo- 
que retiiictur rcnangulukn lAKm ^ectionis ejufdem^ ^ 
retiiietur YFir^ namm vtto Mm , Q^ans mmatur ct- 
sam quadracum femiid^uM&e ih negatrvum > Sc proih^ I * 
de nuyiBm dt pun^Shim afibttiptom in axe }Am Aiito 
HTpefbohB^ io qm haberi pofliftt ordttiatie • Qrdinat^ 
tp^ iit ptmaiir i^faesictim &nt impoflibilcsj & ima- ^ 
giwttist^-y cantdi autem quadratam> quartuni in: iilapro^ ' 

rmone, in qua priores tres termini reales (unt[> r<;a« 
tSt etiam ipfum i f^ mga^vm. 
8xr« Uoc aoiinadvfrfoy paiet jam primo^ cut £U!p- 
iis quidem' fkmr orbc iu fe q>Aim redeat 3 H^perbdU 
vero faabeat bim croSra ot inftcilmm uurinqiic produda« 
Patft edJfm ande orianir <^imcn infigne inter dla- 
mctror eon$t^am primarittmni Hypcrbblse, iSve dlamc^ 
'Vojf fecundaciiir &diaaiedmodi^i^ata# EBipfeos, dm-« 
[tM diamctri Imjuar (efmteafittf ad fieiimetrttm • (num. 
f^^ia j; aHus diametrr» 001^ onlfies» fdlt^ (blr, qiras 
' cdiiciBcnt ir afympootomny ^ngdif qubs axis tranfvcr^ 
fus kcw f occiirruitc pei^lto ipiias ; t^n^ mtm . 
ipfi nuUo mocbr oocarrQar , ftU ta^imunur »1 perl- 
mcttiiar binqrum ramorufH Hyfpttbe^ conjugatat (mL^ 
^ iizM qu» Hypcrbola coojusau cft locus geomctrifus 

a, priore 



3ii t)E tRANSFORMATlONE' 

a priof^ omnino diftinftus. Nam qiucciimqae diximdS 

4e ordinads axi tranfverfos locum habent in ordina'* 

Ut diametronim omnium) cum in Qmnibos juxth num* 

351 debeat eflq redat^lum fub abfciflis ad quadra-» 

tam femiordinatac in conftanti rationc diametrl prima^ 

rix , quac in Hyperbola mutat dire(3:ioner^ ^ ad re€lam 

cktAm » qua parameter dicituc ^ 6c nt paidQi« inferiud 

ikinc demonftrabitur y eam non mutac . Quariiobrcm 

fi per centrum C 9 utiqu» iatercepmm verticibus dta-« 

tnetri , concipiatur ordinata paralldia ordinatis diame* 

tri primae cujufvis » car^[uidelTr-ft|(naginaria eft 9 fed cju^ 

quadratum eft reale» & negativum. Si ea eflet realis « 

tiftc utique analoga diametro conjugatae EUipfeos, qu^ 

cum per centrum tranfeac» & ad pcrimettum EUipfeos 

apfius terminetur » ac &t parallela ordinatis fiias drame« 

ui primac iibi conjugatas^ etiam ipfa eft ordtnata qua> 

dam pertinens ad ipfum centrum . Hinc eruittu: illudr 

femidiametro paraUelaB ordinatis diametri EUipfeos: 01-" 

jufvis terminata: ad ejus perimctrum » adeoque eju$ 

Mnjugatac nihil refpondere analogum > quod reale fit y 

8c percineac ad centrum finitum Hyperbolae « Scd cjus 

quadrato refpondere quadratum quoddam negativum « 

f arametrum pofitivam > Sc^ reiftangulum MOn fo&u* 

vum 9 

812. Porro ob bii|fts <|uac(rati fldgativi afi^dogiani 
cum quadcato poiitivo ftxis conjugati ElUpfeos fadiitn 
c& i uc Geomettas y licet idy ipfum omnino tutn noti 
perfpexerinc 9 femidiamecitos appeUaverim conjogatas 
primariarum > latera ejufmodi quadraci pofitivd conii-» 
deraci > quas cun^ viderenc non tetminari ad peiimc-». 
trum > eas dinerunc iemidiamecros fei^ndarias « lUaff 
fooguntur vice earum , quac immaginarix fisnt i Sc 
qtnz vere analogae eflent r & «flcnc reales 4 Hinc aii' 
fem iUi^ maaiCdfto cotifcquitur i femidiamecros 9 vd 
diaiTiearas fccundarias Hyperbol^ nuUam habere anap' 
logiam cum femidian>etris 9 . vel diametris conjugaii^ 
iSifCws 4 fc4 iUacum quadcaia e(& analoga fccunds' 

tim 



LOCORUM GEOMETRICORUM. 31 j 
rio analogiac generc quaclram barum > nimirum^ubi 
refernir Hyperbola ad EUipfini 3 quadrata femKli:^ 
nietrorum fecunctariarum illius afliit^ienda efle ,' ut 
negativa» dum quadrata femidiamenrorum coniugataruttt 
oqurvis diam(£tri EUipfeos confiderantur, utpofitiva. - 

813. Huc ubi iam delati fumus , prona fient , Sc 
legifcus continuicatis 9 & uniformis Sedtionum Conicar 
rum naturs adittodum confermia plurima , quf vid^ 
rentur olnnem analogiam pervertere . Nimirum in.iis^ 
qux perilinent ad diatretros it>fas fecundarias Hyperb<H 
bc coUatas cum diamctris EUipfeos » difcrepabunt oni* 
nia , ac proprietates earum diverss crunt , & diverfa 
ratione deitioGHrabimtur. Ubi autem earum quadrata 
pccurrent> fervabimr penims analogia , dummodo qua- 
drata .diametrorum iecundariaram Hyperbolx habcan;- 
mr prp negativis • Patebit autem & iUud difcrlmen > 
& h^c conformis ratio , confideratis ipfis Conicarum 
Sedionum Elementis » in quibus , qu^ maximd no* 
tam digna hqc pertincntia arbitrabimur , bic [perfe« 
quentur. 

814, Conftru^lio EUipfeos, quam ex datis binisdia*^ 
nietris dedimus jQum. ^91 , nuUo modo ad Hyperbo^ : 
lam transfcrri poteft : ea vcro , quam pro HyperboW 
dedimus num. 2^9 ad EUipfim pcrtincrc non poteft : 
amb; elegantiffimac funt, & fimpUciflimac, fcd a fein- 
viccm rcmotUcim^ , & penitqs difcrepantes • Axis tratif- 
verfus in EUipfi eft omnium diamctrorum maxima (rk^ 
119 ) y iti Hyperbola omnium primariarum minima 
( nuni. 246 j, ^ m^thodi , quibus ca thcoremata At* 
nioinftrantur a fe invicem difacpant • In EUipfi omnes! 
diamctri terminantur ad cjufdcm ElUpfeos perimctrutn > 
ut diximus: in Hyperbola terminantur omncs primari^ 
tantum , fccundari^ autem ad Hyperbolam conjugatam » 
qu^ aUum locum geomctricam conftimit a priori pr6r* 
fus diftinctura • In quavis EUipfi habentur ( numcr; 
379 } binc diametri conjugat; seqoales , ac vcl pri- 
maria major cfie potcft , quam fua conjugata vel mi- 

I fior : in Hypei'bola»aifi fquilatera fit> fi»npcr in^qualcs funt. 



*\ 



^if 51 tRAK^FOR»tAtib^lS . 
wa prii^aria (num; ^4^) vel fcmpec iat^t ^ vd fhiodl 
^mm fua coHJugaca. 

iit$>, Ipfa rkid) ^a atiem iccMi|iigs|^iim ^ ^ 4uktiie^ 
i&6s friitia];ii« cof)|ag«ca$ dlHmlvitnils^ in fiUipfi^^Hjr^ 
jparbdift difcriineti. i^ dpertiffiind docet > Otttt ^moS' - 
^&Qxa diverfa fiCi licet pritna ifrocitecotilorffiit «ppalreacv 
|S|eque eHim eas definivimits ek ulla idatioiffc ^«HSitmi^ 
tl ^ peritne^iim EllipSebs i 8c Mypetboke, qittilitntA 
riim nuUa faabebatiir 9 kd ^± vik ad hatui ipfan^ano* - 
inaliam dedarandadi aptiffinia i Nimifym pco ^k eM^ *' 
jugaco in £g« ^ i & II afiiimpfimus CX » O tnedias 
intei^ MFi mFi ic diximos ucrobiqple ifitoi Xx aierit 
tonjt^atuni • Vidieibr fiine ba:d definido commufiis efle 
defumpta himirunl ab tadenf ftiatiotid ad rectds ana» * 
logaH MF > ^F • At fe dil^j^tids cottficibrata i coa* ^ 
tfarium erit adthodum itianifeiftiim^ CcM eaim^MF th ^ 
^ura 1 1 habeac leandtm dire Aionem » ae iti fig; | 
9 1» Sc ¥m cocrirarian ^ pacec dterani «inmnijnGiito \ 
ttanfire iti negativam . Hfl^c fi JncbetBr m SlKpfi do^ < 
piex mcdia ;^opofcionalis incer MF & Fs»^ ea tnHf- i 
perbdia haberi notl poctft jaxca nuth; SoS> cuth ndlla *^ 
fic media imef qiiahcicaibm poQtivam > Sc ivegativam i t 
ied binae inveniri peflinc mediae acqu^es quideth ma- 1 
|j;nitudine > fed poficione fDhtrafias all^a pofiuva > alie^ 4 
r^ tiegattira • Si if^mt ki Hyperbola «fiuniuncur medi^i^ < 
CX^ Cx incer MFi i^F ^ jam ^am i^ eonfideraniC, \ 
uc poficiva, adedque ipfa fic cd&fideraca non eft analo^ i 
ga illi mF £Uipfeds ibidem cofiideFacc > ui pofitivae i 
nec proinda il^ medtas analog^ ftihCj 

<i6. Ac pfo diamecris doc^ttgdtis (^vtnii diatnetfi 
i^oterac qutdem iHcid afictthi pro d^iciohe » iic effent 
reccaB per eencmtii dfi<to ^parafielae ordiihi^ts iUihs in 
fo biferiam feccx» quarum Hjuadr^nxm ad quadraratK 
fiue diiiinenri ^rimse efiec 9 tlc ^ft qoaciratam femxocdt- 
0atc ad feccangulum fubabfeiflis» ^tias TifafmffetcoDv * 
ihwis defifkitio • ^cd prietefr quam tjuod ia leimdirm 
ibopolifim incidifiec definifk), quadracb femidiametri k- 
atttMtarile cvadc|ite negativo in Hyper%oia> & ipf^ f^ 

mi* 



r 



iildtthietrp, at diamciroi & ^albgia rkfe fet^ftftda ^^ 
jkt^ imdginark > ptkttt^ eti deifiQitio ^et, gcferjCJikK 
tlttitiflct : i^m. idiiS)t}cter quse^h primAria habet iii H)"^ 
{l^rbtyla ftsam (ecimdaci^i "^^jus e^ ipft 
^4m: tAmeii Iia1>etur cdaftjiili fca ira^b qtiji!ilrati &mid^ 
ditiai» diaj»ie)ti ftcotldalrix ttd rettangulikti fiib ttbfei^ 
6$ a bitu^ e|Ui verticibift ^ fcd e^ ptoprieta^ eft or£^ 
natarum tantUfnfnodo ^ 8f abfciSSkrum ad diaiMttiiM 
pliimriam « Aliam igitur a|>paren^cm tt^nmmmodo an^^ 
logiam teonfeciati fiimuis i quse jpriibo afpctrtu fumhrik 
vidcr£tbr) licet re ipfa $ nuUil ^ffet » cum nimkum ndl- 
la prdrfus liaberi poffet • Nitniniit) in fubfidium voc^ 
vimus ^guraim illam &mdt{ztiA quattlOr Inigttitili bmvt- 
rum Hyperbclarum conjugat^rum ramis y qtias txfai^f.51 
bent figurse jfa» 83, 84-9 St^ ad onicam jEMipfim » trt 8} 
tium. 172 ilttitiimus , relatidtxes faabcit a^modiim ek- 84 
gantes • Diximus igitur nuih« 212 ilkm diame&i to* 
jufvis diametrum conjugatam$ & pofitibnti & magtii- 
tudint de^nitam^ qu^ pdf centrum dufta ordiaatts il« 
liu5 paraliela tflet , & ad pcrimetrttm *terminaretttr iA 
fillipfl ipiius Elli^fbo^ tn Hjrpcrbtda "Sgatx ipfius a ^trs^ 
tnor binarum Hype^bbl^um confugatarum famts leon* 
dufit, qua definitidne i^tis paoeb!fct cotitineri lates Ip- . 
(o^i €um axcm confugamm ttffminari iti Ellipfi ad pqp- 
rimetrum ipfius EHipfeo^ conftarei ex n^ 72 , & in Hy- 
pcrbola id ia ^fa Hyperbalatum tonjngatarum notiotie 
I contincrctur n. 170* 

817. Porro tanta eft^c/us iigurrfe tpiatuor Hypcrbolat- 
rum ramis conclulff faabitudo ad uniCam EUipfiiti , xrt 
ta vcl miniis perito > vel minus cauto Geometrx fad« 
le poffit imponerc , ac fuadcte c}u$ ctiam figuraft pefr- 
hictrum fimfficcm cfre Gcomctrrcum locum , & unic^ 
£flipfi intcgr? rcfpondentem - Nam qiitsvis rcGta tara 
ia quavis EUipfi^ quam in ejufmodifigura percentrum 
ducra , ipfius petimctro dccurrit hinc , & inde in bi* 
tiis ptmctis t^mmmodo, ii himirtm & afymptotorum 
Concurfus cohfidcrentur ^ ut in infinito delitefccQtes » 
ubi fe 8c am ipfis affmptons octo iUa iqaaxnor ramo- 

rum 



^i* DE TRANSF6RMATIONE 
rum cruta conjutigaot : quxvis cx iis ica tcrmlaau itf 
ipfo centro fecatur bifariam.^ quaevis eft diamcM h^ 
bens ordlnataS) quas bifariam fecet» quibus Uceret w^ 
numerare ettam illas IL ia 6g* S) > qu^ dici po&n» 
crdinate afyptotorum altenri parallelas i^altera bi£}iitna 
kaxj juxta ^iun. 240 ^ <^vis babet binas cafl^ntes 
perimetri figuras «rdiQatisi p^rallels^ P^Vtt afympcoto^ 
rum ordinatas illas li 9 qua; nullam habcat nifi ip^ 
fa afymptoto coniiderata pro taogeote». cujus cbmaaus^ 
ita in infinimm recedic >. ut nufquam |am^ ^ : qu^vi^ 
diametrum flbi conjugatam habec paraUelam binis tan^ 
genttbus fi^urr per binos fuos vertices dactis » Demum 
«ant ia Ellipfi % quam in ea iigura quatupr tangentes 
per extrema puncta . diametrorutn conjugatarutn duct^ 
parallelogrammum concinenC) cuj[us area cooftantis eft 
magftitudinis , acqualis nimirum rectao^jlo^ fiib biBis 
axibus , juxta nuili. 469 » ubi illud eciam ad hujufino!» 
di analogiam accedit» quod anguli eius parallelogram- 
mi terminantur in jEIIipfi ad aQam EUip^m iimikm 
(nura.:}75^ , & in Hypcrbola ad afymptotos ( num* 
244) , quas patet communes cffe debere omnibus Hy- 
perbolis (imilibus idcm iiabentibus centrum C> & eanr 
dem directionem axium Mm, Xx <, ac eandem eorun;- 
dem rationem ad fc invicem, & ia eas debere dcfiac- 
re oinnes Hyp^rbolas , ubi axes evane£cant 7 uc adco 
illa; ipf<^ afymptoti confiderari poiHat > taaquam alia 
qusedam Hyperbola illi fimilis , in cuius perimetrq 
id parallelogr^mmum angulos babet termiaatosit ixc ia 
EUipfi. 

.818. At licet tanta flt buius figuras fimilicudo cum 
Ellipfi , difcrimen admodum facile deprehenditur vel 
ex eo, quod eadcm recu ei figare ia quatuor eciam 
punctis poflit occmrcre , ut illa Hh fig. 84 , qu? oc- 
currit ipfi in N, P^ p, //, praetcr quam quod nuUa c 
millc aliis proprictatibus, qu« vel ad.focos, vcl ai 
ordinatas , vel ad latera recta , normales ^ tangentes , 
ac alia cjufmodi pertinent in Ellipfi , locum habec ia 
ramis omnibus eius figura:, fed xitc applicata in biais 

taa-: 



I£>CORUM GEOMETRICORUM, p7 
Uitt32mm6do • Ula vero qtraKrcumquc apparens analo- 
gia» 8c %urarum fimllimdo rnde ortum duxit ^ quod 
ticct ipfas diametri fecundarise Qon (int ia tJyperboIa 
analogaB diametrisEllipftos, earum tamenquadratafutic 
aoaloga fecundo analogix ^enere quadratis harum » 
quifaus & negative fumantur, prorfus refpondetit. Cum 
ipfe diametri vi ejus definitibnis nuUo modo analogac 
fiat y haec ipfa analogiaqviadratorum demon(lraricom-> 
munt demonftratione non potuit defumpta ex ipfa de- 
finicione • ^ndet ea a theoremate enunctato Prop. 7 
&am. 351 9 in qua habetur pro tftraque curva , qua« 
dramm (bmiordinat^ cu)u(Vis diametri primaris efle 
fld rectaDgulum fub hitiis abfeiflis a binis ejiis] verticl- 
hus j ut eft quadratum femidiametri conjugatac adqua^ 
I dramm femidiametri primarias^ Porro rationcm ejus 

> «qaadrati ad tectangukim fub abfcidis conftaritem efte 
communi demonftratione patuit t^um. ^52 ) at eam 

\ caodem efle > quac eft quadraii femidiametri prima- 
i tix , eadem pro utraque curva demonftratione evinci 

non pomit > iM pro Eliipfi ibidem demonftratum eft 
. .cx eo , quod vertices diametri conjugata^ fimt etiam 
. ji ad eandem EUipfim , pro Hypcrbola repctitum eft 

1« numer. 256 > ubi idem longe alia aemonftrattone^ 
r petita videiicet ab afymptotorumnatura», fucraf dcmon* 

> itratom • 

9x9. CaBtcrum demonftrata jam ejufmodi quadrato- 

tixm analogia , ex qua conftat quadratum ejus 3 qa^ 

dicta cft diameter fecundaria iti Hyperbola , effe ejuf- 

^em magnimdinis , ac eft quadratum negativum vcre 

fianalogum quadrato pofitivo diametri conjugata; Ellip- 

y quascumque in ElHpfi pertinebunt tlon adj ipfas 

iametros conjugatas , fed ad earum quadrata 9 erunc 

lommunia Hyperbolx , dummodo in hac quadra- 

im femidiametri fecundaria? fumatur negative » quod 

lane , fi ipfa fecundaria diameter effet analoga diame- 

o EUipfeos , pofitive fumi deberet , cum nimirum & 

icivarum^ dc negativarum quantitatum quadrata fint 

ifitiva • Fit autem idem , uc ubi de quadratis agitur , 

Bofcavifih.TQm.III. .., ^ aV 



Ii6 DE tRANSFORMAtioMt 
illcero jam ncgatlvc acccpco i fummis j^m rdpohck^^ 
diffctcnd;r > quod id fcqucntibiis cxtmplis liiaQifcftQai 

.f ric i . . . . , . 

Siod In EUipfi in fig. 10 quadracum diftaiida^ CF 

V tc^^ ^ ccntro fcquatur (^ nuir. 64 ) differeiltis qnadrft- 

' ^ torum fctniaxium C M « C X > ac iii Hypcrbola tii 

^ /^. id/uramc • Scitiiaxis qUidcm Hypctb(A«i ttanfter- 

iUs iUc finicus Mfln eft aiialogiis feittiaiu iranfverfo £1- 

lipfeos i fed fccutidario atialogijif gchere i idtoqut rt- 

fpedil ipiius iiegatiVus; Ac pofitivum adhiic manct e}us 

qiiadtacum * Scmiaxis conjugatus iUius terminacas ver- 

tice X tiocl eft ahalbgus ullo analdgl^ genere .(ciniaxi 

conjtigacd hujus > (ed illi rcfpbnckt imagiiiaria > atqoe 

impbffibilis quantitaf ^ cujiis cameti quancicakis qyiklri^ 

tuih rcaie sequacur femiaxis conjugaci^ qaadtato ticgiti- 

ve fumpcb^unde fic, uc ubi ipfius CX adlubecur qui^ 

dracmh inElIipfi ^ fubffitui pbflic ih Hyper(k)U fuae^O^ 

Suadraciint tiegative fumptura, qubd eric idem» ac r^^ 
t;{hgiilb MFim Eliipfcos analogum y fcd hegativuih 
Hyperbol^ re^angulum MFm fubfticUcrc ; Ac ut qua-' 
dt^tum CF in fig. 19^ cft diffcrcncia qaadraci CM, Sc 
tcdlianguli MF^ > in figur^ verb io fumma eeruflr 
dcni,^ mutata nimiruiii dirc(5tiohe red^ahguli MFM.ob 
^F mut^tam pbficione,' quae duo chtbremac^ apud £11- 
clickih rcfporidenc pfopofitioni 5 & 6 Libri 1 ,- lied tn^ 
vcra rice confidcrsfta Gebitlecriae indok » uniciun cbieo- 
tein^ func ; ic^ eciam' ibi differchciae, faic fummas qutf- 
«Iraiofum CM 9 CX acquamr illud idem euadcttuui 
CF. 

821. Eocfem prorfus paiflo cum in Ellipfi fiimnia 
q<i;idracoram iemidiamecrorum conjugacarmn , aequetor 
f umnix qiiadraforum axiura , ia Hyperbola asquancur | 
uiter fe eoruridem quadracofum differcncias ; quod 01- ' 
fiiirum quadraco diamccri conjugacx EUipfcos itfpoit 
dec in Hyperbola quadracum quantitticis' immaginanii 
fcd ipfum reale f 8c acquak qu^dracd fcmidiamctri co^' 
jugatac Ellipfeos ne^atiyc fdriipco. 
$2 2« Quod parameni, fcu laterai rt^ oraniuin dia- 

nctro^ 



ibCORUM GEbMETRICOatJKl . , ^f^ 
jji^trorutn in Ellipfii & primarioram in HyperbdlaJiflC 
prorfus analoga , & quidem piimarid ahaiogias gkht^ 
re^func auccm 9 uc jani vidcbimus > ac proitidc pitdJ 
pieUtes dmtics dominuncs habcanc i & cbmmilni e-> 
nunciaciotie > cdnftrudtione^ dcmonftratioric ubiqucg^U' 
Ucaat 9, ex had ipfa qiiadraci femidiamctfi colijiig^taf 
flc^acivc Aimpci confidcratibiie oninitid pt^ofluic ; Latus 
redum cujufpiain diametri diximus geheralitet ( num. 
jjfi } tcrciam coHcitiue prdpbfcibnsiiem pbft diametram 
,iilaii)ii 8t ejus con;ugafam i A^ ed feduci etiatH lacu^ 
jpridcipale i cdnftac ex niitn; 66 i cuni itide F^atedc > 6c 
ia Ellipfii & in Hypefbola ipfdni ctte tcfclum pdft a-^ 
iern tfatirvcrruiii^ 8c cdnjugacismi licec ubl ipTum defi- 
iiivimuj] n. 54 i ufi iFuerimus ea prdpriecacei qiiatnha* 
bet cbinniutiein cum laccfe fecio pfinc^ipdi Patabtlai 
tarencis axe conjugatd i qUbd tiimitiim fit ciibtdd axi 
pefpendicularis per fdciitii dui^ : Redtahgiilutn fub 
diamecrd pfimafia i Sc laccfe ftd:d idbci\ ^quati ^iia» 
(itato diatiifetri fecuiidatias ; t^orfb ubt EUipfis iii Hy-* 
^rBolam abic^ quadfatum diatiktri cdn;ilgate {ediiida- 
ric Hyytfbols ipfiiis liegativc fumptutti eft atialogiini 
^uadfaco diain^tri cdnjugacsel EiliErfeos^ Debef igiciit e- 
vadere negacivum illud redinguliinl i adedqiid . dcbiec 
tvadere Qcgaciviinl alcefiioi cantummodo e binis ejiia 
Wibus ; Evddic atitem ne^aiiva diamccet prithafii i 
^{|U« tiiinirum fefmitiatur ad ramtini oppoficum . Igitiir 
latus redtuni dcbet adhuc feitlaticre pofitivuni i qiiod 
•iiua GOFnnedatuf ttdn ccitti ti qUantitaie imiigitiaria i 
V^ in Hyperbola refpondet didraectd eonjugacaf £Ui{>^ 
/cosi kd cum ejU^ i|Uadracb reali $ tciU elt 4 
^, taj* Pocerac quidcm fic eciam dcfiniri i ut' e0<:c ^uat'^ 
tiiin poft fc^kadgiiliim fub biiiis abfciffis» qusidracum fe^ 
.miotdinacs , ac diametrum primatiam i cuju^ ea.ef{ 
ardinaca > & itl hac defiilicidn^i i t{\xx todtai tedic i 
l^hil aflutrerctur , qudd noil elfcc hoinbgcacum , 8c 
'feale ; Abcdiit auCciU iri riegaclvos pfiitius cerhiinUs bb 
aJcciam abfciffam, ac cctcius ob dlaxrictrUm primaria;n 
ijkuAik^ ui ria^ivasj ac prdindcl mancncc ^ofitivo fc« 

V a Can- 



iio DE TRANSFORMATIONE 
cundo termino, five quadrato femiordinatx, maQetef 
iam poiitivus quartus, ku latus re£lum. £o padtocjas 
poiitiva analogia in ipfa definitione manifefta eflet pei 
/efe • At quoniam ejus relationis ad. ipfas diaraetros 
major eft ufus 5 & eft multo fimplicior determinado 
tertias continue proportionalis poft binas red:as, quam 
quartas poft illa plana ; idcirco hic etiam (implicicati 
pofthabutmus analogiam^ ut fupra num. 7^7« 

824. Caeterum latera reda communibus gauderepro- 
prietatibus in utraque curva» plurimis exemplis patet 9 
qux elementa ipfa perpendentibus pa/Iim occurrent • 

FiyjEodem pa&o latera reda principalia determinantur 

2S0 num. 54 per chordam axi perpendicularem per focum 

jS6 duftam: codem pa£to num. 464 definitur in fig* 17; , 

1S9 174 ex dimidio laterc refto principali VO fubnorm»- 

lis RM a^qualis RD : eodem padbo num. 475 in fig. 

iSo^ ac 181 illa PT 9 abfcifla per perpendiculum MT 

du<%um ex concurfu normalis cum axe tranfverfo in 

radium foc! » a^qualis dimidio lateri redlo principali ; 

eodem padlio num. 495 determinatur in fig. iS6 , 188 

cx quovis latere Tcdto VA quadratum femiordinatx PR 

^ sequale re<5bangulo VRL : eodem pa<£to num. 50:; ia 

f' fig. 189, 191 chorda VH, quam circulus ofculatorab- 

icindit ex diametro primaria du<9:a per pun<5tttm ofcuU> 

irqualis lateri rcGto ejufdem diametri . In iis omnibus 

tc enunciatio, Sc conftru<tlio, Sc demonftratio commijt- 

nis eft. 

825. Quoniam vero ipfasdiametriconjugatas in EUi- 
pfi , & Hyperbola nullam analogiam habent 3 habem 
nutem earum quadrata fecundariam s tam in demon« 
ftrandis theorematis*, quam in folvendis problematts , 
quas pendcnt a diametris ipfis conjugatis, proderit fxpe 
ad earum quadrata recurrere , quorum ope communis ; 
quandoque invcniri poterit & enunciatio , & conftm- 
^^io, ac demonftratio • Exemplum theorematis defuni 
poteft ab illa area parallelogrammi , quod in fuo aih 
gulo continent binrc diametri conj^gats, & ciraia- 
fcribltur Ellipfi^ ac infcribitqr figura; l^perbolica? 40- 

mo- 



LOCORtrM GEOMETRICORUM- jii; 

f&&tm\ I qux area conltaater (quatar i'edangalo fub ; 
fcmiaxibusk Nos aliam lejus demoaftrationem aedimud \ 
pro Ellipfi num» 375 5 aliam pi*o Hyperbola num. 244» 
qaartunxilla prior ad Hyperbolam) pofterior ad. EUip^ 
fim cransferri nullo modo poflunt ^ atque id idcirco , 
qaod in iis nuUa haberi debuerat analogia . Demon^ 
flrationeiti communem nonnlilll exhibent dpe tangen* 
tium> qu^ proprietates communes habent . Nos etiam 
num. 469C6mmUnem ejus demonftrationem haberi pof* 
it oftendimus petitam ex alio communi theoremate 
propoiito num.4665 quod nimirum in fig^ 171 > & 173 
rcftangulum fub pcrpendiculo CL e centro in tangcn* ^^^ 
fcm, & fcmidiametro colajugata CI fqucmt rciftangu- ^^^ 
io fub jjemiaxibus • Id vfto idcirco iSeri potuit > quia 
Imm. 467 in ejus theorehiatis demonftratiotie inven^ 
tiim fuit) aIteraando> quadratum CL ad quadramm fe-^ 
miaxis tranfverfi CV, ut quadramm femlaxis conjuga- 
d CD ad quadratum fcmidiametri conjugat; CI • Qua^ . 
drata adhibita fufir > qua: funt realia > Iicet negativa 
iint. Quadraltum autem femiaxis (^onjugati CD, & fe^ 
midiametri conjugat^ CI in HyperboIa> (i neg-ativ^ ac^ 
Cipiantur , funt analoga iifdem quadratis pofitive fum^ 
pcts in EUipfi , Sc ratio inter ea negativ^ fumpta eft 
cadcm» ac inter ea pofitiv^ fumpta ^ quam ob caufam 
10 lateribus ipforum pofitive confidcratorum ) qu^ tcz^ 
lia funt» mat^fit ratio^ licet in iis non habeatur analo^ 
m . Id femper accidet ^ ubi proportio aliqua complo» 
^tur in Ellipfi binos terminos, qui in Hyperfaola ma* 
)6cant analogi, &: binas diametros, quf in Hyperbola 
fiani iccuhdari^; mancbit proportio, fcd in dcmonftra*. 
itione recurrendum erit ad quadrata,],& ad hunc ipfum 
^feurfum , quem hic inftituimus * 

8a6. Problcmatis exemplum cfle potcft illud , quod 
fkum. 4J6 propo^fuimus i ubi dc quadratuni fcitridiame- 
tri conjugate analogum fecundario analogig genere 9 
& latiis rcftum primario analogif genere anddgum 
adblbuimus i Ibi datis in fig» 160 ♦ 161 binis diametris Con- Fr6o 
jugads Pi» , I* i qu^rebantwK^es . Poteranii qu?ri bin? 16 1 

rcd^ 

] 



/ 



%%t DE TRANSFORMATIONE^ 
fcffiap einrinodi, m qpadracornm fummaj vd ditkx^ 
fia arqiiarefur fummar, vel differenria; quadratorrim dii« 
ti^um CP^ CI , Sc recfsngulum rect^gile fub altcra, 
m Cr , 8c pcrpendiculo c% alcerius vertice demiflro ia 
ipfam^ qiAbu^ definicur conftans parallelogf ammum ; ^ 
folucio ncc obveniQef communis , ncc ita expedita • 
Suftulimus beterogeneas illas diametros eonjugataS) 9^ 
iUis fubilimimus paramctros , qux daris diamctris pri- 
mariii » ^ coniugari$ daneur, & cum dcKmr perqua^ 
drata diametrorum cun-|ugatarum analoga^' funt in utra-» 
que cutva , & quidcm primario analogia? genere ^ ul 
vidimus num. Siz . Eas combtnavimus cum diamctris 
primariis iridcm aqalogis, fcd fccundariogencrc» adeo* 
quc negarivi$ , Idcirco npplicata PS a»qu^i dimicli^p^ 
rametro utrobiquc ex parte curvs coQvexa , adeoqu^ 
ucrobiquc ad^^nd^m pla^gam , nimirum in EUipfi ia 
fig. i6q in dir^um cum fP, in Hypcrboja in kg,.i6x 
a P v^rfus pun^m f mutafum > unde provcnit CS 
fumm^ ibi fcmidiamctri CP> &c dimidisp paramctriP^i 
qux iii iUa pofitiv^ funt ambc> diffeccntia hic altcriu^ 
pofiii vx ab altera negativa ; commuais profluxit f^ eoa^ 
itrucuQ , & dcmonftrtrio, 

817, Quoniam autem diximus num» 8o8, intci: bi- 

f nas rc^las altcram pofirivam, altcram ncgarivs^m uou 
poflTc invcniri unicam mcdiam proporrionalcm > poifc 

r auccm duas magnitudine quidctn a^qualcs fcd dirccrio* 
m comrariasj non crit abs rc proicrrc cxcmplum, 1« 
quo bin^ fcmidiamctri fccundari^ inHyperbola afibm- 
ptar cum dipectiouibus coi^trariis fint mcdi^ proporrio' 
nalcs inter binas rcctas ^ quas tn Elllpfi ambap funtpo- 
iiriv;, Sc habcnt fcmidiamcn^um coajugfttam pr^ mcdiji 
proporrionali » qruarum famcQ ftltcra dirccrioncm fcr- 
vat in Hypcrbdla, ^tcfa mutai , ubi ca proprictas fc-' ^ 
inidiamctri Ellipfeos ad Hypftbolam transfertut ku)Uf 




^ to'a eflc m^iim ptoporcionolcm intcr abfciS*^m a 

tro 



LOCORUM GBOMETRICOHUM . ; 2 5 
taCR, & diftaaiitoii CQ, coQcurfu? tangcmis PC^ 
cum ^itta diait^ecTQ , ^oni^irnus autcni ipfas CR ^ 
CQ,3 qn* iq figura ij? , & 15^ , ubi num, 419 a- 
g^btmr de diarnctrt$ |lH{»reo$ , ytl de diatricttis Hy- 
pcriwl? pfirtta?iis> faccbant M caiudcnii ccntri^ pirtcm » 
debere in fig. 1^9 |aceit ^ partcs oppQfitas . Niuii* 
m ia diansctris priiKia^riis in fe, J53 > & I54 crat 
CI^ a(( CY , Bt Cy ad CCL» & cju^drawra CV ^- 
quafc Dccii^rigulq fiib CR ? &' CQ^* 'rx fig. ;5$ qua- 
rfranwi CV tldgatfvd foinptum fcfoondct quadrato CV 
fifr 15 J s> 154', Hinc rcqtangulum fub CRr » 5f CQ^ 
^kuii cffc ncgativum in fig, 15^1 rcfp^ttu jSg. 153, & 
f54f adcpque cun^ tti itlis candcm litraquc cfittaionroni 
w^teric, in ftac dcbcnt babcrc coi^tr^ria? ; nec crit > 
fi pofitiq ftifttn fpc^^tur^i CR ^ CV, ut CV adCCt; 
ftd C8, ad CVji ut C« ad CCL. U a^«cm ipfumcrui, 
pir cx: finc dtmoiil^ratioms^ pofit? num, 416, Inyctiitai^ 
«ttim CR ad CQ^» Ut quadratum CR ad quadratum 
^y f Primui tcmunus ^ ^ fecundus babcnt^ dirccti^* 
^ coHtra^iam , adcoque ^ tcrtjus , ac quartus de« 
^CQt bab<;rQ ^oiitrariam it:a ^ ut quadratum CV rcfpc- 
Wi pofitivi quad^ati CR pro ncgntivo babcndum fit , 
f*c pro produ<fto cx Cfh & CV , qu? fint vcr? mcdi<5 
««r CR , CQ^? fi dircctia fpcctatur . §c.d cum ipfa 
^rcftio ibl coofidcranda noi>is non cflct , & fol? ma- 
Jttitudincs fpcctarcntur , qu? in GeoiHctria communi 
ufum habcnt ; diximus fcmidiafnctrum ipfam CV mc» 
*W intcr CRi CQ^ibi, ut in diamCtri^ primarirs, 

*2j. Hand niultutn abfimilc ab co cafu cft illud , 
quod di^miR numi to9. in fig, 34« fi fumatur BGp^ 
f^ mcdia imcr ABj UD CQofidcraias ut pofttitas» mu- ^ 
t^la AB ia ncgativam. AB2 , forc non B^L tcrn^ina- 
jam gd H^pcr^iani me^iara inter ABz^ , B^D , (^d 
^^^ il}a$ Ba L habenics difcction^s oppofitai. , altc- 
^^ pofitiyam , altcram, jicgativam , forc mcdias . Si- 
«^ilc quid (^abetitf ^tiam» fi maor quwcdam rclatioquf- 
^^ inter jnventioncm Locorum Gcomctricorum ; ad 
^^ (<!i^nuaatur vcrtex tri^juli faabcntis bafim datam , 

'V^^..^ cujiis 



|l^ DETRANSFQRMATIONE 

Qi^ias.angulorum ad baiim fumma^ vel differeAti^ ^ 
^^r angulo dato, quotum Locprum nu«36^ invenU 
ipus primum efle circuhim » fecundum elFG; Hyperbolant 

F272a?quilateram . Sit in fig. 272« recta data Vu^ fiat angulas 
^VI azqualis fumm^, vel difFerentia; : tum methodoibi- 
4^m expofita fiac circuiusyvP»^ cujus Yu chorda y IWi 
langens, & Hyperbola aequilatera SVT co tus y cujus 

^ .--diamcter V«, tangensp;ti:iterIVi: ac arcus cii^duIiVP^Ni 
& Hyperbola VT 00 tn^ egrefli ab V ad partes I, ac 
deiinen tes in u^ exhibebunt ille fuitimam, faic difieren- 
|iam sequalem angulo i^VI 1 reliquis idem exhibeniibas 
l^efpectu «Vi# x 

,. ^29* Dcmonftratio pro tlypcrSofa i^t expil^iefla eftfau^ 
lufmodi . DuGta ordinata PRj» parallela IVi , & recris 
VP, «P^ erit f num. 260 j quadratunl RP aequale re-» 
ttangulo VR«, adeoque VR ad RP, u^RPad R«; & 
[^roinde fimKia erunt triangula VRP^ PRm ob angulum 
%d Rjcommunera, & angulus R«P , fiye YwP squaiis 
VPR, five alterno PVI i adepque d^iflferentia ipforum 
5V»> P«V eadem,acPV«, PVI, fivedatusangulus«VIi 
qus demonflratio eadcm efl*et in crure ut 00 ^ (him tan-> 
gens per u debeat efle parailela tangemi IVi y -Sc contH 
liere cum uV angulum asqualem ipfi «VI ad. partes op* 
ppfita^, nimirum ahernum« Porro in circulo ductispft» 
jitcr VP\ «P*, angulus PVl chordae Cum tangente ac* 
qjuatar angulo |'«P' in alterno fegmento; adeoque bi-^ 
ni PW, PV« aequantuif foli «VI, tK opOrtebat . At fi 
illa demonflratio Hyperbol^e ad circulum fitstransferei^ 
da, ubi VR' acquifivit direcrionem con«:ariamVR, & 
m negativmn abiit, non erit jam R'P|media interVR'^ 
K'«3 fed binat R'P^ Rjp', quarum altera dkectionefii 
habet alteri oppofitam» erunt medias • £t quidem Aiot 
cx natura circuli, in quo rcctangula VR«* P'R';'3cqtta* 
lia fiini, adcoquc VR' ad RPV ut R^ ad Kui & 06 
#ngulos ad verticem R'oppofitos2equales, anguIusR'PV 
fivc P'VI arqualis angulo R'//f V five ob arcus VF, V/ 
ihterceptos a cborda tangenti parallela aKjuaks > aBqo*' 
c ^ ; lis angulo VuV\^ ut oportebat^ 

$3P ^oi' 



iSDCORUM GtOMmiCCmtJM. ttf 

' it}9* Pbrrohiac aliqaando fieri poteft> Hc adquonja* 
jiian ptobltmztam refcdadonem » qus videnmr unicuttl 
concinete problemaj refpondeam Loca Gcbmctrica dh 
verfs prorfus natdrx y qux diverfi^ eorum partibus fa« 
dsfaeiantj (ingula iingidis . Satb quidem eft manifo 

fiom id dcbere contingere ^ Ubi pofitivotum y Sc nega^ 
oVorum ratio non babeatur • Ubt in problemate pto^ 
pofito num. 676 quaeritur in fig. 23^ fumma fegmen-JFa^:^ 
corum MN , NO, qux recta data £F intcrcipit intet 
ie, & binas parallelas AB, DG datas e recta ductaper 
punctum P datum v fi ea recta debeat occurrete i^ect^ 
£F iti Ni ihter parallelas ipsas , folutio eft admodum 
expedita, quam ibidem dedinms-^ ope foliu^ reccae KI* 
& PN ipfi parallelx • £adem cdmmutlis eric etiam ^ 
ubi puncmm N cadatextta inNai vei Nj, dummodd 
mutaca directione recta intetcepta habeator pro #TCga« 
tiva • Nani fi nulia hegativorum ratio habeatur , tC 
quaeratur reaa ejufmodi 3 in qha MiN^ » 8c OiN% 
£mul fumpta; squenmr rectq datae , probkma erit al^ 
ti/Cmum, & curvas fublimiores requiret . Si enim cx, 
puncto P2 ducta quavis PtOiNa , fumatut NiR fem» 
per a^ualis, & conttaria OaN^ , baud difficulcer de« 
iQonftratur ^ punctum R fore.ad Hyperfoolaih tranfin 
unccm per Pa^ & habentem pro afymptotis bin^ re<» 
irtas parallelas ipfis £F > DG 9 quarum prima] citra £F ^ 
iecunda ultra P jaceat tantundem , quanmm P jacet ul' 
tra £F ^ vd DG • Quod fi ducta per P2 quavis recta Pz 
M3, iii ea fumatur femperMsR recta asqualis dace fum« 
fUf ; puntmm R erit fempet ad Concoidem axe AB*i polo 
P2 defcripum , cum ea ipfa fit e}uscurv^ notio^ dequa 
nobis alibi agendum erit . Quare ubi cx bin; curve (t 
iecuerinc in R, habebitur ex ptima N^R acqualis Na 
O2 , adeoque MsR fumma ipfaruhi MiNi , NaO^ 
^uf ex fecunda erit ^ualis dat^. 

831» Ac ibi ftacim dignofcitur mutatio problematis 
cx fumma confiderata etiam poft mutatum alterumter^ 
minuhi in negativum. Vcrumprima fronicfaciliusquif- 
I^iani habebitprofimpUci ptoblema, quo'iafig«242> da-F.i^a 

ci^i 



S^ij DE TR ANSFORMATIONP ' 

f:t« iii recca indcfinita AD binis punctis A, Dj qu^v 

fur in cadem pund^ttna B ica» nt rcctangvdum fub ikf 

&ts (e)us dtfti|titiis AB 9 DB a pundis A ^ dc D asquoi 

tiir dato rectangulo. Ad ejujs\ gcneralem folutioncfn re^ 

quiritttr figura cdmpofita ex circulo > cujus cliAmeter 

ADv& HypctMa icquiltucra 5 cm)«s AD axis cranf- 

^fus^ • Si ex quoYis punao R dat« ttct£ er^atur 

perpcndicularis KS mcdia ihf;er tatera dati rc^anguli, 

& flpcatur ex S r^cta ipd dats rectae parallela , quai 

Mccffario 0cciu:ret (nnis ramisHypcrbolje in binis^pun- 

fcis L 5 L2 9 & circi^ttm rel fecabit in binis G ^ G « 

Vcl tangtc in uQico y vtl ^ritabit, extra ipfum delata^ 

prout KS fttcrit miQOir » itquaiis, vel mi^|or circiili r^t 

tiio^ iive dimidias AD^ &c bccurfus il(icum^ %ura cm« 

|flf<»ti(e cx iis binis locis fohr^nt prob}em^ , Pestii0is 

(Bttim pcrpcndiculis Lfi y (3B » que crunt ^qualia ipfi 

$R» erit ( num, 6ts } fcctangdiuin quodyis ABG^ ^ 

qaak qiiadcaio fvat BG > Sve BL > adcoque quadrato 

^R > 6t reaangulo dato • Idem autem proU^ma pro« 

poni pc^et etiain faoc pacto «. Invcntrc duas redpro- 

cas bmi^ cbtcis , quarutfi demr fumma , vd diffcrq^ia * 

l^ani data AD ^ft fiimtna A^^y BD, &: diffcrenua ABa« 

BsD> vcl ABj, 9jD> ^ gtterum dati rectangidi latiit 

^ft ad altcram cx ipHs > \n earum ahera ad ejitfdcm 

rectangtji btos altenam ^ yidctur probfeaia (micMmcA 

fc utriiinque 3 cum fummas in difierentias mutet» ma^ 

cata direc^ioM AB m AB^ * Scd idcLcvio ouijunic di« 

viditur in ^m^ pAter fe diverfa , cuin ABx , Cc^ B^Di 

npn pofiit cfle mcdi^ inter illai[ ipfas, imer quas mc* 

diig funt AB> AD : nam ia, pcopornpac (Bias caDtum 

krminiis directioncm mucaoe non poteft ( nuniw 777^. 

Quadratpm qupque eidem RS xquak > ic femper po« 

fkfvmn, asqualc cffcnpnpoceft ntriqii^ rcccuiguio AH)) 

AB2D> nifi fupppfitio p<>&civoruip> adcoque unita^^o* 

Uematis muvrci^ ; cum alterum ex iis ccctangulis tef- 

pcau alterius > raaueate unica iuppoiitipn^ > pegaiit 

vum eifc debeat. 

Tz7^ S^z Bina paritcr Locs^ Geomcjricji ffgurif ^72 fcU 

vunt 



•« - 



LOCORtJM GEPMETRICORIJM • iij 
ym binos cafus problcinatis ^ qui iimul ad iinicinV' 
problema pcrtin^rc vici^rcntuic j ^ (amcn a4 duos p^f 
tincnt intcr fc divcrros « E( quidcm i^ prpblcma cri^ 
quoddam confiplcmcntum corum » q\jx iti Conicaniti| 
SediQnum clcmcntisdcmonfiravimn^ dc figurarum (iml* 
JitiKfine an.iS. Sit infig.zjjfigura/^dircdc, vdjSni* 
inyierfc fimilisiSgurapFABs &qtt9^ratur> an babeaof ali^ 
qao4 pundum P» vcl P* » in quo t>ina bomoJo|;apaiv^ 
h cocant» iivc quod fu pun^um bpmplogum com^ 
miine ^ Ad id mvcnicndum. produ<^a 4/9 donce ocqir» 
rat in V rc^ AF prbduAa? indcfinir^ in I > iumati^p 
y» in cadcm dirc^ionc rcfpcftu f4% in i|ua cft FV X€m 
fpe^ FA , quae ad ipfam FV fu in ratiouc , ia qii^ 
fmt latcra faomologa ^/, AF , $c patct punda Vh foif 
homologa. /am ut pund^umP, vcl P^commune fit, (i^- 
portcbit, rcft» VP, PV, vd irP% PV fmt in: iUa ea^' 
dena ratione » ac anguli IVP, WhV ad cafdcm plaga^' 
in fimilimdinc dirc^la » JVF, VkP' in iavirrra ad ap^ 
pofit^ , «cqualcs fint intcr fe . Quarc fumma augulq»^; / 
rum m, VtiV, vel diffcrcnda PV«, P»V dcbchit ff» 
ftjequalis angulo IV^ dato, qui cftfumma 4ngufcf\)w 
PV« , PVI3 & diffcrcntia Wh ; PVI . Si igJmr wan 
ftruantur bini Loci Gcomcnricij al(cr, adqucm expun^ 
ftis V^ duftx rcaac hP, VP, v<1 «P*, VF fini in Ub\ 
fadone data /^ ad FA? altcr» in quo fumma angvdo^ 
nim PVn^ P»V f vel diffcrcntia PV«, PkV ^ucturda^ 
to angulo IVn, occurfus cjufmodi locorum folyct prp- 
Uema • Porro patct cx num* 28 primum loavn babC'* 
ri, £ in rc(5ta hV produda datis t>ini$ pun^i^ V^ 4^ » 
flternis proportionis armonicx , & rafionc /l . ad FA 
ipfius propqrtionis, invcniautur rcliqua duo 6 f D pcc 
^m. z^y Sc diamctro BP dcfcribatur circulus BPDPt' 
fccundum autcm ^ patct ex num. %6t » fore pro P ^^ 
CHin drculi VPi» habcntis VI pro tangcntt , Vii^ pr# 
Aorda , & pro P* crura VT ^ fu Hypcirbote acqui*' 
later^ babcntis Vh pro diamctro , & ipfam VI * pro 

tangcnte • Qj^arc patct > quo fzHko problcma conftru-^^ . 
wdura fitt 

SSj.Por- 



V 



«8 DE tRANSFO.RMATiOM£ 

Ijii. Porm videbatuc, pro directa , & iriveffa fo 
militudine cadem Ldca (Jeoiiietrica rcquiri debcre , 2i 
divcrfa obtigcrunt . Rcquii^ebataif enitn iri altclrd (aiA^ 
ixia 3 in altero diffcrentia ingulorum ad bafim ac*^ 
qaalis datse , quas ptoblemata cum iin 6g. 272 mu-» 
tent illam eandem RP medlam inter VR , RV in bi* 
»as Kl?- , R>' medias iiUet VR* , R« ^ mutata po^ 
iitibrie VR iri contrariara VR , Lo^i Geometrici re* 
qtdfxd mutarunt naturam . 

' 8:54. Ex tam cxpedita problematis COriftrUctidoe &* 
cUc deduci poteft, fcmper commune punctum invcni- 
ri dcbere iri binis figutis , utcumque fimilibus , idque. 
iinicum , fi inaeqlxales fint, ac iti cafu cqUalitatis pun-»- 
eto D abeurite in infinitum , circulum PBP* abire in 
tcctam » & P invcniri eipeditius , P^ ablre In infini-* 
tum : in cafu autem iriverf? fimilitudinis fectd bifariant 
angulo VP'« pet rcctam indefiriitam, candem, quie ni* 
niiriira cura tecus PV, P'« homologis sequalcs conti' 
nebit angulos ad pattes dppofitas^ fore Gommunem-po-' 
fitionc bomoldgam , in directa vero fimilitudine > (L 
non congtuant directione ipfae VP >. uP pofitione ho* 
molog^ i tiuUas alias per dommutie panctum ducta& 
Comraunes eflepdffej fi Vetd illgcongruant, omaes pcr 
ipfum ducta^ forc pofitidne (^ommunes , uc func omneS'' 
pcr F dudc? in fig. 33, & 34, cum nimirum novcho-».' 
F.jjmologc cum pr^cedentibui eofdem in eandem plagam 
34 toguios continere debcanc $ adeocj[iie intet fe eundcni 
angulura ^ quem illa^ ititer fe . Sed nes jam longius e< 
Vagacos fepcimiis^ Canon ad fe fe vocat . 

835. Canori. T^ Si in qudtmf ft^opornonis CHfHffiam 
tp^inis hirfis utriuslibet t^ationis maneant finiti >, rc^ 
H^HorHni autenh alfer , aheat in nihilum , vfl ita in vn- 
finitumy Ht alterum /altem eju^ extremum nufquam )a» 
yj>; alter abibitpatitir in nihilum^ vei in infinitumco^ 
dem paHo . Quod fi hiiiis extremis manentibus alttr tx 
emremis abeaty^n nihilumy velininfinitum , alter contr^- 
abibit in infinieum^ vel in nihilum , & idem in me-' 
diis cominget , fi bini extremi maneant : ac fi ^ ^««4 



lOCORUM GEOMETRICORUk. jsf 

eodem rtdU , quoddam reHangulnm finiio riBangulo aqua^ 
b mdneatj ac alterum ejfu latus aheat in nihilum\ vel 
in infinitum y alterum contra abibit in infinitum , vel 
in nihilum. 

836. Canonfe hujus partesomncs videntur adraoduiti 
manifcftsB • Adhuc taraen fic accuratius deraonftrantur • 
Si bini termini rationis utriuslibet finiti flnt Sc uhus 
przierca rationis altcrjus evancfcat , vcl fiat infini- 
tus , altcr ipfius non potcft finitus remaaere ; nara i^ 
qoi fupponitur evafifie infinitus, vel abiifie in nihilura» 
adhuc eftct finitus , & invenirctur ex reliquis trlbus 
codcm pafto , quo in Geomeiria , datis tribus re- 
ftis , quarta proportionalis invenitur * Porro curaeo- 
rum racio debeat cflTe finita , non poteft altcr ex iis 
binis terrainis abirc in nihilura , altcro abeunte in In?* 
finitara , ratio enira infiniti ad nihilura non finita ef- 
fct , fed infinitics infioita . Qi^od fi binis cxtrcmisma* 
nentibus finitis, alter ex mediis evanefcat , vel in in- 
finimra abeat, altcr ex ipfiseaderaratione finitusremar 
nerc non poteft , quodeodem argumentoevincitur. Non 
poflunt auteni fimul abire in nihilura , vel in infini- 
vm » ne eorum produdbura , quod xquari debet finito 
produAo extremorura, fiat nihilura, vclinfinitura. £a« 
dem autcm eft demonftratio, fi maneant medii, Sc aU 
ler ex extremis abeat in nihilum, vel in infinitum. 

837. Porro ubi altcr ex tcrminis exttemis , vel me-' 
diis proportionis ita in infininim abfolute recefilt 3 uc 
nufquam jara fit, verura nihilura illji refpondere debetf 
non quantitas quarpiara, qusc dicatur infinttefiraa ordi-^ 
fiis cu)ufcumque. Id quidem multo evidcntius conftabit^ 
nbi nianrfefto deraonftraveriraus , quantiiatcs infinitefii- 
inas, qax in fe ipfis tales fint, nuUas revera efle, fcd 
a noftro cogitandi modo pcnderc tanturaraodo , ut ni- 
mirum indefinite^ non abfolute iafinite pztvx fint. Ac 
kic etiam, fi nomine infiniti abfolutt intelligatur id » 
cujus faltem alter limes , utinredlaaltcrum pundhim » 
ka in iafininim recefiit , ut nufquam fit ; verum ct 
aihilura rcfpondere millc excmplisc Gcometria petitis facilc 

evincL 



i^6 bt f R-ANSFdRMAtiONi 

tvlricipdtcft; fa fig.25+ubieayRadyAi utABa(iR§;^ 

i^ijf^yfccumquc parva fit VR rcfpohdct ferhpcr alicui RP 

264 iiabehd ali^ucm tcrthiAutii P^ hcc Pita rccedit iii in- 

37) Micttnl^ hi ntifqtiam )am fit, hifi ubi R rccidac id Vi 

jiada ^J? dbfoluce iiifinita i 8c VR pcaitiis evaiiefcea- 

ie : Ihaiii accdratd dcmon^ratibne dftciifdra cft f num; 

14^ )i afymf^totuni folailiiti Hyperboia e rc&is omdi^ 

^m ipfi jpat^allelis hiifqiiani pctimetro occurrere; Sicet- 

t^ni iit ng. 264 vidimus ( huni- yiS ) e qiutuor CZi 

CH» ZLi HP proportibndlibus foluni cocuhtibus.om- 

iiihd phmSkis C» Z, adcoqde eVahcfcente pirdffus CZi 

jlbife ih ih^itutri HP ita^ iit P" hufquaiii jani fit; Iti- 

ifcm iti Hgi iii cdm fif CM ad CO» ut po ad CP j 

iadimas ndm; 720 ^ CP hoh excfefccre in infihiturh 

it^i ut hufquairl /ani (ici hifi CM pichitus evaaerceiKci 

^ ^bcat f iri Qi , /. , . , . ^ 

8j^;. Hujus theofethacis fircquentiflimus e/t uAis pec 
iihivtf fahi Geometriam , & in ipfis Conicarum Secbionuoi 
transformaddnibus eafdem rite conteihplanti fafpiflime 
bccufret ; Prima tjds pars ih quovis enam.ahguio rc- 
fai ^^^^^<i c^ m^ifcft^; Ihfig; 12^ cum fit.ER adRQ| 
* ^ I3t EF ad FV i hoh potcli . evariefccre RCX >■ vcl abire 
^g ih ihfinitunij nifiparicer ectam £R evancfcaci velabcalt 
ih Jhfinituiri ; ScCund^e pards cxcmplum cffc potcft rc- 
ftjPius difeiStficis ih infihitum ita ; ut harquam jaim Qc^ 
ubi Ellipfis ih circulucd abij juxca humer; io^; Naiu 
ftft iri jgg; 9 (num; 90' >l CF ^d CM, ut CM ad CE; 
Vbi liutem Eliipfis abif ih circuluimi debec foc:us F a- 
iire in ceritruni ruc,in dg; iS »* eVancfcence pforfus 
CF ..Quire dcbec CE cvader^ ptoffus infinita ica > uc 
Hufquam jani fic ; In ipfa adcem fig- 9 cum rcclangu- 
l^m fub CF^ & C£ arqucc^r^quiadrato^ CM » abcun» 
CF in nihiium,^ abic fimul C£ in infinimm;> quae cras 
^irs cctcii ; Paricer ih Hypcrbola ad afympcocos fdati 
fcftahguiura fub quavis abfciffa i Sc ordioaca aequaitf 
( tLiim» 227 j reccahguld fub aliis quibufvis . Hincoc^ 
dihacii thm foloih abit in ii\finicilm iti, uc cfus vcrtci 
tllfqitatQ ^uh fiit i cum rcci^ ifli ipfam afymptotunl 



LbdiRUM GE0MEfR!c6klrflf . 33« 
|bm ta CttQgriKQS i triD9quc cum abfdfla penlcBi ef^ 

• , ^1^4 OuKMi. t. Si Ima twRd y imi kd qmddmmfHn^ 
ihm c^mfffiehMt , par^di^ fiant i itbd /w»Shm Ud 
in k^mitum rectdu-,^ut nnf^&um )md fii i Mguhii vm% 
^^ ^iW /ufitri' in i^o .finitd rkiimnintej . coniindfMnt i 
Hm^cit ; nc^ is ; qfmm dier^ cpmituiai cwii ^dmm 
frodMffMi cenfiH diket ; ui in diuf reBMi definini ; Si 
vcro i contrmin concurfus iin in in^mtum reced^ » tii 
hufqfinin jsm fii; vd dhpdux ex Atiierd pme OMnejltnti 
k:d JittcirM nbtMt in duos reSos ; ilU i^ reltd ezuuluni 
p4arsUleldi , „ 

840; Hic ctiafri caiion cft admodum maaifeftus > 6c 
eo |am ikpC uQ faraas . Hoc autcrnt pafto facile dc- 
inonftrari peicft cx. prxccdcnii . Iri fig. 14.6 cx piiti^ , . 
E^ cUicatur rccti E3I ^paraUela AB ; quae, xz&Ti AEi"** 
bccfurtac iii I: Ercm< fimlli^ triangula £;AIj BCiAob 
j>traUdasi erique E?t ad EsA.uc AB ad BCi. Abcat 
jini se&a AEiJh AE^ pafaUclam BH: evancfcc^E^I^ 
^ocjue fiCi ^t i&finiti ; ncc cohcurfus Ci, ufquam 
\am crit . Angi^ aatenl .AC2S femper a^qikiMsBjACi 
altetiio cvaiidcet ; cdm ille cvariefcat i ac proifide fi 
pdfianir H iri BD prockiAa ulcra Ci y arigulas ACiH 
2^cccdct ulnra quofcuriiquc limices ^ ad duos red:o» ; 8c 
ceoferi debet id cas dcfiiiehs»' duni AC2B dcarefcit ul«' 
fik ^uofcumquclimites^ &cyane(cit. Ecoritrarid flcciri- 
ciscfus Ci. ita feccffit iri inliriitam^ iKnufquam', fit^^ 6c 
angiilust alter cfi. nullus, adeoquc altet deunit in duos 
icctds; binz tc&x dcbcnt eflc parallclaB ^ Si cmm niSA 
tflcnt» alicubi concurrcrcnt ; dc atigulos oonftitiitrcAt 
fcmos^ ac fimiil duobus rcctis arqualiM; . 

84X. CflSterufni hiac ctiam pMct , ^binis itCM rM* 
itehi»6us paraUclis^ cdncurfiuii abirc ia iainittt«i ita , 
iti &ttfquim fam fit .* Qiiod ipfar utCK^ue in irifiti^. 
tuin productat niifqtiam concuitam « ^iguiiim aut^ih 
^axaUclafimi nuUum efic cx parte finlta» patet juxta 
htivn. 681 ibidcm cx co^ qucM aiiguli ACaB menfufa 
aA fcmidiArciitia aftoum AB,* fiaD , 4«a{ abcuiite Ei 

m 



— -^ 



, «X DE TRANSFORMATIONE 

la £3 9 cvanefdt peoims > adco^ 8c aqgdos parik« 

lanun Cvadit omniao nuUos. 

842. Hujus canonis in traasfbrmationibos locoruni 
feometricorum ufos eft feequeotifliroiis • Sccoada ejos 

^ parce ufi £umus oum. 7^7 ad oftendendam analo^aiB) 
& continoitaiem tho^ieniatis > quo determinaiur an- 
gttlus 9 quem binx tangentes dv&x per exaetna panda 
chordse tranfei^os per cenniam 3 & cotufMes in dltc- 
dricc ibidem oootinent. lovcnimus enim ipfum ial^^ 
pcrbola» fi rice accipiamt^ ex portc ahera cvaacfizere) 
ex altera definere in duos re<3:os » ubi chorda cft ipfe 
axis tranrvetfus, & tangentes fiunt parallelx. Multot^- 
inen frequentius occurrit pars prima , quse pertinet ad 
reccflum pundi ia infinitum». quo ffpiffime v& fanms. 
Ejus ope invenimus num. 41 akerum. axis verdoem ift 
Parabola ita in iDfinkumrccedere^ ut nufquam jasniit, 

|f p ex eo nimirum quod /T. il> qu; io fig. 9 in EUififf 
^Q concurrebant in punfto t dccerminante puni^um m > 
evaferint in fig« 10 parallelac inter fe » pun<9:o i mtf* 
quam jam exiilente. £jus ope invenoim eft nom. 154, 
redias parallelas axi Parabdae, 6c num. 15^ rc&as pa- 
rallelas alterutri afymptoto Hyperbol^ fcmel oecocreflt 
perimetro, altera interfe&iMc ita in ^nitum recedei»; 
tc, ttt nufquam jam fit. 

843, Porro ejus itidem opc admodum Cxpeditetraii^* 
feruntur ad Parabolatn multae e proprietatibus EUip- 
i^os. In Ellipfi omnes diametri convergunt ad caitroit 
( num. 206 }^ centrum in Parabola recedit io infitt* 
tum, cum reccdat vertex m; quare diametri omnes ut 
Parabola debent cvadere parallela^ axi : 8c funr juml 
lium. 2o6 . Radii ^ qui ex altero foco EUipfeos incnD- 
runt in ejus pcrimetrum^ convergunt poft reflexioneni 
ad focum alterum ( num. 202 ) : focus alter in Pa- 
rabola recedit in infinitum cum pcotro^^ vcrtice l^ ^ 
tero •* hinc fi is focus concipiatur efie primo ille 9 ^ 
quem radii convergebant> tum ille a qao prodibinr) 
habebimus ibidem illa bina theorctruta : radii , qv nt 
Parabola cxeunt c foco, abeunt poft reflcxjonein pa? 



LOCORUM GEOMETRICORUM . | j j; 
r^Udi axi : radii ,^ qi(i advtniunt paraUcU axi , sqa^ 
vcrguat poft reflcxioiicm a4 focum • In EUipfi in "ng. 
«75 cxiftcnte VO ^J^tmdiQ latcre rcfta principali rqepiyj 
&2. OC ad ccntri|ni "ttqic^ detcrminat ( i^ijrn. 464? ) 
RD asqualcm Albnormali axis iRM t in Parabola abit 
'Ccntrum C in infinitum; evadit ergo QD parallala VH» 
Utcxhibct %. I7^r& proinde fit I^paequalis ipfi VOt 
& fubnQrmalis asquaiis dimidio lateri rc<^o principali : 
ita autcm fe vcshabct ibidcra. In fig. 186 cxiftentc V4 
sequait lateri redo, r^da M du<^a ad altcrum vetn^ 
«cm « dctcrmiQat RL, c^jus redangqlum cqm VR^*r 
quatqr quadrato re^iiiQrdinatae B^P ': abit ia ParabblaFiS^/ 
punftum »in infinitum : igtmr AL cvaditparallclaV]^, x<$7 
m cxbibct fig. X87 , & proinde RL asquatur lateri re- 
<Ebo VA, & quadratum femiQrdiqatac Rj? acquatur rcr 
dangulo flib abfcifi^ VR^ Sc latci:^ |:c(|;o ; it^ autcm 
{t rcm habcPc conftat ex 11.4^5» , * 

844. Hic rcccffus in infinitum mutat conftruftlo*''^ 
iies omncs, ubi punctum, adquod aliqua ducendacrac, 
TAit in iiifinitum . A^ con^rudio nova fcmper iadle 
dc(iuci poteft » in quam in eo cafi; niigra^ illa prioi;:/, 
Duo autcm funt cafus « Vcl cnim rcctas inde di^ccn* 
dx dabatur alic^uod aliud pqnctum, quod remanet/^ vj:l 
dabamr (bla directio , ut nimirum debueri^ duqi cx il-r 
lo concurfu recta cupiam d^ata^ parallela . In primo jca* 
fu rcs erit cxpcditiflima . Satiserit ex illa punctQ , q^Hod 
Kmmtt j ducere rectam parallelam iUi, it^ qua crai 
punctum> quod abiit in infiniti|m. |ii fcpiia4o aliqop 
m^tificio erit opus^ quo ante conftructianis traasform^^ 
tionem ^cterminetur aliquod cjus rcctas puncmm , quod 
remancat, yel diftantia ab ea r^ct^ , cui pavalleja fit i 
Sc rcs iterun? epdem redibit. 

845. £a cxcmplum pro primo cafu: data in fig, <i^ 
quavis chord« Pp inEIIipfi, ad invcniendam^diametruni* 
Cujus ca fit ordipata, fati^ cftYn. 209) duccr« infig.7iF,7X 
cx foco F rectam FA ip(| gcrpcndicularcm,.qua5alicubii7;^ 
«KTCurrat dh?cctrici in I . Incfe il per cenfruni C dqca» 
lav rccta , eai problemati faciet fatis ,' 8c ipfam iUani 

'i ^fc$vick Tom. niy Z ' chor- 



5H .b£ tRANSFORMA.TiGHSII , 

ehordltn bifariam fecabit itt R. la Parabola m '4^ 
cefiiriiiTi C abiit ifi iilfidinittl itd^ uc nuCq&mx iamifb 
kd rcmattef t . Qnzttd^s crlt cif I duQcrc tc&ssa., 
mxi parallelaiti5 qttdd it>ideni efl' f^iraeftitotn; 

S46. Secundi cafus cxeiiip^uiii defunicmus cx projbli^ 
inatc teftio genetali 1 qaod niiiti. 140 propofuimiis ^, 
€UJus folutio ad omn&s divefros cafus applicata toturo 
hund no^otum ^ clcmemorum of diiicttf do^is cxbibu^ 
fuxti ca^ qiia^ dkimus num. i66 i Sc 767^ tjcnetsliiS 
^imirtini ipfa fbliitid 'fallebat iii binis cafibus i rcdiarutii 
vidclicc'tpdraHelir4iiTi difc^^ici^ & ttanfeiintiuni per fo^ 
Cuin^ qiickl tibs cdcgit bini ipfi ptoblcmata paraculat 
tia fubftitiici^e ^ quas iii piriiilit^ Sc fecunda. piropofitiooe 
pracmi(imiis i ' 

§4^. Propofitidiits tetxia; 'problcma Jlurf.gericfalc cxii 
hujufmcMi i Daiis foco , difeShke , & rationt dnef^ 
mi/fmtc i invinire concurfum reEi^ daia cum Si^iom 
y C^nica . CdAfti:u6lid ptroblcin^tis erat luijufinddi • Stx 
''4*jii fig. 41^ AB ditcdtrix i fociis F.i re<5la dita Hi^ ^ 
qiia^ dirctStfici occurrat iri H . Aflumptd quovil piiaU 
&<i Li Sc ^tdi LG pefpetldiculari id dlrcdtriccm» ca-' 
f>iatur LS ad LG iti faciocic dctcfminatite data • Cea-f. 
tro L^ iiitcfvalld LS fi^it dirduius . Ducatuf rccta LC5, 
l^afall^Iat datae KH ddciifrciis directtrici iti O : tuni pc^ 
O fecta ;tOZ pirallela FHi quaj fialicubi ocCutraicir-^ 
ttilo in T» /, ducaiituf LT^ Lfi & illis parallelac c£ 
t fcctasf, qd^riim* occurlus P ,- p cum tccti data tD& 
tf an t quarfita puncta ^ : . . ^ 

Sj^Sj ^im yefd fi recfta datai nt pafallela dircctrta ri 
piiflctUiTi M ^kit fn infinuum. Hiilc fecta quidem^H^. 
cujus piiticmni F' adhiic feiiiaiiei 4 adbuc habcti^ <fc; 
cehdc^ cx f rccratit parallelam difc(3:fici * Scd evadfc 
firaul cti<*;m LO pafalleli dircctrici , luledque^puatbBr 
O abit ia infiniturii . Dtbct i^huf ctiom.OZ cvadi^A 
pafallcla dircctflci. Ai cjiis fiullum jam' afiud puocrir 
habeinuf ^ undc csi dud poflii/ Binas habcbat dcaetn^ 
nationel i *a? cran- qudd ducenda cflct cx^ ptmcttf Of 
alicrain , QMod dcbtxti cflc parallek HF v Utriqoc** 



<♦ '• 4r 7 • • • ^ 



ierCR,UM GE(»4ETRIC0RUM. ijf , 
ieritiinatip abiiti in unicum paralleltfmuiT) cum dtrectn^ 
ce. Eain igi|ur jam ducpire. hdri .« ^oiTiinius ., nec ejul 
op^ definiirfe ill^ .piincta X> f i & ^'^^^} .LT ; lii qiii- 
bus ducahturparaUelae FP) Fp.; £n {irifnuin.in priind. 
icafu incobitxibdi^ttp ; » v .. ^ ., .... * l 

if^. .QiW3<* iS recta HK tr^nfeat pcc foctini r j cvi^ 
fiiBifcct. aiigjialus FHKi adeoque &: lOZi Tranfibit igi- 
turO^pcr L, & LT>L/.abtbitfit in ipiam OZ, ac FP» 
tjf iis paralklae m ipfairi tlk j quani ^iiofi fecabiiiic , 
iadeoqiie occurfw? i11q$ cuiit SecHonif Conicas, ^nhi«« 
kp ; qiibs peir fua^ iht^ctnpnec debebant deterinina- 
jcc.> indefinitbs relini)iieai ; £ii iiicclmiiibdani iccundi 

fcaftis. •,-,,,, -^ r •. • .,.-■.-. 

850. Ut primb incoftnmbdo iiiedeatii!:^ , fit fig, 274^3^4^ 

tadcQi^ ^/4^^ QyairatHr LI perpendicularis OZ^ five iyj 

hujus. ieUftai^ua abL :. Do^atiir ip&ii, &c ,FR pei^peiidicu- 

lari^ HF ,. occurre^ H^ m. Ri tiiui R£ l^ei-pendiciila- 

kls dit^ciirioi ; FapM, conftabit s-to^ fiinilid triingula 

FRH, ILOi it REH ,.LG0 ob laita bmiiia paial;. 

iela^ fingysl fuigiilis : QiuUre ^rit FR ad RH i iii IL 

Sd td ; «c KH ^ .R£ i itt.LO illG ; adeoqiie e;|: 

iequalitate ordihata FR ad RE;» m IL ad LGi Pbrrc^ 

i^i H aijit id inAdihlni l 6c Bmi FH { KH paralle* 

li^ i icvadit if^ i^R .pei^pendicularii itiahi irectf datai 

KR i quac proiade 4zm etito Um 9 datb F i datiii 

ydein.R£i.& LG ; £cgb datur Hhht U diftahtiare- 

'^ss OZ a i:enfiro Lj.aua data; ctaci t>otbi:it i^ttii ipfa 

^i i & prbWematis.folutib lihC Mibit ^ Ih.fig; i75 fi^ 

pcfeta data KR paralJcK dircciKci . Dacatui^ PR ipfipct- 

i^hdicularia, qua^ prodncatur ufqiie id dii^ectricetn in £ .^. 

Faac^ . circalb , lit priiis i capiatU^ Jbt id tGi ut i:ft FR 

id RE; verfu.s partem utiamlibet in ipfa ^L,.ac peil 

iiikod Z£ pacarelia directridj ad c}iis. .^c^rvchrfius^ T s /. 

tiixA ciirciild^ fi qiii /uirit» diicitm i^adii ^T> li^tiiht. 

*x F rectaf FPi F> ipfis {)ataUclae I qiuic, folvcQt ^r<^; 

i4paia,. Eirittoiwi l^Prid FRi ut JUT id.Lli & pRid 

REi five PDyUM ad LG; adeoque fP ad P ) , uc. 

tr ,• vd LS ^dua m hubnt m^^tiniimiitc^ Psi;ct au- 

12^ tem 



' ;?<J DE TRANSFORMATIONE 
fcm LP flBqualcm LI cxhiWturam rectas LT ; LT* vt. 
I^, Lr' in directumji-adeoqiie folutionem candem. '^ 

851. At hlc ]am conftat 9 circulum 1 qui coHfitifi 
rtronem nobis fuggedic , ncceflTarium non efTc . Satiif 
crit centro F intervallo rectse , qusc ad PD y vel RE 
fit in ratione decerminante 5 invenire punaa . Id Ppipfunt 
f,p cft praeftitum in flg. 9, Ccntro F intervaUo rcctac, qu^ 
ad RE cffec in ratlone dctermiriatue, quaefita funtpun- 
cta Vf: uc aijtem ea intervalla fempcr proefto eflcnc > 
capta cft FVj & Fh ad FB inca ratione, & ductaBpcr 
E , & V , ac « reci^ il , ^G , qux exbibcne RQ^ & 
RO ad RF in racione cadem , adeoque quacfitis^^ntcrvati 
Jis opportunas. ' 

852.£t hoc idem pacco reineclium adhibimm CMfini- 
ctioni prob^emacis generalis nos perduxic ad hujas priV 
^ mi problematis conftruccionem adco fimpliccm , &eic«. 
gantem, & vero ctiam f^cucidam, qoaD SectionumCo^ 
llicarum nacuram &: varias formas > ac propriecatcs caii^ 
mulcas ftatim exhibuit . Poteirant 8c alia parari rcme-^ 
dia. Sed libuic ill^m inire mm> qua& fc prima obtZK 
lit , & ^uac docec, quid agendum fic, ubi dccerm&ift^ 
tio reccfl^ hos deferir ob punccum ejus a(iquod in infi^ 
nitum reccdens , Sc bina& deiermination^s, ut iiic, ia 
linicam eoalefccnies. Jani ad Caumcm 9, quo Sxathi 
^i problematis p^tebic conftruccco* 

S53. Canon. 9, Si bifM reot4 ex eodeme fnnct^ dif^ 
grefsitfi^erfenantury earfmangtdaeznanefcent; Unaalia^ 
qm iis faralleU erant Jingula fingulis » evadent inseiK 
fe paralleU , vel fariter fnperfonentttr ♦ ^od fi in ki^ 
nis trian^uUs jfimiHius vertex utriufque aheaf in tAfieh 
laterihus hafi fufferfofitis \ bina difiantia funeti in quod 
'^it vertex , quod functum fuecedit interfectufni latfi 
rum , a hinis extremis^ iffius hafis tam ad fe imju 
ffm , quam 4ul^fam hafim % Wint iarobiqu§ iH^^adem 
tmane, 

854« Hujus ctiamCanonis ratio eft manifefta. HeM 
cvan€f<bcncQ angulo binaruni rectarum, cvanfcfcit ag^ 
lu9 e^ram 1 ^^9 ipG^ fudC p^r^mcl^ ., Qiiar^ ee cnte 



V 



l:d(J6RUM GEOMEf RICORUM . i^f 
IVadtiht parallcl? jaxta Ctoonem S vcl fi fortc diltafl-» 
% carom fit Dulia^ fuperpoiifuntui: • Itl triangulis Verd 
m\% cum iatera fint fenaper ad fc invicem 5 iScadbaJlmia 
tadem ratione^ utcumque parum vertices diftcrit a ba^ 
fibur^][is> oportct] omhino etiam i u^i jiakn in ipfas 
^'dum) ratio fit utrpbique eadem » ne fcilicet akcr^ 
m ipfd vicrticuin ap{Hllfu tiTftletur per ialtunii ft^i 

85 5> Ubi conftructip illa generalis fig» 41 i appli 4^ 
6atiu iii fig. 48 id Parabola fectis paraiiclis axi j» cum 
ibi ( num; 154) congtuaiit pimcta GOtS > feclia Lr fu* 
perponitar. rect; LO ], evanefcente illius aaguld 01j • 
£ram autem in 6g, 4^ tectoe HK, Fp parallel^ ipfis 
LO 9 L^ ^ Quarc he jam^ parallel; evaduac intcr ie > ni- 
fi forte HK tranfeat &ipfa pef F^ lit Fliki • Inde au- 
tem confcquitur illud pun^tum p * quod pef tineret ad 
HiKt i H3KJ» abif e m infinitum ica, ut nafquamjam 
fit juxta mi54, parallclis nufquam concuffcncibus ; ad 
cx eocicm prorfus fotite profiuit fcceilus in in^nituiil 
alcerius ihterfectionis in fectis altefi afymptdto paralle* 
tis ia tly^fbola juxta m 1561 

856. Qiiod fi in ipfa fig; 41 tfanJeat 1M pdt1^\ fe* 
ct^ OL^OZjrLy LT fu{$efponuhtur ; ad paritef fupcfpo^ 
buotur HF,HK,F/, PF^ & conftructiohem gcncraleni 
firuftfantur^ ut diximus 9.849* At ex hoc Canonetum 
ettani id ipfaHk jahi tfanfeiitlte i^er punctum F, quod 
^it in bafes triahgulorunl JFPH « FpH, erunt fumchda 
pudicta Py f itai ut fit FP ad i^H , «c Fj» ad jtH » ui 
LT, vcl li liimifum L^ ad Ld< 

j «17, Porro id ipfum pr^ftttiftinlii!! iii fig. Jjf» & jg iff.^^ 
in qulbds faita datieftFCC? gcrentc Qvicci illiiis H i l^ 
Q^^ta fiint puitcca P^pica^ ut elfent FP ad PO^ ^ It 
^ ad ^Q^ ut eft in fig. 41 LS ad JL(3 % ^tpminodam 
JUitcm ftccidit^ ut in % ^5 « & 3^ iii^ ip(*a FV jam ih-^ 
venta in primb probienia^e eftct ad FC^ utitifig.4t LS 

fi LO j fcft enim ifai JFV ad FE, ut hlcLS adLG^ & ibt 
£ ad FQ|> ut hk LG ad LOi Qjiare fatis imi inve-^ 
oire puncta P, p ica, uc cffctLP ad PQ^, & jLj> ad pQ^ 
/n.ratipj^ FY ad FQ^; Id auten^ ttaciin patuit adm6- 

^ . 5£ i dum 



t^» DE TRANSFORMATIONK 
^um facJlc pr^ftari, fi Aimpti^ hic in directricc Q^ ^ 
Q^ argualibus QF , ducerentur rccta; pcr G , & Vjak 
per g &: V, quae juxta num. ijo folvenint prpbfeifaa. 
Atque hoc pacto opc hujus Canonis tJ^ illa gencr^i 
fonftructione problematis cunftructiq profluxit. 

858. Canon. 10, Si circuli radius in infinitum, abtMl 
ita 9 ut altero extremo. tnamnte ^ centnm nujquam \am 
jTt feripheria circuti ^ihit in redam lintani \ *& rt' 
Ba linea viceverfa hatenda erit fro ftrifheria Wf«^ 

/ inpnzti . 

859. Hic Canon ^hundc cfcmpnftratu^ , cft a nun» 
f^Jlyiz. Eruitur amcm e^am ex Canonc 8^ rnum^Sjf). 

Si enim in fig. 271 fcijtriira P iia in in^nitum rcccr 
(Tcdat, u; ijturquani jam fit, pian^ai^t autem guairyisiria 

"^ peripheria? punct^i A, I,^ C, trcs ^•adii AP, IP, Q? c- 

^Vadent paralleli, & anguli Afl , APC prprfus cvanc-. 

* fcent . Bini igitur rcliq^i anguli tani ad tafim M , 
guamac|AC, eyadent binis, fcqissqualcs; adcoqucoun 
db ifo.fc^Hfmurn triangulorum APIi *^PC , Cnt xqna;. 
Ics fiimiH finguHs , ficnt ^nguli flngidis, rccti^' «quat 
les . Quarc anguliAPr, APG aeqpalcs ficnt intcr fc t 

' & rccta AI fupcrpohctur r^ct^i AC, abetint; P^pctp' \ 
in AC , & jaccntibus pqnctis A ^i I > Q l^ dj??^^ 
Cumque id in oint\ibqs rc%uis pcriphcriafe punctis lo-. 
fum^ hii^bcvc dcbcat ; paKt omnchi jpciriptieriam >, qo? 
n^aticif in spatiis finitis , in unicam aibitc rcctana pcr- 
^et^dicularcm cuilrbet e re^s , pcr qiiEri c^qtrum ge» 
ffit iq ii^finitUHi.. 
f.i^ 860. fdfujus C%ixonis ufu^ noi^ fcmel o^currit in 4o-J 
24 ftris Cqtiicar^m Sccttonurq ]|Icmeniti$1 • Iq €gaTt ^i 
"^» oftcnfum ef^ ( nqtn. 100), in Ellipfi diftip^ti^ Tf c»-. 
jufvis puncu R eju^ pcriii[ictri a mcp F acqiiaFi diflaih 
tiaB p^rp^n^iculari PD^ pcriphcria crircuK dcfctip^ c»- 
trq ^^c^q ^q'?it<cro/fo(:q /;^*& int^ryallp'a^^^^ 
fi^ FQCu^/i^ Paratiola abit it\ ifl^itym , Ouaft^ 
«. 9\mVi\ lW« ta mw P?rEendiculveir\ rec|e FE , ^. 
|nit-«m i^ Ipf^ni i^ftiH^eam Parj^bole flirforiccm. Et 

■' pcrbf- 



. • 



tOCORXJM GEOMETRICORUM. 559 
pfrbofla aaaiogiam cum dir^ccricc Parabols notavImiK 
;cHarTi num. 192. ^ifgvjttt? i|lc m 'EUipG cavvis vcrsiw 
Ti ad (juarn pl^^Hi ifei fafct cja? ccntrum /, abic la 
'^lciitam, ubi / In P^rabbja ia ii^finittKn rcccdit\ idcm 
vcro, rcgreflfo / in FJypcrbQl^ in %. z^ cx partc op* 
0qfjta, convcxi^tcm ohvcr^ii .foco j, 
' ' 16 1. Eodcm pacto ctiam cijim dcmpaftratum fit na« 
^94 piQ Eliipfi in % ij, .6f pro Hypcrbol^ ia flg.H 
*Vccta:m*CA «^yctw c ccntrq Q ^ Conc^rfum A aot- 
malis FA ciim ^an^ntc ajquaf i fcqiiaxi qraqrv^rjfQ CM} 
jpatct^ in ii$/cQnciirfijim ipfum A normaS^ gpm ^a^^-pz, 
gcntc jafci^ in pcrimc.trp prcuU ^?rcnpti diamfffaMi» * ^ 
iqiii cogcjtjyfus A ii^ Pv^fec)!^ in fig. 65 ( num. J94 ) ,^ 
incidit in rcctam MA normalcm axi AF . Rcs codcm ^ 
"recidit. Circulus^ illc in EUipil \r\ Rg. ^3 obv^rtcrctca* 
"yitatcm foeof , ab^ynic w^n ipfinitiMn in, Pacabola t 
' fOrirct in fig, «J;j ii^ rectatp h>C\ f Y .pcrpcpciiQt^^-^m » 
V, & rcgrcffp » cx |>artc oppofita ?3? infinitq m 6fr «4 
' -In H^p^rbola, jam.ip(i F convcxitatem. o^vgrncw?, 
' ' 8^z. CapoQ.;'li. iJi H^iC reSl^ slterf faltem: nmt^'* 
^wr /f Wr^ if 4 I» infirimm sbenme , «r nufyiipn jam 
'* \/& 9 ifi^nU^ evaMnfy d^bint in i^a infinitoi^feri^ 
^ l^ dJfecHta ilUm ratiqne^m , #1^ ^^»49!^ ultra ^Hefcumq^ li" 
mites accejf^mtx^ dum in infinitum excrefc^^nt ^/ ^ fi 
^/^meali/e^ r^iid Jiterint fentffr i» eanm ratiene^ ^ iUis 
^^ixcrefcenti^HS^ in infinitHm , remaneAnt finita \ tuAehv^ 
4icckrrate eam ratijinm ifjpfff» n M qMm ^Ua tiljr^ iU^ 
fcHmiu^^ limit^ aeeefserant., RatU stuf^ ?. ^ ?^^ ^" 
(edent hinoi quantitaus , dum *» infinitum a^crefcHm » 
* '& qu^^ afteeut^ <icnf^i dihent , uhi j^ai ^nfinita Jint , 
- ffotefir efse ratio aqi^itatif^y ve^ inaqti^itM^ fiiifiit^^ t^ 
excrefcens , aut de^tfcent t*ltr^ V^fymvtf: limt<s; in^i^ 
'' (imen femper ratio, aqftalifatii .9 Mhi differ^ntia i^aanm 
finiita mAneat^ vej nulla^ &4ifm9fi^fff^ m^ehU 
^ finU^y V^l nf^lja; fi bim vfkl^. ^tmn^K ad idem gim^ 
\ ftum ^A aliis hinit fumtU ahierint in ^nfimtum^ ms* 
jkntihus his hinU ptnctis^ & aheuffte W .ptfinUum UU 
^]* immmi Ua^'ut nufquam jam /^« 



i{46 i>&irRANs|OJiyAf rokfi ^ 

SS^. PrilTia thcoremaris pars dcmonftratur a coiSlU 
r huitatis Itge » ^uam Ciim ^bi ubique tam fanctcGeo- 
Inetria fcmt, fcrvare dctet ctiam ibi i ubi ^iiantrtatei* 
in infinitiim ^xcrefliht , qtise ibi ctiami iibi 8t oculos 
faoftrosi, & iticrttcm fiigiuni, dcbeht,.fi poffibiles /untj 
ih co ftatu habere idi> ad qiiod accefTcraht uitira quof- 
cumquc liniitcs > uec pct ftltum illo uniCo momcn to 
tfcnipbris ^lidni tationem liaberc diverfani at> ea, aqui 
tempori^ pi:eced«ntis InteiV^Uci ^uam minlmd , difta- 
faaflt qudm mihimuih ; Fihitas auteni illse , ^qh? im ca- 
rum ratiohfe erarit 3 & remahcilt adKiic finita! , adco* 
que oinhino aHquani tationem habent , debcnt babere 
illamf ad qtiahi acoeSeraht ultra quofcumquc limitcs ; 
/8^4. Pbrro cjufmodi f^tio ppteii cfte acqualitatis, i 
cum pdfliht aptjtialcs perpetuo efle duiii abeunt in in* 
finitum ; Immo ciiain M rationem ^quaiitatis acce- 
dct, lii^t ^arum diffcrentia fitiita maheat , iit videb^- 
|:^^«mu$ patilo infrd 4 Poffunt dutcm Habcre ratiohem fini- 
jj^Q tam quoqufc. Sic fi ih fig. 14 j hi.ancntibu$ puhciis F i 
p, evadant HI, FG pifallcl^^ipfiCE; puhcta I,&Gita 
m infiHihim rccedent , iit hu/quam jdm fiht ^ 6c rect^ 
CI > CG j at HI ; FG babcbuht fcmpct ih rceeflu corurii 
punctorum i^^tionein finitain qiiamcunlque , quam himi- 
irum hiibcbunt tcct^ CH, CF finit^. Ac fi ihccrca ipfa 
^Uoqiie pdhctd F , H a^ovdanfur Utcuhiquc,' fcd rectis 
HI, FG delatis M p^allclifmtith , ihaheaiit alicubi; re-- 
Ct§ ilte CIj CG ccttfcri dcbeircrit atfccut? f ationciTl cari- 
dcm , in qUd tcllhquuntar CH , CI j & fi qii? quanti ta- 
les fint fempety tn ipl^ CI , CG , 8c ipfis abcuntibuJ 
,. iti ihfinitum 9 Ahivi remancint i debebutit iiaberc tuiti 
tdtioiieni iUath, quaraliabcht CH, CF. totclt aiiteta 
ca raub cdaht in ifafinittjm excrefccre. Sic in 6g.^6o^i 
decreftcnte VR hltra^iiofcfimqlie limites , cref^it ul- 
trd quofeuhiquc limitcs t£im RI, <juamR£, ^abeunte 
R ia V^ J^m it^ id infinituih abeunf, ixtli &E nuf^ 
quam fiht; 6c tanien R£ ad RI erit fcmper , ut AHaii 
RI, w AV ad RV, qiafratio crefcit ultra quofcumqac 
iiitutes rd^ hiaiiehtiE; VA^ iiiinuitui* VR ultra qui^ 

' CU117- 



LOCORtJM iSEOMfetRlCORUM . ^ft 

taaxjpt liniites > ac dcmum evanefcic • Eft id quidem ifl> 
gcns quoddam ihiiniti myfterium y uc li^t jam limes l 
iu in infinitCun per omnes magnicudinum gradus c%r» 
aeYccic» ut nufquani jara fit> ^huc tamen ultraipfunt 
tiabeatui: fpatium. infinities magis prbtenfuin quodaitN 
mAos quo R£ extendatur infinities magis , ic quo & 
pbndum £ abdiderii: . At in infinitiim abfoluium . ad*- 
mittitur.^ id quidcm a^d fervandam analogiam omnind 
admitti debet» & efl: fimile ilU inyftecio> quod fupra > 
faum. 7^9. notavimus. 

S65. katim^m autem s<lualitatts ceiiferi debere i 
iibi hiiix quantitates in infinimm abierint , diiFercntiSi 
inanente finita 3 eft adi)uc myfterium*, cum xquaUa e£* 
fe noii poflint, quas difFerentiam Iiabeant ; adhuc ta* 
roea ad analogiam ^rctihendam eft neccITarium , & fiti 
evincitur» fejufmodi quantitates honpoflunt habereuUam 
rationem inacqualitaps utcumqiie parum disjim&am a 
Vatione xqualitatis • Exptimant enim bini termini finir^ 
ti , utcumqut parum ina^quales \i:ationem iliorum , 6c 
erit > divi^endo^ hprum difFcrentia ad minbrem» ut ih 
ianim differentia illa finita» ad minorem ex iis, qu£ 
ponuntur infinitx . Haec igitiir ex {illis ita penderet 9 
ht limitem alictibi deberet habcre pmnino; Vis demon-* 
tlrationis patebit^ in txemplo fequenti • £x namm 
tangentis IP in fig^ 265 eft iii drculo etiam 5 ut i^p^i^^ 
quavis Ellipfi juxta n^m. .411 !> NP ad VO 9 ut NM 
ad M O • Capiatur C M* aequalis CM j & ipfarum 
■^M , MO diffcr^ntiii erit. MM ipfarum vero NP , • 
Jk) difercntia erii NO . Jam vero ipfas NP i PO 
tion poflunt ita in infinitum abire , ut nttfqU^m 
jam fit earum limis P 5 nifi parallcte evadant ^ Sc 

fun<aum t recidat in Q,* Etiam in iUo cafu ipfarum^ 
JP , PO differentia finita eft , nimirum aequalis fi- 
iit« ilii Nd • At ipfaru«h NM , MO diffcrentia «- 
la M'M pcnitus evancfcit > & fit verum nihilum, cou 
(cuntibus penitus in G ipCs M*M ^ lUarum ratio evafc- 
dit accurata aequalitatis ratio . Quarc etiam binaj illafc 
NP, PM licet dificrjMlt per NO » ccaftridebent ad «qualita- 

m 



|4^ DE TRANSFQRMATIONE, 
fiiis rodonecn dcHxZy nec, qux iis propofci9na1e$fi]R|i|| 
jbot^l ia ico cafa noi| babcrej raitionHf kquali^ateai 19: 
-car^tam • 

. B66f Pl^^ramque qaalitiratlbg^ infinitts ajunt refpMiV 
^re qiiamitates infinitefimas > qusfc inafngn^biles fint> 
^nt taiiicn aliqnie, &:rdMienes ad ftpmvicem Jiabeant. 
^[^jiod a nobis aflignari pbilkat ^ Vd non poflint , i^i 
^^ noftro cognofcendi , & dct^rminandi' moclo pcndcti 
^:& aliiid menq^ fiqicae; genus ad aliam magnitudinum 
^" irelation(:m cognofcendam , affignandamauc' devenict , 
,i>fiud ^ aliam ^ quas minoris» vel maoris inequalita-, 
"^^ fa^ionem fecuni fer4t, ufque ad qucrfUajn liniites a 
' ^i n^entis ipfius pcndentes ; Idcirco nos ad eytt^nda^ 
.ftquivocatieties unmur , ubi opus eft 1 vocibu^ , qus 
'«b ipfa affignationc non pendcnt . Ubi d€ ihfinltcfi; 
* «:)?s agemus , ftatuem^ illud , a<j dcmonftrabjrnus , 
)ineai^ fuperficies, folida, qu«; in fe dctcrminata fint, 
^u^ nimiruni fuos alicubi limitcs habeant , finira effe, 
. & ^fittitam inter fe rationem habere . Hic autem ita 
cas voccs adhibcmus 3 ut infinitum di<?amas id, cufus 
Mitcm aliquis limcs nufquarti jarai fit t^ Npn quanrimu^ 
" \an ipfc a nobi^ affign^ti pofllt , an. non . yitumqae 
'Jfemofura fit pundlum P , dumniodo alicubi fit > pim- 
^um I nion crit in Q^> ftee M, M* in'C ^ & 'ipftifO 
^ Cfuidem P, fiye affignart pofiit a nofcis i|la cjus difiaiH 
;;ria'ab O, & N, five non poffit, ita cri; alicubi , ut 
"iiltra ipfum ali^, (iabcantur, five^ut poffitulterius cxcurr 
rere , crcffcnte ipfa illa dift^ntia ; ac eodera pa^Q 
" punfti I ^ M, M^ ita erunt dicubl , ut iHud cum. Q^ 
^ hsce tum^Q non congruanr , (^d diftantiam ab^iis ba- 
; iKant quandam in fe dct^rnHnatatn , jgve ca^ af nbbis 



^unai P alicubi cxift^ntis a puntffcis O .& N , ut ib 
iffifitsai augeri fcmper poffit^ h^cniinui \ nec ullaJ-- 
&him' Gt Qiaxima , tiarum mininis^ ; iMarum. ^fp^ 
<^iam limes quidam fit infioitiim abfolutiuq » ia im 

' - ^ P nuf- 



XOGORtJM GEdMETRIOOTltTM. hI 
w*ttirquaiii jam fit , liarum nihiluni , m quo I $ Qj 
"^ M , N^ a C diftantlam fiabcant pnri|[iiAO nuUani ; 

^^^ catn ipfisf accurat^ coprgriiadV . run^um kutefil fl 
^Jptflquam tuni cfljf , qua phfafi ftlnpcr 13(fi fiimos^ :fd( 
*?natufcftum ex"^0 ', qpbJ clu* ri&ai pjiraHcteij qtBfe 
. ftcumque producanmr , nunqtiatn a4 ie inVic^tl ac(l9. 

^afat nc mtnimo quidetn , utcumque es^iguo hi fe (i^ 



^^rniinaro intervallo nufquam revcra cdncurrunt ; liwi 

^ti finitarvifti m^nitudinum relationibus eHj^CMiilk 

^jiaberi pofnot pro concutrentibus m ipfo ififimto n^ 

'jitq inenti imperviq , liifi forte hofi imperviHno tatt^ 

^fnftfodq ipfum fit 3 fed prd abfttrdb> bdbeiri debe^^^ 

^Ut mox tridebimus, ' * ' v ' 

^' 867. C?terum li r^ftic NP, PQ ^primant ^nit^ 

'tat^s quafcumque^ -qtias inabfolutum infihitum abeunt^ 

'icli&z Npflfifita diflferfotlay rca?vero NM, MQ firit 

'^uantttates finitfc earum r^dbntm' exprimcwts i & ip- 

^i^ NM , Mp hen eyaffrint accura^ ^q^ales V el^ 

^'^iqiaa ipfarum cKfierentiaMM^ m fc d^{(tmtna|^ . Hitifc 

"^crit a|iqua di^antia MC iti fe determin^ , ^Qlli 

^'"Ml ipfi refppndens, & nonfOngrucns cum ^^Q^ am6^ 

" que aliqui tatigehs GF iion cc)ngruchs cum D]B y 8c ^ 

" ^uod pundum P in aliqua. in fe deterttiinata ciiRan^dji 

^«b O ; &'N . 'Qjiare fi ilte NP ,'& PO ad ftitie^ 

, tiem ^qua|itatis non fuerint dclatae utcuhique parum * 

' ab ea diltetit, ppu crunt abfolute iipfiitit^ ii^ $ V\^ 

"jeies P" nufquam jam i}t, *' / 

^ - 868, Ad hbc mjfft^riij^ utcutnque^Y^b^ctlim^^ "^H»^ 

y ict qUaptitatem finitattT quamcumque ttQ^ft^ ablblclti 

'^ uifiQitd^ habefe prorfus pf^ nullajr quanquam c^a ^UiltMI 

'cum abfqluto nihilotdnfuiidi n€h^J^0t* Sic9:(n7f4i^ 

' .fciatum Piarabels li^Kt itifiiiitun) 3 coafti iiBnas coAifidc^ 

^ rare > ut puodum , ut v«rum nifailum ^ipi^au 4bSdkh 

^ 4|e iiifinU^ pcripheti^:^" d^ 'r^tit^tidf 0cixvmk it^vcifti 

ticc ip&us parabol?, "^ - . 

^" ,^ ^69. Poftrema canonis par^ fic demonft^atii^. Vbl W 
^'lii^ pun^ qu^ reiiianent funt n bt N » &*Oineftd^ill 

"f^^a ^B, in qua pun^mP ift ihfiiiium necfflif , ^ 

P«?t 



344 DltRANSFORMAtlOiMfe , 

& pacet ipfamm NP, PO dif&i^miam forc illath KO 
iSnicam. Vel in diverfis tc&is ea. pui1(5l:a funt in 6$^ 

Pzt626it pmOiz Hi C reftarumCP» HP5 qua? evaduntab- 
foluce infinitte» ubi punAura P ica in infiaimm rcoef^ 
i^t$ tu fiufquam jam iit; Sc m hoc «ftcundo cafii, ao^ 
€r quam pufldhim P iUa infiniti txobis impcrvia velai 
voriiigine abforbcatuTi fi radio. PC fiat circuli areusCR 
acourrcns rectac PH in R; ipfatum PC^ PR difFereo^ 
^a erit HR4 Ul^i autem punctum P ii^ infmitat» re^ 
cefferit»^r^us CR debebit cohgruere gsm a>erp€ndicfild 
CV pec canonem xo . Quarc differ^ntia fict HV > qu^- 
quidem vel nuUa tn%% fi nimirum CH fic pcrpendici^ 
laris HP, puncto H congtuente eum V» vcl finitaerit/ 
cxtantibus adhbc punccis Hi& V^ 

870. Hajus cano^is ufus frequens oceurric » fecut^da 

l^^^partis potiffimtim. Num< a5 cum. datis in fig; 6( hinis 

%6^ punctis altcrnis A9 C proportionis harmonicas, jcdata 
ejus racionc i qusrercmus reliqua duo B » D, inveni^ 
SQUs , fi ratio data f^iret ^Bqualitatis $ B quidem abire 
in mcdiam rectapi AC inR i D vero ita infinitum tt* 
ccderc, nc nufquam jam cffec. Rcs eddem rediic, itque 
Jiic in 6gi i6$y Sc reccac RD, CD ad aequalicacts r» 
lioncm delacas ica in illam infinici baracbrum merfi:runc 
punctum £)i ut^ nufquam jam efTet» 6c ip£e quidemab^ 
foluce ii^nicac evddcrent » diffcrcnu4 auccm ipfaroni 
jam cfrcc.nuUa. , J: ^ , 

S7I; ia appUcacionc thtorctiulcttni EUipfcds ,*. vd Hy4 
ptrbete ad PaTabQlam fummus ejus ufus babcri potcft* 
Qncttiiam rccca; Fm 9 in£ in i^. 9^ debent aa|uiiccc 

fi9 radonem aequalicacis , vbi cain Pacabolaim migrat, ver^ 
.!• cex I9p axis cranfverfi in ipfa Parabola in fig. lo* fia^ 
^uati^ }am efl» fit ipf;. ev^dunt abfolucc tnfinita?. Hii^c 
yero ad Parabolam cransfcrtur theorcma pcrcinens ad 
£Uipfim , Sc Hyperbolam propofimm mjm^ 74 ^ quod 
nicairum qiiadraca femiordinacarum RP in fig^ 9 Si 
Si ^ s^em cranfvcrfom fint 9 ut rcaangula MR/ 
fub abfciffis a binis vcrcicibiis • Dutn eae mucanmr ii 
Parabol^ jS^«ra$ iq j abic m in infinicum ita , ot 



LOCORUM GEOMETRICORUM. f^f 
tintnum jam fit • Qaare fi binx affumantur femioiv* 
£fl^ta& 9 binarum mK y quas ad eas pertinent) 8c eva» 
ikmc abfolute infinkas , ratio evadit ratio aequalitacis t 
cam difierentia ipfarum £t« iUa finita diftantia binou ^ 
rum punetorum R . Quare^ tUa rectangula erunt v ue 
feiae abfciflb MR a vertice M , qui in. Parabila 
naoet , Id autcm Ua fe l>abere conftat ; id enim ip» 
ftm eodem numero pariter propofuimus: atque eomeM 
do ex EUipii 9 & Hypcrbola ad Parabolam transfenuP" 
radem proprietas generalius pro diainetris omnibus^ 
propofita humer, 357 , cutn abeunte 10 infinicum- 
ceQtro in Parabda ; fimul cum ipfo cujufvis dtame**'^ 
n-i alter venex in infinitum abeat ^ n^ ufquam fattii 
fe. V -' 

iyz. Ejufdcm canonis ope iUa etiam Parabolac pro^^- 
rrietas , quam num. 200 Hemonftravimus in fig. iy 9 
qaod nimirum foci radius FP aequetur tam diftantiae^F.^9^ 
FI ejufdem foci a normali 9 quam diftantias FCXc}uf-' 65 
dem a tangente computatis in axe , derivari poteft ex^ 
proprietate EUipfeos propofilta num. 1S9 in fig. 6;^ 
quod normaUs fccet F/ in ratione laterum FP j /P *' 
& qiiod concurrentibus normaU , 8c tangente axi a'an& 
vei^o.tn I , & T 5 conftituant proportionem harmo» 
nicam , quatucH' puncta /, I » F , T . Nam ex prim t 
altctnando erit FP ad FI , ut /P ad fl , quae ratib • 
cum abeunte in Parabola punAo / in infinitum , Sc 
temancntibus FP, Fl, evadat ratio a^qualitatis , erit ift 
ca edam FP asqualis FI • Ex fecunda vero eft FT ad 
FI , ut /T ad fl; qua; ratio pariter abeunte / in infi-i*':» 
nitum , Sc mancntibus T y l , evadit ratio aoquaUtatts) 
Kuidie^fit , ut in fig« ^5 rectae FP, Fl, FT aequari dc«- ' 
kaat inter fc. • ' 

S73. £t his quidem eafibus £acile fuit canonembonc - 
appUcare; at artificio aUquo opus erit no«nunquam> nc '^ 
ipTum^ appUcari poflfit § ut ubi pkires quam doar quanti^ 
tates infinit; fiunt ob imius puncti tantummodo reorf* • 
rum in infinimm. In EUipfi (num.419) fuRt coflLtittUcFi))^" 
5CDportianaI<5 in fig. 15J trcs vmx CR» CV^ CQ^ i^j 




, 3¥ DETItAtfSFOJBLMrfri 

Saie abfiuffa a iscftiro , &mi<Hanecci: . &. iubtaiig^ ^ 

cmcro CQBipacaiR. Abcunte ccnfro Cin jnBokvms nti 

i|aiiaParabip4^m:.tnuitatiiCy^c^^ ea pfo|)fietailr alkm.f^ 

ccjGe apt;atmn ipfi;Pa^abdai • .fei iic. ^ariiic. mvi;aix!i qx^ 

w:ic« Cuni fit eiWto.rwP.CRaiCyi «c,CY.aaCQi 

«dc jpaf iter eadern iraxio dii^ ijuccedeiittimi RVi 

a4 IdiAcccttiani confeqiiMtium VQ^;' ££it igitur.VR 

a4 VQ, i ut CA ad CV : Porro abciiate cctttio C iil 

mfinimra ; ac ina)Qeii(ibai R; ^ V».,ea ratio evadttra^ 

jiii^^aDquaiicatifli eTadUfiC iiimr arqualcs <^tiain yRiVQ* 

m paFlboUiii nimiiuini diftaatia VR. vertids V in % 

Z55 i normali a;<iiialis diftantiae VQ^.a. iarigente cooh 

^jiiBAik in ^se ;; five jiiiastangens du|b}a abfciffae i quod, 

aum. 405 de.ipra paraboia dcmonftravimiis demonftra4 

tiQoe pecuiiairt; , , . ; 

, 1^74; Majus aliquod artificiuni reqiuritiir plcrunEiquc^ 

. m>i hoti ^ckilmune aliquod pundum binarum re&aruoi' 

al)it in.iriiinitum» fcd bui^, dngularum fingula> ydad 

ikiiionftrandiim ; diffcrentiam mariefc firiitam ; ci qui- 

|iroflttat.rati6 sequaliiatis> vcl fi difFerentia qao^ttt 1»- 

ia infinitum excreTcat ; ut earum ratio fic.finita ceii^' 

ierida Jity ic a4iiud ratio inxdualitacis i ad irivcnu!ti^'^ 

cltm racioriem ipfam » Sc rubfticuendas quincitate^ finl--" 

ias» qvx e^m ratidncm expriiiianc. Adbiuc. famcQL liqd^ 

deettuit plur^s mccfaodi cxcrcitaco Geomcctte ad idps^- 

Aandum» Sinapro, biriis faiifoe dafibuf csiciiipla alttroiW ^ 

£ia altero <exbibebit fiiwKipTdm che«re|Tia iUiid geaeraH^ i 
ii' qubd n; ip9* propcaffuiiritis prd rc^s omriibus Se^ 2 
itiom Conicx bis occuttei^tibtts, quodiiioc ipfo. arnfict^r 
cranftuliiriqsad redtas aii p^alldias ia Parabola» Ar u»' 
iritibec aiymptoto; iri iJTypt^bbli, ac cjus ope rinveiiimas - 
cbeorcma Alterum ipii fubftitritum pro hifce cafi^ fiair - 
cicidaribus iium» ^c^ Sc pccutiari demonfttacioflc tef€^ ' 
uwittk itimk cx finica Geonktcia: 
j, , ■Mij- Thcoirema letieraie huc reductmr ; Si ia fig. H 
'''^'felbjf quc^nique PLp diiAst pec quoddam puadhinii 
ocoiiic^l ScAioni €>otricae bis iri puridts P» p ; re&ftf^ 
£uia iiifc carum' fogosifeQci^ LP ; Lp inteiitteptiir idiBixd 

pUtl' 



. Id^CKUM GEpMEtklCORUM . '^ 
^lkabdiim » ic iilo^ ' oocurfus ^runi ad fc iovicetn inr^ 
^ne 9 ^VLX dam C<M)ica Se^iiofte , & iprartim riAsu 
CQim indiaitiopibui, erir fetnper condatYS.;. utci|mque 
^^atb puhAo L; A^at ^am . alcera ihterh:<^o $ m p% 
m ihfinitum , qUod aceidtt fo!um fedis axi pir^delit 
ifi E^^pla ^ vel alteri- ^t^fi^pmm in H)rperboIa. i tc^ 
StmgiAiuh PL^ cvHit ih^itucti ti- Sed fi vcAm, If ii 
kd Jbafu iubftitiiattff fe£fca|. qjui^ inj^tasa poiiif^ ti )sm 
ktuBC m eacicia ,ratk>hc i iii c)iid* mutatur ipfa Lp > jafft 
^tiam re<5laiij^uli fui> LV!i Sc fah Jiac fcfta f^tio ad ec*^ 
liqfod iredlan^a manebit i^oiiftaQ^i necinbiiiiau dtittM 
ktiaiic puneti Li ' \, .,..;.. , , . ' 
&:75. JPpfrci 4xhti^s fcft^ Lf infinitoe in Faf abc^a ha^ 
beddaf efuhi pfo asqualibus^>-ife to HyperbpU pi^ prb^ 
portio]|.alibus rectis ^ quje iq quovts anguio) dacaiitlfr 
ti tp£s puaftis L ad iUani ^aQrhipttftam i cid Lj' eifaiij 




itafilm VpU Lp forci «qualem fiktim#i vcl; 4i' 
.ip(armii LAi pd, ciihiqute ixkilt^rp^i Pjp^ debdtt fecaf^i^ 
iaem axis coajtisatusbuaciani;. paicet, ip&m* M« *f8^^ 
K I^. Cum igiti^ tiiutata Ellipfi iQ^Parakbl^) & a^ 
baiadbtts pundispi f[ iti ihfimium^ ftiaAedd^^t^ AL 
tovixiiahebit.^dita iHaram dlflfetetiti^; dc prdiside ra-^ 
iio erxt xquatibtis ; At iii fig; 2^77 .fi ^c ehdfdarP/» 
Py ixiter fe parallda^ occufrant binis afyriipiGftk iri^bi^ 
iiis pua<iii$ H, i&, & H*» iS/, duCOTwrauteni, c«b&u«- 
quibufvis carum puhdis L, L' bintft fe^x U ^ L'P iti 
quovii ahgulo ad afymptmoth O « cruat L6 » L'^V itt 
Ll,* JLV ob u:iaheul(X^um rimilitudtriem « Ab<tstitii>pii auM 
icrtii f^widtis pi f iri- iniJriituni > cua /*, |»-A' fehiper 
(TiiiUiiri. jzi ) «qucntur finitis PH> P*H', cruat If M 
Lh i &t Up^ zd L!&^ in fatione squaUiatis ob differcii^ 
6^ finitam . Igituf & Lp i Vp cn^t i at Ui LT;* 
fciinc iri iii cafibus pro ea cohftantt ratiohe reaarigiite 
fubbiais diftanttii* Lp, Ly a binis bccurfifeus p^f fub* 
ftiati ^fktii dittahtia ib uciicb' occkcftf iti rc^a qua* 

vis 



>. 



34« DETRANSFOkMATlONB 
vls coaftanci in Parabola, yel illi ipfi ^{ymptoio um£« 
nata in Hyperbola in angulo conftand quovis ^ iBo 
ipfo pundo dftto • ^Atque id ipfuro prasftitaimus iHo 
num. 305,^ 6c rite &(^um per finitam GeQmetrlamde* 
monttravimus . 

877. Ltceret hic addere }am Canon^m 12 huic ^^ 
milem pro redis , qu; in iftfinitum decrefcunt ita, ut 
4cnium evatidcant ^ qua* pariter , dum ita decrefoint 
ad rationem aliquam accedunt ulnra quofcumquc limi- 
tes 9 quam finitas quantitates tc<^uirunt ep momenta 
€emporis> quo illae evanefcunt > quae ratio parieer cfle 
poteft vel utcumque inaequalitatisfimtas, vel etiani ao- 
da» vel imminuta in infinitum» cut canoni tota innin 
titur metbodus, quam Nevvtonus appellavit primamna- 
fcentium^ vel ulcimam evanefcentiumr, 8c ex qua me- 
thodus iUa , quam idem «ppellat fluxUmm , ortum du- 
xit • Quod fi quantitates; dum in infinitum decrefiruntt 
i0finitefi~mas dicantur, & harum infinitefimaruni ceru 
"ordines, & gradus deflgnentur , ac ad certos caitones; 
i^digatur eoruhdem ufus, ilia omnis ubetrima fiinedif-. 
ferentiaiis habetur methodus y-^ux calculo potiinmam 
ad)uta tantos fecit tambrevi ii{ emni & pur-a, &mix- 
ta Mathefi univerfa progrefTus , qui quidem/ gradus & 
etiam in quantitatibus in infinhum excrefcmtibus pa^ 
riter confiderentor, habetur quidquid ad methodumin-* 
finitorum pertinet in Geometria. 
• 878. At ea omnia nos alteri tomoedetido, Ciimpri- 
mura per t»mpusHcuerit, refervamus; in promptuenim. 
eft omnis materia: at ^ folidioribus principiis conabi- 
mur ftabilire omnia. Nam nec illud nobis ;fatisfacic ^ 
quod Nevvconus de evanefcentifous quantitatibus haber, 
cas ad quandam rationem dcvenire , neque antequam 
cvanuerint 3 neque poft , fed tum , cum eyanefcun^ % 
tum enim, cum evanefcunt, jam nibil funt> ncqueuK 
lum eft ultimum efib quod acquirant, fed vel funtaK-t 
quid adhuc, quo minus erunt dcinde, vel nihil otnoi* 
no funt. Mulco autem minus illud arridet , quod alii 
ufurpantj qui infiniiefimas quanritaites qontcmplantaf ^ 



LOCQRUM GEOMETRIGORUM ?+9 
^t ^IiqHM» quod in fe dcterminatam fit > & ratio^em ad dQtcm 

Ilabeat minorem quacumque data. Cumenimdatamdicune, 

iQinteI|ig^c>qu2ereapfe data(it|6erifanQ poteric, ue necdat:v 

fitratio I ad loop^^S^tuncratio i ad 2ooominorerit,quacum« 

qtiedapa;^ v^ro iriiteUigant etiam dabikra,quod ver^ intelligunc 

i\y qyueiufmodiquantitatesitiaQjgnabile^ vocant; difficulta^ 

fcm U quidem neqaaquam eludunt • Si enim ita aOlgn abilem , 

^ds^bilem dicuQt>UtanoBlsdiftin<3:e|>erx:ipipo(Iit ipfaearum 

magnitudo per relatiojnem ad menfuras >quas intuemur ) di \d 

a menti^ ipfias pendehit vi qt fupra diximus numer. S66^ 

ita > HC quod refpectu alterius mentis dari , vel affignaui 

iiQn poffit, pofiit ab altera • Cqmque menti^ cu^ufpiam 

vis fine$ bab^ac omflinQ certos; id , quod uni aflSgnabil^ 

fric, a^quc finitum j alteri eeit inaflignabil^f 8c infinitefi- 

inuni , ac fluplum infinitefimi refpedfcu e|ufdem mentls £rit 

^Aitum , $i vero mentis ipfius vim,& perceptionemdiftin- 

dam neqqaqiiam rcfpiciunt ; cur eac quanti tates , quibu^ in U- 

aiverfaGeometi^a,^ Ana^yfipeirpetuo utimuri quarum epe 

tatnlq^g^ dem^nftrationes pecteximus » quarum ordines , & 

gradiv;, ^;: reiauot^es ad/e.fnvlcem tam multas perfequimur , 

afiigiiari QQn pQflint l Cum ratio cujufdam quantitatis ad fini- 

pm quandam dicatur minor quacumque dabili,cum ejus ipfius 

quancitatis dimidium ad eam ipfam quantitatemfinitam> duplo 

gdhuc minorem rationem ti^4C ? lUqd iinun^ e(^ r^ Ii<^uuni , ut 

ubi rattQ »kV^ q^^cnf^iftcdaMdiataff figniiScetur id, quod 

nomen dafinm iq Qeometria pkrtimque exprimit, nimitum de- 

ffr9ffzx74fxt9i,&inf;nitcfimaequantitatesin fe ipfis determina- 

ta?9 qqa yoce ad toUenjas xquivocationes utimur , nuUas fiat : 

fcd in4aitefiin£ di^^ntur eas , quas nos indefiriit^ concipimus , 

quarumt^imir^mmagnitudinem nondefinimus, fed ita par-r 

yam accipiin^^ Ut ad noftrum libitum imminui pofiit, fine uUo 

^oeanobisd^ferniinatQ^ quo nimirum iiGcac demonftrauo- 

fiemdeindercdi|cere,fiopusfitjadabfurdum. £a acoeptione 

infinitefimoriimHabita&rite confirmata, folidiffimas totiu9 

ineclipdi demonftrationes obveniunt , quas non fimul cum ea^ 

rum ufii in curvarum g^neralibus proprietatibus per fimplicem 

^tiam Qeomctriam eriiendis , ac curvarum utiliorum elemen- 

tif iade repeticis eodem iUq tomo perfequemur • 

S7P. Eod^m aiitem pa(^Q & quantltates , qux in infinitum 

Bofcovich. Tom^ IIL -Aa cn- 



y 



55© DE tkAksfORMAtioki 

fekciefcunt 9 accipi indefinite poflTunt , ac demonftrationcs , i(a^ 
t] iia! i nde protiuunt pro ipiis 'curvafuhi^ti^ansfohiiatipBibus i 
futit fdne miilto & folidlores ; & ipfi noftlfaci mctid iiiagis pcr^ 
viaf ^qbatnear^quasexinfinitiabfbluti myfteriii hk adhtbo^ 
miis . Myfteriaenim ipfa jiiHrliti abfoluti extetifi ejufmcidi fuat^ 
fat hos ea ftudiofiffiiilc {^erfequentes deiTibm deduxtriiit id ccn- 
fetidum pbtids iiiitSdffibile prorfus , & irepiignins iiifinitum ab* 
folutliiti in ^uantit^te 9 quam t^ttimiiiodo fintt^ ndftra; metl- 
tiiitipcrvium.Canones, quo^hic pra^fcripfimus ad erdetida^ 
finitarumquantitatum> qu^ pofl; traiisfbrmatidtiem tefidu£ 
AintsreliiitioheiiiiutuaSihabcmfasfrrocertiffiiTiiS iti iisohiiii- 
bus, qdae peititleht ad ipfas qfaantitates finitias refidtias i & btna 
Ipforum Caiionuin gehuitiaftuidamentacenfemfas efic illa j 
qh.-efiiiS lofcis pfbtulimus. UBi nirtiirum punftura aliquod 
irnpludiui^peHnfiriiifaiil^&pliiiquatnihfinitai quahtitaii fufa^ 
Itituttur hegativuiii tjus coitiplehiefaiiim ad iiit^grulTi iiifinituni 
circulum juxta h. 776>fiindat3iehtiim eft ipfd lidhlogeneitas Lch 
cotvin Gedhietticorfam fimplicium; qtiorutn partes bmnes eaf- 
dehlhabenti'eIatiohcsadfeinViceiiii utibidem mohfaiinusi 
ubi vciro pundum nufquAhl jaiH eft j fed iiifiditd corifcipif lir de- 
Incirfum > atqtifc^ obrutunl ^ habctur contiriUitatiS lex j quae c<y 
gliquaatitatesfitiifdspbftipfaiii ttansformitloheiTi fchiaiien» 
ics caitl habfcte ratioriem ; ad qiualn iccfeflciratit ultrd qijofcura- 
quc liitiitcs ipfae^ quasi remaridnt i & ad qu^th acccdei*^ dc- 
buiircritillxctiaiii^qiiasiinirifihitum exCrefcerites cohcipiua-^ 
tfar, nifi alicfa(>i hec^euatidobi^faitipbretituls ante <juam abiblut^ 
ihfinitseetradererif.Glerifcmui autciti abriimpialicfibi dcbcrc 
omninoitaiiuthilnquaih^flcclfaipdffihtohihem illam extea'* 
fioQCmqu£poffibiIiseft;Q^idquidcxiftit> iddmne firiesha- 
bcfe certos i arbitirathur^ultra qfaoi; alii, fcd omii^s itidehi cerni 
& definiti limites Habe^htUr ; ut diei^uhi futuf ii ^tcrhitati^ riii- 
itictris a praeferi ti dle ad quemctiitiqut cfetcf mirisituni, qui> cxti- 
turus eft aliqUaridd, fiditUs td \ fed alifas Haberi poteft,& dmiii- 
no habebitfar ipfb majoi' i Eodeiit ptdrfus j^adto nbbiS perfai- 
fuiii cft i i^ettim quarumcUmqiie exift^ritium nuhi^ttitis l ut ho 
minum, deCeflario feitipci^.finitum fore ; atqufe id iti i ui eo iirf- 
jor altcr habefi femper poffit>qui & ipfc fiiiitUS fif > nec unqasri 
fiitiul poffit eJttttcte toturil id , <|uod i fi fcorfum i^dtc tur j i^ 
teft cxiftctc 4 Kcque ci;iim ficri pdtett, ut fumhia DiVia: Condi- 

toris 



LC^QRUM GEQWjETRICORUM, jj^ 

IfljCi^Omq^ii^pCfni^vircscjjb^un^^^^ &condat q^ae^mn^ 

gl|p coqdc^^ porfit3!<)^in ali^ fup^rAnc (^tic i^e^qua? itidert) con-. 
^at, & vclit > quod nd^qvii^em appcllare focvmj^Ji?titfm in i^fi^ 
ni^tm 9 8c ibi ubcrius cxp^icablrnus, i^bi oanc in^fmitorum tl^eo- 
riamA^fius pcrfequcmur.Eodcmq^irnivum pacf^q» 6^ ret^am 
lincan^ 9 8c (rurvilincl auris cuiurcum<juc trachim , cenfern^s i 
pps poQk fimij^cxiilcrc cum ca omni Ibagjltudtnc , quap, fuc- 
Ce(Iiv$ h.abcrc potc(^ 5 qu? niraiirum quotic(ik^quc cxti tcrlt> fi- 
^itacrit^ alias fe iQngtQrcs pure adhucfofnbUes pofl: fe rclin- 
guet itajuc Wla fit ulti mi^cart^^uae cxiftcrc poiii^tit > Sc ma- 
)cima> qucmadmodmii nuU^i^idem longitudo e(ln:;inima > 
fcd qaacum^u^ detcrniinatalongitudiiic iitcumquc parva, qii^e 
^on fit ^bfolutiim nit^ilum > a]i^ adhuc minorcsj^ &anihilo 
minqsdiftat^tcs habcri^pdffunt, 

* 8?Q,IntcccacoIIigemus hic illa rayfteri^qnas t^ojbisdcmum 
yifa fantmig^are in ycra abfurda , qux qui4?m fun t pjei:aquc . 
^itamus iJI^in npftfac incnti impcrviiini fanc tranfitura. pun- 
^i pcr in^itum ad partcs opj^ofi tas, & nexufn rciJtaf i^trinquC 
(ninfinitumprodiiciaeiiipartibusoppofitis, qui quidcm om-- 
pcm cxcccfit C^tiim humai^x inct\ti$ ; ciret enim , quo is evitri. 
K\ Ppflfet:^cpncipicmdppuiiftum/itiod cx ppppfita partc rcgrcd i- 
tiiirj nQncfic idcm, ac id, quodreceircrat,'CedaIiud, &fo- 
luhS poft intcgnim tc(^ copycrfipaempun(ftuniidem> pcr 
diinidiam infini ti ?irculi qircumfercn tiani ' e\;agatum> redire ' ; 
lice^ id^ ipfiini omn^mLocorum Geohictricorum analogiam 
pervcr^ct, ut facilc otftcndi poffct . Mjttimus rationcm acqua- 
litati^in quaktitatibus mjsequalibus/quac; nimirum differant 
(|ti$niQUte finita s ^um reponi paflit ,pro nthilo habcndam effe 
.quantitaccm finitam^ re(pe<^u infinitarom ; quamquam aliud 
omninocfthaberidebercpronihilo, aliud revera nihileflTo 
guod ad yeram acqualitatem rec^utritur • Mittiraus illa infinira 
(patia cxtenfalong^ ultraaliainfihita, quxconcipiendadixi- 
inu$n.77i', quo nimiriim infiniti illi circuli excurcant alii lon- 
ge uhra^ altosjyqi^orum^ope n^gatiyarquantitates orcf ex tranfi^ 
cu pcr infinitdm retineant rationcm quandatii fini ^m inasqua- 
litatis cujufvis ) ci^tii ea^ npa aUt^r detiibn,l^entur ,H^uaii^ cx fo- 
la aualogia : qus^quj^m iii tanta cxemploi;,um accuratiffnne 
dcmmftratorum multttudine ipfa etiam analogia ingentem 
habcrevtmdebct. 

Aa 2 881, 



;5i pt TRANStORMAttbNfi 
18 !• Hifce omnibus oinifiis j q\xx poQ^hc vel non admitS) 
Vel pro niyfieriis ^uibufdahi baberi liobis im(>er^iis , quid 
illud) qaod finitsst^ &accurata evideatiffihia Geofnet^ia de-^ 
^'^ monftrationc evincitur, ut n.864 lniiimas> in flg. 260 re8;am 
' R£ debere infinities majorem etfe , qdaiti RI, ub) Iii infinittuTl 
cxcrefcun t ? Concipi atut redta V O i ta ih itifirl i iWti ^xteiifa, ut 
ejusyertexnufqtiam jamfit^nimiramutomnem eatn^lifierl 
poteft , cxtenfionem liabeat, quam haberc pottft,& quae utiquii 
a curvarum fibi adjacentitiin defcriptioti^ hoh petidet l Coiici- 
piatur jam ipfi adjacens foUcurta TEr i Nullanl faiic ctit fcg* 
mentum finitutti ipflusredla: VO^quod aliquaiido ordinataRl 
non f upcret in raptu conrinuo panfti R verAis V: Igitiit fi cur- 
va TB/ extenditur fquanlum extendi pottft,tiihtl ex fefta iBi 
tiltraipfamprocurritj&appellcrite RadV, oTditiataipfa W 
f undo jam I dcmerfb iri illis iriflniti latebris 3 atqiie obratoiilli 
VO pariter iilftniti* acqaabitur. Quo igitor proCUrret dltra K% 
ki jpfa RI iniiQiiies fit major? An fecundacarva accedente,i(^ 
fa iUa redl^ VO 3 quac ab iis ^ ut diximus > non pendebat , pro^ 
tehditur , ut nbvam brdinatam RE fibi )am congroentetn exci- 
^iat? Anndnnoftra^tiimammodo memis a fine abftrahentis 
iigthenturii cft &: re(5l:$.illiuSv&catvaK cbminuatioiine &t? 
Nam cx , fi mcamque fkiittt ^icubi funt , nihil abfardi iavol^ 
vunt«Saaeutcamqa6magria? ordinatas finitasRI ali^RE iri 
quavi^rationemajorrefponderepoccft» congrocnti com fotjL 
VO ^^^(^/^infiaitanQnpottfti, ... 

L t:. . 882. C^d in iUo hiam Paraboias , ih fig. ijo t Ucdbit fkjb 
^^^ ibi deprchendere abfardam pdtias gtaviffimum , qitaifi itnt>cr- 
vitiro noftrs ttltfiti mj^fteriurii • Nam ^x ^c^uod re6la DAet^ 
currat fempcr ui(ra ipfamP^taEbol^^tiili congraat eam ipfo i^ 
xe DA2, eruimus fi''^^ I fpatiafn croribus S,T infinirisineetce- 
pmm miiior^rii quaVis finitai rarione rariontem habere ^ airciim 
circuli circumquaquc infiniri •' Atid ipftim fpariiifn admodum 
facile dcaidnftrabitur ma}iii,qaatri ipfias infkitcperipberif fit' 
quadruplum. CerrifSmum thim eftin Gcometria ex /idir 
medis iRvcnas» dia^mctrum ad pcriphetiaim babere tatioltfffi 

majorem, quam i ad ^«camhabcatfanemajorcmtrianviB^ 
7 ^d 22.AthiatusiUt facilc dcinonftramr arqaalisipfiusdKiA 
infinia diamctrp . Ubicritrique enim afTumatur pun^iiiai 6ih 
tangw^nteB^A^iriinfinittim produdlaita, utomaemeaiiha 



. toCORUM GEOMERRlCORtjM. ^^5 

tcat cxtcnfipncra , fi id fieri polTit , quam habcre potcft , diiea-* 
turq^uc rciiSta jpfi^axi DAi par^^^ fempctParaoolae pcnmc-» 
tro occurrct in aliquo pundo I^. Quarc hiatiis illc idcrn taiitim-j 
denei^tcn^itur, fuatitumjpfa circuli. jn8niti diarpctei; , citi 
proinde aequalis crit * Eft igiiiir cadcm ratlo & majori & infiai- 
tiesmmpr , quam i ad ^ , cjjiod cft aDfuraura . Mj^ttcrium erit i 
fi dicacur , aicm DA2 infinities niagis protcndi , quamtangca- 
^mpp4>^PA4. V Etcmidcmidomnmodiccndumerit.; nani 
iXJ3adG?P3> fivcDRacft^, iit radius adkpgenteiii fmjsuii 
G^D&ijquicumabeunteHainHii quo ealu puodtaG, Sc 
Kuammnnitumabbupt, utnufquain lamunt) abcat inrc-- 
j^um , ca ratio tum ey adi t qiianti tatis finiiae ^d infiiiitam,* u^de 
4;?:ui dcbcrct,» axem Parabolaeinfiriituniinfiniticslongidifc^ 
fc, (juam iiifinitam tangchieni \ At aii idcircqircfitapAz infini- 
ttes,magis protcndi potcft , quam PA4. , quod^^rabol^ ipfis ac- 
ccffii,'cujusillacfttangcHsjbae'cv^^^ Q^id^ aliaiudc-i 

fcribercmus Parabolam axc DA^ , tangcntc pA^ ^ Nntp idcir-* 
icoilla, quae ihfinities mihoi: cirat ^ jh&iitics majpr cvadercjt > 
, gis^, Maxi^iara hoc quidem ar^iimchtUin apud nos yim Ha.^ 
Wti &,ci finailimam aHaquamipiurima^.quacprofcm poffcnti 
^itquoddam aliud, quodjamabahhoi^^iprotuliraus indif- 
ftrtatiQQC^ ^i NMtura , ^ ujfu ihfittmrHmy&infini^^^ i 

ubi o{tcndimu5 ^ admifib infinitp abfolutQ in cxtehfionc > pat'^ 
icm Qbyenire «qiialemii.niipo ctiam majqrcm toto, iVccedit aur 
iem & iuud, quod^ ut h.^ 317 vidiinus , ipfini tp ab/oluto poft ip- 
fum , & feiitam aliquam quantitatcnj pro tcrjcia continuc pro-. 
fortionali rsfpdnJct nihilum ablbiutum, noh quae^i^ quan- 
eitas,qa» infini tcfima dici dcbcar ^ & part^s, atquc cxtenfioncixi 
iiiiquiunbicrepoflGt,i4uoa quidem mi^^ ar^ii- 

tncnw cvinci potfeift.Sic in fig.260 flc gcomctrica conftruaiohc f% . 
lavcftig»rcmus tcruam poft Ri,Jk 1?. A,vel poft RE,& RA, abe-*^ 
imtc R in V , & faais Rt* RE abfblute ihnni tis, facilc fahc iii^ 
Vcnirctur ,utramquc jtbirc in ycruhi., & abfoIutun\hihiIuhi ;' 
Porro facilc & illud patct , tcrtias illas poft RI,vcl RE , & ca^- 
dcm RA JFori reciptocc>ut ipfjis RI , FLE . QjUapiobrchi abcliS- 
tibos in infinitum abfoliitum binis J:c(^is , fi ex iSnitaih aUquam 
tationcm habercnt ad fc inviccm , vcl.ut liic etialil m infinitum 
v«l att<aaiir^ *cl irarain^iiam j c ithdctu tffc ijjitcr ipfanihila ra- 



\ 



u^ 



^54. DE TJV.A.NSFPIIMA.TIONE 

tioQeitioporteret^ &itiipra extenfione aliquod nihil deberet^ 
ciTe magis Qihil, immo& inGnlties magis nihil^quam aliudni^. 
niI,qUod (qui<^em,quam pughet cgm nitidiffima iUaiiihiii idca^ 
^ue men ti hupianac cuilibct ^b pr^f^ns fiftit ^'* nenio non vidct . 
§84.^unt quidem ,^qui in in%ito , ajun t> nequelaequaiitatisi^ 
«Kque tottus, & p^tis nomen adttiitti Jjdffg fAi id quidem erii^ 
liondiffiailu;cm,fcd^^^^ iric tcdarguarisj^ 

ut fi Quis omnium idiom^tum ufum adimeret • Debent fane il^ 
layocaDula admitn etiam ibi , cum ijJea> qus nobis g:ariffima 
11$ refpondet nomitiib^s ab infiniti rationc ndn pendcat. Quo- 
Cdmq; uta^is ripmin^,iutra lUam rc^^miquf ^xtendjturiquan- 
tqm cxtehdi potcft Ijnc yllp limiie,nijiilcfle poieft, quo cadem 
tedba^adjefba ipii ad Ia{u$ altcra potius cui^a^quiam altcja&ai^^ 
altcra potitis <:otiditi9ne>quam cum altera, cxcreTcat j^m»& pro* 
dbcatur • Si bina^ujecuinquc iint ejufmodi , ut in altcro (it ^ 
quidqaid h^b^turjn alrcrp^^ pra?ter^^aIiquid,quod in eo non 
bdbctur ;h6C famjiiivc^Qnituh^iS^ifiye in^nitiim > hafoebit id« 
qu6d conciphtius a cum dicimus,majorem eflc, & cufn dicimus^' 
cj&e totum rdpeOfu ifuamni partium,quarum altcra eri t i4>9^^4 ' 
cph^unune eft,altera id^q^^^ M^jus auttcra adhu(; 

f(Mnper irit tbtum fua parteA pars i{ifi tpto^^qualis cffe non po- 
terit > multo vero minuspoterit.pfli^ major • Idem quo^piamci^, 
dem in hoc fenfu ipfo,(iye finitum^fjt,(ivc infinitum V& acquale 
fimul ,& majus, & mintft cflc non potcrit •'5te cflct , accxicn- 
fioucis abfdlvite infini^ac , ut &"feriei abfojute infim {a^ i fi^as in-^^ 
tcr fc diverfa ratiojnj{coniparaveris , cxdem fane ufdem & ma-. 
Ipres fimidcrunti&iqudes, Scmin^^ quod arguinentis 
cvincitur.Uludigititrdicendum quantitatctb nullam' 

cxiftcre pofle > quac ^^ic^ <^on fit » quaiii cjusclcm appdlanpnes 
infinitb non c6nv(enire;& qu2tcamque contradifttoneni iirrol- 
yuntV abfurdi^ dicenda funt>'quac impoilibilem exifteBtiaai 
cvincant/ nonmyfteria tantuftihiodo, qoacfihiitenicntisc»- 
ptum uranfccncljant; . 
'^ SS^. Acnbbisquidcmfconfid^ 

^bilis fit quan^itas infinita, Qcciirrit lUudjqtaiod mfinrmm fum- 
n^am fimplicitatem , & unitatem requhrat, quse a fumnia irtfiirf-. 
ti pcrfedliionc nequaquahii fejungipolfit ,cuantitas atiii^bapar- 
nbus omnmo cotxitarc debeat ,& compofitioiKjm ^xppicat. o\ 
linea in*infinitum ek utraquc parrc excnrrat > inveniiiius in Ma, 

infi- 



^ LOCORUM GEOMETRICORUM. \i$ . - 

sHfinica diftantja c4m quodammodo copulari , atque con)ung?^ 
gc in orbem f cSirc debereV ut & ihfinita Hypd:boI« , ac parabo- 
Ix cruraafeinviccm pcr infihitumXa(5&m divulfe, ibidctn , 
Quodahimodpconjuhgi, hinquamfi ijla !i\finitacliftahtia jimiyi 
cflct iiulla .lltc infinitus hiatiis ParaBolac infig. lyoiiccrscquJ^r 
lis ihfinitae tangenti Vdcbet efie ^uoddam veluti pimftiim Juxti 
I1.741 , quo motus cohtihuus pundri P nequaquam. iiiicrruinr' 
fatur ; jEquivalcft dcbet unico ilK punftov ni^uod ccntriirft 
Ellipfeos, qtlddcrat linicum puhclum , abiiA,cctifendumcft^, 
Omhcs riirairdm diimc&i ih EHipfi ad cchtrum ciohverg;unt,?fc 
ih co*d6njuhgiihtur . Eacclcm ih Parabol^ tcrmihari dcbent^ a^ 
Ihfihitum cruribtisillis infinitis conclu{uhi,qu6d illi uhicd piui> 
"Cto asquiy ale t. Dum puhdum H eft bxtra H2,utravi^ e parte fit^ 
]>uri<JtiittlPeft in siIrcroramo,putictum'Ai,A:5 ultra ipfum.cx- 
Ctil^it : & unicoraomchto,Tcupuri<5kd temporiJ , quoHefti^ 
HijUtruraqueadpartesAidcbcteflrc inbnihicq infihito Q)ar 
tio,inquo,utdixirau^termihai:idebehtomn(:s illx diamctri » 
\q[U? ibidem fecuiii invicem*, & cum aic , & cdm ipfa recta gr- 
rantcfimul fingula? coirc deberenrquodartihlddo, iit etianipa^ 
rallela^um qliarumcumquc cohcurfus in ihfihito quddarnmo* 
^odelirefcitiEnigituritifihitunl fpatium cohdufuni cruribq^ 
infinitisVquodliccta^qualeKitredx totiumn^ue inlnfinituiii 
fcXfthfx y tamcn liriius piihfti pi-orfus iridiyiiiBilis hadiram ijt- 
fcAat,quod cUra illa infinita diftaintia ; quc dcbeat quodarombr 
do evadcrc huUa tiim; cuni ihfihitum atririgitiir ^ mirurri ih rhor 
cium cohfehtit: Nc£inc6Unttilraradddfpirio; qudd rcfpc6i:ii 
infihitae pcripheriae: deberct effc , ut piindtuih quoddahi, cjufmp- 
di unititeiii i fin1'pIiciUtcm,indi vifibiUlatenvmfihiti naiura.rc^ii' 
ijuirit ; fed etiam in univerfi ipfa ihfiriita pcripheria,5c pci^i&am 
in UhiVcrfa infihitsfe vcliiti fph^ioi fdpcrficieiqk* adato quovii 
piihdld quaqiiaVcrfdm eitcridituf in infinitum: Nim ubi Elli^ 
fis pcr parabolim irt Hypcrbolim miglat i cdhcurfus iile fcrai-. 
aiamcororum om!iiurti,qiiae ci partt cava concurrcbant in lihi- 
C6 ccdtroidifFunditUif qliodariimlcido pct' totos illds ciitiili qua- 
^astverfiim irifiniti Jlrcds i qui ihtercipiiiriiuf iis ifymptoto^ 
f ilrii arigtilisjquos diiitfanfvcrfu^ fccat,qiii ad toram iofinitikf 
|k:tiphtriimfurtf,titiidngiiliad quatiiorredtds: Qu?vi«chiiu 
fccta iri Hypcrbola a finito tjd$ ccridro tjgrcifa in iifdem angiilis 

jaccQsia9urntirtpctuiieauinhiaC|Sc ia^^uiffa qaaniptot^i- 



5j^ DE TRA^SFORMATIO "NE 
tflmre^^partccavainiafinltum, atqueiUaci invenic iniiiutuni 
fcntiutn Hypcrbol? analogam finit^ EUipfcos ccntro ,& iHm 
fLxtm conjug^tun;^ offendit^qui pariter finito axi EUipfeos conjii 
gal(( rerpondensHypcrbolaJividi^in binos inQnip^ratnosver^ 
fus fe cavQ$y& fi^is fingulQS focis pr^ditoSfUt axis conjugatusEU 
lipfeosfinitusipsadividit Inidaa^ cjiifmodii fjaitas ftmiellipfe$, 
^HCcedunt igitur i\ arcus infiniti uni^ocentro.in^ivifibili Ellip- 
^os,^ ejus n^t\]rarn afFectat\t.Si jam Uai acccdant Hyperbolr 
cppj^gat^ ramt,ip(i reliquuni ojiincquoct in reliquis afympto- 
torutn ^hgvdis fupcr^ft,in unicum {>.ari^r pqnctum nitentur cp: 
trahere,t^jamdebeatqviod^m^todoio()arta omnis periplieri^ 
qrci^iciripa Hyperbols centrum circiunquaq; infiniti. afTectarc 
^iiici iiidivifibilis punc;i natiJ\ram. Atqud, ipfum pertin,ebit ad o- 
ipnensifupcrf^cietn fphxras pariter infJhit£e>fiHypc^boIaspIanuiu 
cir^ a axcm conjugatum, gyr^ndo intcgram convcrfioincra ab- 
folvat^fi^ il^a Infinita peripheria puncto ^i\odammodGt a^quiva- 
letis^infinit; ipfius fphere fupcrficiem produca^t^qu; tota co ipfo^ 
^uodinfinita fir^un^ci n^ituram^fimplidtatem^indivifibilitatetti 
requirat;c^mq; ca.ra babefc Qmnino t^on poflit^ fed ia immen- 
fumftuger<;d;be^tvex ipfa quantit^tis i\aturafibi inh^rcflt^ t 
copipo^tioncmj^tq} diyifibilitatem» 8c partes^di^m duo ita conr 
trari^iti^rfe<;onj^ngcrc, S^copvilarevclutiftudcc» comradi« 
ftionem ii^voIvat,nccc(re eft » &^ impofi^bilis pmt^ino fi^t • 

886. Atque hoc d^nxum pa<;to licebit cti^oiic geomctricls hi- 
fceme4i(atio;iibusmcntem ^oIlcrc>aQ C^ivinx Immcnfitari^ 
gmplici^t^mfi^mmam ^4^^irari,quae ab Qmni partium com- 
pofitione: aUcnifl[imai(CamfummaNa;ur^ fimplicit^^t^jatqiimi- 
tate fummiii^finitinaturamconjitngii;^ ^ pcrfccttones oinnes 
Hiiro^atquc iiiexpUcabili acxu conjiinctas complectitur^ Iqfiai- 
faiTi ycnerabimiir majeftatcm perculfi j» at^ue attoaiti > ac hcrc* 
bimus admirabuqdi inf^nitan) iUam animo pervolvc ntes men- 
tis iqfinitas yimjqua, Sc hafce ipfas harum curvarum propricta- 
t^s pan multas» tam varias,tam miras, qi^as nds tam (origa ra- 
tiocinatiQhe,tc dcductipne tam molefta perfequitqr ^^ una cun^ 
aliisinfinitis infiniricsmagisarduis^atqi mirificLS4& pMlcherrif 
mis,atqqc ekgantifnmis fublimiortim curvarqm proprictatir 
bus, unicoiQtuiai»acfimplici(Iimacogtutioneperfpicii^ &ft« 
nitus comprchendit , 



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