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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 32, page id 1041163592:0066
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III
32
PTIILOSOPMILA NATORALIS
DEMoxD Corol. 5. Ec quoniam DB, ab lunt ultimo parallelae ee in dupli-
onronom cataratione iplarum AD. Ad: erunt areae ultimae curvilincae AD B,
Adb (ex nacura Parabolx) dux cerrixe partes triangulorum rectili.
neorum AD B, Adb; et legmenta AB, Ab Partes tertix co.
rundem triansulorum. Et inde lx arexe ec haec legmenta crunt in
criplicata ratione cum tangentium AD, Ad: tum chordarum et
arcuum AB, Ab.
Scholium.
Caeterum in his omnibus supponimus angulum contactus nec in-
Enice majorem esse angulis contactuum, quos Circuli continent cum
cansentibus luis, nec iildem infinite minorem; hoc est, curraturam
ad punctum A, nec intinite parvam esse nec infinite magnam, seu
incervallum AY finitae esse magnicudinis. Capi enim potest DB
uc AD3: quo in casu Circulus nullus per punctum A inter tangen-
tem AD ex curram AB duci potest, proindeque angulus contactus
erie intinite minor Circularibus. Ec simili argumento sI HaE DB
successire ut AD AD, AD, AD, ecc. habebitur series an
sulorum concactus persens in infinitum, quorum quilibet poste-
rIOT CA iNFINITE MINOF PPIOTC. EF AI AaC D B (UCCCNIVC UE A D,
ADZ, AD=, ADZ ADE, ADZ, _c. habebitur alia series infinica
angulorum contactus, quorum primus est ejuldem generis cum C ir-
cularibus, secundus intinice major, ec quiliber polterior intinite ma-
jor priore. Scd ec incer duos quolris ex his angulis potest scries
utrinque in infinitum pergens angulorum intermediorum inseri,
quorum quilibet posterior erie infinite major minorre priore. Vt
si inter terminos AD & AD inseratur series ADS, AD7,
AD2 AD AD3 AD, AD AD AD5, etC. EE ruI-
tus inter binos quolris ansulos hujus seriei inseri potest series no-
ra angulorum intermediorum ab inricem intinitis interrallis disse.
rencium. Aleque norie nacura linitem.
Duxe de curris lineis deque superticiebus comprehensis demon-
Araca Cunt, Facile applicantur ad solidorum superticies curras d
Contenta. Praemili vero haec Lemmaca, ue effuserem caedium dedu-
cendi perplexas demonstrationes, more reterum Cheometrarum, ad
absurdum. Concractiores enim redduntur demonstrationes per mee
thodum Indirisibilium. Sed quoniam durior est Indivisibilium hy-
pochesis, ex proxterea methodus illa minus Cicometrica censetur;
malui demonstrationes rerum seguentium ad ulcimas quantitatum
Cra-