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# Title: [Quo Continentur Claudii Ptolemaei, Porphyrii, Manuelis Bryennii Harmonica: Archimedis Arenarius, & Dimensio Circuli; Cum Eutocii Commentario: Aristarchi Samii, de Magnitudinibus & Distantiis Solis & Lunae, Liber: Pappi Alexandrini, Libri Secundi Collectaneorum, hactenus desiderati, Fragmentum ; Graece & Latine Edita, cum Notis ; Accedunt Epistolae nonnullae, rem Mathematicam spectantes; Et Opuscula quaedam Miscellanea] Quo Continentur Claudii Ptolemæi, Porphyrii, Manuelis Bryennii Harmonica: Archimedis Arenarius, & Dimensio Circuli; Cum Eutocii Commentario: Aristarchi Samii, de Magnitudinibus & Distantiis Solis & Lunæ, Liber: Pappi Alexandrini, Libri Secundi Collectaneorum, hactenus desiderati, Fragmentum ; Græce & Latine Edita, cum Notis ; Accedunt Epistolæ nonnullæ, rem Mathematicam spectantes; Et Opuscula quædam Miscellanea (shelfmark Leibn. Marg. 100:3)
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COLL P CTIO.
In Priori Diagrammate, primus cerminus valoris iplorum p, g, r, in prima columna, inre.
nitur diridendo primum cerminum Cummae proxime superioris, per coefticientem secundi termini
ejusdem TuMa (UE = F DEF TO, AUE O,OSF PEr FF, 23) de mutando signum Ouoci.
Ec idem terminus eodem fere modo invenitur in Sccundo Diagrammate. Sed hic pexecipua
difticultas est in inventione primi cermini Radicis. Id quod methodo genetan perticitur: Sed
hoc brevicaris gratia jam praetereo; ut ex alia quaedam qua ad concinnandam operatione spe-
ctant: neque enim hic compendia tradere racat. Sed dicam tantum in sellete, Ouod racix Cl-
jutris Aquationis lemel extracta, pro Regula resolrendi contimiles aequactones aterrari possit;
quodque ex pluribus ejusmodi Regulis, Regulam Generaliorem plerunque ettormare liceat; de
quod Radices omnes, lire limplices sine sire aftectae, modis infinitis extrahi postine; de quorum
simplicioribus icaque semper contulendum est.
Duomodo ex Aiquarionibus sic ad Intinitas Series reductis, Arex et Lonsitudines cuerarum,
Contenta et Superticies Colidorum, rel quorumlibet Sesmentorum figurarum quarumvis, eorum
que Centra Crraricatis determinantur; ec quomodo etiam Curra omnes Mechianica ad ejulmodi
A quationes Intinitarum Serierum reduci possine, indeque Problemata circa illas resolvi perinde
ac s Geometriex essent; nimis longum toret describere. Sulticiat Specimina quadam talium
Problematum recentuisse: Inque iis, brevicatis gratia, liceras A, B, C, D, dcc. pro terminis Se-
riei, licur Cub inicio, nonnunquam usurpabo.
T. Si ex dato Sinu Recto, vel Sinu Verto, Areus desideretur: Sit radtiusr, et sinus rectus v:
Fricque Arcus = ∞ F  + 3 
5X5 Xi v
6X7 rr
Orr 40 r  212 1 t c. Hoc elt, = x +
TXFX N N
2XZXpr
3X3 X X 
4u5 rr
7X7 ax D F _c. Velu siC a diameter, ac N SinuS verTuS; _ erit AreuS
F BKOrI
+
a
343 p 5 d3 + _c. HoC Clt, = Y a  in L + = f 3 XV
= d 3 _ = + 6a 3 40 d 222 d2
5 43
qad 222dd F dec
2. Si vicissim, ex daco Arcu desideretur Sinus: Sit ractius r, e arCUS Z: Fritque FInUS
rectus = 2 -
4X5 rr
HOC cst,
Srr 120 5040r  362880,0 = 88. Hoc c, = 2
62 7, C - eC. F+ sinus Vertus =
B -
22
32
2Z
2Y
24 73 720r 405L61
2KZY 3X4r 526r T 7uB C = _c.
3. Si Areus capiendus lic in ratione data ad alium arcum: P. sto diameter = d, chorda arcus
daci = X, et areus quaeticus ad arcum illum dacum ut n ad J: Fxieque arelis quaeliri Chorda
= n X 
2X3 a V X A + 2 = nn
n n
nn
4X5 au r. VB +
6X7 ao
BX C. f 40- nn
8XG  ~VD +
BT - in
TOntTA V F. + dec.
Obi nota, quod cum n cst numerus impar, leries detiner esse infinita, e cradet eadem qua
Prodic per rulsarem Alsebram ad mulciplicandum dacum ansulum per iltum numerum n.
4. Si in AxC alterUIrO AB, FIlip (eoS ADB (cujus centrum C
e axis alter DTT) decus punctum aliquod E, circa quod recta R. CF
OCcurrenS EIlipEI in C, motu angulari Feratur; _, ex data area
sectoris ElLipeici BF CI, quaratuE recta GF, quxe a puncto G ad
axem normalicer demicritur: FAo BC = 9, DC =r, F. B. = L
ac duplum areae BF. C = Z; F L exit GF
2099 - 991
T20 r H
3 = 2809 +504991 - 225911 91 4 dec
5040 g5 410
Sic itaque Astronomicum illud Repleri Problema relolvi potelt.
5. In eadem Fllipsi, si RatuatuT CD = 1,
BG
CD
= C, _ C F = ~. Erit Areus Ellipticus
2
c
2.23
5 _u f. eec
66
1O x C3
14 rrC
28
30
M m it M