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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)
# Folio 640, page id 867984570:0676
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=867984570&page=676
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 71 lines, status machine
# Lines: 71 recognised, mean confidence 0.88
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
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640
E PIS TOL A R V HI
vel f vel 8, =0; et n = F vet 2; eth series, in quas icta resolvantur, non potuerim in
Fpistola priori; nedum forre complcarerim; intentus, non in omnia particularia enumeranda
sed in illustrandam Mechodum per unam ec alteram in singulis rerum generibus instantiam, qua
ad ostendendam ejus generalitatem sufficere videbatur.
Caeterum haec Theoremata dane Series plusquam uno modo. Nam primum S Bonatur f = 0,
; unde prodit series nobis communicata. Sed s ponatur 208 = f f,
_ n = 2, inde randem obrinemus hanc Seriem L +. 3 - 5 - 7 + 3. + it = 1 =  _c. pro lonei-
cudine Ouadrantalis Arcus, cujus chorda est Onicas: rel, quod perinde clt, hanc L F 15
F 13 = 2 ec. pro lonsicudine Dimidii ejus. Ut has torte, quia a que simplices Tunt ac alTErX,
et magis convergunt, non repudiabitis.
Sed cso rem aliter aestimo. Ilud enim melius quod urilius est, et Problema minori labore solvic.
Sic, quamvis haec aquatio x = ∞ = 2, appareat simplieior hacce y y - 2y 18 - 120 = Y20.
namen in contelto elt polteriorem rerera simpliciorem elle, proprerea quod Radicem ejus y Ceo.
metra Facilius cruit.
Ft ob hanc rationem Series, pro obrinendis Arcubus Circuli, vel (quod eodem recidic) pro
obcinendis Sectoribus Coniearum Sectionum, pro oprimis habeo qux componuntur ex pocectari
bus Sinuum.
Nam liquis vellet per timplex compurum hujus Seriei + = 3 - = - ; = 3. = _c. colligere lon.
sicudinem Ouadrancis ad Viginci tigurarum loca decimalia, opus esser 5000000000 cerminis
Seriei circiter; ad quorum Calculum Milleni Anni reGUITerENEUT. EE res cardius obtineretr
per Pangencem 45 Graduum. Sed, actibito Sinu recto 45 graduum, Ouinquaginta quinque rel
Sexaginta cermini hujus seriei V = X F F E F 2a F ∞ _C. Susicerent: Ouorum computatio Fri-
bus, uc opinor, rel Ouatuor Diebus absolvi pollet.
Et tamen hic non est oprimus modus compurandi cotam Peripheriam. Ntam series ex sinu
recto 30 graduum rel linu verto 60 graduum conslara, muleo cirius dabit arcum Tuum; cujus
Textuplum vel duodecuplum est tora Peripheria. Neque majori labore eruitur area torius Cir-
culi ex segmento cujus Sagitta elt quadrans diamerri. Ejus Computi Specimen, liguidem ad ma-
nus est, rilum fuit apponere; e una adjungere aream Hyperbolae; quae eodem calculo prodit.
Polito Axe trantrerio = r, e sinu verto seu segmenci Saxitra = _; erit Semi segmentum
Hyperbola2
Cierii 5 = a in: 3 a L ~ ~ - = + = _c. Hac aucem series tic in intinitum produeitur,
Fit 2 2t = a. =  = b.  = = c. 3C = = d. 5 a X
cum Hyperbolae2
Circuli
=c. 7
X = f. _c. F.C erit Semi segmen-
a +  - - + 2 - = + L_c. Forumque semi summa = - -
++
P
c
_ semi disterentia - f = f L f _ec. Fis ita praeparatis, SuppOnO N = 4, quadrantem nempe
Axis; e prodit a (= 4) = 0. 25; B (- a 
0001953125: 4 C=
35∞ 9001953725
0.25
FXS) = 003125: C (- B~ 0O3125
278
) = 0000244240625. Et Sic proceao usque
dum venero ad terminum deprestistimum, qui potest ingredi opus. Deinde hos terminos ver
3, 5; 2 D. 12, ec. respectire diritos, dispono in duas Tabulas: Ambisuos, cum primo, in
unam; et Negatiros in aliam; et Acdo ut hic vides,
00833333333333353
62500000000000
27T26736TTTF
5135169390
14462691
195458)
9094
106
6.0002790778571420
3467906605F
634465027
15T290
8676
26265354
1663
0.069b10960504662 8
00002825719389575
Tunc