# Leibniz Legible — https://leibnizlegible.com # Page: https://leibnizlegible.com/page/00067993:0464 # Permalink to this reading: https://leibnizlegible.com/page/00067993:0464?run=44 # Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1 # Folio 231v, page id 00067993:0464 # Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 232r); this page is its left half. # Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00067993&page=464 # Source image: https://digitale-sammlungen.gwlb.de/content/00067993/jpgs/default/00000464.jpg # Model: FoNDUE-GD_v2_ft_Leibniz, run 44, 2026-08-27T14:25:39Z — 87 lines, status machine # Lines: 87 recognised, mean confidence 0.91 # Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German. # Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text. # Wording rule: say "the machine reads it as …", never "Leibniz wrote". # After the first blank line: the recognised lines in reading order, one per line. la similia, AdC. et DEG erro Ad - DC. AC prop. 40 D5 Ad DG= EDCDC. secans exsplementi in e differentram inter tangentem complementi ed sinum complementi, aequatur dff differentiae inter secautem complementi et radium in tangentem complementu. seu seu gh- gf = hg - ha. ADeEGC Ed AC. secans complementi in differentiam radii a sinu, aequatur radio in differentiam secantis complemen. ergo basis AF = vasi A H. Ergo gf- ha. quod jam supra prop. 20 ti et radii. seu momentum radiorum, demto momento finnum. Ba - ob = ag - a. adde prop. 29. DC EG= DG AC, tangens compl. in differentam sec complementi a radio, aequatur differentae randio in differen tiam tangentis complementi a radio cumque posterior terminus aequationis sit qua det summam quadrabilem eg erit quadrabilis prior quoque; Ergo Tangens complementi Coroll. 1. Tangensi complementi in differentiam secantem complementi a radio quadrari potest. Coroll. 2. u Ergo ductus tangentium complementi in secantes complementi quadrari potest Nam ductus Tangentium complementi in differentias secantium complementia radio qui quadrari potest, aequalur dactui tangentium complementi in secantes complementi, demto radio in tangentes complementi, qui etium quad radii ductus seu cylinder, etiam quadrari potest; Ergo et residuum seu ductus tangentium complementi in secantes complementi. Vla similia: ABR et AEH. ergo HE - AB. AH Est autem Eh = Cd. et AR = Ad. et Plum ss le t simile HE AE lo AdC tionalis inter stm tangentem acu dati, et tangentem arcus o este prop dati auctum tangente arcu complementi. Atque ideo quadrata tangent recantium aequantur quadratis tangentium, ductuo tangentium in tangentes complementi auctos. At sup. prop. 11. ostensum esst quadrata secantium, dento cylindio hyperbolie subradio aequari sangentiii scipsum, demt tange quadratis tangen- ttrumdemto ductu tangentium in tanusentes arcus dimidii, se seu demto, cylindio hyperbolico addito b radii Ergu coroll. 1. quadrata summa quadrata tangentium, quadrata secantium, a= quadratis tangentium Hac ita posito patet tangentem arcu quadrato radii in altitudinem, seu a vertice circuli abscissam, arctis. dati in tangentem arcus complementisaequari quadrato radii, quadj am habidimus prop. 33. pp. coer Hasimilia: A. ER et AEH. ergo AR. ER cd AE. Triangulum AEA. aequale et simile Vlo ADC propcto. CdeHE = AC et. sed hoc jam habumus. propieus 3. et passimo. HR AE = AH AR. summa tangentiem, arus composita ex tangente et tangente arcus complementi in radium aequatur secanti in secantem complementi; hoc facie conciliabiscum prop. 36. et 28. AR HE = AEAH, habuimus, vide prop. 32. AH Ila similia: AHR. e AdC. ergo HR. AR Quadrg torad. in arcu prop. 50 HR DC = Ad AR: tangens tangente complementi auctus in tangentem complementi, aequatur quadraeo secantis Ad DC AC trop. 33. 37. complementi. Seu secans complementi est media proportionalis inter tangentem complementi, et tangentem complementi, tangente auctum, confer cum prop. 43. Ergo quadrata secantrum coni– plementi excedunt quadrata tangentium complementi, ductu tangentium complementi in tan- gentes complementi, at ducus tangentium i Ergo Differentia inter quadratum tangentis et 13 Secantis, aequatur differentiae inter quadratum tangentis complementi, et quadratum secantis complementi. eaque differentia est quadrabilis per prop. 43. coroll. 1 HR AC = AD AH: haec jam ex prop. 28. et 35 AR AC= DCAH. habuimus prop. 31. XE. = XH =DEH KHAEKF EF tangens in differentiam inter sinum et Tadium, aequatur Tangentianus semixomplement GF spatis conchogidalis in circulare (qui porp. 12 F Eq Ila similia HKE et EFG. Dento cylindio hyperbolico quad rabilis est, prop. 20) aequatur ductui Tangentium semicamplementi in sinus seu portionem circularem porro ductus. secantium s complementi in tangentes remicom plementi (qui sunt a. posito a radio b simi) in tangentes semicomplementi esiquatur cylindro tangentium complementi demto cylindio tangentium semicomplementi per prop. 38. et 12. quod esto X a erit a, e -+ quaeramus rationem a e ad be jea erit a seu si ergo si ductus secantium complementi in Tangates seui complementi sit ut a ductus arularo erit ut b. Et vicissim ductus secant ni tanu. semiuse sin in tanus. semi compl. Ergo sin in tanus serrigel. Scailt ypl. in tan. somispl. atque ideo sin. in tang. temicompl. = secans coplicituer el. tam Ergo ontunder lanus. complementi - tunu. semicompl. in b = colird. cnchoeid + hop otalcy l. yp. - al qus quadia Coroll. dimensio ductus circularis on fi guram tangentium sernicomplementi supponit dimensionem conmdoeidlis HE GF = HHCEF tangens in disferentiam sinus complementi et tangentae semicomplementi, aequatur langenti semicomplementi in differentiam secantis et sinus complementi. Ergo ef- ce = ed - ef ergo cf - c = d.) et hyperbolae. Corollicuo summa quartarum proportionatium, ad has tres, seu tangentem splementi, tangentem sinum conplementi, aequatur c+ +, ef- Soroll 2 Aliters ratio hyperbalicdeto aliquo quad rabili edet ce. Ergo seu monduta tangens aquantur in secantibus in tangentes semi- antur additaue momentis lange utium arcusdud complementi, additis tangentibus in tangenles semicomplementi. pop. 2 sip (B . tam per pap. 25. Supra ed trum alcus dimidii a prop. 42 prop. 23. prot.LI. prop. 44. Pop. 45. prop. 46. num 2.