# Leibniz Legible — https://leibnizlegible.com # Page: https://leibnizlegible.com/page/00068069:0011 # Permalink to this reading: https://leibnizlegible.com/page/00068069:0011?run=73 # Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 24 # Folio 6r, page id 00068069:0011 # Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 1v); this page is its right half. # Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068069&page=11 # Source image: https://digitale-sammlungen.gwlb.de/content/00068069/jpgs/default/00000011.jpg # Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 105 lines, status machine # Lines: 105 recognised, mean confidence 0.82 # 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. pag.7 Verum enim vero tenalior est varietatis suae pulderrimae Natura resum rerum, aeternarum veritatum parens, id est divina Mens, quam ut omnia sub unum senus compingi patiatur. itaque elegans et mirabile effugium reperit, in illo Analyseos miraculo, idealis mundi monstro, pene inter Ens non eqet non Ens Amphibio, quod radicem imaginariam appellamues. It quoties aties Denominator Fractionis Rationalis habet radices imaginarias, quod infinitis modis continosit Hyperbola quoque cujus opus esset quadratura, fieret imaginaria, construique nullo modo posset. Sed quia quaevis radices imaginariae suas compares habent, oriuntur enim extrahendo radicem quadraticam ex quantitate privativa; extractio autem quadratica omnis duplex est, ut signos. ut notae ejus Y.. praefigi possit + vel -. Hinc ex productio talium radicum imaginariarum debito invicem ductu prodit oritur productum reale quod vel erit ipse denominator, eoque casu quadatura (si n ad numeratorem constantem reducta sit fractio quodratura desiderta notollaturque si placet terminus secundus formulae) quadratura proposita non potesthc reduci ad simpliciorem; vel producatur divisor reali aliquis realis divisor reali dei potest simplicor ab alia simpli simpliciore pendet, qualit Ex. gr. sit formula it denominatoris, ejusque ope quadratur proposita fractio patet denominatoris radices esse XtI, X - 1, X + -1, X - -1, quae in se invicem multiplicatae producnt xt - 1. Et erit per regulam, arta, - a- 1, f-1 -13 XtI ++I, t AHI, -t 1, X - 1 +1- I, -1 - Y.1, - -1 - v. 1, X + f- 1 +1+ S- 1, - 1+- 1, + 1∞1+N - 1, X - L-1 vel erit aequ. 4, uri ve 1 4, un 4,X1 4+1, X+ sp1 4s - 1, X - - 1 vel erit (de vel s non possunt ad Hyperbolem nisi imaginariam X-1 4 Ubi e t revocari. Jungendo ergo tot radices imaginarias inter se, quot ad XV I +1 Axr-1- 1 est ideo hoc loco summando duas posteriores fractiones, nempe tollendam tolter prodibit. xs -1 +, = Bx Y- 1 +1 4Xf- 1 - 4, 4XY- 1 +4 . 4, _XYI x L - 1 - 1, XF- 1 + 1 . Est erga fieret inde id est 2 itI exsi vellemus similiter in unum aggregare axt 4XI. 1 . Et aggregando in unum 2, XXOI 2, XX + 1 2, XX - 1 4, x 4, XtI 2, Xx HI redibit. t-1 +Es quod adeo est ped Unde patet pendere Unde patet (de pendere ex Quadratura Ciuliet Hyperbolae ert et circuli simul. Nam (dx et (dx adeoque et f dx pendere 4- 1 x+1 DXX- 1 ex Quadratura Hyperbolae, dudum constabat: sed (dx pendere X-1 Xx+1 Ex h. Hinc sequitur omnes actionos, qdivisores reales habent simplices primi gradus ua into Theorema, Diametri existente omales X4. 1 1 + 1 - 1 etc. Hinc sequitur omnium tigururum) emubi genominatorin valore ordinatae Are ax vel etiam s t ex Quadratura Circuli a me primum cum quadratura mea Arithmetica ut xte, reduci posse ad Quadraturem Hyperbolae; e cum vero divisores reales habet planos seu, secundi gradus, (qui scilicet ipsimet (termino, ut XX + ae, non habent, radices aequales, alioqui aducunt ad Quadraturas absolutas) ut XX + fx + ag, vel sublato secundo uctum pi domuen tpute a tiut ae e t 27 21. 26.,- 1 s-1 +1 + tiL. 1, + 1 -1 se 1 + (.1 - + 4V -1 1 i 4Y sa = x+1 + x+1 = 4, XX - 1 2, XX - 1 XX + 1 = xX +1 2, Xt-1 uamm ali Rationalio