# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/851708439:0573
# Permalink to this reading: https://leibnizlegible.com/page/851708439:0573?run=74
# Title: [Johannis Wallis S. T. D. Geometriae Professoris Saviliani, in Celeberrima Academia Oxoniensi, De Algebra Tractatus; Historicus & Practicus. Anno 1685 Anglice editus; Nunc Auctus Latine. Cum variis Appendicibus; Partim prius editis Anglice, Partim nunc primum editis] Johannis Wallis S. T. D. Geometriæ Professoris Saviliani, in Celeberrima Academia Oxoniensi, De Algebra Tractatus; Historicus & Practicus. Anno 1685 Anglice editus; Nunc Auctus Latine. Cum variis Appendicibus; Partim prius editis Anglice, Partim nunc primum editis (shelfmarks Leibn. Marg. 100:1/2, Stück 2; Leibn. Marg. 100:1/2)
# Folio 549, page id 851708439:0573
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=851708439&page=573
# Source image: https://digitale-sammlungen.gwlb.de/content/851708439/jpgs/default/00000573.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 54 lines, status machine
# Lines: 54 recognised, mean confidence 0.89
# 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.

Cap. III
AN GAE A R EBI5.
ductum (459R9 + + PgRg = 4 R99 = 259 + 2P9 = 2Rg in 2R4) a
quatur Biquacratis crurum, (549 = P49.)
45. F.Aque hac Tertia methodus Aadendi Bignadrata.
46. Irem; In tali triansulo (cujus angulus in verrice est gradtuum 45,  uter
que ad basim acurus:) Si duplum assreeati quadratorum crurum, ducatur in
quodratum basis: Productum aequatur Biquadratis omnium laterum. Cum enim
lic 599 + P99 = 459R9 f 4 P gR9 = 4R99; erit ereo 594 F P99 
(4R49 =) 94 = 454R9 f + P9R9 = 259 f 2P9 1 2R9.
47. FIxC Theoremata Seorsim demonstrata; sve angulus ad verticem LIE BC2-
duum 135, live graduum 45; ee sire ansulorum ad balin sic alier obcusus, sire
REErque aCutS, hoc c, lire criangulum lic obtusangulum, sive acuranoulum
(quorum urxiris reteramus rectangulum) ad haec generalia rerocemus.
48. In trianeulo rectilineo, cujus aneulus in verrice sit vel graduum 135, vet
eraduum 45; duplum aecresati quadratorum crurum, dempto quadrato basis, du
etum in quadratum basis; aequatur biquadratis crurum. (Per 531, 37,44.) Ouae
est Addirio Biquadratorum.
43. Fdemque duplum agcrecati quadratorum crurum, ductum in quadratum
vasis; aequatur omnium laterum Biquadratis. (ver 6 34, 39, 46.)
50. PorrO; AE quatio illa (529) + AgR9. F + F gR9- + R99 = A99+ P99
(aliaque huic limilia 5 35, 42;) elt squatio Ouadratica ex Radice Plana: Cu
jus raclix ct 2 Rq; et, co efticiens medii termini 2 Ag F 2 F. 9; qua itaque aequa-
TuE TumIX duacum quancitatum quarum rectansutum aequatur termino absoluto
A99 f H. 99.
5F. Si iaque eam tractemus pro more aliarum aequationum ejusdem Forma:
adeoque ab A99 F 2 AgF g F P. gq (quadrato semi coetticiencis) auteramus
cerminum absolutum) A99. F P gq: reliduum est 2 AgF. 9. Cujus radix qua
dratica V 2 A gE g = A F V 2, addita vel dempra semi coesticienti, extribet hujus
aequationis Radicem, A9 f H. g = A F. V 2 = 2 R9 = 9.
52. Sed hujus ambigux aequacionis, manitestum est, radicem Majorem eligen
dam clic, pro calu 520; quoniam angulus in verrice (Srac. 135) est recto ma-
jor; adeoque quactratum balis 99, majus esse debet, quam duo quactrata crurum
Ag = R. 9. Ideoque A9 L F g F AE V 2 = 2 R 9 = 9. HOC cst,
53. Si quadrais crarum continentium ansulum sesquiquadrantalem (seu era-
duum 235) addatur rectaneulum eorundem crurum in Y2 ductum; aecresatum
aequatur quadrato basis.
S. Simuliter oltendeur; quod Aiquaciones 535, Eunc icem quadraciea Xqua.
tiones ex ractice plana 2 R 9: Puta, P99 F A99 = q P gR 9 F + A 9R9 4R99.
ec 594 + F99 = 4SgR9 F 4 FgR9 - 4R99: FF IIla 8 42; 599 + P9c
=45qR9 F + PgRg - 4R99. Sed in his omnibus, maniseltum ct, eli-
sendam este (aequationis ambisux) radicem Minorem (non majorem, ut 552.
eo guod ansulus in verrice (srad. 45.) est recto minor; adeoque quadrarum balis
minus esse debet quam duorum crurum quadrata. Adeoque earum radices TuE
P9 = A9 - P AY2 = 2R9, 859 F F9 -SF. V2 2R4, _cS9 + Pg- SPY2
= 2Rg = 9. Hoe est,
55. Si a quadratis crarum concinenrium anculum eractuum 45 (seu recti semis
Tem) auseratur eorundem crurum rectansulum in V 2 ductum; Residuum aequa
cur quadrato basis.
50. Vel utraque cheoremala HC ComsMEaS; Si quaatals etmi onmi
angulum erad. 135, addatur: vel conrinentium aneulum erad. 45, auferatur: eo
rundemerurum rectansulum in V2 auetum: quod prodit, aequabitur quadrato ba-
Sid. Det 3 35155
57. Vetum.