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# Piece: https://leibnizlegible.com/api/records/19857/text
# Catalogue record: 19857 — [Math. Aufz.]
# Incipit: (1) Visum mihi operae pretium est rationes excogitare, qvibus aeqvalit
# Date: 1697
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=19857&katnr=39639
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 103–104 (from the shelfmark LH 35, 12, 2 Bl. 103-104): 4 pages, canvases 205–208, page ids 00068195:0205 to 00068195:0208
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=205
# Lines: 218 recognised on 4 of 4 pages, mean confidence 0.82
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
# 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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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 103r — 00068195:0205
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 104v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000205.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 76 lines, status machine
sum mihi operae cretium est rationes encogitaro quibu
aequalitates figu
0
om in Elementis Euclideis demonstrata nar
transpositione exhiberi posent. Id em
afficit animo
dum demostrationem redit maxime ocularem. Specimen dare
ve opthaprico illo celeberrimo theoremate, quod continetur propo-
lume 47 ma libni imi Ecti Elementorum Euclidis. processum
ouis rem alsesutus sum nunc exhibeto
sit angulum Rectangulum BAC.  ostendem
paatio ABG2 et ac Cd super interibus AB, AC, aquari quadrato BCk) super
sypoterus BC rescripto descriptis quadratis laterum describamus quadeatum
Hpo quadiatum itu potenuse ad easdem partes, ut scilicet pro bona parte
incidat inquadiat. Cstetum; ata itque communibus demtis tantum ostendendum est
eligum aqualia esse . Ante omnia papet
aderentatus appositum quabrati lateris e punctum i cadere in GE et K in
rad tam Fd. et productas GE, Fd concurrem in H angulo recto et 
dinter d et H per A ducatur LT aequalis e parallela ipsi BI, secans BC
e
in L, et IkK n F,  et prodicta LT incidet in H quae og
facide estendi posent si opus toret. Sed nobis nunc investigaadte mathodi
ale
au
anctis facile patet quadratum ABGE aequari
rectangule BITL. Nam BE) hansponatur in AEH et ex Ap, Efet ABet
B+ C BL A et ex ABIH fit BI TL aequantur
tronsponatur
et ex ABGE. d est B BG+ + AB) Esit NI A A81Ce id est
fit st A sit ACH = ABIEi d e_ AOIH, id est ABI T + ITH. Transpo-
ratur i TH in EBLA, et ex ABI Y + , 7H fit XR ABI Y + BL. A
id est et ex ABGE, fit ABI H: rndrbus iF H transponatur ut BLA
et ex ABIH fit BITL. itaque ex ABG2 sola transpositione fit B1 TL,
verum esse  Theorema quod demonstrandum suscep
equantur ergo.
4) Quoniam igitur aliund scimus totum quadratum BCKI aquar
es restanus. PTL + rectanus. CKTL, aequari duobus quadratis crurum,
uani debere alteri quadrato. ACFd. quod si dacih ipsum. ic
L ostensum aequari uni ABGE, sequitur  rectam petum
fur
franspositios ACFD + Kd M idem est quod CAM + KFC, id est quod
ransposito KFC in HD A) da CAM = HDA, id est (transposito HTK in ALC
quo lnde rursus (transposito HTK in ALC) fiet CAM + A _ ALC. + AdK F id est
 CAM + ALC+ AYXM + kdM, id est CKTL + KdM, aequantur ergo
Fd + KDM et
5) sed ris set elegantior, si nihil quus sit assumi extriasceus, possintus
rectangulo reperirineadem omnes exhiberi eadem omnes partes quae
quadrato. Quod in utroque ventabimus. simulus vestigabimus numerum 
pune puo  anae
podditu cecilieum
BGE et sectanguli BI 7L, Cesieremus primum ab ABG2 abscindi deE debuit BG1
transponi posset in AEH, _fienique ABIH. Rursus ex ABIH, abscindi de debuit 1TH, ut transponi posset in BLA
altransponi potin et formari BI TL cepaa quia autem iTH et AE H communem habent
partem HEAC, patet a amen  ab AEH, abscindi debere HERT, et ab ABIE, abscindi
rabere 1ER ut ex ER et HERF conflatum 17H, transponi possit in BLA. quoniam
autem BGi et AEH idem sunt, tantum dransposita loco tantum mutato; ideo sectionem
ippus AEH per RT, reprosentabimus in BG1 per Q conguenter  Similiter quia ITH et BLA idem sunt
cotantum mutato; ideo sectionem ipsius i7H per ER repraesentabimus in BLA per SA
ingruenter. Quo facto jam efficitur, ut extraneo a HERT ad demonstrandum
equalitatem quadrati et rectanguli hic non sit opus nam hujus vicaria habentur
utroque nempe 14 19QP et ASAL. Unde talis prodit transformatio.
Quadiatum ABGE secetur in duo trapenia ABIR, et 1G2p, duoque triangula  1ER et BPZ
et manente ABIR transferatur 1G2P in ASAL, et 1ER in BSa, et denique BPE in ATR,
atque ex hii iisdem duobus trape jiis ABIZ, et ASAL duobusque triangulis B5A et ATR
fiet rectangulum BITL. Schema talé erit
(ANBIR = ABC
ABGE = 192 = A5AL
1ER = B5
LDPZ = ATK
+praeterea C secabit
Ad in M
(Aeia AEA
ABGC 1GQP ASA
EK
ASA
(BPZ AIR

## Folio 103v — 00068195:0206
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 104r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000206.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 101 lines, status machine
estat ut similiter oculari demonstratione exdibeanus
aequalitatem. quadrati ACFd, et rectanguli CKTL quod est prion
aequalitate paulo demonstratu difficilius sine extrinseci assunta
segmradem  tamen qua ante via, incedendum erit per onvest
gationem segmentorum. Primumigitur ut KFd transferatur in HdA, epod
tet ab ACFd resecari CFdM, restabitque cAM. Rursus HTK, cum transferrit
deleat in ALC resecari debet ab HDA. Cum etiam Denique etiam Kdm ause
in debet ut rest qua translatione babebitur ex AC M DI, fiet
A K T LCM D unde aufer refecari debet KdM ut fial Ck TL. sectiones
dem la port figurae transferendae faectas in faciendas in cino loc
amus et in alio um ergo k FC secta sit pe  deo per MD, ideo
imiliter HdA secetur per vio ductam congruenter. Et cum Hd A secta sit
TK ideo similiter KFC fecetur per FX et per KM ideo similiter
XFC secetur per =Y congruenter positam ipsi TK, et Y congruenter posita
 ipsi KM. et ut KM seravit Var secat alicubi tm, nempe in X; ita
ix secabit alicub. Md ut nempe in T. quia Et quiq YF  in KFC respon
et insi Kdm in HdA id unumque ex alio fit transpositione ipsius kFC in
Na dei sectiones ipsius XdM repraesentando in YF ip, repraesentabitur
W per Bd. Et = Y repraesentabitur per 42, et Y per ZC0. Postremo quia
vel HTK transfertur in ALC idso sectuis sctio ipsius X Z Y per X W
repadtentetur in ALCper No. ita d tandem assecnti sumus desideratum
n
mnes partes quadrati ACFd transpositae constituant rectangulus
Nam CAn manet utrique omnune, 2 L Z+ co transferetur
in A0HG Zd in NLCB, +BNC H in XAM XVAM, et denique 2G qued intrur
XTL
Unde talis pro CKTL, Unde talis prodit transformatio: quadratum  ACF
secetur in duo tria Trapezia BMC ,  DDGZ, ZPCo, et duo tri
angula CAM, ZFC n et manente CAM, transferatur _ ZFCd in AON,
et 398 in ON LC, et D M C( in XVAM, et denique  ZGY Co in KTVX, atque
ex iisdem tribus trapeziis durnus triangulis AX AM, et AN AON tribus
traperiis ONLC, XVAM, K TV. X fiet rectangulum CKTL. Schema
tale erit
ACAM = CAM
FC = AON
ACFd = DBGZ = ONLC = CKTL
DNCI = XVAM
(2YCO = KTIX
postquam ergo quae situm assecuti sumus remoto jam pegmate
omnibusque machiramentis ratiolinationis, rem ipsam velut de integro
aliis sic exhibere licet, hactenus dictis omnibus refecti dissimulatis:
Theorema: Quadrata crurum trianguli rectam
guli simul sumta, aequantur quadrabo Hypotenusae
demonstratio ocularis
sit Triangulum rectangulum BAC uius angulus rectus A, dico quadrata
super AB, et quadratum super AC simul sumta aequari quadrato super BC
constituantur quadrata ABGE super AB, etCACFd super AC; et easdem partes BCKI super
t
Nam ang 1BG = ung CBA, et BI = BC, et BG= BA, erqu. B
unt 1391 et BAC, et ang. 1GB = ang. CAB ergo angulus 1GB rec
BC adet i in GE
ad angulus BGE rees ex constructione ergo recta 1C, et EG cadunt in
gicum act eotur
1GB, a
G ipsi, erunt  id et E in directum, et Gt h eu Ac minor q
g. 1BG = aug.
minor GE(= AC
cadent G+, E
1B = BC et A = BA congruero
et ob ang. BGE reZ
ectam
alem
GerC
ex constructione, et 1GB etiam rectum (quia
aleo
irectum, et 1 in B ectum, majoremque kF (id est AB) ipsa D F, id est A C) cadere, 
in ret FK et eK secabit Ad,  sitque punctum intersectionis M
ectum et c
ob angalum rilectum, e
F, D, X, i
ostendemus  pusto Anm
in
d ttu
oqi di
15 et per sputatur par allela ipsi AC socans AB, BC, in 5 et A
dan
in A + se Sectonergo foi est quadratum ABGC in
per
le
iangula
in ABtK, 1G2P, et 
otidem traperia et triangula prio
scilicet duo ABIR ASAL (congruum ipsi 1GQp) et duo triangula
imiliter rectangulam BITL,
AFR (congrnum ipsi Beg) et BS A (congruum ipsi 1EK) ATR, BSA, quia Cergo
ute et
=ASAL, et BPZ= ATR, et 1EA= BSA, aequalia erunt
13p9 BGt E
Fnspectire prroribus congrua, trape
BIR, et +G2
(Trapez 1G2 = Traper. ASAL -p1IL rectang
ABGEquadrat Triang. BPZ =
triang. IER = Triang. BSAS
Ergo AB GE = B1 7 L.
Unde schema tale segmentorum congruentium n AGEstaBI
(Trapez. ABIR = Trape Z. ABIR
Triang. ATR
et AE lecabu
A
ABR
ABG2 et BITL

## Folio 104r — 00068195:0207
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 103v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000207.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 11 lines, status machine
tat
 prut ostendatur quadratum ACF aequari rectangulo CKTL
quod ut fiat duo sunt carsus per congruentia utriusque segmenta, duo sunt casus
R
el vel enim A7 est major quam CM vel
nonpriore casu sumatur in AT ipsa AV = CM. ex ducatur VX aequalis et parallet
vel enim AB e maj d in non est motor dupla AC, 
vel enim AB non est minor dupla AC, vel est minor. Priore casu etiam nk non
sit minor quam CM, et in  MK poterit sumi MX = CM. parallela per eselet
Et quia in hoccasu Md quoque imafur quam AM sumatur in Md ipsa M A = A  et comsle
eu paralle logramum q. A C congruum ipsi XV pa

## Folio 104v — 00068195:0208
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 103r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000208.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 30 lines, status machine
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