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# Piece: https://leibnizlegible.com/api/records/47343/text
# Catalogue record: 47343 — Leibniz an Hermann
# Incipit: Operae pretium est Harrioti theorema
# Date: 1707-06-24
# Sender: Leibniz — Addressee: Hermann
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=47343&katnr=45488
# Work: https://leibnizlegible.com/work/DE-611-HS-973581
# Title: LBr. 396 · Hermann
# Folios: Bl. 46 (from the shelfmark LBr 396 Bl. 46): 2 pages, canvases 91–92, page ids DE-611-HS-973581:0091 to DE-611-HS-973581:0092
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-973581&page=91
# Lines: 85 recognised on 2 of 2 pages, mean confidence 0.87
# 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.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
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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 46r — DE-611-HS-973581:0091
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-973581/jpgs/default/00000091.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 315, 2026-09-10T22:07:11Z — 54 lines, status machine
Ex literis meisad du. termanuum 29 jun. 1707
operae pretium est Harrioti heorema demonstrari
nam multa inde egregia colligi poterunt Exempli
causa sit quaecunque formula habens meras radices
reales falsa(, 10X + 11X + 12 X  + 13X -3 + eM
et multiplicetur per 20x - 21 prodibit
n. 3
n. 3
n. 5
1020X + 11. 20 X + 12. 20X + 13. 20 X  + 1420X + 5. 20X +1 6. 20X
et
= 10.21
I1. 21
= 12. 21 13.21 14. 2 - 15. 21 E
Cum ergo primus terminus producti sit affirmativus, et
ultimus negativus et per Theorema farrioti, non nisi una
mulatio signorum in producto esse possit; sequitur, uno ex terminis
producti existente affirmativo, omnes praecedentes esse affirmativos
1  uta e tet. er deua ons spento es
Hinc iquanditates 40
14 15 10
2, 13.
ve
131
1
)
gaticis
eam iner se habitudinem habebunt, ut si una velut
1c
sit major quam 20 etiam praecedentes velut , sint
20
majores vel saltem non minores quam 2. Ergo 13 non
1
potest esse minor quam 14. Et cum eae demquantitates
similiter eam inter se habitudinem habeant, ut si unel
 sint
minores vel saltem non majores quam 21. Ergo 13 non
eo
20
14
postaesse major quam 14. Et generaliter fractio prior
non potest esse minor fractione posteriore. jam si
radices formulae X + H X + et (posito 10 =1) po-
nantur esse xta, xib, x+C, X+d. patet fore
I = a +(, + C + etC et 12 = ab + ac + etc et
B = abc + et C et 14 abcd + ets. Et ita porro. Unde
actionem
nasitur teoremagenerale: actin ortam exsnt summa
inferiorum non posse
combinationum proxime
esse minorem fractione alia rimiliter tacta
ex combinationibus altioribus. Nempe sam
er
fractio ex summa binionum, divisa per summam

## Folio 46v — DE-611-HS-973581:0092
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-973581/jpgs/default/00000092.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 315, 2026-09-10T22:07:11Z — 31 lines, status machine
unionum, non potet esse mino) quamfractio
ex summa ternionum divisa ex summa
binionum, et ita porro. Hinc efiam producto
ex summa unionum in summam ternionum
non potest mo esse majus quadrato ex summa
vinionum, et ita simiciter in aliis. ita
elegantia circa combinationes ex Harrioti
theoremate supposita derivabuntur quod si
ciliunde talia de combinationibus deriventur
hinc demonstrari poterit heorema
Harrioti. ed hon sine ambitu. praestabit
tamen aliquam ejus demon rationem haberi
quam nullam qua non sine magno scientia
derectu actenus caremus.
forte simili generali demonstratione generaliter hinc ostendetu
multiplicata formula per veram radicem, augeri numerum mutationum
t est affirmativa in casu consecutionis, negitiva in casu mutationa s
Xa aXX habX + abc
Ie2 est negativa in
lasu mutatinis affir-
mativa in eosu con secu
tionis
b. 9 = Xb. 1 = X
XI.
1.
+. X11 + XX 9 +, X
et XX 1 +cX
+ X11 + XX9 +
C = E. 2.
A
+t