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# Piece: https://leibnizlegible.com/api/records/36240/text
# Catalogue record: 36240 — De progressionibus intervallorum tangentiu a vertice: Schedula tertia
# Incipit: Operae pretium videtur, quae de intervallis tangentium curvae paraboli
# Date: 1673-04
# Akademie-Ausgabe: AA VII,3 N. 017
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=36240&katnr=39655
# AA VII,3: the volume in the Repositorium des Leibniz-Archivs (CC BY-NC 4.0): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-bd-vii-3
# In the Akademie-Ausgabe: AA VII,3, p. 252 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/VII3A.pdf#page=252
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 123–124 (from the shelfmark LH 35, 12, 2 Bl. 123-124): 4 pages, canvases 245–248, page ids 00068195:0245 to 00068195:0248
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=245
# Lines: 222 recognised on 4 of 4 pages, mean confidence 0.81
# 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 123r — 00068195:0245
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 124v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000245.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 69 lines, status machine
curvie semiparabolicae
Opera pretium videfur, quae de intervallis Tangentium parabolae a
puncto in ejus applica axia altitudinis applicataeque aequalium a
communi, ratiocinari coepimus, absolvere, nam si ad progressionem
queundam simmiabilem reverari posset eorum series,  curvae
parabolicae rectam aequalem geometrice descriptam, ac proinde
perbu
Geometricam, non ut hactenus mechanicam tantum parabolae
quadraturam haberemus. Esto altitudo parabolae AB. aequalis applicatae
BC = a, DB = 2a ejus B. 4a. DBC = a. summa quadratorum
Sa Et RqSa = DC. Triangulum DBC. = 2a a2 = a. dividatu
per k5a, et productum duplcetur fiet 1a, RqSa 2=
Ra -Ta2
sa
4a 2a = BE. Intervalli Tangulis De.
A75. A95.
Assumatuo nuncalia altitudo minor quam
a puncto B.
nempe AF. cdjus ratio ad AB sit quaecunque (X, erit AF = (a). et Cumque rat
 quadrata applicatarum sint ut altitudines, et applicata prior BC.
fuerit a. erit ejus quadratum a multiplicetur per  fiet a
c et R4 = = (a erit applicata FG. Ejgs F-  addatur Y. kF: 4C
H ~
a eqe
fet a 4a- aa + 4a a= a a + 4a cum bqus cit. (19 a a +  sG
Triangulum DHG erit 24 a, 2. =  a) divit atm per HG
2
a
Rga
pudidumqu duplicetmr fiet: a seput
Bq + 4a
RgaX
(e90)
gae
e
_agat Rg. da +
Dividatur per HG productumque duplicetur, fiet: Aga
2a
eq
a
= FI Cumque sit ut HR ad AB. ita FI ad BK
Rga
ita stabit regula proportunum HF- 2a dat HB = (a + 4) quid datit FD
A9 a f4a
a ea 1a
L
pq 
R92
2a
Rga +4C =
Rq a + 4
(aa + a ) BK
Rja +44
Ergo si ratio altiludinum sit  1. Ratio intero allommeit.
24 aa+a. - 2 Rq. a + + a
nas aqu +a Rsat R45
si jam ponatur altitudo continue decrescere uniformiter, erit a= infinito e
reliqua infinit - 1X infinit - 2X. infin. 3X. etc et si partes sint
funtae erit esto potent esse a = 10 X. reliqua _9X. 8X. 7X etc.
ergo rationes ea obunt:  3o Z =  crgo 1o - a
Erit ergo a. modo 
1  10 etc. Nole sumum labissener ita inititur
1+ = + 1Z +1+ 2, +1+ Z etc
10d+ a
49 4 + 4
0a + a
A4  +
de
2

## Folio 123v — 00068195:0246
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 124r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000246.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
si in verlere parabol
Inquirere distantias vertiuis, a tangentibus parabolae.
Esto vertex A applicata BC. altitudo AB. apalicata BC tangens
CD. inter vallum tangentis a vertice, AE. calculum ita mibimus
AB esto a = Ad. BC. esto leb. BB= 2a. Dlum
DBC = 2ab _ de BC = b. _ de BD = 4a. et
DB= R9 4a + b Vlum DBC = 2 ab. Dividatur
per Hypotenusam DC. et productum duplicetur, abebimus BF.
2ab Ryta + b  Z quo rursus duplamto, habesi
dimidiato habebimus 2ab R4, 4a + b intervallo
tangentis D c. a vertice A. vel Ag. H a b
244 +6
Esto b 2a. BC. R92 b. BD4 a
H. de BC 2B. H de BD. 164 et Cd = Rq
R9, 2b + 16a. Dlum DB = E2a A9 2b.
doidatur per De fiet productuque duplicetur, fite rursus dimidretur
fiet 272a R42b, R9 2b + 16a  X.
seu
2.
4a 2b saba
= AE. intervallo.
b 164426
d est relinquatur
Et si AB ponatur Ba. erit intervallum Tangentis
a vertice: R9
vel sic. H 4a
Aq Bq bae
ib
vel sic: 442
R4
8a +1
2a
A4
e
9a 3b (7ab9
364+ 3b
ga
3ba
362
41
9a
Et si 1B. ta Rq
n
16a
164c1
164
12451 16a +1
3a M
2
44
Aga , 1. Aq a +1 R(a + 1,
16a 4b (B1a5
644+ 462
scilicet altitudine minima, seu quavis dabrli assiu nabili
minore. Pendere videtur aestimatio illa, qua a quantitale
erit altitudo illa aa. eique aequalis applicata. Jam
Illud tantum quaestro nunc est,  an ratio ipsius a. ad b.
seu applicatae ad altitudinem, investigari possit, sumta
ejus altitudines, in qua applicata fit  altitudini aequalis.
sisto enim altitudinis illius istius ad minimam a. ratio a.
applicatarum quadrata sunt ut altitudines, ergo cum altitudinum
ratu sit X. erit et quadrato cum applicalarum. Quadratum
applicatae altitudini aequalis est a a. quod si dividatur per a
habebitur aa cujus R9. Cerit applicata altitudinis a.
nempe R9 L, a.
ec

## Folio 124r — 00068195:0247
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 123v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000247.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 44 lines, status machine
Unde intelligi potest res mirabilis, appliculam altitudinus
quilibet assignabili minoris esse mediam proportionalem inter altitu
dinem qualibet dabili minerem, et eam, in qua a applicata basi
quae applicatae suae aequalis est. seu quadratum applicatae esse aequale
rectangulo altitudinis applicalae suae aequalis cum altitudine minima.
Hinc data e altitudine applicatae suae aequali, c. erit
d = C. et b =
Rg4.
Ergo intervalla tangentium ita stabunt:
36
R42 a + 1. = Bg12 +a
hg1ofa
latus currae, minus qualibet nca asigatit
Quod si jam sint Triangula infinita, quorum basis sit β.
altitudines intervalla tangentium, Triangula ipsa erunt.
3c ( 4c(
SC B
etc. Quorum infinita simul
2+ a _43 +  144+ a N4 +  sumta, aequantur;
x
Segmento parabolico, seu residuo semiparabolae Triangulo inscripto
adem to quod si ergo praboli Rectangulum Parabolae incircumscriptum
sit 3. et parabola 2. et Triangulum inscriptum 3 erit
seg mertum Parabolicum 2 - 3 seu 4 - =  = Ac
proinde si semiparabolae altitudo et et applicata sit c. cum
Rectangulum futurum sit c. et rarabola et Triangulum 
et Parabola 2C erit segmentum 2C c - 4c 3C
Ergo C- erit omnibus illis Triangulis simul sumtis, dividatur
utraque aequationis pars per c. Erit E=
si tamen B. maximum horum se
ultinum (nam ininfinitum progredi
alioqum licet) definiatur.
EA 
a
+dC
Rtea
et.
c are
ha 4c
14 +1 146+L
t
8+ a 3 2+4 Egsta
ota
et

## Folio 124v — 00068195:0248
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 123r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000248.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 43 lines, status machine
sinus est media proportionalis inter radium, e parallelari faxi
parabolae
ins
15. 1. R9 15
1. 4 ey 12
16.9
16- 16 R9 0
Figura conflata ex quadratis sinnum; aequatur cylindro
cujus basis est quadrans, altitudo radius.
Eadem figura conflata ex quadratis smnum, in est ad
Chemispharis figuram conflatam ex circulis quadrantibus eorundem sinnum
seu ad quadratem) ut quadratum ad quadrantem inscriptum fasum
esto quad rans X radius a. Erit figura conflata ex quadatis
sinuum Xa. At figura conflata ex quadrantibus 2Xa
esset ergo quadratum ad quadrantem, ut 3. ad 2.
Ergo porto quad ato 3. erit Radius A9 3. radio a
cum quadrans fiat ox radio in semiquatrantem peripheriae
ducto, ergo Axa posito quod rato 3. erit Radius
R4 3. quo si dividatur quadrans 2, et productum
duplicetur, habebinus, circumferentram arcum quadrantis
s credere fas est:
2
LR43
2
1 BR43. BR43
.
et arcum Circuli L Radio posito R43. et diametro
16
R43
Rq12
peripheriam
Rg 12. Ergo ratio peripheriae ad diametrum erit:
16 16 quod est absurdu
R43b
Falsum ergofiguram conflatam ex summis quadratis sinuum esse
RqN
ad figurum conflatam ex quadantibus, ut quadculum ad quadrantem.
Ergo valde cavendum ne indivisibilibus abutamur.
16. A17 20 24 19
Y2 16 520 424 420 432 436
4 9 16 25
R 1. 2 3. Ag 5. R9 7. Rg. 9. R_ 11. 13 15. 17, 19. 21 23. 25
10 + 4