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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 155v, page id 00067993:0312
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1 de tmeus
1d, semartundum
a tangens p
i  nar  B
lersaus S
iPiC LAS
propositio decima tertia
in Curva Analytica simplici portio inter tangentis et ordinotae occurue
intercepta, est ad abscissam, ut in aequatione, curvae un relationem ad axem explicante
diquitatium apsis proportionalium ab ordadat abscissis, nen e t
autem vocant
ignitatem
vantitatem ipsam aut et
tungens est ab ordinata
Hoc Theorema ex d
is vertice s choliu
adiam puce contiacia
Hujus. Deu theorematis demonstratio ex doctissima Riccii
H excrecitatione de Maxinus et miminis pete potest. Nobis id explicare
hoc loco suffecerit:  sit curva sinalus
1C. e. CC e 1rens sit curva renalytica simplex a1C1CDC. Axis  opos est productus AB
B+2. 5+ P..
8 + 2: G + 2 1 co: e
eB + eB ( coG + co S- c0 2.
fig. T. cui occurrunt tangentes CD in punctis,  ajo ipsam 1B, d esse ad ipsam
1B A, ut est exponens dignitatis ordnatomi ad coponentem dignitatis absissam
Nempe si curva sit Parabola et  C, sint ut abscissae ipsae AB, exponens
ordinatarum erit 2. abscissarum 1. adeoque BB ipsius ABA dupla si inverso
modo sum in parabola Cubica erit tripla in quadrato quadintica quadruplas
et 0G aos e in Semicubicali, ubi cubi ordinatarmm sunt ut abscissarum quadr
al CB + CQ1 GC0 + eco
et eB-Gc0.  2c0- 
ed. . A
(de ju supra..
Aquod ex ipsa harum
sinamr constitutione
1.B1 D, ad 1BA ut 3 ad 2.
erit
coniderem
Si easdem figuras inverso assmamus odo
et in fig.... fermamus pro axc; axem praecedentis conjugatum, erunt
in Parabola ordinatae, in subduplicata ratione abscissorum, seu ab ordinatae
erunt ut abscissarum  quadrata; in cubi unde exponentes earum erunt ut 1. ad L
adeoque erit 18. D dimidrie ipsius A1B., et in parabola cubica, triens, in semicubicae
ut 2 ad 3
denique in fius. Si figura sit ex reciprocarum
numero, erit idem obtinebit, et in hoc erit discrimen, quod ipsa- B punctum D non
ordinatae sunt ix reciproce in ratione abscissarum, exponbrtes erunt ut 1.
ad 1. seu aequales adeoque A1B aeq. 1BB. Si esset Hyperbola bica, foret
cubi quadrata abscissurum ordinatarum ut abscissae, vel (cuna invense resumtos
quadrata abscissarum ut ordinatae, priore ca casa AIB iu 10. D erit dupla,
versus ord a puncto 1B versus A verticem, sed in contrariam stsmetur+
proportio autem easdema servabit leges ut in hyperbola simplici ubi
posteriore dimidia ipsius A, B.
A aequatio est:  e B o ge coC. quaritur SN. seu  y2, p ad
PII A ABCA. ABCA + ACA + AB CG
AB, y. B Dri B
ser MBC M
AN AC N ABDN, 2M AC M
tausus  BD est ad BC:: ∞ - e ad Co.
NA B D NrisoD
sve Mqem M
cone. Ergo EmpCM
En A BCm e macm
mancm
na Bem - ABC
uca. 116. 8 li  Bo B am.  D aB
c a
iguram a lyticam simplicem quamlibet Quadrare.
sit Aris ordinctarum portio quaelibet, ut 1 1C282B∞C, C a
1B 2B, duabus ordinatis
AAIC L .C. per prop. 2. q
IvIBeD.D : 1C18282CL. C i:  co - e in 0, quia singulae ordinatae ad singulat
AN, 1d 1B2B 2 DVD aeq
Ergo 2Ai Ct 2C
AICLC4
aq
nenpe 10.1
ici qdertiC.: c - e: a
AG
Ergo diff inter 1C1B+BC, C + A1B1C, et ACB. C   ad 1C1B. B EC :: ( - C d, 10
Ergo seu diss inter  (D) + A. B  et RBC
aeq. () - e
jam M AC M + ABC,
portione directicis  (sive axis opdinatarum
tos procedar
etBZBICIC
C1BLBICE
ss preca
ie sre ut ae eadio
Engp PEB
sed methodo quae possit esse universalis
4t q