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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 133r, page id 00068197:0265
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(4
140)
seu trilineum AE BF A ag trap ei AHBE. trilim AHBFA. sed Trap AABC= Emper 15B Ex hilamus 4G
et traper AGBE =   volanus AB _ (seu Xy) - trianus AGH (seu gy: 2) et trianus BHG p Z7:2
Ergo fiet denum  Tuilin. AEBF A6q. Xy - gy: 2 + 72. 2 - frilin. AHBFA et pro trilineo AEBFA substituendo
valorem ex aq. 18 fiet pilin. AABFA = 2 nX . y - = X: yy - xy + 9y: 2-27:2 est autem
Xy C (per 20. 21) +6177- vv, : 2 ergo = xy = 22. 2 (= _ - 7Z + vv. 2 et yy, = 2 1V + vV. 2. adeoque
=Xy + 9y: 2 - Z2: 2 2) (- 77 + vr: 2 + Z7: 4 - Zv: 2 + vr: 4 -) 322: 4 + 3vv. 4 - Zv. 2
Z AXy8- ZX: gy = 2n L. v EtV: iq, +    Z∞ + 6Zv  gZV + vF: 8, : Z. yvrvv; 2
seu 2nxy- Lx : gu ~ 1en Z7. vr Z+V + =84ZV+6ZV +ZV+ V,: 4. 22. 230 + W Ergo
trilineum AHBFA fief  16n 97. vv. ZFD + Z4 4ZV+ BZEBVH V, V B. 3 ( WV L 2V - 2ZV _1ZV,: , 4 - v
examinis probationis causa
Sed nunc his omissis saltem calculum ese prioem insistendo ipsis er reassemamus
(4
u
352
sed hoc prolxius ut verear ne in calculo sit eiratum.
X+ G = nxy sit G = xv:n fiet X + xV. D = n Xxv:n seu rXeX = nV seu x = nv: n+V
et  per 2 et 4 fiet y =  (v: n NV:, n’ + V =) nV:, n + v. et dx =  v + N, Ndv, - n v 3Vdo,: EV
= ( PaV) + n dB = dv: ( v + M= dx. Ego per Set 6 fil Gax = nd v a - 2 nv v7:, (L V + _. Cujusqueritur
summa. Quod ut fiat commodius, ponatur V4+ 13 = co et pier 3Vd a seu v co- n et 3Vav deo seu
v du = deo: 3. Ergo per 9 et 11 ex ag. 7 fiet : ydx = = n deo - 2 n co - 13 de, :  co = 3 n deo C - Z  clleo + 31 la,: c0
= nd dco: C5 - 2ndeo: 3coco= ydx jam (ac: coco  Fisco  Ergo  Et pro co in aeq. B substituendo
adesque pro deo, ponendo 2do, fit S2odo: ct = - 1: 0c adeoque Sdco: a = - 1: 2cdc. Ergo ex aeqe 12
fit (ydx A = nb A- n: 2c0c + 2n. 3co Assumta nb tanquam adjectione
summationis ex propessu determinanda. set jam cosseu n3+ v) = n v: X ex aeg. +. et v Z ny: x ex .
Ergo ex 16e117 fit co = nt y: Xx. Unde ex aep. s fit syx = ab - n X. e y +2n X: 3nty se tbdem
seu sydx nb - X: 2yy _ + 2nXX: 3y. jam si X = 0 fit etiam y -0 ex ag. 1. et Sydx = 0.
Itaque eo casu ex aeq. 19 fil 0 =  nb - 6 00: 2 _ + 2n0: 3 seu 0 = nb. Ergo b = adeoque connesis,
et filt denique sydx = 2nXX: 3y - X. 2yy Ut supra. Huic ergo calculo fide potest
vider non esse necesse po C Ctolli Epli
Generalior fieri potest inquisitio. sit, 22+CV  Inxy posito c esse numerum quentum
sit  xo: n fiei  nX + C Vx = n v. seu x = nEr: n’+ CV fiei y = nV., ut cv
et et dx =  (Cp + n3, nav , =no3Cvdv (2 P + h =) ndo - 2 Vdo,: (2 ((V + R. Ergo
sit Cv+ n= co  fit v = co:c - n:C
 et 3Cv dv = deo fit v du  deo: 3C et ex org. 7 fit n deo: 3c - 2C n. . C - n C devis
yIx Z n vao - 2en v d, : (3)  CV + n3
= ydx = (n deo: 3Cco = 2n: 3cct +2n: 3cc =) ndeo: cc , - 2n: 3cc ade oq fiet
jam co = nV: X et v= hy: v. Ergo  co= nt y: XX
adeoque syax = e, 2nX: 3Cy = X. 27y Hinc si c- 2 fei sydx = nN: 3y - x: 4y
(12
Cydx=
-n3: 2C c02 +2n5: 3CC
Aluud problema iihi ab Hugeuio in eadem Epistota propositum
.cuius
invenire lineam cartesio a Beaunio sropositam Ept. 79. voluminis 3 artesii
subtangens sit yy -xy: a, aut dx: du = + posito t: y = dx: du ~ y - xB: a. fiet
dy = 1DdqZe F u dv et Fydy = _ 10 2dZ f u v dB F 10. 1 Zdv. + 10. 1. vd z 410. na d Z +. 1. 2av 117)
adx = ydy = x dy sit y = 10~ + 11 V + 1a
- xdy = - 10. 20Z 4Z - 11. 21 viv - 11. 202 .1v - 10. 21 vtZ 11o. 22 a d z - 11. 2a dv
c
= 20 adz -21 a dv
alatu fiat 10. 11 - 11. 20 = 10. 11 = 10. 21 et summabilis crit aquatio 7. Ex fit 11. 20 = 10. 2)
=adx=
C
non procedit, quic fieret X = 21y - 1a + 22a quod absurdum. jam enim haberetur lineu
ub si fiat H = 1 et 20 = 1. 2t. fieret y = 10Z + v 4 12
ante applicaoem ad problema. Hinc nec licet  tollere Zdo et naqu nam et sic earum coetticem
ferent aequales inte, se, aclae solicet ipsi a. Tollemus ergi si licet terminos in quibus sunt es
sne y vel V. et fiet 6. 12 - 10. 22 - 20=0 et 11. 12 - 1122 -21 20
Erg
20 = 10. 12 - 10. 22, et jam 22  12 - 21: 1  et fiet 20 = to. 12, 10. 1112 + 10. 21, : 11
e Et et ex 10 et 11, fet 202 (10. 12 - 10. 12) + 10. 21: 1I et fiet y = 10Z + IIV+ 12a et f- 1B9. 27 + 21B + 2. 1
sit = 1 et  =- 1 et 13 = 0 et fiet sed vi sed video rarsus rem non procedere nam fit
sit X = _ 10 + u y + 121 + 13 13 + 14 1t fiet ad . = a1  + 2. 1298 + 3. 127 4 4. 140 el sit 10-8 et l -0
a
-
1 - 12 - 13 et
fit X = 29 22 217: 11 = 1 2a + 12-
yd=
14 = 1: 23. 4 eu
oidutu it ie te, st sti  ee su desteti
4d
et fiet X= _  y +  21 + 2740
Edgat