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# Title: LBr. 57, 1 · Bernoulli, Joh.
# Folio 175r, page id DE-611-HS-958803:0351
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 174v); this page is its right half.
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continuatis hoc modo quousque lituerit (quo magis vero continuantur, eo solutio
epit generalior, et facundior) comparentur utrinque dineasionum ipsius X aequa
lium coeficientes a = 0∞a + b = a + b+ C +e+ +g + R, - btC- b- 26- 3C=4f.
5g- 6, -c+ e = + C+ 3e + 6f. + 109 + 15h, - e + f = - e - 4f - 109 - 20h, - f + q
= + f+59+ 15h, -g+ R = = g - 6h, = h = + h; quibus aite collatis prodibit a =0
f= - 2C - 2 eig = c + e, h = 0, ita ut b, ci e, maneant arbitrarice; calerarum
vero substitutis valoribus reperietur aequatio talis y = bx + Cxx + ex 3 2C
2ex+ C+ exS. quae exprimet naturam curvae ABC G in qua summa segmen
toram erit constans DB + DC; Hac autem inventa altera illa ubi potestatum
famma dari devet, facillime construitur. abscindendo tantum Dc, exDC, DB ex DB,
e
9 ex DG, da ex DA, quae sint ut ery
De
ps
DB, VD
etc: habebitur nova curva abcg, ubi tegmentorum potestatus
c
DbDen seu Dan+ dgn, facient ubique eandem sumnam, prout
requinebatur.
des quam longa vea incesserim; sed alian tunc non quaerebam, contentus scilicet
attigisse scopum; beduo enim post cum talia iterum medilarer, sese ultri ofere
fat facilior ille solvendi modus, quen un anterioribus meis Tibi conunicavi; cu
jusq ope statim incidi in curvam transcendentem, cujus segmenti quadratum
duitum in alterum segmentum producat solidum datum, et quidem sic: Sit ffig: 1.
DB1X: BE, y = Xn
s
FX loco 1+X postem assumere quamcunque aliam quantitatem
compositam), jam si CDx DB9 = 1, erit CD = =x; ergo propter DB, DC:: BE.
CF erit CF = X m - 3 - - n
=1+ X; sed ob analogiam relationis punctorum B et C ad E,
et F, erit C +- - - m - n
E = X = 2m - 2a
= 2m = 2n -
Xx. 1+
XFF, n, ergo XM -3
XF+, n; etgue etiam logarithmi, in = 3 lx + n l, + X =-
+x
=X
2m - 2n L+
+n LXx + unde m -- 2n +3lx - n L1+X + n l xx + 1; si itaque X elevetur ad hanc di
~
mentionem m et muetiplicetur per  it
 habebitur equatio pro carva quae
sita, quae ope logarithmice conseui potest, et per consequens ex earum nuy
mero, quas percurrentes appellavi.
Nunquam negavi veritaten tolutionis per circulum, sed aliud est verum esse
aliud satis facere intentioni proponentis; au credis me non praevidisse circu
lum? licet illum non diserte excluserim, excluditur tanen eo ipso quod nemini
non obvius est; ad de etian quod neque circulus satis faciat, quando segmenta non
in puncto coeunt, sed quando ad auen sunt parallelie, ut in fig. 2. Cum ila
que proponentis scopus sit, acunen solutoris explorare, nemineu putabam fore
qui mihi circulum ut solutionem legitimam qualen ego desiderabam obtrusarcs esset
sed