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# Title: Commercium Epistolicum D. Johannis Collins, Et Aliorum De Analysi Promota (shelfmark Ms IV 379a)
# Folio 68, page id 1882181034:0076
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Ouarebam itaque, quomodo in nis Seriehis, Cx daris quabus primis
Exuris, reliquae derivari possent. Rc inveni quod vetica secunda Agura m,
reliquae producerentur per continuam mulciplicationem cerminorum hujus
n - 0. m-
Serici,
R.
X
m - 2
im--
m-
A _C.
Exempli gratia; Sit (terminus secundus) m== 4; _ erit 4 X m =L hoc
CH 6, terrius terminus; ex 6X = 2 hoc eHt 4, GuartUS; _ 4. X  3, hOc
4
CA T, quincus; et YX  hoc ett o, sextus; quo teries in hoc casu
termminatur.
Hlanc Regulam itaque applicui ad Series interserendas. Et cum, pro
Circulo, secundus terminus eNet = potui m = =; _ prodierune ter-
mini = X = = siVC = =; = 5X = 2 five + 2; + , X  2 sive - -25
 Ac in intinitum. Unde coenovi desideratam Aream sexmenti Circularis
CAe ∞ = 2 3_ eou e ec
Ex eadem ratione prodierunt eriam interserendae arex reliquarum
Curvarum: ut ec area Hyperbolae et caeterarum alternarum in hac Serie
T = au)  T. f au), 2 F an 3 Y F au s, QC.
Ec eadem eM ratio intercalandi alias Series, idque per interralla duo.
rum pluriumre terminorum simul deficientium.
Hic fuit primus meus ingrestus in has meditationes: qui e memoria
fane exciderat, nisi oculos in adverlaria quaedam ante paucas sepcimanas
reculissem.
Ubi vero haec didiceram, mox considerabam terminos , = vu), 7 -  2
7 = _ =  = _ 2, _C. HOC eA, L, T - AX, 1 - 29 X + X4, Y = 3 . X + 3∞4
xec. eodem modo interpolari polle ac areas ab iptis generatas:
ee ad hoc nihil aliud requiri quam omistionem denominatorum 1, 3, 5,
7, ac. in terminis exprimentibus areas; hoc elt, coefficientes termino.
rum quantitatis intercalandae T = ax vel r= vup vel generaliter,  m
prodire per concinuam mulciplicationem terminorum hujus Seriei
m-.
MMX
M - 2
X.
CXm 3_C.
Adeoque (exempli gratia) F = uus valeret X = = _2 - ; x4 -  X6 _C.
FF + = un  valereE Y = 2∞2 + 3x4 + 225 _C. FF F = un s ValerCt L - , xX
5X 4 = 3 3 C.
Sic itaque innotur mihi generalis Keductio Radicalium in intinitas
Series, per Regulam quam posui inirio Existolae prioris, antequam scirem
Extractiones Radicum.
Secd,