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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 5
# Folio 58v, page id 00068255:0116
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Quaerruntur numeri triangulares seu summe naturalium; procedetur deinde ad summas sumarum
Tuanut
a naturalibus seu triangularium, hoc est ad pyramidates, inde aad triangula atue ita poroo. Porro cuq
naturales constent ex columnis, quaeque earum summabitus sepuntim, ac tum demuu  summationes conjun–
ergo ad summas prima volumnae quia constat ex meris o et 1; ideo dissimulatis o, per inde ac si non adessent
pro gressio
gentur. prima et liolumnam
quaerentur summae unitatum, id est porentur pro summis harum unitatum exhibebuntur ipsae columrnae numerorum
porro quia qua
naturalium qui utique nihil aliud sunt quam summae unitatum. Subsribentur ergo hae summae unitatum sedibis,
Sed postea in sedibus Nullitatum supplebuntur loci vacua, scribendo tantum otam in columna propime
Quam maxime compendiosam
 dradice. Hoc jam in nostris naturatium coumnis summandis
praecedentem, ita absoluto est. Habemus ergo methodum generalem
actu ipso tentabimus, et indroabimas a priaca. In progressu, o interpolatam exprino per , et 1 interpolatam per,
col. 1. summatrip (0u 1 11001. 001, 001, (0,1, 001, vo l 1001, 00 11 0011 001, 00t, 00t, 00l e01, out,  rex Oot seu 0011 seu
O,d.1, 1, seu 000b111
600011, 1,t
Colmna (sumanoa X 1 91 0101 91O(0191919101010101V10I0101V10101910101019]010101010101501
0b0 00l, 1, 0000 1, 1, 00001, 1,0000 1, 1, 00001, 1, d0001, 10 0000111111
Ascrpsimus hic interpolationes pro locis vacuis, in sequentibus omittamus, quid semper sabintelligitur nota praecedens inert
00 00000,1, 1, 1, 1,1000080B19191 e t, 000 vD0 D, , H1, 000 , , 1,1
interpolandam; pervodos tamen integras colligemus sub finem cujusque columnae summatrias
D212
col.  Summunda p01100I10011001100100110011OSII0011OI 00HIOBIIVO11OORIDIDOII jo2
10 10 10 10 10 10 10 10 10 10 10 10 10
4 9i 1d
10 8
.o 0d 1,I 1,11000 o1 21 1000 01
10 B1
ot 10 01
anmuda OBODMTOODOMLOOBBTMTLOBBBELHBBBOILLLDBBOIELTOBBBLLLLBDBOIL
1010
eii0
1010
1010
1010
orio oitu
11110
- 1040Co1
0001
111
1010
ouo oiio
0001
0001
OucO1
1110 5000
0611
081
0000
016. l16
QQDDONDOTLILMTISOBODBDOMLMMLLLOOBOBPDBLILIL LOBBBBOSOMMMMII 
-- 01110
0010000000001, 1111, 1110
10101010
01 100110
00oui
210001,
(0vo0von0101010
0000000000i100i
1010101b
01100I
00011
10101010
0110011O
00011110
0000000
ete
Ovouv00000001111110000000, IIIMI
Dovorvovooo
OlL vivvvr.0000000000001
900000DO00D0VONGHITE CCLHVOHBOWOVOOE
10vonv1101010 1010101(0)
vor vovvvm o110o11voooio
vnov0oov D0011100001110
101010 10101010r0
01o 010o11Bo
00o1OOMI
ovovoII
O vor vvor vorr ovver 00090006O0DD0OD(111 119)I F, IILILILII
ol vovvov0vvv 000000011 1111116
haec quidem suficiunt ad legen progressus in infinitum cognoscendam.
da periodi e
naturalium
uimanda sit periodiD
091010101
O(2) (2). Erit columnae summatricis primae periodus 0 (22 H). 1. 01
urad
im periodus, O(a (, 6.1) 1 (35.) O( . , (a (2
erit columna usscunque, quae voretur R.
Usquead R cutside vol, o (). (0 (2t) 1(2-1) 7(27
Sed a (pt ma inclusive erit in infinitum
Columnae summatius f, pericdus erit O (2) 1 (35)
itaque notae cujusqerdi ante (n + 1 mam) columnam summatrilem
acujusque periodi priorum n columnarum
erunt semper edem ejusdem multitudinis nempe 2
et
Sed notae cujusque periodi post n(mam) columnam, variabunt, et
si columna sit ab initio fma erunt notae el 35+1
secundae
tertiae
ete usgad
0(2+2) 1(2) 1)1(2)
(n man plitous O ( t )1 (2
R + , naepesi, OE ut1), (uru codnesit
post hoc
(n + 2) mol  O (ant 1), (4+2)
Et ita porro in dafinitum
puorenita notio communis ex natum lillocavis
supest jam at suommas columnas summ atrices diversarum summandarum combinemus
inter se. Hoc modo:
combinabentur summandae primae summatrix an
secundae
da Etia ata sta Cta
jura
ada atia ita
2da etia ita
een inn 2daa
praberea secnda columna ex A addena  3. secmnda Ca B in C; Sunia ex C in D, etita ex C in E, Et ila porro, pront istsae
Tertiae
quatae
A, B, C, d, E etc longe porrigi possunt sic D cum ex quiaque columnis confletur, potest alicubi habere
requipere adbitionem quinque unitatum, seu 101 ergo 1 dextra scribtur ibi in D, et otrans fertur (sed trustra ) in E,
03101
et ita porri
col. 1. summatrix ((0:1l0 10
col. 1. sumatrs
Ergo Generaliter, si colum
0000doII seu OB 18
sen 0414
00D 1 seu 03. 1.
0000111Iseu 0914
opgioou
O(+ 1)101
3(19) 11. Ou B
0 (20).  NII. Dvor
024) 1(8)
(1612) H. 0r1 (3
sen O (1 1) , (4 ovovee (1)
esu B (16+8) 1(8)
u
032) 1(32
O(2+1) 1()001 (222 1)
)