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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 26v, page id 00067993:0052
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cum enim seriei  reciprocorum trigonalium  + L +  + f etc. summam
a se inventam mihi  investigandam proposuisset Hugenius,
felici inquisitione usus non hujus tantum sed et caeterarum huc propositarum
omnium summam regula generalissima pariter ac simplicissima hic expressam,
inveni.
PA050S1710 XL
DL
2
4 48
summa seriei infinitae ai 
3) Summa seriei infinitae
+ 24 +  +    ete es 
3t summa seriei infinitae L L t   qu B gol u ioete est 3
3 sunt autem numeri 8. 24. 48. 80 dc. quadrati imparium unitate minuti. et
3 numeri X15.C3. 8. 15. 24. 35. 48. 63. 80. etc. quadrati  omnium numero
v tam parium quam imparium eadem unitate minuti.
Nam + = + = + L + +  8c. est _ 2. per prop. ~ 39 11
Ergo
Quae est propositionis pars prior.
iam
Ergo  + g.
24(48
dec est divitis oumbus per 8.
1099 120 +,43 etc. erit 3. 88)
summa scilicet utriusque. Quae est pars propositionis posterior.
SA POSITIOXLII
c (expinit (Crali) cujus euadratum
120185
Aeam nyperbolas inscriptum
primariae est .d
Restato tantum probanda pars ultima de Hyperbola Ut vero symbolismus sane
memorabilis clarius appareat,BCD. deseribature ixcrls
maux
cect hdem quadat schema adhibeamus. sit Hyperbola aequilatera
GCH; cujus centrum A. vertex C potentia seu quadratum inscriptum
ABCD. circa quod etiam circulus describatur. Hoc Quadratum circulo
Htyperbolaeve in scriptum ponatur esse. erit circulis , = +. h, + etc. per
prop. 35. Porro circulo quodammodo respondet spatium Hyperbolicum
CBEHC. quia posito angulum astFAE asymptotorum AF, AE esse rectum
et BC, EH. parallelas asgniptoto AF, ac denique BC aequalem ipsi BE
ita enim etiam BE ipsi AB, et BC ipsi AD aequalis erit; neque aliud spatium
(circa asymptotos) quod circulo melius respondeat aut magis c determinatum
sit praecaeteris; assignari potest. Hujus autem patii qualiilinei aream esse
 + L + L etc. sic ostendo. AM est AB+ BM, et AM rectangulum aequale
AB+ BM
B1B BC. per prop. 26. ad modum prop. 28.
Sch o. pilop. L9
c. eodem modo applicatorum summa scilicet omnium ML summenomnium AB, ocn B omnium B M ad BE ordine normaliter
in punctis M applicatarum. Homnes autem AB, quia semper eadem, applicatae semper in punctis M, dant Hectagulum ABC
vel CDE, id est Quadratum ab AB. et omnes BM vel MN applicatae in M dant triangulum. BEP vel BE. C
putuu carne
24 48 80 inc
+
dc est
12
4 per prop. 30
120
irculo moolismus
 ejasque partum vana.
(2 per prop. 41
aequol. 2 per prop. 36.
es
Pefiti
H demta summa omnium BM, et omnin
Ergo summa omnium M
 seuspatium CBEHC eri B8 - B
semiquuadratum a BE, sive BE. C ia porto es antem e BE hic aequal. AB. Ergo er
quadrato ABCD ex natura Hyperbolae. Ergo M L est Quad. AB. vel Quad. AR
sive ML aequal. AB - BM = BM - BM + BM
AB AB
A
FA
AB.
BV
B dic. per prop. 2Sq ad modum prop. 29
fivin B2LC