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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
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SROPoSITIO XXXVI
Summa seriei infinitae  + 1 + : + L + L etc. est  Numeri
autem 3. 15. 35. 63. 99. sunt quadrati parium, unitate minuti
series  + L + +  dc. sit A.
5 + 1 1 7 + B BC. sit B
3 + = + 2 + = dc erit 1B
A serie A auferatur series B singuli scilicet termini a singulis respondentibus
cu ut crestet Geriese. B. arul siaet 1  angfelas  restabit rC; si abet dufiba
rtabit residuum erit A - 2B. ut si ab  auferas = restabit L. Si ab  auferas 2
restabit; si ab L auferas 2 restabit . etc. Quod si continues, experieris semper
futurum ut termini seriei A, ordine redeant; quemadmodum nullo negotio demonstrari
generaliter potest, ad modum propositionis praecedentis, sed in re clara verbis parco.
Ergo = +  + = += BC. erit A. 2B.
sed  + + a  + = e C. est etiam A - 1.
Habetur ergo aequalitas inter A- -B et A- 1. sive (sublato utrinque) A, inter 2B et 1
vel inter B et 2  2
D
Scholium
Res satis mira videbitur seriei L + A + : + L F L etc. Sammam iniri posse,
eamque esse L: Seriem autem ex ipsa excerptam per saltus nempe + + L + eoc
exprimere magnitudinem circuli cujus quadratum inscriptum sit el
que ut ostendimus prop. 35. Cie ts adtatumi
PRososiZio XXXVII.
Si L vel quod idem est  series i + L + = + L + 
6
923
25
Qradratum circumscriptum, tunc series : + L + L etc. e priore excerpta per saltus,
una quantitate semper omissa, repraesentabit circulum inscriptum.
Nam si Quadratum inscriptum sit , circulus erit +  + L etc. per prop. 35. Quadratum
autem circumscriptum quippe inseripti duplum erit . sed L idem est quod summa
hujus seriei  + s. +  + = +  etc. per prop 36. Quare si haec series
repraesentes Quadratum circum scriptum excerpta ex illa, ut diri repraesentabit
Circulum. Q. E. D.
PROSOSITIO XXXVIII.
Trilineui irculareBquod geaiuti C BOE, quod quadranti AB0E
complmento est, et valet Fet B + + etc.
AB0E
Si Quadratum radii valeat . Quadrans ei inseriptus cvalebit + + L + = etc. 3
et
et Trilineum  _ CBOE, quod quadranti complemento est valebit +
03 7
Name eut Ertully est cade quadentum 
Nam quadrans est ad quadratum radii ut circulus ad quadratum diametri;
id est per prop. 37 ut  +  +  etc. ad , vel per prop. 36 ut
 + h, + L etc. ad  +  + , +  + 9gt4 etc. Quare  pece eseri alurem
+ = + = et c. tunc
14
datie qd si quadratum radii sit , sive  + 7 + 3,  53 39
quadrans erit L + L + = etc. auferatur hic valor a priore restabil 
etia
restabit = + L + L etc. valor Trilinei CBOE, quod restat subtato quadrante ABOE, a
quadrato radii crcumscripto BAEC.
SROSOSLTIO, ReR XXXIX
Summa seriei infinitae = + L +  +  +  + =  c. est 2 numeri autem
1. 3. 6. 10. 15. 21.  &c. sunt trigonales
series 1 + = +  +  BC. sit A
i + 3 + 1 + 1 8c. a sit 2B
E + = +  + L etc erit B
A serie A auferatur series B singuli termini a singulis respondentibus restabit series
A- B, ut si ab 1. auferas: restabit. si ab L auferas L restabit = si ab  auferas  restatit 1 e
idemque semper futurum est ut numeri seriei A ordine redeant quemadmodum generuliter aere
demonstraripote posset ad modum propositionis 34. sedin re clara verbis parco. Ergo
i
3 + 4 3 4c. erit X. b
 epemra
33