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# Title: Qua Continentur, Oratio Inauguralis. Mathesis Universalis; sive, Arithmeticum opus Integrum, tum Numerosam Arithmeticam tum Speciosam complectens. Adversus Meibomii, de Proportionibus Dialogum, Tractatus Elencticus (shelfmarks Leibn. Marg. 84:1, Stück 1; Leibn. Marg. 84:1)
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C AP. 35.
Elementi quini Euclidis declaratio.
29 Si sit MI toia ad toram, sic ablata ad ablaram; erit e reliqua au
religuam ut tora ad totam. P. A0 enim e + e
a =R. Prit ica-
que cum AZ B = R in a +6 =R a F R6, cum A = R a. 2-
deoque e residuum B =- R 6. PC
urrinque aequalirer dividendo)
B
6 = R = a t B FAque HaC
propolicio unirertalis ad prop. 5.
a +B = Ka 16
a -
Fuclides illam sic demonstrat. Ouonsam ue A F B. a +6:: A. _C
erit alterne (Ceu permucando) A FB. A :: a +6. a. Pc divi-
dendo (sire per divitionem rationis) B. R : : 6. a. iterumq5
permucando, B. 6:: A. a:: A + B, a + 6.
Acque colligie hoc nesux leu corollarium, Si compositae mac-
nitudines sint proportionales; e per conversionem rationis proportionales
erunt. Cum enim ostenderit A + B, a + 6:: B. 6. FC permutan.
d0 A F B. B:: a F 6, 6- GUX TMnE qLanCIEaCeS COmPOLICR PEO-
portionales. Sed _ octentum eAt A F B. a F 6: : R. a. adc09:
permucando A + B. A:: a F6. a. gux es convertio ratio-
nis. Ouippe 26d5 definitur Conversio rationis (araspoen noge
sumprio antecedentis ad excessum, quo antecedens ipsam consequentem su
Perat.
Clarius hanc demonstrarionem improbat, propter achibi.
tam alcernationem, quae in Tolis homogeneis obcinet, cum ea-
men convertio racionis eciam ubi bina binis (unt hecerogenea
locum habetu adeogne hanc (ubstitmic. Si A F B. B :: a F 6. C. CrIE
(dividendo) a. B :: a. 6, _ (invertendo) B. A:: 6. a. _ tan-
dem (componendo) At B. A: : a 4 6. 8. quae eAt convertiora-
rionis. Nempe siuc coca prima ad sui parcem ablacam, sIC FOTa
Secunda ad Cui ablatam; erie uc coca prima ad lui reliquam, (C
eora secunda ad lui religuam. Vel brerius, E uc cora ad abla-
eam, sic eora ad ablacam; erit _ coca ad religuam; ut toca ad
reliquam.
MorenouroC deulonuab et
6 +
derit
6 3 a
82