In a 1676 comment, Leibniz writes: ""Aliter demonstrandum quod neque majus quia non potest inveniri pars ejus finita aequalis." I have a two-fold question: (1) is demonstrandum gerund or gerundive? and (2) does it convey a sense of "necessity", as for example in "Quod Erat Demonstrandum" (which was [necessary] to be demonstrated)? The sentence occurs in a marginal note at the end of Leibniz's proof of Theorem 11 (Propositio XI) in his De Quadratura Arithmetica. The marginal note does not appear in all editions. It appears in
Leibniz, Gottfried Wilhelm Gottfried Wilhelm Leibniz—De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis. (German) Edited and with an afterword by Eberhard Knobloch. Translated from the Latin by Otto Hamborg. Dual Latin-German text. For the 1993 Latin edition see [MR1248164]. Klassische Texte der Wissenschaft. [Classical Texts of Science] Springer Spektrum, Berlin, 2016. vii+303 pp. ISBN: 978-3-662-52802-0; 978-3-662-52803-7
There is a French translation by Parmentier that runs as follows:
"pour procéder autrement il faudrait démontrer que cet espace n'est ni plus grand (que l'infini), dans la mesure ou aucune de ses parties ne peut être égale à un espace infini; ni plus petit, dans la mesure où il ne peut être égal à aucune partie d'un autre. It existe une infinite de manieres de réaliser cette démonstration, on peut admettre qu'un espace infini contient toujours une partie finie supérieure à un espace fini donne."
According to this translation, there is no aspect of necessity: Leibniz proposes an idea for a different proof, where one would have to, etc. There is no necessity to provide an alternative proof. See page 97, footnote 1 here: https://books.google.co.il/books?id=fNTUULXHmQ0C
To clarify my question, I am wondering whether the sentence should be translated as
(A) "It has to be demonstrated in another way that, etc."
or rather
(B) "If one were to proceed differently (i.e., provide an alternative proof), one would have to prove, etc."
Is (A) or (B) closer to Leibniz's intention? I was wondering whether insight into the "necessity" aspects of gerundives may shed light on the issue.
The full text of the marginal comment by Leibniz reads:
Aliter demonstrandum quod neque majus quia non potest inveniri pars ejus finita aequalis. Nec minor quia nec pars alterius ipsi aequalis. Idem fieri potest infinitis modis[,] infiniti pars finita assumi potest dato finito major.
The context of the marginal comment is as follows. In the main text, Leibniz gave a proof of Theorem 11 using infinitesimals. In the marginal comment, he discusses an alternative proof using Archimedean exhaustion (without using infinitesimals). According to interpretation (A) above, Leibniz would be dissatisfied with the proof using infinitesimals (therefore an alternative proof must be constructed). According to interpretation (B), Leibniz is satisfied with the infinitesimal proof, and points out in the marginal comment that if one wished to give an alternative proof, one could proceed in such-and-such a manner. So the issue is whether the necessity part of the gerundive is absolute (a different proof must be constructed), or if it is only conditional (if one wants another proof, one must proceed as follows).
