E2P39D
Demonstration — Part II
Latin
Sit A id quod corpori humano et quibusdam corporibus externis commune est et proprium quodque æque in humano corpore ac in iisdem corporibus externis et quod denique æque in cujuscunque corporis externi parte ac in toto est. Ipsius A dabitur in Deo idea adæquata (per corollarium propositionis 7 hujus) tam quatenus ideam corporis humani quam quatenus positorum corporum externorum ideas habet. Ponatur jam humanum corpus a corpore externo affici per id quod cum eo habet commune hoc est ab A; hujus affectionis idea proprietatem A involvet (per propositionem 16 hujus) atque adeo (per idem corollarium propositionis 7 hujus) idea hujus affectionis quatenus proprietatem A involvit, erit in Deo adæquata quatenus idea corporis humani affectus est hoc est (per propositionem 13 hujus) quatenus mentis humanæ naturam constituit adeoque (per corollarium propositionis 11 hujus) hæc idea est etiam in mente humana adæquata. Q.E.D.
English (Elwes 1883)
If A be that, which is common to and a property of the human body and external bodies, and equally present in the human body and in the said external bodies, in each part of each external body and in the whole, there will be an adequate idea of A in God (II. vii. Coroll.), both in so far as he has the idea of the human body, and in so far as he has the ideas of the given external bodies. Let it now be granted, that the human body is affected by an external body through that, which it has in common therewith, namely, A; the idea of this modification will involve the property A (II. xvi.), and therefore (II. vii. Coroll.) the idea of this modification, in so far as it involves the property A, will be adequate in God, in so far as God is affected by the idea of the human body; that is (II. xiii.), in so far as he constitutes the nature of the human mind; therefore (II. xi. Coroll.) this idea is also adequate in the human mind. Q.E.D.
Modern English
Let A be what is common to and characteristic of the human body and certain external bodies, equally present in the human body and in those external bodies, and equally in the part and in the whole of each. There will be an adequate idea of A in God (E2P7C), both insofar as God has the idea of the human body and insofar as God has the ideas of the given external bodies. Now suppose the human body is affected by an external body through that which they have in common, namely A. The idea of this affection will involve the property A (E2P16), and therefore (E2P7C) the idea of this affection, insofar as it involves the property A, will be adequate in God insofar as God is affected by the idea of the human body, that is (E2P13), insofar as God constitutes the nature of the human mind. Therefore (E2P11C) this idea is also adequate in the human mind. Q.E.D.