E2P49D
Demonstration — Part II
Latin
In mente (per propositionem præcedentem) nulla datur absoluta facultas volendi et nolendi sed tantum singulares volitiones nempe hæc et illa affirmatio et hæc et illa negatio. Concipiamus itaque singularem aliquam volitionem nempe modum cogitandi quo mens affirmat tres angulos trianguli æquales esse duobus rectis. Hæc affirmatio conceptum sive ideam trianguli involvit hoc est sine idea trianguli non potest concipi. Idem enim est si dicam quod A conceptum B debeat involvere ac quod A sine B non possit concipi. Deinde hæc affirmatio (per axioma 3 hujus) non potest etiam sine idea trianguli esse. Hæc ergo affirmatio sine idea trianguli nec esse nec concipi potest. Porro hæc trianguli idea hanc eandem affirmationem involvere debet nempe quod tres ejus anguli æquentur duobus rectis. Quare et vice versa hæc trianguli idea sine hac affirmatione nec esse nec concipi potest adeoque (per definitionem 2 hujus) hæc affirmatio ad essentiam ideæ trianguli pertinet nec aliud præter ipsam est. Et quod de hac volitione diximus (quandoquidem eam ad libitum sumpsimus) dicendum etiam est de quacunque volitione nempe quod præter ideam nihil sit. Q.E.D.
English (Elwes 1883)
There is in the mind no absolute faculty of positive or negative volition, but only particular volitions, namely, this or that affirmation, and this or that negation. Now let us conceive a particular volition, namely, the mode of thinking whereby the mind affirms, that the three interior angles of a triangle are equal to two right angles. This affirmation involves the conception or idea of a triangle, that is, without the idea of a triangle it cannot be conceived. It is the same thing to say, that the concept A must involve the concept B, as it is to say, that A cannot be conceived without B. Further, this affirmation cannot be made (II. Ax. iii.) without the idea of a triangle. Therefore, this affirmation can neither be nor be conceived, without the idea of a triangle. Again, this idea of a triangle must involve this same affirmation, namely, that its three interior angles are equal to two right angles. Wherefore, and vice versa, this idea of a triangle can neither be nor be conceived without this affirmation, therefore, this affirmation belongs to the essence of the idea of a triangle, and is nothing besides. What we have said of this volition (inasmuch as we have selected it at random) may be said of any other volition, namely, that it is nothing but an idea. Q.E.D.
Modern English
There is in the mind (E2P48) no absolute faculty of willing and not-willing, but only particular volitions: this affirmation, that negation. Let us then conceive a particular volition, namely, the mode of thinking by which the mind affirms that the three interior angles of a triangle are equal to two right angles. This affirmation involves the concept or idea of a triangle. That is, it cannot be conceived without the idea of a triangle, for to say that concept A must involve concept B is the same as to say that A cannot be conceived without B. Moreover, this affirmation (E2A3) cannot exist without the idea of a triangle. So this affirmation can neither exist nor be conceived without the idea of a triangle.
And in turn the idea of a triangle must involve this same affirmation, namely, that its three angles equal two right angles. Therefore, conversely, this idea of a triangle can neither exist nor be conceived without this affirmation. So (E2D2) this affirmation belongs to the essence of the idea of a triangle and is nothing beyond it. What we have said of this volition, chosen at random, must also be said of every volition: that it is nothing but an idea. Q.E.D.