E2P8S
Scholium — Part II
Latin
Si quis ad uberiorem hujus rei explicationem exemplum desideret, nullum sane dare potero quod rem de qua hic loquor, utpote unicam adæquate explicet; conabor tamen rem ut fieri potest, illustrare. Nempe circulus talis est naturæ ut omnium linearum rectarum in eodem sese invicem secantium rectangula sub segmentis sint inter se æqualia; quare in circulo infinita inter se æqualia rectangula continentur : attamen nullum eorum potest dici existere nisi quatenus circulus existit nec etiam alicujus horum rectangulorum idea potest dici existere nisi quatenus in circuli idea comprehenditur. Concipiantur jam ex infinitis illis duo tantum nempe E et D existere. Sane eorum etiam ideæ jam non tantum existunt quatenus solummodo in circuli idea comprehenduntur sed etiam quatenus illorum rectangulorum existentiam involvunt, quo fit ut a reliquis reliquorum rectangulorum ideis distinguantur.
English (Elwes 1883)
If anyone desires an example to throw more light on this question, I shall, I fear, not be able to give him any, which adequately explains the thing of which I here speak, inasmuch as it is unique; however, I will endeavour to illustrate it as far as possible. The nature of a circle is such that if any number of straight lines intersect within it, the rectangles formed by their segments will be equal to one another; thus, infinite equal rectangles are contained in a circle. Yet none of these rectangles can be said to exist, except in so far as the circle exists; nor can the idea of any of these rectangles be said to exist, except in so far as they are comprehended in the idea of the circle. Let us grant that, from this infinite number of rectangles, two only exist. The ideas of these two not only exist, in so far as they are contained in the idea of the circle, but also as they involve the existence of those rectangles; wherefore they are distinguished from the remaining ideas of the remaining rectangles.
Modern English
If someone wants a fuller illustration of this matter, I can unfortunately offer no example that adequately explains the thing I am speaking of here — it being, as I said, unique. I will, however, try to illustrate it as best I can.
A circle is such by nature that the rectangles formed by any two chords intersecting within it are equal to one another. Accordingly, a circle contains infinitely many equal rectangles. Yet none of them can be said to exist except insofar as the circle exists, and none of their ideas can be said to exist except insofar as they are comprehended in the idea of the circle. Now suppose that from those infinitely many only two, call them E and D, are taken to exist. Their ideas then do not merely exist insofar as they are comprehended in the idea of the circle, but also insofar as they involve the existence of those rectangles; this is what distinguishes them from the ideas of the remaining rectangles.