Let AC be a ray coming from the medium DC into the medium CE, and let the density of the former to the latter be as d to e. It is asked, how should the ray ACB be directed so that it is the easiest path of all, or that (AC x d) + (CB x e) is a minimum. Let DC = l, and EC = m. It is given also that FG = f, and let AD = FC (...) = x, CG = EB = f – x. Therefore, AC = √(l2 + x2) and CB = √(m2 + f2 +x2 – 2fx). 2 It will then be the case that d√(l2 + x2) + e√(m2 + f2 + x2 – 2fx is equal to a minimum. Therefore, through my method of tangents it is the case that: (2dx/(√(l2 +x2)(AC))) + ((2ex -2ef)/ (√(m2 +f2 +x2 +2fx)(BC))) = 0. That is, it will be the case that AC/BC = dx/e(f-x). Now if we suppose that AC and BC are equal, it will be the case that f – x is to x, as d to e. Therefore, if a circle, with its center at C, is described by the ray CA or CB, AD or “x,” – the sine of the angle of incidence – will be to BE or “f – x,” – the sine of the angle of refraction – as e, the density of the medium of refraction, will be to d, the density of the medium of incidence, that is, the sines of the angles will be in reciprocal relation to the mediums or densities. (shrink)
1. Constructing a defence in the case of God is doing something not only for his glory but also for our advantage, in that it may move us to •honour his greatness, i.e. his power and wisdom, as well as to •love his goodness and the justice and holiness that stem from it, and to •imitate these as best we can. This defence will have two parts—a preparatory one and then the principal one. The ﬁrst part studies the •greatness and (...) the •goodness of God separately. The second part concerns these two perfections taken together, including the providence that God extends to all created things and the control that he exercises over creatures endowed with intelligence, particularly in all matters concerning piety and salvation. ·The ﬁrst part will occupy sections 2–39, the second part sections 40–144·. (shrink)
1. A substance is a being that is capable of action. It is either •simple, meaning that it has no parts, or •composite, meaning that it is a collection of simple substances or monads. (Monas is a Greek word meaning ‘unity’ or ‘oneness’.) Any composite thing—any body—is a multiplicity, ·a many, but simple substances are unities, ·or ones·. There must be simple substances everywhere, because without simples there would be no composites—·without ones there could not be manies·. And simple substances (...) are lives, souls, minds—·where there is a simple substance there is life·—and the world’s being full of such substances means that the whole of nature is full of life. (shrink)
Beyond the world, i.e. beyond the collection of ﬁnite things, there is some one being who rules, not only as the soul is the ruler in me (or, to put it better, as the self is the ruler in my body), but also in a much higher way. For the one being who rules the universe doesn’t just •govern the world but also •builds or makes it. He is above the world and outside it, so to speak, and therefore he (...) is the ultimate reason for things. ·That follows because •he is the only extramundane thing, i.e. the only thing that exists out of the world; and •nothing in the world could be the ultimate reason for things. I now explain that second premise·. We can’t ﬁnd in any individual thing, or even in the entire collection and series of things, a sufﬁcient reason why they exist. Suppose that a book on the elements of geometry has always existed, each copy made from an earlier one, ·with no ﬁrst copy·. We can.. (shrink)
In God existence is the same as essence; or—the same thing ·put differently·—it is essential for God to exist. So God is a necessary being, ·a being who exists necessarily·. Created things are contingent, i.e. their existence doesn’t follow from their essence. which comes to..