The Philosophy of Roger Bacon

...because there are two means between them. Wherefore, because of the excellent continuity of it, one mean will not be possible?. For the path of each is equal, and one way of the mean is from one heavy thing and the fourth way from a light thing. But this cannot exist in the same simple body; rather, according to their own species, they are held as heavy and light in a subaltern relationship original: subalternatim; a logical relationship where one is subordinate to the other, since both heavy and light, taken from the whole, create a division in species which is taken in respect to that whole. Whence it is said that there are two means between these [extremes]. And for this, there is the authority of Boethius.

And a similar privation of service to each one in Book 3 of The Consolation of Philosophy Boethius, De Consolatione Philosophiae, Book 3, Poem 9: "Tu numeris elementa ligas" (You bind the elements by numbers) introduces Philosophy speaking and saying: "You bind the elements in a mean." And here he himself teaches the reason because? between all two solid bodies there are two means in the same proportion. And here, if you follow it, there is one. For example, consider 9 and 8 and 6, which are bounded as solid figures... if between 24 and 9 there are two means, [namely] 18 and 12 This refers to the Euclidean/Platonic theorem that between two cube numbers (solids), there are exactly two geometric means. For example, between 8 ($2^3$) and 27 ($3^3$), the means are 12 and 18 ($8:12 = 12:18 = 18:27$), so that the proportion of the solid figures contains all things which he and all the ancient philosophers established. Following another minor conclusion, there are two means between the extreme elements. Which are according to the four

of the world: namely, the elements and the four bodies which are called air air, earth, and heaven and the ether. If, however, it is objected that there are only two types of local motion, namely upward motion and downward motion, and

therefore only two bodies. It must be said that those expositors and commentators who saw this here teach a division through? certain means. But he himself certifies us concerning this, wishing that there are not only two motions diverse in species, but rather they are subaltern meaning they exist in a hierarchy or gradient. For the places of air and fire are diverse in species, because if air were in the sphere of fire, it would descend from that place. Upward motion is double in species, through the division of places into diverse species; and the same is true for water and earth.

If it is objected that the philosophers in Book 3 of the Optics original: De Aspectibus; the Latin title for Ptolemy's work on optics and Alhazen in the first [book] teach that there is no refraction fractio: the bending of light as it passes from one medium to another of rays in the sphere of air, but mostly a division... all refractions that do not fall at right angles. And it is shown that heaven and fire are of a similar nature, since for refraction two things are required: namely, a diversity of species and a diversity of the transparent medium dyaphanum: a substance like air or glass that allows light to pass through. As has been touched upon before, in this proposition there is a diversity of species, but there is no diversity of the transparent medium. On this account, the diversity in species and the transparency are created and [are] the same.