The Philosophy of Roger Bacon

...in the place of the ratio, namely between the whole base and the part of the bases arranged in order. Here it may be evident if one of the bases is the diagonal and the side of the triangle is the side [of the square]. Therefore, for example, the diagonal is the side that touches the square, just as the lateral part is proportional to its diagonal. Therefore, it is given first through that which is not a common? ratio and through which, since just as there is a ratio (?) (?) (?) because it is a ratio and proportional, or known (?) (?) (?) (?) and then, since for the sake of the ratios, he teaches how to investigate the ratio of the first part to the last through that which is proportional to the second doubled for the sake of the first to the last; and note here that it is doubled through the 9th [proposition] of the first [book] and the 26th of the same Bacon is referencing Euclid’s Elements. Book I, Prop 9 deals with bisecting an angle; Prop 26 deals with triangle congruency. For the sake of those things which are known to us, we shall recognize it, and what is the mean medium: a middle term in a mathematical proportion to recognize it. And note further that according to this method, the first and second [parts] are sought, namely the diagonal and the side, through those things which become clear from many [examples]. And note then the simple things from which the ratio is known, if it is held as doubled from the ratio of the second to the first, just as to the third. Moreover, the ratio in all mathematics in the books of Mathematics is that through which the opposite is known, and those two things from which they are composed will be known. Secondly, for that [purpose], the mean for it will be unknown and the third [will be] remote. Similarly, it is so if one is known and the rest [are not]. Likewise, that which is between the means and the extremes will be discordant? or that which is between the extremes and the definition makes known the doubt and the more known [thing] is always known through the 12th [proposition].

Likewise, that which is between the diagonal and the second is composed of two ratios which are two middle equalities, then the diagonal to the second. If that same ratio from the ratio of the diagonal to the side is sub-double The "sub-double" ratio refers to a ratio where the first term is half the second (1:2), so that the diagonal meets in the commensurability commensura: the property of two magnitudes having a common measure of the diagonal to the side—namely to those upon which they are founded—the argument regarding motion could be met otherwise, both through the 9th [proposition] of the third book and through the 12th of the same, and other ways in the same manner. If indeed it is not through this from the plurality, because in the doubled transition? of the point according to its nature to those things which are set forth in the 10th [proposition] of the tenth [book] of Euclid original: "eloy euclidi". Book X of Euclid's Elements is the famous and difficult book dealing with irrational (incommensurable) magnitudes, such as the relationship between the diagonal and side of a square. Just as on account of the world from the highest and... as they are... and as it is from the proof that there are two mobile bodies [capable of] motion, [one] with rectilinear motion motu recto: motion in a straight line, traditionally associated with the four earthly elements and a common one with circular motion In medieval physics, the heavens move in circles, while earthly elements move in straight lines toward or away from the center of the universe. Then, thirdly, it must be considered that mobile bodies with rectilinear motion, which we denote by water, air...