Quaestiones in primum librum Sententiarum, dist. 14-48

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See [*Tractatus De Signis*](https://isidore.co/calibre/browse/book/2977) p. 92/37-38, which cites Book I of Scotus's *Commentary on the Sentences of Peter Lombard* , dist. 31, q. 1 ([pp. 483ff.](https://archive.org/stream/operaomni10duns#page/482/mode/2up) // PDF pp. 491ff.), where Scotus, *contra* Thomists, "requires only a distinction between the things which are the extremes, not between the rationales of founding."* * *Jeff Kalb says on p. 80 of [*Music and Measurement: An the Eidetic Principles of Harmony and Motion*](https://isidore.co/calibre/browse/book/5952):> Nevertheless, the concept of equality, whether phenomenal or operational, remains a *relative* one. There must be an absolute in which this relation itself is grounded.This seems to be exactly the Thomists' view: Scotus believes relations are defined only by the *extrema* (terms of the relation) and Thomists believe the fundament and the terms of the relation are both required for establishing similarity or equality of relations. Where you say "There must be an absolute in which this relation itself is grounded.," you seem to be referring to the fundament ( *fundamentum* ) of the relation (cf. [p. 539 for defns. of fundament](https://archive.org/stream/tractatusdesigni00johnrich#page/538/mode/2up)). * * *cf. p. 23 (PDF p. 31) of Jeff Kalb's [*Music and Measurement: An the Eidetic Principles of Harmony and Motion*](https://isidore.co/calibre/browse/book/5952) ** (St. Cecilia's Feast Day, 2016):> A ratio for Euclid … is a *relation* between magnitudes; it is not a magnitude or a quantity in its own right. (Hence, it is most emphatically not a “rational number”…) Therefore, operations to which magnitudes are “naturally” subject (such as addition of line-segments, multiplication of numbers and its geometrical counterpart…) would appear to be alien intruders once transplanted to the domain of ratios (or, indeed, the domain of proportions, as will happen in algebra when equations are added to, or multiplied by, one another, and so on.) Nonetheless, on the most plausible reading of “compound ratio” in Euclid, we are being asked to allow some such alien operation to be applied to a ratio or, more precisely, to a pair of ratios.Euclid's definition of ratio in [Element. (l. 5.) defins.](https://isidore.co/calibre/browse/book/6071) as a " *similitudo duarum proportionum* " is mentioned in the context of "whether mind-independent relations themselves, precisely as such, are able to found other relations." Scotus thought so (cf. Deely's [*Four Ages of Understanding* ](https://isidore.co/calibre/#panel=book_details&book_id=4151)[pp. 376](https://books.google.com/books?id=zAsjkHJ8aP8C&pg=PA376)-7, esp. where he summarizes, on the first full quoted ¶ on p. 377, Scotus's view: " *unica requirit [Scotus] solum distinctionem inter res, quæ sunt extrema, non inter rationes fundandi.* *D. Thomas utrumque requirit* "; cf. the *Summa* article "[Whether there is equality in God?](https://isidore.co/aquinas/summa/FP/FP042.html#FPQ42A1THEP1)"); St. Thomas did not. cf. John of St. Thomas's [*Tractatus de Signis*](https://isidore.co/calibre/browse/book/2977) [pp. 102](https://archive.org/stream/tractatusdesigni00johnrich#page/102/mode/2up) line 23ff.Euclid:> 3\. A **ratio** (Λόγος) is a sort of relation in respect of size between two magnitudes of the same kind. > 6\. Let magnitudes which have the same ratio be called **proportional** (ἁνάλογον).