"Now, it is possible that an infinite collection of number stands to an infinite collection in
every numerical ratio and even in every non-numerical ratio. And some infinites are more than others and others less than others; thus, the collection of all numbers, both odd and even, is infinite and it is greater than the collection of all the even numbers, even though this too is infinite, for it exceeds it by the collection of all the even numbers, even though this too is infinite, for it exceeds it by the collection of all the odd numbers. The collection of all numbers proceeding without break from unity by doubling is also infinite, as too is the collection of all the [numbers] corresponding to those doubles as their halves, and yet the collection of the halves must be half of the collection of their doubles. In the same way, the collection of all numbers proceeding without break from unity by tripling is triple the collection of all the thirds corresponding to those triples. The same clearly holds for numerical ratios of every kind, since one infinite can stand to another infinite in any one of these ratios." (Robert Grosseteste, "De Luce", cca. 1220)
"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)
"Sound is generated by motion, since it belongs to the class of successive things. For this reason, while it exists when it is made, it no longer exists once it has been made. […] All music, especially mensurable music, is founded in perfection, combining in itself number and sound." (Jean de Muris, "Ars novae musicae", 1319)
"Every measurable thing, except numbers, is imagined in the manner of continuous quantity. Therefore, for the mensuration of such a thing, it is necessary that points, lines and surfaces, or their properties be imagined. For in them, as the Philosopher has it, measure or ratio is initially found, while in other things it is recognized by similarity as they are being referred to by the intellect to the geometrical entities." (Nicole Oresme, "The Latitude of Forms", cca. 1348-1362)
"It is established that every continuum has further parts, and not so many parts finite in number that there are not further parts, and has all its parts actually and simultaneously, and therefore every continuum has simultaneously and actually infinitely many parts." (Gregory of Rimini [Gregorii Ariminensis], "Lectura super primum et secundum sententiarum", cca. 1350)
"A second type of the false position makes use of roots of negative numbers. I will give an example: If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 30 or 40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion. This, however, is closest to the quantity which is truly imaginary since operations may not be performed with it as with a pure negative number, nor as in other numbers. [...] This subtlety results from arithmetic of which this final point is, as I have said, as subtle as it is useless." (Girolamo Cardano, "Ars Magna", 1545)
"There are certain pleasures which only fill the outward senses, and there are others also which pertain only to the mind or reason; but music is a delectation so put in the midst that both by the sweetness of the sounds it moveth the senses, and by the artificiousness of the number and proportions it delighteth reason itself." (John Northbrooke , "Against Dicing", 1577)
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