User:Biff2011/Pedro Ciruelo

Arithmetical and Geometrical
In Ciruelo's work the Cursus, he examines why the adjectives "arithmetical" and "geometrical" to are used to describe and name progressions, sequences, and proportions. Ciruelo is one of few known sources discussing this history. Although one might assume that arithmetic progressions or geometric sequences are named so because of the respective branch of mathematics they are used in, Ciruelo argued differently. He believed that these terms were used to emphasize the process of measurement inherent to such mathematics. This view is supported by Aristotle's use of the term "geometrical proportion" to refer to the metric information contained in mathematical ratios.

On the topic of proportions, Ciruelo's four part Cursus quattuor mathematicarum artium liberalium also contains a treatise on music theory. These writings were exemplary of the "arithmetization of geometry" taking place in mathematics during the 16th century. That is, they represent a shift towards proving geometric theorems solely algebraically, choosing to treat geometric objects as abstract algebraic symbols, rather than physical items in space. The subtle emphasis on the algebra being a "symbol" is dropped in this time period, such that the symbol and the geometric object represented by it are viewed as one in the same. This was of great relevance to music theory, as it is by nature intrinsically geometric, involving the division of strings into intervals. Ciruelo's work is able to explain such concepts without reference to physical, geometric strings. For example, he describes a method with which to break compound intervals down into the sum of smaller intervals solely using numeric ratios. This level of abstaction outlined by Ciruelo and other 16th-century scholars is credited with providing a "better approach to equal temperament [tuning]" around this time period.

Mathematics program
Although there is little administrative evidence of Ciruelo in Paris, Ciruelo and Jacques Lefèvre d'Étaples are considered to be responsible for the expansion of mathematical teaching in Paris in the late 1400s. At the time there was no standardized curriculum for mathematics at the University of Paris. In the late 1400s Ciruelo and Lefèvre each had their own advanced mathematical programs for the university. Ciruelo's program used 5 different mathematical texts on "theoretical arithmetic, practical arithmetic, geometry, and astronomy".

Yet, although Ciruelo was called a "lecturer of mathematics" in Thomas Bradwardine's Arithmetic speculativa, it is unclear whether Ciruelo actually taught at the University of Paris because there was no position with that title during the time. During his time in Paris, Ciruelo wrote 4 mathematical books: two of them being new editions of Thomas Bradwardine's texts, 1 being an original publication, and finally a commentary of Sacrobosco's Sphaera. Ciruelo's works were not as well developed as his contemporaries. His works often did not follow standard mathematical programs, and his books were published in different formats. Ciruelo published his works from Guy Marchant’s workshop. Marchant was particularly interested in Ciruelo works because of his academic background and appreciation for illustrated works. Printing Ciruelo’s works soon became a problem because of the technical limitations at the time. Printers would have to obtain special printing equipment to reproduce a variety of mathematical figures and Arabic numbers.

Ciruelo on the cosmos
During the 16th century, followers of Christianity believed in three different explanations for the possible. These were the natural, preternatural, and supernatural orders. Ciruelo summarizes these orders from a cosmological standpoint.


 * The supernatural order refers to events caused by God's intervention, and was in the form of a miracle. As Ciruelo puts it, "[the supernatural] comes from God who operates miraculously on the course of nature."
 * The preternatural order refers to events caused by the intervention of spirits, demons, and angels in the natural world. These entities were created by God and, although having supernatural powers, were still considered natural beings. This combination of the natural and the supernatural necessitates a middle ground between them: the preternatural.
 * The natural order refers to events caused by beings and materials in the 'sublunar sphere' -- that is, here on earth. According to Ciruelo, the third order usually takes the form of actions committed by living creatures with free-will.

Ciruelo believed that these three orders were the only categories needed to explain events on earth. The natural and supernatural orders were originally proposed by Anselm of Canterbury, but Ciruelo eventually added the preternatural order, seeing it to be a natural consequence of the other two. This extra distinction between the natural and the supernatural outlined the futility of superstitious rituals, according to Ciruelo. Superstitious rituals were aiming to accomplish preternatural feats, and thus, would never be able to come to fruition without the help of some preternatural or supernatural power.