15th Century Was an Interesting

¶ … 15th century was an interesting time for the history of science and mathematics. Two great thinkers of the time was Nicholas of Cusa, known for his thinking around the theory of infinity, and Georg Peurbach, for his groundbreaking philosophy surrounding astronomy.

The Infinite Before Cusa

While Nicholas von Cusa did groundbreaking work on the concept of infinity, his was not the first philosophy to introduce it to the world. According to McFarlane (2004), infinity as a concept in the West first appeared with Anaximander of Miletus, who suggested that all things operate upon the principle of the limitless, which was referred to as the "not-limited." By striving towards the not-limited, all opposites are reconciled, and conflict eliminated.

Philosophers such as Pythagoras however regarded the limitless as abhorrent, as he based all concepts of beauty upon the comprehensible, which can be explicated by number. The infinite has no number, and can therefore be neither conceptualized nor understood. As such, this principle cannot provide any derivation of beauty and should therefore not form part of the philosophy of the cosmos. For Pythagoras, all things were constituted by numbers.

This philosophy was however soon proved to be invalid in itself, or at least incomplete, as the principles of geometry and arithmetic began to distinguish themselves from each other. The necessity for further thought on the subject was subsequently addressed by philosophers such as Zeno and later by Plato, upon whose work Nicholas of Cusa-based much of his thought and writing.

Nicholas of Cusa and the Infinite

According to O'Connor and Robertson (1996), Nicholas of Cusa's main interests were geometry and logic. His interest led him to the stud of the infinite -- both the infinitely large and the infinitely small. As symbolic of the infinite, Cusa also used the circle in his religious teaching to indicate how the truth can be approached but never completely reached -- human knowledge of the universe would never be complete.

In his search for the truth, Cusa became interested in astronomy in 1444 and purchased several books on the subject. Theories he constructed in this regard include the earth moving around the sun, the stars being additional suns for additional inhabited worlds, and the infinity of space. Cusa also published improvements to the Alfonsine Tables, including a practical method to find the planets and stars.

According to O'Connor and Robertson (1996), Nicholas of Cusa was born as Nikolaus Kryffs or Krebs. His father was a wealthy shipper working on Mosel River. The name Cusa was derived from the town Kues, of which the Latin name was Cusanus. Religion and science at the time were very closely linked, and Cusa was ordained as a priest in 1440, and promoted to cardinal in 1448. He became the bishop of Brixon in 1450.

Cusa's appointment to these offices provided him with a platform to indulge his passion for astrology and astronomy. According to McFarlane (2004), for example, Cusa served Popes Eugene VIC, Nicholas V and Pius II as papal legate. In addition to his work on astrology and astronomy, Cusa also invested a large amount of his time in juridical, theological, philosophical and scientific studies.

Cusa showed an equally pioneering spirit in his philosophical work as in his scientific writings. He departed from commonly accepted theories such as those by Aristotle and other scholars revered at t the time. His first work was On Learned Ignorance (De docta ignorantia), and concerned his first thoughts on the finite and infinite. These concepts were then applied to both philosophy, other fields in the humanities, and to astronomy and scientific thought. He pioneered thinking later established by Copernicus and Kepler, regarding the sun as the center of the cosmos and the motion of the planets in a non-circular fashion, respectively.

Nicholas of Cusa was well-educated, and read in various languages. His main influences were Plato and Neo-Platonic philosophers such as Plotinus and Proclus, as well as Eionysius and Meister Eckhart. For Cusa, the concept of God was the ultimate Maximum.

Cusa's main philosophies regarding the Infinite derive from a mystical illumination he received during 1437 when he was travelling from Constantinople. Nicholas regarded this as a gift from God, and used it as a basis for his subsequent philosophies on the infinite. A key philosophy in this regard was Cusa's ideas surrounding opposites; the infinite unites opposites, transcends comparison in terms of the positive and negative. Initially, Cusa mainly applied this philosophy to religion. As seen above, however, it also became a key principle in his philosophical and scientific thought.

He viewed the concept of learned ignorance as an example of a coincidence of opposites: the more aware human beings are of their own ignorance, the more true knowledge they gain. This coincides with his views on the attainment of truth -- the concept of truth is infinite while remaining perpetually fluid -- hence finding an absolute and eternal truth in any science or philosophy of life is impossible.

In addition to his contributions to astronomy, Nicholas of Cusa also made significant progress in the area of mathematics. His infinity concepts for example pioneered possibilities in the calculus and mathematics of the continuum. Nicholas von Cusa died on 11 August 1464 in Todi, the Papal States (now known as Italy).

Georg Peurbach and the Planets

Like Nicholas von Cusa, Georg Peurbach was also responsible for a large amount of progress in the field of astronomy during the 15th century. According to O'Connor and Robertson (2006), this was a time when astrology was still a recognized science among both scientific and social circles. Most citizens at the time believed that the planets influenced the lives and events on earth. As such, the astrologer was a scientist whose aim it was to find the precise position of the sun, moon and planets. These were then connected to the specific time and place of a person's birth in order to draw up astrological charts. Astronomical events such as eclipses and the appearance of comets were also regarded as particularly important events during this time.

Astrology then served as a platform for the development of astronomical science, as exemplified by the life and development of Georg Peurbach. Peurbach was appointed as court astrologer by King Ladislas V of Hungary and Bohemia during 1454. Because the king spent most of his time in Prague and Vienna, Peurbach was also employed as a teacher at the University of Vienna, although his work here focused mostly on the humanities rather than astronomy.

After the death of Ladislas in 1457, Peurbach was appointed as court astrologer for the Holy Roman emperor Frederick III. At the University of Vienna, Peurbach formed a collaborative connection with Regiomontanus, one of his students. This collaboration led to Peurbach's focus on astronomical investigation and the publication of Theoricae Novae Planetarum (New Theories of the Planets). This publication explicated Ptolemy's epicycle theory of the planets, and indicated that the planets were solid crystalline spheres. Peurbach believed that the motion of the planets was controlled by the sun.

Peurbach also made improvements to the Alfonsine Tables by observing eclipses and comets. These observations led him to write several reports and construct astronomical instruments in order to better predict the duration and timing of eclipses and comets. A further significant discovery is the deviation of the compass. According to O'Connor and Robertson (2006), Peurbach devised the earliest tablet dial that included the deviation in about 1451. Other instruments include celestial spheres according to his own astronomical discoveries and inventions and wrote the book Algorismus. The book contained practical calculations with integers and fractions.

Johannes Bessarion, the archbishop of Nicaea, was also keenly interested in astronomical and mathematical science. He commissions Peurbach and Regiomontanus to translate the Almagest from the Greek. At the time of his death Peurbach had completed six of the books for this project, while Regiomontanus continued the work after Peurbach was gone. Ultimately, the work included 13 books, which is viewed as a turning point in scientific thought at the time. Ptolemy's thought was significantly expanded, and there was a general move towards innovation, away from the reverence of antiquity.

Georg Peurbach was the son of Ulrich Aunpekh. He was born on 30 May, 1423 in Peuerbach, Austria. Like von Cusa, he also takes his name from the town in which the family lived, west of Linz. Nothing is known of Peurbach's childhood years, and the first known record of his life is his matriculation in 1446 from the University of Vienna and under the name

Georgius Aunpekh de Pewrbach.

He followed his matriculation by studying, also at Vienna, and attained his Bachelor's degree. According to O'Connor and Robertson (2006), he took 'n standard arts course from the University, which most consisted of the humanities. Although the authors note that it is likely that he also took courses that included mathematics, his expertise in astronomy would have been self-taught. The reason for this estimation is the fact that the last astronomer at Vienna during Peurbach's time was John of Gmunden. He died four years before Peurbach's matriculation, leaving the University without an astronomy lecturer. However, his library and instruments were probably accessible to Peurbach.