Nicholas of Cusa and the concept of infinity

¶ … 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.

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.

While it is known that Peurbach travelled throughout Europe between the years 1448 and 1453, there is no record of the precise dates. At the time, he also had an international reputation as an astronomer of note, despite the fact that he had not publications at the time. He did however lecture in Germany, France and Italy.

After lecturing at Bologna and Padua, these universities offered him permanent appointments as lecturer, but Peurbach turned these down. During his travels he also met the leading Italian astronomer of the time, Giovanni Bianchini, in Ferrara. Bianchini also offered Peurbach a post at an Italian university. Peurbach however remained unwilling to be tied to any specific institution of learning and turned down the offer. In 1453, Peurbach returned to Vienna and received a Master's Degree on 28 February of that year. Peurbach died on 8 April 1461 in Vienna, Austria.

Conclusion

In many ways, the 15th century is a time of pioneering thought regarding astrology and the sciences. An interesting element of this time is that ancient thought was not so much discarded as used as a platform for new work. The most prominent manifestation of this was the connection between astrology and astronomy. As seen above, astrology was very much an accepted science of the time. Rather than attempting to discredit this, scientific pioneers like Cusa and Peurbach accepted and studied it in order to derive philosophies and discoveries of their own. Perhaps this is a viewpoint that today's scientists and philosophers can apply usefully in their own work.