Eratosthenes

Eratosthenes Mathematicians

Summary

In Eratosthenes is a man that not only loved to read everything, but he applied everything to life. Philosophy, mathematics, music, astronomy and other facets of life had interconnecting equations. Here is the man that totally believed in and lived through his beliefs.

Early Years

Eratosthenes was born before in 276 B.C. Now known as modern day Libya, he entered this life in Cyrene, Greece. Not much is known of his childhood, but it would be safe to say he enjoyed preparation to be under some of the masters of his day by the time his education began.

Among his early teachers of his youth was Lysanias, that lived in his hometown and Callimachus (poet, also from Cyrene). Another ancient notable to be a part of his education of his early years was Ariston of Chios, a poet. Ariston was from the Stoic school of philosophy. After a period of time staying in Cyrene, Eratosthenes made his way to Athens where he would attend university.

Another of Eratosthenes’s teachers was a scholar by the name of Callimachus. This gentleman would move on to Alexandria, Egypt to become a librarian at famous Alexandrian University. He had been invited by the Egyptian ruler Ptolemy II (Philadelphus) to administer the great known works of the ancient world; but unlike present day books were made of vellum and papyrus.

When Callimachus passed away in 240 B.C., Eratosthenes would take over in his position, requested Ptolemy III (Euergetes) personally. At this point, Eratosthenes was already in Alexandria working as a tutor for Ptolemy III’s son Philopator. His official role in the University was as third librarian, part of a hierarchy that established privilege and prestige of the scholars. The library was positioned in the temple of the Muses called the Mouseion.

One character aspect of Eratosthenes that is known by scholars was his penchant for coming up short when it came to moving up the ladder of, what was known, success. His friends and colleagues recognized that the man had intellect in many arenas of knowledge (while in Athens he had studied mathematics and philosophy). He acquired a nickname, Beta, which being the second position in the Greek alphabet, meant he fell short of being number one. Penthalos was to be a later reference to this man of positional short-comings, just like an athlete coming in second, but never first. By having that association, Eratosthenes was considered a man of many talents but not keen in any one particular. (Skilled in many and master of none.)

However, considering the ramifications that Eratosthenes life would have on modern mathematics and science; it seems unbelievable that these monikers would have been given to such a man. His would be a future that dwelled into many aspects of geometry, and its relationship to other aspects of life; such as philosophy, science and even music.

The mind of Eratosthenes it is said could be one that never failed to take in new knowledge. This was a man that constantly had to be reading, learning, applying in everything he did. Archimedes purport wrote, “If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.” This meant that everything Eratosthenes read and learned had deeper connections to other aspects of life. As it will be seen, Archimedes was correct in his assessment of this man of learning.

Achievements

Platonicus is a writing of Eratosthenes that deals with the correlation of Plato and Mathematics. A famous Greek philosopher of the day was Theon of Smyrna. Much of the work he dealt with used a considerable amount from the work of Eratosthenes in how prime numbers had a considerable amount of relationship to areas such as music and the study of astronomy. Plato’s philosophy and Eratosthenes adaptations of geometry were integral to Theon’s thesis of the correlations.

Eratosthenes spent a considerable amount of time dealing with prime numbers (natural number greater than one that is only divisible by itself and 1 and leaves no remainder). In the work of studying all natural numbers (including aspects associated with them)Eratosthenes created the “sieve of Eratosthenes which was an application of formulating prime numbers up to any specific limit. The idea of the sieve is that if you were to start with two and attempt the prime number equation on every second number, followed by doing the same with three, then four, etc.; prime numbers could be established. This meant going all the way to infinity; which would require the use of computers today to continue the process.

He did considerable work in the field of geometry, especially in measurements. Along with establishing ideas on the distance from the earth to the sun and the moon; he dealt with the circumference of the earth.

While in his time as the librarian in Alexandria, Eratosthenes developed a method of establishing distances between locations. He was able to see a southern town called Syene from the city of Alexandria that at noon of the longest day of the year, shadows were not cast. By comparing the shadow positions between the two communities, he could establish a relative position of distance. His reasoning developed:

  • Assume that the world is round.
  • The position of sun rays to Earth are parallel as they reach the planet.
  • Measure angle of shadow in Alexandria if Syene has no shadows
  • Distance established.
  • Use this to determine approximate size of the earth.

Of course this meant that units of measure, which were called stades, should establish relative size of the earth. If one knew how to translate a value known as stadium to today the number from Eratosthenes could be compared. Some figured he might have used Pliny’s reasoning that a stadium represented about 157 meters. If that is the case, Eratosthenes assumptions about the earth circumference are fairly close. What is important is that he was using a form of geometry (Euclidian) to use angles to answer these type of questions.

AN apparent forged letter from Eratosthenes to Ptolemy III gives mention of some form of a mechanical apparatus to find line segments as a part of the problem of dealing with the duplication of the cube. The specifics of this aspect of the problem involved considerations of understanding the root of 2. Eratosthenes was doing more tutoring the Egyptian leader’s son; he was also staying in touch with Ptolemy over various questions involving mathematics and geometry.

In The History of Mathematics by Thomas Heath, there is a quote by Theon of Smyrna.

when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an alter double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.

This was a reference to the idea of duplicating the cube. The hope was that builders, designers and other areas of Grecian society would be more able to utilize the science of geometry and mathematics as a whole.

If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging. Do not thou seek to do the difficult business of Archytas‘s cylinders, or to cut the cone in the triads of Menaechmus, or to compass such a curved form of lines as is described by the god-fearing Eudoxus. Nay thou couldst, on these tablets, easily find a myriad of means, beginning from a small base. Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands. Thus may it be, and let any one who sees this offering say “This is the gift of Eratosthenes of Cyrene”.

This is an inscription that was located in a column in Alexandria, said have been established by Eratosthenes in relation to the problem of the cube.(Also in Heath’s book) It is remarkable that such displays that were found of ancient civilizations show the importance and cultural hierchy people, such as Eratosthenes, placed upon the study of mathematics.

Another of the advances of Eratosthenes was the development of the calendar. His computations allowed for the establishment of leap years. Leap years were the idea that every few years (four as we know it today) would require a means of allowing calendars to catch up, based on the idea the earth orbits completely around the sun equals one year. However, every so often another day was needed to bring the calendar on track.

He also formulated a chronology, from a systematic standpoint, of important events. His attempt was to use the conquest of Troy as a basis of giving dates to major historical situations; including political happenings. He also developed a catalog of stars that numbered almost 700 at the time.

Not lost surrounding the life of Eratosthenes is his work with geography. For ages scholars had attempted to understand the nature of the Nile river. This mighty river that supplied the irrigation water of much of the lowlands seemed to create a stir in understanding her highs and lows as she forged ahead.Oftentimes these great scholars were inaccurate in their depictions of the flow of the great river, along withits beginnings. Eratosthenes developed a chart of the river’s route as it travelled to its end destination. He theorized that lakes were where the mighty Nile began its course.

Eratosthenes also developed a thought on the races of the people that occupied what is now known as Yemen. His insights placed four main groups of races into the inhabitants of “Eudaimon Arabia” and their placement around the region. Although his suppositions might have been off; the names he gave for each group of people did not change.

Eratosthenes made major contributions to geography. He sketched, quite accurately, the route of the Nile to Khartoum, showing the two Ethiopian tributaries. He also suggested that lakes were the source of the river. A study of the Nile had been made by many scholars before Eratosthenes and they had attempted to explain the rather strange behavior of the river, but most like Thales were quite wrong in their explanations. Eratosthenes was the first to give what is essentially the correct answer when he suggested that heavy rains sometimes fell in regions near the source of the river and that these would explain the flooding lower down the river. The scholars also debated why flooding occurring up the river. Eratotsthenes established that heavy rains created the higher levels in the lower areas, creating the flooding of the region. A seemingly common sense arrival of an idea, displayed by someone once nicknamed as Beta.

Sense of who he was.

How Eratosthenes died is somewhat obscure. There is a contention that he made have simply starved himself to death when he could no longer see. What is certain is that this great man of Greece played a major role in the development of geometry and its everyday applications. Eratosthenes was an individual who studied under the great minds and took their work a step further. His love of math, science, astronomy, geography and so much more were part of his philosophy. A philosophy that believed in the love of learning and applying.

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