Eratosthenes

Eratosthenes of Cyrene (in ancient Greek Ἐρατοσθένης, Eratosthénēs) (Cyrene, 276 BC-Alexandria, 194 BC) was a Greek polymath: mathematician, astronomer and geographer of Cyrenaic origin. He first conceived of geography as a systematic discipline, developing a terminology that is still in use today. He is best known for being the first person to calculate the diameter and circumference of the Earth, he did so by comparing the altitudes of the Sun noon at two places separated by a north-south distance. His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis (again with remarkable precision). In addition, he may have estimated the distance from the Earth to the Sun and devised inserting an additional day every four years in the calendars, producing the leap year.He Created the first world map , incorporating parallels and meridians based on available geographic knowledge of his time.

Biography

Eratosthenes son of Aglaos, born in 276 BC. C. in Cyrene. He studied in Alexandria and for a time in Athens. He was a disciple of Ariston of Chios, of Lisanias of Cyrene and of the poet Callimachus and also a great friend of Archimedes. In the year 236 B.C. C., Ptolemy III called him to take charge of the Library of Alexandria, since he held until the end of his days. The Suda claims that, after losing his sight, he starved to death at the age of 80; however, Luciano de Samosata Luciano says that he reached the age of 82; Censorino also maintains that he died when he was 82 years old.

Eratosthenes possessed a great variety of knowledge and aptitudes for study: astronomer, poet, geographer and philosopher, his last name was Pentathlos, a name reserved for the winning athlete in the five competitions of the Games Olympics in Antiquity Olympic Games. Suidas affirms that he was also known as the second Plato and various authors say that he was nicknamed Beta, after the second letter of the Greek alphabet, because he ranked second in all branches of the science he cultivated.

Armillary Sphere

Eratosthenes is credited with the invention, around 255 BC. C., of the armillary sphere that was still used in the 17th century. Although he must have used this instrument for various astronomical observations, the only record that it led him to determine the obliquity of the ecliptic remains. He determined that the interval between the tropics (double the obliquity of the ecliptic) was equivalent to 11/83 of the complete Earth circumference, resulting for said obliquity 23 ° 51 & # 39; 19", a figure that would later be adopted by the astronomer Claudio Ptolemy.

According to some historians, Eratosthenes obtained a value of 24° and the refinement of the result was due to Ptolemy himself up to 11/83. Furthermore, according to Eusebius of Caesarea, he deduced that the distance to the Sun was 804,000,000 or 4,080,000 stadia (according to different translations), the distance to the Moon 780,000 stadia and, according to Macrobius, that the diameter of the Sun was 27 times greater than that of Earth. Actually, the diameter of the Sun is 109 times that of the Earth and the distance to the Moon is almost three times that calculated by Eratosthenes, but the calculation of the distance to the Sun, assuming that the stadium used was 185 meters, in the estimation of 804,000,000 stadia is 148,752,060 km, very similar to today's astronomical unit. Despite the fact that the work Katasterismoi, which contains the nomenclature of 44 constellations and 675 stars, is frequently attributed to him, critics deny that it was written by him, which is why it is usually designated Pseudo- Eratosthenes to its author.

Measuring the dimensions of the Earth

In the summer solstice, the solar rays incisely on Siena (Aswan). In Alexandria, further north, measuring the height of a building and the length of the shadow that it projects, you can determine the angle formed with the plane of the ecliptic, in which the Sun and the city of Siena are located, an angle that is precisely the difference of latitude between both cities. Known this, it is enough to measure the circumference arch and extrapolate the result to the complete circumference (360o).

Reconstruction of the centuryXIX (according to Bunbury) of the map of Eratosthenes of the world known at the time.

The main reason for its celebrity is undoubtedly the determination of the size of the Earth. For this he invented and used a trigonometric method, in addition to the notions of latitude and longitude, apparently already introduced by Dicearco, for which he well deserves the title of father of geodesy.

From references obtained from a papyrus in his library, he knew that in Siena (today Aswan, Egypt) on the day of the summer solstice, vertical objects did not cast any shadow and the light illuminated the bottom of the wells; this meant that the city was located just on the line of the Tropic of Cancer, and its latitude was equal to that of the ecliptic that he already knew. Eratosthenes, assuming that Sienna and Alexandria were of the same longitude (actually 3° apart) and that the Sun was so far from the Earth that its rays could be assumed to be parallel, measured the shadow in Alexandria on the same day as the summer solstice at noon, demonstrating that the zenith of the city was 1/50 of the circumference, that is, 7° 12'; that of Alexandria. According to Cleomedes, Eratosthenes used the scaphium or gnomon (a solar protodial) to calculate this quantity.

Later, he took the distance estimated by the caravans that traded between both cities, although he was able to obtain the data in the Library of Alexandria itself, fixing it at 5,000 stadia, from which he deduced that the circumference of the Earth was 250,000 stadia, a result that he later raised to 252,000 stadia, so that 700 stadia corresponded to each degree. It is also claimed that Eratosthenes, to calculate the distance between the two cities, used a regiment of soldiers to take steps of uniform size and count them.

Assuming that Eratosthenes used the Attic-Italian stadium of 184.8 m, which was the one used by the Greeks of Alexandria at that time, the error committed would be 6192 kilometers (15%). However, some argue that he used the Egyptian stadium (300 cubits of 52.4 cm), in which case the calculated polar circumference would have been 39,614 km, compared to the 40,008 km currently considered, that is, a error of less than 1%.

Now, it is impossible for Eratosthenes to come up with the exact measurement of the Earth's circumference due to errors in the assumptions he made. He would have had a considerable margin of error and therefore could not have used the Egyptian stadium:

1. I assumed the Earth is perfectly spherical, which is not true. A degree of latitude does not represent exactly the same distance in all latitudes, but varies slightly from 110.57 km in Ecuador to 111.7 km in the Polos. That is why we cannot assume that 7° between Alexandria and Siena represent the same distance as 7° in any other place throughout the meridian.
2. It supposed that Siena and Alexandria were located on the same Meridian, which is not so, since there is a 3-degree difference between the two cities.
3. The real distance between Alexandria and Siena (today Aswan) is not 924 km (5000 attic-Italian stadiums of 184.8 m per stadium), but 843 km (air distance and between the centers of the two cities), which represents a difference of 81 km.
4. Actually Siena is not located exactly on the parallel of the tropic of cancer (the points where the rays of the sun fall vertically to the earth in the summer solstice). It is actually 72 km (from the city centre). But because the variations of the Earth's axis flow between 22.1 and 24.5 in a period of 41 000 years, 2000 years ago it was 41 km.
5. The measure of the shadow that was projected on the rod of Eratosthenes 2200 years ago should be 7.5° or 1/48 part of a circumference and not 7,2° or 1/50 part. Since at that time there was no trigonometric calculation, to calculate the angle of the shadow, Eratosthenes could have been validated from a compass, to directly measure that angle, which does not allow such a precise measure.

If Eratosthenes' calculation is redone with the distance and exact angular measurement from Alexandria to the geographical place located right at the intersection of the meridian that passes through Alexandria with the parallel of the Tropic of Cancer, a value of 40,074 km is obtained for the Earth's circumference. That represents only 66 km or a 0.16% error of the actual circumference of the Earth as measured by advanced satellites, which is 40,008 km, proving the validity of his reasoning. This slight difference is due to the fact that the distance between Alexandria and the line of the Tropic of Cancer is 1/46 of a circle, but the Earth is not a perfect sphere.

Posidonius redid Eratosthenes' calculation 150 years later and obtained a significantly smaller circumference. This value was adopted by Ptolemy and was probably the one Christopher Columbus based on to justify the feasibility of traveling to the West Indies. With Eratosthenes' measurements, the trip would not have been made, at least at that time and with those means, accepting only scientific certainties. The doctors consulted in Salamanca, at the royal request, based themselves on them to determine that the main objective —to reach China and Japan— was impossible given the distance. Finally, the company was approved by the queen, based on testimonies and charters that were in the possession of Columbus' partners, mentioning lands a short distance to the west of the Azores, due to the strategic and commercial advantages that the project provided and on secondary objectives, such as the condition of Columbus to obtain perks and percentages on the lands that he discovered on the way.

Eratosthenes' work is considered by some to be the first scientific attempt to measure the dimensions of our planet, since other calculations were made and refined centuries later by scholars such as Caliph Al-Mamun and Jean François Fernel.

Math

He owes a procedure, known as the Sieve of Eratosthenes, to quickly obtain all the prime numbers smaller than a given number. The computer version of this procedure (algorithm) has become over the years a standard method for characterizing or comparing the performance of different programming languages.

Eratosthenes also measured the obliquity of the ecliptic (the tilt of the Earth's axis) with an error of only 7' of arc, and created a catalog (now lost) of 675 fixed stars. His most important work was a treatise on general geography called Geographica (Greek Γεωγραφικά, Geographika). In this work Eratosthenes described and mapped his entire known world, even dividing the Earth into five climatic zones: two freezing zones around the poles, two temperate zones, and one zone spanning the equator and the tropics. He laid out grid lines. superimposed on maps representing the Earth's surface. He used parallels and meridians to link all the places in the world. It was now possible to estimate distance from remote locations with this network on the surface of the Earth. More than 400 city names and their locations were displayed on Geographica.

Other work

Eratosthenes was one of the most preeminent scholarly figures of his time, producing works covering a wide area of knowledge before and during his time at the Library. He wrote on many subjects: geography, mathematics, philosophy, chronology, literary criticism, grammar, poetry, and even old comedies. Unfortunately, only fragments of his works remain after the Destruction of the Library of Alexandria. Eratosthenes' poetic work comprises two works: Erigone, repeatedly praised by Longinus, and Hermes, the best known, a poem on an astronomical and geographical subject that deals with the shape of the Earth, its temperature, the different climates and the constellations. He wrote several treatises on moral philosophy and other philosophical works are attributed to him, without certainty. His first work, called Platonikos (platonicus), contemplates Plato's philosophy from a mathematical point of view. According to Theon of Smyrna, a Pythagorean mathematician, Eratosthenes' work studied basic definitions of geometry and arithmetic, and covered topics such as music. His historical productions were closely linked to mathematics, and his most important work in this discipline It was the Chronography, which includes the dates of the most important literary and political events. It is believed that The Olympics, cited by Diogenes Laertius and Athenaeus, were part of the Chronography. He also wrote a treatise On the ancient Attic comedy, of which Architectonicos and Skenographicos are fragments, in which he dealt with decoration, costumes, the declamation and argument of works by Aristophanes and Cratinus, among others. He also studied Homer's work and wrote a biography on the poet's life that has not survived to this day. In the aforementioned Eratosthenica , Bernhardy compiled the list of all the works attributed to Eratosthenes, as well as the fragments of his writings then known to him, with the exception of Katasterismoi .

Invented the mesolabium, one of the first instruments discovered that is a primitive calculator.

Eponymy

The Eratosthenes crater

• The Eratosthenes seamount located in the Mediterranean Sea bears its name.
• The lunar crater Eratosthenes also bears his name.
• The Eratostenian period on the lunar geological time scale is thus called in its honor.
• The asteroid (3251) Eratosthenes also commemorates its name.