Συμμετέχοντας το σχολείο μας στο δίκτυο "Eratosthenes Experiment" για δεύτερη συνεχή χρονιά, πραγματοποίησε στις 19 Μαρτίου 2015 το προγραμματισμένο πείραμα. Το σχολείο μας αντάλλαξε στοιχεία μετρήσεων με το 1ο Δημοτικό Σχολείο Αργοστολίου, το 33ο Δημοτικό Σχολείο Βόλου, το Μειονοτικό Γυμνάσιο - Λύκειο Κομοτηνής και το 6ο Λύκειο Ξάνθης. Παρόλες τις δυσκολίες του καιρού το κρίσιμο διάστημα ο ήλιος μας έκανε τη χάρη και μας βοήθησε στην μέτρηση.
By joining our school network "Eratosthenes Experiment" for the second consecutive year, held on March 19, 2015 the planned experiment. Our school exchange measurement data with the 1st Primary School of Argostoli, the 33rd Primary School of Volos, the Minority Middle School - Lyceum of Komotini and the 6th Lyceum of Xanthi. Despite the difficulties of the weather the period in the sun made us a favor and helped us to measure.
Duke u bashkuar rrjetit shkollor tonë "Eksperiment Eratosthenes" për të dytin vit radhazi, të mbajtur më 19 mars 2015 i eksperiment planifikuar. Këmbimit Shkolla jonë dhënave matje me Shkollën Fillore 1 e Argostoli, Shkollën fillore e 33 Volos, Shkolla e Mesme e Minoriteteve - Liceut të Komotinit dhe Liceut 6 të Xanthi. Me gjithë vështirësitë e motit periudhë në diell na bërë një nder dhe na ndihmoi për të matur.
Λίγα λόγια για το πείραμα ...
Το πείραμα του Ερατοσθένη βασίστηκε στη μέτρηση του ύψους του Ηλίου την ίδια ημερομηνία σε δύο διαφορετικές τοποθεσίες, καθώς και στην πεποίθηση του μεγάλου έλληνα μαθηματικού ότι ο Ηλιος είναι πολύ μακριά από τη Γη, τόσο ώστε οι ακτίνες του να φθάνουν στον πλανήτη μας σχεδόν παράλληλα. Από διηγήσεις ταξιδιωτών ο Ερατοσθένης έμαθε ότι στις 21 Ιουνίου, την ημέρα του θερινού ηλιοστασίου, ο Ηλιος καθρεφτίζεται στην επιφάνεια του νερού των πηγαδιών της πόλης Συήνης, αυτής που σήμερα οι Αιγύπτιοι ονομάζουν Ασουάν. Από την πληροφορία αυτή ο Ερατοσθένης συμπέρανε ότι η Συήνη βρίσκεται πάνω στον τροπικό του Καρκίνου, δηλαδή στον παράλληλο κύκλο με γεωγραφικό πλάτος 23,5 μοίρες. Το χαρακτηριστικό των τόπων που βρίσκονται στον τροπικό του Καρκίνου είναι ότι το μεσημέρι της 21ης Ιουνίου ο Ηλιος βρίσκεται στο ζενίθ, δηλαδή ακριβώς κατακόρυφα προς τα πάνω. Ετσι οι ακτίνες του διαδίδονται κατά μήκος των κατακόρυφων τοιχωμάτων των πηγαδιών, ανακλώνται στην επιφάνεια του νερού και επιστρέφουν προς την επιφάνεια, κάνοντας ορατό το είδωλό του σε έναν παρατηρητή που κοιτάζει από το στόμιο του πηγαδιού.
Το μεσημέρι της ημέρας του θερινού ηλιοστασίου ο Ερατοσθένης μέτρησε το ύψος του Ηλίου στην πόλη στην οποία κατοικούσε, την Αλεξάνδρεια της Αιγύπτου. Η μέτρηση έγινε με τη βοήθεια ενός οβελίσκου, ο οποίος είναι το αρχαιότερο αστρονομικό όργανο στην ιστορία της επιστήμης. Το μήκος της σκιάς που ρίχνει ο οβελίσκος, διαιρεμένο με το ύψος του οβελίσκου, μας δίνει, όπως μάθαμε στο σχολείο, την εφαπτομένη της γωνίας του ύψους του Ηλίου. Η γωνία αυτή, η οποία από τη μέτρηση του Ερατοσθένη προέκυψε 7,2 μοίρες, είναι ίση (ως «εντός-εκτός και επί τα αυτά», όπως θυμούνται οι παλαιότεροι) με την επίκεντρη γωνία που σχηματίζουν δύο ακτίνες της Γης με άκρα τη Συήνη και την Αλεξάνδρεια, υπό την προϋπόθεση ότι οι δύο πόλεις έχουν το ίδιο γεωγραφικό μήκος, βρίσκονται δηλαδή στον ίδιο μεσημβρινό. Επειδή από τη γεωμετρία γνωρίζουμε ότι η απόσταση των δύο πόλεων, η ακτίνα της Γης και η γωνία που μέτρησε ο Ερατοσθένης συνδέονται με τη σχέση απόσταση/ακτίνα = 6,28x(7,2/360), η ακτίνα της Γης βρίσκεται αμέσως αν γνωρίζουμε την απόσταση των δύο πόλεων. Την εποχή του Ερατοσθένη, περί το 250 π.Χ., δεν υπήρχε ακριβής μέθοδος μέτρησης τόσο μεγάλων αποστάσεων. Σύμφωνα με την παράδοση, ο Ερατοσθένης ανέθεσε σε επαγγελματίες βαδιστές να την υπολογίσουν, και το αποτέλεσμά τους το συνέκρινε με τις εκτιμήσεις αρχηγών καραβανιών. Το τελικό του αποτέλεσμα ήταν ότι η απόσταση Αλεξάνδρειας- Συήνης ισούται με 5.000 στάδια, οπότε η ακτίνα της Γης προκύπτει ίση με 252.000 στάδια.
Για να μπορέσουμε να εκτιμήσουμε την ακρίβεια της μέτρησης του Ερατοσθένη, θα έπρεπε να γνωρίζουμε πόσο είναι το μήκος ενός σταδίου σε μέτρα, καθώς και κατά πόσο αληθεύουν οι δύο υποθέσεις του Ερατοσθένη, δηλαδή ότι η Συήνη έχει γεωγραφικό πλάτος 23,5 μοίρες και ότι Συήνη και Αλεξάνδρεια βρίσκονται στον ίδιο μεσημβρινό. Μια ματιά σε έναν σύγχρονο χάρτη δείχνει ότι και οι δύο υποθέσεις ήταν λανθασμένες, αλλά το λάθος δεν ήταν μεγάλο: το γεωγραφικό πλάτος της Συήνης είναι 24,1 μοίρες, ενώ τα γεωγραφικά μήκη των δύο πόλεων διαφέρουν μόνο κατά μία μοίρα. Επομένως η βασική πηγή σφάλματος είναι το μήκος ενός σταδίου σε μέτρα. Θα έλεγε κανείς ότι έχουν διασωθεί πολλά αρχαία στάδια, οπότε δεν έχουμε παρά να μετρήσουμε πόσο μήκος έχει ένα από αυτά. Δυστυχώς τα στάδια δεν είχαν το ίδιο μήκος σε όλες τις περιοχές της αρχαίας Ελλάδας. Αν υποθέσουμε ότι ο Ερατοσθένης εννοούσε αττικά στάδια των 185 μέτρων, τότε το αποτέλεσμά του δίνει για την ακτίνα της Γης 7.400 χιλιόμετρα, τιμή 16% μεγαλύτερη από την πραγματική. Αν όμως εννοούσε αιγυπτιακά στάδια, πράγμα που είναι και το πιθανότερο, τότε κατά τον Ερατοσθένη η ακτίνα της Γης είναι 6.316 χιλιόμετρα, μόλις 1% μικρότερη από την πραγματική, που σήμερα γνωρίζουμε ότι είναι 6.366 χιλιόμετρα!
About the experiment ...
The experiment of Eratosthenes was based on the measurement of the height of the Sun the same day in two different locations, and the conviction of the great Greek mathematician that the Sun is very far from Earth, so that the rays to reach the planet almost parallel. From stories of travelers Eratosthenes learned that on June 21, the day of the summer solstice, the sun is reflected on the surface of the water of the city wells Syinis, that now the Egyptians call Aswan. From this information Eratosthenes concluded that Syini lies on the Tropic of Cancer, that is the parallel circle with latitude 23.5 degrees. The characteristic of sites located on the Tropic of Cancer is that at noon on June 21 the sun is at the zenith, that is exactly vertically upwards. Thus rays propagate along the vertical walls of the wells, are reflected at the water surface and back to the surface, making visible his reflection to an observer looking from the mouth of the well.
At noon on the day of the summer solstice Eratosthenes measured the height of the Sun in the city in which he resided, Alexandria, Egypt. The measurement was made using a tang, which is the oldest astronomical instrument in the history of science. The length of the shadow cast by its obelisk, divided by the height of the obelisk, gives us, as we learned in school, the tangent of the angle of the height of the sun. This angle, which by measuring Eratosthenis showed 7.2 degrees, is equal (as "on-off and on those" as remember older) with the central angle formed by two radii of the Earth with its ends Syini and Alexandria, provided that the two cities have the same longitude and are therefore on the same meridian. Because of the geometry we know that the distance between the two cities, the radius of the Earth and the angle measured Eratosthenes associated with the relationship distance / radius = 6,28x (7,2 / 360), the radius of the Earth is immediately if we know the distance between the two cities. At the time of Eratosthenes, around 250 BC, there was no accurate method of measuring such large distances. According to tradition, Eratosthenes commissioned walkers professionals to calculate, and their effect compared with the estimates Heads caravan. The final result was that the distance of Alexandria Syinis equals 5,000 steps, so the radius of the Earth resulting equal to 252,000 steps.
In order to assess the accuracy of measurement of Eratosthenes, we should know how much is the length of a stage in meters and how true the two cases of Eratosthenes, namely that Syini has latitude 23.5 degrees and that Syini and Alexandria are on the same meridian. A look at a modern map shows that both assumptions were wrong, but the error was not large: the latitude of Syinis is 24.1 degrees, while the longitudes of the two cities differ only by one degree. Therefore the main source of error is the length of a step in meters. One could say that they have preserved many ancient stages, so we only have to measure how long is one of them. Unfortunately stages were not the same length in all regions of ancient Greece. Assuming that Eratosthenes meant Attic steps of 185 meters, the result gives the radius of the Earth 7400 km, price 16% higher than the actual. But if he meant Egyptian stage, which is likely, then by Eratosthenes radius of Earth is 6316 km, only 1% less than the real, we now know that it is 6366 kilometers!
Rreth eksperimentit ...
Eksperimenti i Eratosthenes është bazuar në matjen e lartësisë së Diellit të njëjtën ditë në dy lokacione të ndryshme, si dhe bindja e matematikan i madh grek se Dielli është shumë larg nga Toka, në mënyrë që rrezet për të arritur planet pothuajse paralel. Nga tregimet e udhëtarëve Eratosthenes mësuar se më 21 qershor, ditën e solsticit veror, dielli është pasqyruar në sipërfaqen e ujit të puseve të qytetit Syinis, që tani Egjiptasit quajnë Aswan. Nga ky informacion Eratosthenes arriti në përfundimin se Syini shtrihet në Tropiku i Gaforres, që është rrethi paralele me gjerësi 23,5 gradë. Karakteristikë e vendeve të vendosura në Tropiku i Gaforres është se në mesditë më 21 qershor dielli është në zenit, që është pikërisht vertikalisht lart. Kështu rrezet përhapur përgjatë mureve vertikale të puseve, janë pasqyruar në sipërfaqen e ujit dhe përsëri në sipërfaqe, duke e bërë të dukshme reflektimin e tij për një vëzhgues në kërkim nga goja e mirë.
Në mesditë në ditën e solsticit veror Eratosthenes matur lartësinë e Diellit në qytetin në të cilin ai e banuar, Alexandria, Egjipt. Matja është bërë duke përdorur një erë e fortë, e cila është instrumenti më i vjetër astronomik në historinë e shkencës. Gjatësia e hedhur hije nga obelisku i saj, pjesëtuar me lartësinë e obelisk, na jep, siç kemi mësuar në shkollë, tangjent e kënd e lartë të diellit. Ky kënd, e cila duke matur Eratosthenis tregoi 7,2 gradë, është e barabartë (si "on-off dhe mbi ata" si të kujtuar më të vjetër) me kënd qendror i formuar nga dy radii e Tokës me skajet e saj Syini dhe Alexandria, me kusht që dy qytetet kanë të njëjtin gjatësi dhe janë në të njëjtin meridian. Për shkak të gjeometrisë ne e dimë se distanca mes dy qyteteve, rrezja e Tokës dhe kënd matur Eratosthenes i lidhur me marrëdhënie distancë / rrezja = 6,28x (7,2 / 360), rrezja e Tokës është menjëherë në qoftë se ne e dimë distanca ndërmjet dy qyteteve. Në kohën e Eratosthenes, rreth 250 pes, nuk kishte asnjë metodë të saktë të matur distancat tilla të mëdha. Sipas traditës, Eratosthenes porositur rrethoresh profesionistë për të llogaritur, dhe efekti i tyre në krahasim me vlerësimet kokat karavan. Rezultati përfundimtar ishte se distanca e Aleksandrisë Syinis është e barabartë me 5000 hapa, kështu që rrezja e Tokës rezultuar e barabartë me 252.000 hapa.
Për të vlerësuar saktësinë e matjes së Eratosthenes, ne duhet të dimë se sa është gjatësia e një faze në metra dhe sa e vërtetë të dy rastet e Eratosthenes, domethënë që Syini ka gjerësi 23,5 gradë dhe se Syini dhe Aleksandri janë në të njëjtën meridian. Një vështrim në një hartë moderne tregon se të dy supozimet e kishin gabim, por gabimi nuk ishte i madh: gjerësi e Syinis është 24,1 gradë, ndërsa longitudes e dy qyteteve të ndryshojnë vetëm nga një shkallë. Prandaj burimi kryesor i gabimit është gjatësia e një hap në metra. Dikush mund të thotë se ata kanë ruajtur shumë faza të lashta, kështu që ne vetëm duhet për të matur se sa kohë është një prej tyre. Për fat të keq fazat nuk ishin të njëjta gjatësi në të gjitha rajonet e Greqisë së lashtë. Duke supozuar se Eratosthenes menduar hapat papafingo e 185 metra, rezultat i jep rrezja e Tokës 7400 km, çmim 16% më të larta sesa aktual. Por në qoftë se ai do të thotë fazë egjiptiane, e cila është e mundshme, atëherë me Eratosthenes rrezja e Tokës është 6316 km, vetëm 1% më pak se reales, ne tani e dimë se ajo është 6366 kilometra!
