Eratosthenes was able to do so around 240 BC. BC used a scaph to calculate the size of the earth and thus prove that the earth is round and not flat.

The accuracy of Eratosthenes' calculation has been debated since its creation. Dr. Chris Mathew from the University of Sydney delved into this scientific debate to finally determine how accurately Eratosthenes was able to calculate the circumference of the Earth.

Eratosthenes' measurement of the size of the Earth is considered a milestone in the knowledge of the world.

*Finished scaphe with gnomon and geographical data of Alexandria lasered on the front.*

Dr. Christopher Matthew Lecturer- Ancient History School of Arts & Sciences (NSW) Australia Catholic University, in his second doctorate in Astronomy and Astrophysics at the School of Science at Western Sydney University in Australia, had the project: Repetition of Eratosthenes' experiment and accuracy of that time Calculations carried out.

The research of Dr. Mathew was an investigation into how accurately Eratosthenes could estimate the size of the Earth in the 3rd century BC. Calculate BC.

The results show that Eratosthenes' calculations were incredibly accurate and that he calculated the size of the Earth with an error rate of less than 1 %.

*Precise production of the skaphe based on calculations and drawings using CAD processing on the computer.
Isometric view of the finished scaphe 200x200x100, with 4x adjustment screws for alignment.
*

**Scaphe for carrying out the experiments.**

To carry out the experiments and calculations, Dr. Mathew a scaphe that was built and calculated just like this one by Eratosthenes. Dr. Mathew was featured on the homepage of www.skaphe.de found it.

Werner Schreiner is a manufacturer of scaphes. He has been involved in the calculation and manufacture of sundials for over 30 years. Werner Schreiner is, among other things, the manufacturer of Bernhardt's precision sundial.

*Side view of the scaphe 200x200x100.*

**Scaphe with size 200x200x100.**

However, only scaphes measuring 100x100x50 were produced. To obtain a similar size to ancient finds from Delos with a diameter of 275 mm and a find in Naukratis in Egypt with a diameter of 75 mm, a scaphe measuring 200x200x100 mm had to be produced.

Due to parallels with Eastothene's research, the 200x200x100 scaphe is ideal, as he carried out his experiments with a similar size.

*Conversion of CAD data for labeling on laser machine.
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Order for the production of the scaphe by Dr. Chris Mathew July 2020. The scaphe is to be calculated and manufactured for the Egypt, Alexandria location exactly in front of the library.

**Geographical data:**

31°12'0,36'' North — 31.2001 North

29°55'7.32'' East — 29.9187 East

*Location for calculating the scaphe. Forecourt of the new Great Library of Alexandria (the Biblotheca Alexandrina). The measurement should take place again at this point.*

The original plan was to recreate the measurement (Eratosthenes' experiment) on the summer solstice of 2021 in Alexandria Egypt in the forecourt of the new Great Library of Alexandria (the Biblotheca Alexandrina).

The scaphe was precisely designed for the location of the forecourt of the library in Alexandria.

However, this plan had to be abandoned due to the restrictions on international travel introduced in Australia as a result of the Covid pandemic.

*Preparation of the scaphe for labeling the date lines on the laser machine*

Fortunately, a few hours drive north of Sydney, Australia, lies the coastal town of Crescent Head. Crescent Head is located in the Southern Hemisphere at the same latitude as Alexandria in the Northern Hemisphere (31.2 degrees).

Consequently, it was decided to recreate Eratosthenes' experiment at Crescent Head during the southern summer solstice in December 2021.

*Finished scaphes with date lines using laser marking.
*

The scaphe was mounted on mounts used for a Dobsonian telescope so that it could be angled, rotated and aligned with high accuracy using the included alignment jig (taking into account the magnetic variation at the Crescent Head to ensure the sundial faces “due south.” “ is aligned).

Various calculations were made - taking into account the time zone in which Crescent Head is located, the deviation of the "equation of time" between a sundial and a clock in the same location, and adjustments for daylight saving time - which determined that the sun would be on December 22nd 2021 will peak at 12:46:35 p.m.

*Scaphe with adjustment device and tripod from Dobsonian telescope*

When the sundial was set up, it was immediately noticeable that the tip of the shadow cast by the pointer was exactly on the date line for the summer solstice. This would not have happened if the instrument had not been leveled correctly. It was further observed that the tip of the shadow fell on the intersection of the noon line and the scaphe solstice line at 12:44:31, rather than at 12:46:35 as calculated. This corresponds to a slight misalignment of the instrument by half a degree (well within the margin of error).

Even with a small misalignment, the scaphe was still incredibly accurate. The shadow cast by the pointer was also very sharp, easy to observe, and moved slowly across the surface of the scaphe at midday - making observation easier.

*The tip of the shadow was observed to fall at the intersection of the noon line and the solstice line of the scaph at 12:44:31, rather than at 12:46:35 as calculated. This corresponds to a slight misalignment of the instrument by half a degree.
*

The size of the earth could then be determined using the data determined on the scaphe (e.g. the distance from the base of the gnomon to the intersection of the noon and solstice lines).

*In addition, an adjustment device was manufactured for precise alignment of the scaphes. Consisting of a precision compass for aligning the NS direction and
2x precision spirit levels (longitudinal and transverse directions) for precise leveling of the scaphe. *

**Further comments from Dr. Metthew on observations and calculations with the scaphe.**

One of the best results I achieved while working with the scaphe I made was that I was able to figure out exactly how Eratosthenes was able to determine an angle for the shadow of 1/50 of the circumference (which is a key element of his calculations). I'm pretty sure he did it like this:

There are two important distances on the surface of the sundial that Eratosthenes needed to know. The first is the distance from directly below the gnomon (let's call this point G) to the intersection between the noon line and the summer solstice line (let's call this point S)

The other is the distance from point G below the gnomon along the noon line to the outermost edge of the sundial, i.e. the “horizon” (let’s call this point H)

In Eratosthenes' time, the circle was not divided into 360 degrees as it is today, but into 60 “hexatontads,” each corresponding to 6 degrees.

On the sundial calibrated for Alexandria, the distance from G to S is 1.2 hexacontads. The distance from G to H is 15 hexacontads (or ¼ of a circle).

For a sundial calibrated for Syene (which is believed to be on the Tropic of Cancer), the summer solstice line passes directly under the gnomon because no shadow is cast at the tropic at midday of the summer solstice. For this sundial the distance GS = zero

Eratosthenes then only had to calculate the ratio between the two shadows. In Syene there is no shadow, so this has no influence on the calculation. For the shadow in Alexandria, divide the distance GH by the distance GS and get 15/1.2 = 12.5

Since the distance GH is ¼ of a circle, the result is multiplied by 4 to get the fraction of a full circle: 12.5 x 4 = 50

The distance GS for Syene should be subtracted from this, but it is zero anyway

So the overall result is that the difference between the locations of Alexandria and Syene is 1/50 of a circle.

It is an extremely brilliant mathematical deduction by Eratosthenes, which provides a method that one could use for any two sundials calibrated for any location, and calculate on any day of the year, since one only has to see the making and not the actual shadow, and get the same result.

Eratosthenes then had to calculate the distance between Alexandria and Syene and multiply it by 50 to find the total circumference of the Earth.

Calculating the distance he received and the unit of measurement he used was not easy and takes up about half of my new book (new release 2023)!

The end result, however, is that Eratosthenes was able to use the sundial (and particularly the markings on it) to calculate the size of the Earth with a margin of error of less than 1 %. That's not bad in an age without computers and calculators.

**Further questions for Dr. Matthew on the topic of scaphe.**

What significance did Aristarchus of Samos as the inventor of the scaphe and Eratosthenes of Cyrene have for science then and today? (Mathematics, Astronomy).

Modern mathematics and astronomy owe much to Aristarchus and Eratosthenes. Both were contemporaries and knew each other's work. Aristarchus was a man "versed in all branches of science" and is credited with inventing the hemispherical sundial, which was used by Eratosthenes to calculate the size of the Earth. Aristarchus also worked on optics, trying to calculate the size and distances to the sun and moon and predicting eclipses. Perhaps Aristarchus is best known for promoting a theory that the Earth orbits the Sun (rather than the Sun orbiting the Earth, as was commonly believed at the time). This was a very advanced (and controversial) theory and, although rejected at the time, was not proven until the time of Johannes Kepler 1800 years later. Like Aristarchus, Eratosthenes was a man of many talents – an astronomer, mathematician, historian, geographer, philosopher and literary critic. He was a colleague of Archimedes - who dedicated his work The Method to Eratosthenes - and was born in 245 BC. Appointed the third chief librarian of the Great Library of Alexandria in 500 BC. It was here that Eratosthenes conducted much of his research and work - including his calculation of the size of the Earth's circumference.

What significance did the scaphe have at that time?

The design of the hemispherical skape was a major advance in timekeeping instrumentation in ancient times. Before that, most sundials did not take seasonal hours into account (i.e. longer days in summer than in winter), and few were designed for specific locations - which then made them less accurate when moved. By creating a sundial that was precisely calculated for a specific location, the ancient Greeks created an instrument that was not only a clock but also a calendar. The hemispherical skape was used for hundreds of years. The design was further refined in the later Greco-Roman period, but the fundamentals remained unchanged. This instrument would influence other sundials, clocks, and instruments such as armillary spheres for astronomical observation, which would eventually help usher in a new age of scientific understanding.

The results of this research will be published as a book entitled "Eratosthenes and the Measurement of the Earth's Circumference (c.230BC)", which will be published by Oxford University Press in 2023.

**Address of the university**