Identify Greek letters used in formulas

Identify Greek letters used in formulas

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Could anyone tell me what the names of the Greek letters in the screenshot are? Is the first one alpha? They are used in calculating ICRF rotations. Wikipedia uses a different font for their list of characters, so it's hard to tell.

The first one is alpha (α); the second one is delta (δ). You can study the shapes of the letters here:

Here's what they mean (information from a comment to your question):

Orientation models use three Euler angles to describe body orientation. The first two angles are the right ascension and declination of the north pole of a body as a function of time. The third angle is the prime meridian location (represented by "W"), which is expressed as a rotation about the north pole, and is also a function of time.

What is “M” This and “NGC” That? And Why do Comets Have Weird Names?!

Charles Messier, a french astronomer, wasn’t the first astronomer to catalog deep sky objects, but the catalog he published in 1771 is well known and still very much used.

Messier was a comet hunter, and was frustrated by sighting many fuzzy looking objects that resembled but were not actual comets. As he was looking for objects that moved, he compiled these objects that stayed fixed into a list so he wouldn’t waste any further time on them. Go to this website for Messier’s actual accounts as he cataloged the objects.

There are a total of 110 Messier objects in the night sky. They are among the most well known and brightest of the nebulae, star clusters, and galaxies – They can all be observed through small telescopes under dark skies.

Since Messier observed from France, the list only contains objects visible from the skies he could observe from. While he was still able to cover about 2/3 of the sky, he missed objects that are further south.

Many famous deep sky objects with common names have a Messier number. Here are some well known examples:

M 1 – Crab Nebula
M 7 – Ptolemy Cluster
M 8 – Lagoon Nebula
M 13 – Hercules Globular Cluster
M 31 – Andromeda Galaxy
M 33 – Triangulum Galaxy
M 42 – Orion Nebula
M 44 – Beehive Cluster
M 45 – The Pleiades
M 57 – Ring Nebula
M 81 – Bode’s Galaxy
M 82 – Cigar Galaxy

All the Messier objects have an NGC designation as well, but astronomers will usually use their Messier names when identifying them.

But I see Another Catalog with a “C”

That stands for Caldwell, which was made by Patrick Moore. He used his other surname, Caldwell, because Moore had the same initial as Messier. This was made to compliment Messier’s catalog in an attempt to list the brightest deep sky objects that Messier missed. Because Messier was simply listing objects that could be confused as comets, Moore wanted a catalog that actually was meant for deep sky objects!

It’s simply another catalog like Messier’s, only this one is listed in order from north to south, and also includes objects in the southern hemisphere. Here are some of my favorites:

C13 – Owl Cluster
C14 – The Double Cluster
C49 & C50 – The Rosette Nebula and Cluster
C63 – The Helix Nebula
C76 – The Scorpius Jewel Box
C80 – Omega Centauri Cluster

I sometimes use it to identify respective objects, but most people identify them by their NGC catalog number instead.

NGC stands for New General Catalog

The New General Catalog is much larger and much more comprehensive. It was compiled in the 1880s by John Louis Emil Dreyer, and it was essentially an update to existing catalogs that were published by William Hershel and his son decades earlier. The New General Catalog, with later modifications and updates, is still widely used by astronomers today.

There are 7,840 NGC objects. No, I do not know them all, especially because only a fraction of them have names, and most of them are much dimmer in general than the Messier or Caldwell objects. Often times, I only need to identify them when I see them in long exposures when they previously weren’t visible with just my eyes through a telescope!

What about “IC?”

“IC” stands for Index Catalog, which were supplements to the New Galactic Catalog. These were mainly clusters, nebulae, and galaxies that were discovered thanks to photography after the initial NGC was published. Since most of these were discovered with long exposure photography, chances are you may not see them very well even in large telescopes from a dark sky!

Why Do I see Greek Letters or Numbers Next to Star Names?

They’re simply ways to identify and catalog the notable stars in the sky.

Under a system constructed by German astronomer Johann Bayer, familiar stars in a constellation are designated from brightest to dimmest using the Greek alphabet, with α (Alpha) being the brightest, β (Beta) being the next brightest and so on. For example, while a star may have a formal name, like Aldeberan in Taurus, it will also be designated α Tauri (Alpha Tauri) with Tauri being the genitive Latin name for Taurus. This is just a simple way to catalog the familiar stars you see, especially when they don’t have a formal name. When the Greek letters ran out, Latin letters were used

What about numbers in the case of stars like 61 Cygni? That comes from the Flamsteed designation, named after John Flamsteed. Some, but not all constellations use these designations, but when they do, they’re merely another way of identifying stars. It can be a little confusing, as the numerical order is not from brightest to dimmest (For example, the star Deneb, the brightest in Cygnus, has a Bayer designation of α Cygni, but under Flamsteed’s, it’s given the name 50 Cygni). They instead are listed in ascending order according to their position in Right Ascension…

There are other catalogs as well, but I won’t be going over them. After reading this post, you now have a quick rundown on the most common catalogs that astronomers use when referencing deep sky objects.

If you purchase a computerized go-to telescope, more than likely your hand controller software will have every Messier, Caldwell, NGC, and IC object available at the push of a button!

So… Why Do We See These Numbers and Odd Names for Comets Nowadays? Can’t I just use a Simple Name?

Comets are listed by their type and designated by the year of their discovery followed by a letter indicating the half-month of the discovery and a number indicating the order of discovery. The actual name is in parentheses.

Let’s go over that first letter:

P/ – periodic comet: a shorter orbital period less than 200 years.
C/ – non – periodic comet: longer, more eccentric orbits lasting over 1,000 years.
X/ – a comet that doesn’t have an orbit that was calculated.
D/ – a comet that has broken apart, or been lost.
A/ – once thought to be a comet but is now a minor planet
I/ – an interstellar object from outside our solar system!

Now for that second letter and number.

When it comes to half months, each letter corresponds to either the first half or second half of the month it was discovered. (A and B for January, C and D for February, and so on…) The letters not used in these designations are “I” and “Z,” so after April uses “G” and “H,” May uses “J” and “K” and finally, December uses “X” and “Y.” After that, the number used is the order it was discovered – first to be discovered gets number 1, and so on…

C/2020 F3? A non-periodic comet/ third comet discovered in the second half of March 2020.

Okay.. What About the Names?

Amateur astronomers and major research observatories are ALWAYS pointing their telescopes to the sky and looking for them, whether as part of a team, or on their own. Individuals who discover the comet can still choose to have it named after them, or if I ever have that chance, I’ll name it after my late brother.

There are plenty of comets named after the same individual, such as Robert H. McNaught who has discovered 82 comets, thus many are named after him, even though the most famous “Comet McNaught” will always be C/2006 P1 – the Great Comet of 2007.

One comet got named the way it was because it was discovered by two people independently!

Alan Hale and Thomas Bopp both happened to chance upon a new comet on the same night in July 1995 while viewing from different USA locations. By the time Thomas Bopp had notified the the clearing house for astronomical discoveries via a telegram of all things, Alan Hale had emailed them three times with updated coordinates. Because both men were confirmed to discover it on the same night, it was given the name “Hale-Bopp.”

Nowadays, new comets are discovered much more frequently, that they’re just named after the project observatory, which often have long names abbreviated into simpler acronyms. Here are the names you’ve most likely have seen:

ATLAS – Asteroid Terrestrial-impact Last Alert System
PANSTARRS – Panoramic Survey Telescope and Rapid Response System
ISON – International Scientific Optical Network
LINEAR – Lincoln Near-Earth Asteroid Research
NEOWISE – Near Earth Object Wide-field Infrared Survey Explorer

Yes, you can always use simple names like “Comet Atlas” or “Comet Neowise” when referring to a comet that you heard about. Google searches will most likely get you to that respective comet if you look for it without necessarily using the designation numbers. However, it’s important to remember that there exists more than one NEOWISE, or more than one PANSTARRS, etc…

And if you are presenting about a specific comet that isn’t very well known or because there exists more than one with the same name, then it helps to use the designations to make it easier to identify. Otherwise it can turn into an Abbot and Costello routine of “here’s Comet McNaught, no not that McNaught, this McNaught…”

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Astronomy AnswersMathematical Notation

World Wide Web browsers often cannot show many of the symbols that one often finds in mathematical formulas. A mathematical formula from these web pages can therefore look different in one browser than in another browser. Some examples are shown below, so you can see what certain mathematical constructions from these pages look like in your browser. If your browser cannot display many special cases, then it can be difficult to understand the formulas. You can of course try to use a different browser, or you can study the source code of the page.

I am switching to using MathJax (//, but not all formulas are in MathJax format yet.

A number of special characters and notations:

Table 1: Mathematical Notation

notation meaning
is equivalent to
| | absolute value
(i=1) nsummation for [i] from 1 through [n]
√2 the square root of 2
approximately equal to
< less than
> greater than
⌈ ⌉ rounded up (to a whole number)
⌊ ⌋ rounded down (to a whole number)
[ ] rounded to the nearest whole number
&pi ratio of the circumference to the radius of a circle
infinity an infinitely large number

Special functions are indicated by their full name or by their usual abbreviation (like those used on electronic calculators). I've tried to use a different style for these than for the names of variables. Some examples:

Table 2: Mathematical Functions

() min(x,y) the least (closest to (<-infty>)) of () and ()
() max(x,y) the greatest (closest to (<+infty>)) of () and ()
() sin(x) the sine of angle ()
() cos(x) the cosine of angle ()
(< an(x)>) tan(x) the tangent of angle ()
() arcsin(x) the arc sine of ()
() arccos(x) the arc cosine of ()
() arctan(x) the arc tangent of ()
() arctan(y,x) the angle between the ()-axis and the line from (<(0,0)>) to (<(x,y)>)
() exp(x) () to the power () the natural antilogarithm of ()
() log(x) the decimal (base 10) logarithm of ()
() ln(x) the natural (base ()) logarithm of ()
() sinh(x) the hyperbolic sine of ()
() cosh(x) the hyperbolic cosine of ()
(< anh(x)>) tanh(x) the hyperbolic tangent of ()
() arsinh(x) the hyperbolic arc sine of ()
() arcosh(x) the hyperbolic arc cosine of ()
() artanh(x) the hyperbolic arc tangent of ()
() sgn(x) the sign of (): &minus1 for (), 0 for (), +1 for ()

For the arctan function with one argument and the one with two arguments we have ( anleft( arctan left( frac ight) ight) = an(arctan(y,x)) ), but ( arctanleft( frac ight) ) may differ by a multiple of 180° from ( arctan(y,x) ).

Words that are not names of variables or of special functions are written just like the names of special functions, but only if the formula is displayed as a separate equation (with an equation number). For example, "smaller terms" shows up like this:

In science, very large and very small numbers are often written in exponential notation. In these pages we use the calculator or computer notation with an "e" to introduce the power of 10: 1.2 × 10 5 = 1.2 * 10 5 = 120,000 or ( 1.2×10^ <5>).

Why are Greek letters used in mathematical & scientific equations

There are several reasons why there are many Greek letters that have been adopted into common use for constants in equations.

First, of course, it is necessary to realise that many of our standard letters are widely used, especially for variables: x, y, z are some common examples, but others are used as well.

Many letters from the Greek alphabet are used as constants within equations and formulas. &Pi, &Theta , as well as &alpha, &beta, &theta and the like are widely used and seen seen representing the values or constants for a variety of values.

The roots of the usage of Greek letters comes from the earliest philosophers like Aristotle, and Diophantus and others. They used letters from the Greek alphabet as symbols to represent various variables. Although later civilisations used their own letters, the use of Greek letters tended to be used down the ages - people tended to use what was already established.

Today there are advantages to using Greek alphabet symbols. They are more distinctive than the normal alphabet in everyday use and they are less likely to be confused with the language text within mathematical work being written.

It is really a matter of convenience as well as the reduction of confusion that has lead to the continued use of Greek alphabet symbols being used to represent constants and sometimes variables in equations.

The Greek alphabet symbols and characters have a central position for use as constants within the whole of the scientific arena from physicals and chemistry to more specific areas including electronics and electronics engineering. Wherever in the scientific space, from school to university and research to the application of science, Greek alphabet symbols will be encountered.

Identify Greek letters used in formulas - Astronomy

There are two equations concerning light that are usually taught in high school. Typically, both are taught without any derivation as to why they are the way they are. That is what I will do in the following.

Brief historical note: I am not sure who wrote this equation (or its equivalent) first. The wave theory of light has its origins in the late 1600's and was developed mathematically starting in the early 1800's. It was James Clerk Maxwell, in the 1860's, who first predicted that light was an electromagnetic wave and computed (rather than measured) its speed. By the way, the proof that light's speed was finite was published in 1676 and the first reliable measurements of the speed of light, ones that were very close to the modern value, took place in the late 1850's.

Each symbol in the equation is discussed below. Also, right before the examples, there is a mention of the two main types of problems teachers will ask using the equation. I encourage you to take a close look at that section.

1) &lambda is the Greek letter lambda and it stands for the wavelength of light. Wavelength is defined as the distance between two successive crests of a wave. When studying light, the most common units used for wavelength are: meter, centimeter, nanometer, and Ångström. Even though the offical unit used by SI is the meter, you will see explanations and problems which use the other three. Less often, you will see other units used picometer is the most common one among the less-often used wavelength units. Ångström is a non-SI unit commonly included in discussions of SI units because of its wide usage.

Keep in mind these definitions:

The symbol for the Ångström is Å.

Most certainly, you will need to move easily from one unit to the other. For example, notice that 1 Å = 10¯ 10 meter. This means that 10 Å = 1 nm. So, if you are given an Ångström value for wavelength and a nanometer value is required, divide the Ångström value by 10. If you can't make easy transitions between various metric units, you'd better go back to study and practice that area some more.

2) &nu is the Greek letter nu. It is NOT the letter v, it is the Greek letter nu. It stands for the frequency of the light wave. Frequency is defined as the number of wave cycles passing a fixed reference point in one second. When studying light, the unit for frequency is called the Hertz (its symbol is Hz). One Hertz is when one complete cycle passes the fixed point in one second, so a million Hz is when a million cycles pass the fixed point in one second.

There is an important point to make about the unit on Hz. It is NOT commonly written as cycles per second (or cycles/sec), but only as sec¯ 1 (more correctly, it should be written as s¯ 1 you need to know both ways). The "cycles" part is deleted, although you may see an occasional problem which uses it.

A brief mention of cycle: imagine a wave, frozen in time and space, where a wave crest is exactly lined up with our fixed reference point. Now, allow the wave to move until the following crest is exactly lined up with the reference point, then freeze the wave in place. This is one cycle of the wave and if all that took place in one second, then the frequencey of the wave is 1 Hz.

In any event, the only scientifically useful part of the unit is the denominator and so "per second" (remember, usually as s¯ 1 ) is what is used. The numerator "cycles" is not needed and so its presence is simply understood and, if writing a fraction is necessary, a one would be used, as in 1/sec.

3) c (lower case) is the symbol for the speed of light, the speed at which all electromagnetic radiation moves when in a perfect vacuum. (Light travels slower when passing through objects such as water, but it never travels faster than when in a perfect vacuum.)

Both ways shown below are used to write the value. You need to be aware of both:

The actual value is just slightly less, but the above values are the one generally used in introductory classes. (sometimes you'll see 2.9979 rather than 3.00.) Be careful about using the exponent and unit combination. Meters are longer than centimeters, so there are less of them used above.

Since there are two variables (&lambda and &nu), we can have two types of calculations:

(a) given the wavelength, calculate the frequency use this equation: &nu = c / &lambda

(b) given the frequency, calculate the wavelength use this equation: &lambda = c / &nu

One last comment: you sometimes see the letter f used for frequency, replacing the Greek letter nu. Like this:

Most likely, it will not cause you problems, but I did want to mention it, anyway.

An interesting little light trivia: light travels about one foot every nanosecond. You might try and work out the proper calculation before checking the answer.

Example #1: What is the frequency of electromagnetic radiation having a wavelength of 210.0 nm?

210.0 nm x (1 m / 10 9 nm) = 210.0 x 10 -9 m

We can leave it right there or convert it to scientific notation:

2.100 x 10 -7 m

Either way works fine in the following calculation. Check with your teacher to see if they have a preference. Then, follow their preference.

(2.100 x 10 -7 m) (&nu) = 3.00 x 10 8 m/s

&nu = 3.00 x 10 8 m/s divided by 2.100 x 10 -7 m

&nu = 1.428 x 10 15 s -1

Example #2: What is the frequency of violet light having a wavelength of 4000 Å?

The solution below depends on converting Å into cm. This means you must remember that the conversion is 1 Å = 10¯ 8 cm. The solution:

Notice how I did not bother to convert 4000 x 10¯ 8 into scientific notation. If I had done so, the value would have been 4.000 x 10¯ 5 . Note also that I effectively consider 4000 to be 4 significant figures.

Comment: be aware that the range of 4000 to 7000 Å is taken to be the range of visible light. Notice how the frequencies stay within more-or-less the middle area of 10 14 , ranging from 4.29 to 7.50, but always being 10 14 . If you are faced with this calculation and you know the wavelength is a visible one (say 5550 Å, which is also 555 nm), then you know the exponent on the frequency MUST be 10 14 . If it isn't, then YOU (not the teacher) have made a mistake.

Example #3: What is the frequency of EMR having a wavelength of 555 nm? (EMR is an abbreviation for electromagnetic radiation.)

1) Let us convert nm into meters. Since one meter contains 10 9 nm, we have the following conversion:

555 nm x (1 m / 10 9 nm)

555 x 10¯ 9 m = 5.55 x 10¯ 7 m

2) Inserting into &lambda&nu = c, gives:

(5.55 x 10¯ 7 m) (x) = 3.00 x 10 8 m s¯ 1

x = 5.40 x 10 14 s¯ 1

Example #4: What is the wavelength (in nm) of EMR with a frequency of 4.95 x 10 14 s¯ 1 ?

1) Substitute into &lambda&nu = c, as follows:

(x) (4.95 x 10 14 s¯ 1 ) = 3.00 x 10 8 m s¯ 1

x = 6.06 x 10¯ 7 m

2) Now, we convert meters to nanometers:

Example #5: What is the wavelength (in both cm and Å) of light with a frequency of 6.75 x 10 14 Hz?

The fact that cm is asked for in the problem allows us to use the cm/s value for the speed of light:

(x) (6.75 x 10 14 s¯ 1 ) = 3.00 x 10 10 cm s¯ 1

x = 4.44 x 10¯ 5 cm

I could also have used (1 Å / 10 -8 cm) for the conversion. I have a practice of putting the one with the larger unit (the cm in this case) and then figuring out how many of the smaller unit (the Å) there are in one of the larger unit.

Example #6: Which of the following represents the shortest wavelength?

1) Convert the wavelengths such that they are all the same unit. I choose to convert to nanometers and will start with (a):

(6.3 x 10¯ 5 cm) (10 9 nm / 10 2 cm) = 630 nm

One immediate conclusion is that (b) is not the correct answer.

(3.5 x 10¯ 6 m) (10 9 nm / 1 m) = 3600 nm

(a) is the correct answer.

Greek Alphabet

The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BC. It was the first true alphabet, that is, an alphabet with a symbol for each vowel and consonant, and is the oldest alphabetic script in use today.

Besides writing modern Greek, today its letters are used as mathematical symbols, particle names in physics, as names of stars, in the names of fraternities and sororities, in the naming of supernumerary tropical cyclones, and for other purposes. The Greek alphabet originated as a modification of the Phoenician alphabet and in turn gave rise to the Gothic, Glagolitic, Cyrillic, Coptic, and possibly the Armenian alphabets, as well as the Latin alphabet.

Capital Low-case Greek Name English
Alpha a
Beta b
Gamma g
Delta d
Epsilon e
Zeta z
Eta h
Theta th
Iota i
Kappa k
Lambda l
Mu m
Nu n
Xi x
Omicron o
Pi p
Rho r
Sigma s
Tau t
Upsilon u
Phi ph
Chi ch
Psi ps
Omega o

'Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.'

The wheels of Greek astronomical science

Feature Articles – The wheels of Greek astronomical science Over the past fifty years, the Antikythera Device has gone from being the most anomalous and controversial artefact to one of the most renowned pieces of evidence of the scientific genius of our ancestors – a millennium ahead of its time.
by Philip Coppens

In 1900, a Greek sponge diver called Elias Stadiatos, working off the small Greek island of Antikythera, found the remains of a Greek ship at the bottom of the sea. The wreck was 50 metres long, located 15-25 metres off Point Glyphadia, lying in 43 metres of water. At the time, diving had to be done without the aid of any modern technology currently available to the diving community. It meant the work was highly dangerous. In fact, when the authorities began to remove objects from the wreck, out of the ten divers, one was accidentally killed, while two other divers became permanently disabled. Conditions had vastly improved when Cousteau visited the wreck in 1953, but by that time, the Greek government had long removed everything from the boat.

The rewards of the initial team’s work were marble and bronze statues, gold jewellery, amphorae and other artefacts, all dating from the first century BC, when the ship was believed to have sunk, on what is believed to have been a delivery from Rhodes to Rome. In early 1902, Valerio Stais began sorting through the recovered material, all donated to the Museum of Athens. On May 17, 1902, Stais noticed a calcified lump of bronze that did not fit anywhere, and which looked like a big watch. He guessed it was an astronomical clock and wrote a paper on the artefact. But when it was published, he was labelled ridiculed for even daring to suggest such a thing.

His critics argued that sundials were used to tell the time. A dial mechanism was unknown, even though it was described on what must thus have been a purely theoretical basis. The status quo was that “many of the Greek scientific devices known to us from written descriptions show much mathematical ingenuity, but in all cases the purely mechanical part of the design seems relatively crude. Gearing was clearly known to the Greeks, but it was used only in relatively simple applications.”

So they could do it, but they did not do it. So: had Stais rightfully identified what some called the “most complicated piece of scientific machinery known from antiquity”, or was it too good to be true? The future would tell, but it was for the moment definitely too good to be believed. In 1958, Yale science historian Derek J. de Solla Price stumbled upon the object and decided to make it the subject of a scientific study, which was published the following year in Scientific American. Part of the problem, he felt, was its uniqueness. De Solla stated: “Nothing like this instrument is preserved elsewhere. Nothing comparable to it is known from any ancient scientific text or literary allusion. On the contrary, from all that we know of science and technology in the Hellenistic Age we should have felt that such a device could not exist.” He likened the discovery to finding a jet plan in Tutankhamen’s tomb and at first believed the machine was made in 1575 – a date of the first century BC remained hard to accept – let alone defend.

Still, Price must have realised that whereas its age was a dangerous subject to discuss, it was safe to explore the mechanism and function of the instrument. He thus concluded that the object was a box with dials on the outside and a series of gear wheels inside.

At least 20 gear wheels were preserved, including a sophisticated assembly of gears that were mounted eccentrically on a turntable. The device also contained a differential gear, permitting two shafts to rotate at different speeds. Doors were hinged to the box to protect the dials inside. As to its purpose: the mechanism appeared to have been a device for calculating the motions of stars and planets: a working model of the solar system.

This was not just speculation on his part. Price noted that the front dial was just clean enough to read its function: “It has two scales, one of which is fixed and displays the names of the signs of the zodiac the other is on a movable slip ring and shows the months of the year. Both scales are carefully marked off in degrees. […] Clearly this dial showed the annual motion of the sun in the zodiac. By means of key letters inscribed on the zodiac scale, corresponding to other letters on the parapegma calendar plate, it also showed the main risings and settings of bright stars and constellations throughout the year.” Price knew he had merely postponed the inevitable and would have to tackle its age. Evidence of its ancient origin could be found in the device itself: the Greek inscriptions. Price was helped in this work by George Stamires, a Greek epigrapher. To quote Price: “Some of the plates were marked with barely recognisable inscriptions, written in Greek characters of the first century BC, and just enough could be made of the sense to tell that the subject matter was undoubtedly astronomical.” There was no way back and scientists could only pretend the device and Price’s analysis did not exist – or accept the undeniable truth: it was ancient, it was Greek… embedded belief systems of what the ancients were, could and did would have to be adjusted.

There was also circumstantial evidence, which created a historical framework into which the device fit nicely: similar mechanisms were described by Cicero and Ovid. Cicero, writing in the first century BC – the right timeframe –, mentioned an instrument “recently constructed by our friend Poseidonius, which at each revolution reproduces the same motions of the sun, the moon and the five planets.” He also wrote of a similar mechanism that was said to have been built by Archimedes and which was purportedly stolen in 212 BC by the Roman general Marcellus when Archimedes was killed in the sacking of the Sicilian city of Syracuse. The device was kept as an heirloom in Marcellus’ family. Despite these literary references, scientists were doubtful and Price summed up their thinking: “Even the most complex mechanical devices described by the ancient writers Hero of Alexandria and Vitruvius contained only simple gearing. For example, the taximeter used by the Greeks to measure the distance travelled by the wheels of a carriage employed only pairs of gears (or gears and worms) to achieve the necessary ratio of movement. It could be argued that if the Greeks knew the principle of gearing, they should have had no difficulty in constructing mechanisms as complex as epicyclic gears.”

Still, it was clear that someone had obviously applied the theory and had come up with a practical tool. But who had created the machine? The likely suspect may have been the Greek astronomer, mathematician and philosopher Geminus, a student or late follower of Poseidonius. The latter, of course, was the one whom Cicerco credited with inventing exactly what the device was.

Geminus was a Stoic, from a school founded by Zeno, and lived from 135 to 51 BC, teaching on Rhodes. Rhodes was the centre of astronomical research. Geminus himself not only is known to have defended the Stoic view of the universe, but in particular to defend mathematics from attacks by Sceptic and Epicurean philosophers. The Antikythera device would have been right up his street, as it combined astronomy and proved the powers and the excellence to which applied mathematics could excel: science and mathematics could mimic the motions of the universe.

Most importantly, he lived in the right timeframe. Furthermore, the date for which this calculator was set was the year 86 BC, which some researchers have argued can be seen by the positions of the dials and pointers. 86 BC was an important astronomical year, as five conjunctions of planets in four zodiacal signs occurred that year, an ideal time to set an astronomical calendar. This date has also influenced the dating of the ship wreck, as many believe it will not have been much later – as otherwise the clock would have been reset to an astronomical event at a later date. Many thus argue for a date of 83-81 BC, though others posit dates such as 71 BC, adding that there is no guarantee the device was not idle for a number of years before being transported to Rome. All of this understanding is intriguing, but for one researcher, Maurice Chatelain, one important ingredient was missing: logic. Chatelain argued that “if someone wants to construct an astronomical calculator by using intermeshing gears, the first condition is to find the number of cycles necessary to obtain an exact number of whole days. Some of these cycles are easily found but many are nearly impossible.”

Each gear is a cycle this is how any mechanical clock works: seconds turn to minutes, to hours, and in some clocks to days, if not larger cycles. To make such clocks work, not only the cycles need to be known, but also the ratios between the cycles: how seconds relate to minutes (60:1), minutes to hours (60:1), hours to days (24:1), etc. It is difficult enough to construct such a device for the solar year, but the Antikythera device also incorporated the cycles of the moon and five of the nearest planets. No wonder scientists were sceptical that the device was… a device.

To make the system work, the system would have to be based on days, and thus the cycles would be expressed in full, whole days, with the ratios between the various cycles based upon the day counts of the cycles too. The genius that created the artefact would thus have to be aware of the cycles of the heavenly bodies. This in itself was within the remit of the Greek scientific community – and many generations and civilisations older than that. But a key question was what system was used, as each country had its own. The Greeks used the so-called Metonic cycle of 19 tropical years, but this, Chatelain felt, had no real value in creating a gear calculator. According to Chatelain, only the Egyptian calendar system is suited for being used as a calculator – and he also found it was the one at the basis of the Antikythera machine: “The seemingly complicated Egyptian calendar, based on Sirius, the Sun, and also the Moon, actually works like a charm. Every four years represents exactly 1,461 days which in turn represent 49.474 synodical moon months. This last number has to be multiplied only 19 times to give a number of whole days – 27,759 – equal to 940 months, or 76 Sothic years, which is the cycle of the Rhodes calculator!”

Still, some do not share Chatelain’s enthusiasm for an Egyptian origin. One inscription on the device itself significantly reads “76 years, 19 years”. This refers to the Calippic cycle of 76 years, which is four times the Metonic cycle of 19 years, or 235 synodic (lunar) months. The next line includes the number “223”, which refers to the eclipse cycle of 223 lunar months. Price himself reasoned that “using the [Metonic] cycles, one could easily design gearing that would operate from one dial having a wheel that revolved annually, and turn by this gearing a series of other wheels which would move pointers indicating the sidereal, synodic and draconitic months. Similar cycles were known for the planetary phenomena in fact, this type of arithmetical theory is the central theme of Seleucid Babylonian astronomy, which was transmitted to the Hellenistic world in the last few centuries BC.” Though it was quite clear that all of this knowledge was not Greek in origin, the question remained whether it was Babylonian or Egyptian. Price had injected a new life fluid into the device and major breakthroughs occurred in the last decade of the 20th century. With the arrival of powerful computers, those machines were used to reminisce about what many considered to be the oldest computer – and the latest generation was used to shed light on what some considered to be the “Adam” of the line.

First, a partial reconstruction was built by Australian computer scientist Allan George Bromley (1947–2002) of the University of Sydney, working together with the Sydney clockmaker Frank Percival. This project led Bromley to review Price’s X-ray analysis made in 1973 and to make new, more accurate X-ray images that were studied by Bromley’s student, Bernard Gardner, in 1993.

Later, John Gleave constructed a working replica of the mechanism. According to his reconstruction, the front dial shows the annual progress of the sun and moon through the zodiac… against the Egyptian calendar. But, as if to remain neutral in the Egyptian or Greek debate, he stated that the upper rear dial displays a four-year period and has associated dials showing the Metonic cycle of 235 synodic months (19 solar years). The lower rear dial plots the cycle of a single synodic month, with a secondary dial showing the lunar year of 12 synodic months.

Another reconstruction was made in 2002 by Michael Wright, mechanical engineering curator for the Science Museum in London, working with the above mentioned Allan Bromley. On November 30, 2006, the journal Nature published an article on Wright’s and his team’s analysis of the Antikythera device. It confirmed that the instrument had been used to predict solar and lunar eclipses. The article credited Derek Solla Price, but equally stated that “although Solla Price’s work did much to push forward the state of knowledge about the device’s functions, his interpretation of the mechanics is now largely dismissed.”

The new analysis confirmed that the major structure had a single, centrally placed dial on the front plate that showed the Greek zodiac and an Egyptian calendar on concentric scales. On the back, two further dials displayed information about the timing of lunar cycles and eclipse patterns. Previously, the idea that the mechanism could predict eclipses had only been a hypothesis. The study also revealed some of the complexity of the engineering that had gone into this device. The Moon sometimes moves slightly faster in the sky than at others because of the satellite’s elliptic orbit. To overcome this, the designer of the calculator used a “pin-and-slot” mechanism to connect two gear-wheels that introduced the necessary variations.

The team was also able to decipher more of the text on the mechanism, doubling the amount of text that can now be read. Some of the inscriptions mention the word “Venus” and “stationary”, suggesting that the tool could look at retrogressions of planets.

Wright also believes the device was not a one-off. “The designer and maker of the device knew what they wanted to achieve and they did it expertly they made no mistakes. To do this, it can’t have been very far from their every day stock work.” So it was probably “mass produced” at the time and must have been the product of previous, less fancy clocks. That those earlier models have been lost in the mists of time is understandable, but the big question by which everyone is baffled, is why such clocks did not continue to be build in the centuries that followed… indeed, why it took more than a millennium before a clock of the same technological expertise appeared again. Derek Price © Jeffrey Price Despite acceptance that this is a 1st century BC planetarium, some questions remain. Price pointed out that he himself did not know whether it was operated manually, by turning, or automatically. He said: “I feel it is more likely that it was permanently mounted, perhaps set in a statue, and displayed as an exhibition piece. In that case it might well have been turned by the power from a water clock or some other device. Perhaps it is just such a wondrous device that was mounted inside the famous Tower of Winds in Athens. It is certainly very similar to the great astronomical cathedral clocks that were built all over Europe during the Renaissance.” – 1500 years later. Wright’s team argue that it was manually operated, but this would somewhat work against a mass produced item, for it would require the most work from those people buying it care for the device would be labour intensive. So perhaps Price’s hypothesis that it was to be used within a religious setting is more appealing – though every hypothesis is currently guesswork.

The discovery of the Antikythera Device led to one gigantic realisation: that our everyday clock started as an astronomical showpiece that happened also to indicate the time – and not vice versa, as most believed half a century ago. Gradually, the timekeeping functions of the clocks became more important and the device that showed the cycles of heaven became subsidiary – only to be forgotten, and then reinvented all over again – all wheels inclusive.

Today, the device is worshipped by many as it is seen as the first calculator – computer. Price labelled the Antikythera Device “in a way, the venerable progenitor of all our present plethora of scientific hardware.” It should not come as a surprise then that whereas the original mechanism is displayed in the Bronze collection of the National Archaeological Museum in Athens, accompanied by a replica, another replica is on display at the American Computer Museum in Bozeman, Montana. In substance, it is bronze intellectually, it is a computer.

2. Observational background

Kepler originally investigated the orbit of Mars because that was the task allocated to him by Tycho Brahe (1546-1601), when Kepler joined him in Prague around 1600. In their day – and indeed until comparatively recently – the aim of astronomers was to achieve accurate observations of angles, simply because no other feature could be measured directly. Tycho had amassed a vast store of observations extending over 30 years these are probably the most accurate that would ever be made with the naked eye, since Galileo (1564-1642) had introduced the telescope into astronomy soon afterwards (in 1610). Tycho developed, refined and cross-checked his instruments and sometimes attained an accuracy of 2′ (which is approximately the breadth of a hair held at arm’s length).

It so happens that Mars is the only planet whose noncircularity can be detected without a telescope, and its observability was favoured by four factors:

  • it is an outer planet (and therefore it is seldom viewed close to the Sun)
  • the noncircularity of its path is the greatest of the outer planets
  • it is the nearest to the Earth of the outer planets (so changes in position appear larger)
  • it is the nearest to the Sun of the outer planets (and therefore it makes more frequent circuits, producing more observations).

Important Geometry Definitions

Line Segment

A line segment is a straight line segment which is part of the straight line between two points. To identify a line segment, one can write AB. The points on each side of the line segment are referred to as the endpoints.

A ray is the part of the line which consists of the given point and the set of all points on one side of the endpoint.

In the image, A is the endpoint and this ray means that all points starting from A are included in the ray.

Are there any good books of historical fiction set in Ancient Greece that you would recommend?

Thank you sir for your wonderfully informative website.

Thank you so much for your positive feedback! I always appreciate it when people write comments because that lets me know people are actually reading these articles I invest so much time and effort into writing.

There are many great works of historical fiction set in ancient Greece. I, for one, am currently writing a novel set in ancient Greece and I have been working on it quite a bit these past few weeks. It is, of course, not published yet and I have not settled on an official title. There are, however, many novels set in ancient Greece that have already been published. I recommend this list of works of fiction set in ancient Greece from Wikipedia, which lists many works of fiction set in ancient Greece from many different authors and genres. Of all the modern works of fiction set in ancient Greece, I would reckon that the novels of the British novelist Mary Renault (lived 1905 – 1983) are probably the most famous. Madeline Miller’s recently-published novels based on Greek mythology, The Song of Achilles (2011) and Circe (2018), seem to be quite popular, though.


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