A Short History of Early Astronomy

last update: 5 October 2020

Astronomy is simply the study of everything in space, and that's been defined as everywhere more than 100 km above see level (the Kármán Line). Cosmology is a branch of astronomy concerned with the study of the origin and evolution of the Universe. Astrophysics is another branch of astronomy and is all about trying to ascertain the nature of astronomical objects, rather than their positions and motions in space.

The world of
astronomy is full of misconceptions and myths. Does our Moon have a dark side? No, it does present only one face to Earth, but it's 'far side' is still fully illuminated by the Sun (lunar daytime) and in the shade (lunar night) just as often as on Earth. However a lunar day lasts about 29.5 Earth days (the length of a so-called lunar month), and is divided into a 2-week 'daytime' and a 2-week 'nighttime'. On top of that, because of something called libration we can, over time, actually see about 59% of the Moon's surface.
Many people think that there is no
gravity on the International Space Station (ISS) or on the Moon. There is a big difference between weightlessness and no gravity. Firstly, gravity is one of the four fundamental interactions in physics, and as such it is a force of attraction that exists between any two bodies, no matter their masses, e.g. in space a gravitational lens is where the gravity of astronomical objects will even bend the light traveling from a distant star towards Earth. We saw astronauts taking giant steps on the Moon's surface simply because gravity is only about 16.6% of that on Earth, but gravity always exists anywhere and everywhere in space (although in 'deep space' it can be almost zero). So gravity also exists on the ISS even if objects and the astronauts appear to float freely. What we are actually seeing is weightlessness due to the trajectory of the ISS. The ISS is not floating in space, it is in fact orbiting Earth at about 28,000 kph (so it orbits Earth about 15 times a day). And onboard gravity is only about 10-11% weaker than on the Earth's surface. What we are actually seeing is a pure form of 'free fall' which in the orbiting ISS is identical to weightlessness (the technical term is a micro-g environment). Gravity is still acting but the spaceship and astronauts are constantly falling toward Earth along their orbit so they do not feel their weight because there is no mechanical force pushing back on them. Technically weightlessness occurs when gravitational acceleration is perfectly balanced by the centripetal acceleration. In this sense weightlessness occurs thanks to gravity, and not because it's absent.
We always imagine that
outer space is empty, but it's not. It is what is called a hard vacuum and its very cold (about -270°C), but it is in fact full of particles, molecules, radiation, and even dust. For example, in 2009 it was discovered that clouds of gas and dust in the centre of the Milky Way were full of ethyl formate, which oddly enough is what gives rum its smell and makes up part of the flavour of raspberries. Also it's been calculated that about 40,000 tonnes of that cosmic dust falls on Earth every year. But in fact Earth is actually getting lighter each year because it loses about 95,000 tonnes of hydrogen.


Man was interested in
astronomical phenomena well before he developed a need to measure time intervals or create star charts. However navigation and time-reckoning were certainly the oldest astronomical practices, and it was these topics that would later make astronomy a science (organised knowledge). The periods of the Sun and Moon were natural units of time-reckoning (although the Mexicans also used the Venus-period and the Jupiter-period was used by the Indians). The period of the Sun set the seasons, but lunar phases were more striking and shorter, and thus more practical, i.e. calendars were generally dominated by the Moon and many cultures divided the zodiac into 27 or 28 moon-stations. These were small groups of stars, distant from one another, each representing one day in the lunar month. So the idea was that the Moon, in its course around the celestial sphere, every successive night occupies one group of stars after another. This was an effective system to keep track of the passage of the seasons.

It takes 29 days, 12 hours and 44 minutes for the Moon to orbit the Earth, but it only takes 28 days for the Moon to return to the same fixed-star position. Many ancient cultures gave names to these 28 days (moon-stations), and they all noted the specific location of the Moon as it made its appearance at night during different times of the year. They noticed that the Moon appeared in certain parts of the sky that were also associated with the appearance of the rising Sun. These parts of the sky included twelve areas in which the Sun appeared in certain seasons. Over time these parts of the sky became known as the constellation Pisces for the Spring Equinox, Virgo for the Autumn Equinox, Gemini for the Summer Solstice, and Sagittarius marking the Winter Solstice. Early calendars were lunar, but later ones emphasise the movement of the Sun.

Cosmic Laws

There are two sections in the Picatrix (on magic and astrology) which address moon-stations. The first concerns picking the right moment to carry out certain activities, so it's destined for statesmen, merchants, etc. The second is about what talismans you might need to do something, e.g. congregate fish or remove anger.

Certainly the
lunar period is the oldest calendar unit, and the month was rapidly recognised as a powerful period of nature (e.g. defining when to sow and when to harvest, etc.). So with the seasons and months, it was natural that lunar and solar reckoning were closely coordinated (i.e. most calendars use months and years).
The problem was that the
new and full Moon move up and down the calendar and cannot indicate a determinate seasonal date. The nightly stars, already known through their movement and orientation, offered a better solution. The position of the stars at the same hour of the night regularly change with the seasons. The 'heliacal rising and setting' repeats itself every year at the same date. The so-called 'acronychal' rising or setting mark when the stars rise or set. In some cultures the Sun was also used to fix dates. The length of the shadows of vertical sticks could be used to mark the passing of months, and the setting points of the Sun on the horizon were used to fix the longest and shorter days.

In fact, before the reform of Julius Caesar, the calendars of the Greeks and Romans, with their erratic intercalations, were frequently out of step with the seasons and often gave no reliable indication of the time of year. So we find that classical authors referred to the time of year by means of the rising and setting of various stars and constellations. Hesiod encouraged everyone to begin the harvest at the rising of the Pleiades, and to plough at their setting. Alcaeus encoungage everyone to drink at the rising of Sirius, and Horace assured everyone that a man content with his lot would not be disturbed at the setting of Arcturus or at the rising of Haedus.

Agriculture and the centralised regulation of water (e.g. seasonal flooding of alluvial plains and irrigation) depended upon a knowledge of the calendar and seasons (and that knowledge was often a source of prestige and social power for the ruling classes). Time-reckoning required the development of astronomy. The problem was how to adapt the lunar calendar to the solar year. One period of the Moon is, on average, 29.53059 days, and a solar year is 365.24220 days, i.e. 11 days more than 12 lunar months. So after 3 years the lunar calendar is 33 days behind the solar cycle. So an extra month was needed every 3 years, but looking for a more accurate option, people turned to inserting a 13th month seven times in every 19-year period (i.e. the Metonic cycle). We have moved out of the realm of agriculture and into the needs of religious ceremonies and festivals.

Eudoxus of Cnidus (Greek, ca. 390-337 BC) was the first to give a theoretical explanation of planetary motion. He supposed that every planet was fixed to a sphere revolving around the Earth as centre. To explain the irregularity of planetary motion, he supposed more spheres instead of one, all homocentric, i.e. all turning regularly in different ways about the same common centre. The key was explaining things in terms of perfectly regular circular motions. The idea worked for Jupiter and Saturn, but not so well for Venus and Mars. As the Sun had to be carefully observed for calendar purposes, its irregularities were detected. Observing the equinoxes and the solstices disclosed the inequality of the four seasons.
Aristotle (Greek, 384-322 BC) presented the structure of the Universe as having perfectly radial spherical symmetry. This was a set of spherical shells around a centre, and where simple movements along straight lines were along radii from or to the centre. So all movements were away from the centre, towards the centre, or around the centre. The motion of celestial bodies was not straight and finite, but circular, invariable and eternal. In addition the Earth was at rest in the centre of the Universe. The Universe was seen as finite and spherical. Because the Universe was composite and had a centre it could not be infinite. The sphere revolved always in the same place, and outside the sphere there was "neither void nor place". The Moon was spherical and the stars were assumed to be also spherical and did not move. Aristotle thought the planetary spheres were real crystalline shells, and beyond the shells there was a moving force which ensure daily rotation (his system required 55 solid crystalline spheres including 22 counter-rotating spheres).
There were other competing models, for example
Heraclides (Greek, ca. 390-310 BC) thought that the celestial bodies were stationary and the Earth rotated, and his name has also been associated with an explanation for the orbits of Venus and Mars. His answer was to have them orbit the Sun, and not the Earth. Aristarchus (Greek, ca. 310-230 BC) went one step further, placing the Sun in the centre of the Universe with all the planets orbiting it. He also determined that the ratio of the diameters of Earth and Moon was 57:20. Aristarchus went on to note that volume of the Sun was between 254 and 368 times that of the Earth. It maybe this size difference that suggested to Aristarchus that it was more logical for the Earth to orbit the Sun. In about 323-283 BC Euclid made some assumptions and was for the first time able to concluded that the Sun was about 19 times more remote than the Moon (a statement that was accepted for nearly 2,000 years).
Eratosthenes (Greek, ca. 276-194 BC) is said to have determined the circumference of the Earth to be 39,250 km (as compared to today's 40.075 km). Parallax is an important topic in astronomy and Hipparchus of Nicaea (Greek, ca. 190-120 BC), one of the founders of trigonometry, used the technique to determine the distance of the Moon measured in Earth radii (between 62 and 74 radii). It is said that Hipparchus was the first to construct and catalogue nearly 850 fixed stars, and mark them by longitude and latitude. It was also Hipparchus that gave the 'epicycle' theory its classical form, even if it was Ptolemy who brought the theory to completion.

This webpage has a focus on
telescopes. As man used telescopes to look farther out into the Universe, he started to ask some truly profound questions:-
space homogeneous? Is space the same everywhere, and are the laws of physics the same everywhere?
space isotropic? Is space the same in all directions, or is there some direction that is preferred in some way? For example, is the speed of light the same in all directions?
Are the
laws of physics constant in time?
And finally are the
laws of physics independent of uniform relative motion?

With our present ability to look deep into
space it appears that hydrogen atoms are the same everywhere. And because the light rays we see were emitted a very long time ago, it seems that the laws of physics are indeed constant with time.

The last question on uniform
relative motion lies at the core of relativity. Will an experiment performed on a spaceship travelling at a uniform speed through space yield the same results on a different spaceship travelling at a different uniform speed? Yes, the 'common sense' answer was proposed by Galileo as the "principle of relativity". Newton went one step further when he wrote "The motions of bodies included in a given space are the same amount themselves, whether that space is at rest or moving uniformly forward in a straight line". Meaning that the experiments in our moving spaceships would be the same as if the spaceship was stationary. What that tells us is that inside our spaceship we can't do an experiment to find out how fast we are going, we will have to look outside to find out how fast we are going relative to our surroundings.

Motion and Relative Motion

In order to develop properly the ideas of
Ptolemy, and those who followed him such as Copernicus and Kepler, let's just spend a few moment on the concept of motion, and in particular relative motion. In physics, motion is change with time of a position or orientation of a body. Motion along a line or curve is usually called translation. Motion that changes the orientation of a body is called rotation. In both cases, all points in the body have the same velocity (directed speed) and the same acceleration (time rate of change of velocity). The most general kind of motion combines both translation and rotation. We also have to know that all types of motion are relative to some frame of reference. Even a body at rest (i.e. not in motion) merely means that it is related to a frame of reference that is moving with the body. Something on the Earth's surface might appear at rest to an observer also on the Earth's surface, but the Earth itself is rotating and orbiting the Sun. In many ways we have here the reason why, until after the Middle Ages, people imagined that the Earth was an immovable object (and not a planet). In fact for most of history man thought that their point of view had the power of a law that ruled the Universe. It took man a long time to understand that the way forward was to acquire knowledge so that they could predict the physical behaviour of the Universe. We will see in the transition from the Ptolemaic planetary model to the Copernican model that the Earth changes from being the motionless centre of the Universe to being just another planet in the solar system. After Copernicus we would always aim to eliminate any human definition of the uniqueness in the property of an object, and replace it, or connect it, with a new definition of a universal property. This approach would reduce the number and complexity of the definitions that man had created, and point to properties that still needed to be discovered and/or quantified. A good example is the move to Copernicus's heliocentric model, where he was constantly questioned as to the property of the Sun that made the planets rotate around it. The Sun must have some property that holds it in place, and the planets could be expected to have a lesser value of the same property. But what was it? Newton would substitute the formula for centripetal force into Kepler's third law and find that it was this force that held the planets in their orbits. He would find that the force must decrease with the square of the planets' distance from the Sun. He saw that the same rule applied to the moons of Jupiter, and the Moon orbiting Earth. He would relate this rule to that which conditions the acceleration of bodies on the Earth's surface, and would suggest a single quantitative law, which he would name in Latin 'gravitas'.
So saying that a body is at rest, which means that it is not in
motion, merely means that it is being described with respect to a frame of reference that is moving together with the body. And for most purposes it is often motion relative to the Earth that is important. But is that totally true, if you send a spaceship to the Moon then you need to take account of the motion of the Earth, but if you a drinking a cup of coffee on a train then the motion of the coffee relative to you mouth is important rather than relative to the Earth. The important point is that when making a measurement (position, velocity or acceleration) you need to decide on your frame of reference and the measurements taken are then relative to something else.
Circular motion can be along an arc or circumference of a circle or ellipse, or it can be rotation which has no cumulative displacement, i.e. things just spin or go round and round on a fixed axis.
Acceleration is defined as the time rate of change of velocity. We should remember that constant speed is a scalar quantity and only means a constant magnitude of velocity. We will usually deal with vectors, e.g. displacement, velocity and acceleration, although time is also a scalar.
If the direction of the
velocity changes then that is just as real an acceleration as if the speed changes. And we know a body rotating around a fixed axis at constant speed (uniform circular motion) will create a centripetal (or centre-seeking) acceleration always directed towards the axis of rotation. This centripetal acceleration produces a centripetal force which is also constant in magnitude and directed also towards the axis of rotation. A changing speed of rotation (i.e. nonuniform circular motion) will produce a tangential acceleration in addition to the normal acceleration (centripetal acceleration). We see that centripetal acceleration is present in both uniform and nonuniform circular motion. In nonuniform circular motion there is an additional force acting on the object due to the non-zero tangential acceleration, but the sum of all the forces acting on the object must still equal the centripetal force, and it is this force that keeps the object on its circular path.
We will see on this webpage that many disagreements, etc. boil down to a problem of
frame of reference. Does the Earth stand still and the Sun move across the sky? Or, does the Sun stand still and Earth rotate beneath it, creating the apparent motion of the Sun? In the first case we have a stationary frame of reference, in the second case a moving frame of reference (i.e. the Earth's rotation). One the major arguments against the idea that the Earth rotates was as follows. If the Earth rotates so does a tree on the Earth's surface, and therefore an apple falling from that moving tree can't fall straight down, and must land someplace away from the tree. But the apple does fall straight down, thus the tree can't be moving, so the Earth must be stationary. What they did not know or accept was that an apple always falls from a tree in the same way on a stationary Earth or on an Earth rotating at a constant velocity. The velocity of an object will be described differently in reference frames moving at different, but constant, velocities relative to one another. But, the relative velocity between any two objects remains constant. Events occurring in the same manner independently of the velocity of the reference frame led Galileo and others to suspect that the fundamental principles of physics did not depend upon the constant motion of the reference frame from which they were observed. This principle of relativity is usually written "The laws of physics are the same in all reference frames moving at constant velocities relative to one another", or "Equations describing the laws of physics have the same form in all admissible frames of reference".
How can we make sense of the above, and use an 'astronomical' example? Let's imagine a
sci-fi film in which someone is pushed from an orbital station that continues on in its circular orbit. If our astronaut was pushed directly toward the centre of the Earth, will our astronaut start to rapidly fall to Earth? But what about the fact that our orbital station is travelling over the Earth's surface at more than 7 km/s? Once our astronaut becomes detached from the orbital station do they not simply start to orbit in the same way as the station? But will that little extra push toward Earth change things? Now is the time to commit to a prediction, what do you think?
Well, the first thing to do is to stop thinking of the
astronaut with respect to the Earth as the frame of reference. Surely we need to use the orbital station as our frame of reference when thinking about our poor abandoned astronaut. Our astronaut was travelling along with the orbital station, and suddenly was ejected downwards with an additional velocity. As a first guess our astronaut will start to move downwards in the direction of that initial push. Both the astronaut and the orbital station were in a circular orbit around the centre of the Earth. The ejected astronaut, with his additional velocity directed towards the centre of the Earth will now start in a slightly elliptical orbit with a very, very small eccentricity, one foci located at the centre of the Earth, and the other very, very close to the centre. You can easily imagine that we now have the astronaut in an elliptical orbit which is almost identical to the circular orbit of the orbital station. An elliptical orbit means that it will have apsides, points where the astronaut is nearest to Earth and points furthest from Earth, even if only slightly. So we can see that very slight elliptical orbit of our astronaut will bring him very slightly inside the circular orbit of the orbital station, and then very slightly outside that circular orbit. The elliptical orbit is so shallow that as a very good approximation it looks like a circular orbit with a centre very, very slightly different from the centre of the Earth. The diameter of this circular orbit will be almost equal to the major axis on the elliptical orbit, and we know from Kepler's Third Law that the period of revolution of our astronaut and of the orbital station are almost identical. So we can see this as two almost identical circular orbits with very slightly different centres, and with two trajectories that will intersect at two points equidistant from the apsides. This means that our astronaut will often be at the same altitude as the orbital station, but this might not mean that the two objects on the two trajectories actually cross at the same time. The orbital station on its circular orbit will come to these intersection points every half-orbit, but does our poor astronaut do the same? No, because motion of a body along a slightly elliptical orbit is slightly non-uniform. In accordance with Kepler's Second Law a body passing the perigee (the apsis nearest the Earth) will speed up a little and will arrive at the intersection slightly in front of the orbital station, i.e. the distance between the astronaut and the orbital station will actually be at its maximum. Moving on, a body will pass through the apogee (the apsis farthest from the Earth) and its speed with decrease slightly. This means that the astronaut and the orbital station will come to the common initial position almost simultaneously, the astronaut approaching the orbital station from above. Thus the motion of the astronaut relative to the orbital station is almost a closed trajectory.
But what about those still on the
orbital station, what do they see? Now we are looking at the astronaut from the reference frame of those in the orbital station. We know that we have two circular orbits (one for the orbital station and one very close approximation for the astronaut) with their centres slightly displaced. This displacement is a function of the vectors of the velocities, and even with the best push in the world our astronaut is not going to acquire a velocity of more than 15 m/s, i.e. 0.2% of the orbital velocity (7.5 km/s). This means for an orbit radius of 7,000 km, the astronaut can be up to 56 km from the orbital station. In practice visibility is limited inside an orbital station, so they would see our poor astronaut gradual drop below their field of view. But, during one complete orbit our astronaut will make an almost closed horizontal elliptical loop relative to the orbital station, passing below, behind, and then approached the station from above.
One important point to note is that our
astronaut was 'pushed' exactly in the direction of the centre of the Earth, but what happens if the push included a component parallel to the orbit of the station. The trajectory will no longer be closed, even for very small values of the initial relative velocity. The astronaut will not return to the station, and that little difference will mean that the astronaut will steadily raced from the orbital station.
A related problem to the one above is the
space rendezvous, where a 'deputy' spacecraft needs to rendezvous with a 'chief' spacecraft. The question is how their relative motions evolve with time, and how velocity changes in the local-vertical, local-horizontal component directions, change in relative motion due to impulsive manoeuvres. Even the last point about a 'push' with a parallel component to the orbit corresponds to the situation of a push-back when a landing module leaves an orbital station. The problem of the space rendezvous still posed a problem in the mid-1960's, but has now been mastered. However, as indicated above, the trajectories are counter-intuitive and often misrepresented in popular movies and books.

Relative motion appears in the title "On the Relative Motion of the Earth and the Luminiferous Ether", dated 1887. This publication reported on the so-called Michelson-Morlay experiment, one of the most famous physics experiments ever performed. The experiment was originally conceived as a test for the existence of the aether, a medium that was to carry electromagnetic waves and whose rest frame would realise Newton's absolute space. However the experiment's failure to detect the expected anisotropy of the speed of light in a frame moving through the aether presented physicists with a puzzle eventually solved by Einstein's Special Theory of Relativity.
The original
Michelson-Morlay experiment exploited an interferometer in which a light beam was split into two beams propagating back and forth along perpendicular paths of equal length. It was expected that when the interferometer was rotated interference of the recombined beams would show the effects of an anisotropy of the speed of light caused by the Earth's motion through the aether. The experiment's failure to detect this effect had a massive effect on the world of physics at that time.


Let's draw a line under 'antiquity' with
Ptolemy (Roman Empire, ca. 100-170 AD) who wrote a treatise on astronomy, and who presented his geocentric astronomical models in convenient tables so that people could compute the past, present and future positions of the planets. These tables included also a way to determine both the rising and setting of the stars, and the eclipses of the Sun and Moon. He also included a star catalogue and a list of visible constellations. Ptolemy actually describes an instrument for measuring the parallax of the Moon, although he used the chord of the arc and not the circular arc itself. The results were incorrect but it demonstrated that, in theory, they could deriver the distance of the Sun from observations of the lunar eclipse. And this was fundamental because it showed that through observation one could deduce facts about the natural world.

Ptolemaic Geocentric Modeland

In the Ptolemaic system the Earth is immovable and positioned in the centre of the Universe. Ptolemy argued that all bodies must fall to the centre of the Universe, and since all objects always fall to the centre of the Earth, it must be at the centre of the Universe. He also accepted the following order Earth (centre), Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn.
The ancients had seen some of the
planets oscillate from side to side of the Sun, before being again carried along on their yearly course. It was plausible that the movement of the planets could be described by a large circle about the Earth as centre, along which the centre of a smaller circle moved. Ptolemy accepted that to account for the uniform circular motion of the planets he would need to adopt a system of 'deferents' and 'epicycles'. Deferents were large circles (orbits) and epicycles were small circles whose centres moved around the circumferences of the deferents (so a planet was seen as attached to an epicycle whose centre travelled around a deferent). The idea was that both the deferent and the epicycle each executed their own uniform circular motion, the deferent anti-clockwise and the epicycle clockwise. This approach could make any celestial body appear to move on a circular but eccentric orbit, and it described the appearance of changing velocities of the planets, when they appear to travel slower at the apogee and faster at the perigee. The alternative was to simply use an eccentric deferent or orbit, i.e. off-setting the Earth from the centre of the deferent.
However this 'simple' arrangement of
deferents and epicycles did not account for all the planetary movements, because sometimes planets would 'stop', go back on themselves, then 'stop' again and continue forward (so-called retrograde motion). To describe this, a second type of epicyclic motion was needed. To describe this you need to have the planet travel around the epicycle in the same sense as the epicycle is traveling around the deferent, i.e. both travelling anti-clockwise. An observer on Earth would see the planet slow down, stop, go into retrograde motion, stop again, and then continue in its original sense of motion. By adjusting the speed of the planet on the epicycle, the speed of the epicycle on the deferent, the radius of the epicycle and the radius of the deferent, you can simulate any retrograde motion of any planet in any position.
Now the challenge was to combine the two types of
epicyclic motion (variations in velocity plus retrograde motion). There were two ways to do that. The first was to add another epicycle on top of the first epicycle on the deferent. The second was simply to off-set the Earth from the centre of the deferent. Ptolemy went for this second option because it looked mathematical simpler. Increasingly we are moving away from something that might have a physical explanation, and towards a computational model.
However, there was yet another problem. For this model to lead to satisfactory computational results, we can't accept that the centre of the
epicycle travels around the centre of the deferent at a uniform rate. Unwilling to yield to the need for uniform circular motion (or compound uniform circular motions) Ptolemy simply looked for another point from which the motion of the centre of the epicycle could be uniform. This point was the famous 'punctum aequans' (equant point). We have already seen that the Earth was now located a short distance from the centre of a planets' deferent. The equant was an imaginary point placed on the diameter of the deferent but at a position opposite to that of the Earth (so the Earth and the equant were equidistant to the centre of the deferent). So the planet is conceived as travelling uniformly around its epicycle, while the centre of the epicycle (on its deferent) travels uniformly around the equant. Just to note that the planet is moving anti-clockwise on the epicycle with a constant angular velocity, which is moving anti-clockwise on the deferent with a constant but different angular velocity. Another way to describe this is to consider a radius vector situated at the centre of the deferent, linked to the centre of the epicycle, and sweeping out equal angles in equal times, i.e. with a constant angular velocity. Introducing the equant, we redraw the radius vector from the equant to the centre of the epicycle, and continue to sweep out equal angles in equal times, i.e. with a different but constant angular velocity. It is very difficult to visualise planetary motion with 'deferents' and 'epicycles', and below we have a first example published in 1609 and below that second simplified example.

Systeme Ptolemee 1

Systeme Ptolemee 2

You can see in the above diagram that Earth E is not quite in the centre of the deferent. The epicycle was introduced to explain retrograde motion. Also in theory the deferent and epicycle should rotate in the same anti-clockwise direction both at constant speeds. The epicycle did rotate at a constant speed, but Ptolemy made the centre of the epicycle rotate at a constant speed relative to the equant, and not relative to its centre. This actually meant that a planet would travel around at different speeds, mirroring the fact that we know that planets speed up when closer to the Sun, and then slow down when they are farther away. The equant was the great invention of Ptolemy, but purist did not accept it because it did not respect the principle of uniform circular motion.
And of course the
'deferents' and 'epicycles' were different for each planet as were the positions of the centre of the deferents and the equants. In fact the only common point was the position of the Earth. Also, as far as I understand it, the planes of the deferents of the planets were slightly inclined to the plane of the ecliptic, the apparent path of the Sun with respect to Earth. And the epicycles were inclined so they were always parallel to the deferents.
Finally this general system worked for all the
planets, except Mercury. In this case the only solution was to also place the equant of Mercury on a completely separate auxiliary circle (epicycle).
Ptolemy certainly realised that the planets were much closer to Earth than the fixed stars he nevertheless appeared to believe in the existence of crystalline spheres to which the heavenly bodies were attached. Outside this was the 'premum mobile', or prime mover that provided the motive power for all the spheres.

What we see below is the final model of
Ptolemy for the orbit of Mercury (M1). It is the most complex orbit and I offer it here as an option for those passionately interested in the topic.

 Ptolemy Orbit of Mercury

The observer is on Earth at point E, the equant is at Q, bisecting the eccentricity of U. So in the Ptolemy model for Mercury EU = 2EQ, whereas for the other planets Ptolemy used EU = UQ, and U being the centre of the deferent which bisects the line EQ. For Mercury Ptolemy starts as for the other planets, making the circular epicycle move on a circular deferent, but in order to take in to consideration the much larger eccentricity of Mercury's orbit, he then makes the centre of the circular deferent move on a little epicycle (thus creating the dotted-line oval trajectory).
Mercury, point U is the centre of a little epicycle around which the centre of the deferent O moves. The motion of O around U is determined by speed 𝛄, as is the motion of the centre of the deferent Dv and Dv1 around Q. We can see that the centre of the deferent moves from Dv (corresponding to a fixed deferent with centre U) to Dv1 (corresponding to a moving deferent with centre O). It is this final change to the model that produces a dotted-line trajectory of the centre of deferent which is oval. It also moves the planet Mercury from M to M1.
The dashed-line circle is the
orbit around the fixed equant point Q, which would have produced the dashed-line epicycle with the planet Mercury at position M. However because a little epicycle is placed at U the deferent is actually centred on O and draws out the solid-line circle. And this produces a displacement of solid-line epicycle placing the planet Mercury at position M1. With the little epicycle introduced at point U we now see that the deferent maps out a dotted-line oval trajectory. Remember the motion of O around U is equal to the motion of the centre of the epicycle Dv around equant Q, but in the opposite direction.
In reality there are a number of approximations included in the
Ptolemy model, and these all come together in the orbit of Mercury making it the least accurate.

This rather convoluted and non-intuitive view came about because Ptolemy was trying to summarise some 500 years of ideas and observations about the Sun, the stars and the planets. Also the ancients had "reason and common sense" on their side when they decided that the Earth was at the centre of the Universe and the Sun, Moon and the planets all moved around it with uniform circular motions. Remembering that until the early 17th century astronomy was a branch of mathematics, and physics a branch of philosophy, so it was up to astronomy to use mathematics to account for the movement of astronomical objects in terms of that uniform circular motion. Practically no one doubted the geocentric model of the Universe, so for the best part of 1,400 years everyone was devoted to computing tables and designing instruments that translated Ptolemy's theorems and calculations into almanacs, horoscopes, and planetaria.
Ptolemy's work on astronomy was far from being his only, or most important. He wrote on mathematics, mechanics, optics, but perhaps he was best known for his work on geography with maps, etc. of Europe, Africa and Asia. It is said that Christopher Columbus (Italian, 1451-1506) based his belief that Asia could be reached by travelling westwards because of Ptolemy's work.

A great example of the importance and practical application of the
Ptolemaic planetary model is the building of astronomical clocks, i.e. clocks displaying astronomical information.

Dondi Astrarium

Above we have an astrarium, a form of planetarium, built by Giovanni Dondi (Italian, ca. 1330-1388) only 60 years after the very first mechanical clocks. It included a continuous display of the movements of the Sun, Moon and the planets, but more importantly it included non-circular gears so as to improve the accuracy in reproducing the non-uniform kinematics of the planets as part of the Ptolemaic model. And in addition it reproduced the irregular patterns of the planets coordinates, and included retrograde motion also for Mercury. Not surprisingly it took 16 years to build.

It would be
Kepler in 1601 who would put physics and astronomy together. He would suggest that the speed of a planet was inversely proportional to its distance from the Sun throughout its orbit. This meant that he had decided that the Sun was the cause, and the effect was planetary motion. So we should always remember that no matter how complicated, the approach of Ptolemy corresponded to a world of pre-telescope observational data, and was amazingly effective. Even today we still don't have a precise theory to predict the exact position of the Moon relative to the Earth (on lunar missions corrections were injected based upon radar readings from the lunar surface).

Nicolaus Copernicus

We will take up our story again with
Alvise Cadamosto (Venetian, ca. 1432-1488) who, as an explorer (and slave trader), demonstrated that astronomy had survived the Dark Ages as a practical science. Not only did he explore new regions of West Africa, but he also made detailed and reliable observations of his journeys which were published and widely circulated at the time.
However it was
astrology that had become universally accepted, and even used to determined the best time for bloodletting (Saints Days) and haircutting (waxing of the Moon). I'm personally not a fan because astrologers would always write about "the horrible house of the ill-fated Scorpion", my birth-sign. What was certain was that the stars occupied the attention of every man in the 15th and 16th centuries, more so than any other 'science'. So it's not surprising that many people tried to offer 'world views' at the time, but we will focus on the most famous of them, Nicolaus Copernicus (Polish, 1473-1543). In 1512 he would state the following:-

Wikipedia put the above in a more modern-day language, as:-
Heavenly motions are uniform, eternal, and circular or compounded of several circles (epicycles).
The centre of the Universe is near the Sun.
Around the Sun, in order, are Mercury, Venus, the Earth and Moon, Mars, Jupiter, Saturn, and then the fixed stars.
The Earth has three motions: daily rotation, annual revolution (orbit), and annual tilting of its axis.
Retrograde motion of the planets is explained by the Earth's motion, which in short was also influenced by planets and other celestial bodies around Earth.
The distance from the Earth to the Sun is small compared to the distance to the stars.

The only expression that might not be fully understood is retrograde motion. Firstly, as seen from Earth, most of the time the planets appear from night-to-night to move slowly eastwards against the background of the stars, i.e. moving in the same direction as the Earth's daily rotation. But then for a few months they head west before turning back and resuming their easterly course. This westward motion is called retrograde motion. Initially people were baffled or even shocked by this, but it is just an illusion caused by the motion of the Earth and these planets around the Sun. What is happening is that the Earth is passing the slower moving planets, e.g. as Earth passes Mars, Jupiter, or Saturn, which all orbit slower than Earth, they appear to reverse for a couple of months. So this movement should really be called apparent retrograde motion, because for example Triton, the largest moon of Neptune actually has a retrograde motion in that it orbits in the opposite direction to Neptune's spin. Below we have what is a self-explanatory example.

Jupiter Retrograde

People have equated apparent retrograde motion to when you pass a car, it appears for a short period to be moving backwards with respect to your forward movement, for the rest of the time the other car is moving in the same direct but at a slower speed. The car metaphor is quite useful if we imagine Earth and Mars, on two different oval tracks around the Sun, with Earth on the inside track. In fact Earth makes two orbits in the time it take Mars to make one. So about every 26 months, Earth overtakes Mars and we see the illusion that Mars appears to move backwards, then as Earth moves further along its curved orbit the illusion disappears and we see Mars 'behind us' moving along in a more or less straight line. On top of all that the two tracks are not perfectly aligned, there is a small tilt with respect to each other, so the apparent retrograde motion will look different, sometime its a zig-zag and sometimes its even a loop.
In addition during this
retrograde motion there is a short period when the Sun, Earth and Mars are aligned. They are said to be 'in opposition', and at this time Mars is nearest to Earth. On top of that every 15 years Mars comes nearest to the Sun. In 2018 both effects were aligned and Mars was blazing red and brighter than any of the stars and even brighter than Jupiter. In 2003 Mars came closer to Earth than at any time in over 60,000 years (evidently this was accompanied by the usual Mars hoax).

Copernicus World View

Above we have an example of the planisphere of Copernicus showing the 6 planets revolving around the motionless Sun. The planets orbit around the Sun in circular paths at uniform speeds. Earth and Jupiter are shown with their moons revolving around them.

Copernicus explained that his 'world view' was spherical and that the Earth was a sphere. The sphere turned on its axis in a circle, and celestial bodies also had a circular and uniform motion (so everything could return to the same place in fixed periods). In a wonderful expression of logic Copernicus decided that irregularities were not in the 'best order' and therefore uniform motion might appear irregular because the Earth is not in the centre of the orbit or because of a difference in the Earth's poles. Ptolemy had argued that the Earth was stationary and the heavens rotated. Copernicus argued that this would mean that the heavens must be rotating at a tremendous and violent speed, whereas if the heavens were stationary then the Earth would rotate slowly, and do what spheres did, spin about their axis (and of course an immobile heaven was far more noble and divine). Next came the idea that the Earth was visibly not the centre of the orbits of the planets, and the Sun, Moon and planets, having spherical shapes, must have circular orbits. As far I understand it Copernicus even concluded that if the Earth didn't have a circular orbit it was because it was influenced by 'foreign motions'. And if it was the Sun that gave the Earth the rising and setting of the immobile stars, then the Sun must be in the middle of his 'world view' (solar system). And all the planets must also orbit the Sun ("the most beautiful temple"). Of course Copernicus saw the celestial sphere as a real object, and his ideas were not built upon observations (since observations only showed motion relative to Earth). For Copernicus, man had to accept that the Earth was whirling along contrary to direct experience. However, with his ideas he was able to make observations and compute new tables, producing a new manual of astronomy to replace that of Ptolemy.

Copernicus's starting position is that a planetary theory must resolve the apparent, irregular motion of a planet into separate components that are (firstly) uniform in velocity, and (secondly) circular in figure. Copernicus conceded that the Ptolemaic model represented planetary motion accurately for the purposes of computation, but he objected on the violation of the principle of uniform circular motion and in the role played by the equant. In fact numerical predictions of the Ptolemaic and Copernican theories find them to be both of the same predictive accuracy. And given the data available at the time it is certain that the accuracy of the Copernican theory was in no way superior, and in fact Copernicus never claimed that his theory would be more accurate than the Ptolemaic model. It is said that the Copernican model was simpler since it 'only' has about 30 circles, whereas it was incorrectly claimed that the complete Ptolemaic model included 80-odd circles. However the reality was that to computer the apparent position of a planet viewed from Earth didn't need 80-odd circles, but only about six circles for that specific planet. Whereas with the Copernican model you would need to calculate the positions of both the planet and Earth, since now they are both orbiting the Sun. So the Copernican theory was neither more accurate or simpler. However we are really looking at two different points of view. The Ptolemaic model was useful because it fitted the data, but Copernicus wanted his model to have a physical meaning, i.e. to represent a reality.

Copernicus's only real practical objection to the Ptolemaic model was that the motion of the centre of the epicycle was not uniform with respect to the centre from which it maintains a constant distance. i.e. the centre of the deferent. It is said that he imagined that the motion of a planet was fixed by the revolution of a material sphere or spheres in which the planet was fixed. And as such a sphere could only make simple uniform rotations about its diameter, and could not move with respect to a straight line passing through it (i.e. the equant can't be right). It was on this basis that Copernicus developed his seven postulates that we listed above. The reality is that the first postulate stands by itself, and the second, fourth, fifth, and seventh can be proven if postulates three and six are true.

So what were the advantages of the
Copernican model? Firstly the relative distances of the planets from the Sun could be found using geometry. It is not that easy, because Copernicus wanted his planets to make out nice circular orbits so he still needed epicycles (but not the equant). But if we observe a planet at it greatest angular distance from the Sun, then the sightline to the planet represents a tangent to its circular orbit (meaning that the angle between the radius of planet to the Sun and the sightline is a right angle). So we can now find the distance between a planet and the Sun in terms of the distance from Earth to the Sun (a distance called an Astronomical Unit AU). So the Copernican model yielded the relative radii of the planetary orbits, and when viewed alongside the orbital periods, it clearly showed that orbital periods increased with orbit radii. This appeared reasonable, and hinted clearly at something physical meaningful. Along with the simpler explanation of retrograde motion Copernicus felt that his model was more orderly (the beauty of creation), and he went to great lengths to prove that the Ptolemaic and Copernican models produced the same predictions about the position of the planets.

The reality is that after the Copernicus's book in 1543, only a handful of astronomers had accepted by 1600 that the Earth was no longer immovable. Most admired the work of Copernicus, but most preferred the common-sense geocentric model. Even after the arrival of the telescope in ca. 1609, many astronomers still continued to doubt the Copernican model. We will see that Tycho Brahe will propose a different kind of geocentric model, the Tychonic System, and Johannes Kepler would use Brahe's data to work our the elliptical nature of planetary motion. People wanted to know what power made the Earth move in an orbit? They already 'assumed' to know that the celestial bodies were made of a special 'ethereal' substance not found of Earth, and which had a natural tendency toward rapid circular motion (bit like dark matter or dark energy today), so what moved the Earth? Another problem was that despite the fact that now the Earth was orbiting the Sun there was still no annual parallax on the stars. The implication being that the diameter of the Earth's orbit was itself as nothing compared with stellar distances, i.e. the size of the Universe suddenly became immeasurable large. This would suggest that stars should also be individual points of light, yet with a telescope they all appeared to have fixed widths, which Brahe had actually measured and reported on. We now know that the distant stars are effectively point sources of light, and the apparent widths (Airy Disk) are an artefact of the passage of light waves through a circular aperture such as with a telescope or an iris. Brahe had calculated, based upon simple geometry, that if the stars were to lie at Copernican distances, then they would have widths comparable to that of the Earth's orbital diameter, and would utterly dwarf our Sun in size. 'Copernicans' simply answered that the greater the 'palace' of the stars, the greater was the nature of God.
It was on this basis that
Brahe suggested that the Sun, Moon and stars orbited an immobile Earth (as in the Ptolemaic model), whilst the planets orbited the Sun (as in the Copernican model). This became known as the Tychonic System, with a 'lazy' Earth and the stars laying beyond the planets, i.e. with no annual parallax but still being of reasonable sizes. Mathematically the Copernican model and the Tychonic System were identical. Galileo, with his telescope, saw that Jupiter had moons, and that the Universe could harbour more than one centre of motion. And even when he saw that Venus orbited the Sun, these finding were not understood as proof that the Earth orbited the Sun, because the results were still fully compatible with the Tychonic System.
Later still
Giovanni Battista Riccioli (Italian, 1598-1671) would decide that the Copernican model could not be correct for two reasons. Firstly, as with Brahe, he felt that a rotating Earth should deflect a projectile from a straight line. This deflection would not be observed until the 19th century, when Gaspard-Gustave de Coriolis (French, 1792-1843) finally described such effects. The second reason was that Riccioli, using an improved telescope, had seen the 'diameter' of stars 'shrink'. But since he could still not detect an annual parallax, so the net effect was that stars 'must be just as titanic' as Brahe first suggested. Through the 18th century and into the 19th century a growing majority accepted Copernicanism, but it was not until 1838 that Friedrich Bessel (German, 1784-1846) would finally record the annual stellar parallax. And it was about the same time that George Airy (English, 1801-1892) produced the first full explanation for why stars appear wider than they really are, and Ferdinand Reich (German, 1799-1882) would finally detect the deflection of falling bodies induced by the Earth's rotation. So always remember that proving someone wrong when they are in fact right is a major element of good science.

Alongside the 'simpler' world view of
Copernicus, which remained just as complicated in its detail, there was also a gradual evolution in astronomical tables. Ptolemy had produced his Almagest (ca. 150 AD), and there were the Toledo Tables (1080). The Alfonsine Tables (1483) were used by Copernicus, and Erasmus Reinhold (German, 1511-1553) went on to produce the Prutenic Tables in 1551, and in 1627 there came the Rudolphine Tables. With the advent of moveable-type printing, ephemerides (or almanacs) and calendars were produced in large numbers, and ephemerides contained pre-calculated positions of the planets for several years to come. Johannes Regiomontanus (German, 1436-1476) wrote on arithmetic and algebra, and would also provide daily planetary positions so that horoscopes could be produced with the minimum of effort. His work on plane and spherical trigonometry was only published in 1561, long after his death. In most cases the tables were sufficient for astronomical practice of that time, but mathematicians continued to make them more prefect, thus anticipating future applications. By the 16th century trigonometric formulae combined with tables of the sines and other goniometric functions became the most important auxiliary tool of astronomers. It was also in the 16th century that Arabic numerals found their completion in the introduction of decimal fractions, thanks to the work of Simon Stevin (Flemish, 1548-1620). The invention of logarithms, at the beginning of the 17th century, also brought an enormous saving of time. Initially published by John Napier (Scottish, 1550-1617), they were later adapted to the decimal system by Henry Briggs (English, 1561-1630), and they proved so valuable that they were said to have "lengthened the life of all later astronomers".

Tycho Brahe

Despite present day views on
astrology, in the 16th century it was thought that the stars ruled the Earth. The problem was that the connection was insufficiently well known. The idea was that with better astrology would come better prognostics, and man would be given power over his fate. This was the 'world concept' of Tycho Brahe (Danish, 1546-1601), and to him mathematics main use was for astrology. Tycho was drawn to the supernova of early November 1572 which proved, contrary to Aristotle's doctrine, that changes did occur in the world of the stars. Naturally the fact that it was initial yellow, then changed to red, before disappearing in 1574, left people worried and perplexed. Tycho published his opinion that it was just condensation of thin celestial matter, but the appearance of the new star did re-ignite his interest in astronomy. He was lucky that he was offered the small island of Hven to build Uraniborg, his palace of astronomy. His first step at Uraniborg was to improve the measuring instruments, all constructed to Tycho's design and all built to the utmost precision possible at that time. We are of course still 50 years away from when Pierre Vernier (French, 1580-1637) would invent the vernier scale. It is said that Tycho renovated astronomy by giving careful attention to the differences of a few minutes of arc, thus raising the science to a higher standard of precision. As an example, he measured the inclination of the Moon's orbit, and changed it from a simple 5° to 4° 58½' at full and new Moon and a maximum of 5° 17½' at the quarters.
Tycho's catalogue of stars was the first modern catalogue, and far exceeded that of Ptolemy in accuracy. In 1603 Johann Bayer (German, 1572-1625) published a new star atlas based upon the work of Brahe but which included a new system of star designation where Greek letters were added to prominent stars, a practice that is still in use (the system is called the Bayer designation).
Parallax was used by Tycho in 1577 to measure the passage of a comet. The absence of parallax was sufficient to demolish Aristotle's theory that comets were in the upper layers of the Earth's atmosphere. Tycho had elevated the comet to the rank of an astronomical object. In later life Tycho Brahe was based in Prague, and his assistant was a certain Johannes Kepler (German, 1571-1630).

We have already mentioned that
Tycho Brahe spent some considerable effort in critiquing the Copernican model, and more generally the heliocentric doctrine. And his approach was to base his ideas on improved observational accuracy and a more empirically driven model. He admired Copernicus for avoiding the equant, but he did not accept the heliocentric hypothesis because it was "opposed to physical principles". Tycho objected to Ptolemy's explicit use of non-uniform circular motion and what was later called the equant point, a point around which the epicycle of a planet maintains a constant angular velocity even while it follows a path centred on a different point. Copernicus also had planets moving in non-uniform circular motion but it was hidden in the combination of eccentric circles and epicycles, so there were circles revolving around points other than their centre. The equivalence of Copernicus's eccentrepicyclic model and Ptolemy's equant point was not lost on the specialist astronomer of the day including Kepler. Tycho never alluded to this point, but his focus was on problems in the Copernican model that he could resolve, I.e. the motion of the Earth. For him a moving Earth violated "physical principles", i.e. he felt that a very heavy, dense and opaque body would be hard to rotate whereas the sky would be easier. At the time the Earth was considered as "lazy and ignoble", a nature that did not lend itself to motion, equally Tycho did not ignore the "unquestionable authority of the holy scriptures".
His idea was that
Copernicus's numbers might be right, but the model was wrong, and many of his contemporaries shared this view. His first problem was that the position of the stars were often based on old and imprecise data, so his first step was to "restore to pristine condition" those measurements. His idea was that if the numbers were imprecise, the model would prove incorrect. This resulted in him providing more reliable observations for virtually all the celestial bodies. He then used the errors and imprecision in Copernicus's numbers to cast doubt on his cosmological system (whilst admitting that Copernicus's tools had been primitive). Once he had decided that he could only rely on his own measurements, he ended up in 1588 with his 'geoheliocentric' model, which became known as the Tychonic System. Brahes' criticism boils down to a simple statement. Like many people of that period, he did not like the idea of the Earth moving around the Sun. So he ended up with a model where the planets orbit the Sun, but the Sun orbit the Earth.
Tych's legacy was that he set a new standard for astronomical practice. He laid down clear rules for the construction of accurate instruments, and he insisted that refraction be taken into account. He also attacked those who relied on too few observations, and he constantly stressed the need to verify both observations and the conclusions drawn (although others would later criticise him for a lack of diligence and precision).


calendar had always represented a problem for Christianity. It regulated social life but over time it had lost any direct connection with its past. For example, Easter no longer meant the spring offerings of the new harvest but it had become the yearly commemoration of the crucifixion and resurrection of Christ. Easter had to be fixed long beforehand so that it could be celebrated simultaneously in all the churches of the Orient and the Occident, and they could not wait until the full Moon had been observed. The problem was to adapt lunar phenomena to the Roman solar calendar, and using a seven-day week that runs independently of the Sun and Moon. For example, we had to wait unit 1800 when Carl Friedrich Gauss (German, 1777-1855) finally condensed the entire Easter computation into a couple of simple formulae. It did not help that in the Middle Ages simple pieces of arithmetic were often shrouded in mystery and life was dominated by curious rules that only the powerful and initiated could understand. It was clear that the calendar no longer agreed with reality, e.g. in 1300 the vernal equinox fell on the 13th instead of the 21st of March, and the full Moon now came 3 days earlier than was computed. Finally the Council of Trent (1545-63) charged the Pope to make the necessary reforms, and in 1582, under Gregory XIII, they were introduced. Sudden jumps were made to correct for the fact that the equinox came 10 days too early and the computed full Moon came 3 days late. The reforms were based upon the suggestions of Aloysius Lilius (Italian, ca. 1510-1576), but when the Pope ordered the introduction of the reforms it was ignored by the Protestant countries. Germany introduced a reform that meant that Easter could be a week different in Catholic and Protestant regions. The Gregorian calendar was finally introduced throughout Europe in the 18th century (in England in 1752), but Russia only followed in the 20th century.

Lilius had calculated that the use of the Julian year had introduced, since the Council of Nicaea in 325 AD, a discrepancy of about 10 days for the date of the vernal equinox. He proposed to delete a leap year for the following 40 years, whereas Christopher Clavius (German, 1538-1612) proposed a correction in one go, and this was retained by Pope Gregory and 10 days were deleted from the 4 to 15 October 1582.
So we have been using the
Gregorian calendar since 1582, that's over 400 years. Not everyone adopted it at the same time, and even today many religions and cultures celebrate New Year on a different date. Our calendar is not perfect, months are uneven, and each year dates fall on different days of week. In addition we are going to need an extra day by the year 4909. On top of that we now have leap seconds to contend with.

Catholic Church had encouraged Copernicus to publish his results, but Protestant leaders had rejects them. At the time Protestantism took the Bible as literal, whereas the Catholic Church claimed the right of interpretation. But the rigorous doctrine of the Jesuit Order changed the Catholic Church into a militant power. In 1583 Tycho Brahe had devised a world system where the Sun and Moon described circles about a resting Earth, but the Sun was the centre of all the planets orbits and carried them along with her in her yearly revolution. The celestial sphere carried all along with its daily rotation. Copernicus's book (1543) was studied in detail, but in the 16th century almost no one adhered to the heliocentric system. Academics praised Copernicus but considered his ideas absurd. But not everyone refuted the world view of Copernicus. Thomas Digges (English, ca. 1546-1595) wrote of an infinite world filled mostly with invisible stars, and Giordano Bruno was burned at the stake in 1600 for suggesting that the stars were suns surrounded by planets. William Gilbert (English, 1544-1603) found the Earth to be a large magnet, and Simon Stevin sided entirely with Copernicus calling it the 'true' system. Johannes Kepler was a fervent supporter of the heliocentric system. He published a book in which five regular polyhedrons (the Platonic solids) sat between the orbits of the six planets. This raised Copernicus's theory to a kind of fundamental philosophical truth. Galileo Galilei (Italian, 1564-1642) wrote in 1597 that he agreed with Copernicus but feared becoming an object of derision. Galileo had developed a clear idea, supported by his own experiments, that Copernicus was right, but he still taught the 'false' logic to his students.

Imagine you’re an astronomer from antiquity, exploring the night-sky without the aid of a telescope. At first the planets don’t really distinguish themselves from the stars. They’re a bit brighter than most stars and twinkle less, but otherwise look like stars. In antiquity, what really distinguished planets from stars was their motion through the night-sky. From night to night, the planets gradually moved with respect to the stars. Indeed 'planet' is derived from the ancient Greek for “wandering star”. And planetary motion was not simple. Planets appeared to speed up and slow down as they crossed the night-sky. Planets even temporarily reversed direction, exhibiting 'retrograde motion'. How could that be explained? Ptolemy explained planetary motion using the superposition of two circular motions, a large 'deferent' circle combined with a smaller 'epicycle' circle. Furthermore, each planet's deferent could be offset from the position of the Earth and the steady (angular) motion around the deferent could be defined using a position know as an equant, rather than the position of the Earth or the centre of the deferent. It was rather complex, but it did predicted the positions of planets in the night-sky with an accuracy of a few degrees (sometimes better). And it thus became the primary means of explaining planetary motion for over a millennium.
In 1543, the year of his death, Nicolaus Copernicus started a revolution with the publication of De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres). Copernicus' model for the solar system was heliocentric with the planets circling the Sun rather than Earth. Perhaps the most elegant piece of the Copernican model was its natural explanation of the changing apparent motion of the planets. The retrograde motion of planets such as Mars was merely an illusion, caused by the Earth 'overtaking' Mars as they both orbited the Sun. Unfortunately, the original Copernican model was loaded with Ptolemaic baggage. The Copernican planets still traveled around the solar system using motions described by the superposition of circular motions. Copernicus disposed of the equant, which he despised, but replaced it with the mathematically equivalent epicycle. So calculated planetary coordinates using Ptolemaic and Copernican models of the era produced comparable errors, and furthermore, the original Copernican model was no simpler than the earlier Ptolemaic model. As 16th century astronomers did not have access to telescopes or an understanding of Newtonian physics, it wasn’t obvious to them that the Copernican model was superior to the Ptolemaic model, even though it correctly placed the Sun in the centre of the solar system.

Galileo and the Optical Telescope

It was the optical telescope that changed everything. It origins are lost in time, but in August 1609 Galileo built one himself and offered it to the Doge and the Signoria of Venice for use in war and navigation. It already had a magnification of times eight, and in recompense Galileo was awarded a lifetime appointment to the Padua Chair of Mathematics at a salary of one thousands florins a year. It is said that by November 1609 he had a telescope with a magnification of 20. Once Galileo started to turn his telescope to the Moon and stars, Copernicus's doctrine became the focus of his attention. He saw that the Moon had mountains and valleys like Earth, and was not crystalline. He saw that planets were circular disks, but stars remained just bright points. There were nebulous patches consisting of small stars, and invisible stars became visible. He saw moons orbiting Jupiter, so the Earth was not the centre of all movement. His publication of 1610, 'Sidereus Nuncius', was welcomed by many, but often condemned by the learned authorities. Some considered what was seen in a telescope an illusion, others even declared that they saw nothing. Kepler was an active supporter of Galileo, but had not built a telescope because he thought that the thick air would impede its use. Kepler had poor eyesight and was not an able experimentalist, so he never built a telescope. Magnification of 20 or 30 was needed, but grinding good lenses was a difficult art. However over time, people did start to see what Galileo described, and other discoveries followed (possibly the most important was that the Earth was just another planet). In 1611 Galileo visited the Holy Office where they confirmed his discoveries but not his theory. By then many people started to use telescopes and discover (and claim to have already discovered) new phenomena. The problem was no longer the observation, but the interpretation. Galileo saw the observations as proof of the truth of the heliocentric theory, but many warned him to be more cautious in his claims. The real conflict became theological and Galileo was secretly denounced to the Holy Office. They could not accept that the texts of the Bible were open to interpretation, so the result was inevitable. Galileo must submit to the doctrine of the Bible and all the books of Copernicus must be amended. Galileo could not defend the heliocentric theory, so he turned to attack Aristotle's philosophy of nature. Galileo destroyed Aristotle's doctrine of motion and of physics, but tried to also accept the doctrine of the Bible whilst explain the heliocentric theory. But the Holy Office were not deceived, and Galileo was summoned before the Inquisition. He was condemned and compelled in 1635 to solemnly abjure the heliocentric doctrine. The ban was only lifted in 1822, and after 1835 the works of Copernicus, Kepler and Galileo were removed from the index of prohibited books. If you look at the index it is almost a sign of greatness to be included, e.g. Victor Hugo, Gide, John Stuart Mill, Kant, Gibbon, Voltaire, Defoe, Spinoza, Descartes, etc., so Galileo was in good company.

Nimrud Lens

In the British Museum (BM) there is a small oval piece of ground quartz or rock crystal that is called the Nimrud Lens. It dates from 750-710 BC and was carefully ground and polished, and undoubtedly had optical properties. It was ground flat on one side and convex on the other, and it has a focal length of 12 cm and a magnification of about 3x. The crystal is flawed and the grinding opened several cavities in the lens surface, making is a very poor lens. The BM commentary suggests that it was probably a piece of inlay, and no other evidence of an optical instrument has been found. There are claims that there are several hundred lenses on record as found in the ancient world, and it's been suggested that the optical telescope was invented well before the early 17th century.

Earliest Telescope Galileo's Original Telescope

The oldest optical telescope was found in Delft in 2014 (left), and has been dated to the early part of the 17th century. It is a 10 cm long metal tube holding a very poor set of lenses. Already in 1608 the Dutch government discussed a patent application for an optical telescope, an application that was not granted because the idea was too easy to copy. In fact, news of the invention spread rapidly, and Paolo Sarpi (Venetian, 1552-1623), a friend of Galileo, had already heard about it in November 1608. By April 1609 you could buy little telescopes about 30 cm long in spectacle-maker shops in Paris and London. On the right is one of Galileo's first refracting telescopes (late 1609). Some texts mention that it was on 14 April, 1611 during a demonstration by Galileo that Federico Cesi (Italian, 1585-1630) first coined the word 'telescope'.

There is a suggestion that a certain Sacharias Janssen invented the telescope (spyglass) in in the Netherlands in 1604, but based upon an example from Italy dated 1590 (considered a device of feeble quality).

By 1613
Galileo was recording the separations between Jupiter and its moons to within 0.1 Jovian radii (so approximately 2 seconds of arc), and his drawings placed the moons' positions to within the size of a dot. He was also recording and drawing the position of objects as faint as Neptune, and all with a 'Galilean' telescope that lacked a focal plane in which he could place a measuring reticle.

Galileo Jupiter with 4 moons, and Neptune

However Galileo had no knowledge of wave optics, so he was often measuring diffraction artefacts. It would be more than 150 years later that astronomers would learn that the apparent size of a star was a product of a telescopes' aperture. Above we have an observation of Galileo from 6 January 1613 as compared to planetarium software. We can see Jupiter with its four moons, and on the far right we have Neptune. So he had a remarkable level of skill in seeing and marking the positions of objects, but the size measurements were often affected by artefacts. Anyone looking for more information on the 'Galilean' telescope checkout this reference and this reference.

If you are interested in the key developments of the
optical telescope have a look at the following - Newton's Reflecting Telescope, William Herschel's 40-foot Reflecting Telescope, The Leviathan of Parsonstown, The Hooker Telescope, The Hale Telescope, BTA-6, Keck 1 & 2, Hubble, and the Extremely Large Telescope.


Johannes Kepler

Prague in 1601 Johannes Kepler 'inherited' the work of Tycho Brahe, but his approach was quite different. Kepler still believed in astrology, but in the sense that he thought the planets (and the Sun and Moon) could influence earthly events. He was less interested in the meaning and more interested in what the phenomena was and what caused it. His work was always involved with computing the movement of celestial bodies, but now he designated the Sun as the natural centre of the planetary system. Kepler still saw the planets as playing an active role, but he was asking the right questions, e.g. he spoke of a force proceeding from the Sun but not in the Newtonian sense. And as an example of his willingness to challenge established ideas, he took an idea that was a thousand years old, and said that the natural orbit of heavenly bodies was not a circle, but an oval or ellipse with the Sun in one of the foci. This is the first of Kepler's laws of planetary motion.

The key message of both
Tycho Brahe and Johannes Kepler was to collect data from experiment and observation, then to derive rules and laws that would form a body of science. In 1618 Kepler published the 'Epitome Astronomiae Copernicanae', the first complete manual of astronomy constructed after the new principles. This was the century of great names and great innovations, i.e. Cornelis Drebbel (Dutch, 1572-1633), Christiaan Huygens (Dutch, 1629-1695), Isaac Newton (English, 1642-1726), Antonie Philips van Leeuwenhoek (Dutch, 1632-1723), Pierre Gassendi (French, 1592-1655), Francis Bacon (English, 1561-1626), René Descartes (French, 1596-1650), Robert Hooke (English, 1635-1703), Robert Boyle (Anglo-Irish, 1627-1691), Gottfried Wilhelm Leibniz (German, 1646-1716) and the astronomers Ole Rømer (Danish, 1644-1710), Giovanni Domenico Cassini (Italian, 1625-1712) and Jean Picard (French, 1620-1682).

As someone interested in
all natural phenomena Kepler was always asking questions. Why are there six planets? Why do the speeds of the planets decrease the further away from Earth? Why are they order in that way? He was not a good teacher, but over his life he wrote several books of astronomy, mathematics, and optics. He taught about the Platonic solids, and at one moment in time he imagined a sphere sitting on the outside of each, and he saw them nesting one inside the other. His idea was that the radii of these spheres corresponded to the radii of the planets. He published this (fanciful) idea in 1596, and it generated quite a lot of interest. Importantly, it led him to look at more challenging issues.
Tycho Brahe moved to Prague he recruited Kepler and tasked him with looking at the details of the orbit of Mars. When Brahe died, Kepler admitted that he took advantage of his heirs by "taking the observations" under his care. Kepler also got the old job of Brahe, but then spent the rest of his life trying to actually collect his salary. This is probably why Kepler was perpetually destitute and was known to gnaw on bones and dry crusts of bread (he also had a horror of baths). Kepler was equipped with geometry, spherical trigonometry, and the newly invented logarithms, and he had Brahe's collection of thousands of individual position measurements of where Mars appeared as viewed from Earth (remember that Mars undergoes retrograde motion). Firstly Kepler struggled with circular orbits, then with oval orbits, before finally adopting the ellipse (after 6 years of effort). Kepler's publication in 1909 was monumental because for the first time in 2,000 years he demonstrated that planets did not move in perfectly circular orbits. This was a courageous decision because Galileo had been punished only a few years earlier for, admittedly, an even more revolutionary idea.

If there is one personality in astronomy that it worth reading about, it's Kepler. His scientific achievements did not go hand-in-hand with a calm personal life. He was often sick, known as a very bad teacher, and after the death of his protector in 1611 he was obliged to move from Prague to Linz. His first wife had died, and he actually advertised for a new wife, 'hired' one and went on to have six more children. In 1620 he defended his mother from charges of witchcraft, and he would win the case. The conflict with the family of Tycho Brahe resulted in him deciding to pay to publish Tycho's data, naturally the printing works burnt down. It would appear that Kepler died in 1630 still trying to get paid by another of his employers, and of his 12 children by two wives, only 2 survived.

In addition in his writing Kepler took quite a unique approach. With Copernicus, Galileo and Newton, each just presented only the final product of their work. With Kepler he presented an entire journey, step by step, digressions, blind alleys, mishaps, etc. before arriving at his results. In very simple terms the first thing he did was to drop the epicycles which he thought were absurd, but he kept the equant. He assumed that the distance form the centre of Mars' orbit to the Sun and to the equant were unequal (Ptolemy had assumed that they were equal). So there were four variables, the radius of the circular orbit, the axis containing the nearest and farthest points of Mars on the circle, and the Sun-to-centre and centre-to-equant distances along that axis. It was a trial-and-error approach, and his calculations cover 900 pages. Kepler wrote that he had gone through the calculations 70 times and taken 5 years, when he finally decided that the orbit was not circular. What he did next is considered by many to be an example of pure genius. He decided to try to understand first the orbit of Earth. To do that he took observations of Mars separated by one Mars year (about 687 days) and he computed the distances and longitudes of Earth as if measured from Mars. It showed that the Earth's orbit was not exceptional, as expected it moved faster when close to the Sun, and slower when further away. It was here that he decided that the speed of a planet was inversely propositional to its distance from the Sun, throughout its orbit and therefore the Sun caused planetary motion. This was before Newton, so he had no idea that a moving body will continue to move unless a force acts on it. He thought that a magnetic field of the Sun dragged the planets around. Today we know that the Sun does have a magnetic field and it does rotate, but that does not explain planetary motion.
Kepler had decided that the Sun caused the motion of the planets, and that a planet's speed was inversely related to its distances from the Sun, he worked out how long it took to get to any specific position. Again more as a leap of faith than anything else, he looked at the areas subtended by a series of points on the orbit, and determined that a line joining the Sun and a planet sweeps our equal areas in equal times. The last she, obtaining his first law, took another two years. At this point he had found a circular orbit that fitted Tycho's positional data for Mars and he had a different circular orbit that fitted the distances he had calculated. His first view was that it would fit with a small epicycle, but it also looked like an ellipse. But he decided to try an egg-shaped oval. He repeated his calculations 40 times, and even wrote to a friend saying that with an oval it would not work, but an ellipse would work. However he persisted with his 'idée fixe', the oval, until he had to give up. He then mapped out the curved-shaped orbit inside a circle, and even then it took him some time before realising that what he was looking at was an ellipse. Finally, after 7 years work he had his first law of planetary motion.

So what
Kepler did was to replace a machinery of circles, epicycles, and equants with three laws of planetary motion:-
orbit of a planet is an ellipse with the Sun at one focus
A line joining the
Sun and a planet sweeps our equal areas in equal times
square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.

In the 17th and 18th centuries
astronomy depended upon the development of the optical telescope. Kepler, in 1611, was the first to describe how light rays passed through lenses and formed images. Christiaan Huygens derived the options for lenses by theory, and Giuseppe Campani (Italian, 1635-1715) became the best maker of optical instruments of his age. It was Michael Van Langren (Dutch, 1598-1675) who first mapped the Moon's surface, followed by Johannes Hevelius (Polish, 1611-1687) who is often called the founder of lunar topography.

Different Types of Optical Telescopes

What we often forget is that the
optical telescope might have been important for observing the planets and the stars, but it would become far more important as a measuring instrument. Before the 17th century astronomy was an art form involving the description of geometrical models of the solar system based upon naked-eye observations. With the invention of the optical telescope the models gave way to physically-motivated dynamical theories.
In terms of observational data the
optical telescope provided two key improvements. Firstly, it gathered more light allowing the observer to detect fainter objects or features. Secondly, it's resolution improved, allowing the observer to detect finer details and make measurements with greater precision. The first point lead to new discoveries, whereas the second point led to a better understand of the scale and structure of the Universe. We would have to wait unit the 19th century to see two new advances in physics that would again revolutionise astronomy, namely photography and spectroscopy. To understand this move to measurement we need to quickly review the two early types of optical telescopes.

Early Telescopes

The whole idea of a telescope is to view distant objects, producing an image that is larger than can be seen normally, and to gather more light into the eye, allowing dim objects to be observed with greater magnification and better resolution. The 'Galilean' telescope is in figure (a) and has a convex objective and a concave eyepiece. This produces an upright image and is found in spyglasses and opera glasses. In reality the first (convex) lens produced an image that is to the right of the diverging (concave) lens, thus creating a virtual object for the diverging lens. What the diverting lens does is refract the convergence rays to make them parallel, as if they were coming from an object at infinity. It's the lens in the eye that brings the parallel rays to a focus on the retina. Figure (b) is a simple telescope with two convex tens, and is often called a 'Keplerian' telescope. The object being viewed is very far away (infinity) and the first convex lens produces a smaller image of the object at F0. This is very similar to when a simple convex lens creates an image of the Sun as a small point of light. Now the second convex lens acts as an eyepiece magnifying the first image. The 'Keplerian' telescope had the advantage of allowing a much wider field of view but the image was inverted. It had higher magnifications, but to overcome optical aberration from a simple convex lens it need a longer focal length.

Three Lens

Above we can see that increasing the distance between the first two lens inverts the image created by the first lens, restoring the image to its upright position, which can now be viewed using a third convex lens as an eyepiece.

In 1672
Newton showed that chromatic aberration was intrinsic to glass lenses in that the lens acts like a prism in breaking white light into a rainbow of colours around any astronomical object. In 1668 he designed and built the 'Newtonian' telescope, a type of reflecting telescope which used a concave primary mirror as an objective (light gatherer) and a flat diagonal secondary or focusing mirror.

Refracting and Reflecting Design

Above we have examples of the two basic designs. With the refracting telescope light enters through a lens at the upper end, which focuses the light near the bottom of the telescope. An eyepiece then magnifies the image so that it can be viewed with the eye (later a photographic plate, film or CCD will replace the human eye). With a reflecting telescope the upper end is open, and the light passes through to the mirror located at the bottom of the telescope. The mirror then focuses the light at the top end, where it can be detected. Alternatively, as in the above diagram, a second mirror may reflect the light to a position outside the telescope structure, where it can be viewed or detected.

Newton, James Gregory (Scottish, 1638-1675) had invented the 'Gregorian' telescope, but it was only in 1673 that Robert Hooke built the first example. The design had some advantages over that of Newton, but it was superseded by the 'Cassegrain' telescope. It was Laurent Cassegrain (French, 1629-1693) who (probably) invented the folded two-mirror reflecting telescope in 1672. The main feature is that the optical path folds back on itself, and the focal point is conveniently located behind the primary mirror.


Above we have some different options for the reflecting telescope. With the prime focus option the light is detected where it comes to focus after reflection from the primary mirror. With the Newtonian telescope the light is reflected off to one side by a small secondary mirror. With the Cassegrain telescope the light is reflected by a secondary mirror back through a hole in the primary mirror.

With the
refracting telescope there was a way to reduce the effect of chromatic aberration and that was to reduce the curvature of the lens by making telescopes with very long focal lengths, but this made then cumbersome and unwieldy.

Hevelius 46 m

Possibly the longest 'tube' refracting telescope was 46 metres long and built by Johannes Hevelius before 1673. What are we seeing? What was he seeing might also be an appropriate question. This is a long refractor erected on a beach near Gdańsk. According to the description it was not a perfectly enclosed 'tube'. The square perforated boards that we can see were designed to exclude light entering from the sides, so that the light arriving at the lens (and eye) only came from the part of night-sky under examination. It looks as if the lower part was an enclosed tube, and it was noted that the holes in the tube were narrower and the interior blackened. Some texts suggest that doing away with the tube for longer telescopes had started in the 1660's. Hevelius was a master brewer in Gdańsk and had developed a passion for astronomy. In 1641 he joined three houses together a built an observatory on the roof.

Johannes Hevelius Rooftop Observatory

Hand-grinding his own lenses and creating his own sextants was part of the everyday task of stargazing at the Hevelius home.

With these homemade tools, Hevelius quickly became a master of his science, discovering numerous constellations and comets, extensively documenting the topography of the moon, and observing the phases of mercury and spots on the sun. These discoveries and observations led Hevelius to publish 20 works in Latin detailing his findings, many using his own well-crafted illustrations.

Unsurprisingly, Hevelius’s work and stunning observatory caught the attention of his peers, and he was elected to the Royal Society of London in 1664 (the first Pole in the Society’s history).
Check out

Check out this
excellent summary of what was the unique and interesting life of

In 1684
Christiaan Huygens designed a refracting telescope without a tube. It consisted of …

The reality is that as the 17th century drew to a close, the
long refractors gradually fell out of favour because of diminishing returns for increasing unwieldiness. In the first quarter of the 18th century they were used a few times, but after that they were replaced by the reflecting telescope.

There is an entire website dedicated to The Craig Telescope 'London's Lost Leviathan' about the building of a refracting telescope on Wandsworth Common in London, which turned out to be an "expensive failure".

Newton's design, which required only one curved face, was the most successful at the time, but the Cassegrain design, which involves a short tube and has better image quality, is more common today. Reflecting telescopes could be much shorter, owing to the shorter focal lengths, and they have large apertures, however mirrors of highly reflective speculum alloy (copper and tin) were difficult to grind to a regular curvature and they tarnished easily.


'Galilean' telescope that lacked a focal plane in which he could place a measuring reticle.

From 1609, Galileo Galilei used the recently invented telescope to observe the sun, moon and planets.. He saw the mountains and craters of the moon, and for the first time revealed the planets. to be worlds in their own right. Galileo also provided strong observational evidence that planets. orbitsed the sun.
Galileo’s observations of Venus were particularly compelling. In Ptolemaic models, Venus remains between the Earth and the sun at all times, so we should mostly view the night side of Venus. But Galileo was able to observe the day-lit side of Venus, indicating that Venus can be on the opposite side of the sun from the Earth.

Kepler’s war with Mars
The circular motions of Ptolemaic and Copernican models resulted in large errors, particularly for Mars, whose predicted position could be in error by several degrees. Johannes Kepler devoted years of his life to understanding the motion of Mars, and he cracked this problem with a most ingenious weapon.

planets. (approximately) repeat the same path as they orbits the sun, so they return to the same position in space once every orbital period. For example, Mars returns to the same position in its orbits every 687 days.

As Kepler knew the dates when a planet would be at the same position in space, he could use the different positions of the Earth along its own orbits to triangulate the planets.positions, as illustrated above. Kepler, using Tycho Brahe’s pre-telescopic observations, was able to trace out the elliptical paths of the planets. as they orbitsed the sun.
This allowed Kepler to formulate his
three laws of planetary motion and predict planetary positions with far greater precision than previously possible. He thus laid the groundwork for the Newtonian physics of the late 17th century, and the remarkable science that followed.
Kepler himself
captured the new world view and its broader significance in 1609’s Astronomia nova (New Astronomy):
To me, however the truth is more pious still, and (with all due respect to the Doctors of the Church) I prove philosophically not only that the earth is round, not only that it is inhabited all the way around at the antipodes, not only that is it contemptibly small, but also that it is carried along among the stars.

measuring reticle

Where do we go next?

We know what we want. We want a model of the
Universe that enables man to determine the motion of the planets, celestial bodies, etc. Through to Kepler astronomers imagined the Universe governed by circles (orbits) and spheres, both considered examples of divine perfection. Using only circles and spheres they were constrained to devise extremely complex systems that described the apparent movements of the planets and stars as observed from Earth. They had to account for planets that accelerated and decelerated and even occasionally went back on themselves (retrograde motion). They had to account for the changes in brightness of the planets, suggesting variations in the distances between the planets and Earth. So how could the planets travel in orbits centred on the Earth?
Ptolemy built an elaborate model that explained the motion of the planets (in order Mercury, Venus, Mars, Jupiter, and Saturn) and the Sun and Moon. Earth was in the centre and all the celestial bodies moved in circles. But he had to multiple the number of circles and offset them one against the other. So he adopted a system of 'deferents' and 'epicycles', i.e. planets moved in epicycles, and epicycles revolved around deferents which were offset from the Earth's position (using so-called equant points). So Earth was in fact no longer in the centre of the circles, but Ptolemy's model worked and survived for 1,500 years, longer than any other idea in the history of science.
This in not to say that many people did not criticise the
Ptolemaic model, and they looked constantly at ways to correct and improve on it. But is was only with the massive observational campaign of Tycho Brahe in the late 16th century that people had sufficient new data to justify a major revision to the standard mathematical schemes for predicting the position of the planets as handed down from Ptolemy in his Almagest.

It was
Kepler who wrote in 1619 that geometry "supplied God with patterns for the creation of the world". In his hunt for a simple model to describe his experiments and observations, his overriding objective was to understand the causes for what he saw. It was with this in mind that he formulated his laws of planetary motion, that are still valid today. He adopted the ideas of Copernicus but he noticed that circular orbits were not confirmed in his precise observations. Kepler's idea was that the majesty of the Universe was written in the language of mathematics, and that meant geometric figures. This desire for a harmonious view of the position and movement of the planets finally led him in 1609 to discover that orbits were ellipses not circles. Kepler made the description and calculation of the movement of the planets much easier, but he did not explain the physical cause of those movements. It was up to Newton to propose a universal law of attraction.


histry of measurements with telescopes pdf





A. Pannekoek, A History of Astronomy, 1961
Ann Blair,
Tycho Brahe's Critique of Copernicus and the Copernican System, 1990
Eugene I. Butikov,
Relative Motion of Orbiting Bodies, 2001
Eugene I. Butikov,
Orbital Manoeuvres and Space Rendezvous, 2015

Websites Used

E-Luminesciences, and in particular three webpages on Geometry and the Cosmos