Tag: Kepler

Poincaré, inertia, and a common misconception

A historical, and ongoing, alternative conception

Keith S. Taber

"…and eleventhly Madame Curie…" Henri Poincaré enjoying small talk at a physics conference (image source: 'Marie Curie and Poincaré talk at the 1911 Solvay Conference', Wikipedia)

One of the most fundamental ideas in physics, surely taught in every secondary school science curriculum around the world, is also the focus of one of the most common alternative conceptions documented in science education. Inertia. Much research in the latter part of the twentieth century has detailed how most people have great trouble with this very simple idea.

But that would likely not have surprised the nineteenth century French physicist (and mathematician and philosopher) Henri Poincaré in the least. Over a century ago he had this to say about the subject of Newton's first law, inertia,

"The principle of inertia. A body acted on by no force can only move uniformly in a straight line.

Is this a truth imposed a priori upon the mind? If it were so, how could the Greeks have failed to recognise it? How could they have believed that motion stops when the cause which gave birth to it ceases? Or again that every body if nothing prevents, will move in a circle, the noblest of motions?

If it is said that the velocity of a body can not change if there is no reason for it to change, could it not be maintained just as well that the position of this body can not change, or that the curvature of its trajectory can not change, if no external cause intervenes to modify them?

Is the principle of inertia, which is not an a priori truth, therefore an experimental fact? But has any one ever experimented on bodies withdrawn from the action of every force? and, if so, how was it known that these bodies were subjected to no force?"

Poincaré, 1902/1913/2015

There is quite a lot going on in that quote, so it is worth breaking it down.

The principle of inertia

"The principle of inertia. A body acted on by no force can only move uniformly in a straight line."

Poincaré, 1902/1913/2015

We might today choose to phrase this differently – at least in teaching. Perhaps along the lines that

a body remains at rest, or moving with uniform motion, unless it is acted upon by a net (overall) force

That's a pretty simple idea.

  • If you want something that is stationary to start moving, you need to apply a force to it. Otherwise it will remain stationary. And:
  • If you want something that is moving with constant velocity to slow down (decelerate), speed up (accelerate), or change direction, you need to apply a force to it. Otherwise it will carry on moving in the same direction at the same speed.

A simple idea, but one which most people struggle with!

It is worth noting that Poincaré's formulation seems simpler than the versions more commonly presented in school today. He does not make reference to a body at rest; and we might detect a potential ambiguity in what is meant by "can only move uniformly in a straight line".

Is the emphasis:

  • can only move uniformly in a straight line:
    • i.e., ⟨ can only ⟩ ⟨ move uniformly in a straight line ⟩, or
  • can only move uniformly in a straight line:
    • i.e., ⟨ can only move ⟩ ⟨ uniformly in a straight line ⟩

That is, must such a body "move uniformly in a straight line" or must such a body, if moving, "move uniformly in a straight line"? A body acted on by no force may be stationary.

Perhaps this is less ambiguous in the original French? But I suspect that, as a physicist, Poincairé did not, particularly, see the body at rest as being much of a special case.

To most people the distinction between something stationary and something moving is very salient (evolution has prepared us to notice movement). But to a physicist the more important distinction is between any body at constant velocity, and one accelerating* – and a body not moving has constant velocity (of 0 ms-1!)

*and for a physicist accelerating usually includes decelerating, as that is just acceleration with a negative vale, or indeed positive acceleration in a different direction. These 'simplifications' seem very neat – to the initiated (but perhaps not to novices!)

A historical scientific conception

Poincaré then asks:

Is this a truth imposed a priori upon the mind? If it were so, how could the Greeks have failed to recognise it? How could they have believed that motion stops when the cause which gave birth to it ceases?"

Poincaré, 1902/1913/2015

Poincairé asks a rhetorical question: "Is this a truth imposed a priori upon the mind?" Rhetorical, as he immediately suggests the answer. No, it cannot be.

Science is very much an empirical endeavour. The world is investigated by observation, indeed often observation of the effects of interventions (i.e., experiments).

In this way, it diverges from a rationalist approach to understanding the world based on reflection and reasoning that occurs without seeking empirical evidence.

An aside on simulations and perpetual change

Yet, even empirical science depends on some (a priori) metaphysical commitments that cannot themselves be demonstrated by scientific observation (e.g., Taber, 2013). As one example, the famous 'brain in a vat' scenario (that informed films such as The Matrix) asks how we could know that we really experience an external world rather than a very elaborate virtual reality fed directly into our central nervous system (assuming we have such a thing!) 1

Science only makes sense if we believe that the world we experience is an objective reality originating outside our own minds
(Image by Gerd Altmann from Pixabay)

Despite this, scientists operate on the assumption this is a physical world (that we all experience), and one that has a certain degree of stability and consistency. 2 The natural scientist has to assume this is not a capricious universe if science (a search for the underlying order of the world) is to make sense!

It may seem this (that we live in is an objective physical world that has a certain degree of stability and consistency) is obviously the case, as our observations of the world find this stability. But not really: rather, we impose an assumption of an underlying stability, and interpret accordingly. The sun 'rises' every day. (We see stability.) But the amount of daylight changes each day. (We observe change, but assume, and look for, and posit, some underlying stability to explain this.)

Continental drift, new comets, evolution of new species and extinction of others, supernovae, the appearance of HIV and COVID, increasing IQ (disguised by periodically renormalising scoring), climate change, the expanding universe, plant growth, senile dementia, rotting fruit, printers running out of ink, lovers falling out of love, et cetera,…are all assumed to be (and subsequently found to be) explainable in terms of underlying stable and consistent features of the world!

But it would be possible to consider nothing stays the same, and seek to explain away any apparent examples of stability!

Parmenides thought change is impossible

Heraclitus though everything was in flux

An a priori?

So Poincaré was asking if the principle of is inertia was something that would appear to us as a given; is inertia something that seems a necessary and obvious feature of the world (which it probably does to most physicists – but that is after years of indoctrination into that perspective).

But, Poincaré was pointing out, we know that for centuries people did not think that objects not subject any force would continue to move with constant velocity.

There were (considered to be) certain natural motions, and these had a teleological aspect. So, heavy objects, that were considered mainly earth naturally fell down to their natural place on the ground. 3 Once there, mission accomplished (so to speak), they would stop moving. No further explanation was considered necessary.

Violent motions were (considered to be) different as they needed an active cause – such as a javelin moving through the air because someone had thrown it. Yet, clearly (it was believed), the athlete could only impart a finite motion to the javelin, which it would soon exhaust, so the javelin would (naturally) stop soon enough.

Today, such ideas are seen as alternative conceptions (misconceptions), but for hundreds of years these ideas were largely taken as self-evident and secure principles describing aspects of the world. The idea that the javelin might carry on moving for ever if it was 'left to its own devices' seemed absurd. (And to most people today who are not physicists or science teachers, it probably still does!)

An interesting question is if, and if so, to what extent, the people who become physicists and physics teachers start out with intuitions more aligned with the principles of physics than most of their classmates.

"Assuming that there is significant variation in the extent to which our intuitive physics matches what we are taught in school, I would expect that most physics teachers are among those to whom the subject seemed logical and made good sense when they were students. I have no evidence for this, but it just seems natural that these students would have enjoyed and continued with the subject.

If I am right about this intuition, then this may be another reason why physics is so hard for some of our students. Not only do they have to struggle with subject matter that seems counterintuitive, but the very people who are charged with helping them may be those who instinctively think most differently from the way in which they do."

Taber, 2004, p.124

Another historical scientific conception

And Poincaré went on:

"Or again that every body if nothing prevents, will move in a circle, the noblest of motions?"

Poincaré, 1902/1913/2015

It was also long thought that in the heavens bodies naturally moved spontaneously in circles – a circle being a perfect shape, and the heavens being a perfect place.

Orbital motion – once viewed to be natural (i.e., not requiring any further explanation) and circular in 'the heavens'.
(Image by WikiImages from Pixabay: Body sizes and separations not to the same scale!)

It is common for people to feel that what seems natural does not need further explanation (Watts & Taber, 1996) – even though most of what we consider natural is likely just familiarity with common phenomena. We start noticing how the floor arrests the motion of falling objects very young in life, so by the time we have language to help reflect on this, we simply explain this as motion stopping because the floor was in the way! Similarly, reaction forces are not obvious when an object rests on another – a desk, a shelf, etc – as the object cannot fall 'because it is supported'.

Again, we (sic, we the initiated) now think that without an acting centripetal force, an orbiting body would move off at a tangent – but that would have seemed pretty bizarre for much of European history.

The idea that bodies moved in circles (as the fixed stars seemed to do) was maintained despite extensive observational evidence collected over centuries that the planets appeared to do something quite different. Today Kepler's laws are taught in physics, including that the solar system's orbiting bodies move (almost) in ellipses. ('Almost', as they bodies perturb each other a little.)

But when Kepler tried to fit observations to theory by adopting Copernicus's 'heliocentric' model of the Earth and planets orbiting the Sun (Earth and other planets, we would say), he still struggled to make progress for a considerable time because of an unquestioned assumption that the planetary motions had to be circular, or some combination of multiple circles.

Learners' alternative conceptions

These historical ideas are of more than historical interest. Many people, research suggests most people, today share similar intuitions.

  • Objects will naturally come to a stop when they have used up their imparted motion without the need for any forces to act.
  • Something that falls to the floor does not need a force to act on it to stop it moving, as the ground is in its way.
  • Moons and planets continue in orbits because there is no overall force acting on them.

The vast majority of learners some to school science holding versions of such alternative conceptions.

Read about common alternative conceptions related to Newton's first law

Read about common alternative conceptions related to Newton's second law

The majority of learners also leave school holding versions of such alternative conceptions – even if some of them have mastered the ability to usually respond to physics test questions as if they accepted a different worldview.

The idea that objects soon stop moving once the applied force ceases to act may be contrary to physics, but it is not, of course, contrary to common experience – at least not contrary to common experience as most people perceive it.

Metaphysical principles

Poincaré recognised this.

"If it is said that the velocity of a body can not change if there is no reason for it to change [i.e. the principle of inertia],

could it not be maintained just as well that

the position of this body can not change, or

that the curvature of its trajectory can not change,

if no external cause intervenes to modify them?"

Poincaré, 1902/1913/2015 (emphasis added)

After all, as Poincairé pointed out, there seems no reason, a priori, that is intuitively, to assume the world must work according to the principle of inertia (though some physicists and science teachers whom have been indoctrinated over many years may have come to think otherwise – of course after indoctrination is not a priori!), rather than assuming, say, that force must act for movement to occur and/or that force must act to change an orbit.

Science as an empirical enterprise

Science teachers might reply, that our initial intuitions are not the point, because myriad empirical tests have demonstrated the principle of inertia. But Poincairé suggested this was strictly not so,

"Is the principle of inertia, which is not an a priori truth, therefore an experimental fact? But has any one ever experimented on bodies withdrawn from the action of every force? and, if so, how was it known that these bodies were subjected to no force?"

Poincaré, 1902/1913/2015

For example, if we accept the ideas of universal gravitation, than anywhere in the universe a body will be subject to gravitational attractions (that is, forces). A body could only be completely free of this by being in a universe of its own with no other gravitating bodies. Then we might think we could test, in principle at least, whether the body "acted on by no force can only move uniformly in a straight line".

Well, apart from a couple of small difficulties. There would be no observers in this universe to see, as we have excluded all other massive bodies. And if this was the only body there, it would be the only frame of reference available – a frame of reference in which it was always stationary. It would always be at the centre of, and indeed would be the extent of, its universe.

Poincaré and pedagogic awareness

Poincaré was certainly not denying the principle of inertia so fundamental to mechanics. But he was showing that he appreciated that a simple principle which seems (comes to seem?) so basic and obvious to the inducted physics expert:

  • was hard won in the history of science
  • in not 'given' in intuition
  • is not the only possible basic principle on which a mechanics (in some other universe) could be based
  • is contrary to immediate experience (that is, to those who have not been indoctrinated to 'see' resistive forces sch as friction acting everywhere)
  • could never be entirely demonstrated in a pure form, but rather must be inferred from experimental tests of more complex situations where we will only deduce the principle of inertia if we assume a range of other principles (about the action of gravitational fields, air resistance, etc.)

Poincaré may have been seen as one of the great physicists of his time, but his own expertise certainly did not him appreciating the challenges facing the learner of physics, or indeed the teacher of physics.

Work cited:
Notes

1 With current human technology we cannot achieve this – even the best virtual worlds clearly do not yet come close to the real one! But that argument falls away if 'the real' world we experience is such a virtual reality created by very advanced technology, and what we think of as virtual worlds are low definition simulations being created within that! (After all, when people saw the first jumpy black-and-white movies, they then came out from the cinema into a colourful, smooth and high definition world.) If you have ever awaken from a dream, only to later realise you are still asleep, and had been dreaming of being asleep in the dream, then you may appreciate how such nesting of worlds could work.

Probably no one actually believes they are a brain in a vat, but how would we know. There is an argument that

  • 1) the evolution of complex life is a very slow process that requires a complex ecosystem, but
  • 2) once humans (or indeed non-humans) have the technology to create convincing virtual worlds this can be done very much more quickly, and with much less resource [i.e., than the evolution of the physical world which within which the programmers of the simulations themselves live]. So,
  • 3) if we are living in a phase of the universe where such technology has been achieved, then we would expect there to be a great many more such virtual worlds than planets inhabited by life forms with the level of self-consciousness to think about whether they are in a simulation.4 So,
  • 4) [if we are living in a phase of the universe where such technology has been achieved] we would be much more likely to be living in one of these worlds (a character in a very complex simulation) than an actual organic being. 5

2 That is, not a simulation where an adolescent programmer is going to suddenly increase gravity or add a new fundamental force just to make things more interesting.

3 Everything on earth was considered to be made up of different proportions of the four elements, which in terms of increasing rarity were earth, water, air and fire. The rocks of the earth were predominately the element earth – and objects that were mainly earth fell to their natural place. (Rarity in this context means the inverse of density, not scarcity.)

4 When I was a child (perhaps in part because I think I started Sunday School before I could start 'proper' school), I used to muse about God being able to create everything, and being omniscient – although I am pretty sure I did not use that term! It seemed to me (and, sensibly, I do not think I shared this at Sunday School) that if God knew everything and was infallible, then he did not need to actually create the world as a physical universe, but rather just think what would happen. For God, that would work just as well, as a perfect mind could imagine things exactly as they would be in exquisite detail and with absolute precision. So, I thought I might just be an aspect of the mind of God – so part of a simulation in effect. This was a comforting rather than worrying thought – surely there is no safer place to be than in the mind of God?

Sadly, I grew to be much less sure of God (the creation seems just as incredible – in the literal sense – either way), but still think that, for God, thinking it would be as good as (if not the same as) making it. I suspect some theologians would not entirely dismiss this.

If I am just a character in someone's simulation, I'd rather it was that of a supreme being than some alien adolescent likely to abandon my world at the first sign of romantic interest from a passing conspecific.

5 Unless we assume a dystopian Matrix like simulation, the technology has to be able to create characters (sub-routines?) with self-awareness – which goes some way beyond just a convincing simulation, as it also requires components complex enough to be convinced about their own existence, as well as the reality of the wider simulation!

Acute abstracts correcting Copernicus

Setting the history of science right

Keith S. Taber

I recently read a book of essays by Edward Rosen (1995) who (as described by his publisher) was "the editor and translator of Copernicus' complete works, was the leading authority on this most celebrated of Renaissance scientists". Copernicus is indeed, rightly, highly celebrated (for reasons I summarise below *).

The book was edited by Rosen's collaborator, Erna Hilfstein 1, and although the book was an anthology of reprinted journal articles, none of the chapters (articles) had abstracts. This reflects different disciplinary norms. In the natural and social sciences most journals require abstracts – and some even offer a menu of what should be included – but abstracts are not always expected in humanities disciplines.

Read about the abstract in academic articles

A collection of published papers from various journals – all lacking abstracts

It is not unusual for an academic book to be a compilation of published articles – especially when anthologising a single scholar's work. I was a little surprised to find the different chapters in the same book having different formats and typefaces – it had been decided to reproduce the articles as they had originally appeared in a range of journals (perhaps for authenticity – or perhaps to avoid the costs of new typesetting?)

But it was the absence of article abstracts that most felt odd. The potential reader is given a title, but otherwise little idea of the scope of an article before reading. Perhaps it was my awareness of this 'omission' that led me to thinking that for a number of the chapters it would be possible to offer a very minimal abstract (an acute abstract?) that would do the job! Certainly, for some of these chapters, I thought a sentence each might do.

That is not to dismiss the scholarship that has gone into developing the arguments, but Rosen often wrote on a very specific historical point, set out pertinent ideas from previous scholarship, and then argued for a clear position contrary to some earlier scholars.

So, here are my suggestions for 'acute' abstracts

Six summary encapsulations

Chapter 6: on the priest question

Abstract:

Copernicus has often been described as a priest, but Copernicus was never ordained a priest.

Copernicus was a canon in the Roman Catholic church, but this made him an administrator (and he also acted as physician), but he never became a monk or a priest.

Chapter 7: on the notary question

Abstract:

Copernicus has been described as a 'happy notary' but Copernicus was not a notary.

Although Copernicus had various roles as an administrator, even as something of a diplomat, he never took on the role of a legal notary.

Chapter 8: on the disdain question

Abstract:

Copernicus is sometimes said to have had a dismissive attitude to the common people, but there is no evidence that this was so

A comment of Copernicus on not being concerned with the views of certain philosophers seems to have been misinterpreted.

Chapter 11: on the axioms question

Abstract:

It has been claimed that Copernicus misused the term axioms in his work, but his use was perfectly in line with authorities

Today axioms are usually expected to be the self-evident starting points for developing a deductive argument, but Aristotle's definition of axioms did not require them to seem self-evident.

Chapter 16: on the papal question

Abstract:

It has been claimed that Copernicus' 'Revolutions' was approved by the pope before publication, but the manuscript was never shown to the pope

This seems to be a confusion regarding an anecdote concerning a completely different scholar.

Chapter 17: on the Calvin question

Abstract:

It has been suggested that Calvin was highly critical of Copernicus, but it seems unlikely Calvin had ever heard of him

While Calvin's writing strongly suggest he was committed to a stationary earth and a sun that moved around the earth, there is no evidence he had specifically come across Copernicus.

A manifold chapter

Having noticed how so many of Rosen's articles took one claim or historically contentious idea and developed it in the light of various sources to come to a position, I was a little surprised when I reached Chapter 20, 'Galileo's misstatements about Copernicus', to find that Rosen was dealing with 5 distinct (if related) points at once – several of which he had elsewhere made the unitary focus of an article.

Rather than write my own abstract, I could here suggest a couplet of sentences from the text might have done the job,

"According to Galileo, (1) Copernicus was a priest; (2) he was called to Rome; (3) he wrote the Revolutions by order of the pope; (4) his book was never adversely criticised; (5) it was the basis of the Gregorian calendar. Actually, Copernicus was not a priest; he was not called to Rome; he did not write the Revolutions by order of the pope; the book received much adverse criticism, particularly on the ground that it contradicted the Bible; it was not the basis of the Gregorian calendar."

Rosen, 1958/1995, pp.203-204

I noticed that this was the earliest of Rosen's writings that had been included in the compilation – perhaps he had decided to dispense his ideas more sparingly after this paper?

Actually, there's a lot to be said for abstracts that pithily précise the key point of an article, a kind of tag-line perhaps, acting for a reader as an aide-mémoire (useful at least for readers like me who commonly stare at rows of books thinking 'I read something interesting about this, somewhere here…'). I have also read a lot of abstracts in research journals that would benefit from their own (further) abstracts, so perhaps such acute abstraction might catch on?

* Appendix: A scientific giant

Copernicus is indeed 'celebrated', being seen as one of the scientific greats who helped establish modern ways of thinking about the world – part of what is often perceived as a chain that goes Copernicus – Kelper – Galileo – Newton.

Copernicus is most famous for his book known in English as 'On the Revolutions of the Heavenly Spheres', or just 'Revolutions'. The key point of note is that at a time when it was almost universally agreed that the earth was stationary at the centre of 'the world', i.e., the cosmos, and that everything else revolved around the earth, Copernicus proposed a system that put the sun at the centre and had the earth moving around the sun.

The geocentric model of the cosmos was widely accepted for many centuries
(Image by OpenClipart-Vectors from Pixabay)

From our modern worldview, it is difficult to imagine just how, well yes, revolutionary, that move was (even if Copernicus only moved the centre of the universe from earth to the sun, so our solar system still had a very special status in his system). This is clear from how long it took the new view to become the accepted position, and the opposition it attracted. Newton later realised that strictly the centre of revolution was the centre of mass of the solar system not the sun per se. 2

One problem was that there was no absolute observational test to distinguish between the two models and there were well-established reasons to accept the conventional geocentric model (e.g., we do not feel the earth move, or a great wind as it spins beneath its atmosphere; as the most dense element earth would naturally fall to the centre of the world, beneath water, air, fire, and the ether that filled the heavens {although the Earth was not considered a pure form of the element earth, it was earthy, considered mostly earth in composition 3}; and scriptures, if given a literal interpretation, seemed to suggest the earth was fixed and the sun moved.)

Copernicus' model certainly had some advantages. If the earth is still, the distant sphere with all the fixed stars must be moving about it at an incredible rate of rotation. But if the earth spun on its axis, this stellar motion was just an illusion. 4 Moreover, if everything revolves around the earth, some of the planets behave very oddly, first moving one way, then slowing down to reverse direction ('retrograde' motion), before again heading off in their original sense. But, if the planets are orbiting the sun along with the earth (now itself seen as a planet) but at different rates then this motion can be explained as an optical illusion – "these phenomena…happen on account of the single motion of the earth" – the planets only seem to loop because of the motion of the earth.

Despite this clear improvement, Copernicus model did not entirely simplify the system as Copernicus retained the consensus view that the planets moved in circles: the planets' "motions are circular or compounded of several circles,…since only the circle can bring back the past". With such an assumption the observational data can only be made to fit (either to the heliocentric model or its geocentric alternative) by having a complex series of circles rather than one circle per planet. Today when we call the night sky 'the heavens', we are using the term without implying any supernatural association – but the space beyond the moon was once literally considered as heaven. In heaven everything is perfect, and the perfect shape is a circle.

It was only when Kepler later struggled to match the best observational data available (from his employer Tycho Brahe's observatory) to the Copernican model that, after a number of false starts, he decided to see if ellipses would fit – and he discovered how the system could be described in terms of planets each following a single elliptical path that almost repeated indefinitely.

A well-known story is how by the time Copernicus had finished his work and decided to get it printed he was near the end of his life, and he was supposedly only shown a printed copy brought from the printer as he lay on his deathbed (in 1543). In the printed copy of the book an anonymous foreword/preface 5 had been inserted to the effect that readers should consider the model proposed as a useful calculating system for following the paths of heavenly bodies, and not as a proposal for how the world actually was.

Despite this, the book was later added to the Roman Catholic Church's index of banned works awaiting correction. This only occurred much later – in 1616, after Galileo taught that Copernicus' system did describe the actual 'world system'. But, in the text itself Copernicus is clear that he is suggesting a model for how the world is – "to the best of my ability I have discussed the earth's revolution around the sun" – not just a scheme for calculating purposes. Indeed, he goes as far to suggest that where he uses language implying the sun moves this is only to be taken as adopting the everyday way of talking reflecting appearances (we say 'the sun rises'). For Copernicus, it was the earth, not the sun, that moved.

Sources cited:
  • Copernicus, N. (1543/1978). On the Revolutions of the Heavenly Spheres (E. Rosen, Trans.). Prometheus Books.
  • Rosen, E. (1995). Copernicus and his successors (E. Hilfstein, Ed.). The Hambledon Press.

Notes

1 I discovered from some 'internet research' (i.e., Googling) that Erna was a holocaust survivor, "[husband] Max and Erna, along with their families, were sent first to Płaszów, a slave-labor camp, and then on a death march to Auschwitz".

An article in the Jewish Standard reports how Erna's daughter undertook a charity bike ride "from Auschwitz-Birkenau, the Nazi-run death camp in the verdant Polish countryside, to the" Jewish Community Centre of Krakow (the town where her parents lived before being deported by the Nazis).

2 Newton also wrote as if the solar system was the centre of the cosmos, but of course the solar system is itself moving around the galaxy, which is moving away from most other galaxies…

3 These are not the chemical elements recognised today, of course, but were considered the elements for many centuries. Even today, people sometimes refer to the air and water as 'the elements.'

4 Traditionally, the 'heavenly spheres' were not the bodies such as planets, moons and stars but a set of eight conjectured concentric crystalline spheres that supposedly rotated around the earth carrying the distant stars, Saturn, Jupiter, Mars, the Sun, the Moon, Venus and Mercury.

5 A preface is written by the author of a book. A foreword is written by someone else for the author (perhaps saying how wonderful the author and the work are). Technically then this was a foreword, BUT because it was not signed, it would appear to be a preface – something written by Copernicus himself. Perhaps the foreword did actually protect the book from being banned as, until Galileo made it a matter of very public debate, it is likely only other astronomers had actually scrutinised the long and very technical text in any detail!