Category Archives: History of Astronomy

What Isaac actually asked the apple.

Yesterday on my twitter stream people were retweeting the following quote:

“Millions saw the apple fall, but Newton asked why.” —Bernard Baruch

For those who don’t know, Bernard Baruch was an American financier and presidential advisor. I can only assume that those who retweeted it did so because they believe that it is in some way significant. As a historian of science I find it is significant because it is fundamentally wrong in two different ways and because it perpetuates a false understanding of Newton’s apple story. For the purposes of this post I shall ignore the historical debate about the truth or falsity of the apple story, an interesting discussion of which you can read here in the comments, and just assume that it is true. I should however point out that in the story, as told by Newton to at least two different people, he was not hit on the head by the apple and he did not in a blinding flash of inspiration discover the inverse square law of gravity. Both of these commonly held beliefs are myths created in the centuries after Newton’s death.

Our quote above implies that of all the millions of people who saw apples, or any other objects for that matter, fall, Newton was the first or even perhaps the only one to ask why. This is of course complete and utter rubbish people have been asking why objects fall probably ever since the hominoid brain became capable of some sort of primitive thought. In the western world the answer to this question that was most widely accepted in the centuries before Newton was born was the one supplied by Aristotle. Aristotle thought that objects fall because it was in their nature to do so. They had a longing, desire, instinct or whatever you choose to call it to return to their natural resting place the earth. This is of course an animistic theory of matter attributing as it does some sort of spirit to matter to fulfil a desire.

Aristotle’s answer stems from his theory of the elements of matter that he inherited from Empedocles. According to this theory all matter on the earth consisted of varying mixtures of four elements: earth, water, fire and air. In an ideal world they would be totally separated, a sphere of earth enclosed in a sphere of water, enclosed in a sphere of air, which in turn was enclosed in a sphere of fire. Outside of the sphere of fire the heavens consisted of a fifth pure element, aether or as it became known in Latin the quintessence. In our world objects consist of mixtures of the four elements, which given the chance strive to return to their natural position in the scheme of things. Heavy objects, consisting as they do largely of earth and water, strive downwards towards the earth light objects such as smoke or fire strive upwards.

To understand what Isaac did ask the apple we have to take a brief look at the two thousand years between Aristotle and Newton.

Ignoring for a moment the Stoics, nobody really challenged the Aristotelian elemental theory, which is metaphysical in nature but over the centuries they did challenge his physical theory of movement. Before moving on we should point out that Aristotle said that vertical, upwards or downwards, movement on the earth was natural and all other movement was unnatural or violent, whereas in the heavens circular movement was natural.

Already in the sixth century CE John Philoponus began to question and criticise Aristotle’s physical laws of motion. An attitude that was taken up and extended by the Islamic scholars in the Middle Ages. Following the lead of their Islamic colleagues the so-called Paris physicists of the fourteenth century developed the impulse theory, which said that when an object was thrown the thrower imparted an impulse to the object which carried it through the air gradually being exhausted, until when spent the object fell to the ground. Slightly earlier their Oxford colleagues, the Calculatores of Merton College had in fact discovered Galileo’s mathematical law of fall: The two theories together providing a quasi-mathematical explanation of movement, at least here on the earth.

You might be wondering what all of this has to do with Isaac and his apple but you should have a little patience we will arrive in Grantham in due course.

In the sixteenth century various mathematicians such as Tartaglia and Benedetti extended the mathematical investigation of movement, the latter anticipating Galileo in almost all of his famous discoveries. At the beginning of the seventeenth century Simon Stevin and Galileo deepened these studies once more the latter developing very elegant experiments to demonstrate and confirm the laws of fall, which were later in the century confirmed by Riccioli. Meanwhile their contemporary Kepler was the first to replace the Aristotelian animistic concept of movement with one driven by a non-living force, even if it was not very clear what force is. During the seventeenth century others such as Beeckman, Descartes, Borelli and Huygens further developed Kepler’s concept of force, meanwhile banning Aristotle’s moving spirits out of their mechanistical philosophy. Galileo, Beeckman and Descartes replaced the medieval impulse theory with the theory of inertia, which says that objects in a vacuum will either remain at rest or continue to travel in a straight line unless acted upon by a force. Galileo, who still hung on the Greek concept of perfect circular motion, had problems with the straight-line bit but Beeckman and Descartes straightened him out. The theory of inertia was to become Newton’s first law of motion.

We have now finally arrived at that idyllic summer afternoon in Grantham in 1666, as the young Isaac Newton, home from university to avoid the plague, whilst lying in his mother’s garden contemplating the universe, as one does, chanced to see an apple falling from a tree. Newton didn’t ask why it fell, but set off on a much more interesting, complicated and fruitful line of speculation. Newton’s line of thought went something like this. If Descartes is right with his theory of inertia, in those days young Isaac was still a fan of the Gallic philosopher, then there must be some force pulling the moon down towards the earth and preventing it shooting off in a straight line at a tangent to its orbit. What if, he thought, the force that holds the moon in its orbit and the force that cause the apple to fall to the ground were one and the same? This frighteningly simple thought is the germ out of which Newton’s theory of universal gravity and his masterpiece the Principia grew. That growth taking several years and a lot of very hard work. No instant discoveries here.

Being somewhat of a mathematical genius, young Isaac did a quick back of an envelope calculation and see here his theory didn’t fit! They weren’t the same force at all! What had gone wrong? In fact there was nothing wrong with Newton’s theory at all but the figure that he had for the size of the earth was inaccurate enough to throw his calculations. As a side note, although the expression back of an envelope calculation is just a turn of phrase in Newton’s case it was often very near the truth. In Newton’s papers there are mathematical calculations scribbled on shopping lists, in the margins of letters, in fact on any and every available scrap of paper that happened to be in the moment at hand.

Newton didn’t forget his idea and later when he repeated those calculations with the brand new accurate figures for the size of the earth supplied by Picard he could indeed show that the chain of thought inspired by that tumbling apple had indeed been correct.



Filed under History of Astronomy, History of Mathematics, History of Physics, History of science, Myths of Science, Newton

Superlunar mutability does not imply heliocentricity!

Various people and organisations tweet historical scientific facts or events of the day, one of these is the Mathematical Association of America under the Twitter handle @maanow. Today they tweeted the following:

Tycho Brahe first observed a supernova in the constellation Cassiopeia. It provided important evidence to support the Copernican hypothesis.

Put quite simply the second statement is pure bullshit. Once again we have people confusing cosmology with mathematical astronomy. Aristotelian cosmology divided the cosmos into two spheres. The sublunar sphere, i.e. everything below the moon’s orbit around the earth was mutable, that is subject to change. The superlunar sphere, i.e. everything above the moon’s orbit, was immutable, that is unchanging.

In 1572 a stellar nova became visible from the earth. Cornelius Gemma, the son of Gemma Frisius, made the first recorded observation of it on 9 November. Tycho Brahe first saw it on 11 November. Cornelius, Tycho, and others all observed the nova and determined it to be superlunary, thereby signalling a change in the superlunar sphere contradicting Aristotelian cosmology. However this says absolutely nothing about the astronomical model of the cosmos.

Aristotle’s was not the only geocentric cosmology. Stoic cosmology, which was dominant in the later part of antiquity, rejected Aristotle’s two-sphere model for a cosmos that was homogenous and filled with pneuma. The Stoics who regarded comets as being superlunary also accepted change in the heavens, whilst propagating a geocentric astronomy. Stoic cosmology was experiencing a renaissance in the 16th century even before Tycho began his astronomical observations so the discovery that the nova was superlunary had no implication pro or contra for a heliocentric astronomical model.


Filed under History of Astronomy


Dean Burnett is a neuro-scientist who has a blog on Guardian Science Blogs, Brain Flapping, that is usually fairly humorous and mostly satirical.  However last week he decided to take a pot shot at astrology that was rather banal, displaying, as it does, Burnett’s total ignorance of astrology both past and present. Basic rule, if you wish to send something up successfully you should at least be well informed about that which you are trying to ridicule. As I am currently preparing a public lecture on the history of astrology[1], I was halfway tempted to let loose another of my diatribes about the significant role played by astrology not only in the history of science but also in the cultural, social and political histories of Europe in the last two thousand plus years but in the end I couldn’t be bothered. On that score you will have to content yourselves with this excellent old post by Rebekah “Becky” Higgitt (that’s Dr Higgitt to you son), on misinterpreting astrology, and my response to it here, with a closing coda from historian of astronomy Lester Ness here.

However I can’t resist analysing Dr Burnett’s opening paragraph, which provides the hook for his attack, from a historical perspective. He writes:

It’s worryingly common to see people, even professional organisations, accidently use the term astrology” when they mean “astronomy”. While phonetically similar, they are in fact very different things; astrology is an ancient system of divination, popular in several cultures, that makes predictions as to people’s personality and futures derived from the precise arrangement of planets, stars and other stellar bodies, whereas astronomy is a real thing that actually happens. 

Now, if Dr Burnett were a little better acquainted with the histories of astronomy and astrology then he would be aware that the distinction he is so keen to impress upon his readers was for the majority of the last two thousand years by no means as clear cut as he seems to think.

Both terms go back of course to the ancient Greek words astrologia and astronomia. In the BCE period of Greek history they only used the one term astrologia, which means, “telling of the stars” and was originally applied to what we now refer to as astronomy, which the Greeks started developing around 500 BCE. As the Greek then started to develop, what we term, astrology from its Babylonian roots in the first and second centuries BCE they also referred to it as astrologia, because as Ptolemaeus wrote in the introduction of his astrology textbook, the Tetrabiblos, in the middle of the second century CE, the study of the heavens consists of two parts the first of which determines the movements of the heavenly bodies, which he had dealt with in his Syntaxis Mathematiké (Almagest), and the second of which determines the changes that these movements bring out, the subject of the Tetrabiblos. Two aspects of one discipline so only one name is necessary. Ptolemaeus used the name astronomia, which means, “star arrangement”, and was introduced in the first century CE specifically to describe what we call astrology! From this point onwards both terms, astrologia and astronomia, were used interchangeably to mean both astronomy and astrology.

The first person to differentiate between astrologia and astronomia was the seven-century CE Archbishop of Seville and encyclopaedist, Isidore, who used the terms as we do today, astronomia for our astronomy and astrologia for our astrology.  Although Isidore’s encyclopaedia, the Etymologiae, was highly influential throughout the middle ages his linguistic innovation in the study of the heavens didn’t catch on and writers continued to refer to both disciplines with either term, indiscriminately.  In fact you can find texts where the author will refer to astronomy as astronomia in one sentence only to call it astrologia a couple of paragraphs further on. Just to confuse the issue even further the three terms astrologus, astronomus and mathematicus are used totally synonymously from antiquity up to the seventeenth century. In a famous passage from Augustinus, where he seems to be condemning mathematics, because he warns Christians to beware of the mathematici, he is actually condemning astrology, which he thought worked because it was fuelled by evil.

Even in the seventeenth century, in the thick of the so-called Scientific Revolution, you can still find writers referring to astronomers in an astronomical context as astrologers. It was only in the eighteenth century that the two terms came to have the distinct meanings that we still use today, somewhat ironically, basically the wrong way round as astrologia is in terms of its origins the science and astronomia the system of divination. So should you make the, according to Dr Burnett unforgivable, error of confusing astrology and astronomy sometime in the future at least you now know that history is on your side.

[1] Astrologie: Wissenschaft oder Hokus Pokus? That’s Astrology: Science or Hocus Pocus? (Planetarium, Nürnberg 15 January 2014 7:00 pm). You are welcome to attend if you happen to be in the area.


Filed under History of Astrology, History of Astronomy

Counting the hours

My #histsci-soul-sisterTM, Rebekah “Becky” Higgitt, wrote a charming post on her H-Word Blog to mark the end of European summer time describing the mad scheme of a certain William Willett to introduce the time change in twenty minute increments over several weeks. This reminded me of a local time phenomenon that I’ve not yet blogged about, Der Große Nürnberger Uhr

The time taken for the earth to rotate once upon its axis or for the sun to appear to circle the earth (its irrelevant how you view it) is a given but how one then chooses to divide up this period into smaller, easier to handle units is purely arbitrary. We owe our twenty-four hour day to the ancient Egyptians. They marked the passing of time in the night by the raising of stars; twelve stars being allotted for any given night, thus dividing the night into twelve units. They being normally decimal in their thinking divided the day into ten units. Allotting one unit for twilight at each junction between day and night brought the total to twenty-four.

The ancient Greek astronomers took over the Egyptian solar calendar and their twenty-four hour day dividing the diurnal revolution into twenty-four equally long, or equinoctial, hours as we do now. However most cultures who adopted the twenty-four system before the early modern period divided the night and day each into twelve units producing hours that varied in length depending on the time of year. This variation got larger the further away from the equator the culture was. In the middle of summer daytime hours were very long and night-time ones very short and vice versa in the middle of winter.

Beginning in the fourteenth century the city state of Nürnberg introduced a system of dividing up the day that is a sort of halfway station between the unequal hours of the middle ages and equinoctial hours, the so called ‘Große Uhr’, in English ‘Large Clock’. In this system the number of hours allotted to the day and night changed approximately every three weeks, the number of daytime hours increasing from midwinter (8) to midsummer (16) and then decreasing from midsummer to midwinter. The number of night-time hours doing the opposite.

Date of change 1st half of year Daylight hours Night-time hours Date of change 2nd half of year



7 January



16 November

28 January



26 October

14 February



8 October

3 March



22 September

19 March



5 September

5 April



20 August

23 April



2 August

15 May



11 June

In 1506 the Nürnberger humanists created one of the most complicated sundials in the whole of Europe on the wall of the St Lorenz church in the city.

Sundial on the St Lorenz Church Nürnberg

Sundial on the St Lorenz Church Nürnberg

This sundial shows the time of day in various different variations of hours including of course the Large Nürnberg Clock

The definition on this picture is not good enough to say which lines are which.

The good citizens of Nürnberg continued to use their own unique way of counting the hours right down to the year 1811.


Filed under History of Astronomy, History of science, Mediaeval Science, Renaissance Science

The speed of light, a spin off from longitude research.

When I was growing up in the dim and distant twentieth century spin-off was one of the most frequently used buzz words in the public discussion of science and technology; a spin-off being an unintended and unexpected positive product of scientific or technological research. Politicians would use the term to justify high levels of expenditure on political prestige projects claiming that the voters/taxpayers would benefit through the spin-offs from the research. The example that was almost always quoted by the media was that the non-stick coating for frying pans was a spin-off from the space programme. This is, like many popular stories in the history of science and technology, actually a myth but that is not the subject of this post.

Now spin-offs are not a modern phenomenon but have been turning up ever since humans first began hammering bits of stone to make tools and as the title of this post suggests, the first successful scientific determination of the speed of light was actually a spin-off from a project to find a more accurate way to determine longitude.

The speed of light had been a problem since at least the beginning of the ancient Greek study of optics. The Greeks themselves were split into two camps on the subject. Some like Empedocles, on whose shoulders the beginnings of much of Greek science rests, thought that the speed of light was finite arguing that light was something in motion and therefore required time to travel. Heron of Alexandria, a representative of the geometrical school of optics, thought that the transmission of light was spontaneous, the speed thus infinite, because extremely distant objects such as stars appear instantly when we open our eyes. Down through the ages those writing on optics took one side or the other in the argument. In the seventeenth century two of the most important optical experts Kepler and Descartes both argued for an infinite speed of light . Galileo and Isaac Beeckman, who both thought the speed of light was finite, proposed, and may have carried out, experiments to try and determine the speed of light but were of course defeated by its actually extremely high velocity and their very, very primitive timing devices. The actual solution came from the astronomers and it was Galileo who unwittingly set the ball rolling. [Modified 26.09.2013 I done screwed up again! See comments]

In 1610 Galileo and Simon Marius both discovered the four largest, or Galilean, Moons of Jupiter, Io, Europa, Callisto and Ganymede. The orbits of the four are all relatively short and they disappear and reappear from behind Jupiter in a complex but regular dance. Galileo realised that if one could determine the orbits accurately enough then one could use these disappearances and reappearances (eclipses of the moons) as an astronomical clock in order to determine longitude. One would need to create an accurate table of the time of the eclipses for a given prime meridian then in order to determine the longitude of a given point the cartographer-astronomer only needs to determine the local time of the occurrence of one of the eclipses look in his tables to calculate the time difference and thus the longitude difference to his prime meridian. Having thought up this, actually quite brilliant, idea Galileo, never one to pass up a chance to shine and at the same time earn a fast buck, tried to sell it first to Spain and then to Holland; an interesting combination as the two countries were at war with each other at the time. Both sales pitches failed and Galileo never actually produced the necessary tables. Fast-forward about fifty years to Paris.

Ensconced in the new observatory in Paris and equipped with far superior telescopes to those of Galileo, Europe’s star astronomer, Giovanni Domenico (Jean-Dominique) Cassini took up the task abandoned by Galileo and produced the necessary tables to a high enough degree of accuracy to enable the French astronomer-cartographers to accurately determine longitude. (I should point out that this method is impractical at sea as the accurate telescopic observation of the moons of Jupiter on a rolling ship is well-nigh impossible). This is a pan European story. We started in Germany and Northern Italy then moving on to Paris we now take a short diversion to Denmark to meet Ole Rømer.

Ole Rømer by Jacob Coning c.1700

Ole Rømer
by Jacob Coning c.1700

Ole Rømer was born in Århus on the 25th September 1644. In 1662 he started studying at the University of Copenhagen under the mathematician and physician Rasmus Bartholin. In 1671 the French astronomer-cartographer Jean Picard went to Demark to accurately re-measure the latitude and longitude of Tycho Brahe’s observatory on the island of Hven, using the moons of Jupiter, in order to better integrate Tycho’s observation into those made by the observatory in Paris. Picard took Rømer with him to Hven as an assistant. Much impressed by the young Dane Picard offered to take him back with him to Paris. Given the chance of working at the world’s leading centre for astronomical research at that time, Rømer didn’t hesitate, packed his bags and was soon installed as an assistant to Cassini at the Paris Observatory.

A problem had turned up in the eclipse tables for the moons of Jupiter and Rømer took part in the observation programme to try and determine where the error lay. His observations showed that the period between the eclipse of Io got shorter as Earth got closer to Jupiter and longer as Earth moved away. Over a period of eight years Rømer observed and accurately calculated the delay in the eclipse time, which were in fact due to the finite speed of light and the differences in the distance that the light from Io must travel depending on the relative positions of Earth and Jupiter. On the assumption that this was indeed the cause and that the speed of light was finite Christiaan Huygens calculated it from Rømer’s figures producing the first ever scientific calculation of the speed of light. The figure at about 200 000 km per sec is too low and was not universally accepted as many still believed that the speed of light was infinite. The matter was finally settled as James Bradley discovered stellar aberration in the 1720’s and used it to calculate a more accurate figure.

Rømer returned to Copenhagen in 1681 as professor of astronomy at the university, where he made further minor contribution to the sciences. However he’ll always be chiefly remembered as the man who first determined that the speed of light is finite and produced a measure of that speed.


Filed under History of Astronomy, History of Cartography, History of science

A great read

Mathematician and sci-fi author Michael Flynn has put up a great multi-episode post on his blog The TOF Spot. The whole thing is titled The Great Ptolemaic Smackdown and up till now has six glorious parts (it is to be continued!). It covers the historical transition from geocentrism to heliocentrism centred around the life and time of our favourite villain Galileo Galilei. It is erudite, informative, on the whole historically accurate, beautifully illustrated, entertaining, highly amusing and totally irreverent. If you like the Renaissance Mathematicus style of history of science snark then you’ll love Mr Flynn’s treatment of the topic.

I urge all of my readers to pop on over and take your fill of the #histsci wit and wisdom of the OFloinn, as Mr Flynn coins himself.

1) The Great Ptolemaic Smackdown

2) The Great Ptolemaic Smackdown: Down for the Count

3) The Great Ptolemaic Smackdown: The Great Galileo-Scheiner Flame War of 1611-13

4) The Great Ptolemaic Smackdown: The Down ‘n Dirty Mud Wrassle

5) The Great Ptolemaic Smackdown: Here’s Mud in Yer Eye

6) The Great Ptolemaic Smackdown: Comet Creation

Disclosure: I might just possibly be accused of bias in recommending the OFloinn’s majestic take on seventeenth century astronomy history as he references about a dozen of my deathless blog posts on various aspects of the subject and has this to say about The Renaissance Mathematicus, A treasure trove!  Some items used above. However I can assure my readers that I’m totally immune to flattery (and Michael your check’s in the post).


Filed under History of Astronomy

Counting the days.

All calendars are conventions and a compromise. The natural measures of time the day, the (lunar) month and the year are incommensurable, a long mathematical word meaning you can’t measure the one with the other without a bit left over.  In fact the lengths of those three natural measuring units vary according to how you measure them. Any calendar is an attempt to some how combine those units for practical purposes. The end result is that the ancient world had a wonderful patchwork of lunar calendars, lunar-solar calendars and solar calendars making the comparison of dates between different cultures an arithmetical nightmare.

This turned out to be a major problem for Greek astronomers when they first started to develop their version of mathematical astronomy. Greece itself was at the time a conglomerate of small states each with its own lunar calendar, the Egyptians used a pure solar calendar, the Babylonians used a lunar-solar calendar and everybody else had their own ways of recording time. The Greek astronomers drew their observational data from numerous sources throughout the ancient world and in order to able to use it to make usable mathematical models they somehow needed to reduce all the observations to a common time base. The base that they chose was the Egyptian solar calendar without leap years. This calendar had twelve thirty day months plus five extra days at the end of the year making a three hundred and sixty-five day year. This year slips about one day against the natural or solar year every four years, which however doesn’t really matter for mathematical calculations it being just a mechanism for counting off blocks of 365 days. This became the standard method of recording astronomical observations and events from sometime around the second century BCE up to and beyond Copernicus in the fifteenth century who used this theoretical Egyptian calendar in his De revolutionibus.

Of course this problem didn’t go away with the development of modern astronomy in the Early Modern Period and in the nineteenth century John Herschel introduced the Julian Day Count, which unifies all astronomical observations and events into a single day count starting from twelve midday on 1 January 4713 BCE. As I sit writing this post on 5 August 2013 it is 2456510 JD (Julian Day). To make life simpler for astronomers there are various modified Julian Day systems that chop off the first 240 000 or more days. However the most obvious question is why does the original Julian Day count begin in 4713 BCE?

To answer this question we have to go back to the Early Modern Period and the Bible chronologists. Now the Bible chronologists are a favourite target for biting sarcasm of the Gnu Model Army and other champions of scientism. As I have pointed out in the past in a post about James Ussher the Bible chronologists were being perfectly rational and scientific in the terms of their own period. I also wrote a post pointing out that Isaac Newton the hero of all defenders of scientism was also a Bible chronologist. In both of these posts I emphasised that Bible chronology played a significant role in the development of modern academic history providing a rational historical timeframe for study. A Bible chronologist who did much to modernise the study of history in chronological terms was Joseph Justus Scaliger to whom we owe the starting point of the Julian Day count and who was born in Agen in France on 5 August 1540.

Joseph Justus Scaliger

Joseph Justus Scaliger

Scaliger the son of Italian physician and Aristotelian scholar Julius Caesar Scaliger (1484 – 1558)

Julius Caesar Scaliger

Julius Caesar Scaliger

was a theologian, historian, jurist and philologist of vast intellect who was regarded as one of the leading scholars of Europe in his own lifetime. In his historical work he extended the study of ancient history beyond the Greeks and Romans to include Persian, Babylonian, Egyptian and Jewish history. It was in this study that he developed his concept of the Julian Period in order to be able to compare events between the cultures on a unified time scale.

Scaliger defined something he called the Julian Period of 7980 years, which was a product of three different dating cycles: The 19-year Meton cycle used in lunar-solar calendars, such as the Jewish one, the 28-year solar cycle, which is the number of years it takes for days of the week to repeat in the Julian calendar and the 15-year indication indiction cycle used for dating medieval manuscripts. Working backwards 4731 BCE is the last time that all three cycles were in their respective first years. Scaliger chose this date because it preceded his own calculated date for creation, 3949 BCE. With a unified scale Scaliger could give a Julian year number to any historical event within his researches, thus making cross-cultural comparisons possible. It’s sometimes falsely claimed that Scaliger named his epoch after his father, however this is not true. He stated quite clearly that it is named after the Julian calendar, as his years are Julian years and not Gregorian ones.

John Herschel justified his adoption of Scaliger’s system in his Outline of Astronomy (1848) as follows:

The period thus arising of 7980 Julian years, is called the Julian period, and it has been found so useful, that the most competent authorities have not hesitated to declare that, through its employment, light and order were first introduced into chronology. We owe its invention or revival to Joseph Scaliger, who is said to have received it from the Greeks of Constantinople. The first year of the current Julian period, or that of which the number in each of the three subordinate cycles is 1, was the year 4713 BC, and the noon of the 1st of January of that year, for the meridian of Alexandria, is the chronological epoch, to which all historical eras are most readily and intelligibly referred, by computing the number of integer days intervening between that epoch and the noon (for Alexandria) of the day, which is reckoned to be the first of the particular era in question. The meridian of Alexandria is chosen as that to which Ptolemy refers the commencement of the era of Nabonassar, the basis of all his calculations.

For astronomical purposes Herschel changed the count from years to days and by the end of the nineteenth century it had become universal as a dating method for astronomers and is still in use today. Modern astronomy dates all astronomical events using the Julian Day Count a system of dating that was first conceived by a Renaissance historian and Bible chronologist.


Filed under History of Astronomy, Renaissance Science

Apelles hiding behind the painting

Todays post combines two oft-repeated themes here at the Renaissance Mathematicus. We have another highly significant researcher who, however, does not belong to the pantheon of seventeenth century scientific gods and therefore tends to remain unknown to the non-experts. Our subject is also one more refutation of the widely held belief that in the seventeenth century Catholics in general and the Jesuits in particular were anti-science. Today we will take a look at the first man to be involved in a major scientific dispute with Galileo following the invention of the telescope the Jesuit mathematician, astronomer and optical physicist Christoph Scheiner, who was born in the town of Markt Wald in Bavarian Swabia on 25th July (NS) in either 1573 or 1575.

Christoph Scheinet (artist unknown)

Christoph Scheinet (artist unknown)

Scheiner attended a Jesuit Latin school in Augsburg from 1591 to 1595 before entering the Jesuit Order in Landsberg am Lech in 1595. He was educated in Landsberg am Lech, Augsburg and Ingolstadt obtaining his doctorate in theology, the normal higher degree for a Jesuit, in Ingolstadt in 1609. Over the years he had held various teaching post in Bavaria and in 1610 he was appointed professor of mathematics in Ingolstadt. The chair for mathematics in Ingolstadt was the oldest in Germany and Scheiner had several illustrious predecessors including Peter Apian. Scheiner is notorious amongst historians of astronomy for his more than somewhat heated dispute with Galileo on the nature of the sunspots and who had the honour of having discovered them first. However this was not his only scientific achievement.

One thing that I personally find fascinating is that Scheiner invented the pantograph in 1603 whilst still a student a fact that illustrates his ingenuity as an instrument maker. Although he first published his invention, which he had however already displayed publically on numerous occasions, in 1631. Now, if I had been writing this in the nineteen sixties or seventies I could be fairly certain that all of my readers would know what a pantograph is, however in this age of computer graphics where the most amazing graphical transformations can be accomplished by the most ham-fisted of us with the simple click of a mouse the pantograph like many other drawing instruments has almost disappeared from public awareness.

Scheiner's Pantograph

Scheiner’s Pantograph

A pantograph is an instrument for copying pictures and diagrams either one to one or scaled up or down. On the apparatus pictured above the three points at the bottom are the operative parts of the instrument. The instrument is anchored with the point on the left (XN) whilst the original drawing is traced out with the other hand with the middle point (MH) steering (OZ). The point on the right (PT) thus draws the required copy. By adjusting the length of the arms one can determine whether the copy is one to one, enlarged or reduced.

The thing for which Scheiner should be most well known, but he isn’t, is his work on the optical function of the eye Oculus hoc est: Fundamentum opticum, which he published in 1609 1619. Although there were many competing theories of vision in antiquity the majority opinion was that the image was formed in the crystal lens at the front of the eye. Even Ibn al-Haytham who laid the foundation of modern optics in the eleventh century CE with his synthesis of ancient Greek optical theory believed that the image was formed in the lens. In 1604 in his Astronomiae Pars Optica Johannes Kepler made a radical change in that he claimed that the lens is just that, a lens, and the image in formed, as in a camera obscura (a name he coined), on the retina at the back of the eye. This is, as we now know, the correct solution. However Kepler’s suggestion did not go down particularly well as any such image would of course be inverted; we would see everything upside down! Kepler’s ingenious and correct solution to this problem was to suggest that the brain copes with this situation telling the viewer which side in up. Now Kepler did not offer any form of empirical evidence for his theories and so they remained more than somewhat disputed; enter Scheiner who delivered the empirical proof that Kepler was right and quite a bit more.

Scheiner took bull’s eyes, the real things and not the middle of archery targets or darts boards. And pared them down so that he could see through the retina from outside of the eye. He then projected an image through the lens and could observe its inverted formation on the retina, Kepler vindicated. Unfortunately for Scheiner in most popular presentations of the history of optics this achievement is almost always wrongly attributed to Descartes. Scheiner’s work on vision contains many other important discoveries on the physiology of the eye making him alongside Kepler, Descartes, Gregory and Huygens to one of the important optical researchers of the seventeenth century.

Scheiner's Optics

Scheiner’s Optics

As already stated above Scheiner is most well know for his sunspots dispute with Galileo, which rumbled on for more than twenty years after its initial very heated phase between 1611 and 1613. Scheiner and Galileo both began observing sunspots with a telescope around the same time at the beginning of 1611, although Galileo would later claim he had begun earlier in order to claim priority, whereas unknown to both of them the priority in observation went to Thomas Harriot in England and the priority in publication to Johannes Fabricius in Frisia. The Augsburger businessman and astronomy enthusiast Marcus Welser published two sets of three letters from Scheiner on his sunspot observations under the pseudonym Apelles latenspost tabulam a reference to the fourth century Greek painter Apelles who is said to have displayed his paintings in his shop window whilst hiding behind them to listen to any criticism that might fall from the viewers, hence the title of this post. Scheiner hypothesised that the sunspots were actually small satellites orbiting the sun and casting their shadows on its surface, a theory that preserved the Aristotelian perfection of the heavens. Galileo countered with three letters of his own written in Tuscan and published by the Accademia dei Lincei in which he proved by careful observation and mathematical analysis that the sunspots must be on the surface of the sun, something that Scheiner then came to accept. Welser had declined to publish Galileo’s letters partially because of the gratuitous insults against Scheiner that they contained, some of which even the Lincei removed.

Later in his dispute with another Jesuit Orazio Grassi over the nature of comets, which would eventually lead to the publication of his Il Saggiatore (The Assayer), Galileo even went as far as to falsely accuse Scheiner of plagiarism. Unlike Galileo who basically abandoned his telescopic astronomy in 1613 Scheiner went on to make an extensive telescopic study of the sun, which he published as Rosa Ursina in 1630.

Scheiner Observing the Sun

Scheiner Observing the Sun

In the first of the four books of his masterwork Scheiner very carefully detailed the sunspot observations made by Galileo and himself correctly attributing to each his discoveries, something that did not please Galileo at all. Galileo’s supporters rubbished the Rosa Ursina upon publication and the book being a rather turgid scientific study was a flop although it is without doubt one of the most important astronomical works published in the seventeenth century; it just wasn’t very readable!

Scheiner's Sunspot Observations

Scheiner’s Sunspot Observations

In his infamous Dialogo Galileo compounded his sins again gratuitously insulting Scheiner whilst knowingly claiming several of Scheiner’s discoveries as his own. Galileo was not a nice man.

Initially Scheiner had built his own thirty power Dutch telescope and had, like Harriot, observed the sun at dawn through low clouds. Later at the suggestion of his assistant Johann Baptist Cysat, another Jesuit astronomer of some importance, he began to use filters of coloured glass so as not to damage his eyes. Scheiner was the first astronomer to construct and use Keplerian telescopes, two convex lenses, rather than Dutch or Galilean ones, one convex and one concave lens, for his observations, something that Galileo also criticised him for. The Keplerian or astronomical telescope is in fact superior but more susceptible to spherical aberration. It would be another twenty years before other astronomers, largely influenced by Galileo, followed Scheiner’s lead. For his solar observations Scheiner developed his helioscope a Keplerian telescope mounted on a sophisticated stand, which projected the image of the sun on to a screen.

Scheiner's Helioscope

Scheiner’s Helioscope

Scheiner was a first class astronomer and optician who suffered much under the arrogance and spitefulness of Galileo and who deserves to be much better known than he is.


Filed under History of Astronomy, History of Optics, Renaissance Science

Getting the measure of the earth.

It is generally accepted that the Pythagoreans in the sixth century BCE were the first to recognise and accept that the earth is a sphere. It is also a historical fact that since Aristotle in the fourth century BCE nobody of any significance in the western world has doubted this fact. Of course recognising and accepting that the earth is a sphere immediately prompts other questions, one of the first being just how big a sphere is it? Now if I want to find the circumference of a cricket ball (American readers please read that as a baseball ball) I whip out my handy tape measure loop it around the ball and read of the resulting answer, between 224mm and 229mm (circa 230mm for American readers). In theory I could take an extremely long tape measure and following a meridian (that is a great circle around the globe through the north and south poles) loop it around the globe reading off the answer as before, 40,007.86 km (radius) 6356.8km according to Wikipedia. However it doesn’t require much imagination to realise the impracticality of this suggestion; another method needs to be found.

Famously, the earliest known scientific measurement of the polar circumference was carried out by Eratosthenes in the third century BCE. Eratosthenes measured the elevation of the sun at midday on the summer solstice in one of two cities that he thought to be on the same meridian knowing that the sun was directly overhead at the other.  Knowing the distance between the cities it is a relative simple trigonometrical calculation to determine the polar circumference. Don’t you love it when mathematicians say that a calculation is simple!  He achieved an answer of 250 000 stadia but as there were various stadia in use in the ancient world at this time and we don’t know to which stadia he was referring we don’t actually know how accurate his measure was.

Other astronomers in antiquity used the more common method of measuring a given distance on a meridian, determining the latitude of the ends and again using a fairly simple trigonometrical calculation determining the polar circumference. This method proved to be highly inaccurate because of the difficulties of accurately measuring a suitably long, straight north south section of a meridian. The errors incurred leading to large variations in the final circumference determined.

In the eleventh century CE the Persian scholar al-Biruni developed a new method of determining the earth’s circumference. He first measured the height of a suitable mountain, using another of those simple trigonometrical calculations, then climbing the mountain measuring the angle of dip of the horizon. These measurements were followed by, you’ve guessed it, yet another simple trigonometrical calculation to determine the circumference. Various sources credit al-Biruni with an incredibly accurate result from his measurements, which is to be seriously doubted. For various reasons it is almost impossible to accurately determine the angle of dip and the method whilst theoretically interesting is in practice next to useless.

In the Renaissance Gemma Frisius’ invention of triangulation in 1533 provided a new method of accurately measuring a suitably long north south section of a meridian. The first to apply this method to determine the length of one degree of meridian arc was the Dutch mathematician Willibrord Snel, as described in his Eratosthenes Batavus (The Dutch Eratosthenes) published in 1617. He measured a chain of triangles between Alkmaar and Bergen op Zoom and determined one degree of meridian arc to be 107.395km about 4km shorter than the actual value. Snel’s measurement initially had little impact but it inspired one that was to become highly significant.

The meridian arc measurement inspired by Snel was carried out by the French astronomer Jean-Félix Picard who was born the son of a bookseller, also called Jean, in La Flèche on 21st July 1620.  As is unfortunately all too often the case for mathematicians in the Early Modern Period we know very little about Picard’s background or childhood but we do know that he went to school at the Jesuit College in La Flèche where he benefited from the Clavius mathematical programme as did other La Flèche students such as Descartes, Mersenne and Gassendi. Picard left La Flèche around 1644 and moved to Paris where he became a student of Gassendi then professor for mathematics at the Collège Royal. Whether he was ever formally Gassendi’s student is not known but he certainly assisted in his astronomical observations in the 1640s. In 1648 Picard left Paris for health reasons but in 1655 he returned as Gassendi’s successor at the Collège Royal; an appointment based purely on his reputation, as he had published nothing at this point in his life.

Jean Picard (artist unknown)

Jean Picard (artist unknown)

In 1665 Jean-Baptist Colbert became finance minister of France and began to pursue an aggressive science policy.

Colbert 1666 Philippe de Champaigne

Colbert 1666 Philippe de Champaigne

He established the Académie des sciences in 1666 modelled on its English counterpart the Royal Society but unlike Charles who gave his scientists no financial support Colbert supplied his academicians, of whom Picard was one, with generous salaries. It was also Colbert who motivated his academicians to produce a new, modern, accurate map of France and this was when Picard became a geodesist and cartographer.

Following the methods laid down by Snel Picard made the first measurement of what is now the legendary Paris meridian, which would a hundred years later in extended form become the basis of the metre and thus the metric system. He first measured an eleven-kilometre base line south of Paris between Villejuif and Juvisy across what is now Orly airport using standardised wooden measuring rods.

Southern end of Picard's baseline

Southern end of Picard’s baseline


Northern end of Picard's baseline

Northern end of Picard’s baseline

The straight path that he created became the Avenue de Paris in Villejuif and later the Route National 7 through Orly to Juvisy. From this baseline Picard triangulated northwards through Paris and a little further south. For his triangulation Picard used a theodolite whose sighting telescope was fitted with cross hair. The first ever use of such an instrument. Picard determined one degree of meridian arc to be 110.46 km making the polar radius 6328.9 km.

Picard's triangulation and his instruments

Picard’s triangulation and his instruments

Whilst Picard was out in the field measuring triangles Colbert was hiring Giovanni Domenico Cassini away from Bologna to work in the newly constructed Paris Observatory in what was probably to most expensive scientific transfer deal in the seventeenth century.

Giovanni Cassini (artist unknown)

Giovanni Cassini (artist unknown)

Following the success of Picard’s meridian triangulation he set about using the skills he had developed to map the coastline of France together with Cassini and Philippe de La Hire. The results of their endeavours greatly reduced the presumed size of France provoking Louise XIV’s famous quip that he had lost more territory to the cartographers than he had ever lost to his enemies.

Map showing both old and new French coastlines

Map showing both old and new French coastlines

Picard now began preparations for the accurate mapping of France but died before the project could begin. Cassini took up the reins and the mapping of France became a Cassini family project stretching over four generations.

Picard’s determination of the size of the earth would go on to play a significant role in the history of physics. In 1666 when the young Isaac Newton first got an inkling of the concept of a universal gravity he asked himself if the force that causes an object to fall to the ground (that infamous apple) is the same force that prevents the moon from shooting off at a tangent to its orbit, which it should do according to the law of inertia. Young Newton did a quick calculation on the back of an envelope and determined that it wasn’t.  As we now know they are of course the same force so what went wrong with the young Isaac’s calculations? The size of the earth that he had used in his calculation had been wrong. In the 1680s when Newton returned to the subject and redid his calculation he now took Picard’s value and discovered that his original assumption had indeed been correct. In his Principia Newton uses Picard’s value with acknowledgement.

The difference between Picard’s value for one degree of meridian arc and that determined by Snel led Cassini and his son to hypothesise that the earth is a prolate spheroid (lemon shaped) whereas Newton and Huygens had hypothesised that it is an oblate spheroid (orange shaped) a dispute that I’ve blogged about in the past.


Picard made other important contributions to astronomy and physics and it’s a little bit sad that that today when people hear or read the name Jean Picard they think of a character in a TV science fiction series and not a seventeenth century French astronomer.

[The photos showing the monuments marking the ends of Picard's baseline are taken from Paul Murdin, Full Meridian of Glory: Perilous Adventures in the Competition to Measure the Earth, Copernicus Books, 2009]


Filed under History of Astronomy, History of Cartography, History of Physics, History of science, Uncategorized

“If he had lived, we might have known something”

The title of this post is Newton’s rather surprising comment on hearing of the early death of the Cambridge mathematician Roger Cotes at the age of 33 in 1716. I say rather surprising, as Newton was not known for paying compliments to his mathematical colleagues, rather the opposite. Newton’s compliment is a good measure of the extraordinary mathematical talents of his deceased associate.

Cotes the son of rector from Burbage in Leicestershire born 10th July 1682 is a good subject for this blog for at least three different reasons. Firstly he is like many of the mathematicians portrayed here relatively obscure although he made a couple of significant contributions to the history of science. Secondly one of those contributions, which I’ll explain below, is a good demonstration that Newton was not a ‘lone’ genius, as he is all too often presented. Lastly he just scraped past the fate of Thomas Harriot, of being forgotten, having published almost nothing during his all too brief life, he had the luck that his mathematical papers were edited and published shortly after his death by his cousin Robert Smith thus ensuring that he wasn’t forgotten, at least by the mathematical community.

Cotes was recognised as a mathematical prodigy before he was twelve years old. He was taken under the wing of his uncle, Robert Smith’s father, and sent to St Paul’s School in London from whence he proceeded to Trinity College Cambridge, Newton’s college, in 1699. Newton whilst still nominally Lucasian Professor had already departed for London and the Royal Mint. Cotes graduated BA in 1702. Was elected minor fellow in 1705 and major fellow in 1706 the same year he graduated MA. His mathematical talent was recognised on all sides and in the same year he was nominated, as the first Plumian Professor of Astronomy and Natural Philosophy, still not 23 years old. However he was only elected to this position on 16th October 1707. It should be noted that the newly created Plumian Chair was only the second chair for the mathematical sciences in Cambridge following the creation of the Lucasian Chair in 1663. In comparison, for example, Krakow University in Poland, the first humanist university outside of Northern Italy, already had two dedicated chairs for the mathematical sciences in the middle of the sixteenth century. This illustrates how much England was lagging behind the continent in its promotion of the mathematical sciences in the Early Modern Period.

Cotes election to the Plumian chair was supported by Richard Bentley, Master of Trinity, and by William Whiston, in the meantime Newton’s successor as Lucasian professor, who claimed to be “a child to Mr Cotes” in mathematics but was opposed by John Flamsteed, the Astronomer Royal, who wanted his former assistant John Witty to be appointed. In the end Flamsteed would be proven right, as Cotes was shown to be a more than somewhat mediocre astronomer.

Cotes’ principle claim to fame is closely connected to Newton and his magnum opus the Principia. Newton gave the task of publishing a second edition of his masterpiece to Richard Bentley, who now took on the role filled by Edmond Halley with the first edition. Now Bentley who was a child prodigy, a brilliant linguist and a groundbreaking philologist was anything but a mathematician and he delegated the task of correcting the Principia to his protégé, Cotes in 1709. Newton by now an old man and no longer particularly interested in mathematical physics had intended that the second edition should basically be a reprint of the first with a few minor cosmetic corrections. Cotes was of a different opinion and succeeded in waking the older man’s pride and convincing him to undertake a complete and thorough revision of the complete work. This task would occupy Cotes for the next four years. As well as completely reworking important aspects of Books II and III this revision produced two highly significant documents in the history of science, Newton’s General Scholium at the end of Book III, a general conclusion missing from the first edition, and Cotes’ own preface to the book. Cotes’ preface starts with a comparison of the scientific methodologies of Aristotle, the supporters of the mechanical philosophy, where here Descartes and Leibniz are meant but not named, and Newton. He of course come down in favour of Newton’s approach and then proceeds to that which Newton has always avoided a discussion of the nature of gravity introducing into the debate, for the first time, the concept of action at a distance and gravity as a property of all bodies. The second edition of Principia can be regarded as the definitive edition and is very much a Newton Cotes co-production.

Cotes’ posthumously published mathematical papers contain a lot of very high class but also highly technical mathematics to which I’m not going to subject my readers. However there is one of his results that I think should be better known, as the credit for it goes to another. In fact a possible alternative title for this post would have been, “It’s not Euler’s Formula it’s Cotes’”.

It comes up fairly often that mathematicians and mathematical scientists are asked what their favourite theorem or formula is. Almost invariably the winner of such poles polls is what is known technically as Euler’s Identity

e + 1 = 0


Now this is just one result with x = π of Euler’s Formular:

cosx  + isinx = eix

Where i is the square root of -1, e is Euler’s Number the base of natural logarithms and x is an angle measured in radians. This formula can also be expressed as a natural logarithm thus:

ln[cosx + isinx] = ix

and it is in this form that it can be found in Cotes’ posthumous mathematical papers.

As one mathematics’ author expresses it:

This identity can be seen as an expression of the correspondence between circular and hyperbolic measures, between exponential and trigonometric measures, and between orthogonal and polar measures, not to mention between real and complex measures, all of which seemed to be within Cotes’ grasp.

Put simply, for the non-mathematical readers, this formula is one of the most important fundamental relationships in analysis.

Cotes died unexpectedly on 5th June 1716 of a, “Fever attended with a violent Diarrhoea and constant Delirium”. Despite his important contributions to Newton’s Principia Cotes is largely forgotten even by mathematicians and their ilk so the next time somebody waxes lyrical about Euler’s Formula you can gentle point out to them that it should actually be called Cotes’ Formula.



Filed under History of Astronomy, History of Mathematics, History of Physics, Newton