Category Archives: History of Physics

The emergence of modern astronomy – a complex mosaic: Part XXX

As stated earlier the predominant medieval view of the cosmos was an uneasy bundle of Aristotle’s cosmology, Ptolemaic astronomy, Aristotelian terrestrial mechanics, which was not Aristotle’s but had evolved out of it, and Aristotle’s celestial mechanics, which we will look at in a moment. As also pointed out earlier this was not a static view but one that was constantly being challenged from various other models. In the early seventeenth century the central problem was, having demolished nearly all of Aristotle’s cosmology and shown Ptolemaic astronomy to be defective, without however yet having found a totally convincing successor, to now find replacements for the terrestrial and celestial mechanics. We have looked at the development of the foundations for a new terrestrial mechanics and it is now time to turn to the problem of a new celestial mechanics. The first question we need to answer is what did Aristotle’s celestial mechanics look like and why was it no longer viable?

The homocentric astronomy in which everything in the heavens revolve around a single central point, the earth, in spheres was created by the mathematician and astronomer Eudoxus of Cnidus (c. 390–c. 337 BCE), a contemporary and student of Plato (c. 428/27–348/47 BCE), who assigned a total of twenty-seven spheres to his system. Callippus (c. 370–c. 300 BCE) a student of Eudoxus added another seven spheres. Aristotle (384–322 BCE) took this model and added another twenty-two spheres. Whereas Eudoxus and Callippus both probably viewed this model as a purely mathematical construction to help determine planetary position, Aristotle seems to have viewed it as reality. To explain the movement of the planets Aristotle thought of his system being driven by friction. The outermost sphere, that of the fixed stars drove the outer most sphere of Saturn, which in turn drove the next sphere down in the system and so on all the way down to the Moon. According to Aristotle the outermost sphere was set in motion by the unmoved mover. This last aspect was what most appealed to the churchmen of the medieval universities, who identified the unmoved mover with the Christian God.


During the Middle Ages an aspect of vitalism was added to this model, with some believing that the planets had souls, which animated them. Another theory claimed that each planet had its own angel, who pushed it round its orbit. Not exactly my idea of heaven, pushing a planet around its orbit for all of eternity. Aristotelian cosmology said that the spheres were real and made of crystal. When, in the sixteenth century astronomers came to accept that comets were supralunar celestial phenomena, and not as Aristotle had thought sublunar meteorological ones, it effectively killed off Aristotle’s crystalline spheres, as a supralunar comet would crash right through them. If fact, the existence or non-existence of the crystalline spheres was a major cosmological debate in the sixteenth century. By the early seventeenth century almost nobody still believed in them.

An alternative theory that had its origins in the Middle Ages but, which was revived in the sixteenth century was that the heavens were fluid and the planets swam through them like a fish or flew threw them like a bird. This theory, of course, has again a strong element of vitalism. However, with the definitive collapse of the crystalline spheres it became quite popular and was subscribed to be some important and influential thinkers at the end of the sixteenth beginning of the seventeenth centuries, for example Roberto Bellarmino (1542–1621) the most important Jesuit theologian, who had lectured on astronomy at the University of Leuven in his younger days.


Robert Bellarmine artist unknown Source: Wikimedia Commons

It should come as no surprise that the first astronomer to suggest a halfway scientific explanation for the motion of the planets was Johannes Kepler. In fact he devoted quite a lot of space to his theories in his Astronomia nova (1609).


Astronomia Nova title page Source: Wikimedia Commons

That the periods between the equinoxes and the solstices were of unequal length had been known to astronomers since at least the time of Hipparchus in the second century BCE. This seemed to imply that the speed of either the Sun orbiting the Earth, in a geocentric model, or the Earth orbiting the Sun, in a heliocentric model, varied through out the year. Kepler calculated a table for his elliptical, heliocentric model of the distances of the Sun from the Earth and deduced from this that the Earth moved fastest when it was closest to the Sun and slowest when it was furthest away. From this he deduced or rather speculated that the Sun controlled the motion of the Earth and by analogy of all the planets. The thirty-third chapter of Astronomia nova is headed, The power that moves the planets resides in the body of the sun.

His next question is, of course, what is this power and how does it operate? He found his answer in William Gilbert’s (1544–1603) De Magnete, which had been published in 1600.


William Gilbert Source: Wikimedia Commons

Kepler speculated that the Sun was in fact a magnet, as Gilbert had demonstrated the Earth to be, and that it rotated on its axis in the same way that Gilbert believed, falsely, that a freely suspended terrella (a globe shaped magnet) did. Gilbert had used this false belief to explain the Earth’s diurnal rotation.


It should be pointed out that Kepler was hypothesising a diurnal rotation for the Sun in 1609 that is a couple of years before Galileo had demonstrated the Sun’s rotation in his dispute over the nature of sunspots with Christoph Scheiner (c. 1574–1650). He then argues that there is power that goes out from the rotating Sun that drives the planets around there orbits. This power diminishes with its distance from the Sun, which explains why the speed of the planetary orbits also diminishes the further the respective planets are from the Sun. In different sections of the Astronomia nova Kepler argues both for and against this power being magnetic in nature. It should also be noted that although Kepler is moving in the right direction with his convoluted and at times opaque ideas on planetary motion there is still an element of vitalism present in his thoughts.

Kepler conceived the relationship between his planetary motive force and distance as a simple inverse ratio but it inspired the idea of an inverse squared force. The French mathematician and astronomer Ismaël Boulliau (1605–1694) was a convinced Keplerian and played a central roll in spreading Kepler’s ideas throughout Europe.


Ismaël Boulliau portrait by Pieter van Schuppen Source: Wikimedia Commons

His most important and influential work was his Astronomia philolaica (1645). In this work Boulliau hypothesised by analogy to Kepler’s own law on the propagation of light that if a force existed going out from the Sun driving the planets then it would decrease in inverse squared ratio and not a simple one as hypothesised by Kepler. Interestingly Boulliau himself did not believe that such a motive force for the planet existed.


Another mathematician and astronomer, who looked for a scientific explanation of planetary motion was the Italian, Giovanni Alfonso Borelli (1608–1697) a student of Benedetto Castelli (1578–1643) and thus a second-generation student of Galileo.


Giovanni Alfonso Borelli Source: Wikimedia Commons

Borelli developed a force-based theory of planetary motion in his Theoricae Mediceorum Planatarum ex Causius Physicis Deductae (Theory [of the motion] of the Medicean planets [i.e. moons of Jupiter] deduced from physical causes) published in 1666. He hypothesised three forces that acted on a planet. Firstly a natural attraction of the planet towards the sun, secondly a force emanating from the rotating Sun that swept the planet sideway and kept it in its orbit and thirdly the same force emanating from the sun pushed the planet outwards balancing the inwards attraction.

The ideas of both Kepler and Borelli laid the foundations for a celestial mechanics that would eventually in the work of Isaac Newton, who knew of both theories, produced a purely force-based mathematical explanation of planetary motion.







Filed under History of Astronomy, History of Mathematics, History of Physics, History of science, Mediaeval Science

The emergence of modern astronomy – a complex mosaic: Part XXIX

One of the most well known popular stories told about Galileo is how he dropped balls from the Leaning Tower of Pisa to disprove the Aristotelian hypothesis that balls of different weights would fall at different speeds; the heavier ball falling faster. This event probably never happened but it is related as a prelude to his brilliant experiments with balls and inclined planes, which he carried out to determine empirically the correct laws of fall and which really did take place and for which he is justifiably renowned as an experimentalist. What is very rarely admitted is that the investigation of the laws of fall had had a several-hundred-year history before Galileo even considered the problem, a history of which Galileo was well aware.

We saw in the last episode that John Philoponus had actually criticised Aristotle’s concept of fall in the sixth century and had even carried out the ball drop experiment. However, unlike his impulse concept for projectile motion, which was taken up by Islamic scholars and passed on by them into the European Middle Ages, his correct criticism of Aristotle’s fall theory appears not to have been taken up by later thinkers.

As far as we know the first people, after Philoponus, to challenge Aristotle’s concept was the so-called Oxford Calculatores.


Merton College in 1865 Source: Wikimedia Commons

This was a group of fourteenth-century, Aristotelian scholars at Merton College Oxford, who set about quantifying various theory of nature. These men–Thomas Bradwardine (c. 1300–1349), William of Heytesbury (c. 1313–c. 1372), Richard Swineshead (fl. c. 1340–1354) and John Dumbleton (c. 1310–c. 1349)–studied mechanics distinguishing between kinematics and dynamics, emphasising the former and investigating instantaneous velocity. They were the first to formulate the mean speed theorem, an achievement usually accredited to Galileo. The mean speed theorem states that a uniformly accelerated body, starting from rest, travels the same distance as a body with uniform speed, whose speed in half the final velocity of the accelerated body. The theory lies at the heart of the laws of fall.

The work of the Oxford Calculatores was quickly diffused throughout Europe and Nicole Oresme (c. 1320–1382), one of the so-called Parisian physicists,


Portrait of Nicole Oresme: Miniature from Oresme’s Traité de l’espère, Bibliothèque Nationale, Paris, France, fonds français 565, fol. 1r. Source: Wikimedia Commons

and Giovanni di Casali (c. 1320–after 1374) both produced graphical representation of the theory.


Oresme’s geometric verification of the Oxford Calculators’ Merton Rule of uniform acceleration, or mean speed theorem. Source: Wikimedia Commons

We saw in the last episode how Tartaglia applied mathematics to the problem of projectile motion and now we turn to a man, who for a time was a student of Tartaglia, Giambattista Benedetti (1530–1590). Like others before him Bendetti turned his attention to Aristotle’s concept of fall and wrote and published in total three works on the subject that went a long way towards the theory that Galileo would eventually publish. In his Resolutio omnium Euclidis problematum (1553) and his Demonstratio proportionum motuum localium (1554) he argued that speed is dependent not on weight but specific gravity and that two objects of the same material but different weights would fall at the same speed.


Source: Wikimedia Commons

However, in a vacuum, objects of different material would fall at different speed. Benedetti brought an early version of the thought experiment, usually attributed to Galileo, of viewing two bodies falling separately or conjoined, in his case by a cord.  Galileo considered a roof tile falling complete and then broken into two.

In a second edition of the Demonstratio (1554) he addressed surface area and resistance of the medium through which the objects are falling. He repeated his theories in his Demonstratio proportionum motuum localium (1554), where he explains his theories with respect to the theory of impetus. We know that Galileo had read his Benedetti and his own initial theories on the topic, in his unpublished De Motu, were very similar.

In the newly established United Provinces (The Netherlands) Simon Stevin (1548–1620) carried out a lot of work applying mathematics to various areas of physics. However in our contexts more interesting were his experiments in 1586, where he actually dropped lead balls of different weights from the thirty-foot-high church tower in Delft and determined empirically that they fell at the same speed, arriving at the ground at the same time.


Source: Wikimedia Commons

Some people think that because Stevin only wrote and published in Dutch that his mathematical physics remained largely unknown. However, his complete works published initially in Dutch were translated into both French and Latin, the latter translation being carried out by Willebrord Snell. As a result his work was well known in France, the major centre for mathematical physics in the seventeenth century.

In Italy the Dominican priest Domingo de Soto (1494–1560) correctly stated that a body falls with a constant, uniform acceleration. In his Opus novum, De Proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum. Item de aliza regula (1570) Gerolamo Cardano (1501–1576) demonstrates that two balls of different sizes will fall from a great height in the same time. The humanist poet and historian, Benedetto Varchi (c. 1502–1565) in 1544 and Giuseppe Moletti (1531–1588), Galileo’s predecessor as professor of mathematics in Padua, in 1576 both reported that bodies of different weights fall at the same speed in contradiction to Aristotle, as did Jacopo Mazzoni (1548–1598), a philosopher at Padua and friend of Galileo, in 1597. However none of them explained how they arrived at their conclusions.

Of particular relevance to Galileo is the De motu gravium et levium of Girolamo Borro (1512–1592) one of Galileo’s teachers in Pisa. In a dispute with his colleague Francesco Buonamici (1533–1603), another Pisan professor, Borro carried out experiments in which he threw objects of different material and the same weights out of a high window to test Aristotle’s theory, which he describes in his book. Borro’s work is known to have had a strong influence on Galileo’s early work in this area.

When Galileo started his own extensive investigations into the problem of fall in the late sixteenth century he was tapping into an extensive stream of previous work on the subject of which he was well aware and which to some extent had already done much of the heavy lifting. This raises the question as to what extent Galileo deserves his reputation as the man, who solved the problem of fall.


Galileo Portrait by Ottavio Leoni Source: Wikimedia Commons

We saw in the last episode that his much praised Dialogo, his magnum opus on the heliocentricity contra geocentricity debate, not only contributed nothing new of substance to that debate but because of his insistence on retaining the Platonic axioms, his total rejection of the work of both Tycho Brahe and Kepler and his rejection of the strong empirical evidence for the supralunar nature of comets he actually lagged far behind the current developments in that debate. The result was that the Dialogo could be regarded as superfluous to the astronomical system debate. Can the same be said of the contribution of the Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) (1638) to the debate on motion? The categorical answer is no; the Discorsi is a very important contribution to that debate and Galileo deserves his reputation as a mathematical physicist that this book gave him.


Source: Wikimedia Commons

What did Galileo contribute to the debate that was new? It not so much that he contributed much new to the debate but that he gave the debate the solid empirical and mathematical foundation, which it had lacked up till this point. Dropping weights from heights to examine the laws of fall suffers from various problems. It is extremely difficult to ensure that the object are both released at the same time, it is equally difficult to determine if they actually hit the ground at the same time and the whole process is so fast, that given the possibilities available at the time, it was impossible to measure the time taken for the fall. All of the previous experiments of Stevin et al were at best approximations and not really empirical proofs in a strict scientific sense. Galileo supplied the necessary empirical certainty.

Galileo didn’t drop balls he rolled them down a smooth, wooden channel in an inclined plane that had been oiled to remove friction. He argued by analogy the results that he achieved by slowing down the acceleration by using an inclined plane were equivalent to those that would be obtained by dropping the balls vertically. Argument by analogy is of course not strict scientific proof but is an often used part of the scientific method that has often, as in this case, led to important new discoveries and results.  He released one ball at a time and timed them separately thus eliminating the synchronicity problem. Also, he was able with a water clock to time the balls with sufficient accuracy to make the necessary mathematical calculations. He put the laws of falls on a sound empirical and mathematical footing. One should also not neglect the fact that Galileo’s undoubtable talent as a polemicist made the content of the Discorsi available in a way that was far more accessible than anything that had preceded it.


Galileo’s demonstration of the law of the space traversed in case of uniformly varied motion. It is the same demonstration that Oresme had made centuries earlier. Source: Wikimedia Commons

For those, who like to believe that Catholics and especially the Jesuits were anti-science in the seventeenth century, and unfortunately they still exist, the experimental confirmation of Galileo’s law of fall, using direct drop rather than an inclined plane, was the Jesuit, Giovanni Battista Riccioli(1598–1671).


Giovanni Battista Riccioli Source: Wikimedia Commons

The Discorsi also contains Galileo’s work on projectile motion, which again was important and influential. The major thing is the parabola law that states that anything projected upwards and away follows a parabolic path. Galileo was not the only natural philosopher, who determined this. The Englishman Thomas Harriot (c. 1560–1621) also discovered the parabola law and in fact his work on projectile motion went well beyond that of Galileo. Unfortunately, he never published anything so his work remained unknown.  One of Galileo’s acolytes, Bonaventura Cavalieri (1598–1647),


Bonaventura Cavalieri Source: Wikimedia Commons

was actually the first to publish the parabola law in his Lo Specchio Ustorio, overo, Trattato delle settioni coniche (The Burning Mirror, or a Treatise on Conic Sections) 1632.

This brought an accusation of intellectual theft from Galileo and it is impossible to tell from the ensuing correspondence, whether Cavalieri discovered the law independently or borrowed it without acknowledgement from Galileo.

The only problem that remained was what exactly was impetus. What was imparted to bodies to keep them moving? The answer was nothing. The solution was to invert the question and to consider what makes moving bodies cease to move? The answer is if nothing does, they don’t. This is known as the principle of inertia, which states that a body remains at rest or continues to move in a straight line unless acted upon by a force. Of course, in the early seventeenth century nobody really knew what force was but they still managed to discover the basic principle of inertia. Galileo sort of got halfway there. Still under the influence of the Platonic axioms, with their uniform circular motion, he argued that a homogenous sphere turning around its centre of gravity at the earth’s surface forever were there no friction at its bearings or against the air. Because of this Galileo is often credited with provided the theory of inertia as later expounded by Newton but this is false.

The Dutch scholar Isaac Beeckman (1588–1637) developed the concept of rectilinear inertia, as later used by Newton but also believed, like Galileo, in the conservation of constant circular velocity. Beeckman is interesting because he never published anything and his writing only became known at the beginning of the twentieth century. However, Beeckman was in contact, both personally and by correspondence, with the leading French mathematicians of the period, Descartes, Gassendi and Mersenne. For a time he was Descartes teacher and much of Descartes physics goes back to Beeckman. Descartes learnt the principle of inertia from Beeckman and it was he who published and it was his writings that were Newton’s source. The transmission of Beeckman’s work is an excellent illustration that scientific information does not just flow over published works but also through personal, private channels, when scientists communicate with each other.

With the laws of fall, the parabola law and the principle of inertia the investigators in the early seventeenth century had a new foundation for terrestrial mechanics to replace those of Aristotle.









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

The emergence of modern astronomy – a complex mosaic: Part XXVIII

One of the central problems in the transition from the traditional geocentric astronomy/cosmology to a heliocentric one was that the system that the Early Modern astronomers inherited from their medieval predecessors was not just an uneasy amalgam of Aristotelian cosmology and Ptolemaic astronomy but it also included Aristotle’s (384–322 BCE) theories of terrestrial and celestial motion all tied together in a complete package. Aristotle’s theory of motion was part of his more general theory of change and differentiated between natural motion and unnatural or violent motion.

The celestial realm in Aristotle’s cosmology was immutable, unchanging, and the only form of motion was natural motion that consisted of uniform, circular motion; a theory that he inherited from Plato (c. 425 – c.347 BCE), who in turn had adopted it from Empedocles (c. 494–c. 434 BCE).

His theory of terrestrial motion had both natural and unnatural motion. Natural motion was perpendicular to the Earth’s surface, i.e. when something falls to the ground. Aristotle explained this as a form of attraction, the falling object returning to its natural place. Aristotle also claimed that the elapsed time of a falling body was inversely proportional to its weight. That is, the heavier an object the faster it falls. All other forms of motion were unnatural. Aristotle believed that things only moved when something moved them, people pushing things, draught animals pulling things. As soon as the pushing or pulling ceased so did the motion.  This produced a major problem in Aristotle’s theory when it came to projectiles. According to his theory when a stone left the throwers hand or the arrow the bowstring they should automatically fall to the ground but of course they don’t. Aristotle explained this apparent contradiction away by saying that the projectile parted the air through which it travelled, which moved round behind the projectile and pushed it further. It didn’t need a philosopher to note the weakness of this more than somewhat ad hoc theory.

If one took away Aristotle’s cosmology and Ptolemaeus’ astronomy from the complete package then one also had to supply new theories of celestial and terrestrial motion to replace those of Aristotle. This constituted a large part of the development of the new physics that took place during the so-called scientific revolution. In what follows we will trace the development of a new theory, or better-said theories, of terrestrial motion that actually began in late antiquity and proceeded all the way up to Isaac Newton’s (1642–1726) masterpiece Principia Mathematica in 1687.

The first person to challenge Aristotle’s theories of terrestrial motion was John Philoponus (c. 490–c. 570 CE). He rejected Aristotle’s theory of projectile motion and introduced the theory of impetus to replace it. In the impetus theory the projector imparts impetus to the projected object, which is used up during its flight and when the impetus is exhausted the projectile falls to the ground. As we will see this theory was passed on down to the seventeenth century. Philoponus also rejected Aristotle’s quantitative theory of falling bodies by apparently carrying out the simple experiment usually attributed erroneously to Galileo, dropping two objects of different weights simultaneously from the same height:

but this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small. …

Philoponus also removed Aristotle’s distinction between celestial and terrestrial motion in that he attributed impetus to the motion of the planets. However, it was mainly his terrestrial theory of impetus that was picked up by his successors.

In the Islamic Empire, Ibn Sina (c. 980–1037), known in Latin as Avicenne, and Abu’l-Barakāt Hibat Allah ibn Malkā al-Baghdādī (c. 1080–1164) modified the theory of impetus in the eleventh century.


Avicenne Portrait (1271) Source: Wikimedia Commons

Nur ad-Din al-Bitruji (died c. 1204) elaborated it at the end of the twelfth century. Like Philoponus, al-Bitruji thought that impetus played a role in the motion of the planets.


Brought into European thought during the scientific Renaissance of the twelfth and thirteenth centuries by the translators it was developed by Jean Buridan  (c. 1301–c. 1360), who gave it the name impetus in the fourteenth century:

When a mover sets a body in motion he implants into it a certain impetus, that is, a certain force enabling a body to move in the direction in which the mover starts it, be it upwards, downwards, sidewards, or in a circle. The implanted impetus increases in the same ratio as the velocity. It is because of this impetus that a stone moves on after the thrower has ceased moving it. But because of the resistance of the air (and also because of the gravity of the stone) which strives to move it in the opposite direction to the motion caused by the impetus, the latter will weaken all the time. Therefore the motion of the stone will be gradually slower, and finally the impetus is so diminished or destroyed that the gravity of the stone prevails and moves the stone towards its natural place. In my opinion one can accept this explanation because the other explanations prove to be false whereas all phenomena agree with this one.


Jean Buridan Source

The impetus theory was now a part of medieval Aristotelian natural philosophy, which as Edward Grant pointed out was not Aristotle’s natural philosophy.

In the sixteenth century the self taught Italian mathematician Niccolò Fontana (c. 1500–1557), better known by his nickname, Tartaglia, who is best known for his dispute with Cardanoover the general solution of the cubic equation.


Niccolò Fontana Tartaglia Source: Wikimedia Commons

published the first mathematical analysis of ballistics his, Nova scientia (1537).


His theory of projectile trajectories was wrong because he was still using the impetus theory.


However, he was the first to demonstrate that an angle of 45° for a canon gives the widest range.


His book was massively influential in the sixteenth century and it also influenced Galileo, who owned a heavily annotated copy of the book.

We have traced the path of the impetus theory from its inception by John Philoponus up to the second half of the sixteenth century. Unlike the impetus theory Philoponus’ criticism of Aristotle’s theory of falling bodies was not taken up directly by his successors. However, in the High Middle Ages Aristotelian scholars in Europe did begin to challenge and question exactly those theories and we shall be looking at that development in the next section.







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Filed under History of Islamic Science, History of Mathematics, History of Physics, Mediaeval Science, Renaissance Science

Stylish writing is not necessarily good science

I have become somewhat infamous for writing #histSTM blog posts that are a predominately negative take on the scientific achievements of Galileo Galilei. In fact I think I probably made my breakthrough as a #histsci blogger with my notorious Extracting the Stopper post, deflating Galileo’s popular reputation. I actually got commissioned to write a toned down version of that post for AEON several years later. In my opinion Galileo was an important figure in the evolution of science during the early seventeenth century but his reputation has been blown up out of all proportion, well beyond his actual contributions. To make a simple comparison, in the same period of time the contributions of Johannes Kepler were immensely greater and more significant than those made by Galileo but whereas Galileo is regarded as one of the giants of modern science and is probably one of the three most well known historical practitioners of the mathematical sciences, alongside Newton and Einstein, Kepler is at best an also ran, whose popular image is not even a fraction of that of Galileo’s. This of course raises the question, why? What does/did Galileo have that Kepler didn’t? I think the answer lies in Galileo’s undeniable talents as a writer.

Galileo was a master stylist, a brilliant polemicist and science communicator, whose major works are still a stimulating pleasure to read. If you ask people about Galileo they will more often than not quote one of his well-known turns of phrase rather than his scientific achievements. The two books trope with its ‘mathematics is the language of nature’, which in the original actually reads: Philosophy is written in this grand book, which stands continually open before our eyes (I say the ‘Universe’), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth. Or the much-loved, the Bible shows the way to go to heaven, not the way the heavens go, which again in the original reads: The intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes. It is a trivial truth that Galileo had a way with words.

This cannot be said of Johannes Kepler. I shall probably bring the wrath of a horde of Kepler scholars on my head for saying this but even in translation, Johannes Kepler is anything but an easy read. Galileo even commented on this. When confronted with Kepler’s Dioptrice (1611), one of the most important books on optics ever written, Galileo complained that it was turgid and unreadable. Having ploughed my way through it in German translation, I sympathise with Galileo’s negative judgement. However, in his rejection Galileo failed to realise just how scientifically important the Dioptrice actually was. Nobody in their right mind would describe Kepler as a master stylist or a brilliant polemicist.

I think this contrast in literary abilities goes a long way to explaining the very different popular conceptions of the two men. People read Galileo’s major works or selections from them and are stimulated and impressed by his literary mastery, whereas Kepler’s major works are not even presented, as something to be read by anyone, who is not a historian of science. One just gets his three laws of planetary motion served up in modern guise, as a horribly mathematical side product of heliocentricity.

Of course, a serious factor in their respective notorieties is Galileo’s infamous trial by the Roman Inquisition. This was used to style him as a martyr for science, a process that only really began at the end of the eighteenth and beginning of the nineteenth centuries. Kepler’s life, which in many ways was far more spectacular and far more tragic than Galileo’s doesn’t have such a singular defining moment in it.

Returning to the literary theme I think that what has happened is that non-scientists and non-historians of science have read Galileo and impressed by his literary abilities, his skill at turning a phrase, his adroit, and oft deceitful, presentation of pro and contra arguments often fail to notice that they are being sold a pup. As I tried to make clear in the last episode of my continuing ‘the emergence of modern astronomy’ series although Galileo’s Dialogo has an awesome reputation in Early Modern history, its scientific value is, to put it mildly, negligible. To say this appears to most people as some form of sacrilege, “but the Dialogo is an important defence of science against the forces of religious ignorance” or some such they would splutter. But in reality it isn’t, as I hope I made clear the work contributed nothing new to the on going debate and all that Galileo succeeded in doing was getting up the Pope’s nose.

The same can be said of Il Saggiatore, another highly praised work of literature. As I commented in another post the, oft quoted line on mathematics, which had led to Galileo being praised as the man, who, apparently single handed, mathematized the physical science was actually, when he wrote it, old hat and others had been writing the book of nature in the language of mathematics for at least one hundred years before Galileo put pen to paper but none of them had taken the time to express what they were doing poetically. In fact it took historians of science a long time to correct this mistaken perception, as they also tended to suffer from a serious dose of Galileo adoration. The core of Il Saggiatore is as I have explained elsewhere is total rubbish, as Galileo is arguing against the scientific knowledge of his time with very spurious assertions merely so that he doesn’t have to acknowledge that Grassi is right and he is wrong. An admission that very few Galileo scholars are prepared to make in public, it might tarnish his reputation.

Interestingly one work that deserves its historical reputation Galileo’s Sidereus Nuncius, also suffers from serious scientific deficits that tend to get overlooked. Written and published in haste to avoid getting beaten to the punch by a potential, unknown rival the book actually reads more like an extended press release that a work of science. It might well be that Galileo intended to write a more scientific evaluation of his telescopic observations and discoveries once he had established his priority but somehow, having become something of a scientific superstar overnight, he never quite got round to it. This is once again a failing that most readers tend to overlook, over awed by the very impressive literary presentation.

Much of Galileo’s written work is actually more appearance than substance, or as the Germans say Mehr Schein als Sein, but ironically, there is one major work of Galileo’s that is both literarily brilliant and scientifically solid but which tends to get mostly overlooked, his Discorsi. The experiments on which part of it is based get described by the book itself remains for most people largely unknown. I shall be taking a closer look at it in a later post.







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

The Renaissance Mathematicus Christmas Trilogies explained for newcomers


Being new to the Renaissance Mathematicus one might be excused if one assumed that the blogging activities were wound down over the Christmas period. However, exactly the opposite is true with the Renaissance Mathematicus going into hyper-drive posting its annual Christmas Trilogy, three blog posts in three days. Three of my favourite scientific figures have their birthday over Christmas–Isaac Newton 25thDecember, Charles Babbage 26thDecember and Johannes Kepler 27thDecember–and I write a blog post for each of them on their respective birthdays. Before somebody quibbles I am aware that the birthdays of Newton and Kepler are both old style, i.e. on the Julian Calendar, and Babbage new style, i.e. on the Gregorian Calendar but to be honest, in this case I don’t give a shit. So if you are looking for some #histSTM entertainment or possibly enlightenment over the holiday period the Renaissance Mathematicus is your number one address. In case the new trilogy is not enough for you:

The Trilogies of Christmas Past

Christmas Trilogy 2009 Post 1

Christmas Trilogy 2009 Post 2

Christmas Trilogy 2009 Post 3

Christmas Trilogy 2010 Post 1

Christmas Trilogy 2010 Post 2

Christmas Trilogy 2010 Post 3

Christmas Trilogy 2011 Post 1

Christmas Trilogy 2011 Post 2

Christmas Trilogy 2011 Post 3

Christmas Trilogy 2012 Post 1

Christmas Trilogy 2012 Post 2

Christmas Trilogy 2012 Post 3

Christmas Trilogy 2013 Post 1

Christmas Trilogy 2013 Post 2

Christmas Trilogy 2013 Post 3

Christmas Trilogy 2014 Post 1

Christmas Trilogy 2014 Post 2

Christmas Trilogy 2014 Post 3

Christmas Trilogy 2015 Post 1

Christmas Trilogy 2015 Post 2

Christmas Trilogy 2015 Post 3

Christmas Trilogy 2016 Post 1

Christmas Trilogy 2016 Post 2

Christmas Trilogy 2016 Post 3

Christmas Trilogy 2017 Post 1

Christmas Trilogy 2017 Post 2

Christmas Trilogy 2017 Post 3

Christmas Trilogy 2018 Post 1

Christmas Trilogy 2018 Post 2

Christmas Trilogy 2018 Post 3






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Filed under History of Astronomy, History of Mathematics, History of Physics, History of science, History of Technology, Uncategorized

Mathematics or Physics–Mathematics vs. Physics–Mathematics and Physics

Graham Farmelo is a British physicist and science writer. He is the author of an excellent and highly praised biography of the British physicist P A M Dirac, The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius(Faber and Faber, 2009), which won a couple of book awards. He is also the author of a book Winston Churchill role in British war time nuclear research, Churchill’s Bomb:A hidden history of Britain’s first nuclear weapon programme (Faber and Faber, 2014), which was also well received and highly praised. Now he has published a new book on the relationship between mathematics and modern physics, The Universe Speaks in Numbers: How Modern Maths Reveals Nature’s Deepest Secrets (Faber and Faber, 2019).


I must admit that when I first took Farmelo’s new book into my hands it was with somewhat trepidation. Although, I studied mathematics to about BSc level that was quite a few years ago and these days my active knowledge of maths doesn’t extend much beyond A-Level and I never studied physics beyond A-Level and don’t ask what my grade was. However, I did study a lot of the history of early twentieth century physics before I moved back to the Renaissance. Would I be able to cope with Farmelo’s book? I needn’t have worried there are no complex mathematical or physical expressions or formulas. Although I would point out that this is not a book for the beginner with no knowledge; if your mind baulks at terms like gauge theory, string theory or super symmetry then you should approach this text with caution.

The book is Farmelo’s contribution to the debate about the use of higher mathematics to create advanced theories in physics that are not based on experimental evidence or even worse confirmable through experiment. It might well be regarded as a counterpoint to Sabine Hossenfelder’s much discussed Lost in Math: How Beauty Leads Physics Astray(Basic Books, 2018), which Farmelo actually mentions on the flyleaf to his book; although he obviously started researching and writing his volume long before the Hossenfelder tome appeared on the market. The almost concurrent appearance of the two contradictory works on the same topic shows that the debate that has been simmering just below the surface for a number of years has now boiled over into the public sphere.

Farmelo’s book is a historical survey of the relationship between advanced mathematics and theoretical physics since the seventeenth century, with an emphasis on the developments in the twentieth century. He is basically asking the questions, is it better when mathematics and physics develop separately or together and If together should mathematics or physics take the lead in that development. He investigated this questions using the words of the physicists and mathematicians from their published papers, from public lectures and from interviews, many of which for the most recent developments he conducted himself. He starts in the early seventeenth century with Kepler and Galileo, who, although they used mathematics to express their theories, he doesn’t think really understand or appreciate the close relationship between mathematics and physics. I actually disagree with him to some extent on this, as he knows. Disclosure: I actually read and discussed the opening section of the book with him, at his request, when he was writing it but I don’t think my minuscule contribution disqualifies me from reviewing it.

For Farmelo the true interrelationship between higher mathematics and advanced theories in physics begins with Isaac Newton. A fairly conventional viewpoint, after all Newton did title his magnum opus The Mathematical Principles of Natural Philosophy. I’m not going to give a decade by decade account of the contents, for that you will have to read the book but he, quite correctly, devotes a lot of space to James Clerk Maxwell in the nineteenth century, who can, with justification, be described as having taken the relationship between mathematics and physics to a whole new level.

Maxwell naturally leads to Albert Einstein, a man, who with his search for a purely mathematical grand unification theory provoked the accusation of having left the realm of experiment based and experimentally verifiable physics; an accusation that led many to accuse him of having lost the plot. As the author of a biography of Paul Dirac, Farmelo naturally devote quite a lot of space to the man, who might be regarded as the mathematical theoretical physicist par excellence and who, as Farmelo emphasises, preached a gospel of the necessity of mathematically beautiful theories, as to some extent Einstein had also done.

Farmelo takes us through the creation of quantum mechanics and the attempts to combine it with the theories of relativity, which takes the reader up to the early decades following the Second World War, roughly the middle of the book. Here the book takes a sharp turn away from the historical retelling of the emergence of modern theoretical physics to the attempts to create a fundamental theory of existence using purely mathematical methods, read string theory, M theory, supersymmetry and everything associated with them. This is exactly the development in modern physics that Hossenfelder rejects in her book.

Farmelo is very sympathetic to the mathematicians and physicists, who have taken this path but he is in his account very even handed, letting the critics have their say and not just the supporters. His account is very thorough and documents both the advances and the disappointments in the field over the most recent decades. He gives much emphasis to the fruitful co-operations and exchanges that have taken place between mathematicians and theoretical physicists. I must say that as somebody who has followed the debate at a distance, having read Farmelo’s detailed account I came out of it more sympathetic to Hossenfelder’s standpoint than his.

As always with his books Farmelo’s account is excellently researched, much of the more recent material is based on interviews he conducted with the participants, and very elegantly written. Despite the density of the material he is dealing with, his prose is light and often witty, which makes it easier to grapple with the complex themes he is discussing. I would certainly recommend this book to anybody interested in the developments in modern theoretical physics; maybe to be read together with Hossenfelder’s volume. I would also make an excellent present for any young school leaver contemplating studying physics or one that had already started on down that path.


Filed under Book Reviews, History of Mathematics, History of Physics

Conversations in a sixteenth century prison cell

Science writer Michael Brooks has thought up a delightful conceit for his latest book.* The narrative takes place in a sixteenth century prison cell in Bologna in the form of a conversation between a twenty-first century quantum physicist (the author) and a Renaissance polymath. What makes this conversation particularly spicy is that the Renaissance polymath is physician, biologist, chemist, mathematician, astronomer, astrologer, philosopher, inventor, writer, auto-biographer, gambler and scoundrel Girolamo Cardano, although Brooks calls him by the English translation of his name Jerome. In case anybody is wondering why I listed autobiographer separately after writer, it is because Jerome was a pioneer in the field writing what is probably the first autobiography by a mathematician/astronomer/etc. in the Early Modern Period.


Portrait of Cardano on display at the School of Mathematics and Statistics, University of St Andrews. Source: Wikimedia Commons

So what do our unlikely pair talk about? We gets fragments of conversation about Jerome’s current situation; a broken old man rotting away the end of his more than extraordinary life in a prison cell with very little chance of reprieve. This leads to the visitor from the future, relating episodes out of that extraordinary life. The visitor also picks up some of Jerome’s seemingly more strange beliefs and relates them to some of the equally, seemingly strange phenomena of quantum mechanics. But why should anyone link the misadventures of an, albeit brilliant, Renaissance miscreant to quantum mechanics. Because our author sees Jerome the mathematician, and he was a brilliant one, as the great-great-great-great-great-great-great-great-great-great-great-great-great grandfather of quantum mechanics!


As most people know quantum mechanics is largely non-deterministic in the conventional sense and relies heavily on probability theory for its results. Jerome wrote the first mathematical tome on probability theory, a field he entered because of his professional gambling activities. He even included a section about how to cheat at cards. I said he was a scoundrel. The other thing turns up in his Ars Magna (printed and published by Johannes Petreius the publisher of Copernicus’ De revolutionibus in Nürnberg and often called, by maths historians, the first modern maths book); he was the first person to calculate with so-called imaginary numbers. That’s numbers using ‘i’ the square root of minus one. Jerome didn’t call it ‘i’ or the numbers imaginary, in fact he didn’t like them very much but realised one could use them when determining the roots of cubic equation, so, holding his nose, that is exactly what he did. Like probability theory ‘i’ plays a very major role in quantum mechanics.

What Michael Brooks offers up for his readers is a mixture of history of Renaissance science together with an explanation of many of the weird phenomena and explanations of those phenomena in quantum mechanics. A heady brew but it works; in fact it works wonderfully.

This is not really a history of science book or a modern physics science communications volume but it’s a bit of both served up as science entertainment for the science interested reader, lay or professional. Michael Brooks has a light touch, spiced with some irony and a twinkle in his eyes and he has produced a fine piece of science writing in a pocket-sized book just right for that long train journey, that boring intercontinental flight or for the week in hospital that this reviewer used to read it. If this was a five star reviewing system I would be tempted to give it six.

*  Michael Brooks, The Quantum Astrologer’s Handbook, Scribe, Melbourne & London, 2017


Filed under Book Reviews, Early Scientific Publishing, History of Astrology, History of Astronomy, History of Physics, Renaissance Science, Uncategorized