Category Archives: History of Mathematics

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.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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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.

 

 

 

 

 

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

It’s all a question of angles.

Thomas Paine (1736–1809) was an eighteenth-century political radical famous, or perhaps that should be infamous, for two political pamphlets, Common Sense (1776) and Rights of Man (1791) (he also wrote many others) and for being hounded out of England for his political views and taking part in both the French and American Revolutions.

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Thomas Paine portrait of Laurent Dabos c. 1792 Source: Wikimedia Commons

So I was more than somewhat surprised when Michael Brooks, author of the excellent The Quantum Astrologer’s Handbook, posted the following excerpt from Paine’s The Age of Reason, praising trigonometry as the soul of science:

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My first reaction to this beautiful quote was that he could be describing this blog, as the activities he names, astronomy, navigation, geometry, land surveying make up the core of the writings on here. This is not surprising as Ivor Grattan-Guinness in his single volume survey of the history of maths, The Rainbow of Mathematics: A History of the Mathematical Sciences, called the period from 1540 to 1660 (which is basically the second half of the European Renaissance) The Age of Trigonometry. This being the case I thought it might be time for a sketch of the history of trigonometry.

Trigonometry is the branch of mathematics that studies the relationships between the side lengths and the angles of triangles. Possibly the oldest trigonometrical function, although not regarded as part of the trigonometrical cannon till much later, was the tangent. The relationship between a gnomon (a fancy word for a stick stuck upright in the ground or anything similar) and the shadow it casts defines the angle of inclination of the sun in the heavens. This knowledge existed in all ancient cultures with a certain level of mathematical development and is reflected in the shadow box found on the reverse of many astrolabes.

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Shadow box in the middle of a drawing of the reverse of Astrolabium Masha’Allah Public Library Bruges [nl] Ms. 522. Basically the tangent and cotangent functions when combined with the alidade

Trigonometry as we know it begins with ancient Greek astronomers, in order to determine the relative distance between celestial objects. These distances were determined by the angle subtended between the two objects as observed from the earth. As the heavens were thought to be a sphere this was spherical trigonometry[1], as opposed to the trigonometry that we all learnt at school that is plane trigonometry. The earliest known trigonometrical tables were said to have been constructed by Hipparchus of Nicaea (c. 190–c. 120 BCE) and the angles were defined by chords of circles. Hipparchus’ table of chords no longer exist but those of Ptolemaeus (fl. 150 CE) in his Mathēmatikē Syntaxis (Almagest) still do.

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The chord of an angle subtends the arc of the angle. Source: Wikimedia Commons

With Greek astronomy, trigonometry moved from Greece to India, where the Hindu mathematicians halved the Greek chords and thus created the sine and also defined the cosine. The first recoded uses of theses function can be found in the Surya Siddhanta (late 4th or early 5th century CE) an astronomical text and the Aryabhatiya of Aryabhata (476–550 CE).

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Statue depicting Aryabhata on the grounds of IUCAA, Pune (although there is no historical record of his appearance). Source: Wikimedia Commons

Medieval Islam in its general acquisition of mathematical knowledge took over trigonometry from both Greek and Indian sources and it was here that trigonometry in the modern sense first took shape.  Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850), famous for having introduced algebra into Europe, produced accurate sine and cosine tables and the first table of tangents.

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Statue of al-Khwarizmi in front of the Faculty of Mathematics of Amirkabir University of Technology in Tehran Source: Wikimedia Commons

In 830 CE Ahmad ibn ‘Abdallah Habash Hasib Marwazi (766–died after 869) produced the first table of cotangents. Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī (c. 858–929) discovered the secant and cosecant and produced the first cosecant tables.

Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī (940–998) was the first mathematician to use all six trigonometrical functions.

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Abū al-Wafā Source: Wikimedia Commons

Islamic mathematicians extended the use of trigonometry from astronomy to cartography and surveying. Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (1201–1274) is regarded as the first mathematician to present trigonometry as a mathematical discipline and not just a sub-discipline of astronomy.

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Iranian stamp for the 700th anniversary of Nasir al-Din Tusi’s death Source: Wikimedia Commons

Trigonometry came into Europe along with astronomy and mathematics as part the translation movement during the 11th and 12th centuries. Levi ben Gershon (1288–1344), a French Jewish mathematician/astronomer produced a trigonometrical text On Sines, Chords and Arcs in 1342. Trigonometry first really took off in Renaissance Europe with the translation of Ptolemaeus’ Geōgraphikḕ Hyphḗgēsis (Geographia) into Latin by Jacopo d’Angelo (before 1360–c. 1410) in 1406, which triggered a renaissance in cartography and astronomy.

The so-called first Viennese School of Mathematics made substantial contributions to the development of trigonometry in the sixteenth century. John of Gmunden (c. 1380–1442) produced a Tractatus de sinibus, chodis et arcubus. His successor, Georg von Peuerbach (1423–1461), published an abridgement of Gmunden’s work, Tractatus super propositiones Ptolemaei de sinibus et chordis together with a sine table produced by his pupil Regiomontanus (1436–1476) in 1541. He also calculated a monumental table of sines. Regiomontanus produced the first complete European account of all six trigonometrical functions as a separate mathematical discipline with his De Triangulis omnimodis (On Triangles) in 1464. To what extent his work borrowed from Arabic sources is the subject of discussion. Although Regiomontanus set up the first scientific publishing house in Nürnberg in 1471 he died before he could print De Triangulis. It was first edited by Johannes Schöner (1477–1547) and printed and published by Johannes Petreius (1497–1550) in Nürnberg in 1533.

At the request of Cardinal Bessarion, Peuerbach began the Epitoma in Almagestum Ptolomei in 1461 but died before he could complete it. It was completed by Regiomontanus and is a condensed and modernised version of Ptolemaeus’ Almagest. Peuerbach and Regiomontanus replaced the table of chords with trigonometrical tables and modernised many of the proofs with trigonometry. The Epitoma was published in Venice in 1496 and became the standard textbook for Ptolemaic geocentric astronomy throughout Europe for the next hundred years, spreading knowledge of trigonometry and its uses.

In 1533 in the third edition of the Apian/Frisius Cosmographia, Gemma Frisius (1508–1555) published as an appendix the first account of triangulationin his Libellus de locorum describendum ratione. This laid the trigonometry-based methodology of both surveying and cartography, which still exists today. Even GPS is based on triangulation.

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With the beginnings of deep-sea exploration in the fifteenth century first in Portugal and then in Spain the need for trigonometry in navigation started. Over the next centuries that need grew for determining latitude, for charting ships courses and for creating sea charts. This led to a rise in teaching trigonometry to seamen, as excellently described by Margaret Schotte in her Sailing School: Navigating Science and Skill, 1550–1800.

One of those students, who learnt their astronomy from the Epitoma was Nicolaus Copernicus (1473–1543). Modelled on the Almagest or more accurately the Epitoma, Copernicus’ De revolutionibus, published by Petreius in Nürnberg in 1543, also contained trigonometrical tables. WhenGeorg Joachim Rheticus (1514–1574) took Copernicus’ manuscript to Nürnberg to be printed, he also took the trigonometrical section home to Wittenberg, where he extended and improved it and published it under the title De lateribus et angulis triangulorum (On the Sides and Angles of Triangles) in 1542, a year before De revolutionibus was published. He would dedicate a large part of his future life to the science of trigonometry. In 1551 he published Canon doctrinae triangvlorvm in Leipzig. He then worked on what was intended to be the definitive work on trigonometry his Opus palatinum de triangulis, which he failed to finish before his death. It was completed by his student Valentin Otho (c. 1548–1603) and published in Neustadt an der Haardt in 1596.

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Source: Wikimedia Commons

In the meantime Bartholomäus Pitiscus (1561–1613) had published his own extensive work on both spherical and plane trigonometry, which coined the term trigonometry, Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuous in 1595.

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Source: Wikimedia Commons

This work was republished in expanded editions in 1600, 1608 and 1612. The tables contained in Pitiscus’ Trigonometria were calculated to five or six places, where as those of Rheticus were calculated up to more than twenty places for large angles and fifteenth for small ones. However, on inspection, despite the years of effort that Rheticus and Otho had invested in the work, some of the calculations were found to be defective. Pitiscus recalculated them and republished the work as Magnus canon doctrinae triangulorum in 1607. He published a second further improved version under the title Thesaurus mathematicus in 1613. These tables remained the definitive trigonometrical tables for three centuries only being replaced by Henri Andoyer’s tables in 1915–18.

We have come a long way from ancient Greece in the second century BCE to Germany at the turn of the seventeenth century CE by way of Early Medieval India and the Medieval Islamic Empire. During the seventeenth century the trigonometrical relationships, which I have up till now somewhat anachronistically referred to as functions became functions in the true meaning of the term and through analytical geometry received graphical presentations completely divorced from the triangle. However, I’m not going to follow these developments here. The above is merely a superficial sketch that does not cover the problems involved in actually calculating trigonometrical tables or the discovery and development of the various relationships between the trigonometrical functions such as the sine and cosine laws. For a detailed description of these developments from the beginnings up to Pitiscus I highly recommend Glen van Brummelen’s The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009.

 

[1] For a wonderful detailed description of spherical trigonometry and its history see Glen van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, Princeton and Oxford, 2013

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Filed under History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, History of science, Mediaeval Science, Renaissance Science

Mathematics at the Meridian

Historically Greenwich was a village, home to a royal palace from the fifteenth to the seventeenth centuries, that lay to the southeast of the city of London on the banks of the river Thames, about six miles from Charing Cross. Since the beginning of the twentieth century it has been part of London. With the Cutty Sark, a late nineteenth century clipper built for the Chinese tea trade, the Queen’s House, a seventeenth-century royal residence designed and built by Inigo Jones for Anne of Denmark, wife of James I & VI, and now an art gallery, the National Maritime Museum, Christopher Wren’s Royal Observatory building and of course the Zero Meridian line Greenwich is a much visited, international tourist attraction.

Naturally, given that it is/was the home of the Royal Observatory, the Zero Meridian, the Greenwich Royal Hospital School, the Royal Naval College (of both of which more later), and most recently Greenwich University, Greenwich has been the site of a lot mathematical activity over the last four hundred plus years and now a collection of essays has been published outlining in detail that history: Mathematics at the Meridian: The History of Mathematics at Greenwich[1]

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This collection of essays gives a fairly comprehensive description of the mathematical activity that took place at the various Greenwich institutions. As a result it also function as an institutional history, an often-neglected aspect of the histories of science and mathematics with their concentration on big names and significant theories and theorems. Institutions play an important role in the histories of mathematic and science and should receive much more attention than they usually do.

The first four essays in the collection cover the history of the Royal Observatory from its foundation down to when it was finally closed down in 1998 following several moves from its original home in Greenwich. They also contain biographies of all the Astronomers Royal and how they interpreted their role as the nation’s official state astronomer.

This is followed by an essay on the mathematical education at the Greenwich Royal Hospital School. The Greenwich Royal Hospital was established at the end of the seventeenth century as an institution for aged and injured seamen. The institution included a school for the sons of deceased or disabled sailors. The teaching was centred round seamanship and so included mathematics, astronomy and navigation.

When the Greenwich Royal Hospital closed at the end of the nineteenth century the buildings were occupied by the Royal Naval College, which was moved from Portsmouth to Greenwich. The next three chapters deal with the Royal Naval College and two of the significant mathematicians, who had been employed there as teachers and their contributions to mathematics.

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Another institute that was originally housed at Greenwich was The Nautical Almanac office, founded in 1832. The chapter dealing with this institute concentrates on the life and work of Leslie John Comrie (1893–1950), who modernised the production of mathematical tables introducing mechanisation to the process.

Today, apart from the Observatory itself and the Meridian line, the biggest attraction in Greenwich is the National Maritime Museum, one of the world’s leading science museums and there is a chapter dedicated to the scientific instruments on display there.

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Also today, the buildings that once housed the Greenwich Royal Hospital and then the Royal Naval College now house the University of Greenwich and the last substantial chapter of the book brings the reader up to the present outlining the mathematics that has been and is being taught there.

The book closes with a two-page afterword, The Mathematical Tourist in Greenwich.

Each essay in the book is written by an expert on the topic and they are all well researched and maintain a high standard throughout the entire book. The essays covers a wide and diverse range of topics and will most probably not all appeal equally to all readers. Some might be more interested in the history of the Royal Observatory, whilst the chapters on the mathematical education at the Greenwich Royal Hospital School and on its successor the Royal Naval College should definitely be of interest to the readers of Margaret Schotte’s Sailing School, which I reviewed in an earlier post.

Being the hopelessly non-specialist that I am, I read, enjoyed and learnt something from all of the essays. If I had to select the four chapters that most stimulated me I would chose the opening chapter on the foundation and early history of the Royal Observatory, the chapter on George Biddel Airy and his dispute with Charles Babbage over the financing of the Difference Engine, which I blogged about in December, the chapter on Leslie John Comrie, as I’ve always had a bit of a thing about mathematical tables and finally, one could say of course, the chapter on the scientific instruments in the National Maritime Museum.

The book is nicely illustrated with, what appears to have become the standard for modern academic books, grey in grey prints. There are extensive endnotes for each chapter, which include all of the bibliographical references, there being no general bibliography, which I view as the books only defect. There is however a good, comprehensive general index.

I can thoroughly recommend this book for anybody interested in any of the diverse topic covered however, despite what at first glance, might appear as a somewhat specialised book, I can also recommend it for the more general reader interested in the histories of mathematics, astronomy and navigation or those perhaps interested in the cultural history of one of London’s most fascinating district. After all mathematics, astronomy and navigation are all parts of human culture.

[1] Mathematics at the Meridian: The History of Mathematics at Greenwich, eds. Raymond Flood, Tony Mann, Mary Croarken, CRC Press, Taylor & Francis Group, Bacon Raton, London, New York, 2020.

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Filed under Book Reviews, History of Astronomy, History of Mathematics, History of Navigation

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.

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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.

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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.

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Niccolò Fontana Tartaglia Source: Wikimedia Commons

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

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

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

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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

Christmas Trilogy 2019 Part 2: Babbage, Airy and financing the Difference Engine.

Charles Babbage first announced his concept for his first computer, the Difference Engine, in a Royal Astronomical Society paper, Note on the application of machinery to the computation of astronomical and mathematical tables in 1822.

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Engraving of Charles Babbage dated 1833 Source: Wikimedia Commons

He managed to convince the British Government that a mechanical calculator would be useful for producing numerical tables faster, cheaper and more accurately and in 1823 they advance Babbage £1700 to begin construction of a full scale machine. It took Babbage and his engineer, Joseph Clements, nine years to produce a small working model but costs had spiralled out of control and the government suspended payment at around £17,000, in those days a small fortune, in 1833.

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A portion of the difference engine. Woodcut after a drawing by Benjamin Herschel Babbage Source: Wikimedia Commons

Babbage and Clement had parted in dispute by this time. The next nine years saw Babbage negotiating with various government officials to try and get payment reinstated. Enter George Biddel Airy (1801–1892).

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George Biddell Airy caricatured by Ape in Vanity Fair Nov 1875 Source: Wikimedia Commons

Airy entered Trinity College Cambridge in 1819, graduating Senior Wrangler and Smith Prize man in 1823. He was elected a fellow of Trinity in 1824 and Lucasian Professor of mathematics beating Babbage for the position in 1826. In 1828 he was elected Plumian Professor of astronomy and director of the new Cambridge Observatory. Babbage succeeded him as Lucasian Professor. Airy proved very competent and very efficient as the director of the observatory, which led to him being appointed Astronomer Royal at the Greenwich Observatory in 1835 and thus the leading state scientist and effectively the government scientific advisor. It was in this capacity that the paths of the two Cambridge mathematicians crossed once again[1].

In 1842 Henry Goulburn (1784–1856), Chancellor of the Exchequer in the cabinet of Sir Robert Peel (1788–1850) was asked by Peel to gather information on Babbage’s Difference Engine project, which he would have liked to ditch, preferable yesterday rather than tomorrow. Goulburn turned to Airy as the countries leading scientific civil servant and also because the Royal Observatory was responsible for producing many of the mathematical tables, the productions of which the Difference Engine was supposed to facilitate. Could Airy offer an opinion on the utility of the proposed mechanical calculator? Airy could and it was anything but positive:

Mr Babbage made the approval of the machine a personal question. In consequence of this, I, and I believe other persons, have carefully abstained for several years from alluding to it in his presence. I think he lives in a sort of dream as to its utility.

An absurd notion has been spread abroad, that the machine was intended for all calculations of every kind. This is quite wrong. The machine is intended solely for calculations which can be made by addition and subtraction in a particular way. This excludes all ordinary calculation.

Scarcely a figure of the Nautical Almanac could be computed by it. Not a single figure of the Geenwich Observations or the great human Computations now going on could be computed by it. Indeed it was proposed only for the computation of new Tables (as Tables of Logarithms and the like), and even for these, the difficult part must be done by human computers. The necessity for such new tables does not occur, as I really believe, once in fifty years. I can therefore state without the least hesitation that I believe the machine to be useless, and that the sooner it is abandoned, the better it will be for all parties[2].

Airy’s opinion was devastating Peel acting on Goulburn’s advice abandoned the financing of the Difference Engine once and for all. Even the personal appeals of Babbage directly to Peel were unable to change this decision. Airy’s judgement was actually based on common sense and solid economic arguments. The tables computed by human computers were comparatively free of errors and nothing could be gained here by replacing their labour with a machine that would probably prove more expensive. Also setting up the machine to compute any particular set of tables would first require human computers to determine the initially values for the algorithms and to determine that the approximations delivered by the difference series remained within an acceptable tolerance range. Airy could really see no advantages in employing Babbage’s machine rather than his highly trained human computers. Also any human computers employed to work with the Difference Engine would, by necessity, also need first to be trained for the task.

Airy’s views on the utility or rather lack thereof of mechanical calculators was shared by the Swedish astronomer Nils Seelander (1804–1870) also used the same arguments against the use of mechanical calculators in 1844 as did Urbain Le Verrier (1811–1877) at the Paris Observatory.

Babbage was never one to take criticism or defeat lying down and in 1851 when the working model of the Difference Engine No. 1 was on display at the Great Exhibition he launched a vicious attack on Airy in his book The Exposition of 1851: Views of The Industry, The Science and The Government of England.

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Babbage was not a happy man. By 1851 Airy was firmly established as a leading European scientist and an exemplary public servant and could and did publically ignore Babbage’s diatribe. Privately he wrote a parody of the rhyme This is the House that Jack Built mocking Babbage’s efforts to realise his Difference Engine. Verse seven of This is the Engine that Charles Built reads as follows:

There are Treasury lords, slightly furnished with sense,

Who the wealth of the nation unfairly dispense:

They know but one man, in the Queen’s vast dominion,

Who in things scientific can give an opinion:

And when Babbage for funds for the Engine applied,

The called upon Airy, no doubt, to decide:

And doubtless adopted, in apathy slavish,

The hostile suggestions of enmity knavish:

The powers of official position abused,

And flatly all further advances refused.

For completing the Engine that Charles built.[3]

Today Charles Babbage is seen as a visionary in the history of computers and computing, George Airy very clearly did not share that vision but he was no Luddite opposing the progress of technology out of principle. His opposition to the financing of Babbage’s Difference Engine was based on sound mathematical and financial principles and delivered with well-considered arguments.

[1] The following account is based almost entirely on Doran D. Swade’s excellent paper, George Biddell Airy, Greenwich and the Utility of Calculating Engines in Mathematics at the Meridian: The History of Mathematics at Greenwich, de. Raymond Flood, Tony Mann & Mary Croarken, CRC Press, Boca Raton, London New York, 2019 pp. 63–81. A review of the entire, excellent volume will follow some time next year.

[2] All three quotes are from Airy’s letter to Goulburn 16 September 1842 RGO6–427, f. 65. Emphasis original. Quoted by Swade p. 69.

[3] Swade p. 74 The whole poem can be read in Appendix I of Doran David Swade, Calculation and Tabulation in the Nineteenth Century: Airy versus Babbage, Thesis submitted for the degree of PhD, University College London, 2003, which of course deals with the whole story in great depth and detail and is available here on the Internet.

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The Renaissance Mathematicus Christmas Trilogies explained for newcomers

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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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There is no year zero!

I realise that in writing this post I am wasting my time, pissing against the wind, banging my head against a brick wall and all the other colourful expressions in the English language that describe embarking on a hopeless endeavour but I am renowned for being a pedantic curmudgeon and so I soldier on into the jaws of disappointment and defeat. I shall attempt to explain carefully and I hope clearly why the 31st of December of the year 2019 does not mark the end of the second decade of the 21st century. I know, I know but I must.

The core of the problem lies in the fact that we possess two basic sets of counting numbers, cardinals and ordinals. Now cardinals have nothing to do with the Holy Roman Catholic Church, a family of birds or a baseball team from St. Louis but are the numbers we use to say how many items there are in a group, a collection, a heap or as the mathematician prefer to call it a set. Let us look at a well-known example:

I’ll sing you twelve, O

Green grow the rushes, O

What are your twelve, O?

Twelve for the twelve Apostles

Eleven for the eleven who went to heaven,

Ten for the ten commandments,

Nine for the nine bright shiners,

Eight for the April Rainers.

Seven for the seven stars in the sky,

Six for the six proud walkers,

Five for the symbols at your door,

Four for the Gospel makers,

Three, three, the rivals,

Two, two, the lily-white boys,

Clothed all in green, O

One is one and all alone

And evermore shall be so.

This is the final round of an old English counting song the meaning of several lines of which remain intriguingly obscure. Starting with the fourth line from the top we have a set of 12 Apostles i.e. the original twelve follower of Jesus. One line further in, we have a set of 11, who went to heaven, presumably the Apostles minus Judas Iscariot. And so we proceed, each line refers to a group or set giving to number contained in it.

In everyday life we use cardinal numbers all the time. I bought 6 eggs today. There are 28 children in Johnny’s class. My car has 4 wheels and so on and so forth. The cardinal numbers also contain the number zero (0), which indicates that a particular group or set under discussion contain no items at all. There are currently zero kings of France. We can carry out all the usually simple arithmetical operations–addition, subtraction, multiplication and division–on the cardinal numbers including zero, with the exception that we can’t divide by zero; mathematicians say division by zero is not defined. So if Johnny’s class with its 28 members are joined by Jenny’s class with 27 members for the school trip there will be 55 children on the bus. I’m sure you can think up lots of other examples yourselves.

Ordinal numbers have a different function, there signify the position of items in a list, row, series etc. We also use different names for ordinal numbers to cardinal numbers, so instead of one, two three four…, we say first, second, third, fourth…etc. an example would be, Johnny was the fifth person in his class to get the flu this winter. Now, in the ordinal numbers there is no zero, it would be a contradiction in terms, as it can’t exist. Occasionally when there is an existing ordered list of principles or laws people will talk about the ‘zeroeth’ law, meaning one that wasn’t originally included but that they think should precede the existing ones.

When we talk about years we tend to use the words for cardinal numbers but in fact we are actually talking about ordinal numbers. What we call 2019 CE or AD i.e. two thousand and nineteen is in fact the two thousand and nineteenth year of the Common Era or the two thousand and nineteenth year of Our Lord. Whichever system of counting years one uses, Gregorian, Jewish, Muslim, Persian, Chinese, Hindu or whatever there is and never can be a year zero, it is, as stated abve, a contradiction in terms and cannot exist. Therefore the first decade, that is a group of ten year, in your calendrical system consists of the years one to ten or the first year to the tenth year, the second decade the years eleven to twenty or the eleventh year to the twentieth year and so on. The first century, that is a group of one hundred years, consists of the years one to one hundred or the first year to the one-hundredth year. First millennium, that is one thousand years, consists of the years one to one thousand or the first year to the one-thousandth year.

Going back to our starting point the first decade of the 21st century started on the 1st January 2001 and finished on the 31st December 2010. The second decade started on the 1st January 2011 and will end on the 31st December 2020 and not on 31st December 2019 as various innumerate people would have you believe.

 

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