Category Archives: History of Mathematics

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.


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:


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.


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.


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).


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.


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.


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.


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.


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.


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.


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


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]


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.


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.


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.


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.


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

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.


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.


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).


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.


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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Filed under History of Astronomy, History of Computing, History of Mathematics, History of 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

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.



Filed under Calendrics, History of Mathematics, Myths of Science

Finding your way on the Seven Seas in the Early Modern Period

I spend a lot of my time trying to unravel and understand the complex bundle that is Renaissance or Early Modern mathematics and the people who practiced it. Regular readers of this blog should by now be well aware that the Renaissance mathematici, or mathematical practitioners as they are generally known in English, did not work on mathematics as we would understand it today but on practical mathematics that we might be inclined, somewhat mistakenly, to label applied mathematics. One group of disciplines that we often find treated together by one and the same practitioner consists of astronomy, cartography, navigation and the design and construction of tables and instruments to aid the study of these. This being the case I was delighted to receive a review copy of Margaret E. Schotte’s Sailing School: Navigating Science and Skill, 1550–1800[1], which deals with exactly this group of practical mathematical skills as applied to the real world of deep-sea sailing.

Sailing School001.jpg

Schotte’s book takes the reader on a journey both through time and around the major sea going nations of Europe, explaining, as she goes, how each of these nations dealt with the problem of educating, or maybe that should rather be training, seamen to become navigators for their navel and merchant fleets, as the Europeans began to span the world in their sailing ships both for exploration and trade.

Having set the course for the reader in a detailed introduction, Schotte sets sail from the Iberian peninsular in the sixteenth century. It was from there that the first Europeans set out on deep-sea voyages and it was here that it was first realised that navigators for such voyages could and probably should be trained. Next we travel up the coast of the Atlantic to Holland in the seventeenth century, where the Dutch set out to conquer the oceans and establish themselves as the world’s leading maritime nation with a wide range of training possibilities for deep-sea navigators, extending the foundations laid by the Spanish and Portuguese. Towards the end of the century we seek harbour in France to see how the French are training their navigators. Next port of call is England, a land that would famously go on, in their own estimation, to rule the seven seas. In the eighteenth century we cross the Channel back to Holland and the advances made over the last hundred years. The final chapter takes us to the end of the eighteenth century and the extraordinary story of the English seaman Lieutenant Riou, whose ship the HMS Guardian hit an iceberg in the Southern Atlantic. Lacking enough boats to evacuate all of his crew and passengers, Riou made temporary repairs to his vessel and motivating his men to continuously pump out the waters leaking into the rump of his ship, he then by a process of masterful navigation, on a level with his contemporaries Cook and Bligh, brought the badly damaged frigate to safety in South Africa.

Sailing School004

In each of our ports of call Schotte outlines and explains the training conceived by the authorities for training navigators and examines how it was or was not put into practice. Methods of determining latitude and longitude, sailing speeds and distances covered are described and explained. The differences in approach to this training developed in each of the sea going European nations are carefully presented and contrasted. Of special interest is the breach in understanding of what is necessary for a trainee navigator between the mathematical practitioners, who were appointed to teach those trainees, and the seamen, who were being trained, a large yawning gap between theory and practice. When discussing the Dutch approach to training Schotte clearly describes why experienced coastal navigators do not, without retraining, make good deep-sea navigators. The methodologies of these two areas of the art of navigation are substantially different.

The reader gets introduced to the methodologies used by deep-sea navigators, the mathematics developed, the tables considered necessary and the instruments and charts that were put to use. Of particular interest are the rules of thumb utilised to make course corrections before accurate methods of determining longitude were developed. There are also detailed discussions about how one or other aspect of the art of navigation was emphasised in the training in one country but considered less important in another. One conclusion the Schotte draws is that there is not really a discernable gradient of progress in the methods taught and the methods of teaching them over the two hundred and fifty years covered by the book.

Sailing School003.jpg

As well as everything you wanted to know about navigating sailing ships but were too afraid to ask, Schotte also delivers interesting knowledge of other areas. Theories of education come to the fore but an aspect that I found particularly fascinating were her comments on the book trade. Throughout the period covered, the teachers of navigation wrote and marketed books on the art of navigation. These books were fairly diverse and written for differing readers. Some were conceived as textbooks for the apprentice navigators whilst others were obviously written for interested, educated laymen, who would never navigate a ship. Later, as written exams began to play a greater role in the education of the aspirant navigators, authors and publishers began to market books of specimen exam questions as preparation for the exams. These books also went through an interesting evolution. Schotte deals with this topic in quite a lot of detail discussing the authors, publishers and booksellers, who were engaged in this market of navigational literature. This is detailed enough to be of interest to book historians, who might not really be interested in the history of navigation per se.

Schotte is excellent writer and the book is truly a pleasure to read. On a physical level the book is beautifully presented with lots of fascinating and highly informative illustrations. The apparatus starts with a very useful glossary of technical terms. There is a very extensive bibliography and an equally extensive and useful index. My only complaint concerns the notes, which are endnotes and not footnotes. These are in fact very extensive and highly informative containing lots of additional information not contained in the main text. I found myself continually leafing back and forth between main text and endnotes, making continuous reading almost impossible. In the end I developed a method of reading so many pages of main text followed by reading the endnotes for that section of the main text, mentally noting the number of particular endnotes that I wished to especially consult. Not ideal by any means.

This book is an essential read for anybody directly or indirectly interested in the history of navigation and also the history of practical mathematics. If however you are generally interested in good, well researched, well written history then you will almost certainly get a great deal of pleasure from reading this book.

[1] Margaret E. Schotte, Sailing School: Navigating Science and Skill, 1550–1800, Johns Hopkins University Press, Baltimore, 2019.


Filed under Book Reviews, History of Astronomy, History of Cartography, History of Mathematics, History of Navigation, Renaissance Science, Uncategorized

Calculus for the curious

Some weeks ago I got involved in a discussion on Twitter about, which books to recommend on the history of calculus. Somebody chimed in that Steven Strogatz’s new book would tell you all that you needed to know about the history of calculus. I replied that I couldn’t comment on this, as I hadn’t read it. To my surprise Professor Strogatz popped up and asked me if I would like to have a copy of his book. Never one to turn down a freebee, I naturally said yes. Very soon after a copy of Infinite Powers: The Story of Calculus The Language of the Universe arrived in the post and landed on my to read pile. Having now read it I can comment on it and intend to do so.

For those, who don’t know Steven Strogatz, he is professor of applied mathematics at Cornell University and the successful author of best selling popular books on mathematics.


First off, Infinite Powers is not a history of calculus. It is a detailed introduction to what calculus is and how it works, with particular emphasis on its applications down the centuries, Strogatz is an applied mathematician, presented in a history-light frame story. Having said this, I’m definitely not knocking, what is an excellent book but I wouldn’t recommend it to anybody, who was really looking for a history of calculus, maybe, however, either as a prequel or as a follow up to reading a history of calculus.

The book is centred on what Strogatz calls The Infinity Principle, which lies at the heart of the whole of calculus:

To shed light on any continuous shape, object, motion, process, or phenomenon–no matter how wild and complicated it may appear–reimagine it as an infinite series of simpler parts, analyse those, and then add the results back together to make sense of the original whole.

Following the introduction of his infinity principle Strogatz gives a general discussion of its strengths and weakness before moving on in the first chapter proper to discuss infinity in all of its guises, familiar material and examples for anybody, who has read about the subject but a well done introduction for those who haven’t. Chapter 2 takes us  into the early days of calculus, although it didn’t yet have this name, and introduces us to The Man Who Harnessed Infinity, the legendary ancient Greek mathematician Archimedes and the method of exhaustion used to determine the value of π and the areas and volumes of various geometrical forms. Astute readers will have noticed that I wrote early days and not beginning and here is a good example of why I say that this is not a history of calculus. Although Archimedes put the method of exhaustion to good use he didn’t invent it, Eudoxus did. Strogatz does sort of mention this in passing but whereas Archimedes gets star billing, Eudoxus gets dismissed in half a sentence in brackets. The reader is left completely in the dark as to who, why, what Eudoxus is/was. OK here, but not OK in a real history of calculus. This criticism might seem petty but there are lots of similar examples throughout the book that I’m not going to list in this review and this is why the book is not a history of calculus and I don’t think Strogatz intended to write one; the book he has written is a different one and it is a very good one.

After Archimedes the book takes a big leap to the Early Modern Period and Galileo and Kepler with the justification that, “When Archimedes died, the mathematical study of nature nearly died along with him. […] In Renaissance Italy, a young mathematician named Galileo Galilei picked up where Archimedes had left off.” My inner historian of mathematics had an apoplectic fit on reading these statements. They ignore a vast amount of mathematics, in particular the work in the Middle Ages and the sixteenth century on which Galileo built the theories that Strogatz then presents here but I console myself with the thought that this is not a history of calculus let alone a history of mathematics. However, I’m being too negative, let us return to the book. The chapter deals with Galileo’s terrestrial laws of motion and Kepler’s astronomical laws of planetary motion. Following this brief introduction to the beginnings of modern science Strogatz moves into top gear with the beginnings of differential calculus. He guides the reader through the developments of seventeenth century mathematics, Fermat and Descartes and the birth of analytical geometry bringing together the recently introduced algebra and the, by then, traditional geometry. Moving on he deals with tangents, functions and derivatives. Strogatz is an excellent teacher he introduces a new concept carefully, explains it, and then shows how it can be applied to an everyday situation.

Having laid the foundations Strogatz move on naturally to the supposed founders of modern calculus, Leibnitz and Newton and their bringing together of the strands out of the past that make up calculus as we know it and how they fit together in the fundamental theorem of calculus. This is interwoven with the life stories of the two central figures. Again having introduced concepts and explained them Strogatz illustrates them with applications outside of pure mathematics.

Having established modern calculus the story moves on into the eighteenth century.  Here I have to point out that Strogatz perpetuates a couple of myths concerning Newton and the writing of his Principia. He writes that Newton took the concept of inertia from Galileo; he didn’t, he took it from Descartes, who in turn had it from Isaac Beeckman. A small point but as a historian I think an important one. Much more important he seems to be saying that Newton created the physics of Principia using calculus then translated it back into the language of Euclidian geometry, so as not to put off his readers. This is a widely believed myth but it is just that, a myth. To be fair it was a myth put into the world by Newton himself. All of the leading Newton experts have over the years very carefully scrutinised all of Newton’s writings and have found no evidence that Newton conceived and wrote Principia in any other form than the published one. Why he rejected the calculus, which he himself developed, as a working tool for his magnum opus is another complicated story that I won’t go into here but reject it he did[1].

After Principia, Strogatz finishes his book with a random selection of what might be termed calculus’ greatest hits, showing how it proved its power in solving a diverse series of problems. Interestingly he also addresses the future. There are those who think that calculus’ heyday is passed and that other, more modern mathematical tools will in future be used in the applied sciences to solve problems, Strogatz disagrees and sees a positive and active future for calculus as a central mathematical tool.

Despite all my negative comments, and I don’t think my readers would expect anything else from me, given my reputation, I genuinely think that this is on the whole an excellent book. Strogatz writes well and fluidly and despite the, sometimes, exacting content his book is a pleasure to read. He is also very obviously an excellent teacher, who is very good at clearly explaining oft, difficult concepts. I found it slightly disappointing that his story of calculus stops just when it begins to get philosophical and logically interesting i.e. when mathematicians began working on a safe foundation for the procedures that they had been using largely intuitively. See for example Euler, who made great strides in the development of calculus without any really defined concepts of convergence, divergence or limits, but who doesn’t appear here at all. However, Strogatz book is already 350-pages-long and if, using the same approach, he had continued the story down to and into the twentieth century it would probably have weighed in at a thousand plus pages!

Despite my historical criticisms, I would recommend Strogatz’s book, without reservations, to anybody and everybody, who wishes to achieve a clearer, deeper and better understanding of what calculus is, where it comes from, how it functions and above all, and this is Strogatz’s greatest strength, how it is applied to the solution of a wide range of very diverse problems in an equally wide and diverse range of topics.


[1] For a detailed analysis of Newton’s rejection of analytical methods in mathematics then I heartily recommend, Niccolò Guicciardini, Reading the Principia, CUP, 1999, but with the warning that it’s not an easy read!



Filed under Book Reviews, History of Mathematics

Mathematical aids for Early Modern astronomers.

Since its very beginnings in the Fertile Crescent, European astronomy has always involved a lot of complicated and tedious mathematical calculations. Those early astronomers described the orbits of planets, lunar eclipses and other astronomical phenomena using arithmetical or algebraic algorithms. In order to simplify the complex calculations needed for their algorithms the astronomers used pre-calculated tables of reciprocals, squares, cubes, square roots and cube roots.


Cuniform reciprocal table Source

The ancient Greeks, who inherited their astronomy from the Babylonians, based their astronomical models on geometry rather than algebra and so needed other calculation aids. They developed trigonometry for this work based on chords of a circle. The first chord tables are attributed to Hipparkhos (c. 190–c. 120 BCE) but they did not survive. The oldest surviving chord tables are in Ptolemaeus’ Mathēmatikē Syntaxis written in about 150 CE, which also contains a detailed explanation of how to calculate such a table in Chapter 10 of Book I.


Ptolemaeus’ Chord Table taken from Toomer’s Almagest translation. The 3rd and 6th columns are the interpolations necessary for angles between the given ones

Greek astronomy travelled to India, where the astronomers replaced Ptolemaeus’ chords with half chords, that is our sines. Islamic astronomers inherited their astronomy from the Indians with their sines and cosines and the Persian astronomer Abū al-Wafāʾ (940–998 CE) was using all six of the trigonometrical relations that we learnt at school (didn’t we!) in the tenth century.


Abū al-Wafāʾ Source: Wikimedia Commons

Astronomical trigonometry trickled slowly into medieval Europe and Regiomontanus (1536–1576)  (1436–1476) was the first European to produce a comprehensive work on trigonometry for astronomers, his De triangulis omnimodis, which was only edited by Johannes Schöner and published by Johannes Petreius in 1533.

Whilst trigonometry was a great aid to astronomers calculating trigonometrical tables was time consuming, tedious and difficult work.

A new calculating aid for astronomers emerged during the sixteenth century, prosthaphaeresis, by which, multiplications could be converted into additions using a series of trigonometrical identities:

Prosthaphaeresis appears to have first been used by Johannes Werner (1468–1522), who used the first two formulas with both sides multiplied by two.

However Werner never published his discovery and it first became known through the work of the itinerant mathematician Paul Wittich (c. 1546–1586), who taught it to both Tycho Brahe (1546–1601) on his island of Hven and to Jost Bürgi (1552–1632) in Kassel, who both developed it further. It is not known if Wittich learnt the method from Werner’s papers on one of his visits to Nürnberg or rediscovered it for himself. Bürgi in turn taught it to Nicolaus Reimers Baer (1551–1600) in in exchange translated Copernicus’ De revolutionibus into German for Bürgi, who couldn’t read Latin. This was the first German translation of De revolutionibus. As can be seen the method of prosthaphaeresis spread throughout Europe in the latter half of the sixteenth century but was soon to be superceded by a superior method of simplifying astronomical calculations by turning multiplications into additions, logarithms.

As is often the case in the histories of science and mathematics logarithms were not discovered by one person but almost simultaneously, independently by two, Jost Bürgi and John Napier (1550–1617) and both of them seem to have developed the idea through their acquaintance with prosthaphaeresis. I have already blogged about Jost Bürgi, so I will devote the rest of this post to John Napier.


John Napier, artist unknown Source: Wikimedia Commons

John Napier was the 8th Laird of Merchiston, an independently owned estate in the southwest of Edinburgh.


Merchiston Castle from an 1834 woodcut Source: Wikimedia Commons

His exact date of birth is not known and also very little is known about his childhood or education. It is assumed that he was home educated and he was enrolled at the University of St. Andrews at the age of thirteen. He appears not to have graduated at St. Andrews but is believed to have continued his education in Europe but where is not known. He returned to Scotland in 1571 fluent in Greek but where he had acquired it is not known. As a laird he was very active in the local politics. His intellectual reputation was established as a theologian rather than a mathematician.

It is not known how and when he became interested in mathematics but there is evidence that this interest was already established in the early 1570s, so he may have developed it during his foreign travels. It is thought that he learnt of prosthaphaeresis through John Craig (d. 1620) a Scottish mathematician and physician, who had studied and later taught at Frankfurt an der Oder, a pupil of Paul Wittich, who knew Tycho Brahe. Craig returned to Edinburgh in 1583 and is known to have had contact with Napier. The historian Anthony à Wood (1632–1695) wrote:

one Dr. Craig … coming out of Denmark into his own country called upon John Neper, baron of Murcheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus as ’tis said) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportionable numbers. [Neper is the Latin version of his family name]

Napier is thought to have begum work on the invention of logarithms about 1590. Logarithms exploit the relation ship between arithmetical and geometrical series. In modern terminology, as we all learnt at school, didn’t we:

Am x An = Am+n

Am/An = Am-n

These relationships were discussed by various mathematicians in the sixteenth century, without the modern notation, in particularly by Michael Stefil (1487–1567) in his Arithmetica integra (1544).


Michael Stifel Source: Wikimedia Commons


Michael Stifel’s Arithmetica Integra (1544) Source: Wikimedia Commons

What the rules for exponents show is that if one had tables to convert all numbers into powers of a given base then one could turn all multiplications and divisions into simple additions and subtractions of the exponents then using the tables to covert the result back into a number. This is what Napier did calling the result logarithms. The methodology Napier used to calculate his tables is too complex to deal with here but the work took him over twenty years and were published in his Mirifici logarithmorum canonis descriptio… (1614).


Napier coined the term logarithm from the Greek logos (ratio) and arithmos (number), meaning ratio-number. As well as the logarithm tables, the book contains seven pages of explanation on the nature of logarithms and their use. A secondary feature of Napier’s work is that he uses full decimal notation including the decimal point. He was not the first to do so but his doing so played an important role in the acceptance of this form of arithmetical notation. The book also contains important developments in spherical trigonometry.

Edward Wright  (baptised 1561–1615) produced an English translation of Napier’s Descriptio, which was approved by Napier, A Description of the Admirable Table of Logarithmes, which was published posthumously in 1616 by his son Samuel.


Gresham College was quick to take up Napier’s new invention and this resulted in Henry Briggs (1561–1630), the Gresham professor of geometry, travelling to Edinburgh from London to meet with Napier. As a result of this meeting Briggs, with Napier’s active support, developed tables of base ten logarithms, Logarithmorum chilias prima, which were publish in London sometime before Napier’s death in 1617.


He published a second extended set of base ten tables, Arithmetica logarithmica, in 1624.


Napier’s own tables are often said to be Natural Logarithms, that is with Euler’s number ‘e’ as base but this is not true. The base of Napierian logarithms is given by:

NapLog(x) = –107ln (x/107)

Natural logarithms have many fathers all of whom developed them before ‘e’ itself was discovered and defined; these include the Jesuit mathematicians Gregoire de Saint-Vincent (1584–1667) and Alphonse Antonio de Sarasa (1618–1667) around 1649, and Nicholas Mercator (c. 1620–1687) in his Logarithmotechnia (1688) but John Speidell (fl. 1600–1634), had already produced a table of not quite natural logarithms in 1619.


Napier’s son, Robert, published a second work by his father on logarithms, Mirifici logarithmorum canonis constructio; et eorum ad naturales ipsorum numeros habitudines, posthumously in 1619.


This was actually written earlier than the Descriptio, and describes the principle behind the logarithms and how they were calculated.

The English mathematician Edmund Gunter (1581–1626) developed a scale or rule containing trigonometrical and logarithmic scales, which could be used with a pair of compasses to solve navigational problems.


Table of Trigonometry, from the 1728 Cyclopaedia, Volume 2 featuring a Gunter’s scale Source: Wikimedia Commons

Out of two Gunter scales laid next to each other William Oughtred (1574–1660) developed the slide rule, basically a set of portable logarithm tables for carry out calculations.

Napier developed other aids to calculation, which he published in his Rabdologiae, seu numerationis per virgulas libri duo in 1617; the most interesting of which was his so called Napier’s Bones.


These are a set of multiplication tables embedded in rods. They can be used for multiplication, division and square root extraction.


An 18th century set of Napier’s bones Source: Wikimedia Commons

Wilhelm Schickard’s calculating machine incorporated a set of cylindrical Napier’s Bones to facilitate multiplication.

The Swiss mathematician Jost Bürgi (1552–1632) produced a set of logarithm tables independently of Napier at almost the same time, which were however first published at Kepler’s urging as, Arithmetische und Geometrische Progress Tabulen…, in 1620. However, unlike Napier, Bürgi delivered no explanation of the how his table were calculated.


Tables of logarithms became the standard calculation aid for all those making mathematical calculations down to the twentieth century. These were some of the mathematical tables that Babbage wanted to produce and print mechanically with his Difference Engine. When I was at secondary school in the 1960s I still carried out all my calculations with my trusty set of log tables, pocket calculators just beginning to appear as I transitioned from school to university but still too expensive for most people.


Not my copy but this is the set of log tables that accompanied me through my school years

Later in the late 1980s at university in Germany I had, in a lecture on the history of calculating, to explain to the listening students what log tables were, as they had never seen, let alone used, them. However for more than 350 years Napier’s invention served all those, who needed to make mathematical calculations well.














Filed under History of Astronomy, History of Mathematics, History of Technology, Renaissance Science

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