My mate the HISTSCI_HULK, known to his friends as Hulky, was perusing our email website this morning, which as well as being the repository for our electronic mail is a sort of online newspaper, unfortunately rather close to the gutter press. He had just begun reading an editorial on the heatwaves, caused by the climate crisis, that the world is currently suffering and the resulting high number of deaths caused by heat exhaustion, when he began to splutter, his agitation quickly turning to an outbreak of HISTSCI rage!
“BLEEDIN’ BRAIN DEAD IMBICILE!” he screamed, threatening to throw the computer out of the window.
“DON’T THE IDIOTS LEARN ANTHING ABOUT SCIENCE AND ITS HISTORY AT SCHOOL?” he shouted as I gently prised him away from the computer screen.
“IT’S 2024 CE FOR FUCK’S SAKE! NOT BLEEDIN’ 2024 BCE!” he howled, before storming off to pulverise some rocks.
Cautiously, I looked at the computer monitor to see what had provoked the good beast into such an explosive fury and read the following mindbogglingly ludicrous statement.
„Ein fallender Apfel soll es gewesen sein, der einst den Naturwissenschaftler Isaac Newton auf die zündende Idee brachte, dass es eine Schwerkraft gibt.“
For those of you who don’t read German in English it’s:
“It was supposedly a falling apple that once brought Isaac Newton to the explosive idea that gravity exists.”
This was written by a leading conservative political commentator and not some BRAIN DEAD IMBICILE, to use Hulky’s choice phrase, and has got to be the most stupid take on the Newton’s apple and the theory of gravity story that has ever crossed my path.
Does the author of this totally absurd statement, which is potentially going to be read by literally tens of millions of people, who use this email service, really believe that nobody realised that gravity existed before the young Isaac, sitting in his mother’s garden in Woolsthorpe during an outbreak of the plague in 1665, observed an apple falling from a tree?
Alone the etymology of the word gravity, Latin gravitatum from gravis meaning heavy, used in Latin translations to express Aristotle’s concept that naturally heavy things, i.e. those with gravity, are attracted to the Earth and fall downwards when dropped, should have given our author a clue that people have always known that gravity exists and that Newton did not to realise that gravity exists in 1665 but twenty years later he provided a scientific explanation for why it exists.
At the beginning of the first episode of this series I wrote the following:
In popular histories of science in Europe the history of physics is all too often presented roughly as follows, in antiquity there was Aristotle, whose writings also dominated the Middle Ages, until Galileo came along and dethroned him, following which Newton created modern physics.
We have already seen that already beginning in the sixth century CE with John Philoponus, the impetus theory that would be taken up and developed by both Islamic and medieval European scholars offered a strong alternative to Aristotle’s theory of projectile motion. Philoponus, and many others, also questioned his theories of fall and as we have seen in the fourteenth century, the Oxford Calculatores and the Paris Physicists did quite a lot on the laws of fall that is usually credited to Galileo.
What is very often ignored in that Galileo was very much aware of substantial work done on both projectile motion and fall in the sixteenth century on which he built his own theories.
The first scholar to make an important contribution to the physics of motion during the sixteenth century was Niccolò known as Tartaglia (1499–1557). Although, often referred to as Niccolò Fontana, his actual surname is not known for certain. He came from simple circumstances and suffered much tragedy in his childhood. He was born in Brescia, in the Lombardy, the son of Michele a dispatch rider, who was murdered when he was just six years old. In 1512, French troops invaded Brescia and although his family sought refuge in the cathedral, the French troops entered the building and the young Niccolò was slashed across the face with a sabre slicing open his jaw and palate and leaving him for dead. His mother nursed him back to life but he was left with a speech impediment , which earned him the nickname Tartaglia, the stammerer. He grew a beard to cover his scars.
Source: Wikimedia Commons
Largely self-taught, he moved to Verona around 1517 and then to Venice in 1534. He earned his living teaching practical mathematics in abbacus schools. A Maestro d’abaco or reckoning master Tartaglia was one of the first to transcend the world of practical mathematics that was common for the period and in which mathematicians were viewed not as scholars but as craftsmen, and in many senses became a mathematician in the modern meaning of the term. This transition of mathematicians from craftsmen to scholars was only truly completed a century later thanks largely to the contributions of Kepler and Galileo.
Tartaglia is, of course, best known as the second mathematician after Scipione del Ferro to discovery a general solution for some forms of the cubic equations, a solution wider ranging than that of Scipione. I wrote about this briefly in the last episode and more fully in two earlier posts, here and here, so I won’t repeat it here. It’s also not directly relevant to the topic of this episode.
As well as his work on the cubic equation, Tartaglia also wrote a typical reckoning mater guide to elementary mathematics his General trattato di numeri et misure, 6 pts. (Venice, 1556–1560).
General trattato de’ numeri et misure, 1556 Source: Wikimedia Commons
In the second part of which he includes the triangle of binomial coefficients, known generally as Pascal’s Triangle, who first published it a hundred years later.
Tartaglia’s triangle from General Trattato di Numeri et Misure, Part II, Book 2, p. 69 Source: Wikimedia Commons
One should point out that Peter Apian (1495–1552) had already published it in his Ein newe und wolgegründete underweisung aller Kauffmanns Rechnung in dreyen Büchern, mit schönen Regeln und fragstücken begriffen, in 1527. It was earlier published by al-Karaji (953–1029), by Omar Khayyám (1048–1131). In Arabic it’s known as Kayyám’s Triangle. In China it was first published by Jia Xian (1010–1070) in the early eleventh century, and by Yang Hui (1238–1298) in the thirteenth century were it is known as Yang Hui’s Triangle.
In 1543, Tartaglia produced the first translation of the Elements of Euclid into Italian, which was also the first translation into the vernacular. There was a second edition in 1565 and a third in 1585. It appears that he translated from Latin not Greek and in his second edition he mentions the first translation by Campano, that is the Latin edition of Campanus of Novara ( c. 1220–1296), which was based on the translation of Robert of Chester (12th century) and which became the first printed edition, published by Erhard Ratdolt (1442–1528) in Venice in 1482. However, there is reason to believe that Tartaglia’s translation is actually based on the 1505 Latin translation direct from the Greek of Bartolomeo Zamberti (c.1473–after 1543) published in Venice.
In 1543,Tartaglia also produced a seventy-one page edition of the Latin translation of the works of Archimedes by William of Moerbeke ( between 1215 & 1235–c. 1286), Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi containing Archimedes’ works on the parabola, the circle, centres of gravity, and floating bodies. In 1551, he published an Italian translation of parts of the Archimedes text, part of Book I of De insidentibus aquae. His active interest in Archimedes would have an influence on his one time pupil Giambattista Benedetti (1530–1590), as we will see in a later episode.
Tartaglia’ contribution to the laws of motion were not made in what we would recognise as a work of physics but in what was the first ever mathematical treatise on gunnery or ballistics, which of course is the study of projectile motion. The earliest known mention of gunpowder in European literature is by Roger Bacon in his Opus Major in 1267. Depiction of guns begin to appear in the early fourteenth century. The use of cannons in warfare, particularly during sieges, developed over the fourteenth and fifteenth century. The invention of the gun carriage at the end of the fifteenth century saw the introduction of field artillery and the need for a science of gunnery or ballistics. A need that Tartaglia became the first to fulfil.
Tartaglia’s first published book Nova Scientia (Venice,1537), a tome on ballistics, was in the words of historian Matteo Valleriani:
In 1537, a mathematician from Brescia, Nicolò Tartaglia (1500–1557) published a work entitled Nova scientia. It is this work that established the modern science of ballistics, as characterized by the search for a mathematical understanding of the trajectory of projectiles. Tartaglia’s intentions were to create a science based on axioms and more geometrico, fundamental to the entire subject of mechanics, starting from a limited number of principles and arriving at a series of propositions through a process of rigid deduction. The methodological model Tartaglia intended to follow was the one he was able to extrapolate from works like Euclid’s Elements.
[…]
However, from a wider perspective, more specifically from the perspective of the entire history of the development of mechanics during the Renaissance, Tartaglia’s most important achievement is having demonstrated in 1537 that an exact science of ballistics was possible, based on the application of mathematical and geometrical methods. Challenged by the knowledge and experience of the bombardier, Tartaglia made an enormous contribution to the field of mathematical physics.[1]
Nova Scientia frontispice
The book was very successful and very widely read. There was a second edition published in 1550 with reprints in 1551 and 1558. Further reprints were made in 1562, 1583, and 1606. There were translations into French and English.
Tartaglia wrote a second more widely ranging book including the topic of artillery his Quesiti et Inventioni Diversi published in 1554.
In this work Tartaglia dealt with algebraic and geometric material (including the solution of the cubic equation), and such varied subjects as the firing of artillery, cannonballs, gunpowder, the disposition of infantry, topographical surveying, equilibrium in balances, and statics.[2]
Galileo was influenced by the works of Tartaglia and owned a richly annotated copies of his works on ballistics. These contained the first statement of the theorem that
…the maximum range, for any given value of the initial speed of the projectile, is obtained with a firing elevation of 45°. The latter result was obtained through an erroneous argument, but the proposition is correct (in a vacuum) and might well be called Tartaglia’s theorem. In ballistics Tartaglia also proposed new ideas, methods, and instruments, important among which are “firing tables.”[3]
Fig. 2.1: Representation of a cannon positioned at a 45-degree angle of elevation as verified by means of the bombardier’s quadrant. From Tartaglia 1558.
Although he rejected the theories of Aristotle, Tartaglia’s work was informed by the impetus theory and his projectiles did not fly along parabolic trajectories. Tartaglia’s trajectories were in three segments, first a straight line upwards, then a curve, and finally a fall straight down to earth when the impetus was exhausted and gravity took over.
Nova Scientia 1606 Ballistic curve
Through his work on ballistics Tartaglia had a major impact on the physics of projectile motion in the first half of the sixteenth century. It continued to be a major influence in the practical field of gunnery well into the eighteenth century.
Addendum:
Jacopo Bertolotti, Associated Professor of Physics at the University of Exeter posted the following on social media inspired by this post:
If you ever studied any Physics in school you probably know that the trajectory of an object in a uniform gravitational field will be a parabola. But if the drag is not negligible, the trajectory will be much more skewed, and it will fall almost vertically.
Riccioli as portrayed in the 1742 Atlas Coelestis (plate 3) of Johann Gabriel Doppelmayer. Source: Wikimedia Commons
Amongst other things, Riccioli distinguished himself as the teacher of Giovanni Domenico Cassini (1625–1712) one of the greatest observational astronomers of the seventeenth century. He was also the physicist who gave a direct, empirical physical proof of the laws of fall using pendulums to time balls dropped in the Torre de Asinelli in Bologna. He was the first to realise that if diurnal rotation exist then there must be, what later got named, the Coriolis Effect. He was an excellent physicist and tried to detect it but failed, it in fact proved very difficult to detect, he concluded that there is no diurnal rotation. Together with Francesco Maria Grimaldi (1618–1663) he produced a highly influential map of the Moon. However, he is perhaps best known for his mammoth, encyclopaedic astronomy reference book, his 1651 Almagestum Novum.
This is basically a summation of all astronomical knowledge known in the middle of the seventeenth century including details of how it was acquired, albeit written from the standpoint of a devout Catholic, who rejected the heliocentric hypothesis. Originally planned as a three volume work, Riccioli only ever wrote the first volume but that is fifteen hundred pages long and was originally published in two parts. In its time, it was regarded as an important reference work by astronomers throughout Europe. However, it has not fared well historically.
When I first got interested in the history of astronomy in the late 1960s early 1970s, the standard popular presentation was that Copernicus published his De revolutionibus in 1543 against push back from a reactionary Catholic Church. Kepler added his three laws of planetary motion and Galileo carried the day for heliocentricity with his Dialogo in 1632, for which he was punished by that reactionary Church. Riccioli’s Almagestum Novum, published, as it was, in 1651being geocentric was regarded as an anachronistic relic and not worth wasting time on. Dismissed as some sort of dying gasp by those reactionary Jesuits, it was largely ignored and never translated in English.
Now a new champion has entered the ring and taken up the fight on behalf of Riccioli. Michal J. A. Paszkiewicz has taken on the monumental task of producing an annotated translation of the whole of the Almagestum Novum. On his own admission, he is going to require twenty, (yes, you read that right, twenty) volumes to complete the task and what follows is a review of the first of those planned volumes, Almagestum Novum: History of Astronomy; Giovanni Battista Riccioli; English Translation & Commentary.[1] One can only hope that he lives long enough to complete what is an excellent endeavour both in its scope, as well as in its execution. If you have a serious interest in the history of astronomy in the seventeenth century you should immediately acquire a copy!
An English translation of the Almagestum Novum would in itself be an important addition to the tools available to those interested in the history of seventeenth-century, European astronomy. However, Paszkiewicz takes the Commentary and the very extensive additions that he has made to Riccioli’s text make his translation an even more valuable tool.
The frontispiece of the Almagestum Novum is one of the most famous illustrations in the history of astronomy and Paszkiewicz devotes his third chapter to a very detailed description of the iconography of what he describes as “a marvellous work that is not just a beautiful image, but also a statement or even credo that Riccioli presents as a key to his work.”
Source:Wikimedia Commons
As with any scientific work of the period the Almagestum Novum opens with a dedication to Riccioli’s patron Cardinal Girolamo Grimaldi. In his fourth chapter Paszkiewicz introduces the House of Grimaldi and the Cardinal before in Chapter 5 presenting the translation of the dedication followed in Chapter 6 by the translation of the imprimatur.
We now move on to the preface. Usually, the word preface brings to mind a couple of pages where the author sets out his motivation and aims in writing the book, not so by Riccioli. What we have here is seven sections spread over a total of thirty-nine pages in which Riccioli pays tribute to astronomy, explains its proper use, sketches its origins, outlines its historical progress, names its deficits, presents the new astronomy, and finally on the last two pages names his aims for this work. This preface alone is a fascinating insight into how astronomy was viewed in the middle of the seventeenth century and could be presented as a stand alone publication.
The book proper opens with a timeline of astronomers and astrologers. However, before presenting it Paszkiewicz gives an extensive in-depth analysis of this timeline. He explains the problems of identifying some of the persons, who might be behind a given name. He lists those missing here but are in the following alphabetical biographies and those who are missing from both lists, which includes some surprising omissions. His notes close with an interesting essay on the mythological characters included in this timeline. The following chapter compares the Almagestum Novum timeline with the one Riccioli includes in his Chronicon Magnum timeline from the second volume of his Chronologiae reformatae published in 1669.
He presents both the timeline and the following much more extensive Alphabetical listof astronomers and astrologers in a very useful format of two column, with the original texts in the lefthand column and the translations on the right. Both the timeline and the alphabetical list contain a wealth of information. Here I have a very minor personal quibble. Paszkiewicz refers to Clavius in his translation as Christopher Clavius, which irritates me. Clavius is German and his name is Christoph not Christopher. I personally think that one should not translate names into their equivalent in other languages. The English king who had six wives was named Henry and not Heinrich as all too many German sources have it. The Alphabetical list which closes Paszkiewicz’s first volume of translation and occupies over one hundred and fifty pages.
This is a truly exceptional endeavour and it’s executed excellently. It has excellent explanatory footnotes where required and a useful index. Apart from Riccioli’s fabulous frontispiece and the diagrams of the planetary systems in the opening explanatory material there are no illustrations.
I really can’t recommend this valuable addition to the English language history of astronomy source material highly enough and await eagerly the following volumes, not that I will live long enough to read all of them.
[1] Michal J. A. Paszkiewicz, Almagestum Novum: History of Astronomy; Giovanni Battista Riccioli; English Translation & Commentary, volume 0, version 1.0.0. 1651/2023, Cricetus Cricetus Ltd, 2023.
The standard clichéd narrative of the so-called Scientific Revolution features the mathematisation of the sciences as a central element. Whilst it is in fact historically correct that a change in the status of mathematics and the development of new mathematical disciplines played a significant role in the evolution of the sciences, in particular those disciplines that came to be presented together under the term physics, this process actually began well before the seventeenth century, the traditional period assigned to the Scientific Revolution.
A radical change in the status of mathematics and mathematicians took place in the time span between the High Middle Ages and the Early Modern period. It was not a sudden change but rather a gradual one over a couple of centuries. It was also not a linear progression but rather a stumbling series of changes in different areas. Different branches of mathematics progressed at different speeds, and leading practitioners, at times, questioned and even rejected the progress made by their predecessors. Beginning around fourteen hundred, it wasn’t until about seventeen hundred that mathematics had acquired the form and status that it enjoys today.
On the medieval European universities mathematics was sorely neglected. Although the quadrivium–arithmetic, geometry, music, astronomy– was nominally the second, advanced part of an undergraduate degree it was largely neglected in favour of the works of Aristotle. Arithmetic and music were taught on the basis of the texts from Boethius, which were extremely elementary. In theory geometry fared better being based on the Elements of Euclid, but on paper they only taught the first six of the thirteen books; a fact reflected in the fact that the early vernacular translation were also only the first six book. However, in fact many courses never got further than Book I. Astronomy should have fared better but the normal university course was based on the Sacrobosco’s De sphaera mundi (On the Sphere of the World) from the thirteenth century, a non-technical description of Ptolemaic astronomy. Outside of the universities, in the world of trade and commerce, people did their calculation on the abacus or counting board, using Roman numerals to record the results but not the calculations themselves.
The introduction of the Hindu-Arabic place value, decimal number system shows very clearly the very slow adoption process. In the twelfth century an unknow translator translated al-Khwārizmī’s text on the system into Latin, where it became known by various names, today it is usually referred to as Algoritmi de Numero Indorum. Algorithmi being a corruption of his name and the source of the word algorithm. In the Middle Ages, algorism referred to doing arithmetic using the place value, decimal number system. This first introduction only found use on the universities in Computus, the calculation of the dates of Easter and the other moveable Church festivals.
Page from a Latin translation of Algoritmi de Numero Indorum, beginning with “Dixit algorizmi” Source: Wikimedia Commons
Outside of the universities people continued to use their counting boards and Roman numerals. The abacus or counting board is, of course, a material representation of the place value, decimal number system.
Gregor Reisch Margarita Philosophica 1508 Algorists vs. abacists, depicted in a sketch Source: Wikimedia Commons
The system was introduced a second time by Leonardo of Pisa (c. 1170–c. 1240-1250) in his Liber Abbaci(The Book of Calculation) in 1202. This time the system found more uptake but in the world of commercial arithmetic and not the world of mathematics and even there only slowly.
A page of the Liber Abbaci Source: Wikimedia Commons
The uptake increased in speed with the introduction in Europe of double-entry bookkeeping in the early fourteenth century but here the uptake was also slow, with the authorities in a couple of the North Italian commercial centres banning their use in accounts because they were easier to manipulate after the fact than words or roman numerals. At the end of the fifteenth century, Luca Pacioli (c. 1447–1517) published the first printed account of double-entry bookkeeping in his Summa de arithmetica, geometria, proportioni et proportionalita (Summary of arithmetic, geometry, proportions and proportionality) in 1494, which could be considered the watershed in the acceptance of the Hindu-Arabic number system, although the counting board continued to dominate and the Roman numeral system continued to be widely used. In the course of the sixteenth century the Hindu-Arabic place value, decimal number system finally became dominant, a significant factor in the acceptance being the invention of printing.
Luca Pacioli Summa title page 2nd edition 1523 Source: Wikimedia Commons
We have a similar drawn out process of acceptance for algebra. Taking algebra to be in its origins the theory of equations then the Ancient Egyptians, the Babylonians, the Chinese and the Indians all developed and practiced algebra in one way or another. Only the Ancient Greeks form an exception. Instead of a branch of mathematics that deals with unknown number quantities that can be found by solving some form of equation , the Greek developed a geometric algebra in which the quantities under discussion are not numbers per se but line segments so what we now know as X squared was literally a square with a side length of X. This is why we still talk about quadratic and cubic equations, as for the Greeks, they dealt literally with squares and cubes. Although in the third century CE, Diophantus of Alexandria (c. 200-c. 214–c. 284-298) wrote his Arithmetica with arithmetical problems solved by algebraic equations.
Once again algebra came into medieval Europe with the translation of a work by al-Khwārizmī, his al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah (The Compendious Book on Calculation by Completion and Balancing), by Robert of Chester in 1145, which gave us the word algebra.
A page from al-Khwārizmī’s Algebra Source: Wikimedia Commons
Once again it had little impact and as before it was Leonardo of Pisa’s Liber Abbaci, which included quite a lot of algebra taken from al-Khwārizmī, which successfully launched algebra in Europe but once again not in abstract mathematics but as part of commercial arithmetic. Leonardo’s book led to the setting up of so-called abbacus schools. Small private schools that taught Hindu-Arabic arithmetic and the rudiments of algebra to the apprentices of traders. These started in Northern Italy and gradually spread throughout Europe. The teachers in these schools also wrote and published textbooks, known as abbacus book, explaining the basics of Hindu-Arabic arithmetic and algebra.
The algebra of al-Khwārizmī/Leonardo of Pisa is a rhetorical algebra with no use of symbols and with everything spelt out in words. Gradually a so-called syncopated algebra developed that uses some symbols, often abbreviated words, as had been the case in the works of Diophantus and the Indian mathematician Brahmagupta (c. 598–c. 668 CE), whose Brāhma-sphuṭa-siddhānta was the principle source of al-Khwārizmī’s works on the Hindu-Arabic number system and algebra. A full symbolic algebra evolved over time with various people making contributions. The French mathematician François Viète (1540–1603) is credited with his In artem analyticem isagoge (Introduction to the art of analysis) published in 1591 and largely derived from Diophantus rather that the Arabic stream, as having published the first fully symbolic algebra but this is not strictly true as it is not entirely symbolic.
Almost a hundred years earlier, algebra had begun to cross over from commercial arithmetic to abstract mathematics. The general solution of the quadratic equation had been known since Brahmagupta and Luca Pacioli in his Summa stated that there was no general solution for the cubic equation. Famous last words! The Italian mathematician from the University of Bologna Scipione del Ferro (1465–1526) found a solution to one form of the cubic equation, which however he didn’t publish. After his death his student Antonio Fiore acquired his solution and, as was the common in those days, challenged Niccolò Fontana (c. 1499–1557), known as Tartaglia, to a public contest solving mathematical problems.
Tartaglia, a very talented mathematician, was aware that Fiore has a solution to one form of the cubic and realised that he was on to a hiding to nothing, so he sat down and discovered a much more general solution to the cubic. Come the day of the contest he trounced Fiore and became much feted, retaining however, his solution for himself. The polymath Girolamo Cardano (1501–1576), mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler, who had a much higher social status than Tartaglia, basically bribed him into revealing the secret of his solution. Tartaglia did so only on the condition that Cardano would not publish it until Tartaglia had published it, Cardano promised he would not do so.
However, Cardano travelled to Bologna and discovered Scipione del Ferro’s solution, which predated Tartaglia’s, and so no longer felt himself bound by his promise to Tartaglia. In the meantime, he had found the complete general solution to the cubic equation and his student Lodovico de Ferrari (1522–1565) had discovered the general solution to the quartic equation. In 1545, Cardano’ Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra), containing the general solutions to the cubic and quartic equation but acknowledging the contribution of Tartaglia and Lodovico de Ferrari was published by Johannes Petreius (c. 1497–1550) in Nürnberg.
Ars Magna title page Source: Wikimedia Commons
The Ars Magna has, by those who indulge in such hyperbole, been called the first modern maths book. It certainly established algebra as a branch of mathematics in Europe and not just a tool in commercial arithmetic. A year earlier Petreius had also published the Arithmetica integra by the Lutheran cleric, and mathematician Michael Stifel (1487–1567) a volume of arithmetic and algebra, which went a long way along the route to a symbolic algebra.
Titelblatt von Michaels Stifels “Arithmetica Integra” (1544).
Stifel was building on the work of Christoph Rudolff (1499–1545), whose Behend und hübsch Rechnung durch die kunstreichen regeln Algebre so gemeinicklich die Coss genent werden (Nimble and beautiful calculation via the artful rules of algebra, so commonly called “coss”), which was an advanced abbacus book published in 1525, whereas Stifel crossed the line from commercial arithmetic into mathematics. The word “coss” derived from the Italian “cosa” meant thing or the unknown i.e. that which is to be solved. The early German practitioners of algebra were known as “Cossists.”
Rudolff Behend und hübsch Rechnung Source: Wikimedia Commons
Both the books of Cardano and Stifel established something in mathematics that we considered perfectly normal and commonplace, negative numbers. For most of the history of mathematics, mathematicians have regarded the concept of negative numbers as an oxymoron. How can you have a quantity less than nothing? The Babylonians had a general solution for quadratic equations by about seventeen hundred BCE. However, several historians of mathematics point out, quite correctly, that it wasn’t a general solution because they only recognised positive solutions. Interestingly Brahmagupta, who, as noted above, was the primary source for both the Hindu-Arabic arithmetic and algebra for Arabic mathematicians, had basically the same general solution with both positive and negative solutions that is taught in schools today and gave all the usual rules for manipulation of negative numbers. Although the Arabic mathematicians took over his whole book they generally ignored negative numbers and in their own work did not pass them onto Europe. Leonardo of Pisa made a limited use of negative values in his commercial arithmetic but described them as debts. Interestingly, double-entry bookkeep doesn’t use negative values, income and outgoings both being written as positive values in their respective columns. However, it was through trade and commerce that the concept of negative values began to become accepted in Europe. The plus and minus symbols were originally developed to signify that standard bales of good were over or under weight.
As already noted at the beginning of the previous paragraph, both Cardano and Stifel acknowledge that equations could have both positive and negative solutions, Stifel for quadratic equations and Cardano for quadratic, cubic, and quartic equations. However, Cardano was always careful to write out cubic equations in positive terms, transferring negative terms to the other side of the equals sign to make them positive. Cardano went one step further; he introduced the square root of minus one! It is obvious that if you know the rules for manipulation of both positive and negative numbers that the square of any number, whether positive or negative, is always positive, so how can you have a square root of minus one? Cardano didn’t know but he realised that in the solution of some cubic equations he had impossible terms involving square roots of negative numbers that cancelled out when one calculated further. He had stumbled across conjugate pairs of complex numbers, which when multiplied together become normal numbers. Cardano didn’t like what he had stumbled across but he had the wisdom to retain it.
The next truly significant algebra book was L’Algebra of the Italian mathematician Rafael Bombelli (1526–1572) published in 1572. In this book, Bombelli spells out the rules for manipulating positive and negative numbers and then goes one step further, taking up Cardano’s innovation and running with it. Bombelli gives a comprehensive account of the arithmetic of both imaginary and complex numbers, although he doesn’t use those terms. He refers to ‘i’ as “plus of minus” and ‘-i’ as minus of minus. Of course, the acceptance of imaginary numbers took time as many were suspicious of them. Descartes, who was not really happy with negative numbers intensely disliked imaginaries and he was the person who coined the term imaginary for them. Descartes was not alone in his distrust of negative numbers, as late as the beginning of the nineteenth century, the social reformer, William Frend (1757–1841), father in law to Augustus De Morgan (1806–1871) and one time maths teacher to Ada Lovelace (1815–1852) still rejected negative numbers.
Source: Wikimedia Commons
As already noted above the next important text was Viète’s In artem analyticem isagoge in 1591. Significantly, Viète rejected the Arabic word algebra preferring the Latin term analysis and in doing so emphasised the difference between the methods of proof of synthetic Euclidian geometry and analytic algebra. Interesting is how people at the beginning of the seventeenth century reacted to the newly established analytical mathematics.
The supposedly conservative, Jesuit mathematician Christoph Clavius (1538–1612), who was responsible for introducing mathematics as a major educational element in the Jesuit schools and colleges, for which he wrote the textbooks and trained the teachers, also wrote and published a textbook on Viète’s algebra in 1608, despite the fact that he and Viète had argued strongly about the mathematics of the Gregorian calendar. At the same time the supposedly modern, even revolutionary Galileo (1564–1642) rejected the new analytical mathematics.
The fifteenth century saw a major renaissance of the works of Archimedes in Europe. His method of exhaustion was used by Kepler and others to determine areas and volumes as a primitive form of integral calculus. The determination of areas and volumes and the determination of rates of change, the basis of the differential calculus, became major themes in the mathematics of the seventeenth century. Both problems requiring the completion of infinite sums. Bonaventura Cavalieri (1598–1647), Grégoire de Saint-Vincent (1584–1667), Gilles de Roberval (1602–1675), Pierre de Fermat (1607–1665), Blaise Pascal (1623–1662), René Descartes (1596–1650), Frans van Schooten Jr. (1615–1660), John Wallis (1616–1703), Isaac Barrow (1630–1677), and James Gregory (1638–1675) all made contributions to both sets of problems, their results being collated and formalised by both Gottfried Leibniz (1646–1716) and Isaac Newton (1642–1726) to create what became known as calculus and later the analysis. In the same period Pierre de Fermat and René Descartes independently of each other combined algebra and geometry to create analytical geometry. This was improved on by a collective of mathematicians under the direction of Frans van Schooten Jr. in his expanded Latin translation of Descartes La Géométrie.
Source: Wikimedia Commons
Having followed the path of Hindu-Arabic arithmetic and algebra from their introduction into Europe the twelfth century down to the creation of calculus during the seventeenth century, we now return to the beginning of the fifteenth century to follow the development within Europe of another branch of mathematics, trigonometry.
In 1406 the first Latin translation of Ptolemaeus’ Geographia was made in Europe; this unleashed a new era in cartography, which brough with it increased interest in accurate astronomy to determine longitude and latitude. Around the same time there developed major new interest in navigation and surveying. All of these activities require trigonometry. There was, of course, some trigonometry in medieval Europe but compare to the level of development that trigonometry had seen in Islamic astronomy and geometry it was very low level. This began to change in the first half of the fifteenth century.
First in the so-called first Viennese School of Mathematics, which made major developments in both astronomy and cartography, Johannes von Gmunden (c. 1380–1442), Georg von Peuerbach (1423–1461), and Regiomontanus (1436–1476) began to push the development of trigonometry. This led to new more accurate sine tables from Peuerbach and Regiomontanus’ De Triangulis omnimodis (On Triangles) in 1464, which was however posthumously first edited by Johannes Schöner (1477–1547) and published by Petreius in Nürnberg in 1533.
This was the first account of nearly the whole of trigonometry published in Europe, only the tangent was missing, which Regiomontanus presented separately in his Tabulae directionum profectionumque written in 1467 but again first published in print posthumously in 1490.
Peuerbach and Regiomontanus also in their Epitome in AlmagestumPtolemei, published in Venice in 1496, replaced Ptolemaeus’ cord tables with modern trigonometrical tables.
Frontispiece of ‘Epitome in Ptolemaei Almagestum’
In 1533 in the third edition of the Apian/Frisius Cosmographia, Gemma Frisius (1508–1555) published as an appendix the first account of triangulation in his Libellus de locorum describendum ratione. This laid the trigonometry-based methodology of both surveying and cartography, which still exists today.
Gemma Frisius’s 1533 diagram introducing triangulation into the science of surveying Source: Wikimedia Commons
In the late fourteen hundreds as the Europeans began to explore the world outside of Europe by sea. Trigonometry became an important tool for the navigators on those voyages.
Copernicus’ De revolutionibus, also published by Petreius in Nürnberg in 1543, contained new extensive trigonometrical tables, which had already been published separately by Georg Joachim Rheticus (1514–1574) in an extended and improved form under the title De lateribus et angulis triangulorum (On the Sides and Angles of Triangles) in Wittenberg in 1542.
Rheticus would go on to devoting a large part of his life to improving trigonometrical tables. In 1551 he published Canon doctrinae triangvlorvm in Leipzig.
Source: Wikimedia Commons
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. This work was republished in expanded editions in 1600, 1608 and 1612.
Source: Wikimedia Commons
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.
Going into the seventeenth century trigonometry had grown into a highly developed and significant branch of mathematics. 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:
Source: Wikimedia Commons
Prosthaphaeresis appears to have first been used by Johannes Werner (1468–1522). 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) 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 superseded by a superior method of simplifying astronomical calculations by turning multiplications into additions, logarithms.
Logarithms were developed independently by both Jost Burgi and John Napier (1550–1617) both of whom had been introduced to the method of Prosthaphaeresis. It is based on the relationship between arithmetical and geometrical series:
Am x An= Am+n
Am/An= Am-n
A relationship that had already been discussed by various sixteenth-century mathematicians, most notably by Michael Stifel in hisArithmetica integra.
Today imaginary numbers, trigonometry and logarithms all deliver mathematical functions in analysis that play important roles in physics, but at the beginning of the seventeenth century these still lay in the future. However, with the Hindu-Arabic arithmetic, algebra, the evolving calculus, trigonometry, and logarithms the seventeenth century proto-physicists had a much more extensive and powerful mathematical tool box with which to undertake their investigations than their medieval predecessors had possessed.
Over this series we have tracked the discovery of magnetic variation and the gradual realisation that it was a real phenomenon and not just a malfunction of badly made or adjusted compasses. This early phase was followed by attempts to track and understand the causes of variation. These attempts were initially driven by a belief, or perhaps better desire, that variation was somehow regular and could be used to fulfil the long held desire to find a way to determine longitude at sea. With time it became clear that there was no discernible regularity in magnetic variation; a disappointment that was further increased by the discovery that variation changed not just with geographical location but for any given location over time. All of this seemed to make the concept of systematically and accurately mapping variation a waste of time and effort. So, it comes as somewhat as a surprise that late in the seventeenth century the English government financed a major attempt to do just that.
The man, who suggested a systematic mapping of the magnetic variation of the Atlantic and in fact would go on to carry it out, was Edmond Halley (1656–1741). For most people the name Halley conjures up visions of his eponymous comet but Edmond was a man of many talents and many achievements. He had already had experience of long distant voyaging on the Atlantic, whilst still a student at Oxford. He took up contact with the Astronomer Royal, John Flamsteed (1646–1719) concerning astronomical observations in 1675 and in 1676 Flamsteed help him publish his first paper, A Direct and Geometrical Method of Finding the Aphelia, Eccentricities, and Proportions of the Primary Planets, Without Supposing Equality in Angular Motion, in the Philosophical Transactions of the Royal Society. Flamsteed was engaged in producing a new more accurate star map of the northern celestial hemisphere, so Halley volunteered his services to do the same for the southern celestial hemisphere.
Portrait of Halley (c. 1690) Artist unknown Source: Wikimedia Commons
Halley’s endeavour found favour with King Charles II, who gave orders to the East India Company to transport Halley to the island of St Helena, the only British outpost in the Southern Atlantic, which was under the control of the East India Company. St Helena also had the advantage that it was far enough north that Halley could also observe stars in the northern celestial hemisphere for cross-reference. He sailed in November 1676, just twenty years old and still an undergraduate and upon arrival, in February 1677, set up an observatory on St Helena with a five and a half foot radius sextant with a telescopic sight and a two foot radius quadrant[1]. The weather on St Helena was not good for astronomical observation and the young Halley had a lot of difficulties making the observations he needed, but he persevered and having completed his survey of the stars he left St Helena for the return journey in March 1678. Unfortunately, we have no accounts of either journey, so we don’t know how much Halley learnt during them about marine navigation.
Source: Wikimedia Commons
During his time on St Helena, he made two other important scientific observations. He tried to determine the astronomical unit, the distance between the Earth and the Sun during a transit of Mercury but unfortunately there were not enough observations made by others from other locations on the Earth, so the figure he arrived at was unsatisfactory. Secondly, he had taken a pendulum clock with him as part of his portable observatory and he discovered that he had to shorten the pendulum for it to keep accurate time. This meant that the effect of gravity was different on St Helena to England. This was one of the results that led to the investigation of the shape of the Earth in the middle of the next century.
Remains of Halley’s Observatory on St Helena
Once he was back in England he prepared his acquired observational data for publication in a catalogue of 341 southern stars with the wonderful title:
A Catalogue of the Southern Stars, or a supplement to the Catalogue of Tycho showing the longitudes and latitudes of the Stars, which being close to the southern pole are invisible from the Uraniborg of Tycho, accurately reckoned from measured distances and completely corrected to the year 1677, with those very observations of the heavens produced with great care and a sufficiently large sextant on the island of St Helena which lies in latitude 15° 55’ south and longitude 7° 00’ west of London. A labour so far needed in Astronomy. To which is added a small note concerning things no unwanted in Astronomy.[2]
It was published in November of 1678. Before the catalogue was published, John Seller (1632–1697), the royal hydrographer, had already published a chart of the southern celestial hemisphere based on Halley’s observations and engraved by a James Clark of Clerk (d. 1718). This was dedicated to the king, Charles II and contained Halley’s new constellation Robur Carolinium, Charles’ Oak.
In the top left and right corners are diagrams of the start and finish of the transit of Mercury across the Sun that Halley had observed from St Helena on October 28, 1677.
The catalogue established Halley, still only twenty-two years old as one of the leading astronomers in England. The king supported the suggestion that Oxford University should award him an MA although he hadn’t even completed his undergraduate studies before he had left the university to undertake this scientific expedition.
In 1693, Halley, now firmly established as a leading light in England’s scholarly community, together with a Mr Benjamin Middleton, proposed a much more ambitious marine scientific expedition. The Journal Book of the Royal Society records for 12 April of that year:
The President was pleased to propose a paper lately offered him by Mr. Benjamin Middleton, requesting the assistance of this society to procure for him a small vessel of about 60 Tuns to be fitted out by the Government, but to be victualled and manned at his own proper charges. And this in order to compass the Globe to make observations on the Magneticall Needle, &c. The President in the name of the Society promised to use his endeavours towards obtaining such a vessel.[3]
Nothing is known about Benjamin Middleton other than he was a fellow of the Royal Society.
The Navy Board and the Exchequer both approved the scheme. Already in October Fisher Harding, the master shipwright at Deptford began preparing the ship and the pink, later christened the Paramore (also spelt Paramour) was launched on 1 April 1694.
A pink was any small ship with a narrow stern, having derived from the Dutch word pincke meaning pinched. They had a large cargo capacity and were generally square rigged. Their flat bottoms (and resulting shallow draught) made them more useful in shallow waters than some similar classes of ship. (Wikipedia)
Middleton and Halley’s craft had a length of fifty-two feet, a breadth of eighteen feet, and a depth of seven feet, seven inches.
Halley continued to follow the developments and on 4 June 1696 he was commissioned as master and commander of the Paramore. Halley was free to choose his own crew but requested that they should be enlisted men to better maintain discipline. His crew consisted of a naval surgeon, a chief mate, a crew of ten foremast men and two cabin boys, as well as a carpenter, gunner, and boatswain. At this point Middleton together with his servant still intended to take part in the voyage, making a total ships compliment of twenty. In the end Middleton didn’t take part in the expedition.
On 15 October 1698, the Admiralty issued Halley’s instruction for the voyage, which he had in fact effectively written himself:
Captn. Edmd. Hally Comandr. Of his Mats. Pink Paramour
Instructions for proceeding to Improve the knowledge of the Longitude and Variations of the Compasse. … You are to make the best of your way to the Southward of the Equator, and there to observe on the East Coast of South America and the West coast of Africa, the variations of the Compasse, with all the accuracy that you can, as also the true Scituation both in Longitude and Latitude of the Ports where you arrive. You are likewise to make the like observations at as many Islands in the Seas between the aforesaid Coasts as you can (without too much deviation) bring into your course: and if the Season of the Yeare permit, you are to stand soe farr into the South, till you discover the Coast of the Terra Incognita, [Australia] supposed to lye between Magelan’s Streights and the Cape of Good Hope, which coast you are carefully to lay downe in its true position. In your returne home you are to visit the English West India Plantations, or as many of them as conveniently you may, and in turn to make such observations as may contribute to lay them downe truly in their Geographical Scituation, And in all the Course of your Voyage, you must omit no opportunity of Noteing the variation of the Compasse, of which you are to keep a record in your Iournall.[4]
Halley’s log, which is preserved in the British Museum, contains a full record of the expedition. They set sail from Deptford on Thursday 20 October 1698, anchoring the next day in Gravesend reach, finally setting off on 25 October. This was a fail start as they ran into bad weather and it turned out that the boat leaked and the pumps got clogged with the sand used for ballast. In Portsmouth the ship was repaired and the sand replaced with shingle and on 22 November they set sail again.
The zig-zagged their way south in the Atlantic carrying out their orders but Halley began to have trouble with his crew. On 17 February 1699, he caught his boatswain sailing the wrong course. The crew continued to be querulous. This reached a head on 16 March when the first lieutenant chose to sail his own course rather than that commanded by Halley and tried impolitely to justify his behaviour. In June, Halley had the lieutenant confined to his cabin after the man had told him to his face that he was neither qualified not fit to command the vessel. Halley proved the man wrong by sailing the ship without incident back to England by himself.
The lieutenant, named Harrison, was court marshalled on 3 July 1699. It turned out that Harrison had devised his own scheme for determining longitude, which he had submitted to the Admiralty, the Navy Board, and the Royal Society. It was rejected upon review by experts, one of whom had been Halley for which Harrison had never forgiven him. In particular Harrison rejected Halley’s notion that magnetic variation could be used to determine longitude, the purpose of the Paramour’s expedition.
After some more problems with the appointment of a new crew the Paramour set sail a second time on 27 September 1699. Halley sailed up and down, and back and forth across the Atlantic systematically measuring longitude, latitude, and variation finally returning to Deptford on 10 September 1770, after many adventures but without any major incidents.
As with his earlier expedition to St Helena, Halley was quick to turn his carefully collected data into a form that would be useful to and benefit others. His large Atlantic chart was published in 1701 and like his earlier chart of the southern celestial hemisphere it was dedicated to the king, although now a different king. He displayed the magnetic variation, the measuring of which had been the main aim of the voyages, by connecting points of equal variation with dotted lines. Halley referred to them as curve lines but throughout the eighteenth century they were known as Halleyan lines. The modern technical term is isogonic lines or simply isogons.
The Atlantic Chart was originally titled, “A New and CorrectCHARTShewing theVARIATIONS of the COMPASS in the WESTERN AND SOUTHERN OCEANS as observedin ye year 1700 by his Maties Command by Edm. Halley.[5]
On the chart Halley explained:
The Curve Lines which are drawn over the Seas in this Chart do shew at one View all the Places where the Variations of the Compass is the same; The Numbers to them, shew how many degrees the Needle declines either Eastwards or Westwards from the true North; and the Double Lines passing near Bermudas and the Cape Virde Isles is that where the Needle stands true without Variation.
Later under Halley’s direction the Atlantic Chart was replaced by a World Chart published in 1702 using data collected from many sources. This cart was very popular and went through many editions, both English and foreign. In 1744 a revised updated edition was published and another revised edition was published in 1758.
Halley was not the first to produce a chart isogenic lines. That honour goes to the Italian, Jesuit, mathematician, astronomer, cartographer, and missionary to Indochina, Christoforo Borri (1583–1632), who produced a chart of the Atlantic and Indian Oceans showing the spots of equal magnetic variation. That chart, which has never been published, but was known to Athanasius Kircher (1602–1680), is now in the Vatican Library.
In 1701, Halley undertook a third and final voyage on the Paramore, this time nearer to home, having been commissioned to determine the tides, headlands, and promontories of the English Channel. His chart of the English Channel being published in 1702.
This was the end of his career as a mariner and in 1705, Halley published his Synopsis of the Astronomy of Comets containing his first description of the comet that would make him an almost universally known name.
[1] The accounts of Halley’s observations on St Helena and his later voyages to measure magnetic variations are taken from Colin A. Ronan’s Edmond Halley: Genius in Eclipse, Macdonald &Co., London, 1970.
One area of what would become physics that developed significantly as a direct result of the translations made during the Scientific Renaissance of the twelfth century was optics. The two Islamic authors whose work had the biggest impact were al-Kindi (C. 801–873) and Ibn al-Haytham (c.965–1040). The two Europeans, who initially did most to make people aware of their work in the thirteenth century were Robert Grosseteste (1168–1253) and Roger Bacon (c. 1220–c. 1292). Before going into more detail, it pays to remember that optics here was still the theory of vision and not yet the theory of the behaviour of and properties of light, a transition that first took place in the late sixteenth and early seventeenth centuries. However, with the theories of Ibn a-Haytham as transmitted by Bacon and others with have what would become the first phase of that transition.
Robert Grosseteste was principally a churchman and a theologian but was also a polymath who made contributions to a wide range of subjects. Although Grosseteste became one of the most influential scholars of the thirteenth century very little is known for fact about his birth, up bringing, and education. He is said to have been born of humble parents in Suffolk. He possibly received a liberal arts education in Lincoln. He was active in the household of William de Vere, Bishop of Lincoln, from about 1195. The next real information is from 1225, when he was awarded the benefice of Abbotsley in the diocese of Lincoln. By 1229 at the latest he was lecturing on theology to the Franciscans at Oxford University. Anecdotal evidence says he was Chancellor of the University but his actual title was magister scholium (master of students). Over the years he was awarded various other benefices and a prebend as canon in Lincoln Cathedral. However, in 1232, he resigned all of his benefices retaining only his prebend. In 1235 he was elected Bishop of Lincoln, a compromise candidate following a deadlock under the preferred candidates. He maintained this position to his death and became a highly influential figure in the church.
An early 14th-century portrait of Grosseteste Source: Wikimedia Commons
He wrote extensive commentaries on Aristotle and also, having learnt both Greek and Hebrew, became an extensive translator. Important for his influence on the development of optics was the cosmogony that he developed based on al-Kindī’s De radiis stellarum in his De luce self de inchoation formarum and De motu corporali et luce “that everything in the world … emits rays in every direction, which fill the whole world”, in which optics became central, believing light to be the first form of all things, the source of all generation and motion.
Grosseteste’s analysis a light was geometrical, as he wrote in his De lineis, angulis, etfiguris (“On Lines, Angles, and Figures” c. 1230):
a consideration of lines, angles, and figures is of the greatest utility because it is impossible to gain a knowledge of natural philosophy without them … for all causes of natural effects must be expressed by means of lines, angles, and figures.
Note, this is three hundred years earlier than Galileo’s over hyped quote from IlSaggiatore.
Grosseteste applies this analysis in detail to both reflection and refraction drawing explicitly in his discussion on Aristotle’s Meteorology and Metaphysics, Boethius’ Arithmetic, Euclid’s Elements. Implicit sources include Pseudo-Euclid’s De speculis, Euclid’s Optics, and al-Kindī’s De aspectibus, all of which he quotes in other works. There is also a hint of al-Kindī’s De radiis in Grosseteste’s assertion that the analysis applies to all natural agents (for example, heat), not just light.[1] Grosseteste and later Bacon regarded light as the multiplication of species.
There is minimal evidence that Grosseteste might have read Ibn al-Haytham’s Kitāb al-Manāzir (Book of Optics) translated into Latin by an unknown translator in the late twelfth or early thirteenth century with the title De aspectibus, but he is a convinced supporter of extramission and not al-Haytham’s intromission.
Because light played the central role in Grosseteste’s cosmogony and his major influence in the thirteenth century, optics became part of the quadrivium, that is the mathematical disciplines that formed the second half of the undergraduate curriculum on the newly emerging medieval universities.
The quadrivium (arithmetic, geometry, music and astronomy),
Grosseteste was, in some aspects of his thought, surprisingly modern in his approach to science. He strongly believed in the formulation of laws based on personal observation of nature and the use of mathematical to present and explicate those laws.
Roger Bacon is often falsely presented as a student of Grosseteste, but, although, he was very obviously influenced by Grosseteste there is no evidence that they ever met. As with Grosseteste we have a fragmented biography scratched together out of unprecise references. He was born in Ilchester in Somerset of apparently wealthy parents. There are numerous contradictory birth dates. He studied at Oxford probably shortly after Grosseteste had left. He graduated MA in Oxford and said to have taught there. In 1237 or later, he was invited to the University of Paris, where he lectured on the trivium and quadrivium. Around 1247 he left Paris and went wandering through Europe as an independent scholar and his exact whereabouts are unknown for most of the next decade. Somewhere, during that time he meet and became friend with Adam Marsh (c. 1200–1259) a Franciscan and student of Grosseteste, who probably introduced him to Grosseteste’s work. In 1256/7, he became a Franciscan.
Statue of Bacon at the Oxford University Museum of Natural History Source: Wikimedia Commons
As a friar he was blocked from doing or publishing scientific work and he became rebellious and sort a way out of his predicament, sometime in the 1260s taking up contact with the papal legate Guy de Foulques, Bishop of Narbonne and Cardinal of Sabina (1190–1268) hopping for some sort of preferment. Due to a misunderstanding Guy de Foulques, who in 1265 became Pope Clement IV, thought he had already written and prepared for publication a major work and requested to see it. Bacon sat down and produced in short order his Opus Majus, Opus Minus, and Opus Tertium plus several other works in about a year of intense writing. The three Opus volumes are in effect an encyclopaedia of medieval knowledge with his proposals for improving it.
Bacon wrote three treatises on optics De multiplicatione specierum (on the Multiplication of Species), which was probably composed in the early 1260’s. It is an extensive elaboration on Grosseteste’s De lineis based primarily on al-Haytham and it provides the physical foundation for his account of visual perception in his second and most significant optical work the Perspectiva, which is included in part five of the Opus Majus, so composed around 1267/8. Sometime later he composed his third work on optics, the De speculis comburentibus (On Burning Mirrors).
Bacon’s optic studies, from Opus Majus Source: Wikimedia Commons
Although Grosseteste and Ibn a-Haytham provided the two main pillars on which Bacon’s theories of optics stood, he also consulted a wide range of other sources that included a-Kindī’s De aspectibus and De radiis, Euclid’s Optics and Catoptrics, Tideus’ De speculis and Pseudo-Euclid’s De speculis. More significantly Ptolemaeus’ Optics and book six of Ibn Sina’s Healing.
His views are close to those of Ibn al-Haytham but are consistently more philosophical and less mathematical and that despite Grosseteste’s influence. Another major difference is that although Bacon, like al-Haytham, believed that visual perception was principally the result of light entering the eyes, thus intromission, he still believed in the involvement element of visual rays emitted by the eyes. Also, whereas, al-Haytham saw his De aspectibus as a finished theory of visual perception, Bacon regarded his Perspectiva as an art prospectus for future research.
Through Grosseteste’s belief in the central role of light in the moment of creation, both Grosseteste and Bacon included a strong element of theology in their accounts of light, and perception.
The movement in the development of optics in Europe in the thirteenth century was taken up by John Pecham (c. 1230–1292). Born to humble parents in Patcham in Sussex, he was educated at Lewes Priory before going to Oxford in about 1250 where he became a Franciscan friar. From here he went the University of Paris where he studied under Bonaventure (1221–1274) becoming a lecturer for theology after graduation. He remained in Paris for many years, where he interacted with many of the leading scholars including Thomas Aquinas (c. 1225–1274). Influenced by both Grosseteste and Bacon he was also active in astronomy and optics.
Portrait of Peckham from a 1504 edition of his book, Perspectiva Communis
Around 1270, he returned to England and began teaching at Oxford and was elected provincial minister of the Franciscans in England in 1275. He was soon summoned to Rome and appointed lector sacri palatii,or theological lecturer at the papal palace. In 1279 he was appointed Archbishop of Canterbury by Pope Nicholas III (c. 1225–1280) in opposition to Edward I’s preferred candidate. As archbishop he was a strict taskmaster and, as a result, was not very popular.
Grave of John Pecham in Canterbury Cathedral Source: Wikimedia Commons
He wrote the Perspectiva cummunis with the intention of condensing “into concise summaries the teachings of perspective [perspectiva], which [in existing treatises] are presented with great obscurity.”[2] What he presents is essentially an epitome of al-Haytham’s De aspectibus in three fairly short books. Pecham’s book would go on to become an optics textbook on the medieval university.
Perspectiva, 1556 Source: Wikimedia Commons
The last, and possibly the most important, of the founders of what is know as the perspectivist theory of optics is the Polish scholar Witelo (c. 1230–1280). To quote my own blog post on Witelo’s Perspectiva:
His biography can only be pieced together from scattered comments and references. In his Perspectivahe refers to “our homeland, namely Poland” and mentions Vratizlavia (Wroclaw) and nearby Borek and Liegnitz suggesting that he was born in the area. He also refers to himself as “the son of Thuringians and Poles,” which suggests his father was descended from the Germans of Thuringia who colonized Silesia in the twelfth and thirteenth centuries and his mother was of Polish descent.
A reference to a period spent in Paris and a nighttime brawl that took place in 1253 suggests that he received his undergraduate education there and was probably born in the early 1230s. Another reference indicates that he was a student of canon law in Padua in the 1260s. His Tractatus de primaria causa penitentie et de natura demonum, written in Padua refers to him as “Witelo student of canon law.” In late 1268 or early 1269 he appears in Viterbo, the site of the papal palace. Here he met William of Moerbeke (c. 1220–c. 1286), papal confessor and translator of philosophical and scientific works from Greek into Latin. Witelo dedicated his Perspectiva to William, which suggest a close relationship. This amounts to the sum total of knowledge about Witelo’s biography.
Page from a manuscript of De Perspectiva, with miniature of its author Vitello Source: Wikimedia Commons
If Pecham’s Perspectiva cummunis is the shortened version of Ibn al-Haytham’s De aspectibus, the Witelo’s Perspectiva is the extended version. It is very obvious that his major debt is to al-Haytham’s De aspectibus, although he never mentions him by name. However, Witelo also used Ptolemy’s Optica, Hero’s Catoptrica and the anonymous De speculis comburentibus in composing his Perspectiva, and that he was aware of Euclid’s Optica, the Pseudo-Euclid Catoptrica and other prominent works on optics. Like Pecham’s text, Witelo’s Perspectiva became a standard optics textbook on the medieval European universities. As we shall see in a later episode it also came to play a central role in the transition of optics from a theory of vision to the theory of the behaviour of and properties of light in the Early Modern Period.
First printed edition of Witelo’s Perspectiva printed in Nürnberg by Johannes Petreius 1535
Following this burst of activity in the development of optics in the thirteenth century, now known as the perspectivist theory of optics, it became establish as a discipline within the quadrivium at the medieval university. When Regiomontanus (1436–1476) finally was awarded his magister in 1457, the first lecture course he held at the University of Vienna was on optics. However, the discipline rapidly stagnated and there was little in the way of further developments before the sixteenth century.
One should, however, not assume, as unfortunately far too many do, that because of the strong reception in medieval Europe of Ibn al-Haytham’s intromission theory of vision that it now totally dominated the field. There still existed substantial support for extramission theories, such as that of Plato, throughout the medieval period and into the Renaissance.
I will close by briefly touching on an aspect of the history of optics that has little to do with its role within physics. In the fifteenth century, linear perspective emerged in European art, specifically in Northern Italy. Some historians, most notably art historians, state categorically that linear perspective was first made possible by the perspectivist theory of optics and more specifically that it owes its existence to the work of Ibn al-Haytham. This hypothesis, and as I will show it is a hypothesis not an established fact, is, to say the least, problematic.
We start with the simple fact that the geometry of linear perspective is entirely Euclidian and the optical theories on which it is based come entirely from Euclid’s optics or the more advanced presentation of it by Ptolemaeus. Nothing added to optical theory by Ibn al-Haytham contributes to the optical geometry of linear perspective. Of course, Euclid’s geometrical optics is at the core of the perspectivist theory.
If we turn to the pioneers of linear perspective there is only very minimal proof for the claim. The earliest, the sculptor Lorenzo Ghiberti (1378–1455), who used linear perspective in the panels of the second set of bronze doors he was commissioned to produce for the Florence Baptistry, dubbed the Gates of Paradise by Michelangelo. Ghiberti wrote an unfinished autobiography Commentarii in which he quotes the Italian translation of al-Haytham’s book, Deli Aspecti. This is the only reference to al-Haytham in the milieu.
Gates of Paradise, The Story of Adam and Eve (copy at the Baptistery) Source: Wikimedia Commons
We only know about the experiments in linear perspective of Filippo Brunelleschi (1377–1446) through hearsay and have no direct information from the man himself, so we can’t say what his influences were.
Anonymous portrait from the 2nd half of the 15th century (Louvre, Paris) Source: Wikimedia Commons
The first published account of how to create linear perspective in art was by Leon Battista Alberti (1404–1472) in his book On painting, published in Tuscan dialect as Della Pittura in 1436/6 and in Latin as De pictura first in 1450.
Leon Battista Alberti (1404 -1472), De Pictura
Mark Smith thinks that Alberti’s approach to linear perspective is deduced from his work as an architectural historian surveying ancient building and ruins in Rome using a plain table.
Philippe Danfrie (c.1532–1606) Surveying with a plane table
A very plausible hypothesis in my opinion. Perhaps even more significant is Alberti’s statement that in linear perspective it is irrelevant with you hold an extramission or intromission theory of vision. Ibn al-Haytham biggest contribution was, of course, combining a Euclidian theory of geometrical optics with a pure, light based intromission theory of vision.
All in all not really a solid basis for appointing Ibn al-Haytham the progenitor of linear perspective.
[1] Adapted from A. Mark Smith, From Sight to Light: The Passage from Ancient to Modern Optics, University of Chicago Press, Chicago, 2015 pp. 258–259
I fell in love with numbers some time before I began going to school. I loved arithmetic from day one. I was one of those horrible people who were good at maths at school and actually enjoyed doing it, ending with me doing maths A-level. I studied maths at university to about BSc level before changing to philosophy with an emphasis on history and philosophy of science, where I spent several years working in a research project into history of mathematical logic. During my student years, as a mature student, I worked as evening manager in a culture centre, where amongst other things I was responsible for stocking up the change in the admission cash boxes for various areas and then counting the takings at the end of the evening and preparing the payment for the bank. All of my life I have been very good at mental arithmetic and it should be clear from all of the above that I have spent a lot of time in my life calculating but at times I still find myself counting on my fingers to check something I’ve calculated in my head!
Finger counting is one of the earliest forms of human calculation as Jessica Marie Otis tells us in her By the Numbers: Numeracy, Religion, and the Quantitative Transformation of Early Modern England.[1] As soon as I read the title and description of this book, given my very strong interest in all aspects of mathematics in the Early Modern Period, I knew that I wanted to read it. I just hoped that my expectations would not prove excessive. Exactly the opposite occurred, my expectations were comprehensively exceeded by Jessica Otis’ excellent book, which instantly acquired a slot on my list of all time favourite history of maths books.
The Introduction opens by explain the religion in the title, not a term one would normally expect in a history of arithmetic:
In an almanac written for the year 1659, George Wharton mentioned a biblical passage that would have been familiar to early modern “well-wishers,” “lovers,” and “students” of mathematics: “For indeed all things were made by God, in Numbers, Weights and Measures.” An unknown reader found this passage so interesting that they underlined it in their copy. The implications of this simple statement were enormous. If numbers were indeed the building blocks of God’s Creation, then everything in the world could be understood through the use of numbers, particularly the concrete numerical exercises of weighing and measuring. Furthermore, a knowledge of numbers would be necessary for those who wished to study any aspect of God’s inherently numerical Creation. This biblical passage was not a common subject for Sunday sermons, and the Book of Wisdom in which it can be found is technically part of the Apocrypha. Nonetheless, Wisdom 11:21–God’s creation of the world by numbers, weight, and measure–formed a rallying cry for people who wished to encourage the wider study of mathematics among the general population of early modern England.
We normally think of the history of mathematics in terms of the big names and the major development in the abstract disciplines that cause so many school kids to moan and to ask, when will I ever need this in real life. Names such as al-Khwārizmī and Newton and the disciplines with which they are associated algebra and calculus. Otis’ book deals with that branch of mathematics that almost everybody needs and regularly uses in their everyday lives, simple arithmetic. She examines in depth the transformations that this discipline went through in England in the early modern period, placing those transformations firmly in their cultural, political, social, and religious contexts. This is contextual history at its finest. Otis encapsulates the transformation she describes in her book on the second page of the introduction:
This transformation in numerical practices was complex and wide-ranging in its impact on early modern society and thought. In part, it was a transformation in symbolic systems–the culturally agreed upon symbols and syntax used to represent numbers. It was also a transformation in mathematical education, enabled by increasing literacy rates, and the printing revolution. Most important, it was a transformation in technologies of knowledge, specifically the way the people of early modern England conceived of and used numbers in their daily lives.
Having laid out the path that her research and book take, Otis delivers up six chapters each one of which deals with a different aspect of the modes of presentation, manipulation, methods of calculation of numbers and how they changed and why over the roughly two hundred years the book covers.
Chapter one, “The Dyuers Wittes of Man: The Multiplicity and Materiality of Numbers, delivers up the raw material, the presentation of numbers, which Otis introduces thus:
The main symbolic systems employed during the sixteenth and seventeenth centuries can be roughly divided into three categories based on their most prominent material characteristics: performative, object based, and written.
Performative is both spoken number words, children learning to count verbally intuitively and almost everyone expected to have a least a basic grasp of numbers and counting. The earlier mentioned finger counting is also performative, both in the simple variant, ten fingers-ten numbers, and the more complex systems using body parts to count up to twenty or complex figure manipulations to represent numbers up to one thousand.
Object based can simple be using any objects in place of fingers–peas, stones, twigs–to count and manipulate numbers. However, it also cover the much more sophisticated tallying, with tally sticks, and the use of counting boards. These two methods dominated in the early part of the period covered and were still present at the end. Otis gives an in depth analysis of the uses of both forms of number manipulation and their significance in the period.
There were two written systems, Roman and Arabic numerals. The former dominated at the beginning of the period but they were only used for recording but not for calculation. Otis explains that Arabic numerals were gradually introduced initially only for specific purposes but like the Roman numerals not used for calculations.
Having been introduced to the methods of numerations, in the second chapter, “Finding Out False Reckonings”: Thrust and the Function of Numbers, Otis takes us through the use of numbers in creating accounts, their principal use in the sixteenth century. We first learn about the problems of creating accounts that could be trusted and also could not be altered after the fact. Tally sticks were the only system that was safe from falsification and became dominant. Otis describes the various strategies employed to prevent people altering written accounts, whether in words or numerals. The least safe were those written in Arabic numerals, so they were, on the whole, avoided. We then get taken into the world of creating accounts, where the counting board was the dominant method, with the results of the calculation being mostly written in words or Roman numerals. Otis, however, explains that it was not unusual to find accounts with a mix of the use of the counting board, Roman numerals and Arabic numerals.
If you are doing your accounts using arithmetic then you first have to learn how to do it. Otis’ third chapter, “Set Them To the Cyphering Schoole”: Reading, Writing, and Arithmetical Education, takes the reader on a journey through the world of teaching arithmetic in England in the sixteenth and seventeenth centuries. She describes the fascinating world of the printed, early modern, arithmetic textbooks. Their authors, their scope, their contents, and their methods and how these all change and evolved during the period. She also introduces us to the users of the textbooks through the analysis of their often extensive marginalia. Having learnt about the books, we now learn about the schools, starting with the so-called petty schools, private institutions that taught young pupils the basics of writing and arithmetic and moving on to the grammar schools and apprenticeships for older children Apparently the grammar schools, which concentrated on Latin, Greek and reading classical literature often employed extra teachers to teach their pupils the basics of writing and arithmetic. Otis explains that can be assumed that apprentices were taught the basics of writing and arithmetic as part of their training. She also briefly touches upon the lack of interest in mathematics at the Oxbridge universities.
All of the topics that Otis analyses in her, I repeat, excellent book interest me but I was particularly pleased by her fourth chapter, “According to Our Computations Here”: Quantifying Time. The main quantification of time analysed in detail in this chapter is days, months and years in the form of calendars and in particular the problems engendered by the calendar reform, which took place in the middle of the time period she covers. This was naturally very much a religious problem and more that justifies the religion in the book’s title. This is a topic that I have devoted much effort in studying and I can happily report that I learnt quite a lot of new things through the aspects of the topic that she chose to bring to the fore.
However, in this chapter I also detected the only serious error in her book that I’m aware of. Concerning the date of the Feast of Easter, the principle cause or reason for the calendar reform she writes the following:
In AD 325, the Council of Nicaea had officially fixed Easter to the Sunday following the first full moon after the vernal equinox of March 21.[2]
This is simply factually wrong, to quote Wikipedia[3]:
In 325 an ecumenical council, the First Council of Nicaea, established two rules: independence from the Jewish calendar, and worldwide uniformity. However, it did not provide any explicit rules to determine that date, writing only “all our brethren in the East who formerly followed the custom of the Jews are henceforth to celebrate the said most sacred feast of Easter at the same time with the Romans and yourselves [the Church of Alexandria] and all those who have observed Easter from the beginning.
It is historically important that it did not provide any explicit rules to determine that date, as there followed a, at time bitter, dispute, largely between Alexandria and Rome, who employed different systems, over the correct way to determine the date. A dispute involving religion, astronomy, and mathematics and which was first settled by about the tenth century.
In my opinion the key quote in this chapter is:
During the early modern period, temporal locations were defined using three different conceptions of time–linear, episodic, and cyclical. Linear time fixes a single important event as the center of its chronology and measures duration forward and backward from the event along an imagined time line. Episodic time is similar to linear time but relies on multiple events, which are usually of the same type, and reckons duration by each event only until the next event occurs. That next event then becomes the new chronological reference point, and the process repeats itself. Finally, cyclical time consists of an event or events that repeat in a regular pattern without a clear beginning or end.
She notes that all three systems were used in conjunction with each other during this period and goes into a detailed analysis of the various and varied ways of dating that the people employed. I particularly like the emphasis that she gives to regnal calendars, that is dating according to the year of the reign of a specific monarch, a widespread dating system, right up to the modern period, that usually gets ignored in discussion on calendars and dating.
She includes a long and detailed discussion of the problems engendered by the acceptance by some countries and rejection by other, in particular England, of the calendar reform and how various people, traders for example, dealt with the problem of multiple dates for the same day. A problem exacerbated by the use of different dates for the start of the year. For a nice example of this see my blog post on the dates of Newton’s birth and death. I found totally fascinating the use of fractional notation in dating to cope with the problems, such as “19/29 March 1638/9”,[4] which was new to me.
She devotes a fair amount of space in this chapter to the topic of almanacs, another favourite topic of mine, and the tables of calendars, important dates, regnal calendars, astronomical events and, and that they contained to save people the trouble of calculations and conversions.
The chapter closes with a look at the introduction of the mechanical clock and with it a new type of clock time and the effects that this had. I warmly recommend the whole book, but I would also recommend it for chapter four alone.
In chapter five, “It is Oddes of Many to One”: Quantifying Chance and Risk, we turn away from the world of calendars, dates, and almanacs to address the early modern world of gaming, bets and odds. What we have here is the very gradual emergence of probability theory in gambling but still at the stage of guestimates, not mathematical calculations. Otis takes the readers through the complex and oft heated discussion on God’s providence contra natural order and fortune, which at times led to the condemnation of gaming by puritans. Leaving the world of gaming we enter the world of insurance also based on estimated rather than calculated odds. This progressed from marine insurance against losses at sea, through life insurance, the first lives insured being those of slaves, who were after all maritime trade goods, to fire insurance following the Great Fire of London. These early ventures into insurance being closer to wagers in gaming that anything mathematically calculated. However, towards the end of the period under discussion insurance based on statistical analysis of the mortality bills began to emerge, a process driven by the plague and the list of deaths it engendered.
Chapter six “David’s Arithmetick”; Quantifying the People, takes us further down the road of the newly crystalising statistical analysis. David is King David, who in the Bible conducted a census of his people, and censuses is the topic that open this chapter. After a brief survey of censuses from antiquity Otis takes us through the early use of them in early modern England, mainly for tax purposes, which made them unpopular. She follows the development of censuses to answer specific single questions leading up to the work of John Gaunt and William Petty, who founded modern demographic analysis, initially under Petty’s name Political Arithmetic. Here Otis’ journey through the story of numbers in early modern England ends.
With academic books I usually comment on the academic apparatus, notes, bibliography, etc. The book has them in spades, the text is a mere 160 pages, attached to which are 100 pages of endnotes, bibliography and index. The endnotes are extensive, detailed and highly informative and definitely worth reading parallel to the main text. The forty page bibliography of primary and secondary sources is exhaustive and reading through it, I had the feeling that I would require at least ten years, if not considerably more, to read my way through all of the listed texts. A comprehensive index finishes off the whole.
There are only a handful of greyscale illustrations but it is not a topic that lends itself to lots of pictures. Otis writes fluidly with an easy to grasp style and I found myself ploughing through her excellent text smoothly at high speed, consumed by all the wonderful titbits it contains.
My brief sketches of each chapter barely scratch the surface of the in depth research and detailed analysis that Otis brings to her topic. Research and analysis on the highest academic level but here presented in a way that the non-expert can read and comprehend with ease.
In the Epilogue Otis summarises her own work perfectly:
Numbers were a socially pervasive technology of knowledge, with the power to shape modes of thought. As much as possible, By the Numbers has reconstructed a cognitive element of English culture by looking at not just how people used numbers, but how people though about numbers and how they used them to think about events in their day-to-day lives. In analysing numbers this way, it gives ordinary people a stake in philosophies and technologies of knowledge, and in doing so reconnects some of the revolutionary changes in early modern science with the changes occurring in everyday numeracy. This period witnessed not only the mathematization of elite natural philosophy but also the increasing use of numbers by ordinary men and women to interpret the world around them. Many later eighteenth-century developments in economics and politics, among other fields, had antecedents in early modern ways of thinking about numbers. The men and women of seventeenth-century England understood the potential of numbers as a technology of knowledge, even if they had not yet developed the mathematics to completely explain the world via numbers.
I can only repeat what I’ve already said, this is a truly excellent piece of academic research brought skilfully and lucidly to paper. If you have any interest in numbers, their usage and the evolution of that usage over a wide range of fields in the early modern period then I can’t recommend this book highly enough. I would also recommend it, even for those more interested in the social, religious and political developments in early modern England for the contextual history of the period that it provides.
[1] Jessica Marie Otis, By the Numbers: Numeracy, Religion, and the Quantitative Transformation of Early Modern England, OUP, 2024
[3] I know one shouldn’t quote Wikipedia when criticising an academic work but I know that the Wikipedia statement is correct and it saves me having to search through my books on the topic looking for a passing quote:
Fifteen years! 15 years, that’s 5479 days, including 4 days for leap years, that’s 328,740 minutes, or 19,724,400 incredible seconds the Renaissance Mathematicus has polluted the Internet. I am more than a little flabbergasted, I really do have to repeat what I have said on more than one occasion, when I started this blog I didn’t think I would last more than six weeks. Now 1211 blog posts and more than 11,000 comments later, fifteen years have passed and I wonder where they have gone. I have led a life that was full of change, different jobs, different countries, different courses of study, without finishing any of them and one thing I realised writing this, being a #histsci blogger for fifteen years is the longest period I have stuck to anything in my whole life. I think I might just have found my niche in life.
Fifteen is not a particularly interesting number. It is the product of three and five the first two uneven primes and that’s about it. A fifteen sided polygon is a pentadecagon and an internal angle of 156°, which is constructible with compass and straight edge. Because it has so many edges and vertices it has quite a lot of symmetries.
The symmetries of a regular pentadecagon as shown with colours on edges and vertices. Lines of reflections are blue. Gyrations are given as numbers in the centre. Vertices are coloured by their symmetry positions.(Wikipedia)
It also has three regular star polygons: {15/2}, {15/4}, {15/7}, and three regular star figures: {15/3}, {15/5}, {15/6}
The regular pentadecagon is the Petrie polygon for some higher-dimensional polytopes, projected in a skew orthogonal projection
14-Simplex 14D In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions The simplex is so-named because it represents the simplest possible polytope in any given dimension.
Fifteen years, one and a half decades, is called a “quindecennium” or a “quindecade.” More interesting for a blog that often pokes around in the Middle Ages, it is also known as an indiction. An indiction was a fiscal period of fifteen years, instituted by Constantine in 313 CE (but counting from 1st September 312), used throughout the Middle Ages as a way of dating events, documents etc.
Bible chronologist, Joseph Justus Scaliger (1540–1609) used the indiction in his Julian period, which he used to determine the date of creation, to quote myself:
Scaliger defined something he called the Julian Period of 7980 years, which was a product of three different dating cycles: The 19-year Meton cycle used in lunar-solar calendars, such as the Jewish one, the 28-year solar cycle, which is the number of years it takes for days of the week to repeat in the Julian calendar and the 15-year indiction cycle used for dating medieval manuscripts. Working backwards 4731 BCE is the last time that all three cycles were in their respective first years. Scaliger chose this date because it preceded his own calculated date for creation, 3949 BCE. With a unified scale Scaliger could give a Julian year number to any historical event within his research, thus making cross-cultural comparisons possible.
Indictions seem to have played a role in my life. I was fifteen years and five days old, when my mother died under, for me, traumatic circumstances, as I’ve documented elsewhere. I celebrated my second indiction, my thirtieth birthday, here in Franconia, a couple of weeks after I had decided to stay and settle down here, at least for the foreseeable future, on the holiday trip that originally brought me here. On my sixtieth birthday, the end of my fourth indiction I acquired my first iPad, my birthday present from my siblings. I have another two and a half years till the end of my fifth indiction, I wonder what that will bring.
As is my wont at the end of my bog’s annual journey around the sun, I thank all of those who have engaged with my humble efforts, those who have simply read them, those who have commented, and especially those who have dared to contradict, from whom I often learn something new. Learning is what keeps me focused and stops me going insane.
One area of what would become physics that saw serious developments during the European Middle Ages was motion, both fall and projectile motion. I’ve dealt with various aspects of these developments in earlier blog posts but I’m going to bring the strands together in one episode here.
The first thing to note is that the theories of motion of both Aristotle and John Philoponus re-entered European intellectual intercourse during the Scientific Renaissance in the twelfth century. This was the beginning of the Aristotelian philosophy that became the core of the syllabus on the scholastic medieval university following its elucidation by Albertus Magnus (c. 1200–1280) and his student Thomas Aquinas (c. 1225–1274). However, as I am very fond of repeating, Edward Grant points out that medieval Aristotelian philosophy is not Aristotle’s philosophy. The medieval university scholars changed and developed Aristotle’s ideas and this is very true of his concepts of fall.
Perhaps the most significant development in the study of fall was made by the so-called Oxford Calculatores, a group of scholars at Merton College Oxford, in the first half of the fourteenth century. However, the Calculatores didn’t start from scratch but were influenced by a couple of earlier medieval scholars, who we will briefly look at before turning our attention to Merton College.
Gerard of Brussels (French: Gérard de Bruxelles, Latin: Gerardus Bruxellensis) is a somewhat obscure figure from the early thirteenth century, who is only really known through his book Liber de motu (On Motion), which was composed sometime between 1187 and 1260. In this book, he, to paraphrase the title of a Marshall Clagett essay on his work, reduces curvilinear velocities to uniform rectilinear velocities.
The Liner de motu contains thirteen propositions, in three books. In these propositions the varying curvilinear velocities of the points and parts of geometrical figures in rotation are reduced to uniform rectilinear velocities of translation. The four propositions of the first book relate to lines in rotation, the five of the second to areas in rotation, and the four of the third to solids in rotation, Gerard’s proofs are particularly noteworthy for their ingenious use of an Archimedean-type reductio demonstration, in which the comparison of figures is accomplished by the comparison of their line elements. (Marshall Clagett, DSB)
The second influence comes from Walter Burley (or Burleigh c. 1275–1344/45), who had been a fellow of Merton College until about 1310, following which he spent time in Paris on the Sorbonne before spending seventeen years as a clerical courtier in England then in Avignon. Burley wrote extensive commentaries on the works of Aristotle but perhaps most importantly he was one of the first medieval scholars to recognise the priority of propositional logic over the term or syllogistic logic of Aristotle.
The principal figures of the Calculatores were Thomas Bradwardine (c. 1300–1349), William of Heytesbury (c. 1313–1372/73), Richard Swineshead,(fl. c. 1340–1354) and John of Dumbleton (d. c. 1349).
Merton College in 1865 Source: Wikimedia Commons
Thomas Bradwardine devoted much of his life’s work to theology but it is his work in mathematics and physics that interests us here. He wrote two introductory works on mathematics Geometria speculativa (Speculative Geometry) and Arithmetica speculative (Speculative Arithmetic), which are elementary text books on the subjects.
His application of logical and mathematical analysis to Aristotle’s theories of motion, however, were totally groundbreaking. The emphasis of Bradwardine’s work is on kinetics– The branch of mechanics concerned withmotion of objects, as well as the reason i.e. the forces acting on such bodies (Wiktionary). In his Tractatus de proportionibus velocitatum in motibus, which handles the ratio of speeds in motion, he derived the mean speed theorem– a uniformly accelerated body (starting from rest, i.e. zero initial velocity) travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body (Wikipedia)–which is the key element in the laws of fall and is often falsely attributed to Galileo.
In his Tractatus de proportionibus (1328) Bradwardine also produced important advances in Eudoxus’ theory of proportion anticipating the concept of exponential growth. He also criticised Aristotle’s concepts of the relationship between powers and speeds:
In Book VII of Physics, Aristotle had treated in general the relation between powers, moved bodies, distance, and time, but his suggestions there were sufficiently ambiguous to give rise to considerable discussion and disagreement among his medieval commentators. The most successful theory, as well as the most mathematically sophisticated, was proposed by Thomas Bradwardine in his Treatise on the Ratios of Speeds in Motions. In this tour de force of medieval natural philosophy, Bradwardine devised a single simple rule to govern the relationship between moving and resisting powers and speeds that was both a brilliant application of mathematics to motion and also a tolerable interpretation of Aristotle’s text. (Wikipedia, attributed to Lindberg and Shank without a reference but I think is a paraphrase from Laird, see footnote 1)
In his work Bradwardine referenced Gerard of Brussels Liber de motu and his use of philosophy and logic is obviously influenced by the writings of Walter Burley.
William of Heytesbury, the second member of the group was at heart a logician, whose main work was Regulae solvendi sophismata (Rules for Solving Sophisms). Part IV of this work is devoted to problems concerning the three species of change: local motion or change of place, augmentation or change of size, and alteration or change of quality. In the section on local motion, he distinguished uniform motion, which he defined as motion in which equal distances are transversed at equal speeds from nonuniform motion or difform motion.[1]
On local motion he writes:
For whether it commences from zero degree or from some [finite] degree, every latitude [of velocity], provided that it is terminated at some finite degree, and is acquired or lost uniformly, will correspond to its mean degree. Thus, the moving body, acquiring or losing this latitude uniformly during some given period of time, will traverse a distance exactly equal to what it would traverse in an equal period of time if it were moved continuously at its mean degree. For of every such latitude commencing from rest and terminating at some [finite] degree [of velocity], the mean degree is one-half the terminal degree of that same latitude (Regule [Venice, 1494], fol. 39). (Curtis A. Wilson, DSB)
The third member of the group Richard Swineshead is regarded as the best mathematician of the group and therefore was known as The Calculator. His magnum opus was a series of treatises known as the Liber calculationum (Book ofCalculations) written around 1350. His reputation as a mathematician carried down to the Renaissance and the Early Modern Period, with Julius Caesar Scaliger (1484–1558), Gerolamo Cardano (1501–1576) and Gottlieb Leibniz (1646–1716) all expressing admiration for it.
Richard Swineshead Opus aureum calculationum Source
The emphasis in the Liber calculationum is on logicomathematical techniques rather than on physical theory. What it provides are techniques for calculating the values of physical variables and their changes, or for solving problems or sophisms about physical changes. (John E. Murdoch & Edith Dudley Sylla, DSB)
The fourth member John of Dumbleton, who wrote a sort of general encyclopaedia, Summa logicae et philosophiae naturalis, which amongst other things includes the work of the other three.
His proof of the “Merton mean speed theorem” is interesting and in some ways reminiscent of the geometric method of exhaustion. He also considered how mathematical techniques could be applied to motions other than local motion. (A. G. Molland, DSB)
What we see in general by the Oxford Calculatores is the systematic application of logical and, above all, mathematical analysis to problems that Aristotle, who basically rejected mathematics in his physics, only dealt with philosophically. This approach is something that is widely and falsely believed first took place in the seventeenth century during the so-called scientific revolution. One thing that they did not do, however, is to back up their purely theoretical application of mathematical analysis with any form of experimentation.
The work of the Merton scholars did not take place in a bubble but quickly spread throughout the European academic community and it found people who took it up and propagated in at the university in Paris amongst a group who have become known as the Paris Physicists.
The first Parisian scholar we will look at is Nicole Oresme (1325–1382), a polymath who wrote on a very wide range of topics, became Bishop of Lisieux and a counsellor of King Charles V (1338–1380). Oresme was the thinker who introduced the concept of comparing the celestial motions to a mechanical clock, a concept that would go on to have a long and varied history all the way down to the nineteenth century. Otherwise, his cosmology remained fairly traditional.
Medieval miniature of Nicole Oresme Source: Wikimedia Commons
On the mathematical side he wrote a treatise on the theory of proportions, Proportionespropotionum (The Ratio of Ratios), the starting point for which was Bradwardine’s Tractatus de proportionibus (1328). It is essentially a treatment of fractional exponents conceived as ratios of ratios.
In this treatment Oresme made a new and apparently original distinction between irrational ratios of which the fractional exponents are rational, for example, and those of which the exponents are themselves irrational. In making this distinction Oresme introduced new significations for the terms pars, partes, commensurabilis, and incommensurabilis. Thus pars was used to stand for the exponential part that one ratio is of another.
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It should also be noted that Oresme composed an independent tract, the Algorism of Ratios, in which he elucidated in an original way the rules for manipulating ratios. (Marshall Clagett, DSB)
Oresme also tackled the problem of falling bodies:
In discussing the motion of individual objects on the surface of the earth, Oresme seems to suggest (against the prevailing opinion) that the speed of the fall of bodies is directly proportional to the time of fall, rather than to the distance of fall, implying as he does that the acceleration of falling bodies is of the type in which equal increments of velocity are acquired in equal periods of time. He did not, however, apply the Merton rule of the measure of uniform acceleration of velocity by its mean speed, discovered at Oxford in the 1330’s, to the problem of free fall, as did Galileo almost three hundred years later. Oresme knew the Merton theorem, to be sure, and in fact gave the first geometric proof of it in another work, but as applied to uniform acceleration in the abstract rather than directly to the natural acceleration of falling bodies.
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The mention of Oresme’s geometrical proof of the Merton mean speed theorem brings us to a work of unusual scope and inventiveness, the Tractatus de configurationibus qualitatum et motuum composed in the 1350’s while Oresme was at the College of Navarre. This work applies two-dimensional figures to hypothetical uniform and nonuniform distributions of the intensity of qualities in a subject and to equally hypothetical uniform and nonuniform velocities in time. (Marshall Clagett, DSB)
Oresme’s geometric verification of the Oxford Calculators’ Merton Rule of uniform acceleration, or mean speed theorem. Source: Wikimedia Commons
Oresme’s use of two-dimensional geometrical figures to represent change comes very close to the concept of using a rectangular coordinate system to plot change. For what we would call the vertical axis he used the term latitudo and for the horizontal axis longitudo, terms used in his time primarily for astronomical coordinate systems.
A page from Tractatus de latitudinibus formarum (1505)
He shows that his method of figuring the latitude of forms is applicable to the movement of a point, on condition that the time is taken as longitude and the speed as latitude; quantity is, then, the space covered in a given time. In virtue of this transposition, the theorem of the latitudo uniformiter difformis became the law of the space traversed in case of uniformly varied motion; thus Oresme published what was taught over two centuries prior to Galileo’s making it famous. (Wikipedia, paraphrasing Pierre Duhem & Marshall Clagett)
Our second fourteenth century Parisian is Oresme’s contemporary Jean Buridan (c. 1301–c. 1359/62). Buridan is a rare example of a scholar, who spent his whole life teaching at the university but remained in the arts faculty, never moving up into one of the three higher faculties.
Jean Buridan (Paris ca. 1370, Kraków cod. BJ 1771, fol. 142v) Source: Wikimedia Commons
With Buridan we move away from the Oxford Calculatores and Aristotle’s concepts of fall to his concepts of projectile motion. Buridan rejected Aristotle’s thoughts on the topic and accepted instead the theory of John Philoponus (c. 490– c. 570) via the Islamic scholar Abu’l-Barakāt Hibat Allah ibn Malkā al-Baghdādī (c. 1080–1164/5). In this theory the mover imparts something to the projectile and it is this that keeps it in motion. Abu’l-Barakāt called this mayl qasri (a violent inclination), Buridan gave it the name still used today, impetus.
…after leaving the arm of the thrower, the projectile would be moved by an impetus given to it by the thrower and would continue to be moved as long as the impetus remained stronger than the resistance, and would be of infinite duration were it not diminished and corrupted by a contrary force resisting it or by something inclining it to a contrary motion (Questions on Aristotle’s Metaphysics XII.9: 73ra).
It should be noted that whereas Philoponus thought that which was imparted somehow got used up in flight, Buridan thought, as noted above, that it got slowed down and eventually stopped by the air resistance and gravity. This was an important development in the theory in the direction inertia. Falsely Buridan thought that rotational motion at uniform angular velocity as due to a rotational impetus analogous to the rectilinear impetus Out of this, Buridan thought that it was impetus which was the cause of celestial motion. With no wind resistance or gravity in the celestial sphere impetus would be perpetual.
A pupil of Buridan’s was the Italian, Dominicus de Clavasio (fl. mid-fourteenth century).
He taught arts at Paris during 1349–1350, was head of the Collège de Constantinople at Paris in 1349, and was an M.A. by 1350. Dominicus received the M.D. by 1356 and was on the medical faculty at Paris during 1356–1357. He was astrologer at the court of John II and may have died between 1357 and 1362.
Dominicus is the author of a Practica geometriae written in 1346; a questio on the Sphere of Sacrobosco; a Questiones super perspectivam; a set of questiones on the first two books of the De caelo of Aristotle, written before 1357; and possibly a commentary on Aristotle’s Meteorology. He mentions in the Practica his intention to write a Tractatus de umbris et radiis. (Claudia Kren, DSB)
In his questiones on the first two books of the De caelo he adopted the impetus theory as an explanation of projectile motion as well as of acceleration in free fall.
“When something moves a stone by violence, in addition to imposing on it an actual force, it impresses in it a certain impetus. In the same way gravity not only gives motion itself to a moving body, but also gives it a motive power and an impetus, … .” (Dominicus, De caelo)
His work was clearly influenced by both Oresme and Buridan.
The Polish priest, theologian and philosopher, Joannes Cantius (Polish: Jan z Kęt or JanKantyEnglish: John Cantius, 1390–1473) in his philosophical works also propagated Buridan’s impetus theory.
St. Joannes Cantius Source: Wikimedia Commons
The most important propagator of Buridan’s impetus theory was his student Albert von Rickmersdorf (Latin: Albertus de Saxonia; English: Albert of Saxony c. 1316–1390). Following studies in Prague and Paris, Albert taught as a professor in Paris from 1351 to 1362. In 1353 he was rector of the Sorbonne. From 1363 he was at the court of Pope Urban V in Avignon, who appointed him first rector of the University of Vienna in 1365. In 1366 he was appointed Bishop of Halberstadt, where he remained until his death.
Albert was influenced by both Oresme and Buridan and he followed Buridan in adopting and propagating the theory of impetus. It appears that there was, at least on paper and interchange on the topic between the teacher and his student. In his Quaestiones super libros Physicorum, Albert appears to have referenced an earlier version of Buridan’s work with the same title, as in a later version of the same work Buridan references Albert’s work. One result is that Albert doesn’t use the term impetus, which Buridan only introduced in the later version of his work, but rather a virtus motiva or virtus impressa, an impressed power.
Quaestiones super quatuor libros Aristotelis de caelo et mundo (Published by Hieronymus Surianus, Venedig 1497: Source: Wikimedia Commons
In his own contribution to the theory of impetus Albert added a third stage to the two stage theory of John Philoponus.
Initial stage. Motion is in a straight line in direction of impetus which is dominant while gravity is insignificant
Intermediate stage. Path begins to deviate downwards from straight line as part of a great circle as air resistance slows projectile and gravity recovers.
Last stage. Gravity alone draws projectile downwards vertically as all impetus is spent.
Source: Wikimedia Commons
Both Nicole Oresme and Jean Buridan were significant figures in the world of medieval philosophy and their modifications and development of the Aristotelian theories of motion were influential down to the beginnings of the Early Modern Period
[1] Walter Roy Laird, Change and Motion, in The Cambridge History of Science, Volume 2, Medieval Science, CUP, ppb 2015, p. 429
The whole of this section on pre-twentieth century astronomer is horribly ahistorical and inaccurate:
Likely influenced by criticism of the Ptolemaic cosmology by the Persian and Syrian astronomers, Al-Urdi, Al-Tusi and Ibn al-Shatir – all members of the Maragheh observatory funded in what was then Mongolia, now Iran – Copernicus revived the idea that the Earth orbits around the Sun and not vice versa. Kepler’s laws of the motions of the planets were the first mathematical description of a physical phenomenon, subsequently bolstered by Galileo’s first observations of the heavens through a telescope. Newton developed a whole branch of mathematics to connect those motions to physical laws.
We start with the simple historical fact that whereas al-Tusi (1201–1274) and al-Urdi (d. 1266) both worked in the Maragheh observatory, Ibn al-Shatir (1304–1375) didn’t. He studied astronomy in Cairo and Alexandria and then worked as muwaqqit (timekeeper) of the Umayyad Mosque in Damascus. A major point of the criticism of the astronomy of Ptolemy was his use of the equant point, a bone of contention for many astronomers over the centuries. Al-Tusi developed the so-called Tusi-couple, a mathematical device to help eliminate the equant point. Ibn al-Shatir presented a radical reform of the Ptolemaic planetary models in his kitab nihayat al-sul fi tashih al-usul (The Final Quest Concerning the Rectification of Principles) in which he incorporated the so-called Urdi lemma, which allowed an equant in an astronomic model to be replaced with an equivalent epicycle (the Tusi-couple) that moved around a deferent centred at half the distance to the equant point.
We now turn to Copernicus. The only motivation the Copernicus gave for his attempts to reform Ptolemaic astronomy was like al-Tusi, al-Urdi, Ibn al-Shatir, and many others, the removal of the equant point. We know that this reform led him to replacing the geocentric model of Ptolemy with a heliocentric model. Although there are numerous speculations over it, we don’t actually know what led him to take this step. In Derevolutionibus, Copernicus uses the Tusi-couple in several places including his model of Mercury and in his theory of trepidation, which of course doesn’t exist. Once again although there has been much speculation on the topic there is no known link between Copernicus and those Islamic astronomers, who used the Tusi-couple and certainly absolutely no evidence that that Copernicus was likely influenced by their criticism of the Ptolemaic cosmology.
Kepler’s laws of the motions of the planets were the first mathematical description of a physical phenomenon is simply a mindbogglingly false statement. Archimedes would like a quite word with the author of the piece as would the Oxford Calculatores, as well as several others who provide mathematical descriptions of physical phenomenon before Kepler was even born. Kepler would like to point out that his own law of the spreading of light from a point source predates his laws of planetary motion.
I find the very generalised statement that Kepler’s laws of planetary motion were subsequently bolstered by Galileo’s first observations of the heavens through a telescope rather strange. The telescopic discovery of the Moons of Jupiter, independently by both Galileo and Simon Marius, and the subsequent determination of their orbits, showed that they too conformed with Kepler’s laws. But the rest of the early telescopic astronomical observations made by Galileo and others did nothing to bolster those laws.
The simple sentence, Newton developed a whole branch of mathematics to connect those motions to physical laws, contains a whole collection of historical myths. The reference here is, of course, to calculus but as I have explained in an earlier blog post, Newton and Leibniz did not develop, invent, or discover[1] calculus but rather collated about two thousand years’ worth of work on the topic by numerous predecessors. Secondly, Newton’s work on collating the elements of the calculus had nothing to do with the motions of the planets.
The next is something I have constant battles on social media about with people who “know better,” Newton did not used calculus either in the preparatory work for, or in the final presentation of his Principia, having lost faith in quality of proof using the analytical method. The whole work in done in tradition Euclidian geometry. I have a blog post on this, too.
I do wish that if scientists are going to make statements about the history of science, they would take the time and make the effort to check their facts first.
[1] Choose your own term according to your preferred philosophy of mathematics.
If your philosophy of [scientific] history claims that the sequence should have been A→B→C, and it is C→A→B, then your philosophy of history is wrong. You have to take the data of history seriously.
John S. Wilkins 30th August 2009
Culture is part of the unholy trinity—culture, chaos, and cock-up—which roam through our versions of history, substituting for traditional theories of causation. – Filipe Fernández–Armesto “Pathfinders: A Global History of Exploration”
The Renaissance Mathematicus 