Category Archives: History of Mathematics

The swashbuckling, philosophical alchemist

If you go beyond the big names, big events version of the history of science and start looking at the fine detail, you can discover many figures both male and female, who also made, sometime significant contribution to the gradual evolution of science. On such figure is the man who inspired the title of this blog post, the splendidly named Sir Kenelm Digby (1603–1665), who made contributions to a wide field of activities in the seventeenth century.

Kenelm Digby (1603-1665) Anthony van Dyck Source: Wikimedia Commons

To show just how wide his interests were, I first came across him not through my interest in the history of science, but through my interest in the history of food and cooking, as the author of an early printed cookbook, The Closet of the Eminently Learned Sir Kenelme Digbie Kt. Opened (H. Brome, London, 1669).

Source: Wikimedia Commons

Born 11 June in Gayhurst, Buckinghamshire, in 1603 into a family of landed gentry noted for their nonconformity, he, as we will see, lived up to the family reputation. His grandfather Everard Digby (born c. 1550) was a Neoplatonist philosopher in the style of Ficino, and fellow of St John’s College Cambridge, (Fellow 1573, MA 1574, expelled 1587), who authored a book that suggested a systematic classification of the sciences in a treatise against Petrus Ramus, De Duplici methodo libri duo, unicam P. Rami methodum refutantes, (Henry Bynneman, London, 1580, and what is considered the first English book on swimming, De arte natandi, (Thomas Dawson, London, 1587). The latter was published in Latin but translated into English by Christopher Middleton eight years later. 

Source: Wikimedia Commons
Source: Wikimedia Commons

His father Sir Everard Digby (c. 1578–1606) and his mother Mary Mulsho of Gayhurst were both born Protestant but converted to Catholicism.

Sir Everad Digby artist unknown Source: Wikimedia Commons

His father was executed in 1606 for his part in the Gunpowder Plot and Kenelm was taken from his mother and made a ward first of Archbishop Laud (1573–1645) and later of his uncle Sir John Digby (1508-1653), who took him on a sixth month trip (August 1617–April 1618) to Madrid in Spain, where he was serving as ambassador.

Sir John Digby portrait by Cornelis Janssens van Ceulen Source: Wikimedia Commons

Returning from Spain, the fifteen-year-old Kenelm entered Gloucester Hall Oxford, where he came under the influence of Thomas Allen (1542–1632).

Thomas Allen by James Bretherton, etching, late 18th century Source: wikimedia Commons

 Thomas Allen was a noted mathematician, astrologer, geographer, antiquary, historian, and book collector. He was connected to the circle of scholars around Henry Percy, Earl of Northumberland (1564–1632), the so-called Wizard Earl, through whom he became a close associate of Thomas Harriot (c. 1560–1621). Through another of his patrons Robert Dudley, Early of Leicester, (1532–1588) Allen also became an associate of John Dee (1527–c. 1608). Allen had a major influence on Digby, and they became close friends. When he died, Allen left his book collection to Digby in his will: 

… to Sir Kenelm Digby, knight, my noble friend, all my manuscripts and what other of my books he … may take a liking unto, excepting some such of my books that I shall dispose of to some of my friends at the direction of my executor.

Digby donated this very important collection of at least 250 items, which contained manuscripts by Roger Bacon, Robert Grosseteste, Richard Wallinford, amongst many others to the Bodleian Library.

Digby left Oxford without a degree in 1620, not unusual for a member of the gentry, and took off on a three-year Grand Tour of the continental. In France Maria de Medici (1575–1642) is said to have cast an eye on the handsome young Englishman, who faked his own death and fled France to escape her clutches. In Italy he became accomplished in the art of fencing. In 1623 he re-joined his uncle in Madrid, this time for a nearly a year and became embroiled in the unsuccessful negotiations to arrange a marriage between Prince Charles and the Infanta Maria. Despite the failure of this mission, when he returned to England in 1623, the twenty-year-old Kenelm was knighted by James the VI &I and appointed a Gentleman to Prince Charles Privy Chamber at the time converting to Anglicanism. In 1625 he secretly married his childhood sweetheart Venetia Stanley (1600–1633). They had two sons Kenelm (1626) and John (1627) before the marriage was made public. 

Venetia, Lady Digby by Anthony van Dyck Source: Wikimedia Commons

Out of favour with Buckingham, Digby now became the swashbuckler of the title. Fitting out two ships, the 400-ton Eagle under his command and the 250-ton Barque under the command of Sir Edward Stradling (1600–1644), he set off for the Mediterranean to tackle the problem of French and Venetian pirates, as a privateer, a pirate sanctioned by the crown.

Arbella, previously the Eagle Digby’s flagship

Capturing several Flemish and Dutch prize on route, on 11 June 1628 they attacked the French and Egyptian ships in the bay of Scanerdoon, the English name for the Turkish port of Iskender. Successful in the hard-fought battle, Digby returned to England with both ships loaded down with the spoils, in February 1629, where he was greeted by both the King and the general public as a hero. He was appointed a naval administrator and later Governor of Trinity House. 

The next few years were spent in England as a family man surrounded by a circle of friends that included the poet and playwright Ben Johnson (1572–1637), the artist Anthony van Dyck (1599–1641), the jurist and antiquary John Seldon (1584–1654), and the historian Edward Hyde (1609–1674) amongst many others. Digby’s circle of friends emphasises his own scholarly polymathic interests. His wife Venetia, a notable society beauty, died unexpectedly in 1633 and Digby commissioned a deathbed portrait and from van Dyck and a eulogy by Ben Johnson, now partially lost. 

Venetia Stanley on her Death Bed by Anthony van Dyck, 1633, Dulwich Picture Gallery Source: Wikimedia Commons

Digby stricken by grief entered a period of deep mourning, secluding himself in Gresham College, where he constructed a chemical laboratory together with the Hungarian alchemist and metallurgist János Bánfihunyadi (Latin, Johannes Banfi Hunyades) (1576–1646), where they conducted botanical experiments. 

In 1634, having converted back to Catholicism he moved to France, where he became a close associate of René Descartes (1596–1650). He returned to England in 1639 and became a confidant of Queen Henrietta Maria (1609–1669) and becoming embroiled in her pro-Catholic politics made it advisable for him to return to France.

Henrietta Maria portrait by Anthony van Dyck Source: Wikimedia Commons

Here he fought a duel against the French noble man Mont le Ros, who had insulted King Charles, and killed him. The French King pardoned him, but he was forced to flee back to England via Flanders in 1642. Here he was thrown into goal, however his popularity meant that he was released again in 1643 and banished, so he returned to France, where he remained for the duration of the Civil War.

Henrietta Maria established a court in exile in Paris in 1644 and Digby was appointed her chancellor. In this capacity he undertook diplomatic missions on her behalf to the Pope. Henrietta Maria’s court was a major centre for philosophical debates with William Cavendish, the Earl of Newcastle, his brother Charles both enthusiastic supporters of the new sciences, William’s second wife Margaret Lucas, who had been one of Henrietta Maria’s chamber maids and would go on to great notoriety as Margaret Cavendish prominent female philosopher, Thomas Hobbes, and from the French side, Descartes, Pierre Gassendi (1592–1655), Pierre Fermat (1607–1665), and Marin Mersenne. Digby was in his element in this society.

Margaret Cavendish and her husband, William Cavendish, 1st Duke of Newcastle-upon-Tyne portrait by Gonzales Coques Source: Wikimedia Commons

After unsuccessfully trying to return to England in 1649, in 1653, he was granted leave to return, perhaps surprisingly he became an associate of Cromwell, whom he tried, unsuccessfully, to win for the Catholic cause. He spent 1657 in Montpellier to recuperate, but returned to England in 1658, where he remained until his death. 

He now became friends with John Wallis (1616–1703), Robert Hooke (1635–1703), and Robert Boyle (1627–1691) and was heavily involved in the moves to form a scientific society, which would lead to the establishment of the Royal Society of which he was a founder member. On 23 January 1660/61 he read his paper A discourse concerning the vegetation of plants before the founding members of the Royal Society at Gresham College, which was the first formal publication to be authorised by that still unnamed body. The Discourse would prove to be his last publications, as his health declined, and he died in 1665.

Source: Wikimedia Commons

Up till now the Discourse is the only publication that I’ve mentioned, but it was by no means his only one. Digby was a true polymath publishing works on religion, A Conference with a Lady about choice of a Religion(1638), Letters… Concerning Religion (1651), A Discourse, Concerning Infallibility in Religion (1652). Autobiographical writings including, Articles of Agreement Made Betweene the French King and those of Rochell… Also a Relation of a brave and resolute Sea Fight, made by Sr. Kenelam Digby (1628), and Sr. Kenelme Digbyes honour maintained (1641). Critical writings on Sir Thomas Browne, Observations upon Religio Medici (1642), and on Edmund Spencer, Observations on the 22. Stanza in the 9th Canto of the 2d. Book of Spencers Faery Queen (1643). 

What, however, interests us here are his “scientific” writings. The most extensive of these is his Two Treatises, in One of which, the Nature of Bodies; in the Other, the Nature of Mans Soule, is looked into: in way of discovery, of the Immortality of Reasonable Soules originally published in Paris in 1644 but with further editions published in London in 1645, 1658, 1665, and 1669. Although basically still Aristotelian, this work shows the strong influence of Descartes and contains a positive assessment of Galileo’s Two New Sciences, which was still relatively unknown in England at the time. It also contains a form of mechanical atomism, which, however, is different to those of Epicure or Descartes.

Source

Digby’s most controversial work was his A late discourse made in solemne assembly … touching the cure of wounds by the powder of sympathy, originally published in French in 1658 and then translated into English in the same year. This was a discourse that Digby had held publicly in Montpellier during his recuperation there.

Source

This was a variation on Weapon Salve, an ointment that was applied to the weapon that caused a wound rather than to the wound itself. Digby was by no means the first to write positively about this supposed cure. It has its origins in the theories of Paracelsus and the Paracelsian physician Rudolph Goclenius the Younger (1572–1621), professor at the University of Marburg, first published on it in his Oratio Qua defenditur Vulnus Non Applicato Etiam Remedio, in 1608. In England the divine William Forster (born 1591), the physician and alchemist Robert Fludd (1574–1637), and the philosopher Francis Bacon (1561–1626) all wrote about it before Digby, but it was Digby’s account that attracted the most attention and ridicule. In 1687, an anonymous pamphlet suggested using it to determine longitude. A dog would be wounded with a blade and placed aboard a ship before it sailed. Then every day at noon the weapon salve would be applied to the blade causing the dog to react, thus tell those on board that it was noon at their point of departure. 

Also in 1658, John Wallis dedicated his Commercium epistolicum to Digby who was also author of some of the letters it contained.

John Wallis by Sir Godfrey Kneller Source: Wikimedia Commons

In 1657, Wallis had published his Arithmetica Infinitorum, an important contribution to the development of calculus.

Source

Digby brought the book to the attention of Pierre Fermat and Bernard Frénicle de Bessy (c. 1604 – 1674) in France, Fermat wrote a letter to the English mathematician, posing a series of problems to be solved. Wallis and William Brouncker (1620–1684), who would later become the first president of the Royal Society, took up the challenge and an enthusiastic exchange of views developed between the French and English mathematicians, with Digby acting as conduit for the correspondence. Wallis collected the letter together and published them as his Commercium epistolicum

As already stated, A discourse concerning the vegetation of plants was Digby’s final publication and was to some extent his most interesting. Digby was interested in the question of how to revive dying plants and his approach was basically alchemical. He argued that saltpetre was necessary to the process of revival and that it attracted vital air, which is the food of the lungs. He is very obviously here close to discovering oxygen and in fact he supports his argument with the information that Cornelius Drebbel had used saltpetre to refresh the air in his submarine. In the paper he also hypothesises something very close to photosynthesis. Others such as Jan Baptist van Helmont (1580–1644) were conducting similar investigations at the time. These early investigations would lead on in the eighteenth century to the work of Stephen Hales (1677–1761) and the pneumatic chemists of the eighteenth century. 

Digby made no major contributions to the advancement of science, but he played a central role as facilitator and mediator between groups of philosophers, mathematicians, and scientists promoting and stimulating discussions in both France and England in the first half of the seventeenth century. He also played an important role in raising the awareness in England of the works of Descartes and Galileo. Although largely forgotten today, he was in his own time a respected member of the scientific community.

Digby is best remembered, today, for two things, his paper on the powder of sympathy, which I dealt with above, and his cookbook, to which I will now return. The Closet of the Eminently Learned Sir Kenelme Digbie Kt. Opened was first published posthumously by one of his servants in 1669 and has gone through numerous editions down to the present day, where it is regarded as a very important text on Early Modern food history. However, this was only one part of his voluminous recipe collection. Two other parts were also published posthumously. Choice and experimental receipts in physick and chirugery was first published in 1668 and went through numerous editions and translation by 1700, and A choice collection of rare chymical secrets and experiments in philosophy first published in 1682, which also saw many editions. What we have here is not three separate recipe collections covering respectively nutrition, medicine, and alchemy but three elements of a related recipe spectrum. We find a similar convolute in the work of Katherine Jones, Viscountess Ranelagh (1615–1691), Robert Boyle’s sister, an alchemist/chemist in her own right and an acquaintance of Digby’s. 

There is little doubt in my mind that Sir Kenelm Digby Kt. was one of the most fascinating figures of the seventeenth century, a century rich in fascinating figures. 

As was also believed when he died on his birthday in 1665, his epitaph read

‘Under this Tomb the Matchless Digby lies;

Digby the Great, the Valiant, and the Wise:

The Ages Wonder for His Nobel Parts;

Skill’d in Six Tongues, and Learn’d in All the Arts.

Born on the Day He Dy’d, Th’Eleventh of June,

And that Day Bravely Fought at Scanderoun.

‘Tis Rare, that one and the same Day should be

His Day of Birth, of Death, and Victory.’

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Filed under History of Alchemy, History of Chemistry, History of Mathematics, History of science

NIL deGrasse Tyson knows nothing about nothing

They are back! Neil deGrasse Tyson is once again spouting total crap about the history of mathematics and has managed to stir the HISTSCI_HULK back into butt kicking action. The offending object that provoked the HISTSCI_HULK’s ire is a Star Talk video on YouTube entitled Neil deGrasse Tyson Explains Zero. The HISTSCI_HULK thinks that the title should read Neil deGrasse Tyson is a Zero!

You simple won’t believe the pearls of wisdom that NdGT spews out for the 1.75 million Star Talk subscribers in a video that has been viewed more than one hundred thousand times. If there ever was a candidate in #histSCI for cancellation, then NdGT is the man.

 Before we deal with NdGT’s inanities, we need some basic information on number systems. Our everyday Hindu-Arabic number system is a decimal, that’s base ten, place value number system, which means that the value of a number symbol is dependent on its place within the number. An example:

If we take the number, 513 it is actually:

 5 x 10+ 1 x 101 + 3 x 100

A quick reminder for those who have forgotten their school maths, any number to the power of zero is 1. Moving from right to left, each new place represents the next higher power of ten, 100, 101, 102, 103, 104, 105, etc, etc. As we will see the Babylonians [as usual, I’m being lazy and using Babylonian as short hand for all the cultures that occupied the Fertile Crescent and used Cuneiform numbers] also had a place value number system, but it was sexagesimal, that’s base sixty, not base ten. It is a place value number system that requires a zero to indicate an empty place. There are in fact two types of zero. The first is simply a placeholder to indicate that this place in the number is empty. The second is the number zero, that which occurs when you subtract a number from itself.

Now on to the horror that is NdGT’s attempt to tell us the history of zero:

HISTSCI_HULK: Not suitable for those who care about the history of maths

 NdGT: I pick these based on how familiar we think we are about the subject and then throw in some things you never knew

HISTSCI_HULK: All NildGT throws in, in this video, is the contents of the garbage pail he calls a brain.

NdGT: For this segment, we’re gonna talk about zero … so zero is a number, but it wasn’t always a number. In fact, no one even imagined how to imagine it, why would you? What were numbers for?

Chuck Nice, Star Talk Host: Right, who counts nothing?

NdGT: Right, numbers are for counting … nobody had any use to count zero … For most of civilisation this was the case. Even through the Roman Empire…

 Here NdGT fails to distinguish between ordinal numbers, which label the place that object take in a list and cardinal numbers which how many things are in a collection or set. A distinction that at one point later will prove crucial.

HISTSCI_HULK: When it comes to the history of mathematics NildGT is a nothing

CN: They were so sophisticated their numbers were letters!

In this supposedly witty remark, we have a very popular misconception. Roman numerals were not actually letters, although in later mutated forms they came to resemble letters. Roman numbers are collections of strokes. One stroke for one, two strokes for two, and so one. To save space and effort, groups of strokes are bundled under a new symbol. The symbol for ten was a crossed or struck out stroke that mutated into an X, the symbol for five, half of ten, was the top half of this X that mutated into a V; originally, they used the bottom half, an inverted V.  The original symbol for fifty was ↓, which mutated into an L and so on. As the Roman number system is not a place value number system it doesn’t require a place holder symbol for zero. If Romans wanted to express total absence, they did so in words not numbers, nulla meaning none. This was first used in a mathematical context in the Early Middle Ages, often simply abbreviated to N. 

NdGT: [Some childish jokes about Roman numeral] … I don’t know if you’ve ever thought about this Chuck, you can’t write zero with Roman numerals. There is no symbol for zero.

The Roman number system is not a place value number system but a stroke counting system that can express any natural number, that’s the simple counting numbers, without the need for a zero. The ancient Egyptian number system was also a stroke counting system, whilst the ancient Greeks used an alpha-numerical system, in which letters do represent the numerals, that also doesn’t require a zero to express the natural numbers.

NdGT: It’s not that they didn’t come up with it, it’s the concept of zero was not yet invented. 

HISTSCI_HULK: I wish NildGT had not been invented yet

This is actually a much more complicated statement than it at first appears. It is true, that as far as we know, the concept of zero as a number had indeed not been invented yet. However, the verbal concept of having none of something had already existed linguistically for millennia. Imaginary conversation, “Can I have five of your flint arrowheads?” Sorry, I can’t help you, I don’t have any at the moment. Somebody came by and took my entire stock this morning.” 

Although the Egyptian base ten stroke numeral system had no zero, by about 1700 BCE, they were using a symbol for zero in accounting texts. Interestingly, they also used the same symbol to indicate ground level in architectural drawings in much the same way that zero is used to indicate the ground floor in European elevators. 

Also, the place holder zero did exist during the time of the Roman Empire. The Babylonian sexagesimal number system emerged in the third millennium BCE and initially did not have a zero of any sort. This meant that the number 23 (I’m using Hindu-Arabic numerals to save the bother of trying to format Babylonian ones) could be both 2 x 601 + 3 x 600 = 123 in decimal, or 2 x 602 + 3 x 600 = 7203 in decimal. They apparently relied on context to know which was correct. By about 700 BCE the first placeholder zero appeared in the system and by about 300 BCE placeholder zeros had become standard. 

During the Roman Empire, the astronomer Ptolemaeus published his Mathēmatikē Syntaxis, better known as the Almagest, around 150 CE, which used a weird number system. The whole number part of numbers were written in a ten-base system in Greek alphanumerical symbols, whereas fractional parts were written in the Babylonian sexagesimal number system, with the same symbols, with a placeholder zero in the form of small circle, ō.

HISTSCI_HULK NildGT now takes off into calendrical fantasy land.

NdGT: So, when they made the Julian calendar, that’s the one that has a leap day every four years, … That calendar … that anchored its starter date on the birth of Jesus, so this obviously came later after Constantine, I think that Constantine brought Christianity to the Roman Empire. So, in the Julian calendar they went from 1 BC, BC, of course, stands for before Christ, to AD 1, and AD is in Latin, Anno Domini the year of our Lord 1, and there was no year zero in that transition. So, when would Jesus have been born? In the mythical year between the two? He can’t be born in AD 1 cause that’s after and he can’t be born in 1 BC, because that’s before, so that’s an issue.

CN: I’ve got the answer, it’s a miracle.

The Julian calendar was of course introduced by Julius Caesar in AUC 708 (AUC is the number of years since the theoretical founding date of Rome) or as we now express it in 44 BCE. The Roman’s didn’t really have a continuous dating system, dating things by the year of the reign of an emperor. Constantine did not bring Christianity to the Roman Empire, he legalised it. Both Jesus and Christianity were born in Judea a province of the Roman Empire, so it was there from its very beginnings. For more on Constantine and Christianity, I recommend Tim O’Neill’s excellent History for Atheists Blog. 

To quote myself in another blog post criticising NdGT’s take on the Gregorian calendar

The use of Anno Domini goes back to Dionysius Exiguus (Dennis the Short) in the sixth century CE in his attempt to produce an accurate system to determine the date of Easter. He introduced it to replace the use of the era of Diocletian used in the Alexandrian method of calculating Easter, because Diocletian was notorious for having persecuted the Christians. Dionysius’ system found very little resonance until the Venerable Bede used it in the eight century CE in his Ecclesiastical History of the English People. Bede’s popularity as a historian and teacher led to the gradual acceptance of the AD convention. BC created in analogy to the AD convention didn’t come into common usage until the late seventeenth century CE. [Although BC does occur occasionally in late medieval chronicles.]

As NdGT says Anno Domini translates as The Year of Our Lord, so Jesus was born in AD 1 the first year of our Lord, simple isn’t it. 

I wrote a whole blog post about why you can’t have a year zero, but I’ll give an abbreviated version here. Although we speak them as cardinal numbers, year numbers are actually ordinal numbers so 2022 is the two thousand and twenty second year of the Common Era. You can’t have a zeroth member of a list. The year zero is literally a contradiction in terms, it means the year that doesn’t exist. 

HISTSCI_HULK You can’t count on NilDGT

NdGT: So now, move time forward. Going, it was in the six hundreds, seven hundreds, I’ve forgotten exactly when. In India, there were great advances in mathematics there and they even developed the numerals, early versions of the numerals we now use, rather than Roman numerals. Roman numerals were letters [no they weren’t, see above], these were now symbolic shapes that would then represent the numbers. In this effort was the hint that maybe you might want a zero in there. So, we’re crawling now before we can walk, but the seeds are planted. 

We have a fundamental problem dating developments in Hindu mathematics because the writing materials they used don’t survive well, unlike the Babylonian clay tablets. The decimal place value number system emerged some time between the first and fourth centuries CE. The symbols used in this system evolved over a long period and the process is too complex to deal with here. 

The earliest known reference to a placeholder zero in Indian mathematics can be found throughout a commercial arithmetic text written on birch bark, the Bakhshali manuscript, the dating of which is very problematical and is somewhere between the third and seventh centuries CE. 

The Aryasiddhanta a mathematical and astronomical work by Āryabhaṭa (476–550 n. Chr.) uses a decimal place value number system but written with alphanumerical symbols and without a zero. The Āryabhaṭīyabhāṣya another mathematical and astronomical work by Bhāskara I (c. 600–c. 680 n. Chr.) uses a decimal place value number system with early Hindu numerals and a zero. With the Brāhmasphuṭasiddhānta an astronomical twenty-four chapter work with two chapters on mathematics by Brahmagupta (c. 598–c. 668 n. Chr.) we arrive out our goal. Brahmagupta gives a complete set of rules for addition, subtraction, multiplication, and division for positive and negative numbers, as well as for zero as a number. The only difference between his presentation and one that one might find in a modern elementary arithmetic text is that Brahmagupta tried to define division by zero, which as we all learnt in school is not defined, didn’t we? Far from being “hint that maybe you might want a zero in there” this was the real deal. 

HISTSCI_HULK: NildGT would be in serious trouble with the Hindu Nationalist propagators of Hindu science if they found out about his garbage take on the history of Hindu mathematics.

NdGT: These [sic] new mathematics worked their way to the Middle East. Baghdad specifically, a big trading post from all corners of Europe and Asia, and Africa and there it was. Ideas were put across the table. This was the Golden Age of Islam, major advances were made in all…in engineering, in astronomy, in biology, physiology, and vision. The discovery that vision is a passive phenomenon not active. So, all of this is going on and zero was perfected. They called those numerals Hindu numerals; we today call them Arabic numerals. 

What NdGT doesn’t point out is that the Golden Age of Islam lasted from about 700 to 1600 CE and took place in many centres not just in Baghdad. The Brāhmasphuṭasiddhānta was translated into Arabic by Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (ges. 777 n. Chr.), Muhammad ibn Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (ges. c. 800 n. Chr.), and Yaʿqūb ibn Ṭāriq (ges. c. 796 n. Chr.) in about 770 CE. This meant that Islamicate[1] mathematical scientists had a fully formed correct theory of zero and negative numbers from this point on. They didn’t develop it, they inherited it. 

Today, people refer to the numerals as Hindu-Arabic numerals!

NdGt: So, this is the full tracking because in the Middle East algebra rose up, the entire arithmetic and algebra rose up invoking zero and you have negative numbers, so mathematics is off to the races. Algebra is one of the very common words in English that has its roots in Arabic. A lot of the a-l words, a-l is ‘the’ in Arabic as I understand it. So, algebra, algorithm, alcohol these are all traceable to that period. … So, I’m saying just consider how late zero came in civilisation. The Egyptian knew nothing of zero [not true, see above]. 

The Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850) wrote a book on the Hindu numeral system of which no Arabic text is known, but a Latin translation Algoritmi de Numero Indorum was made in the twelfth century. The word algorithm derives from the Latin transliteration Algoritmi of the name al-Khwārizmī. He wrote a second book al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah (c. 82O), the translation of the title is The Compendious Book on Calculation by Completion and Balancing. The term al-Jabr meaning completion or setting together became the English algebra. 

The first time I heard this section I did a double take. “The entire arithmetic and algebra rose up invoking zero and you have negative numbers, so mathematics is off to the races”, you what! Ancient cultures had been doing arithmetic since at least three thousand years BCE and probably much earlier. I can’t do a complete history of algebra in this blog post but by the early second millennium BCE the Babylonians could solve linear equations and had the general solution to quadratic equations but only for positive solutions as they didn’t have a concept of negative numbers. The also could and did solve some cubic equations. In the middle of the first millennium BCE they had astronomical algorithms to predict planetary orbits, as well as lunar and solar eclipses. Brahmagupta’s work includes the general solution of linear equations, and the full general solution of quadratic equations, as we still teach it today. NdGT’s statement is total rubbish.

Of historical interest in the fact that although Islamicate mathematical scientists acquired negative numbers from Brahmagupta, they mostly didn’t use them, regarding them with scepsis 

HISTSCI_HULK: NildGT is off with the fairies

CN: What is this that I hear about the Mayans and zero?

NdGT: I don’t fully know my Mayan history other than that they really worshipped Venus, so their calendar was Venus based. The calendar in ancient Egypt was based on the star Sirius [something unintelligible about new year]. It’s completely arbitrary when you say the new year’s just began. Pick a date whatever matters in your culture and call it new year. Even today when is the Chinese New Year, it’s late January, February. Everybody’s got a different starter date.

The Mayan culture developed a vigesimal, base twenty, place value number system, which included a placeholder zero, independent of the developments in the Middle East and India. The Dresden Codex, one of the most important Maya written documents contains a mixture of astronomy, astrology, and religion, in which observations of Venus play a central role. The first day of Chinese New Year begins on the new moon that appears between 21 January and 20 February

HISTSCI_HULK: I’d worship Venus, she was a very beautiful lady

CN: The Jewish New Year is another new year that…

NdGT: Everybody’s got another new year. The academic calendar’s got a new year that’s September the first…

I assume that NdGT is referring to the US American academic calendar, other countries have different academic years. In Germany where I live, each German state has a different academic year, in order to avoid that the entire population drive off into their summer holidays at the same time. 

NdGT: …and by the way one quick question you’ve got a hundred dollars in your bank account, and you go and withdraw a hundred dollars from the cash machine and the bank tells you what?

[…]

So, here’s the thing, you have no money left in the bank and that’s bad, but what worse is to have negative money in the bank and so this whole concept of negative numbers arose and made complete sense once you pass through zero. Now instead of something coming your way, you now owe it. The mathematics began to mirror commerce and the needs of civilisation, as we move forward, because we are doing much more than just counting. 

CN: So, this is like the birth of modern accounting. Once you find zero that’s when you’re actually able to have a ledger that shows you minuses and pluses and all that kind of stuff.

One doesn’t need negative numbers in order to do accounting. In fact, the most commonly used form of accounting, double entry bookkeeping, doesn’t use negative numbers; credits and debits are both entered with positive numbers. 

Numbers systems and arithmetic mostly have their origin in accounting. The Babylonians developed their mathematics in order to do the states financial accounting. 

HISTSCI_HULK: There’s no accounting for the stupidity in this podcast

NdGT: So now we’re into negatives and this keeps going with math and you find other needs of culture and civilisation, where whole other branches of math have to be developed and we got trigonometry. All those branches of math where you thought the teacher was just being angry with you giving you these assignments, entire branches of math zero started it all. Where it gives you deeper insights into the operations of nature. 

I said I did a double take when NdGT claimed that arithmetic and algebra first took off when the Islamic mathematicians developed zero and negative numbers, which of course they didn’t, but his next claim completely blew my mind. So now we’re into negatives and this keeps going with math and you find other needs of culture and civilisation, where whole other branches of math have to be developed and we got trigonometry. I can hear Hipparchus of Nicaea (c. 190–c. 120) BCE, who is credited with being the first to develop trigonometry revolving violently in his grave.

HISTSCI_HULK: I could recommend some good books on the history of trigonometry, do you think NildGT can read?

There is another aspect to the whole history of zero that NdGT doesn’t touch on, and often gets ignored in other more serious sources. The ancient cultures that didn’t develop a place value number system, didn’t actually need zero. Almost all people in those cultures, who needed to do and did in fact do arithmetical calculations, didn’t do their calculation by writing them out step for step as we all learnt to do in school, they did them using the oldest analogue computer, the abacus or counting board. The counting board was the main means of doing arithmetical calculation from some time a couple of thousand years BCE, we don’t know exactly when, all the way down to the sixteenth century CE. An experienced and skilled user of the counting board could add, subtract, multiply, divide and even extract square roots much faster than you or I could do the same calculations with paper and pencil. 

The lines or column on a counting board represent the ascending powers of ten in a decimal place value number system, powers of sixty on a Babylonian counting board. During a calculation, an empty line or column represents an implicit zero. In fact, there is one speculative theory that realising this led someone to make that zero explicit when writing out the results of a calculation and that is how the zero came into existence. Normally, when using a counting board only the initial problem and the result are recorded in writing and if one is using a stroke collection, ancient Romans and Egyptians, or an alphanumerical, ancient Greeks, as well as ancient Indian and Arabic cultures before they adopted Hindu numerals, number system, then, as already noted above, you don’t need a zero to express any number. 

This blog post is already far too long but before I close a personal statement. I am baffled as to why a supposedly intelligent and highly educated individual such as Neil deGrasse Tyson chooses to pontificate publicly, to a large international audience, on a topic that he very obviously knows very little about, without taking the trouble to actually learn something about the topic before he does so. Maybe the fact that the podcast is heavily sponsored and littered with commercial advertising is the explanation. He’s just doing in for the money.

His doing so is an insult to his listeners, who, thinking he is some sort of expert, believe the half-digested mixture of half-remembered half-facts and made-up rubbish that he spews out. It is also a massive insult to all the historian of mathematics, who spent their lives finding, translating, and analysing the original documents in order to reconstruct the real history. 

HISTSCI_HULK: If I were a teacher and he had handed this in as an essay, I wouldn’t give him an F, I would give it back to him, tell him to burn it, and give him a big fat ZERO!


[1] Islamicate is the preferred adjective used by historians for mathematics and science produced under Islamic hegemony and published mostly in Arabic. It is used to reflect that fact that those producing it were by no means only Arabs or indeed Muslim

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Filed under History of Mathematics, Myths of Science

Renaissance science – XXVIII

In the last episode of this series, we explored the history of the magnetic compass in Europe and marine cartography from the Portolan chart to the Mercator Projection. We will now turn our attention to the other developments in navigation at sea in the Renaissance. As already stated in the last episode, the need to develop new methods of navigation and the instruments to carry them out was driven by what I prefer to call the Contact Period, commonly called the Age of Discovery or Age of Exploration. The period when the Europeans moved out into the rest of the world and exploited it. 

This movement in turn was motivated by various factors. Curiosity about lands outside of Europe was driven both by travellers’ tales such as The Travels of Marco Polo c. 1300 and The Travels of Sir John Mandeville, which first appeared around 1360, both of which were highly popular throughout Europe, and also by new cartographical representation of the know world, known to the Europeans that is, in particular Ptolemaeus’ Geographia, which first became available in the early fifteenth century. Another development was technological, the development by the Portuguese, who as we shall see led the drive out of Europe into the rest of the world, of a new type of ship, the caravel, which was more manoeuvrable than existing vessels and because of its lateen sails was capable of sailing windward, making it more suitable for long ocean voyages, as opposed to coastal sailing.

The Portuguese invention of the caravel, which was maneuverable and able to undertake ocean voyages, was essential to European maritime exploration. The present image shows the “Caravela Vera Cruz“, navigating the Tagus river, Lisboa. Source: Wikipedia Commons
Depending on the situation, different intervals between tacking can be used. This does not influence the total distance travelled (though may impact the time required). Sailing from point A to point B, path P1 involves more turns but only requires a narrow channel. Path P2 involves fewer turns but a wider channel. Path P3 requires only a single turn but covers comparatively the widest channel. Source: Wikimedia Commons

The final and definitely most important factor was trade or perhaps more accurately greed. The early sailors, who set out to investigate the world outside of Europe, were not the romantic explorers or discoverers, we get taught about in school, but hard-headed businessmen out to make a profit by trade or if necessary, theft. 

The two commodities most desired by these traders, were precious metals, principally gold but also silver and copper, and spices. The metal ore mines of Middle Europe could not fill the demands for precious metals, so other sources must be found. This is perhaps best illustrated by the search in South America, by the Spanish, for the mythical city of gold, El Dorado, during the sixteenth century. Spices had been coming into Europe from the East over the Indian Ocean and then overland, brought by Arab traders, to the port cities of Northern Italy, principally Venice and Genoa, from where there were distributed overland throughout Europe since the eleventh century. The new generation of traders thought they could maximise profits by cutting out the middlemen and going directly to the source by the sea route. This was the motivation of both Vasco da Gama (c. 1460–1524), sailing eastwards, and Christopher Columbus (1451–1506), sailing westward. Their voyages are, however, one end point of a series of voyages, which began with the Portuguese capture of Ceuta, in North Africa, from the Arabs, in 1415.

Having established a bridgehead in North Africa the Portuguese, who were after all situated on the Atlantic coast of the Iberian Peninsula, argued that they could bypass the middleman, their trading partners the Arabs, and sail down the coast to Sub-Saharan West Africa and fetch for themselves, the gold and the third great trading commodity of the Contact Period, slaves, who they had previously bought from Arab traders. It is fair to ask why other countries, further north, with Atlantic coasts did not lead the expansion into unknown territory? The first decades of the Portuguese Atlantic ventures were still very much coastal sailing progressively further down the African coast; other northern European countries, such as Britain did sail north and south along the Atlantic coast, but their journeys remained within Europe. 

Starting in 1520, Portuguese expeditions worked their way down the west coast of Africa until the end of the sixteenth century.

The gradual Portuguese progress down the West Coast of Africa Source: Wikipedia Commons

The Nürnberger Martin Behaim (1459–1507), responsible for the creation of the oldest surviving terrestrial globe and member of the Portuguese Board of Navigation (to which we will return), claimed to have sailed with Diogo Cão, who made two journeys in the 1480s, which is almost certainly a lie. At the time of Cão’s first voyage along the African coast Behaim is known to have been in Antwerp. On his second voyage Cão erected pillars at all of his landing places naming all of the important members of the crew, who were on the voyage, Martin Behaim is not amongst them. 

The two most significant Portuguese expedition were that of Bartolomeu Dias (c. 1450–1500) in 1488, which was the first to round the Cape of Good Hope, actually Diogo Cão’s aim on his two voyages, which he failed to achieve, and, of course, Vasco da Gama’s voyage of 1497, which took him not only up the east African coast but all the way to India with the help of a local navigator. The two voyages also showed that the Indian Ocean was open to the south, whereas Ptolemaeus had shown it to be a closed sea in his Geographia. 

Much earlier in the century the Portuguese had ventured out into the Atlantic and when blown off course by a storm João Gonçalves Zarco (c. 1390 –1471) and Tristão Vaz Teixeira (c. 1395–1480) discovered the archipelago of Madeira in 1420 and one expedition discovered the Azores, 1,200 km from the Portuguese coast in 1427. The Canaries had already been discovered in the early fourteenth century and were colonised by the Spanish in 1402. The Cap Verde archipelago was discovered around 1456. The discovery of the Atlantic islands off the coasts of the Iberian Peninsula and Africa was important in two senses. Firstly, there developed myths about other islands further westward in the Atlantic, which encouraged people to go and look for them. Secondly, by venturing further out into the Atlantic sailors began to discover the major Atlantic winds and currents,, known as gyres essential knowledge for successful expeditions.

The Atlantic Gyres influenced the Portuguese discoveries and trading port routes, here shown in the India Run (“Carreira da Índia“), which would be developed in subsequent years. Source: Wikipedia Commons

Dias could only successfully round the Cape because he followed the prevailing current in a big loop almost all the way to South America and then back past the southern tip of Africa. Sailors crossing the Indian Ocean between Africa and India had long known about the prevailing winds and currents, which change with the seasons, which they had to follow to make successful crossings. The Spanish and the Portuguese would later discover the currents they needed to follow to successfully sail to the American continent and back.

The idea of island hopping to travel westwards in the Atlantic that the discoveries of the Azores and the other Southern Atlantic islands suggested was something already been followed in the North Atlantic by fishing fleets sailing out of Bristol in Southwest England in the fifteenth century. They would sail up the coast of Ireland going North to the Faroe Islands, settled by the Vikings around 800 CE and then onto Iceland, another Viking settlement, preceding to Greenland and onto the fishing grounds off the coast of Newfoundland. This is the route that Sebastian Cabot (c. 1474–c. 1557) would follow on his expedition to North America in the service of Henry VIII. It is also probable that Columbus got his first experience of navigating across the Atlantic on this northern route. 

Columbus famously made his first expedition to what would be erroneously named America in 1492, in an attempt to reach the Spice Islands of Southeast Asia by sailing westward around the globe. This expedition was undertaken on the basis of a series of errors concerning the size of the globe, the extent of the oikumene, the European-Asian landmass known to the Greek cartographers, and the distance of Japan from the Asian mainland. Columbus thought he was undertaking a journey of about 3,700 km from the Canary Islands to Japan instead of the actual 19,600 km! If he hadn’t bumped into America, he and his entire crew would have starved to death on the open sea. Be that as it may, he did bump into America and succeeded in returning safely, if only by the skin of his teeth. With Columbus’ expedition to America and da Gama’s to India, the Europeans were no longer merely coastal sailors but established deep sea and new approaches to navigation had to be found.

The easiest way to locate something on a large open area is to use a geometrical coordinate system with one set of equally spaced lines running from top to bottom and a second set from side to side or in the case of a map from north to south and east to west. We now call such a grid on a map or sea chart, lines of longitude also called meridians, north to south, and lines of latitude also called parallels, east to west. The earliest know presentation of this idea is attributed to the Greek polymath Eratosthenes (c. 276­–c. 195 BCE).

A perspective view of the Earth showing how latitude (𝛟) and longitude (𝛌) are defined on a spherical model. The graticule spacing is 10 degrees.

The concept was reintroduced into Early Modern Europe by the discovery of Ptolemaeus’ Geographia. It’s all very well to have a location grid on your maps and charts but it’s a very different problem to determine where exactly you are on that grid when stuck in the middle of an ocean. However, before we consider this problem and its solutions I want to return to the Portuguese Board of Navigation, which I briefly mentioned above.

Both the Portuguese and the Spanish realised fairly early on as they began to journey out onto the oceans that they needed some way of collecting and collating new geographical and navigation relevant information that their various expeditions brought back with them and also a way of imparting the relevant information and techniques to navigators due to set out on new expeditions. Both countries established official institutions to fulfil these tasks and also appointed official cosmographers to lead these endeavours. Pedro Nunes (1502–1578), who we met in the first episode on navigation, as the discoverer of the loxodrome, was appointed Portugal’s Royal Cosmographer in 1529 and Chief Royal Cosmographer in 1547, a post he held until his death.

Image of Portuguese mathematician Pedro Nunes in Panorama magazine (1843); Lisbon, Portugal. Source: Wikimedia Commons

The practice of establishing official organisations to teach cartography and navigation, as well as the mathematics they needed to carry them out to seamen was followed in time by France, Holland, and Britain as they too began to send out deep sea marine expeditions. 

To determine latitude and longitude are two very different problems and I will start with the easier of the two, the determination of latitude. For the determination of longitude or latitude you first need a null point, for latitude this is the equator. In the northern hemisphere your latitude is how many degrees you are north of the equator. You can determine your latitude using either the Sun during the day or the North Star at night. At night you need to observe the North Star with some sort of angle measuring device then measure the angle that makes to the horizon and that angle is your latitude in degrees. During the day you need to observe the Sun at exactly noon with an angle measuring device then the angle to makes with a vertical plumb line is your latitude. This is only strictly true for the date of the two equinoxes. For other days of the year, you have to calculate an adjustment using tables. For these observations mariners initially used either a quadrant,

Geometric quadrant with plumb bob. Source: Wikimedia Commons

which had been in use since antiquity or a Jacob’s Staff or Cross Staff, the invention of which is attributed to the French astronomer Levi Ben Gershon (1268–1344).

A sailor uses a ‘Jacob’s Staff’ to calculate the angle between a star and the horizon Source

Contrary to many claims, astrolabes were never used on ships for this purpose. However, around the end of the fifteenth century a much-simplified version of the astrolabe, the mariner’s astrolabe began to be used for this purpose. 

Mariner’s astrolabe Source: Wikimedia Commons

Because looking directly into the Sun is not good for the eyes, the backstaff was developed over time. With a backstaff the mariner stands with his back to the Sun and a shadow is cast onto the angle measuring scale. Thomas Harriot (c. 1560–1621) is credited with being the originator of the concept. The mariner John Davis (c. 1550–1605) introduced the double quadrant or Davis quadrant in his book on practical navigation, The Seaman’s Secrets in 1594, a device that evolved over time.

Davis quadrant, made in 1765 by Johannes Van Keulen. On display at the Musée national de la Marine in Paris. Source: Wikimedia Commons
How a Davis Quadrant is used Source includes a video of how to use one

In 1730, John Hadley invented the reflecting octant, which incorporated a mirror to reflect the image of the Sun, whilst the user observed the horizon.

John Hadley Source: Wikimedia Commons
Hadley Octant Source includes video

This evolved into the sextant the device still used today to “shoot the Sun” as it is called. Here we see an evolution of instruments used to fulfil a specific function.

The determination of longitude at sea is a much more difficult problem. First, there is no natural null point, and any meridian can be and indeed was used until the Greenwich Meridian was chosen as the international null point for the determination of longitude at the International Meridian Conference in Washington in 1884. Because the Earth revolves once in twenty-four hours the determination of the difference in longitude between two locations is equivalent to the difference in local time between them, one degree of longitude equals four minutes of time difference, so the determination of longitude is basically the determination of time differences, which is easy to state but much more difficult to carry out.

The various European sea going nations–Spain, Portugal, France, Holland, Britain–all offered financial awards to anybody who could come up with a practical solution for determining longitude at sea. 

In antiquity, the difference in longitude between two locations was determined by calculating the difference in the observation times of major astronomical events such as lunar or solar eclipses. Then, if one had determined the difference in longitude between two given locations and their respective distances from a third location, it was possible to calculate the difference in longitude for the third location geometrically. Using these methods, astronomers, and cartographers gradually built-up tables of longitude for large numbers of towns and cities such as the one found in Ptolemaeus’ Geographia. This method is, of course, not practical for mariners at sea.

Starting in the early sixteenth century, various methods were suggested for determining time differences in order to determine longitude. The Nürnberger mathematicus Johannes Werner (1468 – 1522) in his In hoc opere haec continentur Nova translatio primi libri geographiae Cl’ Ptolomaei … (Nürnberg 1514) proposed the so-called lunar distance method. In this method an accurate table of the position of the Moon relative to a given set of reference stars for a given location for the entire year needs to be created.

Source: Wikimedia Commons

The mariner then has to observe the position of the Moon relative to the reference stars for his local time and then calculate the time difference to the given location from the tables. Unfortunately, because the Moon is pulled all over the place by the gravitational influence of both the Sun and the Earth, its orbit is highly irregular and the preparation of such tables proved beyond the capabilities of sixteenth century astronomers and indeed of seventeenth century astronomers, when the method was proposed again by Jean-Baptiste Morin (1583–1656). There was also the problem of an instrument accurate enough to measure the position of the Moon on a moving ship. It was Tobias Mayer (1723–1762), who first managed to produce accurate tables and Hadley’s octant or rather the sextant that evolved out of it solved the instrument problem. The calculations necessary to determine longitude having measured the lunar distance proved to be too complex and too time consuming for seamen and so Neville Maskelyne produced the Nautical Almanac containing the results pre-calculated in the form of tables and published for the first time in 1766.

Portrait of Nevil Maskelyne by Edward Scriven Source: Wikimedia Commons
Source: Library of Congress Washington

The next solution to the problem of determining longitude suggested during the Renaissance by Gemma Frisius (1508–1555) was the clock, published in his De principiis astronomiae et cosmographiae. (Antwerp, 1530).

Gemma Frisius 17th C woodcut by E. de Boulonois Source: Wikimedia Commons

The mariner should take a clock, capable of maintaining accurate time over a long period under the conditions that prevail on a ship on the high seas, set to the time of the point of departure. By comparing local time with the clock time, the longitude difference could then be calculated. The problem was that although mechanical clocks had been around for a couple of centuries when Gemma Frisius made his suggestion, they were incapable of maintaining the required accuracy on land, let alone on a ship at sea. Jean-Baptiste Morin thought it would never be possible, “I do not know if the Devil will succeed in making a longitude timekeeper but it is folly for man to try.” A view apparently shared by Isaac Newton, when he sat on the English Board of Longitude.

Only when Christiaan Huygens (1629–1695) had the first pendulum clock constructed by Salomon Coster (c. 1620–1659) accord his design in 1657 that Frisius’ idea began to seem realistic.

Christiaen Huygens II (1629-1695) signed C.Netscher / 1671 Source: Wikimedia Commons
Spring-driven pendulum clock, designed by Huygens and built by Salomon Coster (1657),  with a copy of the Horologium Oscillatorium (1673), at Museum Boerhaave, Leiden. Source: Wikimedia Commons

One of Huygens’ clocks was actually sent on sea trials but failed the test. In what is, thanks to Dava Sobel[1], probably the most well-known story in the history of technology John Harrison (1693–1776)

P. L. Tassaert’s half-tone print of Thomas King’s original 1767 portrait of John Harrison, located at the Science and Society Picture Library, London Source: Wikimedia Commons

finally succeeded in producing a clock capable of fulfilling the demands with his H4 in 1761, slightly later than the successful fulfilment of the lunar distance method. In one sense the problem was still not really solved because the H4 was too complex and too expensive for it to be mass produced at a reasonable cost for use in sea transport. It was only really in the nineteenth century, after further developments in clock technology, that the marine chronometer became a real solution to the longitude problem.

Harrison’s “sea watch” No.1 (H4), with winding crank Source: Wikimedia Commons

Back tacking, at the beginning of the seventeenth century with the discovery of the four largest moons of Jupiter another method suggested itself. These moons, Io, Europa, Ganymede, and Callisto, have orbital periods of respectively, 1.77, 3.55, 7.15, and 16.6 days.

A montage of Jupiter and its four largest moons (distance and sizes not to scale) Source: Wikimedia Commons

This means that one or other of them is being fairly often eclipsed by Jupiter. Galileo argued that is one could calculate the orbits accurately enough they could be used as a clock to determine longitude. He tried to sell the idea to the governments of both Spain and the Netherlands without success. The principal problem was the difficulty of observing them with a telescope on a moving ship. Galileo worked on an idea of an observing chair with the telescope mounted on a helmet, but the idea never made it off the paper. Later in the seventeenth century Jean-Dominique Cassini (1625–1712) produced tables of the orbits accurate enough for them to be used to determine longitude and he and Jean Picard (1620–1682) used the method on land to accurately determine the borders of France, leading Louis XVI to famously quip that he had lost more territory to the cartographers than he ever lost to his enemies.

Map showing both old and new French coastlines Source: Wikimedia Commons

In the first part of this account of navigation I described the phenomenon of magnetic variations or declination, which is the fact that that a compass does not point to true north but to magnetic north, which is somewhat removed from true north. I also mentioned that magnetic declination is not constant but varies from location to location. This led to the thought that if one were to map the magnetic inclination for the entire Atlantic one could use the data to determine longitude, whilst at sea. Edmond Halley (1556–1742) did in fact create such a map on a voyage from1699 to 1700. However, this method of determining longitude was never really utilised. 

Portrait of Halley (c. 1690) by Thomas Murray Source: Wikimedia Commons
Halley’s 1701 map showing isogonic lines of equal magnetic declination in the Atlantic Ocean. Source: Wikimedia Commons

Although the methods eventually developed to determine longitude on the high seas all came to fruition long after the Renaissance, they all have their roots firmly planted in the practical science of the Renaissance. This brief sketch also displays an important aspect of the history of science and technology. A lot of time can pass, and very often does, between the recognition of a problem, the suggestion of one or more solutions to that problem, and the realisation or fulfilment of those solutions.

Having gone to great lengths to describe the principal methods suggested and eventually realised for determining longitude, there were others ranging from the sublime to the ridiculous that I haven’t described, there remains the question, how did mariners navigate when far away from the coast during the Early Modern Period? There are two answers firstly latitude sailing and secondly dead reckoning. In latitude sailing, instead of, for example, trying to cross the Atlantic by the most direct course from A to B, the navigator first sails due north or south along the coast until he reaches the latitude of his planned destination. They then turn their ship through ninety degrees and maintain a course along that latitude. This, of course, nearly always means a much longer voyage but one with less risk of getting lost. 

In dead reckoning, the navigator, starting from a fixed point, measures the speed and direction of his ship over a given period of time transferring this information mathematically to a sea chat to determine their new position. The direction is determined with the compass, but the determination of the ship’s speed is at best an approximation, which was carried out in the following manner. A log would be thrown overboard at the front of the ship and the mariners would measure how long it took for the ship to pass the log, and the result recorded in a book, which became known as the logbook. The term logbook expanded to include all the information recorded on a voyage on a sip and then later on planes and even lorries. Of note, the word blog is an abbreviation of the term weblog, a record of web or internet activity, but I’m deviating from the topic.

An example of dead reckoning Columbus’ return voyage Source

The process of measuring the ships speed evolved over time. The log was thrown overboard attached to a long line and using an hourglass, the time how long the line needed to pay out was recorded. Later the line was knotted at regular intervals and the number of knots were recorded for a given time period. This is, of course, the origin of the term knots for the speed of ships and aircraft. Overtime the simple log of wood was replaced with a so-called chip-log, which became standardised:

The shape is a quarter circle, or quadrant with a radius of 5 inches (130 mm) or 6 inches (150 mm), and 0.5 inches (13 mm) thick. The logline attaches to the board with a bridle of three lines that connect to the vertex and to the two ends of the quadrant’s arc. To ensure the log submerges and orients correctly in the water, the bottom of the log is weighted with lead. This provides more resistance in the water, and a more accurate and repeatable reading. The bridle attaches in such a way that a strong tug on the logline makes one or two of the bridle’s lines release, enabling a sailor to retrieve the log. (Wikipedia)

Model of chip log and associated kit. The reel of log-line is clearly visible. The first knot, marking the first nautical mile is visible on the reel just below the centre. The timing sandglass is in the upper left and the chip log is in the lower left. The small light-coloured wooden pin and plug form a release mechanism for two lines of the bridle. From the Musée de la Marine, Paris. Source: Wikimedia Commons

The invention of the log method of determining a ship’s speed is attributed to the Portuguese mariner Bartolomeu Crescêncio at the end of the fifteenth century. The earliest known published account of using a log to determine a ship’s speed was by William Bourne (c. 1535–1582) in his A regiment of the Sea in 1574, which went through 11 English editions up to 1631 and at least 3 Dutch edition from 1594. 

Dead reckoning is a process that is prone to error, as it doesn’t take into account directional drift caused by wind and currents. Another problem was that not all mariners processed the necessary mathematical knowledge to transfer the data to a sea chart. Those mariners, who disliked and rejected the mathematical approach used a traverse board, which uses threads and pegs to record direction and speed of a ship. William Bourne writing in 1571 said:

I have known within these 20 years that them that were ancient masters of shippes hathe derided and mocked them that have occupied their cards and plattes and also the observation of the Altitude of the Pole saying; that they care not for their sheepskinne for he could keepe a better account upon a board.

This blog post is already far too long, so I’ll skip a detailed description of the traverse board, but you can read one here.

We have one last Renaissance contribution to the art of navigation from the English mathematical practitioner, Edmund Gunter (1581–1626), who we have already met as the inventor of the standard English surveyor’s chain in the episode on surveying. Gunter invented the Gunter scale or rule, simply known as the “gunter” by mariners, which he published in his Description and Use of the Sector, the Crosse-staffe and other Instrumentsin 1623. Developed shortly after the invention of logarithms, the scale is usually somewhat more than a half metre long and about 40 mm broad. It is engraved on both sides with various scales or lines. Usually, on the one side are natural line, chords, sines, tangents, rhumbs etc., and on the other scales of the logarithms of those functions. Navigational mathematical problems were then worked through using a pair of compasses. 

Gunter scale front
Gunter scale back Source

Despite its drawbacks, uncertainties, and errors dead reckoning was used for centuries by European mariners to crisscross the oceans and circumnavigate the globe. It continued to be used well into the nineteenth century, long after the perfection of the marine chronometer and the lunar distance method. 

This over long blog post is but a sketch of the contributions made by the Renaissance mathematical practitioners to the development of methods of deep-sea navigation required by the European mariners during the Contact Period, when they swarmed out to investigate the world beyond Europe and exploit it. Those contributions were in the form of theories, publications, instruments, charts, and practical instruction (which I haven’t really expanded upon here). For a more detailed version of the story, I heartily recommend Margaret Scotte’s excellent Sailing School: Navigation Science and Skill, 1550–1800 (Johns Hopkins University Press, 2019).


[1] Sobel’s account of the story is somewhat less than historically accurate and as always, I recommend instead Dunne and Higgitt, Finding LongitudeHow ships, clocks and stars helped solve the longitude problem (Collins, 2014)

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Filed under Early Scientific Publishing, History of Astronomy, History of Mathematics, History of Navigation, Renaissance Science

Renaissance science – XXVII

Early on in this series I mentioned that a lot of the scientific developments that took place during the Renaissance were the result of practical developments entering the excessively theoretical world of the university disciplines. This was very much the case in the mathematical sciences, where the standard English expression for the Renaissance mathematicus is mathematical practitioner. In this practical world, areas that we would now regard as separate disciples were intertwined is a complex that the mathematical practitioners viewed as one discipline with various aspects, this involved astronomy, cartography, navigation, trigonometry, as well as instrument and globe making. I have already dealt with trigonometry, cartography and astronomy and will here turn my attention to navigation, which very much involved the other areas in that list.

The so-called Age of Discovery or Age of Exploration, that is when Europeans started crossing the oceans and discovering other lands and other cultures, coincides roughly with the Renaissance and this was, of course the main driving force behind the developments in navigation during this period. Before we look at those developments, I want to devote a couple of lines to the terms Age of Discovery and Age of Exploration. Both of them imply some sort of European superiority, “you didn’t exist until we discovered you” or “your lands were unknown until we explored them.” The populations of non-European countries and continents were not sitting around waiting for their lands and cultures to be discovered by the Europeans. In fact, that discovery very often turned out to be highly negative for the discovered. The explorers and discoverers were not the fearless, visionary heroes that we tend to get presented with in our schools, but ruthless, often brutal chancers, who were out to make a profit at whatever cost.  This being the case the more modern Contact Period, whilst blandly neutral, is preferred to describe this period of world history.

As far as can be determined, with the notable exception of the Vikings, sailing in the Atlantic was restricted to coastal sailing before the Late Middle Ages. Coastal sailing included things such as crossing the English Channel, which, archaeological evidence suggests, was done on a regular basis since at least the Neolithic if not even earlier. I’m not going to even try to deal with the discussions about how the Vikings possibly navigated. Of course, in other areas of the world, crossing large stretches of open water had become common place, whilst the European seamen still clung to their coast lines. Most notable are the island peoples of the Pacific, who were undertaking long journeys across the ocean already in the first millennium BCE. Arab and Chinese seamen were also sailing direct routes across the Indian Ocean, rather than hugging the coastline, during the medieval period. It should be noted that European exploited the navigation skills developed by these other cultures as they began to take up contact with the other part of the world. Vasco da Gamma (c. 1460–1524) used unidentified local navigators to guide his ships the first time he crossed the Indian Ocean from Africa to India. On his first voyage of exploration of the Pacific Ocean from 1768 to 1771, James Cook (1728–1779) used the services of the of the Polynesian navigator, Tupaia (c. 1725–1770), who even drew a chart, in cooperation with Cook, Joseph Banks, and several of Cooks officer, of his knowledge of the Pacific Ocean. 

Tupaia’s map, c. 1769 Source: Wikimedia Commons

There were two major developments in European navigation during the High Middle Ages, the use of the magnetic compass and the advent of the Portolan chart. The Chinese began to use the magnetic properties of loadstone, the mineral magnetite, for divination sometime in the second century BCE. Out of this they developed the compass needle over several centuries. It should be noted that for the Chinese, the compass points South and not North. The earliest Chinese mention of the use of a compass for navigation on land by the military is before 1044 CE and in maritime navigation in 1117 CE.

Diagram of a Ming Dynasty (1368–1644) mariner’s compass Source: Wikimedia Commons

Alexander Neckam (1157–1219) reported the use of the compass for maritime navigation in the English Channel in his manuscripts De untensilibus and De naturis rerum, written between 1187 and 1202.

The sailors, moreover, as they sail over the sea, when in cloudy whether they can no longer profit by the light of the sun, or when the world is wrapped up in the darkness of the shades of night, and they are ignorant to what point of the compass their ship’s course is directed, they touch the magnet with a needle, which (the needle) is whirled round in a circle until, when its motion ceases, its point looks direct to the north.

This and other references to the compass suggest that it use was well known in Europe by this time.

A drawing of a compass in a mid 14th-century copy of Epistola de magnete of Peter Peregrinus. Source: Wikimedia Commons

The earliest reference to maritime navigation with a compass in the Muslim world in in the Persian text Jawāmi ul-Hikāyāt wa Lawāmi’ ul-Riwāyāt (Collections of Stories and Illustrations of Histories) written by Sadīd ud-Dīn Muhammad Ibn Muhammad ‘Aufī Bukhārī (1171-1242) in 1232. There is still no certainty as to whether there was a knowledge transfer from China to Europe, either direct or via the Islamic Empire, or independent multiple discovery. Magnetism and the magnetic compass went through a four-hundred-year period of investigation and discovery until William Gilbert (1544–1603) published his De magnete in 1600. 

De Magnete, title page of 1628 edition Source: Wikimedia Commons

The earliest compasses used for navigation were in the form of a magnetic needle floating in a bowl of water. These were later replaced with dry mounted magnetic needles. The first discovery was the fact that the compass needle doesn’t actually point at the North Pole, the difference is called magnetic variation or magnetic declination. The Chinese knew of magnetic declination in the seventh century. In Europe the discovery is attributed to Georg Hartmann (1489–1564), who describes it in an unpublished letter to Duke Albrecht of Prussia. However, Georg von Peuerbach (1423–1461) had already built a portable sundial on which the declination for Vienna is marked on the compass.

NIMA Magnetic Variation Map 2000 Source: Wikimedia Commons

There followed the discovery that magnetic declination varies from place to place. Later in the seventeenth century it was also discovered that declination also varies over time. We now know that the Earth’s magnetic pole wanders, but it was first Gilbert, who suggested that the Earth is a large magnet with poles. The next discovery was magnetic dip or magnetic inclination. This describes the fact that a compass needle does not sit parallel to the ground but points up or down following the lines of magnetic field. The discovery of magnetic inclination is also attributed to Georg Hartmann. The sixteenth century English, seaman Robert Norman rediscovered it and described how to measure it in his The Newe Attractive (1581) His work heavily influenced Gilbert. 

Illustration of magnetic dip from Norman’s book, The Newe Attractive Source: Wikimedia Commons

The Portolan chart, the earliest known sea chart, emerged in the Mediterranean in the late thirteenth century, not long after the compass, with which it is closely associated, appeared in Europe. The oldest surviving Portolan, the Carta Pisana is a map of the Mediterranean, the Black Sea and part of the Atlantic coast.

Source: Wikimedia Commons

The origins of the Portolan chart remain something of a mystery, as they are very sophisticated artifacts that appear to display no historical evolution. A Portolan has a very accurate presentation of the coastlines with the locations of the major harbours and town on the coast. Otherwise, they have no details further inland, indicating that they were designed for use in coastal sailing. A distinctive feature of Portolans is their wind roses or compass roses located at various points on the charts. These are points with lines radiating outwards in the sixteen headings, on later charts thirty-two, of the mariner’s compass.

Central wind rose on the Carta Pisana

Portolan charts have no latitude or longitude lines and are on the so-called plane chart projection, which treats the area being mapped as flat, ignoring the curvature of the Earth. This is alright for comparatively small areas, such as the Mediterranean, but leads to serious distortion, when applied to larger areas.

During the Contact Period, Portolan charts were extended to include the west coast of Africa, as the Portuguese explorers worked their way down it. Later, the first charts of the Americas were also drawn in the same way. Portolan style charts remained popular down to the eighteenth century.

Portolan chart of Central America c. 1585-1595 Source:

A central problem with Portolan charts over larger areas is that on a globe constant compass bearings are not straight lines. The solution to the problem was found by the Portuguese cosmographer Pedro Nunes (1502–1578) and published in his Tratado em defensam da carta de marear (Treatise Defending the Sea Chart), (1537).

Image of Portuguese mathematician Pedro Nunes in Panorama magazine (1843); Lisbon, Portugal. Source: Wikimedia Commons

The line is a spiral known as a loxodrome or rhumb lines. Nunes problem was that he didn’t know how to reproduce his loxodromes on a flat map.

Image of a loxodrome, or rhumb line, spiraling towards the North Pole Source: Wikimedia Commons

The solution to the problem was provided by the map maker Gerard Mercator (1512–1594), when he developed the so-called Mercator projection, which he published as a world map, Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata (New and more complete representation of the terrestrial globe properly adapted for use in navigation) in 1569.

Source: Wikimedia Commons
The 1569 Mercator world map Source: Wikimedia Commons.

On the Mercator projection lines of constant compass bearing, loxodromes, are straight lines. This however comes at a price. In order to achieve the required navigational advantage, the lines of latitude on the map get further apart as one moves away from the centre of projection. This leads to an area distortion that increases the further north or south on goes from the equator. This means that Greenland, slightly more than two million square kilometres, appear lager than Africa, over thirty million square kilometres.

Mercator did not publish an explanation of the mathematics used to produce his projection, so initially others could reproduce it. In the late sixteenth century three English mathematicians John Dee (1527–c. 1608), Thomas Harriot (c. 1560–1621), and Edward Wright (1561–1615) all individually worked out the mathematics of the Mercator projection. Although Dee and Harriot both used this knowledge and taught it to others in their respective functions as mathematical advisors to the Muscovy Trading Company and Sir Walter Raleigh, only Wright published the solution in his Certaine Errors in Navigation, arising either of the Ordinarie Erroneous Making or Vsing of the Sea Chart, Compasse, Crosse Staffe, and Tables of Declination of the Sunne, and Fixed Starres Detected and Corrected. (The Voyage of the Right Ho. George Earle of Cumberl. to the Azores, &c.) published in London in 1599. A second edition with a different, even longer, title was published in the same year. Further editions were published in 1610 and 1657. 

Source: Wikimedia Commons
Wright explained the Mercator projection with the analogy of a sphere being inflated like a bladder inside a hollow cylinder. The sphere is expanded uniformly, so that the meridians lengthen in the same proportion as the parallels, until each point of the expanding spherical surface comes into contact with the inside of the cylinder. This process preserves the local shape and angles of features on the surface of the original globe, at the expense of parts of the globe with different latitudes becoming expanded by different amounts. The cylinder is then opened out into a two-dimensional rectangle. The projection is a boon to navigators as rhumb lines are depicted as straight lines. Source: Wikimedia Commons

His mathematical solution for the Mercator projection had been published previously with his permission and acknowledgement by Thomas Blundeville (c. 1522–c. 1606) in his Exercises (1594) and by William Barlow (died 1625) in his The Navigator’s Supply (1597). However, Jodocus Hondius (1563–1612) published maps using Wright’s work without acknowledgement in Amsterdam in 1597, which provoked Wright to publish his Certaine Errors. Despite its availability, the uptake on the Mercator projection was actually very slow and it didn’t really come into widespread use until the eighteenth century.

Wright’s “Chart of the World on Mercator’s Projection” (c. 1599), otherwise known as the Wright–Molyneux map because it was based on the globe of Emery Molyneux (died 1598) Source: Wikimedia Commons

Following the cartographical trail, we have over sprung a lot of developments in navigation to which we will return in the next episode. 

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Filed under History of Cartography, History of Mathematics, History of Navigation, Renaissance Science

The Epicurean mathematician

Continuing our look at the group of mathematician astronomers associated with Nicolas-Claude Fabri de Peiresc (1580-1637) in Provence and Marin Mersenne (1588–1648) in Paris, we turn today to Pierre Gassendi (1592–1655), celebrated in the world of Early Modern philosophy, as the man who succeeded in making Epicurean atomism acceptable to the Catholic Church. 

Pierre Gassendi Source: Wikimedia Commons

Pierre Gassendi was born the son of the peasant farmer Antoine Gassend and his wife Fançoise Fabry in the Alpes-de-Haute-Provence village of Champtercier on 22 January 1592. Recognised early as something of a child prodigy in mathematics and languages, he was initially educated by his uncle Thomas Fabry, a parish priest. In 1599 he was sent to the school in Digne, a town about ten kilometres from Champtercier, where he remained until 1607, with the exception of a year spent at school in another nearby village, Riez. 

In 1607 he returned to live in Champtercier and in 1609 he entered the university of Aix-en-Provence, where his studies were concentrated on philosophy and theology, also learning Hebrew and Greek. His father Antoine died in 1611. From 1612 to 1614 his served as principle at the College in Digne. In 1615 he was awarded a doctorate in theology by the University of Avignon and was ordained a priest in 1615. From 1614 he held a minor sinecure at the Cathedral in Digne until 1635, when he was elevated to a higher sinecure. From April to November in 1615 he visited Paris for the first time on Church business. 

Cathédrale Saint-Jérome de Digne Source: Wikimedia Commons

In 1617 both the chair of philosophy and the chair of theology became vacant at the University of Aix; Gassendi applied for both chairs and was offered both, one should note that he was still only twenty-four years old. He chose the chair for philosophy leaving the chair of theology for his former teacher. He remained in Aix for the next six years. 

When Gassendi first moved to Aix he lived in the house of the Provencal astronomer Joseph Gaultier de la Valette (1564–1647), vicar general of Aix and Peiresc’s observing partner. Whilst living in Gaultier’s house he got to know Jean-Baptiste Morin (1583–1556), who was also living there as Gaultier’s astronomical assistant. Although, in later years, in Paris, Gassendi and Morin would have a major public dispute, in Aix the two still young aspiring astronomers became good friends. It was also through Gaultier that Gassendi came to the attention of Peiresc, who would go on to become his patron and mentor. 

Jean-Baptiste Morin Source: Wikimedia Commons

For the next six years Gassendi taught philosophy at the University of Aix and took part in the astronomical activities of Peiresc and Gaultier, then in 1623 the Jesuits took over the university and Gassendi and the other non-Jesuit professors were replaced by Jesuits. Gassendi entered more than twenty years of wanderings without regular employment, although he still had his sinecure at the Cathedral of Digne.

In 1623, Gassendi left Aix for Paris, where he was introduced to Marin Mersenne by Peiresc. The two would become very good friends, and as was his wont, Mersenne took on a steering function in Gassendi’s work, encouraging him to engage with and publish on various tropics. In Paris, Gassendi also became part of the circle around Pierre Dupuy (1582–1651) and his brother Jacques (1591–1656), who were keepers of the Bibliothèque du Roi, today the Bibliothèque nationale de France, and who were Ismael Boulliau’s employers for his first quarter century in Paris.

Pierre Dupuy Source: Wikimedia Commons

The Paris-Provence group Peiresc (1580–1637), Mersenne (1588–1648), Morin (1583–1656), Boulliau (1605–1694), and Gassendi (1592–1655) are all members of the transitional generation, who not only lived through the transformation of the scientific view of the cosmos from an Aristotelian-Ptolemaic geocentric one to a non-Aristotelian-Keplerian heliocentric one but were actively engaged in the discussions surrounding that transformation. When they were born in the late sixteenth century, or in Boulliau’s case the early seventeenth century, despite the fact that Copernicus’ De revolutionibus had been published several decades earlier and although a very small number had begun to accept a heliocentric model and another small number the Tychonic geo-heliocentric one, the geocentric model still ruled supreme. Kepler’s laws of planetary motion and the telescopic discoveries most associated with Galileo still lay in the future. By 1660, not long after their deaths, with once again the exception of Boulliau, who lived to witness it, the Keplerian heliocentric model had been largely accepted by the scientific community, despite there still being no empirical proof of the Earth’s movement. 

Given the Church’s official support of the Aristotelian-Ptolemaic geocentric model and after about 1620 the Tychonic geo-heliocentric model, combined with its reluctance to accept this transformation without solid empirical proof, the fact that all five of them were devout Catholics made their participation in the ongoing discussion something of a highwire act. Gassendi’s personal philosophical and scientific developments over his lifetime are a perfect illustration of this. 

During his six years as professor of philosophy at the University of Aix, Gassendi taught an Aristotelian philosophy conform with Church doctrine. However, he was already developing doubts and in 1624 he published the first of seven planned volumes criticising Aristotelian philosophy, his Exercitationes paradoxicae adversus aristoteleos, in quibus praecipua totius peripateticae doctrinae fundamenta excutiuntur, opiniones vero aut novae, aut ex vetustioribus obsoletae stabiliuntur, auctore Petro Gassendo. Grenoble: Pierre Verdier. In 1658, Laurent Anisson and Jean Baptiste Devenet published part of the second volume posthumously in Den Hague in 1658. Gassendi seems to have abandoned his plans for the other five volumes. 

To replace Aristotle, Gassendi began his promotion of the life and work of Greek atomist Epicurus (341–270 BCE). Atomism in general and Epicureanism in particular were frowned upon by the Christian Churches in general. The Epicurean belief that pleasure was the chief good in life led to its condemnation as encouraging debauchery in all its variations. Atomists, like Aristotle, believed in an eternal cosmos contradicting the Church’s teaching on the Creation. Atomist matter theory destroyed the Church’s philosophical explanation of transubstantiation, which was based on Aristotelian matter theory. Last but no means least Epicurus was viewed as being an atheist. 

In his biography of Epicurus De vita et moribus Epicuri libri octo published by Guillaume Barbier in Lyon in 1647

and revival and reinterpretation of Epicurus and Epicureanism in his Animadversiones in decimum librum Diogenis Laertii: qui est De vita, moribus, placitisque Epicuri. Continent autem Placita, quas ille treis statuit Philosophiae parteis 3 I. Canonicam, …; – II. Physicam, …; – III. Ethicam, … and his Syntagma philosophiae Epicuri cum refutationibus dogmatum quae contra fidem christianam ab eo asserta sunt published together by Guillaume Barbier in Lyon in 1649,

Gassendi presented a version of Epicurus and his work that was acceptable to Christians, leading to both a recognition of the importance of Epicurean philosophy and of atomism in the late seventeenth and early eighteenth centuries. 

Gassendi did not confine himself to work on ancient Greek philosophers. In 1629,  pushed by Mersenne, the scientific agent provocateur, he wrote an attack on the hermetic philosophy of Robert Fludd (1574–1637), who famously argued against mathematics-based science in his debate with Kepler. Also goaded by Mersenne, he read Descartes’ Meditationes de prima philosophia (Meditations on First Philosophy) (1641) and published a refutation of Descartes’ methodology. As a strong scientific empiricist, Gassendi had no time for Descartes’ rationalism. Interestingly, it was Gassendi in his Objections (1641), who first outlined the mind-body problem, reacting to Descartes’ mind-body dualism. Descartes was very dismissive of Gassendi’s criticisms in his Responses, to which Gassendi responded in his Instantiae (1642). 

Earlier, Gassendi had been a thorn in Descartes side in another philosophical debate. In 1628, Gassendi took part in his only journey outside of France, travelling through Flanders and Holland for several months, although he did travel widely throughout France during his lifetime. Whilst in Holland, he visited Isaac Beeckman (1588–1637) with whom he continued to correspond until the latter’s death. Earlier, Beeckman had had a massive influence on the young Descartes, introducing him to the mechanical philosophy. In 1630, Descartes wrote an abusive letter denying any influence on his work by Beeckman. Gassendi, also a supporter of the mechanical philosophy based on atomism, defended Beeckman.

Like the others in the Mersenne-Peiresc group, Gassendi was a student and supporter of the works of both Johannes Kepler (1571–1630) and Galileo Galilei (1564–1642) and it is here that he made most of his contributions to the evolution of the sciences in the seventeenth century. 

Having been introduced to astronomy very early in his development by Peiresc and Gaultier de la Valette, Gassendi remained an active observational astronomer all of his life. Like many others, he was a fan of Kepler’s Tabulae Rudolphinae (Rudolphine Tables) (1627) the most accurate planetary tables ever produced up till that time. Producing planetary tables and ephemerides for use in astrology, cartography, navigation, etc was regarded as the principal function of astronomy, and the superior quality of Kepler’s Tabulae Rudolphinae was a major driving force behind the acceptance of a heliocentric model of the cosmos. Consulting the Tabulae Rudolphinae Gassendi determined that there would be a transit of Mercury on 7 November 1631. Four European astronomers observed the transit, a clear proof that Mercury orbited the Sun and not the Earth, and Gassendi, who is credited with being the first to observe a transit of Mercury, published his observations Mercvrivs in sole visvs, et Venvs invisa Parisiis, anno 1631: pro voto, & admonitione Keppleri in Paris in 1632.

He also tried to observe the transit of Venus, predicted by Kepler for 6 December 1631, not realising that it was not visible from Europe, taking place there during the night. This was not yet a proof of heliocentricity, as it was explainable in both the Capellan model in which Mercury and Venus both orbit the Sun, which in turn orbits the Earth and the Tychonic model in which the five planets all orbit the Sun, which together with the Moon orbits the Earth. But it was a very positive step in the right direction. 

In his De motu impresso a motore translato. Epistolæ duæ. In quibus aliquot præcipuæ tum de motu vniuersè, tum speciatim de motu terræattributo difficulatates explicantur published in Paris in 1642, he dealt with objections to Galileo’s laws of fall.

Principally, he had someone drop stones from the mast of a moving ship to demonstrate that they conserve horizontal momentum, thus defusing the argument of those, who claimed that stones falling vertically to the Earth proved that it was not moving. In 1646 he published a second text on Galileo’s theory, De proportione qua gravia decidentia accelerantur, which corrected errors he had made in his earlier publication.

Like Mersenne before him, Gassendi tried, using a cannon, to determine the speed of sound in 1635, recording a speed of 1,473 Parian feet per second. The actual speed at 20° C is 1,055 Parian feet per second, making Gassendi’s determination almost forty percent too high. 

In 1648, Pascal, motivated by Mersenne, sent his brother-in-law up the Puy de Dôme with a primitive barometer to measure the decreasing atmospheric pressure. Gassendi provided a correct interpretation of this experiment, including the presence of a vacuum at the top of the tube. This was another indirect attack on Descartes, who maintained the assumption of the impossibility of a vacuum. 

Following his expulsion from the University of Aix, Nicolas-Claude Fabri de Peiresc’s house became Gassendi’s home base for his wanderings throughout France, with Peiresc helping to finance his scientific research and his publications. The two of them became close friends and when Peiresc died in 1637, Gassendi was distraught. He preceded to mourn his friend by writing his biography, Viri illvstris Nicolai Clavdii Fabricii de Peiresc, senatoris aqvisextiensis vita, which was published by Sebastian Cramoisy in Paris in 1641. It is considered to be the first ever complete biography of a scholar. It went through several edition and was translated into English.

In 1645, Gassendi was appointed professor of mathematics at the Collège Royal in Paris, where he lectured on astronomy and mathematics, ably assisted by the young Jean Picard (1620–1682), who later became famous for accurately determining the size of the Earth by measuring a meridian arc north of Paris.

Jean Picard

Gassendi only held the post for three years, forced to retire because of ill health in 1648. Around this time, he and Descartes became reconciled through the offices of the diplomat and cardinal César d’Estrées (1628–1714). 

Gassendi travelled to the south for his health and lived for two years in Toulon, returning to Paris in 1653 when his health improved. However, his health declined again, and he died of a lung complaint in 1655.

Although, like the others in the group, Gassendi was sympathetic to a heliocentric world view, during his time as professor he taught the now conventional geo-heliocentric astronomy approved by the Catholic Church, but also discussed the heliocentric systems. His lectures were written up and published as Institutio astronomica juxta hypotheseis tam veterum, quam Copernici et Tychonis in 1647. Although he toed the party line his treatment of the heliocentric was so sympathetic that he was reported to the Inquisition, who investigated him but raised no charges against him. Gassendi’s Institutio astronomica was very popular and proved to be a very good source for people to learn about the heliocentric system. 

As part of his campaign to promote the heliocentric world view, Gassendi also wrote biographies of Georg Peuerbach, Regiomontanus, Copernicus, and Tycho Brahe. It was the only biography of Tycho based on information from someone, who actually knew him. The text, Tychonis Brahei, eqvitis Dani, astronomorvm coryphaei vita, itemqve Nicolai Copernici, Georgii Peverbachii & Ioannis Regiomontani, celebrium Astronomorum was published in Paris in 1654, with a second edition appearing in Den Hague in the year of Gassendi’s death, 1655. In terms of historical accuracy, the biographies are to be treated with caution.

Gassendi also became engaged in a fierce dispute about astronomical models with his one-time friend from his student days, Jean-Baptiste Morin, who remained a strict geocentrist. I shall deal with this when I write a biographical sketch of Morin, who became the black sheep of the Paris-Provencal group.

Like the other members of the Paris-Provencal group, Gassendi communicated extensively with other astronomers and mathematician not only in France but throughout Europe, so his work was well known and influential both during his lifetime and also after his death. As with all the members of that group Gassendi’s life and work is a good example of the fact that science is a collective endeavour and often progresses through cooperation rather than rivalry. 

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

Renaissance Science – XXV

It is generally acknowledged that the mathematisation of science was a central factor in the so-called scientific revolution. When I first came to the history of science there was widespread agreement that this mathematisation took place because of a change in the underlaying philosophy of science from Aristotelian to Platonic philosophy. However, as we saw in the last episode of this series, the renaissance in Platonic philosophy was largely of the Neoplatonic mystical philosophy rather than the Pythagorean, mathematical Platonic philosophy, the Plato of “Let no one ignorant of geometry enter here” inscribed over the entrance to The Academy. This is not to say that Plato’s favouring of mathematics did not have an influence during the Renaissance, but that influence was rather minor and not crucial or pivotal, as earlier propagated.

It shouldn’t need emphasising, as I’ve said it many times in the past, but Galileo’s infamous, Philosophy is written in this grand book, which stands continually open before our eyes (I say the ‘Universe’), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth, is not the origin of the mathematisation, as is falsely claimed by far too many, who should know better. One can already find the same sentiment in the Middle Ages, for example in Islam, in the work of Ibn al-Haytham (c. 965–c. 1040) or in Europe in the writings of both Robert Grosseteste (c. 1168–1253) and Roger Bacon, (c. 1219–c. 1292) although in the Middle Ages, outside of optics and astronomy, it remained more hypothetical than actually practiced. We find the same mathematical gospel preached in the sixteenth century by several scholars, most notably Christoph Clavius (1538–1612).

As almost always in history, it is simply wrong to look for a simple mono-casual explanation for any development. There were multiple driving forces behind the mathematisation. As we have already seen in various earlier episodes, the growing use and dominance of mathematics was driving by various of the practical mathematical disciplines during the Renaissance. 

The developments in cartography, surveying, and navigation (which I haven’t dealt with yet) all drove an increased role for both geometry and trigonometry. The renaissance of astrology also served the same function. The commercial revolution, the introduction of banking, and the introduction of double entry bookkeeping all drove the introduction and development of the Hindu-Arabic number system and algebra, which in turn would lead to the development of analytical mathematics in the seventeenth century. The development of astro-medicine or iatromathematics led to a change in the status of mathematic on the universities and the demand for commercial arithmetic led to the establishment of the abbacus or reckoning schools. The Renaissance artist-engineers with their development of linear perspective and their cult of machine design, together with the new developments in architecture were all driving forces in the development of geometry. All of these developments both separately and together led to a major increase in the status of the mathematical sciences and their dissemination throughout Europe. 

This didn’t all happen overnight but was a gradual process spread over a couple of centuries. However, by the early seventeenth century and what is generally regarded as the start of the scientific revolution the status and spread of mathematics was considerably different, in a positive sense, to what it had been at the end of the fourteenth century. Mathematics was now very much an established part of the scholarly spectrum. 

There was, however, another force driving the development and spread of mathematics and that was surprisingly the, on literature focused, original Renaissance humanists in Northern Italy. In and of itself, the original Renaissance humanists did not measure mathematics an especially important role in their intellectual cosmos. So how did the humanists become a driving force for the development of mathematics? The answer lies in their obsession with all and any Greek or Latin manuscripts from antiquity and also with the attitude to mathematics of their ancient role models. 

Cicero admired Archimedes, so Petrarch admired Archimedes and other humanists followed his example. In his Institutio Oratoria Quintilian was quite enthusiastic about mathematics in the training of the orator. However, both Cicero and Quintilian had reservations about how too intense an involvement with mathematics distracts one from the active life. This meant that the Renaissance humanists were, on the whole, rather ambivalent towards mathematics. They considered it was part of the education of a scholar, so that they could converse reasonably intelligently about mathematics in general, but anything approaching a deep knowledge of the subject was by and large frowned upon. After all, socially, mathematici were viewed as craftsmen and not scholars.

This attitude stood in contradiction to their manuscript collecting habits. On their journeys to the cloister libraries and to Byzantium, the humanists swept up everything they could find in Latin and/or Greek that was from antiquity. This meant that the manuscript collections in the newly founded humanist libraries also contained manuscripts from the mathematical disciplines. A good example is the manuscript of Ptolemaeus’ Geographia found in Constantinople and translated into Latin by Jacobus Angelus for the first time in 1406. The manuscripts were now there, and scholars began to engage with them leading to a true mathematical renaissance of the leading Greek mathematicians. 

We have already seen, in earlier episodes, the impact that the works of Ptolemaeus, Hero of Alexander, and Vitruvius had in the Renaissance, now I’m going to concentrate on three mathematicians Euclid, Archimedes, and Apollonius of Perga, starting with Archimedes. 

The works of Archimedes had already been translated from Greek into Latin by the Flemish translator Willem van Moerbeke (1215–1286) in the thirteenth century.

Archimedes Greek manuscript

He also translated texts by Hero. Although, he was an excellent translator, he was not a mathematician, so his translations were somewhat difficult to comprehend. Archimedes was to a large extent ignored by the universities in the Middle Ages. In 1530, Jacobus Cremonensis (c. 1400–c. 1454) (birth name Jacopo da San Cassiano), a humanist and mathematician, translated, probably at request of the Pope, Nicholas V (1397–1455), a Greek manuscript of the works of Archimedes into Latin. He was also commissioned to correct George of Trebizond’s defective translation of Ptolemaeus’ Mathēmatikē Syntaxis. It is thought that the original Greek manuscript was lent or given to Basilios Bessarion (1403–1472) and has subsequently disappeared.

Bessarion had not only the largest humanist library but also the library with the highest number of mathematical manuscripts. Many of Bessarion’s manuscripts were collected by Regiomontanus (1436–1476) during the four to five years (1461–c. 1465) that he was part of Bessarion’s household.

Basilios Bessarion Justus van Gent and Pedro Berruguete Source: Wikimedia Commons

When Regiomontanus moved to Nürnberg in 1471 he brought a large collection of mathematical, astronomical, and astrological manuscripts with him, including the Cremonenius Latin Archimedes and several manuscripts of Euclid’s Elements, that he intended to print and publish in the printing office that he set up there. Unfortunately, he died before he really got going and had only published nine texts including his catalogue of future intended publications that also listed the Cremonenius Latin Archimedes. 

The invention of moving type book printing was, of course, a major game changer. In 1482, Erhard Ratdolt (1447–1522) published the first printed edition of The Elements of Euclid from one of Regiomontanus’ manuscripts of the Latin translation from Arabic by Campanus of Novara (c. 1220–1296).

A page with marginalia from the first printed edition of Euclid’s Elements, printed by Erhard Ratdolt in 1482
Folger Shakespeare Library Digital Image Collection
Source: Wikimedia Commons

In 1505, a Latin translation from the Greek by Bartolomeo Zamberti (c. 1473–after 1543) was published in Venice in 1505, because Zamberti regarded the Campanus translation as defective. The first Greek edition, edited by Simon Grynaeus (1493–1541) was published by Jacob Herwegens in Basel in 1533.

Simon Grynaeus Source: Wikimedia Commons
Editio princeps of the Greek text of Euclid. Source

Numerous editions followed in Greek and/or Latin. The first modern language edition, in Italian, translated by the mathematician Niccolò Fontana Tartaglia (1499/1500–1557) was published in 1543.

Tartaglia Euclid Source

Other editions in German, French and Dutch appeared over the years and the first English edition, translated by Henry Billingsley (died 1606) with a preface by John Dee (1527–c. 1608) was published in 1570.

Title page of Sir Henry Billingsley’s first English version of Euclid’s Elements Source Wikimedia Commons

In 1574, Christoph Clavius (1538–1612) published the first of many editions of his revised and modernised Elements, to be used in his newly inaugurated mathematics programme in Catholic schools, colleges, and universities. It became the standard version of Euclid throughout Europe in the seventeenth century. In 1607, Matteo Ricci (1552–1610) and Xu Guanqui (1562–1633) published their Chinese translation of the first six books of Clavius’ Elements.

Xu Guangqi with Matteo Ricci (left) From Athanasius Kircher’s China Illustrata, 1667 Source: Wikimedia Commons

From being a medieval university textbook of which only the first six of the thirteen books were studied if at all, The Elements was now a major mathematical text. 

Unlike his Euclid manuscript, Regiomontanus’ Latin Archimedes manuscript had to wait until the middle of the sixteenth century to find an editor and publisher. In 1544, Ioannes Heruagius (Johannes Herwagen) (1497–1558) published a bilingual, Latin and Greek, edition of the works of Archimedes, edited by the Nürnberger scholar Thomas Venatorius (Geschauf) (1488–1551).

Thomas Venatorius Source

The Latin was the Cremonenius manuscript that Regiomontanus had brought to Nürnberg, and the Greek was a manuscript that Willibald Pirckheimer (1470–1530) had acquired in Rome.

Venatori Archimedes Source

Around the same time Tartaglia published partial editions of the works of Archimedes both in Italian and Latin translation. We will follow the publication history of Archimedes shortly, but first we need to go back to see what happened to The Conics of Apollonius, which became a highly influential text in the seventeenth century.

Although, The Conics was known to the Arabs, no translation of it appears to have been made into Latin during the twelfth-century scientific Renaissance. Giovanni-Battista Memmo (c. 1466–1536) produced a Latin translation of the first four of the six books of The Conics, which was published posthumously in Venice in 1537. Although regarded as defective this remained the only edition until the latter part of the century.

Memmo Apollonius Conics Source: Wikimedia Commons

We now enter the high point of the Renaissance reception of both Archimedes and Apollonius in the work of the mathematician and astronomer Francesco Maurolico (1494–1575) and the physician Federico Commandino (1509-1575). Maurolico spent a large part of his life improving the editions of a wide range of Greek mathematical works.

L0006455 Portrait of F. Maurolico by Bovis after Caravaggio Credit: Wellcome Library, London, via Wikimedia Commons

Unfortunately, he had problems finding sponsors and/or publishers for his work. His heavily edited and corrected volume of the works of Archimedes first appeared posthumously in Palermo in 1585. His definitive Latin edition of The Conics, with reconstructions of the fifth and sixth books, completed in 1547, was first published in 1654.

Maurolico corresponded with Christoph Clavius, who had visited him in Sicily in 1574, when the observed an annular solar eclipse together, and with Federico Commandino, although the two never met.

Federico Commandino produced and published a whole series of Greek mathematical works, which became something like standard editions.

Source: Wikimedia Commons

His edition of the works of Archimedes appeared in 1565 and his Apollonius translation in 1566.

Two of Commandino’s disciples were Guidobaldo del Monte (1545–1607) and Bernardino Baldi (1553–1617). 

Baldi wrote a history of mathematics the Cronica dei Matematici, which was published in Urbino in 1707. This was a brief summary of his much bigger Vite de’ mathematici, a two-thousand-page manuscript that was never published.

Bernadino Baldi Source: Wikimedia Commons
Source: Wikimedia Commons

Guidobaldo del Monte, an aristocrat, mathematician, philosopher, and astronomer

Guidobaldo del Monte Source: Wikimedia Commons

became a strong promoter of Commandino’s work and in particular the works of Archimedes, which informed his own work in mechanics. 

In the midst of that darkness Federico Commandino shone like the sun, for his learning he not only restored the lost heritage of mathematics but actually increased and enhanced it … In him seem to have lived again Archytas, Diophantus, Theodosius, Ptolemy, Apollonius, Serenus, Pappus and even Archimedes himself.

Guidobaldo. Liber Mechanicorum, Pesaro 1577, Preface
Source: Wikimedia Commons

When the young Galileo wrote his first essay on the hydrostatic balance, his theory how Archimedes actually detected the substitution of silver for gold in the crown made for King Hiero of Syracuse, he sent it to Guidobaldo to try and win his support and patronage. Guidobaldo was very impressed and got his brother Cardinal Francesco Maria del Monte (1549–1627), the de’ Medici family cardinal, to recommend Galileo to Ferinando I de’ Medici, Grand Duke of Tuscany, (1549–1609) for the position of professor of mathematics at Pisa University. Galileo worked together with Guidobaldo on various projects and for Galileo, who rejected Aristotle, Archimedes became his philosophical role model, who he often praised in his works. 

Galileo was by no means the only seventeenth century scientist to take Archimedes as his role model in pursuing a mathematical physics, for example Kepler used a modified form of Archimedes’ method of exhaustion to determine the volume of barrels, a first step to the development of integral calculus. The all pervasiveness of Archimedes in the seventeenth century is wonderfully illustrated at the end of the century by Sir William Temple, Jonathan Swift’s employer, during the so-called battle of the Ancients and Moderns. In one of his essays, Temple an ardent supporter of the superiority of the ancients over the moderns, asked if John Wilkins was the seventeenth century Archimedes, a rhetorical question with a definitively negative answer. 

During the Middle Ages Euclid was the only major Greek mathematician taught at the European universities and that only at a very low level. By the seventeenth century Euclid had been fully restored as a serious mathematical text and the works of both Archimedes and Apollonius had entered the intellectual mainstream and all three texts along with other restored Greek texts such as the Mathematical Collection of Pappus, also published by Commandino and the Arithmetica of Diophantus, another manuscript brought to Nürnberg by Regiomontanus and worked on by numerous mathematicians, became influential in development of the new sciences.  

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

OHMS or everything you wanted to know about the history of trigonometry and didn’t know who to ask

When I was a kid, letters from government departments came in buff, manila envelopes with OHMS printed on the front is large, black, capital letters. This acronym stood for, On Her Majesty’s Service and earlier during Liz’s father’s reign (and no I’m not that old, although I was just born in his reign), On His Majesty’s Service, implying that civil servants worked directly for the monarch.  This was, of course, the origin of the title of Ian Fleming’s eleventh James Bond novel, On Her Majesty’s Secret Service

When I started learning trigonometry at school this acronym took on a whole new meaning as a mnemonic for the sine relation in right angle triangles, Opposite over Hypotenuse Means Sine. Recently it occurred to me that we had no mnemonic for the other trigonometric relations. Now in those days or even later when the trigonometry I was taught got more complex, I wasn’t aware of the fact that this mathematical discipline had a history. Now, a long year historian of mathematics, I am very much aware of the fact that trigonometry has a very complex, more than two-thousand-year history, winding its way from ancient Greece over India, the Islamic Empire and Early Modern Europe down to the present day. 

The Canadian historian of mathematics, Glen van Brummelen has dedicated a large part of his life to researching, writing up and publishing that history of trigonometry. The results of his labours have appeared in three volumes, over the years, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton University Press, Princeton and Oxford, 2009, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, Princeton and Oxford, 2013 and most recently The Doctrine of TrianglesA History of Modern Trigonometry, Princeton University Press, Princeton and Oxford, 2021. He describes himself as the “best trigonometry historian, and the worst trigonometry historian”, as he is the only one[1]

A review of these three volumes could be written in one sentence, if you are interested in the history of trigonometry, then these three masterful volumes are essential. One really doesn’t need to say more, but in what follows I will give a brief sketch of each of the books. 

The Mathematics of the Heavens and the Earth: The Early History of Trigonometry delivers exactly what it says on the cover. The book opens with a brief but detailed introduction to the basics of spherical astronomy, because for a large part of the period covered, what we have is not the history of plane trigonometry, that’s the stuff we all learnt at school, but spherical trigonometry, that is the geometry of triangles on the surface of a sphere, which was developed precisely to do spherical astronomy. 

A friendly warning for potential readers this is not popular history but real, hardcore history of mathematics with lots of real mathematical examples worked through in detail. However, given the way Van Brummelen structures his narrative, it is possible to skip the worked examples and still get a strong impression of the historical evolution of the discipline. This is possible because Van Brummelen gives a threefold description of every topic that he elucidates. First comes a narrative, fairly non-technical, description of the topic he is discussing. This is followed by an English translation of a worked example from the historical text under discussion, followed in turn by a technical explication of the text in question in modern terminology. Van Brummelen’s narrative style is clear and straightforward meaning that the non-expert reader can get good understanding of the points being made, without necessarily wading through the intricacies of the piece of mathematics under discussion. 

The book precedes chronologically. The first chapter, Precursors, starts by defining what trigonometry is and also what it isn’t. Having dealt with the definitions, Van Brummelen moves onto the history proper dealing with things that preceded the invention of trigonometry, which are closely related but are not trigonometry. 

Moving on to Alexandrian Greece, Van Brummelen takes the reader through the beginnings of trigonometry starting with Hipparchus, who produced the first chord table linking angles to chords and arcs of circles, Moving on through Theodosius of Bithynia and Menelaus of Alexandria and the emergence of spherical trigonometry. He then arrives at Ptolemy his astronomy and geography. Ptolemy gets the longest section of the book, which given that everything that follows in some way flows from his work in logical. Here we also get two defining features of the book. The problem of calculating trigonometrical tables and what each astronomer or mathematician contributed to this problem and the trigonometrical formulas that each of them developed to facilitate calculations. 

From Greece we move to India and the halving of Hipparchus’ and Ptolemy’s chords to produce the sine function and later the cosine that we still use today. Van Brummelen takes his reader step for step and mathematician for mathematician through the developments of trigonometry in India. 

The Islamic astronomers took over the baton from the Indians and continued the developments both in astronomy and geography. It was Islamic mathematicians, who developed the plane trigonometry that we know today rather than the spherical trigonometry. As with much other mathematics and science, trigonometry came into medieval Europe through the translation movement out of Arabic into Latin. Van Brummelen traces the development in medieval Europe down to the first Viennese School of mathematics, John of Gmunden, Peuerbach, and Regiomontanus. This volume closes with Johannes Werner and Copernicus, with a promise of a second volume. 

In the book itself, the brief sketch above is filled out in incredible detail covering all aspects of the evolution of the discipline, the problems, the advances, the stumbling stones and the mathematicians and astronomers, who discovered each problem, solved, or failed to solve them. To call Van Brummelen comprehensive would almost be an understatement. Having finished this first volume, I eagerly awaited the promised second volume, but something else came along instead.

Having made clear in his first book that the emphasis is very much on spherical trigonometry rather than plane trigonometry, in his second book Van Brummelen sets out to explain to the modern reader what exactly spherical trigonometry is, as it ceased to be part of the curriculum sometime in the modern period. What we have in Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry is a spherical trigonometry textbook written from a historical perspective. The whole volume is written in a much lighter and more accessible tone than The Mathematics of the Heavens and the Earth. After a preface elucidating the purpose of the book there follow two chapters, Heavenly Mathematics and Exploring the Sphere, which lay out and explain the basics in clear and easy to follow steps.

Next up, we have the historical part of the book with one chapter each on The Ancient Approach and The Medieval Approach. These chapters could be used as an aid to help understand the relevant sections of the authors first book. But fear not the reader must not don his medieval personality to find their way around the complexities of spherical trigonometry because following this historical guide we are led into the modern textbook.

The bulk of the book consists of five chapters, each of which deals in a modern style with an aspect of spherical trigonometry: Right Angle Triangles, Oblique Triangles, Areas, Angles and Polyhedra, Stereographic Projection, and finally Navigation by the Stars. The chapter on stereographic projection is particularly interesting for those involved with astrolabes and/or cartography. 

The book closes with three useful appendices. The first is on Ptolemy’s determination of the position of sun. The second is a bibliography of textbooks on or including spherical trigonometry with the very helpful indication, which of them are available on Google Books. The final appendix is a chapter by chapter annotated list of further reading on each topic. 

If you wish to up your Renaissance astrology game and use the method of directions to determine your date of death, which require spherical trigonometry to convert from one celestial coordinate system to another, then this is definitely the book for you. It is of course also a brilliant introduction for anybody, who wishes to learn the ins and outs of spherical trigonometry. 

I bought Van Brummelen’s first book when it was published, in 2009, and read it with great enthusiasm, but experienced a sort of coitus interruptus, when in stopped in the middle of the Renaissance, the period that interested me most. I was consoled by the author’s declaration that a second volume would follow, which I looked forward to with great expectations. Over the years those expectations dimmed, and I began to fear that the promised second volume would never appear, so I was overjoyed when the publication of The Doctrine of Triangles was announced this year and immediately placed an advanced order. I was not disappointed. 

The modern history of trigonometry continues where the early history left off, tracing the developments of trigonometry in Europe from Regiomontanus down to Clavius and Gunter in the early seventeenth century. There then follows a major change of tack, as Van Brummelen delves into the origins of logarithms.

Today in the age of the computer and the pocket calculator, logarithmic tables are virtually unknown, a forgotten relic of times past. I, however, grew up using my trusty four figure log tables to facilitate calculations in maths, physics, and chemistry. Now, school kids only know logarithms as functions in analysis. One thing that many, who had the pleasure of using log tables, don’t know is that Napier’s first tables were of the logarithms of trigonometrical factions in order to turn the difficult multiplications and divisions of sines, cosines et al in spherical trigonometry into much simpler additions and subtractions and therefore Van Brummelen’s detailed presentation of the topic.

Moving on, in his third chapter, Van Brummelen now turns to the transition of trigonometry as a calculation aid in spherical and plane triangles to trigonometrical functions in calculus. There where they exist in school mathematics today. Starting in the period before Leibniz and Newton, he takes us all the way through to Leonard Euler in the middle of the eighteenth century. 

The book now undergoes a truly major change of tack, as Van Brummelen introduces a comparative study of the history of trigonometry in Chinese mathematics. In this section he deals with the Indian and Islamic introduction of trigonometry into China and its impact. How the Chinese dealt with triangles before they came into contact with trigonometry and then the Jesuit introductions of both trigonometry and logarithms into China and to what extent this influenced Chinese geometry of the triangle. A fascinating study and an enrichment of his already excellent book.

The final section of the book deals with a potpourri of developments in trigonometry in Europe post Euler. To quote Van Brummelen, “A collection of short stories is thus more appropriate here than a continuous narrative.” The second volume of Van Brummelen’s history is just as detailed and comprehensive as the first. 

All three of the books display the same high level of academic rigour and excellence. The two history volumes have copious footnotes, very extensive bibliographies, and equally extensive indexes. The books are all richly illustrated with many first-class explanatory diagrams and greyscale prints of historical title pages and other elements of the books that Van Brummelen describes. All in all, in his three volumes Van Brummelen delivers a pinnacle in the history of mathematics that sets standards for all other historians of the discipline. He really does live up to his claim to be “the best historian of trigonometry” and not just because he’s the only one.

Coda: If the potential reader feels intimidated by the prospect of the eight hundred and sixty plus pages of the three volumes described here, they could find a gentle entry to the topic in Trigonometry: A Very Short Introduction (OUP, 2020), which is also authored by Van Brummelen, a sort of Van Brummelen light or Van Brummelen’s greatest hits.

In this he covers a wide range of trigonometrical topics putting them into their historical context. But beware, reading the Very Short Introduction could well lead to further consumption of Van Brummelen’s excellent work. 


[1] This is not strictly true as Van Brummelen has at least two predecessors both of who he quotes in his works. The German historian Anton von Braunmühl, who wrote several articles and a two volume Vorlesung über Geschichte der Trigonometrie (Leipzig, 1900/1903) and the American Sister Mary Claudia Zeller, The Development of Trigonometry from Regiomontanus to Pitiscus (Ann Arbor 1944)

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Filed under History of Astronomy, History of Cartography, History of Islamic Science, History of Mathematics, History of Navigation

The astronomical librarian 

I’m continuing my look at the French mathematician astronomers of the seventeenth century with some of those, who were both members of Nicolas-Claude Fabri de Peiresc’s group of telescopic, astronomical observers, as well as Marin Mersenne’s informal Academia Parisiensis, starting with Ismael Boulliau (1605–1694), who like Peiresc and Mersenne was also a prominent member of the Republic of Letters with about 5000 surviving letters. 

Ismael Boulliau Source: Wikimedia Commons

Boulliau was born in Loudun, France the second son of Ismael Boulliau a notary and amateur astronomer and Susanne Motet on 28 September 1605. The first son had been born a year earlier and was also named Ismael, but he died and so the name was transferred to their second son. Both of his parents were Calvinists. His father introduced him to astronomy and in his Astronomia philolaica (1645) Ismael junior tells us that his father observed both Halley’s comet in 1607 and the great comet of 1618. The later was when Boulliau was thirteen years old, and one can assume that he observed together with his father. 

Probably following in his father’s footsteps, he studied law but at the age of twenty-one he converted to Catholicism and in 1631, aged twenty-six, he was ordained a priest. In 1632 he moved to Paris and began to work for Pierre Dupuy (1582–1651) and his brother Jacques (1591–1656), who were keepers of the Bibliothèque du Roi, today the Bibliothèque nationale de France. Boulliau held this position until the death of the Dupuy brothers and during that time travelled widely in Europe collecting books and manuscripts for the library. 

Pierre Dupuy Source: Wikimedia Commons

Boulliau also enjoyed the patronage of the powerful and influential de Trou family, who were closely connected with the library and who financed his book collecting travels. Following the death of the Dupuy brothers he became employed by the French ambassador to the United Provinces, a member of the de Trou family, a secretary and librarian. In 1666, following a dispute with his employer, he became librarian at the Collège de Laon in Paris. For the last five years of his live he returned to the priesthood in the Abbey St Victor near Paris where he died aged 89. Although Boulliau was an active member of Mersenne’s Academia Parisiensis he never became a member of the Académie des sciences, but he was elected one of the first foreign associates of the Royal Society on 4 April 1667. 

Abbey of St. Victor, 1655 Source: Wikimedia Commons

 Like Peiresc, Boulliau was a polymath with extensive knowledge of a wide range of humanities topics, which was useful in his work as a librarian, but, as with Peiresc, it is scientific activities that are of interest here. He continued to make astronomical observations throughout his life, which were of a high level of accuracy. In his Principia, Newton puts him on a level with Kepler for his determination of the planetary orbits. In Book 3 Phenomenon 4 of Principia Newton writes: 

But of all astronomers, Kepler and Boulliau have determined the magnitude of the orbits from observations with the most diligence. 

Boulliau’s first significant scientific publication was, however, not in astronomy but in optics, his De natura lucis (On the Nature of Light) (1638) based on the discussions he was having with Gassendi on the topic. This work is not particular important in the history of optics but it does contain his discussion of Kepler’s inverse square law for the propagation of light.

Source: Wikimedia Commons

His first astronomical work Philolaus (1639), which places him firmly in the Copernican heliocentric camp but not, yet a Keplerian was next. 

He now changed tack once again with a historical mathematical work. In 1644, he translated and published the first printed edition of Theon of Smyrna’s Expositio rerum mathematicarum ad legendum Platonem utilium a general handbook for students of mathematics of no real significance. Continuing with his mathematical publications. In 1657, he published De lineis spiralibus (On Spirals) related to the work of Archimedes and Pappus on the topic.

Source: Wikimedia Commons

Much later in 1682, he published Opus novum ad arithmeticam infinitorum, which he claimed clarified the Arithmetica infinitorum(1656) of John Wallis (1616–1703).

Source: Wikimedia Commons

All of Boulliau’s work was old fashioned and geometrical. He rejected the new developments in analytical mathematics and never acknowledged Descartes’ analytical geometry. As we shall see, his astronomy was also strictly geometrical. He even criticised Kepler for being a bad geometer. 

Boulliau’s most important publication was his second astronomical text Astronomia philolaica (1645).

Source: Wikimedia Commons

In this highly influential work, he fully accepted Kepler’s elliptical orbits but rejects almost all of the rest of Kepler’s theories. As stated above his planetary hypothesis is strictly geometrical and centres round his conical hypothesis:

“The Planets, according to that astronomer [Boulliau], always revolve in circles; for that being the most perfect figure, it is impossible they should revolve in any other. No one of them, however, continues to move in any one circle, but is perpetually passing from one to another, through an infinite number of circles, in the course of each revolution; for an ellipse, said he, is an oblique section of a cone, and in a cone, betwixt the vertices of the ellipse there is an infinite number of circles, out of the infinitely small portions of which the elliptical line is compounded. The Planet, therefore, which moves in this line, is, in every point of it, moving in an infinitely small portion of a certain circle. The motion of each Planet, too, according to him, was necessarily, for the same reason, perfectly equable. An equable motion being the most perfect of all motions. It was not, however, in the elliptical line, that it was equable, but in any one of the circles that were parallel to the base of that cone, by whose section this elliptical line had been formed: for, if a ray was extended from the Planet to any one of those circles, and carried along by its periodical motion, it would cut off equal portions of that circle in equal times; another most fantastical equalizing circle, supported by no other foundation besides the frivolous connection betwixt a cone and an ellipse, and recommended by nothing but the natural passion for circular orbits and equable motions,” (Adam Smith, History of Astronomy, IV.55-57).

Boulliau’s Conical Hypothesis [RA Hatch] Source: Wikimedia Commons

Boulliau’s theory replaces Kepler’s second law, and this led to the Boulliau-Ward debate on the topic with the English astronomer Seth Ward (1617–1689), the Savilian Professor of astronomy at Oxford University.

Bishop Seth Ward, portrait by John Greenhill Source: Wikimedia Commons

Ward criticised Boulliau’s theory in his In Ismaelis Bullialdi astro-nomiae philolaicae fundamenta inquisitio brevis (1653), also pointing out mathematical errors in Boulliau’s work. 

Boulliau responded to Ward’s criticisms in 1657, acknowledging the errors and correcting but in turn criticising Ward’s model in his De lineis spiralibus. A year earlier Ward had published his own version of Keplerian astronomy in his Astronomia geometrica (1656).

Source: Wikimedia Commons

This exchange failed to find a resolution but this very public debate between two of Europe’s leading astronomers very much raised awareness of Kepler’s work and was factor in its eventual acceptance of Kepler’s elliptical heliocentric astronomy. 

It was in his Astronomia philolaica that Boulliau was the first to form an inverse squared theory of attraction between the sun and the planets. 

As for the power by which the Sun seizes or holds the planets, and which, being corporeal, functions in the manner of hands, it is emitted in straight lines throughout the whole extent of the world, and like the species of the Sun, it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances that is, 1/d2 ​.

Here we see the influence of Kepler’s theory of light propagation, which as noted Boulliau discussed in his De natura lucis. However, having set up this hypothesis Boulliau goes on to reject it. 

… I say that the Sun is moved by its own form around its axis, by which form it was ignited and made light, indeed I say that no kind of motion presses upon the remaining planets … indeed [I say] that the individual planets are driven round by individual forms with which they were provided …

Despite Boulliau’s rejection of his own hypothesis, during Newton’s dispute with Hooke over who should get credit for the theory of gravity, he gives Boulliau the credit in a letter to Edmond Halley.

…so Bullialdus [i.e., Boulliau] wrote that all force respecting ye Sun as its center & depending on matter must be reciprocally in a duplicate ratio of ye distance from ye center, & used that very argument for it by wch you, Sr, in the last Transactions have proved this ratio in gravity. Now if Mr Hook from this general Proposition in Bullialdus might learn ye proportion in gravity, why must this proportion here go for his invention?

In 1667, Boulliau published a final astronomy book, Ad astronomos monita duo in which he was the first to establish the periodicity of the variable star, Mira Ceti.

Source:

His estimate of the period 333 days was only out by a one day. Mira had first been recognised as a variable star by David Fabricius beginning 3 August 1596.

Apart from his publications Boulliau kept Mersenne’s correspondence network alive for another thirty years after Mersenne’s death, communicating with Leopoldo de’ Medici (1617–1675) in Italy, Johannes Hevelius (1611–1687) in Danzig and Christiaan Huygens (1629–1695). Huygens first imparted his discovery of the rings of Saturn to Boulliau and Boulliau distributed Huygens’ System sarturnium (1658) in Paris. Boulliau also distributed Pascal’s Letters D’Amos Dettonville (1658–1659) to English and Dutch mathematicians, his challenge on the mathematics of the cycloid, an important publication in the development of calculus.

Ismael Boulliau is a prime example of a scholar, who didn’t make any major discoveries or develop any major theories himself but still had a very significant influence on the development of science.

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Filed under History of Astronomy, History of Mathematics, History of Optics, History of science, The Paris Provencal Connection

Renaissance Science – XXII

Perhaps surprisingly, land surveying as we know it today, a mathematical discipline utilising complex technological measuring instruments is very much a product of the practical mathematics of the Renaissance. Why surprisingly? Surveying is an ancient discipline that has its origins in humanity becoming settled many thousands of years ago. Ancient monuments such as the pyramids or Stonehenge definitely required some level of surveying in their construction and there are surviving documents from all literate ancient societies that refer to methods or the practice of surveying. 

All surveying uses some aspects of geometry and as Herodotus famously claimed geometry (Greek: geōmetría from geōmétrēs), which literally means measurement of earth or land, had its origins in Egyptian surveying for tax purposes. According to his account, King Sesostris divided all the lands in Egypt amongst its inhabitants in return for an annual rent. However, every year the Nile floods washing away the parts of the plots:

The country is converted into a sea, and nothing appears but the cities, which looked like islands in the Aegean. 

Those whose land had been lost objected to paying the rent, so Sesostris summoned those affected to appear before him.

Upon which, the king sent persons to examine, and determine by measurement the exact extent of the loss: and thenceforth only such a rent was demanded of him as was proportionate to the reduced size of his land. From this practice, I think, geometry first came to be known in Egypt, whence it passed into Greece.

According to legend, both Thales and Pythagoras, are reputed to have learnt their geometry in Egypt.

In all early cultures surveying was fairly primitive with measurements being made with ropes and measuring rods. In Egypt, surveyors were known as rope stretchers (harpedonaptai), the ropes used for measuring being stretched to avoid sagging.

A rope being used to measure fields. Taken from the Tomb of Menna, TT69. (c. 1500–1200 BCE) Source: Wikimedia Commons

Longer distances were either measured by estimation or by pacing. In ancient Egypt and Greece Bematistae (step measurer) where trained to walk with equal length paces and the historical records of Alexander the Great’s campaigns suggest that they were indeed highly accurate. This measuring of distances by pacing in reflected in our word mile, which is the Latin word for a thousand, mille, meaning a thousand paces.

The Latin for surveyor was agrimensores, meaning field measurers. They were also called gromatici after the groma a surveyor’s pole, an early instrument for determining lines at right angles to each other. 

The groma or gruma was a Roman surveying instrument. It comprised a vertical staff with horizontal cross-pieces mounted at right angles on a bracket. Each cross piece had a plumb line hanging vertically at each end. It was used to survey straight lines and right angles, thence squares or rectangles. They were stabilized on the high ground and pointed in the direction it was going to be used. The helper would step back 100 steps and place a pole. The surveyor would tell him where to move the pole and the helper would set it down.

(Lewis, M. J. T., Surveying instruments of Greece and Rome, McGraw Hill Professional, 2001, p. 120)
Staking out a right angle using a groma

Another instrument used for the same purpose was the dioptra. The dioptra was a sighting tube or, alternatively an alidade, that is a rod with a sight at each end, attached to a stand. If fitted with protractors, it could be used to measure angles. Hero from Alexandria wrote a whole book on this instrument and its use but there are doubts that the dioptra in the complex form described by Hero was actually used in field surveying.

Dioptra as described by Hero of Alexandria Source: Wikimedia Commons

The methods used by the Romans in field surveying were described in the works of technical authors such as Sextus Julius Frontinus (c. 40–103 CE) and Gaius Julius Hyginus (c. 64 BCE–17 CE).

All of the surveying described in antiquity was fairly small scale–measuring fields, determining boundaries, laying out military camps, etc–and geometrically centred on squares and rectangles. Cartography was done using astronomical determinations of latitude and longitude, whereby the latter was difficult, and distances estimated or paced. Nothing really changed in Europe during the medieval period. The surveying that was done was carried out using the same methods that the Romans had used. However, during the fifteenth century things began to change substantially and the first question is why?

The rediscovery of Ptolemaeus’ Geographia at the beginning of the fifteenth century, as described here, and the subsequent substantial increase in cartographical activity, as described here, played a major role, but as already stated above Ptolemaic cartography relied almost exclusively on astronomical methods and did not utilise field surveying. However, there was an increased demand for internal accuracy in maps that astronomical methods could not supply. Secondly, changes in land ownership led to an increased demand for accurate field surveying of estates that required more sophisticated methods than those of the agrimensores. Lastly, we have a good example of the knowledge crossover, typical for the Renaissance, as described in Episode V of this series. The surveyors of antiquity were artisans producing practical knowledge for everyday usage. In the Renaissance, university educated scholars began to interest themselves for this practical knowledge and make contributions to its development and it is these developments that we will now look at. 

The biggest change in surveying was the introduction of the simple geometrical figure the triangle into surveying, as Sebastian Münster, one of the most influential cosmographers (today we would say geographer) of the period, wrote in a German edition of his Cosmographia. Beschreibung aller Lender durch Sebastianum Münsterum in 1550:

Every thing you measure must be measured in triangles.

Actually, the theory of similar triangles, as explained in Euclid’s Elements, had been used in surveying in antiquity, in particular to determine the height of things or for example the width of a river. A method that I learnt as a teenager in the Boy Scouts.

What was new as we will see was the way that triangles were being used in surveying and that now it was the angles of the triangles that were measured and not the length of the sides, as in the similar triangles’ usage. We are heading towards the invention and usage of triangulation in surveying and cartography, a long-drawn-out process.

In his Ludi rerum mathematicarum (c. 1445), the architect Leon Battista Alberti describes a method of surveying by taking angular bearings of prominent points in the area he is surveying using a self-made circular protractor to create a network of triangles. He concludes by explaining that one only needs to the length of one side of one triangle to determine all the others. What we have here is an early description of a plane table surveying (see below) and step towards triangulation that, however, only existed in manuscript 

Alberti Ludi rerum mathematicarum 

Münster learnt his geometry from Johannes Stöffler (1452–1531), professor for mathematics in Tübingen, who published the earliest description of practical geometry for surveyors. In his De geometricis mensurationibus rerum (1513),

Johannes Stöffler Engraving from the workshop of Theodor de Brys, Source: Wikimedia Commons

Stöffler explained how inaccessible distances could be measured by measuring one side of a triangle using a measuring rod (pertica) and then observing the angles from either end of the measured stretch. However, most of the examples in his book are still based on the Euclidian concept of similar triangles rather than triangulation. In 1522, the printer publisher Joseph Köbel, who had published the Latin original, published a German version of Stöffler’s geometry book. 

Joseph Köbel Source: Wikimedia Commons

Both Peter Apian in his Cosmographia (1524) and Oronce Fine in his De geometria practica (1530) give examples of using triangles to measure distances in the same way as Stöffler.

Source

Fine indicating that he knew of Stöffler’s book. Apian explicitly uses trigonometry to resolve his triangles rather than Euclidian geometry. Trigonometry had already been known in Europe in the Middle Ages but hadn’t been used before the sixteenth century in surveying. Fine, however, still predominantly used Euclidian methods in his work, although he also, to some extent, used trigonometry.

A very major development was the publication in 1533 of Libellus de locorum describendum ratione (Booklet concerning a way of describing places) by Gemma Frisius as an appendix to the third edition of Apian’s Cosmographia, which he edited, as he would all edition except the first. Here we have a full technical description of triangulation published for the first time. It would be included in all further editions in Latin, Spanish, French, Flemish, in what was the most popular and biggest selling manual on mapmaking and instrument making in the sixteenth and seventeenth centuries.

Source: Wikimedia Commons

1533 also saw the publication in Nürnberg by Johannes Petreius (c. 1497–1550) of Regiomontanus’s De triangulis omnimodis (On triangles of every kind) edited by the mapmaker and globe maker, Johannes Schöner (1477–1547).

Source:

This volume was originally written in 1464 but Regiomontanus died before he could print and publish it himself, although he had every intention of doing so. This was the first comprehensive work on trigonometry in Europe in the Early Modern Period, although it doesn’t cover the tangent, which Regiomontanus handled in his Tabula directionum (written 1467, published 1490), an immensely popular and oft republished work on astrology. 

Regiomontanus built on previous medieval works on trigonometry and the publication of his book introduces what Ivor Grattan Guinness has termed The Age of Trigonometry. In the sixteenth century it was followed by Rheticus’ separate publication of the trigonometrical section of Copernicus’s De revolutionibus, as De lateribus et angulis triangulorium in 1542. Rheticus (1514–1574) followed this in 1551 with his own Canon doctrinae triangulorum. This was the first work to cover all six trigonometric functions and the first to relate the function directly to triangles rather than circular arcs.

Source: Wikimedia Commons

Rheticus spent the rest of his life working on his monumental Opus Palatinum de Triangulis, which was, however, first published posthumously by his student Lucius Valentin Otho in 1596. Rheticus and Otho were pipped at the post by Bartholomaeus Pitiscus (1561–1613), whose Trigonometriasive de solutione triangulorum tractatus brevis et perspicuous was published in 1595 and gave the discipline its name.

Source: Wikimedia Commons

Pitiscus’ work went through several edition and he also edited and published improved and corrected editions of Rheticus’ trigonometry volumes. 

Through Gemma Frisius’ detailed description of triangulation and sixteenth century works on trigonometry, Renaissance surveyors and mapmakers now had the mathematical tools for a new approach to surveying. What they now needed were the mathematical instruments to measure distances and angles in the field and they were not slow in coming.

The measure a straight line of a given distance as a base line in triangulation surveyors still relied on the tools already used in antiquity the rope and the measuring rod. Ropes were less accurate because of elasticity and sagging if used for longer stretches. In the late sixteenth century, they began to be replaced by the surveyor’s chain, made of metal links but this also suffered from the problem of sagging due to its weight, so for accuracy wooden rods were preferred. 

A Gunter chain photographed at Campus Martius Museum. Source: Wikimedia Commons

In English the surveyor’s chain is usually referred to as Gunter’s chain after the English practical mathematician Edmund Gunter (1581–1626) and he is also often referred to erroneously as the inventor of the surveyor’s chain but there are references to the use of the surveyor’s chain in 1579, when Gunter was still a child. 

He did, however, produce what became a standardised English chain of 100 links, 66 feet or four poles, perches, or rods long, as John Ogilby (1600–1676) wrote in his Britannia Atlas in 1675:

…a Word or two of Dimensurators or Measuring Instruments, whereof the mosts usual has been the Chain, and the common length for English Measures 4 Poles, as answering indifferently to the Englishs Mile and Acre, 10 such Chains in length making a Furlong, and 10 single square Chains an Acre, so that a square Mile contains 640 square Acres…’

An English mile of 5280 feet was thus 80 chains in length and there are 10 chains to a furlong. An acre was 10 square chains. I actually learnt this antiquated system of measurement whilst still at primary school. The name perch is a corruption of the Roman name for the surveyor’s rod the pertica. 

To measure angles mapmakers and surveyors initially adopted the instruments developed and used by astronomers, the Jacob staff, the quadrant, and the astrolabe. An instrument rarely still used in astronomy but popular in surveying was the triquetum of Dreistab. The surveyors triquetum consists of three arms pivoted at two points with circular protractors added at the joints to measure angles and with a magnetic compass on the side to determine bearings. 

Surveyors then began to develop variants of the dioptra. The most notable of these, that is still in use today albeit highly modernised, was the theodolite, an instrument with sights capable of measuring angles both vertically and horizontally. The name first occurs in the surveying manual A geometric practice named Pantometria by Leonard Digges (c. 1515–c. 1559) published posthumously by his son Thomas (c. 1546–1595) in 1571.

Leonard Digges  A geometric practice named Pantometria Source

However, Digges’ instrument of this name could only measure horizontal angles. He described another instrument that could measure both vertical and horizontal angles that he called a topographicall instrument. Josua Habermehl, about whom nothing is known, but who was probably a relative of famous instrument maker Erasmus Habermehl (c. 1538–1606), produced the earliest known instrument similar to the modern theodolite, including a compass and tripod, in 1576. In 1725, Jonathan Sisson (1690–1747) constructed the first theodolite with a sighting telescope.

Theodolite 1590 Source:

A simpler alternative to the theodolite for measuring horizontal angles was the circumferentor. This was a large compass mounted on a plate with sights, with which angles were measured by taking their compass bearings.

18th century circumferentor

Instruments like the triquetum and the circumferentor were most often used in conjunction of another new invention, the plane table. Gemma Frisius had already warned in his Libellus de locorum describendum rationeof the difficulties of determining the lengths of the sides of the triangles in triangulation using trigonometry and had described a system very similar to the plane table in which the necessity for these calculation is eliminated. 

Surveying with plane table and surveyor’s chain

The plane table is a drawing board mounted on a tripod, with an alidade. Using a plumb bob, the table is centred on one end of a baseline, levelled by eye or after its invention (before 1661) with a spirit level, and orientated with a compass. The alidade is placed on the corresponding end of the scaled down baseline on the paper on the table and bearings are taken of various prominent features in the area, the sight lines being drawn directly on the paper. This procedure is repeated at the other end of the baseline creating triangles locating the prominent figures on the paper without having to calculate.

Philippe Danfrie (c.1532–1606) Surveying with a plane table

As with the theodolite there is no certain knowledge who invented the plane table. Some sources attribute the invention of the plane table to Johannes Praetorius (1537–1616), professor for mathematics at the University of Altdorf, as claimed by his student Daniel Schwentner (1585–1636). However, there was already a description of the plane table in “Usage et description de l’holomètre”, by Abel Foullon (c. 1514–1563) published in Paris in 1551. It is obvious from his description that Foullon hadn’t invented the plane table himself. 

The plane table is used for small surveys rather than mapmaking on a large scale and is not triangulation as described by Gemma Frisius. Although the Renaissance provided the wherewithal for full triangulation, it didn’t actually get used much for mapping before the eighteenth century. At the end of the sixteenth century Tycho Brahe carried out a triangulation of his island of Hven, but the results were never published. The most notable early use was by Willebrord Snel (1580–1626) to measure one degree of latitude in order to determine the size of the earth in 1615. He published the result in his Eratosthenes batavus in Leiden in 1617. He then extended his triangulation to cover much of the Netherlands.

Snel’s Triangulation of the Dutch Republic from 1615 Source: Wikimedia Commons

In the late seventeenth century Jean Picard (1620–1682) made a much longer meridian measurement in France using triangulation. 

Picard’s triangulation and his instruments

In fourteen hundred European surveyors were still using the same methods of surveying as the Romans a thousand years earlier but by the end of the seventeenth century when Jean-Dominique Cassini (1625–1712) began the mapping of France that would occupy four generations of the Cassini family for most of the eighteenth century, they did so with the fully developed trigonometry-based triangulation that had been developed over the intervening three hundred years. 

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Filed under History of Astronomy, History of Cartography, History of Geodesy, History of Mathematics, History of science, Renaissance Science

Musical, mathematical Minim, Marin Mersenne 

In the seventeenth century, Marin Mersenne (1588–1648) was a very central and highly influential figure in the European intellectual and scientific communities; a man, who almost literally knew everybody and was known by everybody in those communities. Today, in the big names, big events, popular versions of the history of science he remains only known to specialist historian of science and also mathematicians, who have heard of Mersenne Primes, although most of those mathematicians probably have no idea, who this Mersenne guy actually was. So, who was Marin Mersenne and why does he deserve to be better known than he is?

Marin Mersenne Source: Wikimedia Commons

Mersenne was born 8 September 1588, the son of Julien Mersenne and his wife Jeanne, simple peasants, in Moulière near Oizé, a small commune in the Pays de la Loire in North-Western France. He was first educated at the at the nearby College du Mans and then from 1604 to 1609 at the newly founded Jesuit Collège Henri-IV de La Flèche. The latter is important as in La Flèche he would have received the mathematical programme created by Christoph Clavius for the Jesuit schools and colleges, the best mathematical education available in Europe at the time. A fellow student at La Flèche was René Descartes (1596–1650) with whom he would become later in life close friends.

René Descartes at work Source: Wikimedia Commons

However, it is unlikely that they became friends then as Mersenne was eight years older. Leaving La Flèche he continued his education in Greek, Hebrew, and theology at the Collège Royal and the Sorbonne in Paris. In 1611 he became a Minim friar and a year later was ordained as a priest. The Minims are a mendicant order founded in Italy in the fifteenth century. From 1614 to 1618 he taught philosophy and theology at Nevers but was recalled to Paris in 1619 to the newly established house on the Place Royal (now Place des Vosges), where he remained, apart from travels through France, to Holland, and to Italy, until his death. 

View map of an area of Paris near Place Royale, now Place des Vosges, showing the Minim convent where Mersenne lived and the Rue des Minimes, not far from the Bastille, undated, but before 1789 (paris-grad.com) Source: Linda Hall Library

In Paris he was introduced to the intellectual elite by Nicolas-Claude Fabri de Pereisc (1580–1637)–wealthy astronomer, antiquarian, and patron of science–whom he had got to know in 1616. 

Nicolas-Claude Fabri de Peiresc by Louis Finson Source: Wikimedia Commons

Settled in Paris, Mersenne began a career as a prolific author, both editing and publishing new editions of classical works and producing original volumes. In the 1620s his emphasis was on promoting and defending the Thomist, Aristotelian philosophy and theology in which he’d been educated. In his first book, Questiones celeberrimae in Genesim (1623), 

he attacked those he saw as opponents of the true Catholic religion, Platonist, cabbalistic and hermetic authors such as Telesio, Pomponazzi, Bruno and Robert Fludd. His second book, L’impiété des déistes, athées, et libertins de ce temps (1624), continued his attacks on the propagators of magic and the occult. His third book, La Vérité des sciences (1625), attacks alchemists and sceptics and includes a compendium of texts over ancient and recent achievements in the mathematical sciences that he saw as in conformity with his Christian belief. The latter drew the attention of Pierre Gassendi (1592–1655), who became his closest friend. I shall return to their joint activities in Paris later but now turn to Mersenne’s own direct scientific contributions, which began to replace the earlier concentration on theology and philosophy.

Pierre Gassendi Source: Wikimedia Commons

Mersenne’s scientific interests lay in mathematics and in particular what Aristotle, who was not a fan of mathematics, claiming it did not apply to the real world, called the mixed sciences or mixed mathematics i.e., astronomy, optics, statics, etc. Here he compiled to collections of treatises on mixed mathematics, his Synopsis Mathematica (1626) and Universae geometriae synopsis (1644). In his Traité de l’Harmonie Universelle (1627), to which we will return, Mersenne gives a general introduction to his concept of the mathematical disciplines:

Geometry looks at continuous quantity, pure and deprived from matter and from everything which falls upon the senses; arithmetic contemplates discrete quantities, i.e. numbers; music concerns har- monic numbers, i.e. those numbers which are useful to the sound; cosmography contemplates the continuous quantity of the whole world; optics looks at it jointly with light rays; chronology talks about successive continuous quantity, i.e. past time; and mechanics concerns that quantity which is useful to machines, to the making of instruments and to anything that belongs to our works. Some also adds judiciary astrology. However, proofs of this discipline are borrowed either from astronomy (that I have comprised under cosmology) or from other sciences. 

In optics he addressed the problem of spherical aberration in lenses and mirrors and suggested a series of twin mirror reflecting telescopes, which remained purely hypothetical and were never realised.

Source: Fred Watson, “Stargazer: The Life and Times of the Telescope”, Da Capo Press, 2004, p. 115

This is because they were heavily and falsely criticised by Descartes, who didn’t really understand them. It was Mersenne, who pushed Descartes to his solution of the refraction problem and the discovery of the sine law. He wrote three books on optics, De Natura lucis (1623); Opticae (1644); L’Optique et la catoptrique (1651). Although his theoretical reflecting telescopes were published in his Harmonie universelle (1636), see below.

Mersenne also wrote and published collections of essays on other areas of mixed mathematics, mechanics, pneumatics, hydro- statics, navigation, and weights and measures, Cogitata physico-mathematica (1644); Novarum observationum physico- mathematicarum tomus III (1647). 

Mersenne dabbled a bit in mathematics itself but unlike many of his friends did not contribute much to pure mathematics except from the Mersenne prime numbers those which can be written in the form Mn = 2n − 1 for some integer n. This was his contribution to a long search by mathematicians for some form of law that consistently generates prime numbers. Mersenne’s law whilst generating some primes doesn’t consistently generate primes but it has been developed into its own small branch of mathematics. 

It was, however, in the field of music, as the title quoted above would suggest, which had been considered as a branch of mathematics in the quadrivium since antiquity, and acoustics that Mersenne made his biggest contribution. This has led to him being labelled the “father of acoustics”, a label that long term readers of this blog will know that I reject, but one that does to some extent encapsulate his foundational contributions to the discipline. He wrote and published five books on the subject over a period of twenty years–Traité de l’harmonie universelle (1627); Questions harmoniques (1634); Les preludes de l’harmonie universelle (1634); Harmonie universelle (1636); Harmonicorum libri XII (1648)–of which his monumental (800 page) Harmonie universelle was the most important and most influential.

Title page of Harmonie universelle Source: Wikimedia Commons

In this work Mersenne covers the full spectrum including the nature of sounds, movements, consonance, dissonance, genres, modes of composition, voice, singing, and all kinds of harmonic instruments. Of note is the fact that he looks at the articulation of sound by the human voice and not just the tones produced by instruments. He also twice tried to determine the speed of sound. The first time directly by measuring the elapse of time between observing the muzzle flash of a cannon and hearing the sound of the shot being fired. The value he determined 448 m/s was higher than the actual value of 342 m/s. In the second attempt, recorded in the Harmonie universelle (1636), he measured the time for the sound to echo back off a wall at a predetermined distance and recorded the value of 316 m/s. So, despite the primitive form of his experiment his values were certainly in the right range. 

Mersenne also determined the correct formular for determining the frequency of a vibrating string, something that Galileo’s father Vincenzo (1520–1591) had worked on. This is now known as Mersenne’s Law and states that the frequency is inversely proportional to the length of the string, proportional to the square root of the stretching force, and inversely proportional to the square root of the mass per unit length.

The formula for the lowest frequency is f=\frac{1}{2L}\sqrt{\frac{F}{\mu}},

where f is the frequency [Hz], L is the length [m], F is the force [N] and μ is the mass per unit length [kg/m].

Source: Wikipedia

Vincenzo Galileo was also involved in a major debate about the correct size of the intervals on the musical scale, which was rumbling on in the late sixteenth and early seventeenth centuries. It was once again Mersenne, who produced the solution that we still use today.

Although Mersenne is certainly credited and honoured by acoustic researchers and music theorists for his discoveries in these areas, perhaps his most important contribution to the development of the sciences in the seventeenth century was as a networker and science communicator in a time when scientific journals didn’t exist yet. 

Together with Gassendi he began to hold weekly meetings in his humble cell with other natural philosophers, mathematicians, and other intellectuals in Paris. Sometime after 1633 these meetings became weekly and took place in rotation in the houses of the participants and acquired the name Academia Parisiensis. The list of participants reads like an intellectual who’s who of seventeenth century Europe and included René Descartes, Étienne Pascal and his son Blaise, Gilles de Roberville, Nicolas-Claude Fabri de Pereisc, Pierre de Fermat, Claude Mydorge, the English contigent, Thomas Hobbes, Kenhelm Digby, and the Cavendishes, and for those not living in or near Paris such as Isaac Beeckman, Jan Baptist van Helmont, Constantijn Huygens and his son Christiaan, and not least Galileo Galilei by correspondence. When he died approximately six hundred letters were found in his cell from seventy-nine different correspondents. In total 193 scholars and literati have been identified as participants. Here it should be noted that although he tended to reject the new emerging sciences in his earlier defence of Thomist philosophy, he now embraced it as compatible with his teology and began to promote it.

This academy filled a similar function to the Gresham College group and Hartlib Circle in England, as well as other groups in other lands, as precursors to the more formal scientific academies such as the Académie des sciences in Paris and the Royal Society in London. There is evidence that Jean-Baptist Colbert (1619–1683), the French Minister of State, modelled his Académie des sciences on the Academia Parisiensis. Like its formal successors the Academia Parisiensis served as a forum for scholars to exchange views and theories and discuss each other’s work. Mersenne’s aim in establishing this forum was to stimulate cooperation between the participants believing science to be best followed as a collective enterprise.

Mersenne’s role was not restricted to that of convener, but he functioned as a sort of agent provocateur deliberately stimulating participants to take up research programmes that he inaugurated. For example, he brought Torricelli’s primitive barometer to Paris and introduced it to the Pascals. It is thought that he initiated the idea to send Blaise Pascal’s brother-in-law up the Puy de Dôme to measure the decreasing atmospheric pressure.

Blaise Pascal, unknown; a copy of the painting of François II Quesnel, which was made for Gérard Edelinck in 1691. Source: Wikimedia Commons

Although they never met and only corresponded, he introduced Christiaan Huygens to the concept of using a pendulum to measure time, leading to Huygens’ invention of the pendulum clock.

Portrait of Christiaan Huygens (1629-1695) C.Netscher / 1671 Source: Wikimedia Commons

It was Mersenne, who brought the still very young Blaise Pascal together with René Descartes, with the hope that the brilliant mathematicians would cooperate, in this case he failed. In fact, the two later became opponents divided by their conflicting religious views. Mersenne also expended a lot of effort promoting the work of Galileo to others in his group, even offering to translate and publish Galileo’s work in French, an offer that the Tuscan mathematician declined. He did, however, publish an unpublished text by Galileo on mechanics, Les Mechaniques de Galilée.

Although not the author of a big theory or big idea, or the instigators of a big event, Mersenne actually contributed with his activities at least as much, if not more, to the development of science in the seventeenth century as any of the more famous big names. If we really want to understand how science develops then we need to pay more attention to figures like Mersenne and turn down the volume on the big names. 

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Filed under History of Mathematics, History of Optics, History of science, The Paris Provencal Connection