Category Archives: Renaissance Science

Renaissance Science – XLIX

The mathematisation of science is considered to be one of the principle defining characteristics of the so-called scientific revolution in the seventeenth century. Knowledge presentation on the European, medieval universities was predominantly Aristotelian in nature and Aristotle was dismissive of mathematics. He argued that the objects of mathematics were not real and therefore mathematics could not produce knowledge (episteme/scientia). He made an exception for the so-called mixed disciplines: astronomy, geometrical optics, and statics. These were, however, merely functionally descriptive, and not knowledge. So, mathematical astronomy described how to determine the positions of celestial bodies at a given point in time, but it was non-mathematical cosmology that described the true nature of those celestial bodies. Knowledge production and knowledge acquisition was, for Aristotle and those who adopted his philosophy, non-mathematical.

With just a relatively superficial examination, it is very clear that the new knowledge delivered up in astronomy, physics etc in the seventeenth century was very mathematical, just consider the title of Newton’s magnum opus, Philosophiæ Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy), something major had changed and two central questions for the historian of science are what and why? 

When I first started learning the history and philosophy of science several decades ago, there was a standard pat answer to this brace of questions. It was stated that there had been a change in philosophical systems underlying knowledge acquisition, Aristotelian philosophy and been replaced by a mathematical neo-Platonic philosophy; Plato had, according to the legend, famously the dictum, May no one ignorant of Geometry enter here Inscribed above the entrance to his school, The Academy. The belief that the mathematisation of science was driven by a Platonic renaissance was probably strengthened by the fact that the title page of Copernicus’ De Revolutionibus, the book that supposedly signalled the start of the scientific revolution, carried the same dictum. In fact, there is no evidence in Greek literature from the entire time that The Academy was open that the dictum existed. It was first mentioned by John Philoponus (c. 490­–c. 570) after Justinian had ordered The Academy closed in 529. De Revolutionibus is also not in anyway Platonic. 

To be fair to the proposers of the Plato replaced Aristotle thesis, Plato’s philosophy was definitively more mathematical than Aristotle’s and there was a neo-Platonic revival during the Renaissance, but it was more the esoteric and mystical Plato rather than the mathematical Plato, as I’ve already outlined in an earlier episode in this series.

So, what did drive the mathematisation? As already explained in explained in earlier episodes there were major expansions and developments in astronomy, cartography, surveying, and navigation starting in the fifteenth century during the Renaissance. All four disciplines demanded an intensive use of geometry and especially trigonometry. This can be seen in the publication of the first printed edition of Euclid by Erhard Ratdolt (1442–1528) in 1482, which was followed by significant printed translations in the vernacular throughout Europe.

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 trigonometry, Johannes Petreius (c. 1497–1550) published Regiomontanus’ De triangulis omnimodis (On Triangles of All Kinds), edited by Johannes Schöner, in 1533. This was the first almost complete account of trigonometry published in Europe, the only thing that was missing was the tangent, but Regiomontanus had included the tangent in his earlier Tabula directionum, written in 1467 but first published in print in 1490. Regiomontanus’ trigonometry was followed by several important volumes on the topic during the sixteenth century. 


These areas of mathematical development were however for the Aristotelian academics at the universities not scientia and the mathematical practitioners, who did the mathematics were not considered to be academics but mere craftsmen. However, the widening reliance on mathematics in what had become important political areas of Renaissance society did much to raise the general status of mathematics.

Another area where a mathematical subdiscipline was on the advance was algebra, the basis for the analytical mathematics that would become so important in the seventeenth century. Already introduced in the twelfth century, with the translation of al-Khwarizmi’s text on the Hindu-Arabic number system into Latin, Algoritmi de numero Indorum along with the book that gave algebra its name, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah. It initially had minimal impact. However, reintroduced in the next century by Leonardo Pisano, with his Liber Abbaci, and although the acceptance was slow, only beginning to accelerate with the introduction of double entry bookkeeping at the end of the thirteenth century and beginning of the fourteenth. By the sixteenth century many abbaco schools (scuole d’abbaco or botteghe d’abbco) had been established throughout Europe teaching the Hindu-Arabic number system and algebra to apprentices using Libri d’abbaco, (abbacus books). In fact, the first ever printed mathematics book was an abbacus book, the so-called Treviso Arithmetic or Arte dell’Abbaco written in vernacular Venetian and published in Treviso in 1478. 

Once again, however, what was being taught here was not an academic discipline, which could generate scientia, but commercial arithmetic, very useful for an increasing commercial Europe dependent on extensive trade but not for the acquisition of academic knowledge. This began to change slowly in the sixteenth century beginning with the Summa de arithmetica, geometria, proportioni et proportionalita of Luca Pacioli (c. 1447–1517) published in 1496, still a book of practical mathematics and so not academic, but one which contained the false claim that there could not be a general solution of the cubic equation. This led on to Scipione del Ferro (1465–1526) discovering such a solution, Tartaglia (c. 1499–1557) rediscovering it and Gerolamo Cardano (1501–1576) seducing Tartaglia into revealing his solution and then publishing it in his Ars Magna. All of which I have outlined in more detail here. Cardano’s Ars magna, published in Nürnberg in 1545 by Johannes Petreius, has been called the first modern mathematics book, a term I don’t particularly like, but it did bring algebra into the world of academia, although it still wasn’t considered to be knowledge producing.


So, how did the change in status of mathematics on the universities come about and who was responsible for it if it wasn’t Plato? The change was brought about by Italian, humanist scholars in the sixteenth century and the responsibility lies not with a philosopher but with a mathematician, Archimedes. 

Bronze statue of Archimedes in Syracuse Source: Wikimedia Commons

Archimedes of Syracuse (c.287–c. 212 BCE) mathematician, physicist, engineer, and inventor is one of the most well-known figures in the entire history of science. Truly brilliant in a range of fields of study and immersed in a cloud of myths and legends. He is famous for the machines he invented and a legend for alleged machines he constructed to defend his hometown of Syracuse against the Romans. It is these war machines and the myth of his death at the hands of a Roman soldier that dominate the accounts of his life all written posthumously in antiquity. His mathematical work, which is what interest us here, remained largely unknown in antiquity. There only began to become known in the Early Middle Ages but, although translated into Arabic by Thābit ibn Qurra (836–901) and from there into Latin by Gerard of Cremona (c. 1114–1187) and again directly from Geek into Latin by William of Moerbeke (c. 1215–1286) and once more by Iacobus Cremonnensis (c. 1400–c. 1454), his work received very little attention in the Middle Ages.

Beginning, already in the fifteenth century, Renaissance humanist began to seriously re-evaluate the leading Greek mathematicians, in particular Euclid and Archimedes. Euclidian geometry was playing a much greater role in the evolving optics, in particular linear perspective, than it had ever played on the medieval universities. This led, as already noted above, to the publication of the first printed edition of The Elements, by Erhard Ratdolt, in 1482. Interestingly, the manuscript that Ratdolt used for edition was one that Regiomontanus (1436–1476) had brought with him to Nürnberg, where he established the world’s first scientific published endeavour, intending to publish it himself, as he announced in his published catalogue of intended publications. Unfortunately, he died before he could print most of this extensive catalogue of scientific and mathematical texts. 

This catalogue also included a manuscript of the works of Archimedes in the Latin translation of Iacobus Cremonnensis, which also fell foul of the Franconian mathematician’s early death. This manuscript would eventually be published in Basel, together with a Greek original brought from Rome by Willibald Pirckheimer (1470–1530), in a bilingual edition of the works edited by the Nürnberger theologian, humanist, and mathematician, Thomas Venatorius (1488–1551), in 1544. 

Venatorius’ edition of the works of Archimedes Source

The Italian astrologer, astronomer, and mathematician, Luca Gaurico (1475–1558), had published Archimedes’ works On the Parabola and On the Circle in the Latin translation by William of Moerbeke in 1503. Niccolò Fontana Tartaglia, who had published an Italian translation of The Elements in 1543, also published On the ParabolaOn the CircleCentres of Gravity, and On Floating Bodies in the Moerbeke translation in 1543. Later he would publish translation into Italian of these works, some of which appeared posthumously. Unlike, other later, Italian mathematicians, Tartaglia did not incorporate much of Archimedes’ work into his own highly influential, original Nova Scientia (1537), a mathematical work that did deliver, as the title says, scientia or knowledge. It is difficult to say how much Tartaglia was influenced by Archimedes in his approach to physics. 

Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi per Nicolaum Tartaleam … (facsimile) Source

We have already seen in the episode on hydrostatics how Archimedes work On Floating Bodies, informed and influenced the work of both Tartaglia’s one time student, Giambattista Benedetti (1530–1590), and the engineer, Simon Stevin (1448–1620) in the Netherlands in their work on hydrostatics and on the laws of fall. As I have also outlined in great detail, Archimedes work on statics had a major influence on the Italian mathematicians of the so-called Urbino School. Federico Commandino (1509 – 1575), Guidobaldo dal Monte (1545 – 1607), and Bernardino Baldi (1553–1617), the first two also producing and publishing new improved translations of various of Archimedes work. Once again Simon Stevin also produced a major, Archimedes inspired work on statics. 

These mathematicians had, directly inspired, and heavily influenced by the work of Archimedes, now produced, in several fields, work that was indisputably scientia or knowledge by the use of mathematics and thus instigated the turn from Aristotelian philosophical knowledge to the mathematisation of knowledge production and this movement began to spread in the seventeenth century.

The Urbino School, in particular dal Monte, who was his patron, influenced Galileo, who openly declared that he had in his natural philosophy replaced Aristotle with Archimedes. Galileo’s pupils Vincenzo Viviani (1622–1703) and Evangelista Torricelli (1608–1647) followed his lead on this. Stevin’s work, written in Dutch was translated into Latin by Willebrord Snel (1580–1626) and heavily influenced the French natural philosophers such as, Marin Mersenne (1588–1648), who together with Pierre Gassendi (1592–1655) led the informal academic society, the Academia Parisiensis, a weekly gathering from 1633 onwards, which included the most important French, English and Dutch natural philosophers of the period. This group was of course also influenced by the work of Galileo. 

It was not the thoughts of a philosopher, Plato, that pushed Aristotelian philosophy from its throne on the medieval university as the purveyor of factual knowledge of the real world and replaced it with a mathematics-based system, but the work of a mathematician, Archimedes. As the century progressed Euclid and Archimedes would in turn be replaced by the algebra-based analytical mathematics that would eventually develop into calculus, although also here the method of exhaustion first developed by Eudoxus of Cnidus (c. 408–c. 355 BCE), but popularised by Archimedes was the basis of the integral calculus half of the new mathematics.


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

An inventor of instruments

Way back at the beginning of November I wrote what was intended to be the first of a series of posts about English mathematical practitioners, who were active at the end of the sixteenth and the beginning of the seventeenth centuries. I did not think it would be two months before I could continue that series with a second post, but first illness and then my annual Christmas trilogy got in the way and so it is only now that I am doing so. The subject of this post is a man for whom a whole series of mathematical instruments are named, Edmund Gunter (1581–1626).

Unfortunately, as is all to often the case with Renaissance mathematici, we know almost nothing about Gunter’s origins. His father was apparently a Welshmen from Gunterstown, Brecknockshire in South Wales but he was born somewhere in Hertfordshire sometime in 1581. Obviously from an established family he was educated at Westminster School as a Queen’s Scholar i.e., a foundation scholar (elected on the basis of good academic performance and usually qualifying for reduced fees). He matriculated at Christ Church Oxford 25 January 1599 (os). He graduated BA 12 December 1603 and MA 2 July 1606. He took religious orders and proceeded B.D. 23 November 1615. He was appointed Rector of St. George’s, Southwark and of St Mary Magdalen, Oxford in 1615, he retained both appointments until his death. 

Whilst still a student in 1603, he wrote a New Projection of the Sphere in Latin, which remained in manuscript until it was finally published in 1623. This came to the attention of Henry Briggs (1561–1630), who had been appointed professor of geometry at the newly founded Gresham College in 1596, and as such was very much a leading figure in the English mathematical community. Briggs was impressed by the young mathematician befriending him and becoming his mentor. The two men spent much time together at Gresham College discussing topics of practical mathematics. In 1606, Gunter developed a sector, about which later, and wrote a manuscript describing it in Latin, without a known title. This circulated in manuscript for many years and was much in demand. Gunter gave into that demand and finally published it also in 1623.

When the first Gresham professor of astronomy, Edward Brerewood (c. 1556–1613) died 4 November 1613, Briggs recommended Gunter as his successor. However, Thomas Williams another Christ Church graduate, of whom little is known, was appointed just seven days later 11 November 1613. When Williams resigned from the post 4 March 1619, for reasons unknown, Briggs once again supported his friend for the position, this time with success. Gunter was appointed just two days later, 6 March 1619. Like his two rectorships, he retained the Gresham professorship until his death. 

Gresham College, engraving by George Vertue, 1740 Source: Wikimedia Commons

Apparently, he was already spending so much time at Gresham College before being appointed that when the mathematician William Oughtred (1574–1660) visited Henry Briggs there in 1618, he thought that Gunter was already professor there.

In the Spring 1618 I being at London went to see my honoured friend Master Henry Briggs at Gresham College: who then brought me acquainted with Master Gunter lately chosen Astronomical lecturer there, and was at that time in Doctor Brooks his chamber. With whom falling into speech about his quadrant, I showed him my Horizontal Instrument. He viewed it very heedfully: and questioned about the projecture and use thereof, often saying these words, it is a very good one. And not long after he delivered to Master Briggs to be sent to me mine own Instrument printed off from one cut in brass: which afterwards I understood he presented to the right Honourable the Earl of Bridgewater, and in his book of the sector printed six years after, among other projections he setteth down this.

Gunter and Oughtred would go on to become firm friends.

William Oughtred engraving by Wenceslaus Hollar Source: Wikimedia Commons There are apparently no portraits of Briggs or Gunter

We now have the known details of the whole of Gunter’s life and can turn our attention to his mathematical output but before we do so there is an anecdote from Seth Ward (1617–1689), another mathematician and astronomer, concerning a position that Gunter did not get. In 1619, Henry Savile (1549–1622) established England’s first university chairs for mathematics the Savilian chairs for geometry and astronomy at Oxford. Savile’s first choice for the chair of geometry was Edmund Gunter and he invited him to an interview, according to John Aubrey (1626–1697) relating a report from Seth Ward:

[Gunter] brought with him his sector and quadrant, and fell to resolving triangles and doing a great many fine things. Said the grave knight [Savile], “Do you call this reading of geometry? This is showing of tricks, man!”, and so dismissed him with scorn, and sent for Henry Briggs.

So, Henry Briggs became England’s first university professor of geometry and not Edmund Gunter. One should point out that Ward can only have heard the story second hand as he was only two years old in 1619.

In 1614, John Napier (1550–1617) published his Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms, 1614), a new method of simplifying calculations. Edward Wright (1561–1615) produced an English translation, which was published posthumously in 1616. Napier’s logarithms were base:

NapLog(x) = –107ln (x/107)

Henry Briggs travelled all the way to Edinburgh to meet the inventor of this new calculating tool. After discussion with Napier, he received his blessing to produce a set of base ten logarithms. His Logarithmorum chilias prima, which were publish in London sometime before Napier’s death in 1617.

Many people don’t realise that Napier’s logarithmic tables were not straight logarithms but logarithms of trigonometrical functions. These are of particular use for astronomers and navigators. It is almost certainly through Brigg’s influence that Gunter’s first publication was a set of base ten, seven figure logarithmic tables of sines and tangents. His Canon Triangulorum sive Tabulae Sinuum et Tangentium Artificialum was published in Latin in 1620. An English translation was published in the same year. The terms sine and tangent were already in use, but it was Gunter, who introduced the terms cosine and cotangent in this publication. Later, on his scale or rule he introduced the short forms sin and tan.

In 1623, Gunter finally published his New Projection of the Sphere written in his last year as an undergraduate. He also published his most important book, Description and Use of the Sector, the Crosse-staffe and other Instruments. This was one of the most important guides to the use of navigational instruments for seamen and became something of a seventeenth century best seller in various forms. David Waters in his The Art of Navigation say this, ” Gunter’s De Sectore & Radio must rank with Eden’s translation of Cortes’s Arte de Navegar and Wright’s Certain Errors as one of the three most important English books ever published for the improvement of navigation.” [1]

Waters opposite page 360

His various publications were collected into The Works of Edmund Gunter, which went through six editions by 1680. Each edition having extra content by other authors. Isaac Newton (1642-1727) bought a copy of the second edition. The title page of the fifth edition is impressive:

The Workers of Edmund Gunter 5th ed. Title page with diagrams of the sector on the fly leaf

The Works of Edmund Gunter:
Containing the description and Use of the
Sector, Cross-staff, Bow, Quadrant,
And other Instruments.
With a Canon of Artificial Sines and Tangents to a Radius of 10.00000 parts, and the Logarithms from Unite to 100000:
The Uses whereof are illustrated in the Practice of
Arithmetick, geometry, Astronomy, Navigation, Dialling and Fortification.
And some Questions in Navigation added by Mr. Henry Bond, Teacher of mathematicks in Ratcliff, near London.
To which is added,
The Description and Use of another Sector and Quadrant, both of them invented by Mr. Sam. Foster, Late Professor of Astronomy in Gresham Colledge, London, furnished with more Lines, and differing from those of Me. Gunter′s both in form and manner of Working.
The Fifth Edition,
Diligentyl Corrected, and divers necessary Things and Matters (pertinent thereunto) added, throughout the whole work, not before Printed.
By William Leybourne, Philomath.
Printed by A.C. for Francis Eglesfield at the Marigold in St. Pauls Church-yard. MDCLXXIII.

The sector, also known as a proportional compass or military compass, was a major calculating instrument in use from the end of the sixteenth century until the nineteenth century. It is an instrument consisting of two rulers of equal length joined by a hinge. A number of scales are inscribed upon the instrument which facilitate various mathematical calculations. It was used for solving problems in proportion, multiplication, and division, geometry, and trigonometry, and for computing various mathematical functions, such as square and cube roots. Its several scales permitted easy and direct solutions of problems in gunnery, surveying and navigation. The sector derives its name from the fourth proposition of the sixth book of Euclid, where it is demonstrated that similar triangles have their like sides proportional. (Wikipedia)

The sector has many alleged inventors. The earliest was Fabrizio Mordente (1532–c. 1608). The invention is often credited to Galileo (1564–1642), who marketed a very successful variant in the early seventeenth century, including selling lessons and an instruction manual in its use. However, Galileo’s instrument was a development of one created by Guidobaldo dal Monte (1545–1607). It is not known if dal Monte developed the device independently or knew of Mordent’s. Thomas Hood (1556–1620) appear to have reinvented the instrument, a description of which he published in his Making and Use of the Sector, 1596.

Waters opposite page 345

Gunter developed Hood’s instruments adding addition scales, including a scale for use with Mercator’s new projection of the sphere. 


Water opposite page 361
Waters page 361

The French Jewish scholar, Levi ben Geshon (1288–1344), published the first description of the cross staff or Jacob’s staff, used in astronomy, surveying, and navigation, in his Book of the Wars of the Lord (originally in Hebrew but also translated into Latin). 

Gunter image of a cross staff

Gunter’s book also describes the Gunter Quadrant, basically a horary quadrant for telling the time by taking the altitude of the sun but with some additional functions.

Boxwood Gunter-type sector, made by Isaac Carver and owned by George Lason; 1706 Whipple Museum
Illustration of a quadrant from Edmund Gunter’s Works (1653). Image © the Whipple Library.
Modern reproduction of the Gunter Quadrant Source

There is also a description of the crossbow an alternative to the backstaff that never became popular. 

; Navigation: an Astrolabe, a Cross-Staff, and a Back-Staff or Davis’s Sextant; Wellcome Collection;
; Navigation: a Cross-Staff or Cross-Bow, and a Sailor Using the Device; Wellcome Collection;

Gunter’s most popular instrument was his scale. The Gunter scale or rule was a rule containing trigonometrical and logarithmic scales, which could be used with a pair of dividers to carry out calculations in astronomy and in particular navigations. The Gunter scale is basically a sector folded into a straight line without the hinge.Sailors simply referred to the rule as a Gunter. William Oughtred would go on to place two Gunter rules next to each other thus creating the slide rule and eliminating the need for dividers to carry out the calculations.

Gunter scale front side
Gunter scale back side

In 1622, Gunter engraved a new sundial at Whitehall, which carried many different dial plates supplying much astronomical data. At the behest of Prince Charles, he wrote and published an explanation of the dials, The Description and Use of His Majesties Dials in Whitehall, 1624. The sundial was demolished in 1697.

Gunter’s most well-known instrument was his surveyor’s chain, which became the standard English Imperial chain. 100 links and 22 yards (66 feet) long, there are 10 chains in a furlong and 80 chains to a mile. 

Although Gunter invented, designed, and described the use of several instruments, he didn’t actually make any of them. All of his instruments were produced by the London based, instrument maker Elias Allen (c. 1588–1652). Allen was born in Kent of unknown parentage and was apprenticed in 1602 to London instrument maker Charles Whitwell (c. 1568–1611) in the Grocer’s Company, serving his master for nine years. Following Whitwell’s death in 1611, Allen set up his own business. He rapidly became the foremost instrument maker in London, working mostly in brass, but occasionally in silver. He became very successful and made instruments for various aristocratic patrons and both James I and Charles I. Allen also produced the engravings in Gunter’s books, using them also as advertising in his shop.

He worked closely with various mathematicians including both Oughtred and Gunter. His workshop became a meeting place for discussion amongst mathematical practitioners. He was the first London instrument maker, who could make a living from just making instruments without working on the side as a map engraver or surveyor. His master Whitwell subsidised his income as a map engraver. He rose in status in the Grocers’ Company, becoming its treasurer in 1636 and its master for eighteen months in 1637-38. Over the years many of his apprentices became successful instrument maker masters in the own right, most notably Ralph Greatorex (1625–1675), who was associated with Oughtred, Samuel Pepys, John Evelyn, Samuel Hartlib, Christopher Wren, Robert Boyle, and Jonas Moore, the English scientific elite of the time. 

Allen had the distinction of being one of the few seventeenth-century artisans to have his portrait painted. The Dutch artist Hendrik van der Borcht the Younger (1614–1676) produced the portrait, now lost, in about 1640. It still exists as an engraving done by the Bohemian engraver, Wenceslaus Hollar (1607­–1677).

Edmund Gunter was not a mathematician as we understand the term today, but a mathematical practitioner, who exercised a large influence on the practical side of astronomy, navigation, and surveying in the seventeenth century through the instruments that he designed and the texts he wrote explaining how to use them. 

[1] David Walters, The Art of Navigation in England in Elizabethan and Early Stuart Times, Yale University Press, 1958 p. 359

1 Comment

Filed under History of Astronomy, History of Cartography, History of Navigation, Renaissance Science, Uncategorized

Renaissance Science – XLVIII

Using the simplest and widest definition as to what constitutes a scientific instrument, it is literally impossible to say who first created, devised, used a scientific instrument or when and where they did it. My conjecture would be that the first scientific instrument was some sort of measuring device, a rod, or a cord to standardise a unit of measurement, almost certainly taken from the human body: a forearm, the length of a stride or pace, maybe a foot, a unit that we still use today. It is obviously that all the early great civilisation, Indus valley, Yellow River, Yangtze River, Fertile Crescent and so on, definitely used measuring devices, possibly observational devices, instruments to measure or lay out angles, simple compasses to construct circles, all of them probably as much to do with architecture and surveying, as with anything we might now label science.

This is the Royal cubit rod of Amenemope – a 3320-year-old measuring rod which revealed that Egyptians used units of measurement taken from the human body. The basic unit was the cubit – the length from the elbow to the tip of the middle finger, about 45cm. Source: British Museum

Did the early astronomers in China, India, Babylon use some sorts of instruments to help them make their observations? We know that later people used sighting tubes, like a telescope without the lenses, to improve the quality of their observations, did those first astronomers already use something similar. Simple answer, we don’t really know, we can only speculate. We do know that Indian astronomers used a quadrant in their observation of solar eclipses around 1000 BCE. 

Turning to the Ancient Greeks we initially have a similar lack of knowledge. The first truly major Greek astronomer Hipparkhos (c. 190–c. 120 BCE) (Latin Hipparchus) definitely used astronomical instruments but we have no direct account of his having done so. Our minimal information of his instruments comes from later astronomers, such as Ptolemaios (c. 100–c. 170 CE). Ptolemaios tells us in his Mathēmatikē Syntaxis aka Almagest that Hipparkhos made observations with an equatorial ring.

The easiest way to understand the use of an equatorial ring is to imagine a ring placed vertically in the east-west plane at the Earth’s equator. At the time of the equinoxes, the Sun will rise precisely in the east, move across the zenith, and set precisely in the west. Throughout the day, the bottom half of the ring will be in the shadow cast by the top half of the ring. On other days of the year, the Sun passes to the north or south of the ring, and will illuminate the bottom half. For latitudes away from the equator, the ring merely needs to be placed at the correct angle in the equatorial plane. At the Earth’s poles, the ring would be horizontal. Source: Wikipedia

At another point in the book Ptolemaios talks of making observations with an armillary sphere and compares his observations with those of Hipparkhos, leading some to think that Hipparkhos also used an armillary sphere. Toomer in his translation of the Almagest say there is no foundation for this speculation and that Hipparkhos probably used a dioptra. [1]

Ptolemaios mentions four astronomical instruments in his book, all of which are for measuring angles: 

1) A double ring device and

Toomer p. 61

2) a quadrant both used to determine the inclination of the ecliptic.

Toomer p. 62

3) The armillary sphere, which he confusingly calls an astrolabe, used to determine sun-moon configurations. 

Toomer p. 218

4) His parallactic rulers, used to determine the moon’s parallax, which was called a triquetrum in the Middle Ages. 

Toomer p. 245

Ptolemaios almost certainly also used a dioptra a simple predecessor to the theodolite used for measuring angles both in astronomy and in surveying. As I outlined in the post on surveying, ancient cultures were also using instruments to carry out land measuring.

Graphic reconstruction of the dioptra, by Venturi, in 1814. (An incorrect interpretation of Heron’s description) Source: Wikimedia Commons

Around the same time as the armillary sphere began to emerge in ancient Greece it also began to emerge in China, with the earliest single ring device probably being used in the first century BCE. By the second century CE the complete armillary sphere had evolved ring by ring. When the armillary sphere first evolved in India is not known, but it was in full used by the time of Āryabhata in the fifth century CE.

Armillary sphere at Beijing Ancient Observatory, replica of an original from the Ming Dynasty

A parallel development to the armillary sphere was the celestial globe, a globe of the heavens marked with the constellations. In Greece celestial globes predate Ptolemaios but none of the early ones have survived.  In his Almagest, Ptolemaios gives instruction on how to produce celestial globes. Chinese celestial globes also developed around the time of their armillary spheres but, once again, none of the early ones have survived. As with everything else astronomical, the earliest surveying evidence for celestial globes in India is much later than Greece or China.

The Farnese Atlas holding a celestial globe is the oldest known surviving celestial globe dating from the second century CE Source: Wikimedia Commons

In late antiquity the astrolabe emerged, its origins are still not really clear. Ptolemaios published a text on the planisphere, the stereographic projection used to create the climata in an astrolabe and still used by astronomers for star charts today. The earliest references to the astrolabe itself are from Theon of Alexandria (c. 335–c. 414 CE). All earlier claims to existence or usage of astrolabes are speculative. No astrolabes from antiquity are known to have survived. The earliest surviving astrolabe is an Islamic instrument dated AH 315 (927-28 CE).

North African, 10th century AD, Planispheric Astrolabe Khalili Collection via Wikimedia Commons

Late Antiquity and the Early Middle Ages saw a steady decline in the mathematical sciences and with it a decline in the production and use of most scientific instruments in Europe until the disappeared almost completely. 

When the rapidly expanding Arabic Empire began filing their thirst for knowledge across a wide range of subjects by absorbing it from Greek, Indian and Chinese sources, as well as the mathematical disciplines they also took on board the scientific instruments. They developed and perfected the astrolabe, producing hundreds of both beautiful and practical multifunctional instruments. 

As well large-scale astronomical quadrants they produced four different types of handheld instruments. In the ninth century, the sine or sinical quadrant for measuring celestial angles and for doing trigonometrical calculations was developed by Muḥammad ibn Mūsā al-Khwārizmī. In the fourteenth century, the universal (shakkāzīya) quadrant used for solving astronomical problems for any latitude. Like astrolabes, quadrants are latitude dependent and unlike astrolabes do not have exchangeable climata. Origin unknown, but the oldest known example is from 1300, is the horary quadrant, which enables the uses to determine the time using the sun. An equal hours horary quadrant is latitude dependent, but an unequal hours one can be used anywhere, but its use entails calculations. Again, origin unknown, is the astrolabe quadrant, basically a reduced astrolabe in quadrant form. There are extant examples from twelfth century Egypt and fourteenth century Syria.

Horary quadrant for a latitude of about 51.5° as depicted in an instructional text of 1744: To find the Hour of the Day: Lay the thread just upon the Day of the Month, then hold it till you slip the small Bead or Pin-head [along the thread] to rest on one of the 12 o’Clock Lines; then let the Sun shine from the Sight G to the other at D, the Plummet hanging at liberty, the Bead will rest on the Hour of the Day. Source: Wikimedia Commons
Astrolabic quadrant, made of brass; made for latitude 33 degrees 30 minutes (i.e. Damascus); inscription on the front saying that the quadrant was made for the ‘muwaqqit’ (literally: the timekeeper) of the Great Umayyad Mosque of Damascus. AH 734 (1333-1334 CE) British Museum

Islamicate astronomers began making celestial globes in the tenth century and it is thought that al-Sufi’s Book of the Constellations was a major source for this development. However, the oldest surviving Islamic celestial globe made by Ibrahim Ibn Saîd al-Sahlì in Valencia in the eleventh century show no awareness of the forty-eight Greek constellations of al-Sufi’s book.

Islamicate mathematical scholars developed and used many scientific instruments and when the developments in the mathematical sciences that they had made began to filter into Europe during the twelfth century scientific renaissance those instruments also began to become known in Europe. For example, the earliest astrolabes to appear in Europe were on the Iberian Peninsula, whilst it was still under Islamic occupation.  

Canterbury Astrolabe Quadrant 1388 Source Wikimedia Commons
Astrolabe of Jean Fusoris, made in Paris, 1400 Source: Wikimedia Commons

The medieval period in Europe saw a gradual increase in the use of scientific instruments, both imported and locally manufactured, but the use was still comparatively low level. There was some innovation, for example the French Jewish scholar, Levi ben Geshon (1288–1344), published the first description of the cross staff or Jacob’s staff, used in astronomy, surveying, and navigation, in his Book of the Wars of the Lord (originally in Hebrew but also translated into Latin). 

…of a staff of 4.5 feet (1.4 m) long and about one inch (2.5 cm) wide, with six or seven perforated tablets which could slide along the staff, each tablet being an integral fraction of the staff length to facilitate calculation, used to measure the distance between stars or planets, and the altitudes and diameters of the Sun, Moon and stars

A Jacob’s staff, from John Sellers’ Practical Navigation (1672) Source: Wikimedia Commons

Also, the magnetic compass came into use in Europe in the twelfth century, first mentioned by Alexander Neckam (1157–1217) in his De naturis rerum at the end of the century.

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.

Petrus Pereginus (fl. 1269) gave detailed descriptions of both the floating compass and the dry compass in his Epistola de magnete

However, it was first in the Renaissance that a widespread and thriving culture of scientific instrument design, manufacture, and usage really developed. The steep increase in scientific instrument culture was driving by the substantial parallel developments in astronomy, navigation, surveying, and cartography that began around fourteen hundred that I have already outlined in previous episodes of this series. Renaissance scientific instrument culture is too large a topic to cover in detail in one blog post, so I’ll only do a sketch of some major points and themes with several links to other earlier related posts.

Already, the first Viennese School of Mathematics, which was heavily involved in the development of both astronomy and cartography was also a source of scientific instrument design and manufacture.Johannes von Gmunden (c. 1380–1442) had a notable collection of instruments including an Albion, a multipurpose instrument conceived by Richard of Wallingford (1292–1336).

Albion front side Source: Seb Falk’s Twitter feed
Albion rear Source: Seb Falk’s Twitter feed

Georg von Peuerbach (1423–1461) produced several instruments most notably the earliest portable sundial marked for magnetic declination.

Folding sundial by Georg von Peuerbach

His pupil Regiomontanus (1436–1476) wrote a tract on the construction and use of the astrolabe and there is an extant instrument from 1462 dedicated to Cardinal Bessarion and signed IOHANNES, which is assumed to have been made by him. At least eleven other Regiomontanus style astrolabes from the fifteenth century are known.

Regiomontanus style astrolabe Source: Wikimedia Commons

Elements of his design were adopted by both Johannes Stöffler (1452–1531), the first professor of astronomy at the University of Tübingen, and by the Nürnberger mathematicus Georg Hartmann (1489–1564).

Stöffler also made celestial globes and an astronomical clock.

Celestial Globe, Johannes Stöffler, 1493; Landesmuseum Württemberg Source: Wikimedia Commons

Mechanical astronomical clocks began to emerge in Europe in the fourteenth century, but it would not be until the end of the sixteenth century that mechanical clocks became accurate enough to be used as scientific instruments. The earliest clockmaker, who reached this level of accuracy being the Swiss instrument maker, Jost Bürgi (1552–1632)

Bürgi made numerous highly elaborate and very decorative mechanical clocks, mechanised globes and mechanised armillary spheres that were more collectors items for rich patrons rather than practical instruments.

Bürgi Quartz Clock 1622-27
Source: Swiss Physical Society

This illustrates another driving force behind the Renaissance scientific instrument culture. The Renaissance mathematicus rated fairly low in the academical hierarchy, actually viewed as a craftsman rather than an academic. This made finding paid work difficult and they were dependent of rich patrons amongst the European aristocracy. It became a standard method of winning the favour of a patron to design a new instrument, usually a modification of an existing one, making an elaborate example of it and presenting it to the potential patron. The birth of the curiosity cabinets, which often also included collections of high-end instruments was also a driving force behind the trend. Many leading instrument makers produced elaborate, high-class instruments for such collections. Imperial courts in Vienna, Prague, and Budapest employed court instrument makers. For example, Erasmus Habermel (c. 1538–1606) was an incredibly prolific instrument maker, who became instrument maker to Rudolf II. A probable relative Josua Habermel (fl. 1570) worked as an instrument maker in southern Germany, eventually moving to Prague, where he probably worked in the workshop of Erasmus.

 1594 armillary sphere by Erasmus Habermel of Prague.

Whereas from Theon onwards, astrolabes were unique, individual, instruments, very often beautiful ornaments as well as functioning instruments, Georg Hartmann was the first instrument maker go into serial production of astrolabes. Also, Hartmann, although he didn’t invent them, was a major producer of printed paper instruments. These could be cut out and mounted on wood to produce cheap, functional instruments for those who couldn’t afford the expensive metal ones. 

Hartmann astrolabe front
Hartmann astrolabe rear
Paper and Wood Astrolabe Hartmann Source: HSM Oxford

Hartmann lived and worked in Nürnberg, which as I have sketched in an earlier post, was for more than a century the scientific instrument capital of Europe with a massive produce of instruments of all sorts.

One of the most beautiful sets on instruments manufactured in Nürnberg late 16th century. Designed by Johannes Pretorius (1537–1616), professor for astronomy at the Nürnberger University of Altdorf and manufactured by the goldsmith Hans Epischofer (c. 1530–1585) Germanische National Museum

As well as astrolabes and his paper instruments Hartmann also produced printed globes, none of which have survived. Another Nürnberger mathematicus, Johannes Schöner (1477–1547) launched the printed pairs of terrestrial and celestial globes onto the market.

Celestial Globe by Johannes Schöner c. 1534 Source

His innovation was copied by Gemma Frisius (1508–1555), whose student Gerard Mercator (1512–1594) took up globe making on a large scale, launching the seventeenth century Dutch globe making industry. 

Gemma Frisius set up a workshop producing a range of scientific instruments together with his nephew (?) Gualterus Arsenius (c. 1530–c. 1580).  

Astronomical ring dial Gualterus Arsenius Source

In France, Oronce Fine (1494–1555), a rough contemporary, who was appointed professor at the Collège Royal, was also influenced by Schöner in his cartography and like the Nürnberger was a major instrument maker. In Italy, Egnatio Danti (1536–1586) the leading cosmographer was also the leading instrument maker. 

Egnation Danti, Astrolabe, ca. 1568, brass and wood. Florence, Museo di Storia della Scienza Source: Fiorani The Marvel of Maps p. 49

A major change during the Renaissance was the emergence, for the first time in Early Modern Europe, of large-scale astronomical observatories, Wilhelm IV (1532–1592) in Hessen-Kassel beginning in about 1560 and Tycho Brahe (1546–1601) on the Island of Hven beginning in 1575. Both men commissioned new instruments, many of which were substantially improved in comparison with their predecessors from antiquity.

Sternwarte im Astronomisch-Physikalischen Kabinett, Foto: MHK, Arno Hensmanns Reconstruction of Wilhelm’s observatory
Tycho Brahe, Armillary Sphere, 1581 Source
Tycho Brahe quadrant

Their lead was followed by others, the first Vatican observatory was established in the Gregorian Tower in 1580.

View on the Tower of Winds (Gregorian tower) in Vatican City (with the dome of Saint Peter’s Basilica in the background). Source: Wikimedia Commons

In the early seventeenth century, Leiden University in Holland established the first European university observatory and Christian Longomontanus (1562–1647), who had been Tycho’s chief assistant, established a university observatory in Copenhagen 

Drawing of Leiden Observatory in 1670, seen on top of the university building. Source: Wikimedia Commons
Copenhagen University Observatory Source: Wikimedia Commons

As in all things mathematical England lagged behind the continent but partial filled the deficit by importing instrument makers from the continent, the German Nicolas Kratzer (c. 1487–1550) and the Netherlander Thomas Gemini (c. 1510–1562). The first home grown instrument maker was Humfrey Cole (c. 1530­–1591). By the end of the sixteenth century, led by John Dee (1527–c. 1608), who studied in Louven with Frisius and Mercator, and Leonard Digges (c. 1515–c. 1559), a new generation of English instrument makers began to dominate the home market. These include Leonard’s son Thomas Digges (c. 1546–1595), William Bourne (c. 1535–1582), John Blagrave (d. 1611), Thomas Blundeville (c. 1522–c. 1606), Edward Wright (1561–1615), Emery Molyneux (d. 1598), Thomas Hood (1556–1620), Edmund Gunter (1581–1626) Benjamin Cole (1695–1766), William Oughtred (1574–1660), and others.

The Renaissance also saw a large amount of innovation in scientific instruments. The Greek and Chinese armillary spheres were large observational instruments, but the Renaissance armillary sphere was a table top instrument conceived to teach the basic of astronomy.

Armillary Sphere by Carlo Plato, Rome, 1588 Museum of the History of Science

In navigation the Renaissance saw the invention various variations of the backstaff, to determine solar altitudes.

Davis quadrant (backstaff), made in 1765 by Johannes Van Keulen. On display at the Musée national de la Marine in Paris. Source: Wikimedia Commons

Also new for the same purpose was the mariner’s astrolabe.

Mariner’s Astrolabe c. 1600 Source: Wikimedia Commons

Edmund Gunter (1581–1626) invented the Gunter scale or rule a multiple scale (logarithmic, trigonometrical) used to solve navigation calculation just using dividers.

Gunter scale front
Gunter scale back Source

William Oughtred (1574–1660) combined two Gunter scales to produce the slide rule.

New in surveying were the surveyor’s chain,

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

the plane table,

Surveying with plane table and surveyor’s chain

the theodolite

Theodolite 1590 Source:

and the circumferentor.

18th century circumferentor

All of which were of course also used in cartography. Another Renaissance innovation was sets of drawing instruments for the cartographical, navigational etc draughtsmen.

Drawing instruments Bartholomew Newsum, London c. 1570 Source

The biggest innovation in instruments in the Renaissance, and within its context one of the biggest instrument innovation in history, were of course the telescope and the microscope, the first scientific instruments that not only aided observations but increased human perception enabling researchers to perceive things that were previously hidden from sight. Here is a blog post over the complex story of the origins of the telescope and one over the unclear origins of the microscope.

The Renaissance can be viewed as the period when instrumental science began to come of age. 

[1] The information on Ptolemaios’ instruments and the diagrams are taken from Ptolemy’s Almagest, translated and annotated by G. J. Toomer, Princeton Paperbacks, 1998


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

Renaissance Science – XLVII

In a previous post we have seen how hydrostatic, an area of physics first developed by Archimedes in the third century CE, underwent a modernisation and development during the Renaissance. Today we are going to look at another area of physics examined by Archimedes, which was also revived, and developed during the Renaissance, statics. To give a modern definition:

Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. Wikipedia

In antiquity and the Middle Ages, the concept of force did not exist, so we here find the discipline developed around the concept of weight. Statics is one half of the discipline of mechanics from the ancient Greek μηχανική mēkhanikḗ, lit. “of machines” and in antiquity it is literally the discipline of the so-called simple machines: lever, wheel and axel, pulley, balance, inclined plane, wedge, and screw. 

Archimedes (c. 287–c. 212 BCE), whose work on the topic was his On the Equilibrium of Planes (Ancient Greek: Περὶ ἐπιπέδων ἱσορροπιῶν, Romanised: perí epipédōn isorropiôn) was not the first to tackle the subject. His work was preceded by a text known in Latin as the Questiones Mechanicae (Mechanical Problems), which in the Middle Ages was attributed to Aristotle (384­–322 BCE) but is now considered to actually be by one of his followers or by some to be based on the earlier work of the Pythagorean Archytas (c.420–350 BCE). There was also a On the Balance attributed, almost certainly falsely to Euclid (fl. 300 BCE), which won’t play a further role here. Later than Archimedes there was the Mechanica of Hero of Alexandria (c. 10–c. 70 CE), unknown in the phase of the Renaissance we shall be reviewing but discussed along with the work of Archimedes in Book VII of the Synagoge or Collection of Pappus (c. 290–c. 350 CE).

The two major texts are the pseudo-Aristotelian Questiones Mechanicae and Archimedes’ On the Equilibrium of Planes, which approach the topic very differently. The Questiones Mechanicae is a philosophical work, which derives everything from a first principle that all machines are reducible to circular motion. It gives an informal proof of the law of the lever without reference to the centre of gravity. The pseudo-Euclidian on the Balance contains a mathematical proof of the law of the lever, again without reference to the centre of gravity.

In Archimedes’ On the Equilibrium of Planes the centre of gravity plays a very prominent role. In the first volume Archimedes presents seven postulates and fifteen propositions to mathematically using the centre of gravity to mathematical demonstrate the law of the lever. The volume closes with demonstrations of the centres of gravity of the parallelogram, the triangle, and the trapezoid. Centres of gravity are a part of statics because they are the point from which, when a figure is suspended it remains in equilibrium, that is unmoving. In volume two of his text Archimedes presents ten propositions relating to the centres of gravity of parabolic sections. This is achieved by substituting rectangles of equal area, a process made possible by his work Quadrature of the Parabola (Greek: Τετραγωνισμὸς παραβολῆς).

Although already translated from Greek into Latin in the thirteenth century by William of Moerbeke (c. 1220 – c. 1286), On the Equilibrium of Planes remained largely unknown in medieval Europe. Thābit ibn Qurra (c.830 –901) had translated it into Arabic and he wrote two related works, his Kitab fi ‘l-qarastun (Book of the Steelyard)­–a steelyard is a single armed balance– and his Kitab fi sifat alwazn (Book on the Description of Weight) on the equal armed balance. 

Thābit ibn Qurra 

The pseudo-Aristotelian Questiones Mechanicae was well known in the Middle Ages and Jordanus de Nemore (fl. 13th century) developed a scholastic theory of statics in his science of weights (scientia de ponderibus) presented in three texts, the first Elementa super demonstrationem ponderum, which presents the conclusions of Thābit ibn Qurra’s text on the steelyard deriving them from seven axioms and nine propositions. This is the earliest of the three and the only one definitely ascribable to Jordanus. The two later texts are usually attributed to the school of. The second text Liber de ponderibus is a reworking of the Elementa super demonstrationem ponderum. The third De ratione ponderis is a corrected and expanded version of the Elementa. In his work he proves the law of the lever by the principle of work using virtual displacements. Using the same method, the De ratione ponderis also proves the conditions of equilibrium of unequal weights on planes inclined at different angles.

FIRST EDITION of Jordanus’s De ratione ponderis, published by Curtio Troiano from a manuscript copy owned by Tartaglia, who had died in 1557. 

The Questiones Mechanicae went through more than a dozen editions between the end of the fifteenth century and the beginning of the seventeenth. The first Greek edition was in the Aldine edition of the works of Aristotle published in Venice in 1497, which was often reprinted. There were various Latin translations published in Paris, Venice, Rome. The engineers Antonio Guarino and Vannoccio Biringuccio (c. 1480 – c. 1539) both produced Italian translations, published respectively in Moderna 1573, and Rome 1582. The Questiones Mechanicae were also known to the authors of the Renaissance Theatre of Machines books, such as Agostino Ramelli (1531–c. 1610) and there was a strong correlation between the theoretical works on machines such as the Questiones Mechanicae and the works of Jordanus de Nemore and the strong interest in projected new machine designs.  

When we turn to the Renaissance mathematici were meet many of the same names as by the Renaissance revival of hydrostatics. In 1546, Niccolò Tartaglia (c.1500 – 1557) published his Quesiti ed invention diverse (Various Questions and Inventions) in Venice, which referenced some of the contents of the Questiones Mechanicae and the works of Jordanus. As did his student Giambattista Benedetti (1530 – 1590) in his Diversarum speculationum mathematicarum et physicarum liber published in Turin in 1585. 

Tartaglia also published Moerbeke’s Latin translations of both books of Archimedes’ On the Equilibrium of Planes together with his Quadrature of the Parabola and Book I of On Floating Bodies in 1543. 

However, it was the so-called Urbino School, who truly reintroduced and began to modernise Archimedes’ work on statics. Federico Commandino (1509 – 1575) found the Moerbeke translations of Archimedes defective and produced new Latin translations of them as well as a new Latin translation of Pappus’ Synagoge containing parts of Hero’s Mechanica. Convinced that some of Archimedes’ proofs in On Floating Bodies were not well grounded he wrote and published his own Liber de centro gravitatis solidorum (Book on the Centres of Gravity of Solid Bodies) in 1565.

Commandino laid the foundations of the revival in Archimedean mathematical statics in the sixteenth century, but it was his student Guidobaldo dal Monte (1545 – 1607), who using Commandino’s new translations, both published and unpublished, who erected the structure.

Guidobaldo dal Monte Source: Wikimedia Commons

Dal Monte reconstructed the statics of Questiones Mechanicae using an Archimedean mathematical approach with postulates and propositions in his Mechanicorum Liber published in Pesaro in 1577. Under dal Monte’s supervision, the mathematician and explorer, Filippo Pigafetta (1533 – 1604) published an Italian translation Le mechniche in Venice in 1581, indicating the interest in dal Monte’s work. New editions of both were published in 1615 and a German translation appeared in 1629. Dal Monte rejected the earlier concept that all machines could be reduced to circular motion, concentrating in the first instance on the lever and then describing other machines in terms of the lever. He presents detailed analyses of the both the balance and pully systems. 

Source: Wikimedia Commons

It should be noted that for dal Monte the theoretical discipline of mechanics cannot be separated from the study and construction of real machines, in his Mechanicorum Liber, he wrote:

Mechanics can no longer be called mechanics when it is abstracted and separated from machines.

Although he thought that the theoretical study of mechanics should be kept separate from the actual construction of machines. The primary source for dal Monte’s approach is Pappus’ Synagoge, of which he had access to both a Greek manuscript and the manuscript of Commandino’s Latin translation, which Commandino had been unable to publish before his death. In 1588, dal Monte edited and published that Latin translation, bringing Pappus’s synopses of Hero’s Mechanica and Archimedes’ On the Equilibrium of Planes to public attention for the first time in print. In the same year he published his own In duos Archimedis Aequeponderantium Libros Paraphrasis scholiss illustrata, a paraphrase of On the Equilibrium of Planes, both books were published in Pesaro by Hieronymus Concordia, who had also published the Mechanicorum Liber.

 In duos Archimedis Aequeponderantium Libros Paraphrasis scholiss illustrata

A third member of the Urbino School, dal Monte’s student, the mathematician and historian of mathematics, Bernardino Baldi (1553–1617), referenced and amplified the works of Commandino and dal Monte on statics in his own writings, in particular his In mechanica Aristotelis problemata exercitationes published posthumously in 1621.

The man, who broke the connection in statics with the Middle Ages was the Netherland’s engineer and mathematician, Simon Stevin (1548–1620). Stevin had read the Questiones Mechanicae and was aware of the medieval work on statics, but we don’t know how, he had read the relevant works of Archimedes, and Commandino’s Liber de centro gravitas solidorum, but does not seem to have read Papus’ Synagoge, and so was not aware of  Hero’s Mechanica.

Source: Wikimedia Commons

In 1586 he published three books in one volume: De Beghinselen der Weegconst (The Principles of the Art of Weighing), De Weegdaet (The Practice of Weighing), and De Beghinselen des Waterwichts (The Principles of the Weight of Water).

Source: Wikimedia Commons

De Beghinselen der Weegconst consists of two books. Book I has two parts of which the first deals with vertical weights with the law of the balance as central result. The second part deals with oblique weights with the law of the inclined plane as central result.  Book II is devoted to centres of gravity taking Commandino’s work as its starting point. Stevin rejected both circular motion and virtual displacement, the key arguments of the medieval discussion of weights. Regarding the latter he argued that when discussing equilibrium, it was nonsense to start with a discussion of motion, which is what virtual displacement entailed.

Stevin’s proof of the law of the inclined plane involves his famous Clootcrans, wreath of weights:

He derived the condition for the balance of forces on inclined planes using a diagram with a “wreath” containing evenly spaced round masses resting on the planes of a triangular prism (see the illustration on the side). He concluded that the weights required were proportional to the lengths of the sides on which they rested assuming the third side was horizontal and that the effect of a weight was reduced in a similar manner. It’s implicit that the reduction factor is the height of the triangle divided by the side (the sine of the angle of the side with respect to the horizontal). The proof diagram of this concept is known as the “Epitaph of Stevinus”. Wikipedia

From this Stevin derives the parallelogram of forces well before its existence is acknowledge by the mathematician.

Simon Stevin’s illustration [47] of the vector character of a force that is due to the weight G of a mass on a hillside with inclination. As indicated, a force can be decomposed into components. One can add vectors such as D, E, and B as their sum componentwise along Cartesian horizontal and vertical axes or, alternatively, use the parallelogram rule for the D and E (arrows) to obtain B. Because Stevin’s plot is in a twodimensional plane, B can be decomposed into, for instance, the two vectors D and E Source

Although Stevin insisted on writing and publishing in Dutch, his work was translated into Latin by Willebrord Snell (1580–1626) and was well known to the French natural philosophers of the middle of the seventeenth century, who went on to develop the science of mechanics

The work on statics of the Urbino School was well known, widely read and highly influential. In particular dal Monte was Galileo’s first patron and his work influenced the young natural philosopher. In c. 1600 Galileo wrote a manuscript Le manchaniche, which was heavily influenced by dal Monte’s work, but which was first published posthumously. Guidobaldo dal Monte only dealt with statics, keeping it separate from dynamics. Galileo brought statics and dynamics together as mechanics in his Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences), published in 1638, combining the previous works of others with his own experiments and discoveries, opening acknowledging dal Monte’s influence.

Source: Wikimedia Commons

Galileo’s Discorsi was, together with other works such as those of Stevin and Beeckman, one of the foundation stones of modern mechanics.

Leave a comment

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


Due to the impact of Isaac Newton and the mathematicians grouped around him, people often have a false impression of the role that England played in the history of the mathematical sciences during the Early Modern Period. As I have noted in the past, during the late medieval period and on down into the seventeenth century, England in fact lagged seriously behind continental Europe in the development of the mathematical sciences both on an institutional level, principally universities, and in terms of individual mathematical practitioners outside of the universities. Leading mathematical practitioners, working in England in the early sixteenth century, such as Thomas Gemini (1510–1562) and Nicolas Kratzer (1486/7–1550) were in fact immigrants, from the Netherlands and Germany respectively.

In the second half of the century the demand for mathematical practitioners in the fields of astrology, astronomy, navigation, cartography, surveying, and matters military was continually growing and England began to produce some home grown talent and take the mathematical disciplines more seriously, although the two universities, Oxford and Cambridge still remained aloof relying on enthusiastic informal teachers, such as Thomas Allen (1542–1632) rather than instituting proper chairs for the study and teaching of mathematics.

Outside of the universities ardent fans of the mathematical disciplines began to establish the so-called English school of mathematics, writing books in English, giving tuition, creating instruments, and carrying out mathematical tasks. Leading this group were the Welsh man, Robert Recorde (c. 1512–1558), who I shall return to in a later post, John Dee (1527–c. 1608), who I have dealt with in several post in the past, one of which outlines the English School, other important early members being, Dee’s friend Leonard Digges, and his son Thomas Digges (c. 1446–1595), who both deserve posts of their own, and Thomas Hood (1556–1620) the first officially appointed lecturer for mathematics in England.  I shall return to give all these worthy gentlemen, and others, the attention they deserve but today I shall outline the life and mathematical career of John Blagrave (d. 1611) a member of the landed gentry, who gained a strong reputation as a mathematical practitioner and in particular as a designer of mathematical instruments, the antiquary Anthony à Wood (1632–1695), author of Athenae Oxonienses. An Exact History of All the Writers and Bishops, who Have Had Their Education in the … University of Oxford from the Year 1500 to the End of the Year 1690, described him as “the flower of mathematicians of his age.”

John Blagrave was the second son of another John Blagrave of Bullmarsh, a district of Reading, and his wife Anne, the daughter of Sir Anthony Hungerford of Down-Ampney, an English soldier, sheriff, and courtier during the reign of Henry VIII, John junior was born into wealth in the town of Reading in Berkshire probably sometime in the 1560s. He was educated at Reading School, an old established grammar school, before going up to St John’s College Oxford, where he apparently acquired his love of mathematics. This raises the question as to whether he was another student, who benefitted from the tutoring skills of Thomas Allen (1542–1632). He left the university without graduating, not unusually for the sons of aristocrats and the gentry. He settled down in Southcot Lodge in Reading, an estate that he had inherited from his father and devoted himself to his mathematical studies and the design of mathematical instruments. He also worked as a surveyor and was amongst the first to draw estate maps to scale.

Harpsden a small parish near Henley-on-Thames Survey by John Blagrave 1589 Source

There are five known surviving works by Blagrave and one map, as opposed to a survey, of which the earliest his, The mathematical ievvel, from1585, which lends its name to the title of this post, is the most famous. The full title of this work is really quite extraordinary:


Shewing the making, and most excellent vse of a singuler Instrument So called: in that it performeth with wonderfull dexteritie, whatsoever is to be done, either by Quadrant, Ship, Circle, Cylinder, Ring, Dyall, Horoscope, Astrolabe, Sphere, Globe, or any such like heretofore deuised: yea or by most Tables commonly extant: and that generally to all places from Pole to Pole. 

The vse of which Ievvel, is so aboundant and ample, that it leadeth any man practising thereon, the direct pathway (from the first steppe to the last) through the whole Artes of Astronomy, Cosmography, Geography, Topography, Nauigation, Longitudes of Regions, Dyalling, Sphericall triangles, Setting figures, and briefely of whatsoeuer concerneth the Globe or Sphere: with great and incredible speede, plainenesse, facillitie, and pleasure:

The most part newly founde out by the Author, Compiled and published for the furtherance, aswell of Gentlemen and others desirous or Speculariue knowledge, and priuate practise: as also for the furnishing of such worthy mindes, Nauigators,and traueylers,that pretend long voyages or new discoueries: By John Blagave of Reading Gentleman and well willer to the Mathematickes; Who hath cut all the prints or pictures of the whole worke with his owne hands. 1585•

Dig the spelling!
Title Page Source Note the title page illustration is an  armillary sphere and not the Mathematical Jewel

Blagrave’s Mathematical Jewel is in fact a universal astrolabe, and by no means the first but probably the most extensively described. The astrolabe is indeed a multifunctional instrument, al-Sufi (903–983) describes over a thousand different uses for it, and Chaucer (c. 1340s–1400) in what is considered to be the first English language description of the astrolabe and its function, a pamphlet written for a child, describes at least forty different functions. However, the normal astrolabe has one drawback, the flat plates, called tympans of climata, that sit in the mater and are engraved with the stereographic projection of a portion of the celestial sphere are limited in their use to a fairly narrow band of latitude, meaning that if one wishes to use it at a different latitude you need a different climata. Most astrolabes have a set of plates each engraved on both side for a different band of latitude. This problem led to the invention of the universal astrolabe.

Full-page figure of the rete of Blagrave’s Jewel (Peterborough A.8.13) For more illustration from The Mathematical Jewel go here

The earliest known universal astrolabes are attributed to Abū Isḥāq Ibrāhīm ibn Yaḥyā al-Naqqāsh al-Zarqālī al-Tujibi (1029-1100), known simply as al-Zarqālī and in Latin as Arzachel, an Arabic astronomer, astrologer, and instrument maker from Al-Andalus, and another contemporary Arabic astronomer, instrument maker from Al-Andalus, Alī ibn Khalaf: Abū al‐Ḥasan ibn Aḥmar al‐Ṣaydalānī or simply Alī ibn Khalaf, about whom very little is known. In the Biographical Encyclopedia of Astronomers (Springer Reference, 2007, pp. 34-35) Roser Puig has this to say about the two Andalusian instrument makers: 

ʿAlī ibn Khalaf is the author of a treatise on the use of the lámina universal (universal plate) preserved only in a Spanish translation included in the Libros del Saber de Astronomía (III, 11–132), compiled by the Spanish King Alfonso X. To our knowledge, the Arabic original is lost. ʿAlī ibn Khalaf is also credited with the construction of a universal instrument called al‐asṭurlāb al‐maʾmūnī in the year 1071, dedicated to al‐Maʾmūn, ruler of Toledo. 

The universal plate and the ṣafīḥa (the plate) of Zarqalī (devised in 1048) are the first “universal instruments” (i.e., for all latitudes) developed in Andalus. Both are based on the stereographic meridian projection of each hemisphere, superimposing the projection of a half of the celestial sphere from the vernal point (and turning it) on to the projection of the other half from the autumnal point. However, their specific characteristics make them different instruments.

Al-Zarqālī’s universal astrolabe was known as the Azafea in Arabic and as the Saphaea in Europe.

A copy of al-Zarqālī’s astrolabe Source: Wikimedia Commons

Much closer to Blagrave’s time, Gemma Frisius (1508–1555) wrote about a universal astrolabe, published as the Medici ac Mathematici de astrolabio catholico liber quo latissime patientis instrumenti multiplex usus explicatur, in 1556. Better known than Frisius’ universal instrument was that of his one-time Spanish, student Juan de Rojas y Samiento (fl. 1540-1550) published in his Commentariorum in Astrolabium libri sex in 1551.


Although he never really left his home town of Reading and his work was in English, Blagrave, like the other members of the English School of Mathematics, was well aware of the developments in continental Europe and he quotes the work of leading European mathematical practitioners in his Mathematical Jewel, such as the Tübingen professor of mathematics, Johannes Stöffler (1452–1531), who wrote a highly influential volume on the construction of astrolabes, his Elucidatio fabricae ususque astrolabii originally published in 1513, which went through 16 editions up to 1620

or the works of Gemma Frisius, who was possibly the most influential mathematical practitioner of the sixteenth century. Blagrave’s Mathematical Jewel was based on Gemma Frisius astrolabio catholico.

Blagrave’s Mathematical Jewel was obviously popular because Joseph Moxon (1627–1691), England first specialist mathematical publisher, cartographer, instrument, and globe maker republished it under the title:

The catholique planisphaer which Mr. Blagrave calleth the mathematical jewel briefly and plainly discribed in five books : the first shewing the making of the instrument, the rest shewing the manifold vse of it, 1. for representing several projections of the sphere, 2. for resolving all problemes of the sphere, astronomical, astrological, and geographical, 4. for making all sorts of dials both without doors and within upon any walls, cielings, or floores, be they never so irregular, where-so-ever the direct or reflected beams of the sun may come : all which are to be done by this instrument with wonderous ease and delight : a treatise very usefull for marriners and for all ingenious men who love the arts mathematical / by John Palmer … ; hereunto is added a brief description of the cros-staf and a catalogue of eclipses observed by the same I.P.

Engraved frontispiece to John Palmer (ed.), ‘The Catholique Planispaer, which Mr Blagrave calleth the Mathematical Jewel’ (London, Joseph Moxon, 1658); woman, wearing necklace, bracelet, jewels in her hair, and a veil, and seated at a table, on which are a design of a mathematical sphere, a compass, and an open book; top left, portrait of John Blagrave, wearing a ruff; top right, portrait of John Palmer; top centre, an angel with trumpets.
Engraving David Loggan Source: British Museum

John Palmer (1612-1679), who was apparently rector of Ecton and archdeacon of Northampton, is variously described as the author or the editor of the volume, which was first published in 1658 and went through sixteen editions up to 1973.

Following The Mathematical Jewel, Blagrave published four further books on scientific instruments that we know of: 

Baculum Familliare, Catholicon sive Generale. A Booke of the making and use of a Staffe, newly invented by the Author, called the Familiar Staffe (London, 1590)

Astrolabium uranicum generale, a necessary and pleasaunt solace and recreation for navigators … compyled by John Blagrave (London, 1596)

An apollogie confirmation explanation and addition to the Vranicall astrolabe (London, 1597)

None of these survive in large numbers.

Blagrave also manufactured sundials and his fourth instrument book is about this: 

The art of dyalling in two parts (London, 1609)


Here there are considerably more surviving copies and even a modern reprint by Theatrum Orbis Terrarum Ltd., Da Capo Press, Amsterdam, New York, 1968.

People who don’t think about it tend to regard books on dialling, that is the mathematics of the construction and installation of sundials, as somehow odd. However, in this day and age, when almost everybody walks around with a mobile phone in their pocket with a highly accurate digital clock, we tend to forget that, for most of human history, time was not so instantly accessible. In the Early Modern period, mechanical clocks were few and far between and mostly unreliable. For time, people relied on sundials, which were common and widespread. From the invention of printing with movable type around 1450 up to about 1700, books on dialling constituted the largest genre of mathematical books printed and published. Designing and constructing sundials was a central part of the profession of mathematical practitioners. 

As well as the books there is one extant map:

Noua orbis terrarum descriptio opti[c]e proiecta secundu[m]q[ue] peritissimos Anglie geographos multis ni [sic] locis castigatissima et preceteris ipsiq[ue] globo nauigationi faciliter applcanda [sic] per Ioannem Blagrauum gen[er]osum Readingensem mathesibus beneuolentem Beniamin Wright Anglus Londinensis cµlator anno Domini 1596 

This is described as:

Two engraved maps, the first terrestrial, the second celestial (“Astrolabium uranicum generale …”). Evidently intended to illustrate Blagrave’s book “Astrolabium uranicum generale” but are not found in any copy of the latter.
The original is in the Bodleian Library.

When he died in 1611, Blagrave was buried in the St Laurence Church in Reading with a suitably mathematical monument. 

Blagrave is depicted surrounded by allegorical mathematical figures, with five women each holding the five platonic solids and Blagrave (in the center) depicted holding a globe and a quadrant.
The monument was the work of the sculptor Gerard Christmas (1576–1634), who later in life was appointed carver to the navy. It is not known who produced the drawing of the monument. 
Modern reconstruction of the armillary sphere from the cover of The Mathematical Jewel created by David Harber a descendent of John Blagrave

Blagrave was a minor, but not insignificant, participant in the mathematical community in England in the late sixteenth century. His work displays the typical Renaissance active interest in the practical mathematical disciplines, astronomy, navigation, surveying, and dialling. He seems to have enjoyed a good reputation and his Mathematical Jewel appears to have found a wide readership.  


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

Renaissance science – XLVI

One area that is not usually counted among the sciences is cryptography, lying as it does, in this day and age, between, logic, mathematics, and informatics. In earlier times it is perhaps best viewed as a part of logic. Perhaps surprisingly, cryptography underwent a major development during the Renaissance provoked by an earlier development in the hands of Islamicate scholars.

Cryptography means literally hidden writing, coming from the Greek kryptos meaning hidden and graphiameaning write, express in written characters, so codes. We have very little evidence of the use of codes by the ancient Egyptians, Babylonians, or Greeks, although it can be assumed that they did so. It is known that the ancient Greeks sent secret messages by shaving the head of a slave, writing, or tattooing the message on their skull, and then waiting until the hair grew back. Not really encryption and anything but high-speed communication.

The most well-known system of encryption from antiquity is the Caesar cipher, used by Julius Caesar in his correspondence. This is a very simple substitution code in which each letter in the plain text is replaced by the letter so many places before or after it in the alphabet. Up till the Middle Ages, in Europe all codes were some form of simple substitution code.

The action of a Caesar cipher is to replace each plaintext letter with a different one a fixed number of places down the alphabet. The cipher illustrated here uses a left shift of three, so that (for example) each occurrence of E in the plaintext becomes B in the ciphertext: Wikipedia

The first significant work on cryptography was written the Arabic philologist Abu ‘Abd ar-Raḥmān al-Khalīl ibn Aḥmad ibn ‘Amr ibn Tammām al-Farāhīdī al-Azdī al-Yaḥmadī (718–786 CE) known as Al-Farāhīdī or Al-Khalīl. His book Kitab al-Muamma (Book of Cryptographic Messages), which has been lost, presents the use of permutations and combinations to list all possible Arabic words with and without vowel. 

Sculpture of al-Farahidi in Basra Source: Wikimedia Commons

Al-Khalīl’s book influenced the work on cryptography by the Arabic polymath Abū Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (c. 801–873), whose theory of radiations played a significant role in the history of optics. In his book Risāla fī Istikhrāj al-Kutub al-Mu’ammāh (On Extracting Obscured Correspondence), al-Kindī introduced the concept of frequency analysis, which made all simple substitute codes accessible to solution. He wrote: 

One way to solve an encrypted message, if we know its language, is to find a different plaintext of the same language long enough to fill one sheet or so, and then we count the occurrences of each letter. We call the most frequently occurring letter the “first”, the next most occurring letter the “second”, the following most occurring letter the “third”, and so on, until we account for all the different letters in the plaintext sample. Then we look at the cipher text we want to solve and we also classify its symbols. We find the most occurring symbol and change it to the form of the “first” letter of the plaintext sample, the next most common symbol is changed to the form of the “second” letter, and the following most common symbol is changed to the form of the “third” letter, and so on, until we account for all symbols of the cryptogram we want to solve.

The first page of al-Kindi’s manuscript “On Deciphering Cryptographic Messages”, containing the oldest known description of cryptanalysis by frequency analysis. Source: Wikimedia Commons

Ai-Kindī’s explication of frequency analysis meant that a new more complex approach to encryption was necessary, and this was first delivered by the Renaissance polymath Leon Battista Alberti (1404–1472), whom we have already met in this series in the episodes on linear perspective and architecture. Alberti is one of those people, who deserves to be much better known than he is. 

Leon Battista Alberti Source: Wikimedia Commons

Alberti published the earliest known description of a polyalphabetic cipher in his De componendis cifris in 1467.

De componendis cifris frontispiece Source: Wikimedia Commons

In a polyalphabetic cipher not one but several substitution alphabets are used for different sections of the text, changing at random intervals, with a signal in the text to indicate for the reader deciphering the code of the alphabetical change. A polyalphabetic cipher is, in theory, not susceptible to frequency analysis; this, of course, is only true if one doesn’t know the method. In order to implement his method Alberti invented his cipher disc, which he explained in a letter was inspired by the recent invention of the movable type printing press. 

Alberti Cipher disc Source: Wikimedia Commons

How the whole thing is supposed to function is described by Alberti in his De componendis cifris

Chapter XIV. I will first describe the movable index. Suppose that we agreed to use the letter k as an index letter in the movable disk. At the moment of writing I will position the two disks of the formula as I wish, for example juxtaposing the index letter to capital B, with all other small letters corresponding to the capital letters above them. When writing to you, I will first write a capital B that corresponds to the index k in the formula. This means that if you want to read my message you must use the identical formula you have with you, turning the movable disk until the letter B corresponds to the index k. Thus all small letters in the ciphertext will receive the meaning and sound of those above them in the stationary disk. When I have written three or four words I will change the position of the index in our formula, turning the disk until, say, the index k is under capital R. Then I will write a capital R in my message and from this point onward the small k will no longer mean B but R, and the letters that follow in the text, will receive new meanings from the capital letters above them in the stationary disk. When you read the message you have received, you will be advised by the capital letter, which you know is only used as a signal, that from this moment the position of the movable disk and of the index has been changed. Hence, you will also place the index under that capital letter, and in this way you will be able to read and understand the text very easily. The four letters in the movable disk facing the four numbered cells of the outer ring will not have, so to speak, any meaning by themselves and may be inserted as nulls within the text. However, if used in groups or repeated, they will be of great advantage, as I will explain later on.

Chapter XV. We can also choose the index letter among the capital letters and agree between us which of them will be the index. Let us suppose we chose the letter B as an index. The first letter to appear in the message will be a small one at will, say q. Hence, turning the movable disk in the formula you will place this letter under the capital B that serves as an index. It follows that q will take the sound and meaning of B. For the other letters we will continue writing in the manner described earlier for the movable index. When it is necessary to change the set up of the disks in the formula, then I will insert one, and no more, of the numeral letters into the message, that is to say one of the letters of the small disk facing the numbers which corresponds to, let’s say, 3 or 4, etc. Turning the movable disk I will juxtapose this letter to the agreed upon index B and, successively, as required by the logic of writing, I will continue giving the value of the capitals to the small letters. To further confuse the scrutinizers you can also agree with your correspondent that the capital letters intermingled in the message have the function of nulls and must be disregarded, or you may resort to similar conventions, which are not worth recalling. Thus changing the position of the index by rotating the movable disk, one will be able to express the phonetic and semantic value of each capital letter by means of twenty-four different alphabetic characters, whereas each small letter can correspond to any capital letter or to any of the four numbers in the alphabet of the stationary disk. Now I come to the convenient use of the numbers, which is admirable.

I leave it to the reader to decipher Alberti’s instructions.

Alberti was not the only Renaissance scholar to suggest the adoption of a polyalphabetic cipher. The German, Benedictine monk, Johannes Trithemius (1462–1516), born Johann Heidenberg, was Abbot of the Abbey of Sponheim and from 1506 of the St, James’ Abbey in Würzburg.

Tomb relief of Johannes Trithemius by Tilman Riemenschneider Source: Wikimedia Commons

Trithemius was a polymath active as a lexicographer, chronicler, cryptographer, but above all he is known as an occultist. As I have noted in an earlier episode in this series the occult sciences played a significant part in Renaissance thought. Trithemius is considered to have had a major influence on both Paracelsus (c. 1493–1541) and Heinrich Cornelius Agrippa von Nettesheim (1486–1535).

Trithemius’ most famous work was his Steganographia (written c. 1499 but first published in 1606), which was initially thought to be about magic and was placed on the Index Librorum Prohibitorum (List of Prohibited Books) by the Catholic Church in 1609. In fact, the book was written in code and already in 1606, the first two volumes were shown to be about steganography (a word that Trithemius coined) and cryptography. Steganography is: 

The practice of hiding messages, so that the presence of the message itself is hidden, often by writing them in places where they may not be found until someone finds the secret message in whatever is being used to hide it. (def. Wiktionary). 

The third volume was, thought to be really about magic but has comparatively recently also shown to be about cryptography.

Source: Wikimedia Commons

Trithemius also wrote his Polygraphiae libri sex (Six books of polygraphia), the first printed book on cryptography, a further text on steganography, which was published posthumously in 1518.

Source: Wikimedia Commons

This work contains a progressive key polyalphabetic cipher now known as the Trithemius cipher. 

Unlike Alberti’s cipher, which switched alphabets at random intervals, Trithemius switched alphabets for each letter of the message. He started with a tabula recta, a square with 26 letters in it (although Trithemius, writing in Latin, used 24 letters). Each alphabet was shifted one letter to the left from the one above it, and started again with A after reaching Z (see table). Source: Wikimedia Commons

Our third Renaissance cryptographer was Blaise de Vigenère (1523–1596) a French, diplomat, cryptographer, translator, and alchemist.

Blaise de Vigenère Source: Wikimedia Commons

Although, he created a polyalphabetic cipher, the one that bears his name was actually first described by Giovan Battista Bellaso (1505–?) an Italian cryptographer in his La Cifra del Sig. Giovan Battista Belaso published in 1553. Bellaso went on to publish a second book, Novi et singolari modi di cifrare, in 1555 and a third one Il vero modo di scrivere in cifra, in 1565. Vigenère published his polyalphabetic cipher, first in 1586, in his Traicté des Chiffres ou Secrètes Manières d’Escrire. 1586. Both men’s ciphers were based on a so-called auto key but differed in detail. 

Source: Wikimedia Commons

An autokey cipher (also known as the autoclave cipher) is a cipher that incorporates the message (the plaintext) into the key. The key is generated from the message in some automated fashion, sometimes by selecting certain letters from the text or, more commonly, by adding a short primer key to the front of the message. (Def. Wikipedia). 

The Vigenère cipher was thought to be unbreakable and in fact Charles Lutwidge Dodgson (better known as Lewis Carroll), a very competent logician, said that it was unbreakable in 1868, unaware that Charles Babbage had already broken it earlier but had not published his results. The first to publish a general system to solve polyalphabetic ciphers, including the Vigenère cipher, was the German soldier, cryptographer, and archaeologist, Friedrich Wilhelm Kasiski (1805–1881) in his Die Geheimschriften und die Dechiffrir-Kunst (Secret Writing and the Art of Deciphering) in 1863, a publication that, at the time, went largely unnoticed. It was first in the nineteenth century that the art of cryptography evolved past the innovations of the Renaissance cryptographers, Alberti, Trithemius, Vigenère, and Bellaso. 


Filed under Renaissance Science

Renaissance science – XLV

70.8% of the earth’s surface is covered by the world ocean; we normally divide it up–Atlantic Ocean, Pacific Ocean, Indian Ocean, etc.– but they are all interconnected in one giant water mass.

The world ocean Source: Wikimedia Commons

Only 29.2% of the surface is land but, on that land, there are many enclosed seas, lakes, ponds, rivers, and streams so there is even more water. The human body is about 60% water, and humans are sometimes referred to as a water-based life form. The statistics are variable, but a healthy human can exist between one and two months without food but only two to four days without water. Brought to a simple formular, water is life.

When humans first began to settle, they did so on or near sources of water–lake shores, streams, rivers, natural springs. Where there was no obvious water supply people began to dig wells, there are wells dating back to 6500 BCE. As settlements grew the problem of water supply and sewage disposal became important and the profession of water manager or hydraulic engineer came into existence. Channelling of fresh water and sewage disposal, recycling of wastewater etc. Initial all of this was powered by gravity but over time other systems of moving water, such as the bucket water wheel or noria were developed for lifting water from one channel into another, appearing in Egypt around the fourth century BCE.

Close-up of the Noria do Mouchão Portugal Source: Wikimedia Commons

Probably the most spectacular surviving evidence of the water management in antiquity are the massive aqueducts built by Roman engineers to bring an adequate supply of drinking water to the Roman settlements. Alone the city of Rome had eleven aqueducts built between 312 BCE and 226 CE, the shortest of which the Aqua Appia from 312 BCE was 16.5 km long with a capacity of 73,000 m3 per day and the longest the Aqua Anio Novus from 52 CE was 87 km long with a capacity of 189,000 m3 per day. The Aqua Alexandrina from 226 CE was only 22 km long but had a capacity of 120,00 to 320,000 m3 per day.

Panorama view of the Roman Aqueduct of Segovia in 2014 Built first century CE originally 17 kilometres long Source Wikimedia Commons

The simplest water clock or clepsydra, a container with a hole in the bottom where the water was driven out by the force of gravity dates back to at least the sixteenth century BCE.

A reconstruction of the water clock used in ancient Greece (Museum of Ancient Agora/Athens) Figure 5: Water Clock/Clepsydra Source

It evolved over the centuries with complex feedback mechanism to keep the water level and thus the flow constant. Water clocks reach an extraordinary level of sophistication as illustrated by the Astronomical Clock Tower of Su Song (1020–1101 CE) in China

The original diagram of Su’s book showing the inner workings of his clocktower Source: Wikimedia Commons

and the Elephant Clock invented by the Islamic engineer al-Jazari (1136-1206). Al-Jazari invented many water powered devices.  

Al-Jazari’s elephant water clock (1206) Source: Wikimedia Commons

Much earlier the Greek engineer Hero of Alexandria (c. 10–c. 70 CE), as well as numerous devices driven by wind and steam, invented a stand-alone fountain that operates under self-contained hydro-static energy, known as Heron’s Fountain. 

Diagram of a functioning Heron’s fountain Source: Wikimedia Commons

All of the above is out of the realm of engineers. Another engineer Archimedes (c. 287–c. 212 BCE), is the subject of possibly the most well-known story in the history of science, one needs only utter the Greek word εὕρηκα (Eureka) to invoke visions of crowns of gold, bathtubs, and naked bearded man running through the streets shouting the word. In fact, you won’t find this story anywhere in Archimedes not insubstantial writings. The source of the story is in De architectura by Vitruvius (C. 80-70–after c. 15 BCE), so two hundred years after Archimedes lived. You can read the original in translation below:

Vitruvius “Ten Books on Architecture”, Ed. Ingrid D. Rowland & Thomas Noble Howard, (CUP, 1999) p. 108

However, Archimedes did write a book On Floating Bodies, which now only exists partially in Greek but in full in a medieval Latin translation. This book is the earliest known work of the branch of physics known as hydrostatics. It contains clear statement of two fundamental principles of hydrostatics, Firstly Archimedes’ principle:

Any body wholly or partially immersed in a fluid experiences an upward force (buoyancy) equal to the weight of the fluid displaced

Secondly the principle of floatation:

Any floating object displaces its own weight of fluid.

As well these two fundamental principles, he also discovered that a submerged object displaces a volume of water equal to its own volume. This is the discovery that led to the legendary of mythical Eureka incident. A crown of pure gold would have a different displacement volume to one of a gold and silver amalgam. The bath story was, as we will see later, highly implausible because it would be very, very difficult to measure the difference in the displaced volumes of water of the two crowns.

Whilst water management continued to develop through out the Middle Ages, with the invention of every better water mills etc., In the Renaissance the profession hydraulic engineer saw developments in two areas. Firstly, the increase in wealth and the development of residences saw the emergence of the Renaissance Garden. Large ornamental gardens the usually featured extensive and often spectacular water features.

Garden of Villa d’Este Tivoli (1550–1572) Source: Wikimedia Commons

The Renaissance mathematici employed by potentates and aristocrats were often expected to serve as hydraulic engineers alongside their other functions as instrument makers, astrologers etc. Secondly the major increase in mining for precious and semi-precious metals meant ever deeper mines, which brought with it the problem of pumping water out of the mines.

Archimedes’ On Floating Bodies was translated into Latin by William of Moerbeke (c. 1215–1286) in the thirteenth century and no complete Greek manuscript is known to exist. This translation was edited by Nicolò Tartaglia Fontana (c. 1506–1557) and published in print along with other works by Archimedes by Venturino Ruffinelli in Venice in 1543, as Opera Archimedis Syracvsani philosophi et mathematici ingeniosissimi

Opera Archimedis Syracvsani philosophi et mathematici ingeniosissimi1543 Source

The Nürnberger theologian and humanist Thomas Venatorius (1488–1551) edited the first printed edition of the Greek manuscripts of Archimedes, in a bilingual Greek/Latin edition, which was published in Basel by Johann Herwagen in 1544. The Greek manuscript had been brought to Nürnberg by the humanist scholar, Willibald Pirckheimer (1470–1530) from Rome and the Latin translation by Jacopo da Cremona (fl. 1450) was from the manuscript collection of Regiomontanus (1436-1476).


Venatorius claimed, in the foreword to the Archimedes edition to have studied mathematics under Johannes Schöner (1577–1547) but if then as a mature student in Nürnberg and not as a schoolboy. 

A reconstruction of On Floating Bodies was published by Federico Commandino (1509–1575) in Bologna in 1565. 


Tartaglia, who also produced an Italian edition of On floating Bodies, was the first Renaissance scholar to address Archimedes work on hydrostatics. It did not play a major role in his own work, but he was the first to draw attention to the relationship between the laws of fall and Archimedes’ thoughts on flotation. Tartaglia’s work was read by his one-time student, Giambattista Benedetti ((1530–1590), Galileo (1564–1642), and Simon Stevin (1548–1620), amongst other, and was almost certainly the introduction to Archimedes’ text for all three of them. 

Benedetti replaced Aristotle’s concepts of fall in a fluid directly with Archimedes’ ideas in his work on the laws of fall, equating resistance in the fluid with Archimedes’ upward force or buoyancy. This led him to his anticipations of Galileo’s work on the laws of fall. 

Moving onto Simon Stevin, who wrote a major work on hydrostatics, his De Beghinselen des Waterwichts (Principles on the weight of water) in 1586 and a never completed practical Preamble to the Practice of Hydrostatics.


One of Benedetti’s major works, Demonstratio propotionummotuum localiumcontra Aristotilem et omnes philosphos (1554) had been plagiarised by the French mathematician Jean Taisnier (1598–1562) Opusculum perpetua memoria dignissimum, de natura magnetis et ejus effectibus, Item de motu continuo (1562) and it was this that Stevin read rather than Benedetti’s original. Taisnier’s plagiarism was also translated into English by Richard Eden (c. 1520–1576) an alchemist and promotor of overseas exploration. Stevin a practical engineer ignored or rejected the equivalence between the laws of fall and the principle of buoyancy, concentrating instead on the relationship between flotation and the design of ship’s hulls. His major contribution was the so-called hydrostatic paradox often falsely attributed to Pascal. This states that the downward pressure exerted by a fluid in a vessel is only dependent on its depth and not on the width or length of the vessel. 

Of the three, Galileo is most well-known for his adherence to Archimedes. He clearly stated that in his natural philosophy he had replaced Aristotle with Archimedes as his ancient Greek authority, and this can be seen in his work. His very first work was an essay La Bilancetta (The Little Balance) written in 1586, but first published posthumously in 1644, which he presented to both Guidobaldo del Monte (1545–1607) and Christoph Clavius (1538–1612), both leading mathematical authorities, in the hope of winning their patronage. He was successful in both cases.

Galileo Galilei, La bilancetta, in Opere di Galileo Galilei (facsimile) Source:

Realising, that the famous bathtub story couldn’t actually have worked, Galileo tried to recreate how Archimedes might actually have done it. He devised a very accurate hydrostatic balance that would have made the discovery feasible. 

Later in life, when firmly established as court philosopher in Florence, Galileo was called upon by Cosimo II Medici to debate the principles of flotation with the Aristotelian physicist Lodovico delle Columbe (c. 1565–after 1623), as after dinner entertainment. As I have written before one of Galileo’s principal functions at the court in Florence was to provide such entertainment as a sort of intellectual court jester. Galileo was judged to have carried the day and his contribution to the debate was published in Italian, as Discorso intorno alle cose che stanno in su l’acqua, o che in quella si muovono, (Discourse on Bodies that Stay Atop Water, or Move in It) in 1612.


As was his wont, Galileo mocked his Aristotelian opponent is his brief essay, which brought him the enmity of the Northern Italian Aristotelians. Although Galileo’s approach to the topic was Archimedean, he couldn’t explain everything and not all that he said was correct. However, this little work enjoyed a widespread reception and was influential.

Our last Renaissance contribution to hydrostatics was made by Evangelista Torricelli (1608–1647), a student of Benedetto Castelli (1578–1643) himself a student of Galileo, and like Stevin’s work it came from the practical world rather than the world of science.

Evangelista Torricelli by Lorenzo Lippi Source: Wikimedia Commons

Torricelli was looking for a solution as to why a suction pump could only raise water to a hight of ten metres, as recounted in Galileo’s Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) (1638), a major problem for the expanding deep mining industry, which needed to pump water out of its mines. Torricelli in his investigations invented the Torricellian tube, later called the barometer, with which he demonstrated that there was a limit to the height of a column of liquid that the weight of the atmosphere, or air pressure, could support.

Torricelli’s experiment Source: Wikimedia Commons

He also incidentally demonstrated the existence of a vacuum, something Aristotle said could not exist. 

Torricelli’s work marks the transition from Renaissance science to what is called modern science. Building on the work of Benedetti, Stevin, Galileo, and Torricelli, Blaise Pascal (1623–1662) laid some of the modern foundation of hydrodynamics and hydrostatics, having a unit for pressure named after him and being sometimes falsely credited with discoveries that were actually made in the earlier phase by his predecessors. 

Painting of Pascal made by François II Quesnel for Gérard Edelinck in 1691. Source:Wikimedia Commons


Filed under History of Physics, History of Technology, Renaissance Science

Renaissance science – XLIV

This blog post is a modified version of two blog posts from my The emergence of modern astronomy–a complex mosaic series and yes it involves self plagiarism. I wrote it, rather than simply linking, because the content also belongs in this blog post series, which I wish to be complete and autonomous

Short popular presentations of the history of the origins of modern physics usually consist of three sections. In ancient Greece, Aristotle got almost everything wrong. In the Middle Ages, people clung religiously to Aristotle’s wrong theories. Then came Galileo and everything was light! A somewhat, sarcastic exaggeration but pretty close to the truth of what people like to believe and believe is the right verb because it bears little relation to what actually happened. You will note in my little parody that there is no mention of the Renaissance. This is because it just gets subsumed into the amorphous Middle Ages in this version of history. Galileo is always presented as a sort of messiah single handily casting a shining light into the dark reaches of medieval Aristotelianism and bringing forth, in a sort of virgin, birth modern physics. 

In reality, whilst the considerations of what became modern physics are based on the concepts of Aristotle, there were major developments between the fourth century BCE and the early seventeenth century, especially during the Renaissance, changes of which Galileo was well aware and on which he built his, not always correct, contributions. In what follows I’m going to briefly outline the evolution of the theories of motion from Aristotle down to the seventeenth century; the theories of motion that then emerged being the bedrock on which Isaac Newton constructed his physics.

When talking of the history of physics it is important to note that what Aristotle meant with the term, one that he coined, is very different to the modern meaning, one that only began to emerge in the eighteenth century. For Aristotle his ta physika literally means “the natural things” and his physics means the study of all of nature, a study that is also non-mathematical. For Aristotle the objects of mathematics do not describe anything real, so mathematics can not be used to describe the real world. He does allow the use of mathematics in the so-called mixed sciences–astronomy, optics, harmonics (music or more accurately acoustics)–all of which we would include, at least in part, in a general definition of physics but for Aristotle were not part of his ta physika.

Central to Aristotle’s theory of nature was the establishment of the general principle of change that governs all natural bodies. For Aristotle motion is quite simply change of place. Aristotle complicates this simple picture in that he differentiates between celestial and terrestrial motion and between natural and violent or unnatural motion. 

Simplest is his description of natural celestial motion, which is uniform circular motion, a concept that he inherited from Empedocles (c. 494 – c. 434 BCE, fl. 444–443 BCE) via his teacher Plato (c.425 BCE – 348/347 BCE). There is no violent or unnatural celestial motion. Aristotle’s theory of celestial motion is cosmology not astronomy and therefore not mathematical. The attempts to describe that motion mathematically are astronomical and thus not part of Aristotle’s physics.

Unlike celestial motion, both natural and violent terrestrial motion exist. For Aristotle, natural terrestrial motion is always perpendicular to the Earth’s surface and is the result of the four elements-earth, water, air, fire (another concept inherited from Empedocles)–striving to return to the natural places. So, the light elements–air and fire–travel upwards away from the Earth’s surface and the heavy elements–water and earth–fall downwards towards the Earth’s surface. In Latin this indication of heaviness is termed gravitas, object consisting principally of earth and/or water have gravitas and so they fall downwards.

Violent terrestrial motion is any motion that is not natural motion and must be brought about by the application of force. Simplified for something to move, other than falling, it has to be pulled or pushed. For Aristotle, the only contactless motion is the fall of water and earth due to gravitas and the rise of air and fire to their natural place in the world, all other motion requires contact between the object being moved and whoever or whatever is doing the moving. As with much of Aristotle’s philosophy these concepts are based on empirical observation of the real world and are not so wrong, as they are often painted. Aristotle does not have a quantitative law of fall, but he asserts that objects fall at speed proportion to their weight and inversely proportional to the density of the fluid they are falling through. This is often contrasted with Galileo’s “correct” law of fall that all objects fall at the same speed, but Galileo’s law is only valid for a vacuum. 

Source: Wikimedia Commons

Aristotle has major problems with projectile motion. If you throw a ball or shoot and arrow, then according to Aristotle, as soon as the ball leaves your hand or the arrow the bowstring then it should immediately stop moving and fall to the ground, which it very obviously doesn’t. He got round the problem by claiming that the projectile parts the air as it flies, which then rushed round to the back of the object to prevent a vacuum forming and pushes the projectile forwards. An explanation that people found difficult to swallow and I suspect that even Aristotle found it less than satisfactory.

It is exactly here in his theory of projectile motion that Aristotle’s theories of terrestrial motion were first challenged and that already in the sixth century CE by the Alexandrian philosopher John Philoponus (c. 490–c. 570). Philoponus broke with the Aristotelian-Neoplatonic tradition of his own times and subjected Aristotle to severe criticism, writing commentaries on many of Aristotle’s major works and most importantly on Aristotle’s Physics. Whilst in general accepting Aristotle’s concept that for violent movement to take place a force must be applied but supplemented it by writing that in the case of projectiles, they acquired a motive power from the source providing the initial projection, which dissipated over time. 

Philoponus didn’t restrict his concept to projectile motion, he also thought that the planets in their orbits had acquired the same motive when set in motion at the creation. Philoponus also rejects Aristotle’s theory of fall. It is obvious that one stone twice as heavy as another falls twice as fast. He apparently backed this up by doing empirical experiments. Showing that stones of differing weights fall at almost the same speed.

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

Because he fell into disgrace with the Church because of his religious writings, Philoponus’ Aristotle commentaries were little read in the West in the Early Middle Ages. However, they were read by Islamic scholars, such as Ibn Sina (c. 980–1037) (Avicenne), al-Baghdādī (c. 1080–1164), and al-Bitruji (died c. 1204), all adopted and modified Philoponus’ theory of projectile motion.

Avicenne Portrait (1271) Source: Wikimedia Commons

Whether directly from medieval manuscripts or through transmission by the translation movement, Philoponus’ work was known in Europe in the High Middle Ages. Amongst others, Thomas Aquinas (1225–1274) referenced but rejected it. It was the French scholastic philosopher, Jean Buridan (c. 1301–c. 1360), who adopted it and gave it both its final form and its name, impetus. Unlike Philoponus and his Islamic supporters, who thought that the implied motive force simply dissipated spontaneously over time, Buridan argued that the projectile was slowed and eventually brought to a halt by air resistance and gravity opposing its impetus. In Buridan’s more sophisticated version of Philoponus’ theory one can already see the seeds of the theory of inertia. 

Jean Buridan Source

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

Buridan, like Philoponus and al-Bitruji, thought that impetus most the motive force of the planets, there being in the celestial sphere no air resistance of gravity to weaken it. 

Impetus was established as the accepted theory of projectile motion at the beginning of the sixteenth century and it was the theory that Niccolò Fontana (c. 1500–1557), better known by his nickname, Tartaglia, used in his mathematical analysis of ballistics, his Nova scientia (1537), the first such book on the topic.

Tartaglia Source: Wikimedia Commons

Here we have a classic example of Renaissance science, the application of the scientific approach to an artisanal practice, gunnery. Because he was using the theory of impetus and not the theory of inertia, Tartaglia’s theories of the flight path of cannon balls is wrong, but his book was widely read and highly influential, Galileo owned a heavily annotated copy. 

Various projectile trajectories from Tartaglia’s Nova Scientia Source: Wikimedia Commons

Philoponus had also criticised Aristotle’s theory of fall and he was by no means the last medieval scholar to do so. The so-called Oxford Calculatores at Merton College, Thomas Bradwardine (c. 1300–1349), William of Heytesbury (c. 1313–c. 1372), Richard Swineshead (fl. c. 1340–1354) and John Dumbleton (c. 1310–c. 1349)–studied mechanics distinguishing between kinematics and dynamics, emphasising the former and investigating instantaneous velocity.

Merton College in 1865 Source: Wikimedia Commons

They were the first to formulate the mean speed theorem, an achievement usually accredited to Galileo. The mean speed theorem states that a uniformly accelerated body, starting from rest, travels the same distance as a body with uniform speed, whose speed in half the final velocity of the accelerated body. The theory lies at the heart of the laws of fall.

Nicole Oresme (c. 1320–1382), a Parisian colleague of Jean Buridan, in his own work on the concept of motion produced a graphical representation of the mean speed theorem,

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

as did Giovanni di Casali (c. 1320–after 1374), a Franciscan friar, who encountered the mathematical physics of the Oxford Calculatores, whilst working as a lecturer at Cambridge University around 1340. He wrote a treatise on his ideas on motion in 1346, which was published as De velocitate motus alterationis (On the Velocity of the Motion of Alteration) in Venice in 1351. His work on mathematical physics influenced scholars at the University of Padua and possibly later Galileo.

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

The most important work on the theories of motion by a Renaissance scholar is that of Giambattista Benedetti (1530–1590), a one-time pupil of Tartaglia. Addressing the law of fall, Benedetti in his Demonstratio proportionum motuum localium (1554) he argued that speed is dependent not on weight but specific gravity and that two objects of the same material, but different weights would fall at the same speed.


Benedetti brought an early version of the thought experiment, usually attributed to Galileo, of viewing two bodies falling separately or conjoined, in his case by a cord.  Galileo considered a roof tile falling complete and then broken into two. Benedetti’s work was all done within the theory of impetus. Galileo’s first work on the topic, the unpublished De motu, written whilst he was still at the University of Pisa, also assumes the impetus theory and bears a strong resemblance to Benedetti’s work, which raises the question to what extent Galileo was acquainted with it. The opinions of the historians are divided on the topic.

Whereas Galileo almost certainly never threw balls off the Tower of Pisa, Simon Stevin (1548–1620), the mathematical engineer living and working in the newly established United Provinces, actually dropped lead balls of different weights from the thirty-foot-high church tower in Delft and determined empirically that they fell at the same speed, arriving at the ground at the same time. Stevin’s work was translated into both French and Latin and was widely read and highly influential in the France of Descartes, Mersenne, Gassendi et al. 

Anonymous Dutch painter / engraver, 17th century. Collection Leiden University, Icones Leidenses 40. Source: Wikimedia Commons

There is a significant list of Renaissance scholars who reached and published the same conclusion, the Dominican priest Domingo de Soto (1494–1560) in Spain, Gerolamo Cardano (1501–1576), Benedetto Varchi (c. 1502–1565), Giuseppe Moletti (1531–1588) and Jacopo Mazzoni (1548–1598) in Italy. Girolamo Borro (1512–1592) one of Galileo’s teachers in Pisa, actually carried out empirical experiments to test Aristotle’s laws of fall, throwing objects of different material and the same weights out of a high window. 

As can be seen from the above, when Galileo started working on the problems of motion towards the end of the sixteenth century, when he was still very much a Renaissance scientist, he was building on a strong tradition of criticisms and corrections to the Aristotelian theories stretching back to the early Middle Ages but also particularly vibrant in the sixteenth century. As already noted, Galileo earliest unpublished work, De Motu, was firmly entrenched in that tradition. 

Of course, Galileo would go on to make significant advances in both projectile motion and the laws of fall but in the first he was definitely strongly influenced by another Renaissance mathematician, the Urbino aristocrat, Guidobaldo del Monte (1545–1607). Del Monte was one of the influential figures the young Galileo turned too for assistance at the beginning of his career. Impressed by the young Tuscan, Del Monte helped him to his appointment as professor for mathematics at the University of Pisa and again when he moved to the University of Padua. 

Guidobaldo del Monte. Source:Wikimedia Commons

Galileo’s major contribution to the theory of projectile motion was the law of the parabola i.e., that the path of a projectile traces out a parabola. Galileo presents this in his Discorsi in 1638. However, it can be found in a notebook of del Monte’s from 1601, with a description of the proofs for this that are identical to those published by Galileo thirty-seven years later. The charitable explanation is that the two of them made this discovery together during one of Galileo’s visits to del Monte’s estate. The less charitable one is that Galileo borrowed del Monte’s results without acknowledgement, not the only time he would do such a thing.

The English polymath, Thomas Harriot (c.1560–1621) discovered the parabola law independently of del Monte/Galileo, but as with everything else didn’t publish his discovery. Bonaventura Cavalieri (1598–1647) did publish the parabola law, and in fact did so before Galileo, which brought an accusation of plagiarism from Galileo. Whether he borrowed the law from Galileo or discovered it independently is not known.

Bonaventura Cavalieri Source:Wikimedia Commons

On the laws of fall, Galileo carried out his famous series of experiments using an inclined plane to verify what many others had confirmed during the preceding century. Here the problem is that his inclined plane would not give the level of accuracy of the results that he published. This led the historian of science and Galileo expert, Alexander Koryé (1892–1964), to hypothesise that the inclined plane was a purely hypothetical experiment that Galileo never actually carried out. The modern consensus is that Galileo did in fact carry out his experiments but massaged his results to make them fit the required theoretical values. As we have seen the mean speed theorem was already well established, as was the principle that objects of different weights fall at the same speed.

Galileo’s supposed other great contribution was the law of inertia. Moving from impetus to inertia was the major breakthrough in concepts of motion in the history of physics as it turns the whole problem on its head. Whereas Aristotle asked what moves things, the principle of inertia asks what stops them moving. Aristotle takes still stand as the natural state of objects that has to be changed, inertia takes motion of as the natural state of objects that has to be changed. 

It is interesting to note that the supposedly modern scientist Galileo was in this concept trapped in the Greek paradigm of uniform circular motion being natural motion. Because of this, Galileo only defined inertia for circular motion:

“…all external impediments removed, a heavy body on a spherical surface concentric with the earth will maintain itself in that state in which it has been; if placed in a movement towards the west (for example), it will maintain itself in that movement.”[1]

Galileo’s addiction to the concept of uniform circular motion is also clear in his Dialogo, where he completely ignored Kepler’s laws of planetary motion, with their elliptical orbits maintaining the Copernican deferent-epicycle model

The Netherlander, Isaac Beeckman (1588–1637) had independently developed the concept of inertia already in 1614 and unlike Galileo he applied it to both circular and linear motion. Although, like Harriot, Beeckman never published, his work was well known to the physicist in Paris especially Descartes (1596–1650), Mersenne (1588–1648) and Gassendi (1592–1655). Newton (1642–1727) took the principle of inertia from Descarte, and not from Galileo as is often falsely claimed as his first law, and Descarte had it from Beeckman. Beeckman was an archetypal Renaissance scientist, an artisan who turned his attention to empirical experiments and science.

When one looks below the surface of the superficial accounts of the history of physics, it become clear the Renaissance scholar contributed substantially to the development of the theories of motion in the period leading up to the so-called scientific revolution.

[1] Stillman Drake, Discoveries and Opinions of Galileo, Doubleday, New York, 1957, p.113


Filed under History of Physics, Renaissance Science

Renaissance science – XLIII

The world has been full of sound since the Earth first acquired an atmosphere at least 3.8 billion years ago. Sound is the principle means with which humans communicate with each other. Hearing is one of the five senses with which humans perceive and explore the world in which they live. Sound in the form of music is one of the oldest and most widespread art forms that humans have created. All of this being so, it might come as a surprise to realise that the science of sound, acoustics, only really developed in any real sense in the eighteenth and nineteenth centuries. Acoustic derives from the Greek akoutikos = pertaining to hearing, from akoustos = heard, audible. Acoustics as the science of sound first appeared in the 1680s and meaning the acoustic properties of a building in 1885. However, the first tentative steps towards the science of acoustics took place in the Renaissance. Before we begin to examine those steps, we first need to look at what took place in antiquity, as this is the basis on which the Renaissance scholars built their theories. 

As usually this only refers to the developments within Europe, where there was historically no clear distinction between the terms we now use – sound, acoustics, music. Our first reference point in Pythagoras and his music theory. According to the well-known myth/legend/story, Pythagoras was walking past a smithy when he noticed the pitch of the sound made by the hammers varied with their weight and he decided to investigate the phenomenon. Pythagoras supposedly established that the weight of hammers or the length of strings on a monochord, a single stringed musical instrument, that produced a pleasing sound when sounded together, which we would now term harmonious, stood in whole number ratios to each other. So, a string twice as long as a given string will sound a note one octave lower than that of the given string:

In modern parlance, if a string sounds the note C when plucked, a string twice as long will sound a C an octave lower, so a ratio of 2:1. A perfect fourth has a ratio of 4:3 and a perfect fifth one of 3:2. The four numbers 1, 2, 3, 4 sum to ten making the Pythagorean tetractys.

The tetractys (Greek: τετρακτύς), is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music

Tetractys Source: Wikimedia Commons

The Pythagoreans also propagated the theory of the harmony of the spheres, which claimed that the planets in their orbits created a harmonious musical scale, which depending on who’s telling the story is either inaudible to human ears or only audible to the enlightened sage. 

As to be expected Aristotle had a theory on the nature of sound. He correctly attributes it to air pressure variations

Sound takes place when bodies strike the air, . . . by its being moved in a corresponding manner; the air being contracted and expanded and overtaken, and again struck by the impulses of the breath and the strings, for when air falls upon and strikes the air which is next to it, the air is carried forward with an impetus, and that which is contiguous to the first is carried onward; so that the same voice spreads every way as far as the motion of the air takes place. 

—Aristotle (384–322 BCE), Treatise on Sound and Hearing[1]

In On Things Heard, attributed to Strato of Lampsacus (c. 335–c. 269 BCE), in the only extant version made up of long extracts included in the Commentary on Ptolemy’s Harmonics of Porphyry (c. 234–c. 305 CE) it is stated that the pitch is related to the frequency of vibrations of the air and to the speed of sound.

Ancient Greek philosopher Strato of Lampsacus, depicted in the Nuremberg Chronicle Source Wikimedia Commons

Ptolemaeus’ Harmonikon or Harmonics is not so well known as his astronomical or geographical works but did, as we will see, have an influence during the Renaissance and especially on Johannes Kepler. Like the Pythagoreans he argued for music intervals based on mathematical ratios. However, whereas the Pythagoreans saw the perfect fifth 3:2, as the basis of the musical scale, Ptolemaeus preferred perfect fourths 4:3 and octaves. His book closes with general thoughts on the relationships between harmony, the soul, and the planets (harmony of the spheres).

A diagram showing Pythagorean tuning. Source: Wikimedia Commons

Turning from the philosophical and theoretical to the practical the De architectura of Vitruvius addresses the subject of acoustics in theatres. In Book V of his masterwork, Chapter 4 is titled Harmonic Principles. He apologises for using many Greek terms, explaining that there are not any Latin terms. There follows a lengthy discussion on the tones produced by the human voice and the intervals in singing. Chapter 5 is titled The EcheaSounding Vessels in Theatres and discusses the production of bronze vessels designed to produce harmonics and their placement within the theatre to improve the acoustics. 

Manuscript of Vitruvius; parchment dating from about 1390 Source: Wikimedia Commons

Moving on to the education system, because of the Pythagorean association of music with arithmetic, it became part of the quadrivium the mathematical part of the seven liberal arts–arithmetic, geometry, music, astronomy–whereby music was regarded as arithmetic in motion and astronomy was geometry in motion.


The standard text for teaching music as a part of the quadrivium at the medieval university was the De institutione musica by Boethius (c. 480–524 CE), one of the four texts that he wrote on the quadrivium subjects. 

Lady Philosophy and Boethius from the Consolation, (Ghent, 1485) Source: Wikimedia Commons

In “De Musica”, Boethius introduced the threefold classification of music: 

  • Musica mundana – music of the spheres/world; this “music” was not actually audible and was to be understood rather than heard
  • Musica humana – harmony of human body and spiritual harmony
  • Musica instrumentalis – instrumental music
Boethius’ De institutione musica. This is a 15th-century copy of a Latin treatise on the Pythagorean-based theory of ancient Greek music, in which the text reflects an older (10th-century) tradition and the numerous diagrams related to ratio and pitch demonstrate later developments in the tradition 

Boethius’ Musica, Arithmetica, and Geometria were all printed for the first time in 1492 making the Pythagorean arithmetical theory of music widely available in the Renaissance.

As I explained in the episode of this series dealing with Vitruvius and De archtectura, it was also printed and published towards the end of the fifteenth century, not only in Latin but also fairly quickly in various vernacular languages. However, as I also pointed De archtectura was well-known in the Middle Ages but had little impact on medieval architecture. Having said that medieval churches often had acoustic jars, thought to have been inspired by Vitruvius’ enchae, inserted in strategic positions in the walls and flours to improve the acoustics. Unlike Vitruvius’ enchae, which were of bronze, the medieval acoustic jars were made of pottery. 

This brick structure comprises the walls of a 15th-century acoustic passage beneath a set of choir-stalls in a friary church. By placing empty pottery jars at strategic points within this passage the builders hoped to be able to improve the acoustics of the Choir by creating a sort of sound-stage for the singers above. Dr Dave Evans, City of Hull archaeologist (doing the pointing in the photo above)

We know from illustrations of instruments, people playing, singing, or dancing that there was a wide variety of music in the Middle Ages, and this is confirmed from written sources. However, the only form of medieval music from before eight-hundred CE, which we know in detail as music, is plain song or Gregorian chant. After eight hundred there are successive developments within a limited musical framework dominated by so-called modal music. There is a difference between modes and scales, which I’m not going to go into now. There was a definite development in the style of musical composition starting around the beginning of the fifteenth century and is regarded as the musical element of the Renaissance. To look more closely at this would however take us of course.  

For our purposes the important development was initiated by Gioseffo Zarlino (1517–1590) a composer and musical theorist.

1599 painting of Zarlino by an unknown artist Source: Wikimedia Commons

He was the first to prioritise the primacy of triad (a three note sequence) over the interval (two notes) that predominated in Pythagorean theory as central to Boethius. He also argued for various technical changes in intonation against the tradition Pythagorean system. Zarlino’s system as presented in his Istitutioni hamoniche (1558) was basically equivalent to that of Ptolemaeus in his Harmonikon.

Le Istitutioni Harmoniche by Gioseffo Zarlino, 1558 Source Brandeis University

Amongst his pupils was the lutenist, composer, and music theorist Vincenzo Galilei (1520­–1591). Galilei disagreed with his teacher on many issues preferring the Pythagorean system, which he presented in his Dialogo della musica antica et della moderna (1581).

Title page Dialogo della musica antica et della moderna (1581) Source

In defence of his system, he turned from pure theory to practical experiments with vibrating strings making him a pioneer in the systematic study of acoustic. He was assisted in his experiments by his, later more famous, son Galileo, and it is generally argued that Galileo acquired his interest in experimental physics working with his father in his youth. 

The Galileis, father and son produced experimental proof that the ratio of an interval is proportional to string length, varies with the square root of the tension applied. Galileo extended his father’s experimental investigations of vibrating strings, looking at the effect of area of the cross-section, and the density of the material used. He almost certainly did this whilst he was still young, but the results of his father’s and his research were first published in his Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences) in 1638. 

Source: Wikimedia Commons

At the end of the First Day in the Discorsi, Galileo begins a discussion on vibrations (pages 127 to 150 in the original). He starts by discussing his pendulum experiments including his slightly incorrect version of the law of the pendulum, Galileo’s observation of isochronism was accurate only for small swings. On page 138 he moves onto the whole theory of vibrating strings, pitch, consonance, and dissonance. He explains consonance as occurring when the pulses of the two notes frequently coincide on the tympanum of the ear. He also explains that one can demonstrate this visually by setting a set of pendula with strings in the same ratio as vibrating strings, 3:2 for the perfect fifth for example. It is obvious that Galileo has given the subject a lot of thought and carried out lots of experiment. He also describes a way to demonstrate the wave nature of sound:

The same phenomenon is observed to better advantage by fixing the base of the goblet upon the bottom of a rather large vessel of water filled nearly to the edge of the goblet; for if, as before, we sound the glass by friction of the finger, we shall see ripples spreading with the utmost regularity and with high speed to large distances about the glass. I have often remarked, in thus sounding a rather large glass nearly full with water, that at first the waves are spaced with great uniformity, and when, as sometimes happens, the tone of the glass jumps an octave higher I have noticed that at this moment each of the aforementioned waves divides into two; a phenomenon that shows clearly that the ratio involved in the octave [forma dell’ ottava] is two.

(Discorsi pp. 142-143)[2]

The work of Vincenzo and Galileo Galilei on acoustics is a classic example of what defines Renaissance science. They proceed from artisan knowledge, that of a working musician, Galileo was like his father a lutenist, on stringing and tuning an instrument and through careful experimental investigation turn it into scientific knowledge. 

Although, Galileo discusses the factors that affect the pitch of a vibrating string–length, tension, cross-section, density–he doesn’t actually give a formula that combines the various factors; this was left up to the French mathematical scholar, Marin Mersenne ((1588–1648). Mersenne, who I have dealt with more fully here, personally knew Galileo and was a big fan of the Tuscan scholar’s work. He invested much effort into trying to persuade others to read Galileo’s books, which rather contradicts the popular image that Galileo was widely read in the seventeenth century. As I stated in my earlier post on him, Mersenne’s biggest contribution to science was in the field of acoustics, and I’m just going to quote what I wrote there:

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.

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

We now need to go back a little and look at the harmony of the spheres as propagated by both the Pythagoreans and Ptolemaeus, as this too experienced a new lease of life in the Renaissance. Tycho Brahe designed and built his research centre Uraniborg on the Island of Hven in the second half of the 1570s. All the dimensions of the plan and façade of the building as well as the layout of the gardens surrounding it were in the Pythagorean harmonic ratios, as befits a temple to celestial research.

Tycho Brahe’s Uraniborg main building from Blaeu’s Atlas Maior (1663) Source: Wikimedia Commons

The man who took the whole concept of the harmony of the spheres furthest was Johannes Kepler in his monumental Harmonices mundi libri V (1619). As the scholar, who contributed the most to the foundations of modern heliocentric astronomy, much more than Tycho Brahe or Copernicus, on whose work he built, Kepler is usually regarded as one of the early modern scientists, but he was in fact a quintessential Renaissance figure. Blending practical knowledge, mathematics, mysticism, and rock-solid empirical science he constructed a wonderfully bizarre and totally unique synthesis. Ninety nine percent of his work usually simply gets ignored and he gets presented as the man, who discovered the three laws of planetary motion, as if he simply pulled them out of his hat, but Harmonices mundi libri V is the wonderful Renaissance creation, 508 large format pages in the modern English translation, that delivered up the third of the laws Kepler’s Harmonic Law

Kepler had already indicated in his Mysterium cosmographicum (1596) that he intended to fine tune his model of the cosmos, based on the five regular Platonic solids, with music or the harmony of the spheres.

He began working on his theories already in the 1590s, expressing his interest in music theory in his correspondence with Michael Mästlin (1550–1631), Edmund Bruce (fl. 1597-1605) and Herwart von Hohenburg (1533–1622). He wrote to Bruce on 14th December 1599 of his intention to write a work with the title De harmonice mundi listing the contents, which are those that the book would eventually have, although it took him almost twenty years to write it. He saw his book as a direct competitor to or update of Ptolemaeus’ Harmonikon, even going so far as to borrow a Greek manuscript of Ptolemaeus’ book and translating parts of it himself. By following a basic Pythagorean line in his theories, Kepler was rejecting to more modern music theories that had come to dominate in the seventeenth century. 

Books I & II Kepler deals with the construction of geometrical figures. In Book III he outlines Pythagorean music and number theory, as an introduction before presenting his own geometrical theory of consonance and dissonance based on numerical ratios of figures constructable with a straight edge and compass. In Book IV he applies his harmonic theories to his own more than somewhat deviant theories of astrology. It is in Book V that he now applies his theories to his model of the cosmos, beginning with the five Platonic solids then moving on to the motions of the planets, analysing all aspects of the planetary orbits for harmonious ratios.

Illustration from the Harmonice mundi taken from the English translation The Harmony of the World by Johannes Kepler, Translated into English with an Introduction and Notes by E.J. Aston, A.M. Duncan and J.V. Field, American Philosophical Society, 1997

All of this wonderfully bizarre theorising, in the end, delivered up Kepler’s third law of planetary motion his harmony law, perhaps the most important contribution to the mathematical theory of astronomy before Newton.

Newton, often regarded as the founder of much of modern science, was, himself, not immune to the allure of the Pythagorean theory of the harmony of the spheres, using it, as I outlined in an earlier post, to justify differentiating seven colours in the rainbow, seven colours, seven notes of the scale, a colour scheme that we still follow today. 

Penelope Gouk, The harmonic roots of Newtonian science, in John Fauvel, Raymond Flood, Michael Shortland & Robin Wilson eds., Let Newton Be: A new perspective on his life and works, OUP, Oxford, New York, Tokyo, ppb. 1989 p. 118

The Renaissance occupation with music theory yielded some strange fruit.  

[1]  “How Sound Propagates” (PDF). Princeton University Press.

[2] Dialogues Concerning Two New Sciences Galileo Galilei, translated by Henry Crew and Alfonso de Salvio, Dover, NY, 1954 p. 99


Filed under History of Acoustics, Renaissance Science

Renaissance science – XLII

As with much in European thought, it was Aristotle, who first made a strong distinction between, what was considered, the two different realms of thought, theoretical thought epistêmê, most often translated as knowledge, and technê, translated as either art or craft. As already explained in an earlier post in this series, during the Middle Ages the two areas were kept well separated, with only the realm of epistêmê considered worthy of study by scholars. Technê being held to be inferior. As also explained in that earlier post the distinguishing feature of Renaissance science was the gradual dissolution of the boundary between the two areas and the melding of them into a new form of knowledge that would go on to become the empirically based science of the so-called scientific revolution. 

A second defining characteristic of the developing Renaissance science was the creation of new spaces for the conception, acquisition, and dissemination of the newly emerging forms of knowledge. We have followed the emergence of libraries outside of the monasteries, the establishment of botanical gardens as centres of learning, and cabinets of curiosity and the museums that evolved out of them, as centres for accumulating knowledge in its material forms. 

Another, space that emerged in the late Renaissance for the generation and acquisition of knowledge was the laboratory. The very etymology of the term indicates very clearly that this form of knowledge belonged to the technê side of the divide. The modern word laboratory is derived from the Latin laboratorium, which in turn comes from laboratus the past participle of laboare meaning to work. This origin is, of course, clearly reflected in the modern English verb to labour meaning to work hard using one’s hands, and all of the associated words, the nouns labour and labourer etc. It was only around 1600 that the word laboratorium came to signify a room for conducting scientific experiments, whereby the word scientific is used very loosely here. 

Of course, laboratories, to use the modern term, existed before the late sixteenth century and are mostly associated with the discipline of alchemy. Much of the Arabic Jabirian corpus, the vast convolute of ninth century alchemical manuscripts associated with the name Abū Mūsā Jābir ibn Ḥayyān is concerned with what we would term laboratory work. It would appear that medieval Islamic culture did not share the Aristotelian disdain for manual labour. However, in Europe, the practical alchemist in his workshop or laboratory actually working with chemicals was regarded as a menial hand worker. Although, it should be remembered that medieval alchemy incorporated much that we would now term applied or industrial chemistry, the manufacture of pigments or gunpowder, just to give two examples. Many alchemists considered themselves philosophical alchemists, often styling themselves philosopher or natural philosopher to avoid the stigma of being considered a menial labourer. 

The status of artisan had already been rising steadily since the expansion in European trade in the High Middle Ages and the formation of the guilds, which gave the skilled workers a raised profile. After all, they manufacture many of the goods traded. It should also be remembered that the universities were founded as guilds of learning, the word universitas meaning a society or corporation. 

So, what changed in the sixteenth century to raise the status of the laboratorium, making it, so to speak, acceptable in polite society? The biggest single change was the posthumous interest in the medical theories of Theophrastus of Hohenheim (c. 1493–1541), or as he is better known Paracelsus (c. 1493–1541), based on his medical alchemy, known as chymiatria or iatrochemistry, a process that began around 1560. 

Aureoli Theophrasti ab Hohenheim. Reproduction, 1927, of etching by A. Hirschvogel, 1538. Source: Wikimedia Commons

The new Paracelsian iatrochemistry trend did not initially enter the Renaissance university but found much favour on the courts of the European royalty and aristocracy and it was here that the new laboratoria were established by many of the same potentates, who had founded new libraries, botanical garden, and cabinets of curiosity. The Medici, Spanish and Austrian Hapsburgs, and Hohenzollerns all established laboratoria staffing them with their own Paracelsian alchemical physicians. Many of these regal loboratoria resembled the workshops of apothecaries, artisans, and instrument makers. Techné had become an integral part of the European aristocratic court. 

It was in the Holy Roman Empire that the Renaissance laboratory celebrated its greatest success. The most well documented Renaissance laboratory was that of Wolfgang II, Graf von Hohenlohe und Herr zu Langenburg (1546–1610). In 1587, having constructed a new Renaissance residence, he constructed a two-story alchemical laboratory equipped with a forge, numerous furnaces, a so-called Faule Heinz or Lazy Henry which made multiple simultaneous distillations possible, and a vast array of chemical glass ware.

Graf Wolfgang II. zu Hohenlohe-Weikersheim, Portrait by Peter Franz Tassaert in the great hall of the castle in Weikersheim Source: Wikimedia Commons

His library contained more than five hundred books, of which fifteen were about practical chemistry, for example from Georg Agricola (1494–1555), author of De re metallica, Lazarus Ecker (c. 1529–1594), a metallurgist, and books on distillation from Heironymous Brunschwig (c. 1450–c. 1512), thirty-three about alchemy including books from Pseudo-Geber (late 13th early 14th centuries), Ramon Llull (c.1232–1316), Berhard von Trevesian (14th century), and Heinrich Khunrath (c. 1560–1605), sixty-nine books by Paracelsus, and twelve about chemiatria including works by Leonhard Thurneysser (1531–1596), Alexander von Suchten (c.1520–1575) , both of them Paracelsian physicians, and Johann Isaac Hollandus (16th & 17th centuries!), a Paracelsian alchemist and author of very detailed practical chemistry books. The laboratory had a large staff of general and specialised workers but was run by a single laborant for sixteen years.

Wolfgang’s fellow alchemist and correspondent, Friedrich I, Duke of Württemberg (1557–1608) employed ten Laboranten in the year 1608 and a total of thirty-three between 1593 and 1608.

Friedrich I, Duke of Württemberg artist unknown Source: Wikimedia Commons

Friedrich had a fully equipped laboratory constructed in the old Lusthaus of a menagerie and pleasure garden. A Lusthaus was a large building erected in aristocratic parks during the Renaissance and Baroque used for fests, receptions, and social occasions.

New Lusthaus in Stuttgart (1584–1593) Engraving by Matthäus Merian 1616 Source: Wikimedia Commons

He also had laboratories in Stuttgarter Neue Spital and in the Freihof in Kirchheim unter Teckabout 25 kilometres south of Stuttgart, where he moved his court during an outbreak of the plague in 1594. Friedrich was interested in both chymiatria and the production of gold and gave a fortune out in pursuit of his alchemical aim. However, he also used his laboratories for metallurgical research.

Heinrich Khunrath (c. 1560–1605) was a Paracelsian physician, hermetic philosopher, and alchemist. In 159, he published his Amphitheatrum Sapientiae Aeternae (Amphitheatre of Eternal Wisdom) in Hamburg, which contains the engraving by Paullus van der Doort of the drawing credited to Hans Vredeman de Vries (1527–1604) entitled The First Stage of the Great Work better known as the Alchemist’s Laboratory.

Heinrich Khunrath Source. Wikimedia Commons
Amphitheatrum Sapientiae Aeternae title page Source: Wikimedia Commons
The First Stage of the Great Work better known as the Alchemist’s Laboratory. Source: Wikimedia Commons

Khunrath was one of the alchemists, who spent time on the court of the Holy Roman Emperor, Rudolf II, also serving as his personal physician.

Rudolf II portrait by  Joseph Heintz the Elder, 1594. Source: Wikimedia Commons

Rudolf ran several laboratories and attracted alchemists from over all in Europe.

Underground alchemical laboratory Prague Source

John Dee and Edward Kelly visited Rudolf in Prague during their European wanderings. Oswald Croll (c. 1563–1609) another Paracelsian physician, who visited Prague from 1597 to 1599 and then again from 1602 until his death, dedicated his Basilica Chymica (1608) to Rudolf.

Title page Basilica Chymica, Frankfurt 1629 Source: Wikimedia Commons

The Polish alchemist and physician Michael Sendivogius (1566–1623), who in his alchemical studies made important contributions to chemistry, is another who gravitated to Rudolf in Prague in 1593.

19th century representation of the alchemist Michael Sendivogius painted by Jan Matejko Art Museum  Łódź via Wikimedia Commons

His De Lapide Philosophorum Tractatus duodecim e naturae fonte et manuali experientia depromti also known as Novum Lumen Chymicum (New Chemical Light) was published simultaneously in Prague and Frankfurt in 1604 and was dedicated to Rudolf.

Michael Sendivogius Novum Lumen Chymicum 

The German alchemist and physician Michael Maier (1568–1622), author of numerous hermetic texts, served as Rudolf’s court physician beginning in 1609. 

Engraving by Matthäus Merian of Michael Maier on the 12th page of Symbola avreae mensae dvodecim nationvm Source: Wikimedia Commons

Along with Rudolf’s Prague the other major German centre for Paracelsian alchemical research was the landgrave’s court in Kassel. Under Landgrave Wilhelm IV (1532–1592), the court in Kassel was a major centre for astronomical research. His son Moritz (1572–1632) turned his attention to the Paracelsian chymiatria, establishing a laboratory at his court.

Landgrave Moritz engraving by Matthäus Merian from Theatrum Europaeum Source: Wikimedia Commons

Like Rudolf, Moritz employed a number of alchemical practitioners. Hermann Wolf (c. 1565­ 1620), who obtained his MD at the University of Marburg in 1585 and was appointed as professor for medicine there in 1587, served as Moritz’s personal physician from 1597. Another of Moritz’s personal physicians was Jacob Mosanus (1564–1616, who obtained his doctorate in medicine in Köln in 1591. A Paracelsian, he initially practiced in London but came into conflict with the English authorities. He moved to the court in Kassel in 1599. He functioned as Moritz’s alchemical diplomat, building connection to other alchemists throughout Europe. Another of the Kasseler alchemists was Johannes Daniel Mylius (1585–after 1628). When he studied medicine is not known but from 1612 in Gießen he, as a chymiatriae studiosus, carried out chemical experiments with the support and permission of the landgrave. In 1613/14 and 1616 he had a stipend for medicine on the University of Marburg. He was definitely at Moritz’s court in Kassel in 1622/23 and carried out a series of alchemical experiment there for him. How long he remained in Kassel is not known. He published a three volume Opus medico-chymicum in 1618 that was largely copied from Libavius’ Alchemia (see below)

Astrological symbol from Opus medico-chymicum Source: Wikimedia Commons

The most important of Moritz’s alchemist was Johannes Hartmann (1568–1631), Mylius’ brother-in-law.

Johannes Hartmann engraving by Wilhelm Scheffer Source: Wikimedia Commons

Hartmann originally studied mathematics at various Germany universities and was initially employed as court mathematicus in Kassel in 1591. In the following year he was appoint professor for mathematics at the University of Marburg by Moritz’s father, Wilhelm. In the 1590s, together with Wolf and Mosanus he began to study alchemy and medicine in the landgraves’ laboratory. In 1609, Moritz appointed Hartmann head of the newly founded Collegium Chymicum on the University of Marburg and professor of chymetria. Hartmann established a laboratory at the university and held lecture courses on laboratory practice. 

Collected works of Johannes Hartmann Source

The four German chymetria laboratory centres that I have sketched were by no means isolated. They were interconnected with each other both by correspondence and personal visits, as well as with other Paracelsian alchemists all over Europe. Both Croll and Maier although primarily associated with Rudolf in Prague spent time with Moritz in Kassel.

I now turn to Denmark, which in some senses was an extension of Germany. Denmark was Lutheran Protestant, German was spoken at the Danish court and many young Danes studied at German universities. Peder Sørensen (1542–1602), better known as Petrus Severinus, was one of the leading proponents of Paracelsian iatromedicine in Europe. It is not known where Severinus acquired his medical qualifications. In 1571, he became personal physician to King Frederick II until his death in 1588 and retained his position under Christian IV. In 1571, he published his Idea medicinæ philosophicæ, which was basically a simplified and clear presentation of the iatromedical theories of Paracelsus and was highly influential. 


Severinus moved in the same social circles as Tycho Brahe (1546–1601) and the two were friends and colleagues. Severinus’ medical theories had a strong influence on the astronomer and Tycho also became an advocate and practitioner of Paracelsian alchemical medicine.

Portrait of Tycho Brahe at age 50, c. 1596, artist unknown Source: Wikimedia Commons

When Tycho began to construct his Uraniborg on the island of Hven in 1576, he envisaged it as temple dedicated to the muses of arts and sciences. The finished complex was not just a simple observatory but a research institute with two of the most advanced observatories in Europe, a papermill, a printing works and in the basement an alchemical laboratory with sixteen furnaces for conduction distillations and other chemical experiments.

An illustration of Uraniborg. The Tycho Brahe Museum Alchemical laboratory on the left at the bottom

Tycho took his medical research very seriously developing medicines with which he treated colleagues and his family.

In the south of Germany Andreas Libavius (c. 1550–1616) took the opposite path to Severinus, he totally rejected the philosophies of Paracelsus, which he regarded as mystical rubbish, whilst at the same time embracing chymetria. Having received his MA in 1581, somewhat late in life in 1588, he began to study medicine at the University of Basel. In 1591, he was appointed city physician in Rothenburg ob der Taube, later being appointed superintendent of schools. 

Andreas Libavius artist unknown Source: Wikimedia Commons

In 1597, Libavius published his Alchemia, an alchemical textbook, a rarity in a discipline that lived from secrecy. It was written in four sections: what to have in a laboratory, chemical procedures, chemical analysis, and transmutation. Although, Libavius believed in transmutation he firmly rejected the concept of an elixir of life. In the laboratory section of his Alchemia, he contrasted Tycho’s laboratory on Hven, which, being Paracelsian, he viewed as defective with his own vision of an ideal alchemical laboratory.

Source:Wikimedia Commons

Roughly contemporaneous with Libavius, the German physician and alchemist Daniel Sennert (1572­–1637), who played a significant role in the propagation of atomic theory in chemistry, introduced practical laboratory research into his work in the medical faculty of the University of Wittenberg. Sennert represents the beginning of the transition of the laboratory away from the courts of the rulers and aristocrats into the medical faculties of the universities. 

Portrait of Daniel Sennert engraved by Matthäus Merian Source: Wikimedia Commons

During the seventeenth century the medical, alchemical laboratory gradually evolved into a chemical laboratory, whilst remaining a part of the university medical faculty, a transmutation[1] that was largely complete by the early eighteenth century. Herman Boerhaave (1668 – 1738), regarded as one of the founders of modern chemistry in the eighteenth century, his Elementa Chemiae (1732) was one of the earliest chemistry textbooks, was professor of medicine at Leiden University. A generation earlier, Robert Boyle (1627–1691), who ran his own private laboratory, and whose The Sceptical Chymist (1661) was a transitional text between alchemy and chemistry, was still a practicing alchemist, although he rejected the theories of Paracelsus.  

[1] Pun intended

Leave a comment

Filed under History of Alchemy, History of Chemistry, History of medicine, History of science, Renaissance Science