Mathematical aids for Early Modern astronomers.

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


Cuniform reciprocal table Source

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


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

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


Abū al-Wafāʾ Source: Wikimedia Commons

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

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

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

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

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

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


John Napier, artist unknown Source: Wikimedia Commons

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


Merchiston Castle from an 1834 woodcut Source: Wikimedia Commons

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

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

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

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

Am x An = Am+n

Am/An = Am-n

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


Michael Stifel Source: Wikimedia Commons


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

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


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

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


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


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


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

NapLog(x) = –107ln (x/107)

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


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


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

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


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

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

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


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


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

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

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


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


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

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














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

6 responses to “Mathematical aids for Early Modern astronomers.

  1. Todd Timberlake

    I’ve read about prosthaphaeresis (as used in Kepler’s work) on many occasions but I have to say yours is the clearest and most succinct description I’ve seen. The power of equations! (At least for those of us who really DID learn those trig identities … and still use them.)

    When I saw the diagram from Gunter I thought “that looks like a slide rule” and I was glad to see I wasn’t far off. I’m too young to have much slide rule experience (electronic calculators were the norm for me in grade school and personal computers were just coming on the scene – we had a TRS-80), but my dad was an engineer and he still had his old slide rule from his school days, so I learned how to use it.

    By the way, there’s a minor typo in the dates for Regiomontanus (off by a century).

  2. Dick Feynman’s explanation of how Henry Briggs calculated logarithms to base 10 is on pp 22-4 to 22-7 of Volume 1 of ‘The Feynman Lectures on Physics’. There is a free download of the pdf at:

  3. Speed Shifter

    Great blog! Must show it to my mathematics loving friends.

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