Napier’s Logarithms

Only great men have great faults.
François de la Rochefoucauld

John Napier (1550-1617)
John Napier (1550-1617), Marvellous Merchiston

In July 1614 in Edinburgh, Scotland, was published a small book (57 pages of explanatory matter and 90 pages of tables) which will make a key advance in the use of mathematics. The book was Mirifici Logarithmorum Canonis Descriptio (Description of the Marvelous Canon of Logarithms), written by a Scotsman—John Napier. Two years later an English translation of Napier’s original Latin text was published, translated by Edward Wright. What is the history of this remarkable book?

There is an indirect evidence that Napier was occupied with logarithms as early as 1594 (in a 1624 letter from Kepler to Petrus Criigerus). Moreover, some historians claim, that there was another man, who invented logarithms before Napier (around 1588), the Swiss mechanical and mathematical genius Joost Bürgi (1552-1632). Bürgi however published his work (Tafeln arithmetischer und geometrischer Zahlenfolgen mit einer gründlichen Erläuterungen, wie sie zu verstehen sind und gebraucht werden können) 6 years after Napier, as late as 1620 in Prague. Described by Bürgi tables distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms. Bürgi’s method is different from that of Napier and was clearly invented independently. Bürgi was also an important contributor to prosthaphaeresis, a technique for computing products quickly using trigonometric identities, which predated logarithms and was introduced circa 1005 by Ahmad ibn Yunus al-Sadafi al-Misri (c. 950-1009), an important Egyptian Muslim astronomer and mathematician, and introduced in Europe by Nuremberg mathematician Johannes Werner (1468-1522) in the late 15th century.

Interestingly, after the Descriptio, published in 1614, appeared another Napier’s book on the logarithms, known as the Constructio (Mirifici Logarithmorum Canonis Constructio), published after his death in 1619 by Robert Napier (1580-1655), his second son by second marriage. Internal evidence as well as the distinct statement of Robert, make it clear that it was in fact written years before the Descriptio, and it represents in many passages an earlier stratum of thought. In the Preface (Greeting) of Constructio Robert asserts:
You have then (kind Reader) in this little book most amply unfolded the theory of the construction of logarithms, (here called by him artificial numbers, for he had this treatise written out beside him several years before the word Logarithm was invented,) in which their nature, characteristics, and various relations to their natural numbers, are clearly demonstrated.
The Constructio is considered to be the most important of all Napier’s works, presenting as it does in a most clear and simple way the original conception of logarithms.

It seems the concept of a logarithm made it first appearance in ancient Babylonia (just as the concept of the abacus) around 1800 B.C. Baked clay tablets have been found, which contain tables of successive powers of whole numbers. In some of these records a question is asked: “To what power must a certain number be raised in order to yield a given number?” More then 1500 years later the great Archimedes made an observation that is the basis if the modern logarithms—he defined “the order” of a number to be equivalent of the exponent and observed that the addition of orders corresponds to finding their product.

In his book Napier proposed a method, which allows the more complex arithmetical operations as multiplication, division and calculating of a root to be done by means of addition and subtraction. He realized, that all numbers can be expressed in what is now called exponential form, meaning 8 can be written as 23, 25 as 52 and so on. What make logarithms so useful is the fact that the operations of multiplication and division are reduced to simple addition and subtraction. When very large numbers are expressed as a logarithm, multiplication becomes the addition of exponents.

In the preface of the book Napier explains his thinking behind his great discovery—Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.

Napier didn’t mentioned what kind of sources he used, in order to develop the logarithms. It is known, that he was acquainted with a mathematical problem, solved some 25 years before 1614—multiplying two sines together was solved by the method of prosthaphaeresis, which corresponds to the formula:
sin a x sin b = [cos(a - b) - cos(a + b)]/2
As Napier knew of, and used, the method of prosthaphaeresis, it may well have influenced his thinking, because the first logarithms were not of numbers, but were logarithms of sines.

Another factor in the development of logarithms at the beginning of 17th century was that the properties of arithmetic and geometric series had been used since at least Greek times had been studied extensively in the previous century. We now know that any numbers in an arithmetic series are the logarithms of other numbers in a geometric series, in some suitable base. For example, the following series of numbers is geometric, with each number being two times the previous one:
natural numbers 1 2 4 8 16 32 64 128 256 512 1024
And the series below is an arithmetic one whose values are the corresponding base 2 logarithms:
logarithms 0 1 2 3 4 5 6 7 8 9 10
It had long been known that if you take any two numbers in the arithmetic progression, say 3 and 4, their sum, 7, would indicate the position of the term in the geometric series that is the product of the terms in the corresponding positions of the geometric series, e.g., 3 + 4 = 7 and 8 x 16 = 128 (the third times the fourth = the seventh).
This is starting to look very much like our own conception of logarithms as being the powers to which some base number is raised, a concept that was not understood in Napier’s time. Often the use of a good form of notation will suggest some basic mathematical principle. Our use of indices to indicate the power to which a number is being raised seems to have an obvious connection with logarithms, but without this form of notation, the connection is vague at best.
John Napier came at the idea of logarithms not by algebra and indices but by way of geometry. When first thinking about this subject, he used the term artificial number but later created the term logarithm from two Greek words—λoγoς and αριθμός, meaning word, ratio and number respectively. He decided to use this term because his logarithms were based on the concept of points moving down lines in which the velocity of one point was based on the ratio of the lengths of the line on either side of it.

In the end of 1614 one of the most famous English mathematicians of the day, Henry Briggs (1561-1630), who was a Professor of Geometry at Gresham College, London, obtained a copy of Napier’s Descriptio and, by March of the following year wrote that:
Napier, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.

Henry Briggs, Arithmetica Logarithmica from 1624
Henry Briggs, Arithmetica Logarithmica, 1624

Briggs immediately began to popularize the concept of logarithms in his lectures and even began to work on a modified version of the tables. Working together with Napier for some time (he was able to visit Napier twice at his baronial estate, staying there for a month in 1616, and again in 1617), later on Briggs will propose (firstly in his 1617 Logarithmorum Chilias Prima (The First Thousand Logarithms)) that the base of the logarithms should be changed in order to make them easier to use, thus we have 10 based logarithms, and in 1624 published tables (Arithmetica Logarithmica), containing the logs of the numbers from 1 to 20000 and from 90000 to 100000 all calculated to 14 decimal places.

In order to make calculations by means of logarithms, we have to use tables with logarithms. If we have to multiply two numbers, we have to find their logarithms in the table, to add the logarithms, and then to find in the table the number, which logarithm corresponds to the sum of logarithms. But what to do, if this logarithm cannot be find in the table? Let’s for example suppose, that we have only a table with decimal logarithms of the integers up to 1000, but we need a logarithm of a fractional number, e.g. 7,93. As long as
So, needed logarithm can be found by subtraction of 2 (this is the decimal logarithm of 100) from the logarithm of 793. Of course, the usefulness of the decimal logarithms is due to our decimal numbering system.

It was not only Briggs who was impressed by Napier’s Descriptio. In 1617, Johann Kepler first saw the Descriptio in Prague. He was too busy to pay it much attention, as during this period that he was hard at work on his third law of planetary motion, but did acknowledge its existence in a letter to his friend Wilhelm Schickard, the creator of the first mechanical calculator in the world, where he indicated:
A Scottish baron has started up, his name I cannot remember, but he has put forth some wonderful mode by which all necessity of multiplications and divisions are commuted to mere additions and subtractions.

A year later, Kepler wrote to Napier expressing his admiration and letting him know that he must publish the promised Constructio as soon as possible. Unfortunately, Napier had been dead for two years before the letter arrived. The letter may have spurred John’s son, Robert Napier, into putting the finishing touches on his father’s notes and overseeing the publication of the Constructio in 1620. In 1624 Kepler published his book Chilias Logarithmorum ad totidem numeros…, creating the logarithmic tables by a new geometrical procedure, the form thus differing from the logarithms of both Napier and Briggs.

Within twenty years of the time that Briggs’s tables first appeared, the use of logarithms had spread worldwide. From being a limited tool of great scientists like Johann Kepler, they had become commonplace in the schoolrooms of the civilized nations. Logarithms were used extensively in all trades and professions that required calculations to be done. It is hard to imagine an invention that has helped the process of computation more dramatically than has logarithms.

Very soon after popularizing of logarithms began attempts for removing of the nasty end error-prone process of looking into tables and manual adding of numbers. First was a colleague of Brigs and professor of astronomy at Gresham College—Edmund Gunter (1581-1626). Briggs’s work naturally came to the notice of Gunter, who had some earlier experience in the development of calculating instruments, having been one of the major figures in the perfection of an instrument known as a sector. This experience soon led him to realize that the process of adding together a pair of logarithms could be partially automated by engraving a scale of logarithms on a piece of wood and then using a pair of compasses to add together two values in much the same way, as he would have done when using a sector. Not only did this method eliminate the mental work of addition, but it also removed the necessity for the error-prone and time-consuming process of looking up the logarithms in a table. Gunter described his instrument in the book Description and use of the Sector, the Crosse-staffe and other Instruments, published in 1623. Such a rule is frequently referred to as a Gunter Line or Gunter Scale, often combination not only of a logarithmic scale, but also with chords, sines, tangents and rhumbs. A two foot long boxwood ruler inscribed with a variety of scales was a standard navigator’s tool up until the end of the 19th century.

The original Drawing of Gunter Scale from 1624 (upper) and part of a wooden Gunter's Scale (lower image)
The original drawing of Gunter Scale from 1624 (upper) and part of a wooden Gunter Scale (lower image)

Biography of John Napier

John Napier (John Napier frequently signed his name “Jhone Neper, Fear of Merchiston”, but later on we can find his name also as Napeir, Nepair, Nepeir, Napare, Naper, Naipper, Neperus, etc., the only form of Napier that for sure would not have been used in Napier’s lifetime was the present modern spelling “Napier”!) was born on 1 February 1550, in Merchiston Castle (or Tower as it was usually called in those days), in the vicinity of Edinburgh, Scotland, as the first child of Sir Archibald Napier (1534–1608), 7th Laird of Merchiston and Baron of Edenbellie.

Merchiston Castle in Edinburgh in late 18th century
Merchiston Tower (Castle) in Edinburgh in late 18th century

Napiers was one of the most important Scottish families at the time. The second Napier was Controller of the King’s Household and ambassador. The third sat in Parliament, and several received the honor of knighthood. Three of them fell in battle (between them John Napier’s grandfather and great-grandfather). During the 15th century no fewer than 3 Napiers were provosts of Edinburgh.

Sir Archibald Napier fully maintained the repute of his ancestors for energy and sagacity. He had studied the laws, and was a proficient in the mathematics. In 1549, being only 15 years old, Archibald married to Janet Bothwell (1534-1563), a daughter of member of Parliament Francis Bothwell and sister of Adam Bothwell, Bishop of Orkney. The next year was born his heir John, then two other children—Francis and Janet.

Archibald Napier was a justice-depute and was knighted in 1565. After some troubles during the civil war in the 1560s and 1570s, Archibald was back in favor and in 1582 he was appointed Master of the Mint in Scotland, with the sole charge of superintending the mines and minerals within the realm, and this office he held till his death in 1608. The family also owned estates at Lennox and at Menteith and a residence at Gartness. Sir Archibald must have been very wealthy, and John inherited from him an estate sufficient to live well on.

Merchiston Tower (Castle) (engravings from book Modern Athens, published in 1829)
Merchiston Tower (Castle) (engravings from book Modern Athens, published in 1829)

Three years before the birth of John, in 1547, Alexander Napier, John’s grandfather, had been slain while fighting in the army of Mary Queen of Scotland against the English at the Battle of Pinkie. Later John Napier would spend most of his life trying not to get involved in the sectarian strife that swept Scotland.

As was the practice for members of nobility, Napier did not enter school until he was 13, being rather taught at home. When he was 13, in October 1563, he entered St Andrews University (he was registered as Johannes Neaper). His mother arranged for him to live in St Salvator’s College (see the photo below), the oldest of the three endowed collegiate societies of the university, and special arrangements were made for the principal of the college, John Rutherford (1515-1577), the most distinguished teacher of his day in Scotland, to take care of him personally. Rutherford was an able academic and, besides adding to Napier’s anti-Rome views, gave him a good grounding in Latin and the other subjects then taught at the college. Napier wrote many years later that it was in St Andrews that he first became passionately interested in theology, but there he studied also Latin and mathematics.

Unfortunately, on 20 December 1563, only two months after Napier matriculated at St Andrews, his mother Janet died only 29 years old. Later on in 1571 Sir Archibald married to a cousin, Elizabeth Mowbray, by whom he had ten children—(Sir) Alexander (of Lauriston); Archibald (of Wowmet); Walter; William; Susanna; Abellina; Agnes; Helen; Marion; and Elizabeth.

St Salvator College of St Andrews University, photo from 1846
St Salvator College of St Andrews University, photo from 1846

Little is known about John Napier’s early years. One of the few scraps of information that we have is from a letter from the Bishop of Orkney, John’s uncle, to Archibald Napier written when John was eleven years old:
I pray you, schir, to send your son Jhone to the schuyllis; oyer to France or Flandaris; for he can leyr na guid at hame, nor get na proffeitt in this maist perullous worlde …

(Let’s translate to English the old Scots that the Bishop of Orkney actually wrote):
I pray you, sir, to send your son John to school; over to France or Flanders; for he cannot learn well at home nor get profit in this most perilous world—that he may be saved in it;—that he may seek honor and profit as I do not doubt that he will…

For unknown reason, Napier had not remained long enough at St Andrews to get a degree. His name didn’t appear appear in the list of determinantes for the year 1566, and of masters of arts for 1568. Most probably in 1567 Napier was forced to study abroad, having left St Andrews University prematurely (probably after his friendship with a Catholic student—according to Napier’s own account, he had there “contracted a loving familiaritie with a certaine gentleman, a Papist”). Even this was thought inadvisable in such sensitive times. Anyhow, Napier left the University without taking a degree and went abroad for several years.

Absolutely no evidence exists in which European country Napier studied (University of Paris is highly likely and it is also probable that he spent some time in Italy, Germany and the Low Countries), but when he returned home in 1571, he was a scholar competent in Greek, and applied himself closely to the study of the mathematics.

In 1571 Napier returned in Edinburgh, to find his father imprisoned in Edinburgh Tower by the Queen’s party, while the family home at Merchiston was occupied by the forces of the Regent, then besieging the town. The following year, 1572, when Merchiston Castle was bombarded by the guns of Edinburgh Castle, Napier sought refuge on one of the family estates at Gartness in Stirlingshire.

Elizabeth Stirling (1554-1579), the first wife of John Napier, 8th Laird of Merchiston
Elizabeth Stirling (1554-1579), the first wife of John Napier

In 1571, the preliminaries of his marriage were arranged at Merchiston, to Elizabeth Stirling (1554-1579, see the image below), the daughter of a neighboring landowner, Sir James Stirling, 4th Laird of Keir and of Cadder and a friend of his father. The couple married towards the close of 1572. Most of the estates of the Napier family were made over to John Napier and a castle was planned for the estate at Gartness. When the castle was completed in 1574, Napier and his wife took up residence there.

Napier devoted himself to running his estates. This task he took very seriously and, being a great genius as an inventor, he applied his skills to these tasks. He approached agriculture in a scientific way and he experimented with:
… improving and maturing of all sorts of field land with common salts, whereby the same may bring forth in more abundance, both of grass and corn of all sorts, and far cheaper than by the common way of dunging used heretofore in Scotland.

About the end of the year 1579 Elizabeth Stirling died, leaving him one son, Archibald (1576-1645) (who in 1627 was raised to the peerage by the title of Lord Napier), and one daughter, Joan (the twins Agnes and Barbara died in infancy). Next year, 1580, John Napier married again, his second wife being Agnes Chisholm (the second cousin of the Napier’s first wife, from the same Chisholm family that had provided his grandfather’s second wife, daughter of Sir James Chisholm of Cromlix, and great-granddaughter of King James IV), who survived him. By her he had five sons and five daughters. On the death of his father in 1608, Napier and his family moved into Merchiston Castle, where he lived the rest of his life.

Sir Archibald Napier, 1st Lord Napier, 9th Laird Napier of Merchiston (1576 - 1645). Dated 1637. Artist: George Jamesone
Sir Archibald Napier, 1st Lord Napier, 9th Laird Napier of Merchiston (1576 – 1645). Portrait form 1637, Artist: George Jamesone.

Napier’s father had been deeply interested and involved in religious matters, and Napier himself was no different. Because of his inherited wealth, he needed no professional position. He kept himself very busy by being involved with the political and religious controversies of his time. For the most part, religion and politics in Scotland at this time pitted Catholics against Protestants. Napier was anti-Catholic, as evidenced by his 1593 book against Catholicism and the papacy entitled A Plaine Discovery of the Whole Revelation of St. John. This work suggested that the Antichrist of the Book of Revelation was none other than the reigning pope, and urged James VI, the Scottish King, to “purge his house, family and court of all Papists, Atheists and Newtrals.” This attack was so popular that it was translated into several languages and saw many editions. Napier always felt that if he attained any fame at all in his life, it would be because of that book.

After the publication of the Plaine Discovery, Napier seems to have occupied himself with the invention of secret instruments of war. There is a document, dated the 7 June 1596 and signed by Napier, giving a list of his inventions for the defense of the country against the anticipated invasion by Philip of Spain. The document is entitled “Secrett Inventionis, proffitabill and necessary in theis dayes for defence of this Iland, and withstanding of strangers, enemies of God’s truth and religion,” and the inventions consist of:
• a mirror for burning the enemies’ ships at any distance;
• a piece of artillery destroying everything round an arc of a circle;
• a round metal chariot, so constructed that its occupants could move it rapidly and easily, while firing out through small holes in it.
It has been asserted that the piece of artillery was actually tried upon a plain in Scotland with complete success, a number of sheep and cattle being destroyed 🙂

Besides the above-mentioned book, Napier wrote three other books: Mirifici Logarithmorum Canonis Descriptio, Ejusque uses, in utraque Trigonometria; ut etiam in omni Logistica Mathematica, Amplissimi, Facillimi, expeditissimi explicatio. Authore ac Inventore Ioanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, ex ofjicina Andreae Hart Bibliopolae (in 1614, a translation into English by Edward Wright was published in 1616); Rabdologiae, seu Numerationis per virgulas Libei duo: Cum Appendice de expeditissimo Multiplicationis promptuario. Quibus accessit & Arithmeticae Localis Liber unus. Authore & Inventore Ioanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, Excudebat Andreas Hart (published posthumously in 1617) and Mirifici logarithmorum canonis constructio (written before the Descriptio, but published posthumously in 1619 by his second son by second marriage, Robert). In this treatise (which was written before Napier had invented the name logarithm) logarithms are called artificial numbers.

As a person of high energy and curiosity, Napier paid much attention to his landholdings and tried to improve the workings of his estate. Around the Edinburgh area, he became widely known as Marvelous Merchiston for the many ingenious mechanisms he built to improve his crops and cattle. He experimented with fertilizers to enrich his land, invented an apparatus to remove water from flooded coal pits, and bat devices to better survey and measure land.

Napier had a great interest in astronomy, which led to his contribution to mathematics. He was not just a star gazer; he was involved in research that required lengthy and time consuming calculations of very large numbers. Once the idea came to him that there might be a better and simpler way to perform large number calculations, Napier focused on the issue and spent twenty years perfecting his idea.

Edinburgh, St Cuthbert Church, the plaque for Napier
Edinburgh, St Cuthbert Church, the plaque for Napier

It would be surprising if a man of such great an intellect as Napier did not appear rather strange to his contemporaries and, given the superstitious age in which he lived, strange stories began to circulate. Like Johannes Kepler and all his contemporaries Napier believed in astrology, and he certainly also had some faith in the power of magic, for there is extant a deed written in his own handwriting containing a contract between himself and Robert Logan of Restalrig, a turbulent baron of desperate character, by which Napier undertakes to serche and sik out, and be al craft and ingyne that he dow, to tempt, trye, and find out some buried treasure supposed to be hidden in Logan’s fortress at Fastcastle, in consideration of receiving one-third part of the treasure found by his aid.”

Napier first described the decimal point, enabling calculations to be made without the use of complex fractions. He discovered what eventually would be called “Pascal’s Triangle” and placed it in common use long before Pascal was even born.

One of the greatest men of Scotland—John Napier, died on 3 April 1617, apparently of goat, with which he had long been afflicted, and was buried in the graveyard (crypt) of the old church of St Cuthbert’s (formerly known as the West Kirk), then outside the West Port of Edinburgh.