Navigation, Mercantilism, and Logarithms:

Reinterpreting the preface of the English

translation of John Napier’s Descriptio Canonis

                                                                                                                        Zhao Fang

                                                                                                            Professor Hayton

John Napier’s Descriptio Canonis, published in Latin in 1614, has long been recognized as the first mathematical work on the logarithm. Its appearance immediately attracted the attention of two eminent English mathematicians that time---Edward Wright and Henry Briggs. The former translated the book into English; the latter was “concerned with Napier in the change of the logarithms from those originally invented to decimal or common logarithms”[1], and part of his logarithm tables was included in the English translation. The English translation of the Descriptio was published in 1616, one year after the death of the translator, Edward Wright. His son, John Napier, and Henry Briggs wrote three prefaces for the English translation of the book. The prefaces revealed the translator, the original author, and the co-author’s attitudes toward the invention of logarithm. However, its importance was conventionally ignored in the studies on the history of logarithm.  In this paper, I will show how the social and economical trends in the early 17^th century England, especially the advancement in navigation and commercial expansion to colonial market, shaped the three persons’ opinions on the invention of logarithm, which were stated explicitly in the preface of the English translation of  the book.

            Before discussing the contents of the three prefaces, it is necessary to first take a closer look at the situations of English navigation and economy in the late 16^th century and early 17^th century. Perhaps the most important change in English navigational practice during the latter half of the 16^th century was the introduction and further development of navigational technologies founded upon mathematics and astronomy. [2] These new technologies included both instruments and techniques---the physical devices themselves together with the set of skills and knowledge required to use them. The astronomical instruments used to make stellar observation were quite useless apart from the calculations in which those observations were subsequently employed. Likewise, the pilot’s plane charts were meaningless to a traditional, nonmathematical pilot who could not perform geometrical calculations. In the early 17^th century, the introduction of mathematically derived methods of navigation had changed the very idea of what it meant to be a pilot: while the pilots in the 1550s found the route mostly on their personal experience, the pilots in the early 17^th century constantly used astronomical instruments, sea-charts, and manuals to calculate the ship’s position and determine which direction to go. Although the preeminence of mathematical navigation became increasingly obvious to the 17^th century pilots, they had to face two serious problems when performing navigational calculation. The first one was the abounded errors existing in the sea chart, compass, cross-staff, and tables of the declinations of the sun and stars which were then in use. The second one was the difficulties of calculating exact values of trigonometry functions. The pilots demanded more reliable and efficient methods of calculation to ease their surging workload.    

Although England’s maritime power expanded dramatically in the late 16^th century, the process of settling in distant lands did not begin until the 17^th century. From 1603 to 1630, there were at least ten plantations founded by English companies in the America whose locations ranged from New England to Jamaica, while there was only one between 1560 and 1600. The East India Company and Hudson’s Bay Company founded several valuable factories for trade in Africa and India after the turn of the 17^th century.[3] Merchants involved in oversea trades and ship owners started to flourish at around the same time: the value of imports to London was much the same in 1600 as it had been in 1560, but by the early 1660s there was 150,000 tons of English shipping, three times as much as in the early 1570s.[4]    

            The first several decades of the 17^th century England also witnessed the transformation of a landed economy to a more capital oriented economy, together with social structure change. One example is the growth of the Cumbrain iron Industry: from 1606 to 1630, eleven new smelting sites appeared in this region.[5] Landowners who owed or worked iron ore were able to benefit from sales not only to the local industry but also to distant smelting sites. The benefit of participation in trade encouraged noble class to invest their capital on commercial activities. Arithmetical calculations, crucial for financial activities, gradually became one of a daily practice for gentries who were involved in commerce. [6]      

Given all the backgrounds, one shall not find difficulties understanding William Kempe’s comments on the value of mathematics at that time

Take away arithmetic; ye take away the merchant’s eye, whereby he seeth his direction in buying and selling; ye take away the goldsmith’s discretion, whereby he mixeth his metals in due quantities; ye take away the captain’s dexteritie, whereby he embattaileth his army in convenient order; finally ye take away from all sorts of men, the faculty of executing their functions aright.[7]

In the early 17^th century, the navigator, the surveyor, the gunner, the mechanic, and the engineer and even some landlords were all constantly obliged to calculate. However, the existing computing technologies were insufficient to meet the rising need of calculation before the late 1610s. There was no uniform notation system for decimal numbers and fractions; there was not any relatively easy method to calculate the exact value of trigonometry functions; there was no mechanical tool to help pilots or merchants to perform multiplication and division. Under all these social and economical pressure for simplifying arithmetic calculation and avoiding calculation errors, the invention of logarithm immediately received recognition as an extraordinary method to simplify these laborious calculations. This recognition was so influential that even John Napier viewed his invention as a calculating tool and somehow underestimated its theoretical value.

John Napier’ prefaces consisted of two parts, one was the dedication letter to the Prince Charles, which was in the Latin version of the book, and the other part was the preface he wrote for the English translation. In both parts one can perceive that he did not reckon that the invention of logarithm was a theoretical advancement in mathematics, but rather, a method to help calculation. In the dedication letter he stated the function of the logarithms was “helpe the weakness of memory, that by means thereof it is easie to resolve moe Mathematical questions in one houres space,”[8] He also made a statement on how powerful this method was “this new course of Logarithms doth cleane take away all the difficultie that heretofore hath beene in mathematical calculations (which otherwise might have been distasteful to your worthy towardnesse),”[9] It was clear for him that his new invention would be of great help for complicated calculations. In the preface he wrote for the English version, he stated explicitly that the logarithms could serve as “excellent briefer rules”[10] for multiplication, division and square roots. More importantly, he noticed that the people affected of this invention were not only mathematicians and astronomers, but also non academic people who needed to do calculations. In the preface, he said

I thought good heretofore to set forth in Latine for the publique use of Mathematicians. But now come of our Countrey-men in the Island well affected to these studies, and the more publique good, procured a most learned Mathematician to translate the same into our vulgar English tongue…[11]

Latin was the academic language used by university educated mathematicians, however, there were “our Countrey-men in the Island well affected to these studies” who could not read Latin and wanted to read English translation of the book.  These country men were neither professional mathematicians, nor natural philosophy professors in the university. They were the navigators, the surveyors, the gunners, the mechanics, and the engineers who wanted to know how to use logarithms instead of grasping its theoretical value. Jonh Napier did not think that the audience of this book would be limited to the educated elite class. The common people who had to do “troublesome” calculations at that time would also find his invention useful.     

John Napier’s awareness that common people also needed methods to simplify calculations was probably linked with the ongoing social structure transformation mentioned above. As the Baron of Merchiston, his primary interests were politics and theology. However, mathematics played an important role in his mature life. Through his daily experiences of various calculation for the management of his estate, also through the letters his friends sent him, he realized that the heavy burden of doing arithmetic and thus spent lots of time designing various devices to ease the labor.[12] The most famous one was his Rabdologia, also known as Napier’s bone. Without landowners’ gradual involvement in commercial activities and the spread of mercantilism, John Napier might not be able to realize the social demand for quick methods of computing.   

John Napier in fact wrote two books on logarithm: the book The Descriptio contained only an explanation of the use of the logarithms without any account of the manner in which the canon was constructed. There was also a second book The Constructio which stated the mode of construction of the canon for logarithms. The second book was published in 1617 by his son, Robert Napier, under the title Mirifici logarithmorum canonis construction. Although it offered the grounding of the mathematical concept, the earliest English translation of The Constructio appeared in 1889, some 250 years after its publication.[13] The reason why the second book was not immediately translated was that the translator, Edward Wright, did not think logarithm as a new mathematical discovery and thus its nature was relatively insignificant. His view on logarithm as a computing tool was represented in the preface written by his son.

It should be mentioned that in the Renaissance, the preface of a book often had the function of dedicatory prologues and letters. Early modern prefactory apparatus, like the texts that it introduced, typically observed certain formulas of composition, utilized tropes and figures of speech, and most important, directed itself to specific patrons.[14] Unlike most authors at that time who dedicated the book to the church or a political power, the son of Edward Wright, Samuel Wright dedicated this book to “company of merchants of London trading to the East-Indies”, the East Indian Company, in which Edward Wright once served as a mathematician. This occupation, which was not common before the 17th century, and his experience with the navigation, determined that he viewed logarithm as a mathematical calculating tool when translating Napier’s book.

Edward Wright started serving in the East Indian Company in 1589. In that year he was “called forth to the public business of the nation, by the Queen,”[15] and he joined the expedition to the Azores, whose object was to prey upon Spanish commerce. In his three years life on the sea, he got to know the difficulties pilots had with emerging navigational technologies. He also found that the ordinary sea-chart was in many places “like an inextricable labyrinth of error,”[16] After he returned from the sea, he wrote his most important work Certaine Errors in Navigation.  In this book, he discussed his own method of calculating geometrical projection. However, this new method was difficult to carry out because it involved multiple steps of evaluating trigonometry functions and divisions. That was probably why he did not offer a complete table for the results of his method in his book. [17] When John Napier published his work, Edward Wright was at once attracted by how powerful the application of logarithms could be in navigation. His perspective was stated explicitly by his son in the preface “I doubt not but it is apparent enough that he esteemed of it, and intended to have recommended it as a booke of more than ordinary worth, especially to Sea-men.” Once as a mathematician in the sea, he truly understood how crucial it was for pilots to know the logarithms when performing navigational calculation, especially computing the value of trigonometry functions.

  The third preface of the book was written by Henry Briggs, the Professor of geometry in Gresham College, London, under the title the preface to the reader. Henry Briggs was a university professor who did not have direct contact with the seamen and merchants, yet he realized how valuable logarithm would be for common people as well as for scholars.  The contents of this preface revealed his expectations regarding who the readers of the book would be. At the beginning of the preface he said “by the help of them (logarithm) we may attaine to the knowledge and use of the Mathematicks, and especially of Astronomie and Nauigation”[18]. Navigation here appeared again as one field which would benefit from the invention. The rest of the preface was a very comprehensible and brief introduction on how to use the tables in the book. If he thought the book was only going to be used by scholars for academic research, it would not be necessary to include so many details on how to comprehend the table and the notions in the preface; instead, he should probably concentrate more on mathematical principles in the construction of the conans of the logarithms to offer the justification of the new method. In his lectures at Gresham College, he always started with the geometrical origin of the logarithms to show his students that this was a sound method. The reason why he penned so much on the use of the tables was that he assumed that the readers of the English version would not care so much about the theoretical value of the invention; he wrote exactly what the readers were concerned with: how to use the tables to perform calculations. In the later years, he devoted himself to the calculation of the logarithms for natural numbers and finally, in 1624, he published his table containing the logarithms of the numbers from 1 to 20,000 and from 90,000 to 100,000, all calculated to 14 decimal numbers. All subsequent works on logarithms in the 17^th century were concerned with calculating tables, and its value in pure mathematics had been completely ignored. The increasingly usage of new technologies in navigation and surging amount of commercial related calculations in 17^th century England continuously demanded reliable logarithm tables, not its connection with calculus or  analytical geometry. 

            In the history of mathematics, the invention of logarithms was often referred as “the bolt from the blue”[19], a flash in an insightful intellectual’s mind. However, this perspective failed to interpret this invention in the social and economical context of early 17^th century England. In the specific social and economical context, the exigencies of practical problems in navigation and commerce called for a new method to prompt the speed and accuracy of calculation. Once the resolution appeared, the society soon adopted it and developed it in a particular way to meet the social demands.  The Logarithms was introduced into China in the late 17^th century, however, it did not win any public recognition. China did not possess the proper social and economical context for the wide adoption of logarithm: China’s backwardness in navigation technology and government suppression of commercial activities made the preeminence of logarithm inconspicuous. When we reread the prefaces of the English translation of Napier’s work Descriptio Canonis, we should be with careful consideration of the circumstance in which it was written.  By deliberately examining the invention of logarithm and its relations with navigation and mercantilism in 17^th century England, one should recognize how much the scholars’ view on their studies was shaped by the outside world and how people adopted their works.