Mathematics

John Napier

13th Oct '209 views1 mins read

13 October 2020                

Read time: 5 minutes

Introduction

John Napier Scottish mathematician

John Napier (born 1550, Merchiston Castle, near Edinburgh, Scot - died April 4, 1617, Merchiston Castle) was a Scottish mathematician and theological writer who originated the concept of logarithms to aid in calculations. He also invented the "Napier's bones" and made decimal points of common use in arithmetic and mathematics.


Early Life

At the age of 13, Napier entered the University of St. Andrews, but his stay appears to have been short, and he left without taking a degree.

There is little known about Napier’s early life, but it is thought that he traveled abroad, as was then the custom of the sons of the Scottish landed gentry. He was certainly back home in 1571, and he stayed either at Merchiston or Gartness for the rest of his life. He married the following year. A few years after his first wife’s death in 1579, he married again.


Theology And Inventions

Napier’s life was spent amid bitter religious dissensions. In his dealings with the Church of Rome, a passionate and uncompromising Protestant sought no quarter and gave none. It was well known that James VI of Scotland hoped to succeed Elizabeth I to the English throne, and it was suspected that he had sought the help of the Catholic Philip II of Spain to achieve this end. Panic-stricken at the peril that seemed to be impending, the Scottish Church’s general assembly, a body with which Napier was closely associated, begged James to deal effectively with the Roman Catholics. On three occasions, Napier was a member of a committee appointed to make representations to the King concerning the welfare of the church and to urge him to see that “justice be done against the enemies of God’s Church.”

In January 1594, Napier addressed the King a letter that forms the dedication of his Plaine Discovery of the Whole Revelation of Saint John, a work that, while it professed to be of a strictly literary character, was calculated to influence contemporary events. In it, he declared:

 
important notes to remember
 

“Let it be your Majesty’s continuall study to reforme the universall enormities of your country, and first to begin at your Majesty’s owne house, familie and court, and purge the same of all suspicion of Papists and Atheists and Newtrals, whereof this Revelation forthtelleth that the number shall greatly increase in these latter daies.”

The work occupies a prominent place in Scottish ecclesiastical history.

Following this work’s publication, Napier seems to have occupied himself with the invention of secret instruments of war, for in a manuscript collection now at Lambeth Palace, London. There is a document bearing his signature, enumerating various inventions “designed by the Grace of God, and the worke of expert craftsmen” to defend his country. These inventions included two kinds of burning mirrors, a piece of artillery, and a metal chariot from which shot could be discharged through small holes.


Contribution To Mathematics

Napier devoted most of his leisure to studying mathematics, particularly to devising methods of facilitating computation, and it is with the greatest of these, logarithms, that his name is associated. He began working on logarithms probably as early as 1594, gradually elaborating his computational system whereby roots, products, and quotients could be quickly determined from tables showing powers of a fixed number used as a base.

His contributions to this powerful mathematical invention are contained in two treatises: Mirifici Logarithmorum Canonis Descriptio (Description of the Marvelous Canon of Logarithms), which was published in 1614, and Mirifici Logarithmorum Canonis Constructio (Construction of the Marvelous Canon of Logarithms), which was published two years after his death. In the former, he outlined the steps that had led to his invention.

Logarithms are meant to simplify calculations, especially multiplication, such as those needed in astronomy. Napier discovered that the basis for this computation was a relationship between an arithmetical progression—a sequence of numbers in which each number is obtained, following a geometric progression, from the one immediately preceding it by multiplying by a constant factor, which may be greater than unity (e.g., the sequence 2, 4, 8, 16 . . . ) or less than unity (e.g., 8, 4, 2, 1, 1/2 . . . ).

In the Descriptio, besides giving an account of the nature of logarithms, Napier confined himself to an understanding of the use to which they might be put. He promised to explain the method of their construction in later work. This was the Constructio, which claims attention because of the systematic use in its pages of the decimal point to separate the fractional from the integral part of a number. Decimal fractions had already been introduced by the Flemish mathematician Simon Stevin in 1586, but his notation was unwieldy. The use of a point as the separator frequently occurs in the Constructio. Joost Bürgi, the Swiss mathematician, between 1603 and 1611 independently invented a system of logarithms, which he published in 1620. But Napier worked on logarithms earlier than Bürgi and has the priority due to his prior date of publication in 1614.

Although Napier’s invention of logarithms overshadows all his other mathematical work, he made other mathematical contributions. In 1617 he published his Rabdologiae, seu Numerationis per Virgulas Libri Duo (Study of Divining Rods, or Two Books of Numbering by Means of Rods, 1667); in this, he described ingenious methods of multiplying and dividing of small rods known as Napier’s bones, a device that was the forerunner of the slide rule. He also made significant contributions to spherical trigonometry, particularly by reducing the number of equations used to express trigonometrical relationships from 10 to 2 general statements. He is also credited with certain trigonometrical relations—Napier’s analogies—but it seems likely that the English mathematician Henry Briggs had a share in these.


Napier’s Bones

Napier's bones is a manually-operated calculating device created by John Napier

Napier's bones is a manually-operated calculating device created by John Napier for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called 'rabdology’. Napier published his version in 1617 in Rabdologiæ, printed in Edinburgh, dedicated to his patron Alexander Seton.

Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to subtractions which is also the layman definition of multiplication and division. Advanced use of the rods could be made to extract square roots as well. Napier's bones are not the same as logarithms, with which Napier's name is also associated, but are based on dissected multiplication tables.

The complete device usually includes a baseboard with a rim; the user places Napier's rods inside the rim to conduct multiplication or division. The board's left edge is divided into nine squares, holding the numbers 1 to 9. In Napier's original design, the rods are made of metal, wood, or ivory and have a square cross-section. Each rod is engraved with a multiplication table on each of the four faces. In some later designs, the rods are flat and have two tables or only one engraved on them and made of plastic or heavy cardboard. A set of such bones might be enclosed in a carrying case.

A rod's face is marked with nine squares. Each square except the top is divided into two halves by a diagonal line from the bottom left corner to the top right. The squares contain a simple multiplication table. The first holds a single digit, which Napier called the 'single'. The others hold the multiples of the single, namely twice the single, three times the single, and so on up to the ninth square containing nine times the number in the top square. Single-digit numbers are written in the bottom right triangle leaving the other triangle blank, while double-digit numbers are written with a digit on either side of the diagonal.

If the tables are held on single-sided rods, 40 rods are needed in order to multiply 4-digit numbers – since numbers may have repeated digits, four copies of the multiplication table for each of the digits 0 to 9 are needed. If square rods are used, the 40 multiplication tables can be inscribed on 10 rods. Napier gave details of a scheme for arranging the tables so that no rod has two copies of the same table, enabling every possible four-digit number to be represented by 4 of the 10 rods. A set of 20 rods, consisting of two identical copies of Napier's 10 rods, allows calculation with numbers of up to eight digits, and a set of 30 rods can be used for 12-digit numbers.

For learning about the math behind Bones, please visit Link.


Frequently Asked Questions (FAQs)

When did John Napier develop Logarithms?

John Napier, the Scottish mathematician, published his discovery of logarithms in 1614.

What is John Napier famous for?

John Napier is best known as the discoverer of logarithms. He also invented the "Napier's bones" and made the decimal point a common use in arithmetic and mathematics.

What is Napier’s Bones machine used for?

Napier's bones is a manually-operated calculating device created by John Napier of Merchiston, Scotland, to calculate products and quotients of numbers.


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