John Napier
Born: 1550 in Merchiston Castle, Edinburgh,
Scotland
Died: 4 April 1617 in Edinburgh, Scotland
John Napier's father, Archibald Napier, was an important
man in late 16th century Scotland. His family had owned the Merchiston estate
from the 1430s when one of his ancestors acquired the estate, becoming the first
Napare of Merchiston. (We shall comment shortly on the different spellings of
Napier's name.) The family also owned estates at Lennox and at Menteith and a
residence at Gartness. Archibald Napier married Janet Bothwell, the sister of
the Bishop of Orkney, in 1549 when he was only 15 years old. Their son John
Napier was born the following year. Archibald Napier was a justice-depute and
was knighted in 1565. He was appointed Master of the Mint in 1582.
Before continuing we should comment on the spelling of John
Napier. The name John is most easily dealt with as John Napier, and almost
everyone else around his time, used the old spelling "Jhone". His surname
appears in a large variety of different spellings. The forms Napeir, Nepair,
Nepeir, Neper, Napare, Naper, Naipper are all seen but John Napier would most
commonly have been written Jhone Neper at that time. The only form of Napier
that we are sure would not have been used in Napier's lifetime was the present
modern spelling "Napier"!
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, 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 honour
and profit as I do not doubt that he will...
This is a translation of the old Scots that the Bishop of
Orkney actually wrote. For those interested the original version reads:-
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 ...
Napier was educated at St Andrews University, entering the
university in 1563 at the age of 13. His mother arranged for him to live in St
Salvator's College and special arrangements were made for the Principal of the
University, John Rutherford, to take care of him personally. Napier's name
appears on the matriculation roll of St Salvator's College for 1563. Shortly
after Napier matriculated his mother died. We know that Napier spent some time
at St Andrews University and he wrote himself many years later that it was in St
Andrews that he first became passionately interested in theology.
However Napier's name does not appear in the list of those
being awarded degrees in the subsequent years so he must have left St Andrews to
study in Europe before completing a degree. Of other facts we can also be
certain. Napier did not acquire his knowledge of higher mathematics at St
Andrews nor did he acquire his deep knowledge of classical literature there.
Both these must have been acquired during his studies in Europe but no record
exists to show where he studied, although the University of Paris is highly
likely and it is also probable that he spent some time in Italy and the
Netherlands.
By 1571 Napier had returned to Scotland for he was present at
his father's second marriage which took place in that year. It was in 1571 that
Napier himself began to make arrangements for his own marriage but it was at
nearly two years before that took place. In 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 manuring 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.
The above is quoted in [11] without reference to its origin.
Napier took part in the religious controversies of the time. He
was a fervent Protestant and published, what he considered his most important
work, the Plaine Discovery of the Whole Revelation of St. John (1593).
Napier had been a fanatical Protestant from his days as an
undergraduate at St Andrews. He wrote the Plaine Discovery of the Whole
Revelation of St. John according to his preface:-
... for preventing the apparent danger of Papistry arising
within this Island...
In fact there were good reasons why Napier thought that a
change in the religious situation in Scotland might occur, for there had, for
some time, been rumours that Philip of Spain might invade Scotland. The
Plaine Discovery of the Whole Revelation of St. John did gain Napier
quite a reputation, not only within Scotland, but also on the Continent after
the work was translated into Dutch, French and German. Gibson, in [11], remarks
however:-
... I suppose that there are few indeed of the present
generation who have read, or even heard of, the book; whatever its merits may
have been they do not appeal to the modern mind...
Napier's study of mathematics was only a hobby and in his
mathematical works he writes that he often found it hard to find the time for
the necessary calculations between working on theology. He is best known,
however, for his invention of logarithms but his other mathematical
contributions include a mnemonic for formulas used in solving spherical
triangles, two formulas known as Napier's analogies used in solving spherical
triangles and an invention called Napier's bones used for mechanically
multiplying dividing and taking square roots and cube roots. Napier also found
exponential expressions for trigonometric functions, and introduced the decimal
notation for fractions.
Much of Napier's work on logarithms seems to have been done
while he was living at Gartness. The Statistical Account (Vol. xvi, page
108) contains the following:-
Adjoining the mill at Gartness are the remains of an old
house in which John Napier of Merchiston, Inventor of Logarithms, resided a
great part of his time (some years) when he was making his
calculations. It is reported that the noise of the cascade, being constant,
never gave him uneasiness, but that the clack of the mill, which was only
occasional, greatly disturbed his thoughts. He was therefore, when in deep
study, sometimes under the necessity of desiring the miller to stop the mill
that the train of his ideas might not be interrupted.
Napier's discussion of logarithms appears in Mirifici
logarithmorum canonis descriptio in 1614. Two years later an English
translation of Napier's original Latin text was published, translated by Edward
Wright. In the preface of the book Napier explains his thinking behind his great
discovery (we quote from the English translation of 1616 of the original Latin
of 1614):-
Seeing there is nothing (right well-beloved
Students of the Mathematics) 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.
Unlike the logarithms used today, Napier's logarithms are not
really to any base although in our present terminology it is not unreasonable
(but perhaps a little misleading) to say that they are to base 1/e.
Certainly they involve a constant 107 which arose from the
construction in a way that we will now explain. Napier did not think of
logarithms in an algebraic way, in fact algebra was not well enough developed in
Napier's time to make this a realistic approach. Rather he thought by dynamical
analogy. Consider two lines AB of fixed length and A'X of
infinite length. Points C and C' begin moving simultaneously to
the right, starting at A and A' respectively with the same initial
velocity; C' moves with uniform velocity and C with a velocity
which is equal to the distance CB. Napier defined A'C' (=
y) as the logarithm of BC (= x), that is
y = Nap.log x.
Napier chose the length AB to be 107, based
on the fact that the best tables of sines available to him were given to seven
decimal places and he thought of the argument x as being of the form
102.sin X.
The fact that Nap.log 1 does not equal 0 is a major difficulty
which make Nap.logs much less convenient for calculations than our logs. A
change to logs with log 1 = 0 came about in discussions between Napier and
Briggs. Briggs read Napier's 1614 Latin text and, on the 10 March 1615 wrote in
a letter to a friend:-
Napper, lord of Markinston, hath set my head and hands a
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 or made me more
wonder.
In fact Briggs did make the difficult journey from London to
Edinburgh to see Napier in the summer of 1615 (would he have dreamed that now
it takes 4 hours by train, rather than at least 4 days by horse and coach in
those times). A description of their meeting was told by John Marr to William
Lilly who writes the following (see [11]):-
Mr Briggs appoints a certain day when to meet at
Edinburgh; but failing thereof, Merchiston was fearful he would not come.
It happened one day as John Marr and the Lord Napier were speaking of Mr Briggs,
"Oh! John," saith Merchiston, "Mr Briggs will not come now"; at the
very instant one knocks at the gate, John Marr hastened down and it proved
to be Mr Briggs to his great contentment. He brings Mr Briggs
into my Lord's chamber, where almost one quarter of an hour was spent, each
beholding other with admiration, before one word was spoken. At last Mr Briggs
began, -"My Lord, I have undertaken this long journey purposely to see your
person, and to know by what engine of wit or ingenuity you came first to think
of this most excellent help unto astronomy, viz. the Logarithms ...
Briggs had suggested to Napier in a letter sent before their
meeting that logs should be (in our terminology) to base 10 and Briggs had begun
to construct tables. Napier replied that he had the same idea but ([11]):-
... he could not, on account of ill-health and for other
weighty reasons undertake the construction of new tables.
At their meeting Napier suggested to Briggs the new tables should
be constructed with base 10 and with log 1 = 0, and indeed Briggs did construct
such tables. In fact Briggs spent a month with Napier on his first visit of
1615, made a second journey from London to Edinburgh to visit Napier again in
1616 and would have made yet a third visit the following year but Napier died
in the spring before the planned summer visit.
Napier presented a mechanical means of simplifying calculations
in his Rabdologiae published in 1617. He described a method of
multiplication using "numbering rods" with numbers marked off on them. The
reason for publishing the work is given by Napier in the dedication, where he
says that so many of his friends, to whom he had shown the numbering rods, were
so pleased with them that they were already becoming widely used, even beginning
to be used in foreign countries.
Napier's numbering rods were made of ivory, so that they looked
like bones which explains why they are now known as Napier's bones. To multiply
numbers the bones were placed side by side and the appropriate products read
off. Glaisher described how to use Napier's bones in an article he wrote for
Encyclopaedia Britannica and this description is quoted in [10]. Napier's
bones are also described in [5], [15] and [18].
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. Many
traditions suggest that Napier was
... in league with the powers of darkness...
and these are taken seriously in the biased biography [6]
written by Mark Napier, one of John Napier's descendants. Mark Napier suggests
that John Napier deliberately played upon the primitive beliefs of his servants
by going round with a cock which he had covered in soot. Even the Statistical
Account (quoted above) says:-
[Napier] used frequently to walk out in his
nightgown and cap. This, with some things which to the vulgar appear rather
odd, fixed on him the character of a warlock. It was formerly believed and
currently reported that he was in compact with the devil; and that the time he
spent in study was spent in learning the black art and holding conversation
with Old Nick.
Napier, however, will be remembered for making one of the most
important contributions to the advance of knowledge. It was through the use
of logarithms that Kepler was able to reduce his observations and make his breakthrough
which then in turn underpinned Newton's theory of gravitation. In the preface
to the Mirifici logarithmorum canonis descriptio, quoted above, Napier
says he hoped that his logarithms will save calculators much time and free them
from the slippery errors of calculations. Laplace, 200 year later, agreed,
saying that logarithms:-
...by shortening the labours, doubled the life of the
astronomer .