life
tt<N
John Napier and the Invention
of Logarithms, 1614
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John Napier and the Invention
of Logarithms, 1614
A LECTURE
BY
E. W. HOBSON, Sc.D., LL.D., F.R.S.
sadle irian professor of pure mathematics
fellow of Christ's college, Cambridge
Cambridge :
at the University Press
1914
(Eambrtoge:
PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS
EMS
LIB
Q\
4
JOHN NAPIER AND THE INVENTION
OF LOGARITHMS, 1614
In the present year there will be held a
celebration, under the auspices of the Royal
Society of Edinburgh, of the tercentenary of
one of the great events in the history of Science,
the publication of John Napier's " Mirifici
Logarithmorum Canonis Descriptio," a work
which embodies one of the very greatest scien-
tific discoveries that the world has seen. The
invention of Logarithms not only marks an
advance of the first importance in Mathematical
Science, but as providing a great labour-saving
instrument for the use of all those who have
occasion to carry out extensive numerical calcu-
lations it can be compared in importance only
with the great Indian invention of our system
of numeration.
6 JOHN NAPIER
It is almost always extremely instructive to
study in detail the form in which a great dis-
covery or invention was presented by its
originator, and to trace in detail the mode in
which the fundamental ideas connected with the
discovery shaped themselves in his mind, even
when, and just because, later developments or
simplifications may have so transformed the under-
lying principles, and still more the practice, of the
invention, that we have become accustomed to
look at the matter from a point of view, at least
superficially, very different from the original one
of the discoverer. The case of logarithms is very
far from being an exception to this rule ; accord-
ingly I propose to give an account, as concise as
may be, of the conception of a logarithm in the
mind of Napier, and of the methods by which he
actually constructed his table of logarithms.
In order fully to appreciate the nature of the
difficulties of the task accomplished by the genius
of John Napier, some effort of imagination is
JOHN NAPIER 7
required, to be expended in realizing the narrow-
ness of the means available in the early part of
the seventeenth century for the calculation of the
tables, at a time, before the invention of the
Differential and Integral Calculus, when calcula-
tion by means of infinite series had not yet been
invented. Napier's conception of a logarithm
involved a perfectly clear apprehension of the
nature and consequences of a certain functional
relationship, at a time when no general conception
of such a relationship had been formulated, or
existed in the minds of Mathematicians, and
before the intuitional aspect of that relationship
had been clarified by means of the great invention
of coordinate geometry made later in the century
by Ren£ Descartes. A modern Mathematician
regards the logarithmic function as the inverse of
an exponential function ; and it may seem to us,
familiar as we all are with the use of operations
involving indices, that the conception of a loga-
rithm would present itself in that connection as a
8 JOHN NAPIER
fairly obvious one. We must however remember
that, at the time of Napier, the notion of an index,
in its generality, was no part of the stock of ideas
of a Mathematician, and that the exponential
notation was not yet in use.
Summary of the life of Napier.
I must content myself with giving an exceed-
ingly brief account of the external facts of the life
of Napier*.
John Napier^, the eighth Napier of Merchiston,
usually described as Baron, or Fear, of Merchiston,
was born at Merchiston near Edinburgh in 1550,
when his father Archibald Napier was little more
than sixteen years old. John Napier matriculated
at St Andrews in 1563, but did not stay there
* For a full account of the life and activities of Napier the
" Memoirs of John Napier of Merchiston " by Mark Napier,
published in 1834, may be consulted.
t The name Napier was spelled in various ways, several
of which were used by John Napier; thus we find Napeir,
Nepair, Nepeir, Neper, Nepper, Naper, Napare, Naipper.
JOHN NAPIER 9
sufficiently long to graduate, as he departed
previous to 1566 in order to pursue his studies
on the Continent, whence he returned to Mer-
chiston in or before 1571. His first marriage, by
which he had one son Archibald who was raised
to the peerage in 1627 as Lord Napier, and one
daughter, took place in 1572. A few years after
the death of his wife in 1579, he married again.
By his second marriage he had five sons and
five daughters ; the second son, Robert, was his
literary executor. The invasion of the Spanish
Armada in 1588 led Napier, as an ardent Protes-
tant, to take a considerable part in Church politics.
In January 159^ he published his first work "A
plaine discovery of the whole Revelation of
St John." This book is regarded as of consider-
able importance in the history of Scottish theo-
logical literature, as it contained a method of
interpretation much in advance of the age ; it
passed through several editions in English, French,
German, and Dutch.
io JOHN NAPIER
In July 1594, Napier entered into a curious
contract with a turbulent baron, Robert Logan of
Restalrig, who had just been outlawed. In this
contract, which appears to shew that John Napier
was not free from the prevalent belief in Magic,
he agreed to endeavour to discover a treasure
supposed to lie hidden in Logan's dwelling-place,
Fast Castle. Napier was to receive a third part
of the treasure when found, in consideration that
"the said J hone sail do his utter & exact dili-
gens to serche & sik out, and be al craft & ingyne
that he dow, to tempt, trye, and find out the
sam, and be the grace of God, ather sail find the
sam, or than mak it suir that na sik thing hes
been thair ; sa far as his utter trawell diligens and
ingyne may reach."
In a document dated June 7, 1596, Napier
gave an account of some secret inventions he had
made which were "proffitabill & necessary in
theis dayes for the defence of this Hand & with-
standing of strangers enemies of God's truth &
JOHN NAPIER ii
relegion." His activities in this direction were no
doubt stimulated by the fear of the generally
expected invasion by Philip of Spain. It is
interesting to note them, in view of the military
tastes of many of his descendants. The inventions
consisted of a mirror for burning the enemies'
ships at any distance, of a piece of artillery capable
of destroying everything round an arc of a circle,
and of a round metal chariot so constructed that
its occupants could move it rapidly and easily,
while firing out through small holes in it. Napier's
practical bent of mind was also exhibited in the
attention he paid to agriculture, especially on the
Merchiston estate, where the land was tilled by a
system of manuring with salt.
There is evidence that Mathematics occupied
Napier's attention from an early age. From a
MS. that was first published in 1839 under the
title " De Arte Logistica " it appears that his
investigations in Arithmetic and Algebra had led
him to a consideration of the imaginary roots of
12 JOHN NAPIER
equations, and to a general method for the ex-
traction of roots of numbers of all degrees. But,
led probably by the circumstances of the time, he
put aside this work in order to devote himself to
the discovery of means of diminishing the labour
involved in numerical computations. The second
half of the sixteenth century was the time in which
the Mathematicians of the Continent devoted a
great deal of attention to the calculation of tables
of natural trigonometrical functions. The most
prominent name in this connection is that of
Georg Joachim Rheticus, the great computer
whose work has never been superseded, and the
final result of whose labours is embodied in the
table of natural sines for every ten seconds to
fifteen places of decimals, published by Pitiscus in
1613, the year before the publication by Napier
of the discovery which was destined to revolu-
tionize all the methods of computation, and to
substitute the use of logarithmic for that of natural
trigonometrical functions. It was in the early
JOHN NAPIER 13
years of the seventeenth century that Johannes
Kepler was engaged in the prodigious task of
discovering, and verifying by numerical calculation,
the laws of the motion of the planets. In this age
of numerical calculation then Napier occupied
himself with the invention of methods for the '
diminution of the labour therein involved. He
himself states in his " Rabdologia," to which
reference will presently be made, that the
canon of logarithms is "a me longo tempore
elaboratum." It appears from a letter of Kepler
that a Scotsman, probably Thomas Craig, a
friend of the Napier family, gave the astronomer
Tycho Brahe in the year 1594 hopes that an
important simplification in the processes of arith-
metic would become available. There is strong
evidence that Napier communicated his hopes to
Craig twenty years before the publication of the
Canon.
The " Descriptio," of which an account will
be given presently, was as stated at the outset
i 4 JOHN NAPIER
published in 1614. About the same time Napier
devised several mechanical aids for the perform-
ance of multiplications and divisions and for the
extraction of square and cube roots. He published
an account of these inventions in 161 7 in his
" Rabdologia," as he says, " for the sake of those
who may prefer to work with the natural numbers."
The method which Napier calls Rabdologia
consists of calculation of multiplications and
divisions by means of a set of rods, usually called
"Napier's bones." In 1617, immediately after
the publication of the " Rabdologia," Napier
died.
The " Descriptio " did not contain an account
of the methods by which the " wonderful canon "
was constructed. In an "Admonitio" printed at
the end of Chapter 11, Napier explains that he
prefers, before publishing the method of construc-
tion, to await the opinion of the learned world
on the canon ; he says " For I expect the judge-
ment & censure of learned men hereupon, before
JOHN NAPIER 15
the rest rashly published, be exposed to the
detraction of the envious."
The " Mirifici Logarithmorum Canonis Con-
struction' which contains a full explanation of the
method of construction of the wonderful canon,
and a clear account of Napier's theory of loga-
rithms, was published by his son Robert Napier
in 16 1 9. In the preface by Robert Napier it is
stated that this work was written by his father
several years before the word "logarithm" was
invented, and consequently at an earlier date than
that of the publication of the "Descriptio." In
the latter the word " logarithm " is used through-
out, but in the " Constructio," except in the title,
logarithms are called " numeri artificiales." After
explaining that the author had not put the finish-
ing touch to the little treatise, the Editor writes
"Nor do I doubt that this posthumous work
would have seen the light in a much more perfect
& finished state, if God had granted a longer
enjoyment of life to the Author, my most dearly
1 6 JOHN NAPIER
beloved father, in whom, by the opinion of the
wisest men, among other illustrious gifts this
shewed itself pre-eminent, that the most difficult
matters were unravelled by a sure and easy
method, as well as in the fewest words."
Reception of the Canon by Contemporary
Mathematicians.
The new invention attracted the attention
ol British and Foreign Mathematicians with a
rapidity which may well surprise us when we take
into account the circumstances of the time. In
particular the publication of the wonderful canon
was received by Kepler with marked enthusiasm.
In his "Ephemeris" for 1620, Kepler published
as the dedication a letter addressed to Napier,
dated July 28, 161 9, congratulating him warmly
on his invention and on the benefit he had con-
ferred upon Astronomy. Kepler explains how he
verified the canon and found no essential errors
JOHN NAPIER 17
in it, beyond a few inaccuracies near the begin-
ning of the quadrant. The letter was written two
years after Napier's death, of which Kepler had
not heard. In 1624 Kepler himself published a
table of Napierian logarithms with modifications
and additions. The " Descriptio," on its publi-
cation in 1 6 14, at once attracted the attention of
Henry Briggs (1 556-1 630), Fellow of St John's
College, Cambridge, Gresham Professor of Geo-
metry in the City of London, and afterwards
Savilian Professor of Geometry at Oxford, to
whose work as the successor of Napier in the task
of construction of logarithmic tables in an improved
form I shall have to refer later. In a letter
to Archbishop Ussher, dated Gresham House,
March 10, 161 5, Briggs wrote, " Napper, lord of
Markinston, hath set my head & hands a work
with his new & admirable logarithms. I hope to
see him this summer, if it please God, for I never
saw book which pleased me better, or made me
more wonder." Briggs visited Napier, and stayed
H.
1 8 JOHN NAPIER
with him a month in 1615, again visited him in
1 61 6, and intended to visit him again in 16 17,
had Napier's life been spared. Another eminent
English Mathematician, Edward Wright, a Fellow
of Gonville and Caius College, who at once saw
the importance of logarithms in connection with
navigation, in the history of which he occupies a
conspicuous place, translated the " Descriptio,"
but died in 161 5 before it could be published.
The translation was however published in 1618
by his son Samuel Wright.
The contents of the " Descriptio " and of the
" Construction
The " Descriptio " consists of an ornamental
title page, fifty-seven pages of explanatory matter,
and ninety pages of tables. A specimen page of
the tables is here reproduced. The explanatory
matter contains an account of Napier's conception
of a logarithm, and of the principal properties of
logarithms, and also of their application in the
Gr.
P
fnin\
+ 1 —
Smut } Logarithmi \ Differentia \ U^arithmi f | Sinus f
1*6434?
1*67218
1570091
1572964
15787°?
18**1174
18532826
18514*11'
18427295
18408484
18389707
1581*81
1*84453
15873**
1590197
1593069
I59594I
1849*23*
18477984
1 8459772
18441594
18423451
18405341
113*81
124342
124804
18387265
18369223
18351214
j 8 370964
183*2253
i83>35 76
183H933
18296324
18277747
125267
12*731
1 26 196
18259203
18240692
18222213
126662
127127
i"-7*94
9876882
5876427
9875971
9875514
987*056
9874597
0874137
9873677
9873216
1598812
1601684
1604555
1607426
1610297
1613168
1616038
1618909
1621779
18333237
18315294
182^7384
18279*07
18261663
182438*1
18226071
18208323
18190606
18203765
1818*3*1
18166969
18148619
18130301
18112014
128062
128*31
129061
? 29472
129943
130415
60
59
_**
57
56
jj
*4
*3
*2
130888]
1 3 1 3 62
131837
18093758
1807*533
180573 z 8
1624649
1627*19
1630389
76 33~259
1636129
1638999
18172924
1815*273
18137654
18120067
18102*11
18084987
13*313
132790
133268
18039177
18021047
18002948
17984880
17966842
17948835
133747
13-4226
134706
13*187
13*669
136152
9871362
9870897
98704^1
9869964
9869496
9869027 1
51
50
9872754
9872291
9871827 J 49
48
47
46
4*
44
43
9X6^5 57.1
9868087
9867616 I
4*
41
43
9867144
9866671
9866197
39
38
37
9*65722
9865246
9864770
36
3*
34
33
3*
u
30
ttin
t
1641868
1644738
1647607
16*0476
1806749*
18050034
18032604
1801*207
I K
\ 17930859
17912913
17894997
T7I77TT4
136636
137121
137607
138093
9864293
9863815
9863336
98628*6
80
JOHN NAPIER 19
solution of plane and spherical triangles. Napier's
well-known rules of circular parts containing the
complete system of formulae for the solution of
right-angled spherical triangles are here given.
The logarithms given in the tables are those of
the sines of angles from o° to 90 at intervals of
one minute, to seven or eight figures. The table
is arranged semi-quadrantally, so that the loga-
rithms of the sine and the cosine of an angle
appear on the same line, their difference being
given in the table of differentiae which thus forms
a table of logarithmic tangents. It must be
remembered that, at that time and long after-
wards, the sine of an angle was not regarded, as
at present, as a ratio, but as the length of that
semi-chord of a circle of given radius which sub-
tends the angle at the centre. Napier took the
radius to consist of io 7 units, and thus the sine of
90°, called the whole sine, is io v ; the sines of
smaller angles decreasing from this value to zero.
The table is therefore one of the logarithms of
2 — 2
2o JOHN NAPIER
numbers between io 7 and o, not for equidistant
numbers, but for the numbers corresponding to
equidistant angles. It is important to observe
that the logarithms in Napier's tables are not
what we now know under the name of Napierian
or natural logarithms, i.e. logarithms to the
base e. His logarithms are more closely related
to those to the base i/e ; the exact relation
Nap
is that, if x is a number, and Log x its
logarithm in accordance with Napier's tables,
Nap
T qqt % . X
— ^— is the logarithm of — - t to the base
io 7 b io 7
lie ; thus
Nap
, > Log.*
X /l V
io'
io' \ej
Nap
or Log x = i o 7 log e i o 7 — i o 7 log e x.
Napier had no explicit knowledge of the exist-
ence of the number £, nor of the notion of the
base of a system of logarithms, although as we
shall see he was fully cognizant of the arbitrary
JOHN NAPIER 21
element in the possible systems of logarithms.
His choice was made with a view to making the
logarithms of the sines of angles between o° and
90 , i.e. of numbers between o and io 7 , positive
and so as to contain a considerable integral part
The " Constructio " consists of a preface of
two pages, and fifty-seven pages of text. The
conception of a logarithm is here clearly explained,
and a full account is given of the successive steps
by which the Canon was actually constructed. In
this work one of the four formulae for the solution
of spherical triangles, known as Napier's analogies,
is given, expressed in words ; the other three
formulae were afterwards added by Briggs, being
easily deducible from Napier's results.
The decimal point.
Our present notation for numbers with the
decimal point appears to have been independently
invented by Napier, although a point or a half
bracket is said to have been employed somewhat
22 JOHN NAPIER
earlier by Jobst Biirgi for the purpose of separat-
ing the decimal places from the integral part of a
number*. The invention of decimal fractions was
due to Simon Stevin (i 548-1 620), who published
a tract in Dutch, "De Thiende," in 1585, and in
the same year one in French under the title " La
Disme," in which the system of decimal fractions
was introduced, and in which a decimal system of
weights, measures and coinage was recommended.
In the "Rabdologia" Napier refers to Stevin in an
'* Admonitio pro Decimali Arithmetica " in which
he emphasizes the simplification arising from the
use of decimals, and introduces the notation with
the decimal point. By Stevin and others the
notation 94© 1 ©3 ®o® 5 or 94 i'fo'"f",
for example, was used instead of Napier's notation
94*1305. Later on Briggs sometimes used the
notation 941305. It is clear that the notation
introduced by Napier, which was however not
* The decimal point was also employed by Pitiscus in the
tables appended to the later edition, published in 1612, of his
" Trigonometria."
JOHN NAPIER 23
universally adopted until the eighteenth century,
was far better adapted than the more complicated
notation used by Stevin and later writers to bring
out the complete parity of the integral and decimal
parts of a number in relation to the operations of
arithmetic, and to emphasize the fact that the
system of decimal fractions involves only an
extension of the fundamental conception of our
system of notation for integral numbers, that the
value of a digit, in relation to the decimal scale, is
completely indicated by its position.
Napier s definition of a logarithm.
1 1
A T
1 111
a AAA
1
s
ft T x
& Q2 G &
tO CO
Napier supposes that on a straight line TS
which he takes to consist of io 7 units, the radius
of the circle for which the sines are measured, a
point P moves from left to right so that its velocity
is at every point proportional to the distance from
24 JOHN NAPIER
S. He supposes that on another straight line a
point Q moves with uniform velocity equal to that
which P has when at T, and that Q is at T lt when
P is at T. When P has any particular position
Pj in the course of its motion, the logarithm of the
sine or length SP 1 is defined to be the number
representing the length T X Q X , from T x to the
position <2i of Q at the time when P is at P x ,
Thus the logarithm of the whole sine TS(= io 7 )
is o, and the logarithm of any sine less than io 7
is positive and increases indefinitely as the sine
diminishes to zero. Napier recognized that in
accordance with this definition, the logarithm of
a number greater than io 7 , corresponding to Sp lf
where p 1 is the position of P when it has not yet
reached T, will be negative, the corresponding
position of Q being on the left of T x .
Let Q lt Q 2 , Q 3 , <2 4 , ... be a number of positions
of Q such that Q 1 Q i = Q 2 Q 3 =Q 3 Q i = ... and let
P lt P 2> P 3 , ... be the corresponding positions of
P; so that P X P^ P 2 P 3 , P%P» ••• are spaces
JOHN NAPIER 25
described by P in equal times. Napier then
shews by means of special illustrations that
and thus that, corresponding to a series of values,
of T^Q that are in arithmetic progression, there
are a series of values of SP that are in geometric
progression.
The matter may be put in a concise form
which represents the gist of Napier's reasoning,
and of the essential point of which he had a clear
intuition.
P P'
1 > r-J 1 .
PS PS
Let -p-* = -^-p. ; and let p be any point
in P x P it and /' the corresponding point in P t P, ;
so that P,p : pP^P^p' : p'P 3 . The velocity of
the moving point when at/ bears a constant ratio
^=^=-~-p) to its velocity when at /.
As this holds for every corresponding pair of
26 JOHN NAPIER
points p,p' in the two intervals P X P» P 2 P 3 , it is
dear that the motion in P^P % takes place in the
same time as that in PJ\ ; the velocities at all
corresponding points being changed in the same
ratio, that of P x P t to P 2 P a . Hence the result
follows that
P 1 S_P 2 S_P 9 S_
P t S m P t S m P t S m " u
if the points Q lf <2 2 , Q 3 , Q 4 , ... are such that
QiQ*=Q*Q*=Q%Qi=->\ i.e. if the spaces P x P %t
P 2 P 3 , P 3 P 4 , ... are described in equal times. Thus
the logarithms of a set of numbers in geometric
progression are themselves in arithmetic pro-
gression.
In our modern notation, if x = SP, we have
— = r, where V denotes the velocity of P
dt io 7
at T; and if V—'T.Q, -Z=V\ thus -y- = =;
* dt dy io 7
and accordingly Napier's method amounts to an
intuitional representation of the integration of this
differential equation.
JOHN NAPIER 27
The limits of a logarithm.
As no method was available by which a
logarithm could be calculated to an arbitrarily
great degree of approximation, Napier obtained
two limits between which a logarithm must lie,
and his whole method of construction depends
upon the use of these limits, together with corre-
sponding limits for the value of the difference of
the logarithms of two numbers.
— 1 1 1
T P x S
— 1 1 1
ft T, G
Since the velocities of P and Q at T, T x are
the same, and the velocity of P decreases after-
wards whereas that of Q remains constant, it is
clear that TP l < T^Q^. Again let p t T on the left
of T be described in the same time as TP Xi so
that ^7", = 7\ Q v It is then clear that p l T>q l T l .
28 JOHN NAPIER
Nap
If x = P 1 S, hogx= T X Q„ we thus have
Nap
hogx> TP lt or jo 7 —x;
Nap IQ 7
and Log x=y 1 T 1 <f> 1 T, or TP 1 — -, which is
(io 7 -*) — . Thus
' x
j Q 7 Nap
(io 7 — x) — >Logx> io 7 —x ...(i);
these are Napier's limits for a logarithm.
In a similar manner it is shewn that
y—x Nap Nap y — x
\o- > hogx— Logy> io 7 - ...(2),
where x<y; these are the limits which Napier
Nap Nap
employs for Log x — Log jj/.
The results (1) and (2) were given by Napier
in words, no short designation being employed
even for a logarithm.
Napier s construction of the canon.
The first step taken by Napier in the process
of constructing the canon was to form three tables
JOHN NAPIER 29
of numbers in geometric progression. The first
table consists of 101 numbers, of which io 7 is the
first, and of which 1 = is the common ratio ;
io 7
thus (in modern notation) the table consists of the
numbers io 7 ( 1 ; ) , where r has the values o to
\ io 7 /
100. Each number was formed by subtracting
from the preceding one the number obtained by
moving the digits seven places to the right.
First Table
X°V~itf)> '-oto iooj
I OOOOOOO'OOOOOOO
I'OOOOOOO
99999990000000
-9999999
9999998 'OOOOOO I
'9999998
999999 7 '0000003
'9999997
9999996*0000006
to be continued up to
9999900*0004950
30 JOHN NAPIER
The second table consists of the 51 numbers
io 7 (i 1) , where r = o, 1, ... 50. The com-
mon ratio 1 1 is nearly equal to ( 1 7 J ,
that of the last number in the first table, to the
first in that table.
Second Table
{ Io7 ( x -^) r ' "'—»*}
I OOOOOOO'OOOOOO
IOO'OOOOOO
9999900*000000
99*999000
9999800*001000
to be continued up to
9995001*222927
In this table there is an arithmetical error, as
the last number should be 9995001*224804; the
effect of this error on the canon will be referred
to later.
The ratio of the last number to the first is
1 N 50
1 " loV ' Whkh is nearly I ~ 2oW'
JOHN NAPIER 31
The third table consists of 69 columns, and
each column contains 21 numbers. The first
number in any column is obtained by taking
1 — y^ of the first number in the preceding
column. The numbers in any one column are
obtained by successive multiplication by 1 — 275V0 ;
thus the/^ number in the q th column is
First column
I OOOOOOO "oooo
9995000*0000
9990002*5000
9985007*4987
continued to
9900473-5780
Third Table
Second column
9900000-0000 .
9895050*0000 .
9890102-4750 .
9885157-4237 .
continued to
69th column
5048858*8900
5046334-4605
5043811-2932
5041289-3879
continued to
4998609-4034
9801468-8423
The ratio of the last to the first number in
any one column is ( 1 — ^uVo") 20 , which is nearly
i-t4tt or i^fr-
It will be observed that the last number of the
last column is less than half the radius, and thus
corresponds to the sine of an angle somewhat less
than 30°.
32 JOHN NAPIER
In this table there are, speaking roughly,
68 numbers in the ratio ioo : 99 interpolated
between io 7 and \\d ; and between each of these
are interpolated twenty numbers in the ratio
10000 : 9995.
Having formed these tables, Napier proceeds
to obtain with sufficient approximation the loga-
rithms of the numbers in the tables. For this
purpose his theorems (1) and (2) as to the limits
of logarithms are sufficient. In the first table, the
logarithm of 9999999 is, in accordance with (1),
between rooooooi and I'ooooooo, and Napier
takes the arithmetic mean 1*00000005 f° r tne
required logarithm. The logarithm of the next
sine in the table is between 2 "0000002 and
2 •0000000; for this he takes 2-00000010, for
the next sine 3*00000015, and so on.
The theorem (2) is used to obtain limits for
the logarithms of numbers nearly equal to a
number in the first table. In this way the loga-
rithm of 9999900 the second number in the second
JOHN NAPIER 33
table is found to lie between 100*0005050 and
100*0004950; the next logarithm has limits double
of these, and so on. The logarithm of the last
sine in the second table is thus found to lie
between 5000*0252500 and 5000*0247500. The
logarithms of the numbers in the second table
having thus been found to a sufficient degree of
approximation, the logarithm of a number near
one in the second table is found thus : — Let y
be the given sine, x the nearest sine in the table ;
say y < x. Determine z so that — ; = - , then
J y 10 7 x
Nap Nap Nap Nap Nap
Log g = Log z — Log io 7 = Logjj/ — Log .a: ;
Nap
find the limits of Logz by means of the first table,
and when these are found add them to those of
Nap Nap
Log:r, and we thus get the limits of Logy. In
this manner limits are found for the logarithms
of all the numbers in the first column of the
third table; thus those of 9900473*57808 are
100024*9657720 and 100024*9757760, and the
h. 3
34 JOHN NAPIER
logarithm is taken to be 100024*9707740, the
mean of the two limits. The first number in
the second column differs only in the fifth cypher
from the last number in the first column, and thus
its logarithm can be calculated approximately.
The logarithms of all the other numbers in the
table can then be found, since the logarithms of
all the numbers in any one column, or in any one
row, are in arithmetic progression.
When the logarithms of all the numbers in
the third table have thus been calculated, the
table formed by filling them in is called by
Napier his radical table, and is of the form
given on the opposite page.
The radical table being completed, the loga-
rithms in it are employed for the calculation of
the principal table or canon. For this purpose
the logarithms of sines very nearly equal to the
whole sine io 7 are obtained simply by subtracting
the given sine from io ? . The logarithm of a sine
embraced within the limits of the radical table is
<3
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3—2
36
JOHN NAPIER
found thus : — Let x be the given sine, y the
nearest sine in the table ; say x >y ; calculate
(x— y) io ? and divide it either by x or by y, or
by some number between the two ; then add the
result to the logarithm of the table sine.
For the purpose of finding the logarithm of a
sine which is not embraced within the limits of
the radical table, Napier gives a short table
containing the difference of the logarithms of two
sines of which the ratio is compounded of the
ratios 2 : 1, and 10 : 1.
Short Table
Given
Corresponding
Given
Corresponding
proportion
difference of
proportion
difference of
of sines
logarithms
of sines
logarithms
2
to I
6931469-22
8000 to
I 89871934*68
4
>J
13862938-44
I OOOO ,
, 92103369-36
8
51
20794407*66
20000 ,
, 99034838-58
10
J>
23025842-34
40000 ,
, 105966307*80
20
»
2995731156
80000 ,
, 112897777*02
40
»
36888780-78
I OOOOO ,
, 11512921170
80
>)
43820250-00
200000 ,
, 122060680-92
100
»
46051684-68
400000 ,
, 128992150*14
200
»»
52983153-90
800000 ,
1 i359 2 3 6l 9"36
400
»
59914623-12
I OOOOOO ,
• i38i55°54 , o4
800
>>
66846092-34
2000000 ,
, 145086523-26
1 000
H
69077527*02
4000000 ,
, 152017992-48
2000
»
76008996*24
8000000 ,
, 158949461-70
4000
M
82940465-46
lOOOOOOO ,
, 161180896-38
JOHN NAPIER 37
Nap
To calculate this table, Napier found Log io 7
Nap
and Log 500000 by using the radical table ; and
thus 6931469*22 was found as the difference of
the logarithms of numbers in the ratio 2:1. The
difference of logarithms of sines in the ratio 8 : 1
is three times 6931469*22, i.e. 20794407*66. The
sine 8000000 is found by using the radical table
to have for its logarithm 2231434*68, whence by
addition the logarithm of the sine 1000000 is
found to be 23025842*34. Since the radius is ten
times this sine, all sines in the ratio 10 : 1 will
have this number for the difference of their
logarithms. The rest of the table was then
calculated from these determinations.
The logarithms of all sines that are outside
the limits of the radical table could now be deter-
mined. Multiply the given sine by 2, 4, 8, ...
200, ... or by any proportional number in the
short table, until a number within the limits of
the radical table is found. Find the logarithm
38 JOHN NAPIER
of the sine given by the radical table, and add to
it the difference which the short table indicates.
In the manner described, the logarithms of
all sines of angles between o° and 45 s could now
be determined, and the principal table or canon
completed. Napier gave, however, a rule by
which when the logarithms for all the angles not
less than 45 ° are known, the logarithms for all
the angles less than 45 ° can be determined. This
rule we may write in the form
Nap Nap
Log \ io 7 + Log sin x
Nap Nap
= Log sin \x+ Log sin (90 — \x),
cos ^-x &
which follows from the fact that — — — = . 2 ,
sin x sin \x
when we take into account the change in the
definition of a sine.
The accuracy of Napier s Canon.
It has been observed above that a numerical
error occurs in the value of the last number in
the second table. As Napier employed this
JOHN NAPIER 39
inaccurate value in his further calculations, it
produced an error in the greater part of his
logarithmic tables. The effect of this error is
that most of the logarithms are diminished by
about J io~ 6 of their correct values. Napier
himself observes in the " Constructio" that some
of the logarithms he obtained by means of his rule
for finding the logarithms of numbers outside the
limits of the radical table differ in value from
the logarithms of the same numbers found by the
rule for determining the logarithms of sines of
angles less than 45 °. He attributes this dis-
crepancy to defects in the values of the natural
sines he employed, and suggested a recalculation
of natural sines in which io 8 should be the radius.
Owing to these two causes, the last figure in the
logarithms of the canon is not always correct.
The improved system of logarithms.
The special purpose of application to trigono-
metrical calculations accounts for Napier's choice
40 JOHN NAPIER
of the system in which the logarithm of io 7 is
zero, and the logarithms of sines of angles between
o° and 90° are positive. It is, however, clear
that the rule of the equality of the sum of the
logarithms of two numbers and that of their pro-
duct would hold for numbers in general, only
if the logarithm of unity were taken to be zero,
as a number is unaltered by multiplication by
unity. On this account, Napier, in an appendix
to the " Constructio," proposed the calculation of
a system of logarithms in which Log 1 = o, and
Logio=io 10 . This is practically equivalent to
the assumption Log 10= 1, as the former assump-
tion merely indicates that the logarithms are to
be calculated to 10 places of decimals. Briggs
pointed out, in his lectures at Gresham College,
that a system would be convenient, on .which
o should be the logarithm of 1, and io 10 that of the
tenth part of the whole sine (viz. sin 5 44/ 21 "),
which would be equivalent to Log^= io 10 . This
system he suggested to Napier during his visit
JOHN NAPIER 41
to Merchiston in 161 5, when Napier pointed out
that the same idea had occurred to himself, but
that the assumption Log 10= io 10 would lead to
the most convenient system of all, and this was
at once admitted by Briggs.
In the appendix above referred to, Napier
gives some indications of methods by which the
improved logarithms might be calculated. These
depend upon exceeding laborious successive ex-
tractions of fifth and of square roots, which work
he proposed should be carried out by others,
and especially by Briggs. In an "Admonitio"
printed in the " Constructio," Napier remarked
that it is a matter of free choice to what sine or
number the logarithm o is assigned, that it is
necessary frequently to multiply or divide by
the complete sine (sin 90°), and thus that a
saving of trouble arises if the logarithm of this
sine be taken to be zero.
Briggs immediately set about the calculation
of these improved logarithms, and in the following
42 JOHN NAPIER
year, when he again visited Napier, shewed him
a large part of the table which was afterwards
published in 1624. On the death of Napier in
161 7 the whole work of developing the new
invention passed into the skilful hands of Briggs,
who, in the same year, published his " Logarith-
morum Chilias Prima," containing the common or
Briggian logarithms of the first thousand numbers
to 14 places of decimals. In 1624 he pub-
lished the "Arithmetica Logarithmica," a table of
logarithms of the first 20000 numbers and of the
numbers from 90000 to 100000, to 14 places of
decimals. The gap between 20000 and 90000
was fitted up by Adrian Vlacq, who published
in 1628 at Gouda a table of common logarithms
of numbers from 1 to 1 00000, to 10 places of
decimals. Vlacq's tables, although not free from
error, have formed the basis of all the numerous
tables of logarithms of natural numbers that have
been since published.
JOHN NAPIER 43
Other Tables.
A table of logarithms exactly similar to those
of Napier in|the " Constructio " was published in
1624 by Benjamin Ursinus at Cologne. The
intervals of the angles are 10", and the logarithms
are given to 8 places. The first logarithms to
the base e were published by John Speidell in his
"New Logarithmes," in London in 1619; this table
contains logarithmic sines, tangents and secants
for every minute of the quadrant to 5 decimal
places.
Predecessors of Napier.
It is usually the case that the fundamental
conceptions involved in a great new invention
have a history, which reaches back to a time,
often a long time, before that of the inventor.
Although Napier's introduction of logarithms is
justly entitled to be regarded as a really new
44 JOHN NAPIER
invention, it is not an exception to the usual rule.
The notion of an integral power of a ratio was
employed by the Greek Mathematicians. The
nature of the correspondence between a geometric
progression and an arithmetic progression had
been observed by various Mathematicians. In
particular Michael Stifel (i 486-1 567), in his
celebrated " Arithmetica Integra," published in
1544, expressly indicated the relations between
operations with the terms of a geometric and an
arithmetic series, in which the terms are made to
correspond, viz. the relations between multiplica-
tion, division and exponentiation on the one
hand, and addition, subtraction and multiplication
or division by an integer on the other hand.
But no indication was given by Stifel or others
how this correspondence could be utilized for
the purpose of carrying out difficult arithmetical
calculations. There were even given by the
Belgian Mathematician Simon Stevin (1548-
1620) certain special tables for the calculation
JOHN NAPIER 45
of interest, consisting of tables of the values of
ff , and of — V -, — - — ej + ... +
(i+r) n ' i+r (i+rf (i+r) n *
The first of these tables are really tables of
antilogarithms, but there were given no theoretical
explanations which would extend the use of the
tables beyond their special purpose. Napier,
whether he was acquainted with Stifel's work
or not, was the first whose insight enabled him
to develop the theoretical relations between
geometric and arithmetic series into a method
of the most far-reaching importance in regard
to arithmetic calculations in general. On the
theoretical side, Napier's representation by con-
tinuously moving points involved the conception
of a functional relationship between two con-
tinuous variables, whereas Stifel and others
had merely considered the relationship between
two discrete sets of numbers. This was in itself
a step of the greatest importance in the develop-
ment of Mathematical Analysis.
46 JOHN NAPIER
A rival inventor.
No account of the invention of logarithms
would be complete without some reference to
the work of Jobst Biirgi (i 552-1632), a Swiss
watch-maker and instrument-maker, who in-
dependently invented a system of logarithms.
His system was published in 1620, after Napier's
Canon had become known and fully recognized,
in a work entitled "Arithmetische und Geo-
metrische Progress-Tabulen." The table is really
an antilogarithmic table, and consists of a set of
numbers printed red placed in correspondence
with a set of numbers printed black. The red
numbers are o, 10, 20, ... , those of an arithmetic
series, and the corresponding black numbers are
1 00000000, 1 000 1 0000, 1 0002000 1, of a geometric
series ; thus the red numbers are the logarithms of
the black ones divided by io 8 with the base
y i*O0Ol. Biirgi appears to have devised his
JOHN NAPIER 47
system a good many years before he published it,
but kept it secret until he published his tables
six years after the appearance of those of
Napier.
Conclusion.
The system of Biirgi is decidedly inferior to
that of Napier, and the knowledge of the use of
logarithms which was spread in the scientific
world was entirely due to the work of Napier.
The concensus of opinion among men of
Science of all nations has ascribed to Napier
the full honour due to the inventor of the method
which has provided the modern world with a tool
that is indispensable for all elaborate arithmetical
calculations. In the great advance which had
taken place in Mathematical Science during the
half century preceding the publication of the
" Constructio," British Mathematicians had taken
no part. It is very remarkable that, in a country
48 JOHN NAPIER
distracted by political, social, and religious feuds
of the most serious kind, such as Scotland then
was, there should have arisen the first of those
great thinkers who in the course of the seven-
teenth century brought Great Britain to the
highest point of achievement in the domain of
Mathematical Science.
CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS
UC SOUTHERN REGIONAL LIBRARY FACILITY
AA 000139 312 3