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Full text of "John Napier and the invention of logarithms, 1614; a lecture"

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John Napier and the Invention 
of Logarithms, 1614 



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~~h2j°^- m-OWw- <>£^- y *&'<rfy\/bw 



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 
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9876882 

5876427 
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9875514 
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0874137 

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1604555 



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1616038 
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18333237 
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182^7384 



18279*07 
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18226071 
18208323 
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9863336 



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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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00 m CONO 


00 




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CM N CM N 


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00 00 00 00 


ON 


c 


O O VO VO 


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5 
















o 








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O to CM On 


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oo tj- h b\ 


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3 


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Id 


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IfllOmifl 


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i 


piO CO H 


CM 




fO 't lO t>- 


00 






o o o o 


M 






iO u-> U-) IT) 


to 




O in O to 


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O O M M 


o 


J 


M tH M M 


CM 








<J 








-o 


E 


O O O t- 


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c 


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O O to CO 


CM 


o 


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e 


O O J» Tf 


00 


w 


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c 


Q O H t« 


00 






O to O to 


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"2 

2 


o O •■* •■* 


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O to O to 






O ON OnOO 


o 




a 


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£ 


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O cm yo r-» 


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m cm co 


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t-l 


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i2 


o o o ^ 


o 


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£ 


O O O 00 


00 


2 

£ 


O O O On 


r- 


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3 

s 


o o cm r* 


CO 






o o o o 


t^» 




1 


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55 


O On ON On 


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M 





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 



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