Juty 30, 1914) 
NATURE 
J73 
consisted in passing from the idea of figures to the 
geometrical representation of the quantity by a line, | 
the repeated operations being perfectly represented by | 
repeatedly cutting off the same fraction of the dimin- | 
ished length. This led Napier to establish the 
proposition that the logarithms of proportionals are 
‘equally differing.”’ Napier felt fully the importance 
of this proportion, and he literally revelled in it, 
showing how it enabled us to find mean proportionals | 
of all kinds, extract roots, calculate powers. In the 
third stage he boldly applied his principle to con- 
tinuous motion. Napier was now ready to calculate 
his table. We give Lord Moulton’s own ~ words. 
‘*He (Napier) takes the radius and forms from it a 
geometrical series where the reduction between suc- 
cessive terms is one-hundredth. Say he takes 60 
terms of such a series. 
all these terms and he writes them over against the 
number. These are widely separated by intervals 
commencing with 100,000 and diminishing as_ they 
proceed. He then takes each of these numbers as the 
first terms of a geometrical series, where the reduc- | 
tion is 5000 out of the million, i.e., one two- | 
thousandth. He knows the logarithms of all these 
numbers. . . . Thus he has 1200 numbers fairly well 
distributed over the field, and of these he knows the 
logarithms. . . . They are to serve as his measuring 
posts. He therefore takes the table of sines which 
gives the numbers of which he wished to calculate 
the logarithms. Taking each sine he sees where it, 
regarded as a number, comes in the scale. It cannot 
be far from a measuring post. His method enables 
him to make a proper allowance in its logarithm for 
this small difference in fact, and as the logarithm of 
the measuring post is known the logarithm of the 
sine is known also... . I have now given you, as 
I read it, the line of discovery which led up to 
Napier’s table of logarithms. What deeply impresses 
me is his tenacity of aim combined with his recep- | 
tivity of new ideas for attaining it. From first to 
last it was a table of logarithms of sines that he 
proposed to make and he did not permit himself to 
be turned aside from that purpose till it was com- 
pleted. His concepts evidently widened as he _ pro- 
ceeded. . . . As soon as the discovery had actually 
seen the light... . Napier proceeded justifiably to 
destroy the scaffolding which had been so serviceable 
in the erection of the building. For example, the 
plan of taking the radius as the starting point had 
been of inestimable service in keeping up the con- 
tinuity of his methods. Before his tables were pub- 
lished he had seen that this was unnecessary and he 
proclaimed it to be so in the Descriptio. We know 
that at this time he had seen that it would be better 
to start from unity as the number the logarithm of 
which should be zero... . A still more remarkable 
change which he himself proposed was to follow up 
the last proposal by fixing unity as the logarithm of 
10. That this could be safely done could scarcely 
have been seen by him until the completion of his 
work. From the top of the mountain he could see 
how the climb might be made easier by deviations 
which to the climbers might well seem to be courting 
unnecessary difficulty. . . . Napier took twenty years 
to do the work—many of which, probably the greater’ 
part of which, were spent in arriving at his method. 
[t would be sad to think that most of this was wasted 
because the solution came by a lucky chance at the 
last. In my view all these years did their share, 
and I have tried to show how gradual and continuous 
was his progress. As to the greatness of the achieve- 
ment it is needless to speak. Logarithms have played 
well nigh as important a part in mathematical theory 
as in practical work. We know infinitely more of 
their nature than Napier or any man of Napicr’s age 
WO 2335, VOL..93) 
He knows the logarithms of | 
| 
| paper on 
could have done. We have means of calculating them 
so effective that if all the logarithmic tables in the 
world were destroyed the replacing them would be 
the work of a few months. But not all the three 
centuries that have elapsed have added one iota to the 
completeness or the scope of the two and only existing 
systems of logarithms as they were left by the genius 
| of John Napier of Merchiston.”’ 
On the Saturday forenoon the members met in one 
of the class-rooms of the University to discuss chiefly 
historic questions relating to the discovery of log- 
arithms. Prof. Hobson was voted to the chair, and 
Dr. Glaisher opened the discussion by an interesting 
certain aspects of Napier’s work. He 
pointed out how difficult it is for us with our con- 
venient notations and modern notions, to realise what 
a supreme intellectual effort it must have been for 
Napier to do what he did. The problem solved by him 
would be expressed now-a-days in terms of a simple 
differential equation. The interesting view which 
Lord Moulton had brought forward the previous day 
was worth our consideration, although he himself had 
never thought of getting behind the beautiful geo- 
metrical approach given in the Descriptio. 
Prof. Eugene Smith, of New York, read a paper 
on the law of exponents in the works of the sixteenth 
century; Prof. Cajori discussed algebra in Napier’s 
day and the alleged prior inventions of logarithms; 
Lieut. Salih Mourad, of the Turkish navy, gave a 
short account of the introduction of logarithms into 
Turkey; and in a brief note from Dr. Vacca, of Rome, 
it was pointed out that a compound interest rule 
given in an Italian work of the fifteenth century 
virtually contained the approximate calculation of the 
Napierian logarithm of the number 2. Prof. Gibson 
communicated a careful discussion on the question of 
Napier’s logarithms and the change to Briggs’s 
logarithms. These historic papers raised a good deal 
of discussion, in which the authors already named, 
the chairman, and Dr. Conrad Miller took part. Dr. 
Glaisher agreed very emphatically with Prof. Cajori 
that it was dangerous to take information second 
hand. An error carelessly made by one historian was 
copied by others, and once the error got started it 
was difficult to get rid of it. It was not always easy 
to reach first sources. He had, for example, never 
seen a copy of Biirgi’s antilogarithmic table (as it 
would be called now) until the day before, in the 
exhibition, when, through the kindness of the Town 
Librarian of Dantzig, a copy had been placed on view. 
The other side of Napier’s mathematical work was 
represented by a paper by Dr. Sommerville on Napier’s 
rules and trigonometricaily equivalent polygons, with 
extensions to non-euclidean space. 
On the Friday night the Lord Provost of Edinburgh 
and the Town Council gave a brilliant reception in 
the new Usher Hall. On Saturday afternoon the 
members were received at a garden party by the 
governors and headmaster of Merchiston Castle 
School, and were shown the small room at the top: of 
the battlemented tower where Napier used to think 
and work. On Saturday evening the members and 
their friends met for social enjoyment in the hall of 
the University Union. 
A memorial service was held in St. Giles’s Cathedral 
on the Sunday at 3.30 p.m. The officiating clergyman 
was the Rev. Dr. Fisher, of St. Cuthbert’s Parish 
Church, of which church John Napier had been an 
elder, and in the graveyard of which his body lies 
buried. A special feature of the service was the 
presence of the masters and boys of Merchiston Castle 
School. They numbered 260, and filled the transept 
of the Cathedral 
CG: 
G. Knorr. 
