NAP 
lure and fine accompliffiments foon at traced notice, and 
might have railed him to the liigheft offices of the ftate; 
but, declining all civil employment, and the buftle of 
the court, he retired from the world to purfue literary re- 
fearches, in which he made an uncommon progrefs, as 
appears by the l'everal ufeful difcoveries with which he 
afterwards favoured the public. He applied himfelf chiefly 
to the lludy of the mathematics ; but at the fame time did 
not negleCt that of the holy Scriptures ; in both of which 
he difcovered the moll extenfive knowledge, and pro¬ 
found penetration. His “ Plain Difcovery of the Reve¬ 
lation of St.John,” publiffied in the year 1593, affords 
evidence of the mod acute inveftigation 5 though the pro¬ 
grefs of time has difcovered that his calculations con¬ 
cerning certain events had proceeded upon fallacious 
data. This work has been printed abroad in feveral lan¬ 
guages ; and a French edition of it, which appeared at 
.Rochelle in the fame year, was very acceptable to the 
French proteftants, on account of the zeal and erudition 
employed by the author to prove the pope to be antichrill. 
But what principally contributed to give celebrity to 
his name, was his great and fortunate difcovery of loga¬ 
rithms in trigonometry, which by eafe and expedition in 
calculation have fo wonderfully affifled the fcience of af- 
tronomy, and the arts of practical geometry and naviga¬ 
tion. As the accuracy of aftronomical obiervation had 
been continually advancing, it was necefiary that the cor- 
re&nefs of trigonometrical calculation, and of courfe its 
difficulty, ffiould advance in the fame proportion. The 
figns and tangents of angles could not be expreffed with 
fufficient corre&nefs without decimal fractions extend¬ 
ing to five or fix places below unity ; and, when to three 
fuch numbers a fourth proportional was to be found, the 
work of multiplication and divifion became extremely la¬ 
borious. Accordingly, in the end of the fixteenth cen¬ 
tury, the time and labour confumed in fuch calculations 
had become exceffive, and were felt as extremely burthen- 
fome by the mathematicians and aftronotners all over 
Europe. Napier, whofe mind feems to have been pecu¬ 
liarly turned to arithmetical refearches, and who was alfo 
devoted to the ftudy of aftronomy, had early fought for 
the means of relieving himfelf and others from this diffi¬ 
culty. He had viewed the fubjeCt in a variety of lights, 
and a number of ingenious devices had occurred to him, 
by which the tedioufnefs of arithmetical operations might, 
more or lefs completely, be avoided. In the courfe of 
thefe attempts, he did not fail to obferve, that, whenever 
the numbers to be multiplied or divided were terms of a 
geometrical progreffion, the produCt or quotient mull: alfo 
be a term of that progreffion, and imift occupy a place in 
it pointed out by the places of the given numbers, fo that, 
it might be found from mere infpeCtion, if the progreffion 
were far enough continued. If, for inftance, the third 
term of the progreffion were to be multiplied by the le- 
venth, the product muft be the tenth ; and, if the twelfth 
were to be divided by the fourth, the quotient muft be the 
eighth ; fo that the multiplication and divifion of fuch 
terms was reduced to the addition and fubtraCtion of the 
numbers which indicated their places in the progreffion. 
This obfervation, or one very fimilar to it, was made by 
Archimedes, and was employed by that great geometer 
to convey an idea of a number too vaftto becorrefitly ex- 
preffed by the arithmetical notation of the Greeks. Thus 
far, however, there was no difficulty, and the difcovery 
might certainly have been made by men much inferior ei¬ 
ther to Napieror Archimedes. What remained to be done, 
what Archimedes did not attempt, and what Napier com¬ 
pletely performed, involved two great difficulties. It is 
plain, that the refource of the geometrical progreffion 
was fufficient, when the given numbers were terms of 
that progreffion ; but, if they were not, it did not feem 
that any advantage could be derived from it. Napier, 
.however, perceived, (and it was by no means obvious,) 
that all numbers whatfoever might be inferted in the pro¬ 
greffion, and have their places affigned in it. After con- 
Vo-t.XVI. No. xi33. 
1 E R. 529 
ceiving the poffibility of this, the next difficulty was, to 
dil'cover the principle, and to execute the arithmetical 
procefs, by which thefe places were to be afcertained. It 
is in thefe two points that the peculiar merit of his in¬ 
vention confifts; and at a period when the nature of 
feries, and when every other refource of which he could 
avail himfelf were fo little known, his fuccefs argues a 
depth and originality of thought which have been rarely 
furpafled. The way in which he fatisfied himfelf that 
all numbers might be intercalated between the terms of 
the given progreffion, and by which he found the places 
they muft occupy, -was founded on a rnoft ingenious 
fuppofition ; that, of two points defcribing two different 
lines, the one with a conftant velocity, and the other with 
a velocity always increafing in the ratio of the fpace the 
point had already gone over; the firft of thefe would ge¬ 
nerate magnitudes in arithmetical, and the fecond mag¬ 
nitudes in geometrical, progreffion. It is plain, that all 
numbers whatfoever would rind their places among tire 
magnitudes fo generated; and, indeed, this view of tire 
fubjeft is as fimple and profound as any which, after two 
hundred years, has yet prefented itlelf to mathematicians. 
The mode of deducing the refults has been Amplified ; 
but it can hardly be faid that the principle has been more 
clearly developed. The numbers which indicate the 
places of the terms of the geometrical progreffion are called 
by Napier the logarithms of thofe terms. 
Various fyftems of logarithms, it is evident, may be 
conftrufited according to the geometrical progreffion af- 
fumed; and of thefe, that which was firft contrived by 
Napier, though the fimpleft, and the foundation of the 
reft, was not fo convenient for the purpofes of calcula¬ 
tion, as one which foon afterwards occurred, both tu 
himfelf and his friend Briggs, by whom the aCtual calcu¬ 
lation was performed. The new fyftem of logarithms was 
an improvement, practically conlidered ; but in as far as 
it was connected with the principle of the invention, it 
is only of fecondary confideration. The original tables 
had been alfo fomewhat embarralfed by too clofe a con¬ 
nection between them and trigonometry. The new tables 
were free from this inconvenience. See the article Lo¬ 
garithms, vol. xii. p. 885 & feq. 
• It is probable, however, that the greateft inventor in 
fcience was neverabie to do more than to accelerate tire 
progrefs of difcovery, and to anticipate what Time, “ the 
authorofauthors,” would have gradually brought to light. 
Though logarithms had not been invented by Napier, 
they would have been difcovered in the progrefs of the al¬ 
gebraic analyfis, when the arithmetic of powers and ex¬ 
ponents, both integral and fractional, came to be fully 
underftood. The idea of confidering all numbers as 
powers of one given number would then- have readily 
occurred, and the doCtrine of feries would have greatly 
facilitated the calculations which it was necefiary to un¬ 
dertake. Napier had none of thefe advantages, and they 
were all fupplied by the refources of his own mind. In¬ 
deed, as there never was any invention for which the 
ftate of knowledge had lefs prepared the way, there never 
was any where more merit fell to the fliare of the inventor. 
His good fortune, alfo, not lefs than his great fagacity, 
may be remarked. Had the invention of logarithms 
been delayed to the end of the feventeenth century, it 
would have come about without effort, and would not 
have conferred ok the author the high celebrity which 
Napier fo juftly derives from it. In another refpeCt he 
has alfo been fortunate. Many inventions have been 
eclipfed or ohjfcured by new difcoveries ; or they have 
been fo altered by fublequent improvements, that their 
original form can hardly be recognifed,- and, in fome in- 
ftances, has been entirely forgotten. This has almoft al¬ 
ways happened to the difcoveries made at an early pe¬ 
riod in the progrefs of fcience, and before their principles 
were fully unfolded. It has been quite otherwife with 
the invention of logarithms, which came out of the hands 
of the author fo perfect, that it has never received but 
6 T \ one 
