516 
NATURE 
| Oct. 16, 1873 
1 
metike,’ was published by Gellibrand, and other friends of 
Briggs, who died the year before, with a much more detailed 
and popular explanation of the doctrine of logarithms than was 
to be found in the ‘ Arithmetica Logarithmica.’ It is there said 
. . - From this period we may consider the decimal arith- 
metic as fully established, inasmuch as the explanation of it 
began to form an essential part of all books of practical Arith- 
metic. The simple method of marking the separation of the 
decimals and integers by a comma, of which Napier has given a 
solitary example, was not however generally adopted.” ... 
De Morgan (‘‘ Arithmetical Books,” 1847, p. xxiii.) writes : 
“Dr. Peacock mentions Napier as being the person to whom 
the introduction [of the decimal point] is unquestionably ue; a 
position which I must dispute upon additional evidence. The 
inventor of the single decimal distinction, be it point orline, as 
in 123°456, or 123 | 456, is the person who first madet his dis- 
tinction a permanent language ; not using it merely as a 7st in 
the process, to be useful in pointing out afterwards how cnother 
process is to come on, or language is to be applied, but making 
it his final and permanent indication as well of the way of po!nt- 
ing out where thei ntegers end and the fractions begin, as of the 
manner in which that distinction modifies operations. Now 
first I submit that Napier did not do this; secondly, that if he 
did do this, Richard Witt did it before him.” 
De Morgan then states that he has not seen Wright’s transla- 
tion of 1616, but he proceeds to examine Napier’s claim as resting 
on the two examples in the ‘‘ Rabdologia,” in the first of which 
a comma is used, but only inone place. After this examination 
he proceeds, ‘‘I cannot trace the decimal point in this: but if 
required to do so, I can see it more distinctly in Witt, who 
published four years before Napier. But I can hardly admit 
him to have arrived at the notation of the decimal point. . . .”* 
I agree with De Morgan in all that he has stated in the above 
extracts, and do not think that the single instance of the comma 
used in the course of work, and replaced immediately afterwards 
by exponential marks, is a sufficient ground for assigning to 
Napier the invention of the decimal point, or even affords a pre- 
sumption that he made use of it at all in the expression of 
results. 
Still one of the objects of this paper is to claim (provisionally 
of course, till evidence of any earlier use is produced, if such 
there be) the invention of the decimal point for Napier, but not 
on account of anything contained in the ‘‘ Rabdologia.” The 
mathematical works published by Napier in his life-time (he 
died in 1617) were his ‘‘ Mirifici Logarithmorum Canonis De- 
scriptio,” 1614, containing the first announcement of the inven- 
tion of logarithms, and the ‘‘ Rabdologia,” 1617, giving an 
account of his almost equally remarkable (as it was thought at 
the time) invention of numbering rods or ‘‘bones.” In 1619, 
two years after his death, the ‘* Mirifici Logarithmorum Canonis 
Constructio,” containing the method of construction of the canon 
of logarithms was published, edited by his son, and in this work 
the decimal point is systematically used in a manner identical 
with that in which we employ it at the present day. I can find 
no traces of the decimal point in Wright's translation of the 
“* Descriptio,” 1616 ; and, as De Morgan says, the use of the 
decimal separator is not apparent in Witt. The. earliest work, 
therefore, in which a decimal separator was employed seems to 
be Napier’s posthumous work, the “ Constructio” (1619), where 
the following definition of the point occurs on p. 6. ‘In 
numeris periodo sic in se distinctis, quicquid post periodum 
notatur fractio est, cujus denominator est unitas cum tot cyphris 
post se, quot sunt figuree post periodum. Ut 10000000°04 valet 
idem, quod roco000e;4;. Item 25°803, idem quod 25,5", 
{tem 9999998 0005021, idem valet quod 9999998zs}0sG00) et 
sic de ceteris.” On p. 8 we have 10°502 multiplied by 3°216, 
and the result found to be 33°774432; and on pp. 23 and 24 
occur decimals not attached to integers, viz. ‘4999712 and 
"0004950. These show that Napier was in possession of all the 
conventions and attributes that enable the decimal point to com- 
plete so symmetrically our system of notation, viz. (1), he saw 
that a point or separatrix was quite enough to separate integers 
from decimals, and that no signs to indicate primes, seconds, 
&c., were required ; (2), he used ciphers after the decimal point 
and preceding the first significant figure, and (3), he had no 
* In an essay “ On some points in the History of Arithmetic ” (Companion 
to the Almanac for 1851), De Morgan has further discussed the invention of 
the decimal point, but in the same spirit as regards Napier. He seems 
never to have seen Napier’s ‘‘Constructio” of 1619, and the work is very 
rare. The only copy I have heen able to see is that in the Cambridge Uni- 
Yersity Library. 
objection to a decimal standing by itself without any integer. 
Napier thus had complete command over decimal fractions and 
understood perfectly the nature of the decimal point, and I 
believe (except perhaps Briggs) he is the first person of whom 
this can be said. When I first read the “Constructio,” I felt 
some doubt as to whether Napier really appreciated the value 
of the decimal point in all its bearings, as he seemed to have 
regarded it to some extent as a mark to separate figures that | 
were to be rejected from those that were to be retained; but a 
careful examination has led me to believe that his views on the 
subject were pretty nearly identical with those of a modern 
arithmetician. There are perhaps 200 decimal points in the 
book, affording abundant evidence on the subject. 
The claim of Napier to the invention of the decimal point is 
not here noticed for the first time, as both Delambre (‘‘ Hist. de 
l’Astron. mod. t. i. p. 497) and Hutton allude to the decimal 
fractions in the “ Constructio” (though the latter claims priority 
for Pitiscus), and Mr. Mark Napier (‘* Memoirs of John Napier,” 
P: 454) devotes a good deal of space to it. 
Briggs also used decimals, but in a form not quite so conve- 
nient as Napier ; thus, he writes 63°0957379 as 630957379, viz., 
he prints a bar under the decimals: this notation first appears 
without any explanation, in his- ‘‘ Lucubrationes” appended to 
the “ Constructio.”* Briggs used this notation all his life (he 
died in 1631), and he explains itin the ‘‘ Arithmetica Logarith- 
mica’’ of 1624. Oughtred’s symbol first used (as far as I know), 
in his “ Arithmeticz in numeris”...Clavis, 1631, differed only 
from Briggs’s in the insertion of a vertical bar to separate the 
decimals from the integers more completely, thus: 63 | 0957379. 
Oughired’s and Briggs’s notation are essentially the same, the 
improvement of the former being no doubt due to the uncer- 
tainty that sometimes might be felt as to which was the first 
figure above Briggs’s line. 
From an_ inspection of MSS. of Briggs and Oughtred 
(the Birch MSS. contain a letter of Briggs’s to Pell, and 
the Royal Society has a Peter Ramus with many of his 
MS. notes, while the Cambridge University copy of the 
“*Constructio ” is annotated in MS. by Oughtred), it is apparent 
that in writing, Briggs and Oughtred both made the separating 
rectangle in exactly the same way, viz., they wrote it 63 ; 0957379, 
the upright mark usually being just high enough to fix distinctly 
what two figures it was intended to separate, and rarely took the 
trouble to continue the horizontal bar to the end of the decimals, 
if there were many. Thus Oughtred was a follower of Briggs, 
and only made an improvement in the /rin/ed notation, It is 
clear that in writing Briggs’s rectangle was preity nearly as con- 
venient as Napier’s point, and there is every probability that 
Briggs appreciated all the properties of the ‘‘separatrix” as 
clearly as Napier ; but in his 8 pp. of ‘“ Lucubrationes” he has 
left much less to judge by than has Napier. In 1624, as we can 
see from his “ Arithmetica Logarithmetica,” he had full command 
over decimal arithmetic in its present form (except that he used 
the rectangular ‘‘separatrix” instead of the point). Gunter was 
a follower of Napier, and employed the point (but see De Mor- 
gan). In his “ Description and Use of the Sector” (1623), he 
uses the point throughout pretty much as we do at present (e.g. 
Pp: 41 of the ‘‘First Booke of the Crosse-staffe”: ‘*As 4°50 
unto 1°00 :so 1'000 unto 0'222”), except that he calls the 
decimals farts in the text. In Roe’s ‘* Tabulze Logarithmicz, or 
Two Tables of Logarithme ” (1633), the explanatory portion of 
which was written by Wingate, decimal points are used every- 
where ; thus we have (p. 29): “As 1 is to ‘079578 : so is the 
square of the circumference to the superficial content,” and he 
takes the case of circumference 88°75, and obtains by multiplica- 
tion (performed by logarithms) 626°8 for the result. Wingate 
refers for explanation on the decimal point to his arithmetic, 
but I have not seen any edition of this work that was published 
previously to Roe’s tables (Watt gives one, 1630). In his 
“Construction and Use of the Line of Proportion” (1628), Win- 
gate also uses decimals and decimal points. 
On the whole, therefore, it appears that both Napier and 
Briggs saw that a mere separator to distinguish integers from 
decimals was quite sufficient, without any exponential marks 
being attached to the latter; but that Napier used a simple 
point for the purpose, while Briggs employed a bent or curved 
line, for which in print he substituted merely a horizontal bar 
* Acurious blunder is made in Bartholomew Vincent’s reprint of the 
“* Constructio,” Lyons, 1620 (of which there is a copy in the Royal Society’s 
library). The printer, unaware that the position of Briggs’s subscript bars: 
had any meaning, has disposed them symmetrically under all the figures, 
