2OI
portant to discover if possible a way of shortening
them. Napier, as we have said, applied himself
assiduously to this object; and he was probably not
the only person of that age whose attention it occu-
pied. He was, however, undoubtedly the first who
succeeded in it, which he did most completely by
the admirable contrivance which we are now about
to explain.
"When we say that I bears a certain proportion,
ratio, or relation to 2, we may mean any one of two
things; either that one is the half of two, or that it is
less than 2 by1 . If the former be what we mean, we
may say that the relation in question is the same as
that of 2 to 4, or of 4 to 8 ; if the latter, we may
say that it is the same as that of 2 to 3, or of 3 to 4.
Now, in the former case, we should be exemplifying
what is called a geometrical, in the latter, what is
called an arithmetical proportion: the former being
that which regards the number of times, or parts of
times, the one quantity is contained in the other; the
latter regarding only the difference between the two
quantities. We have already stated that the pro-
perty of four quantities arranged in geometrical pro-
portion is, that the product of the second and third,
divided by the first, gives the fourth. But when four
quantities are in arithmetical proportion, the sum of
the second and third, diminished by the subtraction
of the first, gives the fourth. Thus, in the geo-
metrical proportion, I is to 2 as 2 is to 4; if 2 be
multiplied by 2 it gives 4; which divided by I still
remains 4; while, in the arithmetical proportion,
I is to 2 as 2 is to 3 ; if 2 be added to 2 it gives 4;
from which, if I be subtracted, there remains the
fourth term 3. It is plain, therefore, that especially
where large numbers are concerned, operations by
arithmetical must be much more easily performed
than operations by geometrical proportion; for, in
the one case you have only to add and subtract,
while in the other you have to go through the
greatly more laborious processes of multiplication
and division.
"Now it occurred to Napier, reflecting upon this
important distinction, that a method of abbreviating
the calculation of a geometrical proportion might
perhaps be found, by substituting upon certain fixed
principles, for its known terms, others in arithmetical
proportion, and then finding, in the quantity which
should result from the addition and subtraction of
these last, an indication of that which should have
resulted from the multiplication and division of the
original figures. It had been remarked before this
by more than one writer, that if the series of num-
bers I, 2, 4, 8, &c., that proceed in geometrical
progression, that is, by a continuation of geometrical
ratios, were placed under or along side of the series
o, I, 2, 3, &c., which are in arithmetical progression,
the addition of any two terms of the latter series would
give a sum, which would stand opposite to a number
in the former series indicating the product of the two
terms in that series, which corresponded in place to
the two in the arithmetical series first taken. Thus,
in the two lines,
I, 2, 4, 8, 16, 32, 64, 128, 256,
o, 1,  2,  3,  4,  5,  6,  7,  8,          
the first of which consists of numbers in geometrical,
and the second of numbers in arithmetical progres-
sion, if any two terms, such as 2 and 4, be taken
from the latter, their sum 6, in the same line, will
stand opposite to 64 in the other, which is the pro-
duct of 4 multiplied by 16, the two terms of the
geometrical series which stand opposite to the 2 and
4 of the arithmetical. It is also true, and follows
directly from this, that if any three terms, as, for
instance, 2, 4, 6, be taken in the arithmetical series,
the sum of the second and third, diminished by the
subtraction of the first, which makes 8, will stand
opposite to a number (256) in the geometrical series
which is equal to the product of 16 and 64 (the op-
posites of 4 and 6), divided by 4 (the opposite of 2).
"Here, then, is to a certain extent exactly such
an arrangement or table as Napier wanted. Having
any geometrical proportion to calculate, the known
terms of which were to be found in the first line or
its continuation, he could substitute for them at once,
by reference to such a table, the terms of an arith-
metical proportion, which, wrought in the usual
simple manner, would give him a result that would
point out or indicate the unknown term of the geo-
metrical proportion. But, unfortunately, there were
many numbers which did not occur in the upper line
at all, as it here appears. Thus there were not to
be found in it either 3, or 5, or 6, or 7, or 9, or 10,
or any other numbers, indeed, except the few that
happen to result from the multiplication of any of its
terms by two. Between 128 and 256, for example,
there were 127 numbers wanting, and between 256
and the next term (512) there would be 255 not to
be found.
" We cannot here attempt to explain the methods
by which Napier's ingenuity succeeded in filling up
these chasms, but must refer the reader, for full in-
formation upon this subject, to the professedly scien-
tific works which treat of the history and construction
of logarithms. Suffice it to say that he devised a
mode by which he could calculate the proper number
to be placed in the table over against any number
whatever, whether integral or fractional. The new
numerical expressions thus found he called loga-
rithms, a term of Greek etymology, which signifies
the ratios or proportions of numbers. He after-
wards fixed upon the progression, 1, 10, loo, looo,
&c., or that which results from continued multipli-
cation by 10, and which is the same according to
which the present tables are constructed. This im-
provement, which possesses many advantages, had
suggested itself about the same time to the learned
Henry Briggs, then professor of geometry in Gresham
College, one of the persons who had the merit of
first appreciating the value of Napier's invention,
and who certainly did more than any other to spread
the knowledge of it, and also to contribute to its
perfection."1
The invention was very soon known over all
Europe, and was everywhere hailed with admiration
by men of science. Napier followed it up in 1617,
by publishing a small treatise, giving an account of
a method of performing the operations of multiplica-
tion and division by means of a number of small rods.
These materials for calculation have maintained their
place in science, and are known by the appellation,
of Napier's Bones.
In 1608 Napier succeeded his father, when he had
a contest with his brothers and sisters on account of
some settlements made to his prejudice by his father,
in breach of a promise made in 1586, in presence of
some friends of the family, not to sell, wadset, or
dispose, from his son John, the lands of Over Mer-
chiston, or any part thereof. The family disputes
were probably accommodated before June 9, 1613,
on which day John Napier was served and returned
heir of his father in the lands of Over Merchiston.
This illustrious man did not long enjoy the in-
heritance which had fallen to him so unusually late
in life. He died, April 3, 1617, at Merchiston Castle,
and was buried in the church of St. Giles, on the
1 The above account of logarithms, which has the advantage
of being very simple and intelligible, is extracted from the
Library of Entertaining Knowledge.