Logarithms of the ratios of arcs to sines from 0^q.00000
to 0^q.05000, and log sines throughout the quadrant 4 "
Logarithms of the ratios of arcs to tangents from
0^q.00000 to 0^q.05000, and log tangents throughout
the quadrant 4 "
The trigonometrical results are given for every hundred-thousandth of
the quadrant (10´´ centesimal or 3´´.24 sexagesimal). The tables were
all calculated to 14 places, with the intention that only 12 should be
published, but the twelfth figure is not to be relied upon. The tables
have never been published, and are generally known as the _Tables du
Cadastre_, or, in England, as the great French manuscript tables.
A very full account of these tables, with an explanation of the
methods of calculation, formulae employed, &c., was published by
Lefort in vol. iv. of the _Annales de l'observatoire de Paris_. The
printing of the table of natural sines was once begun, and Lefort
states that he has seen six copies, all incomplete, although including
the last page. Babbage compared his table with the _Tables du
Cadastre_, and Lefort has given in his paper just referred to most
important lists of errors in Vlacq's and Briggs's logarithms of
numbers which were obtained by comparing the manuscript tables with
those contained in the _Arithmetica logarithmica_ of 1624 and of 1628.
As the _Tables du Cadastre_ remained unpublished, other tables
appeared in which the quadrant was divided centesimally, the most
important of these being Hobert and Ideler's _Nouvelles tables
trigonométriques_ (1799), and Borda and Delambre's _Tables
trigonométriques décimales_ (1800-1801), both of which are
seven-figure tables. The latter work, which was much used, being
difficult to procure, and greater accuracy being required, the French
government in 1891 published an eight-figure centesimal table, for
every ten seconds, derived from the _Tables du Cadastre_.
_Decimal or Briggian Antilogarithms._--In the ordinary tables of
logarithms the natural numbers are all integers, while the logarithms
tabulated are incommensurable. In an antilogarithmic table, the
logarithms are exact quantities such as .00001, .00002, &c., and the
numbers are incommensurable. The earliest and largest table of this
kind that has been constructed is Dodson's _Antilogarithmic canon_
(1742), which gives the numbers to 11 places, corresponding to the
logarithms from .00001 to .99999 at intervals of .00001.
Antilogarithmic tables are few in number, the only other extensive
tables of the same kind that have been published occurring in
Shortrede's _Logarithmic tables_ already referred to, and in
Filipowski's _Table of antilogarithms_ (1849). Both are similar to
Dodson's tables, from which they were derived, but they only give
numbers to 7 places.
_Hyperbolic or Napierian logarithms_ (i.e. to base e).--The most
elaborate table of hyperbolic logarithms that exists is due to
Wolfram, a Dutch lieutenant of artillery. His table gives the
logarithms of all numbers up to 2200, and of primes (and also of a
great many composite numbers) from 2200 to 10,009, to 48 decimal
places. The table appeared in Schulze's _Neue und erweiterte Sammlung
logarithmischer Tafeln_ (1778), and was reprinted in Vega's
_Thesaurus_ (1794), already referred to. Six logarithms omitted in
Schulze's work, and which Wolfram had been prevented from computing by
a serious illness, were published subsequently, and the table as given
by Vega is complete. The largest hyperbolic table as regards range was
published by Zacharias Dase at Vienna in 1850 under the title _Tafel
der natürlichen Logarithmen der Zahlen_.
_Hyperbolic antilogarithms_ are simple exponentials, i.e. the
hyperbolic antilogarithm of x is e^x. Such tables can scarcely be said
to come under the head of logarithmic tables. See TABLES,
MATHEMATICAL: _Exponential Functions_.
_Logistic or Proportional Logarithms._--The old name for what are now
called ratios or fractions are _logistic numbers_, so that a table of
log (a/x) where x is the argument and a a constant is called a table
of logistic or proportional logarithms; and since log (a/x) = log a -
log x it is clear that the tabular results differ from those given in
an ordinary table of logarithms only by the subtraction of a constant
and a change of sign. The first table of this kind appeared in
Kepler's work of 1624 which has been already referred to. The object
of a table of log (a/x) is to facilitate the working out of
proportions in which the third term is a constant quantity a. In most
collections of tables of logarithms, and especially those intended for
use in connexion with navigation, there occurs a small table of
logistic logarithms in which a = 3600´´ (= 1° or 1^h), the table
giving log 3600 - log x, and x being expressed in minutes and seconds.
It is also common to find tables in which a = 10800´´ (= 3° or 3^h),
and x is expressed in degrees (or hours), minutes and seconds. Such
tables are generally given to 4 or 5 places. The usual practice in
books seems to be to call logarithms logistic when a is 3600´´, and
proportional when a has any other value.
_Addition and Subtraction, or Gaussian Logarithms._--_Gaussian
logarithms_ are intended to facilitate the finding of the logarithms
of the sum and difference of two numbers whose logarithms are known,
the numbers themselves being unknown; and on this account they are
frequently called addition and subtraction logarithms. The object of
the table is in fact to give log (a ± b) by only one entry when log a
and log b are given. The utility of such logarithms was first pointed
out by Leonelli in a book entitled _Supplément logarithmique_, printed
at Bordeaux in the year XI. (1802/3); he calculated a table to 14
places, but only a specimen of it which appeared in the _Supplément_
was printed. The first table that was actually published is due to
Gauss, and was printed in Zach's _Monatliche Correspondenz_, xxvi. 498
(1812). Corresponding to the argument log x it gives the values of log
(1 + x^-1) and log (1 + x).
_Dual Logarithms._--This term was used by Oliver Byrne in a series of
works published between 1860 and 1870. Dual numbers and logarithms
depend upon the expression of a number as a product of 1.1, 1.01,
1.001 ... or of .9, .99, .999....
In the preceding _résumé_ only those publications have been mentioned
which are of historic importance or interest.[14] For fuller details
with respect to some of these works, for an account of tables
published in the latter part of the 19th century, and for those which
would now be used in actual calculation, reference should be made to
the article TABLES, MATHEMATICAL.
_Calculation of Logarithms._--The name logarithm is derived from the
words [Greek: logon arithmos], the number of the ratios, and the way
of regarding a logarithm which justifies the name may be explained as
follows. Suppose that the ratio of 10, or any other particular number,
to 1 is compounded of a very great number of equal ratios, as, for
example, 1,000,000, then it can be shown that the ratio of 2 to 1 is
very nearly equal to a ratio compounded of 301,030 of these small
ratios, or _ratiunculae_, that the ratio of 3 to 1 is very nearly
equal to a ratio compounded of 477,121 of them, and so on. The small
ratio, or _ratiuncula_, is in fact that of the millionth root of 10 to
unity, and if we denote it by the ratio of a to 1, then the ratio of 2
to 1 will be nearly the same as that of a^{301,030} to 1, and so on;
or, in other words, if a denotes the millionth root of 10, then 2 will
be nearly equal to a^{301,030}, 3 will be nearly equal to a^{477,121},
and so on.
Napier's original work, the _Descriptio Canonis_ of 1614, contained,
not logarithms of numbers, but logarithms of sines, and the relations
between the sines and the logarithms were explained by the motions of
points in lines, in a manner not unlike that afterwards employed by
Newton in the method of fluxions. An account of the processes by which
Napier constructed his table was given in the _Constructio Canonis_ of
1619. These methods apply, however, specially to Napier's own kind of
logarithms, and are different from those actually used by Briggs in
the construction of the tables in the _Arithmetica Logarithmica_,
although some of the latter are the same in principle as the processes
described in an appendix to the _Constructio_.
The processes used by Briggs are explained by him in the preface to
the _Arithmetica Logarithmica_ (1624). His method of finding the
logarithms of the small primes, which consists in taking a great
number of continued geometric means between unity and the given
primes, may be described as follows. He first formed the table of
numbers and their logarithms:--
Numbers. Logarithms.
10 1
3.162277... 0.5
1.778279... 0.25
1.333521... 0.125
1.154781... 0.0625
each quantity in the left-hand column being the square root of the one
above it, and each quantity in the right-hand column being the half
of the one above it. To construct this table Briggs, using about
thirty places of decimals, extracted the square root of 10 fifty-four
times, and thus found that the logarithm of 1.00000 00000 00000 12781
91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and
that for numbers of this form (i.e. for numbers beginning with 1
followed by fifteen ciphers, and then by seventeen or a less number of
significant figures) the logarithms were proportional to these
significant figures. He then by means of a simple proportion deduced
that log (1.00000 00000 00000 1) = 0.00000 00000 00000 04342 94481
90325 1804, so that, a quantity 1.00000 00000 00000 x (where x
consists of not more than seventeen figures) having been obtained by
repeated extraction of the square root of a given number, the
logarithm of 1.00000 00000 00000 x could then be found by multiplying
x by .00000 00000 00000 04342....
To find the logarithm of 2, Briggs raised it to the tenth power, viz.
1024, and extracted the square root of 1.024 forty-seven times, the
result being 1.00000 00000 00000 16851 60570 53949 77. Multiplying the
significant figures by 4342 ... he obtained the logarithm of this
quantity, viz. 0.00000 00000 00000 07318 55936 90623 9336, which
multiplied by 2^47 gave 0.01029 99566 39811 95265 277444, the
logarithm of 1.024, true to 17 or 18 places. Adding the characteristic
3, and dividing by 10, he found (since 2 is the tenth root of 1024)
log 2 = .30102 99956 63981 195. Briggs calculated in a similar manner
log 6, and thence deduced log 3.
It will be observed that in the first process the value of the modulus
is in fact calculated from the formula.
h 1
-------- = ---------,
10^h - 1 log(e) 10
the value of h being 1/2^54, and in the second process log10 2 is in
effect calculated from the formula.
1 2^47
log(10) 2 = [2^(10/2^47) - 1] × --------- × ----.
log(e) 10 10
Briggs also gave methods of forming the mean proportionals or square
roots by differences; and the general method of constructing
logarithmic tables by means of differences is due to him.
The following calculation of log 5 is given as an example of the
application of a method of mean proportionals. The process consists in
taking the geometric mean of numbers above and below 5, the object
being to at length arrive at 5.000000. To every geometric mean in the
column of numbers there corresponds the arithmetical mean in the
column of logarithms. The numbers are denoted by A, B, C, &c., in
order to indicate their mode of formation.
Numbers. Logarithms.
A = 1.000000 0.0000000
B = 10.000000 1.0000000
C = [root](AB) = 3.162277 0.5000000
D = [root](BC) = 5.623413 0.7500000
E = [root](CD) = 4.216964 0.6250000
F = [root](DE) = 4.869674 0.6875000
G = [root](DF) = 5.232991 0.7187500
H = [root](FG) = 5.048065 0.7031250
I = [root](FH) = 4.958069 0.6953125
K = [root](HI) = 5.002865 0.6992187
L = [root](IK) = 4.980416 0.6972656
M = [root](KL) = 4.991627 0.6982421
N = [root](KM) = 4.997242 0.6987304
O = [root](KN) = 5.000052 0.6989745
P = [root](NO) = 4.998647 0.6988525
Q = [root](OP) = 4.999350 0.6989135
R = [root](OQ) = 4.999701 0.6989440
S = [root](OR) = 4.999876 0.6989592
T = [root](OS) = 4.999963 0.6989668
V = [root](OT) = 5.000008 0.6989707
W = [root](TV) = 4.999984 0.6989687
X = [root](WV) = 4.999997 0.6989697
Y = [root](VX) = 5.000003 0.6989702
Z = [root](XY) = 5.000000 0.6989700
Great attention was devoted to the methods of calculating logarithms
during the 17th and 18th centuries. The earlier methods proposed were,
like those of Briggs, purely arithmetical, and for a long time
logarithms were regarded from the point of view indicated by their
name, that is to say, as depending on the theory of compounded ratios.
The introduction of infinite series into mathematics effected a great
change in the modes of calculation and the treatment of the subject.
Besides Napier and Briggs, special reference should be made to Kepler
(_Chilias_, 1624) and Mercator (_Logarithmotechnia_, 1668), whose
methods were arithmetical, and to Newton, Gregory, Halley and Cotes,
who employed series. A full and valuable account of these methods is
given in Hutton's "Construction of Logarithms," which occurs in the
introduction to the early editions of his _Mathematical Tables_, and
also forms tract 21 of his _Mathematical Tracts_ (vol. i., 1812). Many
of the early works on logarithms were reprinted in the _Scriptores
logarithmici_ of Baron Maseres already referred to.
In the following account only those formulae and methods will be
referred to which would now be used in the calculation of logarithms.