Chapter II: Front Matter (2)

The first publication of Briggian logarithms on the continent is due to Wingate, who published at Paris in 1625 his _Arithmétique logarithmétique_, containing seven-figure logarithms of numbers up to 1000, and log sines and tangents from Gunter's _Canon_ (1620). In the following year, 1626, Denis Henrion published at Paris a _Traicté des Logarithmes_, containing Briggs's logarithms of numbers up to 20,001 to 10 places, and Gunter's log sines and tangents to 7 places for every minute. In the same year de Decker also published at Gouda a work entitled _Nieuwe Telkonst, inhoudende de Logarithmi voor de Ghetallen beginnende van 1 tot 10,000_, which contained logarithms of numbers up to 10,000 to 10 places, taken from Briggs's _Arithmetica_ of 1624, and Gunter's log sines and tangents to 7 places for every minute.[11] Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.

The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the _Principia_ of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier's _Descriptio_. The calculation of tables of the natural trigonometrical functions may be said to have formed the work of the last half of the 16th century, and the great canon of natural sines for every 10 seconds to 15 places which had been calculated by Rheticus was published by Pitiscus only in 1613, the year before that in which the _Descriptio_ appeared. In the construction of the natural trigonometrical tables Great Britain had taken no part, and it is remarkable that the discovery of the principles and the formation of the tables that were to revolutionize or supersede all the methods of calculation then in use should have been so rapidly effected and developed in a country in which so little attention had been previously devoted to such questions.

For more detailed information relating to Napier, Briggs and Vlacq,
and the invention of logarithms, the reader is referred to the life of
Briggs in Ward's _Lives of the Professors of Gresham College_ (London,
1740); Thomas Smith's _Vitae quorundam eruditissimorum et illustrium
virorum_ (Vita Henrici Briggii) (London, 1707); Mark Napier's _Memoirs
of John Napier_ already referred to, and the same author's _Naperi
libri qui supersunt_ (1839); Hutton's _History_; de Morgan's article
already referred to; Delambre's _Histoire de l'Astronomie moderne_;
the report on mathematical tables in the _Report of the British
Association_ for 1873; and the _Philosophical Magazine_ for October
and December 1872 and May 1873. It may be remarked that the date
usually assigned to Briggs's first visit to Napier is 1616 and not
1615 as stated above, the reason being that Napier was generally
supposed to have died in 1618; but it was shown by Mark Napier that
the true date is 1617.

In the years 1791-1807 Francis Maseres published at London, in six volumes quarto "Scriptores Logarithmici, or a collection of several curious tracts on the nature and construction of logarithms, mentioned in Dr Hutton's historical introduction to his new edition of Sherwin's mathematical tables ...," which contains reprints of Napier's _Descriptio_ of 1614, Kepler's writings on logarithms (1624-1625), &c. In 1889 a translation of Napier's _Constructio_ of 1619 was published by Walter Rae Macdonald. Some valuable notes are added by the translator, in one of which he shows the accuracy of the method employed by Napier in his calculations, and explains the origin of a small error which occurs in Napier's table. Appended to the Catalogue is a full and careful bibliography of all Napier's writings, with mention of the public libraries, British and foreign, which possess copies of each. A facsimile reproduction of Bartholomew Vincent's Lyons edition (1620) of the _Constructio_ was issued in 1895 by A. Hermann at Paris (this imprint occurs on page 62 after the word "Finis").

It now remains to notice briefly a few of the more important events in the history of logarithmic tables subsequent to the original calculations.

_Common or Briggian Logarithms of Numbers._--Nathaniel Roe's _Tabulae
logarithmicae_ (1633) was the first complete seven-figure table that
was published. It contains seven-figure logarithms of numbers from 1
to 100,000, with characteristics unseparated from the mantissae, and
was formed from Vlacq's table (1628) by leaving out the last three
figures. All the figures of the number are given at the head of the
columns, except the last two, which run down the extreme columns--1 to
50 on the left-hand side, and 50 to 100 on the right-hand side. The
first four figures of the logarithms are printed at the top of the
columns. There is thus an advance half way towards the arrangement now
universal in seven-figure tables. The final step was made by John
Newton in his _Trigonometria Britannica_ (1658), a work which is also
noticeable as being the only extensive eight-figure table that until
recently had been published; it contains logarithms of sines, &c., as
well as logarithms of numbers.

In 1705 appeared the original edition of Sherwin's tables, the first
of the series of ordinary seven-figure tables of logarithms of numbers
and trigonometrical functions such as are in general use now. The work
went through several editions during the 18th century, and was at
length superseded in 1785 by Hutton's tables, which continued in
successive editions to maintain their position for a century.

In 1717 Abraham Sharp published in his _Geometry Improv'd_ the
Briggian logarithms of numbers from 1 to 100, and of primes from 100
to 1100, to 61 places; these were copied into the later editions of
Sherwin and other works.

In 1742 a seven-figure table was published in quarto form by Gardiner,
which is celebrated on account of its accuracy and of the elegance of
the printing. A French edition, which closely resembles the original,
was published at Avignon in 1770.

In 1783 appeared at Paris the first edition of François Callet's
tables, which correspond to those of Hutton in England. These tables,
which form perhaps the most complete and practically useful collection
of logarithms for the general computer that has been published, passed
through many editions.

In 1794 Vega published his _Thesaurus logarithmorum completus_, a
folio volume containing a reprint of the logarithms of numbers from
Vlacq's _Arithmetica logarithmica_ of 1628, and _Trigonometria
artificialis_ of 1633. The logarithms of numbers are arranged as in an
ordinary seven-figure table. In addition to the logarithms reprinted
from the _Trigonometria_, there are given logarithms for every second
of the first two degrees, which were the result of an original
calculation. Vega devoted great attention to the detection and
correction of the errors in Vlacq's work of 1628. Vega's _Thesaurus_
has been reproduced photographically by the Italian government. Vega
also published in 1797, in 2 vols. 8vo, a collection of logarithmic
and trigonometrical tables which has passed through many editions, a
very useful one volume stereotype edition having been published in
1840 by Hülsse. The tables in this work may be regarded as to some
extent supplementary to those in Callet.

If we consider only the logarithms of numbers, the main line of
descent from the original calculation of Briggs and Vlacq is Roe, John
Newton, Sherwin, Gardiner; there are then two branches, viz. Hutton
founded on Sherwin and Callet on Gardiner, and the editions of Vega
form a separate offshoot from the original tables. Among the most
useful and accessible of modern ordinary seven-figure tables of
logarithms of numbers and trigonometrical functions may be mentioned
those of Bremiker, Schrön and Bruhns. For logarithms of numbers only
perhaps Babbage's table is the most convenient.[12]

In 1871 Edward Sang published a seven-figure table of logarithms of
numbers from 20,000 to 200,000, the logarithms between 100,000 and
200,000 being the result of a new calculation. By beginning the table
at 20,000 instead of at 10,000 the differences are halved in
magnitude, while the number of them in a page is quartered. In this
table multiples of the differences, instead of proportional parts, are
given.[13] John Thomson of Greenock (1782-1855) made an independent
calculation of logarithms of numbers up to 120,000 to 12 places of
decimals, and his table has been used to verify the errata already
found in Vlacq and Briggs by Lefort (see _Monthly Not. R.A.S._ vol.
34, p. 447). A table of ten-figure logarithms of numbers up to 100,009
was calculated by W. W. Duffield and published in the _Report of the
U.S. Coast and Geodetic Survey for 1895-1896_ as Appendix 12, pp.
395-722. The results were compared with Vega's _Thesaurus_ (1794)
before publication.

_Common or Briggian Logarithms of Trigonometrical Functions._--The
next great advance on the Trigonometria artificialis took place more
than a century and a half afterwards, when Michael Taylor published in
1792 his seven-decimal table of log sines and tangents to every second
of the quadrant; it was calculated by interpolation from the
_Trigonometria_ to 10 places and then contracted to 7. On account of
the great size of this table, and for other reasons, it never came
into very general use, Bagay's _Nouvelles tables astronomiques_
(1829), which also contains log sines and tangents to every second,
being preferred; this latter work, which for many years was difficult
to procure, has been reprinted with the original title-page and date
unchanged. The only other logarithmic canon to every second that has
been published forms the second volume of Shortrede's _Logarithmic
Tables_ (1849). In 1784 the French government decided that new tables
of sines, tangents, &c., and their logarithms, should be calculated in
relation to the centesimal division of the quadrant. Prony was charged
with the direction of the work, and was expressly required "non
seulement à composer des tables qui ne laissassent rien à désirer
quant à l'exactitude, mais à en faire le monument de calcul le plus
vaste et le plus imposant qui eût jamais été exécuté ou même conçu."
Those engaged upon the work were divided into three sections: the
first consisted of five or six mathematicians, including Legendre, who
were engaged in the purely analytical work, or the calculation of the
fundamental numbers; the second section consisted of seven or eight
calculators possessing some mathematical knowledge; and the third
comprised seventy or eighty ordinary computers. The work, which was
performed wholly in duplicate, and independently by two divisions of
computers, occupied two years. As a consequence of the double
calculation, there are two manuscripts, one deposited at the
Observatory, and the other in the library of the Institute, at Paris.
Each of the two manuscripts consists essentially of seventeen large
folio volumes, the contents being as follows:--

Logarithms of numbers up to 200,000 8 vols.

Natural sines 1 "

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.

Since

log(e)(1 + x) = x - ½x² + (1/3)x³ - ¼x^4 + &c.,

we have, by changing the sign of x,

log(e)(1 - x) = -x - ½x² - (1/3)x³ - ¼x^4 - &c.;

whence

1 + x
log(e) ----- = 2(x + (1/3)x³ + (1/5)x^5 + &c.),
1 - x

p - q
and, therefore, replacing x by -----,
p + q
_ _
p | p - q /p - q\³ /p - q\^5 |
log(e) --- = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |,
q |_ p + q \p + q/ \p + q/ _|

in which the series is always convergent, so that the formula affords
a method of deducing the logarithm of one number from that of another.

As particular cases we have, by putting q = 1,
_ _
| p - 1 /p - 1\³ /p - 1\^5 |
log(e) p = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |,
|_ p + 1 \p + 1/ \p + 1/ _|

and by putting q = p + 1,
_ _
| 1 1 1 |
log(e)(p + 1) - log(e)(p) = 2 | ------ + (1/3)--------- + (1/5)----------- + &c. |;
|_ 2p + 1 (2p + 1)³ (2p + 1)^5 _|

the former of these equations gives a convergent series for log(e)p,
and the latter a very convergent series by means of which the
logarithm of any number may be deduced from the logarithm of the
preceding number.

From the formula for log(e)(p/q) we may deduce the following very
convergent series for log(e)2, log(e)3 and log(e)5, viz.:--

log(e)2 = 2( 7P + 5Q + 3R),
log(e)3 = 2(11P + 8Q + 5R),
log(e)5 = 2(16P + 12Q + 7R),

where

1 1 1
P = -- + (1/3) · ------ + (1/5) · ------ + &c.
31 (31)^3 (31)^5

1 1 1
Q = -- + (1/3) · ------ + (1/5) · ------ + &c.
49 (49)^3 (49)^5

1 1 1
R = --- + (1/3) · ------- + (1/5) · ------- + &c.
161 (161)^3 (161)^5

The following still more convenient formulae for the calculation of
log(e)2, log(e)3, &c. were given by J. Couch Adams in the _Proc. Roy.
Soc._, 1878, 27, p. 91. If

10 / 1 \ 25 / 4 \
a = log -- = -log ( 1 - -- ), b = log -- = -log ( 1 - --- ),
9 \ 10 / 24 \ 100 /

81 / 1 \ 50 / 2 \
c = log -- = log ( 1 + -- ), d = log -- = -log ( 1 - --- ),
80 \ 80 / 49 \ 100 /

126 / 8 \
e = log --- = log ( 1 + ---- ),
125 \ 1000 /

then

log 2 = 7a - 2b + 3c, log 3 = 11a - 3b + 5c, log 5 = 16a - 4b + 7c,

and

log 7 = ½(39a - 10b + 17c - d) or = 19a - 4b + 8c + e,

and we have the equation of condition,

a - 2b + c = d + 2e.

By means of these formulae Adams calculated the values of log(e)2,
log(e)3, log(e)5, and log(e)7 to 276 places of decimals, and deduced
the value of log(e)10 and its reciprocal M, the modulus of the
Briggian system of logarithms. The value of the modulus found by Adams
is

Mo = 0.43429 44819 03251 82765 11289
18916 60508 22943 97005 80366
65661 14453 78316 58646 49208
87077 47292 24949 33843 17483
18706 10674 47663 03733 64167
92871 58963 90656 92210 64662

81226 58521 27086 56867 03295
93370 86965 88266 88331 16360
77384 90514 28443 48666 76864
65860 85135 56148 21234 87653
43543 43573 17253 83562 21868
25

which is true certainly to 272, and probably to 273, places (_Proc.
Roy. Soc._, 1886, 42, p. 22, where also the values of the other
logarithms are given).

If the logarithms are to be Briggian all the series in the preceding
formulae must be multiplied by M, the modulus; thus,

log(10) (1 + x) = M (x - ½x² + (1/3)x³ - ¼x^4 + &c.),

and so on.

As has been stated, Abraham Sharp's table contains 61-decimal
Briggian logarithms of primes up to 1100, so that the logarithms of
all composite numbers whose greatest prime factor does not exceed this
number may be found by simple addition; and Wolfram's table gives
48-decimal hyperbolic logarithms of primes up to 10,009. By means of
these tables and of a factor table we may very readily obtain the
Briggian logarithm of a number to 61 or a less number of places or of
its hyperbolic logarithm to 48 or a less number of places in the
following manner. Suppose the hyperbolic logarithm of the prime number
43,867 required. Multiplying by 50, we have 50 × 43,867 = 2,193,350,
and on looking in Burckhardt's _Table des diviseurs_ for a number near
to this which shall have no prime factor greater than 10,009, it
appears that

2,193,349 = 23 × 47 × 2029;

thus

43,867 = (1/50)(23 × 47 × 2029 + 1),

and therefore

log(e) 43,867 = log(e) 23 + log(e) 47 + log(e) 2029 - log(e) 50

1 1 1
+ --------- - ½ ------------ + (1/3) ---------- - &c.
2,193,349 (2,193,349)² (193,349)³

The first term of the series in the second line is

0.00000 04559 23795 07319 6286;

dividing this by 2 ×2,193,349 we obtain

0.00000 00000 00103 93325 3457,

and the third term is

0.00000 00000 00000 00003 1590,

so that the series =

0.00000 04559 23691 13997 4419;

whence, taking out the logarithms from Wolfram's table,

log(e) 43,867 = 10.68891 76079 60568 10191 3661.

The principle of the method is to multiply the given prime (supposed
to consist of 4, 5 or 6 figures) by such a factor that the product may
be a number within the range of the factor tables, and such that, when
it is increased by 1 or 2, the prime factors may all be within the
range of the logarithmic tables. The logarithm is then obtained by use
of the formula

d d² d³
log(e)(x + d) = log(e)x + --- - ½ -- + (1/3) -- - &c.,
x x² x³

in which of course the object is to render d/x as small as possible.
If the logarithm required is Briggian, the value of the series is to
be multiplied by M.

If the number is incommensurable or consists of more than seven
figures, we can take the first seven figures of it (or multiply and
divide the result by any factor, and take the first seven figures of
the result) and proceed as before. An application to the hyperbolic
logarithm of [pi] is given by Burckhardt in the introduction to his
_Table des diviseurs_ for the second million.

The best general method of calculating logarithms consists, in its
simplest form, in resolving the number whose logarithm is required
into factors of the form 1 - .1^(r)n, where n is one of the nine
digits; and making use of subsidiary tables of logarithms of factors
of this form. For example, suppose the logarithm of 543839 required to
twelve places. Dividing by 10^5 and by 5 the number becomes 1.087678,
and resolving this number into factors of the form 1 - .1^(r)n we find
that

543839 = 10^5 × 5(1-.1²8)(1-.1^(4)6)(1-.1^(5)6)(1-.1^(6)3)(1-.1^(7)3)
× (1-.1^(8)5)(1-.1^(9)7)(1-.1^(10)9)(1-.1^(11)3)(1-.1^(12)2),

where 1-1²8 denotes 1-.08, 1-.1^(4)6 denotes 1-.0006, &c., and so on.
All that is required therefore in order to obtain the logarithm of any
number is a table of logarithms, to the required number of places, of
.n, .9n, .99n, .999n, &c., for n = 1, 2, 3, ... 9.

The resolution of a number into factors of the above form is easily
performed. Taking, for example, the number 1.087678, the object is to
destroy the significant figure 8 in the second place of decimals; this
is effected by multiplying the number by 1-.08, that is, by
subtracting from the number eight times itself advanced two places,
and we thus obtain 1.00066376. To destroy the first 6 multiply by 1 -
.0006 giving 1.000063361744, and multiplying successively by 1 -
.00006 and 1 - .000003, we obtain 1.000000357932, and it is clear that
these last six significant figures represent without any further work
the remaining factors required. In the corresponding antilogarithmic
process the number is expressed as a product of factors of the form 1
+ .1^(n)x.

This method of calculating logarithms by the resolution of numbers
into factors of the form 1 - .1^(r)n is generally known as Weddle's
method, having been published by him in _The Mathematician_ for
November 1845, and the corresponding method for antilogarithms by
means of factors of the form 1 + (.1)^(r)n is known by the name of
Hearn, who published it in the same journal for 1847. In 1846 Peter
Gray constructed a new table to 12 places, in which the factors were
of the form 1-(.01)^(r)n, so that n had the values 1, 2, ... 99; and
subsequently he constructed a similar table for factors of the form 1
+ (.01)^(r)n. He also devised a method of applying a table of Hearn's
form (i.e. of factors of the form 1 +.1^(r)n) to the construction of
logarithms, and calculated a table of logarithms of factors of the
form 1 + (.001)^(r)n to 24 places. This was published in 1876 under
the title _Tables for the formation of logarithms and antilogarithms
to twenty-four or any less number of places_, and contains the most
complete and useful application of the method, with many improvements
in points of detail. Taking as an example the calculation of the
Briggian logarithm of the number 43,867, whose hyperbolic logarithm
has been calculated above, we multiply it by 3, giving 131,601, and
find by Gray's process that the factors of 1.31601 are

(1) 1.316 (5) 1.(001)^(4)002
(2) 1.000007 (6) 1.(001)^(5)602
(3) 1.(001)²598 (7) 1.(001)^(6)412
(4) 1.(001)³780 (8) 1.(001)^(7)340

Taking the logarithms from Gray's tables we obtain the required
logarithm by addition as follows:--

522 878 745 280 337 562 704 972 = colog 3
119 255 889 277 936 685 553 913 = log (1)
3 040 050 733 157 610 239 = log (2)
259 708 022 525 453 597 = log (3)
338 749 695 752 424 = log (4)
868 588 964 = log (5)
261 445 278 = log (6)
178 929 = log (7)
148 = log (8)
--------------------------------------------------------
4.642 137 934 655 780 757 288 464 = log(10)43,867

In Shortrede's _Tables_ there are tables of logarithms and factors of
the form 1 ± (.01)^(r)n to 16 places and of the form 1 ± (.1)^(r)n to
25 places; and in his _Tables de Logarithmes à 27 Décimales_ (Paris,
1867) Fédor Thoman gives tables of logarithms of factors of the form 1
± .1^(r)n. In the _Messenger of Mathematics_, vol. iii. pp. 66-92,
1873, Henry Wace gave a simple and clear account of both the
logarithmic and antilogarithmic processes, with tables of both
Briggian and hyperbolic logarithms of factors of the form 1 ± .1^(r)n
to 20 places.

Although the method is usually known by the names of Weddle and Hearn,
it is really, in its essential features, due to Briggs, who gave in
the _Arithmetica logarithmica_ of 1624 a table of the logarithms of 1
+ .1^(r)n up to r = 9 to 15 places of decimals. It was first formally
proposed as an independent method, with great improvements, by Robert
Flower in _The Radix_, _a new way of making Logarithms_, which was
published in 1771; and Leonelli, in his _Supplement logarithmique_
(1802-1803), already noticed, referred to Flower and reproduced some
of his tables. A complete bibliography of this method has been given
by A. J. Ellis in a paper "on the potential radix as a means of
calculating logarithms," printed in the _Proceedings of the Royal
Society_, vol. xxxi., 1881, pp. 401-407, and vol. xxxii., 1881, pp.
377-379. Reference should also be made to Hoppe's _Tafeln zur
dreissigstelligen logarithmischen Rechnung_ (Leipzig, 1876), which
give in a somewhat modified form a table of the hyperbolic logarithm
of 1 + .1^(r)n.

The preceding methods are only appropriate for the calculation of
isolated logarithms. If a complete table had to be reconstructed, or
calculated to more places, it would undoubtedly be most convenient to
employ the method of differences. A full account of this method as
applied to the calculation of the _Tables du Cadastre_ is given by
Lefort in vol. iv. of the _Annales de l'Observatoire de Paris_.
(J. W. L. G.)

FOOTNOTES:

[1] Dr Thomas Smith thus describes the ardour with which Briggs
studied the _Descriptio_: "Hunc in deliciis habuit, in sinu, in
manibus, in pectore gestavit, oculisque avidissimis, et mente
attentissima, iterum iterumque perlegit,..." _Vitae quorundam
eruditissimorum et illustrium virorum_ (London, 1707).

[2] William Lilly's account of the meeting of Napier and Briggs at
Merchiston is quoted in the article NAPIER.

[3] It was certainly published after Napier's death, as Briggs
mentions his "librum posthumum." This _liber posthumus_ was the
_Constructio_ referred to later in this article.

[4] Frisch's _Kepleri opera omnia_, ii. 834. Frisch thinks Bramer
possibly relied on Kepler's statement quoted in the text ("Quibus
forte confisus Kepleri verbis Benj. Bramer...."). See also vol. vii.
p. 298.

The claims of Byrgius are discussed in Kästner's _Geschichte der
Mathematik_, ii. 375, and iii. 14; Montucla's _Histoire des
mathématiques_, ii. 10; Delambre's _Histoire de l'astronomie
moderne_, i. 560; de Morgan's article on "Tables" in the _English
Cyclopaedia_; Mark Napier's _Memoirs of John Napier of Merchiston_
(1834), p. 392, and Cantor's _Geschichte der Mathematik_, ii. (1892),
662. See also Gieswald, _Justus Byrg als Mathematiker und dessen
Einleitung in seine Logarithmen_ (Danzig, 1856).

[5] See Mark Napier's _Memoirs of John Napier of Merchiston_ (1834),
p. 362.

[6] In the _Rabdologia_ (1617) he speaks of the canon of logarithms
as "a me longo tempore elaboratum."

[7] A careful examination of the history of the method is given by
Scheibel in his _Einleitung zur mathematischen Bücherkenntniss_,
Stück vii. (Breslau, 1775), pp. 13-20; and there is also an account
in Kästner's _Geschichte der Mathematik_, i. 566-569 (1796); in
Montucla's _Histoire des mathématiques_, i. 583-585 and 617-619; and
in Klügel's _Wörterbuch_ (1808), article "Prosthaphaeresis."

[8] Besides his connexion with logarithms and improvements in the
method of prosthaphaeresis, Byrgius has a share in the invention of
decimal fractions. See Cantor, _Geschichte_, ii. 567. Cantor
attributes to him (in the use of his prosthaphaeresis) the first
introduction of a subsidiary angle into trigonometry (vol. ii. 590).

[9] The title of this work is--_Benjaminis Ursini_ ... _cursus
mathematici practici volumen primum continens illustr. & generosi Dn.
Dn. Johannis Neperi Baronis Merchistonij &c. Scoti trigonometriam
logarithmicam usibus discentium accommodatam_ ... _Coloniae_ ...
_CI[~C] I[~C]C XIX_. At the end, Napier's table is reprinted, but to
two figures less. This work forms the earliest publication of
logarithms on the continent.

[10] The title is _Logarithmorum canonis descriptio, seu
arithmeticarum supputationum mirabilis abbreviatio_. _Ejusque usus in
utraque trigonometria ut etiam in omni logistica mathematica,
amplissimi, facillimi & expeditissimi explicatio. Authore ac
inventore Ioanne Nepero, Barone Merchistonii, &c. Scoto. Lugduni_....
It will be seen that this title is different from that of Napier's
work of 1614; many writers have, however, erroneously given it as the
title of the latter.

[11] In describing the contents of the works referred to, the
language and notation of the present day have been adopted, so that
for example a table to radius 10,000,000 is described as a table to 7
places, and so on. Also, although logarithms have been spoken of as
to the base e, &c., it is to be noticed that neither Napier nor
Briggs, nor any of their successors till long afterwards, had any
idea of connecting logarithms with exponents.

[12] The smallest number of entries which are necessary in a table of
logarithms in order that the intermediate logarithms may be
calculable by proportional parts has been investigated by J. E. A.
Steggall in the _Proc. Edin. Math. Soc._, 1892, 10, p. 35. This
number is 1700 in the case of a seven-figure table extending to
100,000.

[13] Accounts of Sang's calculations are given in the _Trans. Roy.
Soc. Edin._, 1872, 26, p. 521, and in subsequent papers in the
_Proceedings_ of the same society.

[14] In vol. xv. (1875) of the _Verhandelingen_ of the Amsterdam
Academy of Sciences, Bierens de Haan has given a list of 553 tables
of logarithms. A previous paper of the same kind, containing notices
of some of the tables, was published by him in the _Verslagen en
Mededeelingen_ of the same academy (Afd. Natuurkunde) deel. iv.
(1862), p. 15.

LOGAU, FRIEDRICH, FREIHERR VON (1604-1655), German epigrammatist, was born at Brockut, near Nimptsch, in Silesia, in June 1604. He was educated at the gymnasium of Brieg and subsequently studied law. He then entered the service of the duke of Brieg. In 1644 he was made "ducal councillor." He died at Liegnitz on the 24th of July 1655. Logau's epigrams, which appeared in two collections under the pseudonym "Salomon von Golaw" (an anagram of his real name) in 1638 (_Erstes Hundert Teutscher Reimensprüche_) and 1654 (_Deutscher Sinngedichte drei Tausend_), show a marvellous range and variety of expression. He had suffered bitterly under the adverse conditions of the time; but his satire is not merely the outcome of personal feeling. In the turbulent age of the Thirty Years' War he was one of the few men who preserved intact his intellectual integrity and judged his contemporaries fairly. He satirized with unsparing hand the court life, the useless bloodshed of the war, the lack of national pride in the German people, and their slavish imitation of the French in customs, dress and speech. He belonged to the _Fruchtbringende Gesellschaft_ under the name _Der Verkleinernde_, and regarded himself as a follower of Martin Opitz; but he did not allow such ties to influence his independence or originality.

Logau's _Sinngedichte_ were edited in 1759 by G. E. Lessing and K. W.
Ramler, who first drew attention to their merits; a second edition
appeared in 1791. A critical edition was published by G. Eitner in
1872, who also edited a selection of Logau's epigrams for the
_Deutsche Dichter des XVII. Jahrhunderts_ (vol. iii., 1870); there is
also a selection by H. Oesterley in Kürschner's _Deutsche
Nationalliteratur_, vol. xxviii. (1885). See H. Denker, _Beiträge zur
literarischen Würdigung Logaus_ (1889); W. Heuschkel, _Untersuchungen
über Ränders und Lessings Bearbeitung Logauscher Sinngedichte_ (1901).

LOGIA, a title used to describe a collection of the sayings of Jesus Christ ([Greek: logia Iêsou]) and therefore generally applied to the "Sayings of Jesus" discovered in Egypt by B. P. Grenfell and A. S. Hunt. There is some question as to whether the term is rightly used for this purpose. It does not occur in the Papyri in this sense. Each "saying" is introduced by the phrase "Jesus says" ([Greek: legei]) and the collection is described in the introductory words of the 1903 series as [Greek: logoi] not as [Greek: logia]. Some justification for the employment of the term is found in early Christian literature. Several writers speak of the [Greek: logia tou kuriou] or [Greek: ta kuriaka logia], i.e. oracles of (or concerning) the Lord. Polycarp, for instance, speaks of "those who pervert the oracles of the Lord." (Philipp. 7), and Papias, as Eusebius tells us, wrote a work with the title "Expositions of the Oracles of the Lord." The expression has been variously interpreted. It need mean no more (Lightfoot, _Essays on Supernatural Religion_, 172 seq.) than narratives of (or concerning) the Lord; on the other hand, the phrase is capable of a much more definite meaning, and there are many scholars who hold that it refers to a document which contained a collection of the sayings of Jesus. Some such document, we know, must lie at the base of our Synoptic Gospels, and it is quite possible that it may have been known to and used by Papias. It is only on this assumption that the use of the term Logia in the sense described above can be justified.

"The Sayings," to which the term Logia is generally applied, consist of (a) a papyrus leaf containing seven or eight sayings of Jesus discovered in 1897, (b) a second leaf containing five more sayings discovered in 1903, (c) two fragments of unknown Gospels, the former published in 1903, the latter in 1907. All these were found amongst the great mass of papyri acquired by the Egyptian Exploration Fund from the ruins of Oxyrhynchus, one of the chief early Christian centres in Egypt, situated some 120 m. S. of Cairo.

The eight "sayings" discovered in 1897 are as follows:--

1. ... [Greek: kai tote diablepseis ekbalein to karphos to en tô
ophthalmô tou adelphou sou].

2. [Greek: Legei Iêsous ean mê nêsteusête ton kosmon ou mê eurête tên
basileian tou theou. kai ean mê sabbatisête to sabbaton ouk opsesthe
ton patera].

3. [Greek: Legei Iêsous e[s]tên en mesô tou kosmou kai en sarki
ôpsthên autois, kai euron pantas methuontas kai oudena euron dipsônta
en autois, kai ponei ê psychê mou epi tois huiois tôn anthrôpôn, hoti
typhloi eisin tê kardia autô[n] k[ai] ou ble[pousin]]....

4. [Illegible: possibly joins on to 3] ... [Greek: [t]ên ptôcheian].

5. [Greek: [Leg]ei [Iêsous hop]ou ean ôsin [b, ouk] e[isi]n atheoi kai
h[o]pou e[is] estin monos, [le]gô, egô eimi met aut[ou] egei[r]on ton
lithon kakei heurêseis me, schison to xylon kagô ekei eimi].

6. [Greek: Legei Iêsous ouk estin dektos prophêtês en tê patridi
aut[o]u, oude iatpos poiei therapeias eis tous ginôskontas auton].

7. [Greek: Legei Iêsous polisoi kodomêmenê ep' akron [o]rous hypsêlou
kai estêrigmenê oute pe[s]ein dynatai oute kry[b]ênai].

8. [Greek: Legei Iêsous akoueis [e]is to hen ôtion sou to [de eteron
synekleisas]].

Letters in brackets are missing in the original: letters which are
dotted beneath are doubtful.

1. "... and then shalt thou see clearly to cast out the mote that is
in thy brother's eye."

2. "Jesus saith, Except ye fast to the world, ye shall in no wise find
the kingdom of God; and except ye make the sabbath a real sabbath, ye
shall not see the Father."

3. "Jesus saith, I stood in the midst of the world and in the flesh
was I seen of them, and I found all men drunken, and none found I
athirst among them, and my soul grieveth over the sons of men, because
they are blind in their heart, and see not...."

4. "... poverty...."

5. "Jesus saith, Wherever there are two, they are not without God, and
wherever there is one alone, I say, I am with him. Raise the stone and
there thou shalt find me, cleave the wood and there am I."

6. "Jesus saith, A prophet is not acceptable in his own country,
neither doth a physician work cures upon them that know him."

7. "Jesus saith, A city built upon the top of a high hill and
stablished can neither fall nor be hid."

8. "Jesus saith, Thou hearest with one ear [but the other ear hast
thou closed]."

The "sayings" of 1903 were prefaced by the following introductory statement:--

[Greek: hoi toioi hoi logoi hoi [... hous elalêsen Iê(sou)s ho zôn
k[yrios? ... kai Thôma kai eipen [autois; pas hostis an tôn logôn
tout[ôn akousê thanatou ou mê geusêtai.]

"These are the (wonderful?) words which Jesus the living (Lord) spake
to ... and Thomas and he said unto (them) every one that hearkens to
these words shall never taste of death."

The "sayings" themselves are as follows:--

(1) [Greek: [legei Iê(sou)s· mê pausasthô ho zê[tôn...
heôs an heurê kai hotan heurê [thambêthêsetai
kai thambêtheis basileusei ka[i basileusas
anapaêsetai.]

(2) [Greek: legei I[ê(sous ... tines ...
hoi helkontes hêmas [eis tên basileian ei
hê basileia en oura[nô estin;
ta peteina tou our[anou kai tôn thêriôn ho
ti hypo tên gên est[in ê epi tês gês kai
hoi ichthyes tês thala[ssês houtoi hoi helkon-
tes hymas kai hê bas[ileia tôn ouranôn
entos hymôn [e]sti [kai hostis an heauton
gnô tautên heurê[sei...
heautous gnôsesthe [kai eidêsete hoti huioi
este humeis tou patros tou t[...
gnôs(es)the heautous en[...
kai hu eis este êpto[]

(3) [Greek: [ legei Iê(sou)s
ouk apoknêsei anth[rôpos...
rôn eperôtêsai pa[...
rôn peri tou topou tê[s...
sete hoti polloi esontai p[rôtoi eschatoi kai
hoi eschatoi prôtoi kai [...
sin.]

(4) [Greek: legei Iê(sou)s· [pan to mê empros-
then tês opseôs sou kai [to kekrummenon
apo sou apokalyph(th)êset{ai soi. ou gar es-
tin krypton ho ou phane[ron genêsetai
kai tethammenon ho o[uk egerthêsetai.]

(5) [Greek: [ex] etazousin auton ho[i mathêtai autou kai
[le]gousin; pôs nêsteu[somen kai pôs...
[ ... ] metha kai pôs [ ...
[ ... k]ai ti paratêrês{omen...
[ ... ]n? legei Iê(sou)s; [ ...
[ ... ]eitai mê poeit[e...
[ ... ]ês alêtheias an[ ...
[ ... ]n a[p]okekr[y...
[ ... ma] kari[os] estin [ ...
[ ... ]ô est[i...
[ ... ]in [ ... ]

1. "Jesus saith, Let not him who seeks ... cease until he finds and
when he finds he shall be astonished; astonished he shall reach the
kingdom and having reached the kingdom he shall rest."

2. "Jesus saith (ye ask? who are those) that draw us (to the kingdom
if) the kingdom is in Heaven? ... the fowls of the air and all beasts
that are under the earth or upon the earth and the fishes of the sea
(these are they which draw) you and the kingdom of Heaven is within
you and whosoever shall know himself shall find it. (Strive
therefore?) to know yourselves and ye shall be aware that ye are the
sons of the (Almighty?) Father; (and?) ye shall know that ye are in
(the city of God?) and ye are (the city?)."

3. "Jesus saith, A man shall not hesitate ... to ask concerning his
place (in the kingdom. Ye shall know) that many that are first shall
be last and the last first and (they shall have eternal life?)."

4. "Jesus saith, Everything that is not before thy face and that which
is hidden from thee shall be revealed to thee. For there is nothing
hidden which shall not be made manifest nor buried which shall not be
raised."

5. "His disciples question him and say, How shall we fast and how
shall we (pray?) ... and what (commandment) shall we keep ... Jesus
saith ... do not ... of truth ... blessed is he ..."

_The fragment of a lost Gospel_ which was discovered in 1903 contained originally about fifty lines, but many of them have perished and others are undecipherable. The translation, as far as it can be made out, is as follows:--

1-7. "(Take no thought) from morning until even nor from evening until
morning either for your food what ye shall eat or for your raiment
what ye shall put on. 7-13. Ye are far better than the lilies which
grow but spin not. Having one garment what do ye (lack)?... 13-15. Who
could add to your stature? 15-16. He himself will give you your
garment. 17-23. His disciples say unto him, When wilt thou be manifest
unto us and when shall we see thee? He saith, When ye shall be
stripped and not be ashamed ... 41-46. He said, The key of knowledge
ye hid: ye entered not in yourselves, and to them that were entering
in, ye opened not."

_The second Gospel fragment_ discovered in 1907 "consists of a single vellum leaf, practically complete except at one of the lower corners and here most of the lacunae admit of a satisfactory solution." The translation is as follows:--

... before he does wrong makes all manner of subtle excuse. But give
heed lest ye also suffer the same things as they: for the evil doers
among men receive their reward not among the living only, but also
await punishment and much torment. And he took them and brought them
into the very place of purification and was walking in the temple. And
a certain Pharisee, a chief priest, whose name was Levi, met them and
said to the Saviour, Who gave thee leave to walk in this place of
purification, and to see these holy vessels when thou hast not washed
nor yet have thy disciples bathed their feet? But defiled thou hast
walked in this temple, which is a pure place, wherein no other man
walks except he has washed himself and changed his garments neither
does he venture to see these holy vessels. And the Saviour straightway
stood still with his disciples and answered him, Art thou then, being
here in the temple, clean? He saith unto him, I am clean; for I washed
in the pool of David and having descended by one staircase, I ascended
by another and I put on white and clean garments, and then I came and
looked upon these holy vessels. The Saviour answered and said unto
him, Woe ye blind, who see not. Thou hast washed in these running
waters wherein dogs and swine have been cast night and day and hast
cleansed and wiped the outside skin which also the harlots and
flute-girls anoint and wash and wipe and beautify for the lust of men;
but within they are full of scorpions and all wickedness. But I and my
disciples who thou sayest have not bathed have been dipped in the
waters of eternal life which come from.... But woe unto thee....