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Trigonometria Britannica (1633)
Denis Roegel
To cite this version:
Denis Roegel. A reconstruction of the tables of Briggs and Gellibrand’s Trigonometria Britannica
(1633). [Research Report] 2010. �inria-00543943�
of the tables of Briggs and Gellibrand’s
Trigonometria Britannica
(1633)
Denis Roegel
6 December 2010
1
Briggs’ first tables (1617)
Henry Briggs (1561–1631)
1is the author of the first table of decimal logarithms, published
in 1617, of the first extensive table of decimal logarithms of numbers, and of one of the
first two extensive tables of decimal logarithms of trigonometric functions.
After having been educated in Cambridge, Briggs became in 1596 the first professor of
geometry at Gresham College, London [85, p. 120], [74, p. 20], [92]. Gresham College was
England’s scientific centre for navigation, geometry, astronomy and surveying.
2Briggs
stayed there until 1620, at which time he went to Oxford, having been appointed the first
Savilian Professor of Geometry in 1619 [74, p. 24].
While at Gresham College, Briggs became friends with Edward Wright. He also seems
to have spent time doing research in astronomy and navigation [74, pp. 29–30]. Briggs had
in particular published several tables for the purpose of navigation in 1602 and 1610 [93].
Several of Briggs’ tables were published under the name of others [74, p. 8].
3In 1614, John Napier (1550–1617) published his Mirifici logarithmorum canonis
de-scriptio, the description of his table of logarithms [53, 69]. It is through this work that
Briggs was early exposed to the theory of logarithms. After Napier’s publication, Briggs
went to visit him in Scotland in the summers of 1615 and 1616 and they agreed on the
need to reformulate the logarithms, a task that Briggs took over.
Briggs published his first table of decimal logarithms in 1617 [8, 62]. It was a small
booklet of 16 pages, of which the first page was an introduction, and the remaining 15
pages were tables. Briggs’ table gave the decimal logarithms of the integers 1 to 1000 to
14 places.
2
From numbers to trigonometric functions
In 1624, Briggs published his Arithmetica logarithmica [9]. This work was incomplete,
and it is thought that Briggs had completed a large portion of the interval 20001–90000
in the subsequent years. But it is likely that Vlacq’s own table in 1628 postponed Briggs’
project.
According to Hallowes [37, p. 85], in 1628 Briggs must have felt that he could not
complete both the tables of the logarithms of numbers and the logarithms of trigonometric
functions, and he must then have turned exclusively to the trigonometric functions.
So, Briggs worked on tables of logarithms of trigonometric functions, first introduced
by Gunter in 1620 [35]. Briggs’ tables were completed by Gellibrand and published in
1
Briggs was baptized on February 23, 1560 (old style), which is 1561 new style. He died on January
26, 1630 (old style), which is 1631 new style [45].
2
Gresham College was a very fluctuating institution, and the main reason for its claim to scientific
responsability was the work and influence of Briggs. The flourishing period of Gresham College ended
with the death of Henry Gellibrand in 1636 [2, p. 20].
functions. Although Briggs and Vlacq both used a division of 90 degrees, Briggs divided
the degrees centesimally, whereas Vlacq used the usual sexagesimal division.
3
Briggs’ Trigonometria britannica (1633)
The Trigonometria britannica [10] published in 1633 contains the sines to 15 places,
the tangents and secants to 10 places, the logarithms of the sines to 14 places and the
logarithms of the tangents to 11 places, every hundredth of a degree.
4The introduction to the tables consists of two books, one of explanations [10, pp. 1–
60], and one of applications [10, pp. 61–110], the latter written by Henry Gellibrand
5after Briggs’ death.
6The Trigonometria britannica was published in 1633 by Vlacq in
Holland.
An English translation of the second book of the Trigonometria britannica was
pub-lished in 1658 as part of John Newton’s Trigonometria britannica [56]
3.1
The computation of sines
The first book of the Trigonometria britannica [10, pp. 1–60] is mainly concerned with
the construction of sines.
7In chapter [10, pp. 2–3], Briggs first considered Ptolemy’s method of computing sines.
This method is based on Ptolemy’s theorem according to which the chord of a − b can
be obtained from the chords of a and b.
Briggs’ sine is of course a “line sines,”
8that is the length of the opposite side of a
triangle of which the hypothenuse is some given value such as 10
10. It is however possible
to argue between two interpretations. In the first interpretation, there is indeed a radius
of 10
10. In the second interpretation, the radius is considered to be 1, but given with
10 digits, the position of the unit being omitted. It is the second interpretation which
should be favored, for in chapter 1 of the Trigonometria britannica, Briggs writes that
the radius is taken as one part, and that it is divided in a number of smaller parts [10,
p. 1]. Further computations confirm this interpretation.
94
After Briggs, more accurate tables of sines, tangents and secants were computed by Andoyer to 15
places and with a step of 10
′′[26, p. 178–180], [4]. Andoyer computed also the logarithms of sines and
tangents to 14 places and with a step of 10
′′[26, p. 200–201], [3]. Some authors have computed sines to
a larger number of decimals, but not with degrees, or only for very large steps.
5
Gellibrand (1597–1637) had become professor of astronomy at Gresham college after Edmund
Gunter’s death [85]. Besides completing the Trigonometria britannica, he also worked on the
varia-tion of the magnetic declinavaria-tion.
6
Briggs’
introduction
was
translated
in
English
and
annotated
by
Ian
Bruce,
see
http://www.17centurymaths.com.
7
Most of the present discussion is borrowed from Ian Bruce’s translation of the Trigonometria
bri-tannica, and from Bruce’s article [10, 12].
8
Although some authors take the radius of the circle to be 1, one has to wait for Lardner’s trigonometry
(1826) to see the sines defined as ratios [19, p. 526]. This should be clearly distinguished from authors
such as Prony, who, for reasons of convenience, give the sines as “parts of the radius” (ca. 1795), at the
Briggs writes that the radius is 1. In chapter 5, he considers similarly quintuplication.
The fact that the radius is 1 is also confirmed by the way Briggs notes his
exam-ples of triplication. In his first example, he considers the radius 10000000000 and the
chord of 16
◦, 02783462019. The square of that chord is noted 0077476608112 and its
cube 0021565319604. The subtended chord is then tripled 08350386057 and the cube
00215653196 is subtracted, the result being 08134732861. All these numbers are aligned
as follows by Briggs [10, p. 4]:
10000000000
02783462019
0077476608112
0021565319604
. . .
08350386057
00215653196
08134732861
It would be possible to interpret the various values differently, for instance 02783462019
as 2783462019, but for the reason given above, Briggs means 0.2783462019.
In the sequel, we will always consider that the radius of the circle is 1, but that the
sines, tangents and secants are given in units of smaller parts, for this is what Briggs
meant.
3.1.2
Division of arcs
After having considered the triplication and quintuplication, Briggs considered again
the equations, but now in order to divide an arc into 3 (chapter 4 [10, pp. 5–10]), 5
(chapter 6 [10, pp. 12–18]), and 7 (chapter 7 [10, pp. 19–20]) parts. For instance, if p is
the chord of an angle a, we have seen that the chord of 3a is c(3a) = 3p−p
3, and trisecting
an angle amounts to solve a cubic equation [12, p. 461]. For a division by 5, the equation
is c(5a) = 5p − 5p
3+ p
5. Then we have c(7a) = 7p − 14p
3+ 7p
5− p
7. And so on. Even
sections lead to the equations c(2a) =
p4p
2− p
4, c(4a) =
p16p
2− 20p
4+ 8p
6− p
8, and
so on. The general case is considered in chapter 8 [10, pp. 20–28] and the coefficients of
all these equations can be obtained from a table given by Briggs [10, p. 23]. This table
can easily be extended.
Briggs’ work is certainly partly inspired from François Viète’s Ad angulares sectiones
which has such a table [81, p. 295].
10Viète is in particular explicitely quoted on the cover
of the Trigonometria britannica.
It is interesting to observe that Jost Bürgi also obtained another similar table for the
same purpose, certainly independently, and described it in his “Coss,” probably around
1598 [49, pp. 33–35] [57, p. 77].
Like Bürgi before him [49], Briggs develops a method to find some roots of these equations
by iteration. In the case of a cubic x
3− 3x = a which is handled in the chapter 4 of the
Trigonometria britannica, Briggs considers a first approximation b of a root made of one
significative digit. He then writes L = b + c for a new approximation. Replacing x by L
in the equation and ignoring the terms in c
2and c
3, Briggs obtains an approximation of
c ≈
a−b3+3b 3b2
−3
of which he keeps one significative (non zero) digit. The process continues
until the approximation is accurate enough.
This happens to be exactly the so-called Newton-Raphson method, with the constraint
that only one new digit is obtained at a time. The Newton-Raphson method actually
goes back at least to Viète.
11In modern terms, the method goes as follows [79, 94]: if x0
is an approximation of a root of f(x) = 0, then a new approximation x1
is given by
x1
= x0
−
f (x0)
f
′(x0)
.
If we set f(x) = x
3− 3x − a, then f
′(x) = 3x
2− 3, and we obtain Briggs’ algorithm
in the case of trisection.
The sixth chapter of the Trigonometria britannica is devoted to the quinquisection of
arcs and expounds the same method. Briggs considers the equation x
5− 5x
3+ 5x = a,
a first approximation b of a root, and he obtains a new approximation L = b + c with
c ≈
a−b5+5b3
−5b 5b4+15b2+5
.
3.1.4
The fundamental sines
Starting with the chord of 60
◦which is equal to 1 (in a circle of radius 1), Briggs used
tri-section obtaining c(20
◦), 5-fold multiplication obtaining c(100
◦), bisection (c(50
◦), c(25
◦),
c(12
◦30
′), c(6
◦15
′)), triplication (c(18
◦45
′), c(56
◦15
′)), duplication (c(37
◦30
′), c(75
◦)), and
again triplication (c(112
◦30
′)). By 5-fold multiplication, he obtained c(31
◦15
′), then by
duplication c(62
◦30
′) and c(125
◦), and by triplication c(93
◦45
′). By 7-fold multiplication,
he obtained c(43
◦45
′) and by duplication c(87
◦30
′). Still multiplying c(6
◦15
′) by 11, 13,
17 and 19, he obtained c(68
◦45
′), c(81
◦15
′), c(106
◦15
′), and c(118
◦45
′). The halves of
all these chords are the sines of 3
◦ 18
, 6
◦ 28, 9
◦ 38, . . . , 62
◦ 12, which he obtained accurate to
22 places. These values are given in the chapter 13 of the Trigonometria britannica [10,
p. 42].
3.1.5
Division of the quadrant in 144 parts and first quinquisection
In chapter 12 of the Trigonometria britannica [10, pp. 35–41], Briggs describes his method
of quinquisection using differences.
12This is the same method as that expounded in the
Arithmetica logarithmica, and that we have described elsewhere [67]. The method of
11
On Newton-Raphson’s method, and Viète’s influence, see Whiteside [88, pp. 218–222], [89, p. 665].
Newton appears to have never read Briggs’ works [88, p. 164]. Newton owned Vlacq’s Trigonometria
artificialis
, but not Briggs’ Trigonometria britannica [89, p. 193].
12
decimal places. The remaining values of the quadrant could be found easily with
sin x + sin(60
◦− x) = sin(60
◦+ x).
So, eventually, the quadrant was divided into 144 parts [10, pp. 43–44].
If the sines at interval of 1
◦15
′are taken, that is, if the quadrant is divided into 72
parts, and if the intervals are divided again three times using the quinquisection with
differences, we reach an interval of 0
◦.01. The quadrant is then divided into 9000 parts.
If instead it was desired to give the sines every thousandths of a degree, we can start
with the division into 144 parts and do four quinquisections with differences, which gives
a division of the quadrant in 90000 parts.
Briggs gives an example where the quinquisection is applied to divide an interval of
0.625
◦and he uses up to the 7th differences [10, p. 45]. In the second quinquisection, he
uses up to the 6th differences [10, p. 46], in the third quinquisection he uses up to the 5th
differences [10, p. 47], and in the last quinquisection he stops at the 4th differences [10,
p. 48]. When starting with an interval of 1
◦15
′, Briggs has certainly used only lower
differences.
3.2
The computation of tangents and secants
The computation of the tangents and secants is described in chapter 15 of the
Trigono-metria britannica [10, pp. 50–52]. Once the sines have been computed for the 72 or 144
divisions of the quadrant, Briggs computes the tangents and secants of the same angles
in the first half of the quadrant with:
r
tanb(90
◦− x)
=
sinb
x
sinb(90
◦− x)
(TB, Prop. 1, p. 50)
sinb
x
r
=
r
secb(90
◦− x)
(TB, Prop. 2, p. 50)
where r is the radius and the sines, tangents and secants are expressed in parts of the
radius. We have sinb
x = r sin x, secb
x = r sec x and tanb
x = r tan x. These functions
were not used previously, because the previous equations were true even with the modern
functions. Note that the name ‘cosine’ is not used by Briggs. It was first used by Gunter
in 1620 [35].
Briggs also gave the two propositions:
r
sinb
x
=
secb
x
tanb
x
(TB, Prop. 3, p. 50)
tanb
x
r
=
r
tanb
(90
◦− x)
(TB, Prop. 4, p. 50)
the functions are taken with smaller units.
sec x = tan x + tan
90
◦− x
2
(TB, Prop. 5, p. 50)
sec x + tan x = tan
x +
90
◦− x
2
(TB, Prop. 6, p. 51)
sec x − tan x = tan
90
◦
− x
2
(TB, Prop. 7, p. 51)
2 tan x + tan
90
◦− x
2
= tan
x +
90
◦− x
2
(TB, Prop. 8, p. 51)
Like for the sines, Briggs then applied quinquisection width differences to obtain
inter-vals of 0.01
◦. However, although Briggs does not mention it, it is likely that Briggs only
used quinquisection for the first half of the quadrant and filled the second half using the
above formulæ. For instance, the value of tan(89.
◦99) can be obtained from tan(89.
◦98)
and tan(0.
◦01), tan(89.
◦98) can be obtained from tan(89.
◦96) and tan(0.
◦02), tan(89.
◦96)
can be obtained from tan(89.
◦92) and tan(0.
◦04), etc. There is an accumulation of errors,
but each initial tangent must be computed with these errors in view. It is not known to
what accuracy Briggs computed the tangents and secants, but the printed values have 10
decimal places. Briggs may have compared the quinquisection with the use of the above
formulæ to decide which one was most advantageous.
3.3
The logarithms of sines
The chapter 16 of the Trigonometria britannica [10, pp. 52–55] is devoted to the
compu-tation of the logarithms of the sines.
For their computation, briggs takes the total sine to be 10
15, or more exactly 1, with
15 zeros, or 10
15smaller parts. Its logarithm is taken to be 10. In other words, the
total sine is actually considered to be 10
10for the purpose of the logarithms. This was
already Gunter’s convention in 1620 [35] and might be called the convention of “shifted
logarithms.” For Gunter and Briggs, log sinb
x = log(10
10sin x) = 10 + log sin x.
13The
characteristic is here the number of integer digits of 10
10sin x minus one. Briggs writes
that the whole sine has the characteristic 10, but that the characteristic of the remaining
sines until arcsin 0.1 = 5
◦44
′is 9, then it is 8, and so on.
14Now, Briggs first computes the logarithms of the 72 sines of the quadrant, at intervals
of 1
◦15
′[10, p. 55]. The computation of the logarithms is done using the radix method
described in chapter 14 of Briggs’ Arithmetica logarithmica.
Once the logarithms of these 72 sines are known, quinquisection is used to obtain the
logarithms of most of the other sines of the quadrant. The quinquisection will however
13
This is what Briggs writes, in Ian Bruce’s translation: “the number of places in this table is more
than the characteristic, as we would have the sines themselves more accurate, and finally truly five places
are added on to the sines (...).” I assume that Gunter and Briggs chose this correspondence in order to
make sure that the characteristic has only one digit, except for the total sine.
sinb
θ
sinb
90
◦−
2or in modern terms
sin
θ 2sin θ
=
sin 30
◦sin 90
◦−
θ 2and therefore
log sinb
θ
2
= log sinb
30
◦+ log sinb
θ − log sinb
90
◦−
θ
2
Briggs therefore obtains the logarithms (of the sines) of small angles by the logarithms
(of the sines) of larger angles computed beforehand.
3.4
The logarithms of tangents and secants
In the 17th and last chapter of Briggs’ part in the Trigonometria britannica [10, pp. 56–
60], Briggs describes the computation of the logarithms of tangents and secants. This
chapter partly overlaps chapter 15, probably because Briggs did no longer have the time
to organize it.
Briggs starts by giving a number of properties of the secants and tangents (the first
eight properties being only additive are given in modern terms):
tan x + tan(90
◦− x)
2
= sec(x − (90
◦− x))
(TB, Prop. 1, p. 56)
tan x − tan(90
◦− x)
2
= tan(x − (90
◦− x))
(TB, Prop. 2, p. 56)
sec x + tan x = tan
x +
90
◦− x
2
(TB, Prop. 3, p. 56)
sec x − tan x = tan
90
◦
− x
2
(TB, Prop. 4, p. 56)
tan x + tan
90
◦− x
2
= sec x (TB, Prop. 5, p. 56)
2 tan x + tan
90
◦− x
2
= tan
x +
90
◦− x
2
(TB, Prop. 6, p. 56)
tan(90
◦− x) − tan x = 2 tan((90
◦− x) − x)
(TB, Prop. 7, p. 57)
tan(90
◦tanb
x
r
=
r
tanb(90
◦− x)
(TB, Prop. 9, p. 57)
sinb
x
sinb(90
◦− x)
=
r
tanb(90
◦− x)
(TB, Prop. 10, p. 57)
r
sinb
x
=
secb(90
◦− x)
r
(TB, Prop. 11, p. 57)
sinb
x
tanb
x
=
r
secb
x
(TB, Prop. 12, p. 57)
sinb
x
tanb
x
=
tanb(90
◦− x)
secb
(90
◦− x)
(TB, Prop. 13, p. 58)
r
secb
x
=
tanb(90
◦− x)
secb
(90
◦− x)
(TB, Prop. 14, p. 58)
These properties are normally not needed, but one might guess that they could be
useful for checking the tangents and secants obtained from the formulæ of chapter 15.
Once Briggs has the tangents and secants, he computes the logarithms. Briggs
ex-plains that the logarithms can either be obtained by the radix method described in
chapter 14 of the Arithmetica logarithmica, or preferably by his propositions 10 and 11:
log sinb
x − log sin
b(90◦− x) = log r − log tan
b(90◦− x)
log r − log sinb
x = log secb(90
◦− x) − log r
This somewhat abruptly ends Briggs’ explanations. The second part of the
Trigo-nometria britannica written by Gellibrand contains applications and is not described
here.
4
Vlacq’s Trigonometria artificialis (1633)
Adriaan Vlacq published his Trigonometria artificialis [83] the same year as Briggs’
Tri-gonometria britannica, and one might wonder whether Vlacq copied some values from
Briggs’ table, as he did for Briggs’ Arithmetica logarithmica. This was answered by
Glaisher’s careful analysis who has shown that Briggs’ and Vlacq’s tables had in fact
been constructed independently [31, p. 444], [61].
5
Decimal system
5.1
Centesimal division of the degree
Briggs perceived the advantage of a centesimal division of the right angles, and he made
a step in this direction by dividing the degrees not into minutes, but into hundredths [30,
p. 301]. Glaisher considered that if Vlacq had done the same in his Trigonometria
ar-5.2
Decimal division of the circle
In the chapter 14 of the Trigonometria britannica, Briggs actually gives the sines for a
quadrant, every 2
◦ 14
. He also gives the angle assuming the circle is divided in 100 parts.
Thus, 2
◦ 14
corresponds to 0 hundredths, and 625 thousandths of hundredths.
The division of the quadrant in 25 parts was not followed, except by Mendizábal in
1891 [18].
6
Errors in the tables
An examination of Briggs’ tables reveals that the most important computation errors are
those of the sines of small angles, and of their logarithms. The last three digits of the
first ten sines were given as 313 (+0), 309 (−1), 672 (−2), 085 (−3), 232 (−3), 796 (−4),
461 (−4), 910 (−4), 827 (−4), and 894 (−4), and there are up to four units of error in
the last place. These errors seem to decrease when the angle becomes larger. Tangents,
secants, and their logarithms seem to be computed very accurately, with usually no more
than one or two units of error on the last place. The logarithms of sines also seem to be
accurate for larger values of the angles.
The logarithms of sines have large errors for small angles, and the reason may be the
difference in the number of significative digits. There are always 15 significative digits
in the logarithms of sines, whereas the sines start with only 12 significative digits. A
difference of one unit of the last place in the first sine causes a difference of more than
200 units of the last place in the corresponding logarithm. The actual errors can be
guessed from the four last digits of the first ten logarithms of sines: 8610 (+17), 3540
(+17), 6652 (+8), 4065 (+17), 0453 (+5), 9835 (+8), 6535 (+10), 6969 (+17), 3372 (+3),
0141 (+5). The errors still appear smaller than one might have anticipated, first, because
Briggs certainly used more accurate values for the sines (19 places according to the above
description), and second because there is actually a correspondence error. For instance,
Briggs should not have found 10+log sin 0
◦.10 = 7.24187 71471 0141 given that his value of
the sine is 0.0017453283 65894 and that 10 + log 0.0017453283 65894 = 7.24187714710029
and 10 + log 0.0017453283 65895 = 7.24187714710054. Since a similar observation can be
made for all these first values, it is clear that the sines used by Briggs were not those
printed in the tables, at least not for the beginning, because the logarithms would be
even less accurate than they are.
One might be tempted to make the same observations for the first tangents, as there
is even a larger discrepancy in the number of significative digits, but Briggs did not use
the values of the tangents to compute their logarithms. Instead, as shown previously, he
used previously computed values of the logarithms of the sines, and since the logarithms
of the tangents are only given to 10 decimal places, even errors of several 100 units of the
(1596), since his errors were also due to an insufficient accuracy in the fundamental values.
For more information on Briggs’ errors, see in particular Henri Andoyer [3] and
Fletcher et al. [26, p. 794] who list eleven errors, plus an entire page of errors where
the first digit is wrong, apart from those affecting the last digit or two.
7
Structure of the tables and recomputation
The original Trigonometria britannica contains an introduction of 110 pages, followed by
a section of tables with the frontispice Canones sinvvm tangentivm secantivm et
logari-thmorvm pro sinvbvs & tangentibvs, ad Gradus & Graduum Centesimas, & ad Minuta &
Secunda Centesimis respondentia.
The tables were recomputed using the GNU mpfr multiple-precision floating-point
library developed at INRIA [27], and give the exact values. The comparison of our table
and Briggs’ will therefore immediately show where Briggs’ table contains errors, and this
is of course one of the purposes of this reconstruction. Apart from the change in accuracy,
we have tried to be as faithful as possible to the original tables. We have however added
some values in a few cases where Briggs had left blanks or put obviously incorrect values.
The original tables had for instance log sin 0 = log tan 0 = 0 and we have replaced these
two values by Infinita which was used by Briggs in other places. There are also some
other minor changes related to commas.
8
Acknowledgements
are marked so. We have added notes about the contents of the articles in certain cases.
[1] Juan Abellan. Henry Briggs. Gaceta Matemática, 4 (1st series):39–41, 1952.
[This
article contains many incorrect statements.]
[2] Ian R. Adamson. The administration of Gresham College and its fluctuating
fortunes as a scientific institution in the seventeenth century. History of Education,
9(1):13–25, March 1980.
[3] Marie Henri Andoyer. Nouvelles tables trigonométriques fondamentales contenant
les logarithmes des lignes trigonométriques. . . . Paris: Librairie A. Hermann et fils,
1911.
[Reconstruction by D. Roegel in 2010 [65].]
[4] Marie Henri Andoyer. Nouvelles tables trigonométriques fondamentales contenant
les valeurs naturelles des lignes trigonométriques. . . . Paris: Librairie A. Hermann
et fils, 1915–1918.
[3 volumes, reconstruction by D. Roegel in 2010 [66].]
[5] Évelyne Barbin et al., editors. Histoires de logarithmes. Paris: Ellipses, 2006.
[6] Peter Barlow. A new mathematical and philosophical dictionary; etc. London:
Whittingham and Rowland, 1814.
[7] H. S. Bennett. English books and readers, III: 1603–1640. Cambridge: Cambridge
University Press, 1970.
[8] Henry Briggs. Logarithmorum chilias prima. London, 1617.
[The tables were
reconstructed by D. Roegel in 2010. [62]]
[9] Henry Briggs. Arithmetica logarithmica. London: William Jones, 1624.
[The tables
were reconstructed by D. Roegel in 2010. [67]]
[10] Henry Briggs and Henry Gellibrand. Trigonometria Britannica. Gouda: Pieter
Rammazeyn, 1633.
[An English translation of the introduction was made by Ian Bruce and
can be found on the web.]
[11] Ian Bruce. The agony and the ecstasy — the development of logarithms by Henry
Briggs. The Mathematical Gazette, 86(506):216–227, July 2002.
15
Note on the titles of the works:
Original titles come with many idiosyncrasies and features (line
splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore
seemed pointless to capitalize works according to conventions which not only have no relation with the
original work, but also do not restore the title entirely. In the following list of references, most title
words (except in German) will therefore be left uncapitalized. The names of the authors have also been
homogenized and initials expanded, as much as possible.
Gazette, 88(513):457–474, November 2004.
[13] Evert Marie Bruins. On the history of logarithms: Bürgi, Napier, Briggs, De
Decker, Vlacq, Huygens. Janus, 67(4):241–260, 1980.
[14] Florian Cajori. Historical note on the Newton-Raphson method of approximation.
The American Mathematical Monthly, 18(2):29–32, February 1911.
[15] Moritz Cantor. Vorlesungen über Geschichte der Mathematik. Leipzig:
B. G. Teubner, 1900.
[volume 2, pp. 737–739, 743–748 on Briggs]
[16] Lesley B. Cormack. Charting an empire. Chicago: University of Chicago Press,
1997.
[17] Jean-Charles de Borda and Jean-Baptiste Joseph Delambre. Tables
trigonométriques décimales : ou Table des logarithmes des sinus, sécantes et
tangentes, suivant la division du quart de cercle en 100 degrés, du degré en 100
minutes, et de la minute en 100 secondes précédées de la table des logarithmes des
nombres depuis dix mille jusqu’à cent mille, et de plusieurs tables subsidiaires.
Paris: Imprimerie de la République, 1801.
[18] Joaquín de Mendizábal-Tamborrel. Tables des Logarithmes à huit décimales des
nombres de 1 à 125000, et des fonctions goniométriques sinus, tangente, cosinus et
cotangente de centimiligone en centimiligone et de microgone en microgone pour les
25000 premiers microgones, et avec sept décimales pour tous les autres microgones.
Paris: Hermann, 1891.
[A sketch of this table was reconstructed by D. Roegel [68].]
[19] Augustus De Morgan. On the almost total disappearance of the earliest
trigonometrical canon. Philosophical Magazine, Series 3, 26(175):517–526, 1845.
[reprinted from [20] with an addition]
[20] Augustus De Morgan. On the almost total disappearance of the earliest
trigonometrical canon. Monthly Notices of the Royal Astronomical Society,
6(15):221–228, 1845.
[reprinted in [19] with an addition]
[21] Jean-Baptiste Joseph Delambre. On the Hindoo formulæ for computing eclipses,
tables of sines, and various astronomical problems. The Philosophical Magazine,
28(109):18–25, June 1807.
[22] Jean-Baptiste Joseph Delambre. Histoire de l’astronomie moderne. Paris: Veuve
Courcier, 1821.
[two volumes, see volume 1, pp. 544–545 and volume 2, pp. 76–88 on Briggs’
Trigonometria britannica
]
[23] Jean-Marie Farey et Patrick Perrin. Les logarithmes de Briggs (1624). In La
mémoire des nombres, pages 319–341. IREM de Basse Normandie, 1997.
[The same
article was also published separately in 1995 [24].]
Ltd., 1962.
[2nd edition (1st in 1946), 2 volumes]
[27] Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, and Paul
Zimmermann. MPFR: A multiple-precision binary floating-point library with
correct rounding. ACM Transactions on Mathematical Software, 33(2), 2007.
[28] Carl Immanuel Gerhardt. Geschichte der Mathematik in Deutschland, volume 17 of
Geschichte der Wissenschaften in Deutschland. Neuere Zeit. München:
R. Oldenbourg, 1877.
[pp. 114–116 on Briggs]
[29] David Gibb. A course in interpolation and numerical integration for the
mathematical laboratory, volume 2 of Edinburgh Mathematical Tracts. London:
G. Bell & sons, Ltd., 1915.
[30] James Whitbread Lee Glaisher. Notice respecting some new facts in the early
history of logarithmic tables. The London, Edinburgh and Dublin Philosophical
Magazine and Journal of Science, Series 4, 44:291–303, 1872.
[31] James Whitbread Lee Glaisher. On logarithmic tables. Monthly notices of the
Royal Astronomical Society, 33(7):440–458, 1873.
[32] James Whitbread Lee Glaisher. Report of the committee on mathematical tables.
London: Taylor and Francis, 1873.
[Also published as part of the “Report of the forty-third
meeting of the British Association for the advancement of science,” London: John Murray, 1874.]
[33] James Whitbread Lee Glaisher. On early tables of logarithms and the early history
of logarithms. The Quarterly journal of pure and applied mathematics, 48:151–192,
1920.
[34] Herman Heine Goldstine. A history of numerical analysis from the 16th through the
19th century. New York: Springer, 1977.
[35] Edmund Gunter. Canon triangulorum. London: William Jones, 1620.
[Recomputed
in 2010 by D. Roegel [64].]
[36] Jean-Pierre Hairault. Calcul des logarithmes décimaux par Henry Briggs. In
Barbin et al. [5], pages 113–129.
[37] D. M. Hallowes. Henry Briggs, mathematician. Transactions of the Halifax
Antiquarian Society, pages 79–92, 1962.
[38] Albert Hatzfeld. La division décimale du cercle. Revue scientifique, 48:655–659,
1891.
catalogue of mathematical tables. Part I: Logarithmic tables (A. Logarithms of
numbers), volume XIII of Tracts for computers. London: Cambridge University
Press, 1926.
[40] Samuel Herrick, Jr. Natural-value trigonometric tables. Publications of the
Astronomical Society of the Pacific, 50(296):234–237, 1938.
[41] Christopher Hill. Intellectual origins of the English Revolution revisited. Oxford:
Clarendon press, 1997.
[42] Charles Hutton. Mathematical tables: containing common, hyperbolic, and logistic
logarithms, also sines, tangents, secants, and versed-sines, etc. London: G. G. J.,
J. Robinson, and R. Baldwin, 1785.
[43] G. Huxley. Briggs, Henry. In Charles Coulston Gillispie, editor, Dictionary of
Scientific Biography, volume 2, pages 461–463. New York: Charles Scribner’s Sons.
[44] Graham Jagger. The making of logarithm tables. In Martin Campbell-Kelly, Mary
Croarken, Raymond Flood, and Eleanor Robson, editors, The history of
mathematical tables: from Sumer to spreadsheets, pages 48–77. Oxford: Oxford
University Press, 2003.
[45] Graham Jagger. The will of Henry Briggs. BSHM Bulletin: Journal of the British
Society for the History of Mathematics, 21(2):127–131, July 2006.
[46] Wolfgang Kaunzner. Über Henry Briggs, den Schöpfer der Zehnerlogarithmen. In
Rainer Gebhardt, editor, Visier- und Rechenbücher der frühen Neuzeit, volume 19
of Schriften des Adam-Ries-Bundes e.V. Annaberg-Buchholz, pages 179–214.
Annaberg-Buchholz: Adam-Ries-Bund, 2008.
[47] Johannes Kepler, John Napier, and Henry Briggs. Les milles logarithmes ; etc.
Bordeaux: Jean Peyroux, 1993.
[French translation of Kepler’s tables and Neper’s descriptio
by Jean Peyroux.]
[48] Adrien Marie Legendre. Sur une méthode d’interpolation employée par Briggs,
dans la construction de ses grandes tables trigonométriques. In Additions à la
Connaissance des tems, ou des mouvemens célestes, à l’usage des astronomes et
des navigateurs, pour l’an 1817, pages 219–222. Paris: Veuve Courcier, 1815.
[49] Martha List and Volker Bialas. Die Coss von Jost Bürgi in der Redaktion von
Johannes Kepler. Ein Beitrag zur frühen Algebra, volume 5 (Neue Folge) of Nova
Kepleriana. München: Bayerische Akademie der Wissenschaften, 1973.
[50] Andrei Andreivich Markov. Differenzenrechnung. Leipzig: B. G. Teubner, 1896.
[Translated from the Russian.]
1844.
[A summary is given in the Comptes rendus hebdomadaires des séances de l’Académie des
sciences, 19(2), 8 July 1844, pp. 81–85, and the entire article is translated in the Journal of the
Institute of Actuaries and Assurance Magazine, volume 14, 1869, pp. 1–36.]
[52] Erik Meijering. A chronology of interpolation: from ancient astronomy to modern
signal and image processing. Proceedings of the IEEE, 90(3):319–342, March 2002.
[53] John Napier. Mirifici logarithmorum canonis descriptio. Edinburgh: Andrew Hart,
1614.
[54] John Napier. A description of the admirable table of logarithmes. London, 1616.
[English translation of [53] by Edward Wright, reprinted in 1969 by Da Capo Press, New York. A
second edition appeared in 1618.]
[55] Katherine Neal. Mathematics and empire, navigation and exploration: Henry
Briggs and the northwest passage voyages of 1631. Isis, 93(3):435–453, 2002.
[56] John Newton. Trigonometria Britannica, etc. London: R. & W. Leybourn, 1658.
[not seen]
[57] Ludwig Oechslin. Jost Bürgi. Luzern: Verlag Ineichen, 2000.
[58] Penny cyclopædia. Briggs (Henry). In The Penny cyclopædia of the society for the
diffusion of useful knowledge, volume V, pages 422–423. London: Charles Knight
and Co., 1836.
[59] Alfred Israel Pringsheim, Georg Faber, and Jules Molk. Analyse algébrique. In
Encyclopédie des sciences mathématiques pures et appliquées, tome II, volume 2,
fascicule 1, pages 1–93. Paris: Gauthier-Villars, 1911.
[See p. 54 for remarks on Briggs.]
[60] Jean-Charles Rodolphe Radau. Études sur les formules d’interpolation. Bulletin
Astronomique, Série I, 8:273–294, 1891.
[61] Denis Roegel. A reconstruction of Adriaan Vlacq’s tables in the Trigonometria
artificialis (1633). Technical report, LORIA, Nancy, 2010.
[This is a recalculation of
the tables of [83].]
[62] Denis Roegel. A reconstruction of Briggs’s Logarithmorum chilias prima (1617).
Technical report, LORIA, Nancy, 2010.
[This is a recalculation of the tables of [8].]
[63] Denis Roegel. A reconstruction of De Decker-Vlacq’s tables in the Arithmetica
logarithmica (1628). Technical report, LORIA, Nancy, 2010.
[This is a recalculation of
the tables of [82].]
Technical report, LORIA, Nancy, 2010.
[This is a reconstruction of [3].]
[66] Denis Roegel. A reconstruction of Henri Andoyer’s trigonometric tables
(1915–1918). Technical report, LORIA, Nancy, 2010.
[This is a reconstruction of [4].]
[67] Denis Roegel. A reconstruction of the tables of Briggs’ Arithmetica logarithmica
(1624). Technical report, LORIA, Nancy, 2010.
[This is a recalculation of the tables of
[9].]
[68] Denis Roegel. A sketch of Mendizábal y Tamborrel’s table of logarithms (1891).
Technical report, LORIA, Nancy, 2010.
[This is a sketch of Mendizábal’s table [18].]
[69] Denis Roegel. Napier’s ideal construction of the logarithms. Technical report,
LORIA, Nancy, 2010.
[70] Denis Roegel. The great logarithmic and trigonometric tables of the French
Cadastre: a preliminary investigation. Technical report, LORIA, Nancy, 2010.
[71] Demetrius Seliwanoff. Lehrbuch der Differenzenrechnung. Leipzig: B. G. Teubner,
1904.
[72] Thomas Smith. Vitæ quorundam eruditissimorum et illustrium virorum. London:
David Mortier, 1707.
[Contains a 16-pages separately paginated biography of Briggs
“Commentariolus de vita et studiis clarissimi & doctissimi viri, D. Henrici Briggii, olim geometriæ
in academia Oxoniensi professoris saviliani,” of which a translation is given pp. lxvii–lxxvii of
volume 1 of [77].]
[73] Thomas Sonar. The grave of Henry Briggs. The Mathematical Intelligencer,
22(3):58–59, September 2000.
[74] Thomas Sonar. Der fromme Tafelmacher : Die frühen Arbeiten des Henry Briggs.
Berlin: Logos Verlag, 2002.
[75] Thomas Sonar. Die Berechnung der Logarithmentafeln durch Napier und Briggs,
2004.
[76] D. J. Struik. Vlacq, Adriaan. In Charles Coulston Gillispie, editor, Dictionary of
Scientific Biography, volume 14, pages 51–52. New York: Charles Scribner’s Sons.
[77] Alexander John Thompson. Logarithmetica Britannica, being a standard table of
logarithms to twenty decimal places of the numbers 10,000 to 100,000. Cambridge:
University press, 1952.
[2 volumes]
[78] Glen van Brummelen. The mathematics of the heavens and the Earth: the early
history of trigonometry. Princeton: Princeton University Press, 2009.
[79] Johan Verbeke and Ronald Cools. The Newton-Raphson method. International
Journal of Mathematical Education in Science and Technology, 26(2):177–193, 1995.
Leiden: Bonaventure & Abraham Elzevir, 1646.
[reprinted by Georg Olms Verlag,
Hildesheim & N.Y., 1970]
[82] Adriaan Vlacq. Arithmetica logarithmica. Gouda: Pieter Rammazeyn, 1628.
[The
introduction was reprinted in 1976 by Olms and the tables were reconstructed by D. Roegel in
2010. [63]]
[83] Adriaan Vlacq. Trigonometria artificialis. Gouda: Pieter Rammazeyn, 1633.
[The
tables were reconstructed by D. Roegel in 2010. [61]]
[84] Anton von Braunmühl. Vorlesungen über Geschichte der Trigonometrie. Leipzig:
B. G. Teubner, 1900, 1903.
[2 volumes]
[85] John Ward. The lives of the professors of Gresham College. London: John Moore,
1740.
[pp. 81–85 on Gellibrand and pp. 120–129 on Briggs. The part on Briggs was reprinted in
The Monthly Magazine, vol. 28, no. 190, 1st October 1809, pp. 275–281.]
[86] Derek Thomas Whiteside. Henry Briggs: The binomial theorem anticipated. The
Mathematical Gazette, 45(351):9–12, February 1961.
[87] Derek Thomas Whiteside. Patterns of mathematical thought in the later
seventeenth century. Archive for History of Exact Sciences, 1:179–388, 1961.
[88] Derek Thomas Whiteside, editor. The Mathematical Papers of Isaac Newton:
Volume II, 1667–1670. Cambridge: Cambridge University Press, 1968.
[89] Derek Thomas Whiteside, editor. The Mathematical Papers of Isaac Newton:
Volume IV, 1674–1684. Cambridge: Cambridge University Press, 1971.
[90] Thomas Whittaker. Henry Briggs. In Dictionary of National Biography, volume 2,
pages 1234–1235. London: Smith, Elder, & Co., 1908.
[volume 6 (1886), pp. 326–327, in
the first edition]
[91] J. Hill Williams. Briggs’s method of interpolation; being a translation of the 13th
chapter and part of the 12th of the preface to the “Arithmetica Logarithmica”.
Journal of the Institute of Actuaries and Assurance Magazine, 14:73–88, 1869.
[92] Robin Wilson. The oldest mathematical chair in Britain. EMS Newsletter,
64:26–29, June 2007.
[93] Edward Wright. Certaine errors in nauigation, detected and corrected. London:
Felix Kingston, 1610.
[contains several tables computed by Briggs]
S I N V V M
T A N G E N T I V M
S E C A N T I V M
ET
LOGARITHMORVM
pro SINVBVS & TANGENTIBVS,
ad Gradus & Graduum Centesimas,
.
grad.
Centesimæ Sinus. Tangentes. Secantes. Logarithmi Sinu¯u. Log. Tangent. M: S
, Infinita Infinita : ,, , Infinita Infinita ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, , ,, , , ,,, ,, : ,, , ,, ,
,,, Infinita Infinita ,,, Infinita : , Infinita Infinita , Infinita
,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , , ,, , , ,,, ,, : , , , , ,
