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A reconstruction of the tables of Briggs and
Gellibrand’sTrigonometria 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�
https://hal.inria.fr/inria-00543943https://hal.archives-ouvertes.fr
-
A reconstruction
of the tables of Briggs and Gellibrand’s
Trigonometria Britannica (1633)
Denis Roegel
6 December 2010
This document is part of the LOCOMAT
project:http://www.loria.fr/~roegel/locomat.html
-
I ever rest a lover of all them that love the Mathematickes
Henry Briggs, preface to [54]
1 Briggs’ first tables (1617)
Henry Briggs (1561–1631)1 is the author of the first table of
decimal logarithms, publishedin 1617, of the first extensive table
of decimal logarithms of numbers, and of one of thefirst two
extensive tables of decimal logarithms of trigonometric
functions.
After having been educated in Cambridge, Briggs became in 1596
the first professor ofgeometry at Gresham College, London [85, p.
120], [74, p. 20], [92]. Gresham College wasEngland’s scientific
centre for navigation, geometry, astronomy and surveying.2
Briggsstayed there until 1620, at which time he went to Oxford,
having been appointed the firstSavilian Professor of Geometry in
1619 [74, p. 24].
While at Gresham College, Briggs became friends with Edward
Wright. He also seemsto have spent time doing research in astronomy
and navigation [74, pp. 29–30]. Briggs hadin 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].3
In 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 thatBriggs was early
exposed to the theory of logarithms. After Napier’s publication,
Briggswent to visit him in Scotland in the summers of 1615 and 1616
and they agreed on theneed 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 smallbooklet of 16 pages, of which the first page
was an introduction, and the remaining 15pages were tables. Briggs’
table gave the decimal logarithms of the integers 1 to 1000 to14
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 20001–90000in 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 notcomplete both the tables of the logarithms of
numbers and the logarithms of trigonometricfunctions, and he must
then have turned exclusively to the trigonometric functions.
So, Briggs worked on tables of logarithms of trigonometric
functions, first introducedby Gunter in 1620 [35]. Briggs’ tables
were completed by Gellibrand and published in
1Briggs was baptized on February 23, 1560 (old style), which is
1561 new style. He died on January26, 1630 (old style), which is
1631 new style [45].
2Gresham College was a very fluctuating institution, and the
main reason for its claim to scientificresponsability was the work
and influence of Briggs. The flourishing period of Gresham College
endedwith the death of Henry Gellibrand in 1636 [2, p. 20].
3For more biographical information on Briggs, consult Smith
[72], Ward [85], Sonar [74] and Kaunz-ner [46]. Sonar gives an
overview of Briggs’ works prior to his tables of logarithms
[74].
3
-
1633. That same year, Vlacq published his own tables of
logarithms of trigonometricfunctions. Although Briggs and Vlacq
both used a division of 90 degrees, Briggs dividedthe 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 thelogarithms of the
tangents to 11 places, every hundredth of a degree.4
The 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 Gellibrand5
after Briggs’ death.6 The Trigonometria britannica was published
in 1633 by Vlacq inHolland.
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 withthe construction of sines.7
In 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 canbe obtained from the
chords of a and b.
Briggs’ sine is of course a “line sines,”8 that is the length of
the opposite side of atriangle of which the hypothenuse is some
given value such as 1010. It is however possibleto argue between
two interpretations. In the first interpretation, there is indeed a
radiusof 1010. In the second interpretation, the radius is
considered to be 1, but given with10 digits, the position of the
unit being omitted. It is the second interpretation whichshould be
favored, for in chapter 1 of the Trigonometria britannica, Briggs
writes thatthe 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.9
4After Briggs, more accurate tables of sines, tangents and
secants were computed by Andoyer to 15places and with a step of
10′′ [26, p. 178–180], [4]. Andoyer computed also the logarithms of
sines andtangents to 14 places and with a step of 10′′ [26, p.
200–201], [3]. Some authors have computed sines toa larger number
of decimals, but not with degrees, or only for very large
steps.
5Gellibrand (1597–1637) had become professor of astronomy at
Gresham college after EdmundGunter’s death [85]. Besides completing
the Trigonometria britannica, he also worked on the varia-tion of
the magnetic declination.
6Briggs’ introduction was translated in English and annotated by
Ian Bruce, seehttp://www.17centurymaths.com.
7Most of the present discussion is borrowed from Ian Bruce’s
translation of the Trigonometria bri-tannica, and from Bruce’s
article [10, 12].
8Although 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 authorssuch as Prony, who, for reasons of
convenience, give the sines as “parts of the radius” (ca. 1795), at
thesame time suggesting that there are different definitions
[70].
9Nevertheless, several authors claim that Briggs’ radius was not
1. According to Gerhardt, for in-stance, the radius was 1015 for
the sines and 1010 for the tangents and secants [28, pp.
115–116].
4
-
3.1.1 Multiplication of arcs
In the third chapter of the Trigonometria britannica [10, pp.
3–5], Briggs considers thetriplication of an arc and shows that
c(3a) = 3p−p3, where p = c(a) is the chord of arc aand c(3a) is the
chord of 3a. This formula is only true if the radius of the circle
is 1, andBriggs 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 thechord of 16◦,
02783462019. The square of that chord is noted 0077476608112 and
itscube 0021565319604. The subtended chord is then tripled
08350386057 and the cube00215653196 is subtracted, the result being
08134732861. All these numbers are alignedas follows by Briggs [10,
p. 4]:
100000000000278346201900774766081120021565319604
. . .083503860570021565319608134732861
It would be possible to interpret the various values
differently, for instance 02783462019as 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 thesines, tangents and secants are given in
units of smaller parts, for this is what Briggsmeant.
3.1.2 Division of arcs
After having considered the triplication and quintuplication,
Briggs considered againthe 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 isthe chord of an angle a, we have seen that the chord of 3a is
c(3a) = 3p−p3, and trisectingan angle amounts to solve a cubic
equation [12, p. 461]. For a division by 5, the equationis c(5a) =
5p− 5p3 + p5. Then we have c(7a) = 7p− 14p3 + 7p5 − p7. And so on.
Evensections lead to the equations c(2a) =
√
4p2 − p4, c(4a) =√
16p2 − 20p4 + 8p6 − p8, andso on. The general case is considered
in chapter 8 [10, pp. 20–28] and the coefficients ofall these
equations can be obtained from a table given by Briggs [10, p. 23].
This tablecan easily be extended.
Briggs’ work is certainly partly inspired from François Viète’s
Ad angulares sectioneswhich has such a table [81, p. 295].10 Viète
is in particular explicitely quoted on the coverof the
Trigonometria britannica.
It is interesting to observe that Jost Bürgi also obtained
another similar table for thesame purpose, certainly independently,
and described it in his “Coss,” probably around1598 [49, pp. 33–35]
[57, p. 77].
10An English translation by Ian Bruce is available on
http://www.17centurymaths.com.
5
-
3.1.3 Solving the equations
Like Bürgi before him [49], Briggs develops a method to find
some roots of these equationsby iteration. In the case of a cubic
x3 − 3x = a which is handled in the chapter 4 of theTrigonometria
britannica, Briggs considers a first approximation b of a root made
of onesignificative digit. He then writes L = b+ c for a new
approximation. Replacing x by Lin the equation and ignoring the
terms in c2 and c3, Briggs obtains an approximation ofc ≈ a−b
3+3b3b2−3
of which he keeps one significative (non zero) digit. The
process continuesuntil the approximation is accurate enough.
This happens to be exactly the so-called Newton-Raphson method,
with the constraintthat only one new digit is obtained at a time.
The Newton-Raphson method actuallygoes back at least to Viète.11 In
modern terms, the method goes as follows [79, 94]: if x0is 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) = x3 − 3x− a, then f ′(x) = 3x2 − 3, and we
obtain Briggs’ algorithmin the case of trisection.
The sixth chapter of the Trigonometria britannica is devoted to
the quinquisection ofarcs and expounds the same method. Briggs
considers the equation x5 − 5x3 + 5x = a,a first approximation b of
a root, and he obtains a new approximation L = b + c withc ≈
a−b
5+5b3−5b5b4+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◦)), andagain triplication
(c(112◦30′)). By 5-fold multiplication, he obtained c(31◦15′), then
byduplication 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 ofall these chords are the sines of 3◦ 1
8, 6◦ 2
8, 9◦ 3
8, . . . , 62◦ 1
2, 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 methodof quinquisection using differences.12
This is the same method as that expounded in theArithmetica
logarithmica, and that we have described elsewhere [67]. The method
of
11On 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
Trigonometriaartificialis, but not Briggs’ Trigonometria britannica
[89, p. 193].
12In this chapter, Briggs observes the proportionality between
the second differences and the sines andthis relationship can be
used to construct the differences. About this relationship, see
Delambre [17,p. 47–48], [21], as well as our study of the Tables du
cadastre [70].
6
-
quinquisection is equivalent to Newton’s forward difference
formula, but Briggs didn’tknow Newton’s formula [67].
So, in a first stage (chapter 13 of the Trigonometria britannica
[10, p. 42]), Briggsused this method to compute the sines from 0◦
to 62◦ 1
2, every 0.625◦ = 6
◦15′
5and to 19
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 withdifferences, 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
startwith the division into 144 parts and do four quinquisections
with differences, which givesa division of the quadrant in 90000
parts.
Briggs gives an example where the quinquisection is applied to
divide an interval of0.625◦ and he uses up to the 7th differences
[10, p. 45]. In the second quinquisection, heuses up to the 6th
differences [10, p. 46], in the third quinquisection he uses up to
the 5thdifferences [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
lowerdifferences.
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 144divisions of the
quadrant, Briggs computes the tangents and secants of the same
anglesin 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 theradius. We have sinb x = r sin x, secb x =
r sec x and tanb x = r tan x. These functionswere not used
previously, because the previous equations were true even with the
modernfunctions. Note that the name ‘cosine’ is not used by Briggs.
It was first used by Gunterin 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)
For the remaining part of the quadrant, Briggs used the
following relations with whichthe tangents and secants of the upper
quadrant can be computed from those of the lower
7
-
quadrant. These relations are true whether for the modern
functions, or if the values ofthe 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 onlyused
quinquisection for the first half of the quadrant and filled the
second half using theabove 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 towhat accuracy
Briggs computed the tangents and secants, but the printed values
have 10decimal places. Briggs may have compared the quinquisection
with the use of the aboveformulæ 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 1015,
or more exactly 1, with15 zeros, or 1015 smaller parts. Its
logarithm is taken to be 10. In other words, thetotal sine is
actually considered to be 1010 for the purpose of the logarithms.
This wasalready Gunter’s convention in 1620 [35] and might be
called the convention of “shiftedlogarithms.” For Gunter and
Briggs, log sinb x = log(1010 sin x) = 10 + log sin x.13
Thecharacteristic is here the number of integer digits of 1010 sin
x minus one. Briggs writesthat the whole sine has the
characteristic 10, but that the characteristic of the
remainingsines until arcsin 0.1 = 5◦44′ is 9, then it is 8, and so
on.14
Now, Briggs first computes the logarithms of the 72 sines of the
quadrant, at intervalsof 1◦15′ [10, p. 55]. The computation of the
logarithms is done using the radix methoddescribed in chapter 14 of
Briggs’ Arithmetica logarithmica.
Once the logarithms of these 72 sines are known, quinquisection
is used to obtain thelogarithms of most of the other sines of the
quadrant. The quinquisection will however
13This is what Briggs writes, in Ian Bruce’s translation: “the
number of places in this table is morethan the characteristic, as
we would have the sines themselves more accurate, and finally truly
five placesare added on to the sines (...).” I assume that Gunter
and Briggs chose this correspondence in order tomake sure that the
characteristic has only one digit, except for the total sine.
14In more recent tables, the logarithms of sines were sometimes
rendered positive by adding 10. Thismay take its origin in Gunter
and Briggs’ convention, but it is still slightly different, and
Gunter andBriggs’ values are only similar because they chose the
total sine to mean 1010.
8
-
not work for the first logarithms, because the differences have
a too large variation.Briggs therefore uses the following relation
(which he obtains geometrically):
sinb(
θ2
)
sinb θ=
r/2
sinb(
90◦ − θ2
)
or in modern termssin
(
θ2
)
sin θ=
sin 30◦
sin(
90◦ − θ2
)
and 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. Thischapter
partly overlaps chapter 15, probably because Briggs did no longer
have the timeto organize it.
Briggs starts by giving a number of properties of the secants
and tangents (the firsteight 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
)
= secx (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◦ − x)− 2 tan((90◦ − x)− x) = tan x (TB, Prop. 8, p.
57)
9
-
The following equations involve mean proportionals:
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 beuseful 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 inchapter 14 of the
Arithmetica logarithmica, or preferably by his propositions 10 and
11:
log sinb x− log sinb(90◦− x) = log r − log tanb(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 describedhere.
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 fromBriggs’ table, as he
did for Briggs’ Arithmetica logarithmica. This was answered
byGlaisher’s careful analysis who has shown that Briggs’ and
Vlacq’s tables had in factbeen 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 madea 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-tificialis (1633) [83], the switch to a centesimal division
might have been easier [30,pp. 301–302]. This may well be true, as
Vlacq’s book was much more widespread than
10
-
Briggs’, and became the basis of other tables. The fact that the
quadrant was still 90◦ inBriggs’ system is actually only a minor
point, as the value 90 only plays a marginal rolein the
computations. But converting degrees, minutes and seconds in
fractions of degreesis always cumbersome, and Briggs’ change would
have alleviated these difficulties.
5.2 Decimal division of the circle
In the chapter 14 of the Trigonometria britannica, Briggs
actually gives the sines for aquadrant, every 2◦ 1
4. 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 arethose of the sines of small angles, and of
their logarithms. The last three digits of thefirst 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 inthe 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
morethan one or two units of error on the last place. The
logarithms of sines also seem to beaccurate for larger values of
the angles.
The logarithms of sines have large errors for small angles, and
the reason may be thedifference in the number of significative
digits. There are always 15 significative digitsin the logarithms
of sines, whereas the sines start with only 12 significative
digits. Adifference of one unit of the last place in the first sine
causes a difference of more than200 units of the last place in the
corresponding logarithm. The actual errors can beguessed 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, becauseBriggs certainly
used more accurate values for the sines (19 places according to the
abovedescription), 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 ofthe
sine is 0.0017453283 65894 and that 10+ log 0.0017453283 65894 =
7.24187714710029and 10+ log 0.0017453283 65895 = 7.24187714710054.
Since a similar observation can bemade for all these first values,
it is clear that the sines used by Briggs were not thoseprinted in
the tables, at least not for the beginning, because the logarithms
would beeven less accurate than they are.
One might be tempted to make the same observations for the first
tangents, as thereis even a larger discrepancy in the number of
significative digits, but Briggs did not usethe values of the
tangents to compute their logarithms. Instead, as shown previously,
heused previously computed values of the logarithms of the sines,
and since the logarithmsof the tangents are only given to 10
decimal places, even errors of several 100 units of thelast place
in the logarithms of sines have barely noticeable consequences in
the logarithmsof tangents.
11
-
It is of course interesting to correlate this with Rheticus’
errors in the Opus palatinum(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] andFletcher et al. [26, p. 794] who list eleven errors,
plus an entire page of errors wherethe 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 bya 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-pointlibrary developed at INRIA [27], and give the exact
values. The comparison of our tableand Briggs’ will therefore
immediately show where Briggs’ table contains errors, and thisis 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 addedsome 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 thesetwo values by Infinita which was used
by Briggs in other places. There are also someother minor changes
related to commas.
8 Acknowledgements
It is a pleasure to thank Ian Bruce for his fruitful
interaction.
12
-
References
The following list covers the most important references15
related to Briggs’ tables. Notall items of this list are mentioned
in the text, and the sources which have not been seenare 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. [Thisarticle contains many incorrect
statements.]
[2] Ian R. Adamson. The administration of Gresham College and
its fluctuatingfortunes 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 contenantles 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 contenantles valeurs naturelles des lignes
trigonométriques. . . . Paris: Librairie A. Hermannet 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: CambridgeUniversity Press, 1970.
[8] Henry Briggs. Logarithmorum chilias prima. London, 1617.
[The tables werereconstructed by D. Roegel in 2010. [62]]
[9] Henry Briggs. Arithmetica logarithmica. London: William
Jones, 1624. [The tableswere reconstructed by D. Roegel in 2010.
[67]]
[10] Henry Briggs and Henry Gellibrand. Trigonometria
Britannica. Gouda: PieterRammazeyn, 1633. [An English translation
of the introduction was made by Ian Bruce andcan be found on the
web.]
[11] Ian Bruce. The agony and the ecstasy — the development of
logarithms by HenryBriggs. The Mathematical Gazette,
86(506):216–227, July 2002.
15Note 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 thereforeseemed pointless to
capitalize works according to conventions which not only have no
relation with theoriginal work, but also do not restore the title
entirely. In the following list of references, most titlewords
(except in German) will therefore be left uncapitalized. The names
of the authors have also beenhomogenized and initials expanded, as
much as possible.
The reader should keep in mind that this list is not meant as a
facsimile of the original works. Theoriginal style information
could no doubt have been added as a note, but we have not done it
here.
13
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[12] Ian Bruce. Henry Briggs: The Trigonometria Britannica. The
MathematicalGazette, 88(513):457–474, November 2004.
[13] Evert Marie Bruins. On the history of logarithms: Bürgi,
Napier, Briggs, DeDecker, 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.
Tablestrigonométriques décimales : ou Table des logarithmes des
sinus, sécantes ettangentes, suivant la division du quart de cercle
en 100 degrés, du degré en 100minutes, et de la minute en 100
secondes précédées de la table des logarithmes desnombres 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 desnombres de 1 à 125000, et des fonctions
goniométriques sinus, tangente, cosinus etcotangente de
centimiligone en centimiligone et de microgone en microgone pour
les25000 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 earliesttrigonometrical 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 earliesttrigonometrical 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: VeuveCourcier, 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 Lamémoire des nombres, pages 319–341. IREM de
Basse Normandie, 1997. [The samearticle was also published
separately in 1995 [24].]
[24] Jean-Marie Farey and Patrick Perrin. Les logarithmes de
Briggs. Repères-IREM,21:61–77, October 1995. [This is a separate
publication of [23].]
14
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[25] Mordechai Feingold. The mathematicians’ apprenticeship:
science, universities andsociety in England, 1560–1640. Cambridge:
Cambridge University Press, 1984.
[26] Alan Fletcher, Jeffery Charles Percy Miller, Louis
Rosenhead, and Leslie JohnComrie. An index of mathematical tables.
Oxford: Blackwell scientific publicationsLtd., 1962. [2nd edition
(1st in 1946), 2 volumes]
[27] Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick
Pélissier, and PaulZimmermann. MPFR: A multiple-precision binary
floating-point library withcorrect rounding. ACM Transactions on
Mathematical Software, 33(2), 2007.
[28] Carl Immanuel Gerhardt. Geschichte der Mathematik in
Deutschland, volume 17 ofGeschichte 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 themathematical 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 earlyhistory of logarithmic tables. The London,
Edinburgh and Dublin PhilosophicalMagazine and Journal of Science,
Series 4, 44:291–303, 1872.
[31] James Whitbread Lee Glaisher. On logarithmic tables.
Monthly notices of theRoyal 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-thirdmeeting 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 historyof 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 the19th century. New York: Springer,
1977.
[35] Edmund Gunter. Canon triangulorum. London: William Jones,
1620. [Recomputedin 2010 by D. Roegel [64].]
[36] Jean-Pierre Hairault. Calcul des logarithmes décimaux par
Henry Briggs. InBarbin et al. [5], pages 113–129.
[37] D. M. Hallowes. Henry Briggs, mathematician. Transactions
of the HalifaxAntiquarian Society, pages 79–92, 1962.
[38] Albert Hatzfeld. La division décimale du cercle. Revue
scientifique, 48:655–659,1891.
15
-
[39] James Henderson. Bibliotheca tabularum mathematicarum,
being a descriptivecatalogue of mathematical tables. Part I:
Logarithmic tables (A. Logarithms ofnumbers), volume XIII of Tracts
for computers. London: Cambridge UniversityPress, 1926.
[40] Samuel Herrick, Jr. Natural-value trigonometric tables.
Publications of theAstronomical 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 logisticlogarithms, 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 ofScientific Biography, volume 2, pages 461–463.
New York: Charles Scribner’s Sons.
[44] Graham Jagger. The making of logarithm tables. In Martin
Campbell-Kelly, MaryCroarken, Raymond Flood, and Eleanor Robson,
editors, The history ofmathematical tables: from Sumer to
spreadsheets, pages 48–77. Oxford: OxfordUniversity Press,
2003.
[45] Graham Jagger. The will of Henry Briggs. BSHM Bulletin:
Journal of the BritishSociety for the History of Mathematics,
21(2):127–131, July 2006.
[46] Wolfgang Kaunzner. Über Henry Briggs, den Schöpfer der
Zehnerlogarithmen. InRainer Gebhardt, editor, Visier- und
Rechenbücher der frühen Neuzeit, volume 19of 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 descriptioby 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 laConnaissance des tems, ou des
mouvemens célestes, à l’usage des astronomes etdes 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 vonJohannes Kepler. Ein Beitrag zur frühen Algebra,
volume 5 (Neue Folge) of NovaKepleriana. München: Bayerische
Akademie der Wissenschaften, 1973.
[50] Andrei Andreivich Markov. Differenzenrechnung. Leipzig: B.
G. Teubner, 1896.[Translated from the Russian.]
[51] Frédéric Maurice. Mémoire sur les interpolations, contenant
surtout, avec uneexposition fort simple de leur théorie, dans ce
qu’elle a de plus utile pour lesapplications, la démonstration
générale et complète de la méthode de quinti-section
16
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de Briggs et de celle de Mouton, quand les indices sont
équidifférents, et duprocédé exposé par Newton, dans ses Principes,
quand les indices sont quelconques.In Additions à la Connaissance
des temps ou des mouvements célestes, à l’usagedes astronomes et
des navigateurs, pour l’an 1847, pages 181–222. Paris:
Bachelier,1844. [A summary is given in the Comptes rendus
hebdomadaires des séances de l’Académie dessciences, 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 modernsignal 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: HenryBriggs 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 thediffusion of useful knowledge, volume V,
pages 422–423. London: Charles Knightand Co., 1836.
[59] Alfred Israel Pringsheim, Georg Faber, and Jules Molk.
Analyse algébrique. InEncyclopédie 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. BulletinAstronomique, Série I, 8:273–294,
1891.
[61] Denis Roegel. A reconstruction of Adriaan Vlacq’s tables in
the Trigonometriaartificialis (1633). Technical report, LORIA,
Nancy, 2010. [This is a recalculation ofthe 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 Arithmeticalogarithmica (1628). Technical report, LORIA,
Nancy, 2010. [This is a recalculation ofthe tables of [82].]
[64] Denis Roegel. A reconstruction of Gunter’s Canon
triangulorum (1620). Technicalreport, LORIA, Nancy, 2010. [This is
a recalculation of the tables of [35].]
17
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[65] Denis Roegel. A reconstruction of Henri Andoyer’s table of
logarithms (1911).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 FrenchCadastre: 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 ofScientific Biography, volume 14,
pages 51–52. New York: Charles Scribner’s Sons.
[77] Alexander John Thompson. Logarithmetica Britannica, being a
standard table oflogarithms 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 earlyhistory of trigonometry. Princeton: Princeton
University Press, 2009.
[79] Johan Verbeke and Ronald Cools. The Newton-Raphson method.
InternationalJournal of Mathematical Education in Science and
Technology, 26(2):177–193, 1995.
18
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[80] Erik Vestergaard. Henry Briggs’ differensmetode. Normat —
Nordisk MatematiskTidsskrift, 45(2):49–61, 1997.
[81] François Viète. Ad angulares sectiones theoremata
καθολικώτερα, demonstrata perAlexandrum Andersonum. In Franciscus
van Schooten, editor, Opera mathematica.Leiden: Bonaventure &
Abraham Elzevir, 1646. [reprinted by Georg Olms Verlag,Hildesheim
& N.Y., 1970]
[82] Adriaan Vlacq. Arithmetica logarithmica. Gouda: Pieter
Rammazeyn, 1628. [Theintroduction 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. [Thetables 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 inThe Monthly Magazine,
vol. 28, no. 190, 1st October 1809, pp. 275–281.]
[86] Derek Thomas Whiteside. Henry Briggs: The binomial theorem
anticipated. TheMathematical Gazette, 45(351):9–12, February
1961.
[87] Derek Thomas Whiteside. Patterns of mathematical thought in
the laterseventeenth 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, inthe first edition]
[91] J. Hill Williams. Briggs’s method of interpolation; being a
translation of the 13thchapter 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]
[94] Tjalling J. Ypma. Historical development of the
Newton-Raphson method. SIAMReview, 37(4):531–551, December
1995.
19
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[95] Mary Claudia Zeller. The development of trigonometry from
Regiomontanus toPitiscus. PhD thesis, University of Michigan, 1944.
[published in 1946]
20
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CANONES
S I N V V MT A N G E N T I V M
S E C A N T I V MET
LOGARITHMORVMpro SINVBVS & TANGENTIBVS,
ad Gradus & Graduum Centesimas,
& ad Minuta & Secunda Centesimis respondentia.
21
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Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
, Infinita Infinita :
,, , Infinita Infinita
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
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22
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,,, Infinita Infinita ,,, Infinita : , Infinita Infinita ,
Infinita
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
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,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
grad. .
23
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
24
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
grad. .
25
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, ,
26
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
grad. .
27
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
28
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
grad. .
29
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
30
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
grad. .
31
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, ,
32
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
grad. .
33
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
34
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
grad. .
35
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
36
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
grad. .
37
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, , ,, , , ,,, ,, :
,, , ,, ,
38
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
grad. .
39
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , ,, ,
,, , , ,,, ,, : ,, , ,, ,
,, , , ,,, ,, :,, , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
40
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, : , , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
grad. .
41
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
42
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
grad. .
43
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, ,
44
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
grad. .
45
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
46
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
grad. .
47
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
,, , , ,,, ,, :,, , , ,, ,
,, , , ,,, ,, : ,, , , ,, ,
,, , , ,,, ,, :
,, , , ,, ,
48
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, :,, , , , ,
,, , , ,,, ,, : ,, , , , ,
,, , , ,,, ,, :,, , , , ,
grad. .
49
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
. grad.Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, , ,, , , ,,, ,, :
,, , , ,, ,
50
-
Briggs’ 1633 table (reconstruction, D. Roegel, 2010)
Cente
simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent.
M: S
,, , , ,,, ,, :,, , , , ,
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