A reconstruction of the tables of Briggs and Gellibrand’s ... · the tangents and secants to 10...

HAL Id: inria-00543943https://hal.inria.fr/inria-00543943

Submitted on 6 Dec 2010

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

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

[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

[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

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

[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

[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

[95] Mary Claudia Zeller. The development of trigonometry from Regiomontanus toPitiscus. PhD thesis, University of Michigan, 1944. [published in 1946]

20

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

Briggs’ 1633 table (reconstruction, D. Roegel, 2010)

. grad.Cente

simæSinus. Tangentes. Secantes. Logarithmi Sinuū. Log. Tangent. M: S

, Infinita Infinita :

,, , Infinita Infinita

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

22


Cente


,,, Infinita Infinita ,,, Infinita : , Infinita Infinita , Infinita

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

grad. .

23


. grad.Cente


,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

24


Cente


,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

grad. .

25


. grad.Cente


,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, ,

26


Cente


,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

grad. .

27


. grad.Cente


,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

28


Cente


,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

grad. .

29


. grad.Cente


,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

30


Cente


,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

grad. .

31


. grad.Cente


,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, ,

32


Cente


,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

grad. .

33


. grad.Cente


,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

34


Cente


,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

grad. .

35


. grad.Cente


,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

36


Cente


,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

grad. .

37


. grad.Cente


,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, , ,, , , ,,, ,, :

,, , ,, ,

38


Cente


,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

grad. .

39


. grad.Cente


,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , ,, ,

,, , , ,,, ,, : ,, , ,, ,

,, , , ,,, ,, :,, , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

40


Cente


,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, : , , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

grad. .

41


. grad.Cente


,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

42


Cente


,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

grad. .

43


. grad.Cente


,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, ,

44


Cente


,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

grad. .

45


. grad.Cente


,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

46


Cente


,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

grad. .

47


. grad.Cente


,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

,, , , ,,, ,, :,, , , ,, ,

,, , , ,,, ,, : ,, , , ,, ,

,, , , ,,, ,, :

,, , , ,, ,

48


Cente


,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

,, , , ,,, ,, :,, , , , ,

grad. .

49


. grad.Cente


,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, , ,, , , ,,, ,, :

,, , , ,, ,

50


Cente


,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, :,, , , , ,

,, , , ,,, ,, : ,, , , , ,

A reconstruction of the tables of Briggs and Gellibrand’s ... · the tangents and secants to 10...

Documents

Transcript of A reconstruction of the tables of Briggs and Gellibrand’s ... · the tangents and secants to 10...