# Navigation on Sea: Topics in the History of Geomathematics

### Abstract

In this essay we review the development of the magnet as a means for navigational purposes. Around 1600, knowledge of the properties and behavior of magnetic needles began to grow in England mainly through the publication of William Gilbert's influential book *De Magnete*. Inspired by the rapid advancement of knowledge on one side and of the English fleet on the other, scientists associated with Gresham College began thinking of using magnetic instruments to measure the degree of latitude without being dependent on a clear sky, a quiet sea, or complicated navigational tables. The construction and actual use of these magnetic instruments, called dip rings, is a tragic episode in the history of seafaring since the latitude does not depend on the magnetic field of the Earth but the construction of a table enabling seafarers to take the degree of latitude from is certainly a highlight in the history of geomathematics.

## 1 General Remarks on the History of Geomathematics

Geomathematics in our times is thought of being a very young science and a modern area in the realms of mathematics. Nothing is farer from the truth. Geomathematics began as man realized that he walked across a sphere-like Earth and that this had to be taken into account in measurements and computations. Hence, Eratosthenes can be seen as an early geomathematician when he tried to determine the circumference of the Earth by measurements of the sun's position from the ground of a well and the length of shadows farther away at midday. Other important topics in the history of geomathematics are the struggles for an understanding of the true shape of the Earth which led to the development of potential theory and much of multidimensional calculus, see Greenberg (1995), the mathematical developments around the research of the Earth's magnetic field, and the history of navigation.

## 2 Introduction

The history of navigation is one of the most exciting stories in the history of mankind and one of the most important topics in the history of Geomathematics. The notion of navigation thereby spans the whole range from the ethnomathematics of polynesian stick charts via the compass to modern mathematical developments in understanding the earth's magnetic field and satellite navigation via GPS. We shall concentrate here on the use of the magnetic needle for navigational purposes and in particular on developments having taken place in early modern England. However, we begin our investigations with a short overview on the history of magnetism following Balmer (1956).

## 3 The History of the Magnet

The earliest sources on the use of magnets for the purpose of navigation stem from China. During the *Han* epoche 202-220 we find the description of carriages equipped with compass-like devices so that the early Chinese imperators were able to navigate on their journeys through their enormous empire. These carriages were called *tschinan-tsche*, meaning "carriages that show noon." The compass-like devices consisted of little human-like figures which swam on water in a bowl, the finger of the stretched arm pointing always straight to the south. We do not know nowadays why the ancient Chinese preferred the southward direction instead of a northbound one. In a book on historical memoirs written by *Sse-ma-tsien* (or *Schumatsian*) we find a report dating back to the first half of the second century on a present that imperator *Tsching-wang* gave 1100 before Christ to the ambassadors of the cities of Tonking and Cochinchina. The ambassadors received five "magnetic carriages" in order to guide them safely back to their cities even through sand storms in the desert. Since the ancient Chinese knew about the attracting forces of a magnet they called them 'loving stones.' In a work on natural sciences written by Tschin-tsang-ki from the year 727 we read:

The loving stone attracts the iron like an affectionate mother attracts their children around her; this is the reason where the name comes from.

It was also very early known that a magnet could transfer its attracting properties to iron when it was swept over the piece of iron. In a dictionary of the year 121 the magnet is called a stone "with which a needle can be given a direction" and hence it is not surprising that a magnetic needle mounted on a piece of cork and swimming in a bowl of water belonged to the standard equipment of larger chinese ship as early as the fourth century. Such simple devices were called "bussola" by the Italians and are still known under that name.

A magnetic needle does not point precisely to the geographic poles but to the magnetic ones. The locus of the magnetic poles moves in time so that a deviation has always to be taken into account. Around the year 1115 the problem of deviation was known in China.

The word "magnet" comes from the greek word magnes describing a sebacious rock which, according to the greek philosopher and natural scientist Theophrastos (ca. 371-287 bc), was a forgeable and silver white rock. The philosopher Plato (428/27-348/47 bc) called magnetic rock the "stone of Heracles" and the poet Lucretius (ca. 99-55 bc) used the word "magnes" in the sense of attracting stone. He attributed the name to a place named "Magnesia" where this rock could be found. Other classical greek anecdotes call a shepherd named "Magnes" to account for the name. It is said that he wore shoes with iron nails and while accompanying his sheep suddenly could no longer move because he stood on magnetic rock.

Homer wrote about the force of the magnet as early as 800 bc. It seems typical for the ancient greek culture that one sought for an explanation of this force fairly early on. Plato thought of this force as being simply "divine." Philosopher Epicurus (341-270) had the hypothesis that magnets radiate tiny particles - atoms. Eventually Lucretius exploited this hypothesis and explained the attracting force of a magnet by the property of the radiated atoms to clear the space between the magnet and the iron. Into the free space then iron atoms could penetrate and since iron atoms try hard to stay together (says Lucretius) the iron piece would follow them. The pressure of the air also played some minor role in this theory. Lucretius knew that he had to answer the question why iron would follow but other materials would not. He simply declared that gold would be too heavy and timber would show too large porosities so that the atoms of the magnet would simply go through.

*perpetuum mobile*according to >

*Fig*.

*1*. A magnet is mounted at the tip of a hand which is periodically moving (says Peregrinus) beacuse of iron nails on the circumference.

Fig. 1 The magnetic perpetuum mobile of Peregrinus |

Fig. 2 Elizabeth I (Armada portrait) |

Fig. 3 De Magnete |

Fig. 4 Norman's discovery of the magnetic dip |

Fig. 5 Dip rings: (a)The dip ring after Gilbert in De Magnete (b) A dip ring used in the 17th century |

Fig. 6 Measuring the dip on the terrella in De Magnete |

Peregrinus' work was so influential in Western Europe that even 300 years after his death he is still accepted as the authority on the magnet.

## 4 Early Modern England

In the sixteenth century Spain and Portugal developed into the leading sea powers. Currents of gold, spices, and gemstones regorge from the South Americans into the home countries. England had missed connection. When Henry VIII died in 1547, only a handful of decaying ships were lying in the English sea harbors. His successor, his son Edward VI, could only rule for 6 years before he died young. Henry's daughter Mary, a devout catholic, tried hard to re-catholize the country her father had steered into protestantism and married Philipp II, King of Spain. Mary was fairly brutal in the means of the re-catholization and many of the protestant intelligentsia left the country in fear of their lives. "Bloody Mary" died in 1558 at the age of 47 and the way opened to her stepsister Elizabeth. Within one generation itself Elizabeth I transformed rural England to the leading sea power on Earth. She was advised very well by Sir Walter Raleigh who clearly saw the future of England on the seas. New ships were built for the navy and in 1588 the small English fleet was able to drown the famous Spanish armada - by chance and with good luck; but this incident served to boost not only the feeling of self-worth of a whole nation but also the realization of the need of a navy and the need of efficient navigational tools.

English mariners realized on longer voyages that the magnetic needle inside a compass lost its magnetic power. If that was detected the needle had to be magnetized afresh - it had to be "loaded" afresh. However, this is not the reason why the magnet is called *loadstone* in the English language but, only a mistranslation. The correct word should be *lodstone* - "leading stone" - but that word was actually never used (Pumfrey, 2002).

## 5 The Gresham Circle

In 1592, Henry Briggs (1561-1639), chief mathematician in his country, was elected Examiner and Lecturer in Mathematics at St John's College, Cambridge, which nowadays corresponds to a professorship. In the same year he was elected *Reader of the Physics Lecture founded by Dr Linacre* in London. One hundred years before the birth of Briggs, Thomas Linacre was horrified by the pseudomedical treatment of sick people by hair dressers and vicars who did not shrink back from chirurgical operations without a trace of medical instruction. He founded the *Royal College of Physicians of London* and Briggs was now asked to deliver lectures with medical contents. The Royal College of Physicians was the first important domain for Briggs to make contact with men outside the spheres of the two great universities and, indeed most important, he met William Gilbert (1544-1603) who was working on the wonders of the magnetical forces and who revolutionized modern science only a few years later.

While England was on its way to become the worlds leading sea force the two old English universities Oxford and Cambridge were in an alarming state of sleepyness (Hill 1997, p. 16ff). Instead of working and teaching on the forefront of modern research in important topics like navigation, geometry, astronomy, the curricula were directly rooted in the ancient greek tradition. Mathematics included reading of the first four or five books of Euclid, medicine was read after Galen and Ptolemy ruled in astronomy. When the founder of the English stock exchange (Royal Exchange) in London, Thomas Gresham, died, he left in his last will money and buildings in order to found a new form of university, the Gresham College, which is still in function. He ordered the employment of seven lecturers to give public lectures in theology, astronomy, geometry, music, law, medicine, and rhetorics mostly in English language. The salary of the Gresham professors was determined to be £50 a year which was an enormous sum as compared to the salary of the Regius professors in Oxford and Cambridge (Hill 1997, p. 34). The only conditions on the candidates for the Gresham professorships were brilliance in their field and an unmarried style of life.

Briggs must have been already well known as a mathematician of the first rank since he was chosen to be the first Gresham professor of Geometry in 1596. Modern mathematics was needed badly in the art of navigation and public lectures on mathematics were in fact already given in 1588 on behalf of the East India Company, the Muscovy Company, and the Virginia Company. Even before 1588 there were attempts by Richard Hakluyt to establish public lectures and none less than Francis Drake had promised £20 (Hill 1997, p. 34), but it needed the national shock of the attack of the Armada in 1588 to make such lectures come true.

During his time in Gresham College Briggs became the center of what we can doubtlessly call the Briggsian circle. Hill writes, (Hill 1997, p. 37):

He [i.e. BRIGGS] was a man of the first importance in the intellectual history of his age, … . Under him Gresham at once became a centre of scientific studies. He introduced there the modern method of teaching long division, and popularized the use of decimals.

The Briggsian circle consisted of true copernicans; men like William Gilbert who wrote *De Magnete*, the able applied mathematician Edward Wright who is famous for his book on the errors in navigation, William Barlow, a fine instrument maker and men of experiments, and the great popularizer of scientific knowledge, Thomas Blundeville. Gilbert and Blundeville were protégés of the Earl of Leicester and we know about connections with the circle of Ralegh in which the brilliant mathematician Thomas Harriot worked. Blundeville held contacts with John Dee who introduced modern continental mathematics and the Mercator maps in England, (Hill 1997, p. 42). Hence, we can think of a scientific sub-net in England in which important work could be done which was impossible to do in the great universities. It was this time in Gresham College in which Briggs and his circle were most productive in the calculation of tables of astronomical and navigational importance. In the center of their activities was Gilbert's "Magnetick Philosophy."

## 6 William Gilberts Dip Theory

The role of William Gilbert in shaping modern natural sciences cannot be overestimated and a recent biography of Gilbert (Pumfrey 2002) emphasizes his importance in England and abroad. Gilbert, a physician and member of the Royal College of Physisicans in London, became interested in navigational matters and the properties of the magnetic needle in particular, by his contacts to seamen and famous navigators of his time alike. As a result of years of experiments, thought, and discussions with his Gresham friends, the book *De Magnete, magneticisque corporibus, et de magno magnete tellure; Physiologia nova, plurimis & argumentis, & experimentis demonstrata* was published in 1600. (I refer to the English translation Gilbert (1958) by P. Fleury Mottelay which is a reprint of the original of 1893. There is a better translation by Sylvanus P. Thompson from 1900 but while the latter is rare the former is still in print.) It contained many magnetic experiments with what Gilbert called his *terrella* - the little Earth - which was a magnetical sphere. In the spirit of the true copernican Gilbert deduced the rotation of the Earth from the assumption of it being a magnetic sphere.

Concerning navigation, Briggs, and the Gresham circle the most interesting chapter in *De Magnete* is Book V: *On the dip of the magnetic needle*. Already in 1581 the instrument maker Robert Norman had discovered the magnetic dip in his attempts to straighten magnetic needles in a fitting on a table. He had observed that an unmagnetized needle could be fitted in a parallel position with regard to the surface of a table but when the same needle was magnetized and fitted again it made an angle with the table. Norman published his results already in 1581 (R. Norman - The New Attractive *London, 1581*. Even before Norman the dip was reported by the german astronomer and instrument maker Georg Hartmann from Nuremberg in a letter to the Duke Albrecht of Prussia from 4th of March, 1544, see Balmer 1995, p. 290-292.) but he was not read. In modern notion the phenomenon of the dip is called *inclination* in contrast to the *declination* or *variation* of the needle. A word of warning is appropriate here: in Gilbert's time many authors used the word *declination* for the *inclicnation*. Anyway, Norman was the first to build a *dip ring* in order to measure the inclination. This ring is nothing else but a vertical compass. Already Norman had discovered that the dip varied with time!

However, Gilbert believed that he had found the secrets of magnetic navigation. He explained the variation of the needle by land masses acting on the compass which fitted nicely with the measurements of seamen but is wrong, as we now know.

Concerning the dip let me give a summary of Gilbert's work in modern terms. Gilbert must have measured the dip on his terrella many, many times before he was led to his

**First hypothesis:** *There is an invertible mapping between the lines of latitude and the lines of constant dip.*

Hence, Gilbert believed to have found a possibility of determining the latitude on Earth from the degree of the dip. Let *β* be the latitude and *α* the dip. He then formulated his

**Second Hypothesis:** *At the equator the needle is parallel to the horizon, i.e. α = 0* ^{∘} *. At the north pole the needle is perpendicular to the surface of the Earth, i.e. α = 90* ^{∘}.

He then draws a conlcusion but in our modern eyes this is nothing but another

**Third hypothesis:** *If β = 45* ^{∘} *then the needle points exactly to the second equatorial point.*

*Fig*.

*7*. Gilbert himself writes

Note that in Fig. 7 the equator is given by the line… points to the equator F as the mean of the two poles.(Gilbert 1958, p. 293)

Fig. 7 Third hypothesis

*A*-

*F*and the poles are

*B*(north) and

*C*so that our implicit (modern) assumption that the north pole is always shown ontop of a figure is not satisfied.

From his three hypothesises Gilbert concludes correctly:

**First Conclusion:**

*The*rotation

*of the needle has to be faster on its way from A to L than from L to B.*Or, in Gilbert's words,

… the movement of this rotation is quick in the first degrees from the equator, from

AtoL, but slower in the subsequent degrees, fromLtoB, that is, with reference to the equatorial pointF, towardC. (Gilbert 1958, p. 293)

… it dips; yet, not in ratio to the number of degrees or the arc of the latitude does the magnetic needle dip so many degrees or over a like arc; but over a very different one, for this movement is in truth not a dipping movement, but really a revolution movement, and it describes an arc of revolution proportioned to the arc of latitude.

This is simply the lengthy description of the following

**Second Conclusion:** *The mapping between latitude and dip cannot be linear.*

*β*= 0

^{○}, 45

^{○}, 90

^{○}a construction of the general mapping was sought. It is exactly here where

*De Magnete*shows strange weaknesses, in fact, we witness a qualitative jump from a geometric construction to a

*dip instrument*. Gilbert's geometrical description can be seen in >

*Fig*.

*8*. I do not intend to comment on this construction because this was done in detail elsewhere, (Sonar 2001), but it is

*not*possible to understand the construction from Gilbert's writings in

*De Magnete*. Even more surprising, while all figures in

*De Magnete*are raw wood cuts in the quality shown before, suddenly there is a fine technical drawing of the resulting construction as shown in >

*Fig*.

*9*. The difference between this drawing and all other figures in

*De Magnete*and the weakness in the description of the construction of the mapping between latitudes and dip angles suggest that at least this part was not written by Gilbert alone but by some of his friends in the Gresham circle. Pumfrey speaks of

*the dark secret of De Magnete*(Pumfrey 2002, p. 173ff), and gives evidence that Edward Wright, whose

*On Certain Errors in Navigation*had appeared a year before

*De Magnete*, had his hands in some parts of Gilberts book. In Parsons (1939, pp. 61-67), we find the following remarks:

Wright, and his circle of friends, which included Dr. W. Gilbert, Thomas Blundeville, William Barlow, Henry Briggs, as well as Hakluyt and Davis, formed the centre of scientific thought at the turn of the century. Between these men there existed an excellent spirit of co-operation, each sharing his own discoveries with the others. In 1600 Wright assisted Gilbert in the compilation ofDe Magnete. He wrote a long preface to the work, in which he proclaimed his belief in the rotation of the earth, a theory which Gilbert was explaining, and also contributed chapter 12 of Book IV, which dealt with the method of finding the amount of the variation of the compass. Gilbert devoted his final chapters to practical problems of navigation, in which he knew many of his friends were interested.

Fig. 8 Gilbert's geometrical construction of the mapping inDe Magnete

Fig. 10 Title page

There is no written evidence that Briggs was involved too but it seems very unlikely that the chief mathematician of the Gresham circle should not have been in charge in so important a development as the dip theory. We shall see later on that the involvement of Briggs is highly likely when we study his contributions to dip theory in books of other authors.

## 7 The Briggsian Tables

If we trust (Ward, (1740), pp. 120-129), the first published table of Henry Briggs is the table which represents Gilbert's mapping between latitude and dip angles in Thomas Blundevilles book

The Theoriques of the seuen Planets, shewing all their diuerse motions, and all other Accidents, called Passions, thereunto belonging.

Whereunto is added by the said Master Blundeuile, a breefe Extract by him made, of Magnus his Theoriques, for the better vnderstanding of the Prutenicall Tables, to calculate thereby the diuerse motions of the seuen Planets.

There is also hereto added, The making, description, and vse, of the two most ingenious and necessarie Instruments for Sea-men, to find out therebye the latitude of any place vpon the Sea or Land, in the darkest night that is, without the helpe of Sunne, Moone, or Starre. First inuented by M. Doctor Gilbert, a most excellent Philosopher, and one of the ordinarie Physicians to her Maiestie: and now here plainely set down in our mother tongue by Master Blundeuile.

London

Printed by Adam Islip.

1602.

*popularizers*of scientific knowledge. He did not write for the expert, but for the layman, i.e., the

*young gentlemen*interested in so diverse questions of science, writing of history, map making, logic, seamenship, or horse riding. We do not know much about his life (Campling 1921-1922), but his role in the Gresham circle is apparent through his writings. In

*The Theoriques*Gilbert's dip theory is explained in detail and a step-by-step description of the construction of the dip instrument is given. I have followed Blundeville's instructions and constructed the dip instrument again elsewhere, see Sonar (2002). See also Sonar (2001). The final result is shown in Blundeville's book as in >

*Fig*.

*11*. In order to understand the geometrical details it is necessary to give a condensed description of the actual construction in >

*Fig*.

*8*which is given in detail in

*The Theoriques*. We start with a circle

*ACDL*representing planet Earth as in >

*Fig*.

*12*. Note that

*A*is an equatorial point while

*C*is a pole. The navigator (and hence the dip instrument) is assumed to be in point

*N*which corresponds to the latitude

*β*= 45

^{○}. In the first step of the construction a horizon line is sought, i.e., the line from the navigator in

*N*to the horizon. Now a circle is drawn around

*A*with radius

*AM*(the Earth's radius). This marks the point

*F*on a line through

*A*parallel to

*CL*. A circle around

*M*through

*F*now gives the arc . The point

*H*is constructed by drawing a circle around

*C*with radius

*AM*. The point of intersection of this circle with the outer circle through

*F*is

*H*.

*If the dip instrument is at A, the navigator's horizon point is F. If it is in C, the navigator's horizon will be in H.*Correspondingly, drawing a circle through

*N*with radius

*AM*gives the point

*S*, hence,

*S is the point at the horizon seen from N.*Hence to every position

*N*of the needle there is a

*quadrant of dip*which is the arc from

*M*to a corresponding point on the outer crircle through

*F*. If

*N*is at

*β*= 45

^{○}latitude as in our example we know from Gilbert's third hypotheses that the needle points to

*D*. The angle between

*S*and the intersection point of the quadrant of dip with the direction of the needle is the dip angle. The remaining missing information is the point to which the needle points for a general latitude

*β*. This is accomplised by

*Quadrants of rotation*which implement Gilbert's idea of the needle

*rotating*on its way from

*A*to

*C*. The construction of these quadrants is shown in >

*Fig*.

*13*. We need a second outer circle which is constructed by drawing a circle around

*A*through

*L*. The intersection point of this circle with the line

*AF*is

*B*and the second outer circle is then the circle through

*B*around

*M*. Drawing a circle around

*C*through

*L*defines the point

*G*on the second outer circle. These arcs, and , are the

*quadrants of rotation*corresponding to the positions

*C*and

*A*, of the needle respectively. Assuming again the dip instrument in

*N*at

*β*= 45

^{○}. Then the corresponding quadrant of rotation is constructed by a circle around

*N*through

*L*and is the arc . This arc is now divided in 90 parts, starting from the second outer circle (0

^{○}) and ending at

*L*(90

^{○}). Obviously, in our example, according to Gilbert, the 45

^{○}mark is exactly at

*D*.

Fig. 11 The dip instrument in The Theoriques |

Fig. 12 The Quadrant of dip |

Fig. 13 The Quadrant of rotation |

*Fig*.

*14*. The needle at

*N*points to the mark 45

^{○}on the arc of rotation and hence intersects the quadrant of dip (arc ) in the point

*S*.

*The angle of the arc*

*is the dip angle δ.*

Fig. 14 The final steps |

*β*= 0

^{○}to

*β*= 90

^{○}in steps of 5

^{○}. Each latitude gives a new quadrant of dip, a new quadrant of rotation, and a new intersection point

*R*. The final construction is shown in >

*Fig*.

*15*. However, in >

*Fig*.

*15*the construction is shown in the lower right quadrant instead of in the upper left and uses already the notation of Blundeville instead of those of William Gilbert.

Fig. 15 The construction of Gilbert's mapping |

*Fig*.

*16*and can already be seen in the upper left picture in Blundeville's drawing in >

*Fig*.

*11*. The spiral line consists of all intersection points

*R*.

Fig. 16 The mater of the dip instrument in The Theoriques |

Fig. 17 The quadrant |

*C*of the

*mater*the instrument is ready to use. In order to illustrate its use we give an example. Consider a seamen who has used a dip ring and measured a dip angle of 60

^{○}. Then he would rotate the quadrant until the spiral line intersects the quadrant at the point 60

^{○}on the inner side of the quadrant. Then the line

*A*-

*B*on the quadrant intersects the scale on the mater at the degree of latitude; in our case 36

^{○}, see >

*Fig*.

*18*. However, accurate reading of the scales becomes nearly impossible for angles of dip larger than 60

^{○}and the reading depends heavily on the accuracy of the construction of the spiral line. Therefore, Henry Briggs was asked to compute a table in order to replace the dip instrument by a simple table look-up. At the very end of Blundeville's

*The Theoriques*we find the following appendix, see >

*Fig*.

*19*:

A short Appendix annexed to the former Treatise by Edward Wright, at the motion of the right Worshipfull M. Doctor Gilbert

Fig. 18 Determining the latitude for 60^{○}dip

Fig. 19 The Appendix

Fig. 20 CHAP. XIIII To finde the inclination or dipping of the magneticall needle under the HorizonBecause of the making and using of the foresaid Instrument, for finding the latitude by the declination of the Magneticall Needle, will bee too troublesome for the most part of Sea-men, being notwithstanding a thing most worthie to be put in daily practise, especially by such as undertake long voyages: it was thought meet by my worshipfull friend M. Doctor Gilbert, that (according to M. Blundeuiles earnest request) this Table following should be hereunto adioned; which M. Henry Briggs (professor of Geometrie in Gresham Colledge at London) calculated and made out of the doctrine and tables of Triangles, according to the Geometricall grounds and reason of this Instrument, appearing in the 7 and 8 Chapter of M. Doctor Gilberts fift[h] booke of the Loadstone. By helpe of which Table, the Magneticall declination being giuen, the height of the Pole may most easily be found, after this manner.

With the Instrument of Declination before described, find out what the Magneticall declination is at the place where you are: Then look that Magneticall declination in the second Collum[n]e of this Table, and in the same line immediatly towards the left hand, you shall find the height of the Pole at the same place, unleße there be some variation of the declination, which must be found out by particular obseruation in euery place.

The next page (which is the final page of *The Theoriques*) indeed shows the Table. > *Fig*. *19* shows the Appendix. In order to make the numbers in the table more visible I have retyped the table.

Briggs was at the center of Gilbert's group. At Gilbert's request he calculated a table of magnetic dip and variation. Their mutual friend Edward Wright recorded and tabulated much of the information which Gilbert used, and helped in the production ofDe Magnete. Thomas Blundeville, another member of Brigg's group, and - like Gilbert - a former protégé of the Earl of Leicester, popularized Gilbert's discoveries inThe Theoriques of the Seven Planets(1602), a book in which Briggs and Wright again collaborated.

First Column.

Second Column.

First Column.

Second Column.

First Column.

Second Column.

Heighs of the Pole.

Magnetical declination.

Heighs of of the Pole.

Magnetical declination.

Heighs of the Pole.

Magnetical declination.

Degrees.

Deg.

Min.

Degrees.

Deg.

Min.

Degrees.

Deg.

Min.

1

2

11

31

52

27

61

79

29

2

4

20

32

53

41

62

80

4

3

6

27

33

54

53

63

80

38

4

8

31

34

56

4

64

81

11

5

10

34

35

57

13

65

81

43

6

12

34

36

58

21

66

82

13

7

14

32

37

59

28

67

82

43

8

16

28

38

60

33

68

83

12

9

18

22

39

61

37

69

83

40

10

20

14

40

62

39

70

84

7

11

22

4

41

63

40

71

84

32

12

23

52

42

64

39

72

84

57

13

25

38

43

65

38

73

85

21

14

27

22

44

66

35

74

85

44

15

29

4

45

67

30

75

86

7

16

30

45

46

68

24

76

86

28

17

32

24

47

69

17

77

86

48

18

34

0

48

70

9

78

87

8

19

35

36

49

70

59

79

87

26

20

37

9

50

71

48

80

87

44

21

38

41

51

72

36

81

88

1

22

40

11

52

73

23

82

88

17

23

41

39

53

74

8

83

88

33

24

43

6

54

74

52

84

88

47

25

44

30

55

75

35

85

89

1

26

45

54

56

76

17

86

89

14

27

47

15

57

76

57

87

89

27

28

48

36

58

77

37

88

89

39

29

49

54

59

78

15

89

89

50

30

51

11

60

78

53

90

90

0

It took Blundeville's *The Theoriques* to describe the construction of the dip instrument accurately which nebulously appeared in Gilbert's *De Magnete*. However, even Blundeville does not say a word concerning the computation of the table. Another friend in the Gresham circle, famous Edward Wright, included all of the necessary details in the second edition of his *On Errors in Navigation* (Wright 1610), the first edition of which appeared 1599. Much has been said about the importance of Edward Wright, see for instance (Parsons 1939), and he was certainly one of the first - if not the first - who was fully aware of the mathematical background of Mercator's mapping, see Sonar 2001, p. 131ff).

It is in > Chap. 14 where Wright and Briggs explain the details of the computation of the dip table which was actually computed by Briggs showing superb mastership of trigonometry. We shall now turn to this computation.

## 8 The Computation of the Dip Table

*On Errors in Navigation*we find in chapter XIIII:

Let

OBRbe a meridian of the earth, wherein letObe the pole,Bthe æquinoctal,Rthe latitude (suppose 60 degrees) letBDbe perpendicular toABinB, and equall to the subtenseOB; and drawing the lineAD, describe therwith the archDSV. Then draw the subtenseOR, wherewith (takingRfor the center) draw the linesRSequall toRO, andASequall toAD. Also becauseBRis assumed to be 60 deg. therefore letSTbe parts of the archSTO, and draw the lineRT, for the angle ART shall be the cõplement of the magnetical needles inclinatiõ vnder the Horizon. Which may be found by the solution of the two trianglesOARand RAS, after this manner:

*De Magnete*we can easily see the situation as described by Gilbert. Now the actual computation starts:

First the triangle

OARis given because of the archOBR, measuring the same 150 degr. and consequently the angle atR15 degr. being equall to the equall legged angle atO; both which together are 30 degr. because they are the complement of the angleOAR(150 degr.) to a semicircle of 180 degr.

*OAR*in Figure 21. Since point

*R*lies at 60

^{○}(measured from

*B*) the arc

*OBR*corresponds to an angle of . Hence the angle at

*A*in the triangle

*OAR*is just . Since

*OAR*is isosceles the angles at

*O*and

*R*are identical and each is 15

^{○}.

Fig. 21 The first step |

Fig. 22 Second step |

Secondly, in the triangle ARS all the sides are given

ARthe Radius or semidiameter 10,000,000:RSequal toROthe subtense of 150 deg. 19,318,516: andASequall toADtriple in power toAB, because it is equal in power toABandBD, that isBO, which is double in power toAB.

*ARS*in

*Figure*

*21*is looked at where

*S*lies on the circle around

*A*with radius

*ADand*on the circle around

*R*with radius

*OR*. The segment

*AR*is the radius of the Earth or the "whole sine." Wright takes this value to be 10

^{7}. We have to clarify what is meant by

*subtense*and where the number 19318516 comes from. Employing the law of sines in triangle

*OAR*we get

*O*lies on the circle around

*R*with radius

*OR*as

*S*does we also have

*RO*=

*RS*. Furthermore

*AS*=

*AD*since

*D*as

*S*lies on the circle around

*A*with radius

*AD*.

*Per constructionem*we have

*BD*=

*OB*and using the Theorem of Pythagoras we conclude

*triple in power to AB*: "the square is three times as big as

*AB*." Hence it follows for

*AS*:

Or else thus: The arch

OBbeing 90 degrees, the subtense therofOB, that is, the tangentBDis 14,142,136, which sought in the table of Tangents, shall giue you the angle BAD 54 degr. 44 min. 8 sec. the secant whereof is the lineADthat isAS17,320,508.

*ABD*we know the lenghts of the segments

*AB*and . Hence for the angle at

*A*we get

Now then by 4 Axiom of the 2 booke of Pitisc.

^{1}as the base or greatest sideSR19,318,516 is to y^{e}summe of the two other sidesSAandAR27,320,508; so is the difference of themSX7,320,508 to the segment of the greatest sideSY10,352,762; which being taken out ofSR19,318,516, there remainethYR8,965,754, the halfe whereofRZ4,482,877, is the Sine of the angleRAZ26 degr. 38 min. 2 sec. the complement whereof 63 degr. 21 min. 58 sec. is the angleARZ, which added to the angleARO15 degr. maketh the whole angle ORS, 78 degr. 21 min. 58 sec. wherof make 52 degr. 14 min. 38 sec. which taken out ofARZ63 degr. 21 min. 58 sec. there remaineth the angleTRA11 deg. 7 min. 20 sec. the cõplement whereof is the inclination sought for 78 degrees, 52 minutes, 40 seconds.

If two chords in a circle intersect then the product of the segments of the first chord equals the product of the segments of the other.

*Figure*

*23*the Theorem of chords is

*M*

*S*=

*A*

*S*+

*A*

*B*it follows

*AS*= 17320508,

*AB*= 10

^{7},

*S*

*R*=

*O*

*R*= 19318516, and

*S*

*X*=

*A*

*S*−

*A*

*X*= 7320508. Hence,

*YR*has length

*Y*

*R*=

*S*

*R*−

*S*

*Y*= 8965754.

*Per constructionem*the point

*Z*is the midpoint of

*YR*. Half of

*YR*is

*RZ*= 4482877. From sin ∠

*R*

*A*

*Z*=

*R*

*Z*/

*A*

*R*= 4482877 / 10

^{7}= 0.4482877 we get . In the right-angled triangle

*ARZ*we see from

*Figure*

*24*that .

Fig. 23 Test |

Fig. 24 Test |

Fig. 25 Test |

Fig. 26 Test |

^{○}the angle

*ARO*at

*R*is 15

^{○}since the obtuse angle in the isosceles triangle

*ORA*is . Therefore, . The part

*TRS*of this angle is 60 ∕ 90 of it, hence

*∠TRS*= 52

^{○}14

*′*38

*′′*. We arrive at

*δ*is the complement of the angle

*TRA*,

The Summe and difference of the sides

SAandARbeing alwaies the same, viz. 27,320,508 and 7,320,508, the product of them shall likewise be alwaies the same, viz. 199,999,997,378,064 to be diuided by y^{e}sideSR, that isROthe subtense ofRBO. Therefore there may be some labour saued in making the table of magneticall inclination, if in stead of the said product you take continually but y^{e}halfe thereof, that is 99,999,998,689,032, and so diuide it by halfe the subtenseRO, that is, by the sine of halfe the archOBR. Or rather thus: As halfe the baseRS(that is, as the sine of halfe the archOBR) is to halfe the summe of the other two sidesSA&AR13,660,254, so is half the difference of theẽ 3,660,254 to halfe of the segmentSY, which taken out of half the base, there remainethRZy^{e}sine ofRAZ, whose cõplement to a quadrãt is y^{e}angle sought forARZ.According to this Diagramme and demonstration was calulated the table here following; the first columne whereof conteineth the height of the pole for euery whole degree; the second columne sheweth the inclination or dipping of the magnetical needle answerable thereto in degr. and minutes.

Although we have taken these computations from Edward Wright's book there is no doubt that the author was Henry Briggs as is also clear from the foreword of Wright.

## 9 Conclusion

The story of the use of magnetic needles for the purposes of navigation is fascinating and gives deep insight into the nature of scientific inventions. Gilbert's dip theory and the unhappy idea to link latitude to dip is a paradigm of what can go wrong in mathematical modeling. The computation of the dip table is, however, a brillant piece of mathematics and shows clearly the mastery of Henry Briggs.

## References

- Balmer H (1956) Beiträge zur Geschichte der Erkenntnis des Erdmagnetismus. Verlag H.R. Sauerländer & Co. Aarau
- Campling A (1922) Thomas Blundeville of Newton Flotman, co. Norfolk (1522-1606). Norfolk Archaeol 21: 336-360
- Gilbert W (1958) De Magnete. Dover, New York
- Greenberg JL (1995) The problem of the Earth's shape from Newton to Clairault. Cambridge University Press, Cambridge, UK
- Hill Ch (1997) Intellectual origins of the English revolution revisited. Clarendon Press, Oxford
- Parsons EJS, Morris WF (1939) Edward Wright and his work. Imago Mundi 3: 61-71
- Pumfrey S (2002) Latitude and the Magnetic Earth. Icon Books, UK
- Sonar Th (2001) Der fromme Tafelmacher. Logos Verlag, Berlin
- Sonar Th (2002) William Gilberts Neigungsinstrument I: Geschichte und Theorie der magnetischen Neigung. Mitteilungen der Math. Gesellschaft in Hamburg, Band XXI/2, 45-68
- Taylor EGR (1954) The mathematical practioneers of Tudor & Stuart England. Cambridge University Press, Cambridge, UK
- Ward J (1740) The lives of the professors of Gresham College. Johnson Reprint Corporation London
- Waters DJ (1958) The art of navigation. Yale University Press, New Haven
- Wright E (1610) Certaine errors in navigation detected and corrected with many additions that were not in the former edition as appeareth in the next pages. London

## Footnotes

^{1} |
The Silesian Bartholomäus Pitiscus (1561-1613) authored the first useful text book on trigonometry: Trigonometriae sive dimensione triangulorum libre quinque, Frankfurt 1595, which was published as an appendix to a book on astronomy by Abraham Scultetus. First independent editions were published in Frankfurt 1599, 1608, 1612 and in Augsburg 1600. The first English translation appeared in 1630. |