SYNOPSIS

Many papers, subject matter cartography, and more particularly those concerning the methodology of Claudius Ptolemy^{1} now include the “de rigueur” utilisation of computer programmes, computer generated visualisations and the Star Burst Plots which endeavour to illustrate data comparisons. That this usage of the computer does not of itself appear to offer any additional information to the subject matter and is merely presentational, should perhaps give cause for reconsideration by those authors of its usage.

This paper is not a polemic against the use of computer generated data, which certainly has its place when appropriately used. It is in fact questioning the over complication caused by such usage when simple maps or diagram maps used as the prime research tool would actually explain more.

This paper illustrates by example that the use of simple maps, for that is in fact all that Claudius Ptolemy has bequeathed to us, a fuller understanding of “Geographia” can be simply gained.

INTRODUCTION

WHAT IS THE MARTELOIO?

In the four subsections which follow the Marteloio will be discussed in the manner each of the authors dictated or indicated. They are basically a table of distances which are determined by triangles. These are the triangles of wind directions that are formed in a circle of 100 mile radius, when the 100 miles is the hypotenuse of the triangle. Thus a triangle having a hypotenuse of 100 miles and being formed from the first quarter wind, 11.25 degrees, the distance sailed, the adjacent side would be 98 miles, whilst the distance off course from the zero degree wind would be 20 miles.

Thus they were be tabulated as follows;

The figures when first viewed make little sense as there is a disparity b

between the pairs of distances; columns 1 & 2 are predicated upon the 100 miles distance discussed, but the second pair are predicated upon a 10 mile distance which enables the mariner to calculate an actual course distance sailed, say 65 miles by a substitution of the units given.

The confirmation of that is simply the 8th wind quarter or 90 degree change and is the maximum distance quoted, 100 or 10 miles.

But how did the Marteloio come into being. If the figures for the first two columns of distances are studied they are simply the approximation for the Sine or Cosine of the wind angle multiplied by the 100 mile radius measurement.

Thus we can rewrite the first part of the table;

The second part of the table is merely an approximation when reduced to the tenth part. That is 100 reduced to 10 miles. The actuality and approximations are fully discussed later.

Thus within the following texts, although detail explanation is again necessary and given, the basic concept is simple and manageable by a competent mathematician.

It must be borne in mind that the Sine table is basically an Indian idea which was copied by Arabic scholars and hence Europeans became aware of it. Whether the Marteloio is actually an Arabic or European invention is open to question, with little chance of a resolution. But, these four texts do give an indication of that knowledge.

RAMON LULL: SECTION 1

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There are two texts by Ramon Lull which include navigational references that may be the precursor to the Marteloio table, or even just a poor description of a table or text which describes the Marteloio. Some researchers even consider they were written on a voyage.

ARBOR SCIENTIAE, Page 569 and 570; Of Geometrical Questions.

De Questionibus Geometrie.

Principia Geometriae quae sunt? Solutio. Vade ad rubricam ante-dictum. Quare geometricus ante lineam diametralem considerat, quam triangularem? Solutio. Figura circularis ita divisibilis non est in aequalibis partibus sicut figura triangularis. Quare Geometricus primo multiplcat duo ex uno, & quatuor ex duobus, & octo ex quatuor, quam tria ex uno & sex ex tribus, & duodecim ex sex? Solutio. Illas mensuras quae citius multiplicand numrerum, Geometricus primo considerat, cum ita sit, quod quantitas comtinua ante sit divisibilis in discretam per numerum dualem, quam ternalem. Idcirco Geometri dicunt, quod numrus dualis est ita multiplication suarum mensurarum in duplicado, sicut in arte arithmeticae unius in numerando unum post alium. Marinarii quomodo mesurat miliaria in mari? Solutio. Marinarii considerat quatuor vetos generals, videlicet vetu orietale, occidentale, meridionalem, & ventum septemtrionalem, similiter alios quatuor ventos qui ex primis exeunt, considerant, videlicet grec, exaloch, lebig, & maestre, & centum circulos considerant in quo venti angulos faciunt, diende considerant per ventu orientalem navem euntem centum miliaria a centro quot sunt miliaria usque ad ventum de exaloch,& miliaria duplicant usque ad ducenta miliaria, & cognoscunt quod miliaria sunt multiplicata. Quae sunt ducenta a vento oriental, usque ad ventum de exaloch per multiplicationem miliarium, quae sunt de termino centenario orientis usque ad terminum de exaloch. Et ad hoc instrumentum habent chartam, compassum, acum, & stellam maris.

The translation is as follows;

*“Questions about Geometry; What are the principles of geometry? Solution- go to the above chapter. Why does geometry consider a diametric line before considering a triangular line? Solution- a circular figure does not divide into equal parts as readily as does a triangular figure. Why does the geometer first produce 2 from 1, then 4 from 2, then 8 from 4 before producing 3 from 1, 6 from 3 and 12 from 6? Solution- the geometer first considers the measures which multiply by numbers most readily given that continuous quantity divides into discrete quantity sooner by the number 2 than by the number 3. Consequently geometers say that the number 2 yields the multiplication of measurement by enumerating one thing after another.*

How do mariners measure miles at sea? Mariners consider the 4 general winds, that is to say the eastern, Western, Southern and Northern, and also another 4 winds that lie between them, grec (NE), exaloch (SE) Lebig (SW) and maestre (NW). And they look carefully at the centre of the circle in which the winds meet at angles; they consider when a ship travels by the east Wind 100 miles from the centre, how many miles it would make on the South east (exaloch) wind; and for200 miles they double the number by multiplying and then they know how many miles there are from the end of each 100 miles in an easterly direction to the corresponding point in a south east direction. And for this they have this instrument, and a chart, compasses, needle and Pole star. (Or, to do this they have instruments such as maps, magnetic compass needle and astrolabes!)

In the second text of Ramon Lull, (1305), ARS MAGNA GENERLIS ET ULTIMA, taken from the 1517 printed edition are the following sections; pages 215-217, ‘De Navigatide’ and page 273, ‘De Questionibus Navigationis’. But as they are in Medieval Latin shorthand I have not included a copy of the text or retyped the original.

The translations are as follows;

Article 96; NAVIGATION

“*Navigation is the art by which sailors know how to navigate the sea. Navigation is originally derived from geometry and arithmetic, through motion and its correlatives signified by the second species of rule C, namely the mover, the mobile and the act of moving; and through time and place, because a ship is in one place at one time and in another place at another time. Given that arithmetic and geometry are derived from this art – as we proved in previous articles – it is therefore clear that the art of Navigation originates in this art first of all, and is subsequently derived from geometry and arithmetic. To clarify this, we first draw this figure divided into four triangles, as shown, and consisting of right, acute and obtuse angles.
Supposing that the place where four angles meet is due north, and this is where the ship’s port is, shown by the letter B. From here, a ship wants to sail eastward, but is actually on a course due southeast, so that when it sails 4 miles, those 4 miles due southeast amount to 3 miles of progress eastward; and when the ship sails for 8 miles, this only amounts to 6 miles of eastward progress; and if it sailed 100 miles, it would amount to 75 miles eastward. And thus, the arithmetician calculates by saying that if 4 miles amount to 3, then twice 4 amounts to 6, and if 4 times 4 equal 16, the three times 3 equal 9, and so with other multiplications of this kind, in their different ways. The reason for this is that in motion, first there is a point, followed by a line, then followed by a triangle, and then by a square, by reason of which successive local motion is generated through multiplication. And this is signified by the previous articles on the point and the line. This kind of natural motion and multiplication is unknown to sailors and navigators, although they know what it is to experience it. And to shed further light on this experience, we will provide doctrine about it.
If a ship leaves the port B and wants to sail due east, but is actually on a southward course, it then deviates twice as much as it would if it sailed southeast. This is because southeast is between east and south. And if the ship is on a south westerly course while it wants to sail eastward, it deviates 3 times as much, and if it is on a westward course, it deviates 4 times as much. And here the intellect sees how the ship’s motion is composed of straight and oblique lines. We have shown a method with which sailors can gauge the deviation of their course from the intended destination, now we intend to provide a doctrine and art to enable sailors and navigators to know where a ship is located at sea, and we will show this by an example of calculating distances between different mountains.
Let L be a mountain 4 miles to the east of port B (where the ship is moored). M is another mountain 4 miles southeast from port B and N is a mountain 4 miles south of the port. Let O be another mountain 8 miles south of the port. Now we ask; how far is L from N and from M, and how far is O from L and M?
In answer to the first and second questions, we say that sailors can gauge these distances by multiplying the miles and calculating the deviation. Now if 4 miles amounts to 3 it follows that mountain L in the east is 3 miles from mountain M in the southeast, and 6 miles from mountain N in the south. And if mountain O in the south is 8 miles from the ship, then the sailor reckons that O is proportionately more than twice as far from L as it is from M in the southeast.
In answer to the third question we say that there are 8 miles from port to O, and four miles from L to the port, which shows that L is 9 miles from O.
In answer to the fourth question we say that O is 8 miles from the port, and M is four miles from the port, so that M is 6 miles from O. And this is the solution to the 4th question.
Following the example in which the distances in miles between mountains L, M, N and O were reckoned, this art can be applied by the artist for proportionally reckoning the distances in miles between other mountains, by multiplying miles in triangles and squares; because just as the arithmetician multiplies numbers by calculating that 3 times 3 equals 9 and 4 times 4 equals sixteen, so can the sailor also make his calculations. Thus we have clarified a method by which sailors can determine the position of a ship at sea, by gauging the distances from north and east, south, as well as west, southeast, southwest, northwest and northeast, with respect to the ship’s position. And this doctrine is easy, brief and most useful; it is general and applicable to particulars.
By rule G and by the subject of the elementative we know that if a wind blows from the east, it is more inclined to the southeast than to the northeast, because the southeast is moist and warm, whereas the northeast is cold and moist, since it is caused by the north. And if the wind blows from the southeast, it is more inclined toward the east than toward the south, because the south is hot and dry. And the same can be said of other winds in their different ways.
Then the sailor must consider different qualities of air; cold, gross air heralds the north wind; moist and tenuous air heralds the east wind; warm and subtle air heralds the south wind; dry and cold air heralds the west wind.
The clouds signify the winds by their colours; red clouds herald the east wind; golden clouds herald the south wind; white clouds herald the north wind; black clouds herald the west wind. Clouds composed of several colours herald a mixture of winds, and the prevalence of different colours signifies the prevalence of the winds they stand for.
Rain coming from a direction signifies wind from the same direction. And the same applies to lightning and thunder in their way. A whirlwind at sea signifies wind in circular motion taking on a shape like a snail or conch shell, by whirling around in a circle to raise seawater aloft as if it were fine dust rising from the ground. And a whirlwind’s colours signify different winds in the same way a cloud’s colours do.
We need not deal with magnets and iron in this article on Navigation, because what we know about them from experience is sufficient. Here we need not seek to know why a magnet attracts iron, as this topic does not belong here, but with the natural sciences. Those interested in the natural sciences can look into this for themselves. And the intellect is*

*delighted to reflect on the things said here about navigation, because it has been amply instructed about the art of Navigation.”*

The translations of the Medieval Latin text used by Ramon Lull may be found at, http://lulliquarts.net/cont.htm

Diagram ChMa1D01 – The text of Ramon Lull is simply illustrated by two diagrams (or one combined) indicating that his approximation of the South East wind being at a ratio of 3:4 from the East wind is slightly inaccurate; i.e. 45 degrees becomes 41.41 degrees. In terms of medieval sailing in the Mediterranean Sea, coasting mainly, this deviation over short distances would be quite manageable, actually capable of being ignored and thus a reasonable rule of thumb. When it becomes 100/75 miles however, the deviation is still only 4 miles from the 45 degree course and hardly noticeable with the inaccurate compasses and charts of the day.

Thus it is quite possible to opine that Ramon Lull may have learnt this rule of thumb from a mariner, or simply applied his mathematical skills to the simple problem.

A CURIOSITY; TEXT OR DIAGRAM

However, between Ramon Lull’s 1295 and 1315 texts and the next section there is the distinct possibility that another text or texts and diagrams were available for both Michael of Rhodes (1434) and Andreas Biancho (1436) to draw upon. In 1390 the inventory of the mother of Oberto Foglieto of Genoa an entry reads; “unum martelogium—item carta una pro navegando”. (see Albertis, E A. [1893] page 118). Thus provenance of origin is difficult.

MICHAEL OF RHODES, VENETIAN MARINER: SECTION

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In 1434 Venetian mariner Michael of Rhodes (Michalli da Ruodo) had risen through the ranks of shipboard life and was preparing himself for the interview or examination which would elevate him to the highest position in the Venetian fleet a commoner could hold; that of ‘Homo de Conseio & Armiraio’. He attained and held that rank, 1435-1443.

Whether it was to aid his elevation or not, Michael of Rhodes at this time started to write a personal manuscript which by the time it was completed in the 1440’s was to cover mathematics, navigation, astronomy, astrology and even ship-building techniques. It was as stated a personal document, just one copy, and included a record of his voyages and other information obtained during his travels.

He wrote a second manuscript into which he copied some of his original text information concerning sailing techniques etc., amplifying some of the sailing direction information, but this text is considered as being usurped by Pietro di Versi, who added his name to the text entitled; “Raxon de Marineri”, Taccuino Nautico del XV Secolo (and now published in Venice edited by Annalisa Conterio). The original manuscript is held in the “Biblioteca Nazionale Marciana, Venice, as Ms It., cl., IX., 170 (=5379).

The original of Michael’s text was lost for centuries and when it resurfaced was made available via the ‘Dibner Institute for the History of Science and Technology’.

Therefore, being a personal document we must accept that it is written without long explanations, the author knew the texts, and thus much of the preparatory information on our subject is missing. This text concentrates upon four pages, 47a, 47b, 48a and 48b which discuss navigation and the Marteloio and are thus linked to the work of Ramon Lull which preceded this section and also the work of Andreas Biancho and ??, which follows.

MICHAEL OF RHODES; NAVIGATION; THE METHOD OF MARTELOIO

Within the four pages M of R endeavours to explain a mathematical solution to sailing discrepancies and their correction to attain an intended course or destination. As this is a private document, which may have been written with future publication in mind, it is perhaps neither self-explanatory nor adequately filled with examples which he would have made use of in his career. As will be readily observed they are quite superficial examples of sailing problems.

M of R commences his text with;”*This method called the ‘Marteloio’ for navigating mentally, as I set forth systematically below*”.

This does not mean that the solution to a mis-direction can be ascertained just by thinking about it, (although as will be shown in some instances by a simple thought it could be solved), but, that you must use your mental faculties, a knowledge of simple mathematics, multiplication and division, to solve the mis-directions in sailing.

He continues;” *First is the distance off course; second is the advance; third is the return; fourth the advance on the return, and each of these has 8 daughters. And first;
The distance off course 8, that is; 20, 38, 55, 71, 83, 92, 98, 100*.(called the alargar)

The advance 8, that is; 98, 92, 83, 71, 55, 38, 20, 0

The return 8, that is; 51, 26, 18, 14, 12, 11, 10.2, 10

The advance on the return 8, that is; 50, 24, 15, 10, 6.5, 4, 2.2, 0

The advance 8, that is; 98, 92, 83, 71, 55, 38, 20, 0

The return 8, that is; 51, 26, 18, 14, 12, 11, 10.2, 10

The advance on the return 8, that is; 50, 24, 15, 10, 6.5, 4, 2.2, 0

Thus he has set down a table of distances which can be used to determine the result of mis-direction once the regime has been understood as there is no explanatory text. The tables give a clue to the distance measures used to regulate the mathematics; distance off course and advance are predicated on a distance of 100 miles; return and advance on return are predicated on a distance of 10 miles and are thus subject to rectification if they are used for distances up to 100 miles.

The table is listed in such a way that it does not aid natural usage. Firstly, each of the ‘*daughters*’ must be recognised as the quarter winds of a quadrant; 11.25, 22.50, 33.75, 45, 56.25, 67.5, 78.75 and 90 degrees, and they are listed in that order such that the distance off course of 20 miles, that is 11.25 degrees or 1 quarter would mean that on the original course line the ship would have sailed 98 miles.

It would appear Michael of Rhodes recognised this problem of identification for at the end of his text he sets down the table using the quarter terminology. Thus we can insert that table here to aid the discussion.

TABLE OF MARTELOIO

His text continues;*
“And for an example of this rule, let us say that a land is 100 miles to the east of you, how much do I want to go east southeast, which is 2 quarters so that the land will be north by east of me, which is 7 quarters, and how far would I be from that place? And below is written the way according to this technique”. *Diagram ChMa1D03

*Note well that it is a simple triangle and the winds chosen naturally form the precise opposite wind and thus the return should be simply noted as 100 miles. Thus if it is a simple triangle by drawing a horizontal line east/west and mark off 100 miles; from the west draw a line east southeast, that is 2 quarters or 22.5 degrees and from the east draw a line on the opposite course, actually north by east projected south by west and they will cross naturally, indicating the outward and return courses and distances. What will be noted from this diagram triangle is that both the intended distance and the distance sailed ESE are one and the same. But it is never stated and is therefore a questionable fact of knowledge. Can we prove that fact?*

His text then continues;

“Take the distance off course that you want to remain, which is 7 quarters, which is 98/10, then we will add the 2 quarters that you go and the 7 that you seek which will be 9 (quarters). So here is the return of 9 quarters, which is 10.2. We multiply and we say in this way 98/10, 51/5, and multiplied together, multiplied and divided is 99 48/50 miles.

And we will be that far wide of the place”.

At this juncture it is worth noting that to achieve the figure for 9 quarters it is necessary to count forward 8 and back one quarter, and thus any number over 8 is achieved. The text requires that the table be used to provide the distance measures to calculate the ESE end point position, and he calculates that the ship will be 99 48/50 miles ESE of the start point. That is obviously the 100 miles equivalent of the actual voyage, but not stated!

Here it is worth inserting the corrections to Michael of Rhodes table, as it is an approximation of the true distances. The whole table is predicated upon 100 miles and then 10 miles and is no more than an extrapolation of the cosine ratio’s for the quarter winds;

1st quarter = 100 Cos 11.25 = 98.0785, with 98 actually used. Little error

2nd quarter = 100 Cos 22.5 = 92.388, with 92 actually used. Large error

3rd quarter = 100 Cos 33.75 = 83.147, with 83 actually used. Medium error

4th quarter = 100 Cos 45 = 70.71, with 71 actually used. Little error

5th quarter = 100 Cos 56.25 = 55.56, with 55 actually used. Large error

6th quarter = 100 Cos 67.5 = 38.268, with 38 actually used. Large error

7th quarter = 100 Cos 78.75 = 19.51, with 20 actually used. Large error

Hence it is obvious discrepancies will inevitably arise, and errors accumulate.

Next we read,

*“Then we want to say how many miles we will be away from that place. And wanting to make this calculation, do the opposite of what you have done and say that the distance off course of 2 quarters is 38/10 and the return of 9 quarters is 10.2, and multiply 38/10 times 51/5, multiply and divide, will make 38 38/50 miles. And so we will do all calculations similarly*.”

Thus we now have the full triangle sides with their measurements such that the original west/east is 100 miles, the ESE is 99.96 miles and the NbE is 38.76 miles according to Michael’s figures.

But what he has given as an example is perhaps for ease of mathematics using virtually the same figures (i.e. only 98 becomes 38) and thus is perhaps a spurious example because it indicates the ship sailed c139 miles to cover an original 100 miles; that is too much extra as time at sea which was unnecessary would be costly to merchants.

A simple diagram of triangles will indicate that the voyage could be as little as c104 miles and up to c139 miles. Thus by illustrating the simplest method using the 100 mile distance he has suggested an extraordinarily excessive sailing distance. Did he understand the figures?

The next example given in the text is;

“*And if you want to tack, that is turn, to know how to return to the course and see the advance you have made, it is done in the same way as the advance, as you will see systematically below. And for example, my course is to the east and I cannot go, and we go 100 SEbS. How many miles do I want to go NEbE so that I come to my course, and how much will I have advanced?
This is the method which is the distance off course of 5 quarters which is 83/10 and the return of 3 quarters which is 18. Multiply 83/10 by 18/1 makes 1494 and divide by 10 will be 149 4/10ths miles and this many miles we want to travel to our course.
And if someone asks you how much we will have advanced here is the way. Take the advance on return of 3 quarters, which is 15, and the distance off course of 5 quarters that is 83/10, multiplied and divided will be 124 5/1oths and then add the advance on the distance off course of 5 quarters which is 55, added makes 179 5/10th miles. You have advanced that much.”*

Again by simply using the 100 mile distance sailed the figures are a direct abstraction from the table and the angles chosen, SEbS to NEbE is merely a 90 degree course change which would have produced the simplest reckoning for a mariner. That is 100 divided by Cos 56.25, the 3rd quarter equals 55.557/100 and thus it is 100 x 100 divided by 55.557 = 179.99 or actually 180 miles total advance. Then 179.99 x Cos 33.75 or 179.99 x 83.147/100 = 149.656. Thus in round figures, sail 100 miles SEbS, turn 90 degrees to sail NEbE for 150 miles to your original easterly course and you will have advanced a total of 180 miles.

By not noticing or deliberately avoiding the simplest triangular calculation when the whole Marteloio is predicated on triangles having a 100 mile measurement, the example is oversimplified and requires angles and distances of a less simple nature to fully explain the obvious capabilities of the Marteloio.

The next example is perhaps the corollary to that statement. He continues;

*“And by another calculation, my course is to the west and I cannot go, and we go 100 miles WbS. The wind goes on and we go toward WSW, 100 miles. The wind goes on and we go SWbW 100 miles. The wind goes on and we go SW 100 miles. I ask how many miles I want to go NW so I come to my course and this is the way and how much I will have advanced.
And we will say that the distance off course of 1 quarter will be 20, and of 2 quarters, 38, and of 3 quarters, 55, and of 4 quarters 71. Added all together it will be 184. And then we will say that the return of 4 quarters is 14. Multiply 184/10 with 14/1 makes 2576 divided by 10 makes 257 6/10ths miles. And that is how many miles you will have travelled to get to the course.
And what will I have advanced on the return of 4 quarters? It is 10 multiplied by 184/10 makes 1840 divided by 10 will be 184 miles. And then we will say, what is the advance on the distance off course? From 1 quarter it is 98 and 2 quarters 92, and 3 quarters 83, and 4 quarters 71. Added together make 344 added to 184 makes 528. And this much you will have advanced”.
*Basically little mathematics is required in this example, merely additions and recognising that again the angle of return is 90 degrees to the penultimate course. This is in fact what Michael of Rhodes has stated and must have been aware of as he has said the penultimate course is SW and the return is NW, a 90 degree course change. Thus by adding the advances, 98+92+83+71=344 advance and then the alargars, 20+38+55+71 = 184 alargar, because the return is 90 degrees the alargar must equal the second portion of the advance and give the total of 344 + 184 = 528 miles advance. The return is thus simply 184 x 100/71 = 259 11/71 (259.156 or 184/Cos45).

His final section of the text is altogether a different calculation, one fraught with mis-description leaving too many assumptions and translations to be made. In fact it appears to be an unconsidered addition as the original paragraph first words have been erased.

Having read the text many times to understand its mathematics and in fact comprehend just what he is actually stating has led to the conclusion that it was written whilst studying a diagram and thus each separate section must be carefully analysed sentence by sentence to separate the sections properly and understand the whole.

Here is the complete text, which will be subdivided for ease of explanation;

*“And if someone asks you; [what will I have]. Another calculation, a land is west of me in the evening and I cannot say how many miles we go in the night. In the night we go NW 41 8/10ths miles and in the morning that land is towards the WSW”.
*

It must be evident here that M of R is copying or not concentrating on his text.

He continues;

*“How many miles is the distance now that the land is towards WSW, and how many miles was it in the evening when it was to my west? You should do it this way and say that my distance off course for 2 quarters 38/10 and the return of 6 quarters is 11. Therefore multiply 38 x 11 making 418, divide by 10 will be 41 8/10 miles you will have travelled.”*

At this juncture it is so very apparent that there has been a methodology fiddle; or he is reading from a diagram. Why start by stating the distance and then prove it.

But he continues;

*“And how far away will you be when it stands WSW of which the distance off course 4 quarters is 71/10 and the return of 6 quarters is 11? Multiply 71/10 by 11/1 makes 781, divide by 10 it will be 78 1/10 miles. You will be that far away.”*

Having copied out both sections it now becomes apparent that he has miscopied from a diagram or from notes for the text. If a diagram is studied that sets out both examples then it will be seen that the distance of 78 1/10th miles is in fact the resultant of the 4 quarters when the distance travelled off course is 41 8/10th miles. If the return distance is noted for the first or 2 quarter diagram it is 41.131, which could be 41 8/10th miles by his figures. With the 2 quarter diagram the original distance is 92 miles plus 15.74 miles and for the 4 quarter diagram it is the 100 mile base figure. It would therefore appear that Michael of Rhodes has inadvertently mixed the 2 and 4 quarter figures for his solution. But it is not the end point.

He finishes thus;

“*And to know how far away you were yesterday when it stood to the west, go back the opposite of your course and go SE that which remains to the west, such that it stands WSW you will go 53 1/5th miles. Reduce by a quarter and 39 2/10 or 39 9/10th miles remains and you will be 126 4/5th miles away. Reduce by a quarter there remains 96 2/5th miles. And you were that far away in the evening.”*

This is no doubt a complete misreading of a diagram or misunderstanding of another person’s text. The only point which will allow a reverse voyage of 53.2 miles to attain a WSW position is the apex of the 4 quarter triangle. But the actual WSW position is 58.2 miles from that apex, thus giving a total side of 100 miles from 58.2 + 41.8 = 100 miles the hypotenuse distance of a 71 side triangle.

The distance 126 4/5th miles is only valid from a diagram and not a calculation, but the 96 2/5th miles distance can be assessed via a simple 2 quarter angle against the 1 quarter off course 20 mile distance as the diagram illustrates.

To understand the last section a Tondo e Quadro was constructed, called LINES.

LINES; JUST HOW MANY CAN BE DRAWN?

If we study Portolan Charts, one obvious comment is that they are covered in angled lines, wind direction lines and it appears there are virtually as many as it is possible to draw. If only the necessary lines were drawn it would be just the 32 quarter wind lines projected from a single point across the face of the chart. But every conceivable wind is drawn as many times as it is possible until the cartographer has filled the chart and in fact made it harder to read.

If we reread what Michael of Rhodes has written in the final paragraphs it can be seen that they are totally outwith the first basic premise calculations.

Thus if we construct a diagram of the Marteloio, rather than a table format as is normal we can evaluate any description made of a voyage by wind direction.

Basically we have a single quadrant of winds, that is 8 sections of 11.25 degrees, but if we draw the quadrant not as a quarter circle but as a square there can then be drawn a multitude of lines in 3 or 4 directions. Thus it is possible that a diagram of the Marteloio was used to write the text and not the tables that appear at the beginning and end of the 4 pages of his text. The number of lines is reminiscent of the Portolan chart and when the winds are drawn from both east and west terminals they indicate half quarter wind positions also. Thus the strange distance of 126 4/5th miles can be explained.

CURIOUSITIES WITHIN THE TEXT