Chapter II The Secrets of the Piri Re’is Map
When our investigation started my students and I were amateurs together. My only advantage over them was that I had had more experience in scientific investigations; their advantage over me was that they knew even less and therefore had no biases to overcome.
At the very beginning I had an idea—a bias, if you like—that might have doomed our voyage of discovery before it began. If this map was a copy of some very ancient map that had somehow survived in Constantinople to fall into the hands of the Turks, as I believed, then there ought to be very little in common between this map and the maps that circulated in Europe in the Middle Ages. I could not see how this map could be both an ancient map (recopied) and a medieval one. Therefore, when one of my students said this map resembled the navigation charts of the Middle Ages, at first I was not much interested. Fortunately for me, I kept my opinions to myself, and encouraged the students to begin the investigation along that line.
We soon accumulated considerable information about medieval maps. We were not concerned with the land maps, which were exceedingly crude. (See Figures l and 2.) We were interested only in the sea charts used by medieval sailors from about the 14th Century on.  These “portolan”  maps were of the Mediterranean and Black Seas, and they were good. An example is the Dulcert Portolano of 1339. (Fig. 3.) If the reader will compare the pattern of lines on this chart with that on the Piri Re’is Map (Frontispiece) he will see that they are similar. The only difference is that, while the Dulcert Portolano covers only the Mediterranean and the Black Seas, the Piri Re’is Map deals with the shores of the entire Atlantic Ocean. The lines differ from those on modern maps. The lines do not resemble the modern map’s lines of latitude and longitude that are spaced at equal intervals and cross to form “grids” of different kinds. Instead, some of the lines, at least, on these old maps seem to radiate from centers on the map, like spokes from a wheel. These centers seem to reproduce the pattern of the mariner’s compass, and some of them are decorated like compasses. The radiating “spokes” are spaced exactly like the points of the compass, there being sixteen lines in some cases, and thirty-two in others.
Since the mariner’s compass first came into use in Europe about the time that these charts were introduced, most scholars have concluded that the charts’ design must have been intended to help medieval sailors sail by the compass. There is no doubt that medieval navigators did use the charts to help them find compass courses, for the method is described in a treatise written at the time (89, 179, 200). However, as we continued to study these medieval charts, a number of mysteries turned up.
We found, for example, that one of the leading scholars in the field did not believe that the charts originated in the Middle Ages. A. E. Nordenskiold, who compiled a great Atlas of these charts (146) and also wrote an essay on their history (147), presented several reasons for concluding that they must have come from ancient times. In the first place, he pointed out that the Dulcert Portolano and all the others like it were a great deal too accurate to have been drawn by medieval sailors. Then there was the curious fact that the successive charts showed no signs of development. Those from the beginning of the 14th Century are as good as those from the 16th. It seemed as though somebody early in the 14th Century had found an amazingly good chart which nobody was to be able to improve upon for two hundred years. Furthermore, Nordenskiold saw evidence that only one such model chart had been found and that all the portolanos drawn in the following centuries were only copies—at one or more removes—from the original. He called this unknown original the “normal portolano” and showed that the portolanos, as a body, had rather slavishly been copied from this original. He said:
The measurements at all events show: (1) that, as regards the outline of the Mediterranean and the Black Sea, all the portolanos are almost unaltered copies of the same original; (2) that the same scale of distance was used on all the portolanos (147:24).
After discussing this uniform scale that appears on all the portolanos, and the fact that it appears to be unrelated to the units of measurement used in the Mediterranean, except the Catalan (which he had reason to believe was based on the units used by the Carthaginians), Nordenskiold further remarks:
… It is therefore possible that the measure used in the portolanos had its ultimate origin in the time when the Phoenicians or Carthaginians ruled over the navigation of the western Mediterranean, or at least from the time of Marinus of Tyre … (147:24). 
Nordenskiold inclined, then, to assign an ancient origin to the portolanos. But this is not all. He was quite familiar with the maps of Claudius Ptolemy which had survived from antiquity and had been reintroduced in Europe in the 15th Century. After comparing the two, he found that the portolanos were much better than Ptolemy’s maps. He compared Ptolemy’s map of the Mediterranean and the Black Seas with the Dulcert Portolano (Fig. 4) and found that the superiority of the portolano was evident.
Let us stop to consider, for a moment, what this means. Ptolemy is the most famous geographer of the ancient world. He worked in Alexandria in the 2nd Century A.D., in the greatest library of the ancient world. He had at his command all the accumulated geographical information of that world. He was acquainted with mathematics. He shows, in his great work, the Geographia (168), a modern scientific mentality. Can we lightly assume that medieval sailors of the fourteenth century, without any of this knowledge, and without modern instruments except a rudimentary compass—and without mathematics—could produce a more scientific product?
Nordenskiold felt that there had been in antiquity a geographic tradition superior to the one represented by Ptolemy. He thought that the “normal portolana” must have been in use then by sailors and navigators, and he answered the objection that there was no mention of such maps by the various classical writers by pointing out that in the Middle Ages, when the portolan charts were in use, they were never referred to by the Schoolmen, the academic scholars of that age. Both in ancient and in medieval times the academic mapmaker and the practical navigator were apparently poles apart. (See Figs. 5, 6, 7, 8.) Nordenskiold was forced to leave the problem unsolved. Neither the medieval navigators nor the known Greek geographers could have drawn them. The evidence pointed to their origin in a culture with a higher level of technology than was attained in medieval or ancient times. 
All the explanations of the origins of the portolan charts were opposed by Prince Youssouf Kamal, a modern Arab geographer, in rather violent language:
Our incurable ignorance … as to the origin of the portolans or navigation charts known by this name, will lead us only from twilight into darkness. Everything that has been written on the history or the origin of these charts, and everything that will be said or written hereafter can be nothing but suppositions, arguments, hallucinations … (107:2) 
Prince Kamal also argued against the view that the lines on the charts were intended to facilitate navigation by the compass:
As for the lines that we see intersecting each other, to form lozenges, or triangles, or squares: these same lines, I wish to say, dating from ancient Greek times, and going back to Timosthenes, or even earlier, were probably never drawn … to give … distances to the navigator. …
The makers of portolans preserved this method, that they borrowed from the ancient Greeks or others, more probably and rather to facilitate the task of drawing a map, rather than to guide the navigator with such divisions … (107: 15-16)
In other words, the portolan design was an excellent design to guide a mapmaker either in constructing an original map or in copying one, because of the design’s geometrical character.
Early in our investigation, three of my students, Leo Estes, Robert Woitkowski, and Loren Livengood, decided to take this question—the purpose of the lines on the portolan charts—as their special project. They journeyed to Hanover, New Hampshire, to inspect the medieval charts in the Dartmouth College Library. On their return, one of them, Loren Livengood, said he thought he knew how the charts had been constructed.
The problem was to find out, from the lines actually found on the charts, whether it might be possible to construct a grid of lines of latitude and longitude such as are found on modern maps. In other words, the problem was to see if this portolan system could be converted to the modern one.
Livengood’s approach was simple. Without actually realizing the importance of his choice, he put himself in the position of a mapmaker rather than of a navigator. That is, he saw the problem not as one of finding a harbor, but of actually constructing a map. He had never heard of Prince Kamal, but he was adopting the Prince’s view of the purpose of the lines. The probable procedure of the mapmaker, Livengood speculated, was first to pick a convenient center for his map and then determine a radius long enough to cover the area to be mapped. With this center and radius the mapmaker would draw a circle.
Then he would bisect his circle, again and again, until he had sixteen lines from the center to the periphery at equal angles of 22½° 
The third step would be to connect points on the perimeter to make a square, with four different squares possible.
The fourth step would be to choose one of the squares, and draw lines connecting the opposite points, thus making a map grid of lines at right angles to each other. (Fig. 9)
Now, although the scholars agreed that the portolan charts had no lines of latitude and longitude, it stood to reason that if one of the vertical lines (such as the line through the center) was drawn on True North, then it would be a meridian of longitude, and any line at right angles to it would be a parallel of latitude. Assuming that a projection similar to the famous Mercator projection, in which all meridians and parallels are straight lines crossing at right angles, underlay these maps (see Fig. 10), then all parallel vertical lines would be meridians of longitude, and all horizontal lines would be parallels of latitude. 
Applying this idea to the Piri Re’is Map, we could see that the mapmaker had selected a center, which he had placed somewhere far to the east of the tom edge of our fragment of the world map,  and had then drawn a circle around it. He had bisected the circle four times, drawing sixteen lines from the center to the perimeter, at angles of 22½°, and he had also drawn in all the four possible squares, perhaps with the intention of using different squares for drawing grids for different parts of the map, where it might be necessary to have different Norths.  It was Estes who originally pointed out to us that the portolan design had the potentiality of having several different Norths on the same map.
Now the next question was: Which was the right square for us? That is, which (if any) of the squares that could be made out of the design of the Piri Re’is Map was correctly oriented to North, South, East, and West?
Estes found the solution. Comparing the Piri Re’is Map with a modern map (Figs. 10, 11, 12) he found a meridian on the modem map that seemed to coincide very nearly with a line on the Piri Re’is Map—a line running north and south close to the African coast, in about 20° W longitude, leaving the Cape Verde Islands to the west, the Canaries to the east, and the Azores to the west.
Estes suggested that this line might be our prime meridian, a line drawn on True North. All lines parallel to this (assuming, of course, that the underlying projection resembled in some degree the Mercator projection) would also be meridians of longitude; all lines at right angles would be parallels of latitude. The meridians and parallels thus identified, provisionally, on the Piri Re’is Map, formed a rectangular grid, as shown in Fig. 12.
The only difference between this large rectangular grid actually found on the Piri Re’is Map and the grids of modern maps was that the latter all carry registers of degrees of latitude and longitude, with parallels and meridians at equal intervals, usually five or ten degrees apart. We could convert the Piri Re’is grid into a modern grid if we could find the precise latitudes and longitudes of its parallels and meridians. This, we found, meant finding the exact latitude and longitude of each of the five projection centers in the Atlantic Ocean, through which the lines of Piri Re’is’ grid ran.
At the beginning of our inquiry I had noticed that these five projection centers had been placed at equal intervals on the perimeter of a circle, though the circle itself had been erased (Fig. 11). I had also noticed that converging lines were extended from these points to the center, beyond the eastern edge of the map. This, it seemed to me at the time, was a geometrical construction that should be soluble by trigonometry . I did not then know that, in the opinion of all the experts, there was no trigonometrical foundation to the portolan charts.
Not knowing that there was not supposed to be any mathematical basis for the portolanos, we now made the search for it our main business. I realized from the start that to accomplish this we would have to discover first the precise location of the center of the map, and then the precise length of the radius of the circle drawn by the mapmaker. I was fortunate in having a mathematician friend, Richard W . Strachan, at the Massachusetts Institute of Technology. He told me that, if we could obtain this information for him, he might be able, by trigonometry, to find the precise positions of the five projection points in the Atlantic Ocean on the Piri Re’is Map, in terms of modern latitude and longitude. This would enable us to draw a modern grid on the map, and thus check every detail of it accurately. Only in this way, of course, could we verify the claim of Mallery regarding the Antarctic sector of the map.
The search for the center of the map lasted about three years. We thought from the beginning that the lines extending from the five projection points probably met in Egypt. We used various methods to project the lines to the point where they would meet. Our first guess for the center of the map was the city of Alexandria. This appealed to me because Alexandria was long the center of the science and learning of the ancient world. It seemed likely that, if they were drawing a world map, the Alexandrian geographers might naturally make their own city its center.
However, this guess proved to be wrong. A contradiction appeared. The big wind rose in the North Atlantic looked as if it were meant to lie on the Tropic of Cancer. One of the lines from this center evidently was directed toward the center of the map. But we noticed that this line was at right angles to our prime meridian. This meant, of course, that it was a parallel of latitude. Now, the Tropic of Cancer is at 23½° North Latitude, and therefore the parallel from the wind rose would reach a center in Egypt at 23½° North. But Alexandria is not at that latitude at all. It lies in 31° North. Therefore Alexandria could not be the center of our circle.
We looked at the map of ancient Egypt to find, if we could, a suitable city on the Tropic of Cancer that might serve as a center for the map. (We were still attached to the idea that the center of our map should be some important place, such as a city. Later, we were emancipated from this erroneous notion.)
Looking along the Tropic of Cancer, we found the ancient city of Syene, lying just north of the Tropic, near the present city of Assuan, where the great darn is being built. Now we recalled the scientific feat of Eratosthenes, the Greek astronomer and geographer of the 3rd Century B.C., who measured the circumference of the earth by taking account of the angle of the sun at noon as simultaneously observed at Alexandria and at Syene.
We were happy to change our working theory and adopt Syene as the center of the map. With the help of hindsight, we could now see how reasonable it was to place the center of the map on the Tropic, an astronomically determined line on the surface of the earth. The poles, the tropics, and the equator can be exactly determined by celestial observations, and they have been the bases of mapmaking in all times. Syene, too, was an important city, suitable for a center. A good “proof” of this center for the map was constructed by two students, Lee Spencer and Ruth Baraw. Only at the end of our inquiry did we find that Syene was not, after all, exactly the center.
The matter of the radius caused us much more trouble. At first, there appeared to be absolutely no way of discovering its precise length. However, some of my students started talking about the Papal Demarcation Line—the line drawn by Pope Alexander VI in 1493, and revised the next year, to divide the Portuguese from the Spanish possessions in the newly discovered regions (Fig. 13). On the Piri Re’is Map there was a line running north and south, passing through the northern wind rose and then through Brazil at a certain distance west of the Atlantic coast. This line appeared to be identical, or nearly identical, with the Second Demarcation Line (of 1494), which also passed through Brazil. Piri Re’is had mentioned the Demarcation Line on his map, and we reached the conclusion that this line, if it was the Demarcation Line, could give us the longitude of the northern wind rose and thus the length of the radius of the circle with its center at Syene.
The Papal Demarcation Line of 1494 is supposed to have been drawn north and south at a distance of 370 leagues west of the Cape Verde Islands. Modern scholars have calculated that it was at 46° 30′ West Longitude (140:369). We therefore assigned this longitude to the northern wind rose, and thus obtained our first approximate guess as to the length of the radius of the circle. According to this finding the radius was 79° in length (32½ plus 46½). This result was wrong by 9½°, as we later discovered, but it was close enough for a starter.
At this stage, our findings were too uncertain to justify an attempt to apply trigonometry to the problem. Instead, we tested our results directly on an accurate globe provided by Estes. We made our test by actually drawing a circle, with Syene as the center, and the indicated radius, and then laying out the lines from the center to the perimeter, 22½° apart, beginning with one to the equator. The result seemed pretty good, and we were sure we were on the right track.
It was lucky that we got so far before we discovered that our interpretation of the Demarcation Line on the map was wrong. This fact was finally brought home to us by two other students, John F. Malsbenden and George Batchelder (Fig. 14). They had been bending over the map during one of our long night sessions  when suddenly Malsbenden straightened up and exclaimed indignantly that all our work had been wasted, that the line we had picked out was not the right one. In an inscription on his map which we had overlooked Piri Re’is had himself indicated an entirely different line. It was the first line, the line of 1493, and it did not go through the wind rose at all. The mistake, however, had served its purpose. It was true enough that the line we had picked out on the Piri Re’is Map represented neither line; nevertheless it was close enough to the position of the Demarcation Line of 1494 to give us a first clue to the longitude.
Another error that turned out to be very profitable was the assumption we made, during a certain period of time, that perhaps our map was oriented not to True North, but to Magnetic North. Later, we were to find that many, if not most, of the portolanos were indeed oriented, very roughly, to Magnetic North. Some writers on the subject had argued, as already mentioned, that the lines on the portolan charts were intended only for help in finding compass directions, and were therefore necessarily drawn on Magnetic North. 
In the interest of maximum precision, I wanted to find out how the question of Magnetic North might affect the longitude of the Second Demarcation Line, which now determined our radius. If the Demarcation Line lay at 46° 30′ West Longitude at the Cape Verde Islands, it would, with a magnetic orientation, lie somewhat farther west at the latitude of the northern wind rose, and this would affect the radius. We spent time trying to calculate how much farther west the hne would be. This in turn involved research to discover the amount of the compass declination (the difference between True and Magnetic North) today in those parts of the Atlantic, and speculation as to what might have been the amount of the variation in the days of Piri Re’is or in ancient times. We found ourselves in a veritable Sargasso Sea of uncertainties and frustrations.
Fortunately, we were rescued from this dead end by still another wrong idea. I noticed that the circle drawn with Syene as a center, and with a radius to the intersection of the supposed Second Demarcation Line with the northern wind rose, appeared to pass through the present location of the Magnetic Pole. We then allowed ourselves to suppose (nothing being impossible) that somebody in ancient times had known the location of the Magnetic Pole and had deliberately selected a radius that would pass through it. Shaky as this assumption might have been, it was at least better than the Demarcation Line, since in ancient times nobody could have had an idea of a line that was only drawn in 1494 A.D. The Magnetic Pole is, however, very unsatisfactory as a working assumption because it does not stay in one place. It is always moving, and where it may have been in past times is anybody’s guess.
In the middle of this I read Nordenskiold’s statement that the portolan charts were drawn on True North, and not on Magnetic North (146: 17). In this Nordenskiold was really mistaken, unless he meant that the charts had originally been drawn on True North and then had been reoriented in a magnetic direction. But his statement impressed us, and then I observed, looking again at the globe with our circle drawn on it, that the circle that passed through the Magnetic Pole also passed very close indeed to the True Pole. Now, you may be sure, we abandoned our magnetic theory in a hurry, and adopted the working assumption that perhaps someone in ancient times knew the true position of the Pole, and drew his radius from Syene on the Tropic of Cancer to the Pole. Again, hindsight came to our support. As in the case of the Tropic of Cancer, the Pole was astronomically determined: It was a precisely located point on the earth’s surface.
It appeared to us that we had swum through a murky sea to a safe shore. We had now reached a point where it would be feasible to attempt a confirmation of the whole theory by trigonometry. We were proceeding now on the following asumptions: (1) The center of the projection was at Syene, on the Tropic of Cancer and at longitude 32½° East; (2) the radius of the circle was from the Tropic to the Pole, or 66½° in length, and (3) the horizontal line through the middle projection point on the map (Point III) was the true equator. By comparison with the African coast of the Gulf of Guinea, this line, indeed, appears to be very close to the position of the equator. Nevertheless, this was not merely an assumption but also guesswork. We could not know, either, that the ancient mapmaker had precise information as to the size of the earth, which would be necessary for correctly determining the positions of the poles and the equator. Such assumptions could be only working assumptions, to be used for purposes of experiment and discarded if they proved wrong. They were, however, the best assumptions we had been able to come up with so far, and assumptions we had to have to work with.
We could now give our mathematician, Strachan, the data he required for a mathematical analysis. He calculated the positions of all the five projection centers on the Piri Re’is Map to find their precise locations in latitude and longitude.  He used our assumed equator as his base line of latitude. I have tried to explain this in Fig. 15. Here I have drawn the first radius from the center of the projection to the point of intersection of the assumed equator with the perimeter of the circle. I then have laid out the other radii at angles of 22½° northward and southward. In this way, our assumption that this equator is precisely correct controls the latitudes to be found for the other four projection points. The assumed equator is the base line for latitude, just as Syene is the reference point for longitude.
Strachan initially computed the positions of the five projection points both by spherical and by plane trigonometry. At each successive step, with varying assumptions as to the radius of the projection and the position of its center, he did the same thing, but in every case the calculations by plane trigonometry made sense—that is, plane trigonometry made it possible to construct grids that fitted the geography reasonably well, while the calculations by spherical trigonometry led to impossible contradictions. It became quite clear that our projection had been constructed by plane trigonometry. 
Once we had precise latitudes and longitudes for the five centers on the Piri Re’is map, we could construct a modern type of grid. The total difference of latitude between Point I and Point V, divided by the millimeters that lay between them on our copy of the map (we used a tracing of our photograph of the map), gave us the length of the degree of latitude in millimeters. To check on any possible irregularities we measured the length of the degree of latitude separately between each two of the five points. We followed the same procedure with the longitude, as illustrated in Fig . 16. The lengths of the degrees of latitude and longitude turned out to be practically the same; we thus appeared to have a square grid. In doing this we disregarded the scales actually drawn on the map, since there was no way of knowing when or by whom they had been drawn, or what units of distance they had represented.
The next step was to learn how to draw a grid, not at all an easy task. It was not a particularly complicated task, but it demanded a very high level of accuracy and an extreme degree of patience. Fortunately, one of my students, Frank Ryan, was qualified for the job . He had served in the Air Force, had been stationed at Westover Air Force Base in Massachusetts, and had been assigned to the Cartographic Section of the 8th Reconnaissance Technical Squadron, under a remarkable officer, Captain Lorenzo W. Burroughs. The function of the unit at that time was to prepare maps for the use of the United States Air Force’s Strategic Air Command, known as SAC. Later, it was attached to the 8th Air Force. Needless to say, the personnel of that unit were competent to serve the demanding requirements of the Air Force, as far as mapmaking was concerned, and Frank Ryan had been intensively trained in the necessary techniques. He had had the experience of being drafted into the Air Force: now he had the experience of being drafted again, to draw our grid.
Later Ryan introduced me to Captain Burroughs, and I visited Westover Air Force Base. The captain offered us his fullest cooperation in preparing a draft map with the solution of the projection, and virtually put his staff at our disposal. The co-operation between us lasted more than two years, and a number of officers and men gave us very valuable assistance.  Later both Captain Burroughs and his commanding officer, Colonel Harold Z. Ohlmeyer reviewed and endorsed our work (Note 23).
The procedure for drawing the grid was as follows: All the meridians were drawn parallel with the prime meridian, at intervals of five degrees, and all parallels were drawn parallel with the assumed equator, at intervals of five degrees. These lines did not turn out in all cases to be precisely parallel with the other lines of the big grid traced from the Piri Re’is Map, but this was understandable. The effect might have resulted from warping of the map, or from carelessness in copying the lines from the ancient source map Piri Re’is used. We had to allow for a margin of error here, for we could not be sure that no small errors had crept in when the equator or the prime meridian was recopied. Here, as in other respects, we simply had to do the best we could with what we had. 
When the grid was drawn, we were ready to test it. We identified all the places we could on the map and made a table comparing their latitudes and longitudes on the Piri Re’is Map with their positions on the modern map. The errors in individual positions were noted and averages of them made (Table 1). The Table is, of course, the test of our solution of the Piri Re’is projection.
But I must not get ahead of my story. We found that some of the positions on the Piri Re’is Map were very accurate, and some were far off. Gradually we became aware of the reasons for some of the inaccuracies in the map. We discovered that the map was a composite, made up by piecing together many maps of local areas (perhaps drawn at different times by different people), and that errors had been made in combining the original maps. There was nothing extraordinary about this. It would have been an enormous task, requiring large amounts of money, to survey and map all at once the vast area covered by the Piri Re’is Map. Undoubtedly local maps had been made first, and these were gradually combined, at different times, into larger and larger maps, until finally a world map was attempted. This long process of combining the local maps, so far as the surviving section of the Piri Re’is Map is concerned, had been finished in ancient times. This theory will, I believe, be established by what follows. What Piri Re’is apparently did was to combine this compilation with still other maps—which were probably themselves combinations—to make his world map.
The students were responsible for discovering many of the errors. Lee Spencer and Ruth Baraw examined the east coast of South America with great care and found that the compiler had actually omitted about 900 miles of that coastline. It was discovered that the Amazon River had been drawn twice on the map. We concluded that the compiler must have had two different source maps of the Amazon, drawn by different people at different times, and that he made the mistake of thinking they were two different rivers. We also found that besides the equator upon which we had based our projection (so far as latitude was concerned) there was evidence that somebody had calculated the position of the equator differently, so that there were really two equators. Ultimately we were able to explain this conflict. Other important errors included the omission of part of the northern coast of South America, and the duplication of a part of that coast, and of part of the coasts of the Caribbean Sea. A number of geographical localities thus appear twice on the map, but they do not appear on the same projection. For most of the Caribbean area the direction of North is nearly at right angles to the North of the main part of the map.
As we identified more and more places on our grid, and averaged their errors in position, we found all over the map some common errors that indicated something was wrong with the projection. We concluded that there must still be errors either in the location of the center of the map, in the length of the radius, or both. There was no way to discover these probable errors except by trying out all reasonable alternatives by a process of trial and error. This was time consuming and a tax on the patience of all of us. With every change in the assumed center of the map, or in the assumed radius, Strachan had to repeat the calculations, and once more determine the positions of the five projection points . Then the grid had to be redrawn and all the tables done over. As each grid in turn revealed some further unidentified error, new assumptions had to be adopted, to an accompaniment of sighs and groans. We had the satisfaction, however, of noting a gradual diminution of the errors that suggested that we were approaching our goal.
Among the various alternatives to Syene as the center of the map we tried out, at one stage, the ancient city of Berenice on the Red Sea. This was the great shipping port for Egypt in the Alexandrian Age, and it, too, lay on the Tropic of Cancer. Berenice seemed to be a very logical center for the map because of its maritime importance. We studied the history of Berenice, and everything seemed to point to this place as our final solution. But then, as in an Agatha Christie murder mystery, the favorite suspect was proved innocent. The tables showed the assumption to be wrong, for in this case the errors were even increased. We had to give up Berenice, with special regrets on my part because of the beauty of the name.
Now we went back to Syene, but with a difference. The tables showed that the remaining error in the location of the center of the map was small. Therefore we tried out centers near Syene, north, east, south and west, gradually diminishing the distances, until at last we used the point at the intersection of the meridian of Alexandria, at 30° East Longitude, with the Tropic. This finally turned out to be correct.
Immediately hindsight began to make disagreeable comments. Why hadn’t we thought of this before? Why hadn’t we tumbled to this truth in the beginning? It combined all the most reasonable elements: the use of the Tropic, based on astronomy, and the use of the meridian of Alexandria, the capital of ancient science. Later we were to find that all the Greek geographers based their maps on the meridian of Alexandria.
Remaining errors in the tables suggested something wrong with the radius. We knew, of course, that our assumption that the mapmaker had precise knowledge of the size of the earth was doubtful. It was much more likely that he had made some sort of mistake. We therefore tried various lengths. We shortened the radius a few degrees, on the assumption that the mapmaker might have underestimated the size of the earth, as Ptolemy had. This only increased the errors. Then we tried lengthening the radius. The entire process of trial and error was repeated with radii 7°, 5°, 2°, and 1° too long. Finally we got our best results with a radius extended three degrees. This meant that our radius was not 66.5°, the correct number of degrees from the Tropic to the Pole, but 69.5°. This error amounted to an error of 4½ per cent in overestimating the size of the earth.
A matter of great importance, which we did not realize at all at the time, was that we were, in fact, finding the length of the radius (and therefore the length of the degree) with reference mainly to longitude. I paid much more attention to the average errors of longitude than I did to the errors of latitude. I was especially interested in the longitudes along the African and South American coasts. Our radius was selected to reduce longitude errors to a minimum while not unduly increasing latitude errors. As it turned out, this emphasis on longitude was very fortunate, for it was to lead us to a later discovery of considerable importance.
With regard to the overestimating of the circumference of the earth, there was one geographer in ancient times who made an overestimate of about this amount. This was Eratosthenes. Does this mean that Eratosthenes himself may have been our mapmaker? Probably not. We have seen that the Piri Re’is Map was based on a source map originally drawn with plane trigonometry. Trigonometry may not have been known in Greece in the time of Eratosthenes. It has been supposed that it was invented by Hipparchus, who lived about a century later. Hipparchus discovered the precession of the equinoxes, invented or at least described mathematical map projections, and is generally supposed to have developed both plane and spherical trigonometry (58 :49; 175:86).  He accepted Eratosthenes’ estimate of the size of the earth (184:415) though he criticized Eratosthenes for not using mathematics in drawing his maps.
We must interfere in this dispute between Hipparchus and Eratosthenes to raise an interesting point. Did Hipparchus criticize his predecessor for not using mathematically constructed projections on which to place his geographical data? If so, his criticism looks unreasonable. The construction of such projections requires trigonometry. If Hipparchus himself developed trigonometry, how could he have blamed Eratosthenes for not using it a century before? Hipparchus’ own books have been lost, and we really have no way of knowing whether the later writers who attributed trigonometry to Hipparchus were correct. Perhaps all they meant, or all he meant or said in his works, was that he had discovered trigonometry. He might have discovered it in the ancient Chaldean books whose star data made it possible for him to discover the precession of the equinoxes.
But this is speculation, and I have a feeling that it is very much beside the point. If Hipparchus did in fact develop both plane and spherical trigonometry, the Piri Re’is Map, and the other maps to be considered in this book, are evidence suggesting that he only rediscovered what had been very well known thousands of years earlier. Many of these maps must have been composed long before Hipparchus. But it is not possible to see how they could have been drawn as accurately as they were unless trigonometry was used. (See Note 7.)
We have additional confirmation that the Piri Re’is projection was based on Eratosthenes’ estimate of the size of the earth. The Greeks had a measure of length, which they called the stadium. Greek writers, therefore, give distances in stadia. Our problem has been that they never defined this measure of length. We have no definite idea, therefore, of what the stadium was in terms of feet or meters. Estimates have varied from about 350 feet to over 600. Further, we have no reason to even suppose that the stadium had a standard length. It may have differed in different Greek states and also from century to century.
A great authority on the history of science, the late Dr. George Sarton of Harvard, devoted much attention to trying to estimate the length of the stadium used by Eratosthenes himself at Alexandria in the 3rd Century B.C. He concluded that the “Eratosthenian stadium” amounted to 559 feet (184: l05). 
The solution of the Piri Re’is projection has enabled us to check this. Presumably, it proves the amount of the overestimate of the earth’s circumference to be 4½ per cent (or very nearly that). Eratosthenes gave the circumference of the earth as 252,000 stadia. We checked the length of his stadium by taking the true mean circumference of the earth (24,800 miles), increasing this by 4½ per cent, turning the product into feet, and dividing the result by 252,000. We got a stadium 547 feet long.
Now, if we compare our result with that of Sarton, we see that there is a difference of only 12 feet, or about 2 per cent. It would seem—again by hindsight that we could have saved all our trouble by merely adopting Eratosthenes’ circumference and Sarton’s stadium. We could then have drawn a grid so nearly like the one we have that the naked eye could not have detected the difference.
The next stage, which came very late, was our realization that if Eratosthenes’ estimate of the circumference of the earth was used for drawing Piri Re’is’ source map, and if it was 4½ per cent off, then the positions we had found by trigonometry for the five projection points on the map were somewhat in error both in latitude and longitude. It was now necessary to redraw the whole grid to correct it for the error of Eratosthenes. We found that this resulted in reducing all the longitude errors until they nearly vanished.
This was a startling development. It could only mean that the Greek geographers of Alexandria, when they prepared their world map using the circumference of Eratosthenes, had in front of them source maps that had been drawn without the Eratosthenian error, that is, apparently without any discernible error at all. We shall see further evidence of this, evidence suggesting that the people who originated the maps possessed a more advanced science than that of the Greeks.
But now another perplexing problem appeared. The reduction of the longitude errors left latitude errors that averaged considerably larger. Since accurate longitude is much more difficult to find than accurate latitude, this was not reasonable. There had to be some further undetected error in our projection.
We started looking for this error, and we found one. That is, we found an error. It was not quite the right one; it did not solve our problem, but it helped us on the way. As already mentioned, we had found the positions of the five projection points by laying out a line first from the center of the projection to the intersection of the circle with the line on the Piri Re’is map running horizontally through the middle projection point, Point III, assuming this to be the equator. We had used this assumed equator as our base line for latitude. (See Fig. 15.)
When we laid out the projection in this way, we had not yet realized that the mapmaker was much more likely to have drawn his first radius from the center of the map directly to the pole and not to the equator. (See Fig. 17.) If he did this, since his length for the degree was wrong, then his equator must be off a number of degrees. This required new calculations, and still another grid.
At first, this new grid seemed to make matters worse, especially on the coast of Africa. The equator seemed to pass too near the Guinea coast by approximately five degrees. My heart sank when this result became apparent, but I am thankful that I persisted in redrawing the grid despite the apparent increase in the errors, for the result was a discovery of the very greatest importance.
At first I thought that the African coast (and that of Europe) had simply been wrongly placed too far south on the projection. But I soon saw that if the African coast appeared too far south on the corrected projection, the French coast was in more correct latitude than before. There was simply, I first concluded, an error in scale. Piri Re’is, or the ancient mapmaker, had used too large a scale for Europe and Africa. But why, in that case, though latitudes were thrown out, did longitudes remain correct?
I finally decided to construct an empirical scale for the whole coast from the Gulf of Guinea to Brest to see how accurate the latitudes were relative to one another. The result showed that the latitude errors along the coasts were minor. It was obvious that the original mapmakers had observed their latitudes extremely well. From this it became apparent that those who had originally drawn this map of these coasts had used a different length for the degree of latitude than for the degree of longitude. In other words, the geographers who designed the square portolan grid for which we had discovered the trigonometric solution, had apparently applied their projection to maps that had originally been drawn with another projection.
What kind of projection was it? Obviously it was one that took account of the fact that, northward and southward from the equator, the degree of longitude in fact diminished in length as the meridians drew closer toward the poles. It is possible to represent this by curving the meridians, and we see this done on many modern maps. It is also possible to represent this by keeping the meridians straight and spacing the parallels of latitude farther and farther apart as the distance from the equator increases. The essential point is to maintain the ratio between the lengths of the degrees of latitude and longitude at every point on the earth’s surface.
Geographers will, of course, instantly recognize the projection I have described here. It is the Mercator projection, supposedly invented by Gerard Mercator and used by him in his Atlas of 1569 (Note 5). For a time we considered the possibility that this projection might have been invented in ancient times, forgotten, and then rediscovered in the 16th Century by Mercator (Note 15). Further investigation showed that the device of spreading the parallels was found on other maps, which will be discussed below.
I was very reluctant to accept without further proof the suggestion that the Mercator projection (in the full meaning of that term) had been known in ancient times. I considered the possibility that the difference in the length of the degree of latitude on the Piri Re’is Map might be arbitrary. That is, I thought it possible that the mapmaker, aware of the curvature of the earth, but unable to take account of it as is done in the Mercator projection by spherical trigonometry, had simply adopted a mean length for the degree of latitude, and applied this length over the whole map without changing the length progressively with each degree from the equator.
Strangely enough, shortly after this, I found that, according to Nordenskiold, this is precisely what Ptolemy had done on his maps (see Note 9). In Nordenskiold’s comparison of the maps of the Mediterranean and Black Sea regions as drawn by Ptolemy and as shown on the Dulcert Portolano (Fig. 4), we see that he has drawn the lines of Ptolemy’s projection in this way. This is, of course, another indication of the ancient origin of Piri Re’is’ source map.
This is not quite the end of the story. We shall see, in subsequent consideration of the De Canerio Map of 1502, that the oblong grid, used by Ptolemy and found on the Piri Re’is Map, has its origin in an ancient use of spherical trigonometry.
These successive discoveries finally enabled us to draw a modern grid for most of the Piri Re’is map, as shown in Figure 18.
- Maps in this book, except where it is otherwise indicated, are taken from the Vatican Atlas (139) or that of Nordenskiold (146).
- The term “portolan” or “portolano” apparently derived from the purpose of the sea charts, which was to guide navigators from port to port.
- Marinus of Tyre lived in the 2nd Century A.D. and was the predecessor of the geographer Claudius Ptolemy.
- The Arabs, famous for their scientific achievements in the Early Middle Ages, apparently could not have drawn them either. Their maps are less accurate than those of Ptolemy. (See Fig. 5.)
- My translation from the French.
- These angles could also be bisected, if desired, resulting in thirty-two points on the periphery, at angles of 11¼°.
- See Note 4 and Note 5.
- The complete map included Africa and Asia. It was, according to Piri Re’is, a map “of the seven seas” (see Note 3). In addition to the loss of the eastern part, there was also originally a northern section, which was detached and lost. I am indebted to Dr. Alexander Vietor, of Yale, for this observation.
- Since the earth is round, and the portolan design was apparently based on a flat projection (that is, apparently on plane geometry) which could not take account of the spherical surface, the parallel meridians would deviate further and further from True North the farther they were removed from the center of the map. The portolan design could compensate for this, however, as we shall see in the next chapter, by using different Norths.
- Interest in the map was so keen that the students would come to my apartment in the evening, and sometimes argue until the small hours.
- See the Bibliography, Nos. 89, 116, 143, 179, 199, 200, 223.
- For the final determinations of these positions see Figure 18. For the calculations see Appendix.
- See Note 6 for a comparison of the results in one case.
- See Acknowledgments.
- See also Strachan’s discussion, Note 8.
- However, a knowledge of plane trigonometry has been attributed to Appolonius, an earlier Greek scientist, by Van der Waerden (216). The date of its origin appears, then, unknown.
- That is, there were about 9.45 Eratosthenian stadia to a mile of 5,280 feet, which figures out to 558.88 feet per stadium.