[061129gmd1200]{.underline}

John Hébert:

Good afternoon, and welcome to this special presentation on a segment of
the Waldseemüller map. I\'m John Hébert, chief of the Geography and Map
Division. And down in our way, and I hope in the Library \[of
Congress\], we\'re all excited about a map that we acquired officially
in 2003 that has caused a great deal of excitement in the field; none
more so than that generated by some of the research of the man who is
about ready to talk to you, revealing information about the map and
raising questions about the map that had not been addressed since it was
uncovered some 100 years ago. John Hessler works in the Collection
Management Team of the Geography and Map Division, and I\'ll stop right
there because anyone who\'s followed his research knows that this is
really his excitement, and what he can really contribute. It\'s a great
pleasure to bring to you John Hessler. John.

\[applause\]

John Hessler:

Thank you. In way of introduction to the map, for those who don\'t know,
for over 300 years, arguably two of the most import maps in the history
of cartography \-- the 1507 and the 1516 world maps by Martin
Waldseemüller \-- sat on a shelf in a castle named Wolfegg Castle in
\[Baden-\] Württemberg, Germany. They were discovered there in 1901 in a
single codex by a Jesuit priest, Josef Fischer. Now, the maps were bound
into the single codex by a Nuremberg astronomer and alchemist named
Johann\[es\] Schöner. And Schöner annotated these maps, and I\'ll talk a
little bit about Schöner\'s annotations later on in the presentation.
But the maps were constructed by a small group of cosmographers from
Saint-Dié, France, and the group was composed of three principle people.
One\'s name was Gautier Lud, and he was the canon of the Cathedral
School in Saint-Dié. The other was Mathias Ringman.

Mathias Ringman was born in Alsace. He was trained as an astronomer in
Paris, and he learned Greek in Italy. The third, Martin Waldseemüller,
our friend up here, is basically credited with the creation and the
structure of the way the maps look. Now, there\'s very little
information, very little documentary information that has survived on
any of these two maps, and that has forced different approaches to their
history. And one of the approaches that I\'m going to take is a very
computational approach. First, and I\'m going to try to walk a very thin
line here today between talking to you about the history of this map and
what it means to the history of cartography, and explaining to you some
very, very complex transformational algorithms that I\'ve been using to
study the map structure and what it looks like. I\'m going to begin with
the 1507 world map.

Now, the 1507 world map \-- this is a composite of it \-- is of course
now owned by the Library of Congress, as John pointed out. But the map
has some difficult historical problems associated with it. And two that
I\'m not going to talk about are the sections around the Caspian Sea,
and the area which is called Cathay, which is all the way over where
China and Japan would be. The principle thing I\'m going to talk to you
about today is the outline of South America on this map. Now, the
outline of South America on this map is problematic for several reasons;
first of all, it shows the Pacific Ocean. Now, the Pacific Ocean was not
going to be discovered for several years.

Balboa hadn't crossed the strait, hadn't crossed the isthmus in 1507,
nor had Magellan yet come around the outside \-- the bottom of the map
to discover the Pacific, so no one at this point in time knows there is
a Pacific Ocean. But yet, Waldseemüller portrays the Pacific Ocean on
the map. Now, he talks about his portrayal of the Pacific Ocean in a
book that he wrote called the "Cosmographiae Introductio." And the
"Cosmographiae Introductio" is an extremely important book for the
history of the Americas, and mostly because on the title page it
mentions two maps, and it mentions something \-- it\'s called the Plano,
and the other called the Solido. And the Plano is the 1507 world map
that I just showed you, and the Solido is a globe which was designed by
Waldseemüller.

Now, in this text Waldseemüller describes exactly what he is portraying
on the 1507 world map. This book was in fact meant to be a guidebook to
the 1507 world map. So in one section he says the Earth is now known to
be divided into four parts; the first three are continents and the
fourth part is an island because it has been found to be surrounded by
water on all sides. Now, we don\'t really know whether Waldseemüller had
any empirical evidence, or whether it was a lucky guess when he drew the
Pacific Ocean on the, in 1507 on the world map the way he did. But in
his book he basically tells us that is in fact an island, and it is
surrounded by water. And I think the semantics of the Latin that
Waldseemüller uses here are extremely important. And they imply some
sort of evidence; he uses the words \"now known\" and \"has been
found.\"

Now, those two words seem to imply that he\'s actually working with some
sort of data; he\'s not guessing. Now also on the map itself, in one of
the lower left hand panels, there's this piece of text. And
Waldseemüller basically describes how he did the map, and what he used
for the map. He basically tells us he used several voyages of both
Columbus and Vespucci, and that he used everything, data that he thought
was best to basically carefully draw a map, and to furnish true and
precise geographical knowledge. So these two things kind of lead you to
believe that Waldseemüller was attempting to construct a world map \--

\[laughter\]

\-- that was actually based on real data. Now, if we take a look \--
this is the top of the South American section, and this is the bottom of
the South American section. You\'ll see down at the bottom that the map
continues into the border.

Now, no one is really, knows why Waldseemüller continued this piece down
through the border, but certain people have speculated that it was just
sort of an indication, his way of showing that this continues, that it
continues down further.

So, Waldseemüller was not the only person who portrayed South America as
an island, and a passage that went around South America long before
either Magellan or Balboa had discovered the Pacific Ocean. This is the
globe, a 1515 globe done by Johann\[es\] Schöner. Now, this globe is
done by the very person who bound the 1507 and 1516 maps into the codex
that was found by Fischer. And you\'ll see at the bottom, there is in
fact a passage that goes around South America. And again, we don\'t know
what the source of this was; people have speculated that Schöner used
Waldseemüller as a source. There are other things that both of these
people could have used. There\'s a rare German text called the "Newen
Zeytung" which is, talks about a Portuguese expedition that sails
through the straits and then gets blown back. There\'s some scholarly
debate on whether that was an expedition that happened before or after
1507, but there\'s a lot of things going on at the time that could lead
us to think that Waldseemüller might have had some sort of empirical
evidence for his representation of the coast of South America.

Before I go on to the models which I\'m going to show you about how I\'m
going to kind of give you some notion of what high probability there is
that Waldseemüller did have that, I just want to digress for a moment on
a couple of conceptual issues that I want you to think about as we go
along, and these are kind of important.

There\'s an issue that is occurring in the early 16th century which is
called demonstrative regress. And what demonstrative regress is, is this
question in the history of science of whether the people who were doing
cosmography and doing astronomy and doing cartography were actually
attempting to portray realism, or were they simply just portraying the
symbolic nature of their world? Were they simply just adjusting slightly
to ancient text, were they reacting to ancient text, were they simply
repeating ancient texts? And this is an important issue here, because my
work is going to very much depend on the notion that they\'re doing
realistic work, as opposed to instrumental work. They\'re not simply
making symbolic gestures, they\'re attempting to represent the reality
of their environment and their world.

Now I\'m going to talk about several different techniques of geometric
analysis on both maps; both the 1507 and 1516. One of them is polynomial
warping, the other is thin-plate splines. And on the 1507 I\'m going to
do a little bit of projection modeling to show you how Waldseemüller
came up with the projection structure that he did, and what it really
shows us about Waldseemüller\'s view of the world and what he knew. I\'m
going to start with modeling the projection. And in 1507 \-- for those
of you who don\'t know what a map projection is, if you take a spherical
globe and you want to make a map out of it, you\'ve got to make it flat.
So if you consider taking an orange and attempting to take the peel off
and then flatten it, in order to flatten it you have to tear it or
introduce some sort of distortion. So a map projection is simply a way
of keeping track of what that distortion is. And in Waldseemüller\'s
time, in 1507, he really only had two realistic models to use. And those
were both defined by Ptolemy, a 2nd century geographer who was very,
very influential and important well into the Renaissance period, and
really who Waldseemüller\'s work is somewhat reacting against.

Now, these are two of Ptolemy\'s projections: the first one on top and
the second one below. And the problem that Renaissance cartographers
have with these projections is that they only represent a small portion
of the known world. In fact, Ptolemy\'s projections only do a 180-degree
sweep, and only 10 degrees below the equator. So, to make these larger
was a problem. If you begin trying to map a larger world on this, things
get squashed into the bottom and they just don\'t look right. And most
of the cosmographers and cartographers of the period were attempting to
follow Ptolemy in getting an esthetic representation of the world that
sort of still gave you some notion of its sphericity.

Now, there were many, many people at this time working on this problem.
This particular manuscript is from the "Codex Ambrosiano." And this is
Leonardo da Vinci\'s manuscript, and he\'s working on the same series of
projections and the same type of projections that Waldseemüller was
working on, at the same time. The thing up in the corner is a modern
picture of Leonardo\'s projection.

Now, Waldseemüller did not do a very good job in making a smooth
projection. His projection is choppy, it\'s segmented, it doesn\'t look
right to the modern viewer. It wasn\'t until 1511 \-- and this is a map
by Bernard Sylvanus \-- that the meridians and the lines of the
parallels are smooth, and they really kind of look like a modern map. So
this was in 1511. Now, Waldseemüller was attempting to, like I said,
expand the world to include the discoveries of Vespucci and Columbus,
which is a greatly expanded world from the Ptolemaic models.

And in 1514 a man named Johann\[es\] Werner published a book which was a
commentary on the first book of Ptolemy, which is where the projections
are located. And I was doing a translation of this text for a paper I
was working on, and I came upon these two propositions in this text. And
after I read these propositions over I realized that I could use
Werner\'s work. I could construct a mathematical model based on this
which would allow me to actually redo Waldseemüller\'s projection in a
way that I could actually calculate numbers. I could actually see the
way it moved and the way it changed the shapes of the continents. So
what I did at this point is I did a mathematical model of
Waldseemüller\'s projection using Werner\'s propositions, which are very
geometric propositions, but they were very easy to convert to analytic
geometry. And what I did at that point kind of surprised me. I basically
took the modern map of South America and I projected it on
Waldseemüller\'s model projection.

So on the left, what we have here is we have Waldseemüller\'s projection
model, the graticule or the grid with the modern South America mapped on
it, next to Waldseemüller\'s South America. Now, a lot of scholars have
looked at the linear features. If you look at the world, the old world
those linear features on the side as things that Waldseemüller did not
know, that this was an area that he did not really know what was going
on there, so he simply drew lines. But if you model the modern coast of
South America on this projection, you\'ll see those same linear features
appear, and that the two are quite close. They\'re quite remarkable if
you think that that\'s a 1507 rendition of a coastline that no one was
supposed to have been on the west coast of, compared to this particular,
compared to the modern coastline. Now, this doesn\'t really tell us
much. It\'s a beautiful picture, it\'s very cute, and, you know, it kind
of started me off on this sort of line of research.

Now, what I decided to do at this point was to try and really find out
how close Waldseemüller was; how good was his representation. And
there's a few techniques available to do this. And most of these
techniques were invented back in the 1970s by a Santa Barbara professor
named Waldo Tobler, who did a lot of early computer mapping and
comparison of ancient and modern materials. And some of his, some of his
techniques are still being used. And I\'m going to begin by talking
about a very simple one, which is polynomial warping. And then \-- I\'m
not going to do any elastic bidimensional grid regression, but I\'m then
going to talk about thin-plate spline. And I\'m going to use the
thin-plate spline technique to compare the 1507 and 1516 maps, and then
show you how they are based on very, very different geographic sources,
and how Waldseemüller changed his mind of what the coasts of these areas
looked like in that, in those intervening periods.

Thin plate splines are actually used in medical imaging. Basically it's
CT scans; when you get your base scan, and then the next scan they use a
thin-plate spline technique in order to merge the two together to see if
there are any changes. It\'s never been applied to historical
cartography before. And I\'ve gotten a lot of praise for it and I\'ve
also gotten a lot of flak for it, so the jury\'s still out on whether
the technique is really working or not. The notion that we\'re going to
have when we do these transformations is we are going to take
homologies. And what I mean by that is we\'re going to take points on
one map, usually the old map, and the modern map that we know are
exactly the same. And I\'m going to take those values and I\'m going to
squeeze them together and merge them. And I\'m going to keep track of
the error induced in the modern map when I try and transform the old map
to it, and vice versa.

And we\'re going to start out with something called polynomial warping.
And what polynomial warping is \-- and I\'ll explain it in a little bit
more detail as I show some examples, but basically I\'m going to take
the image of the old map and I\'m going to stretch it and I\'m going to
turn it. I\'m going to twist it, I\'m going to bend it, and then I\'m
going to just basically shove it on top of the new image. And I\'m going
to figure out at that point how much error I\'m inducing into the old
map in order to match the new map.

Now, the warping process is parametric. In other words, the X and Y
coordinates of those points \-- in other words the latitude and
longitude \-- are different. Okay, I\'m going to use different equations
to do each one. So I'm going to move, say, laterally first and then
vertically \-- and then basically I\'m going to, as I said, move these
two things on top of one another. And while I\'m doing that I\'m going
to keep track of certain vectors on the surface, in order to basically
tell how much warping and how much motion I\'m inducing into these
particular, into these particular images. Now, this is a warped image of
the bottom sheet of South America. And I\'m showing you this because
this is called an affine transformation; it\'s linear. The only thing
I\'ve done here is I\'ve rotated points and I\'ve moved points
laterally, I haven\'t done any actual warping. There're two things here;
there\'s a linear side, and there\'s going to be a nonlinear side. The
nonlinear side is going to be more complex, and it\'s also going to be
more interesting. But this is an affine transformation. This is a
nonlinear transformation. As we move up in degrees of the polynomials
that I showed you \-- in other words, as I add more terms; those terms
go out to infinity if I want \-- I induce more and more twisting and
turning into the map images.

And you can see, now that I\'ve turned this it\'s becoming a little bit
closer to looking like the real South America, as I\'m sort of twisting
and turning this thing. Now, when I begin comparing it with the real
coastlines, I can begin to start getting real numbers about how close
Waldseemüller really was. And when I do this for the top of South
America sheet and when I do it from the bottom of South America sheet, I
get a correlation for the top sheet of .72 and of the bottom sheet of
.69. Now, that\'s out of one. One would be perfect zero would be
there\'s nothing matching. Now, those two correlations are not very
high, but in a sort of random experiment, I picked five people and I
gave them the east coast of South America and a grid, and I said, "Draw
South America on this for me as you would know it." And no one\'s
correlations came up as close as Waldseemüller\'s correlations did.

\[laughter\]

So, modern people, knowing what those are, haven't been able to do it.

So, this thing is very inconclusive at this point. I have a very
beautiful picture that shows these linear features, I\'ve got
correlations that suggest maybe there\'s something going on here, but
nothing enough for me to say, \"Wow, this is great, I can say that
Waldseemüller had empirical information.\" So I started searching around
for something different. The unfortunate part of the polynomial warping
process \-- as you saw, it was actually transforming the entire sheet.
So those correlations are based on everything on that sheet as it\'s
being correlated against the known map, so there\'s a lot of error
induced in that. So I thought for a long time about, you know, what else
could I use to prove that Waldseemüller may have had some, some
empirical information? And I was doing a translation of the
"Cosmographiae Introductio," and I came upon this chart. And most
scholars in the history of cartography see a chart like this and they
turn the page.

I see a chart like this and I go right to it, because it has numbers and
I think I can do something with it. And what Waldseemüller is showing us
here is he\'s showing us the distances at each latitude for a degree.
And I found that interesting, because when I graft Waldseemüller\'s
values against the known values \-- the known values are the red line,
and Waldseemüller\'s are the blue and the green line there \-- they are
extremely close. So what I thought to myself at that point was, well,
obviously Waldseemüller knows how to measure distances along lines of
latitude very accurately. So in thinking that I said, "What about the
width of South America? If he can measure lines along latitude lines,
which are the parallels that run across the bottom of the map, how good
was he at measuring the width of South America?" It turns out he was
very good at it. And this is a polynomial regression curve of the coast
of South America on a modern map, and on a, on Waldseemüller\'s map. And
there are a couple of interesting features. The Waldseemüller one is the
green line, and the real South America is the blue line.

And there are a couple things we need to look at here. One, the shape of
the curve is an indication of how close Waldseemüller was to the real
coastline. And you\'ll see that the shape of this curve \-- they\'re
pretty good, they\'re pretty close. The peaks are kind of lining up at
the same point. The width of South America at its widest point is at the
exact place. The points that the two curves intersect are points of 100
percent correlation; in other words, where Waldseemüller and the map
agree identically. It\'s interesting where they agree identically. They
agree one degree off the equator identically. They agree at the place
where Arnica, Chile is, which is the place where South America makes its
westward turn, the sort of bump; it\'s right at that point. And they are
just two or three degrees off at the Tropic of Capricorn, so, kind of
interesting landmarks for extremely high correlations.

So, basically we have really these three things that have led me to
speculate that there was a possibility that Waldseemüller has had
empirical information. We have fairly high correlations and really good
inflection point behaviors on these regression curves. We have
projection modeling which shows linear features in Waldseemüller\'s map
and the modern map based on Waldseemüller\'s projection. And we've got
nonrandom correlations on each of the sheets. So, based on that I
basically concluded that Waldseemüller must have had some indication.
It\'s possible he guessed right. It\'s possible, but at least in my
opinion he didn\'t.

Now I\'m going to talk to you a little bit about, now, the "Carta
Marina." And after I\'m done with the "Carta Marina" I\'m going to take
all the information that I\'ve gathered from both of these maps, and
compare the two maps and kind of speculate on the difference in the
geographical sources that Waldseemüller had.

The "Carta Marina" is a very, very different map. It was done in 1516.
When it was originally bound in, there was a sheet missing. That is
sheet number 10, which is this sheet here, and this is Johann\[es\]
Schöner\'s manuscript drawing of that sheet, which was bound into the
"Carta Marina." You\'ll see there is a series of red grid lines over
this map; these are Schöner\'s annotations and red grid lines. The 1507
map has the same red grid lines over certain areas of the map. This map
is different in a number of respects. One, its projection; this map
appears to be a portolan chart. A portolan chart is a sailing chart;
generally it was done of the Mediterranean in the early period. But they
are different in the fact that they've these compass roses, and the
latitude and longitude lines tend not to exist on them. They tend to be
compass settings. And it\'s been speculated by scholars they were used
for navigation. There\'s some disagreement about that \-- it\'s pretty
hard to sail along a straight line \-- but there are some manuscripts
that show us how to tack back and forth. But it\'s a very, very
different map, and you can see also that it\'s a very, very accurate
map. This is that sheet, Schöner\'s drawing of the sheet number 10, with
on top of it the modern Africa and the South American coast.

And you can see it\'s fairly close. In fact, the "Carta Marina" is much
more of an accurate map in the sense of external accuracies. Internal
accuracy is a little bit different, but it\'s a much better map than the
1507. The 1507 being constructed both of Ptolemaic sources and of modern
sources. This one appearing to be constructed mostly of modern sources.
Now look at the, to prove that the "Carta Marina" was made from a
portolan chart, there were several things that I needed to do. And I
started by looking at, how would I model the "Carta Marina" and improve
on my models on the Waldseemüller map? And I was looking for
mathematical techniques that were much more powerful than the polynomial
warping. And I started really looking at medical imaging and how medical
imaging was done, and thin-plate splines are the place that I wound up.
And they are extremely powerful. I\'m only going to explain them in the
briefest of terms because the algorithms are very complex, but I just
want to get you an idea of how they work.

Now, what a thin-plate spline is, is basically I take all the points on
the map that are homologies. In other words, so all the points on the
modern map and all the points on the old map that I believe are the
same. And what I\'m going to do is I\'m going to find some sort of
distance function. I\'m going to say that every one of these points is
somehow distant from the other point, and I\'m going to build a matrix
of all of these different distances of each of the points on both maps.
And this turns out to be a non-Euclidean metric, which means it\'s very
nonlinear; it doesn\'t have a linear form.

And what I\'m going to do with that is I\'m going to form several very
large matrices; the top one is a distance matrices, and the second one
is a point matrices. And what I do is a whole bunch of very complex
manipulations with these things. I\'ve solved this particular equation
right here, which is very important. The first three terms in that
equation define an affine transformation, which, if you recall in the
first part of the talk, the affine transformation was the linear
transformation. It was the one where there were only angles changed,
where there was only movement of the points in a linear fashion. There
were no non-linear motions or curves induced into the map.

The second part of this thing is the nonlinear portion. So, basically
what I have here is I have a real convenient solution to an equation for
the bringing together of these two maps, that gives me a separation of
both the nonlinear and linear forms of the error that I\'m inducing into
the map. So, basically what I\'m doing is I\'m taking these two sets of
points and I\'m squeezing them together. And I\'m inducing in the map
motion the error that you see on that right hand side. So if I have two
sets of points, one on each map, squeeze them together, I basically
induce an error grid.

Now, this is the Mediterranean sheet from the "Carta Marina," with an
error grid from this type of transformation overlaid on top of it. And
what this really shows is it really shows how different this particular
old map is from the original, the original modern map. If this was
perfectly aligned, all you would have would be a perfectly square grid.
So all the error is induced in the curvature, in the changes that this
grid has. And just for convenience, this is what it looks like without
the map superimposed.

And you\'ll see that where the points are there is more error or less
error depending on how much distance I had to move each of those points.

The difference between this and the technique I showed you before was,
before I was using statistical techniques in order to get a correlation
for a best fit for those two sets of points. In this sense I am placing
these two sets of points exactly on top of one another. There is no best
fit correlation, it is fitting and then I\'m looking at how much error
I\'m inducing into the map. Now, the reason I did this was because I
wanted to prove the "Carta Marina" was a portolan chart. I wanted to
prove that it was derived directly from a portolan chart. And there are
several things about portolan charts which we know throughout history
remain the same. One is that they are oriented to magnetic declination,
or in this case they have rotation. In other words, if you take a
particular map \-- maps are usually oriented toward the north. In order
to get a portolan chart to orient to standard north, nonmagnetic north,
you would have to rotate it. So all of the maps that we see that are
portolan charts all have some sort of rotation about them. And what
these are here, these are graphs of models of magnetic declination for
1400, 1500 and 1600.

These are how the magnetic declination lines \-- magnetic declinations
lines is \--how your compass arrow points \-- throughout these periods.
Now, using the thin-plate spline technique I can separate out the
rotation of whatever area I\'m studying. And when I separate that
rotation out I can put rotation isolines \-- in other words, lines of
constant rotation across the surface of the map I\'m studying. And what
this shows is, it shows a near perfect match to the magnetic declination
lines from around 1522, which, in the era of this technique \-- 1516,
1522 \-- I really don\'t know where to draw the line of what\'s an
accurate rendition and what isn\'t, but it is very, very close. So this
particular map, this "Carta Marina," which is a printed map, appears to
have been derived directly from portolan charts. And I\'ve done this at
other various areas on the map, and I get a fairly good fit matching it
with magnetic declination models. Now, an offshoot of this was, as I
said, that portolan charts all have rotations.

So because I really wanted to see what the difference between
Waldseemüller 's map was, and what the 1516 \-- or the difference
between the 1507 and the 1516 maps, I took a whole bunch of portolan
charts. And I calculated rotation, and that line is the red line. So
basically, through history of portolan charts I calculated the rotation
and graphed the red line; you\'ll see somewhere around 1600 that that
rotation drops off radically. The other thing I did is I took
paleomagnetic data from Sicily. Now, paleomagnetic data is based on lava
flows; the lava flows at a certain time. As it cools, the magnetic field
of the Earth orients the lattice of the iron in a certain direction, so
one can look at historical magnetic declination. And then I graphed
that. So you can see the portolan charts, that the rotation of portolan
charts matches very, very closely the history of magnetic declination in
Europe. And this is kind of a new finding; people have speculated about
this, and this is sort of just an offshoot of this; that in fact
portolan charts do orient themselves to magnetic declination, as
opposed, as opposed to true north.

Now, the 1507 map\'s rotation is well off this curve, and hence I\'ve
concluded that it\'s not based on a portolan source. Its rotation is
very small, and it appears to be based purely on Ptolemaic sources; at
least on Ptolemaic models. The 1516 map, however, is in fact based on
portolan charts, which is something that again had been speculated
about, but had never actually been proved in any substantial way, simply
because there\'s very little documentary evidence to back any of those
notions up.

The other thing I can do with the thin-plate splines \-- and not to bore
the heck out of you with this at this point \-- is in the separation,
and the ability to separate those affine and nonlinear parts very well,
and to be able to separate those parts in the matrix calculations. I can
calculate the error in longitude or latitude at any point on the surface
of the map. And this is a stretch through the Mediterranean. And this is
just a three-dimensional surface with all the points that I used and how
much error is induced in them. And I can do this for Ptolemy, I can do
this for the 1507 map, I can do this for the 1516 map, and it yields me
basically this curve.

And what this shows me is this is the error in longitude for three maps.
The blue line is Ptolemaic, just your standard Ptolemaic Codex \-- I
used the Ulm 1482 \-- the other two lines are both the Waldseemüller \--
the 1507 is the red line swooping upward, and the 1516 is the other
line. And this sort of proves my point; that in fact, that the 1507 map
was based on purely Ptolemaic sources. Its structure is purely
Ptolemaic. Its error across the Mediterranean is Ptolemaic, and comes
basically from map models from the 2nd century. The "Carta Marina" is of
a very different ilk, and has a much more modern \-- if you graph
Mercator\'s error, you will get curves much, much more like that bottom
curve. So, really what you have is you have a technique that\'s kind of
allowed me to look inside the map in a way that really no one has ever
looked before, and to compare the maps in a way that no one has ever
compared them before.

I cannot draw any conclusions based on documentary evidence, because
there really is none, but these techniques as I\'ve continued to improve
them have given me much, much greater results, to the point now where I
am able to take error on particular portolan charts, and I can now
speculate of two or three particular sources for a map. The 1507, I\'ve
been able to look at error profiles in various regions of the 1507 map,
and they\'re almost like a fingerprint because I can compare them to
other maps and see where the error is spliced together, and how these
maps have been spliced together from different geographical sources.

Now, there are a lot of sources of error in doing this, and it\'s really
important to be careful that we don't over, have much more confidence in
our numbers than we should. The scale of these maps, the fact that they
are world maps makes our calculations very difficult. It makes or
calculates to the point where you know your confidence level in them is
low, but as you continue and as you get more and more numbers that seem
to match up, you get a little bit more confidence.

And the maps are all drawn from composite sources. There\'s no one
geographical source, so the error skips around all over the place, and
there's no smooth error curves that we can look at. And the other is
that the projections are not constructed smoothly; they\'re not
constructed like modern maps. They\'re not constructed using very smooth
equations which are differentiable everywhere. They\'ve got
discontinuities; they\'re not smooth in places. And it induces much,
much more error into the calculations than you would have if you were
working with modern materials. So, one has to be very careful before one
makes any real judgments about the work. Now, the problem with all of
this, the problem with both of these maps is that the people who made
them appear on the historical scene somewhere around 1505, and they
disappear by 1518. They leave almost nothing behind.

And the only thing that I\'ve been able to add to this debate, and to
give you is basically probabilities in correlations. Modern historians
are in a very difficult situation. There\'s a lot more people working on
the maps than have been in the past. A lot of the publicity through the
Library of Congress and its encasement of the map and its exhibition of
the map has generated more interest in scholarly circles. There are
several really important conferences next year on the map, and it is the
500^th^ anniversary of the 1507 map. But to a certain extent, historians
are in a situation which really is best said by Ringman. Ringman, in a
dedicatory poem in the "Cosmographiae Introductio," I think sums up our
situation best, which is, \"All this we have given you, but of ourselves
we are silent.\" So, thank you.

\[applause\]

\[end of transcript\]