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Caro Joaquim Alves Gaspar: Viva :-) Many thanks for your comments relative to your interesting thesis and contents there related to the discussion on the measurement of a degree of longitude, and the dispute matters between Portugal and Spain. Well first of all it is obvious that the longitude degree-scale translated into leagues-length of curved surface was not the same, between Portugal and Spain - otherwise they would match nicely in what comes to accept where the Anti-Tordesilhas line would exactly pass by. This was not the case at all - Spain and Portugal claimed different measurements and thus had a very hard dispute to solve. As you say in your thesis - pages 41, 42 - the relevant matter is not exactly the measurement/definition of a league in itself, but how many leagues exist in a longitude degree. I believe these thesis pages are the most relevant ones for the discussion here. Your argument there assumes that the distance between Lisbon and the Molucas islands is known - then it follows a/your correct discussion for the Portuguese more leagues per degree being correct, vs. the smaller Spanish leagues per degree, that would sooner grasp too many degrees, placing such distance to the Molucas already in Spanish Territory, and the Anti-Tordesilhas line too close (say) to Malacca instead (too far West of the Molucas then...). This in the scenario where you measure from East to West, from Lisbon along Europe, Asia - to the other side of the world. In this scenario, the world would then have a bigger perimeter for the Portuguese - Earth being smaller for the Spanish. This was the orientation of my first email, stating so. If you make the measurements however in the opposite direction, from Lisbon towards the Atlantic, then Pacific, etc. - then more leagues per degree help the Spanish and not the Portuguese. Thus the text from David Owens that Joel Kovarsky copied in a former email, refers to more leagues per degree for the Spanish instead. In that case the Earth would be bigger for Spain - and not for Portugal. So my correction to my first email (that perhaps I should not have corrected, but instead explain better the "axioms" of the measurements in my first email) assumes a measurement from East to West, and Owens reference to a longer degree for the Spanish - unlike what you refer in your pages Joaquim. However I must say that I always learned the problem as you state it Joaquim - and that was my first email information content indeed. One needs to be careful though about what quantity is fixed - if the distance Lisbon-Molucas is fixed and accepted both by Portugal and Spain then your pages are correct (doing longitude measurements from Lisbon-West to the Molucas at East of Lisbon). However if that distance is not fixed, if a dispute also exists about how big is indeed the Earth's equatorial perimeter...then one can end up in the exact opposite situation when measuring from Lisbon at East to the Molucas at West (if taking a longer Spanish degree instead, as suggested in the text of Owens/Furlong, etc.). In fact what I recalled was the Spanish degree having less leagues than the Portuguese one for purposes of Molucas/Tordesilhas definition (with the assumption of fixing such distance Lisbon-Molucas, measuring longitude from West to East). However indeed the discussion can be very confusing if one does not clearly defines what is fixed to start the problem with. I think the only thing that could be peacefully fixed is to divide the world in 2 - so 180 degrees for Portugal 180 degrees for Spain. I find it harder to have agreements on the exact distance to the Molucas (thus on the Earth's equatorial perimeter)...and thus also on the length of a degree...assuming both parts could easily agree on making the measurements also from West to East. Indeed King Dom João III knew quite well about how hard (and potentially confusing as it was since Tordesilhas) it would be to convince Spain (or vice-versa) of an absolute geographical-longitude-distance "scale". Thus Portugal goes for the re-negotiations of Tordesilhas (that ended up in the Treaty of Saragossa of 1529) also with the strong argument of older presence in the Molucas/Spice islands area. Portugal was there already for a decade, before the Spanish arrived (in Magellan-lead expedition). The Portuguese Fort of Ternate Island is then built in 1522. Later Spain built a fort just in front, in the neighbor island of Tidore in 1526, that the Portuguese attack at once. If there was no difference at all - as you suggest/prove in your thesis to be the case in the Atlantic latitude degree length - in the longitude degrees of both nations, then there would be no open issues left in the Saragossa Treaty. Yet these issues were indeed left open. Future cartographers/astronomers/cosmographers would settle in in the future within joint Portuguese-Spanish commissions what the real length of a longitude degree was! In fact Portugal was so committed to establish this degree issue clearly that another legal clause ended up in the Treaty of Saragossa - namely, if Portugal turns up to be correct, and not Spain, then all the huge fortune (that almost bankrupted Portugal by then, with the Molucas tax being implemented all over the nation, etc.) of 350 000 golden ducados paid to Spain is to be returned to Portugal !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Nobody accepts to introduce an extra tax to its own people to pay a huge fortune for some islands in the other side of the world, believing they would already belong to Portugal in first place - without making absolutely sure to have such a legal clause guarantee included in the Treaty, once the degree measurement matter would be settled. In favor of the Portuguese - so King Dom João III believed (correctly as we know today - we here including the Spanish, finally). So categorically, obviously, there was a difference - has many other academic papers refer to. Just out of memory I recall now Cmdt. Estacio dos Reis, e,g.. I appreciate your reference to the similarity of the latitude degree used in the Atlantic between Portuguese and Spanish sailors - as you refer to pages 190 - 192 of your thesis. However those pages are not so relevant for the discussion at matter here as are indeed pages 40-41 of your thesis: where the longitude matter is addressed better. In fact it was proved later that the Portuguese were correct - the Molucas were inside the Portuguese Tordesilhas territory and the Anti-Tordesilhas line did not needed to be pushed farthe east some 14 degrees as a result of the Saragossa Treaty. Nevertheless Spain never returned the huge fortune to Portugal - how kind of them... bahh... Well you may wonder why the Portuguese King pushed the line so much (14 degrees!!) further East, when the Molucas (as we know today) were inside Portugal territory for a couple of degrees only...??!!! This may hint for what happened in the pushing of the Tordesilhas line before further West to include more of Brazil. So maybe for the Portuguese the Saragossa Treaty also had the obvious knowledge of Australia included in the game - and not just the Molucas/Moluccas Islands... So nobody can state that the Portuguese and Spanish longitude degrees were the same - since the fact is that Spain was wrong. If the practical use of navigational charts by both Portuguese and Spanish sailors in the Atlantic (as your thesis refers to) may be similar - it seems on the political level, on the planetary scale of deciding the Portuguese and Spanish hemisphere, things do not match indeed. One further topic/changing discussion a bit: I understand your thesis can conclude that somehow based on the magnetic declination issues, some maps of Greenland and Brazil ended up inside the Portuguese Tordesilhas "naturally", and not by political influence, pushed further East not on purpose, as is commonly accepted. However I have to strongly disagree that this denies totally the idea that maps were in fact politically twisted!! No magnetic declination matters can explain a coast of Brazil going vertically from North-to-South, when clearly it does not!! So politically deliberated erroneous charts were categorically made. Therefore, dear Joaquim , I have to say, with all due respect for your interesting work, that I find your conclusions in your thesis on this matter a little bit hasty. You can not generalize your findings applied to several charts to the full cartographic body - where indeed such political twists are so evident. In statistics this is called a "bad sampling" - your sample does not allow you to conclude what you state history-politics wise. You should have limited your comments to the particular set of charts you looked into, given that many other charts are political and not "mathematical". Anyway I find it very refreshing and potentially useful the kind of work you did - but the historical context can not be removed out of your GIS-cartographic related work. In physics, and specially astrophysics, too many times mathematics leads into un-physical solutions. So one must always keep the physics in context, always present. I would say something along these lines applies to some of your conclusions history-politics wise. Knowing (in some cases personally) some members of your thesis juri - I am sure they could have/should have raised such discussion. Finally though there is some symmetry in our planet polar and equatorial perimeters - they are in fact different. Fine, just some 50 km or so, not much, we know today. However in those times, half millennium ago there were still discussions on how oblate our planet would be. Columbus himself is known for making references to our planet having the shape of a pear somehow - instead of a perfect sphere. My point is the measurement of longitude was always more tricky (Galileo himself applied a couple of times to the premium given by several European crowns to whoever could solve the measurement of longitude - and Galileo failed all the times!!). Other than being more tricky to measure longitude, if people back then had false ideas about how much more of a "belly" Earth would have - makes it then clear that to establish the longitude degree scale was not exactly the same as discussing latitudes on the Atlantic. One final comment about some astronomical contents of your thesis: indeed it was not known how to measure longitudes at sea 500 years ago. However it was not impossible - as you repeatedly wrote!! To not know something, to ignore it, is different from being impossible. If one has a good set of stellar positions, taken all over many months and years, side by side with the relative position of the Moon - this is the best astronomical clock/chronometer one can have!! So it is quite possible in fact to establish longitudes at sea - even Eratosthenes of Alexandria knew this quite well!!! However somehow such knowledge was lost - and only a few torquetums survived into the Discoveries time. Either recovered by the Arabs back into the European knowledge after the dark ages of barbarian invasions in Europe, or some other way, Columbus was one of the few people known to have one such torquetum that would allow to establish the longitude based on the Moon and stars relative position along the night. Of course eclipses in the same context of use of the torquetum would be even better, since they occur along the ecliptic plane, with very well known positions relative to the celestial equator. Peter Apian - the German cosmographer, was known to have one other such torquetum. Vespucci supposedly was good at measuring astronomical longitudes as well. Thus the dispute between these Spanish and Portuguese degrees would make the Tordesilhas line pass from Belem do Para in the Amazon river mouth, to Sao Paulo area - according to Spain. For Portugal the Portuguese degree length would make the Tordesilhas line pass instead further south - at the mouth of the La Plata river. The Portuguese believed in their better nautical knowledge clearly. If the Molucas issue was unfortunately lost with all that fortune spent - Portugal did fight with Spain for more than 100 years around Colonia do Sacramento, in nowadays Uruguay, because those would be the Portuguese degree Tordesilhas limit. In the Amazon with the amazing voyage of Pedro Teixeira all changed - Portugal gaining all the Amazon up to the current boarder between Peru and Ecuador. But if not considering Pedro Teixeira epic voyage, the Portuguese Tordesilhas would also move further inside the Amazon Basin to include a bigger chunk of the rain forest. No wonder then how important Vespucci would be - if indeed he had the rare talent/knowledge of measuring astronomical longitudes...despite the risk that Vespucci (a foreigner) would also represent when present in such far south Portuguese water. So measurements of longitude were made indeed near Santos-São Vicente - São Paulo area, to mark the Spanish imposed Tordesilhas. However the Portuguese needed to know also the longitude of La Plata better - since they believe (with reason!!) that to be the correct limit of the Portuguese territory. I wonder if Vespucci also had a torquetum or not?... Votos de maior sucesso - Joaquim. Kind regards, Paulo Afonso
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