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Eratosthenes' calculations of the size of the Earth were very precise and very accepted during Columbus' time. Still, Columbus was convinced that the world was significatively smaller and that getting to Asia from Europe was feasible.

What was the method used by Columbus to calculate the size of the Earth? Maybe more importantly, why were his results accepted?

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Eratosthenes' calculations did turn out to be quite accurate. This was mostly a matter of luck though. He in fact had two major errors, that just happened to cancel each other out. It is also a fact that nobody is sure how big his unit of distance was, and it is only now after the fact that we can take one of the possibilities and say he was only 2% off.

It is also not true that this was universally accepted. Posidonius, via Ptolemy had his own much smaller estimate, which was widely accepted up until around Columbus' day.

In point of fact there were quite a few competing estimates for the size of the earth and the size of the Eurasian land mass (not to mention that the best numbers and maps were likely kept by Portugal as State Secrets). Most people quite sensibly figured the truth was somewhere in the middle of the estimates. However, Christopher Columbus, always a master of self-delusion (can you tell I'm not a fan?), took Earth estimates on the small side and Eurasian estimates on the large side, some even more fanciful theories about the position of Japan and other eastern islands, and convinced himself that Japan was only about 3,000 miles west of the Canaries (in fact, the number is more like 12,000).

His results were not in fact accepted. He was denied in Portugal. The Spanish court's mathematicians argued (correctly) that the guy was way wrong, if not an out-and-out crackpot. However, Portugal had just rounded the tip of Africa, opening the way for a future control of the spice trade. This was so potentially lucrative, that the Spanish Monarchs felt forced to at least try Columbus' idea.

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Well, Columbus case is one hallmark how to cherry-pick your data to come to desirable conclusions.

Columbus began with the values of the best sources available: from the Arabian astronomer al-Farghani. Al-Farghani calculated very carefully that the distance of one degree latitude (north-south) equals 56 2/3 arabian miles (1972 m) which is 111.8 km; the modern value is 111.3 km. This is an error of 0.45 % !

Dilettante Columbus used the value 56 2/3, but instead using the correct Arabic mile he used the Roman one which is only 1481 m long. So you get only 3/4 of the correct distance which is 83.9 km.

Now we need to estimate the distance. 360° is the full circle, so how far away is Japan (Cipangu) from Spain away ? Contemporaries estimated it to 180° which is exactly half-way, but we have no problems with those Arabs anymore. If we use this, we get

Travel distance = 180° * 83.9 km = 15 102 km.

So even with the already flawed measuremants, we have still one big ocean between us. Even starting from the Canaries save us a marginal 10° = 839 km. Given a day's run of approximately 80 km this would have meant 178 days which is exactly half a year without provision. This would have meant certain death.

Now Toscanelli, a cartographer screwed up royally and used an outdated estimate of Marius of Tyre and estimated the east-west dimension of Spain-Chinas east coast to 230° (The correct value is 140°). And making no prisoners, he shifted Japan's east coast another 30° eastwards. So we have 260° of land and the distance shrinks to 100°.

Travel distance = 100° * 83.9 km = 8 390 km = 105 days.

This completely false value is still too big.

What now happened can only be attributed to "The wish is the father of the thought". Cipangu (Japan) must have islands which extend eastwards and naturally Cipangu is on the exact same latitude as the Canaries. And 56 miles could be shortened to 50 miles and there is very likely the island Antillia which could be considered part of Japan. And perhaps it is better that I subtract something from the value to make it look better...

In the end Columbus claimed a distance of approx. 4400 km which he very likely pul...erm...fabricated. Noone has an idea how he came to this completely deluded distance estimate. In the end he was lucky that America was in his way.

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    What remains so tantalizing is that, with all these totally fabricated calculations, he was nearly dead on to actual landfall. It's almost as if he knew something was there, and simply had to justify an economic rationale for exploring it. Jump now to rumours of Norse voyages and Irish monks and Grand Banks secrets of Breton fishermen and all that. – Pieter Geerkens Oct 15 '16 at 19:00
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    @PieterGeerkens - Mr. McEvedy pointed out that he also made near-perfect use of the circular winds in the Atlantic on both his voyage out and back, and said that some have used this to claim that he was doing as you said. However, I believe later events in his life show that where understanding Columbus is concerned, incompetence is always a better presumption than malice. – T.E.D. Oct 16 '16 at 4:07
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    @T.E.D.: The Portuguese had discovered the Atlantic Westerlies & Easterlies wind pattern a generation earlier, so nothing new there - Columbus's knowledge of it was only partial. – Pieter Geerkens Oct 16 '16 at 4:42
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    @PieterGeerkens - Columbus did participate in that as well. I prefer that explanation greatly to the conspiracy theory. – T.E.D. Oct 17 '16 at 13:31
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    I have a small question that is related but isn't worth creating a new topic: Famously, when Colombus reached the Bahamas, he thought he was in India, and he called the locals "Indians". What made him think he was in India and not in Japan (Cepangu) ? – Evargalo Nov 14 '17 at 9:11

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