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Columbus set off on his westward journey to Asia, believing the Earth was much smaller than it is.

There was some apparent disagreement about the size, and Columbus was staking his life on it. Why didn't he make his own estimate, for example as follows:

  1. Find a mountain near the coast, such as this one:

enter image description here

  1. Calculate its height $h$, for example, by climbing it and measuring the angular dimensions and position of something of known size

  2. Sail away from the mountain until half of it disappears

  3. Measure the angular height $\alpha$ of the remaining half.

The Earth's diameter will be $$h \over {2 \alpha ^2}$$


Relevant:

enter image description here

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  • $\begingroup$ It was quite common (before modern tech) to calculate distances on sea from the visible height (knowing Earth's radius, i.e. the opposite of what you ask), so I guess the question here is if sailors at the time used this calculation already. Atmospheric distortion is also an interesting idea here. But it generally means that one can see ships that are much further away, so the Earth looks much less curved and you would calculate an even larger Earth radius; if anything this would have discouraged Columbus. $\endgroup$ Jan 10 at 21:17
  • $\begingroup$ Strongly related: history.stackexchange.com/questions/4337/… $\endgroup$
    – Gae. S.
    Jan 10 at 22:11
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Answering the earlier version of the question first (on Columbus mistakes).

There were two main sources of mistakes: exaggerated size of Asia and underestimate of the size of the Earth.

Columbus was relying on the work of his contemporary geographer Toscanelli, with whom he corresponded (some of this correspondence survives).

Toscanelli estimated the distance (westward) from Lisbon to Beijin as about 5000 (nautical) miles. He also estimated the westward distance from "Antilles" to Japan as 2000 miles. (Existence of "Antilles" was conjectured at that time, as some islands midway where a traveler could make a stop.)

These huge mistakes are mainly due to the exaggerated estimate of the size of Asia, based on Marco Polo. Most contemporary scientists did not believe Marco Polo writings, but Toscanelli did. In fact Marco Polo contradicted Ptolemy, the most comprehensive and reliable author of antiquity, but this was already the time when Ptolemy was beginning to loose his authority:-)

The second, smaller source of error was estimate of the size of the Earth. The difficulty with ancient sources was (and still is) that the exact value of the distance units used by ancient authors was not known. For the length of the (nautical) mile Columbus assumed the value of Alfragan, who stated it as 56 2/3 of (Arabian) miles, but Columbus assumed that Alfragan used Italian mile which gives an underestimate of 75 % of the true value.

Source: A comprehensive account of Columbus calculations and related controversies is given in the book:

Samuel E. Morrison, Admiral of the Ocean Sea. A life of Christopher Columbus, NW Univ. Press, 1983.

Answer to the updated question. You describe an imaginary measurement which could not be performed at the time of Columbus or before. There are several reasons for this. 1. Atmospheric refraction at the horizon is about 30' but it is not constant (depends of the weather conditions), and the knowledge of refraction at that time was very rudimentary. It is discussed by Ptolemy, but he could not really measure it, and I am not aware of any reliable measurement until 17 century. 2. One needs a devise to measure small angles with sufficient accuracy. We are talking of angles of less than 1 degree. There is no evidence that such devises existed until 17 century. (If you read how Archimedes tried to measure the diameter of the Sun directly, you get the idea of the difficulties involved. There was certainly no progress in this since Archimedes to the time of Columbus). 3. "You sail until you see 1/2 of the mountain". How do you measure what distance you sailed? Read the book of Morrison about how they measured sailing distances at the time of Columbus, and see how crude the methods were. 4. There is no evidence whatsoever that anyone climbed high mountains until 18 century, not speaking of measuring their heights. Measuring the height of a mountain is a notoriously difficult task, which could be reliably performed only in 19 century, using high precision instruments (and a lot of expenses). If you don't believe me, just try to measure the height of any mountain with the instruments you can do yourself, with your hands:-)

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I think that observing ships disappear over the horizon would be an imprecise method of calculating the size of the Earth.

[added 01-13-21 I found a quote where Aristotle (384-322 BC) mentioned that the circumference of the Earth was estimated at about 400,000 stades, and thus not large compared to stars.

https://historum.com/threads/loading-of-animals-cannon.191924/page-2#post-3545934 https://bmcr.brynmawr.edu/1996/1996.10.02/ ]

Eratosthenes of Cyrene (c.276 BC-195/94 BC) was the first person to measure the size of the Earth.

The measure of Earth's circumference is the most famous among the results obtained by Eratosthenes,3 who estimated that the meridian has a length of 252,000 stadia, with an error on the real value between -2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres).4 Eratosthenes described his technique in a book entitled On the measure of the Earth, which has not been preserved.

Eratosthenes' method to calculate Earth's circumference has been lost; what has been preserved is the simplified version described by Cleomedes to popularise the discovery.5 Cleomedes invites his reader to consider two Egyptian cities, Alexandria and Syene, modern Assuan:

Cleomedes assumes that the distance between Syene and Alexandria was 5,000 stadia (a figure that was checked yearly by professional bematists, mensores regii).6 He assumes the simplified (but false) hypothesis that Syene was precisely on the Tropic of Cancer, saying that at local noon on the summer solstice the Sun was directly overhead. He assumes the simplified (but false) hypothesis that Syene and Alexandria are on the same meridian.

Under the previous assumptions, writes Cleomedes, you can measure the Sun's angle of elevation at noon of the summer solstice in Alexandria, by using a vertical rod (a gnomon) of known length and measuring the length of its shadow on the ground; it is then possible to calculate the angle of the Sun's rays, which he claims to be about 7°, or 1/50th the circumference of a circle. Taking the Earth as spherical, the Earth's circumference would be fifty times the distance between Alexandria and Syene, that is 250,000 stadia. Since 1 Egyptian stadion is equal to about 157.7 metres,7 the result is roughly 39,425 km, which is 1.5% less than the real circumference, 40,008 km.

Eratosthenes' method was actually more complicated, as stated by the same Cleomedes, whose purpose was to present a simplified version of the one described in Eratosthenes' book. The method was based on several surveying trips conducted by professional bematists, whose job was to precisely measure the extent of the territory of Egypt for agricultural and taxation-related purposes.4

https://en.wikipedia.org/wiki/Earth%27s_circumference#Eratosthenes

Posidonius calculated the Earth's circumference by reference to the position of the star Canopus. As explained by Cleomedes, Posidonius observed Canopus on but never above the horizon at Rhodes, while at Alexandria he saw it ascend as far as 7+1⁄2 degrees above the horizon (the meridian arc between the latitude of the two locales is actually 5 degrees 14 minutes). Since he thought Rhodes was 5,000 stadia due north of Alexandria, and the difference in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth.[10] It is generally thought that the stadion used by Posidonius was almost exactly 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to 24,000 mi (39,000 km), not much short of the actual circumference of 24,901 mi (40,074 km).[10] Strabo noted that the distance between Rhodes and Alexandria is 3,750 stadia, and reported Posidonius's estimate of the Earth's circumference to be 180,000 stadia or 18,000 mi (29,000 km).[11] Pliny the Elder mentions Posidonius among his sources and without naming him reported his method for estimating the Earth's circumference. He noted, however, that Hipparchus had added some 26,000 stadia to Eratosthenes's estimate. The smaller value offered by Strabo and the different lengths of Greek and Roman stadia have created a persistent confusion around Posidonius's result. Ptolemy used Posidonius's lower value of 180,000 stades (about 33% too low) for the earth's circumference in his Geography. This was the number used by Christopher Columbus in order to underestimate the distance to India as 70,000 stades.[12]

https://en.wikipedia.org/wiki/Earth%27s_circumference#Posidonius

Around AD 525, the Indian mathematician and astronomer, Aryabhata wrote Aryabhatiya, in which he calculated the diameter of earth to be of 1,050 yojanas. The length of the yojana intended by Aryabhata is in dispute. One careful reading gives an equivalent of 14,200 kilometers, too large by 11%.[13] Another gives 15,360 km, too large by 20%.[14] Yet another gives 13,440 km, too large by 5%.[15]

https://en.wikipedia.org/wiki/Earth%27s_circumference#Aryabhata

Around AD 830, Caliph Al-Ma'mun commissioned a group of Muslim astronomers led by Al-Khwarizmi to measure the distance from Tadmur (Palmyra) to Raqqa, in modern Syria. They calculated the Earth's circumference to be within 15% of the modern value, and possibly much closer. How accurate it actually was is not known because of uncertainty in the conversion between the medieval Arabic units and modern units, but in any case, technical limitations of the methods and tools would not permit an accuracy better than about 5%.[16]

A more convenient way to estimate was provided in Al-Biruni's Codex Masudicus (1037). In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations, based on the angle between a plain and mountain top, which made it possible for it to be measured by a single person from a single location.[16] From the top of the mountain, he sighted the dip angle which, along with the mountain's height (which he calculated beforehand), he applied to the law of sines formula. This was the earliest known use of dip angle and the earliest practical use of the law of sines.[17] However, the method could not provide more accurate results than previous methods, due to technical limitations, and so al-Biruni accepted the value calculated the previous century by the al-Ma'mun expedition.[16]

https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus's_error

1,700 years after Eratosthenes's death, Christopher Columbus studied what Eratosthenes had written about the size of the Earth. Nevertheless, based on a map by Toscanelli, he chose to believe that the Earth's circumference was 25% smaller. If, instead, Columbus had accepted Eratosthenes's larger value, he would have known that the place where he made landfall was not Asia, but rather a New World.[18]

https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus's_error

I note that Columbus also used estimations of the size of Asia which made it extend much farther east than it really does.

And I point out that in a sense Columbus was not actually wrong about the distance from Europe west across the Atlantic to Asia.

A continent is any of several large landmasses. Generally identified by convention rather than any strict criteria, up to seven geographical regions are commonly regarded as continents. Ordered from largest in area to smallest, these seven regions are: Asia, Africa, North America, South America, Antarctica, Europe, and Australia.3 Variations with fewer continents may merge some of these, for example some systems include Afro-Eurasia, America or Eurasia as single continents. Zealandia, a largely submerged mass of continental crust, has also been described as a continent.

https://en.wikipedia.org/wiki/Continent

Modern geologists know about plate tectonics and continental crust, and may count more or fewer continents than ordinary people do.

But most modern people still think and feel that if you can travel by land from place A to place B, those two places are part of one continent.

And certainly nobody proved that the Americas were not part of Asia during the lifetime of Columbus, or for centuries afterwards. The Pacific Ocean east of Asia could have simply been a gigantic bay extending north into Asia and the Americas could have been vasts southward peninsulas from eastern Asia.

The Bering Strait between Siberia and Alaska was not discovered by Euoprean explorers until 1728, though Euoprean geographers speculated about it earlier.

Even after 1728, it might have seemed possible for a land bridge to extend across the Arctic ocean from Siberia to Canada.

I don't knew when exploratins in Siberia proved the it had a coast line all the way from Norway to the Bering Strait. Nordenskiold in the Vega made the first passage along the Northeast Passage in 1878-79, but the coast must have been surveyed long before then.

The first complete voyage along the Northwest Passage north of Canada was made by Amundsen in Gjoa in 1903-06. But it had been proven by 1854 that there was water all the way during the searches for the lost Franklin Expediiton.

Anyway, there was no reason for anyone to claim as a fact that the Americas were not extensions of Asia until long after the time of Columbus. Many people wanted to believe that Asia and the Americas were separate continents, because that would make it possible to find a Northwest Passage or a Northeast Passage from Europe to east Asia and open up alternate trading routes. So the wish for such passages was the father of the belief that the Americas were separated by water from Asia. And that belief was not proved correct for centuries afterwards.

But has the belief that the Americas are separate from Asia actually been proved?

An ice age is a long period of reduction in the temperature of Earth's surface and atmosphere, resulting in the presence or expansion of continental and polar ice sheets and alpine glaciers. Earth's climate alternates between ice ages and greenhouse periods, during which there are no glaciers on the planet. Earth is currently in the Quaternary glaciation.3 Individual pulses of cold climate within an ice age are termed glacial periods (or, alternatively, glacials, glaciations, glacial stages, stadials, stades, or colloquially, ice ages), and intermittent warm periods within an ice age are called interglacials or interstadials.4

https://en.wikipedia.org/wiki/Ice_age

The Quaternary Glaciation / Quaternary Ice Age started about 2.58 million years ago at the beginning of the Quaternary Period when the spread of ice sheets in the Northern Hemisphere began. Since then, the world has seen cycles of glaciation with ice sheets advancing and retreating on 40,000- and 100,000-year time scales called glacial periods, glacials or glacial advances, and interglacial periods, interglacials or glacial retreats. Earth is currently in an interglacial, and the last glacial period ended about 10,000 years ago. All that remains of the continental ice sheets are the Greenland and Antarctic ice sheets and smaller glaciers such as on Baffin Island.

https://en.wikipedia.org/wiki/Ice_age#Major_ice_ages

So for the last two and a half million years Earth's climate has varied between cold glacial periods and warm interglacial periods. And as far as I can tell the glacial periods tended to last longer than the interglacial periods.

And during the glacial periods the vast amounts of water tied up in ice sheets causes the sea level to drop a considerable distance, thus exposing the sea floor between Alaska and Siberia as dry or ice sheet covered land.

So for the last two and a half millin years a land connection between Siberia and Alaska has been at least a common as a strait between them. And so North and South America could be considered to be part of the same continent as Africa, Europe, and Asia, making Antarctica and Australia the only other real separate continents.

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