Author's blog

On this page, observations of various kinds will appear that are related to big history teaching, as well as to big history in general.
Earlier blogs
- Big history and web site design
- Did Galileo overstate the magnification of his telescope? 
In-class observations
A few years ago, as part of teaching big history for small groups I started developing a series of observations that can be done by students in class. To my surprise, some of these observations turned out not only to be very helpful for teaching big history, but they also led to a few intriguing discoveries. This report is the second of a series of such observations with unexpected, exciting, results.
To improve students’ understanding of navigation in the period of early globalization, most notably the scientific and technological knowledge underpinning it, they measure in class the altitude of the sun using both a homemade quadrant
and a replica of a mariner's astrolabe that I bought in a nautical antique shop in Rotterdam.
These instruments were formerly used to determine one’s latitude, which is a measure of one’s distance between the equator (which is defined as zero degrees), and one of the poles (located at ninety degrees either north or south). One degree along a line connecting the north and south pole equals 60 nautical miles, 69 (statute) miles, or 111 km.
The city of Amsterdam, for instance, is situated at 52 degrees north of the equator. The concepts of latitude (and also longitude, the distance along the equator measured from a certain point) had already been used by Claudius Ptolemy of Alexandria (c.100-170 CE), whose geography was firmly based on the idea that the world was a globe. And Ptolemy had also been familiar with the quadrant for determining celestial and geographic altitudes.
The quadrant and the astrolabe are both used by holding them and lining up two holes on the instrument while aiming at a celestial object, the sun or a star, and then read the resulting angle on the scale. Doing so provides the celestial object’s altitude in degrees with respect to the local horizon.
For observing the sun’s altitude, this lining up of the holes is done by letting the sun shine through one hole and projecting its resulting small but bright image right onto the next hole, and then read the scale.
In contrast to what is stated on many web sites, there is no need at all to look at the sun through the holes, which would damage one’s eyes. At night, however, one would directly observe Polaris, the pole star, by looking through these holes.
Because the sun’s altitude changes throughout the year between +23.5 degrees north and -23.5 degrees south, accurate data about its daily positions are required to calculate one’s latitude.
Measuring Polaris’s rather stable altitude above the North Pole, by contrast, will immediately yield one’s latitude with a precision of about one degree. No greater direct precision is possible because the pole star wobbles a little with respect to the true north. In practice, both sun and star observations were used whenever possible, sometimes as a check on each other.
The technology of using quadrants and astrolabes for measuring latitudes was further developed by the Portuguese after they started sailing southwards on the Atlantic Ocean to reach the African gold coast and later also Asia, both famous for their wealth.
Navigating the Mediterranean had not necessitated using such instruments because of its largely east-west orientation. Sailing it meant moving through changing longitudes, but never experiencing a significant change in latitude.
By sailing southward, however, the Portuguese did encounter a significant change in latitude. They also found that Polaris, situated almost exactly above the North Pole, disappeared below the horizon as soon as they crossed the equator. And in the southern hemisphere there proved to be no suitable stars situated above the South Pole that could be used for determining latitude.
The Portuguese solved this problem by developing more accurate methods to measure the sun’s altitude, preferably at noon when the sun appears the highest in the sky. But because the altitude of the sun changes during the year from +23.5 degrees north to -23.5 degrees south and back, the Portuguese needed to construct tables based on accurate measurements that would provide those data from day to day.
Such documents are known as sun declination tables. The oldest extant Portuguese sun declination table, called Regimento do estrolabio e do quadrante, probably dates from 1509 CE, and was likely a printed version of earlier editions. A copy of the Regimento can be downloaded here as a PDF.
Such tables may have been constructed with careful observations made at the peninsula of Sagres, where the first Portuguese navigational school was founded in 1420 CE by Prince Henry the Navigator (1394-1460 CE).
Although old documents that could have shed a light on what happened at Sagres were lost, it is remarkable that this peninsula is oriented to the south, and thus very suitable for observing the noon sun and stars.
Furthermore, Sagres Point still offers the widest unobstructed view of the horizon over the ocean available in Portugal, and, in consequence, the largest possible reference line for determining celestial altitudes during both day and night over the broadest possible range.
In sum, Sagres was perfectly situated for making sun declination tables. This would have allowed the Portuguese to take the remarkably accurate measurements that appeared in the Regimento.
Such measurements could not possibly have been taken with the small hand-held quadrants and astrolabes used by  mariners. Much larger instruments were needed to achieve such a high precision. Traces of such possible instruments can still be seen at Sagres Point.
The rule for determining latitude, explained here, is: 90 degrees, minus the measured noon sun altitude, plus the noon sun declination of that particular day.

This way of determining latitude at sea remained in use until the 1990s CE while using sextants, which are much more precise than quadrants and astrolabes. And even after NavSat and GPS replaced sextants for everyday use, these instruments still remain in use as an emergency back-up option in case GPS fails.
For sextants, a fix on the horizon or an artificial horizon are required to determine celestial altitudes. This is not needed while using quadrants and astrolabes, because gravity’s action on the instrument provides the required frame of reference.
In Columbus’ time, the sun’s altitude was preferably measured using a quadrant, because it was deemed more precise than the mariner’s astrolabe, mostly because the plumb line can be very thin, and can therefore be read more precisely than the astrolabe’s pointer (usually called alidade), which is a little more complicated to make, and therefore more likely to have a systematic error.
It can be very hard, however, to keep the quadrant’s plumb line sufficiently steady on a rolling ship. The mariner’s astrolabe is more stable, and therefore provides more accurate measurements at sea, which explains why that instrument became the preferred choice on board.
The mariner’s astrolabe was a simplified version of the much more complicated medieval astrolabe, which has been called the world’s first portable computer. Based on earlier Sumerian and Babylonian designs, astrolabes were already known in antiquity, and had reached Western Europe through the Arabs, who had developed them further.
The simplification of the mariner’s astrolabe consisted of retaining only the alidade and the scale in grades, while making it so heavy so that it would be sufficiently stable on board of a rolling ship. Holes in the instrument provided as much room as possible for wind to blow through it to prevent it from swaying. In short, the mariner’s astrolabe was a very rationally-designed and convenient solution based on the best available technology.
In class, students are asked to measure the sun’s altitude with both instruments. They must also estimate the margin of error of their results, most notably by answering the following questions: what is the estimated average error of their measurements? And how large is the estimated systematic error due to the instrument’s lack of perfection? And how would you deal with that to improve your measurements? In doing so, students become familiar with these concepts, which are most important aspects of all systematic observations.
The students’ results tend to coincide with my measurements and estimates: 0.5 to 1 degree of maximum average error of measurement, and 1 to 2 degrees of maximum systematic error. Much like the mariners of old, also the students consider the quadrant more precise than the mariner’s astrolabe, and for the same reason.
If students were able to measure the noon sun altitude –the sun does not always shine in Amsterdam– , they are requested to determine their latitude using sun declination data from the Regimento do estrolabia e do quadrante for that particular day, while comparing these data with modern data obtained from the Internet.
This introduces another problem, namely that the sun declination data provided by the Regimento are off by ten days. The reason for that is that in the meantime there has been a calendar change, from the Julian to the Gregorian calendar.
In 1582 CE, the Catholic church led by pope Gregory XIII struck ten days off the calendar, because it was no longer in phase with the observed solar positions, while a different arrangement with the leap years was made to ensure that this would not happen again for a long time to come.
Going back ten days in the Regimento, students will find data for the solar declination that rather closely conform to the modern predictions. So the Portuguese had done an excellent job!
This small remaining difference can be explained by several factors. First of all, the Gregorian calendar reform took place about 80 years after the sun declination tables were published, which means that the calendar error had increased about half a day by the time the ten days were removed from October of 1582 CE. This difference needs to be subtracted from the table.
In the second place, the noon sun declination for a certain date changes from year to year, because the solar calendar, which is determined by Earth’s orbit around the sun, does not exactly fit day length, which is determined by Earth’s rotation around its axis.
This annually increasing difference is mostly, but not entirely, corrected every four years by adding an extra day to the calendar. As a result, these tables had to be adjusted from year to year. If not, over a period of four years these tables  would go out of whack to a maximum of about 20 minutes, which would add an error of about 35 km (21 miles) distance.
Furthermore, as a result of the decreasing tilt of Earth’s axis over the past centuries, about 1 minute needs to be subtracted from these data. This must have been known to the scientists who constructed the Regimento tables.
And last but not least, all of this does not yet take into account that the sun declination keeps changing during the day. Around the equinox (around March 20 and September 23), for instance,  the sun declination changes as much as 20 minutes in 24 hours.
This means that these data also depend on the place where they were measured, or for which they were calculated to happen. In fact, the precision in stating the daily noon sun declination in degrees and minutes would allow us, in principle, to calculate all of that.
Such an assumption seems reasonable, because the differences per day mentioned in the Regimento very much agree with modern observations and predictions. Checking these old tables and interpreting them in such ways could become a fascinating research project.
While using these data, most students will usually arrive at a latitude of about 52 degrees for Amsterdam. They have, on average, at most an error of about 30 miles (45 km) with respect to their real position.  Not a bad result at all with this ancient technology.
Thanks to the fact that my long-standing research interest into the history of Peru had included its initial conquest, and that I had defended my Ph.D. thesis about Peru on October 12, 1992 CE, in name exactly 500 years after Columbus had stepped ashore for the first time on the other side of the Atlantic Ocean, I had begun studying the accounts of some of these early explorers.
Because the commemoration in 1992 CE had led to a wave of publications about Columbus and his exploits, I had become aware of some very smart recent Spanish scholarship, mentioned below. As a result, I was familiar, among other things, with Columbus’ diaries as well as with some of the scholarly discussions about them.
All of that explains how I got onto the track of trying to find out which specific instruments Columbus had used on his first voyage and what measurements he had reported. This turned out to be a fascinating trail.
After students have determined their latitude using this ancient maritime technology, they are confronted with Columbus’ measurements that he made during his first voyage in 1492-93 CE to what later would become known as the Americas.
According to Columbus’ diary, of which only a partial transcription of a transcription survives, he measured his latitude several times: first on the island of Hispaniola on October 30, 1492 CE, with a quadrant. Columbus' result was 42 degrees, "if my transcription of his letters is not corrupted," wrote Fray Bartolomé de las Casas (1484-1566 CE), who transcribed and edited Columbus' diary (p.85). In reality, however, it was only about 20 degrees.
(All the pages mentioned in this context are from: Consuelo Varela, editor, Cristóbal Colón: Los cuatro viajes; Testamento. Madrid, Alianza Editorial, 1986.)
On November 2, 1492 CE, Columbus used the quadrant at night in the same place (apparently to get a fix on Polaris) and obtained the same result (p.88). On November 21, 1492 CE, along the coast of Cuba, Columbus found the same latitude of 42 degrees with his quadrant (p.102), while reporting very high temperatures.
At this point in the diary, Fray Bartolomé de las Casas expressed further doubt, because he thought, among other things, that at 42 degrees north (about the latitude of modern Boston, MA) it had to be a lot colder in November (even though no Europeans were known to have gone there yet). On December 13, 1492 CE, Columbus measured his latitude on the island of Hispaniola with the quadrant and obtained 34 degrees, while his correct latitude was closer to 19 degrees (p.128).
On February 3, 1493 CE, on the Atlantic Ocean on his way back to Spain, he used both the quadrant and the mariner’s astrolabe to determine latitude, but was unsuccessful because of the high waves. Yet Columbus did note then that Polaris was high in the sky, much like “cabo de Sant Viceinte” (p.183).
Interestingly, Cabo de São Vicente, the most southwesterly cape of Portugal, is situated at 37 degrees latitude, so still 5 degrees lower than three of the four measured latitudes in the Carribean just mentioned. Apparently, during that earlier portion of his trip the pole star had been a lot lower in the sky, which indicates that by that time Columbus must have been aware of the fact that the values that he had reported earlier were erroneous.
How could the so very experienced sailor Columbus get this so very wrong, one wonders, while his instruments would have caused an error of at most 1-2 degrees? What would students think about that? Was Columbus incompetent, or was something else going on?
Spanish scholar Consuelo Varela (1945 CE - ), mentioned above, thought that Columbus probably falsified his data, because Cipangu, Japan, was also at 42 degrees on his map, and Columbus thought that he had reached that island (note 31, p.85). This may be a partial explanation for this situation.
But there may be more to this than that. The Spanish nautical expert and sea captain Luis Miguel Coín Cuenca (1953 CE - ) has argued for decades that Columbus systematically falsified the diary of his first voyage for one single specific reason: namely to make it appear that he had stayed out of territory that the Portuguese could claim.
Dr. Coín Cuenca’s thesis is best documented in his brilliant but virtually unknown book Una travesía de 20 dias a dos rumbos que cambió el mundo, Universidad de Cádiz, 2003.
When Columbus started his first voyage, Coín Cuenca argues, there was a treaty between Spain and Portugal known as the Alcáçovas Treaty, signed in 1479 CE, and confirmed in 1481 CE by the papal bull Æterni regis. This treaty granted all lands south of the Canary Islands to Portugal, and thus allowed the Spanish to go only as far south as the Canary Islands (about 25 degrees north).
According to Coín Cuenca, this is why Columbus reported in his diary that he sailed straight west from the Canary Islands, even though his descriptions of what he witnessed actually fit a more southernly course, while they do not agree with his stated course.
This also explains why, on all his subsequent voyages, Columbus did sail further south. He could do so legitimately, because the earlier treaty had been replaced by papal bulls issued in 1493 CE, and by the Treaty of Tordesillas of 1494 CE, which had redefined this situation in legal terms.
In his book, Coín Cuenca provides ample, very carefully assembled, evidence for his thesis, including his first-hand experience of having sailed this more southernly trajectory in 1990 CE as captain on a carefully-constructed replica of one of Columbus’ ships.
But because Coín Cuenca’s analysis ends when Columbus first reached land in the Americas, he did not discuss Columbus’ later quadrant and astrolabe measurements, which very much support his thesis.
As a result of the Alcáçovas Treaty, Columbus had every reason to make sure that his measurements showed latitudes well north of 25 degrees, so that not only those lands could not be claimed by the Portuguese, but that also a considerable stretch of possible land situated to the south of where he had gone could be claimed by the Spanish. That is why, I think, all his measurements show these large systematic errors.
This may also explain why, in the fragments of his diary that survive, Columbus never reported latitude measurements on his way to the Americas, simply because he knew he was sailing in Portuguese territory.
Even if Columbus indeed sailed west after leaving the Canary Islands, as he reported in his diary, he would have been right on the borderline between Spanish and Portuguese territories, which could easily have led to Portuguese suspicions and claims. So from a political point of view it was a smart move not to report any latitude measurements on his outbound voyage.
This lack of celestial measurements on his way to the Americas has led to the impression that Columbus almost exclusively relied on dead reckoning: determining his position with the aid of measuring his direction by compass and his velocity by other means, which were rather unreliable.
Columbus surely did use dead reckoning all the time, as all mariners before and after him have done until the advent of GPS. Yet on his way home, Columbus did try to measure his latitude, as we saw above. So he was not averse to doing so if he felt the need.
If Dr. Coín Cuenca is right, how would Columbus have determined his (lower) latitude while sailing to the New World? I think that the observation mentioned above of Columbus estimating the altitude of Polaris may indicate that he also did so on his outbound voyage. This would explain how Columbus estimated his latitudes during that portion of his trip without making observations with an astrolabe or a quadrant.
Because such guesses inevitably contained a considerable margin of error, this may also explain why Colombus missed the first Carribean islands, simply because he was a little more to the north than he thought he was. By using an astrolabe, he might have hit them right away.
More in general: in addition to successfully navigating around the globe, carefully measuring latitude and longitude has had great many other profound consequences in human history over the past 500 years, most notably in North America, Australia, and Africa, where many borders have been drawn along such geographically-determined lines (cf. Mark Monmonier: Drawing the Line, 1995, New York, Henry Holt, p.107-8). The Alcáçovas Treaty may have been the first instance in which such lines were drawn.
Furthermore, the problem of knowing where exactly the Tordesillas line should be drawn on the ground on the northern coast of South America may have motivated the Spanish and Portuguese to stay away from each other there in order to avoid conflicts. If so, this may have provided room for other Northern European powers (who did not respect the Tordesillas treaty) to colonize the Guyanas and, in doing so, deeply influence the history of those areas.
What students learn from these observations, apart from doing the measurements themselves, is that one may only be able to judge historical documents well if one tries to place oneself into the other persons’ positions, and that this may be surprisingly easy to do. They also learn that very interesting questions can be raised as a result of very simple and basic observations.
Finally, the change into the Gregorian calendar in 1582 CE may have led to new commercial opportunities. During the sixteenth century, the Dutch overtook the Portuguese and the Spanish by producing mariner’s handbooks that were printed and sold in different languages. The Dutch first copied these manuals and subsequently improved them. This turned out to be an excellent business.
Remarkably, the seafaring Dutch provinces of Holland and Zeeland accepted the Catholic Gregorian calender almost immediately, even though the Dutch were rejecting papal authority by turning Protestant while fighting the Catholic Spanish.
The British, by contrast, only adopted the Gregorian calendar as late as 1752 CE, apparently because of their dislike of papal power. Could this coincide with the fact that in the 17th century and beyond, British mariner’s handbooks appear to have sold not nearly as well as those of their more pragmatic Dutch competitors? And were these different attitudes perhaps connected to the Dutch running a republic in which mercantile interests dominated, while Britain was a mix between royalty and parliament?
All these intriguing questions, to which I do not have clear answers yet, as well as Columbus likely having falsified his latitude measurements, came as a result of the decision to ask students to observe the sun's altitude with ancient instruments and determine their latitude, while comparing those data with information from old documents.
International Big History Association
Un. of Amsterdam big history
Cosmic Evolution
Big History Project
Book: Teaching Big History
Bill Bryson: Short History of Nearly Everything
Other useful stuff on the web
Other big history
How to use the book
Course models
Learning goals and objectives
Teaching tools
Assignments (little big histories)
Answers to FAQs by students
Questions by students and teachers that go beyond the book
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Teaching big history