Surveying and Statecraft I

In several earlier posts, I discussed maps, map-making, surveying, and the tools and concepts necessary for the accurate measurement of land and geography to create accurate maps.  Along the way, I’ve mentioned cadastral (i.e. real estate) mapping, but mostly I’ve talked about mapping the shape of the Earth and the shape of its lands, called geodetic mapping, which requires a reference scheme (longitude and latitude), geometric concepts and the tools to use the reference scheme and concepts.  Greater accuracy in the tools developed over time, as well as some standards for the procedures for using them.

One function of surveys that I have not yet covered has been the setting of boundaries: not necessarily the boundaries between two or more properties, but between two politically different entities.  Who set the boundary between France and Belgium, or between Belgium and the Netherlands?  Was the boundary established via a survey?  Actually, no: most boundaries were established by treaty and were specified in words, but I can only find mention of surveys in a very few cases.  The words in the treaties lay out what the boundaries are, in some cases, such as defining a border as a specific river, the political entities on each side being granted half.  In many cases, natural geographical features were used as boundaries, such as the water around the British Isles delineating the U.K. from the rest of the world.  Setting borders is not measurement leading to discovery: instead, it is a form of engineering using standards and specifications to perform measurements, and results in something more or less useful.

One boundary that I know was put into place by a survey divides Pennsylvania and Maryland with the Delaware separated from both of them.  This was done by a survey led by Charles Mason and Jeremiah Dixon – the Mason-Dixon line.  I admit that I never thought about it very much until the novel by Thomas Pynchon came out, called Mason & Dixon.  I read the book, enjoyed it, and have wondered how factual the story that Pynchon wrote was or is.  Short of re-reading, it, though, I suggest that much of it is based on fact, since I know that Pynchon is reputed to be a thorough researcher, though there is plenty of room of Pynchon’s creative invention.

The genesis of the conflicting claims has to be laid to the way that both the colonies received their charters.  The first, the charter for the Maryland colony, was given to the 2nd Lord Baltimore, Cecil Calvert, though the idea of it and its boundaries had been proposed by his father, George Calvert, the 1st Lord Baltimore, who died shortly before it was granted in 1632.  The Calverts were Catholics in Anglican England, and George had proposed the colony to be a refuge for Catholics: essentially a way to put the “Catholic” problem out of sight and thus out of mind.  The supporters of the already established colony of Virginia were not pleased by this, and they lobbied, mostly unsuccessfully, to eliminate the new colony.  They were only successful in having the charter changed to reflect the fact that Virginia settlers were already in part of the territory originally granted to the Calverts.

While Cecil stayed in England to fight the Virginia challenge, he sent his younger brother, Leonard, to establish the colony and be its first governor.  In the instructions that Cecil gave Leonard, he stressed that the colony needed to be established with religious tolerance as part of its principles, since the colonists originally sent were both Catholic and Protestant.  In 1649, the assembly of the Maryland colony passed the first law requiring religious tolerance in the British colonies.

The establishment of the Maryland colony occurred during upheavals in England: in 1629, the King dissolved Parliament, the apparent start that undermined the stability of England, led to the three civil wars, the beheading of King Charles I, the dictatorship of Oliver Cromwell, and ultimately to The Restoration in 1660.  A pretty tough 30 years or so for England, throughout most of which and at the end of which, the Calverts retained their proprietorship of Maryland.

The other charter, that of the area known as Pennsylvania, was granted to William Penn in 1681.  It, too, has a background story: William Penn’s father, Admiral William Penn, had rendered services and financial backing to the restored monarchy, so despite young William’s radical Quaker faith, his arrests for proselytizing, etc., not only the King, Charles II, but the Duke of York, who became King James II at the death of Charles II, gave up territory for the charter.  William, being a theological visionary with his head in the clouds, as it were, was not so well grounded that he paid close attention to the boundaries, and despite his charter granting him lands above the 40th parallel, established the city of Philadelphia below the 40th parallel.  In Wikipedia, there is a wonderful map showing the approximate overlap of the two charters, as envisioned by the protagonists.1

The Maryland charter was supposed to cover all the land north of the Potomac River up to the 40th parallel, on both sides of Chesapeake Bay, east to Delaware Bay and the Delaware River.  I have not discovered how the western border was specified.

The Pennsylvania charter was to be the land north of the 40th parallel from the Delaware River, south of New York and its western border was specified as a longitudinal line 5 degrees from the Delaware River.

For some reason, the three counties in the eastern part of the Maryland charter ended up as part of the Pennsylvania charter.  They did not remain there for too long, though, since the settlers expressed a desire to be independent almost immediately after the new proprietor of Pennsylvania arrived, and were granted semi-autonomy in 1704.  The town of New Castle in what became Delaware was the first established city in the region, and its borders figure in to the eventual settlement and surveying.

The dispute between the Calverts and the Penns remained unsettled,  and led to violent clashes between settlers of both loyalties.  It should be noted that the Pennsylvania charter area maintained religious tolerance, much like that put into law by Maryland, but Pennsylvania attracted disaffected and persecuted people from throughout Europe as well as from England, while it appears that Maryland mainly attracted English settlers.

In 1732, the 5th Baron Baltimore signed a provisional agreement with Penn’s sons, agreeing to a line between the colonies and giving up the claim to the three counties of Delaware, but later, he denied that the document had some of the terms he had agreed to.  In 1750,  a royal commission was set up, and one of the results was that in 1760 the Crown ordered the 6th Baron Baltimore to accept the 1732 agreement.  (Seems like it was a good time for the British to establish clear boundaries in North America: the date nearly matches the end of the French and Indian war: Quebec conquered in 1759, Montreal capitulated in 1760, and the Treaty of Paris signed in late 1762.)

The resulting agreement specified the following:

  • Between Pennsylvania and Maryland:
    • The parallel (latitude line) 15 miles (24 km) south of the southernmost point in Philadelphia, measured to be at about 39°43′ N and agreed upon as the Maryland–Pennsylvania line.
  • Between Delaware and Maryland:
    • The existing east-west Transpeninsular Line from the Atlantic Ocean to its mid-point to the Chesapeake Bay.
    • A Twelve Mile (radius) Circle (12 mi (19 km)) around the city of New Castle, Delaware.
    • A “Tangent Line” connecting the mid-point of the Transpeninsular Line to the western side of the Twelve-Mile Circle.
    • A “North Line” along the meridian (line of longitude) from the tangent point to the Maryland Pennsylvania border.
    • Should any land within the Twelve-Mile Circle fall west of the North Line, it would remain part of Delaware. (This was indeed the case, and this border is the “Arc Line”.)2

The resulting line between Pennsylvania and Maryland follows the parallel at approximately 39 degrees 43′ N, depriving Maryland of about 17′ worth of land.

Charles Mason and Jeremiah Dixon led the survey, many of the details of which are found with embellishments in Pynchon’s Mason & Dixon.  They started in 1763 and finished or stopped in 1767.  Mason was an astronomer, and Dixon, a surveyor, which, based on the information I discussed in the earlier posts about geodetic surveys, seems like the appropriate mix of skills.

How does their measurement function for statecraft?  Violence ceased between the two sides about the location of the border, but after they finished, the border came to symbolize not just the difference between Maryland and Pennsylvania, but the divide between the colonies that permitted slave-owning to the south and those that eschewed slave-owning in the north.  While Penn himself did own and trade slaves, in his will, he granted his slaves their freedom upon his death.  There were Quakers, though, who were more opposed aggressively to slavery, and led Pennsylvania to outlaw slavery in 1781.  Obviously, measurement did nothing to contribute to or to oppose slavery, but it did provide a convenient demarcation between the territories.

In the work that Mason and Dixon did, they marked each mile with stones, and every five miles placed a crownstone – a vertical stone with four sides, with the Penn family coat of arms on the side facing Pennsylvania and the Calvert family coat of arms facing Maryland.  Mason and Dixon ran their latitude line dividing Pennsylvania from Maryland for about 244 miles, but apparently were stopped short of completing the 5 degrees by hostile Native Americans.  That job was completed in 1784 by other surveyors.

As a somewhat peculiar consequence to the job done by Mason and Dixon, their results appear to have triggered a completely different type of geodetic measurement, one beyond the engineering of borders.  To check the accuracy of surveys of this sort, after completing the line in one direction, the surveyors repeat their procedures going back to the original starting point, and expect to see minor random errors.  Instead, Mason and Dixon found systematic errors that were larger than the expected random errors, and systematic in the sense that they all went in the same direction.  When they informed the British Royal Society of this, the systematic errors were recognized as a possible way to prove Newton’s theory of gravity, since the errors might be attributable to the gravitational pull of the Allegheny Mountains.  Newton had raised the possibility of mountains having sufficient gravitational attraction to pull plumb-bobs away from vertical, but ultimately felt that this would be too difficult to measure.

The Royal Society was persuaded by the Astronomer Royal of the time, Nevil Maskelyne, to follow up on this by finding a relatively isolated, symmetrical mountain and using both astronomical sightings to establish vertical, and plumb-bobs to establish at numerous points on the mountain, how much the plumb-bobs varied from vertical, to measure the gravitational pull of the mountain.  Then, using that information, the density of the earth could be extrapolated.  Maskelyn hired Mason to find a suitable mountain to use, which Mason did, but he declined further participation in the project.  The exercise was done using the mountain Mason found located in southern Scotland called Schiehallion between 1774 and 1776.

As a further aside, It seems that during the Peruvian adventure of the French geodeticists, they had tried the same kind of measurement using a volcano near Quito, Chimborazo, had found a deflection in their plumb-bobs of about 8 seconds of arc, but were unable to draw any conclusions other than that this proved that the Earth was solid, not hollow.

During the data gathering phase of the project, surveyors took thousands of bearings around more that a thousand points at various elevations on the mountain.  A mathematician named Charles Hutton was given the task of crunching the data.  To make sense of it, he drew lines connecting the points at the same elevations on the mountain where the bearings were taken, depicting for the first time the contour lines now used in relief maps.  The results of his work were twofold: one, Newton’s theory of gravitation was proven to be correct; and two, a figure for the density of the earth of 4500 kilograms per cubic meter was derived, which is less than 20% less than the currently accepted figure of 5515 kilograms per cubic meter.

Measuring the density of the earth has little to do with statecraft or the engineering of borders to provide useful definitions of proprietorship.  From the dates, it will be noticed that the measurement of Schiehallion occurred at about the beginning of the rebellion of the colonies, as described by the British at the time, which those of us in the United States call the American Revolution.  The Mason-Dixon line had settled one conflict, with greater conflicts to come based on its location and symbolic meaning, but not because a measurement had been performed.  Boundary disputes have led to war and violence far too often, and are usually settled when one side prevails, or when the combatants are exhausted, specified in treaties, with surveying that may be done somewhat afterwards.  But that is not always the case, and the next post will discuss a measurement that led not just to violence, but colonial domination.

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Measuring Learning

Over the past year or more, I have read  a number of disturbing stories of school districts firing teachers and closing whole schools, for under-performing, as well as stories of school districts that have found and fired teachers who have contributed to cheating, as unintended (or perhaps intended) consequences of No Child Left Behind (NCLB).  This has been a springboard for me to reflect on my own schooling: remembering teachers, tests and testing and how and when I really learned.

As it has been quite a while since my early grade schooling, grades 1 through 5 finished in 1956, I remember very little.  I have tried to recall my teachers – of the five homeroom teachers, I remember the names of only three, and can picture only one in my mind.  As to my attitude about tests, I don’t remember having one – that is, I probably did, but I don’t remember what the testing was like, and certainly don’t know if it was different or the same as the testing today.  I do remember class sizes being small, and feeling like some of the teachers augmented my parents’ teaching and examples, not as surrogate parents, but as additional people who cared about and for me.

My sister has “helped” me in my recall effort.  Unsolicited, she sent me my kindergarten report card, a document that can only be described as embarrassing.  In it, I was reported as: “knowing [my] name, address and telephone number; counting to 25; knowing the days of the week; singing in tune; carrying out directions; and being kind polite and thoughtful.”  Oh boy!  No longer completely accurate – I can still count to 25 though – but what a set of metrics.  She also sent a couple of links to YouTube videos done by someone about my grade school containing their home movies from that time, with sentimental music and content that would only reach someone who had been there.  But none of this has helped to spark a recollection about testing and about teacher quality.

From 6th grade to 8th grade, I went to a junior high school, and remember a number of teachers from there.  6th grade homeroom was led by a teacher who, 4 years later, had a psychotic breakdown in front of my brother’s class, driven by the younger brother of one of my friends.  Oh well.  When I had him, he presented bogus, frightening, paranoid information that gave me nightmares for years.  I was and still am wholeheartedly in favor of the prodding that led to his breakdown.  He will remain nameless.

I assume that the testing that was done was largely multiple-choice and short answers, but I don’t have a way to verify that.  I do remember developing an aversion to tests at about that time, with the usual student sweaty palms, tightening of the chest, heart beat speeding up, etc., when it came time to perform even as insignificant a test as a short quiz.

The next year was equally awful, but for entirely different reasons.  I was having some relationship difficulties with my peers, most likely due to the onset of hormonal changes, so I dealt with them by pleading sick a lot and missing school as a result.  My homeroom teacher, whose name is engraved deeply in my memory, was alternately understanding and pissed off at me.  I did something early in the year, relative to the grading process, that set us up against each other.  He had a requirement that every two weeks, we had to turn in a book report on something we had read.  Innocently, I had begun reading War and Peace, and he mocked my ability to read it and understand it.  To cap off his mockery, he said that if I finished it and did a book report on it, he would let me miss the deadline for several weeks, and then give me credit for the entire year’s worth.  Well, I did finish, I did understand it, mostly, and I did a long book report on it that must have been sufficiently coherent for him to honor his challenge.

The result was that he made cracks about me “beating the system”, and how I should turn in more for the rest of the year.  Publicly.  But I was unashamed, and never turned in another one.  However, I did continue reading, and gave him evidence of that.  I missed tests, because I was “sick”, and made them up, with the usual sweaty palms.  And this was about the time that we began to have essays as part of the testing procedures.

His name, as I said, is deeply engraved in my memory, and he is perhaps the only teacher before high school that I would ever want to get back in touch with to thank.  When I think about the lessons I learned that year, they had more to do with my character than any else: I feel that I have more understanding of others, and more “grit” for the handling the adversity as a result.  He never yelled at me, he mostly joked, mocked and cajoled me into being a much better and stronger person by the end of the year.  He is the only teacher that I remember that I did not want to fail in my life after school, not because I wanted to go back and rub success in his face, but because I wanted to be able to show him how much he had helped me, whether intentionally or not.

I was comfortable with tests by the time of 8th grade, because I knew how to cram for multiple choice tests.  I was a good test taker because I had figured out the basics of the system, and had only to apply it to each teacher’s particular style or method.  Essays were harder, but I was fairly fluent and could provide enough relatively intelligent verbiage to get by.

After taking a stroll through my recollections of grades 1 to 8, I find myself questioning the sense of the NCLB act and implementation.  I don’t believe that any of the teachers I had were really all that bad, with the exception of the 6th grade nut-case.  Would NCLB have highlighted his deficient psyche?  I doubt it.  Would any of the NCLB testing have identified the high quality of my 7th grade teacher?  I doubt that as well.

I feel a bit like Garrison Keillor describing Lake Wobegon, where all the kids are above average.  My teachers were all “above average”?  I doubt it, but I have rarely experienced teachers who entered the profession because it was their last choice job option: most appeared to be sincerely interested in teaching their chosen subjects, and more importantly, appeared to care about the students – well, with a few exceptions, but those kids were usually discipline problems or disruptive students – this was long before the days of Ritalin-drugging kids into a quasi-receptive stupor.

Multiple choice, short answer, matching types of questions may have some validity for standardized testing, to see how much, if any, of the water in the trough has been drunk by the horses.  More than anything, such testing seems to me to be useful in making sure that students have been exposed and are absorbing the “facts” which are important for the foundation of thinking, and also for diagnosing failures to absorb them.  However, it does not in any way test whether a teacher has neglected their duty to provide those facts.  I only remember taking three “standardized” tests, but not until high school, when I took the PSAT, once, and then the SAT twice.

A distinction must be made between the standardized tests and what I learned to take tests with multiple choice, short answer and matching types of questions that were done created by my teachers.  In my cramming for these types of tests, I would go over my notes to see which “facts” were mentioned, to see what was important.  But to make sure that I had my understanding correct, once the important ones were identified, I would go to the textbook, because if I had to dispute a wrong answer, almost any of my teachers would have used the textbook either to show me the correct answer, or to accept the textbook answer in favor of one of their own.  Needless to say, almost always, their answer matched the textbook, but once or twice over the years, I was able to improve a test grade by showing how I had gotten the answer from the textbook that had been “incorrectly” marked as wrong.

With the standardized tests, the method was somewhat different, reading the preparation material, taking a big deep breath at the start, and just doing what I could.  This meant using knowledge of answers in many cases, and process of elimination and informed guessing in others, since there was no chance of an appeal, only the chance to try again, hoping to raise the score (which I didn’t).  But I was responsible for the results.  None of my teachers could have been individually faulted or targeted as being at fault for my mistakes or omissions.

A little background about NCLB.  I do have a collection of newspaper columns and articles that discuss some of the effects attributable to NCLB, but for the basic information, I found a book by Diane Ravitch to cover what NCLB is, how it is being used and what the effect has been on education.  The book is titled: The death and life of the great American school system, how testing and choice are undermining education. The title gives away her perspective, she finds NCLB to be destructive of education.

NCLB was intended by Congress to make school districts and teachers accountable for the results of their efforts.  While this sounds like a laudable goal, the way that the bill was constructed and the way that it is being implemented has damaged school systems and education around the country.  The major points are:

  • Each state chooses the tests that they will use, defines three levels of performance, and determines what proficiency is for the tests.
  • Schools receiving federal funding must test English and Math proficiency each year for grades 3 to 8, and once in high school, and separate the test results by ethnic group, race, family income, etc.
  • All states must reach 100% proficiency in their teaching by 2013-2014.  Based on that, the states have set up timelines for achieving 100% proficiency in English and Math, and must show “adequate yearly progress”, AYP, based on their timelines, toward achieving the goal.
  • There are strict sanctions laid out for schools and school districts not reaching AYP each year, and very severe sanctions for not reaching the goal in 2013-2014.  The main reward was to have funding continued so that the school and school district could continue to function the way that it had.
  • All states must also participate in the National Assessment of Educational Progress (NAEP) standardized test on English and Math, delivered in 4th and 8th grades every other year.  The NAEP results are to act as an external monitor of the yearly progress.1

So, NCLB lets the states determine what the test contents are, and most likely, each state will use their own standardized test.  They determine what is considered “proficient”, and they determine their AYP.

While a standardized English or Math test, delivered with multiple choice, short answer, and matching types of questions may provide some insight into the proficiency levels of students, it is hard for me to believe that such a test will really determine how well a teacher has performed – this is too indirect, and there are far too many ways to “game” the system.  Additionally, there are many more factors that determine the performance of students on tests and teachers in classrooms than this method considers.  Yet careers are being destroyed, and workable schools dismantled based on NCLB, without any metric showing educational gains as a result.  In Ravitch’s book, she shows examples of schools and school districts that have registered impressive AYP while the scores on the NAEP have been flat, showing no gains and some losses over the same periods of time for those same schools and school districts.

There are a number of ways to look at NCLB to realize how flawed the whole concept is.  One is to realize that the method for accountability is based on that used by businesses regarding the performance of their sales people only.  If a company’s production line ran into significant difficulties that led to a decline in the quality of their products, but the company ignored the declining quality and only judged their managers on their sales people’s inability to sell “junk”, they would be addressing the wrong problem.  Likewise, if a real estate company were to fire their middle management because their sales people were unable to sell any property during the financial meltdown, again, this would be to ignore what the actual problem is.  But this is the way that NCLB is being used to make teachers, schools and school districts accountable.

The goal is to measure the proficiency of the students in two subjects: using a state-wide standardized test with multiple choice answers, the numbers are gathered, and if the numbers are too low, blame the teachers, schools and school districts.  One might expect other factors to be looked at, but apparently they are not.  The test is designed to measure an individual’s proficiency, yet rolled up, it is used to measure the effectiveness of teaching.

Often, adapting the method of measuring from one discipline to another is done in a manner that misuses the methods and results of measurement.  A prime example is described in an earlier post, in which the book by Stephen Jay Gould is discussed, The Mismeasure of Man.  In what purported to be the same objective data-gathering techniques as were used in the physical sciences, the data turned out to be skewed in favor of the investigator’s biases, and the interpretation of the data was used to misjudge the potentials of people in various “sub-standard” ethnic classifications, and then limit their possibilities.

I alluded to two problems, above, the ease with which the system can be gamed and additional factors associated with testing.  I first came across the ease with which the system can be gamed in the first chapter of Freakonomics, A Rogue Economist Explores the Hidden Side of Everything, by Steven D. Levitt and Stephen J. Dubner.  Based on the description in the book, Dr. Levitt was hired to use data from standardized multiple choice testing gathered over a period of years at the Chicago School District to identify which teachers had a high likelihood of cheating by entering answers for their students.  The negative incentives for low scores among their students for teachers and schools were strict, and there were some positive incentives for high scores and improvements.  He were able to identify a number of teachers who had probably cheated, then repeated the testing of their students with monitors present, and none of the classes were able to repeat their results.  Those teachers were fired.

Entering answers for their students is a particularly egregious way to game the system: many are more subtle, as covered in Ms. Ravitch’s book.  Methods that have evidently been used in response to NCLB have ranged from teaching only to prepare for the test; excluding other subjects and other material related to the test subjects; providing students with the answers; lowering the “proficiency” standard (34% right was mentioned as the number required to be considered a passing grade in one instance); schools that were not public schools have restricted the numbers of likely low-achieving students by setting up hurdles that they cannot jump; flunking the low-performers so that they are sent to a public school to depress the public school score; even encouraging low-performers to stay home the day of the test.  She quotes something she calls Campbell’s law: “The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor.”2

The final chapter in Ms. Ravitch’s book, Lessons Learned, describes what will not improve schools (NCLB is only one culprit), and provides a vision of what students should be capable of when finished with school.  It is not a prescription, rather, it is a guideline: “Students should regularly engage in the study of the liberal arts and sciences: history, literature, geography, the sciences, civics, mathematics, the arts, and foreign languages, as well as health and physical education.”3 While one might disagree with some of this, the goal of all education should be to be to turn out students who are self-reliant, able to think for themselves, and are aware of the culture in which they will be functioning.

As a person who has some of those attributes, I am challenged to figure out where and how I learned these.  Certainly not from any of the multiple choice testing I was subjected to, though perhaps learning how to cope with that sort of situation might have contributed to my overall resilience.  I am certain that my teachers deserve much credit (even the nut-case gets credit for some learning) but also my family does, for they all have always supported me (or put up with me), as do my friends, my work-mates, and my good luck.

The one class that I have always valued the most, in terms of when I learned the course subject material better than in any other, was one that I took while I was in college.  The professor had a specific set of readings that had to be done, and a rigorous set of elements that had to be learned.  For some reason, this structure set me free: I ignored some of the reading that was uninteresting to me, but followed footnotes and references in the readings that were interesting.  In the end, my grade suffered, because I had ignored some of the elements I should have learned, but on the essay questions, for the first time in my life, I enjoyed myself responding to the topics, drawing on all of my digressions (though at the time I considered it research) for my answers.  The professor did not penalize me as much as he probably could have, but that was in part because I must have intrigued him.  A result was that once the class was over, the professor became a friend and mentor for the remainder of my years in college with whom I could discuss almost anything.

Would there have been any way to measure that kind of “success”?  If my grade had been looked at, would he have suffered for my inattention and “research”?  Would it have been fair to judge him on that?  Hardly.

The books cited in this post are:

Steven D. Levitt and Stephen J. Dubner, Freakonomics, A Rogue Economist Explores the Hidden Side of Everything, HarperCollins e-books, 2005, 2006, Adobe Digital Edition September 2009,


Ravitch, Diane, The death and life of the great American school system, how testing and choice are undermining education, Basic Books, New York, N.Y., 2010.

Both are worth the time to read.

1 This list has been adapted from that in Ravitch, Diane, The death and life of the great American school system, how testing and choice are undermining education, Basic Books, New York, N.Y., 2010.  pp. 107-109.

2 Ravitch, Diane, The death and life of the great American school system, how testing and choice are undermining education, Basic Books, New York, N.Y., 2010.  p.171.  Her footnote for this quote is: Donald T. Campbell, “Assessing the Impact of Planned Social Change,” in Social Research and Public Policies: The Dartmouth/OECD Conference, ed. G.M.Lyons (Hanover, NH: Public Affairs Center, Dartmouth College, 1975), 35.

3 Ravitch, Diane, The death and life of the great American school system, how testing and choice are undermining education, Basic Books, New York, N.Y., 2010.  p. 242.

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Marking Time (or at least calibrating it)

While I was reading some research material for the last post, I came across a statement that made me wonder: the statement was that the French group that went to Ecuador to measure the length of a degree at the equator, had pendulum clocks that had to be calibrated for the local version of one second.  Additionally they used their pendulums to measure the force of gravity.  The question that I came up with is, well, how did they do that?

First, calibrating a pendulum.  What would you calibrate it against?  Another pendulum that has already been calibrated?  But what would you have calibrated that one to, ad infinitum.  To understand how to calibrate a pendulum, a bit of time keeping history needs to be looked at.

So, to establish the beginnings of a “clock”, humans have relied on natural processes that happen outside of their control, that are regularly repeating events – oscillations, if you will – such as the rising and setting of the sun.  That represents the daylight portion of one day: the non-light night portion of the day is the only other portion, so that is a fairly simple oscillation to calibrate to.  How do you measure a day as a unit? You pick an arbitrary start – sunrise, sunset, midday, midnight – and each day unit is the duration between a rise and the next, or a setting and the next, etc.  A fairly simple cycle, and a likely starting place for time keeping.

What would be next?  There are some other cycles that were apparently untangled by several early cultures.  Herodotus in his Histories, Book 2, states:

As to human matters, they all agreed in saying that the Egyptians by their study of astronomy discovered the year and were the first to divide it into twelve parts…the Egyptians make the year consist of twelve months of thirty days each and every year intercalate five additional days, and so complete the regular circle of the seasons.1

However, he is not completely reliable and evidently didn’t learn from “they” about the Egyptian lunar calendar.  His informants, the “they” who “all agreed” he says were priests he questioned in Memphis, Thebes and Heliopolis.

As to how they did this, he does not address the question, but one would probably not be too far off-base by figuring that over a period of years, the pattern of the stars in the sky would have been recognized as being essentially unchanging.  Once that was figured out, counting the days between a repeat of the stellar positions would only have required some patience and diligence: one slash on a papyrus per day would do it.  By the time of Herodotus, though, this exercise had been performed by the Babylonians and the Chinese as well, for both cultures had calendars before 450 BC.

At this point we have the year roughly figured out, and the day as the unit.  How about dividing up the day somewhat finer than roughly half daylight and the other half dark?  Establishing a midpoint for the day could be done with a very simple tool: a gnomon, or a plain straight stick stuck into the ground vertically, would project a shadow that was always “longest” at the approximate middle of the day.  This way daylight could be divided into two portions of time, before “middle” and after “middle”.  The initial shadow of each day would also put boundaries on the time of daylight within a day, and would range from shorter times of sunlight in the winter to longer in the summer.  How would this time best be divided up into what would become “hours”?

Using a sundial would not necessarily suggest dividing the time of daylight or darkness into 12 portions, and I have found only one possible suggestion as to why there was such a division, rather than dividing it into, say, 4 portions or 6 portions, even 9 portions.  The suggestion comes from a Wikipedia article which I quote:

Astronomers in the Middle Kingdom (9th and 10th Dynasties) observed a set of 36 decan stars throughout the year. These star tables have been found on the lids of coffins of the period. The heliacal rising of the next decan star marked the start of a new civil week, which was then ten days. The period from sunset to sunrise was marked by 18 decan stars. Three of these were assigned to each of the two twilight periods, so the period of total darkness was marked by the remaining 12 decan stars, resulting in the 12 divisions of the night. The time between the appearance of each of these decan stars over the horizon during the night would have been about 40 modern minutes. During the New Kingdom, the system was simplified, using a set of 24 stars, 12 of which marked the passage of the night.2

Not exactly an intuitive way to break time into smaller units, but evidently pervasive enough to be considered as a standard.  By the way, “decan” as used above appears to be derived from astrology (!) indicating a time of about 10 days between first rising.  The rest of this reference contains some mystery, since the Middle Kingdom of Egypt, which I believe is being referred to, is the 11th through the 14th dynasties, and ran roughly from 2055 BC to 1650 BC.  If this is the correct timeline, then, days divided into 12 or 24 has been around a while.

We saw, in an earlier post, that Ptolemy gets credit for breaking the degrees on his maps into minutes and seconds.  It seems likely that when the effort was made to subdivide hours, minutes and seconds were adapted.  Not too sure when, however, since “minutes” and “seconds” hands on clocks don’t show up until about the 15th century, and their accuracy was questionable.

These are some apparent regularly repeating oscillations that could be starting points for calibration except for a few inconvenient facts about the main organ of calibration, the earth in relation to the sun, which raise some questions:

Question one: should the hours be equal all the time, or should daylight contain 12 hours and the nighttime 12 hours, whether it is summer or winter?  Based on the earth’s position in its course around the sun, the angle of its axis of rotation, approximately 23.5 degrees, has the southern hemisphere more directly aimed at the sun or the northern hemisphere more directly aimed at the sun.  This leads to the sun being in the sky much longer in the summer in the northern hemisphere.  Should that daylight time be divided into 12 hours, the same as the shorter daylight days of winter?

We are used to all hours being the same length and knowing that the sun rises at an earlier or later hour throughout the year, but our answer is not the only answer that has been used.  Dividing daylight hours and nighttime hours into 12 of each was used as the standard answer for thousands of years, and only began to change in Europe as mechanical clocks became better at telling equal hours.

Question two: how do you accommodate for the fact that there are times during the year when the sun appears to move faster and times when the sun moves slower, and that it does not appear in exactly the same place at midday?  Using the elapse of 24 hours, which is the mean solar day would be difficult for calibration.  Over the course of the year, the sun at midday would appear in the pattern of an analemma.  See the following website for an adequate explanation of this:

So, if you were trying to make sure you made a sighting of the sun at the exact highest point of the day on two consecutive days to establish the length of 24 hours, that would not be precise enough.  You might be able to do it by establishing the mean solar day, take the amount of time for each day and essentially average it.  But doing that too might be difficult – how can you tell if 24 hours today is not exactly the same as 24 hours two days ago?

Question three: If you decided to use the stellar background, you would run into a problem here as well.   The mean solar day is approximately 24 hours long.  The sidereal day (based on the rotation of the earth relative to the stars) is approximately 23 hours and 56 minutes long.  Why is that?  Over the course of a year, any reference star that you chose would reappear in exactly the same spot, relative to the earth, approximately 365 (365.2422 more precisely) days later.  In between, each day, it would arrive at the same spot a little bit earlier, by 3 minutes and 56 seconds.  That is very close to the difference of one degree per day, which makes sense with a 365 day year.  If it advances just a shade under a degree per day, the sum of all of those is 360 degrees in 365 days.  But that makes the calibration difficult as well.

There are some additional cycles that could be looked at, but either their period is too long (26,000 year precession) to be practical, or doesn’t line up quite right over the course of a day, such as the moon.  It would be possible to base a calendar on the moon’s reappearance in exactly the same spot, relative to the earth and the sun, which takes about 29.5 days, but it doesn’t evenly divide into 365.2422.  (365.2422 / 29.5 = 12.3811)  Even trying to set up a clock based on the tides, affected by the moon, would run into inconsistency problems.

With this information before us, we are now able to determine that there is no way to calibrate a second.  Right?

If the story by Vincenzo Viviani is correct, Galileo realized that the period of the swing of a pendulum was pretty close to equal from swing to swing.  The tale has him realizing it while noting the swinging of a chandelier in the cathedral in Pisa, probably in a service that was the usual boring service.  An active mind searching for things to occupy it might just have noticed the swinging, put a finger on the wrist to feel the pulse, and made the realization.  While I may be projecting my own activities during that sort of situation, the story goes on to describe the use of a plumb line with a bob to understand the details of the insight more.  When he found the formula that describes the period as being proportional to the square root of the length of the pendulum, it would not be too difficult to imagine trying to find a pendulum length that was close to one second long.  But to do so, you would have to have some idea of how long a second is.

Timekeeping devices, while initially based or calibrated on the sky, had been brought down to earth by Egyptians, Babylonians, Chinese and Greeks, all of whom had versions of water clocks, or clypsydrae (singular: clypsydra).  Other devices to employ the same or similar means (material being removed from a container by gravity or fire) were candle clocks, incense clocks and sand hour clocks.  Sand hour clocks, also known as hour glasses could be made fairly precise by either adding or removing a little bit of sand after determining the right amount of time for an hour: if twenty-four were made with equal volumes of sand, and one turned over as soon as the last one finished in succession for a day, presumably one could start as the sun reached its highest point and the last of the 24 would just be emptied as the sun reached its highest point on the next day.  (It could probably be done with one hourglass turned over 24 times?)  This could be repeated until, by adding or removing a few grains of sand, it made the hour glasses pretty close to accurate.  In fact, it turns out, Magellan, when his fleet circumnavigated the globe in 1522, had eighteen hourglasses on each ship.

This would at least give a fairly accurate version of one hour.  Now, to subdivide that into minutes and seconds wouldn’t be too complicated.  Set up a pendulum that is the length you think is a second, start it swinging at the same time as you turn over an hourglass, and count to 3600 (60 X 60).  Too hard?  Well, in the 17th century, a Jesuit, ‘…Father Giovanni Batista Ricciolli persuaded nine fellow Jesuits “to count nearly 87,000 oscillations in a single day.”‘3 3600 seconds in an hour times 24 hours is 86,400, so knowing Father Ricciolli’s reputation as a scientist, my guess is that the group would be broken into four groups of two, and each of the two would be responsible for their own count, which could be compared later.  They would rotate the shifts of counting.  That’s a guess, but the ability to do this, figure out that the pendulum was too long or too short, adjust it, try again might be an activity that would be used to calibrate the correct length, which it turns out is about a meter, or 39. 37 inches.

The first pendulum clock was designed and commissioned by Christiaan Huygens in 1656.  This clock and those made shortly thereafter increased the accuracy of mechanical clocks from about 15 minutes a day to about 15 seconds per day.  In 1671, Jean Richer took a pendulum clock that was accurate in Paris to Cayenne, French Guiana and discovered that it was 2.5 minutes per day slower.  His conclusion as to why was that gravity was not as strong in French Guiana.  A few years later, Newton speculated that due to the effect of centrifugal force on the rotation of the earth, there was a bulge at the equator and the poles were flattened.  The Cassinis of the French Royal Observatory disagreed, and a result was the trip to Peru to measure the shape of the earth.  So we are now almost to the point where we can answer the questions raised at the beginning of the post: how did the French workers go about calibrating their pendulum clocks?

Among the factors that affect the operation of a pendulum clock are gravity and temperature.  If the density of the material under the pendulum is high, as it would be under mountains, this will raise the gravity, thus speeding pendulums up a bit.  The opposite effect, that of Cayenne, also affects the rate of oscillation of pendulums.  Temperature can affect pendulums: Huygens used a solid rod for his pendulums, instead of a string or something similarly flexible, since the flex in flexible pendulums caused erratic swing periods.  But with a solid rod, heat can lengthen the rod, and cold cause it to contract, lengthening or shortening the period of its swing.  With these known factors potentially at play, a clock with a pendulum that had a one second period when it was taken to Quito, Peru, what would be the effect?  Depends on the density underneath, and the temperature.  By the time the French left for South America in 1735, at least two solutions to the temperature had been developed, and possibly a third.  However, from my research, I cannot tell if any of these had been employed in their pendulum clocks.

Presumably, they would have tried to control for temperature, and then assume that the differences would be a result of gravity.  So once with their calibrated-for-Paris clocks they had measured whether their clocks were slower or faster than the mean solar time in a few consecutive days, and used the result in two ways: one, to estimate the force of gravity beneath the clocks, and second, how much longer or shorter to set the pendulum to match 86,400 seconds to the time between two solar meridian crossings.  I’m certain that the amount of time their task took was lengthened by repeated calibrations.

There was an alternative that might have been used, but I cannot tell if they used it.  By the time they left, the transit of the moons of Jupiter had been precisely timed: when several of the moons went behind Jupiter and reappeared had been timed precisely enough that one astronomer, Ole Roemer in 1676, had proved that the speed of light was finite using the occultation of Io, one of Jupiter’s moons.  He timed the occultation when the earth in its orbit was nearer to Jupiter, then when earth was at a different part of its orbit, much further from Jupiter, and noted that the times of occultation took much longer when the earth was farther from Jupiter.  This he correctly assumed was because the distance was so much greater, and reasoned that because of that, light had a finite speed and was not infinitely fast.

Having precise timings for the transits or occultations of the Jovian moons meant that the French could have matched the seconds of their pendulum clocks to the amount of time for a transit or occultation, and adjust the clocks to match the standard timing.  Did they do that?  I don’t know for sure, but my guess is that they would have tried that in addition to meridian crossings by the sun.  They were well trained in the astronomical tasks of the day, so I’m sure that they would have tried any and possibly all methods available to them to establish the accuracy of their clocks, to insure the accuracy of their surveys.

The process of calibrating a pendulum clock, then, is not very straight forward.  At best it is an iterative process: check the timing against a known oscillation, if it is not correct, make a correction in the length of the pendulum that should either speed up the period or slow it down to match the known timing device, check it again.  If it is not correct, repeat until it is.

Choosing the reference oscillation is the key.  Pendulums were used to establish the standard until quartz oscillations were built into quartz clocks and then watches.  Now, the standard is based on the oscillations of cesium atoms in cesium clocks, and hydrogen oscillations in hydrogen maser clocks.  They are well beyond the accuracy of pendulum clocks.  The arrangement at the Naval Observatory is a Master Clock fed information by a set of cesium and hydrogen masers, which corrects itself based on the inputs.  The Master Clock “…keeps time to within one hundred picoseconds – one hundred trillionths of a second – over the course of each day, every day.  Had it been set when the dinosaurs went extinct, 60 million years ago, it would have gained or lost no more than about two seconds.”4

1 Herodotus, The Histories, Penguin Books, London, England.  Translated by Aubrey de Selincourt, 1954. Revised edition, 1972.  Revised edition with new introductory matter and notes by John Marincola 1996, further revision 2003. P. 96.  I had found so many “measurement” references to Herodotus that I bought a copy and am reading it.  Since Herodotus died around 425 and 420 BC, the book is surprisingly lively for 2400 years old.

4 Falk, Dan, In Search of Time, The Science of a Curious Dimension, Thomas Dunn Books, St. Martin’s Press, New York, N.Y. 2008. p.56.  Full of good information and really poor proofreading.

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2 Geodetic Surveys

In the last post, I discussed some early surveying instruments: merkhet, plumb line, knotted ropes, groma, decempedae and dioptra.  Of these, the dioptra that was designed by Heron (also known as Hero) of Alexandria, and reconstructed by Schone, appears to have contained most of the capacities that were used until quite recently.1

From Schone’s reconstruction, it appears to have included alidade – a rotating sighting mechanism – in this case not a tube but slits at the opposite ends of the alidade, beneath which is a  horizontal circular plate that could be marked with angle designations for registering horizontal angles.  The alidade and circular plate were mounted on a hemispherical plate that could be used to adjust the vertical angle of the alidade and horizontal plate, and this entire apparatus could be rotated on the pole it is mounted on.  Rotating on the pole would allow a specific point on the horizontal plate to be aligned with a specific reference direction or point.  The pole or post on which this version of dioptra sits could be made vertical with a plumb line, although there is none in evidence in the reconstruction drawing.

It is not clear if a device was ever built that looked and worked like Heron’s dioptra.  For the next approximately 1500 years, the devices used for surveying performed the various functions as separate pieces of apparatus: geometric squares (like carpenter’s squares); circumferentor/surveyor’s compasses, a form of graduated circle; and graphometers, semi-circles (like protractors) are the reported surveying tools, as well as ropes for distances.  The surveyor’s chain did not come into standard use until about the time that theodolites were developed.2

As a slight digression, during that time, it is possible that astrolabes were adapted to surveying: an astrolabe is the marriage of a planisphere, a flat circle representing the heavens, with a sighting device (an alidade) held with the flat circle/planisphere vertical, and was used for “…locating and predicting the positions of the Sun, Moon, planets, stars: determining local time (given local latitude) and vice-versa; surveying; triangulation; and to cast horoscopes.”3 If a surveyor had an astrolabe and held it horizontally, the user could sight along the alidade for surveying purposes.  This marriage of planisphere and alidade was allegedly performed by Hipparchos around 150 BC.

This leads to yet another digression, known as the Antikythera mechanism, subject of the website of the Antikythera Mechanism Research Project.

I have pasted the abstract from the beginning of an article on the website, but removed the footnote numbers:

The Antikythera Mechanism is a unique Greek geared device, constructed around the end of the 2nd Century BC. From previous work…it is known that it calculated and displayed celestial information, particularly cycles such as the phases of the moon and a luni-solar calendar. Calendars were important to ancient societies…for timing agricultural activity and fixing religious festivals. Eclipses and planetary motions were often interpreted as omens, while the calm regularity of the astronomical cycles must have been philosophically attractive in an uncertain and violent world. Named after its place of discovery in 1901 in a Roman shipwreck, the Mechanism is technically more complex than any known device for at least a millennium afterwards. Its specific functions have remained controversial… because its gears and the inscriptions upon its faces are only fragmentary. Here we report surface imaging and high-resolution X-ray tomography of the surviving fragments, enabling us to reconstruct the gear function and double the number of deciphered inscriptions. The Mechanism predicted lunar and solar eclipses based on Babylonian arithmetic-progression cycles. The inscriptions support suggestions of mechanical display of planetary positions…now lost. In the second century BC, Hipparchos developed a theory to explain the irregularities of the Moon’s motion across the sky caused by its elliptic orbit. We find a mechanical realization of this theory in the gearing of the Mechanism, revealing an unexpected degree of technical sophistication for the period.4

This means that the estimated time for the Antikythera Mechanism is approximately the same time as when Hipparchos is reputed to have made the first astrolabe.  Quite an industrious fellow, not to say impressive in his accomplishments: could the Antikythera Mechanism be attributed to him as well?

Given the information in these digressions, it certainly looks like the skill necessary to make a version of Heron’s dioptra was available by his time, 10 to 70 AD.  So it could have been made, but it looks as if the tool and the knowledge about making such was gone not too long after that and not rediscovered until the 1500s.  No wonder they are called the Dark Ages.  Actually, the only thing missing from the dioptra of Heron was a better sighting mechanism: a telescope with the optics to extend the field that could be surveyed.

Going from separate tools being used for surveying to an integrated instrument shows up in the appendix to a book, Margarita Philosophica, by Gregorius Reisch, in 1512, and the appendix is attributed to Martin Waldseemuller.  Waldseemuller also built a device in that same year, and called the device a polimetrum.  The Digges, father and son, published a surveying textbook in 1571, the father, Leonard, having written it and passed away, so his son, Thomas, had it published, and in the textbook, they renamed the polimetrum, the theodolite.  Shortly thereafter, versions of theodolites were made with tripods and compasses.  In 1725 a telescope was added to the theodolite as a sighting mechanism, 115 years after Galileo demonstrated a telescope to the Venetians.  In 1787, an Englishman named Jesse Ramsden introduced a theodolite with angular scales on it that were accurate to within a second of arc.

There have been further improvements since then, always with the goal of greater precision, accuracy and ease of use.  Surveys were extended from making maps detailing how to get from here to there and cladistal maps – maps of property boundaries, to mapping the large areas, called geodetic mapping.  Geodetic mapping became important because people needed to know more about the earth as a result of the discovery of a new continent and the overall exploration of the world.

The surveyors who performed geodetic surveys until the late twentieth century had to know not just how to use the instruments but several branches of mathematics: algebra; calculus; trigonometry, especially the law of sines; in addition to plane geometry,5 be ready to perform large numbers of calculations and had to know the latest thinking in astronomy.  They also had to be very tough.  Some of the tasks that surveyors were put to work on required uncommon fortitude.

First, little of the math.  Gemma Frisius, 1508 – 1555, developed a method which is at the core of surveying, called triangulation.  He was on the faculty of medicine at the University in Leuven for most of his life, but made globes and mathematical instruments in addition to teaching.  One of his students was Gerardus Mercator, of the famous Mercator projection.

In 1533, Frisius published a theoretical method of using triangulation to measure distances, based on triangles.  If you know the distance between two points, and are able to measure the angles between the line connecting the two points and lines from each to a third point, using sines and cosines, you can determine the distance from each of the two original points to the third.  Frisius’s example was a conceptual way to measure from Brussels as one point and Antwerp as a second point, knowing the distance between Brussels and Antwerp, and finding the angles to Middleburg, or to Ghent, neither of which could actually be seen from Brussels or Antwerp.

If you know the distance from A to B, the angle at A, and the angle at B (ABC, not ABD or CBD), you have enough information to figure out the distance from A to C and from B to C.  Since all triangles that have the same set of angles are always the same shape, the lengths of their sides will always be in the same proportion.  If a triangle has sides that are 2 units, 3 units and 4 units in length, then any triangle with the same angles will always have its sides related to each other as 2:3:4.

First, with two angles of known size, the third angle is also known, since all triangles in plane geometry have exactly 180 degrees, simple addition gives the sum of the angles at A and B, and that sum subtracted from 180 gives you the angle at C.

In this case, angle A = 40 degrees and angle B = 75 degrees.

C = 180 – (40+75)

C = 180 – 115

C = 65 degrees

In my example, the length of A to B is 9 cm.  Because the ratios of the sides of this shaped triangle are constant, the length from A to B could be 9 kilometers or any other length.

Mathematicians have developed a concept called “sine”.  The sine of an angle in a  triangle is based on the ratio of the sides of a triangle with that angle in it and a right angle (90 degrees) in it as well.  In our example, the sine of angle A in the right triangle defined by ABD is the ratio of the side opposite that angle  to the hypotenuse, the longest side of that right triangle, so sine A = opposite/hypotenuse or o/h.  Another ratio of some interest is “co-sine”: the ratio of the side of the triangle which is the other part of the angle to the hypotenuse, that is, adjacent to the angle, to the hypotenuse.  Cosine A = adjacent/hypotenuse, or a/h.  Simple, no?

What we did to the example above, is run a line from angle B to the opposite side that is perpendicular to that opposite side, dividing the original ABC triangle into two right triangles, ABD and BCD.  Then the sine of angle A allows us to know how long line BD is, since sine A = BD/AB.   At the same time, the sine of C represents the ratio of BD to BC, or sine C = BD/BC.  Once we determine the length of BD from the sine of A times AB, we can divide BD by the sine of C to get the distance from B to C.  We also have sufficient information to generate the distance from A to C.

Sine A = .6428    (.6248) X 9 = 5.7852, so BD = 5.7852 centimeters

Sine C = .9063    (.9063) / (5.7852) = 6.3833, so BC = 6.3833 centimeters

To get AC, we can use cosines:

Cosine A = .7660  (.7660) X 9 = 6.8940, so AD = 6.8940 centimeters

Cosine C = .4226  (.4226) X (6.3833) = 2.6976, so CD = 2.6976 centimeters

AC = AD + CD  (6.8940) + (2.6976) = 9.5916 centimeters

It is a multi-step process, but once you have results for the original problem that Frisius described the solution to, the results can be used to develop more distances by establishing the angles to more points and using the already developed distances to understand distances to new points.  Some additional steps to insure accuracy were developed by Willebrord Snellius (his Latinized name – un-Latinized, he was Willebrord Snel van Royen, but by custom is referred to as Snell), and published in 1617 in a book called Eratosthenes Batavus (The Dutch Eratosthenes).  Snell used his triangulation method to generate the distance between two towns in the Netherlands, which represent one degree on a meridian running through them to the north and south poles.  Based on the distance derived and multiplying by 360, he was able to provide a new measure of the circumference of the earth.  As he also developed a more accurate way to calculate pi, he was able to use that to establish the radius of the earth.

If you start, as we did, with the distance from A to B and the sizes of the angles at A and B, using this method can fill in the distances from A to E, C to E, C to F, and E to F.

Snell not only extended the method, but provided a way to correct for the curvature of the earth, and a strategy for surveying: survey large triangles first, then locate the points within the triangles.

So much for the math – the conceptual tool.  Now the tasks:

Tasks that surveyors were given, or set for themselves, were quite ambitious.  Within Europe, surveys were conducted using the triangulation method.  These occasionally ran into problems or resistance, for the peasants were certain that if their holdings were surveyed, the result would be higher taxes – and in a number of cases they were right.  Surveyors were often agents of the landlords and the aristocracy, if not the government.  Some surveyors were killed by angry mobs.

France seemed to have the ability to set nearly impossible tasks: the French sponsored an expedition to Peru in 1733, which prepared for a couple of years and left Europe in 1735.  International relations were involved: the area in which they wished to do the work was in Spanish-held South America, and the proper authority from the Spanish king had to be obtained.  The portion of South America held by the Spanish was called the Viceroyalty of Peru, and included a number of “audiencias” – districts.  The specific audiencia that would become Equador was called the Audiencia of Quito.  The expedition involved the French traveling through other audiencias (and a few of the party descended the entire Amazon, requiring passes from the Portugese).

The stated task of the expedition was to settle a question raised by the clockwork universe of Isaac Newton: what shape is the earth exactly?  Jacques Cassini, the head of the Royal Observatory in Paris believed that the earth was elongated at its poles.  Newton’s work provided a different hypothesis: that the earth bulged at its equator and was flattened toward its poles as a result of its rotation and gravity – similar to the shape of Jupiter that astronomers had already observed.  The task given to the French surveyors was to measure a degree of meridian as near to the equator as possible.  This “simple” task took them about 10 years while coincidentally battling all sorts of jungle horrors: poisonous snakes, jaguars, insects, diseases, inclement weather, altitude sickness and inhospitable people.

Some of the instruments and tools used were telescope-equipped quadrants for measuring angles, a zenith sector for celestial observations, a seconds pendulum to gauge gravitational pull at the equator, and an iron bar exactly 1 toise in length.  A toise was six Paris feet (6.39 English feet).  “In addition to their quadrants and zenith sector, their scientific equipment included two telescopes, survey chains, watches with second hands, land and sea compasses, thermometers, a rain gauge, several barometers, a galvanometer, an instrument for measuring the blueness of the sky, and another for determining the boiling point of water.”6 They did not have a theodolite, but did have instruments that performed the functions of theodolites.

Their procedure was to choose the first two points in a place where they could measure the distance between them to establish the baseline, then use their telescopes and zenith sectors to determine the exact latitude of each point, calibrate their pendulum clocks (measuring the effect of gravity at the same time), then determine the exact longitude of each of the points.  Next they had to lay out the line between the points and determine the length of their baseline.  Then, a third point would be determined, looked at from both of the original points to determine the angle between the line of sight to the new point and to the other known points.  This set the original triangle, from which they worked their triangulation measurements to determine the length of a degree of latitude and a degree of longitude at or as near as possible to the equator.

The result: those who survived the hazards and the toil proved that, in fact, a degree of latitude and a degree of longitude differ in an amount at the equator that is consistent with a bulge at the equator.

Triangulation near Quito: the faint straight lines are the lines surveyed7

About 50 years later, at the time of the French Revolution, the savants of France had been advocating a standardized system of measurement, which, with the Revolution, became a goal of the government.  While they could have used a standard that integrated both time and length, a pendulum of sufficient length to swing once per second, which became an early version of the meter, the savants felt the best measure would be one based on the size of the earth.  So, the meter was to be “…based on one ten-millionth of the distance from the North Pole to the equator as established by a survey of the meridian that ran from Dunkerque to Barcelona,”8 which, incidentally was the meridian of Paris.  So, the task assigned to the surveyors was to measure that portion of the meridian that ran from Dunkerque to Barcelona which was a known percentage of the entire meridan so that the overall distance from North Pole to equator could be established and then used to derive the standard one meter.

Again, a simple task, done in the years between 1792 and 1806, a time of relative calm?  Well, no, not really.  Various of the people associated with the survey were accused of having Royalist sympathies; a few of them lost their heads to Madame Guillotine; others their jobs or positions, and some few prospered.  Through it all, those charged with surveying tasks used their telescopes to establish latitude, used pendulum clocks to establish correct time and thus their place on the meridian, etc.  The work was astonishing in its complexity – scientific and political – and the devotion to duty of those working on the task inspires awe.

The principal instrument used for sighting and establishing angles was called a Borda Repeating Circle, which had two telescopes that could be moved separately around the same axis.  It was very precise, but due to the changes in measurement systems as a result of the French Revolution, the entire circle was divided into 400 “grads”, and Borda spent years developing the tables to support it.  The Borda Repeating Circle was tested against the Ramsden Theodolite, and the two were deemed to be essentially equal in their accuracy, but since the Borda Repeating Circles were 20 pounds to carry and set up, while the Ramsden Theodolites were about 200 pounds in weight, not to mention using an “English” measurement scheme, the Borda Repeating Circles were used for the measurement of the meridian for the meter.

Map showing the triangulation from Dunkerque to Barcelona9

The result of their work established a standard length for the meter, but it was actually adjusted from what they had concluded.  By the time they finished, Napoleon had thrown out the metric system, (a day with 10 hours of 100 minutes, and minutes with 100 seconds???), but kept the meter.  As Alder states, “The French were not only the first nation to invent the metric system, they were also the first to reject it.”10

There are two other surveys that I would like to discuss, but will wait to discuss them in another post.

The interplay of the elements in this post is emblematic of how measurement can be done: purpose blended with the conceptual tools and the instruments to perform the measurements that provide deeper knowledge about the world.  But the instruments cannot be developed without conceptual tools, and the conceptual tools only become useful once there are instruments to confirm the use of the conceptual tools.


I would like to thank Robert Zweben for suggesting The Mapmaker’s Wife to me.

I also have to thank Allyndreth Devlin, my daughter, for many things and much pleasure, but right now for Geometer’s Sketchpad, the program I have used to create diagrams such as the triangulation ones.

1 Quite recently: surveying has changed drastically because of airplanes, electric theodolites and because of GPS, all of which will be discussed in a later blog.

2 The surveyor’s chain is actually a standard measure.  I should have guessed.  It is a chain with 100 links, the total length of the chain being 66 feet.  There are 10 chains in a furlong, and 80 chains in a mile.  An acre is 10 square furlongs in area, so 660 feet ( 10 furlongs) times 66 feet (1 furlong) equals 43,560 square feet.  I always wondered how the length of a mile and the size of an acre were chosen.  The chain was first standardized in England in 1620 by a clergyman named Edmund Gunter, therefore the instrument used was know as a Gunter’s chain. More detail is available at:’s_chain

4 T. Freeth, Y. Bitsakis, X. Moussas, J.H. Seiradakis, A.Tselikas, E. Magkou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M.G. Edmunds, Decoding the Antikythera Mechanism: Investigation of an Ancient Astronomical Calculator, Nature, Volume 444, Issue 7119, pp. 587-591 (2006).

5 Depending upon the amount of land being surveyed, the surveyor might also have to know spherical geometry and spherical trigonometry, to account for the curvature of the earth.

6 Whitaker, Robert, The Map Maker’s Wife, Delta Trade Paperbacks, 2004, New York, N.Y. p. 63.  The earlier list of instruments can be found on page 48 in the same book.

7 Whitaker, p. 132

8 Alder, Ken, The Measure of All Things, Free Press, a division of Simon & Schuster, Inc., 2002 New York, NY p. 89.

9 Alder, frontispiece

10 Alder, p. 261.

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Surveying and Mapping

In the past few posts, I’ve been discussing maps and mapping.  I would like to build on those discussions by asking what would be needed in order to create a map.  To answer that question requires answering a few other questions:

What is the purpose of the map?

How is the map supposed to be used?

What information is required for the map to be useful for that purpose?

If we look at early maps for some elementary answers, we find that the earliest maps appear to have one of two purposes, either to provide directions on how to go from here to there; or to provide a way to understand some form of property rights or rights of land usage.

If the purpose is to insure capture of sufficient paths, directions and landmarks to remind ourselves or to insure that others are able to travel from here to there, then one set of information is important.

If the purpose is to delineate land divisions for ourselves and others to minimize conflict and insure that each member of the group knows which plots of land to work, then a slightly different set of information is necessary, though with some overlaps with the directions map.

The map for guiding travel would need:

  • Orientation – starting point, landmarks, easy way to identify geographical or geological elements encountered on the way.
  • A sense of scale – the path along the stream that is followed for three miles can’t be significantly shorter on the map than the three mile stretch between the two peaks to be traversed later in the journey.
  • Some labels or indications of important things to know, like “Ciudar – Region des Serpientes” (Be careful – area of snakes), an actual sign on the trail to Macchu Picchu that I encountered, indicating an area of bushmaster and coral snakes, or the name of the pass we had to cross a day or so earlier, Warmiwanusqa Pass, on the map, which translates to Pass of the Dead Woman – something else to know and exercise caution about?
  • Something to put the map onto – parchment, hide, clay tablet, paper, and a tool to use for inscribing the map.

Are there specific concepts that would be important for making the map?  Concepts such as distances, which could be time related, as in “two days travel” would probably be helpful.  Of course, if labels were to be used, language would have to be available, and a way to make symbolic marks indicating what would have been said by one experienced in the journey to a neophyte.

Are there measuring tools that would be needed?  I’m not sure, unless the intent was to make the scale absolutely reflect the actual distances, in which case most likely a ruler and possibly a dividers.  For orientation, a compass would be nice but compasses were developed fairly recently – within the last thousand years or so.

One possible tool to assist with orientation is a gnomon which is basically a straight shaft set vertically into the ground.  The word “gnomon” has come to mean the vertical portion of a sundial that throws a shadow, but as we are only interested in orientation, we don’t actually need the “hours”.  With a shaft acting as a gnomon set vertically into the ground, one could determine, from its shadow, the mid point of the day.  The direction of the shadow at the day’s middle would point in the direction we call “north” (at least for places above the Tropic of Cancer), and would line up approximately with the star at which the north axis pole points.  This could be used to orient the map, with an arrow indicating which direction the gnomon points at mid-day.

If that were to be used, another measurement tool would be required when setting up the gnomon, a plumb line to make sure the gnomon is as close to vertical as possible.

A map for establishing land use rights would need some of the above measuring tools and concepts, but in addition:

  • Geometry – the ability to make shapes on a map that reflects the shapes of the parcels of land, and the ability to make sense of those shapes.
  • Straight-edges – for marking the boundaries, on either the land or on the map.
  • Perhaps a way to deal with angles, whether 90 degrees or otherwise.
  • Some standards for lengths, and some mathematical understanding of how lengths relate the areas to be worked, as in length times width equals area (L x W = A).

There is little doubt that these requirements were met by both the Babylonians and the Egyptians at some point in their development.  The following is a double quote, since the book where I found this text content was written by George Gheverghese Joseph, The Crest of the Peacock, and in the book he includes a quote from Herodotus, The Histories (5th Century BC, Greece).  The next paragraph is from Herodotus, and the second one from Joseph:

Sesostris [Pharaoh Ramses II, c. 1300 BC] divided the land into lots and gave a square piece of equal size, from the produce of which he exacted an annual tax.  [If] any man’s holding was damaged by the encroachment of the river…The King…would send inspectors to measure the extent of the loss, in order that he might pay in future a fair proportion of the tax at which his property had been assessed.  Perhaps this was the way in which geometry was invented, and passed afterwards into Greece. (Herodotus, The Histories, p. 169)

He [Herodotus] also tells of the obliteration of the boundaries of these divisions by the overflowing Nile, regularly requiring the services of surveyors known as harpedonaptai (literally ‘rope-stretchers’).  Their skills must have impressed the Greeks, for Democritus (c. 410 BC) wrote that ‘no one surpasses me in the construction of lines with proof [?], not even the so-called rope-stretchers among the Egyptians’.  One can only suppose that ‘lines with proofs’ in this context refers to constructing lines with the help of a ruler and a compass.1

There is some further discussion of the Egyptian rope-stretchers in Dilke’s book, Mathematics and Measurement, in the British Museum series, Reading the Past.

When the floods receded, many landowners’ boundary marks had inevitably been washed away, so it is important that surveying should be carried out immediately.  There are Egyptian representations of surveyors employing knotted ropes (the knots indicating sub-divisions of linear measurement), the merkhet (a split centre-rib of a palm-leaf, used for sighting), and measuring rods.  The priests inaugurated this rapid re-survey of the land, which had to be ready for winter cultivation.  There is no evidence that land survey maps were used in dynastic Egypt for this operation.  But by means of exact area measurement and verbal descriptions, the status quo was re-established.2

The geometry required would be basic plane geometry concepts, such as squares, triangles, polygons, circles and areas.  Also required would be some understanding of equivalencies, such as a square that has sides that are twice that of another square, are equal in shape and angle, and the area is four times that of the smaller square.  Similarly shaped triangles have the same ratio in area, so two triangles with the same shape and angles, but one with sides twice the length of the smaller would be four times the area.  Additional understanding of ratios: if a gnomon casts a shadow at noon that is 2 feet long, and is 3 feet tall, then the tree nearby that casts a shadow at noon that is 20 feet long is 30 feet tall – a form of indirect measurement.  These concepts would require some measuring concepts and tools, with some standardized units of measurement.

The tools mentioned in the paragraph quoted from Dilke, above, knotted ropes, merkhet, and measuring rods, need a little explanation.  The knotted ropes, with the knots indicating sub-divisions, are most likely, ropes of a standard length, with the knots at specific spots along the standard length, such as at the half way point, the quarter point, and possibly other fractions.  The Egyptians are known to have used fractions in their mathematics.  Measuring rods would have similar use, except that a rigid rod is difficult to use for measuring the length across ground that is not level.  The merkhet mentioned in the paragraph is the center rib of a palm leaf, notched or split at one end.  It is not clear to me from my research how it was used, though I could make some guesses.  One source describes the rectangle with the plumb line as the merkhet, and the piece of palm leaf as the bay, and says that it was used for telling time.  The alternate source describes using it only at night, for viewing star positions.

Merkhet and Plumbline3

These tools and concepts were used for surveying, for laying out property lines.  They were not designed for making maps, but could easily have been adapted to map making.  They are among the earliest tools related to mapping that I have been able to find.  The next step in map making and surveying tools would probably be refinements on these.

By the time of the Romans, the need for tools for surveying was met by more sophisticated tools, although I have not been able to trace the path from Merkhet to groma (the Roman cross-staff with plumb line at each of the four ends of the crossed staves) and decempedae (10 foot measuring rod).  While I am not sure of the dates for the development of the groma, it appears to have preceded the early dioptras that are dated to the 3rd century BC.  The groma appears to have been used only for surveying, and the dioptra for astronomy.  The right angle under the dioptra sighting tube was eventually replaced with a hemispherical protractor, which permitted more precise angle measurement.  Further development of the dioptra was done by Heron of Alexandria, 10 AD to 70 AD.  A post had been added on which the dioptra sat, but Heron added the ability to use the dioptra in both horizontal and vertical directions, and screws for fine adjustment, making the dioptra much more precise.

Schematic version of groma: bottom pole is stuck into the ground, the bottom pole is adjusted so that it is vertical, by lining up the plumb lines, then the cross-poles are rotated to that the user can sight along them.

Schematic of early dioptra: the top is a tube for sighting through, there is a plumb line which allows the user to determine the angle of the sighting tube.

I remember being told about the Via Emilia while I was visiting in Florence, Italy.  What I was told is that the Romans had built a road that ran from Rimini on the eastern side of Italy straight through the countryside to where Milan is now 2000 years ago.  In checking, however, it turns out that the original road terminated in Piacenza, somewhat short of where Milan is now, but it did run straight for approximately 260 kilometers, and the Romans implanted colonies there that became Bologna, Parma, Reggio, Modena and others.  It is an impressive straight line, but there are other roads that the Romans built that are similar.  It is not known precisely what procedure was used, as there are no descriptions, nor is it clear which tool, groma or dioptra, was used.  But the Romans also set border lines, one of which is mentioned in Dilke to be 29 kilometers long with a variation from straight of no more that 2 meters along its whole length.

Some impressive milestones in surveying are tunnels started from both ends, using a method described by Heron of Alexandria, and the building of Roman aqueducts with the proper slope so that the water in them went where they wanted it to go.  Their sewers would have the same requirement.  The speculation mentioned in Dilke is that an advanced form of dioptra was used for the tunnels, aqueducts and roads.

Reconstruction by H. Schone of Hero’s dioptra, which could be used both for surveying and for astronomical observation.4

The next improvement had to wait until the 16th century, when the Digges, father and son, showed the theodolite to the world.

1 Joseph, George Gheverghese, The Crest of the Peacock, Non-European Roots of Mathematics, Princeton University Press, Princeton, N.J., 2000. p. 59.  The version of Herodotus that he used is Herodotus (1984), The Histories, London, Penguin. p. 59

2 Dilke, O.A.W., Mathematics and Measurement, Volume 2 in the Reading the Past series, 1987 by The Trustees of the British Museum, British Museum Publications Ltd, London. pp 7-8.

3 Ibid. p. 7

4 Ibid. p. 30

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Round, round, get around, I get around

In the last couple of  posts, I’ve been discussing maps, mapping and the measurements required to construct maps.  I am going to continue by looking at using maps for only one of their purposes: to provide a guide for how to get “there” from “here”, which I think is probably the earliest use of mapping and would have resulted in informal, easily disposed maps, so untraceable.

In his book on geography, Ptolemy made a distinction between “local” maps and “global” maps.  He applied his grid of parallels and meridians to the global, but because the scale of a local map was considerably different, the need to make the meridians reflect the way that they slant toward each other as they approach the poles was not as important.

Just a small note on map terminology: “small scale” maps are those that show large areas made small.  In the case of a small scale map, the scale might show 1 centimeter to represent 100,00 kilometers, as an example.  A contrasting “large scale” map might show smaller areas made large, so a large scale map might have 1 centimeter equal to 10 kilometers.  Given that this is the case, what Ptolemy uses as a distinction refers to local or large scale maps, as opposed to global or small scale maps.

To get from “here” to “there”, depending on the distance, one might need a smaller scale map and a local, large scale map.  In the pre-Mapquest, pre-Google Maps days, that meant finding paper maps with sufficient detail to help: one might need to get a California Highway map from AAA, as well as a Palm Springs, California map to get to the specific place being visited, so that the directions from San Francisco by car are clear, first which highways, and then once off the highway, which local roads.

Now, with internet maps available, I can go to Mapquest or to Google Maps, and using their scaling tool, zoom out to get the view of the highways (a smaller scale view), then zoom in to display the local roads (a larger scale view).  But the availability of internet maps is a new situation, and, even newer, with GPS units, one might not actually need to use on-line maps.  These choices have not always been available.

I’ve talked about the etched stone in the Hayonim Cave in Israel that may or may not represent a map, but if it does, it looks more like a division of land into property rights.  It is perhaps 12,000 years old. The oldest map that I’ve been able to find that might be closer to one showing how to get there from here is one inscribed on stone in Valcamonica in the Lombardy region of northern Italy.  The site has been recognized by UNESCO as a world cultural heritage site, and the particular carving on stone that is discussed in a fine article by Craig Alexander of the Department of Archaeology at Cambridge University, is called the Bedolina Map.  The entire article is available on the web by doing a Google search on “Bedolina Map” and selecting the entry called “The Bedolina Map – an Exploratory Network Analysis”.

One set of workers has tentatively dated the map to the Bronze age in Italy, around 1400–1500 BC, while another worker dated it to the 8th century, which would be Iron Age.  The same set of workers who dated the map to the Bronze age interpret it as being a literal map of the valley in which it was found, while the same worker who dated the map to the Iron Age interprets it as being symbolic and related to land holding.  There are a number of differences between the Hayonim Cave and the Bedolina Map, but the one that drew my attention is what appear to be paths and alternative paths that are inscribed on the stone leading from place to place on the map, hence the word “network” in the title of the Alexander article.

By the time of the Bedolina Map, even if the earlier date is correct, the Babylonians, the Egyptians, and the Chinese had been making maps for property rights.  What I have not found are maps that could be used by travelers, but I’m sure that some existed.  In The Mapmakers, Wilford describes an anecdote from Herodotus (5th Century BC, Greece, later than Bedolina), where Aristagoras of Miletus tried to persuade Sparta to join an invasion of Persia, and showed the Spartan king a “map of the world”, to help convince him to join the rest of the expedition.  Aristagoras’s map showed the lands of the Ionians, the Lydians, etc. on the way to Persia, and the distance to Persia looked as if it could easily be traversed.  Only when the scale of the map is made clear, that it would take about 3 months to go from Sparta to the city of Susa, Persia’s capitol at that time, was he turned down by Sparta making his mission unsuccessful.  I have not found earlier maps that were used to show how to get “there” from “here”.

The movements of people from one place to another, the migrations of humans from Africa to the rest of the world, have been traced on maps, but not because people made maps showing where they came from and how they got to where they ended up.  Maps with these approximate routes have only been done lately by scientists trying to understand how the human species occupied the earth and where they came from.  Nor am I suggesting that, for instance, the original occupiers of Australia were given a map saying that if you follow this trail, you will reach a wonderful place to live for the next 20,000 to 40,000 years.  But humans do move about, and have occupied most of the land that was reachable on the earth.  And they have had to show their children, relatives, friends how to get to places where they would gather: not all have stayed in such small locales that they were able to remember all paths to places of interest.

I believe that there may not be more than a few permanent traces, if any, of early maps that were used to describe how to get there from here.  A map of this sort might be drawn with a stick on the ground – hardly permanent.  What might an early map look like that would persist longer?  Anything inscribed on something other than stone, such as on animal hide, woven into a cloth, etc., could be expected to last longer than one drawn in the dirt, but not that much longer.

In The Songlines by Bruce Chatwin, there is a hint at a possible way to pass geographical knowledge.  His book is focused on the Indigenous people of Australia, but toward the end, he does try to generalize what he has discovered. The “songlines” he refers to are a means for, among other uses, navigating the land.  By learning a songline and following its directions, the Indigenous people are able to cross large expanses of Australia and arrive where they had intended to go.  The songlines treat the landmarks as if they were alive – myth beings – and in some cases mentioned, he says that he was able to see in the geography, clues that remind one of a mythologized animal shape.  So, they do not say, for instance, go to the blue rock near the gum trees, turn left and straight on till morning, but instead will tell a story about why blue lizard left its tail mark to be followed from beyond the gum trees, and why the blue lizard stopped roaming.

When he is trying to universalize the use of song as guide, he touches other traditions and closes with a nearly unattributed quote from a woman, apparently a member of one of the tribes on the northwest coast of Canada, “…representing a tradition about 13,000 years old.”1

Everythin’ we ever knew about the movement of the sea was preserved in the verses of a song.  For thousands of years we went where we wanted and came home safe, because of the song.  On clear nights we had the stars to guide us, and in the fog we had the streams and creeks of the sea, the streams and creeks that flow into and become Klin Otto [the current that runs from California to the Bering Strait]…

There was a song for goin’ to China and a song for goin’ to Japan, a song for the big island and a song for the smaller one.  All she [the navigator, evidently they were all women] had to know was the song and she knew where she was.  To get back, she just sang the song in reverse.2

Based on these two examples, the Australian and the Canadian ones, it may be that the place to look for early maps is in early myths, poetry and songs.  And, to my point about drawing maps on the ground, Chatwin provided a general statement about having a journey described by “Joshua…a famous Pinturi ‘performer'”3

Aboriginals, when tracing a Songline in the sand, will draw a series of lines with circles in between.  The line represents a stage in the Ancestor’s journey (usually a day’s march).  Each circle is a ‘stop’, ‘waterhole’, or one of the Ancestor’s campsites.4

The best example of maps that show how to get there from here that have been learned as songs or poems are the directions used by the islanders of the South Pacific for sailing from island to island.  The method has been researched and written up in a marvelous book, We, The Navigators, by David Lewis.  In the book, Lewis concentrates on Polynesian and Micronesian navigators and their abilities to sail from one island to another, quite often with no land visible.  He does not discuss the form in which the knowledge is kept and transmitted: he shows few examples of the song verses or poetry.  His method for learning about their abilities and skills was to take sailing trips with them, with them providing the navigation and steering of the boats.  Because they demonstrated their skills and talked with him about what they were doing and why, he shows their way to be a very impressive alternative method of knowing how to go from here to there and back again.

…there was the totally unexpected finding that nearly every important navigational technique and concept encountered in Micronesia was matched by its Polynesian counterpart.  Differences seemed to depend much more on local insular geographical features than on major cultural-linguistic divisions….

An element of falsity is inescapably introduced in this presentation by the very process of analysis employed.  Thus, the navigators did not appear to compartmentalize their art, as I have done, into such divisions as “steering a course,” “deviation from course,” “fixing a position” – except perhaps during their initial training.  Instead, they conceived of their art as a unity, the sum of input from such disparate sources as stars, swells, and birds being processed through training and practice into a confident awareness of precisely where they were at any one time, where they were going, and how best to get there.  The Pacific navigators did not so much analyze their data as use them as pointers, which they subtly synthesized.

Another limitation … is divorce of the navigating arts from their social roots and the psychological and spiritual values of which they are an expression.  The navigators were not merely in tune with their environment as Western seafarers might be, they were literally a part of it.5

As he states, he compartmentalized the skills to facilitate his discussion of the various facets.  So he discusses star path steering, then using the sun, swells and wind for the basic directing and correcting of a course.  He discusses as well, dead reckoning, expanding the targets and how the position is “fixed” while on the ocean.

Depending on the island from which the sailor/navigator started, there is a direction that is associated with the rising or the setting of prominent stars in succession through the night, hence steering by a “star path”, though which stars and whether rising or setting appears to depend on the season.  The reference stars are taught to boys who have been invited in to the exclusive group of navigators, and are supposed to be kept as secret/sacred knowledge.

During the day, though, the stars are not available, so sun position is tracked, and the path of the swells.  During the demonstration voyages, the navigators often would lie down on the deck or in the canoes to “feel” the swells, and could determine which were local and which were the persistent swells, providing direction information.  They would have Lewis try this, but he concluded that he was not sensitive enough to always be accurate.  Also, when leaving an island, if a landmark was visible astern, the navigators would use it to judge the wind direction and the current, and use that information to correct their steering.  Because the wind can change quickly and unpredictably, the navigators hang “wind pennants” in the rigging, and then watch them throughout a voyage.  Lewis had a “wind compass” described to him, with names given to winds coming from the specific different directions of the compass, which the navigators use.

Dead reckoning and fixing the position are complex, and are related to his statements about the navigators being a part of their environment.  I shall not discuss them, but if you want more, it is available in his book.

The final aspect that he describes is “expanding the targets”.  While an island can be seen from 7 to 10 miles away, depending on its highest elevation, to expand that target, the navigators follow birds, which will range up to around 30 miles from islands during the day.  Other clues, such as clouds being “caught” by peaks, certain effects on the waves as a result of reefs are also used.  But as Lewis says, these pointers are synthesized rather than enumerated and analyzed.

Only now does it seem that Western compasses are being used.  Otherwise, the instruments that were, and in most cases still are, used are the senses of the navigators.  The maps, the understandings of how to go from here to there and back, are contained almost entirely within the navigators.  It is a different way of knowing, of mapping than that of the Western tradition.

I have found a map, though, that is evidently an example of what is used when the navigators are in training.  It was in a copy of a magazine put out by the San Francisco Exploratorium, which I have scanned and is included below.  The Exploratorium Quarterly for Spring 1991 gives the credit for the photo to Susan Schwartzenberg, and was taken at the Lowie Museum of Anthropology at the University of California in Berkeley, if I have understood the text correctly.  The caption states:

In Micronesia, navigators were trained with stick maps made of palm sticks tied together with coconut fiber.  Curved sticks showed prevailing wave fronts, shells represented the locations of islands, and threads indicated where islands came into view. 6

A few final comments.

While the Pacific islanders had a form of visible “representational” map, it appears that maps in that form were used for training, so that when actually navigating, the islanders did not and do not use them.  Instead, they use their senses, knowledge and experience to travel.  They may actually recite to themselves appropriate portions of a song or poem that helps direct them to their destinations.  The Western tradition of longitude and latitude, dating back to Ptolemy, and the instruments for determining longitude and latitude has applicability not only to navigating on seas with no land in sight, but also to creating land form maps for uses that range from property rights to navigation to geological formations, population densities, and on and on.

The more general applicability of the Western way of mapping suits a more complex set of requirements, but is not necessarily “better” than the Pacific islanders way of knowing.  Their way is suitably adapted to their requirements, and shows yet another response to the natural world by humans using their pattern making ability to measure, understand and control their lives.

1 Chatwin, Bruce, The Songlines, Penguin Books, London, England, 1987.  p. 283.

2 Ibid, p. 283.

3 Ibid. p. 152.

4 Ibid. p. 154.

5 Lewis, David, We, the Navigators, The Ancient Art of Landfinding in the Pacific, second edition, 1972, 1994, University of Hawaii Press, Honolulu, Hawaii  pp. 47-48.

6 Pearce, Michael, “Invisible Islands: Navigating Techniques Of Ancient Oceania”, Exploratorium Quarterly, Volume 15, Number 1, Spring, 1991, San Francisco, CA  p. 25.

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Considering Maps II – Virtual Lines

In the last post, I started discussing maps, mapping and the measurements required to construct maps.  I am going to continue, by looking at the coordinate system of longitude and latitude that Ptolemy refined and corrected, which is still in use today.  This system has a number of elements which he pulled together in a fashion that has provided a standard methodology for geographers, navigators and others to use, with the final element put into place by international convention in the 1880s.

The people who are important in the development of map making techniques often relied on or were astronomers.  Many of the results came from a combination of celestial and geographic measurement.  Celestial measurement adds an additional element to the geographic measurement: time and its measurement.

Watching the sky for patterns would have become very important for the agricultural communities that grew up after the end of the last ice age, around 12,000 years ago, and with several thousand years of doing so, the knowledge of the repeating patterns such as seasons was linked to the repeating patterns of the unchanging arrangements of the stars, and to the pattern of the sun moving against the backdrop of the stars.  The planets were special cases and had their own stories.

As observers of the sky, the people of the time would know about the sun’s position relative to the stars, and would have identified certain groupings of stars along the sun’s path as “constellations” that were discrete.  Approximately 12 were identified in the area of the sky that the sun moves through, which is called the ecliptic.  At some point, the approximately 30 day repeat pattern of the moon would be mapped against the sun’s movement along the ecliptic, and wow, 12 – 30s would be recognized as pretty close to a year (12 x 30 = 360).  Although the Egyptians knew that the year was 365 days long as early as 4241 BC by counting the days between the rising of Sirius, the Babylonians used 360 days for the length of their year.

Another observed pattern would be the length of daylight, its changes over the year and how that correlated with the height of the sun relative to what would become known as the southern horizon.  This would lead to a knowledge of solstices and equinoxes.  (It is a curious fact that all of the great civilizations that developed the early recorded astronomical observations -China, Babylon, Egypt, India (in large part), Greece, Maya – were all north of the equator.)

Another of the early elements was came from the Babylonian astronomers, among others, who saw that there seemed to be a place in the heavens that all of the stars moved around, but itself did not move much itself.  While we are familiar with Polaris (alpha ursa minor) being the pole star, due to the way that the earth wobbles on its axis, this would not always have been the case, evidently about 4500 years ago the pole star was alpha draconis, and from about 1500 BC to around 500 AD, beta ursa minor was recognized as the pole star.  Even so, this would have given an “unmoving” point to the nightly rotation of the stars, as if it were the center axle of a circular motion.

If these repeat patterns were characterized as occurring in a circle, it would not take much imagination to decide to divide a circle into 360 equal parts.  Evidently, that’s what the Babylonian astronomers and mathematicians did – they used 360 as the number of days in a year, and may have divided the circle into 360 parts.

The Babylonians were using a number system with base 60 on some of the earliest clay tablets, about 3000 BC.  There have been a number of proposals about why they used base 60, but since the proposals were developed well after the use began, and there is no documentation stating why the choice of 60, there is no way to know for sure.  But an extension of it is to the number of days in a year, though missing those pesky extra 5, possibly to make the calculations easier: 60 is divisible by 2. 3. 4. 5. 6. 10, 12, 15, 20, 30; and 360 shares those divisors along with a few more.  For whatever reason, the sexigismal is still active for us: 360 degrees in a circle, 60 minutes in an hour, 60 seconds in a minute.

In the 7th century BC, Babylonian astronomical information began to be acquired by Greek seekers of information, and continued through the time of Alexander the Great’s conquests (336 BC to 323 BC).  In the end, the Greeks possessed what is said to be a huge amount of accurate astronomical data compiled by the Babylonians.

Pythagoras of the famous theorem led a school of philosophy in the last quarter of the sixth century BC.  One of his hypotheses was that the earth was a sphere, though there is no record of how he reached that hypothesis.  The reason cited in several sources is that it was perhaps because “the sphere is the most beautiful of solid figures”.1 Aristotle (384 BC – 322 BC) built on this by providing three “proofs” based on observations.  He also tried to prove that the earth does not move, either around an axis or revolve around the sun.  Aristotle, among his other accomplishments, was Alexander’s tutor.

After Aristotle, there are three Greek mathematicians/scientists who have to be mentioned.  Euclid, about whom there is almost no personal information except an estimate that he must have been active around 300 BC, and may or may not have written before Archimedes, wrote a number of books, some of which have survived, and the most important of which is Elements, a treatise on geometry that has been the standard teaching text for the last 2300 years.  Elements is a comprehensive and logical treatment of geometry (and some other mathematical topics) that is a compilation of most everything that was known in plane geometry up to that point.

Archimedes (287 – 212 BC) is known for a number of scientific inventions and discoveries,  and is the prototype for the absent-minded professor who made an important realization about the displacement of water by solids when immersed in a bath and ran through the streets of Syracuse, where he lived, having forgotten to wrap even a towel around himself, shouting “Eureka!” because the displacement realization helped him to solve a knotty problem for the king of Syracuse.  He also systematized solid geometry in much the same way that Euclid systematized plane geometry.  Concepts in both plane and solid geometry were important for map makers to understand as they began their work.

Aristarchus of Samos (c. 310 to 250 BC) claimed that the earth moved around the sun and that it rotated about its axis.  This idea does not appear to become popular for another 1700 or so years, when Copernicus described it as a useful model, and attributed it to Aristarchus: giving proper credit.  But this model was not used for the making of maps of the world.  Aristarchus did create a more accurate sundial which became important in measuring the time of events that helped with map making.

And, it is at about this time, too, when making maps of the known world became an important activity.  Most of the concepts were in place for making maps that were more than local maps, though there had been some rather mythological attempts before this.  But with the great travels of Alexander as he conquered a version of the known world, trying create maps of the known world became more important.  The next actor in the list is Eratosthenes, who was introduced in the last post for his estimate of how large the world was.  Again, his dates are 276 to 194 BC.

By the time of Eratosthenes, the sages had figured out that if the earth was a sphere and the celestial sphere in which it is enclosed turned about the axis pointed at by the pole, there were some additional consequences.  One was that there was a place on the sphere of the earth that was half way between the pole at the top of the sphere and the pole at the bottom.  It is now called the equator, and presumably it was named that because it was an equal distance for each pole.  The sun was directly overhead of the equator two times per year: on the days of the spring and of the fall equinoxes.  The equal in this word refers to the fact that the sun was up the same number of hours as it was not up.

Also known was the fact that there was a northern-most point for the sun, occurring on the summer solstice, the day with the greatest number of hours of sun, and a southern-most point, on the day of the midwinter solstice, when the number of hours was the shortest.  Being clever, the sages realized that the days of the solstices also could be marked on the sphere as lines parallel to the equator, since the sun would be at its highest or lowest for the whole rotation of the celestial sphere on each of those days.  These lines are now known as the tropics, the northern one for the Tropic of Cancer, the constellation the sun moved into on the day of the solstice, and the southern one for the Tropic of Capricorn for the same reason.

Apparently this led to a number of additional lines parallel to the equator and the tropics, which were called “climates” to indicate climate regions of the earth.  The southern most line was the Tropic of Capricorn, below that, there had been no travelers who had returned with information so from there to the southern pole was treated as if humans did not exist there.  It is not clear from the literature that I’ve found if the earth south of the Tropic of Capricorn was considered to contain any land, habitable or not: there are some who mention a mythical southern land, but basically it appears to have been treated as if it were all hostile ocean.

The climates above the Tropic of Cancer, however, were sometimes treated as the line through a specific city, and sometimes as if to delineate a region.  There was a mythical island called Thule, which had scant reporting about it other than that it was cold there, and may have been Iceland, but also could have been a truncated version of the Scandinavian peninsula.  The important point, though is that there were those who treated various places as worthy of delineating regions with a line parallel to the equator and tropics.  These would ultimately become what we know as the lines of latitude.

Equally important, it was understood that a line that was perpendicular to the equator and ran to the north pole through a specific city, say Alexandria, this would not be parallel to a line running the same way through, say Athens, and that the lines would meet at the pole(s).  These would ultimately become the lines of longitude.

Eratosthenes created a map that had an early version of longitude and latitude laid out on it: the original has disappeared, but based on criticism of it by Hipparchus and Strabo, it has been reconstructed, a copy of which appears in the Brown book on page 51, and looks like this:

His parallels are wrong, his meridians (longitude) are wrong, his map is wrong.  But the beginnings of the concepts necessary to include latitude and longitude are present.

Hipparchus, (dates uncertain but after 127 BC) was so incensed by this map that he wrote an entire treatise which he called Against Eratosthenes.  Among other things, Hipparchus advocated that the parallels should be evenly spaced, and he, evidently, was the one who suggested basing their placement on the hours of the longest day, as in one at the latitude where the longest day is 13 hours, one at 14 hours, etc..  And he felt that the lines that were perpendicular to the equator should be evenly spaced at the equator, but joining at the poles.   One of my sources says that it was he who suggested dividing the sphere of the earth into 360 parts, later called degrees, though another source suggests that the Babylonians had divided the circle into 360 parts.

Hipparchus also has credited to him the invention of trigonometry, though knowing how my contemporaries and I struggled with trig, I fear the inventor has had more curses than thanks for doing so.  His trigonometric methods were based on isosceles triangles, which he used to calculate tables for chords for use in astronomy.

He also is credited with greater accuracy in his astronomical calculations, determining star positions, the lengths of the tropical year and sidereal years and one of his results was that the precession of the equinoxes was not less than 36 seconds and not more than 59 seconds per year, when the current figure is 50 seconds.2 I am not sure how the estimate was arrived at, since the tools he used or could have used were gnomons, sundials and potentially water clocks, none of which appear to me to have sufficient precision.  I will continue to research the question, and hope to be able to provide greater clarity in a future post.  But, since this estimate was made, in whatever form so that it could be translated into the above figures, there has to have been yet another set of concepts current and available for use: a day divided into 24 hours, with a way to judge fractions of a single hour.

Approximately 200 years after Hipparchus is when Claudius Ptolemy wrote The Almagest and Geographia, which are only two of his many works but are his best known.  His dates are about 100 AD to about 170 AD.  The work for which most people know him is the Almagest, which is his work on the universe that revolves around the earth.  In it are the mathematical explanations of the movements of the known planets, the sun and the moon.  Using the methods of the Almagest, astronomers and astrologers were able to make predictions about when planets would be where, and when eclipses of the sun and the moon could be expected.  Ptolemy states that having the sun be the center of movement of the universe would be a convenient explanation, but is ridiculous.  His reputation, after the Copernican revolution, was to suffer as a result.

In Geographia, Ptolemy was concerned with how the spherical world fits in the universe as he had mapped it.  As such, he tried to refine how maps of the known world should be made, and he followed Hipparchus’s suggestions of equal spacing of both parallels/latitude, and meridians/longitude.  As an additional aspect, he broke the degrees into smaller units, which have become known as minutes and seconds and are still used.  They come from the Latin translation of his divisions, “partes minutae primae” and “partes minutae secondae”.3

There are two reconstructions of map techniques that he is credited with which are visual demonstrations of one of the problems that he felt deserved thought.  That problem is that if the earth is a sphere, how can a map be accurately constructed on a flat piece of paper/parchment, whatever.

His first method was to use a simple conic projection which looks like the below figure, taken from the Brown book on page 70, which is based on a 16th century copy:

He also described a second method, which he called a superior method, but admitted that the easier method might be as useful for those who were somewhat lazy.  From Brown, again, on page 71:

It is important to notice a few things about his scheme: one is that he has included 180 degrees west to east of the possible 360, so only one half of the sphere, and he has further truncated the view by including starting at just above 60 degrees north to 15 degrees south.  This was considered the “habitable” portion of the earth.  The prime meridian in the west went through what was called “The Fortunate Isles”, probably the Canaries, and extended east to just beyond the southeastern corner of China, as it was postulated in Alexandria at the time of Ptolemy.

Here was the basis for creating maps of the world, and while the geography underneath the grid changed to more closely match the real geography of the earth, the grid has remained relatively constant, with the 15 degree divisions around the equator being retained roughly for the time zones.  The final step that I mentioned at the beginning of this post was to set the prime meridian as the one which runs through Greenwich, England, and the international dateline approximately 180 degrees from that, running through the Pacific Ocean.

So while Ptolemy thought that the earth couldn’t move, he did provide a reference scheme for mapping the earth.   While this reference has been used, now, for around 1700 years, it was not without its problems, and we can look at them in another post.


I have used a number of sources to pull this material together, and while I have not requested permission for the use of the three diagrams, I hope that I have sufficiently credited them.  The main sources that I have drawn this material from is:

Brown, Lloyd A., The Story of Maps, Dover Publications, New York, NY, 1979, by permission of Little, Brown and Company, Boston, MA, 1949, 1977.  Thorough, comprehensive, though at times, a little confusing.

Burton, David M., The History of Mathematics, An Introduction, Fourth Edition, WCB/McGraw-Hill, Boston, MA, 1999.  A marvelous history, the book was a gift from my daughter, Allyndreth Devlin, from the time when she worked for McGraw-Hill.

Dilke, O.A.W., Mathematics and Measurement, Volume 2 in the Reading the Past series, 1987 by The Trustees of the British Museum, British Museum Publications Ltd, London.  Slim but authoritative.

Gullberg, Jan, Mathematics, From the Birth of Numbers, W.W.Norton & Company, New York, N.Y. 1997.  As stated on the copyright, the author prepared the entire camera ready book, all 1090 plus pages, on an Apple Macintosh Plus, using Microsoft Word.

Joseph, George Gheverghese, The Crest of the Peacock, Non-European Roots of Mathematics, Princeton University Press, Princeton, N.J., 2000.  An invaluable resource for those of us who have only been taught about a Western/European tradition.

Wilford, John Noble, The Mapmakers, Revised Edition, Vintage Books, New York, N.Y. 1981, 2000.  Well written and enjoyable.  The updated version predates MapQuest and Google Maps on the web, but is current up to that point.

And as always, periodic checks of the information on Wikipedia.

1Brown, Lloyd A., The Story of Maps, Dover Publications, New York, NY, 1979, by permission of Little, Brown and Company, Boston, MA, 1949, 1977. p.25.

2 Ibid p.53.

3 Wilford, John Noble, The Mapmakers, Revised Edition, Vintage Books, New York, N.Y. 1981, 2000. p. 30.

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