The Square of Pegasus

The accuracy to which today’s surveyors locate a point on the surface of the globe, and the speed with which precise measurements are obtained, would have been inconceivable to our antepasados as little as a generation or two ago. Moreover, global positioning can now be made by a technician skilled in the mechanics of operating the equipment, without much knowledge of the underlying geometry. Astronomers and mathematicians have at long last succeeded to impart the surveyor their knowledge in spheroidal geometry, even if only by locking some of it into a computer chip from which it can be retrieved by a push of a button. This is not to say that surveyors are ignorant, only that it took the better part of four millennia to reach that point; and therein lies a story.

If you look at the stars on a clear evening in late September you will notice a little south of the zenith a large square, formed by four stars of second magnitude. They constitute a part of the constellation of Pegasus and the large, almost empty square is known as the “Square of Pegasus”. Markab (Alpha Pegasi) and Algenib, the two stars that form its base, are both located on the fifteenth parallel north of the celestial equator, while Scheat and Alpheratz, the top of the square, are almost on the thirtieth. Actually, Alpheratz (Alpha Andromedae) is already in the neighboring constellation of Andromeda, but that doesn’t matter. An unmistakable object of striking beauty, the ancient astronomers of Mesopotamia saw in it the pointing fingers of the gods and used it as a starting point for mapping the sky.

The Babylonian surveyors too got the message, but they surveyed on the sunbaked floodplains of the rivers and not in the sky. Being godfearing yet practical men, they put that great “winged horse” onto the ground, and in the process turned it into the jackass is has remained to this day. Ignorant of geodesy, they started any land survey by marking a square of standard dimension, from which their survey was expanded in a checkerboard fashion. This original square the Sumerians called iku, a name that was derived from the name of the Square of Pegasus; it can be found in many a cuneiform text designating a unit of area. Their squares did not extend over a large enough area, nor were they accurate enough to let them suspect that there might be something wrong somewhere.

Geometry now had its practical foundation. The square was inscribed by a circle; centerlines and diagonals were added, and it was discovered that basic mathematical shapes could thereby be surveyed long before they could be computed. These basic shapes of square, circle and polygons turned into sacred symbols. – Noticing how the surveyors had succeeded in flattening a curved surface, the mathematicians labored for untold centuries trying to square the circle, (turning a circle into a square of equal area) oblivious to the fact that it cannot be done, but sustained by the hope that somebody would find an end to the myriad decimals of pi. 

Surveyors of the ancient world measured by feet and cubits, the cubit being 1½ foot and consisting of six hands of four fingers each. The emerging sciences of mathematics and geometry were even at that time mastered only by a few, very few, of the top brains of the respective civilizations. Computing the area of an irregular tract of land without dividing it into squares would have been beyond the capability of even the best of land surveyors. However there was no need, for they worked closely with kings and priests, and the gods themselves had given them that beautiful Square of Pegasus.

Ever able to find practical solutions to perplexing problems, the surveyor would double the area of a given unit square of 100, by increasing its sides to 140, assuming that 140 squared is twice 100 squared. Sticklers for accuracy used 99 for the length of their unit square. In order to half the area of that standard square, a square of 70 units was laid out, again assuming that 4,900 was sufficiently close to 5,000 to keep the landowner happy. This lucky discovery led to the creation of septenary units of measurement and contributed to the fascination with the number seven.

The average land surveyor of Greece surveyed by the square and had never heard of Thales and Pythagoras, even if he called himself geometros. The Hebrews continued the tradition. The author of the Book of Revelation described how an angel took him in his dreams to the top of a mountain overlooking Jerusalem. “The angel who spoke with me carried a gold measuring-rod, to measure the city, its wall, and its gates. The city was build as a square …” [Rev. 21:15]. Come on now, John, you would not expect the holy City of David to have been surveyed any other way?

Not that the leading mathematicians throughout history did not try to educate the struggling surveyor. Starting with the ancient Greeks there was no shortage of learned treatises on the mathematics and geometry involved in geodetic land surveying, about which the English-American surveyor John Love as late as 1688 lamented: ”I cannot find in those Books many things of great consequence to be understood by the Surveyor,” upon which observation he promptly wrote his own book “Geodaesia, or the Art of Surveying and Measuring Land Made Easie.” He did not say if anybody read it.

More than a century before John Love, an English part-time mathematician and full-time physician to Queen Mary by the name of Robert Recorde (1510-1558) introduced the English surveyors to algebra, which the Arabs had taken up from their Mesopotamian ancestors and perfected. He wrote a number of books, including in 1551 “Pathewaie to Knowledge”, in which he published an abridgement of Euclid’s “Elements of Geometry”. Knowing how surveyors were operating, he warned them of the damage that could be done with all that knowledge:

“Survayers have cause to make muche of me.
And so have all Lordes that landes do possesse:
But Tennauntes I feare will like me the lesse.
Yet do I not wrong, but measure all truly,
And yelde the full right to everye man justely.
Proportion Geometricall hath no man opprest,
Yf anye bee wronged, I wish it redrest.” 

He need not have worried, John Love testified to that.

Two generations after Recorde his countryman and fellow astronomer Edmund Gunter (1581-1626) introduced surveyors to the cosine and cotangent in his “Canon triangolorum”. This learned tome was lost on most of them, but he had more luck when he provided them with his 66-foot chain. His squares, ten of which would make an acre, were a lot easier to deal with than worrying about the fact that the world is round.    

That was of course by no means the end of it. The basic philosophy behind the congressional township and its subdivision can be found in the Square of Pegasus. A look at many subdivision of sections of the public land surveys, made as late as the recent past, will disclose greater affinity with that sacred Square than with the rules of subdivision established by the General Land Office. Observing many a tape and compass carrying land surveyor, the gods of Sumer would feel right at home, and recognize the concept as well as the methodology.

In the 1930s, officials in what was then the U. S. Coast and Geodetic Survey (today NOAA) thought, that it might be a good idea to make the positions of the brass tablets established on the North American datum of 1927, obtained at great cost by transcontinental triangulation, and laboriously computed on Clarke’s spheroid of 1866, available to the average land surveyor. In order to come up with something “to be understood by the surveyor” the agency established plane-rectangular coordinate systems for each State in the Union. Published in the familiar light blue brochures it was hoped to induce surveyors to tie their
surveys to the national geodetic grid. It could be that someone in the agency thought like Samuel Johnson, who supposedly said on the eve of his second marriage: “Perhaps hope will triumph over experience”.  It didn’t.

Now that we have gotten squared away (pun intented) with that wonderful GPS equipment, it is only fair to be kind to our antepasados and their four-thousand-year-old profession. They surveyed fair and square. “Judge not, that ye not be judged,” sayeth the Lord.