The Geometrical Foot in the Geodesy Classics

  Both Cassini and Mecain & Delambre talk about a geometrical foot that is one 6000th of a minute of arc. Cassini brings it up in De la Grandeur et de la Figure de la Terre. Mecain & Delambre mention it in the Preliminary Discussion of Base du Systeme Metrique, Vol. 1. The latter also cites an earlier article by Gabriel Mouton in Observationes diametrorum (1670).

Some More Geodesy Classics From Republican France

  I found a few more works on geodesy from about the time of the French Revolution. You may need to study the Republican Calendar to make sense of the dates which seems somewhat bizarre now. The three books by Mechain & Delambre form the basis for the definition of the meter.

GEODESY

1787 Legendre,
  Mémoire sur les Opérations trigonométriques,
  dont les résultats dépendent de la figure de la Terre

1799 (AN VII) Delambre & Legendre,
  Methodes analytiques pour la determination d'un arc du meridien

1799 Legendre,
  Methode pour determiner la longueur exacte du quart du Meridien

1806 Mechain & Delambre, Base du systeme metrique decimal Vol. 1

1807 Mechain & Delambre, Base du systeme metrique decimal Vol. 2

1810 Mechain & Delambre, Base du systeme metrique decimal Vol. 3

Calentrier republicain

CALCULUS OF VARIATIONS

1805 Legendre,
  Nouvelles méthodes pour la détermination des orbites des comètes

1809 Gauss,
  Theoria motus corporum coelestium in sectionibus conicis solem ambientium

The last two books are cited as the first publications on the use of least squares to minimize errors. There is some controversy over priority since Legendre published first but Gauss claims to have used the method in 1795.

More Geodesy Classics from the 16th & 17th Century

  Gemma Frisius and Willibrord Snellius were two Netherlanders who wrote about the figure of the Earth and introduced improved trigonometric methods to surveying in the 16th Century. Here are some of their works. The dates indicated are either those of the edition of the book's publication or that of the original document.

1533 Frisius, Libellus de locorum describendorum ratione...

1547 Frisius, De principiis astronomiae et cosmographiae

1574 Apiani & Frisius, Cosmographia

1627 Snell, Doctrinae Triangulorum Canonicae

1617 Snell, Eratosthenes Batauus de tarrae ambitus vera quantitate

More Geodesy Classics from the 17th - 19th Centuries

  Picard's improvement of surveying methods in the 17th Century led to an effort to measure the length of the degree in order to determine the size of the Earth. Here are some more geodesy classics with some of the earlier work.

1671 Picard, Mesure de la Terre

1680 Picard, Voyage D'Uranibourg

1684 Picard, Traite du nivellement

1738 Celsius, De observationibus pro figura telluris

1740 Picard, Degre du Meridien entre Paris et Amiens

1742 Maupertuis, Elements de Geographie

1750 Cesar-Francois Cassini, La meridienne de l'Observatoire royal de Paris

1751 Condamine, Mesure des trois premiers degres du Meridien...

1751 Condamine, Journal du voyage fait par ordre du Roy a l'equateur

1860 Struve, Arc du Meridien, Vol. 1, Vol. 2

The arc of the meridian in Lapland that was first measured by Maupertuis and Celsius was extend from the Arctic Ocean to the Black Sea and is now know as the Struve Geodetic Arc. Some of the station points and markers of this arc still exist and have been recommended as World Heritage sites.

Classics in Geodesy from the 18th & 19th Centuries

  From the measurement of meridional arcs by the French in the early part of the 18th Century it became evident that the Earth was not a perfect sphere but an oblate spheroid instead. It took about 150 years to get accurate values for the equatorial and polar radii of an ellipsoid to represent the figure of the Earth. Here are some of the classic works on the development of modern geodesy.

1720  Cassini, De la Grandeur de la Terre

1738  Maupertuis, La Figure de la Terre (in English)

1740  Picard, Degre du Meridien

1743 Clairaut, Theorie de la Figure de la Terre

1749  Bouguer, La Figure de la Terre

1805  Biot, Traite elementaire d'astronomie physique (in English)

1826  Airy, Figure of the Earth

1830  Everest, Measurement of an Arc of the Meridian

1830  Airy, Figure of the Earth

1841  Bessel, Ueber einen Fehler und der Figur der Erde

1847  Everest, The Meridional Arc of India, Vol. 1, Vol. 2

1858  Clarke, Principal Triangulation, GB Ordinance Survey

1866  Clarke, Figure of the Earth

The 18th Century French units of length may be helpful with the earlier works.

Definition of a Unit of Measure

  When we measure something we need a unit of measurement that is both accessible and convenient. For a unit of length the circumference of the Earth is more accessible than its radius but is too large to be a convenient standard of measurement. So we have to consider its subdivisions. Like the degrees, minutes and seconds that are used to divide the circumference of the circle their corresponding lengths could be used to divide the circumference of the Earth and we arrive at the geometrical mile and foot. The meter is an alternative unit of length. The circumference of the Earth is 129,600,000 geo-ft or 40,000,000 meters. Eliminating the common multiple gives us 81 geo-ft = 25 m and we could define the geo-ft in terms of the meter with 1 geo-ft = 25/81 meters. This can be expressed more exactly the other way around with 1 meter = 3.24 geo-ft. The common foot is well defined with 3.28083 feet per meter and more easily accessible.

  From a practical point of view the conversion factor Δs/Δα for changing an angular distance into ordinary one is more easily observable than the circumference of the Earth so in this sense it is more fundamental. But while a spherical surface is a reasonably good approximation to the Earth's surface, we still have to admit that an ellipsoid is a better one.

Supplemental (May 9): For a spherical globe the geometrical foot can be based on a definition, 1 geo-ft = 0.01 asec.  The physical length of the geo-ft would have to be determined by the measurement of change in angle along a great circle with change in distance. This measurement needs to be made along a meridian since the change in latitude can be determined by astronomical observations. But the problem with this scheme is that the vertical direction is determined by gravity and the true shape of the Earth. The latter can be obtained through measurements of the curvature of the Earth's surface. This is the purpose of a geodetic survey. If we could use a spherical to represent the Earth's surface then the following definitions would be exact instead of just an approximation.

1 geo-ft = 0.01 asec
1 meter = 3.24 geo-ft
As it is there's no real geo-ft since it's definition assumes a conversion factor that in truth is not a constant.

Eratosthenes' Estimate Corrected

  One has to use the geometrical stade for the theoretical model. This gives better results. Eratosthenes' approximation has a 6.5% error with the same Google data. The relative error shouldn't depend on the units that are used. It was the model that was off.

The Origin of the English Mile

  In Faye's Origin of the English Mile it is claimed that Ptolemy confused the Greek foot with the royal or Phileterian foot of Egypt and that led to a reduced size for the English mile. He talks about a 1/6th error but that can also be found in Eratothenes' estimate for the size of the Earth.

It's easy to come up with fractions like this and difficult to assign much credence to them. In the discussion of the Phileterian foot above there is mention of a geometrical foot being commented on by De Morgan who appears to be trivializing the idea. This could also be anti-Roman bias and a move in favor of a clean slate on the size of the Earth.

  The Smithsonian and the American government were also aware of the paper on the origin of the English mile.

Eratosthenes' Estimate of the Earth's Circumference

  Eratosthenes as head of the Museum in Alexandria, Egypt had access to the best geographical information of his time and made an estimate of the Earth's circumference. Although only fragments of his writings now exist we know through later writers such as Cleomedes that from the nearly vertical shadow in a well in ancient Syene (modern Aswan) and the shadow of an obelisk in Alexandria Erathosthenes concluded that the difference in latitude was 1/50th of the circumference of the Earth. Assuming that the distance between Alexandria and Syene was roughly 5000 stades he arrived at an estimate of 250,000 stades for the Earth's circumference. We can check what he did.

For Google Earth positions we can use a spice market in Aswan as the location of the well in ancient Syene and the location of Cleopatra's Needles in front of the Caesarium in ancient Alexandria as the location of the obelisk. From this data we get 1/50.6 for the fraction of the Earth's circumference and 4675 stades as the great circle distance between the two positions. Our estimate of the Earth's circumference is found to be off by 10% from the theoretical value based on 10 stades per arc minute. It is not unreasonable to accept Eratosthenes' estimate in the way that it was presented.

  What happens if we apply a little more mathematics to the problem by including the deviation of Syene from Alexandria's great circle meridian? Instead of 7.59 degrees separation between Alexandria and Syene we have to use 7.08 degrees for the separation in latitude. Our conversion factor from angle to degrees tells us that this separation corresponds to 4372 stades. So our error from the theoretical model is reduced to about 3% and this is approximately the error in the Egyptian foot.

We have to consider the possibility that the Egyptian foot was intended to be a geometrical unit of length but that it's value was slightly off.

Geometrical Units and the Size of the Earth

  This is a convenient time to consider the proportion between the unit sphere and the dimensions of the Earth. How do we convert the angular units into units of length? When the meter was defined it was assumed that the distance from the equator to the poles was 10,000 km. So we can define an angle to length conversion factor, α2λ = 10000 km/90°, and see what we get in terms of the customary civil units of measurement.

 This looks like someone was really clever in defining the nautical mile and the foot but one can do the same thing with the ancient Greek units of measure, the pous, the stadion and the plethron.

The Greek mathematician, Eratosthenes, may have noticed this in ancient times and his rough measurements may have used geometrical units. If this is true then the ancient estimate of the size of the Earth was accurate to about 3% while that of the more modern foot has an accuracy of about 1%. It should be noted that a pous = 300 mm was used by the Phoenicians and the ancient Egyptians.

Best Position Estimate for a Number of Intersecting Great Circles

  If one increases the scale of distance for finding positions the curvature of the Earth needs to be considered. One then has to find the intersection of great circles instead of straight lines on a plane. If all the circles don't meet at a point then a best estimate of the position needs to be made. Since the estimated positions should be closely grouped together a simple average correct to produce a unit vector will do. Consider the following intersection problem in which the intersections of pairs of great circles are used to make rough estimates of the position and an improved estimate found by combining the estimates. One computes the binormals, the angles along the great circle and the intersection points as before.

A plot helps visualize the situation.

At greater magnification we can distinguish the points of intersection and the improved estimate. The average position is close to that found by deviations from the great circles and much easier to calculate.

A simple procedure for finding the angular distance, φ, of a point, q, from a great circle and the arc length, θ, for the closest point along the circle is shown below. It was found by using vector analysis.

The point of intersection is where φ = 0 and q is on the line p. At this point q·b = 0 and one can see how the binormal b can be used to find the intersection point.

Supplemental (May 3): The choice of angles for arc along the great circle and the angular distance of a point away from it may not have been the best ones since the analogy with latitude and longitude could be a little confusing. A better model would replace (θ,φ) with the angles (α,δ) which are used to represent the Right Ascension and Declination on the Celestial Sphere. Like the great circles of constant Right Ascension those of constant arc length along the great circle will converge to a point that is analogous to the Celestial Pole, the pole of rotation. The analogy identifies the great circle with the Celestial Equator.

Finding The Intersection Point For Two Great Circles

  One can use vector analysis to find the points where two great circle cross each other. This is an alternative to using spherical trigonometry or a search for a solution. A great circle on a unit sphere can be specified by a point it passes through and its tangent at that point. A third unit vector at the point and perpendicular to the first two unit vectors is the binormal and can be found by taking the cross product of the unit vectors for the point and its tangent. The binormal points in the direction of the axis of rotation that would move a point on the great circle along it. It is the same everywhere on the great circle.

  The distance of an arbitrary point on the sphere from the great circle is the length of the shortest arc connecting them.  This arc is part of another great circle that is perpendicular to the first and can be defined by their common point and the direction of the binormal of the first great circle. To get to the arbitrary point one travels along two arcs, the first along the great circle and a second perpendicular to it. One can use this procedure to find the perpendicular distance of a point on a second great circle from the first great circle. Where the they meet this distance is zero.

 To set up a problem I chose two arbitrary points on the equator of a unit sphere and arbitrary directions for the intersecting great circles and then computed the vectors for the points and the tangents. The formula for the unknown angle is found by taking the dot product of the initial position vector and the tangent for the second great circle with the binormal of the first. Once the angle is found the intersection point can be found by a simple rotation along the great circle.

A plot shows that the two great circles do indeed meet at this point.