Did Robert Gordon really "Square the Circle" or did he "prove the sphere?"

Ah, 16th century Sir Robert Gordon, 1st Baronet of Nova Scotia, or is it 16th century Robert Gordon of Straloch, cartographer and father of nine sons and six daughters by one wife (whew), about your trip circumnavigating the globe during six days (New York city time).  I think Mr. Gann 'buries a dog' (yet another) of mathematic interest in The Tunnel Thru The Air deserving of an attempt of disinterment.

RG, you left New York City at 7am on July 20, 1932 and completed your circumnavigation six days later at 7am in Montreal.  Check it.  RG left the NYC at the longitude of 74* and arrived at Montreal at 73*56".  He actually traveled somewhat more that the circumference of the earth in exactly 144 hours (NYC time).  So what was the East/West distance you traveled given that you ended up further north than from where you began?

Well, since the earth is a sphere, you certainly didn't travel the 24,901 miles distance around the world which is measured at the equator.  Nope, you had to have traveled less.  As a cartographer and geometer, you know the distance around the world at the latitude of NYC (latitude 40.7069) and at Montreal (45.5000) is far less than 24,901 and can be determined by a simple formula.  So, let's compute those two distances to find the average:

So, on a purely East/West path, you traveled 18,132 miles.  Goodness, I seem to recall finding much that same number elsewhere.  Why yes.  I found the East/West mileage by deriving that pure East/West mileage from separate methods; 1) by punching into an Internet calculator every one of you stops to isolate the East/West component (holding latitude constant) and 2) by computing the distance between each of you stops using the Haversine formula (again holding latitude constant).  When I did that, I found the consensus of the two methods was 18,236 miles (see the column "Cumulative East/West" column and the row "Montreal"):

So, I've proven RG's trip was comprised of around 18,200 East/West miles, but what was his North/SOuth mileage?  Well, we really don't have a simple formula for that, but on the spreadsheet that was just validated according to the Haversine formula and convenient Internet calculators, we have the same calculation for North/South mileage.  It is under the column "Cumulative N/S absolute" and it is 18,164 miles.

Uncanny, rounded we find that the East/West is 18,200 miles and the North/South miles is 18,200 miles.  Ponder that.  Just an accident or did Gann mean to make a point?

Now the question.  Did RG "square the circle" or did he "prove the sphere?"  [No, not circle within the square or square within the circle, something quite different.]  Squaring the circle can be defined as finding the circumference of a circle equal to the perimeter of a square OR, similarly, finding the equivalence of the areas of the circle and the square.  In this case, RG MAY HAVE demonstrated the perimeter squaring of the circle and square.  OR

Perhaps, RG intended to demonstrate the roundness or spherical nature of earth.  He traveled E/S and N/S the same distance….a sphere.

Symbolically, I side with his having squared the circle.  RG never crosses the North or South poles and thus never created a circumnavigation on a N/S basis.  And second, the earth has a maximum circumference of 24,901 (or thereabouts)….  The sphere RG created would be 18,200 both E/W and N/S.

As if to put a point on it, RG quotes Ezekiel 48:35 in the last pages of TTTTA.  He doesn't spell it out, but if you read that scripture, you'll find "The distance all around will be 18,000 cubits. 

So, I believe Gann was teaching squaring the earth's circle at the approximate latitude of New York City….. oh, did I forget?  In 144 NYC hours.  [That might be the context for additional research and another essay.]

[Does it occur to anyone that 24,901 / 18,200 = 1.3682 AND 1.3682^3 = 2.5612 AND that the square root of 2.5612 is equal 1.6003…. an early iteration of Fibonacci's Phi..  Perhaps the inner circle, the square and the outer circle proving the 4th dimension Mr. Cowan?  The "square within the circle.'  The conic counterpart of the pyramid? Another context for additional research and, yet, another essay.]

Jim