looking for a few good ideas

  amongst the irregular verbiage

On matters of size and shape

Around about 200 BC, a Greek called Eratosthenes measured the size of the world.

Eratosthenes hung out in the library at Alexandria, which was basically the richest university town that ever was. A lot of travellers passed through Alexandria (and were consequently "persuaded" to donate their books to the library, in exchange for a copy when they left - but that's another story.)

Eratoshenes got to hear a lot of unusual and interesting tales, one of which was that there was a deep well in Syene where the Sun shone upon the waters at the bottom on the longest day of the year.

Now, the Greeks were pretty cool. They already accepted that the Earth was round, because they were quite good with angles and stuff, and had observed that the path the Sun moves across the sky (the "ecliptic") not only wiggles back and forth with the seasons, but wiggles back and forth in different places, depending on how far North from which you make your observations.

They called these extremes of movement of the ecliptic the "Tropics", because tropos is the Greek word for "turn". These extremes occur when the constellations Cancer and Capricorn appear in the sky, hence became to be called the "Tropic of Cancer", and the "Tropic of Capricorn" respectively.

They also observed that, if you traveled far enough South, you eventually got to a point where the noon-day Sun is directly overhead at a time when the ecliptic is equally between the two tropic extremes - and thus they called the line along the Earth at which this happens, the "Equator".

The Greeks estimated that the angle that the Sun makes with the equator at the tropical extremes was 24 degrees (which was pretty accurate - it's actually 23 1/2).

So when Eratosthenes heard the story about the well at Syene, he knew it must be located on or near the northern edge of the tropics. But he also knew that the Sun was not directly overhead in Alexandria at that time, as he could see shadows cast by the noon-day Sun.

On the longest day of the year, June 21st, when the Sun was directly overhead in Syene, Eratosthenes used a convenient perpendicular obelisk with a known height, and measured the noon-day shadow. Then, using geometry, he calculated that in Alexandria the Sun was at an angle of 7 1/5 degrees, while in Syene, the Sun was presumably at an angle of 0 degrees (being directly overhead). 7.2 degrees is one 50th of a full circle of 360 degrees, which means however far away Syene was from Alexandra, that distance represented one fiftieth of the total circumference of the Earth.

So all it took was for some poor schlub to be sent out to pace off an accurate measurement of the distance between Alexandria and Syene - 5000 stadia - which Eratosthenes then mulitplied by 50, to yield a total Earthly circumference of 250,000 stadia, or 46,000 kilometers.

These days we know the actual circumference is 40,000 km, so he got it pretty much right, despite a number of minor errors that effectively cancelled each other out.

Now, I think that was pretty cool. But this story isn't about Eratosthenes; it's about Claudius Ptolemy.

Claudius Ptolemy lived in Alexandria too, but about 350 years later than Eratosthenes. He was the most reknowned scholar in Alexandria at the time, and is famous for editing two massive tomes  of compiled knowledge called, respectively, "Almagest" - an encyclopaedia; and "Geography", which was essentially an Atlas. In these books he documented and extended many mathematic and scientific principles. For example, he proposed subdividing angles into minutes and seconds rather than using clumsy fractions of a degree; pre-empted Occam by stating that "for the best explanation of any phenomenon one should adopt the simplest one that explains and does not contradict the observations"; developed the principles of drawing maps to a consistent scale; practically invented lattitude and longitude as a mapping co-ordinate system; proposed using detailed, larger-scale maps to show areas about which more detail was known; and invented his own mapping system to get around the problems of projecting a spherical surface area on to a flat piece of paper.

Because Ptolemy was so thorough and generally excellent, his work was very influential, and therefore, along with all the good knowledge, several unfortunate errors and prejudices were perpetuated and accepted into the "common knowledge". One of these was his rejection of a theory of Aristarchus, who 50 years before Eratosthenes' time, had proposed that that the Earth revolved around the Sun. Ptolemy adapted Aristotle's Earth-centered scheme and made it a central premise in his encyclopaedia. It wasn't until the 1500's that Copernicus famously re-introduced the concept of a heliocentric universe.

Ptolemy made another, more consequential error in his Geography volume. He rejected Eratosthene's measurement of the Earth's circumference, and prepared his maps with an alternative estimation that was 3/4 of the size. He also assumed that the breadth of the "known" world extended East to cover 180 degrees - one complete hemisphere - of the Earth.

It was these two errors, propagated through history, that enabled Columbus to seriously propose launching an expedition to sail West in a quest to locate new trade routes to Asia: if he had known the true distance he would have had to sail, he almost certainly would not have attempted it.

Oh, and the book of the month is The Mapmakers, by John Noble Wilford, from which I learned most of the above information.