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The problem with the comments above is that the arc and the chord are measured in different units: the arc in degrees, i.e. 180ths of the semicircle, and the chord in 120ths of the diameter. If the arc length were 60 times the number of radians, then the comments above would be unproblematic.
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The article currently states "For tiny arcs, the chord is to the arc as π is to 3, or more precisely, the ratio can be made as close as desired to π/3 ≈ 1.04719755 by making θ small enough. Thus, for the arc of (1/2)°, the chord is slightly more than the arc."
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More on the geometric theorems: Their precise statements, how they are proved, how they are used in deriving trigonometric identities, how those identities are used in computing chords.
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Most, or maybe all, of the assertions are in the listed references, but I'd have to sort out which is which. I'll probably get to that in the coming weeks if no one beats me to it.
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as π is to 3, or more precisely, the ratio can be made as close as desired to π/3 ≈ 1.04719755 by making θ small enough. Thus, for the arc of (1/2)°, the
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longer than chord since in
Euclidean Geometry the shortest path between two points is a straight line. It should state: For tiny arcs, the
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The details of the description of the meaning of the "sixtieths" column doesn't seem to make a lot of sense. How is it scaled? --
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content on
Knowledge. If you would like to participate, you can edit the article attached to this page, or visit the
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I found a cite for "an earlier table of chords by
Hipparchus gave chords only for arcs that were multiples of 7½°)"
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It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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439:(an earlier table of chords by Hipparchus gave chords only for arcs that were multiples of 7½°).
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Several centuries passed before more extensive trigonometric tables were created.
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When did more extensive tables supersede this one? Which century?
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The parts about the three distinct methods of computing chords.
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This statement has the arc and chord reversed. The arc is
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History of editions of the book including those in Arabic.
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420:Updated 2011-11-03
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