Hipparchus and Chords (Historical Trig Question)

[dkotschessaa]

I am reading Eli Maor's Trigonometric Delights which is a fascinating story of the history of Trigonometry. I have a rather elementary question.

On the early history:

"To be able to do his calculations Hipparchus needed a table of trigonometric ratios, but he had nowhere to turn: no such table existed, so he had to compute one himself. He considered every triangle - planar or spherical - as being inscribed in a circle, so that each side becomes a chord. In order to compute the various parts of the triangle one needs to find the length of the chord as a function of the central angle, and this became the chief task of trigonometry for the next several centuries."

My dumb question is - why chords? Why circles? Looking at the diagram, I can't see anything that relates to the circle itself, other than it's radius, which is merely one of the sides used to determine other sides (it is set at 60). I can't see any reason the circle needs to be there. A triangle on it's own would have worked just fine. Why a circle?

-Dave KA

 

[SpectraCat]

I am reading Eli Maor's Trigonometric Delights which is a fascinating story of the history of Trigonometry. I have a rather elementary question.

On the early history:

"To be able to do his calculations Hipparchus needed a table of trigonometric ratios, but he had nowhere to turn: no such table existed, so he had to compute one himself. He considered every triangle - planar or spherical - as being inscribed in a circle, so that each side becomes a chord. In order to compute the various parts of the triangle one needs to find the length of the chord as a function of the central angle, and this became the chief task of trigonometry for the next several centuries."

My dumb question is - why chords? Why circles? Looking at the diagram, I can't see anything that relates to the circle itself, other than it's radius, which is merely one of the sides used to determine other sides (it is set at 60). I can't see any reason the circle needs to be there. A triangle on it's own would have worked just fine. Why a circle?

-Dave KA

My guess is that it is because they could use that well-defined relationship between circles and triangles to deduce relationships based on the known properties of the two figures, and also test these deductions empirically by making actual constructions (say by sketching them in the sand or on a slate).

For example, a chord subtends a unique central angle. If you make the same chord be one leg of an inscribed triangle, then you can prove that the central angle must be exactly twice as large as the inscribed angle. This then leads to the conclusion that the sum of the angles of a triangle must be 180 degrees (since a complete circle is 360 degrees).

 

[dkotschessaa]

I also realized that an inscribed triangle with the central angle at the center of the circle has two sides of the same length, always. Is this a simplistic way of saying the same thing?

That is, two of the sides are equal to the radius.

 

[SpectraCat]

I also realized that an inscribed triangle with the central angle at the center of the circle has two sides of the same length, always. Is this a simplistic way of saying the same thing?

That is, two of the sides are equal to the radius.

That would be a relation for isosceles triangles, but that is not what is being described here. An inscribed triangle simply means that all of the vertices lie on the circle. http://en.wikipedia.org/wiki/Inscribed_figure

So the Hipparchus relation is much more general. The central angle they refer to is the central angle between radii extending to two vertices of the triangle.