Another example that shows up in mid school is how we measure angle.

Like the Greeks and Babylonians we usually measure it in degrees. Later around 18th century radians started getting used, especially in power series expansions.

In India, historically, angle was measured in the units of length (for a standardized circle). That made functions like sin be a function from length to length.

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kqr 5 days ago | parent [-]

> angle was measured in the units of length

i.e. a variant of radians? A radian is the circular arc whose length is equal to the radius. If we standardise the unit circle then a radian is a length of 1.

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srean 5 days ago | parent [-]

Yes. Similar idea. Sometimes they would also use minutes. Whether this was as a result of contact with the Greeks I do not know. Indian trigonometry however has a different flavour from the trigonometry of Hipparchus.

What Indian mathematicians typically used was a circle with radius 3438 units. Where units would be one of the standard units of length.

Why 3438 you may wonder.

They also wanted to divide the circle into 360 x 60 minutes. For the standard circle they wanted each of those minute arcs to be of 1 unit length. The radius that would accomplish this is (360 x 60)/ 2pi ~= 3438 units.

An angle of 1 minute would then be described as arc length 1 unit on that standardized circle of radius 3438 units.

Indian version of sine and cosine were not expressed as ratios but the corresponding (half) chord for a hypotenuse of 3438 units.

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kqr 5 days ago | parent [-]

Very neat. Alternative-length radians seem quite common. Radians are by definition one radius, but NATO mils are 0.00098 radii (and apparently the Finnish piiru are 0.00105 radii). What you describe are effectively a unit that is 0.0018 radii. Makes sense.

	▲	srean 5 days ago \| parent [-]
		I see. Today I learned about piiru from you. I do dislike the fact the libc sin takes argument in radians. For two reasons. One, the angles in the application are rarely in radians, so they need to be converted before the function call. Two, I would like the standard angles, such as the multiples of 15 degrees to have an accurate representation in 32 bits (or 64 bits). Anyhow this is way off topic.