Calculating the Radius of a Graduated Circle

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In summary, the conversation discusses a problem from a 1895 trigonometry book about finding the radius of a graduated circle using the arc-length formula. The confusion arises from the use of the term "5'" which is clarified to be 5 minutes of arc.
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Ragnarok7
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Hello, I was working problems from a very old trigonometry book, Loney's Trigonometry from 1895. There appears here a problem stating:

The value of the divisions on the outer rim of a graduated circle is 5' and the distance between successive graduations is .1 inch. Find the radius of the circle.

I cannot determine what is meant exactly by a graduated circle, other than that it was a surveying instrument. I am also unsure if the 5' measurement is supposed to mean 5 inches or 5 seconds (I'm thinking the latter). Does anyone have any ideas? Thank you!
 
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  • #2
The expression 5' refers to 5 minutes of arc, or 1/12 of a degree (since there are 60 minutes of arc in a degree). So, I would use the arc-length formula:

\(\displaystyle s=r\theta\)

You are given $s$ and $\theta$, so you can solve for $r$. :D
 
  • #3
Ah! Okay. I should have said minutes, not seconds. I get it now. Not sure why I was confused.

Thanks!
 

FAQ: Calculating the Radius of a Graduated Circle

What is the "Graduated circle" problem?

The "Graduated circle" problem is a mathematical puzzle in which a circle is divided into equal parts, and a specific number of points are placed on the circumference. The goal is to find the minimum number of points needed to divide the circle into the specified number of equal parts.

What is the significance of the "Graduated circle" problem?

The "Graduated circle" problem has been studied for centuries and has practical applications in fields such as geometry, astronomy, and cartography. It also has connections to number theory and other areas of mathematics.

What is the solution to the "Graduated circle" problem?

The solution to the "Graduated circle" problem depends on the number of points and the desired number of equal parts. For example, if the circle is divided into 4 equal parts, the minimum number of points needed is 3. However, there is not a general formula for the solution and it must be found through trial and error or using mathematical methods.

Are there any variations of the "Graduated circle" problem?

Yes, there are variations of the "Graduated circle" problem that involve different shapes, such as squares or triangles, and different numbers of equal parts. These variations can be more challenging and require different approaches to find the solution.

What are some strategies for solving the "Graduated circle" problem?

One strategy for solving the "Graduated circle" problem is to start with a known solution and try to modify it to fit the desired number of points and equal parts. Another strategy is to use mathematical methods, such as geometry or number theory, to find patterns and relationships that can lead to the solution.

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