History of Math and Leonardo's Table of Chords

In summary, the conversation discusses the use of Leonardo's table of chords to calculate arc length in circles of various radii. The table provides the corresponding chord length for each integral arc from 1 to 66 rods, using the Pisan measures of feet, unciae, and points. The conversation also mentions a similar table from Ptolemy and how to rescale the chord length for a circle with a radius of 21. Additionally, there is confusion about the measurement unit "unciae" and its plural form. The conversation concludes with a discussion about finding the arc length for a resized circle.
  • #1
BurtZ
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0
Homework Statement
Use Leonardo's table of chords to solve the following: Suppose a given chord in a circle of diameter 10 is 8 rods, 3 feet, 16 2/7 unciae. Find the length of the arc cut off by the chord.
Relevant Equations
Leonardo's table of chords
I know that arc length is L=rx where x is the central angle in radians. But, that doesn't help me here, because I don't know the central angle, and because I need to use Leonardo's table of chords. I don't understand how the table works.
 
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  • #2
Can you post a picture of the table of chords?
 
  • #3
Office_Shredder said:
Can you post a picture of the table of chords?
1609806191214.png
 
  • #4
Can you describe what each column means?
 
  • #5
I can't read the headings on the table. Wikipedia has something that I believe is similar -- Ptolemy's table of chords - Wikipedia

The part of the circle spanned by the chord can be thought of as being an isosceles triangle. Half of this triangle is a right triangle with hypotenuse 5 (rods?) and a base of half the length of the chord (4 rods, 1.5 ft, 8.5 + 1/7 unciae). That is one weird measurement. I'm guessing that unciae is plural of the Latin for inches, but I seem to remember that there were 12 unciae in a foot, but not sure.

Whatever the measurements are, it should be easy to find all of the angles of this right triangle, and from that you know the angle subtended by the chord, and from that the arc length along the circle.
 
  • #6
Office_Shredder said:
Can you describe what each column means?
I don't know. That's part of what I don't understand.
 
  • #7
In the book it says: "For each integral arc from 1 to 66 rods (and also from 67 to 131) the table gives the corresponding chord, in the same measure, with fractions of the rods not in sixtieths, but in the Pisan measures of feet (6 to the rod), unciae (18 to the foot), and points (20 to the uncia). Leonardo then demonstrated how to use the chord table to calculate arcs to chords in circles of radius other than 21."
 
  • #8
Mark44 said:
I can't read the headings on the table. Wikipedia has something that I believe is similar -- Ptolemy's table of chords - Wikipedia

The part of the circle spanned by the chord can be thought of as being an isosceles triangle. Half of this triangle is a right triangle with hypotenuse 5 (rods?) and a base of half the length of the chord (4 rods, 1.5 ft, 8.5 + 1/7 unciae). That is one weird measurement. I'm guessing that unciae is plural of the Latin for inches, but I seem to remember that there were 12 unciae in a foot, but not sure.

Whatever the measurements are, it should be easy to find all of the angles of this right triangle, and from that you know the angle subtended by the chord, and from that the arc length along the circle.
The chart from Wikipedia is similar but not the same. The column headings are not English (as far as I can tell at least)
 
  • #9
The circle described has radius 5. Your quote kind of says the table is for circles of radius 21? So the first thing is to rescale the length of the chord to compute what it would be if the radius was 21. (You should confirm this 21 radius thing, I'm just guessing from the last sentence)
 
  • #10
Office_Shredder said:
The circle described has radius 5. Your quote kind of says the table is for circles of radius 21? So the first thing is to rescale the length of the chord to compute what it would be if the radius was 21. (You should confirm this 21 radius thing, I'm just guessing from the last sentence)
Yes, it seems that the table is for a circle with radius 21
 
  • #11
So do you know how to compute the length of the corresponding chord for a radius of 21?
 
  • #12
Office_Shredder said:
So do you know how to compute the length of the corresponding chord for a radius of 21?
No...
 
  • #13
I assume it's based on the relationship between radius and chord length? Is there such a relationship?
 
  • #14
Mark44 said:
I'm guessing that unciae is plural of the Latin for inches,
It's plural for inch(es), and for ounce(s) of both weight and fluid.
 
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  • #15
BurtZ said:
I assume it's based on the relationship between radius and chord length? Is there such a relationship?

You just have to blow up for circle by a factor of 21/5 to change the radius from 5 to 21. So all lengths become 21/5 times longer, and areas become ##(21/5)^2## times larger.

So what is the length of the arc when you resize the circle?
 
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  • #16
Office_Shredder said:
You just have to blow up for circle by a factor of 21/5 to change the radius from 5 to 21. So all lengths become 21/5 times longer, and areas become ##(21/5)^2## times larger.

So what is the length of the arc when you resize the circle?
So I need to look for the the arc with a chord length of 168 rods, 64 feet, 75 unciae, 12 points?
 
  • #17
I don't think that's the right value. How did you get it? At first glance it looks like you multiplied by 21 instead of 21/5.
 
  • #18
Office_Shredder said:
I don't think that's the right value. How did you get it? At first glance it looks like you multiplied by 21 instead of 21/5.
I multiplied by 21/5 but when I had fractions I tried to convert them into the other units. I think I messed up there...
 
  • #19
Let me try again - 35 rods, 1 foot, 0 unciae.
Does this make more sense?
You were right - my other values clearly had problems.
 
  • #20
I got 36 rods and 2 feet. I might be messing up the units though.
 
  • #21
Office_Shredder said:
I got 36 rods and 2 feet. I might be messing up the units though.
I checked my work - you are right. I messed up in that last conversion. How do I read the chart now?
 
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  • #22
Ah I see - the answer is either 42 or 90.
 
  • #23
Yep. Remember that there are two arcs, the small one that it cuts off and the big one which makes the rest of the circle. One is length 42, the other is 90.
 
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  • #24
Office_Shredder said:
Yep. Remember that there are two arcs, the small one that it cuts off and the big one which makes the rest of the circle. One is length 42, the other is 90.
That makes a lot of sense! Now it is obvious why there are two numbers :)
 
  • #25
BurtZ said:
That makes a lot of sense! Now it is obvious why there are two numbers :)
One more question - 42 what?
 
  • #26
Whatever units the radius is in. If the radius is 21, then the circumference of the circle is ##2\pi r \approx 132##.

And 90+42=132 of course.

With that said, I realize we forgot one step. You get 42/90 in the circle of radius 21, but the original circle was radius 5, so you need to multiply by 5/21 to get back to the right size.
 
  • #27
Office_Shredder said:
Whatever units the radius is in. If the radius is 21, then the circumference of the circle is ##2\pi r \approx 132##.

And 90+42=132 of course.

With that said, I realize we forgot one step. You get 42/90 in the circle of radius 21, but the original circle was radius 5, so you need to multiply by 5/21 to get back to the right size.
Of course! I really appreciate your help - you walked me through this, but in a way that helped me really understand what I was doing!
 

1. What is the history of math and its significance?

The history of math dates back to ancient civilizations such as the Egyptians, Babylonians, and Greeks. It has played a crucial role in the development of human society, from basic counting and measuring to complex calculations and theories. Math has also been used to solve real-world problems and advance technology.

2. Who is Leonardo and what is his Table of Chords?

Leonardo of Pisa, also known as Fibonacci, was a mathematician from the 13th century. He introduced the Hindu-Arabic numeral system to Europe and is best known for his famous sequence of numbers. His Table of Chords is a mathematical table that relates the chords of a circle to its corresponding angles, which was used for trigonometric calculations.

3. How did Leonardo's Table of Chords contribute to the development of math?

Leonardo's Table of Chords was a significant contribution to the field of trigonometry. It provided a systematic way of calculating the lengths of chords and angles in a circle, which was crucial for navigation, astronomy, and architecture. It also laid the foundation for the development of modern trigonometry and its applications in various fields.

4. What were some challenges faced by Leonardo in creating his Table of Chords?

One of the main challenges faced by Leonardo was the lack of advanced mathematical tools and techniques at the time. He had to rely on basic geometric principles and calculations to create his table. Another challenge was the limited availability of accurate measurements, which affected the accuracy of his calculations.

5. How is Leonardo's Table of Chords still relevant today?

Leonardo's Table of Chords is still relevant today as it forms the basis of modern trigonometry. It is used in various fields such as engineering, physics, and astronomy to calculate angles and distances. It also serves as a historical reference for the development of math and its impact on society.

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