History of Math and Leonardo's Table of Chords

[BurtZ]

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.

 

[Office_Shredder]

Can you post a picture of the table of chords?

 

[BurtZ]

Can you post a picture of the table of chords?

 

[Office_Shredder]

Can you describe what each column means?

 

[Mark44]

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.

 

[BurtZ]

Can you describe what each column means?

I don't know. That's part of what I don't understand.

 

[BurtZ]

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."

 

[BurtZ]

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)

 

[Office_Shredder]

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)

 

[BurtZ]

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

 

[Office_Shredder]

So do you know how to compute the length of the corresponding chord for a radius of 21?

 

[BurtZ]

So do you know how to compute the length of the corresponding chord for a radius of 21?

No...

 

[BurtZ]

I assume it's based on the relationship between radius and chord length? Is there such a relationship?

 

[sysprog]

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.

 

[Office_Shredder]

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?

 

[BurtZ]

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?

 

[Office_Shredder]

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.

 

[BurtZ]

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...

 

[BurtZ]

Let me try again - 35 rods, 1 foot, 0 unciae.
Does this make more sense?
You were right - my other values clearly had problems.

 

[Office_Shredder]

I got 36 rods and 2 feet. I might be messing up the units though.

 

[BurtZ]

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?

 

[BurtZ]

Ah I see - the answer is either 42 or 90.

 

[Office_Shredder]

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.

 

[BurtZ]

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 :)

 

[BurtZ]

That makes a lot of sense! Now it is obvious why there are two numbers :)

One more question - 42 what?

 

[Office_Shredder]

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.

 

[BurtZ]

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!