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Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Archimedes and limit 2

I have main question about limit
Why archimesdes used two polygon
One inscribed in a circle and second circumscribed around the circle
Why he didn't use only one polygon
Can u interpret this gor me in very easy way pleaee
Because i think it enough to use yhe inscribed polygon to measure the perimeter of the circle

Also archimesdes used " Archimedes used a basic geometry including properties of circle and polygon
ratios of sides of triangles to calculate polygon perimeter (would not have used the word "sine" in his work, as this term was not used until much later in history)"

May i know what is these
1) basics geometry
2) properties of circle abd polygon
3) ratios of sides of traingle
Specially the ratios!! What r these
Is there a book ir websites to illustrate these all of them where can i study these and learn about
Please it's my last question about archimesdes i need fullfill illustratetion in details without using sine and cos and modern calculation method
I want to find Pi just like him and at my home

Last edited by Hannibal lecter (2023-06-23 23:06:37)

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Archimedes and limit 2

Imagine I do a calculation with a certain amount of accuracy and get, let's say, 1.9

Then I repeat the calculation using more accuracy and get 1.99

Then again and I get 1.999

I notice that improved accuracy brings me closer and closer to an answer of 2.  It looks like 2 is the 'limit' that I could reach if I could achieve perfect accuracy.  That's what we mean by a limit.

The numbers are getting closer and closer to a certain value and we call that value the limit of the calculation.

There's an old tale about a frog. When it jumps it can go one metre. But this makes it tired so its next jump is only 0.5 m. And the next 0.25 m.  Each time it jumps it only manages to go half as far as the previous time.  Does the frog ever manage to get altogether 2 metres from its starting point.

1 + 0.5 + 0.25 + 0.125 + 0.0625 + .... is always under 2 no matter how far along the calculation we go but we're getting closer and closer.  The limit of the frog's jumps is 2.

Archimedes wanted to find a way to find the circumference of a circle.  He knew how to calculate the perimeter of a regular polygon so he reasoned like this. I'll surround the circle with a regular polygon and find out its perimeter.  I know the circumference must be less than this.  I'll also fit a regular polygon inside the circle and calculate its perimeter.  These two calculations give an upper bound and a lower bound for the circumference. The actual value must be below the upper bound but above the lower bound.

Increasing the number of sides for the polygon produces a shape whose perimeter is closer to the cicumference so we can 'move' the upper bound downwards and the lower bound upwards. So the real circumference is 'sandwiched' between two values that are getting closer and closer to the answer we want.  That's how he was able to find the circumference of a circle.

In this post http://www.mathisfunforum.com/viewtopic.php?id=26709 the poster had to work out the area of various polygons. I posted back a method for this. Calculating the perimeter if you know the distance from the middle of the shape to the edge is shorter but uses the same ideas. 

To understand it you have to know how to do trigonometry.  Here's a link to the MIF page on this.  https://www.mathsisfun.com/algebra/trigonometry.html

It'll take you quite a few days to master all of this.  Please do this and try to avoid posting any new questions until this one is fully resolved.  If you need help with 'trig' of course that's ok.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob