Euclid's Elements - The Application of Areas

In summary, Heath emphasizes the significance of applying areas in his commentary on Euclid's Elements, particularly in Book I Proposition 44. He initially believed the proposition was proven using I.42, but later realizes the importance of also utilizing I.43 to maintain a constant area while changing dimensions. This was due to a misconception about AB being a finite segment rather than an infinite straight line. Therefore, the area must be applied to AB itself and not AB produced, which is why Euclid employs I.43.
  • #1
Hunus
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In Heath's commentary on Euclid's Elements he stresses the importance of the application of areas (Book I Proposition 44) with, "The marvellous ingenuity of the solution is indeed worth of the 'godlike men of old'...".

The proposition, "To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle" seemed to me to be proven with, "Let the parallelogram BEFG be constructed equal to the triangle C, in the angle EBG which is equal to the rectilineal angle D [I.42]", but he goes on to employ I.43 (complements of a parallelogram about its diameter are equal to one another) to be able to change the dimensions of the area, but to hold the area constant.

Now I understand that that is important in and of itself, but I don't see why using [I.42] isn't 'applying' the area to the straight line.
 
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  • #2
I figured out what was confusing me. My misconception was that I believed AB to be an infinite straight line with A and B only denoting the different directions of the line that had not yet been fixed -- in which case the proof would have ended after the construction of the parallelogram BEFG -- but AB is a finite segment and BE is only a production of AB beyond B.

So you have to apply the area to AB itself not AB produced -- which is why Euclid employed I.43.
 

1. What is Euclid's Elements?

Euclid's Elements is a mathematical treatise written by the ancient Greek mathematician Euclid around 300 BC. It is a collection of 13 books that cover various aspects of geometry, including the application of areas.

2. What is the significance of Euclid's Elements?

Euclid's Elements is considered one of the most influential works in the history of mathematics. It is the oldest surviving mathematical textbook and has had a profound impact on the development of geometry and other mathematical fields.

3. What is the focus of "The Application of Areas" in Euclid's Elements?

"The Application of Areas" is the tenth book in Euclid's Elements and is dedicated to the study of plane geometry, specifically the properties and application of different types of areas, such as triangles, parallelograms, and circles.

4. How does Euclid's Elements explain the concept of area?

Euclid defines area as the measure of a two-dimensional figure, or the amount of space it takes up. He also introduces the concept of congruent figures, where two figures have the same shape and size, and explains how to use this concept to calculate areas.

5. What is the most famous theorem in Euclid's Elements that involves the application of areas?

The most famous theorem in "The Application of Areas" is the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

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