What tool did the Greeks use to draw parallel lines?

[musicgold]

Homework Statement

Not a homework problem. I am curious as to how the Greek geometers drew parallel lines using only the compass and the straight age.

Relevant Equations

At the 1.30 mark in the video below, Johnny Ball explains how the Greeks multiplied numbers. As far as I know the Set Square is a later invention; so the Greeks must have used something else.

Thanks

 

[sysprog]

They could e.g. take a perpendicular bisector of a line segment $ab$, creating line segment $cd$, and then take a perpendicular of $cd$, creating $ef$, which would be parallel to $ab$.

Here's an explanation of the transverse angle copy method with a proof:
https://www.mathopenref.com/constparallel.html

 

[Lnewqban]

Two arcs from two points (as far from each other as the length of the available straight edge) on the original straight line, then, draw the second line being tangent to both arcs.

 

[sysprog]

Two arcs from two points (as far from each other as the length of the available straight edge) on the original straight line, then, draw the second line being tangent to both arcs.

This method is fast and simple in practice, but it relies on reckoning and verifying to get the correct tangent, instead of on first establishing points of intersection and then joining them.

 

[Lnewqban]

This method is fast and simple in practice, but it relies on reckoning and verifying to get the correct tangent, instead of on first establishing points of intersection and then joining them.

That is true.

Please, see these other methods:
https://www.mathopenref.com/constparallel.html

https://www.mathopenref.com/constparallelrhombus.html

https://www.mathopenref.com/constparalleltt.html

In building construction layout, any of those methods can be used with a string under similar tension for drawing all the arcs, rather than a compass.

 

[sysprog]

@Lnewqban $-$

If y'ain't cheat'n', y'ain't tryin' hard enough $-$ US Army NCO​

 

[jack action]

As far as I know the Set Square is a later invention; so the Greeks must have used something else.

They could have made a 3-4-5 triangle with rope segments:

​

 

[DaveC426913]

This method is fast and simple in practice, but it relies on reckoning and verifying to get the correct tangent, instead of on first establishing points of intersection and then joining them.

What need for tangents?

 

[sysprog]

What need for tangents?

Your drawing shows 4 arcs, while @Lnewban's tangent is common to 2 arcs:

Two arcs from two points (as far from each other as the length of the available straight edge) on the original straight line, then, draw the second line being tangent to both arcs.

 

[DaveC426913]

Your drawing shows 4 arcs, while @Lnewban's tangent is common to 2 arcs:

Well yeah, but a bad idea is a bad idea.

 

[sysprog]

Of course it's true that you don't need tangents to construct parallel lines, but I wouldn't call the quick method posted by @Lnewqban a bad idea $-$ it's fast and simple, and could be practical for e.g. carpentry or masonry purposes, in that you can set the compass to the distance at which you want the parallel line.

 

[Lnewqban]

Well yeah, but a bad idea is a bad idea.

Not too precise, but not a bad method when you need to field-layout a parallel of a sequence of irregular curves, like when building a sidewalk or road that follows a twisting course.

Please, see:
https://en.wikipedia.org/wiki/Parallel_curve

 

[Office_Shredder]

Is the question how did they make parallel lines in geometric proofs, or how did they do it when like, engineering stuff? I would be surprised if they didn't know to just measure off a distance perpendicular to the line in practice.