# Points & Lines

A reinterpretation into comics of Book 1 of Euclid's Elements.

Inspired by
Oliver Byrne's pioneering 1847 edition of the Elements,
Edward R. Tufte's Envisioning Information ("the notoriously circuitous Euclid"),
Scott McCloud's Understanding Comics: The Invisible Art,
Michael Nielsen's Thought as a Technology essay, and
Molly Bang's Picture This: How Pictures Work.

1. ## Maker of same-sided triangles

Given a line.
Draw on it a same-sided △ triangle.
1. 1.1
Draw a ◯ circle centered on one endpoint with the line as radius.
Compass.
2. 1.2
Draw another ◯ circle centered on the other endpoint with again the line as radius.
Compass.
3. 1.3
Draw lines from the endpoints to the point where the ◯◯ circles cross.
Point-to-point ruler.
4. 1.4
These lines are both radius of the same ◯ circle and so equal.
Circle definition.
5. 1.5
These lines are both radius of the other ◯ circle and so equal too.
Circle definition.
6. 1.6
So we have a △ triangle with all lines equal to one another. Boom.
Equal to the same, equal to each other.
2. ## On-point line-duplicator

Given a point & a line.
Draw on the point another line
equal to the first.
1. 2.1
Draw a line from the point to an endpoint of the line.
Point-to-point ruler.
2. 2.2
Make a same-sided △triangle on this new line.
Prop1: Maker of same-sided △triangles.
3. 2.3
Extend these two sides of the △triangle.
Line-extending ruler.
4. 2.4
Draw a ◯ circle with the original line as radius and centered on an endpoint.
Compass.
5. 2.5
With one △triangle vertex as center, draw a ◯ circle with these 2 lines as radius.
Compass.
6. 2.6
Since these 2 lines are radius of the same ◯ circle, they are equal.
Circle definition.
7. 2.7
Substracting the 2 equal △ triangle sides to the 2 previous lines, the remaining 2 lines are still equal.
Take equal from equals, still equal.
8. 2.8
Now note that these lines are radius of the same ◯ circle and so equal.
Circle definition.
9. 2.9
So the 3 lines are equal to each other. Finally, done!
Equal to same, equal to each other.
3. ## On-line line-duplicator (Measurer!)

Given 2 lines.
Cut from the big one
a part equal to the small.
1. 3.1
Place at an endpoint of the big line
a line equal to the small one.
Prop2: On-point line-duplicator.
2. 3.2
With the same endpoint as center draw a circle with the new line as radius.
Compass.
3. 3.3
These lines are equal because they are both radius of the same circle.
Circle definition.
4. 3.4
But this line was also equal to the small line.
That's how we built it.
5. 3.5
So all lines are equal to each other. Cake!
Equal to same, equal to each other.