Going from memory, you use a straightedge then mark it at a specific intersection. That’s not the same as using a ruler.
That’s no different than using a specific measurement on the ruler (which is how we did it).
I think the issue some have with neusis construction is that it isn’t intrinsically precise. You need to slide the marked ruler so that, by eye, a marked distance intersects a desired element. There is an intuition that an ideal compass and straightedge is infinitely precise. But sliding something about doesn’t, for want of a better phrase, “lock in”, in the same way.
Very hand wavy. Neusis construction isn’t limited by the precision of the markings made on the ruler, these can created with compass and straightedge along with the rest of the construction. The trick is that distances on a straight line so inscribed can’t be arbitrarily lifted up and slid about without it becoming a neusis construction.
But complaining about this is perhaps best left to the ancient Greeks.
As the saying goes, there are no bad Pizzagons nor bad Sextangles…
You can make the reference circle for trisecting using neusis any size, so you can set your compass to line up exactly with a specific mark on the ruler just as easily. So it’s as precise as the compass is (which, let’s face it, is never going to be 100% either)
Yes it is because a ruler has gradations whereas (theoretically) a neusis is infinitely precise. Show me where π/4 is on a ruler.
The specific trisect neusis uses an arbitrary circle. So it can be precisely X ruler ticks in radius.
Show me the relevance of π/4 for the trisection method.
There is a difference between making a mark on a straightedge at a specific point and using a premarked straightedge.
Let the vertex of α be labelled B.
Let A be the point of one of the branches of α such that the distance AB equals pq.
Let the other branch of α be extended past B through D.
Let a circle be drawn with center at B with radius AB.
Let the straightedge S be placed so that:
S passes over A
p lies on the circle at C
q lies on the straight line BD at D.
Show me where in this construction you can make a circle so that you a guarantied that in the last step that the intersection C is at one of the premarked points on your ruler.
Proof taken from here if a visual is needed. To do what you are claiming (just use a ruler) it would have to have both constructible and non-constructible numbers on it. For example, do rulers have the root of cubic polynomials on them?
How about cos 20 degrees?
If you don’t understand that pq can be any arbitrary measurement and is equal to AB = CD*, I can’t really help you. Maybe this will help?:
If I use ten ticks on a marked ruler as my pq, then CD is going to be ten ticks. If I use 20, it’ll be 20. I’m not using the markings on the ruler to measure anything. I’m using them as the delineators of my neusis marking. Which all the rest of the construction uses as its basis.
You do understand that the proof you linked to starts with a measurement on the ruler, right?
You know, the bit you somehow failed to copy, from the start of that proof:
Let there be a neusis ruler S marked by two points p and q.
What about it? You didn’t answer what the relevance of the first one is, this second one is equally irrelevant.
That sounds very rational. I was lucky enough to have an irrational math teacher. I might even have had an imaginary math teacher.
“Imaginary numbers” is a terrible term for them. They should be called “rotational numbers”. All i is, is a left turn. Really, that’s it. If you’ve ever made a left turn, you’ve used imaginary numbers. “i^2 = -1” just means that if you make a left turn, and then make another one, you’ve turned yourself in the opposite direction from where you were going before.
Quick, ask ChatGPT what a good name would be! /S
My vote is for positive numbers to be called “forward numbers,” negative numbers to be called “backward numbers,” and imaginary numbers to be called “sideways numbers.”