Does anyone have any information about the "linea gyrativa"? I came across it in a discussion by Suarez which I am attempt to translate from the Latin.
Roughly, imagine a cylinder of unit length divided into two halves. Draw a spiral from the top of the cylinder round to the bottom of the first half. Then divide the second half into two quarters, and draw a spiral round to the bottom of the first quarter. Then divide the other quarter into two eights, and draw another spiral.
So you have a spiral staircase down the cylinder that gets progressively shallower.
Since there are an infinite number of such cylinders, and since the length of each spiral cannot be less than the circumference of the cylinder, it follows that the spiral is of infinite length.
Yet all on a cylinder of unit length. Suarez says this paradox originated with Buridan.
And is anyone aware of the background to Suarez' discussion? It is in section 5 of the 40th Metaphysical Disputation, about continuity and the continuum. It was written in the late 16th century. Suarez seems to accept the existence of an actual infinity of points or 'continuing indivisibles' that give continuity to the line. He reconciles this with Aristotle by saying
"Cum ergo haec indivisibilia dicuntur esse in continuo in potentia, non opinor esse intelligendum illam dictionem in potentia ut excludit realem existentiam, sed ut excludit realem divisionem.", which I translate as "When these indivisibilia are said to be in the continuum potentially, I do not think the expression 'potentially' should be understood as excluding real existence, but that it excludes real division.
And yet Suarez was a leading philosopher of the Jesuit order, who were supposed to be hostile to such an idea.