"dimensionless points*": I meant to add a note to that. In fact, mathematically there are dimensionless points, and it held back the development of calculus. Although no one is known to have commented on Archimedes' proto- calculus as revealed in the palimpsest revealed twnty years ago or so, they certainly did on the efforts of Galileo amongst others. He used the method of integration by adding up slices to make a shape, other mathematicians objected that you can't make it exact, because you'd have to slice it infinitely thin like a Euclidean line, and no number of dimensionless lines can make up an area. This objection persisted even after Newton and Leibniz had made the use of calculus normal; see Bishop Berkeley's "ghosts of departed quantities". It was only finally sorted out in the 19th century, at least to the satisfaction of mathematicians. Actually I'm buggered if I can see why the solution was better than what went before, but that's maths for you.

But look at it in a different way. Take a square, side 1. Draw a diagonal across it. That's 1/2 + 1/2 = 1, yeah.

Now split one half from the corner of the square to the midpoint of the diagonal. You now have 3 similar triangles, and 1/2 + 1/4 + 1/4 = 1. Still at primary school.

Now take one of the little triangles and repeat, except that the split now goes to the midpoint of the hypotenuse of its hypotenuse: 1/2 + 1/4 + 1/8 + 1/8 = 1. Bright kid.

Repeat ad kalendas graecas. !/2+1/4+1/8+1/16+1/32+1/64+1/128+... lots of little bits .... + 1/8. 1/8? Isn't that nothing? Indeed it is.

Now repeat that process, except each time you divide a triangle, you do the same to all the others. By the same logic, you've now got:
1/8+1/8+1/8+1/8+...1/8... = 1.

Many a muckle macks a mickle, as the muckmen say in Mickleover. And 8 squared of nothings makes 1.

But you didn't have to start with a square with side 1, it could have been any number, well any positive number. So that rather tedious to write out series can be equal to anything.

But infinitesimals are mathematically very useful. I haven't found a use for infinity yet.