Introducing Infinity

IMAGINE DIVIDING THE CIRCLE INTO LOTS OF THIN SEGMENTS, LIKE PIECES OF AN ORANGE.
This reflected an idea that went back as far as the ancient Greek philosopher Antiphon.
Antiphon was a contemporary of Socrates, born in the 5th century BC. He suggested that you could work out the area of a circle by drawing a regular polygon inside it and gradually increasing the number of sides until it was closer and closer to the circle itself. But 15th-century philosopher Nicholas of Cusa went further. He imagined stacking segments of a circle on top of each other, alternating direction. This made something very close to a rectangle that would be πr in height and r in width, making its area πr2. Of course the edges of those segments would never be quite straight unless the indivisibles were infinitely narrow.

UNFORTUNATELY, IF YOU TOOK GALILEO’S ARGUMENT LITERALLY YOU WOULD THINK THAT EVERYTHING WITH WHITE CURLY FUR AND FOUR LEGS WAS A SHEEP – EVEN IF WHAT YOU WERE LOOKING AT WAS A WHITE SOFA WITH A FURRY COVER.
The indivisibles

Around this time, the idea of indivisibles became popular. This was a similar approach to the ancient Greek idea of atoms. Ancient Greek atoms were the result of cutting something up so small that it was no longer possible to cut any further (a-tomos means “uncuttable”). The use of indivisibles involved dividing an object into smaller and smaller pieces, but not necessarily in all three dimensions. Take the example of the area of a circle.

MEAN THAT UNITY CONTAINS IN ITSELF AS MANY SQUARES AS THERE ARE CUBES AND NATURAL NUMBERS.
A common error

The trap Galileo seems to have fallen into is a common one – the assumption that when two things have similar properties they can be equated. This is part of the basis of homeopathy, where the idea of the “law of similars” is that a poison that causes similar symptoms to a disease will cure that disease. Galileo equated infinity and unity because they had similar properties.