apotomes

An apotome is a mathematical term that has seen variations in its meaning throughout history. In ancient Greek mathematics, specifically in Euclid's Elements, an apotome referred to a line segment obtained by subtracting a smaller segment from a larger segment, where the smaller segment is commensurable with the whole but not with the difference. This definition is quite specific and relates to the concept of commensurability, meaning their lengths can be expressed as a ratio of two integers. For example, if a line segment has length $a$, and a smaller segment has length $b$, and $a$ and $b$ are commensurable, but $a$ and $a-b$ are incommensurable, then the remaining segment $a-b$ could be considered an apotome.

Later mathematicians, particularly in the context of algebra, adopted a broader definition. In modern usage, an apotome often refers to the difference between two magnitudes. This can be seen as a generalization of the ancient definition. For instance, it can represent the difference between two square roots, such as $\sqrt{a} - \sqrt{b}$. The significance of apotomes lies in their connection to incommensurable quantities and their appearance in geometric problems that lead to irrational numbers. The term itself originates from the Greek word "apotomein," meaning "to cut off."