Which mathematical functions cannot be performed on the ancient abacus?

The ancient abacus is a manual counting tool that allows users to perform basic arithmetic operations like addition and subtraction, as well as multiplication and division through repeated addition or subtraction. However, it is not designed for more complex mathematical functions that require deeper algebraic manipulation or trigonometric calculations.

The performance of cosine and tangent functions falls into this category of complexity. These trigonometric functions are fundamentally different from basic arithmetic operations and involve geometric interpretations that cannot be represented or calculated directly on an abacus. An abacus operates primarily with integer and basic fractional manipulations, making the execution of trigonometric functions such as cosine and tangent—both of which fundamentally involve angles and ratios of side lengths in triangles—impossible without additional tools or methods.

In essence, the ancient abacus is effective for foundational arithmetic but is limited when it comes to performing advanced mathematical functions that go beyond simple calculations.