Where does exhaustion collide?

In classical mathematics the method of exhaustion finds an area or volume by inscribing a sequence of polygons or solids whose combined size approaches the figure as closely as desired, then arguing that no other value can be correct, a rigorous forerunner of the integral. Heath's edition of The Works of Archimedes presents Archimedes' use of the method, building on Eudoxus, to square the parabola and bound the circle. The referent is a limiting procedure in geometry; the shared word says nothing about dye leaving a bath.[1]

Unit

The limit that the inscribed figures converge to

Recovery means

Not applicable

Measured by

Established by proof that the quantity is squeezed between converging bounds

In logic and computer science proof by exhaustion, or case analysis, establishes a statement by splitting it into finitely many cases and verifying each one, so that checking all of them settles the whole claim. Rosen's Discrete Mathematics and Its Applications treats it as a standard proof technique, noting it is feasible only when the cases are few enough to enumerate. Despite the shared name, this is exhausting a list of cases, not the geometric limit of the method of exhaustion nor the depletion of a dye bath.[2]

Unit

Whether every case has been checked and holds

Recovery means

Not applicable

Measured by

Verified by enumerating the complete set of cases and confirming each

In dyeing, exhaustion is the proportion of dye that has transferred out of the dyebath and onto the fibre, so a bath is 'exhausted' when little dye remains in solution. Burkinshaw's Theoretical Aspects of Textile Coloration describes exhaustion as a function of time and of the equilibrium between fibre and liquor, central to dyeing efficiency and to effluent. The referent is the degree of dye uptake from a bath, unrelated to a mathematical limit or a logical case analysis.[3]

Unit

The fraction of dye taken up from the bath by the fibre

Recovery means

Not applicable

Measured by

Measured from the fall in dye concentration in the bath, often as percent exhaustion over time