Archimedes — "Every magnitude is comparable with every other magnitude of the same kind."
Every magnitude is comparable with every other magnitude of the same kind.
Get This Quote & Author's Image Illustrated On:
Click any product to generate a realistic preview. Up to 3 at a time.
* Initial load can take up to 90 seconds — revising the preview in another color is nearly instant.
Kitchen
Apparel
Other
More from Archimedes
"The sun is a very large body, much larger than the Earth."
"Eureka!"
"No difficulty can be too great for the human mind, if it applies itself with diligence and skill."
"It is easier to make a thousand discoveries than to invent a single new method."
"Give me the place to stand, and I shall move the earth."
Verification
Unverifiable
Found in 1 providers: grok
1 source checked
Understanding this quote
What it means
Any two quantities of the same type — two lengths, two areas, two weights — can always be meaningfully compared. Neither is so tiny nor so vast that comparison becomes impossible. Repeat the smaller quantity enough times and it will eventually exceed the larger. This eliminates infinitely large or infinitely small values from practical mathematics, asserting that measurement between like things is always achievable.
Relevance to Archimedes
Archimedes built his greatest results — computing areas under curves, approximating π to remarkable precision, finding volumes of spheres — by systematically comparing magnitudes through the method of exhaustion. This principle was his operational foundation: he could only prove a curved area matched a rectilinear one by showing both could be squeezed between comparable bounds. His legendary precision in measurement depended entirely on knowing that any two like quantities could be brought into direct relation.
The era
Greek mathematicians had been unsettled by incommensurable magnitudes — quantities like √2 that no ratio of integers could express. Working in 3rd-century BC Syracuse, shortly after Euclid formalized geometric foundations, Archimedes needed to assert that magnitudes of the same kind remain comparable even without a shared integer unit. This grounded the method of exhaustion in rigorous logic, allowing Greeks to calculate areas and volumes with confidence before the formal concept of limits existed.
AI-generated insights based on extensive research and information for context. Factual errors? Email [email protected].