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In special relativity (SR) and general relativity (GR), there are two concepts of time: coordinate time t and proper time τ. Two facts deserve reflection: (1) Clocks measure τ, but τ is assigned a much smaller role in the equations of physics than t. (2) Cosmologists are aware of the Hubble parameter Hθ, but when parameterizing worldlines in spacetime, the parameter τ is preferred over θ = 1/Hθ. Here we show: There is a different description of nature that does not conflict with SR/GR. Euclidean relativity (ER) uses τ as a coordinate, θ as the “cosmic evolution parameter”, and a Euclidean metric. All energy moves through 4D Euclidean space (ES) at the speed c. The laws of physics have the same form in each object’s reference frame. An object’s reference frame is spanned by its proper space and proper time. Each object experiences its 4D motion as proper time τ, which is the length of a 4D Euclidean vector “flow of proper time” τ. Any acceleration rotates an object’s τ and curves its worldline in flat ES. τ proves crucial for objects that are very far away or entangled. Information hidden in θ and τ is not available in SR/GR. ER solves 15 fundamental mysteries, such as the nature of time, the Hubble tension, the wave–particle duality, and the baryon asymmetry. On top, ER declares cosmic inflation, expanding space, dark energy, and non-locality obsolete.