Leonhard Euler published this method in 1768: to solve an equation you cannot solve, just follow the slope it hands you, one small step at a time.
A slope field lets you see solutions you could never write down — every hairline is the local tangent, and a solution is any curve that rides the current.
Each Euler step trusts a single tangent line, so the error piles up at O(h): halve the step, roughly halve the drift.
On a closed orbit plain Euler quietly adds energy every loop, so a ring that should close spirals outward instead. That is numerical instability, not the real solution.
The dashed gold path is RK4 — four slope samples per step, drift so small it is your ruler for the truth. That accuracy is why RK4, not Euler, is the real workhorse.