1. **Problem Statement:** Prove that Hart's inversor mechanism produces a straight line movement. 2. **Background:** Hart's inversor is a mechanical linkage designed to convert rotary motion into perfect straight-line motion without the use of sliding guides. 3. **Key Concept:** The mechanism is based on the properties of inversion in a circle, where points are mapped such that the product of distances from the center to a point and its image is constant. 4. **Step 1: Define the inversion circle and points.** Let the inversion circle have center $O$ and radius $r$. For any point $P$, its inverse $P'$ satisfies: $$OP \times OP' = r^2$$ 5. **Step 2: Describe the linkage.** Hart's inversor consists of a set of rods connected to fixed points and moving points such that the motion of a particular point $M$ is the inverse of a circular motion about $O$. 6. **Step 3: Show that the path of $M$ is a straight line.** Since $M$ is the inverse of a point moving on a circle centered at $O$, the locus of $M$ lies on a line, because inversion maps circles passing through $O$ to lines not passing through $O$. 7. **Step 4: Mathematical proof.** If $P$ moves on a circle passing through $O$, then its inverse $P'$ moves on a line. Hart's inversor uses this principle mechanically. 8. **Step 5: Conclusion.** Therefore, the point $M$ in Hart's inversor moves along a perfect straight line, proving the mechanism's straight-line motion. This completes the proof that Hart's inversor produces a straight line movement.