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How did observations of Uranus lead to the prediction of Neptune?

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A diagram with labeled vector illustrations. Panel A shows vector addition with two green arrows A and B forming a parallelogram, and their sum C as a red diagonal. Panel B shows vector subtraction, with A and negative B forming resultant D. Panel C demonstrates vector components using angles and projections. Panel D uses the right-hand rule with thumbs to show the direction of the cross product E = A x B and its opposite, -E = B x A. — mechanics Figure 1: (A) The vector sum C = A + B = B + A. (B) The vector difference A + (−B) = A − B = D. (C, left) A cos θ is the component of A along B and (right) B cos θ is the component of B along A. (D, left) The right-hand rule used to find the direction of E = A × B and (right) the right-hand rule used to find the direction of −E = B × A.

The prediction of Neptune stands as a landmark achievement in celestial mechanics, demonstrating the predictive power of Newton’s laws of motion and gravitational theory.12

Following the discovery of Uranus in 1781, astronomers observed that the planet’s actual path deviated from the orbit predicted by Newtonian calculations, which accounted for the gravitational influence of the Sun and other known planets. Furthermore, historical "prediscovery" sightings of Uranus dating back to 1690 did not align with its calculated positions.12

Scientists hypothesized that these anomalies were caused by the gravitational pull of an undiscovered, more distant planet. In the 1840s, mathematicians John Couch Adams and Urbain-Jean-Joseph Le Verrier independently calculated the potential mass and position of this unseen body. Based on Le Verrier’s calculations, astronomers Johann Gottfried Galle and Heinrich Louis d’Arrest identified Neptune at the Berlin Observatory on September 23, 1846, within one degree of the predicted location. This discovery provided a profound confirmation of Newtonian physics.123