While Edmond Halley and Alexis Clairaut both made significant contributions to astronomy and physics using Newton's ideas, they did not make a single, joint discovery. They each used Newton's laws of motion and gravity to make independent, important contributions:

Edmond Halley:

* Predicted the return of Halley's Comet: Using Newton's laws of motion and gravity, Halley calculated the orbit of a comet observed in 1682. He predicted its return in 1758, which was confirmed after his death. This was a groundbreaking achievement, showing the power of Newton's laws to predict celestial phenomena.

Alexis Clairaut:

* Calculated the return of Halley's Comet: Clairaut also made a calculation of the return of Halley's comet. He considered the gravitational influence of Jupiter and Saturn, which was a significant advancement. His calculation, published in 1757, was closer to the actual return date than Halley's.

* Theory of the shape of the Earth: Clairaut applied Newton's gravitational theory to explain the Earth's shape, showing how it was flattened at the poles and bulging at the equator.

In Summary:

While Halley and Clairaut both used Newton's ideas to make important contributions to astronomy and physics, their work was mostly independent. Halley is best known for his prediction of Halley's Comet, while Clairaut is known for his refined calculations of the comet's return and his contributions to understanding the Earth's shape.