What is the Kepler problem, position as a function of time, or radius as a function of angle?

In theory of motion the first two quantities that are defined are velocity and acceleration.

Velocity: time derivative of position.
Acceleration: time derivative of velocity.

The reason for using time derivatives is obvious: while we do have control over position of objects in space, we don't have any control over time. Time simply proceeds.

In the case of the Kepler problem:

It isn't enough to have only a formula for the shape of the orbit.

I'm not sure how Kepler arrived at the shape of the orbit of Mars.

Among the methods he used was the following: the true period of the orbit of Mars was well known, since that can be inferred from records that go back many years. When Mars has completed a full revolution it must be back at the same location as before. After a full Mars revolution the Earth is in a completely different location than a Mars year earlier, that provides the means for a parallax calculation.

It is my understanding that with methods like that Kepler was able to reconstruct the shape of the orbit of Mars, allowing him to come to the conclusion that the orbit is an ellipse, with the Sun at one focus.

However, at that stage Kepler had only a rough idea of the velocity of Mars during various stages of its orbit. Kepler could infer that during closest approach to the Sun Mars moved quicker, and that during furthest distance from the Sun Mars moved slower. But at that stage Kepler did not have a formula that would allow him to calculate the position of Mars in advance.

As we know, it was when Kepler formulated the law of areas that he was able to describe the motion of Mars with a better accuracy than had been achieved before.

The law of areas establishes a time relationship.

There is a resemblance between Newton's first law and Kepler's law of areas:

Newton's first law:
An object in free motion will cover equal amounts of distance in equal intervals of time.

Kepler's law of areas: An object in orbital motion will sweep out equal areas in equal intervals of time.

Kepler's second law doesn't narrow down what the shape of the orbit will be. The first law, on the other hand, gives the shape but not a time relationship. In Kepler's time it was the combination of the two that solved the Kepler problem.

As we know, Newton subsequently moved the understanding of celestial motion to a whole new level. For instance, Newton showed that Kepler's law of areas (for planetary motion) is a specific instance of a more general principle. As we know, this more general principle is today known as conservation of angular momentum.