Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth... No force of gravitation is... needed to impress curvature on world-lines; the curvature is inherent in the space...

I tell you that if natural bodies have it from Nature to be moved by any movement, this can only be circular motion, nor is it possible that Nature has given to any of its integral bodies a propensity to be moved by straight motion. I have many confirmations of this proposition, but for the present one alone suffices, which is this. I suppose the parts of the universe to be in the best arrangement, so that none is out of its place, which is to say that Nature and God have perfectly arranged their structure. This being so, it is impossible for those parts to have it from Nature to be moved in straight, or in other than circular motion, because what moves straight changes place, and if it changes place naturally, then it was at first in a place preternatural to it, which goes against the supposition. Therefore, if the parts of the world are well ordered, straight motion is superfluous and not natural, and they can only have it when some body is forcibly removed from its natural place, to which it would then return by a straight line, for thus it appears that a part of the earth does [move] when separated from its whole. I said "it appears to us," because I am not against thinking that not even for such an effect does Nature make use of straight line motion.

If the earth were not round, heavy bodies would not tend from every side in a straight line towards the center of the earth, but to different points from different sides.

Space tells matter how to move
Matter tells space how to curve

If … the motion of the earth were circular, it would be violent and contrary to nature, and could not be eternal, since … nothing violent is eternal.… It follows, therefore, that the earth is not moved with a circular motion.

Of the bodies in the cosmos, some imitate mind and move in orbits; some imitate soul and move in a straight line, fire and air upward, earth and water downward.

The Earth is an irregular oblate spheroid (a sphere flattened at the poles). Measurements of its dimensions and the amount of its flattening are subjects of geodesy. However, for most navigational purposes, assuming a spherical Earth introduces insignificant error. The Earth’s axis of rotation is the line connecting the north and south geographic poles.
A great circle is the line of intersection of a sphere and a plane through its center. This is the largest circle that can be drawn on a sphere. The shortest line on the surface of a sphere between two points on the surface is part of a great circle. On the spheroidal Earth the shortest line is called a geodesic. A great circle is a near enough approximation to a geodesic for most problems of navigation.

The suggestion that the body really wanted to go straight but some mysterious agent made it go crooked is picturesque but unscientific. It makes two properties out of one; and then we wonder why they are always proportional to one another - why the gravitational force on different bodies is proportional to their inertia or mass. The dissection becomes untenable when we admit that all frames of reference are on the same footing. The projectile which describes a parabola relative to an observer on the earth's surface describes a straight line relative to the man in the lift. Our teacher will not easily persuade the man in the lift who sees the apple remaining where he released it, that the apple really would of its own initiative rush upwards were it not that an invisible tug exactly counteracts this tendency. (The reader will verify that this is the doctrine the teacher would have to inculcate if he went as a missionary to the men in the lift.)

[T]he property of being a geodesic is invariant under coordinate transformations... so the path of the photon or particle will... be a geodesic in any spatial coordinate system...

I hold that space cannot be curved, for the simple reason that it can have no properties. It might as well be said that God has properties. He has not, but only attributes and these are of our own making. Of properties we can only speak when dealing with matter filling the space. To say that in the presence of large bodies space becomes curved is equivalent to stating that something can act upon nothing. I, for one, refuse to subscribe to such a view.

First, the line of progress is never straight. For a period a movement may follow a straight line and then it encounters obstacles and the path bends. It is like curving around a mountain when you are approaching a city. Often if feels as though you were moving backwards, and you lose sight of your goal: but in fact you are moving ahead, and soon you will see the city again, closer by.

I ask at what part of its curved motion the moving cause will leave the thing moved and moveable.

...wheresoever the earth may be placed, or whithersoever it may be carried by its animal faculty, heavy bodies will always be carried towards it.

I was almost driven to madness in considering and calculating this matter. I could not find out why the planet would rather go on an elliptical orbit. Oh, ridiculous me! As the liberation in the diameter could not also be the way to the ellipse. So this notion brought me up short, that the ellipse exists because of the liberation. With reasoning derived from physical principles, agreeing with experience, there is no figure left for the orbit of the planet but a perfect ellipse.