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Newton conceived space as being:

Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space

and motion as

Absolute motion is the translation of a body from one absolute place into another: and relative motion, the translation from one relative place into another

But space has no point of origin, each point is alike as another. So no motion can be absolute, every motion must be relative.

Introduce a particle into this space. Then this particle cannot be at rest. For what would it be at rest with? Here I do not mean with another particle, but with regards to space itself.

From this is it possible to deduce that a particle moves in a straight line linearly?

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    Empty space has no points of reference. If we introduce a particle, the concepts of rest and motion are undefined because, as you point out, there's nothing for it to rest or move relative to. So I would say there's no way to determine whether the singleton particle is moving or at rest, either linearly or otherwise. – Tom Barron Apr 16 '14 at 0:43
  • Only when we have at least two particles can we talk about motion or rest. And if the two particles are at rest relative to each other, is the system they comprise moving or at rest? We can't say because there's no point of reference outside the system. – Tom Barron Apr 16 '14 at 0:44
  • We could build up our universe like this, one particle at a time, and, if space is infinite, we can't say whether the universe is moving or not relative to anything outside of it since we haven't allowed for any external points of reference. – Tom Barron Apr 16 '14 at 0:44
  • Even if we don't have particles we still have position; we also know that these positions are connected by continuity; the reason I'm saying that rest is undefined is not that there are no particles - but that there is no point of origin. – Mozibur Ullah Apr 16 '14 at 2:02
  • This is ultimately why the term "rest mass" is being ousted from physics. Straight lines are similarly problematic until one has two particles - as long as something they break the appropriate symmetry. – Lucas Apr 16 '14 at 20:03
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I think you have the key to the problem when you notice that "every motion must be relative."

Newton, in speaking of a particle "at rest," he is referring to the particle in a particular frame of reference. In saying "A particle at rest tends to stay at rest," you could interpret that as saying that a particle that is not moving with respect to some reference objects tends to stay not moving with respect to those same objects.

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  • This is the classical argument; and upto a point I agree - except that there are many equivalent frames; any frame that is moving at constant speed are equivalent. It would be nice rather than having a whole host of frames to have a unique one. I suppose in this framework, I would say although a particle is at rest with one frame, in an equivalent frame, it won't be. Thus we can say in a sense that rest is only a relative term but not an absolute sense (ie doing without coordinates); – Mozibur Ullah May 15 '14 at 13:56
  • I guess one reason as to why this question is interesting is that some notions are always absolute - for example the length of a rod can be calculated without any frames entering the calculation. – Mozibur Ullah May 15 '14 at 13:59
  • I think, in practice, there is only ever one frame of reference that is useful for a given problem. You start with the particle(s) in question, you add in any other things that could apply a force to the particle, and that gives you a sort of canonically unique frame of reference to work in. Any other frame of reference contains extraneous or insufficient forces/particles. – James Kingsbery May 15 '14 at 19:41
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As James Kingsley noted in his answer Rest is a relative notion and not an absolute one. That is can only be defined by reference to an inertial frame, a frame that is not accelerating.

Absolutely, the closest notion to rest is of non-acceleration. That is an object that is not accelerating will be not accelerating in any other frame; and corresponding an object that is accelerating will be accelerating in any other frame.

Physically, it is the distincion between acceleration and non-acceleration that is relevant. This is noted in the definition of (relative) rest, because the very definition of an (inertial) frame is one which is not accelerating.

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Newton was a physicist in need of justifying his completely new science in terms of plausible philosophical garb. You need to read further down the passages Newton wrote to understand what his actual purpose in writing this stuff is. He needs to convince his readers later on that earth is not immovable, and rotates on its axis relative to the stars, or "absolute space". Because already since ancient times people had been arguing whether it is the fixed stars which rotate around the earth or is it the earth rotating. Ptolemy, who otherwise was a very good scientist, came down on the opposite side of the argument. Newton is aiming to demolish him by making allusions to absolute space as opposed to "vulgar perceptions of motion" as he puts it.

Keep in mind however, that there is no philosophical argument which can actually prove the main postulate which Newton needs to convince his reader of, which is that bodies without external force will move in straight lines, uniformly. Now, regarding rotation. He proposes the spinning spheres experiment which ought to prove or disprove the rotation of the earth experimentally.

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If you read the quote in context, it becomes clearer why Newton took the position that absolute space exists. I will explain briefly here but for more detail see "The Discovery of Dynamics" by Julian Barbour. The problem has to do precisely with the law that objects continue to move in a straight line with constant velocity (inertial motion) unless a force acts on them. The question that arises is with respect to what frame will the object move inertially? If you drive down the road and turn your car, then another car moving in a straight line appears to you not to be moving in a straight line. So why should we say that you are accelerating and not the other car?

Descartes proposed a solution to this problem: objects do not experience a force unless they are acted on by adjacent objects but Newton pointed out that this is false with the bucket experiment. Suppose that you take a bucket full of water and tie a rope to the handle of the bucket. Then you twist the rope up a lot and release the bucket. The bucket spins and at this stage of the experiment the surface of the water is flat although the bucket is moving. Later, when the bucket has stopped moving the water will be spinning. So in both cases the motion of the water is not a straight line with respect to the surrounding matter.

Newton concluded that what determines whether an object is moving inertially is its motion with respect to absolute space. He was wrong about that but his argument was not stupid. The issue was not about what he could see but with what explanations actually accounted for what he could see. The problem was not properly solved until Einstein came up with the general theory of relativity, which explained that acceleration is equivalent to not moving in free fall: it is caused by interactions with the gravitational field.

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  • I'm not asking about absolute space - thats a different question. Newtons inertial frame is defined as that in which his first law holds - ie non-accelerating frames. They're still the frames that are used in GR - so his argument was neither wrong nor stupid. – Mozibur Ullah May 15 '14 at 17:11

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