Understanding field representation of force I am reading the book The Evolution of Physics. I have a doubt in the topic "The field as representation". In this topic authors give the example of gravitational force represented as a field. In the following image the small circle represents an attracting body(say sun) and the lines are the well known lines of force of the gravitational field.

It is said that the density of the lines of force in space shows how the force varies with the distance. Let us consider a finite volume $\Delta V$  in the vicinity of sun. Now the number of lines of force passing through this is finite but there are infinite points in this $\Delta V$ volume.
$1.$Is there any gravitational force acting on those points through which no line of force passes.
$2.$ If the gravitational force acts on all the points contained in $\Delta V$ shouldn't there be infinite lines of forces passing through $\Delta V$.  
$3.$ If it is supposed that there are really infinite lines of force passing thru $\Delta V$ then how to decide the density of no of lines won't it be infinite.  

Please cite some canonical references which explains the 3 different points I've mentioned in your answer.  
Thank you.
 A: There are field lines going through every point.  Obviously, it's only possible to draw finitely many of them.  The picture contains an implied "and so forth" --- that is, you are supposed to assume that the infinitely many un-drawn lines look qualitatively like the drawn ones.
Analogy:  Think about a topographic map, which shows lines (or curves) of elevation.  There's a curve labeled "elevation 1000 feet", one labeled "elevation 2000 feet", one labeled "elevation 3000 feet", etc.  Many points are on no curve at all.  You're not meant to conclude that those points have no elevation; you're meant to conclude that they have elevations which can be (more or less) inferred from the curves that are drawn.
A: No field representation of force, the representation you are seeing I.e. the lines of force representation indicates relative field/force strength with respect to distance, as you can see at different distances from the source the field density(no of lines/ area) is different. 
The lines of force representation does not represent where the force reaches or not, it simply designates relative strength and the direction of force on a conventionaly defined unit particle such as point charge in electrostatics used to determine field is conventionally taken to be positive and field lines are drawn in accordance to the direcction of force it feels.
For further reading you can refer to FIELD LINES, it has everything from the definitions to their physicsl significance, and was just a google search of "field line" away !
