Properties of Measuring Points in perspective drawing In his Complete Guide to Perspective Drawing, Craig Attebery defines a central concept in two-point perspective: Measuring Points, which can be used to determine depth in lines going away from the viewer.
Attebery states, without proof, that the distance from Vanishing Points to Measuring Points should be the same as the distance from Vanishing Points to the Station Point (artist’s eye). That is, Measuring Points can be obtained using a compass:

A different theory, still, comes from Joseph D’Amelio’s Perspective Drawing Handbook. From the latter’s definition, Measuring Points would be Vanishing Points themselves. As each set of parallel lines has its Vanishing Point, this would imply each Vanishing Point to have not a single, but infinite Measuring Points:

Both theories, as interpreted above, cannot be right: Vanishing Points have either a single or infinite measuring points. The question is, thus:
How many Measuring Points does each Vanishing Point has? If only one, why should Attebery’s statement hold: that the distance from Vanishing Points to Measuring Points  be the same as the distance from Vanishing Points to the Station Point?
 A: I believe Glarbo is correct that scale is the reason. The books you mention are intended for architectural drawing, where it is important the scales be consistent in both directions. If you choose an arbitrary guideline vanishing point you will evenly subdivide a line, but you won’t be able to guarantee that you’re subdividing at the same scale in both directions of a square grid (which is the context in which I think you usually find the MP derivation). In other words you could correctly subdivide in both directions of a perspective right angle but the subdivisions would not correspond to an equal interval in both directions of a constructed plan.
You can show this by constructing the measuring points from the plan of known evenly subdivided four square as I did in the attached image. Set up basic right angle vanishing points and then reconstruct the guidelines in plan by swinging them down to the picture plane. Then project the plan view guidelines in the perspective and you will see that the guideline vanishing points correspond to the MP that would be produced by swinging an arc up to the picture plane from the line vanishing point as the books you mention show.(Basic Perspective Drawing by John Montague uses the same technique, also without explanation).
The method in the books is a shortcut that allows you to know you’re drawing a square in perspective without having to construct as I have done. diagram
