We have seen the connection between the general theory of verniers and the indeterminate equation of the form pn − qm = r; the equation pn + qm = r might be illustrated in much the same way. But, taking the case originally considered, we observe that it is the same question as before, from a different point of view, if instead of starting with an interval AB between divisions of two contiguous scales, and asking how far we must go to reach a coincidence, we suppose the zeros of two scales to coincide and ask how far we have to go to reach a certain interval, between divisions of the two. The question only differs in so far as it suggests the simultaneous consideration of more than one such interval. But further if, instead of repeated divisions of each kind, we take a circle whose entire circumference = one of the larger divisions, a say, and from a fixed point A on the circumference measure off continually arcs = to the smaller division 6, (figs. 5, 6), the question how far we have to go to reach a certain approximation to the starting point is the same thing in another form, although the arrangement brings to light many new relations, which in their turn throw light on the result. We have the subdivisions of the different a-intervals superposed as it were upon one a-interval, and the circle is thus continually divided into more and more parts until (if a and b are commensurable) exact coincidence with the starting point is reached.

An inhabitant of Flatland would not be in the same position with regard to elementary geometry as a three-dimensioned being. He could only use superposition for figures of the same aspect. I have not met with any discussion of the way in which he would prove, say, Euc. I. 5; the following seems complicated, but at least it shews that the scope of the usual assumption as to superposition can be made narrower.