972 Transactions.—Miscellaneous. 
As to elliptical skew arches, the principles involved in designing these 
are the very same as for segmental arches ; therefore, in applying these, the 
ellipse may be divided into two or more portions, which may be taken as 
continuous segments of circles, or an approximate segment of a circle in 
position and length may be assumed as the actual one, and developed 
accordingly. 
And as to skew arches on curves, these may be designed as if the faces 
were straight, in the same manner as given in this paper, and the correction 
in the lengths of courses applied afterwards to the development of the 
intrados, lengthening those on the convex side and shortening those on the 
concave side. But it is evident that the principle can only be applied safely 
within certain limits, and in so doing, it must always be borne in mind that 
the properties of the lines and sections of a cylinder lie at the basis of the 
design of any skew arch. 
Arr. XXIV.—On the Simplest Continuous Manifoldness of two Dimensions and 
of Finite Extent. By F. W. FRANKLAND. 
[Read before the Wellington Philosophical Society, 11th November, 1876.) 
Amone the most remarkable speculations of the present century is the 
speculation that the axioms of geometry may be only approximately true, 
and that the actual properties of space may be somewhat different from 
those which we are in the habit of ascribing to it. It was Lobatchewsky who 
first worked out the conception of a space in which some of the ordinary 
laws of geometry should no longer hold good. Among the axioms which 
lie at the foundation of the Euclidian scheme he assumed all to be true 
except the one which relates to parallel straight lines. An equivalent form 
of this axiom, and the one now generally employed in works on geometry? 
is the statement that it is impossible to draw more than one straight line 
parallel to a given straight line through a given point outside it. In other 
words, if we take a fixed straight line 4 B, prolonged infinitely in both 
directions, and a fixed point P outside it ; then, if a 
second straight line, also infinitely prolonged in both 
directions, be made to rotate about P, there is only 
one position in which it will notintersect 4 B. Now 
Fig.1 Lobatchewsky made the supposition that this axiom 
should be airia, and that there should be a finite angle through which the 
rotating line might be turned, without ever intersecting the fixed straight 
line A B. And in following out the consequences of this assumption, he 
was never brought into collision with any of the other axioms, but was able 
