ON SPHERICS 
BY M. B. PORTER, SUGAR LAND, TEXAS. 
[Read February 6, 1892.] 
“ Upon this supposition of a positive curvature, the whole of geom¬ 
etry is far more complete and interesting; the principle of duality, 
instead of half breaking down over metric relations, applies to all 
propositions without exception. In fact, I do not mind confessing that 
I personally have often found relief from the dreary infinities of hom- 
aloidal space in the consoling hope that, after all, this other may be the 
true state of things.”—W. K. Clifford, Lecture on Philosophy of the 
Pure Sciences.” 
INTRODUCTORY NOTE. 
The following is the substance of a paper read before the Texas Acad¬ 
emy of Science, Feb. 6, 1892, and is an attempt to outline a synthetic 
treatment of the conic sections, by regarding them as degraded forms 
of the spherical ellipse. The advantages accruing from such a method, 
are, it is believed, greater unity in the conception of the properties and 
generation of these curves, by considering them as different aspects of 
one and the same curve; secondly, as affording a far more definite and 
satisfactory view of the line at infinity, so all important in the theory 
of planimetry, the theory of parallels, and the treatment of plane curves. 
The principle of duality, which, to one unacquainted with Plucker’s 
point and line co-ordination, might seem, in piano, wanting in that 
axiomatic certitude indispensable in a proof, occasions no difficulty in 
spherics, where the quadrantal correspondence of line and point has 
been made use of since the foundation of this branch of geometry. Quad¬ 
rantal reciprocity, though a special form of the principle of duality, is 
of incalculable value in studying the metrics and graphics of the sphere, 
and, in showing that it may be considered, either as a point-aggre¬ 
gate or a line-aggregate, facilitates the realization of polar-reciprocity 
in its most general form. The principle of limits is systematically 
employed, and is to be regarded as an assumption by means of which 
it is possible to define the expressions, length of a curve, area of a warp, 
direction of a curve at a point, etc. No attempt is made to establish 
the graphical properties independently of metrical considerations; but 
rather to connect these as closely as possible. Hence the theory of 
projective rows and pencils is worked out from the more elementary 
properties of the conic. Van Staudt’s symbol f\ is used to indicate 
projective rows and pencils. 
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