EEV. T. P. KIEKMAN ON ATJTOPOLAE POLYEDEA. 
187 
ing B. But this section is the triangle efm, whose sides are em in C and fm in D ; there- 
fore m, any point of gh, is a point of Jcl ; which is absurd, unless U is gh. And if U is 
gh, m may be a point indefinitely near to and the section efm will be the face A, which 
is impossible, because A is no triangle. Wherefore the absurdity remains, and Jd has no 
existence. Therefore P has the evanescible edge AB. 
And if, secondly, e be a triace, AC and BC are edges of C, e'p and e<]^\ and C having 
three summits in A and B, and one or two in gh^ is no triangle ; therefore ej)' is an 
edge of P not in a triangle, which is convanescible, unless e and <p' are in two covertical 
faces besides C and B. Now if they are, A is one of those faces, e being the triace ACB. 
Let G be a face through <p' meeting A on P. The triangular section efm being made as 
above, cuts no face besides C and D ; and divides P into P' containing A, and P" con- 
taining B with its edge ef. Therefore G, being uncut and covertical with A, is entirely 
in P' ; and being uncut and having the summit is entirely in P". Which is absurd ; 
wherefore G has no existence, and P has the convanescible edge ef. 
Thus it is proved that in any case P, being not a pyramid, and having an edge AB, 
not in a triangle, has either a convanescible or an evanescible edge. 
Next, let P have an edge cd, not in a triace. Then will P^ the sympolar of P, have 
an edge CD, the gamic of cd, not in a triangle, and also either a convanescible or an 
evanescible edge g ; wherefore P has the gamic of g, which is either evanescible or con- 
vanescible. 
Therefore in all cases P, not being a pyramid, has a vanescible edge. Q. E. D. 
V. Theorem. Any ^-edral (^-acron P, not a pyramid, can he reduced hy the vanishing of 
an edge to eithar a {-^—lyedral q-acron or a ^-edral {q—l)-acron. 
For P must have either a convanescible edge AB, or an evanescible edge ah. Let it 
have the former (AB). 
If AB convanesces by the union of its summits, P becomes P', losing the faces A 
and B, and receiving two others. A' and B', in their place. A' being A with one edge less, 
and B' being B with one edge less. At the same time the two summits dd! of (AB) 
disappear from P, which in their stead and at theh union, receives a new summit h 
having two edges less than the sum of thehs. All the faces of P containing d or ^ 
remain unchanged, except that each now contains the summit h, whose edges are 
those of d and d! together save two ; and no two of these faces thus brought to have a 
common summit h can coincide, because, by the definition of the convanescible e, they 
had not in P a common edge. Hence all the other edges of P remain undisturbed in the 
result P'. Or if ah be an evanescible edge of P, and evanesces by the revolution of DD' 
its two faces into one plane, giving rise to a new face H, this will have the edges of D 
and D' together save two ; the summit a becomes d wdth one edge less, and h becomes h' 
with one edge less. All the summits of P containing D and D' remain unchanged, 
except that each now contains the face H ; and no two of these summits, thus brought 
into a common face H, coincide, because by the definition of {ah) they were not before 
in any line dravm or drawable on P. Hence all the other edges of P remain unaltered. 
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