In 1894 Sondat published a theorem that the centre of perspectivity and the 2 orthologic centres of any 2 bilogic (perspective as well as orthologic) triangles lie on a line perpendicular to their axis of perspectivity. Thébault (1952) gave an elementary proof of this theorem. Here we give two new proofs, one synthetic and the other analytic, and then deduce the existence of an orthologic Desargues' figure where all the 10 pairs of perspective triangles in it are orthologic. Consequently we arrive at an orthologic Veronese configuration of 15 points and 10 pairs of perpendicular lines studied in 5 different ways.

Let (A), (B) be a special pair of perspective tetrahedra such that the 4 joins of their corresponding vertices are perpendicular to their plane of perspectivity. It has been already established (Mandan (1977a), p. 573) that they are, in general, skew-orthologic such that the perpendiculars from the vertices of one to the corresponding opposite faces of the other lie in a regulus. Here we give a construction of a special pair of semi-orthologic and orthologic tetrahedra, the said regulus degenerating into 2 pairs of intersecting lines and 4 concurrent lines respectively. Following Thébault ((1952), p. 25; (1955), p. 67), we call them semi-bilogic and bilogic respectively.

In 1959, Professor N. A. Court [2] generated synthetically a twisted cubic C circumscribing a tetrahedron T as the poles for T of the planes of a coaxal family whose axis is called the Lemoine axis of C for T. Here is an analytic attempt to relate a normal rational curve rn of order n, whose natural home is an n-space [n], with its Lemoine [n—2] L such that the first polars of points in L for a simplex S inscribed to rn pass through rn anf the last polars of points on rn for S pass through L. Incidently we come across a pair of mutually inscribed or Moebius simplexes but as a privilege of odd spaces only. In contrast, what happens in even spaces also presents a case, not less interesting, as considered here.

The main purpose of this paper is to prove the proposition: “A set of r mutually perspective (m.p.) (s—1)-simplexes have the same [s−2] (say x) of perspectivity, if and only if their centres of perspectivity (c.p.) lie in an [r−2] (say y); there then arises another such set of s m.p. (r—1)-simplexes, having the same rs vertices, which have y as their common [r−2] of perspectivity such that their c.p. lie in x.” The proposition is true in any [k] for k = s−1, s,…, r+s−2 (r ≦ s). The configuration of the proposition in [r+s−2] arises from the incidences of any r+s arbitrary primes therein and is therefore invariant under the symmetric group of permutations of r+s objects, and that in [r+s−3] is self-dual and therefore self- polar for a quadric therein. Some special cases of some interest for r = s are deduced. The treatment is an illustration of the elegance of the Möbius Barycentric Calculus ([15], pp. 136–143; [1], p. 71).

Let S = A0 hellip An be an n-simplex and Aih the foot of its altitude from its vertex Ai to its opposite prime face Si; O, G the circumcentre and centroid of S and Oi, Gi of Si. Representing the position vector of a point P, referred to O, by p, Coxeter [2] defines the Monge pointM of S Collinear with O and G by the relation so that the Monge point Mi of Si is given by If the n+1 vectors a are related by oi be given by Aih is given by Since Aih lies in Si, If Ti be a point on MiAih such that i.e. That is, MTi is parallel to ooi or normal to Si at Ti:. Or, the normals to the prime faces Si of S at their points Ti concur at M. In fact, this property of M has been used to prove by induction [3] that an S-point S of S lies at M. Thus M = 5, M = S or .

The interesting results arising from the study of ‘Four intersecting spheres’ [9] in a solid made the author think of an analogous picture in higher spaces too and thus the present paper arose.

The purpose of this paper is to prove that the altitudes of an n-simplex (a simplexin an n-space) S form an associated set of n+1 lines (see Baker, [4] for n = 4) such that any (n–2)-space meeting n of them meets the (n+1)th too. As an immediate consequence 2 quadrics are associated with S, one touching its primes at the respective feet of its altitudes and the other touching n(n+1) primes, n parallel to each of its altitudes and 2 through each of its (n−2)-spaces. Certain special cases are also mentioned.

An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

A tetrad of Möbius tetrahedra consists of a set of 4 mutually inscribed and therefore circumscribed tetrahedra whose 16 vertices and 16 faces form a Kummer's 166 configuration (5; 11; 12; 21). As pointed out by the refere, fundamental to all work on the 166 figure are the 10 quadrics, called fundamental for the associated Kummer's quartic surface (13). To every quadric F correspond a matrix scheme of the 16 points or planes, arranged in 4 rows or columns, such that the 8 Rosenhain tetrahedra (7) formed of the rows and columns are all self-polar for F. The rows form one and the columns another tetrad of Mobius tetrahedra. Nine new schemes can be derived from one such scheme to make the total ten as explained by Baker (3, p. 133) leading to 80 Rosenhain tetrahedra in all. The 16 nodes (5; 8) or tropes of the Rummer's quartic are the 16 common elements of the 10 schemes such that the nodes and tropes are poles and polars for any one of the 10 quadrics. Each trope touches the quartic along a singular conic through the 6 points of the figure lying therein. The lines tangent to the surface at its each node N generate a quadric cone which is enveloped by the 6 tropes through N (12).

In continuation of the two previous papers (10; 11), this paper was originally written at the Indian Institute of Technology, Kharagpur and revised at the University of Sydney under the advice of Prof. T. G. Room. Although the altitudes of a general simplex S(A) in n-space (n > 2) do not concur as they do for a triangle (n = 2), yet we observe that its Monge point, M (1; 5), is an appropriate analogue of the orthocentre of a triangle such that M coincides with its orthocentre when it is orthogonal (or orthocentric). In consistency with the previous papers (10; 11; 13; 15) we shall call M as the S-point of S(A) and denote it as S as explained in § 1.2. The altitudes of S(A) are all met by the (n − 2)-spaces normal to its plane faces at their orthocentres, each parallel to of them, thus indicating the associated character of the altitudes as discussed separately in 2 other papers (12; 16). Before we introduce an orthogonal simplex and develop its properties in regard to its γ-altitudes and associated hyperspheres, we come across a number of intermediate ones of special interest. Two special types are treated here and the other two are developed in 2 other papers (13; 15).

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.

“The remarkable analogies which exist between the complete quadrilateral and the desmic system of points suggest that it may be possible to extend the properties considered above to spaces of higher dimensions”, remarks Prof. N. A. Court at the end of his paper (2). Here is an attempt in that direction in Euclidean as well as in elliptic 4-space, suggesting extensions in higher spaces. The corresponding figure, called an S-configuration, is discussed.