Warrior Ancient and Medieval Rules

[Craig Scott]

Ewan; I am shocked, shocked to find a "flat earthier" cranial
activity in that brilliant mind of yours (o;

Be well and warmest regards,

Craig

--- In WarriorRules@yahoogroups.com, Ewan McNay <ewan.mcnay@y...>
wrote:
>
> Bzzt. Thankyou for playing, Mr. Downs. However - as the 'at 90
degrees'
> implies - the conditions being described would rather be
irrelevant to one
> another, not opposing.
>
> Or, as the OED would have it: "Of a set of variates: statistically
> independent. Of an experimental design: such that the variates
under
> investigation can be treated as statistically independent.*"
>
> Who said wargaming was not educational?
>
> darnd022263@a... wrote:
>
> > In a message dated 11/10/2005 9:26:09 AM Eastern Standard Time,
> > Turnerm@l... writes:
> > orthogonal.
> >
> > orthogonal
> >
> > At right angles. The term is used to describe electronic signals
that appear
> > at 90 degree angles to each other. It is also widely used to
describe
> > conditions that are contradictory, or opposite, rather than in
parallel or in sync
> > with each other.
>
> * I. General uses.
>
> 1. a. Designating a right angle; (also) having a right angle;
> right-angled; rectangular. Now rare.
>
> 1571 L. DIGGES Geom. Pract. I. Defs. sig. Bjv, Of straight lined
angles
> there are three kindes, the Orthogonall, the Obtuse and the Acute
Angle.
> 1571 L. DIGGES Geom. Pract. I. Defs. sig. Bjv, Eche of those
Angles is an
> Orthogonall or right Angle. 1611 R. COTGRAVE Dict. French & Eng.
Tongues,
> Orthogonal, orthogonall, right cornered. 1612 J. SELDEN in M.
Drayton
> Poly-olbion I. To Rdr. sig. A3, Pythagoras his sacrifice after his
> Geometricall Theorem in finding the Squares of an Orthogonall
Triangles
> sides. 1871 L. COLANGE Zell's Pop. Encycl. II. 531/2
> Orthogonal..Right-angled; rectangular. 1979 W. STYRON Sophie's
Choice
> (2000) 327 The old man..seemed thoroughly exhilaratedthe clean
bright
> orthogonal Mondrians bringing special delight to his technician's
eye.
>
> b. Relating to or involving right angles; at right angles (to
> something else).
>
> 1694 P. A. MOTTEUX Wks. F. Rabelais (1737) v. 235 An Orthogonal
Line. 1878
> H. P. GURNEY Crystallogr. 37 If two symmetral planes intersect at
right
> angles the line in which they cut is called an axis of orthogonal
> symmetry. 1892 A. E. H. LOVE Treat. Math. Theory Elasticity I. 200
Systems
> of surfaces which cut each other everywhere at right angles. Such
systems
> are called orthogonal surfaces. 1953 S. W. AMOS & D. C. BIRKINSHAW
Telev.
> Engin. I. iv. 107 It is important, therefore, that the scanning
beam
> should approach the target strictly at right angles at all points;
in
> other words orthogonal scanning is necessary. 1978 P. W. ATKINS
Physical
> Chem. xv. 476 Two orthogonal (perpendicular) bonds are formed in
this way.
> 1989 B. ALBERTS et al. Molecular Biol. Cell (ed. 2) iv. 171 The
narrow gel
> containing the separated proteins is again subjected to
electrophoresis
> but in a direction orthogonal to that used in the first step.
>
> 2. Math. a. Of a linear transformation of a vector space:
> preserving lengths and angles; leaving unchanged the inner product
of any
> two vectors.
>
> 1852 J. J. SYLVESTER in Cambr. & Dublin Math. Jrnl. 7 57 Then x2 +
y2 + z2
> remains invariable, which is the well-known characteristic of
orthogonal
> transformation. 1893 L. G. WELD Short Course Theory Determinants
ix. 227
> The transformation, in analytical geometry, from one set of axes
to
> another, without changing the origin, is orthogonal. 1941 G.
BIRKHOFF & S.
> MACLANE Surv. Mod. Algebra ix. 222 A linear transformation T is
orthogonal
> if it preserves the absolute value of every vector , so that |T| =
||.
> 1972 F. E. HOHN Introd. Linear Algebra viii. 237 An orthogonal
> transformation of n maps orthogonal vectors onto orthogonal
vectors and
> nonorthogonal vectors onto nonorthogonal vectors. 1998 R. H. HECK
in G. A.
> Marcoulides Mod. Methods Business Res. vii. 195 The goal of
orthogonal
> rotation is to maintain the 90° of separation between the clusters
of
> variables but to get each set of variables clustered on an axis.
>
> b. Of a square matrix: representing an orthogonal
transformation;
> such that the rows (and likewise the columns) are orthonormal when
> considered as vectors; equal to the inverse of its transpose.
> These three properties are equivalent.
>
> 1891 Proc. London Math. Soc. 22 449 (title) On certain properties
of
> symmetric, skew symmetric, and orthogonal matrices. 1907 M. BÔCHER
Introd.
> Higher Algebra xi. 154 An orthogonal transformation. [Note] The
matrix of
> such a transformation is called an orthogonal matrix. 1964 N. N.
HANCOCK
> Matrix Anal. Electr. Machinery ii. 18 The value of the determinant
of an
> orthogonal matrix is necessarily ± 1, but the converse is not
true. 1990
> IMA Jrnl. Numerical Anal. 10 551 k is an m × m orthogonal matrix
> corresponding to a suitable Householder transformation (or a
product of
> Givens transformations).
>
> c. Designating the group of all orthogonal matrices of a
given order.
>
> 1898 Bull. Amer. Math. Soc. 4 196 A linear substitution S on the
marks of
> a Galois Field of order pn..will be called orthogonal if it leaves
> absolutely invariant 12 + 22 +...+ m2... The order of the
orthogonal group
> G on m indices in the GF[2n] is thus [etc.]. 1941 G. BIRKHOFF & S.
MACLANE
> Surv. Mod. Algebra ix. 225 This subgroup of the full linear
group..is
> called the orthogonal group On; it is isomorphic to the group of
all
> orthogonal transformations of the given Euclidean space. 1972 F.
E. HOHN
> Introd. Linear Algebra viii. 252 Show that the set of all linear
operators
> on n of the form Y = UX, where U is orthogonal, constitute a group
(the
> orthogonal group). 1992 G. ELLIS Rings & Fields viii. 138 The
orthogonal
> group..consisting of all n × n complex matrices which are
orthogonal.
>
> d. Of two vectors or functions: perpendicular; having an
inner
> product equal to zero. Of a set of vectors or functions: such that
the
> inner product of any two elements is zero if and only if the two
are
> distinct. Also: involving such factors or functions.
>
> 1913 Proc. London Math. Soc. 12 297 The theory of Fourier series
and of
> other series of orthogonal functions. 1926 E. W. HOBSON Theory of
> Functions of Real Variable (ed. 2) II. x. 754 If {n(x)} be a
complete
> sequence of linearly independent functions for the interval (a,
b), a
> normal orthogonal and complete system of functions {n(x)} can be
so
> determined that n(x) is a linear function of 1(x), 2(x),...n(x).
1941 R.
> V. CHURCHILL Fourier Series iii. 45 The functions einx = cos nx +
i sin nx
> (n = 0, ±1, ±2,...) form a system which is orthogonal on the
interval (-,
> ). 1967 A. A. GOLDSTEIN Constructive Real Anal. iii. 112 We define
an
> inner product space I[a, b] by introducing an inner
product..defined by
> [f, g] = ba f(t) g(t)dt. Two functions f and g in I[a, b] are said
to be
> orthogonal if [f, g] = 0. 1967 A. A. GOLDSTEIN Constructive Real
Anal.
> iii. 115 Two points x and y of [a Hilbert space] H are orthogonal
if [x,
> y] = 0. Similarly, two subspaces M and N of H are said to be
orthogonal if
> [M, N] = 0. 1968 C. G. KUPER Introd. Theory Superconductivity i. 3
> Bardeen, Cooper and Schrieffer (1957) constructed a variational
wave
> function for a ground state with complete electron pairing, and
orthogonal
> functions for low-lying excited states having only a few such
pairs
> broken. 1990 T. PETRIE & J. RANDALL Connections, Definite Forms, &
> Four-manifolds iii. 35 Thus * preserves harmonic forms and so
induces an
> orthogonal splitting H2 = H+ H-.
>
> 3. Statistics. Of a set of variates: statistically
independent. Of an
> experimental design: such that the variates under investigation
can be
> treated as statistically independent. Freq. with to.
>
> 1933 Jrnl. Agric. Sci. 23 110 In an ordinary replicated field
experiment
> of the randomised block or Latin square type the differences of
the means
> of plots receiving the same treatments are taken without
hesitation to be
> true measures of treatment differences, but this is only so
because the
> experiment has been specially arranged so as to be orthogonal.
1950 M. H.
> QUENOUILLE Introd. Statistics iv. 59 The only manner in which we
can
> assume that sex does not enter into the comparison is to choose
the same
> proportion of each sex in each age group. The effect of sex is
then said
> to be `orthogonal' to the effect of age. 1973 Jrnl. Genetic
Psychol. 122
> 45 Implicit in the work..is the concept that creativity and
intelligence
> are relatively orthogonal (i.e., unrelated statistically) at high
levels
> of intelligence. 1982 Lang. & Communication 2 261 Orthogonal
comparisons
> between levels of attribute type were made using Wilcoxon's tests.
1991
> Forestry 64 364 Each additional new variable accounts for the
largest
> amount of remaining variation such that it is independent of
(orthogonal
> to) the previously derived principal components.
>
> II. Special uses.
>
> 4. orthogonal projection, a method of projection employing
rays which
> are at right angles to the plane of projection; (also) a
projection drawn
> or made by such means. orthogonal trajectory Geom., a curve
intersecting
> each of a family of curves at right angles.
>
> 1835 Philos. Trans. Royal Soc. 125 223 We assume, for reasons too
well
> known to need specification here, that the *orthogonal projection
of the
> dipping-needle upon the horizontal plane gives the position of the
> horizontal needle. 1878 R. T. H. BARTLEY tr. P. Topinard
Anthropol. II.
> iii. 264 Orthogonal projections are the only ones which give exact
> measurements applicable to craniometry. 1957 J. EMMONS tr. O.
Benesch
> Rembrandt 69 The Night Watch..shows Rembrandt laying greater
stress on the
> plane of relief, which he boldly cuts through by means of
orthogonal
> projection, with the result that the plane surface of the canvas
is
> seemingly transformed into three-dimensional space. 1816 tr. S. F.
Lacroix
> Differential & Integral Calculus 403 The trajectories in which the
angle
> TMt is a right angle, are called *orthogonal trajectories. 1882 S.
H.
> VINES tr. F. G. J. von Sachs Text-bk. Bot. 951 The planes of the
walls in
> a growing-point are classified thus: a. Periclinal... b.
Anticlinal, those
> which intersect the surface and the periclinal walls at right
angles... If
> the outline..of the growing-point is a parabola..the anticlinals
being the
> orthogonal trajectories of the periclinals, constitute a system of
> confocal parabolas. 1904 Trans. Amer. Math. Soc. 5 56 Such a
system of
> geodesics together with the orthogonal trajectories may be
regarded as
> dividing the surface into infinitesimal squares. 1993 Amer. Math.
Monthly
> 100 127 But because z z2 is conformal, the image H must be an
orthogonal
> trajectory through the new family of ellipsis.
>