Subject: [HM] The History of Horn Angles (3/4)
From: Ken Pringle (kenneth.pringle@studentmail.newcastle.edu.au)
Date: Tue Dec 21 1999 - 21:17:23 EST
( b ) Concerning the title translated _On Contact with a Circle and a
Sphere_ .
We have already observed that a sphere considered in relation to an
outside point as vertex for a circumscribing cone defines an angle, viz.
the vertex angle or generating angle of the cone. The circumscribing
cone is necessarily a right circular cone. A sphere cannot be inscribed
in an oblique circular cone. With a right circular cone and an inscribed
sphere the intercepts between the vertex and points of tangency of the
generators are equal. This provides a basis of stereometric
consideration of conic sections as arising by section of the cone by a
plane entering tangent to an inscribed sphere. Such configuration is
realisable with any plane section through the cone that does not pass
through the vertex. The point of tangency of sectioning plane with
sphere is a focus of the conic section. The generators in the meridian
plane through this point have contact both with the sphere (called the
focal sphere) and its circular section in that plane (called here the
focal circle). The title translated as _On Contact with a Circle and a
Sphere_ might have had these salient features in mind. Related further
considerations in respect of which the title is readily seen to be
highly appropriate will presently be cited.
Involvement of focal spheres leads to recognition of the directrix, as
the line in which the sectioning plane meets the plane of the circle in
which the generators touch the focal sphere. (Following an interest in
relations of different sets of data pertaining to a figure. Given the
cone, specification of the directrix amounts to specification of the
inscribed sphere, the sectioning plane, focus and curve of section. All
consequential upon there being only one inscribed sphere through the
circular section whose plane contains the directrix.) Focus-directrix
treatment of the conic sections is readily attained on the basis of
reference to configuration including focal sphere, focus, and directrix.
Ordinate-abscissa relations (in the antique context not appropriately
thought of as Cartesian equations) can be deduced without much trouble
from the focus-directrix relations. The application of conics to
geometrical problems in antiquity characteristically used the
ordinate-abscissa relations. The focus-directrix treatment as arising in
connection with considerations of focal spheres may however have
contributed importantly to solution by conics of at least one notable
problem.
The configuration of cone, focal sphere, and sectioning plane issues in
a discrete proportion in six terms, all lengths in the focal meridian
plane. This by imposition of certain constraints obviously possible
issues in a continued proportion in four terms. This gives two mean
proportionals between extremes. This might relate to Eudoxus's solution
of the problem of the two mean proportionals as we hear of it from
Eutocius of Ascalon (c. 530 A. D.) (Knorr 1986, pp.52-61). A feature of
the discrete proportion is that none of its terms involves explicit
reference to any point of the conic section associated with the
configuration to which they belong. (The terms are related to a point of
the conic section by considerations of equality of tangents to the focal
sphere as for several points as vertices.) The discrete proportion is
established without having to refer to the conic curve involved.
(Because only one circle can be inscribed in a plane angle so as to have
contact with an arm at a particular point on it, the disposition of the
extremities of the relevant intervals in plane configuration completely
specifies the cone and curve of section and two symmetrically placed
points of the latter, and conversely.) Eutocius says that Eudoxus
claimed to discover his solution by means of curved (kampylai) lines,
but in his proof makes no use of them. Further, that having found a
discrete proportion, he uses it as if it were continued. Eudoxus's
"curved lines" may have been curved lines arising by section of the
right circular cone, i.e. , conic sections. The Menaechmean solution
(Knorr 1986, pp.61-6; 1989, pp.94-100, 114-5) would enter with
characterisation of conic curves by means of ordinate-abscissa
relations.
We turn now to details as to essentials of a possible procedure of
Eudoxus.
Visualise a right circular cone, vertex V and axis VC. A plane not
through V cuts the curved surface in a conic section. A sphere is
inscribed above the sectioning plane and tangent to it. The point of
tangency we call S. In the meridian plane through S circular section FES
is tangent to generators VT, VR at F, E respectively. Circular base TPR
meets the conic curve in P. The generators touch the sphere in circle
FQE. Q is on the generator VP through P.The conic curve passes through
meridian plane VTR at A. SA produced in both directions meets TR in N
and FE produced in X. VC meets TR in C, NX in W. S is a focus of the
conic curve AP. The directrix passes through X perpendicularly to
meridian section TVR. Sphere FEQS is the focal sphere. Circle FES is the
focal circle. The reader able to consult Durell's _Elementary Coordinate
Geometry_ (1960) will find on p.227 a figure corresponding almost
exactly to the specifications.
Referring to figure constructed as specified it is not difficult to
establish discrete proportion
RE : NX = RA : NA = EA : AX .
By considerations of similar triangles, RA : NA = EA : AX. Drawing
through X a line parallel to RE to meet TR produced, we have again by
similar triangles proportion RE : NX = RA : NA.The focus-directrix
relation for the curve is readily deduced, observing that AE = AS, and
that SP = RE, since SP = PQ and PQ = RE (Q on FQE and on generator VP).
Here we use three applications of the principle that a sphere inscribed
in a cone cuts off toward the vertex equal intervals from the generators
(identifying as vertices of cones circumscribed to the focal sphere not
only V but also points A and P). The discrete proportion and
focus-directrix relations possibly were prior to Eudoxus. They might
have been found by Democritus. Note in connection with this possibility
that the lengths compared in the discrete proportion all lie in lines
tangent to the focal circle and except for RA, AX actually touch it. The
focus -directrix relation cannot appear without involving further
lengths AS and SP, PQ. These fall on the focal sphere tangent to it. The
tangencies have operative significance. The relations attained are
reached by attending to considerations of contact with a circle and a
sphere. In the documented history of study of the conic sections the
present use of the focal sphere is not encountered earlier than the
nineteenth century.
From the relations of the intersecting planes defining the directions of
NX, NP we infer that these lines are mutually perpendicular. We have in
NX, NP two mean proportionals, provided only that RA = g NX, NA= g NP, g
a positive number, and RE, (AX / AE)NP are the extremes. (Here
anachronistic elements for convenience have been brought into the
exposition.) EA, AX are in the same proportion to NP, (AX / AE)NP
respectively. Supposing as required, we have proportion RE : NX = NX :
NP. Supposing now <EAX right, we have <NXP = <AEX = <NRA = <VWA. Given
the cone, to determine NX, NP we have only to construct on NX at X in
the plane of section an angle equal to <VWA, complement of generating
angle TVC. But with given extremes RE, b we need to have the generating
angle such that RE : b = EA^3 : AX^3. With these extremes we are able to
state though continued proportion
RE : NX = NX : NP = NP : b = CW : NC (= tan <TVC, taking a little
liberty with the concepts) .
From this we deduce solving relations
NX^2 = RE . NP , NP^2 = b . NX , NX . NP = RE . b .
The connection of these relations to conic curves is not immediately
evident. But if ordinate-abscissa relations have been deduced from
focus-directrix relations for conic curves of the three primary classes,
it will be evident that the relations concern respectively a parabola,
another parabola, and a hyperbola. The ordinate-abscissa relations
holding for the mean proportionals, as derived in connection with a
curve arising by section of the right circular cone, may suggest the
ordinate-abscissa forms corresponding to the focus-directrix
characterisations. Recall here Eutocius's testimony that Eudoxus claimed
to have discovered his solution by means of curved lines.
Using trigonometric concepts (which here is not essential) it is easy to
base the continued proportion on the discrete proportion. Denote <TVC by
letter a, <VWA by letter c. We can apply identities
__ RE__ = __ RE cot a__ = __b tan a __ = tan a
RE cot a b tan a b
by setting NX = RE cot a, NP = b tan a, and supposing a such that tan a
is cube root of RE / b. Given RE anywhere on a generator of the cone,
there is a position of NX such that NX = RE cot a, so long as RE cot a
is not less than RE cos a (the distance between FE and TR), or cot a is
not less than cos a. This is the case (cos a >= sin a cos a) for all
angles a not exceeding a right angle. The position must make <EAX a
right angle. If NX = RE cot a, we have RA : NA = EA : AX = RE : NX = tan
a. This is consistent with tan <RNA = tan <EXA = tan a, or <RNA = <EXA =
a, or consistent with <EAX being a right angle. That <EAX indeed is a
right angle follows when it is shown that angle c in which NX meets axis
VC is complement of angle a. When NX is fitted in between FE produced
and TR so that an extremity falls on each, its orthogonal projection on
axis VC must equal distance RE cos a between FE and TR. We must have RE
cot a cos c = RE cos a, or cot a cos c = cos a, reducing to cos c = sin
a provided that cos a does not vanish, or a is not a right angle. Whence
angle c is complement of angle a. Cutting the cone at right angles to a
generator facilitates realisation of the solving lengths with the conic
curve. Because the appropriate generating angle will have to be found by
means of some independent construction, it is clear that solution
involving essentially only one conic curve really is not possible. The
present remarks are merely to satisfy the reader that the continued
proportion is based by possible constraint on the discrete proportion.
[ end of part 3 / 4 ]
Kenneth W. Pringle
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