Our department of Technical Drawing was quick to point out that these sections of a Torus were once studied by the Greek Perseus and that they bore a close affinity with the well known Secant and Tangent theorem.

Our mathematics department could readily relate them to Prop. 35 of Book 3 of Euclid's Elements:
"If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other."

Our students found that many and varied were the processes that could be modeled using these section curves that came to be known as Cassinian ovals.

Examples of academic sources using Cassinian oval models included :-
-modeling of the shape and volume changes in the red blood cell
-modeling electro-magnetic activity in the case of wires of equal current and direction.

When we turned our attention to the author of these curves we were disappointed to find Encyclopedia Britannica saying that he believed the oval described the varying distances between the earth and the sun.

Despite our best efforts to acquaint students with the elegance and simplicity of the Keplerian elliptical model of the Earth's motion about the Sun, and the fact that it enjoyed widespread academic consensus, the Dept. Heads would in time be cast in the role of "slaves to some Newtonian dogmatism" that supported what they termed "The Kepler Cult" .

By way of "educating" the Department heads who were deemed to be "no longer teachable" and "ever ready to flee from the wheel of Newtonian dogmatism", they presented to us over the next few years an amazing range of sources on the work of Cassini.

They claimed that some very learned people had informed them that Cassini had the academic stature to be taken seriously and not to have his work consigned to the potters field of academic tut tut?

After all was it not Cassini that discovered the Zodiacal light?
Was he not as Chalmer's Concise Dictionary of Scientists (1989) says
"the one who worked out the rotational periods of Jupiter, Mars and Venus"?

Did he not work out the first reasonably accurate figures for the distances of the Earth from the Sun?

Is the Cassini division of the rings about Saturn not named after him?

Is the greatest astronomical undertaking of the decade, the launch of the Cassini probe on the 6-Oct-1997 not named after him?

Should we not seriously consider the work of a man who is reported in the
Transactions of the Royal Society - Abridged version by Hutton/Shaw (1809) Anno 1665 as having
"determined geometrically the apogee and eccentricity of a planet from its true and
mean place; a problem which Kepler had pronounced was impossible to be solved."?

The same source for Anno 1676 comments on "Cassini's superior interpretation of Jupiter's satellites over Galileo"

Such is the quality of research that our students presented us with after the summer recess of 1994.

We felt that these research skills would bear more fruit if they were directed to some study of the Torus (such as Tesla's toroidal transmitter at Wardencliff Tower New Jersey Patent number (1,119,732) for year 1902 ) more worthy of their attention than the trajectory of the Earth about the Sun.

Towards this end we sought an authoritative dismissal of the Cassinian model from a reputable authority and were a little surprised to learn from the University of London Observatory 25/04/95 that the Cassinian oval model "very nearly reflects (or exactly reflects?)" the departure of the Earth's orbit from a circle and that "if the Earth were translated to the origin, the Sun would appear to follow a Cassinian oval about some point F1 in uniform circular motion about the Earth ".

A subsequent communication from the same observatory 04-August-95 informed us that
"it is certainly true that the formulae for Cassinian ovals....resemble terms of higher order expansion of slightly elliptical orbits around circular approximations. For example, the term (Cos 2 theta) would occur in such an expansion."

Some students understood clearly that the varying diameters of the Sun as seen from the orbiting Earth was but the reciprocal of the respective distances of the Sun from the Earth at that point in time, and that any equation describing the varying distances would need to simultanoeusly account for the correspondingly varying angles or what the astronomer would term the ecliptic longtitudes.

They therefore took great delight in pointing out to us that the Paris Observatory -which was even
older than Greenwich- had concerned itself with just this question of the exact geometry of the Earth orbit,
and that this constant-product-oval-model-that-allowed-for-the-precession-of-the-equinoxes was first "shown" to be entirely consistent with observation - this being done by Europe's then foremost astronomer better known as the Paris Observatory's first director and none other than Cassini himself (1625-1712).

Was it not the Table of the eclipses of the satellites of Jupiter that Cassini had published (1666) that was utilized for the determination of the longitudes in the course of numerous worldwide expeditions undertaken by French astronomers?

Was it not Cassini's tables entitled "Ephemerides Bononienses Mediceorem siderum" (1668) that were used by Olaus Roemer (1675) in his demonstration that light had a finite speed?

Was it not Cassini who discovered the Saturnian satellites of Iapetus, Rhea, Tethys and Dione?

What began as a consideration of the Torus was rapidly becoming a study of the diverse applications of its
many cross-sections called Cassinian ovals.

The German teacher arrived back with 2 sources on Cassinian ovals
-"Journal of Geometry" Nov 01 1992 v 45 by H. Martini entitled
Cassini-Kurven in der Lichfeldtheorie.
-Die Cassinische Flache - von - Gottfried Baumberger Bern 1900

One of our students told us that it was Cassini's rather than Kepler's figure that was now accepted as the more accurate for the eccentricity of the Earth's orbit.

The Biology teacher told us that Cassinian ovals are used to model evolutionary processes such as
morphogenetic sequences and gave her source as a book "Solid Shape" copyrighted by the Massachusetts Institute of Technology and authored by Jan J. Koenderink (1990).

In the staff room we were deemed to be the least teachable people in the whole school, and we often got asked questions like "How did you come to accept the theory that the Earth is not flat?."

While staff and students alike got a lot of fun out of all this, outsiders started looking up sources referring to Cassinian ovals and sending them to the students.

The physics teacher brought in a source showing how the Cassinian oval model is used in modeling at the sub-atomic level "Journal: High energy physics and Nuclear physics" vol. 13 no. 5 p 455-8 May 1989
article published by Dai Guangxi and Liu Ximing of the Acad. Sinica Lanzhou, China, and entitled
"Approach to dynamics of fission described by Cassinian ovaloid"

Didn't some dude called Perseus refer to 3 groupings of sections (cyan,green & red) (ie seperate, sagging & bursting) stradling 2 threshold-sections (yellow & blue) (ie lemniscate of Bernoulli & Taut) in all 5 different section-types of the torus, that so freaked him out that he felt he should make 'offerings to the gods?'

What about the following observation that was dumped on us by a drunk:-
Congratulations for being able to ignore a curve that deals in precision, has many natural occurences, involves the simplest of all area laws, enabling distances from one focus to be calculated from uniform angular motion about the other and proposed by Europe's foremost observer.
Stand up with courage to anyone who makes little of a curve that deals in approximations, has few if any natural occurences, involving a mathematically complex area law, conferring no utility whatsoever on the second of its two foci.
Ignore anyone who says that it came to Kepler in a dream. Good luck my friends.

The students would ask us how we felt about the Moon ! for was it not Cassini's own son Jacques (1677-1756) who gave us the 3 empirical laws describing the rotation of the Moon about its center of mass and known to this day as Cassini's laws.

We often wondered about the identity of a mystical sort of man with a foreign accent who on the school's answering machine left a message thanking us for introducing him to the "family of cassinian ovals" and pointing out that the properties of this "talismanic curve of the 4th power" were such that it "could be drawn even if the focal-distances from a chosen origin were IMAGINARY" a fact that would be readily appreciated by students of "types, archetypes, mysticism and ontology".
Whow!!  While none of us knew what that meant it sounded great.

The Engineering Department said that Cassinian ovals could easily be generated by linkages.

Despite the momentum generated by the group we regret to say that we the Department heads could not give ourselves the intellectual mandate to commit the school to investigate any curve other than Kepler's ellipse as a trajectory of Earth-motion about the Sun.

This all changed when the youngest of our group handed us an article that her father had found by one
J. Sivardiere in the 1994 v 15 n 2 edition of the "European Journal of Physics" On page 62 the title of the article was :-"Kepler Ellipse or Cassinian oval?"

We now decided that to test this hypothesis without prejudice we would need to have daily figures for both the ecliptic longitudes and the semi-diameters of the sun as its distance varied from day to day on its apparent and non-circular motion about the Earth.

These figures for the Sun as seen from the orbiting Earth and pre-calculated for 1995 by the authors of the "Astronomical Almanac" were supplied to us by Dunsink Observatory.

Using the Autocad/Autolisp environment we used the Almanac distances and directions to plot the true position of the apparent Sun for every day of 1995.

For subsequent visual comparison purposes a small circle was centered on the exact position of the apparent Sun for each day of the year.

As Bishop Ward and Count Pagan had attempted to do in the case of the simple ellipse we in the case of the Cassinian ellipse anchored the Earth at one focus, letting the focal angle there act as the Real Anomaly while the focal angle at the other focus acted as the Mean Anomaly.

Using this method we superimposed on the Almanac model a Cassinian model likewise derived for every day of the year and taking into account the daily precession of the equinoxes.

How compatible they were was readily seen after just a few of the daily plots.

The Cassinian equation placed the apparent Sun so near to the Almanac position as to be indistinguishable to the naked eye.

Mathematical comparisons between the Almanac and Cassinian models showed a discrepancy of less than 1% in both the longitudes and semi-diameters.

In the case of the Sun or any of its planets we would now like to have, instead of the theoretical projections of the almanacs, real-time and actually recorded positional co-ordinates.

We would like to be advised on what is the best source of RECORDED data to use for modeling purposes, and would welcome any help, advice or information that could be supplied to us on this or indeed on any of the properties or applications of these toroidal sections called Cassinian ovals.

With every good wish to all who read this (From Pádraig Ó Searcaigh)

Ps--------I have just been handed a source entitled "NASA Tech. Paper 3326". The paper deals with
"Systems Design Analysis Applied to Launch Vehicle Configuration" written by Robert Ryan and V. Verderaime of the Marshall Space flight Center. It reads:-
"Tank end closures are usually elliptically shaped, but may not be the best configuration for performance and cost. The lox and fuel tank forward domes are designed primarily for internal pressure. Cassinian domes may provide as much volume as an ellipse in a shorter length and with less discontinuity at the edges for a total vehicle net weight saving."