Part IV of the Summarische Uebersicht


by Tarrajna Dorsey

One approaches a newly discovered island, or a strange planet, or prospective battlefield, with the intent of exploring it fully before settling in. Nor, unless one is a fool, does one settle in permanently in any battlefield, if it is not already the entire world, or at least a continent. A reasonable composer creates such a domain, and does not advertise that kind of artistic "real-estate" for sale unless, and until, he knows how to develop that territory as a whole, and is able to present it only as one for which no significant, functional aspect remains undeveloped.


Lyndon H. LaRouche, Music and Statecraft: How Space is Organized


Note to the Reader: it will be necessary to have a copy of the Summary Overview at hand, as it will be referenced regularly, and sections will be cited, which will not be reproduced in detail here.


After establishing the parameters and nomenclature for the space in which we will be operating in article 3. of the Summary Overview, Gauss leads us in a seemingly curious direction. He first throws out yet another one of his notorious "known theorems," which appears thus:

0=xy¢z"+x¢y"z+x"yz¢-xy"z¢-x¢yz"-x"y¢z
attributing the cause for this to the fact that the three observed positions of the planet (p, p¢, p") lie in a plane with the Sun. We will note that these quantities x, y, z, etc., were given above to represent the perpendicular distances of the three observed positions of the planet from the X, Y, Z planes, respectively. With the premonition that this is a representation of some volume encompassed by the three positions of the planet, we will read on, to return to this theorem a bit later.
Before we proceed, let us momentarily catch our bearings, and recall the characteristics of the space in which we are operating. This requires a constant return to the standpoint of the physical astronomer: feet firmly planted on the surface of the Earth, head tilted up toward the star-spangled heavens, watching intently over long periods of time, and carefully taking note of what is observed. Although the manner in which any noticeable change in the orientation of the stars can be accurately and methodically recorded begs an extensive discussion in and of itself, we must necessarily note on this occasion, that any change registered to our senses in the nighttime sky occurs, as it were, on a sphere. This can be easily demonstrated merely by walking outside, and beginning to contemplate the nature and method of relating the phenomena that you are observing. Everything that you can point to, lies at some distance along that line, from yourself. When you point to another object anywhere around you, what is the change that occurs? The key lies in the motion that you made from the one object, to the other. How would you begin to measure that motion? How could that motion serve as a measure for speaking of the relation of those two objects to one another? And now, an even more interesting question-introduce the element of change in position of certain objects over the course of time, and we are presented with another challenge-how to create a language of measurement among objects which are themselves changing over time? By the way, these are not rhetorical questions.
Now we proceed to another challenge to the mind of man: if Tycho Brahe stood in his observatory at Uraniborg, observing the heavens, and you stand, wherever you are, seemingly at the center of your sphere, do we not have a problem? Are not all of the observations taken by him at the center of his sphere of measurement? Will this not create a tremendous mess in the ability of one astronomer to communicate their observations to another in any precise way? Well, let us try an experiment. Grab a friend, and have them find a star, and point to it. Now have them describe the position of that star to you, which will be a fun experiment in and of itself. Once you are sure that you have located the exact star that they are speaking of, point to it as well. Now look at each other's arms. Move farther apart. Look again. Continue this demonstrative, albeit imprecise, experiment, until you are satisfied with the constancy of the result.
And now, we reach the introduction of the vantage point of the "celestial sphere." In other words, considering that the fixed stars are, relative to both you and your friend, infinitely far away, you are effectively at the center of the same sphere. Hence, spying a planet upon the backdrop of a particular constellation from your vantage point, can be accurately communicated to your fellow astronomer with no trouble at all. But what about the difference in vantage point between the Earth and the Sun? Well, our discovery still holds. Take a look at this:
[ANIMATION COMING SOON]
Hence, all that is measured and discussed will occur as a mapping of phenomena onto our "space," bounded by this celestial sphere. Keep this in mind, as this "spherical compass" will come up again and again as a measure in other physical investigations as well.

Moving forward in article 4., we are presented with nine "magnitudes," which Gauss establishes as maintaining a certain proportionality with one another. To avoid any confusion, we reproduce them here:

y¢z"-y"z¢
y"z-yz"
yz¢-y¢z
z¢x"-z"x¢
z"x-zx"
zx¢-z¢x
x¢y"-x"y¢
x"y-xy"
xy¢-x¢y
Let us begin to investigate the nature of these magnitudes, as a first step toward discovering how Gauss employs them. Gauss states that the aforementioned magnitudes "represent the doubled area of the projection of the triangles whose areas are f, f¢, f" onto the fundamental planes X, Y, Z..." Here lies an intriguing proposition. We are taking the partially known triangles (we know the angles between them, but not their distances from us, r,r¢,r") which approximate the sector swept out by the planet in the interval of time during which we observed it, and projecting this area onto the three planes that we have constructed as a type of measurement. But as anyone can plainly see, merely by observing the change of their shadow over the course of the day, shadowboxing, or attempting to accurately render a 3-D object on a 2-D surface, projection is not always a "straight-forward" matter. In the case of your shadow, although it remains a projection of you onto the surface of the Earth throughout the entire day, it is clearly changing constantly. What is the principle behind the cause of this change? What other factors play a role in varying the type of projection?
Gaspard Monge, a French military officer and expert geometer who founded the renowned French military and scientific academy, the Ecole Polytechnique, published an extensive treatise on projective geometry in 1799, Géométrie Descriptive, where he investigates the entire field of projections of surfaces, perspective, shadows, and shading.1 One of the figures from his work is included here. This treatise served as a foundation for much of Gauss' later work on the mapping of curved surfaces, which will also crop up in the course of the present discussion. Here, we will investigate one aspect of projection that should be placed in our "astronomical toolbox," since it will continue to come into play.
Monge Diagram

Since we are operating within a spherical space, and projecting the areas of these triangles onto the planes created by three orthogonally intersecting great circles in that sphere, let us investigate the characteristics of spherical projection.
Now it is evident that the projections of the triangular sectors onto Gauss' "fundamental planes" will be changed in area by a factor of the cosine of the angle of projection. But why, then, does Gauss say that these projected areas are proportional to the triangles "as the doubled cosine of the inclination?" Remember, we still have not resolved why the nine given magnitudes represent the doubled area of the projection. Let us investigate this latter question, and perhaps this will, in turn, shed some light on the former.
Hence, we have demonstrated that the magnitude xy¢-x¢y is indeed equal to the double area of the projection of the f" triangle onto the Z plane, which will likewise hold true for the other f triangles, and their projections onto the X and Y planes. Now, as we investigated previously, the factor of change in any spherical projection will always be by the cosine of the angle of projection. So, if the areas of the projections are knowable to us when they are doubled, then, to maintain a truthful proportion between them, we must double the factor of change which occurs in the projection; meaning, we must take the doubled cosine of the inclination of the plane to the three axis, in turn. This relationship can be expressed as:

xy¢-x¢y:2f"::cosb:1,
This yields

2(f"cosb)=xy¢-x¢y,
or also

f"cosb= xy¢-x¢y

2
,
since b was designated earlier as the angle of projection. Thus, if we are allowed the old expression, we have killed two birds with one stone!

At this point, as prepared as we can be for the journey ahead, we enter into the labyrinth. The second matter that Gauss states as being "easily concluded" from the proportionality of these nine projected areas, is that, when multiplied by the distances x,x¢,x" etc. respectively, the equations

0=fx+f¢x¢+f"x"
0=fy+f¢y¢+f"y"
0=fz+f¢z¢+f"z"
are produced. Let us do the work, and then ask questions.
If we take the third row and and multiply it by z,z¢,z" respectively, as Gauss directs, then we obtain:

z(x¢y"-x"y¢)+z¢(x"y-xy")+z"(xy¢-x¢y)
or, when multiplied through,

zx¢y"-zx"y¢+z¢x"y-z¢xy"+z"xy¢-z"x¢y.
Does not this look familiar? Indeed, it is the same quantity as the "known theorem" from the beginning of the section, and as mentioned above! Perhaps now, though, our understanding of what this quantity represents has come a bit more into focus, though not yet completely. Now, from what we gathered as the proportion between the f triangles and their projections, we can further set

0=z(2fcosb)+z¢(2f¢cosb)+z"(2f"cosb),
and, dividing out the shared factor of 2cosb, obtain

0=fz+f¢z¢+f"z",
which is the equivalent of the third equation Gauss obtains, as included above. It is easily seen that all of these relationships will hold in the case of the three Earth positions. Thus, now we can ask some questions.
Firstly, what does it mean to multiply the f triangles, or their projections, by a perpendicular height off of the plane of projection (i.e. (x¢y"-x"y¢)·z)? Secondly, why does carrying out the multiplication as Gauss directs yield an identical result to the "known theorem," presented at the outset of this discussion? Well, one thing that we can add to our notes on the matter before we move on, is that multiplying the area of a triangle, or any polygon for that matter, by a height perpendicular to that plane, is like building up a stack of that particular polygon, up to the designated height, which produces a volume. And not just any volume, but a particular class of solid figures, known as prisms. This class encompasses any figure composed of a polygonal base, whose top is identical with the base, though not necessarily parallel to it, creating the condition that the sides of the figure will always assume the form of a parallelogram.
As is the case when working through any paper by Gauss, we must continue to move along, remaining conscious that what is being presented is only a deceptive shadow of an actually truthful thought process, and that we are going to have to keep our eyes peeled for potential "threads" on which to pull, in order to unravel the shrouds veiling Gauss' thoughts.
We next encounter an intense series of algebraic operations, mainly consisting of multiplying a modified version of the equation we found just now, by a number of quantities. At this point, we will not attempt to delve into the technical acrobatics Gauss is throwing our way. There is certainly a reason for what he is doing, but that will take some more work to uncover. At the present moment, we will "fast-forward" to the end result of these acrobatics, in order to first understand the nature of the intended destination. In so doing, we will find ourselves at a much better standpoint for making a judgment upon whether these acrobatics were the actual road upon which Gauss traveled, or merely a front. We see that a whole table is produced, containing many variations of the product of three of the Sun-Earth or Earth-Ceres distances along the plane of the ecliptic (the Z plane), multiplied by a strange-looking quantity in brackets. Where did these come from? What are they? Gauss tells us that, "it is easily recognized that the expression [pp¢p"], multiplied by the product of the three cosines of the latitudes which appear in it, is the sixfold volume of a pyramid, whose apex falls in the center, and whose three base angle points fall on the surface of a sphere described with radius 1, such that they correspond to three geocentric positions of p..."
As previously discussed in the introduction to this section, the Summary Overview originally served as a rough draft of Gauss' method, sent to his contemporary and fellow astronomer, Wilhelm Olbers. Fortunately for us, Olbers had many questions for Gauss concerning his method, providing us another vantage point from which to `triangulate' Gauss' mind. One of the first questions which Olbers asks, is the following: "`One easily recognizes,' you say, `that (pp¢p") is the six-fold volume of a pyramid, etc.' I do not doubt it, but the proof does not immediately present itself to me. Please, give me some hints regarding it. I very much love such translations of analytical expressions to geometric [ones]."
In Gauss' reply to Olbers, he attaches a hastily written exposition of the proof for this statement, which the reader may work through in its entirety if desired. For our purposes, we will point to the highlights of this proof, which lead us to a perhaps surprising place.
Similar to the language of the Summary Overview, Gauss begins his short exposition to Olbers with a triangle PP¢P" in space, and its projection onto three perpendicularly situated planes. The coordinates for the three positions which form the corners of the triangle are, similar to the Summary Overview, x,y,z;x¢,y¢,z¢;x",y"z". Yet again, the ghost of our mysterious theorem appears, in slightly different clothing:

D=xy¢z+x¢y"z+x"yz¢-xy"z¢-x¢yz"-x"y¢z
Using two intriguing theorems, 1) that the sum of the squares of the projections of a triangle onto three perpendicular planes is equal to the square of the area of the original triangle, and 2) that the sum of the squares of the cosines of these three angles of projection is equal to unity, Gauss later shows that

D=±2KL[height of pyramid]·Area DPP¢P"[its base]
and thus concludes that the volume of the pyramid formed between the points PP¢P" and the origin is

=± 1\6 D.

 

[Box on Gauss's proof to Olbers Coming Soon!]

 

Presented with difficulties in working through the foundations of the "proof" that Gauss here provides to Olbers, and unsatisfied with the foundations upon which it is built (seemingly derived from the equation for a plane), I found solace in tracking down the location where I recalled encountering theorem 2), mentioned above. In the very beginning of Gauss' second treatise on curved surfaces2, the same theorem is indeed found. But here, it is found in the context of Gauss' development of the concept of using a sphere as a measure of curvature for any surface. In his words, from the beginning of the paper:
Investigations, in which the directions of various straight lines in space are to be considered, attain a high degree of clearness and simplicity if we employ, as an auxiliary, a sphere of unit radius described about an arbitrary center, and suppose the different points of the sphere to represent the directions of straight lines parallel to the radii ending at these points. As the position of every point in space is determined by three coordinates, that is to say, the distances of the point from three mutually perpendicular fixed planes, it is necessary to consider, first of all, the directions of the axis perpendicular to these planes...
Here, we see that the origins for all of Gauss' work on the measurement of curved surfaces, and the use of what he here calls an "auxiliary sphere," are clearly traced back to his work in astronomy, and his employment of the celestial sphere. Yet again, we are presented with evidence supporting the idea that the roots of man's investigation of the universe are ever to be found in the ancient Pythagorean's investigation of sphaerics, and all developments in scientific method since that time, including Gauss' work, have arisen from similar investigations.
Here, too, is where we finally find a satisfactory reply to Olbers' question on the matter of the pyramid volume. At this juncture we will include the first "axioms" of Gauss' paper on curved surfaces, in order to gain a sense of how Gauss develops the language of the "space" in which he is operating, but it is highly recommended that the reader work through at least the beginning sections of the paper on their own to gain a more complete understanding of the investigation.
It will be advantageous to bring together here some propositions which are frequently used in questions of this kind.
I. The angle between two intersecting straight lines is measured by the arc between the two points on the sphere which correspond to the directions of the lines.
II. The orientation of any plane whatever can be represented by the great circle on the sphere, the plane of which is parallel to the given plane.
III. The angle between two planes is equal to the spherical angle between the great circles representing them, and, consequently, is also measured by the arc intercepted between the poles of these great circles. And, in like manner, the angle of inclination of a straight line to a plane is measured by the arc drawn from the point which corresponds to the direction of the line, perpendicular to the great circle which represents the orientation of the plane.
After this introduction, Gauss builds up to his construction of the spherical pyramid. Here, we include an animation which does not cover everything that Gauss does in the paper, but provides a construction suitable for our present purposes. Some of the variables from the paper on curved surfaces have been replaced with language that will be more familiar to the reader, coming from the standpoint of the Summary Overview.
Or, in Gauss' words from the end of article (2.) in his paper on curved surfaces:
...it is easily seen that the expression ±D represents six times the volume of the pyramid formed by the points L,L¢,L" [pp¢p"] and the center of the sphere. Whence, finally, it is clear that the expression ±1\6D expresses generally the volume of any pyramid contained between the origin of coordinates and the three points whose coordinates are x,y,z;x¢,y¢,z¢;x",y",z".
 
Concerning the Greek investigation which led to the proof that the volume of any pyramid is one third of the volume in which it is inscribed, the reader is challenged to develop their own original proof. My efforts have not yet produced an original proof, though much fun was had in searching for one; I tried to approach it from the knowledge of Johannes Kepler's assertion in the Epitome of Copernican Astronomy, Book IV, that the volume of a regular tetrahedron is \tfrac13 the volume of the cube it is inscribed within. Despite construction of many paper pyramids, and cutting pyramids out of potatoes and other vegetables, my efforts have so far remained fruitless. Curious to dig up the original proofs of this discovery, I found that its origin is attributed to the Greek contemporary of Socrates, Democritus, by Archimedes, in his Method. What is most notable, however, is that the other main discovery which Archimedes attributes to Democritus in the same location, is that "the volume of a cone is one third of that of a cylinder having the same base and equal height."3 How exciting! Clearly, whether originally discovered by Democritus or not, the well-springs of this discovery appear to be located in the context of the general investigations of volumes, which, in the case of Archytas and the doubling of the cube, as we know, had much more profound implications in the development of a higher concept of number. Sir Thomas Heath, the author of the cited History, claims that the discovery of the cone and cylinder would have naturally flowed from that of the pyramid and the prism, simply by increasing the sides of the polygonal base of the prism and pyramid, essentially to infinity. However, any student of the teachings of both Cusa, and one of his foremost students, Lyndon LaRouche, will see that there is a problem of transcendental species here; the same which arises in the matter of Archimedes' method of quadrature for the circle4
Back to the matter of the actual proof, aside from Eudoxus being cited as providing the first actual proof, which seems to be a manifestation of his method of exhaustion, the only other proof I actually located was that of Euclid's, which is found in Book XII, Proposition 7, of his Elements.
Now, let us conclude this journey and return from our long voyage into the distant past and near future, to the matter of the Summary Overview. As desired, we have now gained a more complete concept of the nature of these pyramid volumes, and a general proof for their construction. However, many questions remain. We are now able to see that the multitude of bracketed quantities, multiplied by the cosines of their respective latitudes, do indeed represent the sixfold volume of the pyramid contained by the respective observations, as mapped onto a unit sphere [SEE BOX II BELOW]. This means that these volumes, multiplied by the distances (e.g. dd¢d") would be the volumes of the actual pyramids formed by the positions of the planet, the Earth, and the Sun. But what is the point of having all of these volumes? What do they have to do with finding the actual distances of the planet from the Earth and Sun, which is what we really need, in order to figure out the orbit? And how did Gauss come up with these volumes, anyway, since it does not look at all similar to how he creates them in his paper on curved surfaces? And are these volumes really unique to Gauss' method at all??

With all of these queries buzzing in our mind, we move ahead...


Box II



Footnotes:

1For an English translation of this work, see Heather, J.F. An Elementary Treatise on Descriptive Geometry. J. Weale, 1851. Translated from the French by Barnabé Brisson.
2General Investigations of Curved Surfaces, 1827. Originally translated from the German by James Caddall Morehead and Adam Miller Hiltebeitel in 1902. Edited by Peter Pesic, 2005.
3Heath, Thomas. A History of Greek Mathematics, Vol. I. Dover Publications Inc., New York. 1981.
4For more on the matter of quadrature, see Cusa, Nicolaus. On the Quadrature of the Circle. Translated from the Latin by William Wertz, 1994. http://www.schillerinstitute.org/fid_91-96/941_quad_circle.html. See also, LaRouche, L.H. Let There be a Time of Thanksgiving. http://www.larouchepac.com/files/pdfs/20071210-lpac_thanksgiving.pdf.


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