Originally posted by author:

Suffice it to say that their are many great mathematicians and many small, bur few who define their area of study well.

Manipume is the study and theory of ratio and proportion in all aspects of reality.

Eudoxus is credited with the foundation of this description, and thus may well be the founder of Mathematics.

Originally posted by author:

So when i did not understand i went all over the place. Now i can save you time by linking you to this video.

[/VIDEO]

Originally posted by author:

Originally posted by author:

3d polar coordinate geometry is quite hard to find a very general treatment of it which is not fully wrapped in tensor or vector notation.

Take a unit pole and a variable rod, and attach the rod to the pole so that it is joined in a corner which is free to rotate. This measring device will be the basis of some measurements of regions, and works by pointing the pole at one point on the boundary of the region and moving the rod around the region boundary noting the scalar length of the rod as it moves round and the radian angle measure between the rod and the pole as it moves in this way. I thus record [tex](r, theta)[/tex] but clearly not the rotation of the polar axis( the pole) as the rod traces round the region boundary. For this purpose i have an orthogonal unit that is attached to the pole, thus the pole becomes a right triangle, a Bombelli vector, and i have a third marker of unit length orthogonal to the pole but free to rotate in a plane orthogonal the pole . This marker is also fixed on a point on the boundary of the region if possible or some point relative to which the region is fixed.

Now i can measure pole's axial rotation in radians. This is exactly like using a pair of compasses with the region being traced out by the pencil. Thus we record [tex]r, theta,alpha[/tex] .

We can then write equations that have to distinguish [tex]theta[/tex] and [tex]alpha[/tex] and therefore these require +gates and the cos and other trig functions.

When we look at the role of the roots of unity we find that they do one thing, they rotate the unit magnitude in space. So + and – are π rotations of the unit magnitude. The functions that control this rotation and its rate and nature are the trig functions. The + and the – are +gate modifiers, like mod() and control how units are aggregated. and what quantity.

[tex]r=theta^2+theta*cosalpha+1[/tex] for example.

Divine proportion may help.

I have to say that having read just now Descarte's Geometry in translation, i find that Descartes, While not acknowledging Bombelli fully concurs with Bombelli's use of Neusis and the use of the gnomon or carpenters rule.

`in fact he identifies elegantly the source of this method from the Greek fathers, and also how to employ it in circular or semicircular constructions in which relevant measures are set orthogonally.

There is a lot of neusis in his explanations and proofs and the whole subject is accordingly very dynamic. Not once do the standard cartesian coordinates that i was taught, rigidly fixed in the page and to which every curve or line must conform, appear. instead the form took precedence and the "coordinates" were constructed as and when and where needed by means of a circle construction!

This may not seem of importance to you, but i can assure you that this is very Greek, very natural, and immediately apprehendable by any child who has been taught the rudiments of Euclidean construction. Thus amazingly as if appearing out of nowhere Descartes by proportion is able to sensibly write down proportions and equations necessary to the finding of measurements on many forms.

It is also apparent within his terms of useage that he neither intended to slur or in any way denigrate Bombelli's codification of the "roots of minus one". Within his method these roots were imaginary as they existed only by rotation, translation etc of the form, in short by neusis. In addition some roots existed in multiple form and so needed to be distinguished from those that existed by moving the form hence his terms imaginary and real.

His explanation is full, gentle and above all reasonable and accommodating.

`i of course withdraw any comments made hitherto to his intentions in calling complex numbers imaginary. Indeed if the translation is literal, it seems he used the term complex liberally when dealing with equations which may give rise to these kinds of roots.

He finishes…

" I hope that posterity will judge me kindly not only as to the things i have explained, but also as to those which i have intentionally omitted so as to leave to others the pleasure of discovery.."

As I begin my exploration of Greek geometry and Euclid in particular I find many assumptions of my youth hiding curious facts about my world.

Angles for example come from the idea of bending to form a hook in all languages and etymologies the idea of corner therefore is too static!

The gnomon is a carpenters or artisans measurement tool enshrining orthogonality and was so well known and useful that it was sufficient to represent hold or support a rectangle area or a rectilinear form. The artisans and builders hodder for carrying bricks and tiles was a type of gnomon holding solid forms. The gnomon could be used to cast shadows and Mark off shadow lines, and so was useful for celestial and terrestrial measurement.

The dynamic circle is a curious creature! It hides an infinite number of relations in it's perfect form and links all measurement in some relation one to the other. The drawing of a circle which seems so simple reveals a curious fact about time differentiation which is as intuitive to grasp as Einsteins curious time distortion as speed approaches light speed. For to draw a circle in it's full circumference takes a time t but as the radius of the circle increases the time taken to draw the circle increases. Thus at infinity it takes infinity in time to complete one circle. Thus to the stationary observer the time to complete one revolution appears to slow down.

On the other hand, should the time appear to remain the same then the speed of traversing the circle appears to tend to infinite speed!

Such curiosities do not reside only in these apparent measurements, but commensurately the length of arc diminishes as the radius increases in the sense that should two circles be tangential, that is kissing, then the same distance traveled along each arc to each circle results in a different rotational experience to the traveling observer. Therefore the larger the arc one traverses the smaller the sense of rotation one experiences. This is of course a suitable notion of curvature measure.

It seems also fairly clear now that angle measure was not corner angle but arc distance along a circle or inside of a sphere. Each arc is indivisibly associated with a chord, a proper tie that in many ways linked circle and sphere to the triangle or rather gnomon. Thus the gnomon is a convenient and versatile measuring tool for all fields of measurement especially when linked to the sphere or circle.

These fundamental relations form the basis of all our systems of measurement by hand and eye, and enable our logos response to standardise a spaciometric measurement response to a constant form with a constant set of relations.

That we have over time been able to extend these measures to define evens gustatory and all sensory signal measures through the modular arithmetic models is truly amazing.

I am struck by Descartes geometry which at the last he characterises by the circle, the gnomon and the form these are used to measure. Thus Bombelli's vector as well as his operator took full pride of place in Descarte's geometry, being used to distinguish plane geometry from solid geometry and as Descartes hoped geometries of more exotic descriptions .

Plane geometry is characterised by a straight line intersecting with a circle, solid geometry by a conic curve intersecting with a circle, and I daresay other numerous geometries may be described by how their standard form of curved surface or curve intersects with a circle or sphere.

Originally posted by author:

The nature of all possible geometries and all future geometries will be found in the geometrical objects generated by the relative interaction of at least 2 dynamic spheres. These said geometrical objects will be found to concord with every generalisation or designation of the set of geometrical curves that are now called the Roulettes, but which formerly were named the Trochoids including the cycloid. The concord may one day be shown to be a congruence which after the suggestion of Descartes in his Geometry serves to categorise every geometry we will ever invent.

" …i have found a proof of this that is most wonderful! However the margins of the Forum are too small to hold it….."

Originally posted by author:

Here , dear reader is the origin of logarithms in mathematics.

I have to say, that like Bombelli Euclid aimed at a more popular audience of Artisans, that is artists engineers, building contractors, land surveyors etc. We have been at the mercy of classical scholars who have made his work more high brow than it is!

It still requires the genius of Napier to bring this and Ptolemy's work to the practicalities of logarithms, but it is clear neither would have been able to without this definition by Eudoxus.

I also have to adjust my understanding of ratio and proportion in line with Eudoxus.

[tex]r=theta^2+theta*cosalpha+1[/tex]

r=θ2 +θ×cosα+ 1(posted as BBcode, click Edit to see it as I typed it)