Originally posted by author:

Because of an inate pythagorean attitude scalars were promoted as the solution for all our geometrising. For a very long time even up to now the goal of geometry was to produce a scalar! this is why the √-1 was met with such derisison and incredulity. What √-1 reminds every mathematician and geometer that the goal of geometry is to devise methods of proportioning, that is algorithms that proportion. What √-1 means is that within your method of proportioning you need to rotate!.

Of course we need to do more than rotate to proportion, so there are special terms that must relate to these motile actions i guess, Now i wonder what √-p does where p is any prime? Or even better still[tex] (-p)^{1 over p}[/tex]

Hamilton's Analysis of the algebra of imaginaries for quaternions and Graves for octonions were important but ahead of their time. When i look at Hamiltons's analysis of quaternions i know that the quaternions drop out of a more general formulation. Thus Hamilton's Analysis has many more applications than just Quaternions. I am sure that more of the roots or -1 are covered by his system but he was constrained by the cartesian coordinates to apply what he had developed to the physics of his day. Hamilton struggled for the rest of his life to find applicability and meaning in his system, and so did everybody else!

He found some powerful uses but not fast enough to dominate the analysisi that many physiscists were doing, and not in an easy predictive manner. In fact quaternions are now seen to be superior because they simplify the description of the geometry, just like Descartes system did, but descartes applied his system to the past not the future,and thus establised its merit well before it became an essential tool to analysis and the methododlogy of a generation of geometers and mathematicians.

Originally posted by author:

Among the scalars of unity the primes play an interesting part. They really show that scaling as an operation: that is proportioning and ratioing can be sorted into independent scaling "seeds". So for example if unity is fundamantal then the next fundamental scalar is 2. This means that i can reduce or scale down certan ratios to a "seed" of 2.

Three is the next fundamental scalar because of course bundling in 2" will not capture 3 and similarly i can scale down certain ratios to a seed of 3. It becomes apparrent that the prme numbers are interellated by scaling as 3 can be scaled to 6 as can 2. Systematically going through the scalars to find how scaling works, that is an analysis of scaling reveals these "seed" numbers which are all scalars of unity but which unlike unity scale to pick out only certain numbers.To a geometer this evidence of a kind of mesh, ameshing together of these seed number scales explains the fundamental nature of the unity, and describes in a way how all thingsdeveloped from unity. Of course unity had to be male!

This mesh was related to an actual sieve mesh by Euclid, and a spatial arrangement was used to describe this curious mesh arrangement, by relating it to finding the area by multiplying the side lengths of rectangles. This mesh partitioned rectangles into equivalence classes, and related directly to proportioning as well as to bundling or packing.

So for a while Pythagoras had a nice little thing going there with is theory of unity being fundamental, and worthy of study for that sake alone! Theodorus, shut up!!

What Theodorus showed was that primes or proto arithmetical objects, with the pythagorean theorem did not produce scalars of unity, or eve scalarsof proto aritmetical objects, so even the prot objects were not "proto" in that sense.

My guess is that Theodorus got the short end of the stick if not the sharp end of a dagger! That is how important it was. It took Eudoxus to restore equanimity and maintain"unity" by explaining the arithmoi as scalars the solution of all proportioning, and that these irrational numbers could be proportioned and scaled among themselves. Thus the arithmoi maintained their foundational and "creative" status status to this day. Some like to stand in awe of number in the style of Pythagoras even now.

So geometric measure and the right triangle "rule" with the aid of the arithmoi are seen to give birth to a whole new set of ratios,never before conceived, but children of tha arithmoi.

When negative numbers were introduced, as debt mainly, their geometric nature demanded an origin, which is why the Indians were far advanced in their use and consequences and their sign rules and also their consequence of √-1.

For me this leads to the measure being the fundamental arithmoi: arithmetic and geometric object and as theodorus showed, the unit circle and sphere being the fundamental proportion or scalar, and consequently the mesh of prime measures being spiral in nature.

Of course if a circle or sphere is a proportion this makes an measure necessarily a vector, that is having magnitude and direction, and square rooting a geometrical operation or algorithm. In fact it reveals that all operations are geometrical operations not counting ones. Therefore, and it took a long while to realise this we need to define all operations geometrically and rigorusly.

Descartes began to do this in his Geometry and continued to do so all his geometrical career. Thus Arithmetic was sidelined as a practical case of a more general geometric construction, with a restricted set of operators. However mathematicians clung on to their familiar arithmoi with a religious fervour insisting that all True mathematikos should make them supreme, including their field properties.

Over time algebraic analysis and categorisation has lead to a downgrading of the integroi but has maintained a commensurate high regard for the field properties. However i think we have to find a geoemetrical definition of the field properties or behaviours if they are to survive as algebraic analytical fundamentals.

Thus Bombelli using a set square vector with neusis has in a practical and applicable form all the elements so far alluded to in this story": the integral scales on both perpendicular/ orthogonal sides of the set square the incommensurable third side, the trigonometry, and the neusis, and the √-1.

From this geometrical start, polynomials of all sorts flow, and their inherent nature is that of a vector algebra of geometry. Polynomials are vectors and an algebra of vectors and a training in vector math that was of course not understood in this way, because "number" a translation of arithmoi was in many peoples head, along with the pythagorean and archimedian doctrine.

It was not until Descartes that the doctrine or matheis substantially changed, but the √-1 he could not conceive of. He did not have the "intelligence" of Bombelli, who in his travels may have gleaned that the indians had done work on this, and trusted to his instinct or intuition or intrguing discoveries and meditations. In any case Descartes had little time for it and derided it elegantly. His name for them stuck and in fact inspired an interest in them that lead Euler to find his remarkable equation and gauss to prove the fundamental theorem of polynomials, which i now regard as a fundamental theorem of vector algebra.

You have all heard of [tex]E= m*c^2[/tex]

And of course many here know [tex]e^{i*theta}=cos theta + i*sin theta[/tex]

Which Euler arrived at because of his love of power series polynomials, which we would call infinite vectors today. Just as Einstein showed through polynomial transformations that the "rest" motion of a body is not zero but a scalar of value m*c*c

Euler showed that the infinite vector [tex]e^{i*theta}[/tex] has a form directly related to the trig functions on the unit circle, which describe a continuous series of right angled triangle measures with hypotenuse of unity. These right angled triangle measures relate directly by symmetry to Bombelli's set square and is the reason why i define the set squares as vector models

Formally and fortunately convergent Eulers equation is the definition of unity for all planar vectors. it therefore must be a basis for all planar vectors.

I will have to check but i think this is what Hamilton established in his seminal work on couples.

Hamilton therefore looked for a generalisation to 3d and at the same time a change in the doctrine of numbers up until then, for he could see that the doctrine of imaginaries was a better or more useful mathesis for algebraic geometry than those of former times. He had a hard time but couples at least maintaned the field properties.

He gave up after ten years of looking for a solution in 3 variables, and looked at four variables. By this i mean he attempted to reduce infinite series to an algebraic system using the trig function series (Fourrier analysis) , the x polynomial series without success because he was seeking a scalr value for i*j to make sense in completeness terms. Thus when he eventually abandoned the attempt and moved to 4 variables he had substantially done al the work, and was able to show that an infinite series of quaternions converged to a quaternion.

So using a polynomial vector Hamilton established a general vector math after Cotes, Euler, Graves in the plane and another one After Euler with 4 variables which he took to represent a vector with a scalar time component.

Thus the 3d vector is given by [tex]e^{i*theta+j*psi+k*omega}= cos(M)+L sin(M)[/tex]

"The quantity

[tex]L = (itheta + jpsi + komega)/sqrt (theta^2 +psi^2 + omega^2) satisfies L^2 = -1[/tex] (check it

for yourself). And you can go back to the infinite series to see that

exp Lx = cos x + L sin x, whenever L^2 = -1 and x is real. So let

[tex]M = sqrt (theta^2 + psi^2 + omega^2)[/tex], so [tex]itheta + jpsi + komega = LM.[/tex]

Then

[tex]exp (a + itheta + jpsi+ komega) = exp a *(cos M + L sin M).[/tex]"Hamilton realises that to make sense of these values they have to be applied to arc lengths on the surface of the unit sphere, thus establishing a use for a radian measure.

Originally posted by author:

I have in fact turned up a new hero in the story, a certain Roger Cotes, who prior to Euler took great interest in Napier's logarithms and discovered

[tex]i*theta= ln (costheta +i* sintheta)[/tex]

Which is every bit as wonderful as Euler's because it is one of the first uses of the radian measure that Cotes invented, linking the imaginary surd to the infinite iterative sequence of surds that napier worked with entirely by proportioning, and maybe usisng his napier rods as a geometrical aid. Thus it s doubly satisfying as even though Cotes used quadrature easily and so was at ease with calculating series of finite or infinite length, the result speaks entirely of geometry as fundamental to algebraic understanding.

We often hear the mantra that everthing is connected, everything is one!

As there is no fundamental basis to one it is probably more accurate to say everything is connected through everything being transcendental.

Originally posted by author:

Originally posted by author:

So using a polynomial vector Hamilton established a general vector math after Cotes, Euler, Graves in the plane and another one After Euler with 4 variables which he took to represent a vector with a scalar time component.

Thus the 3d vector is given by [tex]e^{i*theta+j*psi+k*omega}= cos(M)+L sin(M)[/tex]

"The quantity

[tex]L = (itheta + jpsi + komega)/sqrt (theta^2 +psi^2 + omega^2) satisfies L^2 = -1[/tex] (check it

for yourself). And you can go back to the infinite series to see that

exp Lx = cos x + L sin x, whenever L^2 = -1 and x is real. So let

[tex]M = sqrt (theta^2 + psi^2 + omega^2)[/tex], so [tex]itheta + jpsi + komega = LM.[/tex]

Then

[tex]exp (a + itheta + jpsi+ komega) = exp a *(cos M + L sin M).[/tex]"Hamilton realises that to make sense of these values they have to be applied to arc lengths on the surface of the unit sphere, thus establishing a use for a radian measure.

Hamilton immesiately refers to these as a system of equations, thus leading me to note that this 3d vector form is formally a matrix algebra on the 3 x n matrices. I do not know much about Banach Algebras and will look into it, but i suspect a link.

Hamilton spent most of his original presentation showing the constraints and equations necessary to transform from a spherical geometry to a cartesian. In this he reminds me much of the initial apects of tensor theory, which leads me to suspect a link with tensors which would become more obvious the higher the set of ntuples used.

I have to remark that Napiers logarithms along with greek spherical geometry and trigonometry advanced through al Khwarzim is the basis for our notion of algorithm and its incessant iteration. Thus mathematicians of old were well aware of and embraced iteration, but it needed Mandelbrot to point out its geometrical implications outside of the mathematical disciplines. He really simply pointed mathematicians outward instead of inward. Like Bombelli he said: i can.., we can! And he looked at the bogymen of maths and said "you know what, they scale!, And i think you will find them rather beautiful geometrically."

Cartesian coordinates are not a fundamental measure, they are a tool of the cartesian method which invites the use of any other additional tool.

Spherical geometry is the natural measure for the quaternion system, and the measure system i would devise for that is : 1 radial pole for the origin. Then great circles as planar decomposites such that the radius of the great circle has the same magnitude as the spherical pole. Finally 1 radial rod for each great circle. The angle measure betwwen the rods and the pole are radians

Using this as the construction basis i can define the constraint on any appropriate system of spherical measure to suit. This system of measure i will define as what i have envisaged radials to be

an observation

We can rewrite the transformations of the plane using matrix operators, and without quaternions we have to use matrix operators for the transformation of the 3d space.I therefore would expect that moving to generalised coordinates would involve tensors.

Originally posted by author:

I had not suspected, nor even was i taught that logarithms were based on the properies of the trigonometric scalars.

I had just started reading Napier when this point was first adduced by me in puzzlement and then deduced by careful reading of the text, and then boldly stated by Napier!

Therefore to me the mystery is revealed in Napiers invention and method! and what at first seemed strange now seems inevitable and consequential!

By his invention Napier has given a logarithmic basis to all measures of spaciometry,and an alternative decription of the aggregation and disaggregation of bounded quantities/magnitudes. The scalar mesh lso becomes amaen`ble to logarithmic description as do all vectors, matrices and tensors and beyond,

Logarithmic operators must exist and may form an Abelian Group.

I see also the reasoning behind Hamiltons couples as an extension of Napier's explanation of the development of the logarithm of sines!

I may have overstated it but it is a fascinating find to me and explains why the term logarithms is used in distinction to powers in polynomials, a question i have had since a child.

Indeed all things come to those who wait!

" they also serve,

who stand and wait…"Another rather startling insight From Hamilton's work is the notion of vector.

Most simple definitions link magnitude with direction/orientation. Magnitude is an old fashioned word, but mathematicians still use it for a procees of squaring and taking the positive square root of a number or quantity. Quantity is a more recent word but does not have a mathematical algorithm attached. and denotes a magnitude or amount of something. Amount is another word used to describe the same thing.

Mgnitude is actually linked to the word magnify,and then we have size and bigness and even mass which has a physical significance attached. So these terms sit uneasily nestled in the mind with the words matter and space.

I generally use an activity to define these ntions precisely for me. The activity is extension, a proprioceptive action of extending or stretching or reaching with any suitable part of my body.

Extending is what proprioceptivly through kinesthesia gives me the sense of "extension" called in olden times "magnitude". The fisherman's tale is a fable of magnitude that is extension demonstrated by stretching out the arms. Extnsion, therefore ,is the notion of magnitude.

But extension is always directional! so our basic notion is and always has been vectorial!

We multiplex these vectorial sensations to apply to a volume of space. A volume measure then is always spaciometrically rotational built up from multiplexing vectorial senstions.This is our notion of the magnitude of space also called in general magnitude.

The shapes and forms hold for us a memory of these magnitude sensations by projection/perception-recognition. This really is the limit of our sense of magnitude, the rest we progress by imagination, which means we bring the outside into our model of sensation and adopt an observer view proprioceptively. This is possible because the mosel is a vector entity based on active vector measures, This type of model we have begun to call tensors.

So magnitude has always been this tensorial vector matrix or mesh. How do we dimension and parametrise it?

Unity does not exist, so we choose a cultural standard and scale according to the cultural algorithm. Personally i choose a standard in my own body system and scale by actively manipulating, touching and measuring against.

Extension and mensuration then are important dynamic defining activities for my sense of "magnitude", amount, bigness,greatness….

So a complex number is actually a polynomial vector, a measuring action ina combined direction in a "plane". Adjugate means combined by "yoking" together, this is making two separated things work together . The two things are measuring by extending alon ga rule, and rotating around some point to get the measuring in the right direction.

Bombelli was clearly thinking of yoking the √-1 in service with the ordinary numbers, but what he actually did was measure them with a set square during neusis. Bingo! the birth of the model vector!

WEll as i have just explained i think it is the activity of measuring that defines a vector, or rather measuring is a combined vectorial experience in which statement the use of vector means every which way kind of motion!

The every whichway kind of motions "draw out" lines . areas and volumes in space, in short they boundarise spatial regions.

This i think is precisely what Quatternions do. Firstly they make Volume into a vector by shaping it as a triangular pyramid with the apex at the centre of the unit sphere and the face giving direction or orientation on the surface of the unit sphere. So now we have along with "pointing", "facing" as an orientation notion. Deduced ferom that we have the notion of a surface area vector,which make the the other three sides of the pointing pyramid also surface vectors, and the addition of surface vectors being the "missing" face of the closed form they make.

On each surface vector we have the pointing vectors that we are familiar with and vector addition is the missing side of the closed shape they make.

On each pointing vector we have scalar addition. Scalar multiplication scales the pointing vectors, squares the area vectors,and cubes the volume vectors.

Within this vector system there is rotational vector development: so interaction of pointing vectors leads to quantized rotation and magnification or stretching; interaction of area vectors leads to quantized rotation and stretching and twisting and magnification, and finally interaction of volume vectors leads to quantized rotation and tumbling,twisting, bending, and skewing and other spatial deformations as well as magnification. In fact i do not know all the outputs for a volume vector interaction these are just what i can presently imagine.

All this and more i think are measurable by quarenion interactions because they hold all 3 levels of vectors. In my view then octonions should do the same but more smoothly and with more detail.

Octonions would kind of represent what i have been thinking of as relativistic motion of a linked system like a tree: each part is linked but independent of the others or rather partially dependent. The utility of these types of measuring and analysis tools i can only imagine, but hey, let the computer sweat the hard stuff!

Originally posted by author:

[img]http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/RechtwKugeldreieck.svg/356px-RechtwKugeldreieck.svg.png

Spherical trigonometry provides a link to quaternions.As you may read, the application of quaternions to special relativity and topology is fundamental. These topics have taken on the names of their developers but Hamiltons work with his colleagues and supporters has the seminal pole position!

As i said, despite working on them for the rest of his life Hamilton could not find applications fast enough to dominate the academic market place of ideas, but his contention that quaternions are of fundamental importance has proven to be true.

But i wanted to go down a different track when i started to write so Doug Sweetser will have to wait a mo .

from quaternions i have deduced that we have tensors which amount to pointing vectors, then tensors which amount to area vectorsrequiring 2 pointing vectors and an arc radian in the form of an isosceles sector to describe them, and finally what amounts to a volume vector requiring 3 area vectors and a spherical triangle area in the form of an isosceles segment (an orange segment) to describe them.

The shapes have been carefully described because they are important as definitions on the unit sphere. For practical purposes and depending on accuraccy required we can use the chord or the tangent.

So for area vectors it is important that they have a rotation motion in their conception, around some point, just as for pointing vectors. This is also fundamental to a volume vector. The point of rotation for a volume vector has a significance in human vision and art, and that is namely perspective.

The stucy of perspective therefore will give insight into the "workings" of volume vectors and provides insight into their interactions. 3d animation therefore is an engaging way to study quaternions and their interactions and highlights the range of applicability of Hamiltons discovery.

There are other applications i could not think of if i tried!

The Lorentz boost was one thing i had not heard of until recently when Doug wrote the following to me:

ME

> It is satisfying to me that essentially Grassman's ideas were illuminated

> by Hamilton's quaternions, and because of Hamilton's applications

> Grassman's obscure writings became slowly appreciated for doing the same

> thing simpler or more easily. and without the non commutativity.

>

> I can not emphasise how objectionable mathematicians found this property of

> non commutativity. and some work has been done showing that Lewis Carrol was

> so opposed that he incorporated its derision in the Alice stories. The

> success of the Alice stories therefore inveighed against the acceptance of

> non commutativity.

>

> Hamilton clearly had an insight ahead of his time and if it had been

> embraced Einstein may have not been the first to expound on relativity. The

> development of Tensors etc would have taken a different course as would have

> matrices, all of which are shadows of Quaternions.

DOUG

There is a fine reason for Einstein and everyone else to ignore

quaternions. Consider a simple rotation in 3D space. That is easy to

do with quaternions:R => R' = U R U*

where U = (cos (a), I sin (a)), I being a 3-vector.

Minkowski argued that special relativity was just a rotation in

spacetime. The way to write that using quaternions…is missing. Two

guys in 1910 and 1911 figured out how to do this with biquaternions,

tossing in an extra factor of i, but that is cheating. Without a

simple way to do boosts, there is NO reason to use quaternions.

Physicists are justified in ignoring quaternions for this one

technical reason alone.In 1995, an Italian fellow named De Leo figured out how to do this

with infinitesimal rotations. Some college student in Indonesia wrote

me about doing boost, and I recommended the paper. Thing is, I don't

quite get the paper. I expected to see hyperbolic sines and cosines,

but they are not there. In July, I figured it out:R => R' = B R B* + ( (B B R)* – (B* B* R)* )/2

where B = (cosh (a), I sinh (a)), I being a 3-vector.

Not sure why no one else figured this one out, but it is vital on a

technical level. All you need is one strike and people will avoid a

tool.ME

> Scientists, physicists in particular avoided non commutativity by using a

> system that simply ignored those combinations of Hamilton Vectors that

> caused them problems. What you do is show that like Dirac'c Equations the

> odd results have physical meaning. After all Anti matter was ruled out by

> the peers of Dirac at the time.

DOUG

non-commutativity is essential in angular momentum and quantum

mechanics. The put it all in their own way, often with the Pauli

matrices.Good luck in your studies,

DougOriginally posted by author:

[center]I do believe that whatever has been of interest and of great importance in Manipume¨ has come from the study of and spaciometric analysis of motion in space, whether that motion be little or large or in equilibrium.[/center]

There is a taxonomy of plane curves in my mind and an analogous one for curved surfaces.

The plane curve is distinguished by a tool which is a unit circle with a unit tangent. Any curve is measured by this tool and thus classified by whatever modofications are needed to describe it with this tool.

So for example an angle is a curve distinguished from the tangent and lying between it and the circle or intersecting the circle in some chord. Then by degrees i can describe all the polygonal curves until i eventually come to the continuous curves including those that sweep back and intersect the circle like pedal curves etc.. The logarithmic and exponential curves fit nicely in between these dimensions, which if we parametrise will even give a coedinate reference to the curves. Spirals will also fit in nicely

The analogy in 3d a sphere with a tangential plane, allows a classification of the curved surfaces, and of particular interest will be the conic surface and other vorticular surfaces.

I do not know if it can be extended to a classification of volumes, but it would need to include surface distinctions if it can.

Curves that lie entirely within the circle or sphere would form a special group of interest.

Originally posted by author:

What is the fuss about negative numbers and √-1? The greeks based their maths on geometry, and the negative numbers make no difference there.

So negative numbers are a different arithmetic object, that had no geoemetrical meaning until Wallis.Even Descartes coordinates did not deal with negative numbers until Wallis.

Bombellis' secret tool was his set square and his neusis, but he tried many methods including 3d models for measurement.

Geometrically -1 is a square like anyother.

But Bombelli had a way of representing "negative" squares geometrically by placing it under an arm as if in a balance .

Then usisng his set squares he arranges a system which produces a line or square above the line. This i believe is a positive and negative value system that is geometeical. As you can imagine it required fiddling about a bit until it was just right- this is precisely what neusis mrans!

Now as usual in maths you get to a point where your symbols totally confuse you! What the hell does this mesn you say to yourself and you become lost and confused, but however you plod on and somehow get the right answer!

"In modern notation, Cardano's multiplication was (5−√−15)(5+√−15), and applying the rule for brackets this becomes 25−−15=40."

Bombelli simply observed that the rule of signs still applied to the surd bracket.

Now here comes the red herring! Who knows what a surd bracket means? Bombeelli did not neither did Cardano or Tagliatelli, and nither for that matter do we.

What the hell does "radice" mean!? By that i mean, how do you "radice"? Let me simplify the question to get to the point of it–How do you square root?

The "operation" of square rooting, the "algorithm" for finding a root is not taught. We usually skip that and learn them by rote or use tables.

There are geometrical and numerical operations to approximate to a square root. Theodorus spiral is a brilliant example! It exemplifies that geometrically Square rooting is rotating a vector around the circle in a certain way!

Numerical equivalent is to iterate between two values successively "adjusting " the starting point of the calculation and "measuring" how close the "operation" brings you to the desired result.

Whichever way it is looked at the "process" of radice is a fiddling about operation well described by neusis.

The red herring is : it is not -1 that is the "imaginary" number generator it is the surd operation itself. In short the issue has never really been about number, but about "algorismo" how to proceed in calculating. What Bombelli observed is that it does not matter how you calculate the surd the result still obeys the sign rules if it comes back signed.

Negative numbers were and are the issues in general and square rooting them just highlighted how much western mathematicians with their greek traditions hated them a snot being "arithmoi" that is geometrical objects used as magnitudes for arithmetic.

Bombelli showed that geometrically we can distinguish them by drawing them either side of a line. Wallis later refined this into the number line concept and applied it to an extension of cartesian coordinates to graph the same thing.

How do we square root, cube root nth root in general? we iterate a neusis.

It is the "operation" of neusis that makes the square root of -1 into an operator that rotates the plane! Bombelli did nor have a cartesian plane he had a set square instead, and he rotated it. This has come down to us as rotating the plane, and gone on to develop the notion of relativity through other affine transforms.

Theodorus did the same thing earlier, but o course his ideas were buried by Pythagoras or by his adherents. Archimedes was brave enough to ressurect some of them, and perhaps the most useful of them, and that is why Archimedes is such a master of engineering: he recognised the importance and value of neusis to engineers, despite what the academics were pontificating.

Without neusis it is hard to think how one might deal with many geometrical issues and construction issues. Neusis is after all the greek way of saying "trial and error" through iteration till the error becomes indistinguishable.

This is the fundamental notion of Empiriscism and the methodology of science, and the aim of reductionism;with the meta purpose being to synthesise all the perfect parts into the perfect whole.

Originally posted by author:

Euclids proposition fourteen I believe is Bombelli's basic neusis as well as his basic "radice" or square rooting algorithm.

I can see Bombelli's set square and i can see the semi circular rotation required in the construction. Thus a rotation by π is needed but the value is read off at π/2 at the specific corner of the rectangle.

This site more than adequately illustrates my meaning of neusis, but more importantly it demonstrates why Geometry was so loved by the Greeks: motion and transformation through motion of specific elements!

When i learned geometry we had a text book, a set square, protractor, and a pair of compass with a marked ruler. We did not have a java applet or even a film to suggest moving the elements of the figures. Those of us who were good at geometry "spotted" the required basic elements of a proof, and naturally moved the book or the diagram around in space or in our imaginations, others who struggled did not move the elements or even the book!

Nobody ever said this is what you were upposed to do, and in fact are rquired to do in some "proofs". Nobody mentioned "neusis"At primary level or even at secondary level, Nobody mentioned in my day that ruler and compass was in fact frowned on by the "greeks", who desired a "pure reason" for their demonstrations, not an approximation.

Even the Greek academicans were up themselves! Archimedes was an engineer and whatever worked he used! So was Bombelli.

In the development of their abstraction there are many false assumptions that can be pointed out in the greeks "fashionable" derivation of basic theorems, but in this case Neusis is my concern. It was absolutely relied upon until it became so commomplace that it was not referred to, and then it was forgotten and then repudiated as infereior. The fact is that parallel lines are the formalisation of nuesis, along with the circular rotation. Because for everything in Euclids elements one could "replace" neusis with these formal constructions, i think that is probably why it fell out of favour.

However as the master of spirals observed , not every curve is a circle! Archimedes in fact demonstrated the necessity of retaining neusis in geometry by trisecting the angle easily and elegantly, and then he went on to show how a spiral neusis path could do the same thing.

Sometimes engineers and pragmatists have to rescue mathematics from stuffy academicians, you know. Sometimes you have to give it back to the people to play with! A sentiment that resonates with Bombelli, Duerer and other great communicators and innovators.

It is of inerest to me that in this construction of the square root a type of spiral curve is evident, linking back elegantly to Theodorus. The points of intersection G and E can be used as generators of an Archimedian style spiral.

Originally posted by author:

Note to myself:

Pythagoreans constructed the arithmoi from the proto arithmoi. The geometry of space suggested to them that a unity existed, because they could construct, so it seemed any rectilinear figure and reduce it by neusis to a rectangle an from there to a square.

Transformations of this sort they believed happened all the time and that was the explanation of the construction of the cosmos.

The special transformations were parallelogram transformations and triangle transformations between parallel lines,plus transformations by rotation in a circle. These were fundamental because the area magnitude was not changed, meaning matter was not created or lost and therefore there must be a fundamental unit that is unchangeable. They called it the "atom" and it was a unity that scaled up to every other form of matter and it was constantly moving through space, and changing thereby through interactions with other portions of matter.

The geometry with its portions and proportions was an exact model of these atoms it was thought and the proportioning revealed the ratio of the amounts of atoms, the fundamental unity.

By constructing various rectilinear shapes they found they could transform all to rectangular or square forms and one could explain how unity worked to build portions. They found the proto arithmoi as objects that could not be transformed beyond a simple rectangle. They therefore were proto forms that could be developed into other forms, but were irreducible to other forms.

Like one or unity, the unit square they could be scaled to other forms and so with unity formed the prime elements of the cosmos. Finding them became important to fully describe the geometry of the universe.

It soon became apparent that they scaled, but not independently! They crossed each scaled arrangment in a mesh form

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The structure was used to organise the proportions into arithmoi or scalars, which later mathematicians named as integers and fractions and used them as a unified measuring scale. Geometrical figures or arithmoi had become numbers, and numbers went on to lose their scalar significance, until the development of vectors.

Therefore there exists a fundamental group of geometrical figures for every abstract ring, group, field,equivalence class, or number set.

Descartes demonstrated that the basic operations of number had geometric counterparts, not realising that for the west arithmoi are the root source of the number concept and operations, that is greek geometrical thinking produced the scalars and their operations and Eudoxus organised their application in proportioning arguments , based on the neusis and transformations of geometrical forms.

Gradually over time the pythagorean ideal was eroded to allow for the real and hypereal nymbers, but the fact that they were still scalars was being lost at the same time as vectors were being found as fundamental elements of description.

There is no fundamental basis for unity, only a fundamental scalar role, and the mysterious atom has been fractured enough times to support the notion that infinitesimally small divisions may be possible forever.

Using the Planck constants as unity only re emphasise the scalar nature of measurement systems and the vector basis of notFS,that is the motion field.

Now magnitude is a visual and kinaesthetic sensation, allied to direction and called a vector. However we have other sensations which are allied to direction and should also be understood as vectors" taste, smell and hearing. As these are more directly linked to the brain through specific mesh sensors located regionally in the head and magnitude does not seem appropriate measure, but Intensity does.

Therefore a vector is an entity that has intensity and direction and rotation,or magnitude and direction and rotation.

Originally posted by author:

So the research comes full circle.

Bombelli's vector model when generalised to 3d produces a vector shape like Hamilton's Quaternions.

So in fashioning a measure i took design cues from a sextant, a theodolite, and a sundial. I could not get the measure to measure within the body of the unit sphere, so i went for a half sphere. Looking at the yin yang symbol i realised i could fix the measure orthogonal/tangential to a sphere shaped round a yin yang symbol. Then i realised that Bombelli had stumbled onto an n dimensional space!

Firstly because he was using a set square he was utilising the fundamental metric of space and carrying with it trigonometrical relations. Therefore he had everything he needed of the cartesian coordinate system without needing the coordinates. Thus the model vector in its entirety is needed to visualise a vector in 2 space.

In 3 space the model vector is able to rotate freely about a point on one of its corners. The resolution of the model vector is always the hypotenuse.

Now for generalised coordinate vectors the set square is not square, but a general triangle with resolution dependent on the sides of the triangle and the angles between them.

Concentrating on the right triangle it suddenly appears that for each resolution i can choose an orthogonal vector to form with it a resolution2 . Then repeat the process with resolution2 to produce resolution3 and so on. This set of orthogonal vectors are a tangential set of vectors to the curve traced by the resolution vectors. Thus the resolution vectors , the curve and the tangential vectors form an n-dimensional space, in which the tangent n is orthogonal to the space of resolutions n-1 .

Now this system of tangents and resolutions is similar to Theodorus's spiral, and therfore a n dimensional vector space has a tangential envelope which is a spiral form.

By choosing the curves appropriately i could combine n dimension vector spaces to delineate a surface, but i will need to understand better about bundling, nearness of curves, any cross relationships and system constraints or conditions such as Lagrangian or Laplacians. I would need to study Hilbert spaces to see if this has already been done, but nevertheless i have a correspondence between vortices and spirals and n dimensional orthogonal vector spaces.

My spider friends have been building n dimensional vector spaces for a while it seems and the cone spider shows that they can be used to cover surfaces and volumes.

Why mathematics should be a sub branch of computier science.

I think so because maths is about space and motion in space and relative equilibrium on space,and we no longer have to specify that the objects we play with stay still or follow simple curved paths. In fact we can understand dynamic systems better by watching animations of first analyses. What is always forgotten is that the mathematical geniuses of the past and even today were often prodigious calculators,and could quickly compute variations to formulae or propositions to get numerical confirmation of suspicions. Euler for example loved infinite series!

For those of us not so gifted a calculator or better still a computer puts the same intuitive power in our hands. Therefore Runiter, spacetime, mathmatica, mathmaxima are essential ools worth learning.

If one can learn to programme that is good, but the applications most relevant require a concerted effort from many to be recognised as useful.

Originally posted by author:

Maths online giving a historical perspective.We see the growing diffusion of greek scientific methods and ideology, and a move away from arabic rhetoric to symbolic notation.

I find also the theory and definition of "number" arithmoi clearly understood as geometrical objects and being able to measure and scale and be compounds of units.

The essential and mysterious notion of unity is expressed here as existing, existence, being,occupying space and moving.An account of greek Geometry which makes the mistake of separating mathematics out and so failing to perceive that the whole was the greek science.

We find a link back to an older science of the Sumerian, Egyptians and Dravidians.

" Around the year 390, Plato visited Sicily, where he came under the influence of Archytas of Tarentum, a follower of the Pythagoreans. Archytas studied, among other mathematical topics, the theory of those means that are associated with Greek mathematics: the arithmetic, geometric and harmonic means. Plato returned to Athens in 388, and in the next twenty years, his Academy came into existence. The purpose of the Academy was to train young people in the sciences (mathematics, music and astronomy) before they undertook careers as legislators and administrators. The two main interests of the Academy were mathematics and dialectic (the Socratic examination of the assumptions made in reasoning). While Plato regarded the study of mathematics as preparatory to the study of dialectic, he nonetheless believed that the study of arithmetic and plane geometry, as well as the geometry of solids, must form the basis of an education leading to knowledge, as opposed to opinion. Plato’s teaching at the Academy was assisted by Theaetetus, whom we have mentioned above. Eudoxus of Cnidus, a pupil of Archytas and an important contributor to the emerging Greek theory of magnitude and number, also taught from time to time at the Academy. Plato’s role in the teaching at the Academy was probably that of an organizer and systematizer, and he may have left the specialist teaching to others. The Academy may be seen as a place where selected sciences were taught and their foundations examined as a mental discipline, the goal being practical wisdom and legislative skill. Clearly, this has relevance to the nature of university learning nowadays, especially as it relates to the conflict between a liberal education, as espoused by Plato, and vocational education with some special aim or skill in mind.

Plato’s enthusiasm for mathematics is described by Eudemus, writing some time after the death of Plato:

• Plato . . .caused the other branches of knowledge to make a very great advance through his earnest zeal about them, and especially geometry: it is very remarkable how he crams his essays throughout with mathematical terms and illustrations, and everywhere tries to arouse an admiration for them in those who embrace the study

of philosophy.

Aristotle (384-322 BCE), the famous philosopher and logician, came to Athens in 367 and became a member of Plato’s Academy. He remained there for twenty years, until Plato’s death in 347. As we noted above, in Plato’s time, dialectic was of primary importance at the Academy, with mathematics an important prerequisite. Aristotle held that the mathematical method then being developed was to be a model for any properly organized science. Greek mathematics at the time was distinguished by its axiomatic method, and sequence of reasoning, from which irrefutable theorems are derived. Aristotle required that any science should proceed as mathematics does, and the mathematical method should be applied to all sciences.

Aristotle is important for laying down the working method for each demonstrative science. Writing in his Posterior Analytics, he says:

• By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first terms and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight line is, or what a triangle is must be assumed, but the rest must be proved. Now of the premises used in demonstrative sciences some are peculiar to each science and others are common to all . . .Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is meant by ‘odd’ or ‘even’, ‘a square’ or ‘a cube’, and geometry to the question what is meant by ‘the irrational’ or ‘deflection’ or the so-called ‘verging’ to a point.

Aristotle notes that every demonstrative science must proceed from indemonstrable principles; otherwise, the steps of demonstration would be endless. This is especially apparent in mathematics. He discusses the nature of what is an axiom, a definition, a postulate and a hypothesis. It is quite difficult to distinguish between a postulate and a hypothesis. All these terms play a leading role in Euclid’s Elements.

Aristotle’s influence on later European thought was immense. For many centuries, "

Although the word mathematics is used here liberally, it means science not mathematics. This science as Plato enthused was an analytical, dialectical, deductive mathesis, that is, doctrine of praxis. Of course it was applied to space in motion, and motion in space. It was truely a spaciometry, not just a geometry.

Originally posted by author:

The most detailed account of foundational indian science i have read so far.

The information highlights the rhetoric of early thinkers, which put another way is using words to stand for things, relations, operations and aspects and not the customary referent. The other way of thinking of rhetoric is analogous thinking, or metaphor. In this sense there is a connection with Chinese I ching and Nine Chapter formulations .

It is clear that Indian science is different to greek science, preferring to use all aspects and attributes of form in their rhetoric. and enjoying the relationships and poetry of forms in their exposition or exegesis. Thus the rhythm and metre, the arrangement and juxtaposition of rhetoric conveys a major part of the sense of the "advice". Like a song the advice relates analogous things regardless of context, and so meaning of the "advice" can be found in all sorts of contexts. This is the true heart of Algebra, a generality of applicability, an essence of relation.

The rhythmical and metrical nature of indian science refers directly to the process of iteration. Of all the fundamental things we know of being human, and existing in a world outside of human subjectivity, iteration has got to be the one fundamental common action.

While mathematics has lately distinguished itself by complex notation, this has been a move by people who tired of the constant repetition. But the indians enjoyed the rhythm of the repetition and so carried the relationships in hymns and and songs and poems.. This is markedly different to the greek discoursive dialectical style, and is in many ways the source of the clash between greek and indian science.

Originally posted by author:

Technical treatment of number. Dealing with this metaphysical analysis . Again it is a science not a mathematics that the greeks were exploring, a science of space and motion. Arithmoi

In short arithmoi are scalars of a unity. You may choose the unity, and all its properties are scaled. However each unity consists of space which is part of unity and not a scalar. This part can be proportioned but not represented by arithmoi scalars .

Thus arithmoi are an attribute to space which i as a animate attribute and having attributed proceed to count, scale and manipulate. However space itself within these confines is also being manipulated. Arithmoi enable me to scale space,so when the arithmoi become a mouthful i can rescale and calculate at that new scale. Thus the greeks had self similaity built into their scalars.

Their analysis of the space within a unit attributed to it divisibility and aggregation,but no mensuration. However by changing scale for unity mensuration could be effected, but the same condition applied to the space within the unity. Therefore not only did they appreciate self similarity they had the apriori of iteration. Thus the greek idea of space was necessarily nested abd fractal.

The Platonic notion of space was that it was reducible to two fundamentals: unity and extensible/elastic magnitude. Pythagoras had thought that there was a fundamental unity which by definition was fixed and merely scalable upwards and thus was the measure of all things.

Originally posted by author:

A nice treatment of indian sensibilities in their science

Brahmagupta the man who invented negative numbers.

"I……n the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multipliedby zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

The product or quotient of a fortune and a debt is a debt……."

Brahmagupta wroote a hymn to his goddess in which the void was no longer excluded, but reverenced as the opening or unfolding of the universe. His attribution of debt or fortune to numbers is a worshipful insight into the hidden world of his goddess. This kind of attribution was not strange or uncommon in indian sensibilities, as they understood unity to be in and of itself, the qualities and attributes that the space utilised to standardise a measure has. Everything attributed to a unity was a referent to that unity and is scalable with that unity.

So in his consideration of the Astronomy of his time and the origin of the universe he considered the unfolding from the void as the ultimate source of everything,but true to indian philosophy of astrology it unfolded good fortune and bad fortune first in equal measure. This is the meaning of fortune and debt, and why negative numbers are so hated!

Originally posted by author:

Out of respect for Brahmagupta and his goddess it is probably best to explain what he was about.

BG defined unity in the usual customary way in india, replete with all its attributes and potentials. Any advice he would give would have to have applicability in many analogous situations and so it was usual not to constrain a definition unecessarily. The greeks however did not obseve such niceties,attempting instead to grasp the robust core of the essential nature of space. they left a lot of things out.

Having defined unity, in which no measurement could be made because it itself was the thing, the space by which measurement was made, BG observed that unity could be taken from itself to leave the void. No one had done this before because it is understood that unity and all measurement are relative to each other in scalar ratio forms, and all things spatial exist in a form.

That form no matter how small is a unity, thus to have no unity implies that it does not exist. The idea that all is nothingness is neither a greek or indian or ancient idea. In fact although i once thought it was a medieval christian idea it is not a serious idea in any culture, except unphilosophical scientists! They of course blame Newton,but that does not hold up as Newton was a Descartesian.

So for BG unity "offing" itself was not "annihilation" introducing nothingness, but returning to the infinite void from which all things came, and in which is all potentiality in indistinguishable form and activity. The void and therefore 0 was a dynamic state of superpositional potential, anything could happen , and anything could explode out of the void, and frequently did!

So having made what seemed an innocuous observation that unity "subtract" itself returns to the void that is 0 BG advises how to use this entity with regard to our custumary manipulations of unity. The customary manipulations of Arithmoi became called arithmetic, and arithmoi strictly were the integers/scalars above unity. If our unity was a calculus(stone) then our customary manipulations would be called calculation. in any case ar(ea) were the unity of the greeks and the ar(ithmoi) were geometrical shapes and volumes based on those and arithmetic is the manipulation of the ars(areas,volumes)

So what was this new observation to be called? BG may have given it a name, but it like unity was a separate idea to the scalars.

Today we have a candidate name called nullity and infinity, and some axioms based on it. BG saw nullity as a relationship with unity which defined the meaning of our measure and our operators. When doing everyday calculations our operators have one meaning, but when dealing with nullity and unity our operators take on a subtly different meaning.

To start with how can one take unity from itself? you can make a relative motion of unity to another place but unity still exists, just in another place: by this BG advises on relativity.

The only way was if unity was returned to the void by an equal but opposite unity which exists for that purpose: by this BG advises on quantum phenomena.

BG as an example decomposes the void by an astrological measure: fortune and poverty. AS i say this is an example of its applicability that BG advances. He expects fully that others would draw out the other applications, and there are probably others he had in mind, but by aligning his thinking with the astrologers he fulfilled his other hat as a keeper of the astrological knowledge of indian astronomy.

"…..Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-

A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The product of zero multiplied by a debt or fortune is zero.

The product of zero multipliedby zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.