Re: Fractal Foundations of mathematics: Axioms notions and the set FS as a model

For me Newton, De Moivre and Cotes are the most significant triple in modern mathematical science.

N(x,y,z) = xñ + yč + z$

Where n is the unity in honour of Newton
Č is the Cotes root of unity and $ is ths De Moivre root of unity
Č$ = -ñ, č*-ñ= -č and $*-ñ =-$, č²=$ , $²=-č

N²(x,y,z)= x²ñ - z²č +y²$ + 2*( -yzñ + xyč+ xz$)

It is clear that these are not quaternions, but they demonstrate that quaternions are based on the roots of unity modulo 8, these of course being based on roots of unity modulo 6.

However I awake this morning and realise the quaternion 8 group is not commutative. In Normans introduction to algebraic topology he overviews commutative groups, and implies the direct sum of 2 cyclic groups is commutative under addition. This means that within a cyclic group the logarithmic addition commutes with the underlying factorisation in tha case of the roots of unity, but, this is not the case in a direct sum of cyclic groups based on the roots of unity. The logarithmic commutativity does not cross the divide between the two groups. Thus when we identify isomorphic and homeomorphic groups, the distinction jumps out. The direct sum of cyclic 2 and cyclic 4 roots is not isomorphic to the cyclic 8 group of the roots of unity. It may be homeomorphic.
The implication is that we have several cyclic 8 groups which can be bases for the quaternion space, and we may have to explore them all to see which models our reality the best. Then again, some believe the encompassing Clifford algebras are a better model. I do not know enough yet about them to possibly comment, but I suspect finite or even infinite algebras which are improperly distinguished by such small details such as factorisation structure and logarithmic structure are going to be equally misleading.

A model is a model in the end, and we need to enjoy our rich life experiences unencumbered by any ideological constraint, where we can. The scientist/ empiricist cannot entirely exclude the possibility of gods, nether can the believer entirely exclude the possibility of resident systems of motion acting automatically. That both sides frequently do just that is the litmus test of dogmatic belief systems which are homeomorphic to the same degree! There is always a middle way.

The division operation in triples is involved, but my attempt goes as follows:

Originally posted by author:

For me Newton, De Moivre and Cotes are the most significant triple in modern mathematical science.

N(x,y,z) = xñ + yč + z$

Where n is the unity in honour of Newton
Č is the Cotes root of unity and $ is the De Moivre root of unity
Č$ = -n, č*-n= -č and $*-n =-$, č²=$ , $²=-č

N²(x,y,z)= x²ñ - z²č +y²$ + 2*( -yzñ + xyč+ xz$)

It is clear that these are not quaternions, but they demonstrate that quaternions are based on the roots of unity modulo 8, these  being based on roots of unity modulo 6.
As my computer is down at the moment I wonder if anyone would be so kind as to code this up and render a mandelbulb for me. Just replace ñ,ç,$ by 1,i,j using the given transform.
The formula for higher powers I will discuss later.
For the theoretical background and a bit of historical blather see my blog

Modulo 6 for the roots of unity mean the circle is divided into 6 sectors. Thus i can enscribe in one sector an equilateral triangle and that enables me to read off the root of unity.

Actually i have a desktop image that gives me them anyway.
č=(1/2,√3/2)=1/2+i√3/2
conj(č)=(1/2,-√3/2)=1/2-i√3/2=-$

The division in analogy with the dual complex notation is going to be based on the factorisation of x³+y³+z³
The corresponding factorisation is going to be more involved, more polynomial.
let r=(y³+z³)^1/3

x³+y³+z³=(x+r)(x–čr)(x–conj(č)r)

Compare this pattern with the pattern for x²+y²

x²+y²=(x+iy)(x+conj(i)y)

Thus we can expect the same pattern when we factorise r

r=(y³+z³)^1/3={(y+z)(y–čz)(y–conj(č)z)}^1/3

finally, in terms of the axes labels or vectors or even Strecken
r={(y+ñz)(y–čz)(y+$z)}^1/3


x³+y³+z³=(x+ñr)(x–čr)(x+$r)

All coefficients are labeled, but i have chosen not to label the leading coefficient in each bracket with ñ to provide a distinctive mnemonic

Applying these factors appropriately should estabish the denominator  of a division as a real value, i just need to check what state it leaves the numerator in.

Q1 is a unit quaternion. I am going to write <q1> as the associated quaternion unit VECTOR.
Q1 is written in the form exp{(1+L*M)/||q1||} the magnitude of the unit quaternion being of course 1 and that is the magnitude of the quaternion Vector <q1>. The direction of the quaternion is then given in terms of the radians in the De Moivre Cotes Euler formula.

Now the coefficients of the quaternions can be expressed in terms of differentials when considering attraction and repulsion using the following analys. In passing, it is to be observed that time is a ratio of speeds and is therefore classically dimensionless. but in fact time is a differential of motion in space. It cannot be a differential of position in space, because we assume in our reference frames for space, that every position exista instantaneously. This is a reasonable assumption, but in practice we know that knowledge of this takes time to be experienced. Currently we place this delay on the speed of light! It is the speed of light that formally encodes time in motion in space. However we could also encode time in space by using the speed of sound, or some other constant motion/wave phenomenon.

The acceleration due to "gravity" is a different way to encode the notion of time in the motion in space. These constants(assumed or actual) are the basis of any notion of time and make time as a measurement a ratio o motion in compared directions or compared amounts of space covered by motion. This comparison means that if we assume constancy in motion we have a constancy in time, acceleration means that in fact our time is constant only exponentially or logarithmically.

In essence this was what Cotes was discussing with Newton before he died. Newton was too feeble in mind to pursue i by then, but he had hoped Cotes would have pursued it to explain the source of gravity. In a very real sense Cotes had discovered the Quaternion vectors and was about to explore how they unified all measurements and explained gravity. It would have meant that e would have discovered logarithmic time distortion or rather the warping of space in time. Thus he would have confounded his own sensibilities by explaining gravity in terms of a bending of space and time, something he would have found too incredible to believe.

We believe it today, and i am not sure we know why, but we certainly should not assume that space and time are independent of each other in any analysis of possible alternatives. There is no way to differnetiate time from space because they are analogous notions of the same experience: motion.

R1 is defined as ae+bi+cj+dk where a,b,c,d are coefficients from a field R or Q, and e,i,j,k are labels of fixed directions in £d space, with i,j,k being orthogonal to each other and e being a free diection which acts as the axis of rotation. The whole combination is restrained by algebraic rules that do not allow the labels to be dependent on each other individually, and thus these rules make the labels a basis for space in the now established vector/tensor sense. Essentially the rules of the game is everything has to be expressed in terms of these 4 labels in this combination, and this sequence order.

You may justifiably question my notation about the Fields R and Q, because they represent proportion fields rather than Number fields. For convenience i lable the equivalence classes of proportions with the scalar numerals, which is in line with Hamilton's construction of these fields. In this case cardinal numerals are just labels, the ordinal notation to which they are attached form a one to one mapping, through which comparisons of size and order ar passed to the cardinal labels. So in contrast the direction labels are not ordered in this way, they are a one to one mapping to the quaternions and their order etc, and directionality are passed to them throufh this mapping, hence the translucent labeling notation <q1>

||R1||² is defined as a²+b²+c²+d²
and the unit quaternion is defined in general as

<q1>= <R1/||R1||>
Hamilton provides some tools for accessing the terms of a quaternion but i wont invoke those here.

Let a_o= ae+0i+0j+0k the axis quaternion , then

<a₀>=||a₀||* < A >

where < A > is the unit quaternion associated with a₀, that is

< A >=a_o/||a_o||

Now let us suppose that v_o= 0e+bi+cj+dk a general quaternion in the space rotating around the axis vector <a_o>
then
,
<v_o>=||v_o||*< S >

where < S > is the unit quaternion associated with v₀, that is

< S >=v_o/||v_o||

Because these are quaternions < S > labels a quaternion which when squared produces its π radian opposite.
i can now decompose <R1>
<R1> =||a₀||*< A > + ||v₀||*< S >
and i can write these in the Exponential form

spherical<R1>= exp(a)*exp(0)*exp(0)*exp(< S >*||||v₀||)

which is a transform from the linear combination in cartesian coeeficients to a linear combination in spherical or radial coefficients.

Now i can determine the differential coefficients and transform them into a radial force field or a cartesian one. I will naturally derive them in a radial field as this is the fundamental assumption of these phenomenon, nowadays, and the conversion to cartesian is besides complex.

In this form , however the a as opposed to thr ||a|| provides additional properties such as rooting, and also scaling below 1.

Whereas the cartesian form flips the vector from one side to the other, the spherical form actually condenses space when these exponentials are used, It also expands space when a is greater than 1 . Thus the exponential form is clearly the analogue of the differential behaviour of space with regard to force and acceleration etc. The use has to be considered carefully to prepare the mind for the expected results.

The more i study Quaternions the more i understand Grassmann's Comment that Quaternions had nothing to teach him. This was not arrogance, this was understanding that Hamilton had misnamed his find, misdirected attention to four labels that are of themselves meaningless, and obscured the rich infinite fiedl of quaternion vectors, which are at least homeomorphic and more likely isomorphic to Grassmanann's Ausdehunungs groesse. The relation between the quaternions behind the four labels form not only a group structural definition for the quaternion 8 group, but also the Lagrangian constraint on the basis quaternions for the field. The quaternions are Hamilton's vectors, but not Gibbs. The quaternion vectors have the product rules built in by the complex group structure, where Gibbs vectors ignored Grassmann's analytical description and so left out Grassmann's product rules, leaving a gapp for others to fill.

However Clifford was willing to go to the sources and he was thus able to confirm Grassmann's more general but less developed manifold system. His idea of attaching quaternions to a Grassmann manifold was confirmed by Grassmann in "Die Ort der Hamilton'snen Quaternion in der Ausdehnungslehre " , a paper I would kke to read.


F(d)--->∂a--->∂r--->(r₁-r₂)||a₀||*< A > increasing or decreasing depending on whether force is attracting or repelling. Therefore the change in radial distance between two bodies measures the forces of attraction and repulsion, the differentials are indicators of repulsion or attraction.


Now the point about time becomes relevant, time is in the same direction of measurement however one measures, thus ∂x/∂t as a ratio tell you no more than ∂x/∂x without knowing which differentials are being referred to.

I have explained the basic process in forming differentials thus by subtraction of the sequenced data we get the first level of differentials. These differentials are sequenced with respect to distance and thus with respect to time. It makes sense therefore to say these differentials are space time sequenced. What the differentials show is that spacetime is variable! It is variable with respect to motion. And under acceleration space time shrinks, while under deceleration ans repulsion spacetime expands.. Newton was fully aware of relativity, and the consequences of his fluent method. To head off this weirdness at the pass, Newton declared absolute time. It is a very reasonable declaration to his fellows who had an interest in astrology and the development of pendulum clocks. They accepted that their mechanics would always need to be corrected or equated to the divine or celestial mechanics.

Thus every mechanical aspect of the universe was set in absolute time " amber" , fossilised to be passed down to our generation as the ordinance of God. Newton's fluents therefore had only one interpretation with respect to time, despite the dependence between time and distance. Newton consciously fixes, finites, and otherwise distinguishes time from space and motion in space. Before Newton this fundamental distinction was not that clear. Galileo was of the same opinion, but his work was challenged by Descartes who believed extension was the fundamental distinction. That did not just mean distance, it meant every attribute that was inhere in space. It was never believed that time was inhere in space. Time was a god ordained tracking of the motion of planets, and was therefore a spiritual thing not a material one.

Pendulum clocks mixed up motion , time and space, and thus Newton's declaration of absolute time was the right thing to do to sort everything out. He gained a lot of respect for that from all god fearing lovers of science.. His philosophy of quantity however was sufficiently precise and clear for Einstein to realise that these were postulates, not empirical data, axioms not facts. In addition differentials could not be separated in that way.
Differentials are always disposed spatially. The tracking of that spatial motion and disposition is called timing , but it constitutes a mental process not an objective spatial quantity. Hence his notion of space time blends the 2 in a way Newton did not want, because he knew that his method of fluents was not a respecter of quantity. Under its principles all quantities were emergent and expansive or vanishing and contracting, ans he simply did not think time behaved that way. Einstein under the influence of Lorentz felt that it did, but it needed physical data to establish.

So, differentials can actually be added subtracted divided or multiplied, ratioed and proportioned. Each process can tell us something about motion in space. In tis regard Lagranges dictum "there is no time only velocity" is insightful.

Wenn einerseits die rückhaltslose Anerkennung, welche
Preyer den Methoden der Ausdehnungslehre zollte, und
die interessanten Anwendungen, welche er von denselben
machte, eine neue Bürgschaft dafür waren, dass diese
Methoden sich nun endlich doch Bahn brachen, so zei-
tigte dieses Interesse Preyer's noch eine andere schöne
Frucht. Wiederholt hatte er Grassmann aufgefordert,
eine neue Ausgabe seiner „ Ausdehnungslehre von 1844"
zu veranstalten. Das Interesse an derselben in den
Kreisen der gelehrten Welt sei im Wachsen, aber man
kenne das Buch nicht, weil es sehr selten sei. Anfangs
lehnte Grassmann diese Aufforderung ab; noch, meinte
er, sei die Zeit dazu nicht gekommen; aber endlich, im
Sommer 1877, entschloss er sich zu Schritten in dieser
Richtung, welche auch von Erfolg begleitet waren, und
so ist neuerdings dank der Initiative Preyer's die „Aus-
dehnungslehre von 1844" unverändert, nur mit einer
Reihe neuer Anmerkungen und einem Nachtrag über
das Yerhältniss der nicht-euklidischen Geometrie zur Aus-
dehnungslehre wieder auf dem Büchermarkte erschienen.
Diese Auferstehung, welche sein Hauptwerk feierte, war
die letzte grosse Freude seines Lebens, eine Freude,
wie ihm das Schicksal keine schönere bereiten konnte.

In rascher Aufeinanderfolge schrieb er dann noch im
Frühjahr 1877 zwei Arbeiten für die „Mathematischen
Annalen", von denen die erste: „Die Mechanik nach den
Principien der Ausdehnungslehre", noch vor seinem Tode,
die andere: „Der Ort der Hamilton'schen Quaternionen
in der Ausdehnungslehre", kurz nachher erschien. Es
war von je ein Lieblingsgedanke Grassmann's gewesen,
die Mechanik, an deren Hand er die Gesetze der Aus-



* Diese Abhandlung bildet Heft 10 der von Prej'er heraus-
gegebenen „Sammlung physiologischer Abhandlungen" (Jena, Dufft).



12

dehnungslehre gefunden hatte, und deren einfache Be-
griffe mit den Operationsbegriffen derselben in der alier-
nächsten Beziehung stehen, in neuer, einfacher Methode
aufzubauen; aber nicht früher fand er Zeit und Gelegen-
heit dazu. Aus den neuesten Lehrbüchern der Mechanik
ersah er, dass noch heute, nach 37 Jahren, diese Me-
thode ebenso neu und zeitgemäss sei wie damals; daher
unternahm er nun die Arbeit, deren erster Theil we-
nigstens vollendet wurde. In demselben gab er auch
ein Resume seiner früher erwähnten Prüfungsarbeit über
die Theorie der Ebbe und Flut. Ein zweiter Aufsatz
sollte die in dem ersten noch nicht berührten Probleme
der Mechanik durch neue Methoden lösen. In der Ar-
beit über die Quaternionen zeigte er in der Hauptsache,
wie diese Ausdrücke aus einer der 16 Multiplicationen
hervorgehen, die in der oben erwähnten Abhandlung
„Sur les differents genres de multiplication" dargestellt
sind. Beide Arbeiten enthielten wieder eine Fülle neuer
Ideen und interessanter Anregungen. Eine dritte für
das Borchard'sche Journal geschriebene Abhandlung:
„Theorie der Polaren", befindet sich im Druck.

Während Grassmann in dieser Weise noch im letzten
Jahre seines Lebens auf physikalischem und mathema-
tischem Gebiet eine reiche Producta vität entfaltete, mehrte
sich auch die Zahl der Autoritäten, welche theils ihren
Beifall und ihre Bewunderung für seine mathematischen
Leistungen aussprachen, theils durch Benutzung seiner
Methoden ihre Anerkennung bewiesen. Es sind ausser
den oben Genannten in dieser Hinsicht unter andern zu
nennen die Herren Burmester in Dresden , Cläre in New-
haven (Connecticut, Vereinigte Staaten), Cremona in
Pvom, Heye in Strassburg. Auch der Druck der Rig-Yeda-
Uebersetzung ward noch im Sommer 1877 beendet, und
Grassmann hatte die Freude, das stattliche Werk voll-
endet vor sich zu sehen.

Aber während so von allen Seiten Anerkennung und



73

Erfolg zeigten, dass der Stern seines Ruhmes im Auf-
gehen sei, um unwiderstehlich fortschreitend seine Bahn
zu verfolgen, neigten sich seine Tage sichtlich ihrem
Ende zu. Es war zu seinem Leiden noch die Wassersucht
getreten, deren Fortschritt ihn allmählich in immer
engere Banden fesselte. Aber er harrte treu aus in dem
Amte, in welchem ihn ein ungünstiges Schicksal festge-
halten hatte. Als er nicht mehr fähig war, die Treppen
zu steigen, und sich auf die zu ebener Erde gelegenen
Zimmer seiner Amtswohnung beschränken musste, richtete
man ihm das ebenso gelegene physikalische Zimmer des
Gymnasiums als Lehrzimmer ein; dorthin liess er sich
später im Rollstuhl fahren , und gab seine Lehrstunden,
bis die Abnahme seiner Kräfte ihm diese Thätigkeit un-
möglich machte. In Frieden mit seinem Gott entschlief
er sanft in der Mitte der Seinen am frühen Morgen des
26. September 1877. Sein jüngster Sohn Konrad im
Alter von zehn Jahren war ihm nur vier Monate früher
vorausgegangen.

http://etc.usf.edu/lit2go/pdf/passage/1730/history-of-modern-mathematics-007-article-5-quaternions-and-ausdehnungslehre.pdf
https://docs.google.com/viewer?a=v&q=cache:AhO_mPnoyBcJ:robotics.technion.ac.il/publications/grassmann3.pdf+&hl=en&pid=bl&srcid=ADGEESgj70Wx-04EL3wCbKxD1cyXi5yzkoNY63V1zuX4ZwcyDiJmDrW0OLjxrGd3RV676vhx7p82nJ1R_9g8LZ6O9IPavOkMnKc4AaDmpfX80pr-Iakpy0CLKKVSO1FMYo0lkY_Ms6_O&sig=AHIEtbSLs8MIf9Z4jI9GJTeDjTyBb1al-A

http://www.xtec.cat/~rgonzal1/cgga.htm
http://www.ma.utexas.edu/users/villegas/F02/quaternions.txt
https://docs.google.com/viewer?a=v&q=cache:eR1l1mP16j4J:www.unc.edu/~mgood/research/Geometric_Algebra.ppt+&hl=en&pid=bl&srcid=ADGEESgX94snyk0pF4OQ9Jhvlz3ZfWG3-i430rMm8a7bOeTaOKoFwvQCZhqKD5ytmZZub6_guXumD5Rf5HMgQ3OElMOF3N-cEJ6Rgj-C2DF7qmaB73MLMtVD1vxpyWsi4R0dbjMGnAS1&sig=AHIEtbSi0m_IRcoPob-Ivg72UHuR85eG9g

http://www.maths.tcd.ie/pub/ims/bull57/S5701.pdf

https://docs.google.com/viewer?a=v&q=cache:ldPO43arBKMJ:arxiv.org/pdf/0709.2238+&hl=en&pid=bl&srcid=ADGEESjgnDdXWT_uKFaoUaoumjq6FveGgkbF8Oa60GhY6-JmbdVtD8Cosbowg8HzMn2njEJeQitgfJ38e7hUuT36W4z5tsJ5WgbvSAtD2Pq-TBWHJFkV03oA9y7Sezf2JINnb_yE0ljb&sig=AHIEtbQNrIK8vLcR3TdoLmDzLry8mmCDjg
http://home.us.archive.org/stream/mathematicalpap00smitgoog/mathematicalpap00smitgoog_djvu.txt

http://archive.org/search.php?query=grassmann%20%20%20AND%20mediatype%3Atexts

http://m.friendfeed-media.com/c094b9efabadf195187adceede5b78971b36dc64
http://arxiv.org/pdf/gr-qc/0405033
http://www20.us.archive.org/stream/hermanngrassman00unkngoog/hermanngrassman00unkngoog_djvu.txt

The coefficients for contraction and attraction would necessarily be logarithmic while those for expansion and repulsion would necessarily be exponential. As I have constantly pointed out Cotes realised that the logarithmic form of so called imaginsries, but to Newton and Walliss _/-1 was a circular and condensing set of differentials, a perfect model of gravitational attraction, and more besides..

I HAVE TO ADMIT that while deservedly trumpeting the development of the European understanding of the "imaginaries" i have skirted around the British mathematicians of the time in this regard. Bombelli deservedly is the start, Cardano and Taglia highlighting the issue but not resolving it. But it is notrealised that the british mathematicians took the Bombelli start and actually solved the issue!

It starts with Napier, a contemporary of Bombelli and it ends with Cotes.

Napier discovers in the sine tables the inspiration for logarithms. Wallis used his position to build a mathematical and scientific enclave in Britain at Cambridge. He brought across the best information he could get from the Continent to study and teach. He formed the Royal Society and he mentored and nurtured Newton. Barrow brought Back the Rudiments of the classical Curriculum after his travels, laying the ground for studying mathematics in cambridge for centuries.

Wallis obtained a copy of Euclid'd teaching material and translated it into latin, the common academic language, and this he gave to Newton. In the meantime Wallis contributed to the development of the Napierian Logarithms, the algebraic formulation of the conics(possibly his greatest underrated contribution), continued fractions and notions of the number line.

Because of his algebraic conics Wallis made a comment that √-1 would probably be a point on the plane. This was simply because the conics required the use of the sine table values to plot the points, and it is noticeable that -1 occurs in a relationship with +1 and the square root of the radius of 1.

Howeve Wallis did not have any need to take this further, as the mechanics were sufficient to solve his equations.

The challenge of the day became the general solution of all Polynomials of the form a.x²ⁿ+bx+c=0, the conics, and so o course Newton had a view. There was an intense interest in the sines and the geometry or trigonometry of the unit circle. Napiers Logarithms were intensely studied in this regard and the relationship to the circle understood, at least by Newton. De Moivre Studies Newton and found many great insights, including the notion that the imaginary magnitude was related to the sine tables through the logarithms.
Newton was able to get De Moivre into the Royal Society by his exposition of the multinomial equation solutions based on the sines, and cosines. Frankly the Society was amazed and confounded, and let him in in wonderment. It is evident that few in the society had ever read let alone understood Newton's work.
Thus De Moivre Developed a Newtonian insight into the Cotes De Moivre theorem. Newton was not able to get his friend permanent Stipend despite his brilliance, but this never seemed to derail De Moivre from his studies. Cotes on the other hand obtained a position as an astronomer in cambridge and was brought into conversation with Newton through this and a reediting of Newton's Principia. This also brought him into contact and collaboration with De Moivre and together they worked on th Newtonian insight about the connection between the logarithms and the imaginaries.

To Newton the connection was obvious, as it became to Cotes and De Moivre, But i think it is Cotes who made it explicitly clear in the diagram of the unit circle, rather than just in the sine tables

Succinctly put, Newton showed De Moivre that √-1 can be calculated using the sine tables, that is there is an identity for √-1 in the sine tables, but it represents a logarithmic magnitude.

De Moivre found a cognate or factor in some of his multinomial equations cosø+isinø, in various forms. It turned out to be just what Cotes needed to solve the logarithmic spiral, the rhumb line on the surface of a sphere.

http://en.wikipedia.org/wiki/Prosthaphaeresis



With this insight Cotes was able to see that the arc length on the sphere was in a logarithmic relationship to the sine of the laititude. IN FACT he realised that it was the basic idea of prosthapheresis that underlay the logarthm tables! in other words the logarithmic magnitude was in fact the arc length, and in addition the imaginary magnitude was the same, it was an arc length.

Cotes declared this in his Logometria ,and devised the radian to make this easier to apprehend. Unfortunately he died before he could take it much further, and De Moivre had other fish to fry.

Euler may very well have read and been influenced by Cotes Logometria, because he too declares the logarithm of the imaginaries to be the arcs of the circle or sphere some 6 decades later. Bernoulli apparently had only just hit upon the algebraic formulation of the conics that Wallis had devised years before, and he suggested Euler explore this idea.

It os safe to say no one believed Cotes, and even fewer believed Euler. To this day no one accepts that the imaginary magnitudes are in fact arc lengths. Thus they are real magnitudes, and for them Cotes devised the radian. Except Napier had done his work the solution would never have been so simply put.

sin a sin b = ½[cos(a − b) − cos(a + b)]
cos a cos b = ½[cos(a − b) + cos(a + b)]
sin a cos b = ½[sin(a + b) + sin(a − b)]
cos a sin b = ½[sin(a + b) − sin(a − b)]

By using a combination of these formulae Napier was able to see a simplification in the lookup tables that lead to logarithms.

sina*sinb*sinb=-½*sina*cos2b
sina*cosb*cosb=½*sina*cos2b
sina*cosb*sinb=½*sina*sin2b
½*sina*sin2b*cos2b=½*½*sina*sin4b
½*½*sina*sin4b*cos4b=½*½*½*sina*sin8b

The idea is evident in the behaviour of these identites, but requires great familiarity with them to tease out and some insight and motivation. Napier was aware of the multiplicative behaviour of power series 2²*2²=2²⁺²

This was simpler, required only one type of table and had reduced look ups. Unfortunately, their was very little motivation to devise or calculate such tables, and few would even understand what was being calculated. However there was a great need for and understanding of the sine tables and a method of look up calculation that employed them would have an immediate and admiring audience and following.

Napiers remarkable idea was the first he could come up with : find a geometric relationship in the sine tables so that sina²*sina²=sina²⁺²

Now it is clear that by utilising the 2 ideas Napier began to devise his "logarithm" tables. He had to understand what he was actually doing and the wisdom came to him in a dream: The rotating radial. The rotating radial was linked to all the sines, but as the identities above show it involved a little bit of a mixture of all the ratios , plus it involved a little bit more associated with a little bit less of an angle. Bombelli had observed this when he was trying to use neusis to find the geometric means of polynomial equations, and in fact he in blind faith used the idea to define the rules for the √-1 magnitudes.

Because Bombelli was a geometer engineer, using the greek methods of neusis he was familiar with this property of the sine ratios and the cosine ratios, and indeed the identities. The versie and the coversine had been formulated into useful tables for finding square roots.

So where Napier differed is in recognising that by basing the radial on a fixed sine, thus a fixed angle, the identities simplified, as shown above, and he could develop a geometric series as we call it, but what Napier properly calls a ratio between an increasing form and a proportionately decreasing one.

But h went further: The moving radial did not just pick out some of the sines in the sine table, which would have been only partially useful in improving calculation, it picked out all those sines that behaved according to that principle. Thus it was demonstrable that a sine could be picked out by any icreasig form, even forms that increased in extremely small steps.

Arithmoi are and were forms up until Berkely challenged the notion of fluxions and fluens. These were dynamic arihmoi, fluid forms that Newton too saw in dream of dynamic reality. The pragmatists and the german industrialists decided to base magnitude not on form but on its adjective:number, or Zahlen.Dedekind and Cantor no less collaborated in defining number in terms of a cut off an infinite measure, a number line, surprisingly suggested and detailed by Wallis as a useful organising tool for measurement of length. Form was pushed aside for length, and Number as an entity of length was set as the standard.

Thus when Napier thought of his logarithms he could immediateley see real magnitudes changing, and thus had no problem with continuity. Look up tables were always only the approximation reuired to do the job. There was no presentiment of perfection or exactness. Near enough to do the job well would do.

Napiers idea was also aided by the fact that many of the necessary calculations could be done utilising the existing sine tables, and the above identified types of identities.

It tookk him years, but eventually he was able to produce a set of tables based on a fixed sine which involved using the form as the thing looked up by the proportion. That is instead of searching in the sine tables and then finding an angle , the look up ws more straight forward. Whizz down the proportions till you got the figures you needed; nextto them would be a number representing the size of the form associated with that proortion. Finding 2 ,3 or more of these forms and adding them gave a result. You then had to whizz down the form side to locate those figures and next to it you would find the correct sine proportion.

You did not need to find anything else. You merely had to adjust the result to the correct scale.

Napiers idea is firmly based on the greek trigonometry of the circle, and showed that there are many subsets of the sines which form useful tools. It was oly later that it was realised that within the sine ratios there are infinite subsets representing different relations in space, the conic sectional spaces. When Gauss started to look for the so calle "non' Euclidean"geometrie", he did not seem able to acknowledge the fact that it was part of the Eucclidean space, as a subset, and always had been, and in fact spherical geometers had been using it for millenia.

Briggs of course modified Napiers tables , not his work. Once Napier showed how to calculate these useful geometric series it was easy to sell the idea of doing the same for commerce, who typically would want to use "ordinary" numbers. The calculation of the logarithms for the base 10 are entirely Briggs work, but they use the same construction and calculating techniques that napier set out. Briggs in particular used the geometric mean of the √2 as this falls out naturally from the sine tables.

Great labours and great familiarity was developed with this form of calculation, but the construcion was often ignored being replaced by the interpolation formulae, the difference equations. These are easily derived and suggested by the trig identities.

Thus when Cotes went back to basics , he was the first to realise that a set oflogarithms can be constructed using not an arithmoi to compare with the sine, but the curved sector of the very circle in which the sine is located. In addition to this observation he noted that the "imaginary" magnitudes found a natural meaning in this relationship, as ratios of a fixed arc length:π/2

That he was able to relate it to cosx+isinx is due to the differential relationships of the trig ratios, which he explored, but more simply to the tangent relationship to the same sector.

The gematria of the semeia is a fairly modern approach, in that Euclids approached the subject using Epiphaneia.

I have benefited from years of additional thought and application, not the least being developing notions and terminology and etymological drift in word association. So it is natural that i would see the apriori notion of vector in the genatria of the semeion. However i am careful to distinguish the notion as a bicompass bi-vector system in the basic form. This is of course the basis of trigonometric land survey as well as geodesic surveyance. As far as i am concerned the basic vector notion was exploited in this way, and if Hipparchus takes the credit, then it is down to him, but i would of course point out that it is never that cut and dried, and Thales is reputed to have used such a system to calculate distances of "lengths". That may sound "funny" but one has to distinguish the subjective experience from the objective metron, and in a general phraseology to boot!

When i first tackled Monas i noted the kath" in kath'ekastoon, and subsequently derived etymoogically the "individual" notion in ekastoon, the singled out loner from a group. I later went on to derive etymologically the notion of katametria, the comparison by laying down a metron.

oi ontoi is interesting in itself, but the essential thing is that these are existing or real objects, they are objective therefore and not subjective, and interestingly can be "produced" that is manufacured, made sensible to the hand or the senses in general.

Thus to awake and see a connection between kath'ekaston and katametreesai is not surprising. The elasson and the meizonos if anything give a linguistic clue. The kat'ekaston is therefore a laid down singled out object or more directly a metron to be used in katmetreesai .

For individual beings ...... ENUMERATION

The translation is treacherous! but it suffices to highlight the functional link. IN Euclid i do not expect enumeration i expect gematria and as for individual, when placed back into its context the phrase is defining monas as the case when an individual is "laid down" and called "one".

This individual object is a metron and it has form. The form is called arithmos partly because it is graspable "by the hand", and liftable into the air, and partly because it can be joined into a harmonia. From this aspect of the notion we get artois or harmonious fit and perissos which means fits around but not snuggly, just a bit too big!.http://en.wiktionary.org/wiki/ἀριθμός

The real objects that Euclid concentrates on initially, it is a teaching course after all , have a surface, an epiphaneia, This surface can be grasped and manipulated, and consequently it has "edges", perata. However Euclid is concentrating on a particular simbola to represent these edges and so he defines the gramme, introducing the use of drawing in the surface to define exactly what the topic focus is. Having defined gramme he focuses on semeion briefly. He does not define a sumbola for semeion.

Today we are so "hot" on the point, this seems an oversight. But in fact as i have explained at length, there was no need to define a symbol as any symbol for a subjective concept would do and in fact gramme cross producing the required referent. Semeia were not the focus of Euclids topics. Space was and is his focus.

Our first notion of space is therefore in the word "area", which of course is not a needed concept in Euclid, because the arithmoi are real forms. The concepts Euclid develops are meros and pollapleisios, which are derived forms . In deriving these forms Euclid develops the actions of division and multiples followed closely by the notion of sugkeimeia, that is aggregating to cover a form. The notion of subtraction is also inckuded in the development, but it is in the form of perissos , that is close enough, or approximation.

Thus Before any action is the form of space that governs or pertains to the development of that action. Losing sight of this is where the difficulty in mathematics arise. Thus for example , quite naturally, in tesselation with a triangle metron/monas we will find that the form to which the monas is being applied requires the monas to adapt different orientations.

Thus Before any action is the form of space that governs or pertains to the development of that action. Losing sight of this is where the difficulty in mathematics arise.

Think about this point carefully. I start with a large triangukar form. I decide the best way to "measure" that form is to use a smaller, similar triangle as a metron/monas. Why? Maybe i want to cmpare 2 triangular forms in terms of a common measure. This seems to be a common goal for comparison a notion fully utilised by Eudoxus.

Now i make the measurement by tiling. As i tile i notice that some of the triangle tiles have to be placed rotated π radians relative to others. Thus i do not have one metron but 2 which are related by a π radian rotation.

In common everyday use i would ignore this distinction and sum the tiles as one aggregate. This would give me a cipher for the monas, that is a scalar value for the metron. These scalar values would be related to triangle "dots" by the gematria of the Pythagoreans. Thus the Pythaagorean gematria dealt with a geometrical form in dots. It is enough to drive one dotty!(dots,by the way have become a standard symbol of a point, but not a semeion, which is now symbolised by a line vector)

However, when Descartes began to algebraise geometry, he did not include this notion of orientation, but rather the notion of imaginary. It was Bellini, and indeed prior to him Brahmagupta who introduced these spatial orientation relationships. Brahmagupta introduced them in the fullest sense, as balanced notions describing orientations in space signified in astrological and astronomical calculations. His relating of them to Shunya was to balance out the perceived error in the Monad philosophy of the greeks, which was influencing the traditional Brahman view of transformational, dynamic "origin".

Philosophically these differences are apparent only, as the reasoning behind the 2 philosophies is analogical. Cultural differences and insights however do distinguish the 2, and in particular the monadic modulus is more in line with Indian Philosophy of Shunya, that is "Fullness", than greek Philosophy of "Monad" that is "oneness or wholeness". The inherent notion in Shunya is "infinite", something the greeks tended to place "away" from their reasoning, as something which they could not reasonably attain to.

The greeks certainly considered the notion of the infinite, but as a process, Appolonius's famous stick process. but they could not conceive of it as a whole, that is as something essentially mensurable, from which commensurability may be defined.

Brahmagupta, and Indian and Chinese thought, subsequently modified by Buddhist Philosophy, vedic philosophy etc conceived of Brahma as whole, and the void from which the universe sprang as a transforming whole full of infinite potential and dynamic motion. thus we may see that the wholeneess idea unites all philosophies, but the kairos oor proportionality is where they differ. The Greeks defined proportionality through multiple. and essentially so do all philosophies, but Brahmagupta pointed out that muliples come in at least 2 forms, forms that cancel each other out. In fact they return the other to the infinite void, to the shunya of all things.

For Brahmagupta it was pertinent to advise, astrologically, on the fortunate ciphers and the unfortunate ciphers, that is scalars derived from the sifre which was the arabic word for Shunya from which we tended to down play the notion of infinite and play up the notion of nothingness or zero.. The reason is quite nationalistic! the western world did not want any thing to supersede the Greeks!

Bombelli by introducing his "pui" and "meno" began the process of freeing the negative numbers from their astrological gematrial associations, but the negative numbers have never really been liked since their introduction for this very reason: astrologically they mean doom and gloom!

However, Cardano and Tartaglia, introducing a notation for number from the Al Jibre did not understand the spatial significance of the "sign" of a number. Brahmagupta did however, but nobody, apart from possibly Bombelli wanted or could understand what he was advising. In addition rhetoric was the common mode of discourse, and it was Bombelli's innovative use of "signs", that is notation, that enasbled these distinctions to be isolated and pondered. Of course i do not Discount Vieta, nor the whole germanic rationalisation of notation and signage.

The point of importance is that the "sign" of a number has a spatial as well as a gematrial significance, and the spatial significance derives from the same notion from which i derive the notion of a compass vector network, the notion of the semeion.

The presentation of the difficulty as imaginary, or the √-1 has been a red herring for far too long. The fundamental difficulty has been in developing a consistent "vector" notion and notation based on space and gematrial considerations of monads.

We can now do that easily and consistently, and in fact we can see the connection between spherical Trigonometry and the development of a semiotic algebra(gematria) of vectors.

It is always a delight to serendipitously come upon an observation that completes a puzzle picture.

Diophantus it would seem is the first, greek mathematician to utilise a symbol to refer to an "unknown" cipher, and indeed is the most notable early greek in the subject of cipherism/number science or as we call it number theory. This number theory was transported to the Indians who took it to its ultimate sophistication.

However, from the greek and Indian Fathers the arabs constructed a rhetorical Algebra, in which the meaning of an action or a conception was exposed at length. From this style Bombelli is the real tipping point between the arabic rhetoric and the more terse latin/ European notation.

Bombelli not only majored in Algebra but also in a gnomonic geometry . Although he did not take the headlines for the great tussle in solving the cubic and quartic equations, nevertheless he ser out ways to do this with the utmost adroitness and solve up to the quintic and some equations of degree 6. In doing so he introduced at his own devising notation to expedite and simplify and clarify the exposition. His target audience was not the classical scholar and philosopher but the everyday engineer in Italy.

Bombelli therefore had a wider more populist influence thanany have given him credit for, and his translation of the Diophantine papyrus with a colleage not only informed his Algebra but opened up the topic for those that followed, including Pierre de Fermat and Descartes and Napier.

Now Napier is a hugely interesting "curate". For being a religious cleric and a keen observer, he distracted himself with the study of astronomy, science and geometry, number theory, trigonometry etc. He was a widely read man who delved into things with gusto and kept abreast of the latest information. Thus it does not surprise me to find that Napier knew of Bombellis work, and also of Al Kwarzhims and thus of Brahmaguptas. What i cannot say is how much he relied upon them, but he certainly was not ignorant.

Now when Bombelli confessed to using a carpenter square, and thus to a gnomon, the same which i have referred to as a vector, he implied a coordinate system that rotated via neusis. This system again implied was everywhere in astronomy and spherical trigonometry. Hidden within spherical and plane trigonometry is the gnomon, sometimes tantalising references to the ordinate were made, but the full conception was a gnomon. No one had made the connection to an ordered pair of any description.

Thus it is easy to miss this ordered pair in Napiers important work on Logarithms.

At the same time as Bombelli was popularising Algebra in Europe ,Vieta was devising a consistent system of notation. Thus Bombelli was read and admired, but then translated using various notation schemes which gave the glory to the inventors of the notation, not the source. So by some means Napier came to view symbolic algebra as a concise way of mastering the topic, and in doing so imbibed on the importance of gnomonic order in trigonometry and in solving equations.

In one of his investigations he related the arcs of a circle to their gnomons in the style of the Indians, relating the ratios of the limbs of the Gnomon accordingly. In fact at about his time Regiomantus had formally defined the ratios in terms of the gnomon( the half chord) and not the chord as Ptolemy had done.

What Napier then did was to keep the dynamic connection with the movement of the stars through astronomy, thus realising that the ratios were in dynamic relative motion connected by the movement along the arcs. When he laid these relations out in motion sequents he realised that prosthapheresis was evident: that is if he carefully selected the ratios to conform to equal blocks of wood, and thus time he could make a decreasing slope, the beauty of which was that it converted multiplication into addition of these blocks.

It is clearer to us today who are nurtured at the breast on Cartesian Coordinates, but what Napier is somewhat joyously describing is a 3d model of a logarithmic curve derived from the sine ration :the logarithm of the sine!

There was no term to express it and no conception to wrap it neatly all together because no one had realised before Napier that you could "graph" an abstract notion. Napiers sentiment was that it was a dynamic relation between the Ratios (logos) in this case the sine ratios , And the Arithmoi (the articulated forms) in this case wooden blocks of the same breadth and depth, but whose height represented the proportion as calculated, in this case a decreasing height.

If you build yourself such geometrical model you will grasp what Napier was trying to convey. The point is that it was an astoundingly NEW conception, which took the world by storm. In it Napier not only implied graphing using a gnomonic system , but also n orderly recording of the relationships. This order was expounded in the name Logos, Arithmos which means Ratio, block which later Descartes used in his praxis as measure,measure.

It took a while after Vieta for notation to become established and custom and practice to be formalised in standard, agreed definitions, but Napier, Descartes and Wallis are the chief architects of the modern algebraic notation system, and indeed the Cartesian coordinate system with De Fermat. Logos arithmos would now be written(logos,arithmoi) and indeed (antilog,logarithm)

This is the first relation that shows analogy and proportionality do not mean equality but nevertheless solving one part of the analogy can give the analogous solution! This is like magic\1, but today we use the term fractal and analogy. If an analogy holds, that is if we have a homeomorphic relationship, then what we can say and do in one part of the homeomorphism has an analogy in the other part. Napier makes the first steps in function thory also, which is more evident in his work than in Descartes, although Descartes proposes a categorisation that utilises analogy to make its "prophetic" point.


Napier was the first to derive a logarithmic graph/function from the circle. It was soon realised that the sine, cos and tan functions also could be derived from the circle and eventually pi and the exponential functions. Cotes is apparently the first to calculate Eulers constant, because he was using it as a proportional base for his logarithms modelled on Napiers but involving increase not decrease.E does not arise naturally in Napierian logarithms, but rather in compound interest issues. It is simple thing to devise a set of Napierian logarithms based on the compound interest formula to assist in calculation, but clearly there is no calculation benefit. The benefit lies in the differential Calculus that Newton based on it, the calculation of compounded growth. Newton developed his Binomial series from exploring compound interest calculations, and noting that it had the property of differentiating to itself.

Cotes simply calculated what this constant must be to have these properties and found it was less than pi, and opened up the door to complex logarithms.

We have to realise that no one knew what these things were they were discovering, they were just having fun and pleasing their patrons by solving problems. It does not pay to get too serious, because the meaning of you communication is the response in the other party, not what you think. like Hamilton's Quaternions, by taking them seriously Hamilton neglected to consider the more general case.This William Kingdon Clifford did, and ultimately his development of Hamilton's ideas have won the day, vindicating Hamilton in the process of burying his lifes work of latter years!

Ah well! Lets play! For we seem to listen more that way!

One of the things missed in logarithms based on the trig functions is the connection to the arcs of a circle, This is generally missed in the modern treatment using angles. For this reason mathematicians have retained the convention of Cotes in utilising his invention: Radians. Whereas angles are generally situated a the "hook" of an angle radians are situated on the circumference of a circle sector with that "angle/hook" at its centre of rotation. Within that sector a gnomon may be described thus relating arcs, trig functions and logarithmic functions in one go, and as it turns out vectors and tensors too, including contravariant and covariant tensors. the notion of complex numbers i believe are better described as logical relations between vectors in a vector algebra with a group structure and property the most fundamental of which is reflection in a centre of rotation. This curious property has not been sufficiently drawn out, but is the basis of mirror reflection and other curious behaviours exhibited by the complex vector groups. I am happy to call them complex vectors from now on.