If we take into account that we admit a finite ontology, namely the multitude of elements must not include the notions of great “infinity” or small “infinity”, then with an adequate geometry there cannot be mentioned any “straight line” or a “plane”, as they are both ab initio infinite. More than that, phrases like pseudo-notions of “limited straight line”, “limited plane” and, if possible, “limited space” leads to a finitude of the “infinity”.

    In our work An essay about a finite ontology, we have shown that the Universe is finite, and hence there is no “infinite straight line” or “plane”. Thus, the wonderful edifices of D. Hilbert or G. D. Birhoff’s are left without a subject. G. J. Darboux had the same expressions, a bit more hidden, for integrability of some functions and A. L. Vauchy for convergence of rows, as well as many other great mathematicians.

    On the point of view of a finite ontology, a primordial element is the “point”, an adimensional element. A pair of two distinct points, is undetermined and undeterminatory, because through them very many circles and ellipses can be drawn.

    The second primordial element, that cannot be defined, is the “circle”. Three distinct points delimit a circle. I have not used the expression “not situated on the same imaginary line” because there is no such an expression. The largest circle is the one that includes the whole (finite) Universe, and the narrowest circle is the one that includes a v qquant.

    Instead of a circle there can also be arc of a circle or arcs of circles.

    Let us assume that we have in front of us an ordinary coordinates system (rectangular). In this case we shall be able to represent in Mathematics the equation of the circles.

(1)      (x-a)² + (y-b)² = R² or  x² + y² + 2px + 2qy + c = 0, where a= -p and b= -q are coordinates of the centre of the circle, R = the ray of the circle and 

The sphere.

(2)       (x-a)² + (y-b)² +(z-c)² = R²      or     x² + y² + z² + Mx + Ny + Pz + Q

The ellipsis.

(3)            Ax² + xy + 2Bxy + Cy² + 2Dx + 2Ey + F =0                                                       

    (3) is an ellipsis if D·I < O; d < 0

The ellipsoid.

(4)        a₁₁x²  + 2a₁₂xy + a₂₂y² + 2a₁₃xz + 2a₂₃yz + a₃₃z² + 2a₁₄x + 2a₂₄y + 2a₃₄z + a₄₄ = 0 

    The secular equation is the following one: S² + IS² + JS + Δ/δ.                    

   The conditions for this equation to be an ellipsoid are:

   D¹0;  d¹0;   S₁X² + S₂Y² + S₃Z² + Δ/δ = 0, where S₁, S₂ and S₃ are solutions to secondary equation, all of them are positive and Δ/δ < 0.

    What is left now is displaying the axioms. It may happen that some of them be redundant, others to lack something or to have a thing more than it is necessary, but all these shortcomings will not impede on the essence of what I shall next say.

   The point.

   The point, unlike what was told of it by me or by others until now, is something abstract, without contents or dimensions. I shall consider the point as dimensional and substantial entity, as the v type qquantor (written with two qs)x may be. In order to make you have an idea about the small dimensions of a v qquantor, I must reveal you that the whole Universe has 10¹³⁴ v qquantors. But this number is so enormously big – Skewes’s number – , that Hardy discovered in Mathematicsxx.

Axiom 1. Any three distinct points (we did not say linear because the straight line does not exist) delimit one and only circle.

Axiom 2. Any four distinct points delimit one and only sphere.

Axiom 3. All real (existent) points are in a continuous movement.

Axiom 4. We consider any of the points that determine a circle or a different figure as points connected to the circle or to a different figure, because they have given up the movements of the other points.

   The circle.

   First of all we must introduce some new notions. The fact that allows this to us is that the straight line and the plane are not conceivable to us, as they are infinite and infinity is not a number, as analysts would believe. Beside the point, there are the circle, the sphere, the ellipsis and the ellipsoid which are the only finite elements that we shall use.

    Since there are and cannot be infinite elements, elements that are left remain measurable, they need to establish a definite scale of values and a gauge for their dimensions. To those which have already been measured, there will continue to exist for the meter (m), the kilogram (kg), as well as the force (N) – the newton with its multiples and submultiples. Thus, in drawings (Technique) it is recommended that the helping elements for circle, sphere, ellipsis or ellipsoid be used with one or two degree of gauges, without reducing its gauge or of using an analogous comparison.

Definition 1.     Being given two circles c₁ and c₂,

    c₁<c₂  ( c₁ is lesser than c₂),

    c₁£c₂  (c₁ is lesser or at the most equal to c₂),

    c₁>c₂  (c₁ is more than c₂).

    c₁³c₂  (c₁ is more or at least equal to c₂)

    c₁£c₂  Ç c₁>c₂  É c_i = c₂ 

Definition 2.     Being given three points A-B-C we shall be able to define distances as it follows. (We shall measure dimensions according to the proportion 1:100 – that is measuring at the second degree of gauge, where curving reduces).

    V_A,B,C d(A-B-C)>0                            (1:100)

    V_A,B,C d(A-B-C) = d(C-B-A)             (1:100)             and

    V_A,B,C,D,E d(A-C-E) = d(A-B-C) + d(C-D-E)

Axiom 5.     Inside every circle there is a point called the centre of the circle. Distances from the centre to the circle (the periphery of the circle), called radiuses, are equal.

   The opening between two radiuses of a circle is called an angle. The opening between two circle radiuses splits the interior of the circle into two angles. When radiuses are overposed then the angle is said to be of 0º (degrees); when the openings of the radiuses are half the interior, the angle is supposed to be of 180º, and when the opening has gone over the whole interior, the angle is supposed to have 360º. It is in the same way that arcs (parts of the periphery) are noted. To 0, 180 and 360 degrees correspond 0, p and 2p radians.

Definition 3.     V_A,B,O,C,E = m("ABODE) ³ 0 , where O-B-A and O-D-E are radiuses.

Definition 4.     V_{A,B,O,D,E,F,G} = m("ABODE) + m("EDOFG) = m("ABOFG).

Circles and points

    Axiom 6.     Between two circles there the following situations may occur: exterior circles, with a common point, that is tangents (extern or intern), with two points in common, that are secant, they may be interior and concentric, namely parallel.

    Axiom 7.     Every circle has its own individuality, distinct from other circles. Two circles, with two points in common allow each other to take part at their own individuality.

   Three circles may be connected in triangles or connected in chain. For the reason that in the triangle connection there are six points in common and in the chain connection there are only four points, the triangle connection may preferred. It is in the same way that circle agglomerations behave towards (finite) circle chains. Constructions of various circle combinations can be found, for example chain circles alternatively with agglomeration circles.

    Axiom 8.     If upon a sphere or upon a calotte of a sufficiently big ray was drawn a rectangular system of coordinates, then biunivoquely with a square function with f(x,y) = x² + y² + 2Mx + 2Ny + P a circle of coordinates –M and –N and of a radius: .

   The sphere.   As a basic notion, the sphere cannot be defined. Intuitively, the sphere is born when a semicircle rotates (the periphery half of a circle) in a complete rotation 2p around the diameter of the circle (2R). A sphere is determined by four distinct points.

    Axiom 9.     The intersection of two spheres is done on a circle.

    Several spheres may intersect either as a conglomerate, or as a chain.

    Spheres as conglomerates are more numerous than chain ones. With conglomerates they may reach a number of 4R and with chains they may reach 2R of circles. In fact, multiple and multiform combinations of spheres can be done.

   Spherical triangles.     We have the following postulates:

   P.1.     A spherical triangle is modelled upon a spherical calotte.

   P.2.    In a spherical triangle, a random side is smaller than the sum of the other two sides and is bigger than their difference.

   P.3.     The sum of sides of a spherical triangle is less than four straight angles (2p).

   P.4.    Two spherical triangles are equal when each of them has an equal side situated between two respectively equal angles and placed in the same order.

   P.5.    Two spherical triangles are equal when each of them has an equal angle situated between two respectively equal sides and positioned in the same order.

   P.6.    Two spherical triangles are equal when they have three sides respectively equal and situated in the same order.

   P.7.    Two spherical triangles are equal when they have the three angles respectively equal and situated in the same order.

   P.8.     If in a spherical triangle two sides are equal, then the angles opposite them are equal, too.

   P.9.    If in a spherical triangle two angles are equal, then the sides opposite them are equal, too.

   P.10.     The sum of the three angles of a spherical triangle is situated between 2p and 3p.

   P.11.     In a spherical triangle, the sum of two angles is less than the third angle with p³.

   Theorem: The shortest path between two points on a sphere is on the largest arc that passes through the two points.

   The demonstration is simple, if “its smaller part”xxx will be added.

    Solving spherical triangles.

    Elements to work with:the angles A, B, C and their corresponding arcs, namely the sides a, b, c.

1.     All the three sides – a, b, c – are known:

cosA = (cosa-cosb·cosc)/sinb·sinc;  cosB = (cosb-cosc·cosa)/sina·sinc;

cosA = (cosc-cosa·cosb)/sina·sinb.

2.      Two sides and the angle situated between them are known: a, b, C.

cosc = cosa·cosb+sina·sinb·cosC; ctgA=ctga·sinb/sinC;

 ctgB = ctgb·sina/sinC-cosa·ctgc.

 3.      Two angles and the side situated between them are known: a, B, C.

     ctgb = ctga·cosB+sinC·ctgB/sina; ctgc = ctga·cosB+sinB·ctgC/sinA;

 cosA = -sinB·sinC·cosa-cosB·cosC.                     

    4.      The three angles A, B, C are known.

cosa = (cosA+cosB·cosC)/sinb·sinC; cosb = (cosB+cosC·cosA)/sinA·sinC;

                         cosC = (cosC+cosA·cosB)/sinA·sinB.                                                       

5.      Two sides and the angle of one of them are known: a, b, A.

sinB = sinA·sinb/sina.

knowing B, namely sinB.

cosb·cosc+sinb·cosA·sinc = cosA.                                     

the unknown element being cosc or sinc, we shall denote as

cosc = (1-tg²t/2)/(1+tg²t/2) and sinc = (2·tg²t/2)/(1+tg²t/2),  then we obtain the equation: cosb·(1-tg²c/2)+sinb·cosa·(2·tg(c/2)) = cosa·(1+tg²c/2).

    It may be noticed that an easy to solve 2^nd degree equation was obtained, of a tg(c/2) variable .

6.      One side and two angles, one of which is opposite to the side: a, A, B.                            .

sinb = sina·sinB/sinA; sinc = sina·sinC/sinA;

cosA = sinb·sinc·cosa-cosB·cosC.

   The ellipse.     The ellipse is not definable. One property of the ellipse is that it is the geometrical place of any points starting from which the distance to two fixed points, called focuses, is always constant. An ellipse has four vertexes, two of them are acute and two are obtuse. On the construction point of view, the ellipse has a symmetry centre, two symmetry axes, the big axe, 2a and the small axe, 2b. The big and small axes may be oblique or rectangular. One ellipse may be determined by five distinct points.

            The equation of a conic ellipse is:

Ax²+2Bxy+Cy²+2Dx+2Ey+F = 0

 I = A+C. 

In order to represent an ellipse, the equation has to: D·I < 0 and d > 0 .

   The focuses F and F’ of the ellipse are obtained if from the edge of the small axis B we fix the big semiaxe a on the big axe on one side and the other of the small axe.  Starting from the centre of the ellipse O, on the big axe, there are the focuses, on a distance equal to OF = OF’.          

 OF² = BF² – OB²  or  OF² = OA² – OB²; OF=c; c² = a² – b²    

   If the centre of the ellipsis moves to the origin of coordinates O and its axes coincides with coordinate axis, then the equation becomes x²/a²+y²/b²-1 = 0 .

   We hereby reproduce two theorems (without demonstrations) upon a few properties of ellipses.

    The geometric place of the vertex of a straight angle, the sides of which stay tangent to an ellipse, is a circle – the orthoptic circle. (Gaspar Monge).

   Tangents drawn from a (exterior) point to an ellipse are inclined on the focal radiuses MF and MF’ of the point M. (Jean Victor Poncelet).

   Ellipses have co-chain properties, but, due to their formal way of being and of manifesting, chain relationship prevails upon conglomerates. But this default is chased by a great privilege, namely the richness of elements such as a centre, two focuses and the absence of the boring roundness of the circle.

    Two ellipses or one circle and one ellipsis can be exterior to each other, can intersect in two points or in four points, and as lines of border there are arcs of ellipsis or/and arcs of circle and of ellipsis.

   The ellipsoid cannot be defined. On the formal-geometrical point of view, the ellipsoid is a quadrix. The ellipsoid has three reference axes and 6 main focuses. It is therefore a body very rich in elements. It needs nine distinct points and determinations between invariants D, d, J, I, which will become coefficients of the secular equation. The calculus of invariants is to be exposed in the following page.

   When ellipsoids intersect one another or intersect spheres, ellipses will appear.

    The same as with ellipses, ellipsoids have the tendency to form chains and/or to conglomerate in what regards its elements. As we have already shown, the same as with ellipses, with ellipsoids there is a preference for chains, meaning not arborescent forms of chains. In order to avoid this shortcoming, especially in information theory – namely with theory of transmitting information – a symbiosis was adopted between the sphere and the ellipsoid. This way, chains would alternate with conglomerates.

    For the time being, there are few known data upon the way ellipsoid’s elements behave.

    The title of this article A Project of Geometric Axiomatic into a Finite Ontology may seem a bit awkward. I must admit too, it seemed somewhat false, if not paradox-like. How should I admit a finite ontology along with geometry which has as basic elements the point (the tiny, immaterial infinity), the straight line (infinite), and the plane (infinite)? However, I used a categorical discrepancy in order to elude the presented incompatibility. The axioms of Euclid, Hilbert, Birkhoff and many more started from their studies upon the infinite plane, the infinite straight line and the tiny infinite point and they represent axioms of inexistence. It sounds like a logical paradox.

    In my work I have admitted point as something material, a v qquantor. How otherwise understand the notion of absolute void? Instead of a straight line (infinite) I considered the circle and the ellipsis, and I considered the periphery of the (finite) Universe as the largest element. I used the sphere and the ellipsoid as bodies. The greatest (finite) body is the Universe or the Existence.

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