Changes

  | issue =  2019-2

  | issue =  2019-2

  | number = 8

  | number = 8

  | publications = The Theosophist. Vol. 144.10, July 2023, pp.26-34

  | publications = ''The Theosophist'', vol. 144-10, July 2023, pp.26-34

  | info =

  | info = Translated from Russian by V. Bazyukin

  | russian = Малахов П.Н. - Как построить квадратуру круга

  | russian = Малахов П.Н. - Как построить квадратуру круга

  | categories = Theosophy

  | categories = Theosophy

{{Style P-Title level|2|Geometrical Aspect}}

{{Style P-Title level|2|Geometrical Aspect}}

[[File:Квадратура круга2.png|150px|right]]

[[File:Квадратура круга2.png|100px|right]]

Building a square equal in area to a circle is not an easy thing to do, since the features of these geometric figures require different tools to build.

Building a square equal in area to a circle is not an easy thing to do, since the features of these geometric figures require different tools to build.

Now, what must be the side of the square so that its area is equal to the area of the circle? Simple mathematical operations give the answer: a = R√π, ''i.e''., the length of the side depends on π (pi), an irrational and transcendental number, one that does not have a complete expression in the form of a decimal fraction. In practice, this means that such a square cannot be constructed using a ruler and a pair of compasses. There is a purely theoretical solution employing a special curve called quadratrix, which however being also of a transcendental nature,<ref>The quadratrix is described (or defined) analytically and dynamically, rather than using an algebraic formula. See [https://en.wikipedia.org/wiki/Quadratrix Wikipedia].</ref> thus means that it cannot be accurately constructed using physical tools.

Now, what must be the side of the square so that its area is equal to the area of the circle? Simple mathematical operations give the answer: a = R√π, ''i.e''., the length of the side depends on π (pi), an irrational and transcendental number, one that does not have a complete expression in the form of a decimal fraction. In practice, this means that such a square cannot be constructed using a ruler and a pair of compasses. There is a purely theoretical solution employing a special curve called quadratrix, which however being also of a transcendental nature,<ref>The quadratrix is described (or defined) analytically and dynamically, rather than using an algebraic formula. See [https://en.wikipedia.org/wiki/Quadratrix Wikipedia].</ref> thus means that it cannot be accurately constructed using physical tools.

[[File:Квадратура круга с длинами.png|150px|right]]

[[File:Квадратура круга с длинами.png|100px|right]]

Some mathematicians closed the question at this stage, concluding that the problem of squaring the circle is unsolvable. If we proceed from the narrow task of graphically constructing a square equal in size to a circle using a compass and a ruler, then yes, the task is impossible, but geometers were engaged in calculations not only for practical construction and engineering purposes. As philosophers they sought to study the very properties of figures and ways to measure them, and, based on these illustrative examples, tried to delve into invisible relationships and laws, believing those had similar properties.

Some mathematicians closed the question at this stage, concluding that the problem of squaring the circle is unsolvable. If we proceed from the narrow task of graphically constructing a square equal in size to a circle using a compass and a ruler, then yes, the task is impossible, but geometers were engaged in calculations not only for practical construction and engineering purposes. As philosophers they sought to study the very properties of figures and ways to measure them, and, based on these illustrative examples, tried to delve into invisible relationships and laws, believing those had similar properties.

# the ''appearance of physical objects'' — the final clothing in forms.

# the ''appearance of physical objects'' — the final clothing in forms.

[[File:Квадрат с диагоналями (4 плана).png|150px|right]]

[[File:Square with diagonals.png|100px|right]]

The above order is rather arbitrary, since it is difficult for us to imagine any of the aspects in isolation from the rest. Usually, they are presented in the given sequence in order, among other things, to highlight a decrease in the world’s energy component at the expense of the growing predominance of the material component — in other words, the transition of energy into matter, of movements into objects, of force into form, etc. These stages (when thus seen) do not end when the next one appears, but serve as a constant source and filling for each other. As they complement or reveal each other’s features, they can be best illustrated graphically with a square having two crossing diagonals.

The above order is rather arbitrary, since it is difficult for us to imagine any of the aspects in isolation from the rest. Usually, they are presented in the given sequence in order, among other things, to highlight a decrease in the world’s energy component at the expense of the growing predominance of the material component — in other words, the transition of energy into matter, of movements into objects, of force into form, etc. These stages (when thus seen) do not end when the next one appears, but serve as a constant source and filling for each other. As they complement or reveal each other’s features, they can be best illustrated graphically with a square having two crossing diagonals.

The form (or shape) of any figure is an external constraint that determines its properties. The area of a figure is its inner content, meaning, and substance. It is the form that determines the manner in which the meaning will be expressed. The invariability of the area in different forms suggests the invariance of the meaning in its different expressions. This idea, clearly expressed in geometric terms, corresponds to one of the main tenets of Theosophy: all world religions and philosophies have one source and express the same moral and cosmic laws, only rendered in different words. The squaring of the circle helps us understand more deeply the idea of the One Life manifesting itself in different forms. To build a square equal to a circle in area means to express in it the essence and meaning of the circle, i.e., to transfer everything that was contained in the circle to a different form, using appropriate analogies and rendering adequately each feature of the circle to a square.

The form (or shape) of any figure is an external constraint that determines its properties. The area of a figure is its inner content, meaning, and substance. It is the form that determines the manner in which the meaning will be expressed. The invariability of the area in different forms suggests the invariance of the meaning in its different expressions. This idea, clearly expressed in geometric terms, corresponds to one of the main tenets of Theosophy: all world religions and philosophies have one source and express the same moral and cosmic laws, only rendered in different words. The squaring of the circle helps us understand more deeply the idea of the One Life manifesting itself in different forms. To build a square equal to a circle in area means to express in it the essence and meaning of the circle, i.e., to transfer everything that was contained in the circle to a different form, using appropriate analogies and rendering adequately each feature of the circle to a square.

[[File:Разность круга и квадрата.png|150px|right]]

[[File:Разность круга и квадрата.png|100px|right]]

The graphic representation of these figures having the same area, when superimposed on each other, emphasizes one important feature: their shapes go beyond each other’s limits. This means that each of the forms has points that do not belong to the other and thus make this form unique. And yet, the uniqueness of the forms notwithstanding, the content of both figures is the same, their areas (that is, the totality of all points) being equal. This graphic representation leads to the following two interesting conclusions:

The graphic representation of these figures having the same area, when superimposed on each other, emphasizes one important feature: their shapes go beyond each other’s limits. This means that each of the forms has points that do not belong to the other and thus make this form unique. And yet, the uniqueness of the forms notwithstanding, the content of both figures is the same, their areas (that is, the totality of all points) being equal. This graphic representation leads to the following two interesting conclusions:

{{Style P-Title level|2|Theological Aspect}}

{{Style P-Title level|2|Theological Aspect}}

[[File:Круг пунктиром и квадрат.png|150px|right]]

[[File:Круг пунктиром и квадрат.png|100px|right]]

Let us look again — but from a slightly different angle this time — at those parts of the two figures that go beyond each other’s boundaries. We see four truncated parts, called segments in geometry, that remains from the circle. What remains from the square is four triangles, one side of each of which is an arc. The radius of curvature of this arc, being a continuation of the same circle, is equal to the radius of curvature of the segment. Thus, what is external for one figure is internal for the other. Theologically, this can be interpreted as God (the circle) manifesting himself (the circumference) in every being (the square or any other figure) as that being’s inner nature. Also, this means that God, his power transcending our comprehension (the outer segments of the circle), can at the same time be found within our own nature (the arc of the triangle) as well.

Let us look again — but from a slightly different angle this time — at those parts of the two figures that go beyond each other’s boundaries. We see four truncated parts, called segments in geometry, that remains from the circle. What remains from the square is four triangles, one side of each of which is an arc. The radius of curvature of this arc, being a continuation of the same circle, is equal to the radius of curvature of the segment. Thus, what is external for one figure is internal for the other. Theologically, this can be interpreted as God (the circle) manifesting himself (the circumference) in every being (the square or any other figure) as that being’s inner nature. Also, this means that God, his power transcending our comprehension (the outer segments of the circle), can at the same time be found within our own nature (the arc of the triangle) as well.

An interesting geometric definition of God, known to us through Blaise Pascal, enables us to delve deeper into the essence of the divine nature. He said, “God is a circle whose center is everywhere and circumference nowhere.”

An interesting geometric definition of God, known to us through Blaise Pascal, enables us to delve deeper into the essence of the divine nature. He said, “God is a circle whose center is everywhere and circumference nowhere.”

[[File:Окружность нигде.png|150px|right]]

[[File:Окружность нигде.png|100px|right]]

The '''centre of the circle''' is the point of balance, rest or potentiality of everything. It is the source ofall would-be manifestations. This is a laya-centre, a state of non-manifestation, homogeneity (non-differentiation). As any point in space contains the possibility of manifestation, the centre of the circle is everywhere.

The '''centre of the circle''' is the point of balance, rest or potentiality of everything. It is the source ofall would-be manifestations. This is a laya-centre, a state of non-manifestation, homogeneity (non-differentiation). As any point in space contains the possibility of manifestation, the centre of the circle is everywhere.

Together, all these concepts (characteristics of the circle) represent the Divine World, collectively called God. With God beginning to act, a source of energy appears in space, known also as the source of a space curvature or the source of radiation of material particles. In other words, the center of a circle appears, a circle accompanied by a circumference. What makes the circle so good as a symbol is its clearly showing that the power of the Divine Nature can be manifested at every point in space and the degree of that manifestation is determined by an infinite radius — a ray. Any light source that emits its rays in all directions can be taken as an additional illustration of this idea.

Together, all these concepts (characteristics of the circle) represent the Divine World, collectively called God. With God beginning to act, a source of energy appears in space, known also as the source of a space curvature or the source of radiation of material particles. In other words, the center of a circle appears, a circle accompanied by a circumference. What makes the circle so good as a symbol is its clearly showing that the power of the Divine Nature can be manifested at every point in space and the degree of that manifestation is determined by an infinite radius — a ray. Any light source that emits its rays in all directions can be taken as an additional illustration of this idea.

[[File:Квадрат радиуса.png|150px|right]]

[[File:Квадрат радиуса.png|100px|right]]

Now, having described the meaning of the symbols of the circle, if we want to express the idea of the incarnation of the Divine Thought in the manifested world, we have to resort to the radius and, using this value, build a '''square''' by finding a suitable analogy. Thus, the power of the circle will be determined by its radius as much as the power of the square will be determined by the length of its side. By constructing a square with a side equal to the radius, we get a symbol of the realization of the Divine Thought in the world of forms. As becomes obvious from this construction, the Divine Thought cannot be realized in full

Now, having described the meaning of the symbols of the circle, if we want to express the idea of the incarnation of the Divine Thought in the manifested world, we have to resort to the radius and, using this value, build a '''square''' by finding a suitable analogy. Thus, the power of the circle will be determined by its radius as much as the power of the square will be determined by the length of its side. By constructing a square with a side equal to the radius, we get a symbol of the realization of the Divine Thought in the world of forms. As becomes obvious from this construction, the Divine Thought cannot be realized in full