Corpuscular-kinetic theory of M.V. Lomonosov

In the middle of the 18th century , European science was dominated by the theory of calorific , first put forward by Robert Boyle . The basis of this theory was the idea of ​​some fiery (or, alternatively, cold-forming) matter, through which heat and fire are distributed and transmitted.

MV Lomonosov draws the attention of the scientific community that neither the expansion of bodies as they heat up, nor the increase in weight during firing, nor the focus of the sun's rays with a lens can be qualitatively explained by the theory of caloric. The connection of thermal phenomena with changes in mass partly gave rise to the idea that mass increases due to the fact that the material heat penetrates into the pores of bodies and remains there. But, asks M.V. Lomonosov, why does the calorific remain when the body cools, and the heat is lost?

Refuting one theory, MV Lomonosov proposes another, in which, using Occam's razor, he cuts off the superfluous concept of calorific value. Here are the logical conclusions of MV Lomonosov, according to which, "a sufficient basis of heat is":

1. “In the movement of some kind of matter” - since “when the motion ceases, heat also decreases”, and “motion cannot occur without matter”;
2. “In the internal motion of matter”, since it is inaccessible to the senses;
3. “In the internal motion of their own matter” of bodies, that is, “not extraneous”;
4. “In the rotational motion of particles of the body’s own matter”, since “there are very hot bodies without” two other types of motion “internal translational and oscillatory”, for example. the hot stone is at rest (no translational motion) and does not melt (no oscillatory motion of particles).

“Thus, we proved a priori and confirmed a posteriori that the cause of heat is the internal rotational motion of bound matter” ^[3] .

These considerations had a huge resonance in European science. The theory, as expected, was more criticized than accepted by scientists. Basically, criticism was aimed at the following sides of the theory:

1. Particles of MV Lomonosov are necessarily spherical, which is not proved (according to Rene Descartes , before all the particles were cubic , but after they were erased to balls );
2. The statement that the oscillatory motion entails the decay of the body and therefore cannot serve as a source of heat, however, it is well known that the particles of the bells oscillate for centuries and the bells do not crumble;
3. If heat was transferred by the rotation of particles only by transferring the action that a body has to another body, then “b and the pile of gunpowder would not catch fire” from a spark;
4. And since, due to the attenuation of the rotational motion during its transfer from one particle to another, “the heat of Lomonosov merged with that motion; but this would be sad, most important in Russia ” ^[4] .

Rotation Move

All of these dissertations are not only very good, but also very excellent, because he [Lomonosov] writes about physical and chemical materials that are very necessary, which even the most witty people could not understand and could not interpret, that he made with such success that I am absolutely sure in the justice of his explanations. In this case, Mr. Lomonosov must be given justice, which has excellent talent for explaining physical and chemical phenomena. It should be wished that other Academy members were able to produce such a revelation, as Mr. Lomonosov showed. Euler in response to His Excellency Mr. President of 1747. ^[five]

MV Lomonosov claims that all substances consist of corpuscles - molecules , which are "collections" of elements - atoms . In his dissertation “Elements of mathematical chemistry” (1741; unfinished), the scientist gives the following definition: “An element is a part of the body, not consisting of any other smaller and different bodies ... A corpuscle is a collection of elements that form one small mass.”

In a later work (1748) he uses the word "atom" instead of "element", and instead of "corpuscle" - particle ( lat. Particula) - "particle" or "molecule" ( lat. Molecula ). He gives the “Element” a contemporary meaning to it - in the sense of the limit of divisibility of bodies - their last component. The ancients said: "As words consist of letters, so the body of elements." MV Lomonosov’s atoms and molecules (corpuscles and elements) are often also “physical insensitive particles”, which emphasizes that these particles are sensually imperceptible. MV Lomonosov points to the difference between “homogeneous” corpuscles, that is, consisting of “the same number of the same elements connected in the same way”, and “heterogeneous” - consisting of different elements. Bodies consisting of homogeneous corpuscles, that is, simple bodies, he calls beginnings ( lat. Principium ). ^{[1] [2]}

But the scientist does not stop at the structure diagram - the main merit of the kinetic theory of heat of MV Lomonosov is to give the concept of motion a deeper physical significance. Moreover, it is M.V. Lomonosov who has the priority of the idea of internal rotational (“rotary”) particle motion in the context of his theses on the nature of heat, which, to the greatest extent, at all costs of his system, brought the ideas about the structure of matter closer to its current state - no one of his predecessors does not give a similar model; one of the main misconceptions was the opinion that particles are in contact (according to the modern model, they are not in constant contact, but collide, but the “contact” factor can be considered, in accordance with the general notions of time, as the equivalent of the current factors of particle coupling and interaction) , despite the fact that their indivisibility ( "lower limit") does not imply whatsoever structure - the next step was taken only with electron hypothesis ( 1874 ), to be exact - with the formation of ideas about the rotation with mmetrov electron cloud .

His further discussion is about the speed of rotation, the increase of which is expressed by an increase in the temperature of bodies and the environment, which, speculatively, has no limit, at the same time - an imaginary absence of movement - a state of rest, leads most of all to the idea of ​​an absolute zero ("the highest degree of cold ... an amphibious ball does not exist anywhere ”) - to the basis of the second law of thermodynamics ( 1850 ) ^[6] . MV Lomonosov experimentally came close to the exclusion of phlogiston and caloric from the system of natural science views, and to the final "dismantling" of the fluid theory - to the discovery of hydrogen .

In the article “An Experience in the Theory of Air Elasticity” (1748), MV Lomonosov gives a kinetic model of an ideal gas , according to certain provisions, with a number of corrections - corresponding to the one adopted later. According to his hypothesis, particles repel like rotating bodies - and this is a consequence of the fact that the gas is constantly heated to a certain temperature. The scientist demonstrates the relationship between volume and air elasticity (see the Boyle-Mariotte law ); at the same time, he points out that this regularity does not apply to air when it is compressed strongly, the reason for which is the finite size of its molecules, the real idea was applied by J. D. van der Waals in deriving the real gas equation . Considering the heat and light, the scientist in the “Word on the origin of light ...” (1756-1757) comes to the conclusion about the rotational (“rotary”) propagation of the first and the wave (“fluctuating”) particles of the second, the first are absorbed by the “incendiary strong mirror” and the second ones are reflected; in 1771, thermal radiation ("radiant heat") was considered by KV Scheele . The Russian scientist points to the origin of light and electricity as a consequence of the motion of the same ether, which, with certain corrections and taking into account the time-simplified understanding of the phenomenon, can be compared with the provisions of D.K. Maxwell's electromagnetic theory .

The validity of such correspondences can be observed in many sections of the concept of M.V. Lomonosov, anology, these and previous hypotheses are convincingly shown by the outstanding chemist and science historian N. A. Figurovsky . In general, the rotational movement of MV Lomonosov puts at the forefront of his "Natural Philosophy", as one of the fundamental principles of the universe. Despite the speculative and philosophical nature of the logical reasoning of M.V. Lomonosov, according to the prevailing misconception - in the absence of a mathematical evidence base (which, however, is unfair, as we will see later, the scientist made extensive use of the mathematical apparatus; despite the fact that mathematics is not “An absolute guarantee of reliability” - it’s no coincidence that W. Gibbs states: “A mathematician can say whatever he pleases, a physicist must keep at least a bit of common sense” ^[7] ), they are convincing and fair (this was noted by as we see, the mathematician Leonard Euler) and are in good agreement with the discoveries that followed after many decades - like the discovery of his successor - D. I. Mendeleev , who, not knowing the structure of the atom, gave the fundamental law, which subsequently guided those who comprehended this structure. ^{[1] [2] [8]} .

The conclusions of the mechanical theory of heat, confirming it itself, for the first time substantiated the hypothesis of the atomic-molecular structure of matter - atomistics received objective natural science evidence. Corpuscular theory and molecular kinetic views of M.V. Lomonosov are directly related to his understanding of the relevance of the law of conservation of matter and force (or motion). The principle of conservation of force (or motion) for him became the initial axiom in his consideration of arguments in substantiating molecular thermal motion. This principle is regularly applied by him in his early works. In his dissertation “On the action of chemical solvents in general” (1743), he writes: “When a body accelerates the motion of another, it tells him part of its motion; but it cannot communicate part of the movement except by losing exactly the same part. " Similar considerations about the principle of conservation of matter, showing the inconsistency of the theory of calorific. Guided by it, MV Lomonosov criticizes the ideas of R. Boyle about the transformation of fire into a “persistent and significant” substance. In the “Materials for the biography of Lomonosov” in document No. 165 - we see that the scientist writes in December 1756: “ In Chemistry: 1) Between different chemical experiments. of which there is a journal on 13 sheets, experiments were carried out in glass vessels melted in hard-pressed glass to examine whether the weight of metals comes from pure heat. It was found by other experiments that the glorious Robert Bots (a mistake should be read, of course, Boyle), the opinion is false, because without the passage of external air the weight of the burned metal remains to the same extent ... ” In 1774, A. L. Lavoisier published a work in which similar experiments were described; later he formulated and published the law of conservation of matter - the results of the experiments of M.V. Lomonosov were not published, so they became known only after a hundred years. ^{[1] [2] [9]} ..

In a letter to L. Euler, he formulates his "universal natural law" (July 5, 1748). repeating it in the dissertation “The argument about the hardness and liquid of bodies” (1760) ^{[1] [2]} :

Being an opponent of the phlogiston theory, MV Lomonosov, nevertheless, was forced to make attempts to reconcile it with his “corpuscular philosophy” (for example, to explain the mechanism of oxidation and reduction of metals and the “composition” of sulfur - there was no rational understanding of the phenomena, there was no scientific by the theory of combustion - oxygen has not yet been discovered), which was natural in his contemporary universal “ conventionality ” regarding the theory of “weightless fluids” - otherwise he would not only not be understood, but his ideas would not be accepted at all to education. But the scientist is already criticizing G.E. Stahl : “Since the restoration is carried out the same as the calcination, even with stronger fire, no reason can be given why the same fire either penetrates into the bodies, then leaves them” .

The main doubts of MV Lomonosov are related to the issue of weightlessness of phlogiston, which, when removed from calcination from the metal, gives an increase in the weight of the calcination product - in which the scientist sees a clear contradiction to the “universal natural law”. MV Lomonosov operates with phlogiston as a material substance that is lighter than water - essentially indicating that it is hydrogen. In his dissertation On Metallic Shine (1745), he writes: “... When a base metal, especially iron, is dissolved in acid alcohols, flammable vapor escapes from the opening of the flask, which is nothing more than phlogiston released from the friction of the solvent with metal molecules (link to “Thesis on the action of chemical solvents in general”) and carried away by bursting air with thinner parts of alcohol. For: 1) pure vapors of acidic alcohols are non-flammable; 2) the lime of metals destroyed during the loss of combustible vapors cannot be restored at all without the addition of any body abundant in combustible matter. ” A similar conclusion (“combustible air” is phlogiston, later called hydrogen), more than 20 years later came the English scientist G. Cavendish ^[12] , who was sure that his discovery resolves all the contradictions of the phlogiston theory. The identical conclusion of MV Lomonosov in the work “On the metallic luster” (1751) “went unnoticed”, ^{[1] [2] [9]}

With his “Corpuscular Philosophy”, MV Lomonosov not only criticizes the legacy of alchemy and iatrochemistry, but, putting forward the productive ideas that he used in practice, forms a new theory, which was destined to become the foundation of modern science. ^[one]

• "Elements of mathematical chemistry" (1741) ^[2]

The scientist’s first attempt at reconciling the corpuscular (atomic-molecular) doctrine he is developing with chemistry.

Definition

40) ... The corpuscles are heterogeneous when their elements are different and connected in different ways or in different numbers; the infinite variety of bodies depends on it. ...

• "The experience of the theory of insensitive particles of bodies and in general about the causes of particular qualities" (1743-1744) ^[2]

Experience 2

§ 51. Metals and some other bodies dissolve in solvents and separate into very small parts, which are inseparable from solvents but constitute a homogeneous body with them. Volatile bodies disperse through the air and disappear in it. Combustibles decay from the action of fire into intangible particles.

Addition

§ 52. Physical bodies are divided into the smallest parts, separately escaping the sense of vision, so that the bodies are composed of insensitive particles.

Explanation

§ 53. The following proves the astonishing smallness of insensitive physical particles. The cubic line of a Parisian foot of gold weighs approximately 3 grains, and one gran is stretched into a thin sheet of 36 square inches. ... three grains or a cubic line of gold, stretched into such a leaf, equals approximately 108 square inches or 15 552 quadratic lines. ... one cubic line of Parisian foot of gold contains 3,761,479,876,608 cubic particles of gold, the side of which is equal to the thickness of the leaf; Thus, in one cubic grit, the side of which is 1/10 of the line, approximately 3,761,479,876 particles can be contained that are physically separated from each other ... Mr. de Malesier observed tiny creatures with a microscope ^[13] , the value of any of which belonged to the size of aphids, as 1 in 27 million; and since these creatures live, therefore, have the parts and vessels necessary for movement, nutrition and feeling ... it is clear that their bodies are composed of insensitive particles, amazingly small and physically separable, ...

• "The experience of the theory of elasticity of air" (1748) ^[2]

§ 7. ... recall the experience of Roberval , who kept the air tightly compressed for 15 years and finally found its elasticity unchanged ... Therefore, we accept ... that the air particles - those that produce elasticity, trying to move away from each other - are devoid of any physical build-up and organized structure, and in order to be able to endure such tests and produce such amazing effects, they must be extremely durable and not subject to any changes; therefore, they should rightly be called atoms . And since they physically act on material bodies, they themselves must have a stretch .

§ 8. As for the figure of the atoms of air, ... the weight is close to spherical. ... Since, further, hot air heats the cold bodies inside it, it means that its atoms excite a rotational movement in the particles of bodies in contact with it (See “Reflections on the Cause of Heat and Cold”, 1749), which produces heat. ...

• "The theory of electricity, mathematically stated" (1756) ^[2]

Position

§. Light propagates by oscillatory motion.

Evidence

Light cannot be propagated by translational motion (§); therefore, it propagates either rotational or oscillatory (§). Rotation also spreads heat (§); but by the movement by which heat spreads, light does not spread (§). Consequently, light does not propagate either by translational or rotational motion, and therefore only by vibrational. ...

Explanation

§. How natural and how much consonant with nature is this reason for the propagation of light, it becomes quite obvious by analogy: for as in air, etc.

Position

§. The particles that make up the ether are always in contact with neighboring particles.

Proof 1

Light travels through vast spaces at an insensitive moment in time (evident from countless observations and everyday experience); the oscillatory movement by which light travels through the ether cannot otherwise occur, as if one corpuscle hits from another corpuscle ... And the value is infinitesimal, taken an infinitely large number of times, gives tactile and large. Therefore, the propagation time of light would be noticeable if the ether particles were not all in contact; but since the propagation time of light over vast distances is hardly noticeable, then, obviously, the ether particles must be in contact, which was to be proved.

And can we not observe an analogy with what was said almost two hundred years later (at least in the form of a "fluctuating" particle motion as a substrate of ideas about its inherent wave), replacing the "contact" with another equivalent of generality and interaction?

Thus, try to attribute ... in general to all particles, like photons , a dual nature, to endow them with wave and particle properties, interconnected by a quantum of action ...

What is the main task? Essentially, in establishing a certain correspondence between the propagation of a certain wave and the motion of a particle, the quantities describing the wave must be associated with the dynamic characteristics of the particle ...

- Louis de Broglie , Wave mechanics - from the book “Revolution in Physics” (1936) ^[14]

• “A word about the origin of light, representing a new theory of flowers” ​​(1756) ^[2]

A rotary movement of particles to explain and prove all the properties of heat is enough. ...

An incendiary strong mirror coated with black varnish produces a great amount of light at the incendiary point, no less heat, clearly showing that the rotary movement of the ether in black matter is tired, the unsteady unobstructed remains. ...

In every eight minutes, light propagates to the earth from the sun; consequently, at twelve o'clock it will pass from it to this grit of ethereal matter with the axis of eight thousand six hundred and forty million cubic terrestrial half-diameters. ...

I find these things in insensitive initial particles, all the bodies of the components, from the wise architect and omnipotent mechanic, are arranged and established between immutable natural laws. ...

For the approval of this system, I would wish for all the examples from the numerous experiments that I especially performed in the search for multi-colored glass for mosaic art. ...

• "Reasoning on the hardness and fluid of bodies" (1760) ^[2]

In the winter of 1759/1760, I. A. Braun and M. V. Lomonosov managed for the first time to freeze mercury, which until then had been considered a liquid body. Russian and many foreign scientific journals reported this ^[15] . MV Lomonosov, on the basis of his atomic theory, proposes a new model of a solid body, trying to explain the reason for the increase in its volume during heating and the mechanism for changing the state of aggregation. He completely repeats his formulation of the “universal natural law” set forth in a letter to L. Euler (July 5, 1748). In the XX century, this work was translated into German, French and English ^[16] . N. N. Beketov writes on this occasion: "... He expresses such views that even now hardly penetrate the minds of scientists ... Explains on the basis of his dynamic theory of heat ... This view is now accepted already in the forties and fifties of the XIX century" .

§ 13. The following rule follows from this: particles are insensitive, constituting bodies, the larger, the stronger the union, the smaller, the weaker. . When the union consists of particles - balls, then let there be half-diameters of larger particles (Fig. 1) AE, CF, AI, CI = a , half-diameter EB and BF of the particle of compressive matter [is] r . Moreover, it follows from the very addition of the figure that BI is perpendicular to AC ; therefore will${\displaystyle \mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">B</span></span>}\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">I</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">=</span></span>\sqrt{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">[</span></span><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">(</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">+</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">r</span></span>}{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">)</span></span>}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span>{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">]</span></span>}}$ {\ displaystyle {BI} = {\ sqrt {{[} ({a} + {r}) ^ {2} - {a} ^ {2} {]}}}}   . But as AD , DC , AB , BC are equal to each other, there will be a triangle ADC = and ~ ABC ; for this, and BI = DI ; Consequently${\displaystyle \mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">B</span></span>}\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">D</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">=</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}\sqrt{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">[</span></span><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">(</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">+</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">r</span></span>}{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">)</span></span>}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span>{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">]</span></span>}}$ {\ displaystyle {BD} = 2 {\ sqrt {{[} ({a} + {r}) ^ {2} - {a} ^ {2} {]}}}}   = diameter of the union plan of particles A and C. Then let there be p - the periphery of the circle, whose diameter = 1; it will be the most allied plane =${\displaystyle \mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">p</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">=</span></span>\sqrt{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">[</span></span><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">(</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">+</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">r</span></span>}{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">)</span></span>}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span>{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">]</span></span>}}$ {\ displaystyle {p} = {\ sqrt {{[} ({a} + {r}) ^ {2} - {a} ^ {2} {]}}}}   . Finally, let there be a half-diameter of smaller particles, bodies of components, A and C = a - e and a half-diameter of a particle of compressive matter = r . And more so, the same thing happens, as is proved above, it will be BD =${\displaystyle \mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">B</span></span>}\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">D</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">=</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}\sqrt{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">[</span></span><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">(</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">e</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">+</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">r</span></span>}{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">)</span></span>}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">(</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">a</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">e</span></span>}{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">)</span></span>}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">]</span></span>}}$ {\ displaystyle {BD} = 2 {\ sqrt {{[} ({a} - {e} + {r}) ^ {2} - ({a} - {e}) ^ {2} {]}} }}   = the diameter of the union plane of smaller particles, and the plane itself is union = p [( a - e + r ) ² - ( a - e ) ²]; so, the union plane of larger particles to the union plane of smaller particles will be = p [( a + r ) ² - a ²] to p [( a - e + r ) ² - ( a - e ) ²] = ( a + r ) ² - a ² k ( a - e + r ) ² - ( a - e ) ² = r + 2 a to r + 2 ( a - e ). Therefore, the union plane of larger particles will be larger than the union plane of smaller particles; therefore, the larger the particles, the stronger the union; the smaller, the weaker.

§ 14 So, from this it is not difficult to conclude, since many properties differ, which are in the union of particles, according to this rule it is possible to interpret by reasoning different sizes of particles in the mixture. For the sake of this, let the testers of nature cease to marvel and doubt that all special bodies of quality can come from particles that have only a round shape, and especially take into consideration the power of combining particles shown in the Word on the origin of light and colors. Moreover, to take as an example art, which is made of round threads, and especially if they have different thicknesses, countless and different sets of woven and woven items are made with excellent patterns according to their different positions.