Ninul A. S. Tensor Trigonometry. 3rd. ed. – Moscow: Scientific Publisher “Fizmatkniga”, 2025, 320 p. , 8 ill. ISBN 978-5-89155-429-0 / DOI 10. 29039/978-5-89155-429-0-320-01-2025 / WorldCat OCLC 1526562989 All rights reserved. Copyright: © 2025 by Anatoly Ninul On this web-page you may read and upload the book in its digitized copy from a paper original in pdf-A (35. 8Mb) for scientific goals in geometric and physical regions. It is 3rd and last author version in English, updated and added, of 1st edition: Ninul A. S. Tensor Trigonometry. Theory and Applications. – Moscow: Scientific Publisher “Mir”, 2004, 336 p. , 8 ill in Russian. (ISBN-10: 5-03-003717-9, Open Library ID Number: OL27049231M, WorldCat OCLC 255128609) Description Generally, the Tensor Trigonometry, with revealing a tensor nature of angles between two lineors or planars (in particular, vectors or lines), and their functions, added under following developing by Differential Trigonometry of regular curves, is created for its wide applications in various fields of exact and applied sciences. Described in the book are fundamentals of this new math subject with many initial various examples of its successful applications. The main goals of this monograph were to develop, for beginning, a number of algebraic and geometric notions in Theory of Exact Matrices (Part I, Chapters 1-4), and then, on this complete platform, to work out the basic contents of Tensor Trigonometry with bivalent tensor angles of two kinds – projective and motive. They are formed either between two linear subspaces (planars) in the given space or formed by rotation of a linear subspace (planar) in the same space (Part II, Chapters 5-12). Since Tensor Trigonometry may have a lot of applications in mathematical and physical domains, some very important examples of them are exposed in large Appendix in the book. Planimetry includes metric part and trigonometry. In geometries of metric spaces, from the end of XIX age, their tensor forms are widely used. However, the flat trigonometry is remained only in scalar forms in a plane or in a pseudoplane. Tensor trigonometry is development of the flat Scalar Trigonometry from Leonard Euler classic forms into k-dimensional tensor forms (at k ≥ 2) with their vector and scalar orthoprojections in an admissible coordinate base, and with step by step increasing a complexity and opportunities. So, for k = 2, we developed pseudoplane tensor trigonometry with all its trigonometric relations and complete solution of pseudo-Euclidean right triangles. We illustrate some relations with complementary hyperbolic angles onto the book’s front cover. In math analysis we apply operations of orthogonal differentiation and use well-known such for scalar trigonometric functions. In theoretic plan, Tensor Trigonometry, with its binary tensor angles and their functions, complements naturally Analytic Geometry and Linear Algebra. In practical plan, it gives own clear instrument for analysis and solutions of various geometric and physical problems in homogeneous isotropic spaces, such as quasi-Euclidean and pseudo-Euclidean ones (i. e. , as binary spaces with quadratic metrics); and on their perfect hypersurfaces of constant radius-parameter R embedded into them, with their non-Euclidean geometries of spherical and hyperbolic types. In these spaces and hypersurfaces, the elementary kinds of Tensor Trigonometry give very clarity all general laws of tensor trigonometric summing two-step and polysteps rotations or motions (relativistic velocities) in complete forms with polar decompositions of sums into principal (spherical or hyperbolic) and secondary orthospherical ones. In particular, these laws lead geometrically to the non-commutative Pythagorean Theorems for motions. Tensor Trigonometry gives also projective models of non-Euclidean geometries, but as descriptive trigonometric ones, in that number, using corresponding specific spherical-hyperbolic analogies. Exclusive physical subjects for its effective applications are Theory of Relativity and namely Relativistic Quantum Mechanics. Along the way, in Minkowski space-time, the simplest sequential genesis of the relativistic physical tensor of energy-momentum and tensor of momenta have been inferred from our dimensionless trigonometric tensors of motions – hyperbolic and general pseudo-Euclidean ones only using corresponding constant physical multipliers to them. From the trigonometric tensor of momenta, we got the pseudo-Euclidean Pythagorean Theorem of three momenta: for own 4-momentum on a world line with relative scalar total momentum on the time arrow and real 3-momentum in the Euclidean subspace. Note especially, that in Minkowski space-time the three possible axial momenta go into 3x3 Euclidean part of the trigonometric tensors of momenta – hyperbolic or general with polar decomposition of the latter. Possible inertial axial orthospherical rotations are causing initially by axial momenta, but one specific kind from them may relate to the non-inertial Thomas precession with axial rotation at the 3rd Euclidean normal axis. In Ch. 7A we inferred simplest universal formula for the relativistic Thomas precession (arising at non-collinear motions) and for angular deviations in the hyperbolic and spherical geometries. The applications were developed by us till the Differential Tensor Trigonometry of world lines and regular curves in pseudo- and quasi-Euclidean spaces of index 1, in addition to the Frenet-Serret theory in 3D Euclidean space, with absolute and relative differential-geometric parameters of curves, main kinematic and dynamical characteristics of a body moving in space-time along a world line with 4-velocity of Poincaré. The essential feature of our approach is that we use natural frame axes to describe main rotations in the binary spaces or identical motions on their hypersurfaces. So, in result of sequential orthogonal differentiations of the Poincaré 4-velocity along a world line, in general, we obtain three inner accelerations: principal 4-acceleration along 4-pseudonornal and two normal 3-accelerations along sine and cosine 3-binormals. All these accelerations are proportional to corresponding angular differentials, tangential to the concomitant Minkowski hyperboloids II or I, and to corresponding curvatures of a world line. These characteristics are summarized in general ones with their local Relative and Absolute Pythagorean Theorems and in tensor-vector-scalar (“tvs”) forms. They are accompanied by tetrahedron or trihedrons along a world line. For next developing general relativistic (GR) applications of our Tensor Trigonometry, under strict mathematical accompanying, for beginning, we introduced the scalar concept “accelerational potential” P, in addition, to the notion “gravitational potential” P. In its turn, both they generate, as one-to-one with them, the next concepts “accelerational and gravitational cosines of the hyperbolic angle at a world line”, which causing equivalent time dilations. This hyperbolic angle can be variable (so, for hyperbolic motions) and constant (so, for pseudo-screwed motions). Any relative potential P, including arising from acting the hyperbolic motion angle, is proportional with coefficient c2 to difference between the hyperbolic angle cosine and 1. And we named such a purely trigonometric difference by an “energetic coefficient”, since it is equal to k=P/c2=A/E0. In the given world point of N, these scalar accelerational and gravitational potentials P are summarized. Hence these potentials sum or these coefficients k sum determines for an object N its full decrement of time dilation from their influences. Thus, for free relativistic motion of an object N, any from two potentials P is doubling, or any from two coefficients k is doubling. With using such an approach, the simplest explanations of all STR and GR relativistic effects with exact trigonometric or equivalent potential formulas are gotten by us with their clear physical interpretations in Minkowski space-time, moreover, in full agreement with the Law of Energy-Momentum conservation, the 1st Noether Theorem, Relativistic Quantum Mechanics and Higgs Theory. We hope that this will contribute to resolving finally the current acute problem of the theory of relativity compatibility with other fundamental theories, namely, at relativistic velocities and in the presence of gravity. (! ) For example, we inferred that six of real and equal trigonometric coefficients k lead exactly to the well-known “Mercury perihelion relativistic shift” (with its constant hyperbolic motion angle), according to the Gerber formula. From the other hand, the energetic coefficient k gives an opportunity to interpret relativistic effects, including the Thomas precession, also from the point of view of the Law of Energy conservation. In addition, then the frame Universe potential P0=c2 determines Poincaré scale factor “c” to time and light velocity in vacuum. Absolute potential is equal to (P0+P), hyperbolic angle cosine is equal to (P0+P)/ P0. We may suppose logically that in any world point the frame potential P0=c2 is produced by the sum of relative potentials P in it of all material objects and other matter in the Universe. Hence in the Universe the meaning of "c" must depend a few on the place and the time. (! ) Despite the relative proximity of the Sun, its relative potential P contribution to the value of “c” by this cause is negligible, namely, in the vicinity of the Sun surface near 317m/sec, in the Earth orbit near 1. 5m/sec. But the Sun acts with the directed intensity "g" of its gravity field, what leads primary to all GR-effects in the Sun system, for example, to the Mercury perihelion, as was shown above. If the relative potential P=fM/R of the Sun acts at the distance R on the mass “m” of an object N, then it gives to N an additional energy equivalently to ~ mv2/2. However, if the frame Universe potential P0=c2 acts on the mass “m” of an object N or of another matter, then it gives to them a full energy E= P0m=mc2. If m=m0, then it is E0=m0c2. (! ) We see here also immediately, the “m” is gravitational and inertial mass. (! ) The book is intended for researchers in regions of multi-dimensional spaces, analytic geometry, linear algebra with theory of exact matrices, non-Euclidean geometries, theory of relativity with Relativistic Quantum Mechanics, and also to all those who is interested in new knowledges and applications, given by exact sciences. It may be useful for educational purposes on this new math subject in the university departments of algebra, geometry, and physics. 2025 12 05 See in addition on the following inet address of our article in the peer-reviewed scientific Russian journal “Mirovaja Nauka” (“World Science”) in operation since 2017 (on https://rus. science-j. com): https://scholar. google. ru/scholar? hl=ru&as_sdt=0%2C5&q=EQUIVALENT+ACTIONS+OF+POTENTIALS+FROM+ACCELERATION+AND+GRAVITY+ON+TIME+AND+WORLD+LINES+IN+MINKOWSKI+SPACETIME+INSTEAD+OF+ITS+CURVING+&btnG= or immediately in the author personal website: https://ninulas. narod. ru/english. html with direct references: https://ninulas. narod. ru/NinulAS_Article_Mirovaja_Nauka_2026_n. 1_p. 157-199. pdf in English https://ninulas. narod. ru/NinulAS_Article_Mirovaja_Nauka_2025_n. 12_p. 223-264. pdf in Russian This article presents the most general and very useful results that the Tensor Trigonometry could and did give for the Theory of Relativity generally with presence of gravity ang to all Non-Euclidean Geometries. Some of them have already been mentioned very briefly in the Description above. In particular, the author proved that the so-called and unresolved since 1931 problem of creating a hypothetical "Theory of All" (in particular, the unification in one but curved space-time of the General Relativity and the Relativistic Quantum Mechanics) is equivalent to the non-resolving "Circle Squaring" (but with exchange “circle-hyperbola”) and to discrediting the Law of Energy-Momentum Conservation. However, the Theory of Relativity and the Relativistic Quantum Mechanics with action of gravity are strictly uniting in the pseudo-Euclidean space-time of Minkowski with the full correspondence to all Fundamental Principles of material Nature and with derivations of precise and understandable trigonometric formulas for all known and new relativistic gravitational effects and phenomena. Moreover, as consequences from our tensor-trigonometric approach, we explained the generation of fundamental physical relationships E=mc2 and P=mс, but also only in the same flat space-time of Minkowski. As is now known to researchers and readers well versed in a history of Science, similar fundamental relations for energy, momentum and mass first appeared at the end of 1900 in an article by the great scientist and thinker Henri Poincaré (this article was devoted by him to the eminent physicist Hendrik Lorentz in December 11, 1900 on the occasion of the 25th anniversary of his doctorate). We hope that the readers with a free scientific mind and who place the Truth first in Science will show interest in what is written above and, in particular, will read the author's recent article and evaluate it objectively and without prejudice. But oddly enough, in the new era in an academic environment, the grants increasingly rule, rather than great thinkers, as was the case in the past. A rare exception to this is an activity of sir Roger Penrose, who gives and has been daring to pursue new and fruitful ideas for very many years. In a paper form, without having the book "Tensor Trigonometry", one may read it, for instance, in the Russian State Library (RSL. ru), in the main Scientific Libraries of Russia (now >20, beginning from MSU SL Library), and in EU in the most known and oldest mathematical Library “Zentral Universitätsbibliothek Göttingen” as Tensor Trigonometry (2025) and as Tenzornaja Trigonometrija (2004). This paper book can be purchased through the online store “Fizmatkniga” in Moscow. In an electronic form you may read or upload it here and, for instance, in the Internet Archive https://archive. org/details/tensor-trigonometry-3rd. ed. -by-ninul-a-s-moscow-fizmatkniga-2025-320-p and in the E-Library of the Russian State Library RSL. ru, in E-library. ru, in the Google Books, even as p-book and e-book, etc. Finally, we note (! ), that the algebraic part of our investigations concerning theory and solution of algebraic equations, considered in the beginning of Chapter 1 along the way, was subsequently brought by the author to its full logical development till final result in his next mathematical monograph in the frame of another item: Ninul A. S. Optimization of Objective Functions: Analytics. Numerical Methods. Design of Experiments. – Moscow, Fizmatlit, 2009, 336 p. ISBN-13: 978-5-94052-175-4 / Open Library ID Number: OL35648174M See this edition, for example, on the Internet Archive also in the author account: https://archive. org/details/optimizatsija-tselevykh-funktsij-by-ninul-a-s-moscow-fizmatlit-2009-336-p and in the Google Books: https://books. google. ru/books/? id=2PQuEAAAQBAJ This book was filling up also the existing "blind spots" in the important math field as Optimization. But the book is so far only in Russian language. The author will welcome this book's initiative translation into English too by a specialist with interest to this very important mathematical field, even with the use of an advanced electronic translator, but with conservation all of its 600 formulas, 18 figures and diagrams, and 5 tables. Ninul Anatoly Sergeevich, D. Ph. , Member of Math-Net. ru - under the civilian name Lunin A. S. Personal author's web-sites for communications: https://ninul-eng. narod. ru https://ninulas. narod. ru https://ninulas. narod. ru/english. html URL of the author accounts in the Internet Archive: https://archive. org/details/@anatoly_ninul https://openlibrary. org/authors/OL7564314A/Anatoly_Sergeevich_Ninul