## Portrait Gallery

The following portraits of some pioneer elasticians, and some opticians and astronomers who used elasticity to improve the performance of astronomical telescopes and instrumentation, are a personal choice of the author.

Fig. 1 Galileo Galilei (1564-1642) was a universal scientist who, from poor "monocular opera glasses," invented the telescope in 1609 - just four centuries ago - and discovered with it Jupiter's four Galilean satellites. He introduced mathematics in the formulation of the basic laws of dynamics and parabolic trajectories. Galileo's problem of equal strength cantilevers and its scaling law on the flexure of beams marked the birth of the elasticity theory and strength of materials (Discorsi e Dimostrazioni Matematiche, 1638)

PHOTO: (credit: Portrait Galileo Galilei florentino by Ottavio Leoni, 1624, Musee du Louvre)

Fig. 1 Galileo Galilei (1564-1642) was a universal scientist who, from poor "monocular opera glasses," invented the telescope in 1609 - just four centuries ago - and discovered with it Jupiter's four Galilean satellites. He introduced mathematics in the formulation of the basic laws of dynamics and parabolic trajectories. Galileo's problem of equal strength cantilevers and its scaling law on the flexure of beams marked the birth of the elasticity theory and strength of materials (Discorsi e Dimostrazioni Matematiche, 1638)

PHOTO: (credit: Portrait Galileo Galilei florentino by Ottavio Leoni, 1624, Musee du Louvre)

Fig. 2 Robert Hooke (1635-1703) discovered, around 1660, the fundamental law of elasticity which states that the uniaxial stretching of a solid body is proportional to the stress applied in this axial direction. This linear stress-strain relation, which allows perfect characterization of elastic materials, is known as Hooke's law (De Potentia Restitutiva, 1678)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 2 Robert Hooke (1635-1703) discovered, around 1660, the fundamental law of elasticity which states that the uniaxial stretching of a solid body is proportional to the stress applied in this axial direction. This linear stress-strain relation, which allows perfect characterization of elastic materials, is known as Hooke's law (De Potentia Restitutiva, 1678)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 3 Daniel Bernoulli (1700-1782) was a mathematician, physicist, and philosopher particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. In 1742, he suggested to Euler that the plane elastic bent curve of a rod without central stretching could be derived from the principle of minimum total bending energy, thus by rending minimal the sum of the square curvatures along the rod PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 3 Daniel Bernoulli (1700-1782) was a mathematician, physicist, and philosopher particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. In 1742, he suggested to Euler that the plane elastic bent curve of a rod without central stretching could be derived from the principle of minimum total bending energy, thus by rending minimal the sum of the square curvatures along the rod PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 4 Leonhard Euler (1707-1783) was a prolific mathematician also renowned for his work in mechanics, optics, and astronomy. From a suggestion by Daniel Bernoulli he initiated the first calculus of variations of a non-linear integral function as a minimal energy principle to determine the bending deformation of unstretchable circular-section objects. Hence Euler's beautiful contributions to elasticity are the critical load buckling relations of compressed beams and the first theory of elasticity known as theory of elasticae (1744)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 4 Leonhard Euler (1707-1783) was a prolific mathematician also renowned for his work in mechanics, optics, and astronomy. From a suggestion by Daniel Bernoulli he initiated the first calculus of variations of a non-linear integral function as a minimal energy principle to determine the bending deformation of unstretchable circular-section objects. Hence Euler's beautiful contributions to elasticity are the critical load buckling relations of compressed beams and the first theory of elasticity known as theory of elasticae (1744)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 5 C. Augustin de Coulomb (1736-1806) was a physicist who worked with mechanical rupture and friction, elasticity and electrostatics. He formulated the inverse square law for the force between two charges, known as Coulomb's law. Inventing the torsion balance and using thin silk and hair threads, Coulomb developed the first elasticity theory of torsion. He thus derived the magnitude of the electrostatic forces from the torsion angle of a charge pair when introducing a third charge (1780)

PHOTO: (credit: Archives of the Academie des Sciences and Encyclopedia Wikipedia)

Fig. 5 C. Augustin de Coulomb (1736-1806) was a physicist who worked with mechanical rupture and friction, elasticity and electrostatics. He formulated the inverse square law for the force between two charges, known as Coulomb's law. Inventing the torsion balance and using thin silk and hair threads, Coulomb developed the first elasticity theory of torsion. He thus derived the magnitude of the electrostatic forces from the torsion angle of a charge pair when introducing a third charge (1780)

PHOTO: (credit: Archives of the Academie des Sciences and Encyclopedia Wikipedia)

Fig. 6 Thomas Young (1773-1829) was a versatile physicist who brought important contributions on vision, light, elasticity, capillarity, and energy. Illuminating a narrow-slit pair by a distant source point, he proved the wave nature of light. For isotropic materials Young noticed that if a uniaxial stress is applied, then the stress-strain ratio - Young's modulus - is an elasticity coefficient depending only on the material. He also introduced the concept of shear stresses in 1807 PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 6 Thomas Young (1773-1829) was a versatile physicist who brought important contributions on vision, light, elasticity, capillarity, and energy. Illuminating a narrow-slit pair by a distant source point, he proved the wave nature of light. For isotropic materials Young noticed that if a uniaxial stress is applied, then the stress-strain ratio - Young's modulus - is an elasticity coefficient depending only on the material. He also introduced the concept of shear stresses in 1807 PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 7 S. Denis Poisson (1781-1840) was a mathematician, geometer, and physicist. Among the derivative equations bearing Poisson's name those of the first form involve the constancy of a single laplacian (gravitational or electrical potentials). In elasticity, those of the second form -at the fourth derivatives - involve the constancy of the bilaplacian of the flexure to a uniform load. Solving this latter form in the axisymmetric case, Poisson derived the general solutions, thus created the thin plate theory of elasticity (1828) PHOTO: (credit: Archives of the Academie des Sciences)

Fig. 7 S. Denis Poisson (1781-1840) was a mathematician, geometer, and physicist. Among the derivative equations bearing Poisson's name those of the first form involve the constancy of a single laplacian (gravitational or electrical potentials). In elasticity, those of the second form -at the fourth derivatives - involve the constancy of the bilaplacian of the flexure to a uniform load. Solving this latter form in the axisymmetric case, Poisson derived the general solutions, thus created the thin plate theory of elasticity (1828) PHOTO: (credit: Archives of the Academie des Sciences)

Fig. 8 C.L.M. Henri Navier (1785-1836) was an engineer and physicist who specialized in mechanics. He formulated an equation set for the motion of viscous fluids well known as the Navier-Stokes equations, since they were independently derived by G.G. Stokes. Navier founded the basis of the elasticity theory on the 3-D stress-strain relations which he derived from the equilibrium of a volume element (1826). The Navier stress-strain relations were finalized with the shear components by A. Cauchy in 1827-1829

PHOTO: (credit: Ecole des Ponts et Chaussees and The MacTutor History of Mathematics archive)

Fig. 8 C.L.M. Henri Navier (1785-1836) was an engineer and physicist who specialized in mechanics. He formulated an equation set for the motion of viscous fluids well known as the Navier-Stokes equations, since they were independently derived by G.G. Stokes. Navier founded the basis of the elasticity theory on the 3-D stress-strain relations which he derived from the equilibrium of a volume element (1826). The Navier stress-strain relations were finalized with the shear components by A. Cauchy in 1827-1829

PHOTO: (credit: Ecole des Ponts et Chaussees and The MacTutor History of Mathematics archive)

### No portrait of George Green is known

Fig. 9 George Green (1793-1841) was a mathematician and physicist mainly known for Green's divergence theorem which relates the properties of a mathematical function at the surface of a closed volume to other properties inside. Although Poisson, Navier, Cauchy, and Lame used a bi-constant elasticity theory (E, v), they implicitly assumed that v = 1/4 for all materials; this was called the uni-constant theory. Green showed that the number of elastic constants reaches 21 for crystals (1837)

Fig. 10 Gustav R. Kirchhoff (1824-1887) was a mathematician and physicist who made important contributions in the theory of elasticity and, using topology, generalized Ohm's laws to multi-loop circuits. He demonstrated that an electric current flows on a conductor at the velocity of light. In elasticity, after conflicting discussions over decades, Kirchhoff's formulation of the boundary conditions for a free edge when the flexure of a plate is non-axisymmetric were found to be exact; this involves his concept of net shearing force (1850)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 10 Gustav R. Kirchhoff (1824-1887) was a mathematician and physicist who made important contributions in the theory of elasticity and, using topology, generalized Ohm's laws to multi-loop circuits. He demonstrated that an electric current flows on a conductor at the velocity of light. In elasticity, after conflicting discussions over decades, Kirchhoff's formulation of the boundary conditions for a free edge when the flexure of a plate is non-axisymmetric were found to be exact; this involves his concept of net shearing force (1850)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 11 Adhemar J.C. Barre de Saint-Venant (1797-1886) was an engineer and mathematician who significantly contributed to elasticity and fluid mechanics, for which he formulated the equations on open channel flow. In La Torsion de Prismes, he derived the shape of various cross-section beams when large torsions occur, thus creating the large torsion theory. In this textbook he also enounced a general principle of equivalence for the application of the boundary conditions, well known as the Saint-Venant principle (1855) PHOTO: (credit: Archives of the Academie des Sciences, Paris)

Fig. 11 Adhemar J.C. Barre de Saint-Venant (1797-1886) was an engineer and mathematician who significantly contributed to elasticity and fluid mechanics, for which he formulated the equations on open channel flow. In La Torsion de Prismes, he derived the shape of various cross-section beams when large torsions occur, thus creating the large torsion theory. In this textbook he also enounced a general principle of equivalence for the application of the boundary conditions, well known as the Saint-Venant principle (1855) PHOTO: (credit: Archives of the Academie des Sciences, Paris)

Fig. 12 R.F. Alfred Clebsch (1833-1872) was a mathematician and physicist who succeeded B. Riemann in the chair of Gauss in Göttingen. His major works are on abelian functions, elasticity, algebraic geometry and invariant theory. In his Theorie der Elastizität fester Körper, Clebsch finds the first solutions to Galileo's problem for equal constraint cantilevers, and an infinite set of four-term polynomials - Clebsch polynomials - as radial component solutions to the non-axisymmetric loading of circular plates in the thin plate theory (1862)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 12 R.F. Alfred Clebsch (1833-1872) was a mathematician and physicist who succeeded B. Riemann in the chair of Gauss in Göttingen. His major works are on abelian functions, elasticity, algebraic geometry and invariant theory. In his Theorie der Elastizität fester Körper, Clebsch finds the first solutions to Galileo's problem for equal constraint cantilevers, and an infinite set of four-term polynomials - Clebsch polynomials - as radial component solutions to the non-axisymmetric loading of circular plates in the thin plate theory (1862)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 13 Augustus Love (1863-1940) was a mathematician and elastician. He formulated the thick plate theory by introducing the shear components of the flexure in addition to the bending ones. Then he completely solved the problem in the case of axisymmetric plates subjected to a uniform load and simply supported at the edge. Love also devised the first elasticity theory of shells by establishing the fourth derivative equation of the flexure for constant thickness cylinders around 1915 (Mathematical Theory of Elasticity, Sect. 339)

PHOTO: (credit: The MacTutor History of Mathematics archive, University of Saint Andrews)

Fig. 14 Bernhard Schmidt (1879-1935) was an optician and astronomer who in 1928 invented the famous wide field concept with a spherical mirror and an aspherical corrector plate. Schmidt telescopes allowed unprecedented complete photographic mappings of the sky in various bandpasses. Recognized for his great skill in the parabolization of mirrors, Schmidt emphasized the idea that the elastic relaxation method, or stress figuring, would provide much smoother surfaces. Therefore he must be considered the initiator ofActive Optics PHOTO: (courtesy of Erik Schmidt)

Fig. 14 Bernhard Schmidt (1879-1935) was an optician and astronomer who in 1928 invented the famous wide field concept with a spherical mirror and an aspherical corrector plate. Schmidt telescopes allowed unprecedented complete photographic mappings of the sky in various bandpasses. Recognized for his great skill in the parabolization of mirrors, Schmidt emphasized the idea that the elastic relaxation method, or stress figuring, would provide much smoother surfaces. Therefore he must be considered the initiator ofActive Optics PHOTO: (courtesy of Erik Schmidt)

Fig. 15 Stephen P. Timoshenko (1878-1972) was an engineer and mathematician who greatly contributed to the theory of elasticity, elastic stability and buckling, strength of materials, and wrote famous textbooks on these questions. He derived the critical load for the buckling of thin cylindrical shells. These results were applied to improve the strength of very large ships. Timoshenko also established, around 1930, the fourth derivative equation which gave rise to the theory of axisym-metric cylinders with variable thickness

PHOTO: (credit: Stanford University and National Academy of Sciences)

Fig. 15 Stephen P. Timoshenko (1878-1972) was an engineer and mathematician who greatly contributed to the theory of elasticity, elastic stability and buckling, strength of materials, and wrote famous textbooks on these questions. He derived the critical load for the buckling of thin cylindrical shells. These results were applied to improve the strength of very large ships. Timoshenko also established, around 1930, the fourth derivative equation which gave rise to the theory of axisym-metric cylinders with variable thickness

PHOTO: (credit: Stanford University and National Academy of Sciences)

Fig. 16 Andre J.A. Couder (1897-1979) was an optician and elastician who contributed to large telescope optics in writing Lunettes et Télescopes with A. Danjon, discovered Couder's two-mirror anastigmatic telescope, and invented the null test in 1927. He applied the elasticity theory to determine the axial flexure of large mirrors under gravity and improved the image quality by optimized mirror supports. After Schmidt's suggestion that a corrector plate might be aspherized by active optics, Couder solved the theoretical problem (1940) PHOTO: (courtesy of Charles Fehrenbach)

Fig. 16 Andre J.A. Couder (1897-1979) was an optician and elastician who contributed to large telescope optics in writing Lunettes et Télescopes with A. Danjon, discovered Couder's two-mirror anastigmatic telescope, and invented the null test in 1927. He applied the elasticity theory to determine the axial flexure of large mirrors under gravity and improved the image quality by optimized mirror supports. After Schmidt's suggestion that a corrector plate might be aspherized by active optics, Couder solved the theoretical problem (1940) PHOTO: (courtesy of Charles Fehrenbach)

Fig. 17 Eric Reissner (1913-1996) was a mathematician whose research was dedicated to turbulence and aerodynamic wing theory, and elasticity theory. With W. Martin, he published Elementary Differential Equations. Reissner created the theory of shallow shells which is one of the most remarkable achievements in elasticity. Compared to the theory of plates, this also takes into account the stresses and strains in the midsurface of the shell. Reissner's theory relates two bihar-monic functions in a fourth-order equation pair (1946)

PHOTO: (credit: Scripps Institution of Oceanography Archives, UC San Diego Libraries)

Fig. 17 Eric Reissner (1913-1996) was a mathematician whose research was dedicated to turbulence and aerodynamic wing theory, and elasticity theory. With W. Martin, he published Elementary Differential Equations. Reissner created the theory of shallow shells which is one of the most remarkable achievements in elasticity. Compared to the theory of plates, this also takes into account the stresses and strains in the midsurface of the shell. Reissner's theory relates two bihar-monic functions in a fourth-order equation pair (1946)

PHOTO: (credit: Scripps Institution of Oceanography Archives, UC San Diego Libraries)

Fig. 18 Gerhard Schwesinger (1913-2001) was an engineer and elastician who developed the elasticity theory for the determination of the lateral flexure of large astronomical mirrors under gravity. Introducing Fourier series for representing the lateral supporting forces (1954), he obtained the first comparison results for various systems, which thus led to minimal deformation designs. Advances in finite element codes combined with Schwesinger's expertise led to efficient lateral support systems for 8-m monolithic mirrors

PHOTO: (courtesy Raymond N. Wilson, European Southern Observatory)

Fig. 18 Gerhard Schwesinger (1913-2001) was an engineer and elastician who developed the elasticity theory for the determination of the lateral flexure of large astronomical mirrors under gravity. Introducing Fourier series for representing the lateral supporting forces (1954), he obtained the first comparison results for various systems, which thus led to minimal deformation designs. Advances in finite element codes combined with Schwesinger's expertise led to efficient lateral support systems for 8-m monolithic mirrors

PHOTO: (courtesy Raymond N. Wilson, European Southern Observatory)

Fig. 19 Edgar Everhart (1920-1990) was a physicist specializing in atomic collisions, professor at the University of Denver, and director of the associated observatory. He discovered the comets 1964 IX Everhart and 1966 IV Ikeya-Everhart. In elasticity, he independently derived Couder's results for making a Schmidt plate by partial vacuum and spherical figuring tool when the plate is simply supported at its edge. Everhart was the first to apply Active Optics to the complete aspher-ization of a telescope optical surface (1966)

PHOTO: (copyright and courtesy of the University of Denver Penrose Archives Special Collections)

Fig. 19 Edgar Everhart (1920-1990) was a physicist specializing in atomic collisions, professor at the University of Denver, and director of the associated observatory. He discovered the comets 1964 IX Everhart and 1966 IV Ikeya-Everhart. In elasticity, he independently derived Couder's results for making a Schmidt plate by partial vacuum and spherical figuring tool when the plate is simply supported at its edge. Everhart was the first to apply Active Optics to the complete aspher-ization of a telescope optical surface (1966)

PHOTO: (copyright and courtesy of the University of Denver Penrose Archives Special Collections)

Fig. 20 Raymond N. Wilson's (1928- ) inclination in school were towards the humanities, above all history and Latin, not science and above all not mathematics. Through interest in astronomy and telescope making he finally studied physics and specialized in optics. His invention of Active Optics combining optics, mechanics and computer technology has revolutionized modern telescope technology. However, he himself sees his greatest achievement by far as his two books Reflecting Telescope Optics, a standard work for specialists PHOTO: (80th birthday party, courtesy Peter Wilson)

Fig. 20 Raymond N. Wilson's (1928- ) inclination in school were towards the humanities, above all history and Latin, not science and above all not mathematics. Through interest in astronomy and telescope making he finally studied physics and specialized in optics. His invention of Active Optics combining optics, mechanics and computer technology has revolutionized modern telescope technology. However, he himself sees his greatest achievement by far as his two books Reflecting Telescope Optics, a standard work for specialists PHOTO: (80th birthday party, courtesy Peter Wilson)

## Relaxation Audio Sounds Autumn In The Forest

This is an audio all about guiding you to relaxation. This is a Relaxation Audio Sounds with sounds from Autumn In The Forest.

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