Wright Brothers - The history of brothers who achieved the dream to the sky (ライト兄弟 - 大空への夢を実現した兄弟の物語). By Kiyoshi Tomizuka 富塚清 and revised by Akio Yamazaki 山崎明夫 (Miki Press, Tokyo, 2003, ISBN4-89522-322-1), in Japanese language

This book was originally published by Kodansha 大日本雄弁会講談社 in 1956 as a book for education for boys and girls by Prof. Tomizuka, however, the content is so fruitful a work, it is remade to this edition with approval by the deceased family. The description is very academic and meticulous, though, it is very easy to read, beginning from familial background, brothers' work from printing to bicycle business in Dayton, Ohio, aspiration to aircraft, stepwise research and development from toy helicopter, glider to powered aircraft. They studied extensively world-wide information and papers, especially major books and papers obtained from Smithsonian Institute till May 1899. Since they had much experience in bicycle dynamics, they were aware of the importance of stability and controllability during take-off/landing and in-flight. Especially amazing to me was, they figured out and developed DIY wind-tunnel experiment equipment for aerodynamics analysis of the biplane. Canard wing and rudder fin too is their original. These assured the stability and controllability. They were shuttling with gliders and powered plane between Dayton (main factory, home) and Kitty Hawk, North Carolina, circa 1,000 km distance, in 1900, 1901, 1902 and 1903 (first successful powered flight). They selected Kitty Hawk, since constant 25-40kilometer wind is blowing there, according to the inquiry to US Weather Bureau. The Wright Brothers succeeded with long succession of basic research, thorough rationality, and repeated experiments. Their success was not lucky shot by the rural bicycle shop owners, but a result of orthodox R and D. I hope this book is distributed in the library of junior and senior high schools all over Japan.

(Start reading 2024/9/12 and finished 2024/9/14)

The History of Japanese Motorcycles - The rise and fall of Japanese motorcycle manufacturers (日本のオートバイの歴史 - 二輪車メーカー興亡の記録). By Kiyoshi Tomizuka 富塚清 (Miki Press, Tokyo, 2024, ISBN 978-4-89522-819-0)

This book is an invaluable reprinted edition of the original 1980 version by Sankaido Press 山海堂, which is written by Professor Tomizuka of Tokyo Imperial University (1917 graduate of mechanical engineering division of the faculty of engineering, 1893-1988). The description is smart and refined, beginning from Daimler, dawn of the motorcycles in Meiji and Taisho Era, defeat of WW2 and birth of new faces, a new boom in 1955-1965 with Honda, Yamaha, Suzuki and Kawasaki in full flourish. Very interesting. After those history, an age of Japanese automobile industry followed.

(Start reading 2024/6/27 and finished 2024/9/1)

The Living in the final road to Death (死を生きる -訪問診療医がみた709人の生老病死-). By Ohichiro Kobori 小堀?一郎 (Asahi Newspaper Publications, Tokyo, 2024, ISBN 978-4-02-251979-5)

This book is personally presented to me by Dr. Kobori, who is a 1965 graduate of Tokyo University School of Medicine, eleven years senior to me. He was an active duty skilled surgeon of the First Department of Surgery in those days, when we were working in cooperation, he as a surgeon and me as a diagnostic pathologist at the Tokyo Univeristy Hospital around 1980-1990. Dr. Kobori changed his career after age-limit retirement from National International Medical Center to elderly care medicine. This book is full of many case reports with his first-hand experiences.

(Start reading 2024/4/28 and finished 2024/7/15)

The Remains of The Day. By Kazuo Ishiguro (Faber, Great Britain, 1989, ISBN 978-0-571-25824-6)

I picked up this novel by the winner of the Nobel prize in literature who was born in Nagasaki, Japan, in 1954 and moved to Britain at the age of five.

(Start reading 2022/11/3 and finished February 4 2023)

Elementary Differential-Integral Equations (Precalculus) By Magoichiro Watanabe 渡辺孫一郎 (Shoukabou 裳華房, Tokyo, Nippon, first edition 1930)

Since I was collapsed at the Maxwell equations in the preceding book, I return to the basics, precalculus learned at the Faculty of Liberal Arts at Komaba of Tokyo University in 1970 and 1971. I retrieve this old textbook, and drill one question after another, taking up much of my time, including relaxation time at work.

At page 11, Q 8 , My answer.

At page 24, Q 8, what is the length by tangent line of x^2/3 + y ^2/3 = a ^2/3 shut in by x-axis and y-axis My answer.

At page 31, Q 9 My answer.

At page 31, Q 12, My answer.

At page 31, Q 13 My answer.

At page 76, Q 5. Taylor expansion of x/(e^x-1), My answer.

At page 77, Q 11, lim x->0 (tan x/ x)^1/x2,
My answer.



(Start reviewing 2018/2/1)

The Truth (La Vérité in French) - The advantages of concealing it, The drawbacks to revealing it. By Florian Zeller (translated into English by Christpher Hampton) (Faber & Faber, London, UK, 2016, ISBN 978-0-571-32744-7)

I picked up this book after reading column for stage drama in Tokyo on Sankei newspaper (reputedly center right), and really enjoyed reading through on a single cold winter day at Hayama retreat. Paperback version is very cheap, available at Amazon, The Truth.

Characters:

Michel,
Laurence (Michel's twenty years wife),
Paul (Michel's best friend),
Alice (Paul's wife).

A drama full of irony and sarcasm in French style. C'est la vie.

(Start reading and finished 2019/1/19)

The Equations: Icons of knowledge. By Sander Bais (Harvard University Press,Cambridge, MA, USA, 2005, ISBN 10: 0674019679, 13: 978-0674019676)

I picked up this book after reading book review of Nature, page 2006: 440; 423, by Prof. Malcolm Longair.

(Start reading 2018/2/8, resigned at page 37 "Electromagnetics: The Maxwell equations" 2018/2/23, because of mathematics running out of my ability)

Beautiful Geometry. By Eli Maor and Eugen Jost (Princeton University Press, Prinston, New Jersey, USA, 2014, ISBN 10: 0691150990, 13: 978-0691150994)

This book requires pencil, straightedge and compus. I occasionally work out the proof of theorems, as the author suggests (",,, and we leave it to the reader to work out the details" on Page 34).

On page 22: For the three excircles of a Pythagorean triangles, each being externally tangent to one side and to the other two sides extended, r_c = (c + b + a)/ 2 ---> my proof.
On page 34 (Theorem 36): If from a pont P outside a circle we draw a line that cuts the circle at points A and B, the product PA x PB is constant for all possible lines through P and is equal to the square of the length of the tangent line from P to the circle, PA x PB = PT² ---> my proof.

I enjoyed this book with pencil, paper, straight edge ruler and compass, with no problem to understand the beauty and elegance of geometry. Drawing figures for myseft make me more clearly understand the problem. I hope my neural network of the cerebrum reactivate a bit in spite of age 66.

(Start reading 2017/8/19, finished 2018/2/3)

Albert Einstein - The Enduring Legacy of a Modern Genius. By TIME (Managing Editor: Richard Stengel, Writer/Editor: Richard Lacayo) (TIME Books, New York, USA, 2011, ISBN 10: 1-60320-173-4, 13: 978-1-60320-173-5)

In this illustrated biography, we can take a look into his personal life including eventful married life and family life including the fate of his two sons. Pictures of his family members and some of his study rooms of his apartments are very interesting. I am happy that I am not a genius, though I feel reverence for Einstein and his revolutionary discoveries.

(Start reading 2017/4/22, finished 2017/5/20)

Philosophy of Physics: Space and Time. By Tim Maudlin. (Prinston Foundations of Contemporary Philosophy series, Prinston University Press, Prinston, New Jersey, USA, and Oxford, UK, 2012, ISBN 978-0-691-14309-5)

Lexicon
Chapter One, entitled "Classical Accounts of Space and Time": Newton's discussion about "absolute, true, and mathematical" space and time vs. "relative, apparent, and common" space and time is introduced, with explanation of Euclidean geometry and space E³, isometry, affine transformation and topological transformation, including differentiable structure and diffeomorphism (January 2 2014).
Chapter Two, entitled "Evidence for Spatial and Temporal Structure": Aristotle's physics, uniform circular motion, circular inertia by Galileo, uniform motion in a straight line, centripetal force, Newton's thought experiment of motion of water in a spinning bucket and rotating two globes connected by a cord, absolute space and absolute time, arithmetic, geometry and coordinate systems, E³, gauge freedom, linear momentum, inertial frames of reference, rectilinear and orthogonal in Cartesian coordinates, diagrams, translational symmetry and rotational symmetry, homogeneous and isotropic, the Principle of Sufficient Reason (PSR), God' creative act, Principle of Identity of Indiscernibles (PII), Leibniz's theory of monads, Mach's rejection of absolute space and time by Newton because of their unobservable properties (February 1 2014).
Chapter Three, entitled "Eliminating Unobservable Structure": Unobservable absolute velocity, absolute rest, Galilean relativity, Galilean space-time, world-lines, space-time diagram, straight world-line and curved world-line/curvature in space-time diagram, inertial motion, space-time points or events, simultaneity slice, foliation of the space-time, affine structure of space-time = inertial structure, no absolute velocities in Galilean space-time, the origin of the coordinate system should be an inertial trajectory and the spatial origin of the coordinate system should trace out a straight line through space-time in Newtonian mechanics, non-inertial coordinates attached to a rotating body, Newton could not answer "what is your present absolute velocity?", space-time structure is not directly observable, There is no more call to try to eliminate space-time structure from physics than there is to eliminate the postulation of atoms, just because they cannot be directly seen, light and electromagnetism resisting all attempts to account for them in Newtonian terms, overthrown in favor of the Special Theory of Relativity (April 29 2014).
Chapter Four, entitled "Special Relativity": Einstein's two principle (1) the equivalence of all inertial frames and (2) the constancy of the speed of light, but to understand Relativity, we have to expunge all ideas of things having speeds including light, future light-cone and past light-cone, the space-time of Ralativity is called Minkowski space-time, Lorentz coordinates, the way that Lorentz coordinates relate to the geometry of Minkowski space-time is almost the same as the way Cautesian coordinates relate to Euclidean space, Minkowski space-tome is four-dimentional (t, x, y, z),
Invariant Relativistic Interval =
[(T(p) - T(q))² - (X(p) - X(q))² - (Y(p) - Y(q))² - (Z(p) - Z(q))²] ^1/2 which is sometimes called the Minkowski metric, but it is not positive-definite and does not satisfy the triangle inequality, Lorentz transformation, zero-valued Intervals, light-cones, the Law of Light in the theory of relativity "The trajectory of a light ray emitted from an event (in a vacuum) is a straight line on the future light-cones of that event", time-like separated from the origin o (Interval I(p,q) is positive), light-like separated from o (Interval I(p,q) is zero) and space-like separated from o (Interval I(p,q) is imaginary), Limiting Role of the Light-cone, Relativistic Law of Inertia: "The trajectory of any physical entity subject to no external influences is a straight line in Minkowski space-time", no pair of straight lines intersects more than once in Minkowski space-time, Clock Hypothesis: "Clocks measure the Interval along their trajectories", in Relativity accurate clocks are like odometers on cars measuring the length of their trajectory through space-time, Twins Paradox, In Minkowski space-time the Triangle Inequality does not hold for the Interval (anti-Triangle Inequality), Minkowski straightedge, Minkowski compass, hyperboloid of revolution

[Interlude] I struggled with rotation (revolution) of coordinate system problem this weekend July 6 2014. Stress test on my aging cerebrum with back-to-the-basic mathematics at high school level. On page 85, The problem of "the Minkowski equivalent of a Euclidean compass" appeared, and the ringing events make hyperboloid curve, it is written. My thinking process: To simplify the problem, the equation is revised to only-two dimension, i.e., T and X ---> t² = x² + 1 --> t = +/- SQRT (x² + 1), this is y1 and y2 of the next Grapes image. Then I tried -45-degree revolution of the coordinate axes.
[Solution 1] My deviation is basic with simple geometry but it worked. Certainly the original curve is hyperboloid! (y4 in the next image). Using Japanese mathematics free software Grapes, I validated this revolution of the axis. y² = x² + 1 (y1 blue and y2 red) -- (45-dgree rotation of axes)--> xy = 1/2 (y4 purple).
[Solution 2] Secondly I tried solution using polar cordinate.
[Solution 3] Finally my son Takao's lecture on the relationship between x² - y² = 1 and xy = 1/2.

Constructing Lorentz coordinates, co-moving clocks, no simultaneity structures in Minkowski space-time, calibration and synchronization, equal t-slices, coordinate simultaneity, relativity of simultaneity, nonexistence of simultaneity in Relativity, coordinate-based time dilation, Minkowski space-time does not support any objective measure of the speed of anything including that of light, coordinate-based Lorentz-FitzGerald contraction (March 21 2015).
Chapter Five, entitled "The Physics of Measurement": Clock definition - Ideal Clock, a light clock consisting of a ray of light that is reflected back and force between two mirrors, changing a light clock by changing it from one inertial motion to another by suddenly accelerating two mirrors in some direction, physical Lorentz-FitzGerald contraction, John Stewart Bell' thought experiment - the tread between two identically constructed rockets breaks when they are lifted off on congruent trajectories, abstract boosts and physical boosts, rigid rod, equilibrium state (I could not understand this section), "constancy of the speed of light", actual laboratory operations indicating the constancy of the speed of light, Hippolyte Fizeau, Michelson and Morley, interferometry, the Law of Light, the Relativistic Law of Inertia and the Clock Hypothesis, Maxwell's electrodynamics (August 30 2015).
Chapter Six, entitled "General Relativity": flat Minkowski space-time and curved non-Minkowski space-time, the space of constant positive curvature (e.g., the surface of sphere),"locally parallel" lines on the sphere eventually intersect (e.g., lines of longitude on a glove at the north or south pole), hyperbolic space or space of constant negative curvature (e.g., saddle-shaped two-dimensional surface), the sum of internal angles of any triangle is less than two right angles, space of variable curvature - Riemannian spaces, the Weak Equivalence Principle (the inertial mass of a body is proportional to its passive gravitational mass), active gravitational mass, "accelerometer" to make the source of a force visible (a weight free to move in a tube and attached by springs to the ends of the tube), the falling accelerometer will not show any visible effect of acceleration; it will behave just as if it were moving inertially in empty space with no gravity at all, in General relativity there is no "force of gravity", the Strong Equivalence Principle (the outcome of any experiment carried out "in free fall" in a uniform gravitational field will have the same result as if carried out in an inertial laboratory in empty space, and any experiment carried out "at rest" in a uniform gravitational field will have the same result as one carried out in a uniformly accelerating laboratory in empty space), prediction of the "bending of light by a gravitational field" by the Strong Equivalence Principle, confirmed by Arthur Eddington's eclipse expedition in 1919 (the bending of light that passes close to the sun), Einstein's Field Equation G_ab = R_ab - 1/2 Rg_ab = 8 πT_ab, G_ab: Einstein curvature tensor, tidal effect, the field equation essentially says that the way the volume of small balls of test particles (a spherical swarm of particles) in free fall behave in a region is determined by the amount of matter and energy in that region - The more matter and energy, the greater the Einstein curvature and the more volume of the ball will shrink, gravitational waves which produce tidal effects, the distribution of matter and energy constrains the geometry of space-time but does not determine it, the deviations from Special Relativity only become manifest at a larger scale (in a General Relativity setting: OTH, in small enough regions the space-times of General Relativity looks like Minkowski space-time, i.e., all physics is local), black hole, event horizon, singularity, the light cones appear to "tip" (lean) toward the center (black hole) and also "narrow", space time could also begin with a singularity, the Big Bang, an initial singularity in the history of the universe. The Hole Argument, which I could not understand. Chapter Seven, entitled "The Direction and Topology of Time": the geometry of time, Newton's absolute simultaneity, the future-directed and the past-directed, temporally orientable, Mobius strip, "compactified" - closed space-like curves and closed time-like curves (CTCs), Cauchy surface, time travel as a technical problem, direction of time.

(Start reading 2013/12/28, finished 2017/5/14)

Concepts of Force - A Study in the Foundations of Dynamics-. By Max Jammer (Harper and Brothers, New York, USA,1962, originally published in 1957 by Harvard University Press, ISBN not yet born)

I picked up this antique book at the musty but academic shop on Hongo street near Tokyo University a few years back, and left it lay for a while. As I read a few pages lately, I noticed that I can not answer the question what is force in the first place, and understand this is the core issue of physics for a long time. I love physics since high school years, but I decided to go over basics of basics to understand the universe with this classic book written by an Israeli physicist and philosopher of physics. Regarding this genre of philosophy of science, I have read The Philosophy of Science - An Introduction by Stephen Toulmin, Hutchinson Co, London, 1955, with Japanese translation edition in the undergraduate days (faculty of liberal arts at Komaba) from curiosity.

[Abstract] (Personal Memorandum, for translation of the original non-English text in Latin, French or German quoted in the book, I reasoned the context using Google Translate, feel free to refer to its result)

Preface: The modern trend toward eliminating the concept of force from the conceptual scheme of physical science is notified to me. (Up till the present, I have accepted the concept of force rather a priori without any suspicion)
Chapter 1, entitled "The formation of scientific concepts": A conceptual apparatus of objective science consists of two parts, according to the author (1) a hypothetico-deductive system, consisting of concepts, definitions, axioms and theorems, e.g., mathematics by Euclidean geometry (2) using data of sensory experience and "rules of interpretation" or "epistemic correlation", an association between observation and predictive or explanatory power is set up. The concept of force has been constantly revised, that is originally in analogy to human will power, or muscular effort, transforming to a power dwelling in physical things, related to "mass" and "momentum". In ancient Indian philosophy, there is no concept of force. They explained the physical phenomenon without a dynamic agent as force. In the 19th century, the nomenclature of force became more ambiguous when force was used to denote our present-day notion of "energy" and "work". Among the Kantian school, "force" was considered the physical formulation of "cause" and causality. To liberate from the bondage of casuality, some thought faction seeks total elimination of the concept of force from physics. The purpose of this book to critically analyze the development of the concept of force.
Chapter 2, entitled "The conceptions of force in ancient thought": The abstract concept of force as a notion of divinity can be traced back in ancient Egypt to the nineteenth dynasty. Judaism transformed the idea of force into the notion of an ethical power. The Biblical God too is frequently associated with "might", "power", "strength", "vigor", mostly by the expression "dynamis" and "omnipotens". The ancient Persian or Iranian concept of god is that of great force of nature. Thus the concept of force is closely related to religious ideas, an expression of the world soul. (Lately I occasionally heard people saying about Tohoku earthquake and ensuing tsunami "The force of nature is tremendous!", which is one example of personification of force of nature)
Chapter 3, entitled "The development of the concept of force in Greek science": The early cosmologists such as Thales conceived nature as a living being. According to Heraclitus's doctrine, nature is opposing tensions, the battlefields of antagonistic forces. Empedocles assumed force as a regulative agent in nature as love and strife, which he explained as the cause of motion. He interpreted the world as a breathing organism, love as the binding power and strife as driving love to the center (contraction of love), so to speak the cosmic systole and diastole as derived from the physiology of the body. For Plato nature has an immortal living soul, and all motion is spontaneous motion, the principle of life and soul. The ultimate origin of all forces in nature lies in the hidden world-soul. The motion of force is linked again with the divine conception. Aristotles reached radical dichotomy of physical phenomena into celestial and terrestrial processes, that was long afterward discarded with Newton's theory of universal attraction. Aristotles recognized only two kinds of forces, that is force inherent in matter "nature" (natural inherent tendency, e.g., a stone moves downwards on the slope. We may call it energy today, and he does not say that it happens by force!) and force as an emanation from substance, the force of push and pull in constant and direct contact (compulsory motion, e.g., a stone moved upwards by force contrary to inherent tendency. He calls this movement by force!). For the latter, he describes "In all cases of local movement, there will be nothing between the mover and the moved." For Aristotles there is no concept of action at a distance, friction and inertia. The query for an explanation of the connection between tides and the movement of the sun and moon lead the Stoa to a new development. Poseidonius interpreted it as a result of a universal tension, a manifestation of forces pervading all space, who conceives force as an expression linking the two subjects, mutual correspondence of action, a "sympathy": "die Welt ist zu erklären aus dem Kraft." "Sympathy" is originally from medicine in ancient Greek (Hippocrates and Galen), and The Stoa thinks the cosmos as a whole just like the human body. "Sympathy" acts by means of the pneuma, an all-pervading fluid, which is interpreted as a soul (psyche) inherent in the world, which is referred to a world soul (Plato). Poseidonius' concept of force as a "sympathy" is near to the notion action at a distance in 17th and 18th centuries. In the Roman period, the force conception of "sympathy" appears as a mixture of natural science and superstition, the concepts of physical force and occult magical power being indiscriminately intermingled. Astrology employed the concept of “sympathy” when explaining the influence of the moon not only on the tides but also on living beings, in more generalization the influence of the celestial spheres on mundane events, i.e., stars are expression of the omniscience of the world soul. In the Jewish-Alexandrian school of thought, stars and angels were recognized as manifestations of divine powers, all forces in nature are but manifestations of divine power, and the existence of invisible bonds of forces throughout the universe, by uniting all with all, were postulated (Philo). This notion of infinite divine force (dynamis) is taken up by philosophers of the Middle ages in their explanations of celestial motions, including Thomas Aquinas, according to whose doctrine of celestial intelligences "Therefore the motions of these inferior bodies (the earth, terrestrial motions), are related to the motion of the celestial bodies (, and God as the pilot and the celestial body as ship,) as to their cause." (dynamic astrology).
Chapter 4, entitled "Concepts of Force in Preclassical Mechanics": Modern science began when the belief in the divine nature of the celestial motive powers was dispensed with Jean Buridan's theory of momentum and Newton's theory of universal attraction, which does not discriminate between superior, the celestial, and inferior, the terrestrial.
Astrology was introduced into the Muslim world during the 8th century, and it was founded on Aristotelian conceptions of dynamics and concept of force. For example, Albumasar tried to explain how the tides are caused by astrological influence, without specific description of physical nature of influences. However, Alkindi under the influence of his optical investigations conceived force (not only light, heat but also any other type of force!) as an entity propagated by rays (his treatise “On stellar rays”), who expounded the astrological forces of influences by radiation of light from the stars, each transmitting its peculiar light through space. Later Alkindi’s conception of forces greatly influenced Roger Bacon’s treatment of forces. Bacon (1214-1294) employs his central (and obscure) notion “species” to clarify the propagation of forces, which “causes every action in this world”. For the sun, light (force) is species, but species is produced by the agent which excites the potential activity of the medium between the agent and the patient. For Bacon, the transmission of force is some kind of chain reaction (The action at a distance is reduced to a chain of contiguous contact-processes), and the author Max Jammer suggests that the nearest analogy to species is a wave, a form of transmission of energy or force. “Force” does not act at a distance but spreads through the medium according to the specified laws of propagation. Bacon was probably lead to his theory by studying optics (optical speculations). In 13th century, Bonaventure (1221 - 1274, Italian) proposed among the causes of the motion of a heavy body a force of repulsion exerted on it by the celestial spheres. In those days, however, the idea of attraction and repultion was not popular. William of Occam (Occam's razor!, 1299-1348) discarded the concept of species and acknowledged explicitly the concept of action at a distance, rejecting Aristotle's principle of the immediate contact between the mover and the moved.
The mathematical formulation of the law of motion was disputed at that time. Peripatetic (Aristotelian) law of motion was that the velocity v is determined by the ratio of the motive force A to the resistance B, i.e., v = A/B (In the case of equality A=B, non-zero velocity, it's a problem). Toward the end of sixth century a different law was advanced, i.e., v is proportional to A-B. A third law was advanced by Thomas Bradwardine (1290-1349, English scholar/Archibishop of Canterbury), in essense (by the author's interpretation) v = log (A/B) (when A=B, v=0! What a brilliant idea!).
How to define resistive forces? , which belong to two powers: External resistance (such opposing forces as pressure on the mobile in the opposite direction) and internal tendencies (gravity and lightness), which lead to the problem of free fall , "haec quaestio inter omnes physicas quaestiones gravissima" (This question is among the most serious physical problems, page 67). According to Aristotle, either the body moves by itself (only by a living being), or it is moved by something else. Various new theories were developed in the fourteenth century, in a mojority of them explained by an intrinsic principle, an inherent activity, by a force supposed to be seated in the mobile itself, vis mortiva (moving force). Jean Buridan (ca.1300 - after 1358, French priest) rejected intelligence (intelligences motrices, intelligence of movement) as the cause of the celestial spheres, and presented impetus theory, as the God creating the universe, communicating to these spheres an initial impetus and securing the circular motion of the stars, and "on the seventh day resting from the work he has done",,,, (very similar to the 'Intelligent Design' proponent's view, Akio). In the sixteenth century, force was generally accepted as (1) "like attracting like". (2) cosmic sympathy (Neo-Platonic theory), and (3) assumption of higher intelligence as the motive power behind the stars (Peripatetic). Nicolaus Copernicus (1473 - 1543) interpreted gravity as a tendency or "appetition" of the parts to be united with the whole to which they belong - inherent tendency or propensity to be united with (the theory of sympathy or cosmic harmony). According to Girolamo Fracastoro (1478 - 1553), it is "mutual attraction of like to like". Antonius Ludovicus, occasionally hailed as a precursor of Newton in the eighteenth century, concluded "similia similibus conjugantur", like joins like; the faculty of attraction maintains order in the universe. These ideas, still metaphysical and lacking mathematical implications, transformed the mythical conception of a world soul into a more scientific idea, i.e., various parts of the universe, by being parts of one organic whole (Ludovicus follows Galen and Hippocrates), influencing each other mutually. However, the conception of nature as an autonomous, self-sufficient organism with its own and immanent laws had the character of merely occult qualities ("natural magic"). Important exception was Bernardino Telesio (1509-1588, Italian), who reduces all active forces to the force of expansion by heat and the force of contraction by cold (heat is the originator of motion and life, cold of rest and rigidity). However, the great difference between the conception of a universal force and Newton's universal attraction was the complete absence of any mathematical determination. William Gilbert (1544-1603, English doctor and physicist, early supporter of Copernicus) and Johannes Kepler (1571 - 1630) were percursors of the new concept of force of the seventeenth century. Aristotelian differentiation between terrestrial motion in straight lines and celestial motions in circles has to be discarded. Gravity must be a property common to all celestial bodies.
Chapter 5, entitled "The Scientific Conceptualization of Force: Kepler": Johannes Kepler reached a decisive stage of conceptualization of force by his new approach to mathematical formulation, a quantitative definition of force. In a letter of March 28 1605, he conceives the universal nature of gravitational forces and calls gravity a “passivity” rather than an activity. Federicus Chrysogonus (1472-1538, Italian), Patritius (1529-1597, Italian), Definus, Caesareus, and Telesio had already advanced the theory that the moon causes the tides, and in 1607 Kepler generalized these findings: The oceans are attracted by the moon much as all heavy objects, the oceans included, are attracted by the earth, suggestive of Kepler’s concept of the universal character of attraction. Before Galileo’s observation of the moon surface with telescope, Kepler accepted in Mysterium cosmographicum that the physical conditions on the moon and on the earth are alike. In his “first modern book in astronomy” Astronomia nova he introduced “Here is the true doctrine of gravity. Gravity is a mutual attraction among related bodies which tend to unite and conjoin.” With regard to the motive cause of planetary motion, Kepler firstly supposed the existence of (immaterial) animated being (“soul”, anima motrix, species immateriata), but increasingly the conviction becomes stronger that these phenomenon is of an essentially physical nature. Especially, based on Tycho Brahe's exceedingly accurate observational data, he recognized that the planetary velocity is greatest at perihelion and least at aphelion - the determination of the dependence of planetary velocity upon distance - “Kepler’s second law” (as we call today): the radius vector joining the sun and any given planet sweeps out equal areas in equal times, cf. Kepler’s original formulation of this law: the velocity of any given planet is inversely proportional to its distance from the sun (strictly speaking only correct at the apsides where the radius vector is perpendicular to the tangent). Kepler introduced an attractive force as the cause of the fluctuations in speed of the planets, and explained "The planets are magnets and are driven by the sun by magnetic force". He says "When I realized that these motive causes attenuate with the distance from the sun, I came to the conclusion that this force is something coporeal (, that is, mechanical)". His idea of reciprocity, i.e. stone vs. earth, and moon vs. earth, one attracts the other, also the other attracts the one, set the concept of force free from all animistic ingredients. Der Relationsbegriff is necessary to ask for the mathematical formula of the intensity of force. Kepler conjectures first that this intensity is proportional to the reciprocal value of the square of the distance, but soon rejects this suggestion (alas!). From his "second law", that is, rv = constant, he asserts that the force is proportional to the reciprocal of the distance. In spite of this error, Kepler transformed the concept of force from its Platonic form and interpretation to an essentially relational concept. Main oppositions to his theory in those days were mainly based on the tenet that the principle of action must lie in the agent itself, the motion of a planet has to have its cause in the planet itself (Ismael Bulliadus 1605-1694, French astronomer. [Quote from Wikipedia] In this work he strongly supported Kepler's hypothesis that the planets travel in elliptical orbits around the Sun, but argued against the physical theory the latter had proposed to explain them[/Quote]). Bullialdus could not reconcile his revived Aristotelian conceptions with the undeniable fact of the elliptical paths of the planets.
Chapter 6, entitled " 'Force' and the Rise of Classical Mechanics": Galileo have laid the foundation of classical physics, but his contribution may be regarded as complementary to that of Kepler. Galileo studied the kinetic aspects of motion, not delving into the nature of force itself. Still with Gelileo various synonyms for the designation of “force” are found, i.e., forza, potenze, virtu, possanza, momento della potenza, etc. (impetus, ability, energy, moment). Weight is for Galileo a natural inclination of a body to come nearer to the center of the world, saying "things are so constituted that the heavier bodies rest beneath the lighter bodies" (“placuit autem Summae Providentiae in hunc distribuere” --> it was decided Supreme Providence distribute them in this order). He uses moment as “Bodies of equal weights and moved with equal velocities have equal forces and moments; equal weights with unequal velocities have moments in proportion to their velocities”. Originally moment was conceived as the torque of a force (static concept), then becoming transferred to a dynamic idea. In his investigation of the laws of free fall, although he still does not possess a clear definition of mass, he already reduces the action of force to a gradual increase of velocity, and conceive it as an activity from outside. In his Dialogue on the great world systems Galileo related terrestrial gravity (which moves a stone downwards) to the motive forces of the planets (which move Mars, Jupiter, Moon round)(he did not take theory of magnetic forces too seriously). With the recognition of the law of inertia since as early as 1585 (Giambattista Benedetti as part of his impetus theory, 1530 - 1590, Italian) two alternative possibilities occurred: either to conceive force as the cause of change of motion, or to abolish the notion of force altogether. In any case velocity is no longer considered as an indication of the existence of force or of its measure. Rene Descartes (1596 - 1650) chose the second alternative - the rejection of force. For Descartes, all physical phenomena are to be deduced from only two fundamental kinematic assumptions: the law of the conservation of quantity of motion (m1*v1 + m2*v2 = m1*v’1 + m2*v’2) and his theory of swirling ethereal vortices, and rejected any possibility of an action at a distance. He used a better known experiment to demonstrate his vortex theory of gravitation, with two glass cylinders, each containing water and a wooden ball, attached, slightely inclined above the horizontal, to a common axis of rotation. When the system is at rest, the wooden balls will occupy the highest (peripheral) possible places in the cylinders, but when the system spins the balls will be forced toward the center (or axis of rotation). According to Descartes, the water in the cylinder is his vortical medium of ethereal particles and the balls are terrestrial objects. In the first half of the 17th century, the existence of some kind of central force, directed from the sun to the planets, was generally accepted. Giovanni Alphonso Borelli (1608-1679, Italian), after observing the satellites of Jupiter over years, thought the tangential motion along the orbit is accounted by a moveing force, inherent in the planet and decreasing in magnitude with increasing distance from the sun - If moving in a path too distant from the sun, a planet approaches the sun, since its motion of progression gives way to the tendency of approach to the sun. He contributed to the rejection of magnetic forces as the cause of planetary motions. Christian Huygens (1629-1695, Dutch) proved the centrifugal force, that is given by mathematical expression, the formula F = mv^2/r (De vi centrifuga, published 1703 Leiden). He used the concept of conatus, which approximately means effort, exertion, or impulse, to signify the action of a force. Huygens, while studying the properties of centrifugal force, realized the possibility that this force may counterbalance the gravitational force exerted by the sun on the planets and thus keep them in their proper orbits (The idea of regarding this centrifugal force as a fictional or inertial force is of a much later date). Actually Huygens was just one of the numerous scientists on the continent who could not reconcile themselves to the conception of force as an action at a distance. For him and them, such a conception meant to surrender scientific reasoning to occult qualities. He defended a vortex theory similar to that by Descartes in 1668, and spoke about his own vortical theory of gravity in 1689 at the Royal Society of London in the presence of Newton (Discours de la cause de la pesanteur, published 1690 Leiden).
Chapter 7, entitled "The Newtonian Concept of Force": Much has been written on Newton's concept of gravitation but next to nothing on his concept of force in general. His concept of force is intimately related to his profound study of gravitation, because the problem of a dynamical explanation of planetary motions to account for Kepler's three laws was the question of the hour, which imply that the force exerted by the sun on the planets decreased with distance, and an inverse-square law had been suggested by Bullialdus (Hooke mentioned to Wren and Halley the same thing in 1684). Realizing the the weight is a function of its distance from the center of the earth, Newton made a clear distinction between weight and mass, "quantity of matter", as a basic concept in mechanics. On the concept of mass, the definition of momentum, or in Newton's words, "quantity of motion", was no longer a difficult task, with Galileo's principle of inertia being duly considered. The term "force" appears for the first time in Newton's magnum opus Philosophiae naturalis principia mathematica in Definition III: "The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, continues in its present state, whethter it be of rest, or moving uniformly forwards in a right line." Inertia, in Newton's opinion, is some kind of force that is inherent (insita) in matter and latent as long as no other force, impressed upon the body, "endeavors to change its condition." (The innate force of matter is a force of inactivity, inertia. It is resistance to the impressed force of another). In contrast to this "innate force", Definition IV defines "impressed force": "An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.", which (a) is pure action, transient in character, (b) no longer remains in the body once the action is over, (c) may have different origin, as "percussion, pressure, or centripetal force." Since every change must have its cause (Newton's metaphysical principle of causality), the change of motion is an effect and the impressed force its cause, and Cessante cause cessat effectus. Definition V introduces the centripetal force: "A centripetal force is that by which bodies are drawn or impelled or any way tend, towards a point as to a centre.", "Of this sort is gravity." He mentioned in a separate definition not on percussion nor on pressure, but specially on centripetal force! Definition VI: The absolute quantity of a centripetal force is the measure of the same, proportional to the efficacy of the cause that propagates it from the centre through the spaces round about. Definition VII: The accelerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time. Definition VIII: The motive quantity of a centripetal force is the measure of the same, proportional to the motion which it generates in a given time. Accelerating force in Definition VII in the conventional modern meaning is equal to the acceleration. The motive force in Definition VIII corresponds to the force, ditto in conventional meaning as mass times acceleration. Absolute force in Definition VI is ascertainable only by accelerating or motive force, Newton seems to have already discarded this notion. Now, Newton’s three famous axioms of motion (or laws of motion) - Law I: Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by force impressed upon it (the principle of inertia). Law II: The change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which that force is impressed [F ∝ Δ (mv)]. Law III: To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. Newton credits the first two laws of motion to Galileo and Huygens. Newton ultimately undecided position with respect to the real nature of force in general, and to gravitational force in particular, stressing in the discussion after Definition VIII “considering those forces not physically, but mathematically.” Einstein applauded later ”Newton himself was better aware of the weakness inherent in his intellectual edifice than the generations of learned scientists which followed him.” Following the third law of motion, Corollary I: A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately. This formulation is very important because it characterized force as a vectorial quantity. In the following 18th century, this parallelogram theorem was a priori truth for Daniel Bernoulli, however, the problem of its validity occupied the minds of d’Alembert and Denis Poisson as late as 1833, though Leonhard Euler had made it clear that the theorem cannot be probed analytically without assumptions. Simon Stevin (1548/49 - 1620, a Flemish/Flandre mathematician and military engineer) had published the theorem of the triangle of forces, which is equivalent to the parallelogram diagram of forces. In 1687, the year of publication of the Principia, two French contemporaries too proposed the theorem of parallelogram of forces. These coincidences clearly show that the way to add forces was well understood. With regard to the motion resulting from a given force, i.e., uniform motion or accelerative action, the conception of force as a cumulative effect has a history of its own. Da Vinci already embraced this view that the arrow, when released from the bow, is not projected only at the moment of the greatest tension, but also receives additional impulses at the other positions. A sequence of the consecutive impulses as impressed upon the flying arrow is similar to the action of gravity. Newton’s derivation of the parallelogram tacitly assumes that the action of one force on a body is independent of the action of another one, the assumption that is far from self-evident. French elasticians, e.g., Barre de Saint-Venant （1797-1886, French engineer, mathematician, physicist） rejected the validity of parallelogram for the microscopic realm (the actions at imperceptible distance that produce the elasticity). Regarding Newton’s conception of universal attraction, historians of science have argued on two respects. First, whether Newton conceived gravitation as a pure action at a distance or not. Second, direct influence upon Newton by Jacob Böhme (1575-1624, German Christian mystic and theologian), mystic writer of the seventeenth century. [For first question] Newton discusses some vague speculations on an ethereal medium as a possible explanation of gravity around in 1675 and 1678, "from the top of the air to the surface of the earth, and again from the surface of the earth to the centre thereof, the aether is insensibly finer and finer,," and "the bodies are descending to make room for the finer aether to be lodged in the pores,,,", still he says, "my notions ,, are so indigested, that I am not well satisfied myself with them." (in his letter to Boyle in 1678). Newton preferred to treat gravitation independently of whether it is an action at a distance or is caused by contiguous action between ether particles and ordinary matter in the first and second edition of Principia, i.e. "hypotheses non fingo" (Latin for "I feign no hypotheses", "I frame no hypotheses", "I contrive no hypotheses", or "I do not pretend to know the hypotheses", from his General Scholium in 1713) - "Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of power". We have somewhat Newton's agnostic attitude with respect to the cause of gravitation found in the Principia ("whatsoever be the Cause") and several other references. But on the whole, he was opposed to any metaphysical or theological interpretation of gravitation. [For second question] There were many contentions of Newton's indebtedness to Böhme by many authors and biographers on Newton, however, there is no convincing evidence that Newton ever read even one single work of Böhme's. Even Louis T. More, the distinguished biographer of Newton once thought this tradition as established fact, but altered his view and Hobhouse has recently disproved its veracity altogether (My comment: This is one more example of a doubtful history starting to walk on its own as an established fact). In spite of early opposition, as voiced by Leibnitz, the notion of force as action at a distance became a basic concept for the great classical edifice of theoretical mechanics (September 2 2012).
Chapter 8, entitled “The Concept of Force in Theological Interpretations of Newtonian Mechanics”: Many theoreticians of early 18th century viewed action at a distance as a disguise of occult quality and criticized as incomprehensible. As indicated in the previous chapter, Newton himself took a stand of Hypotheses non fingo with regard of the cause of gravitational force. To reconcile this new and successful notion (the motions of the planets and comets can be explained!) with traditional ideas, force and gravitation were conceived as manifestations of divine omnipresence and omnipotence on a metaphysico-theological foundation and to assimilate into the Neo-Platonic thought that was still active in England at the time of Newton. Ralph Cudworth and Henry More are the famous Cambridge Platonists. Cudworth called “Plastick Nature” and More called “Immortality of the Soul” to account for the motion of material substances, both equivalent to a variation of the Neo-Platonic concept of a world soul. Henry More’s conclusion: “Nam sic mobilia omnia moventur a Deo” --> “In fact, all these movable things are moved by God”. More exerted a strong influence on Locke, Newton, and Clarke, and Newton recognized a divine intervention in the past as the first cause that started the motion of the planets and of the comets (“there is a being incorporeal, living, intelligent, omnipresent,,” in Opticks), but little is said about force as an immediate divine operation. To interpret natural phenomena in a theological manner, Newtonian principles were incorporated, assimilated and made use of for apologetic purposes by English humanists. In Astrotheology William Derham speaks of gravitation as “an active quality, impressed on matter by the great Creator.” In addition to many English commentators of theological interpretations of Newton’s conceptions of force and gravitation, P.L. Moreau de Maupertuis insisted that God selected the inverse-square law between other possibilities. It was God’s choice which endows nature with a harmonious unity.
Chapter 9, entitled " 'Dynamism: Leibniz, Bostovich, Kant, Spencer": Leibniz had recognized many points of contact with those of Cambridge Platonists, but considered that the explanation of physical phenomenon should be accounted for by the physical sciences and their mathematical method of reasoning. In 1669 he still adheres to the Cartesian conception of objects as mere extensions (For the meaning of "extension" the Wiki page was helpful for me, which is roughly speaking, the property of "taking up space"), but in his Hypothesis physicae nova in 1671 he emphasized that the pure kinetic point of view by Descartes is too one-sided and must be supplemented by a dynamic principle. For Leibnitz, a moving body is different from a body at rest, its motion is not merely a successive occupancy of different places in space, it is a state of motion at each separate moment, this state of continual change of place involves some effort that must be the outcome of an inherent force or activity, that is inertia. Leibniz concludes his mathematical function of force in its state of action as mv², which is conserved, in "Specimen dynamicum" (1695). The famous controversy regarding whether the appropriate measure of force is the "quantity of motion, mv" (Cartesians) or the "vis viva, mv²" was essentially a mere battle of words, i.e., different concepts under the same name. One half of the vis viva, 1/2 mv², is known today as "kinetic energy", which was referred to by Leibniz unfortunately as "force" (September 30 2012). However, an anticipation of the principle of conservation of energy was at the bottom of this issue. Huygens had in mind a principle of conservation of energy, and had shown that the sum of the products of the masses and the squares of their respective velocities, before the (elastic) collision, is equal to the corresponding expression after the collision (1699). Leibniz maintained consistently that "force" is conserved in the universe. In the case of inelastic collisions, Leibniz says, the loss of "force" is only apparent, "but scattered among the small parts,,, as when men change great money into small." Modern theory of transformation of energy will contend that the decrease of kinetic energy is accounted by the quantity of heat caused by the impact, that is ultimately by the increase of molecular energy. In contrast, Newton's followers were of the opinion that "force" decreases constantly in inelastic collisions, and new "forces" have to be injected from time to time by God to the universe, just like God Almighty winds up his watch (October 7 2012). Leibniz opposes such divine intervention as incompatible with God's power and foresight, and insisted on conservation of "force". For him, "Agere est character substantiarum" (Action is character substance), and "Quod non agit, non existit" ("Whatever does not act, does not exist"). "Quod non agit, non existit" (Whatever does not act, does not exist) is characteristic for Leibniz's dynamism, and in contrast to traditional "Operari sequitur esse" ("Action follows existence"). Leibniz rejected Newton's theory of gravitation and in particular action at a distance. He assumed that motion results only from contact, "Corpus a corpore non moveri, nisi contiguo et modo" --> The body is not moved by the body, unless in a contiguous manner (Leibniz's Impact theory). It was Roger Joseph Boscovich (1711-1787, Croatian physicist, astronomer, mathematician and Jesuit) who advanced the real Leibnizian theory of dynamics. He analyzes whether the change of velocity in the impact of two bodies changes continuously or discontinuously, say, two equal bodies, one with a speed of 6 and the other with 12, moving in the same direction on a straight line, the slower in front of the faster. According to the law of conservation of momentum, both bodies will move on with the same common speed of 9 after impact. Boscovich concludes that velocity changes abruptly, but this violates the law of continuity, which denies the passage from one magnitude to another without passing through intermediate stages. His solution was "Oportet, ante contactum ipsum immediatum incipiant mutari velocitates ipsae." (It is necessary, before the immediate contact, to begin to change their velocities). In the case of the impact of two bodies the force is repulsive, he argues. His theory of conception of force is summarized as: The force of repulsion increases indefinitely as the distance decreases (as immediately before the impact), and at greater distances the repulsive force changes its sign, and becomes attractive, thus accounting for the common phenomenon of gravity and gravitation. According to Boscovich's theory of force, it is the same to assert that contact never takes place, thus his theory reduces contact phenomenon to actions at a distance and eliminates impact as a fundamental concept of mechanics. (Nov 17) After investigating the theory on the force by Kant, Spender and others, the author concluded that the conception of force as the primordial element of physical reality, advanced by Leibniz, Boscovich, Kand and their followers, was not very fruitful and productive for the advancement of theoretical physics, as Thomson and Tait called such dynamic doctrine as "untenable theory." (Dec 22)
Chapter 10, entitled "Mechanistic Theories of Force (Gravitation)": In post-Newtonian concepts of force, there were two major trends of thought, firstly Newton's foundation of classical mechanisms, which religious school of commentators used as a proof of their theology, and secondly the school of dynamism which conceived force as the ultimate essence of physical reality. From the time of Robert Hooke to the present day, innumerable mechanistic theories of gravitation have been published. Newton's Principia did not put a sudden death end to Cartesian physics of ethereal vortex theory, however, with the gradual recognition that Cartesian physics is incapable of accouning for Kepler's planetary laws (Newton's theory could derivate that), including that of John Bernoulli's "central torrent" of radially converging ethereal particles, the number of publications based on vortical mechanics decreases rapidly. In different model to Bernoulli's, Georges-Louis Le Sage (1724 in Geneva - 1803 in Geneva) exposited other mechanistic theory of gravatation in which a homogeneous torrent of ethereal particles are traversing space in straight lines in all directions. A single body placed far away from other matter would remain at rest, however, if a second body is placed nearby, it will "screen" off the particles (screening effects), and the two bodies on the same straight line will approach each other. At first glance his idea looked brilliant and simple, however, difficulties appeared soon, including (1) the situation that three bodies are placed in one straight line, e.g., an eclipse of the moon, and (3) the mass dependence of gravitational force, and was discarded as Sir John Herschel wrote "The hypothesis is,,,, too grotesque to need serious consideration" (1865). Leonhard Euler mentioned the question of gravitation only in two of his works, Dissertatio de magnette (1743) and his Letters to a German Princess (1760), in which he reverted to an ether theory, ",,, though its manner of acting may be unknown to us, than to have recourse to an unimaginable property." Euler adopted this cautious and reserved attitude, because he realized that so far no purely mechanical explanation of gravitation had been afforded. John Herapath's thermogenetic theory of gravitation was one more example of mechanistic explanation of gravitation, reducing gravitation to the effect of temperature differences (1816). In summary, the mechanistic explanation of gravitation was the most engaging and promising problem of the middle nineteenth century, yet, today these papers have little importance. These theories are ignotum per ignotius ( --> unknown by unknown) or devoid of epistemic correlations of experimental verifiability (Jan 5 2013).
Chapter 11, entitled "Modern Criticism of The Concept of Force": The third thought (in addition to dynamism and mechanistic theories) conceives force either as a primitive and irreducible notion or as a purely relational concept, devoid of a separate ontological status and to be defined only operationally (empirical school of thought). John Keill (professor of astronomy at Oxford), one of the first to promulgate Newtonian physics, compared physical forces with unknowns in algebraic equations (quantitates incognitae --> unknown quantities) ",,, as in algebraical equations we denote the unknown quantities by the letter x or y,,,," (Latin original 1702, English version 1726), meaning "We can handle, measure, or compute their quantitative aspects without knowing what they really are." Samuel Clarke, in defense to Leibniz' attack on Newton's conception of attraction, replied ",, all this is nothing but a phenomenon, or actual matter of fact, found by experience. ,,, Philosophers therefore may search after and discover that cause, if they can,,,, But if they cannot discover the cause, is the effect itself, the phenomenon, or the matter of fact even the less true?" (1717, very near to the operational concept of force and gravitation). Soon this issue transcends the technicalities of physics to the problems of the philosophy of science, the evaluation of scientific knowledge as a whole. George Berkeley (Bishop Berkeley, 1685 - 1753) argued in his De motu (1721) and Siris (1744), "Force, gravity, attraction and similar terms are convenient for purposes of reasoning and for computations of motion and moving bodies but not for the understanding of the nature of motion itself" (De motu), and "Force" in his view has the same status in science as the notion of "epicycles" in the Ptolemaic system of astronomy, which may lead to correct results ("explain the motions and appearances of the planets") but they are not part of nature itself ("not principles true in fact and nature"). According to Berkeley, the only objective of physical science is to find out the regularities and uniformities of natural phenomena (and not to supply causal explanations). All that natural science can supply is an account of the relations among symbols and signs; but the signs should not be confounded with the vera causa, the real cause of the phenomena. Real causes are not the subject of physical science. According to Berkeley "We cannot know or measure the forces otherwise than by their effects, that is to say, the motions, which motions only, not the forces, are indeed in the bodies.,,,, But what is said of forces residing in bodies, whether attracting or repelling, is to be regarded only as a mathematical hypothesis, and not as any thing really existing in nature" (Siris). Explanation with use of the term "force" may be valid (which makes it possible to subject the world to human calculations), but not real or "true", he argues. The real causes of motion are agents, can be only a spirit, and the final cause is God. For the philosophically minded of the generation of Hume (1711-1776) and Pierre Louis Moreau de Maupertius (1698-1759), it became more and more obvious that the scientific notion of force has little to do with causal explanation, and connection between "cause" and "effect" in science is not a matter of logical inquiry, but of experimental experience or observation. "Force" is a construct, descriptive names of perceptible and measurable empirical relations. Maupertius declares that forces are "un mot qui ne sert qu'a cacher notre ignorance" (a word that only serves to hide our ignorance), and the concept of force is but an invention to satisfy our desire for explanation (Jan 13 2013). According to David Hume's thesis (British empiricist, 1711-1776), the origin of the concept of causality is found in habit reduced causality to a mere association of perceptions, and the possibility of understanding the nature of an objective connection between cause and effect must be denied. Maupertius simply applied these ideas to the most fundamental phenomenon in mechanics, the impact of two bodies. With the investigations of Boscovich, Kant and Maupertius, the very Newtonian concept of force itself became the object of critical analysis. Yet for Boscovich and Maupertius, the origin of a certain body's motion had still an answer, which is given already by Berkeley: God. The empirical, antimetaphysical attitude in mechanics was culminating toward the end of the nineteenth century in the attempts of Kirchhoff, Hertz, and Mach to eliminate the concept of force from science. This change of opinion can be traced back to Berkeley's criticism of Newton's mechanics and to the Humean analysis of the concept of causality (Jan 26 2013). Leonhard Euler (1707 - 1783), in his Mechanica, proposed to demonstrate that Newtonian mechanics is an apodictic science (apodictic: of the nature of necessary truth or absolute certainty By Merriam-Webster), of necessary truth. His Definition I states "Potency is a force which initiates the motion of a body at rest or which alters its motions," (gravity is such a force). Euler's "potency" corresponds to what was generally called "accelerative force" (vis accelerativa). Jean Le Rond d'Alembert (1717 - 1783), in his Traite de dynamique, consciously ignores causes of motion, and for him all causes other than impact or impulse are only indirectly ascertainable through their effects of accelerating or retarding the motion of bodies. Consequently, he introduces what he calls "accelerative force" (force acceleratrice), φ, by the equation, φ dt = du (t: time, u: velocity)(recall Newton's concept of "accelerating force" in Definition VII is numerically equal to the acceleration). D'Alembert defines "motive force" (force motrice) as the product of accelerative force and mass, although he seems to be unable to supply a strict definition of mass. Lois de Lagrange (1736 - 1813) published monumental Mecanique analytique in 1788, in which mass times accelerative force (acceleration) is his formula for mathematical unification of mechanics. Lazare Carnot (1753 - 1823 French politician, engineer, and mathematician, father of Sadi Carnot) admits in his Principes that there are two different approaches to the study of mechanics; in one the subject is regarded as a theory of forces as the causes of motion, and in the other mechanics is interpreted as the theory of motion itself. According to Carnot, the first approach has the disadvantage of being based on an obscure metaphysical notion, that is force. In order to avoid the metaphysical notion of force and not to distinguish between cause and effect, Carnot prefers the second method of approach, however, his treatment is not always consistent and he introduces the concept of mass as an intuitive notion (like d'Alembert). He defines quantity of motion in the conventional manner, at his disposal the concept of mass, as the product of mass and velocity: mv, and introduces the concept F dt = d(mv) (February 10 2013). Barre de Saint-Venant (1797-1886 French mathematician and physicist), whose ideas show a striking similarity to those of Carnot, i.e., the problem of mechanics should be be answered in terms of distances, times and velocities. He defines mass: The mass of a body is the ratio of two numbers which express how often the body and a standard body contains parts which, if separated and then brought into mutual collision, two by two, communicate to each other opposite equal velocities. He hopes physics will not have any need of an occult and metaphysical nature (concept of force, causes of motion), but based solely on concepts of velocities and their changes. About thirty years later, Mach, Kirchhoff and Hertz carried out that program (February 24). In this gradual last stage of eliminating the concept of force from classical physics, Ferdinand Reech (1805-80, Alsatian teaching in a naval college at Paris) laid the foundation of "the school of the thread" (ecole du fil in French), who took as the starting point the concept of force almost in the Aristotelian sense of push or pull. For them the true and genuine notion of force is that of pressure or traction (March 9). According to his mechanics, the fundamental dynamical phenomenon is the elongation of an elastic thread or string, which is not only the indication for the existence of a force, but also the dynamometric measure of the intensity of the force. Then it is easy to show how forces can be compared and measured relative to each other. But his theory cannot explain "field forces" such as gravitation, electrostatic and magnetic attractions or repulsions, to which he refers as "causes mysterieuses agissantes" (--> acting mysterious causes). Ultimately, the proponents of the "dynamics of strings" tried to construct their system of mechanics from an operational point of view (Although a reinstatement of the notion of force, his system nevertheless shows strong antimetaphysical tendencies, e.g., the term force does not denote any cause of motion). In this respect, Ernst Mach (1838-1916, Austrian-Czech physicist and philosopher) more successfully presented his deanthropomorphic, positivistic or operational doctrine of mechanics that divested it of the concepts of cause and force and adopted the purely functional point of view. Mach's "experimental propositions" were published in 1868 - First definition: The negative inverse ratio of the mutually induced accelerations of any two bodies is called their mass ratio (Akio' comment: In my understanding, m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2' --> m1 * (v1' - v1) = - m2 * (v2' - v2) --> m1/m2 = - Δv2/Δv1), Second definition: The product of the mass value and the acceleration induced in that body is called the moving force. Here, force was only a name to signify the product of mass and acceleration, and any other name would have done as well. This replacement of the cencepts of cause and force by mathematical functions or relational cencepts was, in his view, not only the program of mechanics but that of science as a whole, and he declared "I hope that the science of the future will discard the idea of cause and effect, as being formally obscure" (Popular scientific lectures, 1894) (March 23 2013). Gustav Kirchhoff (1824 - 1887, German physicist) advanced similar views, who wrote "Mechanics is the science of motion; its task is defined as follows: to describe completely and in the simplest manner the motions which take place in nature" (Lectures on mechanics, 1876), and in his view mechanics has to discard the search after causes. To descrive the motions, only the second derivative (d²x/dt²) contributes,,, Why just the second and why not the third and fourth? The second derivative contributes most to the simplification of the description of the motions, he explains. Heinrich Rudolf Hertz (1857 - 1894, German physicist) published his conception of mechanics posthumously in 1894 (Principles of mechanics), and it was strongly influenced by his work in electrodynamics, which leads to his averstion to the traditional concept of force (in Newtonian mechanics). He thought it necessary to account not only for electrodynamic forces, but also of gravitational forces, for all actions at a distance, and finally for all mechanical forces, by some mechanism of concealed masses and motions (just as Maxwell conceived electromagnetic forces as it is). Hertz criticizes the third law of motion, and concludes that what is commonly called centrifugal force (opposing force in the third law), e.g., in a circular path of the stone tied to a string whirling around, is not a force at all (it is inertia). Hertz propsed his new arrangement of the principles of mechanics, starting with only three independent fundamental conceptions: space, time and mass, and force as an independent autonomous conception is avoided (in great resemblance to Kirchhoff's). Hertz proposed single fundamental law: Every natural motion of an independent material system is such that the system follows with uniform velocity the path of minimum curvature (Akio: Actually, I could not understand this law - the path of minimum curvature). With the works of Mach, Kirchhoff and Hertz the logical development of the process of eliminating the concept of force from machanics was completed. On the other hand, a more serious argument in the defense of the concept of force was the appeal to psychology and physiology. Francois Pierre Gonthier Maine de Biran (1766-1824, French philosopher) declares that the origin of the law of causality is found in the fact that the act of our will as cause is followed by the movement of our limbs as effect. According to Brian, the Cartesian "Cogito ergo sum" has to be "Volo ergo sum" (I wish, therefore I am, or I want, therefore I am). Arthur Schopenhauer (1788-1860, German philosopher, the author of The World as Will and Representation in which he claimed that our world is driven by a continually dissatisfied will, continually seeking satisfaction) rejected this view "Will and the action of body are one and indivisable, apprehended in a double manner.", and according to Shopenhauer the notion of force is thus the last one to be questioned. From such reflections on the immediacy of the psychological experience of forces the attack was launched against those who affirm that we know nothing but matter and motion (eliminator of the concept of "force"). William B. Carpenter (1813 - 1885, English physician, invertebrate zoologist and physiologist) argues that the concept of force stands in the same relation to the sensation of muscular effort as the concept of motion to the sensation of visual perception. These arguments seemed to be strong, it did not take long to invalidate them. That the experience of strain or tension (during muscular contraction) is a purely subjective phenomenon and not necessarily a real property of physical nature, in the same sense as light, colour and sound are not necessarily outside the mind, as emphasized by Peter Guthrie Tait (1831 - 1901, Scottish mathematical physicist) in his article "On force", which was followed by many scientists who favorably accepted the ideas advanced by Mach, Kirchhoff, and Hertz. DuBois-Reymond (1818-1896, a German physician and physiologist) pointed out the irresistible tendency to personification and its role as a factor in the formation of our concept of force ",,,What do we gain by saying it is reciprocal atrraction whereby two particles of matter approach each other? ,,, of a hand which gently draws the inert matter to it,," (Untersuchungen ueber thierische Elektrizitaet, 1848-1860, Berlin). Karl Pearson (1857-1936, English mathematician) pointed out in this Grammar of science (1892, London), "Primitive people attribute all motion to some will behind the moving body - the sun carried round by a sun-god -. The notion of force as that which necessitates certain changes or sequences of motion is a ghost of the old spiritualism". Henri Poincare discussed the anthropomorphic origin of the concept of force in this Science and hypothesis. An interesting, but problematic argument in favor of ideas expressed by Mach, Kirchhoff, and Hertz was advanced by Bertrand Russell (1872-1970, a British philosopher, logician, mathematician, historian, and social critic). Russell's denial of the validity of the concept of force hinges on two provocative arguments; first, a limit, a mere number, that is, the second derivative of destance with respect to time, cannot be a real (physical) event, and consequently it cannot be the effect of anything; second, components of acceleration or of forces are merely fictitious.
These are opinions pro and contra the necessity of the concept of force in mechanics (Aug 18 2013).
Chapter 12, entitled "The Concept of Force in Contemporary Science": When viewing the process of eliminating the concept of force from mechanics with Mach, Kirchhoff and Hertz in retrospect; at first, Kepler found the notion of force a convenient concept to explain planetary motion, and it was essentially a methodological device. The concept of force in older prescientific and semiscientific stages became loaded with metaphysical, spiritual and other extrascientific connotations, which seemed to be an instrument to satisfy the human desire for causal explanation. The author asks himself if the whole story of the concept of force in classical mechanics merely the creation of an illusion (the status of an almighty potentate of totalitarian rule over the phenomena in Newtonian dynamics) and its evanescent dissolution ("the king for a day"), and his answer is: No! It played a most constructive role in the advancement of science and therefore wholly justified its existence (August 24). The concept of force in contemporary physics plays the role of a methodological intermediate comparable to the so-called middle term in the traditional syllogism. (Akio's interpretation: A is B, B is C, and therefore A is C. Middle term B drops out. The concept of "force" is the middle term B). A certain body A moves on a certain trajectory B, when surrounded by a given constellation of bodies C, D,,,, which may be gravitating, electrically charged, magnetized,,,, When this configuration is denoted by the symbol X, thus ma = Φ(X). If we replace test body of mass m with another body of mass m', an acceleration will be ma = m'a' = Φ(X). Force is a nominal definition of Φ(X) (force = mass x acceleration). It is an analytic statement with respect to the nominal definition of force. If the function Φ tends to zero (distance variables increase indefinitely), we express the law of inertia (Newton's first law is the boundary condition of the function Φ). Mach's determination of mass as negative inverse ratio of the mutually induced accelerations predisposes mutually acted two bodies, but the question arises whether his approach holds also in the more general cases of n bodies. The author demonstrates n vector equations have determinable αki if n <= 4. Mach's method becomes inoperable as soon as n >= 5. In recognition of the highly abstract character of the concept of force, modern physics saw some justification in modifying its logical status and considering it as based on the concept of work or energy. In consequence, force is defined as the space rate of change of energy: Force is a covalent vector (f1, f2, f3,,,,,fn in an n-dimensional coordinate system) satisfying the equation work = Σ(i=1 to n) f_i rⁱ
In the theory of special relativity, the force is defined as,
f = d(mu)/dt
which allows for a change in mass as well as for a change in velocity. According to special relativity, rest mass m₀ depends on the velocity,
m = m_o/(1 - u²/c²)^1/2
in which if u = 0, m = m_o, and if u approaches the velocity of the light, the denominator tends towards zero, and the value of m increases without bound (this explanation is from Sander Bais: The Equations - icons of knowledge, from Harvard University Press 2005). The component of the relativistic force are given by the rate of change of the momentum components,
f_n = d {m_ox'_n/(1-u²/c²)^1/2}/dt
where x'_n is the time derivative of the coordinate x_n with respect to the local time t, and u the velocity of the moving body.

Here I must humbly admit that to understand the last 8 pages, which discusses general theory of graviation and Riemannian coordinate system for the spece-time continuum, is beyond my current ability, so for the meantime I would cease struggling in the no man's land.


(Start reading 2012/3/27, finished 2013/12/28)

The Selfish Gene (30th Anniversary edition). By Richard Dawkins (Oxford University Press, Oxford, UK, 2006, ISBN 978-0-19-929115-1)

"The gene is the unit in the sense of replicator.
The organism is the unit in the sense of vehicle.
Both are important."
(Introduction to the 30th anniversary edition by the author)

"We are survival machines - robot vehicles blindly programmed to preserve the selfish molecules known as genes."
(Preface to first edition by the author)

",,their longevity, fecundity, and copying-fidelity,,, Genes have no foresight. They do not plan ahead. Genes just are,,,""
(Regarding genes, Chapter 3 Immortal Coils, page 24, line 9)

"At the gene level, altruism must be bad and selfishness good."
(Chapter 3 Immortal Coils, page 36, line 30)

"In the world of Darwinism, winnings are not paid out as money; they are paid out as offspring. To a Darwinian, a successful strategy is one that has become numerous in the population of strategies."
(Chapter 12 Nice guys finish first, page 215, lines 13-16)
It is remarkable that Prof. Dawkins could explain the new hypothetico-deductive system of biology based on gene's eye view of evolution in easy English and with no help of schematic drawings. As a scientist, I agree these genetic theory of evolution - gene as replicator, organism/individual as vehicle -. This book is on pure science, not on morals or ethics. As a pathologist I am a bit perplexed actually with this big picture by the author, or in broader sense by Darwinism, since medical practices seem resisting in vain to the pressure of natural selection, especially in the field of maybe too much highly-sophisticated and expensive treatment on tax-based national health insurance in the current nanny welfare state of developed countries with balooning national budget deficit. It is understandable that new genre of Darwinian Medicine is lately burgeoning (see [News and Analysis] Darwinian Medicine's drawn-out dawn, by Elizabeth Pennisi. Science 2011: 334; 1486-1487, Dec 16 2011).

(Start reading 2011/05/21, finished 2012/3/31)

Flatland. A Romance of Many Dimensions. By Edwin A. Abbott (Dover Publications, New York, 1992, ISBN 978-0-486-27263-4)

This book has been reviewed,

In American Scientist by Colin C. Adams, Thomas T. Read Professor of Mathematics at Williams College in Williamstown, Massachusetts (2010:98 November-December; 498-500), entitled A Forgivably Flat Classic.

in which Dover Thrift Editions "for just two dollers" is quite hesitantly recommended. Indeed, once I have read the two-dollar Dover edition, then I really should reread Flatland in either or both of the annotated editions.

Pungent sarcasm is quite clear on page 8 already, where it is explained how Equal-Sided Triangles (middle class) are born rarely from Isoscleles (lower classes of workmen) in two-dimensional Flatland.

A resident in Flatland Japan would like to reserve comment any more, who may actually feel superior at least to the people of Pointland China and North Korea under single-party Proletarean Dictatorship.

(Start reading 2011/02/26, finished 2011/5/15)

The Calculus of Friendship. By Steven Strogatz (Prinston University Press, New Jersey, 2009, ISBN 987-0-691-13493-2)

This book has been reviewed,

In Sicence by Brie Finegold at the Department of Mathematicsc, Univeristy of California (2009:326 October 30; 669), entitled Correspondece in Flux.

In American Scientist by David T. Kung at the Department of Mathematics and Computer Science, St. Mary's College of Maryland (2010:98 January-February; 85-86), entitled Growing Affinity.

On pages 52-54, in the section of a square wave presented by Fourier series f(x) = 4/π ∑ (k = odd) sin(kx)/x , I grasp the outline by rendering this graph with Grapes 6.52 with (k for 2k-1) to 1 (blue line), 2 (purple line), 3 (green line), 5 (black line) and 50 (red line). When enlarging x = 0 to π, 0.8 < y < 1.2, the trend is obvious. During this derivation, I try twice partial integral to prove ∫ (- π to π) sin (nx) sin x dx = 0 and ∫ (- π to π) sin²x dx = π (my calculation with pencil and paper). Actually my son Takao, mathematics teacher at a high school, informed me of more elegant proof.

On page 56, an exercise to derivate the Fourier series for f(x) = x for x ∈ (-π, π) is suggested, for which I calculate using partial integral as usual, and plot it with adding them step-wise for k = 1 to 5 (red), 20 (black), 50 (green) (ditto enlarged).

Regarding the alternating harmonic series on page 45 and the series of Newton and Mercator on page 60, for which I have no idea to compute them now

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6,,,, = log 2 ("Alternating harmonic series", special case of x = 1 of the below series)
x - x²/2 + x³/3 - x⁴/4 + ,,, = ln (1 + x) (the series of Newton and Mercator)
I manage to navigate to "Euler. The Master of Us All" by William Dunham", pages 23-27, depending on my dim memory. The Gedankengang there is still understandable after a moment of contemplation, but my cerebral capacity to register (memory) is pitiable indeed (2010/12/19).

"The first sign of senility is that a man forgets his theorems,
the second is that he forgets to zip up,
the third sign is that he forgets to zip down."
(Stanislaw Ulam, Polish mathematician)
On Euler's "gamma function" Γ (x) and its product (1/2)! = √ π /2 (square root of pi divided by 2) on pages 57-59, I return to "An Imaginary Tale: The Story of the Square Root of Minus One" by Paul J. Nahin, pages 175-178.

Γ(n+1) = n! = ∫ (x = 0 to ∞) xⁿe^-xdx
Using integration by parts with (xⁿe^-x)' --> Γ (n+1) = n Γ (n)
Γ (1) = [-e^-x](∞/0) = 0 - (-1) = 1
Γ (2) = 1 • 1
Γ (3) = 2 Γ (2) = 2 • 1 = 2!
Γ (4) = 3 Γ (3) = 3 • 2 • 1 = 3!
,,,,,,,,,,,,,,,,,

Graphic presentation of this gamma function is demonstrated with clarity with Grapes (for cross-check), including approximated integral value (2010/12/29).

Regarding the series Σ zⁿ/k (k = 1 to ∞) on page 60, in order to grasp its "radius of convergence" I check the approximation with Grapes with sum: k = 1 to 5 (yellow), 10 (green), 30 (red), 100 (blue) and -In (1-z) (purple dot-line), ditto enlarged.

∫ e^-x*x dx (x=0 to ∞) = √Π/ 2 (≈ 0.8862269254,, ) on page 64 can be approximated on the graph, with an estimated value of 0.88622693 (x=0 to 10) with 40 subdivisions. I too try to check (1/2)! (factorial) on page 65 on Grapes with ∫ x^1/2 e^-x dx (x=0 to 30) with 200 subdivisions, and get an estimated value of 0.8813,,

On so-called camel hump functions on page 67, the result of definite integration Π/4 after applying half-angle identity corresponds well with the approximation on the graph with 40 subdivisions.

In the final paragraph of this chapter, Strogatz refers to a graph of the Γ function, which I render with Grapes, seeking the minimum value for x>0. Certainly, (0.46,,)! = Γ (1.46,,) seems the smallest! (2011/1/9).

sin²ⁿθ on page 81 is graphically displayed (n = 1 to 5).

On ∫ e^-x2 dx = √ π (x = -∞ to ∞) and ∫ sin(x)/x dx = π/2 (x = 0 to ∞) (Dirichlet's discontinuous integral) on pages 86, I return again to "An Imaginary Tale: The Story of the Square Root of Minus One" by Paul J. Nahin, pages 177-178 and 180-182, respectively (2011/1/29).

Wallis' product formula for π on pages 95 and 97 can be brushed up on page 156 of "An imaginary Tale" by Nahin again, which is very understandable on the basis of Taylor expansion of f(x) = sin(x)/x.

To grasp Laplace's method to deal with "sharply peaked" function on page 102, I render with Grapes cosine x (blue dotted line) and e^-x2/2 (red dotted line), and each powered by 100 (almost overlapped on black line excluding near +/- kπ). Area adjacent to 0 is enlarged stepwisely, -1, -2, -3, certainly both functions being overlapped on black line. On the other hand, the two 100-powered functions reveal a little difference near 0.150-0.153 (circa π/20). Anyway, very good approximation near 0, indeed!
This approximation ∫ e^-nx2/2 dx (x = -∞ to ∞) conduces to √ 2π/ √n, on which Prof Strogatz gives Joff (and the reader us) a trial to verify this calculation with use of the polar coordinate trick. I worked hard with paper and pencil and finally succeeded, using vague memory of similar calculation for ∫ e^-t2 dt (t = 0 to ∞) on page 177-178 of Nahin. In fact, when I accomplished this, I wanted to jump up and down in my second house (country house). I hope I am still in the stage of freshman of the university calculus course, not in senility.
As for comparison between original ∫ sinⁿx dx (x = 0 to π/2) and approximated √ (π/2) / √n on page 104, I check with Grapes. When n = 99, the former value is .1256, the latter number is .1260 on this calculation, when n=200, it is .0885, the latter is .0886 with this approximation on MS calculator.

I really love the pathos on page 109-110, you will love it, of college/univeristy politics all over the world (gag order issued).

Regarding when cos x = x on page 125, the overview is caught with this y = cos x/ x graph. When I look into the area where y = 1.000, Grapes too converges on 0.739 as with Joff's calculator (2011/2/11).

The conclusion of this book may be summerized on page 142, saying

I'm starting to realize what it was that he gave me.
He let me teach him.
Before I had any students, he was my student.
Somehow, he knew that's what I needed most. And he let me, and encouraged me, and helped me. Like all great teachers do.

The essense is "flexions of life", certainly.

Corrigendum:
Page 60, line 13. "This series ∑ (k=1 to ∞) = zⁿ/k" --> "This series ∑ (k=1 to ∞) = z^k/k"
Page 64, line 28. "= Π/ √ 2" --> "=√Π/ 2"
Page.97, line 11. ",,=(1/2)²/2!" --> ",,=(1/2!)²/2!"

(Start reading 2010/11/13, finished 2011/2/19)

Dr. Euler's Fabulous Formula Cures Many Mathematical ills. By Paul Nahin (Prinston University Press, New Jersey, 2006, ISBN 0-691-1-11822-1)

This book has been reviewed,

In Nature by Timothy Gowers at Centers for Mathematical Sciences, Univeristy of Cambridge (2006:443 September 14;147), entitled A mathematical tonic.

Obviously, the essense of this book is centered at Euler's identity e^iθ = cos(θ) + i sin(θ)

Finished Introduction, pages 1-12, on December 22 2007.

Finished Chapter 1, entitled Complex Numbers, pages 13-67, on February 16 2008, which covers two-dimensional matrix rotation operator, De Moivre theorems, Ramanujan's sum, Cauchy-Schwartz inequality, regular n-gons and primes, Fermat's last theorem, unique factorization theorem, especially its failure in complex number systems, and Dirichlet's discontinuous integral.
I believe I hardly succeed to do integral dg/dy = - ∫ e^-sysin(s) ds (s: 0 → ∞) to dg/dy = - 1/(1 + y²) on pages 64-65 by double application of integration by parts as the author suggests, i.e.,
firstly when ∫ uv'dx = uv - ∫u'vdx, e^-yx = v' and sin(x) = u (please kindly verify my pencil-and-paper computing).

Finished Chapter 2, entitled Vector Trips, pages 68-91, on March 22 2008.

Finished Chapter 3, entitled The Irrationality of π², pages 92-113, on September 9 and 13 2008, after an uphill battle and reaching page 106 on July 26, August 3 and September 8 2008. In fact, I had rerun the last 10+ pages several times. After finished a few weeks ago, I must return to the previous bivouac point to start again, because I cannot recall the technical details. Walk, Don't run, saying to myself. However, I feel I have traversed the basic thinking process for the demonstration of π²'s irrationality by Adrien-Marie Legendre (1752-1833) and Carl Siegel (1896-1981), i.e., the starting assumption of π² = p/q (p and q are integers) leads to the absurdity. Really a long travel from the starting approximation e^x ≈ - A(x)/B(x) for x ≈ 0 (A and B are two finite polynomials, each with degree n) and the definition of the function R(x) = B(x)e^x + A(x) on page 95, while the polynomial coefficients of A(x) and B(x) are revealed being all integers on page 106.

Finished Chapter 4, entitled Fourier Series, pages 114-187, on March 8 2009. The essence in this chapter is Joseph Fourier's assertion in 1807-1822 that any arbitrary function can be written in the form of infinite trigonometric series, with implication that an infinity of individually odd sine (and even cosine) functions can represent non-odd functions. The first simple example is Euler's first Fourier series (in 1744, historically precedent),

x(t) = (π - t)/2 = ∑ (n=1 to ∞) sin(nt)/n = sin(t) + sin(2t)/2 + sin(3t)/3 + ....
This Euler's series can be elegantly rendered with Grapes 6.52 where this graph is presented as one example of its rendering capability. I too try with n = 1 to 60, which is obviously a periodic function and valid only for a 0-to-π interval.

The essense of Fourier series expansions are more generally summarized as the following two expressions (pages 143-145).

f(t) = ∑(k = -∞ to ∞) c_k e^ikw_ot, w_oT = 2π
c_k = (1/T) * ∫ (t = 0 to period) f(t) e^-ikw_otdt, w_oT = 2π

On page 146, partial integration appears once again. This time I can fairly easily compute this integration with formula, i.e., ∫ uv'dx = uv - ∫u'vdx, when v' = e^at and u = t. After reaching the above infinite trigonometric series through step-bey-step arithmetic operation, which high school student probably can understand, the author presents us with a special case of t = π/2.

π/4 = 1 - 1/3 + 1/5 - 1/7 + ,,,,
This is a famous Leipnitz's series. I wept because the operation process is so clear-cut and the result is so elegant.

A square-wave Fourier series on page 151
f(t) = 4/π [cos(t) - cos(3t)/3 + cos(5t)/5 - ,,,]
can be rendered with Grapes 6.52 quite easily, with input of the y1 = Sum(n,1,50,(4/Pi)*(-1)^(n+1)*cos[(2n-1)*x]/(2n-1)) (you must replace the symbol of multiplication properly in your English version, though), where you can replace 50 with other numbers. I show here one example with n = 1 to 3, 10, 50.
The integral on the bottom of page 152 was grasped firstly by rendering with Grapes for typical examples, Case 1; y = cos (3x) x cos (3x), Case 2; cos(3x) x cos(5x). Then I derivate step-by-step with pencil and paper. The progress of mathematical expression in the lower half of page 154 can be traced as well, although it is a bit complicated.

The Fourier series of the function f(t)= sin(t), which is free of discontinuities and the Gibbs phenomenon, was rendered with Grapes again, with k = 1, k = 1 to 2, to 3, to 4, to 5 in gradual approximation, and those plots are getting more and more like to sin(t) as the terms are increasing; (i) overview, (ii) enlarged view (a) approaching to x=0, y=0, (iii) enlarged view (b) approaching to x=π/2, y=1. The author wrote "a bit of albgebra (which I'll let you do)" is needed to calculation in the middle of page.166, actually its step-by-step computation requires me a patience and concentration.

In the section dealing with Dirichlet's evaluation of Gauss's quadratic sum, Fresnel integral appears during its calculation process (page 179). I check this concept in the Dictionay of Mathematics (Iwanami), when I stumble across relevant Cornu's spiral which is interesting to me, and I render this spiral with Grapes fairly elegantly
x = Igr(s,0,t,cos[Pi/2 x s^2 ]): x = ∫ (s=0 to t) cos (π/2 * s²) ds
y = Igr(s,0,t,sin[Pi/2 x s^2 ]): y = ∫ (s=0 to t) sin (π/2 * s²) ds
0 ≤ t ≤ 5 (increment 0.01))

As an example of the pure mathematical tool of Rayleigh's energy formula, appeared on page 208, the definite integral

∫ (x = 0 to ∞) [sin²(x)]/x²dx = π/2
is derived elegantly using this formula. I depict this function with Grapes 6.52, and integrate up to x=20, whose approximation is probably increasing to π/2. Ditto,

∫ (x = - ∞ to ∞) 1/ [1 + x²] dx = π
with Grapes I approximately integrate on the range (x = -100 to 100).


Chapter 5, entitled Fourier Integrals, pages 188-274, intruded on March 20 2009 and finished on May 4 2010. After treading on the Dirac impulse function δ (t), ummm
δ (t) = 1/2π ∫(ω = - ∞ to ∞) e^iωxdω
I reach the Fourier transform integral pair at page 201-202

F(ω) = ∫(t = - ∞ to ∞) f(t) e^-iωtdt
f(t) = 1/2π ∫(ω = - ∞ to ∞) F(ω) e ^iωtdω


Some exampled functions are rendered by Grapes to grasp the graph and to roughly approximate the integrated value,

Example 1. on page 208-209,
ditto, on page 210,
on page 217,
on page 233,
function in Dirichlet's discontinuous integral on page 235,
energy spectral density (ESD) of an N-cycle sinusoidal burst on page 243.
Example 1. on page 248.
Example 3. Gaussian pulse on page 250 and note 10. on page 367 with integral estimate between -6 and 6. BTW, Root π = 1.7724538509,,,, by my desktop calculator on Windows version 6.0 (service pack 2)
Figure 5.6.1. on page 257, Reciprocal spreading in the time domain and frequency domain (α = 0.2 -blue-, 0.05 -green-, 0.002 -red- , respectively).
Fresnel integrals on page 266 (and 179)(approximately 0.5 on integration width 0 ≤ x ≤ 5).
The behavior of the integrant of the function in the middle of page 273 is clearly shown, 0 at ω = 0 and converging to 0 as ω -> ∞, certainly well behaved.

On page 261, I once again perform double application of partial integration to reach σ_t = T/√2. (please kindly verify my pencil-and-paper computing, if you have time. Hard work for Grandpa-thologist on April 24 2010,,,).

Chapter 6, entitled Electronics and square root (-1), pages 275-, finished on Oct 16 2010. In fact I leaped pages 286-323 because they are too technical concentrating on electrical engineering, and I cannot understand even after many times of hand-to-hand fighting to run through. Finished Biography - Euler: The Man and the Mathematical Physicist, pages 324-345, on November 6 2010. I supposed previously that Euler went through his life politically-free, however, this biography depicts his life being interfered by political ferments in St. Petersburg and in Berlin. He would like to become the president of Berlin Academy of Sciences but could not under the reign of Frederick II "the Great", who preferred French nobility to commoners as Euler . After that frustration, he returned to St.Petersburg, and remained a celebrity there, admired by the Empress Catherine the Great.

I finally finished this book after three years of weekend reading and computation on paper, sometimes trying integration by parts! I think I could understand the basic road map with basic technicality. My concern is I will be unable to recall the details in a short period due to gradual age-related and alcohol-related loss of neuronal cells of my cerebrum, certainly it is unavoidable, but, anyway I feel satisfaction that even within the short period I could get closer to the intelligence of Leonhard Euler, 1707-1783.

Corrigendum:
Page 26, line 15 and 16. My suggestion is why not directly divide the numerator and denominator, respectively, i.e., [2sin(β)cos(β)]/[cos²(β) - sin²(β)], by cos²(β), which will leapfrog to 2tan(β)/[1 - tan²(β)]. (The author divide at first by sin²(β), then,,,)
Page 30, line 5 and 6. "Thus, P(x) + iQ(x) = [ln{2cos(x/2) + i(x/2)}],,," → "Thus, P(x) + iQ(x) = [ln{2cos(x/2)} + i(x/2)],,,"
Page 121, line 3. "t = (u + v)/ 2" → "t = (u + v)/ 2c"
Page 121, line 13. ",,,,,(- 1/2) + [,,,1/2c]" →",,,,,(- 1/2) + [,,,1/2c](1/2c)"
Page 127, line 3. ",,,B x icnπ/l x [,,,]" →",,,B x [,,,]" (Certainly, if this is ∂y/∂t icnπ/l is needed, however, this is y (x, t), I think, though I still have no confidence about my calculation.
Page 143, line 9. "= a₀ + ∑ {a_k/2 + b_k/2i}e^ikω₀t + {a_k/2 - b_k/2i}e^-ikω₀t." → "= a₀ + ∑ [ {a_k/2 + b_k/2i}e^ikω₀t + {a_k/2 - b_k/2i}e^-ikω₀t]."
Page 144, line 5. "e^-nω₀t" → "e^-inω₀t"
Page 144, line 11. ",,,(e^i(k-n)ω₀t/i(k-n)ω₀t|" → ",,,(e^i(k-n)ω₀t/i(k-n)ω₀|"
Page 144, line 19. ",,= {T, k ≠ n: 0, k = n" → ",,= {T, k = n: 0, k ≠ n"
Page 151, line 11. ",,,,, e^-int" → ",,,,, e^int"
Page 159, line 14. "W = ∑ c_k c^*_-k T = T ∑ c_k c^*_-k" → "W = ∑ c_k c^*_k T = T ∑ c_k c^*_k ", and from definition of the absolute value of the imaginary number c_k c^*_k = |c_k|², "W = T ∑|c_k|²"
Page 208, line 13. "= (e^-ωr/2-e^ωr/2)/-iω =" → " = (e^-iωr/2-e^iωr/2)/-iω ="
Page 216, line 3. ",,, = ∫ G(u)G(ω - u) du"→ ",,, = 1/2π x ∫ G(u)G(ω - u) du"
Page 217, line 20. ",,,, + |e ^{-(α - iω)t}/-(α + iω)|^∞₀"→",,,, + |e ^{-(α + iω)t}/-(α + iω)|^∞₀" (Added August 17 2009)
Page 233, line 4. "G(ω) = ∫,,,dt = ∫,,,dt + i ∫,,,dt" → "G(ω) = ∫,,,dt - i ∫,,,dt"

(Start reading 2007/11/17, finished 2010/11/06)

Secrets of the Old One - Einstein, 1905. By Jeremy Bernstein (Copernicus Books/Springer Sciences, New York, New York, 2006, ISBN 0-387-26005-6)

"Everything should be made as simple as possible, but not simpler." - Albert Einstein

Young Albert Einstein, then 26-year-old young patent examiner at the Swiss Federal Patent Office in Bern, wrote four revolutionary papers in 1905 (Einstein's Miracle Year) and submitted hand-written manuscripts to the German journal Annalen der Physik. This is one of the attractive books published in 2005/2006 commemorating the 100th anniversary and reviewed in some major science journals, e.g.

In American Scientist by Peter L. Galison, Pellegrino University Professor of the history of science and physics at Harvard Univeristy (2006:94 May-June;264-266), entitled Events of 1905.

In the introductory chapter, the author Jeremy Bernstein explains his life-long interest in Einstein and his life is derived from General Education Program at Harvard University in the fall of 1947, received as a freshman and taught by Profs. Bernard Cohen and Philipp Frank.

The 1st chapter: The prehistory. I certainly knew the background and thought process of Albert Abraham Michelson's experiment with interferometer in 1880 to 1881 to negieren the hypothesis of the aether, which had been the basis of absolute space and time of Newton's deterministic view of physical world, the diagrams and arithmetic process explained in this chapter make me understand its techinical details plainly. Only triviality to add is that the bottom notes # 5 and 6's approximations can be understood by the knowledge of Taylor expansion.

Finished Chapter 2, entitled Einstein's theory of relativity, page 55-101, on August 4 2007.

"I saw that mathematics was split up into numerous specialties, each of which could easily absorb the short lifetime granted to us. Consequently I saw myself in the position Buridan's ass which was unable to decide upon any specific bundle of hay. This was obviously due to the fact that my intuition was not strong enough in the field of mathematlcs in order to differentiate clearly the fundamentally important, that which is really basic, from the rest of the more or less dispensable erudition." - Albert Einstein (Page 60)

In fact, I too keep on seeking the similar answer after entering the medical education and practice. Although I like pathology general as objective basis of medicine, "Which subspeciality of pathology is more important and fundamental to me?" In this chapter, the basics is explained fairly plainly, using minimal but essential mathematical derivations and schematic drawings, and I think I could understand time dilation, Lorentz contraction, constancy principle, and velocity addition theorem in relativity.

"Travel and stay young." (Page 97, quotation from Professor Philipp Frank)

Yes, I do commute everyday on Shinkansen bullet train at over 150 km high speed for 140 km round trip. I hope I am aging less than the resting colleagues.

Finished Chapter 3, entitled Do atoms exist?, page 103-136, on September 1 2007. I understand that we do accept ATOMS a priori now, however, it was nonobvious at least till circa 1900 to reach the atom hypothesis, which means "without cut" or "indivisible" in the original Greek word. The Gedankengang went through Newton, Robert Boyle, Daniel Bernoulli, Benjamin Franklin, Thomas Young (British polymath and ophthalmologist), John Dalton, Louis Gay-Lussac, Carlo Avogadro, Jan Josef Loschmidt, James Maxwell, and the champion of antiatomist Ernst Mach and till about 1890 Max Planck who changed his mind by the turn of the century through black-body radiation problem. I think the mathematical interlude: the drunkard's walk is well written and understandable, and guide the readers to understand how far a Brownian particle would go in a time. BTW, Robert Brown (1773-1858), born in Scotland, studied to be a doctor, and was a surgeon's mate, then naturalist, and finally discovered his eponymous movement as a microscopist.

Finished Chapter 4, entitled The Quantum, page 137-172, on October 13 2007.

Einstein used to say that the best profession for a physicist would be as a lighthouse keeper, because you could just sit there all day thinking about physics. Of course Einstein became a university professor (page 152, the author's description)

When I read this passage, I must keep a straight face, but would like to suggest two more possibilities: (1) morgue service officer, (2) jail keeper of the district police station. Wilhelm Wien, who later contributed to the mathematical expression of the temperature-dependent spectrum of black-body radiation, was born in Prussia in 1864 as a son of the family farm and wanted to be a farmer, eventually became a physicist. I render changes of black body radiation curves at each temperature with his approximation formula for myself, using shareware program GRAPES 6.52 by Katsuhisa Tomoda. Since my objective is to sensuously experience the outline, I simplified the original Wien formula universal (α) = a (1/α^5) exp(-b/αT) (a and b: constants, T: temperature, α: wave length) to


y = (1/x^5) exp(-1/xT) (T arbitrarily substituted to 1.3, 1.0, 0.7) (Click here to enlarge the graph rendered with Grapes 6.52)
Certainly, it appears to me that the curve at the ultraviolet end (near zero wave length) is exponential, and at long wave lengths proportional to the wave length going to zero, and as the temperature increases, the peak wavelength emitted by the black body decreases.

Index is earnestly prepared, and of assistance to retrieval of the preceding page about the topic.

Corrigendum:
Page 119, line 9. "+ 10a²b³ + b⁵." → "+ 10a²b³ + 5ab⁴ + b⁵."


(Start reading 2007/5/20, finished 2007/10/13)

The Origins of Cauchy's Rigorous Calculus. By Judith V. Grabiner (Dover, Mineola, New York, 2005, ISBN 0-486-43815-5; Republication from The MIT Press, Cambridge, Massachusetts, 1981)

I recognized this book by reading a book review by the current author on Unknown Quantity written by John Derbyshire, which appeared in September-October 2006 issue of American Scientist (2006:471-472). According to the footnote of that article, Judith V. Grabiner, who won the Mathematical Association of America's Haimo Award for Distinguished College or University Teaching in 2003, is the Flora Sanborn Pitzer Professor of Mathematics at Pitzer College in Claremont, California, and the author of many articles on the history of mathematics. This book appears one of them, and is introduced on the back cover as "The text for upper-level undergraduate and graduate students".

Finished Chapter 2, entitled The Status of Foundations in Eighteenth-Century Calculus, pages 16-46, on November 26 2006. I understand the 18 century views of the calculus, and the important role of Lagrange's work, Fonctions analytiques, whose first edition was published in 1797 to teach at the Ecole Polytechnique, and which changed an attitude of mathematicians toward the foundations of the calculus and at the same time did justice to the already existing wealth of results obtained by practically using calculus.

Finished Chapter 3, entitled The Algebraic Background of Cauchy's New Analysis, page 47-76, on January 13 2007.

Finished Chapter 4, entitled The Origins of the Basic Concept of Cauchy's Analysis, page 77-113, on February 10 2007. Cauchy pushed the envelope to solidify the foundations of the analysis with the rigorous definitions of limit, continuity and convergence, using δ-ε logic. I refer to Prof. Dunham's The Calculus Gallery occasionally to cross-check my understanding, e.g., pointwise and uniform continuity, root test and ratio test for convergence. These two books complement one another well, the latter is kinder and gentler to me, though. I would add my recurring impression how useful power series expansion by Taylor and Maclaurin is, not only in Lagrangean definition of the first derived function (p.53) but also in Cauchy's application of analysis to the complex x (complex analysis). I believe Taylor expansion is one of the most elegant formulas in mathematics.

Finished Chapter 5, entitled The Origin of Cauchy's Theory of the Derivative, page 114-139, on March 31 2007, which covered from Lagrange property of the derivative, Taylor's series with Lagrange remainder, and Cauchy's rigorous definition of the derivative with use of delta-epsilon logic.

Finished Chapter 6, entitled The Origins of Cauchy's Teory of the Definite Integral, page 140-163, and Conclusion, on May 5 2007. Cauchy difined the intergral as the limit of sums, not inverse of differentiation, i.e., antiderivative, as Euler, Lagrange, Lacroix, and Laplace liked to think. I can understand the Fundamental Theorem of Calculus once again and much more profoundly than the high school days. The author's warm attention to provide us English translations of several of Cauchy's major contributions to the foundations of the calculus in the Appendix, e.g., intermediate value theorem, mean value theorem, existence of definite integrals, and fundamental theorem of calculus, from Cours d'analyse, Calcul infinitésimal and Oeuvres, is greatly to be appreciated. They are the essense of his revolutionary contribution to the calculus based on clear definitions and rigorous proofs.

Corrigendum:
Page 61, line 27,28. "the ratio between the (n + 1)st and nth terms" is in fact "the ratio between the nth and (n - 1)th terms" This is not error, but the reader would be able to calculate/understand easily with my suggestion.
Page 62, line 1. "(1 + 199)(1 - 3/2n)" → "(1 + 1/199)(1 - 3/2n)" because the first term is 200/199.
Page 62, line 30 - 31. ",,,+ Aμ(1 - m + 1/n) +,," → ",,,+ Aμ[1 - (m + 1)/n] +,," (bracket operation)
Page 62, line 36. "A/1- μ - A/[1 - μ(1 - m + 1/n)]" → "A/(1- μ) - A/[1 - μ(1 - (m + 1)/n)]" (bracket operation)
Page 63, line 11. "abs (A/1 - abs (u))" → "abs (A/1 - abs (μ))"
Page 66, line 16. Formula (3.6). ",, = [[1/(α - a)] + R]/(R - α) = ,," → ",, = [[1/(α - a)] + R]/[R (α - a)] = ,,"
Page 75, line 33. "Gauchy" → "Cauchy"
Page 173, line 8. ",, f [ x₀θ₀(x₁- x₀)],," → ",, f [ x₀ + θ₀(x₁- x₀)],,"
Page 175, line 12. "divided by a,," → "divided by α,,"
Page 197 (Notes), Item 40. "∑ 1/k converged to π²/6"→ "∑ 1/k² converged to π²/6". The harmonic series diverges (Jacob Bernoullis, 1654-1705). ;)


(Start reading 2006/11/3, finished 2007/5/5)

The Calculus Gallery. Masterpieces from Newton to Lebesgue. By William Dunham (Princeton University Press, Princeton, New Jersey, 2005, ISBN 0-691-09565-5)

Upon spotting Prof. Dunham's new book in the book review of major science journals, e.g.,

In Science by Judith V. Grabiner of the Department of Mathematics at Pitzer College in Claremont, California (2005:308 June 24; 1872), entitled Landmarks on the Road to Modern Analysis.
In American Scientist by Victor J. Katz of the Department of mathematics at the University of the District of Columbia (2006:94 Jan-Feb; 83-84) entitled Reading the Masters.

I promptly ordered AMAZ_N to send me a copy. The first two chapters on Newton and Leibniz are traceable, understandable and readable. What a joy to chase the Gedankengang of the early achievements by these pioneers! By Bill｡ﾇs own descriptive words including technical detail, even high school students would be able to reach the height of Leibniz-Gregory series step by step, when I actually dissolved into laughter, because it was so simple and beautiful. Only thing that we lament is that we can understand these achievements only by hindsight. In the current market-driven, competitive and shortsighted, or totally authoritarian environment of science, medicine and culture, the importance of ORIGINAL IDEA is to be advocated once again.

Finished Chapter 8 Liouville, pages 116-127, on April 16 2006. Arithmetic logical process is traceable and understandable up to here. The only point I can't completely figure out is "An induction argument established that (m + r)! >= (m + 1)! + (r - 1) for any whole number R >= 1" in line 16, page 125, into which I am still delving. Does this mean that I should try one by one r = 1, 2, 3, 4,,, and satisfy myself that it seems valid, o.k. no problem?

Finished Chapter 9 Weierstrass, pages 128-148, on May 21 2006, still keep going. To understand the internal structure of Weierstrass's pathological function, which is everywhere continuous but nowhere differentiable, it helps to display it's partial sum on CRT, e.g.,
S₅(x) = cos(pi*x) + cos(21*pi*x)/3 + cos(441*pi*x)/9 + cos(9261*pi*x)/27 + cos(194481*pi*x)/81
with gnuplot (MS-Win Ver.3.4). Then, I narrow x-range stepwise and re-plot (enlarge the graph just like we do with microscope by changing objective lens), voila!, steep rising and falling behaviour of the function continues somewhat recursively. I am provided a glimpse into "Weierstrassian rigor".


Partial sum of Weierstrass's Pathological Function, S₅(x), displayed with gnuplot at low magnification (Crick to enlarge)

Partial sum of Weierstrass's Pathological Function, S₅(x), displayed with gnuplot at middle magnification (Crick to enlarge)

Partial sum of Weierstrass's Pathological Function, S₅(x), displayed with gnuplot at high magnification (Crick to enlarge)
Finished Chapter 11 Cantor, pages 158-169, on July 1 2006.

Finished Chapter 12 Volterra, pages 170-182, on July 22 2006.

Finished Chapter 13 Baire, pages 183-199, on September 9, 2006. Rene Baire (1874-1932) in his doctoral thesis of 1899 proposed and clarified "Any problem relative to the theory of funcition leads to certain questions relative to the theory of set." The Baire category theorem and its applications takes a long time for me to understand, i.e.,ponder the page, ponder the page, again, again, once again. Now, it seem justified for me to advance to the next chapter. But, I would say at this moment how deeply I was impressed to understand that Dirichlet's function,

d (x) = 1 if x is rational, and d (x) = 0 if x is irrational
can be expressed as a function.

D(x) = lim(k→∞)[lim (j→∞) {cosk!πx}^2j]
when if x can be p/q (rational in lowest terms) k!πx is an integer multiple of π,,,,

Finished Chapter 14 Lebesgue, pages 200-219, and Afterword, on October 28 2006. I feel I have grasped the basic concept of Lebesgue integral, after I read about outer content, outer and inner measure, measure zero, measurable set, measurable function, Lebesgue measure, range,, over and over again. Yeah, Dirichlet's function, which is everywhere discontinuous, is nonintegrable for Riemann, but integrable for Lebesgue. Partition not the domain, but its range. The merchant would multiply the value of the currency by the number of the pieces, i.e., a * dimes + b * quarters + c * dollers + and so on, at day's end. Once again, the Dirichlet's function is Lebegue integrable.

This book is extremely well written without discarding the necessary technical details and formula, is readable with endurance and ardor. In the arena of mathematics, we cannot understand without contemplating enough. It took me a year to read through 200 + α pages, but it was a steady and progressive, and priceless journey for me. A joy of life.

Corrigendum:
Page 55, line 16. "Integral of (ln x)⁴ (x 0→1) = [x(ln x)⁴ - 2x(ln x)³ + 12x(ln x)²,,," → "Integral of (ln x)⁴ (x 0→1) = [x(ln x)⁴ - 4x(ln x)³ + 12x(ln x)²,,,"
Page.79, line 9-10. Trigonometry identity "sin(α + β) - sinα = 2 sin(β/2)*cos(α + β/2)". My previous correction about this identity was completely wrong, for which I am very ashamed. The printed formula is correct. My failure then was probably due to too much wine glasses. ;)
Page 210, line37. "If α ≥ 1;" → "If α > 1;"

(Start reading 2005/12/4, finished 2006/10/28)

The Science Absolute of Space. Independent of the Truth or Falsity of Euclid's Axiom XI (which can never be decided a priori). By John (sic) Bolyai and translated from Latin to English by Dr. George Bruce Halsted (The Neomon, Austin, Texas, 1896)

These are facsimiles of Bolyai's original paper included in the Burndy Library's publication in 2004, which was translated by G.B. Halsted to English who was President of Texas Academy of Science then. This paper includes a total of 43 definitions and theorems, §1 to §43, in a space of 44 pages. I read one § and ponder for a while or a day, and advance to the next if I feel confident I could understand. In fact, it takes a fair amount of time to share in my brain 2-dimensional or 3-dimensional images Bolyai fashioned to build his pure logical sequence, and it seemed not necessarily impossible for me till I am finally deadlocked at §27.

(Start reading and studying 2005/9/11; traversed §26 and then stopped 2005/12/04)

János Bolyai. Non-Euclidean Geometry, and the Nature of Space. By Jeremy J. Gray (Burndy Library, Cambridge, Mass., 2004, ISBN 0-262-57174-9: Distributed by the MIT Press)

I found this book by the review entitled "Genius unapprecited" by Brian Hayes, which appeared in the May-June issue of American Scientist (2005:93;275-276) and is recommended as "exceptionally well designed, printed and bound."

I seem to understand now that the parallel postulate is an assumption with no proofs. Euclid's Elements is the only geometry which we are taught in the primary and secondary education and is practical and easy to grasp with intuitions, though, modern geometry embraces the plurality with more abstract thinking process and internal consistency. BUT, dare you explain to the common people around, e.g., to sales representatives visiting your hospital or to the butcher on the street, about Bolyai's New Geometry?

Corrigendum:
Page 125, Figure 25. There are two G in this Cartesian plane, and the lower one on the x-axis should be C.

(Start reading 2005/7/31, almost finished 2005/9/11 just before English translation of Bolyai's 1832 original)

The Riemann Hypothesis. The Greatest Unsolved Problem in Mathematics. By Karl Sabbagh (Farrar, Straus and Giroux, New York, 2002, ISBN 0-374-25007-3)

After finishing Prime Obsession by John Derbyshire 1.5 years earlier (my review, please see below), I return to another book published in the same period, with the same topic but in a quite different approach.

First of all, I recommend to refer to the online book review from an expert of number theory, S. W. Graham, a former program director in the Algebra and Number Theory Program at the National Science Foundation, which is posted at the Mathematical Association of America (MAA) web site.

A long and deep review, entitled The Invisible Man by Enrico Bombieri at the Institute for Advanced Study in Princeton, who is a top mathematician in this field and has asked his father for an algebra book at the age of 8 for Christmas (pp. 87-88), is also available in July-August 2003 issue of American Scientist (2003:91;360-364).

This book is fun to read, revealing interesting and strange characters of contemporary mathematicians. I really hope Louis de Branges has successfully passed his written test for driving in France, and been allowed to drive there in addition to his Indiana driver's lisence (pages 128 and 213).

To understand the details of pair correlation function and "the zeros of the Riemann zeta funcition repel each other on the critical line", which is described in the Chapter 9 "The Princeton Tea Party" (pp.158-159), an article entitled "The Spectrum of Riemannium" by Brian Hayes, which appeared in July-August 2003 issue of American Scientist (2003:91;296-300), greatly helps. The essence is this.

I like the Toolkit 6: Matrices and Eigenvalues, which is concise and understandable even for layperson.

Corrigendum:
Page 190, line 29 "1, -1, -1, 0, -1, 1, 1, 0, 0, 1" --> "1, -1, -1, 0, -1, 1, -1, 0, 0, 1",
because 7 is a prime, µ (7) = -1. BTW, it would be kinder to add the definition µ (1) = 1 (as John Derbyshire's page 249)
Page 206, line 11 "Γ(n) = n x Γ(n -1)" --> "Γ(n) = (n-1) x Γ(n -1)", I confirmed this at mathworld after my own annoyance.
Page 264, line 31-32 "The Riemann zeta function is a mathematical expression involving powers of prime numbers and one unknown quantity, usually denotes by s" --> "The Riemann zeta function is a mathematical expression involving powers of whole numbers (or positive integers) and one unknown quantity, usually denotes by s"

(Start reading 2005/4/30, finished 2005/7/30)

Oliver Twist. By Charles Dickens (Wordsworth Classics, Hertfordshire, Great Britain, 1992, ISBN 1-85326-012-6)

I like British storytellers, such as Somerset Maugham and Dickens because of their straightforward context, language and historical description of the landscape of the street. I have been enjoying this book on commuter trains for the last half year.

Regarding the mediocre argument of whether Dickens harbored anti-Semitism when he painted his Jewish monster Fagin, you had better google-search with "Oliver Twist" and "anti-Semitism". Just like Shakespeare's villain, Shylock, who appeared in The Merchant of Venice, these portraits may be somehow reflecting der Zeitgeist of those days, though, I think it is a complete waste of time to judge by the current standard of ethics and viewpoint without considering the historical context. They were only stereotypical evil persons in the fiction.

In the last Chapter 53 entitled "At Last", Dickens honestly disclosed how he liked his own characterization of Rose Maylie in all the bloom and grace of her early womanhood, Mr Grimwig's and Dr. Lesberne's taste for gardening, planting, fishing and carpentering in the rural retreat, and in the almost final paragraph he concluded,

",, and without strong affection, and humanity of heart, and gratitude to that Being whose code is Mercy, and whose great attribute is Benevolence to all things that breathe, true happiness can never be attained."

I like these simple and pure classic grace and moral, although I do not routinely make a faith-based medical judgement.

After I finished 353 pages with help of dictionary, for example seaching the word "parochial", I came across a classic DVD version of Oliver Twist (David Lean's 1948 UK version), at discount as usual, in which Alec Guinness's remarkable performance as Fagin is famous. It has actually omitted the major sequence in the later half of the original story, as a result there is no appearance of important persona such as Rose Maylie, Mrs Maylie, Harry Maylie, Mr.(Dr.)Lesberne, etc in this motion picture. It would have been quite difficult to compress the whole long story into a 110 minutes movie. In spite of this, this video presents us the realistic dunkel landscape of London (with no electricity!) and the appearance and daily activity of the average people there in the great Victorian period, including moral severity or hypocrisy, middle-class stuffiness, and pompous conservatism, which Dickens described in the text far more sarcastically.

The original text is great in its depth. Any movie cannot compete, whatsoever. (Start reading 2004/11/01, finished 2005/06/20)

The Fabric of the Cosmos. Space, Time, and the Texture of Reality. By Brian Greene (Alfred A. Knopf, New York, 2004, ISBN 0-375-41288-3)

Traversed about a half point (Page 294/ total 493 pages) in December 2004, and finished in May 2005. The book fairly kindly explains the Gedankengang (thought process) of the fundamental physicists, covering the main historical and current theories and topics including Newton's, Mach's and Einsteinian view of the space and time, relativity, entropy, time's arrow, beam-splitter experiment, quantum physics, uncertainty principle, locality and entanglement, inflationary cosmology, curvature of space (positive, flat and saddle), Higgs field, grand unified theory, string and M-theory, eleven dimensions and braneworlds, hidden dimension, Planck length, gravitational waves, LIGO!, Lense-Thirring frame dragging, quantum teleportation, a bit of loop quantum gravity, etc. The text is plainly and contextually presented in Greenish informal and popularizer touch ("You get a sense of sitting down with Greene and chatting over a good cappuccino" - James Trefil -), but never superficially. Before and during my reading, several book reviews appeared in the major science journals, e.g.,

In American Scientist by Lee Smolin at the Perimeter Institute in Waterloo, Canada (2005:93 July-August;371-373), who confessed to be one of the inventors of loop quantum gravity that is generally seen rival to string theory. After praising the effort by Greene in evenhanded tone, his critique is focusing that string theory is an unproven conjecture that has not been verified by experiment, which is rather exaggerated as truths in the book as the reviewer argues, maybe just like the delution by even smartest and best-educated people in the world that Iraq had weapons of mass destruction (WMD).

In Science by James Trefil in the Department of Physics and Astronomy at George Mason University, VA, USA (2004:304 April 9; 212), entitled Untangling the Universe, who praises this book as the best exposition and explanation of early 21st-century research into the fundamental nature of the universe. Regarding the intended target of this book, he says one public for whom this book will be important is that composed of scientists and engineers. He wonders how many readers will be willing to slog through nealy 500 pages of text to get the whole story, though, I would declare here that after 10-months slow march I DID IT!

In Nature by Paul Davies at the Australian Centre for Astrobiology, Macquarie University, Sydney, Australia (2004:428 March 18; 257-258), entitled Shooting time's arrow, in which he mainly comments on time's arrow in the context of Helmholtz' pessimistic prediction on the second law of thermodynamics that the Universe is dying.

As one of medical scientists (physician scientist), I am encouraged by the book and these kind of profound reviews to never give up the earnest will of the pure science, and at the same time to be careful of pseudosicence and fantasies infiltrating into our secular hospital and health care environment. ;)

(Start reading 2004/6/27, finished 2005/4/30)

The Outsider. By Collin Wilson with translation into Japanese by Tsuneari Fukuda and Yasuo Nakamura (Kinokuniya, Tokyo, 1981, published in Japan by arrangement with Victor Gollancz, London, UK)

I purchased this nostalgic book at the antique store located in front of the Tokyo University in the spring 2004, which I was reading but have not finished in early 70's. Age-related restructuring of my neural network appeared to accept the implication more easily now, but it was revealed that the book is still very difficult for me to understand.

The outsider is characteristic of the person with Western creative thoughts, and he pursued or chose the alienated state of his own will, Wilson argues. The author categorizes the people between the bourgeoisie or the snob and the outsider, by the way.

It appears to me that pathologists are sometimes classifiable as Outsider in the medical community in terms of the struggle of those who observe the facts on the basis of scientific objectiveness as alienated intellectuals. We never reject the collaboration between clinicians and us, and our practices are not faith-based, though, of course.

(Start reading 2004/6/10, provisionally finished 2005/1/03)

Abel's Proof. An Essay on the Sources and Meaning of Mathematical Unsolvability. By Peter Pesic (The MIT Press, Cambridge, Massachusetts, 2003, ISBN 0-262-16216-4)

"The inescapable verdict: Niels Abel was guilty of ingenuity in the fifth degree" (Prof. William Dunham, the author of my favorite books read before)

It had been proved in the 16th century in Italy that cubic equation is solvable by the del Ferro-Cardano-Tartaglia method and quartic one, too, is solvable by Ferrari's method, but quintic equation (in the fifth degree) remains problem to solve for next two hundred years. In fact, it was widely believed to be soluvable until almost 1800 (page 67). In 1824, a Norwegian named Abel proved that it is not solvable in radicals, and died five years later at age 26 in poverty.

Actually, I cannot read this book contextually from the beginning, because the explanation is occasionally not logical and rigorous enough. Probably, the author has summed up too much substance into a concise book. Certainly, this is an essay. For example,
(1) "The ratio between the sides of the simplest regular polygons" on pages 48-49 in the context of Kepler's Harmonia Mundi is described superficially, so the average reader as me would not be able to understand the mathematical implications.
(2) Descartes' Rule of Signs on page 53 cannot be understood per se, for which Purplemath was very helpful to me, which I came across after Google search.

I reach Chapter 8 "Seeing Symmetries", where I again struggle with "group theory" including basic concepts as invariant subgroup, commutative and abelian subgroup, "normal" subgroup, "simple" group, etc. During this trek, I stop for days or weeks and deliberate to fill the gaps. But beforehand I must notice the gap. The cleft is there, which I cannot leap over. It must be fixed with my own neuronal network. Walk don't run. Feel joy to think. This concise book compels me. Mathematical details incorporated into sidebar (box) help me to understand the essential minimum.

I finished the main text through Chapter 10 at page 153 on May 1, but have been arrested at page 163 of Appendix A: "Abel's 1824 Paper" (Original paper with author's commentary) since May 22. Quite honestly, the final attack to the highest peak accomplished by ingenuity in the fifth degree seems beyond my ability, so I temporarily withdraw from this book here, leaving 6 pages,,,,,

Corrigendum:
Page 37, Box 2.4, line 17 ",,,let y = u²;" → ",,,let y = u³;"
Page 158, line 20 ", r₂,,,r_k, and, as these,," → ", r₂,,,,r_k, and α, as these,,"
Page159, line 15 ",,,, + p_m-1R^m/m-1[A3],,," → ",,,, + p_m-1R^m-1/m[A3],,,"

(Start reading 2004/1/17, almost finished 2004/6/27)

Prime Obsession. Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. By John Derbyshire (Joseph Henry Press, Washington, D.C., 2003, ISBN 0-309-08549-7)

I picked up this book after reading a long and deep review which appeared on July-August 2003 issue of American Scientist (The Invisible Man by Enrico Bombieri at the Institute for Advanced Study in Princeton. 2003:91;360-364), and in the middle of reading I come across another ones on Oct 9 issue of Nature (Prime time for mathematics by W.T.Towers at University of Cambridge. 2003:425;562) and Oct 3 issue of Science (Prime time for Riemann Hypothesis by Brian Conrey at the American Institute of Mathematics, Palo Alto. 2003:302;60-61). Indeed, this is a very addictive book in terms of clarity and depth.

This remarkable book consists of two parts: Part I: The Prime Number Theorem (PNT) and Part II: The Riemann Hypothesis. The part I guides the journey of mathematics in 19 century in Europe, of course mainly in Germany, from Euler, Gauss, Dirichlet, Riemann, Dedekind and to Hadamard (proof of PNT), explaining its pure mathematical thought process (Gedankengang) and evolution in odd-numbered chapters and its biographical, geopolitical and historical context in the even-numbered chapters, respectively.

As usual, I occasionally receive pleasure from struggling to calculate the basic function values with pencil, paper and Casio scientific calculator in order to countermeasure the age-related decrease in number of neuronal cells in the frontal lobe of my brain which is perhaps now being accelerated by wine consumption, e.g., to calculate the function value of e^z when z is a complex number -1 + 2.141593i (on page 202, in the chapter 13, part II). I recall the power rule x^m x xⁿ = x^m+n, and that glorious ｡ﾈEuler's identity｡ﾉ cos (x) + isin (x) = e^{i x}, but then the batteries of my calculator abruptly die, I search the manual, disassemble the back cover of the calculator, replace with new LR44 batteries,,,,then again calculate to reach the wrong result, consult the manual to reset angular unit from degrees to radians, followed by several cycles of calculation,,,,, Actually, it takes over 30 minutes to reach the correct answer of -0.198766 + 0.30956i.

In Chapter 17, the author explains the algebraic background to understand the Riemann Hypothesis. i.e., field theory and operator theory. I could not understand the algorithm underlying the finite field F₄ (Figure 17-2), even after contemplating for a while. Without reading Chapter 4. "Looking for F₄ " there, I probably could not have grasped the least essense of the field theory. It helps enormously.

I would say that in retrospect, the article which appeared in July-August 2003 issue of American Scientist entitled The Spectrum of Riemanium written by Brian Hayes (2003:91;296-300) is instructive, especially with regard to nearest-neighbor spacings and pair-correlation function, and gives us a backbone context of their thinking process.

German mathematician David Hilbert (1862-1943), when asked what would be the first thing he would say upon awaking after sleeping for five hundred years, replied "I would ask 'Has somebody proved the Riemann hypothesis? '"

The text is written in very plain English, even to non-mother-tongue English speaker as me, and in straight, solid and step-wise logic. Equations and derivations are scattered in proper density. I think this is one of the best mathematics readings currently available. The backbone is tightly designed on mathematics, i.e., THE PERSUIT OF ESSENCE. The only perceived difficulty for non-professional mathematician as me to understand is the concept of "p-adic numbers" and "adele" in Chapter 20.v. Reading proceeds only slowly, 10-20 pages a day in my case till my brain is tired, but with unstoppable interest it progresses steadily in weekends for three months. I really thank John for this introspective pure joy of philosophos (loving wisdom). Merry Christmas and Happy New Year!

"Wir muessen wissen, wir werden wissen" (David Hilbert)

PS: The only typo I noticed in the whole 364-page text is at page 336, line 10; "Figure 11-1" is in fact "Figure 11-2"?

(Start reading 2003/9/20, finished 2003/12/27)

Ignorance. By Milan Kundera with translation from the French by Linda Asher (HarperCollins, New York, 2000, ISBN 0-06-000209-3)

From the beginning, Kundera ponders augumentatively, i.e., etymology of nostalgia, which is composed of nostos the Greek word for return and algos for suffering, and history of Europe's twentieth century, but never tediously. We get the essence, then retrace the psychodynamics of a man and a woman who return to their homeland Czechoslovakia after twenty years of exile in Paris and Denmark. The settings are deeply European indeed, and a final explosion of obscenities between Irena and Josef!

In fact, I too am suffering from "nostalgic insufficiency" (page 74), the disease entity that Kundera defined, i.e., no affection for the dimly visible, feeble past, no disire to return, detachment. I have never wanted to return zu meinem Heimatland Okazaki City. BTW, Ed, do you want ot return to Tennessee? Ummmm

Persona
Irena: A Czech woman who emigrated to France from Czechoslovakia with her first husband Martin in 1969 after Soviet occupation.
Gustaf: A Swede, Irena's second husband after Martin's death and a friend of Martin's in Paris, who had escaped from Sweden and his first wife and family.
Sylvie: Irena's friend in Paris who prompted Irena the Great Return to Prague.
Josef: A Czech veterinarian who emigrated to Denmark, where he was widowed with his wife. This time he met by chance Irena at Paris Airport during a return trip to Czech. In fact, he had met Irena in Prague a long time ago, the encounter he forgot and she remembers,,,(Ashtray!)
Milada: Irena's old friend and a colleague of Martin's in Prague, and Josef's girlfriend in youth. She attempted suicide because of him then and lost one ear from frostbite. Be careful the subject of Chapter 46 is Milada.
N.: Josef's old friend in college, a communist party deputy, Red Commissar, who was exceptionally kind to Josef.

(Start reading 2003/2/11, finished 2003/2/23)

Theory and Problems of Quantum Mechanics (Schaum's Outline Series). By Yoav Peleg, Reuven Pinini, Elyahu Zaarur (McGraw-Hill, New York, 1998, ISBN 0-07-054018-7)

This is a more advanced study guide for quatum mechanics, including more mathematical explanations and problem solutions. I must recall basic concepts and definitions of physics in every section, in this sense, the textbook of physics of high school course must be always at hand with a pile of rough white paper. For an over-50-year-old pathologist, this concentration makes sense in order to decelerate the continuous decrease of IQ, I hope!

Corrigendum: page 6, regarding the basic equation of the Compton effect, line 6, equation (1.3.8) should be "lambda' - lambda = (h/mc) (1 - cos (theta) ) ", not "....(h/mc^2) ...." as printed. The derivation steps from (1.3.7) to (1.3.8) are a simple mistake that even I can notice. The final equation is also confirmed at the site by Professor R.F. Egerton of the Cambridge University, who also presents "...(h/mc)...". Request your comment.

(Start reading 2002/11/17)

An Introduction of Linear Algebra. By Masahiko Saito (Tokyo University Press, Tokyo, 1966)

I could not understand the concept of Eigenvectors and Eigenvalues in Schaum's Quantum Mechanics (page. 13), so I once again returned to the basic textbook of linear algebra used in my freshman or sophomore years at the university, i.e. back to the level of 30 year earlier. It is still understandable step by step, and I am solving the problem one by one. Do you remember that the absolute value of det (a1, a2, a3) is equal to the volume of parallel hexahedron framed by vectors a1, a2, a3? Now, I am still a long way to Eigenvectors and Eigenvalues,,,,

Traversed adjoint matrix (transpose conjugate), Hermitian matrix, unitary matrix and orthogonal matrix at page 63-64 on June 22 2003.

The Quantum Universe. By Tony Hey and Patrick Walters (Cambridge University Press, Cambridge, UK, 1987, ISBN 0-521-31845-9)

On page 1, a quotation from Richard Feynman is posted,

,,I can safely say that nobody understands quantum mechanics.

From my undergraduate sophomore years, I probably had a subliminal desire someday to understand quantum physics after studying medicine and becoming a doctor with sustainable income and some reading time. It was too huge an obstacle to overcome for me those days. Now is the time, but from the beginning that ambition of mine is revealed to be a bit impossible goal. ;) Anyway, I go forward courageously (In fact, I use Koushiro Yoshioka's Inorganic Chemistry, Tokyo University Press, 1968, as a reference book which is still useful since those undergraduate years).

Well, may I ask you from where Schroedinger derived his wave function Ψ (psi) ?

d²Ψ/dx² + d²Ψ/dy² + d²Ψ/dz² = - 8(PI)² * m * (E - V) * Ψ/h²

from which Ψ²dxdydz gives us a probability of finding an electron (probability density) in a microvolume dxdydz (Planck's constant h, mass of a particle m, total energy E, potential energy V). Feynman is quoted as saying "Nowhere. Out of the mind of Schroedinger". As far as I understand, from this wave function the science can explain the atomic structure (nucleus and electron) of hydrogen and other elements, Balmer and other series, Niels Bohr's model of the atom and eigenvalue, and apparently all the atomic physical reality without contradiction, while consistent with Heisenberg's uncertainty principle. But the wave function itself from nowhere. Outrageous, not to mention crazy. Probably, we should not compare this ingenuity with those of Mendel's heredity law and Watson and Click's double helix model, however,,,

In short, in the first half of this book, we learn the basic history, ideas and concepts of quantum physics, e.g., particle-wave duality, uncertainly principle, wave equation, Pauli's exclusion principle, etc. The following half helps us catch up with the contemporary keywords such as neutron stars, supernova and black hole, Lasers, superfluids, Bose-Einstein condensation, quantum electrodynamics and chromodynamics etc. Even if the authors originally toyed other possible titles such as "Quantum Mechanics for Bank Managers", as admitted in the preface, actually I felt sheer difficulties in some paragraphs to grasp the basic concept, when the following web sites were very informative, instructive and helpful,

Kids Scientist (In Japanese), BOHM-AHARANOV EFFECT, Stanford Linear Accelerator Center, Color Charge and Confinement

After reading this book from page 1 to the last, I notice that I understand some of original articles on physics in latest issues of Nature magazine (e.g., Armoretti M, et al: Production and detection of cold antihydrogen atoms. Nature 2002:419;456-459), which I formerly skipped or ignored because of inability to grasp the least essence. Now, I wish I could quest more deeply into quantum physics with a bit more recourse to mathematics and equations.

But, as the Epilogue suggests kindly, I would like to indulge myself with a glass of Cabernet sauvignon for a break now.

If our srnall minds, for some convenience, divide this glass of wine, this universe, into parts - physics, biology, geology, astronomy, psychology, and so on - remember that Nature does not know it! So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all! By Richard Feynman

(Start reading 2002/09/07, Finished 2002/11/10)

A History of Pi. By Petr Beckmann (St. Martin's Press, New York, 1971, ISBN 0-312-38185-9)

After the latest restructuring of mathematics education in Japan, the pupils in the primary schools are being forced to be taught that PI is 3, just 3, integral 3, which has evoked public lamentation and surprise. Up to the generation of mine, the Japanese people has been enjoying ubiquitous and deep understanding of mathematics by rigorous education system, which could have contributed to the post WW2 recovery with industrialization. On the other hand, Hungary is a small and low profile nation, but is known to be a country of geniuses, e.g., John von Neumann, Paul Erdos, Edward Teller, etc, I know, and I once read that the ennoblement of a schoolteacher is an indication of how highly the Hungarians valued education. In this context, I think this deterioration of mathematics education in the primary level is really a serious problem in the long run, and this is a good timing for me to read this rather classic book.

Obviously, this author abhors the totalitarian tyrannies, e.g., Roman Empire, the Third Reich and Soviet, and in the beginning of some chapters he quotes from Bernard Shaw, the famous Irish-born playwright who was too famous for sarcastic remarks, which suggests a razor-sharp and deep-seated love for rationality of Dr. Beckmann. Although facts and opinion are clearly separated in the text, criticism to a new tyranny under the name of "Society" flows underground, which may be originated from his birth and experience in Prague, Czechoslovakia, till 1963, when he settled in Colorado, USA (He died in 1993).

In the text, all the derivational steps in the proof of theorems are not necessarily presented and explained, and I occasionally did this by myself using margins, sometimes I felt "this margin is too narrow to contain (Pierre de Fermat)", though. For example, it took over one hour to reach the formula cot(x/2) = cot(x) + cosec (x) (page 66) from the more basic trigonometry formulas which were almost completely forgotten by me and I searched once again into the dictionary of mathematics (in Japanese, 2nd edition, Iwanami, Tokyo). I was really glad to develop those process with pencil and paper and in fact agony, and am confident that my wine-immersed neocortex still makes it!

In the Old Testament verse,

He made a molten sea ten cubits from brim to brim, round in compass, and a line of thirty cubits did compass it round about. (I King vii, 23)

which implies PI=3. In the dark age whoever tampered with what the Bible said risked torture chamber and the state, the effort by Nehemiah (150 A.D.), a Hebrew Rabbi and the author of Mishnat ha-Middot, Hebrew geometry, who wittingly managed to compromise the secular Archimedean value PI = 3 1/7 and the PI = 3 in the scripture, is quite marvelous and touching to my tears. He insisted that the diameter ten cubits from brim to brim is measured from outer rim to outer rim, and in fact the inner diameter is firstly measured, subtracting the thickness of the walls, one seventh each, to calculate a inner circumference, i.e., (10 - 2 x 1/7) x 3 1/7 = 30.53,,! (Chaper 7. Dusk). Even after the fall of Constantinople (1453) that is regarded as the end of the Middle age, the insane persecution of science continued, e.g., Giordano Bruno burned alive in Rome in 1600, and Galileo Galilei sentenced to life in prison in 1633.

Corrigendum: page 86, line 2 "sin(beta) = sin(theta)/square root [5 + 2 cos(theta)]" → "sin(beta) = sin(theta)/square root [5 + 4 cos(theta)]", where 5 + 4 cos(theta) is derived from [2 + cos(theta)]² + sin²(theta)
Note: Regarding the figure on page 93. As the author's note on page 94 indicates, probably the author noticed this mistake during the proof reading, the supplimentary cord marked as SC is not of an n-sided polygon. The cord should have be drawn from S to B, not C. After this correction, the ratio of the supplimentary cord to the diameter is exactly cosine beta (= 2 r*cos beta/ 2 r).

Oh, Japanese mathematicians appear on the stage of the digit hunters for PI , i.e., Takebe to 41 decimal places in 1722 and Matsunaga (Dr. Beckmann misspelled his name, correction requested) up to 50 decimal places in 1739 (page 102). And now, PI is taught just 3 in the primary schools here in Japan,,,

In the history of PI, Viete's and Wallis formulae were a great mile stone in that PI was expressed in the form of an infinite product, especially Wallis formula involving only rational numbers. John Wallis (1616-1703) was one more example of mathematician who had an educational background of medicine, who had graduated from Cambridge.

The invention of infinite series, binominal theorem, and differential and integral calculus in 17th century had contributed the discovery of Gregory-Leibniz series, Newton｡ﾇs method, Sharp's series and Machin's formula. It was a giant step forward from Archimedean approximation using inscribed and circumscribed polygons.

PI/4 = 4 arctan (1/5) - arctan (1/239)

By this John Machin｡ﾇs formula, which can accelerate the convergence very rapidly compared with Gregory series, he calculated PI to 100 decimal places in 1706. I was pleased, exactly speaking relieved, that, by reviewing trigonometrical formulas and calculus, including derivatives of inverse trigonometrical functions (in fact high school/undergraduate level: Aided by Primary Differential and Integral Calculus by Magoichiro Watanabe, Shokabou, Tokyo, 1969), I could trace from the original Gregory series to Machin｡ﾇs formula, confirming the derivation step by step in the margins using a pencil. The author paid a special attention (p.146) that England revered Isaac Newton not only in death but also in his life time, a marked difference from the way in which other countries treated their scientists, e.g., Galileo, De Moivre, Kepler, Einstein,,,,

The transcendence of PI proved by F. Lindemann (1882) was a grand finale to this journey, and the author summarizes its framework. Here, too, there was a time interval from some genius' suggestion (Euler) to the rigorous proof; in this case it took over one hundred years. So I don't think it is a shame that it took three months for me to finish this charming book, which describes from Babylonians use of PI (2000 B.C.) to IBM 7030's computation of PI (1966).

(Start reading 2002/04/21, Finished 2002/07/26)

Journey through Genius. The Great Theorems of Mathematics. By William Dunham (Penguin Books, John Wiley and Sons, New York, 1991, ISBN 0-14-014739-X)

I seem to be a Prof. Dunham's books maniac, and once again return to this one which is full of sarcastic remarks peculiar to this brutally honest author.

From the beginning of the preface, I can expect the substance in this book, which begins,

"In his autobiography, Bertrand Russell recalled the crisis of his youth,

There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.

Admittedly, few people find such absolute salvation in mathematics, but many appreciate its power and, more critically, its beauty."

Prof. Dunham knocked us out when he insists,

"There is a remarkable permanence about these mathematical landmarks. In other diciplines, the fads of today become the forgotten discards of tommorow."

Hey, Ed, he doesn't talk about oncology in general, and the histologic classification of tumors and those grading/staging schema in particular, does he?

"Contrary to popular belief, imaginary numbers entered the realm of mathematics not as a tool for solving quadratics but as a tool for solving cubics." (page 151)

I am once again deeply affected by this historical context. When you solve the depressed cubic equation X³ - 15 X - 4 = 0 with Cardano's formula, you encounter square root of minus 121 as a part of the solution, but the solution is yielded real number 4! Mathematicians could not ignore the legitimacy of imaginary numbers at this stage. By the way, did you know the great algebraists were burgeoning in the sixteenth century Italia?, e.g., Pacioli, del Ferro, Tartaglia, Gerolamo Cardano, Ferrari and Bombelli.
It is very interesting to know that Cardano too was originally a physician educated at the University of Padua, was refused permission to practice medicine in his home of Milan, and then turned to mathematician. Copernicus studied medicine and law at the University of Cracow and at Bologna, Padua, and Ferrara, and Galileo firstly entered the University of Pisa to study medicine, largely under the influence of his intellectual father who anyway did not object his later redirection to mathematics. Physician-turned-mathematician/physicist are not rare. I am a bit annoyed if this is one more example of "Man doth not live by bread only".

"In answering it, Euler was again aided by his keen sense of pattern recognition." (page 220)

Hmmmm, pattern recognition works also in mathematics! In the process of evaluating the "wilder" series than 1+1/4+1/9+1/16+,,, he returned to the key equation that was equal to sin x/x, provided x is not 0, and extracted the pattern. Later mathematicians point out the folly of those generalizations of the finite products to infinite products, though, the recognition of the pattern oriented him to the right direction anyway. I hope the raison d'etre of Pathology is still located here even in the age of genomics and post-genomics.

The last two chapters tracks down the journey of Georg Ferdinand Ludwig Philip Cantor (1845-1918) and his set theory, transfinite cardinals and "continuum hypothesis", which occurred as a consequence of the foundational questions and discoveries about the infinite and limit by Augustin-Louis Cauchy and Karl Weierstrass. This theoretical ramification, which I several times read on other books and journals (e.g., Cohen PJ, Hersh R. Non-Cantorian set theory. Sci Am pp 104-116, Dec 1967. Dauben JW. Georg Cantor and the origins of transfinite set theory. Sci Am pp112-121, June 1983), is understandable for non-mathematicians if deliberately reading one sentence after another. In fact, for the rest of us who are daily working down to earth for money and power, and can't maintain solid and intense mathematical thinking process for a long time, this temporary understanding of ultra-abstract theorems and hypothesis easily disappear from our neural network, quite regrettably.

In the afterword, Prof. Dunham cites from G.H. Hardy,

"Truly great theorems possess the three characteristics of economy, inevitability, and unexpectedness"

"Once seen, you immediately believe you would have discovered it"
Certainly it is understandable, but only some geniuses could make it with extraordinary elegance. A tremendous difference originated from the basically same 3 billion base pairs of human genome,,,, (Start reading 2001/12/8, Finished 2002/03/23)

An Introduction to Mathematics. By Alfred North Whitehead (Oxford University Press, London, 1948, ISBN 0-19-500211-3)

Back to the basic is my motivation, when I focus on this book in the current climate of uncertainty following the WTC attacks on September 11. Pursuit of truth, uprightness and beauty, in other word abstractness, in mathematics is far and beyond any particular sensations, any particular persons, or any particular nations, that is, perfectly neutral. With regard to this abstract nature as the root, the author explains.

"These three notions, of the variable, of (algebraic) form, and of generality, compose a sort of mathematical trinity which preside over the whole subject." (page 57)

I love the story, in which Greek geometer Menaechmus (375-325 B.C.) replied to Alexander the Great, saying

"In the country there are private and even royal roads, but in geometry there is only one road at all."

The author sarcastically adds, "There are royal roads in science; but those who first tread them are men of genius and not kings."

We can glimpse at the abstractness of mathematics without a thought of utility in the branch of conic sections (chapter 10). At first, I cannot understand or imagine what are the directrices on the conic curves before referring to the dictionary of mathematics (in Japanese, 2nd edition, Iwanami, Tokyo), which are very complex. Kepler was an astronomer but also an able geometer, especially of conic sections which contributed to the enunciation of three laws of planetary motions, and the author says that the idea that success in scientific research demands an exclusive absorption in one narrow line of study is false.

In the final chapter, the author deals with axioms of quantity, which form the preconceived conditions for operations of addition and of judgement of equality, e.g., "Oh, this liver weighs 1,200 grams, which consists of 1,200 of a unit weight, which is larger than 1,190 grams in the continuum of a weight." Whitehead is the co-author, with Bertrand Russell, of that monumental Principia Mathematica, in which they finally proved "1+1=2". The process was so deliberate that it took until page 362 in a chapter of volume 1 entitled "Prolegomena (A preliminary discussion/Prelude!) to Cardinal Arithmetic". It concludes "From this proposition it will follow, when arithmetical addition has been defined, that 1+1 = 2."

I seem to get his message, "It is an error to confine attention to technical processes, excluding consideration of general ideas." (page 2). (Start reading 2001/09/24, Finished 2001/11/23)

The Evolution of Physics. From Early Concepts to Relativity and Quanta. By Albert Einstein and Leopold Infeld (Simon & Schuster, Touchstone Edition, New York, 1938, 1966. ISBN 0-671-20156-5)

In the preface, the authors write, "He (our idealized reader) realized that in order to understand any page he must have read the preceding ones carefully. He knows that a scientific book, even though popular, must not be read in the same way as a novel." Indeed, this bool traces the evolution of the Gedankengang in Physics in a simple and concise manner. I can immerse myself in the world of physics word by word, phrase by phrase, and am leaving the mundane world that is full of greed, lie, and appearance. Do you know that the currency is no more than a rectangular paper with a printed human face? What a giant step it was from Aristotle to Galileo Galirei, from intuitive explanation to speculative thinking consistent with observation!

After the decline of the mechanical view (chapter 2), which was given up by the difficulties to explain electric and magnetic phenomena and the wave theory of light with ether, the results of Oersted, Faraday, Maxwell, and Hertz led to the development of revolutionay ideas and concepts, i.e., FIELD and electromagnetic wave, which is described by field language. The old view of forces acting between material particles apart (Newton's laws) was replaced by the field theory (Maxwell's field laws). In fact, my reading speed is remarkably slowing down in this 3rd chapter "Field, Relativity". Indeed, I must have read the preceding pages carefully. Maxwell's equations are invariant with respect to the Lorentz transformation, by which two assumptions are assured, i.e., (1) the velocity of light in vacuo is the same in all CS (co-ordinate system) moving uniformly, relative to each other, and (2) all laws of nature are the same in all CS moving uniformly, relative to each other. Lorentz transformation considers four-dimentional time-space continuum of our four-dimensional world of events, which cannot be split into three-dimentional spaces and the one-dimentional time-continuum (special relativety theory). The rhythm of a moving clock is slowed down!

The 4th chapter ("Quanta") begins with the introduction that particular quantity changes by jumps, in a discontinuous way, which is called elementary quantum. Light and a beam of electrons behaves like a wave in one phenomenon and like elementary particles in other phenomena, which appears contradictory. Then, the law of probability wave was introduced, which governs crowds not individuals, the law governing the changes in time of probabilities.

Einstein stresses that thought and ideas, not mathematical formulae, are the beginning of every physical theory. He concluded,

"Without the belief that it is possible to grasp the reality with our theoretical constructions, without belief in the inner harmony of our world, there could be no science."

After I read this book from cover to cover, actually it took two months, I am feeling now satisfaction like reaching the summit of a high mountain. Perspective is grand. I also strongly recommend to read the book by the same author "Relativity. The special and the general theory" beforehand, because the understanding of the Lorentz transformation with its mathematical deriviation helped me to follow the context of this book profoundly.(Finished 2001/09/16)

The Man Who Mistook His Wife for a Hat and Other Clinical Tales. By Oliver Sacks (Simon & Schuster, Touchstone Edition, New York, 1998. ISBN 0-684-85394-9)

Erdös on Graphs. His legacy of unsolved problems. By Fan Chung and Ron Graham (A K Peters, Massachusetts, 1998. ISBN 1-56881-079-2)

Actually abandoned in the middle. As an introduction to graph theory for me instead, I found the review by Brian Hayes which appeared in American Scientist valuable (Graph theory in practice: Part I. American Scientist. 2000:88;9-13, Graph theory in practice: Part II. American Scientist. 2000:88;104-109), which discusses stepwise to that graph theory seems useful in exlaining the architecture of the World Wide Web (applications side of the austere formalism!) .

My Brain Is Open. The mathematical journeys of Paul Erdös. By Bruce Schechter (Simon & Schuster, New York, 1998. ISBN 0-684-84635-7)

Corrigendum: page 107, line 12 "45 and 65" --> "45 and 55"

The Man Who Loved Only Numbers. The story of Paul Erdös and the search for mathematical truth. By Paul Hoffman (Hyperion, New York, 1998. ISBN 0-7868-6362-5)

,,,,, My Erdös number is ∞ (infinite or aleph-null). But, the diameter of this collaboration graph centered on Paul Erdös and encompassing almost all contributors to the science literature may be much smaller ("six degrees separation"; See Brian Hayes. Graph theory in practice. American Scientist. 2000:88;9-13)

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