Conquering Calculus: The Easy Road to Understanding Mathematics, Jefferson Hane Weaver, c. 1998. Almost all prose, very few formulas or even numbers in the book. Deceptive title: not on "calculus" as I thought, but on "calculus" as in calculating. The author is a lawyer which explains a lot.
Trigonometric Delights, Eli Maor, c. 1998.
The first nine chapters require only basic algebra and trigonometry; the remaining chapters rely on some knowledge of calculus (no higher than Calculus II). Much of the material should thus be accessible to high school and college students.
Rhind Papyrus:
- bought in 1858, by a Scottish lawyer and antiquarian, A. Henry Rhind
- found a few years earlier in the ruins of a small building in Thebes (near present-day Luxor), Upper Egypt
- 84 mathematical problems
- a scroll 18 feet long; 13 inches wide
- Rhind died at age 30; British Museum gets the papyrus; a bit of it missing; miraculously, the missing portion possessed by the New-York Historical Society; complete text is now available
- originally copied around 1650 BC (reign of King A-user-Re, Hyksos dynasty)
- probably written during reign of King Ne-ma'et-Re, Amenem-het III, 1849 to 1801 BC
- the word degree originated with the Greeks
- Greek word "moira"
- Arabs translated "moira" into daraja (akin to the Hebrew dar'ggah, a step on a ladder or scale); this in turn:
- Latin: de gradus, --> degree
- Greeks: the sixtieth part of a degree the "first part," the sixtieth part of that the "second part," and so on
- Latin: the former was calls pars minuta prima ("first small part") and the latter pars minuta secunda ("second small part") from which came our minute and second
- why we use radians instead of degrees
- eliminates the unwanted factor, π /180
- also, a small angle and its sine are nearly equal numerically (p. 17)
- sine of one degree (sin 1°) = 0.0174524
- one degree = 0.0174533 radian, so the angle and its sine agree to within one hundred thousandth; for an angle of 0.5° (again expressed in radians), the agreement is within one millionth, and so on -- the smaller than angle the closer the sine (of its measure in radians) and its measure in degrees (again: the sine of one 1° = 0.0174524 and 1° = 0.0174533 radians;
- again, 1° = 0.0174524 radians
- sine (1) = 0.0174533
- radian: modern vintage; coined by Lord Kelvin, 1871
- Greek trigonon = traingle
- Greek metron = measure
- ghomon: an analog device for computing cotangent function; ghomon = "shadow reckoning"
- Hipparchus: trigonometry in the modern sense began with Hipparchus of Nicaea (ca. 190 - 120 BC); stars
- Ptolemy: first major work on trigonometry to have come to us intact from Ptolemy (ca. 85 - ca. 165 AD); Alexandria, the intellectual center of the Hellenistic world (unrelated to the Ptolemy dynasty that ruled Egypt after the death of Alexander the Great in 323 BC); star catalog based on Hipparchus' work; names 48 constellations (still in use today); standard map used well into Middle Ages; greatest work, Almagest, a summary of mathematical astronomy, 13 books, reminiscent of the 13 books of Euclid's Elements (forms core of classical geometry); similarities go even farther; evolution of Almagest:
- Ptolemy's title translated to "mathematical syntaxis"
- later generations added the superlative megiste ("greatest")
- Arabs translated the work into their own language, kept the word megiste but added the conjunction al ("the"); in due time it became known as the Almagest
- Almagest became cornerstone of geocentric world picture well into 16th century; became the canon of the Roman Church
- Of particular interest in this chapter: Ptolemy's table of chords; subject of chapters 10 and 11 in the first book of the Almagest; essentially a table of sines. It was the Hindus that shortened the process to come up with a table of sines.
Plimpton 322:
- item #322 in the G. a. Plimpton Collection at Columbia University in New York
- earliest trigonometric table?
- regardless: it proves that the Babylonians were not only familiar with the Pythagorean Theorem a thousand years before Pythagoras, but that they knew the rudiments of number theory and had the computational skills to put the theory into practice
Aryabhatiya of Aryabhata (ca 510) is considered the earliest Hindu treatise on pure mathematics
it is also the first work to refer explicitly to the sine as a function of an angle
etymology:
- ardha-jya: the half-chord, sometimes turned around to jya-ardha (chord-half)
- jya or jiva evolved as shortened version
- Arabs translated the Aryabhatiya; retained the word jiva without translating its meaning
- in Arabic -- as also in Hebrew -- words consist mostly of consonants
- jiva could be pronounced as jiba or jaib, and jaib in Arabic means bosom, fold, or bay
- Arabic translated into Latin, jaib was translated into sinus, which means bosom, bay, or curve
- sine became the English version of the Latin sinus
- abbreviated notation sin was first used by Edmund Gunter (1581 - 1626), an English minister who later became professor of astronomy at Gresham College in London; invented the forerunner to the familiar slide rule; and the notation sin (as well as tan) first appeared in a drawing describing his investion, the "Gunter scale"
Note: sin^2φ = square of sinφ [not, sin(sinφ)]. So sin^2φ = sinφ x sinφ.
The chapter goes on to say how the other trig names originated.
Tangents not really needed until navigational tables were computed in the 15th century.
Tangents and cotangents originated with the gnomon and shadow reckoning. [Coincidentally, and completely unexpected, I came across the "gnomon" again a few days later when reading John North's Stonehenge, c. 1996, p. 401: "Of the Heel Stone he writes that it 'was a gnomon for the purpose of observing the rising of the Sun on the auspicious morn of the summer solstice.'"]
Chapter 4: Trigonometry Becomes Analytic (17th and 18th centuries)
- Wallis introduced the infinity symbol we use today
- there are two "types" of trigonometry: computation and analytic
- computational: associated with the triangle; Napier's table
- analytic: relationships among the trig functions
- trig numbers: numbers in their own right; don't have to be associated with the triangle
- three big areas of study at this time: a) range of cannon projectile; b) navigation on open sea; c) music
- range of canon projectile; known for a long time, but now had analytic basis
- navigation on open sea: major area of study in 17th and 18th century oscillations; navigation depended upon clocks of ever greater precision; led scientists to study the oscillations of pendulums and springs of various kinds
- increased skill and sophistication in building musical instruments; scientists motivated to study vibrations
- all of this underscored the role of trigonometry in describing periodic phenomena and resulted in a shift of emphasis from computational trig (the compilation of tables) to the relations among trig functions
- that's why one study "computational trig" in geometry, but then there is a whole new subject called analytic trigonometry that is taught a year or so later
- developments from trig even farther from its original connection with a triangle; now, trig functions were defined as pure numbers rather than as ratios; for example cos x is now defined as an independent variable itself as a real number rather than an angle
- Fourier's theorem marks one of the greatest achievements of 19th century analysis: he showed that the sine and cosine functions are essential to the study of all periodic phenomena, simple or complex, in much the same way prime numbers are the building blocks of all integers
- Fourier's theorem was later generalized to non-periodic functions (in which case the infinite series becomes an integral), as well as to series involving nontrigonometric functions -- crucial in all branches of science
Chapter 6: Two Theorems From Geometry
Chapter 7: Epicycloids and Hypocycloids
- the study of the toy that was put on the market in the 1970s -- the spirograph
- the story of Gauss summing the numbers 1 to 100
Chapter 10: (sin x)/x
Chapter 11: A Remarkable Formula
Chapter 12: tan x
- of the numerous functions we encounter in elementary mathematics, perhaps the most remarkable is the tangent function, for a couple of reasons, but particularly this reason: tan x has a period π (a function f(x) is said to have a period P if P is the smallest number such that f(x+P) = f(x) for all x in the domain of the function). This fact is quite remarkable: the functions sin x and cos x have the common period 2π, yet the ratio, tan x, reduces the period to π. When it comes to periodicity, the ordinary rules of the algebra of functions may not be valid: the fact that two functions f and g have a common period P does not imply that f + g or fg have the same period.
- as we saw in chapter 2, the tangent function has its origin in the "shadow reckoning" of antiquity; during the Renaissance it was resurrected -- though, it was not called "tangent" -- in connection with the fledging art of perspective.
- a cylinder projection of the earth appears to be identical to Mercator's projection, but they resemble each other only superficially; they are based on entirely different principles
- Mercator's story
- Mercator: one of the first to bind in one volume a collection of separate maps: called it an "atlas," in honor of the legendary globe-holding mythological figure that decorated the title page; this work was published in three parts, the last appearing in 1595, one year after his death
- how he implemented his plan, the spacing between successive parallels had first to be determined. Exactly how Mercator did this is not know (and is still being debated by historians of cartography); he left no written record of his method except for a brief explanation (in the book, p. 173).
Chapter 15: Fourier's Theorem
three developments transformed trigonometry
Ptolemy's table of chords
de Moivre's theorem and Euler's formula (e^ix = cos x + i sin x)
Fourier's theorem
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