When he wrote this, Earth had twice the number of species of plants and animals that now survive. Human population was 3 billion (up from 750,000 200 years earlier p. 211) (and rising by one billion per 12 years, as it has ever since).
Speed of sound in surface ocean at 0ºC is 1543 m/s; in air at 0ºC, 332 m/s. p. 54.
Reef-forming coral won't grow below about 70ºF. p. 59. [Nor will they survive more than tiny concentrations of atmospheric CO2, which dissolves in seawater, becomes acid, and kills coral.] Polyps range from pinhead- to pea-size. They extend their tentacles only at night. p. 62. When a moray bites it doesn't let go: you have to cut its head off. p. 69. Sea urchins walk at night. p. 70. Probably the greatest hazard of the reef is sunburn. p. 75.
Minimum rainfall for a tropical rainforest is about 80 inches/year. p. 101. Zero to 10 inches/year is desert. Deserts lie in latitudes where winds blow toward the equator. p. 123.
Forest and sea are similar in having sunny high layers, grading down to dark depths, with appropriately-adapted communities of living things at each level.
In northern woods, the mosquitoes biting you are apt to be all of the same kind; while in the rain forest, almost every bite will be from a different species of mosquito, if that is any comfort. p. 105.
The ants poured on in their tens of thousands, swept through our snake pit, and left us with skeletons. p. 110.
Some male moths can locate a female of their species a mile away, and fly directly to her. pp. 177, 186.
To a dog, the silently approaching vampire bat would sound like a boiler factory falling in out of the sky. p. 184.
Look at the human brain as a product, a consequence, of the use of tools, the development of culture. pp. 219-228.
Southeast Asia seems to be the original home of a surprising number of domesticated plants and animals. p. 237-239.
This, I think, will be terrific for a middle-school kid. Benjamin expresses enthusiasm for, and enjoyment of, math. Conversational tone, a little humor.
Trigonometry, Pythagorean triples, trig functions, finding heights of trees and mountains, law of cosines, law of sines, Hero's formula, trigonometric identities, radians, graphs of trig functions
Imaginary number i, Euler's number e, e^(i theta) notation, polyhedrons, complex arithmetic, compound interest, exponential functions, logarithms
Calculus: optimization, differentiation, derivatives of products of functions, quotients of functions, polynomials, exponentials, trig functions; chain rule; Taylor series
The sum of the cubes of the first n integers is the square of the sum of the first n integers: Sum[i^3, {i, 1, n}] = (Sum[i, {i, 1, n}])^2 = (1/4)(n^2)(n + 1)^2 https://www.wolframalpha.com/input?i=... p. 12
How to find the day of the week for any date. pp. 65-70.
Sierpinski triangle: a fractal pattern of the odd (black) and even (white) numbers in Pascal's triangle: https://www.wolframalpha.com/input?i=... pp. 92-94. The larger the triangle, the more nearly white it is.
Wilson's theorem: n is a prime number if and only if (n - 1)! + 1 is a multiple of n. p. 144.
If p is an odd prime number, then 2^( p - 1) - 1 is a multiple of p . (Fermat. p. 145) A number that has this property, but is not prime, is called a pseudoprime.
Connecting the midpoints of any quadrilateral always produces a parallelogram. p. 150.
There are over 300 proofs of the Pythagorean Theorem. p. 174. Benjamin gives us five of them.
Three points on a circle, two of them forming the diameter, are a right triangle. Proof p. 185.
Central angle theorem: For any two points X and Y on a circle centered at O, the angle XPY at /any/ point P on the major (larger) arc of the circle, from X around to Y, will be half of the angle XOY. The angle XQY at any point Q on the minor arc of the circle, from X to Y, will be 180 degrees minus the angle XPY. p. 186.
Area of circle of radius r = pi r^2. Proofs pp. 187-188.
Ellipse, semimajor and semiminor axes a and b: (x/a)^2 + (y/b)^2 = 1 p. 189.
Drawing an ellipse: p. 190.
Approximate formula for the circumference of an ellipse, semimajor and semiminor axes a and b: pi( 3a + 3b - sqrt( (3a + b)(3b + a) ) ) Notice that if a = b = r, it's a circle, circumference 2 pi r. Srinivasa Ramanujan (1887-1920). p. 191.
Volume of sphere = (4/3) pi r^3
Area of sphere = 4 pi r^2
Cone of height h, on a circle of radius r, slant height s (s^2 = r^2 + h^2):
Volume of cone = pi r^2 h/3
Area of cone = pi r s p. 192.
Volume of a pizza, radius z, thickness a, V = pi z z a p. 193.
Pythagorean triples: For a right triangle, short sides a and b, hypotenuse c: Where a, b, c are whole numbers, they're called a "Pythagorean triple."
For any two positive integers (m, n), m > n, the numbers a = m^2 - n^2 b = 2mn c = m^2 + n^2 are a Pythagorean triple. Notice that a^2 + b^2 = c^2. p. 205.
Every Pythagorean triple can be created by some choice of (m, n). p. 205.
Law of cosines: for any triangle, sides a, b, c, angle C opposite side c: c^2 = a^2 + b^2 - 2 a b cos C proof p. 216.
Area of any triangle, sides of lengths a, b form angle C: area = (1/2) a b sin C proof p. 217.
Law of sines: for any triangle, sides a, b, c, opposite angles A, B, C: (sin A)/a = (sin B)/b = (sin C)/c proof pp. 217-218.
Hero's formula: for any triangle, sides a, b, c, semiperimeter s = (a + b + c)/2, area of triangle = Sqrt[ s(s - a)(s - b)(s - c) ] p. 219.
Trigonometric identities sin^2 x + cos^2 x = 1 many others p. 225.
For any polyhedron with a number F of flat faces, a number E of straight-line edges, and a number V of vertices, F + V = 2 + E p. 232
Multiplying complex numbers: The magnitude (length) of the product is the product of the magnitudes. The argument (angle) of the product is the sum of the arguments. Dividing complex numbers, divide the lengths and subtract the angles. p. 240.
e^(i theta) = cos theta + i sin theta e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + … cos x = 1 - x^2/2! + x^4/4! - x^6/6! + … sin x = x - x^3/3! + x^5/5! - x^7/7! + … p. 251
Thomas Ray
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