CALCULUS from a Historical Perspective Paul Yiu Department of Mathematics Florida Atlantic University Fall 2009 Chapters 1–49 Tuesday, December 2 Tuesdays 8/25 9/1 9/8 9/15 9/22 9/29 10/6 10/13 Thursdays 8/27 9/3 9/10 9/17 9/24 10/1 10/8 10/15 Tuesdays 10/20 10/27 11/3 11/10 11/17 11/24 12/1 Thursdays 10/22 10/29 11/5 11/12 11/19 *** 12/3 Contents 1 Extraction of square roots 101 2 Convergence of sequences 107 3 Square roots by decimal digits 113 4 a + a + + √a + 115 ··· ··· q 5 An infinitep continued fraction 117 6 Ancient Chinese calculations of π 121 7 Existence of π 125 8 Archimedes’ calculation of π 129 9 Ptolemy’s calculations of chord lengths 131 sin θ 10 The basic limit limθ 0 =1 135 → θ 11 The Arithmetic - Geometric Means Inequality 139 12 Principle of nested intervals 143 13 The number e 147 14 Some basic limits 151 15 The parabola 201 iv CONTENTS 16 Archimedes’ quadrature of the parabola 205 17 Quadrature of the spiral 209 18 Apollonius’ extremum problems on conics 213 19Normalsofaparabola 215 20 Envelope of normal to a conic 217 21 The cissoid 221 22 Conchoids 225 23 The quadratrix 231 24 The lemniscate 233 25 Volumes in Euclid’s Element 301 26 Archimedes’ calculation of the volume of a sphere 305 27 Cavalieri’s principle 309 28 Fermat: Area under the graph of y = xn 311 29 Pascal: Summation of powers of numbers in arithmetic pro- gression 313 30 Pascal: On the sines of a quadrant of a circle 315 31 Newton: The fundamental theorem of calculus 317 32 Newton: The binomial theorem 319 33 Newton’s method of approximate solution of an equation 321 34 Newton’s reversion of series 323 35 Newton: The series for sine and cosine 325 36 Wallis’ product 327 CONTENTS v 37 Newton: Universal gravitation 329 38 Logarithms 401 39 Euler’s introduction to e and the exponential functions 405 40 Euler: Natural logarithms 409 41 Euler’s formula eiv = cos v + i sin v 411 42 Summation of powers of integers 417 43 Series expansions for the tangent and secant functions 421 44 Euler’s first calculation of 1+ 1 + 1 + 1 + 1 + 425 22k 32k 42k 52k ··· 45 Euler: Triangulation of convex polygon 433 46 Infinite Series with positive terms 501 47 The harmonic series 507 48 The alternating harmonic series 511 49 Conditionally convergent series 515 45.0.1 2 ......................... div Chapter 1 Extraction of square roots Heron Heron’s numerical example illustrating the use of his formula ∆= s(s a)(s b)(s c) − − − for the area of a trianglep in terms of its sides a, b, c and semiperimeter a+b+c s = 2 : Let the sides of the triangle be 7, 8, and 9. ...The result [of multiplying s, s a, s b, s c] is 720. Take the square root of this and it− will be− the area− of the triangle. Since 720 has not a rational square root, we shall make a close approxi- mation to the root in this manner. Since the square nearest to 720 is 729, having a root 27, divide 27 into 720; the result is 2 2 26 3 ; add 27; the result is 53 3 . Take half of this; the result is 1 1 5 26 2 + 3 (= 26 6 ). Therefore the square root of 720 will be very 5 5 1 nearly 26 6 . For 26 6 multiplied by itself gives 720 36 ; so that 1 the difference is 36 . If we wish to make the difference less 1 than 36 , instead of 729 we shall take the number now found, 1 720 36 , and by the same method we shall find an approxima- 1 tion differing by much less than 36 . Heron, Metrica, i.8. Let a be a given positive integer. Consider the sequence (an) defined recursively by 1 a an = an 1 + , (1.1) 2 − an 1 − 102 Extraction of square roots with an arbitrary positive initial value a1. The sequence (an) converges to the square root of a regardless of the positive initial value. Example 1.1. Approximations of √2 with initial values a =1: n an 1 1/1 1 2 3/2 1.5 3 17/12 1.4166666666666 4 577/408 1.4142156862745 ··· 5 665857/470832 1.4142135623746 ··· 6 886731088897/627013566048 1.4142135623730 ··· ··· Example 1.2. Approximations of √3 with initial value a1 =2: n an 1 2 2 2 7/4 1.75 3 97/56 1.73214285714286 4 18817/10864 1.73205081001473 ··· 5 708158977/408855776 1.73205080756888 ··· 6 1002978273411373057/579069776145402304 1.73205080756888 ··· ··· Convergence Why does the sequence (an) converge to √a? Note that 1 (i) with any arbitrary positive a1, all subsequent an are greater than √a, (ii) (an) is a decreasing sequence: an 1 a 1 = 1+ 2 < (1+1)=1. an 1 2 an 1 2 − − Since a decreasing sequence bounded from below converges, we write ℓ = limn xn. From (1.1), we have →∞ 1 a ℓ = ℓ + . 2 ℓ Solving this equation, we have ℓ = √a. 1This follows from the arithmetic-geometric means inequality. 103 2 2 1 a an a = an 1 + a − 4 − an 1 − − 1 a 2 = an 1 4 − − an 1 − 2 2 (an 1 a) = − 2− 4an 1 − 2 2 (an 1 a) < − − . 4a Therefore, beginning with a1, by iterating n steps, we obtain a2,..., an+1, with 1 a2 a< (a2 a)2 n+1 − 4a · n − 1 1 2 4 < (an 1 a) 4a · (4a)2 − − . 1 1 1 2 2n < − (a a) 4a · (4a)2 ··· (4a)2n 1 1 − 1 2 2n = 2n 1 (a1 a) (4a) − − This shows that the convergence is very fast. For example, for a =2, if we begin with a1 = 2 and execute n steps, then we have a rational approximation an+1 satisfying 2 1 an+1 2 < 2n 3 . − 2 − To guarantee an accuracy up to, for example, 10 decimal digits be- 1 yond the decimal point, we need to make the error < 1010 . This requires 2n 3 10 2 − > 10 , and it is enough to iterate 5 times. 104 Extraction of square roots Reorganization It is more convenient to calculate the numerators and denominators sep- arately. If we write a = xn , then n yn 2 2 xn 1 xn 1 ayn 1 xn 1 + ayn 1 = an = − + − = − − . yn 2 yn 1 xn 1 2xn 1yn 1 − − − − This leads to two sequences of integers (xn) and (yn) defined recursively 2 2 xn = xn 1 + ayn 1, − − yn = 2xn 1yn 1, − − with arbitrary initial values x1 and y1. Exercise (1) Find a rational approximation r of √5 satisfying r2 3 < 1 . | − | 1030 (2) Construct two integer sequences (x ) and (y ) such that the xn n n yn 2 converges to the square root of 3 . (3) Let a be a positive integer. Consider a sequence (an) defined recursively by 1 a an = an 1 . 2 − − an 1 − (a) Show that the sequence does not converge regardless of the initial value. (b) Show that for every given positive integer ℓ, an initial value can be chosen so that the sequence is periodic with period ℓ. [Hint: rewrite the √a recursive relation by putting an 1 = ]. − tan θ (4) Start with two positive numbers a0 and a0, and iterate according to the rule an = √an 1 + √an 2. − − Does the sequence (an) converge? If so, what is the limit? (No proof is required). Irrationality Why is √2 an irrational number? Here are two simple proofs. 105 √ √ p (1) If 2 is rational, we write 2= q in lowest terms, then a square of side p has the same area as two squares of side q, and for this, the p p square is smallest possible. × q q p Yet this diagram shows that there is a smaller square which also has the same area as two squares. This contradicts the minimality of p. (Ex- ercise: what are the dimensions of the smaller squares involved?) (2) If √2 is rational, then there is a smallest integer q for which q√2 is an integer. Now, the number (q√2 q)√2 is a smaller integer multiple of √2, which is also an integer. This− contradicts the minimality of q. Irrationality of √k n Here is a simple proof that for a positiveinteger n, √n is either irrational 2 p or integral. If √n = q in lowest terms, then both q√n = p and p√n = qn are integers. Since there are integers a and b satisfying ap + bq =1, we have √n = a pn + b qn, an integer. More generally,· for given· positive integers n and k, the number √k n is irrational unless n is the k-power of an integer. This fact follows easily from the fundamental theorem of arithmetic: every positive integer > 1 is uniquely the product of powers of distinct prime numbers. 3 The ladder of Theon Consider an integer a> 1 which is not the square of an integer. Let b be the integer closest to √a (so that b √a < 1 ). | − | 2 2M. Levin, The theorem that √n is either irrational or integral, Math. Gazette, , 60 (1976) 138. Levin p2 has subsequently (ibid., 295) given an even shorter proof: Since the reduced fraction q = p√n = qn is an integer, q must be equal to 1, and √n is an integer. 3See the supplements to Chapter 1 for a more elementary proof.