Henry Sinclair Hall (1848–1934)

Indeholder "Preface", "Preface to present reprint", "Chapter I. Ratio", " Commensurable and incommensurable quantities", " Ratio of greater and less inequality", " a/b = c/d = e/f = ... = ((p*a^n + q*c^n + r*e^n + ... )/(p*b^n + q*d^n + r*f^n + ... )) ^ (1/n)", " (a_1 + a_2 + a_3 + ... + a_n)/(a_1 + a_2 + a_3 + ... + a_n) lies between greatest and least of fractions a_1/b_1, ... a_n/b_n", " Cross multiplication", " Eliminant of three linear equations", " Examples I", "Chapter II. Proportion", " Definitions and Propositions", " Comparison between algebraical and geometrical definitions", " Case of incommensurable quantities", " Examples II", "Chapter III. Variation", " If A ∝ B, then A = mB", " Inverse variation", " Joint variation", " If A ∝ B when C is constant, and A ∝ C when B is constant, then A=mBC", " Illustrations. Examples on joint variation", " Examples III", "Chapter IV. Arithmetical progression", " Sum of n terms of an arithmetical series", " Fundamental formulae", " Insertion of arithmetic means", " Examples IV. a", " Discussion of roots of dn^2 + (2a-d)n - 2s = 0", " Examples IV. b", "Chapter V. Geometrical progression", " Insertion of geometric means", " Sum of n terms of a geometrical series", " Sum of an infinite geometrical series", " Examples V. a", " Proof of rule for the reduction of a recurring decimal", " Sum of n terms of an arithmetico-geometric series", " Examples V. b", "Chapter VI. Harmonical progression. Theorems connected with the progressions", " Reciprocals of quantities in Harmonical progression are in arithmetical progression", " Harmonic mean", " Formulae connecting arithmetical mean, geometrical mean, harmonical mean", " Hints for solution of questions in Progressions", " Sum of squares of the natural numbers", " Sum of cubes of the natural numbers", " Sum notation", " Examples VI. a", " Number of shot in pyramid on a square base", " Pyramid on a triangular base", " Pyramid on a rectangular base", " Incomplete pyramid", " Examples VI. b", "Chapter VII. Scales of notation", " Explanation of systems of notation", " Examples VII. a", " Expression of an integral number in a proposed scale", " Expression of a radix fraction in a proposed scale", " The difference between a number and the sum of its digits is divisible by r - 1", " Proof of rule for "casting out the nines"", " Test of divisibility by r + 1", " Examples VII. b", "Chapter VIII. Surds and imaginary quantities", " Rationalising the denominator of a / (sqrt(b) + sqrt(c) + sqrt(d))", " Rationalising factor of a^(1/p) +/- b^(1/q)", " Square root of a + sqrt(b) + sqrt(c) + sqrt(d)", " Cube root of a + sqrt(b)", " Examples VIII. a", " Imaginary quantities", " sqrt(-a) * sqrt(-b) = - sqrt(ab)", " If a + ib = 0, then a = 0, b = 0", " If a + ib = c + id, then a = c, b = d", " Modulus of product is equal to product of moduli", " Square root of a + ib", " Powers of i", " Cube roots of unity; 1 + w + w^2 = 0", " Powers of w", " Examples VIII. b", "Chapter IX. The theory of quadratic equations", " A quadratic equation cannot have more than two roots", " Conditions for real, equal, imaginary roots", " Sum of roots = -b/a, product of roots = c/a", " Formation of equations when the roots are given", " Conditions that the roots of a quadratic should be (1) equal in magnitude and opposite in sign, (2) reciprocals", " Examples IX. a", " For real values of x the expression ax^2 + bx + c has in general the same sign as a ; exceptions", " Examples IX. b", " Definitions of function, variable, rational integral function", " Condition that ax^2 + 2hxy+ by^2 + 2gx + 2fy + c may be resolved into two linear factors", " Condition that ax^2 + bx + c = 0 and a'x^2 + b'x + c' = may have a common root", " Examples IX. c", "Chapter X. Miscellaneous equations", " Equations involving one unknown quantity", " Reciprocal equations", " Examples X. a", " Equations involving two unknown quantities", " Homogeneous equations", " Examples X. b", " Equations involving several unknown quantities", " Examples X. c", " Indeterminate equations ; easy numerical examples", " Examples X. d", "Chapter XI. Permutations and combinations", " Preliminary proposition", " Number of permutations of n things r at a time", " Number of combinations of n things r at a time", " The number of combinations of n things r at a time is equal to the number of combinations of n things (n-r) at a time", " Number of ways in which m + n + p + ... things can be divided into classes containing m, n, p, ... things severally", " Examples XI. a", " Signification of the terms 'like' and 'unlike'", " Number of arrangements of n things taken all at a time, when p things are alike of one kind, q things are alike of a second kind, etc.", " Number of permutations of n things r at a time, when each may be repeated", " The total number of combinations of n things", " To find for what value of r the expression C(n,r) is greatest", " Ab initio proof of the formula for the number of combinations of n things r at a time", " Total number of selections of p + q + r + ... things, whereof p are alike of one kind, q alike of a second kind, etc", " Examples XI. b", "Chapter XII. Mathematical induction", " Illustrations of the method of proof", " Product of n binomial factors of the form x + a", " Examples XII", "Chapter XIII. Binomial theorem. Positive integral index", " Expansion of (x + a)^n, when n is a positive integer", " General term of the expansion", " The expansion may be made to depend upon the case in which the first term is unity", " Second proof of the binomial theorem", " Examples XLII. a", " The coefficients of terms equidistant from the beginning and end are equal", " Determination of the greatest term", " Sum of the coefficients", " Sum of coefficients of odd terms is equal to sum of coefficients of even terms", " Expansion of multinomials", " Examples XIII. b", "Chapter XIV. Binomial theorem. Any index", " Euler's proof of the binomial theorem for any index", " General term of the expansion of (1 + x)^n", " Examples XIV. a", " Expansion of (1+x)^n is only arithmetically intelligible when x<1", " The expression (x+y)^n can always be expanded by the binomial theorem", " General term of the expansion of (1 - x)^(-n)", " Particular cases of the expansions of (1 - x)^(-n)", " Approximations obtained by the binomial theorem", " Examples XIV. b", " Numerically greatest term in the expansion of (1 + x)^n", " Number of homogeneous products of r dimensions formed out of n letters", " Number of terms in the expansion of a multinomial", " Number of combinations of n things r at a time, repetitions being allowed", " Examples XIV. c", "Chapter XV. Multinomial theorem", " General term in the expansion of (a + bx + cx^2 + dx^3 + ...)^p, when p is a positive integer", " General term in the expansion of (a + bx + cx^2 + dx^3 + ...)^n, when n is a rational quantity", " Examples XV", "Chapter XVI. Logarithms", " Definition. N=a^(log_a N)", " Elementary propositions", " Examples XVI. a", " Common Logarithms", " Determination of the characteristic by inspection", " Advantages of logarithms to base 10", " Advantages of always keeping the mantissa positive", " Given the logarithms of all numbers to base a, to find the logarithms to base b", " log_a(b) * log_b(a) = 1", " Examples XVI. b", "Chapter XVII. Exponential and logarithmic series", " Expansion of a^x. Series for e", " e is the limit of (1 + 1/n)^n, when n is infinite", " Expansion of log_e(1 + x)", " Construction of Tables of Logarithms", " Rapidly converging series for log_e(n + 1) - log_e(n)", " The quantity e is incommensurable", " Examples XVII", "Chapter XVIII. Interest and annuities", " Interest and Amount of a given sum at simple interest", " Present Value and Discount of a given sum at simple interest", " Interest and Amount of a given sum at compound interest", " Nominal and true annual rates of interest", " Case of compound interest payable every moment", " Present Value and Discount of a given sum at compound interest", " Examples XVIII. a", " Annuities. Definitions", " Amount of unpaid annuity, simple interest", " Amount of unpaid annuity, compound interest", " Present value of an annuity, compound interest", " Number of years' purchase", " Present value of a deferred annuity, compound interest", " Fine for the renewal of a lease", " Examples XVIII. b", "Chapter XIX. Inequalities", " Elementary Propositions", " Arithmetic mean of two positive quantities is greater than the geometric mean", " The sum of two quantities being given, their product is greatest when they are equal : product being given, the sum is least when they are equal", " The arithmetic mean of a number of positive quantities is greater than the geometric mean", " Given sum of a, b, c, ...; to find the greatest value of a^m * b^n * c^p", " Easy cases of maxima and minima", " Examples XIX. a", " The arithmetic mean of the m'th powers of a number of positive quantities is greater than m'th power of their arithmetic mean, except when m lies between 0 and 1", " If a and b are positive integers, and a>b, (1 + x/a)^a > (1 + x/b)^b", " if 1 > x > y > 0, ((1+x)/(1-x))^(1/x) > ((1+y)/(1-y))^(1/y)", " a^a * b^b > ((a+b)/2)^(a+b)", " Examples XIX. b", "Chapter XX. Limiting values and vanishing fractions", " Definition of Limit", " Limit of a_0 + a_1 * x + a_2 * x^2 + a_3 * x^3 + ... is a_0 when x is zero", " By taking x small enough, any term of the series a_0 + a_1 * x + a_2 * x^2 + a_3 * x^3 + ... may be made as large as we please compared with the sum of all that follow it; and by taking x large enough, any term may be made as large as we please compared with the sum of all that precede it", " Method of determining the limits of vanishing fractions", " Discussion of some peculiarities in the solution of simultaneous equations", " Peculiarities in the solution of quadratic equations", " Examples XX", "Chapter XXI. Convergence and divergency of series", " Case of terms alternately positive and negative", " Series is convergent if Lim(u_n/u_n-1) is less than 1", " Comparison of Σ(u_n) with an auxiliary series Σ(v_n)", " The auxiliary series 1/1^p + 1/2^p + 1/3^p + ...", " Application to Binomial, Exponential, Logarithmic Series", " Limits of (log n)/n and n*x^n when n is infinite", " Product of an infinite number of factors", " Examples XXI. a", " u-series is convergent when v-series is convergent, if u_n / u_n-1 > v_n / v_n-1", " Series is convergent if Lim (n*(u_n / u_n+1 - 1)) > 1", " Series is convergent if Lim (n*log(u_n / u_n+1)) > 1", " Series Σφ(n) compared with series Σ(a^n*φ(n))", " The auxiliary series Σ(1/(n*log(n)^p))", " Series is convergent if Lim ((n*(u_n / u_n+1 - 1) - 1) * log(n)) > 1", " Product of two infinite series", " Examples XXI. b", "Chapter XXII. Undetermined coefficients", " If the equation f(x)=0 has more than n roots, it is an identity", " Proof of principle of undetermined coefficients for finite series", " Examples XXII. a", " Proof of principle of undetermined coefficients for infinite series", " Examples XXII. b", "Chapter XXIII. Partial fractions", " Decomposition into partial fractions", " Use of partial fractions in expansions", " Examples XXIII", "Chapter XXIV. Recurring series", " Scale of relation", " Sum of a recurring series", " Generating function", " Examples XXIV", "Chapter XXV. Continued fractions", " Conversion of a fraction into a continued fraction", " Convergents are alternately less and greater than the continued fraction", " Law of formation of the successive convergents", " p_n*q_n-1 - p_n-1*q_n = (-1)^n", " Examples XXV. a", " The convergents gradually approximate to the continued fraction", " Limits of the error in taking any convergent for the continued fraction", " Each convergent is nearer to the continued fraction than a fraction with smaller denominator", " pp'/(qq') > or < x^2 as p/q > or < p'/q'", " Examples XXV. b", "Chapter XXVI. Indeterminate equations of the first degree", " Solution of ax - by = c", " Given one solution, to find the general solution", " Solution of ax + by = c", " Given one solution, to find the general solution", " Number of solutions of ax + by = c", " Solution of ax + by + cz = d, a'x + b'y + c'z = d'", " Examples XXVI", "Chapter XXVII. Recurring continued fractions", " Numerical example", " A periodic continued fraction is equal to a quadratic surd", " Examples XXVII. a", " Conversion of a quadratic surd into a continued fraction", " The quotients recur", " The period ends with a partial quotient 2*a_1", " The partial quotients equidistant from first and last are equal", " The penultimate convergents of the periods", " Examples XXVII. b", "Chapter XXVIII. Indeterminate equations of the second degree", " Solution of ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0", " The equation x^2 - N*y^2 = 1 can always be solved", " Solution of x^2 - N*y^2 = -1", " General solution of x^2 - N*y^2 = 1", " Solution of x^2 - n^2*y^2 = a", " Diophantine Problems", " Examples XXVIII", "Chapter XXIX. Summation of series", " Summary of previous methods", " u_n the product of n factors in Arithmetical Progression", " u_n the reciprocal of the product of n factors in Arithmetical Progression", " Method of Subtraction", " Expression of u_n as sum of factorials", " Polygonal and Figurate Numbers", " Pascal's Triangle", " Examples XXIX. a", " Method of Differences", " Method succeeds when u_n is a rational integral function of n", " If a_n is a rational integral function of n, the series Σ(a_n * x^n) is a recurring series", " Further cases of recurring series", " Examples XXIX. b", " Miscellaneous methods of summation", " Sum of series 1^r + 2^r + 3^r + ... + n^r", " Bernoulli's Numbers", " Examples XXIX. c", "Chapter XXX. Theory of numbers", " Statement of principles", " Number of primes is infinite", " No rational algebraical formula can represent primes only", " A number can be resolved into prime factors in only one way", " Number of divisors of a given integer", " Number of ways an integer can be resolved into two factors", " Sum of the divisors of a given integer", " Highest power of a prime contained in n!", " Product of r consecutive integers is divisible by r!", " Fermat's Theorem N^(p-1) - 1 = M(p) where p is prime and N prime to p", " Examples XXX. a", " Definition of congruent", " If a is prime to b, then a, 2a, 3a, ... (b-1)a when divided by b leave different remainders", " φ(abcd...)=φ(a)φ(b)φ(c)φ(d)...", " φ(N) = N(1 - 1/a)(1 - 1/b)(1 - 1/c)...", " Wilson's Theorem : 1 + (p-1)! = M(p) where p is a prime", " A property peculiar to prime numbers", " Wilson's Theorem (second proof)", " Proofs by induction", " Examples XXX. b", "Chapter XXXI. The general theory of continued fractions", " Law of formation of successive convergents", " b_1/(a_1 + b_2/(a_2 + ...)) has a definite value if Lim(a_n * a_n+1 / b_n+1) > 0", " The convergents to b_1/(a_1 - b_2/(a_2 - ...)) are positive proper fractions in ascending order of magnitude, if a >= 1 + b_n", " General value of convergent when a_n and b_n are constant", " Cases where general value of convergent can be found", " b_1/(a_1 + b_2/(a_2 + ...)) is incommensurable, if b_n/a_n < 1", " Examples XXXI. a", " Series expressed as continued fractions", " Conversion of one continued fraction into another", " Examples XXXI. b", "Chapter XXXII. Probability", " Definitions and illustrations. Simple Events", " Examples XXXII. a", " Compound Events", " Probability that two independent events will both happen is pp'", " The formula holds also for dependent events", " Chance of an event which can happen in mutually exclusive ways", " Examples XXXII. b", " Chance of an event happening exactly r times in n trials", " Expectation and probable value", " "Problem of points"", " Examples XXXII. c", " Inverse probability", " Statement of Bernoulli's Theorem", " Proof of formula Q_r = p_r*P_r/Σ(pP)", " Concurrent testimony", " Traditionary testimony", " Examples XXXII. d", " Local Probability. Geometrical methods", " Miscellaneous examples", " Examples XXXII. e", "Chapter XXXIII. Determinants", " Eliminant of two homogeneous linear equations", " Eliminant of three homogeneous linear equations", " Determinant is not altered by interchanging rows and columns", " Development of determinant of third order", " Sign of a determinant is altered by interchanging two adjacent rows or columns", " If two rows or columns are identical, the determinant vanishes", " A factor common to any row or column may be placed outside", " Cases where constituents are made up of a number of terms", " Reduction of determinants by simplification of rows or columns", " Product of two determinants", " Examples XXXIII. a", " Application to solution of simultaneous equations", " Determinant of fourth order", " Determinant of any order", " Notation Σ +/- a_1*b_2*c_3*d_4", " Examples XXXIII. b", "Chapter XXXIV. Miscellaneous theorems and examples", " Review of the fundamental laws of Algebra", " f(x) when divided by x - a leaves remainder f(a)", " Quotient of f(x) when divided by x - a", " Method of Detached Coefficients", " Horner's Method of Synthetic Division", " Symmetrical and Alternating Functions", " Examples of identities worked out", " List of useful formulae", " Examples XXXIV. a", " Identities proved by properties of cube roots of unity", " Linear factors of a^3 + b^3 + c^3 - 3abc", " Value of a^n + b^n + c^n when a + b + c = 0", " Examples XXXIV. b", " Elimination", " Elimination by symmetrical functions", " Euler's method of elimination", " Sylvester's Dialytic Method", " Bezout's method", " Miscellaneous examples of elimination", " Examples XXXIV. c", "Chapter XXXV. Theory of equations", " Every equation of the n'th degree has n roots and no more", " Relations between the roots and the coefficients", " These relations are not sufficient for the solution", " Cases of solution under given conditions", " Easy cases of symmetrical functions of the roots", " Examples XXXV. a", " Imaginary and surd roots occur in pairs", " Formation and solution of equations with surd roots", " Descartes' Rule of Signs", " Examples XXXV. b", " Value of f(x+h). Derived Functions", " Calculation of f(x+h) by Horner's process", " f(x) changes its value gradually", " If f(a) and f(b) are of contrary signs, f(x) = 0 has a root between a and b", " An equation of an odd degree has one real root", " An equation of an even degree with its last term negative has two real roots", " If f(x) = 0 has r roots equal to a, f'(x) = 0 has r-1 roots equal to a", " Determination of equal roots", " f'(x)/f(x) = 1/(x-a) + 1/(x-b) + 1/(x-c) + ...", " Sum of an assigned power of the roots", " Examples XXXV. c", " Transformation of equations", " Equation with roots of sign opposite to those of f(x) = 0", " Equation with roots multiples of those of f(x) = 0", " Equation with roots reciprocals of those of f(x) = 0", " Discussion of reciprocal equations", " Equation with roots squares of those of f(x) = 0", " Equation with roots exceeding by h those of f(x) = 0", " Removal of an assigned term", " Equation with roots given functions of those of f(x) = 0", " Examples XXXV. d", " Cubic equations. Cardan's Solution", " Discussion of the solution", " Solution by Trigonometry in the irreducible case", " Biquadratic Equations. Ferrari's Solution", " Descartes' Solution", " Undetermined multipliers", " Discriminating cubic ; roots all real", " Solution of three simultaneous equations (x/(a+λ)) + (y/(b+λ)) + (z/(c+λ)) = 1, etc.", " Examples XXXV. e", " Miscellaneous Examples", " Answers".

Gennemgang af en hel del matematik der var førsteårs universitetspensum i Århus i 1979 og en hel del, jeg egentlig ikke har set som pensum nogetsteds fordi det ikke bruges så meget længere, fx teorien for løsning af ligninger og for kædebrøker.

Typografien i denne bog er nogenlunde, men hvis man sammenligner med nutidige bøger, får man lyst til at rejse en statue af Donald Knuth.… (mehr)