The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

CONTENTS.

Preface vi

Biographical Sketch of Joseph Louis Lagrange. viii

Lecture I. On Arithmetic, and in Particular Frac- tions and Logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Systems of numeration. — Fractions. — Greatest common divisor. — Continued fractions. — Termi- nating continued fractions. — Converging fractions.

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  Convergents. — A second method of expression.
  
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  A third method of expression. — Origin of con- tinued fractions. — Involution and evolution. — Proportions. — Arithmetical and geometrical pro- portions. — Progressions. — Compound interest.
  
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  Present values and annuities. — Logarithms.
  
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  Napier (1550–1617). — Origin of logarithms.
  
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  Briggs (1556–1631). Vlacq. — Computation of logarithms. — Value of the history of science. — Musical temperament.
  

Lecture II. On the Operations of Arithmetic. 20

Arithmetic and geometry. — New method of subtraction. — Subtraction by complements. —

Abridged multiplication. — Inverted multiplica- tion. — Approximate multiplication. — The new method exemplified. — Division of decimals. — Property of the number 9. — Property of the number 9 generalised. — Theory of remainders. — Test of divisibility by 7. — Negative remainders.

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  Test of divisibility by 11. — Theory of re- mainders. — checks on multiplication and division.
  
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  Evolution. — Rule of three. — Applicability of the rule of three. — Theory and practice.
  
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  Alligation. — Mean values. — Probability of life. — Alternate alligation. — Two ingredients.
  
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  Rule of mixtures. — Three ingredients. — General solution. — Development. — Resolution by continued fractions.
  

Lecture III. On Algebra, Particularly the Resolu- tion of Equations of the Third and Fourth De- gree. 46

Algebra among the ancients. — Diophantus. — Equations of the second degree. — Other problems solved by Diophantus. — Translations of Diophan- tus. — Algebra among the Arabs. — Algebra in Europe. — Tartaglia (1500–1559). Cardan (1501– 1576). — The irreducible case. — Biquadratic equa- tions. — Ferrari (1522-1565). Bombelli. — Theory of equations. — Equations of the third degree. — The reduced equation. — Cardan’s rule. — The generality of algebra. — The three cube roots of a quantity. — The roots of equations of the third

degree. — A direct method of reaching the roots.

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  The form of the roots. — The reality of the roots. — The form of the two cubic radicals. — Condition of the reality of the roots. — Extraction of the square roots of two imaginary binomials.
  
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  Extraction of the cube roots of two imagi- nary binomials. — General theory of the reality of the roots. — Imaginary expressions. — Trisec- tion of an angle. — Trigonometrical solution. — The method of indeterminates. — An independent consideration. — New view of the reality of the roots. — Final solution on the new view. — Office of imaginary quantities. — Biquadratic equations.
  
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  The method of Descartes. — The determined character of the roots. — A third method. — The reduced equation. — Euler’s formulæ. — Roots of a biquadratic equation.
  

Lecture IV. On the Resolution of Numerical Equa- tions. 87

Limits of the algebraical resolution of equations. — Conditions of the resolution of numerical equations.

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  Position of the roots of numerical equations. — Position of the roots of numerical equations. — Application of geometry to algebra. — Representa- tion of equations by curves. — Graphic resolution of equations. — The consequences of the graphic resolution. — Intersections indicate the roots. — Case of multiple roots. — General conclusions as to the character of the roots. — Limits of the real
  

roots of equations. — Limits of the positive and negative roots. — Superior and inferior limits of the positive roots. — A further method for finding the limits. — The real problem, the finding of the roots. — Separation of the roots. — To find a quantity less than the difference between any two roots. — The equation of differences. — Imprac- ticability of the method. — Attempt to remedy the method. — Further improvement. — Final resolution. — Recapitulation. — The arithmeti- cal progression revealing the roots. — Method of elimination. — General formulæ for elimination. — General result. — A second construction for solving equations. — The development and solution. — A machine for solving equations.

Lecture V. On the Employment of Curves in the So- lution of Problems. 115

Geometry applied to algebra. — Method of res- olution by curves. — Problem of the two lights.

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  Various solutions. — General solution. — Min- imal values. — Preceding analysis applied to bi- quadratic equations. — Consideration of equations of the fourth degree. — Advantages of the method of curves. — The curve of errors. — Solution of a problem by the curve of errors. — Problem of the circle and inscribed polygon. — Solution of a second problem by the curve of errors. — Problem of the observer and three objects. — Employment of the curve of errors. — Eight possible solutions
  

of the preceding problem. — Reduction of the pos- sible solutions in practice. — General conclusion on the method of curves. — Parabolic curves. — Newton’s problem. — Simplification of Newton’s solution. — Possible uses of Newton’s problem. — Application of Newton’s problem to the preceding examples.

Appendix 136

Note on the Origin of Algebra.

Index 138