The Philosophy of Mathematics Translated from Cours de Philosophie Positive by W. M. Gillespie

by Comte, Auguste

• ISBN13: 9780486442495
• ISBN10: 0486442497
• Format: Hardcover
• Copyright: 2005-01-27
• Publisher: Dover Publications

Summary

Written by the nineteenth-century French philosophical founder of positivism, this comprehensive map of mathematical science assigns to each part of the complex whole its true position and value. The two-part treatment begins with a general view of mathematical analysis and advances to algebra, continuing with an exploration of geometry's ancient and modern methods.

Table of Contents

INTRODUCTION GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE	17	(1)
THE OBJECT OF MATHEMATICS	18	(7)
Measuring Magnitudes	18	(7)
Difficulties	19	(1)
General Method	20	(1)
Illustrations	21	(4)
1. Falling Bodies	21	(2)
2. Inaccessible Distances	23	(1)
3. Astronomical Facts	24	(1)
TRUE DEFINITION OF MATHEMATICS	25	(1)
A Science, not an Art	25	(1)
ITS TWO FUNDAMENTAL DIVISIONS	26	(9)
Their different Objects	27	(2)
Their different Natures	29	(2)
Concrete Mathematics	31	(2)
Geometry and Mechanics	32	(1)
Abstract Mathematics	33	(2)
The Calculus, or Analysis	33	(2)
EXTENT OF ITS FIELD	35	(10)
Its Universality	36	(1)
Its Limitations	37	(8)
BOOK I. ANALYSIS.
CHAPTER I. GENERAL VIEW OF MATHEMATICAL ANALYSIS	45	(24)
THE TRUE IDEA OF AN EQUATION	46	(7)
Division of Functions into Abstract and Concrete	47	(3)
Enumeration of Abstract Functions	50	(3)
DIVISIONS OF THE CALCULUS	53	(16)
The Calculus of Values, or Arithmetic	57	(4)
Its Extent	57	(2)
Its true. Nature	59	(2)
The Calculus of Functions	61	(8)
Two Modes of obtaining Equations	61	(1)
1. By the Relations between the given Quantities	61	(1)
2. By the Relations between auxiliary Quantities	64	(3)
Corresponding Divisions of the Calculus of Functions	67	(2)
CHAPTER II. ORDINARY ANALYSIS; OR, ALGEBRA.	69	(19)
Its Object	69	(1)
Classification of Equations	70	(1)
ALGEBRAIC EQUATIONS	71	(1)
Their Classification	71	(1)
ALGEBRAIC RESOLUTION OF EQUATIONS	72	(3)
Its Limits	72	(1)
General Solution	72	(2)
What we know in Algebra	74	(1)
NUMERICAL RESOLUTION OF EQUATIONS	75	(4)
Its limited Usefulness	76	(2)
Different Divisions of the two Systems	78	(1)
THE THEORY OF EQUATIONS	79	(1)
THE METHOD OF INDETERMINATE COEFFICIENTS	80	(1)
IMAGINARY QUANTITIES	81	(1)
NEGATIVE QUANTITIES	81	(3)
THE PRINCIPLE OF HOMOGENEITY	84	(4)
CHAPTER III. TRANSCENDENTAL ANALYSIS: ITS DIFFERENT CONCEPTIONS	88	(32)
Preliminary Remarks	85	(4)
Its early History	89	(2)
METHOD OF LEIBNITZ	91	(12)
Infinitely small Elements	91	(2)
Examples:
1. Tangents	93	(1)
2. Rectification of an Arc	94	(1)
3. Quadrature of a Curve	95	(1)
4. Velocity in variable Motion	95	(1)
5. Distribution of Heat	96	(1)
Generality of the Formulas	97	(1)
DeMonstration of the Method	98	(4)
Illustration by Tangents	102	(1)
METHOD OF NEWTON	103	(5)
Method of Limits	103	(1)
Examples:
1. Tangents	104	(1)
2. Rectifications	105	(1)
Fluxions and Fluents	106	(2)
METHOD OF LAGRANGE	108	(12)
Derived Functions	108	(1)
An extension of ordinary Analysis	108	(1)
Example: Tangents	109	(1)
Fundamental Identity of the three Methods	110	(3)
Their comparative Value	113	(8)
That of Leibnitz	113	(2)
That of Newton	115	(2)
That of Lagrange	117	(3)
CHAPTER IV. THE DIFFERENTIAL AND INTEGRAL CALCULUS	120	(31)
ITS TWO FUNDAMENTAL DIVISIONS	120	(1)
THEIR RELATIONS TO EACH OTHER	121	(6)
1. Use of the Differential Calculus as preparatory to that of the Integral	123	(2)
2. Employment of the Differential Calculus alone	125	(1)
3. Employment of the Integral Calculus alone	125	(2)
Three Classes of Questions hence resulting	126	(1)
THE DIFFERENTIAL CALCULUS	127	(8)
Two Cases: Explicit and Implicit Functions	127	(4)
Two sub-Cases: a single Variable or several	129	(1)
Two other Cases: Functions separate or combined	130	(1)
Reduction of all to the Differentiation of the ten elementary Functions	131	(1)
Transformation of derived Functions for new Variables	132	(1)
Different Orders of Differentiation	133	(1)
Analytical Applications	133	(2)
THE INTEGRAL CALCULUS	135	(16)
Its fundamental Division: Explicit and Implicit Functions	135	(1)
Subdivisions: a single Variable or several	136	(1)
Calculus of partial Differences	137	(1)
Another Subdivision: different Orders of Differentiation	138	(2)
Another equivalent Distinction	140	(2)
Quadratures	142	(1)
Integration of Transcendental Functions	143	(1)
Integration by Parts	143	(1)
Integration of Algebraic Functions	143	(1)
Singular Solutions	144	(2)
Definite Integrals	146	(2)
Prospects of the Integral Calculus	148	(3)
CHAPTER V. THE CALCULUS OF VARIATIONS	151	(16)
PROBLEMS GIVING RISE TO IT	151	(3)
Ordinary Questions of Maxima and Minima	151	(1)
A new Class of Questions	152	(4)
Solid of least Resistance; Brachystochrone; Isoperimeters	153	(1)
ANALYTICAL NATURE OF THESE QUESTIONS	154	(1)
METHODS OF THE OLDER GEOMETERS	155	(1)
METHOD OF LAGRANGE	156	(7)
Two Classes of Questions	157	(5)
1. Absolute Maxima and Minima	157	(1)
Equations of Limits	159	(1)
A more general Consideration	159	(1)
2. Relative Maxima and Minima	160	(2)
Other Applications of the Method of Variations	162	(1)
ITS RELATIONS TO THE ORDINARY CALCULUS	163	(4)
CHAPTER VI. THE CALCULUS OF FINITE DIFFERENCES	167	(12)
Its general Character	167	(1)
Its true Nature	168	(2)
GENERAL THEORY OF SERIES	170	(3)
Its Identity with this Calculus	172	(1)
PERIODIC OR DISCONTINUOUS FUNCTIONS	173	(1)
APPLICATIONS OF THIS CALCULUS	173	(6)
Series	173	(1)
Interpolation	173	(1)
Approximate Rectification, &c	174	(5)
BOOK II. GEOMETRY.
CHAPTER I. A GENERAL VIEW OF GEOMETRY	179	(33)
The true Nature of Geometry	179	(2)
Two fundamental Ideas	181	(3)
1. The Idea of Space	181	(1)
2. Different kinds of Extension	182	(2)
THE FINAL OBJECT OF GEOMETRY	184	(6)
Nature of Geometrical Measurement	185	(5)
Of Surfaces and Volumes	185	(2)
Of curve Lines	187	(2)
Of right Lines	189	(1)
THE INFINITE EXTENT OF ITS FIELD	190	(3)
Infinity of Lines	190	(1)
Infinity of Surfaces	191	(1)
Infinity of Volumes	192	(1)
Analytical Invention of Curves, &c	193	(1)
EXPANSION OF ORIGINAL DEFINITION	193	(9)
Properties of Lines and Surfaces	195	(1)
Necessity of their Study	195	(3)
1. To find the most suitable Property	195	(2)
2. To pass from the Concrete to the Abstract	197	(1)
Illustrations:
Orbits of the Planets	198	(1)
Figure of the Earth	199	(3)
THE TWO GENERAL METHODS OF GEOMETRY	202	(10)
Their fundamental Difference	203	(1)
1°. Different Questions with respect to the same Figure	204	(1)
2°. Similar Questions with respect to different Figures	204	(1)
Geometry of the Ancients	204	(1)
Geometry of the Moderns	205	(2)
Superiority of the Modern	207	(2)
The Ancient the base of the Modern	209	(3)
CHAPTER II. ANCIENT OR SYNTHETIC GEOMETRY
ITS PROPER EXTENT	212	(5)
Lines; Polygons; Polyhedrons	212	(1)
Not to be farther restricted	213	(1)
Improper Application of Analysis	214	(2)
Attempted Demonstrations of Axioms	216	(1)
GEOMETRY OF THE RIGHT LINE	217	(1)
GRAPHICAL SOLUTIONS	218	(6)
Descriptive Geometry	220	(4)
ALGEBRAICAL SOLUTIONS	224	(8)
Trigonometry	225	(7)
Two Methods of introducing Angles	226	(1)
1. By Arcs	226	(1)
2. By trigonometrical Lines	226	(1)
Advantages of the latter	226	(1)
Its Division of trigonometrical Questions	227	(1)
1. Relations between Angles and trigonometrical Lines	228	(1)
2. Relations between trigonometrical Lines and Sides	228	(1)
Increase of trigonometrical Lines	228	(2)
Study of the Relations between them	230	(2)
CHAPTER III. MODERN OR ANALYTICAL GEOMETRY
THE ANALYTICAL REPRESENTATION OF FIGURES	232	(5)
Reduction of Figure to Position	233	(1)
Determination of the position of a Point	234	(3)
PLANE CURVES	237	(14)
Expression of Lines by Equations	237	(1)
Expression of Equations by Lines	238	(2)
Any change in the Line changes the Equation	240	(1)
Every "Definition" of a Line is an Equation	241	(4)
Choice of Co-ordinates	245	(6)
Two different points of View	245	(1)
1. Representation of Lines by Equations	246	(1)
2. Representation of Equations by Lines	246	(2)
Superiority of the rectilinear System	248	(1)
Advantages of perpendicular Axes	249	(2)
SURFACES	251	(4)
Determination of a Point in Space	251	(2)
Expression of Surfaces by Equations	253	(1)
Expression of Equations by Surfaces	253	(2)
CURVES IN SPACE	255
Imperfections of Analytical Geometry	258	(1)
Relatively to Geometry	258	(1)
Relatively to Analysis	258