Mathematics
DIVISIONS OF THE CALCULUS 53 _The Calculus of Values, or Arithmetic_ 57 Its Extent 57 Its true Nature 59 _The Calculus of Functions_ 61 Two Modes of obtaining Equations 61 1. By the Relations between the given Quantities 61 2. By the Relations between auxiliary Quantities 64 C...
DIVISIONS OF THE CALCULUS 53 _The Calculus of Values, or Arithmetic_ 57 Its Extent 57 Its true Nature 59 _The Calculus of Functions_ 61 Two Modes of obtaining Equations 61 1. By...
2. CHAPTER II.3. CHAPTER III.METHOD OF LEIBNITZ 91 Infinitely small Elements 91 _Examples_: 1. Tangents 93 2. Rectification of an Arc 94 3. Quadrature of a Curve 95 4. Velocity in variable Motion 95 5. Dist...
4. CHAPTER IV.THEIR RELATIONS TO EACH OTHER 121 1. Use of the Differential Calculus as preparatory to that of the Integral 123 2. Employment of the Differential Calculus alone 125 3. Employme...
5. CHAPTER V.METHOD OF LAGRANGE 156 Two Classes of Questions 157 1. Absolute Maxima and Minima 157 Equations of Limits 159 A more general Consideration 159 2. Relative Maxima and Minima 160...
6. CHAPTER VI.7. CHAPTER I.EXPANSION OF ORIGINAL DEFINITION 193 Properties of Lines and Surfaces 195 Necessity of their Study 195 1. To find the most suitable Property 195 2. To pass from the Concrete to...
8. CHAPTER II.ALGEBRAICAL SOLUTIONS 224 _Trigonometry_ 225 Two Methods of introducing Angles 226 1. By Arcs 226 2. By trigonometrical Lines 226 Advantages of the latter 226 Its Division of tr...
9. CHAPTER III.PLANE CURVES 237 Expression of Lines by Equations 237 Expression of Equations by Lines 238 Any change in the Line changes the Equation 240 Every "Definition" of a Line is an Equ...
10. CHAPTER I.In the historical development of mathematical science since the time of Descartes, the advances of its abstract portion have always been determined by those of its concrete port...
11. CHAPTER II.The _Calculus of direct Functions_, or _Algebra_, is (as was shown at the end of the preceding chapter) entirely sufficient for the solution of mathematical questions, when they...
12. chapter i., page 51), or as they include also exponential or circularfunctions. The names of _Algebraic_ functions and _Transcendental_ functions, commonly given to these two principal groups of analytical elements, are undoubtedly very inappropr...
13. CHAPTER III.We determined, in the second chapter, the philosophical character of the transcendental analysis, in whatever manner it may be conceived, considering only the general nature of...
14. CHAPTER IV.The _calculus of indirect functions_, in accordance with the considerations explained in the preceding chapter, is necessarily divided into two parts (or, more properly, is deco...
15. CHAPTER V.In order to grasp with more ease the philosophical character of the _Method of Variations_, it will be well to begin by considering in a summary manner the special nature of the...
16. CHAPTER VI.The different fundamental considerations indicated in the five preceding chapters constitute, in reality, all the essential bases of a complete exposition of mathematical analys...
17. CHAPTER I._Its true Nature._ After the general exposition of the philosophical character of concrete mathematics, compared with that of abstract mathematics, given in the introductory cha...
18. CHAPTER II.The geometrical method of the ancients necessarily constituting a preliminary department in the dogmatical system of geometry, designed to furnish _general_ geometry with indisp...
19. CHAPTER III._General_ (or _Analytical_) geometry being entirely founded upon the transformation of geometrical considerations into equivalent analytical considerations, we must begin with e...