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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org/license + + +Title: The philosophy of mathematics + +Author: Auguste Comte + +Translator: W. M. Gillespie + +Release Date: May 15, 2012 [EBook #39702] + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF MATHEMATICS *** + + + + +Produced by Anna Hall, Albert László and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) + + + + + + +</pre> + +<p><span class="pagenum"><a name="Page_1" id="Page_1">[Pg 1]</a></span></p> + + +<h1><span class="small">THE</span><br /> +PHILOSOPHY<br /> +<span class="small">OF</span><br /> +MATHEMATICS.</h1> + +<p><span class="pagenum"><a name="Page_2" id="Page_2">[Pg 2]</a></span></p> + + +<p><span class="pagenum"><a name="Page_3" id="Page_3">[Pg 3]</a></span></p> + +<div class="figcenter" style="width: 600px;"> +<a href="images/tree_b.jpg" class="fnanchor"> +<img src="images/tree_s.jpg" width="600" height="282" alt="The science of mathematics" /> +</a> +</div> + + +<p class="center">THE +PHILOSOPHY +OF +MATHEMATICS;</p> + +<p class="center">TRANSLATED FROM THE<br /> +COURS DE PHILOSOPHIE POSITIVE<br /> +<span class="small">OF</span><br /> +<span class="big">AUGUSTE COMTE,</span></p> +<p class="center"><span class="small">BY</span><br /> +W. M. GILLESPIE,<br /> +<span class="small">PROFESSOR OF CIVIL ENGINEERING & ADJ. PROF. OF MATHEMATICS<br /> +IN UNION COLLEGE.</span></p> + +<p class="center">NEW YORK:<br /> +HARPER & BROTHERS, PUBLISHERS,<br /> +82 CLIFF STREET<br /> +1851.</p> +<p><span class="pagenum"><a name="Page_4" id="Page_4">[Pg 4]</a></span></p> + + +<p class="center"> +Entered, according to Act of Congress, in the year one thousand<br /> +eight hundred and fifty-one, by<br /> +<br /> +<span class="smcap">Harper & Brothers</span>.<br /> +<br /> +in the Clerk's Office of the District Court of the Southern District<br /> +of New York.</p> +<p><span class="pagenum"><a name="Page_5" id="Page_5">[Pg 5]</a></span></p> + + + + +<h2><a name="PREFACE" id="PREFACE">PREFACE.</a></h2> + + +<p>The pleasure and profit which the translator +has received from the great work here presented, +have induced him to lay it before his fellow-teachers +and students of Mathematics in a more accessible +form than that in which it has hitherto appeared. +The want of a comprehensive map of the +wide region of mathematical science—a bird's-eye +view of its leading features, and of the true bearings +and relations of all its parts—is felt by every +thoughtful student. He is like the visitor to a +great city, who gets no just idea of its extent and +situation till he has seen it from some commanding +eminence. To have a panoramic view of the +whole district—presenting at one glance all the +parts in due co-ordination, and the darkest nooks +clearly shown—is invaluable to either traveller or +student. It is this which has been most perfectly +accomplished for mathematical science by the +author whose work is here presented.</p> + +<p>Clearness and depth, comprehensiveness and +precision, have never, perhaps, been so remarkably +united as in <span class="smcap">Auguste Comte</span>. He views his subject +from an elevation which gives to each part of +the complex whole its true position and value, +while his telescopic glance loses none of the needful +details, and not only itself pierces to the heart<span class="pagenum"><a name="Page_6" id="Page_6">[Pg 6]</a></span> +of the matter, but converts its opaqueness into +such transparent crystal, that other eyes are enabled +to see as deeply into it as his own.</p> + +<p>Any mathematician who peruses this volume +will need no other justification of the high opinion +here expressed; but others may appreciate the +following endorsements of well-known authorities. +<i>Mill</i>, in his "Logic," calls the work of M. Comte +"by far the greatest yet produced on the Philosophy +of the sciences;" and adds, "of this admirable +work, one of the most admirable portions is that +in which he may truly be said to have created the +Philosophy of the higher Mathematics:" <i>Morell</i>, +in his "Speculative Philosophy of Europe," says, +"The classification given of the sciences at large, +and their regular order of development, is unquestionably +a master-piece of scientific thinking, as +simple as it is comprehensive;" and <i>Lewes</i>, in +his "Biographical History of Philosophy," names +Comte "the Bacon of the nineteenth century," +and says, "I unhesitatingly record my conviction +that this is the greatest work of our age."</p> + +<p>The complete work of M. Comte—his "<i>Cours +de Philosophie Positive</i>"—fills six large octavo volumes, +of six or seven hundred pages each, two +thirds of the first volume comprising the purely +mathematical portion. The great bulk of the +"Course" is the probable cause of the fewness of +those to whom even this section of it is known. +Its presentation in its present form is therefore felt +by the translator to be a most useful contribution +to mathematical progress in this country.<span class="pagenum"><a name="Page_7" id="Page_7">[Pg 7]</a></span> +The comprehensiveness of the style of the author—grasping +all possible forms of an idea in one +Briarean sentence, armed at all points against +leaving any opening for mistake or forgetfulness—occasionally +verges upon cumbersomeness and +formality. The translator has, therefore, sometimes +taken the liberty of breaking up or condensing +a long sentence, and omitting a few passages +not absolutely necessary, or referring to the peculiar +"Positive philosophy" of the author; but he +has generally aimed at a conscientious fidelity to +the original. It has often been difficult to retain +its fine shades and subtile distinctions of meaning, +and, at the same time, replace the peculiarly +appropriate French idioms by corresponding English +ones. The attempt, however, has always +been made, though, when the best course has been +at all doubtful, the language of the original has +been followed as closely as possible, and, when +necessary, smoothness and grace have been unhesitatingly +sacrificed to the higher attributes of +clearness and precision.</p> + +<p>Some forms of expression may strike the reader +as unusual, but they have been retained because +they were characteristic, not of the mere language +of the original, but of its spirit. When a great +thinker has clothed his conceptions in phrases +which are singular even in his own tongue, he who +professes to translate him is bound faithfully to +preserve such forms of speech, as far as is practicable; +and this has been here done with respect +to such peculiarities of expression as belong to the<span class="pagenum"><a name="Page_8" id="Page_8">[Pg 8]</a></span> +author, not as a foreigner, but as an individual—not +because he writes in French, but because he +is Auguste Comte.</p> + +<p>The young student of Mathematics should not +attempt to read the whole of this volume at once, +but should peruse each portion of it in connexion +with the temporary subject of his special study: +the first chapter of the first book, for example, +while he is studying Algebra; the first chapter of +the second book, when he has made some progress +in Geometry; and so with the rest. Passages +which are obscure at the first reading will brighten +up at the second; and as his own studies cover +a larger portion of the field of Mathematics, he +will see more and more clearly their relations to +one another, and to those which he is next to take +up. For this end he is urgently recommended to +obtain a perfect familiarity with the "Analytical +Table of Contents," which maps out the whole +subject, the grand divisions of which are also indicated +in the Tabular View facing the title-page. +Corresponding heads will be found in the body of +the work, the principal divisions being in <span class="smcap">small +capitals</span>, and the subdivisions in <i>Italics</i>. For +these details the translator alone is responsible.</p><p><span class="pagenum"><a name="Page_9" id="Page_9">[Pg 9]</a></span></p> + + + + +<h3>ANALYTICAL TABLE OF CONTENTS.</h3> + +<h5>INTRODUCTION.</h5> + +<ul class="toc"> +<li> <span class="label">Page</span></li> + +<li>GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE <span class="label"><a href="#Page_17">17</a></span></li> +<li><span class="smcap">The Object of Mathematics</span> <span class="label"><a href="#Page_18">18</a></span></li> +<li><ol class="subtoc"> +<li>Measuring Magnitudes <span class="label"><a href="#Page_18">18</a></span></li> +<li><ol class="subtoc"> +<li>Difficulties <span class="label"><a href="#Page_19">19</a></span></li> +<li>General Method <span class="label"><a href="#Page_20">20</a></span></li> +<li>Illustrations <span class="label"><a href="#Page_21">21</a></span></li> +<li><ol class="subtoc"> +<li>1. Falling Bodies <span class="label"><a href="#Page_21">21</a></span></li> +<li>2. Inaccessible Distances <span class="label"><a href="#Page_23">23</a></span></li> +<li>3. Astronomical Facts <span class="label"><a href="#Page_24">24</a></span></li> +</ol></li></ol></li></ol></li> +<li><span class="smcap">True Definition of Mathematics</span> <span class="label"><a href="#Page_25">25</a></span></li> +<li><ol class="subtoc"> +<li>A Science, not an Art <span class="label"><a href="#Page_25">25</a></span></li> +</ol></li> +<li><span class="smcap">Its Two Fundamental Divisions</span> <span class="label"><a href="#Page_26">26</a></span></li> +<li><ol class="subtoc"> +<li>Their different Objects <span class="label"><a href="#Page_27">27</a></span></li> +<li>Their different Natures <span class="label"><a href="#Page_29">29</a></span></li> +<li><i>Concrete Mathematics</i> <span class="label"><a href="#Page_31">31</a></span></li> +<li>Geometry and Mechanics <span class="label"><a href="#Page_32">32</a></span></li> +<li><i>Abstract Mathematics</i> <span class="label"><a href="#Page_33">33</a></span></li> +<li>The Calculus, or Analysis <span class="label"><a href="#Page_33">33</a></span></li> +</ol></li> +<li><span class="smcap">Extent of Its Field</span> <span class="label"><a href="#Page_35">35</a></span></li> +<li><ol class="subtoc"> +<li>Its Universality <span class="label"><a href="#Page_36">36</a></span></li> +<li>Its Limitations <span class="label"><a href="#Page_37">37</a></span></li> +</ol></li> +</ul> +<p><span class="pagenum"><a name="Page_10" id="Page_10">[Pg 10]</a></span></p> + + + +<h4>BOOK I.<br /> +ANALYSIS.</h4> + +<h5>CHAPTER I.</h5> +<ul class="toc"> +<li> <span class="label">Page</span></li> + +<li>GENERAL VIEW OF MATHEMATICAL ANALYSIS <span class="label"><a href="#Page_45">45</a></span></li> +<li><span class="smcap">The True Idea of an Equation</span> <span class="label"><a href="#Page_46">46</a></span></li> +<li><ol class="subtoc"> +<li>Division of Functions into Abstract and Concrete <span class="label"><a href="#Page_47">47</a></span></li> +<li>Enumeration of Abstract Functions <span class="label"><a href="#Page_50">50</a></span></li> +</ol></li> +<li><span class="smcap">Divisions of the Calculus</span> <span class="label"><a href="#Page_53">53</a></span></li> +<li><ol class="subtoc"> +<li><i>The Calculus of Values, or Arithmetic</i> <span class="label"><a href="#Page_57">57</a></span></li> +<li>Its Extent <span class="label"><a href="#Page_57">57</a></span></li> +<li>Its true Nature <span class="label"><a href="#Page_59">59</a></span></li> +<li><i>The Calculus of Functions</i> <span class="label"><a href="#Page_61">61</a></span></li> +<li>Two Modes of obtaining Equations <span class="label"><a href="#Page_61">61</a></span></li> +<li><ol class="subtoc"> +<li>1. By the Relations between the given Quantities <span class="label"><a href="#Page_61">61</a></span></li> +<li>2. By the Relations between auxiliary Quantities <span class="label"><a href="#Page_64">64</a></span></li> +</ol></li> +<li>Corresponding Divisions of the Calculus of Functions <span class="label"><a href="#Page_67">67</a></span></li> +</ol></li> +</ul> +<h5>CHAPTER II.</h5> + +<ul class="toc"> +<li>ORDINARY ANALYSIS; OR, ALGEBRA. <span class="label"><a href="#Page_69">69</a></span></li> +<li><ol class="subtoc"> +<li>Its Object <span class="label"><a href="#Page_69">69</a></span></li> +<li>Classification of Equations <span class="label"><a href="#Page_70">70</a></span></li> +</ol></li> +<li><span class="smcap">Algebraic Equations</span> <span class="label"><a href="#Page_71">71</a></span></li> +<li><ol class="subtoc"> +<li>Their Classification <span class="label"><a href="#Page_71">71</a></span></li> +</ol></li> +<li><span class="smcap">Algebraic Resolution of Equations</span> <span class="label"><a href="#Page_72">72</a></span></li> +<li><ol class="subtoc"> +<li>Its Limits <span class="label"><a href="#Page_72">72</a></span></li> +<li>General Solution <span class="label"><a href="#Page_72">72</a></span></li> +<li>What we know in Algebra <span class="label"><a href="#Page_74">74</a></span></li> +</ol></li> +<li><span class="smcap">Numerical Resolution of Equations</span> <span class="label"><a href="#Page_75">75</a></span></li> +<li><ol class="subtoc"> +<li>Its limited Usefulness <span class="label"><a href="#Page_76">76</a></span></li> +</ol></li> +<li>Different Divisions of the two Systems <span class="label"><a href="#Page_78">78</a></span></li> +<li><span class="smcap">The Theory of Equations</span> <span class="label"><a href="#Page_79">79</a></span></li> +<li><span class="smcap">The Method of Indeterminate Coefficients</span> <span class="label"><a href="#Page_80">80</a></span></li> +<li><span class="smcap">Imaginary Quantities</span> <span class="label"><a href="#Page_81">81</a></span></li> +<li><span class="smcap">Negative Quantities</span> <span class="label"><a href="#Page_81">81</a></span></li> +<li><span class="smcap">The Principle of Homogeneity</span> <span class="label"><a href="#Page_84">84</a></span></li> +</ul> +<p><span class="pagenum"><a name="Page_11" id="Page_11">[Pg 11]</a></span></p> + +<h5>CHAPTER III.</h5> +<ul class="toc"> +<li>TRANSCENDENTAL ANALYSIS: <span class="smcap">its different conceptions</span> <span class="label"><a href="#Page_88">88</a></span></li> +<li><ol class="subtoc"> +<li>Preliminary Remarks <span class="label"><a href="#Page_88">88</a></span></li> +<li>Its early History <span class="label"><a href="#Page_89">89</a></span></li> +</ol></li> +<li><span class="smcap">Method of Leibnitz</span> <span class="label"><a href="#Page_91">91</a></span></li> +<li><ol class="subtoc"> +<li>Infinitely small Elements <span class="label"><a href="#Page_91">91</a></span></li> +<li><i>Examples</i>:</li> +<li><ol class="subtoc"> +<li>1. Tangents <span class="label"><a href="#Page_93">93</a></span></li> +<li>2. Rectification of an Arc <span class="label"><a href="#Page_94">94</a></span></li> +<li>3. Quadrature of a Curve <span class="label"><a href="#Page_95">95</a></span></li> +<li>4. Velocity in variable Motion <span class="label"><a href="#Page_95">95</a></span></li> +<li>5. Distribution of Heat <span class="label"><a href="#Page_96">96</a></span></li> +</ol></li> +<li>Generality of the Formulas <span class="label"><a href="#Page_97">97</a></span></li> +<li>Demonstration of the Method <span class="label"><a href="#Page_98">98</a></span></li> +<li><ol class="subtoc"> +<li>Illustration by Tangents <span class="label"><a href="#Page_102">102</a></span></li> +</ol></li> +</ol></li> +<li><span class="smcap">Method of Newton</span> <span class="label"><a href="#Page_103">103</a></span></li> +<li><ol class="subtoc"> +<li>Method of Limits <span class="label"><a href="#Page_103">103</a></span></li> +<li><i>Examples</i>:</li> +<li><ol class="subtoc"> +<li>1. Tangents <span class="label"><a href="#Page_104">104</a></span></li> +<li>2. Rectifications <span class="label"><a href="#Page_105">105</a></span></li> +</ol></li> +<li>Fluxions and Fluents <span class="label"><a href="#Page_106">106</a></span></li> +</ol></li> +<li><span class="smcap">Method of Lagrange</span> <span class="label"><a href="#Page_108">108</a></span></li> +<li><ol class="subtoc"> +<li>Derived Functions <span class="label"><a href="#Page_108">108</a></span></li> +<li>An extension of ordinary Analysis <span class="label"><a href="#Page_108">108</a></span></li> +<li><i>Example</i>: Tangents <span class="label"><a href="#Page_109">109</a></span></li> +<li><i>Fundamental Identity of the three Methods</i> <span class="label"><a href="#Page_110">110</a></span></li> +<li><i>Their comparative Value</i> <span class="label"><a href="#Page_113">113</a></span></li> +<li>That of Leibnitz <span class="label"><a href="#Page_113">113</a></span></li> +<li>That of Newton <span class="label"><a href="#Page_115">115</a></span></li> +<li>That of Lagrange <span class="label"><a href="#Page_117">117</a></span></li> +</ol></li> +</ul> + +<p><span class="pagenum"><a name="Page_12" id="Page_12">[Pg 12]</a></span></p> + +<h5>CHAPTER IV.</h5> + +<ul class="toc"> +<li>THE DIFFERENTIAL AND INTEGRAL CALCULUS <span class="label"><a href="#Page_120">120</a></span></li> +<li><span class="smcap">Its two fundamental Divisions</span> <span class="label"><a href="#Page_120">120</a></span></li> +<li><span class="smcap">Their Relations to each Other</span> <span class="label"><a href="#Page_121">121</a></span></li> +<li><ol class="subtoc"> +<li>1. Use of the Differential Calculus as preparatory to that of the Integral <span class="label"><a href="#Page_123">123</a></span></li> +<li>2. Employment of the Differential Calculus alone <span class="label"><a href="#Page_125">125</a></span></li> +<li>3. Employment of the Integral Calculus alone <span class="label"><a href="#Page_125">125</a></span></li> +<li><ol class="subtoc"> +<li>Three Classes of Questions hence resulting <span class="label"><a href="#Page_126">126</a></span></li> +</ol></li> +</ol></li> +<li><span class="smcap">The Differential Calculus</span> <span class="label"><a href="#Page_127">127</a></span></li> +<li><ol class="subtoc"> +<li>Two Cases: Explicit and Implicit Functions <span class="label"><a href="#Page_127">127</a></span></li> +<li><ol class="subtoc"> +<li>Two sub-Cases: a single Variable or several <span class="label"><a href="#Page_129">129</a></span></li> +<li>Two other Cases: Functions separate or combined <span class="label"><a href="#Page_130">130</a></span></li> +</ol></li> +<li>Reduction of all to the Differentiation of the ten elementary Functions <span class="label"><a href="#Page_131">131</a></span></li> +<li>Transformation of derived Functions for new Variables <span class="label"><a href="#Page_132">132</a></span></li> +<li>Different Orders of Differentiation <span class="label"><a href="#Page_133">133</a></span></li> +<li>Analytical Applications <span class="label"><a href="#Page_133">133</a></span></li> +</ol></li> +<li><span class="smcap">The Integral Calculus</span> <span class="label"><a href="#Page_135">135</a></span></li> +<li><ol class="subtoc"> +<li>Its fundamental Division: Explicit and Implicit Functions <span class="label"><a href="#Page_135">135</a></span></li> +<li>Subdivisions: a single Variable or several <span class="label"><a href="#Page_136">136</a></span></li> +<li>Calculus of partial Differences <span class="label"><a href="#Page_137">137</a></span></li> +<li>Another Subdivision: different Orders of Differentiation <span class="label"><a href="#Page_138">138</a></span></li> +<li>Another equivalent Distinction <span class="label"><a href="#Page_140">140</a></span></li> +<li><i>Quadratures</i> <span class="label"><a href="#Page_142">142</a></span></li> +<li><ol class="subtoc"> +<li>Integration of Transcendental Functions <span class="label"><a href="#Page_143">143</a></span></li> +<li>Integration by Parts <span class="label"><a href="#Page_143">143</a></span></li> +<li>Integration of Algebraic Functions <span class="label"><a href="#Page_143">143</a></span></li> +</ol></li> +<li>Singular Solutions <span class="label"><a href="#Page_144">144</a></span></li> +<li>Definite Integrals <span class="label"><a href="#Page_146">146</a></span></li> +<li>Prospects of the Integral Calculus <span class="label"><a href="#Page_148">148</a></span></li> +</ol></li> +</ul> + +<p><span class="pagenum"><a name="Page_13" id="Page_13">[Pg 13]</a></span></p> + +<h5>CHAPTER V.</h5> + +<ul class="toc"> +<li>THE CALCULUS OF VARIATIONS <span class="label"><a href="#Page_151">151</a></span></li> +<li><span class="smcap">Problems giving rise to it</span> <span class="label"><a href="#Page_151">151</a></span></li> +<li><ol class="subtoc"> +<li>Ordinary Questions of Maxima and Minima <span class="label"><a href="#Page_151">151</a></span></li> +<li>A new Class of Questions <span class="label"><a href="#Page_152">152</a></span></li> +<li><ol class="subtoc"> +<li>Solid of least Resistance; Brachystochrone; Isoperimeters <span class="label"><a href="#Page_153">153</a></span></li> +</ol></li> +</ol></li> +<li><span class="smcap">Analytical Nature of these Questions</span> <span class="label"><a href="#Page_154">154</a></span></li> +<li><span class="smcap">Methods of the older Geometers</span> <span class="label"><a href="#Page_155">155</a></span></li> +<li><span class="smcap">Method of Lagrange</span> <span class="label"><a href="#Page_156">156</a></span></li> +<li><ol class="subtoc"> +<li>Two Classes of Questions <span class="label"><a href="#Page_157">157</a></span></li> +<li><ol class="subtoc"> +<li>1. Absolute Maxima and Minima <span class="label"><a href="#Page_157">157</a></span></li> +<li>Equations of Limits <span class="label"><a href="#Page_159">159</a></span></li> +<li><ol class="subtoc"> +<li>A more general Consideration <span class="label"><a href="#Page_159">159</a></span></li> +</ol></li> +<li>2. Relative Maxima and Minima <span class="label"><a href="#Page_160">160</a></span></li> +</ol></li> +<li>Other Applications of the Method of Variations <span class="label"><a href="#Page_162">162</a></span></li> +</ol></li> +<li><span class="smcap">Its Relations to the ordinary Calculus</span> <span class="label"><a href="#Page_163">163</a></span></li> +</ul> + +<h5>CHAPTER VI.</h5> + +<ul class="toc"> +<li>THE CALCULUS OF FINITE DIFFERENCES <span class="label"><a href="#Page_167">167</a></span></li> +<li><ol class="subtoc"> +<li>Its general Character <span class="label"><a href="#Page_167">167</a></span></li> +<li>Its true Nature <span class="label"><a href="#Page_168">168</a></span></li> +</ol></li> +<li><span class="smcap">General Theory of Series</span> <span class="label"><a href="#Page_170">170</a></span></li> +<li><ol class="subtoc"> +<li>Its Identity with this Calculus <span class="label"><a href="#Page_172">172</a></span></li> +</ol></li> +<li><span class="smcap">Periodic or discontinuous Functions</span> <span class="label"><a href="#Page_173">173</a></span></li> +<li><span class="smcap">Applications of this Calculus</span> <span class="label"><a href="#Page_173">173</a></span></li> +<li><ol class="subtoc"> +<li>Series <span class="label"><a href="#Page_173">173</a></span></li> +<li>Interpolation <span class="label"><a href="#Page_173">173</a></span></li> +<li>Approximate Rectification, &c. <span class="label"><a href="#Page_174">174</a></span></li> +</ol></li> +</ul> + +<p><span class="pagenum"><a name="Page_14" id="Page_14">[Pg 14]</a></span></p> + + +<h4>BOOK II.<br /> +GEOMETRY.</h4> + +<h5>CHAPTER I.</h5> + +<ul class="toc"> +<li>A GENERAL VIEW OF GEOMETRY <span class="label"><a href="#Page_179">179</a></span></li> +<li><ol class="subtoc"> +<li>The true Nature of Geometry <span class="label"><a href="#Page_179">179</a></span></li> +<li>Two fundamental Ideas <span class="label"><a href="#Page_181">181</a></span></li> +<li><ol class="subtoc"> +<li>1. The Idea of Space <span class="label"><a href="#Page_181">181</a></span></li> +<li>2. Different kinds of Extension <span class="label"><a href="#Page_182">182</a></span></li> +</ol></li> +</ol></li> +<li><span class="smcap">The final object of Geometry</span> <span class="label"><a href="#Page_184">184</a></span></li> +<li><ol class="subtoc"> +<li>Nature of Geometrical Measurement <span class="label"><a href="#Page_185">185</a></span></li> +<li><ol class="subtoc"> +<li>Of Surfaces and Volumes <span class="label"><a href="#Page_185">185</a></span></li> +<li>Of curve Lines <span class="label"><a href="#Page_187">187</a></span></li> +<li>Of right Lines <span class="label"><a href="#Page_189">189</a></span></li> +</ol></li> +</ol></li> +<li><span class="smcap">The infinite extent of its Field</span> <span class="label"><a href="#Page_190">190</a></span></li> +<li><ol class="subtoc"> +<li>Infinity of Lines <span class="label"><a href="#Page_190">190</a></span></li> +<li>Infinity of Surfaces <span class="label"><a href="#Page_191">191</a></span></li> +<li>Infinity of Volumes <span class="label"><a href="#Page_192">192</a></span></li> +<li>Analytical Invention of Curves, &c. <span class="label"><a href="#Page_193">193</a></span></li> +</ol></li> +<li><span class="smcap">Expansion of Original Definition</span> <span class="label"><a href="#Page_193">193</a></span></li> +<li><ol class="subtoc"> +<li>Properties of Lines and Surfaces <span class="label"><a href="#Page_195">195</a></span></li> +<li>Necessity of their Study <span class="label"><a href="#Page_195">195</a></span></li> +<li><ol class="subtoc"> +<li>1. To find the most suitable Property <span class="label"><a href="#Page_195">195</a></span></li> +<li>2. To pass from the Concrete to the Abstract <span class="label"><a href="#Page_197">197</a></span></li> +</ol></li> +<li>Illustrations:</li> +<li><ol class="subtoc"> +<li>Orbits of the Planets <span class="label"><a href="#Page_198">198</a></span></li> +<li>Figure of the Earth <span class="label"><a href="#Page_199">199</a></span></li> +</ol></li> +</ol></li> +<li><span class="smcap">The two general Methods of Geometry</span> <span class="label"><a href="#Page_202">202</a></span></li> +<li><ol class="subtoc"> +<li>Their fundamental Difference <span class="label"><a href="#Page_203">203</a></span></li> +<li><ol class="subtoc"> +<li>1⁰. Different Questions with respect to the same Figure <span class="label"><a href="#Page_204">204</a></span></li> +<li>2⁰. Similar Questions with respect to different Figures <span class="label"><a href="#Page_204">204</a></span></li> +</ol></li> +<li>Geometry of the Ancients <span class="label"><a href="#Page_204">204</a></span></li> +<li>Geometry of the Moderns <span class="label"><a href="#Page_206">206</a></span></li> +<li>Superiority of the Modern <span class="label"><a href="#Page_207">207</a></span></li> +<li>The Ancient the base of the Modern <span class="label"><a href="#Page_209">209</a></span></li> +</ol></li> +</ul> + +<p><span class="pagenum"><a name="Page_15" id="Page_15">[Pg 15]</a></span></p> + +<h5>CHAPTER II.</h5> + +<ul class="toc"> +<li>ANCIENT OR SYNTHETIC GEOMETRY <span class="label"><a href="#Page_212">212</a></span></li> +<li><span class="smcap">Its proper Extent</span> <span class="label"><a href="#Page_212">212</a></span></li> +<li><ol class="subtoc"> +<li>Lines; Polygons; Polyhedrons <span class="label"><a href="#Page_212">212</a></span></li> +<li>Not to be farther restricted <span class="label"><a href="#Page_213">213</a></span></li> +<li>Improper Application of Analysis <span class="label"><a href="#Page_214">214</a></span></li> +<li>Attempted Demonstrations of Axioms <span class="label"><a href="#Page_216">216</a></span></li> +</ol></li> +<li><span class="smcap">Geometry of the right Line</span> <span class="label"><a href="#Page_217">217</a></span></li> +<li><span class="smcap">Graphical Solutions</span> <span class="label"><a href="#Page_218">218</a></span></li> +<li><ol class="subtoc"> +<li><i>Descriptive Geometry</i> <span class="label"><a href="#Page_220">220</a></span></li> +</ol></li> +<li><span class="smcap">Algebraical Solutions</span> <span class="label"><a href="#Page_224">224</a></span></li> +<li><ol class="subtoc"> +<li><i>Trigonometry</i> <span class="label"><a href="#Page_225">225</a></span></li> +<li>Two Methods of introducing Angles <span class="label"><a href="#Page_226">226</a></span></li> +<li><ol class="subtoc"> +<li>1. By Arcs <span class="label"><a href="#Page_226">226</a></span></li> +<li>2. By trigonometrical Lines <span class="label"><a href="#Page_226">226</a></span></li> +</ol></li> +<li>Advantages of the latter <span class="label"><a href="#Page_226">226</a></span></li> +<li>Its Division of trigonometrical Questions <span class="label"><a href="#Page_227">227</a></span></li> +<li><ol class="subtoc"> +<li>1. Relations between Angles and trigonometrical Lines <span class="label"><a href="#Page_228">228</a></span></li> +<li>2. Relations between trigonometrical Lines and Sides <span class="label"><a href="#Page_228">228</a></span></li> +</ol></li> +<li>Increase of trigonometrical Lines <span class="label"><a href="#Page_228">228</a></span></li> +<li>Study of the Relations between them <span class="label"><a href="#Page_230">230</a></span></li> +</ol></li> +</ul> + +<p><span class="pagenum"><a name="Page_16" id="Page_16">[Pg 16]</a></span></p> + +<h5>CHAPTER III.</h5> + +<ul class="toc"> +<li>MODERN OR ANALYTICAL GEOMETRY <span class="label"><a href="#Page_232">232</a></span></li> +<li><span class="smcap">The analytical Representation of Figures</span> <span class="label"><a href="#Page_232">232</a></span></li> +<li><ol class="subtoc"> +<li>Reduction of Figure to Position <span class="label"><a href="#Page_233">233</a></span></li> +<li>Determination of the position of a Point <span class="label"><a href="#Page_234">234</a></span></li> +</ol></li> +<li><span class="smcap">Plane Curves</span> <span class="label"><a href="#Page_237">237</a></span></li> +<li><ol class="subtoc"> +<li>Expression of Lines by Equations <span class="label"><a href="#Page_237">237</a></span></li> +<li>Expression of Equations by Lines <span class="label"><a href="#Page_238">238</a></span></li> +<li>Any change in the Line changes the Equation <span class="label"><a href="#Page_240">240</a></span></li> +<li>Every "Definition" of a Line is an Equation <span class="label"><a href="#Page_241">241</a></span></li> +<li><i>Choice of Co-ordinates</i> <span class="label"><a href="#Page_245">245</a></span></li> +<li>Two different points of View <span class="label"><a href="#Page_245">245</a></span></li> +<li><ol class="subtoc"> +<li>1. Representation of Lines by Equations <span class="label"><a href="#Page_246">246</a></span></li> +<li>2. Representation of Equations by Lines <span class="label"><a href="#Page_246">246</a></span></li> +</ol></li> +<li>Superiority of the rectilinear System <span class="label"><a href="#Page_248">248</a></span></li> +<li><ol class="subtoc"> +<li>Advantages of perpendicular Axes <span class="label"><a href="#Page_249">249</a></span></li> +</ol></li> +</ol></li> +<li><span class="smcap">Surfaces</span> <span class="label"><a href="#Page_251">251</a></span></li> +<li><ol class="subtoc"> +<li>Determination of a Point in Space <span class="label"><a href="#Page_251">251</a></span></li> +<li>Expression of Surfaces by Equations <span class="label"><a href="#Page_253">253</a></span></li> +<li>Expression of Equations by Surfaces <span class="label"><a href="#Page_253">253</a></span></li> +</ol></li> +<li><span class="smcap">Curves in Space</span> <span class="label"><a href="#Page_255">255</a></span></li> +<li>Imperfections of Analytical Geometry <span class="label"><a href="#Page_258">258</a></span></li> +<li><ol class="subtoc"> +<li>Relatively to Geometry <span class="label"><a href="#Page_258">258</a></span></li> +<li>Relatively to Analysis <span class="label"><a href="#Page_258">258</a></span></li> +</ol></li> +</ul> + +<p><span class="pagenum"><a name="Page_17" id="Page_17">[Pg 17]</a></span></p> + + + + +<p class="center">THE<br /> +<span class="big">PHILOSOPHY OF MATHEMATICS.</span></p> + +<h2>INTRODUCTION.</h2> + +<p class="center">GENERAL CONSIDERATIONS.</p> + + +<p>Although Mathematical Science is the most ancient +and the most perfect of all, yet the general idea which +we ought to form of it has not yet been clearly determined. +Its definition and its principal divisions have +remained till now vague and uncertain. Indeed the +plural name—"The Mathematics"—by which we commonly +designate it, would alone suffice to indicate the +want of unity in the common conception of it.</p> + +<p>In truth, it was not till the commencement of the last +century that the different fundamental conceptions which +constitute this great science were each of them sufficiently +developed to permit the true spirit of the whole +to manifest itself with clearness. Since that epoch the +attention of geometers has been too exclusively absorbed +by the special perfecting of the different branches, and +by the application which they have made of them to the +most important laws of the universe, to allow them to +give due attention to the general system of the science.</p> + +<p>But at the present time the progress of the special +departments is no longer so rapid as to forbid the contemplation +of the whole. The science of mathematics<span class="pagenum"><a name="Page_18" id="Page_18">[Pg 18]</a></span> +is now sufficiently developed, both in itself and as to its +most essential application, to have arrived at that state +of consistency in which we ought to strive to arrange its +different parts in a single system, in order to prepare for +new advances. We may even observe that the last important +improvements of the science have directly paved +the way for this important philosophical operation, by impressing +on its principal parts a character of unity which +did not previously exist.</p> + +<p>To form a just idea of the object of mathematical science, +we may start from the indefinite and meaningless +definition of it usually given, in calling it "<i>The science +of magnitudes</i>," or, which is more definite, "<i>The science +which has for its object the measurement of magnitudes.</i>" +Let us see how we can rise from this rough +sketch (which is singularly deficient in precision and +depth, though, at bottom, just) to a veritable definition, +worthy of the importance, the extent, and the difficulty +of the science.</p> + + +<h3>THE OBJECT OF MATHEMATICS.</h3> + +<p><i>Measuring Magnitudes.</i> The question of <i>measuring</i> +a magnitude in itself presents to the mind no other +idea than that of the simple direct comparison of this +magnitude with another similar magnitude, supposed to +be known, which it takes for the <i>unit</i> of comparison +among all others of the same kind. According to this +definition, then, the science of mathematics—vast and +profound as it is with reason reputed to be—instead of +being an immense concatenation of prolonged mental labours, +which offer inexhaustible occupation to our intellectual +activity, would seem to consist of a simple<span class="pagenum"><a name="Page_19" id="Page_19">[Pg 19]</a></span> +series of mechanical processes for obtaining directly the +ratios of the quantities to be measured to those by which +we wish to measure them, by the aid of operations of +similar character to the superposition of lines, as practiced +by the carpenter with his rule.</p> + +<p>The error of this definition consists in presenting as +direct an object which is almost always, on the contrary, +very indirect. The <i>direct</i> measurement of a magnitude, +by superposition or any similar process, is most frequently +an operation quite impossible for us to perform; so +that if we had no other means for determining magnitudes +than direct comparisons, we should be obliged to renounce +the knowledge of most of those which interest us.</p> + +<p><i>Difficulties.</i> The force of this general observation +will be understood if we limit ourselves to consider specially +the particular case which evidently offers the most +facility—that of the measurement of one straight line +by another. This comparison, which is certainly the +most simple which we can conceive, can nevertheless +scarcely ever be effected directly. In reflecting on the +whole of the conditions necessary to render a line susceptible +of a direct measurement, we see that most frequently +they cannot be all fulfilled at the same time. +The first and the most palpable of these conditions—that +of being able to pass over the line from one end of +it to the other, in order to apply the unit of measurement +to its whole length—evidently excludes at once by far the +greater part of the distances which interest us the most; +in the first place, all the distances between the celestial +bodies, or from any one of them to the earth; and then, +too, even the greater number of terrestrial distances, which +are so frequently inaccessible. But even if this first condition<span class="pagenum"><a name="Page_20" id="Page_20">[Pg 20]</a></span> +be found to be fulfilled, it is still farther necessary +that the length be neither too great nor too small, which +would render a direct measurement equally impossible. +The line must also be suitably situated; for let it be one +which we could measure with the greatest facility, if it +were horizontal, but conceive it to be turned up vertically, +and it becomes impossible to measure it.</p> + +<p>The difficulties which we have indicated in reference +to measuring lines, exist in a very much greater degree +in the measurement of surfaces, volumes, velocities, times, +forces, &c. It is this fact which makes necessary the +formation of mathematical science, as we are going to +see; for the human mind has been compelled to renounce, +in almost all cases, the direct measurement of +magnitudes, and to seek to determine them <i>indirectly</i>, +and it is thus that it has been led to the creation of +mathematics.</p> + +<p><i>General Method.</i> The general method which is constantly +employed, and evidently the only one conceivable, +to ascertain magnitudes which do not admit of a direct +measurement, consists in connecting them with others +which are susceptible of being determined immediately, +and by means of which we succeed in discovering +the first through the relations which subsist between +the two. Such is the precise object of mathematical +science viewed as a whole. In order to form a sufficiently +extended idea of it, we must consider that this +indirect determination of magnitudes may be indirect in +very different degrees. In a great number of cases, +which are often the most important, the magnitudes, by +means of which the principal magnitudes sought are to +be determined, cannot themselves be measured directly,<span class="pagenum"><a name="Page_21" id="Page_21">[Pg 21]</a></span> +and must therefore, in their turn, become the subject of a +similar question, and so on; so that on many occasions +the human mind is obliged to establish a long series of +intermediates between the system of unknown magnitudes +which are the final objects of its researches, and +the system of magnitudes susceptible of direct measurement, +by whose means we finally determine the first, +with which at first they appear to have no connexion.</p> + +<p><i>Illustrations.</i> Some examples will make clear any +thing which may seem too abstract in the preceding +generalities.</p> + +<p>1. <i>Falling Bodies.</i> Let us consider, in the first place, +a natural phenomenon, very simple, indeed, but which +may nevertheless give rise to a mathematical question, +really existing, and susceptible of actual applications—the +phenomenon of the vertical fall of heavy bodies.</p> + +<p>The mind the most unused to mathematical conceptions, +in observing this phenomenon, perceives at once +that the two <i>quantities</i> which it presents—namely, the +<i>height</i> from which a body has fallen, and the <i>time</i> of its +fall—are necessarily connected with each other, since they +vary together, and simultaneously remain fixed; or, in +the language of geometers, that they are "<i>functions</i>" of +each other. The phenomenon, considered under this +point of view, gives rise then to a mathematical question, +which consists in substituting for the direct measurement +of one of these two magnitudes, when it is impossible, +the measurement of the other. It is thus, for +example, that we may determine indirectly the depth of +a precipice, by merely measuring the time that a heavy +body would occupy in falling to its bottom, and by suitable +procedures this inaccessible depth will be known<span class="pagenum"><a name="Page_22" id="Page_22">[Pg 22]</a></span> +with as much precision as if it was a horizontal line +placed in the most favourable circumstances for easy and +exact measurement. On other occasions it is the height +from which a body has fallen which it will be easy to ascertain, +while the time of the fall could not be observed +directly; then the same phenomenon would give rise to +the inverse question, namely, to determine the time from +the height; as, for example, if we wished to ascertain +what would be the duration of the vertical fall of a body +falling from the moon to the earth.</p> + +<p>In this example the mathematical question is very simple, +at least when we do not pay attention to the variation +in the intensity of gravity, or the resistance of the fluid +which the body passes through in its fall. But, to extend +the question, we have only to consider the same +phenomenon in its greatest generality, in supposing the +fall oblique, and in taking into the account all the principal +circumstances. Then, instead of offering simply +two variable quantities connected with each other by a +relation easy to follow, the phenomenon will present a +much greater number; namely, the space traversed, +whether in a vertical or horizontal direction; the time +employed in traversing it; the velocity of the body at +each point of its course; even the intensity and the +direction of its primitive impulse, which may also be +viewed as variables; and finally, in certain cases (to +take every thing into the account), the resistance of the +medium and the intensity of gravity. All these different +quantities will be connected with one another, in such a +way that each in its turn may be indirectly determined +by means of the others; and this will present as many +distinct mathematical questions as there may be co-existing<span class="pagenum"><a name="Page_23" id="Page_23">[Pg 23]</a></span> +magnitudes in the phenomenon under consideration. +Such a very slight change in the physical conditions of +a problem may cause (as in the above example) a mathematical +research, at first very elementary, to be placed at +once in the rank of the most difficult questions, whose +complete and rigorous solution surpasses as yet the utmost +power of the human intellect.</p> + +<p>2. <i>Inaccessible Distances.</i> Let us take a second example +from geometrical phenomena. Let it be proposed +to determine a distance which is not susceptible of direct +measurement; it will be generally conceived as making +part of a <i>figure</i>, or certain system of lines, chosen in +such a way that all its other parts may be observed directly; +thus, in the case which is most simple, and to +which all the others may be finally reduced, the proposed +distance will be considered as belonging to a triangle, +in which we can determine directly either another +side and two angles, or two sides and one angle. Thence-forward, +the knowledge of the desired distance, instead +of being obtained directly, will be the result of a mathematical +calculation, which will consist in deducing it +from the observed elements by means of the relation +which connects it with them. This calculation will become +successively more and more complicated, if the parts +which we have supposed to be known cannot themselves +be determined (as is most frequently the case) except in +an indirect manner, by the aid of new auxiliary systems, +the number of which, in great operations of this kind, +finally becomes very considerable. The distance being +once determined, the knowledge of it will frequently be +sufficient for obtaining new quantities, which will become +the subject of new mathematical questions. Thus, when<span class="pagenum"><a name="Page_24" id="Page_24">[Pg 24]</a></span> +we know at what distance any object is situated, the +simple observation of its apparent diameter will evidently +permit us to determine indirectly its real dimensions, +however inaccessible it may be, and, by a series of analogous +investigations, its surface, its volume, even its +weight, and a number of other properties, a knowledge +of which seemed forbidden to us.</p> + +<p>3. <i>Astronomical Facts.</i> It is by such calculations +that man has been able to ascertain, not only the distances +from the planets to the earth, and, consequently, +from each other, but their actual magnitude, their true +figure, even to the inequalities of their surface; and, what +seemed still more completely hidden from us, their respective +masses, their mean densities, the principal circumstances +of the fall of heavy bodies on the surface of +each of them, &c.</p> + +<p>By the power of mathematical theories, all these different +results, and many others relative to the different +classes of mathematical phenomena, have required no +other direct measurements than those of a very small +number of straight lines, suitably chosen, and of a greater +number of angles. We may even say, with perfect +truth, so as to indicate in a word the general range of +the science, that if we did not fear to multiply calculations +unnecessarily, and if we had not, in consequence, +to reserve them for the determination of the quantities +which could not be measured directly, the determination +of all the magnitudes susceptible of precise estimation, +which the various orders of phenomena can offer us, +could be finally reduced to the direct measurement of a +single straight line and of a suitable number of angles.</p><p><span class="pagenum"><a name="Page_25" id="Page_25">[Pg 25]</a></span></p> + + +<h3>TRUE DEFINITION OF MATHEMATICS.</h3> + +<p>We are now able to define mathematical science with +precision, by assigning to it as its object the <i>indirect</i> +measurement of magnitudes, and by saying it constantly +proposes <i>to determine certain magnitudes from others +by means of the precise relations existing between them</i>.</p> + +<p>This enunciation, instead of giving the idea of only an +<i>art</i>, as do all the ordinary definitions, characterizes immediately +a true <i>science</i>, and shows it at once to be composed +of an immense chain of intellectual operations, +which may evidently become very complicated, because +of the series of intermediate links which it will be necessary +to establish between the unknown quantities and +those which admit of a direct measurement; of the number +of variables coexistent in the proposed question; and +of the nature of the relations between all these different +magnitudes furnished by the phenomena under consideration. +According to such a definition, the spirit of +mathematics consists in always regarding all the quantities +which any phenomenon can present, as connected +and interwoven with one another, with the view of deducing +them from one another. Now there is evidently +no phenomenon which cannot give rise to considerations +of this kind; whence results the naturally indefinite extent +and even the rigorous logical universality of mathematical +science. We shall seek farther on to circumscribe +as exactly as possible its real extension.</p> + +<p>The preceding explanations establish clearly the propriety +of the name employed to designate the science +which we are considering. This denomination, which +has taken to-day so definite a meaning by itself signifies<span class="pagenum"><a name="Page_26" id="Page_26">[Pg 26]</a></span> +simply <i>science</i> in general. Such a designation, rigorously +exact for the Greeks, who had no other real science, +could be retained by the moderns only to indicate +the mathematics as <i>the</i> science, beyond all others—the +science of sciences.</p> + +<p>Indeed, every true science has for its object the determination +of certain phenomena by means of others, in +accordance with the relations which exist between them. +Every <i>science</i> consists in the co-ordination of facts; if +the different observations were entirely isolated, there +would be no science. We may even say, in general terms, +that <i>science</i> is essentially destined to dispense, so far as +the different phenomena permit it, with all direct observation, +by enabling us to deduce from the smallest +possible number of immediate data the greatest possible +number of results. Is not this the real use, whether in +speculation or in action, of the <i>laws</i> which we succeed +in discovering among natural phenomena? Mathematical +science, in this point of view, merely pushes to the +highest possible degree the same kind of researches which +are pursued, in degrees more or less inferior, by every +real science in its respective sphere.</p> + + +<h3>ITS TWO FUNDAMENTAL DIVISIONS.</h3> + +<p>We have thus far viewed mathematical science only +as a whole, without paying any regard to its divisions. +We must now, in order to complete this general view, +and to form a just idea of the philosophical character of +the science, consider its fundamental division. The secondary +divisions will be examined in the following chapters.</p> + +<p>This principal division, which we are about to investigate,<span class="pagenum"><a name="Page_27" id="Page_27">[Pg 27]</a></span> +can be truly rational, and derived from the real nature +of the subject, only so far as it spontaneously presents +itself to us, in making the exact analysis of a complete +mathematical question. We will, therefore, having +determined above what is the general object of mathematical +labours, now characterize with precision the +principal different orders of inquiries, of which they are +constantly composed.</p> + +<p><i>Their different Objects.</i> The complete solution of +every mathematical question divides itself necessarily +into two parts, of natures essentially distinct, and with +relations invariably determinate. We have seen that +every mathematical inquiry has for its object to determine +unknown magnitudes, according to the relations between +them and known magnitudes. Now for this object, +it is evidently necessary, in the first place, to ascertain +with precision the relations which exist between +the quantities which we are considering. This first +branch of inquiries constitutes that which I call the <i>concrete</i> +part of the solution. When it is finished, the question +changes; it is now reduced to a pure question of +numbers, consisting simply in determining unknown +numbers, when we know what precise relations connect +them with known numbers. This second branch of inquiries +is what I call the <i>abstract</i> part of the solution. +Hence follows the fundamental division of general mathematical +science into <i>two</i> great sciences—<small>ABSTRACT MATHEMATICS</small>, +and <small>CONCRETE MATHEMATICS</small>.</p> + +<p>This analysis may be observed in every complete +mathematical question, however simple or complicated +it may be. A single example will suffice to make it +intelligible.</p><p><span class="pagenum"><a name="Page_28" id="Page_28">[Pg 28]</a></span></p> + +<p>Taking up again the phenomenon of the vertical fall +of a heavy body, and considering the simplest case, we +see that in order to succeed in determining, by means of +one another, the height whence the body has fallen, and +the duration of its fall, we must commence by discovering +the exact relation of these two quantities, or, to use the +language of geometers, the <i>equation</i> which exists between +them. Before this first research is completed, +every attempt to determine numerically the value of one +of these two magnitudes from the other would evidently +be premature, for it would have no basis. It is not enough +to know vaguely that they depend on one another—which +every one at once perceives—but it is necessary to determine +in what this dependence consists. This inquiry +may be very difficult, and in fact, in the present case, +constitutes incomparably the greater part of the problem. +The true scientific spirit is so modern, that no one, perhaps, +before Galileo, had ever remarked the increase of +velocity which a body experiences in its fall: a circumstance +which excludes the hypothesis, towards which our +mind (always involuntarily inclined to suppose in every +phenomenon the most simple <i>functions</i>, without any other +motive than its greater facility in conceiving them) +would be naturally led, that the height was proportional +to the time. In a word, this first inquiry terminated +in the discovery of the law of Galileo.</p> + +<p>When this <i>concrete</i> part is completed, the inquiry becomes +one of quite another nature. Knowing that the +spaces passed through by the body in each successive second +of its fall increase as the series of odd numbers, we +have then a problem purely numerical and <i>abstract</i>; to +deduce the height from the time, or the time from the<span class="pagenum"><a name="Page_29" id="Page_29">[Pg 29]</a></span> +height; and this consists in finding that the first of these +two quantities, according to the law which has been established, +is a known multiple of the second power of the +other; from which, finally, we have to calculate the value +of the one when that of the other is given.</p> + +<p>In this example the concrete question is more difficult +than the abstract one. The reverse would be the case +if we considered the same phenomenon in its greatest +generality, as I have done above for another object. +According to the circumstances, sometimes the first, +sometimes the second, of these two parts will constitute +the principal difficulty of the whole question; for the +mathematical law of the phenomenon may be very simple, +but very difficult to obtain, or it may be easy to discover, +but very complicated; so that the two great sections +of mathematical science, when we compare them +as wholes, must be regarded as exactly equivalent in extent +and in difficulty, as well as in importance, as we +shall show farther on, in considering each of them separately.</p> + +<p><i>Their different Natures.</i> These two parts, essentially +distinct in their <i>object</i>, as we have just seen, are no less +so with regard to the <i>nature</i> of the inquiries of which +they are composed.</p> + +<p>The first should be called <i>concrete</i>, since it evidently +depends on the character of the phenomena considered, +and must necessarily vary when we examine new phenomena; +while the second is completely independent of +the nature of the objects examined, and is concerned with +only the <i>numerical</i> relations which they present, for which +reason it should be called <i>abstract</i>. The same relations +may exist in a great number of different phenomena,<span class="pagenum"><a name="Page_30" id="Page_30">[Pg 30]</a></span> +which, in spite of their extreme diversity, will be viewed +by the geometer as offering an analytical question susceptible, +when studied by itself, of being resolved once +for all. Thus, for instance, the same law which exists +between the space and the time of the vertical fall of a +body in a vacuum, is found again in many other phenomena +which offer no analogy with the first nor with +each other; for it expresses the relation between the surface +of a spherical body and the length of its diameter; +it determines, in like manner, the decrease of the intensity +of light or of heat in relation to the distance of the objects +lighted or heated, &c. The abstract part, common +to these different mathematical questions, having +been treated in reference to one of these, will thus have +been treated for all; while the concrete part will have +necessarily to be again taken up for each question separately, +without the solution of any one of them being +able to give any direct aid, in that connexion, for the solution +of the rest.</p> + +<p>The abstract part of mathematics is, then, general in +its nature; the concrete part, special.</p> + +<p>To present this comparison under a new point of view, +we may say concrete mathematics has a philosophical +character, which is essentially experimental, physical, +phenomenal; while that of abstract mathematics is purely +logical, rational. The concrete part of every mathematical +question is necessarily founded on the consideration +of the external world, and could never be resolved +by a simple series of intellectual combinations. The abstract +part, on the contrary, when it has been very completely +separated, can consist only of a series of logical +deductions, more or less prolonged; for if we have once<span class="pagenum"><a name="Page_31" id="Page_31">[Pg 31]</a></span> +found the equations of a phenomenon, the determination +of the quantities therein considered, by means of one another, +is a matter for reasoning only, whatever the difficulties +may be. It belongs to the understanding alone +to deduce from these equations results which are evidently +contained in them, although perhaps in a very involved +manner, without there being occasion to consult +anew the external world; the consideration of which, +having become thenceforth foreign to the subject, ought +even to be carefully set aside in order to reduce the labour +to its true peculiar difficulty. The <i>abstract</i> part +of mathematics is then purely instrumental, and is only +an immense and admirable extension of natural logic to a +certain class of deductions. On the other hand, geometry +and mechanics, which, as we shall see presently, constitute +the <i>concrete</i> part, must be viewed as real natural +sciences, founded on observation, like all the rest, +although the extreme simplicity of their phenomena permits +an infinitely greater degree of systematization, +which has sometimes caused a misconception of the experimental +character of their first principles.</p> + +<p>We see, by this brief general comparison, how natural +and profound is our fundamental division of mathematical +science.</p> + +<p>We have now to circumscribe, as exactly as we can +in this first sketch, each of these two great sections.</p> + + +<h3>CONCRETE MATHEMATICS.</h3> + +<p><i>Concrete Mathematics</i> having for its object the discovery +of the <i>equations</i> of phenomena, it would seem at +first that it must be composed of as many distinct sciences +as we find really distinct categories among natural<span class="pagenum"><a name="Page_32" id="Page_32">[Pg 32]</a></span> +phenomena. But we are yet very far from having discovered +mathematical laws in all kinds of phenomena; +we shall even see, presently, that the greater part will +very probably always hide themselves from our investigations. +In reality, in the present condition of the human +mind, there are directly but two great general classes of +phenomena, whose equations we constantly know; these +are, firstly, geometrical, and, secondly, mechanical phenomena. +Thus, then, the concrete part of mathematics +is composed of <span class="smcap">Geometry</span> and <span class="smcap">Rational Mechanics</span>.</p> + +<p>This is sufficient, it is true, to give to it a complete +character of logical universality, when we consider all +phenomena from the most elevated point of view of natural +philosophy. In fact, if all the parts of the universe +were conceived as immovable, we should evidently have +only geometrical phenomena to observe, since all would +be reduced to relations of form, magnitude, and position; +then, having regard to the motions which take place in it, +we would have also to consider mechanical phenomena. +Hence the universe, in the statical point of view, presents +only geometrical phenomena; and, considered dynamically, +only mechanical phenomena. Thus geometry +and mechanics constitute the two fundamental natural +sciences, in this sense, that all natural effects may be conceived +as simple necessary results, either of the laws of +extension or of the laws of motion.</p> + +<p>But although this conception is always logically possible, +the difficulty is to specialize it with the necessary +precision, and to follow it exactly in each of the general +cases offered to us by the study of nature; that is, to +effectually reduce each principal question of natural philosophy, +for a certain determinate order of phenomena, to<span class="pagenum"><a name="Page_33" id="Page_33">[Pg 33]</a></span> +the question of geometry or mechanics, to which we might +rationally suppose it should be brought. This transformation, +which requires great progress to have been previously +made in the study of each class of phenomena, has thus +far been really executed only for those of astronomy, and +for a part of those considered by terrestrial physics, properly +so called. It is thus that astronomy, acoustics, optics, +&c., have finally become applications of mathematical +science to certain orders of observations.<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> But these +applications not being by their nature rigorously circumscribed, +to confound them with the science would be to +assign to it a vague and indefinite domain; and this is +done in the usual division, so faulty in so many other +respects, of the mathematics into "Pure" and "Applied."</p> + + +<h3>ABSTRACT MATHEMATICS.</h3> + +<p>The nature of abstract mathematics (the general division +of which will be examined in the following chapter) is +clearly and exactly determined. It is composed of what is +called the <i>Calculus</i>,<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> taking this word in its greatest extent, +which reaches from the most simple numerical operations +to the most sublime combinations of transcendental +analysis. The <i>Calculus</i> has the solution of all questions +<span class="pagenum"><a name="Page_34" id="Page_34">[Pg 34]</a></span>relating to numbers for its peculiar object. Its <i>starting +point</i> is, constantly and necessarily, the knowledge of the +precise relations, <i>i.e.</i>, of the <i>equations</i>, between the different +magnitudes which are simultaneously considered; +that which is, on the contrary, the <i>stopping point</i> of concrete +mathematics. However complicated, or however indirect +these relations may be, the final object of the calculus +always is to obtain from them the values of the unknown +quantities by means of those which are known. +This <i>science</i>, although nearer perfection than any other, +is really little advanced as yet, so that this object is rarely +attained in a manner completely satisfactory.</p> + +<p>Mathematical analysis is, then, the true rational basis +of the entire system of our actual knowledge. It constitutes +the first and the most perfect of all the fundamental +sciences. The ideas with which it occupies itself +are the most universal, the most abstract, and the +most simple which it is possible for us to conceive.</p> + +<p>This peculiar nature of mathematical analysis enables +us easily to explain why, when it is properly employed, +it is such a powerful instrument, not only to give more +precision to our real knowledge, which is self-evident, but +especially to establish an infinitely more perfect co-ordination +in the study of the phenomena which admit of +that application; for, our conceptions having been so +generalized and simplified that a single analytical question, +abstractly resolved, contains the <i>implicit</i> solution +of a great number of diverse physical questions, the human +mind must necessarily acquire by these means a +greater facility in perceiving relations between phenomena +which at first appeared entirely distinct from one +another. We thus naturally see arise, through the medium<span class="pagenum"><a name="Page_35" id="Page_35">[Pg 35]</a></span> +of analysis, the most frequent and the most unexpected +approximations between problems which at first +offered no apparent connection, and which we often end +in viewing as identical. Could we, for example, without +the aid of analysis, perceive the least resemblance +between the determination of the direction of a curve at +each of its points and that of the velocity acquired by a +body at every instant of its variable motion? and yet +these questions, however different they may be, compose +but one in the eyes of the geometer.</p> + +<p>The high relative perfection of mathematical analysis +is as easily perceptible. This perfection is not due, as +some have thought, to the nature of the signs which are +employed as instruments of reasoning, eminently concise +and general as they are. In reality, all great analytical +ideas have been formed without the algebraic signs having +been of any essential aid, except for working them +out after the mind had conceived them. The superior +perfection of the science of the calculus is due principally +to the extreme simplicity of the ideas which it considers, +by whatever signs they may be expressed; so that +there is not the least hope, by any artifice of scientific +language, of perfecting to the same degree theories which +refer to more complex subjects, and which are necessarily +condemned by their nature to a greater or less logical inferiority.</p> + + +<h3>THE EXTENT OF ITS FIELD.</h3> + +<p>Our examination of the philosophical character of mathematical +science would remain incomplete, if, after having +viewed its object and composition, we did not examine +the real extent of its domain.</p><p><span class="pagenum"><a name="Page_36" id="Page_36">[Pg 36]</a></span></p> + +<p><i>Its Universality</i>. For this purpose it is indispensable +to perceive, first of all, that, in the purely logical +point of view, this science is by itself necessarily and +rigorously universal; for there is no question whatever +which may not be finally conceived as consisting in determining +certain quantities from others by means of certain +relations, and consequently as admitting of reduction, +in final analysis, to a simple question of numbers. +In all our researches, indeed, on whatever subject, our +object is to arrive at numbers, at quantities, though often +in a very imperfect manner and by very uncertain methods. +Thus, taking an example in the class of subjects +the least accessible to mathematics, the phenomena of +living bodies, even when considered (to take the most +complicated case) in the state of disease, is it not manifest +that all the questions of therapeutics may be viewed +as consisting in determining the <i>quantities</i> of the different +agents which modify the organism, and which must +act upon it to bring it to its normal state, admitting, for +some of these quantities in certain cases, values which +are equal to zero, or negative, or even contradictory?</p> + +<p>The fundamental idea of Descartes on the relation of +the concrete to the abstract in mathematics, has proven, +in opposition to the superficial distinction of metaphysics, +that all ideas of quality may be reduced to those of +quantity. This conception, established at first by its +immortal author in relation to geometrical phenomena +only, has since been effectually extended to mechanical +phenomena, and in our days to those of heat. As a result +of this gradual generalization, there are now no geometers +who do not consider it, in a purely theoretical +sense, as capable of being applied to all our real ideas of<span class="pagenum"><a name="Page_37" id="Page_37">[Pg 37]</a></span> +every sort, so that every phenomenon is logically susceptible +of being represented by an <i>equation</i>; as much so, +indeed, as is a curve or a motion, excepting the difficulty +of discovering it, and then of <i>resolving</i> it, which +may be, and oftentimes are, superior to the greatest powers +of the human mind.</p> + +<p><i>Its Limitations</i>. Important as it is to comprehend +the rigorous universality, in a logical point of view, of +mathematical science, it is no less indispensable to consider +now the great real <i>limitations</i>, which, through the +feebleness of our intellect, narrow in a remarkable degree +its actual domain, in proportion as phenomena, in +becoming special, become complicated.</p> + +<p>Every question may be conceived as capable of being +reduced to a pure question of numbers; but the difficulty +of effecting such a transformation increases so much +with the complication of the phenomena of natural philosophy, +that it soon becomes insurmountable.</p> + +<p>This will be easily seen, if we consider that to bring +a question within the field of mathematical analysis, we +must first have discovered the precise relations which exist +between the quantities which are found in the phenomenon +under examination, the establishment of these +equations being the necessary starting point of all analytical +labours. This must evidently be so much the +more difficult as we have to do with phenomena which +are more special, and therefore more complicated. We +shall thus find that it is only in <i>inorganic physics</i>, at +the most, that we can justly hope ever to obtain that +high degree of scientific perfection.</p> + +<p>The <i>first</i> condition which is necessary in order that +phenomena may admit of mathematical laws, susceptible<span class="pagenum"><a name="Page_38" id="Page_38">[Pg 38]</a></span> +of being discovered, evidently is, that their different quantities +should admit of being expressed by fixed numbers. +We soon find that in this respect the whole of <i>organic +physics</i>, and probably also the most complicated parts of +inorganic physics, are necessarily inaccessible, by their +nature, to our mathematical analysis, by reason of the +extreme numerical variability of the corresponding phenomena. +Every precise idea of fixed numbers is truly +out of place in the phenomena of living bodies, when we +wish to employ it otherwise than as a means of relieving +the attention, and when we attach any importance to the +exact relations of the values assigned.</p> + +<p>We ought not, however, on this account, to cease to +conceive all phenomena as being necessarily subject to +mathematical laws, which we are condemned to be ignorant +of, only because of the too great complication of the +phenomena. The most complex phenomena of living +bodies are doubtless essentially of no other special nature +than the simplest phenomena of unorganized matter. If +it were possible to isolate rigorously each of the simple +causes which concur in producing a single physiological +phenomenon, every thing leads us to believe that it would +show itself endowed, in determinate circumstances, with +a kind of influence and with a quantity of action as exactly +fixed as we see it in universal gravitation, a veritable +type of the fundamental laws of nature.</p> + +<p>There is a <i>second</i> reason why we cannot bring complicated +phenomena under the dominion of mathematical +analysis. Even if we could ascertain the mathematical +law which governs each agent, taken by itself, the combination +of so great a number of conditions would render +the corresponding mathematical problem so far above our<span class="pagenum"><a name="Page_39" id="Page_39">[Pg 39]</a></span> +feeble means, that the question would remain in most +cases incapable of solution.</p> + +<p>To appreciate this difficulty, let us consider how complicated +mathematical questions become, even those relating +to the most simple phenomena of unorganized bodies, +when we desire to bring sufficiently near together the abstract +and the concrete state, having regard to all the +principal conditions which can exercise a real influence +over the effect produced. We know, for example, that +the very simple phenomenon of the flow of a fluid through +a given orifice, by virtue of its gravity alone, has not as +yet any complete mathematical solution, when we take +into the account all the essential circumstances. It is +the same even with the still more simple motion of a +solid projectile in a resisting medium.</p> + +<p>Why has mathematical analysis been able to adapt itself +with such admirable success to the most profound study +of celestial phenomena? Because they are, in spite of +popular appearances, much more simple than any others. +The most complicated problem which they present, that +of the modification produced in the motions of two bodies +tending towards each other by virtue of their gravitation, +by the influence of a third body acting on both of them +in the same manner, is much less complex than the most +simple terrestrial problem. And, nevertheless, even it +presents difficulties so great that we yet possess only +approximate solutions of it. It is even easy to see that +the high perfection to which solar astronomy has been +able to elevate itself by the employment of mathematical +science is, besides, essentially due to our having skilfully +profited by all the particular, and, so to say, accidental +facilities presented by the peculiarly favourable constitution<span class="pagenum"><a name="Page_40" id="Page_40">[Pg 40]</a></span> +of our planetary system. The planets which compose +it are quite few in number, and their masses are in +general very unequal, and much less than that of the +sun; they are, besides, very distant from one another; +they have forms almost spherical; their orbits are nearly +circular, and only slightly inclined to each other, and so +on. It results from all these circumstances that the perturbations +are generally inconsiderable, and that to calculate +them it is usually sufficient to take into the account, +in connexion with the action of the sun on each +particular planet, the influence of only one other planet, +capable, by its size and its proximity, of causing perceptible +derangements.</p> + +<p>If, however, instead of such a state of things, our solar +system had been composed of a greater number of +planets concentrated into a less space, and nearly equal +in mass; if their orbits had presented very different inclinations, +and considerable eccentricities; if these bodies +had been of a more complicated form, such as very eccentric +ellipsoids, it is certain that, supposing the same +law of gravitation to exist, we should not yet have succeeded +in subjecting the study of the celestial phenomena +to our mathematical analysis, and probably we should +not even have been able to disentangle the present principal +law.</p> + +<p>These hypothetical conditions would find themselves +exactly realized in the highest degree in <i>chemical</i> phenomena, +if we attempted to calculate them by the theory +of general gravitation.</p> + +<p>On properly weighing the preceding considerations, +the reader will be convinced, I think, that in reducing +the future extension of the great applications of mathematical<span class="pagenum"><a name="Page_41" id="Page_41">[Pg 41]</a></span> +analysis, which are really possible, to the field +comprised in the different departments of inorganic physics, +I have rather exaggerated than contracted the extent +of its actual domain. Important as it was to render +apparent the rigorous logical universality of mathematical +science, it was equally so to indicate the conditions +which limit for us its real extension, so as not to +contribute to lead the human mind astray from the true +scientific direction in the study of the most complicated +phenomena, by the chimerical search after an impossible +perfection.</p> +<p><span class="pagenum"><a name="Page_42" id="Page_42">[Pg 42]</a></span></p><p><span class="pagenum"><a name="Page_43" id="Page_43">[Pg 43]</a></span></p> +<hr class="tb" /> + +<p>Having thus exhibited the essential object and the +principal composition of mathematical science, as well as +its general relations with the whole body of natural philosophy, +we have now to pass to the special examination +of the great sciences of which it is composed.</p> + +<div class="blockquot"><p><i>Note.</i>—<span class="smcap">Analysis</span> and <span class="smcap">Geometry</span> are the two great heads under which +the subject is about to be examined. To these <i>M. Comte</i> adds Rational +<span class="smcap">Mechanics</span>; but as it is not comprised in the usual idea of Mathematics, +and as its discussion would be of but limited utility and interest, it is not +included in the present translation.</p></div> + + + + +<p class="center big">BOOK I.</p> + +<p class="center">ANALYSIS.</p> + +<p><span class="pagenum"><a name="Page_44" id="Page_44">[Pg 44]</a><br /><a name="Page_45" id="Page_45">[Pg 45]</a></span></p> + + +<p class="center big">BOOK I.</p> + +<h1>ANALYSIS.</h1> + + + + +<h2><a name="CHAPTER_I" id="CHAPTER_I">CHAPTER I.</a></h2> + +<h3>GENERAL VIEW OF MATHEMATICAL ANALYSIS.</h3> + + +<p>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 +portion; but it is none the less necessary, in order +to conceive the science in a manner truly logical, to +consider the Calculus in all its principal branches before +proceeding to the philosophical study of Geometry and +Mechanics. Its analytical theories, more simple and +more general than those of concrete mathematics, are in +themselves essentially independent of the latter; while +these, on the contrary, have, by their nature, a continual +need of the former, without the aid of which they could +make scarcely any progress. Although the principal +conceptions of analysis retain at present some very perceptible +traces of their geometrical or mechanical origin, +they are now, however, mainly freed from that primitive +character, which no longer manifests itself except in some +secondary points; so that it is possible (especially since +the labours of Lagrange) to present them in a dogmatic +exposition, by a purely abstract method, in a single and<span class="pagenum"><a name="Page_46" id="Page_46">[Pg 46]</a></span> +continuous system. It is this which will be undertaken +in the present and the five following chapters, limiting our +investigations to the most general considerations upon +each principal branch of the science of the calculus.</p> + +<p>The definite object of our researches in concrete mathematics +being the discovery of the <i>equations</i> which express +the mathematical laws of the phenomenon under +consideration, and these equations constituting the true +starting point of the calculus, which has for its object +to obtain from them the determination of certain quantities +by means of others, I think it indispensable, before +proceeding any farther, to go more deeply than has +been customary into that fundamental idea of <i>equation</i>, +the continual subject, either as end or as beginning, of +all mathematical labours. Besides the advantage of circumscribing +more definitely the true field of analysis, +there will result from it the important consequence of +tracing in a more exact manner the real line of demarcation +between the concrete and the abstract part of +mathematics, which will complete the general exposition +of the fundamental division established in the introductory +chapter.</p> + + + + +<h3><a name="THE_TRUE_IDEA_OF_AN_EQUATION" id="THE_TRUE_IDEA_OF_AN_EQUATION">THE TRUE IDEA OF AN EQUATION.</a></h3> + + +<p>We usually form much too vague an idea of what an +<i>equation</i> is, when we give that name to every kind of +relation of equality between <i>any</i> two functions of the +magnitudes which we are considering. For, though every +equation is evidently a relation of equality, it is far +from being true that, reciprocally, every relation of equality +is a veritable <i>equation</i>, of the kind of those to which, +by their nature, the methods of analysis are applicable.</p><p><span class="pagenum"><a name="Page_47" id="Page_47">[Pg 47]</a></span></p> + +<p>This want of precision in the logical consideration of +an idea which is so fundamental in mathematics, brings +with it the serious inconvenience of rendering it almost +impossible to explain, in general terms, the great and +fundamental difficulty which we find in establishing the +relation between the concrete and the abstract, and which +stands out so prominently in each great mathematical +question taken by itself. If the meaning of the word +<i>equation</i> was truly as extended as we habitually suppose +it to be in our definition of it, it is not apparent what +great difficulty there could really be, in general, in establishing +the equations of any problem whatsoever; for the +whole would thus appear to consist in a simple question +of form, which ought never even to exact any great intellectual +efforts, seeing that we can hardly conceive of +any precise relation which is not immediately a certain +relation of equality, or which cannot be readily brought +thereto by some very easy transformations.</p> + +<p>Thus, when we admit every species of <i>functions</i> into +the definition of <i>equations</i>, we do not at all account for +the extreme difficulty which we almost always experience +in putting a problem into an equation, and which +so often may be compared to the efforts required by the +analytical elaboration of the equation when once obtained. +In a word, the ordinary abstract and general idea +of an <i>equation</i> does not at all correspond to the real +meaning which geometers attach to that expression in +the actual development of the science. Here, then, is a +logical fault, a defect of correlation, which it is very important +to rectify.</p> + + +<p><i>Division of Functions into Abstract and Concrete.</i> +To succeed in doing so, I begin by distinguishing two<span class="pagenum"><a name="Page_48" id="Page_48">[Pg 48]</a></span> +sorts of <i>functions</i>, <i>abstract</i> or analytical functions, and +<i>concrete</i> functions. The first alone can enter into veritable +<i>equations</i>. We may, therefore, henceforth define +every <i>equation</i>, in an exact and sufficiently profound manner, +as a relation of equality between two <i>abstract</i> functions +of the magnitudes under consideration. In order not +to have to return again to this fundamental definition, I +must add here, as an indispensable complement, without +which the idea would not be sufficiently general, that +these abstract functions may refer not only to the magnitudes +which the problem presents of itself, but also to +all the other auxiliary magnitudes which are connected +with it, and which we will often be able to introduce, +simply as a mathematical artifice, with the sole object +of facilitating the discovery of the equations of the phenomena. +I here anticipate summarily the result of a +general discussion of the highest importance, which will +be found at the end of this chapter. We will now return +to the essential distinction of functions as abstract +and concrete.</p> + +<p>This distinction may be established in two ways, essentially +different, but complementary of each other, <i>à +priori</i> and <i>à posteriori</i>; that is to say, by characterizing +in a general manner the peculiar nature of each species +of functions, and then by making the actual enumeration +of all the abstract functions at present known, +at least so far as relates to the elements of which they +are composed.</p> + +<p><i>À priori</i>, the functions which I call <i>abstract</i> are those +which express a manner of dependence between magnitudes, +which can be conceived between numbers alone, +without there being need of indicating any phenomenon<span class="pagenum"><a name="Page_49" id="Page_49">[Pg 49]</a></span> +whatever in which it is realized. I name, on the other +hand, <i>concrete</i> functions, those for which the mode of dependence +expressed cannot be defined or conceived except +by assigning a determinate case of physics, geometry, mechanics, +&c., in which it actually exists.</p> + +<p>Most functions in their origin, even those which are +at present the most purely <i>abstract</i>, have begun by being +<i>concrete</i>; so that it is easy to make the preceding +distinction understood, by citing only the successive different +points of view under which, in proportion as the +science has become formed, geometers have considered +the most simple analytical functions. I will indicate +powers, for example, which have in general become abstract +functions only since the labours of Vieta and Descartes. +The functions <i>x<sup>2</sup></i>, <i>x<sup>3</sup></i>, which in our present analysis +are so well conceived as simply <i>abstract</i>, were, for +the geometers of antiquity, perfectly <i>concrete</i> functions, +expressing the relation of the superficies of a square, or +the volume of a cube to the length of their side. These +had in their eyes such a character so exclusively, that +it was only by means of the geometrical definitions that +they discovered the elementary algebraic properties of +these functions, relating to the decomposition of the +variable into two parts, properties which were at that +epoch only real theorems of geometry, to which a numerical +meaning was not attached until long afterward.</p> + +<p>I shall have occasion to cite presently, for another reason, +a new example, very suitable to make apparent the +fundamental distinction which I have just exhibited; it +is that of circular functions, both direct and inverse, which +at the present time are still sometimes concrete, sometimes<span class="pagenum"><a name="Page_50" id="Page_50">[Pg 50]</a></span> +abstract, according to the point of view under which +they are regarded.</p> + +<p><i>À posteriori</i>, the general character which renders a +function abstract or concrete having been established, the +question as to whether a certain determinate function is +veritably abstract, and therefore susceptible of entering +into true analytical equations, becomes a simple question +of fact, inasmuch as we are going to enumerate all the +functions of this species.</p> + + +<p><i>Enumeration of Abstract Functions.</i> At first view +this enumeration seems impossible, the distinct analytical +functions being infinite in number. But when we +divide them into <i>simple</i> and <i>compound</i>, the difficulty disappears; +for, though the number of the different functions +considered in mathematical analysis is really infinite, +they are, on the contrary, even at the present day, +composed of a very small number of elementary functions, +which can be easily assigned, and which are evidently +sufficient for deciding the abstract or concrete character +of any given function; which will be of the one or the +other nature, according as it shall be composed exclusively +of these simple abstract functions, or as it shall include +others.</p> + +<p>We evidently have to consider, for this purpose, only +the functions of a single variable, since those relative +to several independent variables are constantly, by their +nature, more or less <i>compound</i>.</p> + +<p>Let <i>x</i> be the independent variable, <i>y</i> the correlative +variable which depends upon it. The different simple +modes of abstract dependence, which we can now conceive +between <i>y</i> and <i>x</i>, are expressed by the ten following elementary +formulas, in which each function is coupled<span class="pagenum"><a name="Page_51" id="Page_51">[Pg 51]</a></span> +with its <i>inverse</i>, that is, with that which would be obtained +from the direct function by referring <i>x</i> to <i>y</i>, instead +of referring <i>y</i> to <i>x</i>.</p> + + +<div class="center"> +<table border="0" cellpadding="4" cellspacing="0" summary="functions"> +<tr><td> </td><td align="center">FUNCTION.</td><td align="center">ITS NAME.</td></tr> +<tr><td align="left" rowspan="2">1st couple</td><td align="left">1° <i>y</i> = <i>a</i> + <i>x</i></td><td align="left"><i>Sum.</i></td></tr> +<tr><td align="left">2° <i>y</i> = <i>a</i> - <i>x</i></td><td align="left"><i>Difference.</i></td></tr> +<tr><td align="left" rowspan="2">2d couple</td><td align="left">1° <i>y</i> = <i>ax</i></td><td align="left"><i>Product.</i></td></tr> +<tr><td align="left">2° <i>y</i> = <i>a/x</i></td><td align="left"><i>Quotient.</i></td></tr> +<tr><td align="left" rowspan="2">3d couple</td><td align="left">1° <i>y</i> = <i>x^a</i></td><td align="left"><i>Power.</i></td></tr> +<tr><td align="left">2° <i>y</i> = <i>[aroot]x</i></td><td align="left"><i>Root.</i></td></tr> +<tr><td align="left" rowspan="2">4th couple</td><td align="left">1° <i>y</i> = <i>a^x</i></td><td align="left"><i>Exponential.</i></td></tr> +<tr><td align="left">2° <i>y</i> = <i>[log a]x</i></td><td align="left"><i>Logarithmic.</i></td></tr> +<tr><td align="left" rowspan="2">5th couple</td><td align="left">1° <i>y</i> = sin. <i>x</i></td><td align="left"><i>Direct Circular.</i></td></tr> +<tr><td align="left">2° <i>y</i> = arc(sin. = <i>x</i>).</td><td align="left"><i>Inverse Circular.</i><a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a></td></tr> +</table></div> + +<p>Such are the elements, very few in number, which directly +compose all the abstract functions known at the +present day. Few as they are, they are evidently sufficient +to give rise to an infinite number of analytical +combinations.</p> +<p><span class="pagenum"><a name="Page_52" id="Page_52">[Pg 52]</a></span></p> +<p>No rational consideration rigorously circumscribes, <i>à +priori</i>, the preceding table, which is only the actual expression +of the present state of the science. Our analytical +elements are at the present day more numerous +than they were for Descartes, and even for Newton and +Leibnitz: it is only a century since the last two couples +have been introduced into analysis by the labours of John +Bernouilli and Euler. Doubtless new ones will be hereafter +admitted; but, as I shall show towards the end of +this chapter, we cannot hope that they will ever be greatly +multiplied, their real augmentation giving rise to very +great difficulties.</p> + +<p>We can now form a definite, and, at the same time, +sufficiently extended idea of what geometers understand +by a veritable <i>equation</i>. This explanation is especially +suited to make us understand how difficult it must be +really to establish the <i>equations</i> of phenomena, since we +have effectually succeeded in so doing only when we +have been able to conceive the mathematical laws of +these phenomena by the aid of functions entirely composed +of only the mathematical elements which I have +just enumerated. It is clear, in fact, that it is then +only that the problem becomes truly abstract, and is reduced +to a pure question of numbers, these functions +being the only simple relations which we can conceive +between numbers, considered by themselves. Up to this +period of the solution, whatever the appearances may be, +the question is still essentially concrete, and does not come +within the domain of the <i>calculus</i>. Now the fundamental +difficulty of this passage from the <i>concrete</i> to the <i>abstract</i> +in general consists especially in the insufficiency +of this very small number of analytical elements which<span class="pagenum"><a name="Page_53" id="Page_53">[Pg 53]</a></span> +we possess, and by means of which, nevertheless, in spite +of the little real variety which they offer us, we must +succeed in representing all the precise relations which +all the different natural phenomena can manifest to us. +Considering the infinite diversity which must necessarily +exist in this respect in the external world, we easily +understand how far below the true difficulty our conceptions +must frequently be found, especially if we add +that as these elements of our analysis have been in the +first place furnished to us by the mathematical consideration +of the simplest phenomena, we have, <i>à priori</i>, no +rational guarantee of their necessary suitableness to represent +the mathematical law of every other class of phenomena. +I will explain presently the general artifice, so +profoundly ingenious, by which the human mind has succeeded +in diminishing, in a remarkable degree, this fundamental +difficulty which is presented by the relation of +the concrete to the abstract in mathematics, without, +however, its having been necessary to multiply the number +of these analytical elements.</p> + + + + +<h3><a name="THE_TWO_PRINCIPAL_DIVISIONS_OF_THE_CALCULUS" id="THE_TWO_PRINCIPAL_DIVISIONS_OF_THE_CALCULUS">THE TWO PRINCIPAL DIVISIONS OF THE CALCULUS.</a></h3> + + +<p>The preceding explanations determine with precision +the true object and the real field of abstract mathematics. +I must now pass to the examination of its principal +divisions, for thus far we have considered the calculus +as a whole.</p> + +<p>The first direct consideration to be presented on the +composition of the science of the <i>calculus</i> consists in dividing +it, in the first place, into two principal branches, +to which, for want of more suitable denominations, I will +give the names of <i>Algebraic calculus</i>, or <i>Algebra</i>, and of<span class="pagenum"><a name="Page_54" id="Page_54">[Pg 54]</a></span> +<i>Arithmetical calculus</i>, or <i>Arithmetic</i>; but with the caution +to take these two expressions in their most extended +logical acceptation, in the place of the by far too restricted +meaning which is usually attached to them.</p> + +<p>The complete solution of every question of the <i>calculus</i>, +from the most elementary up to the most transcendental, +is necessarily composed of two successive parts, +whose nature is essentially distinct. In the first, the object +is to transform the proposed equations, so as to make +apparent the manner in which the unknown quantities +are formed by the known ones: it is this which constitutes +the <i>algebraic</i> question. In the second, our object +is to <i>find the values</i> of the formulas thus obtained; that +is, to determine directly the values of the numbers sought, +which are already represented by certain explicit functions +of given numbers: this is the <i>arithmetical</i> question.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a> +It is apparent that, in every solution which is +<span class="pagenum"><a name="Page_55" id="Page_55">[Pg 55]</a></span>truly rational, it necessarily follows the algebraical question, +of which it forms the indispensable complement, +since it is evidently necessary to know the mode of generation +of the numbers sought for before determining their +actual values for each particular case. Thus the stopping-place +of the algebraic part of the solution becomes +the starting point of the arithmetical part.</p> + +<p>We thus see that the <i>algebraic</i> calculus and the <i>arithmetical</i> +calculus differ essentially in their object. They +differ no less in the point of view under which they regard +quantities; which are considered in the first as to their +<i>relations</i>, and in the second as to their <i>values</i>. The +true spirit of the calculus, in general, requires this distinction +to be maintained with the most severe exactitude, +and the line of demarcation between the two periods +of the solution to be rendered as clear and distinct +as the proposed question permits. The attentive observation +of this precept, which is too much neglected, may +be of much assistance, in each particular question, in directing +the efforts of our mind, at any moment of the +solution, towards the real corresponding difficulty. In +truth, the imperfection of the science of the calculus +obliges us very often (as will be explained in the next +chapter) to intermingle algebraic and arithmetical considerations +in the solution of the same question. But, however +impossible it may be to separate clearly the two parts +of the labour, yet the preceding indications will always +enable us to avoid confounding them.</p> + +<p>In endeavouring to sum up as succinctly as possible +the distinction just established, we see that <span class="smcap">Algebra</span> +may be defined, in general, as having for its object the +<i>resolution of equations</i>; taking this expression in its<span class="pagenum"><a name="Page_56" id="Page_56">[Pg 56]</a></span> +full logical meaning, which signifies the transformation +of <i>implicit</i> functions into equivalent <i>explicit</i> ones. In +the same way, <span class="smcap">Arithmetic</span> may be defined as destined +to <i>the determination of the values of functions</i>. Henceforth, +therefore, we will briefly say that <span class="smcap">Algebra</span> is the +<i>Calculus of Functions</i>, and <span class="smcap">Arithmetic</span> the <i>Calculus of +Values</i>.</p> + +<p>We can now perceive how insufficient and even erroneous +are the ordinary definitions. Most generally, the +exaggerated importance attributed to Signs has led to the +distinguishing the two fundamental branches of the science +of the Calculus by the manner of designating in +each the subjects of discussion, an idea which is evidently +absurd in principle and false in fact. Even the celebrated +definition given by Newton, characterizing <i>Algebra</i> +as <i>Universal Arithmetic</i>, gives certainly a very false +idea of the nature of algebra and of that of arithmetic.<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a></p> + +<p>Having thus established the fundamental division of +the calculus into two principal branches, I have now to +compare in general terms the extent, the importance, and +the difficulty of these two sorts of calculus, so as to have +hereafter to consider only the <i>Calculus of Functions</i>, +which is to be the principal subject of our study.</p> +<p><span class="pagenum"><a name="Page_57" id="Page_57">[Pg 57]</a></span></p> + + + +<h3><a name="THE_CALCULUS_OF_VALUES_OR_ARITHMETIC" id="THE_CALCULUS_OF_VALUES_OR_ARITHMETIC">THE CALCULUS OF VALUES, OR ARITHMETIC.</a></h3> + + +<p><i>Its Extent.</i> The <i>Calculus of Values, or Arithmetic</i>, +would appear, at first view, to present a field as vast as +that of <i>algebra</i>, since it would seem to admit as many +distinct questions as we can conceive different algebraic +formulas whose values are to be determined. But a very +simple reflection will show the difference. Dividing functions +into <i>simple</i> and <i>compound</i>, it is evident that when +we know how to determine the <i>value</i> of simple functions, +the consideration of compound functions will no longer +present any difficulty. In the algebraic point of view, +a compound function plays a very different part from that +of the elementary functions of which it consists, and from +this, indeed, proceed all the principal difficulties of analysis. +But it is very different with the Arithmetical Calculus. +Thus the number of truly distinct arithmetical +operations is only that determined by the number of the +elementary abstract functions, the very limited list of +which has been given above. The determination of the values +of these ten functions necessarily gives that of all +the functions, infinite in number, which are considered +in the whole of mathematical analysis, such at least as +it exists at present. There can be no new arithmetical +operations without the creation of really new analytical +elements, the number of which must always be extremely +small. The field of <i>arithmetic</i> is, then, by its nature, +exceedingly restricted, while that of algebra is rigorously +indefinite.</p> + +<p>It is, however, important to remark, that the domain +of the <i>calculus of values</i> is, in reality, much more extensive +than it is commonly represented; for several questions<span class="pagenum"><a name="Page_58" id="Page_58">[Pg 58]</a></span> +truly <i>arithmetical</i>, since they consist of determinations +of values, are not ordinarily classed as such, because +we are accustomed to treat them only as incidental +in the midst of a body of analytical researches +more or less elevated, the too high opinion commonly +formed of the influence of signs being again the principal +cause of this confusion of ideas. Thus not only the +construction of a table of logarithms, but also the calculation +of trigonometrical tables, are true arithmetical operations +of a higher kind. We may also cite as being +in the same class, although in a very distinct and more +elevated order, all the methods by which we determine +directly the value of any function for each particular system +of values attributed to the quantities on which it depends, +when we cannot express in general terms the explicit +form of that function. In this point of view the +<i>numerical</i> solution of questions which we cannot resolve +algebraically, and even the calculation of "Definite Integrals," +whose general integrals we do not know, really +make a part, in spite of all appearances, of the domain +of <i>arithmetic</i>, in which we must necessarily comprise all +that which has for its object the <i>determination of the +values of functions</i>. The considerations relative to this +object are, in fact, constantly homogeneous, whatever the +<i>determinations</i> in question, and are always very distinct +from truly <i>algebraic</i> considerations.</p> + +<p>To complete a just idea of the real extent of the calculus +of values, we must include in it likewise that part +of the general science of the calculus which now bears +the name of the <i>Theory of Numbers</i>, and which is yet +so little advanced. This branch, very extensive by its +nature, but whose importance in the general system of<span class="pagenum"><a name="Page_59" id="Page_59">[Pg 59]</a></span> +science is not very great, has for its object the discovery +of the properties inherent in different numbers by virtue +of their values, and independent of any particular system +of numeration. It forms, then, a sort of <i>transcendental +arithmetic</i>; and to it would really apply the definition +proposed by Newton for algebra.</p> + +<p>The entire domain of arithmetic is, then, much more +extended than is commonly supposed; but this <i>calculus +of values</i> will still never be more than a point, so to +speak, in comparison with the <i>calculus of functions</i>, of +which mathematical science essentially consists. This +comparative estimate will be still more apparent from +some considerations which I have now to indicate respecting +the true nature of arithmetical questions in general, +when they are more profoundly examined.</p> + + +<p><i>Its true Nature.</i> In seeking to determine with precision +in what <i>determinations of values</i> properly consist, +we easily recognize that they are nothing else but veritable +<i>transformations</i> of the functions to be valued; +transformations which, in spite of their special end, are +none the less essentially of the same nature as all those +taught by analysis. In this point of view, the <i>calculus +of values</i> might be simply conceived as an appendix, and +a particular application of the <i>calculus of functions</i>, so +that <i>arithmetic</i> would disappear, so to say, as a distinct +section in the whole body of abstract mathematics.</p> + +<p>In order thoroughly to comprehend this consideration, +we must observe that, when we propose to determine the +<i>value</i> of an unknown number whose mode of formation is +given, it is, by the mere enunciation of the arithmetical +question, already defined and expressed under a certain +form; and that in <i>determining its value</i> we only put its<span class="pagenum"><a name="Page_60" id="Page_60">[Pg 60]</a></span> +expression under another determinate form, to which we +are accustomed to refer the exact notion of each particular +number by making it re-enter into the regular system +of <i>numeration</i>. The determination of values consists +so completely of a simple <i>transformation</i>, that when the +primitive expression of the number is found to be already +conformed to the regular system of numeration, there +is no longer any determination of value, properly speaking, +or, rather, the question is answered by the question +itself. Let the question be to add the two numbers <i>one</i> +and <i>twenty</i>, we answer it by merely repeating the enunciation +of the question,<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> and nevertheless we think that +we have <i>determined the value</i> of the sum. This signifies +that in this case the first expression of the function +had no need of being transformed, while it would not be +thus in adding twenty-three and fourteen, for then the +sum would not be immediately expressed in a manner +conformed to the rank which it occupies in the fixed and +general scale of numeration.</p> + +<p>To sum up as comprehensively as possible the preceding +views, we may say, that to determine the <i>value</i> of +a number is nothing else than putting its primitive expression +under the form</p> + +<p> +<i>a</i> + <i>bz</i> + <i>cz<sup>2</sup></i> + <i>dz<sup>3</sup></i> + <i>ez<sup>4</sup></i> . . . . . + <i>pz<sup>m</sup></i>,<br /> +</p> + +<p><i>z</i> being generally equal to 10, and the coefficients <i>a</i>, <i>b</i>, +<i>c</i>, <i>d</i>, &c., being subjected to the conditions of being whole +numbers less than <i>z</i>; capable of becoming equal to zero; +but never negative. Every arithmetical question may +thus be stated as consisting in putting under such a form +<span class="pagenum"><a name="Page_61" id="Page_61">[Pg 61]</a></span>any abstract function whatever of different quantities, +which are supposed to have themselves a similar form +already. We might then see in the different operations +of arithmetic only simple particular cases of certain algebraic +transformations, excepting the special difficulties +belonging to conditions relating to the nature of the coefficients.</p> + +<p>It clearly follows that abstract mathematics is essentially +composed of the <i>Calculus of Functions</i>, which had +been already seen to be its most important, most extended, +and most difficult part. It will henceforth be the exclusive +subject of our analytical investigations. I will +therefore no longer delay on the <i>Calculus of Values</i>, but +pass immediately to the examination of the fundamental +division of the <i>Calculus of Functions</i>.</p> + + + + +<h3><a name="THE_CALCULUS_OF_FUNCTIONS_OR_ALGEBRA" id="THE_CALCULUS_OF_FUNCTIONS_OR_ALGEBRA">THE CALCULUS OF FUNCTIONS, OR ALGEBRA.</a></h3> + + +<p><i>Principle of its Fundamental Division.</i> We have +determined, at the beginning of this chapter, wherein +properly consists the difficulty which we experience in +putting mathematical questions into <i>equations</i>. It is essentially +because of the insufficiency of the very small +number of analytical elements which we possess, that +the relation of the concrete to the abstract is usually so +difficult to establish. Let us endeavour now to appreciate +in a philosophical manner the general process by +which the human mind has succeeded, in so great a number +of important cases, in overcoming this fundamental +obstacle to <i>The establishment of Equations</i>.</p> + + +<p>1. <i>By the Creation of new Functions.</i> In looking at +this important question from the most general point of +view, we are led at once to the conception of one means of<span class="pagenum"><a name="Page_62" id="Page_62">[Pg 62]</a></span> +facilitating the establishment of the equations of phenomena. +Since the principal obstacle in this matter comes +from the too small number of our analytical elements, the +whole question would seem to be reduced to creating +new ones. But this means, though natural, is really +illusory; and though it might be useful, it is certainly +insufficient.</p> + +<p>In fact, the creation of an elementary abstract function, +which shall be veritably new, presents in itself the +greatest difficulties. There is even something contradictory +in such an idea; for a new analytical element +would evidently not fulfil its essential and appropriate +conditions, if we could not immediately <i>determine its +value</i>. Now, on the other hand, how are we to <i>determine +the value</i> of a new function which is truly <i>simple</i>, +that is, which is not formed by a combination of those +already known? That appears almost impossible. The +introduction into analysis of another elementary abstract +function, or rather of another couple of functions (for each +would be always accompanied by its <i>inverse</i>), supposes +then, of necessity, the simultaneous creation of a new +arithmetical operation, which is certainly very difficult.</p> + +<p>If we endeavour to obtain an idea of the means which +the human mind employs for inventing new analytical +elements, by the examination of the procedures by the +aid of which it has actually conceived those which we +already possess, our observations leave us in that respect +in an entire uncertainty, for the artifices which it has +already made use of for that purpose are evidently exhausted. +To convince ourselves of it, let us consider +the last couple of simple functions which has been introduced +into analysis, and at the formation of which we<span class="pagenum"><a name="Page_63" id="Page_63">[Pg 63]</a></span> +have been present, so to speak, namely, the fourth couple; +for, as I have explained, the fifth couple does not strictly +give veritable new analytical elements. The function +<i>a<sup>x</sup></i>, and, consequently, its inverse, have been formed by +conceiving, under a new point of view, a function which +had been a long time known, namely, powers—when the +idea of them had become sufficiently generalized. The +consideration of a power relatively to the variation of its +exponent, instead of to the variation of its base, was sufficient +to give rise to a truly novel simple function, the +variation following then an entirely different route. But +this artifice, as simple as ingenious, can furnish nothing +more; for, in turning over in the same manner all our +present analytical elements, we end in only making them +return into one another.</p> + +<p>We have, then, no idea as to how we could proceed to +the creation of new elementary abstract functions which +would properly satisfy all the necessary conditions. This +is not to say, however, that we have at present attained +the effectual limit established in that respect by the +bounds of our intelligence. It is even certain that the +last special improvements in mathematical analysis have +contributed to extend our resources in that respect, by +introducing within the domain of the calculus certain definite +integrals, which in some respects supply the place +of new simple functions, although they are far from fulfilling +all the necessary conditions, which has prevented +me from inserting them in the table of true analytical +elements. But, on the whole, I think it unquestionable +that the number of these elements cannot increase except +with extreme slowness. It is therefore not from +these sources that the human mind has drawn its most<span class="pagenum"><a name="Page_64" id="Page_64">[Pg 64]</a></span> +powerful means of facilitating, as much as is possible, +the establishment of equations.</p> + + +<p>2. <i>By the Conception of Equations between certain +auxiliary Quantities.</i> This first method being set aside, +there remains evidently but one other: it is, seeing the +impossibility of finding directly the equations between +the quantities under consideration, to seek for corresponding +ones between other auxiliary quantities, connected +with the first according to a certain determinate law, +and from the relation between which we may return to +that between the primitive magnitudes. Such is, in +substance, the eminently fruitful conception, which the +human mind has succeeded in establishing, and which +constitutes its most admirable instrument for the mathematical +explanation of natural phenomena; the <i>analysis</i>, +called <i>transcendental</i>.</p> + +<p>As a general philosophical principle, the auxiliary +quantities, which are introduced in the place of the primitive +magnitudes, or concurrently with them, in order to +facilitate the establishment of equations, might be derived +according to any law whatever from the immediate +elements of the question. This conception has thus a +much more extensive reach than has been commonly attributed +to it by even the most profound geometers. It +is extremely important for us to view it in its whole logical +extent, for it will perhaps be by establishing a general +mode of <i>derivation</i> different from that to which we +have thus far confined ourselves (although it is evidently +very far from being the only possible one) that we shall +one day succeed in essentially perfecting mathematical +analysis as a whole, and consequently in establishing +more powerful means of investigating the laws of nature<span class="pagenum"><a name="Page_65" id="Page_65">[Pg 65]</a></span> +than our present processes, which are unquestionably susceptible +of becoming exhausted.</p> + +<p>But, regarding merely the present constitution of the +science, the only auxiliary quantities habitually introduced +in the place of the primitive quantities in the +<i>Transcendental Analysis</i> are what are called, 1<sup>o</sup>, <i>infinitely +small</i> elements, the <i>differentials</i> (of different orders) +of those quantities, if we regard this analysis in the +manner of <span class="smcap">Leibnitz</span>; or, 2<sup>o</sup>, the <i>fluxions</i>, the limits of +the ratios of the simultaneous increments of the primitive +quantities compared with one another, or, more +briefly, the <i>prime and ultimate ratios</i> of these increments, +if we adopt the conception of <span class="smcap">Newton</span>; or, 3<sup>o</sup>, +the <i>derivatives</i>, properly so called, of those quantities, +that is, the coefficients of the different terms of their respective +increments, according to the conception of <span class="smcap">Lagrange</span>.</p> + +<p>These three principal methods of viewing our present +transcendental analysis, and all the other less distinctly +characterized ones which have been successively proposed, +are, by their nature, necessarily identical, whether +in the calculation or in the application, as will be explained +in a general manner in the third chapter. As to +their relative value, we shall there see that the conception +of Leibnitz has thus far, in practice, an incontestable +superiority, but that its logical character is exceedingly +vicious; while that the conception of Lagrange, +admirable by its simplicity, by its logical perfection, by +the philosophical unity which it has established in mathematical +analysis (till then separated into two almost entirely +independent worlds), presents, as yet, serious inconveniences +in the applications, by retarding the progress<span class="pagenum"><a name="Page_66" id="Page_66">[Pg 66]</a></span> +of the mind. The conception of Newton occupies nearly +middle ground in these various relations, being less rapid, +but more rational than that of Leibnitz; less philosophical, +but more applicable than that of Lagrange.</p> + +<p>This is not the place to explain the advantages of the +introduction of this kind of auxiliary quantities in the +place of the primitive magnitudes. The third chapter +is devoted to this subject. At present I limit myself to +consider this conception in the most general manner, in +order to deduce therefrom the fundamental division of +the <i>calculus of functions</i> into two systems essentially +distinct, whose dependence, for the complete solution of +any one mathematical question, is invariably determinate.</p> + +<p>In this connexion, and in the logical order of ideas, +the transcendental analysis presents itself as being necessarily +the first, since its general object is to facilitate +the establishment of equations, an operation which must +evidently precede the <i>resolution</i> of those equations, which +is the object of the ordinary analysis. But though it is +exceedingly important to conceive in this way the true +relations of these two systems of analysis, it is none the +less proper, in conformity with the regular usage, to +study the transcendental analysis after ordinary analysis; +for though the former is, at bottom, by itself logically +independent of the latter, or, at least, may be essentially +disengaged from it, yet it is clear that, since +its employment in the solution of questions has always +more or less need of being completed by the use of the +ordinary analysis, we would be constrained to leave the +questions in suspense if this latter had not been previously +studied.</p><p><span class="pagenum"><a name="Page_67" id="Page_67">[Pg 67]</a></span></p> + + +<p><i>Corresponding Divisions of the Calculus of Functions.</i> +It follows from the preceding considerations that +the <i>Calculus of Functions</i>, or <i>Algebra</i> (taking this word +in its most extended meaning), is composed of two distinct +fundamental branches, one of which has for its immediate +object the <i>resolution</i> of equations, when they +are directly established between the magnitudes themselves +which are under consideration; and the other, +starting from equations (generally much easier to form) +between quantities indirectly connected with those of +the problem, has for its peculiar and constant destination +the deduction, by invariable analytical methods, of +the corresponding equations between the direct magnitudes +which we are considering; which brings the question +within the domain of the preceding calculus.</p> + +<p>The former calculus bears most frequently the name +of <i>Ordinary Analysis</i>, or of <i>Algebra</i>, properly so called. +The second constitutes what is called the <i>Transcendental +Analysis</i>, which has been designated by the different +denominations of <i>Infinitesimal Calculus</i>, <i>Calculus of +Fluxions and of Fluents</i>, <i>Calculus of Vanishing Quantities</i>, +the <i>Differential and Integral Calculus</i>, &c., according +to the point of view in which it has been conceived.</p> + +<p>In order to remove every foreign consideration, I will +propose to name it <span class="smcap">Calculus of Indirect Functions</span>, giving +to ordinary analysis the title of <span class="smcap">Calculus of Direct +Functions</span>. These expressions, which I form essentially +by generalizing and epitomizing the ideas of Lagrange, +are simply intended to indicate with precision the true +general character belonging to each of these two forms +of analysis.</p><p><span class="pagenum"><a name="Page_68" id="Page_68">[Pg 68]</a></span></p> + +<p>Having now established the fundamental division of +mathematical analysis, I have next to consider separately +each of its two parts, commencing with the <i>Calculus +of Direct Functions</i>, and reserving more extended developments +for the different branches of the <i>Calculus of +Indirect Functions</i>.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_69" id="Page_69">[Pg 69]</a></span></p> + + + + +<h2><a name="CHAPTER_II" id="CHAPTER_II">CHAPTER II.</a></h2> + +<h3>ORDINARY ANALYSIS, OR ALGEBRA.</h3> + + +<p>The <i>Calculus of direct Functions</i>, or <i>Algebra</i>, is (as +was shown at the end of the preceding chapter) entirely +sufficient for the solution of mathematical questions, when +they are so simple that we can form directly the equations +between the magnitudes themselves which we are +considering, without its being necessary to introduce in +their place, or conjointly with them, any system of auxiliary +quantities <i>derived</i> from the first. It is true that +in the greatest number of important cases its use requires +to be preceded and prepared by that of the <i>Calculus +of indirect Functions</i>, which is intended to facilitate +the establishment of equations. But, although algebra +has then only a secondary office to perform, it has +none the less a necessary part in the complete solution +of the question, so that the <i>Calculus of direct Functions</i> +must continue to be, by its nature, the fundamental base +of all mathematical analysis. We must therefore, before +going any further, consider in a general manner the logical +composition of this calculus, and the degree of development +to which it has at the present day arrived.</p> + + +<p><i>Its Object.</i> The final object of this calculus being the +<i>resolution</i> (properly so called) of <i>equations</i>, that is, the +discovery of the manner in which the unknown quantities +are formed from the known quantities, in accordance +with the <i>equations</i> which exist between them, it +naturally presents as many different departments as we<span class="pagenum"><a name="Page_70" id="Page_70">[Pg 70]</a></span> +can conceive truly distinct classes of equations. Its appropriate +extent is consequently rigorously indefinite, the +number of analytical functions susceptible of entering +into equations being in itself quite unlimited, although +they are composed of only a very small number of primitive +elements.</p> + + +<p><i>Classification of Equations.</i> The rational classification +of equations must evidently be determined by the +nature of the analytical elements of which their numbers +are composed; every other classification would be essentially +arbitrary. Accordingly, analysts begin by dividing +equations with one or more variables into two principal +classes, according as they contain functions of only +the first three couples (see the table in chapter i., page +51), or as they include also exponential or circular functions. +The names of <i>Algebraic</i> functions and <i>Transcendental</i> +functions, commonly given to these two principal +groups of analytical elements, are undoubtedly very inappropriate. +But the universally established division between +the corresponding equations is none the less very +real in this sense, that the resolution of equations containing +the functions called <i>transcendental</i> necessarily +presents more difficulties than those of the equations +called <i>algebraic</i>. Hence the study of the former is as +yet exceedingly imperfect, so that frequently the resolution +of the most simple of them is still unknown to us,<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a> +and our analytical methods have almost exclusive reference +to the elaboration of the latter.</p> +<p><span class="pagenum"><a name="Page_71" id="Page_71">[Pg 71]</a></span></p> + + + +<h3><a name="ALGEBRAIC_EQUATIONS" id="ALGEBRAIC_EQUATIONS">ALGEBRAIC EQUATIONS.</a></h3> + + +<p>Considering now only these <i>Algebraic</i> equations, we +must observe, in the first place, that although they may +often contain <i>irrational</i> functions of the unknown quantities +as well as <i>rational</i> functions, we can always, by +more or less easy transformations, make the first case +come under the second, so that it is with this last that +analysts have had to occupy themselves exclusively in +order to resolve all sorts of <i>algebraic</i> equations.</p> + + +<p><i>Their Classification.</i> In the infancy of algebra, these +equations were classed according to the number of their +terms. But this classification was evidently faulty, since +it separated cases which were really similar, and brought +together others which had nothing in common besides this +unimportant characteristic.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a> It has been retained only +for equations with two terms, which are, in fact, capable +of being resolved in a manner peculiar to themselves.</p> + +<p>The classification of equations by what is called their +<i>degrees</i>, is, on the other hand, eminently natural, for this +distinction rigorously determines the greater or less difficulty +of their <i>resolution</i>. This gradation is apparent +in the cases of all the equations which can be resolved; +but it may be indicated in a general manner independently +of the fact of the resolution. We need only consider +that the most general equation of each degree necessarily +comprehends all those of the different inferior degrees, +as must also the formula which determines the unknown +quantity. Consequently, however slight we may +suppose the difficulty peculiar to the <i>degree</i> which we +<span class="pagenum"><a name="Page_72" id="Page_72">[Pg 72]</a></span>are considering, since it is inevitably complicated in the +execution with those presented by all the preceding degrees, +the resolution really offers more and more obstacles, +in proportion as the degree of the equation is elevated.</p> + + + + +<h3><a name="ALGEBRAIC_RESOLUTION_OF_EQUATIONS" id="ALGEBRAIC_RESOLUTION_OF_EQUATIONS">ALGEBRAIC RESOLUTION OF EQUATIONS.</a></h3> + + +<p><i>Its Limits.</i> The resolution of algebraic equations is +as yet known to us only in the four first degrees, such +is the increase of difficulty noticed above. In this respect, +algebra has made no considerable progress since +the labours of Descartes and the Italian analysts of the +sixteenth century, although in the last two centuries +there has been perhaps scarcely a single geometer who +has not busied himself in trying to advance the resolution +of equations. The general equation of the fifth degree +itself has thus far resisted all attacks.</p> + +<p>The constantly increasing complication which the +formulas for resolving equations must necessarily present, +in proportion as the degree increases (the difficulty +of using the formula of the fourth degree rendering it almost +inapplicable), has determined analysts to renounce, +by a tacit agreement, the pursuit of such researches, although +they are far from regarding it as impossible to +obtain the resolution of equations of the fifth degree, and +of several other higher ones.</p> + + +<p><i>General Solution.</i> The only question of this kind +which would be really of great importance, at least in +its logical relations, would be the general resolution of +algebraic equations of any degree whatsoever. Now, +the more we meditate on this subject, the more we are +led to think, with Lagrange, that it really surpasses the +scope of our intelligence. We must besides observe that<span class="pagenum"><a name="Page_73" id="Page_73">[Pg 73]</a></span> +the formula which would express the <i>root</i> of an equation +of the <i>m<sup>th</sup></i> degree would necessarily include radicals of +the <i>m<sup>th</sup></i> order (or functions of an equivalent multiplicity), +because of the <i>m</i> determinations which it must admit. +Since we have seen, besides, that this formula +must also embrace, as a particular case, that formula +which corresponds to every lower degree, it follows that +it would inevitably also contain radicals of the next +lower degree, the next lower to that, &c., so that, even +if it were possible to discover it, it would almost always +present too great a complication to be capable of being +usefully employed, unless we could succeed in simplifying +it, at the same time retaining all its generality, by +the introduction of a new class of analytical elements of +which we yet have no idea. We have, then, reason to +believe that, without having already here arrived at the +limits imposed by the feeble extent of our intelligence, +we should not be long in reaching them if we actively +and earnestly prolonged this series of investigations.</p> + +<p>It is, besides, important to observe that, even supposing +we had obtained the resolution of <i>algebraic</i> equations +of any degree whatever, we would still have treated +only a very small part of <i>algebra</i>, properly so called, +that is, of the calculus of direct functions, including the +resolution of all the equations which can be formed by +the known analytical functions.</p> + +<p>Finally, we must remember that, by an undeniable +law of human nature, our means for conceiving new +questions being much more powerful than our resources +for resolving them, or, in other words, the human mind +being much more ready to inquire than to reason, we +shall necessarily always remain <i>below</i> the difficulty, no<span class="pagenum"><a name="Page_74" id="Page_74">[Pg 74]</a></span> +matter to what degree of development our intellectual +labour may arrive. Thus, even though we should some +day discover the complete resolution of all the analytical +equations at present known, chimerical as the supposition +is, there can be no doubt that, before attaining this +end, and probably even as a subsidiary means, we would +have already overcome the difficulty (a much smaller one, +though still very great) of conceiving new analytical elements, +the introduction of which would give rise to classes +of equations of which, at present, we are completely +ignorant; so that a similar imperfection in algebraic science +would be continually reproduced, in spite of the real +and very important increase of the absolute mass of our +knowledge.</p> + + +<p><i>What we know in Algebra.</i> In the present condition +of algebra, the complete resolution of the equations +of the first four degrees, of any binomial equations, of +certain particular equations of the higher degrees, and of +a very small number of exponential, logarithmic, or circular +equations, constitute the fundamental methods +which are presented by the calculus of direct functions +for the solution of mathematical problems. But, limited +as these elements are, geometers have nevertheless succeeded +in treating, in a truly admirable manner, a very +great number of important questions, as we shall find in +the course of the volume. The general improvements +introduced within a century into the total system of +mathematical analysis, have had for their principal object +to make immeasurably useful this little knowledge +which we have, instead of tending to increase it. This +result has been so fully obtained, that most frequently +this calculus has no real share in the complete solution<span class="pagenum"><a name="Page_75" id="Page_75">[Pg 75]</a></span> +of the question, except by its most simple parts; those +which have reference to equations of the two first degrees, +with one or more variables.</p> + + + + +<h3><a name="NUMERICAL_RESOLUTION_OF_EQUATIONS" id="NUMERICAL_RESOLUTION_OF_EQUATIONS">NUMERICAL RESOLUTION OF EQUATIONS.</a></h3> + + +<p>The extreme imperfection of algebra, with respect to +the resolution of equations, has led analysts to occupy +themselves with a new class of questions, whose true +character should be here noted. They have busied themselves +in filling up the immense gap in the resolution of +algebraic equations of the higher degrees, by what they +have named the <i>numerical resolution</i> of equations. Not +being able to obtain, in general, the <i>formula</i> which expresses +what explicit function of the given quantities the +unknown one is, they have sought (in the absence of this +kind of resolution, the only one really <i>algebraic</i>) to determine, +independently of that formula, at least the <i>value</i> +of each unknown quantity, for various designated systems +of particular values attributed to the given quantities. +By the successive labours of analysts, this incomplete +and illegitimate operation, which presents an +intimate mixture of truly algebraic questions with others +which are purely arithmetical, has been rendered possible +in all cases for equations of any degree and even of +any form. The methods for this which we now possess +are sufficiently general, although the calculations to which +they lead are often so complicated as to render it almost +impossible to execute them. We have nothing else to +do, then, in this part of algebra, but to simplify the methods +sufficiently to render them regularly applicable, which +we may hope hereafter to effect. In this condition of +the calculus of direct functions, we endeavour, in its application,<span class="pagenum"><a name="Page_76" id="Page_76">[Pg 76]</a></span> +so to dispose the proposed questions as finally to +require only this numerical resolution of the equations.</p> + + +<p><i>Its limited Usefulness.</i> Valuable as is such a resource +in the absence of the veritable solution, it is essential +not to misconceive the true character of these +methods, which analysts rightly regard as a very imperfect +algebra. In fact, we are far from being always able +to reduce our mathematical questions to depend finally +upon only the <i>numerical</i> resolution of equations; that +can be done only for questions quite isolated or truly +final, that is, for the smallest number. Most questions, +in fact, are only preparatory, and intended to serve as an +indispensable preparation for the solution of other questions. +Now, for such an object, it is evident that it is +not the actual <i>value</i> of the unknown quantity which it +is important to discover, but the <i>formula</i>, which shows +how it is derived from the other quantities under consideration. +It is this which happens, for example, in a +very extensive class of cases, whenever a certain question +includes at the same time several unknown quantities. +We have then, first of all, to separate them. By +suitably employing the simple and general method so +happily invented by analysts, and which consists in referring +all the other unknown quantities to one of them, +the difficulty would always disappear if we knew how to +obtain the algebraic resolution of the equations under +consideration, while the <i>numerical</i> solution would then +be perfectly useless. It is only for want of knowing the +<i>algebraic</i> resolution of equations with a single unknown +quantity, that we are obliged to treat <i>Elimination</i> as a +distinct question, which forms one of the greatest special +difficulties of common algebra. Laborious as are the<span class="pagenum"><a name="Page_77" id="Page_77">[Pg 77]</a></span> +methods by the aid of which we overcome this difficulty, +they are not even applicable, in an entirely general manner, +to the elimination of one unknown quantity between +two equations of any form whatever.</p> + +<p>In the most simple questions, and when we have really +to resolve only a single equation with a single unknown +quantity, this <i>numerical</i> resolution is none the less a +very imperfect method, even when it is strictly sufficient. +It presents, in fact, this serious inconvenience of obliging +us to repeat the whole series of operations for the slightest +change which may take place in a single one of the +quantities considered, although their relations to one another +remain unchanged; the calculations made for one +case not enabling us to dispense with any of those which +relate to a case very slightly different. This happens because +of our inability to abstract and treat separately +that purely algebraic part of the question which is common +to all the cases which result from the mere variation +of the given numbers.</p> + +<p>According to the preceding considerations, the calculus +of direct functions, viewed in its present state, divides +into two very distinct branches, according as its +subject is the <i>algebraic</i> resolution of equations or their +<i>numerical</i> resolution. The first department, the only +one truly satisfactory, is unhappily very limited, and will +probably always remain so; the second, too often insufficient, +has, at least, the advantage of a much greater +generality. The necessity of clearly distinguishing these +two parts is evident, because of the essentially different +object proposed in each, and consequently the peculiar +point of view under which quantities are therein considered.</p><p><span class="pagenum"><a name="Page_78" id="Page_78">[Pg 78]</a></span></p> + + +<p><i>Different Divisions of the two Methods of Resolution.</i> +If, moreover, we consider these parts with reference +to the different methods of which each is composed, +we find in their logical distribution an entirely different +arrangement. In fact, the first part must be divided +according to the nature of the equations which we are +able to resolve, and independently of every consideration +relative to the <i>values</i> of the unknown quantities. In +the second part, on the contrary, it is not according to +the <i>degrees</i> of the equations that the methods are naturally +distinguished, since they are applicable to equations +of any degree whatever; it is according to the numerical +character of the <i>values</i> of the unknown quantities; +for, in calculating these numbers directly, without deducing +them from general formulas, different means would +evidently be employed when the numbers are not susceptible +of having their values determined otherwise than +by a series of approximations, always incomplete, or when +they can be obtained with entire exactness. This distinction +of <i>incommensurable</i> and of <i>commensurable</i> roots, +which require quite different principles for their determination, +important as it is in the numerical resolution of +equations, is entirely insignificant in the algebraic resolution, +in which the <i>rational</i> or <i>irrational</i> nature of the +numbers which are obtained is a mere accident of the +calculation, which cannot exercise any influence over the +methods employed; it is, in a word, a simple arithmetical +consideration. We may say as much, though in a less +degree, of the division of the commensurable roots themselves +into <i>entire</i> and <i>fractional</i>. In fine, the case is +the same, in a still greater degree, with the most general +classification of roots, as <i>real</i> and <i>imaginary</i>. All<span class="pagenum"><a name="Page_79" id="Page_79">[Pg 79]</a></span> +these different considerations, which are preponderant as +to the numerical resolution of equations, and which are +of no importance in their algebraic resolution, render more +and more sensible the essentially distinct nature of these +two principal parts of algebra.</p> + + + + +<h3><a name="THE_THEORY_OF_EQUATIONS" id="THE_THEORY_OF_EQUATIONS">THE THEORY OF EQUATIONS.</a></h3> + + +<p>These two departments, which constitute the immediate +object of the calculus of direct functions, are subordinate +to a third one, purely speculative, from which both +of them borrow their most powerful resources, and which +has been very exactly designated by the general name +of <i>Theory of Equations</i>, although it as yet relates only +to <i>Algebraic</i> equations. The numerical resolution of +equations, because of its generality, has special need of +this rational foundation.</p> + +<p>This last and important branch of algebra is naturally +divided into two orders of questions, viz., those which refer +to the <i>composition</i> of equations, and those which concern +their <i>transformation</i>; these latter having for their +object to modify the roots of an equation without knowing +them, in accordance with any given law, providing +that this law is uniform in relation to all the parts.<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a></p> +<p><span class="pagenum"><a name="Page_80" id="Page_80">[Pg 80]</a></span></p> + + + +<h3><a name="THE_METHOD_OF_INDETERMINATE_COEFFICIENTS" id="THE_METHOD_OF_INDETERMINATE_COEFFICIENTS">THE METHOD OF INDETERMINATE COEFFICIENTS.</a></h3> + + +<p>To complete this rapid general enumeration of the different +essential parts of the calculus of direct functions, +I must, lastly, mention expressly one of the most fruitful +and important theories of algebra proper, that relating +to the transformation of functions into series by the aid +of what is called the <i>Method of indeterminate Coefficients</i>. +This method, so eminently analytical, and which +must be regarded as one of the most remarkable discoveries +of Descartes, has undoubtedly lost some of its importance +since the invention and the development of the +infinitesimal calculus, the place of which it might so happily +take in some particular respects. But the increasing +extension of the transcendental analysis, although it +has rendered this method much less necessary, has, on +the other hand, multiplied its applications and enlarged +its resources; so that by the useful combination between +the two theories, which has finally been effected, the use +of the method of indeterminate coefficients has become +at present much more extensive than it was even before +the formation of the calculus of indirect functions.</p> +<p><span class="pagenum"><a name="Page_81" id="Page_81">[Pg 81]</a></span></p> +<hr class="tb" /> + +<p>Having thus sketched the general outlines of algebra +proper, I have now to offer some considerations on several +leading points in the calculus of direct functions, our +ideas of which may be advantageously made more clear +by a philosophical examination.</p> + + + + +<h3><a name="IMAGINARY_QUANTITIES" id="IMAGINARY_QUANTITIES">IMAGINARY QUANTITIES.</a></h3> + + +<p>The difficulties connected with several peculiar symbols +to which algebraic calculations sometimes lead, and +especially to the expressions called <i>imaginary</i>, have been, +I think, much exaggerated through purely metaphysical +considerations, which have been forced upon them, in the +place of regarding these abnormal results in their true +point of view as simple analytical facts. Viewing them +thus, we readily see that, since the spirit of mathematical +analysis consists in considering magnitudes in reference +to their relations only, and without any regard to +their determinate value, analysts are obliged to admit indifferently +every kind of expression which can be engendered +by algebraic combinations. The interdiction of +even one expression because of its apparent singularity +would destroy the generality of their conceptions. The +common embarrassment on this subject seems to me to +proceed essentially from an unconscious confusion between +the idea of <i>function</i> and the idea of <i>value</i>, or, what +comes to the same thing, between the <i>algebraic</i> and the +<i>arithmetical</i> point of view. A thorough examination +would show mathematical analysis to be much more clear +in its nature than even mathematicians commonly suppose.</p> + + + + +<h3><a name="NEGATIVE_QUANTITIES" id="NEGATIVE_QUANTITIES">NEGATIVE QUANTITIES.</a></h3> + + +<p>As to negative quantities, which have given rise to so +many misplaced discussions, as irrational as useless, we +must distinguish between their <i>abstract</i> signification and +their <i>concrete</i> interpretation, which have been almost always +confounded up to the present day. Under the first<span class="pagenum"><a name="Page_82" id="Page_82">[Pg 82]</a></span> +point of view, the theory of negative quantities can be +established in a complete manner by a single algebraical +consideration. The necessity of admitting such expressions +is the same as for imaginary quantities, as above +indicated; and their employment as an analytical artifice, +to render the formulas more comprehensive, is a +mechanism of calculation which cannot really give rise +to any serious difficulty. We may therefore regard the +abstract theory of negative quantities as leaving nothing +essential to desire; it presents no obstacles but those inappropriately +introduced by sophistical considerations.</p> + +<p>It is far from being so, however, with their concrete +theory. This consists essentially in that admirable property +of the signs + and-, of representing analytically +the oppositions of directions of which certain magnitudes +are susceptible. This <i>general theorem</i> on the relation +of the concrete to the abstract in mathematics is one of +the most beautiful discoveries which we owe to the genius +of Descartes, who obtained it as a simple result of properly +directed philosophical observation. A great number +of geometers have since striven to establish directly +its general demonstration, but thus far their efforts have +been illusory. Their vain metaphysical considerations +and heterogeneous minglings of the abstract and the +concrete have so confused the subject, that it becomes +necessary to here distinctly enunciate the general fact. +It consists in this: if, in any equation whatever, expressing +the relation of certain quantities which are susceptible +of opposition of directions, one or more of those quantities +come to be reckoned in a direction contrary to that +which belonged to them when the equation was first established, +it will not be necessary to form directly a new<span class="pagenum"><a name="Page_83" id="Page_83">[Pg 83]</a></span> +equation for this second state of the phenomena; it will +suffice to change, in the first equation, the sign of each +of the quantities which shall have changed its direction; +and the equation, thus modified, will always rigorously +coincide with that which we would have arrived at in +recommencing to investigate, for this new case, the analytical +law of the phenomenon. The general theorem +consists in this constant and necessary coincidence. Now, +as yet, no one has succeeded in directly proving this; we +have assured ourselves of it only by a great number of +geometrical and mechanical verifications, which are, it +is true, sufficiently multiplied, and especially sufficiently +varied, to prevent any clear mind from having the least +doubt of the exactitude and the generality of this essential +property, but which, in a philosophical point of view, +do not at all dispense with the research for so important +an explanation. The extreme extent of the theorem must +make us comprehend both the fundamental difficulties of +this research and the high utility for the perfecting of +mathematical science which would belong to the general +conception of this great truth. This imperfection of theory, +however, has not prevented geometers from making +the most extensive and the most important use of this +property in all parts of concrete mathematics.</p> + +<p>It follows from the above general enunciation of the +fact, independently of any demonstration, that the property +of which we speak must never be applied to magnitudes +whose directions are continually varying, without +giving rise to a simple opposition of direction; in +that case, the sign with which every result of calculation +is necessarily affected is not susceptible of any concrete +interpretation, and the attempts sometimes made to establish<span class="pagenum"><a name="Page_84" id="Page_84">[Pg 84]</a></span> +one are erroneous. This circumstance occurs, +among other occasions, in the case of a radius vector in +geometry, and diverging forces in mechanics.</p> + + + + +<h3><a name="PRINCIPLE_OF_HOMOGENEITY" id="PRINCIPLE_OF_HOMOGENEITY">PRINCIPLE OF HOMOGENEITY.</a></h3> + + +<p>A second general theorem on the relation of the concrete +to the abstract is that which is ordinarily designated +under the name of <i>Principle of Homogeneity</i>. It +is undoubtedly much less important in its applications +than the preceding, but it particularly merits our attention +as having, by its nature, a still greater extent, +since it is applicable to all phenomena without distinction, +and because of the real utility which it often possesses +for the verification of their analytical laws. I +can, moreover, exhibit a direct and general demonstration +of it which seems to me very simple. It is founded +on this single observation, which is self-evident, that the +exactitude of every relation between any concrete magnitudes +whatsoever is independent of the value of the +<i>units</i> to which they are referred for the purpose of expressing +them in numbers. For example, the relation +which exists between the three sides of a right-angled +triangle is the same, whether they are measured by yards, +or by miles, or by inches.</p> + +<p>It follows from this general consideration, that every +equation which expresses the analytical law of any phenomenon +must possess this property of being in no way +altered, when all the quantities which are found in it +are made to undergo simultaneously the change corresponding +to that which their respective units would +experience. Now this change evidently consists in all +the quantities of each sort becoming at once <i>m</i> times<span class="pagenum"><a name="Page_85" id="Page_85">[Pg 85]</a></span> +smaller, if the unit which corresponds to them becomes +<i>m</i> times greater, or reciprocally. Thus every equation +which represents any concrete relation whatever must +possess this characteristic of remaining the same, when +we make <i>m</i> times greater all the quantities which it contains, +and which express the magnitudes between which +the relation exists; excepting always the numbers which +designate simply the mutual <i>ratios</i> of these different +magnitudes, and which therefore remain invariable during +the change of the units. It is this property which +constitutes the law of Homogeneity in its most extended +signification, that is, of whatever analytical functions the +equations may be composed.</p> + +<p>But most frequently we consider only the cases in +which the functions are such as are called <i>algebraic</i>, +and to which the idea of <i>degree</i> is applicable. In this +case we can give more precision to the general proposition +by determining the analytical character which must +be necessarily presented by the equation, in order that +this property may be verified. It is easy to see, then, +that, by the modification just explained, all the <i>terms</i> of +the first degree, whatever may be their form, rational or +irrational, entire or fractional, will become <i>m</i> times greater; +all those of the second degree, <i>m<sup>2</sup></i> times; those of +the third, <i>m<sup>3</sup></i> times, &c. Thus the terms of the same degree, +however different may be their composition, varying +in the same manner, and the terms of different degrees +varying in an unequal proportion, whatever similarity +there may be in their composition, it will be necessary, +to prevent the equation from being disturbed, +that all the terms which it contains should be of the same +degree. It is in this that properly consists the ordinary<span class="pagenum"><a name="Page_86" id="Page_86">[Pg 86]</a></span> +theorem of <i>Homogeneity</i>, and it is from this circumstance +that the general law has derived its name, which, +however, ceases to be exactly proper for all other functions.</p> + +<p>In order to treat this subject in its whole extent, it is +important to observe an essential condition, to which attention +must be paid in applying this property when the +phenomenon expressed by the equation presents magnitudes +of different natures. Thus it may happen that +the respective units are completely independent of each +other, and then the theorem of Homogeneity will hold +good, either with reference to all the corresponding classes +of quantities, or with regard to only a single one or more +of them. But it will happen on other occasions that the +different units will have fixed relations to one another, +determined by the nature of the question; then it will +be necessary to pay attention to this subordination of +the units in verifying the homogeneity, which will not +exist any longer in a purely algebraic sense, and the +precise form of which will vary according to the nature +of the phenomena. Thus, for example, to fix our ideas, +when, in the analytical expression of geometrical phenomena, +we are considering at once lines, areas, and volumes, +it will be necessary to observe that the three corresponding +units are necessarily so connected with each +other that, according to the subordination generally established +in that respect, when the first becomes <i>m</i> times +greater, the second becomes <i>m<sup>2</sup></i> times, and the third <i>m<sup>3</sup></i> +times. It is with such a modification that homogeneity +will exist in the equations, in which, if they are <i>algebraic</i>, +we will have to estimate the degree of each term +by doubling the exponents of the factors which correspond<span class="pagenum"><a name="Page_87" id="Page_87">[Pg 87]</a></span> +to areas, and tripling those of the factors relating +to volumes.</p> +<p><span class="pagenum"><a name="Page_88" id="Page_88">[Pg 88]</a></span></p> +<hr class="tb" /> + +<p>Such are the principal general considerations relating +to the <i>Calculus of Direct Functions</i>. We have now to +pass to the philosophical examination of the <i>Calculus of +Indirect Functions</i>, the much superior importance and +extent of which claim a fuller development.</p> + + + + +<h2><a name="CHAPTER_III" id="CHAPTER_III">CHAPTER III.</a></h2> + +<h3>TRANSCENDENTAL ANALYSIS:<br /> +DIFFERENT MODES OF VIEWING IT.</h3> + + +<p>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 its actual destination as a part of mathematical +science. This analysis has been presented by +geometers under several points of view, really distinct, +although necessarily equivalent, and leading always to +identical results. They may be reduced to three principal +ones; those of <span class="smcap">Leibnitz</span>, of <span class="smcap">Newton</span>, and of <span class="smcap">Lagrange</span>, +of which all the others are only secondary modifications. +In the present state of science, each of these +three general conceptions offers essential advantages which +pertain to it exclusively, without our having yet succeeded +in constructing a single method uniting all these +different characteristic qualities. This combination will +probably be hereafter effected by some method founded +upon the conception of Lagrange when that important +philosophical labour shall have been accomplished, +the study of the other conceptions will have only a historic +interest; but, until then, the science must be considered +as in only a provisional state, which requires the +simultaneous consideration of all the various modes of +viewing this calculus. Illogical as may appear this multiplicity +of conceptions of one identical subject, still, +without them all, we could form but a very insufficient<span class="pagenum"><a name="Page_89" id="Page_89">[Pg 89]</a></span> +idea of this analysis, whether in itself, or more especially +in relation to its applications. This want of system +in the most important part of mathematical analysis will +not appear strange if we consider, on the one hand, its +great extent and its superior difficulty, and, on the other, +its recent formation.</p> + + + + +<h3><a name="ITS_EARLY_HISTORY" id="ITS_EARLY_HISTORY">ITS EARLY HISTORY.</a></h3> + + +<p>If we had to trace here the systematic history of the +successive formation of the transcendental analysis, it +would be necessary previously to distinguish carefully +from the <i>calculus of indirect functions</i>, properly so called, +the original idea of the <i>infinitesimal method</i>, which +can be conceived by itself, independently of any <i>calculus</i>. +We should see that the first germ of this idea is found +in the procedure constantly employed by the Greek geometers, +under the name of the <i>Method of Exhaustions</i>, +as a means of passing from the properties of straight lines +to those of curves, and consisting essentially in substituting +for the curve the auxiliary consideration of an inscribed +or circumscribed polygon, by means of which they +rose to the curve itself, taking in a suitable manner the +limits of the primitive ratios. Incontestable as is this +filiation of ideas, it would be giving it a greatly exaggerated +importance to see in this method of exhaustions +the real equivalent of our modern methods, as some geometers +have done; for the ancients had no logical and +general means for the determination of these limits, and +this was commonly the greatest difficulty of the question; +so that their solutions were not subjected to abstract +and invariable rules, the uniform application of +which would lead with certainty to the knowledge sought;<span class="pagenum"><a name="Page_90" id="Page_90">[Pg 90]</a></span> +which is, on the contrary, the principal characteristic of +our transcendental analysis. In a word, there still remained +the task of generalizing the conceptions used by +the ancients, and, more especially, by considering it in a +manner purely abstract, of reducing it to a complete system +of calculation, which to them was impossible.</p> + +<p>The first idea which was produced in this new direction +goes back to the great geometer Fermat, whom Lagrange +has justly presented as having blocked out the +direct formation of the transcendental analysis by his +method for the determination of <i>maxima</i> and <i>minima</i>, +and for the finding of <i>tangents</i>, which consisted essentially +in introducing the auxiliary consideration of the +correlative increments of the proposed variables, increments +afterward suppressed as equal to zero when the +equations had undergone certain suitable transformations. +But, although Fermat was the first to conceive +this analysis in a truly abstract manner, it was yet far +from being regularly formed into a general and distinct +calculus having its own notation, and especially freed +from the superfluous consideration of terms which, in the +analysis of Fermat, were finally not taken into the account, +after having nevertheless greatly complicated all +the operations by their presence. This is what Leibnitz +so happily executed, half a century later, after some intermediate +modifications of the ideas of Fermat introduced +by Wallis, and still more by Barrow; and he has +thus been the true creator of the transcendental analysis, +such as we now employ it. This admirable discovery +was so ripe (like all the great conceptions of the +human intellect at the moment of their manifestation), +that Newton, on his side, had arrived, at the same time,<span class="pagenum"><a name="Page_91" id="Page_91">[Pg 91]</a></span> +or a little earlier, at a method exactly equivalent, by +considering this analysis under a very different point of +view, which, although more logical in itself, is really +less adapted to give to the common fundamental method +all the extent and the facility which have been imparted +to it by the ideas of Leibnitz. Finally, Lagrange, putting +aside the heterogeneous considerations which had +guided Leibnitz and Newton, has succeeded in reducing +the transcendental analysis, in its greatest perfection, to +a purely algebraic system, which only wants more aptitude +for its practical applications.</p> + +<p>After this summary glance at the general history of +the transcendental analysis, we will proceed to the dogmatic +exposition of the three principal conceptions, in order +to appreciate exactly their characteristic properties, +and to show the necessary identity of the methods which +are thence derived. Let us begin with that of Leibnitz.</p> + + + + +<h3><a name="METHOD_OF_LEIBNITZ" id="METHOD_OF_LEIBNITZ">METHOD OF LEIBNITZ.</a></h3> + + +<p><i>Infinitely small Elements.</i> This consists in introducing +into the calculus, in order to facilitate the establishment +of equations, the infinitely small elements of which +all the quantities, the relations between which are sought, +are considered to be composed. These elements or <i>differentials</i> +will have certain relations to one another, +which are constantly and necessarily more simple and +easy to discover than those of the primitive quantities, and +by means of which we will be enabled (by a special calculus +having for its peculiar object the elimination of these +auxiliary infinitesimals) to go back to the desired equations, +which it would have been most frequently impossible +to obtain directly. This indirect analysis may have<span class="pagenum"><a name="Page_92" id="Page_92">[Pg 92]</a></span> +different degrees of indirectness; for, when there is too +much difficulty in forming immediately the equation between +the differentials of the magnitudes under consideration, +a second application of the same general artifice +will have to be made, and these differentials be treated, +in their turn, as new primitive quantities, and a relation +be sought between their infinitely small elements (which, +with reference to the final objects of the question, will be +<i>second differentials</i>), and so on; the same transformation +admitting of being repeated any number of times, +on the condition of finally eliminating the constantly increasing +number of infinitesimal quantities introduced as +auxiliaries.</p> + +<p>A person not yet familiar with these considerations +does not perceive at once how the employment of these +auxiliary quantities can facilitate the discovery of the +analytical laws of phenomena; for the infinitely small +increments of the proposed magnitudes being of the same +species with them, it would seem that their relations +should not be obtained with more ease, inasmuch as the +greater or less value of a quantity cannot, in fact, exercise +any influence on an inquiry which is necessarily independent, +by its nature, of every idea of value. But +it is easy, nevertheless, to explain very clearly, and in a +quite general manner, how far the question must be simplified +by such an artifice. For this purpose, it is necessary +to begin by distinguishing <i>different orders</i> of infinitely +small quantities, a very precise idea of which +may be obtained by considering them as being either the +successive powers of the same primitive infinitely small +quantity, or as being quantities which may be regarded +as having finite ratios with these powers; so that, to<span class="pagenum"><a name="Page_93" id="Page_93">[Pg 93]</a></span> +take an example, the second, third, &c., differentials of +any one variable are classed as infinitely small quantities +of the second order, the third, &c., because it is +easy to discover in them finite multiples of the second, +third, &c., powers of a certain first differential. These +preliminary ideas being established, the spirit of the infinitesimal +analysis consists in constantly neglecting the +infinitely small quantities in comparison with finite quantities, +and generally the infinitely small quantities of any +order whatever in comparison with all those of an inferior +order. It is at once apparent how much such a +liberty must facilitate the formation of equations between +the differentials of quantities, since, in the place of these +differentials, we can substitute such other elements as we +may choose, and as will be more simple to consider, only +taking care to conform to this single condition, that the +new elements differ from the preceding ones only by quantities +infinitely small in comparison with them. It is +thus that it will be possible, in geometry, to treat curved +lines as composed of an infinity of rectilinear elements, +curved surfaces as formed of plane elements, and, in mechanics, +variable motions as an infinite series of uniform +motions, succeeding one another at infinitely small intervals +of time.</p> + + +<p><span class="smcap">Examples.</span> Considering the importance of this admirable +conception, I think that I ought here to complete +the illustration of its fundamental character by the summary +indication of some leading examples.</p> + + +<p>1. <i>Tangents.</i> Let it be required to determine, for +each point of a plane curve, the equation of which is +given, the direction of its tangent; a question whose +general solution was the primitive object of the inventors<span class="pagenum"><a name="Page_94" id="Page_94">[Pg 94]</a></span> +of the transcendental analysis. We will consider the +tangent as a secant joining two points infinitely near to +each other; and then, designating by <i>dy</i> and <i>dx</i> the infinitely +small differences of the co-ordinates of those two +points, the elementary principles of geometry will immediately +give the equation <i>t</i> = <i>dy</i>/<i>dx</i> for the trigonometrical +tangent of the angle which is made with the axis of the +abscissas by the desired tangent, this being the most simple +way of fixing its position in a system of rectilinear +co-ordinates. This equation, common to all curves, being +established, the question is reduced to a simple analytical +problem, which will consist in eliminating the infinitesimals +<i>dx</i> and <i>dy</i>, which were introduced as auxiliaries, by +determining in each particular case, by means of the equation +of the proposed curve, the ratio of <i>dy</i> to <i>dx</i>, which will +be constantly done by uniform and very simple methods.</p> + + +<p>2. <i>Rectification of an Arc.</i> In the second place, suppose +that we wish to know the length of the arc of any +curve, considered as a function of the co-ordinates of its extremities. +It would be impossible to establish directly the +equation between this arc s and these co-ordinates, while +it is easy to find the corresponding relation between the +differentials of these different magnitudes. The most simple +theorems of elementary geometry will in fact give at +once, considering the infinitely small arc <i>ds</i> as a right +line, the equations</p> + +<p> +<i>ds<sup>2</sup></i> = <i>dy<sup>2</sup></i> + <i>dx<sup>2</sup></i>, or <i>ds<sup>2</sup></i> = <i>dx<sup>2</sup></i> + <i>dy<sup>2</sup></i> + <i>dz<sup>2</sup></i>,<br /> +</p> + +<p>according as the curve is of single or double curvature. +In either case, the question is now entirely within the +domain of analysis, which, by the elimination of the differentials +(which is the peculiar object of the calculus of<span class="pagenum"><a name="Page_95" id="Page_95">[Pg 95]</a></span> +indirect functions), will carry us back from this relation +to that which exists between the finite quantities themselves +under examination.</p> + + +<p>3. <i>Quadrature of a Curve.</i> It would be the same +with the quadrature of curvilinear areas. If the curve is +a plane one, and referred to rectilinear co-ordinates, we +will conceive the area A comprised between this curve, +the axis of the abscissas, and two extreme co-ordinates, +to increase by an infinitely small quantity <i>d</i>A, as the result +of a corresponding increment of the abscissa. The +relation between these two differentials can be immediately +obtained with the greatest facility by substituting for +the curvilinear element of the proposed area the rectangle +formed by the extreme ordinate and the element of the +abscissa, from which it evidently differs only by an infinitely +small quantity of the second order. This will at +once give, whatever may be the curve, the very simple +differential equation</p> + +<p> +<i>d</i>A = <i>ydx</i>,<br /> +</p> + +<p>from which, when the curve is defined, the calculus of +indirect functions will show how to deduce the finite +equation, which is the immediate object of the problem.</p> + + +<p>4. <i>Velocity in Variable Motion.</i> In like manner, in +Dynamics, when we desire to know the expression for +the velocity acquired at each instant by a body impressed +with a motion varying according to any law, we will +consider the motion as being uniform during an infinitely +small element of the time <i>t</i>, and we will thus immediately +form the differential equation <i>de</i> = <i>vdt</i>, in which +<i>v</i> designates the velocity acquired when the body has +passed over the space <i>e</i>; and thence it will be easy to +deduce, by simple and invariable analytical procedures,<span class="pagenum"><a name="Page_96" id="Page_96">[Pg 96]</a></span> +the formula which would give the velocity in each particular +motion, in accordance with the corresponding relation +between the time and the space; or, reciprocally, +what this relation would be if the mode of variation of +the velocity was supposed to be known, whether with respect +to the space or to the time.</p> + + +<p>5. <i>Distribution of Heat.</i> Lastly, to indicate another +kind of questions, it is by similar steps that we are able, +in the study of thermological phenomena, according to +the happy conception of M. Fourier, to form in a very +simple manner the general differential equation which +expresses the variable distribution of heat in any body +whatever, subjected to any influences, by means of the +single and easily-obtained relation, which represents the +uniform distribution of heat in a right-angled parallelopipedon, +considering (geometrically) every other body as +decomposed into infinitely small elements of a similar +form, and (thermologically) the flow of heat as constant +during an infinitely small element of time. Henceforth, +all the questions which can be presented by abstract thermology +will be reduced, as in geometry and mechanics, +to mere difficulties of analysis, which will always consist +in the elimination of the differentials introduced as auxiliaries +to facilitate the establishment of the equations.</p> + +<p>Examples of such different natures are more than sufficient +to give a clear general idea of the immense scope +of the fundamental conception of the transcendental analysis +as formed by Leibnitz, constituting, as it undoubtedly +does, the most lofty thought to which the human +mind has as yet attained.</p> + +<p>It is evident that this conception was indispensable to +complete the foundation of mathematical science, by enabling<span class="pagenum"><a name="Page_97" id="Page_97">[Pg 97]</a></span> +us to establish, in a broad and fruitful manner, +the relation of the concrete to the abstract. In this respect +it must be regarded as the necessary complement +of the great fundamental idea of Descartes on the general +analytical representation of natural phenomena: an +idea which did not begin to be worthily appreciated and +suitably employed till after the formation of the infinitesimal +analysis, without which it could not produce, +even in geometry, very important results.</p> + + +<p><i>Generality of the Formulas.</i> Besides the admirable +facility which is given by the transcendental analysis for +the investigation of the mathematical laws of all phenomena, +a second fundamental and inherent property, perhaps +as important as the first, is the extreme generality of +the differential formulas, which express in a single equation +each determinate phenomenon, however varied the +subjects in relation to which it is considered. Thus we +see, in the preceding examples, that a single differential +equation gives the tangents of all curves, another their +rectifications, a third their quadratures; and in the same +way, one invariable formula expresses the mathematical +law of every variable motion; and, finally, a single equation +constantly represents the distribution of heat in any +body and for any case. This generality, which is so exceedingly +remarkable, and which is for geometers the +basis of the most elevated considerations, is a fortunate +and necessary consequence of the very spirit of the transcendental +analysis, especially in the conception of Leibnitz. +Thus the infinitesimal analysis has not only furnished +a general method for indirectly forming equations +which it would have been impossible to discover in a direct +manner, but it has also permitted us to consider, for<span class="pagenum"><a name="Page_98" id="Page_98">[Pg 98]</a></span> +the mathematical study of natural phenomena, a new +order of more general laws, which nevertheless present a +clear and precise signification to every mind habituated +to their interpretation. By virtue of this second characteristic +property, the entire system of an immense science, +such as geometry or mechanics, has been condensed +into a small number of analytical formulas, from which +the human mind can deduce, by certain and invariable +rules, the solution of all particular problems.</p> + + +<p><i>Demonstration of the Method.</i> To complete the general +exposition of the conception of Leibnitz, there remains +to be considered the demonstration of the logical +procedure to which it leads, and this, unfortunately, is +the most imperfect part of this beautiful method.</p> + +<p>In the beginning of the infinitesimal analysis, the +most celebrated geometers rightly attached more importance +to extending the immortal discovery of Leibnitz +and multiplying its applications than to rigorously establishing +the logical bases of its operations. They contented +themselves for a long time by answering the objections +of second-rate geometers by the unhoped-for solution +of the most difficult problems; doubtless persuaded +that in mathematical science, much more than in any +other, we may boldly welcome new methods, even when +their rational explanation is imperfect, provided they are +fruitful in results, inasmuch as its much easier and more +numerous verifications would not permit any error to remain +long undiscovered. But this state of things could +not long exist, and it was necessary to go back to the +very foundations of the analysis of Leibnitz in order to +prove, in a perfectly general manner, the rigorous exactitude +of the procedures employed in this method, in spite<span class="pagenum"><a name="Page_99" id="Page_99">[Pg 99]</a></span> +of the apparent infractions of the ordinary rules of reasoning +which it permitted.</p> + +<p>Leibnitz, urged to answer, had presented an explanation +entirely erroneous, saying that he treated infinitely +small quantities as <i>incomparables</i>, and that he neglected +them in comparison with finite quantities, "like grains +of sand in comparison with the sea:" a view which would +have completely changed the nature of his analysis, by +reducing it to a mere approximative calculus, which, under +this point of view, would be radically vicious, since +it would be impossible to foresee, in general, to what degree +the successive operations might increase these first +errors, which could thus evidently attain any amount. +Leibnitz, then, did not see, except in a very confused +manner, the true logical foundations of the analysis which +he had created. His earliest successors limited themselves, +at first, to verifying its exactitude by showing the +conformity of its results, in particular applications, to +those obtained by ordinary algebra or the geometry of the +ancients; reproducing, according to the ancient methods, +so far as they were able, the solutions of some problems after +they had been once obtained by the new method, which +alone was capable of discovering them in the first place.</p> + +<p>When this great question was considered in a more +general manner, geometers, instead of directly attacking +the difficulty, preferred to elude it in some way, as Euler +and D'Alembert, for example, have done, by demonstrating +the necessary and constant conformity of the +conception of Leibnitz, viewed in all its applications, +with other fundamental conceptions of the transcendental +analysis, that of Newton especially, the exactitude of +which was free from any objection. Such a general verification<span class="pagenum"><a name="Page_100" id="Page_100">[Pg 100]</a></span> +is undoubtedly strictly sufficient to dissipate any +uncertainty as to the legitimate employment of the analysis +of Leibnitz. But the infinitesimal method is so important—it +offers still, in almost all its applications, such +a practical superiority over the other general conceptions +which have been successively proposed—that there +would be a real imperfection in the philosophical character +of the science if it could not justify itself, and needed +to be logically founded on considerations of another order, +which would then cease to be employed.</p> + +<p>It was, then, of real importance to establish directly +and in a general manner the necessary rationality of the +infinitesimal method. After various attempts more or +less imperfect, a distinguished geometer, Carnot, presented +at last the true direct logical explanation of the method +of Leibnitz, by showing it to be founded on the principle +of the necessary compensation of errors, this being, +in fact, the precise and luminous manifestation of what +Leibnitz had vaguely and confusedly perceived. Carnot +has thus rendered the science an essential service, although, +as we shall see towards the end of this chapter, +all this logical scaffolding of the infinitesimal method, +properly so called, is very probably susceptible of only a +provisional existence, inasmuch as it is radically vicious +in its nature. Still, we should not fail to notice the +general system of reasoning proposed by Carnot, in order +to directly legitimate the analysis of Leibnitz. Here is +the substance of it:</p> + +<p>In establishing the differential equation of a phenomenon, +we substitute, for the immediate elements of the different +quantities considered, other simpler infinitesimals, +which differ from them infinitely little in comparison<span class="pagenum"><a name="Page_101" id="Page_101">[Pg 101]</a></span> +with them; and this substitution constitutes the principal +artifice of the method of Leibnitz, which without it +would possess no real facility for the formation of equations. +Carnot regards such an hypothesis as really producing +an error in the equation thus obtained, and which +for this reason he calls <i>imperfect</i>; only, it is clear that +this error must be infinitely small. Now, on the other +hand, all the analytical operations, whether of differentiation +or of integration, which are performed upon these +differential equations, in order to raise them to finite +equations by eliminating all the infinitesimals which +have been introduced as auxiliaries, produce as constantly, +by their nature, as is easily seen, other analogous errors, +so that an exact compensation takes place, and the +final equations, in the words of Carnot, become <i>perfect</i>. +Carnot views, as a certain and invariable indication of +the actual establishment of this necessary compensation, +the complete elimination of the various infinitely small +quantities, which is always, in fact, the final object of +all the operations of the transcendental analysis; for if +we have committed no other infractions of the general +rules of reasoning than those thus exacted by the very +nature of the infinitesimal method, the infinitely small +errors thus produced cannot have engendered other than +infinitely small errors in all the equations, and the relations +are necessarily of a rigorous exactitude as soon as +they exist between finite quantities alone, since the only +errors then possible must be finite ones, while none such +can have entered. All this general reasoning is founded +on the conception of infinitesimal quantities, regarded as +indefinitely decreasing, while those from which they are +derived are regarded as fixed.</p><p><span class="pagenum"><a name="Page_102" id="Page_102">[Pg 102]</a></span></p> + + +<p><i>Illustration by Tangents.</i> Thus, to illustrate this abstract +exposition by a single example, let us take up again +the question of <i>tangents</i>, which is the most easy to analyze +completely. We will regard the equation <i>t</i> = <i>dy/dx</i>, +obtained above, as being affected with an infinitely small +error, since it would be perfectly rigorous only for the +secant. Now let us complete the solution by seeking, +according to the equation of each curve, the ratio between +the differentials of the co-ordinates. If we suppose +this equation to be <i>y</i> = <i>ax<sup>2</sup></i>, we shall evidently have</p> + +<p> +<i>dy</i> = 2<i>axdx</i> + <i>adx<sup>2</sup></i>.<br /> +</p> + +<p>In this formula we shall have to neglect the term <i>dx<sup>2</sup></i> +as an infinitely small quantity of the second order. Then +the combination of the two <i>imperfect</i> equations.</p> + +<p> +<i>t</i> = <i>dy/dx</i>, <i>dy</i> = 2<i>ax(dx)</i>,<br /> +</p> + +<p>being sufficient to eliminate entirely the infinitesimals, +the finite result, <i>t</i> = 2<i>ax</i>, will necessarily be rigorously correct, +from the effect of the exact compensation of the two +errors committed; since, by its finite nature, it cannot be +affected by an infinitely small error, and this is, nevertheless, +the only one which it could have, according to +the spirit of the operations which have been executed.</p> + +<p>It would be easy to reproduce in a uniform manner +the same reasoning with reference to all the other general +applications of the analysis of Leibnitz.</p> + +<p>This ingenious theory is undoubtedly more subtile than +solid, when we examine it more profoundly; but it has +really no other radical logical fault than that of the infinitesimal +method itself, of which it is, it seems to me, +the natural development and the general explanation, so<span class="pagenum"><a name="Page_103" id="Page_103">[Pg 103]</a></span> +that it must be adopted for as long a time as it shall be +thought proper to employ this method directly.</p> + +<hr class="tb" /> + +<p>I pass now to the general exposition of the two other +fundamental conceptions of the transcendental analysis, +limiting myself in each to its principal idea, the philosophical +character of the analysis having been sufficiently +determined above in the examination of the conception +of Leibnitz, which I have specially dwelt upon because +it admits of being most easily grasped as a whole, and +most rapidly described.</p> + + + + +<h3><a name="METHOD_OF_NEWTON" id="METHOD_OF_NEWTON">METHOD OF NEWTON.</a></h3> + + +<p>Newton has successively presented his own method of +conceiving the transcendental analysis under several different +forms. That which is at present the most commonly +adopted was designated by Newton, sometimes under +the name of the <i>Method of prime and ultimate Ratios</i>, +sometimes under that of the <i>Method of Limits</i>.</p> + + +<p><i>Method of Limits.</i> The general spirit of the transcendental +analysis, from this point of view, consists in +introducing as auxiliaries, in the place of the primitive +quantities, or concurrently with them, in order to facilitate +the establishment of equations, the <i>limits of the ratios</i> +of the simultaneous increments of these quantities; +or, in other words, the <i>final ratios</i> of these increments; +limits or final ratios which can be easily shown to have +a determinate and finite value. A special calculus, which +is the equivalent of the infinitesimal calculus, is then +employed to pass from the equations between these limits +to the corresponding equations between the primitive +quantities themselves.</p><p><span class="pagenum"><a name="Page_104" id="Page_104">[Pg 104]</a></span></p> + +<p>The power which is given by such an analysis, of expressing +with more ease the mathematical laws of phenomena, +depends in general on this, that since the calculus +applies, not to the increments themselves of the proposed +quantities, but to the limits of the ratios of those +increments, we can always substitute for each increment +any other magnitude more easy to consider, provided that +their final ratio is the ratio of equality, or, in other words, +that the limit of their ratio is unity. It is clear, indeed, +that the calculus of limits would be in no way affected +by this substitution. Starting from this principle, we +find nearly the equivalent of the facilities offered by the +analysis of Leibnitz, which are then merely conceived under +another point of view. Thus curves will be regarded +as the <i>limits</i> of a series of rectilinear polygons, variable +motions as the <i>limits</i> of a collection of uniform motions +of constantly diminishing durations, and so on.</p> + + +<p><span class="smcap">Examples.</span> 1. <i>Tangents.</i> Suppose, for example, that +we wish to determine the direction of the tangent to a +curve; we will regard it as the limit towards which would +tend a secant, which should turn about the given point +so that its second point of intersection should indefinitely +approach the first. Representing the differences of the co-ordinates +of the two points by Δ<i>y</i> and Δ<i>x</i>, we would have +at each instant, for the trigonometrical tangent of the angle +which the secant makes with the axis of abscissas,</p> + +<p> +<i>t</i> = Δ<i>y</i>/Δ<i>x</i>;<br /> +</p> + +<p>from which, taking the limits, we will obtain, relatively +to the tangent itself, this general formula of transcendental +analysis,</p> + +<p> +<i>t</i> = <i>L</i>(Δ<i>y</i>/Δ<i>x</i>),<br /> +</p><p><span class="pagenum"><a name="Page_105" id="Page_105">[Pg 105]</a></span></p> + +<p>the characteristic <i>L</i> being employed to designate the limit. +The calculus of indirect functions will show how to deduce +from this formula in each particular case, when the +equation of the curve is given, the relation between <i>t</i> and +<i>x</i>, by eliminating the auxiliary quantities which have +been introduced. If we suppose, in order to complete the +solution, that the equation of the proposed curve is <i>y</i> = <i>ax<sup>2</sup></i>, +we shall evidently have</p> + +<p> +Δ<i>y</i> = 2<i>ax</i>Δ<i>x</i> + <i>a</i>(Δ<i>x</i>)<sup>2</sup>,<br /> +</p> + +<p>from which we shall obtain</p> + +<p> +Δ<i>y</i>/Δ<i>x</i> = 2<i>ax</i> + <i>a</i>Δ<i>x</i>.<br /> +</p> + +<p>Now it is clear that the <i>limit</i> towards which the second +number tends, in proportion as Δ<i>x</i> diminishes, is 2<i>ax</i>. +We shall therefore find, by this method, <i>t</i> = 2<i>ax</i>, as we +obtained it for the same case by the method of Leibnitz.</p> + +<p>2. <i>Rectifications.</i> In like manner, when the rectification +of a curve is desired, we must substitute for the increment +of the arc s the chord of this increment, which +evidently has such a connexion with it that the limit +of their ratio is unity; and then we find (pursuing in +other respects the same plan as with the method of Leibnitz) +this general equation of rectifications:</p> + +<p> +(<i>L</i>Δ<i>s</i>/Δ<i>x</i>)² = 1 + (<i>L</i>Δ<i>y</i>/Δ<i>x</i>)²,<br /> +or (<i>L</i>Δ<i>s</i>/Δ<i>x</i>)<sup>2</sup> = 1 + (<i>L</i>Δ<i>y</i>/Δ<i>x</i>)<sup>2</sup> + (<i>L</i>Δ<i>z</i>/Δ<i>x</i>)<sup>2</sup>,<br /> +</p> + +<p>according as the curve is plane or of double curvature. +It will now be necessary, for each particular curve, to +pass from this equation to that between the arc and the +abscissa, which depends on the transcendental calculus +properly so called.</p><p><span class="pagenum"><a name="Page_106" id="Page_106">[Pg 106]</a></span></p> + +<p>We could take up, with the same facility, by the +method of limits, all the other general questions, the solution +of which has been already indicated according to the +infinitesimal method.</p> + +<p>Such is, in substance, the conception which Newton +formed for the transcendental analysis, or, more precisely, +that which Maclaurin and D'Alembert have presented +as the most rational basis of that analysis, in seeking to +fix and to arrange the ideas of Newton upon that subject.</p> + + +<p><i>Fluxions and Fluents.</i> Another distinct form under +which Newton has presented this same method should be +here noticed, and deserves particularly to fix our attention, +as much by its ingenious clearness in some cases +as by its having furnished the notation best suited to this +manner of viewing the transcendental analysis, and, moreover, +as having been till lately the special form of the calculus +of indirect functions commonly adopted by the English +geometers. I refer to the calculus of <i>fluxions</i> and +of <i>fluents</i>, founded on the general idea of <i>velocities</i>.</p> + +<p>To facilitate the conception of the fundamental idea, +let us consider every curve as generated by a point impressed +with a motion varying according to any law whatever. +The different quantities which the curve can present, +the abscissa, the ordinate, the arc, the area, &c., +will be regarded as simultaneously produced by successive +degrees during this motion. The <i>velocity</i> with which +each shall have been described will be called the <i>fluxion</i> +of that quantity, which will be inversely named its <i>fluent</i>. +Henceforth the transcendental analysis will consist, +according to this conception, in forming directly the +equations between the fluxions of the proposed quantities, +in order to deduce therefrom, by a special calculus,<span class="pagenum"><a name="Page_107" id="Page_107">[Pg 107]</a></span> +the equations between the fluents themselves. What +has been stated respecting curves may, moreover, evidently +be applied to any magnitudes whatever, regarded, +by the aid of suitable images, as produced by motion.</p> + +<p>It is easy to understand the general and necessary +identity of this method with that of limits complicated +with the foreign idea of motion. In fact, resuming the +case of the curve, if we suppose, as we evidently always +may, that the motion of the describing point is uniform +in a certain direction, that of the abscissa, for example, +then the fluxion of the abscissa will be constant, like the +element of the time; for all the other quantities generated, +the motion cannot be conceived to be uniform, except +for an infinitely small time. Now the velocity being +in general according to its mechanical conception, the +ratio of each space to the time employed in traversing it, +and this time being here proportional to the increment of +the abscissa, it follows that the fluxions of the ordinate, +of the arc, of the area, &c., are really nothing else (rejecting +the intermediate consideration of time) than the +final ratios of the increments of these different quantities +to the increment of the abscissa. This method of fluxions +and fluents is, then, in reality, only a manner of +representing, by a comparison borrowed from mechanics, +the method of prime and ultimate ratios, which alone can +be reduced to a calculus. It evidently, then, offers the +same general advantages in the various principal applications +of the transcendental analysis, without its being +necessary to present special proofs of this.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_108" id="Page_108">[Pg 108]</a></span></p> + + + + +<h3><a name="METHOD_OF_LAGRANGE" id="METHOD_OF_LAGRANGE">METHOD OF LAGRANGE.</a></h3> + + +<p><i>Derived Functions.</i> The conception of Lagrange, +in its admirable simplicity, consists in representing the +transcendental analysis as a great algebraic artifice, by +which, in order to facilitate the establishment of equations, +we introduce, in the place of the primitive functions, +or concurrently with them, their <i>derived</i> functions; +that is, according to the definition of Lagrange, +the coefficient of the first term of the increment of each +function, arranged according to the ascending powers of +the increment of its variable. The special calculus of +indirect functions has for its constant object, here as +well as in the conceptions of Leibnitz and of Newton, to +eliminate these <i>derivatives</i> which have been thus employed +as auxiliaries, in order to deduce from their relations +the corresponding equations between the primitive +magnitudes.</p> + + +<p><i>An Extension of ordinary Analysis.</i> The transcendental +analysis is, then, nothing but a simple though very +considerable extension of ordinary analysis. Geometers +have long been accustomed to introduce in analytical investigations, +in the place of the magnitudes themselves +which they wished to study, their different powers, or +their logarithms, or their sines, &c., in order to simplify +the equations, and even to obtain them more easily. +This successive <i>derivation</i> is an artifice of the same +nature, only of greater extent, and procuring, in consequence, +much more important resources for this common +object.</p> + +<p>But, although we can readily conceive, <i>à priori</i>, that +the auxiliary consideration of these derivatives <i>may</i> facilitate<span class="pagenum"><a name="Page_109" id="Page_109">[Pg 109]</a></span> +the establishment of equations, it is not easy to +explain why this <i>must</i> necessarily follow from this mode +of derivation rather than from any other transformation. +Such is the weak point of the great idea of Lagrange. +The precise advantages of this analysis cannot as yet be +grasped in an abstract manner, but only shown by considering +separately each principal question, so that the +verification is often exceedingly laborious.</p> + + +<p><span class="smcap">Example.</span> <i>Tangents.</i> This manner of conceiving the +transcendental analysis may be best illustrated by its application +to the most simple of the problems above examined—that +of tangents.</p> + +<p>Instead of conceiving the tangent as the prolongation +of the infinitely small element of the curve, according to +the notion of Leibnitz—or as the limit of the secants, according +to the ideas of Newton—Lagrange considers it, +according to its simple geometrical character, analogous +to the definitions of the ancients, to be a right line such +that no other right line can pass through the point of +contact between it and the curve. Then, to determine +its direction, we must seek the general expression of its +distance from the curve, measured in any direction whatever—in +that of the ordinate, for example—and dispose +of the arbitrary constant relating to the inclination of the +right line, which will necessarily enter into that expression, +in such a way as to diminish that separation as much +as possible. Now this distance, being evidently equal +to the difference of the two ordinates of the curve and of +the right line, which correspond to the same new abscissa +<i>x</i> + <i>h</i>, will be represented by the formula</p> + +<p> +(<i>f'</i>(<i>x</i>) - <i>t</i>)<i>h</i> + <i>qh<sup>2</sup></i> + <i>rh<sup>3</sup></i> + etc.,<br /> +</p> + +<p>in which <i>t</i> designates, as above, the unknown trigonometrical<span class="pagenum"><a name="Page_110" id="Page_110">[Pg 110]</a></span> +tangent of the angle which the required line makes +with the axis of abscissas, and <i>f'</i>(<i>x</i>) the derived function +of the ordinate <i>f</i>(<i>x</i>). This being understood, it is easy +to see that, by disposing of <i>t</i> so as to make the first term +of the preceding formula equal to zero, we will render the +interval between the two lines the least possible, so that +any other line for which <i>t</i> did not have the value thus +determined would necessarily depart farther from the proposed +curve. We have, then, for the direction of the tangent +sought, the general expression <i>t</i> = <i>f'</i>(<i>x</i>), a result exactly +equivalent to those furnished by the Infinitesimal +Method and the Method of Limits. We have yet to find +<i>f'</i>(<i>x</i>) in each particular curve, which is a mere question +of analysis, quite identical with those which are presented, +at this stage of the operations, by the other methods.</p> + +<p>After these considerations upon the principal general +conceptions, we need not stop to examine some other theories +proposed, such as Euler's <i>Calculus of Vanishing +Quantities</i>, which are really modifications—more or less +important, and, moreover, no longer used—of the preceding +methods.</p> + +<p>I have now to establish the comparison and the appreciation +of these three fundamental methods. Their <i>perfect +and necessary conformity</i> is first to be proven in a +general manner.</p> + + + + +<h3><a name="FUNDAMENTAL_IDENTITY_OF_THE_THREE_METHODS" id="FUNDAMENTAL_IDENTITY_OF_THE_THREE_METHODS">FUNDAMENTAL IDENTITY OF THE THREE METHODS.</a></h3> + + +<p>It is, in the first place, evident from what precedes, +considering these three methods as to their actual destination, +independently of their preliminary ideas, that +they all consist in the same general logical artifice, which +has been characterized in the first chapter; to wit, the<span class="pagenum"><a name="Page_111" id="Page_111">[Pg 111]</a></span> +introduction of a certain system of auxiliary magnitudes, +having uniform relations to those which are the special +objects of the inquiry, and substituted for them expressly +to facilitate the analytical expression of the mathematical +laws of the phenomena, although they have finally to +be eliminated by the aid of a special calculus. It is +this which has determined me to regularly define the +transcendental analysis as <i>the calculus of indirect functions</i>, +in order to mark its true philosophical character, +at the same time avoiding any discussion upon the best +manner of conceiving and applying it. The general effect +of this analysis, whatever the method employed, is, +then, to bring every mathematical question much more +promptly within the power of the <i>calculus</i>, and thus to +diminish considerably the serious difficulty which is usually +presented by the passage from the concrete to the abstract. +Whatever progress we may make, we can never +hope that the calculus will ever be able to grasp every +question of natural philosophy, geometrical, or mechanical, +or thermological, &c., immediately upon its birth, +which would evidently involve a contradiction. Every +problem will constantly require a certain preliminary labour +to be performed, in which the calculus can be of no +assistance, and which, by its nature, cannot be subjected +to abstract and invariable rules; it is that which has +for its special object the establishment of equations, which +form the indispensable starting point of all analytical researches. +But this preliminary labour has been remarkably +simplified by the creation of the transcendental analysis, +which has thus hastened the moment at which the +solution admits of the uniform and precise application of +general and abstract methods; by reducing, in each case,<span class="pagenum"><a name="Page_112" id="Page_112">[Pg 112]</a></span> +this special labour to the investigation of equations between +the auxiliary magnitudes; from which the calculus +then leads to equations directly referring to the proposed +magnitudes, which, before this admirable conception, it +had been necessary to establish directly and separately. +Whether these indirect equations are <i>differential</i> equations, +according to the idea of Leibnitz, or equations of +<i>limits</i>, conformably to the conception of Newton, or, lastly, +<i>derived</i> equations, according to the theory of Lagrange, +the general procedure is evidently always the same.</p> + +<p>But the coincidence of these three principal methods +is not limited to the common effect which they produce; +it exists, besides, in the very manner of obtaining it. In +fact, not only do all three consider, in the place of the +primitive magnitudes, certain auxiliary ones, but, still +farther, the quantities thus introduced as subsidiary are +exactly identical in the three methods, which consequently +differ only in the manner of viewing them. This +can be easily shown by taking for the general term of +comparison any one of the three conceptions, especially +that of Lagrange, which is the most suitable to serve as +a type, as being the freest from foreign considerations. +Is it not evident, by the very definition of <i>derived functions</i>, +that they are nothing else than what Leibnitz calls +<i>differential coefficients</i>, or the ratios of the differential +of each function to that of the corresponding variable, +since, in determining the first differential, we will be +obliged, by the very nature of the infinitesimal method, +to limit ourselves to taking the only term of the increment +of the function which contains the first power of +the infinitely small increment of the variable? In the +same way, is not the derived function, by its nature,<span class="pagenum"><a name="Page_113" id="Page_113">[Pg 113]</a></span> +likewise the necessary <i>limit</i> towards which tends the ratio +between the increment of the primitive function and +that of its variable, in proportion as this last indefinitely +diminishes, since it evidently expresses what that ratio +becomes when we suppose the increment of the variable +to equal zero? That which is designated by <i>dx</i>/<i>dy</i> in the +method of Leibnitz; that which ought to be noted as +<i>L</i>(Δ<i>y</i>/Δ<i>x</i>) in that of Newton; and that which Lagrange has +indicated by <i>f'</i>(<i>x</i>), is constantly one same function, seen +from three different points of view, the considerations +of Leibnitz and Newton properly consisting in making +known two general necessary properties of the derived +function. The transcendental analysis, examined abstractedly +and in its principle, is then always the same, +whatever may be the conception which is adopted, and +the procedures of the calculus of indirect functions are +necessarily identical in these different methods, which in +like manner must, for any application whatever, lead constantly +to rigorously uniform results.</p> + + + + +<h3><a name="COMPARATIVE_VALUE_OF_THE_THREE_METHODS" id="COMPARATIVE_VALUE_OF_THE_THREE_METHODS">COMPARATIVE VALUE OF THE THREE METHODS.</a></h3> + + +<p>If now we endeavour to estimate the comparative value +of these three equivalent conceptions, we shall find in +each advantages and inconveniences which are peculiar +to it, and which still prevent geometers from confining +themselves to any one of them, considered as final.</p> + + +<p><i>That of Leibnitz.</i> The conception of Leibnitz presents +incontestably, in all its applications, a very marked +superiority, by leading in a much more rapid manner, +and with much less mental effort, to the formation of<span class="pagenum"><a name="Page_114" id="Page_114">[Pg 114]</a></span> +equations between the auxiliary magnitudes. It is to its +use that we owe the high perfection which has been acquired +by all the general theories of geometry and mechanics. +Whatever may be the different speculative +opinions of geometers with respect to the infinitesimal +method, in an abstract point of view, all tacitly agree in +employing it by preference, as soon as they have to treat +a new question, in order not to complicate the necessary +difficulty by this purely artificial obstacle proceeding from +a misplaced obstinacy in adopting a less expeditious course. +Lagrange himself, after having reconstructed the transcendental +analysis on new foundations, has (with that +noble frankness which so well suited his genius) rendered +a striking and decisive homage to the characteristic properties +of the conception of Leibnitz, by following it exclusively +in the entire system of his <i>Méchanique Analytique</i>. +Such a fact renders any comments unnecessary.</p> + +<p>But when we consider the conception of Leibnitz in +itself and in its logical relations, we cannot escape admitting, +with Lagrange, that it is radically vicious in +this, that, adopting its own expressions, the notion of infinitely +small quantities is a <i>false idea</i>, of which it is in +fact impossible to obtain a clear conception, however we +may deceive ourselves in that matter. Even if we adopt +the ingenious idea of the compensation of errors, as above +explained, this involves the radical inconvenience of being +obliged to distinguish in mathematics two classes of reasonings, +those which are perfectly rigorous, and those in +which we designedly commit errors which subsequently +have to be compensated. A conception which leads to +such strange consequences is undoubtedly very unsatisfactory +in a logical point of view.</p><p><span class="pagenum"><a name="Page_115" id="Page_115">[Pg 115]</a></span></p> + +<p>To say, as do some geometers, that it is possible in +every case to reduce the infinitesimal method to that of +limits, the logical character of which is irreproachable, +would evidently be to elude the difficulty rather than to +remove it; besides, such a transformation almost entirely +strips the conception of Leibnitz of its essential advantages +of facility and rapidity.</p> + +<p>Finally, even disregarding the preceding important +considerations, the infinitesimal method would no less +evidently present by its nature the very serious defect of +breaking the unity of abstract mathematics, by creating +a transcendental analysis founded on principles so different +from those which form the basis of the ordinary analysis. +This division of analysis into two worlds almost +entirely independent of each other, tends to hinder the +formation of truly general analytical conceptions. To +fully appreciate the consequences of this, we should have +to go back to the state of the science before Lagrange +had established a general and complete harmony between +these two great sections.</p> + + +<p><i>That of Newton.</i> Passing now to the conception of +Newton, it is evident that by its nature it is not exposed +to the fundamental logical objections which are called +forth by the method of Leibnitz. The notion of <i>limits</i> +is, in fact, remarkable for its simplicity and its precision. +In the transcendental analysis presented in this manner, +the equations are regarded as exact from their very origin, +and the general rules of reasoning are as constantly +observed as in ordinary analysis. But, on the other +hand, it is very far from offering such powerful resources +for the solution of problems as the infinitesimal method. +The obligation which it imposes, of never considering<span class="pagenum"><a name="Page_116" id="Page_116">[Pg 116]</a></span> +the increments of magnitudes separately and by themselves, +nor even in their ratios, but only in the limits of +those ratios, retards considerably the operations of the +mind in the formation of auxiliary equations. We may +even say that it greatly embarrasses the purely analytical +transformations. Thus the transcendental analysis, +considered separately from its applications, is far from presenting +in this method the extent and the generality which +have been imprinted upon it by the conception of Leibnitz. +It is very difficult, for example, to extend the theory +of Newton to functions of several independent variables. +But it is especially with reference to its applications +that the relative inferiority of this theory is most +strongly marked.</p> + +<p>Several Continental geometers, in adopting the method +of Newton as the more logical basis of the transcendental +analysis, have partially disguised this inferiority by a serious +inconsistency, which consists in applying to this method +the notation invented by Leibnitz for the infinitesimal +method, and which is really appropriate to it alone. +In designating by <i>dy</i>/<i>dx</i> that which logically ought, in the +theory of limits, to be denoted by <i>L</i>(Δ<i>y</i>/Δ<i>x</i>), and in extending +to all the other analytical conceptions this displacement +of signs, they intended, undoubtedly, to combine the special +advantages of the two methods; but, in reality, they +have only succeeded in causing a vicious confusion between +them, a familiarity with which hinders the formation +of clear and exact ideas of either. It would certainly +be singular, considering this usage in itself, that, +by the mere means of signs, it could be possible to effect<span class="pagenum"><a name="Page_117" id="Page_117">[Pg 117]</a></span> +a veritable combination between two theories so distinct +as those under consideration.</p> + +<p>Finally, the method of limits presents also, though in +a less degree, the greater inconvenience, which I have +above noted in reference to the infinitesimal method, of +establishing a total separation between the ordinary and +the transcendental analysis; for the idea of <i>limits</i>, though +clear and rigorous, is none the less in itself, as Lagrange +has remarked, a foreign idea, upon which analytical theories +ought not to be dependent.</p> + + +<p><i>That of Lagrange.</i> This perfect unity of analysis, +and this purely abstract character of its fundamental notions, +are found in the highest degree in the conception +of Lagrange, and are found there alone; it is, for this +reason, the most rational and the most philosophical of +all. Carefully removing every heterogeneous consideration, +Lagrange has reduced the transcendental analysis +to its true peculiar character, that of presenting a very +extensive class of analytical transformations, which facilitate +in a remarkable degree the expression of the conditions +of various problems. At the same time, this analysis +is thus necessarily presented as a simple extension +of ordinary analysis; it is only a higher algebra. All the +different parts of abstract mathematics, previously so incoherent, +have from that moment admitted of being conceived +as forming a single system.</p> + +<p>Unhappily, this conception, which possesses such fundamental +properties, independently of its so simple and +so lucid notation, and which is undoubtedly destined to +become the final theory of transcendental analysis, because +of its high philosophical superiority over all the +other methods proposed, presents in its present state too<span class="pagenum"><a name="Page_118" id="Page_118">[Pg 118]</a></span> +many difficulties in its applications, as compared with the +conception of Newton, and still more with that of Leibnitz, +to be as yet exclusively adopted. Lagrange himself +has succeeded only with great difficulty in rediscovering, +by his method, the principal results already obtained +by the infinitesimal method for the solution of the general +questions of geometry and mechanics; we may judge +from that what obstacles would be found in treating in +the same manner questions which were truly new and +important. It is true that Lagrange, on several occasions, +has shown that difficulties call forth, from men of +genius, superior efforts, capable of leading to the greatest +results. It was thus that, in trying to adapt his method +to the examination of the curvature of lines, which seemed +so far from admitting its application, he arrived at that +beautiful theory of contacts which has so greatly perfected +that important part of geometry. But, in spite +of such happy exceptions, the conception of Lagrange has +nevertheless remained, as a whole, essentially unsuited +to applications.</p> + +<p>The final result of the general comparison which I +have too briefly sketched, is, then, as already suggested, +that, in order to really understand the transcendental analysis, +we should not only consider it in its principles according +to the three fundamental conceptions of Leibnitz, +of Newton, and of Lagrange, but should besides accustom +ourselves to carry out almost indifferently, according +to these three principal methods, and especially +according to the first and the last, the solution of all important +questions, whether of the pure calculus of indirect +functions or of its applications. This is a course which +I could not too strongly recommend to all those who desire<span class="pagenum"><a name="Page_119" id="Page_119">[Pg 119]</a></span> +to judge philosophically of this admirable creation of +the human mind, as well as to those who wish to learn +to make use of this powerful instrument with success and +with facility. In all the other parts of mathematical science, +the consideration of different methods for a single +class of questions may be useful, even independently of +its historical interest, but it is not indispensable; here, +on the contrary, it is strictly necessary.</p> + +<p>Having determined with precision, in this chapter, the +philosophical character of the calculus of indirect functions, +according to the principal fundamental conceptions +of which it admits, we have next to consider, in the following +chapter, the logical division and the general composition +of this calculus.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_120" id="Page_120">[Pg 120]</a></span></p> + + + + +<h2><a name="CHAPTER_IV" id="CHAPTER_IV">CHAPTER IV.</a></h2> + + +<h3>THE DIFFERENTIAL AND INTEGRAL CALCULUS.<br /> +ITS TWO FUNDAMENTAL DIVISIONS.</h3> + + +<p>The <i>calculus of indirect functions</i>, in accordance with +the considerations explained in the preceding chapter, is +necessarily divided into two parts (or, more properly, is +decomposed into two different <i>calculi</i> entirely distinct, +although intimately connected by their nature), according +as it is proposed to find the relations between the +auxiliary magnitudes (the introduction of which constitutes +the general spirit of this calculus) by means of the +relations between the corresponding primitive magnitudes; +or, conversely, to try to discover these direct +equations by means of the indirect equations originally +established. Such is, in fact, constantly the double object +of the transcendental analysis.</p> + +<p>These two systems have received different names, according +to the point of view under which this analysis +has been regarded. The infinitesimal method, properly +so called, having been the most generally employed for +the reasons which have been given, almost all geometers +employ habitually the denominations of <i>Differential +Calculus</i> and of <i>Integral Calculus</i>, established by +Leibnitz, and which are, in fact, very rational consequences +of his conception. Newton, in accordance with +his method, named the first the <i>Calculus of Fluxions</i>, +and the second the <i>Calculus of Fluents</i>, expressions which +were commonly employed in England. Finally, following<span class="pagenum"><a name="Page_121" id="Page_121">[Pg 121]</a></span> +the eminently philosophical theory founded by Lagrange, +one would be called the <i>Calculus of Derived +Functions</i>, and the other the <i>Calculus of Primitive +Functions</i>. I will continue to make use of the terms of +Leibnitz, as being more convenient for the formation of +secondary expressions, although I ought, in accordance +with the suggestions made in the preceding chapter, to +employ concurrently all the different conceptions, approaching +as nearly as possible to that of Lagrange.</p> + + + + +<h3><a name="THEIR_RELATIONS_TO_EACH_OTHER" id="THEIR_RELATIONS_TO_EACH_OTHER">THEIR RELATIONS TO EACH OTHER.</a></h3> + + +<p>The differential calculus is evidently the logical basis +of the integral calculus; for we do not and cannot +know how to integrate directly any other differential expressions +than those produced by the differentiation of +the ten simple functions which constitute the general elements +of our analysis. The art of integration consists, +then, essentially in bringing all the other cases, as far as +is possible, to finally depend on only this small number +of fundamental integrations.</p> + +<p>In considering the whole body of the transcendental +analysis, as I have characterized it in the preceding chapter, +it is not at first apparent what can be the peculiar +utility of the differential calculus, independently of this +necessary relation with the integral calculus, which seems +as if it must be, by itself, the only one directly indispensable. +In fact, the elimination of the <i>infinitesimals</i> or +of the <i>derivatives</i>, introduced as auxiliaries to facilitate +the establishment of equations, constituting, as we have +seen, the final and invariable object of the calculus of indirect +functions, it is natural to think that the calculus +which teaches how to deduce from the equations between<span class="pagenum"><a name="Page_122" id="Page_122">[Pg 122]</a></span> +these auxiliary magnitudes, those which exist between the +primitive magnitudes themselves, ought strictly to suffice +for the general wants of the transcendental analysis without +our perceiving, at the first glance, what special and +constant part the solution of the inverse question can +have in such an analysis. It would be a real error, though +a common one, to assign to the differential calculus, in order +to explain its peculiar, direct, and necessary influence, +the destination of forming the differential equations, from +which the integral calculus then enables us to arrive at +the finite equations; for the primitive formation of differential +equations is not and cannot be, properly speaking, +the object of any calculus, since, on the contrary, it +forms by its nature the indispensable starting point of any +calculus whatever. How, in particular, could the differential +calculus, which in itself is reduced to teaching the +means of <i>differentiating</i> the different equations, be a +general procedure for establishing them? That which +in every application of the transcendental analysis really +facilitates the formation of equations, is the infinitesimal +<i>method</i>, and not the infinitesimal <i>calculus</i>, which is perfectly +distinct from it, although it is its indispensable complement. +Such a consideration would, then, give a false +idea of the special destination which characterizes the differential +calculus in the general system of the transcendental +analysis.</p> + +<p>But we should nevertheless very imperfectly conceive +the real peculiar importance of this first branch of the +calculus of indirect functions, if we saw in it only a simple +preliminary labour, having no other general and essential +object than to prepare indispensable foundations +for the integral calculus. As the ideas on this matter<span class="pagenum"><a name="Page_123" id="Page_123">[Pg 123]</a></span> +are generally confused, I think that I ought here to explain +in a summary manner this important relation as I +view it, and to show that in every application of the +transcendental analysis a primary, direct, and necessary +part is constantly assigned to the differential calculus.</p> + + +<p>1. <i>Use of the Differential Calculus as preparatory +to that of the Integral.</i> In forming the differential equations +of any phenomenon whatever, it is very seldom that +we limit ourselves to introduce differentially only those +magnitudes whose relations are sought. To impose that +condition would be to uselessly diminish the resources +presented by the transcendental analysis for the expression +of the mathematical laws of phenomena. Most frequently +we introduce into the primitive equations, through +their differentials, other magnitudes whose relations are +already known or supposed to be so, and without the +consideration of which it would be frequently impossible +to establish equations. Thus, for example, in the general +problem of the rectification of curves, the differential +equation,</p> + +<p> +<i>ds</i><sup>2</sup> = <i>dy</i><sup>2</sup> + <i>dx</i><sup>2</sup>, or <i>ds</i><sup>2</sup> = <i>dx</i><sup>2</sup> + <i>dy</i><sup>2</sup> + <i>dz</i><sup>2</sup>,<br /> +</p> + +<p>is not only established between the desired function s and +the independent variable <i>x</i>, to which it is referred, but, at +the same time, there have been introduced, as indispensable +intermediaries, the differentials of one or two other +functions, <i>y</i> and <i>z</i>, which are among the data of the +problem; it would not have been possible to form directly +the equation between <i>ds</i> and <i>dx</i>, which would, besides, +be peculiar to each curve considered. It is the same for +most questions. Now in these cases it is evident that +the differential equation is not immediately suitable for +integration. It is previously necessary that the differentials<span class="pagenum"><a name="Page_124" id="Page_124">[Pg 124]</a></span> +of the functions supposed to be known, which +have been employed as intermediaries, should be entirely +eliminated, in order that equations may be obtained between +the differentials of the functions which alone are +sought and those of the really independent variables, after +which the question depends on only the integral calculus. +Now this preparatory elimination of certain differentials, +in order to reduce the infinitesimals to the +smallest number possible, belongs simply to the differential +calculus; for it must evidently be done by determining, +by means of the equations between the functions +supposed to be known, taken as intermediaries, the +relations of their differentials, which is merely a question +of differentiation. Thus, for example, in the case of rectifications, +it will be first necessary to calculate <i>dy</i>, or <i>dy</i> +and <i>dz</i>, by differentiating the equation or the equations +of each curve proposed; after eliminating these expressions, +the general differential formula above enunciated +will then contain only <i>ds</i> and <i>dx</i>; having arrived at this +point, the elimination of the infinitesimals can be completed +only by the integral calculus.</p> + +<p>Such is, then, the general office necessarily belonging +to the differential calculus in the complete solution of the +questions which exact the employment of the transcendental +analysis; to produce, as far as is possible, the elimination +of the infinitesimals, that is, to reduce in each +case the primitive differential equations so that they shall +contain only the differentials of the really independent +variables, and those of the functions sought, by causing +to disappear, by elimination, the differentials of all the +other known functions which may have been taken as intermediaries +at the time of the formation of the differential<span class="pagenum"><a name="Page_125" id="Page_125">[Pg 125]</a></span> +equations of the problem which is under consideration.</p> + + +<p>2. <i>Employment of the Differential Calculus alone.</i> +For certain questions, which, although few in number, +have none the less, as we shall see hereafter, a very great +importance, the magnitudes which are sought enter directly, +and not by their differentials, into the primitive +differential equations, which then contain differentially +only the different known functions employed as intermediaries, +in accordance with the preceding explanation. +These cases are the most favourable of all; for it is evident +that the differential calculus is then entirely sufficient +for the complete elimination of the infinitesimals, +without the question giving rise to any integration. This +is what occurs, for example, in the problem of <i>tangents</i> +in geometry; in that of <i>velocities</i> in mechanics, &c.</p> + + +<p>3. <i>Employment of the Integral Calculus alone.</i> Finally, +some other questions, the number of which is also +very small, but the importance of which is no less great, +present a second exceptional case, which is in its nature +exactly the converse of the preceding. They are those +in which the differential equations are found to be immediately +ready for integration, because they contain, at +their first formation, only the infinitesimals which relate +to the functions sought, or to the really independent variables, +without its being necessary to introduce, differentially, +other functions as intermediaries. If in these +new cases we introduce these last functions, since, by hypothesis, +they will enter directly and not by their differentials, +ordinary algebra will suffice to eliminate them, +and to bring the question to depend on only the integral +calculus. The differential calculus will then have no<span class="pagenum"><a name="Page_126" id="Page_126">[Pg 126]</a></span> +special part in the complete solution of the problem, which +will depend entirely upon the integral calculus. The +general question of <i>quadratures</i> offers an important example +of this, for the differential equation being then +<i>dA = ydx</i>, will become immediately fit for integration as +soon as we shall have eliminated, by means of the equation +of the proposed curve, the intermediary function <i>y</i>, +which does not enter into it differentially. The same +circumstances exist in the problem of <i>cubatures</i>, and in +some others equally important.</p> + + +<p><i>Three classes of Questions hence resulting.</i> As a +general result of the previous considerations, it is then +necessary to divide into three classes the mathematical +questions which require the use of the transcendental +analysis; the <i>first</i> class comprises the problems susceptible +of being entirely resolved by means of the differential +calculus alone, without any need of the integral calculus; +the <i>second</i>, those which are, on the contrary, entirely +dependent upon the integral calculus, without the +differential calculus having any part in their solution; +lastly, in the <i>third</i> and the most extensive, which constitutes +the normal case, the two others being only exceptional, +the differential and the integral calculus have +each in their turn a distinct and necessary part in the +complete solution of the problem, the former making the +primitive differential equations undergo a preparation +which is indispensable for the application of the latter. +Such are exactly their general relations, of which too +indefinite and inexact ideas are generally formed.</p> +<p><span class="pagenum"><a name="Page_127" id="Page_127">[Pg 127]</a></span></p> +<hr class="tb" /> + +<p>Let us now take a general survey of the logical composition +of each calculus, beginning with the differential.</p> + + + + +<h3><a name="THE_DIFFERENTIAL_CALCULUS" id="THE_DIFFERENTIAL_CALCULUS">THE DIFFERENTIAL CALCULUS.</a></h3> + + +<p>In the exposition of the transcendental analysis, it is +customary to intermingle with the purely analytical part +(which reduces itself to the treatment of the abstract +principles of differentiation and integration) the study of +its different principal applications, especially those which +concern geometry. This confusion of ideas, which is a +consequence of the actual manner in which the science +has been developed, presents, in the dogmatic point of +view, serious inconveniences in this respect, that it makes +it difficult properly to conceive either analysis or geometry. +Having to consider here the most rational co-ordination +which is possible, I shall include, in the following +sketch, only the calculus of indirect functions properly +so called, reserving for the portion of this volume +which relates to the philosophical study of <i>concrete</i> mathematics +the general examination of its great geometrical +and mechanical applications.</p> + + +<p><i>Two Cases: explicit and implicit Functions.</i> The +fundamental division of the differential calculus, or of +the general subject of differentiation, consists in distinguishing +two cases, according as the analytical functions +which are to be differentiated are <i>explicit</i> or <i>implicit</i>; +from which flow two parts ordinarily designated by the +names of differentiation <i>of formulas</i> and differentiation +<i>of equations</i>. It is easy to understand, <i>à priori</i>, the +importance of this classification. In fact, such a distinction +would be illusory if the ordinary analysis was +perfect; that is, if we knew how to resolve all equations +algebraically, for then it would be possible to render +every <i>implicit</i> function <i>explicit</i>; and, by differentiating<span class="pagenum"><a name="Page_128" id="Page_128">[Pg 128]</a></span> +it in that state alone, the second part of the differential +calculus would be immediately comprised in the first, +without giving rise to any new difficulty. But the algebraical +resolution of equations being, as we have seen, +still almost in its infancy, and as yet impossible for most +cases, it is plain that the case is very different, since +we have, properly speaking, to differentiate a function +without knowing it, although it is determinate. The +differentiation of implicit functions constitutes then, by +its nature, a question truly distinct from that presented +by explicit functions, and necessarily more complicated. +It is thus evident that we must commence with the differentiation +of formulas, and reduce the differentiation +of equations to this primary case by certain invariable +analytical considerations, which need not be here mentioned.</p> + +<p>These two general cases of differentiation are also distinct +in another point of view equally necessary, and too +important to be left unnoticed. The relation which is +obtained between the differentials is constantly more indirect, +in comparison with that of the finite quantities, +in the differentiation of implicit functions than in that +of explicit functions. We know, in fact, from the considerations +presented by Lagrange on the general formation +of differential equations, that, on the one hand, the +same primitive equation may give rise to a greater or +less number of derived equations of very different forms, +although at bottom equivalent, depending upon which of +the arbitrary constants is eliminated, which is not the +case in the differentiation of explicit formulas; and +that, on the other hand, the unlimited system of the +different primitive equations, which correspond to the<span class="pagenum"><a name="Page_129" id="Page_129">[Pg 129]</a></span> +same derived equation, presents a much more profound +analytical variety than that of the different functions, +which admit of one same explicit differential, and which +are distinguished from each other only by a constant +term. Implicit functions must therefore be regarded as +being in reality still more modified by differentiation +than explicit functions. We shall again meet with this +consideration relatively to the integral calculus, where +it acquires a preponderant importance.</p> + + +<p><i>Two Sub-cases: A single Variable or several Variables.</i> +Each of the two fundamental parts of the Differential +Calculus is subdivided into two very distinct theories, +according as we are required to differentiate functions +of a single variable or functions of several independent +variables. This second case is, by its nature, +quite distinct from the first, and evidently presents more +complication, even in considering only explicit functions, +and still more those which are implicit. As to the rest, +one of these cases is deduced from the other in a general +manner, by the aid of an invariable and very simple +principle, which consists in regarding the total differential +of a function which is produced by the simultaneous +increments of the different independent variables which +it contains, as the sum of the partial differentials which +would be produced by the separate increment of each +variable in turn, if all the others were constant. It is +necessary, besides, carefully to remark, in connection +with this subject, a new idea which is introduced by +the distinction of functions into those of one variable +and of several; it is the consideration of these different +special derived functions, relating to each variable separately, +and the number of which increases more and<span class="pagenum"><a name="Page_130" id="Page_130">[Pg 130]</a></span> +more in proportion as the order of the derivation becomes +higher, and also when the variables become more numerous. +It results from this that the differential relations +belonging to functions of several variables are, by +their nature, both much more indirect, and especially +much more indeterminate, than those relating to functions +of a single variable. This is most apparent in the +case of implicit functions, in which, in the place of the +simple arbitrary constants which elimination causes to +disappear when we form the proper differential equations +for functions of a single variable, it is the arbitrary functions +of the proposed variables which are then eliminated; +whence must result special difficulties when these +equations come to be integrated.</p> + +<p>Finally, to complete this summary sketch of the different +essential parts of the differential calculus proper, +I should add, that in the differentiation of implicit functions, +whether of a single variable or of several, it is necessary +to make another distinction; that of the case in +which it is required to differentiate at once different +functions of this kind, <i>combined</i> in certain primitive +equations, from that in which all these functions are +<i>separate</i>.</p> + +<p>The functions are evidently, in fact, still more implicit +in the first case than in the second, if we consider +that the same imperfection of ordinary analysis, which +forbids our converting every implicit function into an +equivalent explicit function, in like manner renders us +unable to separate the functions which enter simultaneously +into any system of equations. It is then necessary +to differentiate, not only without knowing how +to resolve the primitive equations, but even without being<span class="pagenum"><a name="Page_131" id="Page_131">[Pg 131]</a></span> +able to effect the proper eliminations among them, +thus producing a new difficulty.</p> + + +<p><i>Reduction of the whole to the Differentiation of the +ten elementary Functions.</i> Such, then, are the natural +connection and the logical distribution of the different +principal theories which compose the general system of +differentiation. Since the differentiation of implicit +functions is deduced from that of explicit functions by +a single constant principle, and the differentiation of +functions of several variables is reduced by another fixed +principle to that of functions of a single variable, the +whole of the differential calculus is finally found to rest +upon the differentiation of explicit functions with a single +variable, the only one which is ever executed directly. +Now it is easy to understand that this first theory, +the necessary basis of the entire system, consists simply +in the differentiation of the ten simple functions, which +are the uniform elements of all our analytical combinations, +and the list of which has been given in the first +chapter, on page 51; for the differentiation of compound +functions is evidently deduced, in an immediate and necessary +manner, from that of the simple functions which +compose them. It is, then, to the knowledge of these +ten fundamental differentials, and to that of the two general +principles just mentioned, which bring under it all +the other possible cases, that the whole system of differentiation +is properly reduced. We see, by the combination +of these different considerations, how simple and +how perfect is the entire system of the differential calculus. +It certainly constitutes, in its logical relations, +the most interesting spectacle which mathematical analysis +can present to our understanding.</p><p><span class="pagenum"><a name="Page_132" id="Page_132">[Pg 132]</a></span></p> + + +<p><i>Transformation of derived Functions for new Variables.</i> +The general sketch which I have just summarily +drawn would nevertheless present an important deficiency, +if I did not here distinctly indicate a final theory, +which forms, by its nature, the indispensable complement +of the system of differentiation. It is that which has +for its object the constant transformation of derived functions, +as a result of determinate changes in the independent +variables, whence results the possibility of referring +to new variables all the general differential formulas +primitively established for others. This question +is now resolved in the most complete and the most simple +manner, as are all those of which the differential +calculus is composed. It is easy to conceive the general +importance which it must have in any of the applications +of the transcendental analysis, the fundamental +resources of which it may be considered as augmenting, +by permitting us to choose (in order to form the differential +equations, in the first place, with more ease) that +system of independent variables which may appear to +be the most advantageous, although it is not to be finally +retained. It is thus, for example, that most of the +principal questions of geometry are resolved much more +easily by referring the lines and surfaces to <i>rectilinear</i> +co-ordinates, and that we may, nevertheless, have occasion +to express these lines, etc., analytically by the aid +of <i>polar</i> co-ordinates, or in any other manner. We will +then be able to commence the differential solution of the +problem by employing the rectilinear system, but only +as an intermediate step, from which, by the general theory +here referred to, we can pass to the final system, +which sometimes could not have been considered directly.</p><p><span class="pagenum"><a name="Page_133" id="Page_133">[Pg 133]</a></span></p> + + +<p><i>Different Orders of Differentiation.</i> In the logical +classification of the differential calculus which has just +been given, some may be inclined to suggest a serious +omission, since I have not subdivided each of its four +essential parts according to another general consideration, +which seems at first view very important; namely, +that of the higher or lower order of differentiation. But +it is easy to understand that this distinction has no real +influence in the differential calculus, inasmuch as it does +not give rise to any new difficulty. If, indeed, the differential +calculus was not rigorously complete, that is, +if we did not know how to differentiate at will any function +whatever, the differentiation to the second or higher +order of each determinate function might engender special +difficulties. But the perfect universality of the differential +calculus plainly gives us the assurance of being +able to differentiate, to any order whatever, all known +functions whatever, the question reducing itself to a constantly +repeated differentiation of the first order. This +distinction, unimportant as it is for the differential calculus, +acquires, however, a very great importance in the +integral calculus, on account of the extreme imperfection +of the latter.</p> + + +<p><i>Analytical Applications.</i> Finally, though this is not +the place to consider the various applications of the differential +calculus, yet an exception may be made for +those which consist in the solution of questions which are +purely analytical, which ought, indeed, to be logically +treated in continuation of a system of differentiation, because +of the evident homogeneity of the considerations +involved. These questions may be reduced to three essential +ones.</p><p><span class="pagenum"><a name="Page_134" id="Page_134">[Pg 134]</a></span></p> + +<p>Firstly, the <i>development into series</i> of functions of +one or more variables, or, more generally, the transformation +of functions, which constitutes the most beautiful +and the most important application of the differential calculus +to general analysis, and which comprises, besides +the fundamental series discovered by Taylor, the remarkable +series discovered by Maclaurin, John Bernouilli, Lagrange, +&c.:</p> + +<p>Secondly, the general <i>theory of maxima and minima</i> +values for any functions whatever, of one or more variables; +one of the most interesting problems which analysis +can present, however elementary it may now have +become, and to the complete solution of which the differential +calculus naturally applies:</p> + +<p>Thirdly, the general determination of the true value +of functions which present themselves under an <i>indeterminate</i> +appearance for certain hypotheses made on the +values of the corresponding variables; which is the least +extensive and the least important of the three.</p> + +<p>The first question is certainly the principal one in all +points of view; it is also the most susceptible of receiving +a new extension hereafter, especially by conceiving, +in a broader manner than has yet been done, the employment +of the differential calculus in the transformation +of functions, on which subject Lagrange has left +some valuable hints.</p> +<p><span class="pagenum"><a name="Page_135" id="Page_135">[Pg 135]</a></span></p> +<hr class="tb" /> + +<p>Having thus summarily, though perhaps too briefly, +considered the chief points in the differential calculus, I +now proceed to an equally rapid exposition of a systematic +outline of the Integral Calculus, properly so called, +that is, the abstract subject of integration.</p> + + + + +<h3><a name="THE_INTEGRAL_CALCULUS" id="THE_INTEGRAL_CALCULUS">THE INTEGRAL CALCULUS.</a></h3> + + +<p><i>Its Fundamental Division.</i> The fundamental division +of the Integral Calculus is founded on the same principle +as that of the Differential Calculus, in distinguishing +the integration of <i>explicit</i> differential formulas, and the +integration of <i>implicit</i> differentials or of differential equations. +The separation of these two cases is even much +more profound in relation to integration than to differentiation. +In the differential calculus, in fact, this distinction +rests, as we have seen, only on the extreme imperfection +of ordinary analysis. But, on the other hand, +it is easy to see that, even though all equations could be +algebraically resolved, differential equations would none +the less constitute a case of integration quite distinct +from that presented by the explicit differential formulas; +for, limiting ourselves, for the sake of simplicity, to the +first order, and to a single function <i>y</i> of a single variable +<i>x</i>, if we suppose any differential equation between <i>x</i>, <i>y</i>, +and <i>dy/dx</i>, to be resolved with reference to <i>dy/dx</i>, the expression +of the derived function being then generally found +to contain the primitive function itself, which is the object +of the inquiry, the question of integration will not +have at all changed its nature, and the solution will not +really have made any other progress than that of having +brought the proposed differential equation to be of only +the first degree relatively to the derived function, which +is in itself of little importance. The differential would +not then be determined in a manner much less <i>implicit</i> +than before, as regards the integration, which would continue +to present essentially the same characteristic difficulty.<span class="pagenum"><a name="Page_136" id="Page_136">[Pg 136]</a></span> +The algebraic resolution of equations could not +make the case which we are considering come within the +simple integration of explicit differentials, except in the +special cases in which the proposed differential equation +did not contain the primitive function itself, which would +consequently permit us, by resolving it, to find <i>dy/dx</i> in +terms of <i>x</i> only, and thus to reduce the question to the +class of quadratures. Still greater difficulties would evidently +be found in differential equations of higher orders, +or containing simultaneously different functions of several +independent variables.</p> + +<p>The integration of differential equations is then necessarily +more complicated than that of explicit differentials, +by the elaboration of which last the integral calculus +has been created, and upon which the others have been +made to depend as far as it has been possible. All the +various analytical methods which have been proposed for +integrating differential equations, whether it be the separation +of the variables, the method of multipliers, &c., +have in fact for their object to reduce these integrations +to those of differential formulas, the only one which, by its +nature, can be undertaken directly. Unfortunately, imperfect +as is still this necessary base of the whole integral +calculus, the art of reducing to it the integration of differential +equations is still less advanced.</p> + + +<p><i>Subdivisions: one variable or several.</i> Each of these +two fundamental branches of the integral calculus is next +subdivided into two others (as in the differential calculus, +and for precisely analogous reasons), according as we +consider functions with a <i>single variable</i>, or functions +with <i>several independent variables</i>.</p><p><span class="pagenum"><a name="Page_137" id="Page_137">[Pg 137]</a></span></p> + +<p>This distinction is, like the preceding one, still more +important for integration than for differentiation. This +is especially remarkable in reference to differential equations. +Indeed, those which depend on several independent +variables may evidently present this characteristic +and much more serious difficulty, that the desired function +may be differentially defined by a simple relation between +its different special derivatives relative to the different +variables taken separately. Hence results the +most difficult and also the most extensive branch of the +integral calculus, which is commonly named the <i>Integral +Calculus of partial differences</i>, created by D'Alembert, +and in which, according to the just appreciation of +Lagrange, geometers ought to have seen a really new +calculus, the philosophical character of which has not yet +been determined with sufficient exactness. A very striking +difference between this case and that of equations +with a single independent variable consists, as has been +already observed, in the arbitrary functions which take +the place of the simple arbitrary constants, in order to give +to the corresponding integrals all the proper generality.</p> + +<p>It is scarcely necessary to say that this higher branch +of transcendental analysis is still entirely in its infancy, +since, even in the most simple case, that of an equation +of the first order between the partial derivatives of a single +function with two independent variables, we are not +yet completely able to reduce the integration to that of +the ordinary differential equations. The integration of +functions of several variables is much farther advanced +in the case (infinitely more simple indeed) in which it +has to do with only explicit differential formulas. We +can then, in fact, when these formulas fulfil the necessary<span class="pagenum"><a name="Page_138" id="Page_138">[Pg 138]</a></span> +conditions of integrability, always reduce their integration +to quadratures.</p> + + +<p><i>Other Subdivisions: different Orders of Differentiation.</i> +A new general distinction, applicable as a subdivision +to the integration of explicit or implicit differentials, +with one variable or several, is drawn from the <i>higher +or lower order of the differentials</i>: a distinction which, +as we have above remarked, does not give rise to any +special question in the differential calculus.</p> + +<p>Relatively to <i>explicit differentials</i>, whether of one variable +or of several, the necessity of distinguishing their +different orders belongs only to the extreme imperfection +of the integral calculus. In fact, if we could always integrate +every differential formula of the first order, the +integration of a formula of the second order, or of any +other, would evidently not form a new question, since, by +integrating it at first in the first degree, we would arrive +at the differential expression of the immediately preceding +order, from which, by a suitable series of analogous +integrations, we would be certain of finally arriving at +the primitive function, the final object of these operations. +But the little knowledge which we possess on integration +of even the first order causes quite another state +of affairs, so that a higher order of differentials produces +new difficulties; for, having differential formulas of any +order above the first, it may happen that we may be able +to integrate them, either once, or several times in succession, +and that we may still be unable to go back to +the primitive functions, if these preliminary labours have +produced, for the differentials of a lower order, expressions +whose integrals are not known. This circumstance +must occur so much the oftener (the number of known<span class="pagenum"><a name="Page_139" id="Page_139">[Pg 139]</a></span> +integrals being still very small), seeing that these successive +integrals are generally very different functions +from the derivatives which have produced them.</p> + +<p>With reference to <i>implicit differentials</i>, the distinction +of orders is still more important; for, besides the +preceding reason, the influence of which is evidently +analogous in this case, and is even greater, it is easy to +perceive that the higher order of the differential equations +necessarily gives rise to questions of a new nature. +In fact, even if we could integrate every equation of the +first order relating to a single function, that would not +be sufficient for obtaining the final integral of an equation +of any order whatever, inasmuch as every differential +equation is not reducible to that of an immediately inferior +order. Thus, for example, if we have given any +relation between <i>x</i>, <i>y</i>, <i>dx/dy</i>, and <i>d</i><sup>2</sup><i>y</i>/<i>dx</i><sup>2</sup>, to determine a function +<i>y</i> of a variable <i>x</i>, we shall not be able to deduce +from it at once, after effecting a first integration, the +corresponding differential relation between <i>x</i>, <i>y</i>, and <i>dy/dx</i>, +from which, by a second integration, we could ascend +to the primitive equations. This would not necessarily +take place, at least without introducing new auxiliary +functions, unless the proposed equation of the second order +did not contain the required function <i>y</i>, together with +its derivatives. As a general principle, differential equations +will have to be regarded as presenting cases which +are more and more <i>implicit</i>, as they are of a higher order, +and which cannot be made to depend on one another +except by special methods, the investigation of which +consequently forms a new class of questions, with respect<span class="pagenum"><a name="Page_140" id="Page_140">[Pg 140]</a></span> +to which we as yet know scarcely any thing, even +for functions of a single variable.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a></p> + +<p><i>Another equivalent distinction.</i> Still farther, when +we examine more profoundly this distinction of different +orders of differential equations, we find that it can be +always made to come under a final general distinction, +relative to differential equations, which remains to be +noticed. Differential equations with one or more independent +variables may contain simply a single function, +or (in a case evidently more complicated and more implicit, +which corresponds to the differentiation of simultaneous +implicit functions) we may have to determine +at the same time several functions from the differential +equations in which they are found united, together with +their different derivatives. It is clear that such a state +of the question necessarily presents a new special difficulty, +that of separating the different functions desired, +by forming for each, from the proposed differential equations, +an isolated differential equation which does not +contain the other functions or their derivatives. This +preliminary labour, which is analogous to the elimination +of algebra, is evidently indispensable before attempting +any direct integration, since we cannot undertake +generally (except by special artifices which are very +rarely applicable) to determine directly several distinct +functions at once.</p> + +<p>Now it is easy to establish the exact and necessary +coincidence of this new distinction with the preceding +<span class="pagenum"><a name="Page_141" id="Page_141">[Pg 141]</a></span>one respecting the order of differential equations. We +know, in fact, that the general method for isolating functions +in simultaneous differential equations consists essentially +in forming differential equations, separately in +relation to each function, and of an order equal to the +sum of all those of the different proposed equations. +This transformation can always be effected. On the +other hand, every differential equation of any order in +relation to a single function might evidently always be +reduced to the first order, by introducing a suitable number +of auxiliary differential equations, containing at the +same time the different anterior derivatives regarded as +new functions to be determined. This method has, indeed, +sometimes been actually employed with success, +though it is not the natural one.</p> + +<p>Here, then, are two necessarily equivalent orders of +conditions in the general theory of differential equations; +the simultaneousness of a greater or smaller number of +functions, and the higher or lower order of differentiation +of a single function. By augmenting the order of +the differential equations, we can isolate all the functions; +and, by artificially multiplying the number of +the functions, we can reduce all the equations to the +first order. There is, consequently, in both cases, only +one and the same difficulty from two different points of +sight. But, however we may conceive it, this new difficulty +is none the less real, and constitutes none the +less, by its nature, a marked separation between the integration +of equations of the first order and that of equations +of a higher order. I prefer to indicate the distinction +under this last form as being more simple, more +general, and more logical.</p><p><span class="pagenum"><a name="Page_142" id="Page_142">[Pg 142]</a></span></p> + + +<p><i>Quadratures.</i> From the different considerations +which have been indicated respecting the logical dependence +of the various principal parts of the integral calculus, +we see that the integration of explicit differential +formulas of the first order and of a single variable is the +necessary basis of all other integrations, which we never +succeed in effecting but so far as we reduce them to this +elementary case, evidently the only one which, by its +nature, is capable of being treated directly. This simple +fundamental integration is often designated by the +convenient expression of <i>quadratures</i>, seeing that every +integral of this kind, S<i>f</i>(<i>x</i>)<i>dx</i>, may, in fact, be regarded +as representing the area of a curve, the equation of which +in rectilinear co-ordinates would be <i>y</i> = <i>f</i>(<i>x</i>). Such a +class of questions corresponds, in the differential calculus, +to the elementary case of the differentiation of explicit +functions of a single variable. But the integral question +is, by its nature, very differently complicated, and +especially much more extensive than the differential +question. This latter is, in fact, necessarily reduced, as +we have seen, to the differentiation of the ten simple +functions, the elements of all which are considered in +analysis. On the other hand, the integration of compound +functions does not necessarily follow from that of +the simple functions, each combination of which may +present special difficulties with respect to the integral +calculus. Hence results the naturally indefinite extent, +and the so varied complication of the question of <i>quadratures</i>, +upon which, in spite of all the efforts of analysts, +we still possess so little complete knowledge.</p> + +<p>In decomposing this question, as is natural, according +to the different forms which may be assumed by the<span class="pagenum"><a name="Page_143" id="Page_143">[Pg 143]</a></span> +derivative function, we distinguish the case of <i>algebraic</i> +functions and that of <i>transcendental</i> functions.</p> + +<p><i>Integration of Transcendental Functions.</i> The truly +analytical integration of transcendental functions is as +yet very little advanced, whether for <i>exponential</i>, or for +<i>logarithmic</i>, or for <i>circular</i> functions. But a very small +number of cases of these three different kinds have as +yet been treated, and those chosen from among the simplest; +and still the necessary calculations are in most +cases extremely laborious. A circumstance which we +ought particularly to remark in its philosophical connection +is, that the different procedures of quadrature +have no relation to any general view of integration, and +consist of simple artifices very incoherent with each other, +and very numerous, because of the very limited extent +of each.</p> + +<p>One of these artifices should, however, here be noticed, +which, without being really a method of integration, +is nevertheless remarkable for its generality; it is +the procedure invented by John Bernouilli, and known +under the name of <i>integration by parts</i>, by means of +which every integral may be reduced to another which +is sometimes found to be more easy to be obtained. +This ingenious relation deserves to be noticed for another +reason, as having suggested the first idea of that transformation +of integrals yet unknown, which has lately +received a greater extension, and of which M. Fourier +especially has made so new and important a use in the +analytical questions produced by the theory of heat.</p> + +<p><i>Integration of Algebraic Functions.</i> As to the integration +of algebraic functions, it is farther advanced. +However, we know scarcely any thing in relation to irrational<span class="pagenum"><a name="Page_144" id="Page_144">[Pg 144]</a></span> +functions, the integrals of which have been obtained +only in extremely limited cases, and particularly by +rendering them rational. The integration of rational +functions is thus far the only theory of the integral calculus +which has admitted of being treated in a truly complete +manner; in a logical point of view, it forms, then, +its most satisfactory part, but perhaps also the least important. +It is even essential to remark, in order to have +a just idea of the extreme imperfection of the integral +calculus, that this case, limited as it is, is not entirely +resolved except for what properly concerns integration +viewed in an abstract manner; for, in the execution, the +theory finds its progress most frequently quite stopped, +independently of the complication of the calculations, by +the imperfection of ordinary analysis, seeing that it +makes the integration finally depend upon the algebraic +resolution of equations, which greatly limits its use.</p> + +<p>To grasp in a general manner the spirit of the different +procedures which are employed in quadratures, we +must observe that, by their nature, they can be primitively +founded only on the differentiation of the ten simple +functions. The results of this, conversely considered, +establish as many direct theorems of the integral calculus, +the only ones which can be directly known. All the +art of integration afterwards consists, as has been said +in the beginning of this chapter, in reducing all the other +quadratures, so far as is possible, to this small number +of elementary ones, which unhappily we are in most +cases unable to effect.</p> + +<p><i>Singular Solutions.</i> In this systematic enumeration +of the various essential parts of the integral calculus, considered +in their logical relations, I have designedly neglected<span class="pagenum"><a name="Page_145" id="Page_145">[Pg 145]</a></span> +(in order not to break the chain of sequence) to +consider a very important theory, which forms implicitly +a portion of the general theory of the integration of differential +equations, but which I ought here to notice separately, +as being, so to speak, outside of the integral calculus, +and being nevertheless of the greatest interest, both +by its logical perfection and by the extent of its applications. +I refer to what are called <i>Singular Solutions</i> +of differential equations, called sometimes, but improperly, +<i>particular</i> solutions, which have been the subject +of very remarkable investigations by Euler and Laplace, +and of which Lagrange especially has presented such a +beautiful and simple general theory. Clairaut, who first +had occasion to remark their existence, saw in them a +paradox of the integral calculus, since these solutions +have the peculiarity of satisfying the differential equations +without being comprised in the corresponding general +integrals. Lagrange has since explained this paradox +in the most ingenious and most satisfactory manner, +by showing how such solutions are always derived +from the general integral by the variation of the arbitrary +constants. He was also the first to suitably appreciate +the importance of this theory, and it is with +good reason that he devoted to it so full a development +in his "Calculus of Functions." In a logical point of +view, this theory deserves all our attention by the character +of perfect generality which it admits of, since Lagrange +has given invariable and very simple procedures +for finding the <i>singular</i> solution of any differential equation +which is susceptible of it; and, what is no less remarkable, +these procedures require no integration, consisting +only of differentiations, and are therefore always<span class="pagenum"><a name="Page_146" id="Page_146">[Pg 146]</a></span> +applicable. Differentiation has thus become, by a happy +artifice, a means of compensating, in certain circumstances, +for the imperfection of the integral calculus. +Indeed, certain problems especially require, by their nature, +the knowledge of these <i>singular</i> solutions; such, +for example, in geometry, are all the questions in which +a curve is to be determined from any property of its tangent +or its osculating circle. In all cases of this kind, +after having expressed this property by a differential +equation, it will be, in its analytical relations, the <i>singular</i> +equation which will form the most important object +of the inquiry, since it alone will represent the required +curve; the general integral, which thenceforth it +becomes unnecessary to know, designating only the system +of the tangents, or of the osculating circles of this +curve. We may hence easily understand all the importance +of this theory, which seems to me to be not as yet +sufficiently appreciated by most geometers.</p> + +<p><i>Definite Integrals.</i> Finally, to complete our review +of the vast collection of analytical researches of which is +composed the integral calculus, properly so called, there +remains to be mentioned one theory, very important in +all the applications of the transcendental analysis, which +I have had to leave outside of the system, as not being +really destined for veritable integration, and proposing, on +the contrary, to supply the place of the knowledge of truly +analytical integrals, which are most generally unknown. +I refer to the determination of <i>definite integrals</i>.</p> + +<p>The expression, always possible, of integrals in infinite +series, may at first be viewed as a happy general +means of compensating for the extreme imperfection of +the integral calculus. But the employment of such series,<span class="pagenum"><a name="Page_147" id="Page_147">[Pg 147]</a></span> +because of their complication, and of the difficulty +of discovering the law of their terms, is commonly of only +moderate utility in the algebraic point of view, although +sometimes very essential relations have been thence deduced. +It is particularly in the arithmetical point of +view that this procedure acquires a great importance, as +a means of calculating what are called <i>definite integrals</i>, +that is, the values of the required functions for certain +determinate values of the corresponding variables.</p> + +<p>An inquiry of this nature exactly corresponds, in transcendental +analysis, to the numerical resolution of equations +in ordinary analysis. Being generally unable to +obtain the veritable integral—named by opposition the +<i>general</i> or <i>indefinite</i> integral; that is, the function which, +differentiated, has produced the proposed differential formula—analysts +have been obliged to employ themselves +in determining at least, without knowing this function, +the particular numerical values which it would take on +assigning certain designated values to the variables. +This is evidently resolving the arithmetical question +without having previously resolved the corresponding algebraic +one, which most generally is the most important +one. Such an analysis is, then, by its nature, as +imperfect as we have seen the numerical resolution of +equations to be. It presents, like this last, a vicious +confusion of arithmetical and algebraic considerations, +whence result analogous inconveniences both in the +purely logical point of view and in the applications. +We need not here repeat the considerations suggested in +our third chapter. But it will be understood that, unable +as we almost always are to obtain the true integrals, +it is of the highest importance to have been able<span class="pagenum"><a name="Page_148" id="Page_148">[Pg 148]</a></span> +to obtain this solution, incomplete and necessarily insufficient +as it is. Now this has been fortunately attained +at the present day for all cases, the determination of +the value of definite integrals having been reduced to +entirely general methods, which leave nothing to desire, +in a great number of cases, but less complication in the +calculations, an object towards which are at present directed +all the special transformations of analysts. Regarding +now this sort of <i>transcendental arithmetic</i> as +perfect, the difficulty in the applications is essentially +reduced to making the proposed research depend, finally, +on a simple determination of definite integrals, which +evidently cannot always be possible, whatever analytical +skill may be employed in effecting such a transformation.</p> + + +<p><i>Prospects of the Integral Calculus.</i> From the considerations +indicated in this chapter, we see that, while +the differential calculus constitutes by its nature a limited +and perfect system, to which nothing essential remains +to be added, the integral calculus, or the simple system +of integration, presents necessarily an inexhaustible field +for the activity of the human mind, independently of +the indefinite applications of which the transcendental +analysis is evidently susceptible. The general argument +by which I have endeavoured, in the second chapter, +to make apparent the impossibility of ever discovering +the algebraic solution of equations of any degree and +form whatsoever, has undoubtedly infinitely more force +with regard to the search for a single method of integration, +invariably applicable to all cases. "It is," says +Lagrange, "one of those problems whose general solution +we cannot hope for." The more we meditate on<span class="pagenum"><a name="Page_149" id="Page_149">[Pg 149]</a></span> +this subject, the more we will be convinced that such a +research is utterly chimerical, as being far above the feeble +reach of our intelligence; although the labours of +geometers must certainly augment hereafter the amount +of our knowledge respecting integration, and thus create +methods of greater generality. The transcendental analysis +is still too near its origin—there is especially too +little time since it has been conceived in a truly rational +manner—for us now to be able to have a correct idea of +what it will hereafter become. But, whatever should be +our legitimate hopes, let us not forget to consider, before +all, the limits which are imposed by our intellectual constitution, +and which, though not susceptible of a precise +determination, have none the less an incontestable reality.</p> + +<p>I am induced to think that, when geometers shall have +exhausted the most important applications of our present +transcendental analysis, instead of striving to impress +upon it, as now conceived, a chimerical perfection, they +will rather create new resources by changing the mode +of derivation of the auxiliary quantities introduced in +order to facilitate the establishment of equations, and +the formation of which might follow an infinity of other +laws besides the very simple relation which has been +chosen, according to the conception suggested in the first +chapter. The resources of this nature appear to me susceptible +of a much greater fecundity than those which +would consist of merely pushing farther our present calculus +of indirect functions. It is a suggestion which I +submit to the geometers who have turned their thoughts +towards the general philosophy of analysis.</p> + +<p>Finally, although, in the summary exposition which +was the object of this chapter, I have had to exhibit the<span class="pagenum"><a name="Page_150" id="Page_150">[Pg 150]</a></span> +condition of extreme imperfection which still belongs to +the integral calculus, the student would have a false idea +of the general resources of the transcendental analysis if +he gave that consideration too great an importance. It +is with it, indeed, as with ordinary analysis, in which a +very small amount of fundamental knowledge respecting +the resolution of equations has been employed with an +immense degree of utility. Little advanced as geometers +really are as yet in the science of integrations, they +have nevertheless obtained, from their scanty abstract +conceptions, the solution of a multitude of questions of +the first importance in geometry, in mechanics, in thermology, +&c. The philosophical explanation of this +double general fact results from the necessarily preponderating +importance and grasp of <i>abstract</i> branches of +knowledge, the least of which is naturally found to correspond +to a crowd of <i>concrete</i> researches, man having +no other resource for the successive extension of his intellectual +means than in the consideration of ideas more +and more abstract, and still positive.</p> +<p><span class="pagenum"><a name="Page_151" id="Page_151">[Pg 151]</a></span></p> +<hr class="tb" /> + +<p>In order to finish the complete exposition of the philosophical +character of the transcendental analysis, there +remains to be considered a final conception, by which +the immortal Lagrange has rendered this analysis still +better adapted to facilitate the establishment of equations +in the most difficult problems, by considering a class of +equations still more <i>indirect</i> than the ordinary differential +equations. It is the <i>Calculus</i>, or, rather, the <i>Method +of Variations</i>; the general appreciation of which will be +our next subject.</p> + + + + +<h2><a name="CHAPTER_V" id="CHAPTER_V">CHAPTER V.</a></h2> + +<h3>THE CALCULUS OF VARIATIONS.</h3> + + +<p>In order to grasp with more ease the philosophical +character of the <i>Method of Variations</i>, it will be well to +begin by considering in a summary manner the special +nature of the problems, the general resolution of which +has rendered necessary the formation of this hyper-transcendental +analysis. It is still too near its origin, and +its applications have been too few, to allow us to obtain +a sufficiently clear general idea of it from a purely abstract +exposition of its fundamental theory.</p> + + + + +<h3><a name="PROBLEMS_GIVING_RISE_TO_IT" id="PROBLEMS_GIVING_RISE_TO_IT">PROBLEMS GIVING RISE TO IT.</a></h3> + + +<p>The mathematical questions which have given birth +to the <i>Calculus of Variations</i> consist generally in the +investigation of the <i>maxima</i> and <i>minima</i> of certain indeterminate +integral formulas, which express the analytical +law of such or such a phenomenon of geometry +or mechanics, considered independently of any particular +subject. Geometers for a long time designated all the +questions of this character by the common name of <i>Isoperimetrical +Problems</i>, which, however, is really suitable +to only the smallest number of them.</p> + + +<p><i>Ordinary Questions of Maxima and Minima.</i> In +the common theory of <i>maxima</i> and <i>minima</i>, it is proposed +to discover, with reference to a given function of +one or more variables, what particular values must be +assigned to these variables, in order that the corresponding<span class="pagenum"><a name="Page_152" id="Page_152">[Pg 152]</a></span> +value of the proposed function may be a <i>maximum</i> +or a <i>minimum</i> with respect to those values which immediately +precede and follow it; that is, properly speaking, +we seek to know at what instant the function ceases +to increase and commences to decrease, or reciprocally. +The differential calculus is perfectly sufficient, as we +know, for the general resolution of this class of questions, +by showing that the values of the different variables, +which suit either the maximum or minimum, must +always reduce to zero the different first derivatives of +the given function, taken separately with reference to +each independent variable, and by indicating, moreover, +a suitable characteristic for distinguishing the maximum +from the minimum; consisting, in the case of a function +of a single variable, for example, in the derived function +of the second order taking a negative value for the maximum, +and a positive value for the minimum. Such +are the well-known fundamental conditions belonging to +the greatest number of cases.</p> + + +<p><i>A new Class of Questions.</i> The construction of this +general theory having necessarily destroyed the chief +interest which questions of this kind had for geometers, +they almost immediately rose to the consideration of a +new order of problems, at once much more important and +of much greater difficulty—those of <i>isoperimeters</i>. It +is, then, no longer <i>the values of the variables</i> belonging +to the maximum or the minimum of a given function +that it is required to determine. It is <i>the form of the +function itself</i> which is required to be discovered, from +the condition of the maximum or of the minimum of a +certain definite integral, merely indicated, which depends +upon that function.</p><p><span class="pagenum"><a name="Page_153" id="Page_153">[Pg 153]</a></span></p> + + +<p><i>Solid of least Resistance.</i> The oldest question of +this nature is that of <i>the solid of least resistance</i>, treated +by Newton in the second book of the Principia, in +which he determines what ought to be the meridian +curve of a solid of revolution, in order that the resistance +experienced by that body in the direction of its axis +may be the least possible. But the course pursued by +Newton, from the nature of his special method of transcendental +analysis, had not a character sufficiently simple, +sufficiently general, and especially sufficiently analytical, +to attract geometers to this new order of problems. +To effect this, the application of the infinitesimal +method was needed; and this was done, in 1695, by +John Bernouilli, in proposing the celebrated problem of +the <i>Brachystochrone</i>.</p> + +<p>This problem, which afterwards suggested such a long +series of analogous questions, consists in determining +the curve which a heavy body must follow in order to +descend from one point to another in the shortest possible +time. Limiting the conditions to the simple fall +in a vacuum, the only case which was at first considered, +it is easily found that the required curve must be +a reversed cycloid with a horizontal base, and with its +origin at the highest point. But the question may become +singularly complicated, either by taking into account +the resistance of the medium, or the change in the +intensity of gravity.</p> + + +<p><i>Isoperimeters.</i> Although this new class of problems +was in the first place furnished by mechanics, it is in +geometry that the principal investigations of this character +were subsequently made. Thus it was proposed +to discover which, among all the curves of the same contour<span class="pagenum"><a name="Page_154" id="Page_154">[Pg 154]</a></span> +traced between two given points, is that whose area +is a maximum or minimum, whence has come the name +of <i>Problem of Isoperimeters</i>; or it was required that +the maximum or minimum should belong to the surface +produced by the revolution of the required curve about +an axis, or to the corresponding volume; in other cases, +it was the vertical height of the center of gravity of the +unknown curve, or of the surface and of the volume +which it might generate, which was to become a maximum +or minimum, &c. Finally, these problems were +varied and complicated almost to infinity by the Bernouillis, +by Taylor, and especially by Euler, before Lagrange +reduced their solution to an abstract and entirely +general method, the discovery of which has put a +stop to the enthusiasm of geometers for such an order of +inquiries. This is not the place for tracing the history +of this subject. I have only enumerated some of the +simplest principal questions, in order to render apparent +the original general object of the method of variations.</p> + + +<p><i>Analytical Nature of these Problems.</i> We see that +all these problems, considered in an analytical point of +view, consist, by their nature, in determining what form +a certain unknown function of one or more variables +ought to have, in order that such or such an integral, +dependent upon that function, shall have, within assigned +limits, a value which is a maximum or a minimum +with respect to all those which it would take if the required +function had any other form whatever.</p> + +<p>Thus, for example, in the problem of the <i>brachystochrone</i>, +it is well known that if <i>y</i> = <i>f(z)</i>, <i>x</i> = π(<i>z</i>), are the +rectilinear equations of the required curve, supposing +the axes of <i>x</i> and of <i>y</i> to be horizontal, and the axis of<span class="pagenum"><a name="Page_155" id="Page_155">[Pg 155]</a></span> +<i>z</i> to be vertical, the time of the fall of a heavy body in +that curve from the point whose ordinate is <i>z<sub>1</sub></i>, to that +whose ordinate is <i>z<sub>2</sub></i>, is expressed in general terms by +the definite integral</p> + +<p> +∫_{<i>z_{2}</i>}<sup><i>z_{1</i></sup>}√(1 + (<i>f'(z))<sup>2</sup></i> + (π'(<i>z</i>))<sup>2</sup>/(2<i>gz</i>))<i>dz.</i><br /> +</p> + +<p>It is, then, necessary to find what the two unknown +functions <i>f</i> and π must be, in order that this integral +may be a minimum.</p> + +<p>In the same way, to demand what is the curve among +all plane isoperimetrical curves, which includes the greatest +area, is the same thing as to propose to find, among +all the functions <i>f(x)</i> which can give a certain constant +value to the integral</p> + +<p> +∫<i>dx</i>√(1 + (<i>f'(x)</i> )<sup>2</sup>),<br /> +</p> + +<p>that one which renders the integral ∫<i>f(x)dx</i>, taken between +the same limits, a maximum. It is evidently always +so in other questions of this class.</p> + + +<p><i>Methods of the older Geometers.</i> In the solutions +which geometers before Lagrange gave of these problems, +they proposed, in substance, to reduce them to the +ordinary theory of maxima and minima. But the means +employed to effect this transformation consisted in special +simple artifices peculiar to each case, and the discovery +of which did not admit of invariable and certain +rules, so that every really new question constantly reproduced +analogous difficulties, without the solutions previously +obtained being really of any essential aid, otherwise +than by their discipline and training of the mind. +In a word, this branch of mathematics presented, then, +the necessary imperfection which always exists when the +part common to all questions of the same class has not<span class="pagenum"><a name="Page_156" id="Page_156">[Pg 156]</a></span> +yet been distinctly grasped in order to be treated in an +abstract and thenceforth general manner.</p> + + + + +<h3><a name="METHOD_OF_LAGRANGE2" id="METHOD_OF_LAGRANGE2">METHOD OF LAGRANGE.</a></h3> + + +<p>Lagrange, in endeavouring to bring all the different +problems of isoperimeters to depend upon a common analysis, +organized into a distinct calculus, was led to conceive +a new kind of differentiation, to which he has applied +the characteristic δ, reserving the characteristic <i>d</i> +for the common differentials. These differentials of a +new species, which he has designated under the name of +<i>Variations</i>, consist of the infinitely small increments +which the integrals receive, not by virtue of analogous +increments on the part of the corresponding variables, as +in the ordinary transcendental analysis, but by supposing +that the <i>form</i> of the function placed under the sign of +integration undergoes an infinitely small change. This +distinction is easily conceived with reference to curves, +in which we see the ordinate, or any other variable of +the curve, admit of two sorts of differentials, evidently +very different, according as we pass from one point to another +infinitely near it on the same curve, or to the corresponding +point of the infinitely near curve produced by +a certain determinate modification of the first curve.<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a> It +is moreover clear, that the relative <i>variations</i> of different +magnitudes connected with each other by any laws +whatever are calculated, all but the characteristic, almost +exactly in the same manner as the differentials. Finally, +<span class="pagenum"><a name="Page_157" id="Page_157">[Pg 157]</a></span>from the general notion of <i>variations</i> are in like manner +deduced the fundamental principles of the algorithm +proper to this method, consisting simply in the evidently +permissible liberty of transposing at will the characteristics +specially appropriated to variations, before or after +those which correspond to the ordinary differentials.</p> + +<p>This abstract conception having been once formed, Lagrange +was able to reduce with ease, and in the most +general manner, all the problems of <i>Isoperimeters</i> to the +simple ordinary theory of <i>maxima</i> and <i>minima</i>. To obtain +a clear idea of this great and happy transformation, +we must previously consider an essential distinction which +arises in the different questions of isoperimeters.</p> + + +<p><i>Two Classes of Questions.</i> These investigations +must, in fact, be divided into two general classes, according +as the maxima and minima demanded are <i>absolute</i> +or <i>relative</i>, to employ the abridged expressions of +geometers.</p> + + +<p><i>Questions of the first Class.</i> The <i>first case</i> is that +in which the indeterminate definite integrals, the maximum +or minimum of which is sought, are not subjected, +by the nature of the problem, to any condition; as happens, +for example, in the problem of the <i>brachystochrone</i>, +in which the choice is to be made between all imaginable +curves. The <i>second</i> case takes place when, on the +contrary, the variable integrals can vary only according +to certain conditions, which usually consist in other definite +integrals (which depend, in like manner, upon the +required functions) always retaining the same given value; +as, for example, in all the geometrical questions relating +to real <i>isoperimetrical</i> figures, and in which, by +the nature of the problem, the integral relating to the<span class="pagenum"><a name="Page_158" id="Page_158">[Pg 158]</a></span> +length of the curve, or to the area of the surface, must +remain constant during the variation of that integral +which is the object of the proposed investigation.</p> + +<p>The <i>Calculus of Variations</i> gives immediately the +general solution of questions of the former class; for it +evidently follows, from the ordinary theory of maxima +and minima, that the required relation must reduce to +zero the <i>variation</i> of the proposed integral with reference +to each independent variable; which gives the condition +common to both the maximum and the minimum: and, +as a characteristic for distinguishing the one from the +other, that the variation of the second order of the same +integral must be negative for the maximum and positive +for the minimum. Thus, for example, in the problem +of the brachystochrone, we will have, in order to determine +the nature of the curve sought, the equation of +condition</p> + +<p> +δ∫_{<i>z_{2}</i>}<sup><i>z_{1</i></sup>}√([1 + (<i>f'</i>(<i>z</i>))<sup>2</sup> + (π'(<i>z</i>))<sup>2</sup>]/(2<i>gz</i>))<i>dz</i> = 0,<br /> +</p> + +<p>which, being decomposed into two, with respect to the +two unknown functions <i>f</i> and π, which are independent +of each other, will completely express the analytical +definition of the required curve. The only difficulty +peculiar to this new analysis consists in the elimination +of the characteristic δ, for which the calculus of variations +furnishes invariable and complete rules, founded, in +general, on the method of "integration by parts," from +which Lagrange has thus derived immense advantage. +The constant object of this first analytical elaboration +(which this is not the place for treating in detail) is to +arrive at real differential equations, which can always +be done; and thereby the question comes under the ordinary<span class="pagenum"><a name="Page_159" id="Page_159">[Pg 159]</a></span> +transcendental analysis, which furnishes the solution, +at least so far as to reduce it to pure algebra if +the integration can be effected. The general object of +the method of variations is to effect this transformation, +for which Lagrange has established rules, which are simple, +invariable, and certain of success.</p> + + +<p><i>Equations of Limits.</i> Among the greatest special +advantages of the method of variations, compared with +the previous isolated solutions of isoperimetrical problems, +is the important consideration of what Lagrange +calls <i>Equations of Limits</i>, which were entirely neglected +before him, though without them the greater part of +the particular solutions remained necessarily incomplete. +When the limits of the proposed integrals are to be fixed, +their variations being zero, there is no occasion for +noticing them. But it is no longer so when these limits, +instead of being rigorously invariable, are only subjected +to certain conditions; as, for example, if the two points +between which the required curve is to be traced are +not fixed, and have only to remain upon given lines or +surfaces. Then it is necessary to pay attention to the +variation of their co-ordinates, and to establish between +them the relations which correspond to the equations of +these lines or of these surfaces.</p> + + +<p><i>A more general consideration.</i> This essential consideration +is only the final complement of a more general +and more important consideration relative to the +variations of different independent variables. If these +variables are really independent of one another, as when +we compare together all the imaginable curves susceptible +of being traced between two points, it will be the +same with their variations, and, consequently, the terms<span class="pagenum"><a name="Page_160" id="Page_160">[Pg 160]</a></span> +relating to each of these variations will have to be separately +equal to zero in the general equation which expresses +the maximum or the minimum. But if, on the +contrary, we suppose the variables to be subjected to any +fixed conditions, it will be necessary to take notice of the +resulting relation between their variations, so that the +number of the equations into which this general equation +is then decomposed is always equal to only the +number of the variables which remain truly independent. +It is thus, for example, that instead of seeking +for the shortest path between any two points, in choosing +it from among all possible ones, it may be proposed to +find only what is the shortest among all those which +may be taken on any given surface; a question the general +solution of which forms certainly one of the most +beautiful applications of the method of variations.</p> + +<p><i>Questions of the second Class.</i> Problems in which +such modifying conditions are considered approach very +nearly, in their nature, to the second general class of +applications of the method of variations, characterized +above as consisting in the investigation of <i>relative</i> maxima +and minima. There is, however, this essential difference +between the two cases, that in this last the +modification is expressed by an integral which depends +upon the function sought, while in the other it is designated +by a finite equation which is immediately given. +It is hence apparent that the investigation of <i>relative</i> +maxima and minima is constantly and necessarily more +complicated than that of <i>absolute</i> maxima and minima. +Luckily, a very important general theory, discovered by +the genius of the great Euler before the invention of +the Calculus of Variations, gives a uniform and very<span class="pagenum"><a name="Page_161" id="Page_161">[Pg 161]</a></span> +simple means of making one of these two classes of +questions dependent on the other. It consists in this, +that if we add to the integral which is to be a maximum +or a minimum, a constant and indeterminate multiple +of that one which, by the nature of the problem, is to +remain constant, it will be sufficient to seek, by the general +method of Lagrange above indicated, the <i>absolute</i> +maximum or minimum of this whole expression. It +can be easily conceived, indeed, that the part of the complete +variation which would proceed from the last integral +must be equal to zero (because of the constant +character of this last) as well as the portion due to the +first integral, which disappears by virtue of the maximum +or minimum state. These two conditions evidently +unite to produce, in that respect, effects exactly +alike.</p> + +<p>Such is a sketch of the general manner in which the +method of variation is applied to all the different questions +which compose what is called the <i>Theory of Isoperimeters</i>. +It will undoubtedly have been remarked in +this summary exposition how much use has been made +in this new analysis of the second fundamental property +of the transcendental analysis noticed in the third chapter, +namely, the generality of the infinitesimal expressions +for the representation of the same geometrical or +mechanical phenomenon, in whatever body it may be +considered. Upon this generality, indeed, are founded, +by their nature, all the solutions due to the method of +variations. If a single formula could not express the +length or the area of any curve whatever; if another +fixed formula could not designate the time of the fall of +a heavy body, according to whatever line it may descend,<span class="pagenum"><a name="Page_162" id="Page_162">[Pg 162]</a></span> +&c., how would it have been possible to resolve +questions which unavoidably require, by their nature, the +simultaneous consideration of all the cases which can be +determined in each phenomenon by the different subjects +which exhibit it.</p> + + +<p><i>Other Applications of this Method.</i> Notwithstanding +the extreme importance of the theory of isoperimeters, +and though the method of variations had at first no +other object than the logical and general solution of this +order of problems, we should still have but an incomplete +idea of this beautiful analysis if we limited its +destination to this. In fact, the abstract conception of +two distinct natures of differentiation is evidently applicable +not only to the cases for which it was created, but +also to all those which present, for any reason whatever, +two different manners of making the same magnitudes +vary. It is in this way that Lagrange himself has made, +in his "<i>Méchanique Analytique</i>," an extensive and important +application of his calculus of variations, by employing +it to distinguish the two sorts of changes which +are naturally presented by the questions of rational mechanics +for the different points which are considered, according +as we compare the successive positions which +are occupied, in virtue of its motion, by the same point +of each body in two consecutive instants, or as we pass +from one point of the body to another in the same instant. +One of these comparisons produces ordinary differentials; +the other gives rise to <i>variations</i>, which, there as every +where, are only differentials taken under a new point of +view. Such is the general acceptation in which we +should conceive the Calculus of Variations, in order suitably +to appreciate the importance of this admirable logical<span class="pagenum"><a name="Page_163" id="Page_163">[Pg 163]</a></span> +instrument, the most powerful that the human mind +has as yet constructed.</p> + +<p>The method of variations being only an immense extension +of the general transcendental analysis, I have no +need of proving specially that it is susceptible of being +considered under the different fundamental points of view +which the calculus of indirect functions, considered as a +whole, admits of. Lagrange invented the Calculus of +Variations in accordance with the infinitesimal conception, +and, indeed, long before he undertook the general reconstruction +of the transcendental analysis. When he +had executed this important reformation, he easily showed +how it could also be applied to the Calculus of Variations, +which he expounded with all the proper development, +according to his theory of derivative functions. +But the more that the use of the method of variations is +difficult of comprehension, because of the higher degree +of abstraction of the ideas considered, the more necessary +is it, in its application, to economize the exertions of the +mind, by adopting the most direct and rapid analytical +conception, namely, that of Leibnitz. Accordingly, Lagrange +himself has constantly preferred it in the important +use which he has made of the Calculus of Variations +in his "Analytical Mechanics." In fact, there does +not exist the least hesitation in this respect among geometers.</p> + + + + +<h3><a name="ITS_RELATIONS_TO_THE_ORDINARY_CALCULUS" id="ITS_RELATIONS_TO_THE_ORDINARY_CALCULUS">ITS RELATIONS TO THE ORDINARY CALCULUS.</a></h3> + + +<p>In order to make as clear as possible the philosophical +character of the Calculus of Variations, I think that I +should, in conclusion, briefly indicate a consideration +which seems to me important, and by which I can approach<span class="pagenum"><a name="Page_164" id="Page_164">[Pg 164]</a></span> +it to the ordinary transcendental analysis in a +higher degree than Lagrange seems to me to have done.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a></p> + +<p>We noticed in the preceding chapter the formation of +the <i>calculus of partial differences</i>, created by D'Alembert, +as having introduced into the transcendental analysis +a new elementary idea; the notion of two kinds of +increments, distinct and independent of one another, +which a function of two variables may receive by virtue +of the change of each variable separately. It is thus +that the vertical ordinate of a surface, or any other magnitude +which is referred to it, varies in two manners +which are quite distinct, and which may follow the most +different laws, according as we increase either the one +or the other of the two horizontal co-ordinates. Now +such a consideration seems to me very nearly allied, by +its nature, to that which serves as the general basis of +the method of variations. This last, indeed, has in reality +done nothing but transfer to the independent variables +themselves the peculiar conception which had been +already adopted for the functions of these variables; a +modification which has remarkably enlarged its use. I +think, therefore, that so far as regards merely the fundamental +conceptions, we may consider the calculus created +by D'Alembert as having established a natural and necessary +transition between the ordinary infinitesimal calculus +and the calculus of variations; such a derivation +of which seems to be adapted to make the general notion +more clear and simple.</p> +<p><span class="pagenum"><a name="Page_165" id="Page_165">[Pg 165]</a></span></p> +<p>According to the different considerations indicated in +this chapter, the method of variations presents itself as +the highest degree of perfection which the analysis of indirect +functions has yet attained. In its primitive state, +this last analysis presented itself as a powerful general +means of facilitating the mathematical study of natural +phenomena, by introducing, for the expression of their +laws, the consideration of auxiliary magnitudes, chosen +in such a manner that their relations are necessarily more +simple and more easy to obtain than those of the direct +magnitudes. But the formation of these differential +equations was not supposed to admit of any general and +abstract rules. Now the Analysis of Variations, considered +in the most philosophical point of view, may be +regarded as essentially destined, by its nature, to bring +within the reach of the calculus the actual establishment +of the differential equations; for, in a great number of +important and difficult questions, such is the general effect +of the <i>varied</i> equations, which, still more <i>indirect</i> +than the simple differential equations with respect to the +special objects of the investigation, are also much more +easy to form, and from which we may then, by invariable +and complete analytical methods, the object of which +is to eliminate the new order of auxiliary infinitesimals +which have been introduced, deduce those ordinary differential +equations which it would often have been impossible +to establish directly. The method of variations +forms, then, the most sublime part of that vast system +of mathematical analysis, which, setting out from the +most simple elements of algebra, organizes, by an uninterrupted +succession of ideas, general methods more and +more powerful, for the study of natural philosophy, and<span class="pagenum"><a name="Page_166" id="Page_166">[Pg 166]</a></span> +which, in its whole, presents the most incomparably imposing +and unequivocal monument of the power of the +human intellect.</p> + +<p>We must, however, also admit that the conceptions +which are habitually considered in the method of variations +being, by their nature, more indirect, more general, +and especially more abstract than all others, the +employment of such a method exacts necessarily and +continuously the highest known degree of intellectual +exertion, in order never to lose sight of the precise object +of the investigation, in following reasonings which +offer to the mind such uncertain resting-places, and in +which signs are of scarcely any assistance. We must +undoubtedly attribute in a great degree to this difficulty +the little real use which geometers, with the exception +of Lagrange, have as yet made of such an admirable +conception.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_167" id="Page_167">[Pg 167]</a></span></p> + + + + +<h2><a name="CHAPTER_VI" id="CHAPTER_VI">CHAPTER VI.</a></h2> + +<h3>THE CALCULUS OF FINITE DIFFERENCES.</h3> + + +<p>The different fundamental considerations indicated in +the five preceding chapters constitute, in reality, all the +essential bases of a complete exposition of mathematical +analysis, regarded in the philosophical point of view. +Nevertheless, in order not to neglect any truly important +general conception relating to this analysis, I think +that I should here very summarily explain the veritable +character of a kind of calculus which is very extended, +and which, though at bottom it really belongs to ordinary +analysis, is still regarded as being of an essentially +distinct nature. I refer to the <i>Calculus of Finite Differences</i>, +which will be the special subject of this chapter.</p> + + +<p><i>Its general Character.</i> This calculus, created by +Taylor, in his celebrated work entitled <i>Methodus Incrementorum</i>, +consists essentially in the consideration of the +finite increments which functions receive as a consequence +of analogous increments on the part of the corresponding +variables. These increments or <i>differences</i>, +which take the characteristic Δ, to distinguish them from +<i>differentials</i>, or infinitely small increments, may be in +their turn regarded as new functions, and become the +subject of a second similar consideration, and so on; from +which results the notion of differences of various successive +orders, analogous, at least in appearance, to the +consecutive orders of differentials. Such a calculus evidently<span class="pagenum"><a name="Page_168" id="Page_168">[Pg 168]</a></span> +presents, like the calculus of indirect functions, +two general classes of questions:</p> + +<p>1°. To determine the successive differences of all the +various analytical functions of one or more variables, as +the result of a definite manner of increase of the independent +variables, which are generally supposed to augment +in arithmetical progression.</p> + +<p>2°. Reciprocally, to start from these differences, or, +more generally, from any equations established between +them, and go back to the primitive functions themselves, +or to their corresponding relations.</p> + +<p>Hence follows the decomposition of this calculus into +two distinct ones, to which are usually given the names +of the <i>Direct</i>, and the <i>Inverse Calculus of Finite Differences</i>, +the latter being also sometimes called the <i>Integral +Calculus of Finite Differences</i>. Each of these would, +also, evidently admit of a logical distribution similar to +that given in the fourth chapter for the differential and +the integral calculus.</p> + + +<p><i>Its true Nature.</i> There is no doubt that Taylor +thought that by such a conception he had founded a calculus +of an entirely new nature, absolutely distinct from +ordinary analysis, and more general than the calculus of +Leibnitz, although resting on an analogous consideration. +It is in this way, also, that almost all geometers have +viewed the analysis of Taylor; but Lagrange, with his +usual profundity, clearly perceived that these properties +belonged much more to the forms and to the notations +employed by Taylor than to the substance of his theory. +In fact, that which constitutes the peculiar character of +the analysis of Leibnitz, and makes of it a truly distinct +and superior calculus, is the circumstance that the derived<span class="pagenum"><a name="Page_169" id="Page_169">[Pg 169]</a></span> +functions are in general of an entirely different nature +from the primitive functions, so that they may give +rise to more simple and more easily formed relations: +whence result the admirable fundamental properties of +the transcendental analysis, which have been already explained. +But it is not so with the <i>differences</i> considered +by Taylor; for these differences are, by their nature, +functions essentially similar to those which have produced +them, a circumstance which renders them unsuitable +to facilitate the establishment of equations, and +prevents their leading to more general relations. Every +equation of finite differences is truly, at bottom, an equation +directly relating to the very magnitudes whose successive +states are compared. The scaffolding of new +signs, which produce an illusion respecting the true character +of these equations, disguises it, however, in a very +imperfect manner, since it could always be easily made +apparent by replacing the <i>differences</i> by the equivalent +combinations of the primitive magnitudes, of which they +are really only the abridged designations. Thus the calculus +of Taylor never has offered, and never can offer, in +any question of geometry or of mechanics, that powerful +general aid which we have seen to result necessarily +from the analysis of Leibnitz. Lagrange has, moreover, +very clearly proven that the pretended analogy observed +between the calculus of differences and the infinitesimal +calculus was radically vicious, in this way, that the formulas +belonging to the former calculus can never furnish, +as particular cases, those which belong to the latter, +the nature of which is essentially distinct.</p> + +<p>From these considerations I am led to think that the +calculus of finite differences is, in general, improperly<span class="pagenum"><a name="Page_170" id="Page_170">[Pg 170]</a></span> +classed with the transcendental analysis proper, that is, +with the calculus of indirect functions. I consider it, on +the contrary, in accordance with the views of Lagrange, +to be only a very extensive and very important branch +of ordinary analysis, that is to say, of that which I +have named the calculus of direct functions, the equations +which it considers being always, in spite of the +notation, simple <i>direct</i> equations.</p> + + + + +<h3><a name="GENERAL_THEORY_OF_SERIES" id="GENERAL_THEORY_OF_SERIES">GENERAL THEORY OF SERIES.</a></h3> + + +<p>To sum up as briefly as possible the preceding explanation, +the calculus of Taylor ought to be regarded +as having constantly for its true object the general theory +of <i>Series</i>, the most simple cases of which had alone +been considered before that illustrious geometer. I +ought, properly, to have mentioned this important theory +in treating, in the second chapter, of Algebra proper, +of which it is such an extensive branch. But, in order +to avoid a double reference to it, I have preferred to notice +it only in the consideration of the calculus of finite +differences, which, reduced to its most simple general +expression, is nothing but a complete logical study of +questions relating to <i>series</i>.</p> + +<p>Every <i>Series</i>, or succession of numbers deduced from +one another according to any constant law, necessarily +gives rise to these two fundamental questions:</p> + +<p>1°. The law of the series being supposed known, to +find the expression for its general term, so as to be able +to calculate immediately any term whatever without being +obliged to form successively all the preceding terms.</p> + +<p>2°. In the same circumstances, to determine the <i>sum</i> +of any number of terms of the series by means of their<span class="pagenum"><a name="Page_171" id="Page_171">[Pg 171]</a></span> +places, so that it can be known without the necessity +of continually adding these terms together.</p> + +<p>These two fundamental questions being considered to +be resolved, it may be proposed, reciprocally, to find the +law of a series from the form of its general term, or the +expression of the sum. Each of these different problems +has so much the more extent and difficulty, as there +can be conceived a greater number of different <i>laws</i> for +the series, according to the number of preceding terms +on which each term directly depends, and according to +the function which expresses that dependence. We may +even consider series with several variable indices, as Laplace +has done in his "Analytical Theory of Probabilities," +by the analysis to which he has given the name +of <i>Theory of Generating Functions</i>, although it is really +only a new and higher branch of the calculus of finite +differences or of the general theory of series.</p> + +<p>These general views which I have indicated give only +an imperfect idea of the truly infinite extent and variety +of the questions to which geometers have risen by means +of this single consideration of series, so simple in appearance +and so limited in its origin. It necessarily +presents as many different cases as the algebraic resolution +of equations, considered in its whole extent; and it +is, by its nature, much more complicated, so much, indeed, +that it always needs this last to conduct it to a complete +solution. We may, therefore, anticipate what must +still be its extreme imperfection, in spite of the successive +labours of several geometers of the first order. We do +not, indeed, possess as yet the complete and logical solution +of any but the most simple questions of this nature.</p><p><span class="pagenum"><a name="Page_172" id="Page_172">[Pg 172]</a></span></p> + + +<p><i>Its identity with this Calculus.</i> It is now easy to +conceive the necessary and perfect identity, which has +been already announced, between the calculus of finite +differences and the theory of series considered in all its +bearings. In fact, every differentiation after the manner +of Taylor evidently amounts to finding the <i>law</i> of +formation of a series with one or with several variable +indices, from the expression of its general term; in the +same way, every analogous integration may be regarded +as having for its object the summation of a series, the +general term of which would be expressed by the proposed +difference. In this point of view, the various problems +of the calculus of differences, direct or inverse, resolved +by Taylor and his successors, have really a very +great value, as treating of important questions relating +to series. But it is very doubtful if the form and the +notation introduced by Taylor really give any essential +facility in the solution of questions of this kind. It +would be, perhaps, more advantageous for most cases, and +certainly more logical, to replace the <i>differences</i> by the +terms themselves, certain combinations of which they +represent. As the calculus of Taylor does not rest on +a truly distinct fundamental idea, and has nothing peculiar +to it but its system of signs, there could never really +be any important advantage in considering it as detached +from ordinary analysis, of which it is, in reality, only an +immense branch. This consideration of <i>differences</i>, most +generally useless, even if it does not cause complication, +seems to me to retain the character of an epoch in which, +analytical ideas not being sufficiently familiar to geometers, +they were naturally led to prefer the special forms +suitable for simple numerical comparisons.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_173" id="Page_173">[Pg 173]</a></span></p> + + + + +<h3><a name="PERIODIC_OR_DISCONTINUOUS_FUNCTIONS" id="PERIODIC_OR_DISCONTINUOUS_FUNCTIONS">PERIODIC OR DISCONTINUOUS FUNCTIONS.</a></h3> + + +<p>However that may be, I must not finish this general +appreciation of the calculus of finite differences without +noticing a new conception to which it has given birth, and +which has since acquired a great importance. It is the +consideration of those periodic or discontinuous functions +which preserve the same value for an infinite series of +values of the corresponding variables, subjected to a certain +law, and which must be necessarily added to the integrals +of the equations of finite differences in order to +render them sufficiently general, as simple arbitrary constants +are added to all quadratures in order to complete +their generality. This idea, primitively introduced by +Euler, has since been the subject of extended investigation +by M. Fourier, who has made new and important +applications of it in his mathematical theory of heat.</p> + + + + +<h3><a name="APPLICATIONS_OF_THIS_CALCULUS" id="APPLICATIONS_OF_THIS_CALCULUS">APPLICATIONS OF THIS CALCULUS.</a></h3> + + +<p><i>Series.</i> Among the principal general applications +which have been made of the calculus of finite differences, +it would be proper to place in the first rank, as the +most extended and the most important, the solution of +questions relating to series; if, as has been shown, the +general theory of series ought not to be considered as constituting, +by its nature, the actual foundation of the calculus +of Taylor.</p> + + +<p><i>Interpolations.</i> This great class of problems being +then set aside, the most essential of the veritable applications +of the analysis of Taylor is, undoubtedly, thus +far, the general method of <i>interpolations</i>, so frequently +and so usefully employed in the investigation of the empirical<span class="pagenum"><a name="Page_174" id="Page_174">[Pg 174]</a></span> +laws of natural phenomena. The question consists, +as is well known, in intercalating between certain given +numbers other intermediate numbers, subjected to the +same law which we suppose to exist between the first. +We can abundantly verify, in this principal application +of the calculus of Taylor, how truly foreign and often inconvenient +is the consideration of <i>differences</i> with respect +to the questions which depend on that analysis. Indeed, +Lagrange has replaced the formulas of interpolation, deduced +from the ordinary algorithm of the calculus of +finite differences, by much simpler general formulas, +which are now almost always preferred, and which have +been found directly, without making any use of the notion +of <i>differences</i>, which only complicates the question.</p> + + +<p><i>Approximate Rectification, &c.</i> A last important +class of applications of the calculus of finite differences, +which deserves to be distinguished from the preceding, +consists in the eminently useful employment made of it +in geometry for determining by approximation the length +and the area of any curve, and in the same way the cubature +of a body of any form whatever. This procedure +(which may besides be conceived abstractly as depending +on the same analytical investigation as the question of +interpolation) frequently offers a valuable supplement to +the entirely logical geometrical methods which often lead +to integrations, which we do not yet know how to effect, +or to calculations of very complicated execution.</p> +<p><span class="pagenum"><a name="Page_175" id="Page_175">[Pg 175]</a></span></p><p><span class="pagenum"><a name="Page_176" id="Page_176">[Pg 176]</a><br /><a name="Page_177" id="Page_177">[Pg 177]</a></span></p> +<hr class="tb" /> + +<p>Such are the various principal considerations to be +noticed with respect to the calculus of finite differences. +This examination completes the proposed philosophical +outline of <span class="smcap">abstract Mathematics</span>.</p> + + +<p><span class="smcap">Concrete Mathematics</span> will now be the subject of a +similar labour. In it we shall particularly devote ourselves +to examining how it has been possible (supposing +the general science of the calculus to be perfect), by invariable +procedures, to reduce to pure questions of analysis +all the problems which can be presented by <i>Geometry</i> and +<i>Mechanics</i>, and thus to impress on these two fundamental +bases of natural philosophy a degree of precision and especially +of unity; in a word, a character of high perfection, +which could be communicated to them by such a +course alone.</p> + + + +<p class="big center">BOOK II.</p> + +<p class="center">GEOMETRY.</p> + +<p><span class="pagenum"><a name="Page_178" id="Page_178">[Pg 178]</a><br /><a name="Page_179" id="Page_179">[Pg 179]</a></span></p> + + + + +<p class="center big">BOOK II.</p> + +<h1>GEOMETRY.</h1> + + + + +<h2><a name="CHAPTER_Ia" id="CHAPTER_Ia">CHAPTER I.</a></h2> + +<h3>GENERAL VIEW OF GEOMETRY.</h3> + + +<p><i>Its true Nature.</i> After the general exposition of the +philosophical character of concrete mathematics, compared +with that of abstract mathematics, given in the introductory +chapter, it need not here be shown in a special +manner that geometry must be considered as a true natural +science, only much more simple, and therefore much +more perfect, than any other. This necessary perfection +of geometry, obtained essentially by the application of +mathematical analysis, which it so eminently admits, is +apt to produce erroneous views of the real nature of this +fundamental science, which most minds at present conceive +to be a purely logical science quite independent of +observation. It is nevertheless evident, to any one who +examines with attention the character of geometrical reasonings, +even in the present state of abstract geometry, +that, although the facts which are considered in it are +much more closely united than those relating to any other +science, still there always exists, with respect to every +body studied by geometers, a certain number of primitive +phenomena, which, since they are not established by any<span class="pagenum"><a name="Page_180" id="Page_180">[Pg 180]</a></span> +reasoning, must be founded on observation alone, and +which form the necessary basis of all the deductions.</p> + +<p>The scientific superiority of geometry arises from the +phenomena which it considers being necessarily the most +universal and the most simple of all. Not only may all +the bodies of nature give rise to geometrical inquiries, as +well as mechanical ones, but still farther, geometrical +phenomena would still exist, even though all the parts +of the universe should be considered as immovable. Geometry +is then, by its nature, more general than mechanics. +At the same time, its phenomena are more simple, +for they are evidently independent of mechanical phenomena, +while these latter are always complicated with the +former. The same relations hold good in comparing +geometry with abstract thermology.</p> + +<p>For these reasons, in our classification we have made +geometry the first part of concrete mathematics; that +part the study of which, in addition to its own importance, +serves as the indispensable basis of all the rest.</p> + +<p>Before considering directly the philosophical study of +the different orders of inquiries which constitute our +present geometry, we should obtain a clear and exact +idea of the general destination of that science, viewed in +all its bearings. Such is the object of this chapter.</p> + + +<p><i>Definition.</i> Geometry is commonly defined in a very +vague and entirely improper manner, as being <i>the science +of extension</i>. An improvement on this would be to say +that geometry has for its object the <i>measurement</i> of extension; +but such an explanation would be very insufficient, +although at bottom correct, and would be far from +giving any idea of the true general character of geometrical +science.</p><p><span class="pagenum"><a name="Page_181" id="Page_181">[Pg 181]</a></span></p> + +<p>To do this, I think that I should first explain <i>two fundamental +ideas</i>, which, very simple in themselves, have +been singularly obscured by the employment of metaphysical +considerations.</p> + + +<p><i>The Idea of Space.</i> The first is that of <i>Space</i>. +This conception properly consists simply in this, that, instead +of considering extension in the bodies themselves, +we view it in an indefinite medium, which we regard as +containing all the bodies of the universe. This notion is +naturally suggested to us by observation, when we think +of the <i>impression</i> which a body would leave in a fluid in +which it had been placed. It is clear, in fact, that, as regards +its geometrical relations, such an <i>impression</i> may +be substituted for the body itself, without altering the +reasonings respecting it. As to the physical nature of +this indefinite <i>space</i>, we are spontaneously led to represent +it to ourselves, as being entirely analogous to the +actual medium in which we live; so that if this medium +was liquid instead of gaseous, our geometrical <i>space</i> +would undoubtedly be conceived as liquid also. This +circumstance is, moreover, only very secondary, the essential +object of such a conception being only to make +us view extension separately from the bodies which manifest +it to us. We can easily understand in advance the +importance of this fundamental image, since it permits +us to study geometrical phenomena in themselves, abstraction +being made of all the other phenomena which +constantly accompany them in real bodies, without, however, +exerting any influence over them. The regular establishment +of this general abstraction must be regarded +as the first step which has been made in the rational +study of geometry, which would have been impossible if<span class="pagenum"><a name="Page_182" id="Page_182">[Pg 182]</a></span> +it had been necessary to consider, together with the form +and the magnitude of bodies, all their other physical +properties. The use of such an hypothesis, which is +perhaps the most ancient philosophical conception created +by the human mind, has now become so familiar to +us, that we have difficulty in exactly estimating its importance, +by trying to appreciate the consequences which +would result from its suppression.</p> + + +<p><i>Different Kinds of Extension.</i> The second preliminary +geometrical conception which we have to examine +is that of the different kinds of extension, designated by +the words <i>volume</i>, <i>surface</i>, <i>line</i>, and even <i>point</i>, and of +which the ordinary explanation is so unsatisfactory.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a></p> + +<p>Although it is evidently impossible to conceive any extension +absolutely deprived of any one of the three fundamental +dimensions, it is no less incontestable that, in +a great number of occasions, even of immediate utility, +geometrical questions depend on only two dimensions, +considered separately from the third, or on a single dimension, +considered separately from the two others. Again, +independently of this direct motive, the study of extension +with a single dimension, and afterwards with two, +clearly presents itself as an indispensable preliminary for +facilitating the study of complete bodies of three dimensions, +the immediate theory of which would be too complicated.<span class="pagenum"><a name="Page_183" id="Page_183">[Pg 183]</a></span> +Such are the two general motives which oblige +geometers to consider separately extension with regard to +one or to two dimensions, as well as relatively to all three +together.</p> + +<p>The general notions of <i>surface</i> and of <i>line</i> have been +formed by the human mind, in order that it may be able +to think, in a permanent manner, of extension in two +directions, or in one only. The hyperbolical expressions +habitually employed by geometers to define these notions +tend to convey false ideas of them; but, examined in +themselves, they have no other object than to permit us +to reason with facility respecting these two kinds of extension, +making complete abstraction of that which ought +not to be taken into consideration. Now for this it is +sufficient to conceive the dimension which we wish to +eliminate as becoming gradually smaller and smaller, +the two others remaining the same, until it arrives at +such a degree of tenuity that it can no longer fix the attention. +It is thus that we naturally acquire the real +idea of a <i>surface</i>, and, by a second analogous operation, +the idea of a <i>line</i>, by repeating for breadth what we had +at first done for thickness. Finally, if we again repeat +the same operation, we arrive at the idea of a <i>point</i>, or +of an extension considered only with reference to its +place, abstraction being made of all magnitude, and designed +consequently to determine positions.</p> + +<p><i>Surfaces</i> evidently have, moreover, the general property +of exactly circumscribing volumes; and in the same +way, <i>lines</i>, in their turn, circumscribe <i>surfaces</i> and are +limited by <i>points</i>. But this consideration, to which too +much importance is often given, is only a secondary +one.</p><p><span class="pagenum"><a name="Page_184" id="Page_184">[Pg 184]</a></span></p> + +<p>Surfaces and lines are, then, in reality, always conceived +with three dimensions; it would be, in fact, impossible +to represent to one's self a surface otherwise than +as an extremely thin plate, and a line otherwise than as +an infinitely fine thread. It is even plain that the degree +of tenuity attributed by each individual to the dimensions +of which he wishes to make abstraction is not +constantly identical, for it must depend on the degree of +subtilty of his habitual geometrical observations. This +want of uniformity has, besides, no real inconvenience, +since it is sufficient, in order that the ideas of surface +and of line should satisfy the essential condition of their +destination, for each one to represent to himself the dimensions +which are to be neglected as being smaller than +all those whose magnitude his daily experience gives him +occasion to appreciate.</p> + +<p>We hence see how devoid of all meaning are the fantastic +discussions of metaphysicians upon the foundations +of geometry. It should also be remarked that these primordial +ideas are habitually presented by geometers in +an unphilosophical manner, since, for example, they explain +the notions of the different sorts of extent in an +order absolutely the inverse of their natural dependence, +which often produces the most serious inconveniences in +elementary instruction.</p> + + + + +<h3><a name="THE_FINAL_OBJECT_OF_GEOMETRY" id="THE_FINAL_OBJECT_OF_GEOMETRY">THE FINAL OBJECT OF GEOMETRY.</a></h3> + + +<p>These preliminaries being established, we can proceed +directly to the general definition of geometry, continuing +to conceive this science as having for its final object the +<i>measurement</i> of extension.</p> + +<p>It is necessary in this matter to go into a thorough<span class="pagenum"><a name="Page_185" id="Page_185">[Pg 185]</a></span> +explanation, founded on the distinction of the three kinds +of extension, since the notion of <i>measurement</i> is not exactly +the same with reference to surfaces and volumes +as to lines.</p> + + +<p><i>Nature of Geometrical Measurement.</i> If we take the +word <i>measurement</i> in its direct and general mathematical +acceptation, which signifies simply the determination +of the value of the <i>ratios</i> between any homogeneous +magnitudes, we must consider, in geometry, that the +<i>measurement</i> of surfaces and of volumes, unlike that of +lines, is never conceived, even in the most simple and the +most favourable cases, as being effected directly. The +comparison of two lines is regarded as direct; that of +two surfaces or of two volumes is, on the contrary, always +indirect. Thus we conceive that two lines may +be superposed; but the superposition of two surfaces, or, +still more so, of two volumes, is evidently impossible in +most cases; and, even when it becomes rigorously practicable, +such a comparison is never either convenient or +exact. It is, then, very necessary to explain wherein +properly consists the truly geometrical measurement of +a surface or of a volume.</p> + + +<p><i>Measurement of Surfaces and of Volumes.</i> For this +we must consider that, whatever may be the form of a +body, there always exists a certain number of lines, more +or less easy to be assigned, the length of which is sufficient +to define exactly the magnitude of its surface or of +its volume. Geometry, regarding these lines as alone +susceptible of being directly measured, proposes to deduce, +from the simple determination of them, the ratio of the +surface or of the volume sought, to the unity of surface, +or to the unity of volume. Thus the general object of<span class="pagenum"><a name="Page_186" id="Page_186">[Pg 186]</a></span> +geometry, with respect to surfaces and to volumes, is +properly to reduce all comparisons of surfaces or of volumes +to simple comparisons of lines.</p> + +<p>Besides the very great facility which such a transformation +evidently offers for the measurement of volumes +and of surfaces, there results from it, in considering it +in a more extended and more scientific manner, the general +possibility of reducing to questions of lines all questions +relating to volumes and to surfaces, considered with +reference to their magnitude. Such is often the most +important use of the geometrical expressions which determine +surfaces and volumes in functions of the corresponding +lines.</p> + +<p>It is true that direct comparisons between surfaces or +between volumes are sometimes employed; but such +measurements are not regarded as geometrical, but only +as a supplement sometimes necessary, although too rarely +applicable, to the insufficiency or to the difficulty of +truly rational methods. It is thus that we often determine +the volume of a body, and in certain cases its surface, +by means of its weight. In the same way, on other +occasions, when we can substitute for the proposed volume +an equivalent liquid volume, we establish directly +the comparison of the two volumes, by profiting by the +property possessed by liquid masses, of assuming any desired +form. But all means of this nature are purely mechanical, +and rational geometry necessarily rejects them.</p> + +<p>To render more sensible the difference between these +modes of determination and true geometrical measurements, +I will cite a single very remarkable example; the +manner in which Galileo determined the ratio of the ordinary +cycloid to that of the generating circle. The<span class="pagenum"><a name="Page_187" id="Page_187">[Pg 187]</a></span> +geometry of his time was as yet insufficient for the rational +solution of such a problem. Galileo conceived +the idea of discovering that ratio by a direct experiment. +Having weighed as exactly as possible two plates of the +same material and of equal thickness, one of them having +the form of a circle and the other that of the generated +cycloid, he found the weight of the latter always +triple that of the former; whence he inferred that the +area of the cycloid is triple that of the generating circle, +a result agreeing with the veritable solution subsequently +obtained by Pascal and Wallis. Such a success evidently +depends on the extreme simplicity of the ratio +sought; and we can understand the necessary insufficiency +of such expedients, even when they are actually practicable.</p> + +<p>We see clearly, from what precedes, the nature of that +part of geometry relating to <i>volumes</i> and that relating to +<i>surfaces</i>. But the character of the geometry of <i>lines</i> is +not so apparent, since, in order to simplify the exposition, +we have considered the measurement of lines as being +made directly. There is, therefore, needed a complementary +explanation with respect to them.</p> + + +<p><i>Measurement of curved Lines.</i> For this purpose, it +is sufficient to distinguish between the right line and +curved lines, the measurement of the first being alone +regarded as direct, and that of the other as always indirect. +Although superposition is sometimes strictly practicable +for curved lines, it is nevertheless evident that +truly rational geometry must necessarily reject it, as +not admitting of any precision, even when it is possible. +The geometry of lines has, then, for its general object, to +reduce in every case the measurement of curved lines to<span class="pagenum"><a name="Page_188" id="Page_188">[Pg 188]</a></span> +that of right lines; and consequently, in the most extended +point of view, to reduce to simple questions of +right lines all questions relating to the magnitude of any +curves whatever. To understand the possibility of such +a transformation, we must remark, that in every curve +there always exist certain right lines, the length of which +must be sufficient to determine that of the curve. Thus, +in a circle, it is evident that from the length of the radius +we must be able to deduce that of the circumference; +in the same way, the length of an ellipse depends +on that of its two axes; the length of a cycloid upon the +diameter of the generating circle, &c.; and if, instead +of considering the whole of each curve, we demand, more +generally, the length of any arc, it will be sufficient to +add to the different rectilinear parameters, which determine +the whole curve, the chord of the proposed arc, or +the co-ordinates of its extremities. To discover the relation +which exists between the length of a curved line +and that of similar right lines, is the general problem of +the part of geometry which relates to the study of lines.</p> + +<p>Combining this consideration with those previously +suggested with respect to volumes and to surfaces, we +may form a very clear idea of the science of geometry, +conceived in all its parts, by assigning to it, for its general +object, the final reduction of the comparisons of all +kinds of extent, volumes, surfaces, or lines, to simple comparisons +of right lines, the only comparisons regarded as +capable of being made directly, and which indeed could +not be reduced to any others more easy to effect. Such +a conception, at the same time, indicates clearly the veritable +character of geometry, and seems suited to show +at a single glance its utility and its perfection.</p><p><span class="pagenum"><a name="Page_189" id="Page_189">[Pg 189]</a></span></p> + + +<p><i>Measurement of right Lines.</i> In order to complete +this fundamental explanation, I have yet to show how +there can be, in geometry, a special section relating to +the right line, which seems at first incompatible with the +principle that the measurement of this class of lines must +always be regarded as direct.</p> + +<p>It is so, in fact, as compared with that of curved lines, +and of all the other objects which geometry considers. +But it is evident that the estimation of a right line cannot +be viewed as direct except so far as the linear unit can +be applied to it. Now this often presents insurmountable +difficulties, as I had occasion to show, for another +reason, in the introductory chapter. We must, then, +make the measurement of the proposed right line depend +on other analogous measurements capable of being effected +directly. There is, then, necessarily a primary distinct +branch of geometry, exclusively devoted to the right +line; its object is to determine certain right lines from +others by means of the relations belonging to the figures +resulting from their assemblage. This preliminary part +of geometry, which is almost imperceptible in viewing +the whole of the science, is nevertheless susceptible of a +great development. It is evidently of especial importance, +since all other geometrical measurements are referred +to those of right lines, and if they could not be determined, +the solution of every question would remain +unfinished.</p> + +<p>Such, then, are the various fundamental parts of rational +geometry, arranged according to their natural dependence; +the geometry of <i>lines</i> being first considered, +beginning with the right line; then the geometry of <i>surfaces</i>, +and, finally, that of <i>solids</i>.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_190" id="Page_190">[Pg 190]</a></span></p> + + + + +<h3><a name="INFINITE_EXTENT_OF_ITS_FIELD" id="INFINITE_EXTENT_OF_ITS_FIELD">INFINITE EXTENT OF ITS FIELD.</a></h3> + + +<p>Having determined with precision the general and +final object of geometrical inquiries, the science must +now be considered with respect to the field embraced by +each of its three fundamental sections.</p> + +<p>Thus considered, geometry is evidently susceptible, +by its nature, of an extension which is rigorously infinite; +for the measurement of lines, of surfaces, or +of volumes presents necessarily as many distinct questions +as we can conceive different figures subjected to +exact definitions; and their number is evidently infinite.</p> + +<p>Geometers limited themselves at first to consider the +most simple figures which were directly furnished them +by nature, or which were deduced from these primitive +elements by the least complicated combinations. But +they have perceived, since Descartes, that, in order to constitute +the science in the most philosophical manner, it +was necessary to make it apply to all imaginable figures. +This abstract geometry will then inevitably comprehend +as particular cases all the different real figures which +the exterior world could present. It is then a fundamental +principle in truly rational geometry to consider, as +far as possible, all figures which can be rigorously conceived.</p> + +<p>The most superficial examination is enough to convince +us that these figures present a variety which is +quite infinite.</p> + + +<p><i>Infinity of Lines.</i> With respect to curved <i>lines</i>, regarding +them as generated by the motion of a point governed +by a certain law, it is plain that we shall have, in<span class="pagenum"><a name="Page_191" id="Page_191">[Pg 191]</a></span> +general, as many different curves as we conceive different +laws for this motion, which may evidently be determined +by an infinity of distinct conditions; although it +may sometimes accidentally happen that new generations +produce curves which have been already obtained. Thus, +among plane curves, if a point moves so as to remain constantly +at the same distance from a fixed point, it will +generate a <i>circle</i>; if it is the sum or the difference of +its distances from two fixed points which remains constant, +the curve described will be an <i>ellipse</i> or an <i>hyperbola</i>; +if it is their product, we shall have an entirely different +curve; if the point departs equally from a fixed +point and from a fixed line, it will describe a <i>parabola</i>; +if it revolves on a circle at the same time that this circle +rolls along a straight line, we shall have a <i>cycloid</i>; +if it advances along a straight line, while this line, fixed +at one of its extremities, turns in any manner whatever, +there will result what in general terms are called <i>spirals</i>, +which of themselves evidently present as many +perfectly distinct curves as we can suppose different relations +between these two motions of translation and of +rotation, &c. Each of these different curves may then +furnish new ones, by the different general constructions +which geometers have imagined, and which give rise to +evolutes, to epicycloids, to caustics, &c. Finally, there +exists a still greater variety among curves of double curvature.</p> + + +<p><i>Infinity of Surfaces.</i> As to <i>surfaces</i>, the figures are +necessarily more different still, considering them as generated +by the motion of lines. Indeed, the figure may +then vary, not only in considering, as in curves, the different +infinitely numerous laws to which the motion of<span class="pagenum"><a name="Page_192" id="Page_192">[Pg 192]</a></span> +the generating line may be subjected, but also in supposing +that this line itself may change its nature; a circumstance +which has nothing analogous in curves, since +the points which describe them cannot have any distinct +figure. Two classes of very different conditions may +then cause the figures of surfaces to vary, while there +exists only one for lines. It is useless to cite examples +of this doubly infinite multiplicity of surfaces. It would +be sufficient to consider the extreme variety of the single +group of surfaces which may be generated by a right line, +and which comprehends the whole family of cylindrical +surfaces, that of conical surfaces, the most general class +of developable surfaces, &c.</p> + + +<p><i>Infinity of Volumes.</i> With respect to <i>volumes</i>, there +is no occasion for any special consideration, since they are +distinguished from each other only by the surfaces which +bound them.</p> + +<p>In order to complete this sketch, it should be added +that surfaces themselves furnish a new general means of +conceiving new curves, since every curve may be regarded +as produced by the intersection of two surfaces. It +is in this way, indeed, that the first lines which we may +regard as having been truly invented by geometers were +obtained, since nature gave directly the straight line and +the circle. We know that the ellipse, the parabola, and +the hyperbola, the only curves completely studied by the +ancients, were in their origin conceived only as resulting +from the intersection of a cone with circular base by +a plane in different positions. It is evident that, by the +combined employment of these different general means +for the formation of lines and of surfaces, we could produce +a rigorously infinitely series of distinct forms in<span class="pagenum"><a name="Page_193" id="Page_193">[Pg 193]</a></span> +starting from only a very small number of figures directly +furnished by observation.</p> + + +<p><i>Analytical invention of Curves, &c.</i> Finally, all +the various direct means for the invention of figures +have scarcely any farther importance, since rational geometry +has assumed its final character in the hands of +Descartes. Indeed, as we shall see more fully in chapter +iii., the invention of figures is now reduced to the +invention of equations, so that nothing is more easy than +to conceive new lines and new surfaces, by changing at +will the functions introduced into the equations. This +simple abstract procedure is, in this respect, infinitely +more fruitful than all the direct resources of geometry, developed +by the most powerful imagination, which should +devote itself exclusively to that order of conceptions. It +also explains, in the most general and the most striking +manner, the necessarily infinite variety of geometrical +forms, which thus corresponds to the diversity of analytical +functions. Lastly, it shows no less clearly that the +different forms of surfaces must be still more numerous +than those of lines, since lines are represented analytically +by equations with two variables, while surfaces give +rise to equations with three variables, which necessarily +present a greater diversity.</p> + +<p>The preceding considerations are sufficient to show +clearly the rigorously infinite extent of each of the three +general sections of geometry.</p> + + + + +<h3><a name="EXPANSION_OF_ORIGINAL_DEFINITION" id="EXPANSION_OF_ORIGINAL_DEFINITION">EXPANSION OF ORIGINAL DEFINITION.</a></h3> + + +<p>To complete the formation of an exact and sufficiently +extended idea of the nature of geometrical inquiries, +it is now indispensable to return to the general definition<span class="pagenum"><a name="Page_194" id="Page_194">[Pg 194]</a></span> +above given, in order to present it under a new point of +view, without which the complete science would be only +very imperfectly conceived.</p> + +<p>When we assign as the object of geometry the <i>measurement</i> +of all sorts of lines, surfaces, and volumes, that +is, as has been explained, the reduction of all geometrical +comparisons to simple comparisons of right lines, we +have evidently the advantage of indicating a general destination +very precise and very easy to comprehend. But +if we set aside every definition, and examine the actual +composition of the science of geometry, we will at first +be induced to regard the preceding definition as much +too narrow; for it is certain that the greater part of the +investigations which constitute our present geometry do +not at all appear to have for their object the <i>measurement</i> +of extension. In spite of this fundamental objection, +I will persist in retaining this definition; for, in +fact, if, instead of confining ourselves to considering the +different questions of geometry isolatedly, we endeavour +to grasp the leading questions, in comparison with which +all others, however important they may be, must be regarded +as only secondary, we will finally recognize that +the measurement of lines, of surfaces, and of volumes, is +the invariable object, sometimes <i>direct</i>, though most often +<i>indirect</i>, of all geometrical labours.</p> + +<p>This general proposition being fundamental, since it +can alone give our definition all its value, it is indispensable +to enter into some developments upon this subject.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_195" id="Page_195">[Pg 195]</a></span></p> + + + + +<h3><a name="PROPERTIES_OF_LINES_AND_SURFACES" id="PROPERTIES_OF_LINES_AND_SURFACES">PROPERTIES OF LINES AND SURFACES.</a></h3> + + +<p>When we examine with attention the geometrical investigations +which do not seem to relate to the <i>measurement</i> +of extent, we find that they consist essentially in +the study of the different <i>properties</i> of each line or of each +surface; that is, in the knowledge of the different modes +of generation, or at least of definition, peculiar to each +figure considered. Now we can easily establish in the +most general manner the necessary relation of such a +study to the question of <i>measurement</i>, for which the +most complete knowledge of the properties of each form +is an indispensable preliminary. This is concurrently +proven by two considerations, equally fundamental, although +quite distinct in their nature.</p> + + +<p><span class="smcap">Necessity of their Study</span>: 1. <i>To find the most suitable +Property.</i> The <i>first</i>, purely scientific, consists in +remarking that, if we did not know any other characteristic +property of each line or surface than that one according +to which geometers had first conceived it, in +most cases it would be impossible to succeed in the solution +of questions relating to its <i>measurement</i>. In fact, +it is easy to understand that the different definitions +which each figure admits of are not all equally suitable +for such an object, and that they even present the most +complete oppositions in that respect. Besides, since the +primitive definition of each figure was evidently not chosen +with this condition in view, it is clear that we must +not expect, in general, to find it the most suitable; +whence results the necessity of discovering others, that +is, of studying as far as is possible the <i>properties</i> of the +proposed figure. Let us suppose, for example, that the<span class="pagenum"><a name="Page_196" id="Page_196">[Pg 196]</a></span> +circle is defined to be "the curve which, with the same +contour, contains the greatest area." This is certainly +a very characteristic property, but we would evidently +find insurmountable difficulties in trying to deduce from +such a starting point the solution of the fundamental +questions relating to the rectification or to the quadrature +of this curve. It is clear, in advance, that the +property of having all its points equally distant from a +fixed point must evidently be much better adapted to +inquiries of this nature, even though it be not precisely +the most suitable. In like manner, would Archimedes +ever have been able to discover the quadrature of the +parabola if he had known no other property of that curve +than that it was the section of a cone with a circular +base, by a plane parallel to its generatrix? The purely +speculative labours of preceding geometers, in transforming +this first definition, were evidently indispensable +preliminaries to the direct solution of such a question. +The same is true, in a still greater degree, with respect +to surfaces. To form a just idea of this, we need only +compare, as to the question of cubature or quadrature, +the common definition of the sphere with that one, no +less characteristic certainly, which would consist in regarding +a spherical body, as that one which, with the +same area, contains the greatest volume.</p> + +<p>No more examples are needed to show the necessity +of knowing, so far as is possible, all the properties of each +line or of each surface, in order to facilitate the investigation +of rectifications, of quadratures, and of cubatures, +which constitutes the final object of geometry. We may +even say that the principal difficulty of questions of this +kind consists in employing in each case the property which<span class="pagenum"><a name="Page_197" id="Page_197">[Pg 197]</a></span> +is best adapted to the nature of the proposed problem. +Thus, while we continue to indicate, for more precision, +the measurement of extension as the general destination +of geometry, this first consideration, which goes to the +very bottom of the subject, shows clearly the necessity +of including in it the study, as thorough as possible, of +the different generations or definitions belonging to the +same form.</p> + + +<p>2. <i>To pass from the Concrete to the Abstract.</i> A +second consideration, of at least equal importance, consists +in such a study being indispensable for organizing +in a rational manner the relation of the abstract to the +concrete in geometry.</p> + +<p>The science of geometry having to consider all imaginable +figures which admit of an exact definition, it necessarily +results from this, as we have remarked, that +questions relating to any figures presented by nature +are always implicitly comprised in this abstract geometry, +supposed to have attained its perfection. But when +it is necessary to actually pass to concrete geometry, we +constantly meet with a fundamental difficulty, that of +knowing to which of the different abstract types we are +to refer, with sufficient approximation, the real lines or +surfaces which we have to study. Now it is for the +purpose of establishing such a relation that it is particularly +indispensable to know the greatest possible number +of properties of each figure considered in geometry.</p> + +<p>In fact, if we always confined ourselves to the single +primitive definition of a line or of a surface, supposing +even that we could then <i>measure</i> it (which, according to +the first order of considerations, would generally be impossible), +this knowledge would remain almost necessarily<span class="pagenum"><a name="Page_198" id="Page_198">[Pg 198]</a></span> +barren in the application, since we should not ordinarily +know how to recognize that figure in nature when +it presented itself there; to ensure that, it would be necessary +that the single characteristic, according to which +geometers had conceived it, should be precisely that one +whose verification external circumstances would admit: +a coincidence which would be purely fortuitous, and on +which we could not count, although it might sometimes +take place. It is, then, only by multiplying as much as +possible the characteristic properties of each abstract figure, +that we can be assured, in advance, of recognizing +it in the concrete state, and of thus turning to account +all our rational labours, by verifying in each case the definition +which is susceptible of being directly proven. This +definition is almost always the only one in given circumstances, +and varies, on the other hand, for the same +figure, with different circumstances; a double reason for +its previous determination.</p> + + +<p><i>Illustration: Orbits of the Planets.</i> The geometry +of the heavens furnishes us with a very memorable example +in this matter, well suited to show the general necessity +of such a study. We know that the ellipse was +discovered by Kepler to be the curve which the planets +describe about the sun, and the satellites about their +planets. Now would this fundamental discovery, which +re-created astronomy, ever have been possible, if geometers +had been always confined to conceiving the ellipse +only as the oblique section of a circular cone by a +plane? No such definition, it is evident, would admit +of such a verification. The most general property of the +ellipse, that the sum of the distances from any of its points +to two fixed points is a constant quantity, is undoubtedly<span class="pagenum"><a name="Page_199" id="Page_199">[Pg 199]</a></span> +much more susceptible, by its nature, of causing the +curve to be recognized in this case, but still is not directly +suitable. The only characteristic which can here +be immediately verified is that which is derived from the +relation which exists in the ellipse between the length of +the focal distances and their direction; the only relation +which admits of an astronomical interpretation, as expressing +the law which connects the distance from the +planet to the sun, with the time elapsed since the beginning +of its revolution. It was, then, necessary that the +purely speculative labours of the Greek geometers on the +properties of the conic sections should have previously +presented their generation under a multitude of different +points of view, before Kepler could thus pass from the +abstract to the concrete, in choosing from among all these +different characteristics that one which could be most +easily proven for the planetary orbits.</p> + + +<p><i>Illustration: Figure of the Earth.</i> Another example +of the same order, but relating to surfaces, occurs in +considering the important question of the figure of the +earth. If we had never known any other property of the +sphere than its primitive character of having all its points +equally distant from an interior point, how would we ever +have been able to discover that the surface of the earth +was spherical? For this, it was necessary previously to +deduce from this definition of the sphere some properties +capable of being verified by observations made upon the +surface alone, such as the constant ratio which exists between +the length of the path traversed in the direction +of any meridian of a sphere going towards a pole, and +the angular height of this pole above the horizon at each +point. Another example, but involving a much longer<span class="pagenum"><a name="Page_200" id="Page_200">[Pg 200]</a></span> +series of preliminary speculations, is the subsequent proof +that the earth is not rigorously spherical, but that its +form is that of an ellipsoid of revolution.</p> + +<p>After such examples, it would be needless to give any +others, which any one besides may easily multiply. All +of them prove that, without a very extended knowledge +of the different properties of each figure, the relation of +the abstract to the concrete, in geometry, would be purely +accidental, and that the science would consequently want +one of its most essential foundations.</p> + +<p>Such, then, are two general considerations which fully +demonstrate the necessity of introducing into geometry a +great number of investigations which have not the <i>measurement</i> +of extension for their direct object; while we +continue, however, to conceive such a measurement as +being the final destination of all geometrical science. In +this way we can retain the philosophical advantages of +the clearness and precision of this definition, and still include +in it, in a very logical though indirect manner, all +known geometrical researches, in considering those which +do not seem to relate to the measurement of extension, +as intended either to prepare for the solution of the final +questions, or to render possible the application of the solutions +obtained.</p> + +<p>Having thus recognized, as a general principle, the close +and necessary connexion of the study of the properties of +lines and surfaces with those researches which constitute +the final object of geometry, it is evident that geometers, +in the progress of their labours, must by no means constrain +themselves to keep such a connexion always in +view. Knowing, once for all, how important it is to +vary as much as possible the manner of conceiving each<span class="pagenum"><a name="Page_201" id="Page_201">[Pg 201]</a></span> +figure, they should pursue that study, without considering +of what immediate use such or such a special property +may be for rectifications, quadratures, and cubatures. +They would uselessly fetter their inquiries by attaching +a puerile importance to the continued establishment of +that co-ordination.</p> + +<p>This general exposition of the general object of geometry +is so much the more indispensable, since, by the very +nature of the subject, this study of the different properties +of each line and of each surface necessarily composes +by far the greater part of the whole body of geometrical +researches. Indeed, the questions directly relating to rectifications, +to quadratures, and to cubatures, are evidently, +by themselves, very few in number for each figure considered. +On the other hand, the study of the properties +of the same figure presents an unlimited field to the activity +of the human mind, in which it may always hope +to make new discoveries. Thus, although geometers have +occupied themselves for twenty centuries, with more or +less activity undoubtedly, but without any real interruption, +in the study of the conic sections, they are far from +regarding that so simple subject as being exhausted; and +it is certain, indeed, that in continuing to devote themselves +to it, they would not fail to find still unknown +properties of those different curves. If labours of this +kind have slackened considerably for a century past, it +is not because they are completed, but only, as will be +presently explained, because the philosophical revolution +in geometry, brought about by Descartes, has singularly +diminished the importance of such researches.</p> + +<p>It results from the preceding considerations that not +only is the field of geometry necessarily infinite because<span class="pagenum"><a name="Page_202" id="Page_202">[Pg 202]</a></span> +of the variety of figures to be considered, but also in virtue +of the diversity of the points of view under the same +figure may be regarded. This last conception is, indeed, +that which gives the broadest and most complete idea of +the whole body of geometrical researches. We see that +studies of this kind consist essentially, for each line or for +each surface, in connecting all the geometrical phenomena +which it can present, with a single fundamental phenomenon, +regarded as the primitive definition.</p> + + + + +<h3><a name="THE_TWO_GENERAL_METHODS_OF_GEOMETRY" id="THE_TWO_GENERAL_METHODS_OF_GEOMETRY">THE TWO GENERAL METHODS OF GEOMETRY.</a></h3> + + +<p>Having now explained in a general and yet precise +manner the final object of geometry, and shown how the +science, thus defined, comprehends a very extensive class +of researches which did not at first appear necessarily to +belong to it, there remains to be considered the <i>method</i> +to be followed for the formation of this science. This +discussion is indispensable to complete this first sketch +of the philosophical character of geometry. I shall here +confine myself to indicating the most general consideration +in this matter, developing and summing up this important +fundamental idea in the following chapters.</p> + +<p>Geometrical questions may be treated according to +<i>two methods</i> so different, that there result from them two +sorts of geometry, so to say, the philosophical character +of which does not seem to me to have yet been properly +apprehended. The expressions of <i>Synthetical Geometry</i> +and <i>Analytical Geometry</i>, habitually employed to designate +them, give a very false idea of them. I would much +prefer the purely historical denominations of <i>Geometry of +the Ancients</i> and <i>Geometry of the Moderns</i>, which have +at least the advantage of not causing their true character<span class="pagenum"><a name="Page_203" id="Page_203">[Pg 203]</a></span> +to be misunderstood. But I propose to employ henceforth +the regular expressions of <i>Special Geometry</i> and +<i>General Geometry</i>, which seem to me suited to characterize +with precision the veritable nature of the two +methods.</p> + + +<p><i>Their fundamental Difference.</i> The fundamental +difference between the manner in which we conceive +Geometry since Descartes, and the manner in which the +geometers of antiquity treated geometrical questions, is +not the use of the Calculus (or Algebra), as is commonly +thought to be the case. On the one hand, it is certain +that the use of the calculus was not entirely unknown +to the ancient geometers, since they used to make continual +and very extensive applications of the theory of +proportions, which was for them, as a means of deduction, +a sort of real, though very imperfect and especially +extremely limited equivalent for our present algebra. +The calculus may even be employed in a much more +complete manner than they have used it, in order to obtain +certain geometrical solutions, which will still retain +all the essential character of the ancient geometry; this +occurs very frequently with respect to those problems of +geometry of two or of three dimensions, which are commonly +designated under the name of <i>determinate</i>. On +the other hand, important as is the influence of the calculus +in our modern geometry, various solutions obtained +without algebra may sometimes manifest the peculiar +character which distinguishes it from the ancient geometry, +although analysis is generally indispensable. I will +cite, as an example, the method of Roberval for tangents, +the nature of which is essentially modern, and which, +however, leads in certain cases to complete solutions,<span class="pagenum"><a name="Page_204" id="Page_204">[Pg 204]</a></span> +without any aid from the calculus. It is not, then, the +instrument of deduction employed which is the principal +distinction between the two courses which the human +mind can take in geometry.</p> + +<p>The real fundamental difference, as yet imperfectly +apprehended, seems to me to consist in the very nature +of the questions considered. In truth, geometry, viewed +as a whole, and supposed to have attained entire perfection, +must, as we have seen on the one hand, embrace +all imaginable figures, and, on the other, discover +all the properties of each figure. It admits, from this +double consideration, of being treated according to two +essentially distinct plans; either, 1°, by grouping together +all the questions, however different they may be, +which relate to the same figure, and isolating those relating +to different bodies, whatever analogy there may +exist between them; or, 2°, on the contrary, by uniting +under one point of view all similar inquiries, to whatever +different figures they may relate, and separating the +questions relating to the really different properties of the +same body. In a word, the whole body of geometry +may be essentially arranged either with reference to the +<i>bodies</i> studied or to the <i>phenomena</i> to be considered. +The first plan, which is the most natural, was that of +the ancients; the second, infinitely more rational, is that +of the moderns since Descartes.</p> + + +<p><i>Geometry of the Ancients.</i> Indeed, the principal characteristics +of the ancient geometry is that they studied, +one by one, the different lines and the different surfaces, +not passing to the examination of a new figure till they +thought they had exhausted all that there was interesting +in the figures already known. In this way of proceeding,<span class="pagenum"><a name="Page_205" id="Page_205">[Pg 205]</a></span> +when they undertook the study of a new curve, +the whole of the labour bestowed on the preceding ones +could not offer directly any essential assistance, otherwise +than by the geometrical practice to which it had +trained the mind. Whatever might be the real similarity +of the questions proposed as to two different figures, +the complete knowledge acquired for the one could not +at all dispense with taking up again the whole of the investigation +for the other. Thus the progress of the mind +was never assured; so that they could not be certain, in +advance, of obtaining any solution whatever, however +analogous the proposed problem might be to questions +which had been already resolved. Thus, for example, +the determination of the tangents to the three conic sections +did not furnish any rational assistance for drawing +the tangent to any other new curve, such as the conchoid, +the cissoid, &c. In a word, the geometry of the +ancients was, according to the expression proposed above, +essentially special.</p> + + +<p><i>Geometry of the Moderns.</i> In the system of the +moderns, geometry is, on the contrary, eminently <i>general</i>, +that is to say, relating to any figures whatever. It +is easy to understand, in the first place, that all geometrical +expressions of any interest may be proposed with +reference to all imaginable figures. This is seen directly +in the fundamental problems—of rectifications, quadratures, +and cubatures—which constitute, as has been +shown, the final object of geometry. But this remark +is no less incontestable, even for investigations which relate +to the different <i>properties</i> of lines and of surfaces, +and of which the most essential, such as the question of +tangents or of tangent planes, the theory of curvatures,<span class="pagenum"><a name="Page_206" id="Page_206">[Pg 206]</a></span> +&c., are evidently common to all figures whatever. The +very few investigations which are truly peculiar to particular +figures have only an extremely secondary importance. +This being understood, modern geometry consists +essentially in abstracting, in order to treat it by itself, +in an entirely general manner, every question relating +to the same geometrical phenomenon, in whatever +bodies it may be considered. The application of the +universal theories thus constructed to the special determination +of the phenomenon which is treated of in each +particular body, is now regarded as only a subaltern labour, +to be executed according to invariable rules, and +the success of which is certain in advance. This labour +is, in a word, of the same character as the numerical calculation +of an analytical formula. There can be no other +merit in it than that of presenting in each case the solution +which is necessarily furnished by the general +method, with all the simplicity and elegance which the +line or the surface considered can admit of. But no real +importance is attached to any thing but the conception +and the complete solution of a new question belonging +to any figure whatever. Labours of this kind are alone +regarded as producing any real advance in science. The +attention of geometers, thus relieved from the examination +of the peculiarities of different figures, and wholly +directed towards general questions, has been thereby able +to elevate itself to the consideration of new geometrical +conceptions, which, applied to the curves studied by the +ancients, have led to the discovery of important properties +which they had not before even suspected. Such is +geometry, since the radical revolution produced by Descartes +in the general system of the science.</p><p><span class="pagenum"><a name="Page_207" id="Page_207">[Pg 207]</a></span></p> + + +<p><i>The Superiority of the modern Geometry.</i> The mere +indication of the fundamental character of each of the +two geometries is undoubtedly sufficient to make apparent +the immense necessary superiority of modern geometry. +We may even say that, before the great conception +of Descartes, rational geometry was not truly constituted +upon definitive bases, whether in its abstract or +concrete relations. In fact, as regards science, considered +speculatively, it is clear that, in continuing indefinitely +to follow the course of the ancients, as did the +moderns before Descartes, and even for a little while afterwards, +by adding some new curves to the small number +of those which they had studied, the progress thus +made, however rapid it might have been, would still be +found, after a long series of ages, to be very inconsiderable +in comparison with the general system of geometry, +seeing the infinite variety of the forms which would still +have remained to be studied. On the contrary, at each +question resolved according to the method of the moderns, +the number of geometrical problems to be resolved +is then, once for all, diminished by so much with respect +to all possible bodies. Another consideration is, that it +resulted, from their complete want of general methods, +that the ancient geometers, in all their investigations, +were entirely abandoned to their own strength, without +ever having the certainty of obtaining, sooner or later, +any solution whatever. Though this imperfection of the +science was eminently suited to call forth all their admirable +sagacity, it necessarily rendered their progress +extremely slow; we can form some idea of this by the +considerable time which they employed in the study of +the conic sections. Modern geometry, making the progress<span class="pagenum"><a name="Page_208" id="Page_208">[Pg 208]</a></span> +of our mind certain, permits us, on the contrary, to +make the greatest possible use of the forces of our intelligence, +which the ancients were often obliged to waste +on very unimportant questions.</p> + +<p>A no less important difference between the two systems +appears when we come to consider geometry in the +concrete point of view. Indeed, we have already remarked +that the relation of the abstract to the concrete +in geometry can be founded upon rational bases only so +far as the investigations are made to bear directly upon +all imaginable figures. In studying lines, only one by +one, whatever may be the number, always necessarily +very small, of those which we shall have considered, the +application of such theories to figures really existing in +nature will never have any other than an essentially +accidental character, since there is nothing to assure us +that these figures can really be brought under the abstract +types considered by geometers.</p> + +<p>Thus, for example, there is certainly something fortuitous +in the happy relation established between the +speculations of the Greek geometers upon the conic sections +and the determination of the true planetary orbits. +In continuing geometrical researches upon the same plan, +there was no good reason for hoping for similar coincidences; +and it would have been possible, in these special +studies, that the researches of geometers should have +been directed to abstract figures entirely incapable of any +application, while they neglected others, susceptible perhaps +of an important and immediate application. It is +clear, at least, that nothing positively guaranteed the +necessary applicability of geometrical speculations. It +is quite another thing in the modern geometry. From<span class="pagenum"><a name="Page_209" id="Page_209">[Pg 209]</a></span> +the single circumstance that in it we proceed by general +questions relating to any figures whatever, we have in +advance the evident certainty that the figures really existing +in the external world could in no case escape the +appropriate theory if the geometrical phenomenon which +it considers presents itself in them.</p> + +<p>From these different considerations, we see that the +ancient system of geometry wears essentially the character +of the infancy of the science, which did not begin +to become completely rational till after the philosophical +resolution produced by Descartes. But it is evident, on +the other hand, that geometry could not be at first conceived +except in this <i>special</i> manner. <i>General</i> geometry +would not have been possible, and its necessity could +not even have been felt, if a long series of special labours +on the most simple figures had not previously furnished +bases for the conception of Descartes, and rendered apparent +the impossibility of persisting indefinitely in the +primitive geometrical philosophy.</p> + + +<p><i>The Ancient the Base of the Modern.</i> From this last +consideration we must infer that, although the geometry +which I have called <i>general</i> must be now regarded as +the only true dogmatical geometry, and that to which +we shall chiefly confine ourselves, the other having no +longer much more than an historical interest, nevertheless +it is not possible to entirely dispense with <i>special</i> geometry +in a rational exposition of the science. We undoubtedly +need not borrow directly from ancient geometry +all the results which it has furnished; but, from the +very nature of the subject, it is necessarily impossible entirely +to dispense with the ancient method, which will +always serve as the preliminary basis of the science, dogmatically<span class="pagenum"><a name="Page_210" id="Page_210">[Pg 210]</a></span> +as well as historically. The reason of this is +easy to understand. In fact, <i>general</i> geometry being +essentially founded, as we shall soon establish, upon the +employment of the calculus in the transformation of geometrical +into analytical considerations, such a manner of +proceeding could not take possession of the subject immediately +at its origin. We know that the application +of mathematical analysis, from its nature, can never commence +any science whatever, since evidently it cannot +be employed until the science has already been sufficiently +cultivated to establish, with respect to the phenomena +considered, some <i>equations</i> which can serve as starting +points for the analytical operations. These fundamental +equations being once discovered, analysis will enable us +to deduce from them a multitude of consequences which +it would have been previously impossible even to suspect; +it will perfect the science to an immense degree, +both with respect to the generality of its conceptions and +to the complete co-ordination established between them. +But mere mathematical analysis could never be sufficient +to form the bases of any natural science, not even to demonstrate +them anew when they have once been established. +Nothing can dispense with the direct study of +the subject, pursued up to the point of the discovery of +precise relations.</p> + +<p>We thus see that the geometry of the ancients will +always have, by its nature, a primary part, absolutely necessary +and more or less extensive, in the complete system +of geometrical knowledge. It forms a rigorously +indispensable introduction to <i>general</i> geometry. But it +is to this that it must be limited in a completely dogmatic +exposition. I will consider, then, directly, in the<span class="pagenum"><a name="Page_211" id="Page_211">[Pg 211]</a></span> +following chapter, this <i>special</i> or <i>preliminary</i> geometry +restricted to exactly its necessary limits, in order to occupy +myself thenceforth only with the philosophical examination +of <i>general</i> or <i>definitive</i> geometry, the only one +which is truly rational, and which at present essentially +composes the science.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_212" id="Page_212">[Pg 212]</a></span></p> + + + + +<h2><a name="CHAPTER_IIa" id="CHAPTER_IIa">CHAPTER II.</a></h2> + +<h3>ANCIENT OR SYNTHETIC GEOMETRY.</h3> + + +<p>The geometrical method of the ancients necessarily +constituting a preliminary department in the dogmatical +system of geometry, designed to furnish <i>general</i> geometry +with indispensable foundations, it is now proper to +begin with determining wherein strictly consists this preliminary +function of <i>special</i> geometry, thus reduced to +the narrowest possible limits.</p> + + + + +<h3><a name="ITS_PROPER_EXTENT" id="ITS_PROPER_EXTENT">ITS PROPER EXTENT.</a></h3> + + +<p><i>Lines; Polygons; Polyhedrons.</i> In considering it +under this point of view, it is easy to recognize that we +might restrict it to the study of the right line alone for +what concerns the geometry of <i>lines</i>; to the <i>quadrature</i> +of rectilinear plane areas; and, lastly, to the <i>cubature</i> of +bodies terminated by plane faces. The elementary propositions +relating to these three fundamental questions +form, in fact, the necessary starting point of all geometrical +inquiries; they alone cannot be obtained except by +a direct study of the subject; while, on the contrary, +the complete theory of all other figures, even that of the +circle, and of the surfaces and volumes which are connected +with it, may at the present day be completely +comprehended in the domain of <i>general</i> or <i>analytical</i> +geometry; these primitive elements at once furnishing +<i>equations</i> which are sufficient to allow of the application<span class="pagenum"><a name="Page_213" id="Page_213">[Pg 213]</a></span> +of the calculus to geometrical questions, which would not +have been possible without this previous condition.</p> + +<p>It results from this consideration that, in common practice, +we give to <i>elementary</i> geometry more extent than +would be rigorously necessary to it; since, besides the +right line, polygons, and polyhedrons, we also include in +it the circle and the "round" bodies; the study of which +might, however, be as purely analytical as that, for example, +of the conic sections. An unreflecting veneration +for antiquity contributes to maintain this defect in method; +but the best reason which can be given for it is the +serious inconvenience for ordinary instruction which there +would be in postponing, to so distant an epoch of mathematical +education, the solution of several essential questions, +which are susceptible of a direct and continual application +to a great number of important uses. In fact, +to proceed in the most rational manner, we should employ +the integral calculus in obtaining the interesting +results relating to the length or the area of the circle, or +to the quadrature of the sphere, &c., which have been +determined by the ancients from extremely simple considerations. +This inconvenience would be of little importance +with regard to the persons destined to study +the whole of mathematical science, and the advantage +of proceeding in a perfectly logical order would have a +much greater comparative value. But the contrary case +being the more frequent, theories so essential have necessarily +been retained in elementary geometry. Perhaps +the conic sections, the cycloid, &c., might be advantageously +added in such cases.</p> + + +<p><i>Not to be farther restricted.</i> While this preliminary +portion of geometry, which cannot be founded on the application<span class="pagenum"><a name="Page_214" id="Page_214">[Pg 214]</a></span> +of the calculus, is reduced by its nature to a +very limited series of fundamental researches, relating to +the right line, polygonal areas, and polyhedrons, it is certain, +on the other hand, that we cannot restrict it any +more; although, by a veritable abuse of the spirit of +analysis, it has been recently attempted to present the +establishment of the principal theorems of elementary geometry +under an algebraical point of view. Thus some +have pretended to demonstrate, by simple abstract considerations +of mathematical analysis, the constant relation +which exists between the three angles of a rectilinear +triangle, the fundamental proposition of the theory +of similar triangles, that of parallelopipedons, &c.; in a +word, precisely the only geometrical propositions which +cannot be obtained except by a direct study of the subject, +without the calculus being susceptible of having +any part in it. Such aberrations are the unreflecting +exaggerations of that natural and philosophical tendency +which leads us to extend farther and farther the influence +of analysis in mathematical studies. In mechanics, +the pretended analytical demonstrations of the parallelogram +of forces are of similar character.</p> + +<p>The viciousness of such a manner of proceeding follows +from the principles previously presented. We have already, +in fact, recognized that, since the calculus is not, +and cannot be, any thing but a means of deduction, it +would indicate a radically false idea of it to wish to +employ it in establishing the elementary foundations of +any science whatever; for on what would the analytical +reasonings in such an operation repose? A labour of this +nature, very far from really perfecting the philosophical +character of a science, would constitute a return towards<span class="pagenum"><a name="Page_215" id="Page_215">[Pg 215]</a></span> +the metaphysical age, in presenting real facts as mere +logical abstractions.</p> + +<p>When we examine in themselves these pretended analytical +demonstrations of the fundamental propositions +of elementary geometry, we easily verify their necessary +want of meaning. They are all founded on a vicious +manner of conceiving the principle of <i>homogeneity</i>, the +true general idea of which was explained in the second +chapter of the preceding book. These demonstrations +suppose that this principle does not allow us to admit the +coexistence in the same equation of numbers obtained by +different concrete comparisons, which is evidently false, +and contrary to the constant practice of geometers. Thus +it is easy to recognize that, by employing the law of homogeneity +in this arbitrary and illegitimate acceptation, +we could succeed in "demonstrating," with quite as much +apparent rigour, propositions whose absurdity is manifest +at the first glance. In examining attentively, for example, +the procedure by the aid of which it has been attempted +to prove analytically that the sum of the three +angles of any rectilinear triangle is constantly equal to +two right angles, we see that it is founded on this preliminary +principle that, if two triangles have two of their +angles respectively equal, the third angle of the one will +necessarily be equal to the third angle of the other. This +first point being granted, the proposed relation is immediately +deduced from it in a very exact and simple manner. +Now the analytical consideration by which this +previous proposition has been attempted to be established, +is of such a nature that, if it could be correct, we +could rigorously deduce from it, in reproducing it conversely, +this palpable absurdity, that two sides of a triangle<span class="pagenum"><a name="Page_216" id="Page_216">[Pg 216]</a></span> +are sufficient, without any angle, for the entire determination +of the third side. We may make analogous +remarks on all the demonstrations of this sort, the sophisms +of which will be thus verified in a perfectly apparent +manner.</p> + +<p>The more reason that we have here to consider geometry +as being at the present day essentially analytical, the +more necessary was it to guard against this abusive exaggeration +of mathematical analysis, according to which +all geometrical observation would be dispensed with, in +establishing upon pure algebraical abstractions the very +foundations of this natural science.</p> + + +<p><i>Attempted Demonstrations of Axioms, &c.</i> Another +indication that geometers have too much overlooked the +character of a natural science which is necessarily inherent +in geometry, appears from their vain attempts, so +long made, to <i>demonstrate</i> rigorously, not by the aid of +the calculus, but by means of certain constructions, several +fundamental propositions of elementary geometry. +Whatever may be effected, it will evidently be impossible +to avoid sometimes recurring to simple and direct observation +in geometry as a means of establishing various +results. While, in this science, the phenomena +which are considered are, by virtue of their extreme simplicity, +much more closely connected with one another +than those relating to any other physical science, some +must still be found which cannot be deduced, and which, +on the contrary, serve as starting points. It may be +admitted that the greatest logical perfection of the science +is to reduce these to the smallest number possible, +but it would be absurd to pretend to make them completely +disappear. I avow, moreover, that I find fewer<span class="pagenum"><a name="Page_217" id="Page_217">[Pg 217]</a></span> +real inconveniences in extending, a little beyond what +would be strictly necessary, the number of these geometrical +notions thus established by direct observation, +provided they are sufficiently simple, than in making +them the subjects of complicated and indirect demonstrations, +even when these demonstrations may be logically +irreproachable.</p> + +<p>The true dogmatic destination of the geometry of the +ancients, reduced to its least possible indispensable developments, +having thus been characterized as exactly as +possible, it is proper to consider summarily each of the +principal parts of which it must be composed. I think +that I may here limit myself to considering the first and +the most extensive of these parts, that which has for its +object the study of <i>the right line</i>; the two other sections, +namely, the <i>quadrature of polygons</i> and the <i>cubature +of polyhedrons</i>, from their limited extent, not being capable +of giving rise to any philosophical consideration of +any importance, distinct from those indicated in the preceding +chapter with respect to the measure of areas and +of volumes in general.</p> + + + + +<h3><a name="GEOMETRY_OF_THE_RIGHT_LINE" id="GEOMETRY_OF_THE_RIGHT_LINE">GEOMETRY OF THE RIGHT LINE.</a></h3> + + +<p>The final question which we always have in view in +the study of the right line, properly consists in determining, +by means of one another, the different elements +of any right-lined figure whatever; which enables us +always to know indirectly the length and position of a +right line, in whatever circumstances it may be placed. +This fundamental problem is susceptible of two general +solutions, the nature of which is quite distinct, the one +<i>graphical</i>, the other <i>algebraic</i>. The first, though very<span class="pagenum"><a name="Page_218" id="Page_218">[Pg 218]</a></span> +imperfect, is that which must be first considered, because +it is spontaneously derived from the direct study +of the subject; the second, much more perfect in the +most important respects, cannot be studied till afterwards, +because it is founded upon the previous knowledge +of the other.</p> + + + + +<h3><a name="GRAPHICAL_SOLUTIONS" id="GRAPHICAL_SOLUTIONS">GRAPHICAL SOLUTIONS.</a></h3> + + +<p>The graphical solution consists in constructing at will +the proposed figure, either with the same dimensions, or, +more usually, with dimensions changed in any ratio whatever. +The first mode need merely be mentioned as being +the most simple and the one which would first occur +to the mind, for it is evidently, by its nature, almost entirely +incapable of application. The second is, on the +contrary, susceptible of being most extensively and most +usefully applied. We still make an important and continual +use of it at the present day, not only to represent +with exactness the forms of bodies and their relative positions, +but even for the actual determination of geometrical +magnitudes, when we do not need great precision. +The ancients, in consequence of the imperfection of their +geometrical knowledge, employed this procedure in a +much more extensive manner, since it was for a long time +the only one which they could apply, even in the most +important precise determinations. It was thus, for example, +that Aristarchus of Samos estimated the relative distance +from the sun and from the moon to the earth, by +making measurements on a triangle constructed as exactly +as possible, so as to be similar to the right-angled +triangle formed by the three bodies at the instant when +the moon is in quadrature, and when an observation of<span class="pagenum"><a name="Page_219" id="Page_219">[Pg 219]</a></span> +the angle at the earth would consequently be sufficient to +define the triangle. Archimedes himself, although he was +the first to introduce calculated determinations into geometry, +several times employed similar means. The +formation of trigonometry did not cause this method to +be entirely abandoned, although it greatly diminished its +use; the Greeks and the Arabians continued to employ +it for a great number of researches, in which we now regard +the use of the calculus as indispensable.</p> + +<p>This exact reproduction of any figure whatever on a +different scale cannot present any great theoretical difficulty +when all the parts of the proposed figure lie in the +same plane. But if we suppose, as most frequently happens, +that they are situated in different planes, we see, +then, a new order of geometrical considerations arise. +The artificial figure, which is constantly plane, not being +capable, in that case, of being a perfectly faithful image +of the real figure, it is necessary previously to fix with +precision the mode of representation, which gives rise to +different systems of <i>Projection</i>.</p> + +<p>It then remains to be determined according to what +laws the geometrical phenomena correspond in the two +figures. This consideration generates a new series of +geometrical investigations, the final object of which is +properly to discover how we can replace constructions in +relief by plane constructions. The ancients had to resolve +several elementary questions of this kind for various +cases in which we now employ spherical trigonometry, +principally for different problems relating to the celestial +sphere. Such was the object of their <i>analemmas</i>, +and of the other plane figures which for a long time supplied +the place of the calculus. We see by this that the<span class="pagenum"><a name="Page_220" id="Page_220">[Pg 220]</a></span> +ancients really knew the elements of what we now name +<i>Descriptive Geometry</i>, although they did not conceive it +in a distinct and general manner.</p> + +<p>I think it proper briefly to indicate in this place the +true philosophical character of this "Descriptive Geometry;" +although, being essentially a science of application, +it ought not to be included within the proper domain of +this work.</p> + + + + +<h3><a name="DESCRIPTIVE_GEOMETRY" id="DESCRIPTIVE_GEOMETRY">DESCRIPTIVE GEOMETRY.</a></h3> + + +<p>All questions of geometry of three dimensions necessarily +give rise, when we consider their graphical solution, +to a common difficulty which is peculiar to them; +that of substituting for the different constructions in relief, +which are necessary to resolve them directly, and +which it is almost always impossible to execute, simple +equivalent plane constructions, by means of which we +finally obtain the same results. Without this indispensable +transformation, every solution of this kind would be +evidently incomplete and really inapplicable in practice, +although theoretically the constructions in space are usually +preferable as being more direct. It was in order to +furnish general means for always effecting such a transformation +that <i>Descriptive Geometry</i> was created, and +formed into a distinct and homogeneous system, by the +illustrious <span class="smcap">Monge</span>. He invented, in the first place, a uniform +method of representing bodies by figures traced on a +single plane, by the aid of <i>projections</i> on two different +planes, usually perpendicular to each other, and one of +which is supposed to turn about their common intersection +so as to coincide with the other produced; in this +system, or in any other equivalent to it, it is sufficient<span class="pagenum"><a name="Page_221" id="Page_221">[Pg 221]</a></span> +to regard points and lines as being determined by their +projections, and surfaces by the projections of their generating +lines. This being established, Monge—analyzing +with profound sagacity the various partial labours of +this kind which had before been executed by a number +of incongruous procedures, and considering also, in a general +and direct manner, in what any questions of that +nature must consist—found that they could always be +reduced to a very small number of invariable abstract +problems, capable of being resolved separately, once for +all, by uniform operations, relating essentially some to +the contacts and others to the intersections of surfaces. +Simple and entirely general methods for the graphical +solution of these two orders of problems having been +formed, all the geometrical questions which may arise in +any of the various arts of construction—stone-cutting, +carpentry, perspective, dialling, fortification, &c.—can +henceforth be treated as simple particular cases of a single +theory, the invariable application of which will always +necessarily lead to an exact solution, which may +be facilitated in practice by profiting by the peculiar +circumstances of each case.</p> + +<p><span class="pagenum"><a name="Page_222" id="Page_222">[Pg 222]</a></span><span class="pagenum"><a name="Page_223" id="Page_223">[Pg 223]</a></span></p> + +<hr class="tb" /> + +<p>This important creation deserves in a remarkable degree +to fix the attention of those philosophers who consider +all that the human species has yet effected as a +first step, and thus far the only really complete one, towards +that general renovation of human labours, which +must imprint upon all our arts a character of precision +and of rationality, so necessary to their future progress. +Such a revolution must, in fact, inevitably commence +with that class of industrial labours, which is essentially +connected with that science which is the most simple, +the most perfect, and the most ancient. It cannot fail +to extend hereafter, though with less facility, to all other +practical operations. Indeed Monge himself, who conceived +the true philosophy of the arts better than any one +else, endeavoured to sketch out a corresponding system +for the mechanical arts.</p> + +<p>Essential as the conception of descriptive geometry +really is, it is very important not to deceive ourselves +with respect to its true destination, as did those who, +in the excitement of its first discovery, saw in it a means +of enlarging the general and abstract domain of rational +geometry. The result has in no way answered to these +mistaken hopes. And, indeed, is it not evident that descriptive +geometry has no special value except as a science +of application, and as forming the true special theory of +the geometrical arts? Considered in its abstract relations, +it could not introduce any truly distinct order of +geometrical speculations. We must not forget that, in +order that a geometrical question should fall within the +peculiar domain of descriptive geometry, it must necessarily +have been previously resolved by speculative geometry, +the solutions of which then, as we have seen, +always need to be prepared for practice in such a way as +to supply the place of constructions in relief by plane +constructions; a substitution which really constitutes the +only characteristic function of descriptive geometry.</p> + +<p>It is proper, however, to remark here, that, with regard +to intellectual education, the study of descriptive geometry +possesses an important philosophical peculiarity, quite +independent of its high industrial utility. This is the +advantage which it so pre-eminently offers—in habituating +the mind to consider very complicated geometrical +combinations in space, and to follow with precision their +continual correspondence with the figures which are actually +traced—of thus exercising to the utmost, in the +most certain and precise manner, that important faculty +of the human mind which is properly called "imagination," +and which consists, in its elementary and positive +acceptation, in representing to ourselves, clearly and easily, +a vast and variable collection of ideal objects, as if +they were really before us.</p> + +<p>Finally, to complete the indication of the general nature +of descriptive geometry by determining its logical +character, we have to observe that, while it belongs to +the geometry of the ancients by the character of its solutions, +on the other hand it approaches the geometry of +the moderns by the nature of the questions which compose +it. These questions are in fact eminently remarkable +for that generality which, as we saw in the preceding +chapter, constitutes the true fundamental character +of modern geometry; for the methods used are always +conceived as applicable to any figures whatever, the peculiarity +of each having only a purely secondary influence. +The solutions of descriptive geometry are then graphical, +like most of those of the ancients, and at the same time +general, like those of the moderns.</p> + +<p><span class="pagenum"><a name="Page_224" id="Page_224">[Pg 224]</a></span></p> + +<hr class="tb" /> + +<p>After this important digression, we will pursue the +philosophical examination of <i>special</i> geometry, always +considered as reduced to its least possible development, +as an indispensable introduction to <i>general</i> geometry. +We have now sufficiently considered the <i>graphical</i> solution +of the fundamental problem relating to the right line—that +is, the determination of the different elements of any +right-lined figure by means of one another—and have +now to examine in a special manner the <i>algebraic</i> solution.</p> + + + + +<h3><a name="ALGEBRAIC_SOLUTIONS" id="ALGEBRAIC_SOLUTIONS">ALGEBRAIC SOLUTIONS.</a></h3> + + +<p>This kind of solution, the evident superiority of which +need not here be dwelt upon, belongs necessarily, by the +very nature of the question, to the system of the ancient +geometry, although the logical method which is employed +causes it to be generally, but very improperly, separated +from it. We have thus an opportunity of verifying, in +a very important respect, what was established generally +in the preceding chapter, that it is not by the employment +of the calculus that the modern geometry is essentially +to be distinguished from the ancient. The ancients +are in fact the true inventors of the present trigonometry, +spherical as well as rectilinear; it being only much +less perfect in their hands, on account of the extreme inferiority +of their algebraical knowledge. It is, then, really +in this chapter, and not, as it might at first be thought, +in those which we shall afterwards devote to the philosophical +examination of <i>general</i> geometry, that it is proper +to consider the character of this important preliminary +theory, which is usually, though improperly, included in +what is called <i>analytical geometry</i>, but which is really +only a complement of <i>elementary geometry</i> properly so +called.</p> + +<p>Since all right-lined figures can be decomposed into +triangles, it is evidently sufficient to know how to determine +the different elements of a triangle by means of one +another, which reduces <i>polygonometry</i> to simple <i>trigonometry</i>.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_225" id="Page_225">[Pg 225]</a></span></p> + + + + +<h3><a name="TRIGONOMETRY" id="TRIGONOMETRY">TRIGONOMETRY.</a></h3> + + +<p>The difficulty in resolving algebraically such a question +as the above, consists essentially in forming, between +the angles and the sides of a triangle, three distinct equations; +which, when once obtained, will evidently reduce all +trigonometrical problems to mere questions of analysis.</p> + + +<p><i>How to introduce Angles.</i> In considering the establishment +of these equations in the most general manner, +we immediately meet with a fundamental distinction +with respect to the manner of introducing the angles +into the calculation, according as they are made to enter +<i>directly</i>, by themselves or by the circular arcs which are +proportional to them; or <i>indirectly</i>, by the chords of +these arcs, which are hence called their <i>trigonometrical +lines</i>. Of these two systems of trigonometry the second +was of necessity the only one originally adopted, as being +the only practicable one, since the condition of geometry +made it easy enough to find exact relations between the +sides of the triangles and the trigonometrical lines which +represent the angles, while it would have been absolutely +impossible at that epoch to establish equations between +the sides and the angles themselves.</p> + + +<p><i>Advantages of introducing Trigonometrical Lines.</i> +At the present day, since the solution can be obtained by +either system indifferently, that motive for preference no +longer exists; but geometers have none the less persisted +in following from choice the system primitively admitted +from necessity; for, the same reason which enabled these +trigonometrical equations to be obtained with much more +facility, must, in like manner, as it is still more easy to +conceive <i>à priori</i>, render these equations much more simple,<span class="pagenum"><a name="Page_226" id="Page_226">[Pg 226]</a></span> +since they then exist only between right lines, instead +of being established between right lines and arcs +of circles. Such a consideration has so much the more +importance, as the question relates to formulas which are +eminently elementary, and destined to be continually +employed in all parts of mathematical science, as well +as in all its various applications.</p> + +<p>It may be objected, however, that when an angle is +given, it is, in reality, always given by itself, and not by +its trigonometrical lines; and that when it is unknown, it +is its angular value which is properly to be determined, +and not that of any of its trigonometrical lines. It seems, +according to this, that such lines are only useless intermediaries +between the sides and the angles, which have +to be finally eliminated, and the introduction of which +does not appear capable of simplifying the proposed research. +It is indeed important to explain, with more +generality and precision than is customary, the great real +utility of this manner of proceeding.</p> + + +<p><i>Division of Trigonometry into two Parts.</i> It consists +in the fact that the introduction of these auxiliary +magnitudes divides the entire question of trigonometry +into two others essentially distinct, one of which has +for its object to pass from the angles to their trigonometrical +lines, or the converse, and the other of which +proposes to determine the sides of the triangles by the trigonometrical +lines of their angles, or the converse. Now +the first of these two fundamental questions is evidently +susceptible, by its nature, of being entirely treated and +reduced to numerical tables once for all, in considering +all possible angles, since it depends only upon those angles, +and not at all upon the particular triangles in which<span class="pagenum"><a name="Page_227" id="Page_227">[Pg 227]</a></span> +they may enter in each case; while the solution of the +second question must necessarily be renewed, at least in +its arithmetical relations, for each new triangle which it +is necessary to resolve. This is the reason why the first +portion of the complete work, which would be precisely +the most laborious, is no longer taken into the account, +being always done in advance; while, if such a decomposition +had not been performed, we would evidently have +found ourselves under the obligation of recommencing +the entire calculation in each particular case. Such is +the essential property of the present trigonometrical system, +which in fact would really present no actual advantage, +if it was necessary to calculate continually the +trigonometrical line of each angle to be considered, or the +converse; the intermediate agency introduced would then +be more troublesome than convenient.</p> + +<p>In order to clearly comprehend the true nature of this +conception, it will be useful to compare it with a still +more important one, designed to produce an analogous +effect either in its algebraic, or, still more, in its arithmetical +relations—the admirable theory of <i>logarithms</i>. +In examining in a philosophical manner the influence +of this theory, we see in fact that its general result is +to decompose all imaginable arithmetical operations into +two distinct parts. The first and most complicated of +these is capable of being executed in advance once for +all (since it depends only upon the numbers to be considered, +and not at all upon the infinitely different combinations +into which they can enter), and consists in considering +all numbers as assignable powers of a constant +number. The second part of the calculation, which must +of necessity be recommenced for each new formula which<span class="pagenum"><a name="Page_228" id="Page_228">[Pg 228]</a></span> +is to have its value determined, is thenceforth reduced +to executing upon these exponents correlative operations +which are infinitely more simple. I confine myself here +to merely indicating this resemblance, which any one can +carry out for himself.</p> + +<p>We must besides observe, as a property (secondary +at the present day, but all-important at its origin) of the +trigonometrical system adopted, the very remarkable circumstance +that the determination of angles by their trigonometrical +lines, or the converse, admits of an arithmetical +solution (the only one which is directly indispensable for +the special destination of trigonometry) without the previous +resolution of the corresponding algebraic question. +It is doubtless to such a peculiarity that the ancients +owed the possibility of knowing trigonometry. The investigation +conceived in this way was so much the more +easy, inasmuch as tables of chords (which the ancients +naturally took as the trigonometrical lines) had been previously +constructed for quite a different object, in the +course of the labours of Archimedes on the rectification +of the circle, from which resulted the actual determination +of a certain series of chords; so that when Hipparchus +subsequently invented trigonometry, he could +confine himself to completing that operation by suitable +intercalations; which shows clearly the connexion of ideas +in that matter.</p> + + +<p><i>The Increase of such Trigonometrical Lines.</i> To +complete this philosophical sketch of trigonometry, it is +proper now to observe that the extension of the same considerations +which lead us to replace angles or arcs of circles +by straight lines, with the view of simplifying our +equations, must also lead us to employ concurrently several<span class="pagenum"><a name="Page_229" id="Page_229">[Pg 229]</a></span> +trigonometrical lines, instead of confining ourselves +to one only (as did the ancients), so as to perfect this +system by choosing that one which will be algebraically +the most convenient on each occasion. In this point of +view, it is clear that the number of these lines is in itself +no ways limited; provided that they are determined by +the arc, and that they determine it, whatever may be the +law according to which they are derived from it, they are +suitable to be substituted for it in the equations. The +Arabians, and subsequently the moderns, in confining +themselves to the most simple constructions, have carried +to four or five the number of <i>direct</i> trigonometrical +lines, which might be extended much farther.</p> + +<p>But instead of recurring to geometrical formations, +which would finally become very complicated, we conceive +with the utmost facility as many new trigonometrical +lines as the analytical transformations may require, +by means of a remarkable artifice, which is not +usually apprehended in a sufficiently general manner. +It consists in not directly multiplying the trigonometrical +lines appropriate to each arc considered, but in introducing +new ones, by considering this arc as indirectly +determined by all lines relating to an arc which is a very +simple function of the first. It is thus, for example, that, +in order to calculate an angle with more facility, we will +determine, instead of its sine, the sine of its half, or of +its double, &c. Such a creation of <i>indirect</i> trigonometrical +lines is evidently much more fruitful than all +the direct geometrical methods for obtaining new ones. +We may accordingly say that the number of trigonometrical +lines actually employed at the present day by +geometers is in reality unlimited, since at every instant,<span class="pagenum"><a name="Page_230" id="Page_230">[Pg 230]</a></span> +so to say, the transformations of analysis may lead us to +augment it by the method which I have just indicated. +Special names, however, have been given to those only +of these <i>indirect</i> lines which refer to the complement of +the primitive arc, the others not occurring sufficiently +often to render such denominations necessary; a circumstance +which has caused a common misconception +of the true extent of the system of trigonometry.</p> + + +<p><i>Study of their Mutual Relations.</i> This multiplicity +of trigonometrical lines evidently gives rise to a third +fundamental question in trigonometry, the study of the +relations which exist between these different lines; since, +without such a knowledge, we could not make use, for +our analytical necessities, of this variety of auxiliary +magnitudes, which, however, have no other destination. +It is clear, besides, from the consideration just indicated, +that this essential part of trigonometry, although simply +preparatory, is, by its nature, susceptible of an indefinite +extension when we view it in its entire generality, while +the two others are circumscribed within rigorously defined +limits.</p> + +<p>It is needless to add that these three principal parts +of trigonometry have to be studied in precisely the inverse +order from that in which we have seen them necessarily +derived from the general nature of the subject; +for the third is evidently independent of the two others, +and the second, of that which was first presented—the +resolution of triangles, properly so called—which must +for that reason be treated in the last place; which rendered +so much the more important the consideration of +their natural succession and logical relations to one another.</p><p><span class="pagenum"><a name="Page_231" id="Page_231">[Pg 231]</a></span></p> + +<p>It is useless to consider here separately <i>spherical trigonometry</i>, +which cannot give rise to any special philosophical +consideration; since, essential as it is by the importance +and the multiplicity of its uses, it can be treated +at the present day only as a simple application of rectilinear +trigonometry, which furnishes directly its fundamental +equations, by substituting for the spherical triangle +the corresponding trihedral angle.</p> + +<p>This summary exposition of the philosophy of trigonometry +has been here given in order to render apparent, +by an important example, that rigorous dependence and +those successive ramifications which are presented by +what are apparently the most simple questions of elementary +geometry.</p> +<p><span class="pagenum"><a name="Page_232" id="Page_232">[Pg 232]</a></span></p> +<hr class="tb" /> + +<p>Having thus examined the peculiar character of <i>special</i> +geometry reduced to its only dogmatic destination, +that of furnishing to general geometry an indispensable +preliminary basis, we have now to give all our attention +to the true science of geometry, considered as a whole, +in the most rational manner. For that purpose, it is +necessary to carefully examine the great original idea of +Descartes, upon which it is entirely founded. This will +be the object of the following chapter.</p> + + + + +<h2><a name="CHAPTER_IIIa" id="CHAPTER_IIIa">CHAPTER III.</a></h2> + +<h3>MODERN OR ANALYTICAL GEOMETRY.</h3> + + +<p><i>General</i> (or <i>Analytical</i>) geometry being entirely +founded upon the transformation of geometrical considerations +into equivalent analytical considerations, we +must begin with examining directly and in a thorough +manner the beautiful conception by which Descartes has +established in a uniform manner the constant possibility +of such a co-relation. Besides its own extreme importance +as a means of highly perfecting geometrical science, +or, rather, of establishing the whole of it on rational +bases, the philosophical study of this admirable conception +must have so much the greater interest in our eyes +from its characterizing with perfect clearness the general +method to be employed in organizing the relations of the +abstract to the concrete in mathematics, by the analytical +representation of natural phenomena. There is no +conception, in the whole philosophy of mathematics +which better deserves to fix all our attention.</p> + + + + +<h3><a name="ANALYTICAL_REPRESENTATION_OF_FIGURES" id="ANALYTICAL_REPRESENTATION_OF_FIGURES">ANALYTICAL REPRESENTATION OF FIGURES.</a></h3> + + +<p>In order to succeed in expressing all imaginable geometrical +phenomena by simple analytical relations, we +must evidently, in the first place, establish a general +method for representing analytically the subjects themselves +in which these phenomena are found, that is, the +lines or the surfaces to be considered. The <i>subject</i> being<span class="pagenum"><a name="Page_233" id="Page_233">[Pg 233]</a></span> +thus habitually considered in a purely analytical +point of view, we see how it is thenceforth possible to +conceive in the same manner the various <i>accidents</i> of +which it is susceptible.</p> + +<p>In order to organize the representation of geometrical +figures by analytical equations, we must previously surmount +a fundamental difficulty; that of reducing the +general elements of the various conceptions of geometry +to simply numerical ideas; in a word, that of substituting +in geometry pure considerations of <i>quantity</i> for all +considerations of <i>quality</i>.</p> + + +<p><i>Reduction of Figure to Position.</i> For this purpose +let us observe, in the first place, that all geometrical +ideas relate necessarily to these three universal categories: +the <i>magnitude</i>, the <i>figure</i>, and the <i>position</i> of the +extensions to be considered. As to the first, there is +evidently no difficulty; it enters at once into the ideas +of numbers. With relation to the second, it must be +remarked that it will always admit of being reduced to +the third. For the figure of a body evidently results +from the mutual position of the different points of which +it is composed, so that the idea of position necessarily +comprehends that of figure, and every circumstance of +figure can be translated by a circumstance of position. +It is in this way, in fact, that the human mind has proceeded +in order to arrive at the analytical representation +of geometrical figures, their conception relating directly +only to positions. All the elementary difficulty is then +properly reduced to that of referring ideas of situation +to ideas of magnitude. Such is the direct destination +of the preliminary conception upon which Descartes has +established the general system of analytical geometry.</p><p><span class="pagenum"><a name="Page_234" id="Page_234">[Pg 234]</a></span></p> + +<p>His philosophical labour, in this relation, has consisted +simply in the entire generalization of an elementary operation, +which we may regard as natural to the human mind, +since it is performed spontaneously, so to say, in all +minds, even the most uncultivated. Thus, when we +have to indicate the situation of an object without directly +pointing it out, the method which we always adopt, +and evidently the only one which can be employed, consists +in referring that object to others which are known, +by assigning the magnitude of the various geometrical +elements, by which we conceive it connected with the +known objects. These elements constitute what Descartes, +and after him all geometers, have called the <i>co-ordinates</i> +of each point considered. They are necessarily +two in number, if it is known in advance in what plane +the point is situated; and three, if it may be found indifferently +in any region of space. As many different +constructions as can be imagined for determining the +position of a point, whether on a plane or in space, so +many distinct systems of co-ordinates may be conceived; +they are consequently susceptible of being multiplied to +infinity. But, whatever may be the system adopted, we +shall always have reduced the ideas of situation to simple +ideas of magnitude, so that we will consider the change +in the position of a point as produced by mere numerical +variations in the values of its co-ordinates.</p> + + +<p><i>Determination of the Position of a Point.</i> Considering +at first only the least complicated case, that of <i>plane +geometry</i>, it is in this way that we usually determine +the position of a point on a plane, by its distances from +two fixed right lines considered as known, which are +called <i>axes</i>, and which are commonly supposed to be<span class="pagenum"><a name="Page_235" id="Page_235">[Pg 235]</a></span> +perpendicular to each other. This system is that most +frequently adopted, because of its simplicity; but geometers +employ occasionally an infinity of others. Thus +the position of a point on a plane may be determined, 1°, +by its distances from two fixed points; or, 2°, by its distance +from a single fixed point, and the direction of that +distance, estimated by the greater or less angle which it +makes with a fixed right line, which constitutes the system +of what are called <i>polar</i> co-ordinates, the most frequently +used after the system first mentioned; or, 3°, by +the angles which the right lines drawn from the variable +point to two fixed points make with the right line which +joins these last; or, 4°, by the distances from that point +to a fixed right line and a fixed point, &c. In a word, +there is no geometrical figure whatever from which it is +not possible to deduce a certain system of co-ordinates +more or less susceptible of being employed.</p> + +<p>A general observation, which it is important to make +in this connexion, is, that every system of co-ordinates is +equivalent to determining a point, in plane geometry, by +the intersection of two lines, each of which is subjected +to certain fixed conditions of determination; a single +one of these conditions remaining variable, sometimes +the one, sometimes the other, according to the system +considered. We could not, indeed, conceive any other +means of constructing a point than to mark it by the +meeting of two lines. Thus, in the most common system, +that of <i>rectilinear co-ordinates</i>, properly so called, +the point is determined by the intersection of two right +lines, each of which remains constantly parallel to a +fixed axis, at a greater or less distance from it; in the +<i>polar</i> system, the position of the point is marked by the<span class="pagenum"><a name="Page_236" id="Page_236">[Pg 236]</a></span> +meeting of a circle, of variable radius and fixed centre, +with a movable right line compelled to turn about this +centre: in other systems, the required point might be +designated by the intersection of two circles, or of any +other two lines, &c. In a word, to assign the value of +one of the co-ordinates of a point in any system whatever, +is always necessarily equivalent to determining a +certain line on which that point must be situated. The +geometers of antiquity had already made this essential +remark, which served as the base of their method of +geometrical <i>loci</i>, of which they made so happy a use to +direct their researches in the resolution of <i>determinate</i> +problems, in considering separately the influence of each +of the two conditions by which was defined each point +constituting the object, direct or indirect, of the proposed +question. It was the general systematization of this +method which was the immediate motive of the labours +of Descartes, which led him to create analytical geometry.</p> + +<p>After having clearly established this preliminary conception—by +means of which ideas of position, and thence, +implicitly, all elementary geometrical conceptions are capable +of being reduced to simple numerical considerations—it +is easy to form a direct conception, in its entire +generality, of the great original idea of Descartes, relative +to the analytical representation of geometrical figures: +it is this which forms the special object of this +chapter. I will continue to consider at first, for more +facility, only geometry of two dimensions, which alone +was treated by Descartes; and will afterwards examine +separately, under the same point of view, the theory of +surfaces and curves of double curvature.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_237" id="Page_237">[Pg 237]</a></span></p> + + + + +<h3><a name="PLANE_CURVES" id="PLANE_CURVES">PLANE CURVES.</a></h3> + + +<p><i>Expression of Lines by Equations.</i> In accordance +with the manner of expressing analytically the position +of a point on a plane, it can be easily established that, +by whatever property any line may be defined, that definition +always admits of being replaced by a corresponding +equation between the two variable co-ordinates of the +point which describes this line; an equation which will +be thenceforth the analytical representation of the proposed +line, every phenomenon of which will be translated +by a certain algebraic modification of its equation. Thus, +if we suppose that a point moves on a plane without its +course being in any manner determined, we shall evidently +have to regard its co-ordinates, to whatever system +they may belong, as two variables entirely independent +of one another. But if, on the contrary, this point is +compelled to describe a certain line, we shall necessarily +be compelled to conceive that its co-ordinates, in all the +positions which it can take, retain a certain permanent +and precise relation to each other, which is consequently +susceptible of being expressed by a suitable equation; +which will become the very clear and very rigorous analytical +definition of the line under consideration, since +it will express an algebraical property belonging exclusively +to the co-ordinates of all the points of this line. +It is clear, indeed, that when a point is not subjected to +any condition, its situation is not determined except in +giving at once its two co-ordinates, independently of each +other; while, when the point must continue upon a defined +line, a single co-ordinate is sufficient for completely +fixing its position. The second co-ordinate is then a<span class="pagenum"><a name="Page_238" id="Page_238">[Pg 238]</a></span> +determinate <i>function</i> of the first; or, in other words, +there must exist between them a certain <i>equation</i>, of a +nature corresponding to that of the line on which the +point is compelled to remain. In a word, each of the +co-ordinates of a point requiring it to be situated on a +certain line, we conceive reciprocally that the condition, +on the part of a point, of having to belong to a line defined +in any manner whatever, is equivalent to assigning +the value of one of the two co-ordinates; which is found +in that case to be entirely dependent on the other. The +analytical relation which expresses this dependence may +be more or less difficult to discover, but it must evidently +be always conceived to exist, even in the cases in +which our present means may be insufficient to make it +known. It is by this simple consideration that we may +demonstrate, in an entirely general manner—independently +of the particular verifications on which this fundamental +conception is ordinarily established for each special +definition of a line—the necessity of the analytical +representation of lines by equations.</p> + + +<p><i>Expression of Equations by Lines.</i> Taking up again +the same reflections in the inverse direction, we could +show as easily the geometrical necessity of the representation +of every equation of two variables, in a determinate +system of co-ordinates, by a certain line; of which such +a relation would be, in the absence of any other known +property, a very characteristic definition, the scientific +destination of which will be to fix the attention directly +upon the general course of the solutions of the equation, +which will thus be noted in the most striking and the +most simple manner. This picturing of equations is one +of the most important fundamental advantages of analytical<span class="pagenum"><a name="Page_239" id="Page_239">[Pg 239]</a></span> +geometry, which has thereby reacted in the highest +degree upon the general perfecting of analysis itself; +not only by assigning to purely abstract researches a +clearly determined object and an inexhaustible career, +but, in a still more direct relation, by furnishing a new +philosophical medium for analytical meditation which +could not be replaced by any other. In fact, the purely +algebraic discussion of an equation undoubtedly makes +known its solutions in the most precise manner, but in +considering them only one by one, so that in this way +no general view of them could be obtained, except as the +final result of a long and laborious series of numerical +comparisons. On the other hand, the geometrical <i>locus</i> +of the equation, being only designed to represent distinctly +and with perfect clearness the summing up of all these +comparisons, permits it to be directly considered, without +paying any attention to the details which have furnished +it. It can thereby suggest to our mind general analytical +views, which we should have arrived at with much +difficulty in any other manner, for want of a means of +clearly characterizing their object. It is evident, for example, +that the simple inspection of the logarithmic +curve, or of the curve <i>y</i> = sin. <i>x</i>, makes us perceive +much more distinctly the general manner of the variations +of logarithms with respect to their numbers, or of +sines with respect to their arcs, than could the most attentive +study of a table of logarithms or of natural sines. +It is well known that this method has become entirely +elementary at the present day, and that it is employed +whenever it is desired to get a clear idea of the general +character of the law which reigns in a series of precise +observations of any kind whatever.</p><p><span class="pagenum"><a name="Page_240" id="Page_240">[Pg 240]</a></span></p> + + +<p><i>Any Change in the Line causes a Change in the +Equation.</i> Returning to the representation of lines by +equations, which is our principal object, we see that this +representation is, by its nature, so faithful, that the line +could not experience any modification, however slight it +might be, without causing a corresponding change in the +equation. This perfect exactitude even gives rise oftentimes +to special difficulties; for since, in our system of +analytical geometry, the mere displacements of lines affect +the equations, as well as their real variations in magnitude +or form, we should be liable to confound them +with one another in our analytical expressions, if geometers +had not discovered an ingenious method designed +expressly to always distinguish them. This method is +founded on this principle, that although it is impossible +to change analytically at will the position of a line with +respect to the axes of the co-ordinates, we can change in +any manner whatever the situation of the axes themselves, +which evidently amounts to the same; then, by +the aid of the very simple general formula by which this +transformation of the axes is produced, it becomes easy +to discover whether two different equations are the analytical +expressions of only the same line differently situated, +or refer to truly distinct geometrical loci; since, in +the former case, one of them will pass into the other by +suitably changing the axes or the other constants of the +system of co-ordinates employed. It must, moreover, be +remarked on this subject, that general inconveniences of +this nature seem to be absolutely inevitable in analytical +geometry; for, since the ideas of position are, as we have +seen, the only geometrical ideas immediately reducible to +numerical considerations, and the conceptions of figure<span class="pagenum"><a name="Page_241" id="Page_241">[Pg 241]</a></span> +cannot be thus reduced, except by seeing in them relations +of situation, it is impossible for analysis to escape +confounding, at first, the phenomena of figure with simple +phenomena of position, which alone are directly expressed +by the equations.</p> + + +<p><i>Every Definition of a Line is an Equation.</i> In order +to complete the philosophical explanation of the fundamental +conception which serves as the base of analytical +geometry, I think that I should here indicate a new +general consideration, which seems to me particularly +well adapted for putting in the clearest point of view this +necessary representation of lines by equations with two +variables. It consists in this, that not only, as we have +shown, must every defined line necessarily give rise to a +certain equation between the two co-ordinates of any one +of its points, but, still farther, every definition of a line +may be regarded as being already of itself an equation of +that line in a suitable system of co-ordinates.</p> + +<p>It is easy to establish this principle, first making a +preliminary logical distinction with respect to different +kinds of definitions. The rigorously indispensable condition +of every definition is that of distinguishing the object +defined from all others, by assigning to it a property +which belongs to it exclusively. But this end may be +generally attained in two very different ways; either by +a definition which is simply <i>characteristic</i>, that is, indicative +of a property which, although truly exclusive, +does not make known the mode of generation of the object; +or by a definition which is really <i>explanatory</i>, that +is, which characterizes the object by a property which expresses +one of its modes of generation. For example, in +considering the circle as the line, which, under the same<span class="pagenum"><a name="Page_242" id="Page_242">[Pg 242]</a></span> +contour, contains the greatest area, we have evidently a +definition of the first kind; while in choosing the property +of its having all its points equally distant from a fixed +point, we have a definition of the second kind. It is, besides, +evident, as a general principle, that even when any +object whatever is known at first only by a <i>characteristic</i> +definition, we ought, nevertheless, to regard it as susceptible +of <i>explanatory</i> definitions, which the farther study +of the object would necessarily lead us to discover.</p> + +<p>This being premised, it is clear that the general observation +above made, which represents every definition +of a line as being necessarily an equation of that line in +a certain system of co-ordinates, cannot apply to definitions +which are simply <i>characteristic</i>; it is to be understood +only of definitions which are truly <i>explanatory</i>. +But, in considering only this class, the principle is easy +to prove. In fact, it is evidently impossible to define the +generation of a line without specifying a certain relation +between the two simple motions of translation or of rotation, +into which the motion of the point which describes it +will be decomposed at each instant. Now if we form the +most general conception of what constitutes <i>a system of +co-ordinates</i>, and admit all possible systems, it is clear +that such a relation will be nothing else but the <i>equation</i> +of the proposed line, in a system of co-ordinates of a nature +corresponding to that of the mode of generation considered. +Thus, for example, the common definition of +the <i>circle</i> may evidently be regarded as being immediately +the <i>polar equation</i> of this curve, taking the centre +of the circle for the pole. In the same way, the elementary +definition of the <i>ellipse</i> or of the <i>hyperbola</i>—as +being the curve generated by a point which moves in<span class="pagenum"><a name="Page_243" id="Page_243">[Pg 243]</a></span> +such a manner that the sum or the difference of its distances +from two fixed points remains constant—gives at +once, for either the one or the other curve, the equation +<i>y</i> + <i>x</i> = <i>c</i>, taking for the system of co-ordinates that in +which the position of a point would be determined by its +distances from two fixed points, and choosing for these +poles the two given foci. In like manner, the common +definition of any <i>cycloid</i> would furnish directly, for that +curve, the equation <i>y</i> = <i>mx</i>; adopting as the co-ordinates +of each point the arc which it marks upon a circle of invariable +radius, measuring from the point of contact of that +circle with a fixed line, and the rectilinear distance from +that point of contact to a certain origin taken on that +right line. We can make analogous and equally easy verifications +with respect to the customary definitions of spirals, +of epicycloids, &c. We shall constantly find that +there exists a certain system of co-ordinates, in which we +immediately obtain a very simple equation of the proposed +line, by merely writing algebraically the condition +imposed by the mode of generation considered.</p> + +<p>Besides its direct importance as a means of rendering +perfectly apparent the necessary representation of every +line by an equation, the preceding consideration seems to +me to possess a true scientific utility, in characterizing +with precision the principal general difficulty which occurs +in the actual establishment of these equations, and in +consequently furnishing an interesting indication with respect +to the course to be pursued in inquiries of this kind, +which, by their nature, could not admit of complete and +invariable rules. In fact, since any definition whatever +of a line, at least among those which indicate a mode of +generation, furnishes directly the equation of that line in<span class="pagenum"><a name="Page_244" id="Page_244">[Pg 244]</a></span> +a certain system of co-ordinates, or, rather, of itself constitutes +that equation, it follows that the difficulty which +we often experience in discovering the equation of a +curve, by means of certain of its characteristic properties, +a difficulty which is sometimes very great, must proceed +essentially only from the commonly imposed condition of +expressing this curve analytically by the aid of a designated +system of co-ordinates, instead of admitting indifferently +all possible systems. These different systems +cannot be regarded in analytical geometry as being all +equally suitable; for various reasons, the most important +of which will be hereafter discussed, geometers think +that curves should almost always be referred, as far as is +possible, to <i>rectilinear co-ordinates</i>, properly so called. +Now we see, from what precedes, that in many cases these +particular co-ordinates will not be those with reference to +which the equation of the curve will be found to be directly +established by the proposed definition. The principal +difficulty presented by the formation of the equation +of a line really consists, then, in general, in a certain +transformation of co-ordinates. It is undoubtedly true +that this consideration does not subject the establishment +of these equations to a truly complete general method, the +success of which is always certain; which, from the very +nature of the subject, is evidently chimerical: but such a +view may throw much useful light upon the course which +it is proper to adopt, in order to arrive at the end proposed. +Thus, after having in the first place formed the +preparatory equation, which is spontaneously derived +from the definition which we are considering, it will be +necessary, in order to obtain the equation belonging to +the system of co-ordinates which must be finally admitted,<span class="pagenum"><a name="Page_245" id="Page_245">[Pg 245]</a></span> +to endeavour to express in a function of these last co-ordinates +those which naturally correspond to the given +mode of generation. It is upon this last labour that it +is evidently impossible to give invariable and precise precepts. +We can only say that we shall have so many +more resources in this matter as we shall know more of +true analytical geometry, that is, as we shall know the +algebraical expression of a greater number of different algebraical +phenomena.</p> + + + + +<h3><a name="CHOICE_OF_CO-ORDINATES" id="CHOICE_OF_CO-ORDINATES">CHOICE OF CO-ORDINATES.</a></h3> + + +<p>In order to complete the philosophical exposition of the +conception which serves as the base of analytical geometry, +I have yet to notice the considerations relating to +the choice of the system of co-ordinates which is in general +the most suitable. They will give the rational explanation +of the preference unanimously accorded to the +ordinary rectilinear system; a preference which has hitherto +been rather the effect of an empirical sentiment of +the superiority of this system, than the exact result of a +direct and thorough analysis.</p> + + +<p><i>Two different Points of View.</i> In order to decide +clearly between all the different systems of co-ordinates, +it is indispensable to distinguish with care the two general +points of view, the converse of one another, which +belong to analytical geometry; namely, the relation of +algebra to geometry, founded upon the representation of +lines by equations; and, reciprocally, the relation of geometry +to algebra, founded on the representation of equations +by lines.</p> + +<p>It is evident that in every investigation of general geometry +these two fundamental points of view are of necessity<span class="pagenum"><a name="Page_246" id="Page_246">[Pg 246]</a></span> +always found combined, since we have always to +pass alternately, and at insensible intervals, so to say, +from geometrical to analytical considerations, and from +analytical to geometrical considerations. But the necessity +of here temporarily separating them is none the +less real; for the answer to the question of method which +we are examining is, in fact, as we shall see presently, +very far from being the same in both these relations, so +that without this distinction we could not form any clear +idea of it.</p> + + +<p>1. <i>Representation of Lines by Equations.</i> Under <i>the +first point of view</i>—the representation of lines by equations—the +only reason which could lead us to prefer one +system of co-ordinates to another would be the greater +simplicity of the equation of each line, and greater facility +in arriving at it. Now it is easy to see that there does +not exist, and could not be expected to exist, any system +of co-ordinates deserving in that respect a constant preference +over all others. In fact, we have above remarked +that for each geometrical definition proposed we can conceive +a system of co-ordinates in which the equation of +the line is obtained at once, and is necessarily found to +be also very simple; and this system, moreover, inevitably +varies with the nature of the characteristic property +under consideration. The rectilinear system could not, +therefore, be constantly the most advantageous for this object, +although it may often be very favourable; there is +probably no system which, in certain particular cases, +should not be preferred to it, as well as to every other.</p> + + +<p>2. <i>Representation of Equations by Lines.</i> It is by no +means so, however, under the <i>second point of view</i>. We +can, indeed, easily establish, as a general principle, that<span class="pagenum"><a name="Page_247" id="Page_247">[Pg 247]</a></span> +the ordinary rectilinear system must necessarily be better +adapted than any other to the representation of equations +by the corresponding geometrical <i>loci</i>; that is to +say, that this representation is constantly more simple +and more faithful in it than in any other.</p> + +<p>Let us consider, for this object, that, since every system +of co-ordinates consists in determining a point by the +intersection of two lines, the system adapted to furnish +the most suitable geometrical <i>loci</i> must be that in which +these two lines are the simplest possible; a consideration +which confines our choice to the <i>rectilinear</i> system. In +truth, there is evidently an infinite number of systems +which deserve that name, that is to say, which employ +only right lines to determine points, besides the ordinary +system which assigns the distances from two fixed lines +as co-ordinates; such, for example, would be that in +which the co-ordinates of each point should be the two +angles which the right lines, which go from that point to +two fixed points, make with the right line, which joins +these last points: so that this first consideration is not +rigorously sufficient to explain the preference unanimously +given to the common system. But in examining in a +more thorough manner the nature of every system of co-ordinates, +we also perceive that each of the two lines, +whose meeting determines the point considered, must +necessarily offer at every instant, among its different conditions +of determination, a single variable condition, which +gives rise to the corresponding co-ordinate, all the rest +being fixed, and constituting the <i>axes</i> of the system, +taking this term in its most extended mathematical acceptation. +The variation is indispensable, in order that +we may be able to consider all possible positions; and<span class="pagenum"><a name="Page_248" id="Page_248">[Pg 248]</a></span> +the fixity is no less so, in order that there may exist +means of comparison. Thus, in all <i>rectilinear</i> systems, +each of the two right lines will be subjected to a fixed +condition, and the ordinate will result from the variable +condition.</p> + + +<p><i>Superiority of rectilinear Co-ordinates.</i> From these +considerations it is evident, as a general principle, that +the most favourable system for the construction of geometrical +<i>loci</i> will necessarily be that in which the variable +condition of each right line shall be the simplest +possible; the fixed condition being left free to be made +complex, if necessary to attain that object. Now, of +all possible manners of determining two movable right +lines, the easiest to follow geometrically is certainly that +in which, the direction of each right line remaining invariable, +it only approaches or recedes, more or less, to +or from a constant axis. It would be, for example, evidently +more difficult to figure to one's self clearly the +changes of place of a point which is determined by the +intersection of two right lines, which each turn around +a fixed point, making a greater or smaller angle with a +certain axis, as in the system of co-ordinates previously +noticed. Such is the true general explanation of the +fundamental property possessed by the common rectilinear +system, of being better adapted than any other to the +geometrical representation of equations, inasmuch as it +is that one in which it is the easiest to conceive the +change of place of a point resulting from the change in +the value of its co-ordinates. In order to feel clearly all +the force of this consideration, it would be sufficient to +carefully compare this system with the polar system, in +which this geometrical image, so simple and so easy to<span class="pagenum"><a name="Page_249" id="Page_249">[Pg 249]</a></span> +follow, of two right lines moving parallel, each one of +them, to its corresponding axis, is replaced by the complicated +picture of an infinite series of concentric circles, +cut by a right line compelled to turn about a fixed +point. It is, moreover, easy to conceive in advance what +must be the extreme importance to analytical geometry +of a property so profoundly elementary, which, for that +reason, must be recurring at every instant, and take a +progressively increasing value in all labours of this kind.</p> + + +<p><i>Perpendicularity of the Axes.</i> In pursuing farther +the consideration which demonstrates the superiority of +the ordinary system of co-ordinates over any other as to +the representation of equations, we may also take notice +of the utility for this object of the common usage of taking +the two axes perpendicular to each other, whenever +possible, rather than with any other inclination. As regards +the representation of lines by equations, this secondary +circumstance is no more universally proper than +we have seen the general nature of the system to be; +since, according to the particular occasion, any other inclination +of the axes may deserve our preference in that +respect. But, in the inverse point of view, it is easy to +see that rectangular axes constantly permit us to represent +equations in a more simple and even more faithful +manner; for, with oblique axes, space being divided by +them into regions which no longer have a perfect identity, +it follows that, if the geometrical <i>locus</i> of the equation +extends into all these regions at once, there will be presented, +by reason merely of this inequality of the angles, +differences of figure which do not correspond to any +analytical diversity, and will necessarily alter the rigorous +exactness of the representation, by being confounded<span class="pagenum"><a name="Page_250" id="Page_250">[Pg 250]</a></span> +with the proper results of the algebraic comparisons. +For example, an equation like: <i>x<sup>m</sup></i> + <i>y<sup>m</sup></i> = <i>c</i>, which, by its +perfect symmetry, should evidently give a curve composed +of four identical quarters, will be represented, on +the contrary, if we take axes not rectangular, by a geometric +<i>locus</i>, the four parts of which will be unequal. +It is plain that the only means of avoiding all inconveniences +of this kind is to suppose the angle of the two +axes to be a right angle.</p> + +<p>The preceding discussion clearly shows that, although +the ordinary system of rectilinear co-ordinates has no constant +superiority over all others in one of the two fundamental +points of view which are continually combined in +analytical geometry, yet as, on the other hand, it is not +constantly inferior, its necessary and absolute greater +aptitude for the representation of equations must cause +it to generally receive the preference; although it may +evidently happen, in some particular cases, that the necessity +of simplifying equations and of obtaining them +more easily may determine geometers to adopt a less +perfect system. The rectilinear system is, therefore, the +one by means of which are ordinarily constructed the +most essential theories of general geometry, intended to +express analytically the most important geometrical phenomena. +When it is thought necessary to choose some +other, the polar system is almost always the one which +is fixed upon, this system being of a nature sufficiently +opposite to that of the rectilinear system to cause the +equations, which are too complicated with respect to the +latter, to become, in general, sufficiently simple with respect +to the other. Polar co-ordinates, moreover, have +often the advantage of admitting of a more direct and<span class="pagenum"><a name="Page_251" id="Page_251">[Pg 251]</a></span> +natural concrete signification; as is the case in mechanics, +for the geometrical questions to which the theory of +circular movement gives rise, and in almost all the cases +of celestial geometry.</p> + +<hr class="tb" /> + +<p>In order to simplify the exposition, we have thus far +considered the fundamental conception of analytical geometry +only with respect to <i>plane curves</i>, the general +study of which was the only object of the great philosophical +renovation produced by Descartes. To complete +this important explanation, we have now to show +summarily how this elementary idea was extended by +Clairaut, about a century afterwards, to the general +study of <i>surfaces</i> and <i>curves of double curvature</i>. The +considerations which have been already given will permit +me to limit myself on this subject to the rapid examination +of what is strictly peculiar to this new case.</p> + + + + +<h3><a name="SURFACES" id="SURFACES">SURFACES.</a></h3> + + +<p><i>Determination of a Point in Space.</i> The complete +analytical determination of a point in space evidently requires +the values of three co-ordinates to be assigned; as, +for example, in the system which is generally adopted, +and which corresponds to the <i>rectilinear</i> system of plane +geometry, distances from the point to three fixed planes, +usually perpendicular to one another; which presents the +point as the intersection of three planes whose direction +is invariable. We might also employ the distances from +the movable point to three fixed points, which would +determine it by the intersection of three spheres with a +common centre. In like manner, the position of a point +would be defined by giving its distance from a fixed point,<span class="pagenum"><a name="Page_252" id="Page_252">[Pg 252]</a></span> +and the direction of that distance, by means of the two +angles which this right line makes with two invariable +axes; this is the <i>polar</i> system of geometry of three dimensions; +the point is then constructed by the intersection +of a sphere having a fixed centre, with two right +cones with circular bases, whose axes and common summit +do not change. In a word, there is evidently, in this +case at least, the same infinite variety among the various +possible systems of co-ordinates which we have already +observed in geometry of two dimensions. In general, +we have to conceive a point as being always determined +by the intersection of any three surfaces whatever, +as it was in the former case by that of two lines: each +of these three surfaces has, in like manner, all its conditions +of determination constant, excepting one, which +gives rise to the corresponding co-ordinates, whose peculiar +geometrical influence is thus to constrain the point +to be situated upon that surface.</p> + +<p>This being premised, it is clear that if the three co-ordinates +of a point are entirely independent of one another, +that point can take successively all possible positions +in space. But if the point is compelled to remain +upon a certain surface defined in any manner whatever, +then two co-ordinates are evidently sufficient for determining +its situation at each instant, since the proposed +surface will take the place of the condition imposed by +the third co-ordinate. We must then, in this case, under +the analytical point of view, necessarily conceive this +last co-ordinate as a determinate function of the two +others, these latter remaining perfectly independent of +each other. Thus there will be a certain equation between +the three variable co-ordinates, which will be permanent,<span class="pagenum"><a name="Page_253" id="Page_253">[Pg 253]</a></span> +and which will be the only one, in order to correspond +to the precise degree of indetermination in the +position of the point.</p> + + +<p><i>Expression of Surfaces by Equations.</i> This equation, +more or less easy to be discovered, but always possible, +will be the analytical definition of the proposed surface, +since it must be verified for all the points of that surface, +and for them alone. If the surface undergoes any change +whatever, even a simple change of place, the equation +must undergo a more or less serious corresponding modification. +In a word, all geometrical phenomena relating +to surfaces will admit of being translated by certain equivalent +analytical conditions appropriate to equations of +three variables; and in the establishment and interpretation +of this general and necessary harmony will essentially +consist the science of analytical geometry of three +dimensions.</p> + + +<p><i>Expression of Equations by Surfaces.</i> Considering +next this fundamental conception in the inverse point of +view, we see in the same manner that every equation of +three variables may, in general, be represented geometrically +by a determinate surface, primitively defined by +the very characteristic property, that the co-ordinates of +all its points always retain the mutual relation enunciated +in this equation. This geometrical locus will evidently +change, for the same equation, according to the +system of co-ordinates which may serve for the construction +of this representation. In adopting, for example, +the rectilinear system, it is clear that in the equation between +the three variables, <i>x</i>, <i>y</i>, <i>z</i>, every particular value +attributed to <i>z</i> will give an equation between at <i>x</i> and <i>y</i>, the +geometrical locus of which will be a certain line situated<span class="pagenum"><a name="Page_254" id="Page_254">[Pg 254]</a></span> +in a plane parallel to the plane of <i>x</i> and <i>y</i>, and at a distance +from this last equal to the value of <i>z</i>; so that the +complete geometrical locus will present itself as composed +of an infinite series of lines superimposed in a series +of parallel planes (excepting the interruptions which +may exist), and will consequently form a veritable surface. +It would be the same in considering any other system +of co-ordinates, although the geometrical construction +of the equation becomes more difficult to follow.</p> + +<p>Such is the elementary conception, the complement of +the original idea of Descartes, on which is founded general +geometry relative to surfaces. It would be useless +to take up here directly the other considerations which +have been above indicated, with respect to lines, and +which any one can easily extend to surfaces; whether +to show that every definition of a surface by any method +of generation whatever is really a direct equation of that +surface in a certain system of co-ordinates, or to determine +among all the different systems of possible co-ordinates +that one which is generally the most convenient. +I will only add, on this last point, that the necessary superiority +of the ordinary rectilinear system, as to the representation +of equations, is evidently still more marked in +analytical geometry of three dimensions than in that of +two, because of the incomparably greater geometrical +complication which would result from the choice of any +other system. This can be verified in the most striking +manner by considering the polar system in particular, +which is the most employed after the ordinary rectilinear +system, for surfaces as well as for plane curves, and for +the same reasons.</p> + +<p>In order to complete the general exposition of the fundamental<span class="pagenum"><a name="Page_255" id="Page_255">[Pg 255]</a></span> +conception relative to the analytical study of +surfaces, a philosophical examination should be made of +a final improvement of the highest importance, which +Monge has introduced into the very elements of this theory, +for the classification of surfaces in natural families, +established according to the mode of generation, and expressed +algebraically by common differential equations, or +by finite equations containing arbitrary functions.</p> + + + + +<h3><a name="CURVES_OF_DOUBLE_CURVATURE" id="CURVES_OF_DOUBLE_CURVATURE">CURVES OF DOUBLE CURVATURE.</a></h3> + + +<p>Let us now consider the last elementary point of view +of analytical geometry of three dimensions; that relating +to the algebraic representation of curves considered in +space, in the most general manner. In continuing to +follow the principle which has been constantly employed, +that of the degree of indetermination of the geometrical +locus, corresponding to the degree of independence of the +variables, it is evident, as a general principle, that when +a point is required to be situated upon some certain curve, +a single co-ordinate is enough for completely determining +its position, by the intersection of this curve with the surface +which results from this co-ordinate. Thus, in this +case, the two other co-ordinates of the point must be conceived +as functions necessarily determinate and distinct +from the first. It follows that every line, considered in +space, is then represented analytically, no longer by a +single equation, but by the system of two equations between +the three co-ordinates of any one of its points. It +is clear, indeed, from another point of view, that since +each of these equations, considered separately, expresses +a certain surface, their combination presents the proposed +line as the intersection of two determinate surfaces.<span class="pagenum"><a name="Page_256" id="Page_256">[Pg 256]</a></span> +Such is the most general manner of conceiving the algebraic +representation of a line in analytical geometry of +three dimensions. This conception is commonly considered +in too restricted a manner, when we confine ourselves +to considering a line as determined by the system +of its two <i>projections</i> upon two of the co-ordinate planes; +a system characterized, analytically, by this peculiarity, +that each of the two equations of the line then contains +only two of the three co-ordinates, instead of simultaneously +including the three variables. This consideration, +which consists in regarding the line as the intersection +of two cylindrical surfaces parallel to two of the +three axes of the co-ordinates, besides the inconvenience +of being confined to the ordinary rectilinear system, has +the fault, if we strictly confine ourselves to it, of introducing +useless difficulties into the analytical representation +of lines, since the combination of these two cylinders +would evidently not be always the most suitable for +forming the equations of a line. Thus, considering this +fundamental notion in its entire generality, it will be +necessary in each case to choose, from among the infinite +number of couples of surfaces, the intersection of which +might produce the proposed curve, that one which will +lend itself the best to the establishment of equations, as +being composed of the best known surfaces. Thus, if +the problem is to express analytically a circle in space, +it will evidently be preferable to consider it as the intersection +of a sphere and a plane, rather than as proceeding +from any other combination of surfaces which could +equally produce it.</p> + +<p>In truth, this manner of conceiving the representation +of lines by equations, in analytical geometry of three dimensions,<span class="pagenum"><a name="Page_257" id="Page_257">[Pg 257]</a></span> +produces, by its nature, a necessary inconvenience, +that of a certain analytical confusion, consisting +in this: that the same line may thus be expressed, with +the same system of co-ordinates, by an infinite number +of different couples of equations, on account of the infinite +number of couples of surfaces which can form it; +a circumstance which may cause some difficulties in recognizing +this line under all the algebraical disguises of +which it admits. But there exists a very simple method +for causing this inconvenience to disappear; it consists +in giving up the facilities which result from this variety +of geometrical constructions. It suffices, in fact, whatever +may be the analytical system primitively established +for a certain line, to be able to deduce from it the +system corresponding to a single couple of surfaces uniformly +generated; as, for example, to that of the two +cylindrical surfaces which <i>project</i> the proposed line upon +two of the co-ordinate planes; surfaces which will evidently +be always identical, in whatever manner the line +may have been obtained, and which will not vary except +when that line itself shall change. Now, in choosing +this fixed system, which is actually the most simple, we +shall generally be able to deduce from the primitive equations +those which correspond to them in this special construction, +by transforming them, by two successive eliminations, +into two equations, each containing only two of +the variable co-ordinates, and thereby corresponding to +the two surfaces of projection. Such is really the principal +destination of this sort of geometrical combination, +which thus offers to us an invariable and certain means +of recognizing the identity of lines in spite of the diversity +of their equations, which is sometimes very great.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_258" id="Page_258">[Pg 258]</a></span></p> + + + + +<h3><a name="IMPERFECTIONS_OF_ANALYTICAL_GEOMETRY" id="IMPERFECTIONS_OF_ANALYTICAL_GEOMETRY">IMPERFECTIONS OF ANALYTICAL GEOMETRY.</a></h3> + + +<p>Having now considered the fundamental conception of +analytical geometry under its principal elementary aspects, +it is proper, in order to make the sketch complete, +to notice here the general imperfections yet presented by +this conception with respect to both geometry and to +analysis.</p> + +<p><i>Relatively to geometry</i>, we must remark that the +equations are as yet adapted to represent only entire +geometrical loci, and not at all determinate portions of +those loci. It would, however, be necessary, in some circumstances, +to be able to express analytically a part of +a line or of a surface, or even a <i>discontinuous</i> line or +surface, composed of a series of sections belonging to distinct +geometrical figures, such as the contour of a polygon, +or the surface of a polyhedron. Thermology, especially, +often gives rise to such considerations, to which +our present analytical geometry is necessarily inapplicable. +The labours of M. Fourier on discontinuous functions +have, however, begun to fill up this great gap, and +have thereby introduced a new and essential improvement +into the fundamental conception of Descartes. But +this manner of representing heterogeneous or partial figures, +being founded on the employment of trigonometrical +series proceeding according to the sines of an infinite +series of multiple arcs, or on the use of certain definite +integrals equivalent to those series, and the general integral +of which is unknown, presents as yet too much +complication to admit of being immediately introduced +into the system of analytical geometry.</p> + +<p><i>Relatively to analysis</i>, we must begin by observing<span class="pagenum"><a name="Page_259" id="Page_259">[Pg 259]</a></span> +that our inability to conceive a geometrical representation +of equations containing four, five, or more variables, analogous +to those representations which all equations of two +or of three variables admit, must not be viewed as an imperfection +of our system of analytical geometry, for it +evidently belongs to the very nature of the subject. +Analysis being necessarily more general than geometry, +since it relates to all possible phenomena, it would be +very unphilosophical to desire always to find among geometrical +phenomena alone a concrete representation of +all the laws which analysis can express.</p> + +<p>There exists, however, another imperfection of less +importance, which must really be viewed as proceeding +from the manner in which we conceive analytical geometry. +It consists in the evident incompleteness of our +present representation of equations of two or of three variables +by lines or surfaces, inasmuch as in the construction +of the geometric locus we pay regard only to the +<i>real</i> solutions of equations, without at all noticing any +<i>imaginary</i> solutions. The general course of these last +should, however, by its nature, be quite as susceptible as +that of the others of a geometrical representation. It +follows from this omission that the graphic picture of the +equation is constantly imperfect, and sometimes even so +much so that there is no geometric representation at all +when the equation admits of only imaginary solutions. +But, even in this last case, we evidently ought to be +able to distinguish between equations as different in +themselves as these, for example,</p> + +<p> +<i>x<sup>2</sup></i> + <i>y<sup>2</sup></i> + 1 = 0, <i>x<sup>6</sup></i> + <i>y<sup>4</sup></i> + 1 = 0, <i>y<sup>2</sup></i> + <i>e<sup>x</sup></i> = 0.<br /> +</p> + +<p>We know, moreover, that this principal imperfection often +brings with it, in analytical geometry of two or of<span class="pagenum"><a name="Page_260" id="Page_260">[Pg 260]</a></span> +three dimensions, a number of secondary inconveniences, +arising from several analytical modifications not corresponding +to any geometrical phenomena.</p> + +<hr class="tb" /> + +<p>Our philosophical exposition of the fundamental conception +of analytical geometry shows us clearly that this +science consists essentially in determining what is the +general analytical expression of such or such a geometrical +phenomenon belonging to lines or to surfaces; and, +reciprocally, in discovering the geometrical interpretation +of such or such an analytical consideration. A detailed +examination of the most important general questions +would show us how geometers have succeeded in actually +establishing this beautiful harmony, and in thus imprinting +on geometrical science, regarded as a whole, its present +eminently perfect character of rationality and of +simplicity.</p> + +<div class="blockquot"><p><i>Note.</i>—The author devotes the two following chapters of his course to +the more detailed examination of Analytical Geometry of two and of three +dimensions; but his subsequent publication of a separate work upon this +branch of mathematics has been thought to render unnecessary the reproduction +of these two chapters in the present volume.</p></div> + + +<p>THE END.</p> + +<div class="footnote"> +<p>FOOTNOTES:</p> + +<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> The investigation of the mathematical phenomena of the laws of heat +by Baron Fourier has led to the establishment, in an entirely direct manner, +of Thermological equations. This great discovery tends to elevate our philosophical +hopes as to the future extensions of the legitimate applications of +mathematical analysis, and renders it proper, in the opinion of author, +to regard <i>Thermology</i> as a third principal branch of concrete mathematics.</p></div> + +<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> The translator has felt justified in employing this very convenient word +(for which our language has no precise equivalent) as an English one, in its +most extended sense, in spite of its being often popularly confounded with +its Differential and Integral department.</p></div> + +<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> With the view of increasing as much as possible the resources and the +extent (now so insufficient) of mathematical analysis, geometers count this +last couple of functions among the analytical elements. Although this inscription +is strictly legitimate, it is important to remark that circular functions +are not exactly in the same situation as the other abstract elementary +functions. There is this very essential difference, that the functions of the +four first couples are at the same time simple and abstract, while the circular +functions, which may manifest each character in succession, according +to the point of view under which they are considered and the manner in +which they are employed, never present these two properties simultaneously. +</p> +<p> +Some other concrete functions may be usefully introduced into the number +of analytical elements, certain conditions being fulfilled. It is thus, for +example, that the labours of M. Legendre and of M. Jacobi on <i>elliptical</i> +functions have truly enlarged the field of analysis; and the same is true of +some definite integrals obtained by M. Fourier in the theory of heat.</p></div> + +<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> Suppose, for example, that a question gives the following equation between +an unknown magnitude x, and two known magnitudes, <i>a</i> and <i>b</i>, +</p> + +<p><br /> +<i>x<sup>3</sup></i> + 3<i>ax</i> = 2<i>b</i>,<br /> +</p> + +<p> +as is the case in the problem of the trisection of an angle. We see at once +that the dependence between <i>x</i> on the one side, and <i>ab</i> on the other, is +completely determined; but, so long as the equation preserves its primitive +form, we do not at all perceive in what manner the unknown quantity is +derived from the data. This must be discovered, however, before we can +think of determining its value. Such is the object of the algebraic part of +the solution. When, by a series of transformations which have successively +rendered that derivation more and more apparent, we have arrived at presenting +the proposed equation under the form +</p> + +<p><br /> +<i>x</i> = ∛(<i>b</i> + √(<i>b<sup>2</sup></i> + <i>a<sup>3</sup></i>)) + ∛(<i>b</i> - √(<i>b<sup>2</sup></i> + <i>a<sup>3</sup></i>)),<br /> +</p> + +<p> +the work of <i>algebra</i> is finished; and even if we could not perform the arithmetical +operations indicated by that formula, we would nevertheless have +obtained a knowledge very real, and often very important. The work of +<i>arithmetic</i> will now consist in taking that formula for its starting point, and +finding the number <i>x</i> when the values of the numbers <i>a</i> and <i>b</i> are given.</p></div> + +<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> I have thought that I ought to specially notice this definition, because +it serves as the basis of the opinion which many intelligent persons, unacquainted +with mathematical science, form of its abstract part, without considering +that at the time of this definition mathematical analysis was not +sufficiently developed to enable the general character of each of its principal +parts to be properly apprehended, which explains why Newton could +at that time propose a definition which at the present day he would certainly +reject.</p></div> + +<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> This is less strictly true in the English system of numeration than in +the French, since "twenty-one" is our more usual mode of expressing this +number.</p></div> + +<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> Simple as may seem, for example, the equation +</p> + +<p><br /> +<i>a<sup>x</sup></i> + <i>b<sup>x</sup></i> = <i>c<sup>x</sup></i>,<br /> + +</p> +<p> +we do not yet know how to resolve it, which may give some idea of the +extreme imperfection of this part of algebra.</p></div> + +<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> The same error was afterward committed, in the infancy of the infinitesimal +calculus, in relation to the integration of differential equations.</p></div> + +<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> The fundamental principle on which reposes the theory of equations, +and which is so frequently applied in all mathematical analysis—the decomposition +of algebraic, rational, and entire functions, of any degree whatever, +into factors of the first degree—is never employed except for functions +of a single variable, without any one having examined if it ought to be extended +to functions of several variables. The general impossibility of such +a decomposition is demonstrated by the author in detail, but more properly +belongs to a special treatise.</p></div> + +<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> The only important case of this class which has thus far been completely +treated is the general integration of <i>linear</i> equations of any order +whatever, with constant coefficients. Even this case finally depends on +the algebraic resolution of equations of a degree equal to the order of differentiation.</p></div> + +<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> Leibnitz had already considered the comparison of one curve with an +other infinitely near to it, calling it "<i>Differentiatio de curva in curvam</i>." +But this comparison had no analogy with the conception of Lagrange, the +curves of Leibnitz being embraced in the same general equation, from which +they were deduced by the simple change of an arbitrary constant.</p></div> + +<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> I propose hereafter to develop this new consideration, in a special work +upon the <i>Calculus of Variations</i>, intended to present this hyper-transcendental +analysis in a new point of view, which I think adapted to extend its +general range.</p></div> + +<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> Lacroix has justly criticised the expression of <i>solid</i>, commonly used by +geometers to designate a <i>volume</i>. It is certain, in fact, that when we wish +to consider separately a certain portion of indefinite space, conceived as gaseous, +we mentally solidify its exterior envelope, so that a <i>line</i> and a <i>surface</i> +are habitually, to our minds, just as <i>solid</i> as a <i>volume</i>. It may also be remarked +that most generally, in order that bodies may penetrate one another +with more facility, we are obliged to imagine the interior of the <i>volumes</i> to +be hollow, which renders still more sensible the impropriety of the word +<i>solid</i>.</p></div> +</div> + + + + + + + + + +<pre> + + + + + +End of Project Gutenberg's The philosophy of mathematics, by Auguste Comte + +*** END OF THIS PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF MATHEMATICS *** + +***** This file should be named 39702-h.htm or 39702-h.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/9/7/0/39702/ + +Produced by Anna Hall, Albert László and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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