,,the ,,philosophy ,,of ,,mathematics4 ,,the ,,science ,,of ,,mathematics4 ---------------------------------------- - ,,abstract ,,mathematics4 ,,concrete ,,mathematics4 ------------- ,,analysis2 or1 ,the ,calculus4 ,,geometry4 ,,mechanics4 ----------------- ---------------- ,ordinary ,transcendental ,synthetic ,analytic ,analysis2 ,analysis2 or ,special or ,general or1 ,calculus or1 ,calculus ,geometry4 ,geometry4 of ,direct of ,indirect ,functions4 ,functions4 ---- ------- -------- ------ ,algebra4 ,calculus ,algebraic4 ,of three of ,trigonometry4 dimensions4 ,variations4 ,arithmetic4 ,differential ,graphical4 ,of two and ,integral ,descriptive dimensions4 ,calculus4 ,geometry4 ,,the ,,philosophy ,,of ,,mathematics2 ,,translated ,,from ,,the ,,cours ,,de ,,philosophie ,,positive ,,of ,,auguste ,,comte1 ,,by ,w4 ,m4 ,,gillespie1 ,,professor ,,of ,,civil ,,engineering ,,adj4 ,,prof4 ,,of ,,mathematics ,,in ,,union ,,college4 ,,new ,,york3 ,,harper ,,brothers1 ,,publishers1 #hb ,,cliff ,,street #ahea4 ,entered1 according to ,act of ,congress1 in the year one thousand eight hundred and fifty-one1 by ,,harper ,,brothers4 in the ,clerk's ,office of the ,district ,court of the ,southern ,district of ,new ,york4 ,,preface4 ,the pleasure and profit which the translator has received from the great work here presented1 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 appeared4 ,the want of a comprehensive map of the wide region of mathematical science--a bird's-eye view of its leading features1 and of the true bearings and relations of all its parts--is felt by every thoughtful student4 ,he is like the visitor to a great city1 who gets no just idea of its extent and situation till he has seen it from some commanding eminence4 ,to have a panoramic view of the whole district--presenting at one glance all the parts in due co-ordination1 and the darkest nooks clearly shown--is invaluable to either traveller or student4 ,it is this which has been most perfectly accomplished for mathematical science by the author whose work is here presented4 ,clearness and depth1 comprehensiveness and precision1 have never1 perhaps1 been so remarkably united as in ,,auguste ,,comte4 ,he views his subject from an elevation which gives to each part of the complex whole its true position and value1 while his telescopic glance loses none of the needful details1 and not only itself pierces to the heart of the matter1 but converts its opaqueness into such transparent crystal1 that other eyes are enabled to see as deeply into it as his own4 ,any mathematician who peruses this volume will need no other justification of the high opinion here expressed2 but others may appreciate the following endorsements of well-known authorities4 ,mill1 in his ,logic1 calls the work of ,m4 ,comte by far the greatest yet produced on the ,philosophy of the sciences2 and adds1 of this admirable work1 one of the most admirable portions is that in which he may truly be said to have created the ,philosophy of the higher ,mathematics3 ,morell1 in his ,speculative ,philosophy of ,europe1 says1 ,the classification given of the sciences at large1 and their regular order of development1 is unquestionably a master-piece of scientific thinking1 as simple as it is comprehensive2 and ,lewes1 in his ,biographical ,history of ,philosophy1 names ,comte the ,bacon of the nineteenth century1 and says1 ,i unhesitatingly record my conviction that this is the greatest work of our age4 ,the complete work of ,m4 ,comte--his ,cours de ,philosophie ,positive--fills six large octavo volumes1 of six or seven hundred pages each1 two thirds of the first volume comprising the purely mathematical portion4 ,the great bulk of the ,course is the probable cause of the fewness of those to whom even this section of it is known4 ,its presentation in its present form is therefore felt by the translator to be a most useful contribution to mathematical progress in this country4 ,the comprehensiveness of the style of the author--grasping all possible forms of an idea in one ,briarean sentence1 armed at all points against leaving any opening for mistake or forgetfulness--occasionally verges upon cumbersomeness and formality4 ,the translator has1 therefore1 sometimes taken the liberty of breaking up or condensing a long sentence1 and omitting a few passages not absolutely necessary1 or referring to the peculiar ,positive philosophy of the author2 but he has generally aimed at a conscientious fidelity to the original4 ,it has often been difficult to retain its fine shades and subtile distinctions of meaning1 and1 at the same time1 replace the peculiarly appropriate ,french idioms by corresponding ,english ones4 ,the attempt1 however1 has always been made1 though1 when the best course has been at all doubtful1 the language of the original has been followed as closely as possible1 and1 when necessary1 smoothness and grace have been unhesitatingly sacrificed to the higher attributes of clearness and precision4 ,some forms of expression may strike the reader as unusual1 but they have been retained because they were characteristic1 not of the mere language of the original1 but of its spirit4 ,when a great thinker has clothed his conceptions in phrases which are singular even in his own tongue1 he who professes to translate him is bound faithfully to preserve such forms of speech1 as far as is practicable2 and this has been here done with respect to such peculiarities of expression as belong to the author1 not as a foreigner1 but as an individual--not because he writes in ,french1 but because he is ,auguste ,comte4 ,the young student of ,mathematics should not attempt to read the whole of this volume at once1 but should peruse each portion of it in connexion with the temporary subject of his special study3 the first chapter of the first book1 for example1 while he is studying ,algebra2 the first chapter of the second book1 when he has made some progress in ,geometry2 and so with the rest4 ,passages which are obscure at the first reading will brighten up at the second2 and as his own studies cover a larger portion of the field of ,mathematics1 he will see more and more clearly their relations to one another1 and to those which he is next to take up4 ,for this end he is urgently recommended to obtain a perfect familiarity with the ,analytical ,table of ,contents1 which maps out the whole subject1 the grand divisions of which are also indicated in the ,tabular ,view facing the title-page4 ,corresponding heads will be found in the body of the work1 the principal divisions being in ,,small ,,capitals1 and the subdivisions in ,italics4 ,for these details the translator alone is responsible4 ,,analytical ,,table ,,of ,,contents4 ,,introduction4 ,page ,,general ,,considerations ,,on ,,mathematical ,,science #ag ,,the ,,object ,,of ,,mathematics #ah ,measuring ,magnitudes #ah ,difficulties #ai ,general ,method #bj ,illustrations #ba #a4 ,falling ,bodies #ba #b4 ,inaccessible ,distances #bc #c4 ,astronomical ,facts #bd ,,true ,,definition ,,of ,,mathematics #be ,a ,science1 not an ,art #be ,,its ,,two ,,fundamental ,,divisions #bf ,their different ,objects #bg ,their different ,natures #bi ,concrete ,mathematics #ca ,geometry and ,mechanics #cb ,abstract ,mathematics #cc ,the ,calculus1 or ,analysis #cc ,,extent ,,of ,,its ,,field #ce ,its ,universality #cf ,its ,limitations #cg ,,book ,i4 ,,analysis4 ,,chapter ,i4 ,page ,,general ,,view ,,of ,,mathematical ,,analysis #de ,,the ,,true ,,idea ,,of ,,an ,,equation #df ,division of ,functions into ,abstract and ,concrete #dg ,enumeration of ,abstract ,functions #ej ,,divisions ,,of ,,the ,,calculus #ec ,the ,calculus of ,values1 or ,arithmetic #eg ,its ,extent #eg ,its true ,nature #ei ,the ,calculus of ,functions #fa ,two ,modes of obtaining ,equations #fa #a4 ,by the ,relations between the given ,quantities #fa #b4 ,by the ,relations between auxiliary ,quantities #fd ,corresponding ,divisions of the ,calculus of ,functions #fg ,,chapter ,,ii4 ,,ordinary ,,analysis2 ,,or1 ,,algebra4 #fi ,its ,object #fi ,classification of ,equations #gj ,,algebraic ,,equations #ga ,their ,classification #ga ,,algebraic ,,resolution ,,of ,,equations #gb ,its ,limits #gb ,general ,solution #gb ,what we know in ,algebra #gd ,,numerical ,,resolution ,,of ,,equations #ge ,its limited ,usefulness #gf ,different ,divisions of the two ,systems #gh ,,the ,,theory ,,of ,,equations #gi ,,the ,,method ,,of ,,indeterminate ,,coefficients #hj ,,imaginary ,,quantities #ha ,,negative ,,quantities #ha ,,the ,,principle ,,of ,,homogeneity #hd ,,chapter ,,iii4 ,,transcendental ,,analysis3 ,page ,,its ,,different ,,conceptions #hh ,preliminary ,remarks #hh ,its early ,history #hi ,,method ,,of ,,leibnitz #ia ,infinitely small ,elements #ia ,examples3 #a4 ,tangents #ic #b4 ,rectification of an ,arc #id #c4 ,quadrature of a ,curve #ie #d4 ,velocity in variable ,motion #ie #e4 ,distribution of ,heat #if ,generality of the ,formulas #ig ,demonstration of the ,method #ih ,illustration by ,tangents #ajb ,,method ,,of ,,newton #ajc ,method of ,limits #ajc ,examples3 #a4 ,tangents #ajd #b4 ,rectifications #aje ,fluxions and ,fluents #ajf ,,method ,,of ,,lagrange #ajh ,derived ,functions #ajh ,an extension of ordinary ,analysis #ajh ,example3 ,tangents #aji ,fundamental ,identity of the three ,methods #aaj ,their comparative ,value #aac ,that of ,leibnitz #aac ,that of ,newton #aae ,that of ,lagrange #aag ,,chapter ,,iv4 ,page ,,the ,,differential ,,and ,,integral ,,calculus #abj ,,its ,,two ,,fundamental ,,divisions #abj ,,their ,,relations ,,to ,,each ,,other #aba #a4 ,use of the ,differential ,calculus as preparatory to that of the ,integral #abc #b4 ,employment of the ,differential ,calculus alone #abe #c4 ,employment of the ,integral ,calculus alone #abe ,three ,classes of ,questions hence resulting #abf ,,the ,,differential ,,calculus #abg ,two ,cases3 ,explicit and ,implicit ,functions #abg ,two sub-,cases3 a single ,variable or several #abi ,two other ,cases3 ,functions separate or combined #acj ,reduction of all to the ,differentiation of the ten elementary ,functions #aca ,transformation of derived ,functions for new ,variables #acb ,different ,orders of ,differentiation #acc ,analytical ,applications #acc ,,the ,,integral ,,calculus #ace ,its fundamental ,division3 ,explicit and ,implicit ,functions #ace ,subdivisions3 a single ,variable or several #acf ,calculus of partial ,differences #acg ,another ,subdivision3 different ,orders of ,differentiation #ach ,another equivalent ,distinction #adj ,quadratures #adb ,integration of ,transcendental ,functions #adc ,integration by ,parts #adc ,integration of ,algebraic ,functions #adc ,singular ,solutions #add ,definite ,integrals #adf ,prospects of the ,integral ,calculus #adh ,,chapter ,v4 ,page ,,the ,,calculus ,,of ,,variations #aea ,,problems ,,giving ,,rise ,,to ,,it #aea ,ordinary ,questions of ,maxima and ,minima #aea ,a new ,class of ,questions #aeb ,solid of least ,resistance2 ,brachystochrone2 ,isoperimeters #aec ,,analytical ,,nature ,,of ,,these ,,questions #aed ,,methods ,,of ,,the ,,older ,,geometers #aee ,,method ,,of ,,lagrange #aef ,two ,classes of ,questions #aeg #a4 ,absolute ,maxima and ,minima #aeg ,equations of ,limits #aei ,a more general ,consideration #aei #b4 ,relative ,maxima and ,minima #afj ,other ,applications of the ,method of ,variations #afb ,,its ,,relations ,,to ,,the ,,ordinary ,,calculus #afc ,,chapter ,,vi4 ,,the ,,calculus ,,of ,,finite ,,differences #afg ,its general ,character #afg ,its true ,nature #afh ,,general ,,theory ,,of ,,series #agj ,its ,identity with this ,calculus #agb ,,periodic ,,or ,,discontinuous ,,functions #agc ,,applications ,,of ,,this ,,calculus #agc ,series #agc ,interpolation #agc ,approximate ,rectification1 c4 #agd ,,book ,,ii4 ,,geometry4 ,,chapter ,i4 ,page ,a ,,general ,,view ,,of ,,geometry #agi ,the true ,nature of ,geometry #agi ,two fundamental ,ideas #aha #a4 ,the ,idea of ,space #aha #b4 ,different kinds of ,extension #ahb ,,the ,,final ,,object ,,of ,,geometry #ahd ,nature of ,geometrical ,measurement #ahe ,of ,surfaces and ,volumes #ahe ,of curve ,lines #ahg ,of right ,lines #ahi ,,the ,,infinite ,,extent ,,of ,,its ,,field #aij ,infinity of ,lines #aij ,infinity of ,surfaces #aia ,infinity of ,volumes #aib ,analytical ,invention of ,curves1 c4 #aic ,,expansion ,,of ,,original ,,definition #aic ,properties of ,lines and ,surfaces #aie ,necessity of their ,study #aie #a4 ,to find the most suitable ,property #aie #b4 ,to pass from the ,concrete to the ,abstract #aig ,illustrations3 ,orbits of the ,planets #aih ,figure of the ,earth #aii ,,the ,,two ,,general ,,methods ,,of ,,geometry #bjb ,their fundamental ,difference #bjc #a4 ,different ,questions with respect to the same ,figure #bjd #b4 ,similar ,questions with respect to different ,figures #bjd ,geometry of the ,ancients #bjd ,geometry of the ,moderns #bjf ,superiority of the ,modern #bjg ,the ,ancient the base of the ,modern #bji ,,chapter ,,ii4 ,,ancient ,,or ,,synthetic ,,geometry ,page ,,its ,,proper ,,extent #bab ,lines2 ,polygons2 ,polyhedrons #bab ,not to be farther restricted #bac ,improper ,application of ,analysis #bad ,attempted ,demonstrations of ,axioms #baf ,,geometry ,,of ,,the ,,right ,,line #bag ,,graphical ,,solutions #bah ,descriptive ,geometry #bbj ,,algebraical ,,solutions #bbd ,trigonometry #bbe ,two ,methods of introducing ,angles #bbf #a4 ,by ,arcs #bbf #b4 ,by trigonometrical ,lines #bbf ,advantages of the latter #bbf ,its ,division of trigonometrical ,questions #bbg #a4 ,relations between ,angles and trigonometrical ,lines #bbh #b4 ,relations between trigonometrical ,lines and ,sides #bbh ,increase of trigonometrical ,lines #bbh ,study of the ,relations between them #bcj ,,chapter ,,iii4 ,,modern ,,or ,,analytical ,,geometry ,page ,,the ,,analytical ,,representation ,,of ,,figures #bcb ,reduction of ,figure to ,position #bcc ,determination of the position of a ,point #bcd ,,plane ,,curves #bcg ,expression of ,lines by ,equations #bcg ,expression of ,equations by ,lines #bch ,any change in the ,line changes the ,equation #bdj ,every ,definition of a ,line is an ,equation #bda ,choice of ,co-ordinates #bde ,two different points of ,view #bde #a4 ,representation of ,lines by ,equations #bdf #b4 ,representation of ,equations by ,lines #bdf ,superiority of the rectilinear ,system #bdh ,advantages of perpendicular ,axes #bdi ,,surfaces #bea ,determination of a ,point in ,space #bea ,expression of ,surfaces by ,equations #bec ,expression of ,equations by ,surfaces #bec ,,curves ,,in ,,space #bee ,imperfections of ,analytical ,geometry #beh ,relatively to ,geometry #beh ,relatively to ,analysis #beh ,,the ,,philosophy ,,of ,,mathematics4 ,,introduction4 ,,general ,,considerations4 ,although ,mathematical ,science is the most ancient and the most perfect of all1 yet the general idea which we ought to form of it has not yet been clearly determined4 ,its definition and its principal divisions have remained till now vague and uncertain4 ,indeed the plural name--,the ,mathematics--by which we commonly designate it1 would alone suffice to indicate the want of unity in the common conception of it4 ,in truth1 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 clearness4 ,since that epoch the attention of geometers has been too exclusively absorbed by the special perfecting of the different branches1 and by the application which they have made of them to the most important laws of the universe1 to allow them to give due attention to the general system of the science4 ,but at the present time the progress of the special departments is no longer so rapid as to forbid the contemplation of the whole4 ,the science of mathematics is now sufficiently developed1 both in itself and as to its most essential application1 to have arrived at that state of consistency in which we ought to strive to arrange its different parts in a single system1 in order to prepare for new advances4 ,we may even observe that the last important improvements of the science have directly paved the way for this important philosophical operation1 by impressing on its principal parts a character of unity which did not previously exist4 ,to form a just idea of the object of mathematical science1 we may start from the indefinite and meaningless definition of it usually given1 in calling it ,the science of magnitudes1 or1 which is more definite1 ,the science which has for its object the measurement of magnitudes4 ,let us see how we can rise from this rough sketch 7which is singularly deficient in precision and depth1 though1 at bottom1 just7 to a veritable definition1 worthy of the importance1 the extent1 and the difficulty of the science4 ,,the ,,object ,,of ,,mathematics4 ,measuring ,magnitudes4 ,the question of measuring a magnitude in itself presents to the mind no other idea than that of the simple direct comparison of this magnitude with another similar magnitude1 supposed to be known1 which it takes for the unit of comparison among all others of the same kind4 ,according to this definition1 then1 the science of mathematics--vast and profound as it is with reason reputed to be--instead of being an immense concatenation of prolonged mental labours1 which offer inexhaustible occupation to our intellectual activity1 would seem to consist of a simple series of mechanical processes for obtaining directly the ratios of the quantities to be measured to those by which we wish to measure them1 by the aid of operations of similar character to the superposition of lines1 as practiced by the carpenter with his rule4 ,the error of this definition consists in presenting as direct an object which is almost always1 on the contrary1 very indirect4 ,the direct measurement of a magnitude1 by superposition or any similar process1 is most frequently an operation quite impossible for us to perform2 so that if we had no other means for determining magnitudes than direct comparisons1 we should be obliged to renounce the knowledge of most of those which interest us4 ,difficulties4 ,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 another4 ,this comparison1 which is certainly the most simple which we can conceive1 can nevertheless scarcely ever be effected directly4 ,in reflecting on the whole of the conditions necessary to render a line susceptible of a direct measurement1 we see that most frequently they cannot be all fulfilled at the same time4 ,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 other1 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 most2 in the first place1 all the distances between the celestial bodies1 or from any one of them to the earth2 and then1 too1 even the greater number of terrestrial distances1 which are so frequently inaccessible4 ,but even if this first condition be found to be fulfilled1 it is still farther necessary that the length be neither too great nor too small1 which would render a direct measurement equally impossible4 ,the line must also be suitably situated2 for let it be one which we could measure with the greatest facility1 if it were horizontal1 but conceive it to be turned up vertically1 and it becomes impossible to measure it4 ,the difficulties which we have indicated in reference to measuring lines1 exist in a very much greater degree in the measurement of surfaces1 volumes1 velocities1 times1 forces1 c4 ,it is this fact which makes necessary the formation of mathematical science1 as we are going to see2 for the human mind has been compelled to renounce1 in almost all cases1 the direct measurement of magnitudes1 and to seek to determine them indirectly1 and it is thus that it has been led to the creation of mathematics4 ,general ,method4 ,the general method which is constantly employed1 and evidently the only one conceivable1 to ascertain magnitudes which do not admit of a direct measurement1 consists in connecting them with others which are susceptible of being determined immediately1 and by means of which we succeed in discovering the first through the relations which subsist between the two4 ,such is the precise object of mathematical science viewed as a whole4 ,in order to form a sufficiently extended idea of it1 we must consider that this indirect determination of magnitudes may be indirect in very different degrees4 ,in a great number of cases1 which are often the most important1 the magnitudes1 by means of which the principal magnitudes sought are to be determined1 cannot themselves be measured directly1 and must therefore1 in their turn1 become the subject of a similar question1 and so on2 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 researches1 and the system of magnitudes susceptible of direct measurement1 by whose means we finally determine the first1 with which at first they appear to have no connexion4 ,illustrations4 ,some examples will make clear any thing which may seem too abstract in the preceding generalities4 #a4 ,falling ,bodies4 ,let us consider1 in the first place1 a natural phenomenon1 very simple1 indeed1 but which may nevertheless give rise to a mathematical question1 really existing1 and susceptible of actual applications--the phenomenon of the vertical fall of heavy bodies4 ,the mind the most unused to mathematical conceptions1 in observing this phenomenon1 perceives at once that the two quantities which it presents--namely1 the height from which a body has fallen1 and the time of its fall--are necessarily connected with each other1 since they vary together1 and simultaneously remain fixed2 or1 in the language of geometers1 that they are functions of each other4 ,the phenomenon1 considered under this point of view1 gives rise then to a mathematical question1 which consists in substituting for the direct measurement of one of these two magnitudes1 when it is impossible1 the measurement of the other4 ,it is thus1 for example1 that we may determine indirectly the depth of a precipice1 by merely measuring the time that a heavy body would occupy in falling to its bottom1 and by suitable procedures this inaccessible depth will be known with as much precision as if it was a horizontal line placed in the most favourable circumstances for easy and exact measurement4 ,on other occasions it is the height from which a body has fallen which it will be easy to ascertain1 while the time of the fall could not be observed directly2 then the same phenomenon would give rise to the inverse question1 namely1 to determine the time from the height2 as1 for example1 if we wished to ascertain what would be the duration of the vertical fall of a body falling from the moon to the earth4 ,in this example the mathematical question is very simple1 at least when we do not pay attention to the variation in the intensity of gravity1 or the resistance of the fluid which the body passes through in its fall4 ,but1 to extend the question1 we have only to consider the same phenomenon in its greatest generality1 in supposing the fall oblique1 and in taking into the account all the principal circumstances4 ,then1 instead of offering simply two variable quantities connected with each other by a relation easy to follow1 the phenomenon will present a much greater number2 namely1 the space traversed1 whether in a vertical or horizontal direction2 the time employed in traversing it2 the velocity of the body at each point of its course2 even the intensity and the direction of its primitive impulse1 which may also be viewed as variables2 and finally1 in certain cases 7to take every thing into the account71 the resistance of the medium and the intensity of gravity4 ,all these different quantities will be connected with one another1 in such a way that each in its turn may be indirectly determined by means of the others2 and this will present as many distinct mathematical questions as there may be co-existing magnitudes in the phenomenon under consideration4 ,such a very slight change in the physical conditions of a problem may cause 7as in the above example7 a mathematical research1 at first very elementary1 to be placed at once in the rank of the most difficult questions1 whose complete and rigorous solution surpasses as yet the utmost power of the human intellect4 #b4 ,inaccessible ,distances4 ,let us take a second example from geometrical phenomena4 ,let it be proposed to determine a distance which is not susceptible of direct measurement2 it will be generally conceived as making part of a figure1 or certain system of lines1 chosen in such a way that all its other parts may be observed directly2 thus1 in the case which is most simple1 and to which all the others may be finally reduced1 the proposed distance will be considered as belonging to a triangle1 in which we can determine directly either another side and two angles1 or two sides and one angle4 ,thence-forward1 the knowledge of the desired distance1 instead of being obtained directly1 will be the result of a mathematical calculation1 which will consist in deducing it from the observed elements by means of the relation which connects it with them4 ,this calculation will become successively more and more complicated1 if the parts which we have supposed to be known cannot themselves be determined 7as is most frequently the case7 except in an indirect manner1 by the aid of new auxiliary systems1 the number of which1 in great operations of this kind1 finally becomes very considerable4 ,the distance being once determined1 the knowledge of it will frequently be sufficient for obtaining new quantities1 which will become the subject of new mathematical questions4 ,thus1 when we know at what distance any object is situated1 the simple observation of its apparent diameter will evidently permit us to determine indirectly its real dimensions1 however inaccessible it may be1 and1 by a series of analogous investigations1 its surface1 its volume1 even its weight1 and a number of other properties1 a knowledge of which seemed forbidden to us4 #c4 ,astronomical ,facts4 ,it is by such calculations that man has been able to ascertain1 not only the distances from the planets to the earth1 and1 consequently1 from each other1 but their actual magnitude1 their true figure1 even to the inequalities of their surface2 and1 what seemed still more completely hidden from us1 their respective masses1 their mean densities1 the principal circumstances of the fall of heavy bodies on the surface of each of them1 c4 ,by the power of mathematical theories1 all these different results1 and many others relative to the different classes of mathematical phenomena1 have required no other direct measurements than those of a very small number of straight lines1 suitably chosen1 and of a greater number of angles4 ,we may even say1 with perfect truth1 so as to indicate in a word the general range of the science1 that if we did not fear to multiply calculations unnecessarily1 and if we had not1 in consequence1 to reserve them for the determination of the quantities which could not be measured directly1 the determination of all the magnitudes susceptible of precise estimation1 which the various orders of phenomena can offer us1 could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles4 ,,true ,,definition ,,of ,,mathematics4 ,we are now able to define mathematical science with precision1 by assigning to it as its object the indirect measurement of magnitudes1 and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them4 ,this enunciation1 instead of giving the idea of only an art1 as do all the ordinary definitions1 characterizes immediately a true science1 and shows it at once to be composed of an immense chain of intellectual operations1 which may evidently become very complicated1 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 measurement2 of the number of variables coexistent in the proposed question2 and of the nature of the relations between all these different magnitudes furnished by the phenomena under consideration4 ,according to such a definition1 the spirit of mathematics consists in always regarding all the quantities which any phenomenon can present1 as connected and interwoven with one another1 with the view of deducing them from one another4 ,now there is evidently no phenomenon which cannot give rise to considerations of this kind2 whence results the naturally indefinite extent and even the rigorous logical universality of mathematical science4 ,we shall seek farther on to circumscribe as exactly as possible its real extension4 ,the preceding explanations establish clearly the propriety of the name employed to designate the science which we are considering4 ,this denomination1 which has taken to-day so definite a meaning by itself signifies simply science in general4 ,such a designation1 rigorously exact for the ,greeks1 who had no other real science1 could be retained by the moderns only to indicate the mathematics as the science1 beyond all others--the science of sciences4 ,indeed1 every true science has for its object the determination of certain phenomena by means of others1 in accordance with the relations which exist between them4 ,every science consists in the co-ordination of facts2 if the different observations were entirely isolated1 there would be no science4 ,we may even say1 in general terms1 that science is essentially destined to dispense1 so far as the different phenomena permit it1 with all direct observation1 by enabling us to deduce from the smallest possible number of immediate data the greatest possible number of results4 ,is not this the real use1 whether in speculation or in action1 of the laws which we succeed in discovering among natural phenomena8 ,mathematical science1 in this point of view1 merely pushes to the highest possible degree the same kind of researches which are pursued1 in degrees more or less inferior1 by every real science in its respective sphere4 ,,its ,,two ,,fundamental ,,divisions4 ,we have thus far viewed mathematical science only as a whole1 without paying any regard to its divisions4 ,we must now1 in order to complete this general view1 and to form a just idea of the philosophical character of the science1 consider its fundamental division4 ,the secondary divisions will be examined in the following chapters4 ,this principal division1 which we are about to investigate1 can be truly rational1 and derived from the real nature of the subject1 only so far as it spontaneously presents itself to us1 in making the exact analysis of a complete mathematical question4 ,we will1 therefore1 having determined above what is the general object of mathematical labours1 now characterize with precision the principal different orders of inquiries1 of which they are constantly composed4 ,their different ,objects4 ,the complete solution of every mathematical question divides itself necessarily into two parts1 of natures essentially distinct1 and with relations invariably determinate4 ,we have seen that every mathematical inquiry has for its object to determine unknown magnitudes1 according to the relations between them and known magnitudes4 ,now for this object1 it is evidently necessary1 in the first place1 to ascertain with precision the relations which exist between the quantities which we are considering4 ,this first branch of inquiries constitutes that which ,i call the concrete part of the solution4 ,when it is finished1 the question changes2 it is now reduced to a pure question of numbers1 consisting simply in determining unknown numbers1 when we know what precise relations connect them with known numbers4 ,this second branch of inquiries is what ,i call the abstract part of the solution4 ,hence follows the fundamental division of general mathematical science into two great sciences--,,abstract ,,mathematics1 and ,,concrete ,,mathematics4 ,this analysis may be observed in every complete mathematical question1 however simple or complicated it may be4 ,a single example will suffice to make it intelligible4 ,taking up again the phenomenon of the vertical fall of a heavy body1 and considering the simplest case1 we see that in order to succeed in determining1 by means of one another1 the height whence the body has fallen1 and the duration of its fall1 we must commence by discovering the exact relation of these two quantities1 or1 to use the language of geometers1 the equation which exists between them4 ,before this first research is completed1 every attempt to determine numerically the value of one of these two magnitudes from the other would evidently be premature1 for it would have no basis4 ,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 consists4 ,this inquiry may be very difficult1 and in fact1 in the present case1 constitutes incomparably the greater part of the problem4 ,the true scientific spirit is so modern1 that no one1 perhaps1 before ,galileo1 had ever remarked the increase of velocity which a body experiences in its fall3 a circumstance which excludes the hypothesis1 towards which our mind 7always involuntarily inclined to suppose in every phenomenon the most simple functions1 without any other motive than its greater facility in conceiving them7 would be naturally led1 that the height was proportional to the time4 ,in a word1 this first inquiry terminated in the discovery of the law of ,galileo4 ,when this concrete part is completed1 the inquiry becomes one of quite another nature4 ,knowing that the spaces passed through by the body in each successive second of its fall increase as the series of odd numbers1 we have then a problem purely numerical and abstract2 to deduce the height from the time1 or the time from the height2 and this consists in finding that the first of these two quantities1 according to the law which has been established1 is a known multiple of the second power of the other2 from which1 finally1 we have to calculate the value of the one when that of the other is given4 ,in this example the concrete question is more difficult than the abstract one4 ,the reverse would be the case if we considered the same phenomenon in its greatest generality1 as ,i have done above for another object4 ,according to the circumstances1 sometimes the first1 sometimes the second1 of these two parts will constitute the principal difficulty of the whole question2 for the mathematical law of the phenomenon may be very simple1 but very difficult to obtain1 or it may be easy to discover1 but very complicated2 so that the two great sections of mathematical science1 when we compare them as wholes1 must be regarded as exactly equivalent in extent and in difficulty1 as well as in importance1 as we shall show farther on1 in considering each of them separately4 ,their different ,natures4 ,these two parts1 essentially distinct in their object1 as we have just seen1 are no less so with regard to the nature of the inquiries of which they are composed4 ,the first should be called concrete1 since it evidently depends on the character of the phenomena considered1 and must necessarily vary when we examine new phenomena2 while the second is completely independent of the nature of the objects examined1 and is concerned with only the numerical relations which they present1 for which reason it should be called abstract4 ,the same relations may exist in a great number of different phenomena1 which1 in spite of their extreme diversity1 will be viewed by the geometer as offering an analytical question susceptible1 when studied by itself1 of being resolved once for all4 ,thus1 for instance1 the same law which exists between the space and the time of the vertical fall of a body in a vacuum1 is found again in many other phenomena which offer no analogy with the first nor with each other2 for it expresses the relation between the surface of a spherical body and the length of its diameter2 it determines1 in like manner1 the decrease of the intensity of light or of heat in relation to the distance of the objects lighted or heated1 c4 ,the abstract part1 common to these different mathematical questions1 having been treated in reference to one of these1 will thus have been treated for all2 while the concrete part will have necessarily to be again taken up for each question separately1 without the solution of any one of them being able to give any direct aid1 in that connexion1 for the solution of the rest4 ,the abstract part of mathematics is1 then1 general in its nature2 the concrete part1 special4 ,to present this comparison under a new point of view1 we may say concrete mathematics has a philosophical character1 which is essentially experimental1 physical1 phenomenal2 while that of abstract mathematics is purely logical1 rational4 ,the concrete part of every mathematical question is necessarily founded on the consideration of the external world1 and could never be resolved by a simple series of intellectual combinations4 ,the abstract part1 on the contrary1 when it has been very completely separated1 can consist only of a series of logical deductions1 more or less prolonged2 for if we have once found the equations of a phenomenon1 the determination of the quantities therein considered1 by means of one another1 is a matter for reasoning only1 whatever the difficulties may be4 ,it belongs to the understanding alone to deduce from these equations results which are evidently contained in them1 although perhaps in a very involved manner1 without there being occasion to consult anew the external world2 the consideration of which1 having become thenceforth foreign to the subject1 ought even to be carefully set aside in order to reduce the labour to its true peculiar difficulty4 ,the abstract part of mathematics is then purely instrumental1 and is only an immense and admirable extension of natural logic to a certain class of deductions4 ,on the other hand1 geometry and mechanics1 which1 as we shall see presently1 constitute the concrete part1 must be viewed as real natural sciences1 founded on observation1 like all the rest1 although the extreme simplicity of their phenomena permits an infinitely greater degree of systematization1 which has sometimes caused a misconception of the experimental character of their first principles4 ,we see1 by this brief general comparison1 how natural and profound is our fundamental division of mathematical science4 ,we have now to circumscribe1 as exactly as we can in this first sketch1 each of these two great sections4 ,,concrete ,,mathematics4 ,concrete ,mathematics having for its object the discovery of the equations of phenomena1 it would seem at first that it must be composed of as many distinct sciences as we find really distinct categories among natural phenomena4 ,but we are yet very far from having discovered mathematical laws in all kinds of phenomena2 we shall even see1 presently1 that the greater part will very probably always hide themselves from our investigations4 ,in reality1 in the present condition of the human mind1 there are directly but two great general classes of phenomena1 whose equations we constantly know2 these are1 firstly1 geometrical1 and1 secondly1 mechanical phenomena4 ,thus1 then1 the concrete part of mathematics is composed of ,,geometry and ,,rational ,,mechanics4 ,this is sufficient1 it is true1 to give to it a complete character of logical universality1 when we consider all phenomena from the most elevated point of view of natural philosophy4 ,in fact1 if all the parts of the universe were conceived as immovable1 we should evidently have only geometrical phenomena to observe1 since all would be reduced to relations of form1 magnitude1 and position2 then1 having regard to the motions which take place in it1 we would have also to consider mechanical phenomena4 ,hence the universe1 in the statical point of view1 presents only geometrical phenomena2 and1 considered dynamically1 only mechanical phenomena4 ,thus geometry and mechanics constitute the two fundamental natural sciences1 in this sense1 that all natural effects may be conceived as simple necessary results1 either of the laws of extension or of the laws of motion4 ,but although this conception is always logically possible1 the difficulty is to specialize it with the necessary precision1 and to follow it exactly in each of the general cases offered to us by the study of nature2 that is1 to effectually reduce each principal question of natural philosophy1 for a certain determinate order of phenomena1 to the question of geometry or mechanics1 to which we might rationally suppose it should be brought4 ,this transformation1 which requires great progress to have been previously made in the study of each class of phenomena1 has thus far been really executed only for those of astronomy1 and for a part of those considered by terrestrial physics1 properly so called4 ,it is thus that astronomy1 acoustics1 optics1 c41 have finally become applications of mathematical science to certain orders of observations4#a ,but these applications not being by their nature rigorously circumscribed1 to confound them with the science would be to assign to it a vague and indefinite domain2 and this is done in the usual division1 so faulty in so many other respects1 of the mathematics into ,pure and ,applied4 ,footnote #a3 ,the investigation of the mathematical phenomena of the laws of heat by ,baron ,fourier has led to the establishment1 in an entirely direct manner1 of ,thermological equations4 ,this great discovery tends to elevate our philosophical hopes as to the future extensions of the legitimate applications of mathematical analysis1 and renders it proper1 in the opinion of author1 to regard ,thermology as a third principal branch of concrete mathematics4 ,,abstract ,,mathematics4 ,the nature of abstract mathematics 7the general division of which will be examined in the following chapter7 is clearly and exactly determined4 ,it is composed of what is called the ,calculus1#b taking this word in its greatest extent1 which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis4 ,the ,calculus has the solution of all questions relating to numbers for its peculiar object4 ,its starting point is1 constantly and necessarily1 the knowledge of the precise relations1 i4e41 of the equations1 between the different magnitudes which are simultaneously considered2 that which is1 on the contrary1 the stopping point of concrete mathematics4 ,however complicated1 or however indirect these relations may be1 the final object of the calculus always is to obtain from them the values of the unknown quantities by means of those which are known4 ,this science1 although nearer perfection than any other1 is really little advanced as yet1 so that this object is rarely attained in a manner completely satisfactory4 ,footnote #b3 ,the translator has felt justified in employing this very convenient word 7for which our language has no precise equivalent7 as an ,english one1 in its most extended sense1 in spite of its being often popularly confounded with its ,differential and ,integral department4 ,mathematical analysis is1 then1 the true rational basis of the entire system of our actual knowledge4 ,it constitutes the first and the most perfect of all the fundamental sciences4 ,the ideas with which it occupies itself are the most universal1 the most abstract1 and the most simple which it is possible for us to conceive4 ,this peculiar nature of mathematical analysis enables us easily to explain why1 when it is properly employed1 it is such a powerful instrument1 not only to give more precision to our real knowledge1 which is self-evident1 but especially to establish an infinitely more perfect co-ordination in the study of the phenomena which admit of that application2 for1 our conceptions having been so generalized and simplified that a single analytical question1 abstractly resolved1 contains the implicit solution of a great number of diverse physical questions1 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 another4 ,we thus naturally see arise1 through the medium of analysis1 the most frequent and the most unexpected approximations between problems which at first offered no apparent connection1 and which we often end in viewing as identical4 ,could we1 for example1 without the aid of analysis1 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 motion8 and yet these questions1 however different they may be1 compose but one in the eyes of the geometer4 ,the high relative perfection of mathematical analysis is as easily perceptible4 ,this perfection is not due1 as some have thought1 to the nature of the signs which are employed as instruments of reasoning1 eminently concise and general as they are4 ,in reality1 all great analytical ideas have been formed without the algebraic signs having been of any essential aid1 except for working them out after the mind had conceived them4 ,the superior perfection of the science of the calculus is due principally to the extreme simplicity of the ideas which it considers1 by whatever signs they may be expressed2 so that there is not the least hope1 by any artifice of scientific language1 of perfecting to the same degree theories which refer to more complex subjects1 and which are necessarily condemned by their nature to a greater or less logical inferiority4 ,,the ,,extent ,,of ,,its ,,field4 ,our examination of the philosophical character of mathematical science would remain incomplete1 if1 after having viewed its object and composition1 we did not examine the real extent of its domain4 ,its ,universality4 ,for this purpose it is indispensable to perceive1 first of all1 that1 in the purely logical point of view1 this science is by itself necessarily and rigorously universal2 for there is no question whatever which may not be finally conceived as consisting in determining certain quantities from others by means of certain relations1 and consequently as admitting of reduction1 in final analysis1 to a simple question of numbers4 ,in all our researches1 indeed1 on whatever subject1 our object is to arrive at numbers1 at quantities1 though often in a very imperfect manner and by very uncertain methods4 ,thus1 taking an example in the class of subjects the least accessible to mathematics1 the phenomena of living bodies1 even when considered 7to take the most complicated case7 in the state of disease1 is it not manifest that all the questions of therapeutics may be viewed as consisting in determining the quantities of the different agents which modify the organism1 and which must act upon it to bring it to its normal state1 admitting1 for some of these quantities in certain cases1 values which are equal to zero1 or negative1 or even contradictory8 ,the fundamental idea of ,descartes on the relation of the concrete to the abstract in mathematics1 has proven1 in opposition to the superficial distinction of metaphysics1 that all ideas of quality may be reduced to those of quantity4 ,this conception1 established at first by its immortal author in relation to geometrical phenomena only1 has since been effectually extended to mechanical phenomena1 and in our days to those of heat4 ,as a result of this gradual generalization1 there are now no geometers who do not consider it1 in a purely theoretical sense1 as capable of being applied to all our real ideas of every sort1 so that every phenomenon is logically susceptible of being represented by an equation2 as much so1 indeed1 as is a curve or a motion1 excepting the difficulty of discovering it1 and then of resolving it1 which may be1 and oftentimes are1 superior to the greatest powers of the human mind4 ,its ,limitations4 ,important as it is to comprehend the rigorous universality1 in a logical point of view1 of mathematical science1 it is no less indispensable to consider now the great real limitations1 which1 through the feebleness of our intellect1 narrow in a remarkable degree its actual domain1 in proportion as phenomena1 in becoming special1 become complicated4 ,every question may be conceived as capable of being reduced to a pure question of numbers2 but the difficulty of effecting such a transformation increases so much with the complication of the phenomena of natural philosophy1 that it soon becomes insurmountable4 ,this will be easily seen1 if we consider that to bring a question within the field of mathematical analysis1 we must first have discovered the precise relations which exist between the quantities which are found in the phenomenon under examination1 the establishment of these equations being the necessary starting point of all analytical labours4 ,this must evidently be so much the more difficult as we have to do with phenomena which are more special1 and therefore more complicated4 ,we shall thus find that it is only in inorganic physics1 at the most1 that we can justly hope ever to obtain that high degree of scientific perfection4 ,the first condition which is necessary in order that phenomena may admit of mathematical laws1 susceptible of being discovered1 evidently is1 that their different quantities should admit of being expressed by fixed numbers4 ,we soon find that in this respect the whole of organic physics1 and probably also the most complicated parts of inorganic physics1 are necessarily inaccessible1 by their nature1 to our mathematical analysis1 by reason of the extreme numerical variability of the corresponding phenomena4 ,every precise idea of fixed numbers is truly out of place in the phenomena of living bodies1 when we wish to employ it otherwise than as a means of relieving the attention1 and when we attach any importance to the exact relations of the values assigned4 ,we ought not1 however1 on this account1 to cease to conceive all phenomena as being necessarily subject to mathematical laws1 which we are condemned to be ignorant of1 only because of the too great complication of the phenomena4 ,the most complex phenomena of living bodies are doubtless essentially of no other special nature than the simplest phenomena of unorganized matter4 ,if it were possible to isolate rigorously each of the simple causes which concur in producing a single physiological phenomenon1 every thing leads us to believe that it would show itself endowed1 in determinate circumstances1 with a kind of influence and with a quantity of action as exactly fixed as we see it in universal gravitation1 a veritable type of the fundamental laws of nature4 ,there is a second reason why we cannot bring complicated phenomena under the dominion of mathematical analysis4 ,even if we could ascertain the mathematical law which governs each agent1 taken by itself1 the combination of so great a number of conditions would render the corresponding mathematical problem so far above our feeble means1 that the question would remain in most cases incapable of solution4 ,to appreciate this difficulty1 let us consider how complicated mathematical questions become1 even those relating to the most simple phenomena of unorganized bodies1 when we desire to bring sufficiently near together the abstract and the concrete state1 having regard to all the principal conditions which can exercise a real influence over the effect produced4 ,we know1 for example1 that the very simple phenomenon of the flow of a fluid through a given orifice1 by virtue of its gravity alone1 has not as yet any complete mathematical solution1 when we take into the account all the essential circumstances4 ,it is the same even with the still more simple motion of a solid projectile in a resisting medium4 ,why has mathematical analysis been able to adapt itself with such admirable success to the most profound study of celestial phenomena8 ,because they are1 in spite of popular appearances1 much more simple than any others4 ,the most complicated problem which they present1 that of the modification produced in the motions of two bodies tending towards each other by virtue of their gravitation1 by the influence of a third body acting on both of them in the same manner1 is much less complex than the most simple terrestrial problem4 ,and1 nevertheless1 even it presents difficulties so great that we yet possess only approximate solutions of it4 ,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 is1 besides1 essentially due to our having skilfully profited by all the particular1 and1 so to say1 accidental facilities presented by the peculiarly favourable constitution of our planetary system4 ,the planets which compose it are quite few in number1 and their masses are in general very unequal1 and much less than that of the sun2 they are1 besides1 very distant from one another2 they have forms almost spherical2 their orbits are nearly circular1 and only slightly inclined to each other1 and so on4 ,it results from all these circumstances that the perturbations are generally inconsiderable1 and that to calculate them it is usually sufficient to take into the account1 in connexion with the action of the sun on each particular planet1 the influence of only one other planet1 capable1 by its size and its proximity1 of causing perceptible derangements4 ,if1 however1 instead of such a state of things1 our solar system had been composed of a greater number of planets concentrated into a less space1 and nearly equal in mass2 if their orbits had presented very different inclinations1 and considerable eccentricities2 if these bodies had been of a more complicated form1 such as very eccentric ellipsoids1 it is certain that1 supposing the same law of gravitation to exist1 we should not yet have succeeded in subjecting the study of the celestial phenomena to our mathematical analysis1 and probably we should not even have been able to disentangle the present principal law4 ,these hypothetical conditions would find themselves exactly realized in the highest degree in chemical phenomena1 if we attempted to calculate them by the theory of general gravitation4 ,on properly weighing the preceding considerations1 the reader will be convinced1 ,i think1 that in reducing the future extension of the great applications of mathematical analysis1 which are really possible1 to the field comprised in the different departments of inorganic physics1 ,i have rather exaggerated than contracted the extent of its actual domain4 ,important as it was to render apparent the rigorous logical universality of mathematical science1 it was equally so to indicate the conditions which limit for us its real extension1 so as not to contribute to lead the human mind astray from the true scientific direction in the study of the most complicated phenomena1 by the chimerical search after an impossible perfection4 99 99 99 99 99 ,having thus exhibited the essential object and the principal composition of mathematical science1 as well as its general relations with the whole body of natural philosophy1 we have now to pass to the special examination of the great sciences of which it is composed4 ,note4--,,analysis and ,,geometry are the two great heads under which the subject is about to be examined4 ,to these ,m4 ,comte adds ,rational ,,mechanics2 but as it is not comprised in the usual idea of ,mathematics1 and as its discussion would be of but limited utility and interest1 it is not included in the present translation4 ,,book ,i4 ,,analysis4 ,,book ,i4 ,,analysis4 ,,chapter ,i4 ,,general ,,view ,,of ,,mathematical ,,analysis4 ,in the historical development of mathematical science since the time of ,descartes1 the advances of its abstract portion have always been determined by those of its concrete portion2 but it is none the less necessary1 in order to conceive the science in a manner truly logical1 to consider the ,calculus in all its principal branches before proceeding to the philosophical study of ,geometry and ,mechanics4 ,its analytical theories1 more simple and more general than those of concrete mathematics1 are in themselves essentially independent of the latter2 while these1 on the contrary1 have1 by their nature1 a continual need of the former1 without the aid of which they could make scarcely any progress4 ,although the principal conceptions of analysis retain at present some very perceptible traces of their geometrical or mechanical origin1 they are now1 however1 mainly freed from that primitive character1 which no longer manifests itself except in some secondary points2 so that it is possible 7especially since the labours of ,lagrange7 to present them in a dogmatic exposition1 by a purely abstract method1 in a single and continuous system4 ,it is this which will be undertaken in the present and the five following chapters1 limiting our investigations to the most general considerations upon each principal branch of the science of the calculus4 ,the definite object of our researches in concrete mathematics being the discovery of the equations which express the mathematical laws of the phenomenon under consideration1 and these equations constituting the true starting point of the calculus1 which has for its object to obtain from them the determination of certain quantities by means of others1 ,i think it indispensable1 before proceeding any farther1 to go more deeply than has been customary into that fundamental idea of equation1 the continual subject1 either as end or as beginning1 of all mathematical labours4 ,besides the advantage of circumscribing more definitely the true field of analysis1 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 mathematics1 which will complete the general exposition of the fundamental division established in the introductory chapter4 ,,the ,,true ,,idea ,,of ,,an ,,equation4 ,we usually form much too vague an idea of what an equation is1 when we give that name to every kind of relation of equality between any two functions of the magnitudes which we are considering4 ,for1 though every equation is evidently a relation of equality1 it is far from being true that1 reciprocally1 every relation of equality is a veritable equation1 of the kind of those to which1 by their nature1 the methods of analysis are applicable4 ,this want of precision in the logical consideration of an idea which is so fundamental in mathematics1 brings with it the serious inconvenience of rendering it almost impossible to explain1 in general terms1 the great and fundamental difficulty which we find in establishing the relation between the concrete and the abstract1 and which stands out so prominently in each great mathematical question taken by itself4 ,if the meaning of the word equation was truly as extended as we habitually suppose it to be in our definition of it1 it is not apparent what great difficulty there could really be1 in general1 in establishing the equations of any problem whatsoever2 for the whole would thus appear to consist in a simple question of form1 which ought never even to exact any great intellectual efforts1 seeing that we can hardly conceive of any precise relation which is not immediately a certain relation of equality1 or which cannot be readily brought thereto by some very easy transformations4 ,thus1 when we admit every species of functions into the definition of equations1 we do not at all account for the extreme difficulty which we almost always experience in putting a problem into an equation1 and which so often may be compared to the efforts required by the analytical elaboration of the equation when once obtained4 ,in a word1 the ordinary abstract and general idea of an equation does not at all correspond to the real meaning which geometers attach to that expression in the actual development of the science4 ,here1 then1 is a logical fault1 a defect of correlation1 which it is very important to rectify4 ,division of ,functions into ,abstract and ,concrete4 ,to succeed in doing so1 ,i begin by distinguishing two sorts of functions1 abstract or analytical functions1 and concrete functions4 ,the first alone can enter into veritable equations4 ,we may1 therefore1 henceforth define every equation1 in an exact and sufficiently profound manner1 as a relation of equality between two abstract functions of the magnitudes under consideration4 ,in order not to have to return again to this fundamental definition1 ,i must add here1 as an indispensable complement1 without which the idea would not be sufficiently general1 that these abstract functions may refer not only to the magnitudes which the problem presents of itself1 but also to all the other auxiliary magnitudes which are connected with it1 and which we will often be able to introduce1 simply as a mathematical artifice1 with the sole object of facilitating the discovery of the equations of the phenomena4 ,i here anticipate summarily the result of a general discussion of the highest importance1 which will be found at the end of this chapter4 ,we will now return to the essential distinction of functions as abstract and concrete4 ,this distinction may be established in two ways1 essentially different1 but complementary of each other1 priori and posteriori2 that is to say1 by characterizing in a general manner the peculiar nature of each species of functions1 and then by making the actual enumeration of all the abstract functions at present known1 at least so far as relates to the elements of which they are composed4 priori1 the functions which ,i call abstract are those which express a manner of dependence between magnitudes1 which can be conceived between numbers alone1 without there being need of indicating any phenomenon whatever in which it is realized4 ,i name1 on the other hand1 concrete functions1 those for which the mode of dependence expressed cannot be defined or conceived except by assigning a determinate case of physics1 geometry1 mechanics1 c41 in which it actually exists4 ,most functions in their origin1 even those which are at present the most purely abstract1 have begun by being concrete2 so that it is easy to make the preceding distinction understood1 by citing only the successive different points of view under which1 in proportion as the science has become formed1 geometers have considered the most simple analytical functions4 ,i will indicate powers1 for example1 which have in general become abstract functions only since the labours of ,vieta and ,descartes4 ,the functions x1 x1 which in our present analysis are so well conceived as simply abstract1 were1 for the geometers of antiquity1 perfectly concrete functions1 expressing the relation of the superficies of a square1 or the volume of a cube to the length of their side4 ,these had in their eyes such a character so exclusively1 that it was only by means of the geometrical definitions that they discovered the elementary algebraic properties of these functions1 relating to the decomposition of the variable into two parts1 properties which were at that epoch only real theorems of geometry1 to which a numerical meaning was not attached until long afterward4 ,i shall have occasion to cite presently1 for another reason1 a new example1 very suitable to make apparent the fundamental distinction which ,i have just exhibited2 it is that of circular functions1 both direct and inverse1 which at the present time are still sometimes concrete1 sometimes abstract1 according to the point of view under which they are regarded4 posteriori1 the general character which renders a function abstract or concrete having been established1 the question as to whether a certain determinate function is veritably abstract1 and therefore susceptible of entering into true analytical equations1 becomes a simple question of fact1 inasmuch as we are going to enumerate all the functions of this species4 ,enumeration of ,abstract ,functions4 ,at first view this enumeration seems impossible1 the distinct analytical functions being infinite in number4 ,but when we divide them into simple and compound1 the difficulty disappears2 for1 though the number of the different functions considered in mathematical analysis is really infinite1 they are1 on the contrary1 even at the present day1 composed of a very small number of elementary functions1 which can be easily assigned1 and which are evidently sufficient for deciding the abstract or concrete character of any given function2 which will be of the one or the other nature1 according as it shall be composed exclusively of these simple abstract functions1 or as it shall include others4 ,we evidently have to consider1 for this purpose1 only the functions of a single variable1 since those relative to several independent variables are constantly1 by their nature1 more or less compound4 ,let x be the independent variable1 y the correlative variable which depends upon it4 ,the different simple modes of abstract dependence1 which we can now conceive between y and x1 are expressed by the ten following elementary formulas1 in which each function is coupled with its inverse1 that is1 with that which would be obtained from the direct function by referring x to y1 instead of referring y to x4 ,,function4 ,,its ,,name4 #ast couple #a y a x ,sum4 #b y a - x ,difference4 #b;d couple #a y ax ,product4 #b y a_/x ,quotient4 #c;d couple #a y xa ,power4 #b y ath rootx ,root4 #dth couple #a y ax ,exponential4 #b y log ax ,logarithmic4 #eth couple #a y sin4 x ,direct ,circular4 #b y arc7sin4 x74 ,inverse ,circular4#c ,footnote #c3 ,with the view of increasing as much as possible the resources and the extent 7now so insufficient7 of mathematical analysis1 geometers count this last couple of functions among the analytical elements4 ,although this inscription is strictly legitimate1 it is important to remark that circular functions are not exactly in the same situation as the other abstract elementary functions4 ,there is this very essential difference1 that the functions of the four first couples are at the same time simple and abstract1 while the circular functions1 which may manifest each character in succession1 according to the point of view under which they are considered and the manner in which they are employed1 never present these two properties simultaneously4 ,some other concrete functions may be usefully introduced into the number of analytical elements1 certain conditions being fulfilled4 ,it is thus1 for example1 that the labours of ,m4 ,legendre and of ,m4 ,jacobi on elliptical functions have truly enlarged the field of analysis2 and the same is true of some definite integrals obtained by ,m4 ,fourier in the theory of heat4 ,such are the elements1 very few in number1 which directly compose all the abstract functions known at the present day4 ,few as they are1 they are evidently sufficient to give rise to an infinite number of analytical combinations4 ,no rational consideration rigorously circumscribes1 priori1 the preceding table1 which is only the actual expression of the present state of the science4 ,our analytical elements are at the present day more numerous than they were for ,descartes1 and even for ,newton and ,leibnitz3 it is only a century since the last two couples have been introduced into analysis by the labours of ,john ,bernouilli and ,euler4 ,doubtless new ones will be hereafter admitted2 but1 as ,i shall show towards the end of this chapter1 we cannot hope that they will ever be greatly multiplied1 their real augmentation giving rise to very great difficulties4 ,we can now form a definite1 and1 at the same time1 sufficiently extended idea of what geometers understand by a veritable equation4 ,this explanation is especially suited to make us understand how difficult it must be really to establish the equations of phenomena1 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 enumerated4 ,it is clear1 in fact1 that it is then only that the problem becomes truly abstract1 and is reduced to a pure question of numbers1 these functions being the only simple relations which we can conceive between numbers1 considered by themselves4 ,up to this period of the solution1 whatever the appearances may be1 the question is still essentially concrete1 and does not come within the domain of the calculus4 ,now the fundamental difficulty of this passage from the concrete to the abstract in general consists especially in the insufficiency of this very small number of analytical elements which we possess1 and by means of which1 nevertheless1 in spite of the little real variety which they offer us1 we must succeed in representing all the precise relations which all the different natural phenomena can manifest to us4 ,considering the infinite diversity which must necessarily exist in this respect in the external world1 we easily understand how far below the true difficulty our conceptions must frequently be found1 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 phenomena1 we have1 priori1 no rational guarantee of their necessary suitableness to represent the mathematical law of every other class of phenomena4 ,i will explain presently the general artifice1 so profoundly ingenious1 by which the human mind has succeeded in diminishing1 in a remarkable degree1 this fundamental difficulty which is presented by the relation of the concrete to the abstract in mathematics1 without1 however1 its having been necessary to multiply the number of these analytical elements4 ,,the ,,two ,,principal ,,divisions ,,of ,,the ,,calculus4 ,the preceding explanations determine with precision the true object and the real field of abstract mathematics4 ,i must now pass to the examination of its principal divisions1 for thus far we have considered the calculus as a whole4 ,the first direct consideration to be presented on the composition of the science of the calculus consists in dividing it1 in the first place1 into two principal branches1 to which1 for want of more suitable denominations1 ,i will give the names of ,algebraic calculus1 or ,algebra1 and of ,arithmetical calculus1 or ,arithmetic2 but with the caution to take these two expressions in their most extended logical acceptation1 in the place of the by far too restricted meaning which is usually attached to them4 ,the complete solution of every question of the calculus1 from the most elementary up to the most transcendental1 is necessarily composed of two successive parts1 whose nature is essentially distinct4 ,in the first1 the object is to transform the proposed equations1 so as to make apparent the manner in which the unknown quantities are formed by the known ones3 it is this which constitutes the algebraic question4 ,in the second1 our object is to find the values of the formulas thus obtained2 that is1 to determine directly the values of the numbers sought1 which are already represented by certain explicit functions of given numbers3 this is the arithmetical question4#d ,it is apparent that1 in every solution which is truly rational1 it necessarily follows the algebraical question1 of which it forms the indispensable complement1 since it is evidently necessary to know the mode of generation of the numbers sought for before determining their actual values for each particular case4 ,thus the stopping-place of the algebraic part of the solution becomes the starting point of the arithmetical part4 ,footnote #d3 ,suppose1 for example1 that a question gives the following equation between an unknown magnitude x1 and two known magnitudes1 a and b1 x #c;ax #b;b1 as is the case in the problem of the trisection of an angle4 ,we see at once that the dependence between x on the one side1 and ab on the other1 is completely determined2 but1 so long as the equation preserves its primitive form1 we do not at all perceive in what manner the unknown quantity is derived from the data4 ,this must be discovered1 however1 before we can think of determining its value4 ,such is the object of the algebraic part of the solution4 ,when1 by a series of transformations which have successively rendered that derivation more and more apparent1 we have arrived at presenting the proposed equation under the form x 7b 7b a77 7b - 7b a771 the work of algebra is finished2 and even if we could not perform the arithmetical operations indicated by that formula1 we would nevertheless have obtained a knowledge very real1 and often very important4 ,the work of arithmetic will now consist in taking that formula for its starting point1 and finding the number x when the values of the numbers a and b are given4 ,we thus see that the algebraic calculus and the arithmetical calculus differ essentially in their object4 ,they differ no less in the point of view under which they regard quantities2 which are considered in the first as to their relations1 and in the second as to their values4 ,the true spirit of the calculus1 in general1 requires this distinction to be maintained with the most severe exactitude1 and the line of demarcation between the two periods of the solution to be rendered as clear and distinct as the proposed question permits4 ,the attentive observation of this precept1 which is too much neglected1 may be of much assistance1 in each particular question1 in directing the efforts of our mind1 at any moment of the solution1 towards the real corresponding difficulty4 ,in truth1 the imperfection of the science of the calculus obliges us very often 7as will be explained in the next chapter7 to intermingle algebraic and arithmetical considerations in the solution of the same question4 ,but1 however impossible it may be to separate clearly the two parts of the labour1 yet the preceding indications will always enable us to avoid confounding them4 ,in endeavouring to sum up as succinctly as possible the distinction just established1 we see that ,,algebra may be defined1 in general1 as having for its object the resolution of equations2 taking this expression in its full logical meaning1 which signifies the transformation of implicit functions into equivalent explicit ones4 ,in the same way1 ,,arithmetic may be defined as destined to the determination of the values of functions4 ,henceforth1 therefore1 we will briefly say that ,,algebra is the ,calculus of ,functions1 and ,,arithmetic the ,calculus of ,values4 ,we can now perceive how insufficient and even erroneous are the ordinary definitions4 ,most generally1 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 discussion1 an idea which is evidently absurd in principle and false in fact4 ,even the celebrated definition given by ,newton1 characterizing ,algebra as ,universal ,arithmetic1 gives certainly a very false idea of the nature of algebra and of that of arithmetic4#e ,footnote #e3 ,i have thought that ,i ought to specially notice this definition1 because it serves as the basis of the opinion which many intelligent persons1 unacquainted with mathematical science1 form of its abstract part1 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 apprehended1 which explains why ,newton could at that time propose a definition which at the present day he would certainly reject4 ,having thus established the fundamental division of the calculus into two principal branches1 ,i have now to compare in general terms the extent1 the importance1 and the difficulty of these two sorts of calculus1 so as to have hereafter to consider only the ,calculus of ,functions1 which is to be the principal subject of our study4 ,,the ,,calculus ,,of ,,values1 ,,or ,,arithmetic4 ,its ,extent4 ,the ,calculus of ,values1 or ,arithmetic1 would appear1 at first view1 to present a field as vast as that of algebra1 since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined4 ,but a very simple reflection will show the difference4 ,dividing functions into simple and compound1 it is evident that when we know how to determine the value of simple functions1 the consideration of compound functions will no longer present any difficulty4 ,in the algebraic point of view1 a compound function plays a very different part from that of the elementary functions of which it consists1 and from this1 indeed1 proceed all the principal difficulties of analysis4 ,but it is very different with the ,arithmetical ,calculus4 ,thus the number of truly distinct arithmetical operations is only that determined by the number of the elementary abstract functions1 the very limited list of which has been given above4 ,the determination of the values of these ten functions necessarily gives that of all the functions1 infinite in number1 which are considered in the whole of mathematical analysis1 such at least as it exists at present4 ,there can be no new arithmetical operations without the creation of really new analytical elements1 the number of which must always be extremely small4 ,the field of arithmetic is1 then1 by its nature1 exceedingly restricted1 while that of algebra is rigorously indefinite4 ,it is1 however1 important to remark1 that the domain of the calculus of values is1 in reality1 much more extensive than it is commonly represented2 for several questions truly arithmetical1 since they consist of determinations of values1 are not ordinarily classed as such1 because we are accustomed to treat them only as incidental in the midst of a body of analytical researches more or less elevated1 the too high opinion commonly formed of the influence of signs being again the principal cause of this confusion of ideas4 ,thus not only the construction of a table of logarithms1 but also the calculation of trigonometrical tables1 are true arithmetical operations of a higher kind4 ,we may also cite as being in the same class1 although in a very distinct and more elevated order1 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 depends1 when we cannot express in general terms the explicit form of that function4 ,in this point of view the numerical solution of questions which we cannot resolve algebraically1 and even the calculation of ,definite ,integrals1 whose general integrals we do not know1 really make a part1 in spite of all appearances1 of the domain of arithmetic1 in which we must necessarily comprise all that which has for its object the determination of the values of functions4 ,the considerations relative to this object are1 in fact1 constantly homogeneous1 whatever the determinations in question1 and are always very distinct from truly algebraic considerations4 ,to complete a just idea of the real extent of the calculus of values1 we must include in it likewise that part of the general science of the calculus which now bears the name of the ,theory of ,numbers1 and which is yet so little advanced4 ,this branch1 very extensive by its nature1 but whose importance in the general system of science is not very great1 has for its object the discovery of the properties inherent in different numbers by virtue of their values1 and independent of any particular system of numeration4 ,it forms1 then1 a sort of transcendental arithmetic2 and to it would really apply the definition proposed by ,newton for algebra4 ,the entire domain of arithmetic is1 then1 much more extended than is commonly supposed2 but this calculus of values will still never be more than a point1 so to speak1 in comparison with the calculus of functions1 of which mathematical science essentially consists4 ,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 general1 when they are more profoundly examined4 ,its true ,nature4 ,in seeking to determine with precision in what determinations of values properly consist1 we easily recognize that they are nothing else but veritable transformations of the functions to be valued2 transformations which1 in spite of their special end1 are none the less essentially of the same nature as all those taught by analysis4 ,in this point of view1 the calculus of values might be simply conceived as an appendix1 and a particular application of the calculus of functions1 so that arithmetic would disappear1 so to say1 as a distinct section in the whole body of abstract mathematics4 ,in order thoroughly to comprehend this consideration1 we must observe that1 when we propose to determine the value of an unknown number whose mode of formation is given1 it is1 by the mere enunciation of the arithmetical question1 already defined and expressed under a certain form2 and that in determining its value we only put its expression under another determinate form1 to which we are accustomed to refer the exact notion of each particular number by making it re-enter into the regular system of numeration4 ,the determination of values consists so completely of a simple transformation1 that when the primitive expression of the number is found to be already conformed to the regular system of numeration1 there is no longer any determination of value1 properly speaking1 or1 rather1 the question is answered by the question itself4 ,let the question be to add the two numbers one and twenty1 we answer it by merely repeating the enunciation of the question1#f and nevertheless we think that we have determined the value of the sum4 ,this signifies that in this case the first expression of the function had no need of being transformed1 while it would not be thus in adding twenty-three and fourteen1 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 numeration4 ,footnote #f3 ,this is less strictly true in the ,english system of numeration than in the ,french1 since twenty-one is our more usual mode of expressing this number4 ,to sum up as comprehensively as possible the preceding views1 we may say1 that to determine the value of a number is nothing else than putting its primitive expression under the form a bz cz dz ez 4 4 4 4 4 pzm1 z being generally equal to #aj1 and the coefficients a1 b1 c1 d1 c41 being subjected to the conditions of being whole numbers less than z2 capable of becoming equal to zero2 but never negative4 ,every arithmetical question may thus be stated as consisting in putting under such a form any abstract function whatever of different quantities1 which are supposed to have themselves a similar form already4 ,we might then see in the different operations of arithmetic only simple particular cases of certain algebraic transformations1 excepting the special difficulties belonging to conditions relating to the nature of the coefficients4 ,it clearly follows that abstract mathematics is essentially composed of the ,calculus of ,functions1 which had been already seen to be its most important1 most extended1 and most difficult part4 ,it will henceforth be the exclusive subject of our analytical investigations4 ,i will therefore no longer delay on the ,calculus of ,values1 but pass immediately to the examination of the fundamental division of the ,calculus of ,functions4 ,,the ,,calculus ,,of ,,functions1 ,,or ,,algebra4 ,principle of its ,fundamental ,division4 ,we have determined1 at the beginning of this chapter1 wherein properly consists the difficulty which we experience in putting mathematical questions into equations4 ,it is essentially because of the insufficiency of the very small number of analytical elements which we possess1 that the relation of the concrete to the abstract is usually so difficult to establish4 ,let us endeavour now to appreciate in a philosophical manner the general process by which the human mind has succeeded1 in so great a number of important cases1 in overcoming this fundamental obstacle to ,the establishment of ,equations4 #a4 ,by the ,creation of new ,functions4 ,in looking at this important question from the most general point of view1 we are led at once to the conception of one means of facilitating the establishment of the equations of phenomena4 ,since the principal obstacle in this matter comes from the too small number of our analytical elements1 the whole question would seem to be reduced to creating new ones4 ,but this means1 though natural1 is really illusory2 and though it might be useful1 it is certainly insufficient4 ,in fact1 the creation of an elementary abstract function1 which shall be veritably new1 presents in itself the greatest difficulties4 ,there is even something contradictory in such an idea2 for a new analytical element would evidently not fulfil its essential and appropriate conditions1 if we could not immediately determine its value4 ,now1 on the other hand1 how are we to determine the value of a new function which is truly simple1 that is1 which is not formed by a combination of those already known8 ,that appears almost impossible4 ,the introduction into analysis of another elementary abstract function1 or rather of another couple of functions 7for each would be always accompanied by its inverse71 supposes then1 of necessity1 the simultaneous creation of a new arithmetical operation1 which is certainly very difficult4 ,if we endeavour to obtain an idea of the means which the human mind employs for inventing new analytical elements1 by the examination of the procedures by the aid of which it has actually conceived those which we already possess1 our observations leave us in that respect in an entire uncertainty1 for the artifices which it has already made use of for that purpose are evidently exhausted4 ,to convince ourselves of it1 let us consider the last couple of simple functions which has been introduced into analysis1 and at the formation of which we have been present1 so to speak1 namely1 the fourth couple2 for1 as ,i have explained1 the fifth couple does not strictly give veritable new analytical elements4 ,the function ax1 and1 consequently1 its inverse1 have been formed by conceiving1 under a new point of view1 a function which had been a long time known1 namely1 powers--when the idea of them had become sufficiently generalized4 ,the consideration of a power relatively to the variation of its exponent1 instead of to the variation of its base1 was sufficient to give rise to a truly novel simple function1 the variation following then an entirely different route4 ,but this artifice1 as simple as ingenious1 can furnish nothing more2 for1 in turning over in the same manner all our present analytical elements1 we end in only making them return into one another4 ,we have1 then1 no idea as to how we could proceed to the creation of new elementary abstract functions which would properly satisfy all the necessary conditions4 ,this is not to say1 however1 that we have at present attained the effectual limit established in that respect by the bounds of our intelligence4 ,it is even certain that the last special improvements in mathematical analysis have contributed to extend our resources in that respect1 by introducing within the domain of the calculus certain definite integrals1 which in some respects supply the place of new simple functions1 although they are far from fulfilling all the necessary conditions1 which has prevented me from inserting them in the table of true analytical elements4 ,but1 on the whole1 ,i think it unquestionable that the number of these elements cannot increase except with extreme slowness4 ,it is therefore not from these sources that the human mind has drawn its most powerful means of facilitating1 as much as is possible1 the establishment of equations4 #b4 ,by the ,conception of ,equations between certain auxiliary ,quantities4 ,this first method being set aside1 there remains evidently but one other3 it is1 seeing the impossibility of finding directly the equations between the quantities under consideration1 to seek for corresponding ones between other auxiliary quantities1 connected with the first according to a certain determinate law1 and from the relation between which we may return to that between the primitive magnitudes4 ,such is1 in substance1 the eminently fruitful conception1 which the human mind has succeeded in establishing1 and which constitutes its most admirable instrument for the mathematical explanation of natural phenomena2 the analysis1 called transcendental4 ,as a general philosophical principle1 the auxiliary quantities1 which are introduced in the place of the primitive magnitudes1 or concurrently with them1 in order to facilitate the establishment of equations1 might be derived according to any law whatever from the immediate elements of the question4 ,this conception has thus a much more extensive reach than has been commonly attributed to it by even the most profound geometers4 ,it is extremely important for us to view it in its whole logical extent1 for it will perhaps be by establishing a general mode of derivation different from that to which we have thus far confined ourselves 7although it is evidently very far from being the only possible one7 that we shall one day succeed in essentially perfecting mathematical analysis as a whole1 and consequently in establishing more powerful means of investigating the laws of nature than our present processes1 which are unquestionably susceptible of becoming exhausted4 ,but1 regarding merely the present constitution of the science1 the only auxiliary quantities habitually introduced in the place of the primitive quantities in the ,transcendental ,analysis are what are called1 #a1 infinitely small elements1 the differentials 7of different orders7 of those quantities1 if we regard this analysis in the manner of ,,leibnitz2 or1 #b1 the fluxions1 the limits of the ratios of the simultaneous increments of the primitive quantities compared with one another1 or1 more briefly1 the prime and ultimate ratios of these increments1 if we adopt the conception of ,,newton2 or1 #c1 the derivatives1 properly so called1 of those quantities1 that is1 the coefficients of the different terms of their respective increments1 according to the conception of ,,lagrange4 ,these three principal methods of viewing our present transcendental analysis1 and all the other less distinctly characterized ones which have been successively proposed1 are1 by their nature1 necessarily identical1 whether in the calculation or in the application1 as will be explained in a general manner in the third chapter4 ,as to their relative value1 we shall there see that the conception of ,leibnitz has thus far1 in practice1 an incontestable superiority1 but that its logical character is exceedingly vicious2 while that the conception of ,lagrange1 admirable by its simplicity1 by its logical perfection1 by the philosophical unity which it has established in mathematical analysis 7till then separated into two almost entirely independent worlds71 presents1 as yet1 serious inconveniences in the applications1 by retarding the progress of the mind4 ,the conception of ,newton occupies nearly middle ground in these various relations1 being less rapid1 but more rational than that of ,leibnitz2 less philosophical1 but more applicable than that of ,lagrange4 ,this is not the place to explain the advantages of the introduction of this kind of auxiliary quantities in the place of the primitive magnitudes4 ,the third chapter is devoted to this subject4 ,at present ,i limit myself to consider this conception in the most general manner1 in order to deduce therefrom the fundamental division of the calculus of functions into two systems essentially distinct1 whose dependence1 for the complete solution of any one mathematical question1 is invariably determinate4 ,in this connexion1 and in the logical order of ideas1 the transcendental analysis presents itself as being necessarily the first1 since its general object is to facilitate the establishment of equations1 an operation which must evidently precede the resolution of those equations1 which is the object of the ordinary analysis4 ,but though it is exceedingly important to conceive in this way the true relations of these two systems of analysis1 it is none the less proper1 in conformity with the regular usage1 to study the transcendental analysis after ordinary analysis2 for though the former is1 at bottom1 by itself logically independent of the latter1 or1 at least1 may be essentially disengaged from it1 yet it is clear that1 since its employment in the solution of questions has always more or less need of being completed by the use of the ordinary analysis1 we would be constrained to leave the questions in suspense if this latter had not been previously studied4 ,corresponding ,divisions of the ,calculus of ,functions4 ,it follows from the preceding considerations that the ,calculus of ,functions1 or ,algebra 7taking this word in its most extended meaning71 is composed of two distinct fundamental branches1 one of which has for its immediate object the resolution of equations1 when they are directly established between the magnitudes themselves which are under consideration2 and the other1 starting from equations 7generally much easier to form7 between quantities indirectly connected with those of the problem1 has for its peculiar and constant destination the deduction1 by invariable analytical methods1 of the corresponding equations between the direct magnitudes which we are considering2 which brings the question within the domain of the preceding calculus4 ,the former calculus bears most frequently the name of ,ordinary ,analysis1 or of ,algebra1 properly so called4 ,the second constitutes what is called the ,transcendental ,analysis1 which has been designated by the different denominations of ,infinitesimal ,calculus1 ,calculus of ,fluxions and of ,fluents1 ,calculus of ,vanishing ,quantities1 the ,differential and ,integral ,calculus1 c41 according to the point of view in which it has been conceived4 ,in order to remove every foreign consideration1 ,i will propose to name it ,,calculus ,,of ,,indirect ,,functions1 giving to ordinary analysis the title of ,,calculus ,,of ,,direct ,,functions4 ,these expressions1 which ,i form essentially by generalizing and epitomizing the ideas of ,lagrange1 are simply intended to indicate with precision the true general character belonging to each of these two forms of analysis4 ,having now established the fundamental division of mathematical analysis1 ,i have next to consider separately each of its two parts1 commencing with the ,calculus of ,direct ,functions1 and reserving more extended developments for the different branches of the ,calculus of ,indirect ,functions4 ,,chapter ,,ii4 ,,ordinary ,,analysis1 ,,or ,,algebra4 ,the ,calculus of direct ,functions1 or ,algebra1 is 7as was shown at the end of the preceding chapter7 entirely sufficient for the solution of mathematical questions1 when they are so simple that we can form directly the equations between the magnitudes themselves which we are considering1 without its being necessary to introduce in their place1 or conjointly with them1 any system of auxiliary quantities derived from the first4 ,it is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the ,calculus of indirect ,functions1 which is intended to facilitate the establishment of equations4 ,but1 although algebra has then only a secondary office to perform1 it has none the less a necessary part in the complete solution of the question1 so that the ,calculus of direct ,functions must continue to be1 by its nature1 the fundamental base of all mathematical analysis4 ,we must therefore1 before going any further1 consider in a general manner the logical composition of this calculus1 and the degree of development to which it has at the present day arrived4 ,its ,object4 ,the final object of this calculus being the resolution 7properly so called7 of equations1 that is1 the discovery of the manner in which the unknown quantities are formed from the known quantities1 in accordance with the equations which exist between them1 it naturally presents as many different departments as we can conceive truly distinct classes of equations4 ,its appropriate extent is consequently rigorously indefinite1 the number of analytical functions susceptible of entering into equations being in itself quite unlimited1 although they are composed of only a very small number of primitive elements4 ,classification of ,equations4 ,the rational classification of equations must evidently be determined by the nature of the analytical elements of which their numbers are composed2 every other classification would be essentially arbitrary4 ,accordingly1 analysts begin by dividing equations with one or more variables into two principal classes1 according as they contain functions of only the first three couples 7see the table in chapter i41 page #ea71 or as they include also exponential or circular functions4 ,the names of ,algebraic functions and ,transcendental functions1 commonly given to these two principal groups of analytical elements1 are undoubtedly very inappropriate4 ,but the universally established division between the corresponding equations is none the less very real in this sense1 that the resolution of equations containing the functions called transcendental necessarily presents more difficulties than those of the equations called algebraic4 ,hence the study of the former is as yet exceedingly imperfect1 so that frequently the resolution of the most simple of them is still unknown to us1#g and our analytical methods have almost exclusive reference to the elaboration of the latter4 ,footnote #g3 ,simple as may seem1 for example1 the equation ax bx cx1 we do not yet know how to resolve it1 which may give some idea of the extreme imperfection of this part of algebra4 ,,algebraic ,,equations4 ,considering now only these ,algebraic equations1 we must observe1 in the first place1 that although they may often contain irrational functions of the unknown quantities as well as rational functions1 we can always1 by more or less easy transformations1 make the first case come under the second1 so that it is with this last that analysts have had to occupy themselves exclusively in order to resolve all sorts of algebraic equations4 ,their ,classification4 ,in the infancy of algebra1 these equations were classed according to the number of their terms4 ,but this classification was evidently faulty1 since it separated cases which were really similar1 and brought together others which had nothing in common besides this unimportant characteristic4#h ,it has been retained only for equations with two terms1 which are1 in fact1 capable of being resolved in a manner peculiar to themselves4 ,footnote #h3 ,the same error was afterward committed1 in the infancy of the infinitesimal calculus1 in relation to the integration of differential equations4 ,the classification of equations by what is called their degrees1 is1 on the other hand1 eminently natural1 for this distinction rigorously determines the greater or less difficulty of their resolution4 ,this gradation is apparent in the cases of all the equations which can be resolved2 but it may be indicated in a general manner independently of the fact of the resolution4 ,we need only consider that the most general equation of each degree necessarily comprehends all those of the different inferior degrees1 as must also the formula which determines the unknown quantity4 ,consequently1 however slight we may suppose the difficulty peculiar to the degree which we are considering1 since it is inevitably complicated in the execution with those presented by all the preceding degrees1 the resolution really offers more and more obstacles1 in proportion as the degree of the equation is elevated4 ,,algebraic ,,resolution ,,of ,,equations4 ,its ,limits4 ,the resolution of algebraic equations is as yet known to us only in the four first degrees1 such is the increase of difficulty noticed above4 ,in this respect1 algebra has made no considerable progress since the labours of ,descartes and the ,italian analysts of the sixteenth century1 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 equations4 ,the general equation of the fifth degree itself has thus far resisted all attacks4 ,the constantly increasing complication which the formulas for resolving equations must necessarily present1 in proportion as the degree increases 7the difficulty of using the formula of the fourth degree rendering it almost inapplicable71 has determined analysts to renounce1 by a tacit agreement1 the pursuit of such researches1 although they are far from regarding it as impossible to obtain the resolution of equations of the fifth degree1 and of several other higher ones4 ,general ,solution4 ,the only question of this kind which would be really of great importance1 at least in its logical relations1 would be the general resolution of algebraic equations of any degree whatsoever4 ,now1 the more we meditate on this subject1 the more we are led to think1 with ,lagrange1 that it really surpasses the scope of our intelligence4 ,we must besides observe that the formula which would express the root of an equation of the mth degree would necessarily include radicals of the mth order 7or functions of an equivalent multiplicity71 because of the m determinations which it must admit4 ,since we have seen1 besides1 that this formula must also embrace1 as a particular case1 that formula which corresponds to every lower degree1 it follows that it would inevitably also contain radicals of the next lower degree1 the next lower to that1 c41 so that1 even if it were possible to discover it1 it would almost always present too great a complication to be capable of being usefully employed1 unless we could succeed in simplifying it1 at the same time retaining all its generality1 by the introduction of a new class of analytical elements of which we yet have no idea4 ,we have1 then1 reason to believe that1 without having already here arrived at the limits imposed by the feeble extent of our intelligence1 we should not be long in reaching them if we actively and earnestly prolonged this series of investigations4 ,it is1 besides1 important to observe that1 even supposing we had obtained the resolution of algebraic equations of any degree whatever1 we would still have treated only a very small part of algebra1 properly so called1 that is1 of the calculus of direct functions1 including the resolution of all the equations which can be formed by the known analytical functions4 ,finally1 we must remember that1 by an undeniable law of human nature1 our means for conceiving new questions being much more powerful than our resources for resolving them1 or1 in other words1 the human mind being much more ready to inquire than to reason1 we shall necessarily always remain below the difficulty1 no matter to what degree of development our intellectual labour may arrive4 ,thus1 even though we should some day discover the complete resolution of all the analytical equations at present known1 chimerical as the supposition is1 there can be no doubt that1 before attaining this end1 and probably even as a subsidiary means1 we would have already overcome the difficulty 7a much smaller one1 though still very great7 of conceiving new analytical elements1 the introduction of which would give rise to classes of equations of which1 at present1 we are completely ignorant2 so that a similar imperfection in algebraic science would be continually reproduced1 in spite of the real and very important increase of the absolute mass of our knowledge4 ,what we know in ,algebra4 ,in the present condition of algebra1 the complete resolution of the equations of the first four degrees1 of any binomial equations1 of certain particular equations of the higher degrees1 and of a very small number of exponential1 logarithmic1 or circular equations1 constitute the fundamental methods which are presented by the calculus of direct functions for the solution of mathematical problems4 ,but1 limited as these elements are1 geometers have nevertheless succeeded in treating1 in a truly admirable manner1 a very great number of important questions1 as we shall find in the course of the volume4 ,the general improvements introduced within a century into the total system of mathematical analysis1 have had for their principal object to make immeasurably useful this little knowledge which we have1 instead of tending to increase it4 ,this result has been so fully obtained1 that most frequently this calculus has no real share in the complete solution of the question1 except by its most simple parts2 those which have reference to equations of the two first degrees1 with one or more variables4 ,,numerical ,,resolution ,,of ,,equations4 ,the extreme imperfection of algebra1 with respect to the resolution of equations1 has led analysts to occupy themselves with a new class of questions1 whose true character should be here noted4 ,they have busied themselves in filling up the immense gap in the resolution of algebraic equations of the higher degrees1 by what they have named the numerical resolution of equations4 ,not being able to obtain1 in general1 the formula which expresses what explicit function of the given quantities the unknown one is1 they have sought 7in the absence of this kind of resolution1 the only one really algebraic7 to determine1 independently of that formula1 at least the value of each unknown quantity1 for various designated systems of particular values attributed to the given quantities4 ,by the successive labours of analysts1 this incomplete and illegitimate operation1 which presents an intimate mixture of truly algebraic questions with others which are purely arithmetical1 has been rendered possible in all cases for equations of any degree and even of any form4 ,the methods for this which we now possess are sufficiently general1 although the calculations to which they lead are often so complicated as to render it almost impossible to execute them4 ,we have nothing else to do1 then1 in this part of algebra1 but to simplify the methods sufficiently to render them regularly applicable1 which we may hope hereafter to effect4 ,in this condition of the calculus of direct functions1 we endeavour1 in its application1 so to dispose the proposed questions as finally to require only this numerical resolution of the equations4 ,its limited ,usefulness4 ,valuable as is such a resource in the absence of the veritable solution1 it is essential not to misconceive the true character of these methods1 which analysts rightly regard as a very imperfect algebra4 ,in fact1 we are far from being always able to reduce our mathematical questions to depend finally upon only the numerical resolution of equations2 that can be done only for questions quite isolated or truly final1 that is1 for the smallest number4 ,most questions1 in fact1 are only preparatory1 and intended to serve as an indispensable preparation for the solution of other questions4 ,now1 for such an object1 it is evident that it is not the actual value of the unknown quantity which it is important to discover1 but the formula1 which shows how it is derived from the other quantities under consideration4 ,it is this which happens1 for example1 in a very extensive class of cases1 whenever a certain question includes at the same time several unknown quantities4 ,we have then1 first of all1 to separate them4 ,by suitably employing the simple and general method so happily invented by analysts1 and which consists in referring all the other unknown quantities to one of them1 the difficulty would always disappear if we knew how to obtain the algebraic resolution of the equations under consideration1 while the numerical solution would then be perfectly useless4 ,it is only for want of knowing the algebraic resolution of equations with a single unknown quantity1 that we are obliged to treat ,elimination as a distinct question1 which forms one of the greatest special difficulties of common algebra4 ,laborious as are the methods by the aid of which we overcome this difficulty1 they are not even applicable1 in an entirely general manner1 to the elimination of one unknown quantity between two equations of any form whatever4 ,in the most simple questions1 and when we have really to resolve only a single equation with a single unknown quantity1 this numerical resolution is none the less a very imperfect method1 even when it is strictly sufficient4 ,it presents1 in fact1 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 considered1 although their relations to one another remain unchanged2 the calculations made for one case not enabling us to dispense with any of those which relate to a case very slightly different4 ,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 numbers4 ,according to the preceding considerations1 the calculus of direct functions1 viewed in its present state1 divides into two very distinct branches1 according as its subject is the algebraic resolution of equations or their numerical resolution4 ,the first department1 the only one truly satisfactory1 is unhappily very limited1 and will probably always remain so2 the second1 too often insufficient1 has1 at least1 the advantage of a much greater generality4 ,the necessity of clearly distinguishing these two parts is evident1 because of the essentially different object proposed in each1 and consequently the peculiar point of view under which quantities are therein considered4 ,different ,divisions of the two ,methods of ,resolution4 ,if1 moreover1 we consider these parts with reference to the different methods of which each is composed1 we find in their logical distribution an entirely different arrangement4 ,in fact1 the first part must be divided according to the nature of the equations which we are able to resolve1 and independently of every consideration relative to the values of the unknown quantities4 ,in the second part1 on the contrary1 it is not according to the degrees of the equations that the methods are naturally distinguished1 since they are applicable to equations of any degree whatever2 it is according to the numerical character of the values of the unknown quantities2 for1 in calculating these numbers directly1 without deducing them from general formulas1 different means would evidently be employed when the numbers are not susceptible of having their values determined otherwise than by a series of approximations1 always incomplete1 or when they can be obtained with entire exactness4 ,this distinction of incommensurable and of commensurable roots1 which require quite different principles for their determination1 important as it is in the numerical resolution of equations1 is entirely insignificant in the algebraic resolution1 in which the rational or irrational nature of the numbers which are obtained is a mere accident of the calculation1 which cannot exercise any influence over the methods employed2 it is1 in a word1 a simple arithmetical consideration4 ,we may say as much1 though in a less degree1 of the division of the commensurable roots themselves into entire and fractional4 ,in fine1 the case is the same1 in a still greater degree1 with the most general classification of roots1 as real and imaginary4 ,all these different considerations1 which are preponderant as to the numerical resolution of equations1 and which are of no importance in their algebraic resolution1 render more and more sensible the essentially distinct nature of these two principal parts of algebra4 ,,the ,,theory ,,of ,,equations4 ,these two departments1 which constitute the immediate object of the calculus of direct functions1 are subordinate to a third one1 purely speculative1 from which both of them borrow their most powerful resources1 and which has been very exactly designated by the general name of ,theory of ,equations1 although it as yet relates only to ,algebraic equations4 ,the numerical resolution of equations1 because of its generality1 has special need of this rational foundation4 ,this last and important branch of algebra is naturally divided into two orders of questions1 viz41 those which refer to the composition of equations1 and those which concern their transformation2 these latter having for their object to modify the roots of an equation without knowing them1 in accordance with any given law1 providing that this law is uniform in relation to all the parts4#i ,footnote #i3 ,the fundamental principle on which reposes the theory of equations1 and which is so frequently applied in all mathematical analysis--the decomposition of algebraic1 rational1 and entire functions1 of any degree whatever1 into factors of the first degree--is never employed except for functions of a single variable1 without any one having examined if it ought to be extended to functions of several variables4 ,the general impossibility of such a decomposition is demonstrated by the author in detail1 but more properly belongs to a special treatise4 ,,the ,,method ,,of ,,indeterminate ,,coefficients4 ,to complete this rapid general enumeration of the different essential parts of the calculus of direct functions1 ,i must1 lastly1 mention expressly one of the most fruitful and important theories of algebra proper1 that relating to the transformation of functions into series by the aid of what is called the ,method of indeterminate ,coefficients4 ,this method1 so eminently analytical1 and which must be regarded as one of the most remarkable discoveries of ,descartes1 has undoubtedly lost some of its importance since the invention and the development of the infinitesimal calculus1 the place of which it might so happily take in some particular respects4 ,but the increasing extension of the transcendental analysis1 although it has rendered this method much less necessary1 has1 on the other hand1 multiplied its applications and enlarged its resources2 so that by the useful combination between the two theories1 which has finally been effected1 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 functions4 99 99 99 99 99 ,having thus sketched the general outlines of algebra proper1 ,i have now to offer some considerations on several leading points in the calculus of direct functions1 our ideas of which may be advantageously made more clear by a philosophical examination4 ,,imaginary ,,quantities4 ,the difficulties connected with several peculiar symbols to which algebraic calculations sometimes lead1 and especially to the expressions called imaginary1 have been1 ,i think1 much exaggerated through purely metaphysical considerations1 which have been forced upon them1 in the place of regarding these abnormal results in their true point of view as simple analytical facts4 ,viewing them thus1 we readily see that1 since the spirit of mathematical analysis consists in considering magnitudes in reference to their relations only1 and without any regard to their determinate value1 analysts are obliged to admit indifferently every kind of expression which can be engendered by algebraic combinations4 ,the interdiction of even one expression because of its apparent singularity would destroy the generality of their conceptions4 ,the common embarrassment on this subject seems to me to proceed essentially from an unconscious confusion between the idea of function and the idea of value1 or1 what comes to the same thing1 between the algebraic and the arithmetical point of view4 ,a thorough examination would show mathematical analysis to be much more clear in its nature than even mathematicians commonly suppose4 ,,negative ,,quantities4 ,as to negative quantities1 which have given rise to so many misplaced discussions1 as irrational as useless1 we must distinguish between their abstract signification and their concrete interpretation1 which have been almost always confounded up to the present day4 ,under the first point of view1 the theory of negative quantities can be established in a complete manner by a single algebraical consideration4 ,the necessity of admitting such expressions is the same as for imaginary quantities1 as above indicated2 and their employment as an analytical artifice1 to render the formulas more comprehensive1 is a mechanism of calculation which cannot really give rise to any serious difficulty4 ,we may therefore regard the abstract theory of negative quantities as leaving nothing essential to desire2 it presents no obstacles but those inappropriately introduced by sophistical considerations4 ,it is far from being so1 however1 with their concrete theory4 ,this consists essentially in that admirable property of the signs and-1 of representing analytically the oppositions of directions of which certain magnitudes are susceptible4 ,this general theorem 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 ,descartes1 who obtained it as a simple result of properly directed philosophical observation4 ,a great number of geometers have since striven to establish directly its general demonstration1 but thus far their efforts have been illusory4 ,their vain metaphysical considerations and heterogeneous minglings of the abstract and the concrete have so confused the subject1 that it becomes necessary to here distinctly enunciate the general fact4 ,it consists in this3 if1 in any equation whatever1 expressing the relation of certain quantities which are susceptible of opposition of directions1 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 established1 it will not be necessary to form directly a new equation for this second state of the phenomena2 it will suffice to change1 in the first equation1 the sign of each of the quantities which shall have changed its direction2 and the equation1 thus modified1 will always rigorously coincide with that which we would have arrived at in recommencing to investigate1 for this new case1 the analytical law of the phenomenon4 ,the general theorem consists in this constant and necessary coincidence4 ,now1 as yet1 no one has succeeded in directly proving this2 we have assured ourselves of it only by a great number of geometrical and mechanical verifications1 which are1 it is true1 sufficiently multiplied1 and especially sufficiently varied1 to prevent any clear mind from having the least doubt of the exactitude and the generality of this essential property1 but which1 in a philosophical point of view1 do not at all dispense with the research for so important an explanation4 ,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 truth4 ,this imperfection of theory1 however1 has not prevented geometers from making the most extensive and the most important use of this property in all parts of concrete mathematics4 ,it follows from the above general enunciation of the fact1 independently of any demonstration1 that the property of which we speak must never be applied to magnitudes whose directions are continually varying1 without giving rise to a simple opposition of direction2 in that case1 the sign with which every result of calculation is necessarily affected is not susceptible of any concrete interpretation1 and the attempts sometimes made to establish one are erroneous4 ,this circumstance occurs1 among other occasions1 in the case of a radius vector in geometry1 and diverging forces in mechanics4 ,,principle ,,of ,,homogeneity4 ,a second general theorem on the relation of the concrete to the abstract is that which is ordinarily designated under the name of ,principle of ,homogeneity4 ,it is undoubtedly much less important in its applications than the preceding1 but it particularly merits our attention as having1 by its nature1 a still greater extent1 since it is applicable to all phenomena without distinction1 and because of the real utility which it often possesses for the verification of their analytical laws4 ,i can1 moreover1 exhibit a direct and general demonstration of it which seems to me very simple4 ,it is founded on this single observation1 which is self-evident1 that the exactitude of every relation between any concrete magnitudes whatsoever is independent of the value of the units to which they are referred for the purpose of expressing them in numbers4 ,for example1 the relation which exists between the three sides of a right-angled triangle is the same1 whether they are measured by yards1 or by miles1 or by inches4 ,it follows from this general consideration1 that every equation which expresses the analytical law of any phenomenon must possess this property of being in no way altered1 when all the quantities which are found in it are made to undergo simultaneously the change corresponding to that which their respective units would experience4 ,now this change evidently consists in all the quantities of each sort becoming at once m times smaller1 if the unit which corresponds to them becomes m times greater1 or reciprocally4 ,thus every equation which represents any concrete relation whatever must possess this characteristic of remaining the same1 when we make m times greater all the quantities which it contains1 and which express the magnitudes between which the relation exists2 excepting always the numbers which designate simply the mutual ratios of these different magnitudes1 and which therefore remain invariable during the change of the units4 ,it is this property which constitutes the law of ,homogeneity in its most extended signification1 that is1 of whatever analytical functions the equations may be composed4 ,but most frequently we consider only the cases in which the functions are such as are called algebraic1 and to which the idea of degree is applicable4 ,in this case we can give more precision to the general proposition by determining the analytical character which must be necessarily presented by the equation1 in order that this property may be verified4 ,it is easy to see1 then1 that1 by the modification just explained1 all the terms of the first degree1 whatever may be their form1 rational or irrational1 entire or fractional1 will become m times greater2 all those of the second degree1 m times2 those of the third1 m times1 c4 ,thus the terms of the same degree1 however different may be their composition1 varying in the same manner1 and the terms of different degrees varying in an unequal proportion1 whatever similarity there may be in their composition1 it will be necessary1 to prevent the equation from being disturbed1 that all the terms which it contains should be of the same degree4 ,it is in this that properly consists the ordinary theorem of ,homogeneity1 and it is from this circumstance that the general law has derived its name1 which1 however1 ceases to be exactly proper for all other functions4 ,in order to treat this subject in its whole extent1 it is important to observe an essential condition1 to which attention must be paid in applying this property when the phenomenon expressed by the equation presents magnitudes of different natures4 ,thus it may happen that the respective units are completely independent of each other1 and then the theorem of ,homogeneity will hold good1 either with reference to all the corresponding classes of quantities1 or with regard to only a single one or more of them4 ,but it will happen on other occasions that the different units will have fixed relations to one another1 determined by the nature of the question2 then it will be necessary to pay attention to this subordination of the units in verifying the homogeneity1 which will not exist any longer in a purely algebraic sense1 and the precise form of which will vary according to the nature of the phenomena4 ,thus1 for example1 to fix our ideas1 when1 in the analytical expression of geometrical phenomena1 we are considering at once lines1 areas1 and volumes1 it will be necessary to observe that the three corresponding units are necessarily so connected with each other that1 according to the subordination generally established in that respect1 when the first becomes m times greater1 the second becomes m times1 and the third m times4 ,it is with such a modification that homogeneity will exist in the equations1 in which1 if they are algebraic1 we will have to estimate the degree of each term by doubling the exponents of the factors which correspond to areas1 and tripling those of the factors relating to volumes4 99 99 99 99 99 ,such are the principal general considerations relating to the ,calculus of ,direct ,functions4 ,we have now to pass to the philosophical examination of the ,calculus of ,indirect ,functions1 the much superior importance and extent of which claim a fuller development4 ,,chapter ,,iii4 ,,transcendental ,,analysis3 ,,different ,,modes ,,of ,,viewing ,,it4 ,we determined1 in the second chapter1 the philosophical character of the transcendental analysis1 in whatever manner it may be conceived1 considering only the general nature of its actual destination as a part of mathematical science4 ,this analysis has been presented by geometers under several points of view1 really distinct1 although necessarily equivalent1 and leading always to identical results4 ,they may be reduced to three principal ones2 those of ,,leibnitz1 of ,,newton1 and of ,,lagrange1 of which all the others are only secondary modifications4 ,in the present state of science1 each of these three general conceptions offers essential advantages which pertain to it exclusively1 without our having yet succeeded in constructing a single method uniting all these different characteristic qualities4 ,this combination will probably be hereafter effected by some method founded upon the conception of ,lagrange when that important philosophical labour shall have been accomplished1 the study of the other conceptions will have only a historic interest2 but1 until then1 the science must be considered as in only a provisional state1 which requires the simultaneous consideration of all the various modes of viewing this calculus4 ,illogical as may appear this multiplicity of conceptions of one identical subject1 still1 without them all1 we could form but a very insufficient idea of this analysis1 whether in itself1 or more especially in relation to its applications4 ,this want of system in the most important part of mathematical analysis will not appear strange if we consider1 on the one hand1 its great extent and its superior difficulty1 and1 on the other1 its recent formation4 ,,its ,,early ,,history4 ,if we had to trace here the systematic history of the successive formation of the transcendental analysis1 it would be necessary previously to distinguish carefully from the calculus of indirect functions1 properly so called1 the original idea of the infinitesimal method1 which can be conceived by itself1 independently of any calculus4 ,we should see that the first germ of this idea is found in the procedure constantly employed by the ,greek geometers1 under the name of the ,method of ,exhaustions1 as a means of passing from the properties of straight lines to those of curves1 and consisting essentially in substituting for the curve the auxiliary consideration of an inscribed or circumscribed polygon1 by means of which they rose to the curve itself1 taking in a suitable manner the limits of the primitive ratios4 ,incontestable as is this filiation of ideas1 it would be giving it a greatly exaggerated importance to see in this method of exhaustions the real equivalent of our modern methods1 as some geometers have done2 for the ancients had no logical and general means for the determination of these limits1 and this was commonly the greatest difficulty of the question2 so that their solutions were not subjected to abstract and invariable rules1 the uniform application of which would lead with certainty to the knowledge sought2 which is1 on the contrary1 the principal characteristic of our transcendental analysis4 ,in a word1 there still remained the task of generalizing the conceptions used by the ancients1 and1 more especially1 by considering it in a manner purely abstract1 of reducing it to a complete system of calculation1 which to them was impossible4 ,the first idea which was produced in this new direction goes back to the great geometer ,fermat1 whom ,lagrange has justly presented as having blocked out the direct formation of the transcendental analysis by his method for the determination of maxima and minima1 and for the finding of tangents1 which consisted essentially in introducing the auxiliary consideration of the correlative increments of the proposed variables1 increments afterward suppressed as equal to zero when the equations had undergone certain suitable transformations4 ,but1 although ,fermat was the first to conceive this analysis in a truly abstract manner1 it was yet far from being regularly formed into a general and distinct calculus having its own notation1 and especially freed from the superfluous consideration of terms which1 in the analysis of ,fermat1 were finally not taken into the account1 after having nevertheless greatly complicated all the operations by their presence4 ,this is what ,leibnitz so happily executed1 half a century later1 after some intermediate modifications of the ideas of ,fermat introduced by ,wallis1 and still more by ,barrow2 and he has thus been the true creator of the transcendental analysis1 such as we now employ it4 ,this admirable discovery was so ripe 7like all the great conceptions of the human intellect at the moment of their manifestation71 that ,newton1 on his side1 had arrived1 at the same time1 or a little earlier1 at a method exactly equivalent1 by considering this analysis under a very different point of view1 which1 although more logical in itself1 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 ,leibnitz4 ,finally1 ,lagrange1 putting aside the heterogeneous considerations which had guided ,leibnitz and ,newton1 has succeeded in reducing the transcendental analysis1 in its greatest perfection1 to a purely algebraic system1 which only wants more aptitude for its practical applications4 ,after this summary glance at the general history of the transcendental analysis1 we will proceed to the dogmatic exposition of the three principal conceptions1 in order to appreciate exactly their characteristic properties1 and to show the necessary identity of the methods which are thence derived4 ,let us begin with that of ,leibnitz4 ,,method ,,of ,,leibnitz4 ,infinitely small ,elements4 ,this consists in introducing into the calculus1 in order to facilitate the establishment of equations1 the infinitely small elements of which all the quantities1 the relations between which are sought1 are considered to be composed4 ,these elements or differentials will have certain relations to one another1 which are constantly and necessarily more simple and easy to discover than those of the primitive quantities1 and by means of which we will be enabled 7by a special calculus having for its peculiar object the elimination of these auxiliary infinitesimals7 to go back to the desired equations1 which it would have been most frequently impossible to obtain directly4 ,this indirect analysis may have different degrees of indirectness2 for1 when there is too much difficulty in forming immediately the equation between the differentials of the magnitudes under consideration1 a second application of the same general artifice will have to be made1 and these differentials be treated1 in their turn1 as new primitive quantities1 and a relation be sought between their infinitely small elements 7which1 with reference to the final objects of the question1 will be second differentials71 and so on2 the same transformation admitting of being repeated any number of times1 on the condition of finally eliminating the constantly increasing number of infinitesimal quantities introduced as auxiliaries4 ,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 phenomena2 for the infinitely small increments of the proposed magnitudes being of the same species with them1 it would seem that their relations should not be obtained with more ease1 inasmuch as the greater or less value of a quantity cannot1 in fact1 exercise any influence on an inquiry which is necessarily independent1 by its nature1 of every idea of value4 ,but it is easy1 nevertheless1 to explain very clearly1 and in a quite general manner1 how far the question must be simplified by such an artifice4 ,for this purpose1 it is necessary to begin by distinguishing different orders of infinitely small quantities1 a very precise idea of which may be obtained by considering them as being either the successive powers of the same primitive infinitely small quantity1 or as being quantities which may be regarded as having finite ratios with these powers2 so that1 to take an example1 the second1 third1 c41 differentials of any one variable are classed as infinitely small quantities of the second order1 the third1 c41 because it is easy to discover in them finite multiples of the second1 third1 c41 powers of a certain first differential4 ,these preliminary ideas being established1 the spirit of the infinitesimal analysis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities1 and generally the infinitely small quantities of any order whatever in comparison with all those of an inferior order4 ,it is at once apparent how much such a liberty must facilitate the formation of equations between the differentials of quantities1 since1 in the place of these differentials1 we can substitute such other elements as we may choose1 and as will be more simple to consider1 only taking care to conform to this single condition1 that the new elements differ from the preceding ones only by quantities infinitely small in comparison with them4 ,it is thus that it will be possible1 in geometry1 to treat curved lines as composed of an infinity of rectilinear elements1 curved surfaces as formed of plane elements1 and1 in mechanics1 variable motions as an infinite series of uniform motions1 succeeding one another at infinitely small intervals of time4 ,,examples4 ,considering the importance of this admirable conception1 ,i think that ,i ought here to complete the illustration of its fundamental character by the summary indication of some leading examples4 #a4 ,tangents4 ,let it be required to determine1 for each point of a plane curve1 the equation of which is given1 the direction of its tangent2 a question whose general solution was the primitive object of the inventors of the transcendental analysis4 ,we will consider the tangent as a secant joining two points infinitely near to each other2 and then1 designating by dy and dx the infinitely small differences of the co-ordinates of those two points1 the elementary principles of geometry will immediately give the equation t dy_/dx for the trigonometrical tangent of the angle which is made with the axis of the abscissas by the desired tangent1 this being the most simple way of fixing its position in a system of rectilinear co-ordinates4 ,this equation1 common to all curves1 being established1 the question is reduced to a simple analytical problem1 which will consist in eliminating the infinitesimals dx and dy1 which were introduced as auxiliaries1 by determining in each particular case1 by means of the equation of the proposed curve1 the ratio of dy to dx1 which will be constantly done by uniform and very simple methods4 #b4 ,rectification of an ,arc4 ,in the second place1 suppose that we wish to know the length of the arc of any curve1 considered as a function of the co-ordinates of its extremities4 ,it would be impossible to establish directly the equation between this arc s and these co-ordinates1 while it is easy to find the corresponding relation between the differentials of these different magnitudes4 ,the most simple theorems of elementary geometry will in fact give at once1 considering the infinitely small arc ds as a right line1 the equations ds dy dx1 or ds dx dy dz1 according as the curve is of single or double curvature4 ,in either case1 the question is now entirely within the domain of analysis1 which1 by the elimination of the differentials 7which is the peculiar object of the calculus of indirect functions71 will carry us back from this relation to that which exists between the finite quantities themselves under examination4 #c4 ,quadrature of a ,curve4 ,it would be the same with the quadrature of curvilinear areas4 ,if the curve is a plane one1 and referred to rectilinear co-ordinates1 we will conceive the area ,a comprised between this curve1 the axis of the abscissas1 and two extreme co-ordinates1 to increase by an infinitely small quantity d,a1 as the result of a corresponding increment of the abscissa4 ,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 abscissa1 from which it evidently differs only by an infinitely small quantity of the second order4 ,this will at once give1 whatever may be the curve1 the very simple differential equation d,a ydx1 from which1 when the curve is defined1 the calculus of indirect functions will show how to deduce the finite equation1 which is the immediate object of the problem4 #d4 ,velocity in ,variable ,motion4 ,in like manner1 in ,dynamics1 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 law1 we will consider the motion as being uniform during an infinitely small element of the time t1 and we will thus immediately form the differential equation de vdt1 in which v designates the velocity acquired when the body has passed over the space e2 and thence it will be easy to deduce1 by simple and invariable analytical procedures1 the formula which would give the velocity in each particular motion1 in accordance with the corresponding relation between the time and the space2 or1 reciprocally1 what this relation would be if the mode of variation of the velocity was supposed to be known1 whether with respect to the space or to the time4 #e4 ,distribution of ,heat4 ,lastly1 to indicate another kind of questions1 it is by similar steps that we are able1 in the study of thermological phenomena1 according to the happy conception of ,m4 ,fourier1 to form in a very simple manner the general differential equation which expresses the variable distribution of heat in any body whatever1 subjected to any influences1 by means of the single and easily-obtained relation1 which represents the uniform distribution of heat in a right-angled parallelopipedon1 considering 7geometrically7 every other body as decomposed into infinitely small elements of a similar form1 and 7thermologically7 the flow of heat as constant during an infinitely small element of time4 ,henceforth1 all the questions which can be presented by abstract thermology will be reduced1 as in geometry and mechanics1 to mere difficulties of analysis1 which will always consist in the elimination of the differentials introduced as auxiliaries to facilitate the establishment of the equations4 ,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 ,leibnitz1 constituting1 as it undoubtedly does1 the most lofty thought to which the human mind has as yet attained4 ,it is evident that this conception was indispensable to complete the foundation of mathematical science1 by enabling us to establish1 in a broad and fruitful manner1 the relation of the concrete to the abstract4 ,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 phenomena3 an idea which did not begin to be worthily appreciated and suitably employed till after the formation of the infinitesimal analysis1 without which it could not produce1 even in geometry1 very important results4 ,generality of the ,formulas4 ,besides the admirable facility which is given by the transcendental analysis for the investigation of the mathematical laws of all phenomena1 a second fundamental and inherent property1 perhaps as important as the first1 is the extreme generality of the differential formulas1 which express in a single equation each determinate phenomenon1 however varied the subjects in relation to which it is considered4 ,thus we see1 in the preceding examples1 that a single differential equation gives the tangents of all curves1 another their rectifications1 a third their quadratures2 and in the same way1 one invariable formula expresses the mathematical law of every variable motion2 and1 finally1 a single equation constantly represents the distribution of heat in any body and for any case4 ,this generality1 which is so exceedingly remarkable1 and which is for geometers the basis of the most elevated considerations1 is a fortunate and necessary consequence of the very spirit of the transcendental analysis1 especially in the conception of ,leibnitz4 ,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 manner1 but it has also permitted us to consider1 for the mathematical study of natural phenomena1 a new order of more general laws1 which nevertheless present a clear and precise signification to every mind habituated to their interpretation4 ,by virtue of this second characteristic property1 the entire system of an immense science1 such as geometry or mechanics1 has been condensed into a small number of analytical formulas1 from which the human mind can deduce1 by certain and invariable rules1 the solution of all particular problems4 ,demonstration of the ,method4 ,to complete the general exposition of the conception of ,leibnitz1 there remains to be considered the demonstration of the logical procedure to which it leads1 and this1 unfortunately1 is the most imperfect part of this beautiful method4 ,in the beginning of the infinitesimal analysis1 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 operations4 ,they contented themselves for a long time by answering the objections of second-rate geometers by the unhoped-for solution of the most difficult problems2 doubtless persuaded that in mathematical science1 much more than in any other1 we may boldly welcome new methods1 even when their rational explanation is imperfect1 provided they are fruitful in results1 inasmuch as its much easier and more numerous verifications would not permit any error to remain long undiscovered4 ,but this state of things could not long exist1 and it was necessary to go back to the very foundations of the analysis of ,leibnitz in order to prove1 in a perfectly general manner1 the rigorous exactitude of the procedures employed in this method1 in spite of the apparent infractions of the ordinary rules of reasoning which it permitted4 ,leibnitz1 urged to answer1 had presented an explanation entirely erroneous1 saying that he treated infinitely small quantities as incomparables1 and that he neglected them in comparison with finite quantities1 like grains of sand in comparison with the sea3 a view which would have completely changed the nature of his analysis1 by reducing it to a mere approximative calculus1 which1 under this point of view1 would be radically vicious1 since it would be impossible to foresee1 in general1 to what degree the successive operations might increase these first errors1 which could thus evidently attain any amount4 ,leibnitz1 then1 did not see1 except in a very confused manner1 the true logical foundations of the analysis which he had created4 ,his earliest successors limited themselves1 at first1 to verifying its exactitude by showing the conformity of its results1 in particular applications1 to those obtained by ordinary algebra or the geometry of the ancients2 reproducing1 according to the ancient methods1 so far as they were able1 the solutions of some problems after they had been once obtained by the new method1 which alone was capable of discovering them in the first place4 ,when this great question was considered in a more general manner1 geometers1 instead of directly attacking the difficulty1 preferred to elude it in some way1 as ,euler and ,d',alembert1 for example1 have done1 by demonstrating the necessary and constant conformity of the conception of ,leibnitz1 viewed in all its applications1 with other fundamental conceptions of the transcendental analysis1 that of ,newton especially1 the exactitude of which was free from any objection4 ,such a general verification is undoubtedly strictly sufficient to dissipate any uncertainty as to the legitimate employment of the analysis of ,leibnitz4 ,but the infinitesimal method is so important--it offers still1 in almost all its applications1 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 itself1 and needed to be logically founded on considerations of another order1 which would then cease to be employed4 ,it was1 then1 of real importance to establish directly and in a general manner the necessary rationality of the infinitesimal method4 ,after various attempts more or less imperfect1 a distinguished geometer1 ,carnot1 presented at last the true direct logical explanation of the method of ,leibnitz1 by showing it to be founded on the principle of the necessary compensation of errors1 this being1 in fact1 the precise and luminous manifestation of what ,leibnitz had vaguely and confusedly perceived4 ,carnot has thus rendered the science an essential service1 although1 as we shall see towards the end of this chapter1 all this logical scaffolding of the infinitesimal method1 properly so called1 is very probably susceptible of only a provisional existence1 inasmuch as it is radically vicious in its nature4 ,still1 we should not fail to notice the general system of reasoning proposed by ,carnot1 in order to directly legitimate the analysis of ,leibnitz4 ,here is the substance of it3 ,in establishing the differential equation of a phenomenon1 we substitute1 for the immediate elements of the different quantities considered1 other simpler infinitesimals1 which differ from them infinitely little in comparison with them2 and this substitution constitutes the principal artifice of the method of ,leibnitz1 which without it would possess no real facility for the formation of equations4 ,carnot regards such an hypothesis as really producing an error in the equation thus obtained1 and which for this reason he calls imperfect2 only1 it is clear that this error must be infinitely small4 ,now1 on the other hand1 all the analytical operations1 whether of differentiation or of integration1 which are performed upon these differential equations1 in order to raise them to finite equations by eliminating all the infinitesimals which have been introduced as auxiliaries1 produce as constantly1 by their nature1 as is easily seen1 other analogous errors1 so that an exact compensation takes place1 and the final equations1 in the words of ,carnot1 become perfect4 ,carnot views1 as a certain and invariable indication of the actual establishment of this necessary compensation1 the complete elimination of the various infinitely small quantities1 which is always1 in fact1 the final object of all the operations of the transcendental analysis2 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 method1 the infinitely small errors thus produced cannot have engendered other than infinitely small errors in all the equations1 and the relations are necessarily of a rigorous exactitude as soon as they exist between finite quantities alone1 since the only errors then possible must be finite ones1 while none such can have entered4 ,all this general reasoning is founded on the conception of infinitesimal quantities1 regarded as indefinitely decreasing1 while those from which they are derived are regarded as fixed4 ,illustration by ,tangents4 ,thus1 to illustrate this abstract exposition by a single example1 let us take up again the question of tangents1 which is the most easy to analyze completely4 ,we will regard the equation t dy_/dx1 obtained above1 as being affected with an infinitely small error1 since it would be perfectly rigorous only for the secant4 ,now let us complete the solution by seeking1 according to the equation of each curve1 the ratio between the differentials of the co-ordinates4 ,if we suppose this equation to be y ax1 we shall evidently have dy #b;axdx adx4 ,in this formula we shall have to neglect the term dx as an infinitely small quantity of the second order4 ,then the combination of the two imperfect equations4 t dy_/dx1 dy #b;ax7dx71 being sufficient to eliminate entirely the infinitesimals1 the finite result1 t #b;ax1 will necessarily be rigorously correct1 from the effect of the exact compensation of the two errors committed2 since1 by its finite nature1 it cannot be affected by an infinitely small error1 and this is1 nevertheless1 the only one which it could have1 according to the spirit of the operations which have been executed4 ,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 ,leibnitz4 ,this ingenious theory is undoubtedly more subtile than solid1 when we examine it more profoundly2 but it has really no other radical logical fault than that of the infinitesimal method itself1 of which it is1 it seems to me1 the natural development and the general explanation1 so that it must be adopted for as long a time as it shall be thought proper to employ this method directly4 99 99 99 99 99 ,i pass now to the general exposition of the two other fundamental conceptions of the transcendental analysis1 limiting myself in each to its principal idea1 the philosophical character of the analysis having been sufficiently determined above in the examination of the conception of ,leibnitz1 which ,i have specially dwelt upon because it admits of being most easily grasped as a whole1 and most rapidly described4 ,,method ,,of ,,newton4 ,newton has successively presented his own method of conceiving the transcendental analysis under several different forms4 ,that which is at present the most commonly adopted was designated by ,newton1 sometimes under the name of the ,method of prime and ultimate ,ratios1 sometimes under that of the ,method of ,limits4 ,method of ,limits4 ,the general spirit of the transcendental analysis1 from this point of view1 consists in introducing as auxiliaries1 in the place of the primitive quantities1 or concurrently with them1 in order to facilitate the establishment of equations1 the limits of the ratios of the simultaneous increments of these quantities2 or1 in other words1 the final ratios of these increments2 limits or final ratios which can be easily shown to have a determinate and finite value4 ,a special calculus1 which is the equivalent of the infinitesimal calculus1 is then employed to pass from the equations between these limits to the corresponding equations between the primitive quantities themselves4 ,the power which is given by such an analysis1 of expressing with more ease the mathematical laws of phenomena1 depends in general on this1 that since the calculus applies1 not to the increments themselves of the proposed quantities1 but to the limits of the ratios of those increments1 we can always substitute for each increment any other magnitude more easy to consider1 provided that their final ratio is the ratio of equality1 or1 in other words1 that the limit of their ratio is unity4 ,it is clear1 indeed1 that the calculus of limits would be in no way affected by this substitution4 ,starting from this principle1 we find nearly the equivalent of the facilities offered by the analysis of ,leibnitz1 which are then merely conceived under another point of view4 ,thus curves will be regarded as the limits of a series of rectilinear polygons1 variable motions as the limits of a collection of uniform motions of constantly diminishing durations1 and so on4 ,,examples4 #a4 ,tangents4 ,suppose1 for example1 that we wish to determine the direction of the tangent to a curve2 we will regard it as the limit towards which would tend a secant1 which should turn about the given point so that its second point of intersection should indefinitely approach the first4 ,representing the differences of the co-ordinates of the two points by y and x1 we would have at each instant1 for the trigonometrical tangent of the angle which the secant makes with the axis of abscissas1 t y_/x2 from which1 taking the limits1 we will obtain1 relatively to the tangent itself1 this general formula of transcendental analysis1 t ,l7y_/x71 the characteristic ,l being employed to designate the limit4 ,the calculus of indirect functions will show how to deduce from this formula in each particular case1 when the equation of the curve is given1 the relation between t and x1 by eliminating the auxiliary quantities which have been introduced4 ,if we suppose1 in order to complete the solution1 that the equation of the proposed curve is y ax1 we shall evidently have y #b;axx a7x71 from which we shall obtain y_/x #b;ax ax4 ,now it is clear that the limit towards which the second number tends1 in proportion as x diminishes1 is #b;ax4 ,we shall therefore find1 by this method1 t #b;ax1 as we obtained it for the same case by the method of ,leibnitz4 #b4 ,rectifications4 ,in like manner1 when the rectification of a curve is desired1 we must substitute for the increment of the arc s the chord of this increment1 which evidently has such a connexion with it that the limit of their ratio is unity2 and then we find 7pursuing in other respects the same plan as with the method of ,leibnitz7 this general equation of rectifications3 7,ls_/x7 #a 7,ly_/x71 or 7,ls_/x7 #a 7,ly_/x7 7,lz_/x71 according as the curve is plane or of double curvature4 ,it will now be necessary1 for each particular curve1 to pass from this equation to that between the arc and the abscissa1 which depends on the transcendental calculus properly so called4 ,we could take up1 with the same facility1 by the method of limits1 all the other general questions1 the solution of which has been already indicated according to the infinitesimal method4 ,such is1 in substance1 the conception which ,newton formed for the transcendental analysis1 or1 more precisely1 that which ,maclaurin and ,d',alembert have presented as the most rational basis of that analysis1 in seeking to fix and to arrange the ideas of ,newton upon that subject4 ,fluxions and ,fluents4 ,another distinct form under which ,newton has presented this same method should be here noticed1 and deserves particularly to fix our attention1 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 analysis1 and1 moreover1 as having been till lately the special form of the calculus of indirect functions commonly adopted by the ,english geometers4 ,i refer to the calculus of fluxions and of fluents1 founded on the general idea of velocities4 ,to facilitate the conception of the fundamental idea1 let us consider every curve as generated by a point impressed with a motion varying according to any law whatever4 ,the different quantities which the curve can present1 the abscissa1 the ordinate1 the arc1 the area1 c41 will be regarded as simultaneously produced by successive degrees during this motion4 ,the velocity with which each shall have been described will be called the fluxion of that quantity1 which will be inversely named its fluent4 ,henceforth the transcendental analysis will consist1 according to this conception1 in forming directly the equations between the fluxions of the proposed quantities1 in order to deduce therefrom1 by a special calculus1 the equations between the fluents themselves4 ,what has been stated respecting curves may1 moreover1 evidently be applied to any magnitudes whatever1 regarded1 by the aid of suitable images1 as produced by motion4 ,it is easy to understand the general and necessary identity of this method with that of limits complicated with the foreign idea of motion4 ,in fact1 resuming the case of the curve1 if we suppose1 as we evidently always may1 that the motion of the describing point is uniform in a certain direction1 that of the abscissa1 for example1 then the fluxion of the abscissa will be constant1 like the element of the time2 for all the other quantities generated1 the motion cannot be conceived to be uniform1 except for an infinitely small time4 ,now the velocity being in general according to its mechanical conception1 the ratio of each space to the time employed in traversing it1 and this time being here proportional to the increment of the abscissa1 it follows that the fluxions of the ordinate1 of the arc1 of the area1 c41 are really nothing else 7rejecting the intermediate consideration of time7 than the final ratios of the increments of these different quantities to the increment of the abscissa4 ,this method of fluxions and fluents is1 then1 in reality1 only a manner of representing1 by a comparison borrowed from mechanics1 the method of prime and ultimate ratios1 which alone can be reduced to a calculus4 ,it evidently1 then1 offers the same general advantages in the various principal applications of the transcendental analysis1 without its being necessary to present special proofs of this4 ,,method ,,of ,,lagrange4 ,derived ,functions4 ,the conception of ,lagrange1 in its admirable simplicity1 consists in representing the transcendental analysis as a great algebraic artifice1 by which1 in order to facilitate the establishment of equations1 we introduce1 in the place of the primitive functions1 or concurrently with them1 their derived functions2 that is1 according to the definition of ,lagrange1 the coefficient of the first term of the increment of each function1 arranged according to the ascending powers of the increment of its variable4 ,the special calculus of indirect functions has for its constant object1 here as well as in the conceptions of ,leibnitz and of ,newton1 to eliminate these derivatives which have been thus employed as auxiliaries1 in order to deduce from their relations the corresponding equations between the primitive magnitudes4 ,an ,extension of ordinary ,analysis4 ,the transcendental analysis is1 then1 nothing but a simple though very considerable extension of ordinary analysis4 ,geometers have long been accustomed to introduce in analytical investigations1 in the place of the magnitudes themselves which they wished to study1 their different powers1 or their logarithms1 or their sines1 c41 in order to simplify the equations1 and even to obtain them more easily4 ,this successive derivation is an artifice of the same nature1 only of greater extent1 and procuring1 in consequence1 much more important resources for this common object4 ,but1 although we can readily conceive1 priori1 that the auxiliary consideration of these derivatives may facilitate the establishment of equations1 it is not easy to explain why this must necessarily follow from this mode of derivation rather than from any other transformation4 ,such is the weak point of the great idea of ,lagrange4 ,the precise advantages of this analysis cannot as yet be grasped in an abstract manner1 but only shown by considering separately each principal question1 so that the verification is often exceedingly laborious4 ,,example4 ,tangents4 ,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 tangents4 ,instead of conceiving the tangent as the prolongation of the infinitely small element of the curve1 according to the notion of ,leibnitz--or as the limit of the secants1 according to the ideas of ,newton--,lagrange considers it1 according to its simple geometrical character1 analogous to the definitions of the ancients1 to be a right line such that no other right line can pass through the point of contact between it and the curve4 ,then1 to determine its direction1 we must seek the general expression of its distance from the curve1 measured in any direction whatever--in that of the ordinate1 for example--and dispose of the arbitrary constant relating to the inclination of the right line1 which will necessarily enter into that expression1 in such a way as to diminish that separation as much as possible4 ,now this distance1 being evidently equal to the difference of the two ordinates of the curve and of the right line1 which correspond to the same new abscissa x h1 will be represented by the formula 7f'7x7 - t7h qh rh etc41 in which t designates1 as above1 the unknown trigonometrical tangent of the angle which the required line makes with the axis of abscissas1 and f'7x7 the derived function of the ordinate f7x74 ,this being understood1 it is easy to see that1 by disposing of t so as to make the first term of the preceding formula equal to zero1 we will render the interval between the two lines the least possible1 so that any other line for which t did not have the value thus determined would necessarily depart farther from the proposed curve4 ,we have1 then1 for the direction of the tangent sought1 the general expression t f'7x71 a result exactly equivalent to those furnished by the ,infinitesimal ,method and the ,method of ,limits4 ,we have yet to find f'7x7 in each particular curve1 which is a mere question of analysis1 quite identical with those which are presented1 at this stage of the operations1 by the other methods4 ,after these considerations upon the principal general conceptions1 we need not stop to examine some other theories proposed1 such as ,euler's ,calculus of ,vanishing ,quantities1 which are really modifications--more or less important1 and1 moreover1 no longer used--of the preceding methods4 ,i have now to establish the comparison and the appreciation of these three fundamental methods4 ,their perfect and necessary conformity is first to be proven in a general manner4 ,,fundamental ,,identity ,,of ,,the ,,three ,,methods4 ,it is1 in the first place1 evident from what precedes1 considering these three methods as to their actual destination1 independently of their preliminary ideas1 that they all consist in the same general logical artifice1 which has been characterized in the first chapter2 to wit1 the introduction of a certain system of auxiliary magnitudes1 having uniform relations to those which are the special objects of the inquiry1 and substituted for them expressly to facilitate the analytical expression of the mathematical laws of the phenomena1 although they have finally to be eliminated by the aid of a special calculus4 ,it is this which has determined me to regularly define the transcendental analysis as the calculus of indirect functions1 in order to mark its true philosophical character1 at the same time avoiding any discussion upon the best manner of conceiving and applying it4 ,the general effect of this analysis1 whatever the method employed1 is1 then1 to bring every mathematical question much more promptly within the power of the calculus1 and thus to diminish considerably the serious difficulty which is usually presented by the passage from the concrete to the abstract4 ,whatever progress we may make1 we can never hope that the calculus will ever be able to grasp every question of natural philosophy1 geometrical1 or mechanical1 or thermological1 c41 immediately upon its birth1 which would evidently involve a contradiction4 ,every problem will constantly require a certain preliminary labour to be performed1 in which the calculus can be of no assistance1 and which1 by its nature1 cannot be subjected to abstract and invariable rules2 it is that which has for its special object the establishment of equations1 which form the indispensable starting point of all analytical researches4 ,but this preliminary labour has been remarkably simplified by the creation of the transcendental analysis1 which has thus hastened the moment at which the solution admits of the uniform and precise application of general and abstract methods2 by reducing1 in each case1 this special labour to the investigation of equations between the auxiliary magnitudes2 from which the calculus then leads to equations directly referring to the proposed magnitudes1 which1 before this admirable conception1 it had been necessary to establish directly and separately4 ,whether these indirect equations are differential equations1 according to the idea of ,leibnitz1 or equations of limits1 conformably to the conception of ,newton1 or1 lastly1 derived equations1 according to the theory of ,lagrange1 the general procedure is evidently always the same4 ,but the coincidence of these three principal methods is not limited to the common effect which they produce2 it exists1 besides1 in the very manner of obtaining it4 ,in fact1 not only do all three consider1 in the place of the primitive magnitudes1 certain auxiliary ones1 but1 still farther1 the quantities thus introduced as subsidiary are exactly identical in the three methods1 which consequently differ only in the manner of viewing them4 ,this can be easily shown by taking for the general term of comparison any one of the three conceptions1 especially that of ,lagrange1 which is the most suitable to serve as a type1 as being the freest from foreign considerations4 ,is it not evident1 by the very definition of derived functions1 that they are nothing else than what ,leibnitz calls differential coefficients1 or the ratios of the differential of each function to that of the corresponding variable1 since1 in determining the first differential1 we will be obliged1 by the very nature of the infinitesimal method1 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 variable8 ,in the same way1 is not the derived function1 by its nature1 likewise the necessary limit towards which tends the ratio between the increment of the primitive function and that of its variable1 in proportion as this last indefinitely diminishes1 since it evidently expresses what that ratio becomes when we suppose the increment of the variable to equal zero8 ,that which is designated by dx_/dy in the method of ,leibnitz2 that which ought to be noted as ,l7y_/x7 in that of ,newton2 and that which ,lagrange has indicated by f'7x71 is constantly one same function1 seen from three different points of view1 the considerations of ,leibnitz and ,newton properly consisting in making known two general necessary properties of the derived function4 ,the transcendental analysis1 examined abstractedly and in its principle1 is then always the same1 whatever may be the conception which is adopted1 and the procedures of the calculus of indirect functions are necessarily identical in these different methods1 which in like manner must1 for any application whatever1 lead constantly to rigorously uniform results4 ,,comparative ,,value ,,of ,,the ,,three ,,methods4 ,if now we endeavour to estimate the comparative value of these three equivalent conceptions1 we shall find in each advantages and inconveniences which are peculiar to it1 and which still prevent geometers from confining themselves to any one of them1 considered as final4 ,that of ,leibnitz4 ,the conception of ,leibnitz presents incontestably1 in all its applications1 a very marked superiority1 by leading in a much more rapid manner1 and with much less mental effort1 to the formation of equations between the auxiliary magnitudes4 ,it is to its use that we owe the high perfection which has been acquired by all the general theories of geometry and mechanics4 ,whatever may be the different speculative opinions of geometers with respect to the infinitesimal method1 in an abstract point of view1 all tacitly agree in employing it by preference1 as soon as they have to treat a new question1 in order not to complicate the necessary difficulty by this purely artificial obstacle proceeding from a misplaced obstinacy in adopting a less expeditious course4 ,lagrange himself1 after having reconstructed the transcendental analysis on new foundations1 has 7with that noble frankness which so well suited his genius7 rendered a striking and decisive homage to the characteristic properties of the conception of ,leibnitz1 by following it exclusively in the entire system of his ,mchanique ,analytique4 ,such a fact renders any comments unnecessary4 ,but when we consider the conception of ,leibnitz in itself and in its logical relations1 we cannot escape admitting1 with ,lagrange1 that it is radically vicious in this1 that1 adopting its own expressions1 the notion of infinitely small quantities is a false idea1 of which it is in fact impossible to obtain a clear conception1 however we may deceive ourselves in that matter4 ,even if we adopt the ingenious idea of the compensation of errors1 as above explained1 this involves the radical inconvenience of being obliged to distinguish in mathematics two classes of reasonings1 those which are perfectly rigorous1 and those in which we designedly commit errors which subsequently have to be compensated4 ,a conception which leads to such strange consequences is undoubtedly very unsatisfactory in a logical point of view4 ,to say1 as do some geometers1 that it is possible in every case to reduce the infinitesimal method to that of limits1 the logical character of which is irreproachable1 would evidently be to elude the difficulty rather than to remove it2 besides1 such a transformation almost entirely strips the conception of ,leibnitz of its essential advantages of facility and rapidity4 ,finally1 even disregarding the preceding important considerations1 the infinitesimal method would no less evidently present by its nature the very serious defect of breaking the unity of abstract mathematics1 by creating a transcendental analysis founded on principles so different from those which form the basis of the ordinary analysis4 ,this division of analysis into two worlds almost entirely independent of each other1 tends to hinder the formation of truly general analytical conceptions4 ,to fully appreciate the consequences of this1 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 sections4 ,that of ,newton4 ,passing now to the conception of ,newton1 it is evident that by its nature it is not exposed to the fundamental logical objections which are called forth by the method of ,leibnitz4 ,the notion of limits is1 in fact1 remarkable for its simplicity and its precision4 ,in the transcendental analysis presented in this manner1 the equations are regarded as exact from their very origin1 and the general rules of reasoning are as constantly observed as in ordinary analysis4 ,but1 on the other hand1 it is very far from offering such powerful resources for the solution of problems as the infinitesimal method4 ,the obligation which it imposes1 of never considering the increments of magnitudes separately and by themselves1 nor even in their ratios1 but only in the limits of those ratios1 retards considerably the operations of the mind in the formation of auxiliary equations4 ,we may even say that it greatly embarrasses the purely analytical transformations4 ,thus the transcendental analysis1 considered separately from its applications1 is far from presenting in this method the extent and the generality which have been imprinted upon it by the conception of ,leibnitz4 ,it is very difficult1 for example1 to extend the theory of ,newton to functions of several independent variables4 ,but it is especially with reference to its applications that the relative inferiority of this theory is most strongly marked4 ,several ,continental geometers1 in adopting the method of ,newton as the more logical basis of the transcendental analysis1 have partially disguised this inferiority by a serious inconsistency1 which consists in applying to this method the notation invented by ,leibnitz for the infinitesimal method1 and which is really appropriate to it alone4 ,in designating by dy_/dx that which logically ought1 in the theory of limits1 to be denoted by ,l7y_/x71 and in extending to all the other analytical conceptions this displacement of signs1 they intended1 undoubtedly1 to combine the special advantages of the two methods2 but1 in reality1 they have only succeeded in causing a vicious confusion between them1 a familiarity with which hinders the formation of clear and exact ideas of either4 ,it would certainly be singular1 considering this usage in itself1 that1 by the mere means of signs1 it could be possible to effect a veritable combination between two theories so distinct as those under consideration4 ,finally1 the method of limits presents also1 though in a less degree1 the greater inconvenience1 which ,i have above noted in reference to the infinitesimal method1 of establishing a total separation between the ordinary and the transcendental analysis2 for the idea of limits1 though clear and rigorous1 is none the less in itself1 as ,lagrange has remarked1 a foreign idea1 upon which analytical theories ought not to be dependent4 ,that of ,lagrange4 ,this perfect unity of analysis1 and this purely abstract character of its fundamental notions1 are found in the highest degree in the conception of ,lagrange1 and are found there alone2 it is1 for this reason1 the most rational and the most philosophical of all4 ,carefully removing every heterogeneous consideration1 ,lagrange has reduced the transcendental analysis to its true peculiar character1 that of presenting a very extensive class of analytical transformations1 which facilitate in a remarkable degree the expression of the conditions of various problems4 ,at the same time1 this analysis is thus necessarily presented as a simple extension of ordinary analysis2 it is only a higher algebra4 ,all the different parts of abstract mathematics1 previously so incoherent1 have from that moment admitted of being conceived as forming a single system4 ,unhappily1 this conception1 which possesses such fundamental properties1 independently of its so simple and so lucid notation1 and which is undoubtedly destined to become the final theory of transcendental analysis1 because of its high philosophical superiority over all the other methods proposed1 presents in its present state too many difficulties in its applications1 as compared with the conception of ,newton1 and still more with that of ,leibnitz1 to be as yet exclusively adopted4 ,lagrange himself has succeeded only with great difficulty in rediscovering1 by his method1 the principal results already obtained by the infinitesimal method for the solution of the general questions of geometry and mechanics2 we may judge from that what obstacles would be found in treating in the same manner questions which were truly new and important4 ,it is true that ,lagrange1 on several occasions1 has shown that difficulties call forth1 from men of genius1 superior efforts1 capable of leading to the greatest results4 ,it was thus that1 in trying to adapt his method to the examination of the curvature of lines1 which seemed so far from admitting its application1 he arrived at that beautiful theory of contacts which has so greatly perfected that important part of geometry4 ,but1 in spite of such happy exceptions1 the conception of ,lagrange has nevertheless remained1 as a whole1 essentially unsuited to applications4 ,the final result of the general comparison which ,i have too briefly sketched1 is1 then1 as already suggested1 that1 in order to really understand the transcendental analysis1 we should not only consider it in its principles according to the three fundamental conceptions of ,leibnitz1 of ,newton1 and of ,lagrange1 but should besides accustom ourselves to carry out almost indifferently1 according to these three principal methods1 and especially according to the first and the last1 the solution of all important questions1 whether of the pure calculus of indirect functions or of its applications4 ,this is a course which ,i could not too strongly recommend to all those who desire to judge philosophically of this admirable creation of the human mind1 as well as to those who wish to learn to make use of this powerful instrument with success and with facility4 ,in all the other parts of mathematical science1 the consideration of different methods for a single class of questions may be useful1 even independently of its historical interest1 but it is not indispensable2 here1 on the contrary1 it is strictly necessary4 ,having determined with precision1 in this chapter1 the philosophical character of the calculus of indirect functions1 according to the principal fundamental conceptions of which it admits1 we have next to consider1 in the following chapter1 the logical division and the general composition of this calculus4 ,,chapter ,,iv4 ,,the ,,differential ,,and ,,integral ,,calculus4 ,,its ,,two ,,fundamental ,,divisions4 ,the calculus of indirect functions1 in accordance with the considerations explained in the preceding chapter1 is necessarily divided into two parts 7or1 more properly1 is decomposed into two different calculi entirely distinct1 although intimately connected by their nature71 according as it is proposed to find the relations between the auxiliary magnitudes 7the introduction of which constitutes the general spirit of this calculus7 by means of the relations between the corresponding primitive magnitudes2 or1 conversely1 to try to discover these direct equations by means of the indirect equations originally established4 ,such is1 in fact1 constantly the double object of the transcendental analysis4 ,these two systems have received different names1 according to the point of view under which this analysis has been regarded4 ,the infinitesimal method1 properly so called1 having been the most generally employed for the reasons which have been given1 almost all geometers employ habitually the denominations of ,differential ,calculus and of ,integral ,calculus1 established by ,leibnitz1 and which are1 in fact1 very rational consequences of his conception4 ,newton1 in accordance with his method1 named the first the ,calculus of ,fluxions1 and the second the ,calculus of ,fluents1 expressions which were commonly employed in ,england4 ,finally1 following the eminently philosophical theory founded by ,lagrange1 one would be called the ,calculus of ,derived ,functions1 and the other the ,calculus of ,primitive ,functions4 ,i will continue to make use of the terms of ,leibnitz1 as being more convenient for the formation of secondary expressions1 although ,i ought1 in accordance with the suggestions made in the preceding chapter1 to employ concurrently all the different conceptions1 approaching as nearly as possible to that of ,lagrange4 ,,their ,,relations ,,to ,,each ,,other4 ,the differential calculus is evidently the logical basis of the integral calculus2 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 analysis4 ,the art of integration consists1 then1 essentially in bringing all the other cases1 as far as is possible1 to finally depend on only this small number of fundamental integrations4 ,in considering the whole body of the transcendental analysis1 as ,i have characterized it in the preceding chapter1 it is not at first apparent what can be the peculiar utility of the differential calculus1 independently of this necessary relation with the integral calculus1 which seems as if it must be1 by itself1 the only one directly indispensable4 ,in fact1 the elimination of the infinitesimals or of the derivatives1 introduced as auxiliaries to facilitate the establishment of equations1 constituting1 as we have seen1 the final and invariable object of the calculus of indirect functions1 it is natural to think that the calculus which teaches how to deduce from the equations between these auxiliary magnitudes1 those which exist between the primitive magnitudes themselves1 ought strictly to suffice for the general wants of the transcendental analysis without our perceiving1 at the first glance1 what special and constant part the solution of the inverse question can have in such an analysis4 ,it would be a real error1 though a common one1 to assign to the differential calculus1 in order to explain its peculiar1 direct1 and necessary influence1 the destination of forming the differential equations1 from which the integral calculus then enables us to arrive at the finite equations2 for the primitive formation of differential equations is not and cannot be1 properly speaking1 the object of any calculus1 since1 on the contrary1 it forms by its nature the indispensable starting point of any calculus whatever4 ,how1 in particular1 could the differential calculus1 which in itself is reduced to teaching the means of differentiating the different equations1 be a general procedure for establishing them8 ,that which in every application of the transcendental analysis really facilitates the formation of equations1 is the infinitesimal method1 and not the infinitesimal calculus1 which is perfectly distinct from it1 although it is its indispensable complement4 ,such a consideration would1 then1 give a false idea of the special destination which characterizes the differential calculus in the general system of the transcendental analysis4 ,but we should nevertheless very imperfectly conceive the real peculiar importance of this first branch of the calculus of indirect functions1 if we saw in it only a simple preliminary labour1 having no other general and essential object than to prepare indispensable foundations for the integral calculus4 ,as the ideas on this matter are generally confused1 ,i think that ,i ought here to explain in a summary manner this important relation as ,i view it1 and to show that in every application of the transcendental analysis a primary1 direct1 and necessary part is constantly assigned to the differential calculus4 #a4 ,use of the ,differential ,calculus as preparatory to that of the ,integral4 ,in forming the differential equations of any phenomenon whatever1 it is very seldom that we limit ourselves to introduce differentially only those magnitudes whose relations are sought4 ,to impose that condition would be to uselessly diminish the resources presented by the transcendental analysis for the expression of the mathematical laws of phenomena4 ,most frequently we introduce into the primitive equations1 through their differentials1 other magnitudes whose relations are already known or supposed to be so1 and without the consideration of which it would be frequently impossible to establish equations4 ,thus1 for example1 in the general problem of the rectification of curves1 the differential equation1 ds dy dx1 or ds dx dy dz1 is not only established between the desired function s and the independent variable x1 to which it is referred1 but1 at the same time1 there have been introduced1 as indispensable intermediaries1 the differentials of one or two other functions1 y and z1 which are among the data of the problem2 it would not have been possible to form directly the equation between ds and dx1 which would1 besides1 be peculiar to each curve considered4 ,it is the same for most questions4 ,now in these cases it is evident that the differential equation is not immediately suitable for integration4 ,it is previously necessary that the differentials of the functions supposed to be known1 which have been employed as intermediaries1 should be entirely eliminated1 in order that equations may be obtained between the differentials of the functions which alone are sought and those of the really independent variables1 after which the question depends on only the integral calculus4 ,now this preparatory elimination of certain differentials1 in order to reduce the infinitesimals to the smallest number possible1 belongs simply to the differential calculus2 for it must evidently be done by determining1 by means of the equations between the functions supposed to be known1 taken as intermediaries1 the relations of their differentials1 which is merely a question of differentiation4 ,thus1 for example1 in the case of rectifications1 it will be first necessary to calculate dy1 or dy and dz1 by differentiating the equation or the equations of each curve proposed2 after eliminating these expressions1 the general differential formula above enunciated will then contain only ds and dx2 having arrived at this point1 the elimination of the infinitesimals can be completed only by the integral calculus4 ,such is1 then1 the general office necessarily belonging to the differential calculus in the complete solution of the questions which exact the employment of the transcendental analysis2 to produce1 as far as is possible1 the elimination of the infinitesimals1 that is1 to reduce in each case the primitive differential equations so that they shall contain only the differentials of the really independent variables1 and those of the functions sought1 by causing to disappear1 by elimination1 the differentials of all the other known functions which may have been taken as intermediaries at the time of the formation of the differential equations of the problem which is under consideration4 #b4 ,employment of the ,differential ,calculus alone4 ,for certain questions1 which1 although few in number1 have none the less1 as we shall see hereafter1 a very great importance1 the magnitudes which are sought enter directly1 and not by their differentials1 into the primitive differential equations1 which then contain differentially only the different known functions employed as intermediaries1 in accordance with the preceding explanation4 ,these cases are the most favourable of all2 for it is evident that the differential calculus is then entirely sufficient for the complete elimination of the infinitesimals1 without the question giving rise to any integration4 ,this is what occurs1 for example1 in the problem of tangents in geometry2 in that of velocities in mechanics1 c4 #c4 ,employment of the ,integral ,calculus alone4 ,finally1 some other questions1 the number of which is also very small1 but the importance of which is no less great1 present a second exceptional case1 which is in its nature exactly the converse of the preceding4 ,they are those in which the differential equations are found to be immediately ready for integration1 because they contain1 at their first formation1 only the infinitesimals which relate to the functions sought1 or to the really independent variables1 without its being necessary to introduce1 differentially1 other functions as intermediaries4 ,if in these new cases we introduce these last functions1 since1 by hypothesis1 they will enter directly and not by their differentials1 ordinary algebra will suffice to eliminate them1 and to bring the question to depend on only the integral calculus4 ,the differential calculus will then have no special part in the complete solution of the problem1 which will depend entirely upon the integral calculus4 ,the general question of quadratures offers an important example of this1 for the differential equation being then d,a ydx1 will become immediately fit for integration as soon as we shall have eliminated1 by means of the equation of the proposed curve1 the intermediary function y1 which does not enter into it differentially4 ,the same circumstances exist in the problem of cubatures1 and in some others equally important4 ,three classes of ,questions hence resulting4 ,as a general result of the previous considerations1 it is then necessary to divide into three classes the mathematical questions which require the use of the transcendental analysis2 the first class comprises the problems susceptible of being entirely resolved by means of the differential calculus alone1 without any need of the integral calculus2 the second1 those which are1 on the contrary1 entirely dependent upon the integral calculus1 without the differential calculus having any part in their solution2 lastly1 in the third and the most extensive1 which constitutes the normal case1 the two others being only exceptional1 the differential and the integral calculus have each in their turn a distinct and necessary part in the complete solution of the problem1 the former making the primitive differential equations undergo a preparation which is indispensable for the application of the latter4 ,such are exactly their general relations1 of which too indefinite and inexact ideas are generally formed4 99 99 99 99 99 ,let us now take a general survey of the logical composition of each calculus1 beginning with the differential4 ,,the ,,differential ,,calculus4 ,in the exposition of the transcendental analysis1 it is customary to intermingle with the purely analytical part 7which reduces itself to the treatment of the abstract principles of differentiation and integration7 the study of its different principal applications1 especially those which concern geometry4 ,this confusion of ideas1 which is a consequence of the actual manner in which the science has been developed1 presents1 in the dogmatic point of view1 serious inconveniences in this respect1 that it makes it difficult properly to conceive either analysis or geometry4 ,having to consider here the most rational co-ordination which is possible1 ,i shall include1 in the following sketch1 only the calculus of indirect functions properly so called1 reserving for the portion of this volume which relates to the philosophical study of concrete mathematics the general examination of its great geometrical and mechanical applications4 ,two ,cases3 explicit and implicit ,functions4 ,the fundamental division of the differential calculus1 or of the general subject of differentiation1 consists in distinguishing two cases1 according as the analytical functions which are to be differentiated are explicit or implicit2 from which flow two parts ordinarily designated by the names of differentiation of formulas and differentiation of equations4 ,it is easy to understand1 priori1 the importance of this classification4 ,in fact1 such a distinction would be illusory if the ordinary analysis was perfect2 that is1 if we knew how to resolve all equations algebraically1 for then it would be possible to render every implicit function explicit2 and1 by differentiating it in that state alone1 the second part of the differential calculus would be immediately comprised in the first1 without giving rise to any new difficulty4 ,but the algebraical resolution of equations being1 as we have seen1 still almost in its infancy1 and as yet impossible for most cases1 it is plain that the case is very different1 since we have1 properly speaking1 to differentiate a function without knowing it1 although it is determinate4 ,the differentiation of implicit functions constitutes then1 by its nature1 a question truly distinct from that presented by explicit functions1 and necessarily more complicated4 ,it is thus evident that we must commence with the differentiation of formulas1 and reduce the differentiation of equations to this primary case by certain invariable analytical considerations1 which need not be here mentioned4 ,these two general cases of differentiation are also distinct in another point of view equally necessary1 and too important to be left unnoticed4 ,the relation which is obtained between the differentials is constantly more indirect1 in comparison with that of the finite quantities1 in the differentiation of implicit functions than in that of explicit functions4 ,we know1 in fact1 from the considerations presented by ,lagrange on the general formation of differential equations1 that1 on the one hand1 the same primitive equation may give rise to a greater or less number of derived equations of very different forms1 although at bottom equivalent1 depending upon which of the arbitrary constants is eliminated1 which is not the case in the differentiation of explicit formulas2 and that1 on the other hand1 the unlimited system of the different primitive equations1 which correspond to the same derived equation1 presents a much more profound analytical variety than that of the different functions1 which admit of one same explicit differential1 and which are distinguished from each other only by a constant term4 ,implicit functions must therefore be regarded as being in reality still more modified by differentiation than explicit functions4 ,we shall again meet with this consideration relatively to the integral calculus1 where it acquires a preponderant importance4 ,two ,sub-cases3 ,a single ,variable or several ,variables4 ,each of the two fundamental parts of the ,differential ,calculus is subdivided into two very distinct theories1 according as we are required to differentiate functions of a single variable or functions of several independent variables4 ,this second case is1 by its nature1 quite distinct from the first1 and evidently presents more complication1 even in considering only explicit functions1 and still more those which are implicit4 ,as to the rest1 one of these cases is deduced from the other in a general manner1 by the aid of an invariable and very simple principle1 which consists in regarding the total differential of a function which is produced by the simultaneous increments of the different independent variables which it contains1 as the sum of the partial differentials which would be produced by the separate increment of each variable in turn1 if all the others were constant4 ,it is necessary1 besides1 carefully to remark1 in connection with this subject1 a new idea which is introduced by the distinction of functions into those of one variable and of several2 it is the consideration of these different special derived functions1 relating to each variable separately1 and the number of which increases more and more in proportion as the order of the derivation becomes higher1 and also when the variables become more numerous4 ,it results from this that the differential relations belonging to functions of several variables are1 by their nature1 both much more indirect1 and especially much more indeterminate1 than those relating to functions of a single variable4 ,this is most apparent in the case of implicit functions1 in which1 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 variable1 it is the arbitrary functions of the proposed variables which are then eliminated2 whence must result special difficulties when these equations come to be integrated4 ,finally1 to complete this summary sketch of the different essential parts of the differential calculus proper1 ,i should add1 that in the differentiation of implicit functions1 whether of a single variable or of several1 it is necessary to make another distinction2 that of the case in which it is required to differentiate at once different functions of this kind1 combined in certain primitive equations1 from that in which all these functions are separate4 ,the functions are evidently1 in fact1 still more implicit in the first case than in the second1 if we consider that the same imperfection of ordinary analysis1 which forbids our converting every implicit function into an equivalent explicit function1 in like manner renders us unable to separate the functions which enter simultaneously into any system of equations4 ,it is then necessary to differentiate1 not only without knowing how to resolve the primitive equations1 but even without being able to effect the proper eliminations among them1 thus producing a new difficulty4 ,reduction of the whole to the ,differentiation of the ten elementary ,functions4 ,such1 then1 are the natural connection and the logical distribution of the different principal theories which compose the general system of differentiation4 ,since the differentiation of implicit functions is deduced from that of explicit functions by a single constant principle1 and the differentiation of functions of several variables is reduced by another fixed principle to that of functions of a single variable1 the whole of the differential calculus is finally found to rest upon the differentiation of explicit functions with a single variable1 the only one which is ever executed directly4 ,now it is easy to understand that this first theory1 the necessary basis of the entire system1 consists simply in the differentiation of the ten simple functions1 which are the uniform elements of all our analytical combinations1 and the list of which has been given in the first chapter1 on page #ea2 for the differentiation of compound functions is evidently deduced1 in an immediate and necessary manner1 from that of the simple functions which compose them4 ,it is1 then1 to the knowledge of these ten fundamental differentials1 and to that of the two general principles just mentioned1 which bring under it all the other possible cases1 that the whole system of differentiation is properly reduced4 ,we see1 by the combination of these different considerations1 how simple and how perfect is the entire system of the differential calculus4 ,it certainly constitutes1 in its logical relations1 the most interesting spectacle which mathematical analysis can present to our understanding4 ,transformation of derived ,functions for new ,variables4 ,the general sketch which ,i have just summarily drawn would nevertheless present an important deficiency1 if ,i did not here distinctly indicate a final theory1 which forms1 by its nature1 the indispensable complement of the system of differentiation4 ,it is that which has for its object the constant transformation of derived functions1 as a result of determinate changes in the independent variables1 whence results the possibility of referring to new variables all the general differential formulas primitively established for others4 ,this question is now resolved in the most complete and the most simple manner1 as are all those of which the differential calculus is composed4 ,it is easy to conceive the general importance which it must have in any of the applications of the transcendental analysis1 the fundamental resources of which it may be considered as augmenting1 by permitting us to choose 7in order to form the differential equations1 in the first place1 with more ease7 that system of independent variables which may appear to be the most advantageous1 although it is not to be finally retained4 ,it is thus1 for example1 that most of the principal questions of geometry are resolved much more easily by referring the lines and surfaces to rectilinear co-ordinates1 and that we may1 nevertheless1 have occasion to express these lines1 etc41 analytically by the aid of polar co-ordinates1 or in any other manner4 ,we will then be able to commence the differential solution of the problem by employing the rectilinear system1 but only as an intermediate step1 from which1 by the general theory here referred to1 we can pass to the final system1 which sometimes could not have been considered directly4 ,different ,orders of ,differentiation4 ,in the logical classification of the differential calculus which has just been given1 some may be inclined to suggest a serious omission1 since ,i have not subdivided each of its four essential parts according to another general consideration1 which seems at first view very important2 namely1 that of the higher or lower order of differentiation4 ,but it is easy to understand that this distinction has no real influence in the differential calculus1 inasmuch as it does not give rise to any new difficulty4 ,if1 indeed1 the differential calculus was not rigorously complete1 that is1 if we did not know how to differentiate at will any function whatever1 the differentiation to the second or higher order of each determinate function might engender special difficulties4 ,but the perfect universality of the differential calculus plainly gives us the assurance of being able to differentiate1 to any order whatever1 all known functions whatever1 the question reducing itself to a constantly repeated differentiation of the first order4 ,this distinction1 unimportant as it is for the differential calculus1 acquires1 however1 a very great importance in the integral calculus1 on account of the extreme imperfection of the latter4 ,analytical ,applications4 ,finally1 though this is not the place to consider the various applications of the differential calculus1 yet an exception may be made for those which consist in the solution of questions which are purely analytical1 which ought1 indeed1 to be logically treated in continuation of a system of differentiation1 because of the evident homogeneity of the considerations involved4 ,these questions may be reduced to three essential ones4 ,firstly1 the development into series of functions of one or more variables1 or1 more generally1 the transformation of functions1 which constitutes the most beautiful and the most important application of the differential calculus to general analysis1 and which comprises1 besides the fundamental series discovered by ,taylor1 the remarkable series discovered by ,maclaurin1 ,john ,bernouilli1 ,lagrange1 c43 ,secondly1 the general theory of maxima and minima values for any functions whatever1 of one or more variables2 one of the most interesting problems which analysis can present1 however elementary it may now have become1 and to the complete solution of which the differential calculus naturally applies3 ,thirdly1 the general determination of the true value of functions which present themselves under an indeterminate appearance for certain hypotheses made on the values of the corresponding variables2 which is the least extensive and the least important of the three4 ,the first question is certainly the principal one in all points of view2 it is also the most susceptible of receiving a new extension hereafter1 especially by conceiving1 in a broader manner than has yet been done1 the employment of the differential calculus in the transformation of functions1 on which subject ,lagrange has left some valuable hints4 99 99 99 99 99 ,having thus summarily1 though perhaps too briefly1 considered the chief points in the differential calculus1 ,i now proceed to an equally rapid exposition of a systematic outline of the ,integral ,calculus1 properly so called1 that is1 the abstract subject of integration4 ,,the ,,integral ,,calculus4 ,its ,fundamental ,division4 ,the fundamental division of the ,integral ,calculus is founded on the same principle as that of the ,differential ,calculus1 in distinguishing the integration of explicit differential formulas1 and the integration of implicit differentials or of differential equations4 ,the separation of these two cases is even much more profound in relation to integration than to differentiation4 ,in the differential calculus1 in fact1 this distinction rests1 as we have seen1 only on the extreme imperfection of ordinary analysis4 ,but1 on the other hand1 it is easy to see that1 even though all equations could be algebraically resolved1 differential equations would none the less constitute a case of integration quite distinct from that presented by the explicit differential formulas2 for1 limiting ourselves1 for the sake of simplicity1 to the first order1 and to a single function y of a single variable x1 if we suppose any differential equation between x1 y1 and dy_/dx1 to be resolved with reference to dy_/dx1 the expression of the derived function being then generally found to contain the primitive function itself1 which is the object of the inquiry1 the question of integration will not have at all changed its nature1 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 function1 which is in itself of little importance4 ,the differential would not then be determined in a manner much less implicit than before1 as regards the integration1 which would continue to present essentially the same characteristic difficulty4 ,the algebraic resolution of equations could not make the case which we are considering come within the simple integration of explicit differentials1 except in the special cases in which the proposed differential equation did not contain the primitive function itself1 which would consequently permit us1 by resolving it1 to find dy_/dx in terms of x only1 and thus to reduce the question to the class of quadratures4 ,still greater difficulties would evidently be found in differential equations of higher orders1 or containing simultaneously different functions of several independent variables4 ,the integration of differential equations is then necessarily more complicated than that of explicit differentials1 by the elaboration of which last the integral calculus has been created1 and upon which the others have been made to depend as far as it has been possible4 ,all the various analytical methods which have been proposed for integrating differential equations1 whether it be the separation of the variables1 the method of multipliers1 c41 have in fact for their object to reduce these integrations to those of differential formulas1 the only one which1 by its nature1 can be undertaken directly4 ,unfortunately1 imperfect as is still this necessary base of the whole integral calculus1 the art of reducing to it the integration of differential equations is still less advanced4 ,subdivisions3 one variable or several4 ,each of these two fundamental branches of the integral calculus is next subdivided into two others 7as in the differential calculus1 and for precisely analogous reasons71 according as we consider functions with a single variable1 or functions with several independent variables4 ,this distinction is1 like the preceding one1 still more important for integration than for differentiation4 ,this is especially remarkable in reference to differential equations4 ,indeed1 those which depend on several independent variables may evidently present this characteristic and much more serious difficulty1 that the desired function may be differentially defined by a simple relation between its different special derivatives relative to the different variables taken separately4 ,hence results the most difficult and also the most extensive branch of the integral calculus1 which is commonly named the ,integral ,calculus of partial differences1 created by ,d',alembert1 and in which1 according to the just appreciation of ,lagrange1 geometers ought to have seen a really new calculus1 the philosophical character of which has not yet been determined with sufficient exactness4 ,a very striking difference between this case and that of equations with a single independent variable consists1 as has been already observed1 in the arbitrary functions which take the place of the simple arbitrary constants1 in order to give to the corresponding integrals all the proper generality4 ,it is scarcely necessary to say that this higher branch of transcendental analysis is still entirely in its infancy1 since1 even in the most simple case1 that of an equation of the first order between the partial derivatives of a single function with two independent variables1 we are not yet completely able to reduce the integration to that of the ordinary differential equations4 ,the integration of functions of several variables is much farther advanced in the case 7infinitely more simple indeed7 in which it has to do with only explicit differential formulas4 ,we can then1 in fact1 when these formulas fulfil the necessary conditions of integrability1 always reduce their integration to quadratures4 ,other ,subdivisions3 different ,orders of ,differentiation4 ,a new general distinction1 applicable as a subdivision to the integration of explicit or implicit differentials1 with one variable or several1 is drawn from the higher or lower order of the differentials3 a distinction which1 as we have above remarked1 does not give rise to any special question in the differential calculus4 ,relatively to explicit differentials1 whether of one variable or of several1 the necessity of distinguishing their different orders belongs only to the extreme imperfection of the integral calculus4 ,in fact1 if we could always integrate every differential formula of the first order1 the integration of a formula of the second order1 or of any other1 would evidently not form a new question1 since1 by integrating it at first in the first degree1 we would arrive at the differential expression of the immediately preceding order1 from which1 by a suitable series of analogous integrations1 we would be certain of finally arriving at the primitive function1 the final object of these operations4 ,but the little knowledge which we possess on integration of even the first order causes quite another state of affairs1 so that a higher order of differentials produces new difficulties2 for1 having differential formulas of any order above the first1 it may happen that we may be able to integrate them1 either once1 or several times in succession1 and that we may still be unable to go back to the primitive functions1 if these preliminary labours have produced1 for the differentials of a lower order1 expressions whose integrals are not known4 ,this circumstance must occur so much the oftener 7the number of known integrals being still very small71 seeing that these successive integrals are generally very different functions from the derivatives which have produced them4 ,with reference to implicit differentials1 the distinction of orders is still more important2 for1 besides the preceding reason1 the influence of which is evidently analogous in this case1 and is even greater1 it is easy to perceive that the higher order of the differential equations necessarily gives rise to questions of a new nature4 ,in fact1 even if we could integrate every equation of the first order relating to a single function1 that would not be sufficient for obtaining the final integral of an equation of any order whatever1 inasmuch as every differential equation is not reducible to that of an immediately inferior order4 ,thus1 for example1 if we have given any relation between x1 y1 dx_/dy1 and dy_/dx1 to determine a function y of a variable x1 we shall not be able to deduce from it at once1 after effecting a first integration1 the corresponding differential relation between x1 y1 and dy_/dx1 from which1 by a second integration1 we could ascend to the primitive equations4 ,this would not necessarily take place1 at least without introducing new auxiliary functions1 unless the proposed equation of the second order did not contain the required function y1 together with its derivatives4 ,as a general principle1 differential equations will have to be regarded as presenting cases which are more and more implicit1 as they are of a higher order1 and which cannot be made to depend on one another except by special methods1 the investigation of which consequently forms a new class of questions1 with respect to which we as yet know scarcely any thing1 even for functions of a single variable4#aj ,footnote #aj3 ,the only important case of this class which has thus far been completely treated is the general integration of linear equations of any order whatever1 with constant coefficients4 ,even this case finally depends on the algebraic resolution of equations of a degree equal to the order of differentiation4 ,another equivalent distinction4 ,still farther1 when we examine more profoundly this distinction of different orders of differential equations1 we find that it can be always made to come under a final general distinction1 relative to differential equations1 which remains to be noticed4 ,differential equations with one or more independent variables may contain simply a single function1 or 7in a case evidently more complicated and more implicit1 which corresponds to the differentiation of simultaneous implicit functions7 we may have to determine at the same time several functions from the differential equations in which they are found united1 together with their different derivatives4 ,it is clear that such a state of the question necessarily presents a new special difficulty1 that of separating the different functions desired1 by forming for each1 from the proposed differential equations1 an isolated differential equation which does not contain the other functions or their derivatives4 ,this preliminary labour1 which is analogous to the elimination of algebra1 is evidently indispensable before attempting any direct integration1 since we cannot undertake generally 7except by special artifices which are very rarely applicable7 to determine directly several distinct functions at once4 ,now it is easy to establish the exact and necessary coincidence of this new distinction with the preceding one respecting the order of differential equations4 ,we know1 in fact1 that the general method for isolating functions in simultaneous differential equations consists essentially in forming differential equations1 separately in relation to each function1 and of an order equal to the sum of all those of the different proposed equations4 ,this transformation can always be effected4 ,on the other hand1 every differential equation of any order in relation to a single function might evidently always be reduced to the first order1 by introducing a suitable number of auxiliary differential equations1 containing at the same time the different anterior derivatives regarded as new functions to be determined4 ,this method has1 indeed1 sometimes been actually employed with success1 though it is not the natural one4 ,here1 then1 are two necessarily equivalent orders of conditions in the general theory of differential equations2 the simultaneousness of a greater or smaller number of functions1 and the higher or lower order of differentiation of a single function4 ,by augmenting the order of the differential equations1 we can isolate all the functions2 and1 by artificially multiplying the number of the functions1 we can reduce all the equations to the first order4 ,there is1 consequently1 in both cases1 only one and the same difficulty from two different points of sight4 ,but1 however we may conceive it1 this new difficulty is none the less real1 and constitutes none the less1 by its nature1 a marked separation between the integration of equations of the first order and that of equations of a higher order4 ,i prefer to indicate the distinction under this last form as being more simple1 more general1 and more logical4 ,quadratures4 ,from the different considerations which have been indicated respecting the logical dependence of the various principal parts of the integral calculus1 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 integrations1 which we never succeed in effecting but so far as we reduce them to this elementary case1 evidently the only one which1 by its nature1 is capable of being treated directly4 ,this simple fundamental integration is often designated by the convenient expression of quadratures1 seeing that every integral of this kind1 ,sf7x7dx1 may1 in fact1 be regarded as representing the area of a curve1 the equation of which in rectilinear co-ordinates would be y f7x74 ,such a class of questions corresponds1 in the differential calculus1 to the elementary case of the differentiation of explicit functions of a single variable4 ,but the integral question is1 by its nature1 very differently complicated1 and especially much more extensive than the differential question4 ,this latter is1 in fact1 necessarily reduced1 as we have seen1 to the differentiation of the ten simple functions1 the elements of all which are considered in analysis4 ,on the other hand1 the integration of compound functions does not necessarily follow from that of the simple functions1 each combination of which may present special difficulties with respect to the integral calculus4 ,hence results the naturally indefinite extent1 and the so varied complication of the question of quadratures1 upon which1 in spite of all the efforts of analysts1 we still possess so little complete knowledge4 ,in decomposing this question1 as is natural1 according to the different forms which may be assumed by the derivative function1 we distinguish the case of algebraic functions and that of transcendental functions4 ,integration of ,transcendental ,functions4 ,the truly analytical integration of transcendental functions is as yet very little advanced1 whether for exponential1 or for logarithmic1 or for circular functions4 ,but a very small number of cases of these three different kinds have as yet been treated1 and those chosen from among the simplest2 and still the necessary calculations are in most cases extremely laborious4 ,a circumstance which we ought particularly to remark in its philosophical connection is1 that the different procedures of quadrature have no relation to any general view of integration1 and consist of simple artifices very incoherent with each other1 and very numerous1 because of the very limited extent of each4 ,one of these artifices should1 however1 here be noticed1 which1 without being really a method of integration1 is nevertheless remarkable for its generality2 it is the procedure invented by ,john ,bernouilli1 and known under the name of integration by parts1 by means of which every integral may be reduced to another which is sometimes found to be more easy to be obtained4 ,this ingenious relation deserves to be noticed for another reason1 as having suggested the first idea of that transformation of integrals yet unknown1 which has lately received a greater extension1 and of which ,m4 ,fourier especially has made so new and important a use in the analytical questions produced by the theory of heat4 ,integration of ,algebraic ,functions4 ,as to the integration of algebraic functions1 it is farther advanced4 ,however1 we know scarcely any thing in relation to irrational functions1 the integrals of which have been obtained only in extremely limited cases1 and particularly by rendering them rational4 ,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 manner2 in a logical point of view1 it forms1 then1 its most satisfactory part1 but perhaps also the least important4 ,it is even essential to remark1 in order to have a just idea of the extreme imperfection of the integral calculus1 that this case1 limited as it is1 is not entirely resolved except for what properly concerns integration viewed in an abstract manner2 for1 in the execution1 the theory finds its progress most frequently quite stopped1 independently of the complication of the calculations1 by the imperfection of ordinary analysis1 seeing that it makes the integration finally depend upon the algebraic resolution of equations1 which greatly limits its use4 ,to grasp in a general manner the spirit of the different procedures which are employed in quadratures1 we must observe that1 by their nature1 they can be primitively founded only on the differentiation of the ten simple functions4 ,the results of this1 conversely considered1 establish as many direct theorems of the integral calculus1 the only ones which can be directly known4 ,all the art of integration afterwards consists1 as has been said in the beginning of this chapter1 in reducing all the other quadratures1 so far as is possible1 to this small number of elementary ones1 which unhappily we are in most cases unable to effect4 ,singular ,solutions4 ,in this systematic enumeration of the various essential parts of the integral calculus1 considered in their logical relations1 ,i have designedly neglected 7in order not to break the chain of sequence7 to consider a very important theory1 which forms implicitly a portion of the general theory of the integration of differential equations1 but which ,i ought here to notice separately1 as being1 so to speak1 outside of the integral calculus1 and being nevertheless of the greatest interest1 both by its logical perfection and by the extent of its applications4 ,i refer to what are called ,singular ,solutions of differential equations1 called sometimes1 but improperly1 particular solutions1 which have been the subject of very remarkable investigations by ,euler and ,laplace1 and of which ,lagrange especially has presented such a beautiful and simple general theory4 ,clairaut1 who first had occasion to remark their existence1 saw in them a paradox of the integral calculus1 since these solutions have the peculiarity of satisfying the differential equations without being comprised in the corresponding general integrals4 ,lagrange has since explained this paradox in the most ingenious and most satisfactory manner1 by showing how such solutions are always derived from the general integral by the variation of the arbitrary constants4 ,he was also the first to suitably appreciate the importance of this theory1 and it is with good reason that he devoted to it so full a development in his ,calculus of ,functions4 ,in a logical point of view1 this theory deserves all our attention by the character of perfect generality which it admits of1 since ,lagrange has given invariable and very simple procedures for finding the singular solution of any differential equation which is susceptible of it2 and1 what is no less remarkable1 these procedures require no integration1 consisting only of differentiations1 and are therefore always applicable4 ,differentiation has thus become1 by a happy artifice1 a means of compensating1 in certain circumstances1 for the imperfection of the integral calculus4 ,indeed1 certain problems especially require1 by their nature1 the knowledge of these singular solutions2 such1 for example1 in geometry1 are all the questions in which a curve is to be determined from any property of its tangent or its osculating circle4 ,in all cases of this kind1 after having expressed this property by a differential equation1 it will be1 in its analytical relations1 the singular equation which will form the most important object of the inquiry1 since it alone will represent the required curve2 the general integral1 which thenceforth it becomes unnecessary to know1 designating only the system of the tangents1 or of the osculating circles of this curve4 ,we may hence easily understand all the importance of this theory1 which seems to me to be not as yet sufficiently appreciated by most geometers4 ,definite ,integrals4 ,finally1 to complete our review of the vast collection of analytical researches of which is composed the integral calculus1 properly so called1 there remains to be mentioned one theory1 very important in all the applications of the transcendental analysis1 which ,i have had to leave outside of the system1 as not being really destined for veritable integration1 and proposing1 on the contrary1 to supply the place of the knowledge of truly analytical integrals1 which are most generally unknown4 ,i refer to the determination of definite integrals4 ,the expression1 always possible1 of integrals in infinite series1 may at first be viewed as a happy general means of compensating for the extreme imperfection of the integral calculus4 ,but the employment of such series1 because of their complication1 and of the difficulty of discovering the law of their terms1 is commonly of only moderate utility in the algebraic point of view1 although sometimes very essential relations have been thence deduced4 ,it is particularly in the arithmetical point of view that this procedure acquires a great importance1 as a means of calculating what are called definite integrals1 that is1 the values of the required functions for certain determinate values of the corresponding variables4 ,an inquiry of this nature exactly corresponds1 in transcendental analysis1 to the numerical resolution of equations in ordinary analysis4 ,being generally unable to obtain the veritable integral--named by opposition the general or indefinite integral2 that is1 the function which1 differentiated1 has produced the proposed differential formula--analysts have been obliged to employ themselves in determining at least1 without knowing this function1 the particular numerical values which it would take on assigning certain designated values to the variables4 ,this is evidently resolving the arithmetical question without having previously resolved the corresponding algebraic one1 which most generally is the most important one4 ,such an analysis is1 then1 by its nature1 as imperfect as we have seen the numerical resolution of equations to be4 ,it presents1 like this last1 a vicious confusion of arithmetical and algebraic considerations1 whence result analogous inconveniences both in the purely logical point of view and in the applications4 ,we need not here repeat the considerations suggested in our third chapter4 ,but it will be understood that1 unable as we almost always are to obtain the true integrals1 it is of the highest importance to have been able to obtain this solution1 incomplete and necessarily insufficient as it is4 ,now this has been fortunately attained at the present day for all cases1 the determination of the value of definite integrals having been reduced to entirely general methods1 which leave nothing to desire1 in a great number of cases1 but less complication in the calculations1 an object towards which are at present directed all the special transformations of analysts4 ,regarding now this sort of transcendental arithmetic as perfect1 the difficulty in the applications is essentially reduced to making the proposed research depend1 finally1 on a simple determination of definite integrals1 which evidently cannot always be possible1 whatever analytical skill may be employed in effecting such a transformation4 ,prospects of the ,integral ,calculus4 ,from the considerations indicated in this chapter1 we see that1 while the differential calculus constitutes by its nature a limited and perfect system1 to which nothing essential remains to be added1 the integral calculus1 or the simple system of integration1 presents necessarily an inexhaustible field for the activity of the human mind1 independently of the indefinite applications of which the transcendental analysis is evidently susceptible4 ,the general argument by which ,i have endeavoured1 in the second chapter1 to make apparent the impossibility of ever discovering the algebraic solution of equations of any degree and form whatsoever1 has undoubtedly infinitely more force with regard to the search for a single method of integration1 invariably applicable to all cases4 ,it is1 says ,lagrange1 one of those problems whose general solution we cannot hope for4 ,the more we meditate on this subject1 the more we will be convinced that such a research is utterly chimerical1 as being far above the feeble reach of our intelligence2 although the labours of geometers must certainly augment hereafter the amount of our knowledge respecting integration1 and thus create methods of greater generality4 ,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 become4 ,but1 whatever should be our legitimate hopes1 let us not forget to consider1 before all1 the limits which are imposed by our intellectual constitution1 and which1 though not susceptible of a precise determination1 have none the less an incontestable reality4 ,i am induced to think that1 when geometers shall have exhausted the most important applications of our present transcendental analysis1 instead of striving to impress upon it1 as now conceived1 a chimerical perfection1 they will rather create new resources by changing the mode of derivation of the auxiliary quantities introduced in order to facilitate the establishment of equations1 and the formation of which might follow an infinity of other laws besides the very simple relation which has been chosen1 according to the conception suggested in the first chapter4 ,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 functions4 ,it is a suggestion which ,i submit to the geometers who have turned their thoughts towards the general philosophy of analysis4 ,finally1 although1 in the summary exposition which was the object of this chapter1 ,i have had to exhibit the condition of extreme imperfection which still belongs to the integral calculus1 the student would have a false idea of the general resources of the transcendental analysis if he gave that consideration too great an importance4 ,it is with it1 indeed1 as with ordinary analysis1 in which a very small amount of fundamental knowledge respecting the resolution of equations has been employed with an immense degree of utility4 ,little advanced as geometers really are as yet in the science of integrations1 they have nevertheless obtained1 from their scanty abstract conceptions1 the solution of a multitude of questions of the first importance in geometry1 in mechanics1 in thermology1 c4 ,the philosophical explanation of this double general fact results from the necessarily preponderating importance and grasp of abstract branches of knowledge1 the least of which is naturally found to correspond to a crowd of concrete researches1 man having no other resource for the successive extension of his intellectual means than in the consideration of ideas more and more abstract1 and still positive4 99 99 99 99 99 ,in order to finish the complete exposition of the philosophical character of the transcendental analysis1 there remains to be considered a final conception1 by which the immortal ,lagrange has rendered this analysis still better adapted to facilitate the establishment of equations in the most difficult problems1 by considering a class of equations still more indirect than the ordinary differential equations4 ,it is the ,calculus1 or1 rather1 the ,method of ,variations2 the general appreciation of which will be our next subject4 ,,chapter ,v4 ,,the ,,calculus ,,of ,,variations4 ,in order to grasp with more ease the philosophical character of the ,method of ,variations1 it will be well to begin by considering in a summary manner the special nature of the problems1 the general resolution of which has rendered necessary the formation of this hyper-transcendental analysis4 ,it is still too near its origin1 and its applications have been too few1 to allow us to obtain a sufficiently clear general idea of it from a purely abstract exposition of its fundamental theory4 ,,problems ,,giving ,,rise ,,to ,,it4 ,the mathematical questions which have given birth to the ,calculus of ,variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas1 which express the analytical law of such or such a phenomenon of geometry or mechanics1 considered independently of any particular subject4 ,geometers for a long time designated all the questions of this character by the common name of ,isoperimetrical ,problems1 which1 however1 is really suitable to only the smallest number of them4 ,ordinary ,questions of ,maxima and ,minima4 ,in the common theory of maxima and minima1 it is proposed to discover1 with reference to a given function of one or more variables1 what particular values must be assigned to these variables1 in order that the corresponding value of the proposed function may be a maximum or a minimum with respect to those values which immediately precede and follow it2 that is1 properly speaking1 we seek to know at what instant the function ceases to increase and commences to decrease1 or reciprocally4 ,the differential calculus is perfectly sufficient1 as we know1 for the general resolution of this class of questions1 by showing that the values of the different variables1 which suit either the maximum or minimum1 must always reduce to zero the different first derivatives of the given function1 taken separately with reference to each independent variable1 and by indicating1 moreover1 a suitable characteristic for distinguishing the maximum from the minimum2 consisting1 in the case of a function of a single variable1 for example1 in the derived function of the second order taking a negative value for the maximum1 and a positive value for the minimum4 ,such are the well-known fundamental conditions belonging to the greatest number of cases4 ,a new ,class of ,questions4 ,the construction of this general theory having necessarily destroyed the chief interest which questions of this kind had for geometers1 they almost immediately rose to the consideration of a new order of problems1 at once much more important and of much greater difficulty--those of isoperimeters4 ,it is1 then1 no longer the values of the variables belonging to the maximum or the minimum of a given function that it is required to determine4 ,it is the form of the function itself which is required to be discovered1 from the condition of the maximum or of the minimum of a certain definite integral1 merely indicated1 which depends upon that function4 ,solid of least ,resistance4 ,the oldest question of this nature is that of the solid of least resistance1 treated by ,newton in the second book of the ,principia1 in which he determines what ought to be the meridian curve of a solid of revolution1 in order that the resistance experienced by that body in the direction of its axis may be the least possible4 ,but the course pursued by ,newton1 from the nature of his special method of transcendental analysis1 had not a character sufficiently simple1 sufficiently general1 and especially sufficiently analytical1 to attract geometers to this new order of problems4 ,to effect this1 the application of the infinitesimal method was needed2 and this was done1 in #afie1 by ,john ,bernouilli1 in proposing the celebrated problem of the ,brachystochrone4 ,this problem1 which afterwards suggested such a long series of analogous questions1 consists in determining the curve which a heavy body must follow in order to descend from one point to another in the shortest possible time4 ,limiting the conditions to the simple fall in a vacuum1 the only case which was at first considered1 it is easily found that the required curve must be a reversed cycloid with a horizontal base1 and with its origin at the highest point4 ,but the question may become singularly complicated1 either by taking into account the resistance of the medium1 or the change in the intensity of gravity4 ,isoperimeters4 ,although this new class of problems was in the first place furnished by mechanics1 it is in geometry that the principal investigations of this character were subsequently made4 ,thus it was proposed to discover which1 among all the curves of the same contour traced between two given points1 is that whose area is a maximum or minimum1 whence has come the name of ,problem of ,isoperimeters2 or it was required that the maximum or minimum should belong to the surface produced by the revolution of the required curve about an axis1 or to the corresponding volume2 in other cases1 it was the vertical height of the center of gravity of the unknown curve1 or of the surface and of the volume which it might generate1 which was to become a maximum or minimum1 c4 ,finally1 these problems were varied and complicated almost to infinity by the ,bernouillis1 by ,taylor1 and especially by ,euler1 before ,lagrange reduced their solution to an abstract and entirely general method1 the discovery of which has put a stop to the enthusiasm of geometers for such an order of inquiries4 ,this is not the place for tracing the history of this subject4 ,i have only enumerated some of the simplest principal questions1 in order to render apparent the original general object of the method of variations4 ,analytical ,nature of these ,problems4 ,we see that all these problems1 considered in an analytical point of view1 consist1 by their nature1 in determining what form a certain unknown function of one or more variables ought to have1 in order that such or such an integral1 dependent upon that function1 shall have1 within assigned limits1 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 whatever4 ,thus1 for example1 in the problem of the brachystochrone1 it is well known that if y f7z71 x 7z71 are the rectilinear equations of the required curve1 supposing the axes of x and of y to be horizontal1 and the axis of z to be vertical1 the time of the fall of a heavy body in that curve from the point whose ordinate is z1 to that whose ordinate is z1 is expressed in general terms by the definite integral zz7#a 7f'7z77 7'7z77_/7#b;gz77dz4 ,it is1 then1 necessary to find what the two unknown functions f and must be1 in order that this integral may be a minimum4 ,in the same way1 to demand what is the curve among all plane isoperimetrical curves1 which includes the greatest area1 is the same thing as to propose to find1 among all the functions f7x7 which can give a certain constant value to the integral dx7#a 7f'7x7 771 that one which renders the integral f7x7dx1 taken between the same limits1 a maximum4 ,it is evidently always so in other questions of this class4 ,methods of the older ,geometers4 ,in the solutions which geometers before ,lagrange gave of these problems1 they proposed1 in substance1 to reduce them to the ordinary theory of maxima and minima4 ,but the means employed to effect this transformation consisted in special simple artifices peculiar to each case1 and the discovery of which did not admit of invariable and certain rules1 so that every really new question constantly reproduced analogous difficulties1 without the solutions previously obtained being really of any essential aid1 otherwise than by their discipline and training of the mind4 ,in a word1 this branch of mathematics presented1 then1 the necessary imperfection which always exists when the part common to all questions of the same class has not yet been distinctly grasped in order to be treated in an abstract and thenceforth general manner4 ,,method ,,of ,,lagrange4 ,lagrange1 in endeavouring to bring all the different problems of isoperimeters to depend upon a common analysis1 organized into a distinct calculus1 was led to conceive a new kind of differentiation1 to which he has applied the characteristic 1 reserving the characteristic d for the common differentials4 ,these differentials of a new species1 which he has designated under the name of ,variations1 consist of the infinitely small increments which the integrals receive1 not by virtue of analogous increments on the part of the corresponding variables1 as in the ordinary transcendental analysis1 but by supposing that the form of the function placed under the sign of integration undergoes an infinitely small change4 ,this distinction is easily conceived with reference to curves1 in which we see the ordinate1 or any other variable of the curve1 admit of two sorts of differentials1 evidently very different1 according as we pass from one point to another infinitely near it on the same curve1 or to the corresponding point of the infinitely near curve produced by a certain determinate modification of the first curve4#aa ,it is moreover clear1 that the relative variations of different magnitudes connected with each other by any laws whatever are calculated1 all but the characteristic1 almost exactly in the same manner as the differentials4 ,finally1 from the general notion of variations are in like manner deduced the fundamental principles of the algorithm proper to this method1 consisting simply in the evidently permissible liberty of transposing at will the characteristics specially appropriated to variations1 before or after those which correspond to the ordinary differentials4 ,footnote #aa3 ,leibnitz had already considered the comparison of one curve with an other infinitely near to it1 calling it ,differentiatio de curva in curvam4 ,but this comparison had no analogy with the conception of ,lagrange1 the curves of ,leibnitz being embraced in the same general equation1 from which they were deduced by the simple change of an arbitrary constant4 ,this abstract conception having been once formed1 ,lagrange was able to reduce with ease1 and in the most general manner1 all the problems of ,isoperimeters to the simple ordinary theory of maxima and minima4 ,to obtain a clear idea of this great and happy transformation1 we must previously consider an essential distinction which arises in the different questions of isoperimeters4 ,two ,classes of ,questions4 ,these investigations must1 in fact1 be divided into two general classes1 according as the maxima and minima demanded are absolute or relative1 to employ the abridged expressions of geometers4 ,questions of the first ,class4 ,the first case is that in which the indeterminate definite integrals1 the maximum or minimum of which is sought1 are not subjected1 by the nature of the problem1 to any condition2 as happens1 for example1 in the problem of the brachystochrone1 in which the choice is to be made between all imaginable curves4 ,the second case takes place when1 on the contrary1 the variable integrals can vary only according to certain conditions1 which usually consist in other definite integrals 7which depend1 in like manner1 upon the required functions7 always retaining the same given value2 as1 for example1 in all the geometrical questions relating to real isoperimetrical figures1 and in which1 by the nature of the problem1 the integral relating to the length of the curve1 or to the area of the surface1 must remain constant during the variation of that integral which is the object of the proposed investigation4 ,the ,calculus of ,variations gives immediately the general solution of questions of the former class2 for it evidently follows1 from the ordinary theory of maxima and minima1 that the required relation must reduce to zero the variation of the proposed integral with reference to each independent variable2 which gives the condition common to both the maximum and the minimum3 and1 as a characteristic for distinguishing the one from the other1 that the variation of the second order of the same integral must be negative for the maximum and positive for the minimum4 ,thus1 for example1 in the problem of the brachystochrone1 we will have1 in order to determine the nature of the curve sought1 the equation of condition zz7#a 7f'7z77 7'7z77_/7#b;gz77dz #j1 which1 being decomposed into two1 with respect to the two unknown functions f and 1 which are independent of each other1 will completely express the analytical definition of the required curve4 ,the only difficulty peculiar to this new analysis consists in the elimination of the characteristic 1 for which the calculus of variations furnishes invariable and complete rules1 founded1 in general1 on the method of integration by parts1 from which ,lagrange has thus derived immense advantage4 ,the constant object of this first analytical elaboration 7which this is not the place for treating in detail7 is to arrive at real differential equations1 which can always be done2 and thereby the question comes under the ordinary transcendental analysis1 which furnishes the solution1 at least so far as to reduce it to pure algebra if the integration can be effected4 ,the general object of the method of variations is to effect this transformation1 for which ,lagrange has established rules1 which are simple1 invariable1 and certain of success4 ,equations of ,limits4 ,among the greatest special advantages of the method of variations1 compared with the previous isolated solutions of isoperimetrical problems1 is the important consideration of what ,lagrange calls ,equations of ,limits1 which were entirely neglected before him1 though without them the greater part of the particular solutions remained necessarily incomplete4 ,when the limits of the proposed integrals are to be fixed1 their variations being zero1 there is no occasion for noticing them4 ,but it is no longer so when these limits1 instead of being rigorously invariable1 are only subjected to certain conditions2 as1 for example1 if the two points between which the required curve is to be traced are not fixed1 and have only to remain upon given lines or surfaces4 ,then it is necessary to pay attention to the variation of their co-ordinates1 and to establish between them the relations which correspond to the equations of these lines or of these surfaces4 ,a more general consideration4 ,this essential consideration is only the final complement of a more general and more important consideration relative to the variations of different independent variables4 ,if these variables are really independent of one another1 as when we compare together all the imaginable curves susceptible of being traced between two points1 it will be the same with their variations1 and1 consequently1 the terms relating to each of these variations will have to be separately equal to zero in the general equation which expresses the maximum or the minimum4 ,but if1 on the contrary1 we suppose the variables to be subjected to any fixed conditions1 it will be necessary to take notice of the resulting relation between their variations1 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 independent4 ,it is thus1 for example1 that instead of seeking for the shortest path between any two points1 in choosing it from among all possible ones1 it may be proposed to find only what is the shortest among all those which may be taken on any given surface2 a question the general solution of which forms certainly one of the most beautiful applications of the method of variations4 ,questions of the second ,class4 ,problems in which such modifying conditions are considered approach very nearly1 in their nature1 to the second general class of applications of the method of variations1 characterized above as consisting in the investigation of relative maxima and minima4 ,there is1 however1 this essential difference between the two cases1 that in this last the modification is expressed by an integral which depends upon the function sought1 while in the other it is designated by a finite equation which is immediately given4 ,it is hence apparent that the investigation of relative maxima and minima is constantly and necessarily more complicated than that of absolute maxima and minima4 ,luckily1 a very important general theory1 discovered by the genius of the great ,euler before the invention of the ,calculus of ,variations1 gives a uniform and very simple means of making one of these two classes of questions dependent on the other4 ,it consists in this1 that if we add to the integral which is to be a maximum or a minimum1 a constant and indeterminate multiple of that one which1 by the nature of the problem1 is to remain constant1 it will be sufficient to seek1 by the general method of ,lagrange above indicated1 the absolute maximum or minimum of this whole expression4 ,it can be easily conceived1 indeed1 that the part of the complete variation which would proceed from the last integral must be equal to zero 7because of the constant character of this last7 as well as the portion due to the first integral1 which disappears by virtue of the maximum or minimum state4 ,these two conditions evidently unite to produce1 in that respect1 effects exactly alike4 ,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 ,theory of ,isoperimeters4 ,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 chapter1 namely1 the generality of the infinitesimal expressions for the representation of the same geometrical or mechanical phenomenon1 in whatever body it may be considered4 ,upon this generality1 indeed1 are founded1 by their nature1 all the solutions due to the method of variations4 ,if a single formula could not express the length or the area of any curve whatever2 if another fixed formula could not designate the time of the fall of a heavy body1 according to whatever line it may descend1 c41 how would it have been possible to resolve questions which unavoidably require1 by their nature1 the simultaneous consideration of all the cases which can be determined in each phenomenon by the different subjects which exhibit it4 ,other ,applications of this ,method4 ,notwithstanding the extreme importance of the theory of isoperimeters1 and though the method of variations had at first no other object than the logical and general solution of this order of problems1 we should still have but an incomplete idea of this beautiful analysis if we limited its destination to this4 ,in fact1 the abstract conception of two distinct natures of differentiation is evidently applicable not only to the cases for which it was created1 but also to all those which present1 for any reason whatever1 two different manners of making the same magnitudes vary4 ,it is in this way that ,lagrange himself has made1 in his ,mchanique ,analytique1 an extensive and important application of his calculus of variations1 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 considered1 according as we compare the successive positions which are occupied1 in virtue of its motion1 by the same point of each body in two consecutive instants1 or as we pass from one point of the body to another in the same instant4 ,one of these comparisons produces ordinary differentials2 the other gives rise to variations1 which1 there as every where1 are only differentials taken under a new point of view4 ,such is the general acceptation in which we should conceive the ,calculus of ,variations1 in order suitably to appreciate the importance of this admirable logical instrument1 the most powerful that the human mind has as yet constructed4 ,the method of variations being only an immense extension of the general transcendental analysis1 ,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 functions1 considered as a whole1 admits of4 ,lagrange invented the ,calculus of ,variations in accordance with the infinitesimal conception1 and1 indeed1 long before he undertook the general reconstruction of the transcendental analysis4 ,when he had executed this important reformation1 he easily showed how it could also be applied to the ,calculus of ,variations1 which he expounded with all the proper development1 according to his theory of derivative functions4 ,but the more that the use of the method of variations is difficult of comprehension1 because of the higher degree of abstraction of the ideas considered1 the more necessary is it1 in its application1 to economize the exertions of the mind1 by adopting the most direct and rapid analytical conception1 namely1 that of ,leibnitz4 ,accordingly1 ,lagrange himself has constantly preferred it in the important use which he has made of the ,calculus of ,variations in his ,analytical ,mechanics4 ,in fact1 there does not exist the least hesitation in this respect among geometers4 ,,its ,,relations ,,to ,,the ,,ordinary ,,calculus4 ,in order to make as clear as possible the philosophical character of the ,calculus of ,variations1 ,i think that ,i should1 in conclusion1 briefly indicate a consideration which seems to me important1 and by which ,i can approach it to the ordinary transcendental analysis in a higher degree than ,lagrange seems to me to have done4#ab ,footnote #ab3 ,i propose hereafter to develop this new consideration1 in a special work upon the ,calculus of ,variations1 intended to present this hyper-transcendental analysis in a new point of view1 which ,i think adapted to extend its general range4 ,we noticed in the preceding chapter the formation of the calculus of partial differences1 created by ,d',alembert1 as having introduced into the transcendental analysis a new elementary idea2 the notion of two kinds of increments1 distinct and independent of one another1 which a function of two variables may receive by virtue of the change of each variable separately4 ,it is thus that the vertical ordinate of a surface1 or any other magnitude which is referred to it1 varies in two manners which are quite distinct1 and which may follow the most different laws1 according as we increase either the one or the other of the two horizontal co-ordinates4 ,now such a consideration seems to me very nearly allied1 by its nature1 to that which serves as the general basis of the method of variations4 ,this last1 indeed1 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 variables2 a modification which has remarkably enlarged its use4 ,i think1 therefore1 that so far as regards merely the fundamental conceptions1 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 variations2 such a derivation of which seems to be adapted to make the general notion more clear and simple4 ,according to the different considerations indicated in this chapter1 the method of variations presents itself as the highest degree of perfection which the analysis of indirect functions has yet attained4 ,in its primitive state1 this last analysis presented itself as a powerful general means of facilitating the mathematical study of natural phenomena1 by introducing1 for the expression of their laws1 the consideration of auxiliary magnitudes1 chosen in such a manner that their relations are necessarily more simple and more easy to obtain than those of the direct magnitudes4 ,but the formation of these differential equations was not supposed to admit of any general and abstract rules4 ,now the ,analysis of ,variations1 considered in the most philosophical point of view1 may be regarded as essentially destined1 by its nature1 to bring within the reach of the calculus the actual establishment of the differential equations2 for1 in a great number of important and difficult questions1 such is the general effect of the varied equations1 which1 still more indirect than the simple differential equations with respect to the special objects of the investigation1 are also much more easy to form1 and from which we may then1 by invariable and complete analytical methods1 the object of which is to eliminate the new order of auxiliary infinitesimals which have been introduced1 deduce those ordinary differential equations which it would often have been impossible to establish directly4 ,the method of variations forms1 then1 the most sublime part of that vast system of mathematical analysis1 which1 setting out from the most simple elements of algebra1 organizes1 by an uninterrupted succession of ideas1 general methods more and more powerful1 for the study of natural philosophy1 and which1 in its whole1 presents the most incomparably imposing and unequivocal monument of the power of the human intellect4 ,we must1 however1 also admit that the conceptions which are habitually considered in the method of variations being1 by their nature1 more indirect1 more general1 and especially more abstract than all others1 the employment of such a method exacts necessarily and continuously the highest known degree of intellectual exertion1 in order never to lose sight of the precise object of the investigation1 in following reasonings which offer to the mind such uncertain resting-places1 and in which signs are of scarcely any assistance4 ,we must undoubtedly attribute in a great degree to this difficulty the little real use which geometers1 with the exception of ,lagrange1 have as yet made of such an admirable conception4 ,,chapter ,,vi4 ,,the ,,calculus ,,of ,,finite ,,differences4 ,the different fundamental considerations indicated in the five preceding chapters constitute1 in reality1 all the essential bases of a complete exposition of mathematical analysis1 regarded in the philosophical point of view4 ,nevertheless1 in order not to neglect any truly important general conception relating to this analysis1 ,i think that ,i should here very summarily explain the veritable character of a kind of calculus which is very extended1 and which1 though at bottom it really belongs to ordinary analysis1 is still regarded as being of an essentially distinct nature4 ,i refer to the ,calculus of ,finite ,differences1 which will be the special subject of this chapter4 ,its general ,character4 ,this calculus1 created by ,taylor1 in his celebrated work entitled ,methodus ,incrementorum1 consists essentially in the consideration of the finite increments which functions receive as a consequence of analogous increments on the part of the corresponding variables4 ,these increments or differences1 which take the characteristic 1 to distinguish them from differentials1 or infinitely small increments1 may be in their turn regarded as new functions1 and become the subject of a second similar consideration1 and so on2 from which results the notion of differences of various successive orders1 analogous1 at least in appearance1 to the consecutive orders of differentials4 ,such a calculus evidently presents1 like the calculus of indirect functions1 two general classes of questions3 #a4 ,to determine the successive differences of all the various analytical functions of one or more variables1 as the result of a definite manner of increase of the independent variables1 which are generally supposed to augment in arithmetical progression4 #b4 ,reciprocally1 to start from these differences1 or1 more generally1 from any equations established between them1 and go back to the primitive functions themselves1 or to their corresponding relations4 ,hence follows the decomposition of this calculus into two distinct ones1 to which are usually given the names of the ,direct1 and the ,inverse ,calculus of ,finite ,differences1 the latter being also sometimes called the ,integral ,calculus of ,finite ,differences4 ,each of these would1 also1 evidently admit of a logical distribution similar to that given in the fourth chapter for the differential and the integral calculus4 ,its true ,nature4 ,there is no doubt that ,taylor thought that by such a conception he had founded a calculus of an entirely new nature1 absolutely distinct from ordinary analysis1 and more general than the calculus of ,leibnitz1 although resting on an analogous consideration4 ,it is in this way1 also1 that almost all geometers have viewed the analysis of ,taylor2 but ,lagrange1 with his usual profundity1 clearly perceived that these properties belonged much more to the forms and to the notations employed by ,taylor than to the substance of his theory4 ,in fact1 that which constitutes the peculiar character of the analysis of ,leibnitz1 and makes of it a truly distinct and superior calculus1 is the circumstance that the derived functions are in general of an entirely different nature from the primitive functions1 so that they may give rise to more simple and more easily formed relations3 whence result the admirable fundamental properties of the transcendental analysis1 which have been already explained4 ,but it is not so with the differences considered by ,taylor2 for these differences are1 by their nature1 functions essentially similar to those which have produced them1 a circumstance which renders them unsuitable to facilitate the establishment of equations1 and prevents their leading to more general relations4 ,every equation of finite differences is truly1 at bottom1 an equation directly relating to the very magnitudes whose successive states are compared4 ,the scaffolding of new signs1 which produce an illusion respecting the true character of these equations1 disguises it1 however1 in a very imperfect manner1 since it could always be easily made apparent by replacing the differences by the equivalent combinations of the primitive magnitudes1 of which they are really only the abridged designations4 ,thus the calculus of ,taylor never has offered1 and never can offer1 in any question of geometry or of mechanics1 that powerful general aid which we have seen to result necessarily from the analysis of ,leibnitz4 ,lagrange has1 moreover1 very clearly proven that the pretended analogy observed between the calculus of differences and the infinitesimal calculus was radically vicious1 in this way1 that the formulas belonging to the former calculus can never furnish1 as particular cases1 those which belong to the latter1 the nature of which is essentially distinct4 ,from these considerations ,i am led to think that the calculus of finite differences is1 in general1 improperly classed with the transcendental analysis proper1 that is1 with the calculus of indirect functions4 ,i consider it1 on the contrary1 in accordance with the views of ,lagrange1 to be only a very extensive and very important branch of ordinary analysis1 that is to say1 of that which ,i have named the calculus of direct functions1 the equations which it considers being always1 in spite of the notation1 simple direct equations4 ,,general ,,theory ,,of ,,series4 ,to sum up as briefly as possible the preceding explanation1 the calculus of ,taylor ought to be regarded as having constantly for its true object the general theory of ,series1 the most simple cases of which had alone been considered before that illustrious geometer4 ,i ought1 properly1 to have mentioned this important theory in treating1 in the second chapter1 of ,algebra proper1 of which it is such an extensive branch4 ,but1 in order to avoid a double reference to it1 ,i have preferred to notice it only in the consideration of the calculus of finite differences1 which1 reduced to its most simple general expression1 is nothing but a complete logical study of questions relating to series4 ,every ,series1 or succession of numbers deduced from one another according to any constant law1 necessarily gives rise to these two fundamental questions3 #a4 ,the law of the series being supposed known1 to find the expression for its general term1 so as to be able to calculate immediately any term whatever without being obliged to form successively all the preceding terms4 #b4 ,in the same circumstances1 to determine the sum of any number of terms of the series by means of their places1 so that it can be known without the necessity of continually adding these terms together4 ,these two fundamental questions being considered to be resolved1 it may be proposed1 reciprocally1 to find the law of a series from the form of its general term1 or the expression of the sum4 ,each of these different problems has so much the more extent and difficulty1 as there can be conceived a greater number of different laws for the series1 according to the number of preceding terms on which each term directly depends1 and according to the function which expresses that dependence4 ,we may even consider series with several variable indices1 as ,laplace has done in his ,analytical ,theory of ,probabilities1 by the analysis to which he has given the name of ,theory of ,generating ,functions1 although it is really only a new and higher branch of the calculus of finite differences or of the general theory of series4 ,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 series1 so simple in appearance and so limited in its origin4 ,it necessarily presents as many different cases as the algebraic resolution of equations1 considered in its whole extent2 and it is1 by its nature1 much more complicated1 so much1 indeed1 that it always needs this last to conduct it to a complete solution4 ,we may1 therefore1 anticipate what must still be its extreme imperfection1 in spite of the successive labours of several geometers of the first order4 ,we do not1 indeed1 possess as yet the complete and logical solution of any but the most simple questions of this nature4 ,its identity with this ,calculus4 ,it is now easy to conceive the necessary and perfect identity1 which has been already announced1 between the calculus of finite differences and the theory of series considered in all its bearings4 ,in fact1 every differentiation after the manner of ,taylor evidently amounts to finding the law of formation of a series with one or with several variable indices1 from the expression of its general term2 in the same way1 every analogous integration may be regarded as having for its object the summation of a series1 the general term of which would be expressed by the proposed difference4 ,in this point of view1 the various problems of the calculus of differences1 direct or inverse1 resolved by ,taylor and his successors1 have really a very great value1 as treating of important questions relating to series4 ,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 kind4 ,it would be1 perhaps1 more advantageous for most cases1 and certainly more logical1 to replace the differences by the terms themselves1 certain combinations of which they represent4 ,as the calculus of ,taylor does not rest on a truly distinct fundamental idea1 and has nothing peculiar to it but its system of signs1 there could never really be any important advantage in considering it as detached from ordinary analysis1 of which it is1 in reality1 only an immense branch4 ,this consideration of differences1 most generally useless1 even if it does not cause complication1 seems to me to retain the character of an epoch in which1 analytical ideas not being sufficiently familiar to geometers1 they were naturally led to prefer the special forms suitable for simple numerical comparisons4 ,,periodic ,,or ,,discontinuous ,,functions4 ,however that may be1 ,i must not finish this general appreciation of the calculus of finite differences without noticing a new conception to which it has given birth1 and which has since acquired a great importance4 ,it is the consideration of those periodic or discontinuous functions which preserve the same value for an infinite series of values of the corresponding variables1 subjected to a certain law1 and which must be necessarily added to the integrals of the equations of finite differences in order to render them sufficiently general1 as simple arbitrary constants are added to all quadratures in order to complete their generality4 ,this idea1 primitively introduced by ,euler1 has since been the subject of extended investigation by ,m4 ,fourier1 who has made new and important applications of it in his mathematical theory of heat4 ,,applications ,,of ,,this ,,calculus4 ,series4 ,among the principal general applications which have been made of the calculus of finite differences1 it would be proper to place in the first rank1 as the most extended and the most important1 the solution of questions relating to series2 if1 as has been shown1 the general theory of series ought not to be considered as constituting1 by its nature1 the actual foundation of the calculus of ,taylor4 ,interpolations4 ,this great class of problems being then set aside1 the most essential of the veritable applications of the analysis of ,taylor is1 undoubtedly1 thus far1 the general method of interpolations1 so frequently and so usefully employed in the investigation of the empirical laws of natural phenomena4 ,the question consists1 as is well known1 in intercalating between certain given numbers other intermediate numbers1 subjected to the same law which we suppose to exist between the first4 ,we can abundantly verify1 in this principal application of the calculus of ,taylor1 how truly foreign and often inconvenient is the consideration of differences with respect to the questions which depend on that analysis4 ,indeed1 ,lagrange has replaced the formulas of interpolation1 deduced from the ordinary algorithm of the calculus of finite differences1 by much simpler general formulas1 which are now almost always preferred1 and which have been found directly1 without making any use of the notion of differences1 which only complicates the question4 ,approximate ,rectification1 c4 ,a last important class of applications of the calculus of finite differences1 which deserves to be distinguished from the preceding1 consists in the eminently useful employment made of it in geometry for determining by approximation the length and the area of any curve1 and in the same way the cubature of a body of any form whatever4 ,this procedure 7which may besides be conceived abstractly as depending on the same analytical investigation as the question of interpolation7 frequently offers a valuable supplement to the entirely logical geometrical methods which often lead to integrations1 which we do not yet know how to effect1 or to calculations of very complicated execution4 99 99 99 99 99 ,such are the various principal considerations to be noticed with respect to the calculus of finite differences4 ,this examination completes the proposed philosophical outline of ,,abstract ,,mathematics4 ,,concrete ,,mathematics will now be the subject of a similar labour4 ,in it we shall particularly devote ourselves to examining how it has been possible 7supposing the general science of the calculus to be perfect71 by invariable procedures1 to reduce to pure questions of analysis all the problems which can be presented by ,geometry and ,mechanics1 and thus to impress on these two fundamental bases of natural philosophy a degree of precision and especially of unity2 in a word1 a character of high perfection1 which could be communicated to them by such a course alone4 ,,book ,,ii4 ,,geometry4 ,,book ,,ii4 ,,geometry4 ,,chapter ,i4 ,,general ,,view ,,of ,,geometry4 ,its true ,nature4 ,after the general exposition of the philosophical character of concrete mathematics1 compared with that of abstract mathematics1 given in the introductory chapter1 it need not here be shown in a special manner that geometry must be considered as a true natural science1 only much more simple1 and therefore much more perfect1 than any other4 ,this necessary perfection of geometry1 obtained essentially by the application of mathematical analysis1 which it so eminently admits1 is apt to produce erroneous views of the real nature of this fundamental science1 which most minds at present conceive to be a purely logical science quite independent of observation4 ,it is nevertheless evident1 to any one who examines with attention the character of geometrical reasonings1 even in the present state of abstract geometry1 that1 although the facts which are considered in it are much more closely united than those relating to any other science1 still there always exists1 with respect to every body studied by geometers1 a certain number of primitive phenomena1 which1 since they are not established by any reasoning1 must be founded on observation alone1 and which form the necessary basis of all the deductions4 ,the scientific superiority of geometry arises from the phenomena which it considers being necessarily the most universal and the most simple of all4 ,not only may all the bodies of nature give rise to geometrical inquiries1 as well as mechanical ones1 but still farther1 geometrical phenomena would still exist1 even though all the parts of the universe should be considered as immovable4 ,geometry is then1 by its nature1 more general than mechanics4 ,at the same time1 its phenomena are more simple1 for they are evidently independent of mechanical phenomena1 while these latter are always complicated with the former4 ,the same relations hold good in comparing geometry with abstract thermology4 ,for these reasons1 in our classification we have made geometry the first part of concrete mathematics2 that part the study of which1 in addition to its own importance1 serves as the indispensable basis of all the rest4 ,before considering directly the philosophical study of the different orders of inquiries which constitute our present geometry1 we should obtain a clear and exact idea of the general destination of that science1 viewed in all its bearings4 ,such is the object of this chapter4 ,definition4 ,geometry is commonly defined in a very vague and entirely improper manner1 as being the science of extension4 ,an improvement on this would be to say that geometry has for its object the measurement of extension2 but such an explanation would be very insufficient1 although at bottom correct1 and would be far from giving any idea of the true general character of geometrical science4 ,to do this1 ,i think that ,i should first explain two fundamental ideas1 which1 very simple in themselves1 have been singularly obscured by the employment of metaphysical considerations4 ,the ,idea of ,space4 ,the first is that of ,space4 ,this conception properly consists simply in this1 that1 instead of considering extension in the bodies themselves1 we view it in an indefinite medium1 which we regard as containing all the bodies of the universe4 ,this notion is naturally suggested to us by observation1 when we think of the impression which a body would leave in a fluid in which it had been placed4 ,it is clear1 in fact1 that1 as regards its geometrical relations1 such an impression may be substituted for the body itself1 without altering the reasonings respecting it4 ,as to the physical nature of this indefinite space1 we are spontaneously led to represent it to ourselves1 as being entirely analogous to the actual medium in which we live2 so that if this medium was liquid instead of gaseous1 our geometrical space would undoubtedly be conceived as liquid also4 ,this circumstance is1 moreover1 only very secondary1 the essential object of such a conception being only to make us view extension separately from the bodies which manifest it to us4 ,we can easily understand in advance the importance of this fundamental image1 since it permits us to study geometrical phenomena in themselves1 abstraction being made of all the other phenomena which constantly accompany them in real bodies1 without1 however1 exerting any influence over them4 ,the regular establishment of this general abstraction must be regarded as the first step which has been made in the rational study of geometry1 which would have been impossible if it had been necessary to consider1 together with the form and the magnitude of bodies1 all their other physical properties4 ,the use of such an hypothesis1 which is perhaps the most ancient philosophical conception created by the human mind1 has now become so familiar to us1 that we have difficulty in exactly estimating its importance1 by trying to appreciate the consequences which would result from its suppression4 ,different ,kinds of ,extension4 ,the second preliminary geometrical conception which we have to examine is that of the different kinds of extension1 designated by the words volume1 surface1 line1 and even point1 and of which the ordinary explanation is so unsatisfactory4#ac ,footnote #ac3 ,lacroix has justly criticised the expression of solid1 commonly used by geometers to designate a volume4 ,it is certain1 in fact1 that when we wish to consider separately a certain portion of indefinite space1 conceived as gaseous1 we mentally solidify its exterior envelope1 so that a line and a surface are habitually1 to our minds1 just as solid as a volume4 ,it may also be remarked that most generally1 in order that bodies may penetrate one another with more facility1 we are obliged to imagine the interior of the volumes to be hollow1 which renders still more sensible the impropriety of the word solid4 ,although it is evidently impossible to conceive any extension absolutely deprived of any one of the three fundamental dimensions1 it is no less incontestable that1 in a great number of occasions1 even of immediate utility1 geometrical questions depend on only two dimensions1 considered separately from the third1 or on a single dimension1 considered separately from the two others4 ,again1 independently of this direct motive1 the study of extension with a single dimension1 and afterwards with two1 clearly presents itself as an indispensable preliminary for facilitating the study of complete bodies of three dimensions1 the immediate theory of which would be too complicated4 ,such are the two general motives which oblige geometers to consider separately extension with regard to one or to two dimensions1 as well as relatively to all three together4 ,the general notions of surface and of line have been formed by the human mind1 in order that it may be able to think1 in a permanent manner1 of extension in two directions1 or in one only4 ,the hyperbolical expressions habitually employed by geometers to define these notions tend to convey false ideas of them2 but1 examined in themselves1 they have no other object than to permit us to reason with facility respecting these two kinds of extension1 making complete abstraction of that which ought not to be taken into consideration4 ,now for this it is sufficient to conceive the dimension which we wish to eliminate as becoming gradually smaller and smaller1 the two others remaining the same1 until it arrives at such a degree of tenuity that it can no longer fix the attention4 ,it is thus that we naturally acquire the real idea of a surface1 and1 by a second analogous operation1 the idea of a line1 by repeating for breadth what we had at first done for thickness4 ,finally1 if we again repeat the same operation1 we arrive at the idea of a point1 or of an extension considered only with reference to its place1 abstraction being made of all magnitude1 and designed consequently to determine positions4 ,surfaces evidently have1 moreover1 the general property of exactly circumscribing volumes2 and in the same way1 lines1 in their turn1 circumscribe surfaces and are limited by points4 ,but this consideration1 to which too much importance is often given1 is only a secondary one4 ,surfaces and lines are1 then1 in reality1 always conceived with three dimensions2 it would be1 in fact1 impossible to represent to one's self a surface otherwise than as an extremely thin plate1 and a line otherwise than as an infinitely fine thread4 ,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 identical1 for it must depend on the degree of subtilty of his habitual geometrical observations4 ,this want of uniformity has1 besides1 no real inconvenience1 since it is sufficient1 in order that the ideas of surface and of line should satisfy the essential condition of their destination1 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 appreciate4 ,we hence see how devoid of all meaning are the fantastic discussions of metaphysicians upon the foundations of geometry4 ,it should also be remarked that these primordial ideas are habitually presented by geometers in an unphilosophical manner1 since1 for example1 they explain the notions of the different sorts of extent in an order absolutely the inverse of their natural dependence1 which often produces the most serious inconveniences in elementary instruction4 ,,the ,,final ,,object ,,of ,,geometry4 ,these preliminaries being established1 we can proceed directly to the general definition of geometry1 continuing to conceive this science as having for its final object the measurement of extension4 ,it is necessary in this matter to go into a thorough explanation1 founded on the distinction of the three kinds of extension1 since the notion of measurement is not exactly the same with reference to surfaces and volumes as to lines4 ,nature of ,geometrical ,measurement4 ,if we take the word measurement in its direct and general mathematical acceptation1 which signifies simply the determination of the value of the ratios between any homogeneous magnitudes1 we must consider1 in geometry1 that the measurement of surfaces and of volumes1 unlike that of lines1 is never conceived1 even in the most simple and the most favourable cases1 as being effected directly4 ,the comparison of two lines is regarded as direct2 that of two surfaces or of two volumes is1 on the contrary1 always indirect4 ,thus we conceive that two lines may be superposed2 but the superposition of two surfaces1 or1 still more so1 of two volumes1 is evidently impossible in most cases2 and1 even when it becomes rigorously practicable1 such a comparison is never either convenient or exact4 ,it is1 then1 very necessary to explain wherein properly consists the truly geometrical measurement of a surface or of a volume4 ,measurement of ,surfaces and of ,volumes4 ,for this we must consider that1 whatever may be the form of a body1 there always exists a certain number of lines1 more or less easy to be assigned1 the length of which is sufficient to define exactly the magnitude of its surface or of its volume4 ,geometry1 regarding these lines as alone susceptible of being directly measured1 proposes to deduce1 from the simple determination of them1 the ratio of the surface or of the volume sought1 to the unity of surface1 or to the unity of volume4 ,thus the general object of geometry1 with respect to surfaces and to volumes1 is properly to reduce all comparisons of surfaces or of volumes to simple comparisons of lines4 ,besides the very great facility which such a transformation evidently offers for the measurement of volumes and of surfaces1 there results from it1 in considering it in a more extended and more scientific manner1 the general possibility of reducing to questions of lines all questions relating to volumes and to surfaces1 considered with reference to their magnitude4 ,such is often the most important use of the geometrical expressions which determine surfaces and volumes in functions of the corresponding lines4 ,it is true that direct comparisons between surfaces or between volumes are sometimes employed2 but such measurements are not regarded as geometrical1 but only as a supplement sometimes necessary1 although too rarely applicable1 to the insufficiency or to the difficulty of truly rational methods4 ,it is thus that we often determine the volume of a body1 and in certain cases its surface1 by means of its weight4 ,in the same way1 on other occasions1 when we can substitute for the proposed volume an equivalent liquid volume1 we establish directly the comparison of the two volumes1 by profiting by the property possessed by liquid masses1 of assuming any desired form4 ,but all means of this nature are purely mechanical1 and rational geometry necessarily rejects them4 ,to render more sensible the difference between these modes of determination and true geometrical measurements1 ,i will cite a single very remarkable example2 the manner in which ,galileo determined the ratio of the ordinary cycloid to that of the generating circle4 ,the geometry of his time was as yet insufficient for the rational solution of such a problem4 ,galileo conceived the idea of discovering that ratio by a direct experiment4 ,having weighed as exactly as possible two plates of the same material and of equal thickness1 one of them having the form of a circle and the other that of the generated cycloid1 he found the weight of the latter always triple that of the former2 whence he inferred that the area of the cycloid is triple that of the generating circle1 a result agreeing with the veritable solution subsequently obtained by ,pascal and ,wallis4 ,such a success evidently depends on the extreme simplicity of the ratio sought2 and we can understand the necessary insufficiency of such expedients1 even when they are actually practicable4 ,we see clearly1 from what precedes1 the nature of that part of geometry relating to volumes and that relating to surfaces4 ,but the character of the geometry of lines is not so apparent1 since1 in order to simplify the exposition1 we have considered the measurement of lines as being made directly4 ,there is1 therefore1 needed a complementary explanation with respect to them4 ,measurement of curved ,lines4 ,for this purpose1 it is sufficient to distinguish between the right line and curved lines1 the measurement of the first being alone regarded as direct1 and that of the other as always indirect4 ,although superposition is sometimes strictly practicable for curved lines1 it is nevertheless evident that truly rational geometry must necessarily reject it1 as not admitting of any precision1 even when it is possible4 ,the geometry of lines has1 then1 for its general object1 to reduce in every case the measurement of curved lines to that of right lines2 and consequently1 in the most extended point of view1 to reduce to simple questions of right lines all questions relating to the magnitude of any curves whatever4 ,to understand the possibility of such a transformation1 we must remark1 that in every curve there always exist certain right lines1 the length of which must be sufficient to determine that of the curve4 ,thus1 in a circle1 it is evident that from the length of the radius we must be able to deduce that of the circumference2 in the same way1 the length of an ellipse depends on that of its two axes2 the length of a cycloid upon the diameter of the generating circle1 c42 and if1 instead of considering the whole of each curve1 we demand1 more generally1 the length of any arc1 it will be sufficient to add to the different rectilinear parameters1 which determine the whole curve1 the chord of the proposed arc1 or the co-ordinates of its extremities4 ,to discover the relation which exists between the length of a curved line and that of similar right lines1 is the general problem of the part of geometry which relates to the study of lines4 ,combining this consideration with those previously suggested with respect to volumes and to surfaces1 we may form a very clear idea of the science of geometry1 conceived in all its parts1 by assigning to it1 for its general object1 the final reduction of the comparisons of all kinds of extent1 volumes1 surfaces1 or lines1 to simple comparisons of right lines1 the only comparisons regarded as capable of being made directly1 and which indeed could not be reduced to any others more easy to effect4 ,such a conception1 at the same time1 indicates clearly the veritable character of geometry1 and seems suited to show at a single glance its utility and its perfection4 ,measurement of right ,lines4 ,in order to complete this fundamental explanation1 ,i have yet to show how there can be1 in geometry1 a special section relating to the right line1 which seems at first incompatible with the principle that the measurement of this class of lines must always be regarded as direct4 ,it is so1 in fact1 as compared with that of curved lines1 and of all the other objects which geometry considers4 ,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 it4 ,now this often presents insurmountable difficulties1 as ,i had occasion to show1 for another reason1 in the introductory chapter4 ,we must1 then1 make the measurement of the proposed right line depend on other analogous measurements capable of being effected directly4 ,there is1 then1 necessarily a primary distinct branch of geometry1 exclusively devoted to the right line2 its object is to determine certain right lines from others by means of the relations belonging to the figures resulting from their assemblage4 ,this preliminary part of geometry1 which is almost imperceptible in viewing the whole of the science1 is nevertheless susceptible of a great development4 ,it is evidently of especial importance1 since all other geometrical measurements are referred to those of right lines1 and if they could not be determined1 the solution of every question would remain unfinished4 ,such1 then1 are the various fundamental parts of rational geometry1 arranged according to their natural dependence2 the geometry of lines being first considered1 beginning with the right line2 then the geometry of surfaces1 and1 finally1 that of solids4 ,,infinite ,,extent ,,of ,,its ,,field4 ,having determined with precision the general and final object of geometrical inquiries1 the science must now be considered with respect to the field embraced by each of its three fundamental sections4 ,thus considered1 geometry is evidently susceptible1 by its nature1 of an extension which is rigorously infinite2 for the measurement of lines1 of surfaces1 or of volumes presents necessarily as many distinct questions as we can conceive different figures subjected to exact definitions2 and their number is evidently infinite4 ,geometers limited themselves at first to consider the most simple figures which were directly furnished them by nature1 or which were deduced from these primitive elements by the least complicated combinations4 ,but they have perceived1 since ,descartes1 that1 in order to constitute the science in the most philosophical manner1 it was necessary to make it apply to all imaginable figures4 ,this abstract geometry will then inevitably comprehend as particular cases all the different real figures which the exterior world could present4 ,it is then a fundamental principle in truly rational geometry to consider1 as far as possible1 all figures which can be rigorously conceived4 ,the most superficial examination is enough to convince us that these figures present a variety which is quite infinite4 ,infinity of ,lines4 ,with respect to curved lines1 regarding them as generated by the motion of a point governed by a certain law1 it is plain that we shall have1 in general1 as many different curves as we conceive different laws for this motion1 which may evidently be determined by an infinity of distinct conditions2 although it may sometimes accidentally happen that new generations produce curves which have been already obtained4 ,thus1 among plane curves1 if a point moves so as to remain constantly at the same distance from a fixed point1 it will generate a circle2 if it is the sum or the difference of its distances from two fixed points which remains constant1 the curve described will be an ellipse or an hyperbola2 if it is their product1 we shall have an entirely different curve2 if the point departs equally from a fixed point and from a fixed line1 it will describe a parabola2 if it revolves on a circle at the same time that this circle rolls along a straight line1 we shall have a cycloid2 if it advances along a straight line1 while this line1 fixed at one of its extremities1 turns in any manner whatever1 there will result what in general terms are called spirals1 which of themselves evidently present as many perfectly distinct curves as we can suppose different relations between these two motions of translation and of rotation1 c4 ,each of these different curves may then furnish new ones1 by the different general constructions which geometers have imagined1 and which give rise to evolutes1 to epicycloids1 to caustics1 c4 ,finally1 there exists a still greater variety among curves of double curvature4 ,infinity of ,surfaces4 ,as to surfaces1 the figures are necessarily more different still1 considering them as generated by the motion of lines4 ,indeed1 the figure may then vary1 not only in considering1 as in curves1 the different infinitely numerous laws to which the motion of the generating line may be subjected1 but also in supposing that this line itself may change its nature2 a circumstance which has nothing analogous in curves1 since the points which describe them cannot have any distinct figure4 ,two classes of very different conditions may then cause the figures of surfaces to vary1 while there exists only one for lines4 ,it is useless to cite examples of this doubly infinite multiplicity of surfaces4 ,it would be sufficient to consider the extreme variety of the single group of surfaces which may be generated by a right line1 and which comprehends the whole family of cylindrical surfaces1 that of conical surfaces1 the most general class of developable surfaces1 c4 ,infinity of ,volumes4 ,with respect to volumes1 there is no occasion for any special consideration1 since they are distinguished from each other only by the surfaces which bound them4 ,in order to complete this sketch1 it should be added that surfaces themselves furnish a new general means of conceiving new curves1 since every curve may be regarded as produced by the intersection of two surfaces4 ,it is in this way1 indeed1 that the first lines which we may regard as having been truly invented by geometers were obtained1 since nature gave directly the straight line and the circle4 ,we know that the ellipse1 the parabola1 and the hyperbola1 the only curves completely studied by the ancients1 were in their origin conceived only as resulting from the intersection of a cone with circular base by a plane in different positions4 ,it is evident that1 by the combined employment of these different general means for the formation of lines and of surfaces1 we could produce a rigorously infinitely series of distinct forms in starting from only a very small number of figures directly furnished by observation4 ,analytical invention of ,curves1 c4 ,finally1 all the various direct means for the invention of figures have scarcely any farther importance1 since rational geometry has assumed its final character in the hands of ,descartes4 ,indeed1 as we shall see more fully in chapter iii41 the invention of figures is now reduced to the invention of equations1 so that nothing is more easy than to conceive new lines and new surfaces1 by changing at will the functions introduced into the equations4 ,this simple abstract procedure is1 in this respect1 infinitely more fruitful than all the direct resources of geometry1 developed by the most powerful imagination1 which should devote itself exclusively to that order of conceptions4 ,it also explains1 in the most general and the most striking manner1 the necessarily infinite variety of geometrical forms1 which thus corresponds to the diversity of analytical functions4 ,lastly1 it shows no less clearly that the different forms of surfaces must be still more numerous than those of lines1 since lines are represented analytically by equations with two variables1 while surfaces give rise to equations with three variables1 which necessarily present a greater diversity4 ,the preceding considerations are sufficient to show clearly the rigorously infinite extent of each of the three general sections of geometry4 ,,expansion ,,of ,,original ,,definition4 ,to complete the formation of an exact and sufficiently extended idea of the nature of geometrical inquiries1 it is now indispensable to return to the general definition above given1 in order to present it under a new point of view1 without which the complete science would be only very imperfectly conceived4 ,when we assign as the object of geometry the measurement of all sorts of lines1 surfaces1 and volumes1 that is1 as has been explained1 the reduction of all geometrical comparisons to simple comparisons of right lines1 we have evidently the advantage of indicating a general destination very precise and very easy to comprehend4 ,but if we set aside every definition1 and examine the actual composition of the science of geometry1 we will at first be induced to regard the preceding definition as much too narrow2 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 measurement of extension4 ,in spite of this fundamental objection1 ,i will persist in retaining this definition2 for1 in fact1 if1 instead of confining ourselves to considering the different questions of geometry isolatedly1 we endeavour to grasp the leading questions1 in comparison with which all others1 however important they may be1 must be regarded as only secondary1 we will finally recognize that the measurement of lines1 of surfaces1 and of volumes1 is the invariable object1 sometimes direct1 though most often indirect1 of all geometrical labours4 ,this general proposition being fundamental1 since it can alone give our definition all its value1 it is indispensable to enter into some developments upon this subject4 ,,properties ,,of ,,lines ,,and ,,surfaces4 ,when we examine with attention the geometrical investigations which do not seem to relate to the measurement of extent1 we find that they consist essentially in the study of the different properties of each line or of each surface2 that is1 in the knowledge of the different modes of generation1 or at least of definition1 peculiar to each figure considered4 ,now we can easily establish in the most general manner the necessary relation of such a study to the question of measurement1 for which the most complete knowledge of the properties of each form is an indispensable preliminary4 ,this is concurrently proven by two considerations1 equally fundamental1 although quite distinct in their nature4 ,,necessity ,,of ,,their ,,study3 #a4 ,to find the most suitable ,property4 ,the first1 purely scientific1 consists in remarking that1 if we did not know any other characteristic property of each line or surface than that one according to which geometers had first conceived it1 in most cases it would be impossible to succeed in the solution of questions relating to its measurement4 ,in fact1 it is easy to understand that the different definitions which each figure admits of are not all equally suitable for such an object1 and that they even present the most complete oppositions in that respect4 ,besides1 since the primitive definition of each figure was evidently not chosen with this condition in view1 it is clear that we must not expect1 in general1 to find it the most suitable2 whence results the necessity of discovering others1 that is1 of studying as far as is possible the properties of the proposed figure4 ,let us suppose1 for example1 that the circle is defined to be the curve which1 with the same contour1 contains the greatest area4 ,this is certainly a very characteristic property1 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 curve4 ,it is clear1 in advance1 that the property of having all its points equally distant from a fixed point must evidently be much better adapted to inquiries of this nature1 even though it be not precisely the most suitable4 ,in like manner1 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 base1 by a plane parallel to its generatrix8 ,the purely speculative labours of preceding geometers1 in transforming this first definition1 were evidently indispensable preliminaries to the direct solution of such a question4 ,the same is true1 in a still greater degree1 with respect to surfaces4 ,to form a just idea of this1 we need only compare1 as to the question of cubature or quadrature1 the common definition of the sphere with that one1 no less characteristic certainly1 which would consist in regarding a spherical body1 as that one which1 with the same area1 contains the greatest volume4 ,no more examples are needed to show the necessity of knowing1 so far as is possible1 all the properties of each line or of each surface1 in order to facilitate the investigation of rectifications1 of quadratures1 and of cubatures1 which constitutes the final object of geometry4 ,we may even say that the principal difficulty of questions of this kind consists in employing in each case the property which is best adapted to the nature of the proposed problem4 ,thus1 while we continue to indicate1 for more precision1 the measurement of extension as the general destination of geometry1 this first consideration1 which goes to the very bottom of the subject1 shows clearly the necessity of including in it the study1 as thorough as possible1 of the different generations or definitions belonging to the same form4 #b4 ,to pass from the ,concrete to the ,abstract4 ,a second consideration1 of at least equal importance1 consists in such a study being indispensable for organizing in a rational manner the relation of the abstract to the concrete in geometry4 ,the science of geometry having to consider all imaginable figures which admit of an exact definition1 it necessarily results from this1 as we have remarked1 that questions relating to any figures presented by nature are always implicitly comprised in this abstract geometry1 supposed to have attained its perfection4 ,but when it is necessary to actually pass to concrete geometry1 we constantly meet with a fundamental difficulty1 that of knowing to which of the different abstract types we are to refer1 with sufficient approximation1 the real lines or surfaces which we have to study4 ,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 geometry4 ,in fact1 if we always confined ourselves to the single primitive definition of a line or of a surface1 supposing even that we could then measure it 7which1 according to the first order of considerations1 would generally be impossible71 this knowledge would remain almost necessarily barren in the application1 since we should not ordinarily know how to recognize that figure in nature when it presented itself there2 to ensure that1 it would be necessary that the single characteristic1 according to which geometers had conceived it1 should be precisely that one whose verification external circumstances would admit3 a coincidence which would be purely fortuitous1 and on which we could not count1 although it might sometimes take place4 ,it is1 then1 only by multiplying as much as possible the characteristic properties of each abstract figure1 that we can be assured1 in advance1 of recognizing it in the concrete state1 and of thus turning to account all our rational labours1 by verifying in each case the definition which is susceptible of being directly proven4 ,this definition is almost always the only one in given circumstances1 and varies1 on the other hand1 for the same figure1 with different circumstances2 a double reason for its previous determination4 ,illustration3 ,orbits of the ,planets4 ,the geometry of the heavens furnishes us with a very memorable example in this matter1 well suited to show the general necessity of such a study4 ,we know that the ellipse was discovered by ,kepler to be the curve which the planets describe about the sun1 and the satellites about their planets4 ,now would this fundamental discovery1 which re-created astronomy1 ever have been possible1 if geometers had been always confined to conceiving the ellipse only as the oblique section of a circular cone by a plane8 ,no such definition1 it is evident1 would admit of such a verification4 ,the most general property of the ellipse1 that the sum of the distances from any of its points to two fixed points is a constant quantity1 is undoubtedly much more susceptible1 by its nature1 of causing the curve to be recognized in this case1 but still is not directly suitable4 ,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 direction2 the only relation which admits of an astronomical interpretation1 as expressing the law which connects the distance from the planet to the sun1 with the time elapsed since the beginning of its revolution4 ,it was1 then1 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 view1 before ,kepler could thus pass from the abstract to the concrete1 in choosing from among all these different characteristics that one which could be most easily proven for the planetary orbits4 ,illustration3 ,figure of the ,earth4 ,another example of the same order1 but relating to surfaces1 occurs in considering the important question of the figure of the earth4 ,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 point1 how would we ever have been able to discover that the surface of the earth was spherical8 ,for this1 it was necessary previously to deduce from this definition of the sphere some properties capable of being verified by observations made upon the surface alone1 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 pole1 and the angular height of this pole above the horizon at each point4 ,another example1 but involving a much longer series of preliminary speculations1 is the subsequent proof that the earth is not rigorously spherical1 but that its form is that of an ellipsoid of revolution4 ,after such examples1 it would be needless to give any others1 which any one besides may easily multiply4 ,all of them prove that1 without a very extended knowledge of the different properties of each figure1 the relation of the abstract to the concrete1 in geometry1 would be purely accidental1 and that the science would consequently want one of its most essential foundations4 ,such1 then1 are two general considerations which fully demonstrate the necessity of introducing into geometry a great number of investigations which have not the measurement of extension for their direct object2 while we continue1 however1 to conceive such a measurement as being the final destination of all geometrical science4 ,in this way we can retain the philosophical advantages of the clearness and precision of this definition1 and still include in it1 in a very logical though indirect manner1 all known geometrical researches1 in considering those which do not seem to relate to the measurement of extension1 as intended either to prepare for the solution of the final questions1 or to render possible the application of the solutions obtained4 ,having thus recognized1 as a general principle1 the close and necessary connexion of the study of the properties of lines and surfaces with those researches which constitute the final object of geometry1 it is evident that geometers1 in the progress of their labours1 must by no means constrain themselves to keep such a connexion always in view4 ,knowing1 once for all1 how important it is to vary as much as possible the manner of conceiving each figure1 they should pursue that study1 without considering of what immediate use such or such a special property may be for rectifications1 quadratures1 and cubatures4 ,they would uselessly fetter their inquiries by attaching a puerile importance to the continued establishment of that co-ordination4 ,this general exposition of the general object of geometry is so much the more indispensable1 since1 by the very nature of the subject1 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 researches4 ,indeed1 the questions directly relating to rectifications1 to quadratures1 and to cubatures1 are evidently1 by themselves1 very few in number for each figure considered4 ,on the other hand1 the study of the properties of the same figure presents an unlimited field to the activity of the human mind1 in which it may always hope to make new discoveries4 ,thus1 although geometers have occupied themselves for twenty centuries1 with more or less activity undoubtedly1 but without any real interruption1 in the study of the conic sections1 they are far from regarding that so simple subject as being exhausted2 and it is certain1 indeed1 that in continuing to devote themselves to it1 they would not fail to find still unknown properties of those different curves4 ,if labours of this kind have slackened considerably for a century past1 it is not because they are completed1 but only1 as will be presently explained1 because the philosophical revolution in geometry1 brought about by ,descartes1 has singularly diminished the importance of such researches4 ,it results from the preceding considerations that not only is the field of geometry necessarily infinite because of the variety of figures to be considered1 but also in virtue of the diversity of the points of view under the same figure may be regarded4 ,this last conception is1 indeed1 that which gives the broadest and most complete idea of the whole body of geometrical researches4 ,we see that studies of this kind consist essentially1 for each line or for each surface1 in connecting all the geometrical phenomena which it can present1 with a single fundamental phenomenon1 regarded as the primitive definition4 ,,the ,,two ,,general ,,methods ,,of ,,geometry4 ,having now explained in a general and yet precise manner the final object of geometry1 and shown how the science1 thus defined1 comprehends a very extensive class of researches which did not at first appear necessarily to belong to it1 there remains to be considered the method to be followed for the formation of this science4 ,this discussion is indispensable to complete this first sketch of the philosophical character of geometry4 ,i shall here confine myself to indicating the most general consideration in this matter1 developing and summing up this important fundamental idea in the following chapters4 ,geometrical questions may be treated according to two methods so different1 that there result from them two sorts of geometry1 so to say1 the philosophical character of which does not seem to me to have yet been properly apprehended4 ,the expressions of ,synthetical ,geometry and ,analytical ,geometry1 habitually employed to designate them1 give a very false idea of them4 ,i would much prefer the purely historical denominations of ,geometry of the ,ancients and ,geometry of the ,moderns1 which have at least the advantage of not causing their true character to be misunderstood4 ,but ,i propose to employ henceforth the regular expressions of ,special ,geometry and ,general ,geometry1 which seem to me suited to characterize with precision the veritable nature of the two methods4 ,their fundamental ,difference4 ,the fundamental difference between the manner in which we conceive ,geometry since ,descartes1 and the manner in which the geometers of antiquity treated geometrical questions1 is not the use of the ,calculus 7or ,algebra71 as is commonly thought to be the case4 ,on the one hand1 it is certain that the use of the calculus was not entirely unknown to the ancient geometers1 since they used to make continual and very extensive applications of the theory of proportions1 which was for them1 as a means of deduction1 a sort of real1 though very imperfect and especially extremely limited equivalent for our present algebra4 ,the calculus may even be employed in a much more complete manner than they have used it1 in order to obtain certain geometrical solutions1 which will still retain all the essential character of the ancient geometry2 this occurs very frequently with respect to those problems of geometry of two or of three dimensions1 which are commonly designated under the name of determinate4 ,on the other hand1 important as is the influence of the calculus in our modern geometry1 various solutions obtained without algebra may sometimes manifest the peculiar character which distinguishes it from the ancient geometry1 although analysis is generally indispensable4 ,i will cite1 as an example1 the method of ,roberval for tangents1 the nature of which is essentially modern1 and which1 however1 leads in certain cases to complete solutions1 without any aid from the calculus4 ,it is not1 then1 the instrument of deduction employed which is the principal distinction between the two courses which the human mind can take in geometry4 ,the real fundamental difference1 as yet imperfectly apprehended1 seems to me to consist in the very nature of the questions considered4 ,in truth1 geometry1 viewed as a whole1 and supposed to have attained entire perfection1 must1 as we have seen on the one hand1 embrace all imaginable figures1 and1 on the other1 discover all the properties of each figure4 ,it admits1 from this double consideration1 of being treated according to two essentially distinct plans2 either1 #a1 by grouping together all the questions1 however different they may be1 which relate to the same figure1 and isolating those relating to different bodies1 whatever analogy there may exist between them2 or1 #b1 on the contrary1 by uniting under one point of view all similar inquiries1 to whatever different figures they may relate1 and separating the questions relating to the really different properties of the same body4 ,in a word1 the whole body of geometry may be essentially arranged either with reference to the bodies studied or to the phenomena to be considered4 ,the first plan1 which is the most natural1 was that of the ancients2 the second1 infinitely more rational1 is that of the moderns since ,descartes4 ,geometry of the ,ancients4 ,indeed1 the principal characteristics of the ancient geometry is that they studied1 one by one1 the different lines and the different surfaces1 not passing to the examination of a new figure till they thought they had exhausted all that there was interesting in the figures already known4 ,in this way of proceeding1 when they undertook the study of a new curve1 the whole of the labour bestowed on the preceding ones could not offer directly any essential assistance1 otherwise than by the geometrical practice to which it had trained the mind4 ,whatever might be the real similarity of the questions proposed as to two different figures1 the complete knowledge acquired for the one could not at all dispense with taking up again the whole of the investigation for the other4 ,thus the progress of the mind was never assured2 so that they could not be certain1 in advance1 of obtaining any solution whatever1 however analogous the proposed problem might be to questions which had been already resolved4 ,thus1 for example1 the determination of the tangents to the three conic sections did not furnish any rational assistance for drawing the tangent to any other new curve1 such as the conchoid1 the cissoid1 c4 ,in a word1 the geometry of the ancients was1 according to the expression proposed above1 essentially special4 ,geometry of the ,moderns4 ,in the system of the moderns1 geometry is1 on the contrary1 eminently general1 that is to say1 relating to any figures whatever4 ,it is easy to understand1 in the first place1 that all geometrical expressions of any interest may be proposed with reference to all imaginable figures4 ,this is seen directly in the fundamental problems--of rectifications1 quadratures1 and cubatures--which constitute1 as has been shown1 the final object of geometry4 ,but this remark is no less incontestable1 even for investigations which relate to the different properties of lines and of surfaces1 and of which the most essential1 such as the question of tangents or of tangent planes1 the theory of curvatures1 c41 are evidently common to all figures whatever4 ,the very few investigations which are truly peculiar to particular figures have only an extremely secondary importance4 ,this being understood1 modern geometry consists essentially in abstracting1 in order to treat it by itself1 in an entirely general manner1 every question relating to the same geometrical phenomenon1 in whatever bodies it may be considered4 ,the application of the universal theories thus constructed to the special determination of the phenomenon which is treated of in each particular body1 is now regarded as only a subaltern labour1 to be executed according to invariable rules1 and the success of which is certain in advance4 ,this labour is1 in a word1 of the same character as the numerical calculation of an analytical formula4 ,there can be no other merit in it than that of presenting in each case the solution which is necessarily furnished by the general method1 with all the simplicity and elegance which the line or the surface considered can admit of4 ,but no real importance is attached to any thing but the conception and the complete solution of a new question belonging to any figure whatever4 ,labours of this kind are alone regarded as producing any real advance in science4 ,the attention of geometers1 thus relieved from the examination of the peculiarities of different figures1 and wholly directed towards general questions1 has been thereby able to elevate itself to the consideration of new geometrical conceptions1 which1 applied to the curves studied by the ancients1 have led to the discovery of important properties which they had not before even suspected4 ,such is geometry1 since the radical revolution produced by ,descartes in the general system of the science4 ,the ,superiority of the modern ,geometry4 ,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 geometry4 ,we may even say that1 before the great conception of ,descartes1 rational geometry was not truly constituted upon definitive bases1 whether in its abstract or concrete relations4 ,in fact1 as regards science1 considered speculatively1 it is clear that1 in continuing indefinitely to follow the course of the ancients1 as did the moderns before ,descartes1 and even for a little while afterwards1 by adding some new curves to the small number of those which they had studied1 the progress thus made1 however rapid it might have been1 would still be found1 after a long series of ages1 to be very inconsiderable in comparison with the general system of geometry1 seeing the infinite variety of the forms which would still have remained to be studied4 ,on the contrary1 at each question resolved according to the method of the moderns1 the number of geometrical problems to be resolved is then1 once for all1 diminished by so much with respect to all possible bodies4 ,another consideration is1 that it resulted1 from their complete want of general methods1 that the ancient geometers1 in all their investigations1 were entirely abandoned to their own strength1 without ever having the certainty of obtaining1 sooner or later1 any solution whatever4 ,though this imperfection of the science was eminently suited to call forth all their admirable sagacity1 it necessarily rendered their progress extremely slow2 we can form some idea of this by the considerable time which they employed in the study of the conic sections4 ,modern geometry1 making the progress of our mind certain1 permits us1 on the contrary1 to make the greatest possible use of the forces of our intelligence1 which the ancients were often obliged to waste on very unimportant questions4 ,a no less important difference between the two systems appears when we come to consider geometry in the concrete point of view4 ,indeed1 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 figures4 ,in studying lines1 only one by one1 whatever may be the number1 always necessarily very small1 of those which we shall have considered1 the application of such theories to figures really existing in nature will never have any other than an essentially accidental character1 since there is nothing to assure us that these figures can really be brought under the abstract types considered by geometers4 ,thus1 for example1 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 orbits4 ,in continuing geometrical researches upon the same plan1 there was no good reason for hoping for similar coincidences2 and it would have been possible1 in these special studies1 that the researches of geometers should have been directed to abstract figures entirely incapable of any application1 while they neglected others1 susceptible perhaps of an important and immediate application4 ,it is clear1 at least1 that nothing positively guaranteed the necessary applicability of geometrical speculations4 ,it is quite another thing in the modern geometry4 ,from the single circumstance that in it we proceed by general questions relating to any figures whatever1 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 them4 ,from these different considerations1 we see that the ancient system of geometry wears essentially the character of the infancy of the science1 which did not begin to become completely rational till after the philosophical resolution produced by ,descartes4 ,but it is evident1 on the other hand1 that geometry could not be at first conceived except in this special manner4 ,general geometry would not have been possible1 and its necessity could not even have been felt1 if a long series of special labours on the most simple figures had not previously furnished bases for the conception of ,descartes1 and rendered apparent the impossibility of persisting indefinitely in the primitive geometrical philosophy4 ,the ,ancient the ,base of the ,modern4 ,from this last consideration we must infer that1 although the geometry which ,i have called general must be now regarded as the only true dogmatical geometry1 and that to which we shall chiefly confine ourselves1 the other having no longer much more than an historical interest1 nevertheless it is not possible to entirely dispense with special geometry in a rational exposition of the science4 ,we undoubtedly need not borrow directly from ancient geometry all the results which it has furnished2 but1 from the very nature of the subject1 it is necessarily impossible entirely to dispense with the ancient method1 which will always serve as the preliminary basis of the science1 dogmatically as well as historically4 ,the reason of this is easy to understand4 ,in fact1 general geometry being essentially founded1 as we shall soon establish1 upon the employment of the calculus in the transformation of geometrical into analytical considerations1 such a manner of proceeding could not take possession of the subject immediately at its origin4 ,we know that the application of mathematical analysis1 from its nature1 can never commence any science whatever1 since evidently it cannot be employed until the science has already been sufficiently cultivated to establish1 with respect to the phenomena considered1 some equations which can serve as starting points for the analytical operations4 ,these fundamental equations being once discovered1 analysis will enable us to deduce from them a multitude of consequences which it would have been previously impossible even to suspect2 it will perfect the science to an immense degree1 both with respect to the generality of its conceptions and to the complete co-ordination established between them4 ,but mere mathematical analysis could never be sufficient to form the bases of any natural science1 not even to demonstrate them anew when they have once been established4 ,nothing can dispense with the direct study of the subject1 pursued up to the point of the discovery of precise relations4 ,we thus see that the geometry of the ancients will always have1 by its nature1 a primary part1 absolutely necessary and more or less extensive1 in the complete system of geometrical knowledge4 ,it forms a rigorously indispensable introduction to general geometry4 ,but it is to this that it must be limited in a completely dogmatic exposition4 ,i will consider1 then1 directly1 in the following chapter1 this special or preliminary geometry restricted to exactly its necessary limits1 in order to occupy myself thenceforth only with the philosophical examination of general or definitive geometry1 the only one which is truly rational1 and which at present essentially composes the science4 ,,chapter ,,ii4 ,,ancient ,,or ,,synthetic ,,geometry4 ,the geometrical method of the ancients necessarily constituting a preliminary department in the dogmatical system of geometry1 designed to furnish general geometry with indispensable foundations1 it is now proper to begin with determining wherein strictly consists this preliminary function of special geometry1 thus reduced to the narrowest possible limits4 ,,its ,,proper ,,extent4 ,lines2 ,polygons2 ,polyhedrons4 ,in considering it under this point of view1 it is easy to recognize that we might restrict it to the study of the right line alone for what concerns the geometry of lines2 to the quadrature of rectilinear plane areas2 and1 lastly1 to the cubature of bodies terminated by plane faces4 ,the elementary propositions relating to these three fundamental questions form1 in fact1 the necessary starting point of all geometrical inquiries2 they alone cannot be obtained except by a direct study of the subject2 while1 on the contrary1 the complete theory of all other figures1 even that of the circle1 and of the surfaces and volumes which are connected with it1 may at the present day be completely comprehended in the domain of general or analytical geometry2 these primitive elements at once furnishing equations which are sufficient to allow of the application of the calculus to geometrical questions1 which would not have been possible without this previous condition4 ,it results from this consideration that1 in common practice1 we give to elementary geometry more extent than would be rigorously necessary to it2 since1 besides the right line1 polygons1 and polyhedrons1 we also include in it the circle and the round bodies2 the study of which might1 however1 be as purely analytical as that1 for example1 of the conic sections4 ,an unreflecting veneration for antiquity contributes to maintain this defect in method2 but the best reason which can be given for it is the serious inconvenience for ordinary instruction which there would be in postponing1 to so distant an epoch of mathematical education1 the solution of several essential questions1 which are susceptible of a direct and continual application to a great number of important uses4 ,in fact1 to proceed in the most rational manner1 we should employ the integral calculus in obtaining the interesting results relating to the length or the area of the circle1 or to the quadrature of the sphere1 c41 which have been determined by the ancients from extremely simple considerations4 ,this inconvenience would be of little importance with regard to the persons destined to study the whole of mathematical science1 and the advantage of proceeding in a perfectly logical order would have a much greater comparative value4 ,but the contrary case being the more frequent1 theories so essential have necessarily been retained in elementary geometry4 ,perhaps the conic sections1 the cycloid1 c41 might be advantageously added in such cases4 ,not to be farther restricted4 ,while this preliminary portion of geometry1 which cannot be founded on the application of the calculus1 is reduced by its nature to a very limited series of fundamental researches1 relating to the right line1 polygonal areas1 and polyhedrons1 it is certain1 on the other hand1 that we cannot restrict it any more2 although1 by a veritable abuse of the spirit of analysis1 it has been recently attempted to present the establishment of the principal theorems of elementary geometry under an algebraical point of view4 ,thus some have pretended to demonstrate1 by simple abstract considerations of mathematical analysis1 the constant relation which exists between the three angles of a rectilinear triangle1 the fundamental proposition of the theory of similar triangles1 that of parallelopipedons1 c42 in a word1 precisely the only geometrical propositions which cannot be obtained except by a direct study of the subject1 without the calculus being susceptible of having any part in it4 ,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 studies4 ,in mechanics1 the pretended analytical demonstrations of the parallelogram of forces are of similar character4 ,the viciousness of such a manner of proceeding follows from the principles previously presented4 ,we have already1 in fact1 recognized that1 since the calculus is not1 and cannot be1 any thing but a means of deduction1 it would indicate a radically false idea of it to wish to employ it in establishing the elementary foundations of any science whatever2 for on what would the analytical reasonings in such an operation repose8 ,a labour of this nature1 very far from really perfecting the philosophical character of a science1 would constitute a return towards the metaphysical age1 in presenting real facts as mere logical abstractions4 ,when we examine in themselves these pretended analytical demonstrations of the fundamental propositions of elementary geometry1 we easily verify their necessary want of meaning4 ,they are all founded on a vicious manner of conceiving the principle of homogeneity1 the true general idea of which was explained in the second chapter of the preceding book4 ,these demonstrations suppose that this principle does not allow us to admit the coexistence in the same equation of numbers obtained by different concrete comparisons1 which is evidently false1 and contrary to the constant practice of geometers4 ,thus it is easy to recognize that1 by employing the law of homogeneity in this arbitrary and illegitimate acceptation1 we could succeed in demonstrating1 with quite as much apparent rigour1 propositions whose absurdity is manifest at the first glance4 ,in examining attentively1 for example1 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 angles1 we see that it is founded on this preliminary principle that1 if two triangles have two of their angles respectively equal1 the third angle of the one will necessarily be equal to the third angle of the other4 ,this first point being granted1 the proposed relation is immediately deduced from it in a very exact and simple manner4 ,now the analytical consideration by which this previous proposition has been attempted to be established1 is of such a nature that1 if it could be correct1 we could rigorously deduce from it1 in reproducing it conversely1 this palpable absurdity1 that two sides of a triangle are sufficient1 without any angle1 for the entire determination of the third side4 ,we may make analogous remarks on all the demonstrations of this sort1 the sophisms of which will be thus verified in a perfectly apparent manner4 ,the more reason that we have here to consider geometry as being at the present day essentially analytical1 the more necessary was it to guard against this abusive exaggeration of mathematical analysis1 according to which all geometrical observation would be dispensed with1 in establishing upon pure algebraical abstractions the very foundations of this natural science4 ,attempted ,demonstrations of ,axioms1 c4 ,another indication that geometers have too much overlooked the character of a natural science which is necessarily inherent in geometry1 appears from their vain attempts1 so long made1 to demonstrate rigorously1 not by the aid of the calculus1 but by means of certain constructions1 several fundamental propositions of elementary geometry4 ,whatever may be effected1 it will evidently be impossible to avoid sometimes recurring to simple and direct observation in geometry as a means of establishing various results4 ,while1 in this science1 the phenomena which are considered are1 by virtue of their extreme simplicity1 much more closely connected with one another than those relating to any other physical science1 some must still be found which cannot be deduced1 and which1 on the contrary1 serve as starting points4 ,it may be admitted that the greatest logical perfection of the science is to reduce these to the smallest number possible1 but it would be absurd to pretend to make them completely disappear4 ,i avow1 moreover1 that ,i find fewer real inconveniences in extending1 a little beyond what would be strictly necessary1 the number of these geometrical notions thus established by direct observation1 provided they are sufficiently simple1 than in making them the subjects of complicated and indirect demonstrations1 even when these demonstrations may be logically irreproachable4 ,the true dogmatic destination of the geometry of the ancients1 reduced to its least possible indispensable developments1 having thus been characterized as exactly as possible1 it is proper to consider summarily each of the principal parts of which it must be composed4 ,i think that ,i may here limit myself to considering the first and the most extensive of these parts1 that which has for its object the study of the right line2 the two other sections1 namely1 the quadrature of polygons and the cubature of polyhedrons1 from their limited extent1 not being capable of giving rise to any philosophical consideration of any importance1 distinct from those indicated in the preceding chapter with respect to the measure of areas and of volumes in general4 ,,geometry ,,of ,,the ,,right ,,line4 ,the final question which we always have in view in the study of the right line1 properly consists in determining1 by means of one another1 the different elements of any right-lined figure whatever2 which enables us always to know indirectly the length and position of a right line1 in whatever circumstances it may be placed4 ,this fundamental problem is susceptible of two general solutions1 the nature of which is quite distinct1 the one graphical1 the other algebraic4 ,the first1 though very imperfect1 is that which must be first considered1 because it is spontaneously derived from the direct study of the subject2 the second1 much more perfect in the most important respects1 cannot be studied till afterwards1 because it is founded upon the previous knowledge of the other4 ,,graphical ,,solutions4 ,the graphical solution consists in constructing at will the proposed figure1 either with the same dimensions1 or1 more usually1 with dimensions changed in any ratio whatever4 ,the first mode need merely be mentioned as being the most simple and the one which would first occur to the mind1 for it is evidently1 by its nature1 almost entirely incapable of application4 ,the second is1 on the contrary1 susceptible of being most extensively and most usefully applied4 ,we still make an important and continual use of it at the present day1 not only to represent with exactness the forms of bodies and their relative positions1 but even for the actual determination of geometrical magnitudes1 when we do not need great precision4 ,the ancients1 in consequence of the imperfection of their geometrical knowledge1 employed this procedure in a much more extensive manner1 since it was for a long time the only one which they could apply1 even in the most important precise determinations4 ,it was thus1 for example1 that ,aristarchus of ,samos estimated the relative distance from the sun and from the moon to the earth1 by making measurements on a triangle constructed as exactly as possible1 so as to be similar to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature1 and when an observation of the angle at the earth would consequently be sufficient to define the triangle4 ,archimedes himself1 although he was the first to introduce calculated determinations into geometry1 several times employed similar means4 ,the formation of trigonometry did not cause this method to be entirely abandoned1 although it greatly diminished its use2 the ,greeks and the ,arabians continued to employ it for a great number of researches1 in which we now regard the use of the calculus as indispensable4 ,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 plane4 ,but if we suppose1 as most frequently happens1 that they are situated in different planes1 we see1 then1 a new order of geometrical considerations arise4 ,the artificial figure1 which is constantly plane1 not being capable1 in that case1 of being a perfectly faithful image of the real figure1 it is necessary previously to fix with precision the mode of representation1 which gives rise to different systems of ,projection4 ,it then remains to be determined according to what laws the geometrical phenomena correspond in the two figures4 ,this consideration generates a new series of geometrical investigations1 the final object of which is properly to discover how we can replace constructions in relief by plane constructions4 ,the ancients had to resolve several elementary questions of this kind for various cases in which we now employ spherical trigonometry1 principally for different problems relating to the celestial sphere4 ,such was the object of their analemmas1 and of the other plane figures which for a long time supplied the place of the calculus4 ,we see by this that the ancients really knew the elements of what we now name ,descriptive ,geometry1 although they did not conceive it in a distinct and general manner4 ,i think it proper briefly to indicate in this place the true philosophical character of this ,descriptive ,geometry2 although1 being essentially a science of application1 it ought not to be included within the proper domain of this work4 ,,descriptive ,,geometry4 ,all questions of geometry of three dimensions necessarily give rise1 when we consider their graphical solution1 to a common difficulty which is peculiar to them2 that of substituting for the different constructions in relief1 which are necessary to resolve them directly1 and which it is almost always impossible to execute1 simple equivalent plane constructions1 by means of which we finally obtain the same results4 ,without this indispensable transformation1 every solution of this kind would be evidently incomplete and really inapplicable in practice1 although theoretically the constructions in space are usually preferable as being more direct4 ,it was in order to furnish general means for always effecting such a transformation that ,descriptive ,geometry was created1 and formed into a distinct and homogeneous system1 by the illustrious ,,monge4 ,he invented1 in the first place1 a uniform method of representing bodies by figures traced on a single plane1 by the aid of projections on two different planes1 usually perpendicular to each other1 and one of which is supposed to turn about their common intersection so as to coincide with the other produced2 in this system1 or in any other equivalent to it1 it is sufficient to regard points and lines as being determined by their projections1 and surfaces by the projections of their generating lines4 ,this being established1 ,monge--analyzing with profound sagacity the various partial labours of this kind which had before been executed by a number of incongruous procedures1 and considering also1 in a general and direct manner1 in what any questions of that nature must consist--found that they could always be reduced to a very small number of invariable abstract problems1 capable of being resolved separately1 once for all1 by uniform operations1 relating essentially some to the contacts and others to the intersections of surfaces4 ,simple and entirely general methods for the graphical solution of these two orders of problems having been formed1 all the geometrical questions which may arise in any of the various arts of construction--stone-cutting1 carpentry1 perspective1 dialling1 fortification1 c4--can henceforth be treated as simple particular cases of a single theory1 the invariable application of which will always necessarily lead to an exact solution1 which may be facilitated in practice by profiting by the peculiar circumstances of each case4 99 99 99 99 99 ,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 step1 and thus far the only really complete one1 towards that general renovation of human labours1 which must imprint upon all our arts a character of precision and of rationality1 so necessary to their future progress4 ,such a revolution must1 in fact1 inevitably commence with that class of industrial labours1 which is essentially connected with that science which is the most simple1 the most perfect1 and the most ancient4 ,it cannot fail to extend hereafter1 though with less facility1 to all other practical operations4 ,indeed ,monge himself1 who conceived the true philosophy of the arts better than any one else1 endeavoured to sketch out a corresponding system for the mechanical arts4 ,essential as the conception of descriptive geometry really is1 it is very important not to deceive ourselves with respect to its true destination1 as did those who1 in the excitement of its first discovery1 saw in it a means of enlarging the general and abstract domain of rational geometry4 ,the result has in no way answered to these mistaken hopes4 ,and1 indeed1 is it not evident that descriptive geometry has no special value except as a science of application1 and as forming the true special theory of the geometrical arts8 ,considered in its abstract relations1 it could not introduce any truly distinct order of geometrical speculations4 ,we must not forget that1 in order that a geometrical question should fall within the peculiar domain of descriptive geometry1 it must necessarily have been previously resolved by speculative geometry1 the solutions of which then1 as we have seen1 always need to be prepared for practice in such a way as to supply the place of constructions in relief by plane constructions2 a substitution which really constitutes the only characteristic function of descriptive geometry4 ,it is proper1 however1 to remark here1 that1 with regard to intellectual education1 the study of descriptive geometry possesses an important philosophical peculiarity1 quite independent of its high industrial utility4 ,this is the advantage which it so pre-eminently offers--in habituating the mind to consider very complicated geometrical combinations in space1 and to follow with precision their continual correspondence with the figures which are actually traced--of thus exercising to the utmost1 in the most certain and precise manner1 that important faculty of the human mind which is properly called imagination1 and which consists1 in its elementary and positive acceptation1 in representing to ourselves1 clearly and easily1 a vast and variable collection of ideal objects1 as if they were really before us4 ,finally1 to complete the indication of the general nature of descriptive geometry by determining its logical character1 we have to observe that1 while it belongs to the geometry of the ancients by the character of its solutions1 on the other hand it approaches the geometry of the moderns by the nature of the questions which compose it4 ,these questions are in fact eminently remarkable for that generality which1 as we saw in the preceding chapter1 constitutes the true fundamental character of modern geometry2 for the methods used are always conceived as applicable to any figures whatever1 the peculiarity of each having only a purely secondary influence4 ,the solutions of descriptive geometry are then graphical1 like most of those of the ancients1 and at the same time general1 like those of the moderns4 99 99 99 99 99 ,after this important digression1 we will pursue the philosophical examination of special geometry1 always considered as reduced to its least possible development1 as an indispensable introduction to general geometry4 ,we have now sufficiently considered the graphical solution of the fundamental problem relating to the right line--that is1 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 algebraic solution4 ,,algebraic ,,solutions4 ,this kind of solution1 the evident superiority of which need not here be dwelt upon1 belongs necessarily1 by the very nature of the question1 to the system of the ancient geometry1 although the logical method which is employed causes it to be generally1 but very improperly1 separated from it4 ,we have thus an opportunity of verifying1 in a very important respect1 what was established generally in the preceding chapter1 that it is not by the employment of the calculus that the modern geometry is essentially to be distinguished from the ancient4 ,the ancients are in fact the true inventors of the present trigonometry1 spherical as well as rectilinear2 it being only much less perfect in their hands1 on account of the extreme inferiority of their algebraical knowledge4 ,it is1 then1 really in this chapter1 and not1 as it might at first be thought1 in those which we shall afterwards devote to the philosophical examination of general geometry1 that it is proper to consider the character of this important preliminary theory1 which is usually1 though improperly1 included in what is called analytical geometry1 but which is really only a complement of elementary geometry properly so called4 ,since all right-lined figures can be decomposed into triangles1 it is evidently sufficient to know how to determine the different elements of a triangle by means of one another1 which reduces polygonometry to simple trigonometry4 ,,trigonometry4 ,the difficulty in resolving algebraically such a question as the above1 consists essentially in forming1 between the angles and the sides of a triangle1 three distinct equations2 which1 when once obtained1 will evidently reduce all trigonometrical problems to mere questions of analysis4 ,how to introduce ,angles4 ,in considering the establishment of these equations in the most general manner1 we immediately meet with a fundamental distinction with respect to the manner of introducing the angles into the calculation1 according as they are made to enter directly1 by themselves or by the circular arcs which are proportional to them2 or indirectly1 by the chords of these arcs1 which are hence called their trigonometrical lines4 ,of these two systems of trigonometry the second was of necessity the only one originally adopted1 as being the only practicable one1 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 angles1 while it would have been absolutely impossible at that epoch to establish equations between the sides and the angles themselves4 ,advantages of introducing ,trigonometrical ,lines4 ,at the present day1 since the solution can be obtained by either system indifferently1 that motive for preference no longer exists2 but geometers have none the less persisted in following from choice the system primitively admitted from necessity2 for1 the same reason which enabled these trigonometrical equations to be obtained with much more facility1 must1 in like manner1 as it is still more easy to conceive priori1 render these equations much more simple1 since they then exist only between right lines1 instead of being established between right lines and arcs of circles4 ,such a consideration has so much the more importance1 as the question relates to formulas which are eminently elementary1 and destined to be continually employed in all parts of mathematical science1 as well as in all its various applications4 ,it may be objected1 however1 that when an angle is given1 it is1 in reality1 always given by itself1 and not by its trigonometrical lines2 and that when it is unknown1 it is its angular value which is properly to be determined1 and not that of any of its trigonometrical lines4 ,it seems1 according to this1 that such lines are only useless intermediaries between the sides and the angles1 which have to be finally eliminated1 and the introduction of which does not appear capable of simplifying the proposed research4 ,it is indeed important to explain1 with more generality and precision than is customary1 the great real utility of this manner of proceeding4 ,division of ,trigonometry into two ,parts4 ,it consists in the fact that the introduction of these auxiliary magnitudes divides the entire question of trigonometry into two others essentially distinct1 one of which has for its object to pass from the angles to their trigonometrical lines1 or the converse1 and the other of which proposes to determine the sides of the triangles by the trigonometrical lines of their angles1 or the converse4 ,now the first of these two fundamental questions is evidently susceptible1 by its nature1 of being entirely treated and reduced to numerical tables once for all1 in considering all possible angles1 since it depends only upon those angles1 and not at all upon the particular triangles in which they may enter in each case2 while the solution of the second question must necessarily be renewed1 at least in its arithmetical relations1 for each new triangle which it is necessary to resolve4 ,this is the reason why the first portion of the complete work1 which would be precisely the most laborious1 is no longer taken into the account1 being always done in advance2 while1 if such a decomposition had not been performed1 we would evidently have found ourselves under the obligation of recommencing the entire calculation in each particular case4 ,such is the essential property of the present trigonometrical system1 which in fact would really present no actual advantage1 if it was necessary to calculate continually the trigonometrical line of each angle to be considered1 or the converse2 the intermediate agency introduced would then be more troublesome than convenient4 ,in order to clearly comprehend the true nature of this conception1 it will be useful to compare it with a still more important one1 designed to produce an analogous effect either in its algebraic1 or1 still more1 in its arithmetical relations--the admirable theory of logarithms4 ,in examining in a philosophical manner the influence of this theory1 we see in fact that its general result is to decompose all imaginable arithmetical operations into two distinct parts4 ,the first and most complicated of these is capable of being executed in advance once for all 7since it depends only upon the numbers to be considered1 and not at all upon the infinitely different combinations into which they can enter71 and consists in considering all numbers as assignable powers of a constant number4 ,the second part of the calculation1 which must of necessity be recommenced for each new formula which is to have its value determined1 is thenceforth reduced to executing upon these exponents correlative operations which are infinitely more simple4 ,i confine myself here to merely indicating this resemblance1 which any one can carry out for himself4 ,we must besides observe1 as a property 7secondary at the present day1 but all-important at its origin7 of the trigonometrical system adopted1 the very remarkable circumstance that the determination of angles by their trigonometrical lines1 or the converse1 admits of an arithmetical solution 7the only one which is directly indispensable for the special destination of trigonometry7 without the previous resolution of the corresponding algebraic question4 ,it is doubtless to such a peculiarity that the ancients owed the possibility of knowing trigonometry4 ,the investigation conceived in this way was so much the more easy1 inasmuch as tables of chords 7which the ancients naturally took as the trigonometrical lines7 had been previously constructed for quite a different object1 in the course of the labours of ,archimedes on the rectification of the circle1 from which resulted the actual determination of a certain series of chords2 so that when ,hipparchus subsequently invented trigonometry1 he could confine himself to completing that operation by suitable intercalations2 which shows clearly the connexion of ideas in that matter4 ,the ,increase of such ,trigonometrical ,lines4 ,to complete this philosophical sketch of trigonometry1 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 lines1 with the view of simplifying our equations1 must also lead us to employ concurrently several trigonometrical lines1 instead of confining ourselves to one only 7as did the ancients71 so as to perfect this system by choosing that one which will be algebraically the most convenient on each occasion4 ,in this point of view1 it is clear that the number of these lines is in itself no ways limited2 provided that they are determined by the arc1 and that they determine it1 whatever may be the law according to which they are derived from it1 they are suitable to be substituted for it in the equations4 ,the ,arabians1 and subsequently the moderns1 in confining themselves to the most simple constructions1 have carried to four or five the number of direct trigonometrical lines1 which might be extended much farther4 ,but instead of recurring to geometrical formations1 which would finally become very complicated1 we conceive with the utmost facility as many new trigonometrical lines as the analytical transformations may require1 by means of a remarkable artifice1 which is not usually apprehended in a sufficiently general manner4 ,it consists in not directly multiplying the trigonometrical lines appropriate to each arc considered1 but in introducing new ones1 by considering this arc as indirectly determined by all lines relating to an arc which is a very simple function of the first4 ,it is thus1 for example1 that1 in order to calculate an angle with more facility1 we will determine1 instead of its sine1 the sine of its half1 or of its double1 c4 ,such a creation of indirect trigonometrical lines is evidently much more fruitful than all the direct geometrical methods for obtaining new ones4 ,we may accordingly say that the number of trigonometrical lines actually employed at the present day by geometers is in reality unlimited1 since at every instant1 so to say1 the transformations of analysis may lead us to augment it by the method which ,i have just indicated4 ,special names1 however1 have been given to those only of these indirect lines which refer to the complement of the primitive arc1 the others not occurring sufficiently often to render such denominations necessary2 a circumstance which has caused a common misconception of the true extent of the system of trigonometry4 ,study of their ,mutual ,relations4 ,this multiplicity of trigonometrical lines evidently gives rise to a third fundamental question in trigonometry1 the study of the relations which exist between these different lines2 since1 without such a knowledge1 we could not make use1 for our analytical necessities1 of this variety of auxiliary magnitudes1 which1 however1 have no other destination4 ,it is clear1 besides1 from the consideration just indicated1 that this essential part of trigonometry1 although simply preparatory1 is1 by its nature1 susceptible of an indefinite extension when we view it in its entire generality1 while the two others are circumscribed within rigorously defined limits4 ,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 subject2 for the third is evidently independent of the two others1 and the second1 of that which was first presented--the resolution of triangles1 properly so called--which must for that reason be treated in the last place2 which rendered so much the more important the consideration of their natural succession and logical relations to one another4 ,it is useless to consider here separately spherical trigonometry1 which cannot give rise to any special philosophical consideration2 since1 essential as it is by the importance and the multiplicity of its uses1 it can be treated at the present day only as a simple application of rectilinear trigonometry1 which furnishes directly its fundamental equations1 by substituting for the spherical triangle the corresponding trihedral angle4 ,this summary exposition of the philosophy of trigonometry has been here given in order to render apparent1 by an important example1 that rigorous dependence and those successive ramifications which are presented by what are apparently the most simple questions of elementary geometry4 99 99 99 99 99 ,having thus examined the peculiar character of special geometry reduced to its only dogmatic destination1 that of furnishing to general geometry an indispensable preliminary basis1 we have now to give all our attention to the true science of geometry1 considered as a whole1 in the most rational manner4 ,for that purpose1 it is necessary to carefully examine the great original idea of ,descartes1 upon which it is entirely founded4 ,this will be the object of the following chapter4 ,,chapter ,,iii4 ,,modern ,,or ,,analytical ,,geometry4 ,general 7or ,analytical7 geometry being entirely founded upon the transformation of geometrical considerations into equivalent analytical considerations1 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-relation4 ,besides its own extreme importance as a means of highly perfecting geometrical science1 or1 rather1 of establishing the whole of it on rational bases1 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 mathematics1 by the analytical representation of natural phenomena4 ,there is no conception1 in the whole philosophy of mathematics which better deserves to fix all our attention4 ,,analytical ,,representation ,,of ,,figures4 ,in order to succeed in expressing all imaginable geometrical phenomena by simple analytical relations1 we must evidently1 in the first place1 establish a general method for representing analytically the subjects themselves in which these phenomena are found1 that is1 the lines or the surfaces to be considered4 ,the subject being thus habitually considered in a purely analytical point of view1 we see how it is thenceforth possible to conceive in the same manner the various accidents of which it is susceptible4 ,in order to organize the representation of geometrical figures by analytical equations1 we must previously surmount a fundamental difficulty2 that of reducing the general elements of the various conceptions of geometry to simply numerical ideas2 in a word1 that of substituting in geometry pure considerations of quantity for all considerations of quality4 ,reduction of ,figure to ,position4 ,for this purpose let us observe1 in the first place1 that all geometrical ideas relate necessarily to these three universal categories3 the magnitude1 the figure1 and the position of the extensions to be considered4 ,as to the first1 there is evidently no difficulty2 it enters at once into the ideas of numbers4 ,with relation to the second1 it must be remarked that it will always admit of being reduced to the third4 ,for the figure of a body evidently results from the mutual position of the different points of which it is composed1 so that the idea of position necessarily comprehends that of figure1 and every circumstance of figure can be translated by a circumstance of position4 ,it is in this way1 in fact1 that the human mind has proceeded in order to arrive at the analytical representation of geometrical figures1 their conception relating directly only to positions4 ,all the elementary difficulty is then properly reduced to that of referring ideas of situation to ideas of magnitude4 ,such is the direct destination of the preliminary conception upon which ,descartes has established the general system of analytical geometry4 ,his philosophical labour1 in this relation1 has consisted simply in the entire generalization of an elementary operation1 which we may regard as natural to the human mind1 since it is performed spontaneously1 so to say1 in all minds1 even the most uncultivated4 ,thus1 when we have to indicate the situation of an object without directly pointing it out1 the method which we always adopt1 and evidently the only one which can be employed1 consists in referring that object to others which are known1 by assigning the magnitude of the various geometrical elements1 by which we conceive it connected with the known objects4 ,these elements constitute what ,descartes1 and after him all geometers1 have called the co-ordinates of each point considered4 ,they are necessarily two in number1 if it is known in advance in what plane the point is situated2 and three1 if it may be found indifferently in any region of space4 ,as many different constructions as can be imagined for determining the position of a point1 whether on a plane or in space1 so many distinct systems of co-ordinates may be conceived2 they are consequently susceptible of being multiplied to infinity4 ,but1 whatever may be the system adopted1 we shall always have reduced the ideas of situation to simple ideas of magnitude1 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-ordinates4 ,determination of the ,position of a ,point4 ,considering at first only the least complicated case1 that of plane geometry1 it is in this way that we usually determine the position of a point on a plane1 by its distances from two fixed right lines considered as known1 which are called axes1 and which are commonly supposed to be perpendicular to each other4 ,this system is that most frequently adopted1 because of its simplicity2 but geometers employ occasionally an infinity of others4 ,thus the position of a point on a plane may be determined1 #a1 by its distances from two fixed points2 or1 #b1 by its distance from a single fixed point1 and the direction of that distance1 estimated by the greater or less angle which it makes with a fixed right line1 which constitutes the system of what are called polar co-ordinates1 the most frequently used after the system first mentioned2 or1 #c1 by the angles which the right lines drawn from the variable point to two fixed points make with the right line which joins these last2 or1 #d1 by the distances from that point to a fixed right line and a fixed point1 c4 ,in a word1 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 employed4 ,a general observation1 which it is important to make in this connexion1 is1 that every system of co-ordinates is equivalent to determining a point1 in plane geometry1 by the intersection of two lines1 each of which is subjected to certain fixed conditions of determination2 a single one of these conditions remaining variable1 sometimes the one1 sometimes the other1 according to the system considered4 ,we could not1 indeed1 conceive any other means of constructing a point than to mark it by the meeting of two lines4 ,thus1 in the most common system1 that of rectilinear co-ordinates1 properly so called1 the point is determined by the intersection of two right lines1 each of which remains constantly parallel to a fixed axis1 at a greater or less distance from it2 in the polar system1 the position of the point is marked by the meeting of a circle1 of variable radius and fixed centre1 with a movable right line compelled to turn about this centre3 in other systems1 the required point might be designated by the intersection of two circles1 or of any other two lines1 c4 ,in a word1 to assign the value of one of the co-ordinates of a point in any system whatever1 is always necessarily equivalent to determining a certain line on which that point must be situated4 ,the geometers of antiquity had already made this essential remark1 which served as the base of their method of geometrical loci1 of which they made so happy a use to direct their researches in the resolution of determinate problems1 in considering separately the influence of each of the two conditions by which was defined each point constituting the object1 direct or indirect1 of the proposed question4 ,it was the general systematization of this method which was the immediate motive of the labours of ,descartes1 which led him to create analytical geometry4 ,after having clearly established this preliminary conception--by means of which ideas of position1 and thence1 implicitly1 all elementary geometrical conceptions are capable of being reduced to simple numerical considerations--it is easy to form a direct conception1 in its entire generality1 of the great original idea of ,descartes1 relative to the analytical representation of geometrical figures3 it is this which forms the special object of this chapter4 ,i will continue to consider at first1 for more facility1 only geometry of two dimensions1 which alone was treated by ,descartes2 and will afterwards examine separately1 under the same point of view1 the theory of surfaces and curves of double curvature4 ,,plane ,,curves4 ,expression of ,lines by ,equations4 ,in accordance with the manner of expressing analytically the position of a point on a plane1 it can be easily established that1 by whatever property any line may be defined1 that definition always admits of being replaced by a corresponding equation between the two variable co-ordinates of the point which describes this line2 an equation which will be thenceforth the analytical representation of the proposed line1 every phenomenon of which will be translated by a certain algebraic modification of its equation4 ,thus1 if we suppose that a point moves on a plane without its course being in any manner determined1 we shall evidently have to regard its co-ordinates1 to whatever system they may belong1 as two variables entirely independent of one another4 ,but if1 on the contrary1 this point is compelled to describe a certain line1 we shall necessarily be compelled to conceive that its co-ordinates1 in all the positions which it can take1 retain a certain permanent and precise relation to each other1 which is consequently susceptible of being expressed by a suitable equation2 which will become the very clear and very rigorous analytical definition of the line under consideration1 since it will express an algebraical property belonging exclusively to the co-ordinates of all the points of this line4 ,it is clear1 indeed1 that when a point is not subjected to any condition1 its situation is not determined except in giving at once its two co-ordinates1 independently of each other2 while1 when the point must continue upon a defined line1 a single co-ordinate is sufficient for completely fixing its position4 ,the second co-ordinate is then a determinate function of the first2 or1 in other words1 there must exist between them a certain equation1 of a nature corresponding to that of the line on which the point is compelled to remain4 ,in a word1 each of the co-ordinates of a point requiring it to be situated on a certain line1 we conceive reciprocally that the condition1 on the part of a point1 of having to belong to a line defined in any manner whatever1 is equivalent to assigning the value of one of the two co-ordinates2 which is found in that case to be entirely dependent on the other4 ,the analytical relation which expresses this dependence may be more or less difficult to discover1 but it must evidently be always conceived to exist1 even in the cases in which our present means may be insufficient to make it known4 ,it is by this simple consideration that we may demonstrate1 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 equations4 ,expression of ,equations by ,lines4 ,taking up again the same reflections in the inverse direction1 we could show as easily the geometrical necessity of the representation of every equation of two variables1 in a determinate system of co-ordinates1 by a certain line2 of which such a relation would be1 in the absence of any other known property1 a very characteristic definition1 the scientific destination of which will be to fix the attention directly upon the general course of the solutions of the equation1 which will thus be noted in the most striking and the most simple manner4 ,this picturing of equations is one of the most important fundamental advantages of analytical geometry1 which has thereby reacted in the highest degree upon the general perfecting of analysis itself2 not only by assigning to purely abstract researches a clearly determined object and an inexhaustible career1 but1 in a still more direct relation1 by furnishing a new philosophical medium for analytical meditation which could not be replaced by any other4 ,in fact1 the purely algebraic discussion of an equation undoubtedly makes known its solutions in the most precise manner1 but in considering them only one by one1 so that in this way no general view of them could be obtained1 except as the final result of a long and laborious series of numerical comparisons4 ,on the other hand1 the geometrical locus of the equation1 being only designed to represent distinctly and with perfect clearness the summing up of all these comparisons1 permits it to be directly considered1 without paying any attention to the details which have furnished it4 ,it can thereby suggest to our mind general analytical views1 which we should have arrived at with much difficulty in any other manner1 for want of a means of clearly characterizing their object4 ,it is evident1 for example1 that the simple inspection of the logarithmic curve1 or of the curve y sin4 x1 makes us perceive much more distinctly the general manner of the variations of logarithms with respect to their numbers1 or of sines with respect to their arcs1 than could the most attentive study of a table of logarithms or of natural sines4 ,it is well known that this method has become entirely elementary at the present day1 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 whatever4 ,any ,change in the ,line causes a ,change in the ,equation4 ,returning to the representation of lines by equations1 which is our principal object1 we see that this representation is1 by its nature1 so faithful1 that the line could not experience any modification1 however slight it might be1 without causing a corresponding change in the equation4 ,this perfect exactitude even gives rise oftentimes to special difficulties2 for since1 in our system of analytical geometry1 the mere displacements of lines affect the equations1 as well as their real variations in magnitude or form1 we should be liable to confound them with one another in our analytical expressions1 if geometers had not discovered an ingenious method designed expressly to always distinguish them4 ,this method is founded on this principle1 that although it is impossible to change analytically at will the position of a line with respect to the axes of the co-ordinates1 we can change in any manner whatever the situation of the axes themselves1 which evidently amounts to the same2 then1 by the aid of the very simple general formula by which this transformation of the axes is produced1 it becomes easy to discover whether two different equations are the analytical expressions of only the same line differently situated1 or refer to truly distinct geometrical loci2 since1 in the former case1 one of them will pass into the other by suitably changing the axes or the other constants of the system of co-ordinates employed4 ,it must1 moreover1 be remarked on this subject1 that general inconveniences of this nature seem to be absolutely inevitable in analytical geometry2 for1 since the ideas of position are1 as we have seen1 the only geometrical ideas immediately reducible to numerical considerations1 and the conceptions of figure cannot be thus reduced1 except by seeing in them relations of situation1 it is impossible for analysis to escape confounding1 at first1 the phenomena of figure with simple phenomena of position1 which alone are directly expressed by the equations4 ,every ,definition of a ,line is an ,equation4 ,in order to complete the philosophical explanation of the fundamental conception which serves as the base of analytical geometry1 ,i think that ,i should here indicate a new general consideration1 which seems to me particularly well adapted for putting in the clearest point of view this necessary representation of lines by equations with two variables4 ,it consists in this1 that not only1 as we have shown1 must every defined line necessarily give rise to a certain equation between the two co-ordinates of any one of its points1 but1 still farther1 every definition of a line may be regarded as being already of itself an equation of that line in a suitable system of co-ordinates4 ,it is easy to establish this principle1 first making a preliminary logical distinction with respect to different kinds of definitions4 ,the rigorously indispensable condition of every definition is that of distinguishing the object defined from all others1 by assigning to it a property which belongs to it exclusively4 ,but this end may be generally attained in two very different ways2 either by a definition which is simply characteristic1 that is1 indicative of a property which1 although truly exclusive1 does not make known the mode of generation of the object2 or by a definition which is really explanatory1 that is1 which characterizes the object by a property which expresses one of its modes of generation4 ,for example1 in considering the circle as the line1 which1 under the same contour1 contains the greatest area1 we have evidently a definition of the first kind2 while in choosing the property of its having all its points equally distant from a fixed point1 we have a definition of the second kind4 ,it is1 besides1 evident1 as a general principle1 that even when any object whatever is known at first only by a characteristic definition1 we ought1 nevertheless1 to regard it as susceptible of explanatory definitions1 which the farther study of the object would necessarily lead us to discover4 ,this being premised1 it is clear that the general observation above made1 which represents every definition of a line as being necessarily an equation of that line in a certain system of co-ordinates1 cannot apply to definitions which are simply characteristic2 it is to be understood only of definitions which are truly explanatory4 ,but1 in considering only this class1 the principle is easy to prove4 ,in fact1 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 rotation1 into which the motion of the point which describes it will be decomposed at each instant4 ,now if we form the most general conception of what constitutes a system of co-ordinates1 and admit all possible systems1 it is clear that such a relation will be nothing else but the equation of the proposed line1 in a system of co-ordinates of a nature corresponding to that of the mode of generation considered4 ,thus1 for example1 the common definition of the circle may evidently be regarded as being immediately the polar equation of this curve1 taking the centre of the circle for the pole4 ,in the same way1 the elementary definition of the ellipse or of the hyperbola--as being the curve generated by a point which moves in such a manner that the sum or the difference of its distances from two fixed points remains constant--gives at once1 for either the one or the other curve1 the equation y x c1 taking for the system of co-ordinates that in which the position of a point would be determined by its distances from two fixed points1 and choosing for these poles the two given foci4 ,in like manner1 the common definition of any cycloid would furnish directly1 for that curve1 the equation y mx2 adopting as the co-ordinates of each point the arc which it marks upon a circle of invariable radius1 measuring from the point of contact of that circle with a fixed line1 and the rectilinear distance from that point of contact to a certain origin taken on that right line4 ,we can make analogous and equally easy verifications with respect to the customary definitions of spirals1 of epicycloids1 c4 ,we shall constantly find that there exists a certain system of co-ordinates1 in which we immediately obtain a very simple equation of the proposed line1 by merely writing algebraically the condition imposed by the mode of generation considered4 ,besides its direct importance as a means of rendering perfectly apparent the necessary representation of every line by an equation1 the preceding consideration seems to me to possess a true scientific utility1 in characterizing with precision the principal general difficulty which occurs in the actual establishment of these equations1 and in consequently furnishing an interesting indication with respect to the course to be pursued in inquiries of this kind1 which1 by their nature1 could not admit of complete and invariable rules4 ,in fact1 since any definition whatever of a line1 at least among those which indicate a mode of generation1 furnishes directly the equation of that line in a certain system of co-ordinates1 or1 rather1 of itself constitutes that equation1 it follows that the difficulty which we often experience in discovering the equation of a curve1 by means of certain of its characteristic properties1 a difficulty which is sometimes very great1 must proceed essentially only from the commonly imposed condition of expressing this curve analytically by the aid of a designated system of co-ordinates1 instead of admitting indifferently all possible systems4 ,these different systems cannot be regarded in analytical geometry as being all equally suitable2 for various reasons1 the most important of which will be hereafter discussed1 geometers think that curves should almost always be referred1 as far as is possible1 to rectilinear co-ordinates1 properly so called4 ,now we see1 from what precedes1 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 definition4 ,the principal difficulty presented by the formation of the equation of a line really consists1 then1 in general1 in a certain transformation of co-ordinates4 ,it is undoubtedly true that this consideration does not subject the establishment of these equations to a truly complete general method1 the success of which is always certain2 which1 from the very nature of the subject1 is evidently chimerical3 but such a view may throw much useful light upon the course which it is proper to adopt1 in order to arrive at the end proposed4 ,thus1 after having in the first place formed the preparatory equation1 which is spontaneously derived from the definition which we are considering1 it will be necessary1 in order to obtain the equation belonging to the system of co-ordinates which must be finally admitted1 to endeavour to express in a function of these last co-ordinates those which naturally correspond to the given mode of generation4 ,it is upon this last labour that it is evidently impossible to give invariable and precise precepts4 ,we can only say that we shall have so many more resources in this matter as we shall know more of true analytical geometry1 that is1 as we shall know the algebraical expression of a greater number of different algebraical phenomena4 ,,choice ,,of ,,co-,,ordinates4 ,in order to complete the philosophical exposition of the conception which serves as the base of analytical geometry1 ,i have yet to notice the considerations relating to the choice of the system of co-ordinates which is in general the most suitable4 ,they will give the rational explanation of the preference unanimously accorded to the ordinary rectilinear system2 a preference which has hitherto been rather the effect of an empirical sentiment of the superiority of this system1 than the exact result of a direct and thorough analysis4 ,two different ,points of ,view4 ,in order to decide clearly between all the different systems of co-ordinates1 it is indispensable to distinguish with care the two general points of view1 the converse of one another1 which belong to analytical geometry2 namely1 the relation of algebra to geometry1 founded upon the representation of lines by equations2 and1 reciprocally1 the relation of geometry to algebra1 founded on the representation of equations by lines4 ,it is evident that in every investigation of general geometry these two fundamental points of view are of necessity always found combined1 since we have always to pass alternately1 and at insensible intervals1 so to say1 from geometrical to analytical considerations1 and from analytical to geometrical considerations4 ,but the necessity of here temporarily separating them is none the less real2 for the answer to the question of method which we are examining is1 in fact1 as we shall see presently1 very far from being the same in both these relations1 so that without this distinction we could not form any clear idea of it4 #a4 ,representation of ,lines by ,equations4 ,under the first point of view--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 line1 and greater facility in arriving at it4 ,now it is easy to see that there does not exist1 and could not be expected to exist1 any system of co-ordinates deserving in that respect a constant preference over all others4 ,in fact1 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 once1 and is necessarily found to be also very simple2 and this system1 moreover1 inevitably varies with the nature of the characteristic property under consideration4 ,the rectilinear system could not1 therefore1 be constantly the most advantageous for this object1 although it may often be very favourable2 there is probably no system which1 in certain particular cases1 should not be preferred to it1 as well as to every other4 #b4 ,representation of ,equations by ,lines4 ,it is by no means so1 however1 under the second point of view4 ,we can1 indeed1 easily establish1 as a general principle1 that the ordinary rectilinear system must necessarily be better adapted than any other to the representation of equations by the corresponding geometrical loci2 that is to say1 that this representation is constantly more simple and more faithful in it than in any other4 ,let us consider1 for this object1 that1 since every system of co-ordinates consists in determining a point by the intersection of two lines1 the system adapted to furnish the most suitable geometrical loci must be that in which these two lines are the simplest possible2 a consideration which confines our choice to the rectilinear system4 ,in truth1 there is evidently an infinite number of systems which deserve that name1 that is to say1 which employ only right lines to determine points1 besides the ordinary system which assigns the distances from two fixed lines as co-ordinates2 such1 for example1 would be that in which the co-ordinates of each point should be the two angles which the right lines1 which go from that point to two fixed points1 make with the right line1 which joins these last points3 so that this first consideration is not rigorously sufficient to explain the preference unanimously given to the common system4 ,but in examining in a more thorough manner the nature of every system of co-ordinates1 we also perceive that each of the two lines1 whose meeting determines the point considered1 must necessarily offer at every instant1 among its different conditions of determination1 a single variable condition1 which gives rise to the corresponding co-ordinate1 all the rest being fixed1 and constituting the axes of the system1 taking this term in its most extended mathematical acceptation4 ,the variation is indispensable1 in order that we may be able to consider all possible positions2 and the fixity is no less so1 in order that there may exist means of comparison4 ,thus1 in all rectilinear systems1 each of the two right lines will be subjected to a fixed condition1 and the ordinate will result from the variable condition4 ,superiority of rectilinear ,co-ordinates4 ,from these considerations it is evident1 as a general principle1 that the most favourable system for the construction of geometrical loci will necessarily be that in which the variable condition of each right line shall be the simplest possible2 the fixed condition being left free to be made complex1 if necessary to attain that object4 ,now1 of all possible manners of determining two movable right lines1 the easiest to follow geometrically is certainly that in which1 the direction of each right line remaining invariable1 it only approaches or recedes1 more or less1 to or from a constant axis4 ,it would be1 for example1 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 lines1 which each turn around a fixed point1 making a greater or smaller angle with a certain axis1 as in the system of co-ordinates previously noticed4 ,such is the true general explanation of the fundamental property possessed by the common rectilinear system1 of being better adapted than any other to the geometrical representation of equations1 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-ordinates4 ,in order to feel clearly all the force of this consideration1 it would be sufficient to carefully compare this system with the polar system1 in which this geometrical image1 so simple and so easy to follow1 of two right lines moving parallel1 each one of them1 to its corresponding axis1 is replaced by the complicated picture of an infinite series of concentric circles1 cut by a right line compelled to turn about a fixed point4 ,it is1 moreover1 easy to conceive in advance what must be the extreme importance to analytical geometry of a property so profoundly elementary1 which1 for that reason1 must be recurring at every instant1 and take a progressively increasing value in all labours of this kind4 ,perpendicularity of the ,axes4 ,in pursuing farther the consideration which demonstrates the superiority of the ordinary system of co-ordinates over any other as to the representation of equations1 we may also take notice of the utility for this object of the common usage of taking the two axes perpendicular to each other1 whenever possible1 rather than with any other inclination4 ,as regards the representation of lines by equations1 this secondary circumstance is no more universally proper than we have seen the general nature of the system to be2 since1 according to the particular occasion1 any other inclination of the axes may deserve our preference in that respect4 ,but1 in the inverse point of view1 it is easy to see that rectangular axes constantly permit us to represent equations in a more simple and even more faithful manner2 for1 with oblique axes1 space being divided by them into regions which no longer have a perfect identity1 it follows that1 if the geometrical locus of the equation extends into all these regions at once1 there will be presented1 by reason merely of this inequality of the angles1 differences of figure which do not correspond to any analytical diversity1 and will necessarily alter the rigorous exactness of the representation1 by being confounded with the proper results of the algebraic comparisons4 ,for example1 an equation like3 xm ym c1 which1 by its perfect symmetry1 should evidently give a curve composed of four identical quarters1 will be represented1 on the contrary1 if we take axes not rectangular1 by a geometric locus1 the four parts of which will be unequal4 ,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 angle4 ,the preceding discussion clearly shows that1 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 geometry1 yet as1 on the other hand1 it is not constantly inferior1 its necessary and absolute greater aptitude for the representation of equations must cause it to generally receive the preference2 although it may evidently happen1 in some particular cases1 that the necessity of simplifying equations and of obtaining them more easily may determine geometers to adopt a less perfect system4 ,the rectilinear system is1 therefore1 the one by means of which are ordinarily constructed the most essential theories of general geometry1 intended to express analytically the most important geometrical phenomena4 ,when it is thought necessary to choose some other1 the polar system is almost always the one which is fixed upon1 this system being of a nature sufficiently opposite to that of the rectilinear system to cause the equations1 which are too complicated with respect to the latter1 to become1 in general1 sufficiently simple with respect to the other4 ,polar co-ordinates1 moreover1 have often the advantage of admitting of a more direct and natural concrete signification2 as is the case in mechanics1 for the geometrical questions to which the theory of circular movement gives rise1 and in almost all the cases of celestial geometry4 99 99 99 99 99 ,in order to simplify the exposition1 we have thus far considered the fundamental conception of analytical geometry only with respect to plane curves1 the general study of which was the only object of the great philosophical renovation produced by ,descartes4 ,to complete this important explanation1 we have now to show summarily how this elementary idea was extended by ,clairaut1 about a century afterwards1 to the general study of surfaces and curves of double curvature4 ,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 case4 ,,surfaces4 ,determination of a ,point in ,space4 ,the complete analytical determination of a point in space evidently requires the values of three co-ordinates to be assigned2 as1 for example1 in the system which is generally adopted1 and which corresponds to the rectilinear system of plane geometry1 distances from the point to three fixed planes1 usually perpendicular to one another2 which presents the point as the intersection of three planes whose direction is invariable4 ,we might also employ the distances from the movable point to three fixed points1 which would determine it by the intersection of three spheres with a common centre4 ,in like manner1 the position of a point would be defined by giving its distance from a fixed point1 and the direction of that distance1 by means of the two angles which this right line makes with two invariable axes2 this is the polar system of geometry of three dimensions2 the point is then constructed by the intersection of a sphere having a fixed centre1 with two right cones with circular bases1 whose axes and common summit do not change4 ,in a word1 there is evidently1 in this case at least1 the same infinite variety among the various possible systems of co-ordinates which we have already observed in geometry of two dimensions4 ,in general1 we have to conceive a point as being always determined by the intersection of any three surfaces whatever1 as it was in the former case by that of two lines3 each of these three surfaces has1 in like manner1 all its conditions of determination constant1 excepting one1 which gives rise to the corresponding co-ordinates1 whose peculiar geometrical influence is thus to constrain the point to be situated upon that surface4 ,this being premised1 it is clear that if the three co-ordinates of a point are entirely independent of one another1 that point can take successively all possible positions in space4 ,but if the point is compelled to remain upon a certain surface defined in any manner whatever1 then two co-ordinates are evidently sufficient for determining its situation at each instant1 since the proposed surface will take the place of the condition imposed by the third co-ordinate4 ,we must then1 in this case1 under the analytical point of view1 necessarily conceive this last co-ordinate as a determinate function of the two others1 these latter remaining perfectly independent of each other4 ,thus there will be a certain equation between the three variable co-ordinates1 which will be permanent1 and which will be the only one1 in order to correspond to the precise degree of indetermination in the position of the point4 ,expression of ,surfaces by ,equations4 ,this equation1 more or less easy to be discovered1 but always possible1 will be the analytical definition of the proposed surface1 since it must be verified for all the points of that surface1 and for them alone4 ,if the surface undergoes any change whatever1 even a simple change of place1 the equation must undergo a more or less serious corresponding modification4 ,in a word1 all geometrical phenomena relating to surfaces will admit of being translated by certain equivalent analytical conditions appropriate to equations of three variables2 and in the establishment and interpretation of this general and necessary harmony will essentially consist the science of analytical geometry of three dimensions4 ,expression of ,equations by ,surfaces4 ,considering next this fundamental conception in the inverse point of view1 we see in the same manner that every equation of three variables may1 in general1 be represented geometrically by a determinate surface1 primitively defined by the very characteristic property1 that the co-ordinates of all its points always retain the mutual relation enunciated in this equation4 ,this geometrical locus will evidently change1 for the same equation1 according to the system of co-ordinates which may serve for the construction of this representation4 ,in adopting1 for example1 the rectilinear system1 it is clear that in the equation between the three variables1 x1 y1 z1 every particular value attributed to z will give an equation between at x and y1 the geometrical locus of which will be a certain line situated in a plane parallel to the plane of x and y1 and at a distance from this last equal to the value of z2 so that the complete geometrical locus will present itself as composed of an infinite series of lines superimposed in a series of parallel planes 7excepting the interruptions which may exist71 and will consequently form a veritable surface4 ,it would be the same in considering any other system of co-ordinates1 although the geometrical construction of the equation becomes more difficult to follow4 ,such is the elementary conception1 the complement of the original idea of ,descartes1 on which is founded general geometry relative to surfaces4 ,it would be useless to take up here directly the other considerations which have been above indicated1 with respect to lines1 and which any one can easily extend to surfaces2 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-ordinates1 or to determine among all the different systems of possible co-ordinates that one which is generally the most convenient4 ,i will only add1 on this last point1 that the necessary superiority of the ordinary rectilinear system1 as to the representation of equations1 is evidently still more marked in analytical geometry of three dimensions than in that of two1 because of the incomparably greater geometrical complication which would result from the choice of any other system4 ,this can be verified in the most striking manner by considering the polar system in particular1 which is the most employed after the ordinary rectilinear system1 for surfaces as well as for plane curves1 and for the same reasons4 ,in order to complete the general exposition of the fundamental conception relative to the analytical study of surfaces1 a philosophical examination should be made of a final improvement of the highest importance1 which ,monge has introduced into the very elements of this theory1 for the classification of surfaces in natural families1 established according to the mode of generation1 and expressed algebraically by common differential equations1 or by finite equations containing arbitrary functions4 ,,curves ,,of ,,double ,,curvature4 ,let us now consider the last elementary point of view of analytical geometry of three dimensions2 that relating to the algebraic representation of curves considered in space1 in the most general manner4 ,in continuing to follow the principle which has been constantly employed1 that of the degree of indetermination of the geometrical locus1 corresponding to the degree of independence of the variables1 it is evident1 as a general principle1 that when a point is required to be situated upon some certain curve1 a single co-ordinate is enough for completely determining its position1 by the intersection of this curve with the surface which results from this co-ordinate4 ,thus1 in this case1 the two other co-ordinates of the point must be conceived as functions necessarily determinate and distinct from the first4 ,it follows that every line1 considered in space1 is then represented analytically1 no longer by a single equation1 but by the system of two equations between the three co-ordinates of any one of its points4 ,it is clear1 indeed1 from another point of view1 that since each of these equations1 considered separately1 expresses a certain surface1 their combination presents the proposed line as the intersection of two determinate surfaces4 ,such is the most general manner of conceiving the algebraic representation of a line in analytical geometry of three dimensions4 ,this conception is commonly considered in too restricted a manner1 when we confine ourselves to considering a line as determined by the system of its two projections upon two of the co-ordinate planes2 a system characterized1 analytically1 by this peculiarity1 that each of the two equations of the line then contains only two of the three co-ordinates1 instead of simultaneously including the three variables4 ,this consideration1 which consists in regarding the line as the intersection of two cylindrical surfaces parallel to two of the three axes of the co-ordinates1 besides the inconvenience of being confined to the ordinary rectilinear system1 has the fault1 if we strictly confine ourselves to it1 of introducing useless difficulties into the analytical representation of lines1 since the combination of these two cylinders would evidently not be always the most suitable for forming the equations of a line4 ,thus1 considering this fundamental notion in its entire generality1 it will be necessary in each case to choose1 from among the infinite number of couples of surfaces1 the intersection of which might produce the proposed curve1 that one which will lend itself the best to the establishment of equations1 as being composed of the best known surfaces4 ,thus1 if the problem is to express analytically a circle in space1 it will evidently be preferable to consider it as the intersection of a sphere and a plane1 rather than as proceeding from any other combination of surfaces which could equally produce it4 ,in truth1 this manner of conceiving the representation of lines by equations1 in analytical geometry of three dimensions1 produces1 by its nature1 a necessary inconvenience1 that of a certain analytical confusion1 consisting in this3 that the same line may thus be expressed1 with the same system of co-ordinates1 by an infinite number of different couples of equations1 on account of the infinite number of couples of surfaces which can form it2 a circumstance which may cause some difficulties in recognizing this line under all the algebraical disguises of which it admits4 ,but there exists a very simple method for causing this inconvenience to disappear2 it consists in giving up the facilities which result from this variety of geometrical constructions4 ,it suffices1 in fact1 whatever may be the analytical system primitively established for a certain line1 to be able to deduce from it the system corresponding to a single couple of surfaces uniformly generated2 as1 for example1 to that of the two cylindrical surfaces which project the proposed line upon two of the co-ordinate planes2 surfaces which will evidently be always identical1 in whatever manner the line may have been obtained1 and which will not vary except when that line itself shall change4 ,now1 in choosing this fixed system1 which is actually the most simple1 we shall generally be able to deduce from the primitive equations those which correspond to them in this special construction1 by transforming them1 by two successive eliminations1 into two equations1 each containing only two of the variable co-ordinates1 and thereby corresponding to the two surfaces of projection4 ,such is really the principal destination of this sort of geometrical combination1 which thus offers to us an invariable and certain means of recognizing the identity of lines in spite of the diversity of their equations1 which is sometimes very great4 ,,imperfections ,,of ,,analytical ,,geometry4 ,having now considered the fundamental conception of analytical geometry under its principal elementary aspects1 it is proper1 in order to make the sketch complete1 to notice here the general imperfections yet presented by this conception with respect to both geometry and to analysis4 ,relatively to geometry1 we must remark that the equations are as yet adapted to represent only entire geometrical loci1 and not at all determinate portions of those loci4 ,it would1 however1 be necessary1 in some circumstances1 to be able to express analytically a part of a line or of a surface1 or even a discontinuous line or surface1 composed of a series of sections belonging to distinct geometrical figures1 such as the contour of a polygon1 or the surface of a polyhedron4 ,thermology1 especially1 often gives rise to such considerations1 to which our present analytical geometry is necessarily inapplicable4 ,the labours of ,m4 ,fourier on discontinuous functions have1 however1 begun to fill up this great gap1 and have thereby introduced a new and essential improvement into the fundamental conception of ,descartes4 ,but this manner of representing heterogeneous or partial figures1 being founded on the employment of trigonometrical series proceeding according to the sines of an infinite series of multiple arcs1 or on the use of certain definite integrals equivalent to those series1 and the general integral of which is unknown1 presents as yet too much complication to admit of being immediately introduced into the system of analytical geometry4 ,relatively to analysis1 we must begin by observing that our inability to conceive a geometrical representation of equations containing four1 five1 or more variables1 analogous to those representations which all equations of two or of three variables admit1 must not be viewed as an imperfection of our system of analytical geometry1 for it evidently belongs to the very nature of the subject4 ,analysis being necessarily more general than geometry1 since it relates to all possible phenomena1 it would be very unphilosophical to desire always to find among geometrical phenomena alone a concrete representation of all the laws which analysis can express4 ,there exists1 however1 another imperfection of less importance1 which must really be viewed as proceeding from the manner in which we conceive analytical geometry4 ,it consists in the evident incompleteness of our present representation of equations of two or of three variables by lines or surfaces1 inasmuch as in the construction of the geometric locus we pay regard only to the real solutions of equations1 without at all noticing any imaginary solutions4 ,the general course of these last should1 however1 by its nature1 be quite as susceptible as that of the others of a geometrical representation4 ,it follows from this omission that the graphic picture of the equation is constantly imperfect1 and sometimes even so much so that there is no geometric representation at all when the equation admits of only imaginary solutions4 ,but1 even in this last case1 we evidently ought to be able to distinguish between equations as different in themselves as these1 for example1 x y #a #j1 x y #a #j1 y ex #j4 ,we know1 moreover1 that this principal imperfection often brings with it1 in analytical geometry of two or of three dimensions1 a number of secondary inconveniences1 arising from several analytical modifications not corresponding to any geometrical phenomena4 99 99 99 99 99 ,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 surfaces2 and1 reciprocally1 in discovering the geometrical interpretation of such or such an analytical consideration4 ,a detailed examination of the most important general questions would show us how geometers have succeeded in actually establishing this beautiful harmony1 and in thus imprinting on geometrical science1 regarded as a whole1 its present eminently perfect character of rationality and of simplicity4 ,note4--,the author devotes the two following chapters of his course to the more detailed examination of ,analytical ,geometry of two and of three dimensions2 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 volume4 ,,the ,,end4