Comte - The positive philosophy. Vol. 1

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In the section on the Integral Calculus, I noticed D’Alembert’s creation of the Calculus of partial differences, in which Lagrange recognized a new calculus. This new elementary idea in transcendental analy- sis,—the notion of two blinds of increments, distinct and independent of each other, which a function of two variables may receive in virtue of the change of each variable separately,—seems to me to establish a natural and necessary transition between the common infinitesimal calculus and the calculus of variations. D’Alembert’s view appears to me to approximate, by its nature, very nearly to that which serves as a general basis for the Method of Variations. This last has, in fact, done nothing more than transfer to the independent variables themselves the view already adopted for the functions of those variables; a process which has remarkably extended its use. A recognition of such a derivation as this for the method of variations may exhibit its philosophical character more clearly and simply; and this is my reason for the reference.

The Method of Variations presents itself to us as the highest degree of perfection which the analysis of indirect functions has yet attained. We had before, in that analysis a powerful instrument for the mathematical study of natural phenomena, inasmuch as it introduced the consideration of auxiliary magnitudes, so chosen as that their relations were necessarily more simple and easy to obtain than those of the direct magnitudes. But we had not any general and abstract rules for the formation of these differential equations; nor were such supposed to be possible. Now, the Analysis of Variations brings the actual establishment of the differential equations within the reach of the Calculus for such is the general effect, in a great number of important and difficult questions, of the varied equations, which still more indirect than the simple differential equations, as regards the special objects of the inquiry, are more easy to form: and, by invariable and complete analytical methods employed to eliminate the new order of auxiliary infinitesimals introduced, we may deduce those ordinary differential equations which we might not have been able to establish directly. The Method of Variations forms, then the most sublime part of that vast system of mathematical analysis which, setting out from the simplest elements of algebra organizes, by an uninterrupted succession of ideas, general methods more and more potent for the investigation of natural philosophy. This is incomparably the noblest and most unquestionable testimony to the scope of the human intellect. If, at the same time, we bear in mind that

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the employment of this method exacts the highest known degree of intellectual exertion, in order never to lose sight of the precise object of the investigation in following reasonings which offer to the mind such uncertain resting places, and in which signs are of scarcely any assistance we shall understand how it may be that so little use has been made of such a conception by any philosophers but Lagrange.

We have now reviewed Mathematical analysis, in its bases and in its divisions, very briefly, but from a philosophical point of view, neglecting those conceptions only which are not organized with the great whole, or which, if urged to their limit, would be found to merge in some which have been examined. I must next offer a similar outline of Concrete Mathematics. My particular task will lore to show how,—suppos- ing the general science of the Calculus to be in a perfect state,—it has been possible to reduce, by invariable procedures, to pure questions of analysis, all the problems of Geometry and Mechanics, and thus to invest philosophy with that precision and unity which can only thus be attained, and which constitute high perfection.

Chapter III

General View of Geometry

We have seen that Geometry is a true natural science;—only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Every body studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone.

The scientific eminence of Geometry arises from the extreme generality and simplicity of its phenomena. If all the parts of the universe were regarded as immovable, geometry would still exist; whereas, for the phenomena of Mechanics, motion is required. Thus Geometry is the more general of the two. It is also the more simple, for its phenomena are independent of those of Mechanics, while mechanical phenomena are always complicated with those of geometry. The same is true in the comparison of abstract thermology with geometry. For these reasons, geometry holds the first place under the head of Concrete Mathematics.

Instead of adopting the inadequate ordinary account of Geometry, that it is the science of extension, I am disposed to give, as a general description of it, that it is the science of the measurement of extension. Even this does not include all the operations of geometry, for there are

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many investigations which do not appear to have for their object the measurement of extension But regarding the science in its leading questions as a whole, we may accurately say that the measurement of lines, of surfaces, and of volumes, is the invariable aim,—sometimes direct, though oftener indirect,—of geometrical labours.

The rational study of geometry could never have begun if we must have regarded at once and together all the physical properties of bodies, together with their magnitude and form. By the character of our minds eve are able to think of the dimensions and figure of a body in an abstract way. After observation has shown us, for instance, the impression left by a body on a fluid in which it has been placed, we are able to retain an image of the impression, which becomes a ground of Geometrical reasoning. We thus obtain, apart from all metaphysical fan- cies,—an idea of Space. This abstraction, now so familiar to us that we cannot conceive the state we should be in without it, is perhaps the earliest philosophical creation of the human mind.

There is another abstraction which must made before we can enter on geometrical science. We must conceive of three kinds of extension, and learn to conceive of them separately. We cannot conceive of any space, filled by any object, which has not at once volume, surface, and line. Yet geometrical questions often relate to only two of these; frequently only to one. Even when all three are to be finally considered, it is often necessary, in order to avoid complication, to take only one at a time. This is the second abstraction which it is indispensable for us to practice,—to thinly of surface and line apart from volume; and again, of line apart from surface. We effect this by thinking of volume as becoming thinner and thinner, till surface appears as the thinnest possible layer or film: and again, we think of this surface becoming narrower and narrower till it is reduced to the finest imaginable thread; and then we have the idea of a line. Though we cannot speak of a point as a dimension, we must have the abstract idea of that too: and it is obtained by reducing the line from one end or both, till the smallest conceivable portion of it is left. This point indicates, not extension, of course, but position, of the place of extension. Surfaces have clearly the property of circumscribing volumes; lines, again, circumscribe surfaces; and lines, once more, are limited by points.

The Mathematical meaning of measurement is simply the finding of the value of the ratios between any homogeneous magnitudes: but geometrically, the measurement is always indirect The comparison of two

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lines is direct; that of two surfaces or two volumes can never be direct. One line may be conceived to be laid upon another: but one volume cannot be conceived of as laid upon another, nor one surface upon another, with any convenience or exactness. The question is, then, how to measure surfaces and volumes.

Whatever be the form of a body, there must always be lines, the length of which will define the magnitude of the surface or volume. It is the business of geometry to use these lines, directly measurable as they are, for the ascertainment of the ratio of the surface to the unity of surface, or of the volume to the unity of volume, as either may be sought. In brief, the object is to reduce all comparisons of surfaces or of volumes to simple comparisons of lines. Extending the process, we find the possibility of reducing to questions of lines all questions relating to surfaces and volumes, regarded in relation to their magnitude. It is true that when the rational method becomes too complicated and difficult, direct comparisons of surfaces and volumes are employed: but the procedure is not geometrical. In the same way, the consideration of weight is sometimes brought in, to determine volume, or even surface; but this device is derived from mechanics, and has nothing to do with rational geometry.

In speaking of the direct measurement of lines, it is clear that right lines are meant. When we consider curved lines, it is evident that their measurement must be indirect, since we cannot conceive of curved lines being laid upon each other with any precision or certainty. The procedure is first to reduce the measurement of curved to that of right lines; and consequently to reduce to simple questions of right lines all questions relating to the magnitude of any curves whatever. In every curve, there always exist certain right lines, the length of which must determine that of the curve, as the length of the radius of a circle gives us that of the circumference; and again, as the length of an ellipse depends on that of its two axes.

Thus, the science of Geometry has for its object the final reduction of the comparisons of all kinds of extent to comparisons of right lines, which alone are capable of direct comparison, and are, moreover, eminently easy to manage.

I must just notice that there is a primary distinct branch of Geometry, exclusively devoted to the right line, on account of occasionable insurmountable difficulties in making the direct comparison; its object is to determine certain right lines from others by means of the relations

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proper to the figures resulting from their assemblage. The importance of this is clear, as no question could be solved if the measurement of right lines, on which every other depends, were left, in any case, uncertain. The natural order of the parts of rational geometry is therefore, first the geometry of line, beginning with the right line; then the. geometry of surfaces; and, finally, that of volumes.

The field of geometrical science is absolutely unbounded. There may be as many questions as there are conceivable figures, and the variety of conceivable figures is infinite. As to curved fines, if we regard them as generated by the motion of a point governed by a certain law, we cannot limit their number, as the variety of distinct conditions is nothing short of infinite; each generating new ones, and those again others. Surfaces, again, are conceived of as motions of lines; and they not only partake of the variety of lines, but have another of their own, arising from the possible change of nature in the line. There can be nothing like this in lines, as points cannot describe a figure. Thus, there is a double set of conditions under which the figures of surfaces may vary: and we may say that if lines have one infinity of possible change, surfaces have two. As for Volumes, they are distinguished from one another only by the surfaces which bound them; so that they partake of the variety of surfaces, and need no special consideration under this head. If we add the one further remark, that surfaces themselves furnish a new means of conceiving of new curves, as every curve may be regarded as produced by the intersection of two surfaces, we shall perceive that, starting from a narrow ground of observation we can obtain an absolutely infinite variety of forms, and therefore an illimitable field for geometrical science

The connection between abstract and concrete geometry is established by the study of the properties of lines end surfaces. Without multiplying in this way our means of recognition, we should not know, except by accident, how to find in nature the figure we desire to verify. Astronomy was recreated by Kepler’s discovery that the ellipse was the curve which the planets describe about the sun, and the satellites about their planet. This discovery could never hare been made if geometers had known no more of the ellipse than as the oblique section of a circular cone by a plane. All the properties of the conic sections brought out by the speculative labours of the Greek geometers, were needed as preparation for this discovery, that Kepler might select from them the characteristic which was the true key to the planetary orbit. In the same way,

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the spherical figure of the earth could not have been discovered if the primitive character of the sphere had been the only one known,—viz., the equidistance of all its points from an interior point. Certain properties of surfaces were the means used for connecting the abstract reasonings with the concrete fact. And others again, were required to prove that the earth is not absolutely spherical, and how much otherwise. The pursuit of these labours does not interfere with the definition of Geometry given above, as they tend indirectly to the measurement of extension. The great body of geometrical researches relates to the properties of lines and surfaces; and the study of the properties of the same figure is so extensive, that the labours of geometers for twenty centuries have not exhausted the study of conic sections. Since the time of Descartes, it has become less important; but it appears as far as ever from being finished. And here opens another infinity We had before the infinite scope of lines, and the double infinity of surfaces and now we see that not only is the variety of figures inexhaustible, but also the diversity of the points of view from which each figure may be regarded.

There are two general Methods of treating geometrical questions. These are commonly called Synthetical Geometry and Analytical Geometry. I shall prefer the historical titles of Geometry of the Ancients and Geometry of the Moderns. But it is, in my view, better still to call them Special Geometry and General Geometry, by which their nature is most accurately conveyed.

The Calculus was not, as some suppose, unknown to the ancients, as we perceive by their applications of the theory of proportions. The difference between them and us is not so much in the instrument of deduction as in the nature of the questions considered. The ancients studied geometry with reference to the bodies under notice, or specially: the moderns study it with reference to the phenomena to be considered, or generally. The ancients extracted all they could out of one line or surface, before passing to another; and each inquiry gave little or no assistance in the next. The moderns, since Descartes, employ themselves on questions which relate to any figure whatever. They abstract, to treat by itself, every question relating to the same geometrical phenomenon, in whatever bodies it may be considered. Geometers can thus rise to the study of new geometrical conceptions, which, applied to the curves investigated by the ancients, have brought out new properties never suspected by them. The superiority of the modern method is obvious at a glance. The time formerly spent, and the sagacity and effort employed,

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in the path of detail, are inconceivably economized by the general method used since the great revolution under Descartes. The benefit to Concrete Geometry is no less than to the Abstract; for the recognition of geometrical figures in nature was merely embarrassed by the study of lines in detail; and the application of the contemplated figure to the existing body could be only accidental, and within a limited or doubtful range: whereas, by the general method, no existing figure can escape application to its true theory, as soon as its geometrical features are ascertained. Still, the ancient method was natural; and it was necessary that it should precede the modern. The experience of the ancients, and the materials they accumulated by their special method, were indispensable to suggest the conception of Descartes, and to furnish a basis for the general procedure. It is evident that the Calcttlus cannot originate any science. Equations must exist as a starting-point for analytical operations. No other beginning can be made than the direct study of the object, pursued up to the point of the discovery of precise relations.

We must briefly survey the geometry of the ancients, in its character of an indispensable introduction to that of the moderns. The one, special and preliminary, must have its relation made clear to the other,—the general and definitive geometry, which now constitutes the science that goes by that name.

We have seen that Geometry is a science founded upon observation, though the materials furnished by observation are few and simple, and the structure of reasoning erected upon them vast and complex. The only elementary materials, obtainable by direct study alone, are those which relate to the right line for the geometry of lines, to the quadrature of rectilinear plane areas, and to the cubature of bodies terminated by plane faces. The beginning of geometry must be from the observation of lines, of flat surfaces angularly bounded, and of bodies which have more or less bulk, also angularly bounded. These are all; for all other figures, even the circle, and the figures belonging to it, now come under the head of analytical geometry. The three elements just mentioned allow a sufficiency of equations for the calculus to proceed upon. More are not needed; and we calmot do with less. Some have endeavoured to extend analysis so as to dispense with a portion of these facts; but to do so is merely to return to metaphyslcal practices, in presenting actual facts as logical abstractions. The more we perceive Geometry to be, in our day, essentially analytical, the more careful we must be not to lose sight of the basis of observation on which all geometrical science is founded. When

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we observe people attempting to demonstrate axioms and the like, we may avow that it is better to admit more than may be quite necessary of materials derived from observation, than to carry logical demonstration into a region where direct observation will serve us better.

There are two ways of studying the right line—the graphic and the algebraic. The thing to be done is to ascertain, by means of one another, the different elements of any right line what ever, so as to understand, indirectly, a right line, under any circumstances whatever. The way to do this is, first, to study the figure, by constructing it, or otherwise directly investigating it; and then, to reason from that observation. The ancients, in the early days of the science, made great use of the graphic method, even in the form of Construction; as when Aristarchus of Samos estimated the distance of the sun and moon from the earth on a triangle constructed as nearly as possible in resemblance to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature, and when therefore an observation of the angle at the earth would define the triangle. Archimedes himself, though he was the first to introduce calculated determinations into geometry, frequently used the same means. The introduction of trigonometry lessened the practice; but did not abolish it. The Greeks and Arabians employed it still for a great number of investigations for which we now consider the use of the Calculus indispensable.

While the graphic or constructive method answers well when all the parts of the proposed figure lie in the same plane, it must receive additions before it can be applied to figures whose parts lie in different planes. Hence arises a new series of considerations, and different systems of Projections. Where we now employ spherical trigonometry, especially for problems relating to the celestial sphere, the ancients had to consider how they could replace constructions in relief by plane constructions. This was the object of their analemmas, and of the other plane figures which long supplied the place of the Calculus. Then were acquainted with the elements of what we call Descriptive Geometry, though they did not conceive of it in a distinct and general manner.

Digressing here for a moment into the region of application, I may observe that Descriptive Geometry, formed into a distinct system by Monge, practically meets the difficulty just stated, but does not warrant the expectations of its first admirers, that it would enlarge the domain of rational geometry. Its grand use is in its application to the industrial arts;—its few abstract problems, capable of invariable solution, relat-

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ing essentially to the contacts and intersections of surfaces; so that all the geometrical questions which may arise in any of the various arts of construction,—as stone-cutting, carpentry, perspective, dialling, fortification, etc.,—can always be treated as simple individual cases of a single theory, the solution being certainly obtainable through the particular circumstances of each case. This creation must be very important in the eyes of philosophers who think that all human achievement, thus far, is only a first step towards a philosophical renovation of the labours of mankind; towards that precision and logical character which can alone ensure the future progression of all arts. Such a revolution must inevitably begin with that class of arts which bears a relation to the simplest, the most perfect, and the most ancient of the sciences. It must extend, in time, though less readily, to all other industrial operations. Monge, who understood the philosophy of the arts better than any one else, himself indeed endeavoured to sketch out a philosophical system of mechanical arts, and at least succeeded in pointing out the direction in which the object must be pursued. Of Descriptive Geometry, it may further be said that it usefully exercises the students’ faculty of Imagination,—of conceiving of complicated geometrical combinations in space; and that, while it belongs to the geometry of the ancients by the character of its solutions, it approaches to the geometry of the moderns by the nature of the questions which compose it. Consisting, as we have said, of a few abstract problems, obtained through Projections, and relating to the contacts and intersections of surfaces, the invariable solutions of these Problems are at once graphical, like those of the ancients, and general, like those of the moderns. Yet, as destined to an industrial application, Descriptive Geometry has here been treated of only in the way of digression. Heaving the subject of graphic solution, we have to notice the other branch,—the algebraic.

Some may wonder that this branch is not treated as belonging to General Geometry. But, not only were the ancients, in fact, the inventors of trigonometry,—spherical as well as rectilinear,—though it necessarily remained imperfect in their hands; but algebraic solutions are also no part of analytical geometry, but only a complement of elementary geometry.

Since all right-lined figures can be decomposed into triangles, all that we want is to be able to determine the different elements of a triangle by means of one another. This reduces polygonometry to simple trigonometry.

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The difficulty lies in forming three distinct equations between the angles and the sides of a triangle. These equations being obtained, all trigonometrical problems are reduced to mere questions of analysis.— There are two methods of introducing the angles into the calculation. They are either introduced directly, by themselves or by the circular arcs which are proportional to them: or they are introduced indirectly, by the chords of these arcs, which are hence called their trigonometrical lines. The second of these methods was the first adopted, because the early state of knowledge admitted of its working, while it did not admit the establishment of equations between the sides of the triangles and the angles themselves, but only between the sides and the trigonometrical lines.—The method which employs the trigonometrical lines is still preferred, as the more simple, the equations existing only between right lines, instead of between right lines and arcs of circles.

To meet the probable objection that it is rather a complication than a simplification to introduce these lines, which have at last to be eliminated, we must explain a little.

Their introduction divides trigonometry into two parts. In one, we pass from the angles to their trigonometrical lines, or the converse: in the other we have to determine the sides of the triangles by the trigonometrical lines of their angles, or the converse. Now, the first process is done for us, once for all, by the formation of numerical tables, capable of use in all conceivable questions. It is only the second, which is be far the least laborious, that has to be undertaken in each individual case. The first is always done in advance. The process may be compared with the theory of logarithms, by which all imaginable arithmetical operations are decomposed into two parts—the first and most difficult of which is done in advance.

We must remember, too, in considering the position of the ancients, the remarkable fact that the determination of angles by their trigonometrical lines, and the converse, admits of an arithmetical solution, without the previous resolution of the corresponding algebraic question. But for this, the ancients could not have obtained trigonometry. When Archimedes was at work upon the rectification of the circle, tables of chords were prepared: from his labours resulted the determination of a certain series of chords: and, when Hipparchus afterwards invented trigonometry, he had only to complete that operation by suitable intercalations. The connection of ideas is here easily recognized.

For the same reasons which lead us to the employment of these