of these facts; but to do so is merely to return to meta- physical practices, in presenting actual facts as logical abstractions. The more we perceive Geometry to be, in our day, essentially analytical, the more cai-eful we must lie not to lose sight of the basis of observation on which all geometrical science is founded. When 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 100 POSITIVE PHILOSOPHY.
carry logical demonstration into a region where direct observation "vvill serve us better.
There are two ways of studying the right thTri'^'h/Hne ^^"® — ^^^® 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 solutions ^^' fi^^t' t^ 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 observa- tion 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 emjiloyed it still for a great number of in- vestigations for which we now consider the use of the Calculus indispensable.
While the grajjhic 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 Pro- jections. Where we now employ sj^herical trigonometry, especially for problems relating to the celestial sphere, the ancients had to consider how they could replace constrnc- tions in relief by plane constructions. This was the object of their analemvnas, and of the other plane figures which long su])plied the place of the Calculus. They were ac- quainted 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 ai)pli- DESCRIPTIVE GEOMETRY. 101 cation, I may observe that Descriptive G-eo-.
metry, formed into a distinct system by Geometry^ 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 in- dustrial arts; — its few abstract problems, capable of invariable solution, relating 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 circum- stances 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 towai-ds a philo- sop>hical 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 endeavoui'ed to sketch out a philosophical system of mechanical arts, and at least suc- ceeded iu 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 geo- metrical 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. Consist- ing, 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 modei'ns. Yet, as destined to an industrial aj^plication, Descriptive Geometry has 102 POSITIVE PHILOSOPHY.
}iere been treated of only in the way of digression. Leaving the subject of graphic solution, we have to notice the other branch,^ — the algebraic...,. Some may wonder that this branch is not Solutions treated as belonging to G-eneral Geometry.
But, not only were the ancients, in fact, the inventors of trigonometry, — spherical as well as recti- linear,— though it necessarily remained imj)erfect in their hands; but algebraic solutions are also no part of ana- lytical 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 polygonometiy to simple trigonometry. Ti •, The difficulty lies in form iup- three distinct '^ equations between the angles and the sides of a triangle. These equations being obtained, all trigono- metrical 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 j)ropor- tional to them: or they are inti'oduced indirectly, by the chords of these arcs, which ai-e hence called their trigono- metrical lines. The second of these methods was the first adopted, because the early state of knowledge admitted of its woi'king, 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 trigono- metrical lines. — The method which employs the trigono- metrical lines is still j^ref erred, 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 com- plication 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 TRIGONOMETRY. 103 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 by 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 jjosition 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 jirevious resolution of the corresponding algebraic question. But for this, the ancients could not have obtained trigono- metry. When Archimedes was at Avork upon the rectifi- cation 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 trigo- nometry, 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 lines, we must employ several at once, instead of confining ourselves to one, as the ancients did. The Arabians, and after them the moderns, attained to only four or five direct trigonometrical lines altogether; whereas it is clear that the number is not limited. — Instead, how- ever, of plunging into deep complications, in obtaining new direct lines, we create indirect ones. Instead, for instance, of directly and necessarily determining the sine of an angle, we may determine the sine of its half, or of its double, — taking am' line relating to an arc which is a very simple function of the first. Thus, we may say that the number of trigonometrical lines actually employed by modern geometers is unlimited through the augmentations we may obtain by analysis. Special names have, however, been given to those indirect lines only which refer to the complement of the primitive arc, — others being in much less frequent use.
Out of this device arises a third section of trigono- 104 POSITIVE PHILOSOPHY.
metrical knowledge. Having introduced a new set of lines, — of auxiliary magnitudes — we have to determine their relation to the first. And this study, though pre- paratory, is indefinite in its scope, while the two other departments are strictly limited.
The three must, of course, be studied in just the reverse order to that in which it has been necessary to exhibit them. First, the student must know the relations between the indirect and direct trigonometrical lines: and the re- solution of triangles, properly so called, is the last process.
Spherical trigonometry requires no special notice here, (all-important as it is by its uses,) — since it is, in our day, simply an application of rectilinear trigonometry, through the substitution of the corresponding trihedral angle for the spherical triangle.
This view of the philosophy of trigonometry has been given chiefly to show how the most simple questions of elementary geometry exhibit a close dependence and regular ramification.
Thus have we seen what is the peculiar character of Special Geometry, strictly considered. We see that it constitutes an indispensable basis to General Geometry. Next, we have to study the philosophical character of the true science of Geometry, beginning with the great original idea of Descartes, on which it is wholly founded.
Modern, or Analytical Geometry.
General or Analytical Geometry is founded upon the transformation of geometrical considerations into equiva- lent analytical considerations. Descartes established the constant possibility of doing this in a uniform manner: and his beautiful conception is interesting, not only from its carrying on geometrical science to a logical perfection, but from its showing us how to organize the relations of the abstract to the concrete in Mathematics by the ana- lytical representation of natural phenomena. Analytical The first thing to be done is evidently to representation find and fix a method for expressing analyti- of ligures. cally the subjects which aft'ord the phenomena. If we can regard lines and surfaces analytically.
MODERN OR ANALYTICAL GEOMETRY. 105 we can so regard, heiieefortli, the accidents of tliese suhjects.
Here occurs the difficulty of reducing all geometrical ideas to those of number: of substituting considerations of quantity for all considerations of quality. — In dealing with this difficulty, we must observe that all geometrical ideas come under three heads: — the magnitude, the Position figure, and the position of the extensions in question. The relation of the first, magnitude, to numbers is immediate and evident: and the other two are easily brought into one; for the figure of a body is nothing else than the natural position of the jjoints of whirh it is com- posed: and its position cannot be conceived of irrespective of its figure. We have therefore only to establish the one relation between ideas of position and ideas of magnitude. It is upon this that Descartes has established the system of General Geometry.
The method is simply a carrying out of an operation which is natural to all minds. If we wish to indicate the situation of an object which we cannot point out, we say how it is related to objects that are known, by assigning the magnitude of the different geometrical elements which connect it with known objects.
Those elements are what Descartes, and all other geo- meters after him, have called the co-ordinates of the point considered. If we know in what plane the point is situated, the co-ordinates are two. If the point may be anywhere m space, the co-ordinates cannot be less than three. They may be multiplied without limit: but whether few or many, the ideas of position will have been reduced to those of magnitude, so that we shall represent the displacement of a point as produced by pure numerical variations in the values of its co-ordinates. — The simplest case of all, that of plane geometry, is when we determine the ^^osition of a point on a plane by considering its distances from two fixed right lines, supposed to be known, and generally concluded to be perpendicular to each other. These are called axes. Next, there may be the less simple process of determining the position by the distances from fixed points; and so on to greater and greater complications. But, from some system or other of co-ordinates being always employed.
106 POSITIVE PHILOSOPHY.
the question of position is always reduced to that of magnitude.
,^.,.. It is clear that our only way of marking •jj^ the position or a point is by the intersection of two lines. When the point is determined by the intersection of two right lines, each parallel to a fixed axis, that is the system of rectilinear co- ordinates,— the most common of all. The polar system of co-ordinates exhibits the point by the travelling- of a right line round a fixed centre of a circle of variable radius. Again, two circles may intersect, or any other two lines: so that to assign the value of a co-ordinate is the same thing as to determine the line on which the point must be situated. The ancient geometers, of course, were like our- selves in this necessary method of conceiving of position: and their geometrical loci were founded upon it. It was in endeavouring to form the process into a general system tliat Descartes created Analytical Geometry. — Seeing, as we now do, how ideas of position, — and, through them, all elemen- tary geometrical ideas, — can be reduced to ideas of number, we learu what it was that he effected.
„i ^ Descartes treated only geometry of two dimensions m his analytical method: and we will at first consider only this kind, beginning with Plane Expression Curves. Lines must be expressed by equa- of Hnes by tions; and again, equations must be expressed Equations. \)y Hues, when the relation of geometrical con- ceptions to numbers is established. — It comes to the same thing whether we define a line by any one of its properties, or supply the corresponding equation between the two variable co-ordinates of the point which describes the line. If a point describes a certain line on a plane, we know that its co-ordinates bear a fixed relation to each other, which may be exj^ressed by an appropriate equation. If the point describes no certain line, its co-ordinates must be two variables independent of each other. Its situation in the latter case can be determined only by giving at once its two co-ordinates, independently of each other: whereas, in the former case, a single co-ordinate suffices to fix its position. The second co-ordinate is then a determinate function of ANALYTICAL GEOMETRY. 107 the first; — that is, there exists between them a certain equation of a nature corresponding to that of the line on which the point is to be found. The co-ordinates of the point each require it to be on a certain line: and again, its being on a certain line is the same thing as assigning the value of one of the two co-ordinates; which is then found to be entirely dependent on the other. Thus are lines analytically expressed by equations.
By a converse argument may be seen the Expression geometrical necessity of representing by a of equations certain line every equation of two variables, ^^y lines, in a determinate system of co-ordinates. In the absence of any other known property, such a relation would be a very chai'acteristic definition; and its scientific effect would be to fix the attention immediately uj^on the general course of the solutions of the equation, which will thus be noted in the most striking and simple manner. There is an evident and vast advantage in this picturing of ecj^uations, which reacts strongly upon the perfecting of analysis itself. The geometrical loc^ls stands before our minds as the represen- tation of all the details that have gone to its preparation, and thus renders comparatively easy our conception of new general analytical views. This method has become entirely elementary in our day; and it is employed when we want to get a clear idea of the general character of the law which runs through a series of particular observations of any kind whatever.
Recurring to the representation of lines by Chanf,^e in the equations, which is our chief object, we see line changes that this representation is, by its nature, so ^1^*^ equation, faithful, that the line could not undergo any modification, even the slightest, without causing a corresponding change in the equation. Some special difiiculties arise out of this perfect exactness; for since, in our system of analytical geometry, mei'e displacements of lines affect equations as much as real variations of magnitude or form, we inight be in danger of confounding the one with the other, if geo- meters had not discovered an ingenious method exj^ressly intended to distinguish them always. It must be observed that general inconveniences of this nature appear to be strictly inevitable in analytical geometry; since, ideas of 108 POSITIVE PHILOSOPHY.
position being the only geometrical ideas immediately re- ducible to numerical considerations, and conceptions of form not being referrible to them but by seeing in them relations of situation, it is impossible that analysis should not at first confound phenomena of form with simjDle phenomena of position; which are the only ones that equations express directly.
Every defini- To complete our description of the basis of tion of a line analytical geometry, it is necessary to point IS an equation, q^^ iha,t not only must every defined line give rise to a certain equation between the two co-ordinates of any one of its points, but ever'y definition of a line is itself an equation of that line in a suitable system of co- ordinates.
Considering, first, what a definition is, we say it must distinguish the defined object from all others, by assigning to it a property which belongs to it alone. But this pro- ]')erty may not disclose the mode of generation of the object, in which case the definition is merely characteristic; or it may express one of its modes of generation, and in that case the definition is explanatory. For instance, if we say that the circle is the line which in the same form contains the largest area, we offer a characteristic definition; whereas if we choose its property of having all its points equally distant from a fixed point, we have an explanatory defini- tion. It is clear moreover that the characteristic definition always leaves room for an explanatory one, which further study must disclose.
It is to explanatory definitions only that Avhat has been said of the definition of a line being an equation of that line can apply. We cannot define the generation of a line without specifying a certain relation between the two simple motions, of translation or of rotation, into which the motion of the point which describes it will be decom- posed at each moment. Now, if we form the most general conception of Avhat a system of co-ordinates is, and if we admit all possible systems, it is clear that such a relation can be nothing else than the equation of the proposed line, in a system of co-ordinates of a corresponding nature to that of the mode of generation considen^d; as in the case of the circle, the common definition of which may be re- ANALYTICAL GEOMETRY. 109 gardecl as being the polar equation of that curve, taking the centre of the circle for the pole.
This view not only exhibits the necessary representation of every line by an equation, but it indicates the general difficulty which occurs in the establishment of these equa- tions, and therefore shows us how to proceed in inquiries of this kind which, by their nature, do not admit of in- variable rules. Since every explanatory definition of a line constitutes the equation of that line, it is clear that when we find difficulty in discovering the ec|uation of a curve by means of some of its characteristic properties, the difficulty must proceed from our taking up a designated system of co-ordinates, instead of admitting indifferently all possible systems. These systems are not all equally siiitable; and, in regard to curves, geometers think that they should almost always be referred, as far as possible, to rectilinear co-ordi- nates. Now, these particular co-ordinates are often not those with reference to which the equation of the curve will be found to be established by the proposed definition. It is in a certain transformation of co-ordinates then that the chief difficulty in the formation of the equation of a line really consists. The view I have given does not furnish us with a complete and certain general method for the establishment of these equations; but it may cast a useful light on the course which it is best to pursue to attain the end proposed.
The choice of co-ordinates — the preference ^...of that system which may be most suitable co-ordinate'^ to the case — is the remaining point which we have to notice.
First, we must distiuguish vei'y carefully the two views, the converse of each other, which belong to analytical geometry, viz. the relation of algebra to geometry, founded on the representation of lines by equations, and, recipro- cally, the relation of geometry to algebra, founded on the picturing of equations by lines. Though the two are necessarily combined in every investigation of general geometry, and we have to pass from the one to the other alternately, and almost insensibly, we must be able to separate them here, for the answer to the question of 110 POSITIVE PHILOSOPHY.
metliod wliicli we are considering is far from being the same under tlie two relations: so that without this distinc- tion we could not form any clear idea of it.
In the case of the representation of lines by equations, the first object is to choose those co-ordinates which afford the greatest simplicity in the equation of each line, and the greatest facility in aiTiving at it. There can be no constant preference here of one system of co-ordinates. The recti- linear system itself, though often advantageous, cannot be always so, and may be, in turn, less so than any other. But it is far otherwise in the converse case of the repre- sentation of equations by lines. Here the rectilinear system is always to be preferred, as the most simple and trustworthy. If we seek to determine a point by the inter- section of two lines, it must be best that those lines should be the simplest possible; and this confines our choice to the rectilinear system. In constructing geometrical loci, that system of co-ordinates must be the best in which it is easiest to conceive the change of place of a point resulting from the change in the value of its co-ordinates; and this is the case with the rectilinear system. Again, there is great advan- tage in the common usage of taking the two axes perpen- dicular to each other, when 2:)ossible, rather than with any other inclination. In rej^resenting lines by equations, we must take any inclination of the axes which may best suit the particular question; but, in the converse case, it is easy to see that rectangular axes permit iis to represent equations in a more sinijile, and even in a more faithful manner. For if we extend the geometrical lociis of the equation into the several imequal regions marked out by oblique axes, we shall have differences of figure which do not correspond to any analytical diversity; and the accuracy of the x'epresen- tation will be lost.
On the whole then, taking together the two points of view of analytical geometry, the ordinary system of recti- linear co-ordinates is superior to any other. Its high aptitude for the representation of equations must make it generally preferred, though a less perfect system may answer better in particular cases. The most essential theories in modern geometry are generally expressed by the rectilinear system. The polar system is preferred next RECTILINEAR AND POLAR CO-ORDINATES. Ill to it, both because its opposite character enables it to solve iu the simplest way the equations which are too compli- cated for management under the first; and because j^olar co-ordinates have often the advantage of admitting of a more direct and natural concrete signification. This is the case in Mechanics, iu the geometrical questions arising out of the theory of circular movement, and in almost all questions of celestial geometry.
Such was the field of the labours of Descartes, his conception of analytical geometry being designed only for the stady of Plane Curves. It was Clairaut who, about a century later, extended it to the study of Surfaces and Curves of double curvature. The conception having been explained, a very brief notice vsdll suffice for the rest.
With regard to Surfaces, the determination Determination of a jjoint in space requires that the values of of a point in three co-ordinates should be assigned. The •'^pace. system generally adoj^ted, which corresj^onds with the rectilinear system of plane geometry, is that of distances from the point to three fixed planes, usually perpendicular to each other, Avhereby the point is presented as the inter- section of three ])lanes whose direction is invariable. Be- yond this, there is the same infinite variety among possible systems of co-ordinates, that there is in geometry of two tlimensions. Instead of the intersection of two lines, it must be that of three surfaces which determines the point; and each of the three surfaces has, in the same way, all its conditions constant, except one, which gives rise to the corresponding co-ordinates, whose peculiar geometrical effect is thus to compel the point to be situated upon that sm-face. Again, if the three co-ordinates of a point are mutually independent, that point can take successively all possible positions in space; but, if its position on any sur- face is defined, two co-ordinates suffice for determining its situation at any moment; as the proposed surface will take