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 partsthe 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 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, however, 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 any 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

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 preparatory, 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 resolution 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 equivalent 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 analytical representation of natural phenomena.

Analytical The first thing to be done is evidently to representation find and fix a method for expressing analytiof figures. cally the subjects which afford the phenomena. If we can regard lines and surfaces analytically,

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we can so regard, henceforth, the accidents of these subjects.

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 points of which it is composed: 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 geometers 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 in 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 position 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,

the question of position is always reduced to that of magnitude. Position of a point.

It is clear that our only way of marking the position of 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 coordinates, 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 ourselves 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 that Descartes created Analytical Geometry.-Seeing, as we now do, how ideas of position, and, through them, all elementary geometrical ideas,- -can be reduced to ideas of number, we learn what it was that he effected.

Plane Curves.

Descartes treated only geometry of two dimensions in his analytical method: and we will at first consider only this kind, beginning with Plane Expression Curves. Lines must be expressed by equaof lines by tions; and again, equations must be expressed Equations. by lines, when the relation of geometrical conceptions 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 expressed 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

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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.

Expression of equations by lines.

By a converse argument may be seen the geometrical necessity of representing by a certain line every equation of two variables, in a determinate system of co-ordinates. In the absence of any other known property, such a relation would be a very characteristic definition; and its scientific effect would be to fix the attention immediately upon 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 equations, which reacts strongly upon the perfecting of analysis itself. The geometrical locus stands before our minds as the representation 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.

Change in the line changes the equation.

Recurring to the representation of lines by equations, which is our chief object, we see that this representation is, by its nature, so faithful, that the line could not undergo any modification, even the slightest, without causing a corresponding change in the equation. Some special difficulties arise out of this perfect exactness; for since, in our system of analytical geometry, mere displacements of lines affect equations as much as real variations of magnitude or form, we might be in danger of confounding the one with the other, if geometers had not discovered an ingenious method expressly 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