Two general
Methods.

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.

Special or ancient, and general or modern Geometry.

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, 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

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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 Calculus 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.

Geometry of

the ancients.

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 cannot 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 metaphysical 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 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.

Geometry of

the right line.

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 whatever, so as to understand, indirectly, a right line, under any circumstances whatever. The way to do this

Graphical solutions.

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 constrnctions 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. They 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 appli

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Descriptive
Geometry.

cation, 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, 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 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. Leaving the subject of graphic solution, we have to notice the other branch, the algebraic.

Some may wonder that this branch is not Algebraic solutions. 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.

The difficulty lies in forming three distinct Trigonometry. 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