925 



SURFACE, SURFACES, THEORY OF. 



SURFACE OF THE EARTH. 



920 



SURFACE, SURFACES, THEORY OF. For the mere definition 

 of surface, see SOLID, &c. We are here to speak of that branch of 

 algebraic geometry which considers the generation and properties of 

 curve surfaces following an assigned law. 



If three planes, each at right angles to the other two, be taken as 

 the planes of CO-OHDIXATES, the position of any point is determined 

 so soon aa ito co-ordinates, or distances from the three planes, are given 

 in sign and magnitude. If the co-ordinates of a point be .T, y, z, and 

 if between these one equation exists, (j% y, z) = 0, any point may be 

 chosen in the plane of x and y, by means of given values of x and y, 

 and the corresponding value or values of z may be found from the 

 equation. The locus of all the points whose position can be ascertained 

 by determining one of the co-ordinates from this equation, the other 

 two being taken at pleasure, is a surface of which $ (x, y, 2) = is 

 called the equation, and the modes of proceeding are pointed out 

 in all works on algebraic geometry. The applications of the differ- 

 ential calculus depend on the principles explained in TAXCIENT : the 

 graphical use of the whole method depends mostly on descriptive geo- 

 metry, whether formally known under that name or not. 



Surfaces are distinguished algebraical]}' by the nature and order 

 of their equations. Thus we have surfaces of the first order, in which 

 the equation is of the first degree (this class contains the plane only) ; 

 surfaces of the second order, which will be classified in the next 

 article ; and go on. 



Surfaces are also distinguished by their mode of generation, and 

 some of the principal cases are as follows : 



1. Cglinrlriml surfaea are generated by a straight line infinitely 

 prcxluced in both directions, which moves so a always to be parallel to 

 a given line, and to have one of ita points on a given curve. 



mieal mrfatet are generated by a straight line infinitely pro- 

 duced in both directions, which always passes through a given point 

 or vertex, and has one point in a given curve. The common 

 CVLIXDER and CONE would be described in this science as a r'ujht 

 rirnilnr cylinder and a ri;/lit circular cone. The cylindrical surfaces 

 thi-mstlven are only an extreme case of the conical ones, being what 

 tin- latter become when the vertex is removed to an infinite 



.nee. 



- arfartt of revolution are generated by the rotation of a curve 

 about an axis, relatively to which it always retains one position. The 

 uniMiiion cone and cylinder, the SPHEKE, and others of the greatest 

 practical use, are contained in this class. 



4. Tubular ttcrfaca are generated by a circle of given radius, which 

 moves with ito centre on a given curve, and its plane at right angles to 

 the tangent of that curve. When the given curve is a circle, the 

 tut. 'ilar surface is a common ring. 



5. Ruled furfacet (the turfaccs regUes of the French writers) are 

 those which are described by the motion of a straight line, which 

 neither remains parallel to a given line nor always passes through a 

 given point. This includes, among many others, the whole class of 

 crmoirial ntr/acci, made by a straight line which moves parallel to a 

 given plane, and always passes through a straight line perpendicular 

 to that plane, and also through a given curve. The surface of a spiral 

 staircase, as it would be if there were no steps but only a gradual 

 ascent, is an instance. 



8. Developable mrfacft are those which can be unwrapped on a plane 

 without any doubling of parts over one another, or separation ; that is, 

 without being rumpled or torn. The only familiar instances are the 

 cylinder and cone. 



SURFACE OF KLKCTKHTY. [POLARIZATION OP LIGHT.] 



SURFACE OF THK EARTH. Geology, by teaching us to look 

 upon the form and distribution of land and sea, the features of hills and 

 valleys, and the various deposits of peat, silt, gravel, Ac., as effects of 

 physical agencies, some of which are no longer in operation upon those 

 areas where once they predominated, confers upon the surface of the 

 earth an interest much greater than that which belongs merely to 

 i il combinations, or even to agricultural utility and commercial 

 i. in I 'mformity.lnequality, height, depth, and area, every the 

 least peculiarity of form, whatever in remarkable in any part of the 

 surface of the land or bed of the sea these are effects of causes 

 h require to be traced out before the problem of the physical 

 history of the globe can be considered as resolved. Geology was pro- 

 nounced by Sir C. Lyell, early in his career of research, to be the 

 science of surface*. 



ffifial Drpotitt. If the stratified and unstratified rocks which 

 compose the skeleton of the earth were laid bare to our view, the 

 aspect of the globe would be far more nigged than it is now. The 

 valleys would in many cases lose their soft and easy curvatures and 

 accordant slope*, in angular fractures and irregular chasms; the 

 mountains and hills would lose those sloping buttress-like banks, com- 

 posed of fallen materials, which connect the broken ridges above with 

 the level expanse below ; a sterner aspect would belong to the now 

 sinuous lines of sea-coast; and an almost general barrenness would 

 overspread the inland surface. 



The soil, gravel, clay peat, and other substances, which by their 

 accumulation mask the features of the interior rocks, constitute a 

 peculiar class of phenomena which have been much, and yet not suffi- 

 ciently, studied by geologists. It is certain that without a more exact 

 appreciation of the cause* which have permitted the aggregation of the 



" superficial deposits " already named, our analysis of the processes 

 whereby the earth has been made fit for the residence of man, and 

 adapted to its present usea, must be very imperfect. 



Soil is often supposed to be merely the disintegrated parts of the 

 subjacent rocks, and this is sometimes really the case ; trap rocks, for 

 example, of which the felspar and the hornblende become decomposed 

 by the atmosphere, yield a soil often remarkable for fertility, and uncou- 

 taminated with foreign ingredients. But the soils which cover clays 

 and limestones and sandstones are seldom of this simple origin. The 

 bases of these soils may be generally derived from the subjacent strata, 

 but they usually contain foreign ingredients. The soil on the chalk 

 and limestone hills of England is often sandy, sandstones are covered 

 by loam, and clays overspread with pebbles. The effect of this 

 admixture of foreign substances with the disintegrated parts of the 

 native rock is usually favourable to fertility. 



We may often understand the cause of these admixtures by consider- 

 ing the effect of rains and currents of water on the sloping surface of 

 the earth. These effects arrive at a maximum in particular vales and 

 plains, into which many streams enter after flowing over strata of 

 different kinds. In such vales the soil is in fact a mixture of 

 calcareous, argillaceous, and arenaceous parts, and its indigenous plants 

 are correspondingly varied, and include many which are not found 

 growing together on any one of the soils which are here mixed 

 together. [SoiL.] 



To watery agency, acting under the actual circumstances of physical 

 geography, we may also ascribe many even extensive accumulations of 

 gravel and sand which lie along the sides of valleys and in hollows of 

 hills, or on the slopes of mountains ; and it requires sometimes only 

 the postulate, that in particular valleys inundations have formerly 

 reached higher levels than at present, to apply the same explanation 

 to terraces of gravel and sand now considerably above the actual 

 flood-mark, but sloping parallel to the general inclination of the 

 valley. 



The beds of old lakes, often consisting of layers of shelly marl, with 

 bones of existing or extinct quadrupeds, the surfaces of silt which lie 

 along the actual and ancient actuaries of rivers, and often conceal 

 buried forests and subterranean peat, present no difficulty as to their 

 origin. For the processes by which peat grows and trees are buried, 

 and marshy land is saved from the sea, and lakes are filled up, are at 

 this day in action. To all such peat or turf moors, subterranean 

 forests, marsh and fen land, drained lake-beds, and sand and gravel, 

 the title of alluvial depogifa is very commonly given. Generally, they 

 require no supposition of extensive changes of physical geography 

 produced by violent disturbances of nature, but seem to be clearly and 

 perfectly explicable by causes still in action, though perhaps not in the 

 same situations on the earth's surface. But there are other gravels, 

 sands, and clays, to which this explanation cannot be applied without 

 calling in aid great changes of physical geography, or physical processes 

 not seen in daily operation ; such as extensive displacement and change 

 of level of hind and sea ; unusual floods of water ; surprising altera- 

 tions of climate, or movement of glaciers in situations where now 

 the snow and ice of the coldest winter melt with the first breath of 

 spring. These phenomena were classed under the title of diluvial 

 deposit*, at a time when their origin was very generally ascribed to 

 violent floods of water, and the title was retained even by geologists 

 who did not admit this hypothesis. They are now generally included 

 by geologists under the designations of boulder-clay, boulder formation, 

 and northern drift; the latter term having been applied to them, 

 because, in Europe and in North America, where only they have yut 

 been distinctly recognised, they have evidently been brought from 

 regions to the north of those in which they are now found. At tho 

 same time, according to Sir C Lyell, " the bulk of the mass in each 

 locality consists of the ruins of subjacent or neighbouring rocks ; so 

 that it is red in a region of red sandstone, white in a chalk country, 

 and gray or black in a district of coal or coal-shale." They belong 

 both to the newer pleiocene or pleistocene and to the post-pleiocene 

 series of present geological nomenclature. 



These so-called diluvial deposits are commonly admitted or assumed 

 to be of older date than those called alluvial, and, taken in a collective 

 sense, they are so, but this is the least important circumstance 

 characteristic of their history. The conditions of their accumulations 

 are remarkable. 



1 . It is often seen that thick deposits of clay, sand, and pebbles, or 

 large fragments of rock, lie on the very summits of hills (as abun- 

 dantly on the hills which adjoin the valley of the Thames). 



8. Fragments of rocks quite unlike those of the vicinity lie in 

 valleys, on hills (as on the Saleve near Geneva), and even on islands (as 

 on Staffa)., 



3. These fragments are found solitary, or buried in clay, sand, or 

 gravel, and sometimes in enormous abundance, as in Huntingdonshire, 

 near Birmingham, in Holderness, and other parts ; and they are such 

 that no stones of like nature occur anywhere in the natural drainage of 

 the country where the gravel is accumulated, nor within 20, 50, or 

 even 100 miles of the spot. 



4. The fragments (often called boulders, and also erratic blocks) 

 appear thus in several cases to have been transported from particular 

 parts of the country, over elevated ground, across the natural valleys 

 and ranges of hills, but yet are, in some cases, distributed in a manner 



