CHAP. III. DEGREES IN DIFFERENT LATITUDES.

113

resulting from actual measurement made with all possible care and precision, by commissioners of various nations, men of the first eminence, supplied by their respective governments with the best instruments, and furnished with every facility which could tend to ensure a successful result of their important labours.*

It is evident from a mere inspection of the second and fourth columns of this table that the measured length of a degree increases with the latitude, being greatest near the poles, and least near the equator. Let us now consider what interpretation is to be put upon this conclusion, as regards the form of the earth.

(173.) Suppose we held in our hands a model of the earth smoothly turned in wood, it would be, as already observed, so nearly spherical, that neither by the eye nor the touch, unassisted by instruments, could we detect any deviation from that form. Suppose, too, we were debarred from measuring directly across from surface to surface in different directions with any instrument, by which we might at once ascertain whether one diameter were longer than another; how, then, we may ask, are we to ascertain whether it is a true sphere or not? It is clear that we have no resource, but to endeavour to discover, by some nicer

*The first three columns of this table are extracted from among the data given in Professor's Airy's excellent paper "On the Figure of the Earth," in the Encyclopædia Metropolitana.

means than simple inspection or feeling, whether the convexity of its surface is the same in every part; and if not, where it is greatest, and where least. Suppose, then, a thin plate of metal to be cut into a concavity at its edge, so as exactly to fit the surface at A: let this

B

now be removed from A, and applied successively to several other parts of the surface, taking care to keep its plane always on a great circle of the globe, as here represented. If, then, we find any position, B, in which the light can enter in the middle between the globe and plate, or any other, C, where the latter tilts by pressure, or admits the light under its edges, we are sure that the curvature of the surface at B is less, and at C greater, than at A.

(174.) What we here do by the application of a metal plate of determinate length and curvature, we do on the earth by the measurement of a degree of variation in the altitude of the pole. Curvature of a surface is nothing but the continual deflection of its tangent from one fixed direction as we advance along it. When, in the same measured distance of advance, we find the tangent (which answers to our horizon) to have shifted its position with respect to a fixed direction in space, (such as the axis of the heavens, or the line joining the earth's centre and some given star,) more in one part of the earth's meridian than in another, we conclude, of necessity, that the curvature of the surface at the former spot is greater than at the latter; and, vice versa, when, in order to produce the same change of horizon with

CHAP. III.

MERIDIONAL SECTION OF THE EARTH. 115

respect to the pole (suppose 1°), we require to travel over a longer measured space at one point than at another, we assign to that point a less curvature. Hence

we conclude that the curvature of a meridional section of the earth is sensibly greater at the equator than towards the poles; or, in other words, that the earth is not spherical, but flattened at the poles, or, which comes to the same, protuberant at the equator.

(175.) Let NA B D E F represent a meridional section of the earth, C its centre, and NA, BD, GE,

arcs of a meridian, each corresponding to one degree of difference of latitude, or to one degree of variation in the meridian altitude of a star, as referred to the horizon of a spectator travelling along the meridian. Let n N, a A, b B, dD, g G, e E, be the respective directions of the plumb-line at the stations N, A, B, D, G, E, of which we will suppose N to be at the pole and E at the equator; then will the tangents to the surface at these points respectively be perpendicular to these directions; and, consequently, if each pair, viz. n N and a A,

b B and d D, g G and e E, be prolonged till they intersect each other (at the points x, y, z), the angles Nx A, By D, G≈ E, will each be one degree, and, therefore, all equal; so that the small curvilinear arcs NA, BD, GE, may be regarded as arcs of circles of one 'degree each, described about x, y, z, as centres. These are what in geometry are called centres of curvature, and the radii x N or x A, y B or y D, ≈ G or ≈ E, represent radii of curvature, by which the curvatures at those points are determined and measured. Now, as the arcs of different circles, which subtend equal angles at their respective centres, are in the direct proportion of their radii, and as the arc NA is greater than BD, and that again than GE, it follows that the radius Nx must be greater than By, and By than E z. Thus it appears that the mutual intersections of the plumb-lines will not, as in the sphere, all coincide in one point C, the centre, but will be arranged along a certain curve, xyz (which will be rendered more evident by considering a number of intermediate stations). To this curve geometers have given the name of the evolute of the curve NABDGE, from whose centres of curvature it is constructed.

(176.) In the flattening of a round figure at two opposite points, and its protuberance at points rectan gularly situated to the former, we recognize the distinguishing feature of the elliptic form. Accordingly, the next and simplest supposition that we can make respecting the nature of the meridian, since it is proved not to be a circle, is, that it is an ellipse, or nearly so, having NS, the axis of the earth, for its shorter, and EF, the equatorial diameter, for its longer axis; and that the form of the earth's surface is that which would arise from making such a curve revolve about its shorter axis NS. This agrees well with the general course of the increase of the degree in going from the equator to the pole. In the ellipse, the radius of curvature at E, the extremity of the longer axis is the least, and at that of the shorter axis, the greatest it admits, and the

CHAP. III. EXACT DIMENSIONS OF THE EARTH.

117

form of its evolute agrees with that here represented.* Assuming, then, that it is an ellipse, the geometrical properties of that curve enable us to assign the proportion between the lengths of its axes which shall correspond to any proposed rate of variation in its curvature, as well as to fix upon their absolute lengths, corresponding to any assigned length of the degree in a given latitude. Without troubling the reader with the investigation, (which may be found in any work on the conic sections,) it will be sufficient to state that the lengths which agree on the whole best with the entire series of meridional arcs which have been satisfactorily measured, are as follow†:

The proportion of the diameters is very nearly that of 298 299, and their difference of the greater, or a very little greater than goō

2

(177.) Thus we see that the rough diameter of 8000 miles we have hitherto used, is rather too great, the ex-. cess being about 100 miles, or th part. We consider it extremely improbable that an error to the extent of five miles can subsist in the diameters, or an uncertainity to that of a tenth of its whole quantity in the compression just stated. As convenient numbers to remember, the reader may bear in mind, that in our latitude there are just as many thousands of feet in a degree of the meridian as there are days in the year (365): that, speaking loosely, a degree is about 70 British statute miles, and a second about 100 feet; and that the equatorial circumference of the earth is a little less than 25,000 miles (24,899).

(178.) The supposition of an elliptic form of the earth's section through the axis is recommended by its * The dotted lines are the portions of the evolute belonging to the other quadrants.

† See Profess. Airy's Essay before cited,