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length, is not beyond the limits of very exact measurement, and could, in fact, be measured (if we knew its exact termination at each extremity) within a very few feet, or, indeed, inches, by methods presently to be particularized.

(166.) Supposing, then, we were to begin measuring with all due nicety from any station, in the exact direction of a meridian, and go measuring on, till by some indication we were informed that we had accomplished an exact degree from the point we set out from, our problem would then be at once resolved. It only remains, therefore, to enquire by what indications we can be sure, Ist, that we have advanced an exact degree ; and, 2dly, that we have been measuring in the exact direction of a great circle.

(167.) Now, the earth has no landmarks on it to indicate degrees, nor traces inscribed on its surface to guide us in such a course. The compass, though it affords a tolerable guide to the mariner or the traveller, is far too uncertain in its indications, and too little known in its laws, to be of any use in such an operation. We must, therefore, look outwards, and refer our situation on the surface of our globe to natural marks, external to it, and which are of equal permanence and stability with the earth itself. Such marks are afforded by the stars. By observations of their meridian altitudes, performed at any station, and from their known polar distances, we conclude the height of the pole; and since the altitude of the pole is equal to the latitude of the place (art. 95.), the same observations give the latitudes of any stations where we may establish the requisite instruments. When our latitude, then, is found to have diminished a degree, we know that, provided we have kept to the meridian, we have described one three hun. dred and sixtieth part of the earth's circumference.

(168.) The direction of the meridian may be secured at every instant by the observations described in art. 137.; and although local difficulties may oblige us to deviate in our measurement from this exact direcCHAP. III. LENGTH OF A DEGREE OF LATITUDE. 111

tion, yet if we keep a strict account of the amount of this deviation, a very simple calculation will enable us to reduce our observed measure to its meridional value.

(169.) Such is the principle of that most important geographical operation, the measurement of an arc of the meridian. In its detail, however, a somewhat modified course must be followed. An observatory cannot be mounted and dismounted at every step; so that we cannot identify and measure an exact degree neither more nor less. But this is of no consequence, provided we know with equal precision how much, more or less, we have measured. In place, then, of measuring this precise aliquot part, we take the more convenient me. thod of measuring from one good observing station to another, about a degree, or two or three degrees, as the case may be, apart, and determining by astronomical observation the precise difference of latitudes between the stations.

(170.) Again, it is of great consequence to avoid in this operation every source of uncertainty, because an error committed in the length of a single degree will be multiplied 360 times in the circumference, and nearly 115 times in the diameter of the earth con. cluded from it. Any error which may affect the astronomical determination of a star's altitude will be especially influential. Now there is still too much uncertainty and fluctuation in the amount of refraction at moderate altitudes, not to make it especially desirable to avoid this source of error. To effect this, we take care to select for observation, at the extreme stations, some star which passes through or near the zeniths of both. The amount of refraction, within a few degrees of the zenith, is very small, and its fluctuations and uncertainty, in point of quantity, so excessively minute as to be utterly inappreciable. Now, it is the same thing whether we observe the pole to be raised or depressed a degree, or the zenith distance of a star when on the meridian to have changed by the same quantity. If at one station we observe any star to pass through the zenith, and at the other to pass one degree south or north of the zenith, we are sure that the geographical latitudes, or the altitudes of the pole at the two stations, must differ by the same amount.

(171.) Granting that the terminal points of one degree can be ascertained, its length may be measured by the methods which will be presently described, as we have before remarked, to within a very few feet. Now, the error which may be committed in fixing each of these terminal points cannot exceed that which may be committed in the observation of the zenith distance of a star, properly situated for the purpose in question. This error, with proper care, can hardly exceed a single second. Supposing we grant the possibility of ten feet of error in the measured length of one degree, and of one second in each of the zenith distances of one star, observed at the northern and southern stations, and, lastly, suppose all these errors to conspire, so as to tend all of them to give a result greater or all less than the truth, it will appear, by a very easy proportion, that the whole amount of error which would be thus entailed on an estimate of the earth's diameter, as concluded from such a measure, would not exceed 544 yards, or about the third part of a mile, and this would be large allowance.

(172.) This, however, supposes that the form of the earth is that of a perfect sphere, and, in consequence, the lengths of its degrees in all parts precisely equal. But when we come to compare the measures of meridional arcs made in various parts of the globe, the results obtained, although they agree sufficiently to show that the supposition of a spherical figure is not very remote from the truth, yet exhibit discordances far greater than what we have shown to be attributable to error of observation, and which render it evident that the hypothesis, in strictness of its wording, is unten. able. The following table exhibits the lengths of a degree of the meridian (astronomically determined as above described), expressed in British standard feet, as



resulting from actual measurement made with all pos. sible 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 en. sure a successful result of their important labours, *

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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

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

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