« ΠροηγούμενηΣυνέχεια »
daily rotation about his axis in 9 hours and 56 minutes; the centrifugal force exerted upon him is therefore considerably greater than that which affects the earth; and the density of Jupiter is less than that of the earth. Hence the figure of Jupiter ought to be much more oblate or compressed at the poles than the earth is. This was found to be the case by an astronomical admeasurement of his diameters. The equatorial diameter of Jupiter is to the polar as 13; 12, and is longer than the polar by about 6230 miles. Saturn and Mars exhibit the same oblate-spheroidal appearance. The other planets do not offer facilities for ascertaining the effect of a centrifugal force upon them. With respect to the moon, her motion about her axis is too slow (being performed in 29 days) to produce enough centrifugal force to make a difference in her diameters, arising from this cause, observable. She has, it is true, a spheroidal form, but this is owing to the attraction of the earth, which is four times greater than what is caused by her motion of rotation.
The horizontal parallaxes of the moon furnish another though subordinate proof of the earth's spheroidal form. If the earth were a perfect sphere, these parallaxes would be the same for all places upon the earth's surface-if a spheroid, they would be different at different places; and this is actually the case; so that the same heavenly body which by her eclipses indicates the earth to be round, by her parallaxes shows it to be not truly spherical.
But the more direct evidence of the existence of a centrifugal force and of the earth's ellipticity, and the means of determining the amount of it, are to be looked for in the experiments made with the pendulum, and in the measurement and comparison of the lengths of degrees of the meridian in different latitudes.
The nature of the evidence afforded by pendulum experiments may be explained by a reference to a few very obvious principles.
The centrifugal force caused by the earth's rotation is (we have seen) greatest at the equator and decreases towards the poles; this centrifugal force, either in its whole quantity or in part, acts in a direction opposite to that of gravity, and therefore, being greatest at the equator, it diminishes the force of gravity at the equator by the greatest
quantity. Hence, bodies are lightest at the equator, and their weight gradually increases as we proceed towards the poles, where it is greatest. The pendulum of a clock performs its vibrations, or swinging motion to and fro, by being acted upon by the force of gravity.
If, therefore, two pendulums Sp, Sp (fig. 11), be taken of the same length, and the same substance or density, and be hung from the same point S, and SP be the vertical position of both, and they be made to fall from equal distances from P, it is evident they will. move through the equal spaces Pp and Pp in exactly the same time, because the force which causes their descent (namely that of gravity) is equal in both cases. But if one of these pendulums (Sp) be made to swing at the equator (being let fall from the same height as before), and the other (Sp') at some place between the equator and the poles (say at Paris), the time of Sp arriving at P will be longer than the time in which Sp will move through the same space; because the force of gravity, which causes and accelerates the motions of both, is less at the equator than at Paris. But if we shorten the equator pendulum by some determinate quantity, it may be made to perform its vibrations in the same time with the Paris pendulum; for it will then have to describe a similar are of a smaller circle or a less space, and a less accelerating force will enable it to describe this space in the same time. Such is the effect of the centrifugal force upon the vibrations of the pendulum in different latitudes. The existence of it was first detected by Richter, a French astronomer, who, in the year 1672, was sent to make astronomical observations in the island of Cayenne, which is not quite 5° north of the equator. Sir Isaac Newton has, in his Prin
cipia, described the particulars of the discovery. He says, that when Richter was, in the month of August, observing the transits of the fixed stars over the meridian, he found his clock to go slower than it ought, in respect of the mean motion of the sun, at the rate of 2' 28" a day. Therefore, setting up a simple pendulum to vibrate in seconds, which were measured by an excellent clock, he observed the length of that simple pendulum; and this he did over and over every week for ten months together; and upon his return to France, comparing the length of that pendulum with the length of the pendulum beating seconds at Paris, he found it shorter by 14 line. In accounting for this difference in the length of the two pendulums, Newton allowed th of a line as the utmost that could be attributed to the extension of the pendulum by the heat of the climate; the difference, or 1 line by which this pendulum was shorter than the Paris one, was made necessary by the less gravity of bodies at and near the equator. From this fact he obtained the same conclusions he had before deduced from theory alone, namely, that the equatorial diameter of the earth was greater than the polar by the 229th part of the whole diameter. Since that time observations upon the lengths of pendulums beating seconds in different latitudes, have been made with great assiduity by scientific men of all countries; but recent experiments tend to show that the earth's ellipticity is not so great as; the fraction is the value which results from the latest investigations.
Length and Measurement of Degrees
upon the Earth's Surface.
THE remaining evidence of the earth's ellipticity is the different lengths of degrees of the meridian arc in different latitudes.
A degree of a meridian is that portion of it which must be travelled over, in order to change the altitude of any particular star, by the 360th part of the imaginary meridian circle in the heavens: if the spaces travelled over in different parts of the same terrestrial meridian, in order to produce this change in the altitude of a star, be not equal
A line is a small French measure equal to the twelfth part of an inch.
to one another, the terrestrial meridian cannot have the same curvature in every part, and is therefore not a circle; and consequently, the figure of the earth on the surface of which the meridian is traced cannot be a perfect sphere. Now it has been found by trial, that to raise the pole star by a quantity equal to a celestial degree, an observer must travel over a greater and increasing space as he proceeds from the equator to the pole. Hence it follows, that the degrees of a meridian line on the earth, or degrees of latitude, gradually increase from the equator to the pole; the meridian has, therefore, less curvature at the poles than at the equator, and the earth upon which it is traced is not a perfect sphere, but is flattened at the poles.
It is not to be immediately concluded from this that the earth is a regular oblate-spheroid; but it has been justly remarked, that, though it is only by experiment that the true figure of the meridian can be discovered, it has been found necessary to assume hypothetically (or by way of supposition), that its figure is the curve next in simplicity to the circle, viz. the ellipse, and also to suppose that the earth is a spheroid generated by the revolution of this ellipse about its shorter axis; for, in many complex cases, this mode of getting near the truth by probable suppositions, has been found the simplest and most convenient to be pursued; the only caution to be observed, is to submit the supposition first made to the test and correction of actual experiment. This caution has been carefully attended to in the matter we are discussing, by the measurement and comparison of degrees at various parts of the earth's surface.
In the measurement of a degree or of an arc of a meridian, many difficulties present themselves in the way of an actual and mechanical measurement. The general features of a country are such as to make any attempt of this kind unadvisable; a great number of almost conjectural allowances must be admitted into such a plan of operations, which forbid our placing much confidence in the result. The first modern measurement, having any just claim to accuracy, was, however, made in this manner. This was the measurement by Norwood, in 1635. The arc measured was that part of the meridian which lies between London and York. The difference of the latitudes of these cities was first ascertained; this gave the
number of degrees in the arc to be measured; the distance between the two cities was then actually measured; and the turnings and windings of the road, and the ascents and descents, were allowed for afterwards. The length of a degree thus determined was 122,399 English yards; which, notwithstanding the extreme liability of this method to error, is not very far from the truth; according to the latest determinations, the length of a degree between these latitudes is 121,660 yards. The only
other instance of the actual measurement of an arc of the meridian is that of Messrs. Mason and Dixon. They measured an arc of the meridian of 179,359.313 English yards in length, in the state of Pennsylvania. An aecount of this measurement is given in the Philosophical Transactions for the year 1768. The other and more accurate mode of finding the length of a degree, is a combination of actual measurement and of trigonometrical calculations founded upon it. All geodesical* operations (as they are called) are now conducted according to this method. Two places are selected which lie under the same meridian, or nearly so; the difference of their latitudes, which gives the number of degrees in the arc to be measured, is then ascertained with the utmost precision. A base line of a few miles in extent, and at some little distance from the meridian arc, is then very carefully measured; this is the only actual measurement which need be made. The extremities of this base line are connected with the extremities of the meridian arc, by imaginary triangles; the sides of which are not measured, but, by the aid of the base line, and by means of the angles of the triangles, which are all ascertained by an instrument for measuring angles, are determined by trigonometrical computation. This mode of ascertaining the length of the meridian will, however, be set in a clearer light by following the steps of the process in the subjoined figure.
Let (fig. 12) M m represent a meridian arc; the difference of latitudes of the two extremities, M and m, being found, the length of a degree in the latitude of M and m will be the length of the whole are divided by the number of degrees contained in it.
A level plain is then to be selected,
be connected with the two extremities of the arc M m, by a series of triangles. For this purpose convenient stations are fixed upon, such that the three stations situate in the three angles of every triangle may be visible to each other. Let C, D, E, be the stations fixed upon, these are supposed to be connected together, and with the points M, m, by the imaginary lines which form the various triangles ABC, BC m, ACD, CDE, and DEM. Then the angles by which the two stations C and B, appear to be separated from each other when viewed from the station A, is observed. This observation gives the angle C A B of the triangle ABC; the other angles of this triangle are observed and determined in the same manner, and the side A B, which is the base line, being known from measurement, the other two sides A C, B C, may be computed by plane trigonometry. By this means we obtain a side of each of the triangles B C m, AC D, and are enabled to continue the process without measuring any more sides. The angles of these triangles are measured as in the case of the first, and their sides are ascertained in like manner by trigonometry; and by proceeding in a similar way in the resolution of the whole series of triangles, the sides and angles of all are determined. The remaining step in the field proceedings is to ascertain the inclination of the lines M D and B m, to the meridian arc; astronomy affords
• From two Greek words, which combined, signify the means of doing this. From the data
a dividing or apportioning of the earth,
furnished by these operations, the length of the arc M m is determined in the following manner:-from M and D, draw the lines Ma perpendicular to D a, parallel with the meridian line, meeting each other in a; D b, A b, A c, B c, m d, B d, are also drawn so as to be respectively perpendicular to and parallel with the meridian. Then it is evident that the length of M m is equal to the sum of the lengths of a D, 6 A, c B, B d, which are found thus:-the inclination of MD to the meridian having been already determined by an astronomical observation, the angle D Ma in the right-angled triangle D Ma is known from it, and the side M D is also known, so that D a (which is equal to M D x sin. D M a) may at once be computed by trigonometrical tables. In a similar manner the sides b A, c B, d B are computed, and the sum of the whole gives the length of the meridian arc M m, and the length of a degree is the length of the whole are divided by the number of degrees contained in it.
Picard was the first person who measured an arc of the meridian by this method. The operation was performed in the year 1670; the arc commenced near Paris, and extended northward; the result of the measurement gave, as the length of a degree in latitude 4940, 121,627 yards, which differs only 35 yards from what is now considered as the most exact length; an accuracy which is justly supposed to be quite
Hounslow-heath with a steel chain of exquisite workmanship. The same base had been measured three years before by General Roy, with glass rods, and the two measurements (in a length of five miles) differed only 23 inches. The French base was measured with rods of platina, that in Lapland with rods of iron, and an allowance was made for the changes of temperature affecting the length of the rods in the course of the operation. In a previous measurement in Lapland, the French astronomers, in order to guard against the extreme contracting effect of cold upon metals, employed rods of deal; this was the more necessary in that measurement, as it was performed in the depth of winter, and the frozen surface of a river was selected for the base line, with a view to obtain as level a plain as possible. It is usual also, in order to prove the correctness of the geodesical process, to measure, towards the conclusion, what is called a base of verification. We have already stated that all the sides of the series of triangles (with the exception of the base line AB, which is a side in the first triangle) are not measured but computed: to verify all the previous steps in the process, the length of one of the sides of the triangles, as it has been deduced from computation, is compared with its length determined by actual measurement. The side of the triangle thus measured is called a base of verification, and is taken as far distant from the first base as circumstances will admit. In the French operations the base of verification was distant between four and 500 miles from the first base, and was 7 miles in length, and yet the difference between its computed length and that obtained from its actual measurement did not amount to 12 inches.
From an inspection of the table before given, it appears that the length of a degree from the equator to the pole increases the curvature therefore diminishes, and the earth is not a sphere but is flattened at the poles, and the polar diameter is less than the equatorial; and although the various modern measurements may not, on a comparison one with another, agree in giving to the difference of the two diameters precisely the same value, yet they all ascertain the fact of the polar diameter being less than the equatorial, and that a degree increases towards the poles; and this establishes the oblate-spheroidal figure of the earth.
The value of the compression or the fraction expressing the difference between the two diameters, as deduced from a comparison of the lengths of a meridional degree in different latitudes, determined by the most approved measurements, has been lately shown by Professor Airy, in a paper in the last volume of the Philosophical Transactions to be that is, the polar diameter is less than the equatorial by the 278.6th part of the whole diameter.
Operations are now being carried on, on the continent, which have for their object the more precise determination of the fraction of ellipticity, and of the compression of the earth. The measurement of an are of the parallel of latitude 45°, of 15° or 16° in extent has been already accomplished. One extremity of this are is at Marennes, on the west coast of France, and a little to the north of the Garonne, and traversing France, Piedmont, and the northern parts of Italy, its other extremity is at Fiume, in the Austrian dominions, and on the eastern shores of the Adriatic. The value of the ellipticity as deduced from these operations is We have already stated that the pendulum experiments give. This similarity in the results afforded by such very different kinds of investigation is a strong argument in favour of the general correctness of both.
The mean degree of a meridian or the degree the length of which is as much greater than that of a degree at the equator, as it is less than that of a degree at the poles, is in latitude 45°, which is the mean latitude between the equator and the poles. Its length, according to the French measurement, is 60759.4 fathoms, or 12158.8 yards. The circumference of the elliptic meridian is found by multiplying the mean degree by 360, and is equal to 24855.84 miles. The circumference of the equator is 24896.16 miles, and is not quite 41 miles longer than the elliptic meridian.
The French measurement, in 1792, was undertaken with a view to obtain a standard measure of length, to serve as the basis of a new system of weights and measures. According to this new system, the unit, or first element of linear measure, is called a metre; and the metre was declared to be equal to ten millionth part of the quadrant of the meridian-which is a fixed and un
alterable quantity in nature. The quadrant of the meridian was by this measurement found to be 5,130,740 toises, or 10,936,578 English yards: the French metre, or the ten millionth part of this quantity, would accordingly be 1,093,578 yards, or 39.37 inches, nearly. This method of obtaining a standard of measure is not, perhaps, so good as that which consists in observing the length of the pendulum, which, in a certain latitude, beats seconds of mean time. For the length of this pendulum is ultimately ascertained by a reference to the equable motion of the earth upon her axis, and is, therefore, ascertainable without the aid and use of any linear measure whatever; whereas, in the very act of determining the French standard, or the quadrant of the meridian, some linear measure already in use must be employed; and thus the very basis of their new system is expressed in terms of that in the place of which it is sub
The importance of possessing the true length of a degree of the meridian, is not confined to investigations having for their object the determination of the figure of the earth. Upon the simple fact of the length of a degree, seemed to depend the overthrow or establishment of the theory of Universal Gravitation. The particulars connected with the discovery of a principle productive of such various effects in nature, is not the less interesting in that it illustrates the secret dependency of parts of science apparently the most distinct, and the assistance which each in its place is calculated to afford to the rest.
The corner-stone of the whole system of Universal Gravitation is, that the force which causes a heavy body to descend to the surface of the earth, is the same that retains the moon in her orbit, and makes her deflect from a straight line, or bend towards the earth. All that was requisite to establish the identity of the forces by which these two effects were produced, was to prove, that the quantity of effect produced in a certain time upon the moon in thus deflecting from a straight line, (taking into consideration the law by which the force varied, and the distance of the moon,) was in due proportion to the effect produced by the force of gravity, in the same time, upon a falling body at the It is evident, surface of the earth. therefore, that the determination of this question depended upon, and would in