Weights on Earth

PROPOSITION XX. PROBLEM IV: Find and compare together the weights of bodies in the different regions of our earth.

Because the weights of the unequal legs of the canal of water ACQqca are equal; and the weights of the parts proportional to the whole legs, and alike situated in them, are one to another as the weights of the wholes, and therefore equal betwixt themselves; the weights of equal parts, and alike situated in the legs, will be reciprocally as the legs, that is, reciprocally as 230 to 229. And the case is the same in all homogeneous equal bodies alike situated in the legs of the canal. Their weights are reciprocally as the legs, that is, reciprocally as the distances of the bodies from the centre of the earth. Therefore if the bodies are situated in the uppermost parts of the canals, or on the surface of the earth, their weights will be one to another reciprocally as their distances from the centre. And, by the same argument, the weights in all other places round the whole surface of the earth are reciprocally as the distances of the places from the centre; and, therefore, in the hypothesis of the earth’s being a spheroid are given in proportion.

Whence arises this Theorem, that the increase of weight in passing from the equator to the poles is nearly as the versed sine of double the latitude; or, which comes to the same thing, as the square of the right sine of the latitude; and the arcs of the degrees of latitude in the meridian increase nearly in the same proportion. And, therefore, since the latitude of Paris is 48° 50′, that of places under the equator 00° 00′, and that of places under the poles 90°; and the versed sines of double those arcs are 11334,00000 and 20000, the radius being 10000; and the force of gravity at the pole is to the force of gravity at the equator as 230 to 229; and the excess of the force of gravity at the pole to the force of gravity at the equator as 1 to 229; the excess of the force of gravity in the latitude of Paris will be to the force of gravity at the equator as 1 × {\displaystyle \scriptstyle \times } 11334⁄20000 to 229, or as 5667 to 2290000. And therefore the whole forces of gravity in those places will be one to the other as 2295667 to 2290000. Wherefore since the lengths of pendulums vibrating in equal times are as the forces of gravity, and in the latitude of Paris, the length of a pendulum vibrating seconds is 3 Paris feet, and 8½ lines, or rather because of the weight of the air, 85⁄9 lines, the length of a pendulum vibrating in the same time under the equator will be shorter by 1,087 lines. And by a like calculus the following table is made.

Latitude of the place. Length of the pendulum Measure of one degree in the meridian. Deg. Feet Lines. Toises. 0 3 . 7,468 56637 5 3 . 7,482 56642 10 3 . 7,526 56659 15 3 . 7,596 56687 20 3 . 7,692 56724 25 3 . 7,812 56769 30 3 . 7,948 56823 35 3 . 8,099 56882 40 3 . 8,261 56945 1 3 . 8,294 56958 2 3 . 8,327 56971 3 3 . 8,361 56984 4 3 . 8,394 56997 45 3 . 8,428 57010 6 3 . 8,461 57022 7 3 . 8,494 57035 8 3 . 8,528 57048 9 3 . 8,561 57061 50 3 . 8,594 57074 55 3 . 8,756 57137 60 3 . 8,907 57196 65 3 . 9,044 57250 70 3 . 9,162 57295 75 3 . 9,258 57332 80 3 . 9,329 57360 85 3 . 9,372 57377 90 3 . 9,387 57382 By this table, therefore, it appears that the inequality of degrees is so small, that the figure of the earth, in geographical matters, may be considered as spherical; especially if the earth be a little denser towards the plane of the equator than towards the poles.

Now several astronomers, sent into remote countries to make astronomical observations, have found that pendulum clocks do accordingly move slower near the equator than in our climates. And, first of all, in the year 1672, M. Richer took notice of it in the island of Cayenne; for when, in the month of August, he was 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, fitting 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 at Paris (which was 3 Paris feet and 83⁄5 lines), he found it shorter by 1¼ line.

Afterwards, our friend Dr. Halley, about the year 1677, arriving at the island of St. Helena, found his pendulum clock to go slower there than at London without marking the difference. But he shortened the rod of his clock by more than the 1⁄8 of an inch, or 1½ line; and to effect this, be cause the length of the screw at the lower end of the rod was not sufficient, he interposed a wooden ring betwixt the nut and the ball.

Then, in the year 1682, M. Varin and M. des Hayes found the length of a simple pendulum vibrating in seconds at the Royal Observatory of Paris to be 3 feet and 85⁄9 lines. And by the same method in the island of Goree, they found the length of an isochronal pendulum to be 3 feet and 65⁄9 lines, differing from the former by two lines. And in the same year, going to the islands of Guadaloupe and Martinico, they found that the length of an isochronal pendulum in those islands was 3 feet and 6½ lines.

After this, M. Couplet, the son, in the month of July 1697, at the Royal Observatory of Paris, so fitted his pendulum clock to the mean motion of the sun, that for a considerable time together the clock agreed with the motion of the sun. In November following, upon his arrival at Lisbon, he found his clock to go slower than before at the rate of 2′ 13″ in 24 hours. And next March coming to Paraiba, he found his clock to go slower than at Paris, and at the rate 4′ 12″ in 24 hours; and he affirms, that the pendulum vibrating in seconds was shorter at Lisbon by 2½ lines, and at Paraiba, by 3⅔ lines, than at Paris. He had done better to have reckoned those differences 1⅓ and 25⁄9: for these differences correspond to the differences of the times 2′ 13″ and 4′ 12″. But this gentleman’s observations are so gross, that we cannot confide in them.

In the following years, 1699, and 1700, M. des Hayes, making another voyage to America, determined that in the island of Cayenne and Granada the length of the pendulum vibrating in seconds was a small matter less than 3 feet and 6½ lines; that in the island of St. Christophers it was 3 feet and 6¾ lines; and in the island of St. Domingo 3 feet and 7 lines.

And in the year 1704, P. Feuillé, at Puerto Bello in America, found that the length of the pendulum vibrating in seconds was 3 Paris feet, and only 57⁄12 lines, that is, almost 3 lines shorter than at Paris; but the observation was faulty. For afterward, going to the island of Martinico, he found the length of the isochronal pendulum there 3 Paris feet and 510⁄12 lines.

Now the latitude of Paraiba is 6° 38′ south; that of Puerto Bello 9° 33′ north; and the latitudes of the islands Cayenne, Goree, Gaudaloupe, Martinico, Granada, St. Christophers, and St. Domingo, are respectively 4° 55′, 14° 40″, 15° 00′, 14° 44′, 12° 06′, 17° 19′, and 19° 48′, north. And the excesses of the length of the pendulum at Paris above the lengths of the isochronal pendulums observed in those latitudes are a little greater than by the table of the lengths of the pendulum before computed. And therefore the earth is a little higher under the equator than by the preceding calculus, and a little denser at the centre than in mines near the su face, unless, perhaps, the heats of the torrid zone have a little extended the length of the pendulums.

For M. Picart has observed, that a rod of iron, which in frosty weather in the winter season was one foot long, when heated by lire, was lengthened into one foot and ¼ line. Afterward M. de la Hire found that a rod of iron, which in the like winter season was 6 feet long, when exposed to the heat of the summer sun, was extended into 6 feet and ⅔ line. In the former case the heat was greater than in the latter; but in the latter it was greater than the heat of the external parts of a human body; for metals exposed to the summer sun acquire a very considerable degree of heat. But the rod of a pendulum clock is never exposed to the heat of the summer sun, nor ever acquires a heat equal to that of the external parts of a human body; and, therefore, though the 3 feet rod of a pendulum clock will indeed be a little longer in the summer than in the winter season, yet the difference will scarcely amount to ¼ line. Therefore the total difference of the lengths of isochronal pendulums in different climates cannot be ascribed to the difference of heat; nor indeed to the mistakes of the French astronomers. For although there is not a perfect agreement betwixt their observations, yet the errors are so small that they may be neglected; and in this they all agree, that isochronal pendulums are shorter under the equator than at the Royal Observatory of Paris, by a difference not less than 1¼ line, nor greater than 2⅔ lines. By the observations of M. Richer, in the island of Cayenne, the difference was 1¼ line. That difference being corrected by those of M. des Hayes, becomes 1½ line or 1¾ line. By the less accurate observations of others, the same was made about two lines. And this dis agreement might arise partly from the errors of the observations, partly from the dissimilitude of the internal parts of the earth, and the height of mountains; partly from the different heats of the air.

I take an iron rod of 3 feet long to be shorter by a sixth part of one line in winter time with us here in England than in the summer. Because of the great heats under the equator, subduct this quantity from the difference of one line and a quarter observed by M. Richer, and there will remain one line 1⁄12, which agrees very well with 187⁄1000 line collected, by the theory a little before. M. Richer repeated his observations, made in the island of Cayenne, every week for ten months together, and compared the lengths of the pendulum which he had there noted in the iron rods with the lengths thereof which he observed in France. This diligence and care seems to have been wanting to the other observers. If this gentleman’s observations are to be depended on, the earth is higher under the equator than at the poles, and that by an excess of about 17 miles; as appeared above by the theory.