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search of truth, an imitation of the method of the geometers will carry a man further that all the dialectical rules. Their analysis is the proper model we ought to form ourselves upon, and imitate in the regular disposition and gradual progress of our enquiries, and even he who io ignorant of the nature of mathematical analysis uses a method somewhat analogous to it.
The composition of the geometers on their method of demonstratiog truth, already found out, viz. by definitions of words agreed upon, by self-evident truths, and propositions that have been already demonstrated, is practicable in other subjects, though not to the same perfection; the natural want of evidence in the things theinselves not allowing it; but it is imitable to a considerable degree. I dare appeal to some writings of our own age and nation, the authors of which have been mathematically inclined. I shall add no more on this head, but that one who is accustomed to the methodical systems of truths, which the geometers have reared up in the several branches of those sciences which they have cultivated, will hardly bear with the confusion and disorder of other sciences, but endeavour, as far as be can, to reform them.
Thirdly, mathematical knowledge adds a manly vigoor to the mind, frees it from prejudice, credulity, and superstition. This it does two ways, Ist. by accustoming us to examine, and not to take things upon trust: 2ndly: by giving us a clear and extensive knowledge of the system of the world, which, as it creates in us the most profound reverence of the al"mighty and wise Creator to it frees us from the mean and narrow thoughts, which ignorance and sepertition are apt to beget.
The mathematics are friends to religion, inasmuch as they charm the pagsions, res train the impetuosity of imagivation, and parge the mind from error and prejudice. Vice is error, confusion, and false reasoning, and all truth is more or less opposite to it. Besides, mathematical studies may serve' for a pleasant entertainment for those hours, which young men are apt to throw away 'upon their vices; the delightfulness of them being such, as to make solitude not only easy but desirable.
What I have said may, serve to recommend mathematics for acquiring a vigorous constitution of mind; for which purpose they are as useful, as exercise is for procuring health and strength to the body.
I proceed now to show their vast extent and usefulness in other parts of knowledge. And here it might spffice to tell you, that mathematics is the science of quantity, or the art of reasoning about things that are capable of more and less, and that the most part of the objects of our knowledge are such ; as matter, space, number, time, motion, gravity, &c. We have but imperfect ideas of things without quantity, and as imperfect a one of quantity itsell, without the help of mathematics. All the visible works of God Almighty are made in number, weight, and measure ; therefore to consider them, we ought to understand arithmetic, geometry, and staties; and *the greater advances we make in those arts, the more capable we are of considering such things as are the ordinary objects of our conception. Bat this will further appear from particulars.
[To be continued.]
ON THE LENGTHS OF PENDULUMS, VIBRATING SECONDS IN DIF. FERENT LATITUDES ; AND THE DESCENT OF HEAVY BODIES.
[By John Baines, jan.)
“ PHILOSOPHERS,” says General Roy, " are not yet agreed in opi. njon with regard to the exact figure of the earth ; some contending that it is an ellipsoid; others a spheroid; and some that it has no regular figure, that is, not such as would be generated by the revolution of a curve around its axis.". But if the earth were in a state of fluidity at the commencement of its diurnal rotation, its figure, by the laws of gravity, must be that of an oblate spheroid. Accordingly the measurements of the French mathematicians in Lapland, in France, and at the equator, prove that the earth's equatorial diameter is greater than its axis. Colonel Mudge, who conducted the most extensive geodesic operations that were ever carried on in this country, makes the earth's equatorial diametep 41,897,040 English feet, and the polar 41,616,084.* Now, (by art. 217, Simpson's Fluxions, Davis's edition,) the periodical time in which a body will revolve round the earth at its surface, when its velocity is just a counterpoise for the force of gravity there, is equal to the square root of the quotient of the earth's equatorial diameter in feet, divided by the space in feet which a heavy body falls from rest at the earth's surface, in the first second of time, multiplied by the circumference of a circle whose diameter is I. Hence, in the present example, 741897040=1645X3.141592—1614.0004 X3.141592=50708-53 seeds.
But in circles having the same radii, the forces are inversely as the squares of the periodical times, (Ibid. art. 137,) and as the earth revolves on its axis in 23hrs. 56m. 48.=86164 seconds, we have the force of gra. vity to the centrifugal force of a body at the equator arising from the earth's rotation, as 1 1
1 to For as 1 : which, by reduction, becomes &$ 5070-531 86164) 288.76 1155:4, extremely pear. I
75A" Put A= esetben (by Simpson's Fluxions, art. 397), 288.76
4 will express the difference between the earth's equatorial diameter and its polar, the polar diameter being considered as unity. This, in numbers, is 004328854+.000006413=004335267; hence, the earth's axis is to its equatorial diameter as 1 : 1•004335267, or as 692:695, extremely near. But the force of gravity at the equator is to that at either of the poles, inversely as the equatorial diameter is to the axis, that is, as 692: 695 ; consequently the diminution of gravity from the poles to the equator is
• Although the earth's polar diameter is here mentioned, it is not for the purpose of making use of it. It will be seen, by the following process, that the ratio of tbe earth's diameters is found independently of any consideration but that of gravity,
og5ths of the gravity at the poles. Hence it appears that gravitation is different in different latitudes ; and this difference arises from two causes, viz. the spheroidical figure of the earth, by which bodies on different parts of its surface are pot equally acted upon by the attractive power, in consequence of their pot being equally distant from its centre ; and the centrifugal force arising from the rotation of the earth on its axis, by which the power of gravity is unequally diminished from the poles to the equator. However when the spheroid differs but little from a sphere, the diminution will be nearly as the square of the co-sine of the latitude. (Marrat’s Mechanics, art. 449.) Therefore as 1 : 387525 (=the square of the co-sine of the latitude of London 51° 30')::3 : 1:162575, the diminution of gravitation from the poles to London; hence 693.837425 will express the force of gravity at London, when that at the poles is expressed by 695, and that at the equator by 692.
These things being premised, we will proceed to determine the lengths of pendulums vibrating seconds at the earth's surface in different latitudes. Now it has been found by experiment that a pendulum vibrating seconds in the latitude of London, is 39•125 inches long, and that a body falls from rest in the first second of time 16+1 feet. At Pello, lat. 66° 48' N. M. de Maupertius found that the length of the seconds pendulum was 441•17 French lines, equal to 39.1824 English inches; and at Paris, latitude 48° 50' 10' N. 440.57 lines, equal to 39•1291 English inches.
But because the length of the seconds' pendulum, in any latitude, is directly as the force of gravity there, (Marrat’s Mechanics, art. 345,) we can, by calculation, find the length of a pendulum vibrating seconds in any latitude, from the experiments which have been made. Thus,
As 693.837425 : 39•125::695: 39:19056 inches, the length of the seconds' pendulum at the poles ; and
A: 693-837425 : 39125.692 : 93-02 139 inches, the length of the pendulum vibrating seconds at the equator.
Again, because the space descended from rest by a heavy body in the Arst second of time is directly as the length of the seconds' pendulam, we have
As 39•125 :1671::39.19056 : 1611025 feet per second, at the poles.
On these principles we have calculated the following Table of the lengths of the pendulum vibrating seconds, and the space fallen from rest by a heavy body in the first second of time on the earth's surface, for every third degree of latitude from the equator to the poles ; the first in inches and latter in feet.
Length of the Space fallen
Names of Places in or near the specified lat.
Names of Places in or near the specified lat.
by a heavy
39.0431 | 16.0497
Goree; Sana, Arabia; Manilla, Philippines.
Deductions. 1. The length of the seconds' pendalam is 16917, or nearly one-sixth of an inch greater at the poles than at the equator.
2. A heavy body falls •16951 of a foot, or 24 inches more in the first second of time at the poles than it does at the equator,
3. Pendulums which vibrate seconds at the earth's surface will lose time, when removed to places considerably above it; aud this is evidently owing to the diminution of the force of gravity. Thus, a pendulum which measures true time at the earth's surface, will lose one minute per day, when removed to the summit of a inountain whose perpendicular height is 14,600 feet. Hence, an allowance ought to be made when the ratio of the earth's diameters is calculated from observations made on the pendulum in different latitudes, if the elevation of the places of observation above the level of the sea be not the same,
4. If the earth revolved on its axis ja lh. 24 m. 308. and '32 thirds, bodies on its surface would weigh pothing, but be as liable to fall off as to riop on. If it revolved in more time than the above, they would stop on ; but if it revolved in less, they would fly off.
5. If the earth’s rotation on its axis were stopped, the weight of bodies
4 at the equator would be increased
th, or 287.76 1151
part of the whole ; but at the poles, their weights would not be altered. Moreover, as the weight of a body at any place is always proportioval to the force of gravity there, it follows, that a body which weighs one poand or 16 ounces at the poles, would only weigh 15.93095 ounces at the equator, while the earth is revolving on its axis, and 15.98631 ounces, if its rotation should be stopped.
Question 4. By W. Godward. It is required to find that number whose 6th power being taken from its 5th, shall have the greatest remainder possible ?
Question 6. By Aaron Arch, York. To construct the plane triangle there are given the base, the vertical angle, and the ratio of a line from the vertex intersecting the base in a given angle, to the difference between the segments of the base made by the intersecting line.
QUESTION 5. By the same. Three men agree to drink a quart of ale out of the same tankard: it is required to determine the divisions of the vessel made by the surfaces of the liquor at the time each man ceased to drink, supposing the height 7] inches.
Tre following verses were written by a learned friend of mine, a Physician, now no