of the pendulum at the equator being unity, and the time of vibration in all of them the same: viz.

The lengths of pendulums oscillating in the same time being proportional to the gravitating forces by which they are caused to vibrate, it is evident from the preceding series that the force of gravity increases from the equator to the poles; since, in order that the pendulum may make each of its vibrations in the same time, in different latitudes, it is necessary to increase its length, as the latitude augments. This increase in the length of the pendulum is nearly proportional to the square of the sine of latitude, as may easily be proved by a comparison of the numbers in the preceding table with each other; and it affords another proof of the diurnal revolution of the Earth on its axis. These numbers also represent the proportional weights of the same mass of matter, if transported respectively into these latitudes. And likewise, as the spaces through which heavy bodies descend, when at full liberty to obey the action of this force, are always proportional to the forces by which they are put in motion, the relation of these spaces will also be expressed by the preceding numbers.

It is, however, only the ratio of the absolute lengths of the pendulum that is given for the respective latitudes in the above table, and which may be found for any other latitude by means of the square of its sine. In order to find the absolute lengths, one of them must be given. Now it has been determined by numerous experiments, that, in the latitude of London, the length of the seconds pendulum is very nearly equal to 39 English inches. Hence by means of

this length, that for any other latitude may be found by proportion.

But as the length of the pendulum will necessarily vary according to the species of time in which its vibrations are estimated, it becomes the first subject of consideration to fix upon the species of time in which these vibrations are to be reckoned. That which corresponds to the length above specified is the second of mean time. By this we are enabled to ascertain the length of the pendulum, acted upon by the same force of gravity, or, which is the same thing, situated on the same parallel of latitude. Under these circumstances, the lengths of simple pendulums are to each other as the squares of their times of vibration; for we have

and by omitting the quantities that are common to the last two terms of this proportion, we have

t: t':: wl: √l',

or t2: t'2:: 1: t.

Then by substituting the numbers in this proportion, gives

1: 99727:39 125: 38.9116,

which is the length of the pendulum vibrating sidereal seconds in the latitude of London; and for any other latitude its length may be found by the principles above stated.

In general terms; if we denote the latitude by L, the corresponding length of the seconds pendulum by 7, and that of the seconds pendulum at the equator by l'; it has been found, from a great number of observations, that the value of I is expressed by the following formula: viz.

1=1+1608 sin2 L.

By substituting for the sin2 L its value

1

cos 2L

2

1=1+(1608 ×

or 10804 (1-cos 2L).

Now the value of l' has been found to be 39.0265 English inches, and hence, by substitution,

1=39-0265 +08040804 cos 2L =
39-1069 - ·0804 cos 2L,

in which value 39.1069 is the length of the seconds pendulum at 45° of latitude; and it will be observed that when the latitude, or the value of L, is less than 45°, the latter member will be subtractive, but when it exceeds 45° it will be additive; for then the cos 2L becomes negative; and consequently the product of that cos by -0804 must be added to the number 39-1069. If we take the latitude of 51° 30′ as an example, which is nearly that of London, this formula gives the length of the seconds pendulum for that latitude 39.124986 inches; agreeing with the length as found by experiments.

This formula is analogous to that for the increase in the degrees of latitude, and the ellipticity of the Earth as deduced from each is very nearly the same; that from the pendulum being 30, and that given by the measurement of degrees. The length of a pendulum may also be found from the number of vibrations it makes in a given time, as well as from the time of each vibration, for these are evidently in the reciprocal ration of the times: we have, therefore,

1 1 t: t':: n'n'

where n and n' denote the numbers of vibrations cor

responding to t and t'. Hence, by substituting these values for t and t'in the preceding formula,

The numbers being substituted in this formula, will immediately give the corresponding length of the pendulum. Thus, if it were required to find the length of the pendulum that would vibrate half seconds in the latitude of London, we have

The Naturalist's Diary

For OCTOBER 1819.

Farewel to SPRING's enchanting reign,
And SUMMER'S smiling charms, adieu !
Farewel, awhile; ye soon again

Shall all your countless sweets renew.
Now welcome AUTUMN's yellow leaf,
Ye hollow gales of Autumn blow;
'Tis your's to calm the bosom's grief,
And sympathetic peace bestow.
Bid me redeem life's wasted morn,
For WINTER's bitter blasts are nigh;
And soon from earthly objects torn,
This frame in kindred dust shall lie.
To HIM then prostrate let me bow,
To Him whose arm is strong to save;
Who can eternal SPRING bestow,

That Spring which blooms beyond the grave.

As the spring and summer seasons have their distinguishing excellencies, so it is, in an especial manner, with respect to autumn. The reviving freshness of the spring is long past, and the summer is declin

ing; autumn succeeds, and its rich blessings may be considered as pleasing to the sight, gratifying to the palate, and cheering to the spirits.

How agreeable is it to behold the quick succession in the productions of nature during the spring, summer, and autumnal seasons! And, notwithstanding, before the commencement of autumn, some fruits and many flowers are gone, yet we scarcely miss them, so quickly are they succeeded by others, the greater part of which are much more important; for the autumnal fruits are those chiefly which are preserved for future use.

In this season, likewise, as the summer vegetables decline, they are succeeded by others no less salubrious and pleasant to the taste; while at the same time our tables are supplied with a variety of animal food, too often abused by the sin of gluttony, but which, in the temperate use of it, is adapted to nourish us, and excite our lively gratitude to Him who giveth us all things liberally to enjoy.'

The general state of the weather toward the close of autumn has a tendency to revive the natural spirits of those whose constitutions have been debilitated by the preceding heats. A great part of the day during the summer is too sultry for exercise; but, as autumn advances, the air becomes more temperate, and the evenings, particularly, are serene and pleasant.

The groves now lose their leafy honours; but, before they are entirely tarnished, an adventitious beauty, arising from that gradual decay which loosens the withering leaf, gilds the autumnal landscape with a temporary splendour, superior to the verdure of spring, or the luxuriance of summer. The infinitely various and ever-changing hues of the leaves at this season, melting into every soft gradation of tint and shades, will long continue to engage the imitation of the painter, and the contemplation of the poet and the philosopher. How pleasant is it,