that the breadth of the illuminated part of the planet's disc, as seen from the Earth, is always the versed sine of the exterior angle v'VS of the triangle SEV formed by the lines supposed to join the centres of these three bodies. But the exterior angle 'VS is equal to the sum of the two interior angles VSE and VES: and hence we have this easy rule for finding the breadth of the enlightened part of the disc of either Venus or Mercury, as seen from the Earth.

Add together the angles formed at the Sun and the Earth by the three lines supposed to join these two bodies to each other, and to the planet, and multiply the natural versed sine of this sum by 6, and the result will be the breadth of the illuminated part of the planet's apparent disc, in digits, or 12th parts of its diameter.

It may be necessary to observe, for the information of such of our readers who have not made much progress in the study of astronomy, that the angle at the Sun is equal to the difference of the heliocentric longitudes of the Earth and the planet; and that at the Earth is the difference of the geocentric longitudes of the Sun and the planet. These are to be found in the Nautical Almanac for the time required. The longitude of the Earth is found by adding 6 signs to the longitude of the Sun, as given in the Nautical Almanac, when that longitude is less than 6 signs, and subtracting them when it is greater than that quantity: thus, if I denote the longitude of the Sun, and l' that of the Earth, we shall have l′ = 1 + 63., as l is less or greater than 6 signs. If the longitude of the Sun, therefore, was 2. 12° 42′, that of the Earth will be 8'. 12° 42'; and if that of the Sun was 9. 3° 7', the longitude of the Earth would be 3o. 3° 7'. It may also be observed, that when the arc of which the versed sine is required is greater than 90°, which is the

case when that angle exceeds 3 signs, this versed sine is found by subtracting the versed sine of its supplement from 2. To find the versed sine of 4. 15° 20', we have 6.-(4. 15° 20')= 1. 14° 40', or 44° 40', the versed sine of which is 28879; and therefore, 2-288791·71121 = the versed sine of 4. 15° 20′ or 135° 20′, as required.

Let it be required to ascertain the illuminated part of Venus on the 1st of January 1819. Then the heliocentric longitude of Venus, taken from the Nautical Almanac for that day, is 3'. 13° 50', and the longitude of the Sun 9o. 10° 19′, consequently that of the Earth is 3'. 10° 19; the geocentric longitude of Venus at the same time, also, is 9. 0° 56'; therefore,

Heliocentric longitude of Venus
Longitude of the Earth

35. 13° 50'

subtract 3 10 19

Exterior angle at Venus

Sum O 12 54

Hence the natural versed sine of 12° 54', multiplied by 6,·02524 × 6 = 0d-15144, the breadth of the enlightened part as stated above, under the head of Occurrences.

Again, for the 1st of July 1819, we have
The heliocentric longitude of Venus
The longitude of the Earth

1s. 3° 28'

9 8 45

Now 180°-141° 12′ 38° 48', or 141° 12', the versed sine of which is 220662; and this taken from 2 leaves 1.779338, for the versed sine of 141° 12'; and which multiplied by 6, gives 10-67603 digits for the enlightened part of the disc of Venus at the time required.

The appearance of Venus, however, will be rendered more palpable by the following delineations. Let ABDE, Figs. 2 and 3, be the projected hemisphere, or disc of Venus, and EF the breadth of the enlightened part as found by the preceding method, as that is less or greater than the radius of the circle ABDE.

Then having described the circle, and drawn the diameters AD and BE at right angles to each other, set off EF, and describe semi-ellipses through the three points A, E, and D, having AD for their transverse, and CE for their semi-conjugate diameters, and they will be correct representations of Venus at these two stages of her illumination.

Now, if it were required to find the ratio between the dark and enlightened part of the planet's apparent disc, since, from the nature of the ellipse and circle, there is always a constant ratio between CF and CE or CB; and as the semi-ellipse and the semicircle may be conceived to be made up of an indefinite number of these lines, or rather small rectangles, the sum will have the same ratio as each pair; and consequently the ellipse and circle are also in the same proportion to each

other; whence the ratio between the dark and enlightened parts becomes known.

But as the diameter of the planet is supposed to be divided into 12 digits, and the breadth of the enlightened part is generally expressed in the same divisions, the ratio between these areas may readily be expressed in numbers. For by finding the area of the semi-ellipse of which both the transverse AD and the semiconjugate axis CF are given, and either subtracting it from the area of the semicircle, or adding it to that area, as EF is less or greater than EC, and the result will be the area of the enlightened part, expressed in square digits: and this area subtracted from that of the whole circle will give the area of the dark part of the disc in the same terms; and consequently the ratio between them is not only known, but also expressed in the most familiar terms.

The same reasoning applies equally to Mercury as to Venus; but as the former planet is so near the Sun as to present few opportunities for ob servation, on account of the intensity of the solar light, the calculation is seldom applied to him, and we have therefore not included the appearances of this planet under the head of Astronomical Occurrences.

[To be continued.]

The Naturalist's Diary

For JANUARY 1819.

Now Cacias sends

His cutting winds, his storms of hail, and snow,
Or binds the pregnant earth in icy chains;
The frost-nipt laurel, and the blighted bay,
Which lately shone in all the pride of health,
Stand withered monuments of his dire reign. ̧

THE most intense cold in England is usually felt in the month of January, and the weather is

either bright with frost or foggy with much snow 1. The inclemency of the season now compels the numerous tribes of birds to quit their retreats in search of food. The red-breast (sylvia rubecula) begins to sing. The robin has been frequently celebrated for his social disposition in entering the habitations of man; the following lines give a pleasing picture of him in the garden:

Attendant on my garden's winter cares,
When robin red-breast sees me with my spade
About to turn the earth, to dig the roots
Of celery or of endive, winter's salad,
He comes, and settles almost at my foot;
And, as I turn the earth-worm to his sight,
He picks it out, retreats a step or two,
And, having swallowed it, he comes again
And asks another; and, when twilight's shades
Have driven other birds to seek their roost,
Before my study window on some spray,
Or on the pointed summit of yon roof,
He chaunts his even song, and then retires.

J. PLUMPTRE2.

This is the season when the ice-houses of gentlemen in the country, and the cellars of confectioners in London, are filled with ice. Some account of the construction of an ice-house may be acceptable to our readers. The situation of an ice-house ought to be towards the south-east, on account of the advantage of the morning sun in expelling the damp air, which is far more prejudicial to it than warmth The best soil on which such a house can be erected, is a chalk-hill, or declivity, as it will take away the waste water, without the aid of any artificial drain; but where such land cannot be procured, a loose stony

Of this phenomenon, and its important services to vegetation, we have already spoken at large in our former volumes. 2 For some interesting particulars of the robin, consult our last volume, p. 21.