diameters, include. Two such lines, AB and A c, will form, together with the sun's diameter BC, a triangle, of which the angle measured by the instrument will be the vertical angle B A C, and the sun's diameter the base. Now the sun's actual diameter, the base of this triangle, must be supposed to remain always the same; and if the earth moved in a circle of which the sun was the centre, the two sides A B and A C of the triangle would always remain the same whenever the observation was made, and therefore the vertical angle B A C would always remain the same; that is, the sun's apparent diameter, as measured by the instrument which has been described, would always remain the same. Now it does not ;—the earth's motion is, therefore, not in a circle whose centre is the sun.

A

B

The sun's apparent diameter on the 31st of December 1828*, was 32′ 35-6′′, or 32.5933′′; and on the 2nd of July 1829 it was 31′ 31′′, or 31.5167'+. Now let A and a represent the positions of the eye of the observer on those two days. It then follows, by the rules of trigonometry, that since B A C and Bac are very small, BC may be considered as the arc of a circle, and that

a

BA: Ba:: < вас : в АС
:: 31-5167: 32.5933;

or, B A B a :: 315167: 325933

:: 1000 : 1034.

Thus, if we suppose the whole distance to the sun on the 31st of December to have been divided into 1000 equal parts, on the 2nd of July its distance will have been increased by 34 of those parts. Now, between these two periods, it will be found that the sun has apparently moved through about 180 degrees of the heavens; or, in other words, that the earth has described 180 degrees about the sun: so that a line, D A, drawn from the centre of the sun to the earth on the 31st of December, and one, Da, drawn to it on the 2nd of July, make with one another an angle of 180°, or are in the same right line. Suppose a a in the accompanying figure to represent this line; take As so as to contain 1000 equal parts, and a s 1034 of these parts; then, A, s, and a will represent the relative positions of the earth and sun at these dates. Knowing that the earth was in the position A on the 31st of December, we can tell what

A

B

S

C

D

angle a line drawn from it to the sun in its position в at any other period, say the 1st of February, makes with s a; that is, we can find the angle AS B; in point of fact, this angle is the number of degrees measured upon the ecliptic between the sun's positions on those two days. Observing also the sun's apparent diameter, we can compare its distance on the 1st of February with that on the 31st of December; thus, taking

A s as before to contain 1000 parts, we can find how many of these parts are contained by BS. Thus then we shall have determined the relative positions, A B. of the earth in respect to the sun on the 31st of December and the 1st of February, and we may proceed similarly to ascertain the positions, CD, of the earth in respect to the sun at the commencement of each month in the year. Of these distances we shall thus discover this remarkable property, one of the laws of Kepler, that they all lie in the circumference of a figure called an ellipse; of which curve the characteristic property is this: that if from two given points within it, called its foci, there be drawn two lines to any point in its circumference or periphery, the sum of these two lines will be the same wherever that point may be situated. Of this ellipse the sun will be found to occupy one of the foci.

The form of the earth's orbit being thus ascertained to be an ellipse, a question at once arises as to the nature of its motion in that ellipse; is it uniform, as it might be supposed to be, if it revolved in a circle of which the sun was the centre? or is it in any way modified by the eccentricity of its orbit? The motion of the earth is not uniform; for if it were, the angles which it describes, in equal times, in different parts of its orbit, would be inversely as its distances at those times. Thus, if the

8

earth, when at its least distance from the sun, s, orin its perihelion, described in a day an arc M N, equal to that, PQ, which it described in a day when at its greatest distance, or its aphelion, then the angle PSQ would be to the angle м s N, in the ratio of мs to s p.

Now it is ascertained by observation, that when it is in its aphelion, the earth moves in its orbit (that is, the sun moves in the ecliptic) through 57.192 in 24 hours, and that in its perihelion it moves through 61·165′ in that time. Also its distance in aphelion we have shown to be to its distance in perihelion as 1034 to 1000; it follows, then, that if the earth's movement in its orbit were uniform, the ratio of 57·192′ to 61.165', or of 1000 to 1069, should equal that of 1000 to 1034,-which it does not. It follows, therefore, that the earth's motion in its orbit is not uniform. But what law governs it? what relation exists between the angle it describes in a given time, and the distance at which it describes it?

156

1000

Between the ratios we have just been stating there exists this remarkable relation. The distances are inversely as 1000 to 1034; and the angles as 1000 to 1069. Now if this last ratio is squared, it will become that of 1000000 1069156; and dividing both terms of it by 1000, it becomes 1000 1069, omitting the small fraction in the last term. Now this ratio 1000 1069 is precisely that of the angles. The ratio of the angles is therefore equal to that of the squares of the distances taken inversely. Or, in other words, the angles described in the same time at aphelion and perihelion are inversely as the squares of the distances at which they are described. Whence it follows that if the angle described in aphelion be multiplied by the square of its distance, the product shall equal the angle described in perihelion multiplied by the square of its distance.

Now this law does not only obtain with regard to the motion of the earth in aphelion and perihelion, but in every other position of its orbit. If the angle described in a day, for instance, in any part of its orbit, be multiplied by the square of its distance, the product shall equal the angle described in a day in any other part of its orbit, multiplied by the square of its distance on that day.

In the following table will be found the angles observed to be described by the earth on the first days of the successive months of the year; and annexed to each is its distance on that day from the sun, in terms of the mean distance, which is taken as 10,000.

Now if the square of each of these distances be multiplied by the corresponding angle, the product will be found to be throughout the same. Generally, therefore, wherever the earth may be situated in its orbit, the angle it describes in a given time, a day for instance, being multiplied by the square of its distance, will always be the same quantity, viz. 59.128.

S

a

R

P

This is a very remarkable law. It was discovered by Kepler, and is the observed fact on which the whole of Newton's physical theory of the universe is made to rest. Kepler did not, however, leave his law in this form in which it first occurred to him. Let us suppose P and Q to represent positions of the earth at the interval of a very small portion of time; an hour for instance, or a minute. Also let s be the sun. The line PQ being exceeding small when compared with SP and so, may be considered a straight line, and SPQ a rectilineal triangle. Draw the perpendicular QR. This may be considered to be a portion of a circle described from the centre s, at the distance są or SP. QR will therefore equal the product of SP by the angle PSQ (SPX PSQ). Now the area of the triangle PSQ is equal to one-half the product of SP by QR (SPXQR); therefore the area of this triangle is equal to half the product of SP by the product of SP and PSQ, or to SPX PSQ. Thus, then, the small triangular area PSQ swept over by the line SP, called the radius vector, in one minute of time, is equal to half the product of the angle PSQ by the square of the distance SP. And if this product be the same in every portion of the earth's orbit, it follows that the area swept over by the radius vector of the earth in every minute is the same, and therefore the area swept over in 60 minutes in one part of the orbit, is the same as that swept over in 60 minutes of any other portion of the orbit, and thus that the space or area swept over by the radius vector in one hour, in one day, or one week or month, in any one portion of the earth's orbit, is the same as the area swept over in any other. Now we have shown that the product of the angle described in a day by the square of the distance is the same everywhere. And precisely the same observations will prove the same fact in reference to the angle described in a minute.

It

follows then, generally, that the area thus described by the earth in a given time in one portion of its orbit, is precisely the same as that described in the same time in any other portion. This is called the law of the equal description of areas. And it was in this form that it was promulgated by Kepler. We shall show, hereafter, that it results from this fact that the deflection of the earth from the rectilinear path which it would otherwise have in space, is produced by a force acting always towards the 'sun, and not at different periods towards different points in space, or at the same time towards different centres. It points out, therefore, with certainty the sun as the controlling power in the earth's motion, distinguishing it in this respect from all the other material existences which people space, but establishing nothing as to the law by which its influence upon it is governed.

The earth is in perihelion, or at its nearest distance to us, in December; it would seem, therefore, that at this season our weather should be hotter than at any other. It is at its greatest distance in July; about that time we should therefore expect the coldest weather. We know the contrary of this to be the case. How is this to be accounted for? The variations of the seasons have been explained as dependant, not upon actual variations in the distance of the source of light and heat, but upon the relative obliquities of the directions in which the rays of light and heat 'are received, and on the relative lengths of the periods during which they are received; but unquestionably these causes, although in themselves they sufficiently explain our alternations of heat and cold, admit of being modified in their results by other causes, and especially by a variation in the distance of the sun. We might, for instance, be brought so much nearer to the sun in winter than in summer, as to make the temperature constant. And unquestionably, our actual variation of distance is such as would produce this effect in some degree, were it not for another cause tending in a great degree to modify this. The earth moves faster round the sun, or with a greater angular velocity, when it is nearer to the sun, than when it is more distant.

T

S

Ꭱ

So that in traversing a given distance off in space it has not so much time to receive heat when near the sun as when more distant. This will be evident from a mere inspection of the diagram. Let PQ and TR be portions of the earth's orbit, described by it in the same time, say one month, the one near its aphelion, and the other near its perihelion; then, by Kepler's law of the equal description of areas, the areas RST and PSQ are equal to one another; and the two distances SP and so being greater than ST and SR, it is evident that the angle PSQ must be less than the angle RST, in order to make up this equality. Now the law by which the light and heat of the sun is communicated to the same body when situated at different distances is this; if at any distance you multiply the quantity (anyhow measured) of light and heat which it receives in a given time by the square of its distance, the product will be the same as though you multiplied the quantity of light and heat which it receives in the same given time at any

other distance, by the square of that distance. This is commonly expressed by saying that quantities of light and heat are inversely as the squares of the distances. But we have shown that if we multiply the angle described in any given time by the square of the distance, that product will equal a similar product taken in any other part of the orbit. The quantity of heat received in any given time varies, then, according to precisely the same law that the angle described in that time varies,-the relation of both to the distance is the same. And thus the quantity of heat received by the earth in describing the same angle is always the same. Thus, if Ps and os be produced

R

T

B

P

to T and R, since the vertical angles at S are equal, there is as much heat received by the earth in moving from P to Q, as in moving from T to R. Or drawing the straight line ASB, the earth receives precisely as much heat from the sun whilst describing the space APB during the summer months, as whilst describing the portion BRA, which is its path in winter. Thus, that modification of the seasons which would otherwise be produced by our different distances from the sun is altogether got rid of, and the causes we have assigned for these phenomena exercise their full influence.

We have now described the form of the earth's orbit. Its position, too (which, not being a circle, but an oblong, is of importance), is easily known thus :-The earth comes to its perihelion on the 31st of December, as is known from the fact of its diameter being then the greatest, also the sun appears always in the opposite quarter of the heavens to that occupied by the earth. Ascertaining, then, what is the opposite place of the sun in the ecliptic on the 31st of December, and measuring from this 180 degrees, we know the precise position of the earth at that time, and, therefore, of the perihelion. The opposite quarter of the heavens is the aphelion; and thus we know which way in space the length of the elliptic orbit lies, and of course which way its breadth lies.

No. VIII.

THE DIMENSIONS OF THE EARTH'S ORBIT.

KNOWING the form and position of the earth's orbit, it remains now only to fix its actual dimensions. The general method by which the distance of the sun from the earth is found the reader will readily understand. Let A and B be two places on the earth's surface, which are on the same meridian of longitude. And suppose that at the same instant two observers ascertain the angular distance of the sun from the zeniths z and z' of these two places. These angular distances will be the angles ZAS and Z'BS, which will, therefore, be known. Also the latitudes of the places of observation being known, the angle AC B, which is the sum, or difference of these latitudes, is known; and the dimensions of the earth being known, the radii CA and Cв are known. Now having these quantities given, we can determine, by the rules of trigonometry, all the others which concern the quadrilateral figure ACBS; as will be evident to any one acquainted with that science. It may be, however,