a attraction, by means of Barlow's plate, as utterly useless on board an iron steamer. Upon this subject, however, as we said before, we intend to speak more fully in a future number: it is sufficient to say here, that Captain Johnson appears to have made but few experiments with it, owing to the “ unfavourable state of the weather.” It is very true that unfavourable weather is disadvantageous for good experiments for mathematical investigations of the laws which reign through physical phenomena ; but we do really think, that, from the circumstance of the knowledge itself being only required in bad weather, the present was an advantageous opportunity thrown away ; inasmuch as the amount of discrepancy to be expected in times when the compass is our only trust, is, in a practical point of view, infinitely more important than a knowledge, however accurate, of the laws which prevail when the compass is never recurred to by a properly educated sailor. It is surely of greater consequence to know how far we may safely trust our instruments, at the time when they are our only trust, than to ascertain their laws in a state of comparative repose, and when they are altogether unnecessary to our safety. Finally, we feel it our bounden duty to our readers and to the public, to express emphatically, our conviction of the extreme danger of this class of vessels, for the purposes of navigation by means of the compass,—or, in other words, for venturing out of the mouth of a river. It is neither our wish, nor our interest, to discourage the progress of the arts and manyfactures of our country; but it is both our duty as scientific journalists, and our spontaneous feeling as men, to warn our countrymen against the dangers to which they may expose themselves, by stretching beyond its due limits, the application of any product of our manufacturing ingenuity. We are fully aware of the advantages which belong to the use of iron, in the construction of iron steamers; but we also wish our readers to be fully impressed with a sense of their recklessness in daring all its dangers. We should nor fulfil our duties towards them, did we not distinctly tell them, that the iron vessel is in precisely the same condition with respect to the compass, as if no compass had ever existed,—or, in some respects, even a worse condition,-since they may be led to trust to a guide that cannot guide them aright, and neglect those slight glimpses of indication which may in some small degree assist them. Such, at least, is our view of the matter; and if we are wrong, we hope we need not say, that we shall be extremely glad to be enlightened on the subject, by those who see better than we can. We do not think it necessary to give here Captain Johnson's determination of the best position for a sailing compass on board the Garryowen, as, even abating all other objections to the use of this kind of vessel, we have given good reasons why it could not be depended on for any other vessel similarly built,much less for one in which a different disposition of materials may be adopted. On the care bestowed upon the experiments, as well as on the evident honesty of their record, we cannot speak too highly; and we are glad to find such men are found by the Admiralty, to be entrusted with this class of expeditions. We cannot, however, in respect to its scientific value, but regret that a more complete apparatus was not furnished to him, and that a more suitable period, both as to date and extent, was not selected for the purpose. . 456 A POPULAR COURSE OF ASTRONOMY. No. VII. THE ELLIPTICAL FORM OF THE EARTH's ORBIT.—KEPLER'S LAWS. The earth revolves continually upon an axis within itself, and continually in an orbit about the sun. Hence result the alternations of day and night, the difference of the duration of the solar from that of the sidereal day, and the different times of the rising of the same fixed stars,of which phenomena the two last constitute an apparent annual revolution of the sun in the heavens, inasmuch as they would result from such a revolution. The axis about which the rotatory motion of the earth takes place remains always parallel to the same line in space; and hence result the phenomena of the seasons. But all these phenomena will be equally well accounted for by a revolution of the earth round the sun, in whatever orbit that revolution may take place,—and no inquiry on which we have hitherto entered indicates with any certainty, what is the real form of the earth's orbit. Provided the earth go completely round the sun in the space of a year, it matters not, so far as the facts which have hitherto been stated are concerned, whether the motion of its centre be in a circle, in an ellipse, or in a spiral,—or, in fact, whether it be in a square or an oblong. We are now about to indicate the means by which the real form of the earth's orbit is ascertained, the nature and law of its motion in that orbit, and the actual dimensions of the orbit. It will then be explained how the apparent motion of the sun, the duration of the seasons, and the length of the year, are modified by these facts. First, then, as to the form of the earth's orbit. It is not a circle, for then the sun would at all times of the year appear of the same size to We judge of the dimensions of objects by the angles which lines, drawn from the extremities of them, subtend at the eye. The conclusions we thus draw, we modify, however, by that which we know of the distance of the objects. Thus two objects, AB and CD, may subtend the same angle, CED. us. С A E D B Rays of light come to us in straight lines. If, therefore, an instrument having two arms, which can be made to include any given angle between them, be placed so that one of these arms is in the direction of a ray coming from one point of the edge of the sun's disk, and the other in the direction of a ray coming from a point on the edge diametrically opposite to this, then these two arms of the instrument will include between them precisely the same angle which two lines drawn from opposite points of the sun, or opposite extremities of one of its B D A B A : Ba :: diameters, include. Two such lines, AB and ac, will form, together with the sun's diameter B c, 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 BAC 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. 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 ZBAC : BAC :: 31:5167 : 32:5933; or, BA : Ba :: 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 angle a line drawn from it to the sun in its position B 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 B D A P M a A s as before to contain 1000 parts, we can find how many of these parts are contained by B s. 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, c D, 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 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 msn, in the ratio of ms 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? 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. N : 156 1000 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. Let us sup a 不 R P а 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. 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. pose 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 po being exceeding small when compared with s P and sq, may be considered a straight line, and spQu 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 (SP XPSQ). Now the area of the triangle psa is equal to one-half the product of sp by QR (ISPXQR); therefore the area of this triangle is equal to half the product of sp by the product of sp and PsQ, or to spo X 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 |