A Treatise on Astronomy, Descriptive, Physical and Practical

CHAP. V.

How to mark

time on the

moon's path.

the

must commence be

5 o'clock.

As this curve touches the disc before 5 and after 7, it shows that, in that latitude, on the day in question, the sun will rise before 5 in the morning, and set after 7 in the evening. If the declination of the sun had been as much south as now north, the point d would have been 12 at noon, and all the hours would have been on the upper part of the ellipse, which is not now represented.

From C, as in the general eclipse, set off the distance Cn equal to the moon's latitude, and, through the point n, draw the moon's path at right angles to CL.

As the ellipse represents the sun's path on the disc, and as the point (12) refers, of course, to apparent noon, and not to mean noon, therefore, we will mark off the time on the moon's path corresponding to apparent time.

When the moon's center passes the point n, it is at ecliptic conjunction, apparent time, at Boston, or it must be considered the apparent time corresponding to any other meridian for which the projection may be intended.

The ecliptic, apparent time, Greenwich, is 8h. 49 m. Os.
For the longitude of Boston, subtract

Conjunction, apparent time, at Boston,

4 44

The moon's hourly motion from the sun is 27′ 39′′: take this distance from the scale, in the dividers, and make the small scale ab, which divide into 60 equal parts; then each In this case, part corresponds with a minute of the moon's motion from the ellipse sun, and the distance ab will correspond with one hour of the moon's motion along its path. At 4 h. 4 m. 44 s. the moon's tween 4 and center will be at the point n; the sun's center, at the same time, will be just beyond the point 4 on the ellipse; and, as the distance between these two points is greater than the sum of the semidiameters of sun and moon, therefore the eclipse will not then have commenced; but the moon moves rapidly along its path, and, at 5 o'clock, the center of the moon will be at the point marked 5 on the moon's path, and the center of the sun will be at the point marked 5 on the ellipse; and these two points are manifestly so near each other, that the limb of the moon must cover a part of that of the sun, show

CHAP. V.

more exact

time.

ing that the eclipse must have commenced prior to that time. To find the time of commencement more exactly, let the hour To find the on the moon's path be subdivided into 10 or 5-minute spaces, and take the sum of the semidiameter of the sun and moon in your dividers from the scale CD, and, with the dividers thus open, apply one foot on the moon's path and the other on the sun's path, and so adjust them that each foot will stand at the same hour and minute on each path as near as the eye can decide. The result in this case is 4h. 28 m. The end of the eclipse is decided by the dividers in the same manner, and, as near as we can determine, must take place at 6 h. 44m.

the time of

greatest ob

To find the time of greatest obscuration, we must look How to find along the moon's path, and discover, as near as possible, from what point a line drawn at right angles from that path will scuration. strike the sun's path at the same hour and minute; the time, thus marked on both paths, will be the time of greatest obscuration.

In this case it appears to be 5 h. 40 m., and the two centers are very nearly together; so near, that we cannot decide on which side of the sun's center the moon's center will be, without a trigonometrical calculation.

tude of the

To show a representation of an eclipse at any time during How to find its continuance, we must take the semidiameter of the sun in the magnithe dividers from the scale; and, from the point of time on eclipse. the sun's path, describe the sun; and, from the same point of time on the moon's path, describe a circle with the radius of the moon's semidiameter; the portion of the sun's diameter eclipsed, measured by the dividers, and compared with the whole diameter, will give the magnitude of the eclipse as near as it can be determined by projection.

The results of this projection are as follows:

From the projection the two centers are nearer together

than the difference of the semidiameter of the sun and moon,

CHAP. V. and the moon's diameter being least, the eclipse will be annular, as represented in the projection.

A very easy and important problem.

The above results are, probably, to be relied upon to within three minutes.

We have now done with the projection, as far as the particular locality, Boston, is concerned; but, in consequence of the facility of solution, we cannot forbear to solve the following problem: In the same parallel of latitude as Boston, find the longitude where the greatest obscuration will be exactly at 2 P. M. apparent time.

From the point 2, in the ellipse, draw a line at right angles to the moon's path, and that point must also be 2h. on the moon's path; running back to conjunction, we find it How solved. must take place at 1 h. 50 m.; but the conjunction for Greenwich time is 8 h. 49 m., the difference is 6 h. 59 m., corresponding to 104° 45′ west longitude; we further perceive that the sun would there be about 9 digits eclipsed on the sun's southern limb.

General

equations to

Now, admitting this construction to be on mathematical principles (as it really is, except the variability of the elements), we can determine the beginning and end of a local eclipse to great accuracy, by the application of ANALYTICAL

GEOMETRY.

Let CD and C be two rectangular co-ordinates, then 'aid in com- the distance of any point in the projection from the center can be determined by means of equations.

puting all the circumstances of an

Let x and y be the co-ordinates of any point on the sun's path or elliptic curve, and X and Y the co-ordinates of any one place. point on the moon's path, then we have the following equa

eclipse as seen at any

tions:

(1) y=p sin. L cos. D+p cos. L sin. D cos. ¿ Į solar (2) x=p cos. L sin. t

(3) Y=d+hi sin. B
(4) Xhicos. B

lunar co-ordinates.

In these remarkable equations, p is the semidiameter of projection, L the latitude, D the sun's declination, t the time from apparent noon, d the difference in declination between

sun and moon at the instant of conjunction in right ascension, h the moon's hourly motion from the sun, i the interval of time from conjunction in right ascension-minus, if before conjunction-plus, if after; and B is the angle LC, or the angle which the moon's path makes with CD.

In the equations, x and X are horizontal distances. In equation (1), the plus sign is taken when the hours are on the upper side of the ellipse, as in winter; when on the lower side,take the minus sign.

CHAP. V.

In equation (3), the plus sign is taken when the motion of Explanation the moon is northward, and the minus sign when southward. of the sym. The sin. t, or cos. t, means the sin. or cos. of an arc, corresponding to the time at the rate of 15° to one hour.

bols.

The solar and lunar co-ordinates, or equations (1), (2), The symbol (3), and (4), are connected together by the following equations; the minus sign applies to forenoon, the plus sign to time of conafternoon :

expresses the

junction in right ascen. sion.

To apply these equations, and, of course, the former ones, i, the interval of time from conjunction must be assumed, and, as the time of conjunction is known, t thus becomes known; d, h, and B, are known by the elements; therefore, x, y, and X, Y, are all known. But the distance between any two points referred to co-ordinates, is always expressed by

√(xX)2+(yun Y)2.

When an eclipse first commences, or just as it ends, this expression must be just equal to the semidiameter of the sun and moon; and if, on computing the value of this expression, it is found to be less than that quantity, the sun is eclipsed; if greater, the sun is not eclipsed; and the result will show how much of the moon's limb is over the sun, or how far asunder the limbs are, and will, of course, indicate what change in the time must be made to correspond with a contact, or a particular phase of the eclipse.

For an eclipse absolutely central, and at the time of being central, the last expression must equal zero; and, in that

CHAP. V. case, x=X, and y=Y. In cases of annular eclipses, to find

the time of formation or rupture of the ring, the expression must be put equal to the difference of the semidiameters of sun and moon. In short, these expressions accurately, efficiently, and briefly cover the whole subject; and we now close by showing their application to the case before us

By the projection we decided that the beginning of the eclipse would be at 4 h. 28 m., apparent time at Boston. Call this the assumed or approximate time, and for this instant we will compute the exact distance between the center of the sun and the center of the moon, and if that distance is equal to the sum of their semidiameter, then 4h. 28 m. is, in fact, the time, otherwise it is not, &c.

h. m. S.

An accurate Conjunc. in R. A., app. time, Boston, 4 13 21

computation

for the begin. Assume i equal to

ning of the Therefore, t is equal to

eclipse as