Sun's longitude from the beginning of Aries is there 4S. 2° 42' 23", that is 2° 42' 23" from the beginning of Cancer; Thus From the Sun's Longitude or place S 0 4 2 42 23 3 0 0 0 Remains the Sun's distance from the 21 solstice of Cancer. 2 42 23 Or, 32° 42′ 23": each sign containing 30 degrees. 4. To find the Sun's declination. Enter Table XIV. with the signs and degrees of the Sun's true place, viz. 4S, 2o and making proportion for the 42' 23", take out the Sun's declination answering to his true place, and it will be found to be 19° 38' 8" North. 5. To find the Moon's latitude. This depends on her distance from her ascending node, which is the same as the Sun's distance from it at the time of New Moon and with this the Moon's latitude is found in Table XVI. Now we have already found that the Sun's equated distance from the ascending node, at the time of New Moon in July 1748, is 5S. 25° 29' 23". See the 189th page. Therefore, enter Table XVI. with 5 signs at the bottom, and 25 and 26 degrees at the right hand counted upward, and take out 26 13', the latitude for 5S. 25°; and 20° 59', the latitude for 5 S. 26' : and by making proportion between these latitudes for the 29* 23" by which the Moon's distance exceeds the 25th. degree; her true latitude will be found to be 23'36" North Ascending. 6. To find the Moon's true horary motion from the Sun. With the Moon's anomaly, viz. OS. 10° 56′ 56", enter Table XVII. and take out the Moon's horary motion; which, by making proportion in that table, will be found to be 30' 14". Then, with the Sun's anomaly, 25°, take out his horary motion 2' 23" from the same table; and subtracting the latter from the former, there will remain 27' 51" for the Moon's true horary motion from the Sun. 7. To find the angle of the Moon's visible path with the Ecliptick. This, in the projection of eclipses, may be always rated. at 5° 35', without any sensible errour. 8, 9. To find the semidiameters of the Sun aud Moon. These are found in the same Table, and by the same arguments, as their horary motions.-In the present case the Sun's anomaly gives his semidiameter 15' 51", and the Moon's anomaly gives her diameter 14' 56". 10. To find the semidiameter of the Penumbra. Add the Moon's semidiameter to the Sun's, and their sum will be the semidiameter of the penumbra, viz. 30' 47". Now collect these elements, that they may be found the more readily when they are wanted in the construction of this Eclipse. D. H. M. S. 1. True time of New Moon in July, 1748, 2. Sun's diameter of Earth's disk, 3. Sun's distance from the nearest solstice, 5. Moon's latitude North descending, 8. Sun's semidiameter, 9. Moon's semidiameter, 10. Semidiameter of the penumbra, 14 11 19 58 } TO PROJECT AN ECLIPSE OF THE SUN GEOMETRICALLY. Make a scale of any convenient length, as A. C. (Fig. 1.) and divide it into 60 equal parts, reckoning each part to be one minute, or the sixtieth part of a degree. Then, take the semidiameter of the Earth's disk, 54 minutes, 33 seconds, (or 541) from the scale, in your compasses; and with that extent, set one foot in the end C of the scale as a centre; and with the other foot describe the semicircle A D B, for the circumference of the northern half of the Earth's illuminated disk, or surface, because we live on the north side of the Equator; continue the line A C to B; so A C B shall be a portion of the Ecliptick, equal to the diameter of the Earth, as seen from the Sun, or Moon at that time. Upon the centre C, raise the straight line C D H, perpendicular to A C B; and call the line CD H, the axis of the ecliptick. Being provided with a good sector, open it to the radius C A in the line of chords; and taking from thence the chord of 234 degrees in your compass, set it off both ways from D to G and to E, in the periphery of the semi-disk. [But, as much the greater number of those into whose hands this work may fall, are not supposed to be thoroughly skilled in the use of Mathematica! Instruments, we shall pursue somewhat a different method; which, in point of simplicity and precision, is no less preferable:] Or: Divide the quadrants A D and D B, each into 90 equal parts, for degrees, beginning at D. Then connect the points E and G (which are distant 234 degrees on each side of D) with the straight line EF G; in which the North pole P of the Earth's disk will always be found. When the Sun is in Aries, Taurus, Gemini, Cancer, Leo, and. Virgo, the North pole of the Earth is enlightened by the Sun: but while the Sun is in the other six signs, the South pole is enlightened, and the North pole is in the dark. And when the Sun is in Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini; the northern half of the Earth's axis C XII P lies to the right hand of the axis of the ecliptick, as seen from the Sun; and to the left hand, while the Sun is in the other six signs. The order, and the names of the Signs, the months and days of the year, in which the Sun appears to enter these Signs, are as follows: (7.) (8.) (9.) Libra, Sept. 23, (10,) (11,) (12,) Scorpio, Sagittarius, Capricornus, Aquarius, Pisces, October, November, December, January, February, 23, 22, 21 20, 19. Open the sector, till the radius (or distance of the two 90's) of the signs be equal to the length of D G, and take the sine of the Sun's distance from the solstice (32° 42' 23") as nearly as you can guess, in your compasses, from the line of sines, and set off that distance from F P, in the line E F G, because the Earth's axis lies to the left hand of the axis of the ecliptick, as seen from the Sun in the month of July. Or; Set one foot of the compasses in the point F, where the line EFG intersects the axis of the ecliptick C D H; and, having extended the other foot from F to E, or from F to G, describe the semicircle E H G, and divide its quadrant H E into 90 equal parts or degrees.-If the Earth's axis had lain to the right hand from the axis of the ecliptick, the quadrant II G must have been divided into 90 degrees, and not the quadrant H E. As the Sun is 32 degrees 42 minutes 25 seconds, (which may be estimated 323 degrees) from the nearest (or summer) solstice, which is the first point of Cancer, on the noon of the 14th of July 1748, draw the right line I P, parallel to H D, from 323 degrees of the quadrant H E till it meets the line E F G at P, then from P to C, draw the right line P C; so PC shall be the northern half of the Earth's axis, and P the North pole. As the Sun is on the North side of the Equator in July, and consequently nearer the point of the heaven just over London (or the vertex of London) than the Equator is; subtract his declination, 19 degrees 38 minutes (neglecting the 8 seconds) from the Latitude of London, 51 degrees 30 minutes, and the remainder will be 31 degrees 52 minutes, for the Sun's distance from the vertex of London on the noon of July the 14th. From the point k (in the right hand side of the semicircle A D B) at 31 degrees 52 minutes, counted upward from B, draw kl, parallel to CD and taking the extent k in your compasses, set it from C to XII on the Earth's axis C P. So, the point XII shall be the place of London, at the instant when it is noon at that place on the 14th. of July 1748. Add the Sun's declination 19° 38', to the Latitude of London 51° 30', and the sum will be 71 degrees 8 minutes, for the Sun's distance from the vertex of London on the 14th of July at midnight. Therefore, From 71° 8', counted upward from B to m in the right hand side of the semicircle A D B, draw the right line m n parallel to C D. Then, taking the extent m n in your compasses, set it from C towards or beyond P on the Earth's axis C P, as it happens to reach short of P or beyond it but in the present case, it reaches so little above P, that we may reckon C P, to be its whole extent and so, the point P shall represent the place or situation of London at midnight, beyond the illuminated part of the Earth's disk, as seen from the Sun; and consequently, in the dark part thereof. Divide the part of the Earth's axis between XII and P into two equal parts, XII K and P K ; then, through the point K, draw the right line VI K VI (till it meets, on each side, the periphery of the disk) perpendicular to the Earth's axis C XII K P. Now, to draw the parallel of latitude of any given place, as suppose London, or the path of that place on the Earth's enlightened disk, as seen from the Sun, from Sun-rise till Sun-sct, proceed as follows Subtract the Latitude of London, 51° 30', from 90° 00', and there will remain 384 for its colatitude, which take in your compasses, from the line of chords, making C A or CB radius; Or, From 38 degrees, counted upward from B tov in the semicircle A D B, draw the right line v w; and, having taken its length in your compasss, set off that extent both ways from K in the Earth's axis, to VI and VI, in the line VI K VE The compasses being opened from K to VI, set one foot in Kas a centre, and with the other describe the semicircle VI 7 8 9 10 11 12 1 2 3 4 5 VI, and divide it into 12 equal parts, Then, from these points of division (7 89, &c.) draw the dotted lines 7 a, 8 b, 9 c, 10 d, &c. all parallel to the Earth's axis C XII P, as in the figure. With the small extent P K as radius, describe the semicir. cle P 6 5 4 3 2 1 XII, and divide the lower quadrant into 6 equal parts as in the points 1, 2, 3, 4, 5, 6; because the Sun has -North declination. But if the Sun had South declination, the other quadrant must have been so divided. Through the said division points of the quadrant XII 1 2 3 4 &c. draw the right lines XI 1 XI, X 2 X, IX 3 IX, VIII 4 VIII VII 5 VII, all parallel to the right line VI K VI; and through the points where these lines meet the former parallel lines 7 a, 8 b, 9 c, 10 d, &c. draw the elliptical curve VI VII VIII IX X XI XII I II III IV V VI, which may be done by hand, from point to point; and set the hour-letters to these points where the right lines meet in the curve, as in the figure.* or, This curve shall represent the parallel of Latitude of London, the path which London (by the Earth's motion on its axis) appears to describe on the Earth's disk, as seen from the Sun on the 14th of July, from VI in the morning untill VI in the evening and the points VI, VII, VIII, IX, &c. in the curve shall be the point of the disk where London would be at each of these hours respectively, as seen from the Sun. If the Sun's declination had been as far South as it was North, the diurnal path of London would have been on the upper side of the line VI K VI; that is the ellipse, of which the curve VI VII, VIII, IX, X, &c. is a part, would have been complete, and must have been regulated by dividing the upper quadrant P 6 (of the small semicircle) into 6 equal parts, and drawing lines parallel to VI K VI, as before, till they meet the intercepting lines drawn through the division points of the quadrant PC. The points in which this elliptical curve would touch the periphery of Earth's disk, would denote the instant of the Sun's rising, and of setting at the given place. MakeC A or C B the radius of a line of chords on the sector, and take therefrom the chord of 5 * 35 ', the angle of the Moon's visible path with the Ecliptick: Or, From the point M, viz. at 5 degrees 35 minutes, to the right *N. B. The hour letters on the right hand side of XII, towards VI (in the Figure) viz. XI X IX VIII VII, is an errour in sculpture; it ought to be I II III IV V &c. The reader is therefore required, to correct this trivial mistake in projecting Eclips es. |