Imagens da página
PDF
ePub
[graphic]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

be easy to find all the requifites by scale and compaffes, by measuring them; or rather by calculation, in the feveral right-angled plain triangles, contained in the fcheme. Thus, to find when the moon first touches the penumbra at L; in the right angled triangle SPK, there is given SP, and SK (the fum of the radii SI, and PB), to find PK. Which being known, the time of the moon's paffing through it will be known, by the moon's horary motion from the fun.

(617.) To find when the moon first enfers the dark shadow of the earth in D: In the right ang. led triangle SPI, there is given SP, and SI, (the fum of the radii SD, PB) to find Pi; and confequently the time of half the duration in the fhadow. (618.) To find the digits or 12th parts of the moon eclipted. Here no the part eclipsed is = 12 no 6no Sa+Po-SP; and 2Po Po

or

is the number of

digits eclipsed. IN TOTAL ECLIPSES of the moon, the earth's fhadow often reaches farther than the moon. And then more than 12 digits are said to be eclipfed, fuppofing the moon's difk to be produced fo far.

(619) To find the time when the moon wholly enters into the dark fhadow BED, follow the fame method as when it entered into the penumbra GQF. This will be evident, by fuppofing GQL the dark shadow. In that cafe S1 will be the difference of the femidiameters of the moon and dark fhadow. The times of paffing through PI, PK, &c. being known, and the time of the middle of the eclipfe at P; the beginning and ending will be known.

(620.) VIII. Hence, if the moon or circle CB, never touches the circle GQF, there will be no eclipfe, not even by the penumbra. And if the fame circle never touches the circle BDE, there will be no part of the moon totally eclipted. And if the whole circle CBo enter into the circle BED, the whole moon will be totally eclipsed; and that is when SP is lefs than the difference of the femidiameters SD and PB. If the point S be in the node, then P falls upon S, and the eclipfe is central. When only a part of the circle CBo goes into the circle BED, the eclipfe is a partial one, as in this figure.

(621.) IX. The time of the eclipfe being known for any particular place, it is easy to know if it be vifible at that place, by knowing if the moon be rifen. Or the place will be known where the moon is vertical; and therefore it will be vifible to all places, within a quadrant's diftance from it. (622.) X. If the fpectator live in the place, (or in the fame longitude) which the tables are calculated for; he will fee the eclipfe at the time determined by the calculation. If not, he will fee it an hour fooner for every 15° difference of longi. tude, that he lives weft from it, And fo much later, if he lives eastward; that is, in the way of reckoning time. But in regard to abfolute time, it is feen from all places at the same instant.

(623.) EXAMPLE. TO FIND the TIME of the LuNAR ECLIPSE, December 13, 1769; its DURATION and DIGITS ECLIPSED.

1. The mean time of the fyzygies, by the tables,

[blocks in formation]

3. Hence the moon is 6° 51′ 22′′ paft the defcending node; that is S is 6° 51′22′′. Therefore & A= 27′32′′, and AS = 6° 23′ 50′′. Therefore the angle SMA = 84° 22′ 28′′.

4. Hence drawing the ecliptic RS, and SM perpendicular to it, and equal to 37′ 58′′ from a fcale of minutes, as in Fig. 3. Plate XXX. and making the angle SMA 84° 22'. We find the perpendicular SP 37′ 47′′, and MP3′ 43′′. And therefore, the horary motion of the moon from the fun being 35′ 33′′, PM will be palled over in 6'17". And fince this is before the oppofition at M, this time must be deducted from the time of oppofition. And the time of the middle of the eclipfe will be Dec. 12d. 18h. 26m. 345.

5. The fun's apparent femidiameter 16′ 20′′
His horizontal parallax

The moon's apparent semidiameter 16
Her horizontal parallax
61

6. Hence the radius BP = 16′ 48′′.
Radius SD 44 59.
Radius SF = 17 39:

[blocks in formation]
[ocr errors]

48 7

[merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

tion of the moon from the fun. For by this fcale, the feveral hours and minutes may be marked along the line Ak, by which it will appear at what time the centre of the moon is at any given point. For the time is known when the moon is at M, and from thence the points at each hour and minute are exfily found. And this conftruction, with only right lines and circles, will be exact enough in a Erge figure; for the beft lunar tables give the times of the phafes of an eclipfe no nearer than to 4 or 5 minutes of time; and therefore fuch a Conftruction is fufficient to anfwer the purpofe. Hence it may be obferved, that no eclipfe of the Inoon can laft above hours, from the moon's first touching the earth's penurabra, to its laft leaving it. For SK=94′ 27′′ 94.45, and the horary motion is 35′ 33′′35-55 and 94-452.66 =

35.55

[blocks in formation]

(625.) The refraction of the carth's atmofphere, in lunar eclipfes, makes the fhadow lefs; by bringing the rays, which terminate the fhadow, fooner to a point. And hence comes that red colour of the moon even in total eclipfes. But that light muft be very dim, by reafon a great number of the rays are ftopt and loft in the earth's atmoSphere.

(626.) The circles terminating the fhadow and the penumbra BED and GQF, cannot be diftinguifhed. For the darknefs from BED, diminishes by infenfible degrees, to GQF, being darken at E, and, lightest at Q, where it vanifhes infenfibly.. And therefore the moon does not appear to be eclipfed till fhe is a good way within the penumbra. For that reafon, there may happen eclipfes of the moon which cannot be difcovered as fuch. (627) All lunar tables fhew the moon's place in eclipfes, more truly in the fyzygies than in the quadratures, or any other place. For the times of the fyzygies, and the moon's place, have been more accurately obferved in eclipfes, than at any other time; and from thence the moon's theory has been deduced. Befides, many of the inequalities ceafe in the fyzygies, but have sensible effects in other places; becoming greater, as the moon is further from the fyzygies; being greateft in the quadratures. Whence the lunar tables do not determine the moon's place truly in the quadratures. And her place calculated from thete tables is not fo exact in the quadratures as in the fyzygies. 1628.) Several inequalities depend on the atpect of the nodes and the fun; but these ceafe when the nodes are in the fyzygies. When the moon and the nodes are in the fyzygies; the moon's place, theb wanting fewer equations, as being fubject to fewer inequalities, will be more correct, than

when he is in other places, where there are more and greater inequalities, and more equations. From hence more errors will happen out of the fyzygies than in them.

V. TO FIND the WAY of the Moos from the Sus, in a SOLAR ECLIPSE, fuppofing the OBSERVER

at REST.

(629.) Let HZO, in PLATE XXXI. fig. 2, be the meridian of the place, HO the horizon, EC the equinoctial, EL the ecliptic, Z the zenith, P the pole, S and M the places of the fun and mcca in conjunction, PSD the fun's meridian. Having found the fun's diftance from the node, S, and the moon's latitude SM, &c. take & A to 2 & as the fun's horary motion to the moon's borŢ motion; then SA is known. Draw MA; then a the spherical triangle ASM, right angled at S there is given SA, SM; to find the angle &MA; AM being the moon's way from the fun.

(63c.) But, as the eye of the obferver is in motion, by the rotation of the earth, which gives n apparent motion to the moon, contrary to that dị the obferver, we must find the quantity and dare tion of that motion. As the obferver is card eaf ward, toward the point C, the apparent c tion of the moon caufed thereby will be in the line CS. And to determine the polition of CS in refpect of AM or SM, leveral fphericle triangles must be refolved, as follows:

(531.) In the right-angled triangle EDS, there is given ES and angle E to fird DS and angle ESD or ASP; or thefe may be eatier had from the f tronomical tables. And in the triangle ZFS, there is given PS (the complement of DS), the angle ZIS (from the time of the day), and ZP the complement of the latitude; to find ZS, and angks PZS and ZSP. Then ZSP and ASP being knows, ZSA will be known. And MSA being a ngt angle, ZSM will be known. In the right argad triangle CFS, there is given CF, the meature of the angle FZC (the difference between the ange PZS and the right angle CZP), and SF the com plement of ZS; to find CS, and the angle CSF or BSZ. Then BSZ and ZSM being known, PSM will be known. And SMA being known, its fepplement SMB is known, and confequently the a gle SBM.

(632.) To find the quantity of the uction That along AM is already known; and to find the apparent motion along SB. The line of 1-* (the horary motion of a point in the equinoctio is 259 to the radius 1. And if b be the moet le horizontal parallax, then the radius of the earth appears at the moon under the angle b, and ther fore 15 of the equinoctial appears under the age of 259b; this then is the herary metion of a pest in the equinetial, viewed directly from the moon, And the moon's apparent motion feen from 14 point in the equinoctial is the very fame. But this motion is to be diminished upon two a counts. 1. Becaufe it is leis in a parallel dice, in proportion to the cone of the latitude. And 2. Upon account of the obliquity of the moti", when not perpendicular to the rays of the ar and this will be as the line of CS, the fun's direc

[ocr errors]
[ocr errors]

from the eaft or weft point of the horizon. There-
fore to find the quantity of this motion.

To the logarithm of .2556
Add the cofine latitude
And the line of CS.

Then the fum abating twice radius, is the loga-
rithm of this apparent horary motion. Then this
motion is to be compounded with the motion a-
long AMB as follows.

(633.) Let AS, PLATE XXXI. fig. 3, be a portion of the ecliptic, SB the way of the apparent motion, MA the moon's way from the fun. Draw NM parallel to SB; and let MN be the horary motion along SB or MN, and MI the horary motion of the moon from the fun. Then complete the paralelogram NMIQ; draw the diagonal MQR, which is the direction of the motion, compounded of the obfervers and the moon's motions, and MQ is the total apparent horary motion, fuppofing the obferver at reft. Then in the plain triange QMI, there is given MI, and 1Q (or MN), and the angle MIQ=MES; to find the angle QMI, and fide MQ or the abfolute horary motion. And the angles QMI and IMS being known, QMS is known.

(634.) If the fun be in the eastern hemifphere, in which cafe the concave fide of the eaftern hemifphere is here projected (in fig. 2.), then the moon's motion from the fun is from M towards A, and the other apparent motion from S towards B, or from M towards N. But if the fun is in the western hemifphere, this projection represents the convex fide of the fphere; and then the moon moves from the fan, in direction AM, and the other apparent motion is from S toward C, being contrary.

VI. TO CALCULATE SOLAR ECLIPSES. (635.) The eclipfes of the fun are more difficult to calculate, than thofe of the moon; the latter being clear of parallaxes, which the former are incumbered with, which gives a great deal of trouble. But a great part of it may be avoided by using projections instead of calculations. The rules are, (636.) I. Find the true time of the conjunction, and the places of the fun and moon at that time. (637) II. Having found the way of the moon from the fun by projection or calculation; find, by the aftronomical tables, the moon's horizontal parallax, her apparent diameter, and horary motion; auo the fun's apparent diameter, and horary motion. But, to avoid a great deal of calculation, if the sphere be projected by a large fcale, it will give all the requifites with fufficient exactnefs, by measuring the feveral angles and fides, without any calculation, or very little. And here it is best to project the concave fide, and then every thing appears as it is in nature.

(638.) III. Find the moon's parallax of altitude, by making as rad: cof. altitude: : fo the moon's horizontal parallax: to her parallax of aititude Vt or Mm. fig. 4. Then find her parallax of latitude Ma, and longitude Ss, or mn, and from thence her apparent latitude and longitude is known.

(639) IV. Draw the line SL, fig. 5. for the ecliptic, and from a large fcale of minutes, erect SM perp. to LS, and equal to the apparent latitude; make she angle SMR, as found in the laft prob. and

draw & MR for the moon's apparent path. From S let fall SP perpendicular to MR, and SP will be the leaft ditance of the centres of the fun and moon, or the middle of the eclipfe. From the centre S with the radius equal to the minutes contained in the fun's femidiameter, defcribe the circle ABC for the fun. And from the centre P, with the radius equal to the moon's femidiameter, defcribe the circle AOCD for the moon. If thefe circles do not interfect, there will be no eclipfe. But if they interfect, an eclipfe mult neceffarily happen.

(640.) V. Then P is the place of the moon in the middle of the eclipfe. Make SI and SK equal to the fum of the femidiameters of the fun and moon; and the moon's centre will be at I when the moon first touches the fun, or at the moon's centre, at the end of it. In the triangle PSI, there is given SI, SP; to find PI PK, which reduced to time, by help of the moon's apparent horary motichews half the duration of the cclipfe; and confequently we shall have the beginning and end. (641.) VI. And to find the quantity no, or the digits eclipfed; we have no S + Po-SP, number of digits.

and

6no

Po

(642.) VIII. The time found being mean time, it mult be reduced to the common or apparent time, by the equation of time. And it the given place be not that for which the tables are made, and fo much time, if the place lic eastward, to the time of conjunction, as aniwers to the difference of meridians; or fubtract it, if it lie weitward.

(643.) EXAMPLE.

To FIND the TIME of the SUN'S ECLIPSE, June 4, 1769, its DURATION and DIGITS ECLIPSED at LONDON.

1. By the tables the mean time of the conjunction is found to be June 2d 20a 41. And hence the true time of conjunction is June 34 20 272 43". And their places are 2' 13° 51' 25". And the moon's lat. 55 32 north. The moon's motion from the fun 35′ 47′′

29

In fig. 2, and 3, Plate XXXI, the angle AMS 84° 47. ZSM 35° 20'. CSF = 5° 18'. SBM 43° 49'. SF 42 16', CF = 3° 34′. CS = 42° 24'. The angle QMI = 8° 25′. SMQ = 92° 52'. MN or IQ= 6° 38'. MQ = 31° 25′. Affe The moon's horizontal parallax 60′ 58′′ Her apparent diameter Her horary motion The fun's diameter His horary motion

33 32

38 19

31 41

2 23

3. In fig. 4. the moon's parallax in altitude Mm is 45′ 09"; her parallax in latitude Ma, 38′ 05′′ i her remaining latitude Sn, 17 26"; her parallax in longitude Ss, 24' 13"; which is increafed 13 much.

4. Draw SL for the ecliptic, as in fig. 5; at any point S, erect the perp. MS equal to 17' 26", the moon's apparent latitude; through M draw the moon's way & MR, making the angle SMR = 92° 52', Draw SP perp. MR, which here falls

very

m

[ocr errors]

(651.) II. From a large scale of minutes, take the moon's horizontal parallax in the compafles, and at any point C, in the right line BD (which reprefents the ecliptic in Plate XXXII. fig. a,) defcribe the circle ABED, for the earth's disk, or the earth's flat face as it appears at a diftance, in a line drawn to the fun. Draw CM perpendicular to CD, and equal to the latitude of the moca upwards, if north. Make the angle CMG equa to that which the moon's way makes with a cir cle of latitude; acute to the right hand, if the tend to the node; or obtufe, if the be paft it; and drawing FMG, it will be the way of the cettre of the moon's fhadow upon the earth. From C let fall CH perpendicular to FG. Then at H will be the middle of the earth's eclipfe.

very near M. From the centre S, with the radius SA 15' 50" defcribe the circle ABC for the fun. And with the radius MD = 16` 46′′, and centre P, defcribe the circle ADCO for the moon. 5. Hence PI or PK 27′ 33′′. And the time of moving through IP or PK, at the rate of 31' 20" an hour, is 52 455, for the femiduration.By reafon of the parallax (24' 13"), the is paft the apparent conjunction; the difference being what the parallax caufes, which comes to 47 235. Therefore the middle of the eclipfe is fo much fooner, being at 3d 19h 41m 20. This reduced to apparent time is 3d 19h 43 275, for the middle. 6. The digits eclipsed are 5, nearly. (644.) In this example, the concave fide of the fphere is projected, which fuits beft to the appear ance of the heavens. And the figures are drawn upon that fuppofition. It appears from the procefs that the moon is advancing to her defcending node, and therefore has north latitude. And by the pofition of that part of the ec.ic, her parallax in longitude, advances her fo much for ward, viz. 24′ 13". And therefore the is fo much paft the apparent conjunction. Hence we gain thefe feveral particulars, as to the eclipse: (645.) I. The begin. June 4d 6h 53 42s morn. middle, - 4 7 43 27 end, 4 8 46 12 I 45 30

[ocr errors]

total duration, digits eclipfed 51, on the upper fide of the fun, towards the left; as appears by the figure. (646.) II. Hence the POSITION of the HORNS at C and A, are eafily found in the middle of the eclipfe. For they are in a position parallel to RI, the moon's way.

(647.) III. The MIDDLE of the eclipfe will not be at the fame time in all places of the fame longitude. For the parallax of longitude will be different in different places.

(548.) No eclipfe of the fun can last above Two HOURS. For SI or SA + MD = 32′ 26′′ = 32.6 and the horary motion = 344735.78. And 32.6

35.78

= .91 = 541 minutes, for the femiduration, (649.) If it were not for the parallax, eclipfes of the fun would be as eafily calculated, as thofe of the moon. And in order to get the parallax, the angle ZSM and SP must be known fig. 2, which occafions the refolving several spherical triangles before they can be had. Likewife it may be obfer. ved, that the apparent way of the moon is ftrictly curve line, concave towards S, which arifes from the parallel of latitude being a curve, and the moon being out of its plane. Likewife the moon's apparent velocity is fomething greater at the beginning than at the end.

VI. RULES for CALCULATING a GENERAL ECLIPSE of the SUN.

(650.) The elements neceffary for this are. I. The fun and moon's place, and the time, at the true conjunction; 2. The moon's latitude, hori zontal parallax, diameter, and horary motions, 3. The fun's declination, diameter, and horary potion; And 4, the angle the moon's way makes with a circle of latitude.

(652.) III. With the centre H, and radius HO, equal to the fum of the femidiameters of the fa and moon, defcribe the circle QOR, which wi be the moon's penumbra. Also defcribe a fm.l circle round the centre H, whofe radius is the difference of the fun and moon's femidiameter, that little circle will be the dark fhadow of the moon. Then all the countries of the earth_contained in the fegment VAW will be fucceffivery eclipfed, by the penumbra, as the fhadow moves along the tract FG; while the other fegment VEW fuffer no eclipfe at all. All places in the line st, will be totally eclipfed, as the dark for dow, or the small circle at H paffes fucceffively over them. But this circle, or dark fhadow, be ing very small, a total eclipfe at any place conti nues but a fmall time. Sometimes the fun's kmidiameter exceeds the moon's; and then there will be no dark circle, or total eclipfe, but a l cid ring will appear about the moon in these pla ces; and this is called an annular eclipfe. Te difference between the femidiameters of the in and moon is fo little, that no total eclipfe lafts a bove 4 minutes.

(653.) IV. Draw CF, CG fum of the femidiameters of the fun and moon, and the moon's parallax; then the moon's fhadow will touch the earth at L and K where the eclipfe begins and ends. In the triangle CFH, there is given CF, CH; to find FH = HG, which converted into time, gives half the duration, or half the time that the moon's fhadow is upon the earth. Alfo, NO measured, fhews how far the eclipfe reaches. Or CO me fured does the fame. It may be fufficient to mee fure all thefe by the scale without calculation.

(654.) V. To find the POLE. Draw the arch AP, making the angle KAP equal to the fun's longitude, and AP, the diftance of the poles of the equator and ecliptic, 23°; then P is the pol For AP is a part of the folftitial colure, and pales through Cancer and Capricorn. And CAP is what the fun wants of Cancer, therefore PAK is what is paft Aries. Through P draw CPT. And here we may fuppofe that the pole P is fixed, during the time of an eclipfe. Then in the right ange fpherical triangle APT, there is given AP angle A; to find AT or angle ACP. In this tr angle PT is the fun's declination, and APF a CPK his right afcenfion from Cancer. Here 500 that any place in the line CT is in the fun's

[ocr errors]
« AnteriorContinuar »