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Les cartes, planches, tableaux, etc., peuvent 6tre film6s d des taun de reduction diffdrents. Lorscfue le document est trop grand pour Atre reproduit en un seul cliche, il est film6 A partir de Tangle supdrieur gauche, do gauche d droite, et de haut en bas, en prenant le nombre d'images ••^cessaire. Les diagrammes suivants illustrent la m^thode. rata lelure. J 32X 1 2 3 1 2 3 4 5 6 UNIVERSny OF WESTERN ONTARIO LIBRAr^Y select LIBRARY ^r^"-"'- -"' "r-- X / M E A S U II i; M K N I' THE SUN'S DISTANCK. -. "--v-.-f- -■• JOHN IIAKRIS. •!® .^ •^^^ ^ ILflnbait : N TRU13XER & CO., .)7 .^L- .)•». LVElWATE II ILL . y.. ■"PiMPiPMiVP*»^n ifi ^ A <5 '- ASTRONOMICAL LECTURES. MEASUREMENT OF THE SUN'S DISTANCE. JOHN HARRIS. N. TRUBNER & CO., 57 & 59, LUDGATE HILL. SEPTEMBER, 1876. AH rights reserved. ■A II 86 V- LECTURE FIFTH. i TMl E F A C E . This fifth lecture, or fifth division of the serioa to which it belongs, differs iu one respect from those preceding it, inasmuch as we have not on this occasion to call in ques- tion or to condemn the present doctrine on the particular subject of which it treats. It is true that, not long since, an error of about four million miles in the estimate of the sun's distance which had been accepted and agreed to by astronomers for many- years previously, was discovered ; and that it is only quite recently the correction of the error has been made and adopted. But this error was of the nature of a mistake which in combining the separate observations of several different observers was almost unavoidable, if it happened that any one of those observers, upon whose collective reports the general computation was based, had from misfortune, or want of due care, fallen into error and furnished a report which, being accepted as trustworthy and being in fact incorrect, vitiated the whole. Such a circumstance does not necessarily prove, nor in any degree evidence, the method itself to be unsound in principle or unreliable in practice. But it does afford 'i^ ■ VI PREFACE. as evidence that the conditions necessitated b}'- the particular method in question practically occasion a liability to error in the collective result, and it suggests very pointedly the desirableness of practical astronomy being in posses- sion of some other method, or methods, equally reliable as to ihe sound theoretical character of the basis, and of such a nature in themselves that the individual observer, usmg due care and diligence, will be independent^ of the want tif care or correctness on the part of other observers. M^ irticular to error jintedly , posses- reliable , and of bserver, den*^ of >f other INDEX TO THE SEVERAL METHODS. PAGE Method First.— By observation of the relativo velocities of earth's orbital motion round the sun, and of earth's equa- torial surface in rotation on the polar axis 9 Mktiiod Seconb. — By observation of the angle of moon's illumiui'tion 12 MetuoD Thiud. — By observation of the time occupied by the entire disc of the moon in passing the sun's centre, as seen from the centre of the earth 15 Method Fourt'II. — By observation of the time occupied by the diameter of the earth in its orbital ascent or descent through a definite aogle); or, observation of the vertical angle subtended by earth's polar diameter, at the distance of the suu, showu in its ascent or descent through the equatorial plane of the dun 18 Method Fifth.— By observation of the angle of obliquity in the path of a (so-called) solar spot in its motion upon and with the surface of the sun 21 Method Sixth. — By comparative observation of the angle of incidence of the sun's light on an object upon the earth's surface, and the angle of incidence of th'i sun's light bounding and containing the earth's shadow ... 25 Method Seventh.— By comparative observation of the angle subtended by the sun's diameter as seen from a station on the equatorial surface of the earth, and as seen at the same time from the earth's ccutre 28 ^xmr PLATES PAGE FrontispifX'E. — The Transit of Venus according to the perpendicular axis theory. Fig. 1. — Solar illumination of the moon 12 Fig. 2. — The angle of moon's illumination 14 Fig. 3. — The moon's diameter passing over the sun's centre, as seen from the centre of the earth ..... 1 G Fig. 4. — The differential angle of moon's synodic period . . 17 Fig. 5. — The descent of a definite part ot the earth's polar diameter through tlie equatorial plane of the sun . 20 Fig. 6. — The axial rotation of the sun, b} terrestrial observa- tion of a solar spot 22 Fjg. 7. — Measurement of the sun's dictance by comparing the angles of incidence of the sun's rays converging (a) to the earth's centre, and (6) to the extremities of the earth's diameter, respectively 25 Fig. 8. — Comparative angular value of the sun's diameter as seen from the nearest point on the earth's surface and, at the same time, from the earth's centre, respectively 29 THE MEASUREMENT OF THE SUN'S DISTANCE. PAGE o the . . 12 . . 14 Dutro, IG . . 17 polar suu . 20 ierva- . 22 g the nities ber as irface 3utre, 25 29 The methods at present known in practice, and which have been utilized with more or less success for the purpose of ascertaining the distance of the sun from the earth are two, that of geocentric parallax (or zenith meridional observation), and the transit of the sun's disc by the planet Venus. Our present purpose is to suggest, explain and illus- trate, so far as may be necessary for the elucidation oi this preliminary explanation, certain methods which we believe to be essentially new and hitherto unpractised. (1.) If, by direct observation, the earth's velocity of revolution in its orbit relatively to the velocity of its rotation can be ascertained with certainty and precision, such information, together with the data already known, will enable the distance of the sun from the earth to be readily computed. The mode of observation by which we propose to obtain this information is by utilizing the retrograde or backward movement of a station on the equator in its rotation round the earth's axis, when on that side of the earth nearest to the sun, compared with the advance of the earth itself in its orbital revolution during the same time. To explain the proposed mode of proceeding, let 10 MEASUREMENT OF SUN 8 DISTANCE ,'K US suppose the station of observation to bo situated on the equator and that the earth's rotation through 00 degrees is to be subjected to the comparative observation. Knowing the time at which the meridian of the observatory will pass the sun, the observation is to commence two hours before that time. The situation of the sun as seen from the earth's centre, and as seen from the station on the equator, is to be recorded, and also the exact time of the sun's centre transitting the meridian of the observatory. And, again, the situa- tion of the sun as seen from the earth's centre, and from the observer's station, four hours later than the time of the first observation, is to be carefully determined. Having thus ascertained by direct observation the parallactic displacement of the sun consequent upon the compound motion of the observatory, we shall be able to deduce the linear velocity in miles of the earth in its orbital revolution around the sun, because we already know the linear value in miles of the arc through which the retrograde or reverse motion of the earth's rotation has carried the observatory, and we have ascertained the value of the chord of that arc as a part of the orbital circle of the earth's annual revolution. "VVe know also in time and in angular measurement the quantity of orbital arc traversed by the earth during the observation. For example : the observation being made for 60 degrees of the earth's rotation, we will assume that 9' 43" is the parallactic displacement of the sun between the first observation and the last. Now the orbital arc moved through by the earth in 24 hours is a little less than one degree, viz., 59' 10", so that the one-sixth of this quan- i -• 'i^il BY FIRST METHOD. 11 tity, viz., 9' 52" is the displacement of the sun caused by the earth's orbital progress, and which would be the difference between the two observations, if both were made from the earth's centre. But the actual displacement shown by the direct observations from the station on the equator is (by the supposition) 9' 43". The 9" of difference is therefore due to the reverse or retrograde movement of the station in consequence of the earth's rotation. This 9", however, represents the chord of the terrestrial arc, and not the arc itself. Now it is the terrestrial arc which we must compare with the solar arc to obtain the linear velocity of the one from the known linear velocity of the other. Therefore, as 9'43" : 9".427 : : 62 : 1. That is, the linear velocity in the orbital revolution of the earth itself is 62 times that of the station on the equator due to the rotation of the earth. Now since if the linear velocities were equal the angular velocities would inversely measure the comparative lengths of the radii, we should have accordingly S60° : 59' 10" * as the proportion of the greater length which the sun's radial distance would have, compared with the radius of the earth, if the linear veloci- ties were equal. But the linear velocity of the earth's orbital motion being determined as 62 times greater than that of the equatorial surface due to the earth's rotation, the computation, taking the radius of the earth at 4,000 miles, will be 4,000x365x62=90^ millions of miles as the sun's distance from the earth. The same observation also directly furnishes the geo- centric horizontal parallax of the sun. For, taking the preceding example, since the chord of the arc of 60° equals the radius of the circle, the difference between the * Or 3G5 : 1. 12 MEASUREMENT OF SUN's DISTANCE 1 m observed angle of parallactic displacement and 9' 52* {which diflference we have assumed in the foregoing as 9") is the geocentric parallax of the sun. Therefore, from this quantity', which results immediately from the observed displacement of the sun, and from the known magnitude of the earth and velocity of the earth's rotation, the distance of the sun from the earth can be readily deter- mined in the usual manner.* (2.) By the angle of the moon's illumination. In the accompanying figure (fig. 1), the moon is repre- sented at A, in quadrature ; that is, in the situation relatively to the sun and earth which she occupies when the one-fourth of her orbital revolution is completed. Now in her position at A, when so situated, rather more than the one-half of the moon's disc, as viewed from the earth, is illuminated by the sun, which is obviously a consequence of the sun's light striking the moon at an angle, with a line joining the centres of the earth and moon, rather less than a right angle. For if, as shown at B, on the other side of the figure, the moon be so situated in her orbit that the direction of the sun's rays forms a right angle with the line joining the centres of the earth and moon exactlj^ one half only of the moon's hemisphere will be illuminated.f * The number of seconds contained in the circle are 1,21)0,000, which, divided by 9 = 144,000 ; which, multipHed by 4,000 miles (the length of tlie earth's semi-diameter) = .576,000,000 as the circle of the earth's orbital circle. The distance of thf' aun from the earth, which is the radius ot that circle, equals, therefore, 91^ million railes. t To simplify the explanation we are here leaving out of con- sideration, for the moment, the great comparative magnitude of the sun, and consequent extension of the illuminated surface of the moon beneath the equator. — Hce page 14. sun's distance -By the An^le of Moom illumindLion. H-i- ^mimm^mimmmmnm^mimmm^ k1 A' If BY SECOND METHOD. 13 The moon's distance from the earth has been long since ascertained by means of geocentric parallax, and it may be assumed that the distance is now known with an approximation to precision. Since a ray of light from the sun to the moon is equivalent to a line joining the sun and moon, the careful astronomical observation of the angle of the moon's illumination at quadrature (by measuring the magnitude of that part of the moon's hemisphere illuminated in excess of the semi-hemisphere), will furnish the angular distance of the moon from the earth in terms of the earth's orbital circle ; or, in other words, it will determine an arc of that circle in linear measurement equal to the distance between the earth and the moon, of which distance the metrical value in miles is clready known. But so soon as the value of a circle, or of any definite fraction of the perimeter of a circle, in terms of the linear metrical standard, is ascertained, the length of the radius in terms of that standard becomes known. Therefore the distance of the earth from the sun, which is the radius of the earth's orbital circle, will become correctly known so soon as the angular illumin- ation of the moon at quadrature has been carefully measured, and accurately determined. ' To measure the angle of illumination it is not necessary, however, that the moon should be at the place of quadra- ture. By observing with exactitude the angular situation • at which 16' more than the one-half of the moon's hemi- •' sphere is illuniinated, the difference between that angle and quadrature will furnish the angle subtended by an arc of the earth's orbital circle contained in the inter- vening space, of known metrical value, between the moon 14 MEASUREMENT OF SUN S DISTANCE and the earth.* For oxamplo, lot us suppose in fig. 2, the moon to be at tliat place A of its orbit, where a lino joining the centres of the earth and moon is exactly at right angles to a line joining the centres of the moon and sun. We have first to consider that the diameter of tho sun as seen from the earth subtends an angle of 32' ; and since the moon, at that place in her orbit, is at the same distance as the earth from the sun, the angular magni- tude of tho sun's diameter as there seen from the moon will be the same. Consequently, since the moon's entire diameter as seen from the sun subtends an angle of only about 2 ", the sun's rays, impinging upon the moon's globe in a converging cone, will strike about 16' beyond the central circle which, posited horizontally to a line joining the centres of the sun and moon, divides the moon's globe into equal hemispheres. Viewed, therefore from the earth, the moon will appear illuminated to an angular distance of 16' below the equator, or, in other words, the whole of the moon's upper hemisphere and 16' of tho lower hemisphere will be illuminated by the sun's light. Now let the moon move onwards to the place of quadra- ture at B. We will assume that astronomical measure- ment determines the angle A E B as 9'. It follows that 16' + 9 =25' of the moon's dark hemisphere will now be illuminated. And, because the line S A is perpendicular to the line A E, and the line S E perpendicular to the line E B, it follows that the angle E S A also contains 9' of arc, which arc belongs to the circle of the earth's ** Other situations of the moon may be made available for the same purpose, only that the more directly the required data are furnished by the observation, the more correct, caieris jxiribus, will be the result. sun's distancf. By tJiH Anglo of Moon.--: illuniinaliot. %2 i^qp BY THIRD METHOn. ir> orbital revolution and in known to equal in metrical value 60 times the radius of the earth. Therefore, taking the earth's radius as before at 4,000 miles, wo have 0' of the earth's orbital circle, equal to 240,000 miles, which furnishes the radial distance of the sun from tho earth aa nearly 92 million miles.* ft (3.) In consequence of the earth's progressive orbital advance during tho time occupied in its diurnal rota- tion, the earth having completed a siderial rotation has to overtake the sun by a space which is a known definite fraction of tho circle bounding the earth's sphere. Similarly in the moon's revolution around the earth, the arc of difference between the siderial and synodic revolu- tion is 1 known definite fraction of the moon's orbital circle (as the earth's satellite). By ascertaining the time occupied by the earth with the velocity of its orbital revolution in moving through the same arc, the distance of the sun may be ascertained ; or, in other words, if we can ascertain the linear value of this differential angle compared with the similar angle of the earth's orbital circle, of which it is a consequent (and with which angle it is necessarily equal) the distance of the sun will become known. Now the moon itself as seen from the earth sub- tends an angle sufficiently large to admit of very accurate appreciation as a definite fraction of its own orbital circle, and of the difierential angle belonging to that circle, of which (diff. angle) the value in terms of the earth's orbital circle is required. If, therefore, we can measure the value of the moon's diameter in terras of the earth's 16 MEASUREMENT OF SUN S DISTANCE orbital circle, we can therefrom compute the value of the differential angle, and hence obtain the sun's distance. The Conditions of this method mav also he stated as follows : since we know the time occupied by the kood in completing a revolution around the earth, and we know the fraction of that orbital circle contained in the moon's disc, if we ascertain the time required by the earth with the velocity of its orbital revolution around the sun to pass through the angle subtended, at the dis- tance of the moon, by the moon's diameter, we shall thereby obtain knowledge of the comparative linear velocity of the earth around the sun, to that of the moon in its orbit around the earth ; from which data we can compute thv3 sun's distance. ' ........ To ascertain the time occupied by the earth in passing through an arc equal to that fraction of the circle of the moon's orbit, made apparent to us and defined by the appari- tion of the illuminated moon, as the angle subtended by the moon's diameter, an occultation of the sun by the moon (fig. 3) affords the most favourable opportunity. The centre of the sun, if the occultation be puch that the centre of the moon will pass over the sun's centre, or any clearly defined spot so situated on the sun's disc that the equator of the moon will pass over it, will equally well answer the purpose of the observation, which is in the first place to ascertain the apparent time occupied by the earth in passing through a fraction of its orbital circle equal to the angular value of the moon's di- ameter. Now the apparent time thus observed would be the actual time of the earth's velocity if the moon were at rest ; but, in fact, as the earth in its solar I [lie of the distance, stated as tha E300D and we ted in the d by the Q around t the dis- we shall ^■e linear the moon a we can I passing le of the e appari- inded by he moon be centre entre of Y clearly equator I answer :he first by the orbital an's di- l would if the ts solar tn 'x (-- CC o o ai > 0) 3 O o cO ^« ^ mmmmmf^ ■IB ,n I sun's distance By Llfie iutiar differenLial MetJiod. J B^.4 BY THIRD METHOD. 17 orbit is moving with a certain velocity in one direction/ the moon in its (tcrestrial) orbit is moving, with a lesser velocity, in the opposite direction. Therefore, since tho moon, instead of remaining at rest for the earth to pass it, has, in part, taken itself out of the way, the apparent time required by the earth to pass through the arc of the lunar c'rcle equal to tho moon's diametei is, by so much, less than the actual time. The velocity of the moon's motion in its orbit is known, and we can estimate with precision the time required by the moon to pass through an arc of its orbital circle equal to its own diameter. To deduce the actual time of the earth's velocity in passing the moon from the apparent time of the observation, we have, therefore, to add to the observed time a fraction propor- tional to the time which the moon requires to pass through an arc of its orbit equal to its own diameter. As an illustra- tion, let us assume the observed time of the passage of the moon's dianeter across the sun's centre, or over the solar spot, to be 1' 56". Now the time required by the moon to pass through the arc of its own diameter is, by computa- tion, 59 minutes. Therefore, to 1' 56" we have to add 1' 56"-=- (59 -^ 1'94) ; so that the actual time comes out very nearl)'-2'. Referring to the figure (fig.4) — in which S repre- sents the Sun, 8 e the solar radius-vector of the Earth, m rii' the Moon's orbit, and E, or e^, the Earth — the angular situation of the moon and earth in relation to the sun after 28 days' orbital progress, is shown on the right of the figure. The moon having then completed one sidereal revolution, and the earth having completed the thirteenth part of her orbital circle, we have the arcs subtending respectively the two equal angles, E S e 18 MEASUREMENT OP SUN\s DISTANCE ' 't. ;;l and Sem, proportional to each other in the ratios of the distances each to the other of their respective centres of revolution — to wit, the distances of the sun from the earth and of the earth from the moon. Now since the earth's orbital velocity carries it through 32 minutes of the moon's orbit in 2 minutes of -time, the time occupied by it in moving through the diflferential arc of 27° 41' (of the moon's orbit) would be 104 minutes 5 seconds. But the time occupied by the earth's velocity in moving through the greater arc (27° 41' of its own orbit) is 28 days. There- fore the greater arc is proportional to the lesser as 387^ to 1. Consequently the sun's distance is 387^ times that of the moon from the earth, and, taking the moon's distance as 238,800 miles, the sun's distance equals about 92i million miles. The same result may be also arrived at by inversely estimating from the relative velocities. For, taking the proportion of a very little less than 2' to 59', as 29| to 1, since the angular velocity of the moon is 13 times greater than that of the earth, we shall obtain the proportion of the greater distance if we multiply 29-j x 13, which gives 386 times the radius of the moon's orbit for the distance of the sun from the earth (equal to about 92 million miles). (4.) By the ascent and descent of the earth in its orbital revolutio.i. This method consists in choosing three sta- tions, one of them on the equator, and of the others — one in high latitude in the northern hemisphere, and one in similar high latitude in the southern hemisphere. The longitude of the stations to be respectively such that each, when passing through the plane of the solar equator, will have the sun on its meridian. The vertical distance BY FOURTH METHOD. 19 between these stations (measured as the chord of the vertical arc joining three places having the same longitude in common, and, respectively, having the same latitude as each of the stations of observation) being known, it is required to determine, by observation, the exact place of the sun in the ecliptic at the time when the meridian of each station successively transits the centre of the sun. We then have the vertical quantity contained in a definite small angular section of the ecliptic measured in the known metrical value of the vertical linear distance between the terrestrial stations. Hence the linear value of the earth's orbital circle and therefrom, of the sun's distance becomes readily determinable. Such observations would be preferably made near the time of the equinox, or when the sun is not, at most, more than two months from one of the nodes. And it is to be observed that, since the earth's vertical velocity is greatest at and near to the time of crossing the nodal plane of the sun, and for a brief period at each of the solstices becomes nothing, the quantity directly obtained by this method would hav6 to be rectified accordingly, in order to get the average of the vertical motion throughout the entire orbit or semi-orbit. Such rectification, however, the angular velocity and magnitude of the angle being known, and the nature of the orbit, as a circle or ellipse posited obliquely, being apprehended, would present no difficulty. To illustrate this method, the following considerations may be stated : — 45° of the earth's vertical motion com- bine with and occupy the same time as 180° of the earth's horizontal motion, i.e., the vertical motion is to the horizontal as one to four. ^ ■■'!'9*' m 20 MEASUREMENT OF SUN S DISTANCE The circle of the earth's rotation equals by time very nearly one degree of its orbital horizontal motion. One degree of the earth's orbital horizontal motion equals by time \ degree of its vertical motion. If, therefore, it be found that the entire vertical angle between the two extreme stations of the northern and southern hemispheres, which we will suppose to be 4,000 miles apart (measured as the chord of the vertical arc joining the latitudes of the two stations), contains 9 seconds, then, since the earth's horizontal orbital motion will occupy about 3"^ 40^ in moving through 9", the vertical motion will occupy 14™ 40^* in moving through the same angle. But 14™ 40* represents about one-hundredth of the earth's circlp of rotation, therefore the longitude of the two extreme stations would require to be such respectively that they would be 3° 36' apart. The meridian of the equatorial station would have such longitude as to be situated equidistantly between the meridians of the two extreme stations, 1° 48' from each.* (See fig. 5.) Instead, however, of thus confining the linear measur- ing distance to 4,000 miles of the earth's diameter, 12,000 miles might be made available for the purpose. Reference to the figure (fig. 5) will, it is thought, make the manner , of the intended application sufficiently in- telligible for the present purpose — viz., by taking the extreme stations c and e in such high latitudes, north and south respectively, as to be 6,000 miles apart (measured by the chord of the vertical arc joining the '" In strictly computing the required difference of longitude, an ^ allowance would have to be made for the onward progress of the «?f earth in its orbit. * + * § (U '-C T. UJ ■ J^ O c3 z w < "s C/3 !> O f C/5 ^ z ^ 3 W ■rs a; .^ IB s >*N, in i"'S| I 8J, ' f "I BY FIFTH METHOD. 21 latitudes, as before.) . . Commencing the compound ob- servation when the southern station reaches the nodal plane of the sun's equator, and completing the observation when the northern station reaches the same plane. The parallactic displacement of the sun would thus furnish the measurement of a definite sectional arc of the ecliptic. The distance of longitude between the meridians of the stations, as determined by computation, may be rectified by repeated experiment (i.e., by shifting the localities of the stations as required) until an indefinite approxima- tion to accuracy is obtained. (5.) A method of measuring the sun's distance quite distinct from the foregoing, but allied thereto, because dependent upon the vertical motion of the earth in its orbit, may be described as characteristically consisting in the astronomical observation of the phenomena called solar spots, which are sometimes seen to traverse the sun's disc. It is well known that at two seasons of the year only, namely at the summer and winter solstices, in June and De- cember, are the spots seen to cross the sun's disc horizon- tally. At other seasons of the year, the path of the spot in crossing the sun is oblique, either ascending or descending as the season is approaching the summer or the winter solstice. It has been already pointed out elsewhere that the apparent obliquity of tiie paths of the solar spots har- monizes perfectly with the perpendicular axis theory, and and may indeed be considered to constitute a part of the demonstration of the truth of that theory. The apparent obliquity is, we consider, certainly an effect of the vertical orbital motion of the earth combined with its orbital 22 MKASURKMENT OF 8UN .S DISTANCE horizontal motion. To apply this method, however, wo must assumo that the velocity of the spots in traversing tho circle of the sun's equator has been correctly ascertained, or, to speak with more particularity, that tho velocity of tho sun's rotation, by which the spots are carried around it cquatorially, has been so ascertained. Then since such rotation, assuming it to exist, must bo certainly in tho same direction as that of the earth's orbital revolution, the apparent motion of tho spots in circulating around the sun, as seen from the earth's centre, must be the resultant of the difference in the velocities of tho two motions, for if the velocity of the sun's rotation were such that a spot on the solar equator was carried around with the same angular velocity as that of tho earth's orbital revolution, the appearance to the terrestrial ob- server (from the earth's centre) would ho that of a spot motionless and constantly occupying tho same situation on the sun's surface. Now if the angular velocity of the sun's rotation were so grout as one complete rotation in 24 days, this velocity being 15 times greater than that of the earth's orbital re^ olution, and the velocity of the earth's vertical motion being only one-fourth of its hori- zontal orbital motion, the obliquity produced in the appa- rent motion of the spots across the sun's disc would be scarcely appr . .'able, because the deviation from a per- fectly horizoJLiTal plane would be so small in amount. But if we reflect that in consequence of the circumferential cur- vature of the sun's globe (as of every other globe) not more than, at most, about 90 degrees of the hemispherical sur- face would present the spot at such a visual angle as to bo visible from the earth, it will become apparent that an ^evcr, wo Tsiug the ertainod, Jlocity of I around ICO such ' ill the 'olution, around bo the ho two ro such around earth's 'al ob- a spot nation 3f the ion in 1 that f the hori- ippa- Idbe per- Bnt cur- lore 3ur- ) be an 1 ^ .^^S_^ROMr/^ /k Mg Fig. 6. to y '-i-J BY FIFTH METHOD. 23 angular velocity of rotation by the sun, equal to about six times that of the earth's orbital motion, would bring the spot into the field at the one side of the sim's disc, and take it out of the field at the opposite side, within about twelve days ; because, beyond the '!mit of the 90 de- grees, the foreshortening angular effect would become so great as to present the spot almost edgewise, and so ren- der it invisible, whether approaching on the one side or receding at the opposite. This interpretation of the facts may become more readily appreciable by aid of the accompanying figure (fig. 6). In twel''^ clays, which is about the average or most usual time that a spot remains visible, the earth will have ad- vanced nearly 12^ in its orbit, that is, in the same direc- tion in whioh the rotation of the sun, or of the sun's surface, carries the spot. This advance will evidently occasion an extension of the solar arc throughout which the spot will be visible ; so that if, supposing the earth had remained stationary in its orbit, the visual solar arc would have been about 78°, it will have extended to about 9C" io consequence of the orbital advance of the earth. Hence, to complete the circuit of the sun's equator, the spot will occupy 48 days (say about 50 days), which may be considered to measure the angular velocity of the sun's axial rotation. It is not, however, to be inferred in such assumption that, supposing the spot to remain existent and unchanged, it will reappear at the sun's eastern limb at the expiration of about 36 days. Such inference would overlook the continuance of the earth's orbital motion, which, in that time, would add nearly '66° to the 270°, making a total of 306°, throughout which arc, equivalent 24 MEASUREMENT OF SUN's DISTANCE to about 41 days in time, the spot would be occulted by the sun's globe.* One of the most distinctly observed phenomena belong- ing to the solar spots is the apparent obliquity of their paths across the sun's disc, with exception of two semi- annual periods in the year when those paths form straight lines. Another observed phenomenon is that the direction of the angle of obliquity during five months of the year is inverted during the other five months. Now this appa- rent obliquity is, we consider, certainly attributable to the vertical ascent and descent of the earth's orbital path. But if such be the cause, it follows (1) that the angle of obliquity of the path will necessarily measure the angular velocity of the spot in its revolution around t'l'^ i' ind therefore of the sun's rotation ; and (2) that the obliquity of the path of the spot, i.e., the amount of its deviation from a horizontal line, may be utilized as a means of measuring the sun's distance in diameters of the sun, for the vertical amount of that deviation is parallax of the ** The most direct and probably the best method of deducing the period of the sun's rotation from the observed velocity of a solar spot would be simply to determine accurately the angular progress of the spot, as seen from the earth's centre, when near the central part of the sun's disc during one complete rotation of the earth, i. c, during 24 hours. Now the earth, during the 24 hours, will have moved round the sun in the direction of its rota tion nearly one degree ("9863 of a degree) ; therefore the observed angle, with addition of chis terrestrial quantity, will be that part of the circle of the sun's equator moved through in its rotation during 24 hours. The time, therefore, occupied by a complete rotation of the sun will be simply 24 hours multiplied by the number of times the observed angular quantity increased by addition of the angle moved through by the earth, is contained in 360". J - ■ ;od by )long- their semi- aight jction year appa- o the path, ^le of ••ular md luity ition 18 of , for ' the icing of a jular neai' mof e 24 fota rved part tion )lete the by incd o X z f; < ■p 1- -c w a ~ CO c . z 3 i' w ,- 1 ■ ' i'- 1 ■^ i y'l \ BY SIXTH METHOD. '^D spot as projected on the suii's disc, occasioned by the ascent and descent of the earth in its orbital revolution.* Let us suppose the observed parallax of the spot, takin;^ the extreme limit of displacement, north and soutli, on the sun's disc to amount to 13 minutes : we shall then have the proportion.. As 13' : 47° : : the Sun's radius: the number of times that radius is contained in the radius of the Earth's orbit. Fur the purpose of illustration we will assume the sun's radius at 420,000 miles. We then have 420,000 X 217 = 91 millions of miles as the radius of the earth's orbit or distance of the earth from the sun. (6.) A method of measuring the sun's distance quite distinct in character from the preceding — for the pre- ceding methods are all dynamical in character, whereas the method now about to be described is statical in character — may be termed the geometrical method. In it a knowledge of the sun's distance is attained by observing the relative angles subtend, \ by the diameters of the sun, the earth, and the moon, respectively, when seen directly or indirectly from several points of view. In Fig. 7, let ^ represent the sun ; E the earth ; and MM' the moon at the two opposite extremities of its orbit, viz., at conjunction and at opposition. The sun's light •■' Supposing the spot to remain permanent or unchanged for some considerable time, and that the angular velocity of the sun's rotation were only the same as that of the eai th's orbital revolu- tion, the effect would then be that the spot would, to the terrestrial observer, appear to ascend and descend vertically on the sun's disc throughout an angle of 2U° beneath and 231° above the solar equator, i. c, throughout an angle of 47'' (45°). 26 MEASUREMENT OF SUn's DISTANCE shining past the splierical earth, projects its shadow as a (lurk cone to the point x. The angle axb h therefore the angle subtended by the sun's diameter as seen from the point x. Now since the linear value of the earth's diameter is known, and the distance of the moon from the earth at opposition is known, and the breadth of the shadow, at g h, is ascertained from the time occu- pied by the moon in passing through it,— the distance E x~(i.e., the length of the earth's shadow)— becomes also known ; consequently the value of the angle c x d, or axh, which is the same, is known. Since the point /is on the surface of the earth, from which the astronomer views the sun, the angle a / ft is the observed angle subtended by the sun's diameter. It is at once evident that the angle afbk greater than the angle e x d-, consequently if the lines a: c and a: ^ be produced in- definitely, they must eventually intercept the lines /a and fh in the points a and b, at the two extremities of the sun's diameter. If, therefore, we ascertain by observation the exact value of the angle a /ft, at the time the moon is in opposition, we shall have the means of readily com- puting the distance 8 E oi the sun from the earth. To illustrate this by example, we have the distance ^ ^ of the earth's shadow ascertained to be equal to 218 semi- diameters of the earth, and, hence, the value of the angle Exck determined as 15' 46^ (Now the angle of the sun's diameter, as observed at different times of the year, is supposed to vary from 31' 32".0 to 32' 36".2, but. with respect to an actual difference in the distance of the sun, we must either decline to accept this reported great variation, as correct only in respect to an apparent BY SIXTH METHOD. 27 variation in the sun's magnitude when viewed at different seasons from the same locality, or from places situated nearly in the same latitude, and presumable erroneous in fact, in respect to observation made from the earth's centre through an atmosphere at all seasons in the same condition as to temperature, density, and humidity ; or, otherwise, we must accept the assumption that the earth, in the course of each annual revolution, approaches and recedes from the sun through a space equal in linear extent to nearly six times the diameter of the moon's orbit.) For the present purpose we will assume the average angular value of the sun's semi-diameter to be determined as 15' 55". The difference between this angle (15' 55") and the angle Exc (15' 4G"), which equals 9", is the angle subtended by the earth's semi-diameter as seen from the sun, and, therefore, since the actual length, or linear value in miles, of the earth's semi-diameter is known, we have ascertained the value of a definite fraction of the earth's orbital circle, and hence the distance of the sun from the earth, which is the radius of that circle, becomes known. For instance, if 9^ be the ascertained difference between the angles, then, since 1 degree contains 400 times 9", multiplying 400 by 4,000 miles as the linear value of the earth's semi- diameter, we obtain 92,800,000 miles as the distance of the sun from the earth.* This method furnishes the angular value of the earth's diameter as seen from the sun, which is already obtained with an approximation to correctness by means of the geo- * Taking the value of tae earth's radius at .3,950 miles, which is very nearly the actual estimate, the distance will be 91,640,000 mi es- These computations arc, however, mainly intended to illustrate the methods. 28 MEASIJUF.MKNT OF SUN S niSTANCK centric parallax of tho sun. It is a question whether or not this method now proposed is susceptible of a greater degree of exactitude. As an entirely independent and distinct method, however, it cannot fail to possess some considerable degree of interest and utility. Instead of measuring the angle of the shadow behind the earth, the same angle may be indirectly obtained by measuring the breadth of the sun's light at any known definite distance between the earth and the sun. This may be readily explained by reference to the figure ; for instance, if the moment when tho limb of the advancing moon commences to interpose itself between the sun and earth on the one side be exactly determined, and also tho moment of the conclusion of the egress of the moon on the other side of tho orbit, so as to ascertain the exact time occupied by the moon in traversing the angular space of the sun's diameter (viewed from the earth), tho required value of the angle at the distance of the moon's semi-orbit would become known. More favourable for this purpose would be a transit of the sun by one of the inferior planets. Records of carefully observed transits of Venus might be made available perhaps to determine in this manner the precise value of the angle of the earth's shadow, supposing the same records to include the angular valu ~ of the sun's diameter as seen at the time of the transit. (7.) A method, which may be considered as allied to the preceding, coL/ats in measuring by astronomical observation the apparent value of the sun's diameter, as seen, on the one hand, from a station on the earth's surface when the sun's centre is over the meridian of that station ; SUNS DISTANCE By the differenUal An^e al'liartiis Had lUS. Eg. 8. BY SEVKNTH METHOD. 29 and, on the other hand, as Hoon, at the same timo, from the «arth'a contro. The manner in which it is proposed to obtain the com- parative angles, is by means of the transit instrument, by which, having first ascertained with precision the angle subtended by the semi-diameter of the sun when the sun's centre is over the meridian of the station, the timo elapsing until the extremity of the sun's diameter (i. c, the edge of the sun's limb) is over tho station, is to be carefully observed, which time will measure the angle subtended by the semi-diameter of the sun as viewed from the earth's centre. This last angle must be evidently less than tho former, and by the dliforonco the distance of tho sun may bo determined, because the metrical value in miles of the earth's semi-diameter is already known, and if the two lines, tho inner of w^ Ich has a greater obli- quity than the outer, be produced until they eventually meet, tho point of interception will be the sun's distance, and must be directly proportional to the semi-diameter of the earth, in a ratio determined by the observed angle.* The principle of this last method is fundamentally the same as that explained in the case of the earth's shadow, and it is possible that the convenience, directness and simplicity of this method, notwithstanding the delicacy and extreme accuracy of observation requisite, may render it preferable and practically more advantageous • Instead of tho semi-diameter, the eutire diameter of the snu may of course be observed. A slight correction would be tlieo- retically required as an allowance for orbital motion of the earth ; the effect of which would tend to increase tlie time, and which would be accordingly corrected by deduction. The quantity, however, would be extremely minute (about the 1400th of a degree of orbit,) and insufficient to be appreciable in the practical application of the method. ! !l 30 MEASUREMENT OF SUN's DISTANCE BY SEVENTH METHOD, than either of the methods previously described or hitherto practised. To illustrate this method : let the angle of the sun's semi-diameter viewed from the station be assumed as 16', and viewed from the earth's centre as .15' 59"-96, the dilference of one-twenty-fifth of a second on the earth's radius, taking the metrical value of that radius, as before, at 4,000 mileo; would give about 96 million miles as the sun's distance.* In the figure (fig. 8) the enormous ex- aggeration of the ratio of the earth's radius to the sun's distance (by representing the sun near to the earth) renders the basis of the method more distinctly apparent. * The computation is . . GO x 2 5 X 16 = 24,000 ; whioh x 4,000 = 96 millions. WEiiTHEiMEn, Lea & Co., Circus Place, Finsbury Circui.