A Discourse concerning the Menstrual Parallax, Arising from the Mutual Gravitation of the Earth and Moon; It's Influence on the Observations of the Sun and Planets; With a Method of Observing It: By J. Smeaton, F. R. S.

XXIV. A Discourse concerning the Menstrual Parallax, arising from the mutual Gravitation of the Earth and Moon; its Influence on the Observations of the Sun and Planets; with a Method of observing it: By J. Smeaton, F. R. S. Read May 12, 1768. It is demonstrated by Sir Isaac Newton in the Principia, that it is not the Earth's center, but the common center of gravity of the Earth and Moon, that describes the ecliptic; and that the Earth and Moon revolve in similar ellipses, about their common center of gravity. The same great author has also investigated, from the different rise of the tides, when the Moon is in conjunction or opposition to the Sun, to those which happen when the Moon is in her quadratures; that the quantity of matter in the Earth is to that in the Moon, as 39.78 to 1; from whence, and the known distance of the Earth and Moon, it would follow, that the common center of gravity of the two bodies falls without the surface of the Earth, by one half of its semidiameter: that is, that the center of the Earth describes an epicycle round the common center of gravity once a month, whose diameter is three semidiameters of the Earth. Dr. Gregory, in his astronomy, has laid hold of this circumstance, in order to prove the relative gravity of the Earth and Moon, by observation; which is the subject of his 60th proposition of the fourth book; book; in which he has demonstrated, that if an observer on the Earth makes a correct observation on the Sun's place, when the Moon is in one quadrature, it will differ from a like observation, taken in the opposite quadrature (according to a mean elliptic motion) by an angle which the diameter of this epicycle will subtend at the Sun. The same learned author has also shewn, in the scholium to the same proposition, that this quantity, or parallax, will be twice greater to Mars in opposition, and three times greater to Venus, in her inferior conjunction with the Sun. The difference thus produced in the apparent place of the Sun, and of all the primary planets being governed by the Moon, and having its period the same, may perhaps be not unaptly called the menstrual parallax. Now if, with Sir Isaac Newton, the relative gravities of the Earth and Moon are taken between the proportion of 39 and 40 to one; the menstrual parallax of the Sun will come out $13''$ upon the radius of the Earth's epicycle, and will affect the solar observations at the opposite quadratures, by double that quantity, viz., $26''$: in like manner, the mean distance from the Earth of Mars in opposition, being to the Sun's mean distance, as 1 to 2. 1; and the least distance of Mars from the Earth, to the Sun's mean distance, as 1 to 2 $\frac{1}{2}$, the menstrual parallax of Mars will affect the observation upon him in that situation, by $56''$ and $73'' \frac{1}{2}$, respectively. The mean distance of Venus from the Earth, in her inferior conjunction, being to that of the Sun as $3 \frac{1}{2}$ to 1 nearly, and not very variable, on account of of the orbit of Venus being almost circular; the menstrual parallax would affect the place of Venus, in that situation, by a quantity not less than $92''$; and in all other situations in proportion to her distance; which also holds with respect to all the rest of the planets. These disturbing quantities are by no means to be dispensed with, in the nice and critical state that astronomical observations and calculations have arrived at, in consequence of the discoveries of Dr. Bradley, who may be said to have given a basis to astronomy; however, could we rely upon the data, on which Sir Isaac's investigation of the relative gravity of the Earth and Moon is founded, we should have nothing to do but to apply an equation to the particular cases, according to the diameter of the epicycle, as deduced from the relative gravity; but whoever considers the great obstructions that the water of the sea meets with in its motion to obey the influence of the Moon; the great difficulty in ascertaining the true height of the tides, from the many disturbing causes intervening; and the many uncertainties, and want of coincidence, that have attended, and must attend, such observations; must confess, that this matter does not seem capable of such a determination from that quarter, as the present state of astronomy requires. Accordingly, since the time of Dr. Gregory, those great astronomers Dr. Bradley, De la Caille, and others, have applied themselves to determine the quantity of the menstrual parallax from solar observations: but though these have given cause to suppose that the relative gravity of the Earth and Moon are not above $\frac{2}{3}$ of the quantity deduced from the tides; yet, as the observation observation of these small angles principally depends upon the observation of the Sun's right ascension (which, depending on the measure of time, is less capable of exact observation, than if depending on divided instruments); the deductions thence drawn seem still wanting of that certainty which the subject demands; and if to this we add, from a deduction of Mr. Maskelyne, that the relative gravity of the Earth and Moon is as 76 to 1, derived from the effect that the Moon produces in the nutation of the Earth's axis; the relative gravity, and consequently the parallaxes thereon depending, will be reduced to almost one half of those resulting from Sir Isaac's determination. It is true, that the quantity of effect of the menstrual parallaxes will not be great, if computed upon Mr. Maskelyne's induction, for as much as that the common center of gravity will be considerably within the Earth's surface; yet, even in that case, the Sun's transit over the meridian, when the Moon is in one quadrature, will differ nearly one second of time from that observed in the opposite quadrature; and though De la Caille and Mayer have formed equations depending on the Moon, to be applied to the equation of time; yet, if we are at an uncertainty, whether the maximum of this equation is one second, two thirds of a second, or half a second of time, each way, we are still under a material difficulty; for though these differences are so small, that it is not easy to determine them exactly from solar observations; yet, as they are capable of creating a sensible difference in these observations, they will, so long as they remain undetermined, prevent that solidity and firmness ness to the solar observations, which is the more necessary as they are the foundation of all the rest: but with respect to those planets, that in their periods come nearer to us than we to the Sun, the observations upon them will be affected by a greater uncertainty. The determination of the menstrual parallax is of still more importance, as it is a necessary consideration in the determination of the Sun's parallax; and this, whether deduced from Mars or Venus, as I shall presently shew more particularly; but first I must state the quantity of the menstrual parallax, according to the best data yet known, by a contrary process; and, taking the mean quantity of the Sun's parallax, according to the determination of Mr. Short, at $8''\frac{1}{8}$, and the relative gravities of the Earth and Moon, according to Mr. Maskelyne, as 76 to 1, and the mean distance of their centers equal to $60\cdot\frac{1}{2}$ semidiameters; we shall then have the distance of the Earth's center from the center of gravity, at $\frac{4}{5}$ of the Earth's semidiameter (that is, $\frac{1}{5}$ of that semidiameter within the Earth's surface) and the menstrual parallax equal to $\frac{3}{10}$ of the Sun's parallax; consequently about $7''$; and the double menstrual parallax, or vacillation, arising from the whole diameter of the epicycle, $14''$; the mean menstrual parallax of Mars in opposition, $29''\frac{1}{2}$; the greatest, $38''\frac{1}{2}$; and that of Venus $49''$; from hence it follows, that, was a person to attempt the Sun's parallax, by the diurnal motion of the Earth, applied as a basis to Mars in opposition, as has formerly been tried; and should the Moon be at new or full at the same time, the change of place of the Earth's center, in its own epicycle, would amount to an angle seen from Mars. of $1''.3$ nearly; that is, in case the interval between the observations was eight hours, and Mars at his mean distance; but if Mars was not at his nearest distance, this change would in the same time amount to $1''.7$ nearly. In like manner, if a transit of Venus happens near the new or full Moon (as will be the case next year), the time of the transit will be affected by a change of place, such as the Earth's center will describe in its epicycle, during the time of the whole transit, if the beginning and end are observed in the same place; or during the difference of absolute time, at which the transit appears to begin or end to different observers in distant meridians. Thus, when the same observer sees the beginning and end in the same place, the base described by that observer, from the Earth's diurnal motion, must be corrected by the space described by the Earth's center, in the circumference of its epicycle, during that time; which, if it be supposed of seven hours, will amount to an angle of $1''.9$, seen from Venus: but, where the beginning or end is seen by different observers in distant meridians, as the difference of absolute time can hardly amount to above 15 minutes, the change of place of the Earth's center will for that time be but small; however, at the rate beforementioned, it will for 15 minutes affect the parallactic angle seen from Venus, by about $\frac{7}{46}$ of a second; and the parallax of the Sun, by about $\frac{1}{46}$ part of the whole: but this proportional part will remain the same, whether the distance of meridians be such as produce a greater or less difference of absolute time than 15 minutes*. * If an error of $\frac{1}{46}$ part of the whole may be supposed in the observation for determining the Sun's parallax by the transit From what has been said, I suppose it will appear, that the effects of the menstrual parallax are worthy of consideration; and that nothing has been yet executed, whereby it has received a determination sufficiently accurate; for, in regard to observations upon the Sun, the whole quantity is too small to be minutely observed in right ascension: and with respect to the application to Mars and Venus, as suggested by Dr. Gregory, I do not know that anything has been done; and indeed no wonder, as the theory of the motion of Mars and Venus has not been as yet so critically reduced to computation, as to render their parallaxes (though in themselves much greater) deducible with equal certainty as that of the Sun. What I therefore have now to propose, is a method of observing the menstrual parallaxes of Mars and Venus, without laying any undue stress upon the theory of their motions. The first opportunity of making an observation for this purpose, will be at the next opposition of Mars; which, according to the Nautical Almanack, will happen the 26th of October next, in the morning; I will therefore endeavour to illustrate this matter by taking that as an example. The distance of Mars from the Earth will then be somewhat less than the mean distance, that is, as 1 to 2.2; and consequently his double menstrual parallax, according to Mr. Maskelyne, will be near $31''$ in the point of opposition. Now, as the Moon of Venus, a neglect of the menstrual parallax may make it $\frac{1}{\pi}$ part of the whole. will be at full, not above 12 hours preceding that opposition, the Moon will be nearly in the most favourable situation for the purpose. For this end, let an accurate observation be made upon the place of Mars at the following times, viz. first, near the time of the new Moon, preceding Mars's opposition; or more properly at the nearest opportunity, to the time of the Moon's opposition to Mars; which will happen in the night, between the 12th and 13th of October: secondly, let the place of Mars be observed when the Moon is nearest her quartile with Mars; that is, between the 19th and 20th of the same month: thirdly, let an observation on Mars be made when the Moon is in conjunction with Mars, the nearest to his opposition with the Sun; that is, between the 25th and 26th of ditto: fourthly, let Mars again be observed when the Moon has moved on to her quartile with Mars, viz. between the 31st of October, and 1st of November: and fifthly and lastly, let the place of Mars be observed, when the Moon has again got to her opposition with Mars, which happens between the 7th and 8th of November. Now it is manifest, that, when the Moon is in conjunction or opposition to Mars, the center of the Earth, the center of Mars, and the common center of gravity of the Earth and Mars, will be nearly in a right line, and consequently, that an observer will then see Mars, in the same place in the heavens, as if the common center of gravity was the same as the center of the Earth; therefore, then the place of Mars will be unaffected by a menstrual parallax; and such will be the first, third, and fifth of the observations above propounded. It is equally evident, that when the Moon is in quartile with Mars, and moving towards a conjunction, an observer, at the Earth's center, will see Mars more backward in the ecliptic, than if seen from the common center of gravity, by $15'' \frac{1}{2}$; and that, when the Moon is in her opposite quartile with Mars, and moving from her conjunction, that then an observer at the Earth's center, will see Mars advanced in his orbit more forward by $15'' \frac{1}{2}$, than if seen from the common center of gravity; and the one observation checked with the other, will, according to a mean elliptic motion, differ by the quantity of $31''$; and such will be the second and fourth observations above propounded. Now, from the first, third, and fifth, observations, three points of Mars's orbit will be given; which, by the help of the theory of Mars's motion in an elliptic orbit, whose aphelion, eccentricity, and nodes, are known sufficiently near for this purpose; the intermediate places of Mars may be inferred with the requisite degree of accuracy: and particularly, as the two intermediate observations, viz. the second and fourth, will be nearly at equal intervals of time between the three others: from hence it follows, that the difference between the inferred, or computed places, at the quartiles, and the observed places at those times, will be the menstrual parallax required. It is to be noted, that the times above specified are the most favourable for the observation; and could those be made uninterruptedly from weather, there would be the less occasion for any other: but, as much as possible to prevent disappointments of this kind, it will be right to begin the observations, a month preceding ing making the proper observations, at the conjunctions, quartiles, and oppositions, of the Moon with Mars, which will be the means of supplying such observations, as may happen to prove abortive before the opposition of Mars, and also, in case any of the observations to be made after that opposition shall prove deficient, the observations may be carried on for a month or competent time afterwards. As a further security against disappointments, as well as cheque, it will also be advisable to make the proper observations, the night preceeding and subsequent to those in which the quartiles, conjunctions, &c. happen; for, as the quantities will not differ considerably from those obtained on the days specified, with proper allowances they may be brought in support and confirmation of the former. In like manner, when Venus is moving towards her inferior conjunction with the Sun, as will happen next year, the same observations may be made with respect to her; and continued for a necessary time, to get observations of the place of Venus; viz. the first, when the Moon is in conjunction or opposition with Venus: a second, when the moon is in her quartile with Venus: a third, in conjunction or opposition: a fourth, when the moon is in her opposite quartile to the former: and a fifth, again in conjunction or opposition: the same opportunity will also offer when Venus is moving from her inferior conjunction with the Sun, and becomes a morning star. In regard to the observation of Venus, it is remarked by astronomers, that she is to be seen with a good transit telescope, when she is within a few degrees of the Sun; but, as she is three times nearer the Earth, than than the Sun's mean distance, when her elongation is $25^\circ$ in the inferior part of her orbit, it is plain, that the necessary observations may easily be made, when her menstrual parallax will be at a medium, three times greater than the Sun's; and consequently amounting for the whole difference to $42''$. To avoid embarrassment in description, I have hitherto supposed, that all the observations are made in the meridian; in which case the right ascensions will be the same as they would appear from the center of the Earth; and consequently, the planet's longitudes thence deduced, nearly the same: but 'tis easy to see, that if the quartile observations are made when the planets are considerably to the east or west of the meridian, and so chosen, that the place of the observer be further distant from the common center of gravity, than the center of the Earth is from that center, that the base of the observations will be considerably enlarged. Thus, in our latitude, supposing that the quartile observations are made four hours before and four hours after the planet passes the meridian, this will produce an enlargement of the basis by one of the Earth's semidiameters: and as the whole base or diameter of the epicycle comes out, according to Mr. Maskelyne, no more than $1.6$ of the Earth's semidiameters; the base will, according to this method, come out $2.6$; and consequently, at the next opposition, the menstrual parallax of Mars will be thereby enlarged to $50''$, the greatest to $62'' \frac{1}{2}$, and that of Venus at a mean, to $74'' \frac{1}{2}$. It must however be acknowledged, that no kind of observations of the places of the planets are of equal validity with those taken with the best instruments in the the meridian; those taken with micrometers perhaps not excepted: for however accurately small distances can be measured by the micrometer of Mr. Dollond, yet, as these measures can hardly be reduced to the ecliptic, without having the difference of declination or right ascension from other means (except two stars making somewhat near a right angle with the planet should appear within the field of view at once); and as in all these cases the rectification of the places of the stars themselves ultimately depends on meridian observations we may perhaps be allowed to say, that in the most favourable cases of the micrometer, the determinations thence to be drawn, are not superior to meridian observations, and in less favourable cases, must be inferior: however, as the micrometer observations out of the meridian give an opportunity of repetition as often as we please; and the observations for rectification of the stars concerned, can be repeated in the meridian, as often as we please also; it must be equally allowed, that when these kind of observations are taken, not too near the horizon, when proper stars offer for this purpose, and the whole skilfully managed; these kind of observations fall but little short of those taken immediately in the meridian. I cannot therefore hesitate to recommend, that the quartile observations be taken out of the meridian, as well as in it: in the first place, by Dollond's micrometer, if stars offer in proper positions; and if not, secondly, by taking differences of right ascension and declination between the planet and the stars, by the common micrometer, in case proper stars offer themselves for this purpose: but as it frequently happens, that no proper stars offer themselves to micrometers of either kind; and this is still more likely to happen in the observations of Venus, which will be chiefly in the day light; I beg leave to offer (what to me is) a new method of observation out of the meridian; and which, though I esteem it not equal to micrometer observations of either kind, I apprehend will fall so little short thereof, and prove to much superior to any other method now in practice in these cases, that I hope I shall on this occasion be excused, in giving a particular description thereof: but, as it is a general method of observing out of the meridian, I shall reserve it by way of appendix. In the next observation of Mars, it has been stated, that, in the meridian observations alone, the menstrual parallax, according to the smallest estimation, may be expected to amount to $31''$ in longitude; which, turned into right ascension, will make about $2''$ of time: now, if it may be allowed, that a well-practised observer can take the time of a transit to $\frac{1}{4}$ part of a second, over a single wire, if he has three wires, or more, as usual, the mean of the three should be within $\frac{1}{2}$ part of a second; or within $\frac{1}{2}$ part of the whole quantity in question: it is however a matter of chance, whether the mean of three may or may not be within $\frac{1}{3}$ part of the whole; and as equal errors may be committed in the observations of the transits of the stars, wherewith the right ascensions of the planets in question are compared; which it is an equal chance, whether they tend to correct or increase the errors committed in the former; yet if, as has already been proposed, the observations are continued for two or three months, instead of one; and observations, taken the day preceding and subsequent to to the days of conjunction, quartile, and opposition; and this as well out of the meridian as in it; we can hardly doubt but that, if the weather should favour, so many cheques would be formed, that, from the next opposition of Mars alone, the affair may be brought within a 24th part of the whole; and, if to this be added the force of such determinations, as may be drawn from observations on Venus, before and after her transit over the Sun next year, it can hardly be doubted, but that those three will bring us within a single second of a degree, subtended from the nearest planet; and these conclusions will be further strengthened by future observations; as two years will scarcely pass without affording one or more opportunities of this kind. As I meant not to embarrass myself with exact computations, I have constantly supposed the distance of the common center of gravity from the center of the Earth, to be a fixed quantity; whereas it will vary in the same proportion as the Moon's distance varies; but, as this and many other minutiae will properly enter the computation, when the observations are made, I must beg leave to refer them to the learned in this science. Austhorpe, April 17, 1768. J. Smeaton.