Two lectures on the parallax and distance of the sun, as deducible from the transit of Venus. Read in Holden-Chapel at Harvard-College in Cambridge, New-England, in March 1769. / By John Winthrop, Esq. Hollisian Professor of the Mathematics and Philosophy at Cambridge, and F.R.S. ; [Three lines in Latin] ; Published by the general desire of the students.

The Use of the TRANSIT of VENUS.

LECTURE I.

A TRANSIT of VENUS UNDER THE SUN is the most uncommon, and the most important phaenomenon, that the whole compass of astronomy affords us. So un|common is it, that it can never happen above twice in any century; in others, but once; and in some centuries it cannot happen at all. And the importance of it is such, as to supply us with a certain and complete solution of a very curi|ous Problem, which is inaccessible any other way. On both accounts it well deserves a very particular attention.

THE phaenomenon itself is, the passage of the planet Venus across the face of the Sun, from east to west, in the form of a round, black spot. As we shall have the opportunity in a few months of viewing one of these phaenomena, if the wea|ther permit, I shall adapt my description of them to the particular circumstances of this Transit. This planet then, having approached the Sun on his eastern side, and arrived at his limb or edge,

will, by being opake, make a small, black notch upon it; which, as soon as it is big enough to be visible, is called the first external contact. The notch will immediately grow bigger, by the planet's advancing, and in about 9 minutes her centre will be got on the Sun's limb, and the appearance will be that of a little black semi-circle, which will occupy about 3½ degrees on the circumference of the Sun. When the pla|net is advanced above half, her circular figure will cause an appearance as of a luminous horn or cusp on each side of it, with the points turned towards each other, and gradually ap|proaching, and in 9 minutes more Venus will be wholly within the Sun's disk, and the two luminous cusps running together at their points, will form a narrow thread of light behind the planet. The Sun's limb now recovers its light, where it had been interrupted by the body of the planet. This is called the immersion, or the first internal contact. In about 3 hours the planet will be in the middle of its course, and nearest to the Sun's centre, from which it will be distant to the north about ⅓ part of the Sun's diameter; its own diameter being 1/32 part of the Sun's. Then approaching the western side of the Sun, the thread of light between their limbs will grow narrower till it breaks, and Ve|nus again touches the Sun's limb, forming, what is called, the beginning of emersion, or the last internal contact. The light of the Sun's limb will now again be interrupted, and two lumi|nous

cusps formed, as before, which will recede from each other as the planet moves off from the Sun. In about 18 minutes, the last external contact will be formed, the impression made by the planet upon the Sun's limb having become gradually less & less, till it is quite impercepti|ble.—These several phases will be very conspicu|ous with any telescope, and even with the most ordinary perspective; but few eyes are sharp enough to discern so small an object as Venus divested of her radiance; her apparent diameter being less than a minute of a degree.

YOU see what are the appearances in a Transit of Venus.—The theory of it is easily under|stood.—Venus circulates round the Sun, the same way as the Earth does, that is, from west to east, but in a smaller orbit, and in a shorter period. When Venus is in the same degree of the eclip|tic with the Sun, she is said to be in conjunction with the Sun; and as her orbit is inclosed within the Earth's, it is plain, that a conjunction may happen two different ways; either by the pla|net's being beyond and above the Sun, and so in the remotest part of its orbit from us, which is called a superior conjunction; or being below the Sun, and nearest to us, and this is called an inferior conjunction.—The planet, having past a superior conjunction, will appear to move quicker to the east than the Sun does, whose apparent velocity is the same as the Earth's real velocity in its orbit; and thus will depart farther and

farther from him, till she has attained to the distance of 48 degrees; and this is her greatest elongation from the Sun, her whole orbit being included within two right lines drawn from the Earth, at the distance of 48° east and west of the Sun. She will now appear stationary, in respect of the Sun; and after this, while she is describ|ing the nearest part of her orbit contained be|tween the two right lines just mentioned, will seem to move with a retrograde motion, or the contrary way to what she did before; ap|proaching the Sun from east to west, and so will pass through an inferior conjunction with him. Then running off to the distance of 48° west from the Sun, she will again appear stationary; and after this, will be seen to resume her direct or proper motion eastward, and will at length arrive at another superior conjunction.—In the course of such a revolution, she puts on all the phases of the moon. Near the superior con|junction she appears round like the Full moon, but small; after this, waning, like the moon after the Full; tho' her diameter grows larger, because she is coming nearer the earth. At her greatest elongation she resembles the moon in the Quarters; and from thence to the inferior conjunction is cuspidated or horned, like the moon from the last Quarter to the Change; that hemisphere, which is enlightened by the Sun, being turned more and more from the Earth, till in the conjunction itself, it is wholly turned from us. Her diameter now appears largest.—

In all this time, being on the eastern side of the Sun, she sets after him; and for her extraordi|nary lustre, is called, by way of eminence, the Evening-Star. She passes thro' the same series of changes, but in an inverted order, from the inferior conjunction to the superior just as the moon does from the Change to the Full; and in this interval, being on the western side of the Sun, and rising before him, she is called the Morning-Star.—Such are the phaenomena of Venus.—It may be observed by the way, that those of Mercury are altogether similar; and that they demonstrate beyond all doubt, that these planets perform their revolutions round the Sun, and not round the Earth.

FROM what has been said 'tis obvious to col|lect, that it is only in the inferior conjunctions that Transits can happen.—If the orbit of Ve|nus lay in the same plane as that of the Earth, or the ecliptic, there would be a Transit at every inferior conjunction; for in this case, the planet would pass directly between the Sun and the Earth; and only her unilluminated side be|ing turned towards us, she must appear like a black spot upon the Sun's disk. But the fact is otherwise. The orbit of Venus is inclined to the ecliptic in a certain angle, tho' but a small one, and intersects it in two opposite points, called the nodes. At one of them, the planet crosses the ecliptic from south to north, which therefore is called the ascending node; the

other, where she is moving from north to south, is called the descending node. 'Tis only at those inferior conjunctions which happen so near one of the nodes as to make Venus's latitude, or her distance from the ecliptic, less than the apparent semidiameter of the Sun, that we can have a Transit; at all others, Venus will pass clear of the Sun, either to the north or south; and that, at so much greater distance as she is farther from her nodes. And for this effect, the Sun must be within 2 degrees of a node; which are he goes over in 2 days. In this age and the next, the nodes being situated near the middle of Gemini and Sagittarius, where the Sun is the 4th of June, and 7th of December, Transits can hap|pen only at those times of the year; that is, be|tween the 2d and 7th of June; and between the 5th and 10th of December. I shall confine myself at present to those which happen in June, when the planet is at its descending node, since it is with these we are more immediately concerned.—The conjunctions succede one ano|ther after an interval of about 584 days, or 1 year and 7 months; as is easily computed from the period of the Earth, which is nearly 365 days; and that of Venus, which is 224. When a Transit happens in the beginning of June, there cannot be another at the same node, till there is another inferior conjunction in the beginning of June, within the limits of the four days just specified. This will not be, till the interval of 584 days, repeated some number

of times, will make nearly some number of years. To find when this may happen, reduce the ratio, 584:365, to its least terms, and you will find it to be as 8:5; which shews that 5 of these intervals make 8 years: More accurately, they make 8 years wanting 2 days, 7 hours; which defect falls within the limits above assigned. This then is the shortest peri|od in which one Transit can follow another.— The several conjunctions in this period succede in the following order; June 6; January 11; August 14; March 25; October 25; and June 3. And this order is repeated in the next peri|od; only the conjunctions happen 2 days soon|er. As the last of these Transits comes earlier in the year, and so the planet is not got so far to the east as it was at the first, while the node re|mains nearly in the same place, it necessarily happens that the planet now describes a differ|ent path from what it did before. By calculation we find that its path the last time is near 20 mi|nutes more northerly than the first. And as the semidiameter of the Sun is near 16 minutes, if at the first of these conjunctions Venus past more than 4 minutes to the South of the Sun's centre, this last more northerly path will still fall within the Sun's disk on its northern half, and there will also be a Transit at the last of the conjunctions. This was the case in the Transit of June 1761, when Venus past near 10 minutes South of the Sun's centre; which is the reason why we shall have another in June

this present year, in which the planet will pass 10 minutes North of the Sun's centre. But if in any Transit Venus pass within 4 minutes South of the Sun's centre▪ or to the North|ward of it, in the conjunction 8 years after she will pass quite to the North of the Sun's disk, and there will be no Transit. Accordingly, in the conjunction which will happen June 1, 1777, Venus will pass near 30^m north of the Sun's centre, and 14^m from his northern limb.— Thus you see that there cannot be another Transit after two consecutive periods of 8 years. Let us inquire when there can be another.

FROM the foregoing account of the order in which the conjunctions succede one another, it appears that the conjunction which happens nearest after the 3d of June, in the course of the year, is on the 12th of August. There will therefore be no other Transit at the des|cending node, till the conjunction which falls on the 12th of August, by anticipating at the slow rate of 2 days, 7 hours, in 8 years, has got back to the 7th of June; which is 66 days. This cannot be done in less than 29 of these periods of 8 years, or 232 years, which make 235 years from the preceding Transit. Nor will it be done then, unless at the preceding Transit Ve|nus past above 5^m north of the Sun's centre; otherwise it will take 8 years more to ac|complish it. Upon these grounds we find, that beside the two in 1761 and 1769, no Transit

has happened at this node, in the beginning of June, since the year 1526, nor will again till 2004.—The foregoing remark clears up what at first sight appears so paradoxical, that there should be two Transits at the same node in 8 years, and no more in so long a period.

ANOTHER may indeed happen before the year 2004, at the opposite node. For, agreable to the account of the order in which the con|junctions succede one another, we shall find that, after a Transit on the 3d of June, the next conjunction will fall on the 9th of January; which is but a month after the time of year when Transits may happen at the ascending node. The conjunctions going back somewhat quicker at this node, viz. 2 days and about 11 hours in 8 years, this space of a month may be gone over in 13 of these periods, that is, in 104 years; or, reckoning from the preceding Transit, in 105 years and an half. According|ly, the next Transit will be on the 9th of De|cember in the year 1874; which will be fol|low'd by another in 8 years, and then there will be no more in December till the year 2117. Thus the 20th century will pass without a Transit.

So extremely rare are these phaenomena. And in fact, till that which was observed in 1761, there never had been but one seen since the cre|ation. This was in December (N. St.) in the

year 1639; and it was seen by two persons only; Jeremiah Horrockes, a young English Astrono|mer, of an admirable genius, who was the only person that predicted it, and a friend of his, William Crabtree. And when this of next June is past, the present race of mortals may take their leave of these Transits; for there is not the least probability, that any one who sees this, will ever see another.

ON account of their rarity alone, they must afford an exquisite entertainment to an astrono|mical taste. But this is not all. There is ano|ther circumstance which strongly recommends them. They furnish the only adequate mean of solving a most difficult Problem,—that of deter|mining the true distance of the Sun from the Earth. This has always been a principal object of astronomical inquiry. Without this, we can never ascertain the true dimensions of the solar system and the several orbs of which it is com|posed, nor assign the magnitudes and densities of the Sun, the planets and comets; nor, of conse|quence, attain a just idea of the grandeur of the works of GOD. Such as are unacquainted with this science, are apt to look upon every thing that can be advanced with relation to this inqui|ty, as at best but mere conjecture. 'Tis how|ever true, that methods have been proposed for this purpose, which shew the great acuteness of the inventors, and are founded on mathematical demonstration. The grand defect of them was,

that they required observations which could not be made with sufficient exactness. And this is the reason why so capital a point has not as yet been duly settled.—This important use of the Transits of Venus was first pointed out by the sagacious Dr. Halley in the year 1691, and large|ly explained by him in the Philosophical Trans|actions in 1716, when that great Astronomer, despairing, by reason of his age, to reach so distant a period, most earnestly left it in charge to the Astronomers who should be on the stage, that with this view they would strenuously and with all their might apply themselves to make observations, in distant parts of the world, of the Transit in 1761; but especially, that of the present year, on account of a peculiar advantage attending it, which we shall hereafter explain.— It will not be easy to give a distinct account of the several steps in the method of applying these phaenomena to this purpose, without the use of diagrams, which are not sutable to this place. I shall, notwithstanding, endeavor to convey to you as clear an idea as I can, of the general method; and, in order to it, shall trace things from their first principles.

TO find the distance of one object from ano|ther, the first way that presents itself is, to repeat a known measure successively till you have gone over the whole space between the objects. But this supposes one of them to be accessible, from the other. If it is not, the method is imprac|ticable,

and we must have recourse to some other.—The properties of a plane triangle sup|ply us with another. If two angles of a tri|angle are given, the third is given too; and the proportion of the sides is also given; and there|fore, if one of the sides be given in magnitude, the others are given in magnitude also; as ap|pears from Euclid's Data. In other words, If we can observe two angles of a triangle, and measure the length of one side, we can find the length of the other sides.—To apply this to the case of measuring a distance on the land or sea. Suppose we observ'd an object to bear due north from us, but having moved one mile to the east, we find the object now bears N. W. Here, by drawing lines thro' our stations and the object, a triangle is formed, two of whose angles are known;—the angle at the first stati|on, which is 90°; and the angle at the object, which is 45°. One side is also known, viz. the distance between the stations. From these Data, it is easy to find the distance of the ob|ject, and that with the same degree of accuracy as the observations can be made. On the pre|sent suppositions, the distance of the object from the first station is equal to the distance between the stations, that is, one mile. If the object were moved, provided its motion were known, its distance might be found on the same princi|ples. This method is rendered universal by trigonometry, and may be extended to astrono|mical as well as geographical disquisitions. The

known distance between the stations is usually called the base; and the angle at the object, the parallax. This is a greek word, which signifies a change; and is used to denote that change in the apparent place of an object, which is occasi|oned by changing the observer's station.

CONCERNING this change or parallax, we must make a few Remarks, which will be of use to us in the sequel.

1. That the parallax is always made the con|trary way to that which the observer has moved. As, if he move East, the object seems to have moved West; if he move North, that moves South; if he upward, that downward, &c.

2. The parallactic angle is so much the less, as the base is more obliquely situated in respect of the distance of the object.—It is also less,

3. As the base is less than that distance.

THE first of these Remarks is evident from the nature of parallax; the two last, from the properties of a triangle.

FROM the last Remark it appears, that this method is subject to limitations in practice;— owing to the limits which nature has prescribed to our senses. For when the base is the same. the parallax is less, as the distance of the object

is greater; and vice versa. Therefore if the base be very small in respect of the distance of the object, the parallactic angle may be so small, that it may be extremely difficult to observe it. Thus, if we should attempt to find the distance of an object, which was 20 miles or more, by changing our station only 5 feet, the parallactic angle would not be quite 10 seconds; which is so small that it can scarcely be observed, even with very nice instruments. In such small angles, the convergence of the lines is so slow, that a small error in observing the angle will make a great difference in the distance of the point of intersection.

IT is plain enough, that our hopes of finding the distances of the heavenly bodies, with any certainty, must be built on observations of their parallaxes. If the diameter of the Earth bear any sensible proportion to the distance of an heavenly body, that body must be subject to a parallax, of some quantity or other; that is, it must appear in different points of the starry heaven, when view'd from different parts of the Earth. For this reason, Astronomers reduce all observations, wherever made, to the centre of the Earth, which they consider as the only true point of view. The semidiameter of the Earth, then, is the base which subtends the parallactic angle. Indeed, if we compare observations made in opposite parts of this globe, then the whole diameter of it will become the base, and

will subtend the sum of the two parallaxes. And this is the greatest base we can possibly get, for investigating these parallaxes.

THE immediate effect of parallax is, to make a star appear below its true place, according to our first Remark; because the Observer is above the centre of the Earth. But while parallax alters the position of a star in respect to the horizon, it must alter it, to the other circles of the sphere too, as the equator, ecliptic, &c. and thus it alters the times of the star's arrival at those several circles. When a star is in the horizon, its parallax is greatest; that semidia|meter of the Earth, at the end of which the ob|server is placed, being then directly opposite to the parallactic angle, and perpendicular to the distance of the star. So that the horizontal parallax of any star is equal to the angle under which the semidiameter of the Earth would appear, if seen from that star. As the star rises, this semidiameter becoming more oblique, the parallax diminishes, according to the second Remark. And if the star comes to the zenith, the parallax vanishes; the star then appearing in the same place as if it were viewed from the centre of the Earth. For the contrary reason, when a star is descending towards the horizon, its parallax must increase.

HENCE it appears, that if one person could observe the place of the Sun or a planet, in the starry heaven, when in his zenith, and another

at the same time, at the distance of 90°, observe it in his horizon, the difference of the two ob|serv'd places would be the horizontal parallax sought. But this method, especially with re|gard to the Sun, tho' the most obvious of all, is attended with insuperable difficulties. And in|deed, so are all other methods of attempting the Sun's parallax, immediately. But, happily, we are under no necessity of confining ourselves to this. As we know the proportion of the pa|rallaxes of the several planets, if we can discover the real quantity of any one of them, we dis|cover all the rest. We are therefore at liberty to choose that which we find most convenient; and, according to our 3d Remark, that planet will be most convenient, which comes nearest to the Earth. This limits the inquiry to Mars, of the superior planets; and Venus, of the inferior. Mars, in his opposition to the Sun, at his mean distance, is but little more than half so far from us as the Sun is, their distances being then near|ly as 11 to 21; and therefore his parallax then is almost twice as great as the Sun's. But if he be in his perihelion at the same time,* 1.1 he comes considerably nearer, his distance being to the Sun's as about 8 to 21; and so his parallax is proportionally greater.—Venus in her inferior conjunction is but little more than one quarter so far from us as the Sun is, and therefore her parallax almost four times as great as his. This therefore is the most advantageous circumstance.

But a Transit of Venus is the most favor|able conjuncture of all, because the limbs of the Sun afford the best terms with which to compare the planet; and instead of trying to observe the parallactic angles, which are extremely small, it is much better to observe the differences of time, occasion'd by them, which are much more sensible. For Venus on the Sun moves 1 second of a degree in 15 seconds of time. Now a second of time is very perceptible; but a second of a degree can scarcely be discerned; much less can 1/15 part of a second be so. The best observations will be, when the planet is in contact with the Sun's limbs, at its immersion and emersion; the moments of which may be determined with great accuracy, if the air be clear, by such as are furnished with good astronomical instruments, and are expert in the use of them.

IF it be enquired, whether the Transits at one node have any advantage above those at the other, the answer is, that, caeteris paribus, those in December, at the ascending node, are pre|ferable. The reason is, Venus is then about 1/1•• part nearer to us than in June. But this ad|vantage, tho' real, may be much overbalanced by other circumstances, which we shall mention when we come to explain the manner in which the parallax may be deduced from such observa|tions. But this must be the subject of another discourse.

Read, 1st March 1769.

LECTURE II.

IN the conclusion of the last discourse we re|marked that a Transit of Venus is the most favorable conjuncture that can offer, for investigating the parallax of the Sun and planets;—a fundamental element in astronomy, which necessarily enters into the theory of those bodies; and yet is extremely difficult to be settled with exactness by its being so very small, as al|most to escape the most critical observation. It was farther remarked, that the best method for this investigation is, to observe the differences of time occasioned by parallax, at the ingress and egress of the planet. Our next business then is to inquire, in what manner the times of these phases will be affected by parallax in differ|ent parts of the Earth.—They will be affected both by the difference of Longitude, and the difference of Latitude. There is accordingly, a parallax in Longitude, and a parallax in Lati|tude* 1.2. The former alters the time, as the Observer

is farther East or West; the latter, as he is far|ther North or South. Their effects, tho' blended together, must be examined separately.—We begin with the parallax in Longitude.

LET us suppose an Observer on such a place of the Earth, as that he can see the first external contact of Venus, when the Sun is in his Ze|nith, and therefore at Noon: He sees this con|tact without parallax, that is, at the same instant of time as he would, if he could see it from the centre of the Earth.—Suppose another Obser|ver West from the former, and under a meridian 6 hours distant. To him, the Transit will begin at Sunrise, in the lower half of the Sun. Both the Sun and Venus being depress'd by parallax, and Venus more than the Sun, because nearer; at the time when the contact is formed to the first Observer, it will not yet be formed to the second; but Venus's western limb will be as far distant below the Sun's eastern limb, as her parallax excedes the Sun's; and the contact can|not happen, till the planet has gone over this distance, which is called her parallax from the Sun. The contact therefore will not happen at 6 o'clock, but some time after. And if the two Observers should compare their observations; instead of finding the difference of the observed times of contact to be the same as the difference of their meridians, viz. 6 hours, as they would, if there were no parallax, they will find it to be less than 6 hours.—Suppose a third Observer

6 hours East from the first. He will see the Transit begin at Sunset. As the Sun's eastern limb, on which Venus enters, is now the upper|most, 'tis evident that Venus, depress'd by pa|rallax, will be brought into contact with this limb, sooner than if there were no parallax. To this Observer, therefore, the contact will happen before 6 o'clock; and upon a comparison of this observation with the first, the difference of the times will here also be found less than 6 hours, which is the proper difference of me|ridians; and as much less, as it was in the fore|going case.—But if the Western and Eastern observations should be compared together, the difference of their times from 12 hours would be double of the preceding difference from 6 hours, when each of them was compared with the Zenith observation.—The same thing is to be said of the first internal contact; as also, of the two contacts at the egress. Each of these phases will happen later to those who see them in the East, and sooner to those who see them in the West, than to those who see them in the zenith, or indeed, near the meridian.—You see what a difference is made in the beginning and end, by the parallax in Longitude.

FROM hence it follows, in the second place, that the whole duration of the Transit will be alter'd by this parallax in Longitude. If our Western Observer, who sees the beginning in the East, and therefore later, could travel so far Eastward as to see the end in the West, and

therefore sooner, it is plain that the duration of the Transit would be shortened to him. The rotation of the Earth produces the same effect: It transports an Observer from West to East, du|ring the continuance of the Transit; so that he, who saw the beginning in the East, may see the end in the West. On this account, the Transit will be shorter to those parts of this globe, which have the beginning of it in the Morning, and the end in the Afternoon, than it would be as seen from the centre of the Earth. And it is not necessary that the observation should be made in a place where the middle of the Transit happens just at noon: It is sufficient, if it be within an hour or two, before or after it. The contraction of the duration will be much the same, if the Latitude be the same. The du|ration will be most contracted, on this account, from the northern tropic, near which the Sun will then be, to the equator. From these li|mits, either to the north or south, it will still be contracted, tho' not so much; because the de|grees of the parallels of Latitude, which measure the base of the parallactic triangle, or the space thro' which the Observer is carried eastward, grow less as you advance toward either of the poles.

THE reverse will take place with respect to such parts of the Earth as have the middle of the Transit not far from midnight, and therefore see the be|ginning in the West, and the end in the East; which will be the case with places so far North

as to have their night shorter than the duration of the Transit. For, the beginning in the West being accelerated by parallax, and the end in the East, retarded, it is manifest that the whole duration must be protracted, and be longer than it would be, as seen from the centre of the Earth.—Or the thing may be considered in this light. Where the middle is near noon, the Ob|server is carried, by the rotation of the Earth, with a motion contrary to that of Venus; the former being from West to East, the latter from East to West. This accelerates Venus's apparent motion on the Sun. But where the middle is near midnight, the Observer, being on the op|posite side of the Earth, is carried in a direction contrary to the former, and therefore the same way as Venus; and this retards her apparent motion on the Sun. These different effects of contrary motions have been well illustrated by an ingenious writer, Mr. Ferguson (Astron. p. 39.)

If two birds, says he, fly along side of a ship under sail, in contrary directions to each other, the bird which flies contrary to the motion of the ship will pass by its whole side sooner than the other bird will, which flies the same way as the ship sails.

The protraction of duration in these northern parts on account of the paral|lax in Longitude, will be diminished as you ad|vance toward the north pole, because the pa|rallels of Latitude diminish. In the pole it|self, the protraction on this account would be nothing; and it will be greatest, in that Lati|tude

where the night is but little shorter than the Transit, and where consequently, the be|ginning might be seen a little before Sunset, and the end a little after Sunrise; as it may be in some places in the Latitude of 60° N. If these lengthened durations, observed where the middle is not far from midnight, be compared with those shorten'd ones, where the middle is not far from noon, the differences will be much greater than if either of them were compared with the true duration as seen from the Earth's centre.

THE foregoing differences may be altered, and made greater or less, by the parallax in Latitude, which is the third particular we have to speak of. A person who sees Venus in his Zenith, sees her without parallax, and therefore in the same place on the Sun as if the view were taken from the centre of the Earth. One to the Southward of the former, will see Venus more Northerly on the Sun; and one to the Northward, more Southerly; according to the first Remark we made concerning parallax. For this reason, if a plane be conceived to pass thro' the centre of the Earth and that little arc of Venus's orbit which falls within the Sun's disk, all the inhabitants of the Earth, when to the Southward of that plane, see Venus with a Northern parallax of Latitude; that is, they see her more Northerly on the Sun, than she would appear, from the centre of the Earth; and all to the Northward of it see her with a

Southern parallax; but all in the plane it self, see her without any parallax of Latitude. Such a plane, about the place which is in the middle of the Earth's enlighten'd disk, where the Sun is in the Zenith, will cross the surface of the Earth just within the tropic of Cancer; on the Western side of that place, it will pass more Southerly; and on the Eastern, in general, more Northerly;—by reason of the incli|nation of Venus's orbit to the equator.—The effect of this parallax is, to make the planet ap|pear to pass at different distances from the Sun's centre, and consequently to describe chords of different lengths on his disk; and thus it will vary the duration of the Transit. The variations on this account will be so much greater, caeteri paribus, as the path of Venus on the Sun is farther from his centre; because the farther the chords of a circle are from the centre, so much quicker they alter in length.—The change of duration by this parallax will not be made the same way as by the parallax in Longitude, if the Declination of the Sun and the Latitude of Venus are of different denominations, one North, the other South; but when they are of the same deno|mination, the change will be made the same way; and the difference of duration in distant parts of the Earth will be more considerable in the latter case than in the former. In June the Declination of the Sun is North. If Venus past South of the Sun's centre, the inhabitants of the Southern part of our globe, who would see

her about noon with a Northern parallax of Latitude, would see her passing nearer to the Sun's centre, describing a longer chord, and consequently continuing longer within the Sun. But to the inhabitants of this part of the globe, the duration would be shortened by the parallax in Longitude; as we shew'd under the second article.—On the other hand, where the middle was near midnight, and where the duration would be protracted by the parallax in Longi|tude, the inhabitants being in a high North La|titude, would see Venus with a considerable Southern parallax of Latitude, which would carry her farther from the Sun's centre, and by making her describe a shorter chord on his disk, would shorten the duration of her Transit. Thus, in the supposition, that Venus past South of the Sun's centre, the parallaxes in Longitude and Latitude would counteract one another; one would shorten the duration in the same parts of the Earth where the other lengthens it; which would diminish the difference between them. This was the case in 1761.—But in the Transit of the present year, the contrary will happen. Venus will now pass North of the Sun's centre; and therefore most of the inha|bitants to the Southward of the tropic of Cancer, who have the middle of the Transit within an hour or two of noon, and so have the duration of it contracted by the parallax in Longitude, will see the planet, with a Northern parallax in Latitude, in a more Northerly path,

that is, passing farther from the Sun's centre, and consequently describing a shorter chord, and having a shorter duration within the Sun. And tho' the parallax of Longitude is not quite so great in the high Southern Latitudes, as between the equator and tropic of Cancer, yet the paral|lax of Latitude is so much greater as to more than make amends for it.—On the other hand, where the middle is near midnight, and the du|ration protracted by the parallax of Longitude, there the inhabitants, on account of their high North Latitude, must see Venus with a consider|able Southern parallax of Latitude. Her path will therefore to them appear not so far to the North, but nearer the Sun's centre, and conse|quently longer. In the case therefore of the present Transit, the parallaxes of Longitude and Latitude conspire to produce the same effects; both of them shorten the duration in the Southern parts of the Earth, and both lengthen it in the Northern. This is the peculiar ad|vantage, hinted in the preceding discourse, which the next Transit will have above the last, for as|certaining the exact quantity of the Sun's paral|lax. The general method of doing which you may now easily understand.

FROM what has been said, it appears that there are two ways in which the observations of this celebrated phaenomenon are applicable to the present research. Observations of only the beginning or end, made in different places,

may be compared; or, observations of the whole duration may be compared. It will be easy to obtain many more observations of the former kind than of the latter, because a great number of places on the Earth are situated, so as to see one or other of these phases, which are not able to see them both. You perceive that pa|rallax differs in its operation in various parts of the Earth, both in kind and in degree: In some, it accelerates; in others, it retards; and that, more or less, according to the dif|ferent situation of the places. And in each place, its effect will be greater or less, in the same proportion as the Sun's total or hori|zontal parallax is greater or less.

IN the former way of proceding, the ob|served times of the beginning or end must be compared with the difference of the meridi|ans where the observations are made; and any excess or defect found between them is owing to the operation of parallax; and from the quantity of this excess or defect, the quantity of the Sun's parallax may be calculated. The method of calculation is indirect, or tentative. The Sun's parallax is assumed to be of a certain quantity, namely, that which from former observations we know to be, either the truth, or at lest, very near it; and from hence, the effects of parallax both in Longitude and La|titude must be computed for each place of observation; the particular process of doing

which, may more properly be explained else|where. The total effect of both parallaxes being thus found by computation, so as to get the difference of times when the phases of the Transit should happen in each place; if the ob|served difference comes out the same as the cal|culated, it will be certain the Sun's parallax was rightly assumed; but if the observed difference turns out greater or less than the calculated, it will follow that the true quantity of the Sun's parallax is greater or less than that which was assumed. To explain this by an example. The difference of meridians between the Royal Ob|servatory at Greenwich and this, is 4^h 44′. By calculation we find, that assuming the Sun's pa|rallax to be 8″, 55, as it was found to be nearly by the last Transit, the external and internal contacts, at the beginning of the next, will be accelerated by parallax 7 minutes of time at Greenwich; happening there but a little before Sunset, when parallax has its greatest effect. With us, where it will begin about half an hour after II, when the Sun is about 54° high in the western hemisphere, the contacts will be ac|celerated by parallax but 4 minutes. Therefore, if observations be made in both places and com|pared, the times should be sound to differ but 4^h 41′; which falls short of the difference of meridians, by 3′. This is on the supposition that the Sun's parallax is 8″, 55; a parallax of this quantity being just sufficient to produce here this defect of 3 minutes. A greater parallax

would produce a greater defect; and a less pa|rallax, a less defect. If therefore the observed difference of the times should be found greater or less than 4^h 41′, and consequently its defect from the difference of meridians greater or less than 3 minutes, the inference must be, that the Sun's true parallax is greater or less than 8″, 55, in the same proportion. Whereas should this de|fect appear to be exactly 3 minutes, we might be assured that the Sun's parallax is exactly 8″, 55. Thus shall we obtain the true value of the Sun's parallax.—In this example, which is designed purely for illustration, I have taken no notice of the seconds of time. But it is of the highest im|portance, that in making the observations they should be attended to with the most critical ac|curacy. And herein lies one great advantage of these Transits above every other phaenomenon that can be used for finding the parallaxes of the planets, namely, that the observations can be made with much greater exactness. For the times of Venus's contacts with the Sun's limb may be determined, by experienced Observers and in favorable circumstances, within a very few seconds; as was intimated before.

As the two places in this instance differ not much in Latitude, which is North in both; and the observations in both must be made in the Afternoon, and so the operation of parallax is alike in both, viz. to accelerate the beginning, there is a difference of only 3 minutes of time,

occasioned by the parallax. If the Greenwich observation should be compared with one made in South latitude, and in a place so far West as to have the beginning happen in the Morning, where the effect of parallax will be contrary, and retard the beginning, the difference would be more considerable. But to take the utmost advantage of these observations, we ought to make the difference of time as great as possible; and therefore, the afternoon observation should be made in some place where the beginning happens not long before Sunset, but yet long enough to have the Sun of a sufficient height; and where the planet will be seen to enter on the highest point of the Sun; for then parallax will have its full effect in accelerating the con|tacts, by depressing the planet directly downward upon the Sun's limb. This place I find to be near the latitude of 50° N▪ and about a quarter of an hour East of Greenwich. But Greenwich itself, or indeed any part of Great-Britain or Ireland, will serve very conveniently for this ob|servation. The corresponding observation, to be compared with this, should be made, if pos|sible, in a place where the beginning happens not long after Sunrise, and the planet enters on the lowest point of the Sun; for then parallax has its full effect in retarding the contacts, by depressing the planet directly downward from the Sun's limb. The place where this will hap|pen, is nearly opposite or antipodes to the former, in the Southern part of the great South

Sea, not far from which is marked on our globes a tract of land but little known, called New Zealand. If an observation could be made there, it would be found to differ from the Greenwich observation about 14 minutes of time; the difference being double of what it would be, if each were compared with an ob|servation in the Zenith, which might be made on the western coast of Mexico, just within the tropic.—All this is as applicable to the egress of the planet, as to its ingress; Only the stations must be different. The egress will happen soonest, in the lowest point of the setting Sun, in the South sea, near the tropic of Capricorn, and meridian of California; and latest, in the highest point of the rising Sun, in Arabia; the difference being about 14 minutes, as before.— To dismiss this head, it is plain, that in this way of investigating the Sun's parallax, it is abso|lutely necessary to have the Longitudes of the places of observation determined with the most rigorous exactness; for the success of the whole depends on comparing the observed dif|ferences of time with the known difference of meridians.

THE other way of deducing the Sun's paral|lax is, by comparing observations of the whole duration of the Transit, made in distant places. From the preceding discourse it appears that to the inhabitants of the Southern parts of this globe, who have the middle of the Transit about

Noon, as will be the case in the South Sea, the duration of it will be shortened by both paral|laxes; and that to the inhabitants of the Northern parts, who have the middle of the Transit in the night, or when the Sun is near the North part of the meridian, below the pole, the duration will be lengthened by both. This will be the case in Lapland. But here again, such stations should be chosen as will make the greatest dif|ference in the duration. The best places for the latter observations will be near the polar circle, rather within it, and about 20° or 30° East from London; between the bottom of the gulph of Bothnia and the North Cape. There, the duration between the internal contacts will be about 5^h 54′; and this is the longest that will be any where seen. And the best for the former will be in the southern part of the South sea, not far from the opposite meridian. It is not in|deed known, whether there be any land situated in that part of the globe. Nor is there any known place, that can see the end of the Transit, nearer to it than Mexico, in the Latitude of 20° N. and in that place there will be hardly any parallax of Latitude. Mexico being very near the plane passing thro' the centres of the Earth and Venus. But by the parallax of Longitude alone, the duration will be made about 18 mi|nutes shorter there than in Lapland. If an ob|servation could be procured in the Latitude of 55° S. where the parallax of Latitude as well as that of Longitude will have a great effect, the du|ration

would not be above 5^h 30′; that is, no less than 24 minutes shorter than in Lapland; which is so considerable a difference, that the Sun's parallax must be ascertained by it to vastly greater exactness than can be done any other way. And in this way, there is no need of being scrupulously nice as to the Longitudes of the places of observation: A considerable error in them, even of several degrees, will only make the difference of a few seconds in the duration.

THERE is yet a third way of deducing the parallax, and that is, from observations of the least distance of Venus from the Sun's centre, at the middle of her Transit. But this distance being useful, chiefly as it serves to determine the total duration, I need not enlarge upon it.—I may add, however, that if an observation could be made of this least distance, in the Latitude of 63° N. and Longitude of 60° E. from London, where the middle will happen at Sunrise; and another in the opposite place of the Earth, where it will happen at Sunset, the comparison of these two distances would give the double of Venus's parallax from the Sun, immediately and without calculation. The former place is in Siberia, the latter in the unknown southern parts of our globe, and probably inaccessible. An observation at the former might, notwith|standing, be compared, to advantage, with a

similar one made in any high southern Latitude. —But enough of this.* 1.3

IN consequence of Dr. Halley's recommen|dation, great endeavors were used to have the Transit in 1761 duly observed. The approach of this interesting and long-expected phaenomenon excited the curiosity of every one who had the least relish for Science; & engaged Astronomers, where|ever situated, to employ all their skill and care in making so critical and delicate an observation. The most learned Societies & the first Monarchs in Europe exerted their influence and power with emulation on this singular occasion. It is beside the present purpose to give a detail of the several observations that were made: They may all be seen in the Philosophical Transactions. I shall content my self with mentioning those which were made at the greatest distances from the Royal Observatory at Greenwich.—The most Northern place the Transit was observ'd at, was in EUROPE, namely, Tornea in Lapland; almost under the polar circle.—In ASIA, it was observed at Tobolsk, the capital of Siberia, by a French Astronomer, who performed a journey thither of 4000 miles from Paris, at the instance of the Imperial Academy of Sciences at Petersburg, and

under the Auspices of the Czarina. This place was farthest East. It was observed besides at Madrass, which was farthest South-east, under the direction of the East-India Company of London. The French King also commissioned two Members of his Royal Academy of Sciences, to make the observation in the East-Indies.—In AFRICA, it was observed only at the Cape of Good Hope: It would have been so at St. Helena too, had not clouds pre|vented; by Astronomers sent to those places by the Royal Society, at the expence of his late Majesty K. GEORGE II. The Cape of Good Hope was the farthest South.—In AMERICA, it was observed only at St. John's Newfoundland; and that at the expence of the Province of the Massachusetts-Bay. And this place was the farthest West.

THESE observations and all the others were collated, and the proper computations and con|clusions made from them, with incredible in|dustry and patience, by the late ingenious Mr. Short of London, F. R. S. tho' he was able to make but little use of the second method, viz. by the total duration; the greatest difference ob|served, which was that between Tobolsk and Madrass, amounting only to 2 minutes, 50 seconds. In a Paper published in Phil. Transact. for the year 1763, he has given the result of the whole; which is, that the parallax of the Sun, on the day of the last Transit, was 8″, 56.

But in a Letter he was pleased to write me af|terward, dated London 23 July, 1766, he has the following words.

I have, since the pub|lication of that Paper on the Sun's parallax which you were pleased to approve of, col|lected the observations of the late Transit made at many more places than I then had; so that by a comparison of 43 places with the observation at the Cape [of Good Hope] the mean result of the Sun's parallax, by the in|ternal contacts, is 8″, 545; and by the ex|ternal contacts, 8″, 551.

ACCORDING to this last determination of Mr. Short, I find the Sun's parallax at his mean distance from the Earth to be 8″, 68; and con|sequently this mean distance to be 23,763 se|midiameters of the Earth. And as, by the latest measures, a mean semidiameter of the Earth is 3,957 miles, the mean distance of the Earth from the Sun is 94,030,000 miles nearly. And in this we must acquiesce till the next Transit; at which time, if the Astronomers dispersed in the several parts of this globe, prove successful in their observations, we shall be able to determine the Sun's distance to a greater de|gree of precision, and from thence ascertain the just dimensions of the whole system. For,

CARE has been taken in due season to station Astronomers in some of the most suitable places of the Earth. We are told that the Czarina

has sent out eight companies to the most northern parts of her empire* 1.4, where the whole duration will be visible and of the greatest length; and that the French King has sent Observers to other remote parts; tho' we have not yet learned the particular places of their destination. The Royal Society with the aid of the Go|vernment have sent several into the South sea, who, it is to be hoped, will be able to observe the duration where it will be shortest. It was also proposed to send some into Hudson's Bay; but whether they actually went, I am not able to say* 1.5.—The beginning may be seen in Great-Britain and Ireland, and all over America; and the end, in the East-Indies; but the begin|ning happening before Sunrise in the East Indies, and the end after Sunset with us, will prevent observations of the whole duration. All Africa and the biggest part of Europe will be entirely deprived of this curious spectacle.

IT were greatly to be wished that we could obtain an observation of the whole duration, in this quarter of the world. If made with ex|actness, it would be a valuable addition to the stock of this sort of observations, which, after the utmost exertions, it is to be feared will be but very small; as there are but few places, at least that are accessible, in which both the be|ginning and end will be visible.* 1.6 The nearest place to us at which an observation of this sort could be made, is Lake Superior, where the end will happen a little before Sunset. An ex|pedition thither, I am sensible, would be at|tended with difficulties; but in such an ex|pedition, what are the difficulties, which love of Science in Persons possessed of wealth and power, and resolution in the undertakers, cannot surmount? * 1.7

I CONCLUDE with the emphatical words of the Rev. Mr. Hornsby, the present Savilian Professor of Astronomy at Oxford, who, in a dissertation on this subject address'd to the Royal Society about three years ago, (Phil. Trans. 1765. p. 326) having shewn the necessity of sending Observers into the South seas, which has accordingly been done, adds,

An opportu|nity of observing another Transit of Venus will not again offer itself till the year 1874. It behoves us therefore to profit as much as possible by the favorable situation of Venus in 1769,* 1.8 when, we may be assured, the several Powers of Europe will again contend which of them shall be most instrumental in

contributing to the Solution of this grand Problem, [finding the parallax.] Posterity must reflect with infinite regret upon their negli|gence or remisness; because the loss cannot be repaired by the united efforts of industry, genius, or power.

Read 15th March 1769.

THE END.

Notes

• * 1.1

  As he was 13th Aug. 1766; but will not be again in many years.
• * 1.2

  In Geography, Longitude and Latitude are referred to the Equator; in Astronomy, to the Ecliptic. This distinction was here past|over, to avoid embarassing the subject; and there was the less need to insist upon it, because in the present case, the former necessarily involves the latter.
• * 1.3

  My subject did not lead me to speak of the use of Transits in determining Longitudes; tho' those of Venus, and of Mercury too, of which last there are 13 or 14 in 100 years, are certainly one of the best means in the world for this purpose, when once the parallax of the Sun is exactly determined.
• * 1.4

  It is a fortunate circumstance for those Northern Mis|sionaries, who are gone to unknown parts of the Russian empire, that there is to be an eclipse of the Sun on the morn|ing of the Transit. This, if they have weather to observe it, will enable them at once to settle their Longitudes; proper phaenomena for which, seldom occur in those re|gions of perpetual day.—Extremely unfortunate it would have been, had this eclipse happened but 6 hours sooner: It would then have intercepted their view of the Transit.
• * 1.5

  The equal-altitude and Transit-Instruments for these several astronomical Missions, as well those from France and Russia as from Britain, were all made by the accu|rate hand of Mr. Bird of London, who also made the Instrument of that kind which lately arrived here for the use of this College.
• * 1.6

  The only places which can see the whole of the Transit, are, the South sea, those parts of America which lie N. W. of a circle drawn thro' the northwest side of the gulph of Mexico, Lake Huron, the south end of James's Bay, the east end of Hudson's straits and the south part of Greenland: Also, all within 67° of the N. pole, Kam|shatka, Japan, the Ladrone Islands and New Zeland.
• * 1.7

  Extract of a Letter to the Author from a Gentleman of the first distinction in the literary world, dated Lon|don, July 2, 1763.

  The Reverend Mr. Ma••ke|lyne [Astronomer-Royal at Greenwich] wishes much that some of the Governments in North-America would send an Astronomer to Lake Superior to observe this Transit. I know no one of them more likely to have a Spirit for such an undertaking, or a Person and Instruments suitable, than the Massachu|setts.—If your health and strength were sufficient for such an Expedition, I should be glad to hear you had undertaken it. Possibly you may have an Eléve that is capable. The fitting you out to observe the former Transit, was a public Act for the benefit of Science, that did your Province great honor.

  Had it been possible, I would have undertaken this af|fair with the utmost alacrity; but my state of health ab|solutely forbidding it, I was obliged, to my great mortifi|cation, to decline all thoughts of it; and could only en|deavor to procure an observation by some other means. Accordingly, I made a proposal for that purpose▪ and persued it, as far as I could, and had the pleasure to find that it met with the approbation and countenance of many Gentlemen of Rank and Fortune. But since the above Discourse was read, I find that in literary expeditions as well as others▪ there may be such things as insurmountable difficulties.—A perfect observation was not likely to be obtained: An imperfect one would be of little service.— The proposal failed of being carried into execution.
• * 1.8

  This situation, from the principles laid down in p. 29. I find to be more favorable for the investigation of paral|lax than any from the year 1526 till 2125.—Six hun|dred years!