[]THE COPERNICUS EXPLAIN'D.

PROB. I.

TURN this Circle till the given Year, exactly correſponds to the beginning of Aries, in the outmoſt Cir⯑cle; but ſo, that for the ſecond Year after Leap Year you turn it farther one third; for the third Year two thirds of a Century's Motion forwards; and for the Leap Year one third backwards, beyond that beginning of Aries.

[13] Thus, if you would Rectifie this Circle to the time of the great Solar Eclipſe this Year, which is the third after Biſſex⯑tile, you would put a Point that is a little beyond 1701, two thirds of a Century's Motion backwarder than the beginning of Aries; by which means this Circle will be entirely Rectify'd to the time.

PROB. II.

THIS is readily done, by bring⯑ing its beginning to the time aſ⯑ſign'd in the Arch; and is ſo eaſie and obvious, as to need no farther Expli⯑cation. Only it muſt be noted, that this Place of the Perihelion, with regard to the Earth, is the Place of the Aphe⯑lion, [14] with regard to the Sun; or the Place where when the Sun is, the Earth is moſt remote from it, whence every Anomaly is to be begun.

Thus if you add to 3 ſ. 7 d. 40 m. the Place of the Perihelion, at the Commencement of this Century, or 1701, thoſe 12 m. which is its Motion in 14 Years; you will have 3 ſ. 7 d. 52 m. for the Place of the Perihelion at that time. To which if you bring the beginning of this Circle, you will have this whole Circle, with the Earth's true Anomaly, or Place thereon, exactly rectify'd to the time of that Eclipſe.

PROB. III.

LOOK in the proper Table for the place of the Moon's Apogee, at the beginning of the Century aſ⯑ſign'd, and accordingly fix it: Count along the Spiral that is upon it the Years, Months, and part of a Month of that Century, and lay one of your Threads over that Place; then remove the Apogee to that Thread; this will exactly ſhew its mean Place. After this, obſerve how far the Sun is from the Apogee or Perigee, and whether it have lately gone paſt, or is going to⯑wards one of 'em, and turn the Circle accordingly forward or backward one [16] third of that Diſtance. For in the former Caſe that quantity is to be ſubſtracted from, in the other added to the former Place, in order to have the true Place. I mean this only as we reckon 45 d. for the Limit: And remember, that the fame Equation is to be ſubſtracted, or added, for any Number that is equally diſtant from that Limit; as for 40 and 50, 60 and 30, 70 and 20, 80 and 10, and ſo in all other Caſes whatſoever. Only obſerve, that if neareſt the Limit you take conſiderably leſs, and remoteſt from it a very little more than one third, you will ſtill more exactly Rectifie this Circle.

Thus if you look for A.D. 1701. you will find the mean Place of the Apogee then to be 11^ſ 8° 18′ and if you turn the Circle forward to Apr. 22. you will have the mean Place of the Apogee then. And if by turning ſtill forwards, you add the third part of the Sun's Diſtance, or 7° to the former Place, you will have the true Place of the Apogee at that time, viz. 6 ſ. 27 d. 23 m.

PROB. IV.

LOOK in the proper Table for the Place of the Aſcending Node, at the beginning of the aſſign'd Century, and there fix it. Count along the Spi⯑ral thereon backwards, the Years, Months, and part of a Month of that Century, and lay your Thread over that Place: Then bring the beginning of the Spiral, or Aſcending Node, to the Thread; which will give you the mean Place of that Node. Then turn the Circle forward a 20th part of the Sun's diſtance from either Node, if it have lately paſs'd it; or a 20th part backward, if it have not yet paſs'd it: This will fix the Nodes to their true [18] Places at that time. I mean this alſo as we reckon here 45° the Limit; and remember, that the Equation is to be ſubſtracted or added for any Number that is equally diſtant from that Limit: Only Obſerve, that if neareſt that Li⯑mit you take conſiderably leſs, and remoteſt from it ſomewhat more than 1/20, you will ſtill more exactly Rectifie this Circle.

Thus, If you look for A.D. 1701. you will find the mean Place of the Aſcending Node to be 4^ſ 27° 24′. Then turn the Circle backward to Apr. 22d, and you will have the mean Place of that Node there. And if by turning ſtill back⯑ward, you ſubſtract a 20th part of the Sun's diſtance, or 27 [...], you will have the true Place of the Node at that time, viz. 7 ſ. 21 d. 9 m.

PROB. V.

LOOK in the proper Table for the Mean Place of the Moon, at the beginning of the Century aſſign'd, and there fix the Circle. Then look along the outward Edge of this Circle for the compleat number of Years of that Cen⯑tury, and laying your Thread exactly over that Year, remove the Circle ſo far. Then count along the Spiral upon it, the number of Days and Parts till the time aſſign'd; over which laying your Thread, remove the Circle ſo far. This rectifies the Circle of the Moon's Mean Motion for that time.

[20] Thus if you look for A.D. 1701. you will find the Mean Place of the Moon to be 10 ſ. 15d. 20 m. whither bring the begin⯑ning of the Circle accordingly. Lay then a Thread over the Number 14, and bring the former Place thereto: Then is this Cir⯑cle intirely rectify'd at that time.

Note, That the Moon's Motion in Hours is not diſtinguiſh'd here on the Spiral; be⯑cauſe it may be better eſtimated on the Ecliptick it ſelf.

Note farther, That when you go upwards, as in the Years before the Chriſtian Aera, contrary to what is uſual, you add the Mean Motions of the Node, and ſubſtract thoſe of the Apogee and Moon it ſelf to and from the Epocha, to prepare the Numbers for the Inſtrument. But then, if you thereupon chuſe proper Numbers for the beginning of each Century before, as well as after the beginning of the Chriſtian Aera, you will have no farther Difficul⯑ties in the uſe of the Inſtrument it ſelf. Thus, if for the beginning of the 431ſt Year before Chriſt, you take 500 Years backwards for the Epocha of the Centu⯑ry; and 69 compleat Years after it for the beginning of the Year it ſelf, you will have no farther difficulty in the uſe of this Inſtrument in that caſe.

Note alſo, That the three laſt of theſe Cir⯑cles will as well ſerve for a Century backwards as forwards, from any Root, or Epocha. I mean, if the known number [21] of Degrees for any number of Years backwards in the Century, be taken on the ſide of the Epocha contrary to that it belong'd to, if the Years had been forwards; as every one will eaſily find upon the leaſt Conſideration and Trial.

Note farther, That to make this Inſtru⯑ment as ready as poſſible for the preſent Century, the Epochae of the Places of the Moon's Apogee and Aſcending Node, and of the Moon it ſelf, at the beginning of it are noted by the following Marks upon the outmoſt Circle, ☉ ☊ ☽, with Points at the oppoſite Places at the ſame time: And the like Marks may be made for any other Century, as occaſſion ſhall re⯑quire.

Note laſtly, That our Time is here always the Mean or Equal Time; which the more curious may correct by the known Equation-Table: But ſince this laſt is little more than a quarter of an Hour different from the other at the utmoſt, and almoſt always much leſs, it is ra⯑ther too nice to deſerve much Conſideration in ſuch an Inſtrument.

PROB. VI.

REctifie the two outmoſt moveable Circles. Look what Sign, De⯑gree, and part of a Degree correſponds to the Time aſſign'd. Lay your two Threads over the firſt of Aries, and over that Place; and count the ſame Number along the Earth's Orbit, which you find belongs to that Number on the Eclip⯑tick. This correſponds to the True Place of the Sun in the Ecliptick for that time.

Thus, if you look for 21 h. 42 m. upon the Ecliptick on Apr. 22d. you will find over againſt it about 43 d. 30 m. This rec⯑kon'd along the Earth's Orbit, reaches from 260 d. 10 m. which is over againſt the Equinox, to 302 d. 35 m. over againſt which laſt Number ſtands 12 d. 15 m. of Taurus; which is therefore the Sun's True Place in the Ecliptick at that time.

PROB. VII.

REctifie the third and fifth moveable Circles. Lay one of your Threads over the Moon's Mean Place, and the other over the Moon's Apogee. Count along the Moon's Orbit the ſame Number which is intercepted between the Threads on the Ecliptick. A Thread laid over that Place, correſponds with the Moon's True Place in the Ecliptick.

Thus, if you look for 111 d. 21 h. 42 m. on the fifth moveable Circle, you will find it is diſtant from the Moon's Apogee, along the Ecliptick, about 192 d. 40 m. which Number, taken in the Moon's own Orbit, on the third moveable Circle, correſponds to 12 d. 15 m. of Taurus, the True Place of the Moon at that time. But it is here to be noted, that when the Moon's Apogee is either in Conjunction or Oppoſition with the Sun, the Eccentricity of the Moon's Orbit is [24] the greateſt; and the Points or Marks neareſt the Apogee are the true ones: (which is nearly the Caſe in this Ex⯑ample, and accordingly made uſe of in it) and when it is in the Octants, that Excentricity is in a Mean, and the Lines and Numbers themſelves are right; but when it is in the Quadratures, the Points or Marks neareſt the Perigee are the true Places: And ſo in all other in⯑termediate Poſitions whatſoever. Which Circumſtances being conſider'd, and pro⯑portionably allow'd for in all Caſes, this Problem will be exactly ſolv'd.

PROB. VIII.

REctifie the firſt, ſecond, third and fifth moveable Circles, as before directed. After that find the True Pla⯑ces of the Sun and Moon for the time given, as before. If they are either the ſame, or directly oppoſite, you have the time already; if not, carry your [25] Threads that lie over the two Places re⯑ſpectively, along their own Orbits, for⯑wards or backwards; ſo that you carry the Moon's 13 Degrees and a quarter to the Sun's one; or a Day's Motion of one, to a Days Motion of the other; and ſo all along proportionally, i.e. ſomewhat above half a Degree to an Hour, till the Moon's Place overtakes the Sun's. For that Place where they are coincident, gives you the Conjun⯑ction or Oppoſition, with the New or Full Moon.

Thus, If you look for the Sun's Place at Noon, Apr. 21. which is, Aſtronomical⯑ly, 21 compleat Days in that Month, you will find it about 11 d. 20 m. in Taurus. And if, in like manner, you look for the True Place of the Moon at the ſame time, you will find it about 28 d. 40 m. in Aries: Whence if you carry both the Threads ſo far as corre⯑ſponds to 21 h. 42 m. you will perceive that the Moon does there overtake the Sun; which ſhews, that at that very time, or Apr. 22 d. about a quarter be⯑fore Ten in the Morning, the Sun and Moon will be in Conjunction, and it will be New Moon. But Obſerve, that if you do not at all know before-hand, at what time of any Month the Conjunction or Oppoſition will fall, you muſt find the diſtinct Places of the Sun and Moon at the beginning of the Month, and ſo go [26] downwards, as to the Days, as well as Hours, and parts of an Hour, till they are coincident; as has been juſt now di⯑rected; and you will thereby find the Day, Hour and part of an Hour of the next Conjunction or Oppoſition. You may alſo if you pleaſe, find the True Places of the Sun and Moon, to that laſt time, over again, as before directed; eſpecially where there is a deſign to examin an E⯑clipſe at the ſame time; for the grea⯑ter Accuracy. But Note, that if you on⯑ly intend to find the time of Conjun⯑ction or Oppoſition, in order to an Eclipſe, you need not examin any Month or Days, but only ſuch as are near one of the Nodes, or within the Limits of ſuch E⯑clipſes; which Limits will be ſtated un⯑der the next Problem.

PROB. IX.

ALL the Circles firſt rectify'd, and the Time and Place of the True Conjunction or Oppoſition found, as already directed, Lay one of your Threads over that Place, and the other over the neareſt Node. If that Diſtance be leſs than 16 Degrees and an half, and it be a Conjunction, there will be an Eclipſe of the Sun ſomewhere: If it be greater, there will be no Eclipſe. If that Diſtance be leſs than 10 d. 38 m. and it be an Oppoſition, there will be an Eclipſe of the Moon: If it be more, there will be no Eclipſe.

[28] Thus, it happening this Year 1715. that the Sun and Moon are in Conjunction, about a quarter before Ten o' Clock, on the next 22 d of April, and that at 8 d. diſtance from the Deſcending Node; it is evident there will at that time happen an Eclipſe of the Sun. And there hap⯑pening an Oppoſition October 31ſt, about a quarter paſt 4 in the Morning, with⯑in 7 Degrees and an half of the ſame Node, 'tis evident there will then be an Eclipſe of the Moon alſo.

Note here, that ſeveral other Circum⯑ſtances of Eclipſes, may be diſcover'd by the Solution of this Problem. Thus, be⯑cauſe the Conjunction for the next Solar Eclipſe happens before the Moon has reach'd its Deſcending Node, 'tis plain, it muſt belong to the Northern Parts more than to the Southern. Becauſe it is in a ſort of a Mean Diſtance between that deſcending Node and the Limit, it will be a great and Total Eclipſe, and the greateſt in the Northern Temperate Zone. Becauſe it happens near the Aphelion of the Earth, and near the Perigee of the Moon, when the appa⯑rent Diameters of the Sun is ſmaller, and of the Moon greater than uſual, it will not be Annular, but Total, and that for ſome few Minutes alſo; I mean along that Line, which the Cen⯑ter of the Penumbra deſcribes upon the Earth. And becauſe the Solar Eclipſe this Year happens here in the Day⯑time, and the Lunar in the Night, they will both be viſible; I mean, in [29] caſe the Cloud cauſe no interruption. So that theſe Circumſtances do not need Particular Problems for their Solution.

PROB. X.

BRing London to the graduated ſide of the Meridian, and turn the Hour Circle till 12 o' Clock is pointed to by its Index: Then turn the Globe, with its Hour Circle, ſo far forward or backward, as the right Aſcenſion be⯑longing to the Diſtance of the Place of the Conjunction, from the beginning of Cancer or Capricorn, does require; reckoning ſtill 15 Degrees to an Hour, and 1 Degree to 4 Minutes. Hold the Globe in that Poſition, and bring 12 in the Hour Circle, to the graduated edge of the Meridian again, and there fix it faſt, to move with the Globe all along afterward. By this means the Index will tell you the true time, as it is coun⯑ted [30] at the Meridian of London, during the whole time of the Eclipſe. And Note, that the Diſtance from Cancer or Capricorn it ſelf, if increas'd or dimini⯑ſhed one 13th part, or one Degree in 13, according as the Poſition of the Sun and Moon ſhall require, will nearly give the Right Aſcenſion; reckoning the Limit 45, as under the 3d and 4th Problems, and increaſing or diminiſhing the Equation as there alſo. But Note, that this Diſtance it ſelf will never err 2 d. 30 m. or 10 Minutes in time, from the Truth; nay, will uſually be much nearer the ſame, even without any Allowance for that Correction at all.

Thus if you bring London to the gradu⯑ated edge of the Meridian and the Hour of 12 to the Index; and remove the Globe, with its Hour Circle, backward from Eaſt to Weſt 50 Degrees, i.e. 3 Hours 20 Minutes; and then turn the Hour of 12 to the Index again, and there fix it, you will have the Globe, with its Hour Circle, rectify'd for the time of the next Total Eclipſe of the Sun, and every part of it will agree to that Hour which the Index ſhews on the Hour Circle.

PROB. XI.

TAKE the eleventh part of the Di⯑ſtance of the Place of the Con⯑junction from the neareſt Node, for the Latitude of the Moon at the middle of the Eclipſe. Remove the inner Cir⯑cles; and elevate the Globe that ſtands beneath them, North or South, as the Caſe ſhall require. By the ſide Screws [32] place it in the Center or Axis of your Circles. Lay your Rule, with its Glaſs of 12 Circles (then ſet over its middle Point) and its Lamp or Candle, ſo that the Node may lie in a Poſition ſui⯑table to that of the Moon's Orbit at that time (which is eaſily known from the Obſervations under the laſt Problem); and ſo, that the middle Point, or that under the Center of the Glaſs, may be directed to the Place of the Conjuncti⯑on in the Ecliptick. Turn your Globe, rectify'd as before, till the Index points to the time of the True Conjunction, already found; That is very nearly the time of the middle of the general E⯑clipſe. Then turn your Globe, and draw your Glaſs backwards any num⯑ber of Minutes equally; ſo that an Hour in the Hour Circle may ever cor⯑reſpond to an Hour in the Path of the Moon; and till the Shadow of the edge of the Circular Glaſs begins to touch the neareſt Place of the Globe; for that is the Time, and that the Place, when and where the general Eclipſe begins. Car⯑ry both Motions forward, and obſerve, whether the Central bright Spot does any where touch the Globe; if it does not, the Eclipſe is no where Total, and the Circles caſt on the Globe will ſhew [33] the number of Digits eclips'd: If it does touch, note the Time and Place when and where it does ſo; for that is the Time, and that the Place of the entry of the Central Shadow, or total Dark⯑neſs upon the Earth. Do the like as to the Central Spot, or other edge of the Circles going off the Globe afterwards; this will, in like manner, ſhew the Time and Place of the end of Total Darkneſs, and of the whole Eclipſe reſpectively And by this means all the other Circum⯑ſtances of the general Eclipſe of the Sun, may be moſt eaſily and readily diſcover'd and exhibited to the Eye, with the great⯑eſt Pleaſure and Satisfaction.

Thus, in the next great Solar Eclipſe, con⯑ſider'd in general, you will find that it will begin in 17 d. of North Latitude, and about 90 d. weſtward from London; and this about 21 m. after 7, and will end in the Latitude of 40, about 98 d. to the Eaſt of London; and this at 3 m. after 12. That it will be a Total Eclipſe for about 150 Miles in breadth, and that the Center of the Penumbra will go near the Lizard Point, Briſtol, Stamford and Boſton; and ſo by Stockholm and Arch⯑angel, into Ruſſia, Siberia, and Eaſt Tartary.

Note, that if inſtead of a Lamp, or Candle, you make uſe of the Sun it ſelf; either by placing the Inſtrument ſo, that the Rays [34] may themſelves fall parallel to the Axis of the Glaſs, or that they may be inclin'd to it by a reflecting Speculum, or Look⯑ing-glaſs, it will be better.

Note alſo, that if with a Pencil, put thro' the Axis of the Glaſs, you draw a Line; or with a ſharp Pin or Needle therein, make Points upon the Globe, as you move it and the Glaſs together, you will have the Path-way ef the Center transferr'd upon the Globe; and may thereby exact⯑ly find all the Places where the Eclipſe will be Total, or Annular, or Central; and may eaſily ſee, from the like Lines drawn, or Marks made at any number of Digits diſtance, how broad the Path of any kind of partial Eclipſe will be alſo over the whole Earth; which how enter⯑taining and uſeful a Sight it may be to all the Curious, I leave to their own Tri⯑al and Determination.

Note alſo, That the Problem, thus ſolv'd, includes one main part of that famous Diſcovery of Sir Chriſtopher Wren's, Dr. Halley's, and Mr. Flamſteed's, publiſh'd by the laſt, and nam'd, The Conſtruction of Solar Eclipſes, without the tedious Method of Calculation. Only what they are forc'd to do with no ſmall pains, and in no ſhort timegeometrically, is here done with great eaſe and quickneſs and an exact imitation of the Originals in Nature, and ſo with a great deal more Pleaſure and Satisfaction.

PROB. XII.

THIS Problem is, in effect, alrea⯑dy ſolv'd, under the former Pro⯑blem: It being as eaſie to obſerve when the Shadow of the edge of the Glaſs-Circle firſt touches, or laſt leaves any Place upon the Globe; which is the beginning and ending of the Eclipſe there: when the bright erect Line croſ⯑ſes it, which is its middle there: How [36] near any Circle of Digits comes to it then: Which is the number of Digits Eclipſed there: With other the like Circumſtances of that Place, as it was before to obſerve the ſame for the Eclipſe in general. Nor is there any occaſion for farther Directions.

Thus the next great Solar Eclipſe will be⯑gin here at London, April 21. about 8 h. 7 m. in the Morning; its middle will be about 9 h. 13 m. its end about 10 h. 24 m. and it will be intirely, or very nearly Total: tho' if it be Total here, it cannot be ſo for much more than a ſingle Minute of Time; as Dr. Halley's particular Map, fitted for this Eclipſe, will eaſily ſhow.

Note, that if in the Solution of this, and the foregoing Problem, you add for Eaſt, and ſubſtract for Weſt Longitude, reſol⯑ved into Time, you will have the Mo⯑ment of each of theſe Appearances at any other Place alſo.

Note alſo, that the Problem, thus ſolv'd, includes the remaining part of the fore⯑mention'd famous Diſcovery of the Con⯑ſtruction of Solar Eclipſes; and that not only as done with the like greater eaſe, quickneſs and pleaſure, but princi⯑pally that with the ſame operation, and without any new trouble, it diſcovers eve⯑ry thing for all particular Places at once; which neither that Method of [37] Conſtruction, nor any other Method of Aſtronomy whatſoever could pretend to before. And yet all this is here done only by ſuch a cloſe imitation of Na⯑ture, as is in it ſelf moſt eaſie and obvi⯑ous, and what one would now imagin ſhould have come into the Thoughts of Aſtronomers firſt of all, before any other Contrivance whatſoever.

PROB. XIII.

TAKE away the Globe, and ſet the Dark Circle in its Place. Take alſo the Glaſs of 12 Circles away, and put the Map of the Moon, with its 6 Circles, in its Place. Then do in all things with this Map of the Moon, as you were order'd to do with the Glaſs-Circle [39] Circle before. By this means you will have all thoſe Phaenomena of Lunar E⯑clipſes ſolv'd, with greater eaſe, and the like pleaſure, which were before re⯑preſented in the Solar. Nor is there any new difficulty in the Application. Only Note, that this Map of the Moon being too ſmall to have the Names of its ſeveral Seas and Lands engraven on it, it will be convenient to have withal a larger Map of the Moon, with thoſe Names, as an Explication of the other; which accordingly is here pro⯑vided, and plac'd on one ſide of the dark Circle, which is us'd with it. And Note farther, that thoſe who have a mind to be here very particular and exact, may uſe the larger Map of the Moon inſtead of the other, in caſe they join with it a dark Circle of 16 Inches and a quarter in Diameter: For the ſame Proportion which the Diameter of our ſmaller Moon of two Inches and an half, bears to that of our dark Cir⯑cle of ſix Inches and two fifths, does the Diameter of that Circle, or of the larger Moon bear to that of a Circle of ſixteen Inches and two fifths Diameter. Whence its plain, the larger Moon, and that largeſt Circle, will more exactly exhibit [40] ſuch Lunar Eclipſes than the other. Tho' I believe there will be but few, who will not be ſatisfy'd with the ſmaller dark Circle; ſo I have not provided the larger: Which yet Mr. Senex, or Mr. Hudſon, the exact Engraver and Fra⯑mer of this Inſtrument, will readily procure for any that deſire them But then the ſeveral Circles concern'd muſt be ever ſo adjuſted, that the perpen⯑dicular of their Centers may ſtill be the ſame with the Latitude of the Moon at the middle of the Eclipſe; and that as meaſur'd by a Scale, which is ſo much larger than ours, as their Diameters are larger.

Thus we ſhall find by this Inſtrument, that there will be an Eclipſe of the Moon this Year, Octob. 31. that it will not be a Total, but a Partial Eclipſe; that the Digits eclips'd will be 8, and this on the North ſide of the Moon's Body: That the Eclipſe will begin about 2 h. 58 m. and end about 4 h. 40 m. in the Morn⯑ing; and that Mount Aetna will enter the Shadow about 3 h. 8 m. and emerge about 5 h. 17 m.

[41] Note thoſe that would exactly imitate Na⯑ture in theſe Lunar Eclipſes, muſt let the Globe ſtand, and have a Lamp or Cir⯑cle of Light, larger in Diameter than the Globe, through whoſe Shadow the Map of the Moon muſt paſs. They ought alſo to have different Maps of the Moon out of Hevelius, to fit the diffe⯑rent Librations of the Moon, or the ſmall variety there is in that Face which is ex⯑pos'd to us in different Eclipſes. But ſince this Method would be much more troubleſome, and but little more advan⯑tageous or entertaining, I choſe the for⯑mer eaſie way of exhibiting theſe Lunar Eclipſes by this Inſtrument, leaving the other to thoſe among the Curious, who ſhall think fit to beſtow any extraordina⯑ry coſt and trouble about it.

But Note, That the Diameters of the Pe⯑numbra, and of the Moon, are here ſtill fitted to their Mean Quantity, when the Earth and Moon are about the ſhor⯑ter Axis of their Ellipſes: So that when they are near their Aphelion and Apogee, their Perihelion and Perigee; or the Earth its Aphelion, and the Moon its Perigee; the Earth its Perihelion, and the Moon its Apogee; ſome ſmall Diffe⯑rences will ariſe in the Buſineſs of E⯑clipſes; which yet may eaſily be allow'd for on a little conſideration. Thus the next viſible Eclipſe of the Sun happening ſomewhat near the Earth's Aphelion, and nearer the Moon's Perigee, as the Poſition of their Orbits at that time in this Inſtrument will ſhew; the Moon will appear a little larger, and the Sun [42] a little ſmaller than ordinary; whereby the Sun will be about 3 Minutes and 3 quarters under a total Eclipſe, all along the Central Path of the Moon's Shadow, and the 12 Digits on your Glaſs will not include the whole. In this Caſe you are to ſup⯑poſe all the Circles 2 thirds of a Digit inlarg'd, and the Central Digit ſo much broader. Thus the next inviſible Eclipſe of the Sun, Octob. 16. happening ſome⯑what near the Earth's Perihelion, and nearer the Moon's Apogee; the Moon will appear ſomewhat leſs, and the Sun ſomewhat greater than ordinary; ſo that this Eclipſe will be only Annular, and not Total, and the Digits Eclipſed will not be quite 12. In this caſe you are to ſuppoſe all the Circles diminiſh'd 2 thirds of a Digit, and the Central Digit ſo much narrower; and the like Allowances are to be made in the Numbers upon the Path of and perpendicular to the Moon's Shadow, in the Diameter of the Moon's Map, in the Duration of Eclipſes, &c. which muſt be left to every curious Perſon's own eſti⯑mation; only with this intimation, that theſe Differences from the Standards here given are ever ſo ſmall, and commonly ſo inſenſible, that they may be well look'd on as almoſt perfectly inconſiderable in the uſe of this or the like Inſtruments.

Yet the Latitude of the Moon will ſome⯑times well deſerve an allowance; and may be corrected by taking not much a⯑bove a 12th part of the diſtance from the Node, near the Moon's Perigee, and al⯑moſt a 10th near its Apogee, inſtead of that 11th part, which is the uſual ſtan⯑dard: [43] whence at this next Eclipſe the real Latitude of 44 m. 10 ſ. will be here repreſented by 41 m. 40 ſ. on our Scale; and ſo proportionally in all Caſes what⯑ſoever.

PROB. XIV.

BY the help of the proper Tables, which are publiſh'd in my Aſtro⯑nomy, find the true Place of each Pla⯑net, as is there directed: From thence you may ſet them accordingly by this Inſtrument.

Thus for Example, Let us compute the He⯑liocentrick Place of Mercury to the Time of the next great Solar Eclipſe. See Aſtron. Lect. Edit. Lat. p. 302, 303. 304. Eng. p. 337, 340.

[45]

But becauſe this Method is not wholly free from the trouble of Calculation, you may frequently help your ſelf by ſome good Ephemeris; ſuch as is annually ſet out by Mr. Parker; whence you may rea⯑dily transfer the Heliocentrick Places of the Planets into your Inſtrument. Accor⯑dingly, by either Method, at the Time of the next Solar Eclipſe the ſix Primary Planets Heliocentrick Places will appear to be theſe:

PROB. XV.

HAving found, as before, the Helio⯑centrick Places of the other Pla⯑nets, and particularly that of the Earth, noted all by ſeveral ſmall Spheres, and having allow'd for their proper Eccen⯑tricities in the ſticking of thoſe Spheres, Lay one of your Threads from the Earth to the Planet, and the other Thread laid parallel thereto from the Center, gives you the Geocentrick Place of that Planet for the time aſ⯑ſigned.

[47] Thus, at the forementioned Time, the five Primary Planets Geocentrick Places will be nearly theſe:

PROB. XVI.

THIS is eaſily done when the Pla⯑ces themſelves of the Primary Planets, with that of the Moon, are once found by the former Solution. Nor is there then any difficulty in noting the ſeveral Angles of diſtance 120°, 90° and 60°, which make the Trine, Quar⯑tile and Sextile Aſpect. But this Pro⯑blem looks too like the Fooleries of Aſtrology, to deſerve any nicer Explica⯑tion.

PROB. XVII.

HAving found the Heliocentrick and Geocentrick Places of the Planets, and ſet ſmall Spheres to repreſent 'em: From the Sphere repreſenting any Pla⯑net lay two Rules or Threads, ſo as to touch the Earth's Orbit on both ſides. If the Earth be conſiderably without that mixtilinear Triangle, the Planet is Di⯑rect; if conſiderably within it, it is Re⯑trograde; if about the Limit, it is Statio⯑nary: or at leaſt lately was, or ſoon will be ſo. Nor can you by this means abſolutely determin, near thoſe Limits, whether the Planet be Direct, Statio⯑nary, or Retrograde. But then, by finding its Geocentrick Place two ſeve⯑ral times within a few Days of one ano⯑ther, and obſerving whether the Planet [50] at the latter time be farther, or in the ſame Place, or not ſo far in the Eclip⯑tick as it was at the former time, you may entirely determin the Problem. Nor is the Caſe of the Inferior Planets much different from that of the Superi⯑or ones, as to this matter. Only Note, that the Rules or Threads muſt be laid from the Earth to touch their Orbits; which is the Station, or Limit of the Di⯑rection and Retrogradation of thoſe Pla⯑nets, and is call'd their utmoſt Elongation; and that the Inferior Planets Poſition near thoſe Limits, are here correſpon⯑dent to the Earth's Poſition in the Su⯑perior. Nor is this a Caſe of ſuch dif⯑ficulty, as to require any nicer Conſide⯑ration.

Thus, at the time of the foremention'd E⯑clipſe, we ſhall find by this Method, that Saturn and Mars will be Retrograde; and that Jupiter, Venus and Mercury will be direct.

PROB. XVIII.

FIND the Place of Saturn or Jupi⯑ter in the Ecliptick, for the given time, as before. Then, by the proper Ta⯑bles of their Satellits Motions about them, when they ſhall be made; (for we have only one Set of ſuch Tables yet pub⯑liſh'd, and render'd fit for our purpoſe; I mean thoſe of Caſſini for Jupiter's inner⯑moſt) fix the beginning of their Orbits, or the ſmall Spheres ſtuck there to re⯑preſent thoſe Satellits: Place alſo, or imagin a Lamp, or Candle, at the di⯑ſtance of about 160 Feet for Saturn, and about the ſame number of Feet for Jupi⯑ter. By this means you will have a true and noble Repreſentation of theſe Sy⯑ſtems of Secondary Planets at any time whatſoever.

PROB. XIX.

THE time of Oppoſition is the time of the Middle of any Eclipſe; and by taking away and adding the half duration of that Eclipſe, you have the time of the Immerſion and Emerſion. 'Tis plain therefore, that finding the Oppoſition, does, in effect, find the Immerſions, Emerſions, and intire Du⯑rations of thoſe Eclipſes alſo.

Note, that the time of half Duration in Ju⯑piter's innermoſt, which is the moſt re⯑markable Satellit in this matter, is near⯑ly 1 h. 6 m. and ſo the whole Duration about 2 h. 12. m. perpetually.

[53] Note farther, that only one of theſe Appear⯑ances, the Immerſion or Emerſion of a Satellit, is generally viſible at the ſame time; viz. the Immerſion from the Con⯑junction of the Primary Planet with the Sun till its Oppoſition; and the Emerſion from its Oppoſition till its Conjunction.

Note alſo, that till we have more exact and ſuitable Tables of the reſt of theſe Secon⯑dary Planets, it will be proper, two Nights ſucceſſively, to obſerve the Poſition of as many as we can of them; and from thoſe Obſervations to fix their Pla⯑ces, with relation to their Primary ones, to thoſe times. For ſince their Orbits, on our Inſtrument, gives their ſeveral Peri⯑ods exactly enough; from thoſe Periods their future Places and Poſitions may be found by two ſuch Obſervations, for a great while; and ſuch Poſitions may be with pleaſure enough compar'd with a great number of other Obſervations af⯑terwards. Nor do we wholly want an Aſtronomical Method, as to the moſt uſe⯑ful of theſe Secondary Planets, which is the innermoſt about Jupiter; ſince the Tables neceſſary for placing the ſame right, and for the Eclipſes for ſome Years to come, tho' too large for this ſmall Ma⯑nual, are publiſh'd in my Aſtrono⯑my, with full Directions for their uſe al⯑ſo; to which I muſt refer the curious Reader.

[54] Note farther, that we are generally to look for only one of Saturn's Satellites; there being few Glaſſes that can ſhew us any more; and none but thoſe of the famous Caſſini that can diſcover the two inner⯑moſt.

Note here, that Jupiter's Planets are all in or near the Plain of his Equator, which is near the Plain of the Ecliptick it ſelf; and that they are hardly at all Eccentrical. That Saturn's moſt viſible Planet, diſcover'd by Hugenius, is the fourth in order, or the outmoſt but one: as alſo that its Planets are in or near the Plain of its Ring, which is about 31 deg. inclin'd to the Ecliptick; and that they do not any of them appear to be very much Eccentrical neither.

And note, that our Terreſtrial Globe may be ſo contriv'd, that it may be alſo us'd with an Horizon, as any other Terreſtrial Globe may; and by conſequence, thoſe who buy its Celeſtial Fellow, may at the ſame time have a Pair of Globes, as well as a Co⯑pernicus; which will at once ſave al⯑moſt all the Charges of one Globe; and afford a Foundation for the underſtanding of both the principal parts of Aſtronomy alſo, I mean the Doctrine of the Sphere, to which the Globes; and the Theory of the Planets, to which our Copernicus does immediately belong.

[55] Note laſtly, That all who purchaſe this Co⯑pernicus, and deſire to have it explain'd more diſtinctly to them, according to the Directions in this Paper, may apply them⯑ſelves to the Author; who will endea⯑vour to make the ſeveral Parts and Uſes of it eaſie and familiar to them.