Meridians of the Mater and shall be so divided by the Parallels of the Mater as the Meridians are divided by them But my advice is that you divide your Ecliptique the first way and you may use this for proof of your work at last 3 The rest of the lineaments of the Reet are the Azimuths to be drawn as the Meridians of the Mater and the Almicanters to be drawn as the Parallels Onely you shall need to draw but half the Almicanters and the Azimuths but half way leaving one half of your Reet viz. E C B D blank and void of them In drawing these Azimuths and Almicanters you shall be carefull to skip over the border of the Ecliptique leaving it fair that the graduations thereof with their figures set to every tenth degr and the characters of the Signes may be more distinctly seen Also you shall do well if you make a border to the Axtree line on the Northside that is toward D and let this border be of the same breadth from A to B the breadth not exceeding one fifth of an inch in a Reet of a foot Diameter upon which border you may make a scale of degrees setting figuresin it to every tenth Almicanter This will be a great strength and Ornament to your Reet Below the Horizon C D likewise you shall make a Limb or border for the Horizon to receive his graduations this may be a quarter or three tenths of an inch broad where the Reet is a foot in Diameter and upon this border you shall set figures at every tenth Azimuths and shall number them both wayes from the Center and from the Meridian 4 You shall inscribe so many of the fixed Stars as your Reet may well receive Which to do you must know their Right Ascensions or Culminations and also their Declinations for which purpose I have given you a Table of 110 of the more notable fixed Stars which may best be inserted in your Reet with their Right Ascensions and Declinations calculated to the year of our Lord 1671. which may serve for 40 years before and after without any considerable error To inscribe them you shall first number the Right Ascension of the Star from ♈ 0 that is from D upon the Limb of the Reet toward B and at the end of that number fix your Label which by this time should be made and pinned on the Center then from the Limb count inwards upon the Label the Stars Declination and at the end of that number make a prick in your Reet close to the edge of the Label there is the Stars place Then with your Graver you shall make there the shape of a Star with 4 5 or 6 points according as the magnitude of the Star deserves and let one point be longer then the rest and let it point outward from the Center if the Stars Declination be North but inward toward the Center if his Declination be South and let the end of his long point called Apex be in the very true place of the Star But if your Label be yet unmade then take the measure of the Stars Declination with your Compasses upon any of the four Semidiameters of the Reet measuring it from the Limb inwards then lay a ruler from the Center to the Right Ascention of the Star and where the ruler cuts the Limb of the Reet there set one foot of your Compasses opened as before and with the other make a prick toward the Center close to the edge of your ruler and there is the Stars place in your Reet 5 Lastly you shall cut out all the spaces of this Reet which may be spared remembring alwaies that you leave uncut the borders of the Ecliptique Horizon and Axtree line and be very carefull that you cut not into the Center of your Reet but leave breadth sufficient about the Center to hold the Center-pin which must joyn Mater Reet and Label together This remembred you shall cut out two third parts of the spaces of the Almicanters beginning from the Horizon C D and cutting out the breadth of two degrees after which you shall leave the breadth of one degree and then cut our again the breadth of two degrees and so forward But for the greater strength and ornament of your Reet and for ease in numbring the Azimuths you shall at every 15th Azimuth leave a string of the breadth of one degree whole from the Horizon to the Pole A or at every thirtieth Azimuth leave such a string going quite through and at every other fifteenth the string may be cut off when it comes within ten degr of the Pole because there the spaces of the Azimuths be very narrow and close together And where among those Almicanters and Azimuths you have any Star you must contrive to leave him standing and to set by him his name or some figure by which you may know him again But you are to content your self with four Stars on this side the Horizon because you will want convenient room On the other side you may have more and room also to writ their names upon strings or branches left for that purpose which you may contrive into some voluntary lettess-work wherein you shall not much regard uniformity of the Quadrants but to make the Reet as open as you can provided you leave it of sufficient strength Paste this on fol. 17. so as it may by open while the first 7. Chapters are reading To cut out the Reet in pastbord is much easier if you be provided of sharp knives and chesills fitted for your purpose A Table of the Right Ascensions and Declinations of 110 of the more notable fixed Stars calculated from Tycho his Tables rectified for the year of our Lord 1671.  As mi. D. mi.   Andromeda her Head 357. 54. 27. 18. N. 2. Mirach Girdle 12. 49. 33. 55. N. 2. Foot Alamath 25. 57 40. 44. N. 2. Perseus his side Algenib 44. 16. 48. 36. N. 2. Meadusaes head Algol 41. 46. 39. 39. N. 3. Henerichus right shoulder 85. 53. 44. 56. N. 2. Left shoulder Alhabot 73. 07. 45. 37. N. 1. Left elbow 69. 29. 43. 15. N. 4. The Kids 69. 57. 40. 33. N. 4. The Kids 70. 51. 40. 42. N. 4. The great Wain The Wheels 160. 18. 58. 08. N 2. The great Wain The Wheels 160. 48. 63. 32. N 2. The great Wain The Wheels 173. 59. 55. 33. N 2. The great Wain The Wheels 179. 48. 58. 51. N 2. The Horses 189. 53. 57. 47. N. 2. The Horses 197. 37. 56. 41. N. 2. The Horses 203. 37. 51. 00. N. 2. The little rain the Pole Star 07. 53. 87. 34. N. 2. Last Wheel 231. 14. 73. 15. N 3. Dragons tongue 254. 36. 54. 55. N. 4. Head first 260. 46. 52. 34. N. 3. Head last Ras Aben. 267. 15. 51. 36. N. 3. Tail 167. 15. 71. 05. N. 3. Bootes Arcturus 210. 13. 20. 58. N. 1. Engonasi's Head 254. 12. 14. 50. N. 3 Ophiucus Head 259. 55.
Projection The Concurrent Circles meeting in the Poles A and B are Meridians Those Meridians are 180 in number and divide the Equator C D into 360. degrees because every one of them cutteth it twice that is once in each Hemisphear By these are numbred the Right Ascensions of the Stars and Planets and the hours and minutes of Day and Night for every 15 of these Meridians numbred from the Limb is an hour Circle as hath been shewed Book 1.6 they are numbred from D to C that is from Septentrio to Meridies 1.2.3 c. for the Morning hours and back again from C to D in like manner for the Afternoon the Axeltree line A B falling out to be the six a clock line both ways By those Meridians also are numbred the Longitudes of Towns and Countries in Geography The Circles or Semicircles crossing these Meridians are the Parallels of Declination they are lesser Circles whose propertie it is to divide the Sphear into unequal parts In the midst of them lies the Equator C D being here a straight line and cutting the Axtree-line A B at Right Angles in the Center E the Parallels are greatest near the Equator and from thence they lessen toward the Poles they are 180 in number i. e. 90 on each side the Equator save that the two extream Parallels are reduced to two points in the Poles By these Parallels are numbred the Declinations of the Stars in Astronomie and the Latitudes of Towns and Countries in Geography And this name and use have the Circles of the Mater always in the Meridional Projection The Ecliptick always standeth for it self when it is used which is onely in the first Mode of this Projection But the Circles of the Reet have divers names and uses in the divers Modes of this Projection which here follow 1 The first mode of the Meridional Projection The point A of the Reet in which the Concurrent Arches meet is called the Vertex of the Reet Set the Vertex of the Reet to the Latitude of your place so shall the Vertex be Zenith and the Concurrent Arches there meeting shall be Azimuths called also Vertical Circles and Circles of Position passing from Zenith to Nadir and dividing the Horizon into 360 degr as the Meridians on the Mater pass from Pole to Pole and divide the Equinoctial The Semicircles crossing these Azimuths shall be Almicanters or Circles of Altitude The Diameter crossing the Axeltree of the Reet at Right Angles shall be the Horizon or Finiter whose Graduations are set to him in a border below the Center and from him are the Almicanters reckoned upward to the Zenith The Azimuths may be reckoned from the North or South Semicircles of the Meridian or from the Axtree line of the Reet which is the East or West Azimuth commonly called the Prime Vertical When I bid you set the Vertex of the Reet to the Latitude of your place you must first know what your Latitude is It is the nearest distance of your place from the Terrestrial Equinoctial numbred in degrees and minutes of a great Circle The Latitude of London is 31 degr 32 min. North. The Latitude of Ecton or Northampton is 52 degr 15 minutes or very near And how to get the Latitude of those or any other place shall be shewed Book 4.11 The Latitude had number the degrees thereof upon the Ring from C or Meridies where the Equator cutteth the Meridian toward A or Oriens which in this Projection is the North Pole because we in England have North Latitude At the end of this number see for London 51. degrees 32. minutes from the Equator Northward set the Vertex of the Reet so this Vertex representeth the Zenith or point in the Heaven which is just over your head in which point all the Azimuths meet and through which also passeth the Meridian of your place which here is represented by the outmost Circle of the Mater or the innermost Circle of the Ring Now is the upper Semicircle of your Meridian divided into four notable parts From the Zenith Southward to the Equator is the Latitude 51. degrees 32 minutes from thence to the Horizon is the complement of the Latitude 38. degrees 28. minutes making up a Quadrant againe from the Zenith Northerly to the Pole is the complement of Latitude 38. degr 28. minutes as before and from thence to the North of the Horizon is the Elevation of the Pole above your Horizon which is always equall to the Latitude of your place for where in a right Sphear the Polesly in the Horizon and have on Elevation there the Equator passeth through the Zenith and if you go from such a Country Northward till the Pole be Elevated one degree the Equator shall there decline from your Zenith one degree Southward because the Equator keeps always the distance of 90 degrees from the Poles And this distance of the Zenith of your place from the Equator is called by Geographers Latitude and is always equal to the Elevation of your Pole So that it is all one whether you set the Vertex 51. degrees 32. min. above the Equator or set the North point of the Horizon 51. degrees 32. minutes below the North Pole Now the Vertex of the Reet set to the Latitude and consequently the Pole mounted to his due Elevation your Planisphear is in a right mode and posture speedily to resolve all questions concerning the Diurnall motion as the Suns longitude Declination Right Ascension the Ascensionall differences with the Semidiurnall Arch or length of the day the Suns Altitude Azimuth and Amplitude the hour and minute of the day the beginnings endings and duration of twilight and such like and that with so great facility that having onely the Longitude of the Sun with the Ephemeris on the Ring shall give you for asking and therewith either the Altitude Azimuth or Houre one of them you may see all the rest at the first view without changing the posture of your Instrument as shall appear in the fourth book 2 The second Mode of the Meridional Projection Set the Zenith or Vertex of the Reet to the North Pole of the Ecliptick or which is all one set the Horizon line of the Reet in the Ecliptick line of the Mater so the Azimuth shall in this posture become Circles of Longitude and the Almicanters Circles of Latitude And in this Mode your Planisphear is fitted to resolve all Questions of the Longitude Latitude Right Ascension and Declination of the Stars 3 The third Mode of the Meridional Projection Number the Altitude of Culmen Caeli that is the Southing point of the Ecliptick in the Ring from the North Pole toward Meridies if the Ascendant be a North Signe or toward Septentrio if the Ascendant be a South Signe To the end of this numeration palce the Finiter Reckon also upon the Finiter from the Center toward Septentrie the Amplitude of the Ascendant the Meridian cutting there gives you
find in the next rule 2. Another way Mark what is the Right Ascension of the point proposed being counted from the next Equinoctial point as of ♉ 9 degr the Right Ascension is 36.36 min. count so many degrees in the Arctick circle from the Axeltree at the end of this number is the Pole of the Ecliptick Lay the Label to him and you shall make a Quadrantal Triangle whose Sides shall be equal to the Angles of the former Triangle which was made of the Longitude Declination and Right Ascension of the point proposed for the Right Angle you have a Radius or Quadrant of the Axis for the Angle of the greatest Declination between the Equator and Ecliptick 23 ½ you have the arch of a Meridian between the Pole of the Equator and the Pole of the Ecliptick for the angle sought you have the arch of the Label between the Pole of the Ecliptick and the Center 71.20 minutes as before the least angle of this Quadrantal Triangle is at the Center and you shall find his measure in the Limb 14.32 minutes that is the measure of the least Side of the former Triangle viz. the Declination of the point proposed Here you see If the Declination had been given you should have set it in the Limb between the Pole and the Label and so had you made the same Quadrantal Triangle and might have found on the Label between the Arctick Circle and the Center the measure of the angle sought and likewise in the Arctick Circle between the Label and the Axtree-line the Right Aseension though it be more then was required The reason hereof you may learn from Book 3.7 CHAP. X. To find the point of the Ecliptick in which the Longitude and Right Ascension have greatest difference Move the Label on the Polar circle till you find the degrees of the Label between the Polar circle and the Limb to be equal to the degr of the Limb between the Label and the Pole so have you a Rectangled aeqaicrurall Triangle made by the Limb Label and the Meridian 46 ¼ like to that in the second Variety Book 3.10 Here the angle B at the Pole between the 46 ¼ Meridian and the Limb is equal to the Longitude of the point sought 46¼ and either Leg is equal to the Declination thereof 16 ¼ Therefore I conclude that when the Sun is 46 ¼ in Longitude that is in ♉ 16 ¼ then his Longitude hath furthest out run the Right Ascension Subtract now the Right Ascension of ♉ 16 ¼ which is 43 ¾ out of the Longitude 46 ¼ there remains 2 deg ½ which being converted into Time is 10 min. the greatest inequality of Ascension in a Right Sphear CHAP. II. To find the Latitude of your Place or the Elevation of the Pole above your Horizon by the Meridional Altitude and Declination of the Sun Meridional Projection GEographers call the distance of a place from the nearest point of the Equator upon Earth the Latitude of that Place as the Latitude of London is 51 deg 32 min. from the Equator Northward the Latitude of St Thomas Island upon the coast of Africk is 0 deg 0 min. because the middle of that Island lyeth under the Equator And because the Latitude of your Place and the Elevation of the Pole above your Horizon are alwaies equal therefore the Elevation of the Pole is oft called Latitude of the Place or Latitude simply and so for brevity sake we shall often call it But when we speak of the Latitude of the Moon or Stars you must understand Astronomers thereby mean their distance from the neerest point of the Ecliptick To find the Latitude of your Place get the Suns Declination by the 6 or 7th and his Meridian Altitude by the second of this Book Then find the parallel of the Suns Declination North or South as the Declination is and where it toucheth the Limb here Meridian there is the point where you observed the Sun at Noon set the South end of the Finiter so many degr below this point as the Meridian Altitude had then is your Finiter set to your Latitude and you shall find the measure of it between the Equator and the Zenith which is properly the Latitude and the same measure shall you find between the North point of the Finiter and the North Pole where it is more properly called the Elevation of the Pole Example June 20 1651. I observed the Meridian Altitude of the Sun here at Ecton four miles Eastward from Northampton 60 degr 59 min. the Longitude of the Sun was then ♋ 8 degr 19 min. ½ his Declination 23 degr 14 min. Northward Therefore having found in the Limb the point where the Parallel 23 degr 14 min. toucheth above the Equator I put the South end of the Finiter 60 degr 59 min. below that point toward the South Pole which done I see the North Pole Elevated above the Finiter 52 degr 15 min. and the Zenith of my Horizon likewise to be removed from the Equator Northward 52 degr 15 min. which is the Latitude of Ecton Note that you may best observe the Latitude when the Sun is near the Summer Tropick for then you shall not be troubled with Refraction and then the Declination varyeth slowly which varyeth almost one minute every hour near the Equinoctial CHAP. XII To do the same by the Meridian Altitudes of the Stars about the Poles MAny of the Stars near the Northern Pole may be seen with us twice in the Meridian in one Winters Night that is one while above the Pole and 12 hours after again below the Pole As for Example the Pole-star called Alrucabe about December 18 will be in the Meridian above the Pole at 6 of the clock at Night and at 6 next morning he will be in the Meridian below the Pole Observe both the Meridian Altitudes and add them together half that sum is the Elevation of the Pole Example I observed at Ecton the greatest Altitude of the Pole-star to be 54 deg 45 min. and his least Altitude 49 degr 45 min. the sum is 104 deg 30 min. the half 52 degr 15 min. the Latitude of Ecton and here I have gotten also the Pole-stars distance from the Pole and consequently his Declination which is the complement thereof for the Latitude being subducted from the greater Altitude leaves the Stars distance from the Pole 2 degr 30 min. and consequently shewes his Declination to be 87 degr 30 min. which is 39 min. more then Gemma Frisius observed it Anno Dom. 1547. for in our age the Pole-star approcheth about 1 min. nearer the Pole in every 3 years Note that these Stars which are distant from the Pole less then the Latitude and more then the complement thereof have their less Meridian Altitude in the North part of the Meridian and their greater Meridian Altitude in the Southern part of the Meridian beyond the Zenith Wherefore for them you shall take the complement of
010352 16 02867 010402 17 03057 010456 18 03249 010514 19 03443 010576 20 03639 010641 21 03838 010711 22 04040 010785 23 04244 010863 23.30 04348 010904 24 04452 010946 25 04663 011033 26 04877 011126 27 05095 011223 28 05317 011325 29 05543 011433 30 05773 011547 31 06008 011666 32 06248 011791 33 06494 011923 34 06745 012062 35 07002 012207 36 07265 012360 37 07535 012521 38 07812 012690 39 08097 012867 40 08390 013054 41 08692 013250 42 09004 013456 43 09325 013673 44 09656 013901 45 10000 014142 46 10355 014395 47 10723 014662 48 11106 014944 49 11503 015242 50 11917 015557 51 12348 015890 52 12799 016242 53 13270 016616 54 13763 017013 55 14281 017434 56 14825 017882 57 15398 018360 58 16003 018870 59 16642 019416 60 17320 02000 61 18040 020626 62 18807 021300 63 19626 022026 64 20503 022811 65 21445 023662 66 22460 024585 66.30 22998 025078 67 23558 025693 68 24750 026694 69 26050 027904 70 27474 029238 71 29042 030715 72 30776 032360 73 32708 034203 74 34874 036279 75 37320 038637 76 40107 041335 77 43314 044454 78 47046 048097 79 51445 052408 80 56712 057587 81 63137 063924 82 71153 071852 83 8144â— 082055 84 9514â— 095667 85 11430 114737 86 14300 143355 87 19081 191073 88 28636 286537 89 57289 572986 90 Â Â CHAP. IIII. To find the Centers of the Parallels six several wayes THe first way but the worst for our purpose as was said before for the Meridians is by the fifth Proposition of the fourth book of Euclid to find the Center of the Circle circumscribing the Triangle made by the three points given 2 A better way is by profers Take this upon trust that as you found the Centers of all the Meridians in the Equator so shall you find the Centers of all the Parallels in the Axtree line prolonged and by making like profers as you were taught for the Centers of the Meridians Chap. 3. you may quickly find the Centers of the Parallels 3 A third way You must consider that the Axtree line represents the East Meridian as well as the Axis of the world which is a common Diameter to all the Meridians Also that every Parallel cuts the East Meridian as it doth the rest in two points Equidistant from the Equinoctial and two Equidistant also from the Poles Therefore having one point already given in the Axtree line within the fundamental Circle where the Parallel shall cut number the distance from this point to the next Pole and number also the same distance again beyond the Pole in the Axtree-line prolonged being divided also as you were taught to divide the Equator line Chap. 3. and at the end of this number shall the Parallel out the Axtree line again And the middle between these two sections is the Center For example the 50th Parallel is 40 degr distant from the Pole Count therefore in the Axtree line prolonged 40 degr beyond the Pole and there is the utter end of this Parallels Diameter which if you part in two the middle at G is the Center 4 If from the point given where the Parallel cuts the great Meridian you raise a Tangent line this Tangent shall cut the Axtree line in the Center of the Parallel Example The said 50th Parallel cuts the great Meridian at H there I raise the Targent H G perpendicular to the Radius E H. And this Tangent as you see cuts the Axtree line in G the Center of the Parallel 5 Hence ariseth a fifth way For it appears by this figure that the Tangent of the Parallels distance from the Pole is equal to his Semidiameter and that the Secant of his distance from the Pole is equal to the distance of his Center from the Center of the great Meridian For here E H is Radius H B an arch of 40 degr H G the Tangent thereof and Semidiameter of the Parallel E G the Secant thereof and the distance of the Center of the Parallel from the Center of the Meridian And all this is evident by the structure in the Scheam Wherefore making E H Radius take from your Scale or Sector with your Compasses the Secant of the Parallels distance from the Pole and set it from E in the Axtree line and it shall end in the Center of the Parallel Or take the Tangent of the Parallels distance from the Pole and set it from the point of his Section with the Meridian toward the extension of the Axtree line and where the end of it just toucheth the Axtree line there is the Center 6 For want of a Sector or other fit Scales of Tangents and Secants you may do thus Set one foot of your Compass in the Center E and extend the other upon the Diameter of the Equator or Axtree line to twice so many degr as your Parallel is distant from the Pole That distance is the very Tangent you seek For example for the 40th Parallel from the Pole I number from E toward D 80 degr to 8. now E 8 is the Tangent of 40 degr though it contain just twice so many degr of the Circle foreshortned in this projection as hath been shewed Chap. 3. Sect. 3. and so if you will have the Secant of 40 degr take with your Compasses the length from 8 where the Tangent ends to A. and that is the Secant to be used as was taught in the last Section Thus have you wayes enough for finding the Centers of the Meridians and Parallels And you may have occasion in the making of the Instrument to use most of them one time or other However the knowledge of them is both pleasant and usefull for the right understanding of this and other Projections of the Sphear as also for the examination of your work when you shall chance to doubt of it CHAP. V How to draw the straighter Meridians and Parallels whose Semidiameters are very long IT may trouble you very much to draw those Meridians and Parallels which lie near to the Diameters because they be arches of great Circles and require Compasses larger then you can well get or manage when you have gotten them Till you come to the 80th Meridian from the Limb a Beam-compass of a yard long will reach if your Mater be not above a foot Diameter and a longer Beam you cannot well manage for it will be apt to tremble with it's own weight and draw double lines though it be made very thick and massie But the 89th Meridian will require a Beam-compass of almost ten yards long For his Semidiameter will contain the Semidiameter of the great Meridian 57 times Therfore to draw the 10 last Meridians and the 10 last Parallels you may help your self one of these wayes 1. Guido Vbaldus hath devised an Instrument for this purpose consisting of three rulers in form of an obtuse Triangle The description and use thereof you may see in Blagr l. 4. c 2 3. and in Vbaldus his book
De Theorica Astrolabij But though it be an Ingenious device yet I have found by experience that it is a ticklish Instrument and hardly managed for which reason I have hanged it by 2 The Bow now commonly used is an Instrument not so artificial but more tractable and steddy then the former It is made of too steel rulers the shorter of them must be of good substance as three quarters of an inch in heighth and as much in breadth that it may be stiff and lie flat the length must be somewhat more then the Diameter of your Instrument The other may be an inch longer of the same heighth but much narrower that it may be bent out with screws into an arch of any Circle required which ruler so bent being laid to the three points given you may by it draw the arch required as easily as you draw a straight line by a straight ruler The stiff ruler carries the screws and it must have rivets by which the bending ruler may be staied at both ends while it is bent by the screws See the figure CHAP. VI. How to draw the Tropiques and Polar Circles and to finish the Mater BEsides the 180 Parallels aboy ementioned you have four more to draw before the Mater is finished viz. the two Tropiques and the two Polar Circles of which the Northern is called the Arctique and the other the Antarctique Circle How to draw these you are sufficiently instructed Chap. 4. if you know but their Declination for they be Parallels The Tropique of Cancer declineth from the Equator toward the North Pole 23 degr 30 min. and the Tropique of Capricorn declines as much toward the South Pole The Arctique Circle declines Northward 66 degr 30 min. and the Antarctique as much Southward And these being drawn after the manner of the other Parallels you have drawn all the lineaments of the Mater And the better to adorn and distinguish them you shall with your Graver hatch every fifteenth Meridian for they are hour lines The South arch of the great Meridian A C B is the hour of Noon and his North arch A D B the hour of Midnight These need not be hatched being the Semicircles of the great Meridian or fundamental Circle which contains all but the Axtree line A E B which is the hour line of the sixes and the rest of the hour lines counted from him both waies would be hatched on both fides to shew like a ragged staff for distinction sake Also every fifth Meridian not being a fifteenth you shall make a pricked line not punching it with a round point lest you make your plate warp but making many short strokes cross the line with your Graver which will be more conspicuous Every tenth Parallel also would be a ragged line and the intermediate fifths pricked lines likewise the Tropiques and Polar Circles would be pricked lines Also if your plate be large you may set figures to the hour lines and to every tenth Meridian at the Equator but if your plate be smal the divisions of the Label applied upon the Equator may supply the lack of them CHAP. VII Of the Reet or Nets HAving shewed you what belongs to the Fabrique of the first part of this Instrument called the Mater A few words more will instruct you how to make the Reet whose lineaments are for the most part the same The Reet is a round plate of metal or pastboard like unto the Mater but of less Diameter it must be well planished and polished and the thinner the better if it hold working it would not be thicker then a shilling being of a foot Diameter It is called the Reet or Rete that is the Net because it must be pierced through and made like unto a Net or Lettess that the lincaments of the Mater may be perceived through it If we had a transparent metall much labour might here be saved A clear Lanthorn horn may serve for a smal Instrument but for large Instruments it is best to have it either of fine pastboard or if you will go to the cost of metal cancelled as shall be taught 1 For the delineation of the Reet first draw your fundamenttall Circle equal to the fundamental Circle of the Mater leaving a border or Limb without of such breadth as may receive the graduations of the Circle and figures set to them which breadth may be three tenths of an inch where the Reet is a foot in Diameter draw likewise two Diameters A B and C D crossing one another in the Center E at right Angles and dividing the Circle into his four Quadrants which you shall subdivide again into 90 degr apeece as you did in the Mater 2 You shall inscribe two arches which shall represent the Semicircles of the Ecliptique which shall meet at the points C and D of the Equator and the middle points of these arches shall be found in the Diameter A B thus The Diameter A B being divided as before you were taught to divide the Diameters of the Mater number from A toward the center E 23 degr 30 min. and there make the point F for ♑ and likewise number from B toward E 23 degr 30 min. and there set the point K. for ♋ then join the points C F D in one arch and the points C K D in another arch as is taught Chap. 3 and your Ecliptique is drawn But now you must make him a narrow Limb inward toward the Center to receive the scale of his degr and the characters of the Signes And to divide him you shall do thus Number in the Axtree line A B from F inwards 90 degr there is the Pole of the arch C F D to this Pole fasten one end of your ruler having an ey-lid-hole in the edge for that purpose and carrying about the other end over the several degr of the Semicircle C A D you shall cut the arch C F D into his correspondent degrees As if you lay the ruler from C to 10 degr in the Limb toward A it shall cut the Ecliptique in ♎ 10 and so of the rest Likewise for the other Semicircle C K D find his Pole 90 degr from K toward F and A and from that Pole by like reason you shall divide the Semicircle C K D by the divisions of the Semicircle C B D. This is the best way Or you way divide the Ecliptique by a Table of Right Ascensions thus Lay your ruler from the Center E to 27 deg 54 min. in the Limb which is the Right Ascension of ♉ 0. to be counted from D towards B and the ruler shall at the same time cut the Ecliptique in ♉ 0 to which that Right Ascension belongs and so for any other deg or you may defer the dividing of the Ecliptique till you have finished and cut out the Reet and then if you set the line C D of the Reet in A B the Axtree line of the Mater the Ecliptique will lie among the
place the known Angle at the Pole as well as at the Zenith and it may be needfull so to do when the Angle C of your Triangle would otherwise fall under the Limb of the Zodiaque Note also that the Angle C may sometime fall under the Finiter where the Azimuths faile As if you had set the Angle 167 degrees at the Pole the opposite side 105 degrees 41 minutes had been set in an Azimuth and C had been beyond the Finiter your remedie in this case is to set Nadir in the place of Zenith so shall C fall among the Azimuths just as you would have him Example Set 37 degrees 45 minutes between the Pole and Nadir a b count the Angle given at the Nadir b 167 degrees 9 minutes and his supplement 12 degrees 51 minutes for A C count a C the supplement thereof and you shall find b C 111 ½ whose supplement is B C 68 ½ Note thirdly that if the angle given in this chapter be a cute then if you place the known Angle at the Zenith the Parallel may cross the Azimuth twice or if you place the known Angle at the Pole the Almicanter taken to find out the opposite side may cross the Meridian twice and so it may be doubtfull in which intersection the Angle C shall be found That you may discover if you examine which agrees best with the other parts of the Triangle being turned or if you reduce this Triangle to two Right-angled Triangles by letting fall a Perpendicular Of which see the last Chapter CHAP. XVII PROB. 4. Two Angles and the Side comprehended between them being given to find the rest SEt the side given between the Pole and Zenith on the Limb then count one Angle among the Meridians the other among the Azimuths and where the Meridian and Azimuth bounding the said Angles meet there is the point of the Angle C and all is known but the Angle C which you may find also if you turn the Triangle Example In the 14th Chap. The Angle A was 45 B 113 ½ the side A B comprehended 40. Having set the Zenith 40 from the Pole I seek the 45th Meridian from A B and the 113 ½ Azimuth from A B and where they cross is C. Now may I number A C by the Parallels 70 and B C by the Almicantars 46 ½ C may now be found by any of the 3 former Problemes if you turn the Triangle and set C at the Pole or at the Zenith CHAP. XVIII PROB. 5. Two Angles and a Side opposite to one of them given to find the rest SEt the Angles given as A and B at the Pole and Zenith the known side as B C in an Azimuth Count among the Meridians the Angle opposite to the known side and having found the Meridian that boundeth him lay a finger or a bodkin point thereon then count the other Angle among the Azimuths and when you come to the Azimuth that boundeth him because that Azimuth maketh the known side of your Triangle you shall number his length from the Zenith and at the end therof make a prick then turn about the Reet till this prick in the Azimuth touch the Meridian before found and then is your Triangle formed on the Planisphear and all is known but the Angle C to be found as in the former Chapters Example Let be given A 45 degrees B 113 ½ B C 46 ½ I count from A B to the 45th Meridian upon which I lay my finger that he get not away for he must make my side A C then I look the 113 ½ Azimuth from A B to stand for the given side and because his length given is 46 ½ therfore in this 113 ½ Azimuth at 46 ½ below the Zenith I make a prick then I turn the Reet till this prick touch the 45th Meridian there at that touch must C stand thence to the Pole is the side A C 70 and on the Limb I have the side A B 40. C is to be had by turning the Triangle as in every of the former Problemes CHAP. XIX PROB. 6. Three Angles given to find the Sides THis Case comes very seldome in use Yet that our Method of Trigonometry by the Planisphear may be compleat and that no Probleme that is soluble may be left here unresolved I shall shew the solution of this Probleme also Mr Blagrave it seemes never attempted this contenting himself that he had found the way to resolve this Probleme in Rectingled Triangles which also he had once given over as impossible Blagr Book 5 24. For resolving this Probleme it is to be known that if you go to the Poles of the 3 great Circles wherof your Triangle is made these Poles shall be the angular points of a second Triangle and the two lesser sides of this second Triangle shall be equal to the two lesser Angles of your first Triangle the greatest side of the second Triangle shall be the supplement of the greatest Angle of the first Triangle that is shall have as many degrees and minutes as the greatest Angle of the first Triangle wanted of 180 degr see Piâ—â—scus Trigonometry Lib. 1. Prop. 61. This second Triangle therfore all whose sides are known from the Angles of the first you shall resolve by the first Probleme of Oblique angled Spherical Triangles Chap. 14. And having by that Probleme found the Angles of this second Triangle know that the 2 lesser Angles of the second Triangle shall be severally and respectively equal to the two lesser sides of the first Triangle and the least Angle to the least side the middle Angle to the middle side and the greatest Angle of this second Triangle being subtracted out of 180 degr shall leave you the greatest side of your first Triangle Because A in the first Triangle is 45 degr therefore in the second Triangle B C subtendeth A shall be 45 degr And because C in the first Triangle is 38 degr 51 min. therefore in the second Triangle the side A B which subtendeth C shall be 38 degr 51 min. And because B the greatest Angle in the first Triangle is 113 ½ therefore in the second Triangle the side A C which subtendeth B shall be the supplement thereof viz. 66 ½ Write now upon the sides of this second Triangle the quantities of the sides so is your second Triangle ready to be resolved by the first Probleme of Oblique-angled Triangles whereby you shall find the Angles of the second Triangle as I have expressed them in the Scheam A 46 26 min. C 40 B 110 degrees Now lastly I say these Angles of the second Triangle thus found give me the sides of the first Triangle which I seek in this manner In the second Triangle In the first Triangle A is 46.26 Therefore B C is 46.26 C is 40.00 Therefore A B 40.00 B is 110.00 Therefore A C 70.00 Supplement of 110 degrees And thus by all the Angles given we have found out all the sides which was required CHAP.
him in his proper lines Go to the Mater of your Planisphear and take him there in the first Projection There number 60 the Suns Longitude in the Ecliptick line of the Mater from the Center outward Where 60 endeth there is C of your Triangle and the Meridian that meets you there is C A the arch of Declination follow him to the Equator and you shall find by his graduation he is 20 degr 12 min. Long. thence turn in the Equator to the Center and you make B A the Right Ascension 57 degr 48 min. so have you the true picture of your Triangle in his proper place Observe your Triangle now and you may see A is a right angle for at such angle all the Meridians cut the Equator B is 23 ½ for such an angle the Ecliptick dayly maketh with the Equator as the arch in the Limb comprehended between them shewes Now take for given any of the three Sides and you have the rest Take the Longitude for given and be it 60 degr as before or 70 degr or what you will and you may find the Declination and Right Ascension as before Let the Right Ascension be given then setting a needles point in the end thereof A you may thence in a Meridian trace out the Declination C A to the Ecliptick and the Longitude B C thence to the Center every Side being divided into his whole parts or degrees If the Declination be given say Because the 20th Parrallel almost must cut off C A the arch of Declination in C therefore I follow the Parallel 20 ⅕ to the place where he cutteth the Ecliptick and there comes the Meridian that serves my turn and I may go down by him to the Equator as you would go down a ladder counting the rounds or degrees as you go and so on round my Triangle and I need no more For observe it when you will in the use of this Planisphear if you can find the way to go round your Triangle you have all the Sides measured to your hand and evermore one Angle also most commonly two and the angle C onely left unknown But admit the Sun be in ♌ 0 then is his Longitude 120 degrees and he is come back from the Solstice in your Planisphear as many degr as he wanted of it before Here the Triangle is equal to the former and resolved in like manner The Declination is the same as before But the arches of Longitude and Right Ascension in the Triangle are supplements of the true Longitude and Right Ascension shewing what the Sun wants of the Longitude and Right Ascension 180 in ♎ 0. wherefore subtract the Base of the Triangle 57 degr 48 min. from a Semicircle or 180 degr and you shall leave 122 degr 12 min. the Right Ascension of ♌ 0. or number in the Equator from the Center the way in which the Right Ascension hath increased that is first to the Limb which here is Colurus Solstitiorum 90 degr then back again to A the Right angle of your Triangle and you have 32 degr 12 min. to be added thereto The Sum is 122 degr 12 min. the Right Ascension as before If you observe this Example you will easily perceive that when the Sun is past ♎ 0. the Triangle will be on the other side the Center and between ♎ and ♑ you must add to the Right Ascension and Longitude found within the Triangle 180 degr and in the last Quadrant between ♑ and ♈ where the Right Ascension again increaseth inwards you must add 270 degr to the complement of Right Ascension found in the Triangle and take the sum or else subduct the Right Ascension found in the Triangle from 360 degr and take the residue for the Right Ascension CHAP. VII To do the same in the second Projection more easily IN the second Projection where the Center is the Pole of the World and the Limb Equator you shall find the Ecliptick fairly drawn upon the Reet and distinguished into his quarters and degrees Remember now from the former chap. that the Ecliptick Equator and a Meridian must make your Triangle and know that the Label supplieth the place of the Meridians If the Longitude or Right Ascension be given lay the Label on the degree given in the Ecliptick for Longitude or in the Limb of the Reet for Right Ascension and your Triangle is made and you may presently see your desire If the Declination be given consider in what quarter of the Ecliptick the Sun is then number the Declination given upon the Label inwards and where the numbring ends make a prick on your Label then move the Label into the quarter where the Sun is and lay the prick on the Ecliptick there and your Triangle is made wherein you may see the Longitude and Right Ascension desired This needeth no Example CHAP. VIII To find the Angle at the Sun made between the Ecliptick and Meridian THis is the angle C of the former Triangle and is the onely part which cannot be found in the former posture of the Triangle neither in chap. 6 nor 7 but is easily had by conversion of the Triangle as you may remember out of the third Book Take the Triangle of chap. 6 and make the Cathetus Base for this turn and by the 1 or 2 Problemes of Rectangled Triangles you may find this angle to be 77 degr 43 min. CHAP. IX To find the said angle of the Ecliptick with the Meridian by the Longitude Declination or Right Ascension divers other wayes IN the Meridional Projection do thus If you have the Longitude given count the distance of the Sun in that Longitude from the next Equinoctial point and count so many degrees in the Arctick Circle from the Limb inwards to the end of this numbring lay the Label and between the Label and Equator you have upon the Limb the lesser angle made between the Ecliptick and Meridian the greater angle is the supplement thereof Also between the Arctick Circle and the Limb you may find the Declination on the Label which is more then was required If you have the Declination given count it on the Label inwards and make a prick where the number ends then turn this prick upon the Arctick Circle and the Label sheweth the lesser angle in the Limb as before Example I would know what angle the Meridian that cutteth the Sun in ♉ 9 degr maketh with the Ecliptick I number therefore in the Arctick Circle from the Limb inwards 39 deg and to the 39th degr I say the Label and it sheweth in the Limb the angle sought 71 degr 20 min. and in the Label the Declination of ♉ 9 degr viz. 14. 32 minutes this is a good way But that the Label at this 39th degr cutteth the Pole of the Ecliptick as Mr. Blagrave saith Book 3 40. is not true either Mr. Blagrave or the Printer here mistakes For the Pole of the Ecliptick lies 14. 24 minutes nearer the Axletree as you shall
of the third House the 32 ½ ♠1. Thus have I the degrees of the Ecliptique in the beginning of 6 Houses and the 6 Houses opposxe begin with the same degrees of the opposite Signes CHAP. L. Another way to find what degree of the Ecliptique is in the beginning of every House and thereby to set a Figure more easily then by the former Chapter The figure of the Heavens March 29. 1652. H. 10. 32. a.m. Your Houses being thus distinguished on the Reet get the degree of Culmination and the Altitude thereof by Chapter 46. then set the Zenith under the North Pole so much as the Altitude of the Culmination comes to and if the Ascendent be a North Signe let the Pole be toward your left Hand and contrary if it be a South Signe so shall the Axis of the Reet be Horizon and the Pole Culmen Caeli Next get the Ascendent by the 47th and his Amplitude by the 16th this Amplitude you shall number in the Axletree of the Reet from the Center alwayes to your left Hand or toward Septentrio and mark what Meridian there cuts the Axletree of the Reet in that degree of Amplitude that Meridian shall be your Ecliptique for this time follow him up to the Pole and you trace out the arch of the Ecliptique from the Ascendent to mid-heaven and if you go down in his match to the like degree of Amplitude on the other side of the Center there is the Western arch of the Ecliptique from the mid-heaven to the Descendent and here you may see every degree of the Ecliptique above the Horizon and in what House it is without any more coursing after them Example ♓ 25 ½ was Culminating his Altitude 36 degrees the Ascendent had ♋ 24. whose Amplitude is 36 ⅓ Setting the Zenith therefore 36 degrees to the right Hand under the Pole I number in the Axtree-line of the Reet from the Center to my left Hand the Amplitude of the Ascendent 36 ⅓ there cometh the 23. Meridian from the Center who must serve for the Ecliptique Now because it is troublesome to number the degrees of the Signes backward I will begin at the Descendent 36 ⅓ from the Center on the other side and say Here is ♑ 24 degrees descending because ♋ 24. was ascending hence I count on toward the Culmen till I come to the Azimuth 19 ½ which is the Domifyer of the 12th and 8th Houses and here I say begins the 8th House in ♒ 13. for there are but 19 degrees from the Descendent hither hence I count to the 47th Azimuth the Domifyer of the 9th and 11th Houses and there I count ♓ 1 degree for the beginning of the 9th House hence I number on to the Pole and there I happen on ♓ 25 ½ the Culmen and beginning of the 10th House Thence I number on the other side of the Maters Axtree in the twenty third Meridian toward the Ascendent and I find the 47th Azimuth cuts ♉ 7 degrees for the beginning of the 11th House but the 19 ½ Azimuth which should shew me the 12th House is cut off by the Finitor and I am left to seek him else where And to find him I need but turn about my whole Planisphear the Reet unmoved and make the other Pole Culmen for this turn and then I find among the Azimuths that peece of my Ecliptique which I wanted in the former posture and I may reckon on him between the Ascendent and the Azimuth 19 ½ 29 ½ and thereby see that ♊ 24 ½ is in the beginning of the 12th House so have I 6. of my Houses and may by them find the other 6 as was shewed Chapter 48. and set them down as in the Figure CHAP. LI. A third way to set a Figure with less labour LEt the Meridians and Azimuths here change their offices in which they served in the former Chapter that is let the 19 ½ and 47th Meridian on both sides the Axis of the Mater be Domifyers and let the 23 Azimuth be Ecliptique and to that purpose set the Zenith above the Pole according to the Altitude of Culmen 36 degrees and make the Axis of the Mater Horizon Then beginning as you did before at the Descendent go up in the Ecliptique till you come to the Meridian 19 ½ and follow the Almicantar that there cutteth to the Limb and there make a mark for the 8th House then mark where the same Ecliptique cuts the next Domifyer the 47th Meridian and follow the Alm̄icantar from that point to the Limb prick there the 9th House the Zenith is the 10th thence go toward the Ascendent and do in like manner making pricks for the 11th and 12th Houses also in the Limb of the Reet at the end of that Almicantar which cutteth the beginning of the Houses in the Ecliptique Then in the Zodiaque of the Ring look the degree of Culmination and set the Zenith of the Reet to it and the Label laid to these pricks shall shew you presently in the Zodiaque the degrees for the beginning of every House CHAP. LII How to place any Star or Planet in his proper House IN the Equinoctial Projection get the Stars Hour distance from the Meridian thus Lay the Suns place to the hour proposed then turn the Label to the Star or to his Right Ascension if he be not in the Reet and it shall shew in the Limb how many hours and minutes the Star is past or short of the Meridian get also the Stars Declination North or South by the Reet or by the 35. or some other Chapter and where the Parallel of the Stars Declination crosseth the hour of the Star in the Mater there is his place for this turn therefore having made a prick with ink for him there set the Zenith line to your Latitude and having your Domifying Azimuths marked upon the Reet as Chapter 49. 30. you shall presently see in what House the Star is Example 1652. March 29 10. hours 32 minutes before noon I would know in what House the Pleiades are The hour of Lucida Pleiadum for that time is 8. 14 minutes after midnight the Declination is 23 degrees North. I number therefore in the 23. Parallel of North Declination from the Atree of the Mater to the Meridian 33 ½ there is the place of Lucida Pleiadum where I prick him down and setting the Zenith line to the Latitude I find the 39 Azimuth or Circle of Position cuts him by which I see he is 8 degrees from the beginning of the 11th House for that begins at Azimuth 47 as appeares Chapters 49 50. CHAP. LIII To find the division of the Houses according to Campanus CAmpanus begins the Houses at every 30th degree of the East Azimuth accounting from the Ascendent in the Sequel of the Signes as was said Chapter 48. Therefore if you will use his way set the Zenith line to the Latitude and the Finitor shall become the East Azimuth and every 30th Azimuth from the
Place now found First I reckon up from the Culmen Caeli to Nonagesimus gradus in the Limb 4 degrees for so much the Nonagesimus gradus is West of the Meridian and thence back again I tell to the 10th Meridian from the Limb which maketh the 14th Azimuth from Medium Caeli in which Azimuth I observed Jupiter in that Meridian used here for the 14th Azimuth I reckon the Altitude of Jupiter from he Equinoctial line 61 degrees and at that Altitude I make therein a prick for the place of ♃ And imediately I see this prick standeth a quarter of a degree above the Finitor shewing the Latitude of ♃ 15 minutes North and it is cut by almost the 5th Azimuth from the Limb which sheweth me that the Longitude of ♃ is 4. 50 minutes less then the Longitude of Nonagesimus gradus and therefore that ♃ is in ♋ 0. 40 minutes CHAP. LXI The Latitude and Azimuth of a Star and either the Ascendent Nonagesimus gradus or the Culmination given to find his Longitude YOur Planisphear being set in the Nonagesimal Projection as in the former Chapter seek the Meridian that serveth for the Azimuth of the Star and mark where it cutteth the Almicantar serving for the Parallel of the Stars Latitude The Azimuth cutting there shewes the Longitude which you shall reckon from the Ascendent or Descendent or Nonagesimus gradus whose Longitudes are known as was shewed in the former Chapter Example Suppose ♃ his Azimuth observed 14 degrees from South Westward as Chapter 60. and suppose his Latitude known 15 minutes North Though I know the Tables make Jupiters Latitude here divers minutes less that matters not to our purpose here I say where the 10th Meridian which by the former Chapter is the 14. Azimuth in this posture of my Planisphear cutteth the Almicantar 0 ¼ serving for Jupiters Latitude there cutteth an Azimuth which gives me Jupiters Longitude as in the former Chapter CHAP. LXII To find the Parallactical Angle that is what Angle the Azimuth maketh with any point of the Ecliptique by the Altitude of that point and of the Nonagesimus gradus NUmber on the Label from the Center the Complement of the Altitude of the point proposed which may be known by Chapter 56. and at the end of it make a prick and having by Chapter 55. or otherwise the Altitude of Nonagesimus gradus turn the prick you made on the Label to touch the Almicantar which is Complement of that Altitude then in the Limb of the Reet between the Finitor and the Label is the quantity of the angle Example In Chapter 56. the Altitude of the Sun in the middle of the Eclipse which hapned March 29. 1652. 10. hours 32 minutes a. m. was 41. 47 minutes and the Altitude of Nonagesimus gradus 41. 58 minutes wherefore I make a prick on the edge of the Label at 41 ¾ counted from the Limb or I count the Complement hereof from the Center and make the prick and having turned that prick to the fourty second Almicantar from the Zenith I find the Label shewing 87 degrees in the Limb of the Reet the quantity of the angle But because the Label here cutteth the Almicantar so slope that you can hardly observe the just point of Intersection I will shew you another way The Complement of the Altitude of Culmen Caeli the distance of the point proposed from Culmen and the Complement of Altitude of the said point make a Triangle whose 3 sides are all known or may be known by the Chapters foregoing Therefore by the first Probleme of Obliquangled Triangles you may find the angle CHAP. LXIII To find the Parallax of Altitude of the Sun or Moon To get the Parallax for any Altitude proposed you must first get the Horizontall Parallax out of some Astronomical Tables for it varyes according to the Planets distance from the Earth which is not alwayes the same yet the Suns Horizontal Parallax you may alwayes reckon to be about 3 minutes and the Moons Horizontal Parallax to be at the least 50 minutes and at the most 68 minutes This had I number the Horizontal Parallax in the Limb from the Equinoctial line and thereto lay the Label and number the Altitude of the Planet on the Label from the Limb and the Parallel that cuts that Altitude shewes the Parallax desired And note here that for every minute of the Horizontal Parallax you may reckon 5 10 or 20 times so many so that your Label rise not beyond 10 degrees in the Limb so shall you attain the minutes more exactly Example March 29. 1652. the Altitude of the Sun in the middle of the Eclipse was 41. 47 minutes and his Horizontal Parallax according to Lantsbergius 2 minutes 18 seconds for which I number 2 degrees 18 minutes from the Equinoctial line and thereto set the Label and so I find the 43 ¾ degrees of the Label to cut the Parallel 1 degree 30 minutes which I am to accompt 1 minute 30 seconds the Suns Parallax for this Altitude Likewise the Moons Horizontal Parallax according to Lantsbergius was then 62 minutes her Altitude the same with the Suns or very near I set the Label therefore to make an angle of 6 degrees 12 minutes with the Equinoctial and so I find the Parallel 4 ⅔ cutting the 41 ¾ degree of the Label which shewes the Parallax of the ☽ in that Altitude 46 ½ accounting every degree 10 minutes as here I had appointed them to signifie CHAP. LXIV The Parallactique Angle and the Parallax of Altitude given to find the Parallax of Longitude and Latitude IF the Azimuth or Circle of Altitude make no angle with the Ecliptique but be co-incident with it as where the Ecliptique cuts the Zenith then doth the Parallax of Altitude vary the Longitude only and so much as the Parallax of Altitude is so much is the apparent Longitude of the Planet greater then the true Longitude in the Eastern Quadrant of the Ecliptique and so much lesser in the Western If the Azimuth make a Right angle with the Ecliptique which it may do only in Nonagesimo gradu then doth the Parallax of Altitude vary the Latitude onely and so much as the Parallax of Altitude is so much must be added to the apparent North Latitude or subducted from the apparent South Latitude to make the true Latitude of the Planet North or South If the Azimuth cut the Ecliptique with Oblique angles as most commonly it hapneth to do then doth the Parallax of Altitude vary both the Longitude and Latitude And the nearer the Planet is to the Nonagesimus gradus the greater is the Parallax of Latitude and the Parallax of Longitude less and contrarily the further the Planet is from Nonagesimus gradus the greater is the Parallax of Longitude and the Parallax of Latitude the less The Parallax of Altitude is alwayes the Hypotenusa and the Parallax of Longitude and Latitude are the legs of a smal Rectangled Spherical Triangle
PROBLEMES And to make all sorts of SVN DYALS very easily by the PLANISPHEAR CHAP. I. The Preface Of the kinds of Dyals ALthough Gnomoniques pertain to Astronomy yet I think it not amiss for the ease of the Reader in finding them to place the Gnomonical Problemes in a distinct Book by themselves Suns Dyals may be reduced to two sorts Some shew the hour by the Altitude of the Sun as Quadrants Rings Cylinders c. for the making whereof you must know the Suns Altitudes for every day or at least every 10th day of the year and for every hour of those dayes which Altitudes you may find immediately upon this Planisphear as in a Table made to your hand for any Latitude by Book 3.25 and so make them of any shape according to your mind The other sort shew the hour by the shadow of a Gnomon or Style Parallel to the Axis of the World and of those I treat cheifly in this Book Those be all Projections of the Sphear upon a plain which lies Parallel to some Horizon or other in the World And if upon such a plain the Meridians onely be projected they shall suffice to shew the hour without projecting the other Circles as the Ecliptique the Equator with his Parallels of Declination the Horizon with his Almicantars and Azimuths which are sometimes drawn upon Dyals more for ornament then for-necessity CHAP. II. Theorems premised FOr the better understanding of the reason of Dyals these Theorems would be known 1. That every plain whereupon any Dyal is drawn is part of the plain of great Circle of the Heaven which Circle is an Horizon to some Country or other that the Center of the Dyal represents the Center of the Earth and World and the Gnomon which casteth the shade representeth the Axis and ought to point directly to the two Poles And if upon the Center of the Dyal you fasten a Label with Sights of equal Altitude and keeping your eye in the line of the Sights turn this Label round you shall thereby describe in the Heavens that great Circle wherein your Dyal-plain lies and see where it cuts our Horizon and how much it is Elevated above it on one side and depressed on the other 2. That those Dyal-plains Geometrically are not in the very plains of great Circles for then they should have their Centers in the Center of the Earth from which they are removed almost 4000. Miles and in truth they lie in the plains of Circles Parallel to the said Horizons but so near them that Optically they seem to be the plains of those Horizons because the Semidiameter of the Earth beareth so smal proportion to the Suns distance that the whole Earth may be taken for one point or Center without any perceivable error 3. That as all great Circles of the Sphear so every Dyal-plain hath his Axis which is a straight line passing through the Center of the plain and making right angles with it and at the ends of the Axis be the two Poles of the plain whereof that above our Horizon is called the Pole Zenith and the other the Pole Nadir of the Dyal 4. That every Dyal-plain hath two faces or sides and look what respect or situation the North Pole of the World hath to the one side the same hath the South Pole to the other and these two sides will alwayes receive 24 hours so that what one side wanteth the other side shall have and the one is described in all things as the other 5. That as Horizons so Dyal-plains are with respect to the Equator divided into 1. Parallel or Equinoctial 2. Right 3. Oblique 6. A Parallel or Polar Dyal-plain maketh no angles with the Equator but lies in the plain of it or Parallel to it Such Dials are Scioterica Orthognomonica that is have the Gnomon erected on the plain at Right angles as the Axis of the World is upon the plain of the Equator because the Axis and Poles of the Dyal be here all one with the Axis and Poles of the World and the hour lines here meet all at the Center making equal angles and dividing the Dyal Circle into 24. equal parts as the Meridians do the Equator in whose plain the Dyal lies 7. A right Horizon or Dyal-plain cutteth the Equator at right angles and so cutteth through both the Poles of the World Therefore such Dyals are Paralielognomonical that is have the Gnomon Parallel to the plain and so the hour lines and the hour lines all Parallel one to another because their plains though infinitely extended will never cut the Axis of the World Yet have those Dyals a Center though not for the meeting of the hour lines viz. through which the Axis of the Dyal Circle passeth cutting the plain at right angles and cutting also near enough for the projecting of a Dyal the Center of the World 8. An Oblique Horizon or Dyal-plain cutteth the Equator at Oblique angles such Dyals are Scalenognomonical that is have for their Gnomon the side of a Triangle whose angles vary according to the more or less Obliquity of the said Horizon and the Gnomon shall alwayes make an angle with the plain of so many degrees as the Axis of the World maketh with the plain or as either of the Poles of the World is Elevated above the plain 9. Every Oblique Horizon is divided by the Meridians or Hour Circles of the Sphear into 24. unequal parts which parts are alwayes lesser as they are scarer to the Meridian of that Horizon or plain and greater as they are further off and on both sides the Meridian of the plain the hour Circles which are equally distant in Time are also equally distant in Space whence it is that the divisions of one Quadrant of your Dyal plain being known the division of the whole Circle is likewise known 10. The Hour-lines in an Oblique Dyal are the Sections of the plains of the Hour-circles of the Sphear with the Dial plain And because the plains of great Circles do alwayes cut one another in halfs by Diameters which are straight lines passing through the common Center therefore lines drawn from the Center of the Dyal to the Intersections of the Hour-circles with the great Circle of the plain shall be those very Sections and the very Hour-lines of the Dyal 11. Every Dyal-plain being an Horizon to some place in the Earth as was said Theorem 1. hath his proper Meridian which is the Meridian cutting through the â—oles of the plain and making Right angles with the plain If the Poles of the Dyal-plain lie in the Meridian of our place then is the Meridian of the plain all one with the Meridian of the place and the Gnomon or Style shall stand erected upon the Noon-line or line of 12 a clock as in all direct Dyals but if the plain decline then shall the substylar or line wherein the Gnomon standeth which is the Meridian of the plain vary from the Noon-line which is the
being placed upon the sole of the Window shall supply the use of the Nodus in the Gnomon and the beams of the Sun being Reflected by this Glass or Water shall shew the Hours upon the Ceeling The Planisphear shall help you to make this Dyal two wayes If the Window Decline not much from the South you may make it most easily the First way But if it Decline much and so the lines fall much upon the partition Walls or if you would adorn this Dyal with the Parallels or other Circles you shall use the Second way The First way is this Draw a Meridian line upon the Floor by Book 4.3 so that it may point upon the Perpendicular which you shall imagine to fall from the Nodus upon the plain of the Floor prolonged And this may be most easily done if you hang a Plumb-line in the Window dnecuy over the Nodus of place of the Glais for the shadow which that Plumb-line gives upon the Floor at Noon is the Meridian line sought and by a Ruler or a line stretched upon it you may prolong it as far as you shall need Then let a Plumb line fall from the Ceeling upon this Meridian line of the Floor and behind it Northward or Southward place your Ey so that the Plumb-line may hide the Meridian line of the Floor from your Ey then keeping your head steddy cast you Ey up to the Ceeling and direct One to make two points at a good distance in the line upon the Ceeling which the Plumb-line now covereth from your Ey and by these points you shall draw a straight Meridian on the Ceeling Then having fastned one end of a Line at Nodus let Another stretch this line up to the Meridian on the Ceeling and let him move his hand nearer or further in the Meridian till you find by a Quadrant that this line pointeth up Northward as many degrees as the Elevation of the Equator is in your Country and then you shall cause him to make a point where the line toucheth the Meridian of the Cieling and through that point you shall draw the Equinoctial line of your Dyal cutting the said Meridian at Right angles The length of the thred from the Nodus to the point in the Meridian where the Equinoctial cuts him is Radius of the Equinoctial to that Radius you shall find the Tangents of 15 30 45 60 75. as you found the Co-tangents Chapter 27. knowing that the Co-tangents of 80 and 70. be the Tangents of 10 and 20 and so of the rest and beginning in the Meridian make pricks in the Equinoctial line at the end of the Tangent of 15. Eastward for 1. and Westward for 11. and at the end of the Tangent of 30. prick Eastward 2. and Westward 10. c. Then by Chapter 9. seek what angles the Hour lines of a Vertical Dyal make at the Center which in our Latitude are 1.11.58 minutes 2.24.32 minutes 3.38.20 minutes 4.53.52 minutes 5.71.17 minutes and with the Complements of these angles shall these Hour lines cross the Equinoctial so the Hour line of 1. shall Incline to the Meridian on the South side the Equinoctial line and shall make his lesser angle with the Equinoctial 78.02 minutes and the rest as in the Figure The Second way is this Fit a plain smooth Board about a foot Square to lie level from the fole of the Window inwards and near the outer edge thereof make a Center in the board in the very place of Nodus or a little under it remembring that the Nodus or Center of the Glass must be set so much higher then this board as the Center of your Quadrant is placed higher in the Projecting of the Dyal Upon that Center taken in the board describe as much of a Circle as you may with the Semidiameter of your Quadrant which Circle shall be Horizon Draw here from the Center to the Horizon inwards a Meridian line by Book 4.3 and where it cuts the Horizon begin to graduate the Horizon into degrees of Azimuths both wayes which you may speedily do by transferring the graduations of your Quadrant or so much as you shall need to this Horizon Next you must devise to make your Quadrant stand firm and upright upon one of his straight sides which I will call his foot for this time and that you may thus do Take a short peece of a Ruler or sinal Transom and saw in one side of it a notch Perperdicularly in which notch you may stick fast or wedge the heel or the toe of your Quadrant in such sort that his foot may come close to the board and the other straight side or leg may stand Perpendicular upon it Those things prepared put your Planisphear in the Meridional Projection with the Finitor at your Latitude and first observe there the Altitudes of the Sun in the Meridian which in Latitude 52.15 minutes you shall find in the Tropique of ♋ 61.15 minutes in the Equator 37.45 minutes and in the Tropique of ♑ 14.15 minutes Now having stuck a short needle in the Center of the Horizon close to which you must alwayes keep the Center of your Quadrant set the foot of your Quadrant in the Meridian line of the Board and from the Center of your Quadrant extend a thred by 14.15 minutes of Altitude straight on to the Cieling the thred only touching the plain of the Quadrant and making no angle with it but held Parallel and where the thred thus extended touches the Cieling make a point then the Quadrant unmoved extend the thred by 61.15 minutes of Altitude and make another point as before and between these two points draw a straight line and that shall be your Meridian and shall be long enough for your use then extend the thred by 37.45 minutes of Altitude and where it touches this Meridian cross the Meridian at Right angles with an infinite line which shall be the Equator Then seek upon your Planisphear for one a clock and you shall find in the Tropique of ♑ the Suns Azimuth 14. and his Altitude 13.06 In the Tropique of ♋ his Azimuth 27½ and his Altitude 59.04 minutes therefore setting the foot of the Quadrant in the Azimuth 14. from the Meridian Eastward I extend the thred by 13.06 of Altitude and make a prick in the Cieling and again setting the foot of the Quadrant in Azimuth 27 ½ and extending the thred by 59.04 minutes of Altitude I make another prick in the Cieling and the straight line which I shall draw between these two pricks shall be all the Hourlines of One and so of the rest And if you be minded to have the other Parallels drawn you may find points for them as you have done for the Tropiques and by those points draw them And note that two points made in the Cieling for the same Hour line in any two Parailels or in the Equator and any Parallel shall suffice to direct the line though it is best
less then the tenth part of an inch for one minute And beyond 30. or 40. degrees this Instrument would not be used because the Ey cannot see both the Sights of the Transom at once without rolling from one to another whereby the Center of Vision is changed 3. Your Ey is better fixed and shadowed by this Ey-sight then when the end of the Index is placed by guess upon the Cheek-bone The inconvenience here is no more then what is found in all Cross-staffes of what form soever And that is they are subject to some errour by reason of the Eccentricity of the Ey For the visual Beams meet within the Ey at a depth uncertain and they are also refracted in the Superficies of the apple of the Ey the apple of the Ey also is not of the same convexity nor of the same breadth in all Men and it is contracted in a bright Air and dilated in a darker Air as you shall soon find if you go about to observe the Diameter of the Moon by this Instrument without correction of the Eccentricity for you shall alwayes find the apparent Diameter too great and much greater in the Night then in the Day Thus November 18. 1653. I observed the Moons Diameter 32. minutes 06. seconds in the Day Time and that Night I observed it 58. minutes by reason of the dilatation of the apple of my Ey in the Night This errour may be rectified two wayes The First is by examining the observations made with your Cross-staff by some other Instrument which is not subject to like errour As for Example I have devised to fasten an arch of a Circle containing 20. or 30. degrees to the end of a Ruler of 6. or 7. foot and fit to it a Label with Sights then having observed by my Cross-staff the length of Orions Girdle I will set my other Instrument to it turning the arch toward me that I may manage the Label better and noting the difference of the observations I will find how to correct my Staff in that posture an another time and so by many observations I may frame a Table to correct the Eccentricity throughout but my Table perhaps will not serve to correct the eccentricity every Mans Ey neither will a Table made for the Night serve me in the Day The other way is most exact and certain for all Men. Make another Transom in all points like the first but shorter by half and let the divisions thereof be into half-inches this Transom must ride upon the Index with a socket between the long Transom and your Ey Now when you observe set the Sights of the short Transom to the like number of half inches as the Sights of the long Transom stand at whole inches and when you have placed your Ey-sight so that you see the Stars upon the edges of the Sights of the long Transom draw your short Transom till you see the Stars by his Sights in like manner at once then look what number is cut by the short Transom the double thereof is the Co-tangent of the angle and look what the number cut by the Ey-sight wants of that double so much is the Eccentricity of your Ey in that place This way is shewed by that Excellent Mathematician Mr Edward Wright in Chapter 15. of his Treatise of Errours in Navigation FINIS A Catalogue of Eclipses Observed since the Year of our Lord 1637. FIrst At Coventree whose Longitude is more West then London 1. degree 29. minutes of space Latitude 52. 28. minutes My especial friends Dr John Twysden and Mr Samuel Foster late Professor of Astronomy in Gresham Colleige and my self all together observed the totall and great Eclipse of the Moon which hapned in the Year 1638. on Tuesday December 11. before Noon The totall obscuration began 1. hour 07. minutes The time of emergence observed by the Altitude of Benenaes was 2. hours 41. minutes so the totall Obscuration continued 1. hour 34. minutes during the greatest part of which time the Moon was quite lost though the Skie was clear When the Moon began to recover light she was in the foremost foot of Apollo between the two Stars of the third Magnitude a line drawn between those Stars did cut off the lower part of the Moons body to ⅙ of her Diameter and setting the distance of the Stars in 12. parts the Moon had gone 7 ½ of those parts toward the Easterly Star which is in Calce Apollinis Hence I compute the apparent Longitude of the Moon at the time of emergence ♊ 29. 36. minutes 19 seconds and her apparent Latitude 0. 44. minutes South 2. At Easton Macodit whose Longitude is West from London 0. 43. minutes of space that is almost 3 minutes of Time the Latitude 52. 13. minutes Anno Dom. 1641. upon Fryday October 8. I observed the end of the totall Eclipse of the Moon when Lyra had Altitude 48. 48. minutes that is at 8 hours 38. minutes 08. seconds after Noon 3. At Ecton whose Longitude is West from London 45. minutes of space or 3. minutes of Time Latitude 52. 15. minutes Anno Dom. 1645 upon Munday Angust 11. I observed the Eclipse of the Sun ending when the Center of the Sun was in Azimuch 0. 55. minutes past the South that is 0. hours 2 ½ minutes after Noon This Eclipse Hevelius observed to end at Danizick at 1. hour 53 minutes as he writes in his Selenographia 4. At Ecton aforesaid Anno Dom. 1649. upon Wednesday May 16. before Noon I observed in the company of Mr Samuel Sillesby late Fellow of Queens Colleige in Cambridge the totall Eclipse of the Moon The beginning when the right Knee of Ophiucus was in Azimuth 7. 42. minutes past South that is 1. hour 08. minutes a.m. The totall obscuration began when the Azimuth of the said Star was 20 degrees Westward that is at 1. hour 55. minutes 44. seconds By the Medicaean Tables it should happen to be totally obscured at Uraniburg 2. hours 46. minutes 23. seconds and at Ecton 1. 53. minutes 23. seconds By Lantsbergius Tables at Ecton 1. hour 40. minutes 48. seconds 5. At Ecton Anno Dom. 1649. October 25. current Afternoon I observed by a Telescope the Eclipse of the Sun The Digits Eclipsed and the Time were as followeth Dig. H. min sec Dig. Hour 0. ⅛ 0. 41.56 4. 1.47.28 1.  49.48 3. 2.03.28 2.  59.44 2. 15.32 3. 1. 09.44 1. 22.40 4.  26.12 0. 31.04 4. ⅛ 33.32   6. At Easton Macodit Anno Domi. 1651 2. on Munday March 15. in the Morning I observed with Dr Twysden that the Moon was Eclipsed about one Digit when Alkair was in Azimuth 79. 40. minutes from the South Eastward More we could not see for Clouds 7. At Ecton Anno Dom. 1652. on Munday March 29. before Noon I observed the great Eclipse of the Sun by a Telescope and a minute-watch Rectified by the Azimuth of the Sun taken both before and
after in the company of half a score Gentlemen and Ministers my Neighbours as followeth Di. mi. Ti. mi. sec Digits Time 0.03 9.21.12 11.00 10.35 ½ 1.00 9.27 10.00 10.42 ½ 2.00 9.31.08 9.00 10.48 ½ 3.00 9.37 8.00 10.55 4.00 9.44 7.00 11.01 5.00 9.50 6.00 11.06 ½ 6.00 9.55 5.00 11.11 ¾ 7.00 10.00 4.00 11.19 8.00 10.06 ½ 3.00 11.24 ½ 9.00 10.11.28 2.00 11.31 10.00 10.18 1.00 11.35 ½ 11.00 10.25 0.00 11 42½ 11.22 ½ 10.32.04  And though this Eclipse was so great yet we could read in the time of the greatest darkness within Dores notwithstanding that the Window was covered with a Blanket 8. At Ecton Anno Dom. 1652. on Tuesday September 7. current the Moon rose Eclipsed about 10. Digits and while 8. Digits were yet darkned all the dark part of the Moon was visible of a Dusk and Tawny colour this Eclipse ended when the double Star in Cornu ♑ wanted in Azimuth 6. 30. minutes of the South that is at 7. hours 51. minutes 52. seconds but the Moon was not free of the Penumbra till 7. minutes after 9. At Ecton Anno Dom. 1654. on Wednesday August 2. current before Noon I observed the great Eclipse of the Sun by a Telescope and a Minute-watch sufficiently Rectified by the Azimuth of the Sun in the company of many learned Men my Neighbours and friends as followeth Di. T. mi. Di. Time 0. 7.47 10 ¼ 1. 7.52 ½ 10. 9.00 2. 7.58 ½ 9. 9.09 3. 8.04 8. 9.18 4. 8.09 7. 5. 8.15 6. 9.31 6. 8.20 ¾ 5. 9.38 7. 8.28 4. 9.45 ½ 8. 8.34 3. 9.51 ¼ 9. 8.40 ½ 2. 10. 8.49 1. 10.03 ½ 10 ¼  0. 10.09 10. At Ecton Anno Dom. 1654. on Thursday August 17. I observed the Eclipse of the Moon by a Telescope and a Minute-watch Rectified by the Azimuth of the first Star in the Horn of ♑ as followeth Time After Noon mi.  9. 47 ½ I saw the Penumbra invading the Moon with my bare Ey 9. 54. I saw the Penumbra invading through my Telescope 10. 15 ½ Shadow 3 minutes deep 10. 25. Shadow 4. minutes deep Yet I could discern all the Limb. 10. 45. Shadow more then 4. minutes deep Yet the Moons Limb all seen 11. 05. Yet the darkness is more on the East side shadow is 5. minutes deep and the Limb is lost in the shadow 11. 11. All the Limb seen again and the shadow seems but 3. minutes deep and just under the Moon so that the East and West side of the are darkned alike 11. 22. The shadow little above 1. minute deep in my Glass 11. 25. The shadow half a minute deep by my Glass 11. 27. The shadow gone in my Glass But the Penumbra still covers almost ⅓ of the Moons Diameter 11. 30. The shadow is here gone in the judgement of my naked Ey but the Penumbra is seen still 11. 35. The Moon as clear as at 9.47 ½ but yet the lower quarter of the Moon is much dusker then the rest of her body 11. At Ecton Anno Dom. 1655 6. upon Tuesday January 1. afternoon I observed the Eclipse of the Moon by a Minute-watch Rectified by the Southing of the Stars Clouds often hindred but thus I observed Ho. mi.  6.43 ½ The Moon growes dusk on the East side 6.49 ½ More dusk yet all the Limb is seen 6.51 ½ Here I judge the Moon to touch the Vmbra 6.53 ½ The Limb begins to be lost in the shadow so far as I can discern both with the Telescope and without it 7.00 ½ ☽ darkned 2. Digits by estimation 7.07 ½ Almost 4. Digits 7.34 ½ Almost 7. Digits here the Clouds thicken 8.29 ½ ☽ darkned about 10. Digits yet almost all the Moon is perceivable through the shadow 8.36 ½ About 10. Digits yet almost all the Limb perceivable 9.11 ½ About 8 Digits 9.23 ½ About 5 ½ Digits 9.28 ½ About 4. Digits 9.39 ½ About 3. Digits 9.51 ½ Here I judge the end The Limb of the ☽ is all restored yet the West side of the Moon looks duskish for 3. or 4. minutes longer 12. At Ecton Anno Dom. 1657. on Munday June 15. the Moon rose Eclipsed I observed the end thereof by the Azimuth of Antares to be 16. minutes after 10. 13. At Ecton Anno Dom. 1057. on Thursday December 10. I observed the Eclipse of the Moon ending when she was apparently 34 degrees high and me thought I discerned the Penumbra till her Altitude 35. it was a thick flying mist no Star but Jupiter could be seen with us all the time of this Eclipse about one third at the most of the Moons Diameter was darkned on the North side From the first Ecliptical opposition mentioned in this Catalogue to this last is the space of a Metonique Year These Observations are faithfully reported as I made them I could have strained some of them to a better Harmony if I would have forged any thing or used my own judgement upon them but I rather leave them to the judgement of the learned Readers especially such as have accustomed themselves to Celestial Observations FINIS The Rudiments of Astronomy Put into plain Rhythmes The Constellations of the Fixed Stars THe Army of the Starry Skie Declares the Glory of God most high Seen and perceived of all Nations In eight and fortie Constellations First neer unto the Northern Pole The Dragon and two Beares do Role Whose hinder parts and Tailes contain The lesser and the greater Wain The Hair the Bear-ward and the Crown And then comes Hercules kneeling down And next below a place doth take Great Serpentarius with his Snake Under the Harp of Orpheus The Eagle and Antinous The Silver Swan her Wings doth spread Above the Dart and Dolphins head Then Pegasus comes on amain Andromeda followes in her Chain The Triangle below her stands And at her feet in Perseus hands The Gorgons Head Above are seen Her Parents Cepheus with his Queen Cassiope Not far below Heniochus his Goat doth show On his left shoulder in his hand He doth the stormy Kids command Here in the Zodiaque begins The Ram the Bull the Loving Twins The Crab the Lion and Virgin Tender The Ballance Scorpion and Bow bender Goat Waterman then Fishes twain Shall bring you round to th' Ram again Fifteen Images appear In the Southern Hemisphear The Monstrous Whale before the rest Eridanus scarce wers his brest Over the Hare Orion bright Sparkles in a Winters night Then comes the great Dog at whose tayl The famous Argo spreads her sayl Above the little Dog doth flame For whom the Latines had no name Long Hydra on her tail alow Carries the Pitcher and the Crow The Centaure holds the Wolfe by th' heel The Altar and Ixions Wheel Are never seen of us but here The Southern Fish brings up the rear The Planets UNder those fixed Stars above Seven Planets in their Orbes do move The high'st is Saturn Thirty Year He spends in Compassing his Sphear Twelve Jupiter