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-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
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
to take your points in the Tropiques at the largest distance as I have here done if there be room enough on the Cieling But because it often happens that part of your Dyal falls beside the Cieling and the plain of the Cieling and of the Walls is often interrupted and made Irregular by Beams Wal-plates Corrishes Wainscot Chimney-peeces and such like bodyes I will â—hew you one absolute device to carry on your Hour lines over all Extend the thred for any Hour line to the Tropique of Cancer â—n the Cieling as you where taught before and fix it there and extend another thred in like manner to the Tropique of Copricorn where ever it shall happen as perhaps beyond the middle beam or quite beyond the Cieling upon the Wall and fix that thred also Then place your Ey so behind these threds that one of them may cover the other and at the same instant where the upper line to your Sight or Imagination cuts the Cieling Beams Wall or any Regular or Irregular body above the end of the lower line there shall the Hour line pass from Tropique to Tropique direct any By-stander to make marks as many as you shall need and by these marks draw the Hour line according to your desire If the arch of the Horizon between the Tropiques be within view of your Window you shall draw the same on the Wall to bound the Parallels the Horizons Altitude you know is nothing and therefore he will be a level line and the Suns Azimuth when he riseth commonly called Amplitude and Ortive Latitude is in Cancer 40.40 minutes East Northward and in Capricorn as much Southward and these will be reflected to the contrary coasts on the Dyal The end of the Fifth Book A breif Description Of a CROSS-STAFF THe Cross-staff consisteth of two Rules joyned by a socket or else pinned in the form of a Romane T and three Sights or more The longer Ruler is called Radius Index and the Yard as A B of which I call A the neer end B the further end The breadth would be ¾ of an inch the depth an inch and half the length 70 or 80. inches and every of those inches would be divided by Parallels and Diagonal lines into 100. equal parts The shorter Ruler E F is called the Transom it would be half an inch or three quarters both in breadth and depth and in length about 2. foot for the Sights there if I may advise you would never be set above 20 inches asunder This Transom would be divided into whole inches onely beginning in the midst at B in the visual line ☉ B. and numbred to 10 both wayes The Sights C and D must have sockets at the bottom through which the Transom must pass so that the Sights may be set to any division of the Transom The Vanes or tops of those Sights must have onely two edges on their sides visible to your ey namely those edges which touch the Transom and the two other edges must be pared away The middle Sight at B would have half his head cut away and a shoulder left as in the Figure and a tenon at the bottom fitted to a mortess made in the middle of the Transom that you may stick him in and take him out when you please for to this mortess you shall do well to fit two other moveable Sights very narrow for observing the Diameter of the Moon or the distance of Stars which are very neer one may be about half an inch broad and the other about a quarter This Cross-staff is exactly made by Mr. Anthony Thomson in Hosier lane London When you would use this Staff you shall first set the Sights of the Transom to like inches as at 10 and 10. if the angle be great or at 5 and 5. as in the Figure they are placed alwayes set them at whole inches and at like numbers on both sides from the middle of the Transom and choose to place those Sights so that your Ey-sight may be far distant from them in observing for so you may the more distinctly observe the minutes and seconds of the angle inquired Then resting the further end of the Index upon a Wall or some device fitted for that purpose put the neer end over your right shoulder and setting your Ey to the Ey-hole slip the Index backward or forward till you see the objects by the sides of the Sights of the Transom and mark what number the backside of the Ey-sight cutteth upon the Index for that shall give you the angle sought in this manner Example The Sights of the Transom being set at 5 and 5 that is 10. inches asunder I observed two Steeples by their edges and the Ey-sight then cut upon the Index 6625. that is inches 66 ¼ from the Transom I say therefore As C B 500. to B G 6625. so C B Radius or 100000. to B G the Co-tangent of half the angle Here I have no more to do then to divide 662500000. by 500. or 6625000. by 5. which is an easy work and the Quotient 1325000. is the Co-tangent of 4. degrees 18. minutes 57. seconds 43. thirds for half the angle Note here that if the Sights had stood at 10 and 10. then had the number 6625. been the very Co-tangent of half the angle and remembring that your Radius on the Transom hath but 1000 actual parts go to the Canon and cutting off so many places as may leave the Radius there but 1000. you shall find your number 6625 to be the Co-tangent of 8. 35 minutes Note also that you may observe the angle between the middle Sight and one of the other and then you find the Co-tangent of the whole angle to that Radius to which your Sight is set on the Transom as to the Radius 200. 300. or any other even hundred to 1000. Note further that you must evermore observe neer the tops of your Sights that the visual lines may run above the Transom as much as the Ey is placed above the plain of it He that will may have room to set several Scales of degrees and minutes to several Radiusses as one to the Radius 300. another to 500. another to 700. by which the very degrees and minutes may be presently had without recourse to the Tables To me the Scale of equal parts is in stead of all The Commodities of this disposition of the Staff are these 1. It is better managed when it rests upon the shoulder and the Ey-sight being made to move while the Transom and his Sights stand Fixed shall save you much labour of coursing up and down from one end of the Staff to the â—â—â—er in observing 2. The Ey-sight being made to shew the angle by the length of the Co-tangents shall alwayes give you large differences insomuch that if your Staff be but 6. foot long you may observe to Seconds and Thirds in lesser angles and till you come beyond 20. degrees your Sight shall seldom move
square to the line A B but if any of these have failed you shall never bring the three points into an arch while the foot of your compass â—andeth in the line C D. Therefore in such case set one foot in A and draw with the other foot a short arch crossing the line C D then set the standing foot in B and with the running foot cross the short arch last drawn where these arches cross will be the center by which you may draw an arch cutting A and B and if it cut 2 also you have your desire But if this arch over reach 2 widen your Compass if it come short of 2 the middle point narrow your Compass and try again in like manner till you can compass all the three points in the same arch 3. A third way I suppose you may know that every Meridian cuts the Equator twice viz. in two opposite points distant 180 degr one from another as for example the Meridian which cutteth the Equator in 60 degr of Right Ascension cuts it again in the opposite degr viz. 240. Now if you can find these two points in the Equator line C D the center will be in the just middle betwixt them One of these points is already given within the fundamental Circle the other without is thus found Prolong your Equator infinitely beyond your plate both wayes and divide the extension thereof by like reason as you divided his Diameter viz. as by a Ruler laid from A to the several deg of the Quadrant B C you devided the Semidiameter E C into 90 degr so keeping still one end of your Ruler fixed at A and carrying about the other end thereof to the severall degrees of the Quadrant C A you may divide the excurrence of the line E C into 90 degr more and so E C and his exccurrence or continuation will be half the Equator divided into his degr and E D with his excurrence on the other side will be the other half divided by like reason And thus the whole Equator is projected in one straight line and divided into degr also Then having a point given within your fundamental circle through which the sixtieth Meridian must pass viz the 60 deg of the Semidimeter E C or E D number thence over the center to 180 deg and there is the point where the other semicircle must cross and the middle between those points is the center But because the two points taken in the quadrant A C are very near together especially towards A and the Ruler also will cross the prolongation of the line E C very obliquely you may therefore do better to divide this line into his degr by a Scale of Tangents for if upon the Equinoctial line D E C you prick down from E both wayes the Tangents of the half degr in order from 0 to 90 those pricks shall be the whole degr of the Equinoctial line in this Projection to be numbred from E both wayes to 180 deg where the Tangent becomes infinite Thus taking A E for Radius E D is the Tangent of 45 deg by the structures yet the arch or Diameter E D is a Quadrant or 90 deg of the Equinoctial in this Projection because the Tangents of the half deg of the Quadrant E A I measure out the whole degr of the line E D as was above-said 4 If you consider well what hath been said you will find or you may take it here upon trust that for the 90 Meridians to be drawn between C and E half the centers will be found in the opposite Semidiameter E D and the other half without D in the said Semidiameter prolonged And that every second division of the line E D from E toward D and forwards shall be the centers of the Meridians which cut the Semidiameter C E. As for example to draw every fifth Meridian from C to E you take every tenth deg from E toward D for the centers And further if you would not be at more trouble then needs to divide the extension of the Diameter beyond the fundamental circle you shall but do thus Begin with the crookedest Meridians first whose centers are within the fundamental Circle and first pitching one foot of your Compass in the point 1 near E extend the other foot beyond the center to 2 there is the center from which you shall draw the first Meridian A 1 B and also turning about your Compass you shall make a marke in the extension beyond D at 1 where the other Semicircle of this Meridian would cross the Equator So for the next Meridian in the line C E marked with 2 your center is beyond E at 4 and after you have drawn his arch A 2 B marke with your compass his other crossing at 2 beyond D and so with one labour you shall both draw the 45 crookeder Meridians and also make the out lying division of the line E D prolonged of which division every second or even number will be a center to some of the straighter Meridians This is a very good and easie way and this way Mr Blagrave alwaies used 5 Or lastly You may frame a decimal Scale of 1000 or 10000 parts equal to the Semidiameter of the Mater by which Scale with the help of the Cannon of Triangles you may presently find the length of any S ne Tangent or Secant you shall desire Now look what inclination any Meridian hath to your fundamental Circle that is what angle they make between them the Secant of that inclination is the Semidiameter of that Meridian and the Tangent of the same inclination is the distance of his center from E the center of the Mater For example the Meridian A 2 B his inclination is 20 deg for the angle C A 2 and likewise the arch C 2 which measures it is 20 degr the S cant of 20 degr is 10641. by the Cannon of Triangles which every Mathematician ought to have at hand Take with your Compasses from your decimal Scale 10641. and setting one foot in A with the other foot cross the Semidiameter E D in that crossing is the center or take with your Compasses 3639. the Tangent of 20 degr and set it from E toward D and it shall give you the same center at 4. For A E being Radius E 4 is the Tangent and A 4 the Secant of 20 deg by the structure And if you like to work this way it will help you much to have a short Cannon of Tangents and Secants of whole degr of the Quadrant gathered into one page which Cannon for your ease is here annexed A Table of Tangents and Secants to every degree of the Quadrant Degr Tan. Secan 1 00174 010001 2 00349 010006 3 00524 010013 4 00699 010024 5 00374 010038 6 01051 010055 7 01227 010075 8 01405 010098 9 01583 010124 10 01763 010154 11 01943 010187 12 02125 01022â— 13 02308 010263 14 02493 010306 15 02679
♋ at Meridies and the rest in like order Then draw another Scale without this upon the Ring consisting of two spaces In the inner space shall be the Dayes of the Year in the outer space which must be a little larger shall be the Names of the moneths in their order And to divide this Scale rightly you shall do thus Go to some Eshemeris for the Leap Year that next comes viz. 1660 or rather for some Leap Year about 20 Years hence that your Scale may serve without any Prosthapheresis for 40 Years to come without sensible error and beginning your year with March look where the Sun was on the 29 of February at Noon which you shall find to be ♓ 20 degr 47 min for the Year 1660. Therefore laying the Label or a Ruler from the center to ♓ 20 degr 47 min. in the inner Scale strike a long stroke through your outward Scale and from thence begin your Year writing from thence toward the right hand March 1660. Then lay the Label to ♓ 21 degr 47 min which is the Suns place on the first day of March at noon the same year and where it cuts the outer Scale mark the first day of March and so the rest in order And to the first day of every moneth you shall set his proper Letter which belongs to him in the Calender as to the first of March you shall set D and to the first of April G c. and when you have done December you must take the Suns place for January and February out of the next years Ephemeris viz. 1661 and note that the space for the last day of the year Febr. 28 will fall out to be less by a fourth part then the rest by reason that the Sun wants almost 6 hours to finish his Circle which he finishes in dayes 365 5 hours 48 minutes And for this cause these Scales will serve you to find the Suns place at noon for any day in a like year that is every fourth year accounted hence either backwards or forwards which year shall evermore be accounted to begin from Febr. 29. and may be accounted the first year after Leap year because the Intercalation was February 25 next before Then for the year next following viz 1661. beginning March 1 and being second from the Bissextile or Leap year these Scales shall give you the place of the Sun at six hours after noon and the third year from Bissextile 1662 beginning as before March 1 these Scales shall give you the Suns place 12 hours after noon or the midnight following And the fourth year 1663 being Bissextile these Scales shew the place of the Sun at 18 hours after noon the next year 1664 being the first after Bissextile and beginning as aforesaid March 1 is the very same year for which your Scale is made and gon for that year your Scale shewes the Suns place at noon again But because the Julian years are bigger then the true Solar years by almost 12 mi. of time that is near a quarter of an hour in which time the Sun moves 27 sec 13 thirds 37 fourths therefore when you have found the Suns place by the former Scale any year after 1660 look how many years are passed since 1660 and so many times you must add 27 sec 13 thirds 37 fourths that is almost half a minute to the Suns place found and for years past â—ou must subtract as much that you may find the Suns place exactly This Prosthapheresis in 2 or 3 years is scarce considerable in an Instrument but in 10 years there will be 4 minutes 32. seconds and in 20 years 9. minutes 5. seconds to be added after 1660. and as much to be subtracted in like number of years preceding the year 1660. to which this Scale is supposed to be framed This Ephemeris or Calender M. Blagrave would have on the back-side where hee would also have a Ruler with Sights to take the Altitude of the Sun or Stars But this will be found incommodious in many respects both in the framing and in the using and therefore I advise that nothing be set on the back-side but the Tables of the Prime Epact and Cycle of the Sun thereby to find the age of the Moon her Conjunctions and Oppositions and the moveable Feasts for ever Of which see Chap. 11. CHAP. X. Of the Label and Sights THe Label is a Ruler slit in the midest and the half of it cut away to the Head where it is pinned to turn upon the Center and reaching to the outside of the Limb. The Fiducial edge thereof which pointeth upon the Center must be graduated like to the semidiameters of the Mater and Reet into 90 degrees to be numbred either inward or outward The fashion of it may be understood by the figure without more words To this Label you may fit Sights either fixed or moveable as you like best for observing Altitudes and Azimuths but for taking Azimuths you had need have one tall Sight at least half as long as the Label and then it had need be moveable to take off at pleasure For taking the Altitude of the Sun I have made a pair of moveable Sights to slip up and down upon the edge of the Planisphear having on the backside springing plates of brass to pintch them close and make them stick where you set them These are commonly to be set at C and D the ends of the Equinoctial line At A in the Limb and in the Circle next unto the inner edge which boundeth the strokes of the severall degrees you shall drill a small hole through which you may put a thred to hang a plummet on The Sun then shining through the Sights placed at C and D the plumb-line shall shew his Altitude in the semicircle B C A you beginning to number from B and observing where the plumb-line crosseth the Circle in which the hole for hanging the plumb-line was made And here you must remember that because the plumb-line falleth not from the Center of the Planisphear but from a point in the circumference about A therefore the space of two degrees must be taken but one degree so that if the Plumb-line fall 20 degr below B toward C the Suns Altitude is 10 degrees as you may see demonstrated Euclid 3.20 and Pitisc Trigonem 1.53 And thus you may observe the Suns Altitude neer the Horizon as exactly as by a Quadrant whose semediameter were equal to the diameter of your Planisphear But if the Altitude exceed 30 or 40. degrees then will the Plumb-line cut the limb too slope and have too much play to your trouble For remedy whereof you shall remove the Sight at D towards A some degrees as for example 60 degrees by which means you shall abate the Suns Altitude 30 degrees which 30 degrees must be added to the Altitude observed as for example the Sights are placed one at C the other 60 degrees above D toward A and the
the arch of the Ecliptick from the Ascendant to the Midheaven and his match taken so many degrees on the other side the Center gives the other arch of the Ecliptick from the Midheaven to the Descendant The rest of the Meridians and the Parallels are in this Mode of no use The Almicanters and Azimuth of the Reet here shew you the Altitude and Azimuth of every degree of the Ecliptick at one view CHAP. II Of the Equinoctial Projection shewing the Northern or Southern Hemisphears THe Equinoctial Projection representeth the Northern or Southern Hemisphear projected upon the plain of the Equator Here the Limb or outmost Circles of the Mater and Reet are Equator The eye-point is the North or South Pole which you will by turns Which Poles are here expressed on the Center of the Equator because the Sphear is pictured on a plain or flat The Axtree line of the Mater A B is Colurus Equinoctiorum the Diameter C D crossing him is Colurus Solstitiorum But contrary on the Reet the Axletree is Colurus Solstitiorum and the Finiter Colurus Equinoctiorum The Colurus Solstitiorum on the Mater is also the Meridian of your place and therefore is marked with Septentrio and Meridies and the ends of the Axtree with Oriens and Occidens The rest of the Meridians being all straight lines meeting in the Poles or Center are casily supplyed by the Label and so may the Parallels also being Concentrick with the Equator For if you lay the Label on the 15. degree in the Limb from Meridies toward Occidens the fiduciall edge of the Label there designeth the 15 Meridian or the One a clock line the North Quadrant of the said Meridian proceeding from the Center now the North Pole outward to the Limb or Equinoctial and the South Quadrant returning in the same line from the Equinoctial to the Center now the South Pole and if you remove the Label 180 degrees from One a clock of the day there it shall designe One a clock at night made by the other Semicircle of the same Meridian which joyneth with his match in the Center without any angle that is into the same straight line and so of the rest And for the Parallels if you set the point of your Compass or a needles point in the 23. degree ½ of the Label and turn about the Label with the point it shall describe a Circle which will serve for both the Tropicks and so may you make any other of the parallels I do not advise you to draw the Meridians and Parallels in this form least you cumber your Instrument but I shew you how you may represent any of them in a moment when ocasion requireth The Meridians of the Mater that were so called in the Meridional Projection are here turned into the severall Horizons of the World And the Parallels here serve only to graduate those Horizons Out of these Horizons choose your own Horizon and distinguish him if you will that you may readily find him when you shall looke for him Your Horizon is thus inquired Because the Elevation of the Pole at Northampton is 52. degrees 15. minutes therefore from the Center now North Pole number in the Meridian line Northward 52. degrees 15. minutes and there cutteth the the North Semicircle of our Horizon or there you may Imagine him between the 52 and 53 Horizons and the Southern Semicircle thereof lies 52 degrees 15 minutes on the other side the Center towards Meridies This may seeme strange that the North and South points of the Horizon which in the Sphear are unequally distant from the North Pole viz. the one but 51. degrees 15. minutes and the other 127. degrees 45. minutes the supplement thereof should be equally distant in this Projection But the reason is because the Center is both North and South Pole here at pleasure and the Northern and Southern Hemisphears are both here represented by turns Carry this in your head and then lay the Eabel upon the South part of the Meridian and number thereon from the Center now North Pole outward to the Equator at the Limb 90. degrees thence number backward toward the Center now the South Pole the Elevation of the Equator which is always complement of the Elevation of the Pole and is here 37. degrees 45 minutes there is the Southern point of the Horizon and is distant from the Center now South Pole onely 52 degrees 15 minutes but from the Center being North Pole 127. degrees 45. minutes and from the Northern point of the Horizon before found just 180. degrees as it is in the Sphear Having found the North arch of your Horizon 52. degr 15. min. behind the Center count as many degrees and minutes forward in the Meridian before the Center toward Meridies and the arch crossing there shall be his match to make up the whole Circle and so may you find your whole Horizon upon the Mater whatsoever your Latitude be Here you must remember that Stars which have Northern Declination rise and set upon the Northern arch of the Horizon and those which have Southern declination upon the Southern arch Remember also that many Stars between the Tropicks which have Northern Latitude have nevertheless Southern Declination and contrary many which have Southern Latitude have Northern Declination The lineaments of the Reet serving you in this Projection are onely the Ecliptick and the fixed Stars the Almicanters and Azimuths here are of no use The Meridians and Parallels are supplyed by the Label for the Reet as well as for the Mater And whereas the Ecliptick here seemes to be irregular seeing the Solstitial points of Cancer and Caprcorn are not distant 180 degrees as they should be you must imagine that the Southern arch of the Ecliptick is Projected by the eye placed in the North Pole and for the Northern arch the eyes place in the South Pole and the Center serveth for both the Poles alike as hath been shewed number therefore as you were taught for the Horizon in this Projection For the reason of the draught of the Horizon and of the Ecliptick in this Projection is the same CHAP. III. Of the Nonagesimal Projection shewing the Eastern and Western parts of the Sphear being divided by the Azimuth of the Nonagesimus gradus NUmber in the Limb from the Equinoctial line toward the Pole the Altitude of the Nonagesimus gradus which is the highest degree of the Ecliptick and thereto set the Finitor turning the Almicanters either to the North or to the South as your work proposed shall require Now is the Finiter Ecliptick his point at the Limb-is Nonagesimus gradus The Center of the Planisphear is Ascendant and Descendant the East and west points of the Horizon are here distant from the Center as much as the Amplitude of the Ascendant cometh to to be counted from the Center upon the Eqinoctial line of the Mater which here stands for Horizon the Meridians and Parallels of the Mater are here
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
30 degrees I say here begins â™’ and going on 26. 22. minutes further I say thus far is the Star gone in Longitude Now here cuts the Finiter by this account the Azimuth 35 â…“ from the Limb in this Azimuth I number the Stars Latitude by the Almicantars 22.07 ½ and at the end of that number in the said Azimuth I prick the Stars place And here I see the 8th Parallel of North Declination upon the Mater cutteth him and the Meridian 51 â…“ from the Limb shewing the excess of his Right Ascension above 270. which I kept before Therefore I conclude the Right Ascension of Eniph Alpharats Anno Dom. 1600 was 321. 20 minutes and his Declination 8. deg North. Past this on fol. 102 so as it may ly open while that Chapter is Reading Another way to place the Stars in the Mater by their Declination and Horary-distance from the Meridian See hereafter Chapter 52. CHAP. XXXVI The Latitude and Declination of a Star given to find his Longitude and Right Ascension SEt your Planisphear in the second Mode of the Meridional Projection turning the Zenith Northward or Southward as the Stars Latitude hapneth to be North or South Then look where the Parallel of the Stars Latitude in the Reet cutteth the Parallel of the Stars Declination on the Mater the Azimuth cutting that intersection sheweth the Longitude of the Star and the Meridian there cutting sheweth his Right â—scension Example The Declination of Spica â™ Anno Dom. 1670. will be 9 ½ South the Latitude was always 1. 59 minutes South Now where the second Almicantar cutteth the 9 ½ Parallel of South Declination there passeth the 19th ¼ Azimuth from the Axis toward my left hand shewing Spica's Longitude ♎ 19 ¼ and the 17th Meridian from the Axis to which I add a Semi-circle because ♎ 0. is at the Center and I make 197 degrees the Right Ascension of Spica for 1670. CHAP. XXXVII The Longitude and Latitude of two Stars given to find their Distance MAke one of the Poles of the Mater to be Pole of the Ecliptique for this turn and set the Star which hath most Latitude at his distance in the Limb and turn the Zenith to him count thence by the Meridians the difference of Longitude till you come to the other side of your Triangle and in that side number either the Latitude from the Equator or his complement from the Pole at the end of this number is the other Star and the Azimuth passing from him to the Zenith shewes the distance This is done by the second Probleme of Obliquangled Triangles Book 3.15 Example In Tycho'es Tables for 1600. Aldebarans Longitude is ♊ 4.12 ½ Latitude 5. 31. min. A. Sirius Longitude ♋ 8. 35 ½ Latitude 39. 30 ½ A. Difference of Longitude 34. 23. I number therefore 39. 30 minutes ½ the Latitude of Sirius from the Equator in the Limb or the Complement thereof from the Pole all is one there I set the Zenith to stand for Sirius then because Aldebaran is distant from Sirius in Longitude 34. 23. minutes I take the 34 ½ Meridian from the Zenith and where the 5 ½ Parallel cutteth him there say I is Aldebaran and C of my Triangle and the Azimuth passing thence to the Zenith measureth the distance of the Stars 46 degrees almost CHAP. XXXVIII The Declination and Right Ascension of any two Stars given to find their distance DO here with the Right Ascension and Declination as you should do with the Longitude and Latitude by the former Chapter for the case is like and requireth the same manner of working CHAP. XXXIX The Declination of a Star or Planet and his distance from a known Star given to find his Right Ascension BEcause this Case is the converse of the precedent and soluble by the first Probleme of Obliquangled Triangles Book 3. 14. an Example or two shall suffice Past this on fol. 105 so as it may ly open while that Chapter is Reading A B the distance of Mandibulae from the Pole 87. 20 minutes I set between the Pole and Nadir in the Limb because B C will reach beyond the Finitor For A C the distance of the Stars I seek the 13th Parallel from the Pole And For B C I seek the 94. 50 minutes Almicantar counted from the Nadir that is the 5th almost above the Finitor and where the said Parallel and Almicantar cross there is Cor Caeti and C of my Triangle through it there cutteth the Azimuth 10 â…” shewing the Difference of the Right Ascension of the Stars which difference I subtract out of the Right Ascension of Mandibula because he was further East and there remaineth the Right Ascension of Cor Caeti 30. 16 minutes or rather 13 minutes And I have here also numbred by the Meridians the angle A at Mandibula 120 degr though un-required Another Example January 7. 1656 7 I observed by my Brass Quadrant of 12 inches in Radius the Meridian Altitude of Jupiter 56. 20 minutes out of which subtracting the height of the Equator here at Ecton 37. 45 minutes I found his Declination 18. 35 minutes North his distance then from Lucida Pleiadum I observed by my Cross-staff 5. 12 minutes and from Aldebaran 10.07 minutes The Complement of Decli of Lucida Pleiadum is 67.00 mi. The Complement of ♃ his Declination was observed 71. 25. And these two Complements with the distance of ♃ and Lucida Pleiadum 5. 12 minutes make a Triangle soluble by the first Probleme of Obliquangled Triangles whereby you may find the angle of the difference of Right Ascension of Lucida Pleiadum and ♃ is 2. 56 minutes which added to the Right Ascension of Lucida Pleiadum because ♃ was East-ward maketh 54. 44 minutes the Right Ascension of Jupiter CHAP. XL. The Latitude of a Star or Planet and his distance from a known Star given to find his Longitude DO here with the Longitude and Latitude as you were taught to do with the Right Ascension and Declination in the former Chapter CHAP. XLI To find the distance of two Stars by their Altitudes and their difference of Azimuth observed at the same time THe Complements of the Altitudes are the distances of the Stars from the Zenith Set one of the Stars at the Pole and set the Zenith as much from him in the Limb as the Complement of his Altitude comes to then considering what difference of Azimuth the Stars had take the Azimuth of like distance from the Limb beginning from that side of the Limb where the Pole aforesaid is and in that Azimuth reckon from the Finitor the Altitude of the other Star or the Complement of his Altitude from the Zenith all is one at the end thereof is C and the other Star and the Meridian that passeth from him to the Pole shewes the distance of the Stars This case is so like that of Chapter 37. that he who knowes one may know the other also CHAP. XLII To find the Angles of
follow this Azimuth to the Finitor there is Nonagesimus gradus and the Altitude thereof 42 degrees counted from the Limb here Horizon the Azimuth thereof lies in the Limb between the Finitor and the Meridian 36 ⅓ as before equal to the Amplitude of the Ascendent I number also from ♓ 25 ½ in the Meridian 23. 41 minutes to the left hand still and there I have ♈ 19. 11 minutes the Suns place which cuts on the Label 41 ⅔ for the Altitude of the Sun there and the Label at the same time cutteth in the Limb about 29. from South East-ward for the Azimuth of the Sun and after the same manner you have before you the Altitude and Azimuth of every other degree of the Ecliptique for the time proposed CHAP. LVIII To do the same by the Nonagesimal Projection if the Altitude of Nonagesimus gradus be first given instead of the Altitude of Culmen Caeli SEt your Planisphear in the Nonagesimal Projection by Book 2.3 that is make the Limb now to represent the Circle of Longitude or Azimuth for it is both which cutteth the Nonagesimus gradus and make the Equinoctial line here to be Horizon and from the Equinectial line number in the Limb the Altitude of Nonagesimus gradus and thereto set the Finitor so shall the Finitor be Ecliptique the Nonagesimus gradus at the Limb the Ascendent and Descendent at the Center and because the Equinoctial line is Horizon in this Projection therefore the Meridians become Azimuths and the Parallels Almicantars shewing the Altitude and Azimuth of every degree of the Ecliptique if you reckon as you ought in this manner Reckon in the Equinoctial line here Horizon from the Center the Amplitude of the Ascendent to the right Hand if it be a North Signe and contrarily if it be a South Signe Where this Amplitude ends is the East point from whence you shall reckon all your Azimuths Count thence to the Limb and back again if need be in the said Equinoctial line till you have made 90 degrees there is your Meridian as far distant from the Limb as the East point was from the Ascendent Follow this Meridian to the Finitor and there he shewes you Culmen Caeli and the Parallel there cutting shewes the Altitude thereof Now may you find every degree of the Ecliptique above the Horizon if you know but what Ascends or Descends or Culminates and of every such degree the Parallels shew you the Altitude and the Meridians shew his Azimuth if you begin your numbring from the East or South Azimuth Example When ♋ 24 degrees was Ascending as in the Example before used as by consequence ♈ 24. in Nonagisimo gradu ♂ was in ♉ 4. 45 minutes and had but 3. or 4. minutes South Latitude I would know ♂ his Altitude and Azimuth setting go the Finitor above the Equinoctial line 42 degrees which is the Altitude of Nonagesimus gradus I say because the Nonagesimus gradus at the end of the Finitor in the Limb is ♈ 24. therefore I must count back 10. 45 minutes toward the Ascendent for Mars and there the Parallel 41 degrees with 10 minutes cutteth the Finitor for the Altitude of ♂ and the 14th Meridian East-ward from the Limb gives me his Azimuth which if I begin to reckon from the East point falleth out to be almost the 40th Azimuth from the East Mars his Latitude here is not regarded CHAP. LIX The Nonagesimus gradus and his Altitude and Azimuth given as in the former Chapter How in the same Projection to get the Altitude and Azimuth of any Planet or Star by his Longitude and Latitude YOur Palnisphear set as in the former Chapter you shall number the Longitude of the Star upon the Finitor here Ecliptique beginning at the Descendent or Nonagesimus gardus and in the Azimuth serving his Longitude count his Latitude by the Almicantars at the end of which account is the Stars place for this time The Parallel cutting there shewes his Altitude and the Meridian cutting there shewes his Azimuth if you count from the East point as you were taught in the former Chapter Example Lucida Pleiadum was in Longitude ♉ 25. 10 minutes Latitude 4 degrees 00 minutes North. Therefore from the Nonagesimus gradus ♈ 24. I number in the Finitor toward the Ascendent 31. 10 minutes and there is the Longitude of Lucida Pleiaedum in the Azimuth that cuts here I go up Northward 4 degrees and there I make a prick for Lucida Pleiadum Now the Parallel 38 ½ shewes me his Altitude and the 48th ½ Meridian from the Center shewes me that Lucida Pleiadum is gone 48 ½ in Azimuth from the Ascendent but from the East point onely 12 degrees 10 minutes CHAP. LX. The Altitude and Azimuth of any Star taken and either the Ascendent Nonagesimus gradus or Culmen Caeli known How by the same Nonagesimal Projection to find the Stars Longitude and Latitude IF you know either the Ascendent Nonagesimus gradus or Culmen Caeli you have enough to put your Planisphear in the Nonagesimal Projection by the former Chapters And your Planisphear so set you shall seek out the Meridian which standeth for the Azimuth in which you observe the Star and therein number from the Equinoctial line the Altitude observed the Azimuth and Almicantar cutting there shew the Longitude and Latitude of the Star inquired If the Azimuths reach not the place of the Star turn the Reet half round and let the Zenith and Nadir points change places and your turn is served Example Febr. 13 1657 8. I observed somewhat near that ♃ was gone West-ward from the Meridian in Azimuth 14 degrees and that his Altitude was 61 degrees Sirius was then in the Meridian by which I have the Ascendent Culmen and Nonagesimus gradus any or all of them given For when in the Equinoctial Projection I bring Sirius to the Meridian line it is all one as if I had set the Suns place to the hour of the Night by Chapter 46. and I see there Culminates with Sirius ♋ 7. 10 minutes whose Meridian Altitude by the 46. is 61. 5 minutes and I see ♎ 5 ½ ascending in my Horizon and ♈ 5 ½ descending therefore ♋ 5 ½ is Nonagesimus gradus which is 90 degrees distant both from the Ascendent and Descendent his Altitude by Chapter 55. 61. 10 minutes almost Therefore I set the Finitor 61. 10 minutes above Meridies as Chapter 58. and in the Finitor at the Limb I count ♋ 5 ½ Nonagesimus gradus thence I go inwards in the Finitor 1. 40 minutes where I come to ♋ 7. 10. the degree of Culmination this degree is cut by the 4th Meridian from the Limb whereby I learn that this 4th Meridian will be the Meridian of my place and that the Amplitude of the Nonagesimus gradus and likewise of the Ascendent is 4 degrees Now to place ♃ in the Mater I count his Azimuth first beginning from the Meridian of my
London from Jerusalem CHAP. LXX The Latitude and distance of two Places given to find the difference of Longitude THe Triangle will stand as in the former Chapter there by two sides and the angle comprehended you sought the third side by Probleme 2. Obliquangled Triangles here by three sides given you seek an angle by Probleme 1 Obliquangled Triangles Make the Pole Pole and set the Zenith to the Latitude of one of the places as you did London Chapter 69. 38. 28 minutes from the Pole then number the Complement of the Latitude of the other place from the Pole by the Parallels and the distance of the two places from the Zenith by the Almicantars and where the last Parallel and last Almicantar meet is C of your Triangle see Book 3. 14. Now count how many Meridians there be between C and the Limb so many degrees is the angle at the Pole sought for the difference of Longitude Example Having the distance of London from the Pole 38. 28 minutes and of Jerusalem from the Pole 58. 5 minutes and the distance of London from Jerusalem 2300. common English Miles of which 60. make a degree I set the Zenith for London 38. 28 minutes from the Pole in the Limb then because Jerusalem is distant from the Pole 58. 5 minutes I go to the 58th Parallel from the Pole and lay one finger or the point of a bodkin on him and because London is distant from Jerusalem 38 degrees 20 minutes I count from the Zenith to the Almicantar 38. 20 minutes now where this Almicantar crosseth the Parallel last found there is C of the Triangle and the place of Jerusalem and you may see that you must cross 46. Meridians before you can go thence to the Zenith in the Limb which sheweth that the angle at the Pole for the difference of Longitude is 46. CHAP. LXXI To find what degree of the Ecliptique Culminates in another Country at any time proposed if the difference of Longitude be known IN the Equinoctial Projection Bring the Suns place to the hour proposed by help of the Label and in the Noon-line you see presently what degree Culminates in your Country as Chapter 46 is shewed Now to know this for another Town set the Label so many degrees from the Noon-line as the difference of Longitude requires and that Eastward if the place proposed be East or Westward if it bear West and so the Label shall cut the degree of Culmination for the place proposed Example If it be demanded what degree is Culminating at Jerusalem March 10. at 10. a clock before noon I will set the Suns place ♈ 0. to the hour and I see upon the Noon line which is our Meridian there Culminates ♒ 28 almost Now for the Meridian of Jerusalem I must lay the Label 46 degrees Eastward that is from Meridies towards Oriens and look what Star or degree of the Ecliptique is then cut by the Label that is then Culminating in the Meridian at Jerusalem as here I find ♈ 17 ½ for in this Projection the Label lay him where you will is a Meridian CHAP. LXXII To find what a Clock it is in another Country by knowing the hour at Home and the difference of Longitude THis is done easily enough without an Instrument for if you turn the difference of Longitude into hours and minutes and add the same to your hours for any place which lies Eastward or subtract the same for any place which lies Westward you shall make the hour of the place Example The difference of Longitude between London and Jerusalem is 46. or being converted into time 3 hours 4 minutes therefore adding this to the time at London I say when it is noon at London it is 4 minutes past 3 a clock after noon at Jerusalem and when it is 2 a clock at London it is 5. and 4 minutes at Jerusalem If you will do it by the Planisphear you shall do it in the Equinoctial Projection thus Whereas the Limb of your Rect is graduated into 360 degrees if you distinguish the hours also at every 15th degree beginning at the Zenith which shall be 12 and numbring thence in the Limb of your Reet to your right hand or Westward 1 2 3 c. then shall you need to do no more but set the Zenith to the difference of Longitude East or West from your Meridian as the strange place happeneth to be situate for then the Label laid to the hour of your Country in the Limb of the Mater shall shew the hour of the other Country in the Limb of the Reet And so the Zenith being laid to 60 degrees Westward which is the Meridian of the Isle of Barbados the Label laid to Meridies shall cut in the Limb of the Reet 8 of the clock before noon which sheweth me that when it is noon with us it is at Barbados but eight in the morning The end of the Fourth Book CHAP. LXXIII The Longitude and Latitude of one Place known and the Rumb and distance of a second Place to find both the Longitude and Latitude of the second Place SEt the Zenith to the Latitude of the first Place then seek the Azimuth which serveth for the Rumb of the second Place and in that Azimuth count his distance from the Zenith where this distance ends there is the second Place whole Latitude is shewn you by the Parallel which cutteth him and the Meridian cutting there also shews his Longitude Example Let Z be London and because Jerusalem beareth from London almost S b E or 77 ½ from South Eastward therefore I choose the Azimuth 77 ½ Z K. therein I number Jerusalems distance from London Z I 2300. miles or minutes that is 38. 20. minutes Now in the Triangle Z P I I may find P I the complement of Jerusalems Latitude 58. 05. minutes and Z P I the difference of Longitude 46 which must be added to the Longitude of London to make the Longitude of Jerusalem CHAP. LXXIV The Latitudes and Distance of two Places given to find the Rumb and the difference of Longitude COunt in the Meridian from P the Pole the complement of the Latitude of the first place and thereto set Z the Zenith Count also from P the complement of the Latitude of the second place and lay your finger on the Parallel at which your number ends Count also from Z the distance of the places in Degrees and Minutes and note the Almicantar at which this number ends where this Almicantar crosseth the aforesaid Parallel there is C of your Triangle but here marked I Look what Azimuth cutteth here it sheweth the Rumb and the Meridian that cutteth here if you count his distance from the Limb shews the difference of the Longitude of the places This is so plain from Chapter 69 70 and 73 that it needeth no example The same Scheam serveth these 4 Chapters The end of the Fourth Book The Fifth Book Shewing the way to resolve all GNOMONICAL
of the Place East or West The Reclination is the distance of his Poles from the Zenith and Nadir of your Place Inclination is the nearest distance of the Poles of the plain from your Horizon And whatsoever the Reclination of the upper face of a plain is the Inclination of the lower face is the Complement thereof CHAP. IX How to make the Vertical Dyal IN the Meridional Projection the Finitor being set to the Latitude of your Place you shall see the Limb which is your Meridian and the Axtree-line which is the sixt hour-circle dividing the Finitor into 4 Quadrants and the rest of the Meridians dividing every Quadrant alike Mark now at what degree numbred from the Limb every hour-circle that is every 15th Meridian being a ragged or blacker line cutteth the Finitor at the same distance shall the same hour-circle cut the Limb of your Dyal in the plain Lastly for the Gnomon set your Compasses to the Chord of the arch of the Poles Elevation in the Limb that is measure in the Limb from the Pole to the Finitor and setting that distance in the Circle of your Dyal from 12. either way make a point through which if you draw a deleble line from the Center you have between this line and the line of 12. the angle of your Gnomon by which when you have shaped him you must set him upright over the 12 a clock line with the point of the said angle at the Center and all is done CHAP. X. How to make the South and North Horizontal Dyal THis is usually called the Erect Direct Dyal and belongs to an upright Wall looking full North or South and the plain of it lies in the East Azimuth which on the Planisphear in the Meridional Projection is represented by the Axis of the Reet The Finitor set to the Latitude as in the former Chapter mark where the hour Circles cut the Axis of the Reet which is the proper Horizon of this Dyal you shall find the first cutteth 9. 20 minutes from the Meridian the second 19. 30 minutes the third 31. 30 minutes the fourth 46. 45 minutes the fifth 66. 24 minutes the sixt 90. And you shall see the North Pole depressed under this plain as much as is the Complement of our Latitude and the South Pole as much Elevated above it 1. Wherefore for the South Dyal draw an Horizontal line about the top of your Dyal plain which shall be the hour of Sixes from the midst whereof let fall a Perpendicular which shall be both the Vertical and the Meridian both of the Place and of the Plain wherein the Gnomon must stand Elevated 37. 45. minutes or the Complement of your Latitude toward the South Pole Another way Because the Almicantars may oft obscure the Intersections of the Hour Circles with the Axis you may avoid that inconvemence if you reduce this Dyal to a Vertical Dyal For the South Horizontal Dyal being the very Vertical Dyal of those People that live 90 degrees Southward from us that is in South Latitude 37. 45 minutes if you set the Finitor to the Latitude 37. 45 minutes you shall see the sections of the Hour Circles with the Finitor more â—pparently and thereby make your Dyal 2. For the North face Imagine you had for you Gnomon a wyre thrust aslope through the Center of the plain from the South side Northward and you will presently conceive that in the North Dyal the Horizontal or 6 a clock line will be lowest and that the Gnomon will turn upwards toward the North Pole as much as he turned downwards on the other side and that all the hours save 4 5 and 6. in the Morning and 6 7 and 8. at Night may be left out in our Latitude because the Sun shineth no longer upon it and those hour-hour-distances you may find and set off from the 12 a clock line or from the 6 a clock line as you did the hours of like distance in the South face Another general and pleasant way to delineate the opposite face of any Dyal see hereafter in the end of the 12th Chapter CHAP. XI How to Observe the Declination of any Declining Plain A B is a Wall or plain declining East by the arch S p to which E B or W A are equal for so much as the Wall bendeth from the East Azimuth so much doth his Pole at p decline or bend from the Meridian 1. Now to find how much any plain declineth and so in what Azimuth he lies one good way is this when the Sun begins to inlighten the Wall or when he leaves it then is the Sun in the same Azimuth with the Wall take at that instant his Altitude and thereby get his Azimuth according to Book 4.27 and that is the Azimuth of the Wall 2. Another way First draw upon the Wall an Horizontal line by Chapter 3. then your Planisphear being fastned to a Square board as in Chapter 4. set one side of the board to that Horizontal line or Parallel to it and fix there your board and Planisphear level by the help of a Square set under him like a bracket the place your Label and Sights in one of the Diameters of your Planisphear and mark when the Sun comes into the line of the Label casting the shadow of one Sight upon the other if the Label be then in the Diameter which is Parallel to the Wall then is the Sun at that time in the Azimuth of the Wall if the Label be in the other Diameter which is Perpendicular to the Wall then the Sun coming to it is in the Azimuth of the Pole of the plain Now having the hour or the Altitude of the Sun get his Azimuth by 4.27 the same is the Azimuth of the Wall or plain if the Label were Parallel to the Wall or the same is the Azimuth of the Pole of the plain that is the very Declination if the Label stood Perpendicular to the Wall 3. Another way If you have not time to watch till the Sun come into the Azimuth of the Wall or the Vertical of it which cutteth the Pole thereof then get the Suns Azimuth by the said Book 4.27 when you can and at the same time Observe by your Label the Suns Horizontal distance from the Pole of the plain and by comparing these together you may easily gather the Declination of the Wall as in Example I observed the Sun to be gone West from the Pole of the plain 70 degrees and by the Altitude of the Sun then taken I found his Azimuth 60 degrees here I reason thus The Sun is gone from the Pole and Vertical of the Wall 70 degrees and from the Meridian but 60 degrees therefore the Meridian lies between the Pole of the plain and the Sun and because ☉ p is 70. and ☉ S 60. therefore S p the Declination of the plain is 10 degrees the difference of 70. and 60 and the Declination is East for the Sun is neerer to the
and Mars in twain Sets forward and comes round again Then in one Year the Sun displaies Three hundred sixty and five dayes And near a quarter which in four Encompassings makes one day more Between the Sun and us there fly Fair Venus and swift Mercury These alwayes near the Sun we find Not far before nor far behind The Moon 's the lowest who in seven And twenty dayes goes round the Heaven And above two dayes more do run Before she overtakes the Sun So twenty nine and an half in all Do make a Moneth Synodical These Planets make their course to th' East Though they be faster hurled West And six degrees the rest may stray Beside the Suns Ecliptique way The Circles of the Sphear SIx greater Circles mark you shall Which equally divide this Ball. Just in the midst between the Poles From East to West th' Equator rolles Th' Ecliptique cuts him and doth slide Scarce twenty four degrees aside Horizon even with the ground From Stars below our sight doth bound Meridian upright doth rise Parting the East and Western Skies Two Colures through the Poles do run Quarrring the Circle of the Sun One where the Spring and Fall begin Th' other where longest dayes come in Four lesser Circles mark them well Are to th' Equator Parallel Two Tropiques bound the Suns high way Shewing the Long'st and Shortest day The Arctique Circle curs the Beares Th' Antarctique opposite appeares Meridians half twenty four For Hours and for Degrees ninescore Through both the Poles o th World do pass And th' Equinoctial down right cross And ninescore Parallels hath that line By which Stars North and South decline Th' Ecliptique hath his Longitudes And Parallels of Latitudes For Stars but in Geography The Towns beside th' Equator lie Over our Head and under Feet The ninescore Azimuths do meet And here as many Parallels Of Altitude Horizon tells Longitudes and Meridians all And Azimuths great Circles call But all their Parallels in Heaven Being lesser cut the Globe uneven Degrees three hundred and threescore Hath every Circle and no more When I consider thy Heavens the work of thy Fingers the Moon and the Stars which thou hast ordained What is Man that thou art mindfull of him Or the Son of Man that thou visitest him Ps 8. Errata Some Faults have been committed between the Writer and the Printer the cheif whereof the Reader is desired to amend as followeth pag. and line Faults Amendments 2 3 4. c. to pag. 30. in the Title The first Book of the Fabrique of the Planisphere The first Book Of the Fabrique of the Planisphear 31 and 32. in the Title The second Book of the Projections of the Sphear The second Book Of the Projections of the Sphere 1. 13. mossie massie 2. 7. Declination Delineation 3. ant Declination Delineation 4. 16. look up look upon 4. 36. eye beam eye-beame 5. 13. Euclid 4 5. Euclid 4.5 22. required of your Compass over reach required If your Compass reach short 5. 23. if it reach short if it over-reach 6. 39. structures structure 8. secant 67. 25693. 25593.  The 5. last Tangents want a place You must add a Cypher to each of them 9. 16. two so 12. 18. all but all But 13. 07. working it working It 16. 19. foure fewer 17. 18. Alamath Alamach 21. Henerichus Heniochus 17. antop little rain little Waine 18. 8. brow Crowne 18. 30. Praecepe Praesepe 19. 16. Bedalgieure Bedalgieuze 23. Alhaber Alhabor 20. 6. round the inner circle or edge of this Ring it must round The inner circle or edge of this Ring must 20. 14. naile screwes male screwes 17. small screwes female screwes 19. bare beare 22. 30. is made and gon for that year your scale is made And so for that year your scale 24. 9. but one degree but for one degree 25. 7.  put out the marks of Parenthesis 26. 8 year Henr. 3. year of Henr 3. 23. Periodus Periodus 28. alwayes upon alwayes upon 35. thus set thus set 28. 1. and 5. Grostons Grastons 30. 3 second Meridional second or the Meridional 33 6. set for London namely for London 33. â—1 on Elevation no Elevation 34. â—◠〈◊〉 the which the 9. Azimuth Azimuthes 37. 6. the eyes place the eye is placed 41. 3. Center B A Center B A 48. 4. either way either way 22. A C C A 50. 16. Zenith of Zenith of 32. Zenith and B Zenith A and B 53. 12. 12 and 13 number 12th and 13th numbred 56. 8. these sides the sides 20. subâ—endeth A which subâ—endeth A 62. 17. fall falls 63. 7 9. wayes rayes ult of deleatur 64. 10. min. at 70 min. and at 70 11. between 8 degr 34. min. between 18 and 24 min. 12. Here Refraction is as the Sun Her Refraction is as the Sun 's 65 1. your Meridians your Meridian 66. 30. require enquire 67. 3. Michals Michaels 68. 39. Long long 73. 6 CHAP. II CHAP XI 74. 20. Alrucabe Alrucaba 75. 8. Alrucabe Alrucaba 75. 12. first made first mode 76. 29. prick here prick here 8â— 16 17. by Declin by their Declin 82. 12. her Declin his Declin antep sta Star 86. 16 17 18 19. Pleiades Riseth setteth Pleiades Rise set 86. 30. to be least to be lost 87. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 88. 4. could happen could not happen â— 14. note know 17. Asera Asera 91. â—1 Duet Deut. 99. 21. 23 degrees 23d degree 102. 6 and 30. Eniph Alph. Eniph Alph. 23. 35 â…“ 56 â…“ 105. 8. Stars I Stars I â— Caeti Ceti 19. 120 deg 125 degr 110. pen. by Oblique Problemes by Probl 2 Obliqu 111. 25. in 39 ½ in all 39 ½ 114. 17. grees setting grees Setting 11. Houses also Houses also 31. 49 30 50 51. 118. 6 7. 49 50 50 51. 24. and so and to 119. 1 Astrologers Astrologie 17. futurus futurus 122. 29. no man no men 123. 3. princeps Nero princeps Nero 4. citherae citharae 10. dereliquit Nero dereliquit Nero 12. persuesum persuasum 27. se nore temerè 128. 26. as by and by 29. setting go setting therefore 130. 34. Jupiter in that Meridian Iupiter In that Meridian 139. 6. Christ time Christs time 17. Ticius Tacitus 141. 6. 4 5. 11. 4. 5 11. 145. 13. Suns Dyals Sun Dials 147. 5. or Equinoctial deleatur 19. so the hour lines to the hour-lines 154. in the scheam the letter I is wanting at the lower end of the hour-hour-line of 11.  157. 17. with an extension with any extension 174. 32. precrucem per crucem 176. 11. by the arch by R T the arch 180. 9. Declination plain declining plain 181. 20. pre per 184. 27. the Vertical of my Dial and also deleatur 185. 28. and so and to 188. 9. Tumiture furniture 190. 7. you use you may use 192. in the scheme the prickt line last save one should be put out  193. an t a Vertical plain a Vertical or a South Horizontal