THE USE OF THE General Planisphere, CALLED THE Analemma, In the Resolution of some of the Chief and most useful PROBLEMS OF ASTRONOMY. By Dr. John Twysden. LONDON, Printed by J. Gain, for Walter Hayes, Mathematical-Instrument Maker; and are to be sold at his House at the Cross Daggers in Moorfields, 1685. The EPISTLE TO THE READER. Courteous Reader, I Did not think to have appeared any more in the World in a Subject of this Nature, being now far declined in Years; and in that regard, more fit to have employed the short Remains of my now languishing Taper in such Meditations as might make me ready to appear before that Tribunal, where those who shall be thought Worthy, shall know the truth of these things, not by Discourse and Connexion of Consequences, but by that Intuition which shall proceed from the Author of all Causes and Things, even God himself. But the Desire of Friends hath overruled my own Inclinations, and prevailed with me to present to thy View and Perusal these ensuing Treatises. The first contains the Uses of the Analemma, an Old Instrument, but in my Judgement, a very good one, and by which most Problems in the Solution of Plain or Spherical Triangles, are Resolved with great Facility, and without much cumber or use of Compasses. 'Tis an Instrument easily made, there being only the great Meridian, which is a Circle, the other Meridian's are Ellipses, which by Elliptical Compasses fitly placed, may be as easily and truly drawn as Circles are with a Beam or Bow; the other are all straight Lines, and divided Sinically. I have one newly drawn by me by Hand, upon a Board pasted over with Pasteboard, between nine or ten Inches Diameter, which I have made use of in the Solution of many Astronomical Problems, with as much Accuracy as can be expected from an Instrument of that size. Let not any man believe that I go about here to commend the Use of Instruments any way to stand in Competion with the natural way of Computation by the Canon, but only as Assistants to those, who being engaged in great Computations, may easily commit an Error, which an Instrument well made will soon discover. I shall not describe the manner of making it, that being sufficiently done by many others. Mr. Blagrave in his Mathematical Jewel finds fault, that the Meridian's towards the Noon-Line are very narrow and close together. 'Tis true, they are so, the manner of this Projection of the Sphere requiring it: but to compensate this, the Reader may consider, that the Hour-Spaces farther from the Meridian, enlarge themselves where his grow narrower: nay, where they are broadest, the Parallels in the Mater, and Almicanters in the Rete will sometimes, run in the same Line, so that it will be troublesome to find the Meridian you desire. I would not be thought to speak this in the least manner to lessen the Honour of Mr. Blagrave in his most excellent Invention of applying a Rete to the Mater of Gemm-Friseus his Astrolabe; by which Addition the Triangle lies open to your eye, or the careful Embellishing, I should rather say, new Moulding, or Framing it, by my very Learned Friend, and Skilful Mathematician Mr. John Palmer, Rector of Ecton, and Archdeacon of Northampton. But to show the Difficulty in the making it, where besides the cutting out the Rete, the Circles must exactly agree to one another, or else the Instrument will be of little Value; which to observe, will require a greater Care than you will easily persuade a Workman to take. I have lately had two made for me in Brass, of about eight Inches Diameter, the one an Analemma, the other Blagrave's Jewel, both very well made by my very good Friend Mr. Walter Hays, who doth employ and direct the best, or as good Workmen as any are, to perform what his Age or Infirmity makes him unable to perform with his own Hands. In this he is much to be commended, that you will very seldom find him out of Doors, and to be sought in Alehouses or Taverns, where many a good Workman spends much of his time. The Uses now given thee, were by me applied to this Instrument many years since, when I was at Paris, about the Year 1645, which hath made me in some of the Examples make use of the Elevation of the Pole of that Place. Many others I have applied to other Latitudes, because by various Examples the Precepts will, I think, be made more easy. I have added two other Tractates, the one called, The Planetary Instrument, by which the Places of all the Planets, except the Moon, may be easily found out, and sooner than the Place of any one of them can be found out by the Tables. The other is a Nocturnal; by which the Hour of the Night may be very accurately known by any Star in the Meridian. The first was the Contrivance of my Worthy Friend Mr. Palmer; the other is the Invention of my very good Friend Mr. Foster, sometime Professor of Astronomy in Gresham-Colledge; both whose Memories I must ever honour, as Persons to whose Conduct I must acknowledge to owe a great part of that Knowledge I have attained in these Sciences. Thus, Reader, thou hast my Design and End in the Publication of these Trifles, which is no other than to Confirm those that are Learned in their further Search into them, and to facilitate others in the Practice of these most Useful Sciences; in the Commendation of which I will not now enlarge myself, who shall be glad to hear these Papers please any; if they have other Fortune, let them pass among those Idle Pamphlets, under the Burden of which the Press now daily labours. diagram of a planisphere W: Hayes fe: THE USE OF THE General Planisphere CALLED THE Analemma, In the Resolution of some of the Chief and most Useful PROBLEMS of ASTRONOMY. PROBLEM I. The Sun's Place being given, to find his Declination. COUNT what Parallel from the Equinoctial doth cut the Place of the Sun in the Ecliptic, for that is the Declination required. So in ♉ 00° 00′ you will find the 11° 30′ Parallel cuts the Ecliptic. PROB. II. Contrarily, the Declination being given, to find the Sun's Place. Count 11° 30′ among the Parallels, and that will cut 00° 00′ of ♉. PROB. III. The Place of the Sun being given, to find his Right Ascension. Look what Meridian, reckoning from the Centre, doth pass through the Sun's place, for that in Degrees is the Rig●… Ascension required. So will the Right Ascension for 〈◊〉 beginning of ♉ be found 27° 54′. So likewise f●… 〈◊〉 beginning of ♏; to which adding 180, because 〈…〉 Southern Sign, the Right Ascension will be found 207° ●… PROB. IU. The Elevation of the Pole, and Degree of the Ecliptic being given, to find, 1st. The Ortive Latitude. 2dly. The Rising and Setting of the Sun. 3dly. The Semidiurnal Arch. Place the Finitor to the Latitude, then see what degree thereof doth cross the Sun's Parallel, for the Degree in the Ecliptic given, for that is the Ortive Latitude sought, being reckoned from the Centre, or East point in the Instrument. So in the Latitude of 49 I find the Ortive Latitude about 38 in the Tropic of Cancer. 2dly, The Meridian there passing, shows the Sun's Rising to be very near four in the Morning. And 3dly. The Semidiurnal Arch to be 8 hours, and so the longest day there 16 hours. PROB. V The same things being given, to find the Ascensional Difference. The Horizon, as before, being set to the Latitude, count from the Centre to the Meridian that cuts it in the Sun's Place for that day; that is to say, in the Sun's Parallel for the place of the Eccliptick given. So shall you find the Ascensional Difference for the beginning of ♉ in the foresaid Latitude of 49° to be 13° 32′. PROB. VI To find the Obliqne Ascension. First, find the Right Ascension by the third Problem, and the Ascensional Difference by the preceding one. In Northern Signs subduct the Ascensional Difference out of the Right Ascension, but in Southern Signs add it. So shall the Sum, or Difference give you the Obliqne Ascension required. So in the beginning of ♉ I find the Sun's Right Ascension 27 54, and in the Latitude of 49° 00′, his Ascentional Difference 13° 32′, their Difference is 14° 22′, which is the Obliqne Ascension for the beginning of ♉. So the Obliqne Ascension for the beginning of ♏, will be found 221° 26′ by adding 13° 32′ to 207° 54′, the Right Ascension before found. PROB. VII. To find the Obliqne Ascension of any other Point in the Ecliptic, not reckoning from the Aequinoctial points, by which you may know whether the said Sign doth ascend Right, or Obliquely. By the preceding Problem find the Obliqne Ascension for the beginning and end of ♉, or any other Arch propounded, then subducting the lesser out of the greater, the residue is the Obliqne Ascension required. So in the Latitude of 49° 00′ I desire to know what degrees of the Aequinoctial do in an Obliqne Sphere ascend between the beginning and end of ♉, I find the obliqne Ascension for the beginning of ♉ to be 14° 22′, and of the latter end 32° 45′, their Difference is 18° 23′, the Obliqne Ascension required. Where note, because the remaining Sum is in the Example less than the given Arch, the said Sign doth ascend Obliquely, contrarily if it had been greater. PROB. VIII. To find the Hour of the day, the Altitude of the Sun being first observed. Set the Ruler to the Latitude, and remove the Vertical till it cross the Sun's Parallel in the Altitude observed, the Meridian there passing shows the hour of the day. So in the beginning of ♉ in the Latitude of 52° 00′ I observed the ☉ Altitude 27° 18′, and the 60 th'. Meridian doth cross the Sun's Parallel in that Altitude, therefore it was then eight of the Clock in the Morning. Thus also may you make a Table of Altitudes for every hour of the day, by removing the Vertical from hour to hour, and observing the Degrees of Altitude cut off by the several Meridian's. PROB. IX. To find what Degree of the Ecliptic is in the Meridian at any hour given. Find the Right Ascension by the third Problem, then turn the hour given from the Meridian into degrees and minutes of the Aequinoctial, add unto these the Right Ascension found, then examine what point in the Ecliptic doth answer to those Degrees of Right Ascension, which is done by following the Meridian, which shows the Right Ascension up to the Ecliptic, for that part of the Ecliptic is in the Meridian. So in the beginning of ♉, and at the hour of nine in the Morning, I find the Right Ascension 27° 54′, the hours from the Meridian last past 21° 00′ in degrees 315°, to which I add 27° 54′, it makes 342° 54′ which is the Right Ascension for the 11 th'. Degree of ♓, or near, which was the degree of the Ecliptic then in the Meridian, which will be ♓ 11° 30′. PROB. X. To find the Sun's Azimuth at any Altitude given. Place the Finitor to your Latitude, and remove the Vertical till it intersect the Sun's Parallel in the Altitude given, then applying it over in the Aequinoctial-line, the Meridian which intersects the Vertical in the given Altitude, shows the Sun's Azimuth, or Distance from the South or East, as you shall please to account it. So in the Latitude of 48° 50′, the Sun's Altitude being 31° 40′ in the Tropic of ♋. Then moving the Vertical till it cross the Parallel of ♋ in that Altitude, I there find it, and then moving the Horizon over in the terms of the Aequinoctial, I find that 88° 2′ Meridian doth cross the Vertical in the Altitude given, which is the Azimuth from the South. Here you are to take notice, that those which were before Meridian's are now become Vertical Circles in this manner of working. PROB. XI. To make an Horizontal DIAL. Set the Horizon or Finitor to the Latitude of that place for which you desire to make your Dial, and then observe the several Angles made by the Meridian's upon the Horizon, which express upon your Plain, and let your Style make an Angle equal to the Latitude of your Place for which you make your Dial. This needs no Example. PROB. XII. To make an Upright Vertical DIAL. Set the Ruler to your Latitude, and the movable Vertical in the Centre of your Instrument, then observe the several Angles made between the 12 of Clock line and the other Meridian's, not upon the Horizon, as you did before; but upon the Vertical, and those shall be the hour spaces required. So in the Latitude of 49° the angle of 11 will be 9° 57′, and also for 1; 20° 44′ for 10 h. and 2 h. 33° 16′ for 9 h. and 3 h. and so the rest. PROB. XIII. To make a Upright Declining DIAL. For the placing the Style, and drawing the Hour-lines there must necessarily be resolved a Sperical Rect-angled Triangle, in which there will be given, beside the Right Angle, the Compliment of your plains Declination (which I always count from the Pole of the Plain to the North or South Points) and thirdly, the Compliment of your Latitude; by help whereof you may find, 1st. The Elevation of the Pole above the Plain. 2dly. The distance of the Substyler from the Vertical in your Plain. And 3dly. The Difference of Longitude, or the Angle comprehended between the Meridian of the Plain, and Meridian of the Place. All which will be made plain by the Figure following: in which, The Scheme following represents a South Plain Declining from the North Westward 55° 30′ in the North Latitude 46° 12′. scheme The Latitude is 46° 12′. Let ENWS be the Horizon of the place and the letters as they are in their order, mark, the East, North, and West and South points. Let then a plain BC be given, the pole whereof will be D, the Meridian of the Plain which passeth always through P, the pole of the world, and D the pole of the plain DPR, the Declination DZN the Compliment thereof NZC, the Triangle to be resolved PZR, in which are given; 1st. PZR the Compliment of the Declination: 34° 30′. 2dly. PZ the Compliment of the Latitude 43° 48′. 3dly. PRZ the Right Angle. Things required are, 1 saint. PR the Elevation of the Pole above the Plain. 2 dly. ZR the distance of the Substylar from the Vertical, or the angle made between the Substylar and the Meridian, or line of 12 h. 3 dly. ZPR, the Difference of Longitude, or the angle comprehended between the Meridian of the Plain, and of the Place. Now these things are thus found at one Operation by this Instrument. 1st. Reckon from the Aequinoctial towards the Pole, the Compliment of the Declination of your Plain, which in our Example will be 34° 30′, at that point place the Horizontal Ruler, then from the Centre of your Instrument towards the Limb, reckon upon the Horizontal Ruler, the side given PZ the Compliment of your Latitude, and the Parallel there intersecting being reckoned from the Aequinoctial, shall be PR the Elevation of the Pole above your Plain. 2dly. The Instrument standing in the same situation, count from the Limb of your Instrument towards the Centre among the Meridian's, the Compliment of your Latitude, for the Meridian, there intersecting the Horizon, shall, upon the Degrees of the Horizon give you ZR, the Distance of the Substylar from the Meridian of the Place. 3dly. the Parallel of Declination in the common Intersection of the foresaid Meridian with the Horizon, being numbered from the Pole of the World, doth give you the Angle sought, ZPR, or the difference of Longitude. But because it may some times fall out, that the said Intersection will be more than 23° 30′, and in little Instruments the Parallels not drawn beyond the Tropics, which are 23°½ from the Aequinoctial: in that case you may thus help yourself: at the common Intersection make a little spot of ink or red Ochre with your pen, and then putting the Horizon back to the terms of the Aequinoctial, run the movable Vertical to the said mark, the degrees of which shall show you the Angle sought, counted from the Pole of the world; or otherwise place one foot of your compasses in the said Intersection, and take the nearest Distance between that and the Aequinoctial, which being applied over, either in the limb, or Vertical, shall give from the Aequinoctial the Angle required. EXAMPLE. In the Latitude of 46° 12′ a South Plain declines from the North Westward 55° 30′ I place the Finitor to 34 degrees, 30 minutes upon the Limb, which is the Compliment of my Declination, and is represented by the Arch NC in the Scheme equal to WD, and from the Centre reckon the Compliment of the Latitude for which the Declining Dial is made, Viz. 43° 48′, and there I find the Parallel 23° 05′ to meet or intersect 43° 28′ which is PR, the Elevation of the Pole above the Plain. 2dly. In the same situation of the Instrument, reckoning the Compliment of my Latitude among the Meridian's, from 12 of the Clock, or first Meridian, I find the 43° 48′ Meridian doth cut the Horizon in 38° 19′, which is the Distance of the Substylar from the Meridian of the Place, Viz. ZR. 3dly, and lastly, I find from the Pole, the 63° 37′ Parallel doth pass through that Meridian in its common Intersection with the Horizon, or ZPR the Difference of Longitude. These things being thus found, Viz. ° ′ The Elevation of the Pole above the Plain, 23 05 The Angle of the Substylar, 38 19 The Difference of Longitude, 63 37. The hour spaces are thus found. By the Difference of Longitude you may know that the Substylar falls four hours' 3° 37′ from the Meridian, place therefore to 3° 37′ from the Meridian the Horizontal Ruler upon the Limb, and then reckon from the Centre among the Meridian's the Elevation of the Pole above your Plain, and look what Parallel doth intersect that Meridian, and the Horizon; for that counted from the Equinoctial, is the Angle between the Substylar and the first hour toward the Meridian, which in our Example will be found to be 1° 25′ then remove the Horizon 15° higher, that is to 18° 37′ upon the Limb, and you shall find 7° 31′ among the Parallels intersect the Horizon in the 23° 5′ Meridian, and so forth for all the hours on that side of the Substylar. In like manner for the hours on the other side; set the Fini or to 11° 23′ in the Limb, and you shall find the Angle up●n the ●lain to be 4° 30′, then removing it 15° higher, you shall fin 10° 58′, and consequently the rest as they follow. Lastly, Because the Plain is a South Plain Declining from South Eastward, that is, from the North Westward, the Style must be placed among the Morning hours on the Western side, and the Dial will be as followeth. Morning Hours. h. ° ′ 8 1 25 9 7 15 10 14 35 11 23 57 12 18 19 1 62 48 7 4 30 6 10 38 5 19 01 4 30 28 diagram of a dial plane PROB. XIV. To draw the Hour-lines upon a Reclining Plain, whose Face looketh directly toward the North or South. Let a Plain be given, reclining from the Zenith toward the North 20° in the Latitude of 49° 00′. Here I set the Finitor to the Latitude, and the movable Vertical to the Degree of Reclination, reckoning from the Zenith towards that Pole towards which the Reclination is. As in our Example of 20° Reclination towards the North Pole, the Meridian's there intersecting denote the Horary Angles, and the Degrees between the Pole and the Vertical, that is to say, in our Example 22° 00′ is the Elevation of the Pole above the Plain. This is very little different from the way of making an upright Vertical Dial, and therefore needs no Example. diagram of a spherical triangle PROB. XV. In any Spherical Triangle whatever, having two sides given with the Angle comprehended to find the rest. In the foregoing Triangle BAC let BASILIUS be 60, AC 50, and the angle BAC 30. Reckon on the Limb from the Aequator toward the North Pole, and on that side of your Instrument that is on your Right Hand, one of the sides given, Viz. AC 50°, and there place the Horizontal Ruler, and then among the Parallels from the North count the other side AB 60°, and among the Meridian's from the Left Hand, there count the Angle given, Viz. 30° BAC, & at the common Intersection of that Meridian with the Parallel, place the Movable Vertical; this being done, apply the Horizontal Ruler over in the terms of the Aequinoctial, so shall the Degree in the Vertical, which was before observed, viz. 26½, in the common Intersection of the Meridian with the Parallel, give you among the Parallels, reckoning from the Pole the Base 26°½, and among the Meridian's, from the utmost on the Right Hand towards the Left you shall find the Angle C. 103. Now for the Angle B, count AB 60° in the Limb, and there place the Horizon, and count the side AC 50° among the Parallels, and among the Meridian's towards the Right Hand, count the Angle given 30°, and apply the Vertical to the common Intersection, then apply the Horizon as before over in the Aequinoctial; so among the Meridian's from the utmost on the Right Hand towards the Left, you shall in the common Intersection find the Angle unknown B 59½ So have you now in your Triangle the three sides and two Angles, the third Angle may be found either by letting fall a Perpendicular, or continuation of the sides to a Semicircle PROB. XVI. To find when the Twilight gins and ends. The Twilight in the Morning gins when the Sun is distant 18 degrees from the Horizon, & continues till it Rise; and at Night gins at Sun Set, and continues till he is 18 degrees below the Horizon, and the● Darkness gins. To find this, you must know the Place of the Sun, and his Declination. The Sun's Place may be known by most Almanacs, or by a Scale, which may be annexed to any convenient Place in the Instrument. The 2 d. to wit, the ☉ is Declination, is found by the first Problem. Example First. If the ☉ be in the Aequinoctial, he hath no Declination; if you therefore place the Finitor to the Latitude of your Place, move your Vertical till the 18 th'. degree thereof crosses the Aequinoctial, and mark what Meridian that is, counting from the Centre of your Planisphere, which in the Latitude of 52 degrees, you will find to be the 30 th'. Meridian; so that you may conclude the Twilight gins two hours before ☉ Rise, and at Night ends two hours after ☉ Set. But if the ☉ have Declination, you must observe that Meridian which crosseth the 18 th'. degree of the Vertical in the Parallel of Declination. Example. Suppose the ☉ to be in ♊ 00° 00′ in the Latitude of 52° 00′. When the ☉ hath 20° of Declination, and observe the Intersection made by the 18 th'. degree of the Vertical, and the 20 th'. Parallel of Declination, and there you may observe the 33 Meridian, or thereabout, from ☉ Rising, passing which showeth the Twilight will then end, to wit, 2 hours and 12′ before the ☉ should rise, but at this time you will find the ☉ is up, which teaches you there is then no Twilight at all. So in the Winter, the ☉ Sets so much before 8 h. that it will not come to be 18 degrees below the Horizon till ☉ Rise, and will continue so till the ☉ cometh back to ♌, there will be no dark time in that Latitude, if the Sky be not Cloudy. PROB. XVII. Of Spherical Rect-angled Triangles in all their Varieties. I have in the third and forth Problems foregoing, shown you the Solution of some Questions, wherein the Triangle hath been Rect-angled and Spherical. I shall now handle the Use of them more fully, and show you how three parts, that is, two beside the Rectangle, which is always known, being given; the other parts sought, either Sides or Angles may be by this Planisphere many ways concluded, as will appear in Practice. 1. In the Scheme. ° ′ BC 60 00 Data. BA 57 48 Rad. 2 Or, ° ′ BC 60 00 Data. CA 20 12 Rad. diagram of a triangle The Data in the Triangle in our following Geniture are 3° 00′ ☉ Long. 1½ ☉ Declin. with the Rect. and the Angle at C will be found 66° 32′, which will be demonstrated by the Scheme, Nº 4. The Proportions in Trigonometry are thus: BC Rad. BASILIUS. C Scheme 1. Or, CA Scheme 2. PC Rad. PD.C. That is, As the Cousin of the Declination given is to the Radius, so is the Cousin of the greatest Declination to the Angle sought. In the Triangle BAC Rect-angled, at A the Longitude, BC is 60° 00′; the Right Ascension may be found by Problem 3 d. 57.48, and the Angle at B 23° 30′. Now to find the Declination, do thus: Choose out the 57° 48′ Meridian, and then move the Horizontal Ruler till the 60 th'. degree from the Centre meets the 57° 48′ Meridian, which you will find will be 20° 12′ the Declination sought, if you count the Parallels from the Aequinoctial till the Intersection is made in the 57° and 48′ Meridian: or if you have the Declination given, bring down the Horizontal Ruler to the Aequinoctial, and you will find the 57° 48′ of the Ruler will be cut by the 57° 48′ Meridian. diagram of a triangle Though I have a very low opinion of Astrology, especially as our Genethliacall-men, or rather Fortune-Tellers use it; yet that I may show the excellent use of this Planisphere, and because it may be useful to know the figure of the Heavens in Eclipses, and the Ingress of the Sun into the Aequinoctial points: I shall now show you how to place the 12 Houses with the Signs to them belonging. In this I shall take for Example ageniture long since past, and not agreeable to any Example before given, that by variety of Operations the Reader may better comprehend the use of the Instrument. PROB. XVIII. How to erect a Figure of the Heavens. First for the time propounded you must first find the Sun's place, that is, in what Degree of the Ecliptic the Sun is in at the time proposed; and afterward, in the following Example; I shall teach you to find the Medium-Coeli, or that degree of the Ecliptic that is in the Meridian. EXAMPLE. Suppose a Geniture to be upon the ninth day of December, at six of the Clock in the Evening, in the Year 1571. The place of the Sun was then found 27° 17′ of ♐, and his Right Ascension will be found 267° Add to this 90° which is six hours after Noon in the degrees of the Aequinoctial, which will give you the Right Ascension of the Mid-Heaven, Viz. 357° to which will answer 27° or rather 26° 46′ ♓; for the Medium-Coeli, and his opposite 27° or 26° 46′ of ♍ for the Imum-Coeli, so you have the 10th and 4th House found, Your next work will be to find the degree of the Ecliptic, which will be in the Ascendant for the Latitude [Lat. 53° 00′] of the place of Birth. Now by adding 90° to the Right Ascension of the Mid-Heaven, you will have the Obliqne Ascension of the Ascendant: In our Example the Right Ascension of the Mid-Heaven is 357°, to which, by adding 90 the Sum will be 447; out of which take 360, the residue will be 87°, which is the Obliqne Ascension of the Ascendant. Now to know to what Degree of the Eccliptick this belongs, must be known by a Table of Obliqne Ascensions fitted to your Latitude, which you may make by the fifth and sixth Problem, but it is better to take it out of the Table at the end of this Treatise, where the Obliqne Ascension may be found for every Degree of the Eccliptick for several Latitudes; by which you may see, that in 53° of Latitude, the Obliqne Ascension 87° will be between the 25 or 26° of ♋, the Ascendant will be in ♋ between 25 and 26°. The Ascending Signs, ♑ ♒ ♓ ♈ ♉ ♊ Descendant. ♋ ♌ ♍ ♎ ♏ ♐ By this you may know the four Cardinal Points or Angles, to wit, the 10 th'. the 1st. the 4 th'. and the Seventh. By which all Questions of Life, Parents, Marriage, and Preferment, or Honour are (according to their Rules) foretold PROB. XIX. In order to find the Cusps of the other Houses, divers Spherical Triangles are to be Resolved. We had before showed how to find the Angle between the Meridian and the Eccliptick, to be 66°½, by Prob. 7. which we shall now find another way. The Degree culminating, or the Medium-coeli was ♓ 26½, and consequently its Distance from ♈ was 3°½; count therefore 3°½ among the Meridian's, from the utmost towards the Centre, then remove the Horizontal Ruler downward towards the Antarctic Pole, the Fiducial Edge of the Vertical remaining fixed at the Centre, till such time as the Vertical shall cross the 3½ Meridian in the Arctic Circle, that is, in the 23½ Parallel from the Pose, and in the Limb you shall meet the 66°½, which is the Angle between the Meridian and the Eccliptick. 2dly. Things standing all in this Posture, look from the extremity of the Vertical Ruler to the 3½ Meridian, there you will find 1½ crossing, which is the Declination of the Mid-Heaven Southward; take this out of 37° the Compliment of your Latitude, there rests 35° 30′, which is the Altitude of the Mid-Heaven, or the Degree culminant, because the Declination is Southward, otherwise it must have been added. 3dly. Number 35° 30′ from the extremity of the Vertical, and this will, among the Parallels from the Pole, show about 42°½, or thereabout, which is the Angle made between the Eccliptick and the Horizon. 4thly. Count among the Meridian's from the utmost to that point, and you shall find the 61st. Meridian there, which is the degree of the Eccliptick between the Meridian and the Horizon. Lastly, You shall find the same Degrees of the Eccliptick, (Viz. 61) if you count the 35½ or thereabout, from the Centre for those Degrees of the Ruler will show you 61, as before. Thus we have obtained, ° 1. The Degree Culminant, ♓ 26 ½ 2. The Angle between the Meridian and the Eccliptick. 66 ½ 3. The Distance of the Point Culminant from ♈ 03 ½. 4. The Declination of the Mid-Caeli. 01 ½. 5. The Altitude or the Mid-Caeli. 35 ½. 6. The Angle between the Eccliptick & the Horizon, 42 ½. 7. The same Degrees, that is to say, of the Eccliptick and Horizon two ways, 61 ½. All these things will be made plain by the following SCHEMES. scheme Nº. 1 scheme Nº. 2 1. The Degree Culminant, that is, the Mid-Caeli was before found between 26 and 27° of ♓, near the 27°. 2. The Ascendant was found between 25 and 26° of ♋. scheme Nº 3 scheme Nº. 4 3. The Angle between the Meridian and Eccliptick represented in the Scheme, No. 2, by the angle at L, may be found by the Precept of the 15 th'. Problem, and will be 66½, and the Triangle BAC in the Scheme, No. 4, is equipollent to that wrought by the Analemma, where you have the same Data, Viz. The Right Angle at A, the greatest Declination at B, the present Declination at the Parallel 1°½, and the third Meridian, and where they cross, will be 66°½ from the Aequinoctial. The Proportion in Trigonometry will be, As Cousin of the Declination given is to the Radius, so will the Cousin of the Declination be to Angle sought. Sch. No. 5. As BC. BAC ∷ Rad. to C. 4. In the Scheme, No. 1. I seek the Angle at 11, which is the Angle between the Meridian and Circle of Position for the 9 th'. & 11 th'. Houses. As HAE Sin. to DAE Tang. ∷ Rad. to the Tang. of the Angle at H. which will be found 44.47. scheme Nº 5 In the Scheme, No. 1. in the Rect-angled Triangle PRO Rect-angled at R. PR, the Elevation of the Pole above that Circle of Position, is found by this Proportion. As Rad. to the Lat. So the Angle at ☉ will be to the side PR. Rad. to OPEN ∷ POR. PR.. But this Work will be shortened by the Table annexed, where you have the Elevation of the Pole given to several Latitudes. 6. I seek the Altitude of the Mid-Caeli in the Scheme, No. 2. HL. There are given HAE the Compliment of the Latitude. 2dly. 1°½ the Declination of the Mid-Coeli in our Example. Now because this is a Southern Sign, I take 1°½ from 37° the Compliment of my Latitude, the Residue will be 35° 30′, the Altitude of the Mid-Coeli in our Example. 7. I still want No. 2. LG, which is the Degree of the Eccliptick between the Meridian and the Circle of Position. In the Obliqueangled Triangle HLG you have given HL, the height of the Mid-Caeli, with the two Angles at O and L, by which LG may be found by Trigonometry, according to the Rules delivered by Artists, by letting fall a Perpendicular either within or without the Triangle, when two Angles and the adjacent side is given. In this manner you may set a Figure of the 12 Houses, which will be near as the Figure adjoining. diagram of the 12 zodiacal houses A Table of the Houses Lati. Locorum. Undecim. & 3ae, nec non 9ae 74 Duodecimae & secundae, nec non 8ae & 6ae Domorum. 46 27.22 41.53 47 28.11 52.53 48 29.02 43.53 49 29.54 44.55 50 30.47 45.55 51 31.41 46.56 52 32.37 47.57 53 33.34 48.59 54 34.32 50.01 55 35.42 51.03 FINIS. Advertisement. FOrasmuch as the Practice of Astronomy depends much upon the exact making of the Instruments; These are to give notice, that these, and all other Instruments for the Mathematical Practice, are accurately Made and Sold by Mr. Walter Hayes, at the Cross-Daggers in Moorfields, next Door to the Popes-Head-Tavern, London; where they may be furnished with Books to show the Use of them: As also with all sorts of Maps, Globes, Sea-Plats, Carpenters-Rules, Post, and Pocket-dials' for any Latitude, at Reasonable Rates. diagram of a planetary instrument W: Hayes fecit. THE Planetary Instrument. OR THE Description and Use of the Theories of the Planets: drawn in true Proportion, either in one, or two Plates, of eight Inches Diameter; by Walter Hayes, at the Cross-Daggers in Moorfields. Being excellent Schemes to help the Conceptions of Young Astronomers; and ready Instruments for finding the Distances, Longitudes, Latitudes, Aspects, Directions, Stations, and Retrogradations of the Planets; either Mechanically, or Arithmetically; with Ease and Speed. By Mr. John Palmer, Rector of Ecton, and Archdeacon of Northampton. The DESCRIPTION. THE first Plate (which I call Saturn's Plate) contains the Theories of ♄ ♃ ♂ ♁: also short, but sufficient Tables of their Anomalies; and a Scale for measuring their Distances in Semidiameters of the Earth. The second Plate (which I call Mars' Plate) contains the Theories of ♂ ♁ ♀ ☿, with like Tables of their Anomalies, and Scale of Distances. The Sun is in the Centre of the Plate. The other Planets have their several Eccentrics & Orbits. These should be Ellipses, but Circles will serve sufficiently, especially for Instruments. Mr. S. Foster disposed these Planets in four Plates, and added thereto other Devices, to be seen in a Book published since his Death. Here they are all contrived in two Plates, or two sides of one Plate: and whereas Mr. F. supposed the Apheliums' and Nodes movable, in these Thories they are fixed, according to Mr. Street's Hypothesis: by which means, though they be framed to the end of 1680, yet not only for this Age, but (with allowance of the Procession of the Aequinoctial) they may serve perpetually. The Aphelium of a Planet is the point of his Eccentric, which is furthest from ●he Sun, and from Aphelium is the Anomaly counted. The Anomaly is the circular Distance of a Planet from his Aphelium. But though the Anomalies be equal, yet their Divisions in every Eccentric are unequal, because they are made to contain the Aggregates of the Anomalies, and Prosthaphorese of the Orb compounded together. Where you see two or three pricks on one side the Orbit, and ☊ on the other side, there the Planet goes into North Latitude, and at the opposite Point over the Centre, is the Place of ☋, where he goes into South Latitude. The Use of the THEORIES. This shall be shown in three Examples only, which may suffice. But, Note 1. That I begin the Years and Days 24 hours later than Mr. Street; for I count the last day of December to end in the Noon of the Circumcision; which is the old way: and to that Account these Theories and Tables are fitted: and all Years and Days here are counted Complete. Note 2. That in gathering the Anomalies out of the Tables, if the same exceed a Circle, (or 360°) you must by Subtraction, or Division, cast away all whole Circles, take the Remainder for the Anomaly sought. The Numbers in the Tables are Degrees, and Centesimal parts, and for the Diurnal Motion, another Figure is added, to make the Parts Millesimal. In the Tables, A. stands for Anni, that is, Years: D. stands for Days: Incl. stands for Inclination; which is set down in Degrees and Minutes. EXAMPLE. I. 1675. April 1. I saw Mars above the foremost foot of Apollo, and he seemed to be much diminished in Magnitude. First out of the Table for ♁. For ♂. write out for 1672 194.59. 259.51. for two Years more 359.75. 191.27. 359.75. 191.27. for 90 days the product of 986 by 90 is 88.74. 524 in 90 47.16. 1002.83. 689.21. for 2 Circles deduct 720. deduct 360. Anomaly of ♁ 282.83. Anomaly of ♂ 329.21. Now in the Earth's Orbit, at 283, make a prick with Ink for ♁; for there is ♁ for this time: and likewise prick ♂ in his Orbit at 329. Lay a Ruler from the Place of ♁ over the ☉ (in the Centre) and it shall cut in the Limb ♈ 21° 38′, the ☉ is Longitude. Again, lay a Ruler from ♁ to ♂ (I mean, the pricks set for them) and know, that a Line Parallel to your Ruler, passing through the ☉ (or Centre) will cut in the Limb the Longitude of ♂. Take therefore with your Compasses the nearest distance of the Centre from the Ruler, and let one foot slide along the Ruler from ♁ to ♂, and beyond him; and let the other foot, keeping even pace with his fellows, pass from the Centre to the Limb, and so it shall touch in the Limb ♊ 29½, the Longitude of ♂. Another way. Mark well the Triangle made by your two Pricks and the Centre, that is, by ♁ ♂ & ☉. Measure the sides upon your Scale, and you shall find ☉ ♁ 3500 Semid. of ♁ ☉ ♂ 5700 ♁ ♂ 6000 Now if you have 2 Thirds from the Centre, and lay one upon ♁, and the other upon ♂, the Arch of the Limb between them, is the Measure of the Angle at the ☉, (or of Commutation) and is here 77° 42′. With this Angle and the Sides comprehending it (which are 35 and 57, as before) you may by Pitiscus his third Axiom, Calculate the other Angles, and find Ang. at ♁ (or Elongation) 67°. 41′, and Ang. at ♂ (or Paralloxis Orbis) 34° 37′. The Elongation of ♂ (67° 41′) added to the Long. of ☉ (♈ 21° 38′) makes the Long. of ♂, 89.17. that is ♊ 29.17. Another way. Transfer your Triangle upon Paper, and there, by help of a Scale of Chords, or a small Quadrant, and Compasses, you may easily find all the Angles very near the truth; Viz. Ang. add ☉ 77° 42′. Ang. add ♁ 67° 54′. Ang. add ♂ 34° 24′. Note, That the reason of ♂ his Diminution is the Increase of his Distance from the Earth; for you may measure it upon the Plate 6000: but in his ☍ he may be distant but 1320, and never above 2350. For the Latitude of ♂, lay one third from the Centre to ☊, and another third to ♂, the Arch of the Limb intercepted by the Thirds (76.10.) is Argumentum Latitudinis. Now as the Radius to the Tang. of 1. 52′, the Inclination of ♂: So is the Sine of 76. 10′ to the Tang. of 1. 49′; the North Latitude of ♂ seen at the Sun. And as ♁ ♂ to ☉ ♂; so is the Tang. of the Lat. at the Sun, to 1. 44′; the Tang. of Lat. seen at the Earth. EXAMPLE II. 1677. Octob. 28. (being St. Simon and Jude's) at Noon. ● seek ♉ Place. ♁ ☿ 1672. 194.59 155.83 A. 4. (or 4 Years) 359.98 218.90 Days 300 295.80 147.60 850.37 522.33 Subtract the Circles 720. 360. Anom. of ♁ 130.37 Anom. of ☿ 162.33. Prick the ♁ and ☿ in their Orbits, at the end of these Anomalies, and you shall see the Prick for ☿ fall in the very Node at ☊; and laying a third, or Ruler from the Centre to ☿ or ♁, it shall cut them both, and show that ☿ is in a Corporal Conjunction with ☉. This ♂ ☉ ☿ would be observed: for by the help of fit Glasses, ☿ may be seen in the ☉ for several hours; and according to the best Tables, he shall pass within 4 or 5 minutes of the ☉ is Centre in North Lat. EXAMPLE III. 1673. May 25. In the day time I saw ♀ with a Telescope, horned like the ☽ at 3 or 4 days old; and though she was so much waned, she appeared bigger and brighter than at any time since she came last out of the Sunbeams. ♁ ♀ 1672. 194.59 62.54. 144 days. 141.98 230.69 Anom. ♁ 336.57 Anom. ♀. 293.23 Prick these Planets in their Anomalies, as before was taught. Lay a Ruler from ♁ over the Centre, and it shall cut in the Limb the Long. of the ☉, ♊ 14. 11′. The Ruler thus lying, draw a third from the Centre over ♀. Now between the Ruler and the third is the Angle of Commutation (163°) and there adjoineth to it the Supplement thereof (17°) which in your Triangle is Angulus ad ☉, and is measured by the Limb. Lay your Ruler from ♁ to ♀, and the Parallel Line made, or imagined to be made, with your Compasses through the Centre, will cut ♋ 17°; the Long. of ♀, and the Arch between this and the ☉ is Place before found, is the Elongation of ♀ from ☉ Eastwards, 32. 49′. And the Sum of the Commutation and Elongation taken out of 180, leaves the Angle at ♀ 130. 11′. Another way. In the Triangle ♁ ♀ ☉, you may take all the sides in your Compasses, and measure them upon the Scale, that is, ☉ ♁ 3520. ☉ ♀ 2450. and ♁ ♀ 1370. Then either by Protraction find the Angles: or, the Angle of Commutation being known (17°) and the sides including by Ax. 3. Pitisci, you may compute the Angle at the ♁ 32. 49′ and the Angle at ♀ 130. 11′. This Angle at ♀ measureth her Waxing and Waning. Let the Radius be 100, the Diameter of ♀ 200, the Angle being 130. 11′, the versed Sine thereof (165) measureth the dark part of the Diameter; the residue (35) is light: So ♀ is Waned 165/200 of her Diameter; that is almost 10 Digits; and yet she seems much bigger than when she was Full: because 2 Digits of light in her present Distance (of 1370) contain more Seconds of light than her full Disk could contain; when coming from the ☉, she was distant about 6000, as you may measure upon the Plate. How these Plates may be also useful for Observing Altitudes, Azimuths, Declinations, and Inclinations of Plains, etc. They who have any Skill in the Mathematics, may easily discern without further Admonition. FINIS. The Description and Use of the NOCTURNAL; By Mr. Samuel Foster, late Reader of Astronomy in Gresham-Colledge. With the Addition of a Ruler, showing the Measures of Inches and other Parts of most Countries, compared with our English one's; Being useful for all Merchants & Tradesmen. THIS Nocturnal is made of two Plates; the thick Plate (which I call the Mater) and a Movable Plate, representing the Aequinoctial. On the Mater, the Circle doth represent the Eccliptick. All the rest of the Writing, is the Names of as many of the F●xed Stars as the bigness of the Instrument will give leave. To these must be added an Index or Label, fastened at the Centre, to cut the several Circles upon the Instrument. The Use of the Nocturnal. 1. SET the Label to the Sun's Place in the Zodiac, and the Hour of Twelve in the Aequinoctial to the Star, whose time of coming to the Meridian you inquire after; and then look what hour and minute is cut by the Label in the Aequinoctial, for that is the hour of the Day or Night that the same Star will come to the South Part of the Meridian. But you must observe, that the hours are marked in the Aequinoctial in this manner, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Now the Difficulty lieth, in finding whether the minutes you shall find cut by the Label in the Aequinoctial, doth belong to the upper row of hours, Viz. 12, 1, 2, 3, 4, 5, or to the under row, Viz. 6, 7, 8, 9, 10, 11; and whether from Noon, or from Midnight: In order to this you must know in what Sign the Star is that you observe, and take notice how far it is distant from the Place where the ☉ is that day; if it be not above three whole Signs, the Minute cut by the Label, belongeth to the upper row of hours to be accounted from Noon; and if the Distance of the Star, and of the ☉ be four, five, or six Signs, than the said Minute cut by the Label belongeth to the under row of hours, accounted also from Noon: but if the Distance of the ☉ and Star be 7, 8, or 9 Signs, than the Minute belongeth to the upper row of hours accounted from Midnight. Lastly, if the Distance of the ☉ and Star be 10, 11, or 12 Signs then the Minute belongeth to the under row of hours, accounted from Midnight. All which beforesaid shall be made clear by Examples. Example the first. The ☉ being in the beginning of ♌, when will Spica ♍ come to the Meridian? Set the Label to the beginning of ♌, and the hour 12 in the Aequinoctial to Spica ♍ then will the Label cut the 59 th'. Minute after 4, or after 10; now this Star being in ♎, which is not above three Signs from ♌, it must be after 4 of the Clock from Noon. I conclude then that the ☉ being in the beginning of ♌, the Spica ♍ will come to the Sonth at 4 h. 59′ past Noon. Example II. When will the same Star come to the Meridian, the ☉ being in the 10 th'. degree of ♊? The Label being set to the 10 of ♊, and 12 to the Star, as before, the Label shall cut the 35 Minute after 2 or 8; now it must be after 8, because the ☉ is above three Signs distant from the Star, and yet not seven Signs; so Spica ♍ will come to the Meridian at 8 h. 35′ past Noon. Example III. When will the same Spica ♍ come to the Meridian, the ☉ being in ♓ the 5 th'. Degree? The Label being set to the 5° of ♓, shall cut 41′ after 2, or 8; but it must be 2, and after Midnight past, because the distance of the ☉ and the Star is above six whole Signs, and not nine. Example iv Working after the same manner, you will find that the same Star will come to the Meridian at 9 h. 58′ past Midnight, the ☉ being in the 20° 00′ of ♏. I take the lower row of hours, and say, that 'tis after Midnight, because the ☉ is above nine Signs distant from the Star. NB. These Precepts are fitted to an Instrument made for 1671. Additions to the Instrument, in Brass, made by Mr. R. Aug. 1st. 1684. Calculated for the Year 1700, which will make some little difference in the aforesaid Precepts. IF in this Instrument you set down to the several Stars their respective several Declinations, and by adding either an A, or B, according to the Declination of either Austral or Boreal, you shall have the height of the Star when it cometh to the Meridian, Viz. by adding the Declination to the height of the Aequinoctial, when the said Declination is Northward, and by taking the Declination from the height of the Aequinoctial when the Declination is Southward. diagram of a nocturnal As for Example. Suppose I desire to know when Cor ♌ shall come to the Meridian, what will be his Altitude in the Latitude of London 51° 30′. The height of the Aequinoctial is 38° 30′, to which add the Stars North Declination, 13° 02′ 38° 30′ 13 02 the Sum is 51° 32′ the Altitude required. So the Altitude of the Spica ♍ in the Meridian will be found to be 28° 57′ in the same Latitude; for the height of the Aequinoctial is 38° 30′; from which take the Stars South Declination 9° 33′, the Remainder is 28° 57′. I have so contrived this Instrument, that by making two little square holes in the Movable Plate, the first showeth you in what Sign the Star is, which is absolutely necessary to be known, to judge of the distance between the ☉ and the Star (as you have been taught before) and the second shows the Magnitude of the Star. To know at any time proposed, what Point of the Eccliptick is in the Meridian. Suppose the ☉ to be in the beginning of ♉, I desire to know what Degree of the Eccliptick shall be in the Meridian at 15′ past Five in the Afternoon. I lay the hour given to the Sun's Place, and then I find over against the 12 a Clock line of the Aequinoctial, 15° 20′ of ♋; and that is the Degree that was then in the Meridian. To know when any of the Planets shall come to the Meridian. The Planets, because of their continual changing of Place, cannot be set fixed in this Nocturnal: Nevertheless, if at any time you desire to know their time of coming to the Meridian, you must look in some Ephemeris for the Place of the Planet, and according as you find it, set it with Black-Lead on your Instrument, which if it be in Brass, shall be easily put out. The Planet thus set, shall be as a Fixed Star, and its time of coming to the Meridian found out, as that of any of the Fixed Stars. But Note, that if it be the Moon that you observe, you must allow about a degree for every two hours passed since Noon; and thus you shall have her true Place; for the Ephemeris gives you her Place only at Noon. For Example. When will the Moon come to the Meridian on January the 1st. 1684/5? The ☉ is then in ♑ 22° 5′, and the Moon in ♈ 10° 12′. Now placing ●…e Moon on my Instrument in ♈ 10° 12′, I find that the Moon shall come to the Meridian at a little past 5 in the Afternoon: and because there are five hours passed since Noon, I must for these five hours allow two degrees and a half to the Moon's Place, and so set it to ♈ 13° 00′▪ which being done, I shall find the Moon's true hour of coming to the Meridian, and that is at about 5 h. 15′ past Five in the Afternoon. Hitherto is the Instrument general to all those that live on this side the Aequioctial; and may serve to any Intelligent Man that shall have South Declination. But besides, I have made two little Windows in the Movable Plate, but the Figures of them are Calculated for the Meridian of London, or any other Place that is under the same Latitude of 51° 30′. The first Window shows the Semi-Nocturnal Arch of the Star in Hours and Minutes; and the Use of it is to know the time of the Stars Rising and Setting, as also how long it continues above the Horizon. First. For the Rising, take the Semi-Nocturnal Arch from the time of the Stars coming to the Meridian, and the Remainder gives you the time of the Stars Rising. So the ☉ being in the beginning of ♊, the Spike of the Virgin comes to the Meridian at 9 h. 18′ after Noon, from which take the Stars 5 11, Semi-Nocturnal Arch, there remains 4 07, which is the time of the Stars Rising in the Afternoon. Secondly, For the Setting, add the Semi-Nocturnal Arch to the time of coming to the Meridian, and the Sum gives the time of the Stars Setting. So on the same day, the ☉ being in the beginning of ♊, the Spike of the Virgin coming to the Meridian at 9 h. 18′ if you add to it the Star's Semi-nocturnal Arch, 5 11′ the Sum is 14 h. 29′ past Noon, or 2 h. 29′ past Midnight. Thirdly, For the time of the Stars being above the Horizon, double the Semi-Nocturnal Arch, and the Sum is the time of the Star's being above the Horizon. The other Window showeth the Star's Amplitude in Degrees and Minutes, which is counted f●… the East towards the North, when the Star's Declination is North; and from the East to South, when the Declination is South: Where note, that the Stars Set at the same Distance from the West that they Rise from the East. This Instrument was first invented by Mr. Samuel Foster, and given to me, drawn upon Pasteboard by his own hand, which is still in my Power; but the Additions to it were put in by an Ingenious Gentleman of the French Nation, and by him drawn in Brass, which I received from him, and will keep for his Sake. The following Table is made to insert all the Stars expressed there according to their Right Ascensions, which is fourfold as great as the true is, the Nature of the Instrument requiring it to be so; because the Aequinoctial, which should be divided into twenty four hours, is divided but into six hours. A Table A. R. As Rec. 4. Decli. Semi-Diurnal Arch. Amplit. ° ′ h. mi. Lucid. Comae Beren. ♎ 182 45 731 00 30 06 8 48 53 30 Lucid. Lyrae ♑. 276 42 1106 48 38 32 24 00 00 00 Syrius. . 98 00 392 00 16 15 4 33 26 43 Vindemiatrix. ♎. 191 53 767 32 2 35 7 05 20 30 Spica Virgins. ♎. 197 23 789 32 9 33 5 11 15 27 protion. ♋. 110 57 443 48 6 00 6 30 9 40 Aquila. ♑. 294 06 1176 24 8 07 6 41 13 07 Luc. cap. Arieties. ♈. 27 38 110 32 22 03 8 04 37 05 Arcturus. ♏. 210 34 842 16 20 49 7 55 34 49 Cauda Delphin. ♒. 304 30 1218 00 10 14 6 54 16 35 Austra lanx ♎. ♏. 218 37 874 28 14 45 7 17 24 09 Cap. Medus. ♉. 42 15 169 00 39 47 12 00 00 00 Bo. lanx. ♎. ♏. 225 16 901 04 8 14 5 18 13 18 Luc. Hydr. ♌. 138 16 553 04 7 22 5 22 11 53 Luc. Pleiad. ♉. 52 26 209 44 23 10 8 11 39 12 Luc. Coron. Sep. ♏. 230 31 922 4 27 45 8 46 48 25 Os Pega. ♒, 322 28 1289 52 8 31 6 44 13 46 Med. nox. col. Serp: ♏ 232 27 929 48 7 25 6 38 11 52 Bo. Fron. Scor. ♏. 237 02 948 08 18 57 4 18 31 27 Antares ♐. cor ♏. 242 50 971 20 25 42 3 30 44 00 Cor Leonis. ♌. 148 08 592 32 13 02 7 08 21 15 Luc. colli Leonis. ♌. 150 51 603 24 21 21 7 58 35 48 Luc. colli Peg. ♑. 336 30 1346 00 09 10 6 47 14 50 In basi Crater. ♍. 161 10 644 40 16 33 4 32 27 14 Marchab. Pega. ♓. 342 30 1370 00 13 37 7 11 22 13 Rigel. ♎. 75 07 300 28 8 33 5 16 13 49 Sin. Hum. Orion. ♊. 77 17 309 08 6 03 6 31 9 45 Cing. Orion. ♊. 80 18 321 12 1 24 5 54 2 15 Caput Ophiuci. . 260 16 1041 04 12 49 7 06 20 52 Cauda Leonis. ♍. 173 28 693 52 16 13 7 26 26 39 Seq. Hum. Orion. ♊. 84 48 339 12 7 20 6 37 11 50 Cufpis Sagit. ♐. 266 00 1064 00 30 22 2 50 54 18 Cap. Andromed. ♓. 358 16 1433 04 27 28 8 44 47 48 Extreme. Ala Pegas. ♓. 359 30 1438 00 13 32 7 10 22 05 Aldeban Tauri. ♊. 64 43 258 52 15 53 7 24 26 05 The 5. 10 15 20 25 ° ′ ° ′ ° ′ ° ′ ° ′ ° ′ ♈ ♎ 00 00 18 20 36 44 55 12 73 48 92 36 ♉ ♏ 111 36 130 48 150 16 170 04 190 08 210 32 ♊ ♐ 231 12 252 08 273 24 294 52 316 28 338 12 ♋ ♑ 360 00 381 48 403 32 425 08 446 36 467 48 ♌ ♒ 488 48 509 28 529 52 549 56 569 44 589 12 ♍ ♓ 608 24 627 24 646 12 664 48 6●3 16 701 40 FINIS. diagram of a scale ruler In the diagonal Scale you have London foot Divided into 1000 Equal parts, Whereof (France) Paris Foot is 1: 068 Lions Ell 3: 976 Boloine el 2 076 The XVII Provinces Amsterdam foot 0: 942 Amsterdam el 2: 269 Antwerp foot 0: 940 Brill foot 1: 103 Dort foot 1: 184 Leyden foot 1: 133 Leyden el 2: 260 Lorain foot 0: 958 Mecalin foot 0: 919 Middleburg foot 0 991 Germany Strashurg foot 0: 920 Bremen foot 0: 964 Cologne foot 0: 954 Francfort Menain foot 0: 948 Francfort Menain el 1: 826 Hamburg el 1 905 Leipsig el 2: 260 Lubeck el 1 90● Noremberg foot 1: 006 Noremberg el 2: 227 Bavaria foot 0: 954 Vienna foot 1 053 Spain & Portugal Spainsh or Castil palm ●: 751 Spanish Vare or rod 3: 004 Spanish foot 1: 001 Lisbon Vare 2: 750 Gibraltar Vare 2: 760 Toledo foot 0: 899 Toledo Vare 2: 685 Italy Roman foot on the Monum of Cossutius 0: 967 Roman foot on the Monum of Statelius 0 972 Roman foot for building w of 10 make the Cauna 0 722 Bononia foot 1 204 Bononia el 2 113 Bononian Perch w of 500 to a Mile 12: 040 Florence Brace or ell 1: 913 Naples Palm 0: 861 Naples Brace 2: 100 Naples Cauna 6: 880 Genoa Palm 0 830 Manlua foot 1 569 Milan Calamus 6 544 Parma Cubit 1: 866 Venice foot 1 162 Other Places Danzick foot 0: 944 Danzick el 1: 903 Copenhagen foot 0: 965 Prague foot 1: 026 Riga foot 1: 831 China cubit 1 016 Turin foot 1 062 Cairo cubit 1 824 Persian Arash 3 197 Turkish Pike at Constantinop: the greater 2: 200 The Greek foot 1 007 Moutons universal foot 0 675 A Pendulum of which length will Vibrate 〈◊〉 times in a minute, A Pendulum of 3 foot 268 par●s long will Vibrate 60 times in a minu●… Ex: per me Ionas▪ Moor Tabula Ascensionum Obliquarum ad Latitudinem 51 deg. 00 min. ° ′ ♈ ♉ ♊ ♋ ♌ ♍ ♎ ♏ ♐ ♑ ♒ ♓ 0 0 00 13 21 30 46 57 31 95 10 137 33 180 00 222 27 264 50 302 29 329 14 346 39 1 0 25 13 50 31 29 58 37 96 33 138 59 181 24 223 52 266 12 303 34 329 56 347 08 2 0 50 14 20 32 13 59 44 97 56 140 24 182 49 225 17 267 34 304 38 330 38 347 37 3 1 16 14 50 32 57 60 51 99 19 141 50 184 03 226 43 268 56 305 41 331 19 348 05 4 1 41 15 20 33 42 61 59 100 42 143 15 185 38 228 0● 270 18 306 44 331 59 348 34 5 2 07 15 50 34 27 63 08 102 06 144 40 187 03 229 34 271 39 307 46 332 58 349 02 6 2 32 16 21 35 13 64 18 103 30 146 06 188 27 230 59 272 59 308 47 333 16 349 30 7 2 58 16 53 36 00 65 29 104 54 147 31 189 52 232 25 274 19 309 47 333 54 349 58 8 3 24 17 24 36 48 66 40 106 18 148 56 191 16 233 52 275 39 310 46 334 32 350 25 9 3 50 17 56 37 36 67 52 107 42 150 21 192 41 235 17 276 58 311 44 335 10 350 53 10 4 16 18 28 38 25 69 04 109 07 151 46 194 ●6 236 42 278 17 312 42 335 47 351 20 11 4 42 19 01 39 15 70 17 110 32 153 11 195 30 238 08 279 35 313 39 336 23 351 47 12 5 08 19 34 40 05 71 30 111 57 154 36 196 55 239 33 280 52 314 35 336 59 352 14 13 5 34 20 07 40 56 72 44 113 22 156 01 198 20 240 58 282 10 315 30 337 35 352 41 14 6 00 20 40 41 48 73 59 114 47 157 26 199 45 242 2● 283 28 316 25 338 11 353 08 15 6 26 21 14 42 41 75 15 116 12 158 50 201 10 243 48 284 45 317 19 338 46 353 34 16 6 52 21 40 43 35 76 32 117 37 160 15 202 34 245 13 ●86 01 318 12 339 2● 354 00 17 7 19 22 25 44 30 77 50 119 02 161 40 203 59 246 38 287 16 319 04 339 53 354 26 18 7 46 23 01 45 25 79 08 120 27 163 05 205 24 248 0● 288 3● 319 55 340 26 354 52 19 8 13 23 37 46 21 80 25 121 52 164 30 206 4● 249 28 289 43 320 45 340 59 355 18 20 8 42 24 13 47 18 81 43 123 18 165 54 208 14 250 53 29● 56 321 35 341 32 355 44 21 9 07 24 50 48 16 83 02 124 43 167 19 209 39 252 18 292 ●8 322 24 342 ●4 356 10 22 9 35 25 28 49 14 84 21 126 0● 168 44 211 04 253 42 293 20 323 1● 342 36 356 36 23 10 02 26 06 50 13 85 41 127 35 170 08 212 ●9 255 ●6 294 31 324 ●● 343 07 357 ●● 24 10 30 26 44 51 13 87 01 129 01 171 32 213 54 256 30 295 42 324 4● 343 39 357 28 25 10 58 27 22 52 14 88 21 130 26 172 57 215 20 257 54 296 52 325 33 344 10 357 53 26 11 26 28 01 53 16 89 42 131 52 174 22 216 45 259 18 298 01 326 18 344 4● 358 19 27 11 55 28 41 54 19 91 04 133 17 175 47 218 10 260 41 299 09 327 ●3 345 1● 358 44 28 12 23 29 22 55 22 92 26 134 43 177 11 219 36 262 04 300 16 327 47 3●● 4● 35● 10 29 12 52 30 04 56 26 93 48 136 08 178 36 221 01 263 27 301 23 328 31 3●6 1● 3●● 35 30 13 21 30 46 57 31 95 10 137 33 180 00 222 27 264 50 302 29 329 14 34● 3● 360 ●● Tabula Ascensionum Obliquarum ad Latitudinem 51 deg. 30 min. ° ′ ♈ ♉ ♊ ♋ ♌ ♍ ♎ ♏ ♐ ♑ ♒ ♓ 0 0 00 13 04 30 12 56 48 94 3● 137 15 180 00 222 45 265 24 303 12 329 48 346 56 1 0 24 13 32 30 54 57 54 95 05 138 42 181 25 224 10 26● 47 304 17 330 29 347 ●5 2 0 49 14 01 31 38 59 01 97 24 140 08 182 50 225 36 268 9 305 21 331 11 347 53 3 1 14 14 30 32 21 60 08 98 46 141 34 184 15 227 02 269 32 306 24 331 51 348 21 4 1 3● 15 01 33 06 61 16 100 10 143 00 185 40 228 48 270 54 307 27 332 39 348 49 5 2 04 15 30 33 50 62 25 101 35 144 26 187 0● 229 54 272 16 308 29 333 09 349 16 6 2 29 16 00 34 35 63 35 102 59 145 52 188 30 231 20 273 37 309 30 333 47 349 44 7 2 54 16 31 35 22 64 46 104 23 147 1● 189 5● 232 46 274 57 310 30 334 25 350 11 8 3 19 17 02 36 08 65 57 105 48 148 43 191 2● 234 13 276 17 311 29 335 2 350 38 9 3 45 17 33 36 37 67 10 107 13 150 09 192 4● 235 39 277 36 312 27 335 40 351 05 10 4 10 18 05 37 46 68 22 108 38 151 3● 194 12 237 05 278 56 313 24 336 17 351 32 11 4 36 18 37 38 35 69 35 110 0● 153 00 195 37 238 32 280 14 314 2● 336 52 351 58 12 5 01 19 1● 39 26 70 4● 111 29 154 2● 197 02 239 57 281 32 315 16 337 27 352 25 13 5 26 19 42 4● 16 72 0● 112 5● 155 51 198 28 241 23 282 51 316 11 338 2 352 51 14 5 52 20 14 41 08 73 19 114 21 157 16 199 53 242 49 284 8 317 6 338 38 353 17 15 6 17 20 48 42 01 74 35 115 46 158 44 201 16 244 14 285 25 317 59 339 12 353 43 16 6 43 21 22 42 54 75 52 117 11 160 07 202 44 245 39 286 41 318 52 339 46 354 8 17 7 09 21 58 43 49 77 0● 118 37 161 32 204 09 247 6 287 57 319 44 340 18 354 34 18 7 35 22 33 44 44 78 28 120 0● 162 58 205 35 248 31 289 11 320 34 340 50 354 59 19 8 0● 23 08 45 39 79 46 121 2● 164 23 207 00 249 57 290 25 321 25 341 23 355 24 20 8 28 23 43 46 26 81 04 122 55 165 48 208 26 251 22 291 38 322 14 341 55 355 50 21 8 55 24 2● 47 33 82 24 124 21 167 14 209 51 352 47 292 50 323 3 342 27 356 15 22 9 22 24 58 48 31 83 43 125 47 168 39 211 17 254 12 294 3 323 52 342 58 356 41 23 9 49 25 35 49 30 85 03 127 14 170 04 212 43 255 37 295 14 324 38 343 29 357 6 24 10 16 26 13 50 30 86 23 128 40 171 30 214 08 257 1 296 25 325 25 344 0 357 31 25 10 40 26 51 51 31 87 44 130 06 172 54 215 34 258 25 297 35 326 10 344 30 357 36 26 11 11 27 30 52 33 89 06 131 32 174 20 217 00 259 59 298 44 326 54 344 59 358 21 27 11 3● 28 09 53 36 90 2● 132 58 175 45 218 26 261 14 299 52 327 39 345 35 358 46 28 12 07 28 49 54 39 91 51 134 24 177 10 219 52 262 36 300 59 328 22 345 59 359 11 29 12 35 29 31 55 43 93 13 135 50 178 35 221 18 264 1 302 6 329 6 346 28 359 36 30 13 04 30 12 56 48 94 36 137 15 180 00 222 45 265 24 303 12 329 48 346 56 360 0 Tabula Ascensionum Obliquarum ad Latitudinem 52 deg. 00 min. ° ′ ♈ ♉ ♊ ♋ ♌ ♍ ♎ ♏ ♐ ♑ ♒ ♓ 0 00 00 12 48 29 4● 56 1● 94 06 137 0● 180 9 223 ● 265 54 303 4● 330 1● 347 12 1 00 24 13 16 30 24 57 17 95 3● 138 2● 181 25 224 26 267 1● 304 5● 330 5● 347 4● 2 00 48 13 45 31 7 58 24 96 5● 139 5● 182 5● 225 5● 268 4● 305 5● 331 3● 348 7 3 1 13 14 14 31 50 59 3● 98 18 141 20 184 16 227 1● 270 3 307 1 332 1● 348 35 4 1 37 14 43 32 34 60 39 99 42 142 47 185 42 228 45 271 26 308 4 332 58 349 2 5 2 02 15 12 33 18 61 48 101 9 144 1● 187 8 230 12 272 48 309 6 333 3● 349 29 6 2 26 15 42 34 3 62 58 102 32 145 40 188 3● 231 38 274 9 310 7 334 1● 349 56 7 2 51 16 13 34 49 64 09 103 57 147 6 189 59 233 5 275 4● 311 7 334 5● 350 23 8 3 15 16 43 35 36 65 20 105 22 148 32 191 25 234 3● 276 5● 312 6 335 29 350 49 9 3 40 17 14 36 24 66 32 106 47 149 5● 192 51 235 58 278 1● 313 4 336 6 351 16 10 4 5 17 45 37 12 67 45 108 12 151 24 194 17 237 25 279 3● 314 1 336 42 351 42 11 4 30 18 16 38 1 68 59 109 38 152 5● 195 42 238 ●2 280 4● 314 57 337 1● 352 8 12 4 55 18 48 38 51 70 13 111 4 154 16 197 8 240 18 282 8 315 52 337 52 352 33 13 5 20 19 20 39 42 71 28 112 3● 155 42 198 34 241 45 283 26 316 47 338 2● 352 59 14 5 45 19 52 40 34 72 44 113 5● 157 8 200 ● 243 11 284 4● 317 41 339 1 353 25 15 6 10 20 25 41 26 74 0 115 23 158 54 201 26 244 37 286 0● 318 34 339 3● 353 50 16 6 35 20 59 42 19 75 17 116 49 160 ● 202 52 246 4 287 1● 319 26 340 8 354 15 17 7 1 21 3● 43 13 76 34 118 15 161 26 204 1● 247 30 288 32 320 18 340 4● 354 40 18 7 26 22 08 44 8 77 52 119 42 162 52 205 44 248 56 28 4● 321 ● 341 1● 355 5 19 7 52 22 43 45 3 79 11 121 8 164 18 207 1● 250 22 291 1 321 5● 341 4● 355 3● 20 8 18 23 18 45 59 80 30 122 35 165 43 208 36 251 48 292 15 322 48 342 15 355 55 21 8 44 23 54 46 56 81 50 124 2 167 9 210 2 253 1● 293 2● 323 36 342 4● 356 20 22 9 11 24 31 47 54 83 10 125 28 168 3● 211 28 254 38 294 4● 324 2● 343 17 356 45 23 9 37 25 08 48 53 84 31 126 55 170 1 212 5● 256 3 295 51 325 11 344 18 357 9 24 10 4 25 45 49 53 85 51 128 22 171 27 214 20 257 28 297 ● 325 57 344 18 357 34 25 10 31 26 23 50 54 87 12 129 48 172 52 215 47 258 53 298 12 326 42 344 48 357 58 26 10 58 27 2 51 56 88 3● 131 15 174 18 217 1● 260 18 299 21 32● 26 345 17 358 23 27 11 25 27 41 52 59 89 5● 132 41 175 44 218 4● 261 42 300 2● 328 10 345 46 358 47 28 11 53 28 21 54 2 91 2● 134 8 177 9 220 6 263 6 301 3● 328 53 346 15 359 12 29 12 2● 29 01 55 6 92 43 135 34 178 35 221 33 264 30 302 4● 329 3● 346 44 359 36 30 12 48 29 42 56 11 94 ●6 137 00 180 9 223 0 265 54 303 49 330 18 347 12 360 0 Tabula Ascensionum Obliquarum ad Latitudinem 53 deg. 00 min. ° ′ ♈ ♉ ♊ ♋ ♌ ♍ ♎ ♏ ♐ ♑ ♒ ♓ 0 0 0 0 12 14 28 34 54 46 92 58 136 26 180 0 223 34 267 2 305 14 331 26 347 46 1 0 23 12 41 29 15 55 52 94 23 137 54 181 26 225 1 268 27 306 20 332 6 348 13 2 0 40 13 8 29 57 56 59 95 48 139 22 182 53 226 29 269 51 307 25 332 45 348 40 3 1 09 13 36 30 39 58 6 97 13 140 49 184 20 227 56 271 15 308 28 333 24 349 6 4 1 32 14 4 31 22 59 14 98 38 142 17 185 47 229 24 272 38 309 30 334 2 349 32 5 1 56 14 32 32 6 60 23 100 4 143 44 187 14 230 52 274 0 310 31 334 40 349 58 6 2 19 15 1 32 51 61 33 101 30 145 12 188 40 232 19 275 22 311 31 335 17 350 24 7 2 43 15 30 33 36 62 44 102 56 146 39 190 7 233 47 276 44 312 30 335 53 350 50 8 3 6 15 59 34 22 63 56 104 22 148 7 191 34 235 15 278 5 313 29 336 29 351 15 9 3 30 16 29 35 8 65 9 105 48 149 34 193 1 236 43 279 26 314 27 337 4 351 40 10 3 54 16 5● 35 55 66 22 107 15 151 1 194 28 238 11 280 47 315 24 337 39 352 5 11 4 17 17 2● 36 43 67 36 108 42 152 29 195 55 239 19 282 7 316 21 338 13 352 30 12 4 41 18 ● 37 32 68 51 110 9 153 56 197 22 241 6 283 26 317 16 338 47 352 55 13 5 5 18 31 38 22 70 6 111 36 155 23 198 49 242 34 284 45 318 10 339 20 353 19 14 5 29 19 2 39 11 71 22 113 4 156 50 200 16 244 1 286 3 319 3 339 53 353 43 15 5 53 19 34 40 5 72 3● 114 32 158 17 201 43 245 28 287 21 319 55 340 26 354 7 16 6 17 20 7 40 57 73 5● 115 59 159 44 203 10 246 56 288 38 320 47 340 58 354 31 17 6 41 20 4● 41 50 75 1● 117 26 161 11 204 37 248 24 289 54 321 38 341 29 354 55 18 7 5 21 13 ●2 ●● 76 34 118 54 162 38 206 4 249 51 291 9 322 28 342 0 355 19 19 7 30 21 47 43 39 77 5● 120 21 164 5 207 31 251 18 292 24 323 17 342 31 355 43 20 7 55 22 2● 44 3● 79 13 121 49 165 32 208 59 252 45 293 38 324 5 343 1 356 6 21 8 20 22 56 45 3● ●0 3● 123 17 166 59 210 26 254 12 294 51 324 52 343 31 356 30 22 8 45 23 3● 46 31 81 5● 124 45 168 26 211 53 255 38 296 4 325 38 344 1 356 54 23 9 10 24 7 47 3● 83 16 126 13 169 53 213 21 257 4 297 26 326 24 344 30 357 17 24 9 36 24 4● ●8 2● 84 38 127 41 171 20 214 48 258 30 298 27 327 9 344 50 357 41 25 10 2 25 2● 49 29 86 ● 129 8 172 46 216 16 259 56 299 37 327 54 345 28 358 4 26 10 28 25 58 50 3● 87 22 130 36 174 13 217 43 261 22 300 46 328 38 345 56 358 28 27 10 54 26 3● 51 3● ●8 45 132 4 175 40 219 11 262 47 301 54 329 21 346 24 358 51 28 11 2● 27 1● 52 3● 90 9 133 31 177 7 220 38 264 12 303 1 330 3 346 52 359 14 29 11 47 27 5● 53 4● 91 33 134 59 178 34 222 6 265 37 304 8 330 45 347 19 359 37 30 12 14 28 34 54 46 92 58 136 26 180 0 223 34 267 2 305 14 331 26 347 46 360 0