engraving of the horizontal quadrant (mathematical instrument) This instrument or any other for the Mathematical arts are made in silver or brass by Elias Allen or john Allen near the Savoy in the strand The making, description, and use of a small portable Instrument for the Pocket (or according to any Magnitude) in form of a mixed Trapezia thus Called a horizontal Quadrant. Composed and prodused solely for the benefit and use of such which are studious of Mathematical Practice Written and delivered by Delamain student and Teacher of the Mathematics. Attribuit nullo praescripto tempore vitae usuram nobis ingenijque Deus. mathematical diagram London printed for Richard Hawkins and are to be sold at his shop in Chancery lane near Sergeants' Inn 1632. TO The right Honourable and his much honoured Lo. Thomas Lo. Brudenel, Baron of Stauton. Right honourable, and my very good Lo. YOur singular knowledge in all excellent, and solid Literature, and your ever Heroic, and Noble disposition to the best kinds of Learning, are not unknown unto the world; And amongst other studies in your Lo. minority at the University, you took no little affection, to the Mathematical Arts, as by your Lo. own Manu-scripts and excellent Books in your Lo. great Library I have often seen; Besides, not only by mine own sundry conferences with your Lo. but also by the relation of others of more mature judgement I have been amply informed in these your L. more aged years not only of your continued love to these Arts, but also that your knowledge in them far exceeds many of the Nobility of this kingdom. Now my L. when I caused the subject of this Tractat to be made for your Lorenzo last Summer (I mean your Lo. horizontal Quadrant) I had not then any intention so soon to have written publicly upon it; But, having then but declared unto your Lo. the excellent and abundant use of the Instrument by the heads of the propositions then slightly compiled, (fare exceeding the Instrumental way in this nature, that either Nobility, Gentry, or others are now acquainted with in this kingdom, for a recreative Instrument, as well for the copious use thereof, as its great facility, and expedition in operation) your Lorenzo then encouraging me to the publishing of it for a general end; many Months after I considered thereon: and drew it up into a Body, and thus accommodated it, as I here present it now unto your Lo. favourable censure, and Patronage (to be sheltered under the wings of your Lo. clemency against all calumniators and malevolents) as belonging especially to your Lo. sith you were the sole motive to this work, and had both the use of the Instrument before it came thus to a public view, and the first Quadrant that ever was made common in this kind: accept therefore favourably I beseech your Lo. this small mite of my labours, as from the hands of one of your poorest servants (yet true affectionate) who shall always acknowledge your Lo. Nobleness towards him, and ever rest Your Lo. most humbly devoted to Honour and serve you De la main. To the Reader. BEfore I show the Projection, describe the particulars, and deliver the uses of this horizontal Quadrant, it will not be impertinent, for the satisfaction of some, to give the Reader to understand what moved me unto it, how it exceeds other Instruments, and whence it might be drawn, and projected. Now the ingenious aptness for Invention, and accommodating of things in a fair and expedite course for Mathematical Practices, of that late professor of Astronomy Mr. Gunter is not unknown unto many about this kingdom, satisfying many of his friends (according to his free and noble affection) by Transcripts in that of more solid matter, but such of vulgar Practices he hath publicly made manifest for the use of all such as affect those Studies. In which work many years passed I took occasion to consider the Scheme, or Diagramme of the fourth Projection in his Book of the Sector Page the 64. & 65. which according as he says is after the manner of the old Concave hemisphere (but being in truth a natural Projection of the visible Hemisphere, that is, one Moiety of the Globe, projected on a Plain) which Diagram and Projection is now challenged by my Reverend good friend Master Oughtred, and it should seem that Master Gunter had the Original of it from his labours, & invention, who composed and made the same so, for more than thirty years past, as appears by his own Writings, & Manuscripts upon that Protection shown me in the time of the Printing of this Tractat upon my horizontal Quadrant, whose excellent knowledge in Mathematical Learning may evidently confirm it: which Projection the said Master Oughtred gave to the late Bishop of Winchester, Doctor Bilson, for more than 20. years past, and to some others of very good quality. And it may also, by a Letter from that most famous and admired Geometer, Master Henry Brigges unto Master Oughtred dated from Gresham College june, the 2. Anno. 1618. be collected that the said Master Gunter had the first overture of that fourth Projection, from the said Master Oughtred, in which letter are these words: Master Gunter doth here send you the Print of an horizontal Dial of his drawing after your Instrument; And afterwards the said Projection was also presented by Master Gunter to many Noble Personages, and in particular to the Right Honourable the Earl of Bridgewater, causing it to be cut in Brass, in such a form a● I have placed at the end of this Tractat, some uses of which Dial are extant, viz. the 2. 18. 21. and 34. Pro. of the Index or Table following. Now having considered divers Pocket Instruments (that many men are practised in) & looked into sundry Projections, amongst which that of Gemmafrisius (there drawn in the Book of the Sector) is of admirable use, yet making a more serious quaere, & contemplating more intensively upon that Diagram, drawn and specified in that 64. and 65. Pages of the Sector (aforesaid) I found it fare to exceed all others in the Multiplicity, and excellency of performance. If I should add unto it a Calendar of time, and an Index graduated with an Axis, and Perpendiculars to be erected upon it at pleasure: & referring only the Trapeziall form, it should be fitted fare to exceed any portable Instrument for the Pocket, ever yet produced in respect of the general uses of it: in resolving such ordinary Propositions which are practised in Astronomy, upon the Globe, Sphere, Hemisphere, Quadrants of all sorts, Astrolabe of Frisius, Blagrave, and others for facility, expedition, or certainty, (like Magnitudes considered) for in these Instruments for several times, and several Propositions, there must be divers rectifications of the parts belonging to these Instruments, and that diversely by reason of their diversity: By this horizontal Quadrant, the former redifications are avoided, Contemplation & the eye being only the Index, the aptness, & fitness of the parts, and lines so naturally projected, or described as they are upon the plain of the Instrument (being a part of the Horizon the Parallels Meridian's, & Vertical Circles, that are contained or may be described in our Latitude sufficiently necessary) induceth any one in the understanding of the uses of it that is but indifferently versed in the linaments and principles of the Globe, what to speak, and what to answer in a Proposition without farther direction: And having had this horizontal Quadrant for many years past, as a Pocket Instrument, divers about this Kingdom being importunate with me for to have it, or to publish the use of it, seeing its great facility, and expedition, in comparison of such Pocket Instruments as are now used, here, or in foreign parts: I was willing at last after I had given order for the making of four of these Instruments in Silver for several Noble Personages, to disburden myself of Transcribing the uses of the Instrument, and Tables for the making of it, to satisfy those which were importunate, and to let others that are studious in Mathematical Practices also participate of it. Now, what I have delivered upon the accommodating of the Instrument thus, the making thereof, with the uses that I have delivered in this Tractat upon it following: I acknowledge due to none Inferior assistant, but to mine own Industry, search and labour, and that 64. 65. and 66. Pages of the Book: of the Sector before specified in which is only shown the 2. 3. 19 22. 25. and 30. Propositions of the Index, or Table following, as uses of the said Projection. But I have extended them to many more, and abundantly, and plentifully supplied the obscurity of that Scheme, or Diagramme there drawn (as for a general good) in the use of this Horizantal Quadrant. I deliver therefore first the making of it, first by the Sector (somewhat different from that of Master Gunter's) secondly by Geometry, and lastly I show a third way, how it may be Projected and made by my Mathematical Ring, and by Numbers, which I have Calculated and accommodated to that end in Tables, for more exactness. Part of the general scope, and use of which Instrument I deliver in the Index, or Table following. An Index, or Table of the uses of the horizontal Quadrant. Viz of the Horizon. Line of Shadows. Calendar. Parallels. Aequator. Ecliptic. Hourelines. Index. 1 By the Horizon to show. 1. The Sun, or Stars Altitude at any time. Pag. 53. 2. At any Day of the year, how fare the Sun riseth, or setteth from the true East or West. Pag. 28. 3. The Sun's Azimuth, and Altitude, at any hour, for any day, Pag. 62. 4. The Meridian line, upon any appearance of the Sun. Pag. 55. 5. The uncertainty of time, by noting the Shadow of things, Pag. 63. 6. The Site of a Building, or Costing of a Place. Pag. 57 7. The Variation of the Needle. Pag. 59 8. The Declination of a Wall, or Plain, the Sun shining thereon, Pag. 71. 9 The Inclination of a Plain, and to place a Plain horizontal, Pag. 89. 2. By the line of shadows is had. 10. At what hour in any Day of the year the shadow of an Altitude is equal, double, triple, etc. unto it. Pag. 35. 11. Instantly the hour of the day, the Azimuth and Altitude of the Sun, with the Meridian line, without observation or sight of the Sun, by knowing the Proportion between the length of a shadow upon a horizontal Plain & that which casts the shadow. Pag. 67. 12. At any hour, an Altitude of the Sun, or Azimuth, what Proportion shadows have to their Bodies. Pag. 37. 13. Whether and Altitude be above, or below the jewel of the eye, & how much. 14. The height of an Altitude, accessible, or in accessible. Pag. 100 101. 15. The measure of any Part of Altitude not approchable. Pag. 102. 3. By the Calendar is known. 16. The inequality of Time, in equal Months, or equal number of Days Pag. 44. 17. What number of days will make the day and hour longer, or shorter at any time. Pag. 43. 18. The hour of the Sun rising, setting, with the length of the day at any time, Pag. 23. 19 What days are alike in length, & what day the Sun rising in the one, shall be the Sun setting in the other, Pag. 24. 20. The inequality of Time between day break, and Sun rising. Pag. 41. 4. By the Parallels to search out. 21. At any day the Sun's declination. Pag. 24 22. The Latitude of a Place, or hight of the Pole above the Horizon, Pag. 60. 23. At what hour in any day, the Sun's Azimuth, and Altitude will be equal, and how much the Altitude & Azimuth will be. Pag. 42. 5. By the Aequator is seen 24. The Sun's equal motion, right Ascention, and obliqne Ascension, Pag. 26. 6. By the Ecliptic to give 25. The Sun's Place at any time of the year, Pag. 25. 26. The Degree of the Aequator in the Horizon, by supposing the degree of the Ecliptic in the Horizon, Pag 46. 27. The Degree of the Ecliptic in the Horizon, by supposing the degree of the Aequator in the Horizon. Pag. 47. 28. The degree of Medium Coeli, or the degree of the Ecliptic in the Meridian, by supposing the degree of the Ecliptic in the Horizon, Vel contra. Pag. 47. 29. The Horoscope, or the degree ascendant ' or descendant, and the Nonagessima degree at any hour. Pag. 49. 30. What Angle the Ecliptic makes with the Horizon, or the Altitude of the Nonagessima degree and what Azimuth it is in at any hour, Pag. 50. 7. By the hour lines to find 31. The hour of the Day, and Azimuth of the Sun, Pag. 54. 32. The hour of the day agreeable to any Altitude, or Azimuth, Pag. 39 33. The Sun's Difference of Ascension for any day, Pag. 23. 34. The Quarter of the year, and day of the month, hour of the day, Meridian-line, and Azimuth of the Sun, if it were forgotten: Pag. 64. 8. By the Index adjoined with other lines you have. 35. At what hour, & Altitude, the Sun will be due East, at any day of the year, Pag. 27. 36. The Sun's Azimuth, & hour without observation, Pag. 58. 37. The time of daybreak, or end of Twilight, for any day in the year, Pag. 30. 38. The height or Depression of the Sun in the Meridian, for any day in the year, here or for any Latitude, Pag. 29. 39 The Sun's depression & Azimuth, at any hour of the Night assigned, Pag. 40. 40. The hour of the day to our Antipodes, by supposing the sun's depression under the Horizon, Pag. 42. 41. What hour, & Altitude the sun cometh upon a declining wall, any day in the year, & how long the sun shineth thereon, Pag. 32. 42. At what hour and Altitude the Sun must have, to be opposite or Perpendicular to a Declination Plain, Pag. 33. 43 The declination of a wall, by seeing the sun beginning to shine thereon, or going from it, Pag. 69. 44. The hour & Altitude of a stars coming to the Meridian at any Night in the year assigned, Pag. 74. 45. The time of the rising, setting, & continuance of a star above the Horizon, & in what part of the Hemisphere they may be seen, with the Azimuth and Altitude thereof at any hour, Pag. 78. 46 In what part of the Horizon a star riseth or setteth, and at what hour & Altitude it will be due East or West, Pag. 80. 47. What Azimuth any star is in, upon any appearance thereof, with the hour of the night, Pag. 82. 48. How to measure the Quantity of an Angle, or to find the distance of two Stars, Pag. 85. 49. How to measure Distances, and Bredths, Pag. 87. 50. How to take the Circuit of a Figure, or the survey of a Place. Pag. 93. Much more I might have laid open upon the use of this Instrument, as the making of horizontal, direct, declining, cilindrical, & Ring dial's, the distance of the hours, substiler & style hight, Stoflerius Astrolabe, Master Gunter's Quadrant, with many other Instruments, now used, but let these be sufficient for the present; the ingenious, may easily add unto that which I have delivered, & therefore I show first how to project the Instruments, than the Description, and lastly how these uses, are compendiously contracted, and operated. The Tables for making of the Horizon●●●● 〈…〉 The Table for the describing of the Parallels. G A B C F E D 58 20 ♑ 37 30 76732 162173 401078 70 00 57 49 ●3 37 15 76041 158854 393750 75 45 56 45 22 36 45 74673 152576 379826 75 15 55 44 2● 36 15 73323 146736 366795 74 45 54 42 20 35 45 71989 141292 354573 74 15 53 43 19 35 15 76673 136205 343084 73 45 52 41 18 34 45 ●9372 131445 332263 73 15 51 46 17 34 15 68087 126982 322052 72 45 50 50 16 33 45 66817 122791 312399 72 15 49 55 15 33 15 65562 118848 303259 71 45 49 0● 1● 32 45 64322 115134 294590 71 15 48 09 13 32 15 63095 111630 286356 70 45 47 17 12 31 45 61881 108321 278523 70 15 46 28 1● 31 15 60681 105190 271061 69 45 45 38 10 30 45 59493 102226 263945 69 15 44 50 9 30 15 58318 99415 257149 68 45 44 03 8 29 45 57154 96748 250651 68 15 43 18 7 29 15 56002 94215 244432 67 45 42 34 6 28 45 54861 91805 238472 67 15 41 50 5 28 15 53731 89512 232756 66 45 41 08 4 27 45 52612 87327 227267 66 15 40 27 3 27 15 51503 85244 221991 65 45 39 47 2 26 45 50404 83256 216916 65 15 39 08 1 26 15 49314 81358 212030 64 45 38 29 0 25 45 48234 79498 207231 64 15 37 53 1 25 15 47163 77808 202719 63 45 37 17 2 24 45 46100 76148 198396 63 15 36 ●● 3 24 15 45046 74558 194162 62 45 36 08 4 23 45 44001 73033 190068 62 15 25 36 5 23 15 42963 71573 186109 61 45 35 03 6 22 45 41933 7017● 182275 61 15 34 33 7 22 15 40911 68825 178562 60 45 34 02 8 21 45 39895 67538 174963 60 15 33 32 9 21 15 38887 66292 171472 59 4● 33 03 10 20 45 37886 65099 168084 59 15 32 36 11 20 15 36891 63951 164994 58 45 32 08 12 19 45 35903 62847 161598 58 15 31 42 13 19 15 34921 61784 158490 57 54 31 17 14 18 45 33945 60761 155467 57 15 30 52 15 18 15 32975 59775 152525 56 45 30 28 16 17 45 32010 58825 149660 56 15 30 05 17 17 15 31050 57909 146869 55 45 29 42 18 16 45 30096 57026 144149 55 15 29 19 19 16 15 29147 56174 141496 54 45 28 5● 20 15 45 28202 55353 138908 54 15 28 36 21 15 15 27263 54559 136382 53 45 28 16 22 14 45 26327 53794 133916 53 15 27 57 23 14 15 25396 53055 131506 52 45 27 47 69 14 00 24932 52695 130522 52 30 The Table for drawing the hour lines T. 125717 P. 34921 H I 1 1745 46 103553 2 3492 47 107236 3 5240 48 111061 4 6992 49 115063 5 8748 50 119175 6 10510 51 123489 7 12278 52 127994 8 14054 53 132704 9 15838 54 137638 10 17632 55 142814 11 19438 56 148256 12 21255 57 153986 13 23086 58 160033 14 24932 59 166427 7 15 26794 60 173205 10 16 ●●● ● ●● The Table fo● 〈…〉 I F M A 1 36 37 22 40 5 4● 13 4 2 36 19 22 07 5 02 14 1● 3 35 58 21 33 4 24 14 5 4 35 38 20 58 3 4● ●5 2 5 35 20 20 24 3 0● ●6 0 6 34 58 19 49 2 31 16 3● 7 34 36 19 14 1 53 17 1 8 34 14 18 39 1 14 17 4● 9 33 50 18 02 0 37 18 2 10 33 27 17 28 0 02 18 5● 11 33 03 16 51 0 40 19 3 12 32 38 16 15 1 18 20 0● 13 32 13 15 39 1 56 20 3● 14 31 47 15 03 2 34 21 1● 15 31 19 14 25 3 11 21 4● 16 30 53 13 49 3 50 22 17 17 30 26 13 12 4 27 22 49 18 29 58 12 36 5 06 23 2● 19 29 28 11 58 5 44 23 53 20 29 00 11 20 6 20 24 2● 21 28 29 10 43 6 59 24 54 22 28 00 10 06 7 35 25 25 23 27 29 9 29 8 12 25 56 24 27 00 8 50 8 50 26 25 25 26 29 8 12 9 27 26 56 26 25 56 7 35 10 03 27 26 27 25 25 6 57 10 40 27 58 28 24 53 6 20 11 16 28 24 29 24 21 11 53 28 50 30 23 46 12 29 29 17 31 23 13 13 07 The Table for the inserti●● 〈…〉 0 1 2 3 ● 1 45 00 42 00 39 49 37 34 3 2 26 33 25 28 24 27 23 30 2 3 18 26 17 53 17 21 16 51 1 4 14 02 13 43 13 23 13 06 8 5 11 19 11 06 10 53 10 41 1 6 9 27 9 19 9 10 9 01 ● 7 8 07 8 01 7 54 7 48 7 8 7 07 7 02 6 58 6 52 6 9 6 20 6 16 6 12 6 08 6 10 5 43 15 3 49 20 2 51 40 1 26 The Table for drawing, and div●●●●● 〈…〉 ♑. 26794. ♋. 1880 K L M N 1 0 55 2604 46 2 1 50 3200 47 3 2 45 4893 48 4 3 40 6408 49 5 4 35 8016 50 6 5 31 9658 51 7 6 25 11246 52 8 7 21 12899 53 9 8 10 14350 54 10 9 12 16196 55 11 10 07 17842 56 12 11 03 19528 57 13 11 57 21164 58 14 12 53 22872 59 15 13 48 24562 60 16 ● ●● Of the Making of the horizontal Quadrant by the Sector. 1. How to describe the Parallels. FIrst, according to any Semidiameter as Z N. or Z S. describe a Circle representing the Horizon, and draw the line S N. for the Meridian: Divide the half Meridian Z N. and Z S. into 90. gr. according to the Tangents of half their Arkes, by the help of the sins on the edge of the Sector: or the semidiameters may be divided into such parts, or points as are required concerning the Projection, thus. Consider what parallels you would describe, and how much they are distant from the Zenith in their intersections, in the Meridian both towards the South and North of the Zenith (for every parallel in an obliqne Sphere, in his intersection with the Meridian, is farther from the Zenith in one part, than in an other) Then if the semidiameter Z N. be placed over in the sign Compliment of half that distance, from the Zenith, the parallel Sine of the former half taken from the Sector: shall from show the intersection in the Meridian with that parallel. gr. m. gr. m. So if the paralles were the Tropic of ♋ whose distances of intersections in the Meridian (according to the Lat. of 51. gr. 30. m.) from Z. towards S. the South is 28. 0. The half of these Arks are 14.00. A N. the North 105.0, 52.30. B Aequator S. the South 51.30. 25.45. C N. the North 128.30 64.15. D Tropic of ♑ S. the South 75.30. 37.30. E N the North 152.0. 76.00. F Now if the semidiameter Z. S. be placed in the Sine Compliment of A. viz. 76. gr. and then the parallel Sine of A taken, viz. 14. gr. it will reach from Z. to ♋. the intersection of the Tropic of ♋ with the South part of the Meridian, but if the semidiameter Z. S. be placed over in the Sine Compliment. B. viz. 37. gr. 30. m. & then the parallel Sine of B. viz. 52. gr. 30. m. being taken it will reach from Z to V the other intersection of the Tropic of 69 with the Meridian below the Pole, the Middle between this V and 69. will be at 1. which is the Centre of that Tropic: In like manner may be found the intersections, and Centres of the other parallels with the Meridian's, and so may be described. 2. To describe the hour lines Secondly, seeing the Lat. is 51. gr. 30. take the Semidiameter Z S. and fit it over in the Sine Compliment of it, viz. 38. gr. 30. then the parallel sine of 51. gr. 30. m. will reach from Z. to T. the centre of the hour of 6. E P W. but if the Radius Z S. be fitted over in the Sine Compliment of half 38. gr. 30. viz. 70. gr. 45. m. and the parallel sine of half 38. gr. 30. viz. 19 gr. 15. m. be taken, it will reach from Z. to P. the Pole, then upon T. erect a perpendicular to the line P T. viz. 2. T. 10. which shall serve for the finding of the Centres of the Meridian's, or hour Circles passing through the Pole P. now seeing that T P. is the nearest distance in the right line 2. 10. unto P. the right line P. T. shall be Radius, to a Circle, and the line 2. T. 10. shall be a Tangent line to that Circle. Now the Radius of a Circle being known, the Tangent of any Angle, or Ark, may be also known, according to the Natural projection and congruity of lines, but because in this first direction we would apply it solely to the Sector: the centre of the Meridian's or hours may be had by the help of the sins thereon thus. Consider what hour, or Meridian circle from the hour of 6. viz. E, P, W. you would describe, for than if the Radius P, T. be fitted over in the Sine compliment of it and the parallel Sine of the hour Angle Taken, it will show from T, in the line 2. 10. the centre of that Meridian, or hour circle: so if the hour circle of 5. or 7. were to be described, whose hour Angle at P, the Pole is 15. gr. fit the Radius or semediameter T, P. over in the Sine compliment of it viz. 75. gr. for then the Sine perallell of 15. gr. being taken will reach from T, to 5. and from T, to 7, the Centre of the hour circles of 5 and 7. If therefore one foot of the Cumpasses be placed in 5. and then extended to P. the Pole, you may describe the hour Circle of 5. and placed in 7. you may draw the hour Circle of 7. and so may be described the rest of the Meridian's. and hour Circles. 3 To describe and divide the Ecliptic Thirdly, to describe the Ecliptic, consider the distances between the Zenith Z. and the Tropics of ♑. and 69. according to the former Lat. of 51. gr. 30. which will be Z, ♑ 75. gr. and Z. ♋. 28. gr. then take the semideameter Z, S. and fit it over in the Sine of those Arkes, than the parallel Sines of the Compliments of those Arkes will show from Z. the distances of the Centres of these Tropical points, so the Centre of the Southern semicircle of the Ecliptic, will be near the Pole P, viz. at ♑. and the Centre of the Northern semicircle of the Ecliptic will be below the Pole at ♋. Therefore placing one foot of the Cumpasses in ♋. below the Pole, and extending the other foot to ♋. above the Pole you may describe the semicircle E, ♋. W. and placing one foot in ♑. near the Pole, you may describe the semicircle, E, ♑, W. Now for the dividing of the Ecliptic; this Mr. Gunter delivers so obstrusly in his 66, page of the Sector. That if a man had not more fundamental Mathematical Doctrine, than his Book teacheth, he should never attain unto it: Consider therefore first, what right Angle Triangles there are made by this Ecliptic, Equator and Meridian's, viz, ♈ B ♓. or ♈ B ♉: ♈ R ♒. or ♈ R ♊, etc. and get the right ascension of these Arkes of the Ecliptic hat you intent to divide, so ♈, B. is the right ascension of the Arkes ♈, ♉, and ♈, ♓. and ♈, R. is the Right asention of Arkes ♈, ♊, and ♈, ♒, from which ground the Table S. is calculated according to the Arkes in the Ecliptic in the Table R. Now to find the Centres of those Meridian's which may divide the Ecliptic according to the Right ascension here calculated answerable to R. S. 5 4.45 10 9.11 15 13.48 20 18.27 25 23.09 30 27.54 35 32.42 40 37.35 45 42.31 50 47.33 55 52.38 60 57.48 65 63.03 70 68.21 75 73.43 80 79.07 85 84.32 90 90.00 the Arkes of the Ecliptic from ♈ or ♎ it nothing differeth from the instruction of the describing of the hour Circles, in the second derection: for if I would intersect the Ecliptic in the beginning of ♉. ♓. ♍. or ♏. the distance of either of those singcs from ♈ or ♎ is 30. gr. against which in the Table, S. is 27. gr. 54. m. Now if the semidiamiter P T. be fitted over in the sine Compliment of this 27. gr. 54. m. viz. 68 gr. 6. m. and then the parallel sine of 27. gr. 54. m. being taken it will reach from T. to 30. in the line 2. T. 10. if therefore one foot of the Compasses be placed in 30. towards 10. and the other foot extended to P. you may intersect the Ecliptic in ♉. and ♓. and then the Compasses placed in the other 30. you may intersect the other part of the Ecliptic in ♍ and ♏. and so may you divide the rest of the Ecliptic. Of the making of the horizontal Quadrant Geometrically. FIrst, having described a Cirrle at pleasure as before, E. S. W. N. draw a line to pass by the centre as S. N. and cross it at right Angles, with the line E. W. in Z. then let the semcircles W S E and W N E. be divided from W. each of them into 180. gr. or rather upon E. we may describe a Quadrant at pleasure, as C D. and augmenting it unto ω, divide the Quadrant D C. from D. into the usual divisions of a Quadrant, and so from D. unto ω, insert or protract the same divisions, then having considered as before the Latitude of the place, and distance of the parallels from Z. the Zenith, towards S. the South, and also towards N. the North, in the Meridian as in the former Table there is specified. Account the distance of the parallels from the Zenith towards S. the South, in the semicircle W S E. but those towards N the North, in the semicircle W N E. from W. so supposing the Latitude as before to be 51. gr. 30. 1 To describe the parallels. m. the distance between the Zenith & the Tropic of 69. towards the South, is 28. gr. which account from W. to F. but rather half of it from D. to F. then consider the distance between the Zenith and the other part beyond the Pole, viz. 105. gr. number this from W. to G. but rather half of it from D. to G. and laying a ruler upon E. F. and E. G. the Meridian A B. may be intersected in 69. and V the middle, between which will be at 1. the Centre of the Tropic: in like manner the distance of the Aequator from the Zenith towards the South is 51. gr 30. reckon it from W. to H. or half of it from D. to H. but the distance of the Aequater from the Zenith towards the North beyond the Pole is 128. gr. 30. m. which I account from W. to I. or half of it from D. to I, then laying a ruler upon E. H. and E. I. the Meridian A B. may be intersected in Q and Y. the half distance between Q and Y. will be at 2. the Centre of the Aequater: In like manner may the Meridian A B. be divided into any of the rest of the divisions, and the parallels also described: But if a ruler be fastened to move upon E. then may you softly move the ruler from D. towards ω. and as it passet by the degrees according to the Column B. of the Tables following, beginning at the bottom, so the edge of the ruler shall show the intersections that the parallels of declination between the Tropickes do make, with the Meridian Z. S. then move the Ruler softly along from D. towards C. as it passeth by the degrees in the Column G. beginning at the bottom, so the edge of the Ruler shall intersect the Meridian A. B. in the Centres of those parallels. Secondly, account the Latitude from D to M. and half the Compliment of the Lat. from D to R. and laying a ruler upon E M. and E R. the Meridian, S N. shall be intersected in T. and P. P. representing the Pole of the world, 2 To describe the hour lines and T the centre of the hour of 6. then unto the line T P. upon the point T. erect a perpendicular 2. 10. and according to the semidiameter P T. describe a semicircle α T ς. divide the Quadrants T α and T ς. from T. each of them into 90. gr. then lay a ruler upon P. and the several hour Arkes in the Quadrants, T α. and T ς. intersect the line 2. T. 10. in the hour points, 2. 3. 4. 5. 7. 8. 9 10. etc. then placing the Compasses in T. and extending the other foot to P. you may describe the hour Circle of 6. but placing it in 5. and extended to P. you may describe the hour Circle of 5. the same extent placed in 7. will describe the hour Circle of 7. and so of the rest: but if a ruler be fastened to move on P. as it passeth by the degrees of the hours in the Quadrants from T. so the edge of the Ruler shall intersect the line 2. 10. in the Centres of those hours from T. 3. To describe and divide the Ecliptic. Thirdly, to describe the Ecliptic, consider the Altitude of each Tropic above the Horizon, according to the Latitude given, which was 51. gr. 30. m. So the Altitude of ♑. is 15. gr. and that of 69. is 62. gr. In the Quadrant D C. account those degrees from D. viz. D y and D δ. lay a ruler upon E. and those several points, so may the Meridian S. N. be intersected in the points ♑. and 69. which are the Centres of the semicircles of the Ecliptic, therefore placing one foot of the Compasses in 69. below the Pole, and extending the other foot to 69. above the Pole, you may describe the Northerens semicircle E. 69. W. and placing one foot of the Compasses in ♑. near the Pole, and extending the other foot to ♑. near S. you may describe the Southern semicircle E ♑ W. those semicircles of the Ecliptic may be divided Geometrically, without the help of the Table of right ascension, but for more expedition we may use them thus. In the Quadrants T α. and T ς. account the degrees of the right ascension for such divisions of the Ecliptic as you intent to have, suppose the beginning of ♉ ♓ or ♍ ♏ the distances of the beginning of any of these signs, fromithe Equinoctial points are equal the one unto the other. viz. 30. gr which find in the Table under R. so right against it under S. is 27. gr. 54. m. this account from T. towards α. and ς. and laying a ruler upon P. and those degrees intersect the Tangent line 2. 10. in 30. and then placing one foot of the Compasses in 30. towards 10. and extending the other foot to P. you may intersect the Ecliptic in the beginning of ♉. ♓. & so in ♍ ♏. the points required: In the likemanner you may divide the other part of the Ecliptic. So the Centres of the degrees of the Ecliptic may be sooner had, if a ruler be placed upon P. and then to move thereon, Now as it passeth by the degrees of the Columes I.O. in tha Tables following from T. in the several Quadrants: so the Ruler shall Intersect the line 2. to in the Centres of those Arkes answerable to the columes K N. the degrees of the Ecliptic. 3. By my Ring it may be otherwise projected: In which it exceeds any Instrument whatsoever for facility, and expedition, for where there is many proportionals required in any service, there the use of the Ring is most excellently made manifest, they being instantly declared at once, which in some sort I have delivered in the use of my Appendix upon plain Triangles, or it may be drawn from that of Proportionating the Fort, to the Fort, or the Building, to the Building, Pag. the 2. and 3. LEt a Circle be described according to any capacity, Constructi. as before, and crossed with diametral lines at right Angles E Z W. and S Z N. then take the semidiameter E Z. and divide it into 10. parts, & supdivide each of those parts into 10. or 100 (according to the capacity of the scale) as A. or more accurately according to the same Radius make a diagonal seal, then consider the distance of the parallels from the Zenith according to the Latitude you intent, as admit 51. gr. 30. m. to be the Latitude as before. Take the half of those distances (according to the first directions) by which is made the Column B. which are the half distances of every degree between the Tropickes and the Zenith, then move the Tangent of 45. gr. unto the parts of the Radius or semidiameter, viz. 10000 in the Circle of Numbers, so right against the Tangent of any one of those degrees in the Column B. Of the calculating of the Numbers, to describe the parallels. in the movable is the Number of equal parts in the fixed, by which is made the Column C. (or they may be extracted out of the Tables of natural Tangents.) Further if we consider the distance between the Zenith and the other intersections of those parallels, with the Meridian beyond the Pole, and take half of those distances we may make the Column D. then moving the movable softly along, as the Tangent of any degree of the Column D. in the movable, passeth by the parts of the scall, viz. 10000 in the fixed (on the Circle of Numbers) so the Tangent of 45. gr. in the movable, shall point out in the Circle of Numbers, the distance between Z. and those parallels beyond the Pole. From these directions are calculated the Numbers in the Column E. or they may be also taken from the Table of natural Tagents as before: The Numbers of the Column C and E. serve to find the distances of the intersections of the parallels in the Meridian from Z. & to describe those parallels, note that at the bottom of the Columes C. & E. are the Numbers, 2493. & 13032. take 2493. from the diagonal scale, and protract it from Z. towards S. viz. at ♋. then take from the same scale also, 13032. and protract it from Z. to ♋. below N. divide the space between ♋. and ♋. into two equal parts which will be at 1. near P. so have you the Centre of the Tropic of ♋. extend the Compasses therefore from 1. to ♋. then may you describe that Tropic, viz. δ. ♋. ζ. In like manner may you draw the other parallels, but for more ease we may take half of the differences of the Numbers in the Column C and the Column E. and so may we have the Column F. and then with greater expedition we may protract the Centres of these parallels, from Z. for if 5269. which is at the bottom of the Column F. (& between the former two Numbers) be taken from the scale, and protracted from Z. it will reach unto 1. the Centre as before, and so any Number in the Column F. is the distance of the Centres from Z. of his opposite Number in the Column B. or A. by which Columes C. and F. you may describe all the parallels, between the Tropics from degree to degree. But for more exactness it were convenient R. T. 1 1.36 2 3.13 3 4.44 4 6.26 5 8.03 6 9.40 7 11.17 8 12.55 9 14.33 10 16.12 11 17.51 12 19.31 13 21.11 14 22.52 15 24.34 16 26.07 17 28.01 18 29.46 19 31.32 20 33.20 21 35.09 22 37.00 23 38.53 23.30 39.50 to show in what points of the Ark δ. E. π. the parallels of declination intersect it, if truly described, and may be found by Page 57 of the Appendix upon the Ring, where is shown to find what Amplitude belongeth to the Sun's declination, for any Latitude by moving the Sine of the Compliment of the Latitude, viz. this of London, 38. gr. 30. m. unto the sine of 90. in the fixed, so every degree of declination in the movable, shall point out the Amplitude in the fixed by which is had the Numbers of the Column T. the sun's Amplitude belonging to the declination of the Column R. Now if the Quadrants E S. EN. WS. W N. be divided from E & W. a ruler laid upon the Centre Z. and to pass by the degrees in the several Quadrants, answerable to the degrees of the Column T. the Arkes' δ π. and ζ ς. shall be noted in such points as the parallels of the declination should intersect. Secondly, move the Tangent of the Latitude in the movable viz. 51 gr. 30. m. unto the former part of the Radius or scale. viz. 10000 in the Circle of Numbers in the fixed, so the Tangent of 45. gr. in the movable shall point out 125717. in the Circle of Numbers in the fixed, which taken from the scale A. and protracted from Z. to T. it shall be the Centre of the hour of 6. upon T. erect a perpendicular 2. T. 10. serving for the Centres of the other hours: then move the Tangent of 45: gr. to the parts of the scale, viz. 10000 in the Circle of Numbers, and consider the distance between the Zenith and the Pole, viz. 38. gr. 30. m. the Tangent of half of it in the movable doth point out in the Circle of Numbers 34921. which taken also from the diagonal scale, and protracted from Z. will reach to P. the Pole, through which all the hours must be drawn, and the Centres of which hours in the line 2.10. from T. may be had thus: which two numbers 125717. & 34921. I place over the Columes H. and I. Of the calculating of the distances of the Centres of the hours According to the distance P.T. make a scale B. (or rather a diagonal scale) to contain 10000 parts, then move the Tangent of 45. gr. to the parts of this scale in the Circle of Numbers, viz. 10000 so every degree in the movable amongst the Tangents unto 45. gr. doth point out in the Circle of Numbers, the distances of the Centres of those degrees from T. in the line 2. 10. by which the Column H. is made, then moving the movable softly along as the Tangent of any degree in the movable above 45. gr. passeth by the parts of the scale B. viz. 10000 in the Circle of Numbers, so the Tangent of 45. gr. in the movable, passeth by the distance of the Centres of those degrees from T. in the Circle of Numbers in the fixed, above 10000 by which is made up the rest of the Column H. viz. I. by help of which Column H. and I. the hours may be thus drawn. Of the describing of the hours Mark, what Numbers are against the hours in the Column H and I. for if those Numbers be taken from the scale B. and protracted from T. in the line 2.10. they shall be the Centres of those hours: so in the Column H. against the hour of 7. or 5. is 2679, which take from the scale B. and protract it from T. to 7. and from T. to 5. in the line 2. 10. then placing one foot of the Compasses in 7. and extending the other foot to P. describe the hour of 7. and one foot of the Compasses at the same extent being placed in 5. shall also describe the hour of 5. In like manner may be protracted from T. out of the Column H. I. the Centres of the other hours with their intermediats, and so also described. But here note, that it were convenient to find the Intersections of the hour lines (and their intermediate degrees) with the Horizon as before was delivered of the intersections of the parallels of Declination with the Horizon, and it may be drawn from my Ring thus. Of the finding the intersection of the hour with the Horizon. Move the Tangent of 45. gr. in the A B A B A B 1 1.16 12 15.12 23 28.29 2 2.33 13 16.26 24 29.38 3 3.50 14 17.40 25 30.48 4 5.16 15 18.54 26 31.56 5 6.23 16 20.7 27 33.4 6 7.39 17 21.20 28 34.12 7 8.55 18 22.33 29 35.19 8 10.11 19 23.45 30 36.25 19 11.27 20 24.57 31 37.21 10 12.42 21 26.08 32 38.36 11 13.57 22 27.18 33 39.41 movable unto the sine of the Latitude, viz. 51. gr. 30. m. in the fixed, then right against the Tangent of any degree from the hour of 6. in the fixed, is the Tangent of the degrees of the intersection of the hours, and the intermediate degrees with the Horizon in the movable: from which direction is this Table drawn, and is only for these degrees which intersect the Horizon in the Calendar, but it might have been extended further. The Application of which is thus: Account in the Limb of the Instrument from E. (the point of East,) any degree in the Column B. and lay a Ruler thereto, and to the Centre Z. so the intersection thereof in the Horizon shall show the intersection that the hour line, or degree opposite thereunto in the Column A. maketh with the Horizon. In like manner I might have delivered the Tables of the intersection of the hour lines with the parallels of declination, which would serve of great use in large Instruments, to describe these degrees, which are near the hour of 12. Now to describe the Ecliptic, consider as before the height of the Tropics above the Horizon, in the Latitude given, viz. 51. gr, 30. m. so ♑ will be 15. above the Horizon, and ♋ will be 62. gr. high. Then move the Tangent of 45. unto the parts of the scale A. in the Circle of Numbers in the fixed. viz. 10000 so right against the Tangent of 15. gr. in the movable is 2679. the distance of the Centre of the Southern semicircle of the Ecliptic from Z. which I place in the Column over M. and against ♑. then move the movable softly along until the Tangent of 62 gr. be right against 10000 To describe the Ecliptic in the Circle of Numbers, so the Tangent of 45. gr. in the movable, shall point out 18807. on the Circle of Numbers in the fixed: The distance of the Centre of the Northern semicircle of the Ecliptic, from Z. which I place in the Column over P. against ♋. if these numbers be taken from the scale A. and protracted from Z. they will reach from Z. to ♑. and from Z. to ♋, and so placing one foot of the Compasses in ♑. near the Pole, and extending the other foot to ♑. near S. you may describe the Southern semicircle of the Ecliptic E, ♑. W. and placing one foot of the Compasses in ♋. below the Pole, and extending the other foot to ♋. above the Pole: you may describe the Northern part of the Ecliptic E, ♋, W. and those semicircles of the Ecliptic may be divided as followeth. Move the Tangent of 45. unto the Sine of 66. and 30. so right against the Tangent of the degrees of the Sun's Longitude in the Ecliptic in the movable, How to make the Table to divide the Ecliptic are the Tangents of the degrees of the Sun's right ascension in the fixed, or they may be had by resolving of a Triangle, in which there will be 9●. several operations, but by this Ring they are given at one rectification, and only by a glance of the eye: for proportionals either in Sines or Tangents are had by the Ring, with the same expedition that Numbers are had, As by the use of the Circles of Sines and Tangents upon the projection of this Ring, in divers particulars is declared in the Appendix upon the use of the Ring; and so according to the former Construction is made the Columes L. and O. for 45. being brought to 66. gr. 30. m. as before, right against 10. gr. in the movable, is 9 gr. 11. m. in the fixed, against 20. gr. in the movable, is 18. gr. 28. m. in the fixed, and so of the rest. Then move the Tangent of 45. gr. to the parts of the scale B. viz. 10000 in the Circle of Numbers, so right against the Tangent of the Arkes in the Column L. in the movable are the distances of the Centres of those Arks, from T. in the Circle of Numbers in the fixed, and so is made the Column M. & if you move the movable softly along as the Tangent of any degree in the Column O. passeth by 10000 the parts of the scale B. so the Tangent of 45. in the movable, passeth by the distances of the Centres of those degrees from T. in the Circle of Numbers in the fixed, by which is made the Column P. or they may be had from the Table of natural Tangents. Then by the scale B. protract the Numbers, To divide the Ecliptic. out of the Column M. and P. from T. in the line 2. 10. for they shall be the Centres of those degrees of the Ecliptic, which are opposite unto them, viz. in the Columes K. and N. so if I would intersect the Ecliptic, in the beginning of ♉. ♓. ♍. or ♏. each being distant from ♈. 30. gr. which I seek in the Column. K. and find right against it in the Column M. 5294. which I take from the scale B. and protract it from T. to 30. in the line 2. 10. Now placing one foot of the Compasses in 30. next 10. and extending the other foot to P. the Ecliptic may be intersected in the points of ♉. and ♓. and placed in 30. towards 2. the same extent will Intersect the Ecliptic in ♍. and ♏. In like manner may the Centres of the rest of the degrees of the Ecliptic be protracted in the line, 2. 10. from T. out of the Columes M. and P. and so all the Ecliptic divided from degree, to degree: but this may be otherwise done. Besides that which is delivered touching the drawing of the Parallels, Ecliptic, and Hour lines, there remains yet how to put on the Calendar, to graduate the Index, and to draw, and divide the line of Shadows. This may be easily done from the Table R. Calculated, and accommodated to that purpose for the year 1640, and may sufficiently serve for many years after, without any sensible error. Having divided the Quadrants, Construction. E.S. and E.N. (as before into the usual degrees of a Quadrant,) lay a ruler upon the Centre Z. and account the degrees from the point E. in the Quadrant towards N. and S. out of the Table R. according to the several Columes of the Table R. and Intersect the Quadrants, How to inscribe the Calendar with small short lines, so shall the Ark of the Horizon of the Instrument from E. be divided into the usual days of the Month, which is the Calendar and the beginning of these divisions, may be at the 10. of March, and so going on to the 11. of june, and then again to begin from the 10. of March, and go on unto the 10. of December, and these days may be noted upon the inside of the horizontal Ark with short lines from E. as before, and at every Month may be placed a representative letter for that Month, and every 10. and 5. day of every Month, may be noted with a small stroke somewhat longer than the rest, to help the memory the readier to number. In like manner may the rest of the days of the Calendar be intersected in the out side of the horizontal Ark, towards the Limb, beginning at the 13. of September, and so going on to the 11. of june, than again from the 13. of September, and going on unto the 10. of December, and these Months may be also noted with significant letters, appropriate to each Month, and each 10. and 15. day of the Month, may be also denoted as before, with a stroke somewhat longer than the rest, according to the Scheme against Page the 1. How to graduate the Index for the Instrument. Let the Index be equal to the semidiameter, Z. E. and then may it be divided out of the Table Q. by the help of the scale A. beginning at the Centre: the Index being divided, and placed on the Centre of the Instrument at Z. it shall help to put on, and divide the line of shadows. as followeth. Lay the edge of the Index to A. in the Limb which is near the 10. of December, and move it to any degree in the Table S. and account the like degree in the Index, and then make a mark upon the plain of the Instrument where that degree toucheth, and so go on from point to point, until the whole line be described and divided, according to the Table S. This line might be placed between the Calendar and the Limb, or in a Quadrant, etc. But I have caused it to be described as is seen upon the Scheme against Page the 1. for expedition and conveniency. ☞ mathematical diagram Let A. B. C. D. be a plain, divide the length A. B. within half an Inch of the higher end, and an Inch of the lower end, in to 3. equal parts which suppose the line M.N. then divide each part into half so the line M. N. shall be divided into 6. equal parts, the middle of which will be at Z. Then take 4. of these parts for Radius, and on Z. describe the obscure Ark, A. E. B. and upon Z. erect a perpendicular to the line M. N. to cut the Ark A. B. in E. now from E. to A. protract 40. gr. and from E. to B. protract 50. gr. so the Angle E. Z. M. shall be 90. and also A.Z.B. shall be 90. Now having made a scale of Z.E. like to the scale A. according to the former directions) then out of the Column C. and by help of the said scale A. from Z. you may protract Z. ♋. 2493. Z.Q. 4823. and Z. ♑ 7673. and from the Column F. you may protract the distances of the Centres of those intersections from Z. viz. Z. 1.5269. Z. 2.7949. and Z. 3.16217. and so placing the Compass in these Centres, you may describe the Equator, and both Tropics. But if Z. M, and Z. N. be divided according to the scale A. then from Z. you may account the intersections of the parallels, and distances of the Centres, and so describe the parallels with greater expedition, and so shall you have the Scheme or Trapeziall form of the Instrument, B. A. ♑. ♋. and may be finished according to that against Page the 1. by the Tables and directions here calculated, and delivered to that end. Now to augment the Instrument to any proportion assigned, as if between the Tropickes were supposed to be 10. Inches, the Radius might be found out, or if the Radius were 4. foot, (which is according to mine own Instrument:) what distance might there be between the Tropickes: the proportion would be as 516. to 1000 so the breadth to the scale, or as 1000 to 516. so the scale to the breadth: therefore by the Ring, bring 516. in the movable, to 1000 in the fixed, so right against any Radius in the fixed, is the distance between the Tropickes in the Movable, or against the distance assigned for the Tropickes in the Movable, is the measure of the Radius or Scale, in the fixed: So if ♑. ♋. be allotted to be 10. Inches, for the distance between the Tropickes, the Scale, or Radius, of the Instrument should be 19 4/10. fere: but if the scale or Radius were 4. foot, or 48. Inches, than the distance between the Tropickes of ♑, and ♋. will be near 24 77/100. Inches. Thus for the making of the Instrument, the description of which followeth. mathematical diagram The Description of the horizontal Quadrant. THe form of this Instrument is like a mixed Trapezia, as appears against Page. 1. where of two sides are right, and the other two sides are Circular, which falleth out to be so from the nature of the Projection, and that part which I have thought most convenient for use, and is fully sufficient for that which I have delivered upon it; and may be made of any plain Material, but fittest in Brass, or Silver: the several parts of which Instrument are five, viz. the Back, the Face, the Sights, the Index, and the divisions, and lines projected on the Face. First, the Back of the Instrument, is a part of Gemma Frisius projection, whose particular description and admirable use I intent here after as God shall give life and Ability to make manifest. Secondly, the face of the Instrument, is that upon which the Index, and sights are placed on. Thirdly, the Sights are the small pieces of Brass in which there is in each a little hole to look through, or the sun beams to pass through, and are fastened upon the Face of the Instrument; one of which Sights is near the Centre of the Instrument, and the other is near the Circumference thereof. Fourthly, the Index, is the movable piece of brass, fastened at the Centre, upon which also two other sights may be placed, the edge of this Index is divided and noted thus. 10. 20. 30. 40. 50. 60. 70. 80. 90. which are called the degrees of the Index, and there is adjoined unto it three small plates to be rectified as occasion requires, one of which is called the Axis, and the other two are perpendiculars. Fiftly, the lines described on the Face of the Quadrant are sixfold. Viz. The Limb and its Parallels. The Calendar and its divisions. The Aequator and its Parallels. The Ecliptic and its divisions. The Hour lines and their intermediates. The line of Shadows and its divisions. First, the Limb is the outmost Circle, which is divided into 140. gr. and noted at every 10. degree thus, 10. 20. 30. 40. 50. 60. 70. 80. 90. and each of these degrees is divided into parts according to the Capacity of a degree in the Instrument. Secondly, the next parallel line to the Limb is the Horizon or Calendar, which is noted with letters thus, I.A. S.O. N.D. I.F. M.A. M.I. of which I. in the first place stands for july, A. for August, S. for September, O. for October. etc. then again on the inside, I. stands for january, F. for February, etc. each letter representing its Month, and each of those Months is divided into days by small short lines, whereof the 10. and 15. day of every Month is signified by Numbers, or else by a line somewhat longer than any of the others, to help the memory the readier to Number, and for more promptness of finding the day of the Month, in the Calendar as occasion requireth. Thirdly, The Equater is that line that meeteth with the tenth of March, and the thirteenth of September in the Calendar, and is divided into degrees, and numbered thus, 10. 20. 30. 40. 50. 60. 70. 80. 90. and the parallels to the Equator are these lines which are on each side of it, every 5. degree of which being noted thus, 5. 10. 15. 20. the outmost of those Parallels on each side of the Equator are the two Tropickes that which is nearest the Centre, is called the Tropic of ♋, and that which is farthest of, is called the Tropic of ♑, those two Tropics, the Calendar, and the hour of 12. comprehend the whole Projectin: and here note farther that these parallels are called parallels of the day of the Month, as well as the parallels of the Sun's Declination, according as they shall be used, and farther below the Tropic of Cancer is a graduation of the common hours of a horizontal Dial: some use of whsch is shown, by pro. 36. Fourthly, the Ecliptic on the instrument is represented by two quarters of the Ecliptic which crosseth the former parallels, and meeteth with the Equator, in the Horizon or Calendar, in the former 10. of March, and 13. of September: that Quarter which is towards the Centre of the instrument, serves for the Northern semicircle of the Ecliptic, and that which is farther from the Centre serves for the Southern semicircle of the Ecliptic; & each of these semicircles is divided into the Signs of the Zodiac, & charactered accordingly thus, ♈. ♉. ♊. ♋. ♌. ♍. ♎. ♏. ♐. ♑. ♒. ♓. Of which the first 6. Signs are called Northern signs and are in the Northern semicircle, & the other 6. Southern signs & are in the Southern Semicircle. And each of those signs is divided into 30. gr. and if the Instrument be large, each of these degrees may be divided into 6. or 12. divisions more, So every division shall accordingly contain 10. or 5. Minute's. Fiftly, the hour lines are those that cross the Aequator and his parallels, and are noted, or numbered in the Tropic of ♋ with numeral Characters thus four V VI VII. VIII. IX. X. XI. XII. And are the forenoon hour notes: those hourelines serve also for the afternoon hours, and are noted likewise with Arithmetical figures, for the hours in the afternoon thus, 1. 2. 3. 4. 5. 6. 7. 8. each of those hours is divided into 3. parts, each part being 20. minutes: and each of those parts is subdivided again into 5. parts, so that each part containeth 4. minutes, and so the whole hour is diveded into 15. parts or degrees, each part or degree being 4. minutes as afore, and so the whole hour shall contain 60. minutes or parts: and here note that these Hour circles with their intermediates are also called Meridian's or degrees of measure, and are Numbered by ten in the Aequator, from the meeting of the Aequator with the Ecliptic, as before thus, 10. 20. 30. 40. 50. 60. 70. 80. 90. Sixtly, the line of shadows is that which makes a spherical Equilateral Triangle upon the plain of the Instrument, the basis of which is the Horizon, or Calendar and one of whose legs is below the Tropic of ♋ and the other crosseth the Tropic and parallels, and meeteth with the Calendar near in the 10. of December: both of those equal sides are called the line of shadows, and are divided alike into 10. unequal divisions, and each of those divisions again is divided into 10. other divisions, and again each of them into other 10. (if the Instrument be large.) The first Capital 10. divisions are noted with Arithmetical figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9 10. of which 1. is at the very meeting of the two lines, not fare from the Centre, which signifieth Equal: the figure of 2. Double: the figure of 3. Triple, the figure of 4. Quadruple: the figure of 5. Quintuple, &c, Thus for the making, and description of the Instrument, the use of it now followeth. Of the uses of the horizontal Quadrant, specified in the Index or Table, formerly delivered. OF which some have relation to the observation or appearance of the Sun, others without observation, or sight of the Sun. The uses of these which are known without seeing the Sun, are 30. of the said Index or Table, as followeth, viz. the 2. 9 10. 12. 13. 14. 15. 16. 17. 18. 19 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 32. 33. 35. 37. 38. 39 40. 41. and 42. of which 13. of them will be shown only by knowing the day of the Month, viz. the 18. 33. 21. 19 25. 24. 35. 2. 38. 37. 20. 41. and 42. as followeth. The day of the Month known to find. 1. The hour of Sun rising, setting, and length of the day. 2. The Sun's difference of Ascention. 3. The Sun's Declination. 4. What days are alike in length, and what day the Sun rising in the one, shall be the Sun setting in the other. 5. The Sun's place, or degree in the Ecliptic. 6. The Sun's right Ascension, and obliqne Ascension. 7. The hour, and Altitude of the Sun's coming East, or West. 8. The distance of the Sun's rising, or setting, from the East, or West. 9 The height, or depression of the Sun in the Meridian here, or for any Latitude. 10. The time of day break, and end of twilight. 11. The inequality of time, between day break, and Sun rising. 12. The hour, and Altitude of the Sun's coming upon any declining wall. 13. At what hour, and Altitude, the Sun must have to be opposite or perpendicular, to a declining wall. First, to find the time of Sun Pro. 1 rising, or setting, and length of the day, for any day of the year. Seek the day of the Month in the Calendar, Constru ¦ ctio. 1 and the hour line that meeteth therewith, showeth the time of Sun rising, or setting. So if the day of the Month were the 13. Exam. of October, the parallel that meeteth therewith is the hour, viz. 7. of the clock, at which time the Sun riseth: the same hour is noted also with 5. which is the time of Sun setting that day, this doubled makes 10. the length of the day required. Secondly, to find the difference Pro. 2 of Ascention, for any day of the year. Mark what Meridian meeteth with the day Constru ¦ ctio. 2 of the Month in the Calendar: as suppose the day to be the former 13. of October, which is the hour line of 7. and 5. (as before) and account the Numbers of Meridian's to the hour of 6. so have you 15. gr. or an hour, which is the difference of Ascension for the 13. day of October required. Pro. 3 Thirdly, to find the Sun's declination for any day. Constru ¦ ctio. 3 Mark what parallel of Declination meeteth with the day of the Month in the Calendar, and account how many degrees it is from the Equinoctial, so have you the Sun's Declination for that day. Exam. So if the day were the last of August, the parallel that meeteth therewith is the 5th. from the Equator, and somuch is the Sun's declination, that day, viz. 5. gr.. North declination. Pro. 4 Fourthly, to find what days in the year are alike in length, and what day the Sun rising in the one, shall be the Sun setting in the other. Constru ¦ ctio. 4 For the first, note that the days between the 10th of December and the 11th. of june, are days of Increase, and the rest are days of Decrease. Now right against any day of decrease in the Calendar, is the day of increase, which days are equal one to the other. Exam. So the 19 day of May, is against the 4. of july, at which time the Sun riseth and setteth alike without sensible error, viz. 4. of the clock, and therefore those days are of equal length, and so of others. For the second, to find what day the sun rising in the one, shall be the sun setting in the other. Admit the day to be the 18th. of February and according to the first pro. find the time of Sun rising, which is at 40. m. after 6. Exam. of the clock Constru ¦ ctio. 5 for that day, and the setting 20. m, after 5. then mark what day of the Month the hour line of 20 m, after 5. in the forenoon, meeteth with the Calendar, which will be the 23. of August, so the 18th. day of February the Sun did set at the same hour that it did rise, the 23. day of August. Fiftly, to find the sun's place, or Pro. 5 degree, for any day of the year. Note where the parallel of the day of the Month crosseth the Ecliptic, that is the Suns Constru ¦ ctio. 6 place. So the former parallel of the 13th. of October meeteth with the Ecliptic in the beginning of ♏, and ♓, but which of these is the Sun's place, the quarter of the year may easily tell you, viz. ♏ which is the Sun's place or the degree in the Ecliptic for that day. Pro. 6 Sixtly, to find the Sun's right ascension, and obliqne ascension at any time. Constru ¦ ctio. 7 Consider what Meridean meeteth with the Sun's place in the Ecliptic for the day given, and mark the number of Meridian's in the Equator (for the Meridian's are numbered in the Equator, as is said before in the description) so have you the Sun's right Ascension: but here note that the degrees in the Ecliptic are numbered forward and backward in the Ecliptic unto 360. gr. upon this Instrument: so are the right Ascensions of those degrees also numbered forward and backward in the Aequator: for the right ascension of any degree in the Ecliptic, is that degree of the Aequator which is opposite unto it, (the succession of the signs considered) so if the Sun were in the beginning of ♏, the right Ascension is near 208. degrees, for the Meridian that passeth by the beginning of ♏, is accounted in the Equator from ♈, and is within 6. m, of 28. gr. Now the right Ascension of ♋, is 90. gr. and the beginning the of ♎, is 180. gr. and from the beginning of ♎, to the beginning ♏, is within 6. m, of 28. gr. as before, all is which put together makes near 208. gr. the right Ascension of the Sun the 13th. day of October. To find the Sun's Obliqne Ascension at any time. Note that the difference of Ascension, is the Constru ¦ ctio. 8 difference always between the right Ascension of the Sun, and the obliqne Ascension thereof: therefore the right Ascension known by the last direction, & the difference of Ascension by the second direction, the obliqne Ascension is easily had, by Addition, or substraction thus. If the Sun be in a Southern sign than the obliqne Ascension, is greater than the right Ascension, by so much as the difference of Ascention comes to: but if the Sun be in a Northern sign, the obliqne Ascension is so much less: which difference of Ascension as before by the 2 Pro: for the said 13th. of October was 15. gr. this ad unto the right Ascension of the beginning of ♏, viz. 208. gr. makes 223. gr. the Sun's obliqne Ascension for the beginning of ♏, on the 13th. day of October; but if the Sun had been in the beginning of ♉, the obliqne ascension would have been only near 13. gr. viz. 12. gr. 54. m. Seventhly, to find the sun's Altitude, Pro. 7 and hour of the suns coming East, or West, any day of the year above the Horizon. Here note that this Proposition holds in use Constru ¦ ctio. 9 only for that time of the Suns being in the Northern signs that is from the 10th. of March to the 13th. of September: therefore lay the Index to the East, or Equinoctial point noted with E. or 40. & 50. in the Limb: so have you instantly at once without farther rectification both the Altitude and hour of the Suns coming East or West, above the Horizon for all or any of the days above specified: so the parallel of any day of the Month meeting with the edge of the Index gives the Sun's Altitude in the Index, and the Meridian meeting therewith shows the hour. Exam. So if it were the second of May, or the 22. of july, the parallel belonging to those days meets with the Index near about, 23. gr. 17. m, and there also meets with that point, the hour line of 7, and 5. which showeth that when the Sun is 23. gr. 17. m, high either upon the second of May or the 22. of july; then the Sun will be due East or West, and that will happen to be at 7 of the clock in the forenoon, and 5. of the clock in the afternoon. Pro. 8 Eightly, to find the distance of the suns rising, or setting, any day of the year, from the East, or West, called the sun's Amplitude. Constru ¦ ctio. 12 Lay the Index to the day of the Month, for the time given, & the edge of it in the Limb of the Instrument shall show the Amplitude required. So if the day were the 13th. Exam. of October the number of degrees from the points of East, noted with 40. 50. unto the Index is 18. gr. 40. m. which is the Sun's Amplitude for the given day, viz. the 13th. day of October. Ninthly, to know the suns Meridional Pro. 9 Altitude, or the sun's depression under the Horizon, at Midnight here, or in any Latitude, for any day in the year. Lay the Index unto the hour of 12, and where Constru ¦ ctio. 11 the parallel of the day of the Month meeteth that therewith shall be the Sun's Meridional Altitude. So if it were the 13th. day of October, as before, Exam. the parallel, for that day is 11. gr. and a half from the Equator South: this crosseth the Index in 27. gr. which is the Sun's Meridional Altitude that day. Now for the Sun's depression at midnight, here is to be noted, that any degree of the Ecliptic is at any time so much below the Horizon, as his opposite degree in the Ecliptic, is above the Horizon at the same time. Constru ¦ ctio. 12 Therefore where the contrary parallel of the Sun viz. 11. gr. and a half North, meeteth with the Index in the hour of 12. that shall be the Sun's Meridional depression at midnight, the said 13th. day of October. Pro. 10 Tenthly, to find the time of daybreak, and end of twilight, with the Position of the sun under the Horizon for any time. This proposition, hath reference to the Sun's depression under the Horizon, for it is said to be day break or twilight to end, when the Sun is, 18. gr. under the Horizon: therefore the Construction in this will be thus. Constru ¦ ctio. 13 Account 18. gr. on the Index then move the Index until that degree meet with the Contrary parallel of Declination for the day given, so the Meridian or Houreline that meeteth therewith shall be the hour of day break required. Exam. So if the day were the 10. of April, the parallel of Declination for that day is North II. gr. and a half which I seek out one the other side of the Aequator viz. II. gr. and a half South Declination, and Mark where the 18th. gr. of the Index meeteth therewith, for there also is the hour of day break viz. with in 20. m. of 3. in the Morning, and 20. m. past 9 for the end of twilight the said 10th. of April, also the Index in the Horizon at that Instant showeth the position of the Sun under the Horizon viz. near 48 gr. 10. m. to the North of the East: but if the day had been the 13th of October the hour of daybreake had been 2 minutes before 5. and twilight would have ended 2. m. after 7. Eleventhly, to find the inequality Pro. 11 of time, between day break and Sun rising, for any day of the year assigned. By the first Construction, for the days given Constructio. 14 find the time of Sun rising, and by the former 13th Construction the hour and time of day break belonging to those days: then compare the time between the Sun rising, and day break of the one, with that of the other, so the difference of those two, shall be the difference of time required. Example. H. M. H. M. So on the tenth of March the Sun rising is at 6. 00 The time between day break and Sun rising is. 2. 0 the difference is, 12. m. day break is at 4. 00 Docemb. the Sun rising it at 8. 12 2. 12 day break, is at 6. 00 the difference is. 1. ho. 22. m. May the Sun rising is at 4. 11 3. 34 day break, at 12, 37 So the difference of time between day break and Sun rising the 10th of December is near a quarter of an hour longer than that of the 10th of March; but more than an hour and half longer between day break, and Sun rising the 10th of May, than the 10th of March. Pro. 12 twelvely to find the hour, and Altitude of the sun's coming upon a Declining wall any day of the year. ☞ Seeing the declinations of Plains, or Walls, are accounted from the points of East or West in the Horizon, as the sun's Amplitude is the numbering of them therefore shall be alike, in the Limb of the Instrument. Now admit the Declination of a Plain or Wall, to be 22. gr. the operation Constru ¦ ctio. 15 would be thus. The Index being set thereto, you may instantly see at what hour the Sun will come upon the Plain, for any day in the year; for where the parallel of the day of the Month crosseth the Index amongst the hourelines, (which Index represents the Plain) that is the hour of the Suns coming upon the Plain and the degrees in the Index gives the Sun's Altitude. Examp. So if the Sun were in the Tropic of ♋ the Tropic meeteth with the Index almost within 5 m. of 9 in the Morning, and at that time the Sun cometh upon the Plain, and there the Tropic cuts also the Index in 45. gr.. 40. m. which is the sun's Altitude at that time that the Sun will glance or begin to shine upon the Plain. As for the time of the sun's continuance on the Plain (as is specified in the Index or Table) account the Declination, on the other side of the East point, and lay the Index thereto, so the edge of it in the Tropic of ♋, will point out at what hour the Sun goes of the Plain viz. at 6. of the clock & 38. m, near, if the declination were West, (as here it is supposed) which added to the time of the suns coming on the Plain, makes 9 hours 33. m, & so long the sun shines on the Plain. Thirteenthly, to find at what Pro. 13 hour, and Altitude the sun must have to be opposite, or perpendicular to a declining Plain, any day in the year. Let a Plain decline from the East point towards the South 22. Example. gr. account this in the Limb Constru ¦ ctio. 16 from the hour of 12. and lay the Index thereto, so the parallel that crosseth the Index, doth show the Sun's Altitude, and the Meridian meeting therewith, gives the hour, at which time the Sun will be opposite to the Plain; so have you at one instant for every day in the year, at what hour and Altitude, the Sun will be opposite to the Plain. gr. m. As admit the days were these. Decem. the tenth, the Sun's place at which time is in ♑ and the hour of the Suns being opposite to the plain, that day would be at 40-m past. 1 and the Sun's Altitude at that time would be, 12.12 March. ♈ 14 m. past. 1. 36.25 june ♋ 48 m. past 12 60.45 Thus touching the resolution of the former 13 uses of the aforesaid Table, or Index which had reference only to the knowledge of the day of the Month, there are 13. other uses of the foresaid Index, or Table viz. the 10. 12. 32. 39 40. 23. 17. 16. 26. 27. 28. 29. 30. Which have no dependence upon the sight of the Sun, of which the 6 first are resolved, only by knowing the day of the Month, and the other 7. are as followeth. viz by knowing the day of the Month to find. 1. At what hour the shadow of an Altitude is equal, double, triple, etc. unto it. 2. At any hour and Altitude of the Sun, or Azimuth, what proportion shadows have to their bodies. 3. The hour of the day agreeable to any Altitude, or Azimuth. 4. The Sun's depression and Azimuth at any hour of the night Assigned. 5. The hour of the day to our Antipodes, by suposing the Sun's Depression under the Horizon. 6. At what hour in any day the Sun's Azimuth and Altitude will be equal, and how much the Altitude and Azimuth will be. To find. 7. What number of days will make the day an hour longer, or shorter at any time. 8. Th● inequality of time in equal months, or equal number of days. 9 The degree of the Aequator in the Horizon, by supposing any degree of the Ecliptic in the Horizon. 10. The degree of the Ecliptic in the Horizon by supposing the degree of the Equator in the Horizon. 11. The degree of Medium Coeli, or the degree of the Ecliptic in the Meridian, by supposing the degree of the Ecliptic in the Horizon, vel contra. 12. The Horoscope, or the degree ascendant, or descend●nt, and the Nonagessima degree at any hour. 13. What Angle the Ecliptic makes with the Horizon, or the Altitude of the Nonagessima degree, & what Azimuth it is in at any hour. First, to find the Proportion of Pro. 14 shadows to their Altitudes at any time. As if it were required the 20 of April, Declaratio. at what hour of the day, and how high the Sun must be either in the forenoon or afternoon, that the shadow of a man or any Altitude, shall be equal unto his height double, triple, quadruple Quintuple etc. Constru ¦ ctio. 17 Lay the Index unto the numbers in the line of shadows viz. to 1. 2. 3. 4. 5. etc. and wheresoever any of those divisions in the line of shadows meet with the Index amongst the degrees; there it showeth what height the Sun must have, to make the shadows equal, double, triple etc. to the Altitude. Exam. So laying the Index upon 1 in the line of shadows, it meeteth with 45. gr. in the Index: & so high the Sun must be to make the shadow of a man or any thing equal to his height upon an horizontal plain: then move the Index to and fro, until the said 45. gr. in the Index meet with the parallel of the day Month, viz the 20. of April, so the hour line that meeteth therewith, is the hour of the day that the shadow of a Man: or other Altitudes, will be equal to his height or Altitude, viz near 10. of the Clock in the forenoon, or 2. of the Clock, in the afternoon. forenoon. afternoon. gr. m. ho. m. ho. m. And according to the same directions when shadows are Double. The Altitude would be. 26. 33 and the hour the said 20. of April would be, 7. 37 4. 23 Triple. 18. 26 6. 43 5. 17 Quadruple. 14. 2 6. 16 5. 44 Quintuple. 11. 19 5. 58 6. 02 Sextuple. 9 27 5. 46 6. 14 Septuple. 8. 7 5. 37 6. 23 Octuple. 7. 7 5. 31 6. 29 N●nocuple. 6. 20 5. 25 6. 35 Decuple. 5. 43 5. 21 6. 39 Vigecuple. 2. 51 5. 2 6. 58 Secondly, to find what proportion Pro. 15 shadows have to their bodies at any hour in the day, Azimuth, or Altitude of the Sun asigned. If the hour be known, or supposed, move the Constru ¦ ctio. 18 Index until it meet with the hour in the parallel of the day of the Month, so the intersection of that parallel with the Index is the sun's Altitude, and the edge of the Index, in the Limb, will show the Sun's Azimuth, then move the Index until the degree of Altitude intersect the line of shadows, so shall you have the proportion of shadows, to their bodies required. So if on the 11th. of April at, 7. Exam. of the Clock in the forenoon, (if the Sun shine,) it were required what proportion the shadow of a man shall bear to his height, or the shadow of an Altitude to the Altitude, the parallel that belongeth to the given day is near 12. gr. Mark where this parallel meeteth with the given hour of 7. and bring the Index to it; so have you the Sun's height at that hour viz 18. gr. 26. m, and the edge of the Index in the Limb of the Instrument, shall give the Azimuth viz 4. gr. from the East: then move the Index, until the degree of the Sun's Altitude viz 18. gr. 26. m, meet with the line of shadows which will be in 3, which showeth that at 7. of the Clock in the forenone the said 11th. of April, the shadow of a man, or the shadow of an Altitude, shall be Triple to his height: the like will be at 5. of the Clock in the afternoon, for equal distances of the Sun from the Meridian the same day, without sensible error, will give equal Altitudes of the sun, and equal Altitude of the sun doth produce equal shadows upon horizontal Plains Constru ¦ ctio. 19 Secondly, if the Position or Azimuth of the Sun be known or supposed, which admit 4 gr. from the East towards the South. Lay the Index unto it in the limb, & mark what degree in the Index the parallel meeteth with, which is with 18. gr. 26. m, so have you the sun's Altitude in the Index: then move the Index until the degree meet with the Line of shadows; so have you the proportion of shadows required at that instant, viz Triple as before. Constru ¦ ctio. 20 Thirdly, if the Sun's height be known or supposed, which admit 18. gr. 26. m, account it in the Index, and move the Index until that degree meet with the line of shadows; so where it intersecteth the line of the shadows, there you have the proportion of shadows to their bodies at that instant of time required, which will be triple as before; so the 10th. of April if the hour be 7. or the Altitude 18. g. 26. m, or the Azimuth 4. gr. from the East toward the South, the proportion of shadows to their bodies will be Triple. Thirdly to find the hour of the Pro. 16 day agreeable to any Altitude or Azimuth, for any day of the year Proposed. For the first account the Sun's Altitude in the Constru ¦ ctio. 21 Index, and move it to and fro until that degree meet with the parallel of the day of the Month: so the Meridian that passeth by that point, shall be the hour required. Thus if the day were the tenth of March, Exam. the Sun being that day in the Aequinoctial, & the Altitude supposed to be 32. gr. 37. m, this I seek out upon the Index and move the Index till that degree meet with the Aequator; so the Meridian or hour Circle that passeth thereby is the hour viz. 10. of the Clock in the forenoon or 2. of the Clock in the afternoon, and if you move the Index softly along as the degrees of the Sun's Altitude in the Index intersect the Aequator (and so of any parallel:) so the Meridian that meeteth therewith is the hour of the day agreeable to that Altitude. For the second, to find the hour Constru ¦ ctio. 22 of the day agreeable to any Azimuth. As suppose it were 36. gr. 35. m, Exam. from the South. Move the Index in the Limb unto this Azimuth known or supposed; so where the Index crosseth the parallel for the day given, there the Meridian that meeteth therewith, shows the hour of the day viz 10. of the Clock in the forenoon or 2 in the afternoon as before. And if you move the Index softly along, as the Index passeth by any Azimuth in the Limb: so the edge of the Index shall intersect the parallel of declination for the day of the Month, in the hour of the day agreeable to that Azimuth: by which proposition and the last, Glasses may be easily placed to burn according to the Sun's Azimuth, or hour assigned. Pro. 17 Fourthly, to find the sun's depression, & position under the Horizon, at any hour of the night, with the hour of the day to our Antipodes, by supposing the sun any number of degrees under the Horizon. Constru ¦ ctio. 23 By the 11th Construction it is said that any degree of the Ecliptic, is as much below the Horizon at any time, as his opposite degree is above the Horizon at the same time: therefore if the Index be laid to the like parallel, on the contrary side of the Aequator, that meeteth with the given hour the intersection in the Index shall show you the degree of the Sun's depression under the Horizon at that hour. So if at 10 of the Clock at night the said 13th of October it were required to find the Sun's depression under the Horizon, Exam. consider the declination, or the sun's parallel for that day, which is 11. gr. and a half South, which declination I seek in the other side of the Aequator, and mark where it meeteth with the hour of 10. unto which I lay the Index, so the edge thereof in the Limb showeth the sun's Azimuth to be ne'er 42 gr. 30. m, from the South, and the parallels intersection that meeteth with the Index, gives the Sun's depression, viz. near 43. gr. and so much is the Sun below the Horizon, and in that position the 13th of October at 10. of the Clock at night. But if it were required at what hour of the Night the Sun would touch the vertical Circle of East and West under the Horizon. Lay the Index to the point of East and mark Constru ¦ ctio. 24 where about the Contrary palle● meeteth with the Index, for there you have both the hour and the degree of the sun's depression. So the day being as before the 13th of October, Exam. & the declination south 11. gr. and a half, this account amongst the North declinations & it meeteth with the Index in 38 m, past 6. the hour of the Suns being West, and with all the sun's depression, at the same time is near 14. gr. and 50. m. Pro. 18 Fiftly, to find the hour of the day to our Antipodes, by supposing the sun's depression under the Horizon. Constru ¦ ctio. 25 Consider the declination for the day, and move the Index to and fro until the degree of the sun's depression in the Index, meeteth with the like parallel or the other side of the Aequator, so the hour that meeteth therewith is the hour of the day to our Antipodes. Exam. So if on the 20th. of April, we should suppose the sun to be 13. gr. under the Horizon, & desire to know the hour to our Antipodes, the parallel of declension for that day is 15. gr. North, Now in the Index account 13. degrees and move it to and fro until the said thirteenth degree in the Index meet with the 15th. parallel of South declination, so the Meridian that meeteth therewith is the hour of the day to our Antipodes, within 2 m, of 9 at night. Pro. 19 Sixtly, to find at what hour in any day, the suns Azimuch and Altitude will be equal, and how much the Altitude and Azimuth will be. Construction. Move the Index, to & fro until the edge of the Index meet with the parallel belonging to that day, in the same Number of degrees that the end of the Index in the Limb from the point of East doth; so have you the degree of the Sun's Azimuth, and Altitude equal the one to the other, and the Meridian meeting with the Index in the parallel of the given day, showeth at what hour that Azimuth, and Altitude will be equal. So admit the sun to be in the Tropic of ♋, Exam the Index being moved to and fro until there be like degrees in the Index, and in the Limb, which will be near 16. gr. 45. m, and there the hour that meeteth therewith is 12. m, after 6. in the forenoon, at which hour the eleventh of june, the Sun's Azimuth, and Altitude, will be equal viz. near 16. gr. 45. m, as before. Seventhly, to find what number Pro. 20 of days any time of the year, will make the day an hour longer or shorter. Account 7. gr. and a half amongst the Meridian's Construc ¦ tio. 27 from the given day in the Calendar, and note the day of the Month against it, than number the days between that day and the given day, and you have the answer. So if the day were the last of February, Exam. or the first of March, consider the Suns setting that day by the Instrument, which is 40. m, past 5. this doubled makes the length of the day, 11. hours 20. m, then from the last of February account 7. gr. and a half and it will point out the fifteenth of March at which time the Sun seateth at 10. m, past 6 which doubled makes 12 hours 20. m, so the fifteenth of March, the length of the day is an hour longer than it was the first of March; and the difference of time only but 15. days, but if the number of days were accounted to or from the Suns entering into the Tropical points, it will be more than 35. days before the day will be an hour longer or shorter. Exam. So if from the tenth of june we should account 7. gr. and a half amongst the Meridian's from that Meridian that meeteth with the tenth of june, it would fall out at the 16th. day of july, at which time the day will be an hour shorter than it was the tenth of june, and the interval of time more than twice as much as the former viz. 39 days. Pro. 21 Eightly, to find the inequality of time, in equal Mounthes or equal number of days. This proposition at the first seems as a Paradox, yet by this Instrument may easily be resolved, and so consequently from Mathematical principles demonstrated, not only the inequality of equal Months, but also the inequality of Natural days. Now a day natural according to the general definition is one revolution of the Aequator or primum mobile, that is from sun rising to sun rising: or it is the time wherein the sun passeth by the Meridian, and cometh to the Meridian again, commonly taken for 24. hours: but be cause that in that intervale of time the sun passing from the Meridian and cometh to the Meridian again, the Sun moves according to his Natural motion (secundum antiquiorum traditionem) near a degree more or less; therefore the Natural day shall be some what longer or shorter than 24. hours, viz. by so much as the difference of right ascension of that degree of the Ecliptic comes to that the sun is in, and seeing the degrees of the Ecliptic amongst themselves have not the same difference of right Ascension that the other degrees have, (notwithstanding the degrees of the Ecliptic amongst themselves being equal the one to the other) the sun's motion ender those degrees being sometimes quicker, and sometimes slower, it will necessarily follow that the sun will move more or less until the sun can touch the Meridian, which is the limit or term of the suns diurnal revolution as before: this difference and inequality of time in natural days may by calculation be given from day to day, but because it is so insensible little in a day, hardly by an Instrument of this nature can be seen, but by a number of days, compared with another number of days it will evidently appear. So, Exam. if it were required how much the Month of December is longer than the Month of March, in the first of which months the sun's motion is quicker, being about the Perigeum then at other Constru ¦ ctio. 28 times, now both of which months have equal number of days, viz. 31. Find the right ascension for the beginning and ending of Mar. viz. 350.0. the difer. of right ascension for the Month of Mar. is 29.30. the difference between these is 5. gr. 19 30. beginning and ending of Dece. viz. 257. ¾ the difer. of right ascension for the Month of Dece. is 34. 30. 292. ¼. which 5. gr. being converted into time by allowing 4 minutes to a degree makes about 20. m, and so much is the Month of December longer than the Month of March, notwithstanding both of these Months containing equal number of days. Pro. 22 Ninthly, to find the degree of the Aequator in the Horizon, by suppossing the degree of the Ecliptic in the Horizon. Notatio. If the degree given be in the Northern part of the Ecliptic, the obliqne Ascension is less than the Constru ¦ ctio. 29 right Ascension vel contra. Get therefore first the right Ascension of the point given by the sixth Pro. and the difference of Ascension by the 2. Pro. for that taken from the right Ascension gives the degree of the Equinoctial in the Horizon, but if the given degree had been in a Southrens sign, the difference of Ascention must be added to the right Ascension, so have you the degree of the Aequator in the Horizon. Tenthly, to find the degree of Pro. 23 the Ecliptic in the Horizon by supposing the degree of the Aequator in the Horizon. This is but the Converse of the former, only Constru ¦ ctio. 30 consider the correspondent quarters of the Equinoctial to these of the Ecliptic. Eleventhly, to find the degree of Pro. 24 Medium Coeli, or the degree of the Ecliptic in the Meridian, by sup sing any degree of the Ecliptic in the Horizon. Seek the degree of the Aequator in the Horizon, Constru ¦ ctio. 31 by the 22. Pro. subtract, 90. from it (if the Number be too little add a whole Circle to it) than the degree of the Ecliptic opposite to the remainder, is the Answer, but note that if the remainder be between 270. and 360. the opposite point belongs to the last Quarter of the Ecliptic, if the remainder be between 180. and 270. then it respects the 3 quarter of the Ecliptic, if the remainder be between 90. and 180. it hath reference to the second Quarter. etc. But if the degree of the Ecliptic in the Horizon were required by knowing the degree of the in Ecliptic the Meridian. This, is only but the converse of the former, & Constru ¦ ctio. 23 is thus performed first, seek the right Ascension of the given degree of Medium Coeli, & add thereto 90. gr. by accounting it from the former right Ascension, & note the suns place opposite thereto for the difference of Ascension of this last degree being subtracted from the former degree of the Aequator in the Horizon, if it be a degree of the Southrens signs (otherwise Add) gives the degree of the Ecliptic in the Horizon demanded. Twelfthly, to find the Horoscope Pro. 25 or the degree Ascendant, or descendant and the Nonagessima degree at any hour. First, note the right Ascension for the day given Constru ¦ ctio, 33 according to the 6. Pro. which is the degree of the Aequator in the Meridian, at 12. of the Clock, unto which degree add 90. so have you the degree of the Aequator in the Horizon at 12. of the Clock. Then consider how many hours the given hours wants of 12. or is past 12. which converted into measure and accounted Eastward, or Westward, according to the hour given from the former points of the Aequator in the Horizon at 12. will give the degree of the Aequator in the Horizon at the hour proposed, then by the 23. Pro. I seek out the degree of the Ecliptic in the Horizon answerable to the degree of the Aequator so have you the degree Ascendant, from which account 90. gr. or 3 signs, so have you the degree of the Nonagessima point in the Horizon, but if you reckon 6. signs from the Ascendant, you have the desendant degree of the Ecliptic in the West of the Horizon. Pro. 26 Thirteenthly, to find what Angle the Ecliptic makes with the Horizon, or the Altitude of the Nonagessima degree of the Ecliptic, above the Horizon, and what Azimuth it is in at any hour. According to the last Pro. finde the degree Constru ¦ ctio. 34 Ascendant, and the Nonagessima degree, then by the 24. Pro. find what degree of the Ecliptic is in the Meridian, Answerable to the degree of the Ecliptic in the Horizon, so shall you know on which side of the Meridian the Nonagessima degree is, & how far from the Meridian, then if that Index be laid upon the hour of 12, where the parallel of the Nonagessima degree crosseth it, that should be the height of it, if it were in the Meridian; account therefore from the Meridian or hour of 12. in the Aequator, the number of degrees between the Nonagessima degree, and the degree of the Ecliptic in the Meridian, & mark where that Meridian meeteth with the parallel of the of the Nonagessima degree, lay the Index thereto, so have you the Altitude of the Nonagessima degree in the Index, and the Azimuth in the Horizon, or Limb of the Instrument. There are yet the 48. 49. 50. 9 13. 14. and 15th. uses of the said Index or Table, which have no relation to the sun's sight or observation in there operation, and resolutions, and should have followed these 26. uses that have been delivered: but I refer them to the end of this Tractat; as for these uses of the Instrument which depend upon the Sun's sight, or observation they are these 13. following viz. the 1. 31. 4. 6. 36. 7. 22. 3. 5. 34. 11. 43. and 8th. viz. to show 1. The Sun, or stars Altitude above the Horizon at any time. 2. The hour of the day, and Azimuth of the sun. 3. The Meridian Line upon any appearance of the Sun. 4. The sit of a building, or costing of a place. 5. The Sun's Azimuth, and hour without Observation. 6. The variation of the Needle. 7. The Latitude of a place, or height of the Pole above the Horizon. 8. The Sun's Azimuth, and Altitude at any hour. 9 The uncertainty of time, by noting the shadow of things. 10. The Quarter of the year and day of the Month, with the hour, Azimuth, and the Meridian line. 11. Instantly the hour of the day, the Azimuth, and Altitude of the Sun: with the Meridional line, without observation or sight of the Sun, by knowing the Proportion between the length of a shadow upon a Horizontal Plain, and that which did cast the shadow. 12. The Declination of a Wall, by seeing the Sun beginning to shine thereon or going from it. 13. The Declination of a Wall, the Sun shining thereon. First, how to observe the Sun, or Pro. 27 stars Altitude above the Horizon at any time. Lift up the edge of the Instrument to the eye, Constru ¦ ctio. 35 so that the sight which is at the Limb or Circumference of the Quadrant be next the eye, and the Index to hang perpendicular and to play easily by the side thereof: then move the Quadrant up and down until you may through both sights see the Centre or middle of the Sun, or star: so the Index in the Limb shall fall upon the degrees of the Sun or stars Altitude above the Horizon at that time. Or without looking at the sun, the Altitude thereof may be thus found: hold the Quadrant Constru ¦ ctio. 36 that the Index may hang perpendicular, or be vertical as before, then move about the Instrument until the edge of it be opposite to the body of the Sun. Now supposing the Instrument to hang thus upon his Centre, softly lift up the edge thereof which is towards the Sun, until you see the beams of the sun pass through both sights, than the Index in the Limb shall give the sun's Altitude as before. Pro. 28 Secondly, how to find the hour of the day, and Azimuth of the Sun, upon any appearance of the Sun. Constru ¦ ctio. 37 By the last Pro. observe or take the sun's Altitude and account it on the Index, then seek for the parallel of the day of the Month for the day present, & move the Index until that degree of Altitude in the edge of the Index meet with the parallel of the day, so the Meridian that meeteth with that degree of Altitude in the Index, shall be the hour of the day required, & the edge of the Index in the Limb of the Instrument, shall likewise show the Sun's Azimuth belonging to that hour. So if upon the last of August the Sun's Altitude in the forenoon should be observed and found to be 30. Exam. gr. & a half, seek this Altitude out upon the Index & move the Index until the degree of Altitude meet with the parallel for the day of the Month given, viz. the fift parallel from the Aequator Northward so the houreline that meeteth also with the 30. gr. & a half in the Index, is the hour viz. near 9 & that shall be the hour of the day at that instant, & the edge of the Index in the Limb cutteth near 35. gr. and 30. m, from the point of East, towards the South, and so much is the Sun's Azimuth at that time. Thirdly, how to find the Meridian Pro. 29 line, and the true points of North, & South, East, and West upon any appearance of the Sun. According to the 27. Pro. first observe the Constru ¦ ctio. 38 Sun's Altitude above the Horizon, and by the last Construction find the Sun's Azimuth agreeable to that Altitude: let the Index and rest at that degree, and erect the prependicular at the end of the Index, then holding the plain or face of the Quadrant parallel to the Horizon, move the Instrument Circular, until the shadow of the said perpendicular fall by the side of the Index, and so the hoverline of 12, or the edge of the Instrument which is parallel unto it (which is the North and south edge of the Instrument) shall represent the Meridian line, and pointeth out the North and South in the Horizon of the world by the terms thereof, and the other strait edge of the Instrument which is perpendicular unto that edge is the (East and West edge of the Instrument) and denoteth or showeth the line of East, and West in the Horizon, of the world. But this may be more accurately done if you place the back of the Instrument down upon an horizontal plain, and the edge of the Index being at the degree of the sun's Azimuth observed, and the perpendicular erected at the end of the Index as before: then moving the Instrument as it so lieth until the shadow of the perpendicular fall by the side of the Index, so the Meridian of the Instrument, shall be in the Meridian of the World, and every point and degree in the Limb of the Instrument shall point out, and be opposite, and represent his like degree in the Horizon of the world. Constru ¦ ctio. 39 But here note that this Construction serves only but for the forenoon observation; for if the practice be in the afternone, the way to find the Meridian line may be thus. Having found the sun's Azimuth as before, lay the Index upon the hour line of 12. and erect the perpendicular at the end thereof, and move the Instrument about Circular, until the shadow of the said perpendicular fall by the side of the Index: for than if the edge of the Index be moved unto the sun's Azimuth before known, the edge of the Index shall represent the Meridian line, & 90. gr. farther shall be the point of East, and the Centre of the Instrument the point of West, therefore if upon the plain that the Instrument lies upon, you make a mark at the edge of the Index which is in the Meridian as before, and another mark right under the Centre and so place the North and South edge of the Instrument unto these two points: then every degree in the Horizon. or Limb of the Instrument, shall point out as before his opposite or ●ike degree in the Horizon of the world. Fourthly, how to find the sit Pro. 30 of a Building, or Costing of a place. By the last Pro. find out or draw the Meridian Constru ¦ ctio 40 line,. and place the North and South edge of the Instrument unto it: if the Building or Place lie in the Eastern semicircle of the world (but if it lie in the Western semicircle, then let the East & West edge of the Instrument be placed upon the Meridian line) so the eye being over the Centre of the Instrument, and beholding the place, let the Index be moved until it be also with the visual line observed by the eye, that is opposite to the place, so the edge of the Index, from the Cardinal points of the Instrument in the Limb, viz. from the East or West, North or South, shall show the bearing of that place from you, in respect of the Cardinal points of the world in the Horizon: but if two sights be placed at the Index (which is according to the description thereof) then may you observe the place through the sights of the Index by letting the Instrument rest, and moving the Index to and fro until you see the object, so the edge of the Index in the limb, shall point out the bearing or Position of the place from you in degrees from the East, West, North, or South, & accounting 11. gr. and ¼ as often as you can in those degrees, observed: you have the point of the Compass which the place, or object bears from you. Pro. 31 Fiftly, to find the sun's Azimuth, and hour without observation. The Meridian line being drawn first upon a Constru ¦ ctio. 41 plain according to the former directions, consider if it be in the forenoon or afternoon; if in the forenoon, then let the North, and South edge of the Instrument be placed unto the Meridian line, but if it be in the afternoon, than set the edge of East, & West of the Instrument, unto the Meridian line, and let the Instrument rest there, then erect the perpendicular at the end of the Index, & move the Index about until the shadow of the perpendicular fall by the side of the Index, so the edge of the Index will amongst the degrees in the Limb show the Sun's Azimuth at that time, and where the edge of the Index meeteth with the parallel of the day of the Month, that is the hour of the day at that time. But if the Axis be rectified then there is no need of a Meridian line to be drawn, for this Instrument will with great facility find out his own Meridian, by moving it to and fro until the shadow of the perpendicular which is over the Centre of the Instrument, intersect the same hour in the Parallel of the day of the Month, that the Axis doth amongst the Common hours▪ so that hour shall be the hour of the day for that instant, and the shadow of the said perpendicular, cutting the Limb, or extended unto it, doth there show the Sun's Azimuth, and so the Meridian of the Instrument at that position, shall be in the Meridian of the world required. Sixtly, to find the variation Pro. 32 of the needle. By the twenty nineth Pro. upon an even Plain parallel to the Horizon draw the Meridian line, & Constru ¦ ctio. 42 place the North & South line of the Card directly over the said Meridian line, so the Number of degrees that the Needle cutteth in the Card from the North and South line of the Card, that shall be the variation of the Needle required; otherwise it may be found thus: near unto the Centre of the Index, upon the Index may a small Brosse pin be so placed that it may be erected perpendicular to the Centre of the Instrument and half an inch above it. Let a Needle by placed upon this pin, then lay the East, and West edge of the Instrument to the Meridian line, & when the Needle resteth, move the Index, until the edge of it be directly under the Needle so the edge of the Index; in the Limb of the Instrument, shall point out or show the Needle's variation required. Pro. 33 Seventhly, to find the Latitude of a place, or the Poles hight above the Horizon. Constru ¦ ctio. 43 First, draw the Meridian line upon some plain by help of the 38. Construction, then erect the prependicular at the end of the Index, and place the North and South edge of the Instrument, to the Meridian line so drawn upon the plain, and move also the Index until the edge thereof touch the hour of 12. let the Instrument rest at this position, then mark diligently about noon or 12. of the Clock when the shadow of the perpendicular doth fall by the edge of the Index, for then the sun is in the Meridian, at which time according to the 27. Pro. observe or take the sun's height (which is his Meridian Altitude, for that day) and by the 3. Pro. find the Sun's declination agreeable to that day, and add it to the Sun's Meridional Altitude observe (if it be South declination, otherwise subtract it from the former Meridional Altitude,) so have you the height of the Aequinoctial above the Horizon, that taken from 90. gives the depression of the South Pole under the Horizon, which is always equal to the elevation of the North Pole above the Horizon. So if upon the tenth of April, Exam. the Meridian Altitude should be found to be 50. gr. the Declination belonging to that day by the 3. Pro. is 11. gr. and a half North, which being subtracted (according to the former directions) leaves 38. gr. 30. m, the height of the Equinoctial above the Horizon: & that taken from 90. leaves 51. gr. 30. m the depression of the South Pole under the Horizon: or the elevation of the North Pole above the Horizon, for the height of the Equinoctial known, the Compliment thereof is always the Latitude of the place, or height of the Pole: and here note generally that the height of the Pole and Equinoctial together, do always make a Quadrant or 90. gr. therefore the height of one of them being known, the height of the other is also known, and further here note that if the sun have North Declination, the sun is so much higher than the Equinoctial at none that day, by so much as his Declination, cometh to, but if the Sun have South Declination, than the Sun is lower than the Equinoctial that day at noon, by so much as his Declination cometh to, by which you may easily gether when to add, or subtract the sun's Declination to, or from the suns Meridianall Altitude to get the height of Aequator, which known the Poles height cannot be unknown. Pro. 34 Eightly, to find the sun's Azimuth and Altitude for any hour. Constru ¦ ctio. 44 Mark where the parallel for the day of the Month meeteth with the given hour, and bring the edge of the Index thereto, so the degree that the edge of the Index cutteth in the Limb of the Instrument that shall be the Sun's Azimuth, and the degree that the hour cutteth in the Index, that shall be the Sun's Altitude required. So, if upon the tenth of December at nine of the Clock in the Morning, Exam. the Sun's Azimuth and Altitude were required, mark first where the Tropic of Capricorn (which is the parallel, for that day given) meeteth with the given hour of nine, and bring the Index thereto, so the edge of it in the Limb pointeth out near 40. gr. and a half, & so much is the Sun's Azimuth, from the South, at nine of the Clock in the forenoon, the said tenth of December, and the hour line meeting with the Index, showeth near 5. gr. 25. m. so much is the sun's Altitude at that time; now if you move the Index softly along, as the edge of it passeth by any hour for any day of the year, so the edge of the Index in the Limb of the Instrument showeth the sun's Azimuth, and the intersection of the parallel with the Index shall show the Sun's Altitude belonging to that hour. Ninthly, to show the uncertainty Pro. 35 of time, by noting the shadow of things. It is usually noted by some, that when the shadow of the edge of a Window, Door, Wall, or such like, shall touch such or such marks, that it shall be then such, or such an hour of the day, and so constantly to hold for all the year, this observation is fare from truth, and the principals of Astronomy (and may be easily contradicted by such which have but indifferent judgement in the Nature of shadows, and the Sun's passages by the Meridian's and vertical Circles of the Heavens, for by how much greater the propinquity of the Sun's approachment is unto the Zenith, or vertical point, by so much the more shall the hour or time be various in one and the same Azimuth. So in the last Pro. the Azimuth of the Sun the tenth of December, at nine of the Clock in the forenoon, was found to be 40. gr. and a half, Exam and the Sun's distance from the Zenith, at that time was near 84. gr. 35. m, Now admit the Sun's distance from the Zenith the tenth of june were but 32. gr. 35. m, the Sun being in the same Azimuth, the hour would be half an hour past 10. For the Index being laid to the hour of 9 in the Tropic of ♑. (which is the Sun's parallel, for the said tenth of December,) and it cutteth the Constru ¦ ctio. 45 parallels of the Sun's Motion in the inequality of time, and so the compliment of the former 32. gr 35. m, in the Index, meeteth with the Tropic of ♋, (which is the Sun's parallel for the tenth of june) in half an hour past 10. so that it evidently appears, that the shadow of a perpendicular thing on the tenth of December, denoting the hour of the day to be 9 of the Clock, the same shadow the tenth of june, shall represent half an hour past 10. so the error shall be an hour and a half: but if you move the Index unto the hour of 9 belonging to the tenth of june, the Index shall point you out in the Limb near 68 gr. of Azimuth for that hour, which at 9 of the Clock the tenth of December, was but 40. gr. & an half, so the difference of Azimuth in one and the same hour, shall be 27. gr. and a half, & the time as before, an hour and a half: which differences are sufficient to confirm the point. Tenthly, to find the Quarter of Pro. 36 the year, and day of the month, if it were forgotten. Constru ¦ ctio. 46 At any appearance of the Sun by the 27. Pro. take the Sun's Altitude, then place the North and South edge of the Instrument unto the Meridian line formerly, drawn (if in the forenoon) otherwise place the East, and West, edge of Instrument to the Meridian-line, and erect the prependicular at the end of the Index, then move the Index to and fro until the shadow of the prependicular fall by the side of the Index, so the parallel that meeteth with the degree of the Suns observed Altitude, in the edge of the Index, parallel in the Calendar that shall show the day of the Month required. So if, Exam. upon a certain day in the year the sun's Altitude were observed and found to be 36. gr. having placed the edge of the Instrument to the Meridian line, and rectified the Index, then move the Index, until the shadow of the prependicular fall by the edge of the Instrument, let the Instrument rest at this position, and account the former 36. gr. upon the Index, which degree meeteth with the hour in the Aequator, and also that intersecteth the Calendar, in the tenth of March, & the thirteenth of September, but which of these days is the day of the Month, the next day's observation of the Sun upon the same hour will help you, for if the sun's Altitude besound to be greater than the day of the month inquired after it was the tenth of (March, because the sun from the tenth of December unto the eleventh of june, doth every day at one & the same hour, ascend,) but if the Sun's Altitude be found to be less than the former day's observation specified was, than the day required, was the thirteenth of September, because that from the eleventh of june, unto the tenth of December, the sun's Altitude every day doth sensibly diminish at one and the same hour. ☞ But here is to be noted that if there be no Meridional line, than the prependicular over the Constru ¦ ctio. 47 Centre and the Axis of the Index being erected, place down the back of the Instrument upon an horizontal plain, and move the Instrument to and fro, until the shadow of the Axis meet with the same hour below the Tropic, amongst the common hours, that the shadow of the prependicular over the Centre of the Instrument meeteth with on the face of the Instrument, for then the parallel that crosseth or meeteth with the shadow of the prependicular, and the hour, will in the Calendar show the day of the Month required, and so then the Meridian of the Instrument shall be in the Meridian of the world, and every point or degree in the Horizon of the Instrument, it shall point out his like, or opposite degree in the Horizon of the world. Constru ¦ ctio, 48 Or otherwise it may be done thus, take the Sun's Altitude, then immediately by some Watch, clock, or Sundial, learn the hour of the day, and move the Index to and fro, until the Sun's Altitude in the Index, meet with the former hour, so the parallel that meeteth therewith, shall show the day of the Month in the Calendar required, then having the day of the Month, you have the Quarter of the year, for from the tenth of March unto the eleventh of june, is the Spring quarter, from the eleventh of june, to the thirteenth of September, is the Summer quarter, from the thirteenth of September, to the tenth of December is the Autummuall quarter, and from the tenth of December, unto the tenth of March, is the Winter Quarter. eleventhly, to find the hour of Pro. 37 the day, the Azimuth and Altitude of the Sun, with the Meridional line without observation, or sight of the sun, by knowing the proportion between the length of a shadow upon a Horizontal Plain, and that which cast the shadow. First, let the thing that casteth the shadow, or Constru ¦ ctio. 59 something equal in length unto it, be divided into ten equal parts, and each of those parts subdived into ten other equal parts, (which thing so divided shall represent a common scale,) then measure the shadow with the scale, and mark how often the shadow is longer than the scale, and the decimal part if there be any, so have you the proportion between the shadow, and that which did cast the shadow, and then is it resolved accor- to the converse of the fifteenth Pro., mathematical diagram Exam. Admit some one: upon the 12. of February, or on the ninth of October, holding a staff prependicular as A B, or supposing it to be part of the Coin of a House, or edge of a Window or such like should cast a shadow, as B, C, which being noted, or drawn and having divided the staff, or thing as before, and should then measure the shadow, as B, C, by the said staff or scale, and find it to be contained therein three times, and 6, parts or 6, decimals, the porportion of the Gnomon, or scale, A B, to the shadow B C, would be as 1 to 3. and 6/10. Construc ¦ tio. 50 Move therefore the Index to and fro, until the edge of it mere with 3. and 6/10, in the line of shadows; so have you the degree of the Sun's Altitude at that instant in the Index, viz. 15. gr. and ½ then seek out the parallel for the 12. of February, or the ninth of October (the day given) which is near the tenth degree from the Aequator South, move the Index, until the former 15. gr.. and ½ in the Index, meeteth with the said parallel for the day, so have you the hour belonging to that time, which will be near 42 m, past 8. in the Morning, or 18. m, past 3 in the after noon, and the edge of the Index in the Limb of the Instrument showeth the sun's Azimuth also at that instant, viz, near 39 gr. 12. m, from the East toward the South. Now for the Meridional line, this may be done Constru ¦ ctio. 51 at any time after, if the Azimuth be not forgotten: for if the Centre of the Instrument be laid down upon any part of the shadow B C, and so the Instrument to be moved upon his Centre until the said shadow B C, formerly drawn, cut the edge of the Limb, in the aforesaid Azimuth of 39 gr. 30. m, than the Meridian of the Instrument shall be in the Meridian of the world, and if that shadow were from a Window, or Building, the position of the Instrument, shall denote the position of the Window or Building. Twelfthly, to find the Declination Pro. 38 of a Wall; by seeing the sun beginning to shine thereon, or going from it. By the 27. Pro. take the height of the Sun, and Constru ¦ ctio. 52 by the 28. Pro. find the Sun's Azimuth for the Altitude, so the Azimuth thus found shall be the declination of the Plain required: for the declination of any perpendicular Plain, is accounted from the points of East, West, North or South, in the Horizon, as the Sun's Azimuth is: therefore whatsoever Plain is in the plain of any virticall Circle, that Plain is as far from any of the Cardinal points of the Horizon, as the sun is at that time, & so the Sun, being in that virticall Circle, shall necessarily glance upon the Plain: and therefore look what the Sun's Azimuth is at that instant, such shall be the Declination of the Plain required. Thirteenthly, to find the Declination Pro. 39 of a Plain, upon any appearance of the Sun. mathematical diagram Thus for the Construction of the aforesaid 13. uses which did depend upon the sun's observation, the 48. 49. 50. 9 13. 14. and 15th. uses of the Index or Table against Page the first, should have followed; but before I speak of them it will not be inconvenient that I apply the Instrument unto the resolution of the 44. 45. 46. & 47. uses of the aforesaid Index or Table, which have reference to night observation, upon such Stars which are, or may be placed on the face of the Instrument, between the two Tropickes, or under the Tropic of Cancer, according to there Declinations, and right Ascensions: which are these following. The names of the Stars Decli. Rec: The names of the stars Decli. Rec. G. M. H. M. G.M. G.M. H. M G. M. Ex. Alae Pegasi 13.9. N. 23.54. 1. 30. Co● Hydrae. 7.5. A 9 10. 42.45. pri. ♈. ●1. 40. N. 1.46. 26. 30. Cor Leonis. 13.45. B. 9 48. 33.00. Oculus. ♉. 15.42. N. 4.15. 63.45. Cauda ♌. 16.38. B 11.30 7.26. pri singuli. Ori. 0. 17. S. 5. 13. 78.15. Spica. ♍. 9 ●0. M. 13. 5. 16.15. Canis maior. 16. 1● S. 6. 30. 82.15. Arcturus. 21.10. B. 14. 0. 29. 30. Canis minor. 6. 9 N. 7.20. 70.00. Aquila. 8. 00. S. 19.32 66.45. Much may be said upon the uses of these stars, but for brevity I only delivere these four examples following. 1. First, for any night of the year, to find at what hour, and Altitude any of the said stars will be in the Meridian, (that so they may be known.) 2. To know at any day, at what hour any of these stars riseth, or setteth, with their time of continuance above the Horizon, and in what part of the Hemisphere, they may be seen with their Azimuth, and Altitude at any hour. 3. Thirdly, to find in any night at what part of the Horizon, any of the aforesaid stars riseth, or setteth, and at what hour, and Altitude they will be due East, or West. 4. Fourthly, upon the sight or appearance of any of the said Stars, to find the Azimuth thereof: and the hour of the night. Pro. 40 First in any night, to find at what hour and Altitude, any of the aforesaid stars will be in the Meridian. Constru ¦ ctio. 54 By the sixth Pro: find the Sun's rigth Ascention for the day given, which converted into time by allowing for every 15. degrees an hour, and for every degree 4. m, then subtract this right Ascension of the Sun, from the stars right Ascension, so the remainder or difference of time, shall show how many hours the stars shall come later to the Meridian than the Sun: but if the subtraction cannot be made, then add 24. hours to it & you have the Answer, Exam. so, if upon the sixth of February, it were required to find at what hour any of the aforesaid stars will be in the Meridian, or due South, first therefore by the said sixth Pro. I find the suns right Ascension for the day given viz, 330. gr. which contains three nineties or 270. each 90. gr. being six hours, and so the whole 270. gr. makes 18. hours, and the other 60. gr. at 15. gr. to an hour makes 4. hours more all which put together makes 22. hours: so the right Ascension of the Sun the sixth of February, is near 330. gr. as before, or 22. hours H. M. H. M. Which 22. hours taken from the right ascension of the aforesaid Stars, viz. from. Ex. Ala Pegasi. 23.54. there remains. 1. 54. P. the time of the stars being in the Meridian. Pri. ♈. 1. 46. 3. 46. P. Oculus. ♉. 4. 15. 6. 15. P. Orion singu. 1. 5. 13. 7. 13. P. Canis Maior 6. 30. 8. 30. P. Canis Minor. 7. 20. 9 20. P. Cor Hydra. 9 10. 11. 10. P. Cor Leonis. 9 48. 11. 48. P. Cauda. ♌. 11.40. 1. 40. A. Spica verginis. 13.05. 2. 05. A. Arcturus 14.00. 4. 00. A. Aquila. 19.32. 9 32. A. For seeing that 22. hours the Sun's right Ascension, is greater than the right Ascension of any of the stars afore specified, subtract this 22, hours from 24. hours, rest 2. hours, which added to the right Ascension of each Star before delivered, you have the hour of the Stars coming to the Meridian: hence you may gather which of those stars, are out of observation for that time, viz. Alae Pegasi, Pri. ♈, and Aquila, which come to the Meridian in the day time: but if the day given had been the 26th of july, the right Ascension of the Sun, that day is near 135. gr. or 9 hours. H.M. H.M. which 9 hours taken from the right Ascension of the aforesaid stars viz, from. Ex. Alae Pegasi. 23. 54. leaves. 2. 54. A. the time of the stars coming to the Meridian. Pri. ♈. 1. 46. 04.46. A. Oculus ♉. 4. 15. 07. 15. A. Orion Singu. 1. 5. 31. 08. 13. A. Caenis Maior. 6. 30. 09.30. A. Canis Minor. 7. 20. 10. 20. A. Cor hidra. 9 10. 00.10. P. Cor Leonis. 9 48. 00.48. P. Cauda. ♌. 11. 40. 02.40. P. Spica vergenis. 13.5. 04. 05. P. Acturus. 14. 00. 05. ●0 P. Aquila. 19.32. 10.32. P For the right Ascension of the Sun being but 9 hours take it from the right ascension of Cor. hidra which is 9 hours 10. m, rest 10. m, which showeth that Cor. Hydra comes to the Meridian 10. m, later than the Sun that day, that is, 10. m, after 12 and so the rest, whose rightascention is greater than the Suns. But for these stars, whose right Ascension is less than the said 9 hours, subtract this 9 hour from 24. hours, rest 15. hour (or rather subtract it from 12. rest 3. hours) this add unto the right ascension of any of the aforesaid stars, as suppose Canis Minor makes 22. hours 20. m, which showeth that Canis minor, will come to the Meridian. 22. hours 20, m, later that day than the sun: therefore this, 22. hours and 20. m, being considered according to an hourly account showeth, that Canis Minor will come to the Meridian at 10. of the clock and 20. m: of the next day (the right ascension of the Internal of time being neglected) or for brevity add the aforesaid 3. hours unto the right ascension of these Stars, whose right ascensions are lesser than the Suns, so have you the Meridional hour required. Hence may be gathered that Alae Pegasi, Pri. ♈ and Aquila, are only for observation that night, the other stars are out of observation, and will come to the Meridian, in the day time. Lastly, to find the Meridional Altitude of any of these stars, lay the edge of the Index unto the hour line, of 12, so the parallel of the stars declination that crosseth the edge of the Index, shall there show you in the Index, the Meridional Altitude of the star required. Pro. 41 Secondly, to know at any day, at what hour any of the stars (inscribed on the Instrument) riseth or setteth, with their time of continuance above the Horizon, & in what part of the hemisphere, they may be seen, with their Azimuth, and Altitude at any hour. Constru ¦ ctio. 55 By the last direction find the hour of the stars being in the Meridian, then mark what hour the parallel of the declination of any star intersecteth the Horizon or Calendar, so have you the hour or time of the stars rising or setting, and the number of hours, from that point of the stars rising in the Horizon, unto the Meridian being doubled, gives the countinuance of the stars above the Horizon, required. So if upon the 6th. of February, it were demanded at what hour Oculus ♉. Exam. would ascend, & how long it would continue above the Horizon. By the last proposition, get the hour of the stars being in the Meridian, which is at 6. of the Clock and 15. minutes at night, and mark the Number of hours between the Meridian, and that point where the parallel of Oculus ♉, meeteth with the Calendar, which is 7. hours 24, minutes, this doubled makes 14. hours 48. m, and so long will Oculus ♉, be above the Horizon. But if from the said 7, hours and 24. m, the said 6, hours 15. m, be taken, there will rest 1. hour 9 m, and so much before 12. of the clock at noon, doth Oculus ♉ rise, that is 51. m, after 10, of the Clock, and so consequently if the said 7. hours and 24. m, be added unto the hour of the stars being in the Meridian, viz. 6. of the Clock and 15. m, as before, the said star will set at 39 m, past 1, in the Morning. Lastly, if at any hour between the rising of the star, and the setting thereof, it be required at what Position and Altitude the star is in. It is thus done. Account to the given hour, from the hour of the star rising, setting, or being in the Meridian, Constru ¦ ctio. 56 (in the parallel of the stars declination) and lay the Index thereto, so the edge of it in the Limb of the Instrument, shall show the stars Azimuth or Position, and where the parallel of the stars Declination crosseth the edge of the Index, that shall be the stars Altitude, at that hour. So if on the said 6th. of February, at 11. Exam. of the Clock at night, it were required in what Position, or Azimuth Oculus ♉, was in, and also how high above the Horizon: I make, or suppose the hour of 12. to be the aforesaid 6. of the Clock and 15. m, (for at that hour as before Oculus ♉ was in the Mridian) and from thence in the stars parallel of Declination, I account until I come unto 11. of the Clock, viz. that is 4. hours, and 45. m, from 12. and lay the Index thereto, so the edge of the Index in the Limb, pointeth out 4. gr: 24. m, and so fare Oculus ♉, is distant from the West at 11. of the Clock at night, and the parallel of the stars Declination meeteth with the Index in 24. gr, near, which is the stars Altitude, at that hour required. Pro. 42 Thirdly, to find in any night of the year, in what part of the Horizon any of the stars on the instrument riseth or setteth, and at what hour, and Altitude a star will be due East, or West. Constru ¦ ctio. 57 For the first, Mark where the parallel of the stars declination crosseth the Horizon, or Calendar, Lay the edge of the Index hereto, so the number of degrees between the edge of the Index, and the point of East or West, upon the limb of the Instrument, showeth the distance of the stars rising from the East or West. So if it were required in what part of the Horizon Oculus ♉ riseth, Exam. mark where the parallel of the stars Declination crosseth the Horizon, and lay the edge of the Index thereto, so it cutteth the Limb of the Instrument from the East near 26. gr. and so fare Oculus ♉, riseth from the East towards the North. For the second to find the time of a stars coming East, or west. By the 40th. Pro. consider at what hour the star Constru ¦ ctio. 58 is in the Meridian, then lay the edge of the Index to the point of East and West, and account in the parallel of the stars Declination the number of hours between the edge of the Index, and the hour of 12. which being taken from the hour of the stars being in the Meridian, gives the hour of the stars coming East, but added unto the hour of the stars being in the Meridian, shows the hour of the stars being West. So if it were demanded at what hour, Exam. upon the 6th. of February, Cor ♌, would be due East or West, and what Altitude the star should then have. First, lay the edge of the Index, to the point of East and West, & wheresoever the parallel of the stars declination crosseth the edge of the Index that shall be the stars Altitude, viz. near 17. gr 45. m, then account the number of hours in the parallel of the stars Declination between the edge of the Index, and the hour of 12. which is near 5. hours and 12. m, which taken from the hour of the stars being in the Meridian,) which by the 40th. Pro. was at 11. of the clock & 48. m, at night) rests 6 hours, and 36. m: but if the said 5. hours and 12. m, be added unto the said 11. hours and 48. m, it makes 17. hours, from which 12. being taken leaves 5, hours. So upon the 6th. of February, Cor ♌ shall be due East, at 36. m, past 6, at night, and due West, at 5, of the Clock in the Morning, and the Stars Altitude, being either East or West, is near 17. gr. 45. m, as was required. Pro. 43 Fourthly, upon the sight or appearance of any of the aforesaid stars, to find the Azimuth thereof, and the hour of the night. By the 40th. Pro. for the day given find the Constru ¦ ctio. 59 hour of the stars coming to the Merid●● then by the 27. Pro. take the stars height 〈◊〉 account that height in the Index, the● 〈…〉 Index until the degree of the starred 〈…〉 the Index, meet with the parall● 〈…〉 Declination, so the edge of the 〈…〉 showeth the stars Azimuth, and the Meridian that meeteth with the degree of the Altitude, in the Index shall show you the hour that the star wants to be in the Meridian, or is past the Meridian, which added, or subtracted from the hour of the stars being in the Meridian, gives the hour of the night required. So if the day were the 26th. of july, Exam. and if Aquila, should be observed to be on the West of the Meridian, 29. gr. 20. m, high above the Horizon, this I seek out upon the Index, and move the Index to and fro until the said, 29. gr. 20. m, meet with the parallel of Declination, of Aquila, so the edge of the Index, in the Limb doth point out the stars Azimuth from the South, viz. 63. gr. 12. m, and the Meridian that meeteth with the aforesaid degree of Altitude, is the time of the stars distance from the Meridian, viz. near 3. hours and 28. m, this added unto the hour of Aquilas being in the Meridian, which by the 40th. Pro. was at 10. of the Clock & 32. m, at night, makes 14. hours, or 2 of the Clock, in the Morning, so if Aquila were observed the 26th. of july, to be 29. gr. 20. m, high to the West of the Meridian, than the Position or Azimuth of that star from the Meridian, was 63. gr. 12. m, and the hour at that instant, was at 2. of the Clock in the Morning. Thus touching the resolution of the aforesaid 44. 45. 46. and 47th. Pro. of the aforesaid Index or Table, which did belong to Astronomical observations, the last uses now follow, viz. 48. 49. 50. 9 13. 14. and 15th. uses of the Index or Table, which are only proper to Geometrical Practices, viz. to show 1. How to measure the Quantity of an Angle, or to take the distance of two Stars. 2. How to measure distances and bredthes. 3. How to take the Circuit of a figure, or the survey of a Place. 4. The inclination of a Plain, or to Place a Plain horizontal. 5. Whether an Altitude be in the Point of libration, or above, or below the level of the eye, and how much. 6. How much the height of an Altitude is above the eye, which is accessable, or in accessable. 7. How to measure any Part of an Altitude, which is not approachable. First, how to observe or find the Pro. 44 measure of an Angle, or take the distance of two stars by the Instrument. Let the Istrument be placed upon some Rest, Constru ¦ ctio. 60 which may be so accommodated that the Instrument, may be elevated, depressed, or be placed horizontal, as occasion requires, then erect the sights of the Index, & place the edge of the Index upon the hour line of 12. the Index so placed look through the sights thereof and moving the Instrument upon his Rest to and fro, until you see the mark or Star, that makes the Angle or distance required. Then screw fast the Instrument to the socket, and move about the Index, until through the sights thereof you see the other mark or star, so the number of degrees between the edge of the Index and the hour of 12. in the Limb of the Instrument shall be the measure of the Angle, or distance of the two Stars sought for. Let E and D be two marks or Stars, Exam. and let the Angle E A D, or distance E D, be required. The Instrument, A B M, being placed upon his Rest G H I K, observe one of the marks or stars as D, through the sights (admit A B,) so the visual line shall be A B D, then having made fast the Instrument, move the Index A B, until through the sights of it you see the other mark or star, E, which suppose to be in the visual line, A C E, so the Ark of the Limb of the Instrument B C, shall be the distance between the two Stars, ED, or the measure of the Angle E A D, Corollarie. required. Now Infinite are the uses of knowing the Quantity of Angleses, in the Copious and vast Body of Mathematical Practices, therefore from the multitude of Examples that might be raised upon them, or extracted from them, I will only instance for brevity, upon these two plain, and ordinary ones following. mathematical diagram Secondly, how to measure distances, Pro. 45 and Breadths. Let G F L H, Exam. represent part of the Perimeter of a Fort, and let it be required, that standing at some convenient place without Musket shot, as admit at Y, the distance between the points of the Bulwark viz. F & L, as also the measure of the face of either of the Bulwarks viz. F E, or M L, with the length of the Curtain D N, and all the distances from Y, viz. Y F, Y E, Y M, and Y L, were required. Or suppose O P Q R, were 4 places, whose several distances the one from the other as from O to P, then from P to Q, and from Q to R, & also the several distances from Y viz. Y O, Y P, Y Q and Y R, were demanded. The Construction and resolution upon either of these is alike, therefore Constru ¦ ctio. 61 we will instance upon the latter. Place the Instrument upon his Rest, at Y, and the edge of the Index, upon the hour of 12. then looking through the sights of the Index, upon some mark taken at pleasure in the field, which, admit to be S, then observe the first mark R, so have you the Angle S Y R, which suppose 52. gr. 30. m, then look to Q, so have you the Angle SYQ, which let be 63. gr. 15. m. then look to P, so ha●e you the Angle SYP, admit to be 74. gr. 15. m, and lastly look to O, so have you the Angle S Y O, 123. gr. 15. m: for these Angleses are taken with great facility, when once the Instrument is rectified as in the first direction is specified, for you need not but move the Index, Circular from object to object, so the Arkes of the Limb of the Instrument as before, from the hour of 12, unto the edge of the Index, shall show the measure of the several Angles observed. Thus at Y, place up a Mark, and in the visual line, Y S, and measure a certain distance at pleasure, as admit to S, and suppose it were found to be 900. foot (or 300. yards) then placing the Instrument at S, upon his Rest, and laying the edge of the Index to the hour of 12. I move the Instrument about, until through the sights of the Index I may see the mark which was set up at my, last station, then make fast the Instrument, and observe O, so have you the Angle Y S O, which suppose to be 26. gr. 50. m, then look to P, so have you the Angle Y S P, which let be 55. gr. 50. m, then look to Q, so have you the Angle Y S Q, which admit to be 60. gr. 15. m, lastly observe R, so have you the Angle Y S R, 87. gr. In like manner may you observe the Angles at the Fort from the stations Y, and S, formerly specified all which observations may be placed down in Tables, as here under appears, which may be called the Tables of observed Angles. mathematical diagrams Ang: G. M. Distances Ang: G.M. Distances. Y. S. 900. R Y S. 52. 30 S R. 1099. 5 S Y L. 55. 45. S L. 1071. 0 Q Y S. 63. 15 S Q. 963. 6 S Y M. 68 15 S M. 1182. 2 P Y S. 74. 15 S P. 1132. 2 S Y E. 89. 10. S E. 1364. 4 O Y S. 123. 50 S O. 1526. 1 S Y F. 104.00 S F. 1459. 5 Y S O. 26. 50. Y O. 829. 2 Y S F. 39 15. Y F. 971. 6 Y S P. 55. 50. Y P. 973. 2 Y S E. 49. 35. Y E. 1039. 2 Y S Q. 60. 15. Y Q. 937. 0 Y S M. 66. 45. Y M. 1169. 4 Y S R. 87. 00. Y R. 1384. Y S L. 79. 30. Y L. 1268. 2 Now touching the resolution of the point, there is a triple way of operation, viz. either Arithmetical, Instrumental, or Geometrical, each of which being sufficiently facile, to such which are versed in the documents of Mathematical Practices, but the later because it is more vulgar, and easiest to be apprehended, I will instance here upon: which is that part of Geometry, commonly called Pretraction, a thing so common that almost every one that hath any entrance in Geometry, can perform it according to the ordinary way they are instructed in. But to facilitate that kind of practice, I advice such as affect this kind of Practice to use the Protracter, which I use, which is a plain then sector, having a small hole at the Centre, whose two legs from the Centre are made with a sharp edge, but so that they lie flat upon the Paper, & each of them to be divided into 100 or 1000 divisions with a Quadrantall Ark or more divided and fastened also at the end of one of the legs, but so that the Quadrantall, Ark have also an edge to lie flat upon the Paper, and to slide in at the other leg, so shall it be accommodated and made a fit, & apt Instrument to find the Quantity of Angleses, in a Plot, or to protract Angles, for service, as followeth. So to search out the distance of O P Q R, Exam. the one from the other, or all the distances from Y as was required: by the help of the former Angles of observations and Protracter, it may be done thus. mathematical diagram Thirdly, how to take the Circuit Pro. 46 of a figure, or the survey of a Place. Let ABCD, Exam. be a plain to be raised as Fortifiers have it, or a field to be plotted as surveyors account it, or a Figure whose Perimeter is required, as Geometritians treat of it. The Plain Table, Notatio. may be held best for this service, as such would have it, whose learning is altogether versed therein. But any Instrument shall be able to do this Service, that can but accuratly take or measure any Angle, (not that we reject that, but make use of this for the present) and therefore in this action, it were no loss of time to make a preambulation about the field, to view the several windings and turnings thereof, and what Angles with greatest Conveniency, and expedition are to be observed, and what might be omitted, and at the Angles of consequence there to set up some mark, and upon those Angles to fabricate the whole work: for here especially is to be noted, that the more Angles that are observed in any practice, by way of Circumscribing a Field, or Campaigne, the greater, and more evident shall the error be in the Conclusion. mathematical diagram So in the Figure A B C D, Exam. there is eleven Angles and as many sides, now if at every Angle, an observation should be made, it would be more. subject to error (as before) then if less Angleses were observed, therefore in this Diagramme fewer Angles of observation may be fully sufficient to raise that Plain, Take the Plot of the Field, or give the Perimeter of that Figure, Therefore suppose the noted Angles of Consequence to be, A, B, C, D, Q, the work may be then thus. Place the Instrument, upon his rest at A, and observe the Angle Q AB, which suppose 32. gr. 10 m, than measure Q A, with a decimal Chain (or such like,) which suppose to be 5. Chains, note this in a piece of paper, then take the Instrument up, and measure the line, A B, but first only A E, which suppose to be 11. Chains and 60 Links, which is written down thus. 11. 60. Then measure the distance from E to the Angle F, which admit to be 2. 20, Lastly, go one with the measure A B, which suppose to be 15. 00, the Angle of observation, and measures thus taken may be noted down one against another, as in the Table following, then place the Instrument Constru ¦ ctio. 63 upon his rest at B, and observe the Angle: ABC, which note down also and measure the distance, B G, and G H, and then going on with G B, to G I, and mark and measure B I, and then measure also I K, and so go on with B C, which measures are all placed down as appears in the Table of Angles and measures following: In like matter perform the rest of the work until you come to Q, and so all the Angles and measures will be according to the Table here under specified. The Table of Angleses. The Table of measures. G.M. 5. 00. 32.10. 11. 60. 2.20. 15. 00. 80.10. 6. 00. 1.00. 9 20. 2.00. 16. 40. 79.30. 6. 90. 1. 29. 8. 00. 74. 45. 3. 70. 1. 15. 5. 40. mathematical diagram But it had been fully sufficient (by help of this Protractor) to have plotted the aforesaid Plain, by knowing the former Measures, and two Angles of observation only. Pro. 47 Fourthly, how to find the Inclination of a Plain, or to Elevate a Plain unto an Angle assigned, and to Place a Plain horizontal. Constru ¦ ctio. 65 For the first, Set the East and West, edge of the Instrument unto the Plain, then if the edge of the Index in the Limb of the Instrument, cut the point of East or West, the Plain is vertical, and doth not Incline, but if the Index fall from the points, look how many degrees it is from the points of East or West in the Limb of the Instrument so much is that Inclination of the Plain. For the Second, to Elevate a Plain, to an Angle assigned. This is only the same with the former, but may be applied to several uses, as to try the mount, or to mount a Piece of Ordinance at any Random: or to place Burning glasses (or others) at several Angles, to receive each others Reflexon, and that the point of concourse, or inflammation in such Glasses may be in the Radius or beam of the Sun, or that the point of inflammation, the representative Image, or the extensive Elumination may be projected to a point assigned. For the Third, to rectify a Plain horizontal. Constru ¦ ctio. 66 Place the North and South edge of the Instrument, unto the under face of the Plain, and then mark if the edge of the Index, cut the points of East or West in the Limb of the Instrument, for then the Plain is horizontal, but if it swarm from that point, than it is not Horizontal, but the Plain is to be raised, or depressed, until by several trials in sundry parts of the Plain, you see the edge of the Index fall upon the points of East or West, for than shall it be truly horizontal: Otherwise you may rectify the Plain horizontal, by operating upon the upper face of it, if you set a Cube upon the plain, and then placing the East and West edge of the Instrument unto the side of the Cube, for then the observation will be as the former, and therefore, accommodated & concluded accordingly. Pro. 48 Fiftly, to find whether an Altitude be in the Point of libration, or above, or below the level of the eye, and how much. Declaration. Let C B and X, be three several objects, and let their several situations be required. Constru ¦ ctio. 67 First, let the Instrument hang upon a rest perpendicular, and let it be held steadfast that the Index may be vertical, and play easily by the side of the Instrument, then looking through the sights of it, lift the Instrument up and down, until you see your mark, which suppose first C, and admit the Index should cut 5. in the line of shadows, which showeth that C, is higher than the eye by the 5th. part of the distance of the bassis of the object, from the eye, supposed at A. Secondly, if through the sights of the Instrument you see the second object, B and the Index falling upon no part in the line of shadows, than it showeth that the point B, is level with the eye, for if in any observation the Index fall between the beginning of the line of shadows, (which is near the beginning of December) and the sight next the eye, it argueth that the object is higher than the eye, but if the Index fall beyond the beginning of the line of shadows, than the object is lower than the eye. Thirdly, if you observe the object X, (the Constru ¦ ctio. 68 eye being at A,) then if in observing the Mark X, through the sights of the Instrument, the Index shall fall beyond the beginning of the line of shadows, that is from the Calendar number the degrees, in the Limb from the edge of the Index unto that point, and account the same backward from the point in the Limb that is opposite to the beginning of the line of shadows, and lay the edge of the Index unto it, then suppose the Index, in the line of shadows intersect 8. which showeth that the point X is lower than the level of the eye, by the eight part of the distance from you to the mark. Now if the distance should be 100 foot, than the point X, shall be below the horizontal line, or line of level A B, 12. foot and ½ which is the ⅛ part of 100 the distance before specified. Sixtly, how to find the height of Pro. 49 an Altitude above the level of of the eye, either Accessible, or inaccessible. Let, B C be an Altitude and the eye at, Declaratio. A distant from the Basis of B, 100 foot. If through the sights of the Instrument the summet of the Altitude B C, viz. C, be seen, and the Index falling upon 5, in the line of shadows, Constru ¦ ctio. 69 it argueth the Altitude B C, to be the 5th. part of the distance, viz. of A. B. which is 20. foot. Or let the Altitude of G, be sought out, whose Basis cannot be seen Admit, the first station be made at A and seeing the summet of the Altitude G, the Index should cut 3 in the line of shadows, it Argueth that the distance to the Basis of the Altitude, is triple to the Altitude, then if I should go nearer to the Altitude, viz. at D, and should observe the summet or top of the Altitude G, and that the Index should fall upon 1. in the line of shadows, than it showeth that the distance from D, to the Basis of the Altitude is equal to the Altitude. Now suppose that between D and A were 80. foot it should seem that the Altitude observed should be 40. foot, for if at D. the distance to the Altitude be equal to the Altitude, & the distance from A, to the Altitude, be Triple to the Altitude, than the distance from D to the Altitude is the ⅓ of the distance ARE, & so AD, shall be double to DR, therefore half the distance AD, viz. 40. foot is the Altitude required. Pro. 50 Seventhly, to measure any part of an Altitude which is not approachable. Declaratio. Let G H, a part of an Altitude be required to be measured. First, search out the height G R, as before 40. Constru ¦ ctio. 70 foot, then admit standing at A and looking to H, through the sights, the Index should cut 4, which shows the distance from A, to be Quadruple to the Altitude of H R, and if coming nearer the Altitude 80. foot, viz. at D, I should observe H again, through the sights of the Instrument, and find the Index to cut 1, and ⅓, in the line of the shadows, than the distance from D, to the Altitude H R, viz. D R, should contain the Altitude H R, once, and a third part of the Altitude, now seeing that D R, is 1 and ⅓, therefore H R, shall be 1, but the observation at A shown the distance from A, to the Altitude H R, to be Quadruple, and seeing that D R, is 1, and 1 part of 3, therefore A D, must be 2, and 2 parts of 3, which makes A R, the whole distance to be 4, or Quadruple to H R, but if A B, 2, and 2 parts be 80. foot then D R, being 1, and 1, part shall be 40 foot, and if D R, 1, and ⅓, be 40. foot, than H R, (which was 1 should be but 30. foot, & so consequently H R,) taken from G R, there shall remain G H, 10. foot, the measure of the part of the Altitude required. In like manner might we apply the Instrument to the measuring of Bredths and distances: but that which is delivered may serve for the present, and as fully sufficient for the Ingenious. Conclusion. I might have Annexed unto this Tractat the demonstration of this Projection, which might have satisfied those which are more learned, but to show them it would be impertinent, seeing the thing lies so obvious: for others, it would not be respected or regarded, seeing the making, and practical use of the Instrument, principally & Totally they look after, which I have plentifully delivered. Now by way of Comparison it is said in the description of Master Gunter's Quadrant, that if a Quadrant were made (as he there relateth) unto a foot semidiameter, it should show the Azimuth unto a degree, & the hour unto a minute. It is most probable that if this Horlzontal Quadrant have the same semidiameter, it shall show the hour unto half a minute, and the Azimuth unto 3 m. And if in this Tractat I have been too obscure (which I have avoided as much as possible I could,) I entreat the Reader to excuse me. I confess I might more Methodically have digested it, and more abundantly Amplified it, howsoever the affectionate I persuade myself will not spurn at that which I have delivered; as for the Malevolent I way not: my few hours would not permit me to make a long premeditation of so great a facility. But if any one desire to say more upon this horizontal Quadrant, than I have done: I have made way for him, and unuailed the subject, to help his sight. From my house in Chancery-Lane, january, Anno. 1631. Deus donat & digerit. FINIS. mathematical diagram Motus Perpetuus Solis Distinguit Tempora VIGILATE Cuncta sciaterists solaria ducta figuris, Cernitur Hoc varijs exuperare modis: Solis, et ascendens, occasus, et ortus, et auster, Azimuth Hinc, mensis, longa videnda dies; Horaque Solaris via, Sol quodcunque per umbras Praestitit, Horologî mobilis orbis habet. QVAERE VT IVENIa● mathematical diagram