An Appendix concerning the Description and Use of a small Portable Quadrant, for the more easie finding of the Hour and Azimuth, and Other Astronomical and Geometrical Conclusions.CHAP. I.

A Table of Right Ascensions.  
Gra.  Aries  Taurus  Gemini  
Gr.  M.  Gr.  M.  Gr.  M.  
0  0  0  27  54  57  48 
5  4  35  32  42  63  3 
10  9  11  37  35  68  21 
15  13  48  42  31  73  43 
20  18  27  47  33  79  7 
25  23  9  52  38  34  32 
30  27  54  57  48  90  0 
As the Right Ascension of the first Point of ♉ being 27 gr. 54 m. you may lay a Ruler to the Center A and 27 gr. 54 m. in the Quadrant BC, the Point where the Ruler crosseth the Ecliptick shall be the first Point of ♉. In like manner, the Right Ascension of the first Point of ♊ bein 57 gr. 48 m. if you lay a Ruler to the Center A and 57 gr. 48 m. in the Quadrant, the Point where the Ruler crosseth the Ecliptick shall be the first Point of ♊: And so for the rest. But the Lines of distinction between Sign and Sign may be best drawn from the Center G.
Gr.  Parts 
1  176 
2  355 
3  537 
4  723 
5  913 
6  1106 
7  1302 
8  1503 
9  1708 
10  1917 
11  2130 
12  2348 
13  2571 
14  2799 
15  3032 
16  3290 
17  3514 
18  3763 
19  4019 
20  4281 
21  4550 
22  4825 
23  5104 
Trop.  5258 
Ho.  M.  Ho.  M.  Ho.  M.  
Pegasus Wing*  March 8  23  56  1  06  13  17 
Arcturus*  October 14  14  00  30  07  21  8 
Lions Heart*  August 7  9  50  32  28  13  42 
Bulls Eye*  May 16  4  18  64  18  15  46 
Vultures Heart*  January 1  19  35  66  26  8  3 
But before you draw the particular Lines, you are to fit four Tables under your Latitude.
First, a Table of Meridian Altitudes, for division of the Circle of Days and Months, which may be thus made. Consider the Latitude of the Place, and the Declination of the Sun for each Day of the Year. If the latitude and Declination be alike, both North, or both South, add the Declination to the Complement of the Latitude; if they be unlike, one North, and the other South, subtract the Declination from the Complement of the Latitude, the Remainder will be the Meridian Altitude belonging unto the Day.
Thus in out Latitude of 51 gr. 30 m. Northward, whose Complement is 38 gr. 30 m. the Declination upon the tenth day of June will be 23 gr. 30 m. Northward; wherefore I add 23 gr. 30 m. unto 38 gr. 30 m. the sum of both is 62 gr. for the Meridian Altitude at the tenth of June. The Declination upon the tenth of December will be 23 gr. 30 m. Southward, wherefore I take these 23 gr. 30 m. out of 38 gr. 30 m. there will remain 15 gr. for the Meridian Altitude at the tenth of December; and in this manner you may find the Meridian Altitude for each Day of the Year, and set them down in a Table.
Dies.  0  5  10  15  20  25  30  
Month.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M. 
January  16  31  17  24  18  26  19  37  20  57  22  24  23  58 
February  24  17  25  59  27  45  29  35  31  29  33  25  
March  34  35  36  33  38  32  40  30  42  27  44  22  46  15 
April  46  37  48  26  50  11  51  50  53  25  54  53  56  15 
May  56  15  57  29  58  35  59  33  60  22  61  2  61  32 
June  61  36  61  54  62  0  61  58  61  45  61  22  60  49 
July  60  40  60  6  59  14  58  13  57  4  55  48  54  25 
August  54  7  52  36  50  59  49  17  47  31  45  41  43  26 
September  43  26  41  30  39  33  37  36  35  38  33  41  31  46 
October  31  46  29  53  28  3  26  16  24  35  22  59  21  29 
November  21  12  19  51  18  39  17  36  16  43  16  0  15  28 
December  15  28  15  5  15  0  15  2  15  17  15  44  16  22 
The Table being made, you may inscribe the Months, and days of each Month into your Quadrant, in the space left below the Tropick. For, lay the Ruler unto the Center A, and 16 gr. 31 m. in the Quadrant BC, there may you draw a Line for the end of December and the beginning of January; then laying your Ruler to the Center A, and 24 gr. 17 m. in the Quadrant, there draw the end of January and beginning of February, and so the rets, which may be noted with J, F, M, A, M, J, &c. the first Letters of each Month, and will here fall between 15 gr. and 62 gr. The second Table which you are to fit, may serve for the drawing and dividing of the Horizon. For drawing of the Horizon,
So in our Latitude of 51 gr. 30 m. we shall find the Horizon to cut the Tropick in 33 gr. 9 m. wherefore if you lay the Ruler to the Center A, and 33 gr. 9 m. in the Quadrant, the Point where the Ruler crosseth the Tropick shall be the Point where the Horizon crosseth the Tropick. And if you find a Point H in the Line AC, whereon setting the Compasses, you may bring the Point at E and this Point in the Tropick both into a Circle, the Point H shall be the Center, and the Ark so drawn shall be the Horizon.
Then for the division of this Horizon,
So in our Latitude of 51 gr. 30 m. we shall find 7 gr. 52 m. belonging to 10 gr. in the Horizon, and 15 gr. 54 m. belonging to 20 gr. And so the rest, as in the Table.
Ho.  Gr.  M.  Ho.  Gr.  M  Ho.  Gr.  M  Ho.  Gr.  M  Ho.  Gr.  M  Ho.  Gr.  M 
0  0  0  15  11  51  30  24  19  45  38  2  60  53  35  75  71  5 
0  47  12  39  25  11  39  1  54  41  71  19  
1  34  13  27  26  4  40  0  55  48  73  33  
2  21  14  16  26  57  41  0  56  56  74  48  
3  8  15  4  27  50  42  0  58  4  76  3  
5  3  55  20  15  54  35  28  43  50  43  0  65  59  13  80  77  18 
4  42  16  43  29  37  44  1  60  22  78  33  
5  29  17  33  30  32  45  3  61  31  79  49  
6  17  18  22  31  27  46  5  62  41  81  5  
7  4  19  12  32  22  47  8  62  52  82  21  
10  7  52  25  20  2  40  33  18  55  48  11  70  65  3  85  83  37 
8  38  20  53  34  14  49  14  66  15  84  53  
9  27  21  44  35  10  50  19  67  27  86  10  
10  14  22  36  36  7  51  24  68  39  87  26  
11  2  23  27  37  4  52  29  69  52  88  43  
15  11  51  30  24  19  45  38  2  60  53  35  75  71  5  00  90  0 
Wherefore you may lay the Ruler to the Center A, and 7 gr. 52 m. in the Quadrant BC, the Point where the Ruler crosseth the Horizon shall be 10 gr. in the Horizon; and so for the rest. But the Lines of distinction between each fifth Degree will be drawn from the Center M.
The third Table for drawing of the Hourlines must be a Table of the Altitude of the Sun above the Horizon at every Hour, especially when he commeth to the Equator, the Tropick, and some other intermediate Declinations.
If the Sun be in the Equator, and so have no Declination,
Thus in our Latitude of 51 gr. 30 m. at six Hours from the Meridian the Sun will have no altitude, at five the Altitude will be 9 gr. 17 m. at four 18 gr. 8 m. at three 26 gr. 7 m. at two 32 gr. 37 m. at one 36 gr. 58 m. at Noon it will be 38 gr. 30 m. equal to the Complement of the Latitude.
If the Sun have Declination, the Meridian Altitude will be found as before, for the Table of Days and Months,
Thus in our Latitude the Declination of the Sun being 23 gr. 30 m. the Altitude will be found to be 18 gr. 11 m. the Declination being 11 gr. 30 m. the Altitude will be 9 gr.
So in our Latitude, and one Hour from the Meridian, this fourth Ark will be found to be 52 gr. 28 m. at two 55 gr, 26 m. at three 60 gr. 39 m. at four 68 gr. 22 m. and at five Hours from the Meridian 78 gr. 22 m.
Then consider the Declination of the Sun, and the Hour proposed; if the Latitude and Declination be both alike, as with us in North Latitude, North Declination, and the Hour fall between Noon and six, take the Declination out of the fourth Ark, the remainder shall be our fifth Ark.
But if either the Hour fall between six and midnight, or the Latitude and Declination shall be unlike, add the Declination unto the fourth Ark, and the sum of both shall be your fifth Ark: or if the sum shall exceed 90 gr. you may take the Complement unto 180 gr. This fifth Ark being known,
Thus in our Latitude of 51 gr. 30 m. Northward, the Sun having 23 gr. 30 m. of North Declination, if it shall be required to find the Altitude of the Sun for seven in the Morning: here (because the Latitude and declination are both alike to the Northward, and the Hour proposed falleth between Noon and six) you may take 23 gr. 30 m. the Ark of the Declination, out of 78 gr. 22 m. the fourth Ark belonging to the fifth Hour from the Meridian, so there will remain 54 gr. 52 m. for your fifth Ark: Then working according to the Canon, you shall find,
If in the same Latitude and Declination it were required to find the Altitude for five in the Morning, here the Hour falling between six and Midnight, if you add 23 gr. 30 m. unto 78 gr. 22 m. the sum will be 101 gr. 52 m. and the Complement to 180 gr. will be 78 gr. 8 m. for the fifth Ark. Wherefore,
Conferatur Arcus DH cum Arcus Declinationis DS, ita dabitur Arcus HS, cujus Compl. Est SRO prim dr. Arcus quantum. Unde erit,
Ut Cosi. PR,  Hoc est,  Ut Sin. DR, 
Ad Cosi. PZ:  ad Sin. EZ:  
Ita Cosi. SR,  Ita Sin. HS,  
ad Cosi SZ.  Ad Sin. AS. 
Hins forte prastabit vacare HS Arcum quintus, iat secunda operatio instituetur per solos sinus.
Vel si libet subtractionem sinis quarti Arcus evitare, inveniatur Angulum OHD quod fieri potest variis modis. Nam,
Invento ut cunque Angulo ad H, erit in Rectangulo HAS,
If in the same Latitude of 51 gr. 30 m. Northward, the Sun having 23 gr. 30 m. of South Declination, it were required the Altitude for nine in the Morning: Here, because the latitude and Declination are unlike, the one North, and the other South, you may add 23 gr. 30 m. the Ark of Declination, unto 60 gr. 39 m. the fourth Ark belonging to the third Hour from the Meridian; so shall you have 84 gr. 9 m. for your fifth Ark. Wherefore,
And so by one or the other of these means you may find the Altitude of the Sun for any Point of the Ecliptick at all Hours of the Day, and set them down in such a Table as this.
Hour  ♋  ♊ ♌  ♉ ♍  ♈ ♎  ♓ ♏  ♒ ♏  ♑  
Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  
12  62  0  58  42  50  0  38  30  27  1  18  18  15  0  
11  1  59  43  56  34  48  12  36  58  25  40  17  6  13  52 
10  2  53  45  50  55  43  12  32  37  21  51  13  38  10  30 
9  3  45  42  43  6  36  0  26  7  15  58  8  12  5  15 
8  4  36  41  34  13  27  31  18  8  8  33  1  15  
7  5  27  17  24  56  18  18  9  17  0  6  
6  6  18  11  15  40  9  0  0  0  
5  7  9  32  6  50  11  37  
4  8  1  32  21  40 
Lastly, You may find what Declination the Sun hath when he riseth or setteth at any Hour.
And so in the Latitude of 51 gr. 30 m. you shall find that when the Sun riseth, either at 5 in the Summer, or seven in the Winter, his Declination is 11 gr. 37 m. when he riseth at four in the Summer, or eight in the Winter, his Declination is 21 gr. 40 m. which may also be set down in the Table.
That done, you may there see, that in this Latitude the Meridian Altitude of the Sun in the beginning of ♋ is 62 gr, in ♊ 58 gr. 42 m. in ♉ 50 gr. in ♈ 38 gr. 30 m. &c. But the beginning of ♋ and ♑ is represented by the Tropick TD, drawn at 23 gr. 30 m. of Declination, and at the beginning of ♈ and ♎, by the Equator EF. If you draw an occult parallel between the Equator and the Tropick, at 11 gr. 30 m. of Declination, it shall represent the beginning of ♉, ♍, ♏ and ♓, if you draw another occult Parallel through 20 gr. 12 m. of Declination, it shall represent the beginning of ♊, ♌, ♐ and ♒.
Then you may lay a Ruler to the Center A, and 62 gr. in the Quadrant BC, and note the Point where it cosset the Tropick of ♋; then move the Ruler to 58 gr. 52 m. and note where it cosset the Parallel of ♊; then to 50 gr. and note where it crosseth the Parallel of ♉; and again to 38 gr. 30 m. noting where it crosseth the Equator: so the Line drawn through these Points shall shew the Hour of 12 in the Summer, while the Sun is in ♈, ♉, ♊, ♋, ♌, or ♍. In like manner, if you lay the Ruler to the Center A, and 27 gr. in the Quadrant, and note the Point where it crosseth the Parallel of ♓; then move to 18 gr. 18 m. and note where it crosseth the Parallel of ♒; and again to 15 gr. noting where it cosset the Tropick of ♑; the Line drawn through these Points shall shew the Hour of 12 in the Winter, while the Sun is in ♎, ♏, ♐, ♌, ♒, and ♓; so may you draw the rest of these Hourlines: only that of 7, from the Meridian in the Summer, and 5 in the Winter, will cross the Line of Declination at 11 gr. 37 m. and that of 8 in the Summer, and 4 in the Winter, at 21 gr. 40 m.
The fourth Table for drawing of the Azimuthlines must likewise be fitted for the Altitude of the Sun above the Horizon at every Azimuth, especially when he cometh to the Equator, the Tropicks, and some other intermediate Declination. If the Sun be in the Equator, and so have no Declination,
Thus in our Latitude of 51 gr. 30 m. at 90 gr. from the Meridian the Sun will have no Altitude; at 80 gr. the Altitude will be 7 gr. 52 m. at 70 gr. it will be 15 gr. 30 m. at 60 gr. it will be 21 gr. 41 m.
If the Sun have Declination, the Meridian Altitude will be easily found as before, for the Table for Days and Moths. And for all other Azimuths,
When the Latitude and Declination are both alike in all Azimuths from the prime Vertical unto the Meridian, add this fourth Ark unto the Ark of Altitude at the Equator.
When the Latitude and Declination are both alike, and the Azimuth more than 90 gr. distant from the Meridian, take the Altitude at the Equator out of this fourth Ark. When the Latitude and Declination are unlike, take this fourth Ark out of the Ark of the Altitude at the Equator, so shall you have the Altitude of the Sun belonging to the Azimuth.
Thus in our latitude of 51 gr. 30 m. Northward, if it were required to find the Altitude of the Sun in the Azimuth of 60 gr. from the Meridian, when the Declination is 23 gr. 30 m. Northward, you may find the Altitude at the Equator belonging to this Azimuth to be 21 gr. 41 m. by the former Canon; and by this last Canon you may find the fourth Ark to be 28 gr. 15 m. Then because the latitude and Declination are both alike to the Northward, if you add them both together, you shall have 49 gr. 56 m. for the Altitude required.
OM  90. 60. 
ME  Comp. Lat. 
OA  Com. Azim. 
AB  Alt. Æqua. 
EZ  Lat. 
ZB  Com. AB. 
DS  Declin. 
SB  Arc. 4. 
Lat. 50 Gr. 00 M.  
Merid.  10  20  30  40  50  60  70  80  90  
♋  63  30  63  14  62  23  60  54  58  42  55  32  51  25  46  2  31  17  31  22 
♊  60  12  59  54  59  0  57  23  55  1  51  43  47  18  41  40  34  47  26  48 
♉  51  31  51  9  0  3  48  10  45  23  41  34  36  38  30  30  23  12  15  5 
♈  40  0  39  34  38  15  36  0  32  44  28  20  22  45  60  0  8  17  0  0 
♓  28  30  28  0  26  27  23  50  20  5  15  6  8  52  1  30  6  38  
♒  19  48  19  14  17  31  14  37  10  27  4  57  1  43  9  40  18  13  
♑  16  30  14  54  14  7  11  6  6  46  1  5  6  58  14  2  22  43  
Lat. 51 Gr.  
♋  62  30  62  14  61  22  59  54  57  40  54  35  50  27  45  8  38  33  30  53 
♊  59  12  58  54  57  59  56  23  54  0  50  43  46  22  41  51  34  0  26  23 
♈  39  0  38  34  37  14  35  3  31  49  27  30  22  2  15  29  8  0  0  0 
♓  27  30  27  1  25  29  22  55  9  13  14  20  8  17  1  10  6  43  
♒  18  48  18  14  16  33  13  43  9  38  4  17  2  18  9  53  18  0  
♑  15  30  14  54  13  10  10  12  5  58  0  25  6  23  14  10  22  33  
Lat. 52 Gr.  
♋  61  30  61  14  60  22  58  52  56  38  54  33  49  29  44  14  37  58  30  24 
♊  58  12  58  54  56  28  56  22  53  0  49  43  44  25  40  0  33  28  26  0 
♉  49  30  49  9  48  3  46  11  43  26  39  44  34  58  29  6  22  15  14  40 
♈  38  0  37  35  36  17  34  5  30  54  26  40  21  20  14  57  7  44  0  0 
♓  26  30  26  1  24  31  22  0  18  22  13  26  17  42  0  48  6  46  
♒  17  48  17  16  15  36  12  48  8  49  3  37  2  45  10  6  18  0  
♑  14  30  14  56  12  12  9  18  5  10  0  13  6  49  14  19  22  30  
If the Declination had been 23 gr. 30 m. to the Southward, you should then have taken this fourth Ark out of the Ark at the Equator; which because it cannot here bee done, it is a sign that the Sun is not then above the Horizon: But if you take the Ark at the Equator out of this fourth Ark, you shall have 6 gr. 34 m. for the Altitude of the Sun when he is in the Azimuth of 60 gr. from the North, and 120 gr from the South part of the Meridian. The like reason holdeth for the rest of these Altitudes, which may be gathered, and set down in a Table.
Lastly, when the Sun riseth or setteth upon any Azimuth, to find his Declination.
Azimuths  ♋  ♊ ♌  ♉ ♍  ♈ ♎  ♓ ♏  ♒ ♐  ♑  
Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  
0  62  0  58  42  50  0  38  30  27  0  18  18  15  0 
10  61  43  58  24  49  38  38  4  26  30  17  45  14  25 
20  60  51  57  28  48  33  36  46  25  9  16  5  12  41 
30  59  52  55  52  46  40  34  34  22  27  13  15  9  45 
40  57  20  53  29  43  55  31  21  18  48  9  14  5  34 
50  54  3  50  12  40  11  27  5  13  58  3  57  0  6 
60  49  56  45  53  35  23  21  41  8  0  
70  44  40  40  25  29  27  15  13  1  0  
80  38  11  33  46  21  29  7  52  
90  30  38  26  10  14  25  0  0  
100  22  27  18  2  6  45  6  12  
110  14  14  9  58  12  18  
120  6  34  2  50  18  8 
And thus in our Latitude of 51 gr, 30 m,. when the Azimuth is 80 gr. from the meridian, the Declination will be found to be 6 gr. 12 m. if the Azimuth be 70 gr. the Declination will be found 12 gr. 18 m. if 60 gr. then 18 gr. 8 m. And so the rest, which may be also set down in the Table.
That done, if you would draw the Line of East or West, which is 90 gr. from the Meridian, lay the Ruler to the center A, and 30 gr. 38 m. numbred in the Quadrant from C to ward B, and note the Point where it crosseth the Tropick of ♋; then move the Ruler to 26 gr. 10 m. and note where it crosseth the Parallel of ♓; then to 14 gr. 45 m. and note where it crosseth the Parallel of ♉; then to 0 gr. 0 m. and you shall find it crosseth the Equator in the Point F: so a Line drawn through these Points shall shew the Azimuth belonging to East and West. The like reason holdeth for all the rest.
These Lines being thus drawn, if you set two Sights upon the Line AC, and hang a Thread and Plummet on the Center A, with a Bead upon the Thread, the Foreside of the Quadrant shall be finished.
On the Backside of the Quadrant you may place the Nocturnal described before in the Use of the Sector, which consisteth of two parts.
The one is an HourPlane, divided equally according to the 24 Hours of the Day, and each Hour into Quarters, or Minutes, as the Plane will bear. The Center represents the North Pole; the Line drawn through the Center from XII to XII stands for the Meridian, and the lower XII stands for the Hour of XII at midnight.
The other part is a Rundle for such Stars as are near the North Pole, together with the twelve Months, and the Days of each Month, fitted to the Right Ascension of the Sun and Stars, in this manner.
First consider where the Sun will be at the beginning of the 5, 10, 15, 20, 25, 30, and if you will, every day of each Month, and find the Right Ascension belonging to the place of the Sun, as I shewed before.
For example: The Sun at midnight, the las of December, or beginning of January, will be communibus annis about 20 gr. 40 m. of ♑, whose Right Ascension is 292 gr. 20 m. At midnight, the last of January, or beginning of February, he will be about 22 gr. 12 m. of ♒, whose Right Ascension is 314 gr. 35 m. and so the rest, which may be set down in a Table.
That done, consider the Longitude and Latitude of the Stars, and thereby find their Right Ascension and Declination as I shewed before, and set them down in a Table. These Tables thus made, let the uppermost part of the Rundle be made even with the innermost Circle of the HourPlane, and a convenient space allowed to contain the divisions for the Days, and names of the Months. Then lay the Center of this Rundle upon the Center of some other Circle divided into 360 gr. and by the Center and 292 gr. 20 m. in that Circle, draw a Line for the beginning of January: In like manner, by the Center and 324 gr. 35 m. draw a Line for the end of January and beginning of February; and so the rest of the days of each Month.
For the Inscription of the Stars, let one of the Lines from the Center, as that at the beginning of July, or rather let a movable Index be divided from the Center toward the inward Circle of the Months into 40 gr. more or less, which may be done for speed equally, but for exactness in such manner as the Semidiameter of the General Astrolabe was divided before in the Use of the Sector. So laying the Index to the Right Ascension in the outward Circle, you may prick down the Stars by their Declination in the Index.
For example: If the Right Ascension of the Polestar be 6 gr. 18 m. and his Declination 87 gr. 10 m. having set the Center of the Index both to the Center of the Rundle and of the other Circle, turn the Index to 6 gr. 28 m. in that outward Circle, and prick down the Star by 87 gr. 20 m. in the edge of the Index, that is, at the distance of 2 gr. 40 m. from the Pole. The like reason holdeth for the rest of the Stars, which may be distinguished according to their Magnitudes, and then be reduced into their Forms, as in the Example. So the Quadrant will be fitted both for Day and Night.
Zurück zum Inhaltsverzeichnis  Nächstes Kapitel 
© Rainer Stumpe URL: http://www.rainerstumpe.de 