The Third Book of the CrossStaffCHAP. XIX.

Azimuths.  An. Zen.  Tang.  
Gr.  M.  In.  Pa.  
South  0  0  0  0 
SbE  11  15  1  99 
SSE  22  30  4  14 
SEbS  33  45  6  68 
SE  45  0  10  00 
SEbE  56  15  14  97 
ESE  67  30  24  14 
EbS  78  45  50  27 
East.  90  0  Infinit. 
In like manner, in the first Example of the Declining Plane, where the Style standeth according to the Declination 24 gr. 20 m. distant from the South toward the West, the next Point of SbW is but 13 gr. 5 m. distant from the Style; and the second of SSW onely 1 gr. 50 m. and the third of SWbS is again 9 gr. 25 m. and the rest in their order. Wherefore having before found the length of the Style to be 6 Inches 80 parts, extend the Compasses from the Tangent of 45 gr. unto 6.80 parts in the Line of Numbers, the same extent will reach from the Tangent of 24 gr. 20 m. unto 3.07 in the Line of Numbers, for the length of the Tangentline between the Style and the South; and from the Tangent of 13 gr. 5 m. unto 1.58 for the Point of SbW: and so for the rest, as in this Table.
Azimuths.  An. Zen.  Tang.  
Gr.  M.  In.  Pa.  
SEbE  80  35  41  00 
SE  69  20  18  03 
SEbS  58  5  10  91 
SSE  46  50  7  25 
SbE  35  35  4  86 
South  24  20  3  07 
SbW  13  5  1  58 
SSW  1  50  0  22 
The Foot of the Style  
SWbS  9  25  1  13 
SW  20  40  2  57 
SWbW  31  55  4  24 
WSW  43  10  6  37 
WbS  54  25  9  50 
West  65  40  15  02 
WbN  76  55  19  26 
WNW  88  10  22  45 
That done, if you take these Parts out of a Line of Inches, and prick them down in the Horizontalline on either side of the Style, drawing Right Lines perpendicular to the Horizon through these Intersections, but so as they may be contained between the Horizontal and the Tropicks, the Lines so drawn shall be the Azimuths required.
In all other Planes inclining to the Horizon, these Vertical Circles will meet in a Point; but that Vertical Point being more or less distant from the Foot of the Style, the Angles at this Point will be unequal.
The Vertical Point, wherein all Vertical Lines do meet, will be always in the Meridian, directly under or over the top of the Style; and the Angle between the perpendicular side of the Style, and the Vertical line, will be equal to the Inclination of the Plane to the Horizon. Wherefore,
Thus in the first Example of the Declining Inclining Plane, where the upper Face of the Plane looking Southwest, the Declination was 24 gr. 20 m. the Inclination 36 gr. and you may suppose AB the length of the Style to be 6 Inches; if you extend the Compasses from the Tangent of 45 gr. unto the Tangent of 36 gr. the same extent will reach in the Line of Numbers from 6.00 unto 4.36, for the distance AV, between A the Foot of the Style and V the Vertical Point.
So the same extent of the Compasses as before will reach in the Line of Numbers from 6.00 unto 8.26 for the distance AH between the Foot of the Style and the Horizontalline.
Then may you take 4 inch. 36 cent. and pricking them down from A the Foot of the Style, unto V the Vertical Point in the Meridian, draw the Line VA, which being produced, shall cut the Horizon in the Point H with Right Angles, and be that particular Azimuth which is perpendicular to the Plane.
Or, if you take 8 inch. 26 cent. and prick them down in the former Line VA, produced from A unto H, and so draw the Horizontalline through H, perpendicular unto VH, which Horizontalline being produced, will cross the Equator in the same Point wherein the Equator crosseth the Hourline of 6, unless there be some former error.
The Angles at the Zenith depend on the Declination of the Plane, as in our Example, where the Style standeth according to the Declination 14 gr. 20 m. distant from the South toward the West, the Azimuth of 10 gr. from the Meridian Eastward will be 24 gr. 20 m. the Azimuth of 10 gr. Westward will be onely 14 gr. 20 m. distant from the Style; and so the rest in their order.
Or if you would rather describe the common Azimuths, the Point of SbE will be 35 gr. 35 m. the Point of SbW 13 gr. 5 m. distant from the Style; and so the rest in their order. Then,
Wherefore the Inclination of the Plane in our Example being 36 gr. extend the Compasses from the Sine of 90 gr. unto the Sine of 54 gr. the same extent shall reach in the Line of Tangents from 24 gr. 20 m. unto 20 gr. 5 m. for the Angle HVa at the Vertical Point, between the Line VH, drawn through A the Foot of the Style, and the South. Again the same extent will reach from the Tangent of 13 gr. 5 m. unto 10 gr. 38 m. for the Angle belonging to SbW; and so the rest, as in this Table.
Azimuths.  An. Zen.  Ang. V.  
Gr.  M.  Gr.  M.  
SEbE  80  35  78  25 
SE  69  20  65  0 
SEbS  58  5  52  25 
SSE  46  50  40  40 
SbE  35  35  30  3 
South  24  20  20  5 
SbW  13  5  10  39 
SSW  1  50  1  29 
Style.  0  20  
SWbS  9  25  7  38 
SW  20  40  16  58 
SWbW  31  55  26  45 
WSW  43  10  37  11 
WbS  54  25  48  30 
West  65  40  60  48 
WbN  76  55  73  58 
WNW  88  10  87  42 
These Angles being known, if on the Center V, at the Vertical Point, you describe an occult Circle, and therein inscribe the Chords of these Angles from the Line VH, and then draw Right Lines through the Vertical Point, and the Terms of those Chords, the Lines so drawn shall be the Azimuths required. The like reason holdeth for drawing of the Azimuths upon all other Inclining Planes, whereof you have another Example in the Diagram belonging to the Meridian Incliner, as before.
Or, for further satisfaction, you may find where each Azimuthline shall cross the Equator.
Extend the Compasses from the Sine of 90 gr. unto the Sine of our Latitude 51 gr. 30 m. the same Extent will reach in the Line of Tangents from 10 gr. unto 7 gr. 50 m. for the Intersection of the Equator with the Azimuth of 10 gr. from the Meridian,. Again the same extent will reach from 20 gr. unto 15 gr. 54 m. for the Azimuth of 20 gr. And so the rest, as in this Table.
Azim.  Equat.  Azim.  Equat.  
Gr.  M.  Gr.  M.  Gr.  M.  Gr.  M.  
10  0  7  50  11  15  0  51  
20  0  15  54  22  30  19  58  
30  0  24  20  33  45  27  36  
40  0  33  18  45  0  38  2  
50  0  43  0  56  15  49  30  
60  0  53  35  67  30  62  6  
70  0  65  3  78  45  75  44  
80  0  77  18  90  0  90  0  
90  0  90  0 
By which you may see that the Azimuth 90 gr. distant from the Meridian, which is the Line of East and West, will cross the Equator at 90 gr. from the Meridian, in the same Point with the Horizontalline and the Hour of 6 and that the Azimuth of 45 gr. will cross the Equator at 38 gr. 2 m. from the Meridian; that is, the Line of SE will cross the Equator at the Hour of 9 and 28 m. in the Morning, and the Line SW at 2 ho. 31 m. in the Afternoon: And so the rest, whereby you may examine your former Work.
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