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The distinction of Plains, with Rules for know∣ing of them.
ALL plain Superficies are either hori∣zontal, or such as make Angles with the Horizon.
Horizontal plains are those, that lie up∣on an exact level, or flat.
Plains, that make Angles with the Hori∣zon are of three sorts.
1. Such as make right angles with the Horizon, generally known by the name of erect, or upright plains.
2. Such as make acute angles with the ho∣rizon, or have their upper edge leaning to∣ward you, usually termed inclining plains.
3. Such as make obtuse angles with the horizon, or have their upper edge falling from you, commonly called reclining plains.
All these three sorts are either direct, viz. East, West, North, South. Or else
All plain Superficies whatsoever are comprized under one of these terms. But before we treat of the affections, or deli∣neation of Dials for them; it will be requisire to acquaint you with the nature of any plain, which may be found by the fol∣lowing Problems.
To finde the reclination of any Plain.
Apply the outward ledge of the move∣able piece to the Plain with the head up∣ward,
and reckoning what number of de∣grees the thread cuts on the limb (beginning your account at 30. on the loose piece, and continuing it toward 60/0 on the moveable piece) you have the angle of reclination. If the thread falls directly on 60/0 upon the moveable piece, its an horizontal; if on 30. on the loose piece, its an erect plain.
To finde the inclination of any Plain.
Apply the outward ledge of the fixed piece to the plain, with the head upward, and what number of degrees the thread cuts upon the limb of the loose piece, is the com∣plement of the plains inclination.
To draw an Horizontal Line upon any Plain.
Apply the proportional side of the Instru∣ment to the plain, and move the ends of the fixed piece upward, or downward, until the thread falls directly on 60/0. upon the loose piece; then drawing a line by the outward
ledge of the fixed piece, its horizontal, or paralel to the horizon.
To draw a perpendicular Line upon any Plain.
When the Sun shineth hold up a thread with a plummet against the plain, and make two points at any distance in the shadow of the thread upon the plain, lay a ruler to these points, and the line you draw is a per∣pendicular.
To finde the declination of any Plain.
Apply the outward ledge of the fixed piece to the horizontal line of your plain, holding your instrument paralel to the hori∣zon. This done, lift up the thread and plum∣mer, until the shadow of the thread fall di∣rectly upon the pin hole on the fixed piece (where you hang the thred to take altitudes) Then observe how many degrees the shadow of the thread cuts in the limb, either from the right hand, or from the left hand 0/60. upon the loose piece; and immediately ta∣king
the altitude of the Sun. By lib. 1. cap. 2. Probl. 9, 10, 11. finde the Suns Azimuth from South. And,
When you make this observation in the morning, these Cases determine the decli∣nation of the plain.
When the shadow of the thread upon the limb falls on the right hand 0/60 on the loose piece, take the difference of the sha∣dow, and Azimuth (by subtracting the lesse out of the greater) and the residue or re∣main is the plains declination. From South toward East, when the Azimuth is greater than the shadow. From South toward West, when the shadow is greater than the azi∣muth, when the shadow and azimuth are equal, its a direct South plain. When the difference is just 90. its a direct East, when above 90. subtract the difference from 180. and the remain is the declination from North toward East.
When the shadow falls on the left hand 0/60. Adde the azimuth and shadow toge∣ther,
that sum is the plains declination; from South toward East, when under 90; if it be just 90, its a direct East. If above 90. subtract it from 180. the remain is the de∣clination from North toward East. When the sum is above 180. subtract 180 from it, and the remain is the declination from North toward West.
When the shadow falls upon 0/60. the azimuth is the plains declination. When un∣der 90, its South-East, when equal to 90. direct East, when above 90. subtract it from 180. the remain is the declination from North toward East.
If you make the observation afternoon, the following Cases will resolve you.
When the shadow falls on the left hand 0/60. the difference 'twixt the shadow and azimuth is the declination; when the shadow is more than the azimuth it declines South∣East, when less, South-West. When the sha∣dow and azimuth are equal, its a direct South plain, when their difference is equal to 90.
its a direct West; when the difference ex∣ceeds 90. subtract it from 180. the remain is the declination North-West.
When the shadow falls on the right hand 0/60. take the sum of the shadow and azi∣muth, and that is the declination from South toward West, when under 90. when just 90. its a direct West plain; when more than 90. subtract it from 180. the remain is the decli∣nation North-West; when the sum is above 180. subtract 180 from it, and the remain is the declination from North toward East.
When the shadow falls upon 0/60. the azimuth is the quantity of declination. From South toward West, when under 90. when equal to 90. its direct West; when more than 90. subtract it from 180. The remain is the declination from North toward West.
To draw a Meridional Line upon a Horizontal Plain.
Draw first a circle upon the plain, and holding up a thread and plummet (when the Sun shines) so that the shadow of the thread may pass through the center of the circle, make a point in the circumference where the shadow intersects it. At the same time finding the Suns Azimuth from South, by a line of chords, set it upon the limb of the circle from the intersection of the shadow toward the South, and it gives the true South point. Wherefore laying a ruler to this last point and the center, the line you draw is a true Meridian.
The affections of all sorts of Plains.
The affections of an horizontal Plain.
1. THe style, or cock of every horizon∣tal Dial, is an angle equal to that latitude for which the Dial is made.
2. The place of the style is directly upon the meridional, or twelve a clock line, and the angular point must stand in the center of the hour lines.
3. The rule for drawing the hour lines before six in the morning, is to draw the re∣spective hour lines afternoon beyond the center; or for the hour lines after six at evening, draw the respective hour lines in the morning beyond the center.
4. To place an horizontal Dial upon the plain; first draw a Meridian line upon the plain by Cap. 1. Probl. 6. and lay in the line of 12. exactly thereon, with the angular point of the style toward the South, fasten the Dial upon the plain.
The affections of erect, direct South and North Plains.
1. The style in both these is an angle equal to the complement of the latitude for which the Dial is made to stand upon the Meridian line, or perpendicular of your plain, with the angular point in the center of the hour lines, and that point in South al∣wayes upward, in North alwayes downward.
2. To prick off the Dial from your paper draught upon the plain, lay the hour line of 6. and 6. upon the plains horizontal; and applying a ruler to the center, and each hour line, transmit the hour lines from your paper draught to the plain.
The affections of erect, direct East and West Plains.
1. In both these the style may be a pin or plate, equal in length or heighth to the ra∣dius of the tangents, by which you draw the Dial.
2. The style in both of them is to stand directly upon the hour of six, and perpen∣dicular to the plain.
The affections of erect declining Plains.
1. These are of two sorts, either such as admit of centers to the hour lines and style, or such as cannot with conveniency (because of the lowness of the style, and nearness of the hour lines to each other) be drawn with a center to those lines. Of this latter sort are all such plains, whose style is an angle less than 15 degrees. For where the angle of the style is more than 15 degrees; those Dials may be drawn with a center to the hour lines.
2. In all erect declining plains with cen∣ters the Meridian is the plains perpendicu∣lar. In those that admit not of centers in their delineation, the meridian is parallel to the plains perpendicular.
3. In all declining plains the substile, or line whereon the style is to stand, must be placed on that side the meridian, which is contrary to the coast of declination; and al∣so in such decliners as admit of centers, the
angle 'twixt 12. and 6. is to be set to the contrary coast to that of declination.
4. In all decliners, without centers, the inclination of meridians is to be set from the substyle toward the coast of declination.
5. The proportions in all erect decliners, for finding the height of the style, the di∣stance of the substyle from the meridian, the angle of twelve and six, with the incli∣nation of meridians (all which may be wrought either by the canon, or exactly enough for this purpose by the instrument) are as followeth.
As the radius is to the cosine of the lati∣tude, so is the cosine of declination, to the sine of the styles height.
As the radius is to the sine of declination, so is the co-tangent of the latitude to the tangent of the substyle from the Meridian.
As the co-tangent of the latitude is to the radius, so is the sine of declination, to the co-tangent of six from twelve.
As the sine of the latitude is to the radius, so is the tangent of declination to the tangent of inclination of Meridians.
6. All North decliners with centers have the angular point of the style downward, and all South have it upward.
7. All North decliners without centers have the narrowest end of the style down∣ward, all South have it upward.
8. In all decliners without centers take so much of the style as you think convenient, but make points at its beginning and end up∣on the substyle of your paper draught, and transmit those points to the substyle of your plain, for direction in placing your style thereon.
9. In all North decliners the Meridian, or inclination of Meridians is the hour line of twelve at mid-night: in South decliners,
at noon, or mid-day. This may tell you the true names of the hour lines.
10. In transmitting these Dials from your paper draught to your plain, lay the hori∣zontal of your paper draught, upon the hori∣zontal line of the plain, and prick off the hours and substyle.
The affections of direct reclining Plains inclining Plains.
1. The difference 'twixt your co-latitude, and the reclination inclination is the elevation, or height of the style.
2. When the reclination inclination exceeds your co-latitude, the contrary pole is elevated so much as the excess. Ex. gr. a North re∣cliner, or South incliner 50. d. in lat. 52. 30. min. the excess of the reclination inclination, to your co-latitude is 12. d. 30. min. and so much the North is elevated on the recliner, and the South pole on the incliner.
3. When the reclination inclination is equal to the co-latitude, its a polar plain.
1. The Sum of your co-latitude, and the reclination inclination is the styles elevation.
2. When the reclination inclination is equal to your latitude, its an equinoctial plain, and the Dial is no more than a circle divided in∣to 24. equal parts, having a wyer of any convenient length placed in the center per∣pendicular to the plain for the style.
3. When the reclination inclination is greater than your latitude, take the summe of the reclination inclination of your co-latitude from 180. and the residue, or remain is the styles height. But in this case the style must be set upon the plain, as if the contrary pole was elevated, viz. These North recliners must have the center of the style upward, and the South incliners have it downward.
Note. In all South re-in-cliners North re-in-cliners, for their delineation the styles height is to be called the co-latitude, and then you may draw them as erect direct plains, for South, or North (as the former rules shall give them) in that latitude, which is the comple∣ment of the styles height.
1. In all East and West recliners incliners, you may refer them to a new latitude, and new declination, and then describe them as erect declining plains.
2. Their new latitude is the complement of that latitude where the plain stands, and their new declination is the complement of their reclination inclination: But to know which way you are to account this new declination, re∣member all East and West recliners are North-East, and North-West decliners. All East and West incliners are South-East, and South-West decliners.
3. Their new latitude and declination known, you may by Sect 4. par. 5. finde the
substyle from the Meridian, height of the style, angle of twelve and six, and inclina∣tion of Meridians, using in those proporti∣ons the new latitude and new declination in∣stead of the old.
4. In all East and West recliners incliners with centers, the Meridian lies in the horizontal line of the plain; in such as have not cen∣ters, its paralel to the horizontal line.
The affections of declining reclining Plains inclining Plains.
The readiest way for these, is to refer them to a new latitude, and new declination, by the subsequent proportions.
As the radius is to the cosine of the plains declination, so is the co-tangent of the reclination inclination to the tangent of a fourth ark.
In South recliners North incliners get the difference 'twixt this fourth ark, and the latitude of
your place, and the complement of that dif∣ference is the new latitude sought. If the fourth ark be less than your old latitude, the contrary pole is elevated; if equal to your old latitude, its a polar plain.
In South incliners North recliners the difference 'twixt the fourth ark, and the complement of your old latitude is the new latitude. If the fourth ark be equal to your old co-lati∣tude, they are equinoctial plains.
As the radius is to the cosine of the reclination inclination, so is the sine of the old decli∣nation to the sine of the new.
As the radius is to the tangent of the old declination, so is the sine of reclination inclination, to the co-tangent of the angle of the meri∣dian with the horizontal line of the plain.
This gives the angle for its scituation. Ob∣serve, in
| North incliners | less than a polar, the Meridian lyes. | above | That end of the Ho∣rizontall Line con∣trary to the Coast of Decli∣nation. |
| below | |||
| South recliners | more than a polar, the Meri∣dian lyes. | below | That end of the Ho∣rizontall next the Coast of Declinati∣on. |
| above |
| North recliners | less than an equino∣ctial, the Meridian lyes | above | That end of the Hori∣zontalnext the Coast of Decli∣nation. |
| below | In North recliners this Meri∣dian is 12. at mid∣night. | ||
| equal to an equinoctial the Meri∣dian descends below the Hori∣zontal at that end contrary to the coast of Declination, and the sub∣style lies in the hour line of six. | |||
| South incliners | more than an equino∣ctial, the Meridian lyes. | below | That end of the hori∣zontal con∣trary to the declination. |
| above | In South in∣cliners this Meridian is only useful for drawing the Dial, and placing the sub∣style, for the hour lines must be drawn through the center to the lower side. |
After you have by the former proportions and rules found the new latitude, new decli∣nation, the angle and scituation of the meri∣dian, your first business in delineating of the Dial will be (both for such as have cen∣ters, and such as admit not of centers) to set off the meridian in its proper coast and quantity. This done, by Sect. 4. Paragr. 5. of this Chapter, finde the substyles distance from the Meridian, the height of the style, angle of twelve and six (and for Dials with∣out centers, the inclination of Meridians) in all those proportions, using your new la∣titude and new declination, instead of the old, and setting them off from the Meridi∣an, according to the directions in Paragr. 3. and 4. you may draw the Dials by the fol∣lowing rules, for erect declining plains. In placing of them, lay the horizontal line of your paper draught upon the horizontal line of your plain, and prick off the substyle and hour lines.
Only observe. That such South-East, or South-West recliners, as have the contrary pole elevated must be described as North∣East, and North-West decliners, and such North-East, and North-West incliners as have the contrary pole elevated, must be described as South-East, and South-West
decliners, which will direct you which way to set off the substyle, and hour line of six from the Meridian in those oblique plains, which admit of centers, or the substyle from the Meridian, and the inclination of Meridians from the substyle in such as admit not of Centers.
4. Declining polar plains must have a pe∣culiar calculation for the substyle and incli∣nation of Meridians, which is thus.
As the radius is to the sine of the polar plains reclination, so is the tangent of decli∣nation to the co-tangent of the substyles di∣stance from the horizontal line of the plain.
As the radius is to the sine of the latitude, so is the tangent of declination to the tan∣gent of inclination of Meridians.
5. The reclining declining polar hath the substyle lying below that end of the horizon∣tal line that is contrary to the coast of de∣clination. The inclining declining polar
hath the substyle lying above that end of the horizontal line, contrary to the coast of declination.
The delineation of Dials for any plain Superficies.
HEre it will be necessary to premise the explication of some few terms and symbols, which for brevity sake we shall hereafter make use of. Ex. gr.
Rad. denotes the radius, or sine 90. or tangent 45.
Tang. is the tangent of any arch or num∣ber affixed to it.
Cos. is the cosine of any arch or number of degrees, or what it wants of 90. Ex. gr. cos. 19. is what 19. wants of 90. that is 71.
Co-tang- is the co-tangent of any ark or number affixed to it, or what it wants of 90. Ex. gr. co-tangent 30. is what 30. wants of 90. that is 60.
=This is a note of equality in lines, numbers, or degrees. Ex. gr. AB=CD. That is, the line AB. is equal unto, or of the same length as the line CD. Again, ABC=EFG. that is the angle ABC. is of the same quantity,
or number of degrees, as the angle EFG. Once more AB=CD=FG=tang. 15. That is AB. and CD. and FG. are all of the same length, and that length is the tangent of 15. d.
♒ This is a note of two lines being paralel unto, or equidistant from each other. Ex. gr.
FI. ♒ RS. that is the line FL. is paralel un∣to, or equi-distant from the line RS.
To delineate an horizontal Dial. See Fig. 9.
First draw the square BCDE. of what quantity the plain will permit. Then make AF=AG=HD=HE=sine of the latitude, and AH=Radius. Enter HD. in tang. 45. and keeping the Sector at that gage, set off HI=HK=tang. 15. and HO=HL=tang. 30. Again, enter FD. in tang. 45. and set off FQ=GN=tang. 15. and FP=GM=tang. 30. This done,
Draw AQ. AP. AD. AO. AI. for the hour lines of 5. 4. 3. 2. 1. afternoon. Again,
Draw AK. AL. AE. AM. AN. for the hour lines of 11. 10. 9. 8. 7. before noon.
The line FAG. is for six and six. In the same manner you may prick the quarters of an hour, reckoning three tangents, and 45.
minutes, for a quarter. How to draw the hour lines before, and after six, was menti∣oned, Chap. 2. Sect. 1.
To describe an erect, direct South Dial. See Fig. 10.
Draw ABCD. a rect-angle parallello∣gram. Then make AE=EB=CF=FD=cos. of your latitude. And EF=AC=BD=sine of your latitude. Enter CF. in tang. 45. and lay down FK=FL=tang. 15. and FI=FM= tang. 30. Again, enter AC. in tang. 45. and lay down AG=BO=15. and AH=BN=tang. 30. with a ruler draw the lines EG. EH. EC. EF. EK. for the hour lines of 7. 8. 9. 10. 11. in the morning. and EO. EN. ED. EM. EL. for 5. 4. 3. 2. 1. afternoon, the line AEB. is for six and six. The line EF. for twelve.
The description of a direct North-Dial differs nothing from this, only the hour lines from Sun rise to six in the morning, and from six in the evening, until Sun set, must be placed thereon, by drawing the respective morning and evening hours beyond the cen∣ter as in the horizontal. See Fig. 11.
To describe an erect, direct East Dial. See Fig. 12.
Having drawn ABCD. a rectangle para∣lellogram, fix upon any point in the lines AB. and CD. for the line of six, provided the distance from that point to A. being en∣tred radius in the line of tangents, the di∣stance from thence to B. may not exceed, nor much come short of the tangent 75. This point being found, enter the distance from thence to A. (which we shall call 6. A.) ra∣dius in the line of tangents, and keeping the sector at that gage, lay down upon the lines AB. and CD. 6. 11.=tang. 75. 6. 10=tang. 60. and 6. 9=6. A.=tang. 45. and 6. 8.= 6.=4.=tang. 30. Lastly, 6. 7=6. 5=tang. 15. draw lines from these points on AB. to the respective points on CD. and you have the hours.
To place it on the plain, draw the angle DCE.=co-latitude, and laying ED. on the horizontal line of the plain, prick off the hours.
The same rules serve for delineation of a West Dial, only as this hath morning, that must be marked with afternoon hours.
To describe an erect declining Dial, having a Center. See Fig. 14.
Draw the square BCDE, and make AC=AK of quantity what you please. Again, draw AG. 12. ♒ CE. and KHF. ♒ AG. 12. By a line of chords, set off the angles of the substyle, style, and hour of six from twelve; having first found these angles by Chap. 2. Sect. 4. Paragr. 5. This done, make a mark in A 6. where it intersects KF- as at H. Then enter AK. in the secant of the plains declination, and keeping the Sector at that gage, take out the secant of the lati∣tude, which place from A to G. upon the the line A. 12. and again, from H. (which is the intersection of the paralel FK. with the line of six) unto F. This done, lay a ruler to the points F. and G. and draw a line un∣til it intersects CE. as FG. 3. Lastly,
Enter GF. in tang. 45. and set off GL=GN=tang. 15. and GM=GO=tang. 30. Again, enter HF. in tang. 45. and set off HR=tang. 15. and HP=tang. 30. A ruler laid to these points, and the center, you may draw the hours lines from six in the morning unto
three afternoon. For the other hour lines, do thus, Produce the line EC. and likewise HA. beyond the center, until they intersect each other as at S. Then setting off ST=HR. and S. 4.=HP. you have the hour points af∣ter three in the afternoon, until six, although none are proper beyond the hour line of four; only by drawing them on the other side the center, they help you to the hour lines before six in the morning.
To describe an erect declining Plain without a Center. See Fig. 12.
The delineation of these Dials is the most difficult of any, and therefore I shall be the larger in their description.
1. By Chap. 2. S. 4. Paragr. 5. finde the angles of the style, substyle, and inclination of Meridians.
2. Having found the inclination of Me∣ridians, make the following Table for the distance of every hour line and quarter from the substyle. Where I take for example a North plain declining East 72. d. 45. min. in lat. 52. 30. min.
| Imcl.mer. | hou. | quart. |
| 76=30. | 12 | 00 |
| 20=15 | 3 | ... |
| 16=30 | 4 | 00 |
| 12=45 | 4 | . |
| 09=00 | 4 | .. |
| 05=15 | 4 | ... |
| 01=30 | 5 | 00 |
| 02=15 | 5 | . |
| 06=00 | 5 | .. |
| 09=45 | 5 | ... |
| 13=30 | 6 | 00 |
| 17=15 | 6 | . |
| 21=00 | 6 | .. |
| 24=45 | 6 | ... |
| 28=30 | 7 | 00 |
| 32=15 | 7 | . |
| 36=00 | 7 | .. |
| 39=45 | 7 | ... |
| 43=30 | 8 | 00 |
| 47=15 | 8 | . |
| 51=00 | 8 | .. |
| 54=45 | 8 | ... |
| 58=30 | 9 | 00 |
| 62=15 | 9 | . |
| 66=00 | 9 | .. |
| 69=45 | 9 | ... |
| 73=30 | 10 | 00 |
The manner of drawing this Table is thus. The inclination of Meridians (which because its a North decliner, is twelve at midnight.) I finde 76. d. 30. min. Now considering the Sun never riseth till more than half an hour af∣ter three in this lati∣tude, I know that one quarter before four is the first line proper for this plain: Therefore reckoning 15. d. for an hour, or 3. d. 45. m. for a quarter of an hour, I finde three hours, three quarters (the distance of a quarter before four from midnight) to an∣swer 56. d. 15. min. which being subtract∣ed from 76. d. 30. min. the inclination of Me∣ridians there remains
20. d. 15. m. for the distance of one quar∣ter before four from the substyle. Again, from 20. d. 15. min. subtract 3. d. 45. min. (the quantity of degrees for one quarter of an hour) and there remains 16. d. 30. min. for the distance of the next line from the substyle, which is the hour of four in the morning. Thus for every quarter of an hour continue subtracting 3. d. 45. min. until your residue or remain be less than 3. d. 45. min. and then first subtracting that residue out of 3. d. 45. min. This new residue gives the quantity of degrees for that line on the other side the substyle. Now when you are passed to the other side of the substyle, continue adding 3. d. 45. min. to this last remain for every quarter of an hour, and so make up the table for what hours are proper to the plain.
3. Draw the square ABCD. of what quan∣tity the plain will admit, and make the angle CAG equal to the angle of the substyle with twelve. Again, cross the line AG. in any two convenient points, as E. and F. at right angles by the lines KL. and CM.
4. Take the distance from the center unto 45. the radius to the lesser lines of tangents, which is continued to 76. on the Sector side, enter this distance in 45. on the larger lines of tangents, and keeping the Sector at that
gage, take out the tang. 20. d. 15. min. (which is the distance of the first line from the substyle) set this from 73. d. 30. min. (the distance of your last hour line from the substyle, as you see by the Table) toward the end upon the lesser line of tangents, and where it toucheth as here at 75. 05. call that the gage tangent.
5. Enter the whole line KL. in the gage tangent, which in this example is 75. d. 05. min. and keeping the Sector at that gage, take out the tangent 73. 30. min. which is your last hour, and set from L. on the line KL. unto V. Again, take out the tangent 20. d. 15. min. which is your first line, and set it from K. towards V. and if it meet in V. it proves the truth of your work, and a line drawn through V. paralel unto AG. is the true substyle line. Then keeping the Sector at its former gage, set off the tangents of the hours, and quarters (as you finde them in the Table) from V. towards K. and from V. towards L. making points for them in the line KL. Lastly, enter the radius of your tan∣gents to these hour points in the radius of se∣cants, and set off the secant of the styles height from V. to T. Thus have you the hour points and style on one line of contin∣gence. To mark them out upon the other line do thus.
Set the radius to the hour points upon the former line of contingence, from h. to p. on the line ChM. and entring hV. as Radius in the line of tangents, take out the tangent of the styles height, and set from p. to r. Again, enter hr. radius in your line of tan∣gents, and keeping the Sector at that gage, take out the tangents for each hour, and quarter, according to the table, and lay them down from h. to the proper side of the sub∣style toward C. or M. and applying a ruler to the respective points on KL. and CM. draw the lines for the hours and quarters. Lastly, enter hr. radius on the lines of se∣cants, and taking out the secant of the styles height, set it from h. to S. and draw the line ST. for the style.
To describe a direct polar Dial. See Fig. 15.
Draw BCDE. a rectangle paralellogram, from the middle of BC. to the middle of DE. draw the line 12. or substyle, appoint what place you please in BC. or CD. for the hour point of 7. in the morning, and 5. after∣noon. Then,
Enter 12. 7=12. 5. In the tangent 75. and set off 12. 1.=12. 11.=tang. 15. and 12. 2.=12. 10=tang. 30. and 12. 3.=12. 9.=tang.
45. Lastly, 12. 4.=12. 8.=tang. 60. From these points draw the hour lines of 7. 8. 9. 10. 11. 12. 1. 2. 3. 4. 5. which are all the hours proper for these plains.
To draw a declining Polar. See Fig. 16.
1. By Chap. 2. Sect. 6. Par. 4. finde the inclination of Meridians, and distance of the substyle from the horizontal.
2. By Chap. 3. Sect. 5. Par. 2. make a Table for the distance of the hour points from the substyle.
3. Draw the square BCDE. Set off the angle CAG. for the substyle, and cross that substyle line at right angles in any two con∣venient places, as at H. and K. with the lines PHS. and RKT. for contingent lines.
4. Take any convenient length for your styles height, and enter it radius in your line of tangents, keeping the Sector at that gage, prick off the hours from the substyle (by your table) upon both the contingent lines. Draw lines by the points in both contingents, and you have the hours:
For all other declining reclining inclining plains, it would be needless (I presume) to insist up∣on the description of them: Sith so much
hath been already mentioned, Chap. 2. S. 6. that there can scarcely be any mistake, unless through meer wilfulness, or grandnegligence.
To determine what hour lines are proper for any plain Superficles. By projection of the Sphere. See Fig. 17.
DRaw the Circle NESW. representing the Horizon, and crossing it into qua∣drants N. is North. S. South, E. East, W. West, NS. the Meridian (which let be infi∣nitely produced) Z. the center represents the Zenith. To finde the pole set half the co-tangent of the latitude from Z. toward N. it gives the point P. for the pole or the point through which all the hour lines must pass.
The Suns declination in Cancer subtracted from the latitude, and the tangent of half the remain set from Z. to Π. gives the inter∣section of Cancer with the Meridian. Again, adde the complement of the Suns delineati∣on in Cancer, unto the complement of your latitude, and the tangent of half that sum set from Z. to ψ, gives the diameter of that tropick, half ψ, is the radius to describe it. Half the tangent of your latitude set from Z.
to Q. gives that point for the intersection of the equator with the Meridian; and the co-secant of the latitude set from oe. toward N. gives the point ζ. the center of the equator.
Adde the Suns greatest declination (or his declination in Capricorn) to the latitude, and the tangent of half that sum set from Z. toward S. gives the point φ, where Capri∣corn intersects the Meridian. Subtract the Suns declination in Capricorn from your la∣titude, and that remain subtract from 180. the tangent of half this last remain, set from Z. toward N. gives the point T. the diameter of Capricorn, and half the distance T φ. is the radius to describe it.
Set the secant of the latitude from P. to∣wards S. it gives the point H. the center of the hour line of six cross the line ZSH. at right angles in the point H. Then entring PH. Radius on the lines of tangents, set off the hour centers both wayes from H. reckon∣ing 15. d. for an hour. Lastly, setting one point of your Compasses in these center points, extend the other to P. and with that radius describe the hour lines. Thus have you the sphere projected, the following Se∣ctions will determine the hours for all plains.
To determine the hour lines for erect direct plains. Fig. 17.
The line NS. represents an erect East, and West plain. That side next W. is West, the other side next E. is East, where you may see that the Sun shines upon the East until twelve, or noon, and at that time comes up∣on the West. The fine WE. represents a di∣rect North and South plain, the side next N. is North, the other next S. is South, where the North cuts the tropick of Cancer (which in the hour lines you finde 'twixt 7. and 8. in the morning; and again 'twixt 4. and 5. afternoon) is the time of the Suns going off, and coming on that plain. Where the South cuts the equator, which is in the points of six, and six is the time of the Suns going off, and coming on that plain.
To determin the hour lines for direct reclining inclining Plains. Fig. 17.
NBS. on the convex side is a West in∣cliner, where it cuts Capricorn, is the time of the Suns coming on that plain, afternoon. On the concave side its an East recliner, where it
cuts Cancer, is the time of the Suns going off that plain, afternoon.
NCS. On the convex side is an East in∣cliner, where it cus Capricorn, is the time of the Suns going off in the morning. On the concave side it is a West recliner, where it cuts Cancer, is the time of the Suns coming on in the morning.
WDE. On the convex side is a South in∣cliner, where until D. reach below oe. it hath all hours from six to six, and until D. reacheth below φ. it may have the twelve a clock line. But when D. reacheth below oe. draw a paralel of declination to pass through the point D. and the intersection of that paralel with the limb of the circle NE SW. doth among the hour lines, shew the time of the Suns coming upon that plain in the morning, and going off again afternoon, when D. reacheth below φ. the intersection of the ark WDE. with the tropick of Capricorn, shews the time of the Suns going off that plain before noon, and coming on again, af∣ternoon. And the intersection of the tropick of Capricorn with the limb shews the first hour in the morning the Sun comes on, and the last hour afternoon, that it staves upon that plain. The convex side of WDE. is a north recliner, where it cuts Cancer, is the
time of the Suns going off in the morning, and coming on again afternoon.
WFE. On the convex side is a North in∣cliner, where it cuts Cancer, is the time of the Suns going off in the morning, and com∣ing on afternoon. On the concave side is a South recliner, where until F. reach beyond P. it enjoyes the Sun only from six to six. When F. reacheth beyond P. where the ark cuts Cancer, you finde how much before six in the morning the Sun comes on, or after six at evening it goes off.
To draw any of these arks, Ex. gr. the ark NBS. do thus. Set the tangent of half the reclination inclination from Z. on the line ZW. and it gives the point B. produce ZE. and set the co-secant of the reclination inclination from B. towards E. which reacheth to G. then G. is the center. & GB. the radius to draw that ark.
Note. The Semidiameter of the circle SENW. is radius to all the tangents, and se∣cants, which you make use of for placing any oblique plain upon the Scheme.
To determin the hour lines for erect declining plains. Fig. 18.
For South-East, or North-West plains.
By a line of chords set the angle of decli∣nation on the limb from W. toward S. as H. lay a ruler to HZ. and draw HZK. which on that side next NW. represents the North∣West, and where the line cuts Cancer, you have the time of the Suns coming on after∣noon, and staying until Sun set. But if it cut Cancer twice, then in the morning hours it shews what time the Sun goes off this plain, having all hours from Sun rise to that time, and in the evening hours you have the time of the Suns coming on again, and staying till sun set. That side HZK. next SE. represents a South-East. Where the line cuts the equator in the evening hours, is the time of the Suns going off, where it cuts Cancer, in the morning hours, is the time of the Suns coming on that plain. For North∣East, or South-West, set the declination by a line of chords from E. towards S. as L. lay a ruler to ZL. and draw the line LZR. which on that side next NE. is North-East, where it cuts Cancer, is the time of the Suns going off in the morning. If it cuts Cancer twice, you have in the evening hours the time of the Suns coming on again, and staying until Sun-set.
On that side next SW. is the South-west. Where it cuts the equator, or Capricorn in the morning hours, is the time of the Suns coming
on, where it cuts Cancer in the evening hours, is the time of the Suns going off.
To determin the hours of declining reclining Plains inclining Plains. Fig. 18.
First, set in the plain according to its decli∣nation. By Sect. 3. Ex. gr. LZR. a North¦East, or South-West declining 50. d. 00. min. This done Cross the line LZR. representing the declination of the plain, at right angles in the point Z. as CZBG. Then for North∣East incliners, or South-West recliners, set half the tangent of the reclination inclination from Z. toward C. 〈◊〉〈◊〉 T. and set the co-secant of the reclination inclination from T. toward B. as TG. Then G. is the center, and GT. the radius to describe the ark RTL. Whose convex side represents a North-East incliner, where it cuts Libra or Capricorn, is the time of the Suns going off in the morning; if it cuts Cancer twice, the intersection of Cancer with the evening hours shews what time the Sun comes again upon such a plain afternoon, and continues till Sun setting. The concave side is a South-West recliner, where it cuts
Cancer in the morning hours, is the time of the Suns coming on, in case it intersects Can∣cer twice in the evening hours, you have the time that the Sun goes off.
For a North-East recliner, or South-West incliner, set the point T. from Z. toward B. and the point G. from T. (so placed) toward C. and draw the ark on that side RZL. to∣ward B. whose convex side will represent a South-West incliner, and where the ark cuts the equator or Capricorn, you have the time of the Suns coming on that plain. The con∣cave side is a North-East recliner, where the ark cuts Cancer, is the time for the Suns go∣ing off that plain. When the ark cuts Cancer twice, the Sun comes on again before it sets.
For a North-West recliner, or South-East incliner. Enter the declination by Sect. 3. as HZK. Cross it in the point Z. at right angles, as OZD. set half the tangent of the recli∣nation incli∣nation from Z. toward O. as V. and the co∣secant of the reclination inclination from V. toward D. as F. then is F. the center, and FV. the radius to draw the ark HVK. Where it cuts Cancer the hour lines, tell you the time of the Suns going off in the morning, and en∣tring again afternoon, upon the North-West
recliner. Where it cuts the equator you have the time of the Suns going off the South-East, where it cuts Cancer in the morning hours is the time of the Suns coming on that plain.
For a North-West incliner, or South-East recliner, set the point V. from Z. toward D. and the point F. set from V. (so placed) to∣ward O. and draw the ark on that side Z. next D. Then where the convex side cuts Cancer, you have the time of the Suns going off in the morning; and coming on again af∣ternoon upon the North-West incliner. Where the concave side cuts the equator, you have the time of the Suns going off the South-East recliner; where it intersects Cancer, is the time of his coming on that plain in the morning.
Note. All the precedent rules about plains are appropriated to us that live in Northern Hemisphere, In case any one would apply them to the South Hemisphere: What is here called North, there name South, and what we here term South, there call North, and the rules are the same.
—Si quid novisti plenius istis, promptius istis, rectius istis, Candidus imperti: Sinon, His u••ere mecum.