The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.

CHAP. III.

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.

Sect. 1.

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.

Sect. 2.

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.

Sect. 3.

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.

Sect. 4.

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.

Sect. 5.

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.

Sect. 6.

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.

Sect. 7.

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.