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. II.

Some uses of the quadrantal side of the Instrument.

PROBL. 1.

To find the altitude of the Sun or any Star.

HAng the thread and plummet upon the pin at the beginning of the line of sines on the fixed piece, and (having two sights in two holes parallel to that line) raise the end of the fixed piece, toward the Sun until the rayes pass through the sights (but when the Sun is in a cloud, or you take the altitude of any Star, look along the outward ledge of the fixed piece, until it be even with the mid∣dle of the Sun or Star) then on the limb the thread cuts the degree of altitude, if you reckon from 0/60 on the loose piece toward the head of the movable piece.

PROBL. 2.

The day of the Moneth given, to find the Suns place, declination, ascensional difference, or time of rising and setting, with his right ascension.

The thread laid to the day of the Moneth gives the Suns place in the line of signs, rec∣koning according to the order of the Moneths (viz. forward from March the 10th. to June, then backward to December, and forward again to March 10.) In the limb you have the Suns declination, reckon∣ing from 60/0 on the movable piece towards the head for North, toward the end for South declination. Again, on the line of right ascensions, the thread shews the Suns right ascension, in degrees, or hours, (accor∣ding to the making of your line) counting from Aries toward the head, and so back a∣gain according to the course of the signs un∣to 24 hours, or 360 degrees. Lastly on the line of hours you have the time of Sun rising and setting, which turned into degrees (for the time from six) gives the ascensional dif∣ference. Ex. gr. in lat. 52. deg. 30 min. for which latitude I shall make all the examples.

The 22 day of March I lay the thread to the day in the Moneths, and find it cut in the Signs 12 deg. 20 min. for the Suns place, on the limb 4 deg. 43 min. for the Suns declina∣tion North. In the line of right ascensions it gives 46 min. of time, or 11 deg. 30 min. of the circle. Lastly, on the line of hours it shews 28 min. before six for the Suns rising; or which is all, 7 deg. for his ascensional dif∣ference.

PROBL. 3.

The declination of the Sun or any Star given to find their amplitude.

Take the declination from the scale of al∣titudes, with this distance setting one point of your Compasses at 90 on the line of Azimuth, apply the other point to the same line it gives the amplitude, counting from 90 Ex. gr. at 10 deg. declination, the amplitude is 16 deg. 30 min. at 20 deg. declination, the amplitude is 34 deg.

PROBL. 4.

The right ascension of the Sun, with his ascen∣sional difference, given to find the oblique ascension.

In Northern declination, the difference betwixt the right ascension and ascensional difference, is the oblique ascension. In Sou∣thern declination take the summ of them for the oblique ascension, Ex. gr. at 11 deg. 30 sec. right ascension, and 6 deg. 30 sec. ascensional difference. In Northern declination the ob∣lique ascension will be 5 deg. in Southern 18 deg.

PROBL. 5.

The Suns altitude and declination, or the day of the Moneth given to find the hour.

Take the Suns altitude from the Scale of altitudes, and laying the thread to the decli∣nation in the limb (or which is all one, to the day in the Moneths) move one point of the compasses along the line of hours (on that side the thread next the end) until the other point just touch the thread; then the for∣mer

point shews the hour; but whether it be before or after noon, is left to your judg∣ment to determine. Ex. gr. The 22 day of March, or 4 deg. 43 min. North declination, and 20 deg. altitude, the hour is either 47 minutes past 7 in the morning, or 13 minutes past 4 afternoon.

PROBL. 6.

The declination of the Sun, or day of the Moneth, and hour given to find the altitude.

Lay the thread to the day or declination, and take the least distance from the hour to the thread, this applyed to the line of alti∣tudes, gives the altitude required. Ex. gr. The 5 day of April or 10 deg. declination North, at 7 in the morning, or 5 afternoon, the altitude will be 17 deg. 10 sec. and better.

PROBL. 7.

The declination and hour of the night, given to find the Suns depression under the horizon.

Lay the thread to the declination on the limb; but counted the contrary way, viz. from 60/0 on the movable piece toward the

head for Southern; and toward the end for Northern declination. This done take the nearest distance from the hour to the thread, and applying it to the line of altitudes, you have the degrees of the Suns depression. Ex. gr. at 5 deg. Northern declination, & 8 hours afternoon, the depression is 13 deg. 30 min.

PROBL. 8.

The declination given to find the beginning and end of twilight, or day-break.

Lay the thread to the declination counted the contrary way, as in the last Problem, and take from your Scale of altitudes 18 deg. for twilight, and 13 deg. for day-break, or clear light; with this run one point of the Com∣passes along the line of houres (on that side next the end) until the other will just touch the thread, and then the former point gives the respective times required. Ex. gr. At 7 deg. North declination, day breaks 8 mi∣nutes before 4: but twilight is 3 houres 12 minutes in the morning, or 8 hours 52 mi∣nutes afternoon.

PROBL. 9.

The declination and altitude of the Sun or any Star, given to find their Azimuth in Northern declination.

Lay the thread to the altitude numbred on the limb of the moveable piece from 60/0 toward the end (and when occasion requires, continue your numbring forward upon the loose piece) and take the declination from your line of altitude; with this distance run one point of your Compasses along the line of Azimuths (on that side the thread next the head) until the other just touch the thread, then the former point gives the Azimuth from South. Ex. gr. at 10 deg. declination North, and 30 deg. altitude, the Azimuth from South is 64, deg. 40 min.

PROBL. 10.

The Suns altitude given to find his Azimuth in the aequator.

Lay the thread to the altitude in the limb, counted from 60/0 on the loose piece to∣ward the end, and on the line of Azimuths

it cuts the Azimuth from South. Ex. gr. at 25 deg. altitude the Azimuth is 53 deg. At 30 deg. altitude the Azimuth is 41 deg. 30 min. fere.

PROBL. 11.

The declination and altitude of the Sun, or any Star given to find the Azimuth in Southern declination.

Lay the thread to the altitude numbred on the limb from 60/0 on the moveable piece toward the end, and take the declination from the Scale of altitudes; then carry one point of your Compasses on the line of Azi∣muths (on that side the thread next the end) until the other just touch the thread, which done, the former point gives the Azimuth from South. Ex. gr. at 15 deg. altitude and 6 deg. South declination the Azimuth is 58 deg. 30 min.

PROBL. 12.

The declination given to find the Suns altitude at East or West in North declination, and by con∣sequent his depression in South declination.

Take the declination given from the Scale of altitudes, and setting one point of your Compasses in 90 on the line of Azimuths, lay the thread to the other point (on that side

90 next the head) on the limb it cuts the al∣titude, counting from 60/0 on the moveable piece. Ex. gr. at 10 deg. declination the al∣titude is 12 deg. 40 min.

PROBL. 13.

The declination and Azimuth given to find the altitude of the Sun or any Star.

Take the declination from the Scale of al∣titudes; set one point of your Compasses in the Azimuth given, then in North declina∣nation turn the other point toward the head, in South toward the end; and thereto laying the thread, on the limb you have the altitude, numbring from 60/0 on the moveable piece toward the end. Ex. gr. At 7 deg. North declination, and 48 deg. Azimuth from South, the altitude is 35 deg. but at 7 deg. declination South, and 50 deg. Azimuth the altitude is onely 18 deg. 30 min.

PROBL. 14.

The altitude, declination, and right ascension of any Star with the right ascension of the Sun given, to find the hour of the night.

Take the Stars altitude from the Scale of

altitudes, and laying the thread to his decli∣nation in the limb, find his hour from the last Meridian he was upon, as you did for the Sun by Probl. 5. If the Star be past the South, this is an afternoon hour; if not come to the South, a morning hour; which keep.

Then setting one point of your Compasses in the Suns right ascension (numbred upon the line twice 12 or 24 next the outward ledge on the fixed piece) extend the other point to the right ascension of the Star num∣bred upon the same line, observing which way you turned the point of your Compas∣ses, viz. toward the head or end. With this distance set one point of your Compasses in the Stars hour before found counted on the same line, and turning the other point the same way, as you did for the right ascensi∣ons, it gives the true hour of the night. Ex. gr. The 22 of March I find the altitude of the Lions heart 45 deg. his declination 13 d. 40 min. then by Probl. 5. I find his hour from the last Meridian 10 houres, 5 min. The right ascension of the Sun is 46 m. of time, or 11 d. 30 m. of the Circle, the right ascension of the Lions heart, is 9 hour 51 m. fere, of time, or 147 deg. 43 m. of a circle; then by a line of twice 12, you may find the true hour of the night, 7 hour 13 min.

PROBL. 15.

The right ascension and declination of any Star, with the right ascension of the Sun and time of night given, to find the altitude of that Star with his Azimuth from South, and by conse∣quent to find the Star, although before you knew it not.

This is no more than unravelling the last Problem. 1 Therefore upon the line of twice 12 or 24, set one point of your Compasses in the right ascension of the Star, extending the other to the right ascension of the Sun upon the same line, that distance laid the same way upon the same line, from the hour of the night, gives the Stars hour from the last Meridian he was upon. This found by Probl. 5. find his altitude as you did for the Sun. Lastly, having now his declination and altitude by Probl. 8. or 10. according to his declination, you will soon get his Azimuth from South. This needs not an example.

By help of this Problem the Instrument might be so contrived, as to be one of the best Tutors for knowing of the Stars.

PROBL. 16.

The altitude and Azimuth of any Star given to find his declination.

Lay the thread to the altitude counted on the limb from 60/0 on the moveable piece toward the end, setting one point of your Compasses in the Azimuth, take the nearest distance to the thread; this applyed to the Scale of altitudes gives the declination. If the Azimuth given be on that side the thread toward the end, the declination is South; when on that side toward the head, its North.

PROBL. 17.

The altitude and declination of any Star, with the right ascension of the Sun, and hour of night given to find the Stars right ascension.

By Probl. 5. or 14. find the Stars hour from the Meridian. Then on the line twice 12, or 24, set one point of your Compasses in the Stars hour (thus found) and extend the other to the hour of the night. Upon the same line with this distance set one point of your Compasses in the right ascension of the

Sun, and turning the other point the same way, as you did for the hour, it gives the Stars right ascension.

PROBL. 18.

The Meridian altitude given to find the time of Sunrise and Sunset.

Take the Meridian altitude from your particular Scale, and setting one point of your Compasses upon the point 12 on the line of hours (that is the pin at the end) lay the thread to the other point, and on the line of hours the thread gives the time required.

PROBL. 19.

To find any latitude your particular Scale is made for.

Take the distance from 90, on the line of Azimuth unto the pin at the end of that line, or the point 12: this applyed to the parti∣cular Scale, gives the complement of that la∣titude the Instrument was made for.

PROBL. 20.

To find the angles of the substile, stile, inclinati∣on of Meridians, and six and twelve, for exact declining plains, in that latitude your Scale of altitudes is made for.

Sect. 1.

To find the distance of the substile from 12, or the plains perpendicular.

Lay the thread to the complement of de∣clination counted on the line of Azimuths, and on the limb it gives the substile counting from 60/0 on the moveable piece.

Sect. 2.

To find the angle of the Stile's height.

On the line of Azimuths take the distance from the Plains declination to 90. This ap∣plyed to the Scale of altitudes gives the an∣gle of the stile.

Sect. 3.

The angle of the Substile given to find the incli∣nation of Meridians.

Take the angle of the substile from the Scale of altitudes, and applying it from 90 on the Azimuth line toward the end; the fi∣gures shew the complement of inclination of Meridians.

Sect. 4.

To find the angle betwixt 6 and 12.

Take the declination from the Scale of al∣titudes, and setting one point of your Com∣passes in 90 on the line of Azimuths, lay the thread to the other point and on the limb it gives the complement of the angle sought, numbring from 60/0 on the moveable piece toward the end.

This last rule is not exact, nor is it here worth the labour to rectifie it by another sine added; sith you have an exact propor∣tion for the Problem in the Treatise of Dial∣ling Chap.2. Sect. 5. Paragr. 4.