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, Arithmetic, Geometry, Geography, Topography, Navigation, dialing, etc. are speedily and exactly resolved.  
By J. T.  
LONDON, Printed for William Tompson, Bookseller at harbour in Leicestershire. 1667.   

To the Reader.  
ALL that is intended in this Treatise, is to acquaint thee with an Instrument, that is both portable and general, of no great price, easy carriage, yet of a speedy and Accurate dispatch in the most difficult Problems in Astronomy, etc. The lines for the most part have been formerly published by Mr. Gunter, the famous Mr. Foster, Mr. White, etc. The reduction of the 28. Cases of Spherical Triangles unto II. Problems, I first learned from the reverend Mr. Palmers Catholic Planisphere. Many of the proportions in the Treatise of dialing are taken from (though first compared with the Globe) my worthy Friend (to whom I am indebted in all the Obligations of Civility, and without whose encouragement this had never adventured the public Test) Mr. John Collins. The applying Mr. Foster's Line of Versed Sins unto the Sector was first published by Mr. John Brown, Mathematical Instrument-Maker, at the Sphere and Sundial in the Minories, London, Anno 1660. who bath very much assisted me, by making, adding unto, and giving me freely the perusal of many Instruments, according to any directions for Improvement, that was proposed to him. After this account, what hath been my part in this Work, I hazard to thy censure; and when I see others publish a more convenient, speedy, accurate, and general Instrument, I assure them to have as low thoughts of this, as themselves. But here is so large a Catalogue of Erratas as would stagger my confidence at thy pardoning, had they not been irrevocably committed before I received the least notice of them. The Printer writing me word (after I had corrected so much as came to my sight) that he could alter no Mistakes until the whole Book was printed: By which means he enforced me to do penance in his Sheets, for his own Crimes: Did not one gross mistake of his become my purgation, viz. in lib. 2. (throughout Chap. 3.) where instead of the note of equality (marked thus =) he hath inserted the Algebraick note of Subtraction, or Minoration (marked thus-)  
Nor hath the Engraver come behind the Composer, who so miserably mangleth Fig. 13. that (at first sight) it would endanger branding of a man's Brains to spell the meaning thereof, either in itself, or in reference to the Book. All that I can help thee herein, is this; Whereas the Book mentions that Figure for an East Dyal, if you account it (as now cut) a West Dyal, and alter the names of the hours, by putting Figures for the afternoon, in the place of those there for the morning; you will then have a true West Dyal of that Figure.  
The correction of Punctations would be an endless task; for I find some to be resolved, ever since Valentine, to recreate themselves at Spurn-point.  
What other material mistakes are in the Book (which ought to be corrected before reading thereof) you will find mentioned in the Errata. Farewell.  
March 29. 1667. J. T.   

Errata.  
PAge 4. line 12. signs, r. sins. p. 8. l. 5. seconds, r. secants. p. 12. l. 9 all, r. alone. p. 13. l. 9, and 10. sec. r. min. p. 14. l. 17. sec. r. min. p. 22. l. 14. any, r. what. p. 23. l. 3. exact, r. erect. p. 24. l. ult. add lib. 2. p. 25. l. 2. signs, r. sins. l. 5. sign. r. sine. p. 35. l. 3. a mark, r. an ark. p. 37. l. 22. 20. r. 22. and 42. r. 20. p. 44. l. 5. At, r. At. p. 49. l. 1. divided by, r. dividing. p. 62. l. 15. deal, a. In lib. 1. chap. 9 the pages are false numbered. But in chap. 9 p. 62. l. 11. next, r. exact. l. 15. gauger, r. gauge. l.24.the, r. what. p. 73.l. 5. whereas, r. where I. p. 80. l. 13. wherein, r. whereof. l.22.pont, r. point. p. 91. l.1. the co-tangent, r. half the co-tangent. l. 19. L. r. P. p. 92. l. 21. PZ. r. PS. p. 93. l. 12. NSP, r. NSZ. 6.angle r. ark. throughout page 96. Fig. 3. r. Fig. 7. and Fig. 4. r. Fig. 8. p. 98. l. 13. serve, r. screw. p. 99. l.12. places, r. plates. l. 13. proportion, r. perforation. l.9.serve, r. screw. l.4.serve, r. screw. 21.serve, r: screw. p. 111. l. 20. lay in, r. laying. l.14.of, ●● and. p. 128. l. 3. Fig. 12. r. Fig. 13. l. 22. ED. r. EC.  
In Fig. 12. C. r. A. at the end of the line G.   

To the Right Honourable, The Lord SHERARD, Baron of Letrim.  
My Lord,  
SInce the trifling Treatise of an Almanac hath usurped a custom to opinion some Honourable Name to the Patronage of the Author's Follies; had we not certain evidence from the uncertainty of their Predictions, that their Brains (like their great Oracles the Planets) are often wand'ring; it might be deemed a Crime, beyond the benefit of the Clergy, to prefix before any Book, a dedication to a Noble Person: Or when I read the unreasonableness of others in those Addresses, imploring their Patrons to be their Dii Tutelares, and prostrate their reputes to the unmannerly mangling of every Censurist, under the notion of protecting (that is adopting) the Author's Ignorance, or negligence; it's enough to tempt the whole world to turn Democritians, and hazard their spleens in laughing at such men's madness. My present design is only to give your Lordship my observance of your Commands about the Description and Improvement of the Sector; and wherein I have erred (through mistake, or defect) I despair not, but from your Honour I shall meet with a pardon of Course to be granted unto  
Your Lordship's most humble Servant J. T.   

THE SEMICIRCLE ON A SECTOR.  

LIB. I.  

CHAP. I.  
A Description of the Instrument, with the several Lines inscribed thereon.  
THe Instrument consists of three Rulers, or Pieces; two whereof are joined together by a River that may open and shut to any Angle, in fashion of the Sector; or to use a courser comparison, after the manner of Compasses. The third piece is loose, or separable from them, to be put into the Tenons at the end of the inward ledge of the joined pieces, and thereby constituting an aequilateral triangle.  
On these Rulers (after this manner put together) we take notice (for distinction sake) of the sides, ledges, ends, and pieces.  
The sides are thus differenced, one we call the quadrantal, the other the proportional, or sector side.  
The ledges are distinguished by naming one the inward, the other the outward ledge.  
The ends are known in terming one the head (viz. where the two pieces are riveted together) the other the end.  
The pieces are discovered by styling one the fixed piece, viz. that which hath the rivet upon it; the second the movable piece, which turns upon that rivet, and the last the loose piece, to be put into the tenons, as before expressed.  
The quadrantal side of the joined pieces is easily discerned, having the names of the Months stamped on the movable piece, and Par. Scale on the fixed piece. The quadrantal side of the loose piece is known by the degrees on the inward and outward limb.  
These directions are sufficient to instruct you how to put the Instrument together.  
Imagining the Instrument thus put together, the lines upon the quadrantal side are these.  
First on the fixed piece, next the outward limb, is a line of 12 equal parts; and each of those parts divided into 30 degrees, marked from the end towards the head with ♈ ♉ ♊, etc. representing the 12 Signs of the Zodiac; the use of this line, with the help of those under it, was intended to find the hour of the night by the Moon.  
The next line to this is a line of twice 12, or 24 equal parts, each division whereof cuts every 15th degree of the former line: and therefore if the figures were set to every 15th degree on this side the former line, this second line would be useless, and the former perform its office more distinctly. This line was intended an assistant for finding the hour by the Moon; but is very ready to find the hour by any of the fixed Stars.  
The third line is a line of 29● equal parts, serving for the days of the Moon's age; in order to find the hour of the night by the Moon. But the operation is so tedious, and far from exactness, that I have no kindness for it; and should place some other lines in the room of this and the former, did I not resolve to impose upon no man's fantasy.  
The fourth line is a line of altitudes for a particular latitude, noted at the end, Par. Scale, etc. This helpeth to find the hour and azimuth of the Sun, or any fixed Star very exactly.  
The fifth line is a line of natural sins; at the beginning whereof there is a pin, or else an hole to put a pin into, whereon to hang the thread and plummet for taking of altitudes. To this line of sins may be joined a line of tangents to 45 degr. The use of the sins alone, is to work proportions in signs. The use of the sins with the tangent line may be for any proportion in trigonometry; but that I leave to liberty.  
The sixth line, and last on this side the fixed piece is a line of versed sins, numbered from the centre at the head to 180 at the end.  
On the quadrantal side of the movable piece, the first line next the inward edge, is a line of versed sins answering to that on the fixed piece. The use of these versed sins is various at pleasure.  
The second line from the inward edge, is a line of hours and Azimuths serving to find the hour by the Sun or Stars, or the Azimuth of the Sun, or any fixed Star from the South.  
The third and fourth lines are lines of Months, marked with the respective names, and each Month divided into so many parts as it contains days.  
The fifth line is a line of signs marked ♈ Taurus; Gemini;, etc. each sign being divided into 30 degr. and proceeding from ♈, or Aries (which answers to the tenth of March) in the same order as the Months. The use of this line, with help of the Months, is to find the Sun's place in the Zodiac.  
The sixth line is a line of the Sun's right ascension, commonly noted by hours from 00 to 24. but better if divided by degrees or sins from 00 to 360, and both ways proceeding backward and forward as the signs of the Zodiac, or days of the Month.  
Lastly the outward edge or limb of the movable and loose piece both, is graduated unto 180 degrees, or two quadrants; whose centre is the pin, or pin-hole before mentioned, at the beginning of the sins on this side the fixed piece. The perpendicular is at 0/60 upon the loose piece; from whence reckoning along the outward edge of the loose piece, till it intersects the produced line of sins at the end of the fixed piece, you have 90 degrees. Or counting from 0/60 on the loose piece, and continuing it along the degrees of the outward limb of the movable piece, until they intersect the produced line of sins on the fixed piece at the head, you have again 90 degrees, which complete the Semicircle. The use of this line is for taking of altitudes, counting upon the former 90. degr. when you hold the head of the fixed piece toward the Sun: and numbering upon this latter, when you hold (which is best because the degrees are largest) the end of the fixed piece toward the Sun. There are other ways of numbering these degrees for finding of the Azimuth, etc. which shall be mentioned in their proper places.  
On the quadrantal side of the loose piece, the inward edge or limb is graduated unto 60 degrees (or twice 30, which you please) whose centre is a pin at the head. The use of this is to find the altitude of the Sun, or any Star, without thread and plummet; or to perform some uses of the Crossstaff. This is for large rules or instruments, and therefore not illustrated here.  
In the empty spaces upon the quadrantal side may be engraven the names of some fixed Stars, with their right ascensions and declinations.  
On the proportional side the lines issuing from the centre are the same upon the fixed and movable piece, but happily transplaced (thanks to the first contriver) after this manner: The line that lies next the inward edge on the fixed piece hath his fellow or correspondent line toward the outward edge on the movable piece; by which means these lines all meeting at the centre, stand all at the same angle, and give you the freedom from a great deal of trouble, in working proportions by sins and tangents, or laying down any sine or tangent to any Radius given, etc.  
The lines issuing from the centre toward the outward edge of the movable piece, whose fellow is next the inward edge of the fixed piece, is a line of natural sins on the outward side, marked at the end S, and on the inward side a line of lines or equal parts, noted at the end L; the middle line serving for both of them.  
The lines issuing from the centre next the inward edge of the movable piece, whose fellows are toward the outward edge of the fixed piece, are lines of natural tangents, which on the outward, side of the line is divided to 45 the Radius; and on the inward side of the lines (the middle line serving for both) at a quarter of the former Radius from the centre is another Radius noted 45 at the beginning, and continued to the tangent 75. These lines are noted at the end T. The use of these you will find Chap. 3,4,5.  
Betwixt the lines of sins and tangents, both upon the fixed and movable pieces, is placed a line of seconds, continued unto 60, and marked at the end. Se.  
Next to the outward edge on the fixed and movable piece (which is best discerned when those pieces are opened to the full length) is a line of Meridian's divided to 85; whose use is for Navigation, in describing Maps or Charts, etc.  
In the vacant spaces you may have a line of chords, sins, and tangents, to any Radius the space will bear; and what other any one thinks best of, as a line of latitudes and hours, etc.  
On the proportional side of the loose piece are lines for measuring all manner of solids, as Timber, Stone, etc. likewise for gauging of Vessels either in Wine or Ale measure.  
On the outward ledge of the movable and fixed piece, both (which in use must be stretched out to the full length) is a line of artificial numbers, sins, tangents, and versed sins. The first marked N, the second S, the third T, the fourth US.  
On the inward ledge of the movable piece is a line of 12 inches divided into halfs, quarters, half-quarters. Next to that is a pricked line, whose use is for computing of weight and carriages. Lastly a line of foot measure, or a foot divided into ten parts; and each of those subdivided into ten or twenty more.  
On the inward ledge of the loose piece you may have a line of circumference, diameter, square equal, and square inscribed.  
There will still be requisite sights, a thread and plummet. And if any go to the price of a sliding Index to find the shadow from the plains perpendicular, in order to taking a plains declination, and have a staff and a ballsocket, the Instrument is completed with its furniture. Proceed we now to the uses.  
Only note by the way that Mr. Brown hath (for conveniency of carrying a pair of Compasses, Pen, Ink, and Pencil) contrived the fixed piece and movable both to be hollow, and then the pieces that cover those hollows do, one supply the place of the loose piece for taking altitudes; the other (being a sliding rule) for measuring solids, and gauging Vessels without Compasses.   

CHAP. II.  
Some uses of the quadrantal side of the Instrument.  

PROBL. 1.  
To find the altitude of the Sun or any Star.  
Hung the thread and plummet upon the pin at the beginning of the line of sins 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 rays 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 middle 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 Month given, to find the Sun's place, declination, ascensional difference, or time of rising and setting, with his right ascension.  
The thread laid to the day of the Month gives the Sun's place in the line of signs, reckoning according to the order of the Months (viz. forward from March the 10th. to June, then backward to December, and forward again to March 10.) In the limb you have the Sun's declination, reckoning 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 shows the Sun's right ascension, in degrees, or hours, (according to the making of your line) counting from Aries toward the head, and so back again according to the course of the signs unto 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 difference. 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 Months, and find it cut in the Signs 12 deg. 20 min. for the Sun's place, on the limb 4 deg. 43 min. for the Sun's declination 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 shows 28 min. before six for the Suns rising; or which is all, 7 deg. for his ascensional difference.   

PROBL. 3.  
The declination of the Sun or any Star given to find their amplitude.  
Take the declination from the scale of altitudes, 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 ascensional difference, given to find the oblique ascension.  
In Northern declination, the difference betwixt the right ascension and ascensional difference, is the oblique ascension. In Southern declination take the sum 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 oblique ascension will be 5 deg. in Southern 18 deg.   

PROBL. 5.  
The Sun's altitude and declination, or the day of the Month given to find the hour.  
Take the Sun's altitude from the Scale of altitudes, and laying the thread to the declination in the limb (or which is all one, to the day in the Months) 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 former point shows the hour; but whether it be before or after noon, is left to your judgement 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 Month, 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 applied to the line of altitudes, 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 Sun's 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 Sun's 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 daybreak.  
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 daybreak, or clear light; with this run one point of the Compasses along the line of hours (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 minutes before 4: but twilight is 3 hours 12 minutes in the morning, or 8 hours 52 minutes 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 numbered on the limb of the movable piece from 60/0 toward the end (and when occasion requires, continue your numbering 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, than 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 Sun's 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 toward 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 numbered on the limb from 60/0 on the movable piece toward the end, and take the declination from the Scale of altitudes; then carry one point of your Compasses on the line of Azimuths (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 Sun's altitude at East or West in North declination, and by consequent 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 altitude, counting from 60/0 on the movable piece. Ex. gr. at 10 deg. declination the altitude 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 altitudes; set one point of your Compasses in the Azimuth given, then in North declinanation turn the other point toward the head, in South toward the end; and thereto laying the thread, on the limb you have the altitude, numbering from 60/0 on the movable 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 only 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 declination 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 Sun's right ascension (numbered 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 numbered upon the same line, observing which way you turned the point of your Compasses, 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 ascensions, it gives the true hour of the night. Ex. gr. The 22 of March I find the altitude of the Lion's heart 45 deg. his declination 13 d. 40 min. then by Probl. 5. I find his hour from the last Meridian 10 hours, 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 Lion's 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 consequent 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 movable piece toward the end, setting one point of your Compasses in the Azimuth, take the nearest distance to the thread; this applied 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 Sun-rise 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 applied to the particular Scale, gives the compliment of that latitude the Instrument was made for.   

PROBL. 20.  
To find the angles of the substile, stile, inclination of Meridian's, 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 compliment of declination counted on the line of Azimuths, and on the limb it gives the substile counting from 60/0 on the movable piece.   

Sect. 2.  
To find the angle of the style's height.  
On the line of Azimuths take the distance from the Plains declination to 90. This applied to the Scale of altitudes gives the angle of the stile.   

Sect. 3.  
The angle of the Substile given to find the inclination of Meridian's.  
Take the angle of the substile from the Scale of altitudes, and applying it from 90 on the Azimuth line toward the end; the figures show the compliment of inclination of Meridian's.   

Sect. 4.  
To find the angle betwixt 6 and 12.  
Take the declination 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 and on the limb it gives the compliment of the angle sought, numbering from 60/0 on the movable piece toward the end.  
This last rule is not exact, nor is it here worth the labour to rectify it by another sine added; sith you have an exact proportion for the Problem in the Treatise of Dialling Chap. 2. Sect. 5. Paragr. 4.     

CHAP. III.  
Some uses of the Line of natural signs on the Quadrantal side of the fixed piece.  

PROBL. 1.  
How to add one sign to another on the Line of Natural Sins.  
TO add one sine to another, is to augment the line of one sine by the line of the other sine to be added to it. Ex. gr. To add the sine 15 to the sine 20, I take the distance from the beginning of the line of sins unto 15, and setting one point of the Compasses in 20, upon the same line, turn the other toward 90, which I find touch in 37. So that in this case (for we regard not the Arithmetical, but proportional aggregate) 15 added to 20, upon the line of natural sins, is the sine 37 upon that line, and from the beginning of the line to 37 is the distance I am to take for the sum of 20 and 15 sins.   

PROBL. 2.  
How to subtract one sine from another upon the line of natural sins.  
The substracting of one sine from another, is no more than taking the distance from the lesser to the greater on the line of sins, and that distance applied to the line from the beginning, gives the residue or remainder. Ex. gr. To subtract 20 from 37 I take the distance from 20 to 37 that applied to the line from the beginning gives 15 for the sine remaining.   

PROBL. 3.  
To work proportions in sins alone.  
Here are four Cases that include all proportions in sins alone.  

CASE 1.  
When the first term is Radius, or the Sine 90.  
Lay the thread to the second term counted on the degrees upon the movaeble piece from the head toward the end, then numbering the third on the line of sins, take the nearest distance from thence to the thread, and that applied to the Scale from the beginning gives the fourth term. Ex. gr.  
As the Radius 90 is to the sine 20, so is the sine 30 to the sine 10.   

CASE 2.  
When the Radius is the third term.  
Take the sine of the second term in your Compasses, and enter it in the first term upon the line of sins, and laying the thread to the nearest distance, on the limb the thread gives the fourth term. Ex. gr.  
As the sine 30 is to the sine 20, so is the Radius to the sine 43. 30. min.   

CASE 3.  
When the Radius is the second term.  
Provided the third term be not greater than the first, transpose the terms. The method of transposition in this case is, as the first term is to the third, so is the second to the fourth, and then the work will be the same as in the second case. Ex. gr.  
As the sine 30 is to the radius or sine 90, so is the sine 20 to what sine; which transposed is  
As the sine 30 is to the sine 20, so is the radius to a fourth sine, which will be found 43, 30 min. as before.   

CASE 4.  
When the Radius is none of the three terms given.  
In this case when both the middle terms are less than the first, enter the sine of the second term in the first, and laying the thread to the nearest distance, take the nearest extent from the third to the thread: this distance applied to the scale from the beginning gives the fourth. Ex. gr.  
As the sine 20 to the sine 10, so is the sine 30 to the sine 15.  
When only the second term is greater than the first, transpose the terms and work as before.  
But when both the middle terms be greater than the first, this proportion will not be performed by this line without a parallel entrance or double radius; which inconveniency shall be remedied in its proper place, when we show how to work proportions by the lines of natural sins on the proportional or sector side.  
These four cases comprising the method of working all proportions by natural sins alone, I shall propose some examples for the exercise of young practitioners, and therewith conclude this Chapter.    

PROBL. 4.  
To find the Sun's amplitude in any Latitude.  
As the cousin of the Latitude is to the sine of the Sun's declination, so is the radius to the sine of amplitude.   

PROBL. 5.  
To find the hour in any Latitude in Northern Declination.  
Proport. 1. As the radius to the sine of the Sun's declination, so is the sine of the latitude to the sine of the Sun's altitude at six. By Probl. 2. subtract this altitude at six from the present altitude, and take the difference. Then  
Proport. 2. As the cousin of the latitude is to that difference, so is the radius to a fourth sine. Again  
Proport. 3. As the cousin of the declination to that fourth sine, so is the radius to the sine of the hour from six.   

PROBL. 6.  
To find the hour in any Latitude when the Sun is in the Equinoctial.  
As the cousin of the latitude is to the sine of altitude, so is the radius to the sine of the hour from six.   

PROBL. 7.  
To find the hour in any latitude in Southern Declination.  
Proport. 1. As the radius to the sine of the Sun's declination, so is the sine of the latitude to the sine of the Sun's depression at six; add the sine of depression to the present altitude by Probl. 1. Then  
Proport. 2. As the cousin of the latitude is to that sum, so is the radius to a fourth sine. Again,  
Proport. 3. As the cousin of declination is to the fourth sine, so is the radius to the sine of the hour from six.   

PROBL. 8.  
To find the Sun's Azimuth in any latitude in Northern Declination.  
Proport. 1. As the sine of the latitude to the sine of declination, so is the radius to the sine of altitude at East, or West. By Probl. 2. subtract this from the present altitude, then,  
Proport. 2. As the cousin of the latitude is to that residue, so is the radius to a fourth sine. Again,  
Proport. 3. As the cousin of the altitude is to that fourth sine, so is the radius to the sine of the Azimuth from East or West.   

PROBL. 9  
To find the Azimuth for any latitude when the Sun is in the Equator.  
Proport. 1. As the cousin of the latitude to the sine of altitude, so is the sine of the latitude to a fourth sine.  
Proport. 2. As the cousin of altitude to that fourth sine, so is the radius to the sine of the Azimuth from East, or West.   

PROBL. 10.  
To find the Azimuth for any latitude in Southern Declination.  
Proport. 1. As the cousin of the latitude to the sine of altitude, so is the sine of the latitude to a fourth. Having by Probl. 4. found the Sun's amplitude, add it to this fourth sine by Probl. 1. and say  
As the cousin of the altitude is to the sum, so is the radius to the sine of the Azimuth from East or West. The terms mentioned in the 5th. 7th. 8th. 10th. Problems are appropriated unto us that live on the North side the Equator. In case they be applied to such latitudes as lie on the South side the Equator. Then what is now called Northern declination, name Southern, and what is here styled Southern declination, term Northern, and all the proportion with the operation is the same.  
These proportions to find the hour and Azimuth, may be all readily wrought by the lines of artificial sins, only the addition and substraction must always be wrought upon the line of natural sins.    

CHAP. IU.  
Some uses of the Lines on the proportional side of the Instrument, viz. the Lines of natural Sins, Tangents, and Secants.  

PROBL. 1.  
To lay down any Sine, Tangent, or Secant to a Radius given. See Fig. 1.  
IF you be to lay down a Sine, enter the Radius given in 90, and 90 upon the lines of Sines, keeping the Sector at that gage, set one point of your Compasses in the Sine required upon one line, and extend the other point to the same Sine upon the other Line: This distance is the length of the Sine required to the given Radius. Ex. gr. Suppose A. B. the Radius given, and I require the Sine 40. proportional to that Radius. Enter A. B. in 90, and 90 keeping the Sector at that gage, I take the distance, 'twixt 40 on one side, to 40 on the other, that is, C. D. the Sine required.  
The work is the same, to lay down a Tangent to any Radius given, provided you enter the given Radius in 45, and 45, on the line of Tangents. Only observe if the Tangent required be less than 45. you must enter the Radius in 45. and 45 next the end of the Rule. But when the Tangent required exceeds 45. enter the Radius given in 45, and 45 'twixt the centre and end, and keeping the Sector at that Gage, take out the Tangent required. This is so plain, there needs no example.  
To lay down a Secant to any Radius given, is no more than to enter the Radius in the two pins at the beginning of the line of Secants; and keeping the Sector at that Gage, take the distance from the number of the Secant required on one side, to the same number on the other side, and that is the Secant sought at the Radius given.  
The use of this Problem will be sufficiently seen in delineating Dial's, and projecting the Sphere.   

PROBL. 2.  
To lay down any Angle required by the Lines of Sines, Tangents, and Secants. See Fig. 2.  
There are two ways of protracting an Angle by the Line of Sines, First if you use the Sins in manner of Chords. Then having drawn the line A B at any distance of your Compass, set one point in B, and draw a mark to intersect the Line B A, as E F. Enter this distance B F in 30, and 30 upon the Lines of Sines, and keeping the Sector at that Gage, take out the Sine of half the Angle required, and setting one point where F intersects B A, turn the other toward E, and make the mark E, with a ruler draw B E and the Angle E B F is the Angle required, which here is 40. d.  
A second method by the lines of Sines is thus, Enter B A Radius in the Lines of Sines, and keeping the Sector at that Gage, take out the Sine of your Angle required with that distance, setting one point of your Compasses in A, sweep the ark D, a line drawn from B by the connexity of the Ark D, makes the Angle A B C 40 d. as before.  
To protract an Angle by the Lines of Tangents is easily done, draw B A the Radius upon A, erect a perpendicular, A C, enter B A in 45, and 45 on the Lines of Tangents, and taking out the Tangent required (as here 40) set it from A to C. Lastly, draw B C, and the Angle C B A is 40 d. as before.  
In case you would protract an Angle by the Lines of Secants. Draw B A, and upon A erect the perpendicular A C, enter A B in the beginning of the Lines of Secants, and take out the Secant of the Angle, with that distance, setting one point of your Compasses in B, with the other cross the perpendicular A C, as in C. This done, lay a Ruler to B, and the point of intersection, and draw the Line B C. So have you again the Angle C B A. 40. d. by another projection. These varieties are here inserted only to satisfy a friend, and recreate the young practitioner in trying the truth of his projection.   

PROBL. 3.  
To work proportions in Sines alone, by the Lines of natural Sins on the proportional side of the Instrument.  
The general rule is this. Account the first term upon the Lines of Sines from the Centre, and enter the second term in the first so accounted, keeping the Sector at that Gage, account the third term on both lines from the Centre, and taking the distance from the third term on one line to the third term on the other line, measure it upon the line of Sins from the beginning, and you have the fourth term. Ex. gr.  
As the Radius is to the Sine 30, so is the Sine 40 to the Sine 18. 45.  
There is but one exception in this Rule, and that is when the second term is greater than the first; yet the third lesser than the first, and in this case transpose the terms, by Chap 3. Probl. 3. Case 3.  
But when the second term is not twice the length of the first, it may be wrought by the general Rule without any transposition of terms. Ex. gr.  
As the Sine 30 is to the Sine 50, so is the Sine 20 to the Sine 31. 30. min.  
And by consequent, when the third term is greater than the first, provided it be not upon the line, double the length thereof, it may be wrought by transposing the terms, although the second was twice the length of the first. Ex. gr.  
As the Sine 20 is to the Sine 60, so is the Sine 42, to what Sine? which transposed is,  
As the Sine 20 is to the Sine 42, so is the Sine 60 to the Sine 35. 30.  
This case will remove the inconveniency mentioned, Chap. 3. Probl. 3. Case 4. of a double Radius. I intended there to have adjoined the method of working proportions by natural Tangents alone, and by natural Sins, and Tangents, conjunctly: But considering the multiplicity of proportions when the Tangents exceed 45. I suppose it too troublesome for beginners, and a needless variety for those that are already Mathematicians. Sith, both may be eased by the artificial Sins and Tangents on the outward ledge, where I intent to treat of those Cases at large, and shall in this place only annex some proportions in Sines alone, for the exercise of young beginners.   

PROBL. 4.  
By the Lines of Natural Sins to lay down any Tangent, or Secant required to a Radius given.  
In some Cases, especially for dialing, your Instrument may be defective of a Tangent, or Secant for your purpose, Ex. gr. when the Tangent exceeds 76, or the Secant is more than 60. In these extremities use the following Remedies. First, for a Tangent.  
As the cousin of the Ark is to the Radius given, so is the sine of the Ark to the length of the Tangent required. Secondly, for a Secant.  
As the cousin of the Ark is to the Radius given, so is the Sine 90 to the length of the Secant required.   

PROBL. 5.  
The distance from the next Equinoctial Point given to find the Sun's declination.  
As the Radius to the sine of the Sun's greatest declination, so is the sine of his distance from the next Equinoctial Point to the sine of his present declination.   

PROBL. 6.  
The declination given to find the Sun's Equinoctial Distance.  
As the sine of the greatest declination is to the sine of the present declination, so is the Radius to the sine of his Equinoctial Distance.   

PROBL. 7.  
The Altitude, Declination, and Distance of the Sun from the Meridian given to find his Azimuth.  
As the cousin of the altitude, to the cousin of the hour from the Meridian, so is the cousin of declination to the sine of the Azimuth.    

CHAP. V.  
Some uses of the Lines of the Lines, on the proportional side of the Instrument.  

PROBL. 1.  
To divide a Line given into any Number of equal parts. See Fig. 3.  
SUppose A B a Line given to be divided into nine equal parts. Enter A B in 9, and 9 on the lines of lines, keeping the Sector at that gage, take the distance from 8, on one side, to 8 on the other, and apply it from A upon the line A B, which reacheth to C; then is C B a ninth part of the line A B. By this means you may divide any line (that is not more than the Instrument in length) into as many parts as you please, viz. 10, 20, 30, 40, 50, 100, 500, etc. parts according to your reckoning the divisions upon the lines, Ex. gr. The line is actually divided into 200 parts, viz. first into 10, marked with Figures, and each of those into twenty parts more. Again, if the line represents a 1000, than every figured division is 100, the second or shorter division is 10, and the third or shortest division is 5. In case the whole line was 2000, than every figured division is 200, every smaller or second division is 20, every third or smallest division is 10, etc. Suppose I have any line given, which is the base of a Triangle, whose content is 2000 poles, and I demand so much of the Base as may answer 1750 poles. Enter the whole line in 10, and 10 at the end of the lines of lines, and keep the Sector at that gage. Now the whole line representing 2000 poles, every figured division is 200; therefore 1700 is eight and an half of the figured divisions, and 50 is five of the smallest divisions more (for in this case every smallest division is 10, as was before expressed) wherefore setting one point of the Compasses in 15 of the smallest divisions beyond 8 on the Rule. I extend the other point to the same division upon the line on the other side, and that distance is 1750 poles in the base of the Triangle proposed. How ready this is to set out a just quantity in any plat of ground, I shall show in a Scheam, Chap. 12.   

PROBL. 2.  
To work proportions in Lines, or Numbers, or the Rule of three direct by the Lines of Lines.  
Enter the second term in the first, and keeping the Sector at that gage, take the distance 'twixt the third on one line, to the third on the other line, that distance is the fourth in lines, or measured upon the line from the centre, gives the fourth in numbers, Ex. gr.  
As 7 is to 3, so is 21 to 9   

PROBL. 3.  
To work the Rule of Three inverse, or the back Rule of Three by the Lines of Lines.  
In these proportions there are always three terms given to find a fourth, and of the three given terms two are of one denomination (which for distinction sake I call the double denomination;) and the third term is of a different denomination from those two, which I therefore call the single denomination, of which the fourth term sought must also be. Now to bring these into a direct proportion, the rule is this. When the fourth term sought is to be greater than the single denomination (which you may know by sight of the terms given) say,  
As the lesser double denomination is to the greater double denomination, so is the single denomination to the fourth term sought. The work is by Probl. 2.  
If 60 men do a work in 5 days, how long will 30 men be about it?  
As 30 is to 60, so is 5 to 10. The number of days for 30 men in the work.  
Again, when the fourth term is to be less than the single denomination, say,  
As the greater double denomination is to the lesser double denomination, so is the single denomination to the fourth term sought.  
If 30 men do a work in 5 days, how long shall 60 be doing of it?  
As 60 is to 30, so is 5 to 2½. The time for 60 men in the work.   

PROBL. 4.  
The length of any perpendicular, with the length of the shadow thereof given, to find the Sun's Altitude.  
At the length of the shadow upon the lines of lines, is to the Tangent 45, so is the length of the perpendicular numbered upon the lines of lines, to the tangent of the Sun's altitude.   

PROBL. 5.  
To find the Altitude of any Tree, Steeple, etc. at one station.  
At any distance from the object (provided the ground be level) with your Instrument, look to the top of the object along the outward ledge of the fixed piece, and take the angle of its altitude. This done, measure by feet or yards, the distance from your standing to the bottom of the object. Then say,  
As the cousin of the altitude is to the measured distance numbered upon the lines of lines, so is the sine of the altitude to a fourth number of feet or yards (according to the measure you meeted the distance) to this fourth, add the height of your eye from the ground, and that sum gives the number of feet or yards in the altitude.    

CHAP. VI  
How to work proportions in Numbers, Sins, or Tangents, by the Artificial Lines thereof on the outward ledge.  
THe general rule for all of these, is to extend the Compasses from the first term to the second (and observing whether that extent was upward or downward) with the same distance, set one point in the third term, and turning the other point the same way, as at first, it gives the fourth. But in Tangents when any of the terms exceeds 45, there may be excursions, which in their due place I shall remove.  

PROBL. 1.  
Numeration by the Line of Numbers.  
The whole line is actually divided into 100 proportional parts, and accordingly distinguished by figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, and then, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 So that for any number under 100, the Figures readily direct you, Ex. gr. To find 79 on the line of numbers, count 9 of the small divisions beyond 70, and there is the point for that number. Now as the whole line is actually divided into 100 parts, so is every one of those parts subdivided (so far as conveniency will permit) actually into ten parts more, by which means you have the whole line actually divided into 1000 parts. For reckoning the Figures impressed, 1, 2, 3, 4, 5, 6, 7, 8, 9, to be 10, 20, 30, 40, 50, 60, 70, 80, 90, and the other figures which are stamped 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, to be 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000 You may enter any number under 1000 upon the line, according to the former directions. And any numbers whose product surmount not 1000, may be wrought upon this line; but where the product exceeds 1000, this line will do nothing accurately: Wherefore I shall willingly omit many Problems mentioned by some Writers to be wrought by this line, as squaring, and cubing of numbers, etc. Sith they have only nicety, and nothing of exactness in them.   

PROBL. 2.  
To multiply two numbers, given by the Line of Numbers. The proportion is this.  
As 1 on the line is to the multiplicator, so is the multiplicand to the product. Ex. gr.  
As 1 is to 4, so is 7, to what?  
Extend the Compasses from the first term, viz. I unto the second term, viz. 4. with that distance, setting one point in 7 the third term, turn the other point of the Compasses toward the same end of the rule, as at first, and you have the fourth, viz. 28. There is only one difficulty remaining in this Problem, and that is to determine the number of places, or figures in the product, which may be resolved by this general rule. The product always contains as many figures as are in the multiplicand, and multiplicator both, unless the two first figures of the product be greater than the two first figures in the multiplicator, and then the product must have one figure less than are in the multiplicator, and multiplicand both. Ex. gr. 47 multiplied by 25, is 2175, consisting of four figures; but 16 multiplied by 16, is 240, consisting of no more than three places, for the reason before mentioned. I here (for distinction sake) call the multiplicator the lesser of the two numbers, although it may be either of them at pleasure.   

PROBL. 3.  
To work Division by the Line of Numbers.  
As the divisor is to 1, so is the dividend to the quotient.  
Suppose 800 to be divided by 20, the quotient is 40. For,  
As 20 is to 1, so is 800 to 40.  
To know how many figures you shall have in the quotient, take this rule. Note the difference of the numbers of places or figures in the dividend and divisor. Then in case the quantity of the two first figures to the left hand in your divisor be less than the quantity of the two first figures to the left hand in your dividend, the quotient shall have one figure more than the number of difference: But where the quantity of the two first figures of the divisor is greater than the quantity of the two first figures of the dividend, the quotient will have only that number of figures noted by the difference. Ex. gr. 245 divided by 15. will have two figures in the quotient; but 16 divided 〈…〉 ●●ve only one figure in the quotient.   

PROBL. 4.  
To find a mean proportional 'twixt two Numbers given by the line of Numbers.  
Divide the space betwixt them upon the line of numbers into two equal parts, and the middle point is the mean proportional. Ex. gr. betwixt 4 and 16, the mean proportional is 8. If you were to find two mean proportionals, divide the space 'twixt the given numbers into three parts. If four mean proportionals divide it into five parts, and the several points 'twixt the two given numbers, will show the respective mean proportionals.   

PROBL. 5.  
To work proportions in Sines alone, by the Artificial Line of Sines.  
Extend the Compasses from the first term to the second, with that distance set one point in the third term, and the other point gives the fourth. Only observe that if the second term be less than the first, the fourth must be less 〈…〉 or if the second term exceed 〈…〉 fourth will be greater than the third. This may direct you in all proportions of sins and tangents singly or conjunctly, to which end of the rule to turn the point of your Compasses, for finding the fourth term. Ex. gr.  
As the sine 60 is to the sine 40, so is the sine 20 to the sine 14. 40. Again,  
As the sine 10 is to the sine 20, so is the sine 30 to the sine 80.   

PROBL. 6.  
To work proportions in Tangents alone by the Artificial Line of Tangents.  
For this purpose the artificial line of tangents must be imagined twice the length of the rules, and therefore for the greater conveniency, it is doubly numbered, viz. First from 1 to 45, which is the radius, or equal to the sine 90: In which account every division hath (as to its length on the rule) a proportional decrease. Secondly, it's numbered back again from 45 to 89, in which account every division hath (as to its length on the line) a proportional increase. So that the tangent 60 you must imagine the whole length of the Rule; and so much more as the distance from 45 unto 30 or 60 is. This well observed, all proportions in tangents are wrought after the same manner of extending the Compasses from the first term to the second, and that distance set in the third, gives the fourth, as was for sins and numbers. But for the remedying of excursions, sith the line is no more than half the length, we must imagine it. I shall lay down these Cases.  

CASE 1.  
When the fourth term is a tangent exceeding 45, or the Radius. Ex. gr.  
As the tangent 10 is to the tangent 30, so is the tangent 20, to what?  
Extending the Compasses from 10 on the line of tangents to 30, with that distance I set one point in 20, and find the other point reach beyond 45, which tells me the fourth term exceeds 45, or the radius; wherefore with the former extent, I set one point in 45; and turning the other toward the beginning of the line, I mark where it toucheth, and from thence taking the distance to the third term, I have the excess of the fourth term above 45 in my compass: wherefore with this last distance setting one point in 45, I turn the other upon the line, and it reacheth to 50, the tangent sought.   

CASE 2.  
When the first term is a tangent exceeding 45, or the Radius. Ex. gr.  
As the tangent 50 is to the tangent 20, so is the tangent 30, to what?  
Because the second term is less than the first, I know the fourth must be less than the third. All the difficulty is to get the true extent from the tangent 50 to 20. To do this, take the distance from 45 to 50, and setting one point in 20, the second term, turn the other toward the beginning of the line, marking where it toucheth, extend the Compasses from the point where it toucheth to 45, and you will have the same distance in your Compasses as from 50 to 20, if the line had been continued at length unto 89 tangents, with this distance, set one point in 30 the third term, and turn the other toward the beginning (because you know the fourth must be less) and it gives 10 the tangent sought.   

CASE. 3.  
When the third term is a Tangent exceeding 45, or the Radius.  
As the tangent 40 is to the tangent 12, 40 min. so is the tangent 65, to what?  
Extend the Compasses from 40 to 12 d. 40 min. with distance, setting one point in 65. turn the other toward 45, and you will find it reach beyond it, which assures you the fourth term will be less than 45. Therefore lay the extent from 45 toward the beginning, and mark where it toucheth, take the distance from that point to 65, and laying that distance from 45 toward the beginning it gives 30, the tangent sought.  
These Cases are sufficient to remove all difficulties. For when the second term exceeds the Radius, you may transpose them, saying, as the first term is to the third; so is the second to the fourth, and then it's wrought by the third Case.  
I suppose it needless to add any thing about working proportions by sins and tangents conjunctly, sith, enough hath been already said of both of them apart, in these two last Problems; and the work is the same when they are intermixed. Only some proportions I shall adjoin, and leave to the practice of the young beginner, with the directions in the former Cases.    

PROBL. 7.  
To find the Sun's ascensional difference in any Latitude.  
As the co-tangent of the latitude is to the tangent of the Sun's declination, so is the radius to the sine of the ascensional difference.   

PROBL. 8.  
To find at what hour the Sun will be East, or West in any Latitude.  
As the tangent of the latitude is to the tangent of the Sun's declination, so is the radius to the cousin of the hour from noon.   

PROBL. 9  
The Latitude, Declination of the Sun, and his Azimuth from South, given to find the Sun's Altitude at that Azimuth.  
As the radius to the cousin of the Azimuth from south, so is the co-tangent of the latitude, to the tangent of the Sun's altitude in the equator at the Azimuth given. Again,  
As the sine of the latitude is to the sine of the Sun's declination, so is the cousin of the Sun's altitude in the equator (at the same Azimuth from East or West) to a fourth ark.  
When the Azimuth is under 90, and the latitude and declination is under the same pole, add this fourth ark to the altitude in the equator. In Azimuths exceeding 90, when the latitude and declination is under the same pole, take the equator altitude out of the fourth ark. Lastly, when the latitude and declination respect different poles, take the fourth ark out of the equator altitude, and you have the altitude sought.   

PROBL. 10.  
The Azimuth, Altitude, and Declination of the Sun, given to find the hour.  
As the cousin of declination is to the sine of the Sun's Azimuth, so is the cousin of the altitude to the fine of the hour from the Meridian.  
Proportions may be varied eight several ways in this manner following. 

1. As the first term is to the second, so is the third to the fourth.  
2. As the second term is to the first, so is the fourth to the third.  
3. As the third term is to the first, so is the fourth to the second.  
4. As the fourth term is to the second, so is the third to the first.  
5. As the second term is to the fourth, so is the first to the third.  
6. As the first term is to the third, so is the second to the fourth.  
7. As the third term is to the fourth, so is the first to the second.  
8. As the fourth term is to the third, so is the second to the first.    
By thesse's any one may vary the former proportions, and make the Problems three times the number here inserted. Ex. gr. To find the ascensional difference in Problem 10, of this Chapter, which runs thus.  
As the co-tangent of the latitude is to the tangent of the Sun's declination, so is the radius to the sine of ascensional difference. Then by the third variety you may make another Problem, viz.  
As the radius is to the co-tangent of the latitude, so is the sine of the Sun's ascensional difference to the sine of his declination. Again, by the fourth variety you may make a third Problem, thus,  
As the sine of the Sun's ascensional difference is to the tangent of the Sun's declination, so is the radius to the co-tangent of the latitude.  
By this Artifice many have stuffed their Books with bundles of Problems.    

CHAP. VII.  
Some uses of the Lines of Circumference, Diameter, Square Equal, and Square Inscribed.  
ALL these are lines of equal parts, bearing such proportion to each other, as the things signified by their names. Their use is this, Any one of them given in inches or feet, etc. to find how much any of the other three are in the same measure. Suppose I have the circumference of a Circle, Tree, or Cylinder given in inches, I take the same number of parts (as the Circle is inches) from the line of circumference, and applying that distance to the respective lines, I have immediately the square equal, square inscribed, and diameter, in inches, and the like, if any of those were given to find the circumference. This needs no example.  
The conveniency of this line any one may experiment in standing timber; for taking the girth, or circumference with a line, find the diameter; from that diameter abate twice the thickness of the bark, and you have the true diameter, when it's barked, and by Chap. 9 Probl. 5. you will guests very near at the quantity of timber in any standing Tree.   

CHAP. VIII.  
To measure any kind of Superficies, as Board, Glass, Pavement, Walnscot, Hangings, Walling, Slating, or Tyling, by the line of Numbers on the outward ledge.  
THe way of accounting any number upon, or working proportions by, the line of numbers, is sufficiently shown already, Chap. 6. which I shall not here repeat, only propose the proportions for these Problems, and refer you to those directions.  

PROBL. 1.  
The breadth of a Board given in inches, to find how many inches in length make a foot at that breadth, say,  
As the breadth in inches is to 12, so is 12 to the length in inches for a foot at that breadth. Ex. gr. At 8 inches breadth you must have 18 inches in length for a foot.   

PROBL. 2.  
The breadth and length of a Board given to find the content.  
As 12 is to the length in feet and inches, so is the breadth in inches to the content in feet. Ex. gr. at 15 inches breadth, and 20 foot length, you have 25 foot of Board.   

PROBL. 3.  
A speedy way to measure any quantity of Board.  
The two former Problems are sufficient to measure small parcels of Board. When you have occasion to measure greater quantities, as 100 foot, or more, lay all the boards of one length together, and when the length of the boards exceeds 12 foot, use this proportion.  
As the length in feet and inches is to 12, so is 100 to the breadth in inches for an 100 foot. Ex. gr. At 30 foot in length 40 inches in breadth, make an 100 foot of board (reckoning five score to the hundred.) This found with a rule or line, measure 40 inches at both ends in breadth, and you have 100 foot. When one end is broader than another, you may take the breadth of the overplus of 100 foot at both ends, and taking half that sum for the true breadth of the overplus by Probl. 2. find the content thereof.  
When your boards are under 12 foot in length, say,  
As the length in feet and inches is to 12, so is 50 unto the breadth in inches for 50 foot of board, and then you need only double that breadth to measure 100 foot as before. In like manner you may measure two, three, four, five, a hundred, etc. foot of board speedily, as your occasion requires.   

PROBL. 4.  
To measure Wainscot, Hangings, Plaster, etc.  
These are usually computed by the yard, and then the proportion is.  
As nine to the length in feet, and inches, so is the breadth or depth in feet and inches to the content in yards, Ex. gr. At 18 foot in length, and two foot in breadth, you have four yards.   

PROBL. 5.  
To measure Masons, or Slaters Work, as Walling, Tyling, etc.  
The common account of these is by the rood, which is eighteen foot square, that is 324 square foot in one rood, and then the proportion is.  
As 324 to the length in feet, so is the breadth in feet to the content in roods.  
Ex. gr. At 30 foot in length, and 15 foot in breadth, you have 1 rood 3/10 and better, or one rood 126/124 parts of a rood.    

CHAP. IX.  
The mensuration of Solids, as Timber, Stone, etc. by the lines on the proportional side of the loose piece.  
THese two lines meeting upon one line in the midst betwixt them (for distinction sake) I call one the right, the other the left line, which are known by the hand they stand toward when you hold up the piece in the right way to read the Figures.  
The right line hath two figured partitions. The first partition is from 3, at the beginning to the letters Sq. every figured division representing an inch, and each subdivision quarters of an inch.  
The next partition is from the letters Sq. unto 12, at the end, every figured division signifying a foot, and each sub-division the inches in a foot. The letters T R, and T D, are for the circumference and Diameter in the next measuring of Cylinders. The letters R and D. for the measuring of Timber, according to the vulgar allowance, when the fourth part of the girt is taken, etc. The letters A and W are the gauger points for Ale and Wine Measures. Lastly, the figures 12 'twixt D and T D. are for an use, expressed Probl. 2.  
The left line also hath two figured partitions, proceeding first from 1 at the beginning to one foot, or 12 inches, each whereof is subdivided into quarters. From thence again to 100, each whereof to 10 foot, is subdivided into inches, etc. and every foot is figured. But from 10 foot to 100, only every tenth foot is figured; the sub-divisions representing feet. The method of working proportions by these lines (only observing the sides) is the same as by the line of Numbers, viz. extending from the first to the second, etc.  

PROBL. 1.  
To reduce Timber of unequal breadth and depth to a true Square.  
As the breadth on the left is to the breadth on the right, so is the depth on the left, to the square on the right line.  
At 7 inches breadth, and 18 inches depth, you have 11 inches ¼ and better for the true square.   

PROBL. 2.  
The square of a Piece of Timber given in Inches, or Feet, and Inches to find how much in length makes a Foot.  
As the square in feet and inches on the right is to one foot on the left, so is the point Sq. on the right to the number of feet and inches on the left for a foot square of Timber.  
At 18 inches square, 5 inches ¼ and almost half a quarter in length makes a foot. When your Timber (if it be proper to call such pieces by that name) is under 3 inches square, account the figured divisions on the right line from the letters Sq. to the end, for inches, and each sub-division twelve parts of an inch. So that every three of them makes a quarter of an inch. Then the proportion is, as the inches and quarters square on the right is to 100 on the left, so is the point 12 'twixt D, and T D, on the right, to the number of feet in length on the left to make a foot of Timber.  
As 2 inches ½ square you must have 23 foot 6 inches, and somewhat better for the length of a foot of Timber.   

PROBL. 3.  
The square and length of a plece of Timber given to find the content.  
As the point Sq. on the right is to the length in feet and inches on the left, so is the square in feet and inches on the right to the content in feet on the left.  
At 30 foot in length, and 15 inches square, you have 46 foot ½ of Timber.  
At 20 foot in length, and 11 inches square you have 16 foot, and almost ¼ of Timber.  
When you have a great piece of Timber exceeding 100 foot (which you may easily see by the excursion upon the rule) then take the true square, and half the length, finde the content thereof by the former proportion, and doubling that content, you have the whole content.   

PROBL. 4.  
The Circumference, or girth, of a round piece of Timber, being given, together with the length, to find the content.  
As the point R. on the right, is to the length in feet and inches on the left, so is the circumference in feet and inches on the right, to the content in feet on the left.  
At 20 foot in length, and 7 foot in girth, you have 60 foot of Timber for the content. This is after the common allowance for the waste in squaring; and although some are pleased to quarrel with the allowancer, as wronging the seller, and giving the quantity less, than in truth it is; yet I presume when they buy it themselves, they scarcely judge those Chips worth the hewing, and have as low thoughts of the overplus, as others have of that their admonition.  
If it be a Cylinder that you would take exact content of, then say,  
As the point T R, on the right, is to the length on the left, so is the girt on the right to the exact content on the left.  
At 15 foot in length, and 7 foot in girt, you have 59 foot of solid measure.  
The Diameter of any Cylinder given, you may by the same proportion find the content, placing the point D, instead of R, in the proportion for the usual allowance, and the point T D, for the exact compute.   

PROBL. 5.  
To measure tapered Timber.  
Take the square or girt at both ends, and note the sum and difference of them. Then for round Timber, as the point R. on the right, is to the length on the left, so is half the sum of the girt at both ends on the right, to a number of feet on the left. Keep this number, and say again,  
As the point R. on the right, is to the third part of the former length on the left, so is half the difference of the girts on the right, to a number of feet on the left; which number added to the former, gives the true content. The same way you may use for square Timber, only setting the feet and inches square instead of the girt, and the point Sq. instead of the point R. At 30 foot in length, 7 foot at one end, and 5 at the other in girt, half the sum of the girts is 6 foot, or 72 inches; the first number of feet found 67, half the difference of the girts is 1 foot, or 12 inches, the third part of the length 10 foot; then the second number found will be 7 foot, one quarter and half a quarter. The sum of both (or true content) 74 foot, one quarter, and half a quarter.  
For standing Timber, take the girt about a yard from the bottom, and at 5 foot from the bottom, by Chap. 7, set down these two diameters without the bark; and likewise the difference 'twixt them. Again, by Chap. 6. Probl. 4. find the altitude of the tree, so far as it bears Timber (or as we commonly phrase it, to the collar) this done, you may very near proportion the girt at the collar and content of the tree, before it falls.  
In case any make choice of the hollow contrivance mentioned, Chap. 1. they need no compasses in the mensuration of any solid; provided the lines for solid measure, and gauging vessels, be doubly impressed (only in a reverted order, one pair of lines proceeding from the head toward the end, and the other pair from the end toward the head) upon the sliding cover, and its adjacent ledges.  
This done, the method of performing any of the Problems mentioned in this Chapter, is easy. For whereas you are before directed to extend the Compasses from the first term to the second; and with that distance setting one point in the third term, the other point gave the fourth, or term sought. So here, observing the lines as before, slide the cover until the first term stand directly against the second; then looking for the third on its proper line, it stands exactly against the fourth term, or term sought on the other line.  
Only note, that when the second term is greater than the first, it's performed by that pair of lines proceeding from the head toward the end: But when the first term is greater than the second, it is resolved by that pair of lines which is numbered from the end toward the head.    

CHAP. X.  
The Gage Vessels, either for Wine, or Ale, Measure.  

PROBL. 1.  
The Diameter at Head, and Diameter at Boung, given in Inches, and tenth parts of an Inch, to find the mean Diameter in like measure.  
TAke the difference in inches, and tenth parts of an inch, between the two diameters. Then say by the line of numbers,  
As 1 is to 7, so is the difference to a fourth number of inches, and tenth parts of an inch. This added to the Diameter at head, gives the mean Diameter. Ex. gr.  
At 27 inches the boung, and 19 inches two tenths at the head, the difference is 7 inches, 8 tenths. The fourth number found by the proportion will be 5 inches, 4 tenths, and one half, which added to the diameter at the head, giveth 24 inches, 6 tenths, and one half tenth of an inch for the mean diameter.   

PROBL. 2.  
The length of the Vessel, and the mean Diameter given in Inches, and tenth parts of an Inch, to find the content in Gallons, either in Wine or Ale measure.  
Note first, that the point A. on the right is the gauge point for Ale measure, and the point W. on the right, is the gauge point for Wine measure. Then say,  
As the gauge point on the right to the length in inches, and tenth parts of an inch on the left, so is the mean diameter in inches and tenth parts of an inch on the right, to the content in gallons on the left.    

CHAP. XI.  
Some uses of the Lines on the inward ledge of the movable piece.  
THe Line of inches and foot measure do by inspection (only) reduce either of the measures into the other. Ex. gr. Three on the line of inches stands directly against 25 cents of foot measure. Or 75 cents of foot measure is directly against 9 on the inches.  
Another use of these lines (welcome perhaps to them that delight in instrumental computations) is to know the price of carriage for any quantity, etc. by inspection.  
For this purpose, the line of inches represents the price of a pound, every inch being a penny, and every quarter a farthing. The pricked line is the price of an hundred pound at five score and twelve to the hundred, every division signifying a shilling.  
The line of foot measure is the price of an hundred pound at five score to the hundred, every division standing for a shilling. Ex. gr.  
At 3 pence the pound on the inches, is 28 shillings on the pricked line, and 25 shillings on the foot measure, for the price of an hundred, the like for the converse. Otherwise, the price of a pound being given, the rate of an hundred is readily computed without the rule. For considering the number of farthings is in the price of a pound, twice that number of shillings, and once that number of pence, is the price of an hundred, reckoning five score to the hundred; of twice that number of shillings, and once that number of groats, is the price of an hundred, at five score and twelve to the hundred. Ex. gr. At three half pence the pound, the number of farthings is six. Therefore, twice six shillings, and once six pence (that is 125. 6d.) is the price of an hundred, at five score to the hundred. Again, twice six shillings, and once six groats (that is 14s.) is the price of an hundred, at five score and twelve to the hundred. But of this enough, if not too much already.   

CHAP. XII.  
To divide a plot of Ground into any proposed quantities. See Fig. 4  
SUppose A B C D E F G H I K L, a plot of ground, containing 54 acr. 2 roods, 28 poles, from the point A, I am required to shut off 18 acres, next the side B C. Draw A D. and measure the figure A B C D, which is 14 acr: 2 r. 3 p. that is 3 acr. 1. r. 37 p. or 557 poles too little. Draw again A F, and measure A D F, which is 1309. poles. Then by Chap. 5. Probl. 1. entering D F, the base of the triangle in 1309 on the lines of lines, and taking out 557, set it from D to E, and draw A E. So have you the figure A B C D E 18 acres. Again, from the point G, I would set off 20 acres next the side A E, draw A G, and measure A E F G 15 acr. or. 2 pol. whereof want 4 acr. or. 38 pol. that 678 poles, then draw G K, and measure G A K 1113 poles. Lastly, by Chap. 5. Probl. I. enter A K, the base of the triangle in 1113 upon the lines of lines, and taking out 678, set it from A, and draw G L. So have you the figure A E F G L, twenty acres.  
How ready the instrument would be for surveying with the help of a staff, Ballsocket, and Needle, is obvious to any one that considers its graduated into 180 degrees.   

CHAP. XIII.  
So much of Geography as concerns finding the distance of any two places upon the Terrestrial Globe.  
HEre are three Cases, and each of those contains the same number of propositions.  

CASE 1.  
When the two places differ in latitude only.  

PROP. 1.  
When one place lies under the Equator, having no Latitude.  
The latitude of the other place turned into miles, (reckoning 60 miles, the usual compute, for a degree) is the distance sought.   

PROP. 2.  
When both places have the same pole elevated, viz. North, or South.  
Take the difference of their Latitudes, and reckoning 60 miles for a degree (as before) you have their distance.   

PROP. 3.  
When the two places have different poles elevated, viz. one North, the other South.  
Add two latitudes together, and that sum turned into miles is the distance.    

CASE 2.  
When the two places differ in Longitude only.  

PROP. 1.  
When neither of them have any Latitude, but lie both under the Equator.  
Their difference of Longitude turned into miles (as before) is their distance.   

PROP. 2.  
When the two places have the same Pole elevated.  
The proportion is thus.  
As the Radius is to the number 60, so is the cousin of the common latitude to the number of miles for one degree of longitude.  
Multiply this number found by the difference of longitude, and that product is the distance in miles.   

PROP. 3.  
When the two places have different poles elevated.  
As the Radius is to the cousin of the common latitude, so is the sine of half the difference of longitude to the sine of half the distance.  
Wherefore this sine of half the distance doubled and turned into miles, is the true distance.    

CASE 3.  
When the two places differ in Longitude and Latitude both.  

PROP. 1.  
When one of the places lies under the equator, having no Latitude.  
As the radius is to the cousin of the difference of longitude, so is the cousin of the latitude to the cousin of the distance.   

PROP. 2.  
When both the places have the same pole elevated.  
As the radius is to the cousin of the difference in longitude, so is the co-tangent of the lesser altitude, to the tangent of a fourth ark.  
Subtract this fourth ark out of the compliment of the lesser latitude, and keep the remain. Then,  
As the cousin of the fourth ark is to the cousin of the remain, so is the sine of the lesser latitude to the cousin of the distance.   

PROP. 3.  
When the two places have different poles elevated.  
As the radius is to the cousin of the difference in longitude, so is the co-tangent of either latitude to the tangent of a fourth ark.  
Subtract the fourth out of the latitude not taken into the former proportion, and note the difference. Then,  
As the cousin of the fourth ark is to the cousin of this difference, so is the sine of the latitude first taken, to the cousin of the distance.     

CHAP. XIV.  
Some uses of the Instrument in Navigation, or plain Salling.  
HEre it will be necessary to premise the explication of some terms, and adjoin two previous proportions.  
1. The Compass being a circle, divided into 32 equal parts, called rumbs; one point or rumb is 11 d. 15 min. of a circle from the meridian: two points or rumbs is 22 d. 30 min. etc. of the rest.  
2. The angle which the needle, or point of the compass under the needle, makes with the meridian, or North and South line is called the course or rumb; but the angle which it makes with the East, and West line, or any parallel, is named the compliment of the course or rumb.  
3. The departure is the longitude of that Port from which you set sail.  
4. The distance run, is the number of miles, or leagues (turned into degrees) that you have sailed.  
5. When you are in North latitude, and sail Northward, add the difference of latitude to the latitude you sailed from; and when you are in North latitude, and sail Southward, subtract the difference of latitude from the latitude you sailed, and you have the latitude you are in.  
The same rule is to be observed in South latitude.  
6. To find how many miles answer to one degree of longitude in any latitude.  
As the radius is to the number 60, so is the cousin of the latitude to the number of miles for one degree.  
7. To find how many miles answer to one degree of latitude on any rumb.  
As the cousin of the rumb from the Meridian, is to the number 60, so is the radius to the number of miles.  
The most material questions in Navigation are these four.  
First, To find the course. Secondly, The distance run. Thirdly, The difference of latitude. Fourthly, The difference in longitude; and any two of these being given, the other two are readily found by the Square and Index.  
These two additional rulers were omitted in the first Chapter of this Treatise; sith they are only for Navigation, and large Instruments of two, or three foot in length, which made me judge their description most proper for this place: because, such as intend the Instrument for a pocket companion, will have no use of them.  
The square is a flat rule, having a piece, or plate fastened to the head, that it may slide square, or perpendicular to the outward ledge of the fixed piece. It hath the same line next either edge on the upper side, which is a line of equal parts, an hundred, wherein is equal to the radius of the degrees on the outward limb of the movable piece.  
The Index is a thin brass rule on one side, having the same scale as the square. On the other side is a double line of tangents, that next the left edge, being to a smaller; that next the right edge, to a larger radius.  
For the use of these rulers, you must have a line of equal parts adjoining to the line of sins on the fixed piece, divided into 10 parts, stamped with figures, each of those divided into 10 parts more; so that the whole line is divided into 100 parts, representing degrees. Lastly, let each of those degrees be subdivided into as many parts, as the largeness of your scale will permit, for computing the minutes of a degree.  
The Index is to move upon the pin on the fixed piece (where you hang the thread for taking altitudes) and that side of the Index (in any of the four former questions) must be upward, which hath the scale of equal parts.  
The square is to be slided along the outward ledge of the fixed piece. Then the general rules are these.  
The difference of latitude is accounted on the line of equal parts, adjoining to the sins on the fixed piece.  
The difference of longitude is numbered on the square.  
The distance run is reckoned upon the Index.  
The course is computed upon the degrees on the limb from the head toward the end of the movable piece.  
But when any would work these Problems in proportions, let them note,  
The distance run, difference of longitude, and difference of latitude, are all accounted on the line of numbers; the rumb or course is either a sine or tangent.  
This premised. I shall first show how to resolve any Problem by the square and Index; and next adjoin the proportions for the use of such as have only pocket Instruments.  

PROBL. 1.  
The course and distance run given, to find the difference of latitude, and difference of longitude.  
Apply the Index to the course reckoned on the limb from the head, and slide the square along the outward ledge of the fixed piece, until the fiducial edge intersect the distance run on the fiducial edge of the Index. Then at the point of Intersection you have the difference of longitude upon the square, and on the line of equal parts on the fixed piece, the square shows the difference of latitude. The proportion is thus,  
As the radius is to the distance run, so is the cousin of the course to the difference of latitude. Again,  
As the radius is to the distance run, so is the sine of the course to the difference in longitude.   

PROBL. 2.  
The course and difference of latitude given to find the distance run, and difference in longitude.  
Slide the square to the difference of latitude on the line of equal parts upon the fixed piece, and set the Index to the course on the limb. Then at the point of intersection of the square and index, on the square is the difference of longitude, on the index the distance run. The proportion is thus.  
As the cousin of the course is to the difference of latitude, so is the radius to the distance run. Again,  
As the radius is to the sine of the course, so is the distance run to the difference of longitude.   

PROBL. 3.  
The course and difference in longitude given, to find the distance run, and difference of latitude.  
Apply the Index to the course on the limb, and the difference of longitude on the square to the fiducial edge of the Index. Then at the point of intersection you have distance run on the index, and upon the line of equal parts on the fixed piece, the square shows the difference of latitude. The proportion is thus.  
As the sine of the course is to the difference of longitude, so is the radius to the distance run. Again,  
As the radius is to the distance run, so is the cousin of the course to the difference of latitude.   

PROBL. 4.  
The distance run, and difference of latitude given to find the course, and difference in longitude.  
Slide the square to the difference of latitude on the line of equal parts on the fixed piece, and move the index until the distance run numbered thereon, intersect the fiducial edge of the square; then at the point of intersection you have the difference of longitude on the square, and the fiducial edge of the Index on the limb shows the course.  
As the distance run is to the difference of latitude, so is the radius to the cousin of the course. Again,  
As the radius is to the distance run, so is the sine of the course to the difference of longitude.   

PROBL. 5.  
The distance run, and difference of longitude given to find the course, and difference of latitude.  
Apply the distance run numbered on the fiducial edge of the index, to the difference of longitude, reckoned on the fiducial edge of the square. Then on the line of equal parts upon the fixed piece, the square shows the difference of latitude, and the index shows the course on the limb. The proportion is thus.  
As the distance run is to the difference of longitude, so is the radius to the sine of the course. Again,  
As the radius is to the distance run, so is the cousin of the course, to the difference of latitude.   

PROBL. 6.  
The difference of latitude, and difference of longitude given, to find the course, and distance run.  
Apply the square to the difference of latitude on the scale of equal parts upon the fixed piece, and move the index until its fiducial edge intersect the difference of longitude, reckoned on the square. Then at the point of intersection you have the distance run upon the index, and the fiducial edge of the index upon the limb shows the course. The proportion is thus.  
As the difference of latitude is to the difference of longitude, so is the radius to the tangent of the course. Again,  
As the sine of the course is to the radius, so is the difference of longitude to the distance run.   

PROBL. 7.  
Sailing by the Ark of a great Circle.  
For this purpose the tangent lines on the index will be a ready help, using the lesser for small, and the greater tangent line, for great latitudes. The way is thus,  
Account the pont 60/0, on the outward limb of the movable piece to be the point, or port of your departure; thereto lay the fiducial edge of the index, and reckoning the latitude of the Port you departed from upon the index, strike a pin directly touching it, into the table your instrument lies upon. This pin shall represent the Port of your departure. Therefore hanging a thread, or hair, on the centre, whereon the index moves; and winding it about this pin. Count the difference of longitude 'twixt the port of your departure, and the Port you sail toward, from 60/0 on the movable piece toward 0/60, on the loose piece; and thereto laying the same fiducial edge of the index, reckon the latitude of this last Port upon the index, directly touching of it, strike down another pin upon the table, and draw the thread straight about this pin fastening it thereto. This done, the thread betwixt the two pins represents the ark of your great Circle; and laying the fiducial edge of the index to any degree of difference of longitude accounted from 60/0 on the movable piece, the thread shows upon the index what latitude you are in, and how much you have raised, or depressed the pole since your departure. On the contrary, laying the latitude you are in (numbered upon the index) to the thread, the index shows the difference of longitude upon the limb; counting from 60/0 on the movable piece. So, that were it possible to sail exactly by the ark of a great Circle, it would be no difficulty to determine the longitude in any latitude you make.  
But I intent not a treatise of Navigation; wherefore let it suffice, that I have already shown how the most material Problems therein, may easily, speedily, and (if the instrument be large) exactly, be performed by the instrument without the trouble of Calculation, or Projection.    

CHAP. XV.  
The Projection and Solution of the sixteen Cases in right angled Spherical Triangles by five Cases. See Fig. 5.  
THe fundamental Circle N B Z C, is always supposed ready drawn, and crossed into quadrants, and the diameters produced beyond the Circle.  

CASE 1.  
Given both the sides Z D, and D R, to project the Triangle.  
By a line of chords, prick off Z D, upon the limb, and draw the diameter D A E. Again, by a line of tangents, set half co-tangent D R, upon A D, from A to R, then have you three points, viz. N R Z, to draw that ark, and make up the triangle. The centre of which ark always lies on AC, (produced beyond C, if need requires) and is found by the intersection of the two arks made from R, and Z.   

CASE 2.  
Given one side Z D, and the hypothenuse Z R, to project the Triangle.  
Prick off Z D, and draw D A E, by Case 1. Again, set half the co-tangent Z R, on the line A Z, from A to F, and the tangent Z R, set from F to P, with the extent F P, upon the centre P, draw the ark V F I, and where it intersects the diameter D A E, set R; then have you three points N R Z, to draw that ark, as in the former Case.   

CASE 3.  
Given the Hypothenuse Z R, and the Angle D Z R, to project the Triangle.  
Prick half the co-tangent D Z R, from A to S, and the secant D Z R, from S to T, upon the centre T, with the extent T S, draw the ark N S Z. Again, by Case 2, draw the ark H F I, where these two arks intersect each other, set R. Lastly, lay a ruler to A R, and draw the diameter DRAE, and your triangle is made.   

CASE 4.  
Given one side Z D, and its adjacent Angle D Z R, to project the triangle.  
Prick off Z D, and draw D A E, by Case 1. Again, by Case 3, draw the ark N S Z, where this ark intersects the diameter DAE, set R, and your triangle is made. Or,  
Given the side D R, and its opposite angle D Z R, you may project the triangle.  
Draw the ark N S Z, as before. Again, take half the co-tangent D R, and with that extent upon A, the centre, cross the ark N S Z, setting R, at the point of intersection. Lastly, lay a ruler to A R, and draw the diameter D R A E, which makes up your triangle.  
The triangle projected in any of the four former Cases, to measure any of the sides, or angles, do thus.  
First, the 〈◊〉 Z D, is found by applying it to a line of chords. Secondly, R A, applied to a line of tangents, is half the co-tangent D R. Thirdly, A S, applied to a line of tangents is the co-tangent D Z R. Fourthly, set half the tangent D Z R, from A to L, then is L the pole point, and laying a ruler to L R, it cuts the limb at V, and the ark Z V, upon the line of chords, gives the hypothenuse Z R. Fifthly, prick Z D from C to K, a ruler laid from R to L, cuts the limb at G, then G K, upon a line of chords, is the quantity of Z R D.   

CASE. 5.  
The two oblique angles, D Z R, and Z R D given, to present the Triangle. See Fig. 6.  
This is no more than turning the former triangle. Thus,  
Draw the ark N S Z, by Case 1, and set half the tangent of that ark from A to L. Again, set half the co-tangent D R Z, from A to F, and the secant of D R Z, from F to L, upon the centre A, with the extent A P. Draw the ark P G, and with the extent F P, from L, cross the ark P G in G. Lastly, upon the Centre G, with the extent G L. Draw the ark R D F L, and your triangle is made.  
The triangle projected you may measure off the sides and hypothenuse. Thus,  
First, the hypothenuse Z R, is measured by a line of chords. Secondly, a ruler laid to L D, cuts the limb at H, and Z H, upon a line of chords, is the measure of the ark Z D. Thirdly, draw A G, and set half the tangent D R Z, from A to V, apply a ruler to V D, it cuts the limb at E, then R E, upon a line of chords, measure the ark R D.  
Note. The radius to all the chords, tangents, and secants, used in the projection, and measuring, any ark or angle, is the semidiameter of the fundamental circle.    

CHAP. XVI.  
The projection and solution of the 12 Cases in oblique angled spherical triangles in six Cases. See Fig. 7.  
THe fundamental circle N H Z M, is always supposed ready drawn, and crossed into Quadrants, and the Diameters produced beyond the Circle.  

CASE 1.  
The three sides, Z P, P Z, and Z S, given, to project the Triangle.  
By a line of chords prick off Z P, and draw the diameter P C T, crossing it at right angles in the centre with A C E, set half the co-tangent P S, from C to G, and he secant P S from C to R, upon the centre R, with the extent R G, draw the the ark FGL. Again, set half the co-tangent Z S, from C to D, and the tangent Z S, from D to O, with the extent O D, upon the centre O, draw the ark B D P, mark where these two arks intersect each other as at S. Then have you three points T S P, to draw that ark, and the three points N S P, to draw that ark, which make up your triangle.   

CASE 2.  
Given two sides Z S, and Z P, with the comprehended Angle P Z S, to project the Triangle.  
Prick off Z P, and draw PCT, and AECE, and the ark B D P, by Case 1. Again, set the tangent of half the excess of the angle P Z S above 90, from C to W, and co-secant of that excess from W to K, upon the centre K, with the extent K W; draw the ark N W Z, which cuts the ark B D P in S. Then have you the three points T S P, to draw that ark which makes up the triangle.   

CASE 3.  
Two Angles S Z P, and Z P S, with the comprehended side Z P, given, to project the Triangle.  
Prick off Z P, and draw the lines P C T, and A C E, by Case 1, and the angle NWZ. by Case 2. Lastly, set half the co-tangent ZPS from C to X, and the secant Z P S, from X to V, upon the centre V, with the extent V X, draw the ark T X S P, and the triangle is made.   

CASE 4.  
Two sides, ZP, and PS, with the Angle opposite to one of them SZP, given, to project the Triangle.  
Prick off ZP, and draw PCT, and AECE, by Case 1. and the angle SZP by Case 2. Lastly, by Case 1. draw the ark FGL, and mark where it intersects NWZ, as at S, then have you the three points TSP, to draw that ark, and make up the triangle.   

CASE 5.  
Two Angles SZP, and ZPS, with the side opposite to one of them ZS, given, to project the Triangle.  
Draw the ark BDP, by Case 1. and the ark NWZ, by Case 2. at the intersection of these two arks, set S, with the tangent of the angle ZPS, upon the centre C. sweep the ark VΔI. Again, with the secant of the ark ZPS upon the centre S, cross the ark VΔI, as at the points V and I. Then in case the hypothenuse is less than a quadrant (as here) the point V, is the centre, and with the extent US, draw the ark TSP, which makes up the triangle. But in case the hypothenuse is equal to a quadrant, Δ, is the centre; if more than a quadrant, I, is the centre; in which cases the extent from Δ, or I, to S, is the semidiameter of the ark TSP.   

CASE 6.  
Three Angles ZPS, and PZS, and ZSP, given, to project the Triangle. See Fig. 7. and 8.  
The angles of any spherical triangle may be converted into their opposite sides, by taking the compliment of the greatest angle to a Semicircle for the hypothenuse, or greatest side. Wherefore by Case 1. make the side ZP, in Fig. 4. equal to the angle ZSP, in Fig. 3, and the side ZS, in Fig. 4. equal to the angle ZPS, in Fig. 3. and the side PS, in Fig. 4. equal to the compliment of the angle PZS, to a Semicircle in Fig. 3. Then is your triangle projected where the angle ZPS in Fig. 4. is the side ZS, Fig. 3. Again, the angle ZSP, Fig. 4. is the side ZP, in Fig. 3. Lastly, the compliment of the angle PZS to a Semicircle in Fig. 4. is the measure of the hypothenuse, or side P S, in Fig. 3.  
The Triangle being in any of the former Cases projected, the quantity of any side or angle may be measured by the following rules.  
First, The side Z P, is found by applying it to a line of chords. Secondly, CX, applied to a line of tangents, is half the co-tangent of the angle ZPS. Thirdly, CW applied to a line of tangents, is half the co-tangent of the excess of the angle SZP, above 90. Fourthly, set half the tangent of the angle ZPS, from C, to Π, a ruler laid to ΠS, cuts the limb at F; then PF, applied to a line of chords, gives the side PS. Fifthly, take the compliment of the angle PZS, to a Semicircle, and set half the tangent of that compliment from C, to λ, a ruler laid to λS, cuts the limb at B, and ZB, applied to a line of chords, gives the side ZS. Sixthly, a ruler laid to Sλ, cuts the limb at L. Again, a ruler laid to SΠ, cuts the limb at φ, and L φ, applied to a line of chords, gives the angle ZSP.    
The end of the first Book.   

An Appendix to the first Book.  
THe sights which are necessary for taking any Altitude. Angle, or distance (without the help of Thread or Plummet) are only three, viz. one turning sight, and two other sights, contrived with chaps, so that they may slide by the inward or outward graduated limbs. The turning sight hath only two places, either the centre at the head, or the centre at the beginning of the line of sins on the fixed piece; to either of which (as occasion requires) it's fastened with a sorne. The centre at the head serving for the graduations next the inward limb of the loose piece. And the centre at the beginning of the line of sins serving for all the graduations next the outward limb of the movable and loose piece both. Yet because it is requisite to have pins to keep the loose piece close in its place. You may have two sights more to supply their place (which sometimes you may make use of) and so the number of sights may be five, viz. two sliding sights, one turning sight, and two pin sights, to put into the holes at the end of the fixed and movable piece, to hold the tenons of the loose piece close jointed. Every one of these sights hath a fiducial (or perpendicular) line, drawn down the middle of them, from the top to the bottom, where this line toucheth the graduations on the limb, is the point of observation.  
The places of these sights have an oval proportion, about the middle of them, only leaving a small bar of brass, to conduct the fiducial line down the oval cavity, and support a little brass knot (with a sight hole in it) in the middle of that bar, which is ever the point to be looked at. There are two ways of observing an altitude with help of these sights. The one when we turn our face toward the object. This is called a forward observation in which you must always set the turning sight next your eye. This way of observation will not exactly give an altitude above 45 degrees.  
The other way of observing an altitude is peculiar to the Sun in a bright day, when we turn our back toward the Sun. This is termed a backward observation; wherein You must have one of the sliding sights next Your eye, and the turning sight toward the Horizon. This serves to take the Sun's altitude without thread, or plummet, when it is near the Zenith.  

PROBL. 1.  
To find the Sun's altitude by a forward observation.  
Serve the turning sight to the centre of those graduations you please to make use of (whether on the inward or outward limb) and place the two sliding sights upon the respective limb to that centre; this done, look by the knot of the turning sight (moving the instrument upward or downward) until you see the knot of one of the sliding sights directly against the Sun, then move the other sliding sight, until the knot of the turning sight, and the knot of this other sliding sight be against the horizon; then the degrees intercepted 'twixt the fiducial lines of the sliding sights on the limb, show the altitude required.   

PROBL. 2.  
To find the distance of any two Stars, etc. by a forward observation.  
Serve the turning sight to either centre, and apply the two sliding sights to the respective limb (holding the instrument with the proportional side downward) and applying the turning sight to your eye, so move the two sliding sights either nearer together, or further asunder, that you may by the knot of the turning sight see both objects even with the knots of their respective sliding sights, then will the degrees intercepted 'twixt the fiducial lines of the object sights on the limb show the true distance. By this means you may take any angle for surveying, etc.   

PROBL. 3.  
To find the Sun's Altitude by a backward Observation.  
Serve the turning sight to the centre at the beginning of the line of sins, and apply one of the sliding sights to the outward limb of the loose piece, and the other to the outward limb of the movable piece; and turning your back toward the Sun, set the sliding sight upon the movable piece next your eye; and slide it upward or downward toward the end, or head, until you see the shadow of the little bar, or edge, of the sight on the loose piece fall directly on the little bar on the turning sight; and at the same time the bar of the sight next your eye, and the bar of the turning sight to be in a direct line with the Horizon. Then will the degrees on the limb intercepted 'twixt the fiducial lines of the sliding sights (if you took the shadow of the bar) or 'twixt the fiducial line of the sliding sight next your eye, and the edge of the other sliding sight (when you took the shadow of the edge) be the true altitude required.    

ΣΚΙΟΓΡΑΦΙΑ, OR, The Art of dialing for any plain Superficies.  
LIB. II.  

CHAP. I.  
The distinction of Plains, with Rules for knowing of them.  
ALL plain Superficies are either horizontal, or such as make Angles with the Horizon.  
Horizontal plains are those, that lie upon an exact level, or flat.  
Plains, that make Angles with the Horizon 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 horizon, or have their upper edge leaning toward 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 

Declining  
From South, toward East, or West.  
From North, toward East, or West.    
All plain Superficies whatsoever are comprised under one of these terms. But before we treat of the affections, or delineation of Dial's for them; it will be requisire to acquaint you with the nature of any plain, which may be found by the following Problems.  

PROBL. 1.  
To find the reclination of any Plain.  
Apply the outward ledge of the movable piece to the Plain with the head upward, and reckoning what number of degrees the thread cuts on the limb (beginning your account at 30. on the loose piece, and continuing it toward 60/0 on the movable piece) you have the angle of reclination. If the thread falls directly on 60/0 upon the movable piece, it's an horizontal; if on 30. on the loose piece, it's an erect plain.   

PROBL. 2.  
To find 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 compliment of the plains inclination.   

PROBL. 3.  
To draw an Horizontal Line upon any Plain.  
Apply the proportional side of the Instrument 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 parallel to the horizon.   

PROBL. 4.  
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 perpendicular.   

PROBL. 5.  
To find the declination of any Plain.  
Apply the outward ledge of the fixed piece to the horizontal line of your plain, holding your instrument parallel to the horizon. This done, lift up the thread and plumber, until the shadow of the thread fall directly upon the pin hole on the fixed piece (where you hang the thread 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 taking the altitude of the Sun. By lib. 1. cap. 2. Probl. 9, 10, 11. find the Sun's Azimuth from South. And,  
When you make this observation in the morning, these Cases determine the declination of the plain.  

CASE 1.  
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 shadow, and Azimuth (by subtracting the less out of the greater) and the residue or remain 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 azimuth, when the shadow and azimuth are equal, it's 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.   

CASE 2.  
When the shadow falls on the left hand 0/60. Add the azimuth and shadow together, that sum is the plains declination; from South toward East, when under 90; if it be just 90, it's a direct East. If above 90. subtract it from 180. the remain is the declination from North toward East. When the sum is above 180. subtract 180 from it, and the remain is the declination from North toward West.   

CASE 3.  
When the shadow falls upon 0/60. the azimuth is the plains declination. When under 90, its Southeast, 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.   

CASE 4.  
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 Southeast, when less, South-West. When the shadow and azimuth are equal, it's a direct South plain, when their difference is equal to 90. it's a direct West; when the difference exceeds 90. subtract it from 180. the remain is the declination North-West.   

CASE 5.  
When the shadow falls on the right hand 0/60. take the sum of the shadow and azimuth, 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 declination North-West; when the sum is above 180. subtract 180 from it, and the remain is the declination from North toward East.   

CASE 6.  
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.    

PROBL. 6.  
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 centre of the circle, make a point in the circumference where the shadow intersects it. At the same time finding the Sun's 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 centre, the line you draw is a true Meridian.    

CHAP. II.  
The affections of all sorts of Plains.  

Sect. 1.  
The affections of an horizontal Plain.  
1. THe style, or cock of every horizontal 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 centre of the hour lines.  
3. The rule for drawing the hour lines before six in the morning, is to draw the respective hour lines afternoon beyond the centre; or for the hour lines after six at evening, draw the respective hour lines in the morning beyond the centre.  
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.   

Sect. 2.  
The affections of erect, direct South and North Plains.  
1. The style in both these is an angle equal to the compliment 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 centre of the hour lines, and that point in South always upward, in North always 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 centre, and each hour line, transmit the hour lines from your paper draught to the plain.   

Sect. 3.  
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 height to the radius 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 perpendicular to the plain.   

Sect. 4.  
The affections of erect declining Plains.  
1. These are of two sorts, either such as admit of centres 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 centre 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 Dial's may be drawn with a centre to the hour lines.  
2. In all erect declining plains with centres the Meridian is the plains perpendicular. In those that admit not of centres 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 also in such decliners as admit of centres, the angle 'twixt 12. and 6. is to be set to the contrary coast to that of declination.  
4. In all decliners, without centres, 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 distance of the substyle from the meridian, the angle of twelve and six, with the inclination of meridians (all which may be wrought either by the canon, or exactly enough for this purpose by the instrument) are as followeth.  

To find the Styles height above the Substile.  
As the radius is to the cousin of the latitude, so is the cousin of declination, to the sine of the styles height.   

To find the Substyles distance from the Meridian.  
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.   

To find the angle of twelve and six.  
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.   

To find the inclination of Meridian's.  
As the sine of the latitude is to the radius, so is the tangent of declination to the tangent of inclination of Meridian's.  
6. All North decliners with centres have the angular point of the style downward, and all South have it upward.  
7. All North decliners without centres have the narrowest end of the style downward, all South have it upward.  
8. In all decliners without centres take so much of the style as you think convenient, but make points at its beginning and end upon 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 Meridian's is the hour line of twelve at midnight: in South decliners, at noon, or midday. This may tell you the true names of the hour lines.  
10. In transmitting these Dial's from your paper draught to your plain, lay the horizontal of your paper draught, upon the horizontal line of the plain, and prick off the hours and substyle.    

Sect. 5.  
The affections of direct reclining Plains inclining Plains.  

For South Recliners, and North Incliners.  
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 recliner, 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, it's a polar plain.   

For South incliners, and North recliners.  
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, it's an equinoctial plain, and the Dial is no more than a circle divided into 24. equal parts, having a wire of any convenient length placed in the centre perpendicular to the plain for the style.  
3. When the reclination inclination is greater than your latitude, take the sum 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 centre 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 compliment of the styles height.   

For direct East and West recliners incliners.  
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 compliment of that latitude where the plain stands, and their new declination is the compliment of their reclination inclination: But to know which way you are to account this new declination, remember all East and West recliners are North-East, and North-West decliners. All East and West incliners are Southeast, and South-West decliners.  
3. Their new latitude and declination known, you may by Sect 4. par. 5. find the substyle from the Meridian, height of the style, angle of twelve and six, and inclination of Meridian's, using in those proportions the new latitude and new declination instead of the old.  
4. In all East and West recliners incliners with centres, the Meridian lies in the horizontal line of the plain; in such as have not centres, its parallel to the horizontal line.    

Sect. 6.  
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.  

1. To find the new Latitude.  
As the radius is to the cousin 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 compliment of that difference 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, it's a polar plain.  
In South incliners North recliners the difference 'twixt the fourth ark, and the compliment of your old latitude is the new latitude. If the fourth ark be equal to your old co-latitude, they are equinoctial plains.   

2. To find the new declination.  
As the radius is to the cousin of the reclination inclination, so is the sine of the old declination to the sine of the new.   

3. To find the angle of the Meridian with the Horizontal Line of the Plain.  
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 meridian with the horizontal line of the plain. This gives the angle for its situation. Observe, in North incliners less than a polar, the Meridian lies. above That end of the horizontal Line contrary to the Coast of Declination.   below  South recliners more than a polar, the Meridian lies. below That end of the horizontal next the Coast of Declination.   above  North recliners less than an equinoctial, the Meridian lies above That end of the Horizontalnext the Coast of Declination.   below In North recliners this Meridian is 12. at midnight.  equal to an equinoctial the Meridian descends below the Horizontal at that end contrary to the coast of Declination, and the substyle lies in the hour line of six.   South incliners more than an equinoctial, the Meridian lies. below That end of the horizontal contrary to the declination.   above In South incliners this Meridian is only useful for drawing the Dial, and placing the substyle, for the hour lines must be drawn through the centre to the lower side.  
After you have by the former proportions and rules found the new latitude, new declination, the angle and situation of the meridian, your first business in delineating of the Dial will be (both for such as have centres, and such as admit not of centres) to set off the meridian in its proper coast and quantity. This done, by Sect. 4. Paragr. 5. of this Chapter, find the substyles distance from the Meridian, the height of the style, angle of twelve and six (and for Dial's without centres, the inclination of Meridian's) in all those proportions, using your new latitude and new declination, instead of the old, and setting them off from the Meridian, according to the directions in Paragr. 3. and 4. you may draw the Dial's by the following 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 Southeast, 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 Southeast, 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 centres, or the substyle from the Meridian, and the inclination of Meridian's from the substyle in such as admit not of Centres.  
4. Declining polar plains must have a peculiar calculation for the substyle and inclination of Meridian's, which is thus.   

To find the Angle of the Substile with the Horizontal Line of the Plain.  
As the radius is to the sine of the polar plains reclination, so is the tangent of declination to the co-tangent of the substyles distance from the horizontal line of the plain.   

To find the inclination of Meridian's.  
As the radius is to the sine of the latitude, so is the tangent of declination to the tangent of inclination of Meridian's.  
5. The reclining declining polar hath the substyle lying below that end of the horizontal line that is contrary to the coast of declination. The inclining declining polar hath the substyle lying above that end of the horizontal line, contrary to the coast of declination.     

CHAP. III.  
The delineation of Dial's 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 number affixed to it.  
Cos. is the cousin 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 parallel unto, or equidistant from each other. Ex. gr.  
FI. ♒ RS. that is the line FL. is parallel unto, 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 OF = 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. A 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 mentioned, Chap. 2. Sect. 1.   

Sect. 2.  
To describe an erect, direct South Dial. See Fig. 10.  
Draw ABCD. a rect-angle parallellogram. Then make A = EBB = CF = FD = cos. of your latitude. And OF = 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 AGNOSTUS = BOY = 15. and AH = BN = tang. 30. with a ruler draw the lines EGLANTINE. EH. EC. EF. EKE. 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 centre as in the horizontal. See Fig. 11.   

Sect. 3.  
To describe an erect, direct East Dial. See Fig. 12.  
Having drawn ABCD. a rectangle paralellogram, fix upon any point in the lines AB. and CD. for the line of six, provided the distance from that point to A. being entered radius in the line of tangents, the distance 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.) radius 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 Centre. 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 latitude, which place from A to G. upon the the line A. 12. and again, from H. (which is the intersection of the parallel FK. with the line of six) unto F. This done, lay a ruler to the points F. and G. and draw a line until 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 centre, 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 centre, until they intersect each other as at S. Then setting off ST st HR. and S. 4. = HP. you have the hour points after 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 centre, they help you to the hour lines before six in the morning.   

Sect. 5.  
To describe an erect declining Plain without a Centre. See Fig. 12.  
The delineation of these Dial's is the most difficult of any, and therefore I shall be the larger in their description.  
1. By Chap. 2. S. 4. Paragr. 5. find the angles of the style, substyle, and inclination of Meridian's.  
2. Having found the inclination of Meridian's, 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. how. quart. 76 = 30. 12 00 20 = 15 3 ... 16 = 30 4 00 12 = 45 4 . 09 = 00 4 .. 05 = 15 4 ... 01 = 30 5 ... 13=30 6 00 02 = 15 5 . 06 = 00 5 .. 09 = 45 5 ... 13 = 30 6 ... 28=30 7 00 17 = 15 6 . 21 = 00 6 .. 24 = 45 6 ... 28 = 30 7 ... 43=30 8 00 32 = 15 7 . 36 = 00 7 .. 39 = 45 7 ... 43 = 30 8 ... 58=30 9 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 Meridian's (which because its a North decliner, is twelve at midnight.) I find 76. d. 30. min. Now considering the Sun never riseth till more than half an hour after three in this latitude, 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 find three hours, three quarters (the distance of a quarter before four from midnight) to answer 56. d. 15. min. which being subtracted from 76. d. 30. min. the inclination of Meridian's there remains 20. d. 15. m. for the distance of one quarter 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 quantity 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 centre 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. parallel 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 find 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 tangents to these hour points in the radius of secants, 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 contingence. 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 entering 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 tangents, 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 substyle 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 secants, 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. afternoon. 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. find the inclination of Meridian's, 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 convenient 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 upon the description of them: Sith so much hath been already mentioned, Chap. 2. S. 6. that there can scarcely be any mistake, unless through mere wilfulness, or grandnegligence.    

CHAP. IU.  
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 quadrants N. is North. S. South, E. East, W. West, NS. the Meridian (which let be infinitely produced) Z. the centre represents the Zenith. To find 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 Sun's declination in Cancer subtracted from the latitude, and the tangent of half the remain set from Z. to Π. gives the intersection of Cancer with the Meridian. Again, add the compliment of the Sun's delineation in Cancer, unto the compliment of your latitude, and the tangent of half that sum set from Z. to ψ, gives the diameter of that tropic, 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 centre of the equator.  
Add the Sun's 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 Capricorn intersects the Meridian. Subtract the Sun's declination in Capricorn from your latitude, 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. towards S. it gives the point H. the centre of the hour line of six cross the line ZSH. at right angles in the point H. Then entering PH. Radius on the lines of tangents, set off the hour centres both ways from H. reckoning 15. d. for an hour. Lastly, setting one point of your Compasses in these centre points, extend the other to P. and with that radius describe the hour lines. Thus have you the sphere projected, the following Sections will determine the hours for all plains.  

Sect. 1.  
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 upon the West. The fine WE. represents a direct North and South plain, the side next N. is North, the other next S. is South, where the North cuts the tropic of Cancer (which in the hour lines you find '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.   

Sect. 2.  
To determine the hour lines for direct reclining inclining Plains. Fig. 17.  
NBS. on the convex side is a West incliner, 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 incliner, where it cousin 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 incliner, 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 parallel of declination to pass through the point D. and the intersection of that parallel with the limb of the circle NE sweet. doth among the hour lines, show 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 tropic of Capricorn, shows the time of the Suns going off that plain before noon, and coming on again, afternoon. And the intersection of the tropic of Capricorn with the limb shows 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 incliner, where it cuts Cancer, is the time of the Suns going off in the morning, and coming on afternoon. On the concave side is a South recliner, where until F. reach beyond P. it enjoys the Sun only from six to six. When F. reacheth beyond P. where the ark cuts Cancer, you find 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 centre. & GB. the radius to draw that ark.  
Note. The Semidiameter of the circle SENW. is radius to all the tangents, and secants, which you make use of for placing any oblique plain upon the Scheme.   

Sect. 3.  
To determine the hour lines for erect declining plains. Fig. 18.  
For Southeast, or North-West plains. By a line of chords set the angle of declination 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 afternoon, and staying until Sun set. But if it cut Cancer twice, then in the morning hours it shows 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 Southeast. 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 Sunset.  
On that side next sweet. 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.   

Sect. 4.  
To determine the hours of declining reclining Plains inclining Plains. Fig. 18.  
First, set in the plain according to its declination. 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 centre, 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 shows 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 Cancer 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. toward 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 concave side is a North-East recliner, where the ark cuts Cancer, is the time for the Suns going off that plain. When the ark cuts Cancer twice, the Sun comes on again before it sets.  
For a North-West recliner, or Southeast 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 reclination inclination from Z. toward O. as V. and the co-secant of the reclination inclination from V. toward D. as F. then is F. the centre, and FU. the radius to draw the ark HUK. Where it cuts Cancer the hour lines, tell you the time of the Suns going off in the morning, and entering again afternoon, upon the North-West recliner. Where it cuts the equator you have the time of the Suns going off the Southeast, where it cuts Cancer in the morning hours is the time of the Suns coming on that plain.  
For a North-West incliner, or Southeast recliner, set the point V. from Z. toward D. and the point F. set from V. (so placed) toward 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 afternoon upon the North-West incliner. Where the concave side cuts the equator, you have the time of the Suns going off the Southeast 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.     
FINIS.