is 13 deg 30 min. PROBL. 8. The declination given to find the beginning and end of twilight or day-break Lay the thread to the declination counted the contrary way as in the last Problem and take from your Scale of altitudes 18 deg for twilight and 13 deg for day-break or clear light with this run one point of the Compasses along the line of houres on that side next the end until the other will just touch the thread and then the former point gives the respective times required Ex. gr At 7 deg North declination day breaks 8 minutes before 4 but twilight is 3 houres 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 numbred on the limb of the moveable piece from 60 0 toward the end and when occasion requires continue your numbring forward upon the loose piece and take the declination from your line of altitude with this distance run one point of your Compasses along the line of Azimuths on that side the thread next the head until the other just touch the thread then the former point gives the Azimuth from South Ex. gr at 10 deg declination North and 30 deg altitude the Azimuth from South is 64 deg 40 min. PROBL. 10. The Suns altitude given to find his Azimuth in the aequator Lay the thread to the altitude in the limb counted from 60 0 on the loose piece 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 numbred on the limb from 60 0 on the moveable piece toward the end and take the declination from the Scale of altitudes then carry one point of your Compasses on the line of 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 Suns 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 moveable 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 numbring from 60 0 on the moveable piece toward the end Ex. gr At 7 deg North declination and 48 deg Azimuth from South the altitude is 35 deg but at 7 deg declination South and 50 deg Azimuth the altitude is onely 18 deg 30 min. PROBL. 14. The altitude declination and right ascension of any Star with the right ascension of the Sun given to find the hour of the night Take the Stars altitude from the Scale of altitudes and laying the thread to his 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 Suns right ascension numbred upon the line twice 12 or 24 next the outward ledge on the fixed piece extend the other point to the right ascension of the Star numbred 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 Lions heart 45 deg his declination 13 d. 40 min. then by Probl. 5. I find his hour from the last Meridian 10 houres 5 min. The right ascension of the Sun is 46 m. of time or 11 d. 30 m. of the Circle the right ascension of the Lions heart is 9 hour 51 m. fere of time or 147 deg 43 m. of a circle then by a line of twice 12 you may find the true hour of the night 7 hour 13 min. PROBL. 15. The right ascension and declination of any Star with the right ascension of the Sun and time of night given to find the altitude of that Star with his Azimuth from South and by 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 moveable piece toward the end setting one point of your Compasses in the Azimuth take the nearest distance to the thread this applyed to the Scale of altitudes gives the declination If the Azimuth given be on that side the thread toward the end the declination is South when on that side toward the head its North. PROBL. 17. The altitude and declination of any Star with the right ascension of the Sun and hour of night given to find the Stars right ascension By Probl. 5. or 14. find the Stars hour from the Meridian Then on the line twice 12 or 24 set one point of your Compasses in the Stars hour thus found and extend the other to the hour of the night Upon the same line
with this distance set one point of your Compasses in the right ascension of the Sun and turning the other point the same way as you did for the hour it gives the Stars right ascension PROBL. 18. The Meridian altitude given to find the time of Sunrise and Sunset Take the Meridian altitude from your particular Scale and setting one point of your Compasses upon the point 12 on the line of hours that is the pin at the end lay the thread to the other point and on the line of hours the thread gives the time required PROBL. 19. To find any latitude your particular Scale is made for Take the distance from 90 on the line of Azimuth unto the pin at the end of that line or the point 12 this applyed to the particular Scale gives the complement of that latitude the Instrument was made for PROBL. 20. To find the angles of the substile stile inclination of Meridians and six and twelve for exact declining plains in that latitude your Scale of altitudes is made for Sect. 1. To find the distance of the substile from 12 or the plains perpendicular Lay the thread to the complement of declination counted on the line of Azimuths and on the limb it gives the substile counting from 60 0 on the moveable piece Sect. 2. To find the angle of the Stile 's height On the line of Azimuths take the distance from the Plains declination to 90. This applyed to the Scale of altitudes gives the angle of the stile Sect. 3. The angle of the Substile given to find the inclination of Meridians Take the angle of the substile from the Scale of altitudes and applying it from 90 on the Azimuth line toward the end the figures shew the complement of inclination of Meridians 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 complement of the angle sought numbring from 60 0 on the moveable piece toward the end This last rule is not exact nor is it here worth the labour to rectifie it by another sine added sith you have an exact 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 adde one sign to another on the Line of Natural Sines TO adde 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 adde the sine 15 to the sine 20 I take the distance from the beginning of the line of sines unto 15 and setting one point of the Compasses in 20 upon the same line turn the other toward 90 which I finde 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 sines 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 summe of 20 and 15 sines PROBL. 2. How to substract one sine from another upon the line of natural sines The substracting of one sine from another is no more than taking the distance from the lesser to the greater on the line of sines and that distance applyed to the line from the beginning gives the residue or remainer Ex. gr To substract 20 from 37 I take the distance from 20 to 37 that applyed to the line from the beginning gives 15 for the sine remaining PROBL. 3. To work proportions in sines alone Here are four Cases that include all proportions in sines 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 numbring the third on the line of sines take the nearest distance from thence to the thread and that applyed 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 sines 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 applyed 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 tearms be greater than the first this proportion will not be performed by this line without a paralel entrance or double radius which inconveniency shall be remedied in its proper place when we shew how to work proportions by the lines of natural sines on the proportional or sector side These four cases comprizing the method of working all proportions by natural sines alone I shall propose some examples for the exercise of young practitioners and therewith conclude this Chapter PROBL. 4. To finde the Suns amplitude in any Latitude As the cosine of the Latitude is to the sine of the Suns declination so is the radius to the sine of amplitude PROBL. 5. To finde the hour in any Latitude in Northern Declination Proport 1. As the radius to the sine of the Suns declination so is the sine of the latitude to the sine of the Suns altitude at six By Probl. 2. substract this altitude at six from the present altitude and take the difference Then Proport 2. As the cosine of the latitude is to that difference so is the radius to a fourth sine Again Proport 3. As the
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 Meridians divided to 85 whose use is for Navigation in describing Maps or Charts c. In the vacant spaces you may have a line of chords sines and tangents to any Radius the space will bear and what other any one thinks best of as a line of latitudes and hours c. On the proportional side of the loose piece are lines for measuring all manner of solids as Timber Stone c. likewise for gaging 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 sines tangents and versed sines The first marked N the second S the third T the fourth VS 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 prick'd 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 compleated with its furniture Proceed we now to the uses Onely 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 gaging 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 HAng the thread and plummet upon the pin at the beginning of the line of sines on the fixed piece and having two sights in two holes parallel to that line raise the end of the fixed piece toward the Sun until the rayes pass through the sights but when the Sun is in a cloud or you take the altitude of any Star look along the outward ledge of the fixed piece until it be even with the 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 Moneth given to find the Suns place declination ascensional difference or time of rising and setting with his right ascension The thread laid to the day of the Moneth gives the Suns place in the line of signs reckoning according to the order of the Moneths viz. forward from March the 10th to June then backward to December and forward again to March 10. In the limb you have the Suns declination 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 shews the Suns 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 Moneths and find it cut in the Signs 12 deg 20 min. for the Suns place on the limb 4 deg 43 min. for the Suns 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 shews 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 summ of them for the oblique ascension Ex. gr at 11 deg 30 sec. right ascension and 6 deg 30 sec. ascensional difference In Northern declination the oblique ascension will be 5 deg in Southern 18 deg PROBL. 5. The Suns altitude and declination or the day of the Moneth given to find the hour Take the Suns altitude from the Scale of altitudes and laying the thread to the declination in the limb or which is all one to the day in the Moneths move one point of the compasses along the line of hours on that side the thread next the end until the other point just touch the thread then the former point shews the hour but whether it be before or after noon is left to your judgment to determine Ex. gr The 22 day of March or 4 deg 43 min. North declination and 20 deg altitude the hour is either 47 minutes past 7 in the morning or 13 minutes past 4 afternoon PROBL. 6. The declination of the Sun or day of the Moneth and hour given to find the altitude Lay the thread to the day or declination and take the least distance from the hour to the thread this applyed to the line of 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 Suns depression under the horizon Lay the thread to the declination on the limb but counted the contrary way viz. from 60 0 on the movable piece toward the head for Southern and toward the end for Northern declination This done take the nearest distance from the hour to the thread and applying it to the line of altitudes you have the degrees of the Suns depression Ex. gr at 5 deg Northern declination 8 hours afternoon the depression
cosine of the declination to that fourth sine so is the radius to the sine of the hour from six PROBL. 6. To finde the hour in any Latitude when the Sun is in the Equinoctial As the cosine of the latitude is to the sine of altitude so is the radius to the sine of the hour from six PROBL. 7. To finde the hour in any latitude in Southern Declination Proport 1. As the radius to the sine of the Suns declination so is the sine of the latitude to the sine of the Suns depression at six adde the sine of depression to the present altitude by Probl. 1. Then Proport 2. As the cosine of the latitude is to that summe so is the radius to a fourth sine Again Proport 3. As the cosine of declination is to the fourth sine so is the radius to the sine of the hour from six PROBL. 8. To finde the Suns 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. substract this from the present altitude then Proport 2. As the cosine of the latitude is to that residue so is the radius to a fourth sine Again Proport 3. As the cosine 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 finde the Azimuth for any latitude when the Sun is in the Equator Proport 1. As the cosine of the latitude to the sine of altitude so is the sine of the latitude to a fourth sine Proport 2. As the cosine of altitude to that fourth sine so is the radius to the sine of the Azimuth from East or West PROBL. 10. To finde the Azimuth for any latitude in Southern Declination Proport 1. As the cosine of the latitude to the sine of altitude so is the sine of the latitude to a fourth Having by Probl. 4. found the Suns amplitude adde it to this fourth sine by Probl. 1. and say As the cosine 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 applyed 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 finde the hour and Azimuth may be all readily wrought by the lines of artificial sines only the addition and substraction must alwayes be wrought upon the line of natural sines CHAP. IV. Some uses of the Lines on the proportional side of the Instrument viz. the Lines of natural Sines 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 center 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 Dyals 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 wayes of protracting an Angle by the Line of Sines First if you use the Sines 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 satisfie 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 Sines on the
piece to be put into the tenons as before expressed The quadrantal side of the joyned pieces is easily discerned having the names of the Moneths 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 ♈ ♉ ♊ c. representing the 12 Signs of the Zodiack 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 dayes of the Moons 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 mans phantasie The fourth line is a line of altitudes for a particular latitude noted at the end Par. Scale c. 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 sines 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 sines may be joyned a line of tangents to 45 degr The use of the sines alone is to work proportions in signs The use of the sines 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 sines numbred 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 sines answering to that on the fixed piece The use of these versed sines 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 Moneths marked with the respective names and each Moneth divided into so many parts as it contains dayes The fifth line is a line of signs marked ♈ Taurus Gemini c. 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 Moneths The use of this line with help of the Moneths is to find the Suns place in the Zodiack The sixth line is a line of the Suns right ascension commonly noted by hours from 00 to 24. but better if divided by degrees or sines from 00 to 360 and both wayes proceeding backward and forward as the signs of the Zodiack or dayes of the Moneth 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 sines 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 sines 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 sines on the fixed piece at the head you have again 90 degrees which compleat 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 numbring 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 wayes of numbring these degrees for finding of the Azimuth c. 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 Cross-staff 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 sines and tangents or laying down any sine or tangent to any Radius given c. 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 sines 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 sines and tangents both upon the fixed and movable pieces is placed a
proportional side of the Instrument The general rule is this Account the first term upon the Lines of Sines from the Center 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 Center 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 Sines 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 adjoyned the method of working proportions by natural Tangents alone and by natural Sines 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 Sines and Tangents on the outward ledge where I intend 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 Sines to lay down any Tangent or Secant required to a Radius given In some Cases especially for Dyalling 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 cosine 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 cosine 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 finde the Suns declination As the Radius to the sine of the Suns 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 finde the Suns 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 finde his Azimuth As the cosine of the altitude to the cosine of the hour from the Meridian so is the cosine 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 c. 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 then 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 then every figured division is 200 every smaller or second division is 20 every third or smallest division is 10 c. 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 shew 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 alwayes three terms given to finde 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 dayes how long will 30 men be about it As 30 is to 60 so is 5 to 10. The number of dayes 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 dayes 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 finde the Suns Altitude At the length of the shadow upon the lines of lines is to the Tangent 45 so is the length of the perpendicular numbred upon the lines of lines to the tangent of the Suns altitude PROBL. 5. To finde the Altitude of any Tree Steeple c. 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 cosine of the altitude is to the measured distance numbred 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 adde 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 Sines 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 finde 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 c. 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 determin the number of places or figures in the product which may be resolved by this general rule The product alwayes 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 〈…〉 â—â—ve only one figure in the quotient PROBL. 4. To finde 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 finde 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 sines 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 numbred 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 numbred back again from 45 to 89 in which account every division hath as to its length on the line a proportional encrease 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 sines 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 finde 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 finde 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 adde any thing about working proportions by sines 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 adjoyn and leave to the practice of the young beginner with the directions in the former Cases PROBL. 7. To finde the Suns ascensional difference in any Latitude As the co-tangent of the latitude is to the tangent of the Suns declination so is the radius to the sine of the ascensional difference PROBL. 8. To finde 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 Suns declination so is the radius to the cosine of the hour from noon PROBL. 9. The Latitude Declination of the Sun and his Azimuth from South given to finde the Suns Altitude at that Azimuth As the radius to the cosine of the Azimuth from south so is the co-tangent of the latitude to the tangent of the Suns altitude in the equator at the Azimuth given Again As the sine of the latitude is to the sine of the Suns declination so is the cosine of the Suns 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 adde 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 finde the hour As the cosine of declination is to the sine of the Suns Azimuth so is the cosine of the altitude to the fine of the hour from the Meridian Proportions may be varied eight several wayes 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 any one may vary the former proportions and make the Problems three times the number here inserted Ex. gr To finde 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 Suns 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 Suns 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 Suns ascensional difference is to the tangent of the Suns 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 c. to finde 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 finde 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 finde 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 guess very near at the quantity of timber in any standing Tree CHAP. VIII To measure any kinde 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 shewn 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 finde 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 finde 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 over-plus of 100 foot at both ends and taking half that sum for the true breadth of the over-plus by Probl. 2. finde 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 c. foot of board speedily as your occasion requires PROBL. 4. To measure Wainscot Hangings Plaister c. 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 c. 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 c. 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 c. 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 sub-divided into quarters From thence again to 100 each whereof to 10 foot is sub-divided into inches c. 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 c. 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 finde 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 finde 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 sinde 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 finde 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 over-plus 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 finde 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. finde 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 easie 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 numbred from the end toward the head CHAP. X. The Gauge 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 finde 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 gives 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
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. c. 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 complement 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 North-ward adde 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 finde how many miles answer to one degree of longitude in any latitude As the radius is to the number 60 so is the cosine of the latitude to the number of miles for one degree 7. To finde how many miles answer to one degree of latitude on any rumb As the cosine 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 finde 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 moveable 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 adjoyning to the line of sines 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 sub-divided 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 adjoyning to the sines on the fixed piece The difference of longitude is numbred 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 moveable 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 shew how to resolve any Problem by the square and Index and next adjoyn the proportions for the use of such as have only pocket Instruments PROBL. 1. The course and distance run given to finde 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 cosine 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 cosine 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 finde 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 cosine of the course to the difference of latitude PROBL. 4. The distance run and difference of latitude given to finde 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 numbred 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 cosine of the course Again As the radius is to the distance run so is the sine of the course to the difference of
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 center 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 Center 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 alwayes 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 center with AE C E set half the co-tangent P S from C to G and he secant P S from C to R upon the center 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 center 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 center 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 AE 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 center 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 center C. sweep the ark VΔI Again with the secant of the ark ZPS upon the center 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 center and with the extent VS draw the ark TSP which makes up the triangle But in case the hypothenuse is equal to a quadrant Δ is the center if more than a quadrant I is the center 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 complement 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 complement 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 complement 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 applyed to a line of tangents is half the co-tangent of the angle ZPS Thirdly CW applyed 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 Πa ruler laid to ΠS cuts the limb at F then PF applyed to a line of chords gives the side PS Fifthly take the complement of the angle PZS to a Semicircle and set half the tangent of that complement from C to λ a ruler laid to λS cuts the limb at B and ZB applyed 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 SΠcuts the limb at φ and L φ applyed 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 chops so that they may slide by the inward or outward graduated limbs The turning sight hath only two places either the center at the head or the center at the beginning of the line of sines on the fixed piece to either of which as occasion requires it s fastened with a sorne The center at the head serving for the graduations next the inward limb of the loose piece And the center at the beginning of the line of sines serving for all the graduations
next the outward limb of the moveable 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 moveable piece to hold the tenons of the loose piece close joynted 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 wayes 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 alwayes 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 Suns altitude without thread or plummet when it is near the Zenith PROBL. 1. To finde the Suns altitude by a forward observation Serve the turning sight to the center 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 center 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 shew the altitude required PROBL. 2. To finde the distance of any two Stars c. by a forward observation Serve the turning sight to either center 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 c. PROBL. 3. To finde the Suns Altitude by a backward Observation Serve the turning sight to the center at the beginning of the line of sines and apply one of the sliding sights to the outward limb of the loose piece and the other to the outward limb of the moveable piece and turning your back toward the Sun set the sliding sight upon the moveable 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 Dyalling 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 comprized under one of these terms But before we treat of the affections or delineation of Dials 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 finde the reclination of any Plain Apply the outward ledge of the moveable 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 moveable piece you have the angle of reclination If the thread falls directly on 60 0 upon the moveable piece it s an horizontal if on 30. on the loose piece it s an erect plain PROBL. 2. To finde the inclination of any Plain Apply the outward ledge of the fixed piece to the plain with the head upward and what number of degrees the thread cuts upon the limb of the loose piece is the complement 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 paralel 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 finde the declination of any Plain Apply the outward ledge of the fixed piece to the horizontal line of your plain holding your instrument paralel to the horizon This done lift up the thread and plummer until the shadow of the thread fall directly upon the pin hole on the fixed
Styles height above the Substile As the radius is to the cosine of the latitude so is the cosine of declination to the sine of the styles height To finde 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 finde 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 finde the inclination of Meridians As the sine of the latitude is to the radius so is the tangent of declination to the tangent of inclination of Meridians 6. All North decliners with centers have the angular point of the style downward and all South have it upward 7. All North decliners without centers have the narrowest end of the style downward all South have it upward 8. In all decliners without centers take so much of the style as you think convenient but make points at its beginning and end 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 Meridians is the hour line of twelve at mid-night in South decliners at noon or mid-day This may tell you the true names of the hour lines 10. In transmitting these Dials from your paper draught to your plain lay the 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 wyer of any convenient length placed in the center perpendicular to the plain for the style 3. When the reclination inclination is greater than your latitude take the summe of the reclination inclination of your co-latitude from 180. and the residue or remain is the styles height But in this case the style must be set upon the plain as if the contrary pole was elevated viz. These North recliners must have the center of the style upward and the South incliners have it downward Note In all South re-in-cliners North re-in-cliners for their delineation the styles height is to be called the co-latitude and then you may draw them as erect direct plains for South or North as the former rules shall give them in that latitude which is the complement 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 complement of that latitude where the plain stands and their new declination is the complement of their reclination inclination But to know which way you are to account this new declination remember all East and West recliners are North-East and North-West decliners All East and West incliners are South-East and South-West decliners 3. Their new latitude and declination known you may by Sect 4. par 5. finde the substyle from the Meridian height of the style angle of twelve and six and inclination of Meridians using in those proportions the new latitude and new declination instead of the old 4. In all East and West recliners incliners with centers the Meridian lies in the horizontal line of the plain in such as have not centers its paralel 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 finde the new Latitude As the radius is to the cosine of the plains declination so is the co-tangent of the reclination inclination to the tangent of a fourth ark In South recliners North incliners get the difference 'twixt this fourth ark and the latitude of your place and the complement of that 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 complement 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 finde the new declination As the radius is to the cosine of the reclination inclination so is the sine of the old declination to the sine of the new 3. To finde 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 scituation Observe in North incliners less than a polar the Meridian lyes above That end of the Horizontall Line contrary to the Coast of Declination   below  South recliners more than a polar the Meridian lyes below That end of the Horizontall next the Coast of Declination   above  North recliners less than an equinoctial the Meridian lyes 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 lyes 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 center to the lower side After you have by the former proportions and rules found the new latitude new declination the angle and scituation of the meridian your first business in delineating of the Dial will be both for such as have centers and
such as admit not of centers to set off the meridian in its proper coast and quantity This done by Sect. 4. Paragr 5. of this Chapter finde the substyles distance from the Meridian the height of the style angle of twelve and six and for Dials without centers the inclination of Meridians 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 Dials 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 South-East or South-West recliners as have the contrary pole elevated must be described as North-East and North-West decliners and such North-East and North-West incliners as have the contrary pole elevated must be described as South-East and South-West decliners which will direct you which way to set off the substyle and hour line of six from the Meridian in those oblique plains which admit of centers or the substyle from the Meridian and the inclination of Meridians from the substyle in such as admit not of Centers 4. Declining polar plains must have a peculiar calculation for the substyle and inclination of Meridians which is thus To finde 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 finde the inclination of Meridians As the radius is to the sine of the latitude so is the tangent of declination to the tangent of inclination of Meridians 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 Dials for any plain Superficies HEre it will be necessary to premise the explication of some few terms and symbols which for brevity sake we shall hereafter make use of Ex. gr Rad. denotes the radius or sine 90. or tangent 45. Tang. is the tangent of any arch or number affixed to it Cos. is the cosine of any arch or number of degrees or what it wants of 90. Ex. gr cos 19. is what 19. wants of 90. that is 71. Co-tang is the co-tangent of any ark or number affixed to it or what it wants of 90. Ex. gr co-tangent 30. is what 30. wants of 90. that is 60. = This is a note of equality in lines numbers or degrees Ex. gr AB = CD That is the line AB is equal unto or of the same length as the line CD Again ABC = EFG that is the angle ABC is of the same quantity or number of degrees as the angle EFG Once more AB = CD = FG = tang 15. That is AB and CD and FG. are all of the same length and that length is the tangent of 15. d. â™’ This is a note of two lines being paralel unto or equidistant from each other Ex. gr FI. â™’ RS. that is the line FL. is paralel 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 AF = AG = HD = HE = sine of the latitude and AH = Radius Enter HD in tang 45. and keeping the Sector at that gage set off HI = HK = tang 15. and HO = HL = tang 30. Again enter FD. in tang 45. and set off FQ = GN = tang 15. and FP = GM = tang 30. This done Draw AQ AP. AD. AO AI. for the hour lines of 5. 4. 3. 2. 1. afternoon Again Draw AK AL. AE AM. AN. for the hour lines of 11. 10. 9. 8. 7. before noon The line FAG is for six and six In the same manner you may prick the quarters of an hour reckoning three tangents and 45. minutes for a quarter How to draw the hour lines before and after six was 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 AE = EB = CF = FD = cos of your latitude And EF = AC = BD = sine of your latitude Enter CF. in tang 45. and lay down FK = FL = tang 15. and FI = FM = tang 30. Again enter AC in tang 45. and lay down AG = BO = 15. and AH = BN = tang 30. with a ruler draw the lines EG EH EC EF. EK for the hour lines of 7. 8. 9. 10. 11. in the morning and EO EN ED. EM EL. for 5. 4. 3. 2. 1. afternoon the line AEB is for six and six The line EF. for twelve The description of a direct North-Dial differs nothing from this only the hour lines from Sun rise to six in the morning and from six in the evening until Sun set must be placed thereon by drawing the respective morning and evening hours beyond the center 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 entred 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 Center See Fig. 14. Draw the square BCDE and make AC = AK of quantity what you please Again draw AG. 12. â™’ CE. and KHF. â™’ 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 paralel 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 center you may draw the hours lines from six in the morning unto three afternoon For the other hour lines do thus Produce the line EC and likewise HA. beyond the center until they intersect each other as at S. Then setting off ST = HR and S. 4. = HP you have the hour points 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 center they help you to the hour lines before six in the morning Sect. 5. To describe an erect declining Plain without a Center See Fig. 12. The delineation of these Dials is the most difficult of any and therefore I shall be the larger in their description 1. By Chap. 2. S. 4. Paragr 5. finde the angles of the style substyle and inclination of Meridians 2. Having found the inclination of Meridians make the following Table for the distance of every hour line and quarter from the substyle Where I take for example a North plain declining East 72. d. 45. min. in lat 52. 30. min. Imcl mer hou quart 76 = 30. 12 00 20 = 15 3 ... 16 = 30 4 00 12 = 45 4 09 = 00 4 05 = 15 4 ... 01 = 30 5 00 02 = 15 5 06 = 00 5 09 = 45 5 ... 13 = 30 6 00 17 = 15 6 21 = 00 6 24 = 45 6 ... 28 = 30 7 00 32 = 15 7 36 = 00 7 39 = 45 7 ... 43 = 30 8 00 47 = 15 8 51 = 00 8 54 = 45 8 ... 58 = 30 9 00 62 = 15 9 66 = 00 9 69 = 45 9 ... 73 = 30 10 00 The manner of drawing this Table is thus The inclination of Meridians which because its a North decliner is twelve at midnight I finde 76. d. 30. min. Now considering the Sun never riseth till more than half an hour 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 finde 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 Meridians 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 center unto 45. the radius to the lesser lines of tangents which is continued to 76. on the Sector side enter this distance in 45. on the larger lines of tangents and keeping the Sector at that gage take out the tang 20. d. 15. min. which is the distance of the first line from the substyle set this from 73. d. 30. min. the distance of your last hour line from the substyle as you see by the Table toward the end upon the lesser line of tangents and where it toucheth as here at 75. 05. call that the gage tangent 5. Enter the whole line KL in the gage tangent which in this example is 75. d. 05. min. and keeping the Sector at that gage take out the tangent 73. 30. min. which is your last hour and set from L. on the line KL unto V. Again take out the tangent 20. d. 15. min. which is your first line and set it from K. towards V. and if it meet in V. it proves the truth of your work and a line drawn through V. paralel unto AG. is the true substyle line Then keeping the Sector at its former gage set off the tangents of the hours and quarters as you finde them in the Table from V. towards K. and from V. towards L. making points for them in the line KL Lastly enter the radius of your 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 entring hV. as Radius in the line of tangents take out the tangent of the styles height and set from p. to r. Again enter hr. radius in your line of 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 =