a Foot shut are drawn usually just 5 degrees assunder or rather the two innermost Lines on each Leg are always just one degree from the inside so that if you put a Center-pin in the Line of Tangents just against the Sine of 30 it makes the two innermost Lines that come from the Center just 2 degrees assunder which is easie to remember either in adding or substracting as followeth two wayes 1. Take the Latteral Sine of 30 viz. the measure from the Center to 30 the Compasses so set set one Point in the Center-pin in the Tangents just against 30 and turn the other till it cut the common Line in the Line of Sines on the other Leg and there it shall shew what Angle the two innermost-Lines make counting from the end toward the Head and two degrees less is the Angle the Sector stands at both on the in-side and out-side the Legs being parallel which Number must nearly agree with what the in-side of the Leg cuts on the Head-semicircle or there is a mistake As thus for Example Suppose I open the Rule at all adventures and taking the Latteral Sine of 30 from the Sines on the Sector-side and putting one Point of the Compass in the Center on the Tangents right against the Sine of 30 on the other Leg or the beginning of the Secants on the same Leg and turning the other Point to the Line of Sines on the other Leg it cuts the Sine of 60 on the innermost Line that comes from the Center then I say that the Lines of Sines and Tangents are just 30 degrees assunder and the in-side or out-side of the Legs but 28 viz. two degrees less as a glance of your eye to the Head will plainly shew 2. This way will serve very well for all Angles above 20 and under 80 But for all under 20 and above 80 to 120 this is a better way Open the Rule to any Angle at pleasure and take the distance parallelly that is across from one Leg to the other between the Center-pin at 30 in the Sines and that in the Tangents right against it and measure it latterally from the Center and it shall shew the Sine of half the Angle the Sines and Tangents stand at and one degree less is the Sine of half the Angle the Sector stands at Example Suppose that opening the Sector at adventures or to the Level of any thing I would know the Angle it stands at I take the parallel Distance between the two Centers and measuring it latterally from the Center I find it gives me the Sine of 51 degrees viz. the half Angle the Lines stand at or 50 the Angle the Rule stands at which doubled is 102 for the Lines or 100 for the Legs of the Sector as a glance of the eye presently resolves by the inner-edge of the Moving-leg and the divided semi-circle 3. On the contrary Would you set the Legs or Lines to any Angle take the half thereof latterally or one degree less in the half for the Legs and make it a Parallel in the two Centers and the Sector is so set accordingly Example I would set the Legs to 90 degrees or a just Square take out the Latteral Sine of 44 one degree less than 45 the half of 90 and make it a Parallel in the two Centers abovesaid and you shall find the Legs set just to a Square or Right-Angle as by looking to the Head you may nearly see At the same time if you take Latteral 30 and lay it from the Center according to the first Rule you shall see a great deficiency therein as above is hinted Use V. The Day of the Month being given to find the Suns Declination true place in the Zodiack Right Ascention Ascentional Difference or Rising and Setting 1. Lay the Thred to the Day of the Month in the upper Line of Months where the length of the Dayes are increasing or in the lower-Line when the Dayes are decreasing according to the time of the year then in the Line of degrees you have his Declination wherein note that if the Thred lie on the right hand of 600 then the Suns Declination is Northwards the contrary-way is Southwards Also on the Line of the Sun 's Right Ascention you have his Right Ascention in degrees and hours counting one Hour for 15 degrees as the Months proceed from March the 10th or Equinoctial the Right Ascention being then 00 and so forward to 24 hours or 360 degrees as the Months and Dayes proceed Again on the Line o◠the Sun 's true place you have the sign and degree of his place in the Ecliptick Aries or the Equinoctial-point being the place to begin and then proceeding forward as the Months and Dayes go Lastly on the Hour-line you have the Ascentional-difference in degrees and minutes counting from 6 or the Suns Rising counting as the morning hours proceed or his Setting counting as the afternoon hours proceed Of all which take two or three Examples 1. For March the 12th lay the Thred to the Day and extend it streight then on the Line of degrees it sheweth near 1 degree or 54 minutes Northward 2. The Suns Right Ascention is in time 8 minutes and better or in degrees 2 deg 5 minutes 3. The Suns Place is 2 degrees and 16 minutes in Aries ♈ 4. The Ascentional Difference is 1 degree and 10 minutes or the Sun riseth 4 minutes before and sets 4 minutes after 6. Again for May the 10th the Thred laid thereon cuts in the degrees 20 deg 9 min. for Northern Declination and 57 deg 24 min. or 3 hours 52 min. Right Ascention and 29 37 in ♉ Taurus for his true place and 27 12 for difference of Ascentions or riseth 11 minutes after 4 and sets 49 minutes after 7. Again on the last of October or the 21 of Ianuary near the Declination is 17 22 Southwards the Right Ascention for October 31 is 225 53 for Ianuary 21 314 21 The true place for October 31 is ♠Scorpio 18 deg 22 min but for Ianuary 21 ♒ Aquarius 11 52 according as the Months go to the end at ♑ and then back again but the Ascentional difference and Rising and Setting is very near the same at both times viz. 23 10 and Riseth 32 minutes and more after 7 and Sets 28 minutes less after 4. Use VI. The Declination of the Sun or a Star given to find his Amplitude Take the Declination being counted on the particular Scale of Altitudes between your Compasses and with this distance set one foot in 90 on the Azimuth-Line the other Point applied to the same Line shall give the Amplitude counting from 90. Example The Declination being 12 North the Amplitude is 19 deg 15 min. Northwards Or the Declination being 20 South the Amplitude is 34 deg 10 min. Southwards Use VII The Right Ascention and Ascentional-difference being given to find his Oblique-Ascention When the Declination is North then the difference between the
Right Ascention and the Ascentional-difference is the Oblique-Ascention But in Southern declinations the sum of the Right Ascention and difference of Asscentions is the Oblique Ascention Example On or between the 25 and 26 of Iuly the Oblique-Ascention is by Substraction 112 15 On the 30th of October the Oblique-Ascention is 337 45 by Addition Use VIII The Day of the Month or Sun's Declination and Altitude being given to find the Hour of the Day Take the Suns Altitude from the particular Scale of Altitudes setting one Point of the Compasses in the Center at the beginning of that Line and opening the other to the degree and minute of the Sun's Altitude counted on that Line then lay the Thred on the Day of the Month or Declination and there keep it Then carry the Compasses set at the former distance along the Line of Hours perpendiculer to the Thred till the other Point being turned about will but just touch the Thred the Compasses standing between the Thred and the Hour 12 then the fixed Point in the Hour-Line shall shew the hour and minute required but whether it be the Fore or Afternoon your judgment or a second observation must determine Example On the first of August in the morning at 20 degrees of Altitude you shall find it to be just 52 minutes past 6 but at the same Altitude in the afternoon it is 7 minutes past 5 at night in the Latitude of 51 32 for London Use IX The Suns Declination and Altitude given to find the Suns Azimuth from the South-part of the Horizon First by the 4th Use find the Suns Declination count the same on the particular Scale and take the distance between your Compasses then lay the Thred to the Suns Altitude counted the same way as the Southern-Declination is from 600 toward the loose-piece and when need requires on the loose-piece then carry the Compasses along the Azimuth-line on the right-side of the Thred that is between the Thred and the Head when the Declination is Northward and on the left-side of the Thred that is between the Thred and the End when the Declination is Southward So as the Compasses set to the Declination as before and one Point staying on the Azimuth-line and the other turned about shall but just touch the Thred at the nearest distance then I say the fixed-Point shall in the Azimuth-line shew the Suns-Azimuth required Example 1. The Sun being in the Equinoctial and having no Declination you have nothing to take with your Compasses but only lay the Thred to the Altitude counted from 600 toward the loose-piece and in the Azimuth-line it cuts the Azimuth required Example At 25 degrees high you shall find the Suns Azimuth to be 54 10 at 32 degrees high you shall find 38 20 the Azimuth Again At 20 degrees of Declination take 20 from the particular Scale and at 10 degrees of Altitude lay the Thred to 10 counted as before then if you carry the Compasses on the right-side for North-Declination you shall find 109 30 from South but if you carry them on the left-side for South-Declination you shall find 38 30 from South The rest of the Vses you shall have more amply afterwards CHAP. VI. The Use of the Line of Numbers on the Edge and the Line of Lines on the Quadrantal-side or on the Sector-side being all as one HAving shewed the way of Numeration on the Lines as in Chapter the first Also to add or substract one Line or Number to or from one another as in Chapter 4th Explanation the 9th I come now to work the Rules of Multiplication and Division and the Rule of Three direct and reverse both by the Artificial and Natural-Lines and first by the Artificial being the most easie and then by the Natural-lines both on the Sector and Trianguler Quadrant being alike and I work them together First because I would avoid tautology Secondly because thereby is better seen the harmony between them and which is best and speediest Thirdly because it is a way not yet as I know of gone by any other And last of all because one may explain the other the Geometrical Figure being the same with the Instrumental-work by the Natural way Sect. I. To multiply one Number by another 1. By the Line of Numbers on the Edge Artificially thus Extend the Compasses from 1 to the Multiplicator the same extent applied the same way from the Multiplicand will cause the other Point to fall on the Product required Example Let 8 be given to be multiplied by 6 If you set one Point of the Compasses in 1 either at the beginning or at the middle or at the end it matters not which yet the middle 1 on the Head-leg is for the most part the most convenient and open the other to 6 or 8 it matters not which for 6 times 8 and 8 times 6 are alike but yet you may mind the Precept if you will the same Extent laid the same way from 8 shall reach to 48 the Product required which without these Parenthesis is thus The Extent from 1 to 6 shall reach the same way from 8 to 48. Or The Extent from 1 to 8 shall reach the same way from 6 to 48. the Product required By the Natural-Lines on the Sector-side or Trianguler Quadrant with a Thred and Compasses the work is thus 1. For the most part it is wrought by changing the terms from the Artificial way as thus The former way was as 1 to 6 so is 8 to 48 or as 1 to 8 so is 6 to 48 but by the Sector it is thus As the Latteral 6 taken from the Center toward the end is to the Parallel 10 10 set over from 10 to 10 at the end counted as 1 so is the Parallel-distance between 8 8 on the Line of Lines taken a-cross from one Leg to the other to the Latteral-distance from the Center to 48 the Product required Or shorter thus As the Latteral 8 to the Parallel 10 So is the Parallel 6 to the Latteral 48. See Figure I. 2. Another way may you work without altering the terms from the Artificial way as thus by a double Radius Take the Latteral-Extent from the Center to 1 or from 10 to 9 if the beginning be defective make this a Parallel in 6 6 then the Latteral-Extent from the Center to 8 of the 10 parts between Figure and Figure shall reach across from 48 to 48 as before See Fig. II. The same work as was done by the Sector is done by the Line of Lines and Thred on the Quadrant-side that if your Sector be put together as a Trianguler Quadrant you may work any thing by it as well as by the Sector in this manner or by the Scale and Compass as in the Figure I. and first as above Sector-wise Take the Extent from the Center to 6 latterally between your Compasses set one Point in 10 and with the other lay the Thred in the nearest distance
or from 12 as it is figured Example On April 20 at 30 deg 20 min. of Altitude Latitude 51-32 the hour will be found to be just 2 hours from 6 or just 8. Again On the 10th of November at 8 deg 25 min. high it is just 3 hours from 6 or 9 a clock in the forenoon or 3 afternoon Or somewhat differing thus Take the sine of the sum or difference of the Suns present Altitude and Altitude at 6 and make it a = in the co-sine of the Latitude and lay the Thred to the nearest distance then take out the = Secant of the declination beyond 90d and make it a = sine of 90 and laying the Thred to the nearest distance on the degrees it shall shew the hour from 6 required First by Use 25 find the Suns height at 6 or depression in Winter then by the former 2d find the sum or difference between the Altitude at 6 and the Suns present Altitude but if you have Tables of Natural Sines and Tangents then in Winter add the Natural Sines of the two Altitudes together and in Summer substract the lesser out of the greater and find the Ark of difference more exactly Then As the Co-sine of the Latitude to the Secant of the Declination counted beyond 90 as much forward as from 90 to the Co-sine of the Suns Declination So is the Sine of the sum or difference to the hour from 6 required Or else â—hus As the Co-sine of the Latitude to the Sine of the sum or difference So is sine of 90 to a 4th Then As the Co-sine of the Suns declination to that 4th So is sine 90 to the hour from 6. By the Sector Take the secant of the Suns declination make it a = in the co-sine of the Latitude then take out the = sine of the sum or difference and turn it twice from the Center latteraâ—ly and it shall be the sine of the hour from 6 required Example April 20 the Suns Declination is 15 degrees and the Suns Height at 6 then is 11 deg 42 min. now the Natural sine of 11-42 20278 taken from the Natural sine of 30 deg 20 min. 50502 the Suns present Altitude the residue is 30224 the sine of 17 deg 35 min. and a half Then The Secant of 15 made a = sine of 38-28 and the Sector so set the = sine of 17-35 ½ turned latterally twice from the Center shall reach to 30 the sine of 2 hours from 6 the hours required Use XXVII Having the Latitude the Suns Declination and Altitude to find the Suns Azimuth Take the Declination from the particular Scale of Sines for the particular Latitude the Instrument is made for Then count the given Altitude on the degrees from 600 toward the loose-piece and sometimes on the loose-piece also and thereunto lay the Thred then carry the Compasses so set along the Azimuth-line on the right-side of the Thred in Northern-declinations and on the left-side in Southern-declinations till the other foot turned about will but just touch the Thred then the fixed-point shall stay at the Suns true Azimuth required Take two or three Examples 1. First When the Sun is in the Equinoctial and hath no Declination then there is nothing to take between your Compasses but just to lay the Thred to the Suns Altitude counted from 600 on the loose-piece toward the End then 〈◊〉 the Azimuth-line it cuts the Azimuth from the South required Example At 00 degrees high the Azimuth is 90 from South and at 10 degrees high it is 77-5 at 20 high the Azimuth is 62-45 at 30 degrees high it is 43-30 at 34 degrees high it is 32 degrees of Azimuth from South and at 38-28 degrees high it is just South 2. Secondly at 10 degrees of Declination Northward and 20 degrees of Altitude take 10 degrees from the particular Scale and lay the Thred to the Suns present Altitude as before and carry the Compasses on the right-side of the Thred on the Azimuth-line till the other foot being turned about will but just touch it then shall the Point rest at 80 degrees 42 min. of Azimuth from the South 3. But if the Declimation be the same to the Southwards and the Altitude also the same then carry the Compasses on the left-side of the Thred on the Azimuth-line till the other foot turned about will but just touch it and you shall find the Point to stay at 41 deg 10 min. the true Azimuth from the South required Note That any thing as thick as the Rule laid by the Rule and the Thred drawn over it will keep the Thred steady till you get the nearest distance more truly First by the 18th Use find the Suns Altitude in the Vertical Circle or Circle of East and West thus Take the sine of the Suns Declination and set one foot in the sine of Latitude lay the Thred to ND and in the degrees you shall have the Altitude at East and West required Which Vertical Altitude in Summer or Northern Declinations you must substract out of the Suns present Altitude or take the lesser from the greater to find a difference but in Winter you must add this depression in the Vertical Circle to the Suns present Altitude to get a sum which must be done on a Line of Natural Sines or by the TABLE of Natural Sines as before in the Hour by laying it over or under the Center and taking from that noted Point to the Suns present Altitude all that day Then take the distance from the Center to the Tangent of the Suns present Altitude on the loose-piece which is the Secant of the Suns present Altitude and lay it from the Center on the Line of Sines and note the place then take the distance from 60 on the loose-piece to the co-tangent of the Latitude by counting 10 20 30 c. from 60 toward the moveable-leg between your Compasses then setting one Point on the Secant of the Suns Altitude last found and noted on the Line of Sines and with the other lay the Thred to the nearest distance and there keep it by noting what degree day of the month or hour minut or Azimuth it cuts Then take the distance on the Sines from the sine of the Suns Vertical Altitude to his present Altitude for a difference in Summer Or The distance from a Point made beyond the Center equal to the sine of the Suns Vertical depression to the Suns present Altitude for a sum in Winter Then having this distance of sum or difference for Winter or Summer between your Compasses carry one Point parallelly on the Line of Sines till the other being turned about shall just touch the Thred at the ND the place where the Point stayeth shall be the Azimuth from East or West as it is figured from the Center or from North or South counting from 90. Which work in brief may be sufficiently worded thus As co-tangent
the complement thereof to 12 Hours and the Stars Right-Ascention and the hour of Rising the Thred cuts and add them into one sum and the sum if under 12 is the time of his Rising in common hours or if you add the hour of Setting that the Thred sheweth it shall give his setting Example If you lay the Thred to 15-48 the Declination of the Bulls Eye in the Hour-line it cuts 4 hours 36 min. for Rising or 7-24 for his Setting then if you work for April the 23d the Suns Right Ascention then is 2-44 and the complement thereof to 12 is 9-16 and the Stars Right-Ascention is 4 hours and 16 minutes and the Hour cut is 4-36 for Rising and the three Numbers viz. 9-16 the complement of the Suns Right Ascention and 4-16 the Stars Right Ascention and 4-36 the Hour of Rising the Thred cuts being added makes 18-8 from which taking 12 rest 6-8 the time that the Bulls Eye Riseth on April 23 and if you add 7-24 the time of Setting that the thred cuts there comes forth 8-56 viz. one hour and 32 min. after the Sun To find the time of a Stars coming to South Substract the Right Ascention of the Sun from the Right Ascention of the Star increased by 24 when you cannot do without and the remainder if less than 12 is the time between 12 at noon and 12 at night but if the remainder be more than 12 it is the time between mid-night and mid-day following Example The Lyons-Heart whose Right Ascention is 9-50 will come to the South on March 10 at 9-48 the Suns Right Ascention being then only 2 minuts By the Line of 24 hours say or work thus Extend the Compasses from the Suns Right Ascention to the Stars Right Ascention that distance laid the same way from 12 at the middle or at the beginning shall reach to the time of the Stars coming to South To find the time of the Stars continuance above the Horizon First find what the Suns semi-diurnal-Ark is having the same declination and that doubled is the whole time of continuance Or if you shall add and substract it to or from the time of the Stars coming to South you shall find the time of Setting or Rising Or else By laying the Thred to the Stars Declination it sheweth the Ascentional difference in this Latitude which added in those Stars that have North declination or substracted in Southern to 6 hours gives the semi-diurnal Ark of the Star above the Horizon Example The Eye 's Ascentional-difference is one hour and 24 minuts which added to 6 hours because of Northern declination makes 7-24 for the semi-diurnal-Ark or 14â— 48′ for the whole time of being above the Horizon Note That to work this for other Latitudes the Suns Ascentional-difference is to be found for that Latitude you are in and the Operation is general for all places To find a Meridian Line by the Sun On any flat Horizontal-Plain set up a streight Wyre in the Center of a Circle or hold up a Thred or Plummet till the shadow of the Thred cut the Center and any where in the Circumference which two Points you must note then immediately take the Suns Altitude and find the Suns Azimuth and count so many degrees in the Circle the right way as the Suns Azimuth comes to from the Points of the shadow marked in the Circumference and draw that Line for a true Meridian-line This Work is best done before 10 in the morning and after two afternoon or in the night by two Plumb-lines set in a right-Line with the North-Star at a right scituation Use XLIII To find the Hour of the Night by the Fixed Stars First find the Stars Altitude by looking along the Fixed or Moveable-leg to the middle of the Star letting the Thred with a weighty Plummet play evenly by the degrees between your Thumb and Fore-finger to the end you may command the Thred and know whether it playeth well or no by feeling Then Take the Altitude found from the particular Scale of Sines and laying the Thred over the Stars declination which for readiness sake is marked with 1 2 3 4 5 6 7 8 9 10 11 12 according to the Figures set to the 12 Names of the 12 Stars on the Rule and then carrying the Compasses as you do in finding the hour by the Sun you shall find how much the Star wants or is past the Meridian which is called the Stars-Hour And note That if the Star be past the South it is an aftâ—rnoon hour if not come to the South a morning hour which you must remember Also knowing the Suns Right Ascention set one Point of the Compasses in the Suns Right Ascention counted in the Line of twice 12 or 24 hours on the outward-leg of the fixed-piece next to the particular Scale of Sines and open the other to the Stars Right Ascention noting which way you turn the Compasses for the same Extent applied the same way from the Stars hour last found shall shew the true hour of the night required Example Suppose on the 10th of Ianuary I should observe the Altitude of the Bulls Eye to be 20 degrees if you take 20 degrees the Altitude from the particular Scale and lay the Thred on 15-48 the Stars declination Northward and measure from the Hour-scale the nearest distance to the Thred you shall find the Compass-point to stay at 6-49 on the East-side of the Meridian suppose Also The Suns Right Ascention the same day is 8 hours and 12 minuts Then The Extent from 8 hours 12 minuts on the Line of twice 12 hours the Suns Right Ascention to 4-16 the Stars Right Ascention shall reach the same way from 6-49 the Stars hour to 2-53 the true hour Use XLIV To find the Hour of the Night by the Moon First by an Almanack or Ephemerides find the Moons Age and true Place for the present time then by laying the Thred on the Moons place you may have her Right Ascention and also the Suns Right Ascention and by the Moons Altitude taken from the particular Scale and the Thred laid over the Moons place you find what the Moon wants or is past coming to South which is called the Moons hour Then by the Line of 24 Hours say As the Suns Right Ascention is to the Moons Right Ascention So is the Moons hour last found to the true hour Example Suppose that on the 8th of Ianuary about 40 min. after 3 there is a New Moon then note That the Suns true place is the Moons true place and consequently their Right Ascentions and the Moons Hour and Altitude is the same with the Suns Therefore As 8 hours 04 min. the Suns Right Ascention is to 8-04 the Moons Right Ascention So is the Moons hour at any Altitude to the Suns true hour Again Suppose that on the 1st quarter-Quarter-day the Moon being gone 90 degrees from the Sun to find her place Then do thus Set one Point in the Moons
place the Change-day and open the other to the beginning or the end of the Line of 24 hours Then The same Extent applied the contrary way from 6 hours or 7 dayes and a half the Moons Age shall give 28 deg 58 min. ♈ to which you must add 7 degrees and 30 minuts the Suns place between and the sum shall be the Moons true place required viz. 6-28 degrees in ♉ Example If the Moon Change on the 8th day the First Quarter being 7 dayes and a half after will be on the 15th day later at night then the difference between the Sun and Moons Right Ascention will be found to be near 6 hours for the Suns Right Ascention Ianuary 15 is 8-32 and the Moons Right Ascention the same day being about 8 degrees and a half in ♉ is 2 hours and 28 minuts if you take the distance between them on the 24 hours it is near 6 hours which is the difference of time between the Moon and the Suns hour Again For the Full Moon on the 22 day near 4 hours after noon the Moons Age being 14 dayes ¾ if you add 12 hours or 6 signs to the Moons place aâ— the Change you shall find ♋ 29-0 to which if you add 14-45 the dayes between the New and Full you shall find ♌ 13 deg 45 min. for the Moons place the Suns Right Ascention the 22 day is 9 hours and the Moons the same day at 1 afternoon is 9 hours also or rather 12 difference so that the Suns hour and the Moons is equal only one is North and the other South Again For the Last Quarter 22 ¼ dayes or 18 hours added and 22 degrees also together makes â™ 22 deg 11 min. for the Moons place by help of which to find the Moons hour by her Altitude above the Horizon found by observation Or Without regarding the Sun or Moons Right Ascention having her true Age and Hour Say thus As 12 on the Line of 24 hours is to the Moons Age in the Line of her Age So is the Moons hour to the true hour For The Extent from 12 in the middle to the Moons Age under or over the middle shall reach the same way on the same Line from the Moons hour to the true hour The like work serves to find the hour of the night by any Planets as Saturn Mars or Iupiter which are seen to shine very brave and bright in Winter evenings and having learned their Place by their distance from the fixed Stars or by the Ephemerides then their Altitude and Place will find their hour from the Meridian and the comparing their Right Ascentions with the Suns gives the true hour as before in the Fixed-Stars Use XLV To find the Moons Place and Declination without the Ephemerides somewhat near First observe when the Moon is in the Meridian and then find her Altitude and take the same from the particular Scale between your Compasses then set one Point in the hour 12 and lay the Thred to ND and on the degrees it shall shew the Moons declination and in the Line of the Suns Place the Moons present Place counting her Progress orderly from the last change-Change-day or New Moon when she was with the Sun Otherwise thus Observe what Hour the Moon sheweth on any Sun-dial at the same instance by the Fixed Stars or other wayes find the true Hour Then The Extent from the Moons Hour to the the true Hour shall reach the same way from 12 to the Moons Age right against which is her coming to South at which time you may find her true Altitude and so come by her Declination Yet again for her Age and Place according to Mr. Street and Mr. Blundevil Add the Epact the Month and Day of Month in one sum counting the Months from March by calling March the first Month April the second c. then that sum if under 30 is the Moons Age but if the sum be above 30 then substract 30 and the remainder is the Moons Age when the Month hath 31 dayes but if the Month hath but 30 or less than 30 dayes then substract but 29 and the remainder is the Moons Age. Or thus Add to the Epact for the present year and in Ianuary 0 in February 2 in March 1 in April 2 in May 3 in Iune 4 in Iuly 5 in August 6 in September 8 in October 8 in November 10 in December 10 and the sum if under 30 or the excess above 30 added to the day of the Month abating 30 if need be gives the Moons Age that day but substracted from 30 leaves the day of her Change in that Month or from the 〈…〉 â—onth Example July 10. 1668. The Epact that year is 26 and the Number for Iuly is 5 the Excess above 30 is 1 which added to any day of the Month as to 10 gives 11 for the Moons Age Iuly 10. 1668. Then for the Moons Place Multiply the Moons Age by 4 and the Product divided by 10 the Quotient giveth the signs and the remainder multiplied by 3 gives the degrees which you must add to the Suns place that day to find out the Moons place for that day of her Age. Example On Iuly 10. 1668 the Moons Age is 11 which multiplied by 4 makes 44 and 44 divided by 10 gives 4 signs in the Quotient and 4 the remainder multiplied by 3 makes 12 degrees more which added to Cancer 29 degrees the Suns place on the 10th day of Iuly makes 11 degrees in Sagittarius the Moons place the same day propè verum Or rather by the Rule thus on the Line of 24 hours by particular Scale having the Moons place to find her Age by the Line of 24 hours The Extent from the Suns true place to the Moons true place shall reach the same way from 0 day to the day of her Age. Or contrarily having the Moons true Age to find her true Place The Extent from 0 day old to the Moons true Age shall reach the same way from the Suns true Place to the Moons Or having the Moons true Place at the New Moon to find her Place any day of her Age after The Extent from ♈ to the Moons true Place at the Change shall reach the same way from the day of her true Age to her true Place adding as many degrees to the Number found as the Moon is dayes old Then Having her Place and Age it is easie to find the Moons Hour and then her true Hour but I fear I spend herein too much time on an uncertain subject Use XLVI The Right Ascention and Declination of any Star with the Suns Right Ascention and the Hour of the Night given to find the Altitude and Azimuth of that Star and thereby to know the Star if you knew it not before Set one Point of the Compasses in the Stars Right Ascention found in the Line of twice 12 hours and open the other to the Suns Right Ascention found in the same Line then this
THE Description and Use OF THE TRIANGULER-QUADRANT BEING A Particular and General Instrument useful at Land or Sea both for Observation and Operation More Universally useful Portable and Convenient than any other yet discovered With its Uses in Arithmetick Geometry Superficial and Solid Astronomy Dyalling Three wayes Gaging Navigation In a Method not before used By Iohn Brown Philomath London Printed by Iohn Darby for Iohn Wingfield and are to be sold at his house in Crutched-Fryers and by Iohn Brown at the Sphear and Sun-Dial in the Minories and by Iohn Sellers at the Hermitage-stairs in Wapping 1671. To the Reader FRiendly Reader Thou hast once more presented to thy view a further Improvement and use of the Sector under the name of the Trianguler Quadrant so called from the shape thereof In the year 1660 it was my lot first to apply and improve this former Contrivance of Mr. Samuel Foster on a Quadrant to a joynt Rule or Sector and did in 1661 publish my present Thoughts thereof in a small Discourse under the name of the Ioynt Rule Since then through my perswasions and assistance another Piece was published 1667 by I. T. under the name of the Semi-Circle on a Sector But neither of these that is to say neither my own nor his spoke what I would have it speak neither have I hopes ever to produce a Discourse either for method or matter worthy or becoming so excellent universal and useful an Instrument for the most Mathematical Occasions being for acurateness conveniency cheapness and universality before all others For 1. If it is made of Wood if the Wood keep but streight it is as true to be made use of as of Metal 2. It may be made of any Radius or bigness and yet in little Room in comparison of other Quadrants 3. More convenient to use wheâ— large than other Quadrants 4. As to the Projection for 〈◊〉 and Azimuth particularly using onâ—ly two Lines of Natural Sines thâ— Thred and Compasses for those twâ— difficult and many more easie Proâ—positions 5. The neat Conveniency of greater and a less Radius doublâ— treble or quadruple one to another 6. The convenient Contrivance that happens to it of three Instruments in one viz. A Sector Quadrant and Gunter's Rule all three conveniently in one The consideration of these things and the love and willingness I alwayes had to the communicating of them to others hath put me on this hard task of writing this Collection of the use thereof Wherein I do most heartily beg thy Pardon and Acceptance to accept in good part the willing endeavours of my poor Ability which I doubt not but to have from most that know me For first my insufficiency â—n the Tongues Arts and Sciences Secondly my Meanness and Poverty in the World for these Imployments which take up so much of a mans time and ability to perform them to purpose my plead my excuse for first here is the Product of more than Two years Improvement of more than vacant Hours with the great disadvantage of taking three Weeks at times to do that which three Dayes together might have as well if not much better performed And at last to calâ— the Assistance of two others to unâ—dertake the Charge thereof to midâ—wife it into the World Thus as Widows Mites are accepted which are offered in sinceriâ—ty so I hope will mine though atâ—tended with much disorder as 〈◊〉 Method more uncouthness as 〈◊〉 Stile and Matter What it is it 〈◊〉 as at first Composing for I could neâ—ver get Time nor Liberty from 〈◊〉 daily Trade and Calling to tranâ—scribe it twice Yet was it not done at any timâ— carelesly but with good will and free intent of plainness and usefulâ—ness for the publick good of others as well as my own recreation 〈◊〉 delight The Gunters Rule the Quadrant and Sector I need not commend they are so well known already but this I will add a better Contrivance and more general hath not yet to my knowledge been produced nor a Discourse where the use of all the Three together hath so been handled nor many more Examples though Mr. Windgate and Mr. Patridge have done sufficiently for the Gunters-Lines and Mr. Gunter for the Sector and Mr. Collins with the Quadrant and all of them distinctly far beyond this yet this Discourse of all the Three together may give content to some others as well as to me The Discourse of Dialling is gathered from Mr. Wells and yet those that shall read Mr. Wells and this may often-times think otherwise for I assure you I saw not one leaf of his Book all the while it was doing but I hope it may please in moderate sort and ordinary capacity both for plainness convenience and variety The cutting of the Regular Bodies I learned from Mr. Iohn Leake and the way is ready convenient and exact and worthy of remembrance The Theorems from Mr. Thomas Diggs as in its due place is observed The way of Measuring Superficies and Solids from Mr. Gunter and my constant Experience in those Imployments and the Learner may here be supplied with what is often complained on viz. the Interpretation of Hard-words as much as I could call to mind or think to be convenient for that purpose In the 15th Chapter I have gathered many Cannons from Mr. Collins his Workes and applied them to the Trianguler Quadrant and been more large than needs in some places yet I hope to the content of some inquiring Persons The business of Navigation I fear may prove most defective for my part I never yet saw Graves-end much less the Streights of Gibralter but for Observation and Operation the Instrument will do as well as any if well made and applied So for the present I rest and remain ready to serve you in and supply defects by well making of these Instruments at the Sphear and Sun-Dial in the Great Minories Iohn Browne The Argument of the Book and the Authors Apologie AT length my pains hath brought to pass the things I long intended And doubt not but in every place hereafter 't may be mended To me it hath been of great use to others more likewise Therefore let no man it abuse before he doth advise One Part thereof hath had renown with Artists far and near The other Part I strive to crown with use and plainness here Although my Parts and Time be small to hold forth Arts aright Yet have I plainly set forth all seemed useful in my sight And though I have not seen so far as some perhaps might see I doubt not but that some there are will pleased with it be For first the Tyroes young may find some terms to be explained Which when well fixed in his mind time quickly will be gained In the next place Mechanicks mean that have small time to spare But yet may have a Love extream to Mathematicks fair And others that of wordly Means have little to afford For various Mathematick Theams this having they are
there being 30 degrees in one Sign Fiftly Next above this is a Kalender of Months and Dayes every single Day being exprest and three or more Letters of the name of every Month being set in the Month and also at the beginning of each Month and every 10th day noted with a Prick on the top of the Line representing it as is usual in such work Sixtly Next over the Months is the Line to find the Hour and Azimuth in a particular Latitude Put alwayes on smaller Instruments and very rarely on large Triangular Quadrants for Sea Observations the lowest Margent whereof and next the Months is numbred from the end toward the Head with 10 20 30 40 50 60 70 80 90 100 110 120 130 near the Head Center For the Semi-diurnal Ark of the Suns Azimuth and in the Margent next above this with 4 5 6 7 8 9 10 11 12 near the end for the Morning hours then the other way viz. toward the Head on the other-side the Hour Line with 1 2 3 4 5 6 7 8 for the Afternoon hours Seventhly On the same Quadrantal-side and Moveable-leg on the spare places beyond the Months toward the end is set an Almanack and the Names of 12 or more Stars to find the hour of the Night which 12 Stars are noted with 1 2 3 4 5 6 7 8 9 10 11 12. among the degrees in small Figures as in the Figure Eightly Next of all to the in-side is the Line of Natural versed Sines drawn to the Center with his correspondent Line on the other or Head-leg Exprest sometimes in a pricked Line for want of room Ninthly On the Head-leg and next to the versed Sines last mentioned is first the Line of Equal Parts or Line of Lines and on the same common Line wherein is the Center is the Line of Natural Sines whose length is equal to the measure from the center at C to 600 on the moveable-leg so that the Line of degrees is a Tangent and the measure from C to any Tangent a Secant to the same Radius of the Natural Lines of Sines and Lines Also beyond the Center C on the same common middle Line is another smaller Line of Natural Sines whose length is equal to the measure from C to 60 on the loose-piece then if you count from the Center pin at 60 on the loose-piece toward the end of the movable-leg they shall be Tangents to the same Radius and the measure from the Center C to those Tangents shall be Secants to the same Radius which may be well to be ordered to a third or fourth part of the former from the Center downwards These two Lines of Sines are best figur'd with their Sines and Cosines the other way with a smaller figure and the Line of Lines from the Center downward from 1 to 10 where 90 is which Lines of Sines may be called a general Scale for all Latitudes Tenthly Next to this toward the outer-edge is another Line of Natural Sines fitted to the particular Line of Hour and Azimuths for one particular Latitude noted Pert. Scale of Altitudes or Sines Eleventhly Next to this is the Line of 29 ½ for so many dayes of the Moon 's age in short Rules of the whole length but in longer not being easily known by the single strokes and Figures annexed to those strokes Twelfthly Next the outer-edge is a Line of 24 hours 360 degrees or 12 Signs or in most Rules inches also used together with the former Line of 29 ½ and as a Theory of the Sun and Moon and ready way of finding the Hour by the Moon or fixed Stars Thirteenthly To this Instrument also belongs a Thred and Plummet and Sights as to other Quadrants and a pair of Compasses as to other Sectors a Staff and Ball socket also if you will be curious and accurate And for large Instruments for Sea a Square and an Index which makes it a perfect sinical Quadrant and two sliding sights also which makes it a fore and back-staff and bow as will appear more at large afterward Some Uses of the Trianguler Quadrant for Land and Sea Observations and Operations CHAP. I. Numeration on the Lines graduated on the Instrument IN the first place it will not be amiss to hint a few words as to the reading the Lines or more properly Numeration on the Lines wherein take notice That all Lines of Equal Parts or Lines applicable to Arithmetick as the Line of Lines the Line of Numbers the Line of Foot-measure and the like wherein Fractions of Numbers are requisite they are most commonly accounted in a Decimal way and as much as may be the small divisions are numbred and counted accordingly But in the Lines of Sines Tangents Secants and Chords being Lines belonging properly to a Circle in regard that the Sexagenary Fraction is still in use the intermediate Divisions are as much as may be fitted to that way of account viz. by whole degrees where they come close together or the Line of no great use And if more room is to half degrees or 30 minuts and sometimes to quarters of degrees or 15 minuts but toward the beginning of the Line of Natural Sines or the end of the Natural Tangents and Secants where the degrees are largest they are divided to every 10th minute in all large Rules as by considering and accounting you may plainly perceive Take two or three Examples of each kind 1. First On the Line of Lines to find the Point that represents 15. In the doing of this or any the like you must consider your whole Scale Radius or length of the Line may be accounted as 1 as 10 as 100 as 1000 or as 10000 and no further can be applicable to any ordinary Instrument Wherein observe That if the whole Line be one then the long stroke by every Figure doth represent one tenth of that Integer and the next shorter without Figures are hundredth parts of that one Integer and a 1000th part is estimated in smaller Instruments and sometimes exprest in larger But the hundredth thousand part is alwayes to be estimated by the eye in all Instruments whatsoever 2. But if the whole Line of Lines shall represent 10 as it usually doth and as it is figured then the long stroke at every Figure is 1 and the next longer are tenths and the shortest are hundred parts and the thousand parts as near as can be estimated 3. But if the whole Line represents a hundred as here in our present Example then the long stroke by every Figure represents 10 and every shorter stroke is one and the shortest strokes are tenths and the hundredth parts as much as can be estimated 4. But if the whole Line shall represent a 1000 then the long stroke by the Figure shall represent a hundred and every shorter 10 and every of the shortest strokes is one Integer and a 10th part as near as can be estimated 5. But lastly if the whole Line represent 10000 then
shall be the sine of the Suns declination required for that distance from the next Equinoctial-point by the 1 st Rule abovesaid By the Sector Take 23-31 from the Sines make it a = in the sine of 90 then the = sine of the Suns distance from the next Equinoctial-point shall be the sine of the Suns declination Example as before Rule the 1 st Use XIII The Suns Declination being given to find his true place or distance from ♈ or ♎ the two Equinoctial Points Lay the Thred to the Declination counted in the degrees from 600 and in the Line of the Suns place is his true place required Example When the Suns declination is 12 degrees Northward the dayes increasing then the Sun will be 31 deg and 23 min. from ♈ or 1 deg 23 min. in ♉ his true place required As Sine of 23-31 the Suns greatest declination to Sine of 90 So Sine of 12-00 the Suns present declination to Sine of Suns distance from ♈ or ♎ 31-23 Which by considering the time of the year gives his true place by looking on the Months and Line of Suns place on the Quadrantal-side Take the Sine of the present declination make it a = Sine in the greatest declination laying the Thred to ND and on the degrees the Thred shall give the Suns distance from ♈ or ♎ required Example as before Make Sine of the given Suns declination a = Sine in the Suns greatest declination then = Sine of 90 measured from the Center is the = Sine of the Suns distance from ♈ or ♎ required or count 30 deg for one sign and the Center for the next Equinoctial-point and 90 for the two Tropicks of Cancer and Capricorn ♋ ♑ Use XIV The Suns place or Day of the Month and greatest Declination given to find his Right Ascention from the same Equinoctial Lay the Thred to the day of the Month or place given and in the Line of the Suns Right Ascention you have his Right Ascention in degrees or hours and minutes counting 4 minuts for every degree Example On the 9th of April near night the Sun being then entring ♉ the Suns Right Ascention will be 1 hour 52 min. or 28 degrees of Right Ascention distant from ♈ As the Sine of 90 to the Sine complement of the Suns greatest declination or C.S. of 23-31 counting backwards from 90 which will be at the Sine of 66-29′ So is the Tangent of the Suns distance from the next Equinoctial-point to the Tangent of the Suns Right Ascention from the same Equinoctial-point Take the co-sine of the greatest declination from the Center downwards being the sine of 66-29′ make it a = sine of 90 laying the Thred to ND and note what degree and minuit it cuts for this is fixed to this Proportion Then take the Tangent of the Suns distance from the next Equinoctial-point from the Center at 600 on the degrees toward the End and lay it on the sines from the Center downwards and note the Point where it stayeth for the ND from thence to the Thred shall be the Tangent of the Suns Right Ascention required Note That if the Suns distance from ♈ or ♎ be above 45 degrees then the Tangents on the loose-piece are to be used instead of the Tangents on the moveable-leg Or by Sines only thus Or Take Sine of the present Suns declination make it a = in the Sine of the Suns greatest declination and lay the Thred to ND then take = Co-sine of the Suns greatest declination and make it a = in Co-sine of the Suns present declination and lay the Thred to ND and in the degrees it cuts the Suns Right Ascention required Make Co-sine of 23-31 viz. the right Sine of 66-29 a = sine of 90 then the = Tangent of the Suns distance from ♈ or ♎ is the = Tangent of the Suns Right Ascention from the same Point of ♈ or ♎ as at 30 from ♈ it is 28 degrees or 1 hour and 52 minuts from ♈ neer Use XV. Having the Suns Right Ascention and greatest Declination to find the Angle of the Ecliptick and Meridian As Sine 90 to Sine 23-31 So is the Co-sine of the Suns Right Ascention to the Co-sine of the Angle of the Ecliptick and Meridian Lay the Thred to 23-31 counted on the degrees from the Head then count the Co-sine of the Right Ascention from the Center downward or the Sine from 90 upwards and take the ND from thence to the Thred and measure it from the Center and it shall reach to the Co-sine of the Angle required Example The Right Ascention being 30 degrees or 2 hours the Angle shall be 69-50 Make the right sine of 23-31 a = sine of 90 then the = co-sine of 30 viz. = 60 shall make the sine of 69-50 the Angle of the Ecliptick and Meridian Use XVI Having the Latitude and Declination of the Sun or Stars to find the Suns or Stars Amplitude at rising or Setting Take the Suns declination from the particular Scale of Sines and lay it from 6 in the hour or Azimuth-line and it shall give the Amplitude from South as it is figured or from East or West counting from 90 observing to turn the Compasses the same way from 90 or 6 as the declination is Northward or Southwards Example The Suns declination being 10 degrees Northward the Suns Amplitude or Line is 106-12 from the South or 16-12 from the East-point As co-sine of the Latitude to S. 90 So is S. of the Suns declination to S. of the Amplitude Take the Sine of the Suns declination make it a = in the co-sine of the Latitude and lay the Thred to the nearest distance and on the degrees the Thred shall shew the true Amplitude required Make the right Sine of the Suns declination a = in co-sine latitude then = 90 taken and measured from the Center gives the Amplitude or Line Use XVII Having the same Amplitude and Declination to find the Latitude As S. of the Suns Amplitude to S. the Suns Declination So is S 90 to Co-sine Latitude Take the sine of the Suns declination set one Point in the Sine of the Suns Amplitude lay the Thred to ND and on the degrees it sheweth the complement of the Latitude required Example The Declination being 20 degrees and the Amplitude 33-15 the complement of the Latitude will be 38-28 counting from the Head toward the End Make the right Sine of the Suns Declination a = sine in the Suns Amplitude then the = sine of 90 shall be the co-sine of the Latitude required Use XVIII Having the Latitude and Suns Declination to find his Altitude at East or West commonly called the Vertical-Circle or Azimuth of East or West Take the Suns Declination from the particular Line of Sines set one Point in 90 on the
Azimuth-line and lay the Thred to the ND and on the degrees it sheweth the Altitude required counting from 600 toward the End As S. latitude S. of 90 So S. of Suns declination to S. Suns height at East or West Take the sine of the Suns declination make it a = in the sine of the latitude and lay the Thred to ND and on the degrees it shall shew the Suns Altitude at East and West required Example Declination 10. Latitude 51-32 the Altitude is 12 degrees and 50 minuts As S. of the Suns Declination to = S. of Latitude So is the = S. of 90 to S. of Vertical Altitude Use XIX Having the Latitude and Suns Declination to find the time when the Sun will be due East or West Having gotten the Altitude by the last Rule take it from the particular Sine then lay the Thred to the Suns declination counted on the degrees then setting one Point in the Hour-line so as the other turned about shall but just touch the Thred and the Compass-point shall stay at the hour and minuit of time required As Tangent latitude to Sine 90 So is the Tangent of the Suns declination to Co-sine of the hour Or As sine 90 to Tangent Suns declination So is Co-tangent-latitude to Co-sine of the hour from noon Example Latitude 51-32 declination 10 the Sun will be due East at 6-32 and West at 5-28 Take the Tangent of the Latitude on the loose-piece counting from 60 towards the moveable-leg or else from 600 on the moving-leg or degrees according as the Latitude is above or under 45 degrees and lay it from the Center downwards and note the Point where it ends Then take from the same Tangent the Tangent of the Suns declination and setting one foot in the Point last noted lay the Thred to ND then the = sine of 90 shall be the sine of the hour from 6. Or by the Sines only work thus Take the sine of the Suns declination make it a = in sine of the latitude lay the Thred to ND then take ND from the Co-sine latitude to the Thred then set one foot in the Co-sine of the Suns declination lay the Thred to ND and on the degrees it gives the hour from noon as it is figured or the hour from 6 counting from the head counting 4 minuts for every degree Make the small Tangent of the Latitude if above 45 taken from the Center a = sine of 90 then the Tangent of the Suns declination taken from the same small Tangent and carried Parallely till it stay in like Sines shall be the Sine of the hour from 6. Or as before by Sines only Make sine Declination a = sine Latitude then take = Co-sine Latitude and make it a = Co-sine of the Suns Declination then take = 90 and lay it from the Center it gives the Sine of the hour from 6. Use XX. Having the Latitude and Suns Declination to find the Ascentional Difference or the Suns Rising and Setting and Oblique Ascention Lay the Thred to the Day of the Month or to the Suns Declination or true Place or to his Right Ascention for the Thred being laid to any one of them is then also laid to all the rest then in the Azimuth-line it cuts the Ascentional difference if it you count from 90 or the Suns Rising as you count the morning hours or his Setting counting the afternoon hours The Oblique Ascention is found out for the six Northern signs or Summer half-year by substracting the Suns difference of Ascentions out of the Suns Right Ascention But for the other Winter-half year or six Southern signs it is found by adding the Suns difference of Ascentions to his Right Ascention this sum in Winter and the remainder as above-said in Summer shall be the Suns Oblique Ascention required As Co-tangent Lat. to sine 90 To is the Tangent of the Suns declination to the sine of the Suns Ascentional difference Take the co-tangent latitude from the loose or moveable-piece as it is above or under 45 degrees make it a = in sine 90 lay the Thred to ND then take the Tangent of the Suns declination from the same Tangents and carry it = till it stay in the parts that the other foot turned about will but just touch the Thred which parts shall be the Sine of the Suns Ascentional difference required Or â—hus by Sines only Make the sine of Declination a = Co-sine of the Latitude lay the Thred to ND then take the = sine of Latitude make it a = in Co-sine of the declination and lay the Thred to ND and on the degrees it shall cut the Suns Ascentional-difference required which being turned into time by counting 4 minuts for every degree and added to or taken from 6 gives the Suns Rising in Summer or Winter Make the Co-tangent Latitude a = sine of 90 then take Tangent of the Suns declination and carry it = till it stay in like parts viz. the Sine of the Suns Ascentional difference required Example otherwise As sine 90 to = Tangent 38-28 So is = Tangent of 23-31 the Suns greatest declination to the sine of the Suns greatest Ascentional difference 33 deg and 12 min. Use XXI The Latitude and Suns Declination given to find the Suns Meridian Altitude When the Latitude and Declination is both alike viz. both North or both South then substract the Declination out of the Latitude or the less from the greater and the remainder shall be the complement of the Suns Meridian Altitude But when they be unlike then add them together and the sum shall be the complement of the Meridian Altitude The contrary work serves when the complement of the Latitude and Declination is given to find the Meridian Altitude Lay the Thred to the Declination counted on the degrees from 600 the right way toward the Head for North and toward the End for South declination Then Take the nearest distance from the Center-prick at 12 in the Hour-line to the Thred this distance measured on the Particular-line of Sines shall shew the Suns Meridian-Altitude required Use XXI The Latitude and Hour from the midnight Meridian given to find the Angle of the Suns Position viz. the Angle between the Hour and Azimuth-lines in the Center of the Sun As Sine 90 to Co-sine of the Latitude So is the Sine of the Hour from Midnight to the sine of the Angle of Position Example As Sine 90 to Co-sine Latitude 38-28 So is the Co-sine of the Hour from midnight 120 for which you must use 60 to 32-34 the Angle of Position Take the distance from the Hour to the 90 Azimuth on the Hour-line and measure it in the particular sines and it shall shew the Angle of Position required This holds in the Equinoctial Take Co-sine Latitude make it a = in sine 90 then take
out the = Co-sine of the Hour from the Meridian and it shall be the sine of the Suns Position Make Co-sine Latitude a = sine 90 then = Co-sine of the Hour shall be sine of the Suns Position Note The Angle of the Suns Position may be varied and it is generally the Angle made in the Center of the Sun by his Meridian or Hour-circle being a Circle passing thorow the Pole of the World and the Center of the Sun and any other principal Circle as the Meridian the Horizon or any Azimuth the Anguler-Point being alwayes the Center of the Sun Use XXII The Suns Declination given to find the beginning and end of Twi-light or Day-break Lay the Thred to the Declination on the degrees but counted the contrary way viz. South-declination toward the 〈…〉 North-declination toward the 〈◊〉 then take 18 degrees from the particulâ—◠〈◊〉 Sines for Twi-light or 13 degrees for Day-break or clear light Then carry this distance of 18 for Twi-light or 13 for Day-break along the Line of Hours on that side of the Thred next the End till the other Foot turned about will but just touch the Thred then shall the Point shew the time of Twi-light or Day-break required Example The Suns Declination being 12 degrees North the Twi-light continues till 9 hours 24 minuts or it begins in the morning at 38 minuts after 2 but the Day-break is not till 22 minuts after 3 in the morning or 38 minuts after 8 at night and last no longer To work this for any other place where the Latitude doth vary do thus Find the Hour that answers to 18 degrees of Altitude in as much Declination the contrary way and that shall be the time of Twi-light or at 13 degrees for Day-break according to the Rules in the 26th Use where the way how is largely handled to the 33d Use both wayes generally Use XXIII To find for what Latitude your Instrument is particularly made for Take the nearest distance from the Center on the Head-leg to the Azimuth-line on the moveable-leg this distance measured on the particular Scale of Sines shall shew the Latitude required or the Extent from 0 to 90 on the Azimuth-line shall shew the complement of the Latitude being measured as before Use XXIV Having the Meridian Altitude given to find the time of Sun Rising or Setting true Place or Declination Take the Suns Meridian Altitude from the particular Scale and setting on Point in ☉ on the Azimuth-line lay the Thred to the ND and on the Hour-line it sheweth the time of Rising or Setting and on the degrees the Declination and the rest in their respective Lines Example The Meridian Altitude being 50 the Sun riseth at 5 and sets at 7. Use XXV The Latitude and Declination given to find the Suns height at 6. Lay the Thred to the Day of the Month or Declination then take the ND from the Hour-point of 06 and 6 to the Thred and that distance measured on the particular Scale of Sines shall be the Suns Altitude at 6 in Summer time or his depression under the Horizon in the Winter time As sine of 90 to sine of the Suns Declination So is sine Latitude to sine of the Suns Altitude at 6. Count the Suns declination on the degrees from 90 toward the End and there lay the Thred then the least distance from the sine of the Latitude to the Thred measured from the Center downwards shall be the sine of the Suns Altitude at 6. Make the sine of the Declination a = sine of 90 then the = sine of the Latitude shall be the sine of the Suns height at 6. Example Latitude 51-32 Declination 23-31 the height at 6 is 18 deg 13 min. Use XXVI Having the Latitude the Suns Declination and Altitude to find the Hour of the Day Take the Suns Altitude from the particular Scale of Sines between the Compasses then lay the Thred to the Day of the Month or Declination then carry the Compasses along the Line of Hours between the Thred and the End till the other Point being turned about will but just touch the Thred and then the fixed Point shall shew the true hour and min. required in the Fore or After-noon if you be in doubt which it is then another Observation presently after will determine it Example May 10th at 30 degrees of Altitude the hour will be 32 minuts after 7 in the Morning or 28 minuts after 4 in the Afternoon This Work being somewhat more difficult than the former I shall part it thus 1. First to find the Hour the Sun being in the Equinoctial Take the sine of the Suns Altitude make it a = Co-sine of the Latitude lay the Thred to ND and on the degrees it shall give the Hour from 12 as it is figured counting 15 degrees for an hour or from 6 counting from the Head at 90. Example Latitude 51-30 Altitude 20 the hour is 8 12′ in the forenoon or 3-48′ in the afternoon The same by Artificial Sines Tangents As Co-sine Latitude to sine 90 So is the sine of the Suns Altitude to sine of the hour from 6. Make S. ☉ Altitude a = S. in ☉ Latitude then take out = S. 90 and it shall be the sine of the hour from 6. 2. The Latitude Declination and Altitude given to find the Hour at any time First by the 25th Use find the Suns Altitude or depression at 6 then in Summer-time lay this distance from the Center downwards and in Winter-time lay it upwards from the Center toward the End of the Head-leg and note that Point for that day or degree of Declination for by taking the distance from thence to the Suns Altitude on the General Scale you have added or substracted the Altitude at 6 to or from the present Altitude For by taking the distance from that noted Point over or under the Center to the Suns present Altitude you have in Summer the difference between the Suns present Altitude and his Altitude at 6. And in Winter you have the sum of the present Altitude and the Altitude at 6. This Operation is plainly hinted at in the 4th Chapter and 9th and 10th Section which being understood take the whole Operation in shorter terms thus Count the Suns Declination from 90 toward the end and thereunto lay the Thred the nearest distance from the sine of the Latitude to the Thred is the Suns height or depression at 6 In Winter use the sum of in Summer the difference between the Suns Altitude at 6 and his present Altitude with this distance between your Compasses set one Point in the co-sine of the Latitude lay the Thred to ND then take the ND from 90 to the Thred then set one foot in the Co-sine of the Suns declination and lay the Thred to ND and on the degrees it gives the hour required from 6 counting from 90
which shall be the versed Sine of the Hour from the North Meridian or mid-night Or If you take the distance from the difference to the Co-altitude and carry that = till it stay in like sines it shall be the hour from noon counting the Center 12 at noon the middle at 90 the two sixes and 180 at the end for 12 at night Use XXXV Having the Latitude the Suns Altitude and Distance from the Elevated Pole to find his true Azimuth from South or North by Natural versed Sines First Of the Co-altitude and Co-latitude find the sum and difference by Addition and Substraction Secondly Count the sum and difference from the Center and take the distance between them with Compasses on the versed Sines Thirdly Make it a = versed sine of 180 and so keep the Sector Fourthly Take the distance between the sum and the Suns distance from the Pole counting the Center the Elevated Pole and 90 the Equinoctial and carry it = till it stay in like parts which shall be the Azimuth from South Or If you take the distance from the difference to the Suns distance from the Pole and carry it as before it shall stay at the versed sine of the Azimuth from the North part of the Horizon These five general wayes of finding the Hour and Azimuth are not all needful to be learned by every one but to delight the ingenious and to hold forth the usefulness of the Instrument and to supply defects that at some times may happen by Excursions and as a four-fold Testimony to shew the harmony in several wayes of Operation the first particular way and this last by versed Sines being most easie and comprehensive of any other Use XXXVI To work the last without the Line of versed Sines Note That if for want of room the versed Sines be set but on one Leg then it is to be laid at the nearest distance instead of like parts after the manner of using the Thred on the General Quadrant Also If you have it not at all then the Azimuth-line for the particular Latitude and if that be too large the little Line of Sines beyond the Center will supply this defect very well thus First Turn the Radius or whole length of that Line of Sines two times from the Center downwards which in Sea-Instruments will most conveniently stay at 30 on the large Line of Sines or general Scale as was hinted in the 28th Use being just 4 times as much one as the other For a Point representing 180 of versed Sines to set the Compasses in when you lay the Thred to ND and to take any versed sine above 90 degrees this being premised the Operation is thus Example Lat. 51-32 ☉ Dist. from Pole 80 ☉ Alt. 25 to find the Hour The sum of Co-lat 38-28 and 80 is 118-28 And the difference is 41-32 Now in regard the sum is above 90 count the Center 90 10 on the smaller Sines 100 and 20 on the same Sines 110 and 28 deg 28 min 118 deg 28 min. turn this distance the other way from the Center downwards and note that place for the Point representing the sum on the versed sines Then The Extent between this sum and the difference 41-32 as the smaller figures reckon it being taken between your Compasses set one Point in 180 the Point first found and lay the Thred to ND and there keep it or observe where it cuts then taking the distance between the versed sine of the difference counted as the small figures are reckoned and the sine of the Suns Altitude 25 as the greater figures are reckoned from the Center toward the End and carrying this Extent parallelly along the greater Line of sines till the other Point will but just touch the Thred at ND Then I say the measure from that Point to the Center measured on the small sines as versed sines shall be the versed sine of the Hour required viz. 62 from South or 7 hours 52 minuts from mid-night This Rule or Use is longer far in wording than the Operation need be in working for if you shall approve of this way the adding of two brass Center-pins will shew you the two Points most used very readily and the Thred is sooner laid than the Legs can be opened or shut and the Instrument keeps its Trianguler form as it is in during the time of Observation Use XXXVII Having the Latitude Suns Declination and Hour to find his Altitude This Problem being not of such use as the contrary viz. having the Altitude to 〈◊〉 the hour it shall suffice to hint only two â—ayes the most convenient And First by the Particular Quadrant Lay the Thred to the Day or Declination then the ND from the Hour to the Thred measured in the particular Scale of Altitudes shall shew the Suns Altitude required Secondly by the versed Sines 1. First of the Co-latitude and Suns distance from the Pole find the sum and difference 2. Take the distance between them and make it a = versed sine of 180 by setting the Sector or laying the Thred to ND 3. Then take the = versed sine of the Hour and lay it latterally from the sum and it shall give the complement of the Altitude required This work is the same both by the Sector or General Quadrant as is shewed in Use â—he 36th and is nothing else but a backward working but the Altitude at any Azimuth is not so to be done To do the same by the Natural-Sines First having the Latitude and the Suns Declination find the Suns Altitude or Depression at 6 and note the Point either below or above or in the Center as is largely shewed in Use the 26th where the Altitude is given to find the hour in any Latitude Then Lay the Thred to the Hour counted in the degrees either from 12 or 6 Then Take the ND from the Co-sine of the Suns Declination and make it a = in the sine of 90 laying the Thred to the ND then the ND from the sine complement of the Latitude to the Thred shall reach from the noted Point for the Suns Altitude or Depression at 6 to the Suns Altitude required Example Latitude 51-32 Declination 23-71 a 8 or 4 viz. 2 hours from 6 Southwards the Altitude will be found to be 36-42′ 1. For the Altitude at 6 at any time of the year say As the sine of 90 to sine of the Latitude So is the sine of the Suns Declination to the sine of the Suns Altitude at 6. 2. For the Suns Altitude at any hour or quarter in Aries or Libra the Equinoctial As sine 90 to Co-sine Latitude So is the sine of the Suns distance from 6 in degrees to sine of the Suns Altitude 3. For the Suns Altitude at all other hours or times of the year As sine 90 to Co-tangent Latitude So is sine of the Suns distance from 6 to the Tangent of a 4th Ark in the Tangents Which
Extent shall reach in the same Line from the true hour of the Night to the Stars hour from the Meridian then laying the Thred to the Stars Declination the ND from the Stars hour in the Line of hours to the Thred measured on the particular Scale of Altitudes gives the Stars Altitude then by his Declination and Altitude you may soon find his Azimuth by Use 27. And if the Instrument be neatly fixed to a Foot to set North and South and turn to any Azimuth and Altitude you may find any Star at any time convenient and visible Use XLVII The Altitude and Azimuth of any Star being given to find his Declination Lay the Thred to the Altitude on the degrees counted from 600 toward the end then setting one Point on the Stars Azimuth counted in the Azimuth Line and take the ND from thence to the Thred which distance measured from the beginning of the particular Scale of Altitudes shall give the Declination If the Compasses stand on the right-side of the Thred then the Declination is North if on the left it is South according as you work for the Suns Azimuth in a particular Latitude Use XLVIII The Altitude and Declination of any Star with the Right Ascention of the Sun and the true Hour of the Night given to find the Right Ascention of that Star First by the 43d Use find the Stars Hour viz. How many hours and minuts it wants of coming to or is past the Meridian then the Extent of the Compasses on the Line of 24 hours on the Head-leg from the Stars hour to the true hour shall reach the same way from the Suns Right Ascention to the Stars Right Ascention on the Line of twice 12 or 24 hours Use XLIX To find when any Fixed-Star cometh to South by the Line of twice 12 or 24 hours In Use 42 Section 4 you have the way by Substraction with its Cautions But by the Line of twice 12 or 24 hours work thus Count the Suns Right Ascention on that Line and take the distance from thence to the next 12 backward viz. that at ♈ at the beginning of the Line when the Suns Right Ascention is under 12 hours or to the next 12 in the middle of the Rule at ♎ when the Suns Right Ascention is above 12 hours which is nothing but a rejecting 12 for more conveniency Then The same Extent laid the same way from the Stars Right Ascention shall reach to the Stars coming to South Or The Extent from the Sun to the Stars Right Ascention shall reach the same way from 12 to the Stars coming to South Example for the Lyons-Heart August 20. The Suns Right Ascention the 20th of August is 10 hours 36 minuts the Right Ascention of the Lions-Heart is 9 hours and 50 min. Therefore The Extent from 10 hours 35 min. to the beginning shall reach the same way from 9 hours 50 min. by borrowing 12 hours because the Suns Right Ascention is more than the Stars to 11 hours 13 min. of the next day viz. at a quarter past 11 or at 11 hours and 13 min. the same day where you may observe that the remainder being above 12 if you add 24 hours the time of Southing is between mid-night and mid-day next following Use L. To find what two dayes in the year are of equal length and the Suns Rising and Setting Lay the Thred on any one day in the upper Line of Months and Dayes and at the same time the Thred cuts in the lower-Line of Months the day that is answerable to it in length rising setting and declination and other requisites Example The 1st of April and the 21 of August are dayes of equal length and the Suns Rising and Setting is the same on both those dayes only in the upper-Line the dayes are increasing in length and in the lower-Line they are decreasing Use LI. To find how many degrees the Sun is under the Horizon at any Hour the Declination and Hour being given Count the Suns Declination on the degrees the contrary way viz. for North Declination count from 600 toward the end and count for Southern Declination toward the Head and thereunto lay the Thred then take the nearest distance from the hour given to the Thred this distance measured in the particular Scale of Altitudes shall shew the Suns Depression under the Horizon at that hour Example January the 10th at 8 at Night how many degrees is the Sun under the Horizon On that Day and Hour the Suns Declination is about 20 degrees South then if I lay the Thred to 20 degrees of Declination North and take the nearest distance from 8 to the Thred that distance I say measured in the particular Scale gives 34 degrees and 9 min. for the Suns Depression under the Horizon of 8 afternoon To do this in other Latitudes you are to find the Suns Altitude at 8 in Northern Declination by Use 37. CHAP. XVII The use of the Trianguler Quadrant in finding of Heights and Distances accessable or inaccessable Use I. To find an Altitude at one Station FIrst The Trianguler Quadrant being rectified and fixed to a Ball and Socket and three-legged-staff being necessary in these Operations to perform them exactly especially for Distances look up to the object as you would to a Star and observe what degree and minut the Thred cuts and set it down Also observe the place where you stand at the time of Observation and the distance from your Eye to the ground and the place on the object that is level with your eye also as the playing of the Thred and Plummet will plainly shew Also you must have the measure from the place where you stood observing to the Point exactly right under the object whose height you would have in Feet Yards Perch or what you please to Integers and Fractions in Decimals if it may be Also Note That in all Right-Angle-Triangles one Acute Angle is alwayes the complement of the other so that observing or finding one by Observation by consequence you have the other by taking that from 90. These things being premised the Operation followes by the Artificial Numbers Sines and Tangents and also by the Natural Note also by the way That in regard the complement of the Angle observed is frequently used if you count the degrees the contrary way that is to say from the Head you shall have the complement required as hath been oftentimes hinted before Then As the sine of the Angle opposite to the measured side is to the measured side counted on the Numbers So is the sine of the Angle found to the Altitude or Height required on Numbers Example at one station Standing at C I look up to B the object whose Height is required and I find the Thred to fall on 41 degrees and 45 minuts but if you count from the Head it is 48-15 the complement thereof as in the Figure you see Also the measure from C to A is found to be
what ibid. Circles of Position what ibid. Of Terms in Astronomy What a Sphear is Page 50 Of ten Points and ten Circles of the Sphear Page 51 The 2 Poles of the World or Equinoctial ibid. The 2 Poles of the Zodiack Page 52 The 2 Equinoctial-points ibid. The 2 Solstitial-points Page 53 The Zenith and Nadir Page 54 The Horizon the Meridian the Equinoctial the Zodiack the 2 Colures the 2 Tropicks and 2 Polar Circles Page 55 56 58 Hours Azimuths Almicanters Declination Latitude Longitude Right Ascention Page 59 60 Oblique Ascention Difference of Ascentions Amplitude Circles and Angles of Position what they are Page 61 62 To rectifie the Trianguler Quadrant Page 63 To observe or find the Suns Altitude Page 64 To try if any thing be level or upright Page 66 To find what Angle the Sector stands at at any opening or to set the Sector to any Angle required Page 67 68 The day of the Month given to find the Suns Declination true Place Right Ascention or Rising and Setting by inspection only Page 71 To find the Suns Amplitude and difference of Ascentions and Oblique Ascention Page 73 To find the Hour of the Day Page 74 To find the Suns Azimuth Page 75 The use of the Line of Numbers and the use of the Line of Lines both on the Trianguler Quadrant and Sector one after another in most Examples To multiply one Number by another Page 78 A help to Multiply truly Page 85 A crabbed Question of Multiplication Page 90 Precepts of Reduction Page 94 To divide one Number by another Page 95 A Caution in Division Page 97 To 2 Lines or Numbers given to find a 3d in Geometrical proportion Page 98 Any one side of a Figure being given to find all the rest or to find a proportion between two or more Lines or Numbers Page 99 To lay down any number of parts on a Line to any Radius Page 100 To divide a line into any number of parts Page 102 To find a Geometrical mean proportion between two Lines or Numbers three wayes Page 104 To make a Square equal to an Oblong Page 107 Or to a Triangle ibid. To find a Proportion between unlike Superficies Page 108 To make one Superficies like another Superficies and equal to a third Page 109 The Diameter and Content of a Circle being given to find the Content of another Circle by having his Diameter Page 111 To find the Square-root of a Number ibid. To find the Cube-root of a Number Page 113 To find two mean Proportionals between two Lines or Numbers given Page 116 The Diameter and Content of a Globe being given to find the Content of another Globe whose Diameter also is given Page 118 The proportion between the Weights and Magnitudes of Metals Page 119 The Weight and Magnitude of a body of one kind of Metal being given to find the Magnitude of a body of another Metal of equal weight Page 121 The magnitudes of two bodies of several Metals having the weight of one given to find the weight of the other Page 122 The weight and magnitude of one body of any Metal being given and another body like unto the former is to be made of any other Metal to find the diameters or magnitudes of it Page 123 To divide a Line or Number by extream and mean proportion Page 124 Three Lines or Numbers given to find a fourth in Geometrical proportion Page 128 The nature reason of the Golden Rule Page 129 The Rule of Three inversed with several Cautions and Examples Page 132 The double and compound Rule of Three Direct and Reverse with Examples Page 139 The Rule of Fellowship with Examples Page 148 The use of the Line of Numbers in Superficial measure and the parts on the Rule Page 154 The breadth given in Foot-measure to find the length of one Foot Page 156 The bredth given in Inches to find how much in length makes one Foot ibid. The bredth given to find how much is in a Foot-long Page 157 Having the length and bredth given in Foot-measure to find the Content in Feet ibid. Having the bredth given in Inches and length in Feet to find the Content in Feet Page 158 Having the length bredth given in Inches to find the content in superficial Inches Page 160 Having the length bredth given in Inches to find the Content in Feet superficial Page 161 The length and bredth of an Oblong given to find the side of a Square equal to it Page 163 The Diameter of a Circle given to find the Circumference Square equal Square inscribed and Content Page 164 The Content of a Circle given to find the Diameter or Circumference Page 166 167 Certain Rules to measure several figures Page 108 A Segment of a Circle given to find the true Diameter and Area thereof Page 169 A Table to divide the Line of Segments Page 170 The use of it in part Page 171 The measuring of Triangles Tapeziaes Romboides Poligons and Ovals Page 172 173 A Table of the Proportion between the Sides and Area's of regular Poligons and the use thereof for any other Page 174 175 To make an Oval equal to a Circle and the contrary two wayes Page 175 176 The length and bredth of any Oblong Superficies given in Feet to find the Content in Yards Page 177 The length and bredth given in feet and parts to find the Content in Rods Page 179 The nearest way to measure a party Wall Page 180 To multiply and reduce any length bredth or thickness of a Wall to one Brick and a half at one Operation Page 183 Examples at six several thicknesses Page 184 To find the Gage-points for this reducing Page 185 At one opening of the Compasses to find how many Rods Quarters and Feet in any sum under 10 Rods Page 186 The usual and readiest equal wayes to measure Tileing and Chimnyes Page 187 Of Plaisterers-work or Painters-work Page 188 Of particulars of work usually mentioned in a Carpenters-Bill with Cautions Page 189 190 At any bredth of a House to find the Rafters and Hip-rafters length and Angles by the Line of Numbers readily Page 191 The price of one Foot being given to find the price of a Rod or a Square of Brick-work or Flooring by inspection Page 193 At any length of a Land given to find how much in bredth makes one Acre Page 194 A useful Table in measuring Land and the use thereof in several Examples Page 196 197 The length and bredth given in Perches to find the Content in Squares Perches Poles or Rods Page 200 The length and bredth in Perches to find the Content in Acres ibid. The length and bredth given in Chains to find the content in square Acres Quarters and Links Page 201 To measure a Triangle at once without halfing the Base or Area ibid. To reduce Statute-measure or Acres to Customary and the contrary ibid. A Table to make Scales to do it by measuring or inspection with Examples Page