Selected quad for the lemma: world_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
world_n	circle_n	point_n	pole_n	2,440	5	11.6304	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A29761	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 John Brown, philomath.	Brown, John, philomath.	1671 (1671)	Wing B5041; ESTC R15524	175,812	507

two Lines drawn equidistant from and parallel to the Hour-line of 12. The Contingent or Touch-line in this way of Dialling with Centers is a Line drawn parallel to the Hour-line of 6 but in those without Centers it is drawn alwayes perpendiculer to the Substile and so may it be also if you please in those with Centers also The Vertical Line on the Plain is the same with the Perpendiculer-line on the Plain being perpendiculer to the Horizontal-line By the word Nodus is meant a Knot or Ball on the Axis or Stile of the Dial to make a black-shaddow on the Dial to trace out the Suns motion in the Heavens or sometimes an open or hollow-place in the Stile to leave a light-place to do the same office But by Apex is meant the same thing when the Top-end or Point of an upright Stile shall shew the Hour and Suns place as the Spot doth in Celing-Dials where the Hours and Quarters are all of one length and distinguished by their tullours or greatness only The Perpendiculer height of the Stile is nothing else but the nearest distance from the Nodus or Apex to the Plain The Foot of the Stile is properly right under the Nodus or Apex at the nearest distance The Vertical-Point is a Point only used in Recliners and Incliners being a Point right over or under the Apex and yet in the Meridian being let fall from the Zenith by or through the Apex or Nodus to the Plain in the Meridian-line The Axis of the Horizon is only the measure from the Apex to the Vertical-point last spoken to being the Secant of the complement of the Reclination to the Radius of the Perpendiculer height of the Stile Erect is when Plains are upright as all Walls are intended to be Direct is when the Dial-plain beholdeth one of the Four Cardinal Points of the Horizon as South or North East or West that is to say when the Pole of the Plain being 90 degrees every way from the Plain doth lie precisely in one of those Four Cardinal Azimuths Which in an Erect and Direct-Plain will be in the Horizon Declining and Reclining or Inclining-Plaines are as the upper or under-side of Roofs at any Oblique Scituation from the Cardinal Points of the Horizon Oblique is only a wry slanting crooked contrary to direct right plain flat or perpendiculer and applied variously as to the Sphear to Triangles to Dial-plains to Discourse and Conversation Circles of Position or rather Semi-circles making 12 Houses are Circles whose Pole or Meeting-point is in the Meridian and Horizon of every Country dividing the Aequinoctial into 12 equal parts being then called Houses when used in Astrologie and some times drawn on Sun-Dials But when they are used in Astronomy they require a more near account as to degrees and minutes Of certain Terms in Astronomy and Spherical Definitions of Points and Lines in the Sphear NOt to be curious in this matter a Sphear may be understood to be a united Spherical Superficies or round Body contained under one Surface in the middle whereof is a Point or Center from whence all Lines drawn to the Circumference are equal Or you may conceive a Sphear to be an Instrument consisting of several Rings or Circles whereby the sensible motion of the Heavenly Bodies are conveniently represented For the better Explanation whereof Astronomers have contrived thereon viz. on the Sphear ten imaginary Points and ten Circles which are usually drawn on Globes and Sphears besides others not usually drawn but apprehended in the fancy for Demonstrations-sake in Spherical Conclusions The ten Points are the two Poles of the World the two Poles of the Zodiack the two Aequinoctial Points the two Solstitial Points the Zenith and Nadir The ten Circles are The Horizon the Meridian the Aequinoctial the Zodiack the two Colures viz. that of the Equinoxes and that of the Solstices the Tropick of Cancer and the Tropick of Capricorn and the two Polar-Circles viz. The Artick or North the Antartick or South polar-Polar-Circle The first six are great Circles cutting the Sphear into two equal parts And the four last are lesser Circles dividing the Sphear unequally All which Points and Circles shall be represented by the Figure of the Analemma from whence the Trianguler Quadrant is derived as a general Instrument and also by the Horizontal projection of the Sphear fitted for London being better for the fancy to apprehend the Mystery of Dialling one thing mainly intended in this Discourse Of the 10 Points in the Sphear THe two Poles of the World are the two Points P and P in the Analemma being directly opposite one to another about which two Points the whole frame of the Heavens moveth from East to West one of which Poles may alwayes be seen by us called the Artick or North-Pole represented in the particular Scheam by the Point P. The other being not seen is not represented in the particular Scheam but the Line PEP in the general Scheam drawn from Pole to Pole is called the Axis or Axeltree of the World because the whole Sphear appears to move round about it The Poles of the Zodiack are two Points diametrically opposite also upon which Points the Heavens move slowly from West to East represented by the two Points I and K 23 degrees and 31 minutes distant from the two former Poles in the Analemma and by the Point PZ in the Horizontal projection but the other Pole of the Zodiack cannot be represented in that particular Scheam The Equinoctial Points are the Points of Aries and Libra to which two Points when the Sun cometh along the Ecliptick it maketh the Dayes and Nights equal in all places at Aries March 10th or 11th to Libra about the 13th of September where the Spring and Autumn begins being represented in the Analemma by the Point ● and in the particular Scheam by the Points E and W. The two Solsticial Points are represented one by the Point ♋ and the other by the Point ♑ in both Scheams to which Points when the Sun cometh along the Ecliptick it makes the Dayes in Cancer ♋ longest in Capricorn ♑ shortest ♋ being about the 11th of Iune and ♑ about the 11th of December The Zenith is an imaginary Point right over our heads being every way 90 degrees distant form the Horizon in which Point all Azimuth Lines do meet represented by the Points Z in both Scheams The Nadir is an imaginary Point under our feet directly opposite to the Zenith represented by the Point N in the Analemma but not in the particular Scheam because it is not seen at any time Of the Circles of the Sphear THe Horizon is twofold viz. Rational and Sensible The Rational Horizon is an imaginary great Circle of the Sphear every where 90 degrees distant from the Zenith and Nadir Points cutting or dividing the whole Sphear into two equal parts

the one called The upper or visible Hemisphear the other the lower or invinsible Hemisphear This Rational Horizon is distinguished also into Right Oblique and Parallel-Horizon 1. The Right Horizon is when the two Poles of the World lie in the Horizon and the Equinoctial at Right Angles to it which Horizon is peculiar to those that live under the Equinoctial who have their Dayes and Nights alwayes equal and all the Stars to Rise and Set and the Sun to pass twice in a year by their Zenith-point thereby making two Winters and two Summers Their Winters being in Iune and December and their Summers in March and September 2. The Oblique Horizon is when one Pole-point is visible and the other not having E●evation above and depression below the North or South part of the Horizon according to the Latitude of the place in which Horizon when the Sun cometh to the Equinoctial the Dayes and Nights are only then equal and the nearer the Sun comes to the visible Pole the Dayes are the longer and the contrary also some Stars never set and some never rise in that Horizon And all Horizons but two are in a strict sense Oblique-Horizons viz. The Right Horizon already spoken to And The Parallel Horizon is that Horizon which hath the Equinoctial for its Horizon and one of the Pole-points for its Zenith peculiar only to those Inhabitants under the Pole if any be there In which Horizon ona half of the Sphear doth only alwayes appear and the other half alwayes is hid and the Sun for one half year doth go round about like a Skrew making it continual Day and the other half year is continual Night and cold enough which Circle in the Analemma is represented by the Line HES but in the particular Scheam by the Circle NESW The Meridian is a great Circle which passeth through the two Pole-points the Zenith and Nadir and the North and South-points of the Horizon and is called Meridian becuse when the Sun or Stars cometh to that Circle it maketh Mid-day or Mid-night which is twice in every 24 hours Also all places North and South have the same Meridian but places that lie Eastward or Westwards have several Meridians Also when the Sun or Stars come to the South or North-part of the Meridian their Altitudes are then highest and lowest And the difference of Meridians is the difference of Longitudes of Places noted by the Circle ZHNS in the Analemma and NZ ♋ S in Horizontal-projection The Equinoctial is a great Circle every where 90 degrees distant from the two Poles of the World dividing the Sphear into two halfs called the North and South Hemisphear and is called also the Aequator because when the Sun passeth by it twice a year it makes the day and nights equal in all places noted by WAEE and AEEAE in both The Zodiack or Signifer is another great Circle that divides the Sphear and Equinoctial into two equal parts whose Poles are the Poles of the Zodiack being 90 degrees from it and it inter-sects the Equinoctial in the two Points of Aries and Libra and one part of it doth decline Northward and the other Southward 23 degrees 31 minutes as the Poles of the Zodiack decline from the North and South-Poles of the World The breadth of this Zodiack or Girdle is counted 14 or 16 degrees to allow for the wandring of Luna Mars and Venus the middle of which breadth is the Ecliptick-Line because all Eclipses are in or very near in this Line And this Circle is divided into 12 Signs and each Sign into 30 degrees according to their Names and Characters ♈ Aries ♉ Taurus ♊ Gemini ♋ Cancer ♌ Leo ♍ Virgo ♎ Libra ♏ Scorpio ♐ Sagittarius ♑ Capricornius ♒ Aquarius ♓ Pisces 6 being Northern and the 6 latter Southern The two Colures are only two Meridians or great Circles crossing one another at Right Angles the one Colure passing through the Poles of the World and the Points of Aries and Libra there cutting the Equinoctial and Ecliptick And the other Colure passeth by the Poles of the World also and cuts the Ecliptick in ♋ and ♑ making the Four Seasons of the year that is the equal Dayes called the Equinoctial-Colure and the unequal Dayes in Iune and December called the Solsticial-Colures represented in the Analemma by ZP ♋ NS and PEP and in the particular Scheam by WPE and NPS the Solsticial-Colure The lesser Circles are the Tropicks of ♋ and ♑ being the Lines of the Suns motion in the longest and shortest dayes noted in the Scheams by ♋ ☉ ♋ and 6 ♋ E and ♑ ♑ and W ♑ 6 to which two Circles when the Sun cometh it is on the 11th of Iune and the 11th of December making the Summer and Winter Solstice polar- The Polar Circles are two Circles drawn about the Poles of the World as far off as the Poles of the Zodiack are viz. 23 degrees 31 minutes That about the North-Pole is called the Artick and that about the South the Antartick being opposite thereunto shewed in the Analemma by II and KK and by the small Circle about P in the particular Scheam Of the other Circle imagined but not described on Sphears or Globes 1. HOurs are great Circles passing through the two Poles and cutting the Equinoctial in 24 equal parts as the Lines P1 P2 P3 c. in the Particular and P ☉ H in the Analemma such also are degrees of Longitude and Meridians the Meridian being the hour of 12. 2. Azimuths are great Circles passing through or meeting in the Zenith and Nadir-points numbred and counted on the Horizon from the Four Cardinal Points of North and South East and West according to Four 90ties or 180 degrees or according to the 32 Rombs or Points of the Compass as Z ☉ A and ZE the Azimuth of East and West being called the prime Virtical viz. SE WZ 3. Almicanters or Circles of Altitude are lesser Circles all parallel to the Horizon counted on any Azimuth from the Horizon to the Zenith to measure the Altitude of the Sun Moon or Stars above the Horizon being the Portion of some Azimuth between the Center of the Sun or Star and the Horizon commonly called its Altitude above the Horizon showed by A ☉ in the Analemma and SAE in the particular Scheam 4. Parallels of Declination are parallels to the Equinoctial as the Almicanters were parallel to the Horizon as ♋ ☉ ♋ the greatest Declination or Circle of ♋ These parallels have the 2 Poles of the World for their Centers and in respect of the Sun or Stars are called degrees of Declination but in respect of the Earth degrees of Latitude being the Arch on the Meridian of any place between the Pole and Horizon as 4 ♋ 4 in the Particular and HP in the Analemma 5. Parallels of Latitude in respect of the Stars are Lines drawn parallel to the Ecliptick as the

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-Equinoctial-points ibid. The 2 solstitial-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

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-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 particul●● 〈◊〉 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

Distance Let CA be the Altitude given and AB the distance required Then I standing at C observe the Angle CAB by setting the end of the Head-leg to my eye and the Head-end downwards and set down as the Thread cuts numbring both wayes for the Angle at C and at B his complement Then say As the Angle at B 30 deg 40 minutes counted on the Sines to 105 the height of the Tower So is 59 deg 20 min. the Angle at C on the Sines to 176 the distance required on the Numbers Also note by the way That if you take an Altitude at two stations as suppose at E and at B if the Angle observed at B be found to be the half of the Angle at E as here in Figure VIII the Angle at E being 61-20 and the Angle at B 30-40 the just half thereof then I say that the distance between the two stations is equal to the Hypothenusa EC at the first station viz. EB is equal to EC which being observed say As the sine of 90 to 120 on the Numbers So is 61-20 on the Sines to 105 the height required on the Numbers A further proof hereof take in this following Figure IX Let AB be a breadth of a Wall or Fort not to be approached unto then by the degrees on the in-side of the loose-piece to find that breadth one way is thus Put two Pins into the two holes in the Head and Moving-leg or set the sights there in large Instruments then move nearer or further from the objects till your eye fixed at the rectifying Point can but just see the marks A and B by the two Pins in each Leg which will only be at the mark C at an Angle of 60 degrees for so the Rule is made to that Angle then the Instrument being still fixed at C look backward in a right Line from the middle of the loose-piece and rectifying Point toward D putting up a mark either in or over or beyond the Point D and also be sure to leave a mark at C the first place of observation Then remove the sights to 15 degrees the half of 30 counting from the middle and go back in a right Line from C toward D till you can just see the marks by the two sights set at 15 degrees each way for then I say that the measure between the two stations C and D shall be exactly equal both to AB the breadth required and also to CB or CA the Hypothenusaes then having the sides CB and CD and the Angles BCE and CBE and BDC it is easie to find all the other Sides and Angles by the Rules before rehearsed by the Lines of Artificial Nmmbers and Sines For As the Sine of 15 degrees the Angle at D viz. BDC to 108 on the Numbers So also is the Angle at B viz. DBC 15 to 108 on the Sines and Numbers So also is the Sine of 150 the Angle at C viz. DCB to BD 208 ½ on the Numbers Note also That if the Angles of 60 and 30 be inconvenient then you may make use of 52 and 26 or 48 and 24 or 40 and 20 or any other and the half thereof and then the measured distance and the Hypothenusa BC at the nearest station will alwayes be equal but not equal to the breadth at any other Angle except 30 and 60 as in the Figure But having the Angles and those Sides you may soon find all the others by the Artificial Numbers Sines and Tangents by the former directions The End of the First Part. The Table or Index of the things contained in this Book TRianguler Quadrant why so called Page 2 The Lines on the ou●ter-Edg N. T. S. VS Page 2 The Line on the inner-edge I. F. 112 Page 3 The Lines on the Sector-side L.S.T. Sec. Page 3 Lesser Sines Tangents and Secants Page 5 The Lines on the Quadrant-side Page 6 The 180 degrees of a semi-Circle variously accounted as use and occasion requires Page 7 60 Degrees on the Loose-piece as a fore-Staff for Sea-Observation Page 7 The Line of right Ascentions Page 8 The Line of the Suns true Place ibid. The Months and Dayes ibid. The Hour and Azimuth-line for a Particuler Latitude Page 9 Natural versed Sines ibid. Lines and Sines or the general Scale of Altitudes for all Latitudes Page 10 The particular Scale of Altitudes or Sines for one Latitude only Page 11 The Degreees of a whole Circle 12 Signs 12 Inches or 24 Hours and Moons Age ibid. The Appurtenances to this Instrument ibid. Numeration on Decimal-lines Page 12 Three Examples thereof Page 13 Numeration on Sexagenary Circular-lines with Examples thereof Page 17 How Right Sines Versed Sines and Chords are counted on the Rule Page 20 Of a Circle Diameter Chord Right Sine Sine Complement or Co-sine Versed Sine Tangent Secant what it is Page 23 24 Two good Notes or Observations Page 25 Of the division of a Circle ibid. What a Radius is Page 26 What an Angle a Triangle Acute Right or Obtuse Plain or spherical Angle is Page 26 27 Parallel-lines and Perpendiculer-lines what they are ibid. The usual Names of Triangles ibid. Of four sided Figures and many sided Page 28 Terms in Arithmetick as Multiplicator Product Quotient c. what they mean Page 29 Geometrical Propositions Page 31 To draw a right Line ibid. To raise a Perpendiculer on any line ibid. To let fall a Perpendiculer any where Page 33 To draw Parallel-lines Page 34 To make one Angle equal to another Page 35 To divide a Line into any number of parts ib. To bring any 3 Points into a Circle Page 36 To cut any two Points in a Circle and the Circle into two equal parts Page 37 A Segment of a Circle given to find the Center and Diameter Page 38 A Segment of a Circle given to find the length of the Arch Page 39 To draw a Helical-line and to find the Centers of the Splayes of Eliptical arches and Key-stones Page 41 To draw an Oval ibid. Explanation of Terms particularly belonging to this Instrument Radius how taken Page 41 Right Sines how taken and counted Page 42 Tangents Secants and Chords how taken ib. Sine complement or co-sine Tangent complement or co-tangent how taken and counted on this Instrument ibid. Latteral Sine and Tangent Page 43 Parallel Sine and Tangent ibid. Nearest Distance what it means ibid. Addition on Lines ibid. Substraction on Lines Page 44 Of Terms used in Dialling Plain and Pole of the Plain Page 45 Declination Reclination and Inclination of a Plain what it is Page 46 What the Perpendiculer-line and Horizontal-line of a Plain are ibid. Meridian-line Substile-line and Stile-line Angle of 12 and 6 and the Inclination of Meridians what they are Page 47 Parallels and Contingent-lines what Page 48 Vertical-line and Point what ibid. Nodus Apex and foot of the Stile what ibid. Axis of the Horizon what Page 49 Erect Direct what ibid. Declining Reclining or Inclining