The triangular quadrant, or, The quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as Davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by John Brown ...
         Brown, John, philomath.
      
       
         
           1662
        
      
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         ocm 13117453
         97765
         
           
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             The triangular quadrant, or, The quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as Davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by John Brown ...
             Brown, John, philomath.
          
           [2], 24, [1] p. : ill.
           
             To be sold at [his, i.e. Brown's] house, or at Hen. Sutton's ...,
             [London] :
             1662.
          
           
             Added illustrated t.p.
             Place of publication suggested by Wing.
             Reproduction of original in the British Library.
          
        
      
    
     
       
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         eng
      
       
         
           Quadrant.
           Dialing.
           Mathematical instruments.
        
      
    
     
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           THE
           TRIANGULAR
           QUADRANT
           :
           OR
           
             The
             QUADRANT
             on
             a
             SECTOR
             .
          
        
         
           Being
           a
           general
           Instrument
           
             For
             Land
             or
             Sea
             Observations
             .
          
        
         
           Performing
           all
           the
           Uses
           of
           the
           ordinary
           Sea
           Instruments
           ;
           as
           
             Davis
             Quadrant
             ,
             Forestaff
             ,
             Crossstaff
             ,
             Bow
             ,
          
           With
           more
           ease
           ,
           profitableness
           ,
           and
           conveniency
           ,
           and
           as
           much
           exactness
           as
           any
           or
           all
           of
           them
           .
        
         
           
             Moreover
             ,
          
           It
           may
           be
           made
           a
           particular
           ,
           and
           a
           general
           
             Quadrant
          
           for
           all
           latitudes
           ,
           and
           have
           the
           
             Sector
          
           lines
           also
           .
        
         
           To
           which
           is
           added
           a
           
             Rectifying
             Table
             ,
          
           to
           find
           the
           Suns
           true
           Declination
           to
           a
           minute
           or
           two
           ,
           any
           day
           or
           hour
           of
           the
           4
           years
           :
           Whereby
           to
           find
           the
           latitude
           of
           a
           place
           by
           a
           
             Meridian
             ,
          
           or
           any
           two
           other
           altitudes
           of
           the
           Sun
           or
           Stars
           .
        
         
           First
           thus
           Contrived
           and
           made
           by
           
             Iohn
             Browne
          
           at
           the
           
             Sphere
          
           and
           
             Dial
          
           in
           the
           
             Minories
             ,
          
           and
           to
           be
           sold
           at
           〈◊〉
           house
           ,
           or
           at
           
             Hen.
          
           Sutton's
           in
           
             Thredneedle-street
          
           behind
           the
           
             Exchange
             .
          
           1662.
           
        
      
       
         
         
           triangular quadrant
        
         
           THE
           TRIANGULAR
           QUADRANT
           :
           Being
           a
           
             GENERAL
             INSTRUMENT
          
           for
           Observations
           at
           
             Land
          
           or
           
             Sea
             ,
          
           performing
           all
           the
           Uses
           of
           all
           ordinary
           
             Sea
             Instruments
          
           for
           
             Observations
             ,
          
           with
           more
           speed
           ,
           ease
           and
           conveniencie
           than
           any
           of
           them
           all
           will
           do
           .
        
         
           Contrived
           and
           made
           by
           
             Iohn
             Browne
          
           at
           the
           
             Spheare
          
           and
           
             Sun-dial
          
           in
           the
           
             Minories
             ,
          
           and
           sold
           there
           or
           at
           Mr
           
             H.
             Suttons
          
           behind
           the
           
             Exchange
             .
          
           1662.
           
        
      
    
     
       
         
         
         
           THE
           Description
           ,
           and
           some
           uses
           of
           the
           
             Triangular
             Quadrant
             ,
          
           or
           the
           Sector
           made
           a
           Quadrant
           ,
           or
           the
           use
           of
           an
           excellent
           Instrument
           for
           observations
           at
           Land
           or
           Sea
           ,
           performing
           all
           the
           uses
           of
           the
           
             Forestaff
             ,
             Davis
             quadrant
             ,
             Bow
             ,
             Gunter's
             crosstaff
             ,
             Gunter's
             quadrant
             ,
          
           and
           
             sector
             ;
          
           with
           far
           more
           convenience
           and
           as
           much
           exactness
           as
           any
           of
           them
           will
           do
           .
        
         
           
             The
             Description
             .
          
           
             First
             ,
             it
             is
             a
             jointed
             rule
             (
             or
             Sector
             )
             made
             to
             what
             Radius
             you
             please
             ,
             but
             for
             the
             present
             purpose
             it
             is
             best
             between
             24.
             and
             36
             inches
             Radius
             ,
             and
             a
             third
             peice
             of
             the
             same
             length
             ,
             with
             a
             tennon
             at
             each
             end
             ,
             to
             make
             it
             an
             Equilateral
             Triangle
             ;
             from
             whence
             it
             is
             properly
             called
             a
             
               Triangular
               Quadrant
               .
            
          
           
             ●●Secondly
             ,
             as
             to
             the
             lines
             graduated
             thereon
             ,
             they
             may
             be
             more
             or
             lesse
             as
             your
             
             use
             and
             cost
             will
             please
             to
             command
             ,
             but
             to
             make
             it
             compleat
             for
             the
             promised
             premises
             ,
             these
             that
             follow
             are
             necessary
             thereunto
             .
          
           
             
               1.
               
               A
               Line
               of
               
                 degrees
                 ,
              
               of
               twice
               90.
               degrees
               on
               the
               moveable
               leg
               ,
               and
               outer
               edge
               of
               the
               cross
               piece
               :
               for
               quadrantal
               and
               back
               observations
               .
            
             
               2.
               
               Such
               another
               Line
               of
               60.
               degrees
               ,
               for
               forward
               observations
               on
               the
               inside
               of
               the
               cross
               piece
               .
            
             
               3.
               
               On
               the
               moveable
               leg
               ▪
               a
               Kallender
               of
               Months
               ,
               and
               dayes
               in
               2
               lines
               .
            
             
               4.
               
               Next
               to
               them
               ,
               a
               Line
               of
               the
               Suns
               place
               in
               degrees
               .
            
             
               5.
               
               Next
               to
               that
               ,
               a
               Line
               of
               the
               Suns
               right
               Assention
               ,
               in
               degrees
               or
               hours
               .
            
             
               6.
               
               Next
               above
               the
               Months
               on
               the
               same
               Leg
               ,
               an
               hour
               and
               Azimuth
               Line
               ,
               fitted
               to
               a
               particular
               Latitude
               ,
               as
               
                 London
                 ,
              
               or
               any
               other
               place
               ,
               for
               all
               the
               uses
               of
               
               Gunter's
               Quadrant
               as
               you
               may
               find
               in
               the
               former
               discourse
               (
               called
               a
               Joynt-rule
               .
               )
            
             
               7.
               
               On
               the
               head
               leg
               ,
               and
               same
               side
               ,
               a
               particular
               scale
               of
               altitudes
               ,
               for
               the
               particular
               Latitude
               .
            
             
               8.
               
               Next
               to
               that
               a
               general
               scale
               of
               Altitudes
               ,
               for
               all
               Latitudes
               .
            
             
               9.
               
               A
               Line
               of
               360
               degrees
               ,
               divided
               so
               as
               to
               serve
               for
               360
               degrees
               of
               12
               Signs
               ;
               and
               24
               hours
               ,
               (
               and
               in
               foot
               and
               2
               foot
               rules
               for
               inches
               also
               .
               )
            
             
               
               10.
               
               A
               Line
               of
               29½
               laid
               next
               to
               the
               former
               ,
               serving
               to
               find
               the
               Moons
               coming
               to
               South
               ,
               and
               her
               age
               ,
               and
               place
               ,
               and
               the
               time
               of
               the
               Night
               ,
               by
               the
               fixed
               Stars
               .
            
             
               11.
               
               A
               perpetual
               Almanack
               ,
               and
               the
               right
               Assension
               ,
               and
               declination
               of
               several
               fixed
               Stars
               .
            
             
               12.
               
               A
               Line
               of
               Lines
               next
               the
               inside
               ,
               may
               be
               put
               on
               without
               Trouble
               ,
               or
               incumbering
               one
               another
               ,
               all
               these
               on
               one
               side
               .
            
          
           
             Secondly
             ,
             on
             the
             other
             side
             may
             be
             put
             the
             Lines
             of
             
               natural
               Signs
               ,
               Tangents
            
             and
             
               Secants
               ,
            
             to
             a
             single
             ,
             double
             ,
             and
             treble
             Radius
             ,
             and
             by
             this
             means
             more
             then
             a
             
               Gunters
               Sector
               ,
            
             (
             the
             particular
             Lines
             being
             inscribed
             between
             the
             general
             Lines
             .
             )
          
           
             Thirdly
             on
             the
             other
             edge
             there
             may
             be
             Artificial
             
               Numbers
               ,
               Tangents
               ,
               Signs
               ,
            
             and
             
               versed
               Signs
               ,
            
             and
             by
             this
             means
             it
             is
             a
             
               Gunters
            
             Rule
             or
             a
             
               Crosstaff
               .
            
          
           
             Fourthly
             ,
             on
             the
             insides
             ,
             
               inches
               ,
               foot-measure
               ,
            
             and
             a
             Line
             of
             112.
             parts
             ,
             and
             a
             large
             
               meridian
               Line
               ,
            
             or
             the
             like
             :
             as
             you
             please
             .
          
           
             Lastly
             ,
             two
             sliding
             nuts
             with
             points
             in
             them
             ,
             fitted
             to
             the
             Cross
             piece
             ,
             makes
             it
             a
             proper
             beam
             Compass
             to
             use
             in
             working
             by
             the
             Numbers
             Signs
             and
             Tangents
             on
             the
             edge
             ,
             or
             flat
             side
             ,
             also
             it
             must
             have
             four
             or
             five
             sights
             ,
             a
             Thred
             ,
             and
             Plummet
             ,
             and
             Compasses
             ,
             as
             other
             instruments
             have
             ,
             thus
             much
             for
             the
             Description
             ,
             the
             uses
             follow
             .
          
           
             
             
               An
               
                 Advertisement
                 .
              
            
             
               First
               ,
               for
               the
               better
               understanding
               and
               brevity
               sake
               ,
               there
               are
               ten
               things
               to
               be
               named
               and
               described
               ,
               as
               followeth
               ;
               1.
               the
               
                 head
                 leg
                 ,
              
               in
               which
               the
               brass
               Revet
               is
               fixed
               ,
               and
               about
               which
               the
               other
               turn
               ;
               2.
               the
               
                 moveable
                 leg
                 ,
              
               on
               which
               the
               Months
               and
               Days
               must
               always
               be
               ;
               3.
               the
               
                 Cross
                 piece
                 ,
              
               that
               is
               fitted
               to
               the
               head
               ,
               and
               moveable
               Leg
               ,
               by
               the
               two
               Tennons
               at
               the
               end
               ;
               4.
               the
               
                 quadrant
                 side
              
               where
               the
               degrees
               and
               Moneths
               are
               ,
               for
               observation
               .
               5.
               the
               other
               or
               
                 Sector
                 side
              
               for
               operation
               ,
               6thly
               the
               
                 head
                 center
                 ,
              
               being
               Center
               to
               the
               degrees
               on
               the
               inside
               of
               the
               crosse
               peece
               ,
               for
               a
               forward
               observation
               as
               with
               the
               forestaffe
               .
               7
               The
               other
               Center
               ,
               near
               the
               end
               of
               the
               head
               leg
               ,
               being
               the
               Center
               to
               the
               moveable
               leg
               ,
               for
               backward
               observations
               ,
               (
               as
               the
               Davis
               quadrant
               is
               used
               ,
               and
               the
               bow
               )
               which
               you
               may
               call
               the
               
                 foot
                 center
                 ,
              
               or
               
                 leg
                 center
              
               for
               backward
               observations
               .
            
             
               8thly
               the
               sights
               ,
               as
               first
               the
               
                 turning
              
               or
               eye
               
                 sight
                 ,
              
               which
               is
               alwayes
               ,
               set
               on
               one
               of
               the
               centers
               ,
               with
               a
               screw
               to
               make
               it
               fast
               there
               ,
               which
               I
               call
               the
               
                 turning
                 sight
                 ,
              
               9thly
               the
               
                 Horizon
                 sight
              
               that
               cuts
               the
               degrees
               of
               altitude
               ,
               and
               sometimes
               is
               next
               to
               the
               eye
               ,
               and
               sometimes
               remote
               from
               the
               eye
               ,
               yet
               called
               the
               
                 horizon
              
               (
               slideing
               )
               
                 sight
                 ,
              
               10thly
               .
               the
               
                 object
              
               or
               
                 shaddow
                 sight
              
               of
               which
               there
               may
               be
               3
               for
               convenience
               sake
               ,
               as
               two
               fixed
               
               and
               one
               moveable
               to
               slide
               as
               the
               horizon
               sight
               doth
               :
               the
               other
               two
               do
               serve
               also
               to
               pin
               the
               crosse
               peece
               ,
               and
               the
               two
               legs
               together
               ,
               through
               the
               two
               tennons
               ,
               all
               whose
               names
               in
               short
               take
               thus
               :
            
             
               1.
               
               The
               
                 head
              
               leg
               :
               2.
               the
               
                 moveable
                 leg
                 :
              
               3.
               the
               
                 crosse
                 peece
                 :
              
               4.
               the
               
                 quadrant
                 side
                 :
              
               5.
               the
               
                 sector
                 side
                 ,
              
               6.
               the
               
                 head
              
               (
               or
               forward
               )
               
                 center
                 .
              
               7
               the
               
                 leg
              
               (
               or
               backward
               )
               
                 center
                 .
              
               8.
               the
               
                 turning
                 sight
                 .
              
               9.
               the
               (
               slideing
               )
               
                 horzon
                 sight
                 .
              
               10.
               the
               
                 object
              
               (
               or
               shaddow
               )
               
                 sight
                 .
              
               of
               which
               there
               be
               3.
               all
               differing
               according
               to
               your
               use
               and
               occasions
               :
               one
               to
               slide
               to
               any
               place
               ,
               the
               other
               2.
               to
               be
               put
               into
               certain
               holes
               .
               nigher
               ,
               or
               further
               off
               :
               as
               will
               afterwards
               largely
               appear
               .
            
          
        
         
           
             THE
             USES
             :
          
           
             
               
                 I.
              
               To
               find
               the
               suns
               declination
               ,
               true
               place
               ,
               right
               assention
               ,
               and
               rising
               ,
               the
               day
               of
               the
               moneth
               being
               given
               .
            
             
               First
               open
               the
               rule
               to
               an
               angle
               of
               60.
               degrees
               ,
               which
               is
               alwayes
               done
               when
               the
               cross
               peece
               is
               fitted
               into
               the
               Mortesse
               holes
               ,
               and
               the
               pins
               of
               the
               
                 object
                 sights
              
               put
               in
               the
               holes
               through
               the
               tennons
               ,
               or
               else
               by
               the
               second
               Chapter
               of
               the
               Joynt-rule
               :
               then
               extend
               a
               thred
               from
               the
               center
               pin
               in
               the
               head
               leg
               ,
               to
               the
               day
               of
               the
               month
               ,
               &
               on
               the
               degrees
               it
               cuts
               the
               suns
               declination
               ,
               in
               the
               line
               of
               right
               assention
               his
               right
               assention
               ,
               in
               the
               line
               of
               true
               place
               his
               true
               place
               ,
               and
               in
               the
               hour
               line
               his
               true
               rising
               and
               setting
               ,
               
               in
               that
               latitude
               the
               line
               is
               m●de
               for
               :
               Example
               ,
               on
               the
               first
               of
               May
               I
               would
               know
               the
               former
               questions
               ,
               the
               rule
               being
               set
               by
               the
               crosse
               peece
               ,
               and
               the
               thred
               on
               the
               
                 leg
                 center
              
               pin
               ;
               and
               drawn
               straight
               and
               laid
               over
               May
               1.
               it
               cuts
               in
               the
               degrees
               18.
               4.
               north
               declination
               ,
               and
               20.
               58.
               in
               ♉
               Taurus
               for
               his
               place
               ,
               and
               3
               hours
               14.
               minutes
               right
               assention
               in
               time
               ,
               or
               48.
               32.
               in
               degrees
               :
               and
               the
               rising
               of
               the
               sun
               that
               day
               is
               at
               4.
               23
               ,
               and
               sets
               at
               5.
               37
               ,
               in
               51.
               32
               latitude
               .
            
             
               The
               finding
               of
               hour
               and
               azimuth
               ,
               either
               particularly
               ,
               or
               generally
               ,
               with
               other
               Astronomical
               propositions
               ,
               are
               spoken
               enough
               of
               before
               in
               the
               Joynt-rule
               ,
               and
               in
               all
               other
               authors
               that
               write
               of
               the
               Sector
               ,
               or
               
                 Gunter's
              
               rule
               ,
               so
               that
               all
               I
               shall
               speak
               of
               now
               ,
               shall
               be
               onely
               what
               was
               forgot
               in
               the
               first
               part
               ,
               and
               what
               is
               new
               as
               to
               the
               using
               the
               instrument
               in
               sea
               observations
               .
            
          
           
             
               
                 II.
              
               To
               find
               the
               Suns
               or
               a
               Stars
               Altitude
               ,
               by
               a
               forward
               Observation
               .
            
             
               Skrew
               the
               turning
               sight
               to
               the
               
                 head
                 Center
                 ,
              
               and
               set
               that
               
                 object
                 sight
                 ,
              
               whose
               holes
               answer
               to
               the
               Sliding
               
                 horizon
                 sight
                 ,
              
               in
               the
               hole
               at
               the
               end
               of
               the
               
                 head
                 leg
                 ,
              
               and
               put
               the
               horizon
               sight
               on
               the
               crosse
               peece
               next
               the
               inside
               ;
               Then
               holding
               the
               crosse
               peece
               with
               your
               right
               hand
               ,
               and
               the
               turning
               sight
               close
               to
               your
               eye
               ,
               and
               the
               moveable
               leg
               against
               your
               body
               ,
               with
               your
               thumb
               
               on
               the
               right
               hand
               thrust
               upwards
               ,
               or
               pull
               downwards
               ,
               the
               horizon
               sight
               :
               till
               you
               see
               the
               sun
               through
               the
               object
               sight
               ,
               and
               the
               horizon
               through
               the
               horizon
               sight
               ,
               then
               the
               degrees
               cut
               by
               the
               middle
               of
               the
               horizon
               sight
               ,
               on
               the
               crosse
               peece
               shall
               be
               the
               true
               altitude
               required
               :
            
          
           
             
               
                 III.
              
               To
               perform
               the
               same
               another
               way
               .
            
             
               If
               your
               instrument
               be
               parted
               ,
               that
               is
               to
               say
               the
               crosse
               peece
               from
               the
               other
               ,
               and
               an
               altitude
               be
               required
               to
               be
               had
               quickly
               ,
               then
               set
               the
               two
               object
               sights
               ,
               in
               two
               holes
               at
               the
               end
               of
               the
               line
               of
               naturall
               signs
               ,
               then
               set
               the
               head
               of
               the
               rule
               to
               your
               eye
               ,
               so
               as
               the
               sight
               of
               the
               eye
               may
               be
               just
               over
               the
               Center
               ,
               then
               open
               or
               close
               the
               Joynt
               ,
               till
               you
               see
               the
               horizon
               through
               one
               sight
               ,
               and
               the
               sun
               or
               star
               through
               the
               other
               ,
               then
               is
               the
               sector
               set
               to
               the
               angle
               required
               ,
               to
               find
               which
               angle
               do
               thus
               ,
               take
               the
               parallel
               sign
               of
               30
               and
               30
               ,
               and
               measure
               it
               from
               the
               Center
               ,
               and
               it
               shall
               reach
               to
               the
               sign
               of
               half
               ●he
               angle
               required
               .
            
             
               Example
               .
            
             
               Suppose
               I
               had
               observed
               an
               altitude
               ,
               and
               the
               distance
               between
               30
               and
               30
               ,
               should
               reach
               from
               the
               center
               to
               10.
               degrees
               on
               the
               signs
               ,
               then
               is
               the
               altitude
               of
               the
               sun
               20.
               degrees
               for
               10
               doubled
               is
               20.
               
            
          
           
             
               
                 IIII.
              
               To
               find
               the
               suns
               Altitude
               by
               a
               back
               observation
               ,
            
             
               Skrew
               the
               Turning
               sight
               to
               the
               leg
               center
               ,
               
               (
               or
               center
               to
               the
               degrees
               on
               the
               moveable
               leg
               )
               and
               put
               one
               of
               the
               object
               sights
               ,
               in
               the
               hole
               by
               00.
               on
               the
               outer
               edge
               of
               the
               crosse
               peece
               ,
               and
               set
               the
               edge
               of
               it
               just
               against
               the
               stroke
               of
               00
               ,
               or
               you
               may
               use
               the
               sliding
               object
               sight
               and
               set
               the
               edge
               or
               the
               middle
               of
               that
               ,
               to
               the
               stroke
               of
               00
               ,
               as
               you
               shall
               Judge
               most
               convenient
               ;
               and
               the
               horizon
               sight
               to
               the
               moveable
               leg
               ,
               then
               observe
               in
               all
               respects
               as
               with
               a
               Davis
               quadrant
               ,
               till
               looking
               through
               the
               small
               hole
               of
               the
               horizon
               sight
               ,
               you
               see
               the
               crosse
               bar
               and
               button
               ,
               in
               the
               turning
               sight
               ,
               cut
               the
               horizon
               :
               and
               at
               the
               same
               instant
               the
               shadow
               of
               the
               edge
               or
               middle
               of
               the
               object
               or
               shadow
               sight
               ,
               fall
               on
               the
               middle
               of
               the
               turning
               sight
               ,
               by
               sliding
               the
               horizon
               sight
               higher
               or
               lower
               ,
               then
               the
               middle
               stroke
               of
               the
               horizon
               sight
               ,
               shall
               cut
               on
               the
               moveable
               leg
               ,
               the
               suns
               true
               altitude
               required
               .
               As
               f
               〈…〉
               t
               stay
               at
               50
               degrees
               ,
               then
               is
               the
               sun
               50
               degrees
               above
               the
               horizon
               .
            
          
           
             
               V.
               But
               if
               the
               sun
               be
               near
               to
               the
               Zenith
               or
               90
               degrees
               high
               ,
               then
               it
               will
               be
               convenient
               to
               move
               the
               object
               sight
               ,
               to
               a
               hole
               or
               two
               further
               as
               suppose
               at
               10
               ,
               20
               ,
               30
               degrees
               more
               ,
               toward
               the
               further
               end
               of
               the
               crosse
               peece
               and
               then
               observe
               as
               you
               did
               before
               in
               all
               respects
               ,
               as
               with
               a
               Davis
               quadrant
               ,
               and
               then
               whatsoever
               degrees
               the
               horizon
               sight
               cuts
               ,
               you
               must
               ad
               so
               much
               to
               it
               ,
               as
               you
               set
               the
               object
               sight
               forwards
               ,
               as
               suppose
               30
               ,
               and
               
               the
               horizon
               sight
               stay
               at
               60
               ,
               then
               I
               say
               60
               ,
               and
               30
               ,
               makes
               90
               :
               the
               true
               altitude
               required
               .
               Note
               that
               by
               this
               contrivance
               ,
               let
               the
               altitude
               be
               what
               it
               will
               ,
               you
               shall
               alwayes
               have
               a
               most
               steady
               observation
               :
               with
               the
               instrument
               leaning
               against
               your
               brest
               ,
               a
               considerable
               thing
               ,
               in
               a
               windy
               day
               ,
               when
               you
               may
               have
               a
               need
               of
               an
               observation
               in
               southern
               voyages
               ,
               when
               the
               sun
               is
               near
               to
               the
               zenith
               at
               a
               Meridian
               observation
               .
            
          
           
             
               
                 VI.
              
               To
               find
               the
               suns
               distance
               from
               the
               zenith
               ,
               by
               observing
               the
               other
               way
               ,
               the
               sun
               being
               not
               above
               60
               degrees
               high
               ,
               or
               30
               from
               the
               zenith
               .
            
             
               Set
               the
               turning
               sight
               as
               before
               on
               the
               leg
               Center
               ,
               then
               set
               an
               object
               sight
               in
               one
               of
               the
               holes
               in
               the
               line
               on
               the
               head
               leg
               ,
               nigher
               or
               further
               of
               ,
               the
               turning
               sight
               :
               as
               the
               the
               brightnesse
               or
               dimnesse
               of
               the
               sun
               will
               allow
               to
               see
               a
               shadow
               ,
               then
               looking
               through
               the
               small
               hole
               on
               the
               horizon
               sight
               ,
               till
               you
               see
               the
               horizon
               cut
               by
               the
               crosse
               bar
               of
               the
               great
               hole
               ,
               in
               the
               turning
               sight
               ,
               turning
               the
               foreside
               of
               that
               sight
               ,
               till
               it
               be
               fit
               to
               receive
               the
               shadow
               of
               the
               middle
               of
               the
               object
               sight
               ;
               then
               the
               degrees
               cut
               by
               the
               horizon
               sight
               ,
               shall
               be
               the
               suns
               true
               distance
               from
               the
               zenith
               ,
               or
               the
               complement
               of
               the
               Altitude
               .
            
          
           
             
               VII
               .
               Note
               that
               by
               adding
               of
               a
               short
               peece
               about
               9
               inches
               long
               on
               the
               head
               leg
               ,
               whereon
               
               to
               set
               the
               slideing
               shadow
               sight
               ,
               you
               may
               obtain
               the
               former
               convenience
               of
               all
               angles
               ,
               this
               way
               also
               ,
               at
               a
               most
               steady
               and
               easie
               manner
               of
               observation
               ;
               but
               note
               whatsoever
               you
               set
               forwards
               on
               that
               peece
               ,
               must
               be
               substracted
               from
               that
               the
               sight
               sheweth
               ,
               and
               the
               remainder
               shall
               be
               the
               suns
               distance
               from
               the
               zenith
               required
               .
            
             
               As
               suppose
               you
               set
               forward
               30
               degrees
               ,
               and
               the
               horizon
               sight
               should
               stay
               at
               40
               ,
               then
               30
               from
               40
               rest
               10
               ,
               the
               suns
               distance
               from
               the
               zenith
               required
               ;
               thus
               you
               see
               ,
               that
               by
               one
               and
               the
               same
               line
               ,
               at
               one
               manner
               of
               figuring
               ,
               is
               the
               suns
               altitude
               ,
               or
               coalitude
               acquired
               and
               that
               at
               a
               most
               certain
               steady
               manner
               of
               observation
               .
            
          
           
             
               
                 VIII
                 .
              
               To
               find
               an
               observation
               by
               thred
               and
               plummet
               ,
               without
               having
               any
               respect
               to
               the
               horizon
               ,
               being
               of
               good
               stead
               in
               a
               misty
               or
               cloudy
               day
               at
               land
               or
               sea
               .
            
             
               Set
               the
               rule
               to
               his
               angle
               of
               60
               degrees
               by
               putting
               in
               the
               crosse
               peece
               ,
               then
               skrew
               the
               turning
               sight
               to
               the
               head
               center
               ,
               then
               if
               the
               sun
               or
               star
               be
               under
               30
               degrees
               high
               ,
               set
               the
               object
               sight
               in
               the
               moveable
               leg
               ,
               then
               looking
               through
               the
               small
               hole
               in
               the
               turning
               sight
               ,
               through
               the
               object
               sight
               ,
               to
               the
               middle
               of
               the
               star
               or
               sun
               ,
               as
               the
               button
               in
               the
               crosse
               bar
               will
               neatly
               shew
               ;
               then
               the
               thread
               and
               plummet
               ,
               hanging
               on
               the
               leg
               center
               pin
               ,
               and
               playing
               evenly
               by
               the
               moveable
               leg
               ,
               shall
               shew
               the
               true
               alti●ude
               
               of
               the
               sun
               ,
               or
               star
               required
               counting
               the
               degrees
               as
               they
               are
               numbred
               ,
               for
               th
               :
               north
               declinations
               from
               60
               toward
               the
               head
               with
               10
               20
               ,
               As
               if
               the
               thred
               shall
               play
               upon
               70
               10
               then
               is
               the
               altitude
               10
               degrees
               .
            
          
           
             
               IX
               But
               if
               the
               sun
               or
               star
               be
               above
               30
               degrees
               high
               ,
               then
               the
               object
               sight
               must
               be
               set
               to
               the
               hole
               in
               the
               end
               of
               the
               head
               leg
               :
               then
               looking
               as
               before
               ,
               and
               the
               thred
               playing
               evenly
               by
               the
               moveable
               leg
               ,
               shall
               shew
               the
               true
               altitude
               required
               ,
               as
               the
               degrees
               are
               numbred
               .
            
             
               Note
               that
               if
               the
               brightnesse
               of
               the
               sun
               should
               offend
               the
               eye
               ,
               you
               may
               have
               a
               peice
               of
               green
               ,
               blew
               ,
               or
               red
               glasse
               ,
               fixed
               on
               the
               turning
               sight
               ,
               or
               else
               remove
               the
               object
               sight
               nearer
               to
               the
               turning
               sight
               ,
               and
               then
               let
               the
               sun
               beams
               pierce
               through
               both
               the
               small
               holes
               ,
               according
               to
               the
               usuall
               manner
               and
               the
               thred
               shall
               shew
               the
               true
               altitude
               required
               .
               Note
               also
               if
               the
               thred
               be
               apt
               to
               slip
               away
               from
               his
               observed
               place
               ,
               as
               between
               25
               and
               40
               it
               may
               :
               note
               a
               dexterious
               handling
               thereof
               will
               naturally
               shew
               you
               how
               to
               prevent
               it
               :
            
          
           
             
               
                 X.
              
               To
               find
               a
               latitude
               at
               Sea
               by
               forward
               meridian
               Observation
               or
               Altitude
               .
            
             
               Set
               the
               moving
               object
               Sight
               to
               the
               Suns
               declination
               ,
               shewed
               by
               the
               day
               of
               the
               Month
               ,
               and
               rectifying
               Table
               ,
               and
               skrew
               the
               turning
               sight
               to
               the
               leg
               center
               ,
               and
               the
               Horizon
               sight
               to
               the
               moveable
               leg
               ,
               or
               the
               
               outside
               of
               the
               Crosse
               piece
               ,
               according
               as
               the
               Sun
               is
               high
               or
               low
               (
               but
               note
               all
               forward
               observations
               respecting
               the
               Horizon
               ,
               ought
               to
               be
               under
               45
               degrees
               high
               ,
               for
               if
               it
               be
               more
               it
               is
               very
               uncertain
               ,
               by
               any
               Instrument
               whatsoever
               ,
               except
               you
               have
               a
               Plummet
               and
               then
               the
               Horizon
               is
               uselesse
               )
               then
               observe
               just
               as
               you
               do
               in
               a
               forward
               observation
               ,
               moving
               the
               Horizon
               sight
               till
               you
               see
               the
               Sun
               through
               the
               Horizon
               sight
               ,
               and
               the
               Horizon
               through
               the
               object
               sight
               ,
               or
               the
               contrary
               .
               (
               moving
               not
               that
               sight
               that
               is
               set
               to
               the
               day
               of
               the
               Month
               or
               Declination
               ,
               )
               then
               whatsoever
               the
               moving
               sight
               shall
               shew
               ,
               if
               you
               add
               30
               to
               it
               ,
               it
               shall
               be
               the
               latitude
               of
               the
               place
               required
               ;
               observing
               the
               difference
               in
               North
               and
               South
               Latitudes
               ;
               that
               is
               ,
               setting
               the
               sight
               to
               the
               proper
               declination
               ,
               either
               like
               ,
               or
               unlike
               ,
               to
               the
               latitude
               -
            
             
               
                 Example
                 .
              
               Suppose
               on
               the
               10.
               of
               
                 March
              
               when
               the
               Suns
               declination
               is
               0
               —
               10.
               
               North
               ,
               as
               in
               the
               first
               year
               after
               leap
               year
               it
               will
               be
               ,
               set
               the
               stroke
               in
               the
               middle
               of
               the
               moving
               object
               sight
               to
               10
               of
               North
               Declination
               ,
               and
               the
               Horizon
               sight
               on
               the
               moveable
               leg
               ,
               then
               move
               it
               higher
               or
               lower
               ,
               till
               you
               see
               the
               Horizon
               through
               one
               ,
               and
               the
               Sun
               through
               the
               other
               ,
               then
               the
               degrees
               between
               ,
               is
               the
               Suns
               meridian
               altitude
               ,
               if
               it
               be
               at
               Noon
               ,
               as
               suppose
               it
               stayed
               at
               21
               30
               ▪
               then
               by
               counting
               the
               degrees
               between
               ,
               
               you
               shall
               find
               them
               come
               to
               38.
               40.
               then
               if
               you
               add
               30.
               to
               21.
               30.
               it
               makes
               51.
               30.
               the
               Latitude
               required
               ,
               for
               if
               you
               do
               take
               0
               10′
               minutes
               from
               38.
               40.
               there
               remains
               38.
               30.
               the
               complement
               of
               the
               Latitude
               .
               Note
               ,
               that
               this
               way
               you
               may
               take
               a
               forward
               observation
               ,
               and
               so
               save
               the
               removing
               of
               the
               ●urning
               sight
               .
            
             
               Note
               also
               ,
               That
               when
               the
               Horizon
               sight
               shall
               stay
               about
               the
               corner
               ,
               you
               may
               move
               the
               object
               sight
               10.
               or
               20.
               degrees
               towards
               the
               head
               ,
               and
               then
               you
               must
               add
               but
               20.
               or
               10
               degrees
               to
               what
               the
               sight
               stayed
               at
               ;
               or
               if
               you
               shall
               set
               the
               sight
               the
               other
               way
               10
               or
               20.
               degr
               .
               then
               you
               must
               add
               more
               then
               30
               so
               much
               .
               As
               suppose
               in
               this
               last
               observation
               ,
               it
               had
               been
               the
               latitude
               of
               45
               or
               50
               degrees
               ,
               then
               you
               shall
               find
               the
               sight
               to
               play
               so
               neer
               the
               corner
               ,
               that
               it
               will
               prove
               inconvenient
               ,
               then
               suppose
               instead
               of
               0
               10.
               
               I
               set
               it
               to
               20
               degrees
               10′
               North
               declination
               ,
               which
               is
               20.
               degrees
               added
               to
               the
               declination
               ,
               then
               the
               Suns
               height
               being
               the
               same
               as
               before
               ,
               the
               sight
               will
               stay
               at
               41.
               30.
               to
               which
               if
               you
               add
               10
               degrees
               ,
               it
               doth
               make
               51.
               30.
               as
               before
               ;
               here
               you
               must
               add
               but
               10
               degrees
               ,
               because
               you
               increased
               the
               declination
               20.
               degrees
               ;
               but
               note
               by
               the
               same
               reason
               ,
               had
               you
               set
               it
               to
               19.
               50.
               
               South
               declination
               ,
               then
               it
               had
               been
               diminished
               20
               degrees
               ,
               and
               then
               instead
               of
               30
               you
               must
               add
               50
               ▪
               to
               1
               -
               30.
               the
               place
               where
               
               the
               sight
               would
               have
               stayed
               .
               Thus
               you
               see
               you
               may
               very
               neatly
               avoid
               this
               inconvenience
               ,
               and
               set
               the
               sights
               to
               proper
               and
               steady
               observations
               ,
               at
               all
               times
               of
               observation
               .
            
          
           
             
               
                 XI
                 .
              
               To
               find
               the
               latitude
               by
               a
               backward
               Meridian
               Observation
               at
               Sea.
               
            
             
               This
               is
               but
               just
               the
               converse
               of
               the
               former
               ,
               for
               if
               you
               set
               one
               sight
               to
               the
               declination
               ,
               either
               directly
               ,
               (
               or
               augmented
               or
               diminish'd
               as
               before
               ,
               when
               the
               moving
               sight
               shall
               stay
               ,
               about
               the
               corner
               of
               the
               
                 Triangular
                 Quadrant
              
               )
               then
               the
               other
               being
               slipt
               to
               and
               fro
               ,
               on
               the
               outside
               of
               the
               Crosse
               peece
               ,
               till
               the
               shadow
               of
               the
               outer
               edge
               ,
               shall
               fall
               on
               the
               middle
               of
               the
               turning
               sight
               ,
               then
               30
               just
               ,
               or
               more
               or
               lesse
               added
               ,
               to
               that
               number
               the
               moving
               sight
               stayed
               at
               ,
               (
               according
               as
               you
               set
               the
               first
               Horizon
               sight
               to
               the
               declination
               )
               shall
               be
               the
               true
               latitude
               required
               .
               
                 Example
                 .
              
               Suppose
               on
               the
               same
               day
               and
               year
               as
               before
               ,
               at
               the
               same
               Noon
               time
               ,
               I
               set
               my
               Horizon
               sight
               to
               just
               10′
               of
               North
               declination
               ,
               you
               shall
               find
               the
               moving
               sight
               to
               stay
               at
               21.
               30.
               neer
               to
               the
               corner
               ,
               now
               if
               the
               Sun
               shine
               bright
               ,
               and
               will
               cast
               the
               shadow
               to
               the
               turning
               sight
               ,
               then
               set
               the
               Horison
               sight
               at
               the
               declination
               ,
               forward
               10
               or
               20.
               degrees
               ,
               then
               the
               moving
               sight
               coming
               lower
               you
               ,
               add
               but
               20.
               or
               to
               that
               it
               shall
               stay
               at
               ,
               and
               the
               summe
               shall
               be
               the
               latitude
               .
            
             
             
               But
               it
               is
               most
               likely
               that
               it
               will
               be
               better
               to
               diminish
               it
               20.
               degrees
               ,
               then
               the
               moving
               sight
               will
               stay
               about
               2
               -
               30.
               on
               the
               Crosse
               piece
               ,
               and
               so
               much
               the
               better
               to
               cast
               a
               shadow
               ;
               for
               if
               you
               look
               through
               the
               Horizon
               and
               turning
               sight
               to
               the
               Horizon
               ,
               you
               shall
               find
               the
               shadow
               of
               the
               former
               edge
               of
               the
               moving
               shadow
               sight
               ,
               to
               stay
               at
               2
               -
               30.
               to
               which
               if
               you
               add
               20.
               the
               degrees
               diminished
               ,
               and
               30.
               it
               makes
               51
               30.
               the
               latitude
               required
               as
               before
               .
            
             
               Note
               also
               for
               better
               convenience
               of
               the
               shadow
               sight
               ,
               when
               you
               have
               found
               the
               true
               declination
               ,
               as
               before
               is
               taught
               ,
               set
               the
               moving
               object
               sight
               to
               the
               same
               ,
               on
               the
               Crosse
               piece
               ,
               counted
               from
               00.
               towards
               the
               head
               leg
               ,
               for
               like
               latitudes
               and
               declinations
               ;
               and
               the
               other
               way
               for
               unlike
               latitudes
               and
               declinations
               ,
               then
               observing
               as
               in
               a
               back
               observation
               ,
               wheresoever
               the
               sight
               shall
               stay
               ,
               shall
               be
               the
               complement
               of
               the
               latitude
               required
               .
               If
               you
               add
               or
               diminish
               consider
               accordingly
               .
            
             
               Note
               likewise
               ,
               when
               the
               declination
               is
               nere
               the
               solstice
               ,
               and
               the
               same
               way
               as
               the
               latitude
               is
               ,
               and
               by
               diminishing
               ,
               or
               otherwise
               the
               moving
               sight
               shall
               fall
               beyond
               00
               on
               the
               Crosse
               piece
               ;
               Then
               having
               added
               30.
               and
               the
               degrees
               diminished
               together
               ,
               whatsoever
               the
               sight
               shall
               stay
               at
               beyond
               00
               must
               be
               taken
               out
               of
               the
               added
               sum
               ,
               and
               the
               remainder
               shall
               be
               the
               latitude
               required
               .
            
             
             
               
                 Example
                 .
              
               Suppose
               on
               the
               11.
               of
               
                 Iune
                 ,
              
               in
               the
               latitude
               of
               51
               30
               north
               ,
               for
               the
               better
               holding
               sake
               I
               diminish
               the
               declination
               30
               degrees
               ,
               that
               is
               in
               stead
               of
               setting
               it
               to
               23.
               32.
               north
               declination
               ,
               I
               set
               it
               to
               6
               :
               28
               south
               then
               the
               sun
               being
               62
               degrees
               high
               will
               stay
               at
               08.
               30.
               beyond
               00.
               the
               other
               way
               now
               30
               to
               be
               added
               ,
               and
               30
               diminished
               ,
               makes
               60
               ,
               from
               which
               take
               8.
               30
               ,
               rest
               51
               30
               the
               latitude
               required
               .
            
          
           
             
               
                 XII
                 .
              
               To
               find
               a
               latitude
               with
               thred
               and
               plummet
               ,
               or
               by
               an
               observation
               made
               without
               respecting
               the
               Horizon
               .
            
             
               Count
               the
               declination
               on
               the
               cross
               peece
               (
               and
               let
               00
               be
               the
               equinoctiall
               and
               let
               the
               declination
               which
               is
               the
               same
               with
               the
               latitude
               be
               counted
               toward
               the
               moveable
               leg
               ,
               and
               the
               contrary
               the
               other
               way
               ,
               as
               with
               us
               in
               north
               latitude
               ,
               north
               declination
               is
               toward
               the
               moveable
               leg
               ,
               and
               south
               declination
               the
               contrary
               and
               contrarily
               in
               south
               latitudes
               )
               and
               thereunto
               set
               the
               middest
               of
               the
               sliding
               object
               or
               horizon
               sight
               ,
               then
               is
               the
               small
               hole
               on
               the
               turning
               sight
               and
               the
               small
               hole
               on
               the
               horizon
               sight
               ,
               two
               holes
               whereby
               the
               sun
               beams
               are
               to
               pierce
               to
               shine
               one
               on
               the
               other
               :
               then
               shall
               the
               thred
               shew
               you
               the
               true
               latitude
               of
               the
               place
               required
               .
               
                 Example
                 .
              
            
             
               Suppose
               on
               the
               11
               of
               
                 Decem.
              
               1663
               ,
               at
               noon
               I
               observe
               the
               noon
               altitudes
               set
               the
               middle
               of
               the
               Horizon
               sight
               to
               23
               ▪
               32
               counted
               
               from
               00.
               toward
               the
               head
               leg
               end
               ,
               then
               making
               the
               sun
               beams
               to
               peirce
               through
               the
               hole
               of
               this
               ,
               and
               the
               turning
               sight
               ,
               you
               shall
               find
               the
               plummet
               to
               play
               on
               51.
               30
               ,
               the
               latitude
               required
               ,
               holding
               the
               turning
               sight
               toward
               the
               sun
               .
            
             
               Note
               also
               that
               here
               also
               you
               may
               avoid
               the
               inconvenience
               of
               the
               corner
               ,
               or
               the
               great
               distance
               between
               the
               sights
               ,
               by
               the
               remedy
               before
               cited
               ,
               in
               the
               back
               and
               forward
               observation
               .
            
             
               For
               if
               you
               move
               it
               toward
               the
               head
               leg
               ,
               then
               the
               thred
               will
               fall
               short
               of
               the
               latitude
               ,
               if
               toward
               the
               moveable
               leg
               then
               it
               falls
               beyond
               the
               latitude
               ,
               as
               is
               very
               easie
               to
               conceive
               of
               :
            
             
               Thus
               you
               see
               all
               the
               uses
               of
               the
               forestaff
               ,
               and
               quadrant
               ,
               and
               Mr.
               
                 Gunter's
              
               bow
               are
               plainly
               and
               properly
               applyed
               to
               this
               Triangular
               quadrant
               ,
               that
               the
               same
               will
               be
               a
               Sector
               is
               easie
               to
               perswade
               you
               to
               believe
               ,
               and
               that
               all
               the
               uses
               of
               a
               
                 Gunter's
              
               quadrant
               ,
               are
               performed
               by
               it
               ,
               is
               fully
               shewed
               in
               the
               use
               of
               the
               Joynt-rule
               ,
               to
               which
               this
               may
               be
               annexed
               ,
               the
               numbers
               signs
               and
               tangents
               and
               versed
               signs
               makes
               it
               an
               excellent
               large
               
                 Gunter's
              
               rule
               ,
               and
               the
               cross
               peece
               is
               a
               good
               pair
               of
               large
               compasses
               to
               operate
               therewith
               ;
               lastly
               ,
               being
               it
               may
               lie
               in
               so
               little
               roome
               it
               is
               much
               more
               convenient
               for
               them
               ,
               with
               whom
               stowage
               is
               very
               precious
               ,
               so
               I
               shall
               say
               no
               more
               as
               to
               the
               use
               
               of
               it
               ,
               all
               the
               rest
               being
               fully
               spoke
               to
               in
               other
               Authors
               ,
               to
               whom
               I
               refer
               you
               :
               only
               one
               usefull
               proposition
               to
               inure
               you
               to
               the
               use
               of
               this
               most
               excellent
               instrument
               ,
               which
               I
               call
               the
               
                 Triangular
                 Quadrant
                 .
              
            
             
               Note
               that
               in
               finding
               the
               latitude
               ,
               it
               is
               necessary
               to
               have
               a
               table
               of
               the
               suns
               declination
               for
               every
               of
               the
               four
               years
               ,
               viz.
               for
               the
               leap
               year
               and
               the
               1
               ,
               2
               and
               3d.
               after
               ,
               now
               the
               table
               of
               the
               suns
               declination
               whereby
               the
               moneths
               are
               laid
               down
               ,
               is
               a
               table
               that
               is
               calculated
               as
               a
               mean
               between
               all
               the
               4
               years
               ,
               and
               you
               may
               very
               well
               distinguish
               a
               minute
               on
               the
               rule
               ;
               now
               to
               make
               it
               to
               be
               exact
               I
               have
               fitted
               this
               rectifying
               table
               for
               every
               Week
               in
               the
               year
               ,
               and
               the
               use
               is
               thus
               :
               hang
               the
               thred
               on
               the
               center
               pin
               ,
               and
               extend
               the
               thred
               to
               the
               day
               of
               the
               moneth
               ,
               and
               on
               the
               degrees
               is
               the
               suns
               declination
               ,
               as
               near
               as
               can
               be
               for
               a
               common
               year
               ,
               then
               if
               you
               look
               in
               the
               rectifying
               table
               for
               that
               moneth
               ,
               and
               week
               you
               seek
               for
               you
               shall
               find
               the
               number
               of
               minutes
               you
               must
               add
               to
               or
               substract
               from
               the
               declination
               found
               for
               that
               day
               and
               year
               :
               Example
               ,
               suppose
               for
               
                 April
              
               10.
               1662.
               the
               second
               after
               leap
               year
               ,
               the
               rule
               sheweth
               me
               11.
               45
               ,
               from
               which
               the
               rectifying
               table
               saith
               I
               must
               substract
               3′
               then
               is
               the
               true
               declination
               11
               42
               ,
               the
               like
               for
               any
               other
               year
               .
               Note
               further
               that
               the
               space
               of
               a
               day
               in
               the
               suns
               swiftest
               motion
               being
               so
               much
               ,
               you
               
               may
               consider
               the
               hour
               of
               the
               day
               also
               ,
               in
               the
               finding
               of
               a
               latitude
               ,
               by
               an
               observation
               taken
               of
               the
               Meridian
               ,
               as
               anon
               you
               shall
               see
               that
               as
               the
               instrument
               is
               exact
               ,
               so
               let
               your
               arithmetical
               calculation
               be
               also
               :
               by
               laying
               a
               sure
               foundation
               to
               begin
               to
               work
               upon
               ,
               then
               will
               your
               latitude
               be
               very
               true
               also
               .
            
             
               
                 
                   A
                   Rectifying
                   Table
                   for
                   the
                   Suns
                   declination
                   .
                
                 
                   
                      
                  
                   
                     D
                  
                   
                     1
                     year
                  
                   
                     2
                     year
                  
                   
                     3
                     year
                  
                   
                     Leap
                     year
                  
                
                 
                   
                     Ianuary
                  
                   
                     7
                  
                   
                     Sub
                     5
                  
                   
                     Sub
                     2
                  
                   
                     Add
                     1
                  
                   
                     Add
                     4
                  
                
                 
                   
                     15
                  
                   
                     s
                     6
                  
                   
                     s
                     2
                  
                   
                     a
                     2
                  
                   
                     a
                     5
                  
                
                 
                   
                     22
                  
                   
                     s
                     7
                  
                   
                     s
                     3
                  
                   
                     a
                     1
                  
                   
                     a
                     6
                  
                
                 
                   
                     30
                  
                   
                     s
                     7
                  
                   
                     s
                     3
                  
                   
                     a
                     2
                  
                   
                     a
                     7
                  
                
                 
                   
                     Februar
                  
                   
                     7
                  
                   
                     s
                     8
                  
                   
                     s
                     3
                  
                   
                     a
                     2
                  
                   
                     a
                     7
                  
                
                 
                   
                     15
                  
                   
                     s
                     8
                  
                   
                     s
                     4
                  
                   
                     a
                     2
                  
                   
                     a
                     8
                  
                
                 
                   
                     22
                  
                   
                     s
                     9
                  
                   
                     s
                     3
                  
                   
                     a
                     2
                  
                   
                     a
                     7
                  
                
                 
                   
                     March
                  
                   
                     1
                  
                   
                     s
                     4
                  
                   
                     s
                     1
                  
                   
                     a
                     7
                  
                   
                     Sub
                     11
                  
                
                 
                   
                     7
                  
                   
                     s
                     3
                  
                   
                     Ad
                     3
                  
                   
                     Add
                     9
                  
                   
                     Sub
                     9
                  
                
                 
                   
                     15
                  
                   
                     Add
                     3
                  
                   
                     ad
                     1
                  
                   
                     Sub
                     9
                  
                   
                     Add
                     9
                  
                
                 
                   
                     22
                  
                   
                     a
                     2
                  
                   
                     Sub
                     4
                  
                   
                     Sub
                     9
                  
                   
                     ad
                     8
                  
                
                 
                   
                     30
                  
                   
                     a
                     1
                  
                   
                     Sub
                     4
                  
                   
                     Sub
                     9
                  
                   
                     ad
                     8
                  
                
                 
                   
                     April
                  
                   
                     7
                  
                   
                     a
                     2
                  
                   
                     Sub
                     3
                  
                   
                     s
                     8
                  
                   
                     a
                     8
                  
                
                 
                   
                     15
                  
                   
                     a
                     2
                  
                   
                     s
                     3
                  
                   
                     s
                     8
                  
                   
                     a
                     7
                  
                
                 
                   
                     22
                  
                   
                     a
                     1
                  
                   
                     s
                     3
                  
                   
                     s
                     8
                  
                   
                     a
                     6
                  
                
                 
                   
                     30
                  
                   
                     a
                     1
                  
                   
                     s
                     3
                  
                   
                     s
                     6
                  
                   
                     a
                     5
                  
                
                 
                   
                     May
                  
                   
                     7
                  
                   
                     a
                     0
                  
                   
                     s
                     2
                  
                   
                     s
                     6
                  
                   
                     a
                     5
                  
                
                 
                   
                     15
                  
                   
                     a
                     0
                  
                   
                     s
                     2
                  
                   
                     s
                     5
                  
                   
                     a
                     3
                  
                
                 
                   
                     22
                  
                   
                     a
                     0
                  
                   
                     s
                     1
                  
                   
                     s
                     3
                  
                   
                     a
                     3
                  
                
                 
                   
                     30
                  
                   
                     a
                     0
                  
                   
                     s
                     1
                  
                   
                     s
                     2
                  
                   
                     a
                     1
                  
                
                 
                   
                     Iune
                  
                   
                     7
                  
                   
                     a
                     0
                  
                   
                     s
                     1
                  
                   
                     s
                     1
                  
                   
                     ad
                     0
                  
                
                 
                   
                     15
                  
                   
                     a
                     0
                  
                   
                     s
                     0
                  
                   
                     s
                     0
                  
                   
                     Sub
                     0
                  
                
                 
                   
                     22
                  
                   
                     s
                     1
                  
                   
                     s
                     0
                  
                   
                     ad
                     2
                  
                   
                     s
                     2
                  
                
                 
                   
                     30
                  
                   
                     s
                     1
                  
                   
                     Ad
                     1
                  
                   
                     ad
                     3
                  
                   
                     s
                     3
                  
                
                 
                   
                     
                     Iuly
                  
                   
                     7
                  
                   
                     Sub
                     2
                  
                   
                     Add
                     1
                  
                   
                     Add
                     3
                  
                   
                     Sub
                     4
                  
                
                 
                   
                     15
                  
                   
                     s
                     1
                  
                   
                     a
                     1
                  
                   
                     a
                     5
                  
                   
                     s
                     5
                  
                
                 
                   
                     22
                  
                   
                     s
                     2
                  
                   
                     a
                     1
                  
                   
                     a
                     5
                  
                   
                     s
                     6
                  
                
                 
                   
                     30
                  
                   
                     s
                     2
                  
                   
                     a
                     2
                  
                   
                     a
                     6
                  
                   
                     s
                     7
                  
                
                 
                   
                     August
                  
                   
                     7
                  
                   
                     s
                     3
                  
                   
                     a
                     2
                  
                   
                     a
                     7
                  
                   
                     s
                     8
                  
                
                 
                   
                     15
                  
                   
                     s
                     3
                  
                   
                     a
                     2
                  
                   
                     a
                     7
                  
                   
                     s
                     8
                  
                
                 
                   
                     22
                  
                   
                     s
                     3
                  
                   
                     a
                     3
                  
                   
                     a
                     8
                  
                   
                     s
                     9
                  
                
                 
                   
                     30
                  
                   
                     s
                     3
                  
                   
                     a
                     3
                  
                   
                     a
                     9
                  
                   
                     s
                     9
                  
                
                 
                   
                     September
                  
                   
                     7
                  
                   
                     s
                     3
                  
                   
                     a
                     3
                  
                   
                     Add
                     9
                  
                   
                     Sub
                     9
                  
                
                 
                   
                     15
                  
                   
                     Add
                     3
                  
                   
                     Sub
                     3
                  
                   
                     Sub
                     9
                  
                   
                     Add
                     9
                  
                
                 
                   
                     22
                  
                   
                     a
                     2
                  
                   
                     s
                     4
                  
                   
                     s
                     9
                  
                   
                     a
                     8
                  
                
                 
                   
                     30
                  
                   
                     a
                     2
                  
                   
                     s
                     4
                  
                   
                     s
                     9
                  
                   
                     a
                     8
                  
                
                 
                   
                     October
                  
                   
                     7
                  
                   
                     a
                     3
                  
                   
                     s
                     3
                  
                   
                     s
                     9
                  
                   
                     a
                     8
                  
                
                 
                   
                     15
                  
                   
                     a
                     2
                  
                   
                     s
                     3
                  
                   
                     s
                     8
                  
                   
                     a
                     7
                  
                
                 
                   
                     22
                  
                   
                     a
                     2
                  
                   
                     s
                     2
                  
                   
                     s
                     7
                  
                   
                     a
                     8
                  
                
                 
                   
                     30
                  
                   
                     a
                     2
                  
                   
                     s
                     2
                  
                   
                     s
                     7
                  
                   
                     a
                     8
                  
                
                 
                   
                     November
                  
                   
                     7
                  
                   
                     a
                     1
                  
                   
                     s
                     1
                  
                   
                     s
                     6
                  
                   
                     a
                     7
                  
                
                 
                   
                     15
                  
                   
                     a
                     1
                  
                   
                     s
                     2
                  
                   
                     s
                     5
                  
                   
                     a
                     5
                  
                
                 
                   
                     22
                  
                   
                     a
                     1
                  
                   
                     s
                     3
                  
                   
                     s
                     4
                  
                   
                     a
                     3
                  
                
                 
                   
                     30
                  
                   
                     a
                     0
                  
                   
                     s
                     2
                  
                   
                     s
                     3
                  
                   
                     a
                     2
                  
                
                 
                   
                     December
                  
                   
                     7
                  
                   
                     0
                  
                   
                     s
                     1
                  
                   
                     s
                     1
                  
                   
                     Add
                     1
                  
                
                 
                   
                     15
                  
                   
                     Sub
                     1
                  
                   
                     s
                     0
                  
                   
                     s
                     0
                  
                   
                     Sub
                     1
                  
                
                 
                   
                     22
                  
                   
                     s
                     1
                  
                   
                     s
                     0
                  
                   
                     Add
                     1
                  
                   
                     s
                     3
                  
                
                 
                   
                     30
                  
                   
                     s
                     3
                  
                   
                     s
                     0
                  
                   
                     Add
                     2
                  
                   
                     s
                     5
                  
                
              
            
          
           
             
             
               The
               Declination
               of
               the
               Sun
               being
               given
               ,
               or
               rather
               the
               Suns
               Distance
               from
               the
               Pole
               ,
               and
               the
               Complement
               of
               two
               Altitudes
               of
               the
               Sun
               ,
               taken
               at
               any
               time
               of
               the
               day
               ,
               knowing
               the
               time
               between
               :
               to
               find
               the
               Latitude
               .
            
             
               Suppose
               on
               the
               11.
               of
               
                 Iune
                 ,
              
               the
               Sun
               being
               66
               degrees
               ,
               29′
               distant
               from
               the
               North
               Pole
               ,
               and
               the
               Complement
               of
               one
               altitude
               be
               80.
               30.
               and
               the
               Complement
               of
               another
               altitude
               44.
               13.
               and
               the
               time
               between
               the
               two
               observations
               just
               four
               hours
               ;
               then
               say
               ,
            
             
               
                 
                   
                     As
                     the
                     Sine
                     of
                  
                   
                     90
                     00
                  
                
                 
                   
                     To
                     Sine
                     of
                     Suns
                     dist
                     .
                     f.
                     the
                     Pole
                  
                   
                     66
                     28
                  
                
                 
                   
                     So
                     is
                     the
                     Sine
                     of
                     ½
                     time
                     betw
                     .
                  
                   
                     30
                     00
                  
                
                 
                   
                     To
                     the
                     Sine
                     of
                     ½
                     the
                     3d.
                     side
                  
                   
                     27
                     17½
                  
                
                 
                   
                      
                  
                   
                     27
                     17½
                  
                
                 
                   
                     of
                     a
                     Triangle
                     as
                     A
                     B.
                     
                  
                   
                     54
                     35
                  
                
              
               
                 
                   
                     The
                     side
                     A
                     P
                  
                   
                     66
                     28
                  
                
                 
                   
                     The
                     side
                     P
                     A
                  
                   
                     66
                     28
                  
                
                 
                   
                     The
                     side
                     A
                     B
                  
                   
                     54
                     35
                  
                
                 
                   
                     whole
                     sum
                  
                   
                     187
                     31
                  
                
                 
                   
                     half
                     sum
                  
                   
                     93
                     46
                  
                
                 
                   
                     The
                     differ
                     .
                     betw
                     ½
                     sum
                     and
                     AP
                     side
                     op
                     .
                     to
                     inqu
                     .
                     Triangle
                  
                   
                     27
                     18
                  
                
                 
                   
                     Then
                     say
                     ,
                     as
                     S.
                     of
                     90
                  
                   
                     90
                     00
                  
                
                 
                   
                     To
                     S.
                     of
                     Suns
                     dist
                     .
                     from
                     Pole
                  
                   
                     66
                     28
                  
                
                 
                   
                     
                     So
                     is
                     the
                     Sine
                     of
                     A
                     B
                  
                   
                     54
                     35
                  
                
                 
                   
                     To
                     the
                     Sine
                     of
                     a
                     fourth
                     Sine
                  
                   
                     48
                     23
                  
                
                 
                   
                     Then
                     as
                     that
                     4
                     to
                     Sine
                     of
                     ½
                     sum
                  
                   
                     93
                     46
                  
                
                 
                   
                     So
                     is
                     S.
                     of
                     the
                     difference
                  
                   
                     27
                     18
                  
                
                 
                   
                     To
                     a
                     seventh
                     Sine
                  
                   
                     37
                     44½
                  
                
                 
                   
                     or
                     the
                     versed
                     Sine
                     of
                     P
                     A
                     B
                  
                   
                     77
                     02
                  
                
              
               
                 
                   
                     Then
                     to
                     find
                     Z
                     A
                     B
                     Z
                     B
                     is
                  
                   
                     80
                     30
                  
                
                 
                   
                     And
                     Z
                     A
                     is
                  
                   
                     44
                     13
                  
                
                 
                   
                     The
                     former
                     side
                     A
                     B
                     is
                  
                   
                     54
                     35
                  
                
                 
                   
                     Sum
                  
                   
                     179
                     18
                  
                
                 
                   
                     half
                     sum
                  
                   
                     89
                     39
                  
                
                 
                   
                     difference
                  
                   
                     09
                     09
                  
                
                 
                   
                     As
                     S.
                     of
                     90
                  
                   
                     90
                     00
                  
                
                 
                   
                     To
                     Sine
                     of
                     A
                     B
                  
                   
                     54
                     35
                  
                
                 
                   
                     So
                     is
                     S.
                     of
                     Z
                     A
                  
                   
                     44
                     13
                  
                
                 
                   
                     To
                     a
                     4th
                     .
                     Sine
                  
                   
                     34
                     38
                  
                
              
               
                 
                   
                     As
                     S.
                     4th
                     .
                  
                   
                     34
                     38
                  
                
                 
                   
                     To
                     S.
                     of
                     ½
                     sum
                  
                   
                     89
                     39
                  
                
                 
                   
                     So
                     is
                     S.
                     diff
                     .
                  
                   
                     09
                     39
                  
                
                 
                   
                     To
                     a
                     7th
                     .
                     Sine
                  
                   
                     16
                     15¼
                  
                
                 
                   
                     or
                     to
                     the
                     vers
                     .
                     sine
                     of
                     Z
                     A
                     B
                  
                   
                     116
                     07
                  
                
              
            
             
               Then
               if
               you
               take
               P
               A
               B
               from
               Z
               A
               B
               there
               will
               remain
               Z
               A
               P
               39
               05
               Then
               say
               again
               by
               the
               rule
               as
               before
               ,
            
             
             
               
                 
                   
                     As
                     the
                     sine
                     of
                  
                   
                     90
                     00
                  
                
                 
                   
                     To
                     Co-sine
                     of
                     Z
                     A
                     P
                  
                   
                     50
                     55
                  
                
                 
                   
                     So
                     is
                     the
                     Tang
                     of
                     A
                     Z
                  
                   
                     44
                     13
                  
                
                 
                   
                     To
                     the
                     Tangent
                     of
                     A
                     C
                  
                   
                     37
                     04
                  
                
                 
                   
                     which
                     taken
                     from
                     A
                     P
                  
                   
                     66
                     28
                  
                
                 
                   
                     remaineth
                     P
                     C
                  
                   
                     29
                     24
                  
                
                 
                   
                     Then
                     lastly
                     say
                     ,
                  
                   
                      
                  
                
                 
                   
                     As
                     the
                     Co-sine
                     of
                     A
                     C
                  
                   
                     52
                     56
                  
                
                 
                   
                     To
                     the
                     Co-sine
                     of
                     C
                     P
                  
                   
                     60
                     36
                  
                
                 
                   
                     So
                     is
                     the
                     Co-sine
                     of
                     Z
                     A
                  
                   
                     45
                     47
                  
                
                 
                   
                     To
                     the
                     Co-sine
                     of
                     Z
                     P
                  
                   
                     51
                     30
                  
                
              
            
             
               The
               Latitude
               required
               to
               be
               found
               .
               This
               Question
               or
               any
               other
               may
               be
               wrought
               by
               the
               Sines
               and
               Tangents
               and
               versed
               sines
               on
               the
               rule
               ,
               But
               if
               you
               would
               know
               more
               as
               concerning
               this
               or
               any
               other
               ,
               you
               may
               be
               fully
               satisfied
               by
               Mr.
               
                 Euclid
                 Spidal
              
               at
               his
               Chamber
               at
               a
               Virginal
               Makers
               house
               in
               
                 Thred-needle
                 Street
                 ,
              
               and
               at
               the
               
                 Kings
                 head
              
               neer
               
                 Broadstreet
              
               end
               .
            
          
        
         
           
             Vale.
             
          
        
         
           FINIS
           .
        
      
       
         
         
           
        
         
           The
           Triangle
           explained
           .
        
         
           S
           P
           Z
           N
           a
           Meridian
           circl
           .
           Ae
           Ae
           the
           Equinoctial
           .
           ♋
           B
           A
           ♋
           the
           ,
           tropick
           of
           
             Cancer
             .
          
           P
           B
           5
           P
           the
           hour
           circle
           of
           Five
           ,
           
             ante
             ,
             mer.
          
           P
           A
           9
           P
           the
           hour
           circle
           of
           Nine
           ,
           
             ante
             ,
             mer.
             
          
        
         
           
             
               
                 Z
                 B
                 the
                 Suns
                 coaltitude
                 at
                 5
                 —
              
               
                 80
                 —
                 30
              
            
             
               
                 Z
                 A
                 the
                 Suns
                 coaltitude
                 at
                 9
                 —
              
               
                 44
                 —
                 13
              
            
             
               
                 P
                 A
                 &
                 P
                 B
                 the
                 Suns
                 distance
                 from
                 the
                 Pole
                 on
                 the
                 2
                 h.
                 li.
                 of
                 5
                 &
                 9
              
               
                 66
                 —
                 28
              
            
             
               
                 5-9
                 the
                 equinoctial
                 time
                 between
                 the
                 two
                 Observations
                 —
              
               
                 60
                 —
                 00
              
            
             
               
                 B
                 A
                 in
                 proper
                 measure
                 is
                 as
                 found
                 by
                 the
                 first
                 measure
                 working
                 —
              
               
                 54
                 —
                 35
              
            
             
               
                 Z
                 C
                 a
                 perpendicular
                 on
                 A
                 P
                 from
                 Z
              
               
                 29
                 —
                 24
              
            
             
               
                 P
                 Z
                 the
                 complement
                 of
                 the
                 latitude
                 that
                 was
                 to
                 be
                 found
                 —
              
               
                 38-30