A collection of centers and useful proportions on the line of numbers by John Brown ...
         Brown, John, philomath.
      
       
         
           1670
        
      
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         97768
         
           
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             A collection of centers and useful proportions on the line of numbers by John Brown ...
             Brown, John, philomath.
          
           16 p.
           
             s.n.,
             [London :
             1670]
          
           
             Caption title.
             Place and date of publication suggested by Wing.
             Reproduction of original in the British Library.
          
        
      
    
     
       
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         eng
      
       
         
           Mathematics -- Formulae.
           Weights and measures -- Tables.
        
      
    
     
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           A
           Collection
           of
           
             Centers
          
           and
           useful
           Proportions
           on
           the
           Line
           of
           Numbers
           ,
           by
           
             Iohn
             Brown
          
           at
           the
           
             Sphear
          
           and
           
             Sun-Dial
          
           in
           the
           
             Minories
             .
          
        
         
           
             In
             Multiplication
             .
          
           
             AS
             1
             to
             the
             Multiplicator
             ,
             so
             is
             the
             Multiplicand
             to
             the
             Product
             ,
             consisting
             of
             as
             many
             Figures
             ,
             as
             the
             Multiplicator
             and
             Multiplicand
             ,
             and
             sometimes
             of
             one
             less
             ,
             when
             the
             2
             first
             Figures
             of
             the
             Product
             are
             greatest
             .
             And
             more
             than
             4
             figures
             in
             the
             Product
             ,
             must
             be
             adjusted
             by
             multiplication
             of
             the
             last
             figures
             by
             the
             Pen
             or
             Head.
             
          
        
         
           
             In
             Division
          
           
             As
             the
             Divisor
             to
             1
             ,
             so
             is
             the
             Dividend
             to
             the
             Quotient
             ,
             containing
             as
             many
             figures
             as
             the
             Divisor
             may
             be
             set
             times
             under
             the
             Dividend
             by
             the
             Pen
             ,
             and
             the
             fraction
             remaining
             is
             a
             Decimal
             Fraction
             ,
             and
             reduced
             to
             the
             Vulgar
             Fraction
             ,
             by
             setting
             the
             sa●e
             extent
             the
             contrary
             way
             from
             the
             Decimal
             Fraction
             ,
             gives
             the
             Vulgar
             Fraction
             .
          
        
         
           
             Of
             mean
             Proportions
          
           
             The
             Arithmetical
             mean
             ,
             is
             the
             half
             sum
             of
             the
             two
             extreams
             added
             .
          
           
           
             The
             Geometrical
             mean
             ,
             is
             the
             Square
             Root
             of
             the
             Product
             of
             the
             two
             extreams
             multiplyed
             together
             :
             Or
             the
             middle
             between
             the
             two
             extreams
             ▪
             counted
             on
             a
             line
             of
             Numbers
             .
          
           
             The
             Musical
             is
             thus
             found
             ;
             As
             the
             Arithmetical
             mean
             ,
             to
             one
             extream
             ,
             so
             is
             the
             other
             extream
             to
             the
             musical
             mean
             proportion
             required
             .
          
        
         
           
             Of
             Square
             and
             Cube
             Roots
             .
          
           
             The
             middle
             between
             1
             ,
             and
             any
             other
             Number
             more
             or
             less
             than
             1
             ,
             is
             the
             Square
             Root
             of
             that
             Number
             ,
             being
             measured
             on
             a
             fit
             Line
             of
             Numbers
             ;
             the
             exact
             third
             part
             between
             1
             and
             any
             other
             Number
             ,
             more
             or
             less
             then
             1
             ,
             counting
             from
             1
             is
             the
             Cube
             Root
             of
             that
             Number
             ,
             counted
             on
             a
             true
             Line
             of
             Numbers
             .
          
           
             The
             exact
             fourth
             part
             ,
             between
             1
             and
             any
             other
             Number
             ▪
             counted
             on
             a
             Line
             of
             Numbers
             ,
             from
             1
             ,
             is
             the
             Byquadrat
             Root
             of
             that
             Number
             ,
             
               &c.
               
            
          
        
         
           
             Of
             Reduction
             .
          
           
             As
             one
             Denominator
             to
             his
             Numerator
             ,
             so
             is
             the
             other
             Denominator
             to
             the
             inquired
             Numerator
             .
          
           
             Or
             ,
             As
             one
             Denominator
             to
             the
             other
             ;
             so
             is
             one
             Numerator
             to
             the
             other
             required
             .
          
           
             
               Note
               ▪
            
             That
             the
             Line
             of
             Numbers
             and
             Pence
             together
             ,
             reduceth
             Decimals
             of
             a
             pound
             ,
             to
             shillings
             and
             pence
             ,
             and
             the
             contrary
             ,
             by
             inspection
             only
             .
          
           
           
             Thus
             ,
             ,001
             is
             neer
             9
             farthings
             ⅗
             ▪
             ,0●●●
             just
             2
             
               s.
            
             or
             1
             ,
             10th
             ,
             of
             
               al.
            
             or
             20
             
               s.
            
             ,0125
             is
             2
             
               s.
            
             6
             
               d.
            
             and
             1,532
             is
             1
             
               l.
            
             10
             
               s.
            
             7
             
               d.
            
             2
             
               f.
            
             ¾
             and
             so
             for
             any
             other
             ;
             for
             1
             is
             1
             
               l.
            
             ,05
             is
             10
             
               s.
            
             and
             just
             against
             ,032
             is
             7
             
               d.
            
             2
             
               f.
            
             ¾
             the
             exact
             Reduction
             .
          
        
         
           
             In
             the
             Rule
             of
             Three
             Direct
             .
          
           
             As
             the
             first
             term
             ,
             to
             the
             second
             ,
             so
             is
             the
             third
             to
             the
             fourth
             required
             .
          
           
             Or
             ,
             As
             a
             greater
             to
             a
             less
             ,
             so
             is
             another
             greater
             to
             another
             less
             ,
             extending
             the
             same
             way
             ,
             from
             the
             third
             to
             the
             fourth
             ,
             as
             from
             the
             first
             to
             the
             second
             .
          
           
             And
             so
             the
             contrary
             ,
             As
             a
             lesser
             term
             to
             a
             greater
             ,
             so
             is
             another
             less
             to
             another
             greater
             extending
             the
             same
             way
             .
          
        
         
           
             In
             Superficial
             Measure
             .
          
           
             1.
             
             As
             the
             breadth
             in
             inches
             to
             12
             ,
             so
             is
             12
             to
             the
             inches
             long
             to
             make
             1
             superficial
             foot
             .
          
           
             2.
             
             As
             the
             breadth
             in
             foot
             measure
             to
             1
             ,
             so
             is
             1
             to
             the
             length
             in
             foot
             measure
             ,
             to
             make
             1
             foot
             of
             Superficial
             measure
             .
          
           
             3.
             
             As
             1
             to
             the
             length
             in
             feet
             ,
             so
             is
             the
             breadth
             in
             feet
             to
             the
             content
             in
             feet
             .
          
           
             4.
             
             As
             12
             to
             the
             breadth
             in
             inches
             ,
             so
             is
             the
             length
             in
             feet
             to
             the
             content
             in
             feet
             .
          
           
             5.
             
             As
             1
             inch
             to
             the
             breadth
             in
             inches
             ,
             so
             is
             the
             length
             in
             inches
             to
             the
             content
             in
             inches
             .
          
           
             6.
             
             As
             144
             to
             the
             breadth
             in
             inches
             ,
             so
             is
             
             the
             length
             in
             inches
             to
             the
             content
             in
             feet
             .
          
        
         
           
             Of
             Land-measure
             .
          
           
             1.
             
             As
             9
             to
             the
             length
             in
             feet
             ;
             so
             is
             the
             breadth
             in
             feet
             to
             the
             content
             in
             yards
             .
          
           
             2.
             
             As
             
               14f
               .
            
             0625
             the
             feet
             in
             a
             Square
             Ell
             to
             the
             breadth
             in
             feet
             ,
             so
             is
             the
             length
             in
             feet
             ,
             to
             the
             content
             in
             Superficial
             Ells.
             
          
           
             3.
             
             As
             1
             Perch
             to
             the
             breadth
             in
             Perches
             ,
             so
             is
             the
             length
             in
             Perches
             ,
             to
             the
             content
             in
             like
             Perches
             ;
             16
             foot
             and
             a
             half
             to
             a
             Perch
             .
          
           
             4.
             
             As
             160
             (
             the
             Perches
             in
             1
             Acre
             )
             to
             the
             breadth
             in
             Perches
             ,
             so
             is
             the
             length
             in
             Perches
             to
             the
             content
             in
             Acres
             .
          
           
             5.
             
             As
             10
             to
             the
             breadth
             in
             Chains
             and
             Links
             ,
             so
             is
             the
             length
             in
             Chains
             and
             Links
             ,
             to
             the
             content
             in
             Acres
             ,
             (
             because
             10
             Square
             Chain
             is
             1
             Acre
             )
             .
          
           
             6.
             
             As
             the
             length
             of
             a
             customary
             Pearch
             ,
             is
             to
             the
             length
             of
             the
             Statute
             Perch
             ,
             so
             is
             the
             content
             in
             Statute
             Acres
             known
             ,
             to
             the
             content
             in
             customary
             Acres
             required
             ,
             at
             two
             repetions
             or
             turning
             the
             Compasses
             twice
             .
          
           
             7.
             
             To
             find
             by
             what
             Scale
             a
             Field
             is
             plotted
             ,
             measure
             by
             a
             Scale
             as
             near
             as
             you
             can
             estimate
             ,
             then
             the
             middle
             between
             the
             
               Area
               ,
            
             found
             by
             the
             estimated
             Scale
             ,
             and
             the
             Area
             set
             down
             in
             the
             Plot
             ;
             and
             the
             
               Area
            
             found
             by
             the
             estimated
             Scale
             ,
             shall
             reach
             f●om
             the
             estimated
             Scale
             to
             the
             true
             Scale
             by
             which
             the
             draught
             was
             plotted
             .
          
           
           
             8.
             
             As
             the
             length
             of
             any
             Oblong
             in
             Perches
             to
             160
             ,
             so
             is
             1
             Acre
             to
             the
             breadth
             in
             Perches
             required
             to
             make
             160
             Perch
             or
             1
             Acre
             .
          
           
             Or
             ,
             As
             the
             length
             in
             Chains
             to
             1
             ,
             so
             is
             1
             to
             the
             breadth
             in
             Chains
             for
             1
             Acre
             .
          
           
             Or
             ,
             As
             the
             length
             in
             feet
             to
             43560
             ,
             (
             the
             feet
             contained
             in
             a
             Square
             Acre
             )
             so
             is
             1
             to
             the
             breadth
             in
             feet
             to
             make
             up
             1
             Acre
             .
          
        
         
           
             Of
             Triangles
             .
          
           
             1
             As
             2
             to
             the
             Base
             ,
             or
             longest
             side
             of
             any
             Triangle
             ,
             so
             is
             the
             Perpendicular
             to
             the
             
               Area
            
             of
             the
             Triangle
             required
             ,
             in
             like
             parts
             .
          
           
             2
             The
             sides
             given
             ,
             to
             find
             the
             Perpendicular
             ,
             square
             each
             side
             severally
             ,
             then
             add
             the
             greatest
             ,
             and
             least
             Squares
             together
             ,
             and
             from
             the
             half
             Sum
             of
             them
             ,
             substract
             the
             remaining
             Square
             ,
             noteing
             the
             difference
             or
             remainder
             ;
             half
             this
             Remainder
             divide
             by
             the
             longest
             side
             ,
             and
             the
             quote
             sheweth
             where
             the
             Perpendicular
             will
             fall
             on
             the
             Base
             .
          
           
             Then
             square
             the
             Quotient
             ,
             and
             substract
             it
             from
             the
             least
             Square
             first
             added
             ,
             then
             the
             Square
             Root
             of
             the
             remainder
             is
             the
             Perpendicular
             required
             for
             that
             Triangle
             .
          
           
             3.
             
             To
             find
             the
             
               Area
            
             by
             the
             sides
             only
             ,
          
           
             Add
             the
             3
             sides
             together
             ,
             to
             get
             the
             sum
             and
             then
             the
             half
             sum
             ,
             and
             then
             find
             the
             difference
             between
             the
             half
             sum
             and
             each
             side
             severally
             .
          
           
             Then
             multiply
             one
             difference
             by
             another
             
             difference
             ,
             and
             then
             this
             Product
             by
             the
             remaining
             difference
             ;
             and
             lastly
             ,
             this
             second
             Product
             ,
             by
             the
             half
             Sum
             ,
             after
             the
             manner
             of
             continual
             Multiplication
             .
          
           
             Then
             Lastly
             ,
             the
             square
             Root
             of
             this
             last
             Product
             ,
             is
             the
             Area
             or
             Content
             of
             the
             Triangle
             required
             ▪
             The
             three
             sides
             6
             ,
             8
             ,
             and
             10
             ,
             have
             24
             for
             the
             Area
             .
          
           
             4
             All
             irregular
             many
             sided
             figures
             ,
             are
             reduced
             to
             Triangles
             ,
             by
             drawing
             of
             Lines
             ,
             and
             then
             measured
             by
             the
             first
             Rule
             of
             Triangles
             ,
             or
             two
             Triangles
             at
             once
             ,
             by
             one
             common
             Base
             and
             two
             Perpendiculars
             .
          
           
             5
             Regular
             Polligon
             ,
             of
             5
             ,
             6
             ,
             7
             ,
             8
             ,
             or
             more
             sides
             ,
             are
             measured
             at
             one
             Multiplication
             ,
             thus
             ,
             As
             1
             to
             the
             Radius
             or
             Measure
             from
             the
             Center
             to
             the
             middle
             of
             one
             side
             ,
             so
             is
             the
             half
             Sum
             of
             the
             measure
             of
             the
             sides
             to
             the
             content
             .
          
        
         
           
             Of
             the
             Circle
             and
             Elipsis
             .
          
           
             1
             As
             10
             ,
             as
             a
             fixed
             Diameter
             ,
             to
             any
             other
             Diameter
             ;
             so
             is
             31,416
             the
             fixed
             circumference
             where
             10
             is
             Diameter
             ,
             to
             the
             circumference
             required
             .
          
           
             2
             As
             31,416
             the
             fixed
             circumference
             ,
             to
             any
             other
             circumference
             ,
             so
             is
             10
             a
             fixed
             Diameter
             to
             the
             Diameter
             required
             ▪
          
           
             3
             As
             10
             to
             8,862
             ,
             the
             fixed
             side
             of
             a
             square
             equal
             to
             a
             Circle
             ,
             whose
             Diameter
             is
             10
             ▪
          
           
             so
             is
             any
             other
             Diameter
             given
             ,
             to
             his
             proportional
             Square
             equal
             required
             .
          
           
           
             4
             As
             10
             to
             7
             ●71
             the
             side
             of
             the
             inscribed
             Square
             ,
             so
             is
             any
             other
             Diameter
             to
             his
             proportional
             side
             of
             the
             inscribed
             square
             required
             .
          
           
             5
             As
             8,862
             ,
             to
             10
             ,
             so
             is
             the
             side
             of
             any
             other
             square
             equal
             to
             his
             Diameter
             required
             .
          
           
             6
             As
             10
             to
             any
             other
             Diameter
             ,
             so
             is
             78,540
             the
             Area
             of
             a
             Circle
             whose
             Diameter
             is
             10
             ,
             twice
             repeated
             to
             the
             Area
             required
             .
          
           
             7
             As
             31,416
             to
             any
             other
             Circumference
             given
             ,
             so
             is
             78,540
             the
             fixed
             Area
             ,
             to
             the
             Area
             required
             at
             twice
             turning
             the
             Compasses
             .
          
           
             8
             The
             half
             distance
             on
             a
             line
             of
             Numbers
             ,
             between
             78,540
             and
             any
             other
             Area
             given
             ,
             shall
             reach
             from
             10
             the
             fixed
             Diameter
             ,
             to
             the
             Diameter
             required
             .
          
           
             And
             from
             31,416
             the
             fixed
             circumference
             to
             the
             circumference
             required
             .
          
           
             And
             from
             8,862
             the
             fixed
             side
             of
             the
             square
             equal
             ,
             to
             the
             side
             of
             the
             square
             equal
             required
             .
          
           
             And
             Lastly
             ,
             from
             7,071
             to
             the
             side
             of
             the
             square
             inscribed
             in
             the
             Circle
             required
             .
          
           
             9
             A
             mean
             Proportion
             Geometrical
             ,
             between
             the
             longest
             and
             shortest
             Diameters
             of
             an
             Elipsis
             ,
             or
             Oval
             ,
             is
             the
             diameter
             of
             a
             circle
             ,
             equal
             in
             Area
             to
             the
             Oval
             .
          
           
             Then
             having
             the
             Diameter
             ,
             you
             have
             the
             rest
             .
          
           
             10
             As
             100
             the
             square
             of
             10
             a
             fixed
             Diameter
             ,
             is
             to
             the
             Rectangle
             Figure
             made
             of
             the
             longest
             and
             shortest
             Diameters
             of
             a
             Oval
             ;
             so
             is
             78,540
             the
             Area
             for
             10
             ,
             to
             the
             Area
             of
             the
             Oval
             or
             Elipsis
             required
             .
          
           
           
             12
             As
             452
             ,
             to
             355
             ,
             or
             as
             14
             to
             11
             ,
             so
             is
             the
             square
             of
             any
             Circles
             Diameter
             to
             the
             Area
             .
          
           
             13
             As
             1
             to
             the
             Radius
             of
             a
             Circle
             ,
             so
             is
             the
             semicircle
             to
             the
             content
             of
             a
             Circle
             .
          
           
             14
             As
             1
             to
             the
             Radius
             of
             a
             Circle
             ,
          
           
             so
             is
             half
             the
             arch
             of
             any
             portion
             ,
             to
             the
             Area
             or
             Superficial
             Content
             of
             that
             Portion
             ,
             proceeding
             to
             the
             Center
             ,
             as
             a
             Quadrant
             ,
             a
             Sextance
             ,
             or
             the
             like
             part
             .
          
           
             15
             For
             other
             segments
             find
             the
             Diameter
             of
             the
             Circle
             or
             the
             Radius
             and
             length
             of
             the
             Arch
             ,
             by
             measuring
             :
             Or
             thus
             ,
             First
             ,
             for
             the
             Diameter
             of
             the
             whole
             Circle
             ,
             Square
             half
             the
             Diameter
             ,
             or
             Chord
             of
             the
             segment
             ,
             divide
             the
             Product
             by
             the
             segments
             Altitude
             or
             Side
             ,
             then
             the
             quote
             and
             sine
             added
             together
             shall
             be
             the
             Diameter
             required
             ,
             and
             half
             the
             Diameter
             is
             Radius
             .
          
           
             2
             And
             the
             Square
             Root
             of
             the
             Sum
             of
             the
             Segments
             Altitude
             ,
             and
             half
             the
             Segments
             Chord
             ,
             is
             the
             Chord
             of
             half
             the
             Segments
             Arch.
             
          
           
             3.
             
             Lastly
             ,
             the
             length
             of
             the
             Arch
             of
             the
             whole
             Segment
             is
             equal
             to
             twice
             the
             Chord
             of
             the
             half
             Segments
             Arch
             ,
             more
             1
             third
             part
             of
             the
             difference
             between
             them
             ,
             and
             the
             whole
             Segments
             Diameter
             .
          
           
             Then
             measure
             the
             whole
             Segment
             as
             before
             by
             Radius
             and
             half
             the
             Arch
             ,
             and
             then
             the
             Triangle
             substracted
             remains
             the
             Area
             of
             the
             Segment
             required
             .
          
        
         
           
           
             Of
             a
             Globe
             ,
             or
             Sphear
             ,
             and
             his
             Segments
             ,
             the
             Diameter
             and
             Circumference
             is
             the
             same
             as
             in
             a
             Circle
             ;
             the
             rest
             thus
             ,
          
           
             1
             As
             1
             to
             the
             Diameter
             in
             inches
             ,
             so
             is
             the
             Circumference
             in
             inches
             ,
             to
             the
             superficial
             Content
             round
             about
             the
             Sphear
             in
             inches
             likewise
             .
          
           
             2
             Or
             as
             10
             to
             the
             Diameter
             ,
             so
             is
             31,416
             twice
             ,
             to
             the
             Superficial
             Content
             round
             about
             .
          
           
             3
             As
             1
             to
             the
             Diameter
             ,
             so
             is
             the
             Diameter
             twice
             to
             the
             Cube
             of
             the
             Diameter
             ;
             then
             as
             21
             to
             11
             ,
             so
             is
             the
             Cube
             of
             the
             Diameter
             to
             the
             solidity
             .
          
           
             4
             Or
             the
             extent
             from
             10
             to
             any
             other
             Diameter
             ,
             being
             thrice
             repeated
             from
             523,60
             ,
             (
             the
             solid
             content
             of
             a
             Globe
             of
             10
             inches
             Diameter
             )
             gives
             the
             solid
             content
             required
             .
          
           
             5
             The
             extent
             from
             1
             to
             1,90986
             ,
             the
             Arithmetical
             Complement
             of
             523,5987
             ,
             the
             Area
             of
             that
             Globe
             whose
             Diameter
             is
             10
             ,
             shall
             reach
             from
             the
             Solid
             Con●ent
             ,
             to
             the
             Cube
             of
             the
             Diameter
             ,
             whose
             Cube
             Root
             ●s
             the
             Diameter
             required
             .
          
           
             6
             As
             any
             Gauge
             point
             ,
             for
             Wine
             or
             Beer
             Gallons
             ,
             to
             the
             Sphears
             Diameter
             ,
             so
             is
             two
             third
             parts
             of
             the
             Sphears
             Diameter
             twice
             ,
             to
             the
             solid
             content
             according
             to
             the
             Gauge
             Point
             .
          
           
             7
             As
             1
             to
             the
             Segments
             Diameter
             in
             Inches
             ,
             so
             is
             the
             ●egments
             half
             Altitude
             ,
             less
             by
             1
             ,
             8th
             thereof
             twice
             ●epeated
             ,
             to
             the
             neer
             solidity
             in
             inches
             required
             .
          
           
             8
             As
             1
             to
             the
             Sphears
             Circumference
             ,
             so
             is
             the
             Seg●ents
             Altitude
             ,
             to
             the
             Superficial
             Content
             of
             the
             sphe●cal
             part
             of
             the
             Segment
             ▪
          
           
             9
             As
             the
             Gauge
             Point
             for
             Wine
             or
             Ale
             Gallons
             ,
             to
             ●●e
             Segments
             D●ameters
             ,
             so
             is
             the
             Segment
             half
             Altitude
             ●ore
             by
             one
             eighth
             of
             the
             half
             Altitude
             twice
             repeated
             〈◊〉
             the
             Segments
             near
             Content
             in
             like
             Gallons
             ,
             to
             the
             ●auge
             Point
             .
          
           
           
             10
             As
             for
             Solid
             Angles
             of
             a
             Sphere
             ,
             they
             are
             reduced
             to
             Cones
             or
             Pyramids
             ,
             and
             so
             measured
             ,
             but
             for
             the
             Superficies
             of
             Spherical
             Triangles
             ,
             by
             Mr.
             
               Iohn
               Leak
            
             thus
             ,
          
           
             As
             1
             to
             the
             excess
             of
             the
             three
             Angles
             above
             180
             degso
             is
             Radius
             ,
             to
             the
             Superficial
             Content
             of
             the
             Spherical
             Triangle
             .
          
        
         
           
             Of
             Cones
             ,
             and
             Pyramids
             ,
             and
             Prismes
             .
          
           
             As
             1
             to
             the
             Area
             of
             the
             Base
             ,
             so
             is
             one
             third
             of
             the
             Perpendicular
             Altitude
             ,
             to
             the
             Solidity
             .
          
           
             1
             A
             Cone
             is
             best
             Gauged
             thus
             ,
          
           
             As
             the
             Gauge
             Point
             for
             round
             Vessels
             ,
             to
             the
             Cones
             Diame●er
             at
             the
             Base
             ,
             so
             is
             one
             third
             of
             the
             Cones
             Perpendicular
             Altitude
             twice
             ,
             to
             the
             Content
             in
             Wine
             or
             Beer
             Gallons
             :
             as
             the
             Gage
             Point
             was
             .
          
           
             2
             To
             get
             the
             Perpendicular
             height
             ,
          
           
             Square
             half
             the
             Diameter
             of
             the
             Base
             ;
             also
             square
             the
             Hypothenuse
             or
             slaunt
             height
             ,
             then
             take
             the
             lesser
             square
             from
             the
             greater
             ,
             and
             the
             Square
             Root
             of
             the
             residue
             ,
             is
             the
             Perpendicular
             height
             of
             the
             Cone
             required
             .
          
           
             3
             For
             Prismes
             and
             Pyramids
             say
             ,
          
           
             As
             the
             Gauge
             Point
             for
             square
             Vessels
             to
             the
             side
             of
             the
             Square
             ,
             equal
             to
             the
             Base
             of
             any
             Prisme
             or
             Piramid
             ,
             so
             is
             one
             third
             part
             of
             the
             Perpendicular
             Altitude
             twice
             ,
             to
             the
             content
             in
             Gallons
             ,
             according
             to
             the
             Gauge
             Point
             used
             .
          
           
             Or
             ,
             As
             the
             Gauge
             Point
             for
             Square
             Vessels
             is
             to
             the
             breadth
             of
             any
             Prisme
             ,
             so
             is
             the
             length
             to
             a
             4th
             .
             Again
             ,
             As
             the
             same
             Gauge
             Point
             to
             the
             4th
             .
             so
             is
             one
             third
             of
             the
             depth
             ,
             to
             the
             content
             in
             Gallons
             acording
             to
             the
             Gauge
             Point
             used
             .
          
           
           
             
               
                 A
                 Table
                 of
                 Cube
                 inches
                 and
                 Gauge
                 Points
                 for
                 these
                 Vessels
                 .
              
               
                 
                   Names
                   of
                   the
                   Vessels
                   .
                
                 
                   Cube
                   Inches
                
                 
                   GP
                   Round
                   ▪
                
                 
                   GP
                   Square
                
              
               
                 
                   Wine
                   Gallon
                   at
                
                 
                   231
                   0
                
                 
                   17
                   1485
                
                 
                   15
                   199
                
              
               
                 
                   Corn
                   Gallon
                   at
                
                 
                   272
                   25
                
                 
                   18
                   6168
                
                 
                   16
                   500
                
              
               
                 
                   Ale
                   Gallon
                   at
                
                 
                   288
                   00
                
                 
                   19
                   1485
                
                 
                   16
                   972
                
              
               
                 
                   Ale
                   Gallon
                   at
                
                 
                   282
                   00
                
                 
                   18
                   9468
                
                 
                   16
                   793
                
              
               
                 
                   Firkin
                   at
                   282
                   Ale
                
                 
                   2256
                   00
                
                 
                   53
                   5960
                
                 
                   47
                   498
                
              
               
                 
                   Firkin
                   at
                   282
                   Beer
                
                 
                   2538
                   00
                
                 
                   56
                   846
                
                 
                   48
                   552
                
              
               
                 
                   Kilder
                   .
                   at
                   282
                   Ale
                
                 
                   4512
                   00
                
                 
                   75
                   798
                
                 
                   67
                   180
                
              
               
                 
                   Kilder
                   .
                   at
                   282
                   Beer
                
                 
                   5076
                   00
                
                 
                   80
                   395
                
                 
                   7
                   246
                
              
               
                 
                   Barrel
                   at
                   282
                   Ale
                
                 
                   9024
                   00
                
                 
                   107
                   191
                
                 
                   95
                   000
                
              
               
                 
                   Barrel
                   at
                   282
                   Beer
                
                 
                   10152
                   00
                
                 
                   113
                   690
                
                 
                   100
                   58
                
              
               
                 
                   Barrel
                   at
                   288
                   Ale
                
                 
                   9216
                   00
                
                 
                   108
                   380
                
                 
                   96
                   000
                
              
               
                 
                   Barrel
                   at
                   288
                   Beer
                
                 
                   10368
                   00
                
                 
                   114
                   88
                
                 
                   101
                   823
                
              
               
                 
                   Corn
                   Bushel
                   at
                   272
                   1
                   ,
                   4th
                   ,
                   for
                   a
                   Gallon
                   ;
                
                 
                   2178
                   00
                
                 
                   52
                   6666
                
                 
                   46
                   668
                
              
               
                 
                   Coal
                   Bushel
                   at
                   280
                   ●
                   ,
                   4th
                   ,
                   to
                   a
                   gallon
                   ,
                
                 
                   2246
                   00
                
                 
                   53
                   500
                
                 
                   47
                   395
                
              
            
          
           
             The
             first
             column
             is
             the
             Number
             of
             Cube
             inches
             con●ained
             in
             any
             of
             these
             Vessels
             ,
             and
             serve
             as
             Divisors
             to
             bring
             any
             great
             Number
             of
             Inches
             ,
             to
             Gallons
             ,
             Barrels
             ,
             or
             Bushels
             ,
             accordingly
             .
          
           
             The
             Second
             is
             the
             Gauge
             Points
             for
             round
             Vessels
             ,
             be●ng
             the
             Diameters
             of
             Circles
             ,
             whose
             Content
             at
             1
             inch
             deep
             ,
             contains
             a
             Gallon
             ,
             or
             Barrel
             ,
             
               &c.
               
            
          
           
             The
             Third
             column
             are
             Square
             Roots
             of
             the
             First
             Co●umn
             ,
             and
             thus
             used
             as
             Gauge
             Points
             for
             Square
             Vessels
             .
          
           
             1
             As
             the
             Gauge
             Point
             to
             the
             side
             of
             the
             Square
             of
             any
             ●quare
             Vessel
             ,
             10
             is
             the
             depth
             twice
             to
             the
             content
             in
             Gallons
             or
             Barrels
             ,
             according
             to
             the
             Gauge
             Point
             used
             .
          
           
             2
             As
             the
             Gauge
             Point
             to
             the
             breadth
             of
             any
             square
             Vessel
             ,
             so
             is
             the
             length
             to
             a
             4th
             .
          
           
           
             Again
             ,
             As
             the
             Gauge
             Point
             to
             the
             4th
             so
             is
             the
             depth
             to
             the
             content
             as
             the
             Gauge
             Point
             was
             .
          
           
             3.
             
             As
             the
             Gauge
             Point
             for
             a
             round
             Vessel
             is
             to
             the
             equated
             Diameter
             ,
             so
             is
             the
             length
             twice
             ,
             to
             the
             content
             as
             the
             Gauge
             Point
             was
             .
          
           
             4.
             
             The
             most
             ready
             Rule
             for
             all
             ordinary
             Casks
             for
             the
             mean
             Diameter
             is
             ,
             as
             10
             to
             7
             ,
             so
             is
             the
             difference
             of
             Diameters
             to
             a
             fourth
             ,
             to
             be
             added
             to
             the
             least
             Diameter
             to
             make
             a
             mean.
             
          
           
             
               Note
               ,
            
             That
             in
             Cask
             neer
             a
             Cyllender
             ,
             you
             may
             say
             as
             10
             to
             8,50
             ;
             in
             Cask
             like
             to
             Cones
             ,
             say
             as
             10
             to
             5,30
             .
             In
             Cask
             like
             a
             Parabolick
             Spindle
             ,
             or
             Oval
             ,
             say
             ,
             as
             10
             to
             6.
             
             But
             for
             very
             much
             swelling
             Cask
             ,
             say
             ,
             as
             10
             to
             7,3
             ;
             or
             7,4
             ,
             or
             7,5
             ,
             so
             is
             the
             difference
             of
             Diameters
             to
             a
             4th
             ,
             to
             add
             to
             the
             lesser
             Diameter
             ,
             to
             make
             a
             mean
             Diameter
             .
          
        
         
           
             Of
             Square
             Solid
             or
             Timber
             Measure
             .
          
           
             1
             A
             mean
             Proportion
             Geometrical
             ,
             or
             the
             middle
             between
             the
             breadth
             and
             thickness
             ,
             measured
             on
             a
             line
             of
             Numbers
             ,
             is
             the
             side
             of
             the
             square
             equal
             .
          
           
             2
             As
             the
             inches
             square
             to
             12
             ,
             so
             is
             12
             twice
             ,
             to
             the
             inches
             long
             to
             make
             1
             foot
             .
          
           
             3
             As
             12
             to
             the
             breadth
             in
             inches
             ,
             so
             is
             the
             depth
             in
             inches
             to
             a
             4th
             .
             Again
             ,
             as
             the
             4th
             .
             to
             12
             ,
             so
             is
             12
             to
             the
             inches
             long
             to
             make
             1
             foot
             .
          
           
             4.
             
             For
             small
             timber
             ,
             say
             ,
             As
             1
             to
             the
             breadth
             ,
             so
             is
             the
             depth
             in
             inches
             to
             a
             fourth
             .
          
           
             Again
             ,
             As
             that
             4th
             .
             to
             12
             ,
             so
             is
             12
             to
             the
             feet
             and
             parts
             long
             ,
             to
             make
             1
             foot
             .
          
           
             5.
             
             As
             12
             to
             the
             inches
             square
             ,
             so
             is
             the
             inches
             square
             to
             the
             quantity
             in
             1
             foot
             in
             inches
             .
          
           
             6.
             
             As
             12
             to
             the
             breadth
             in
             inches
             ,
             so
             is
             the
             depth
             in
             inches
             ,
             to
             the
             quantity
             in
             a
             foot
             in
             inches
             .
          
           
             7.
             
             As
             12
             to
             the
             inches
             square
             ,
             so
             is
             the
             feet
             long
             twice
             ,
             to
             the
             content
             in
             feet
             .
          
           
             8.
             
             As
             12
             to
             the
             breadth
             in
             inches
             ,
             so
             is
             the
             depth
             in
             
             inches
             to
             a
             4th
             .
             Again
             ,
             As
             12
             to
             that
             4th
             .
             so
             is
             the
             length
             in
             feet
             ,
             to
             the
             content
             in
             feet
             .
          
           
             9.
             
             As
             1
             to
             the
             bredth
             in
             inches
             ,
             so
             is
             the
             depth
             in
             inches
             to
             a
             4th
             .
             Again
             ,
             As
             1
             to
             tha
             4th
             .
             so
             is
             the
             length
             in
             inches
             ,
             to
             the
             content
             in
             Cube
             inches
             required
             .
          
           
             10.
             
             As
             1
             to
             the
             thickness
             in
             inches
             ,
             so
             is
             the
             bredth
             in
             inches
             to
             a
             4th
             .
             Again
             ,
             As
             144
             to
             the
             4th
             .
             in
             inches
             ,
             so
             is
             the
             length
             in
             feet
             ,
             to
             the
             conteet
             in
             feet
             .
          
        
         
           
             Of
             round
             Timber
             measure
             ,
             by
             the
             Diameter
             .
          
           
             1
             As
             the
             Diameter
             in
             inches
             to
             the
             13,54
             ,
             (
             the
             Diameter
             ,
             when
             one
             foot
             long
             makes
             1
             foot
             of
             Timber
             )
             so
             is
             12
             twice
             ,
             to
             the
             length
             in
             inches
             to
             make
             1
             foot
             .
          
           
             2.
             
             As
             13,54
             ,
             to
             the
             Diameter
             in
             inches
             ,
             so
             is
             12
             twice
             to
             the
             quantity
             in
             a
             foot
             long
             in
             inches
             ,
             or
             so
             is
             12
             twice
             ,
             to
             the
             feet
             in
             a
             foot
             long
             .
             (
             With
             foot
             measure
             use
             1,128
             .
             )
          
           
             3.
             
             As
             13,54
             ,
             to
             the
             Diameter
             in
             inches
             ,
             so
             is
             the
             length
             in
             feet
             twice
             ,
             to
             the
             content
             in
             feet
             .
          
           
             4.
             
             As
             1,128
             to
             the
             Diameter
             in
             inches
             ,
             so
             is
             the
             length
             in
             inches
             twice
             ,
             to
             the
             content
             in
             inches
             .
          
           
             5.
             
             As
             1,128
             ,
             to
             the
             Diameter
             in
             feet
             and
             parts
             ,
             so
             is
             the
             length
             in
             feet
             and
             parts
             ,
             to
             the
             content
             in
             feet
             and
             parts
             .
          
        
         
           
             By
             the
             Circumference
             .
          
           
             1.
             
             As
             the
             Circumference
             in
             inches
             to
             147,36
             (
             the
             inches
             about
             ,
             when
             one
             inch
             long
             makes
             a
             foot
             )
             so
             is
             1
             twice
             ,
             to
             the
             inches
             long
             to
             make
             a
             foot
             .
          
           
             2.
             
             As
             the
             Circumference
             in
             inches
             to
             42,54
             (
             the
             inches
             about
             ,
             when
             12
             inches
             long
             makes
             one
             foot
             )
             so
             ●s
             12
             twice
             ,
             to
             the
             inches
             long
             to
             make
             1
             foot
             :
             Or
             so
             ●s
             1
             twice
             to
             the
             feet
             long
             to
             make
             one
             foot
             of
             Timber
             .
          
           
             3.
             
             As
             42,54
             to
             the
             Circumference
             in
             inches
             ,
             so
             is
             12
             twice
             ,
             to
             the
             inches
             in
             a
             foot
             long
             :
             Or
             so
             is
             1
             ●wice
             to
             the
             feet
             in
             a
             foot
             long
             .
          
           
           
             4.
             
             As
             42,54
             ,
             to
             the
             inches
             about
             ,
             so
             is
             the
             length
             in
             feet
             twice
             ,
             to
             the
             content
             in
             feet
             .
          
           
             5.
             
             As
             3,545
             the
             feet
             about
             ,
             when
             1
             foot
             in
             length
             is
             a
             foot
             of
             Timber
             )
             to
             the
             girt
             in
             feet
             ,
             so
             is
             the
             length
             in
             feet
             twice
             ,
             to
             the
             content
             in
             feet
             .
          
           
             6.
             
             As
             3,545
             ,
             to
             the
             girt
             in
             inches
             ,
             so
             is
             the
             length
             in
             inches
             twice
             ,
             to
             the
             content
             in
             solid
             inches
             .
          
        
         
           
             To
             measure
             Brickwork
             and
             reduce
             it
             .
          
           
             1.
             
             Note
             272
             foot
             1
             fourth
             is
             a
             rod
             of
             Brickwork
             ,
             at
             a
             Brick
             and
             half
             thick
             ,
             and
             the
             center
             for
             it
             is
             1.
             
          
           
             
               Centers
               for
               other
               thicknesses
               are
               thus
               found
               :
            
          
           
             2.
             
             As
             half
             a
             Brick
             ,
             at
             05
             ,
             or
             as
             (
             1
             Brick
             at
             1,2
             Bricks
             at
             2,2
             and
             a
             half
             at
             2,5,3
             Bricks
             at
             3
             ,
             )
             is
             to
             1,5
             ;
             so
             is
             1
             the
             Center
             for
             a
             Brick
             and
             half
             to
             3
             the
             Center
             for
             half
             a
             Brick
             required
             .
          
           
             3.
             
             As
             the
             Center
             for
             any
             thickness
             ,
             is
             to
             the
             length
             in
             feet
             of
             any
             Brick-wall
             ,
             so
             is
             the
             bredth
             in
             feet
             to
             the
             content
             in
             feet
             ,
             at
             a
             Brick
             and
             a
             half
             thick
             ,
          
           
             4.
             
             As
             272
             ,
             1
             fourth
             to
             1
             ,
             so
             is
             any
             number
             of
             feet
             to
             the
             Rods
             ,
             and
             Quarters
             ,
             and
             Decimals
             ;
             and
             the
             same
             extent
             laid
             the
             contrary
             way
             ,
             from
             the
             Decimal
             Fraction
             gives
             the
             odd
             feet
             .
          
           
             For
             the
             extent
             from
             272
             ,
             1
             fourth
             to
             1
             ,
             laid
             the
             same
             way
             from
             1528
             foot
             ,
             gives
             5
             Rod
             and
             a
             half
             and
             11
             parts
             over
             ,
             then
             the
             same
             extent
             laid
             the
             contrary
             way
             from
             11
             ,
             gives
             30
             foot
             over
             ,
             in
             all
             5
             Rod
             and
             half
             30
             foot
             .
          
           
             5.
             
             As
             272,1
             fourth
             the
             foot
             in
             one
             Rod
             ,
             to
             5
             
               l.
            
             the
             price
             of
             one
             Rod
             ,
             so
             is
             1528
             foot
             to
             28
             
               l.
            
             1
             
               s.
            
             the
             price
             of
             1528
             foot
             at
             5
             
               l.
               per
            
             Rod.
             
          
           
             6.
             
             As
             0666
             ,
             a
             point
             for
             once
             and
             half
             ,
             is
             to
             the
             breadth
             of
             any
             roo
             ,
             so
             is
             the
             length
             ,
             to
             the
             content
             in
             feet
             on
             both
             sides
             (
             being
             measured
             in
             feet
             .
             )
          
        
         
           
             For
             Digging
             of
             Earth
             in
             Square
             Yards
          
           
             1st
             .
             As
             9
             to
             the
             length
             in
             feet
             ,
             10
             is
             the
             bredth
             to
             the
             
             superficial
             Yards
             .
             Again
             ,
             As
             3
             foot
             the
             depth
             of
             one
             Yard
             ,
             to
             the
             content
             in
             superficial
             Yards
             ,
             so
             is
             the
             depth
             11
             feet
             ,
             to
             the
             content
             in
             solid
             Yards
             .
          
           
             2ly
             .
             1.
             
             Or
             plainly
             thus
             ,
             As
             ●
             to
             the
             length
             in
             feet
             ,
             so
             the
             breadth
             in
             feet
             ,
             to
             the
             superficial
             ●eet
             .
          
           
             2.
             
             As
             1
             to
             the
             superficial
             feet
             ,
             so
             is
             the
             depth
             in
             feet
             to
             the
             solid
             content
             in
             feet
             .
          
           
             3.
             
             As
             27
             the
             feet
             in
             1
             yard
             to
             1
             ,
             so
             is
             the
             solid
             content
             in
             feet
             ,
             to
             the
             content
             in
             solid
             yards
             .
          
        
         
           
             Of
             Simple
             Interest
          
           
             1st
             .
             As
             100
             is
             to
             6
             
               l.
            
             the
             Interest
             due
             for
             100
             
               l.
            
             in
             1
             year
             ,
             so
             is
             any
             other
             sum
             to
             his
             proportional
             interest
             in
             1
             year
             .
          
           
             2ly
             .
             For
             months
             count
             thus
             ,
             If
             6
             be
             12
             months
             or
             1
             year
             ,
             then
             5
             is
             10
             months
             ,
             4
             is
             8
             months
             ,
             3
             is
             6
             months
             ,
             
               &c.
            
             decreasing
             less
             than
             1
             year
             ;
             so
             likewise
             12
             is
             2
             year
             ,
             18
             is
             3
             year
             ,
             
               &c.
               
            
             So
             that
             for
             30
             months
             ,
             take
             15
             the
             half
             of
             30
             ▪
             
               &c.
               
            
             As
             100
             to
             15
             the
             interest
             of
             100
             
               l.
            
             in
             30
             months
             ,
             so
             is
             any
             other
             sum
             to
             his
             simple
             interest
             in
             the
             like
             time
             at
             6
             
               per
               Cent.
               
            
          
           
             3ly
             .
             For
             principle
             and
             interest
             either
             increase
             or
             present
             worth
             ,
             thus
             at
             any
             time
             .
          
           
             If
             106
             is
             1
             years
             principle
             and
             increase
             ,
             112
             is
             2
             years
             ,
             118,3
             years
             :
             thus
             every
             tenth
             on
             the
             line
             of
             Numbers
             :
             is
             2
             months
             :
             thus
             ,
          
           
             What
             is
             the
             increase
             of
             125
             
               l.
            
             interest
             and
             principle
             in
             40
             months
             ,
             or
             its
             present
             worth
             being
             due
             40
             months
             to
             come
             .
          
           
             As
             100
             to
             120
             ,
             so
             is
             125
             to
             150
             
               l.
            
             the
             principle
             and
             increase
             in
             40
             months
             .
          
           
             And
             if
             the
             same
             extent
             be
             laid
             the
             contrary
             way
             from
             125
             ,
             it
             gives
             104
             
               l.
            
             4
             
               s.
            
             the
             present
             worth
             of
             125
             
               l.
            
             due
             40
             months
             to
             come
             at
             6
             
               per
               Cent.
            
             simple
             interest
             .
          
           
             4ly
             .
             1.
             
             For
             dayes
             thus
             ,
             As
             365
             ,
             the
             dayes
             in
             a
             year
             to
             the
             rate
             ;
             so
             is
             the
             number
             of
             dayes
             ,
             to
             the
             interest
             of
             200
             
               l.
            
             in
             so
             many
             dayes
             ,
             as
             a
             4th
             .
          
           
             2.
             
             As
             this
             4th
             ,
             with
             100
             added
             to
             it
             ,
             is
             to
             100
             ,
             so
             
             is
             the
             sum
             propounded
             to
             the
             present
             worth
             ;
             if
             the
             sum
             be
             pounds
             ,
             shillings
             and
             pence
             ,
             reduce
             it
             to
             a
             Decimal
             Fraction
             ,
             and
             the
             years
             to
             dayes
             :
             Note
             also
             the
             same
             extent
             laid
             the
             other
             way
             ,
             gives
             the
             principal
             and
             increase
             in
             so
             many
             dayes
             .
          
        
         
           
             Equation
             of
             Payments
          
           
             1.
             
             By
             the
             last
             rule
             find
             all
             the
             present
             worths
             ,
             at
             the
             times
             to
             comes
             ,
             then
             say
             ,
             as
             all
             the
             present
             worths
             to
             all
             the
             payments
             ,
             so
             is
             1
             
               l.
            
             to
             a
             fourth
             ,
             from
             which
             taking
             a
             unite
             ,
             the
             remainder
             is
             the
             interest
             of
             1
             
               l.
            
             the
             time
             sought
             ,
             which
             divided
             by
             the
             inetrest
             of
             1
             
               l.
            
             for
             1
             day
             ,
             
               viz.
            
             000164384
             ,
             the
             quote
             is
             the
             number
             of
             dayes
             required
             .
             
               Example
               ,
            
          
           
             If
             220
             
               l.
            
             be
             due
             at
             a
             years
             end
             ,
             the
             present
             worth
             is
             207
             
               l.
            
             11
             
               s.
            
             6
             
               d.
            
             for
             the
             extent
             from
             106
             to
             100
             laid
             from
             220
             ,
             gives
             207
             
               l.
            
             11
             
               s.
            
             6
             
               d.
            
             But
             if
             it
             should
             be
             paid
             at
             55
             
               l.
               per
            
             quarter
             ,
             when
             should
             it
             be
             paid
             at
             once
             to
             be
             equal
             between
             Debtor
             and
             Creditor
             ?
          
           
             As
             
               
                 
                   101,5
                
                 
                   3
                   months
                
              
               
                 
                   103,0
                
                 
                   6
                   months
                
              
               
                 
                   104,5
                
                 
                   9
                   months
                
              
               
                 
                   106,0
                
                 
                   12
                   months
                
              
            
             to
             1003
             ∷
             So
             is
             55
             to
             
               
                 55,20
              
               
                 53,41
              
               
                 52,63
              
               
                 51,90
              
            
             The
             sum
             of
             them
             212,14
          
           
             Then
             as
             212
             
               l.
            
             14
             
               s.
            
             the
             sum
             of
             the
             present
             worths
             ,
             to
             220
             the
             sum
             of
             the
             payments
             ,
             so
             is
             1
             to
             103,70
             ,
             from
             which
             tak●ng
             100
             ,
             rests
             03,70
             ,
             the
             interest
             of
             100
             
               l.
            
             the
             time
             sought
             .
             Then
             the
             03,70
             ,
             divided
             by
             00016438
             the
             interest
             of
             1
             
               l.
            
             1
             day
             ,
             hath
             in
             the
             quotient
             225
             dayes
             the
             true
             time
             to
             pay
             the
             220
             
               l.
            
             at
             once
             .
          
        
         
           
             Compound
             Interest
          
           
             The
             extent
             from
             100
             to
             106
             ,
             the
             principal
             and
             increase
             in
             one
             year
             ,
             repeated
             as
             many
             times
             as
             there
             be
             years
             ,
             gives
             the
             principle
             and
             increase
             of
             any
             sum
             ,
             and
             being
             turned
             as
             many
             times
             the
             decreasing
             way
             gives
             the
             present
             worth
             .
          
        
         
           FINIS