Gaging promoted an appendix to stereometrical propositions / by Robert Anderson.
         Anderson, Robert, fl. 1668-1696.
      
       
         
           1669
        
      
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             Gaging promoted an appendix to stereometrical propositions / by Robert Anderson.
             Anderson, Robert, fl. 1668-1696.
          
           33, [1] p.
           
             Printed by J.W. for Joshua Coniers,
             London :
             1669.
          
           
             "Faults escaped in the impression of Stereometrical proposition": p. [34]
             Reproduction of original in the Cambridge University Library.
          
        
      
    
     
       
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           Mathematics -- Early works to 1800.
        
      
    
     
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           GAGING
           PROMOTED
           .
           AN
           APPENDIX
           TO
           Stereometrical
           Propositions
           .
        
         
           
             By
          
           ROBERT
           ANDERSON
           .
        
         
           
             LONDON
             ,
          
           Printed
           by
           
             I.
             W.
          
           for
           
             Ioshua
             Coniers
             ,
          
           at
           the
           Raven
           in
           
             Ducklane
             ,
          
           1669.
           
        
      
    
     
       
         
         
         
           Gaging
           Promoted
           AN
           APPENDIX
           TO
           Stereometrical
           Propositions
           .
        
         
           
             I.
             Note
             .
          
           
             AS
             an
             Abstract
             from
             the
             undoubted
             Axioms
             of
             Geometry
             ,
             it
             is
             generally
             observed
             ,
             that
             in
             a
             Rank
             of
             numbers
             ,
             having
             equal
             difference
             ,
             the
             second
             differences
             of
             the
             squares
             of
             those
             numbers
             are
             equal
             ;
             the
             third
             differences
             of
             the
             cubes
             of
             those
             numbers
             are
             equal
             :
             and
             so
             in
             order
             in
             the
             higher
             powers
             .
             Thus
             ,
             
             
               
                 In
                 Squares
              
               
                 1
              
               
                 
                   1
                
                 
                   1
                
                 
                    
                
                 
                    
                
              
               
                 
                   3
                
              
               
                 
                   2
                
                 
                   4
                
                 
                   2
                
              
               
                 
                   5
                
              
               
                 
                   3
                
                 
                   9
                
                 
                   2
                
              
               
                 
                   7
                
              
               
                 
                   4
                
                 
                   16
                
                 
                   2
                
              
               
                 
                   9
                
              
               
                 
                   5
                
                 
                   25
                
                 
                    
                
              
               
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
              
            
             
               
                 2
              
               
                 
                   1
                
                 
                   1
                
                 
                    
                
                 
                    
                
              
               
                 
                   8
                
              
               
                 
                   3
                
                 
                   9
                
                 
                   8
                
              
               
                 
                   16
                
              
               
                 
                   5
                
                 
                   25
                
                 
                   8
                
              
               
                 
                   24
                
              
               
                 
                   7
                
                 
                   47
                
                 
                   8
                
              
               
                 
                   32
                
              
               
                 
                   9
                
                 
                   81
                
                 
                    
                
              
               
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
              
            
          
           
             Observe
             in
             the
             first
             of
             these
             Examples
             ,
             in
             the
             first
             collum
             are
             the
             numbers
             of
             a
             progressition
             ,
             having
             equal
             difference
             ,
             to
             wit
             ,
             a
             unite
             .
             In
             the
             second
             colum
             ,
             the
             squares
             of
             those
             numbers
             .
             In
             the
             third
             colum
             ,
             the
             first
             differences
             .
             In
             the
             fourth
             colum
             ,
             the
             second
             differences
             ,
             to
             wit
             ,
             2
             ,
             2
             ,
             2.
             
             In
             the
             second
             Example
             ,
             in
             the
             first
             colum
             are
             a
             Rank
             of
             numbers
             ,
             having
             equal
             difference
             ,
             to
             wit
             ,
             2.
             
             In
             the
             second
             colum
             ,
             their
             squares
             .
             In
             the
             third
             colum
             ,
             the
             first
             difference
             .
             In
             the
             fourth
             colum
             ,
             the
             second
             difference
             .
          
        
         
           
             II.
             Note
             .
          
           
             Hence
             it
             follows
             ,
             that
             by
             the
             help
             of
             such
             differences
             the
             table
             of
             squares
             may
             be
             calculated
             :
             thus
             ,
             in
             the
             first
             Example
             ,
             the
             sum
             of
             1
             and
             3
             ,
             is
             4
             ;
             the
             square
             of
             2.
             
             The
             sum
             of
             2
             ,
             3
             and
             4
             ,
             is
             9
             ;
             the
             square
             of
             3.
             
             The
             sum
             of
             2
             ,
             5
             and
             9
             ,
             is
             16
             ;
             the
             square
             of
             4.
             
             The
             sum
             of
             2
             ,
             7
             and
             16
             ,
             is
             25
             ;
             the
             square
             of
             5.
             
             The
             sum
             of
             2
             ,
             11
             and
             36
             ,
             is
             49
             ;
             the
             square
             of
             7.
             &c.
             
          
        
         
           
           
             III.
             Note
             .
          
           
             Like
             plain
             numbers
             are
             in
             the
             same
             proportion
             one
             to
             another
             ,
             that
             a
             square
             number
             is
             in
             ,
             to
             a
             square
             number
             :
             
               Euclide
            
             the
             26
             Proposition
             of
             the
             Eighth
             Book
             .
             Therefore
             the
             second
             difference
             in
             such
             a
             Rank
             of
             plane
             numbers
             are
             equal
             .
             Further
             ,
             what
             planes
             and
             solids
             are
             either
             equal
             or
             proportionable
             to
             such
             Ranks
             may
             be
             gradually
             calculated
             ;
             as
             in
             the
             last
             .
          
        
         
           
             IV.
             Note
             .
          
           
             
               
                 
                   1
                
                 
                   1
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   7
                
              
               
                 
                   2
                
                 
                   8
                
                 
                   12
                
                 
                    
                
              
               
                 
                   19
                
                 
                   6
                
              
               
                 
                   3
                
                 
                   27
                
                 
                   18
                
              
               
                 
                   37
                
                 
                   6
                
              
               
                 
                   4
                
                 
                   64
                
                 
                   24
                
              
               
                 
                   71
                
                 
                    
                
              
               
                 
                   5
                
                 
                   125
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
              
            
             
               
                 
                   1
                
                 
                   1
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   26
                
              
               
                 
                   3
                
                 
                   27
                
                 
                   72
                
              
               
                 
                   98
                
                 
                   48
                
              
               
                 
                   5
                
                 
                   125
                
                 
                   120
                
              
               
                 
                   218
                
                 
                   48
                
              
               
                 
                   7
                
                 
                   343
                
                 
                   186
                
              
               
                 
                   386
                
                 
                    
                
              
               
                 
                   9
                
                 
                   729
                
                 
                    
                
              
               
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
              
            
          
           
             In
             the
             first
             Example
             ,
             in
             the
             first
             colum
             are
             the
             numbers
             in
             a
             Rank
             having
             equal
             difference
             ,
             to
             wit
             ,
             a
             unite
             .
             In
             the
             second
             colum
             ,
             the
             cubes
             of
             those
             numbers
             .
             In
             the
             third
             colum
             ,
             the
             first
             differences
             of
             those
             cubes
             .
             In
             the
             fourth
             colum
             ,
             the
             second
             differences
             .
             In
             the
             fifth
             colum
             ,
             the
             third
             differences
             ,
             to
             wit
             ,
             6
             ,
             6.
             
             The
             like
             in
             the
             second
             Example
             .
          
        
         
           
             V.
             Note
             .
          
           
             Hence
             it
             follows
             ,
             that
             the
             table
             of
             cubes
             may
             be
             
             made
             thus
             :
             In
             the
             first
             Example
             ,
             1
             and
             7
             ,
             is
             8
             ;
             the
             cube
             of
             2.
             
             The
             sum
             of
             8
             and
             19
             ,
             is
             27
             ;
             the
             cube
             of
             3.
             
             The
             sum
             of
             18
             ,
             19
             and
             27
             ,
             is
             64
             ;
             the
             cube
             of
             4.
             
             The
             sum
             of
             6
             ,
             18
             ,
             37
             and
             64
             ,
             is
             125
             ;
             the
             cube
             of
             5.
             
             The
             sum
             of
             6
             ,
             24
             ,
             61
             and
             125
             ,
             is
             216
             ;
             the
             cube
             of
             6.
             
             The
             sum
             of
             6
             ,
             30
             ,
             91
             and
             216
             ,
             is
             343
             ;
             the
             cube
             of
             7.
             
             The
             like
             in
             the
             second
             Example
             .
          
        
         
           
             VI.
             Note
             .
          
           
             Like
             solid
             numbers
             are
             in
             the
             same
             proportion
             one
             to
             another
             ,
             that
             a
             cube
             number
             is
             in
             ,
             to
             a
             cube
             number
             ▪
             
               Euclide
            
             the
             XXVII
             Prop.
             of
             the
             Eighth
             Book
             .
             Therefore
             the
             third
             differences
             in
             such
             a
             Rank
             of
             solid
             numbers
             are
             equal
             :
             further
             ,
             such
             planes
             and
             solids
             as
             are
             either
             equal
             or
             proportionable
             to
             such
             Ranks
             ,
             may
             be
             gradually
             calculated
             ,
             as
             in
             the
             last
             .
          
        
         
           
             VII
             .
             Note
             .
          
           
             If
             a
             Rank
             of
             Squares
             ,
             whose
             Roots
             have
             equal
             differences
             ,
             be
             multiplied
             by
             any
             number
             ,
             the
             second
             differences
             of
             such
             a
             Rank
             of
             proucts
             are
             equal
             .
             Let
             the
             number
             multiplying
             be
             10.
             
          
           
             
               
                 
                   0
                
                 
                   0
                
                 
                   00
                
                 
                    
                
                 
                    
                
              
               
                 
                   10
                
              
               
                 
                   1
                
                 
                   1
                
                 
                   10
                
                 
                   20
                
              
               
                 
                   30
                
              
               
                 
                   2
                
                 
                   4
                
                 
                   40
                
                 
                   20
                
              
               
                 
                   50
                
              
               
                 
                   3
                
                 
                   9
                
                 
                   90
                
                 
                   10
                
              
               
                 
                   70
                
              
               
                 
                   4
                
                 
                   16
                
                 
                   160
                
                 
                   20
                
              
               
                 
                   90
                
              
               
                 
                   5
                
                 
                   25
                
                 
                   250
                
                 
                    
                
              
               
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
              
            
          
           
             In
             the
             first
             colum
             are
             the
             numbers
             bearing
             equal
             difference
             .
             In
             the
             second
             colum
             are
             the
             squares
             of
             those
             numbers
             .
             In
             the
             third
             colum
             the
             products
             .
             In
             the
             fourth
             colum
             the
             first
             differences
             .
             In
             the
             fifth
             colum
             the
             second
             differences
             and
             they
             equal
             .
          
        
         
           
           
             VIII
             .
             Note
             .
          
           
             If
             unto
             such
             a
             Rank
             of
             Products
             ,
             as
             in
             the
             last
             ,
             there
             be
             added
             a
             Rank
             of
             Cubes
             ,
             whose
             Roots
             are
             equal
             to
             the
             Roots
             of
             the
             Squares
             ,
             the
             third
             differences
             of
             such
             a
             Rank
             will
             be
             equal
             .
          
           
             
               
                 
                   0
                
                 
                   00
                
                 
                   0
                
                 
                   00
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   11
                
              
               
                 
                   1
                
                 
                   10
                
                 
                   1
                
                 
                   11
                
                 
                   26
                
                 
                    
                
              
               
                 
                   37
                
                 
                   6
                
              
               
                 
                   2
                
                 
                   40
                
                 
                   8
                
                 
                   48
                
                 
                   32
                
              
               
                 
                   69
                
                 
                   6
                
              
               
                 
                   3
                
                 
                   90
                
                 
                   27
                
                 
                   117
                
                 
                   38
                
              
               
                 
                   107
                
                 
                   6
                
              
               
                 
                   4
                
                 
                   160
                
                 
                   64
                
                 
                   224
                
                 
                   44
                
              
               
                 
                   151
                
                 
                    
                
              
               
                 
                   5
                
                 
                   250
                
                 
                   125
                
                 
                   375
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
              
            
          
           
             In
             the
             first
             colum
             are
             the
             numbers
             having
             equal
             difference
             .
             In
             the
             second
             colum
             the
             Products
             of
             their
             squares
             by
             a
             given
             number
             .
             In
             the
             third
             colum
             the
             cube
             of
             the
             numbers
             in
             the
             first
             colum
             .
             In
             the
             fourth
             colum
             the
             sum
             of
             the
             products
             and
             cubes
             .
             In
             the
             fifth
             colum
             their
             first
             difference
             .
             In
             the
             sixt
             colum
             the
             second
             differences
             .
             In
             the
             seventh
             colum
             the
             third
             differences
             which
             are
             equal
             .
          
        
         
           
             IX
             .
             Note
             .
          
           
             Let
             a
             constant
             number
             be
             added
             to
             a
             Rank
             of
             Products
             ,
             so
             that
             one
             of
             the
             numbers
             multiplying
             
             be
             a
             constant
             number
             ,
             and
             the
             other
             of
             the
             numbers
             be
             the
             squares
             of
             numbers
             having
             equal
             difference
             ,
             and
             this
             Rank
             of
             sums
             be
             added
             to
             a
             Rank
             of
             cubes
             ,
             whose
             roots
             are
             the
             same
             with
             the
             roots
             of
             the
             squares
             ;
             such
             a
             compounded
             Rank
             will
             have
             their
             third
             difference
             equal
             .
             Thus
             ,
             
               
                 
                   1
                
                 
                   8
                
                 
                   10
                
                 
                   1
                
                 
                   19
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   37
                
              
               
                 
                   2
                
                 
                   8
                
                 
                   40
                
                 
                   8
                
                 
                   56
                
                 
                   32
                
                 
                    
                
              
               
                 
                   69
                
                 
                   6
                
              
               
                 
                   3
                
                 
                   8
                
                 
                   90
                
                 
                   27
                
                 
                   125
                
                 
                   38
                
              
               
                 
                   107
                
                 
                   6
                
              
               
                 
                   4
                
                 
                   8
                
                 
                   160
                
                 
                   64
                
                 
                   232
                
                 
                   44
                
              
               
                 
                   151
                
                 
                    
                
              
               
                 
                   5
                
                 
                   8
                
                 
                   250
                
                 
                   125
                
                 
                   383
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
                 
                   8
                
              
            
          
           
             In
             the
             first
             colum
             are
             the
             numbers
             having
             equal
             difference
             .
             In
             the
             second
             colum
             is
             constant
             number
             to
             be
             added
             .
             In
             the
             third
             colum
             are
             the
             Rank
             of
             products
             ,
             that
             is
             ,
             the
             squares
             of
             the
             numbers
             in
             the
             first
             colum
             multiplied
             by
             a
             given
             number
             .
             In
             the
             fourth
             colum
             are
             the
             cubes
             of
             the
             numbers
             in
             the
             first
             colum
             .
             In
             the
             fifth
             colum
             are
             the
             sum
             of
             the
             numbers
             in
             the
             second
             ,
             third
             and
             fourth
             colums
             .
             In
             the
             sixth
             colum
             are
             the
             first
             differences
             of
             their
             sums
             .
             In
             the
             seventh
             colum
             are
             the
             second
             differences
             .
             In
             the
             eighth
             colum
             the
             third
             differences
             ,
             and
             they
             equal
             .
          
        
         
           
             X.
             Note
             .
          
           
             In
             a
             Rank
             of
             numbers
             ,
             having
             equal
             difference
             ,
             and
             equal
             in
             number
             ;
             if
             the
             third
             part
             of
             the
             cubes
             of
             each
             of
             these
             numbers
             ,
             be
             substracted
             from
             the
             products
             of
             the
             squares
             of
             each
             of
             these
             numbers
             ,
             in
             half
             the
             greatest
             number
             of
             that
             Rank
             ,
             the
             remainders
             
             will
             be
             a
             Rank
             of
             numbers
             ,
             equal
             to
             all
             the
             squares
             in
             the
             several
             portions
             of
             one
             fourth
             of
             a
             sphere
             ,
             whose
             diameter
             is
             equal
             to
             the
             greatest
             number
             in
             that
             Rank
             ,
             and
             the
             third
             differences
             of
             this
             Rank
             of
             portions
             are
             equal
             ;
             but
             the
             first
             and
             second
             differences
             will
             increase
             and
             decrease
             ,
             differently
             one
             from
             another
             .
             Thus
             ,
             
               
                 
                   0
                
                 
                   00
                
                 
                   00
                
                 
                   00
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   99
                
              
               
                 
                   3
                
                 
                   09
                
                 
                   108
                
                 
                   99
                
                 
                   162
                
                 
                    
                
              
               
                 
                   261
                
                 
                   54
                
              
               
                 
                   6
                
                 
                   72
                
                 
                   432
                
                 
                   360
                
                 
                   108
                
              
               
                 
                   369
                
                 
                   54
                
              
               
                 
                   9
                
                 
                   243
                
                 
                   972
                
                 
                   729
                
                 
                   54
                
              
               
                 
                   423
                
                 
                   54
                
              
               
                 
                   12
                
                 
                   756
                
                 
                   1728
                
                 
                   1152
                
                 
                   00
                
              
               
                 
                   423
                
                 
                   54
                
              
               
                 
                   15
                
                 
                   1125
                
                 
                   2700
                
                 
                   1575
                
                 
                   54
                
              
               
                 
                   369
                
                 
                   54
                
              
               
                 
                   18
                
                 
                   1944
                
                 
                   3888
                
                 
                   1944
                
                 
                   108
                
              
               
                 
                   261
                
                 
                   54
                
              
               
                 
                   21
                
                 
                   3087
                
                 
                   5292
                
                 
                   2205
                
                 
                   162
                
              
               
                 
                   96
                
                 
                    
                
              
               
                 
                   24
                
                 
                   4608
                
                 
                   6912
                
                 
                   2304
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
              
            
          
           
             In
             the
             first
             colum
             are
             the
             numbers
             having
             equal
             difference
             .
             In
             the
             second
             colum
             are
             the
             pyramides
             adscribed
             within
             the
             cubes
             of
             the
             numbers
             in
             the
             first
             colum
             .
             In
             the
             third
             colum
             are
             the
             products
             of
             the
             squares
             of
             the
             numbers
             in
             the
             first
             colum
             ,
             by
             half
             the
             greatest
             number
             in
             the
             first
             colum
             .
             In
             the
             fourth
             colum
             are
             the
             differences
             of
             the
             numbers
             in
             the
             second
             and
             third
             colums
             ,
             that
             is
             ,
             all
             the
             squares
             in
             several
             portions
             of
             one
             fourth
             of
             a
             sphere
             ,
             whose
             diameter
             is
             24.
             
             In
             the
             fifth
             colum
             are
             the
             first
             differences
             .
             In
             the
             sixt
             colum
             are
             their
             second
             differences
             .
             In
             the
             seventh
             colum
             are
             the
             third
             differences
             ,
             and
             they
             equal
             .
          
        
         
           
           
             XI
             .
             Note
             .
             The
             Application
             or
             Vse
             of
             the
             Preceeding
             Notes
             .
          
           
             The
             application
             or
             Use
             may
             be
             ,
             to
             calculate
             Pyramides
             and
             cones
             ,
             either
             the
             whole
             or
             their
             parts
             ▪
             as
             also
             to
             calculate
             the
             parabolick
             and
             hyperbolick
             conoides
             ,
             either
             the
             whole
             ,
             or
             their
             frustums
             ;
             yet
             also
             ,
             to
             calculate
             the
             sphere
             or
             spheroide
             ,
             either
             the
             whole
             or
             their
             portions
             or
             Zones
             ,
             and
             that
             gradually
             ,
             that
             is
             ,
             to
             find
             the
             solidity
             upon
             every
             inch
             or
             foot
             .
          
           
             
               To
               find
               the
               solidity
               of
               a
               parabolick
               Conoide
               upon
               every
               two
               inches
               .
            
             
               To
               do
               which
               ,
               consider
               the
               Diagram
               of
               the
               18
               Prop.
               of
               my
               
                 Stereometrical
                 Prop.
              
               Let
               PA
               be
               16
               ;
               AR
               12
               ;
               therefore
               AV
               or
               PH
               will
               be
               9
               ;
               for
               it
               ought
               to
               be
               as
               PA
               ,
               is
               to
               AR
               ;
               so
               is
               AR
               ,
               to
               AV.
               Let
               the
               axis
               AP
               be
               divided
               into
               eight
               equal
               parts
               ;
               
                 viz.
              
               2
               ,
               4
               ,
               6
               ,
               8
               ,
               10
               ,
               12
               ,
               14
               ,
               16.
               
               Let
               there
               be
               planes
               drawn
               parallel
               to
               the
               base
               ,
               through
               every
               one
               of
               these
               divisions
               ,
               though
               in
               the
               Diagram
               there
               is
               not
               so
               many
               .
               From
               P
               to
               the
               first
               Q
               suppose
               to
               be
               2
               ,
               its
               square
               4
               ;
               the
               half
               thereof
               2
               ,
               which
               multiplied
               by
               9
               equal
               to
               PH
               ,
               the
               Product
               will
               be
               18
               ;
               that
               is
               ,
               the
               Prism
               QZGIHP
               ;
               equal
               to
               all
               the
               squares
               in
               the
               portion
               of
               the
               conoid
               QOP
               .
               Let
               from
               P
               ,
               to
               the
               second
               Q
               be
               4
               ,
               its
               square
               16
               ,
               the
               half
               is
               8
               ;
               which
               
               multiplied
               by
               9
               ,
               the
               Product
               is
               72
               ;
               equal
               to
               the
               Prism
               QZFIHP
               ,
               equal
               to
               all
               the
               squares
               in
               the
               portion
               of
               the
               conoid
               QOP
               .
               Let
               from
               P
               to
               the
               third
               Q
               be
               6
               ,
               its
               square
               36
               ,
               the
               half
               of
               it
               is
               18
               ,
               which
               multiplied
               by
               9
               ,
               the
               product
               is
               162
               ;
               the
               Prism
               QZEIHP
               ;
               equal
               to
               all
               the
               squares
               in
               the
               third
               portion
               of
               the
               conoid
               QOP
               .
            
             
               Having
               obtained
               two
               portions
               ,
               the
               rest
               may
               be
               obtained
               thus
               :
               having
               obtained
               the
               second
               difference
               ,
               which
               is
               36
               ,
               we
               may
               proceed
               to
               find
               the
               rest
               by
               the
               
                 Seventh
                 Note
                 ;
              
               thus
               ,
               add
               36
               to
               54
               ,
               and
               it
               makes
               90
               :
               which
               added
               to
               72
               ,
               the
               sum
               is
               162
               ,
               equal
               to
               all
               the
               squares
               in
               that
               portion
               ,
               and
               so
               in
               order
               ;
               36
               ,
               90
               and
               162
               ;
               the
               sum
               is
               288.
               36
               ,
               126
               and
               288
               ;
               the
               sum
               is
               450.
               36
               ,
               162
               and
               450
               ,
               the
               sum
               is
               648
               ,
               &c.
               
            
             
               
                 
                   
                     0
                  
                   
                     00
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     8
                  
                
                 
                   
                     2
                  
                   
                     18
                  
                   
                     36
                  
                
                 
                   
                     54
                  
                
                 
                   
                     4
                  
                   
                     72
                  
                   
                     36
                  
                
                 
                   
                     90
                  
                
                 
                   
                     6
                  
                   
                     162
                  
                   
                     36
                  
                
                 
                   
                     126
                  
                
                 
                   
                     8
                  
                   
                     28●
                  
                   
                     36
                  
                
                 
                   
                     62
                  
                
                 
                   
                     10
                  
                   
                     45●
                  
                   
                     36
                  
                
                 
                   
                     198
                  
                
                 
                   
                     12
                  
                   
                     648
                  
                   
                     36
                  
                
                 
                   
                     234
                  
                
                 
                   
                     14
                  
                   
                     882
                  
                   
                     36
                  
                
                 
                   
                     270
                  
                
                 
                   
                     16
                  
                   
                     1152
                  
                   
                      
                  
                
                 
                   
                      
                  
                
                 
                   
                     1
                  
                   
                     2
                  
                   
                     3
                  
                   
                     4
                  
                
              
            
             
               In
               the
               first
               colum
               are
               the
               parts
               of
               the
               altitude
               of
               the
               conoid
               .
               In
               the
               second
               colum
               are
               all
               the
               squares
               in
               several
               portions
               of
               one
               fourth
               of
               a
               conoid
               .
               In
               the
               third
               colum
               the
               first
               differences
               .
               In
               the
               fourth
               colum
               the
               second
               differences
               .
               These
               portions
               in
               the
               second
               colum
               may
               be
               reduced
               to
               circular
               portions
               ,
               thus
               ,
               as
               14
               is
               to
               11
               ;
               so
               are
               all
               the
               squares
               in
               these
               portions
               to
               the
               portions
               themselves
               .
            
          
           
             
               I.
               Scholium
               :
            
             
               The
               use
               of
               this
               gradual
               calculation
               may
               be
               thus
               :
               Suppose
               a
               Brewers
               Copper
               be
               in
               form
               of
               a
               parabolick
               
               conoid
               ;
               the
               quantity
               of
               liquor
               therein
               contained
               may
               be
               found
               ,
               thus
               ,
               having
               calculated
               a
               table
               upon
               every
               inch
               ,
               or
               two
               inches
               ,
               or
               as
               is
               thought
               convenient
               ;
               then
               having
               a
               straight
               Ruler
               divided
               equally
               into
               inches
               ,
               putting
               the
               Ruler
               into
               the
               liquor
               to
               the
               bottom
               of
               the
               Copper
               ,
               see
               how
               many
               inches
               of
               the
               Ruler
               is
               wet
               ;
               with
               the
               number
               of
               wet
               inches
               enter
               the
               first
               colum
               of
               your
               table
               ,
               and
               in
               the
               next
               colum
               are
               the
               number
               of
               cubick
               inches
               which
               that
               portion
               contains
               ;
               the
               number
               of
               cubick
               inches
               thus
               found
               ,
               being
               divided
               by
               the
               number
               of
               cubick
               inches
               in
               a
               Gallon
               ,
               the
               quotient
               shews
               the
               number
               of
               Gallons
               in
               that
               portion
               of
               the
               Copper
               .
            
          
           
             
               II.
               Scholium
               :
               To
               compose
               several
               works
               into
               one
               .
            
             
               As
               14
               ,
               is
               to
               11
               ;
               so
               are
               all
               the
               squares
               in
               one
               fourth
               of
               the
               conoid
               ,
               to
               one
               fourth
               of
               the
               conoid
               it self
               .
               because
               this
               one
               fourth
               ought
               to
               be
               divided
               by
               the
               number
               of
               cubick
               inches
               in
               a
               Gallon
               ,
               suppose
               it
               288
               ,
               to
               shew
               the
               number
               of
               Gallons
               in
               each
               portion
               ,
               we
               may
               multiply
               14
               by
               288
               ,
               that
               is
               4032.
               
               Then
               as
               4032
               is
               to
               11
               ;
               so
               are
               all
               the
               squares
               in
               one
               fourth
               of
               the
               conoid
               ,
               to
               the
               Gallons
               in
               that
               one
               fourth
               .
               Further
               ,
               because
               this
               one
               fourth
               ought
               to
               be
               multiplied
               by
               4
               ,
               to
               reduce
               it
               to
               a
               whole
               conoid
               ;
               therefore
               ,
               divide
               the
               constant
               divisor
               ,
               that
               is
               ,
               4032
               ,
               by
               4
               ,
               and
               it
               will
               be
               1008.
               
               Then
               ,
               as
               1008
               ,
               is
               to
               11
               ;
               so
               are
               those
               several
               portions
               in
               the
               second
               colum
               of
               the
               last
               table
               ,
               to
               the
               
               number
               of
               Gallons
               in
               those
               several
               portions
               of
               a
               parabolick
               conoid
               .
               By
               such
               compositions
               may
               the
               Practitioner
               compose
               constant
               divisors
               or
               dividends
               ,
               which
               will
               much
               breviate
               the
               work
               ;
               this
               is
               onely
               for
               an
               Example
               .
            
             
               Every
               parabolick
               conoid
               hath
               its
               second
               differences
               equal
               .
               To
               find
               the
               second
               differences
               ,
               work
               thus
               ,
               Square
               one
               of
               the
               equal
               segments
               of
               the
               axis
               ,
               and
               multiply
               that
               Square
               by
               the
               Parameter
               ,
               that
               product
               will
               be
               the
               second
               difference
               .
               In
               this
               Example
               ,
               the
               equal
               segment
               of
               the
               axis
               is
               2
               ,
               the
               square
               of
               it
               4
               ;
               which
               multiplied
               by
               the
               Parameter
               9
               ,
               the
               product
               is
               36
               ,
               the
               second
               difference
               .
               Half
               of
               the
               second
               difference
               ,
               is
               always
               the
               first
               of
               the
               first
               difference
               .
               Half
               36
               the
               second
               difference
               ,
               is
               18
               ,
               the
               first
               of
               the
               first
               difference
               ,
               &c.
               
               Here
               note
               ,
               this
               36
               is
               the
               second
               difference
               of
               one
               fourth
               of
               all
               the
               squares
               in
               a
               parabolick
               conoid
               ;
               if
               36
               be
               multiplied
               by
               4
               ,
               it
               makes
               the
               second
               difference
               144
               ;
               whose
               half
               is
               72
               ,
               the
               first
               of
               the
               first
               differences
               .
               Or
               ,
               the
               first
               differences
               are
               found
               by
               taking
               half
               the
               difference
               of
               the
               squares
               of
               any
               two
               segments
               ,
               which
               multiplied
               by
               the
               Parameter
               ,
               thereby
               the
               first
               differences
               are
               obtained
               .
               Thus
               ,
               to
               find
               the
               first
               difference
               answerable
               to
               the
               Segments
               6
               and
               8
               ;
               the
               Square
               of
               8
               ,
               is
               64
               ;
               the
               Square
               of
               6
               ,
               is
               36
               ;
               the
               difference
               of
               those
               Squares
               is
               28
               ,
               whose
               half
               is
               14
               ;
               which
               multiplied
               by
               the
               Parameter
               9
               ,
               the
               product
               is
               126
               ;
               the
               first
               difference
               answerable
               betwixt
               6
               and
               8.
               
            
          
        
         
           
           
             XII
             .
             Note
             .
             To
             find
             the
             solidity
             of
             an
             hyperbolick
             conoid
             gradually
             ,
             to
             wit
             ,
             upon
             every
             three
             inches
             .
          
           
             For
             the
             performance
             of
             which
             ,
             take
             notice
             of
             the
             XVI
             Prop.
             of
             
               Stereom
               .
               Prop.
            
             in
             that
             Diagram
             ,
             Let
             AM
             equal
             to
             AB
             be
             9.
             
             Let
             ML
             equal
             to
             AF
             ,
             be
             6.
             
             Let
             AE
             be
             15
             :
             therefore
             FE
             will
             be
             9.
             
             Let
             the
             rest
             of
             the
             construction
             be
             as
             in
             that
             Proposition
             .
             Let
             from
             M
             to
             the
             first
             K
             be
             3
             ,
             whose
             square
             is
             9
             ,
             whose
             half
             is
             4½
             ,
             the
             area
             KHM
             ;
             which
             being
             multiplied
             by
             ML
             ,
             6
             ;
             the
             product
             will
             be
             27
             ,
             the
             prism
             KHNOLM
             .
             Because
             FE
             ,
             FC
             and
             FL
             are
             equal
             ,
             that
             is
             ,
             each
             of
             them
             9
             :
             Therefore
             ,
             the
             first
             pyramid
             ONXILM
             will
             be
             9.
             
             Then
             this
             prism
             and
             pyramid
             being
             added
             ,
             will
             make
             36
             ,
             the
             whole
             prism
             KHXILM
             ,
             equal
             to
             all
             the
             squares
             in
             the
             portion
             KZM.
             Let
             from
             M
             to
             the
             second
             K
             be
             6
             ,
             whose
             square
             is
             36
             ,
             its
             half
             18
             ,
             the
             area
             KHM
             ;
             which
             being
             multiplied
             by
             ML
             ,
             6
             ;
             the
             product
             will
             be
             108
             ,
             equal
             to
             the
             prism
             KHNOLM
             .
             The
             cube
             of
             6
             ,
             is
             216
             ;
             a
             third
             part
             is
             72
             ,
             the
             pyramid
             ONXIL
             ,
             this
             prism
             and
             pyramid
             being
             added
             together
             ,
             the
             sum
             will
             be
             180
             ;
             the
             prism
             KHXILM
             :
             equal
             to
             all
             the
             squares
             in
             the
             portion
             KZM.
             Let
             from
             M
             to
             A
             ,
             be
             9
             ;
             its
             square
             81
             ,
             the
             half
             40½
             ,
             equal
             to
             the
             area
             ABM
             ,
             which
             being
             multiplied
             by
             ML
             ,
             6
             ;
             the
             product
             will
             be
             243
             :
             the
             cube
             of
             9
             ,
             is
             729
             ;
             a
             third
             part
             thereof
             is
             243
             ;
             equal
             to
             the
             pyramid
             FCDEL
             :
             this
             prism
             and
             pyramid
             being
             added
             together
             ,
             is
             486
             ;
             the
             
             whole
             prism
             ABDELM
             ,
             equal
             to
             all
             the
             squares
             of
             the
             portion
             AZM
             .
          
           
             These
             three
             portions
             being
             obtained
             ,
             they
             may
             be
             continued
             by
             the
             VIII
             
               Note
               ,
            
             thus
             :
             
               
                 
                   0
                
                 
                   00
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   36
                
                 
                    
                
              
               
                 
                   3
                
                 
                   46
                
                 
                   108
                
              
               
                 
                   144
                
                 
                   54
                
              
               
                 
                   6
                
                 
                   180
                
                 
                   162
                
              
               
                 
                   306
                
                 
                   54
                
              
               
                 
                   9
                
                 
                   486
                
                 
                   216
                
              
               
                 
                   522
                
                 
                   54
                
              
               
                 
                   12
                
                 
                   1008
                
                 
                   270
                
              
               
                 
                   792
                
                 
                   54
                
              
               
                 
                   15
                
                 
                   1800
                
                 
                   324
                
              
               
                 
                   1116
                
                 
                   54
                
              
               
                 
                   18
                
                 
                   2216
                
                 
                   378
                
              
               
                 
                   1494
                
                 
                   54
                
              
               
                 
                   21
                
                 
                   4410
                
                 
                   432
                
              
               
                 
                   1926
                
                 
                    
                
              
               
                 
                   24
                
                 
                   6336
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
              
            
             For
             if
             the
             third
             differences
             which
             are
             equal
             ,
             and
             in
             this
             Example
             is
             54
             ,
             be
             added
             to
             the
             first
             of
             the
             second
             differences
             ,
             being
             108
             ,
             it
             makes
             162
             ,
             and
             by
             such
             additions
             ,
             the
             second
             differences
             in
             the
             fourth
             colum
             are
             made
             .
             Further
             ,
             by
             adding
             these
             second
             differences
             to
             the
             first
             of
             the
             first
             differences
             which
             is
             36
             ,
             it
             makes
             144
             ,
             &c.
             
             So
             the
             numbers
             in
             the
             third
             colum
             are
             made
             .
             Yet
             further
             ,
             by
             adding
             these
             first
             differences
             to
             the
             first
             number
             in
             the
             second
             colum
             ,
             the
             Rank
             of
             portions
             of
             such
             a
             conoid
             is
             made
             .
          
           
             Then
             ,
          
           
             By
             making
             use
             of
             the
             directions
             in
             the
             first
             and
             second
             
               Scholiums
               ,
            
             the
             number
             of
             Gallons
             are
             obtained
             .
             The
             parabolick
             and
             hyperbolick
             conoides
             may
             well
             be
             made
             use
             of
             for
             Brewers
             Coppers
             ;
             the
             parabolick
             ,
             when
             the
             crown
             is
             somewhat
             blunt
             ;
             but
             the
             hyperbolick
             conoid
             when
             the
             crown
             is
             more
             sharp
             .
          
        
         
           
           
             XIII
             .
             Note
             .
             To
             calculate
             a
             Sphere
             gradually
             ,
             to
             wit
             ,
             upon
             every
             three
             Inches
             .
          
           
             Consider
             the
             XV
             Prop.
             of
             
               Stereom
               .
               Prop.
            
             Let
             ED
             equal
             to
             EF
             ,
             be
             24.
             
             The
             rest
             of
             the
             construction
             as
             in
             that
             Prop.
             Let
             from
             E
             ,
             to
             the
             first
             R
             be
             3
             ,
             whose
             square
             is
             9
             ;
             whose
             half
             is
             4½
             ,
             the
             area
             RXE
             ,
             which
             being
             multiplied
             by
             24
             ,
             the
             product
             will
             be
             108
             ;
             the
             prism
             KHXREF
             .
             The
             cube
             of
             3
             ,
             is
             27
             ;
             a
             third
             part
             thereof
             is
             9
             ,
             the
             pyramid
             KHOIF
             ;
             this
             pyramid
             taken
             from
             the
             former
             prism
             ,
             leaves
             the
             prism
             RXOIFE
             ,
             99
             :
             equal
             to
             all
             the
             squares
             in
             the
             portion
             RQE.
             From
             E
             ,
             to
             the
             second
             R
             ,
             6
             ;
             its
             square
             36
             ,
             the
             half
             18
             ,
             which
             multiplied
             by
             24
             ,
             makes
             432
             ;
             the
             prism
             RXHKFE
             .
             The
             cube
             of
             6
             ,
             is
             216
             ,
             a
             third
             part
             of
             it
             is
             72
             ;
             the
             pyramid
             KHOIF
             :
             this
             pyramid
             being
             taken
             from
             that
             prism
             ,
             there
             rest
             360
             ;
             the
             prism
             RXOIFE
             ,
             equal
             to
             all
             the
             squares
             in
             the
             portion
             RQE.
             Let
             from
             E
             to
             the
             third
             R
             be
             9
             ;
             its
             square
             81
             ,
             the
             half
             thereof
             40½
             ,
             the
             area
             RXE
             ,
             this
             area
             being
             multiplied
             by
             24
             ,
             the
             product
             will
             be
             972
             ,
             the
             prism
             KHYREF
             :
             the
             cube
             of
             9
             ,
             is
             729
             ,
             a
             third
             part
             of
             it
             is
             243
             ,
             the
             pyramid
             KHOIF
             :
             this
             pyramid
             being
             substracted
             from
             that
             prism
             ,
             the
             remainder
             is
             729
             ;
             the
             prism
             RXOIFE
             ,
             equal
             to
             all
             the
             squares
             in
             the
             third
             portion
             RQE.
             Having
             obtained
             these
             three
             portions
             ,
             the
             rest
             may
             be
             found
             by
             their
             third
             difference
             ,
             according
             to
             the
             X.
             
               Note
               .
            
          
           
           
             
               
                 
                   0
                
                 
                   00
                
                 
                   0
                
                 
                   0
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   99
                
                 
                    
                
              
               
                 
                   3
                
                 
                   09
                
                 
                   108
                
                 
                   99
                
                 
                   162
                
              
               
                 
                   261
                
                 
                   54
                
              
               
                 
                   6
                
                 
                   72
                
                 
                   432
                
                 
                   360
                
                 
                   108
                
              
               
                 
                   369
                
                 
                   54
                
              
               
                 
                   9
                
                 
                   243
                
                 
                   972
                
                 
                   729
                
                 
                   54
                
              
               
                 
                   423
                
                 
                   54
                
              
               
                 
                   12
                
                 
                   576
                
                 
                   1728
                
                 
                   1152
                
                 
                   00
                
              
               
                 
                   423
                
                 
                   54
                
              
               
                 
                   15
                
                 
                   1125
                
                 
                   2700
                
                 
                   1575
                
                 
                   54
                
              
               
                 
                   369
                
                 
                   54
                
              
               
                 
                   18
                
                 
                   1944
                
                 
                   3888
                
                 
                   1944
                
                 
                   108
                
              
               
                 
                   261
                
                 
                   54
                
              
               
                 
                   21
                
                 
                   3087
                
                 
                   5292
                
                 
                   2205
                
                 
                   162
                
              
               
                 
                   99
                
                 
                    
                
              
               
                 
                   24
                
                 
                   4608
                
                 
                   6912
                
                 
                   2304
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
                 
                   6
                
                 
                   7
                
              
            
          
           
             The
             numbers
             in
             the
             seventh
             colum
             are
             the
             third
             differences
             ,
             and
             they
             equal
             ;
             the
             numbers
             in
             the
             sixt
             colum
             are
             the
             second
             differences
             ,
             and
             are
             composed
             by
             substracting
             the
             numbers
             in
             the
             seventh
             from
             the
             first
             and
             last
             numbers
             in
             the
             sixt
             colum
             ;
             the
             numbers
             in
             the
             fifth
             colum
             are
             the
             first
             differences
             ,
             and
             are
             composed
             by
             adding
             those
             numbers
             in
             the
             sixt
             colum
             to
             the
             first
             and
             last
             of
             those
             in
             the
             fifth
             colum
             ;
             the
             numbers
             in
             the
             fourth
             colum
             are
             all
             the
             squares
             in
             several
             portions
             of
             one
             fourth
             of
             a
             sphere
             whose
             diameter
             is
             24
             ,
             those
             portions
             are
             made
             by
             adding
             the
             numbers
             in
             the
             fifth
             colum
             to
             the
             numbers
             in
             the
             fourth
             ,
             thus
             ,
             261
             ,
             and
             99
             ,
             is
             360.
             369
             ,
             and
             360
             ,
             is
             729.
             423
             ,
             and
             729
             ,
             is
             1152
             ,
             &c.
             
          
           
             Then
             making
             use
             of
             the
             first
             and
             second
             scholium
             the
             number
             of
             gallons
             are
             obta●ned
             .
             Or
             if
             it
             be
             made
             ,
             as
             14
             ,
             is
             to
             11
             ,
             so
             is
             54
             ,
             to
             a
             fourth
             
             number
             ,
             with
             that
             fourth
             number
             proceede
             to
             make
             tables
             of
             the
             second
             and
             first
             differences
             ,
             and
             then
             the
             table
             of
             portions
             it selfe
             .
             Every
             sphere
             hath
             its
             third
             differences
             equall
             .
             To
             find
             the
             third
             difference
             ,
             doe
             thus
             .
             Cube
             one
             of
             the
             equal
             segments
             of
             the
             axis
             and
             multiply
             that
             cube
             by
             2
             ,
             and
             that
             product
             will
             be
             the
             third
             difference
             ,
             thus
             ,
             the
             cube
             of
             three
             is
             27
             ,
             which
             multiplyed
             by
             2
             ,
             the
             product
             is
             54
             ;
             the
             third
             difference
             of
             all
             the
             squares
             in
             one
             fourth
             of
             a
             sphere
             .
             Here
             note
             ,
             that
             it
             is
             to
             be
             understood
             ,
             that
             the
             axis
             of
             the
             sphere
             is
             equally
             divided
             into
             an
             equal
             number
             of
             segments
             ;
             so
             then
             ,
             if
             the
             number
             of
             segments
             in
             the
             semiaxis
             ,
             less
             by
             one
             ;
             be
             multiplyed
             by
             the
             third
             difference
             ,
             it
             gives
             the
             first
             of
             the
             second
             differences
             .
             Thus
             ,
             the
             number
             of
             segments
             in
             the
             semiaxis
             is
             4
             ,
             then
             4
             less
             1
             ,
             is
             3
             ;
             which
             being
             multiplyed
             by
             54
             ,
             the
             product
             is
             162
             :
             the
             first
             of
             the
             second
             differences
             .
          
           
             To
             find
             the
             third
             difference
             in
             one
             fourth
             of
             all
             the
             squares
             in
             a
             spheroid
             ,
             do
             thus
             :
             The
             axis
             being
             divided
             as
             above
             in
             the
             sphere
             ;
             cube
             the
             difference
             betwixt
             two
             Segments
             ,
             which
             being
             multiplyed
             by
             2
             ,
             makes
             a
             product
             ;
             then
             ,
             as
             the
             square
             of
             the
             semiaxis
             ,
             is
             to
             the
             square
             of
             the
             other
             semidiameter
             ;
             so
             is
             that
             former
             product
             to
             a
             fourth
             number
             ,
             which
             will
             be
             the
             third
             difference
             .
             For
             the
             second
             differences
             ,
             use
             the
             Rules
             given
             for
             the
             sphere
             .
          
        
         
           
             XIV
             .
             Note
             .
          
           
             To
             calculate
             a
             pyramid
             or
             cone
             gradually
             .
             To
             find
             the
             third
             difference
             in
             a
             pyramid
             work
             thus
             ,
             
             the
             Altitude
             of
             the
             pyramid
             being
             equally
             divided
             ▪
             cube
             the
             difference
             of
             the
             two
             segments
             ,
             which
             being
             doubled
             ,
             makes
             a
             number
             ;
             then
             ,
             as
             the
             square
             of
             the
             Altitude
             of
             the
             pyramid
             ,
             is
             to
             the
             area
             of
             the
             base
             of
             that
             pyramid
             ;
             so
             is
             that
             former
             number
             ,
             to
             the
             third
             difference
             of
             that
             pyramid
             .
          
           
             To
             find
             the
             second
             differences
             in
             a
             pyramid
             :
             As
             the
             difference
             of
             two
             of
             the
             segments
             of
             the
             Altitude
             ,
             is
             to
             the
             following
             segment
             ;
             so
             is
             the
             third
             difference
             ,
             to
             the
             second
             difference
             answerable
             to
             that
             segment
             .
          
           
             To
             find
             the
             first
             differences
             in
             a
             pyramid
             .
             Cube
             two
             of
             the
             segments
             ,
             and
             take
             a
             third
             part
             of
             their
             difference
             .
             Then
             ,
             as
             the
             square
             of
             the
             Altitude
             of
             the
             pyramid
             ;
             is
             to
             the
             area
             of
             the
             base
             of
             that
             pyramid
             ;
             so
             is
             that
             former
             difference
             ;
             to
             the
             first
             difference
             answerable
             to
             those
             two
             segments
             .
          
           
             Let
             there
             be
             a
             pyramid
             whose
             Altitude
             is
             10
             ,
             and
             one
             side
             of
             the
             base
             is
             40
             ,
             and
             the
             other
             side
             5
             ;
             therefore
             the
             area
             of
             the
             base
             is
             200.
             
             Let
             the
             Altitude
             be
             divided
             into
             five
             equall
             parts
             ,
             and
             to
             calculate
             accordingly
             .
             To
             find
             the
             third
             difference
             ,
             the
             cube
             of
             2
             ,
             is
             8
             ;
             whose
             double
             is
             16.
             
             Then
             ,
             as
             100
             the
             square
             of
             the
             Altitude
             ,
             is
             to
             200
             the
             area
             of
             the
             base
             ;
             so
             is
             16
             ,
             to
             the
             third
             difference
             32.
             
             To
             find
             any
             of
             the
             second
             differences
             at
             demand
             ,
             to
             find
             the
             second
             difference
             answerable
             to
             8.
             
             As
             2
             ,
             the
             difference
             betwixt
             the
             segments
             6
             ,
             and
             8
             ,
             is
             to
             8
             ;
             so
             is
             the
             first
             difference
             32
             ,
             to
             128
             the
             second
             difference
             answerable
             to
             8.
             
             The
             second
             differences
             are
             in
             proportion
             one
             to
             another
             ,
             as
             their
             answering
             segments
             ;
             as
             2
             ,
             is
             to
             
             3
             ▪
             2
             ;
             so
             is
             8
             ,
             to
             128.
             
             To
             find
             any
             of
             the
             first
             differences
             ,
             cube
             the
             two
             Segments
             ,
             to
             wit
             ,
             2
             and
             4
             ,
             and
             the
             cubes
             will
             be
             8
             and
             64
             ;
             then
             take
             8
             from
             64
             and
             the
             Remainder
             is
             56
             ,
             a
             third
             part
             is
             18⅔
             .
             then
             ,
             as
             the
             square
             of
             the
             Altitude
             100
             ,
             is
             to
             the
             area
             of
             the
             base
             200
             ;
             so
             is
             18⅔
             ,
             to
             37⅓
             ,
             the
             first
             difference
             ,
             answering
             to
             2
             and
             4.
             
             Then
             by
             a
             continuall
             adding
             of
             the
             third
             difference
             to
             the
             second
             differences
             they
             are
             made
             ,
             and
             by
             adding
             the
             first
             of
             the
             second
             differences
             to
             the
             first
             of
             the
             first
             differences
             and
             so
             in
             order
             the
             first
             differences
             are
             made
             .
             Lastly
             by
             adding
             the
             first
             differences
             the
             Segments
             of
             the
             pyramid
             ,
             are
             made
             according
             to
             the
             III.
             Note
             ;
             or
             thus
             .
          
           
             
               
                 
                   0
                
                 
                   0
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   5⅓
                
                 
                    
                
              
               
                 
                   2
                
                 
                   5⅓
                
                 
                   32
                
              
               
                 
                   37⅓
                
                 
                   32
                
              
               
                 
                   4
                
                 
                   42⅔
                
                 
                   64
                
              
               
                 
                   101⅓
                
                 
                   32
                
              
               
                 
                   6
                
                 
                   144··
                
                 
                   96
                
              
               
                 
                   197⅓
                
                 
                   32
                
              
               
                 
                   8
                
                 
                   341⅓
                
                 
                   128
                
              
               
                 
                   325⅓
                
                 
                    
                
              
               
                 
                   10
                
                 
                   666⅔
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
              
            
          
           
             The
             numbers
             in
             the
             fifth
             colum
             are
             the
             third
             differences
             ,
             the
             first
             number
             in
             the
             fourth
             colum
             being
             found
             by
             the
             Rule
             before
             given
             ,
             all
             the
             numbers
             in
             that
             fourth
             colum
             may
             be
             made
             by
             adding
             the
             third
             difference
             ,
             thus
             ,
             to
             32
             adde
             32
             ,
             the
             summe
             is
             64.
             adde
             32
             ,
             to
             64
             ;
             the
             summe
             is
             96.
             adde
             32
             ,
             to
             96
             ;
             the
             summe
             is
             128.
             
             The
             first
             number
             in
             the
             third
             colum
             being
             found
             by
             the
             Rule
             above
             ,
             then
             5⅓
             added
             to
             32
             ;
             the
             summe
             is
             37⅓
             .
             
             adde
             64
             ,
             to
             37⅓
             ;
             the
             summe
             is
             101⅓
             .
             adde
             96
             ,
             to
             101⅓
             ,
             the
             sum
             is
             197⅓
             .
             adde
             128
             ,
             to
             197⅓
             ;
             the
             sum
             is
             325⅓
             .
             further
             ,
             adde
             the
             first
             of
             the
             third
             colum
             ,
             to
             the
             first
             of
             the
             second
             colum
             ;
             thus
             ,
             adde
             5⅓
             ,
             to
             0
             ;
             the
             sum
             will
             be
             5⅓
             ,
             adde
             57⅓
             ,
             to
             5⅓
             ,
             the
             sum
             is
             42⅔
             ,
             adde
             101⅓
             ,
             to
             42⅔
             ;
             the
             sum
             is
             144.
             adde
             197⅓
             ,
             to
             144
             ;
             the
             sum
             is
             341⅓
             ,
             adde
             325⅓
             ,
             to
             341⅓
             ;
             the
             sum
             is
             666⅔
             .
          
           
             If
             it
             be
             to
             calculate
             a
             cone
             whose
             diameters
             of
             the
             base
             are
             40
             and
             5.
             
             Let
             it
             it
             be
             made
             ,
             as
             14
             ,
             is
             to
             11
             ;
             so
             is
             32
             ,
             to
             the
             third
             difference
             of
             the
             same
             cone
             .
             Then
             proceede
             with
             the
             third
             difference
             to
             make
             the
             second
             and
             first
             ;
             and
             lastly
             ,
             the
             table
             it self
             .
          
        
         
           
             XV.
             Note
             .
          
           
             The
             calculation
             of
             frustum
             pyramides
             whose
             bases
             are
             unlike
             ,
             To
             the
             performance
             of
             which
             consider
             the
             third
             case
             of
             the
             second
             proposition
             of
             
               Stereom
               .
               Prop.
               
            
          
           
             Every
             such
             solide
             hath
             its
             third
             differences
             equall
             ,
             but
             the
             second
             and
             first
             differences
             will
             be
             complicated
             according
             to
             the
             IX
             .
             Note
             .
          
           
             To
             find
             the
             third
             difference
             proper
             to
             the
             pyramid
             BCDHF
             ,
             Let
             the
             construction
             and
             numbers
             be
             the
             same
             as
             in
             that
             diagram
             ,
             and
             let
             it
             be
             to
             calculate
             it
             upon
             every
             two
             inches
             ,
             thus
             .
             The
             cube
             of
             2
             ,
             is
             8
             ;
             the
             double
             thereof
             is
             16
             ,
             Then
             ,
             as
             the
             square
             of
             the
             Altitude
             40
             ,
             that
             is
             1600
             ,
             is
             to
             the
             area
             of
             the
             base
             BCDH
             ,
             336
             ;
             so
             is
             16
             ,
             to
             336
             /
             100.
             by
             the
             Rule
             delivered
             in
             the
             
             14
             
               Note
               ,
            
             the
             first
             of
             the
             second
             differences
             is
             336
             /
             100.
             and
             the
             first
             of
             the
             first
             differences
             is
             56
             /
             100.
             
          
           
             The
             solide
             HDEGVF
             hath
             its
             second
             differences
             equall
             by
             the
             VII
             .
             
               Note
               .
            
          
           
             To
             find
             its
             first
             and
             second
             differences
             .
             The
             square
             of
             2
             ,
             is
             4.
             which
             multiplyed
             by
             FV
             ,
             26
             ;
             the
             product
             will
             be
             104.
             then
             ,
             as
             40
             the
             Altitude
             ,
             is
             to
             HD
             ,
             28
             ;
             so
             is
             104
             ,
             to
             7280
             /
             100.
             the
             second
             difference
             .
             Therefore
             the
             first
             of
             the
             first
             differences
             will
             be
             3640
             /
             100.
             
          
           
             To
             find
             the
             second
             differences
             of
             the
             solide
             ABHOIF
             the
             square
             of
             2
             is
             4
             ,
             which
             multiplyed
             by
             IF
             ,
             30
             ;
             the
             product
             is
             120
             ,
             then
             ,
             as
             40
             ,
             the
             Altitude
             ;
             is
             to
             OA
             ,
             12
             :
             so
             is
             120
             ,
             to
             36
             ,
             the
             second
             difference
             .
             Therefore
             the
             first
             of
             the
             first
             differences
             are
             18.
             
             For
             the
             complication
             of
             these
             differences
             .
             
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                    
                
              
               
                 
                   ●6
                   /
                   100
                
                 
                   336
                   /
                   100
                
                 
                   3●6
                   /
                   100
                
                 
                   in
                   the
                   pyramid
                   BCDHF
                
              
               
                 
                   3640
                   /
                   100
                
                 
                   7280
                   /
                   100
                
                 
                    
                
                 
                   in
                   the
                   prism
                   HGEDFV
                
              
               
                 
                   18
                
                 
                   36
                
                 
                    
                
                 
                   in
                   the
                   prism
                   ABHOIF
                
              
               
                 
                   5496
                   /
                   100
                
                 
                   11216
                   /
                   100
                
                 
                   336
                   /
                   100
                
                 
                   their
                   summe
                   .
                
              
            
          
           
             Rejecting
             the
             denominators
             they
             may
             be
             written
             Thus
             ,
             
               
                 
                   5496
                
                 
                   11216
                
                 
                   336
                
              
            
          
           
             Because
             the
             denominators
             are
             Rejected
             ,
             therefore
             the
             two
             last
             figures
             toward
             the
             Right
             hand
             are
             decimals
             .
          
           
           
             
               
                 
                   0
                
                 
                    
                
                 
                    
                
                 
                    
                
                 
                    
                
              
               
                 
                   161496
                
                 
                    
                
              
               
                 
                   2
                
                 
                   161496
                
                 
                   11216
                
              
               
                 
                   172712
                
                 
                   336
                
              
               
                 
                   4
                
                 
                   334208
                
                 
                   11552
                
              
               
                 
                   184264
                
                 
                   336
                
              
               
                 
                   6
                
                 
                   518472
                
                 
                   11888
                
              
               
                 
                   196152
                
                 
                   336
                
              
               
                 
                   8
                
                 
                   714624
                
                 
                   12224
                
              
               
                 
                   208376
                
                 
                   336
                
              
               
                 
                   10
                
                 
                   923000
                
                 
                   12560
                
              
               
                 
                   220936
                
                 
                   336
                
              
               
                 
                   12
                
                 
                   1143936
                
                 
                   12896
                
              
               
                 
                   233832
                
                 
                   336
                
              
               
                 
                   14
                
                 
                   1377768
                
                 
                   13232
                
              
               
                 
                   247064
                
                 
                   336
                
              
               
                 
                   16
                
                 
                   1624832
                
                 
                   13568
                
              
               
                 
                   260632
                
                 
                   336
                
              
               
                 
                   18
                
                 
                   1885464
                
                 
                   13904
                
              
               
                 
                   274536
                
                 
                   336
                
              
               
                 
                   20
                
                 
                   2160000
                
                 
                   14240
                
              
               
                 
                   288776
                
                 
                   336
                
              
               
                 
                   22
                
                 
                   2448776
                
                 
                   14576
                
              
               
                 
                   303352
                
                 
                   336
                
              
               
                 
                   24
                
                 
                   2752128
                
                 
                   14912
                
              
               
                 
                   318264
                
                 
                   336
                
              
               
                 
                   26
                
                 
                   3070392
                
                 
                   15248
                
              
               
                 
                   333512
                
                 
                   336
                
              
               
                 
                   28
                
                 
                   3403904
                
                 
                   15584
                
              
               
                 
                   349096
                
                 
                   336
                
              
               
                 
                   30
                
                 
                   3753000
                
                 
                   15920
                
              
               
                 
                   365016
                
                 
                   336
                
              
               
                 
                   32
                
                 
                   4118016
                
                 
                   16256
                
              
               
                 
                   381272
                
                 
                   336
                
              
               
                 
                   34
                
                 
                   4499288
                
                 
                   16592
                
              
               
                 
                   397864
                
                 
                   336
                
              
               
                 
                   36
                
                 
                   4897152
                
                 
                   16928
                
              
               
                 
                   414792
                
                 
                   336
                
              
               
                 
                   38
                
                 
                   5311944
                
                 
                   17264
                
              
               
                 
                   432056
                
                 
                    
                
              
               
                 
                   40
                
                 
                   5744000
                
                 
                    
                
              
               
                 
                    
                
                 
                    
                
              
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
              
            
          
           
           
             The
             construction
             of
             the
             table
             may
             be
             thus
             ;
             the
             numbers
             in
             the
             first
             colum
             are
             the
             third
             differences
             ▪
             The
             first
             number
             in
             the
             fourth
             colum
             is
             the
             complicated
             second
             difference
             ,
             and
             the
             other
             number
             in
             that
             fourth
             colum
             are
             made
             thus
             ,
             to
             the
             first
             11216
             ,
             adde
             336
             ;
             the
             sum
             is
             11552.
             
             Then
             to
             that
             11552
             ,
             adde
             336
             ;
             the
             sum
             is
             11888
             ,
             &c.
             
          
           
             The
             first
             number
             in
             the
             third
             colum
             is
             complicated
             from
             the
             first
             complicated
             difference
             and
             a
             parallelipepidon
             whose
             base
             is
             the
             plane
             RIFV
             ,
             and
             the
             Altitude
             the
             first
             Segment
             of
             the
             Altitude
             of
             the
             frustum
             ,
             thus
             ,
             the
             plane
             RIFV
             ,
             is
             780
             ;
             which
             being
             doubled
             is
             1560
             ;
             then
             ,
             156000
             more
             5496
             is
             161496
             ;
             the
             first
             of
             the
             first
             differences
             ,
             then
             161496
             more
             11216
             ,
             is
             172712.
             
             Further
             ,
             172712
             more
             11552
             ,
             is
             184264.
             
             Yet
             further
             184264
             more
             11888
             ,
             is
             196152
             ,
             &c.
             
          
           
             The
             numbers
             in
             the
             second
             colum
             are
             made
             thus
             ,
             the
             first
             number
             in
             the
             second
             colum
             ,
             is
             the
             same
             as
             the
             first
             number
             in
             the
             third
             colum
             ,
             then
             ,
             161496
             more
             172712
             ,
             is
             334208
             ,
             and
             334208
             more
             184264
             ,
             is
             518472
             ,
             &c.
             
          
           
             Then
             makeing
             use
             of
             the
             first
             and
             second
             scholium
             ,
             the
             quantity
             of
             Liquor
             that
             such
             vessels
             contain
             may
             easily
             be
             obtained
             .
          
        
         
           
             XVI
             .
             Note
             .
          
           
             To
             calculate
             Elliptick
             solides
             whose
             bases
             are
             unlike
             .
             The
             calculation
             of
             such
             solides
             are
             the
             same
             as
             in
             the
             15
             ,
             note
             for
             if
             the
             first
             ,
             second
             and
             third
             complicated
             differences
             be
             found
             ,
             then
             makeing
             use
             of
             this
             propotion
             as
             14
             ,
             is
             to
             11
             ;
             so
             is
             336
             ;
             to
             the
             third
             difference
             .
          
           
           
             And
          
           
             As
             14
             ,
             is
             to
             11
             ;
             so
             is
             11216
             ,
             to
             the
             first
             of
             the
             second
             differences
             .
          
           
             Further
          
           
             As
             14
             ,
             is
             to
             11
             ;
             so
             is
             161496
             ,
             to
             the
             first
             of
             the
             first
             differences
             ,
             then
             proceede
             to
             make
             the
             table
             it self
             ,
             as
             in
             the
             15
             note
             .
             Or
             make
             use
             of
             the
             secund
             scholium
             of
             the
             11
             note
             and
             you
             will
             have
             the
             quantity
             in
             Gallons
             .
          
           
             Or
          
           
             Such
             Elliptick
             solides
             may
             be
             calculated
             by
             the
             12
             note
             :
             for
             every
             such
             Elliptick
             solide
             is
             equall
             to
             a
             frustum
             hyperbolick
             conoide
             whose
             circular
             bases
             of
             the
             conoide
             ,
             are
             equall
             to
             the
             Elliptick
             bases
             of
             the
             Elliptick
             solide
             ;
             and
             the
             Altitude
             of
             one
             frustum
             is
             equall
             to
             the
             Altitude
             of
             the
             other
             .
          
        
         
           
             XVII
             .
             Note
             .
          
           
             Every
             hyperbolick
             conoid
             hath
             its
             third
             differences
             equal
             .
          
           
             To
             find
             the
             third
             ,
             second
             and
             first
             differences
             in
             an
             hyperbolick
             conoid
             ,
             and
             consequently
             to
             calculate
             that
             conoid
             gradually
             .
             In
             the
             forementioned
             diagram
             of
             the
             17.
             prop.
             Stereom
             .
             Prop.
             Let
             GM
             ,
             the
             Transverse
             diameter
             be
             12.
             
             ML
             ,
             the
             parameter
             6.
             
             MA
             ,
             the
             axis
             of
             the
             conoid
             24.
             
          
           
             
             
               To
               calculate
               the
               solidity
               of
               this
               conoid
               vpon
               every
               three
               Inches
               .
            
             
               
                 To
                 find
                 the
                 third
                 difference
                 of
                 this
                 conoid
                 .
              
            
             
               Take
               the
               difference
               of
               two
               of
               the
               Segments
               ,
               to
               wit
               ,
               3
               ;
               whose
               cube
               is
               27
               :
               whose
               double
               is
               54.
               
               Then
               ,
               as
               GM
               ,
               12
               ;
               is
               to
               ML
               ,
               6
               :
               so
               is
               54
               ,
               to
               27.
               
               The
               third
               difference
               of
               all
               the
               squares
               in
               one
               fourth
               of
               that
               conoid
               proper
               to
               that
               pyramid
               FCDEL
               .
            
             
               By
               the
               Rule
               in
               the
               last
               note
               the
               first
               of
               the
               second
               differences
               is
               27.
               
            
             
               For
               the
               first
               of
               the
               first
               differences
               ,
               worke
               thus
               ;
               take
               the
               first
               Segment
               which
               is
               3
               ;
               whose
               cube
               is
               27
               ;
               a
               third
               part
               is
               9
               ,
               then
               ,
               as
               GM
               ,
               12
               ;
               is
               to
               ML
               ,
               6
               :
               so
               is
               9
               ,
               to
               4½
               .
               the
               first
               of
               the
               first
               differences
               proper
               to
               the
               pyramid
               FCDEL
               .
            
             
               The
               second
               and
               first
               differences
               of
               all
               the
               squares
               in
               the
               fourth
               of
               this
               conoid
               ,
               is
               complicated
               from
               the
               second
               and
               first
               differences
               of
               the
               pyramid
               FCDEL
               ,
               and
               the
               second
               and
               first
               differences
               of
               the
               prism
               ABCFLM
               .
               Every
               such
               prism
               hath
               it
               second
               difference
               equall
               .
            
             
               
                 To
                 find
                 the
                 second
                 and
                 first
                 difference
                 of
                 the
                 prism
                 ABCFLM
                 .
              
            
             
               Square
               the
               difference
               of
               two
               of
               the
               Segments
               of
               the
               axis
               ,
               to
               wit
               ,
               3
               ▪
               that
               is
               9
               ,
               which
               being
               multiplyed
               by
               the
               parameter
               ML
               ,
               6
               ;
               the
               product
               is
               54
               ,
               the
               second
               difference
               .
               The
               first
               of
               the
               first
               d●fferences
               of
               every
               such
               prism
               is
               half
               of
               the
               
               second
               difference
               ;
               therefore
               the
               first
               of
               the
               first
               differences
               is
               27.
               
            
             
               
                 To
                 complicate
                 these
                 differences
                 .
              
            
             
               
                 
                   
                     1
                  
                   
                     2
                  
                   
                     3
                  
                   
                     differences
                  
                
                 
                   
                     4½
                  
                   
                     27
                  
                   
                     27
                  
                   
                     in
                     the
                     prism
                     FCDEL
                     .
                  
                
                 
                   
                     27
                  
                   
                     54
                  
                   
                      
                  
                   
                     in
                     the
                     prism
                     ABCFLM
                     .
                  
                
                 
                   
                     31½
                  
                   
                     81
                  
                   
                     27
                  
                   
                     in
                     all
                     the
                     squares
                     of
                     one
                     fourth
                     of
                     that
                     hyperbolick
                     conoid
                     .
                  
                
              
               
                 
                   
                     0
                  
                   
                     0
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     31½
                  
                   
                      
                  
                
                 
                   
                     3
                  
                   
                     31½
                  
                   
                     81
                  
                
                 
                   
                     112½
                  
                   
                     27
                  
                
                 
                   
                     6
                  
                   
                     144
                  
                   
                     108
                  
                
                 
                   
                     220½
                  
                   
                     27
                  
                
                 
                   
                     9
                  
                   
                     364½
                  
                   
                     135
                  
                
                 
                   
                     355½
                  
                   
                     27
                  
                
                 
                   
                     12
                  
                   
                     720
                  
                   
                     162
                  
                
                 
                   
                     517½
                  
                   
                     27
                  
                
                 
                   
                     15
                  
                   
                     1237½
                  
                   
                     189
                  
                
                 
                   
                     706½
                  
                   
                     27
                  
                
                 
                   
                     18
                  
                   
                     1944
                  
                   
                     216
                  
                
                 
                   
                     922½
                  
                   
                     27
                  
                
                 
                   
                     21
                  
                   
                     2866½
                  
                   
                     243
                  
                
                 
                   
                     1165½
                  
                   
                      
                  
                
                 
                   
                     24
                  
                   
                     4032
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                      
                  
                
                 
                   
                     1
                  
                   
                     2
                  
                   
                     3
                  
                   
                     4
                  
                   
                     5
                  
                
              
            
             
               In
               the
               first
               colum
               are
               the
               third
               differences
               .
               In
               the
               fourth
               colum
               the
               second
               differences
               .
               In
               the
               third
               colum
               the
               first
               differences
               .
               In
               the
               second
               colum
               the
               portions
               of
               all
               the
               squares
               of
               one
               fourth
               of
               an
               hyperbolick
               conoid
               ,
               upon
               every
               three
               
               inches
               ,
               whose
               Transverse
               diameter
               is
               12
               ,
               and
               parameter
               is
               6
               ,
               and
               axis
               is
               24.
               
            
             
               The
               construction
               of
               this
               table
               is
               the
               same
               as
               the
               former
               ;
               thus
               ,
               81
               more
               27
               ;
               is
               108.
               more
               27
               ;
               is
               135.
               more
               27
               ;
               is
               162.
               &c.
               
            
             
               31½
               .
               more
               81
               ;
               is
               112½
               .
               more
               108
               ;
               is
               220½
               .
               &c.
               
            
             
               0
               more
               31½
               ;
               is
               31½
               .
               more
               112½
               ;
               is
               144.
               more
               220½
               .
               is
               364½
               .
               more
               355½
               ;
               is
               720.
               
            
             
               Here
               remember
               that
               the
               Transverse
               diameter
               is
               found
               ,
               by
               the
               9
               of
               the
               23
               Proposition
               .
               of
               
                 Stereom
                 .
                 Prop.
              
               Also
               the
               parameter
               found
               by
               the
               converse
               of
               the
               first
               part
               of
               the
               11
               Prop.
               of
               
                 Stereom
                 .
                 Prop.
              
               The
               parameter
               of
               the
               parabolike
               conoid
               is
               found
               ,
               by
               the
               converse
               of
               the
               9
               Prop.
               Of
               
                 Stereom
                 .
                 Prop.
                 
              
            
          
        
         
           
             XVIII
             .
             Note
             .
          
           
             
               Cautions
               Concerning
               Reduction
               .
               1
            
             
               If
               it
               be
               to
               calculate
               pyramids
               whether
               Regular
               or
               Irregular
               ,
               whole
               or
               frustums
               ;
               the
               third
               ,
               second
               and
               first
               differences
               are
               to
               be
               found
               as
               above
               :
               then
               Reduce
               those
               differences
               into
               Gallons
               and
               parts
               of
               a
               Gallon
               ,
               or
               Barrells
               ,
               or
               parts
               of
               a
               barrels
               ;
            
             
               Thus
            
             
               Suppose
               288
               cubick
               inches
               make
               one
               Gallon
               ,
               and
               36
               Gallons
               make
               one
               Barrell
               .
            
             
               Then
               ,
            
             
               If
               the
               measure
               be
               taken
               in
               inches
               ,
               divide
               the
               third
               ,
               second
               and
               first
               differences
               by
               288
               ,
               and
               so
               there
               will
               be
               three
               quotients
               in
               Gallons
               or
               parts
               
               of
               a
               Gallon
               ,
               then
               with
               those
               three
               quotients
               proceede
               to
               make
               the
               table
               of
               solid
               Segments
               ,
               and
               that
               table
               will
               be
               in
               Gallons
               or
               parts
               of
               a
               Gallon
               .
               If
               it
               be
               to
               calculate
               a
               table
               in
               Barrells
               multiply
               288
               by
               36
               and
               the
               product
               will
               be
               10368
               the
               number
               of
               cubick
               inches
               in
               one
               Barrell
               .
               Then
               divide
               the
               third
               ,
               second
               and
               first
               differences
               by
               10368
               ,
               there
               will
               be
               three
               quotients
               in
               Barrells
               or
               parts
               of
               a
               Barrell
               :
               Then
               with
               these
               three
               quotients
               proceede
               to
               make
               the
               table
               of
               solides
               Segments
               .
            
             
               That
               table
               being
               so
               made
               will
               be
               in
               Barrells
               or
               parts
               of
               a
               Barrell
               .
            
          
           
             
               2
               ▪
            
             
               To
               calculate
               Cones
               and
               Elliptick
               solids
               ,
               whether
               the
               whole
               or
               their
               frustums
               .
            
             
               Haveing
               found
               their
               third
               second
               and
               first
               differences
               ,
               as
               above
               ,
               and
               it
               be
               to
               calculate
               them
               in
               cubick
               inches
               ,
               Let
               it
               be
               made
               as
               14
               ,
               is
               to
               11
               ;
               so
               is
               the
               third
               difference
               ,
               to
               a
               fourth
               ,
            
             
               And
               ,
            
             
               As
               14
               ,
               is
               to
               11
               ;
               so
               is
               the
               second
               difference
               ,
               to
               a
               fourth
               ,
            
             
               Further
               ,
            
             
               As
               14
               ,
               is
               to
               11
               ;
               so
               is
               the
               first
               difference
               ,
               to
               a
               fourth
               with
               these
               three
               number
               thus
               found
               ,
               proceede
               to
               make
               the
               table
               of
               solid
               Segments
               ,
               and
               that
               table
               will
               be
               in
               cubick
               inches
               .
            
             
               To
               calculate
               these
               solids
               in
               Gallons
               .
            
             
               Multiply
               14
               by
               288
               the
               product
               will
               be
               4032.
               
            
             
               Then
               ,
            
             
               As
               4032
               ,
               to
               11
               ;
               so
               is
               the
               third
               difference
               ,
               to
               a
               fourth
               .
            
             
             
               And
               ,
            
             
               As
               4032
               ,
               to
               11
               ;
               so
               is
               the
               second
               difference
               ,
               to
               a
               fourth
               .
            
             
               Further
               ,
            
             
               As
               4032
               ,
               to
               11
               ;
               so
               is
               the
               first
               difference
               ,
               to
               a
               fourth
               .
            
             
               With
               these
               three
               numbers
               thus
               found
               ,
               proceed
               to
               make
               the
               table
               of
               solid
               Segments
               .
               So
               that
               table
               will
               be
               in
               Gallons
               .
            
             
               To
               calculate
               these
               solids
               in
               Barrells
               .
            
             
               Suppose
               288
               cubick
               inches
               makes
               one
               Gallon
               ,
               and
               36
               gallons
               makes
               one
               Barrell
               ,
               then
               multiply
               288
               ,
               36
               and
               14
               one
               into
               another
               and
               they
               make
               145152.
               
            
             
               Then
               ,
            
             
               As
               145152
               ,
               is
               to
               11
               ;
               so
               is
               the
               third
               difference
               ,
               to
               a
               fourth
               .
            
             
               And
               ,
            
             
               As
               145152
               ,
               is
               to
               11
               ;
               so
               is
               the
               second
               difference
               ,
               to
               a
               fourth
               .
            
             
               Further
               ,
            
             
               As
               145152
               ,
               is
               to
               11
               ;
               so
               is
               the
               first
               difference
               ,
               to
               a
               foruth
               .
            
             
               With
               these
               three
               numbers
               thus
               found
               make
               the
               table
               of
               solid
               Segments
               :
               that
               table
               will
               be
               in
               Barrells
               .
            
          
           
             
               III.
               
            
             
               Having
               found
               the
               third
               ,
               second
               and
               first
               differences
               of
               all
               the
               squares
               of
               one
               fourth
               of
               a
               sphere
               ,
               spheroid
               and
               hyperbolick
               Conoid
               ,
               as
               in
               the
               12
               and
               13
               notes
               and
               the
               second
               and
               first
               differences
               of
               all
               the
               squares
               of
               one
               fourth
               of
               a
               parabolick
               conoid
               as
               in
               the
               11
               note
               :
               they
               may
               be
               Reduced
               to
               Circular
               differences
               .
            
             
             
               Thus
               ▪
            
             
               As
               14
               ,
               is
               to
               11
               ;
               so
               is
               the
               third
               difference
               ,
               to
               a
               fourth
               .
            
             
               And
               ,
            
             
               As
               14
               ,
               is
               to
               11
               ,
               so
               is
               the
               second
               difference
               ,
               to
               a
               fourth
               .
            
             
               Further
               ,
            
             
               As
               14
               ,
               is
               to
               11
               ;
               so
               is
               the
               first
               difference
               ,
               to
               a
               fourth
               .
            
             
               With
               these
               numbers
               thus
               found
               make
               a
               table
               ,
               of
               solid
               Segments
               of
               cubical
               inches
               of
               one
               fourth
               of
               any
               of
               these
               solids
               .
            
             
               These
               solid
               Segments
               ought
               to
               be
               multiplyed
               by
               four
               ,
               to
               reduce
               them
               to
               solid
               Segments
               of
               a
               whole
               sphere
               ,
               spheroid
               ,
               hyperbolick
               and
               parabolick
               conoid
               :
               but
               to
               shun
               that
               work
               divide
               14
               ,
               by
               four
               ,
               and
               then
               find
               the
               new
               differences
               ;
               but
               because
               14
               cannot
               be
               just
               divided
               by
               four
               ,
               therefore
               divide
               14
               ,
               by
               two
               ,
               and
               multiply
               11
               ,
               by
               two
               ,
               and
               then
               work
               ;
               Thus
               ,
            
             
               Then
               ,
            
             
               As
               7
               ,
               to
               22
               ;
               so
               is
               that
               third
               difference
               ,
               to
               a
               fourth
               .
            
             
               And
               ,
            
             
               As
               7
               to
               22
               ;
               so
               is
               that
               second
               difference
               ,
               to
               a
               fourth
               .
            
             
               Further
               ,
            
             
               As
               7
               ,
               to
               22
               ;
               so
               is
               that
               first
               difference
               ,
               to
               a
               fourth
               .
            
             
               With
               these
               numbers
               thus
               found
               ,
               proceed
               to
               make
               tables
               as
               is
               taught
               in
               those
               Notes
               :
               tables
               so
               made
               ,
               will
               be
               tables
               of
               solid
               Segments
               of
               those
               solids
               ,
               in
               cubick
               inches
               .
            
             
               To
               calculate
               these
               solids
               in
               Gallons
               .
            
             
             
               Multiply
               288
               by
               14
               ,
               whose
               product
               is
               4032
               ;
               one
               fourth
               thereof
               is
               1008
               ;
            
             
               Then
               ,
            
             
               As
               1008
               ,
               is
               to
               11
               ;
               so
               is
               the
               third
               difference
               ,
               to
               a
               fourth
               .
            
             
               And
            
             
               As
               1008
               ,
               is
               to
               11
               ;
               so
               is
               the
               second
               difference
               ,
               to
               a
               fourth
               .
            
             
               Further
               ,
            
             
               As
               1008
               ,
               is
               to
               11
               ;
               so
               is
               the
               first
               difference
               ,
               to
               a
               fourth
               .
            
             
               Tables
               being
               made
               ,
               with
               numbers
               thus
               found
               ;
               according
               to
               the
               former
               directions
               in
               the
               sphere
               ,
               spheroid
               ,
               hyperbolick
               and
               parabolick
               conoids
               ,
               will
               be
               tables
               of
               solid
               Segments
               of
               a
               whole
               sphere
               ,
               spheroid
               ,
               hyperbolick
               and
               parabolick
               conoid
               ,
               in
               Gallons
               or
               parts
               thereof
               .
            
             
               To
               calculate
               these
               solids
               in
               Barrells
               .
            
             
               Multiply
               4032
               by
               36
               ,
               the
               product
               will
               be
               145152
               ,
               one
               fourth
               thereof
               will
               be
               36288
               ;
            
             
               Then
               ,
            
             
               As
               36288
               ,
               is
               to
               11
               ;
               so
               is
               the
               third
               difference
               ,
               to
               a
               fourth
               .
            
             
               And
               ,
            
             
               As
               36288
               ,
               is
               to
               11
               ;
               so
               is
               the
               second
               difference
               ,
               to
               a
               fourth
               .
            
             
               Further
               ,
            
             
               As
               36288
               ,
               is
               to
               11
               ;
               so
               is
               the
               first
               difference
               ,
               to
               a
               fourth
               .
            
             
               Tables
               being
               made
               ,
               with
               numbers
               thus
               found
               ,
               according
               to
               the
               former
               directions
               ,
               will
               be
               tables
               of
               solid
               Segments
               in
               Barrells
               .
               &c.
               
            
             
             
               Then
               ,
            
             
               Using
               a
               Rod
               or
               Ruler
               equally
               divided
               into
               inches
               as
               in
               scholium
               the
               first
               ,
               the
               number
               of
               Gallons
               or
               Barrells
               may
               speedily
               be
               obtained
               .
            
             
               As
               for
               the
               just
               magnitude
               of
               the
               Gallon
               ,
               it
               i●
               not
               my
               businesse
               to
               dispute
               ;
               that
               being
               determined
               by
               custom
               or
               Authority
               :
               I
               took
               288
               onely
               for
               Example
               sake
               .
            
          
        
         
           
             XIX
             .
             Note
             .
             In
             a
             Rank
             of
             numbers
             having
             equal
             differences
             .
          
           
             Let
             the
             first
             term
             in
             the
             Rank
             be
             Z
             ,
             its
             square
             ZZ
             .
             the
             second
             term
             2Z
             ,
             its
             square
             4ZZ
             ,
             therefore
             the
             first
             of
             the
             first
             differences
             is
             3ZZ
             ,
             the
             third
             term
             in
             that
             Rank
             3Z
             ,
             its
             square
             9ZZ
             ,
             then
             9ZZ
             ,
             Less
             4ZZ
             ,
             the
             second
             of
             the
             first
             differences
             5ZZ
             ,
             therefore
             5ZZ
             Less
             3ZZ
             the
             second
             difference
             will
             be
             2ZZ
             .
          
           
             Further
             ,
          
           
             The
             fourth
             term
             in
             that
             Rank
             is
             4Z
             ,
             its
             square
             is
             16ZZ
             ,
             then
             16ZZ
             Less
             9ZZ
             the
             third
             of
             the
             first
             differences
             7ZZ
             ;
             again
             ,
             7ZZ
             Left
             5ZZ
             the
             second
             difference
             is
             2ZZ
             .
          
           
             Hence
             it
             follows
             ,
          
           
             That
             the
             second
             difference
             is
             equall
             to
             the
             square
             of
             the
             first
             term
             doubled
             .
          
           
             Or
             also
             ,
          
           
             The
             second
             difference
             is
             equall
             to
             the
             squar●
             
             of
             the
             difference
             of
             two
             of
             the
             terms
             ,
             (
             in
             order
             taken
             )
             doubled
             .
          
           
             By
             the
             same
             method
             we
             find
             that
             the
             third
             difference
             in
             a
             Rank
             of
             cubes
             are
             equall
             ,
             and
             the
             third
             difference
             is
             equal
             to
             the
             first
             term
             multiplyed
             by
             6.
             
          
           
             Or
             ,
          
           
             The
             third
             difference
             is
             equal
             to
             the
             cube
             of
             the
             difference
             of
             two
             of
             the
             terms
             ,
             taken
             in
             order
             ,
             multiplyed
             by
             6.
             
          
           
             The
             index
             and
             equal
             difference
             ,
             of
             every
             power
             agrees
             ;
             to
             wit
             ,
             the
             index
             of
             the
             square
             is
             2
             ,
             and
             the
             second
             differences
             are
             equal
             .
             The
             index
             of
             the
             cube
             is
             3
             ,
             and
             the
             third
             differences
             are
             equal
             .
             The
             index
             of
             the
             square
             squared
             is
             4
             ,
             and
             the
             fourth
             differences
             are
             equal
             .
             &c.
             
          
           
             The
             equal
             difference
             of
             every
             power
             ,
             is
             complicated
             from
             the
             index
             of
             that
             power
             ,
             and
             the
             equal
             difference
             of
             the
             next
             Lesser
             power
             .
          
           
             Let
             the
             Rank
             be
             in
             naturall
             order
             ,
             Thus
             ;
             1
             ,
             2
             ,
             3
             ,
             4.
             &c.
             
          
           
             The
             indices
             of
             the
             powers
             ,
             Thus.
             
          
           
             
               
                 
                   1
                
                 
                   2
                
                 
                   3
                
                 
                   4
                
                 
                   5
                
              
               
                 
                   Z
                
                 
                   ZZ
                
                 
                   ZZZ
                
                 
                   ZZZZ
                
                 
                   ZZZZZ
                
              
            
          
           
             A
             unity
             the
             equal
             difference
             in
             that
             naturall
             Rank
             ,
             whose
             square
             is
             1
             ,
             which
             multiplyed
             by
             2
             the
             index
             of
             the
             square
             the
             product
             is
             2
             ,
             the
             equal
             difference
             in
             the
             squares
             .
             3
             ,
             the
             index
             of
             the
             cube
             multiplyed
             by
             2
             the
             equal
             difference
             in
             the
             squares
             ,
             
             the
             product
             is
             6
             ,
             the
             equal
             difference
             in
             the
             cubes
             ▪
             4
             the
             index
             of
             the
             square
             squared
             multiplyed
             by
             6
             the
             product
             is
             24
             the
             equall
             difference
             in
             the
             square
             squared
             ,
             &c.
             
          
           
             If
             the
             Rank
             be
             in
             order
             thus
             ,
             2
             ,
             4
             ,
             6
             ,
             8
             ,
             &c.
             
          
           
             2
             the
             equal
             difference
             of
             this
             Rank
             whose
             square
             is
             4
             ;
             multiplyed
             by
             2
             the
             index
             of
             the
             square
             the
             product
             is
             8
             ;
             the
             equal
             difference
             of
             the
             squares
             in
             such
             a
             Rank
             .
             Because
             the
             equal
             difference
             of
             the
             Rank
             is
             2
             ,
             therefore
             the
             indices
             are
             to
             be
             doubled
             ,
             &c.
             
          
           
             And
             the
             equall
             difference
             of
             the
             powers
             in
             such
             a
             Rank
             will
             be
             8
             ,
             48
             ,
             384
             ,
             &c.
             
          
        
         
           
             XX.
             Note
             .
             For
             the
             more
             easier
             calculation
             of
             the
             second
             sections
             of
             the
             sphere
             and
             spheroid
             ;
             worke
             ,
             Thus.
             
          
           
             From
             the
             double
             of
             the
             superficies
             of
             the
             triangle
             BZN
             ,
             substract
             the
             superficies
             of
             the
             triangles
             BZGD
             and
             NZPA
             ,
             the
             Remainder
             will
             be
             the
             superficies
             of
             the
             triangles
             BZPA
             and
             NZGD
             ,
             the
             areas
             of
             these
             two
             triangles
             being
             substracted
             from
             the
             area
             of
             the
             traingle
             NZB
             ,
             the
             Remainder
             will
             be
             the
             superfice
             of
             the
             triangle
             ZGDAP
             .
          
           
             FINIS
             .
          
        
      
    
     
       
         
         
           By
           
             Iohn
             Baker
             ,
          
           living
           in
           Barmonsey-street
           in
           Southwark
           ,
           over
           against
           the
           
             Princes-Armes
             ,
          
           is
           Taught
           Arithmetick
           ,
           both
           in
           whole
           numbers
           and
           fractions
           ,
           Decimal
           ,
           Logarithmetical
           and
           Algebraical
           ,
           Geometry
           ,
           Trigonometry
           ,
           Astronomy
           ,
           the
           use
           of
           Globes
           ,
           Navigation
           ,
           Measuring
           ,
           Gageing
           ,
           Dyalling
           ,
           &c.
           
           Also
           the
           Construction
           and
           use
           of
           all
           the
           usual
           lines
           put
           upon
           Rules
           or
           Scales
           ,
           He
           also
           teacheth
           how
           to
           find
           the
           (
           Length
           and
           )
           Spreading
           of
           a
           Hip-rafter
           ,
           only
           by
           a
           Line
           of
           Chords
           of
           singular
           use
           for
           Carpenters
           ,
           a
           way
           not
           as
           yet
           vulgarly
           known
           amongst
           Workmen
           .
        
      
       
         
           Faults
           Escaped
           in
           the
           Impression
           of
           Stereometrical
           Propositions
           .
        
         
           Page
           1
           ,
           line
           18
           ,
           for
           and
           Z
           Read
           and
           H.
           p.
           10
           ,
           l.
           23
           ,
           for
           56
           ,
           r.
           58.
           p.
           34
           ,
           l.
           25
           ,
           after
           RI
           put
           .
           p.
           43
           ,
           for
           297232
           ,
           r.
           297432.
           p.
           45
           ,
           for
           18
           ,
           r.
           V432
           ▪
           p.
           46
           ,
           l.
           1.
           for
           XVI
           ,
           r.
           XVII
           .
           and
           l.
           21
           ,
           for
           AE
           ,
           16
           ;
           r.
           AF
           ,
           6
           ;
           p.
           48
           ,
           l.
           6
           ,
           for
           634.
           r.
           624
           ,
           p.
           51.
           l.
           22
           ,
           for
           diameter
           ,
           r.
           semidiameter
           .
           p.
           58
           ▪
           l.
           25
           ,
           for
           ZB
           ,
           r.
           XB
           ▪
           p.
           63
           ,
           l.
           1
           ,
           for
           XIII
           .
           r.
           XXIII
           .
           p.
           100.
           l.
           17
           ,
           for
           parameter
           ,
           r.
           diameter
           .
           p.
           102
           and
           103
           for
           as
           4
           to
           3
           ,
           r.
           as
           3
           to
           2.
           p
           ,
           105.
           l.
           vlt.
           for
           Z+2
           ▪
           r.
           Z-2
           .
           p.
           106
           ,
           l.
           4
           ,
           for
           Z
           =
           ●
           r.
           Z-⅗
           .
           and
           l.
           25
           ,
           for
           89
           ,
           r.
           98.
           
        
         
           p.
           7
           against
           12
           in
           the
           first
           col
           .
           in
           the
           sec.
           r.
           576.
           and
           in
           the
           fifth
           colum
           f.
           96
           r.
           99.
           p.
           13.
           in
           the
           second
           col
           .
           f.
           46.
           r.
           36.
           and
           in
           the
           same
           col
           .
           f.
           2216
           ,
           r.
           2916.